Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research

Major Accomplishments in Composite Materials and Sandwich Structures

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I.M. Daniel Editors

•

E.E. Gdoutos

•

Y.D.S. Rajapakse

Major Accomplishments in Composite Materials and Sandwich Structures An Anthology of ONR Sponsored Research

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Editors I.M. Daniel Dept. Civil & Environmental Engineering Northwestern University 2137 Tech Drive Evanston IL 60208 Catalysis Bldg. USA [email protected]

Y.D.S. Rajapakse Office of Naval Research Solid Mechanics Program Arlington VA 22203-1995 USA [email protected]

E.E. Gdoutos Dept. of Civil Engineering Lab. for Applied Mechanics Democritus University of Thrace 671 00 Xanthi Greece [email protected]

ISBN 978-90-481-3140-2 e-ISBN 978-90-481-3141-9 DOI 10.1007/978-90-481-3141-9 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009930153 c Springer Science+Business Media B.V. 2009  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book contains a collection of major research contributions over the last decade in the area of composite materials and sandwich structures supported by the Office of Naval Research (ONR) under the direction of Dr. Yapa D.S. Rajapakse. The Solid Mechanics Research Program at ONR supports research in mechanics of high performance materials for the effective design of durable and affordable Naval structures. Such structures operate in severe environments, and are designed to withstand complex multi-axial loading conditions, including highly dynamic loadings. The effective design of these structures requires an understanding of the deformation and failure characteristics of structural materials, and the ability to predict and control their performance characteristics. The major focus is on mechanics of composite materials and composite sandwich structures. The program deals with understanding and modeling the physical processes involved in the response of glass-fiber and carbon-fiber reinforced composite materials and composite sandwich structures to static, cyclic, and dynamic, multi-axial loading conditions, in severe environments (sea water, moisture, temperature extremes, and hydrostatic pressure). This anthology consists of 30 chapters written by ONR contractors and recognized experts in their fields and serves as a reference and guide for future research. The topics covered in the book can be divided into three major themes:  Mechanical and failure behavior of composite materials and structures under

static and dynamic loading  Mechanical and failure behavior of sandwich materials and structures under static

and dynamic loading  Constituent materials, including fiber, polymer matrix materials, and core

materials The various topics discussed within each theme are as follows: Mechanical and failure behavior of composites materials and structures under static and dynamic loading The static behavior is discussed in four chapters as follows: “Accelerated Testing for Long-Term Durability of Various FRP Laminates for Marine Use,” by Y. Miyano and M. Nakada, develops an accelerated testing v

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methodology based on the time-temperature superposition principle for the prediction of long-term fatigue life of various FRP laminates for marine use under various temperature and water environments. “Carbon Fiber – Vinyl Ester Interfacial Adhesion Improvement by the Use of an Epoxy Coating,” by F. Vautard, L. Xu and L.T. Drzal, investigates the origins of low interfacial adhesion of carbon fiber – vinyl ester composites. It was found that curing volume shrinkage is the main cause of poor adhesion, and that an engineered interphase consisting of a partially cross-linked epoxy sizing that could chemically bond to the carbon fiber and form an interpenetrating network with the vinyl ester matrix, improves the interface adhesion. “A Physically Based Cumulative Damage Formalism,” by R.M. Christensen, derives a general cumulative damage methodology from the relation of crack growth rate as a power law of the stress intensity factor. The methodology applies in the case of creep to failure under variable stress history as well as for cyclic fatigue to failure under variable stress amplitude history. The advantages of the developed methodology over the Linear Cumulative Damage theory are shown by several examples. “Delamination of Composite Cylinders,” by P. Davies and L.A. Carlsson, studies the delamination resistance of filament wound glass epoxy cylinders for a range of winding angles and fracture mode ratios using beam fracture specimens, and the influence of delamination damage on the failure of externally pressurized composite cylinders. It was found that the delamination fracture resistance increases with increasing winding angle and shear fracture, and that the cylinder strength is insensitive to the presence of single delaminations, and that impact damage causes reduction in failure pressure. The dynamic behavior is discussed in four chapters as follows: “Modeling of Progressive Damage in High Strain-Rate Deformations of FiberReinforced Composites,” by R.C. Batra and N.M. Hassan, develops a mathematical model for analyzing high strain-rate deformations of fiber-reinforced composites subjected to shock loads. The formulation of the problem includes evolution of damage due to fiber breakage, fiber/matrix debonding, matrix cracking and delamination. Energies dissipated in these failure modes are computed and the effect of various parameters is examined. The mathematical model is validated by comparing computed results with experimental findings. A Figure of Merit is introduced to characterize the performance of laminated composites subjected to impact loads. “Post-Impact Fatigue Behavior of Woven and Knitted Fabric CFRP Laminates for Marine Use,” by I. Kimpara and H. Saito, studies the damage evolution in carbon-fiber-reinforced laminates under post-impact fatigue and water environment. The damage was characterized by non-destructive and direct observation methods. Evidence of interfacial degradation caused by water absorption was found. “Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures,” by R. Massab`o, deals with the interaction of multiple damage mechanisms in multilayered structures under static and dynamic loading. This work establishes a link between material and structural performance and gives basic insight for improvements in the survivability of ship structures via material and structural design.

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“A Review of Research on Impulsive Loading of Marine Composites,” by M. Porfiri and N. Gupta provides a general overview of the work performed on the effect of blast waves on marine materials and structures. Mechanical and failure behavior of sandwich materials and structures under static and dynamic loading This theme deals with mechanical and failure behavior of sandwich materials and structures under static and dynamic loading. The static case is discussed in ten chapters as follows. “Failure Modes of Composite Sandwich Beams,” by I.M. Daniel and E.E. Gdoutos studies the failure modes in axially loaded composite sandwich columns and sandwich beams under bending and shear. Failure modes include facesheet failure, facesheet debonding, indentation failure, core failure, and facesheet wrinkling. The transition from one failure mode to another for varying loading or state of stress and beam dimensions was discussed. Experimental results were compared with analytical predictions. “Localised Effects in Sandwich Structures with Internal Core Junctions: Modelling and Experimental Characterisation of Load Response, Failure and Fatigue,” by M. Johannes and O.T. Thomsen, addresses the modelling and experimental characterisation of sandwich structures with internal core junctions under in-plane and transverse shear loading. These local effects lead to an increase of facesheet bending stresses and core shear and transverse normal stresses. The influence of the local effects on the failure of sandwich structures under quasi-static and fatigue loading is investigated. “Damage Tolerance of Naval Sandwich Panels,” by D. Zenkert focuses on damage tolerance modeling and testing of sandwich panels for marine applications. It presents a review of testing methods for extracting properties and data required for damage tolerance assessment of core materials. Some typical damage types are defined and modeled with the objective of predicting their effect on load bearing capacity. The use of such models in providing a systematic damage assessment scheme for composite sandwich ship structures is presented. “Size Effect on Fracture of Composite and Sandwich Structures,” by E.E. Gdoutos and Z.P. Baˇzant reviews the work performed on the scaling and size effect in the failure of advanced composites, foams and sandwiches. The size effect is found to be essentially deterministic caused by energy release due to stress redistribution. The size effect of the above materials is presented in six sections for fiber-composite laminates subjected to tension, compression and flexural loading, to closed-cell polymeric foams and to sandwich panels under eccentric compression and with skin imperfections. For each case, experimental results, the size effect law and concluding remarks are given. “Elasticity Solutions for the Buckling of Thick Composite and Sandwich Cylindrical Shells Under External Pressure,” by G. Kardomateas presents elasticity solutions for buckling problems of thick orthotropic cylindrical shells and sandwich

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shells with orthotropic constituent materials under uniform external pressure. It is shown that the predictions of shell theory can produce in many cases highly nonconservative results on the critical loads. The results of the study can be used to assess the accuracy of the classical and existing improved shell theories for thick composites and sandwich shell constructions. “An Improved Methodology for Measuring the Interfacial Toughness of Sandwich Beams,” by Q. Bing and B.D. Davidson, presents a modified peel test and a modified method of data reduction based on beam theory for determining the interfacial toughness of sandwich structures with composite facesheets. From experiments and nonlinear finite element analyses, it is shown that this approach can be used for accurate determination of the interfacial fracture toughness of sandwich structures under various environmental conditions. “Structural Performance of Eco-Core Sandwich Panels,” by K. Shivakumar and H. Chen studies the performance of sandwich beams made of a fire-resistant core material, Eco-Core, and glass/vinyl ester facesheets. “The Use of Neural Networks to Detect Damage in Sandwich Composites,” by D. Serrano, F.A. Just-Agosto, B. Shafiq and A. Cecchini, presents a neural network based approach using a combined vibration and thermographic technique for the detection of damage in sandwich composites. “On the Mechanical Behavior of Advanced Composite Material Structures,” by J. Vinson, presents an overview of the research performed on the determination of mechanical properties of composite and sandwich materials. “Application of Acoustic Emission Technology to the Characterization and Damage Monitoring of Advanced Composites,” by E.O. Ayorinde, uses acoustic emission technology for monitoring and characterization of damage in sandwich composites. The dynamic loading case of the same theme is discussed in seven chapters as follows: “Ballistic Impacts on Composite and Sandwich Structures,” by S. Abrate presents a critical review of the existing literature on ballistic impact of composites and sandwich structures. Models for predicting the ballistic limit and the extent of damage are described. It is noted that the experimental data can be fitted by the LambertJonas equations to obtain a good estimate of the ballistic limit from results of penetrating impacts. “Performance of Novel Composites and Sandwich Structures under Blast Loading,” by A. Shukla, S.A. Tekalur, N. Gardner, M. Jackson and E. Wang, uses a high speed imaging technique to study the damage modes and mechanisms under air blast loading of different composite material systems, including traditional two-dimensional woven laminated composites, layered composites, and sandwich panels. It is observed that layering of glass fiber composites with a soft interlayer provides better blast resistance. Also good blast resistance properties are obtained by constructions using poly-urea and glass fiber composites and sandwich materials made by sandwiching a soft layer between woven composite skins. It was found that, the gradation of materials in the sandwich can help mitigate the blast damage.

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“Single and Multi-site Impact Response of S2-Glass/Epoxy Balsa Wood Core Sandwich Composites,” by U.K. Vaidya and L.J. Deka, deals with an experimental study supported by finite element analysis of the single-site and multi-site impact damage response of S2-Glass/epoxy balsa wood sandwich composites. A progressive failure model based on Hashin’s criteria is used to predict failure. The effect of projectile diameter, impact location and constituent material properties on the impact damage is investigated. “Real-Time Experimental Investigation on Dynamic Failure of Sandwich Structures and Layered Materials,” by L.R. Xu and A.J. Rosakis, studies the generation and evolution of dynamic failure modes in model sandwich specimens and layered materials subjected to out-of-plane low-speed impact by using high-speed photography and dynamic photoelasticity. Shear-dominated interfacial cracks that propagate with intersonic speeds, even under moderate impact speeds, constitute the dominant dynamic failure mode. The interfacial cracks kinked into the core layer and branched at high speeds causing brittle core fragmentation. “Characterization of Fatigue Behavior of Composite Sandwich Structures at SubZero Temperatures,” by S.M. Soni, R.F. Gibson and E.O. Ayorinde studies the effect of temperature over the range of 22ı C to 60ı C on the failure modes under static and cyclic loading and on the fatigue lives of carbon/epoxy and glass/epoxy composite sandwich beams. It was observed that fatigue failures at the subzero temperatures are catastrophic and without any early significant warning, whereas they were preceded by loss of stiffness at room temperature. “Impact and Blast Resistance of Sandwich Plates,” by G.J. Dvorak, Y.A. BaheiEl-Din and A.P. Suvorov studies the response of conventional and modified sandwich plate designs under static, impact and blast loads. In the modified plate designs, ductile interlayers are inserted and bonded between the exposed outer facesheet and the core. “Modeling Blast and High Velocity Impact of Composite Sandwich Panels,” by M.S. Hoo Fatt, L. Palla and D. Sirivolu presents analytical models for the prediction of the deformation and failure of composite sandwich panels subjected to blast and high velocity projectile impact. Constituent materials, including fiber, polymer matrix, and core materials The third theme dealing with mechanical and failure behavior of constituent materials is discussed in five chapters as follows: “Effect of Nanoparticle Dispersion on Polymer Matrix and Their Fiber Nanocomposites,” by M.F. Uddin and C.T. Sun investigates the effect of dispersion of nanoparticles on mechanical properties of nanocomposites via conventional sonication, sol-gel and modified sonication methods. It was found that the nanocomposites fabricated via a sol-gel method have improved and consistent properties compared to nanocomposites fabricated by the other methods. The silica/epoxy nanocomposites were used to make fiber reinforced composites using the VARTM

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process, with significant improvements in compressive strength, tensile strength and modulus, fracture toughness, and impact resistance. “Experimental and Analytical Analysis of Mechanical Response and Deformation Mode Selection in Balsa Wood,” by M. Vural and G. Ravichandran presents experimental and analytical results for the mechanical behavior of balsa wood over its entire density range from 55 to 380 Kg/m3 for quasi-static (103 s1 / and high strain rate (103 s1 / loading using a split Hopkinson bar. Emphasis is given to relation of the results to the microstructure of balsa wood. It is found that the compressive strength of balsa wood increases with relative density, and that the failure of low-density specimens is governed by elastic and/or plastic buckling, while in higher density specimens kink band formation and end-cap collapse dominate. Analytical models are proposed to predict the quasi-static compressive strength of balsa wood under uniaxial loading. “Mechanics of PAN Nanofibers,” by M. Naraghi and I. Chasiotis studies the mechanical behavior of low-cost polymeric polyacrylonitrile (PAN) nanofibers, aimed at establishing fabrication-structure-property interrelations. It is shown that PAN nanostructures are strongly susceptible to surface processes that control their mechanical response which depends on time and strain rate. It is also shown that the elastic modulus and tensile strength of PAN nanofibers depend strongly on fiber diameter. “Characterization of Deformation and Failure Modes of Ordinary and Auxetic Foams at Different Length Scales,” by Fu-pen Chiang, investigates the size effect on the mechanical properties of polyurethane foams with and without nanoparticles, crack tip deformation field and crack propagation characteristics using a multiscale digital speckle photography technique developed by the author. “Fracture of Brittle Lattice Materials: A Review,” by I. Quintana-Alonso and N.A. Fleck reviews the failure mechanisms of elastic brittle lattice materials used in sandwich construction. We are very thankful to the authors for their excellent contributions to this volume of composite materials and sandwich structures supported by the Office of Naval Research (ONR). A special word of thanks goes to Mrs. Nathalie Jacobs of Springer for her kind and continuous collaboration and support and the esthetic appearance of this book. March 2009

I.M. Daniel E.E. Gdoutos Y.D.S. Rajapakse

Contents

Part I Composite Materials and Structures: Mechanical and Failure Behavior Accelerated Testing for Long-Term Durability of Various FRP Laminates for Marine Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yasushi Miyano and Masayuki Nakada

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Carbon Fiber–Vinyl Ester Interfacial Adhesion Improvement by the Use of an Epoxy Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Frederic Vautard, Lanhong Xu, and Lawrence T. Drzal A Physically Based Cumulative Damage Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Richard M. Christensen Delamination of Composite Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Peter Davies and Leif A. Carlsson

Part II Composite Materials and Structures: Dynamic Effects Modeling of Progressive Damage in High Strain–Rate Deformations of Fiber-Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Romesh C. Batra and Noha M. Hassan Post-Impact Fatigue Behavior of Woven and Knitted Fabric CFRP Laminates for Marine Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 Isao Kimpara and Hiroshi Saito Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133 Roberta Massab`o

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A Review of Research on Impulsive Loading of Marine Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169 Maurizio Porfiri and Nikhil Gupta

Part III Sandwich Materials and Structures: Mechanical and Failure Behavior Failure Modes of Composite Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197 Isaac M. Daniel and Emmanuel E. Gdoutos Localised Effects in Sandwich Structures with Internal Core Junctions: Modelling and Experimental Characterisation of Load Response, Failure and Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229 Martin Johannes and Ole Thybo Thomsen Damage Tolerance of Naval Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279 Dan Zenkert Size Effect on Fracture of Composite and Sandwich Structures . . . . . . . . . . . . .305 Emmanuel E. Gdoutos and Zdenˇek P. Baˇzant Elasticity Solutions for the Buckling of Thick Composite and Sandwich Cylindrical Shells Under External Pressure . . . . . . . . . . . . . . . . . . .339 George A. Kardomateas An Improved Methodology for Measuring the Interfacial Toughness of Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .365 Qida Bing and Barry D. Davidson Structural Performance of Eco-Core Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . .381 Kunigal Shivakumar and Huanchun Chen The Use of Neural Networks to Detect Damage in Sandwich Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407 David Serrano, Frederick A. Just-Agosto, Basir Shafiq, and Andres Cecchini On the Mechanical Behavior of Advanced Composite Material Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .431 Jack Vinson Application of Acoustic Emission Technology to the Characterization and Damage Monitoring of Advanced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441 Emmanuel O. Ayorinde

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Part IV Sandwich Materials and Structures: Dynamic Effects Ballistic Impacts on Composite and Sandwich Structures . . . . . . . . . . . . . . . . . . . .465 Serge Abrate Performance of Novel Composites and Sandwich Structures Under Blast Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .503 Arun Shukla, Srinivasan Arjun Tekalur, Nate Gardner, Matt Jackson, and Erheng Wang Single and Multisite Impact Response of S2-Glass/ Epoxy Balsa Wood Core Sandwich Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .541 Uday K. Vaidya and Lakshya J. Deka Real-Time Experimental Investigation on Dynamic Failure of Sandwich Structures and Layered Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .571 L. Roy Xu and Ares J. Rosakis Characterization of Fatigue Behavior of Composite Sandwich Structures at Sub-Zero Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605 Samirkumar M. Soni, Ronald F. Gibson, and Emmanuel O. Ayorinde Impact and Blast Resistance of Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .625 George J. Dvorak, Yehia A. Bahei-El-Din, and Alexander P. Suvorov Modeling Blast and High-Velocity Impact of Composite Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .661 Michelle S. Hoo Fatt, Leelaprasad Palla, and Dushyanth Sirivolu

Part V Constituent Materials Including Fiber, Polymer Matrix, and Core Materials Effect of Nanoparticle Dispersion on Polymer Matrix and their Fiber Nanocomposites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .693 Mohammed F. Uddin and Chin-Teh Sun Experimental and Analytical Analysis of Mechanical Response and Deformation Mode Selection in Balsa Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .717 Murat Vural and Guruswami Ravichandran Mechanics of PAN Nanofibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .757 Mohammad Naraghi and Ioannis Chasiotis

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Characterization of Deformation and Failure Modes of Ordinary and Auxetic Foams at Different Length Scales. . . . . . . . . . . . . . . . . . .779 Fu-pen Chiang Fracture of Brittle Lattice Materials: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .799 Ignacio Quintana-Alonso and Norman A. Fleck Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .817

Contributors

Serge Abrate Southern Illinois University, Carbondale, Illinois 62901-6603, USA Emmanuel O. Ayorinde Mechanical Engineering Department, Wayne State University, Detroit, Michigan 48202, USA Yehia A. Bahei-El-Din Center for Advances Materials, The British University in Egypt, El Shorouk City, Egypt Romesh C. Batra Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA Zdenˇek P. Baˇzant Department of Civil Engineering and Material Science, Northwestern University, Evanston, Illinois 60208-3109, USA Qida Bing Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, New York 13244, USA Leif A. Carlsson Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida 33431, USA Andres Cecchini College of Engineering, University of Puerto Rico, PO Box 9045, Mayaguez, Puerto Rico 00681, USA Ioannis Chasiotis Aerospace Engineering, University of Illinois at Urbana Champaign, Urbana, Illinois 61801, USA Huanchun Chen Center for Composite Materials Research. Department of Mechanical and Chemical Engineering, North Carolina A&T State University, Greensboro, North Carolina 27411, USA Fu-pen Chiang Department of Mechanical Engineering, Stony Brook University, Stony Brook, New York 11794-2300, USA Richard M. Christensen Stanford University, Stanford, California 94305-2047, USA and Lawrence Livermore National Laboratory, California 94551-0808, USA Isaac M. Daniel Robert McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208, USA Barry D. Davidson Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, New York 13244, USA

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Contributors

Peter Davies Materials and Structures Group IFREMER, 29280 Plouzan´e, France Lakshya J. Deka Department of Material Science and Engineering, The University of Alabama at Birmingham, Birmingham, Alabama 35294-4461, USA Lawrence T. Drzal Composite Materials and Structures Center, Michigan State University, East Lansing, Michigan 48824, USA George J. Dvorak Robert McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208, USA Norman Fleck Engineering Department, Trumpington Street, Cambridge University, Cambridge CB2 1PZ, United Kingdom Nate Gardner Department of Mechanical Engineering and Applied Mechanics, Dynamic Photo Mechanics Laboratory, University of Rhode Island, Kingston, Rhode Island 02881, USA Emmanuel E. Gdoutos School of Engineering, Democritus University of Thrace, GR-671 00 Xanthi, Greece Ronald F. Gibson Mechanical Engineering Department, University of NevadaReno, MS312 Reno, Nevada, USA Nikhil Gupta Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Brooklyn, New York 11201, USA Noha M. Hassan Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA Michelle S. Hoo Fatt Department of Mechanical Engineering, The University of Akron, Akron, Ohio 44325-3903, USA Matt Jackson Department of Mechanical Engineering and Applied Mechanics, Dynamic Photo Mechanics Laboratory, University of Rhode Island, Kingston, Rhode Island 02881, USA Martin Johannes Department of Mechanical Engineering, Aalborg University, Pontoppidanstræde 101, DK-9220 Aalborg East, Denmark Frederick A. Just-Agosto College of Engineering, University of Puerto Rico, PO Box 9045, Mayaguez, Puerto Rico 00681, USA George Kardomateas School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0150, USA Isao Kimpara Research Laboratory for Integrated Technological Systems, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan, Ishikawa 924-0838, Japan Roberta Massab`o Department of Civil, Environmental and Architectural Engineering, University of Genova, Via Montallegro 1, 16145 Genova, Italy

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Yasushi Miyano Materials System Research Laboratory, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan, Ishikawa 924-0838, Japan Masayuki Nakada Materials System Research Laboratory, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan, Ishikawa 924-0838, Japan Mohammad Naraghi Aerospace Engineering, University of Illinois at Urbana Champaign, Urbana, Illinois 61801, USA Leelaprasad Palla Department of Mechanical Engineering, The University of Akron, Akron, Ohio, 44325-3903, USA Maurizio Porfiri Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Brooklyn, New York 11201, USA Ignacio Quintana-Alonso Engineering Department, Trumpington Street, Cambridge University, Cambridge CB2 1PZ, United Kingdom Guruswami Ravichandran Graduate Aeronautical Laboratories, California Institute of technology, Pasadena, California 91125, USA Ares Rosakis Graduate Aeronautical Laboratories, California Institute of technology, Pasadena, California 91125, USA Hiroshi Saito Research Laboratory for Integrated technological Systems, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan, Ishikawa 924-0839, Japan David Serrano College of Engineering, University of Puerto Rico, PO Box 9045, Mayaguez, Puerto Rico 00681, USA Basir Shafiq College of Engineering, University of Puerto Rico, PO Box 9045, Mayaguez, Puerto Rico 00681, USA Kunigal Shivakumar Center for Composite Materials Research. Department of Mechanical and Chemical Engineering, North Carolina A&T State University, Greensboro, North Carolina 27411, USA Arun Shukla Department of Mechanical Engineering and Applied Mechanics, Dynamic Photo Mechanics Laboratory, University of Rhode Island, Kingston, Rhode Island 02881, USA Dushyanth Sirivolu Department of Mechanical Engineering, The University of Akron, Akron, Ohio 44325-3903, USA Samirkumar M. Soni Caterpillar Corporation, 27th Street and Pershing Road, Decatur, Illinois, 62526, USA Chin-Teh Sun School of Aerospace and Astronautics, Purdue University, West Lafayette, Indiana 47906, USA Alexander P. Suvorov McGill University, Montreal, Quebec, Canada

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Srinivasan Arjun Tekalur Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan 48864, USA Ole Thybo Thomsen Department of Mechanical Engineering, Aalborg University, Pontoppidanstræde 101, DK-9220 Aalborg East, Denmark Mohammed F. Uddin School of Aerospace and Astronautics, Purdue University, West Lafayette, Indiana 47906, USA Uday K. Vaidya Department of Material Science and Engineering, The University of Alabama at Birmingham, Birmingham, Alabama 35294-4461, USA Frederic Vautard Composite materials and Structures Center, Michigan State University, East Lansing, Michigan 48824, USA Jack Vinson Department of Mechanical Engineering, University of Delaware, 126 Spencer Lab., Newark, DE 19716, USA Murat Vural Mechanical, Materials and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, Illinois 60616, USA Erheng Wang Department of Mechanical Engineering and Applied Mechanics, Dynamic Photo Mechanics Laboratory, University of Rhode Island, Kingston, Rhode Island 02881, USA Lanhong Xu Composite materials and Structures Center, Michigan State University, East Lansing, Michigan 48824, USA L. Roy Xu Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee 37235, USA Dan Zenkert Department of Aeronautical and Vehicle Engineering Kungliga Tekniska H¨ogskolan, SE-10044 Stockholm, Sweden

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Accelerated Testing for Long-Term Durability of Various FRP Laminates for Marine Use Yasushi Miyano and Masayuki Nakada

Abstract The prediction of long-term fatigue life of various FRP laminates combined with resins, fibers and fabrics for marine use under temperature and water environments were performed by our developed accelerated testing methodology based on the time-temperature superposition principle (TTSP). The base material of five kinds of FRP laminates employed in this study was plain fabric CFRP laminates T300 carbon fibers/vinylester (T300/VE). The first selection of FRP laminate to T300/VE was the combinations of different fabrics, that is flat yarn plain fabric T700 carbon fibers/vinylester (T700/VE-F) and multi-axial knitted T700 carbon fibers/vinylester (T700/VE-K) for marine use and the second selection of FRP laminates to T300/VE was the combinations with different fibers and matrix resin, that is plain fabric T300 carbon fibers/epoxy (T300/EP) and plain fabric E-glass fibers/vinylester (E-glass/VE). These five kinds of FRP laminates were prepared under three water absorption conditions of Dry, Wet and Wet C Dry after molding. The three-point bending constant strain rate (CSR) tests for these FRP laminates at three conditions of water absorption were carried out at various temperatures and strain rates. Furthermore, the three-point bending fatigue tests for these specimens were carried out at various temperatures and frequencies. The flexural CSR and fatigue strengths of these five kinds of FRP laminates strongly depend on water absorption as well as time and temperature. The mater curves of fatigue strength as well as CSR strength for these FRP laminates at three water absorption conditions are constructed by using the test data based on TTSP. It is possible to predict the long term fatigue life for these FRP laminates under an arbitrary temperature and water absorption conditions by using the master curves.

Y. Miyano () Materials System Research Laboratory, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan, Ishikawa 924-0838, Japan e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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1 Introduction Recently, polymer matrix composites reinforced with carbon fibers have been used in primary structures of airplanes, ships, spacecrafts, etc., which require highly reliable long-term operation. Therefore, it is essential to assess the reliability of the composite structures subject to long-term creep and fatigue loadings and environmental conditions (temperature, moisture absorption, etc.). The polymer matrix composites exhibit mechanical behaviors significantly dependent on time and temperature under operation because of the viscoelastic behavior of polymer matrix resin. The viscoelastic behavior has been observed not only above glass transition temperature Tg but also below Tg [1–5]. Previously, authors of this paper have developed an accelerated testing methodology (ATM) to predict the long-term creep and fatigue life of the polymer matrix composites based on the time-temperature superposition principle (TTSP) held for the viscoelastic behavior of matrix resin [6–10]. The ATM using the TTSP enables us to describe the long-term life by means of master curves covering wide ranges of loading and environmental conditions, including load duration, temperature, frequency of load cycles, stress amplitude ratios, etc. The ATM has been applied for various composites and their joint structures, and demonstrated with a great success as a robust and power methodology for the long-term life prediction. Some theoretical explanations were made to backup their experimental observations [11–14]. In the present paper, we revisited and summarized the earlier efforts of the ATM for the long-term creep and fatigue life prediction of the various polymer matrix composites and its structures. Then, the prediction of long-term fatigue life of five kinds of FRP laminates for marine use under temperature and water environments are performed based on the ATM. These FRP laminates were prepared under three water absorption conditions of Dry, Wet and Wet C Dry after molding. The threepoint bending constant strain rate (CSR) tests for five kinds of FRP laminates at three conditions of water absorption were carried out at various temperatures and strain rates. It is possible to calculate the creep strength from the CSR strength determined from relatively easy tests under CSR loading. Furthermore, the three-point bending fatigue tests for these specimens were carried out at various temperatures and frequencies. The time, temperature, and water absorption dependencies of flexural fatigue strength as well as flexural CSR strength for these FRP laminates are discussed by our developed accelerated testing methodology.

2 Accelerated Testing Methodology 2.1 Procedure of ATM The ATM has been established with three following conditions: (A) the same TTSP is applicable for both non-destructive viscoelastic behavior and destructive strength

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properties of matrix resin and their composites, (B) linear cumulative damage (LCD) law is applicable to the strength by the monotonic loading, and (C) the fatigue strengths exhibit linear dependence on the stress ratio of the cyclic loadings. A key component of the ATM is the empirical observation (A), which was pioneered by the authors of this paper, and demonstrated its applicability for various polymeric composite materials and their structures. With the condition (B), it is possible to calculate the creep strength master curve from the CSR strength master curve determined from relatively easy tests under CSR loading at several elevated temperatures and a fixed strain rate. With the condition (C), the fatigue strength at any stress ratios can be interpolated with those between zero and unity, which are obtained by the creep- and the fatigue-strength master curves, respectively. Therefore, the fatigue strength under any arbitrary combinations of frequency, temperature, and stress ratio can be determined with the two test results for obtaining (i) CSR strength measured at a single strain rate and several elevated temperatures and (ii) fatigue strength measured at zero stress ratio, a single frequency, several stress levels and several elevated temperatures. The detail procedure of the ATM is illustrated in Fig. 1. Firstly, change in modulus of the viscoelastic matrix resin is measured over time at a constant temperature. The tests repeat for several elevated temperatures, which results in several modulus curves with the function of time. Time-temperature shift factors (TTSFs) are then determined by shifting the viscoelastic modulus curves at the several temperatures into time scale to form a master curve of the modulus at a reference temperature. The TTSF is thus the measure of the acceleration of the life of matrix resin by means of the elevated temperatures. The next step is to obtain the creep strength master curves. This step consists of two parts. The first part is to determine CSR strength master curve of the composites from the CSR loading tests conducted at a single strain rate and several elevated temperatures by using condition (A) and the second part is to convert CSR strength master curve to the creep strength master curve of

Fig. 1 Accelerated testing methodology for polymer composites

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the composites by using condition (B). Thirdly, the master curves of fatigue strength of the composites at zero stress ratio are determined by conducting the fatigue tests at several stress levels, a single frequency, a single stress ratio (zero stress ratio) and several elevated temperatures by using condition (A). In this step, CSR strength master curve is used as the fatigue strength master curve at the number of cycles to failure Nf D 1=2. Finally, the creep and fatigue strengths at any arbitrary frequency, stress ratio and temperature are obtained from the master curves of creep strength and fatigue strength at zero stress ratio by using condition (C).

2.2 Applicability of ATM The ATM has successfully been applied to many different kinds of composite materials and their structures subject to various loading conditions. Table 1 lists the materials and loading types as well as their reference article numbers for the condition (A) [8, 11, 15–29]. Table 2 lists the same for the conditions (B) and (C) [8, 11, 20, 24, 27, 28]. Table 3 lists the applicability of the conditions (A), (B) and (C) for various composite joint structures subject to tensile loading [30–34]. As the tables show, the conditions are satisfied for almost all composites made of PAN-based carbon fibers and thermosetting resin matrix regardless of fiber architectures (unitape and fabric). Furthermore, they are satisfied for tensile fatigue strengths of adhesively bonded and bolted joints. Therefore, the ATM can be used for the long-term life prediction of these materials and their joint structures using the creep and fatigue strength master curves. It should be noted that not all materials satisfy the conditions. For example, it was observed that the composites with PEEK resin and some pitch-based carbon fibers failed to satisfy the conditions [7, 20, 27, 28]. The reasons of failure might be due to the crystallization of PEEK resin during loading and viscoelastic behaviors of the high-modulus pitch-based carbon fibers. Therefore, the present ATM cannot be used for the life prediction for these materials. Excluding these few cases, the ATM can be used for many available composite materials in general.

2.3 Theoretical Verification of TTSP The failure of composites is dependent on the type of loads. Therefore, the applicability of the condition (A) for the TTSP can be explained differently for the different type of loads. For example, the time and temperature dependence of failure of unidirectional FRP subjected to longitudinal tensile and compressive loading in the direction parallel to the fiber alignment is caused by the non-destructive viscoelastic behavior of the polymer matrix resin, which has been observed for both CSR loading and constantly applied creep loading [11–13]. The former can further be explained by cumulative damage of fibers, which can be explained by a

Conditions (A) Reference CSR Creep Fatigue T400/828  [8, 11]   – HR40/828  – LT [15]  – UD4  –  [16] T300/828  – LC  [17]  T400/828  [18] LT    – UD [19]  – –  UD Epoxy  Fortafil510/Cape2002 LB  LB  [7]  UD  T300/PEEK –  TB  [20]  – SW  [6] LB  T400/3601   [21]  – –  PW  LB T300/828 PAN T800S/3900-2B   [22]  QIL  LB  UT500/#135  [22] –  T800S/TR-A33  –  [23]  – IM600/PIXA-M –  – [24] PI QIL LB  MR50K/PETI-5 – –  – [24] – QIL  –  G40-800/5260 [24] – BMI LB   T300/VE  [25]   Vinylester PW [21] – T700/VE   LB  – NCF   – XN05/828 –  – [26]  – LT UD 4 XN50/828 –  – [26] Pitch Epoxy XN40/25C   [7]  XN70/25C – [27]    – UD LB  YS15/25P – [28]  [28]  XN05/25P –   – E-glass/VE   [25]  Glass Vinylester PW LB  WE18W/VE –  – – [29] 4  Note:  UD: Unidirectional UD : Strand UD : Ring  SW: Satin Woven  PW: Plain Woven  QIL: Quasi-isotropic laminates  NCF: Non Crimp Fabric  LT: Longitudinal Tension  LB: Longitudinal Bending  LC: Longitudinal Compression  TB: Transverse Bending Fiber/Matrix

Loading direction

Carbon

Type

Accelerated Testing for Long-Term Durability

Table 1 Applicability of condition A for FRP Deformation Fiber Matrix Dc E0

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Table 2 Applicability of conditions B and C for FRP Fiber

Matrix

Type

UD

T400/828 T300/828 T400/828

UD

T300/PEEK

4

UD

Carbon

Epoxy

QIL PAN Pitch Glass

Vinylester Epoxy Vinylester

Fiber/Matrix

SW PW UD PW

T800S/3900-2B UT500/#135 T800S/TR-A33 T400/3601 T300/VE XN40/25C E-glass/VE

Loading direction LT LT LB TB LB LB LB LB LB

Conditions (B) (C)    –            –        

Reference [8] [16] [18] [7] [20] [22] [23] [6] [25] [7] [25]

Note:  UD: Unidirectional UD4 : Strand UD : Ring  SW: Satin Woven  PW: Plain Woven  QIL: Quasi-isotropic laminates  LT: Longitudinal Tension  LB: Longitudinal Bending  TB: Transverse Bending

Table 3 Applicability of conditions A, B and C for FRP joints

Conditions (A) (B) (C) Conical shaped joint of GFRP/metal    Brittle adhesive joint of GFRP/metal    Ductile adhesive joint of GFRP/metal    Bolted joint of GFRP/metal  – – Bolted joint of CFRP/metal    FRP joint system

Reference [30] [31] [32] [33] [34]

Rosen’s model [35], while the latter by microbuckling of fibers. Meanwhile, the failure of the unidirectional composites subjected to transverse tensile loading in the transverse direction to the fiber alignment is caused by the failure of the polymer matrix formed by microcracks [14]. In all cases, the failure is caused by both non-destructive viscoelastic behavior and destructive strength properties of the matrix material. Table 4 summarizes the various failure mechanisms under different loading types. As stated earlier, the condition (B) for the LCD law is used in obtaining the creep strength master curves from the CSR strength master curve. The applicability of the condition (B) can also be explained by a crack kinetics theory by Christensen and Miyano [36, 37]. The crack kinetics theory showed that the rate of decrease in the creep strength over time is the same as that of CSR loading strength, so that the master curve of the former can be obtained by horizontal shifting of the latter. It was independently checked that the amount of the horizontal shifting is quantitatively equal to the shifting amount by the LCD law [6].

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Table 4 Failure mechanisms of unidirectional composites subject to various loading types

Longitudinal tension

Longitudinal compression

Transverse tension

3 Experimental Procedures 3.1 Preparation of Specimens The base material of five kinds of FRP laminates employed in this study was plain fabric CFRP laminates T300 carbon fibers/vinylester (T300/VE). The first selection of FRP laminate to T300/VE was the combinations of different fabrics, that is flat yarn plain fabric T700 carbon fibers/vinylester (T700/VE-F) and multi-axial knitted T700 carbon fibers/vinylester (T700/VE-K) for marine use and the second selection of FRP laminates to T300/VE was the combinations with different fibers and matrix resin, that is plain fabric T300 carbon fibers/epoxy (T300/EP) and plain fabric E-glass fibers/vinylester (E-glass/VE) shown in Fig. 2. These FRP laminates were formed by the resin transfer molding (RTM) except T300/EP which was formed by conventional hand lay up shown in Fig. 3. The fiber volume fraction was approximately 51%. The thickness of the laminates was approximately 2 mm. These FRP laminates were prepared under three water absorption conditions of Dry, Wet and Wet C Dry after molding. Dry specimens by holding the cured specimens at 150ı C for 2 h in air, Wet specimens by soaking Dry specimens in hot water of 95ı C for 120 h and Wet C Dry specimens by dehydrating the Wet specimens at 150ı C for 2 h in air were respectively prepared as shown on Table 5. Figure 4 shows water content versus soaking time at 95ı C and the soaking condition of 95ı C and 120 h was determined to Wet specimens. Figure 5 shows water content in resin and FRP for Wet and Wet C Dry specimens. The water absorption of all FRP laminates increases with Wet condition of hot water of 95ı C for 120 h. The water absorption of neat vinylester resin and its CFRP laminates returns to 0% with Wet C Dry condition by re-drying and that of T300/EP and E-glass/VE does not return to 0%.

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Fig. 2 Five kinds of FRP laminates by combinations of matrix resin, fiber and fabric

Fig. 3 Constitution of five kinds of FRP laminates combined with different resins and fibers Table 5 Conditions for Dry, Wet, and Wet C Dry specimens

Specimen Dry Wet Wet C Dry

In air In water In air As cured As cured C 95ı C  120 h As cured C 95ı C  120 h C 150ı C  2 h

3.2 Tests Figure 6 shows the configuration of three point bending test and Table 6 shows the test conditions. To evaluate the viscoelastic behavior of vinylester (VE) and epoxy (EP) as matrix neat resin, the three-point bending creep tests for the neat vinylester and epoxy resins prepared at Dry, Wet and WetCDry conditions were carried out under various temperatures using an creep testing machine with temperature chamber.

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Fig. 4 Water content versus soaking time

Fig. 5 Water content in resin and FRP for Wet and Wet C Dry conditions Fig. 6 Configuration of three point bending test

The creep compliance Dc was calculated from the deflection ı at the center of specimen using the following equation: Dc D

4bh3 • P0 L 3

(1)

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Table 6 Test conditions Loading Deflection rate V Frequency f type (mm/min) (Hz) Creep for neat resin — — CSR for FRP 0:02 , 2, 200 — Fatigue for FRP — 0:02 , 2  Test conditions for confirming of applicability of TTSP. Table 7 Tests for neat resin

Stress ratio R (min =max ) — — 0.05

VE neat resin Dry  Creep compliance Wet  Wet C Dry 

Temperature T .ı C) 25  150 25  160 25  140

EP neat resin   

where P0 is the applied constant load (58.8N), L is the span (50 mm), and b and h are the width (25 mm) and the thickness (3.0 mm) of specimen, respectively. The three-point bending CSR tests for five kinds of FRP laminates at Dry, Wet and Wet C Dry conditions were carried out at various temperatures and strain rates. The span is L D 60 mm, and the width and thickness are b D 15 mm and h D 2:0 mm, respectively. The CSR tests were conducted at three loading-rates V D 0:02; 2;200 mm=min and various constant temperatures T using an universal testing machine with temperature chamber. The flexural CSR strength s is calculated from the maximum load Ps by s D

3Ps L 2bh2

(2)

Furthermore, the three-point bending fatigue tests for these specimens were carried out at various constant temperatures T and two loading frequencies f D 2 and 0.02 Hz using an electro-hydraulic servo testing machine with temperature chamber. The stress ratio R (D minimum stress/maximum stress) was 0.05. The length, width, thickness of specimen, and span are the same to those for the flexural CSR tests. The flexural fatigue strength f is defined by maximum applied load Pmax for the number of cycles to failure Nf . f D

3Pmax L 2bh2

(3)

In order to prevent the dryness of specimens at Wet condition during creep, CSR, and fatigue tests, the specimens were wrapped by a vinyl bag with including distilled water in the bag. Additionally, the basic mechanical properties of all materials under all conditions were measured. Finally, all items of tests are gathered on Tables 7 and 8.

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Table 8 Tests for FRP Temperature dependence Tensile CSR Dry strength Compressive Dry CSR strength Interlaminar Dry shear strength

T300/VE

T700/VE-F

T700/VE-K

T300/EP

E-glass/VE





–

–







–

–

–





–





Time and temperature dependence Flexural CSR Dry  strength Wet  Wet C Dry 









 

 

 

 

Dry









–

Wet Wet C Dry

 

 

– 

– –

– –

Dry











Wet Wet C Dry

 

 

 

 

 

Flexural creep strength

Flexural fatigue strength

4 Results and Discussion The creep compliance of neat resins and the flexural CSR and fatigue strength of various FRP laminates are discussed by using the test data.

4.1 Creep Compliance The left sides of left graphs of Figs. 7 and 8 show the creep compliance Dc versus testing time t at various temperatures T for Dry, Wet and Wet C Dry specimens of VE and EP resin. The master curves of Dc versus the reduced time t 0 at a reference temperature T0 were constructed by shifting Dc at various constant temperatures along the log scale of t and the log scale of Dc . Since the smooth master curve of Dc for each specimen can be obtained as shown in the right sides of each graph, the TTSP is applicable for each Dc . From these master curves, it is cleared that Dc increases with water absorption and returns perfectly to that of Dry specimen by re-drying after water absorption. The horizontal time-temperature shift factor aT o .T / and the vertical temperature shift factor bT o .T / at a reference temperature T0 plotted in Figs. 7 and 8 are respectively defined by

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Fig. 7 Master curve of creep compliance for neat vinylester resin and shift factors

log aT0 .T / D log t  log t 0 log bT0 .T / D log DC .t; T /  log DC .t 0 ; T0 /

(4) (5)

The vertical shift amounts are very small comparing with the horizontal shift amounts and the vertical shift should be done for obtaining reliable horizontal shift factors.

4.2 Flexural CSR Strength The left side of each graph in Fig. 9 shows the flexural CSR strength s versus time to failure ts at various temperatures T for Dry, Wet and Wet C Dry specimens of five kinds of FRP laminates, where ts is the time period from initial loading to maximum load during testing. The master curves of s versus the reduced time to failure ts0 were constructed by shifting s at various constant temperatures along the log scale of ts and the log scale of s using the same time-temperature shift factors

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Fig. 8 Master curve of creep compliance for neat epoxy resin and shift factors

and a half of the temperature shift factors for Dc of matrix resin shown in Figs. 7 and 8. The reason why the shift amount to the log scale of s is a half of that to the log scale of Dc is mentioned after. Since the smooth master curve of s for each specimen can be obtained as shown in the right side of each graph, the TTSP for Dc of matrix resin is also applicable for the s of corresponding FRP laminates. It is cleared from Fig. 9 that the s for all five FRP laminates strongly decreases with increasing time and temperature and that these s decreases with water absorption and returns to that of Dry specimens by re-drying after water absorption except that of GFRP laminates (E-glass/VE). The s of Wet C Dry specimens of E-glass/VE does not return to that of Dry specimens. Figure 10 shows the flexural CSR strength versus the creep compliance of matrix resin for the same conditions of time, temperature and water absorption for five FRP laminates. The degradation of flexural CSR strength for all CFRP laminates except GFRP laminates (E-glass/VE) is uniquely determined by the creep compliance of matrix resin. Therefore, the degradation rate of flexural CSR strength of these CFRP laminates is determined only by increasing of time, temperature and water absorption and is independent upon fiber constitutions which are the type, volume fraction and weaves. The slope is approximately 0.5 shown in each graph of this figure. This indicates that the trigger of failure is the microbuckling of carbon

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T300/VE

T700/VE-F

T300/EP

T700/VE-K

E-glass/VE

Fig. 9 Master curves of flexural CSR strength at Dry, Wet and Wet C Dry conditions

fibers in the compression side of specimen shown by the following equation based on Dow’s theory [38]: 1 (6) log s D log K 0  log DC 2 where s is the CSR strength of CFRP laminates and Dc is the creep compliance of matrix resin. Actually, the fracture appearance indicates that the fracture mode for these CFRP laminates is the compressive fracture of warp carbon fibers in the compression side of specimen for all condition tested as shown in Fig. 11. Therefore, this is the reason why the vertical shift amount for s is a half of that for Dc as mentioned above. The fracture mode for T300/EP laminates and E-glass/VE is the

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T300/VE

T700/VE-F

T300/EP

T700/VE-K

E-glass/VE

Fig. 10 Flexural CSR strength versus creep compliance of matrix resin

tensile fracture in the tension side of specimen at room temperatures T D 25ı C. However the fracture mode at high temperatures is the compressive fracture in the compression side of specimen same to that for T300/VE laminates.

4.3 Flexural Fatigue Strength To construct the master curve of flexural fatigue strength f , we need the reduced frequency f 0 in addition to the reduced time to failure tf 0 , each defined by f 0 D f  aT0 .T /;

tf 0 D

where Nf is the number of cycles to failure.

Nf tf D 0 aT0 .T / f

(7)

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T300/VE

T700/VE-F

T300/EP

T700/VE-K

E-glass/VE

Fig. 11 Fracture appearances of specimens after flexural CSR test at 25ı C at Dry condition

The f versus Nf at frequency f D 2 Hz at various temperatures were measured for Dry, Wet and WetCDry specimens of five kinds of FRP laminates. For examples, the f versus Nf curves at various temperatures for Dry specimen are shown in Fig. 12. By converting f and Nf into f 0 and tf0 using Eq. (7), the time-temperature shift factors aT o .T / and temperature shift factors bT o .T / of the creep compliance of matrix resin for each specimen shown in Figs. 7 and 8, the f versus tf0 for each f 0 were constructed for Dry, Wet and Wet C Dry specimens of five kinds of FRP laminates shown in Figs. 13 and 14. The curves consisted by solid circles in these graphs show the master curves of CSR strengths which can be considered as the fatigue strength at stress ratio R D 0 and Nf D 1=2. Each curve consisted by hollow circles in these graphs shows the curve of fatigue strength f versus reduced time to failure tf0 at each reduced frequency f 0 to diverge from the master curve of CSR strength. In order to confirm the applicability of TTSP for fatigue strength, we predicted the f –Nf curves at f D 0:02 Hz and compared them with the test results. The

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T300/VE

T700/VE-F

T700/VE-K

T300/EP

E-glass/VE

Fig. 12 f versus Nf curves at frequency 2 Hz for Dry specimen

predicted f from fatigue master curves for all FRP laminates agree well with experimental ones, therefore, the TTSP for the creep compliance of matrix resin also holds for fatigue strength of the corresponding FRP laminates. It is cleared from Figs. 13 and 14 that the f of all five FRP laminates strongly decreases with time to failure, temperature and water absorption and that the f of four kinds of FRP laminates except GFRP laminates (E-glass/VE) decreases scarcely with Nf although that of E-glass/VE decreases strongly with Nf . And the degradation rate to time and temperature for the fatigue strength f of these CFRP laminates is very similar to that for CSR strength. The f of all FRP laminates also decreases with water absorption and that returns to that of Dry specimens by re-drying after water absorption except that of T300/EP in the range of long time and that of

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a

b

T700/VE-F

T300/VE

c

T700/VE-K

Fig. 13 Master curves of flexural fatigue strength for (a) T300/VE, (b) T700/VE-F and (c) T700/VE-K

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Fig. 14 Master curves of flexural fatigue strength for (a) T300/VE, (b) T300/EP and (c) E-glass/VE

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Y. Miyano and M. Nakada T300/VE

T300/EP

E-glass/VE

Dry

Wet + Dry

Fig. 15 Side view of Dry and Wet C Dry specimens for T300/VE, T300/EP and E-glass/VE

T300/VE

T300/EP

E-glass/VE

Dry

Wet + Dry

Fig. 16 Fracture appearance of Dry and Wet C Dry specimens for T300/VE, T300/EP and E-glass/VE after fatigue tests

E-glass/VE in all range of time employed. The f of Wet C Dry specimens of T300/EP and E-glass/VE does not return to that of Dry specimens and shows irreversible behavior. It can be explained from Figs. 15 and 16 that the reason why the f of T300/EP and E-glass/VE shows irreversible behavior by the process of wet and dry is due to the chemical degradation of interface between fiber and resin. These figures show the side view of Dry and Wet C Dry specimens for T300/VE, T300/EP and E-glass/VE and the fracture appearance of these FRP laminates after fatigue tests. The interface between fiber and resin in E-glass/VE dissolve chemically during the process of wet and dry as shown in Fig. 15. The interfacial fracture between fiber

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and resin can be observed in Wet C Dry specimens of T300/EP and E-glass/VE after fatigue tests as shown in Fig. 16 and it is presumed that the interfacial fracture occurs by the chemical degradation of interface.

5 Conclusions The prediction of long-term fatigue life of five kinds of FRP laminates combined with matrix resin, fiber and fabric for marine use under temperature and water environments were performed by our developed accelerated testing methodology based on the time-temperature superposition principle (TTSP). The three-point bending CSR and fatigue tests for five kinds of FRP laminates at three conditions of water absorption were carried out at various temperatures and loading rates. As the results, the flexural fatigue strength of three kinds of CFRP laminates with vinylester resin as matrix strongly depends on water absorption as well as time and temperature, however scarcely depends on the number of cycles to failure. The mater curves of fatigue strength for these CFRP laminates are constructed by using the test data based on TTSP. The fatigue strength of these CFRP laminates decreases with water absorption and that returns to the initial fatigue strength by re-drying after water absorption. It is possible to predict the long term fatigue life for these CFRP laminates under an arbitrary temperature and water absorption conditions by using the master curves. Furthermore, it is clear that the degradation rate of fatigue strength of these CFRP laminates is determined only by increasing of time, temperature and water absorption and is independent upon fiber constitutions which are the type, volume fraction and weaves. On the other hand, CFRP laminates with epoxy resin as matrix and GFRP laminates with vinylester resin as matrix chemically change by the process of water absorption and re-drying and the flexural fatigue strength of these FRP laminates decrease with this process.

Navy Relevance The proposed methodology has effectively combined the effects of time, temperature and water absorption on the strength and life of composite materials. It can be confirmed that the methodology is applicable to the innovative CFRP laminates for marine use. Acknowledgments The authors thank the Office of Naval Research for supporting this work through an ONR award (N000140110949) with Dr. Yapa Rajapakse as the program manager of solid mechanics. The authors thank Professor Richard Christensen at Stanford University as the consultant of this project and Toray Industries, Inc. as the supplier of CFRP laminates. All of experimental data were measured by the staffs and graduate students of author’s laboratory, Kanazawa Institute of Technology. The authors thank these staffs and graduate students, Dr. Naoyuki Sekine, Dr. Junji Noda, Mrs. Kumiko Saito, Ms. Jun Ichimura, Mr. Eiji Hayakawa and Mr. Takahito Uozu.

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References 1. Aboudi J and Cederbaum G (1989) Analysis of Viscoelastic Laminated Composite Plates, Compos Struct 12: 243–256 2. Sullivan J (1990) Creep and Physical Aging of Composites, Compos Sci Technol 39: 207–232 3. Gates T (1992) Experimental Characterization of Nonlinear, Rate Dependent Behavior in Advanced Polymer Matrix Composites, Exp Mech 32: 68–73 4. Rotem A and Nelson HG (1981) Fatigue Behavior of Graphite-Epoxy Laminates at Elevated Temperatures, In: Fatigue of Fibrous Composite Materials, ASTM STP 723: 152–173 5. Kharrazi MR and Sarkani S (2001) Frequency-Dependent Fatigue Damage Accumulation in Fiber-Reinforced Plastics, J Compos Mater 35: 1924–1953 6. Miyano Y, Nakada M, McMurray MK and Muki R (1997) Prediction of Flexural Fatigue Strength of CFRP Composites Under Arbitrary Frequency, Stress Ratio and Temperature, J Compos Mater 31: 619–638 7. Miyano Y, Nakada M and Muki R (1999) Applicability of Fatigue Life Prediction Method to Polymer Composites, Mech Time Depend Mater 3: 141–157 8. Miyano Y, Nakada M, Kudoh H and Muki R (1999) Prediction of Tensile Fatigue Life Under Temperature Environment for Unidirectional CFRP, Adv Compos Mater 8: 235–246 9. Miyano Y, Tsai SW, Christensen RM and Kuraishi A (2001) Accelerated Testing for the Durability of Composite Materials and Structures, Long Term Durability of Structural Materials (Durability 2000), Elsevier, 265–276 10. Miyano Y, Tsai SW, Christensen RM and Muki R (2002) Accelerated Testing Methodology for the Durability of Composite Materials and Structures, Proceedings of the 5th Composites Durability Workshop, Paris, pp. 1–14 11. Nakada M, Miyano Y, Kinoshita M, Koga R, Okuya T and Muki R (2002) Time-Temperature Dependence of Tensile Strength of Unidirectional CFRP, J Compos Mater 36: 2567–2581 12. Miyano Y, Nakada M, Kinoshita M, Koga R and Okuya T (2001) Time-Temperature Dependence of Tensile Strength of Unidirectional CFRP, Durability Analysis of Composite Systems 2001: 169–173 13. Nakada M and Miyano Y (2007) Accelerated Testing for Long-Term Durability of Various FRP Laminates for Marine Use, Proc ICCM-16, Kyoto, WeFM1-02 14. Miyano Y, Kanemitsu M, Kunio T and Kuhn AH (1986) Role of Matrix Resin on Fracture Strengths of Unidirectional CFRP, J Compos Mater 20: 520–538 15. Muki R, Nakada M, Watanabe N and Miyano Y (2004) Influence of Fiber Stiffness on Time-Temperature Dependent Tensile Strength of Unidirectional CFRP, Proc 2004 SEM International Congress and Exposition on Experimental and Applied Mechanics, Costa Mesa, 32 16. Nakada M, Yoshioka K and Miyano Y (2008) Prediction of Long-Term Creep Life for Unidirectional CFRP, Proc 6th International Conference on Mechanics of Time Dependent Materials, Monterey, 88 17. Miyano Y, Nakada M, Watanabe N, Murase T and Muki R (2003) Time-Temperature Superposition Principle for Tensile and Compressive Strengths of Unidirectional CFRP, Proc 2003 SEM Annual Conference & Exposition on Experimental and Applied Mechanics, Charlotte, 147 18. Miyano Y, Nakada M, Kudoh H and Muki R (2000) Prediction of Tensile Fatigue Life for Unidirectional CFRP, J Compos Mater 34: 538–550 19. Miyano Y, Sekine N, Ichimura J and Nakada M (2004) Fatigue Life Prediction of CFRP Laminates Under Temperature and Moisture Environments, Proc 2004 SEM International Congress and Exposition on Experimental and Applied Mechanics, Costa Mesa, 36 20. Nakada M, Maeda M, Hirohata T, Morita M and Miyano Y (1996) Time and Temperature Dependencies on the Flexural Fatigue Strength in Transverse Direction of Unidirectional CFRP, Proceedings of the International Conference on Experimental Mechanics at Singapore, Singapore, pp. 492–497

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21. Nakada M and Miyano Y (2009) Accelerated Testing for Long-Term Fatigue Strength of Various FRP Laminates for Marine Use, Compos Sci Technol, 69: 805–813 22. Miyano Y, Nakada M and Nishigaki K (2006) Prediction of Long-Term Fatigue Life of Quasiisotropic CFRP Laminates for Aircraft Use, Int J Fatigue 28: 1217–1225 23. Nakada M, Hamagami Y, Sekine N and Miyano Y (2003) Time-Temperature Dependence of Flexural Behavior of CFRP Laminates for Aircraft Use, Proc the 8th Japan International SAMPE Symposium, Tokyo, pp. 77–80 24. Hamagami Y, Sekine N, Nakada M and Miyano Y (2003) Time and Temperature Dependence Flexural Strength of Heat-Resistant CFRP Laminates, JSME Int J, Series A46: 437–440 25. Miyano Y, Nakada M and Sekine N (2005) Accelerated Testing for Long-Term Durability of FRP Laminates for Marine Use, J Compos Mater 39: 5–20 26. Watanabe N, Koga R, Nakada M, Miyano Y and Muki R (2003) Time-Temperature Dependent Tensile Behavior of Unidirectional CFRP, Proc ICCM-14, San Diego, 842 27. Nakada M, Miyano Y, Daicho N and Takemura S (1998) Time and Temperature Dependence on the Flexural Fatigue Behavior of Unidirectional Pitch-Based Carbon Fiber Reinforced Plastics, Proc ACCM-1, Osaka, 442 28. Nakada M, Miyano Y, Ikeda M and Takemura S (1999) Time and Temperature Dependence of Flexural Fatigue Strength for Pitch-Based CFRP, Proc ICCM-12, Paris, 257 29. Nakada M, Kosho S and Miyano Y (2001) Time-Temperature Dependence on Flexural Behavior of GFRP with Different Surface Treatment for Glass Fiber, Proc ICCM-13, Beijing, 1473 30. Miyano Y, Nakada M and Muki R (1997) Prediction of Fatigue Life of a Conical Shaped Joint System for Fiber Reinforced Plastics Under Arbitrary Frequency, Load Ratio and Temperature, Mech Time Depend Mater 1: 143–159 31. Miyano Y, Tsai SW, Nakada M, Sihn S and Imai T (1997) Prediction of Tensile Fatigue Life for GFRP Adhesive Joint, Proc ICCM-11, Gold Coast, VI: 26–35 32. Miyano Y, Nakada M, Yonemori T, Sihn S and Tsai SW (1999) Time and Temperature Dependence of Static, Creep, and Fatigue Behavior for FRP Adhesive Joints, Proc ICCM-12, Paris, 259 33. Sekine N, Nakada M, Miyano Y and Tsai SW (2001) Time-Temperature Dependence of Tensile Fatigue Strength for GFRP/Metal and CFRP/Metal Bolted Joints, Proc ICCM-13, Beijing, 1610 34. Sekine N, Nakada M, Miyano Y, Kuraishi A and Tsai SW (2003) Prediction of Fatigue Life for CFRP/Metal Bolted Joint Under Temperature Conditions, JSME Int J A46: 484–489 35. Rosen BW (1964) Tensile Failure of Fibrous Composites, AIAA J 2: 1985–1991 36. Christensen R and Miyano Y (2006) Stress Intensity Controlled Kinetic Crack Growth and Stress History Dependent Life Prediction with Statistical Variability, Int J Fract 137: 77–87 37. Christensen R and Miyano Y (2007) Deterministic and Probabilistic Lifetimes from Kinetic Crack Growth-Generalized Forms, Int J Fract 143: 35–39 38. Dow NF and Gruntfest IJ (1960) Space Sciences Laboratory, Structures and Dynamics Operation, T.I.S.R60SD389

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Carbon Fiber–Vinyl Ester Interfacial Adhesion Improvement by the Use of an Epoxy Coating Frederic Vautard, Lanhong Xu, and Lawrence T. Drzal

Abstract With the use of composites expanding into larger structural applications, vinyl ester matrices which are not dependent on an autoclave cure and are more environmentally resistant to water absorption are being investigated. The degree of adhesion between the fiber and matrix has been recognized to be a critical factor in determining the performance of fiber-reinforced composites. The mechanical properties of carbon fiber–vinyl ester composites are low compared to carbon fiber– epoxy composites, partly because of lower interfacial adhesion. The origins of this limitation were investigated. The influence of preferential adsorption of the matrix constituents on the interfacial adhesion was not significant. However, the high cure volume shrinkage was found to be an important factor. An engineered interphase consisting of a partially cross-linked epoxy sizing that could chemically bond to the carbon fiber and form an interpenetrating network with the vinyl ester matrix was found to sharply improve the interfacial adhesion. The mechanisms involved in that improvement were investigated. The diffusion of styrene in the epoxy coating decreased the residual stress induced by the volume shrinkage of the vinyl ester matrix. The optimal value of the thickness was found to be a dominant factor in increasing the value of the interfacial shear strength according to a 2D non-linear finite element model.

1 Introduction Since carbon fiber reinforced composites offer great mechanical properties with a low density, they have been of interest to many fields, such as military equipment, transportation, and sport and recreation goods. Carbon fiber composites are especially used in the aerospace and aeronautics industries when the required F. Vautard, L. Xu, and L.T. Drzal () Composite Materials and Structures Center, Michigan State University, East Lansing, MI 48824, USA e-mail: [email protected]

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mechanical properties have to be outstanding. However, with the price of carbon fibers decreasing their uses increase, spreading into a wider range of applications. Specific demands related to mechanical properties, resistance to chemicals and environment, process and cost of manufacture have lead to an investigation of the use of different types of matrices. Vinyl ester resins are widely used, particularly because of their high resistance to moisture absorption and corrosion. Nevertheless, the mechanical properties of carbon fiber–vinyl ester composites currently cannot compete with the mechanical properties of carbon fiber–epoxy composites, due to their poor mechanical interfacial properties and low interfacial adhesion [1–3]. It has been established that the physico-chemical and mechanical properties of a composite material are not only dependant on the characteristics of the reinforcement material and the matrix, but also on the properties of the interface [4, 5], which depend on the conditions used during the manufacturing [6]. Factors influencing interfacial adhesion are mechanical interlocking [7, 8], physical [9] (dispersive [10], polar [11], acid–base [12, 13]) and chemical interactions [14, 15], the presence of defects [16] (lack of cohesion, voids), and remaining thermal stress [17]. In order to highlight which of those parameters had the largest influence in the case of carbon fiber–vinyl ester composites, the influence of adsorption of the matrix constituents on the carbon fiber surface and the impact of the cure volume shrinkage were also evaluated. An easy process that leads to an improvement of interfacial adhesion was suggested by the use of a slightly cross-linked epoxy coating. The mechanisms that generate that enhancement are characterized and explained.

2 Materials and Methods 2.1 Materials The carbon fibers used in this experiment were surface treated and unsized high strength PolyAcryloNitrile (PAN) AS4 carbon fibers (Hexcel Inc). The vinyl ester resins were Derakane 411-C50 and Derakane 510A-40 provided by Ashland Co. and Fuchem 891 provided by Shangai Fuchem Chemicals Co. The resins were all DiGlycidyl Ether of Bisphenol A (DGEBA) epoxy based vinyl ester. The initiators were Cumene HydroPeroxide (CHP) or Methyl Ethyl Ketone Peroxide (MEKP), supplied by Witco Chemicals Corp and Sigma-Aldrich. Cobalt Naphtenate (CoNap) and DiMethyl Aniline (DMA) were supplied by Sigma-Aldrich and were used as a promoter and an accelerator respectively. The epoxy coating was made of DGEBA cured by Jeffamine T-403, an aliphatic polyether triamine. DGEBA and Jeffamine T-403 were from Shell and Huntsman. Pure styrene and pure Bisphenol A based vinyl ester monomer were kindly donated by Huntsman and Reichhold Inc. respectively. The formula of the chemicals that were used is given in Fig. 1. The physico-mechanical and mechanical properties at full cure (post curing at 120ı C) of the different resins are given in Table 1.

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Fig. 1 Formula of chemicals used in this study Table 1 Composition and thermo-mechanical properties of the vinyl ester resins used in this study Properties Derakane 411-C50 Derakane 510A-40 Fuchem 891 DGEBA vinyl ester molecular 907 1,223 1,500–2,000 weight (g  mol1 / Styrene concentration (wt%) 50 38 35 Tensile modulus (GPa) 3.38 3.40 3.12 Tensile strength (MPa) 79 86 75 Tensile elongation at break (%) 5.5 4–5 4.0 Barcol hardness 35 40 33 102 113 92 Heat distortion temperature (ı C)

2.2 Methods 2.2.1

Microindentation Test

In order to measure the InterFacial Shear Strength (IFSS), the microindentation method, first proposed by Mandell et al. [18], was used. The test was run on individually selected fibers on a polished cross-section of a composite, using a specially

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designed device called the Interfacial Testing System (ITS) [19]. Composite specimens were prepared by embedding a small piece of composite in a metallographic specimen, with the fibers normal to the specimen surface. The surface was polished with sandpaper, using an increasing value of grit and then polished with 1 and 0:05 m alumina particles water solutions. A diamond-tipped stylus mounted on the objective lens holder of a microscope was used to apply a force to single fibers in their surrounding matrix. The value of the IFSS was calculated using an empirical equation (1) based on a finite element analysis and adjusted by empirical data. More details about the equipment and the procedure can be found elsewhere [20]. fg IFSS D A 2 df

s 0:875696

Gm dn  0:01862 ln  0:026496 Ef df

! (1)

In Eq. (1) fg is the load applied on the cross section of the selected fiber, Gm is the shear modulus of matrix, Ef is the tensile modulus of the fiber, dn is the distance between the selected fiber and the nearest fiber, df is the diameter of the fiber and A is a conversion factor.

2.2.2

Thermogravimetric Analysis (TGA)

A High Resolution TGA Thermogravimetric Analyzer (TA Instruments), controlled by Thermal Advantage software was used to measure weight loss as a function of temperature. The ramp was 25ı C min1 from room temperature to 650ı C, with a resolution of 4ı C. The data were analyzed using Universal Analysis 2000 Software.

2.2.3

Dynamic Mechanical Thermal Analysis (DMTA)

A DMTA 2980 Dynamic Mechanical Analyzer (TA Instruments), controlled by Thermal Advantage software, was used to measure the change in the loss and storage modulus as a function of temperature. The ramp was 10ı C  min1 from room temperature to 150ı C. The data were analyzed using Universal Analysis 2000 Software.

2.2.4

Environmental Scanning Electron Microscopy (ESEM)

A Philips Electroscan Environmental Scanning Electron Microscope (ESEM) was used for the observation of the quality of the carbon fiber coating process and the fracture profile of composites. The acceleration voltage was 20 kV and the current ˚ No conductive coating was deposited at the surface in the filament was set to 1.83 A. of the samples.

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3 Preferential Adsorption of Some Constituents of the Matrix on the Carbon Fiber Surface and Its Influence on Interfacial Adhesion 3.1 Evidence of Preferential Adsorption of Some Constituents of the Matrix on the Carbon Fiber Surface Weitzsacker et al. [21] looked for all the reactions that could occur between the components of a vinyl ester resin and the surface of carbon fibers. Evidence of a reaction was seen in the case of pure vinyl ester monomer, DiMethyl Aniline (DMA) and Cobalt Naphthenate (CoNap). After an exposure of the fibers to CoNap, cobalt was detected by X-ray Photoelectron Spectroscopy (XPS) with an atomic concentration of 2.6% at the surface of the fibers, in the form of metallic cobalt or cobalt oxide strongly physisorbed, but not covalently bonded to the fiber surface. When both CoNap and Methyl Ethyl Ketone Peroxide (MEKP) were exposed to the carbon fiber surface, a significant change in the surface composition was revealed with the detection of cobalt at the fiber surface again (1–3%). When MEKP, CoNap, and DMA or Cumene Hydroperoxide (CHP), CoNap, and DMA were mixed together, an important change in the surface composition, a doubling in the surface oxygen concentration (up to 40%) as well as a significant increase in the concentration of cobalt (8%), was detected. In that case, cobalt strongly adhered to the carbon surface as a cobalt oxide. An important parameter that has to be taken into account for the preferential adsorption on the carbon fiber surface is the surface tension of the constituents of the matrix. Compared to the vinyl ester monomer (around 40 mN  m1 ), the styrene monomer has a lower surface tension (34 mN  m1 ) and is likely to be preferentially adsorbed at the carbon fiber surface, leading to the creation of a layer enriched in styrene. The adsorption of some constituents of the matrix on the carbon fiber surface can induce two phenomena. First, the composition of the matrix could be out of stoichiometry, which can affect the cross-link density, the structure of the microgels, and then the mechanical properties of the matrix at the fiber–matrix interphase. Secondly, the adsorption of the catalysts and promoters can react with the reactive sites of the carbon fiber surface, which can lead to a decrease in the level of interactions between the carbon fiber surface and the matrix.

3.2 Influence on Interfacial Adhesion The preferential adsorption on the carbon fibers surface generates a depletion of the corresponding compound in the interphase. To assess the influence of a variation of the matrix (Derakane 411-C50) composition on its mechanical properties and interfacial adhesion with the carbon fiber surface, the concentration of the initiator

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(CHP), promoter (CoNap), and accelerator (DMA) were individually decreased. The associated elastic modulus E was measured by a tensile test according to ASTM D 638-89. Ten samples were tested for each composition. The shear modulus G was calculated using Eq. (2), assuming that the value of the Poisson’s coefficient  remained unchanged and equal to 0.36. GD

E 2 .1 C /

(2)

The glass transition temperature was determined from DMTA analysis. Three samples were tested for each composition. Interfacial adhesion was measured by the microindentation test. Ten single fibers were tested for each composition.

3.2.1

Influence of the Concentration of the Initiator

The concentration of CHP was decreased (Table 2) and neither the mechanical properties nor the glass transition temperature was changed significantly (Fig. 2). The value of IFSS given by the microindentation test was not influenced by an eventual modification of the shear modulus of the matrix but only by a change of interfacial adhesion. The maximum value in IFSS was obtained for 1.5 wt% (Fig. 3). The microstructure of vinyl ester resins was shown to consist of packed highly cross-linked

Table 2 Composition of matrix, decrease of the initiator concentration Composition 1 Composition 2 Composition 3 CHP (wt%) 2.0 1.5 1.0 CoNap (wt%) 0.3 0.3 0.3 DMA (wt%) 0.1 0.1 0.1

Fig. 2 Thermo-mechanical properties of the matrix in function of the concentration of the initiator

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Fig. 3 Interfacial shear strength in function of the concentration of the initiator Table 3 Composition of matrix, decrease of the promoter concentration Composition 1 Composition 2 Composition 3 CHP (wt%) 2.0 2.0 2.0 CoNap (wt%) 0.3 0.2 0.1 DMA (wt%) 0.1 0.1 0.1

Composition 4 2.0 0.0 0.1

Fig. 4 Thermo-mechanical properties of the matrix in function of the concentration of the promoter

microgels separated by some less cured material [22]. It is hypothesized that the concentration of initiator has some impact on that microstructure that interacts with the carbon fiber surface. 3.2.2

Influence of the Concentration of the Promoter

The decrease of the concentration of CoNap (Table 3) did not induce any change of the mechanical properties and the glass transition temperature of the resin (Fig. 4). The value of interfacial adhesion did not show any noticeable change (Fig. 5).

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Fig. 5 IFSS in function of the concentration of the promoter Table 4 Composition of matrix, decrease of the accelerator concentration Composition 1 Composition 2 Composition 3 CHP (wt%) 2.0 2.0 2.0 CoNap (wt%) 0.3 0.3 0.3 DMA (wt%) 0.1 0.075 0.5

Composition 4 2.0 0.3 0.0

Fig. 6 Thermo-mechanical properties of the matrix in function of the concentration of the accelerator

3.2.3

Influence of the Concentration of the Accelerator

The decrease of DMA concentration (Table 4) led to a slight decrease of the tensile and shear moduli (Fig. 6). Nevertheless, the value of the IFSS did not show any noteworthy change (Fig. 7). Considering that a decrease of the initiator, promoter, and accelerator did not sharply modify the thermo-mechanical properties of the matrix and the interfacial shear strength, it was concluded that a preferential adsorption of some of those constituents at the surface of the carbon fiber (especially CoNap and DMA) was not responsible for the low adhesion between carbon fibers and vinyl ester resins.

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Fig. 7 IFSS in function of the concentration of the accelerator

4 Influence of Cure Volume Shrinkage on Interfacial Adhesion It is well known that vinyl ester resins can undergo cure volume shrinkage that can reach 10%, whereas epoxy resins generally undergo only 3–4% cure volume shrinkage [23–25]. In order to assess the cure volume shrinkage of Derakane 411C50, a lab-made dilatometer was manufactured [26]. The cure volume shrinkage depended on the resin and the type of initiator. The cure volume shrinkage of Derakane 411-C50 was compared to the cure volume shrinkage of Fuchem 891. Both resins are DGEBA based vinyl ester. Their properties are listed in Table 1. The compositions were suggested by the manufacturer, using CHP for Derakane 411-C50 and MEKP for Fuchem 891. In the case of the use of CHP for Fuchem 891, the concentration was fixed so that it was proportional to the concentration of carbon–carbon double bonds contained in the monomer and styrene, taking the concentration of MEKP as a reference (Table 5). The concentration of MEKP was reciprocally adjusted the same way for Derakane 411-C50. With the same types of initiator and thermal history, the cure volume shrinkage of Fuchem 891 was much less than the other resin formulations. It was also much smaller with the use of MEKP at room temperature in comparison to the cure volume shrinkage with CHP. The cure volume shrinkage is related to the composition of the resin (molecular weight of vinyl ester monomer, type of initiator). The thermal history of the curing process also had an obvious influence. For both resins, it was much higher when the resin was cured at a high temperature in comparison with the room temperature cure of the same composition (Table 5). For systems cured at a high temperature, the modulus was measured by nanoindentation and the associated shear modulus was calculated. The shear modulus of the samples was higher for the samples cured at high temperatures (Fig. 8). It was obvious that the samples cured at room temperature were not fully cured. A DMTA analysis revealed that the glass transition of Fuchem 891 samples cured at room temperature with MEKP was very low .45ı C/ and that a ramp to 150ı C increased it to 89ı C. When a higher temperature was chosen for the cure, the fraction conversion of the monomers was higher, which led to a denser network and better mechanical properties.

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Table 5 Cure volume shrinkage as a function of the composition of resin Type of resin Fuchem 891 Derakane 411-C50 Composition CHP 1.40 – – 2.00 2.00

–

(wt%)

MEKP CoNap Thermal program

– 0.21

Cure volume Shrinkage

2.00 0.10

2.00 0.30

– 0.30

– 0.10

2.85

1 h room temperature C1 h90ı C C1:5 h 125ı C

Room temperature

1 h room temperature C1 h90ı C C1:5 h125ı C

Room temperature

1 h room temperature C1 h90ı C C1:5 h 125ı C

5.85

0.54

7.18

3.68

8.20

1.73

Fig. 8 Shear modulus of vinyl ester resins in function of cure volume shrinkage

The best interfacial adhesion with AS4 carbon fibers was obtained with Fuchem 891, which had the lowest cure volume shrinkage (Fig. 9). For both resins, a trend showing that the interfacial shear strength was decreasing with increasing cure volume shrinkage was found. The cure volume shrinkage increased the Von Mises effective stress, which had a negative effect on interfacial adhesion [12]. It was shown that the cure volume shrinkage was a main parameter affecting interfacial adhesion in carbon fiber–vinyl ester composites. The composition of the resin, the choice of initiator, the molecular weight of the vinyl ester monomer and the thermal cure profile had to be optimized in order to minimize the shrinkage. Alternatively, another way of improving interfacial adhesion could be accomplished by inserting a third phase between the fiber and the matrix. This phase had to be ductile to absorb the stress created by the shrinkage of the matrix and maintain adhesion to the carbon fiber surface. Therefore to counteract the residual stress created by the high cure volume shrinkage of vinyl ester resins a slightly cross-linked epoxy coating was applied to the carbon fiber surface.

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Fig. 9 Interfacial adhesion in function of cure volume shrinkage

5 Improvement of Interfacial Adhesion by the Use of an Epoxy Coating 5.1 Optimization of the Coating Process A fiber sizing tower system was used to coat the carbon fibers with an epoxy coating. The epoxy coating was made of Epon 828 C Jeffamine-T403 (43 phr). The mix was left to cure for 30 min at room temperature before being dissolved in acetone. A spool of surface treated and unsized AS4 carbon fibers was unwound and the tow went through a bath of the epoxy coating dissolved in acetone and was dried by successive exposure to two tower-shaped furnaces. The temperature inside the furnaces was fixed at 60ı C and the time spent by the fibers in the furnaces was roughly 1 min. The thickness of the coating was measured by TGA. The difference in mass of the coated carbon fiber before and after the 600ı C exposure corresponded to the mass of the epoxy coating, which enabled the calculation of its thickness (Fig. 10). The thickness of the coating was directly proportional to the concentration of the sizing solution. It was found that the time spent by the fibers in the bath, which was proportional to the rotation of the winding device, did not have a major impact on the thickness of the coating (Fig. 11). The quality of the coating (uniformity, absence of resin bridges between single fibers) was checked by ESEM. When the concentration of the sizing solution was 1 wt%, the surface of the fiber was not evenly coated, with some parts totally uncovered. A concentration of 3 wt% gave a coating that appeared to be evenly spread, without creating bridges between single fibers (Fig. 12). It was also found that a concentration higher than 4 wt% (that is to say a coating thickness greater than 100 nm) resulted in significant bridging between fibers within the tow, producing a tow that was hard and could not be processed. The final composition of 2.5 wt% sizing solution produced a coating of 80 nm.

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F. Vautard et al. 200

Weight (mg)

199 198 197 196 195 194 193 0

100

200

300

400

500

600

700

Temperature (C)

Fig. 10 TGA analysis of AS4 carbon fibers coated by epoxy 300

Coating thickness (nm)

250 200 150 100 50 0 0

2

4

6

8

10

12

Sizing solution concentration (wt%) V = 0.43 rpm

V = 0.93 rpm

V = 0.72 rpm

V = 1.35 rpm

Fig. 11 Thickness of the coating as a function of the sizing solution concentration and the speed of rotation of the winding device

5.2 Interactions Between the Epoxy Coating and the Components of the Vinyl Ester Matrix The approach used to study the influence of the preferential adsorption of the matrix components on the carbon fibers surface on IFSS reported earlier in this paper was used to assess the influence of a difference in the diffusion of the initiator, promoter, accelerator, styrene and vinyl ester monomer in the epoxy coating. To assess

Carbon Fiber–Vinyl Ester Interfacial Adhesion Improvement

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Fig. 12 ESEM picture of fibers coated with an epoxy coating (concentration of sizing solution D 3 wt%)

Fig. 13 Influence of the concentration of initiator on interfacial adhesion with the use of an epoxy coating

the influence of a variation of the matrix composition on interfacial adhesion, the concentration of initiator (CHP), promoter (CoNap), and accelerator (DMA) were individually decreased and the same compositions of matrix were used. The values of interfacial adhesion were compared to the values obtained without the epoxy coating. The interfacial adhesion was also investigated with pure styrene and pure vinyl ester monomer.

5.2.1

Influence of the Concentration of the Initiator

A significant improvement of fiber–matrix adhesion as measured by the IFSS was obtained with the insertion of the epoxy coating between the carbon surface and the vinyl ester matrix (Fig. 13). There was a maximum of interfacial adhesion for a concentration of initiator of 1.5 wt% with the use of an epoxy coating.

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F. Vautard et al.

Influence of the Concentration of the Promoter

An improvement of interfacial adhesion was revealed with the use of an epoxy coating (Fig. 14). No particular trend was noted as a function of the concentration of the promoter.

5.2.3

Influence of the Concentration of the Accelerator

A consistent increase of IFSS was found with the use of an epoxy coating (Fig. 15). No particular trend was revealed as a function of the concentration of the accelerator. All in all, an obvious improvement of interfacial adhesion was created by the insertion of an epoxy coating between the carbon fiber surface and the vinyl ester matrix. No particular influence of the concentration of initiator, promoter, or accelerator was found. A possible difference in the diffusion of those components into the epoxy coating was not responsible for the improvement.

Fig. 14 Influence of the concentration of the promoter on interfacial adhesion with the use of an epoxy coating

Fig. 15 Influence of the accelerator concentration on interfacial adhesion with the use of an epoxy coating

Carbon Fiber–Vinyl Ester Interfacial Adhesion Improvement

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41

Influence of Monomers

Styrene and pure vinyl ester monomer were polymerized with the same thermal treatment. The concentrations of initiator, promoter, and accelerator were adjusted to match the vinyl ester resin composition 1 in Table 2 and be proportional to the content of carbon–carbon double bonds. The shear modulus of styrene and pure vinyl ester was calculated with the values of the modulus measured by a nanoindentation test. The shear modulus of polymerized pure vinyl ester was higher than the shear modulus of polymerized styrene (Fig. 16). The shear modulus of Derakane 411-C50 was not an average value of the pure styrene and pure vinyl ester shear moduli, as it could be expected if the rule of mixture was used, but it was similar to the value of the shear modulus of polymerized styrene. That could be the reason why the interfacial shear strength with AS4 carbon fibers was lower in the case of polymerized styrene (Fig. 17). With epoxy coated fibers, the values of the IFSS were ranked in a reverse order, pure styrene giving the highest value. The difference of interfacial adhesion between vinyl ester monomer and styrene was significant. The

Fig. 16 Shear modulus of polymerized styrene, vinyl ester and Derakane 411-C50

Fig. 17 IFSS of polymerized styrene, vinyl ester and Derakane 411-C50 with coated and uncoated AS4 carbon fibers

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value corresponding to Derakane 411-C50 was intermediate, showing that styrene had a major influence on interfacial adhesion of Derakane 411-C50 when an epoxy coating was spread at the surface of the carbon fibers. The epoxy coating had a negative impact on the interfacial adhesion with polymerized vinyl ester. The two monomers interacted totally differently towards the epoxy coating. The vinyl ester monomer did not diffuse in the epoxy coating, leading to a weak interface. On the contrary, the styrene monomer seemed to spread easily in the epoxy coating and enabled the creation of interpenetrated networks, leading to a strong epoxy coating-matrix interphase. The interdiffusion between epoxy and styrene made the epoxy coating swell, which most probably counteracted the cure volume shrinkage during the polymerization of styrene. The use of an epoxy coating increased the level of interfacial adhesion in the case of Derakane 411-C50 also, which revealed that the styrene contained in that resin diffused prefentially in the epoxy coating and blocked the effect of the matrix shrinkage.

5.3 Influence of the Cure Volume Shrinkage with the Use of an Epoxy Coating The impact of the cure volume shrinkage on interfacial adhesion with carbon fibers was assessed in the case of epoxy coated fibers as well. For samples coated with epoxy, the value of interfacial shear strength measured by the microindentation test remained the same, while it decreased with an increase of the cure volume shrinkage in the case of uncoated fibers (Fig. 18). The coating counteracted the effect of the cure volume shrinkage. When styrene diffused in the epoxy coating, the latter expanded, which annealed the residual stress induced by the cure volume shrinkage. The chemistry and the mechanical properties of both vinyl ester resins being similar after curing, it was assumed that the physical interactions with the carbon fiber surface and with the epoxy coating were similar. A general trend for the IFSS as a function of the cure volume shrinkage could be highlighted for coated and

Fig. 18 Influence of cure volume shrinkage on interfacial adhesion for coated and uncoated fibers

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43

Fig. 19 Assessment of the respective contributions of cure volume shrinkage and of the creation of covalent bonds at the interface on IFSS

uncoated samples (Fig. 19). The respective contributions of cure volume shrinkage and the creation of covalent bonds with the carbon fiber surface could be determined. Indeed, the difference between coated and uncoated samples with minimum shrinkage gave an assessment of the contribution of covalent bonds created between the epoxy and the carbon fibers surface. Covalent bonds do not exist between carbon fibers and vinyl ester resins. For a high volume shrinkage resin (Derakane 411-C50), the effect of residual stress was responsible for more than 50% of the lack of interfacial adhesion between coated and uncoated fibers. Optimum fiber-matrix adhesion can be obtained by properly selecting an initiator and adjusting its concentration. The initiator has indeed a direct influence on the mechanical properties and the microstructure of the matrix [27]. An application of a lightly cross-linked amine-cured epoxy polymer on the carbon fiber surface creates a beneficial interphase between the carbon fiber and the vinyl ester matrix, resulting in an increase of the interfacial adhesion.

5.4 Qualitative Assessment of the Use of an Epoxy Coating on the Mechanical Properties of a Carbon Fiber–Vinyl Ester Composite Cured at High Temperature In order to confirm that low interfacial adhesion in carbon fiber–vinyl ester composites was due to the residual stress induced by the cure volume shrinkage and not by the preferential adsorption of the promoter or the accelerator on the carbon fiber surface only the initiator was added to the resin. Derakane 510A-40, a bromated DGEBA based vinyl ester resin, was mixed with 2 wt% of CHP and cured for 2 h at 150ı C and 2 h at 170ı C. A coating of DGEBA mixed with JeffamineT-403 (43 phr) was coated on the surface of AS4 carbon fibers. The thickness of the coating was 60 nm and it was left for 10 h at room temperature before making contact

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with the matrix. Three cylinder-shaped composites with a diameter of 6 mm and a fiber volume fraction of 50% were made with coated and uncoated fibers. A microindentation test was used for each composite. An obvious improvement of the IFSS was obtained with the epoxy coating again (Fig. 20). A three point flexural test was used to evaluate the cylinder-shaped composites, with a span of 30 mm between the support noses and a vertical speed of 5 mmmin1 for the loading nose. The fracture profile was observed by ESEM. For composites made with uncoated fibers, the fracture profile was typical of a low interfacial adhesion (Fig. 21). The surface had a brush aspect with long and clean protruding fibers. Interface failure could be seen at high magnification. The fracture profile obtained with epoxy coated fibers clearly showed improved interfacial adhesion, with protruding fibers that were short and covered with some pieces of matrix (Fig. 22). Nevertheless, some protruding fibers were still long and clean. The fracture mechanism appeared to be a mixed mode of interfacial and matrix failures. The improvement of interfacial adhesion through the use of this epoxy coating was responsible for that improvement.

Fig. 20 Improvement of interfacial adhesion with the use of an epoxy coating for a carbon fiber– vinyl ester composite cured at high temperature

Fig. 21 ESEM pictures of the fracture profile of a composite with uncoated fibers

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Fig. 22 ESEM pictures of the fracture of a composite with epoxy coated fibers

Fig. 23 Finite element model based on a four phases cylinder

5.5 Determination of the Optimal Thickness of the Coating by a Finite Element Analysis Using an epoxy coating improved the interfacial adhesion between carbon fibers and vinyl ester matrix. It is important to optimize the properties of that coating in order to get the best mechanical properties for the interphase and the composite. The influence of the coating thickness on the value of the IFSS measured by a microindentation test was evaluated with the aid of a finite element model. This finite element model was applicable when the fiber volume fraction was between 0.3 and 0.5 [20]. The microindentation specimen assembly was modeled as a combination of four concentric cylindrical tubes representing the fiber, the interphase, the matrix and the composite, as shown in Fig. 23. The indenter was modeled as a rigid half-sphere. The finite element analysis of this cylindrical representative volume element required only a two-dimensional axisymmetric modeling. Because of the

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symmetry, only half of the cross-sectional plane was analyzed. Fixed boundary conditions were assumed at the sides of the composite. Along the axisymmetric axis (z-axis) and the bottom boundary (r-axis), roller constraints were applied. Interface elements were used between the indenter and the fiber to prevent interpenetration of the two. Coulomb friction was assumed in the indenter-fiber contact zone. A compressive load was applied to the top surface of the indenter. The boundary between the fiber and the interphase, the boundary between the interphase and the matrix, and the boundary between the matrix and the composite were all set to be continuous, which assumed perfect bonds between materials side by side. All of the materials were assumed to be homogeneous, isotropic, and elastic and no residual stress was considered. The width and length of the two-dimensional model were 20 and 100 m respectively. The interface thickness, which corresponded to 36% of fiber volume fraction, was 0:16 m. The interphase was divided into eight layers of 0:02 m. One layer of the interphase was set to have the same properties as DGEBA. The other seven layers were set to have the same properties as Derakane 411-C50. The thickness of the matrix was 2:33 m. The mechanical properties of the materials used in the model are listed in Table 6. The composite properties were calculated based on the rule of mixture for a transversely isotropic composite. The models were meshed using IDEAS software. Axisymmetric four-node elements were used. The boundary conditions were set up using IDEAS software. Contact loading and calculation were carried out using ABAQUS software. The finite element model was a simulation of the real microindentation test. The simulated values of IFSS showed that the optimal thickness for the epoxy coating was 80 nm (Fig. 24). The IFSS decreased sharply for a coating thicker than 100 nm. The tensile moduli of the pure epoxy sizing and vinyl ester matrix were 2.1 and 3.38 GPa, respectively. The epoxy coating was then softer than the vinyl ester matrix. Therefore, as the sizing became thicker, the ability of the interphase to transfer stress to the fiber was affected. This finite element analysis did not take into account the inter-diffusion between the coating and the matrix, which would certainly lead to a more efficient transfer of stress to the reinforcement. Some experimental data were necessary to characterize that zone of interdiffusion. The thickness of the zone presenting interdiffusion was characterized by the nanoindentation scratch technique. Table 6 Material properties for the baseline model used in the finite element model Fiber Interphase Matrix Composite Tensile modulus E (GPa) 241 2.1 3.38 85 Shear modulus G (GPa) 96.5 0.92 1.24 32 Poisson ratio  0.250 0.356 0.356 0.318

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Fig. 24 Simulated IFSS in function of the thickness of the epoxy coating

5.6 Determination of the Thickness of the Interdiffusion Zone by a Nanoindentation Scratch Test The nanoindentation scratch test is a technique that determines the Coefficient Of Friction (COF) between a nanoindenter and a material. With the same vertical load applied to the indenter, the harder the scratched material, the shallower the penetration of the indenter in that material, and the lower the COF. This technique can identify the different phases of a multi-phase material, each phase having a different COF. In the case of an interdiffusion of polymers, the value of the COF changes continuously between the values corresponding to the COF of the pure polymers, thereby is highlighting the extent of the zone of interdiffusion. One factor to be considered with this technique is that the indenter can create an artifact by enlarging the interface formed when the sizing meets the vinyl ester matrix. The amplitude of the artifact depends on the size of the indenter. The artifact was characterized by performing the same test with an epoxy–vinyl ester interface without any interdiffusion. This “reference” interface was made by pouring some vinyl ester on an epoxy layer that was post-cured in an oven to make sure that the cross-linking was complete and that it did not undergo interdiffusion. The sample surfaces were microtomed with a diamond knife. The length of the scratch was 50 m and its depth was around 700 nm. As expected, the width of the coating-matrix interphase was much larger in the case of an interdiffusion (Fig. 25). The indenter created artificially an apparent interphase of 4:5 m when there was no interdiffusion. The thickness of the interphase that underwent interdiffusion was 7 m. The real thickness of the zone of interdiffusion between the epoxy coating and the vinyl ester matrix was then estimated to be 2:5 m.

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Fig. 25 Scratching profile for epoxy–vinyl ester interphases with and without interdiffusion (reference interphase)

6 Conclusion The present study identified cure volume shrinkage as being the main parameter at the origin of the poor level of adhesion existing in carbon fiber–vinyl ester composites. The cure volume shrinkage was dependent on the composition of the resin (molecular weight of the monomer, content of styrene, nature and concentration of the initiator) and the thermal program used to cure it. Preferential adsorption of some components of the matrix such as cobalt naphtenate and dimethyl aniline did not generate a noticeable loss of the mechanical properties of the matrix in the interphase or a loss of interfacial adhesion with the surface of carbon fibers. The improvement of the interfacial mechanical properties with the use of an epoxy coating was demonstrated both by the measurement of InterFacial Shear Strength by the microindentation test and the observation of the fracture profile of composites broken by a flexural test. The mechanism that led to the improvement was characterized. The styrene contained in the vinyl ester resin diffused into the epoxy coating, resulting in an interpenetrating network, which counteracted the stress created by the cure volume shrinkage at the interface. A nanoindentation scratch test estimated the thickness of the interdiffusion zone to be 2–3 m. Finally, a finite element analysis showed that the optimized thickness for the epoxy coating was 80 nm. Acknowledgement Ashland, Huntsman and Hexcel Co. are sincerely thanked for having donated samples of vinyl ester resins (Derakane 411-C50 and Derakane 510A-40), Jeffamine T-403 and AS4 carbon fibers respectively. Financial support for this research from the US Office of Naval Research (Y. Rajapakse) and Florida Atlantic University (R. Granata) is gratefully acknowledged.

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References 1. Kang HM, Yoon TH, Bump M, Riffle JS (2001) Effect of solubility and miscibility on the adhesion behavior of polymer-coated carbon fibers with vinyl ester resins. J Appl Polym Sci 79: 1042–1053 2. Kim IC, Yoon TH (2000) Enhanced interfacial adhesion of carbon fibers to vinyl ester resin using poly(arylene ether phosphine oxide) coatings as adhesion promoters. J Adhes Sci Technol 14: 545–559 3. Robertson MAF, Bump MB, Vergese KE, McCartney SR, Lesko JJ, Riffle JS, Kim IC, Yoon TH (1999) Designed interphase regions in carbon fiber reinforced vinyl ester matrix composites. J Adhes 71: 395–416 4. Schultz J, Nardin M (1990) Interfacial adhesion, interphase formation and mechanical properties of single fiber polymer based composites. In: Ishida H (ed) Controlled Interphases in Composites Materials, Elsevier, New York 5. Schultz J, Nardin M (1994) Some physico-chemicals aspects of the fibre–matrix interphase in composite materials. J Adhes 45: 59–71 6. Nardin M, Asloun EM, Schultz J (1990) Physico-chemical interactions between carbon fibers and PEEK. In: Ishida H (ed) Controlled Interphases in Composite Materials, Elsevier, New York 7. Jennings CW (1972) Surface roughness and bond strength of adhesives. J Adhes 4: 25–38 8. Drzal LT, Sugiura N, Hook D (1997) The role of chemical bonding and surface topography in adhesion between carbon fibers and epoxy matrices. Compos Interfaces 5: 337–354 9. Schultz J, Lavielle L, Martin C (1987) The role of the interface in carbon–fiber epoxy composites. J Adhes 23: 45–60 10. Raghavendran VK, Drzal LT, Askeland P (2002) Effect of surface oxygen content and roughness on interfacial adhesion in carbon fiber–polycarbonate composites. J Adhes 16: 1283–1306 11. Owens DK, Went RC (1969) Estimation of the surface free energy of polymers. J Appl Polym Sci 13: 1741–1747 12. Fowkes FM (1987) Role of acid–base interfacial bonding in adhesion. J Adhes Sci Technol 1: 7–27 13. Fowkes FM (1984) Acid–base contribution to polymer-filler interactions. Rubber Chem Technol 57: 328–344 14. Hook KJ, Agrawal RK, Drzal LT (1990) Effects of microwave processing on fiber-matrix adhesion. 2. Enhanced chemical bonding epoxy to carbon-fibers. J Adhes 32: 157–170 15. Pisanova E, M¨ader E (2000) Acid–base interaction and covalent bonding at a fiber–matrix interface: contribution to the work of adhesion and measured adhesion strength. J Adhes Sci Technol 14: 415–436 16. Bikerman JJ (1961) The Science of Adhesive Joints, Academic Press, New York 17. Paiva MC, Nardin M, Bernardo CA, Schultz J (1997) Influence of thermal history on the results of fragmentation tests on high-modulus carbon-fiber/polycarbonate model composites. Compos Sci Technol 57: 839–843 18. Mandell JF, Grande DH, Tsiang T-H, McGarry FJ (1986) Modified microdebonding test for direct in situ fiber/matrix bond strength determination in fiber composites. Composite Materials: Test and Design ASTM STP 893, American Society for Testing and Materials, pp. 87–108 19. The Interfacial Testing System (ITS). The Dow Chemical Company. Freeport, TX. 20. Ho H, Drzal LT (1996) Evaluation of interfacial mechanical properties of fiber reinforced composites using the microindentation method. Compos A 27: 961–971 21. Weitzsacker CL, Xie M, Drzal LT (1997) Using XPS to investigate fiber/matrix chemical interactions in carbon-fiber-reinforced composites. Surf Interface Anal 25: 53–63 22. Brill RP, Palmese GR (2000) An investigation of vinyl-ester-styrene bulk copolymerization cure kinetics using Fourier transform infrared spectroscopy. J Polym Sci 76: 1572–1582 23. Launikitis MB (1982) Vinyl ester resins. In: Lubin G (ed.), Handbook of Composites, Van Nostrand Reinhold, New York

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24. Penn LS, Chiao TT (1982) Epoxy resins. In: Lubin G (ed.), Handbook of Composites, Van Nostrand Reinhold, New York 25. Xu L, Drzal LT (2003) Influence of cure volume shrinkage of the matrix resin on the adhesion between carbon fiber and vinyl ester resin. Proc of the 26th Annual Meeting of the Adhesion Soc (Feb 23–26, 2003 Myrtle Beach, SC) 26. Xu L (2003) Interfacial engineering of the interphase between carbon fibers and vinyl ester resin. Ph.D. thesis, Michigan State University 27. Li P, Yu Y, Yang X (2008) Effects of initiator on the cure kinetics and mechanical properties of vinyl ester resins. J Appl Polym Sci 109: 2539–2545

A Physically Based Cumulative Damage Formalism Richard M. Christensen

Abstract A general cumulative damage methodology is derived from the basic relation specifying crack growth rate (increment) as a power law function of the stress intensity factor. The crack is allowed to grow up to the point at which it becomes unstable, thereby determining the lifetime of the material under the prescribed stress program. The formalism applies for the case of creep to failure under variable stress history as well as for cyclic fatigue to failure under variable stress amplitude history. The formulation is calibrated by the creep rupture lifetimes at constant stress or the fatigue cycle lifetimes at constant stress amplitude. No empirical (non-physical) parameters are involved in the basic formulation; everything is specified in terms of experimentally determined quantities. Several examples are given showing the inadequacy of Linear Cumulative Damage while the present nonlinear damage accumulation method overcomes these deficiencies. The present method is extended to admit probabilistic conditions.

1 Introduction Damage accumulation in materials is very important, but very challenging to characterize in a meaningful and reliable manner. As the possible damage accumulates, the remaining lifetime under future loads becomes more limited. The ultimate goal is to be able to predict the remaining lifetime as the past history of loading induces a growing state of damage. More succinctly, the common purpose is to be given a complete loading spectrum and then predict how far into the loading sequence the material can remain coherent before suffering catastrophic failure. The conditions under which these damage/lifetime conditions remain as the determining factor is creep failure conditions and fatigue conditions. The creep failure case corresponds to polymers under ambient and also high temperature conditions, R.M. Christensen () Stanford University, Stanford, CA 94305-2047, USA Lawrence Livermore National Laboratory, Livermore, CA 94551-0808 e-mail: [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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as well as metals under high temperature conditions. The cyclic fatigue case corresponds to virtually all materials. The present investigation will consider both cases. Most of the derivation will be formed under the creep to failure condition and then it will be shown how through notational changes the results can be converted to the cyclic fatigue condition. Often the case of creep rupture is referred to as static fatigue. The most common approach to such problems is to recognize that cracks under fatigue conditions usually grow in a manner with the rate of growth expressed as stress level (stress intensity factor) to some exponent. This is widely known as the Paris law, Paris [1], and has been verified for many materials over many decades of change on log scales. This power law form is then used to predict the number of load cycles until the crack reaches a pre-selected, unacceptable size. Such matters are discussed in two excellent sources, Suresh [2] and Kanninen and Popelar [3]. Particular models relate the rate of crack growth to nonlinear functions of the stress intensity factor. Such models include those of Wheeler [4], Willenborg et al. [5] and Elber [6]. Chudnovsky and Shulkin [7] give a somewhat different approach, which still results in typical lifetime forms. In a different approach, a strength evolution methodology has been developed by Reifsnider, most recently in Reifsnider and Case [8]. The controlling form involves an integral representation with a power law time weighting function in the integrand. All of these models have a reasonable physical basis, and the work to be given here, although different, shares the same general background. Another general approach is that of Linear Cumulative Damage, LCD. In this method increments of damage, expressed as fractions of lifetime at particular stress levels, are linearly added together to express total damage and thereby the lifetime. This method is also known as the Palmgren-Miner Law, Palmgren [9], Miner [10]. The method is completely empirical, but quite widely used because of its simplicity and utility. However, LCD is widely acknowledged to be inadequate. This is partially based upon its empirical nature and partly based upon its prediction of unsatisfactory results. LCD has been discussed by Stigh [11], referring to it as the Life-Fraction Rule. In particular LCD was shown to mathematically be related to the continuum damage formulation introduced by Kachanov [12], under certain special conditions. The present work is motivated by all of the approaches just described. In particular the power law form for crack growth will be used as providing a solid, physical approach for the method. The damage and life prediction forms will then be converted to integral forms superficially similar to those of LCD. However, detailed comparisons with LCD will be made in order to highlight its shortcomings and to display how the present methodology overcomes these shortcomings. The starting point here will be that of work recently given by Christensen and Miyano [13]. This previous work had many of the features to be included in the present approach, but it did not successfully compare with typical data for creep rupture conditions in some ranges. In a second paper Christensen and Miyano [14] corrected this deficiency in data modeling, but only in the special case of creep rupture. The work to be given here, [15], also corrects this deficiency, but does so more generally than just for the special case of creep rupture.

A Physically Based Cumulative Damage Formalism

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2 Kinetic Crack Based Cumulative Damage and Life Prediction Initially a state of growing damage is taken to be (highly) idealized as the self similar growth of a single crack. This idealization will be generalized later in the derivation. For the central crack problem start with the widely used power law form expressing the crack growth rate as a function of the stress intensity factor as p  a D . a/r

(1)

where crack size a(t) and stress  .t/ are functions of time, r is the power law exponent and  is a constant. Separate variables in (1) and integrate to get r

1 r2

a1 2 .t /  ao

Z  r t r D 1  . /d  2 o

(2)

where ao is the initial crack size and  . / is the given stress history. Relation (1) is appropriate to the creep conditions that occur for polymers and for metals at high temperature. Rewrite (2) as 

a ao

1 r2

Z  r  r2 1 t r 1D 1  . /d  ao 2 o

(3)

The procedure here is to let the crack grow under the imposed load until it reaches the size at which it becomes unstable at the stress level that exists at that time. To carry out this process, a further relation between a(t) and  . / is needed, which must come from the failure event. In the previous work, Christensen and Miyano [13], the critical stress intensity factor at time t was used to provide the needed relationship. This then gave p p  .t/ a.t / D i ao

(4)

where i is the instantaneous static strength which characterizes failure for the virgin material with initial crack size ao . Although this procedure is appealing, since it follows classical fracture mechanics lines, it did not yield entirely satisfactory results. The corresponding creep rupture times, at constant stress, did give the power law lifetime region controlled by exponent, r, but the transition region at higher stress levels than in the power law region did not provide a good match with data. This difficulty will be shown later as Eq. (17). At this point a more general procedure is needed than that followed previously. Instead of using (4), take the crack size at failure instability as a DF ao



 i

 ; at failure

(5)

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where F() is some function yet to be determined. The form (5) rather than (4) is needed to generalize beyond the idealized single sharp crack in order to account for such effects as crack interaction, crack coalescence, complex non-ideal conditions at crack tips, other types of damage beyond that of idealized cracks such as de-bonding in the case of polymeric fiber composites, and a wide variety of other possible non-ideal effects. In the following work, the growing state of damage will continue to be referred to as that of a growing crack, but with the understanding this growing damage state may take a much more complicated form than that of the idealized single crack. Nevertheless, the basic kinetic relation (1) leading to (3) will be retained and used. The concept of the intrinsic static strength, i , also will be retained and will play a pivotal role in the developments ahead. Combining (3) and (5) gives r

1  F 1 2



 i

 

r 2

 r 1 Z t  r . /d   1 ao2

(6)

o

The inequality in (6) applies for times less than the failure time wherein the crack has not yet reached the critical size for failure. The equality in (6) is at the time then given by t. It is convenient to express (6) in terms of non-dimensional variables. Let nondimension stress be given by  Q D (7) i and non-dimensional time by tQ D

t t1

(8)

where t1 D 

r 2

1  r 1  1 ao2 ir

(9)

Henceforth t1 will be treated as a quantity to be determined from data. On log Q versus log tQ scales different values of i gives shifts in the vertical direction while t1 gives shifts in the horizontal direction. With (7)–(9) the form (6) becomes r

Q  1  F 1 2 ./

Z

tQ

Q r . /d 

(10)

o

Now, let r

f .Q / D 1  F 1 2 ./ Q giving (10) as

Z

tQ o

  Q r . /d   f Q .tQ/

(11)

(12)

A Physically Based Cumulative Damage Formalism

55

where the equality applies at the failure event giving the lifetime. Determine f ./ Q from the basic creep rupture behavior at constant stress. Take the given spectrum of creep rupture life times as tQ D tQc .Q /; at constant stress

(13)

Using (13) in (12) at equality gives f ./ Q as Q f .Q / D Q r tQc ./

(14)

As the final step in this procedure, substitute (14) into (12) and write the inequality and equality in separate forms as tQ < Lifetime Z tQ   Q r . /d  < Q r .tQ/tQc Q .tQ/ ƒ‚ … „ „o ƒ‚ … Critical crack size for Actual crack size at time Qt

(15)

stress at time Qt

tQ D Lifetime Z tQ   Q r . /d  D Q r .tQ/tQc Q .tQ/

(16)

o

As noted under (15) the actual crack size has not yet reached the critical size. Thereafter, the crack does reach the critical size and the lifetime tQ is determined from (16). Next the problem of determining exponent r in (16) is taken up. Rarely does one have direct data on the crack growth rate for a particular material. It would be advantageous to be able to determine r directly from tQc .Q /. If instead of the procedure just developed, the simplified expression (4) had been used, the resulting creep function would have been found to be given by tQc D

1 1  2 Q r Q

(17)

This result from Christensen and Miyano [13] gives the power law range result for low stress as 1 tQc ! r (18) Q however, (17) does not otherwise give a good fit with data. Thus in this case, the power law exponent in the kinetic crack expression (1) is the same as the power law exponent in the lifetime expression (18). The more general result (16) is still taken to have this same behavior such that r in (16) corresponds to the inverse slope in the lifetime power law region, when it exists as in Fig. 1a. The case in Fig. 1a is now generalized to the case of a power law region as in Fig. 1b. Finally the form

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Fig. 1 Determination of “r” from creep rupture

in Fig. 1c with an inflection point is simply the limiting case of Fig. 1b type. Thus exponent r in (16) is determined by the inverse slopes in power law regions as shown in Fig. 1a–c. The cumulative damage relation (15) and the lifetime relation (16) along with the determination of r from tQc .Q / as shown in Fig. 1 are the main results of this derivation. It is to be expected that the critical crack size at time tQ in (16) depends upon the stress level at that time. Lower stress levels require larger crack sizes. There is a simple conceptual test that can be applied to the result (16). As discussed by Christensen and Miyano [13] the lifetime results for constant strain rate testing, CSR, gives lifetimes that are shifted on log scales from those for creep rupture. To test this behavior on (16) take the stress history as Q D ˇ

(19)

where ˇ is a constant. Substitute this into the equality form of (12), and integrate to get r C1 f .Q .tQ// (20) tQrC1 D ˇr

A Physically Based Cumulative Damage Formalism

57

Now eliminate ˇ from (20) using (19) to get tQ D .r C 1/

f .Q .tQ// Q r .tQ/

(21)

From (14) this can be written as tQ D .r C 1/tQc .Q /

(22)

Thus using (22) the lifetime for CSR can be written as log tQCSR D log tQc C log.r C 1/

(23)

The shifting property relating creep rupture and CSR to failure is found to be preserved by the general form (16). At this point it is useful to compare the present cumulative damage result (16) with that from linear cumulative damage, LCD. The LCD form is Z o

tQ

d D1 tc ..// Q

(24)

whereas the present form, rewritten from (16) is 1   Q r .tQ/tQc Q .tQ/

Z

tQ

Q r . /d  D 1

(25)

o

Note that if the front factor in (25) were arbitrarily moved to inside the integral at time  , the result would be identically that of LCD, (24). But there is no justification for this transposition, it would destroy the kinetics inherent in the present derivation. LCD, (24), can only be viewed as a directly postulated damage relation which is totally empirical. Lifetime relation (25) follows from the physical derivation given here.

3 A Special Form There is a special form for creep rupture tQc .Q / of the type shown in Fig. 1a that affords considerable advantage in applications, Christensen and Miyano [14]. This is given by .1  Q p /q tQc D (26) Q r

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where p, q, and r are parameters, with r being that for the power law range. With (26) the form (16) becomes  LCD is given by

1

  q 1  Q p tQ Z o

tQ

Z

tQ

Q r . /d  D 1

(27)

o

Q r . /d  D1 Œ1  Q p ./q

(28)

As a first application of the present results and LCD consider the case of residual instantaneous strength after the loading of the material for a specified amount of time. The problem is as shown in Fig. 2. The material is loaded at stress Q 1 for time tQ1 which is less than that which would cause creep rupture, but nevertheless does impart some damage to the material. Relation (27) then gives  r  1 Q 1 tQ1 D 1 .1  Q R p /q

(29)

where Q R is the residual strength. Solving (29) for Q R gives   p1 r 1 ; Present Q R D 1  Q 1q tQ1q

(30)

It is easiest to reason the LCD result directly from (24). Up to time tQ1 in Fig. 2 the integral has some value say ˛ which is less than one. The following instantaneous loading to Q R must generate a value 1  ˛ to satisfy the equality in (24). For this integral to have a finite value over a vanishing time interval requires that tQc .Q R / D 0. This latter result only occurs for Q R D 1, so Q R D 1; LCD R D i

Fig. 2 Instantaneous residual strength

(31)

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59

Fig. 3 Creep rupture example, Eq. (26): p D 15; q D 100, and r D 10

As a simple example take p D 15; q D 100; r D 10

(32)

The resulting creep rupture lifetime function is shown in Fig. 3. Its time variation is spread over many decades, which is typical for many materials. At Q 1 D 0:5 and t1 D 12 tQc .Q 1 / it is found from (30) that Q R D 0:718; Present while Q R D 1; LCD Thus the present residual strength prediction is considerably above the previous load level, but still much less than Q D 1. In contrast LCD is unable to account for the previous accumulation of damage, and simply predicts the undamaged value of Q R D 1. LCD is extremely unconservative in this case.

4 Cyclic Fatigue The previous results for creep damage type of cumulative effect can be converted to the case of cyclic fatigue by suitable notation and terminology changes. Let f be the frequency in cycles per unit time and n be the number of cycles. Then the elapsed time is n (33) tD f

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Some materials show a frequency dependence, so the present method will be developed for fixed frequency and fixed form of the variation within one cycle. The starting point is to define and take as given the standard fatigue relation between constant stress Q and the number of cycles to failure. Write this as n D N.Q /; constant stress

(34)

Where n is the cycles to failure and Q is the non-dimensional stress Q D

 i

where i is the instantaneous static strength corresponding to the maximum tensile stress within the cycle of variation. With these notational changes the previous lifetime creep damage form (16) converts to the cyclic fatigue case as Z

n

Q r . /d D  r .n/N.Q .n//

(35)

o

where stress is allowed to have a variable amplitude so long as the wave form and frequency are unchanged. Variable in (35) is the past history variable for n. Exponent r in (35) is determined by the same method as in the creep damage case of Fig. 1. Any of the three forms in Fig. 1 are admissible with the case of Fig. 1a illustrated in Fig. 4. The term on the left side of (35) represents the current crack size while the term on the right side is the critical size at the stress existing at n number of cycles. The equality in (35) is for the lifetime n in number of cycles.

Fig. 4 Fatigue counterpart of Fig. 1

A Physically Based Cumulative Damage Formalism

61

5 Probabilistic Generalization For extension to probabilistic conditions the terminology of the creep damage case will be used. It will be understood that the same extension applies to fatigue conditions through the change of notation given in the previous section. Rewrite the deterministic form (16) for the creep case as Z tQ

o

 . / i

r



 .tQ/ d D i

r

 tc

 .tQ/ i

 (36)

The instantaneous static strength, i , will be generalized to a probabilistic form through any particular distribution function. With this form then i can be written as i D .k/

(37)

where .k/ is a function of the quantile of failure, k, 0  k  1. Function .k/ is determined by the distribution function, such as a Weibul form or any other one. Let the probabilistic, non-dimensional time to failure be given by . Now, the probabilistic form of (36) is given by Z o





./ Q

.k/

r



Q ./ d D

.k/

r

tQc



Q ./

.k/

 (38)

Given the stress history and k, this relation can be solved for the probabilistic time to failure, . The result is particularly simple for the creep rupture case itself. Write the resulting probabilistic time to failure  as a function of the applied stress level and .k/. Then   Q Dh (39a)

.k/ Now use log scales for stress and time giving log  D g .log Q  log .k//

(39b)

Where g() is a function found from h() after the conversion to log scales. This form reveals a vertical shift along the log Q axis with all curves emanating from a single master curve. The above procedure was illustrated by Christensen and Miyano [13] for the case of using Weibul statistics to represent i . The resulting lifetimes can be shown to be of Weibul form within the power law range but they are not of Weibul form outside of it. Christensen and Miyano [14] provided further experimental verification for these related forms.

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6 Examples Several examples will be given to illustrate the difference between the present approach and LCD. The creep rupture form given by (26) will be used with parameter values in (32). Examples of two and three stress step levels will be used. For two stress levels as shown in Fig. 5, the present form (27) gives tQ D tQ1 C

.1  Q 2 p /q  tQ1 Q 2 r



Q 1 Q 2

r (40)

For LCD form (28) gives tQ D tQ1 C

.1  Q 2 p /q  tQ1 Q 2 r



Q 1 Q 2

r 

1  Q 2 p 1  Q 1 p

q (41)

The last term in parenthesis in (41) is the only difference in the two expressions. In some cases it will be found to be a crucial difference. Three step examples are also readily formed from (27) and (28). The specific examples are as follows Example 1 Q 1 D 0:5 up to tQ1 D 500 Then Q 2 D 0:7 to failure. Lifetimes: tQ D 504:7 Present, tQ D 511:2 LCD Example 2 Q 1 D 0:85 to tQ1 D 0:0005 Then Q 2 D 0:5 to failure, Lifetimes: tQ D 1020:8 Present, tQ D 83:5 LCD Example 3 Q 1 D 0:5 to tQ1 D 500

Fig. 5 Two step load

A Physically Based Cumulative Damage Formalism

63

Then Q 2 D 0:7 to tQ2 D 504 Then Q 1 D 0:5 to failure. Lifetimes: tQ D 909:2 Present, tQ D 839:2 LCD Example 4 Q 1 D 0:5 to tQ1 D 10 Then Q 2 D 0:8 to tQ2 D 10:15 Then Q 1 D 0:5 to Failure, Lifetimes: tQ D 1;004:5 Present, tQ D 430:1 LCD These examples show significant differences between the present method and LCD. It is not easy to prescribe simple rules by which LCD is or is not acceptable except to say that for increasing loads the LCD results appear to be satisfactory.

7 Extended Life Examples Now some examples will be given for which the LCD results exhibit the most egregious shortcomings. The examples are for cases where at a constant load the material is taken right up to the point of incipient failure. But just an instant before failure the load is reduced to a specified level. The problem is to determine the remaining lifetime at the reduced load level. Use the same material specification through tQc .Q / in (26) with parameter values in (32). Take Q 1 D 0:6 up to incipient failure and then reduce the stress levels to Q 2 D 0:5, 0.4, or 0.3 and find the remaining total lifetimes. At Q D 0:6 the creep rupture lifetime is tQc D 157:8 Q 2 D 0:5; tQ D tQc C tQ D 201:7 Using expression (40) the present lifetime extensions are found as Q 2 D 0:4; Q 2 D 0:3;

tQ D 594:9 tQ D 7;938:0

In the third case the lifetime is extended by a factor of 50. When the load level is taken to incipient failure, the LCD method predicts that any additional load at any level will produce immediate failure, as can be verified from (41) with tQ1 D tQc . Thus for LCD tQ D 0

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R.M. Christensen

LCD is unable to account for the fact that the reduced stress level requires that the crack obtain an increasing size before it can reach the criticality condition at the reduced stress level.

8 Conclusions The main results of the present work are the lifetime forms (16) for creep failure and (35) for fatigue. Exponent r in (16) is evaluated from creep rupture at constant stress Q as shown in Fig. 1, with a similar operation for the case of fatigue. These tQc ./ forms accommodate prescribed variable stress amplitudes, but in the case of fatigue the frequency and wave forms are taken to be unchanged. In the non-dimensional stress (7) and non-dimensional time (8), i and t1 are calibrating factors that allow vertical and horizontal shifts along the log axes. The basic failure property inputs are the creep rupture life times tQc .Q / in (16) and fatigue lifetimes N .Q / in (35) both at constant stress amplitude. There are no empirical parameters involved in the theory and final lifetime forms. Several conclusions can be drawn from the general structure of the theory and the examples that have been considered. Comparing LCD with the present more complete and physically based methodology shows that (i) LCD is acceptable for monotonically increasing loads (stress). (ii) LCD is not satisfactory for the residual strength problem. After a given stress history, but before failure, the LCD residual instantaneous static strength, Q R is undiminished from the virgin material value of Q R D 1. The present method gives a reduced value for Q R based upon the damage accumulated in the past stress history. (iii) The following conclusions apply for the extended life problem where the stress history is taken up to incipient failure, but instantaneously reduced to a lower stress level in order to prolong the life of the material. LCD gives no additional or extended lifetime after the decrease in load level. The present method gives an extended lifetime beyond the point at which the load is suddenly reduced. This is perhaps the most fundamental difference between the two methods with LCD being unrealistically conservative. In the other examples that have been considered, some cases showed a good comparison between the two methods while others showed a poor comparison, sometimes different by an order of magnitude or more. In general it must be said that LCD gives an unacceptable result in an unacceptable number of cases. The general conclusion is that this new cumulative damage formalism is no more difficult to apply than Linear Cumulative Damage, but it has a physical interpretation based upon crack growth and it readily admits generalization to probabilistic conditions.

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65

References 1. Paris PC (1960) The growth of cracks due to variations in loads. Ph.D. thesis, Lehigh University, Bethlehem, Pennsylvania 2. Suresh S (1998) Fatigue of materials, 2nd edn., Cambridge University Press, Cambridge 3. Kanninen MF, Poplar CH (1985) Advanced fracture mechanics, Oxford University Press, Oxford 4. Wheeler OE (1972) Spectrum loading and crack growth. J Basic Eng 94:181–185 5. Willenborg J, Engle RM, Wood H (1971) A crack growth retardation model using an effective stress intensity concept. In: Technical Report TFR 71-701. North American Rockwell 6. Elber W (1970) Fatigue crack closure under cyclic tension. Eng Fract Mech 2:37–45 7. Chudnovsky A, Shulkin Y (1996) A model for lifetime and toughness of engineering thermoplastics. In: Jordan et al. (eds.), Inelasticity and damage in solids subject to microstructural Change, Univ. Newfoundland 8. Reifsnider KL, Case SC (2002) Damage tolerance and durability in material systems, WileyInterscience, New York 9. Palmgren A (1924) Die Lebensdauer von Kugellagren. Zeitscrift des vereins deutscher ingenieure 68:339–164 10. Miner MA (1945) Cumulative damage in fatigue. J Appl Mech 12:159–164 11. Stigh U (2006) Continuum damage mechanics and the life-fraction rule. J Appl Mech 73: 702–704 12. Kachanov EL (1958) On the time to failure under creep conditions. Izv. Akad. Nauk SSR, Otd Tekhn. Naukn. Nauk 8:26–31 13. Christensen RM, Miyano Y (2006) Stress intensity controlled kinetic crack growth and stress history dependent life prediction with statistical variability. Int J Fract 137:77–87 14. Christensen RM, Miyano (2007) Y Deterministic and probabilistic lifetimes from kinetic crack growth-generalized forms. Int J Fract 143:35–39 15. Christensen RM (2008) A physically based cumulative damage formalism. Int J Fatig 30: 595–602

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Delamination of Composite Cylinders Peter Davies and Leif A. Carlsson

Abstract The delamination resistance of filament wound glass/epoxy cylinders has been characterized for a range of winding angles and fracture mode ratios using beam fracture specimens. The results reveal that the delamination fracture resistance increases with increasing winding angle and mode II (shear) fraction .GII =G/. It was also found that interlaced fiber bundles in the filament wound cylinder wall acted as effective crack arresters in mode I loading. To examine the sensitivity of delamination damage on the strength of the cylinders, external pressure tests were performed on filament-wound glass/epoxy composite cylinders with artificial defects and impact damage. The results revealed that the cylinder strength was insensitive to the presence of single delaminations but impact damage caused reductions in failure pressure. The insensitivity of the failure pressure to a single delamination is attributed to the absence of buckling of the delaminated sublaminates before the cylinder wall collapsed. The impacted cylinders contained multiple delaminations, which caused local reduction in the compressive load capability and reduction in failure pressure. The response of glass/epoxy cylinders was compared to impacted carbon reinforced cylinders. Carbon/epoxy is more sensitive to damage but retains higher implosion resistance while carbon/PEEK shows the opposite trend.

1 Introduction Composite cylinders are widely used, designed to resist either internal pressure (e.g. fluid transport and storage, cooling systems) or external pressure (e.g. underwater applications). One of the main concerns with these structures is the propagation of P. Davies () Materials & Structures Group, IFREMER, 29280 Plouzan´e, France e-mail: [email protected] L.A. Carlsson Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431, USA e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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P. Davies and L.A. Carlsson

interlaminar defects, produced during manufacture or service. Understanding the factors affecting the delamination resistance of the laminate used in the cylinders is essential if their safe working envelope is to be defined. The analysis and test methods developed here are necessary for conducting a fracture mechanics analysis of the severity of defects in composite cylinders, and hence provide the basis for a more rational approach to both NDT after manufacture (acceptance/rejection criteria) and to in-service inspection. Several tests have been developed to characterize the delamination resistance of composite materials, see e.g. recent publications of Davies [1], but such tests suffer from several drawbacks when cylindrical composite structures are considered. The standard test methods are developed for high-modulus unidirectional composites. Very little work has been reported on other types of composite systems and lay-ups, although cylindrical structures commonly involve low-modulus glass fibers and angle-ply laminates. Furthermore, almost all previous work has considered fracture specimens with rectangular (flat) cross-section, while specimens machined from cylinders have curvature in one direction. Riddle and Beckwith [2] tested mode I and mode II specimens machined into flat coupons from thick carbon/epoxy cylinders. Davies and Rannou [3] performed DCB tests on curved glass/epoxy specimens cut from cylinders and analyzed the data using compliance calibration which neglects several details of the curved laminate specimens that may influence measured toughness values. The delamination of composite tubes under impact loading has been reported by various authors. Alderson and Evans [4, 5] presented experimental data from a large impact test programme on glass/epoxy cylinders. Curtis et al. [6] studied the influence of impact damage on burst strength of thin walled tubes under internal pressure loading and noted that high energy indentation reduced burst strength by 60%. Christoforou et al. [7] impacted thin wall carbon/epoxy cylinders and also noted significant strength loss under internal pressure. Few data are available to enable the influence of impact damage on cylinders subjected to external pressure to be evaluated. A recent study by Gning et al. [8] presented data on implosion of glass/epoxy filament wound cylinders while Davies et al. presented some test data for carbon reinforced epoxy and thermoplastic composite cylinders [9]. The objective of this Chapter is to review our work on the delamination resistance of filament wound composite cylinders and the influence of delamination damage on the failure of externally pressurized composite cylinders. Pressure tests on cylinders without implanted defects and with impact damage and implanted defects to simulate fabrication and in-service damage, were performed, and the relation to measured delamination toughness is discussed.

2 Materials and Specimens Delamination specimens were machined from composite cylinders consisting of E-glass fibers in an epoxy resin. The internal diameter of the cylinders was 160 mm and the nominal wall thicknesses were 6 and 12 mm (12 and 24 layers). The lay-ups

Delamination of Composite Cylinders

69

Fig. 1 Illustration of procedure to obtain delamination beam specimens from filament wound composite cylinders

of the cylinders were Œ˙™6 and Œ˙™12 , where ™ D 30ı ; 55ı and 85ı . A 58 mm long and 13 m thick, release agent coated aluminum film, was inserted at the mid-plane of the cylinders during the filament winding process to define a starter delamination crack, see Fig. 1. The film insert was wrapped around the circumference of the cylinder to enable machining of multiple test specimens from each cylinder. After filament winding, the cylinders were cured at 160ı C for 3 h. The average fiber volume fraction was 0.61 for all cylinder lay-ups. Beam fracture specimens of a nominal length of 200 mm and a nominal width of 18 mm were cut from the cylinder wall for the subsequent delamination tests as schematically illustrated in Fig. 1. The beam axis was parallel to the cylinder axis producing straight beams with a curved cross-section. Figures 2–4 show the DCB, ENF and MMB cylinder specimens and loading principles. In order to accommodate the curved cross-section of the beam fracture specimens, contoured aluminum loading tabs were fitted to the DCB and MMB specimen and contoured loading pins and supports were attached to the ENF and MMB test fixtures as shown in Figs. 2–4. Further experimental details are provided in Refs. [10–12]. Glass/epoxy cylinders of lay-up Œ˙55n were manufactured for the external pressure tests using the filament winding process. Cylinder internal diameters were 55 and 175 mm, with wall thicknesses of 6.5 and 19 mm. The fiber volume fraction was 0.68. E-glass fibers in these cylinders were impregnated with the same epoxy resin as the cylinders used for delamination specimens. After winding, the cylinders were cured at 125ı C for 7 h. In order to simulate fabrication defects in the form of delaminations that may arise during cure of thick cylinders due to exothermic heating, 50 mm square aluminum foil layers of 13 m thickness, coated with release agent on both sides, were introduced at different thickness locations during filament

70

P. Davies and L.A. Carlsson

Fig. 2 DCB cylinder specimen

Fig. 3 ENF cylinder specimen

winding into some of the 55 mm diameter cylinders. The wall consisted of 12 layers and defects were placed between the third and fourth (referred to as 1/4 thickness), sixth and seventh (mid-thickness), and ninth and tenth layers (3/4 thickness) where ply #1 is the inner layer and ply #12 is the outer layer of the cylinder. Carbon fibre (T700) reinforced epoxy cylinders, (the same epoxy resin as for the glass reinforced cylinders), were filament wound with the same dimensions and cure cycle as for the glass/epoxy cylinders. Carbon (AS4)/PEEK (poly-ether-ether-ketone) cylinders were produced with the same dimensions by tape laying.

Delamination of Composite Cylinders

71

Fig. 4 MMB cylinder specimen

3 Delamination Fracture Testing DCB, ENF and MMB specimens obtained from the glass/epoxy cylinders were tested. In addition, flat unidirectional DCB, ENF and ADCB [13] specimens were prepared and tested. The ADCB specimen is a DCB specimens where only one arm is loaded. This produces a mixed mode ratio GI =GII D 4=3 [13]. Five or six replicates were used for each test. All fracture tests were performed from the end of the insert without precracking. The specimens were loaded in displacement-control at a slow cross-head speed, between 0.5 and 1 mm/min, to allow for visual monitoring of the crack propagation. Displacement of the cross-head was measured with a LVDT, and a real-time load (P) versus displacement (•) response was recorded. The onset of crack growth from the starter insert was determined by the inspection of the specimen edge with a traveling microscope during the test and observation of the P  • response on the chart recorder. Fracture testing of the curved DCB specimens employed the incremental method proposed by Wilkins et al. [14]. Data reduction for the flat specimens was performed using the experimental compliance calibration for the DCB specimen [15, 16]; beam theory for the ENF specimen [17, 18], and large displacement corrected beam theory for the ADCB specimen [19]. Energy release rate expressions were derived for the DCB, ENF and MMB cylinder beam specimens to allow for fracture toughness determination [4–6]. Such analysis incorporates the effect of curvature of the beam cross section, asymmetry of the delaminated region due to lay-up and thickness, and elastic foundation

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P. Davies and L.A. Carlsson

Fig. 5 Impact set-up

Tower

Anti-rebound device Cradle

effects for the finite beam geometries. The experimental compliance calibration method [15, 16] was used to reduce the delamination fracture toughness of the DCB and ENF specimens. Compliance calibration of the MMB test is not possible and beam theory [12] was used to reduce the mixed mode delamination fracture toughness, Gc .

4 Impact Testing of Cylinders Impact damage was introduced into cylinders using steel hemispherical impacters, dropped in the radial direction from different heights at mid-length of horizontally arranged cylinders placed in a cradle, Fig. 5. Two series of tests were performed on 55 mm diameter cylinders, in the first a 34 mm diameter impacter was used with 400 and 600 g masses dropped from 2 m. For the second series of tests, of the 55 mm diameter cylinders, mass was 1.6 kg, and drop height was varied. 175 mm diameter cylinders were impacted with a 100 mm diameter steel hemispherical impacter, of mass 10.9 kg. In all cases rebound impacts were prevented. Impact damage was characterized using ultrasonic C-scan equipment in the transmission mode, before implosion testing. This provides a projected damage area. A series of impacted cylinders was also sectioned, in order to measure throughthickness damage and total delaminated area by microscopy. A fluorescent dye penetrant was employed to improve contrast in carbon reinforced composites.

5 External Pressure Tests of Cylinders A series of external pressure tests was performed on the cylinders to examine the influence of different types of delamination. Two series of pressure tests were performed on the 55 mm diameter cylinders. For the first series, the cylinders ends were

Delamination of Composite Cylinders

a

73

b

c

Fig. 6 Cylinders before implosion tests. (a) Small diameter original set-up, (b) small diameter second series with aluminium end-caps, (c) large diameter cylinder

machined and they were simply placed between two thick steel end blocks, Fig. 6a. Silicon compound was applied around the end surfaces to maintain sealing at low pressures. This set-up was used for the tests on glass/epoxy cylinders with and without implanted defects and impacted with the 34 mm diameter impacter. In a second more recent series of tests on 55 mm diameter cylinders made from various composite materials, the composite ends were sealed by specially designed aluminium end caps, Fig. 6b. This change in boundary conditions resulted in higher implosion pressures for the second series of tests on glass/epoxy cylinders, values increasing from 900 to over 1,100 bars. The large cylinders were equipped with tapered end plugs, Fig. 6c, designed to provide a progressive contact with the inside of the cylinder as pressure increases. All three series were loaded to failure in IFREMER’s 2,400 bar pressure tank, Fig. 7. The cylinder length was chosen to be sufficiently short (twice the internal diameter) to minimize the possibility for global buckling of the cylinders. The principal stresses in pressurized cylinders are the axial and hoop stress. The magnitude of the hoop stress is twice that of the axial stress. The apparent strength expressed as

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P. Davies and L.A. Carlsson

Fig. 7 Hyperbaric test facility, large glass/epoxy cylinder before test

hoop stress at failure is calculated using the expression for thin wall pressurized cylinders [20]: pR (1) ™™ D h where p is the external failure pressure, R is the mean radius and h is the wall thickness. The failure mode of cylinders subjected to hydrostatic pressure testing depends on the thickness-to-radius ratio h/R. A large amount of testing has resulted in the identification of two domains, with low h/R values corresponding to thin wall tubes which buckle, and thick wall cylinders with high h/R values for which material (compression) failure is observed, Fig. 8. Various formulations have been applied to predict the failure pressures; for buckling an expression developed by Mistry et al. [21] has been shown to give good agreement with glass/epoxy test data. Strength predictions based on material failure are more controversial. Various failure criteria have been proposed [22], but a maximum hoop stress criterion has been used with some success by the authors. The transition region is not clearly defined but all the cylinders tested within this study are well within the thick wall domain, with an h/R ratio around 0.2. In order to examine the loading of the inner sublaminate of the delaminated region a hoop oriented strain gage was bonded at mid-section to the inner wall of the 55 mm diameter cylinders, and a reference gage was bonded to the diametrically opposite inner surface of the cylinder. The 175 mm diameter cylinders were instrumented with up to 40 strain gages on the inner wall, as will be described below.

Delamination of Composite Cylinders

75 Cylinder failure modes

1500

Implosion pressure, bars

BUCKLING

MATERIAL FAILURE

1000

h/R = 0.2 500 Max. Hoop stress Buckling (Mistry[21])

0 0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

h/R Fig. 8 Failure modes, external pressure tests on glass/epoxy ˙55ı cylinders

6 Results and Discussion 6.1 Delamination Fracture Test Results The glass/epoxy composite cylinder DCB specimens displayed stable load– displacement .P  •/ response. The crack propagation load increased with increased crack length, corresponding to “R-curve behavior”. Interlaced fiber bundles intersecting the crack path resisted the crack growth, and subsequent crack growth was observed at another interface. In addition to the interlaced fibers, it was found that pull-out of bridged fibers contributed to the fracture resistances and R-curve behavior. The ENF and MMB specimen also displayed crack jumping and formation of secondary cracks, although their inherent instability made such observations more difficult. Figure 9 shows delamination fracture toughness, Gc for the cylinder laminates plotted versus the mode II fraction, GII =G. Also included in the graph are the Gc (NL) data for the flat unidirectional glass/epoxy composite. The initiation toughness increases with increasing mode II fraction .GII =G/ and angle ™ at the ˙™ interface. The implications of the toughness trends shown in Fig. 9 for the integrity of a cylinder containing a delamination would require extensive analysis. Such analysis is not currently available. The analysis of composite cylinders by Kardomateas and Chung [23], however, shows that a necessary condition for delamination growth is buckling of the sublaminate/s of the delaminated region. Once the delamination buckles, it would tend to propagate in a direction governed by the local balance between available strain energy release rate and fracture resistance at the given mode mixity (Fig. 9). Considering the consequence of increasing toughness with increasing angle, ™, in the angle-ply laminates, Fig. 9, for the cylinders, it is recognized that a delamination

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P. Davies and L.A. Carlsson

Fig. 9 Mixed mode delamination fracture toughness for flat unidirectional glass/epoxy laminates and angle-ply glass/epoxy cylinder specimens

in a Œ˙556 cylinders would propagate in the hoop direction, because the ˙35ı interface is less tough (assuming the driving force, G, is the same for both the axial and hoop directions). It is also expected that the larger stiffness of the wall laminate in the hoop direction than in the axial direction would promote mode I dominated loading of the crack front as discussed by Whitcomb [24]. This would further contribute to extend the delamination in the hoop direction since the toughness is decreasing with increased mode I fraction, see Fig. 9.

6.2 Influence of Impact The series of 55 mm diameter glass/epoxy, carbon/epoxy and carbon/PEEK specimens was impacted at different energies with a 50 mm diameter impacter. Impact energy is defined as mgh, with h the drop height. Figure 10 shows examples of C-scan projected damage maps for the three materials. It is clear that at iso-energy the damage is restricted to a smaller region near the impact point in the carbon/ PEEK cylinders, compared to more extensive delamination in the two epoxy matrix composites. The damage area is larger in the carbon/epoxy than the glass/epoxy. Figure 11 shows examples of damage revealed by sectioning after impact. There is quite extensive delamination in the carbon/epoxy at all the layer interfaces, delaminations and angled cracks in the glass/epoxy composite, and evidence of minor damage in the thermoplastic C/PEEK composite.

6.3 External Pressure Test Results Figure 12 shows the response of the hoop strains versus applied pressure for the Œ˙556 cylinders without a defect, and with an implanted delamination at 1/4, midthickness, and 3/4 locations.

Delamination of Composite Cylinders Fig. 10 Damage maps obtained by ultrasonic C-scan inspection

77

8J

16 J

24 J

Glass/epoxy

Fig. 11 Sections through impacted cylinders, three materials, 16 J impact energy, all 6.5 mm wall thickness. Arrows indicate impact point. (a) Glass/epoxy, (b) carbon/epoxy, (c) carbon/PEEK

C/Epoxy

C/PEEK

a

Glass/epoxy

b

Carbon/epoxy

c

Carbon/PEEK

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P. Davies and L.A. Carlsson

Fig. 12 Hoop strain versus pressure response for Œ˙556 cylinders without and with implanted 50  50 .mm/ delaminations at instrumented (i) and opposite (o) locations

The cylinders displayed a linear strain response. For the cylinders without defect and 3/4 defect location, the two inner wall strain gages displayed almost identical responses. For the cylinders with an implanted delamination at mid-thickness and 1/4 locations, the strain gages bonded to the sublaminate of the delaminated region responded differently than those on the opposite wall (without defects). For the 1/4 location defect a pressure of 75 bars was required before the inner sublaminate of the delaminated region of the cylinder experienced compression strain. For the cylinder with a defect at mid-wall thickness, the load transferred to the delaminated inner wall at 50 bar pressure. As pressure is increased, the outer part of the cylinder wall then presses against the inner delaminated part, and there is no further divergence of the strain responses. Figure 13 shows a summary of the hoop strength test results for the first series of tests on Œ˙556 glass/epoxy cylinders without and with delamination. The presence of single delaminations has no significant influence on the implosion pressure. For cylinders impacted by a hard object with energy above 5 J, sectioning and examination of the impacted region revealed that multiple delaminations occurred through the wall thickness, Fig. 11. The summed area of the delaminated surfaces after an 8 J impact, determined from sections in the axial and hoop directions, was found to be very similar to the area of the 5050 .mm/ artificial defect .2;500 mm2 /. When these impacted cylinders were pressure tested, local failure occurred, and there was a 15% to 20% reduction in failure pressure, see Fig. 13. These results suggest that for the Œ˙55ı 6 glass/epoxy cylinders examined here, impact damage is likely to be more harmful in service than fabrication defects. It is interesting to examine how the results from these small cylinders translate to larger cylinders. Three cylinders scaled up by a factor of about 3 (diameter and wall thickness) were tested. The first was tested to failure without damage, the two others were impacted at 110 and 220 J energies respectively. All cylinders were extensively

Delamination of Composite Cylinders

79

Fig. 13 Hoop stress at failure for externally pressurized Œ˙556 55 mm diameter cylinders

Fig. 14 Strain gaging of inner wall of large glass/epoxy cylinder. Ring of 12 biaxial gages at mid-section. Additional gages were placed along the cylinder axis to check strain uniformity

instrumented with strain gages, including a ring of 12 gages in the hoop direction at every 30ı on the inner wall at mid-section, Fig. 14. Figure 15 shows an example of hoop and axial strains measured at mid-section on the inner walls of the 175 mm diameter cylinders, compared with the values from two gages in a 55 mm diameter cylinder. These indicate similar responses, though the latter are a little stiffer in the hoop direction. Figure 16 shows the mid-section hoop strains recorded during the pressure tests for undamaged and 220 J impact damaged cylinders. The undamaged cylinder deforms uniformly to pressures over 800 bars (Fig. 16a).The damage introduced at the higher impact energy clearly promotes premature buckling. The damaged cylinder however, shows non-uniform strain even at 200 bars, and this develops rapidly to

80

P. Davies and L.A. Carlsson Hoop stress (MPa) 0

200

400

600

Hoop microstrain

0

–5000

–10000 55mm –15000

175mm

–20000

Fig. 15 Mid section hoop strains, 55 and 175 mm diameter undamaged cylinders under external pressure

result in failure below 500 bars (Fig. 16b). Final failure occurs by local buckling at the point of impact, as shown in Fig. 17. Scaling of the mechanical behaviour of composites is a complex subject. Scaling laws using the non-dimensional analysis or …-theorem of Buckingham [25] were applied to composites by Morton [26]. These are based on the following assumptions: – The sub-scale model and the structure are made of the same material. – The model is a geometrical copy of the structure. Swanson et al. [27] used this approach to study scale effects during impact of thin wall carbon/epoxy tubes, Davies applied it to glass/epoxy tubes [28]. In order to introduce the same degree of impact damage in cylinders which are scaled dimensionally by a factor of , (3 here), the impact energy should be scaled by 3 . So if we plot the residual pressure versus impact energy for the large cylinders and versus scaled-up impact energy (impact energy  27) for the small cylinders, the results should superpose. Figure 18 shows the normalized implosion pressure plotted versus this scaled impact energy, from tests on small and large diameter cylinders. The trends are similar but the drop in strength is more pronounced in the larger cylinders. There are many possible reasons for this; the scaling is not exact. As shown by Fig. 15, the smaller cylinders are slightly stiffer than the large ones. Residual stresses may also develop during fabrication [29]. Their level depends on wall thickness, cure conditions and radius of curvature and they may affect damage development. Finally, the use of alternative materials, carbon/epoxy and carbon/PEEK, has been investigated. This choice has a small influence on the implosion strength of undamaged cylinders (the carbon/PEEK strength was about 15% lower than the other two) but the impact damage observed in Figs. 10 and 11 results in very different implosion strengths for the three materials. Figure 19 shows the normalized results from implosion tests after impact.

Delamination of Composite Cylinders

a

microstrain

81

0

Glass/epoxy Cylinder Undamaged Circumferential Gauges Central Section

0 330

30

–2000 –4000 –6000

300

60

–8000

angle, degrees

–10000 –12000 –14000 270

400 bar

90

–16000

700 bar 800 bar 939 bar 240

120

210

150 180 microstrain

b

0

Glass/epoxy Cylinder 220 J impact Circumferential Gauges Central Section

0 330

–1000

30

–2000 –3000 300

60

–4000

angle, degrees

–5000 –6000 –7000 270

90

–8000

200 bar 300 bar 400 bar 479bar

120

240

150

210 180

Fig. 16 Mid section hoop strains, 175 mm diameter glass/epoxy cylinders. (a) No damage, (b) 220 J

It is interesting to note that the carbon/epoxy cylinder, which showed the most extensive damage after impact (Figs. 10 and 11), shows the smallest reduction in residual strength. The carbon/PEEK, on the other hand, in spite of its significantly better impact damage resistance, is most sensitive to this damage when tested under external pressure. A reason for this may be found by observing the cylinders after implosion, Fig. 20.

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Fig. 17 Imploded cylinder after 220 J impact

Scaled cylinders 120

Residual strength, %

100 80 60 40 20

175 mm diameter 55 mm diameter

0 0

50

100

150

200

250

300

350

Scaled impact energy, J

Fig. 18 Influence of impact energy on residual strength of glass/epoxy cylinders under external pressure loading

Delamination of Composite Cylinders

83

Implosion pressure after impact

% undamaged strength

100 80 60 40 Glass/epoxy C/PEEK

20

C/Epoxy

0 0

5

10

15

20

25

Impact energy, J

Fig. 19 Residual strength after impact for three 55 mm diameter cylinder materials, normalized with respect to strength of undamaged cylinders

Fig. 20 55 mm diameter impacted cylinders after implosion. (a) Glass/epoxy, (b) carbon/epoxy, (c) carbon/PEEK

a

Glass/epoxy

b

Carbon/epoxy

c

Carbon/PEEK

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The carbon/PEEK failure and glass/epoxy failures are local, centred on the impact point, while carbon cylinders show a global catastrophic implosion. It is clear that residual implosion strength is not directly proportional to impact damage dimensions, each material has a specific compression response which must be examined. The influence of local imperfections on biaxial compressive strength must be fully understood if predictive models are to be used to optimize underwater performance of composite cylinders.

7 Conclusions An experimental study on glass/ epoxy angle-ply laminate specimens machined from composite cylinders was conducted over a wide range of mode ratios. Specifically, Œ˙™6 and Œ˙™12 laminates with mid-surface delaminations were considered, where ™ D 30ı ; 55ı and 85ı . The initiation value of mixed mode fracture toughness, Gc increased with increased mode II fraction and ply angle. Cylinders loaded under external pressure were insensitive to single artificial delaminations, but multiple delaminations of similar total area produced by impact caused reductions in the implosion pressure. The insensitivity of the failure pressure to a single delamination is attributed to the absence of buckling of the delaminated sublaminates before the cylinder wall collapsed. The impacted cylinders contained multiple delaminations, and such regions are likely to buckle locally, which would cause local reductions of the compressive strength and failure pressure. Changing fibre or matrix may improve performance but a tougher matrix will not necessarily result in improved implosion performance after impact. Acknowledgments This research has been conducted within a cooperative program between the French Oceanographic Research Organization, IFREMER, and the Department of Mechanical Engineering at Florida Atlantic University (FAU). The participation of FAU in this program is sponsored by the Office of Naval Research (ONR) with Dr. Yapa D. S. Rajapakse as the program monitor. The research of F. Ozdil and Xiaoming Li is greatly appreciated. Additional results from IFREMER studies and the European EUCLID RTP 3.8 project are also presented. Technical support in the pressure test programme from members of the Materials and Structures group at IFREMER, P. Warnier, E. Person, L. Riou, A. Deuff and J.J. Le Roy, is gratefully acknowledged. Thanks are due to Mr. Josh Kahn, Ms. N. Carr, and Mr. M. Farooq for help with preparation of this chapter.

References 1. Davies P (2008) Review of standard procedures for delamination resistance testing, Chapter 3 in Delamination behaviour of composites, editor S. Sridharan, Woodhead, Cambridge, England 2. Riddle RA, Beckwith SW (1986) Development of Test and Analysis Methods for Thick-Wall Graphite/Epoxy Filament Wound Composite Materials Fracture Toughness, ASTM STP 893: 64–83 3. Davies P, Rannou F (1995) The effect of defects in Tubes, Appl Compos Mater 1: 333–349

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4. Alderson KL, Evans KE (1992) Failure mechanisms during the transverse loading of filamentwound pipes under static and low velocity impact conditions, Composites 23(3): 167–173 5. Evans KE, Alderson KL, Marks PR (1992) Modelling of the transverse loading of filament wound pipes, Comput Struct 45(5/6): 1089–1095 6. Curtis J, Hinton MJ, Li S, Reid SR, Soden PD (2000) Damage, deformation and residual burst strength of filament wound composite tubes subjected to impact or quasi-static indentation, Compos Part B 31: 419–433 7. Christoforou AP, Swanson SR, Ventrello SC, Beckwith SW (1987) Impact damage in carbon/epoxy composite cylinders, Proc 32nd Int SAMPE Symp April: 964–973 8. Gning PB, Tarfaoui M, Collombet F, Riou L, Davies P (2005) Damage development in thick composite tubes under impact loading and influence on implosion pressure: experimental observations, Compos Part B Eng 36(4): 306–318 9. Davies P, Riou L, Mazeas F, Warnier P (2005) Thermoplastic composite cylinders for underwater applications, J Thermoplastic Compos Mater 18(5): 417–431 10. Ozdil F, Carlsson LA (2000) Characterization of mode I delamination growth in glass/epoxy composite cylinders, J Compos Mater 34: 398–419 11. Ozdil F, Carlsson LA, Li X (2000) Characterization of mode II delamination growth in glass/epoxy composite cylinders, J Compos Mater 34: 274–298 12. Ozdil F, Carlsson LA (2000) Characterization of mixed mode delamination growth in glass/epoxy composite cylinders, J Compos Mater 34: 420–441 13. Vanderkley PS (1981) Mode I–Mode II delamination fracture toughness of a unidirectional graphite/epoxy composite, MS thesis, Texas A&M University, December 14. Wilkins DJ, Eisenmann JR, Camin RA, Margolis WS, Benson RA (1982) Characterizing delamination growth in graphite/epoxy, ASTM STP 775: 168–183 15. Berry JP (1962) Determination of fracture surface energies by the cleavage technique, J Appl Phys 34: 62–68 16. Ewalds HL, Wanhill RJH (1989) Fracture Mechanics, Edward Arnold, London 17. Russell AJ, Street KN (1982) Factors affecting the interlaminar fracture energy of graphite/epoxy laminates, Proc ICCM 4: 227–286 18. Carlsson LA, Gillespie Jr JW, Pipes RB (1986) On the analysis and design of the end-notched flexure (ENF) specimen for mode II testing, Compos Mater 20: 594–604 19. Wang Y, Williams JG (1992) Corrections for mode II fracture toughness specimens of composite materials, Compos Sci Tech 43: 251–256 20. Gere JM, Timoshenko SP (1997) Mechanics of Materials (4th edn.), PWS, Boston, MA 21. Mistry J, Gibson G, Wu Y-S (1992) Failure of composite cylinders under combined external pressure and axial loading, Compos Struct 22(4): 193–200 22. Hinton MJ, Soden PD, Kaddour AS (1996) Strength of composite laminates under biaxial loads, Appl Compos Mater 3: 151–162 23. Kardomateas GA, Chung CB (1992) Thin film modeling of delamination buckling in pressure loaded laminated cylindrical shells, A1AA J, 30: 2119–2123 24. Whitcomb JD (1986) Parametric analytical study of instability-related delamination growth, Compos Sci Technol 25: 19–48 25. Buckingham E (1914) On physically similar systems; illustrations of the use of dimensional equations, Phys Rev 4: 345 26. Morton J (1988) Scaling of impact-loaded carbon-fiber composites, AIAA J August: 989 27. Swanson SR, Smith NL, Qian Y (1991) Analytical and experimental strain response in impact of composite cylinders, Compos Struct 18: 95–108 28. Davies P (1999) Scale and size effects in the mechanical characterization of composite and sandwich materials, Proc ICCM-12, Paris 29. Casari P, Jacquemin F, Davies P (2006) Characterization of residual stresses in wound composite tubes, Compos Part A, 37(2): 337–343

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Modeling of Progressive Damage in High Strain–Rate Deformations of Fiber-Reinforced Composites Romesh C. Batra and Noha M. Hassan

Abstract We use the theory of internal variables, or equivalently of continuum damage mechanics, to develop a mathematical model involving three variables to describe the evolution of progressive damage in high strain–rate deformations of fiber-reinforced composites. The degradation of material parameters with the damage is considered. Values of material parameters in the postulated evolution laws of internal variables are determined from the test data. The delamination mode of failure is simulated by hypothesizing a damage surface in terms of transverse normal and transverse shear stresses acting on an interface between two adjoining layers. When the stress state at a point on an interface lies on this surface, delamination is assumed to ensue from that point. Initial-boundary-value problems are numerically solved to validate the mathematical model by comparing computed results with test findings. A Figure of Merit, equal to the percentage of work done by external forces dissipated by all failure mechanisms, is introduced to characterize the performance of laminated composites under shock loads.

1 Introduction The failure mechanisms and processes on a micro-mechanical scale vary with the type of load, and are closely related to the ply-stacking sequence and properties of constituents, i.e., the matrix, the fiber, the interface, and the interphase between the fiber and the matrix (i.e., the treatment of the fiber prior to embedding it in the matrix, sometimes called the sizing of the fiber). Micro-level failure mechanisms include fiber breakage, fiber buckling, fiber splitting, fiber pullout, fiber/matrix R.C. Batra () Department of Engineering Science and Mechanics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061 e-mail: [email protected] N.M. Hassan Assistant Professor, Industrial Engineering Department, Alhosn University, P.O. Box: 38772, Abu Dhabi, UAE e-mail: [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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debonding, matrix cracking, and void nucleation, growth and coalescence in the matrix. Modeling each of these mechanisms though feasible is computationally too expensive to be of any practical value. Furthermore, the determination of values of material parameters for the micro-level failure modes from the test data is a formidable, if not an impossible, task. It is very likely that different failure mechanisms interact with each other in a rather complex manner; and the initiation of one failure mode may ensue another failure mode. Nearly a decade ago a world-wide exercise was completed in which investigators provided with the same test data used their failure theories to predict the outcome of 14 experiments. Out of the 19 failure theories tested, only five theories gave encouraging results [1]. Since then there may have been more theories proposed but they, including the present one, have not undergone the same rigorous testing as the previous 19. Here we use three internal variables to represent damage accumulated in a fiber (fiber fracture, buckling, kinking, splitting, etc.), matrix (cracking, and void nucleation, growth and coalescence), and fiber/matrix debonding. Thus the damage induced due to micro-level mechanisms has been lumped into these three variables. The degradation of the material moduli of the laminated composite with the evolution of damage is considered by using the Mechanics of Materials approach. The damage accumulated at a material point is assumed to be irreversible in the sense that values of the three damage variables cannot decrease. A material point is assumed to have failed (i.e., it has no load carrying capacity) when all three internal variables at that point equal 1. The two adjoining layers are assumed to delaminate at a material point when the transverse shear and the transverse normal stresses at that point lie on the postulated damage surface. Values of material parameters for the AS4/PEEK composite are determined by using test data available in the literature. The mathematical model is validated by comparing computed results with experimental findings for test configurations different from those used to ascertain values of material parameters. The Figure of Merit is introduced to characterize the performance of laminated composites subjected to impact loads. The work reported herein is taken from Refs. [2–4]. There is an enormous literature on modeling damage in fiber-reinforced polymeric composites. We admit to not having leaned all of it and are thus unable to summarize it succinctly.

2 Progressive Damage Model Since composites fail at relatively small values of strains, one can use either the spatial or the referential description of motion to describe lamina’s deformations; here we use the referential description of motion and rectangular Cartesian coordinates with the X1 -axis aligned along fibers. To use the theory of internal variables for describing damage evolution in rate dependent bodies, we introduce an ordered triplet  D f m ; f ; d g of three internal variables, and the triplet ! D fYm ; Yf ; Yd g of corresponding conjugate forces. We assume that

Damage in High Strain–Rate Deformations of Fiber-Reinforced Composites

!.E ; EP ; ; G ; / D !e .E ; 0; ; 0; / C !ne .E ; EP ; ; G ; /;

91

(1)

where !e denotes the value of the conjugate force ! at zero strain rate and zero temperature gradient, E ; EP ; and G denote, respectively, the Green–St. Venant strain tensor, its rate, the temperature rise, and the temperature gradient. Henceforth, we consider isothermal processes, neglect effects of temperature rise and temperature gradient, assume that the body is initially stress free, and its response to mechanical deformations can be represented by a neo-Hookean constitutive relation. Thus the second Piola–Kirchhoff stress tensor S e is a linear function of the strain tensor E , and e S˛ˇ D C˛ˇı Eı ; e D Ye !.i/

.i /

D

1 @C˛ˇı E˛ˇ Eı ; 2 @ .i /

i D m; f; d;

(2)

where elasticities C˛ˇı D Cı˛ˇ D Cˇ˛ı are functions of . Equation (2) accounts for all geometric nonlinearities. We assume that a lamina reinforced with unidirectional fibers is transversely isotropic with the axis of transverse isotropy perpendicular to the plane of the lamina. Thus there are five independent elastic constants. The tensors C ; S e and E are transformed to global coordinate axes by using tensor transformation rules [5, 6]. By using the Mechanics of Materials approach to derive the dependence of material parameters upon damage variables [5], we get f

E1C D E1 V f .1  f / C E1m V m ; f

(3)

m

1 V V ; D f C m E .1  m/ E2C E2 2

(4)

1 Vm Vf C m; D f C G12 G12 G12 .1  d /

(5)

where E1 equals Young’s modulus in the axial (or the fiber) direction, E2C the transverse modulus, superscripts f , m and C signify quantities for the fiber, the matrix and the composite respectively, and V f equals the volume fraction of the fiber. The C C effective Poisson’s ratios 13 and 12 are independent of the damage variables, and their values for the composite are derived from those of the constituents by the rule of mixtures. The effective shear modulus in the X2 X3 -plane is taken to be independent of the debonding damage variable. When f and m tend to 1.0 Young’s moduli of the composite along the X1 -axis, the X2 -axis and the X3 -axis approach, respectively, the axial Young’s modulus of the matrix multiplied by the volume fraction of the matrix, zero and zero. Similarly, in the limit of d equaling 1.0, the shear modulus of the lamina in the X1 X2 -plane approaches zero. Thus at failure, material properties of the lamina have been severely degraded. Even though a damage variable affects only one of the elastic moduli in the corresponding material principal direction, it influences all moduli of the lamina when either global axes do not

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coincide with the material principal directions or applied tractions are not along the fibers or both. It is assumed that the damage at a material point does not increase while the material there is unloading as indicated by a decrease in a suitable scalar measure of either stresses or strains. We postulate that f D f .Y f /; m D m .Y m /; d D d .Y d /. The functional dependence determined from the test data of AS4/PEEK with V f D 0:6, and used in this work, is given below: f

f D Af .1  e .Bf Y / /;   Am Bm C Cm .Y m /Dm m ;

D Bm C .Y m /Dm   Ad Bd C Cd .Y d /Dd d :

D Bd C .Y d /Dd

(6) (7) (8)

Values of material parameters for the PEEK matrix, the AS4 carbon fibers, and constants Af , Bf , Am , Bm , Cm , Dm , Ad , Bd , Cd , and Dd are listed in Tables 1 and 2. Constants in Eq. (6) expressing the damage variable due to fiber breakage have different values for tensile and compressive loading along the fiber direction. For fibers not aligned with the loading axis, we analyze the problem in the global coordinate system and subsequently compute the axial strain E11 along the fiber. f The sign of E11 determines which values of Af , Bf and Ycrit to use in Eq. (6). We follow a similar procedure for selecting appropriate values of Am , Bm , Cm , Dm and m d Ycrit in Eq. (7) and of Ad , Bd , Cd , Dd and Ycrit in Eq. (8). Even though f , m and d depend only upon Y f , Y m and Y d , respectively, all failure modes interact with each other because the degradation of a material elastic parameter affects all modes of deformation. Failure criterion: It is assumed that the failure due to fiber breakage, matrix crackf m and ing and fiber/matrix debonding occurs when Y f , Y m and Y d equal Ycrit , Ycrit f d m d Ycrit , respectively. Constants Ycrit , Ycrit and Ycrit are values of conjugate variables when the corresponding damage variables equal 1.0, and depend upon materials of the fiber and the matrix, sizing of fibers, and possibly on the fabrication process. Their values, determined from the experimental data, for the AS4/PEEK composite used herein are listed in Table 2. Delamination: We simulate delamination by allowing cracks to propagate in an interface between adjoining laminas. Delamination ensues at a material point when the transverse normal and the transverse shear stresses there lie on the damage surface

Table 1 Values of material parameters of the fiber and the matrixa

Poisson’s ratio Young’s modulus (GPa) Shear modulus (GPa) Mass density .g=cm3 / a

Matrix (PEEK) 0.356 6.14 2.264 1.44

Carbon fiber (AS4) 0.263 214 84.7 1.78

The fiber and the matrix are assumed to be isotropic.

Damage in High Strain–Rate Deformations of Fiber-Reinforced Composites Table 2 Values of constants in Eqs. (6)–(9) Damage properties Tension Fiber breakage Af 1.931 GPa Bf 1.931497 f 0.0075 GPa Ycrit

93

Compression 0.197 GPa 558 0.007535 GPa

Matrix cracking

Am Bm Cm Dm m Ycrit

1:356  1010 GPa 0.00193 0.37239 GPa 0.43665 0.0005 GPa

Fiber/matrix debonding

Ad Bd Cd Dd d Ycrit

0.1437 GPa 0.00762 1.0022 GPa 0.37714 5:48  102 GPa

Interfacial strengtha

Œ33  Œ13  Œ23 

0.078 GPa 0.157 GPa 0.157 GPa

0.01207 GPa 174 241 GPa 0.195 0.011 GPa

a

Values for the ultimate interfacial strength were obtained from a composite data base site http://composite.about.com/library/data/blcas4apc2-1.htm.

 Dd D

33 Œ33 

2

 C

13 Œ13 

2

 C

23 Œ23 

2 D 1;

.33  0/

(9)

where [x] denotes the ultimate value of the quantity x, and  is the Cauchy stress tensor. For ¢33 < 0, the failure envelope (9) is modified to  Dd D

13 Œ13 

2

 C

23 Œ23 

2 D 1;

.33 < 0/

(10)

In this case there is no separation of adjoining plies, but there is relative sliding between them similar to that in mode-II failure. For ¢13 D ¢23 D 0, Eq. (9) represents mode-I failure, newly created surfaces are taken to be traction free, and points on them are checked for non-inter-penetration during subsequent deformations. Values of Œ33 ; Œ13  and Œ23  for the AS4/PEEK composite are listed in Table 2. Strain rate effect: We postulate the following functional dependence of conjugate damage variables Ym and Yd upon EP 22 and EP 12 respectively: 0

Ym

    11=Dm P 22 Bm Am C m 1  s m log10 E 0 EP 22 B C    D@ A P E 22 Cm  m 1  s m log10 EP 0 22

(11)

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0 B Y D@ d







P

12 Bd Ad C d 1  s d log10 E 0 EP 12    P 12 Cd  d 1  s d log10 E EP 0

 11=Dd C A

(12)

12

0 0 and EP 12 represent, respectively, values of the reference transverse norHere EP 22 mal and the reference shear strain rates. Y f is assumed to be independent of strain rate because the experimental stress–strain curve for the AS4/PEEK in longitudinal tension and compression is insensitive to the axial strain rate [7, 8]. Using Vogler and Kyriakides’s [7] test data for transverse compression and in-plane shear, thevalue of the material parameter s m determined by plotting  P 22 0 m 1  s log10 E D 1:6  105 =s is versus m at Y m D 0:006 and setting EP 22 0 EP 22 0 D 105 =s found to be 0.0361. Similarly, the data plotted for Y d D 0:003 with EP 12 i gave s d D 0:0013. The material constant, Ycrit .i D f; m; d /, was assumed to be strain rate independent because failure strains reported in Refs. [7, 8] do not exhibit any obvious dependence upon the strain rate. For a lack of test data, the interfacial strengths Œ33 ; Œ13  and Œ23  of the composite are assumed to be strain rate independent.

3 Implementation of the Damage Model 3.1 Brief Description of the Numerical Technique A weak form of equations expressing the balance of linear momentum is derived by using the Galerkin approximation thereby reducing nonlinear partial differential equations to nonlinear ordinary differential equations (ODEs). These are integrated with respect to time t by using the subroutine LSODE (Livermore Solver for Ordinary Differential Equations) that adaptively adjusts the time step size in order to compute the solution within the prescribed accuracy. A finite element (FE) code based on the afore-stated formulation of three-dimensional problems and using eight-node brick elements has been developed in Fortran. Degrees of freedom at each node are three components of displacement, and three components of velocity since LSODE integrates first order ODEs. Domain integrals involving integration on an element e that appear in the weak formulation of the problem are evaluated by using the 2  2  2 Gauss quadrature rule. During the time integration of the coupled ODEs, absolute and relative error tolerances in LSODE are each set equal to 1  109 , and MF assigned the value 10. The parameter MF determines the integration method in LSODE, and MF D 10 implies the use of the Adam–Moulton method. After having found nodal displacements, values of conjugate variables and damage parameters f , m and d at each integration point are determined, and are used to update elastic constants for computing results at the next time step. Thus progressive damage is computed and used to degrade material elasticities.

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95

3.2 Simulation of Material Failure When an internal variable f , m or d equals 1.0 or the corresponding conjugate variable Y f , Y m or Y d equals its critical value, then the material there is taken to have failed due to fiber breakage, matrix cracking or fiber/matrix debonding, respectively. Even if the material at all eight integration points within an element has failed, that element is not removed from the analysis. Once all elastic constants at each one of the eight integration points in an element have been reduced to zero, all stress components in that element will subsequently be 0, and for all practical purposes that element will represent a hole or a void. In order to simulate either sliding or crack initiation and propagation due to delamination, we assume that when the stress state at a node N lies on the failure envelope represented by Eqs. (9) and (10), an additional node N coincident with N but not connected to it is added there. The nodal connectivity of elements sharing the node N is modified in the sense that one or more of these elements is now connected to the newly added node N rather than the node N. However, no new element is created in this process. We note that delamination may ensue simultaneously at several nodes. If subsequent deformations of the body move nodes N and N apart and create new surfaces, then these surfaces are taken to be traction free. The non-inter-penetration of nodes N and N into the material is enforced by connecting these two nodes with a two-node spring element that is weak in tension but stiff in compression.

3.3 Energy Dissipation In order to assess structure’s resistance to impact loads we use Eqs. (13)–(17) to compute, respectively, the work done due to applied loads, energies dissipated due to fiber breakage, matrix cracking, fiber/matrix debonding, and delamination. We find the work done to deform the body via Eq. (18), and the kinetic energy of the body at the terminal value of the time t via Eq. (19). The summation in Eq. (13) is over all nodes of the laminate where surface tractions F are applied, u equals the nodal displacement and v the velocity of a material particle. Equation (17) follows from the balance of energy. Zt X d u ni Fin dt: (13) Work done by external forces D dt n 0

Energy; E

fb

Zt Z ; dissipated due to fiber breakage D

Yf

d f dV dt dt

0 V

Z f Z D 0 V

Y f d f d V: (14)

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Energy; E

mc

Zt Z

Ym

; dissipated due to matrix cracking D

d m dV dt dt

0 V m

Z Z

Y m d m dV:

D

(15)

0 V

Zt Z

db

Energy; E ; dissipated due to fiber=matrix debonding D 0

d d dV dt dt

V

d Z Z

D 0

Yd

Y d d d dV:

V

(16) Energy; E d l ; dissipated in delamination D Work done by external forces –Work done to deform the body –Kinetic energy –Energy dissipated due to other three failure modes Work done to deform the body; W

Kinetic energy; K t D

Z

def

Zt Z D

S˛ˇ 0

dE˛ˇ dV dt: dt

(17)

(18)

V

  2 vx C v2y C v2z dV : 2

(19)

V

3.4 Verification of the Code As described in Ref. [2] the code was verified by using the method of fictitious body forces [9]. That is, one assumes an expression for the displacement field, finds the fictitious body force needed to satisfy the balance of linear momentum, and the corresponding initial and boundary conditions. The initial-boundary-value problem with the fictitious body force is then analyzed numerically. If the computed solution agrees well with the assumed analytical solution of the problem, then the code has been verified. Also, computed results for simple problems such as wave propagation in a bar were compared with their analytical solutions. It ensures that the code gives an accurate numerical solution of the governing equations.

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3.5 Validation of the Mathematical Model Referring the reader to Ref. [2] for details, values of material parameters for the AS4/PEEK composite were derived from the test data reported in Refs. [7, 8]. Subsequently, the impact by a 2.54 cm diameter hemispherical hardened steel ball dropped from a known height on a laminated composite plate studied experimentally by Schoeppner and Abrate [10] was numerically simulated with the computer code. The pressure between the steel ball and the laminate over a circular contact surface is found from Hertz’s theory, and is assumed to increase linearly in time. This assumption of circular contact surface is reasonable since each layer has been taken to be transversely isotropic with the axis of transverse isotropy perpendicular to the lamina. Fiber orientations in the nine layers starting from the top layer equaled 45ı ; 45ı ; 0ı ; 0ı ; 90ı ; 0ı ; 0ı ; 45ı ; 45ı . The thickness in the five simulations equaled 1.27, 4, 7, 11, and 13 mm. For each thickness, the laminate was divided into 20  20  9 eight-node brick elements with a finer mesh in the impacted area. Figures 1a, 1b, 1c exhibit time histories of evolution of the three damage variables at the laminate centroid, the damage threshold load (DTL) versus the laminate thickness, and fringe plots of the three damage variables at 1:48 s after impact. The DTL is the load at which damage increases quickly, and the load drops rapidly. We took it to be the load when all of the three damage variables have reached 1 at the specimen centroid. The DTL for laminates of different thicknesses has been normalized by the DTL for the 7-mm thick laminate since for it the experimental and the numerical values equaled 12.5 kN. As depicted in Fig. 1b, the numerically computed DTL matches well with the experimental one for different thicknesses of the laminate.

Fig. 1a Time histories of evolution of the three damage variables, f , m and d , at the centroid of 1.27 mm thick specimen

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Fig. 1b Comparison of the dependence upon the laminate thickness of the numerically computed and the experimentally observed (Schoeppner and Abrate [10]) normalized damage threshold load

Fig. 1c Fringe plots of the damage variables at time D 1:48 s

For a 1.27 mm thick laminate, time histories of evolution of the three damage variables, f ; m and d , at the laminate centroid are shown in Fig. 1a. The matrix cracking occurs soon after the laminate is loaded; it first increases slowly followed by a rather very gradual increase and at about 0:38 s it increases instantaneously to the maximum value of 1. The fiber/matrix debonding begins at t D 0:14 s and increases rapidly in the beginning followed by a gradual increase to 1.0 at t D 0:8 s. The fiber breakage damage mode initiates at 0:6 s and slowly increases to 0.2 in 0:5 s when it quickly approaches 1. In the absence of test data for local deformations of the laminate, detailed comparisons between test findings and numerical solutions could not be made. Fringe plots depicted in Fig. 1c reveal that the damage due to each failure mode is concentrated in a small region around the laminate centroid; the region of fiber breakage is the smallest in size and that of the fiber/matrix debonding the largest.

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Fig. 2a Comparison of the computed axial stress versus axial strain curve of AS4/PEEKwith experimental data of Weeks and Sun [11] for two balanced ŒC30ı =  30ı 2s plies at strain rate of 0.01/s

The second validation exercise involved simulating impact tests of Weeks and Sun [11] on 8  8  8 mm 32-ply Œ30ı =  30ı 16s and Œ60ı =  60ı 16s composites in a split Hopknison bar at nominal strain rates of 300/s and 1,000/s, respectively. We assume that deformations are symmetric about the midsurface, and pressure loads on the two opposite bounding surfaces of the specimen can be replaced by prescribing the uniform axial velocity on these surfaces that increases from 0 to the steady value in 0:1 s; the steady value is such that the maximum nominal strain rate equals that in the experiments. Computed results for these two configurations are compared in Figs. 2a, 2b with the experimental data [10]. It is evident that in every case, two computed results are close to the corresponding experimental ones with a maximum difference of 4% in the ultimate stress, and of 20% in the failure strain.

4 Parametric Studies on a Typical Laminated Composite A schematic sketch of the problem studied is exhibited in Fig. 3a. It involves studying transient deformations of a 22022010 mm AS4/PEEK composite panel with fiber volume fraction V f D 0:6. The four-ply panel, clamped at all four edges, is divided into eight-node brick elements with finer elements in the central portion. The load due to air blast explosion is simulated by applying a time-dependent pressure field P .r; t / on the top surface of the specimen; the exponential decay of the peak pressure at r D 0 with time t, and at any instant its spatial variation are exhibited

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Fig. 2b Comparison of the computed axial stress versus axial strain curve of AS4/PEEKwith experimental data of Weeks and Sun [10] for ŒC30ı =30ı 16s laminate at strain rate of 300/s, and ŒC60ı =60ı 16s laminate at strain rate of 1,000/s

Fig. 3a Schematic sketch of the problem studied

in Figs. 3b, 3c The spatial variation of P derived from the experimental work of T¨urkmen and Mecitolu [12] is taken to be P .r; t / D .0:0005r4 C 0:01r3  0:0586r2  0:001r C 1/P .0; t/;

(20)

where r is the distance, in cm, from the specimen center.

4.1 Effect of Mesh Size For one loading, we used the following four FE meshes: 20  20  4 (1,600 elements, 2,205 nodes), 20  20  8 (3,200 elements, 3,969 nodes), 40  40  4 (6,400

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Fig. 3b Variation at r D 0 of the pressure with time

Fig. 3c For a fixed value of time t, pressure distribution over specimen’s top surface

elements, 8,405 nodes), and 40  40  8 (12,800 elements, 15,129 nodes), with each mesh being fine in the central portion of the specimen. Results were computed until 220 s. The maximum tensile and the maximum compressive principal stresses found with the four meshes differed at most by 11% and 13.7% respectively. Values of W def calculated using Eq. (18) were found to be 378, 408, 397 and 405 J and differ at most by 7%. These numbers give an estimate of the error in results presented and discussed below with the 20  20  4 FE mesh. Because of the fewer computational resources needed for analysis with this mesh, one can quickly find variables to which the impact damage is most sensitive. If desired, subsequent high fidelity computations can be performed with successively finer meshes to improve the design of damage resistant panels.

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We describe below how the lay-up sequence, target thickness, and the fiber volume fraction influence damage induced in the composite laminate.

4.2 Lay-Up Sequence We first analyze deformations of two-ply, and then four-ply laminates. For the twoply laminated composite, we have studied the following eight lay-up sequences: 0ı =90ı , 90ı =0ı , 45ı =  45ı , 45ı =45ı , 0ı =45ı , 45ı =0ı , 90ı =45ı , and 45ı =90ı ; these are called lay-ups 1 through 8, respectively, in the following discussion. Two-ply laminates: For the eight stacking sequences, Fig. 4a shows the energy dissipated in the four failure modes; it is evident that for each lay-up of the two plies most of the input energy due to work done by applied loads is dissipated in delaminating the two layers, and the least in breaking the fibers. The 45ı =0ı sequence has the most energy dissipated due to delamination, and the energy dissipated due to matrix cracking is essentially the same for the 0ı =90ı , 90ı =0ı , 45ı =45ı and 45ı =45ı laminates. However, the energy dissipated due to matrix cracking in the 0ı =45ı , 45ı =0ı , 90ı =45ı and 45ı =90ı laminates is nearly 20% of that in the other four laminates studied. Figure 4b depicts time history of the E d l for the 45ı =90ı laminate. After the initial parabolic rise, the energy dissipated increases essentially linearly. Four-ply laminates: For the four-ply laminates, we examined the following nine sequences: [0ı =45ı =90ı =45ı ], Œ45ı =0ı =45ı =90ı , Œ90ı =45ı =0ı =45ı , Œ45ı =90ı =45ı =0ı , Œ0ı =45ı =45ı =90ı , Œ0ı =90ı =45ı =45ı , Œ0ı =90ı =45ı =45ı ,

Fig. 4a Energy dissipated in the four failure modes for different stack up sequences

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Fig. 4b Evolution of energy dissipation due to delamination for [45/90] stack-up sequence

Fig. 5 For the nine stack-up sequences of four-ply laminates, energy dissipated in different failure modes

Œ0ı =45ı =45ı =90ı  and Œ0ı =45ı =90ı =45ı ; fiber orientations in plies are given in going from the top layer to the bottom layer. Time histories of the deflection of the specimen centroid, not included herein but given in Ref. [13], indicated that deflections for the Œ0ı =90ı =45ı =45ı  and Œ0ı =90ı =45ı =45ı  laminates are nearly the same, and they are smaller than those for the other seven lay-ups. The centroidal deflections for the Œ0ı =45ı =45ı =90ı  and Œ0ı =45ı =45ı =90ı  laminates are larger than those for the remaining seven laminates. From plots of Fig. 5, we deduce that the E d l is maximum for the ı Œ0 =45ı =90ı =45ı  laminate, and for this laminate it is significantly more than the E f b ; E mc and E db . However, for the Œ45ı =0ı =45ı =90ı  laminate, the E d l

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is negligible as compared to the energy dissipated in other modes of failure, and is the least of the E d l ’s for the nine laminates. The E mc is very high for the Œ45ı =0ı =45ı =90ı  and Œ45ı =90ı =45ı =0ı  laminates; it is comparable to the E d l for several other laminates. All four failure modes can be delayed by placing the 45ı plies at the bottom of the laminate to resist fiber fracture, and the 0ı or the 90ı plies at the top of the laminate to resist delamination. Time histories of the evolution of the three damage variables at the laminate centroid, plotted in Fig. 6a, b, c reveal that the fiber/matrix debonding damage developed the last for the Œ0ı =90ı =45ı =45ı  and the Œ0ı =90ı =45ı =45ı  laminates, and the earliest for the Œ0ı =45ı =45ı =90ı  composite (Fig. 6a). The time of initiation of the matrix cracking damage variable for the Œ0ı =90ı =45ı =45ı  and the Œ0ı =90ı =45ı =45ı  laminates is the maximum, and it is minimum for the Œ45ı =90ı =45ı =0ı  and the Œ45ı =0ı =45ı =90ı  laminates (Fig. 6b). The fiber breakage damage variable developed the last for the Œ0ı =45ı =45ı =90ı  and the Œ0ı =45ı =45ı =90ı  laminates and soonest for the Œ0ı =45ı =90ı =45ı  and Œ0ı =45ı =90ı =45ı  laminates (Fig. 6c). In each case, except for delamination, the failure mode that initiated first caused the maximum energy dissipation.

4.3 Target Thickness When studying the effect of target thickness in resisting a shock load, we assumed that it is comprised of four uniform 0ı plies, and varied the thickness of each layer. In each case results were computed for load duration of 220 s, and the maximum centroidal deflection occurred at t D 220 s. Results plotted in Fig. 7 show that the E d l decreases exponentially and is maximum for the thinnest target; however, the W def is not a monotonic function of the target thickness – it first increases with an increase in the target thickness, and then decreases as the target is made thicker. For the 25-mm and thicker targets, the applied load is not sufficient to cause rapid deformations of material particles; thus the total K t of the target is miniscule as compared to the W ef nearly all of which is used to deform the laminate.

4.4 Fiber Orientation In order to examine the effect of the fiber orientation on the impact resistance of the composite structure all four plies were assumed to have the same fiber orientation, and seven different fiber orientations, ™, namely, ™ D 0ı , 10ı , 30ı , 45ı , 60ı , 75ı and 90ı , were considered. Figure 8 exhibits the energy dissipated in each one of the failure modes as a function of the fiber orientation angle. For all fiber orientation angles, the energy dissipated due to matrix cracking is miniscule as compared to that in any of the other three damage mechanisms; this is mainly due to low values of the elastic moduli of the matrix. With an increase in the fiber orientation angle from

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Fig. 6 For different stack-up sequences, time histories at the specimen centroid of (a) fiber breakage, (b) matrix cracking, and (c) debonding damage variable

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Fig. 7 Energy dissipated in different failure modes versus the target thickness

Fig. 8 Energy dissipated in different failure modes versus the fiber orientation angle

0ı to 45ı , the energy dissipated due to delamination increases but energies dissipated due to fiber/matrix debonding and matrix cracking decrease. An examination of the delamination initiation time revealed that it ensued earliest at 128 s for fiber orientation of 0ı or 90ı , and latest at 140 s for fiber orientation of 45ı .

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Fig. 9 Delaminated area as seen from the top surface of the composite; Left figure has fibers oriented at 0ı , and the right figure has fibers oriented at 45ı

4.5 Delamination Figure 9 shows the delaminated areas in all layers as seen from the top as if the composite were transparent. The area was approximated as rectangles, and its magnitude was calculated to be 6,977 and 4; 550 mm2 for fiber orientations of 0ı and 45ı respectively. This method of calculating the delaminated area excludes overlapping delaminated areas, and underestimates the total delaminated area. Whereas one can ascertain in the numerical work delamination among adjoining layers, it is not clear how to do so experimentally.

4.6 Figure of Merit For different fiber orientations, the percentage of work done by external forces dissipated in all failure modes is shown in the bar chart of Fig. 10; a high value of this ratio signifies that more of the total work done by external forces is dissipated in all of the failure modes, and the composite is more effective in resisting explosive loads. This ratio can be used to define the Figure of Merit of the composite. Thus clamped uni-directional AS4/PEEK composites with fiber orientations between 30ı and 60ı are equally effective in resisting dynamic loads. We note that the total work done by external forces is virtually independent of the fiber orientation angle. With an increase in the fiber orientation angle from 0ı to 45ı , the energy required to deform the body decreases and the kinetic energy increases monotonically. The fraction of the total work done by external forces dissipated due to various failure mechanisms has the maximum value of nearly 22% for fiber orientations of 30ı and 60ı ; thus plies with clamped edges and fiber orientations of 30ı to 60ı are good choices for optimizing the energy dissipation due to all failure

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Fig. 10 For different fiber orientation angles, percentage of the work done by external forces dissipated in all failure modes

modes. For these fiber orientations in the AS4/PEEK laminated composite clamped at all edges subjected to a pressure load on the top surface, the energy dissipated due to delamination exceeds that in each of the other three failure modes considered.

4.7 Remarks From a thorough examination of results computed for the present problem, one can draw the following conclusions regarding the order and the location of initiation of different failure modes in a clamped AS4/PEEK laminated composite subjected to severe pressure loads on the top surface: (i) fiber/matrix debonding at edges of the bottom and the top surfaces that are perpendicular to fibers, (ii) matrix cracking at the centroid of the bottom surface, and at edges of the top surface that are perpendicular to fibers, (iii) fiber breakage at edges of the top surface that are parallel to fibers, and then at the centroid of the bottom surface, (iv) debonding at centroids of the bottom and the top surfaces, (v) fiber breakage at the centroid of the top surface, (vi) fiber breakage at sides of the bottom surface, and (vii) matrix cracking at the centroid of the top surface. This information can be exploited in the design of laminates for attaining a designated functional objective.

4.8 Limitations of the Model The three internal variables used here to simulate matrix cracking, fiber breakage, and fiber/matrix debonding homogenize the effect of damage induced locally, enable

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one to compute energies dissipated in different failure modes, and allow for the interaction among various failure modes. However, one cannot compute the length of cracks, sizes of fiber/matrix debonded zones, and fiber kinking, etc. In the analyses of various problems, the loading wave is taken to be plane which is the case for a large value of the stand-off distance. Also, the spatial variation of the pressure may not match well with that assumed here. Values of material parameters given in Table 2 are for the AS4/PEEK composite, and all of the results have been computed for it. One should follow the procedure outlined in Ref. [13] to find values of material parameters for other unidirectional fiber reinforced composites and then study their response to impact loads. The mathematical model and results presented herein are not valid for woven and/or stitched composites, and for composites reinforced with randomly distributed short fibers. The delamination criterion is neither based on the energy release rate nor on the critical stress intensity factor. The energy dissipated during delamination is computed from the balance of total energy and a knowledge of energies dissipated in other deformation modes.

5 Conclusions We have developed a mathematical model for analyzing transient deformations of a composite subjected to shock loads, and a modular computer code, in Fortran, to find numerically an approximate solution of the pertinent initial-boundary-value problem. The problem formulation includes evolution of damage due to fiber breakage, fiber/matrix debonding, matrix cracking, and delamination. Energies dissipated in these failure modes are computed, and the effect on them of various parameters examined. The conclusions summarized below are based on results presented in Refs. [2–4] and [13] but not necessarily included here. It is found that approximately 15% of the total work done by external forces is dissipated in the four failure modes. Both for the clamped 0ı and the 45ı laminated composites, the energy dissipated due to delamination for laminates with clamped edges is nearly twice of that for laminates with simply supported edges. About 43% of the energy input into the structure is used to deform it, and 42% is converted into the kinetic energy. For simply supported laminates, these proportions strongly depend upon the fiber orientation angle. The fiber orientation influences when and where each failure mode initiates and its direction of propagation. Debonding between fibers and the matrix occurs along the fibers rather than in a direction perpendicular to the fibers. For clamped edges, the debonding damage variable starts from the edges perpendicular to the fibers and propagates, along the fibers, towards the center; it propagates in the thickness direction instantaneously most likely due to thin laminates we have studied. Matrix cracking damage variable initiates first at the center of the back surface, where there are high tensile stresses developed, and propagates faster along the fibers than in the transverse direction. Fiber breakage is concentrated at points near specimen’s

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centroid that are along the X2 -axis. For all fiber orientations, the energy dissipated due to matrix cracking is miniscule as compared to that in any of the other three damage modes. The fraction of the total work done by external forces dissipated due to various failure mechanisms has the maximum value of nearly 22% for fiber orientations of 30ı and 60ı , of which 10% is due to delamination. The stacking sequence also strongly influences energies dissipated in different failure modes. The target thickness plays a role in determining which failure mode is dominant. The fraction of energy dissipated due to delamination failure mode decreases exponentially with an increase in the target thickness, and has the maximum value for the thinnest target. Varying constituents’ properties affects the initiation time of the damage modes. Increasing fiber’s Young’s modulus results in slightly different rates of evolution of the fiber/matrix debonding damage variable, and delays the initiation of the matrix cracking damage. Decreasing Young’s modulus delays the initiation of the matrix cracking damage variable. Increasing fiber’s shear modulus delays the initiation of the fiber breakage variable and enhances the initiation of the fiber/matrix debonding damage variable. Increasing the matrix shear modulus reduces noticeably specimen’s centroidal deflection, enhances both the time of initiation and the rate of growth of the fiber/matrix debonding damage variable, and delays the initiation of the damage due to fiber breakage. An increase in the fiber volume fraction decreases affinely the total work done by external forces, decreases quadratically the kinetic energy, and has virtually no effect on the energy required to deform the body. Acknowledgments This work was partially supported by the ONR grant N00014-06-1-0567 to Virginia Polytechnic Institute and State University (VPI&SU) with Dr. Y. D.S. Rajapakse as the program manager. Views expressed herein are those of authors, and neither of the funding agency nor of VPI&SU.

References 1. Soden PD, Kaddour AS, Hinton MJ (2004) Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos Sci Technol 64: 321–327 2. Hassan NM, Batra RC (2008) Modeling damage development in polymeric composites. Compos B 39: 66–82 3. Batra RC, Hassan NM (2008) Blast resistance of unidirectional fiber reinforced composites. Compos B 39: 513–536 4. Batra RC, Hassan NM (2007) Response of fiber reinforced composites to underwater explosions. Compos B 38: 448–458 5. Jones R (1999) Mechanics of Composite Materials. Taylor & Francis, Philadelphia, PA 6. Batra RC (2006) Elements of Continuum Mechanics, AIAA, Reston, VA 7. Vogler T, Kyriakides S (1999) Inelastic behavior of an AS4/PEEK composite under combined transverse compression and shear Part I: Experiments. Int J Plast 15(8): 783–806 8. Kyriakides S, Arseculeratne R, Perry E, Liechti K (1995) On the compressive failure of fiber reinforced composites. Int J Solids Struct 32(6–7): 689–738 9. Batra RC, Liang XQ (1997) Finite dynamic deformations of smart structures. Comput Mech 20: 427–438

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10. Schoeppner G, Abrate S (2000) Delamination threshold loads for low velocity impact on composite laminates. Compos A: Appl Sci Manuf 31: 903–915 11. Weeks CA, Sun CT (1998) Modeling non-linear rate-dependent behavior in fiber-reinforced composites. Compos Sci Technol 58(3–4): 603–611 12. T¨urkmen HS, Mecitolu Z (1999) Dynamic response of a stiffened laminated composite plate subjected to blast load. J Sound Vib 221(3): 371–389 13. Hassan NM (2006) Damage development in static and dynamic deformations of fiberreinforced composite plates. Ph.D. dissertation, Virginia Tech

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Post-Impact Fatigue Behavior of Woven and Knitted Fabric CFRP Laminates for Marine Use Isao Kimpara and Hiroshi Saito

Abstract In this study, the damage evolution behavior was evaluated. Damage observation was conducted by the integration of non-destructive and direct observation methods. Target reinforcements were T300-3k plain woven fabric (PW) and T700S-12k multi-axial knitted fabric (MA). Impact damage distribution in the CFRP laminate was observed precisely, and three-dimensional damage model was constructed. Compression after impact (CAI) and post impact fatigue (PIF) performances were evaluated. The effect of water absorption on these performances was also evaluated. The effect of water absorption on CAI and PIF performances were small in PW CFRP laminates. Conversely, PIF properties of water-absorbed MA drastically decreased than that of dry ones. CAI strength was not affected by water absorption. PIF performance of dry MA CFRP was fairly higher than that of the others. From the precise observation, some evidences of interfacial deterioration caused by water absorption were confirmed in both PW and MA CFRP laminates.

1 Introduction Fiber reinforced plastics (FRP) have been applied to the large-scale structures, such as aerospace structures, marine ships, automobiles and so on. Recently FRP has been used as not only secondary structural materials but also primary ones. In addition, with the development of monocoque and large-scale structures, the trend of preform and molding techniques has been changed. For instance, in addition to a conventional ‘solid’ prepreg for autoclave, new style preforms, which are suitable for the low-cost and large-scale structures with ‘liquid’ molding, have been developed, for example the multi-axial knitted fabric for the vacuum assisted resin transfer molding (VARTM) [1, 2]. In VARTM method, resin is impregnated into reinforcements by atmospheric pressure. Therefore the expensive apparatus such as a I. Kimpara and H. Saito () Research Laboratory for Integrated Technological Systems, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan, Ishikawa 924-0838, Japan e-mail: [email protected]; [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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pressurized resin pot is not necessary unlike conventional RTM. Besides, although VARTM is one of the closed molding techniques, the opposite side of a rigid mold is covered by plastic film so-called bagging film. Because of these two reasons, VARTM method is paid attention as the comparatively low-cost molding method for large-scale structures. However, in the case that FRP is used in large-scale structures like airplane, incidence of damage caused by external factor such as debris or hail should be estimated. Accordingly, a lot of research works have been done about the lowenergy impact damage and the evaluation of the residual strength of FRP [3–8, 10, 17, 21, 24]. Various approaches have been used to observe the impact damage. For example, non-destructive observation methods, such as radiography [9–14] or ultrasonic C-scan [13–25], have been often used. Optical microscopic observation [11, 12, 14–17, 26, 27] also has been conducted to clarify the actual damages. Or direct observation of internal damage such as de-ply technique [9,13–15,18] has been used. Besides, the effects of environments, such as water absorption, should be considered in the large-scale structures. Although the effects of water on polymers and composites have been studied over the past several decades [28–32], research work about the combination effects of such mechanical damage and water absorption is a little. Especially the damage mechanism has not been discussed based on the precise observation. In this study, the impact damage and its evolution behavior in CFRP laminates were discussed. Target materials are two types of CFRP laminates which are reinforced by plain woven T300B and multi-axial knitted T700S. The damage mechanism within the candidate CFRP laminates was evaluated based on the original precise damage observation method which was named the Integrated Damage Evaluation Methodology (IDEM). Based on IDEM, the differences of damage mechanism between two types of CFRP were compared under dried and saturated water absorption conditions.

2 Materials and Testing Methods 2.1 Materials and Molding Method The materials used in this study were following two types of carbon fiber preforms; one was a basic type of plain woven fabric, T300B-3K (shown in Fig. 1a), and the other was a newly developed multi-axial knitted fabric, T700S-12K. Both of them were supplied by Toray Industries, Inc. The innovative multi-axial knitted fabric is expected to apply to the thick and large structural component such as marine ships, because it has much thickness than the conventional fabrics. Additionally, good mechanical property is expected by its non-crimp configuration and smooth warping performance. Multi-axial knitted fabric consists of several unidirectional or random-mat layers joined by polyester knitting fibers as shown in Fig. 1b.

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a

b

T300B-3K plain woven fabric

T700S-12K multi-axial knitted fabric

Fig. 1 Reinforcement fabric used in this study

Fig. 2 Schematic of vacuum assisted resin transfer molding (VARTM) setup

CFRP laminates were molded by vacuum assisted resin transfer molding (VARTM). VARTM is a suitable molding method for large-scale structures, because resin is impregnated into reinforcements by atmospheric pressure, unlike conventional RTM with the pressurized resin pot and the rigid closed mold. Therefore, low-cost and high-quality products can be fabricated by this method. Figure 2 shows a schematic image of VARTM setup.

2.2 Specimens and Impact Test Geometry of specimen was 150 mm in length and 43 mm in width as shown in Fig. 3. Thickness was 2.5 mm for multi-axial knitted CFRP, and 2.0 mm for plain woven CFRP. Width of specimen was decided by the size of test fixture. Here, the plain woven CFRP and the multi-axial knitted CFRP were abbreviated as “PW” and “MA”, respectively. A drop weight with a steel semi-spherical tip was used as an impactor. The weight of the impactor was 1,113.5 g. The diameter of semi-spherical tip was 16 mm. Fixture was two steel plates which have cutout holes of 30 mm in diameter as shown in Fig. 4. Specimens were mounted between these plates, and drop weight was subjected to the center of specimen. 1 J of impact energy an unit thickness was applied to the specimen.

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Fig. 3 Geometry of coupon specimen

Fig. 4 Impact testing machine and its fixture

2.3 Compression After Impact (CAI) Test and Post-Impact Fatigue (PIF) Test Compression after impact (CAI) and post-impact fatigue (PIF) tests were conducted. Geometry of specimen for both CAI and PIF tests were same coupon specimens as shown in Fig. 3. Compression and fatigue loads were applied by electro-hydraulic R EHF-EB100kN-20L, Shimadzu Co., Ltd.). fatigue testing machine (Servopulser Figure 5 shows a special fixture for CAI test. Gauge length of specimen was 50 mm, and test speed was 1.0 mm/min.

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Fig. 5 CAI test fixture

In PIF tests, stress ratio, R, was 1, that is, tension-compression test. Stress amplitude in the PIF test was decided on the basis of CAI strength, and their levels were as follows; 88%, 85% and 80% for multi-axial knitted CFRP, and 80%, 70% and 60% for plain woven CFRP, respectively.

2.4 Water Absorption Condition The effect of water absorption on CAI and PIF strength of CFRP laminates was evaluated by the specimens immersed in water. Same level coupon specimens as well as normal dry CAI and PIF tests were used for water absorbed CAI and PIF tests. In order to absorb water to the specimen up to saturation water content, the accelerated test was conducted. Both multi-axial knitted and plain woven CFRP specimens were soaked in a water bath at 95ı C for 120 h up to saturation water content of 0.6 wt%. After absorption of water, impact energy (1 J an unit thickness) was applied to these specimens. Specimens after soaking were named as “Wet”, as contrasted with “Dry” ones. In order to keep the water content of Wet specimen through the PIF test, gauge region of specimen was covered by plastic bag and filled with water inside of it, as shown in Fig. 6. Stress amplitude in PIF tests was 80%, 70% and 60% for both multi-axial knitted and plain woven CFRP.

3 Approach to Evaluate Damages The impact damage and its evolution behavior of CFRP laminate was evaluated three-dimensionally based on the original precise damage observation method which was named the Integrated Damage Evaluation Methodology (IDEM),

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Fig. 6 “Wet” specimen for PIF test

Fig. 7 Methodology of the three-dimensional characterization of the impact damage inside laminate

in which both non-destructive and direct observation methods were integrated. Figure 7 shows the methodology of the three-dimensional characterization of the impact damage inside laminate. Based on the IDEM, the differences of damage mechanism between two types of CFRP were compared under dried and saturated water absorption conditions. Damage evolution behavior was observed by the ultrasonic scanning machine (UT-2000, Toray Engineering Co., Ltd.). Frequency of transducer was 7.5 MHz.

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After observing the damage by ultrasonic, specimens were cut off in the multiple cross-sections and were observed by optical blue-ray microscope (VL2000D, LaserTech Co., Ltd.). Each cross-section was ground and polished by the precision polishing machine (TegraPol-15, Struers Co., Ltd.). The mirror-finished crosssection was scanned using the blue-ray microscope.

4 Impact Damage of CFRP Laminates 4.1 Plain Woven CFRP Laminate Ultrasonic C-scan images are shown in Fig. 8. Center of these figures is impact point. Near the surface of impact-side (Fig. 9a and b) showed the cross-shaped damage. Damaged area became smaller along the through-thickness direction. Figure 9 shows the cross-sectional images of PW CFRP laminates. Delaminations and cracks are clearly shown in these images. It is clear that the damaged area containing delaminations and cracks decreases with separating from the impact point. Both ultrasonic C-scan images and cross-sectional images were compared in Fig. 10. In this figure, cross-sectional images were divided in three-parts as upper, middle and lower, and each part was compared with C-scan image. The damage distribution observed by optical microscope corresponded with that observed by ultrasonic C-scan.

a

b

0.2 mm under impacted surface

d

c

0.6mm

1.0mm

e

1.4mm

1.8mm

Fig. 8 Ultrasonic C-scan images of impact damage in plain woven CFRP laminate

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a

Cross-section at the impact point

b

Cross-section that was 1.5mm away from the impact point

c

At 3.0mm

d

e

At 4.5mm

At 6.0mm

Fig. 9 Cross-sectional photographs of plain woven CFRP laminate

impact point 1.5mm 3.0mm 4.5mm 6.0mm impact point 1.5mm 3.0mm 4.5mm 6.0mm impact point 1.5mm 3.0mm 4.5mm 6.0mm

upper

middle

lower

Fig. 10 Comparison of the damage distribution images observed by optical microscope and ultrasonic scanning

4.2 Multi-axial Knitted CFRP Laminate Ultrasonic C-scan images are shown in Fig. 11. The impact point is the center of these images. Until the depth of 0.25 mm, damage was not detected. Then two

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a

c

b

0.75mm

0.25mm under impacted surface

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1.25mm

e

d

1.75mm

2.25mm

Fig. 11 Ultrasonic C-scan images of impact damage in multi-axial knitted CFRP laminate

fan-shaped damages which are symmetric with respect to the impact point each other were recognized at the depth of 0.75 mm. At the depth of 1.75 mm, damages spread in the 90ı direction. Accordingly, the damage area detected by ultrasonic C-scan spread along the direction of fiber bundle which constitutes each layer. Figure 12 shows cross-sectional photographs. From these images, it is clear that long delamination occurred between 0ı and 90ı layers. These delaminations were generated at the end of the transverse cracks. Both ultrasonic C-scan images and cross-sectional images observed by optical microscope were compared in Fig. 13. In this figure, cross-sectional images were divided in three-parts as upper, middle and bottom, and each part was compared with ultrasonic C-scan images. Consequently correspondence between the damage distribution observed by optical microscope and ultrasonic C-scan images was recognized. However, especially in upper and middle parts, delamination area observed in cross-sections was larger than that in ultrasonic C-scan images. This fact means that the damage, especially delamination, observed by ultrasonic C-scan was comparatively thick ones, and thin delamination was not detected. Further between of upper and middle delaminations, multiple transverse cracks existed, and they must affect on the damage tolerance performance of CFRP laminate.

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a Cross-section at the impact point

b Cross-section that was 2.0mm away from the impact point

c 4.0mm

d 6.0mm

Fig. 12 Cross-sectional photographs of multi-axial knitted CFRP laminate

Fig. 13 Comparison between damage areas obtained by the cross-sectional observation and the ultrasonic C-scan

4.3 Three-Dimensional Characterization of Impact Damage Within CFRP Laminates Figure 14 shows the three-dimensional damage distribution image constructed by both the ultrasonic C-scan and cross-sectional images of PW and MA CFRP laminates. In the case of PW CFRP laminate, transverse cracks extend radially from

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a

b

Plain woven CFRP laminate

Multi-axial knitted CFRP laminate

Fig. 14 Three-dimensional damage distribution images of the impact damaged CFRP laminates

the impact point to the backside. Especially, transverse cracks near the impact point spread in the circular shape around the impact point. Delamination forms a crossshape in the upper layers, and gradually becomes large except in the backside layer, where the delamination forms circular shape. In the case of MA CFRP laminate, transverse cracks extend radially from the impact point to the backside, as well as PW. Delamination spread by turns in 0ı and 90ı directions along the orientation angle of each layer. Delamination occurred between 0ı and 90ı fiber bundles, and the direction of delamination depends on the direction of the fiber beneath the delamination.

5 Compressive Strength and Fatigue Strength of Impact Damaged CFRP Laminates 5.1 Effect of Water Absorption on Post-Impact Fatigue Properties Figure 15 shows the compressive strength of plain woven CFRP and multi-axial knitted CFRP. The compressive strengths of PW and MA were decreased in 59% and 66% by the impact damage, respectively. On the other hand, CAI strengths of Wet PW and MA were 4.4% and 7.7% lower than that of Dry ones, respectively. Therefore, it is clear that the CAI strength was drastically affected by the impact damage, and that the effect of water absorption was little. Figure 16 shows the result of PIF tests of both PW and MA CFRP laminates. All S–N curves showed approximately inverse proportional relationship. Both CAI and PIF strengths of MA Dry were higher than that of the other specimens. Especially, the slope of approximate curve of MA was smaller than that of PW, that is, the decrease rate of fatigue life against applied stress amplitude was the most small.

124 700

Compressive strength [MPa]

Fig. 15 Compressive strength after impact of plain woven and multi-axial knitted CFRP under dry and wet condition

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600 500 400 300 200 100 0 intact PW PW Dry PW Wet intact MA MA Dry

MA Wet

Fig. 16 Fatigue strength under water absorption condition of plain woven and multi-axial knitted CFRP laminates

Therefore, it is obvious that the PIF performance of MA Dry was fairly higher than that of the others. However, all of water absorbed specimens showed almost same level of PIF performance.

5.2 Damage Evolution Mechanism in Plain Woven CFRP Laminates For the observation of damage evolution, PIF tests were stopped just before failure. The cycles of failure were estimated by S–N curves. Figure 17a shows ultrasonic scanning images of both Dry and Wet specimens before and after PIF tests,

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(i) Dry specimen

N/Nf = 0 (impact only)

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(ii) Wet specimen

Multi-axial knitted CFRP

Fig. 17 Damage evolution shown in ultrasonic C-scan images of CFRP laminates

respectively. Load was applied in the vertical direction in these images. From these images, there was practically no damage evolution in the PIF tested specimen, compared to the specimen which had only impact damage. In fact, plain woven CFRP specimens showed sudden failure at the impact point in the final stage of PIF tests, and almost no signs were observed before failure. Figure 18 shows cross-sectional photographs of both Dry and Wet specimens before and after PIF tests. Cross-sectional photographs were taken along the loading direction, which is the vertical direction in the ultrasonic scanning images shown in Fig. 17a. Although it is clear that the amount of delamination in Wet specimen was larger than that of Dry specimen, the increase of such delamination was confirmed not in the in-plane direction but in the through-thickness direction. Therefore, the characteristic damage morphology in plain woven CFRP is the increase of delaminations not in the in-plane direction but in the through-thickness direction. Figures 19 and 20 show the magnified images of cross-sections of Dry specimen at both N=Nf D 0 (impact only) and N=Nf D 0:9. Compared to the cross-section at N=Nf D 0 in Fig. 19, multiple transverse cracks (indicated as arrows in Fig. 19) and evolution of delamination initiated from the tip of these cracks were observed at N=Nf D 0:9. In addition, at the tip region of delamination in Fig. 20, evolution of delamination was stopped, and crack propagation occurred

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a

N/Nf = 0

N/Nf = 0.9 Dry condition

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N/Nf = 0

N/Nf = 0.9 Wet condition

Fig. 18 Damage evolution behavior observed in cross-sectional photographs of plain woven CFRP

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Dry condition, N/Nf = 0.9

Fig. 19 Damage evolution morphology in plain woven CFRP in dry condition (at the center) (Arrows indicate the transverse cracks)

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a

Dry condition, N/Nf = 0

b

Dry condition, N/Nf = 0.9

Fig. 20 Damage evolution morphology in plain woven CFRP in dry condition (at the tip of delaminations) (Arrows indicate the transition from delamination to crack)

at the tip of delamination. It is thought that these cracks were induced by the stress concentration at the tip of delaminations and that delaminations were transformed into these cracks through PIF test. It is thought that this phenomenon is the reason why the delamination does not spread in the in-plane direction through fatigue loading. Figure 21 shows the magnified images of cross-section at N=Nf D 0 (impact only). In dry condition, it is obvious that the delamination mostly occurred between ‘crossed’ 0ı =90ı fiber bundles. On the other hand, the delaminations of Wet specimen were induced between ‘parallel’ fiber bundles which were aligned in the same direction (indicated as arrows in Fig. 21), in addition to the interface between ‘crossed’ fiber bundles. This fact shows an evidence of interfacial degradation caused by water absorption. Consequently, the delamination tended to increase not in the in-plane direction but in the through-thickness direction, through the PIF test both in Dry and Wet conditions. Therefore, the “apparent” damage evolution, which could be confirmed by non-destructive evaluation methods, was restrained through the PIF test as shown in ultrasonic scanning images. Generally, it is well known that the CAI and PIF

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a

Dry condition, N/Nf = 0

b

Wet condition, N/Nf = 0

Fig. 21 Effects of water absorption on the initial damage of plain woven CFRP (Arrow indicates the delamination between fiber bundles aligned in same direction)

performances are in inverse proportion to the damage area. Thus, the results obtained by the damage observations support the results of S–N curves in PIF test, in which almost the same performances were shown in Dry and Wet conditions.

5.3 Damage Evolution Mechanism in Multi-axial Knitted CFRP Laminates Figure 17b shows the ultrasonic scanning images before and after the PIF test. Load was applied in the vertical direction in these images. It is clear that the damage propagated in the loading direction in both Dry and Wet specimens. However, this delamination occurred at the back side (especially between the outermost 0ı layer and the adjacent layer) of specimen. Therefore it is premature to determine that only this large delamination is the damage propagated by fatigue. Thus it is not enough to observe damages only with ultrasonic scanning. Final failure occurred in horizontal direction centering on the impact point. Figure 22 shows cross-sectional photographs of both Dry and Wet specimens before and after PIF tests. Cross-sectional photographs were taken along the loading direction, which is the vertical direction in the ultrasonic scanning images shown in Fig. 17b. The increase of transverse cracks was dominant damage morphology in the

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a

N/Nf = 0

N/Nf = 0.9 Dry condition

b

N/Nf = 0

N/Nf = 0.9 Wet condition

Fig. 22 Damage evolution behavior observed in cross-sectional photographs of multi-axial knitted CFRP

cross-section of both Dry and Wet specimen in PIF tests. On the other hand, delamination growth was little, except the delamination induced between the outermost 0ı layer and the adjacent layer in the opposite side of impact-side. Figure 23 shows magnified images of cross-section of both Dry and Wet specimens at N=Nf D 0:9 (almost final stage in PIF test). Through the PIF test, multiple transverse cracks occurred in both Dry and Wet specimens. As shown in the cross-sections (in Fig. 22) and the ultrasonic scanning images (in Fig. 17b), delaminations induced by impact mainly exist between 0ı and 90ı layers. These delaminations were aligned in the loading direction in the second and the fourth layers, and in the width direction (the direction perpendicular to the cross-sections in Fig. 22) in the third layer. Referring to the cross-sectional photograph at N=Nf D 0:9 in Dry condition in Fig. 22, and also referring to its magnified image in Fig. 23a, the delamination growth induced by fatigue was not clear, except the delamination between the outermost 0ı layer and the adjacent layer. In contrast, referring to the cross-sectional photograph at N=Nf D 0:9 in Wet condition in Fig. 22 and its magnified image in Fig. 23b, it is clear that delaminations

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a

Dry condition, N/Nf = 0.9

b

Wet condition, N/Nf = 0.9

Fig. 23 Differences of damage evolution behavior between dry and wet conditions

were propagated between 0ı and 90ı fiber bundles in the third layer and between 0ı fiber bundles in the second and the third layers, as shown by arrows in Fig. 23b. It is thought that the existence of these delaminations proved the effect of water absorption on the interfacial degradation between fiber and matrix. In addition, delaminations in the vicinity of 0ı fiber bundles, which are the main load sustainable members, should cause the deterioration of both the compressive performance and the PIF performance of entire specimen. Therefore, it is thought that the large decrease in S–N curves in the PIF test was caused by the incidence of these “critical” delaminations.

5 Conclusions In this study, the damage evolution behavior of T300-3k plain woven (PW) and T700S-12k multi-axial knitted (MA) CFRP laminates under post-impact fatigue (PIF) and water environment were evaluated. Especially, existing research works were mostly done by means of non-destructive observation of damage evolution mechanism. In this study, although between different specimens, the effect of water absorption on the damage evolution and the fatigue strength was able to be clarified by direct cross-sectional observation.

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The effect of water absorption on the performances of compression after impact (CAI) and PIF were small in PW CFRP laminates. Conversely, PIF properties of water-absorbed MA CFRP laminates drastically decreased than that of dry ones. CAI strength was not affected by water absorption. PIF performance of dry MA CFRP was fairly higher than that of the others. However the delamination area of PW CFRP was almost unchanged through the PIF test, the transverse cracks and delaminations were increased not in the in-plane direction but in the through-thickness direction. Especially in Wet condition, the delaminations were induced between ‘parallel’ fiber bundles which were aligned in the same direction, in addition to the delaminations induced between ‘crossed’ 0ı =90ı fiber bundles. Generally, it is well known that the CAI and PIF performances are in inverse proportion to the damage area. Thus, the results obtained by the damage observations support the results of S–N curves in PIF test, in which almost the same performances were shown in Dry and Wet conditions. Delamination area of MA CFRP was almost unchanged through the PIF test however increase of the number of transverse cracks was recognized. In Wet condition, it is clear that delaminations were propagated between 0ı and 90ı fiber bundles in the third layer and between 0ı fiber bundles in the second and the third layers. These delaminations in the vicinity of 0ı fiber bundles should cause the deterioration of both the compressive performance and the PIF performance of entire specimen. Therefore, it is thought that the large decrease in S–N curves in the PIF test was caused by the incidence of these “critical” delaminations. Acknowledgments The authors thank the Office of Naval Research for supporting this work through an ONR award (N000140110949) with Dr. Yapa Rajapakse as the program manager of solid mechanics. The authors thank Professor Richard Christensen at Stanford University as the consultant of this project and Toray Industries, Inc. as the supplier of CFRP laminates. All of experimental data were measured by the graduate students of author’s laboratory, in Kanazawa Institute of Technology. The authors thank these graduate students, Mr. Teppei Kimura and Norihiko Ikeda.

References 1. Brouwer WD, van Herpt ECFC, Labordus M (2003) Vacuum injection moulding for large structural applications. Compos Part 34: 551–558. 2. TECABS general presentation: 5th Framework Programme of European Community for RTD and Demonstration Activities (1998–2002). (http://www.tecabs.org/). 3. Abrate S (1991) Impact on laminate composite materials. ASME J Appl Mech Rev 44(4): 155–189. 4. Demuts E (1985) Assessment of damage tolerance in composites. Compos Struct 4: 45–58. 5. Ding YQ, Yan Y, Mcllhagger R (1995) Effect of impact and fatigue loads on the strength of plain weave carbon–epoxy composites. J Mater Process Technol 55: 58–62. 6. Lauder AJ, Amateau MF, Queeney RA (1993) Fatigue resistance of impact damaged specimen vs. machined hole specimens. Composites 24: 443–445. 7. Mahinfalah M, Skordahl RA (1998) The effects of hail damage on the fatigue strength of a graphite/epoxy composite laminate. Compos Struct 42: 101–106.

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8. Chen AS, Almond DP, Harris B (2002) Impact damage growth in composites under fatigue conditions monitored by acoustography. Int J Fatig 24: 257–261. 9. Palazotto A, Maddux GE, Horban B (1989) The use of stereo X-ray and deply techniques for evaluating instability of composite cylindrical panels with delaminations. Exp Mech 29(2): 144–151. 10. Mitrovic M, Hahn HT, Carman GP, Shyprykevich P (1999) Effect of loading parameters on the fatigue behavior of impact damaged composite laminates. Compos Sci Tech 59: 2059–2078. 11. Kinsey A, Saunders DEJ, Soutis C (1995) Post-impact compressive behaviour of low temperature curing woven CFRP laminates. Composites 26: 661–667. 12. Symons DD, Davis G (2000) Fatigue testing of impact-damaged T300/914 carbon-fibre-reinforced plastic. Compos Sci Technol 60: 379–389. 13. Soutis C, Curtis PT (1996) Prediction of the post-impact compressive strength of CFRP laminated composites. Compos Sci Technol 56: 677–684. 14. Gao SL, Kim JK (1998) Three-dimensional characterization of impact damage in CFRPs. Key Eng Mater 141–143: 35–54. 15. Cantwell WJ (1988) The influence of target geometry on the high velocity impact response of CFRP. Compos Struct 10: 247–265. 16. Davies GAO, Hitchings D, Zhou G (1996) Impact damage and residual strengths of woven fabric glass/polyester laminates. Compos Part A 27: 1147–1156. 17. Symons DD (2000) Characterisation of indentation damage in 0/90 lay-up T300/914 CFRP. Compos Sci Technol 60: 391–401. 18. Bibo GA, Hogg PJ, Backhouse R, Mills A (1998) Carbon-fibre non-crimp fabric laminates for cost-effective damage-tolerant structures. Compos Sci Technol 58: 129–143. 19. Cantwell WJ, Curtis PT, Morton J (1984) Impact and subsequent fatigue damage growth in carbon fibre laminates. Int J Fatig 6: 113–118. 20. Hosur MV, Murthy CRL, Ramamurthy TS, Shet A (1998) Estimation of impact-induced damage in CFRP laminates through ultrasonic imaging. NDT and E Int 31(5): 359–374. 21. Tai NH, Yip MC, Lin JL (1998) Effects of low-energy impact on the fatigue behavior of carbon/ epoxy composites. Compos Sci Technol 58: 1–8. 22. Zhang X, Davies GAO, Hitchings D (1999) Impact damage with compressive preload and post-impact compression of carbon composite plates. Int J Impact Eng 22: 485–509. 23. Kim JK, Sham ML (2000) Impact and delamination failure of woven-fabric composites. Compos Sci Technol 60: 745–761. 24. Margueres PH, Maraghni F, Benzeggagh ML (2000) Comparison of stiffness measurements and damage investigation techniques for a fatigued and post-impact fatigued GFRP composite obtained by RTM process. Compos Part A 31: 151–163. 25. Melin LG, Schon J, Nyman T (2002) Fatigue testing and buckling characteristics of impacted composite specimens. Int J Fatig 24: 263–272. 26. Kumar P, Rai B (1993) Delaminations of barely visible impact damage in CFRP laminates. Compos Struct 23: 313–318. 27. Shikhmanter L, Cina B, Eldror I (1995) Fractography of CFRP composites damaged by impact and subsequently loaded statically to failure. Composites 26: 154–160. 28. Selvarathinam AS, Weitsman YJ (1999) A shear-lag analysis of transverse cracking and delamination in cross-ply carbon-fibre/epoxy composites under dry, saturated and immersed fatigue conditions. Compos Sci Technol 59: 2115–2123. 29. Weitsman YJ, Guo YJ (2002) A correlation between fluid-induced damage and anomalous fluid sorption in polymeric composites. Compos Sci Technol 62: 889–908. 30. Tang R, Guo YJ, Weitsman YJ (2004) An appropriate stiffness degradation parameter to monitor fatigue damage evolution in composites. Int J Fatig 26: 421–427. 31. Matsuda S, Hojo M, Ochiai S (1999) Effect of water environment on Mode II delamination fatigue in interlayer-toughened CFRP. JSME Int J 42(3): 421–428. 32. Kootsookos A, Mouritz AP (2004) Seawater durability of glass- and carbon-polymer composites. Compos Sci Technol 64: 1503–1511.

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures Roberta Massab`o

Abstract The chapter deals with the problem of the interaction of multiple damage mechanisms in multilayered and sandwich structures subject to static and dynamic loading conditions. The effects of the interaction on the mechanical behavior of composite systems are studied by means of one-dimensional models based on particularizations of the theories of bending of beams and plates. The study shows that the interaction of multiple delaminations in laminated and multilayered systems subject to out of plane loadings profoundly affects crack growth characteristics and different key properties; controlled delamination fracture, via material and structure design, can improve impact and damage tolerance and energy absorption.

1 Introduction The chapter summarizes main results of the work performed by the writer and her collaborators in projects funded by the U.S. Office of Naval Research over the last 4 years. The theme of the research is the investigation of the effects of the interaction of multiple damage mechanisms on the mechanical response of composite and sandwich structures. Static and dynamic loading conditions that are typical of the severe environments where naval structures operate are considered; they include low and high velocity impact, due for instance to weapons or collisions, and blast, due to air or underwater explosions. The work gives basic insight for an effective design of marine composite materials and systems with enhanced mechanical properties, e.g. survivability, damage/impact resistance and tolerance and energy absorption.

R. Massab`o () Department of Civil, Environmental and Architectural Engineering, University of Genova, Via Montallegro, 1, 16145 Genova, Italy e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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The main part of the work deals with the response of multiply delaminated composite plates and will be presented in Sections 2–5. Multiple delamination is a dominant damage and failure mechanism of laminated and multilayered systems subject to dynamic loadings. Often the delaminations are undetectable on the surface and may significantly reduce the stiffness of the structure and its ability to sustain repeated loadings; or they may grow catastrophically, leading to structural failures. The behavior of laminated and multilayered systems in the presence of a single delamination has been extensively studied since the early work of Kanninen [1, 2]; important conclusions on the response under static, dynamic and fatigue loading have been drawn and standard tests have been proposed to derive quasi-static fracture properties under single and mixed mode conditions. Recently, a large effort of the composite community has been directed to investigate the response of systems reinforced through the thickness, for instance by stitching, z-pinning or braiding, that have a great potential for a damage and impact tolerant design (a through-thickness reinforcement creates large scale bridging conditions that control interfacial fracture also leading to crack arrest or substantially modifying the mode of failure). The problem of the interaction of multiple delaminations in composite structures has been considered only quite recently in Refs. [3–10] where the response of systems subject to quasi-static loading conditions has been investigated both theoretically and experimentally. The paper by Andrews et al. [10] is the first to report results on the interaction effects of multiple delaminations in systems subject to dynamic loadings. Section 2 of the chapter introduces the theoretical approach of the research, whose main feature is to refer to simple model systems, namely multilayered beams and plates deforming in cylindrical bending, that have analytical or semi-analytical solutions for special configurations and loading conditions. The models that have been formulated within the program will be recalled along with a method proposed to effectively decompose the modes of fracture in the presence of shear in homogeneous and orthotropic beams. Section 3 describes static and dynamic interaction effects of multiple delaminations on fracture parameters in laminated and multilayered systems. Section 4 focuses on the effects of the interaction of multiple delaminations on the macrostructural response and crack growth characteristics. Section 5 shows applications of the models that highlight how controlled delamination fracture, via material/structure design, can effectively improve mechanical performance against dynamic loadings. The last section of the chapter, Section 6, presents preliminary results of current work on the problem of the interaction of multiple damage mechanisms in sandwich systems and focuses on the indentation response of sandwich beams with skin damage.

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2 Modeling Multiple Delamination Fracture in Laminated and Multilayered Systems 2.1 Theoretical Approach Different modeling techniques have been used in the literature to study brittle and cohesive delamination fracture in materials and systems subject to static and dynamic loading conditions (see Refs. [11,12] for reviews). Among the fully numerical approaches, the Finite Element Method has been applied extensively with special crack tip elements able to reproduce the singular crack tip fields of Linear Elastic Fracture Mechanics and cohesive zone (interface) elements able to describe process regions; adaptive or initially elastic interfaces have been used in combination with 2D, 3D and beam and plate elements. Several authors, including the writer and her collaborators, have formulated models that have the capacity of analyzing multiple delaminations [8–10, 13–16], however few have proceeded to do so, perhaps because the complexity of the problem leads to burdensome analyses and makes it difficult to draw general and insightful conclusions. The recent paper by Andrews et al. [10] is the first to report a study on the dynamic interaction effects of multiple delaminations. In the work performed for the ONR, simple model systems have been examined, namely beams or plates deforming in cylindrical bending, which imply plane strain conditions and translational invariance of crack shapes in the direction of the bending axis, subject to static and dynamically applied loads with loading histories specified a priori (Fig. 1a). The problems have been modelled with particularizations of the theories of bending of beams and plates. For special configurations,

a

q

x y

z

b

P hU h

c h

hL aL

P

h

a L/2

aU

Fig. 1 (a) Schematic of a multiply delaminated composite plate subject to an out of plane load and deforming in cylindrical bending; (b) cantilever beam with two delaminations subject to a transverse point force; (c) cantilever beam with a main crack and a damaged area represented by an array of two small delaminations

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e.g. quasi-static or steady state dynamic fracture with piecewise linear cohesive laws and a limited number of delaminations, these models have useful analytical or semianalytical solutions and the effects of the interaction of multiple cracks can be easily singled out. The geometries studied are different to those expected in the important engineering problem of laminated plates subjected to a point impact load, where induced delaminations typically show a staircase or pine-tree configuration that is partly controlled by the ply lay-up (e.g., Ref. [17]). However, the simple model systems represent many laboratory tests: a logical path to predicting cracking in cases of general symmetry is first to deduce quasi-static and dynamic fracture properties, e.g. fracture energies or cohesive laws, from plane tests, which are easier to analyze, and then use those laws in a fully three-dimensional simulation of the field case. To study multiple quasi-static delamination fracture in homogeneous plates, e.g. unidirectionally reinforced laminates or quasi-isotropic laminates with a large number of layers, a one-dimensional model has been formulated in Refs. [8, 9, 18]. The model decomposes the laminated plate (Fig. 1a) into multiple uncracked beam segments, which are defined by the crack tips and all sections where there is a change in the state of contact/cohesion along a delamination surface. The sub-laminates are then treated as Timoshenko beams and the possibility of relative rotations (root rotations) between each of the two sublaminates forming a delamination tip and the intact sublaminate ahead of the delamination tip (e.g. the grey area in Fig. 1a) is included to account for elastic near tip deformations (Fig. 1b). The root rotations are defined as linear functions of the crack tip stress resultants through compliance coefficients and the crack tip energy release rates and stress intensity factors are then calculated following the method proposed by Andrews and Massab`o [19]. The procedure is recalled in Section 2.2. To study multiple dynamic delamination fracture in inhomogeneous multilayered systems a one-dimensional model that represents the plate as a set of Timoshenko beams joined by cohesive interfaces has been formulated in Refs. [10,18]. The cohesive interfaces are used to represent the nonlinear process of material rupture, crack shielding due to cohesive or bridging mechanisms acting along the crack surfaces and the elastic resistance to interpenetration of delaminated sublaminates. The solutions of the approximate models recalled above have been checked for some reference problems by means of accurate two dimensional solutions: static finite element results obtained by the research team [8, 16], dynamic finite element solutions from the literature [20] and semi-analytical results obtained by means of the weight function method and using the approximate weight functions derived by the writer and her collaborators for mode I and mode II orthotropic double cantilever beams [21, 22].

2.2 Energy Release Rate and Stress Intensity Factors in Homogeneous Orthotropic Beams As a part of the research performed within the ONR project, semi-analytical expressions have been derived for the energy release rate and the stress intensity factors

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

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b h1

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h2

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Δ ϕ0,1 h1

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V M1 1 N1 N2 M2 V

h1 h2

M0

h0

2

V0

N0

Δx→0

Δ ϕ0,2

Fig. 2 (a) Edge cracked beam subject to generalized end forces; (b) relative rotations of the beam arms at the crack tip cross section; (c) crack tip stress resultants (normal and shear forces and bending moments) (Note that the reference system differs from that used in Fig. 1)

of edge cracked homogeneous and orthotropic beams subject to arbitrary generalized end forces (Fig. 2a). The expressions are accurate for long and short cracks and have been derived in order to be used within the theoretical model formulated in Refs. [9, 18] to study multiple delamination fracture in homogeneous beams. The expressions can be used to perform mode decomposition on solutions obtained by means of other 1D or 2D models and are relevant to many practical problems, including the interfacial fracture of multilayered systems and the decohesion of thin films and coatings from substrates made of the same material. Following the work of Li et al. [23] for isotropic bi-material layers, the derivation extends the method proposed by Suo [24] for axial forces and bending moments in order to include the contribution of the shear forces. The shear contribution to the fracture parameters depends on the shear deformations along the layer and the elastic near tip deformation of the material. Li et al. [23] derived semi-analytical expressions for the fracture parameters that depend on the crack tip stress resultants, the elastic constants and five numerically-determined constants globally describing the effect of shear. In Andrews and Massab`o [19] analogous constants have been derived for orthotropic beams and defined by semi-analytical expressions that highlight their physical significance and allow separation of the different contributions. The derivation is based on the assumption that the near tip deformation can be described by means of relative rotations between the cross sections of the different sub-beams at the crack tip (root rotations), Fig. 2b: '0;1 D '0  '1 and '0;2 D '0  '2

(1)

where the 'i (for i D 0; 1; 2) are the bending rotations of the beam segments i . The root rotations depend linearly on the crack tip stress resultants (Fig. 2c) through compliance coefficients, a’s, that have been derived numerically for a wide range of orthotropic materials and are presented in Table They depend p 18.1 in Ref. [19]. p on the orthotropy ratios,  D Ez =Ex and  D Ex Ez =.2Gxz /  zx xz , Poisson ratios, zx and xz , the shear coefficient, S D 5=6, and the relative position of the crack, ˜ D h1 = h2 : '0;i

1 D Ex h1



aiM M C aiN N C aiVS VS C aiVD VD h1

 (2)

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where M , N , VS and VD define self equilibrated systems of stress resultants depending on the crack tip stress resultants of Fig. 2c as follows: 1 M D M1   3 M0 ; 1 C 1

N D N1 C

1 6 1 M0 N   ; 3 1 0 h1 1C 1 C 1

VS D V0 ; VD D V2

(3)

and Ex ; Ez and Gxz are longitudinal, transverse and shear moduli, respectively. The expressions are accurate for a,c > cmi n , (Fig. 2a) where cmi n D hi œ1=4 .i D 0; 1; 2/ collectively represents all stress resultants for uncertainties lower than ˙2%. The energy release rate was determined by an application of the J-integral to a path that follows the cross sections immediately preceding and following the crack tip: " 2   Ni2 Mi2 1 X Vi2 C C 2Vi '0;i GD C 2 i D1 Ex h3i =12 S Gxz hi Ex hi N02 M02 V02    Ex h30 =12 S Gxz h0 Ex h0

 (4)

and depends on the crack tip stress resultants (Fig. 2c), the elastic constants, the shear coefficient and the crack tip relative rotations of Eq. (2). The expression highlights the contribution of the shear forces that enter the solution associated to the shear deformations along the layer (second and sixth terms) and the near tip deformation through the root rotations (fourth term). In addition, it shows that the near tip deformation does not affect the bending-moment and normal-force components of G, which confirms the steady-state energy balance of Suo [24]. For a degenerate orthotropic DCB specimen .¡ D 1/ with thickness 2h, longitudinal Young’s Modulus E subject to transverse end forces P , the expressions (4) and (2) yield:  2  a 2

GDCB Eh 1=4 h D 12 1 C 0:673 P2 h a

(5)

that in the case of isotropy ( D Ez =Ex D 1) matches very well, with only a difference in the third significant digit, with existing results for the DCB specimen. The expression of the energy release rate for a degenerate orthotropic ENF specimen turns out to be:

 2 9  a 2 GENF Eh 1=4 h D 1 C 0:208 P2 16 h a again in excellent agreement with existing results.

(6)

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

139

The stress intensity factors were derived in Ref. [19] by extending the methods in Refs. [23, 24] and are defined by the following expressions: " fM M sin.M C !/ 3=8 fN N cos.!/ C p KI D  1=4 3=2 h1 1C h1 2 (7a)  fVD VD sin.VD C !/ fVS VS sin.VS C !/ C p C p h1 h1 KII

" fM M cos.M C !/ 1=8 fN N sin.!/ D C p 1=4  3=2 h1 1C h1 2  fV VD cos.VD C !/ fVS VS cos.VS C !/  D p  p h1 h1

(7b)

which depend on the crack tip stress resultants (Fig. 2c) through Eq. (3), the elastic constants and orthotropy ratios, and the root rotation compliances through the functions fM ; fN ; fVD ; fVS ; M ; VD ; VS (defined in Eq. (15) in Ref. [19]). The additional constant ¨ was previously determined by Suo in Ref. [24] for isotropic and degenerate orthotropic materials as a function of the relative position of the crack and is given by ! D 52:1ı  3ı ; the constant is adequate for a wide range of orthotropic materials, which includes 1  ¡  5. The semi-analytic expressions in Eqs. (7a) and (7b) are very accurate also for beams with short cracks, provided the crack tip stress resultants are calculated accounting for the localized crack tip root rotations (Eq. (2)) in all statically indeterminate problems. In the following sections, predictions of energy release rates and stress intensity factors by means of the model formulated in Refs. [10, 18], which includes root rotations, experimental results and accurate two-dimensional finite element results will be compared for different multiple delamination systems. The comparisons highlight the accuracy of the expressions (7a) and (7b) also when the crack tips are closed to each other.

3 Interaction Effects of Multiple Delaminations on Fracture Parameters 3.1 Amplification and Shielding of the Energy Release Rate The presence of other delaminations induces phenomena of amplification or shielding of the energy release rate of a delamination, namely increase or decrease, respectively, relative to the value for the crack when it is present alone in the body [3–5, 8–10]. Relevant features of this behavior are described in the following using the reference case of a cantilever beam subject to a transverse point force applied quasi statically at the free end (Fig. 1b).

140

R. Massab`o

a 1000

FEM Proposed Model Euler Bernoulli Model Single crack limit

800 UEh 2

P

aU >aL

600 aU = varied aL = 5h hU = h/ 3 hL = h/3 λ = 1, ρ =1

400 200 0

0

b

1

1000

P

3

4

FEM Proposed Model Euler Bernoulli Model Single crack limit

800 UEh 2

2

5

6

7

8

9 aU /h

aU >aL

600 aU = varied aL = 5h hU = h/ 3 hL = h /3 λ = 0.05, ρ =5

400 200 0

0

1

2

3

4

5

6

7

8

9 aU /h

Fig. 3 Dimensionless diagrams of the energy release rate of the upper delamination in the homogeneous cantilever beam of Fig. 1b, as a function of its length, in the presence of a lower delamination of fixed length. (a) Isotropic material; (b) highly orthotropic material, e.g. a unidirectionally reinforced boron–epoxy composite (œ and ¡ orthotropy ratios defined in Section 2.2 of the main text). The curves correspond to different models: FEM is an accurate two-dimensional finite element solution [8], proposed model is based on beam theory and accounts for shear effects and root rotations [9] and Euler Bernoulli model is based on elementary beam theory, which neglects shear and near tip deformations [8] (E D longitudinal Young’s modulus; frictionless contact modelled using a Winkler foundation).

The diagram in Fig. 3a refers to a homogeneous and isotropic beam with two frictionless delaminations (Fig. 1b). The diagram depicts the energy release rate of the upper delamination as a function of its length when the length of the lower delamination is kept fixed. The solution is compared with that in the absence of the lower crack (thin line). A strong amplification is observed when aU < aL ; this interaction effect is long range and occurs also when the two delamination tips are far apart. In addition, a sharp transition in the energy release rate is observed when the two delaminations reach the same length. The response of a highly orthotropic material (Fig. 4b), e.g. a unidirectionally reinforced boron–epoxy composite, shows similar

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

141

Fig. 4 Diagram of the energy release rate of the main delamination (located at a distance 2=3h from the upper surface) in the homogeneous and isotropic cantilever beam of Fig. 1c, as a function of its length, when the delamination crosses the damaged area (small delaminations spaced h=4 from the upper and lower surfaces). FEM is an accurate 2D finite element solution [9], proposed model is based on beam theory and accounts for shear effects and root rotations [9], interface model represents the beam using Timoshenko beams joined by cohesive interfaces [9] and Euler Bernoulli model is based on elementary beam theory [8]. (E D Young’s modulus; frictionless contact modelled using a Winkler foundation)

features but for the sudden transition of the isotropic case that in the orthotropic case occurs over a larger distance as dictated by orthotropy rescaling [24]. In homogeneous beams the occurrence of amplification or shielding is controlled by the crack spacing and the relative lengths of the delaminations and behavioral maps have been constructed in Ref. [8] that synthesize this behavior for two delamination systems. Multiply delaminated systems subject to in-plane loading (compression after impact problem) show similar characteristics [3, 4]. The presence of a damaged area, for instance an array of small delaminations, also has important effects on the energy release rate of a main delamination that is propagating through the array (Fig. 1c). In this case a phenomenon of amplification followed by shielding is typically observed as the main delamination passes through the damage region (Fig. 4). This behavior, which is short range since the amplification and shielding phenomena are observed only when the tip of the main delamination is close or inside the damaged area, is similar to that of infinite homogeneous media where a main crack interact with a cloud or an ordered array of microcracks [25].

3.2 Interaction Effects on Mode Ratio In addition to amplify or shield the local stress fields, the presence of other delaminations modifies the relative amount of mode II to mode I of a delamination when it is present alone in the body, mainly because it favours the occurrence of extensive contact between the delamination surfaces [3, 8].

142

R. Massab`o

a 100 aU >aL a = varied U

tan–1 (KIIU/KIU)

90

aL = 5h hU = h/3 hL = h/3 λ = 1, ρ =1

80 70 FEM Proposed Model Euler Bernoulli Model Single crack limit

60 50 40

b

0

1

2

3

4

5

6

7

8 9 aU /h

100 aU >aL

tan–1 (KIIU/KIU)

90 80 70

aU = varied aL = 5h hU = h/ 3 hL = h/ 3 λ = 0.05, ρ =5

FEM Proposed Model Euler Bernoulli Model Single crack limit

60 50

0

1

2

3

4

5

6

7

8 9 aU /h

Fig. 5 Diagrams of the relative amount of mode II to mode I, measured by the phase angle, of the upper delamination in the homogeneous beam of Fig. 1b as a function of its length. The lower delamination is kept at a fixed length. (a) Isotropic material; (b) highly orthotropic material. The thick curves correspond to different models: FEM is an accurate two-dimensional solution in Ref. [8], proposed model is based on beam theory and accounts for shear effects and root rotations [9], Euler Bernoulli model is based on elementary beam theory, which neglect shear and near tip deformations [8] (frictionless contact modelled using a Winkler foundation)

Figure 5a and b refer to the multiply delaminated beam of Fig. 1b and depict the phase angle ‰ D tan1 .KII =KI / of the upper delamination, as a function of its length, in the presence of a lower delamination of fixed length in (a) an isotropic and (b) a highly orthotropic material (KI and KII are mode I and mode II stress intensity factors). The solution is compared with that of the upper delamination when it is alone in the beam (thin lines). For all configurations where aU < aL , the lower delamination induces contact at the tip of the upper delamination so that the upper crack is under virtually pure mode II conditions and the phase angle is  90ı . When the two delaminations reach the same length there is a sharp transition, the mode I component increases, even above that of the single delamination solution; the

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

143

interaction effect slowly disappears when the upper crack lengthens. The interaction effects are important over a wider range of crack lengths in the highly orthotropic material, as dictated by orthotropy rescaling [24]. The comparison with accurate two-dimensional FE solutions in Fig. 5a and b highlights the accuracy of the beam theory model formulated in Refs. [9, 18] that accounts for shear deformations and root rotations, as well as the limitations of models based on the elementary beam theory, which neglects root rotations and shear effects, especially when dealing with orthotropic materials (see Section 2).

3.3 Coupling of Interaction and Dynamic Effects When the load is applied rapidly, as in the case of high velocity impact or blast, the propagation of the stress waves and the forced and free (after load removal) vibrations of the structural systems yield variations over time of the response variables, leading for instance to amplification (or shielding) of the energy release rate of the delaminations, namely increase (or decrease) relative to the values for the same systems under quasi static conditions. In multiply delaminated structures subject to out of plane loading, dynamic amplification of the energy release rates of the cracks couples with amplification phenomena caused by the interaction of the delaminations in a manner that depends on the layup of the material, the relative length and position of the delaminations and shape and duration of the dynamic excitation. Dynamic effects in clamped–clamped homogeneous beams with single delaminations subject to time dependent out of plane forces are presented first to highlight differences between cases where the deformation field remains mode II at all times and cases where the beams deform in mixed mode conditions. The effects of coupling between interaction and dynamic effects are described later in the section.

3.3.1

Dynamic Response of Beams with Single Stationary Delaminations

Figures 6a and b highlight dynamic effects on the energy release rate of a central mid-plane delamination in a homogeneous clamped–clamped beam subject to a transverse (a) step force with finite rise time and (b) triangular pulse force; in this system the deformation field remains essentially mode II at all times as a consequence of the symmetry. Variation over time of the dynamic energy release rate of the stationary delamination, normalized with respect to the quasi static value corresponding to the maximum applied load, is shown as a function of the (a) rise time and (b) pulse duration. The corresponding response spectra, shown in Fig. 7a and b (curves labeled 1 delamination) show the variation, with load (a) rise time or (b) duration, of the maximum value of the dynamic energy release rate normalized to the static value corresponding to the maximum applied load.

144

R. Massab`o

a

5 tR = 0

4 3

a = L/2 h1 = h/ 2 tR =

P(t)/2 h1

h

3t1 4

a

tR = t1

L

2

P

plane of symmetry

Load

static

t1 4 t tR = 1 2

tR =

1

tR

0 0

1

2

3

4

t

Time

t/t1

b

2

tP = t1 /4 t P = t1

1.5

a = L/2 h1 = h/ 2

P(t)/2 h1

h

tP = 2t1

a

1

L

static

P

plane of symmetry

Load

0.5 0 0

1

2

3

4

5 t/t1

tP /2

tP

t

Time

Fig. 6 Time histories of the dynamic energy release rate of a stationary mid-plane delamination of length a D L=2 in a homogeneous and isotropic clamped–clamped beam subject to (a) a step force with rise time tR and (b) a triangular pulse force with duration time tp . The energy release rate is normalized to the static value corresponding to the maximum applied load. The first dimensionless natural vibration period of the beam is t1 cL = h D 4:245.L= h/2 . The problem remains mode II at all times (cL D longitudinal wave speed; frictionless contact modelled using a Winkler foundation) (Adapted from Ref. [10])

Figures 8 depicts dynamic effects in a homogeneous clamped–clamped beam with a single, central, off-center delamination subject to a transverse triangular pulse force; in this system the deformation field is mixed mode and variations in the mode ratio with time are controlled by the loading conditions. Exemplary time histories of the mode I and mode II components of the energy release rate are presented in Fig. 8a and compared with the values corresponding to the maximum applied force for quasi static loading (dashed lines). Three characteristic regimes are identified. During the loading phase (forced vibrations), the mode II component of the energy release rate experiences a large dynamic amplification, while the mode I component remains at or below the static value because the applied load constrains the upper and lower delaminated arms to displace without separation (this behavior characterizes also the response of systems where the applied load is maintained, e.g. step loads with finite rise time). After the load has been completely removed (initial free vibration phase), the mode II component remains amplified while large oscillatory amplification of the

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

a

5

1 delamination 2 delaminations 3 delaminations

4

a = L/2

P(t)/2 h a

3

static

plane of symmetry

L 2

P Load

max

1 0

145

0

0.5

1

1.5

2

2.5

tR Time

3

t

tR/t1

b

2.5 a = L/2

P(t)/2

2 h max

1.5

a

L

static

1

Load

P 1 delamination 2 delaminations 3 delaminations

0.5 0 0

1

2

3

4 tp/t1

plane of symmetry

tp/2

tp t

Time

Fig. 7 Response spectra showing dynamic amplification factors of the energy release rate versus normalized (a) rise time and (b) pulse duration for systems of 1, 2 and 3 equal length and equally spaced brittle delaminations in a homogeneous and isotropic clamped–clamped beam subject to (a) a step force with a finite rise time and (b) a triangular pulse force (frictionless contact modelled using a Winkler foundation; t1 D first period of vibration) (Adapted from Ref. [10])

mode I component also occurs as the now-unconstrained arms vibrate out of phase due to mismatch in bending stiffness. The beam then enters a third regime of behavior characterized by hammering, i.e., chaotic localized impacts at different points along the crack surfaces and at different time steps. Both components of the energy release rate undergo high frequency oscillations with large amplitudes. In reality, the large oscillations might be smoothed by energy dissipation mechanisms, such as heat generation, microcracking or friction. The three phases of behavior also lead to distinct characteristics in the time history of the vertical displacements of the two arms at the loading point (Fig. 8b).

3.3.2

Dynamic Response of Beams with Multiple Stationary Delaminations

In multiply delaminated systems, the dynamic effects described above strongly couple with interaction effects in all cases but when the system deforms under mode II conditions and the load is kept for a sufficiently long time. In this latter case, which

146

R. Massab`o

a 60

tpcL h = 311

L = 10h a = 5h h1=h/ 3

50 40

P(t)/2 h1

h

W a

II

Eh 30 P2 20

L plane of symmetry

P

I

Load

II-static

10 I-static

0

0

200

b

400

600 tcL/h

800

1000

1200

tp/2

tp

t

1000 500 lower arm

0 wE P –500

upper arm Free vibration: out of phase

Loading: in phase

–1000 –1500 0

200

400 tcL/h

600

800

Fig. 8 (a) Time histories of (a) the energy release rate and (b) the load point displacements of a clamped–clamped beam with a stationary off-center delamination subject to a triangular pulse force of duration tp D 3=4t1 , with t1 cL = h D 415 the natural vibration period (calculated assuming constrained contact). The material is homogeneous, isotropic, and brittle (E D Young’s modulus; cL D longitudinal wave speed) and contact is frictionless. Shape and duration of the load are plotted in dotted lines with an arbitrary scale in the ordinate (Adapted from Ref. [10])

characterizes the response of homogeneous beams with equal length and equally spaced cracks subject to step loads, dynamic and interaction effects are approximately uncoupled and the response of systems with a single delamination can be rescaled to describe the response of systems with multiple delaminations. In particular, the maximum amplification factor of the energy release rate due to dynamic effects remains approximately constant on varying the number of delaminations, as shown in the response spectrum of Fig. 7a for systems with 1, 2 and 3 equally spaced cracks. This behavior is observed also for a pulse load of sufficiently long duration (Fig. 7b). Figure 9a and b highlight coupling between interaction and dynamic effects in a homogeneous clamped–clamped beam with two unequally spaced cracks with (a) unequal and (b) equal lengths. The diagram in Fig. 9a depicts the time history of the mode II component of the energy release rate in the upper crack (solid line). The result is compared with the quasi-static solution corresponding to the maximum value of the applied pulse force and with the quasi-static and dynamic solutions

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

a

40

hU = h / 3 aU = 2.5h

UEh

P(t)/2 hU

IIU

single crack limit IIU

hL = h / 3 aL = 5h

30

147

h

hL

tp = t1

L = 10h

20

aL

IIU static

P2

L Load

10 0

plane of symmetry

P

IIU static single crack limit

aU

0

b 8

0.5

1.5 t / t1

2

2.5

tp /2

tp

h

hU = h / 6 hL = h / 2

hL

tpCL / h = 398.1

P

t

P(t)/2 hU

a = 2.5h

L = 10h

UEh 2

1

a L

4 UII static

UI static

=0

plane of symmetry

P Load

2 UI

UII

0 0

200

400 tcL / h

600

800

tp/2

tp t

Time

Fig. 9 (a) Time history of the mode II energy release rate component of the upper delamination in the presence (solid lines) and in the absence (dashed lines) of a lower delamination in a clamped–clamped beam with two unequal length delaminations subject to a triangular pulse force with duration equal to the first vibration period t1 . (b) Time history of the energy release rate components of the upper delamination in a clamped–clamped beam with two equal length unequally spaced delaminations subject to a triangular pulse force with tp D t1 . The material is homogeneous and isotropic, contact is frictionless (E D Young’s modulus; cL is the longitudinal wave speed) (Adapted from Ref. [10])

for the crack if it were alone in the beam. Figure 9b depicts mode I and mode II components of the energy release rate of the upper crack when the system is subject to a quasi static load (dashed line) and to a triangular pulse force (solid lines). Under quasi-static conditions, the upper delamination is under pure mode II conditions because of crack tip contact generated by the presence of the lower crack (interaction effect). When the load is applied dynamically, the three regimes of behavior, which have been observed previously in Fig. 8, are present and in the free vibration phase both mode I and mode II components of the energy release rate are well above the static values.

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R. Massab`o

4 Interaction Effects of Multiple Delaminations on Crack Growth Characteristics and Macrostructural Behaviour The phenomena of amplification and shielding of the crack tip fields, which are a consequence of the interaction of multiple delaminations in beams and plates subject to static and dynamic out of plane loading, have important effects on crack growth characteristics and macrostructural behavior.

4.1 Local Instabilities and Strengthening Mechanisms Important macrostructural effects of the amplification and shielding phenomena are local instabilities in the load versus displacement curves that describe the global response of the structural elements. The experimental results (thick line in Fig. 10a) obtained by Robinson et al. [7] by testing a double cantilever beam with two implanted delaminations clearly highlight this effect that can be understood and explained by means of the theoretical model formulated in Ref. [9], (thin line in Fig. 10a). The load versus load-point displacement curves shown in Fig. 10a presents an initial strain softening branch when the edge crack starts to propagate unstably at (A). At (B) the tip of the edge delamination reaches the left tip of the internal delamination and a snap-back instability occurs with a sudden drop in the critical load to point (C); this drop corresponds to a sharp change (amplification) in the energy release rate similar to that described in Fig. 3a. The edge crack then continues to propagate stably (with increasing applied load) until its tip reaches approximately the center of the internal delamination at (D) when also the energy release rate of the right tip of the internal crack becomes critical (amplification) and the internal delamination starts to propagate. Simultaneous propagation of the two cracks then continues for the remainder of the test. Other local instabilities have been observed in different systems. The load versus load-point displacement curve shown in Fig. 10c describes the response of the multiply delaminated beam of Fig. 1b (geometrical data are given in the figure). A local snap-back instability can be observed when the lower delamination starts to propagate first at (A). The propagation is then unstable up to point (B) where the delamination reaches the upper delamination and arrests due to a sharp change (shielding) of the energy release rate. The lower delamination can then be made to propagate only by increasing the applied load to the level of point (C), after which the delamination continues to propagate unstably. The load versus displacement curve of Fig. 10d is similar to that of Fig. 10c. However in this case the new critical load due to the sharp negative change (shielding) of the energy release rate is higher than the initial load for delamination propagation leading to a local snap-through instability. This is an interesting example of strengthening mechanism generated only by the presence of other delaminations in the system.

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

a

b

60

Width = 20 mm

Experimental (Robinson et al. 1999)

A

50

1.59 mm

Displacement Increment

Load (N)

B D

30

Hexcel HTA913 carbon epoxy

20

C

10 0

5

EX = 115 GPa vxy = .29 Ey = 8.5 GPa GIc= 330 J/m2 Gxy = 4.5 GPa GIIc= 800 J/m2

10 15 20 25 30 End Displacement (mm)

c

35

1.86 mm

20mm 20mm 40 mm

Proposed Model

40

0

149

40

0

A

100

B-C D

3.18 mm

100 mm

200 300 400 0

20

40

60

80 100 120 140 160 180

Crack tip position (mm)

d P

hU = .25h hL = .4h

Eh cr

aL = 2.5h hL = 10h aU = 5h

P

A cr

C B

Eh

A

C

B hU = .25h hL= .25h aL = 6h L= 10h aU = 5.5h

w(x = L) crh / E

w(x = L) crh

/E

Fig. 10 (a) Diagram of the critical reaction load for crack propagation versus load-point displacement in a double cantilever beam with two unequally spaced implanted delaminations tested by Robinson et al. [7]. The material is a carbon–epoxy laminate. The experimental results are compared with the theoretical results in Ref. [9] (crack growth criterion: GI =GIcr C GII =GIIcr D 1, with GI and GII mode I and mode II energy release rate components and GIcr and GIIcr intrinsic fracture energies). (b–c) Dimensionless diagrams of the critical load versus load-point deflection in the two crack system of Fig. 1b for different lengths and positions of the delaminations. (crack growth criterion based on the global energy release rate with Gcr the intrinsic fracture energy) (Adapted from Ref. [8])

4.2 Stability of the Equality of Length of Systems of Equal Length Delaminations An interesting problem when dealing with multiple delamination fracture is the stability of the equality of length of systems of equal length delaminations. This problem has important consequences on the post-critical response of structural systems and key properties, such as damage and impact tolerance and energy absorption (see Section 5 on this regard). The equality of length of a system of equal length delaminations is said to be stable if the system recovers the condition of equal length after the lengths of one or more cracks of the systems have been perturbed. Suemasu and Majima [5] investigated an axisymmetric system of equally spaced, penny-shaped delaminations in

150

R. Massab`o

a homogeneous, clamped, circular plate subject to an out-of-plane point force and observed the stability of the equality of length. Andrews [8] examined systems of equally and unequally spaced delaminations in homogeneous beams with different boundary conditions and subject to quasi-static out-of-plane loads. The problem was solved in closed form by means of a model based on elementary beam theory. The results are summarized in the following.

4.2.1

Equally Spaced and Equal Length Cracks in Homogeneous Beams (Static Loading)

Equally spaced and equal length cracks in beams subject to out-of-plane loads always propagate together since the energy release rate for the simultaneous propagation of all cracks of the system is higher than the energy release rate corresponding to the propagation of one or just a few cracks. In addition, if the length of one or a few cracks of the systems is perturbed with a positive (or negative) perturbation, the energy release rate for the propagation of the perturbed cracks will always be lower (or higher) of the energy release rate for the simultaneous propagation of the remaining cracks and the equality of length of the system is stable [8].

4.2.2

Equal Length and Unequally Spaced Cracks in Homogeneous Beams (Static Loading)

A system of equal length and unequally spaced cracks does not always grow selfsimilarly, even in the absence of length perturbations. In homogeneous systems, this behavior is controlled by the crack spacing alone. The behavioral map of Fig. 11 refers to a system of two delaminations in a clamped–clamped beam subject to

1

P/2 hU h

0.8

hU 0.6 0.4

> (I)

L

B

>

U, L

plane of symmetry

(II)

0.2

Fig. 11 Map of regions of different energy release rate for the clamped–clamped beam with two equal length cracks shown in the inset

a

U

h

hL

L

>

U

(III) 0 0

0.2

0.4

0.6 hL > h

0.8

1

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

151

a concentrated force at the midspan. The map depends on the through-thickness position of the delaminations and shows regions in which the energy release rate for the propagation of one of the cracks of the system and for their simultaneous propagation have different ordering. When the position of the delaminations falls into the central grey region, the energy release rate for the simultaneous propagation of the cracks is higher than those for the propagation of only one of the two cracks. In addition, in this region, which includes the equally spaced case, the equality of length is stable. On the other hand, when the position of the delaminations falls outside the grey region, the energy release rate for the single propagation of only one of the cracks of the system is always higher (even when the crack lengths are perturbed) than the energy for the simultaneous propagation. If the material is not homogeneous, the condition for the simultaneous propagation of equal length delaminations will differ from that of Fig. 11 and depend on the layup of the material and the flexural rigidity of the sublaminates comprised by the delaminations. For each fixed layup, maps similar to that of Fig. 11 can be constructed.

4.2.3

Dynamic Loading Conditions

The problem of the stability of the equality of length of systems of equal length delaminations subject to dynamically applied loads has been studied by Andrews et al. [10] by means of a theoretical model that represents the multiply delaminated plate as a set of Timoshenko beams joined by cohesive interfaces (see Section 2). The model solution is numerical and the problem of the stability of the equality of length in homogeneous systems has been studied referring to beams subject to transverse point loads, with a wide range of through-thickness positions and lengths of the delaminations, boundary conditions and loading shapes, magnitude and duration. The results essentially confirm a behavior similar to the quasi static case but enriched in important ways by inertial effects.

Equally Spaced and Equal Length Cracks in Homogeneous Beams In systems of equally spaced and equal length delaminations, dynamic conditions do not change the stability conditions from the quasi-static case and the equality of length of the delaminations is stable with respect to length perturbations. Figure 12 depicts the time history of delamination growth in a clamped–clamped beam with three equally spaced delaminations subject to a step force. Phases of delamination growth at nearly constant velocity are separated by intervals of arrest, which in some cases can be permanent [10]. During all phases of propagation and arrest, the delaminations maintain approximately equal lengths. Instances of one or two delaminations growing slightly ahead of the others lead to the eventual growth of the shorter delaminations. If the length of one or more of the delaminations is perturbed prior to loading, the shorter delaminations grow initially to restore the equality of length.

152

R. Massab`o 9 8

a 7 h

L = 10h ao = 5h

P Load

IIc = 2 Ic = 100

P2

time

Eh P(t)/2 h

6

a L

5 150

200

250

300

tcL / h

Fig. 12 Crack propagation history in the system of three equally spaced delaminations in a clamped–clamped beam subject to a step loading (homogeneous, isotropic, perfectly brittle material; cL , longitudinal wave speed; fracture criterion: GI =GIc C GII =GIIc D 1, with GI and GII mode I and mode II energy release rate components and GIc and GIIc intrinsic fracture energies) (Adapted from Ref. [10])

Delamination Configurations Falling into the Stable Quasi-static Domain The exemplary diagrams in Fig. 13 show that when the through-thickness positions of the delaminations fall into the stable quasi-static domain (Fig. 11 for a two-delamination system) which includes the case of equally spaced cracks, two different fracture regimes are predicted depending on whether the initial propagation occurs during the forced or the free vibration phases. The transition is controlled by the magnitude and duration of the excitation. The results in Fig. 13 are for a triangular pulse force with different durations acting on a stable configuration of two delaminations. Figure 13a refers to a load duration equal to the first period of vibration of the beam. The response of the system is stable and the cracks maintain approximately equal lengths for the entire simulation similarly to a system of equally spaced delaminations. However, important differences arise: during the loading phase, which includes the onset of propagation, the energy release rates of the two delaminations are quite different (they were approximately equal in equally spaced systems). The lower delamination, which has a higher energy release rate, starts to propagate first; a small increment of its length is then enough to create a large amplification of the energy release rate of the upper crack that is sufficient for it to begin to propagate and simultaneous propagation ensues. Higher values of the dimensionless applied load, or, which is the same, lower values of the dimensionless fracture energy, lead to similar responses: crack growth occurs during loading or shortly after unloading and the cracks propagate simultaneously. Differences in the responses are found only when the critical fracture energy is higher than the maximum energy release rate that occurs during the loading phase

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

a

b P(t)/2 hU

Load

hL a

tp/2

L L = 10h

plane of symmetry

a = 5h

1

P

> (I) B > U

hU 0.6 h

tp t

hU hL

h

0.8

P

153

a

L U, L

0.4

Time

hU = h / 6 hL = 0.3h

0

L

(II)

0.2 0

0.2

>

U

(III) 0.4

0.6

0.8

1

hL / h

c

IIcr

=

Icr

=

cr

d

10 9

5.7

a h

Upper crack Lower crack

Upper crack Lower crack

5.5

8

tpcL / h = 438

a h 7 6

cr

= 30

tpcL / h = 109

5.4

P2

5.3

Eh

cr

= 30

P2 Eh

5.2 5.1

5

5

4

0

60

120

180

240

300 360 tcL / h

4.9 0

100 200 300 400 500 600 700 800 900

tcL / h

Fig. 13 (a, b) Geometrical data and position of the cracks in the behavioral map of Fig. 11. (c) Crack propagation history in the clamped–clamped beam with two equal length delaminations subject to a pulse force with duration equal to the first period of vibration of the beam. (d) Crack propagation history in the system when the duration of the pulse force is 1/4 of the first natural period of vibration. Shape and duration of the load are plotted in dotted lines with an arbitrary scale in the ordinate. (homogeneous, isotropic, perfectly brittle material; cL , longitudinal wave speed; fracture criterion based on the global energy release rate with Gcr the intrinsic fracture energy) (Adapted from Ref. [10])

so that crack growth starts in the free vibration phase as in the example presented in Fig. 13b. In this regime crack growth is slow, sporadic and characterized by numerous intervals of localized growth of one of the cracks, arrest, and simultaneous propagation of both cracks, driven by the repeated and sudden changes of the conditions of the system. The solution in Fig. 13 does not account for damping mechanisms that would occur and may be important in real structures; these mechanisms would additionally slow the growth process. The response for pulse forces with triangular or non-triangular shapes acting on stable configurations of the cracks (grey pocket of Fig. 11) is always similar to that described above, apart from positions very close to the boundary. The response for a step loading is stable and similar to that obtained for pulse loads acting on equally spaced delaminations since the free vibration regime is absent.

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R. Massab`o

Delamination Configurations Falling into the Unstable Quasi-static Domain For configurations of a two-delamination system in the unstable domain of Fig. 11, three different dynamic fracture regimes are predicted depending on the dimensionless magnitude of the excitation. The first regime, for dimensionless loads below a first limit value and above the threshold for crack propagation, is characterized by the growth of only one of the cracks; the second regime, for intermediate values of the applied dimensionless load, is characterized by the growth of one of the cracks followed, at a later time and reduced speed, by the other crack; the third regime, for loads above a second limit value, is characterized by the essentially simultaneous propagation of both cracks. The limiting load values depend on the shape and duration of the excitation and the length and position of the cracks. Longer durations generally favor growth of a single crack. Exemplary details of dynamic crack growth in a clamped–clamped plate in the three regimes are presented in Fig. 14. Similar dynamic responses are predicted for all crack configurations falling outside the grey pocket in the quasi-static stability map of Fig. 11. Similarly, changing the length of the pulse or sustaining the load over a long time (e.g., step load) does not modify the stability conditions of the system. a P(t)/2 hL a L

L = 10h, a=5h; hU = 0.1h, hL=0.3h Icr

=

IIcr

=

0.8

Load

h

1

P

hU t

tp/2

plane of symmetry

tp

U

>

L

hU 0.6 h 0.4

(I) > U, B

0.2

(II)

0

0.2

>

U

(III)

c

d

e

10 8 a 6 h 4 2 0

10 8 6 4 2 0

10 8 6 4 2 0

10 8 6 4 2 0

0

400

200

0.4

0.6

0.8

1

hL/h

b

200

L

0

cr

tPcL / h = 425.7 = first period of vibration

0

L

400

0

200

P

cr Eh

400

0

200

400

P

cr Eh

= 1.0 or

tcL/h P crEh

cr Eh 2

= 0.183 or

P =30

P crEh

cr Eh 2

= 0.223 or

P =20

crEh

2

= 0.447 or

P =5.0

crEh

2

P =1.0

Fig. 14 (a) Geometrical data and position of the crack configuration in the stability map of Fig. 11 (b–e) Crack propagation histories of a two delamination system in a clamped–clamped beam subject to a triangular pulse force for different magnitudes of the dimensionless applied load (lower crack, thick lines; upper crack, thin lines. Shape and duration of the load are plotted in dotted lines with an arbitrary scale in the ordinate; homogeneous, isotropic, perfectly brittle material; cL D longitudinal wave speed; fracture criterion based on the global energy release rate with Gcr the intrinsic fracture energy) (Adapted from Ref. [10])

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

155

4.3 Crack Growth Characteristics in Systems of Unequal Length Cracks The response of structures with unequal length delaminations depends on the geometry of the system, layup of the material, length and position of the delaminations and loading conditions. However, the special configurations of homogeneous beams with equally spaced cracks share a common feature that has been highlighted first in the analytical studies in Ref. [8]. Equally spaced, unequal length delaminations loaded quasi-statically tend to grow so that the equal length configuration is first approached and then maintained throughout propagation. This behavior is observed in two-crack systems where the relative lengths of the two cracks, U D .aU  aL / =aU and L D .aL  aU / =aL , are smaller than 50%. Similar responses have been observed in unequally spaced systems of delaminations where the crack position falls into the shaded area of Fig. 11; for these systems behavioral maps have been constructed that define maximum relative lengths corresponding to equal length propagation [8]. The quasi static results obtained for equally spaced cracks have been confirmed by numerical analyses under dynamic loading conditions in Ref. [10]. Figure 15 refers to a clamped–clamped beam with two equally spaced delaminations with unequal lengths, aU D 0:5aL , subject to a triangular pulse load with duration equal to the first period of vibration.

a

b P(t)/2

t

9

hL

8 7

a L

plane of symmetry

L = 10h, aL = 5h, aU = 0.65aL; hU = h/3, hL = h/3 P

cr=

20

P2 Eh

Icr

=

IIcr

Upper Crack Lower Crack

=

cr

6

a 5 h 4 3

tpcL / h = 426

Load

10

hU

2 1 0

tp/2 Time

tp

0

50

100

150

200

250

300

350

tcL / h

Fig. 15 (a) Two equally spaced and unequal length brittle delaminations in a clamped–clamped beam subject to a triangular pulse force of duration tp D t1 (with t1 the first natural vibration period t1 cL = h D 426). (b) Crack propagation history. Homogeneous, isotropic, brittle material; E D Young’s modulus, cL D longitudinal wave speed; fracture criterion based on the global energy release rate with Gcr the intrinsic fracture energy. (shape and duration of the load are plotted with an arbitrary scale in the ordinate) (Adapted from Ref. [10])

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R. Massab`o

5 Improving Mechanical Performance Through Controlled Delamination Fracture Controlled delamination fracture can be a useful tool to improve mechanical performance and key properties, such as damage and impact tolerance and energy absorption, of composite laminates and multilayered systems. One way of controlling delamination fracture is by creating large scale bridging conditions, for instance through the insertion of a through-thickness reinforcement. A throughthickness reinforcement is applied by stitching or weaving continuous fiber tows or by inserting discontinuous short rods (z-pins) [26–28]. The through-thickness reinforcement create large regions of bridging along the faces of the delaminations (large scale bridging) and develop tractions that oppose the relative crack displacements. These tractions typically shield the crack tip from the applied load thereby reducing the driving force for crack propagation or even suppressing crack growth. The efficacy of this technology in controlling delamination fracture in systems subject to quasi static loading conditions is widely recognized in the literature (see Refs. [29–32] for papers by the writer). The work recently performed by the writer and her collaborators as a part of the ONR program deals with the problem of dynamic delamination fracture in the presence of large scale bridging and results will be presented in Section 5.2. Another way of controlling delamination fracture in multilayered systems is by designing the material and the layup so that the cracks will be forced to form along predefined planes that ensure the propagation of all cracks of the system. Multiple delamination growth has a number of advantages on the propagation of a single dominant crack. This behavior, which was first noted in Refs. [10, 33], is described in Section 5.1.

5.1 Energy Absorption Through Multiple Delamination Fracture The conclusions drawn in Sections 4.2 and 4.3 on the stability of the equality of length of systems of equal length delaminations and on the crack growth characteristics of systems of equally spaced but unequal length delaminations, suggest the possibility that homogeneous systems designed so that cracks will form at equal through-thickness spacing may have better characteristics in terms of energy absorption and damage tolerance than systems where cracks form at different through thickness spacing. In order to investigate this problem, the schematics shown in Fig. 16 have been examined in Ref. [33]. Clamped–clamped homogeneous and isotropic beams with two central delaminations are loaded quasi-statically, under displacement control, up to predefined values of the transverse midspan displacement, wst . Crack growth is prevented during the quasi static loading phase. The cracks are then allowed to propagate dynamically through the specimen while the midspan displacement is kept fixed at wst . The effects of different crack spacings on the solution have been examined referring to

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

a

b

c h3

w(t)

h a

w(t)

h h5

w(t) wst

157

quasi-static

a

L

L

equally spaced cracks

unequally spaced cracks h3 = 0.1h, h5 = 0.3h

time

Fig. 16 Delaminated beams with (a) equally and (b) unequally spaced cracks loaded under displacement control. (c) Loading history crack growth is prevented during the quasi-static loading phase

a system (a) of equally spaced cracks hU D hL D h=3 (for which simultaneous growth of the cracks occurs when the beam is subject to a transverse time dependent load, Section 4) and a system (b) of unequally spaced cracks, hU D h=10 and hL D 0:3h (for which localized growth of the lower crack typically occurs when the beam is subject to a transverse time dependent load, Section 4). The maximum transverse displacements applied at the midspan, wst , are chosen so that the strain energy accumulated during the loading phase, L, is the same in the two systems. For the examples shown here the ratio Gcr h=L D 0:086 has been chosen, with Gcr the intrinsic fracture energy. With the assumed loading conditions the energies expended through multiple dynamic delamination fracture in the two systems can be compared and the effects of localized versus diffuse crack propagation highlighted. The diagrams in Fig. 17 show time histories of crack growth and energy expended into the creation of new fracture surfaces (energy adsorbed). In the unequally spaced crack system only the lower crack propagates at a speed (approximately 20% of the longitudinal wave speed) higher than the speed of the simultaneously propagating cracks of the system of equally spaced cracks (10% of the longitudinal wave speed). The lower crack of system (b) then quickly approaches the fixed boundary of the specimen while the upper crack remains subcritical. The energy expended into the creation of new surfaces is similar in the two systems during the time when the localized propagation occurs in the system (b) (up to 50% of the strain energy L initially put into the system), since the two propagating cracks of system (a) advance at a much lower speed. However, at later times the two cracks of (a) continue to advance toward the fixed boundary of the beam so that the relative energy absorbed in the system of equally spaced cracks at the end of the simulation is almost 80% of L. The chosen ratio between the intrinsic fracture energy and the strain energy put into the system during the quasi static loading phase, Gcr h=L D 0:086, is such to induce dynamic growth in the regime of small to moderate crack speeds. For higher values of the ratio Gcr h=L (leading to lower crack speeds) the behavior would be similar to that observed in Fig. 17. On the other hand, for lower values of Gcr h=L (leading to higher crack speeds), the initial propagation of a single delamination in systems of unequally spaced cracks would be typically followed by the propagation of the other crack of the system (Fig. 14). In these cases the energy expended into the creation of new crack surfaces could be similar to that of the equally spaced crack system.

158

R. Massab`o

a

12 lower crack

10 8 a h

6

upper crack

4

L = 10h a0 = 5h

unequally spaced cracks equally spaced cracks

2

Gcr h / L = 0.086

0 0

50

100

150

200 tcL / h

250

b 0,8 L = 10h a0 = 5h

0,7 Energyexp L

0,6

Gcr h / L = 0.086

0,5 0,4 0,3 0,2

unequally spaced cracks equally spaced cracks

0,1 0 0

50

100

150

200 tcL / h

250

Fig. 17 (a) Time history of crack growth in the equally spaced (thick line) and unequally spaced (thin line) systems of Fig. 16. (b) Time history diagram of the energy expended into the creation of new crack surfaces, normalized with respect to the strain energy put into the system. Homogeneous, isotropic, perfectly brittle material, cL D longitudinal wave speed; fracture criterion based on the global energy release rate with Gcr the intrinsic fracture energy (Adapted from Ref. [33])

The results presented above and in Section 4 refer to the specific problem of homogeneous beams with pre-existing multiple delaminations subject to point loads. While they cannot be extended to describe other, perhaps more realistic, dynamic problems, they allow drawing some general conclusions. First, it appears from the results presented that if the material is designed so that cracks will form along predefined planes, the equal length configuration can be approached and simultaneous propagation of the cracks ensue. In homogeneous beams the delamination architecture that ensure simultaneous propagation of multiple cracks is the equally spaced

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

159

one while different architectures will be needed in inhomogeneous beams. Simultaneous propagation of multiple cracks typically occurs at a speed lower than that of a single dominant crack. Consequently, when multiple cracks propagate the size of the damage area will be smaller or, at a parity of size, the energy dissipated in the creation of new surfaces higher.

5.2 Damage and Impact Tolerance Large scale bridging mechanisms acting along the crack surfaces of delaminated beams loaded quasi-statically strongly affect fracture characteristics, failure modes and macrostructural behaviour. These mechanisms may be created by the action of a through-thickness reinforcement in the form of stitching or z-pinnings. Large scale bridging effects are well documented in the literature and a basic understanding of quasi-static large scale bridging delamination fracture has been developed that includes the derivation of the characteristic length scales of the problem [29, 34]. On the other hand, the problem of the influence of such mechanisms on the characteristics of dynamically propagating cracks and the mechanical response of structures loaded dynamically has been studied only recently in the literature [10, 35, 36] and a number of open questions still exist. Brandinelli and Massab`o [36] show that the presence of a through-thickness reinforcement can improve the free vibration response of beams with stationary delaminations by suppressing out-of-phase vibrations of the delaminated arms. A study performed by Sridhar et al. [35] on the influence of large scale bridging mechanisms on steady state delamination fracture in wedge loaded mode I DCB specimens, shows that the delamination resistance is considerably enhanced by a typical through thickness reinforcement for small to moderate crack speeds (up to 0.1–0.2 the longitudinal wave speed). For higher velocities the kinetic energy dominates the overall energetics and the relative effect of the reinforcement on the delamination resistance becomes insignificant. Andrews et al. [10] recently studied the response of beams with single and multiple stationary delaminations subject to dynamic loads and showed that imposing a large scale bridging mechanism along the wake of the delaminations reduces the driving force for crack propagation during the loading phase and strongly modifies the free motions that arise after the load has been removed. In the absence of bridging mechanisms, large amplification of the mode I component of the energy release rate and hammering of the crack faces can occur due to out of phase vibrations of the delaminated sub-laminates (behavior already discussed in Fig. 8). Bridging mechanisms similar to those produced by a typical through thickness reinforcement reduce amplification of the mode I component and prevent hammering so that the energy release rates tend to show smooth oscillations associated with waves propagating on the scale of the whole specimen [10]. In systems with dynamically propagating delaminations the behaviour is strongly controlled by the crack velocity as noted by Sridhar et al. [35] for steady state growth

160

R. Massab`o

a 12

L = 10h a0 = 5h

10

Gcr h / L = 0.086 a h

8

lower crack

6 upper crack

4 unequally spaced bridged cracks equally spaced bridged cracks no bridging

2 0

0

20

40

60

80

100

120

140 160 tcL / h

b 0,8

L = 10h 0,7 a0 = 5h

Energyexp 0,6 Gcr h / L = 0.086 L

0,5 0,4 0,3 0,2

unequally spaced bridged cracks equally spaced bridged cracks no bridging

0,1 0

0

20

40

60

80

100

120

140 160 tcL / h

Fig. 18 (a) Time history of crack growth in the equally spaced (thick solid line) and unequally spaced (thin solid line) systems of Fig. 16. (b) Time history diagram of the energy expended into the creation of new crack surfaces, normalized to the strain energy initially put into the system. The diagrams compare the response of systems with and without a through-thickness reinforcement (material data are given in the main text). Homogeneous, isotropic material; cL D longitudinal wave speed; fracture criterion based on the global energy release rate

in single delamination systems. When multiple delaminations are present the effects produced by the large scale bridging conditions combine with the interaction effects leading to some interesting features. Figure 18a and b show time history of crack growth and energy absorbed into the creation of new surfaces in the multiply delaminated beams of Fig. 16 when large scale bridging mechanisms such as those produced by a typical through thickness reinforcement are present. The bridging tractions are T N D k N wN and T S D k S wS , with wN and wS the mode I and mode II relative crack displacements and k N ; k S D 0:01 E= h. (The values assumed

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

161

for k S and k N corresponds to a typical stitched, carbon–epoxy quasi-isotropic laminate as determined by Cox et al. [37], Turrettini [38] and Massab`o et al. [30].) The curves corresponding to the systems in the absence of a through-thickness reinforcement are shown in the figure with dotted lines. The figures confirm that in the reinforced system the equally spaced cracks propagate together as occurs in the unreinforced system (at a speed of approximately 0:1cL , with cL the longitudinal wave speed). The large scale bridging mechanisms reduce crack speed in the unequally spaced system and, after a phase of growth at reduced speed, lead to crack arrest in both the reinforced and the unreinforced systems. This indicates that the damage and impact tolerance of the material can be substantially improved by a through-thickness reinforcement when crack growth occurs in the low to moderate crack speed regime. In addition, the diagram (b) shows that the presence of large scale bridging conditions minimizes differences in the responses of the systems with equally and unequally spaced cracks.

6 Indentation Response of Composite Sandwich Beams in the Presence of Skin Damage The schematic shown in Fig. 19a has been examined in Campi and Massab`o [39,40] in order to investigate the influence of skin imperfections (delaminations, flaws, voids) or thickness variations on the indentation response of composite sandwich beams or plates deforming in cylindrical bending. The composite beam is assumed

a

P

b t

z

x

c

h H

t d

a

a

d

y

y

P

b

q

n delaminations

F t

z

wcr w

qcr s a

K

F

s d

d

a

σcr b

y

Fig. 19 (a) Indentation of a composite sandwich beam with areas of damage or thickness variation. (b) Damaged beam-column, with Young’s modulus Ef , on a Winkler nonlinear foundation and constitutive law of the foundation

162

R. Massab`o

to have homogeneous and isotropic/orthotropic linear-elastic faces and a homogeneous, compressive yielding core. Following the analytical approach often used in the literature [41, 42], the sandwich is modeled as an Euler-Bernoulli beam/beamcolumn with Young’s modulus Ef (the upper face) resting on a Winkler nonlinear foundation (the core) that is elastic in tension and elastic-ideally plastic in compression, with elastic modulus K and yield strength qcr D cr b (Fig. 19b). Compressive forces F are introduced, following the approach in Ref. [42], to account for different boundary conditions; when global bending effects are negligible, for instance if the sandwich beam of Fig. 19a is continuously supported by a rigid plane, then F D 0; when global bending is not negligible, for instance if the sandwich beam of Fig. 19a has finite length L and is simply supported at the ends, then F ¤ 0 defines the compressive force generated in the upper face by the bending moment produced by the concentrated force P, namely F D P ˇL= h, with ˇ a coefficient accounting for the end restraints and the distribution of the stresses in the beam cross section (e.g. ˇ D 1=4 for a simply supported beam with a core of negligible bending stiffness). When F ¤ 0 the problem is treated as the stability of a damaged beam on a nonlinear Winkler foundation. Two damaged areas, which are symmetrically located with respect to the applied load, are considered and the response is studied on varying the size, a, position, d , and entity of the damage (Fig. 19). The entity of the damage is measured by a coefficient n that reduces the moment of inertia of the beam to Ifdel D If = .n C 1/2 , with If the moment of inertia of the intact face sheet; if the damage were multiple frictionless delaminations free to slide on each other and root rotations were negligible, n would coincide with the number of delaminations [8].

6.1 Continuously Supported Sandwich Beam .F D 0/ When global bending effects are negligible, the indentation response of the damaged beam is characterized by an initial linear elastic phase up to the indentation load PcrDel for which the load-point displacement equals the critical displacement for core yielding w.z D 0/ D cr b=K (Fig. 19). The linear elastic phase is followed by a strain hardening phase and the beam then fails by mechanisms other than indentation, e.g. compressive failure of the face (Fig. 20a shows with thin lines the load versus load-point displacement curve of an intact beam resting on a rigid-ideally plastic foundation, for which PcrDel D 0). Upper and lower bound solutions for the indentation load and the load versus load-point displacement curves are defined by the solutions of the intact and the fully delaminated beams. The limiting indentation loads are: Pcr D 2cr b= .intact beam/ p PcrLim D Pcr = n C 1 .fully damaged/

(8a) (8b)

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

a 2 PDel ⎛ σ cr β L⎞13 σcr bt ⎜⎝ Ef h ⎟⎠ 1,5

163

undamaged face sheet, F = 0 undamaged face sheet, F  0

Pmax 1 0,5

PmaxLim damaged face sheet, n = 1,F  0

0 0,00

0,10

0,20

0,30

0,40

0,50 1

b

PcrDel

c1 PmaxDel

Pcr

Pmax

1

PcrLim=Pcr

0.6 d/ d/ d/ d/

0.4 0.2 0 0

L = 0.75 L = 0.5 L = 0.1 L=0

0.1 0.2 0.3 0.4 0.5 0.6

n+1

w0 ⎛σcr ⎞ 3 ⎛ β L⎞ 3 ⎜⎝ h ⎟⎠ t ⎜⎝ Ef ⎟⎠ d = (π2/12)1/ 3

0.6 0.4

n=1 d ≠0 a Λ

0,60 4

d = 0.2 d = 0.1 d =0

0.2 0 0

0.1 0.2 0.3 0.4

n=1 d ≠0 0.5 a ⎛ σcr β L⎞ 1/3 t ⎜⎝ Ef h ⎟⎠

Fig. 20 (a) Dimensionless diagram of the load versus load-point displacement response of sandwich beams with a rigid-perfectly plastic core and different boundary conditions. (b) Dimensionless diagram of the indentation load as a function of the semi-length of the damage on varying its distance from the applied load (continuously supported beam, F D 0). (c) Dimensionless diagram of the indentation collapse load as a function of the semi-length of the damage on varying its distance from the applied load (beam with end constraints). PcrDel is the load at first yielding of 1=3  the core; Pmax Del is the collapse load for indentation and dN D d=t cr ˇL= hEf

where ƒ is the half wavelength p of the oscillations of the transverse displacement in an intact beam, ƒ D  4 4Ef If =K. The transition between the two limiting solutions is controlled by the size, position and entity of the damage and depicted in Figs. 20b and c.

6.2 Sandwich Beam with End Restraints .F ¤ 0/ When the upper face of the sandwich beam is in compression, the initial elastic response is nonlinear, due to second order effects, and strain hardening and is followed by a strain softening phase (thick curves in Fig. 20a). The maximum indentation load, PmaxDel , can be defined analytically for a rigid-plastic foundation [42]. Upper and lower bound solutions for the indentation load and the load versus deflection

164

R. Massab`o

curves correspond to the fully intact and the fully delaminated beams. The maximum loads are:

2 1=3 h  2 Ef cr .intact beam/ (9a) Pmax D bt ˇL 12 Pmax Lim D Pmax = .n C 1/2=3 .fully damaged/

(9b)

The transition between limiting solutions is again controlled by the size, position and entity of the damage and is depicted in Fig. 20c. When the core is elastic-perfectly plastic, the maximum load for the indentation of a damaged sandwich beam varies between that of a beam with a rigid-plastic core and a beam with of a fully elastic core on varying the dimensionless group ˇL cr .n C 1/1=2 [40]. The maximum load for a beam with a fully ˘N D h Ef .t /3 elastic core corresponds to the load for face wrinkling.

6.3 Characteristic Lengths Characteristic lengths have been derived that give general indications on the expected indentation response of damaged sandwich beams [40]. A first characteristic length defines the minimum size of the damage that has the strongest effect on the indentation response, which then coincides with that of a fully damaged beam. For a sandwich beam continuously supported by a rigid plane, this length is a function of the half wavelength of the oscillations of the transverse displacement of a fully damaged face sheet, ƒDel , which depends on the relative flexural stiffness of the face and the core and the entity of the damage through n: amin1 D 0:75ƒDel D 0:75

q 4

p 4Ef If =K= n C 1

(10)

For a sandwich beam with localized end restraints and a rigid-plastic core the length depend on material and geometrical properties and the entity of the damage and is given by:  1=3  1=3 (11) Ef =cr h=ˇL amin2 D t  2 =12 .n C 1/2 A second characteristic length defines the minimum distance of damage areas from the applied load above which the damage has no effect on the response, which then coincides with that of an undamaged beam. For a sandwich beam continuously supported by a rigid plane, this length depends on the half wavelength of the oscillations of an intact face sheet, ƒ, and is given by: q (12) dmin1 D 0:75ƒ D 0:75 4 4Ef If =K

Dynamic Interaction of Multiple Damage Mechanisms in Composite Structures

165

For a sandwich beam with end restraints and a rigid-plastic core is given by:  1=3  1=3 Ef =cr h=ˇL dmin2 D t  2 =12

(13)

This length is unaffected by the entity (n) and size (a) of the damage and depends on geometrical and material properties only. In a continuously supported sandwich beam with a Divinycell H100 foam core and glass fiber–epoxy face sheets, with Young’s modulus of the core Ec D 120 MPa, cr D 1:45 MPa, Ef D 30 GPa, c D 20 mm and t D 4 mm: amin1 D 60 mm (corresponding to a 30% reduction of the indentation load when n D 1) and dmi n1 D 40 mm. In a simply supported supported sandwich beam with a Ec D 500 MPa, cr D 1:45 MPa, Ef D 30 GPa, c D 30 mm, t D 1:48 mm and L D 400 mm: amin2 D 30 mm (corresponding to a 40% reduction of the collapse load for indentation when n D 1) and dmin2 D 25 mm (calculated with the assuming K D Ec b=c). The characteristic lengths have been derived for a sandwich beam and they can be easily adapted to describe the response of plates under plane strain conditions parallel to the y–z plane and the response of circular plates in axial-symmetric conditions. The indentation response of damaged plates with different geometries (boundary conditions and damage areas) will differ, even substantially, from that of sandwich beams, however the characteristic lengths derived above define conservative lower bound solutions. For instance, in a plate under cylindrical bending conditions but where the damage is not through-width, the characteristic lengths d give a conservative estimate of the minimum distance of the damage that has no influence on the solution.

7 Conclusions The work performed by the writer and her collaborators within ONR funded projects over the last 4 year has been summarized in this chapter. The research deals with the problem of the interaction of multiple damage mechanisms in laminated and sandwich structures subject to static and dynamic loading conditions. The effects of static and dynamic interactions on the mechanical behavior and key properties, such as damage and impact tolerance and energy absorption, have been investigated. Main focus during the first years research has been on the interaction of multiple delaminations in laminated beams and plates. Most recent work deals with the interaction of different damage mechanisms in sandwich systems. The work establishes a link between material and structural performance and give basic insight for improvements in the survivability of ship structures via material and structural design. Some of the relevant results and conclusions are listed below. – Approximate one dimensional models based on the theories of bending of beams and plates can accurately describe the delamination fracture response of composite systems subject to static and dynamic loadings provided shear effects and near tip deformations are properly accounted for. Stress intensity factors calculated

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with the simplified semi-analytical method proposed by the writer for homogeneous and orthotropic beams accurately describe the crack tip fields. – Delamination damage often occurs in the form of multiple delaminations, which are the typical outcome of severe dynamic loading, such as blast, impact and shock. The presence and interaction of multiple delaminations have important effects on structural response and performance. The presence of other cracks can lead to accelerated growth of a crack, due to phenomena of amplification. The possibility of acceleration implies that design and life predictions based on solutions for a single crack cannot be safe. – Concepts on multiple delamination fracture can be used to maximize the energy absorption capabilities of laminates and multilayered systems. Results presented in this chapter show that the energy absorbed during failure depends significantly on whether conditions favor multiple delaminations propagating with equal lengths or a single delamination growing dominantly. These ideas have not been explored before and the formulation of design principles for optimal energy absorbing structures and for through-thickness reinforced structures with optimal performance is the topic of current and future research. – Progress toward standardized design methods for through-thickness reinforced systems, including the design of standard experiments, the formulation of design principles and advances in modelling, will be possible only if correct fracture mechanics is used. In addition, since delamination is often the result of dynamically applied loads, inertia effects should be accounted for and considerations of the regimes of crack speed where a through-thickness reinforcement may or not be effective should be included. The formulation of experimental/computational methodologies to derive the bridging traction law of a through-thickness reinforcement in the dynamic regime and to identify its rate dependency is one of the topics of current research. One important contribution of the work performed for the ONR is to provide enough understanding of the expected delamination behavior in such tests that specimen sizes, notch sizes, and loading histories can be chosen to assure that a test yields the desired data. Key issues include choosing tests that ensure the presence of multiple delaminations so that their effect on cohesive laws in conventional and through-thickness reinforced composite laminates can be measured. Acknowledgments The author would like to acknowledge support by the U.S. Office of Naval Research through contract no. N00014-05-1-0098, administered by Dr. Yapa D.S. Rajapakse. The work presented in this chapter has been done in collaboration with the following researchers: B.N. Cox, M.G. Andrews, A. Cavicchi and F. Campi.

References 1. Kanninen, M.F. (1974) Dynamic analysis of unstable crack propagation and arrest in the DCB test specimen, Int J Fracture 10(3): 415–430 2. Kanninen, M.F. (1973) Augmented double cantilever beam model for studying crack propagation and arrest, Int J Fracture 9(1): 83–92

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3. Larsson, P.L. (1991) On multiple delamination buckling and growth in composite plates, Int J Solids Struct 27(13): 1623–1637 4. Suemasu, H. (1993) Postbuckling behaviors of composite panels with multiple delaminations, J Compos Mater 27(11): 1077–1096 5. Suemasu, H. and Majima, O. (1996) Multiple delaminations and their severity in circular axisymmetric plates subjected to transverse loading, J Compos Mater 30(4): 441–463 6. Zheng, S. and Sun, C.T. (1998) Delamination interaction in laminated structures, Eng Fract Mech 59(2): 225–240 7. Robinson, P., Besant, T. and Hitchings, D. (1999) Delamination growth prediction using a finite element approach. 2nd ESIS TC4 Conference on Polymers and Composites, Les Diablerets, Switzerland 8. Andrews, M.G., Massab`o, R. and Cox, B.N. (2006) Elastic interaction of multiple delaminations in plates subject to cylindrical bending, Int J Solids Struct 43: 855–886 9. Andrews, M.G. and Massab`o, R. (2008) Delamination in flat sheet geometries in the presence of material imperfections and thickness variations, Compos Part B-Eng 9: 139–150, special issue on Marine Composites 10. Andrews, M.G., Massab`o, R., Cavicchi, A. and Cox, B.N. (2009) Dynamic interaction effects of multiple delaminations in plates subject to cylindrical bending, Int J Solids Struct 46: 1815– 1833 11. Tay, T.E. (2003) Characterization and analysis of delamination fracture in composites: An overview of developments from 1990 to 2001, Appl Mech Rev 56(1): 1–31 12. Yang, Q. and Cox, B.N. (2005) Cohesive zone models for damage evolution in laminated composites, Int J Fracture 133(2): 107–137 13. Williams, T.O. and Addessio, F.L. (1997) A general theory for laminated plates with delaminations, Int J Solids Struct 34: 2003–2024 14. Williams, T.O. and Addessio, F.L. (1998) A dynamic model for laminated plates with delaminations, Int J Solids Struct 35: 83–106 15. Zhou, Z., Reid, S.R., Soden, P.D. and Li, S. (2001) Mode separation of energy release rate for delamination in composite laminates using sublaminates, Int J Solids Struct 38: 2597–2613 16. Alfano, G. and Crisfield, M.A. (2001) Finite element interface models for the delamination analysis of laminated composites: Mechanical and computational issues, Int J Num Meth Eng 50(7): 1701–1736 17. Case, S.W. and Reifsnider, K.L. (1999) MRLife12 Theory Manual – A strength and life prediction code for laminated composite materials, Materials Response Group, Virginia Polytechnic Institute and State University, Blacksburg VA 18. Andrews, M.G. (2005) The static and dynamic interaction of multiple delaminations in plates subject to cylindrical bending, Dissertation, Ph.D. Degree, Northwestern University, Evanston, IL 19. Andrews, M.G. and Massab`o, R. (2007) The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers, Eng Fract Mech 74: 2700–2720 20. Camacho, G.T. and Ortiz, M. (1996), Computational modelling of impact damage in brittle materials, Int J Solids Struct 33(20–22): 2899–2938 21. Massab`o, R., Brandinelli, L. and Cox, B.N. (2003) Mode I weight functions for an orthotropic double cantilever beam, Int J Eng Sci 41: 1497–1518 22. Brandinelli, L. and Massab`o, R. (2006) Mode II weight functions for isotropic and orthotropic double cantilever beams, Int J Fracture 139: 1–25 23. Li, S., Wang, J. and Thouless, M.D. (2004) The effects of shear on delamination in layered materials, J Mech Phys Solids 52(1): 193–214 24. Suo, Z.G. (1990) Delamination specimens for orthotropic materials, J. Appl Mech-T ASME 57(3): 627–634 25. Brencich, A. and Carpinteri, A. (1996) Interaction of a main crack with ordered distributions of microcracks: a numerical technique by displacement discontinuity boundary elements, Int J Fracture 76(4): 373–389

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26. Evans, D.A. and Boyce, J.S. (1989) Transverse reinforcement methods for improved delamination resistance, Int SAMPE Symp Expos 34: 271–282 27. Freitas, G., Magee, C., Dardzinski, P. and Fusco, T. (1994) Fiber insertion process for improved damage tolerance in aircraft laminates, J Adv Mater 25(4): 36–43 28. Dransfield, K., Baillie, C. and Mai, Y.-W. (1994) Improving the delamination resistance of CFRP by stitching – a review, Compos Sci Technol 50: 305–317 29. Massab`o, R. and Cox, B.N. (1999) Concepts for bridged mode II delamination cracks, J Mech Phys Solids 47(6): 1265–1300 30. Massab`o, R., Mumm, D. and Cox, B.N. (1998) Characterizing mode II delamination cracks in stitched composites, Int J Fracture 92(1): 1–38 31. Rugg, K.L., Cox, B.N. and Massab`o, R. (2002) Mixed mode delamination of polymer composite laminates reinforced through the thickness by z-fibers, Compos Part A-Appl, 33(2): 177–190 32. Massab`o, R. (2008) Single and multiple delamination in the presence of nonlinear crack phase mechanisms. In: Sridharan (ed.), Delamination Behavior of Composites, Woodhead, Cambridge, pp. 515–558 33. Massab`o, R. (2007) Dynamic interaction of multiple damage mechanisms in multilayered systems, Proceedings of the Italian Conference of Theoretical and Applied Mechanics, Brescia, CDrom 34. Suo, Z., Bao, G. and Fan, B. (1992) Delamination R-curve phenomena due to damage, J Mech Phys Solids 40: 1–16 35. Sridhar, N., Massab`o, R., Cox, B.N. and Beyerlein, I. (2002) Delamination dynamics in through-thickness reinforced laminates with application to DCB specimen, Int J Fracture 118: 119–144 36. Brandinelli, L. and Massab`o, R. (2003) Free vibrations of delaminated beam-type structures with crack bridging, Compos Struct 61: 129–142 37. Cox, B.N., Massab`o, R., Mumm, D.R., Turrettini, A. and Kedward, K. (1997) Delamination fracture in the presence of through-thickness reinforcement, Proceedings of the 11th International Conference on Composite Materials, Gold Coast, Australia, ed. M. L. Scott, Woodhead, Melbourne, pp. 159–177 38. Turrettini, A. (1996) An investigation of the mode I and mode II stitch bridging laws in stitched polymer composites, Masters thesis, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 39. Campi, F. and Massab`o, R. (2008) Influence of skin damage on the indentation behavior of sandwich beams, Proceedings of the 8th International Conference on Sandwich Structures, ICSS 8, Porto, Portugal, pp. 75–85, FEUP 40. Campi, F. and Massab`o, R. (2009) Indentation and face-wrinkling of a damaged sandwich beam with a compressive yielding core and cohesive interfaces, Proceedings of the ICF12, Ottawa, Canada 41. Zingone, G. (1968) Limit analysis of a beam in bending immersed in an elasto-plastic medium, Meccanica 3: 48–56 42. Steeves, C.A. and Fleck, N.A. (2004) Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part. I: analytical models and minimum weight design, Int J Mech Sci 46: 561–583

A Review of Research on Impulsive Loading of Marine Composites Maurizio Porfiri and Nikhil Gupta

Abstract Impulsive loading conditions, such as those produced by blast waves, are being increasingly recognized as relevant in marine applications. Significant research efforts are directed towards understanding the impulsive loading response of traditional naval materials, such as aluminum and steel, and advanced composites, such as laminates and sandwich structures. Several analytical studies are directed towards establishing predictive models for structural response and failure of marine structures under blast loading. In addition, experimental research efforts are focused on characterizing structural response to blast loading. The aim of this review is to provide a general overview of the state of the art on analytical and experimental studies in this field that can serve as a guideline for future research directions. Reported studies cover the Office of Naval Research-Solid Mechanics Program sponsored research along with other worldwide research efforts of relevance to marine applications. These studies have contributed to developing a fundamental knowledge of the mechanics of advanced materials subjected to impulsive loading, which is of interest to all Department of Defense branches.

1 Introduction Impulsive loading conditions, such as those generated by in-air or underwater explosions, can be detrimental to materials and eventually lead to structural failure. Detonation of conventional explosives produces a region of highly compressed gas that rapidly expands to occupy a large volume and various solid residues from the explosive or its casing. The rapid expansion of gas after detonation almost instantaneously compresses the surrounding medium into a shock wave that rapidly propagates in all directions from the explosion. The shock wave that is generated by M. Porfiri () and N. Gupta Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201 e-mail: fmporfiri;[email protected]

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the rapid release of gas from an explosive is called blast wave. For in-air explosions, the rush of air caused by the net motion of the gas is called blast wind [1]. Underwater blast has long been a major concern for naval engineers in designing marine structures [2, 3]. Apart from structural damage, transmission of blast waves through naval structures can cause serious harms to sailors [4, 5]. The continuous threats presented by mines, torpedoes, and bombs to ships and boats demand a deep insight into the effects of blast loading on materials used in marine structures. The design of more resilient and safer marine structures demands an improved understanding of reflection and transmission of blast waves, energy dissipation mechanisms, and failure initiation and propagation in materials and structures. It is particularly important to understand the effects of blast loading on advanced composites that are increasingly being used in modern marine structures. This literature review aims at determining the present state of knowledge of the effects of blast waves on materials and structures. The review focuses on marine structures and applications specifically relevant to the United States Navy. Studies supported by the Office of Naval Research form the core of this review. Worldwide studies of relevance to the Navy are also reviewed to develop a comprehensive picture of the present state of knowledge in the area of impulsive response of materials used in marine applications. This review can potentially aid in finding critical knowledge gaps and research issues that need to be addressed in future research efforts.

2 Outline A comprehensive study of the response of advanced materials under impulsive loading conditions requires understanding the formation of blast waves; the propagation of blast waves in fluidic (air and water) environments; the interaction of blast waves with materials and structures; the energy dissipation and blast wave attenuation; the mechanical response and damage of materials; and the failure of structures due to material damage. Theoretical and experimental studies aiming at understanding the material or structural behavior under impulsive loading conditions are summarized here, including fluid–structure interactions. Other relevant aspects such as formation and propagation of blast waves have been extensively studied in the aerodynamics and fluid mechanics literature and are omitted from the present review, see for example [6–12]. This chapter is organized as follows: Section 3 reviews results of experimental research on marine materials and structures, including experimental test methods and experimental results. Section 4 presents mathematical models and numerical results of marine panels and structures to blast loading. Section 5 summarizes the reviewed efforts in a tabular form.

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3 Experimental Studies Blast tests are classically divided into two categories: underwater and in-air, which are also referred as UNDEX and INEX, respectively. Both test environments are of significance to the Navy, and different experimental methods and apparatus are required for conducting these tests.

3.1 Underwater Tests 3.1.1

Test Procedures and Instrumentations

Underwater tests are commonly performed either on flat panels in laboratory environments or on full-scale structures. Compared to the full-scale tests, laboratory tests offer several advantages, which include limited cost, considerable accuracy, acquisition of scientific data, and ease of implementation [13]. The advantages related to the use of flat panels include the simplicity in the test specimen geometry and the experimental procedure. Flat panels are reasonably good experimental benchmarks for assessing the mechanical response of marine structures to impulsive loading, because the curvature of the impacting wave front is generally considerably larger than the size of the structural panel. Damage parameters extracted from the specimen level can be extended to the structural level using scaling techniques [14]. The schematic of a typical underwater explosion test setup is illustrated in Fig. 1 [14]. The test assembly is mounted in a shock tank and is held horizontally by a metallic wire rope. The horizontal configuration avoids snapping of the rope that can occur in a vertical configuration after the explosion. To experimentally reproduce

Fig. 1 A schematic of the underwater explosion test setup for the plate from Ref. [14]

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the boundary conditions of marine structural panels, the test plate is mounted on the fixture by welding the internal edges or by locking the specimen with a bolt ring. Both configurations provide good performance but the latter one has the advantage of facilitating easy removal of the plate. The tube behind the plate is filled with air, in the so-called air-backed configuration, or with water, in the so-called waterbacked configuration. The principal dynamic parameters collected to characterize the test results are the fluid pressure and the acceleration and strain at the mid-point of the test panel. The static parameters are the plastic strain, the deflection, and the percentage thinning of the test panel. Usually, after firing analysis, additional techniques are used to measure the plate deflection and thickness. Numerous alternative experimental setups are also used by various research groups, some of which are reviewed here. In the Dyno experimental setup, the specimen is positioned in the center of a cover plate and a thick metal plate with a central hole is placed flush with the specimen as shown in Fig. 2 [15]. In order to create the pit, a water filled cardboard cylinder is added around the assembly. The positions of the strain gage, gage column, and gage tray are visible in Fig. 2 [15]. An alternative technique relies on suspending a shock rig midway beneath a pair of floating pontoons [16].

Fig. 2 Dyno test system from Ref. [15]

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In other cases, the experiments are conducted in a shock tank by simply detonating a charge at a standoff distance from the specimen assembly [17, 18]. In some studies, impact by a foam projectile is used as a means of simulating blast loads [19, 20]. In other studies [21, 22], underwater blast is simulated by impacting a piston, connected to a water filled tube, by a steel projectile.

3.1.2

Response of Marine Structural Materials to Underwater Blast Loading

The vast majority of available studies are focused on evaluating the effects of underwater blast loading on the damage propagation and the failure mechanisms of marine structures. Glass fiber reinforced polymeric (GFRP) composites are relevant in the fabrication of a wide variety of maritime vessels and submarines, both in the form of laminates and foam core sandwich composites. Vinyl ester matrix GFRP laminates containing either chopped strand mat or woven rovings are studied in the series of works [23–25]. It is identified that the specimen backing conditions in the test rig have a great influence on the composite damage resistance. Water-backed specimens present higher shock resistance than air-backed ones, since the water sensibly reduces the bending loads on the laminate after the shock wave reaches the specimen [25]. The air-backed laminates show cracking in the matrix resin starting at a material-dependent threshold pressure. Cracks always appear in the rear surface of the laminates. Depending on the fiber direction, the cracks can change direction or be arrested. At pressures higher than the threshold pressure, the air-backed laminates show permanent deformation or fracture. From the experimental data, one can evince that the formation of cracks within the polymer have marginal effect on residual tensile properties. This suggests that fibers carry the larger part of the tensile load through bending, thereby improving the shock resistance [25]. Fatigue and flexural properties are also observed to deteriorate in a similar manner as a result of underwater blast. In particular, these properties remain unchanged for low values of pressure even for composite containing some pre-existing cracks in the resin matrix. However, over a threshold pressure, the fatigue and flexural strengths are reduced, and structural damage such as delamination and fiber breakage is observed [23, 24]. The effect of foam core materials and adhesives on the failure modes of marine sandwich composites under blast loading is studied in Ref. [13]. Two different types of tests are performed on 2  1 m2 flat panels and on full-scale 3  3 m2 panel in sea water. These comparative studies can be potentially useful in material and size selection of GFRP/foam sandwich composites. Sandwich structures composed of stainless steel face sheets and pyramidal truss cores, as shown in Fig. 3 [15], are experimentally and numerically studied for underwater blast loading in Ref. [15]. Through experimental analyses conducted using a Dyno test system, it is found that the presence of a truss core significantly mitigates blast propagation when compared to solid homogenous plates.

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Fig. 3 Truss core sandwich structure tested under blast loading from Ref. [15]

Several research efforts are focused on underwater blast testing of metallic marine structures, including aluminum 6061-T6 panels [18], mild steel panels [17], and mild steel bolts [26]. The response of Aluminum 6061-T6 panels to explosive charges at different standoff distances is investigated in Ref. [18]. As the distance between the panel and the explosive charge increases, the peak pressure measured on the panel surface decreases. Therefore, the maximum strain, strain rate, and acceleration experienced by the center of the panel increase as the standoff distance decreases. From numerical results obtained using USA/DYNA, it appears that a simple elastoplastic constitutive model for the aluminum plate without accounting for strain hardening effects can relatively well predict the panel response. Experiments conducted by Ramajeyathilagam and Vendhan [17] show that different modes are involved in the failure of mild steel panels under blast loading, including large deformation, partial tensile tearing, complete tensile tearing with increased mid-point deflection, central rupture with partial tearing, and combined tensile and shear failure, at different Shock Factors (SFs), that are affected by the explosive charge weight and the standoff distance. Some of these failure modes at different impact energy levels are shown in Fig. 4 [17]. The numerical results obtained using CSA/GENSA (DYNA 3D) show that strain rate effects play a crucial role in predicting the dynamic response of mild steel panels under blast loading conditions. Indeed, when strain rate effects are discarded in the finite element formulation, permanent deformations are over-predicted by 50–60%, whereas a simple average-based model for strain rate effects yields fairly accurate deformation predictions. Failure and plastic deformation of mild steel bolts for marine applications under blast loading conditions are studied in Ref. [26]. The tensile failure of bolts under different loading conditions including blast loading, quasi static loading, and highstrain rate loading are compared. Blast loading on the bolt is generated through an explosion along the shank of the bolt while keeping the head and threads fixed. It is found that at low strain rates threads fail at a considerably lower stress than the failure strength of the shank, whereas as the strain rate increases the threads fail at a stress almost equal to the failure strength of the shank.

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Fig. 4 Various deformation and failure modes in steel plates under blast loading from Ref. [17]

3.2 In-Air Tests 3.2.1

Test Procedures and Instrumentations

Several recent studies on in-air blast testing of various materials, including marine composites, are available. Commonly used blast test methods use compressed air shock tubes, where a shock wave is generated by rupture of one or two diaphragms.

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Fig. 5 Shock tube instrumentation and data acquisition system at Polytechnic Institute of New York University

Pressure transducer Clamps Explosive charge

Specimen

Fig. 6 Schematic of a controlled explosion tube from Ref. [28]

A shock tube based test instrumentation is shown in Fig. 5. The tube consists of three sections: driver, intermediate, and driven, separated by two diaphragms. The pressurization of the driver section causes the first diaphragm to rupture, which generates a shock wave. This wave travels in the intermediate section and ruptures the second diaphragm, which helps in better controlling the shape and energy of the shock wave. The resulting wave travels along the length of the tube and interacts with the specimen present at the other end of the tube. Pressure transducers can be mounted along the length of the tube and strain gauges can be mounted on the specimens. Some studies use high speed cameras for real time monitoring of the specimen deformation and failure pattern [27]. The shock tube facilitates the study of interactions between shock waves and specimens. However, debris and trailing blast wind associated with blast waves are generally not replicated with these tests. To simulate a complete blast wave, a shock tube variant, called an explosion tube, is used. In the explosion tube, an explosive charge is ignited in a controlled environment as schematically shown in Fig. 6 [28].

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An alternative method to generate blast waves in shock tubes consists of igniting a mixture of liquefied petroleum gas and oxygen [29]. Shock test equipment is used in a number of recent studies. Use of pressure transducers and strain gauges allows real time monitoring of loading conditions and specimen response. Shock tubes used by various research groups normally range from 1 to 6 inches in diameter [27]. The size of the specimens tested in these tubes can vary from the diameter of the tube to any larger size. The specimens can be either clamped on all sides, on two sides, or supported from the back. 3.2.2

Response of Marine Structural Materials to In-Air Shock and Blast Loading

Shock testing of glass fiber/vinyl ester and carbon fiber/vinyl ester laminated composites is carried out to determine the pressure for damage onset, profile, and extent in Ref. [28]. The results for both types of laminates are shown in Fig. 7 [28] for several input shock pressures. Carbon fiber/vinyl ester laminates are found to sustain significant delamination damage above a threshold pressure. Glass fiber/vinyl ester laminates perform better and damage does not result in any significant fracture. Blast tests on similar specimens using a controlled explosion tube suggest that glass fiber/vinyl ester laminates suffer progressive failure whereas carbon fiber/vinyl ester laminates show rapid damage. In glass fiber/vinyl ester specimens, extensive damage to the front face is observed where matrix burned off and extensive fiber fracture occurred. Explosive loading of similar composites is reported in Ref. [30], where it is found that damage extent is lower in adhesively bonded laminates compared to clamped laminates. In Ref. [27], it is observed that adding a layer of polyurea considerably enhances the blast mitigation capabilities of glass fiber/vinyl ester laminated composites. The same study also shows that a sandwich composite containing polyurea between two laminated face sheets shows superior blast mitigation capabilities compared to the plates or polyurea coated laminates. It is observed in the testing of stiffened laminated plates that most part of the blast wave reflects back from the specimen. The load increases sharply and then exponentially decays. Curved glass fiber/epoxy and carbon fiber/epoxy laminates are studied for blast properties in Ref. [31]. The higher modulus of carbon fiber/epoxy laminates results in lower strain compared to glass fiber/epoxy laminates. Finite element analysis overestimates the peak strain levels for these composites. The finite element analysis is conducted using commercial ANSYS software. The discrepancy in the finite element and experimental results is attributed to the difference in the actual and simulated loading conditions. Experimentally, only the central part of the plate is loaded first and then the load spreads over the entire plate, whereas blast pressure is applied to the whole plate at the same time in simulations. Such discrepancies can be reduced by modifying the loading conditions in accordance to the experimental observations. Glass fiber/epoxy composites are also studied in Ref. [32]. Experimental results from quasi-static and impact indentation tests on thick filament wound glass/epoxy

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a

b

Fig. 7 Failure modes in (a) glass fiber and (b) carbon fiber reinforced laminates from Ref. [28]

tubes are presented. The effects of impact damages on the composites’ implosion resistance are studied. It is found that low impact energies generate intralaminar cracks that result in a large drop in implosion pressure resistance. Glass fiber/polyester composites comprising woven rovings are fabricated and tested for ship deck applications in Ref. [33]. In these experiments, two edges of

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Fig. 8 Schematic representation of a three-dimensional (a) woven fabric and (b) laminate from Ref. [34]

the specimen are clamped while the other two edges are kept free. The in-air blast tests reveal that the initial failure is in the form of interlaminar debonding and fiber breakage near the clamped boundaries of the plate. The observation of fiber breakage near the clamped edge is consistent with other studies [34]. In the same study, steel I-beam stiffeners are used to strengthen the composite plates. The performance of such composites is superior and the fracture of the composite face plate is not observed. Some laminates fabricated using 3-D woven fabrics are also tested for shock response in Ref. [34]. The concepts of 3-D fabrics and laminates composed of such fabrics are shown in Fig. 8 [34]. It is observed that the visual damage starts at a pressure as low as 2.58 MPa but the fiber breakage starts around 5.65 MPa. The lower density fabrics (93 and 98 oz=yd2 ) generally perform better than the heavier fabrics (100 and 190 oz=yd2 ) in compression testing of specimens after blast loading. The damage profile in a laminate containing 93 oz/yd fabric is shown in Fig. 9 [34] for various loading levels. In general, 3-D woven fabrics provide an opportunity to strengthen laminates in the out-of-plane direction. Internal debonding and fracture can be delayed by selecting fabrics of appropriate strength. Hollow glass particle filled composites are studied for shock properties in Refs. [35–38]. It is observed that the size and density of hollow particles play an important role in determining the velocity of a shock wave in the material. Epoxy matrix composites containing higher density and larger diameter particles have higher blast wave velocity [38]. Similar results are obtained in Ref. [39]. Although no visible damage is sustained by the specimens at lower blast pressure, the specimens can have internal damage due to the fracture of hollow particles as seen in Fig. 10 [39]. The blast wave pressure is found to decrease rapidly in the material due to the energy absorption in the fracture of hollow microspheres [35]. The pressure reduces by as much as 40% when the volume fraction of glass spheres is increased in the composite. Experimental results are compared with numerical simulation to better understand the energy absorption mechanism in the composite [36]. It appears that the interaction of the blast wave front with hollow particles plays an important role in defining the blast propagation velocity and energy absorption capabilities

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Fig. 9 Blast damage to a three-dimensional woven laminate at (a) 2.5 MPa, (b) 4.79 MPa, (c) 5.65 MPa, (d) 7 MPa, and (e) 8.15 MPa from Ref. [34]

Fig. 10 (a) A syntactic foam panel subjected to shock loading showing no signs of fracture and (b) internal damage observed using ultrasonic c-scan technique

of the composite. The presence of a void in the center of the particle helps in redistributing the kinetic energy of the shock wave. If the particle does not fracture then the shock front tends to sweep around the particle, which further reduces its energy. The entrapped air released by the particle fracture generates back pressure that tends to reduce the blast wave velocity. This reduction in energy can be tailored by means of the size of the particle, the size of the cavity, and inclusion material. These composites have been optimized for other properties such as vibration damping, elastic stiffness, and thermal conductivity, which are of relevance to marine

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applications [40–43]. Lightweight porous composites provide new mechanisms of energy dissipation of shock mitigation that are not available in fiber reinforced composites. Such composites can be especially attractive for marine applications. There are only a few published studies on the response of thermoplastic composites to blast loading. Laminates of unidirectional glass fiber/polypropylene and glass fabric/polypropylene laminates attached to aluminum 2024-T3 alloy sheets are studied in Ref. [44]. This class of material assembly allows energy dissipation through additional mechanisms such as aluminum plate spalling and petalling. The direction of fibers in the polymer matrix laminate plays a role in determining the direction of petalling. Preferential petalling is found to be along the 0ı direction. The interface between two dissimilar materials causes reflection of blast waves. These kinds of hybrid laminates provide opportunities for tailoring the blast response of marine panels. However, corrosion due to potential differences, debonding under static loading, and difficulty in joining such plates can be limiting factors in practical applications. The understanding of blast mitigation mechanisms by means of multiple fractures and the presence of interfaces can yield important information that can be used in designing metallic or polymeric composite materials having high blast mitigation capabilities.

4 Theoretical and Computational Studies Theoretical and computational studies on blast loading of marine structures can be grouped into two major categories depending on the scale of the system under consideration. More specifically, a large amount of research is oriented towards modeling of plate/shell-like panels in response to blast loading; whereas, a few research efforts are directed towards the analysis of deformations of full-scale marine structures to blast loading. Deformations of representative marine panels are analyzed using plate and shell theories along with ad-hoc developed computational mechanics suites or commercial finite element codes. Efforts devoted towards full-scale structures are mostly conducted using commercially available finite element codes.

4.1 Analysis of Marine Panels 4.1.1

Panel Response to Free Field Blast Loading

Blast loading on marine panels is traditionally modeled following the approach developed in Ref. [45]. Within this approach, blast loading is described as a timedependent uniform pressure acting on the panel, whose peak value exponentially

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decays as time increases. More specifically, the pressure is expressed as P .t / D Pmax e t= , where Pmax indicates the peak pressure, t is the time variable, and  is the exponential rate of decay of the impulsive loading. The peak pressure and the rate of decay can be both related to the standoff distance R and the explosive charge weight W through  1=3 A1  1=3 A2 W 1=3 W ;  D K2 W (1) Pmax D K1 R R where the constants K1 , K2 , A1 , and A2 depend on the explosive charge. As the blast wave expands outward from the detonation source, the pressure accelerates the fluid molecules along with it. This results in a “tensile” condition in the fluid behind the shock wave, whereby the pressure in the water drops below the ambient pressure. Therefore, after some elapsed time, the blast wave profile deviates from the above defined exponential curve and eventually crosses the zero gauge pressure level into the so-called “negative” pressure region. This phenomenon can be described using the modified Friedlander exponential decay equation [46]. In Ref. [46], the modified Friedlander exponential decay equation is used to analyze the response of homogeneous panels to blast loading. Using classical plate theory and modal analysis, a single-degree-of-freedom solution is computed for the response of simply supported plates to blast loadings. Results are compared with finite element simulations conducted using ADINA in both linear and nonlinear geometric and material conditions. The single-degree-of-freedom model is able to accurately replicate the deflection time profile of the plate predicted using a linear finite element solution. Nevertheless, finite element results indicate that the membrane stretching of the plate can substantially contribute to the reduction of the plate central deflection and that early plastic behavior of the material can significantly increase it. The modified Friedlander equation is also used in Ref. [47] to study the behavior of homogenous cylindrical shell structures. Reduced order models are developed by using the shell bending modes and a one-dimensional finite element mesh. Effect of beam stiffeners is also considered. Results show that even a single bending mode can yield an accurate response of shells subjected to blast loading. The effect of stiffening elements on the dynamic response of panels to blast loading is also studied in Ref. [48]. The blast loading is modeled using a simplified version of Cole exponential decay equation, consisting of a single triangular pulse. Unstiffened plates are modeled using von-Karman plate theory and their response to blast loading is studied using modal analysis, whereas stiffened plates are analyzed using finite element codes DYNA3D and ABAQUS/Explicit. A parametric study to characterize the effects of in-plane boundary conditions, stiffener buckling, and initial imperfections is also presented in Ref. [48]. In Ref. [49], the modified Friedlander exponential decay equation is used to analyze the response of laminated composite plates to blast loading. The predictions of different plate theories in computing the response of laminated composites to blast loading are compared. More specifically, higher order, first order, and classical plate theories are used to predict the linear response of moderately thick laminated composites to blast loading.

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In the sequence of works [50–52], the response of marine structures to blast loading is studied following an alternative experimental-numerical approach. The pressure profile on the marine structure is reconstructed from in-situ experiments rather than assumed from a pre-defined model. It is worth noting that experimentally measured pressure profiles agree well with the modified Friedlander exponential decay rate and provide an excellent resource for validating analytical models for both stiffened and unstiffened plates. In Ref. [52], a comparison between experimental findings and numerical predictions from finite element ADINA computer codes based on linear classical plate theory and von-Karman plate theory are presented for a square test plate made of steel with length of 508 mm and thickness of 3.4 or 1.5 mm and for a full-scale stiffened plate with in-plane dimensions of 4:572:44 m and thickness equal to 6.35 mm. Finite element results using linear classical plate theory are in close agreement with experimental findings for the unstiffened square plate, whereas membrane stretching seems to be relevant in characterizing the response of the full-scale stiffened panel. In Ref. [52], an in-depth overview of finite element procedures that can be used to enhance the accuracy of the numerical solutions with respect to experiments is presented. In Ref. [53], response of fiber reinforced laminated composites to underwater explosions is studied. The mechanics of the composite material is described using the comprehensive nonlinear model developed in Ref. [54]. This model uses continuum damage mechanics to describe fiber breakage, matrix cracking, fiber/matrix debonding, and delamination/sliding. The underwater blast is modeled using the classical exponential decay equation in Ref. [45]. A thorough parametric study of the effect of geometric and material properties of the composites on the composite energy dissipation mechanisms is conducted using an in-house developed three dimensional finite element code in Fortran. For a rectangular four-ply laminate, the fiber breakage is found concentrated in the neighborhood of the specimen’s centroid; matrix cracking initiates at the center of the specimen’s back surface; fiber/matrix debonding occurs along the fiber; and delamination/sliding does not significantly contribute to the composite energy absorption. In addition, it is found that stacking sequence of plies strongly influences the energy absorption, and that the thickness of the plies determines which failure mode is dominant. In Ref. [55], the transient response of circular plates with varying support configurations to blast loading is studied. The blast loading is modeled as a uniform pulse and large plastic deformation of the material are accounted for. The finite element analysis is conducted using ANSYS computer code. Numerical results on homogeneous clamped plates are compared with experimental findings and different failure mechanisms, including large plastic deformation, tensile tearing, and shear tearing are investigated. It is determined that the presence of a ring support improves the plate energy absorption and allows for larger plastic deformations. Additional failure mechanisms, including shear hinges, are studied in Ref. [56].

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Fluid–Structure Interaction in Sandwich Composites

In the studies reviewed in the previous section, interactions between the shock wave and the structural panel are not considered, and the impulsive loading on the structure is treated as a separate input to the structural model. However, reflection and transmission phenomena are particularly relevant in marine composites, especially sandwich structures, where panel deformations along the thickness direction may be considerable. In Ref. [57], the uniform pressure acting in the front of a panel is represented as the summation of three distinct contributions: the free-field pressure due to the incident wave, the pressure reflected by the front face, and the pressure transmitted into the panel, reflected by the rear face, and transmitted back to the fluid. The incident pressure is equivalent to the pressure term introduced by Ref. [45], whereas the remaining terms can be determined by solving a one-dimensional wave propagation problem. For underwater blasts, the presence of these competing terms may lead to two cavitation zones: one located in the neighborhood of the structure and the other away from it. In Ref. [58], the approach of Ref. [57] is adapted to the analysis of the response of sandwich composites to in-air and underwater explosions. The structure is modeled using classical laminated theory and accounting for membrane stretching through the von-Karman nonlinearity. Anisotropy of face sheets and core material is also considered. For underwater explosions, it is shown that the presence of a weak core significantly alters the time history of the pressure distribution due to the blast wave. In addition, core material properties can be potentially tailored to maximize the delay between the shock wave impact and the cavitation phenomena. For in-air explosions, the presence of the core does not significantly alter the pressure time history with respect to the modified Friedlander exponential decay equation. Results are applied to sandwich structures comprising graphite/epoxy face sheets and PVC foam core material. A similar framework is adopted in Ref. [59] to analyze slamming impact of sandwich composite hulls. In addition, the effect of fluid–structure interactions on the failure of thick cylindrical composite shells is studied in Ref. [60]. In Ref. [61], the response of sandwich panels to underwater blast is modeled using a distributed one-dimensional model for both the fluid and the composite. More specifically, a one-dimensional linearized acoustical formulation, see for example [3], is used to describe the fluid motion in terms of its pressure and velocity along the direction of the plate thickness. The panel transverse motion is modeled as a vibrating segmented bar. Results show that during loading, the face sheets of composites significantly oscillate leading to considerable strains in the transverse direction that can potentially lead to structural delamination. Fluid–structure interactions can also be relevant for homogenous panels located in reentrant corner confinements as illustrated in a numerical multiphysics study by Ref. [52]. An extension of the one-dimensional model of Ref. [61] to describe crushing of foam core materials in sandwich composites is presented in Ref. [62]. In Ref. [62], the foam core crushing is described by assigning an elastoplastic constitutive behavior to the core material until its densification limit. The fluid–structure

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interaction maps (core strength vs. a non-dimensional fluid–structure interaction parameter) developed in this study can be useful in designing sandwich composites for marine applications. These models are considered applicable to any kind of core material that shows perfectly plastic behavior. Therefore, they are applicable to foams and truss structures. The coupling between core compression and plate bending/stretching is discussed in Ref. [63]. A similar analysis is proposed in Ref. [64] to study uniaxial foam core crushing in sandwich composites. In contrast with Ref. [62], fluid–structure interactions are described using the recent results of Ref. [65] that account for nonlinear compressibility effects that are not included in the linearized acoustical formulation. The findings presented in Ref. [62] are extended to three-dimensional geometries in Refs. [66, 67]. An example of truss core sandwich composites tested under blast loading is shown in Ref. [15]. Underwater blast resistance of metallic sandwich panels is studied in Ref. [66] using an improved fluid–structure interaction model that accounts for fluid cavitation. Based on this approach, the core dynamic strength (yield strength of the core divided by the peak pressure of free field impulse) plotted against the fluid–structure interaction parameter can be divided into four domains as shown in Fig. 11 [66]. These domains include a region where the core has very low strength and crushes completely before cavitation occurs (Domain I) and a region where the core has very high strength and is resistant to crushing (Domain IV). The remaining two intermediate domains are of greater interest, since they feature partial core crushing. For in-air blast, Ref. [67] shows that the fluid–structure interaction enhances the performance of the sandwich structure. However, such performance enhancements are lower than those obtained in the underwater blast conditions. Numerical investigations on the role of core materials in blast performance of unsupported sandwich structures are presented in Ref. [68].

Fig. 11 Fluid–structure interaction parameter plotted against dynamic core crushing strength from Ref. [66], where all the symbols are defined

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Strong and rigid cores, such as honeycomb, are found to have relatively poor blast performance and show considerable back face displacement. Softer core materials are found to improve the performance of the sandwich structure, which is confirmed by other researchers through theoretical modeling based on fluid–structure interaction [69]. Similar approaches have been adopted in the analysis of composite response to wave slamming loading [70–72].

4.2 Analysis of Full-Scale Marine Structures In Ref. [73], the transient dynamics of full-scale steel and aluminum petrol boats subjected to underwater explosions are presented. Elastoplastic behavior and large deformations are considered in the finite element solution as shown in Fig. 12 [73]. In addition, fluid–structure interaction is accounted for by coupling the finite element structural analysis with a boundary element formulation of the fluid dynamics problem. More specifically, the so-called doubly asymptotic approximation is used to determine the scattered wave pressure acting on the ships’ wetted surface. Shock severity is measured in terms of the keel shock factors, commonly abbreviated as

a

b

c

d

Fig. 12 Deformation of a full-scale patrol-boat model subjected to underwater shock from Ref. [73]

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KSF. This factor depends on the weight of the explosive charge W , the standoff distance from the charge to the target R, and the angle between the normal to the water surface and a line passing through the p charge location and the ship keel . More specifically, KSF D 0:5.1 C cos. // W =R. Ship dimensions and general specifications used in the finite element solution are derived from a real design. For a given shock factor and fixed , it is found that small charges or long standoff distances, generally lead to damage in the ship hull, whereas larger charges or shorter standoff distances, may potentially damage ship equipment. A similar approach is adopted in Ref. [74] to study the transient response of a glass–epoxy composite submersible hull subjected to underwater blast. In Ref. [75], gas bubble behavior, bubble pulse loading, and fluid bulk cavitation are also accounted for in the blast response of full-scale ships. Ship structure and fluid region in the neighborhood of the hull are studied using LS-DYNA as shown in Fig. 13 [75]. Underwater Shock Analysis (USA) code is used to characterize the fluid response away from the ship structure. That is, the boundary element code USA based on the doubly asymptotic approximation is used to model the presence of the infinite fluidic medium surrounding the finite element meshed fluid–structure system as a radiation boundary. The simultaneous use of finite element and boundary element methods to describe the fluid response allows for modeling bulk and hull cavitations. Simulated velocity at both the keel and bulkhead are compared with available experimental results on ship shock test and show a very good agreement at early time. Significant discrepancies occur as time increases. A similar approach based on LS-DYNA/USA computer codes is presented in Refs. [76, 77] for the analysis of deformations of sub-sea oil pipelines and stiffened shells to underwater explosion shocks, respectively. In Ref. [76] numerical results from LS-DYNA/USA computer codes are compared with a closed-form solution based on the classical Euler-Bernoulli beam theory under low-frequency excitations. Analytical and numerical results are used to develop a probabilistic failure analysis that can be potentially extended to ship structures. In Ref. [77], smeared finite

Fig. 13 Coupled finite element model of a ship and surrounding fluid from Ref. [75]

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element models of stiffened panels are proposed to develop accurate and computationally efficient procedures for analyzing dynamic response of marine structures to underwater blast loading conditions. As an alternative to traditional boundary and finite element methods, the Lattice Boltzmann Method has recently been used in Ref. [78] to analyze fluid flow for fluid–structure interaction problems.

5 Closing Remarks This work is an attempt to summarize the noticeable worldwide research efforts in the area of blast loading of marine materials and structures. It is inevitable to miss some of the important contributions in the related fields in such an effort, which is regretted. A coarse classification of the discussed research efforts is reported in Table 1.

Table 1 Table of references directly relevant to mechanics of advanced materials and structures. For selected papers, specimen dimensions are reported for the reader’s use Test Specimena dimensions conditions Article Specimen type Study type b Sandwich Num. analysis Underwater blast 220  220  10 mm3 [53] Virginia, USA composite C theory [71] Glass fiber Theory C exp. Wave slamming 440  440  3 mm3 Taiwan reinforced laminates [21, 62]b UK Sandwich Theory C exp. Underwater blast composites [22] Plates Exp. C num. Underwater blast do D 152 mm; t D 1:84 mm Illinois analysis [32] Glass–epoxy Exp. Underwater di D 55 mm France tubes impact test [74] Glass–epoxy Num. analysis Underwater blast Singapore laminate [55] Circular plates Num. analysis In-air blast do D 120 mm; India t D 2 mm [46] Plates Num. analysis In-air blast Maryland, C theory USA [13] Sandwich Exp. Underwater blast 3;000  3;000  Australia composites 25 mm3 [16] Steel plate Exp. C num. Underwater blast 780  780  3 mm3 Australia analysis [51] Stiffened steel Num. analysis In-air blast Alberta, plate Canada [50] Stiffened plate Num. analysis In-air blast Alberta, Canada (continued)

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Table 1 (continued) Test conditions

Specimena dimensions

Article

Specimen type

Study type

[52] Alberta, Canada [18] Taiwan [47] British Columbia, Canada [77] California, USA [44] UK

Steel plates

Exp. C num. analysis

In-air blast

4:57  2:44 m2 ; t D 6:35 mm

Aluminum plate

Exp. C num. analysis Num. analysis C theory

Underwater blast

1;0001;00010 mm

Stiffened plates and cylinders

Num. analysis

Underwater blast

Fiber reinforced thermoplastic core–aluminum skin sandwich Three-dimensional woven composites Aluminum 6061-T6 beams Full-scale ship

Exp.

In-air blast

220  220  5 mm

Exp.

In-air blast

3003006:25 mm3

Theory

In-air blast

Num. analysis

Underwater blast

Metallic sandwich plates

Num. analysis C theory

Underwater blast

Thick laminates

Theory

In-air blast

[34]b Rhode Island, USA [56] UK [73] Taiwan [66]b California, USA [49] Virginia, USA [58]b Virginia, USA

[48] UK [64] Maryland, USA [61] Sweden [60]b [69]b UK and California, USA [26] Australia

Stiffened cylindrical shell

PVC core Theory graphite–epoxy skin thick sandwich Stiffened plates Num. analysis C theory Sandwich Theory composites

In-air blast

2;000  200 mm2

In-air and underwater blasts

Underwater blast In-air blast

Sandwich composites Laminated composite Metallic sandwich plates

Num. analysis C theory Num. analysis C theory Num. analysis C theory

Underwater blast

Mild steel bolt

Exp.

Underwater blast

Underwater blast Underwater blast

di D 6 mm, 90 mm length (continued)

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Table 1 (continued) Article [24] Australia [23] Australia [25] Australia [70] Spain and UK [59] [14] India and Korea [17] India [35] Portugal [36] Portugal [37] Portugal [38] Portugal [75] California, USA [33] Alberta, Canada

Specimen type GFRP

Study type Exp.

Test Specimena dimensions conditions Underwater blast 270  70  8 mm3

Glass reinforced laminates Glass reinforced plastic laminates Laminates

Exp.

Underwater blast 270  70 mm2

Exp.

Underwater blast 270  70  8 mm3

Theory

Wave slamming

Sandwich composites Steel plates

Theory C num. Wave slamming analysis Exp. Underwater explosion

Mild steel plate

Num. analysis C exp. Exp. Exp. Exp. Exp. Num. analysis

In-air blast In-air blast In-air blast In-air blast Underwater blast

Exp.

In-air blast

2700  4900 mm2

Exp.

In-air blasts

600  900  2 mm3

Theory

In-air blast

Num. analysis C exp. Num. analysis C exp.

In-air blast

90  90 mm2

In-air blast

Num. analysis C exp. Theory

Underwater implosion In-air blast

Radius of curvature 25 mm, t D 1:2 mm do D 76:2 mm; t D 0:762 mm

Exp. C num. analysis

Underwater blast 250  250  93 mm3

Num. analysis C theory

Underwater blast

Syntactic foam Syntactic foam Syntactic foam Syntactic foam Full-scale ship

Stiffened and unstiffened glass fiber reinforced composite [27]b Rhode Fiber reinforced Island, USA laminate [63]b UK Sandwich composites [29] Turkey Stiffened laminates [31] Turkey Laminated composite shells [10]b Rhode Thin glass spheres Island, USA Sandwich [67]b Massachussets, composites USA [15]b California, Truss core steel USA sandwich composite [76] Singapore Sub-sea pipelines a b

300  250  4 mm3

Underwater blast 550  450  2 mm3

do D outer diameter, di D internal diameter, and t D thickness. ONR supported research.

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Acknowledgments This work is supported by the Office of Naval Research grant N00014-071-0419 with Dr. Y.D.S. Rajapakse as the program manager. Views expressed herein are those of authors, and not of the funding agency. The authors acknowledge Mr. G. Tagliavia for his help in collecting literature and Dr. S.D. Peterson for his careful review of the manuscript.

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Failure Modes of Composite Sandwich Beams Isaac M. Daniel and Emmanuel E. Gdoutos

Abstract The overall performance of sandwich structures depends in general on the properties of the facesheets, the core, the adhesive bonding the core to the skins, as well as geometrical dimensions. Sandwich beams under general bending, shear and in-plane loading display various failure modes. Their initiation, propagation and interaction depend on the constituent material properties, geometry, and type of loading. Failure modes and their initiation can be predicted by conducting a thorough stress analysis and applying appropriate failure criteria in the critical regions of the beam. This analysis is difficult because of the nonlinear and inelastic behavior of the constituent materials and the complex interactions of failure modes. Possible failure modes include tensile or compressive failure of the facesheets, debonding at the core/facesheet interface, indentation failure under localized loading, core failure, wrinkling of the compression facesheet, and global buckling. In the present work failure modes of sandwich beams were studied. Facesheet materials were typically unidirectional and carbon fabric/epoxy and glass fabric/vinylester. Core materials discussed include four types of a closed-cell PVC foam (Divinycell H80, H100, H160 and H250, with densities of 80, 100, 160 and 250 kg=m3 , respectively) and balsa wood. The facesheet and core materials were fully characterized mechanically. The various failure modes were studied separately and both initiation and ultimate failure were determined. Following initiation of a particular failure mode, this mode may trigger and interact with other modes and final failure may follow a different failure path. The transition from one failure mode to another for varying loading or state of stress and beam dimensions was discussed. Experimental results were compared with analytical predictions.

I.M. Daniel () Robert R. McCormick School of Engineering and Applied Science Northwestern University Evanston, IL 60208, USA e-mail: [email protected] E.E. Gdoutos School of Engineering, Democritus University of Thrace, GR-67100 Xanthi, Greece e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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1 Introduction Composite sandwich construction is widely used in aerospace and marine applications where there is need for lightweight structures with high in-plane and flexural stiffnesses [1, 2]. The basic concept of these structures is the separation of relatively stiff, strong and thin facesheets by a lightweight and thicker flexible core. The overall performance of sandwich structures depends on the material properties of the constituents (facesheets, adhesive and core), geometric dimensions and type of loading. The proper design and application of sandwich construction depends on a thorough characterization and understanding of not only the sandwich constituent materials, but also of the structure as a whole under quasi-static and dynamic loadings. Commonly used materials for facesheets are composite laminates and metals, while cores are made of metallic and non-metallic honeycombs, cellular foams, balsa wood and lattice structures. The facesheets carry almost all of the bending and in-plane loads and the core helps to stabilize the facesheets and defines the flexural stiffness and out-of-plane shear and compressive behavior. A number of core materials, including aluminum honeycomb, various types of closed-cell PVC foams, a polyurethane foam, foam-filled honeycomb and balsa wood, have been characterized under uniaxial and biaxial states of stress. The deformation and failure behavior of composite sandwich beams has been described in many publications by the authors and associates [3–13]. Under quasistatic loading conditions, possible failure modes include tensile or compressive failure of the facesheets, debonding at the core/facesheet interface, indentation failure under localized loading, core failure, wrinkling of the compression facesheet, and global buckling. A general review of failure modes in composite sandwich beams was given by Daniel et al. [8]. Individual failure modes in sandwich columns and beams have been discussed by Daniel et al. [5], Gdoutos et al. [9, 11] and Abot et al. [10,13]. Of all the factors influencing failure initiation and mode, the properties of the core material are the most predominant [14]. Failure modes under dynamic loading were also studied and reviewed elsewhere [15]. In the present work, failure modes were investigated experimentally in axially loaded composite sandwich columns and sandwich beams under bending and shear. Failure modes observed and studied include facesheet failure, facesheet debonding, indentation failure, core failures, and facesheet wrinkling. The transition from one failure mode to another for varying loading or state of stress and beam dimensions was discussed. The work reviewed is primarily experimental, but comparisons with model predictions are described whenever available.

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2 Sandwich Materials Investigated 2.1 Facesheet Materials Facesheet materials discussed in this chapter are carbon/epoxy and glass/vinylester composite laminates. The carbon/epoxy was used in unidirectional form (AS4/3501– 6) and textile form (AGP370–5H/3501–6S). The latter reinforcement was a fiveharness satin weave fabric made of the same fibers as the unidirectional material. The carbon fabric material has the same fiber count in both the warp and fill directions. Both of these materials were obtained in prepreg form and used to fabricate laminates by the autoclave process. Facesheets cut from these laminates were bonded to cores to form sandwich panels. The glass fabric reinforcement was an eight-harness satin weave glass fabric (Style 1581). The resin was a styrene crosslinked vinylester (411–350 from Dow Chemical). The unidirectional material (AS4/3501–6) was characterized statically under tension, compression and in-plane shear. In the longitudinal (fiber) direction, the material is linearly elastic initially and then, it displays a characteristic stiffening nonlinearity in tension and a softening nonlinearity in compression [3]. The stress–strain behavior of the carbon/epoxy and the woven-glass/vinylester composites under tension, compression and shear is shown in Figs. 1 and 2. Measured mechanical properties are listed in Tables 1, 2 and 3.

2.2 Core Materials As mentioned before, the core material controls the response and failure of sandwich structures under various loading conditions [14]. Core materials investigated in this study include four types of a closed-cell PVC foam (Divinycell H80, H100, H160 and H250, with densities of 80, 100, 160 and 250 kg=m3 , respectively), an aluminum honeycomb (PAMG 8.1–3/16 001-P-5052, Plascore Co.), a polyurethane foam, a foam-filled honeycomb, and balsa wood. All core materials were characterized in uniaxial tension, compression and shear along the in-plane and through-thethickness directions. Stress–strain curves of foam cores under in-plane tension and compression, and under compression and shear in the through-thickness direction are shown in Figs. 3–6. Results are summarized in Table 4. The mechanical properties of the foam cores tend to vary almost linearly with their densities [16]. Initially, the behavior under compression is linearly elastic when the air inside the cells is being compressed and the cell walls are being bent and stretched [16, 17]. Later, a plateau is observed due to the formation of plastic hinges in the foam cells (plastic collapse). When the cells collapse almost entirely, there is a densification process, i.e., a sharp increase in the stress with a slope similar to that of the bulk polymer (PVC material in this case). These foams also exhibit similar linear elastic behavior under tension initially,

200

I.M. Daniel and E.E. Gdoutos Tension UD

Stress, s1 (GPa)

Compression UD 200 Tension WF

1

Compression WF

0 0

0.3

0.6

0.9

0.12

0.15

100

Stress, s1 (ksi)

300

2

0 0.18

Strain, e1 (%) 100 12

Unidirectional

152

Woven Fabric 60

8

12.7 x 1.3 40

4

Stress, t6 (ksi)

Stress, t6 (MPa)

80

20 0 0

0.5

1

1.5

2

2.5

3

0 3.5

Strain, g6 (%)

Fig. 1 Stress–strain curves under tension, compression and shear of unidirectional (AS4/ 3501–6) and fabric (AGP370–5H/3501–6S) carbon/epoxy composites (top: tension and compression; bottom: in-plane shear)

followed by plastic collapse, but at a much higher stress than under compression. It is also observed that the higher the density, the higher the difference between in-plane and through-thickness (1- and 3-directions) properties [18]. Balsa wood is a natural and also highly anisotropic material and because of its natural origin, comes in the form of sheets made from small pieces bonded together. Balsa wood is much stiffer in the grain direction than in the in-plane direction and that gives this material an enormous advantage in resisting impact loads, compared to other cores of similar density [16]. It also exhibits high shear stiffness and strength in the grain direction. Stress–strain curves for balsa wood are shown in Figs. 7 and 8. Results are included in Table 4. Polyurethane is another quasi-isotropic material like PVC, although its mechanical properties are lower than those of PVC. The foam-filled honeycomb is a

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201

400 50 40 40

Compression

30

200 Tension

150

12.7 x 1.6

20

Stress, s1 (ksi)

Stress, s1 (GPa)

300

100 10

12.7 x 1.6 0

0

0.5

1

1.5

2

0

Strain, e1 (%) 50 6 152 Shear

30

4 12.7 x 1.6

20

Stress, t6 (ksi)

Stress, t6 (MPa)

40

2 10

0

0 0

0.5

1

1.5 Strain, g6 (%)

2

2.5

3

Fig. 2 Stress–strain curves under tension, compression, and in-plane shear of wovenglass/vinylester

composite core material designed to take advantage of both the honeycomb and the foam characteristics. The honeycomb provides stiffness in the transverse direction and the foam gives it the necessary continuity. The particular material used here is a phenolic-impregnated paper honeycomb filled with polyurethane. The core materials (honeycomb or foam) were provided in the form of 25.4 mm (1 in.) thick plates. The honeycomb core was bonded to the top and bottom facesheets with FM73 M film adhesive and the assembly was cured under pressure in an oven following the recommended curing cycle for the adhesive. The foam cores were bonded to the facesheets using a commercially available epoxy adhesive (Hysol EA 9430) [3].

202 Table 1 Properties of unidirectional AS4/3501–6 carbon/epoxy

I.M. Daniel and E.E. Gdoutos Property Density, , kg=m3 Fiber volume ratio, Vf Longitudinal modulus, E1 , GPa Transverse modulus, E2 , GPa Major Poisson’s ratio, 12 In-plane shear modulus, G12 , GPa Longitudinal compressive strength, F1c , MPa Longitudinal tensile strength, F1t , MPa In-plane shear strength, F12 , MPa

Value 1,610 0.63 147 10 0.27 7.0 1,725 2,300 76

Table 2 Properties of carbon fabric/epoxy (AGP370– 5H/3501–6S)

Property Density, , kg=m3 Fiber volume ratio, Vf Warp modulus, E1 , GPa Fill modulus, E2 , GPa Major Poisson’s ratio, 12 In-plane shear modulus, G12 , GPa Warp tensile strength, F1t , MPa Fill tensile strength, F2t , MPa Warp compressive strength, F1c , MPa Fill compressive strength, F2c ; MPa In-plane shear strength, F12 , MPa

Value 1,600 0.62 77 75 0.06 6.5 960 860 900 900 71

Table 3 Properties of glass/vinylester composite

Property Density, , kg=m3 Fiber volume ratio, Vf Warp modulus, E1 , GPa Fill modulus, E2 , GPa Major Poisson’s ratio, 12 In-plane shear modulus, G12 , GPa Warp tensile strength, F1t , MPa Warp compressive strength, F1c , MPa In-plane shear strength, F12 , MPa

Value 1,800 0.50 21.5 20.5 0.14 4.6 260 310 34

Two core materials, Divinycell H100 and H250 were fully characterized under multiaxial stress conditions [18]. A series of biaxial tests were conducted including constrained strip specimens in tension and compression with the strip axis along the through-thickness and in-plane directions; constrained thin-wall ring specimens in compression and torsion; thin-wall tube specimens in tension and torsion; and thinwall tube specimens under axial tension, torsion and internal pressure. From these tests and uniaxial results in tension, compression and shear, failure envelopes were constructed. It was shown [18] that the failure envelopes were described well by the Tsai–Wu [19] criterion as shown in Fig. 9.

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8 Divinycell H250

1

6 0.8

150

0.6

Divinycell H160

4

Divinycell H100

0.4

Divinycell H80

Stress, s1 (ksi)

Stress, s1 (MPa)

12.7 x 25.4

2 0.2 0

0 0

2

4

6

8

10

Strain, ε1 (%)

Fig. 3 Stress–strain curves of PVC foam cores under in-plane tension

5

Stress, s1 (MPa)

4

Divinycell H250

0.6

Divinycell H160

0.4

75 3 2 Divinycell H100

0.2

Stress, s1 (ksi)

25.4 x

1 Divinycell H80 0

0 0

2

4 6 Strain, ε1 (%)

8

10

Fig. 4 Stress–strain curves of PVC foam cores under in-plane compression

The Tsai–Wu criterion for a general two-dimensional state of stress on the 1–3 plane is expressed as follows f1 1 C f3 3 C f11 12 C f33 32 C 2f13 1 3 C f55 52 D 1 where f1 D

1 F1t

f11 D



1 ; f3 F1c

D

1 F3t



1 F3c

1 1 1 1 ; f33 D ; f13 D  .f11 f33 /1=2 ; f55 D 2 F1t F1c F3t F3c 2 F5

(1)

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I.M. Daniel and E.E. Gdoutos 8

25.4 x 25.4

6

Divinycell H250

75

4

0.8 0.6

Divinycell H160

0.4 2

Divinycell H100

Stress, s3 (ksi)

Stress, s3 (MPa)

1

0.2

Divinycell H80 0

0 0

2

4

6

8

10

Strain, ε1 (%)

Fig. 5 Stress–strain curves of PVC foam cores under through-thickness compression

5

0.7 Divinycell H250

Stress, t13 (MPa)

0.5 3

0.4

Divinycell H160 2

0.3

Divinycell H100

0.2 1

Stress, t13 (ksi)

0.6

4

Divinycell H80 0.1

0 0

20

40

60

80

0 100

Strain, γ13 (%)

Fig. 6 Stress–strain curves of PVC foam cores under through-thickness shear

F1t ; F1c ; F3t ; F3c D tensile and compressive strengths in the in-plane (1, 2) and out-of-plane (3) directions F5 D shear strength on the 1–3 plane Setting 5 D k F5 , Eq.1 is rewritten as f1 1 C f3 3 C f11 12 C f33 32 C 2f13 1 3 D 1  k 2

(2)

It was assumed that the failure behavior of all core materials can be described by the Tsai–Wu criterion. Failure envelopes of all core materials constructed from the values of F1t , F1c and F5 are shown in Fig. 10. Note that the failure envelopes of

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205

Table 4 Properties of typical sandwich core materials Divinycell Divinycell H80 H100 Property 3 Density, , kg=m 80 100 In-plane modulus, E1 , 77 95 MPa 77 95 In-plane modulus, E2 , MPa Out of plane modulus, 110 117 E3 , MPa 18 25 Transverse shear modulus, G13 , MPa In-plane compressive 1.0 1.4 strength, F1c , MPa 2.3 2.7 In-plane tensile strength, F1t , MPa 1.0 1.4 In-plane compressive strength, F2c , MPa Out of plane 1.4 1.6 compressive strength, F3c , MPa 1.1 1.4 Transverse shear strength, F5 , MPa

Divinycell H160 160 140

Divinycell H250 250 255

Balsa Wood CK57 150 110

140

245

110

250

360

4600

26

73

60

2.5

4.5

0.8

3.7

7.2

1.2

2.5

4.5

0.8

3.6

5.6

9.7

2.8

4.9

3.7

Out-of-plane Compression

8

1 6 4

0.5

2

Stress, s1, s3 (ksi)

Stress, s1, s3 (MPa)

10

In-plane Compression In-plane Tension

0 0

2

4 6 Strain, ε1, ε3 (%)

8

0 10

Fig. 7 Stress–strain curves of balsa wood core under in-plane tension and compression and through-thickness compression

all Divinycell foams are elongated along the ¢1 -axis, which indicates that these materials are stronger under normal longitudinal stress than in-plane shear stress. Aluminum honeycomb and balsa wood show the opposite behavior. For all materi-

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I.M. Daniel and E.E. Gdoutos

Fig. 8 Stress–strain curve of balsa wood core under through-thickness shear

Fig. 9 Failure envelopes predicted by the Tsai–Wu criterion for Divinycell H250, for k D 0, 0.8 and 1, and experimental results .k D £13 =F13 D £5 =F5 /

als, the most critical combinations of shear and normal stress fall in the second and third quadrants (the failure envelopes are symmetrical with respect to the ¢1 -axis).

3 Facesheet Failure The tension and compression facesheets may fail under uniaxial stress. In the case of composite facesheets, compressive failure is more likely than tensile failure be-

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207 t5 (MPa)

Fig. 10 Failure envelopes for various core materials based on the Tsai–Wu failure criterion for interaction of normal and shear stress

6

Aluminum Honeycomb

4

Balsa

H250

H160 2 H100 H80

0 –4

–2

0

2

4

–2

Polyurethane

6

s1 (MPa) Foam Filled Honeycomb

–4

1.5 12 10

1 Tension Facings

Compression Facings

6

17.8 0.5

P/2

8

P/2

4

Moment (kip-in)

Moment (kN-m)

Experimental Model (Linear Strain) Model (Constant Strain)

2.7 2 2.6

40.6

0 0

0.2

0.4

0.6

0.8 1 Strain, e1 (%)

1.2

1.4

1.6

0 1.8

Fig. 11 Experimental and predicted moment–strain curves for two facesheets of composite sandwich beam under four-point bending (dimensions shown are in cm)

cause the material is appreciably weaker in compression than in tension. This type of failure occurs in beams under pure bending or bending and low shear with cores of sufficiently high stiffness in the through-thickness direction. Facesheet failures were observed in sandwich beams with carbon/epoxy facesheets and aluminum honeycomb core loaded in four-point bending [3].

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Fig. 12 Moir´e fringe pattern in aluminum honeycomb core corresponding to longitudinal displacements (12 lines/mm; 300 lines/in.)

Figure 11 shows plots of the applied bending moment versus strain on the outer surfaces of the facesheets. As expected, the curves show the same stiffening and softening characteristics on the tension and compression sides as the carbon/epoxy facesheet material (Fig. 11). Failure was governed by the compressive strength of the facesheet which in this case reached a value of 1930 MPa (280 ksi), higher than any recorded value for this material under direct compression. The ultimate compressive strain recorded was 1.6%. This is attributed to the support provided to the skin by the core which suppresses the tendency for buckling. The strain variation through the thickness was checked by embedding strain gages at the interfaces between the facings and the core and by using moir´e gratings on the core. The moir´e pattern corresponding to axial displacements on the core consists of fringes in the form of hyperbolas, which is consistent with a linear strain variation through the thickness (Fig. 12). The horizontal displacement has the form u .x; z/ D cxz and the strain is

(3)

@u D cz (4) @z since u .0; z/ D 0 (along vertical axis of symmetry). The linear strain variation through the thickness of the beam was also corroborated by the embedded strain gage readings. The experimentally obtained stress–strain relations of the facesheet material in tension and compression (Fig. 11) and that of the honeycomb core were used to obtain moment–strain relations. In the modeling, two cases were considered, one by assuming linear variation of the strain through the facesheet thickness and one by assuming constant strain. The nonlinear stress–strain relations of the facesheet material were also taken into account. Results were compared with the experimental "1 .z/ D

Failure Modes of Composite Sandwich Beams

209

ones in Fig. 11. The agreement is satisfactory. The small discrepancies observed are not due to the model but rather to the difficulty in obtaining reliable stress–strain curves in direct longitudinal compression. In general, for the widely used cores which have much lower stiffness than the facesheet material, the contribution of the core is negligible. For relatively thin skins and relatively low core stiffness, compressive failure of the facesheet is satisfactorily predicted by the simple relationship   (5) M Š Ff c hf hf C hc where M D applied bending moment at failure Ff c D compressive strength of facing material hf ; hc D facing and core thicknesses, respectively However, if the core stiffness in the through-the-thickness direction is not sufficiently high, another mode of failure, facesheet wrinkling, could occur. This failure mode will be discussed later on in more detail.

4 Facesheet Debonding Facesheet debonding may develop during fabrication of sandwich panels or may be caused by external loading such as impact. Debonding reduces the stiffness of the structure and makes it susceptible to buckling under in-plane compression. Facesheet/core debonding failures and interfacial cracking have been studied by many investigators over the last two decades by means of experimental, numerical and analytical methods [20–28]. Debonding failures are not typically observed in many sandwich beam specimens under usual quasi-static loading configurations. In the case of foam cores no debonding was observed under quasi-static loading due to the relatively high interface fracture toughness. Under impact, delamination failures of the compressive face sheet have been observed, but no interfacial debonding [15]. Beams with aluminum honeycomb cores showed some premature debonding failure in some cases due to the very small bonded area of the honeycomb cross section. The effect of debonding in double cantilever beam specimens made of aluminum facesheets and PVC foam cores (Fig. 13) was studied by Gdoutos and Balopoulos [29]. The analysis used linear elastic fracture mechanics of interfacial cracks in conjunction with finite elements. The strain energy release rate for interfacial crack growth is given by Gint D



 1  2 1 1 C KI C KII2 2 E1 E2

(6)

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I.M. Daniel and E.E. Gdoutos

Fig. 13 Double cantilever sandwich beam specimen

  where Ej D Ej = 1  j2 for plane strain, and Ej D Ej for plane stress, and for crack growth in a monolithic elastic material by Gvol D

KI2 C KII2 E

(7)

The interfacial crack may propagate along the interface or kink into one of the adjoining materials. The angle of initial crack propagation, ˝, is given, according to the maximum tangential (hoop) stress criterion, by: 0q 1 2 B 1 C 8 .KII =KI /  1 C (8) ˝ D 2 tan1 @ A 4KII =KI Kinking of the interfacial crack into the core occurs when the following inequality is satisfied: !   max G G ˝ > (9) GI; cr Gcr . / int core

The critical strain energy release rate for the core material in mode I, GI;cr , and the critical interfacial strain energy release rate, Gcr . /, as a function of mode mixity, are determined experimentally. They are characteristic parameters of the core and the interface, respectively. Values of GInt;cr and GI;cr are shown in Table 5. On the other hand, the values of energy release rate for crack growth in the core and along

Failure Modes of Composite Sandwich Beams

211

Fig. 14 Initial meshing

the interface depend on the applied loads and the geometry of the sandwich plate, and are determined numerically. Table 5 Values of GInt;cr and GI;cr for various core materials

Materials Divinycell H60 Divinycell H80 Divinycell H100 Divinycell H250

GInt;cr  Nmm=mm2 0.28 0.45 0.78 1.55

GI;cr  Nmm=mm2 0.10 0.22 0.30 1.00

We consider a sandwich double cantilever beam (DCB) specimen of length 152.4 mm (6 in.) and width 25.4 mm (1 in.) loaded by a concentrated load at a distance 25.4 mm (1 in.) from its end (Fig. 13). The beam is made of aluminum 2024 T3 facings of thickness 1 mm and a PVC foam (Divinycell H) core of thickness 25.4 mm (1 in.). The core is bonded to the facings by epoxy adhesive of thickness 0.3 mm. Four different PVC core materials, H60, H80, H100, and H 250, were studied. An interfacial crack of length 51.1 mm (2 in.) is introduced between the core and the adhesive at the loaded end of the specimen. Propagation of the interfacial crack is studied under condition of plane strain. The model of the sandwich DCB specimen is shown in Fig. 14. It is composed of seven topological regions. Each region is divided into regular and transition subregions. Sub-region boundaries are then subdivided into segments of appropriate number and proportions, and meshing is done automatically by boundary extrapolation, using Q8 and T6 elements for regular and transition sub-regions, respectively. The initial model contains 1501 elements (986 Q8 and 515 T6), of which 282 discretize the upper face, 114 the upper layer of adhesive, 97 the lower layer of adhesive, 97 the lower facing and 911 the core. For the prediction of the crack trajectory we use the interface toughness values for normal adhesion (Table 5). It is obtained that first the interfacial crack kinks into the core and then curves back toward the interface (Fig. 15). For intermediate values of the distance x from the crack tip .3 mm < x < 30 mm/, we obtain the following results:  The crack after a small depth h1 becomes parallel to the interface (as shown

in Fig. 15).  KI;int , KII;int , and KI;core vary linearly with the distance x from the crack tip.

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Fig. 15 Initial crack path trajectory

 Gint and Gcore vary linearly with x and are almost independent of the core

properties. Regarding the sub-interfacial crack propagation into the core we obtain that the crack becomes parallel to the interface at a constant depth h1 . An explanation of the constant value of h1 and the linear variation of stress intensity factors with the distance from the crack tip x can be obtained by noting that the debonded part of the specimen (above the crack) can be considered as a thin cantilever beam .l=d  25/, elastically supported by the foam core, and subjected to a dominant bending moment varying linearly with x and to a relatively small (constant) shear force. Thus, the Table 6 Values of critical distance h1

H 60 H 80 H 100 H 250

E (GPa) 0.059 0.087 0.107 0.308

h1 (mm) 1.01 0.70 0.65 0.20

Eh1 (N/mm) 59.6 60.9 69.6 61.6

Failure Modes of Composite Sandwich Beams

213

near-tip stress field is linearly proportional to x and, hence, the crack propagates in a self-similar manner parallel to the interface. The strain energy release rate can be determined by differentiating the work of the applied load with respect to the distance from the crack tip and is constant during crack propagation. Values of h1 for various core materials are shown in Table 6. The core stiffness appears to be the main factor that influences the value of the asymptotic depth h1 . Indeed, it can be obtained from Table 6 that the product Eh1 is almost constant and equal to 60 N/mm for the three PVC foam materials H60, H80 and H250. For H100 it takes the value 70 N/mm. Thus, the depth h1 is inversely proportional to the modulus of elasticity of the core material. Under such conditions and for a critical applied load, debonding propagates along the interface only when the adhesion between the interface and the core is weak. Otherwise, the crack kinks into the core and after a small initial curved path it propagates parallel to the interface at a depth h1 . The value of h1 is inversely proportional to the modulus of elasticity of the core. This behavior is independent of the core thickness, which is an order of magnitude larger than the thickness of the facing and the adhesive. Away from boundary effects (e.g., concentrated loads, beam supports, crack kinking, etc.) both stress intensity factors and strain energy release rate can be approximated as linear functions of the crack length.

5 Core Failures The core is primarily selected to carry the shear loading. Core failure by shear is a common failure mode in sandwich construction [30–36]. In short beams under three-point bending the core is mainly subjected to shear, and failure occurs when the maximum shear stress reaches the critical value (shear strength) of the core material. In long-span beams the normal stresses become of the same order of magnitude as, or even higher than the shear stresses. In this case, the core in the beam is subjected to a biaxial state of stress and fails according to an appropriate failure criterion. It was shown earlier that failure of the PVC foam core Divinycell H250 can be described by the Tsai–Wu failure criterion [18, 37]. For a sandwich beam of rectangular cross section with facesheets and core materials displaying linear elastic behavior, subjected to a bending moment, M, and shear force, V, the in-plane maximum normal stress, ¢, and shear stress, £, in the core, for a low stiffness core and thin facesheets are given as [34] PL D C1 b d 2 D where

P C2 b hc



Ec Ef



hc hf

(10) (11)

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I.M. Daniel and E.E. Gdoutos

M D

PL ; C1

V D

P C2

(12)

P D applied concentrated load L = length of beam Ef , Ec D Young0 s modulus of facing and core material, respectively hf , hc D thickness of facings and core, respectively d D distance between centroids of the facings b D beam width C1 , C2 D constants depending on loading configuration (C1 D 4, C2 D 2 for three-point bending; C1 D C2 D 1 for cantilever beam) The maximum normal stress, ¢, for a beam under three-point bending occurs under the load, while for a cantilever beam under end loading it occurs at the support. The shear stress, £, is constant along the beam span and through the core thickness, as verified experimentally [3, 8]. When the normal stress in the core is small relative to the shear stress, it can be assumed that core failure occurs when the shear stress reaches a critical value. Furthermore, failure in the facesheets occurs when the normal stress reaches its critical value, usually the facesheet compressive strength. Under such circumstances we obtain from Eqs. (10) and 11 that failure mode transition from shear core failure to compressive facesheet failure occurs when Ff L DC hf Fcs

(13)

where Ff D facing strength in compression or tension Fcs D core shear strength C D constant (C D 2 for a beam under three-point bending; C D 1 for a cantilever beam under an end load) When the left hand term of Eq. 13 is smaller than the right hand term, failure occurs by core shear, whereas in the reverse case failure occurs by facing tension or compression. The deformation and failure mechanisms in the core of sandwich beams have been studied experimentally by means of moir´e gratings and photoelastic coatings [3–9, 12–14]. Figure 16 shows moir´e fringe patterns in the core of a sandwich beam under three-point bending for an applied load that produces stresses in the core within the linear elastic range [3]. The moir´e fringe patterns corresponding to the u (horizontal) and w (through-the-thickness) displacements away from the applied load consist of nearly parallel and equidistant fringes from which it follows that "x D

@u Š 0; @x

"z D

@w Š 0; @z

xz D

@u @w C D constant @z @x

(14)

Failure Modes of Composite Sandwich Beams

215

Thus, the core is under nearly uniform shear stress. This is true only in the linear range as will be illustrated below. Figure 17 shows photoelastic coating fringe patterns for a beam under three-point bending for various values of applied load P. The fringe pattern for a low applied load (2.3 kN) is nearly uniform, indicating that the shear strain (stress) in the core is constant. This pattern remains uniform up to an applied load of 3.3 kN which corresponds to an average shear stress in the core of 2.55 MPa. This is close to the proportional limit of the shear stress–strain curve of the core material (Fig. 6). For higher loads, the core begins to yield and the shear strain becomes highly nonuniform peaking at the center and causing plastic flow. The onset of core failure in beams is directly related to the core yield stress in the thickness direction. A critical condition for the core occurs at points where shear stress is combined with compressive stress. The deformation and failure of the core is obviously dependent on its properties and especially its anisotropy. Honeycomb and balsa wood cores are highly anisotropic with much higher stiffness and strength in the thickness direction, a desirable property. Figure 18 shows the damaged region of a beam with glass/vinylester facesheets and balsa wood core loaded in three-point bending. It appears that a crack was initiated near the upper facesheet/core interface and propagated parallel to it. The crack traveled for some distance and then turned downwards along the cell walls of the core until it approached the lower interface. It then traveled parallel to the interface towards the support point. Core failure is accelerated when compressive and shear stresses are combined. This critical combination is evident from the failure envelope of Fig. 9. The criticality of the compression/shear stress biaxiality was tested with a cantilever sandwich beam loaded at the free end. The isochromatic fringe patterns of the birefringent coating in Fig. 19 show how the peak birefringence moves towards the fixed end of the beam at the bottom where the compressive strain is the highest and superimposed on the shear strain. Plastic deformation of the core, whether due to shear alone or a combination of compression and shear, degrades the supporting P

z x

W

U

Fig. 16 Moir´e fringe patterns corresponding to horizontal and vertical displacements in sandwich beam under three-point bending (12 lines/mm, 300 lines/in.; Divinycell H250 core)

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I.M. Daniel and E.E. Gdoutos P

Birefringent coating

Birefringent coating

38 cm

• Uniform shear at low loads

γ P = 2.3 kN (510 lb)

γ

• Nonlinear shear as core yields

P = 4.0 kN (890 lb)

γ • Core yielding precipitates facesheet wrinkling P = 5.3 kN (1182 lb)

Fig. 17 Isochromatic fringe patterns in birefringent coating of sandwich beam under three-point bending (Divinycell H250 core)

role of the core and precipitates other more catastrophic failure modes, such as facesheet wrinkling.

6 Indentation Failure Indentation failure in composite sandwich beams occurs under concentrated loads, especially in the case of soft cores. Under such conditions, significant local deformation takes place of the loaded facesheet into the core, causing high local stress concentrations. The indentation response of sandwich panels was first modeled by Meyer-Piening [38] who assumed linear elastic bending of the loaded facesheet resting on a Winkler foundation (core). Soden [39] modeled the core as a rigid-perfectly plastic foundation, leading to a simple expression for the indentation failure load. Shuaeib and Soden [40] predicted indentation failure loads for sandwich beams with glass-fiber-reinforced plastic facesheets and thermoplastic foam cores. The problem was modeled as an elastic beam, representing the facesheet, resting on an elastic– plastic foundation representing the core. Thomsen and Frostig [41] studied the local bending effects in sandwich beams experimentally and analytically. The indentation failure of composite sandwich beam was studied by Gdoutos et al. [9]. Other works on the indentation problem are discussed in Refs. [42, 43].

Failure Modes of Composite Sandwich Beams

217

Fig. 18 Cracking in balsa wood core of sandwich beam under three-point bending near support

P Birefringent Coating

P = 2.1 kN (474 lb)

P = 2.4 kN (532 lb)

P = 2.44 kN (549 lb)

P = 2.47 kN (555 lb)

P = 2.48 kN (558 lb)

P = 2.47 kN (554 lb)

Fig. 19 Isochromattic fringe patterns in birefringent coating of cantilever sandwich beam under end loading

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For linear elastic behavior, the core is modeled as a layer of linear tension/compression springs. The stress at the core/facesheet interface is proportional to the local deflection w  D kw

(15)

where the foundation modulus k is given by [9] k D 0:64

Ec q 3 Ec =Ef hf

(16)

and where Ef and Ec are the facesheet and core moduli, respectively, and hf is the facesheet thickness. Initiation of indentation failure occurs when the core under the load starts yielding. The load at core yielding was calculated as Pcy D 1:70 cy bhf

q 3

Ef =Ec

(17)

where cy D yield stress of the core, and b D beam width. Core yielding causes local bending of the facesheet which, combined with global bending of the beam, results in compression failure of the facesheet. The compressive failure stress in the facesheet is related to the critical beam loading Pcr as follows 2 9 Pcr Pcr L   C (18) f D Ff c D 2 2 16b hf Fcc 4bhf hf C hc where hc is the core thickness, L the span length, b the beam width, and Fcc , Ff c the compressive strengths of the core (in the thickness direction) and facesheet materials, respectively. In the above equation, the first term on the right hand side is due to local bending following core yielding and indentation and the second term is due to global bending. The onset and progression of indentation failure is illustrated by the moir´e pattern for a sandwich beam under three-point bending (Fig. 20). Figure 21 shows load displacement curves for beams of the same dimensions but different cores. The displacement in these curves represents the sum of the global beam deflection and the more dominant local indentation. Therefore, the proportional limit of the load–displacement curves is a good indication of initiation of indentation. The measured critical indentation loads in Fig. 21 were compared with predicted values using Eq. 18 which can be approximated as [9] Pcr Š

p 4 bhf Ff c cy 3

(19)

Thus, the critical indentation load is proportional to the square root of the core material yield stress. The results obtained are compared in Table 7 below. The

Failure Modes of Composite Sandwich Beams

219

P 1 mm 25.4 mm

Moiré Film

1 mm 356 mm

P=320 N (72 lb)

P=574 N (129 lb)

P=814 N (182 lb)

P=926 N (208 lb)

P=1059 N (238 lb)

P=1081 N (242 lb)

Fig. 20 Moir´e fringe patterns in sandwich beam with foam core corresponding to vertical displacements at various applied loads (11.8 lines/mm grating; carbon/epoxy facesheet; Divinycell H100 core)

Deflection, wA (in) 0

0.2

0.4

0.6

3

2

0.6

P

Divinycell H160

A

127

127

25 25

0.4

Divinycell H100 1

0.2

Load, P (kips)

Load, P (kN)

Divinycell H250

Divinycell H80 0

0 0

5

10

15

20

Deflection, wA (mm)

Fig. 21 Load versus deflection under load of sandwich beam under three-point bending (carbon/ epoxy facesheets, Divinycell H250 core)

220 Table 7 Critical indentation loads for sandwich beams with different cores under three-point bending

I.M. Daniel and E.E. Gdoutos Indentation load (N) Measured Calculated

H80

H100

H160

H250

1,050 1,370

1,250 1,500

2,150 2,000

2,900 2,380

approximate theory with the assumption of rigid-perfectly plastic behavior overestimates the indentation failure load for soft cores, but it underestimates it for stiff cores.

7 Facesheet Wrinkling Failure Wrinkling of sandwich beams subjected to compression or bending is defined as a localized short-wave length buckling of the compression facesheet. Wrinkling may be viewed as buckling of the compression facesheet supported on an elastic or elastoplastic continuum, the core. It is a common failure mode of sandwich beams, leading to loss of the beam stiffness. The wrinkling phenomenon is characterized by the interaction between the core and the facesheet of the sandwich panel. Thus, the critical wrinkling load is a function of the stiffnesses of the core and facesheet, the geometry of the structure, and the applied loading. A large number of theoretical and experimental investigations have been reported in the literature on wrinkling of sandwich structures. Some of the early works were presented and compiled by Plantema [44] and Allen [30]. Hoff and Mautner [45] tested sandwich panels in compression and gave an approximate formula for the wrinkling stress, which depends only on the elastic moduli of the core and facesheet materials. Benson and Mayers [46] developed a unified theory for the study of both general instability and facesheet wrinkling simultaneously for sandwich plates with isotropic facesheets and orthotropic cores. This theory was extended by Hadi and Matthews [47] to solve the problem of wrinkling of anisotropic sandwich panels. More studies on the wrinkling of sandwich plates are found in references [48–57]. The critical wrinkling stress was given by Hoff and Mautner [45] is p cr Š c 3 Ef 1 Ec3 Gc13 (20) where Ef 1 ; Ec3 D Young’s moduli of facesheet and core, in the axial and through thickness directions, respectively Gc13 D Shear modulus of core on the 1–3 plane c D Coefficient, usually in the range of 0.5–0.9 In the relation above, the core moduli are the initial ones while the material  is in the linear range. After the core yields and its stiffnesses degrade Ec0 ; Gc0 , it does not provide adequate support for the facesheet, thereby precipitating facesheet wrinkling. The reduced critical stress after core degradation is

Failure Modes of Composite Sandwich Beams

221

p 3

cr Š c Ef Ec0 Gc0

(21)

Heath [58] proposed a simple expression for facesheet wrinkling in sandwich plates with isotropic components, modified here for a one-dimensional beam in terms of the facesheet modulus along the axis of the beam and the through-thickness modulus of the core. The critical wrinkling stress is given by

1=2 2 hf Ec3 Ef 1 (22) cr D 3 hc Sandwich columns were subjected to end compression and strains were measured on both faces. The stress versus strain curves for three core materials, aluminum honeycomb, Divinycell H100 and Divinycell H250, are shown in Fig. 22. The wrinkling stress is defined as the stress at which the strain on the convex side of the panel reaches a maximum value. Note that the panel with the honeycomb core failed by facesheet compression and not by wrinkling. The measured failure stress of 1,550 MPa is much lower than the critical wrinkling stresses of 2,850 and 2,899 MPa predicted by Eq. 20 with c D 0:5 and Eq. 22, respectively. The columns with Divinycell H100 and H250 foam cores failed by facesheet wrinkling, as seen by the shape of the stress–strain curves of Fig. 22. The measured wrinkling stresses at maximum strain for the Divinycell H100 and H250 cores were 627 and 1,034 MPa, respectively, and are close to the values of 667 and 1,170 MPa predicted by Eq. 22. These stresses, would agree with predictions by Eq. 20 for values of c D 0:834 and c D 0:663, respectively. Photographs of these columns after failure are shown in Fig. 23. Figure 24 shows moment versus strain results for two different tests of sandwich beams with Divinycell H100 foam cores under four-point bending. Evidence of wrinkling is shown by the sharp change in recorded strain on the compression facesheet, indicating inward and outward wrinkling in the two tests. In both cases the critical wrinkling stress was cr D 673 MPa. Heath’s relation above (Eq. 22), selected because of the lack of shear interaction between core and facesheet due to pure bending, predicted a critical wrinkling stress of 667 MPa which is very close to the experimental value. Sandwich beams were also tested in three-point bending and as cantilever beams. The moment–strain curves shown in Fig. 25 illustrate the onset of facesheet wrinkling. Critical stresses obtained from the figure for the maximum moment for specimens 1 and 2 are cr D 860 and 947 MPa, respectively. The average of these two experimental values, cr D 903 MPa, agrees with the prediction of Eq. 20 for c D 0:58. In the case of the short beam (specimen 3), core failure preceded wrinkling. The measured wrinkling stress for specimen 3 was 517 MPa. The core shear stresses at wrinkling for specimens 2 and 3 are 3.20 and 4.55 MPa, respectively, which shows that the core material for specimen 2 is in the linear elastic region, whereas for specimen 3 it is close to the yield point. Equation 21 with c D 0:5 predicts the measured wrinkling stress for a reduced core shear modulus of 0 D 21:2 MPa. Gc13

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Fig. 22 Compressive stress–strain curves for sandwich columns with different cores

Fig. 23 Failure of sandwich columns

8 Failure Mode Interaction From the above discussion it is obvious that failure of sandwich beams depends on loading conditions, geometrical configuration and material properties. In the case of cantilever beams with end loading or beams under three-point bending this is

Failure Modes of Composite Sandwich Beams

223

0.6

4

Buckling

Moment (kip-in)

Moment (kN-m)

5

Buckling (Composite Facing Failure)

Compression Facing

0.4 (No Composite Facing Failure) 3

17.8 P/2

P/2

2

0.2

2.7 40.6

1

2.

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.8

0.7

Strain, e1 (%)

Fig. 24 Facesheet wrinkling in sandwich beam under four-point bending (Divinycell H100 foam core, dimensions are in cm)

Fig. 25 Facesheet wrinkling failure in sandwich beams with Divinycell H250 cores

23 cm Specimen 1

Specimen3

30.5 cm

11.4 cm

2

Moment (N-m)

600

1

6

4

400

3 2

200

0

0

0.5

1.0

Moment (kip-in)

Specimen 2

0 1.5

Strain (%)

illustrated by varying the span length. For short spans, core failure occurs first and then it triggers facesheet wrinkling. For long spans, facesheet wrinkling may occur before any core failure. Core failure initiation can be predicted by calculating the state of stress in the core and applying the Tsai–Wu failure criterion. This yields a curve for critical load (at core failure initiation) versus span length. On the other

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hand, in the absence of core failure, facesheet wrinkling can be predicted by Eq. 20 and expressed in terms of a critical load as a function of span length. Figure 26 shows the critical load versus span length for initiation of the two failure modes discussed above for a beam in three-point bending with Divinycell H250 foam core. The span length for transition of failure mode initiation from core failure to facesheet wrinkling is 35 cm.

9 Conclusions Failure modes were investigated experimentally in composite sandwich columns under compression and sandwich beams under bending. Failure modes observed and studied include facesheet failure, facesheet debonding, indentation failure, core failures, and facesheet wrinkling. The transition from one failure mode to another for varying loading or state of stress and beam dimensions was also discussed. For relatively thin skins and relatively low core stiffness, compressive failure of the facesheet is a typical failure mode under bending. However, if the core stiffness in the through-the-thickness direction is not sufficiently high, facesheet wrinkling could occur. Facesheet debonding may develop during fabrication of sandwich panels or may be caused by external loading such as impact. This mode of failure is not common under normal quasi-static loading conditions. When interfacial debonding starts, the crack will most probably propagate into the core. The initiation of failure in composite sandwich beams is heavily dependent on properties of the core material. Plastic yielding or cracking of the core occurs when

8

Pcrit, (kN)

6 Core Failure 4 P 25 mm

2 L

Facing Wrinkling

25 mm

0 0

25

50 L (cm)

Fig. 26 Critical load versus span length for initiation of core failure and facesheet wrinkling

75

Failure Modes of Composite Sandwich Beams

225

the critical yield stress or strength (usually shear) of the core are reached. Core failure by shear is a common failure mode in sandwich construction. Plastic deformation of the core, whether due to shear alone or a combination of compression and shear, degrades the supporting role of the core and precipitates other more catastrophic failure modes, such as facesheet wrinkling. The critical indentation load is proportional to the square root of the core material yield stress. The available approximate theory with the assumption of rigid-perfectly plastic behavior overestimates the indentation failure load for soft cores, but it underestimates it for stiffer cores. The critical stress for facesheet wrinkling is proportional to the cubic root of the product of the core Young’s and shear moduli in the thickness direction. The ideal core should be highly anisotropic with high stiffness and strength in the thickness direction. In the case of beams subjected to bending and shear, the type of failure initiation depends on the relative magnitude of the shear component. When the shear component is low (long beams), facesheet wrinkling occurs first while the core is still in the linear elastic range. The critical stress at wrinkling can be predicted satisfactorily by an expression given by Hoff and Mautner and depends only on the facesheet and core moduli. When the shear component is relatively high (e.g., short beams), core shear failure takes place first and is followed by compression facesheet wrinkling. Wrinkling failure follows but at a lower than predicted critical stress. The predictive expression must be adjusted to account for the reduced core moduli. Acknowledgement The work discussed in this chapter was sponsored by the Office of Naval Research (ONR). The authors are grateful to Dr. Y.D.S. Rajapakse of ONR for his support, encouragement and cooperation.

References 1. Vinson JR (2001) Sandwich structures. App Mech Rev 54(3): 201–214. 2. Zenkert D (1995) An introduction to sandwich construction. London: Engineering Materials Advisory Services. 3. Daniel IM, Abot JL (2000) Fabrication, testing and analysis of composite sandwich beams. Comp Sci Tech 60(12–13): 2455–2463. 4. Daniel IM, Gdoutos EE, Abot JL, Wang KA (2001) Effect of loading conditions on deformation and failure of composite sandwich beams. ASME, IMECE 2001/AMD-25412. 5. Daniel IM, Gdoutos EE, Abot JL, Wang KA (2001) Core failure modes in composite sandwich beams. ASME, 2001, IMECE, AD-Vol. 65/AMD-Vol. 249: 293–303. 6. Gdoutos EE, Daniel IM, Wang KA, Abot JL (2001) Nonlinear behavior of composite sandwich beams under three-point bending. Exp Mech 41(2): 182–188. 7. Daniel IM, Gdoutos EE, Wang KA (2002) Failure of composite sandwich beams. Adv Comp Lett 11(2): 49–57. 8. Daniel IM, Gdoutos EE, Wang KA, Abot JL (2002) Failure modes of composite sandwich beams. Int J Dam Mech 11: 309–334. 9. Gdoutos EE, Daniel IM, Wang KA (2002) Indentation failure in composite sandwich structures. Exp Mech (42): 426–431.

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10. Abot JL, Daniel IM, Gdoutos EE (2002) Contact law for composite sandwich beams. J Sand Struct Mater (4): 157–173. 11. Gdoutos EE, Daniel IM, Wang KA, Abot JL (2003) Compression facing wrinkling of composite sandwich structures. Mech Mater (35): 511–522. 12. Daniel IM, Gdoutos EE, Abot JL, Wang KA (2003) Deformation and failure of composite sandwich structures. J Thermopl Comp Mater (16): 345–364. 13. Abot JL, Daniel IM, Schubel PM (2003) Damage progression in glass/vinylester balsa wood sandwich beams. Sixth Int Conf Sand Struct (ICSS6) Ft. Lauderdale, FL. 14. Daniel IM (2008) The influence of core properties on failure of composite sandwich beams. Proc Eighth Int Conf Sand Struct (ICSS8), Porto, Portugal. 15. Daniel IM (2009) Impact response and damage tolerance of composite sandwich structures. In Dynamic Failure of Materials and Structures, Shukla A, Ravichandran G, and Rajapakse YDS (eds), Springer. 16. Abot, JL (2000) Fabrication, testing and analysis of composite sandwich beams. Ph.D. Thesis, Northwestern University, Evanston, IL. 17. Gibson, LJ, Ashby MF (1999) Cellular solids, structure and properties. Cambridge University Press, Cambridge. 18. Gdoutos EE, Daniel IM, Wang KA (2002) Failure of cellular foams under multiaxial loading. Comp Part A 33: 163–176. 19. Tsai SW, Wu EM (1971) A general theory of strength for anisotropic materials. J Comp Mat 5: 58–80. 20. Prasad S, Carlsson LA (1994) Debonding and crack kinking in foam core sandwich beams. 1. Analysis of fracture specimens. Eng Fract Mech 47(6): 813–824. 21. Prasad S, Carlsson LA (1994) Debonding and crack kinking in foam core sandwich beams. 2. Experimental investigation. Eng Fract Mech 47(6): 825. 22. Grau DL, Qiu XS, Sankar BV (2006) Relation between interfacial fracture toughness and mode-mixity in honeycomb core sandwich composites. J Sand Struct Mat 8(3): 187–203. 23. Minakuchi S, Okabe Y, Takeda, N (2007) Real-time detection of debonding between honeycomb coreand facesheet using small diameter FBG sensorembedded in adhesive layer. J Sand Struct Mat 9(1): 9–33. 24. Berggreen C, Simonsen BC, Borum KK (2007) Experimental and numerical study of interface crack propagation in foam-cored sandwich beams. J Comp Mat 41(4): 493–520. 25. Jakobsen J, Bozhevolnaya E, and Thomsen OT (2007) New peel stopper concept for sandwich structures. Comp Sci Tech 67: 3378–3385. 26. Aviles F, Carlsson LA (2008) Analysis of the sandwich DCB specimen for debond characterization. Eng Fract Mech 75(2): 153–168. 27. Østergaard RC, Sørensen BF, Brøndsted P (2007) Measurement of interface fracture toughness of sandwich structures under mixed mode loadings. J Sand Struct Mat 9: 445–466. 28. Berggreen C, Simonsen BC, Borum KK (2007) Experimental and numerical study of interface crack propagation in foam-cored sandwich beams. J Comp Mats 41: 493–520. 29. Gdoutos EE, Balopoulos V (2008) Kinking of interfacial cracks in sandwich beams. Proc Eighth Int Conf Sand Struct (ICSS8), Porto, Portugal. 30. Alen HG (1969) Analysis and design of structural sandwich panels. Permanon, London. 31. Hall DJ, Robson BL (1984) A review of the design and materials programme for the GRP/foam sandwich composite hull of the RAN minehunter. Comp 15: 266–276. 32. Zenkert D, Vikstr¨om M (1992) Shear cracks in foam core sandwich panels: nondestructive testing and damage assessment. J Comp Tech Res 14: 95–103. 33. Zenkert D (1995) An introduction to sandwich construction. Chameleon, London. 34. Daniel IM, Gdoutos EE, Abot JL, Wang K-A (2001) Core failure modes in composite sandwich beams. Contemporary Research in Engineering Mechanics, Kardomateas GA, Birman V (eds) AD-65, AMD-249: 293–303. 35. Sha JB, Yip TH, Sun J (2006) Responses of damage and energy of sandwich and multilayer beams composed of metallic face sheets and aluminum foam core under bending loading. Metal Mats Trans A 37: 2419–2433.

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36. Bezazi A, El Mahi A, Berthelot J-M, Bezzazi, B (2007) Experimental analysis of behavior and damage of sandwich composite materials in three point bending. Part I Static tests and stiffness degradation at failure studies. Strength Mat 37: 170–177. 37. Gdoutos EE, Daniel IM, Wang KA (2001) Multiaxial characterization and modelling of a PVC cellular foam. J Therm Comp Mat 14: 365–373. 38. Meyer-Piening HR (1989) Remarks on higher order sandwich stress and deflection analysis. Proc First Int Conf Sand Constr: 107–127. 39. Soden PD (1996) Indentation of composite sandwich beams. J Strain Anal 31: 353–360. 40. Shuaeib FM, Soden PD (1997) Indentation failure of composite sandwich beams. Comp Sci Tech 57: 1249–1259. 41. Thomsen OT, Frostig Y (1997) Localized bending effects in sandwich panels: photoelastic investigation versus high-order sandwich theory results. Comp Struct 37: 97–108. 42. Petras A, Sutcliffe MPF (2000) Indentation failure analysis of sandwich beams. Comp Struct 50: 311–318. 43. Anderson T, Madenci E (2000) Graphite/epoxy foam sandwich panels under quasi-static indentation. Eng Fract Mech 67:329–344. 44. Plantema FJ (1966) Sandwich constructions. Wiley, New York. 45. Hoff NJ, Mautner SE (1945) The buckling of sandwich-type panels. J Aero Sci 12: 285–297. 46. Benson AS, Mayers J (1967) General instability and face wrinkling of sandwich plates unified theory and applications. AIAA J 5: 729–739. 47. Hadi BK, Matthews FL (2000) Development of Benson–Mayer theory on the wrinkling of anisotropic sandwich panels. Comput Struct 49: 425–434. 48. Vonach WK, Rammerstorfer FG (2000) The effect of in-plane core stiffness on the wrinkling behavior of thick sandwiches. Acta Mech 141: 1–10. 49. Vonach WK, Rammerstorfer FG (2000) Wrinkling of thick orthotropic sandwich plates under general loading conditions. Arch Appl Mech 70: 338–348. 50. Fagerberg L (2004) Wrinkling and compression failure transition in sandwich panels. J Sand Struct Mat 6: 129–144. 51. Leotoing L, Drapier S, Vautrin A (2004) Using new closed-form solutions to set up design rules and numerical investigations for global and local buckling of sandwich beams. J Sand Struct Mat 6: 263–289. 52. Birman V, Bert CW (2004) Wrinkling of composite-facing sandwich panels under biaxial loading. J Sand Struct Mat 6: 217–237. 53. Fagerberg L, Zenkert D (2005) Effects of anisotropy and multiaxial loading on the wrinkling of sandwich panels. J Sand Struct 7: 177–194. 54. Fagerberg L, Zenkert D (2005) Imperfection-induced wrinkling material failure in sandwich panels. J Sand Struct 7: 195–237. 55. Meyer-Piening H-R (2006) Sandwich plates: stresses, deflection, buckling and wrinkling loads – a case study. J Sand Struct 8: 381–394. 56. Hayman B, Berggreen C, Pettersen R (2007) The effect of face sheet wrinkle defects on the strength of FRP sandwich structures. J Sand Struct 9: 377–404. 57. Lopatin AV, Morozov EV (2008) Symmetrical facing wrinkling of composite sandwich panels. J Sand Struct 10: 475–497. 58. Heath WG (1960) Sandwich construction, Part 2: the optimum design of flat sandwich panels. Aircraft Eng 32: 230–235.

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Localised Effects in Sandwich Structures with Internal Core Junctions: Modelling and Experimental Characterisation of Load Response, Failure and Fatigue Martin Johannes and Ole Thybo Thomsen

Abstract The objective is to provide an overview of the mechanisms which determine the occurrence and severity of localized bending effects in sandwich structures. It is known from analytical and numerical modelling that local effects lead to an increase of the face bending stresses as well as the core shear and transverse normal stresses. The modelling and experimental characterisation of local effects in sandwich structures will be addressed based on the simple and generic case of sandwich structures with internal core junctions under general shear, bending and in-plane loading conditions. The issue of failure and fatigue phenomena induced by the presence of core junctions will be discussed in detail, with the inclusion of recent theoretical and experimental results.

1 Introduction Structural sandwich panels can be considered as a special type of composite laminate where two thin, stiff, strong and relatively dense face sheets, which are often by themselves composite laminates, are separated by and bonded to a thick, lightweight and compliant core material. Such sandwich structures have gained widespread acceptance as an excellent way to obtain extremely lightweight components and structures with very high bending stiffness, high strength and high buckling resistance [1]. At the same time sandwich structures are notoriously sensitive to failure by the application of concentrated loads, at points or lines of support, and due to localized bending effects induced in the vicinity of points of geometric and material discontinuities [1–6]. The reason for this is that, although sandwich structures are

M. Johannes and O.T. Thomsen () Department of Mechanical Engineering, Aalborg University, Pontoppidanstræde 101, DK-9220 Aalborg East, Denmark e-mail: [email protected]; [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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Fig. 1 Sandwich panel with geometrical and material discontinuities in the form of typical internal sub-structures

well suited for the transfer of overall bending and shearing loads, local effects as mentioned above induce severe through-thickness shear and normal stresses. These through-thickness stress components can be of significant magnitude, and may in many cases approach or exceed the allowable stresses in the core material as well as in the interfaces between the core and the face sheets [2–6]. In addition, localized bending effects may induce in-plane stress concentrations in the face sheets, which, depending on the loading situation and the boundary conditions, may exceed the “globally” induced stresses, and thereby seriously endanger the structural integrity. The vast majority of failures in sandwich structures, due to either static overloading or fatigue loading conditions, are caused by localized effects as described above [1]. Local effects may be induced in sandwich structures due to geometric and material discontinuities associated with various types of internal sub-structures including inserts and backing plates (load introduction), edge inserts, core junctions and transmissions from sandwich to monolithic laminate configurations as shown in Fig. 1. Local effects that are caused by the mismatch of elastic properties of adjoining materials in a sandwich core junction (Fig. 1) display themselves by local face sheet bending at the junction in combination with a rise of the in-plane stresses in the sandwich faces as well as of the shear and through-the-thickness stresses in the adjacent cores [2]. Depending on the type of loading and the configuration of the sandwich structure, the stress concentrations may cause local fracture of the core materials, but local face failure or interface failure are also possible scenarios. The influence of such local effects on the structural integrity has been studied under both quasi-static and fatigue loading conditions for sandwich structures subjected to transverse shear loading in Refs. [3, 4]. In this paper the modelling and experimental characterisation of local effects in sandwich structures will be addressed based on the simple and yet very practical case of sandwich structures with internal core junctions (see Fig. 1) under in-plane and transverse shear loading conditions. The issue of failure and fatigue phenomena provoked by the presence of core junctions will be discussed in some detail, with the inclusion of recent theoretical and experimental results. The results presented in this paper is largely based on the work presented in Refs. [2–6] in general and on the work presented by Johannes et al. [7] and Johannes and Thomsen [8].

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2 Prediction of Failure in Sandwich Structures with Core Junctions A large number of studies concerning the failure behaviour of sandwich structures as well as the material and failure behaviour of sandwich constituent materials have been published in recent years. The general failure modes of sandwich structures under various static load cases have been described in textbooks [1, 9] or in more recent studies [10, 11]. For out-of plane loading, also the failure due to local effects at core junctions has been addressed by various studies [3, 4, 12], and for the special case of rigid inserts there are some design guidelines available for design engineers [13]. However, for the in-plane loading case the focus has generally been on sandwich structures loaded in compression, and the corresponding failure modes such as compressive face sheet failure, core shear instability, face sheet wrinkling or global buckling of the sandwich. The failure associated with local effects at core junctions in sandwich structures subjected to in-plane loading has received little attention. To the knowledge of the authors, only one study carried out parallel to this one has covered that topic [14].

2.1 Failure Criteria for Sandwich Core Materials In general, for prediction of failure in sandwich structures, knowledge about the deformation and failure behaviour of the constituent materials is crucial. The relevant material data can be obtained relatively easily for most face sheet materials (except for compression failure of FRP), but proper data for polymer core materials are more difficult to determine, as many of them show nonlinear load-deformation behaviour. Material suppliers generally provide only linear elastic properties, and tend furthermore to state conservative ultimate stress and strain values rather than actual mean values or even tested data from each production batch. For some materiR there are practically none of these problems, als, such as PMI foams (Rohacell), while for others, such as PVC foam cores, all of these problems apply. In addition, there may be a significant scatter of the material properties, making failure prediction even more difficult. Moreover, many sandwich core materials, such as honeycombs, wood and certain polymer foam core materials, display anisotropic material behaviour. Most polymer foam materials show different behaviour in tension and compression, and a particular behaviour under multiaxial loading. This adds complexity both to materials testing as well as to the establishing of accurate and reliable failure criteria. For cellular sandwich core materials, several studies have treated the failure characterisation taking multiaxial stress states into account [15–20]. The criterion treated in Refs. [15, 16] (often referred to as GAZT criterion) is based on analyses of idealised foams to develop a yield surface. An experimental study by Deshpande and Fleck [17] found that the GAZT criterion with a predicted hydrostatic tensile

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strength an order of magnitude greater than the uniaxial tensile strength does not agree with experimental data from triaxial loading. Instead they proposed a phenomenological yield surface to relate the mean stress and the deviatoric stress, and the hydrostatic tensile strength was found to be comparable to the uniaxial tensile strength [17]. Another phenomenological criterion for PVC foams assuming isotropic properties was proposed by Christensen et al. [18]. Gdoutos et al. showed in their experiments [19, 20] that PVC foams are anisotropic and proposed to use the Tsai-Wu criterion instead. A review of failure criteria for foam materials makes it apparent that a direct comparison of the criteria is generally not possible, as different materials were tested or simply due to the variation of the properties of a single material from study to study. A brief comparison of the failure criteria used in connection with the FEA later in this paper is given in the following. The simplest and most common criteria for isotropic materials are the maximum principal stress criterion, the maximum shear stress (or Tresca) criterion and the von Mises criterion. The first one is most commonly used for brittle materials, and the two latter ones for ductile materials that show a range of plastic deformation. The maximum shear stress criterion is probably the most common when estimating the failure load of a sandwich loaded in transverse shear using classical sandwich theory, as the stress state in the sandwich core is almost completely dominated by shear stresses. For more complex stress states other criteria have to be used, as the two latter criteria do not account for failure due to hydrostatic stress states. As mentioned in the previous section, for brittle foams, as e.g. PMI foams, the maximum principal stress criterion was reported to be appropriate [12]. However, for the more ductile PVC foam materials this appears not to be appropriate. Thus, the two more general criteria by Christensen et al. and Tsai-Wu are presented in more detail in the following. The criterion proposed by Christensen et al. [18] assumes isotropic material properties and is formulated as a polynomial expansion in the invariants of the stress tensor ¢ij . The explicit form of the criterion for foam cores reads   1 1 f3 i i C f33  f5 i2i C f5 ij ij D 1 2 2

(1)

with an additional tensile cutoff for foam materials that show brittle tensile fracture given by (2) 33  F3t and the constants f1 D f3 D

1 1 1 1  ; f11 D f33 D ; f5 D 2 F3tp F3c F3tp F3c F13

(3)

In Eqs. (2) and (3) the material properties F3t , F3c and F13 are the experimentally determined yield/failure stresses for uniaxial tension, uniaxial compression and for simple shear, respectively. Failure is defined as yielding for compression and shear,

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and as brittle fracture for tension. The strength values correspond to the maximum stress values measured for tension and shear and the peak stress before the core crushing plateau for compression (see Ref. [18] for details on the testing). The parameter F3tp in Eq. (3) is defined as the “phantom tensile strength” that would exist if there was no brittle fracture “cutoff” behaviour. It cannot be determined from a simple material test, but has to be found by curve fitting of failure data from multiaxial testing. In Ref. [18] data are provided in the ¢33 –¢13 plane for Divinycell H200 closed cell PVC foam. The Tsai-Wu criterion, as proposed by Gdoutos et al. [19, 20] for the prediction of failure of polymer foam core materials, does not assume isotropic, but rather orthotropic or more specifically transversely isotropic material behaviour. The general form of the criterion reads 2 2 2 f1 11 C f3 33 C f11 11 C f33 33 C 2f13 11 33 C f5 13 D1

(4)

with 1 1 1 1 1  ; f3 D  ; f11 D F1t F1c F3t F3c F1t F1c 1 1 1 D f13 D  .f11 f33 /1=2 ; f5 D 2 F3t F3c 2 F13

f1 D f33

(5)

where F1t and F1c are the uniaxial tensile and compressive strengths in the in-plane one-direction, F3t and F3c are the uniaxial tensile and compressive strengths in the through-thickness three-direction, and F13 is the shear strength in the 1–3 plane, respectively. The strength values correspond to the maximum stress values measured for tension and shear and the core crushing plateau stress for compression (see Refs. [19, 20] for details on the testing). In Refs. [19, 20] data are given for Divinycell H100 and H250 closed cell PVC foams in the ¢11 –¢13 and the ¢11 –¢33 planes. For comparison, the failure envelopes based on the two multiaxial criteria are plotted together in Fig. 2, using data sheet values for H200 foam [21], as no experimental material data were available for the same material from Refs. [19–21]. The plots show the similar nature of the two criteria, but also the very predictions of brittle tensile failure. Where the Tsai-Wu failure envelope touches the experimentally determined uni-axial tensile strength value in the ¢11 –¢13 plane, the unconfined envelope proposed by Christensen et al. [18] (drawn with a dashed line in Fig. 2) extends far over this value to the “phantom tensile strength”. For tensile stresses, the failure envelopes by Christensen et al. [18] are confined by the brittle tensile “cutoffs” (see the vertical and horizontal lines in Fig. 2). For details and a comparison with experimental data the reader is referred to Refs. [18–20]. Besides the disadvantage of determining the “phantom” tensile strength by curve fitting of experimental data, the anisotropy of many foam materials cannot be captured with the Christensen criterion [18]. For completeness it should be mentioned that it is also possible to imagine the existence of a compressive buckling cap due to buckling of

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3.5

σ13 [MPa]

3 2.5 2 1.5 1 0.5 0 –6

–4

–2

0

2 σ33 [MPa]

4

6

8

10

12 Christensen Tsai-Wu

10 8

σ33 [MPa]

6 4 2 0 –2 –4 –6 –6

–4

–2

0

2

4

6

8

10

12

σ11 [MPa]

Fig. 2 Failure envelopes in the ¢11 –¢13 (top) and ¢11 –¢33 plane (bottom)

the cell walls under compressive stresses, as proposed in Ref. [20]. However, no sufficient data are available to draw a conclusion on the accuracy of the various criteria relative to each another. Firstly, because different materials were used in the different studies, and secondly due to the substantial scatter of the material properties of some of the materials. The scatter of the material data for the H gradeDivinycell PVC foams is well known and is explicitly stated in the Divinycell H-grade manual [21], where the density range of the foams is given as 90% to 115% of the nominal density. As the foam core material properties scale with relative density, a similar variation of

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the modulus and strength properties can be expected. To be precise, Zenkert et al. [22] showed that the foam properties can be described by means of a power law as xN D ˛ Nn

(6)

where the exponent n was found from fitting experimental data. In Eq. (6), xN is a mechanical property of the foam normalised with the value of the bulk polymer material that the cell edges and faces are made of. The exponent n varies according to the structure of the foam between n D 1 and n D 2, and Zenkert et al. [22] found that the tensile stress–strain relations of a group of both PVC foams (Divinycell H60, H100, H200) and PMI foams (Rohacell 51WF, 110WF, 200 WF) can be scaled with an exponent n D 1:1. This suggests that the other stiffness and strength properties scale with approximately the same exponent.

3 Core Junctions in Sandwich Panels Subjected to In-Plane Loading This chapter focuses on local effects induced at a junction between two different core materials in a sandwich panel/beam subjected to tensile in-plane loading, a secondary load case which also occurs commonly in practice. A preliminary investigation of this was presented by Bozhevolnaya et al. [5, 6]. In this chapter the influence of the local effects on the failure behaviour under quasi-static as well as fatigue loading conditions will be studied for five typical sandwich configurations. In the quasi-static tests, the specimens were subjected to loading up to failure. The failure loads and failure modes are discussed and elaborated on based on results obtained from linear and nonlinear finite element modelling (FEM). In the fatigue tests, the focus is on observation of failure initiation and development. For two sandwich configurations, the fatigue life of the specimens will be compared to that of a set of reference specimens without a core junction. For full details see Ref. [7].

3.1 Test Specimens Five different sandwich beam configurations have been used to examine the influence of the local effects caused by core junctions in practice. Two designs of dog bone shaped specimens as shown in Fig. 3 were used in the experiments. Each specimen comprised four core sections, and the dog bone shape was used to ensure that failure is confined to a specific gauge section of the specimen. The gauge section is the middle part of the specimen with nominal width w, where two different core materials form a so called butt junction. At the ends of the specimen, plywood or aluminium was used as an edge stiffener for a proper load introduction. The specimen configurations are specified in Table 1, where hf and hc denote

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Fig. 3 Schematic of the dog bone shaped specimens used in the tensile tests; top: specimen for configuration 1a/b; bottom: specimen for configuration 2a/b/c [7]

Table 1 Considered sandwich beam configurations [7] Specimen Face Core Core hf config. material material 1 material 2 [mm] 1a Aluminium Divinycell Divinycell 0.4 7075-T6 H60 H200 1b Aluminium Rohacell Rohacell 0.4 7075-T6 51WF 200WF 2a GFRP, NCF Divinycell Divinycell 1.0 Œ0=90s H60 H200 1.0 2b GFRP, NCF Divinycell Plywood H200 Œ0=90s 1.0 2c GFRP, NCF Divinycell H60 Œ0=90s

hc [mm] 25

w [mm] 10

nstatic 3

nfatigue 5

25

10

3

–

25

20

3

8

25

20

6

5

25

20

–

4

the face and core thicknesses, respectively, and nstati c and nfatigue are the number of specimens used for the quasi-static and fatigue testing, respectively. In the reference configuration 2c only one core material was used, i.e. there was no core junction in the gauge section. Two techniques were employed to manufacture the specimens of configurations 1a/b and 2a/b/c, respectively. For configurations 1a/b, aluminium strips were bonded to previously manufactured core layers. The core layers consisted of the two

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polymeric foam core materials as described in Table 1 and two aluminium blocks as edge stiffeners, bonded together. The bonding of the face sheets to the core layer was done such that there was adhesive only between the face sheets and the foam cores, whereas bolted fastenings joint the face sheets and the aluminium blocks, simply in order to allow for a reuse of the aluminium blocks. For all bonds, an Aralditer 2011 epoxy adhesive was used. To bond the face sheet strips to the core layer, the specimens were stacked and a uniform pressure applied to all specimens simultaneously. After curing of the resin, the specimens were machined to the dog bone shape. The specimens of configurations 2a/b/c were produced by liquid resin vacuum infusion. The lay-up of the face sheets consisted of 2 layers of a bidirectional stitched non crimp fabric with an areal weight of 650 g=m2 . A previously manufactured core plate was placed on the lay-up of the lower face, and the lay-up of the upper face was placed on top of the core layer. The core plate consisted of the two core materials as described in Table 1 and two plywood sections as edge stiffeners, bonded together. The complete lay-up was bagged and through a one-step vacuum infusion process a sandwich plate was produced, from which sandwich beams were cut out and then machined to the dog bone shape.

3.2 Material Properties The material data were partly taken from data sheets provided from the manufacturers, partly obtained by materials testing at the manufacturer and partly by in-house materials testing. The material data is essential for the numerical modelling of the sandwich specimens presented later on. An overview over the elastic and strength properties of the sandwich constituent materials is given in Table 2. The ultimate stress and strain values are given in terms of engineering stress and strain. The “characteristic distance” is a fracture mechanics property that will be used for the failure prediction later on. A description of the “characteristic distance” and the corresponding failure criteria will be provided in the finite element modelling section of this paper. The sandwich constituent materials display very different mechanical behaviour. Where the GFRP is known to display approximately linear (normal) stress–strain behaviour over the whole range of deformations, the aluminium and the Divinycell foam cores show nonlinear material behaviour at higher strains. This will be taken into account in the nonlinear FEM. Nonlinear stress strain data for the H60 and H200 foam core materials were made available from DIAB AB, and data for the aluminium were obtained from in-house tensile tests according to EN 10002-1. Uniaxial stress–strain curves for the aluminium and the GFRP are given in Figs. 4 and 5, respectively, and stress–strain curves for the Divinycell H60 and H200 core materials are given in Figs. 6 and 7, respectively [21]. From the Rohacell materials it is generally known that the material behaviour is almost linear elastic until failure [23]. Figure 6 shows the results of the tensile tests performed on the plywood and it can be seen that the behaviour of the plywood can also be approximated well by a linear curve fit.

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Table 2 Material data of the sandwich constituent materials [21, 23] Material E-modulus Poisson’s Yield/ultimate Ultimate [MPa] ratio stress [MPa] strain [%] a a a Aluminium 71,700 0.33 480 /550 12.70a 7075-T6 GFRP, NCF, 24,200a 0.12c –/465a 2.45a Œ0=90s b DIAB H60 (linear) 60 0.32 –/1.4 5.7 0.32 –/1.2 5.7 DIAB H60 37d (nonlinear) Rohacell 51WF 75 0.32 –/1.6 3.0 DIAB H200 (linear) 310 0.32 –/4.8 4.7 DIAB H200 359d 0.32 –/7.3 4.7 (nonlinear) Rohacell 200WF 350 0.38 –/7.0 3.5 Plywood 9,200a 0.32e –/53.0a 0.63a

Characteristic distance [mm] – – 1.37

0.62 1.16

0.38 0.60e

a

In-house material testing. Fibre volume fraction approx. 50% (determined by burn tests and by weighing). Calculated with classical laminate theory. d Initial E-modulus. e Simplifying assumption. b c

700

Stress [MPa]

600 500 400 300 200 True stress-strain 100

Eng. stress-strain

0 0

2

4

6 8 Strain [%]

10

12

14

Fig. 4 Uniaxial stress–strain curve of aluminium; blue: engineering stress–strain; red: true stress– strain

The stress strain curves resulting from the material testing are given in engineering stress and strain. As the aluminium is exposed to very high stress/strain levels due to large plastic deformation, the engineering stress and strain data were converted into true/natural stress and strain for the further use in the nonlinear FEM. The conversion can be justified by the fact that the aluminium does not undergo a volume change during deformation. In Fig. 4 both the engineering and the true/natural stress and strain are plotted. For the Divinycell materials the assumption of a constant volume during large deformations is not meaningful because of the mechanics

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500

Stress [MPa]

400 300 200 100 0 0

0.5

1

1.5 Strain [%]

2

2.5

3

3

4

5

6

4

5

6

Fig. 5 Uniaxial stress–strain curve of GFRP 1.4

Stress [MPa]

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

Strain [%]

Fig. 6 Uniaxial stress–strain curve of DIAB H60 [21] 8 7 Stress [MPa]

6 5 4 3 2 1 0 0

1

2

3 Strain [%]

Fig. 7 Uniaxial stress–strain curve of DIAB H200 [21]

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Stress [MPa]

50 40 30 20 10 0 0.0

0.1

0.2

0.3

0.4 0.5 Strain [%]

0.6

0.7

0.8

Fig. 8 Uniaxial stress–strain curve of plywood; blue: test results; red: linearized

of cellular polymer materials (yielding depends on both deviatoric and hydrostatic stresses). On the account of this the material curve for the nonlinear FEM was simply formulated in engineering stresses and strains as measured in the material tests for the H60 and H200. It has to be noted that for the linear FE analyses exclusively the linear elastic values that are mostly from data sheets as given in Table 2 will be used as material input, whereas in the nonlinear model the whole stress–strain curves as shown in Figs. 4 to 8 will be considered. The experimentally obtained curves are fitted by piecewise linear curves as indicated by the markers in Figs. 4 to 8. It should also be noted that those nonlinear material data represent exemplary stress–strain curves from the materials tests at the manufacturer, and their Young’s modulus and ultimate stress show some deviation from the datasheet values. For both the linear and nonlinear modelling all materials were assumed to be isotropic, which is a substantial simplification for the GFRP face sheets and the plywood. However, it is considered as reasonable for the face sheets, as the nominal stress state in the faces of a sandwich beam is that of a membrane with in-plane normal stresses in the longitudinal direction dominating. For a narrow sandwich beam section the stress state in the face sheets can be approximated as a plane stress state (close to a unidirectional stress state). This means that the assumption of isotropic material properties of the GFRP is justified for the areas of the sandwich away from the core junction where the nominal stress state is dominant. In the area of the local effects near the junction the local stresses need special consideration and a layered model of the composite may be more accurate. The assumption of isotropy causes inaccuracies for two reasons. The first one is the error due to erroneous/inaccurate material properties in the through-thickness direction. However, even in the area of local effects the through-thickness stresses are very small compared to the in-plane stresses, so that the deviation of the presumed material data compared to the real through-thickness data does not have a big influence. The second and most important source of inaccuracies is the difference in the flexural stiffness of a material

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model assuming isotropy and that of the real layered material. For the given layup the isotropic material will have a lower flexural stiffness as compared to the real layered material. On the other hand, there is a resin rich layer on each side of the GFRP laminate of the specimens, which in turn reduces the difference in the flexural stiffness and thereby the inaccuracies in stiffness predictions associated with the assumption of isotropy. As the overall/global stiffness of the face sheets is modelled accurately and the local stiffness is modelled reasonably well, the influence on the core stresses should be minor. Accordingly, the error made by the simplified material model is believed to be acceptable in all cases where the failure of the sandwich is driven by the core material. The resulting stresses in the face sheets would obviously be different on the layer/lamina level for a layered model. However, a layered model would not be more accurate with respect to the strength of the GFRP faces, as the strength of the GFRP was measured as an overall strength of the laminate, and no strength data are available for the single layers. For the plywood core, there is no such justification for a simplified material model, as there are both transverse and through-thickness stresses present in a sandwich core, but as no multiaxial data were available for the plywood, isotropic material behaviour was assumed for simplicity. In this case, a simplifying assumption was also made for the value of the Poisson’s ratio. Due to their manufacturing process the DIAB PVC core materials show a slightly anisotropic behaviour as well, with a somewhat higher stiffness and strength and lower ultimate strain in the through-thickness direction than in the in-plane directions. This is stated by the manufacturer and has been shown in independent tests [10, 11, 17]. However, a full multiaxial characterisation of the material was not available for the types of foam used in this study, and thus isotropic material behaviour was assumed for simplicity. The Rohacell foams are produced with a different process and are practically completely isotropic so that they can be described well by an isotropic linear elastic material model [23].

3.3 Experimental Investigation – Part 1: Quasi-static Tests Quasi-static tests to failure were carried out to assess the influence of the local effects on the static failure behaviour and the failure load. The tests were carried out on a Schenck Hydropuls servohydraulic test machine, and a double-hinged fixture as shown in Fig. 9 was used to avoid bending and ensure a pure in-plane loading. Three or six specimens, respectively, of each configuration (see Table 1) were subjected to axial tensile loading until final failure. The tests were run in loadcontrolled mode (due to control irregularities observed in displacement-controlled mode) with a load rate of 0.1 kN/s, and the load and crosshead displacement data were recorded. One specimen of each configuration was equipped with strain gauges on the outer face surface on each side of the specimen, as indicated in Fig. 1. This was done to ensure there was no bending loading, and to allow for a rough correlation of the FEA strain data with the displacement and strain data from the

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Fig. 9 Experimental set-up with double-hinged loading fixture

experiment within the linear range of deformations. Video recordings and in some cases high speed video recordings were taken during the tests. Figure 11 show the distinctively different load–displacement curves of the tests with configurations 1a/b and 2a/b, respectively. Except from the differences in the failure loads the individual load–displacement curves of the specimens were very similar and very close to the average within each configuration, so that only one representative curve of each configuration is shown in Figs. 10 and 11. Configurations 1a/b correspond to a case where the face material is more ductile than the core, and in configurations 2a/b the opposite is the case. For all configurations, the load–displacement behaviour is dominated by the stiff face sheets. For configurations 1a/b failure was dominated by plastic deformation of the aluminium faces once their yield stress was reached, leading to failure of the compliant foam core at its ultimate strain and followed by final face failure. The core failure is indicated in Fig. 10 and appears for configuration 1b as a small load drop and/or displacement jump in the curve. After the first core failure the proceeding behaviour was dependent on the type of core. For configuration 1a the core failure was immediately accompanied by consecutive face failure, whereas for configuration 1b the face sheets continued to yield a little longer. This is because of the more ductile behaviour and therefore a higher ultimate strain of the DIAB foams compared to the brittle Rohacell foams (see Table 2). The Rohacell 51WF foam fails at a lower strain and load level where a redistribution of the load into the face sheets is still possible. At the higher ultimate strain of the DIAB H60 the load level is higher, the face sheets are in a more progressed stage of yielding and the ability for load redistribution is limited. In all cases the core failure was visible by a crack opening

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Core + face failure at junction

4500 4000 Final face failure at junction

Load [N]

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3000 2500 2000 1500 1000

Configuration 1a

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Configuration 1b

0 0

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Displacement [mm]

Fig. 10 Load–displacement curves of tests with configurations 1a and 1b

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Fig. 11 Load–displacement curves of tests with configurations 2a and 2b

in the compliant core almost parallel to the core junction. Two of three specimens of configuration 1a, and all specimens of configuration 1b failed in the vicinity of the junction. It is assumed that the stress concentrations at the junction may have caused a pronounced local face yielding in this area, leading to subsequent core failure and final failure close to the junction. Evidence for the local yielding influenced by the local effects is that yielding starts at a load of about 3,700 N for both configurations, which corresponds to a nominal face stress of about 450 MPa. This is only about 94% of the nominal yield strength of 480 MPa, and an indication that in some areas

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the local face stress may indeed have reached 480 MPa at that load as a consequence of the local effects. The same applies also to the final failure load. With respect to the failure prediction discussed later in the paper, the course of failure described above indicates that nonlinear modelling is necessary to describe the failure behaviour accurately. It further indicates that for the first failure of the core a strain criterion may be a meaningful approach, and that in particular the first principal (crack opening) strain/stress deserve closer attention. From the load–displacement plot in Fig. 11 it can be seen that for configuration 2a/b the sandwich deformations, again reflecting the face sheet deformations, show roughly linear behaviour up to final failure when the ultimate strain of the GFRP of about 2.5% is reached. In this range of strains also the foam core material behaves almost linearly, which indicates that for configurations 2a/b the linear FEM may be sufficient to predict the failure behaviour. For configuration 2a, the core junction did not seem to have an influence on the failure behaviour, and the specimens simply failed by tensile face failure at arbitrary locations in the gauge section. For configuration 2b with the junction between plywood and H200, failure was observed in the core/core interface two times, it was observed in the H200 foam two times, and twice both the H200 and Plywood materials failed practically at the same time. Closer examination of the fractured interfaces showed that material of the other core could be found on both sides. For the cases of failure in the H200 foam highspeed video recordings showed that failure initiated at the tri-material corner of the core junction. For the cases of failure of both H200 and plywood high-speed video recordings were not available, and it cannot be concluded clearly where the failure initiated. However, it can be stated that the core junction had an influence on the failure behaviour. Table 3 gives a summary of the observed failure loads, failure modes and failure locations of the quasi-static tests. Figure 12 shows images from video recordings showing the typical failure modes of each specimen configuration. Table 3 Average failure loads, failure modes and failure location in the quasi-static tests Location of first Location of final First failure load failure (distance Final failure load failure (distance from junction) from junction) [N] ˙ std. dev. Specimen [N] ˙ std. dev. 1a D Final failure Compliant core, at 4;444 ˙ 06 Face sheet, at junction .4 mm; junction .8 mm; >20 mm; 2:5 mm/ >20 mm; 1 mm/ 1b 4;085 ˙ 375 Compliant core, at 4;262 ˙ 86 Face sheet, at junction (15 mm, junction (0 mm, 5.5 mm, 7 mm) 12.5 mm, 8.5 mm) 2a D Final failure D Final failure 17;831 ˙ 1;195 Face sheet, arbitrary locations .>20 mm/ 2b 11;801 ˙ 742 Compliant/stiff 16;392 ˙ 879 Face sheet, at core/interface junction (0 mm (0 mm for all for all specimens) specimens)

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Config. 1a

Config. 1b

Config. 2a

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Config. 2b

Fig. 12 Pictures of typical failure modes in the quasi-static tests

As the failure occurred directly at or in the vicinity of the junction for several of the tested sandwich configurations it is clear that the local effects at the core junction/discontinuity have an influence on the failure behaviour under in-plane tensile loading. As the configurations differ greatly in their deformation and failure behaviour, however, linear and nonlinear finite element modelling are used in the following to explain the failure behaviour as described above, as well as to propose criteria for failure prediction of sandwich beams with core junctions under in-plane tensile loading.

3.4 Finite Element Analyses (FEA) Finite element modelling of sandwich beam sections with core junctions has been used to evaluate the local stresses and strains at the core junctions numerically. Both linear and nonlinear FE modelling was used. The nonlinear modelling adopted geometrical nonlinearity (large deformations and rotations) as well as nonlinear material behaviour (e.g. material plasticity). Both two dimensional (2D) and a three dimensional (3D) models were used in the initial stage of the study. However, the analyses conducted later on were limited to 2D plane stress modelling for simplicity, see Ref. [7] for details. The material combinations and face and core thicknesses were chosen in accordance with the sandwich configurations used in the experiments (see Table 1). Isoparametric two-dimensional eight-node PLANE 183 elements in ANSYS were used for the finite element mesh of the 2D model. The mesh was refined at the core junction to obtain sufficiently small element edge lengths that ensure convergence of the results. The resulting element size adjacent to the core junction was 1/32 mm for the linear model, and 1/8 mm for the nonlinear model in order to reduce the computational effort to a reasonable level. The bond between the face sheets and the core and between the two core materials at the junction was assumed to be perfect and

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Fig. 13 Sketch of a sandwich core junction and the 2D FE mesh at the tri-material corner

very thin, and hence no additional layers of adhesive were considered in the analysis. Figure 13 shows a sandwich section with a core junction and the corresponding finite element mesh at the tri-material corner of face sheets and cores. Instead of modelling the entire sandwich specimen with the load being introduced via the rigid outer core sections, a 200 mm long sandwich section with the tri-material corner as indicated in the left of Fig. 13 was modelled. Symmetry with respect to the y–z-plane (see Figs. 1 and 13) was chosen as the boundary condition for the beam end with the compliant core, and tensile pressure loads in accordance with the nominal stresses were applied on the face and the core at the other end of the beam, as indicated in Fig. 13. The symmetry boundary condition is not used as a geometric representation of the real structure, but it simply corresponds to a zero displacement in the x-direction and a constraint of rotation around the z-axis. For the linear model only the elastic material properties as given in Table 2 were used as material data, whereas for the nonlinear model the piece-wise linear stress strain curves given in Figs. 4 to 8 were input into the model. For the linear model the loads were chosen to be equal to the failure loads observed in the quasi-static tests, but results can be scaled as it is a linear analysis. For the nonlinear model, the loads were incrementally increased by the FE program until convergence of the results was not achieved any more. Non-convergence can generally occur for several reasons. In this case it indicated that the stresses/strains of one of the nonlinear materials approached the ultimate stress/strain, and the final part of the stress–strain curve with very little incremental stiffness was reached. That point marks the load bearing capacity for that material and the first failure in the structure. However, neither voids nor cracks were created in the model when one of the materials exceeded its ultimate stress/strain, and the stress level for that material remained constant. Thus both the linear and nonlinear modelling will only be meaningful to simulate (indicate) failure initiation, whereas the following progressive development of plastic yielding and finally failure cannot be predicted accurately by the adopted approach. The results of the FEA were analysed in various ways. First, the predicted face stresses along the face outer surfaces, along the face/core interfaces and the core stresses were read out on various paths along the length direction (x-axis) of the sandwich beams to obtain an impression about the characteristics of the local effects. Figures 14 and 15 show the face and core stresses, respectively, as calculated by the linear model for sandwich configuration 1a at the experimentally observed

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Face Surface Face/ Core Interface

Stress in faces σx [MPa]

560 550

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Nominal stress σx H60: 541.34 MPa

540 530

Nominal stress σx

520

H200: 489.38 MPa

510 500 490 480 –30 –25 –20 –15 –10 –5 0 5 10 15 Distance across junction [mm]

20

25

30

Fig. 14 Face stresses for sandwich configuration 1a at the failure load (linear model)

Core stresses close to interface [MPa]

2.5 2 1.5

Nominal stress σx H200: 2.116 MPa

1 Nominal stress σx H60: 0.453 MPa 0.5 0 –0.5 –1

σx σy τxy σ1 σ2 –1.5 –30 –25 –20 –15 –10 –5 0 5 10 15 20 25 30 Distance across junction [mm]

Fig. 15 Core stresses for sandwich configuration 1a at the failure load (linear model)

average failure load. The stresses ¢x , ¢y and £xy denote the normal stresses and the shear stress in the global x–y coordinate system, as shown in Figs. 1 and 13, and ¢1 and ¢2 denote the first and second principal stresses, respectively. The plots show the characteristics of the local effects at the junction. The larger contraction of the compliant core than of the stiff core under in-plane tensile loading, and the necessary redistribution of stresses due to the stiffness mismatch, leads to

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local bending of the face sheets at the core junction (see Fig. 13) which is associated with a local rise of the face normal stresses (see Fig. 14) and the core shear and through-thickness normal stresses (see Fig. 15). It has to be noted that as the tri-material corner is inherently associated with a stress singularity within the framework of linear elasticity [24], the core stresses predicted by the linear model are infinitely high at the tri-material corner and thus the path for evaluation of the core stresses cannot be located directly at the face/core interface. Thus, for the linear model the core stresses are assessed at a characteristic distance away from the interface. The approach of a characteristic distance was proposed by Ribeiro-Ayeh and Hallstr¨om [12] together with a point stress criterion to examine the failure of brittle sandwich foam core materials at bi-material interfaces, and of fracture of cracks and wedge shaped notches in PVC foam by Hallstr¨om and Grenestedt [25]. According to Refs. [12, 25], the characteristic distance is assumed to be a material constant which can be calculated from the characteristic distance is assumed to be a material constant which can be calculated from the mode 1 fracture toughness KIc and the tensile strength t of a material as follows: 1 rch D 2 



KIc ¢1

2 (7)

The values for the characteristic distance in Table 2 were calculated using material data given in Refs. [12, 22, 25]. As an alternative approach, following the work by Bozhevolnaya et al. [5, 6], the characteristic distance can be related to the microstructure of the material. In this case, as failure in the core material occurred in the compliant core, the characteristic distance was chosen as the average cell size acell D 0:6 mm [5, 6] of the H60 foam. Both approaches were compared for the failure prediction. In case of the Rohacell 51WF foam, the characteristic distance equals to about 0.6 mm in both cases. As no data were available for the plywood, a value of 0.6 mm was used here as well for simplicity. For the failure prediction it was decided to use a point stress criterion, tying up on the work of Ribeiro-Ayeh and Hallstr¨om [12], Hallstr¨om and Grenestedt [25], Bozhevolnaya et al. [5, 6] and Jacobsen et al. [14]. For the nonlinear model the first principal stress and first principal strain were used; for the linear model the first principal stress was considered as well as a “simplistic” ultimate strain criterion. The analysis procedure was to read out the stresses of the core components into a so called element table. For the linear model the elements within the characteristic distance around the tri-material corner were excluded hereby. From the element tables the maximum stress or strain could easily be read out. The location of the corresponding element was then checked in a contour plot. Table 4 gives an overview of the stresses in each sandwich constituent at the failure loads predicted by the different failure criteria using the stress results of the linear FE model. The bold entries in the table indicate the component that reaches its ultimate stress first and is predicted to fail first. Table 5 shows an estimation of the failure load by only considering the nominal stresses far away from the core junction, i.e. without considering the local effects.

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Table 4 Failure loads and stresses predicted by linear FEA and point stress criteria Material dependent distance Distance acell D 0:6 mm

Config. 1a 1b 2a 2b

Exp. failure load [N] 4,444 4,085 17,831 11,801

Predicted failure load [N] 4,323 4,379 11,921 10,320

Maximum principal stresses [MPa] Face sheet 550 550 310 459

Compl. core 0.84 1.15 1.40 4.80

Stiff core 2.47 3.25 4.11 –

Predicted failure load [N] 4,323 4,379 10,406 8,939

Maximum principal stresses [MPa] Face sheet 550 550 271 397

Compl. core 0.95 1.15 1.40 4.62

Stiff core 2.72 2.98 4.05 53.0

Table 5 Failure loads and stresses predicted from nominal stresses and “simplistic” strain ( the nominal face stresses refer to the stresses in the sandwich section with the compliant core) Prediction from nominal stresses Prediction from “simplistic” ultimate strain

Config.

Exp. Predicted failure failure load [N] load [N]

1a 1b 2a 2b

4,444 4,085 17,831 11,801

4,515 4,544 17,388 17,388

Nominal stresses [MPa] Face sheet 550 550 422 375

Compl. core 0.46 0.58 1.05 4.80

Stiff core 2.15 2.41 4.80 28.73

Predicted failure load [N] 4,580 4,560 19,000 21,600

Nominal stresses [MPa] Face sheet 535 520 465 465

Compl. core 1.2 1.6 0.8 6.0

Stiff core 2:5 2:5 6.0 35.7

It also shows an estimation based on a “simplistic” ultimate strain criterion. This criterion is based on the nominal state as well, but gives the failure load according to the component reaching its ultimate strain first. Here the nonlinear stress–strain data (see Figs. 4 to 8) were used for reading out the stresses and calculating the failure load. As an example, for configuration 1a/b the stresses in the face sheet are higher in the section with the compliant core than in the stiff core and thus face yielding will occur in this section. The compliant cores fail first as their ultimate strains of 5.7% and 3.0% (H60 and 51WF), respectively, are lower than that of the aluminium. From the material data of the H60/51WF foams a stress of 1.2/1.6 MPa, and from the data of the aluminium a stress of 535/520 MPa (using the engineering stress/strain data) is read out. Using the geometric data of Table 1, and assuming equal strain in face sheets and core, the failure load for the first failure is then calculated to 4,580/4,560 N. The stress in the stiff core at that load can only be estimated, e.g. to about 2.5 MPa when assuming that no yielding has occurred in the specimen section with the stiff core. This phenomenological criterion reflects the real failure occurrence well, but it does not consider the local effects or the multiaxial stress state in the yielding zone. The bold entries in the table indicate the component that reaches its ultimate stress/strain first and is predicted to fail first. From Table 5 it is seen that the prediction using the nominal stresses gives a relatively accurate failure load for the final failure of all configurations where failure is

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dominated by the face sheets. However, it overestimates the loads for the first failure for 1a/b and 2b, as it does not account for the local effects. For configuration 2a the predicted failure load is very close, but the prediction for the failure location is not correct. The shortcomings of the nominal stress criterion are that it neglects the local effects and the nonlinear material behaviour that can lead to major discrepancies when the local effects are important as for configuration 2b. The “simplistic” ultimate strain criterion has the advantage that the material-wise nonlinearity is taken into account, and thus the prediction for the location of the first failure is better for configuration 1a/b and 2a. However, the failure loads are overestimated as the local effects are neglected, leading to a particularly big discrepancy for configuration 2b. From Table 4 it can be seen that the maximum stress criteria using the stress data of the linear FEA also fail to predict the failure for configurations 1a and 1b. The predicted failure load is lower as for the prediction from the nominal stresses, as it considers the local stress rise at the junction as seen in Figs. 14 and 15, but failure is predicted to occur in the face sheet as the material nonlinearity is not taken into account. For configuration 2a the maximum stress criteria overestimate the local effects and predict failure at a drastically reduced load due to the local effects. However, this was not observed in the experiments where only face failure occurred, possibly because local core yielding can relieve the stress concentrations at the junction. The results are practically the same with the different characteristic distances. For configuration 2b where the local effects caused premature failure in the experiments the failure is predicted correctly, and especially the failure load from the material dependent distance is close to the experimentally observed failure load. Thus, for the case of nearly linear and brittle material behaviour, as for configuration 2b, it can be concluded that a maximum principal stress criterion evaluated at a material dependent characteristic distance with stress data from linear FEM can be used for reasonably accurate failure prediction. However, it is not feasible to extrapolate results obtained for the case of a bi-material interface between two brittle materials, where the adopted material dependent characteristic distance is based on a fracture mechanics approach [12, 25], to the case of a trimaterial corner with more complex loading and nonlinear material behaviour. Table 6 presents the results of the nonlinear modelling. Here, the failure behaviour of configurations 1a and 1b is modelled much better, and the failure in the compliant core is predicted correctly. The failure loads are relatively close to the experimentally observed values, with a difference of 100 N between the maximum principal stress and the maximum principal strain criterion. The different results from the stress and strain criteria can be explained by the internal procedure of the FE program to calculate the stiffness matrix. With increasing load, the stiffness matrix is updated using the stress–strain data that was input as the material model, i.e. the stiffness matrix is dependent on the stress/strain level. However, if a multiaxial stress/strain state is present, the strains are converted into an equivalent strain to allow for a comparison with the uniaxial stress–strain data of the material model. In ANSYS, this is done according to the following relation (according to von Mises yield criterion):

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Table 6 Failure loads and stresses/strains predicted by nonlinear FEA and point stress/strain criteria Maximum principal stress criterion Maximum principal strain criterion

Config.

Exp. Predicted failure failure load [N] load [N]

1a 1b 2a 2b

4,444 4,085 17,831 11,801

4,450 4,250 4,250 11,800

1 "e D 1 C eff

Maximum principal stresses [MPa] Face sheet

Compl. core

Stiff core

Predicted failure load [N]

540 475 475 310

1.3 1.4 1.4 6.3

4.3 3.4 3.4 69

4,550 – 1,7900 11,800

r

Maximum principal stresses [MPa] Face sheet

Compl. Stiff core core

5.0 – 2.4 1.5

5.9 – 4.5 2.6

1 ."1  "2 /2 C ."2  "3 /2 C ."3  "1 /2 2

1.0 – 2.0 0.8

(8)

where eff is the effective Poisson’s ratio eff D

material Poisson0 s ratio for elastic strains 0:5 for plastic strains

(9)

It is anticipated that this procedure can lead to stresses/strains in the model that are actually above the (uniaxial) ultimate stress/strain of a material, as seen in Table 6. A certain tolerance has thus to be taken into account when interpreting the data. As an example, for configuration 2a the failure is predicted in the compliant core, but the stresses in the face and the stiff core are very close to the ultimate stress as well. According to the maximum principal strain criterion the failure is predicted in the face sheet as seen in the experiment. For configuration 2b both criteria predict failure in the stiff core. The major difference between the stress and the strain criterion is the location in each component where the failure is predicted to occur. For the stress criterion this is generally around the tri-material corner, whereas for configuration 1a/b the maximum strains are in the centre axis of the sandwich. Figure 16 shows the core stresses and strains along the beam centre axis as calculated by the nonlinear FEA model for sandwich configuration 1a at the predicted failure load. It can be seen that the ultimate strain is reached for the compliant H60 core, whereas the stresses are subcritical at the beam centre. Contrary, at the face/core interface the predicted stresses are determinative for the failure. Unfortunately it is not entirely clear from the experiments where the failure initiated and therefore it is not possible to make a final assessment about the criteria. It has also to be noted that the von Mises approach for calculation of the equivalent strain as given in Eqs. (8) and (9) is generally not suitable for cellular materials, as the deformation and failure behaviour of foams depends on both deviatoric and hydrostatic stresses. This may cause further inaccuracies in the calculation of the stresses close to the core junction due to regions with a hydrostatic stress state. As the stresses for the nonlinear model were evaluated directly at the face/core interface and not at a characteristic distance

7%

3.5

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4%

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1 First principal strain

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First principal strain [%]

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First principal stress 0% –30 –25 –20 –15 –10 –5

0 0

5

10

15

20

25

30

Distance across junction [mm]

Fig. 16 Core stresses and strains in the centre axis of sandwich configuration 1a at the failure load (nonlinear model)

from the interface, the calculation of the equivalent strain and the effective stress as well as the available plasticity model and material data are the major factors of influence for the failure prediction in the nonlinear model. Overall, the nonlinear FEA model captures the nonlinear material behaviour and failure behaviour much better than the linear FEA model. However, it has to be concluded that some issues are still unclear and warrant further investigation. It is essential to consider the multiaxial stress/strain state in the core material. Even though substantial work has been done on describing the failure envelopes of some foam core materials, the data is still sparse and the integration into an FE model is not straightforward, in particular when considering the plasticity. Finally it has to be noted that the material properties of some of the used materials show substantial variations from one production batch to another and thus limit the accuracy of any model.

3.5 Experimental Investigation – Part 2: Fatigue Tests The scope of the fatigue tests was to examine whether the local effects influence the failure behaviour under repetitive loading, and to compare the failure behaviour to that of the quasi-static tests. The tests were run with a sinusoidal load, a frequency of 3 Hz and a loading ratio R D 0:1. Table 7 gives an overview over the chosen fatigue loads and the resulting failure behaviour and fatigue life times. For configuration 1a, no failure could be observed in connection with the core junction. The chosen load was well below the regime where face yielding could occur and this might explain the different failure behaviour compared to static loading conditions.

Sandwich Structures with Internal Core Junctions: Load Response, Failure and Fatigue Table 7 Fatigue test results Fatigue load Fdyn =Fstat;max Config. Fdyn ŒN 1a 2,700 0.6 2a 7,100 0.4 7,100 0.4 8,900 0.5 11,590 0.65 2b 7,100 0.6 7,100 0.6 2c 7,100 (0.4)

Fatigue life ˙ std. dev 27; 877 ˙ 3; 691 17; 282 ˙ 1; 026 19,687 7,900 593 4,654 10; 558 ˙ 10; 945 65; 380 ˙ 37; 725

Config. 2a

Failure mode Face failure Face failure Face failure Face failure Face failure Face failure Face failure Face failure

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Failure location Arbitrary (5 of 5/5) Junction (3 of 6/8) Arbitrary (1 of 6/8) Junction (1 of 6/8) Junction (1 of 6/8) Junction (1 of 3/5) Arbitrary (2 of 3/5)  cf. text

Config. 2b

Fig. 17 Pictures of failure observed in the fatigue tests

For configuration 2a, an accumulation of failure at the core junction was noticed, which had not been the case for quasi-static loading. During the fatigue tests, a gradual whitening of the face sheets was observed, the intensity of which was most pronounced at the core junctions, meaning that the local effects possibly led to an increased damage accumulation in this area. Figure 17 shows images of the course of failure in the fatigue tests. Regarding the tests with configuration 2a to 2c it should be noted that a part of the specimens failed in the transition radii of the dog bones. As it is not clear whether this type of failure was a result of the transition or the core junction between the edge stiffener and the foam core (see Figure 1) these specimens were not considered for evaluation. The total number of specimens of one specimen configuration is stated after the slash in Table 7, and the number of evaluated specimens is stated before the slash. A part of the tests with configuration 2a were run at different load levels, because an appropriate load level had to be established in the beginning. Altogether, the results clearly indicate that fatigue damage accumulation is connected to the stress concentrations at the core junction for sandwich configuration 2a. For configuration 2b a similar course of failure as for configuration 2a was observed for one specimen that failed at the core junction (see Fig. 17). However,

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two specimens failed at arbitrary places in the beam gauge sections and two in the transition radii, thus preventing a solid conclusion due to the limited statistical significance. Configuration 2c was used as reference to compare with configurations 2a and 2b with respect to failure behaviour and fatigue life. Three specimens of configuration 2c failed in the transition radius and one specimen had not failed after 105 cycles, when the test was stopped. The fatigue life for this specimen was taken as 105 cycles into calculation of the average fatigue life, together with the fatigue lives of the specimens that failed in the transition. Even though the recorded fatigue lives have limited statistical significance, the comparison of configurations 2a, 2b and 2c with respect to the fatigue lives associated with failure at and away from the core junction indicates that there is a correlation between localised failure at core junctions and a reduced fatigue life.

3.6 Discussion and Conclusions (In-Plane Loading) The influence of local effects on the failure behaviour of sandwich beams with core junctions has been studied experimentally for tensile in-plane loading under both quasi-static and fatigue loading conditions. An attempt was made to predict the failure loads and failure modes based on results from both linear and nonlinear finite element modelling using a point stress criterion originally proposed for the fracture of brittle foam materials. Additionally, simplistic ultimate stress and strain failure criteria were used for the failure assessment. In the quasi-static tests, the failure modes were dependent of the particular material configuration studied, whereas in the fatigue tests visible failure was always confined to the face sheets. Depending on the sandwich configuration, premature failure accumulating at the core junction was observed in the experiments for quasistatic as well as fatigue loading conditions. For a part of the tested configurations, the failure mode shifted from core failure in the quasi-static tests to face failure in the fatigue tests. In the fatigue tests, failure at the core junction was accompanied by a reduced fatigue life. The accuracy of the models used for failure prediction in the quasi-static case was found to depend strongly on the particular sandwich configuration. For specimens using materials with linear (brittle) material behaviour the failure modes and the corresponding failure loads predicted by the linear model agreed reasonably well with the experimental observations. For the cases with distinctly non-linear material behaviour the predictions based on the linear elastic FE model did not correlate well with the observations. The nonlinear FE model was able to give better results in that case, but various uncertainties remain with respect to the nonlinear FE modelling. In particular the multiaxial stress/strain state at the core junction and the plasticity models used for the foam core materials warrant further examination.

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4 Core Junctions in Sandwich Panels Subjected to Transverse Shear Loading In this chapter the focus is on local effects induced at junction between two different core materials in sandwich panels/beams subjected to transverse shear loading, which is perhaps the most important load case for sandwich structures. A preliminary investigation of this was presented by Bozhevolnaya et al. [2–6]. In this chapter the influence of the local effects on the failure behaviour is studied for six typical sandwich configurations with and without core junctions loaded in transverse shear under quasi-static as well as fatigue loading conditions. In the quasi-static tests, the specimens are subjected to loading up to failure. The failure loads and failure modes will be discussed based on results obtained from linear finite element analyses (FEA) in connection with various failure criteria. Failure initiation and development is the focus for both quasi-static and fatigue experiments, and highspeed video recordings are used to document the observed failure behaviour. For full details see Ref. [8].

4.1 Sandwich Test Specimens Six sandwich beam configurations were considered in the experimental investigation. All configurations contained GFRP face sheets using a bidirectional stitched non crimp fabric (NCF) with an areal weight of 650 g=m2 . The lay-up was symmetric and balanced using either two or four layers of the NCF with a stacking sequence Œ0=90S or Œ0=902S , respectively. The resulting face sheet thickness was 1 or 2 mm, respectively. The core layer was either 10 or 25 mm thick, and contained two different core materials in three sections as shown in Fig. 18, forming two core junctions between the different cores. Configurations 4 and 6 contained only one core material and they were meant as reference specimen types without a core junction. The width of the sandwich beams was 30 mm in all cases.

Fig. 18 Schematic of sandwich beam specimens with core junctions; top: butt junction; bottom: 2ı  45ı double scarf junction

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Table 8 Considered sandwich configurations

Config. 1 2 3 4 5 6

Core material 1 Divinycell H60 Divinycell H60 Divinycell H60 Divinycell H60 Divinycell H200 Divinycell H200

Core material 2 Divinycell H200 Aluminium

Core junction shape Butt junction

Core stiffness hc lcore2 ratio Gcore1 =Gcore2 hf [mm] [mm] [mm] [MPa/MPa] 2.0 25 140 22/90 [24]

Butt junction

2.0

25

140

Aluminium 2ı  45ı junction No junction

2.0

25

150

2.0

25

–

Aluminium Butt junction

1.0

10

100

1.0

10

–

No junction

22/27,000 [24, 27] 22/27,000 [24, 27] (1/1) 90/27,000 [24, 27] (1/1)

The sandwich configurations are specified in Table 8. The table also shows the associated stiffness ratios, i.e. the ratio of the shear moduli of the core materials that determine the magnitude of the local effects at the junction [2]. The specimens were manufactured by liquid resin vacuum infusion. The dry fibre lay-up of the face sheets was placed on the laminating table, a previously manufactured core plate was placed on the lay-up of the lower face, and the lay-up of the upper face was placed on top of the core layer. The core plate consisted of the two core materials stated in Table 8. The whole lay-up was then bagged and a sandwich plate was produced by a one-step vacuum infusion process. The sandwich specimens were cut from the manufactured plates and subsequently machined to their final width.

4.2 Experimental Results – Part 1: Quasi-static Tests Quasi-static tests to failure were carried out to assess the influence of the local effects on the static failure behaviour, and to obtain the relevant load levels for the fatigue tests. The tests were carried out on a Schenck Hydropuls servohydraulic test machine using a three point bending set-up as shown in Fig. 19. The set-up consisted of a lower fixture with two rollers and an upper fixture with a loading bolt. The specimen was centred with respect to the loading bolt and the span between the rollers was set to 440 mm. Small polymer tabs were attached to the specimen at the supports and the loading point to avoid local indentation failure. The tests were run in displacement-controlled mode at a constant displacement rate of 6 mm/min. Four specimens of each configuration were tested at room temperature. The load and crosshead displacement data were recorded and high-speed video recordings were taken of the tests. Figure 20 shows load–displacement curves from the test series with configurations 4 and 6.

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Fig. 19 Three point bending setup for transverse shear loading (shown with a configuration 6 specimen and a medium sized Divinycell H200 foam loading tab under the central loading bolt) 1400 1200

Load [N]

1000 800 600 400 200 0 0

5

10

15

20

25

30

35

40

Displacement [mm] 1400 1200

Load [N]

1000 800 600 400 200 0 0

5

10

15

20

25

30

35

40

45

50

55

60

Displacement [mm]

Fig. 20 Typical load displacement curves from the quasi-static tests; top: configuration 4 (representing 1–4); bottom: configuration 6 (representing 5–6)

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The top chart of Fig. 20 shows the deformation behaviour that is representative for practically all the tests with configurations 1–4. After an initial linear domain the specimen deformation continues at a constant load until failure occurs at very large displacements. In the nonlinear domain the deformation behaviour is dominated by shear deformation and eventually local compression of the compliant H60 core. The curves were obtained with specimen configuration 4 using a relatively thick tab of H200 foam under the central loading bolt in addition to the small polymer tab that was used in the other tests. Without this tab, the high compressive stresses at the loading point lead to core crushing and subsequent face wrinkling. Alternatively, a medium sized foam tab was used which also lead to local face failure. It was thus decided to evaluate only those three tests of configuration 4 where a thick tab was used, and to use the same set-up also in the fatigue tests. Configurations 1–3 all showed similar deformation behaviour as configuration 4, but there was no need for an additional tab due to the stiff core material in the centre of the sandwich beams. All beams of configurations 1–3 failed due to core shear close to the centre/middle of the Divinycell H60 core section, somewhat closer to the supports, and for the beams of configuration 4 core shear failure occurred some distance inwards (towards beam midspan) from this location. For one specimen of configuration 2, delamination of the lower face sheet from the aluminium core was observed prior to core shear failure. Images from high-speed video recordings showing the typical core shear failure modes of specimen configurations 1–4 are presented in Fig. 21.

Config. 1

Config. 2

Config. 3

Config. 4

Fig. 21 Images of typical failure behaviour in the quasi-static tests, configurations 1–4

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The recordings were taken at a frame rate of 1,000 frames per second (fps), except the image shown for configuration 2 that was taken at 100 fps only. In particular from the images of configuration 1 and 3 it can be seen that the crack in the core did not initiate at the core junction or the support, leading to the conclusion that in all configurations the failure mode was core shear, without influence of the core junction. This contradicts the hypothesis presented in Ref. [12], where the failure was associated with the stress concentrations at the core junction. With the given evidence it can be assumed that for sandwich configurations 1–4 irregularities and local density variations in the Divinycell H60 core material in combination with nonlinear geometric effects determine the location of core shear failure under quasistatic transverse shear loading. From the high-speed video recordings there is no evidence that the local effects at the core junctions are connected to the core failure. The bottom chart in Fig. 20 shows the deformation behaviour that is representative for practically all the tests with configurations 5–6. Due to the thinner and stiffer core, the deformation behaviour of configuration 5–6 is very much dominated by bending and to a much lower extent by core shear, as can be seen from the deformation pattern in Fig. 19 in comparison to the patterns displayed in Fig. 21. The load–displacement behaviour is almost linear until failure. The curves were obtained with configuration 6 using an additional medium sized foam tab of H200 foam at the loading point. Failure occurred as compressive face failure at the beam midspan. In this configuration it was not really necessary to use the tab with respect to a local indentation failure, as the same failure behaviour occurred also when using the small polymer tab only. However, the additional foam tab was considered useful to increase the credibility of the measured failure loads. Thus, it can be assumed that the foam tab reduced the local stress concentrations at the load introduction, and that the measured failure load using the tab corresponds to an almost purely compressive face sheet failure. The specimens of configurations 5 failed either due to compressive face sheet failure at the core junction or by delamination of the upper face sheet from the aluminum core. Images from high-speed video recordings showing the typical failure modes of specimen configurations 5–6 are presented in Fig. 22. Whether the failure at the core junction for configuration 5 indicates a significance of the local effects is not entirely clear, as the junction is also the location of the highest nominal face stresses (the highest bending moment occurs at the beam midspan, but due to the rigid aluminium core the face normal stresses are much lower in the section of the aluminium core; contrary to that, for configuration 6 both the highest bending moment and the highest face normal stresses occur at the midspan). Thus, the failure loads have to be examined further to explore this. The experimental failure loads were determined as the maximum loads from the load– displacement curves. The average maximum load of each specimen configuration was also used as a basis to calculate the loads for the subsequent fatigue tests. The average failure loads are given in Table 9. They were determined from four specimens for each specimen batch. As can be seen from Table 9 one to three batches of specimens were produced and tested for every sandwich configuration. This was done to examine the variability

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Config. 5

Config. 6

Fig. 22 Images of typical failure behaviour in the quasi-static tests, configurations 5–6

from batch to batch. This variability could either result from the manufacturing process, and thus changes in the face sheet properties, or from variations of the core properties. As was found from separate tests of the face sheet material, the elastic properties of the face sheets hardly varied with the manufacturing process, even if for example a small leak in the vacuum bagging caused an increased amount of air bubbles in the laminate. The low variability of the face sheet properties is confirmed by practically constant initial stiffness properties of the specimens of configurations 1–4 and 5–6, which are dominated by the face sheet stiffness. Whether the low variability also applies to the strength properties cannot be deduced from this, but this is suggested by the very similar failure loads of specimen batches 8-1 and 9-1. Thus, it is assumed that the high variability of the failure loads from batches 1–3 origins (to a high extent) from the variability of the core material properties, even though material from one delivery (but not necessarily one core plate) was used. An explanation for this was given in Section 3. The results for configurations 1–4 in Table 4 can be interpreted in different ways. One way could be to take the overall average of the failure loads of batches 1–3 of configuration 1, which gives a load of 1,203 N. This load could be compared to configurations 2–3 and lead to an assessment/evaluation of the designs of the core junctions. However, considering the variability of the failure loads it is obvious that this makes little sense. Instead, with the average load of configuration 1 it can be seen, that except for configuration 3 the failure loads of configurations 1–4 all scatter around 1,200 N. This correlates very well with the failure load of 1,140 N predicted for core shear failure by classical sandwich theory [1]. Together with the findings from the high-speed video recordings it is thus concluded that for configurations 1–4 there is no indication that the local effects at the core junctions exerted an influence on the failure behaviour. In other words, failure occurred due to core shear for configurations 1–4.

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Table 9 Average failure loads, failure modes and failure location in the quasi-static tests Failure load [N] Core Core Junction Specimen Failure ˙ std. Config. material 1 material 2 shape batch mode/location dev. 1

Divinycell H60

Divinycell H200

Butt 1 junction

942 ˙ 11

2

1;222 ˙ 69

3

1;446 ˙ 44

2

Divinycell H60

Aluminium Butt 4 junction

1;136 ˙ 69

3

Divinycell H60

Aluminium junction 2  45

6

1;459 ˙ 18

4

Divinycell H60

5

1;167 ˙ 11

Aluminium Butt 7 junction

1;192 ˙ 37

No junction

5

6

Divinycell H200

Divinycell H200

No

8-2

991 ˙ 99

9-2

954 ˙ 45

8-1

1; 074 ˙ 177

9-1

1;022 ˙ 98

junction

Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Compr. face failure at junction Delamination face/core 2 Delamination face/core 2 Compr. face failure at midspan Compr. face failure at midspan

Configurations 5 and 6 were created as a very different sandwich design, where a very stiff and strong foam core was used to change the deformation behaviour from shear dominated to bending dominated and to prevent core shear failure and trigger face sheet failure. To overcome the shortcomings of a comparison across specimen batches and to ensure as direct comparability as possible, specimen batches 8 and 9 were produced, where each batch included specimens of both configuration 5 and 6. This means the sandwich plates manufactured with vacuum infusion each contained a core plate with and without core junctions. (Sub-)batches 8-2 and 92 denote the specimens of batch 8 and 9 with core junction and (sub-)batches 8-1 and 9-1 the ones without core junctions. Unfortunately, the specimens of batch 8-2 and 9-2 did not fail by compressive face failure, but by delamination between the

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upper face sheet and the aluminium core. This is considered as the consequence of a bad bonding between the components rather than as the typical failure mode. The reason for the bad bonding is not clear, as the surface treatment of the aluminium insert (degreasing with acetone, sandblasting and again degreasing with acetone before the vacuum infusion) was the same for batches 7–9. A pickling treatment may have been more appropriate, but the equipment for this was not available, and the bonding quality of batch 7 proved to be sufficient. As a consequence of the observed delamination failures the intended direct comparisons of batches 8-1 versus 9-1 and 8-2 versus 9-2 was not possible. However, the indirect comparison across specimen batches between 7 and 8-1 versus 9-1 should be possible, as the results of batches 8-1 and 9-1 have a considerably lower variability than for configurations 1–4. This is because the face sheet properties dominate the failure behaviour of configurations 5–6. Assuming that this low variability is associated with the strength of the face sheets, the failure loads of configuration 5 and 6 can be compared with respect to the deviation from the predicted failure loads according to classical sandwich theory [1]. With an average experimentally observed failure load of 1,192 N for configuration 5 the deviation of about 30% is bigger than for configuration 6 with an average failure load of 1,048 N and about 20% deviation, even though local indentation at the loading point for configuration 6 may have reduced the failure load compared to an ideal transverse shear loading without indentation effects. This indicates that the rise in face stresses at the core junction reduces the failure load, and that the local effects did play a role for this sandwich configuration.

4.3 Finite Element Analyses (FEA) Finite element analyses (FEA) of the sandwich beams were used to calculate the stresses in the sandwich components away from the core junctions and the local stresses at the junctions. An idealised 2D finite element model of a sandwich beam was made using the commercially available FEA program ANSYS (ANSYS Version 10 with Academic Teaching Advanced license). The material combinations and face and core thicknesses were chosen in accordance with the sandwich configurations used in the experiments (see Table 8). Isoparametric two-dimensional eight-node PLANE 183 elements were used for the finite element mesh. A sequential refinement of the mesh was used at the core junctions to obtain sufficiently small element edge lengths (1/32 mm) to ensure convergence of the results. The bond between the face sheets and the core and between the two core materials at the junction was assumed to be perfect and relatively thin, and hence an additional layer of adhesive was not considered in the analysis. Figure 23 shows a sandwich beam with a core junction as well as the corresponding finite element mesh at the tri-material corner of face sheets and cores. Only half of the beam as seen in Fig. 18 was modelled, as it is symmetrical about its centre. Thus, symmetry boundary conditions were applied to the centre axis (parallel to the y-axis in Fig. 23, i.e. with the x–z plane acting as the symmetry

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Fig. 23 Sketch of a sandwich with core junction and the FE mesh at the tri-material corner

plane). A simple support was used at the left beam end. The transverse force was distributed on the nodes around the loading point (see Fig. 23). The loading tabs used in the experiments were included in the model. This is of minor importance for the configurations with a stiff core section in the middle of the beam, but relevant for configurations 4 and 6, where both the stress concentrations at the core junction and at the loading point influence the failure prediction. The computations were run using the assumptions of plane stress and linear elastic material properties. Geometric nonlinearities were taken into account. The simplification of a 2D plane stress model was used for a number of reasons. Most importantly, there was a limitation to the number of elements in the FE program (ANSYS Version 10 with Academic Teaching Advanced license). The discussion concerning 2D versus 3D FEA modelling was discussed in Section 3.4, and will not be repeated here. The assumption of linear elastic material properties is valid for the GFRP, which display approximately linear stress–strain behaviour over the whole range of deformations [7]. However, the Divinycell PVC foam cores show nonlinear material behaviour at higher strains. In previous studies on the failure behaviour of sandwich structures with core junctions subjected to in-plane tensile loading [7,14] an attempt was made to take the material nonlinearities of the Divinycell H-grade foams into account. This was done using the engineering stress and strain data from uniaxial tensile tests. The failure behaviour of the specimens reported in Refs. [7, 14] was dominated by large in-plane strains, so that this approach could be justified. However, this does not work for the case of general multiaxial stress states. Relating multiaxial stress and strain data to uniaxial material data as in Refs. [7, 14] requires the calculation of equivalent stresses and strains. For the nonlinear elastic-plastic model used in Refs. [7, 14] this was done based on the von Mises hypothesis, which is not physically meaningful in this context because of the behaviour of cellular polymer materials. Thus, for polymer closed cell foams yielding and failure depends on both deviatoric and hydrostatic stresses, as mentioned in Chapter 6, and the assumption of volume constancy during yielding is incorrect. To the knowledge of the authors there is no common agreement about which material model is appropriate to model the nonlinear behaviour of Divinycell PVC foams. In fact, it is not clear to which extent the nonlinear behaviour observed in the materials testing results from plasticity, nonlinear elasticity or some combination of nonlinear (visco-)elastic-plastic behaviour. On the account of the lack of data and of a reliable

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Table 10 Material data of the sandwich constituent materials Material GFRP, NCF, Œ0=90s (homogenised) GFRP, NCF, [0/90]s, (0ı layer) GFRP, NCF, [0/90]s, (90ı layer) DIAB H60 DIAB H200 Aluminium alloy

E-Modulus [MPa]

G-Modulus [MPa]

Poisson’s ratio

Ultimate stress [MPa]

Characteristic distance [mm]

Data source

Ex DEy D 24;200

–

xz D0:12

F1;t DF1;c D 465

–

In house

Ex D38;000 Ey D9;000

Gxy D4;500

xy D0:30

F1;t DF1;c D770 –

a

Ex DEy D9;000 Gxy D3;400

xy D0:30

F1;t DF1;c D170 –

a

60 310 71,700

0.32 0.32 0.33

F13 D0:7 F13 D3:3 –

22 90 –

1.37 1.16 –

[21] [21] [26]

a

Back-calculated with classical laminate theory from the overall properties obtained in tensile tests (see Section 4.4).

material model the PVC foams are modelled as linear elastic materials, which is in fact the common assumption for calculations in industrial applications. The material data used in the FEA are given in Table 10 (foam, face sheets and the aluminium insert). The GFRP face sheets were modelled with their layered structure using different material properties for the 0ı and 90ı layers. For configurations 1–4, where the face sheets are thicker and the failure of the sandwich was driven by the core material, a simplified isotropic model could have been used with only minor errors. However, in the case of configurations 5–6 where failure occurred in the face sheets it was considered necessary to model the faces as layered structures (laminates). Material properties were only available for the entire laminate in the 0ı direction obtained by tensile testing. The properties are given in Table 10. The properties of the single layers were then back calculated using the assumptions from classical lamination theory, both for the elastic and the strength properties. The loads used in the FEA were chosen to be equal to the failure loads observed in the quasi-static tests, but results can be scaled as it is a linear analysis. The FEA results were analysed in two ways. First, the predicted face stresses as well as the core stresses were read out on various paths along the length direction (x-axis) of the sandwich beams (see Fig. 23) to obtain an impression about the characteristics of the local effects. The paths were placed along the face outer surfaces and along the face/core interfaces. Figures 24 and 25 show the stresses in the upper face and the core obtained in this way, calculated at the experimentally observed average failure load for sandwich configurations 2 and 5, respectively. The stresses ¢x , ¢y and £xy denote the normal stresses and the shear stress in the global x–y coordinate system as shown in Figs. 18 and 23. Note that the plot for the core stresses is split up using two different scales for the stresses in the foam core and the aluminium insert. Note also that the core stresses were evaluated on a path parallel to the face/core interface, not directly in the interface, but at a characteristic distance rch away from the interface. This approach will be discussed later on in more detail.

Sandwich Structures with Internal Core Junctions: Load Response, Failure and Fatigue

Stress in faces σx [MPa]

50

265

Face surface Face/core interface

0 –50 –100 –150 –200

80

100

120

140

160

180

200

220

240

Stresses at face/core interface [MPa]

x-coordinate [mm] 1

0.5

τxy σx

60 40

τxy σx σy

σy 20

0 0 –0.5

–20

–40 –1 120 140 160 180 180 190 200 x-coordinate [mm] x-coordinate [mm]

Fig. 24 Face (top) and core stresses (bottom) for sandwich configuration 2 at the failure load

The plots show the characteristics of the local effects at the core junctions. Under transverse loading, the difference in shear stiffness of the compliant core and the stiff core leads to different shear angles on the left and the right side of the junction and consequently to local bending of the face sheets at the junction, along with local tension or compression of the adjacent cores. This is accompanied by a local variation of the in-plane stresses in the sandwich faces and of the shear and normal stresses in the core, as shown in Figs. 24 and 25. The extent of the area affected by the local effects decreases with increasing stiffness mismatch of the adjoining cores, and it increases with increasing face sheet thickness (as this increases the flexural stiffness of the faces). It is observed from Figs. 24 and 25 that even though the absolute and relative increase of the face stress compared to the nominal stress is much higher for configuration 2 than for configuration 5, the face stress is still well below the strength value (see the strength properties of the 0ı layer in Table 10), whereas the core shear stress reaches the shear strength. Contrary to this, for configuration 5 the local increase of

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Stress in faces σx [MPa]

0

–200 –300 –400 –500 –600 80

Stresses at face/core interface [MPa]

Face surface Face/core interface

–100

5

2.5

100 120 140 160 180 200 220 240 x-coordinate [mm]

τxy σx σy

200

σx

100

0

0

–2.5

–100

–5 140 160 180 200 x-coordinate [mm]

τxy

–200 200

σy

210 220 x-coordinate [mm]

Fig. 25 Face and core stresses for sandwich configuration 5 at the failure load

the face stress at the junction is small, but local face failure was the failure mode observed in the experiments. Another failure mode observed in the experiments was delamination between the face and the aluminium core. This was explained by a bad bonding of the face to the aluminium, which could in fact be seen from the fracture surface. However, the high tensile transverse stresses in the core indicated in Fig. 25 may also partly explain this type of failure. To evaluate the core stresses more systematically, including the use of the failure criteria discussed earlier, a second way of analysing the data was used. The stress components were read out into a so called element table. They were then postprocessed by programming the failure criteria of interest in this study: ! ! ! !

Maximum shear stress Maximum principal stress Christensen criterion (see discussion in Chapter 5) Tsai-Wu criterion (see discussion in Chapter 5)

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In addition, for the face sheets the stresses were evaluated using the third principal stress, which corresponds to a face failure of the upper (compressed) face. The maximum shear stress is defined as (¢1 –¢3 /=2 and is a simple and classical criterion to determine failure in sandwich cores. The maximum principal stress criterion was used for its simplicity, due to its physical meaning (crack opening effect of tensile stresses; the uniaxial tensile strength of the foam is one of the more accurately determined strength properties) and to follow up on the work of Ribeiro-Ayeh and Hallstr¨om [12], Bozhevolnaya and Thomsen [3, 4], Johannes et al. [7] and Jacobsen et al. [14]. The Christensen and Tsai-Wu criteria were discussed in detail in Chapter 5. For the core and the upper face sheet, failure indices were calculated according to these criteria at the experimentally observed failure load. These values were added to the element tables, from where they could be read out and visualised in contour plots. When evaluating the core stresses close to the core junction it has to be taken into account that the tri-material corners, i.e. the points where the face material and the core materials meet (see Fig. 23), represent a stress singularity within the framework of linear elasticity [24]. This means that the stresses at the tri-material corners are predicted to be infinitely high, so that the FE model cannot provide meaningful stress data in the near vicinity of the tri-material corners. Thus, the elements within a certain “characteristic distance” around the tri-material corner were excluded from the evaluation of the core stresses and the calculation of the failure indices. Accordingly, the path at the face/core interface for the evaluation of the core stresses shown in Figs. 24 and 25 was not located directly at the interface, but at a characteristic distance away from it, as discussed in Section 3.4 (Eq. (7)). The values for the characteristic distance in Table 10 were calculated using material data given in Ref. [22]. It should be noted that the study in Ref. [12] found the characteristic distance approach in connection with the point stress criterion to be reasonable for brittle PMI foam. For the semi-brittle PVC foams used in this study it was shown in Ref. [7] that the approach does not work as well. As an alternative to this, the characteristic distance could be related to the microstructure of the material and e.g. be chosen as the average cell size of the foams as in Refs. [3, 4] and as discussed in Section 3.4 herein. Table 11 shows the failure indices calculated by the various failure criteria using the stress results of the linear FE model at the respective experimental failure loads. The entries in bold indicate the component/criterion that is relevant for the failure prediction stated in the right column. The entries in italics in the table indicate values obtained at the characteristic distance from the core junction, whereas the other values were obtained away from the core junction. In some cases, higher failure indices were observed due to the local indentation at the load introduction area and the supports. However, as there was no significance of these effects in the experimental tests, the elements in those areas were excluded from the evaluation. A special case in this context was configuration 6, where failure occurred by face failure at the beam mid-span. For this configuration, the local stresses at the loading point may have had an influence on the failure behaviour by reducing the failure load compared to an ideal transverse shear loading without local indentation. Thus,

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Table 11 Failure indices according to various failure criteria at the average experimental failure load Predicted failure Face Failure load [N] mode/location Config. sheet Core 1 Min. Max. third first Max. princ. princ. shear stress stress Christensen Tsai-Wu stress 1 0.29 1.08 0.49 1.23 1.27 1,203 Core shear failure in core 1 2 0.25 1.03 0.47 1.14 1.14 1,136 Core shear failure in core 1 3 0.31 1.32 0.60 1.90 1.91 1,459 Core shear failure in core 1 4 0.42 1.04 0.48 1.20 1.21 1,167 Core shear failure in core 1 5 0.75 0.90 0.68 1.19 1.36 1,192 Failure of core 1 at junction 6 0.87 0.70 0.50 0.82 0.90 1,048 Core/face failure at beam mid-span

for configuration 6 the increased failure indices in the load introduction area are the ones stated in Table 11. For configuration 1–4 the highest failure indices were calculated for the shear dominated area in core 1 (see Fig. 18), where failure was observed in the experiments. Accordingly, the maximum shear stress criterion predicts failure very well. The failure is dominated by shearing of the H60 core. The maximum first principal stress criterion does not predict failure accurately. The Christensen and Tsai-Wu criteria predict the failure in line with the maximum shear stress criterion. Note that these criteria do not scale linearly with the failure load as the other criteria. The high failure indices for configuration 3 result from the high failure load of this configuration, which is believed to be at the upper boundary of the experimental variation caused by the variation of the foam material properties as discussed in Chapter 5. Figures 26 and 27 show contour plots of the failure index according to the Tsai-Wu criterion for configurations 2 and 5, respectively. For configuration 5 the highest values were obtained close to the core junction. Except from the maximum principle stress criterion, all criteria predict core failure close to the upper tri-material corner. However, no core failure was observed in the experiments. The explanation for this is that the Christensen and Tsai-Wu criteria predict failure mainly due to a rise of a compressive stress ¢x at the upper tri-material corner (see Fig. 25). As the strength values corresponding to compressive core failure are defined as the yield stress or plateau stress under compressive loading for these criteria, this failure can practically not be observed directly in the experiments. It may be failure in the sense that the structure/core has taken irreversible damage (core yielding), but it does not necessarily represent the initiation of a crack that

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Fig. 26 Contour plot of failure index according to Tsai-Wu criterion for configuration 2

Fig. 27 Contour plot of failure index according to Tsai-Wu criterion for configuration 5

propagates and leads to failure of the whole structure. However, it may have an influence on the stability of the face sheet close to the junction, and thus support local face sheet wrinkling and the compressive face failure observed in the experiments. It has to be noted that there is a corresponding rise of a tensile core stress ¢x at the lower tri-material corner. The failure indices by Christensen and Tsai-Wu are lower in this area, as the tensile strength properties of the foam are higher than the compressive strength properties, and they do not suggest failure. The maximum first principal stress failure index is highest close to the lower tri-material corner, but the value indicates that a tensile core failure will not occur before a compressive failure of the face sheet. It should be noted that there is no clear zone of constant stress as for configurations 1–4, as the stress field is dominated by the bending deformation rather than by shear deformation. For configuration 6, apart from the local indentation effects, the failure is predicted as face failure according to the minimum third principal stress criterion, which correlates well with the experimental results. With respect to the failure predictions for the sandwich structures with local effects investigated in this study it is concluded that the multi-axial criteria do not provide more information on the relevance of critical spots in the structure than the simple stress based criteria. In particular, the effect of compressive stresses is overestimated due to the definition of the compressive strength as the yield or core crushing strength. For sandwich configurations 1–4 with shear dominated behaviour the estimations from classical sandwich theory together with a simple maximum shear stress criterion were sufficient to predict failure correctly. The FEA and the

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experiments showed that contrary to what was suggested in previous studies [3, 4] for similar configurations, the local effects did not affect the failure behaviour under transverse loading. With respect to the FEA, however, it has to be noted that a larger characteristic distance was used than in Refs. [3, 4], which consequently lead to lower predicted stresses. For sandwich configuration 5, the investigation suggests that local effects did determine the failure behaviour, as indicated by the experimental and FEA. With respect to the FEA it is concluded that the failure prediction strongly depends on the choice of the characteristic distance for the evaluation of stresses. Unfortunately, there are no systematic studies available for PVC foam cores analogue to the studies for PMI foams in Ref. [12], which suggest that the characteristic distance approach is reliable.

4.4 Experimental Results – Part 2: Fatigue Tests The first objective of the fatigue tests was to examine whether it is possible to observe and record fatigue failure behaviour that is associated with local effects at core junctions. Further it should be examined if there is a change in fatigue life from a specimen without a core junction to a specimen with a core junction, from a core junction with low stiffness ratio to one with a higher stiffness ratio, or from one with a simple butt shape to one with an optimised shape, respectively. The fatigue tests were run in load-control using the same setup as used in the quasi-static tests. The loading was sinusoidal with a loading ratio R D Ffat;min =Ffat;max D 0:1 and a frequency of 3 Hz for configurations 1–4, and of 1.5 Hz for configurations 5–6. The maximum load Ffat;max was chosen relative to the failure loads in the quasistatic tests. Except for configuration 6 at least two different relative load levels were used. Special care was taken to ensure an approximately constant room temperature throughout the tests by means of an air-conditioning system. It is known from initial experiments and Burman and Zenkert [27] that a temperature variation of more than 5–10ı C can greatly alter the fatigue life of the Divinycell PVC foams and specimens made thereof. Four specimens for each specimen batch and load level were tested in fatigue loading, and the peak and valley load and displacement data were recorded. The high-speed video camera was set-up to be triggered when a certain displacement limit was exceeded thus recording the failure of the specimen. For configurations 1–4 the fatigue failure behaviour was as described in Ref. [27]. Very small cracks or irregularities formed in the areas of maximum shear between the centre loading bolt and the outer support rolls, generally in the centreline of the sandwich beam (the “fracture process zone”, as described in Ref. [27]). The reduction in stiffness during the test was very small until the small initial cracks formed a macroscopic horizontal crack. The final failure occurred when this crack kinked towards the face sheets and propagated rapidly along the face/core interface. The last stage of failure from macro-crack formation to final failure happened within a very short time (less than 5% of the fatigue life). Several attempts were made to track also initial fatigue failure and damage formation, but it was impossible

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to correlate this initial failure to a change in displacement. Thus, the high-speed video recording captured only the shearing of the accumulated macro-crack and the final failure. In addition to the problem of triggering the camera for the initial failure, the resolution was not sufficient to identify any small initial cracks. However, it is believed that there is an indication whether there is a connection between the local effects at the core junction and the location where the macro-crack is formed and where the crack-kinking takes place. Figure 28 shows images of the high-speed video recordings showing the typical final failure of specimen configurations 1–4. The images of the final failure correspond to the failure behaviour as described above and in more detail in Ref. [27]. The size of the fracture process zone is related to the area of maximum shear stresses and thus dependent on the loading conditions [27]. For the given specimen type and loading conditions it had about constant size. It should be noted, though, that care has to be taken when measuring the size on the failed specimens. Although not common, it is also possible that the cracks do not kink away from the end of the fracture process zone, but from a place within (as can be seen in the middle picture of Fig. 28), so that the length between the kinked cracks does not necessarily correspond to the size of the fracture process zone. Judging from the failure pictures it

Config. 1

Config. 2

Config. 3

Config. 4

Fig. 28 Images of typical final failure behaviour in the fatigue tests (configurations 1–4)

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has to be concluded that for sandwich configurations 1–4 failure was not associated with the local effects at the core junctions. This means that unlike to the tests conducted with in-plane tensile loading [5–7] and Section 3.5 herein it was found that the local stress concentrations at the junctions did not influence the fatigue failure behaviour under transverse shear loading conditions. Contrary to this, the fatigue failure behaviour of sandwich configuration 5 was influenced by the local effects. For some of the specimens damage accumulation was observed in the upper face sheet at the core junction by a gradual whitening of the GFRP, that eventually lead to final face failure at the junction, as shown in Fig. 29. For other specimens, failure occurred by delamination of the upper face sheet from the aluminium core, as shown in Fig. 30. Table 12 gives a summary of the failure modes and the fatigue lifetimes observed in the fatigue tests. Note that each line corresponds to a set of four specimens. As with the quasi-static tests the results (Table 9) in Table 12 show significant scatter. The observed failure modes and locations are also practically the same as in the quasi-static tests. Table 12 shows an interesting tendency in the recorded fatigue lives of configurations 1–4. At the same load level Ffat;max =Fstat;max the fatigue lives of the specimens with core junction (configs. 1–3) were shorter than those recorded for the specimens without a core junction (config. 4). Moreover, the recorded fatigue lives of the specimens with a higher stiffness ratio at the core junction (config. 2–3) were lower than the ones with a lower stiffness ratio (config. 1), and the specimens with the optimised 2ı  45ı shape (config. 3) displayed higher fatigue lives than the ones with the butt junction (config. 2). This could be interpreted as a correlation

Fig. 29 Local face failure at the core junction in the fatigue tests (configuration 5)

Fig. 30 Face/core delamination in the fatigue tests (configuration 5)

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Table 12 Fatigue test results

Config. 1

2

3

4

5

6

a b

Specimen batch 1

Fatigue load [N] 720

Relative fatigue load Ffat =FŒN 0.76

Loading frequency [Hz] 3

2

720

0.59

3

3

1080

0.75

3

3

720

0.50

3

4

850

0.75

3

4

570

0.50

3

5

1,100

0.75

3

5

730

0.50

3

6

870

0.75

3

6

580

0.50

3

7

890

0.75

1.5

7

590

0.50

1.5

8-2

815

0.68a /0.84b

1.5

9-2

815

0.68a /0.84b

1.5

8-1

630

0.60

1.5

9-1

630

0.60

1.5

Fatigue life [cycles] ˙ Std. dev. (rel. std. dev) 29;658 ˙ 7;506 (25%) 297;250 ˙ 32;739 (11%) 31;790 ˙ 11;271 (35%) 494;591 ˙ 15;515 (3%) 22;985 ˙ 6;352 (28%) 479;787 ˙ 84;738 (18%) 26;464 ˙ 3;262 (12%) 315;424 ˙ 25;001 (8%) 58;674 ˙ 11;629 (20%) 920;755 ˙ 137;688 (15%) 3;476 ˙ 1;709 (49%) 152;636 ˙ 137;043 (90%) 86 ˙ 70 (81%) 129 ˙ 126 (98%) 21;204 ˙ 23;002 (108%) 11;737 ˙ 8;011 (68%)

Failure mode and location Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Core shear failure of core 1 Compr. face failure at junction Delamination face/core 2 Delamination face/core 2 Delamination face/core 2 Compr. face failure at midspan Compr. face failure at midspan

Relative to the static failure load of batch 7 (failure by compressive face failure). Relative to the static failure load of batch 8-2/9-2 (failure by face/core delamination).

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between the stress concentrations at the core junctions and the fatigue lives, as it was proposed in Ref. [4]. However, there is no indication for this from the failure behaviour described earlier and neither a statistical significance of the differences in fatigue lives. Further, differences between configurations 1–3 and configuration 4 may also have been influenced by the change in the overall stiffness and thus the different deformation patterns of the specimens. For configurations 5–6 the relevant failure mode was face failure. Thus, in order to compare those configurations, the fatigue loads should be chosen in a way that provided a normalisation of the nominal face stresses at the failure locations of each configuration, i.e. the core junction for configuration 5 and the beam midspan for configuration 6, respectively. The scaling factor for the fatigue load for configuration 6 relative to the one of configuration 5 considered the relevant bending length of each configuration, i.e. 170 mm for configuration 5 and 220 mm for configuration 6, and was calculated to 170=220 0:77. Choosing the fatigue load of configuration 6 to 77% of the fatigue load for configuration 5 would provide the same nominal stress at the location of failure and allow for a direct comparison of fatigue lives. Accordingly, the fatigue load for specimen batches 8-1 and 9-1 of configuration 6 was supposed to be 0:77  890 N D 690 N. Unfortunately it turned out that this was not possible due to control problems of the testing machine and a load of 630 N had to be chosen instead as the highest possible load without control problems. Alternatively, trial tests were run in displacement control, but the constant displacement lead to a decreasing load during time. With this restriction, the fatigue load of specimen batches 8-2 and 9-2 of configuration 5 was then chosen as 815 N using the same scaling methodology relative to batches 8-1 and 9-1, as the comparison within the specimen batches 8 and 9 was the most promising approach. However, for batches 8-2 and 9-2 the failure occurred by face/core delamination and this again prevents a clear conclusion with respect to the influence of the local effects on the fatigue lifetimes. A very interesting general observation can be made from the results in Tables 9 and 12 with respect to whether to choose the fatigue load as an absolute or a relative load. From Table 9 it can be seen that specimen batches 1 and 3 of configuration 1 differed by as much as 50% in their average static failure load. At the same relative load level of Ffat;max =Fstat;max D 0:75 they yielded about the same fatigue lives, whereas at the same absolute load Ffat;max D 720 N for specimen batch 3 yielded a much higher fatigue life. This indicates that the variation in the core material properties that determines the static failure loads also significantly influences the fatigue lifetimes. This also suggests that the approach to test each specimen batch quasi-statically and to determine a relative fatigue load is meaningful to compare results across specimen batches. For sandwich configurations where the face sheet stresses are relevant for the failure behaviour, the normalisation method described above is recommended for future studies to examine the influence of local effects.

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4.5 Discussion and Conclusions (Transverse Shear Loading) This chapter has examined failure and fatigue associated with local effects occurring in the vicinity of junctions between different core materials in sandwich beams subjected to transverse shear loading. Typical sandwich beam configurations with glass-fibre reinforced plastic (GFRP) face sheets and core junctions between polymer foams of different densities and rigid aluminium were tested under quasi-static and fatigue loading conditions. In addition, reference specimens without a core junction were examined. The six specimen configurations can be separated in two distinctive groups with respect to their failure behaviour. The one group represents the most common sandwich configurations using low density semi-brittle PVC foam core. The deformation behaviour was dominated by shear deformations, and for all specimens of this group failure occurred by core shear in both the quasi-static and the fatigue tests. Here, the inherent core shear stresses under transverse loading outcompetes the local effects with respect to being most critical in terms of causing failure (cracking). For the sandwich configurations used in this paper, high-speed video recordings document the core shear failure and proof that failure was not associated with the local effects for this group of specimens. The sandwich configurations in the second group comprised a relatively thin and very strong core material and thin face sheets. The deformation behaviour was dominated by bending deformations, and for the specimens of this group the dominant failure mode was face failure. Results from FEA indicated face or core failure depending on the failure criterion. It was shown that multiaxial failure criteria proposed for foam core materials have some shortcomings when they are used for predicting the failure of sandwich structures with stress concentrations. The experimental results indicate that the local effects did have an influence on the failure behaviour for this group of specimens. Thus, the stress concentrations in the face sheets at the core junctions lead to reduced failure loads under quasi-static loading and to fatigue damage accumulation and reduced fatigue lives under fatigue loading conditions. As a consequence of this, even though the local increase in the face stresses at core junctions can be rather small (a few percent), the local face stress increases should be taken it into account for the failure prediction for sandwich configurations where failure can be expected to occur in the face sheets and not by core shear.

5 Summary and Conclusions The paper has provides and overview of the mechanisms which determine the occurrence and severity of localized bending effects in sandwich structures with internal core junctions. The paper discusses to distinct cases, sandwich panels/beams with internal core junctions subjected to in-plane loading or transverse shear loading. For both cases it is demonstrated that local effects are induced due to the material and

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geometric discontinuity at a core junction. These local effects lead to an increase of the face bending stresses as well as the core shear and transverse normal stresses. Moreover, the factors that influence the amplitude and extension of the local effects are discussed for both load cases, and the influence of the local effects on the failure under both quasi-static and fatigue loading conditions are investigated experimentally as well as through extensive FE analysis studies. It is shown that local effects induced by internal core junctions in sandwich panels/beams do influence the static and fatigue failure behaviour under certain conditions, which are discussed in detail. Acknowledgement The work presented was supported by: – The Innovation Consortium “Integrated Design and Processing of Lightweight Composite and Sandwich Structures” (abbreviated “KOMPOSAND”) funded by the Danish Ministry of Science, Technology and Innovation and the industrial partners Composhield A. – S, DIAB ApS (DIAB Group), Fiberline Composites A/S, LM Glasfiber A/S and Vestas Wind Systems A/S. – US Navy, Office of Naval Research (ONR), Grant/Award No. N000140710227: “Influence of Local Effects in Sandwich Structures under General Loading Conditions & Ballistic Impact on Advanced Composite and Sandwich Structures”. The ONR program manager was Dr. Yapa Rajapakse. The support received is gratefully acknowledged.

References 1. Zenkert D (1997) An introduction to sandwich constructions, EMAS, London 2. Bozhevolnaya E, Thomsen OT, Kildegaard A, Skvortsov V (2003) Local effects across core junctions in sandwich panels. Compos B: Eng 34: 509–517 3. Bozhevolnaya E, Thomsen OT (2005) Structurally graded core junctions in sandwich beams: Quasi static loading conditions. Compos Struct 70: 1–11 4. Bozhevolnaya E, Thomsen OT (2005) Structurally graded core junctions in sandwich beams: Fatigue loading conditions. Compos Struct 70: 12–23 5. Bozhevolnaya E, Lyckegaard A (2006) Local effects at core junctions of sandwich structures under different types of loads. J Compos Struct 73(1): 24–32 6. Bozhevolnaya E, Lyckegaard A, Thomsen OT (2005) Localized effects across core junctions in sandwich beams subjected to in-plane and out-of-plane loading. Appl Compos Mater 12: 135–147 7. Johannes M, Jakobsen J, Thomsen OT, Bozhevolnaya E (2008) Examination of the failure of sandwich beams with core junctions subjected to in-plane loading. Compos Sci Technol. Available online (doi: 10.1016/j.compscitech.2008.09.012) 8. Johannes M, Thomsen OT (2009) Examination of the failure of sandwich beams with core junctions subjected to transverse shear loading. J Sand Struct Mater. First published on June 24 (doi:10.1177/1099636209105379) 9. Allen HG (1968) Analysis and design of structural sandwich panels, Pergamon, London 10. Daniel IM, Gdoutos EE, Wang K-A, Abot JL (2002) Failure modes of composite sandwich beams. Int J Damage Mech 11(4): 309–334 11. Daniel IM, Gdoutos EE, Abot JL, Wang K-A (2003) Deformation and failure of composite sandwich structures. J Thermopl Mater 16: 345–344 12. Ribeiro-Ayeh S, Hallstr¨om S (2003) Strength prediction of beams with bimaterial butt-joints. Eng Fract Mech 70: 1491–1507

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13. ECCS – European Cooperation for Space Standardization (2004) Space Engineering Insert Design Handbook ECSS-E-30-06 14. Jacobsen J, Johannes M, Bozhevolnaya E (1993) Failure prediction of in-plane loaded sandwich beams with core junctions. Compos Struct 82: 194–200 15. Gibson LJ, Ashby MF, Zhang J, Triantafillou TC (1989) Failure surfaces for cellular materials under multiaxial loads. I: modelling. Int J Mech Sci 31(9): 635–663 16. Triantafillou TC, Zhang J, Shercliff TL, Gibson LJ, Ashby MF (1989) Failure surfaces for cellular materials under multiaxial loads. II: Comparison of models with experiment. Int J Mech Sci 31(9): 665–678 17. Deshpande VS, Fleck NA (2001) Multi-axial yield behaviour of polymer foams. Acta Mater 49: 1859–1866 18. Christensen R, Freeman D, DeTeresa S (2002) Failure criteria for isotropic materials, applications to low-density types. Int J Solids Struct 39(4): 973–982 19. Gdoutos EE, Daniel IM, Wang K-A (2001) Multiaxial characterization and modeling of a PVC cellular foam. J Thermopl Compos Mater 14: 365–373 20. Gdoutos EE, Daniel IM, Wang K-A (2002) Failure of cellular foams under multiaxial loading. Compos A 33: 163–176 R Technical Manual H-grade 21. DIAB AB (2003) Sweden, Divinycell 22. Zenkert D, Shipsha A, Burman M (2006) Fatigue of closed cell foams. J Sand Struct Mater 8: 517–538 R (2006) 23. R¨ohm GmbH & Co. KG, Data CD – ROHACELL 24. Pageau SS, Joseph PF, Biggers Jr SB (1994) The order of stress singularities for bonded and disbonded three-material junctions. Int J Solids Struct 31(21): 2979–2997 25. Hallstr¨om S, Grenestedt JL (1997) Mixed mode fracture of cracks and wedge shaped notches in expanded PVC foam. Int J Fract 88: 343–358 26. Matweb Online Materials Information Resource, www.matweb.com [viewed 15/01/2008] 27. Burman M, Zenkert D (1997) Fatigue of foam core sandwich beams – 1: undamaged specimens. Int J Fatigue 19: 551–561

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Damage Tolerance of Naval Sandwich Panels Dan Zenkert

Abstract This paper is a review of activities concerning various damage tolerance modelling and testing aspects of sandwich panels for typical Naval applications. It starts with a review of testing methods for primarily core materials and how to extract properties and data required for damage tolerance assessment. Next some typical damage types are defined and how they are modelled with the aim of predicting their effect on load bearing capacity. The paper then describes in brief how such models can used in the context of providing a systematic damage assessment scheme for composite sandwich ship structures.

1 Introduction and Background Sandwich panels have been used to construct hulls and superstructures of Naval ships for quite some time. These comprise of composite face sheets (glass or carbon) separated by a low density core, usually polymeric foam. The main driver for using composite sandwich structures in ship hulls is otherwise low structural weight. Ships are often built in short or very short series, implying that using composites can be cost effective too. Other important features are the non-corrosive material that leads to much lower maintenance costs in the lifetime of the structure. Maintenance costs in the order of 15% of those of a similar steel ship have been reported. In Naval applications, a non-magnetic hull material is very advantageous and sometimes necessary. Other important features of composite sandwich hulls are low acoustic and thermal signatures. Through the development of various naval sandwich ships, one has also found that sandwich panels are inherently blast resistant compared to steel or single skin structures.

D. Zenkert () Department of Aeronautical and Vehicle Engineering, Kungliga Tekniska H¨ogskolan, SE-10044 Stockholm, Sweden e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

279

280

a

D. Zenkert

b

Fig. 1 (a) The Landsort class mine counter measure vessel and (b) Visby class corvette (Courtesy of FMV)

Historically, sandwich structures were first used in mine counter measure vessels, mainly designed for underwater blast protection UNDEX, such as the Landsort class depicted in Fig. 1a. Having relatively low speed, these vessels became very much over-designed for normal sea loads and thus very damage tolerant under normal usage. However, ships with other mission objective, be it high speed ferries or naval craft, where speed and agility are the main priority, low weight becomes the most important design driver. More recent Naval ship structures are utilising carbon composite face sheets for increased weight saving, such as the Visby class corvette, Fig. 1b. This has also enabled the reduction of signatures, particularly radar signatures but electromagnetic shielding can also be obtained. Optimization methods are often used to minimize the structural weight. A minimum weight design is then sought after, while fulfilling constraints on stiffness, strength, natural frequencies, etc. The materials are then utilized highly even in normal usage. This implies that the hull structure will be designed for sea loads rather than extreme load cases such as underwater explosion and are thus designed towards more narrow limits. Another complicating factor appears in the design of more advanced composite sandwich hull structures using carbon composite face sheets. Carbon fiber laminates are much stiffer than their glass counterparts, while only giving limited increases in strength, if any. Thus, glass composite structures were mostly designed for stiffness whereas carbon composite marine structures to a larger extent must be designed for strength. Localized damage generally affects the global stiffness very little whereas the strength can be reduced significantly. Strength driven components in ship hulls will be much more affected by any kind of damage, in-service damage or manufacturing defects. Damage requires special attention both in the design process and in the operation of the structure. Historically, damage is dealt with implicitly by means of incorporating safety factors on the design loads. Damage is thus treated as any other unknown in the design process, such as material variability, fatigue, etc. However, large safety factors will induce a weight penalty and increased cost. As summarized in Ref. [1], the possible damage types in composite sandwich structures can be divided into two categories; manufacturing flaws and in-service damage. In the former category there are those concerned with the face sheets, such as

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281

Dry-zones and/or voids Delaminations Wrinkles or misalignment of fibres Poor curing

For the sandwich, manufacturing flaws are

Face/core debonds

Voids and inclusions in the core

Lack of bond between core sheets In the latter category, in-service flaws, the main worry is that of contact damage (dynamic and quasi-static), resulting in face sheet damage, delamination, skin/core debonds, core crushing or even complete penetration. This paper summarizes the efforts by the author concerning modeling and testing of composite sandwich damage, however limited to damage related to the core material and face/core interfaces. It will be completed with a summary of a suggested damage tolerance assessment scheme as outlined in Refs. [1, 2].

2 Fracture of Foam Core Materials Many damage tolerance models are based on fracture mechanics, i.e., the damage is assumed to have some sort of geometry so that stress field must be described as being singular. The sought after properties are fracture toughness for various modes of loading to be used as material data input in subsequent modelling of damage. The mode I fracture toughness of polymer foam cores used in sandwich structures is described in e.g. [3]. Here, two different specimen types were used; the cracked three-point bending specimen and centre cracked specimen. The former is shown in Fig. 2a. The fracture toughness under mode II loading was later pursued [4]. The specimen type used for that purpose is shown in Fig. 2b. It is a sandwich specimen where the starter crack is located in the middle of the core by placing a thin Teflon film in between two foam blocks. The crack propagation will extend in mode I, with a path indicated in Fig. 2b. The most important results from these studies are; The fracture toughness measurements are basically independent of specimen size, specimen width and crack length. The apparent mode II toughness is basically the same as the mode I fracture

a

b

P

a

P

a

Fig. 2 (a) The cracked three-point bend specimen and (b) the cracked sandwich beam specimens. Crack propagation path indicated by dashed lines

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Fig. 3 (a) The double modified cantilever specimen and (b) the cracked sandwich beam (CSB) specimen

toughness. It was also found that the fracture toughness scales with relative density in the same way as the tensile strength. For Divinycell and Rohacell foams these relations are ten D C1 N1:1 ; KIc D C2 N1:1 (1) where N is the relative density of the foam (foam density divided by the density of the solid polymer). In Ref. [5], Shipsha et al. performed a comparative study of fatigue crack growth in foams using both compact tension specimens and a modified double cantilever specimen. The latter is depicted in Fig. 3a. This also measure the mode I fracture toughness, but with the crack extending along the face/core interface. Actually, the measured fracture toughness is still the same as when measured using the specimens shown in Fig. 2 at least as long as the crack grows in the core, along the interface. This is usually the case unless the interface is very weak. In order measure true mode II crack initiation and propagation one should use a specimen that simulates crack propagation along a face/core interface. The cracked sandwich beam (CSB) specimen was first proposed by Carlson [6] and then used in Ref. [7] to measure interfacial fracture toughness. The interfacial fracture toughness is significantly higher than the mode I toughness, mainly due to the fact that the crack is forced to grow in mode II. Østergaard et al. [8] recently devised a testing method to measure the interface toughness at various mode mixities, ranging from pure mode I to pure mode II. They found that the toughness remain approximately equal to the mode I fracture toughness as long as the mode II component is relatively small (mode mix down to 30ı ) and then increases. In Refs. [5,7], tests were also performed using fatigue loading in order to measure fatigue crack propagation, both in mode I and mode II. The important findings in these papers are; The crack propagation rates in mode I is very high in structural foams and, as expected, the mode II propagation rates are lower than in mode I.

3 Disbonds in Sandwich Beams Debonds can occur either from manufacturing or in cases from in-service impact loads. In cases where two foam core blocks are bonded together to build the required thickness, debonds can occur as depicted in Fig. 4a. A more common place would be at the face/core interface, as shown in Fig. 4b.

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Fig. 4 (a) Sandwich beam with mid-plane core debond and (b) sandwich beam with face/core interface debond. Schematic test-up above and photos of failure modes below

Mid-plane debonds in sandwich beams, Fig. 4a, was studied in Ref. [9]. The debond was modelled analytically by means of beam theory where the energy release rate could be calculated by means closed form solutions for the total potential energy of the beam. The model was validated with FE-analyses and the correlation was very good. For validation, beams were manufactured with simulated debonds by means of thin Teflon films, and the beams were tested in four-point bending with the debond located between one inner and outer support, i.e., in the shear region. In order to predict the failure loads, the fracture toughness as measured using specimens of the type shown in Fig. 2 were used. The failure mode, Fig. 4a, indicates that the crack propagates in mode I. In a similar study [10], interface debonds were studied. The same type of analytical model was derived, however using a simplified fracture mechanics approach. Again, the model was validated with FE-simulations with very good correlation. The failure mode prediction has to be performed for both crack fronts. The left crack front in Fig. 4b exhibits the same type of loading as the CSB specimen depicted in Fig. 3b. The crack “wants” to grow up towards the face sheet and is thus forced to grow along the interface and it is then assumed to start growing when the stress intensity equals the mode II interface toughness. The right crack tip, on the other hand, exhibits almost the same type of loading, but the shear stress has the opposite sign. The crack “wants” to grow in the mode I direction, which in this case is downwards into the core material, which will happen when the stress intensity equals the mode I toughness. Since the latter case predicts a lower initiation load this is what occurs, as seen in Fig. 4b. A somewhat different damage type is a flawed butt-joint, as shown in Fig. 5. In the production of large sandwich structures, the core is often used as mould. The foam core is cut and bonded together to the correct shape or geometry using adhesive of putty. A large number of butt-joints are thus created, bonding the core blocks edge-to-edge. A possible manufacturing defect is when this bonding fails leaving a cavity between the core blocks.

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P

P

Fig. 5 Sandwich beam with flawed butt-joint; schematic of test set-up and photo of failure mode

In Refs. [11–13] this damage type was modelled and experimentally validated. The modelling was performed using FE-simulations in order to calculate so called generalised stress intensity factors. For a crack, the stress field has a so called square root singularity. The geometry of the butt-joint damage can be treated by a generalised fracture mechanics approach by computing the stress-field close to the material corner created by the flaw. The stress fields for a crack and a corner are given in Eq. (2) where  is the so-called strength of singularity. For a crack in homogeneous media,  equals 0.5, but for a corner it takes other values, smaller than 0.5.  / Kr 

(2)

The actual value of  was computed using a theoretical approach in Ref. [14] using potential theory. For the configurations tested it was found that  was 0.64 and 0.69, implying that the strength of the singular stress field is weaker than for a sharp crack. These values were validated by using high-fidelity FE-calculations from which the stress intensity factor K was also calculated. In order to make failure load predictions for beams of the kind depicted in Fig. 5 one also needs the corresponding value of the fracture toughness and this has to be measured for a specimen configuration having the same singularity as the tested flawed butt-joint. This was done using modified block shear test [14]. Testing of beams with flawed butt-joints was then performed in four-point bending. The failure predictions were similar to that of the other beam damage types. The load at which the stress intensity K equals the measured fracture toughness was taken as the failure load. Two different material combinations (both having glass reinforced face sheets, but with different cores), each with two different widths of the butt-joint were tested. The predictions were in all cases very good resulting in a residual strength in order of 50% of the undamaged configuration. Both interface debonds and flawed butt-joints were also tested in fatigue load using the same type of material configurations and test set-up [15, 16]. In a similar study [17] also beam configurations with honeycomb cores were tested in fatigue. The result of this was that the effect of the damage was the same in fatigue as for quasi-static loading, implying the SN-curve for an undamaged sandwich beam could simply be reduced with a factor to obtain the SN-curve for the damaged case, and that the factor equals the static strength reduction.

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4 Impact Damage in Sandwich Beams Perhaps one of the most common threats to structures in general is that of foreign object impact. For sandwich structures this threat is particularly important since a sandwich panel usually has thin face sheet in order to save weight. However, in the impact scenario not only the face sheet can suffer from damage but also the underlying core. Face sheet damage could be matrix cracking, delamination and fiber failures. Of particular interest is the damage occurring in the core at an impact event. Depending on the foam core type, two different failure types occur as defined in Fig. 6. The first one implies that the core is compressed and densified, but not cracked. Upon unloading the core “holds down” the face sheet leaving a residual dent in the face sheet. The second scenario occurs for more brittle cores. At unloading after the impact event, the core fractures in tension leaving a cavity in the core resembling a crack. However, the zone of crushed core, as indicated by the dashed line in Fig. 6, is larger than the cavity. The size of the residual dent can be modelled in the quasi-static impact case [18, 19]. This analysis model is based on the assumption that the core crushes progressively so that the strain in the core is either linear elastic or at the densification strain. The dynamic impact event creating a core cavity (Fig. 6b) requires more elaborate analyses [20]. One very important aspect in the modelling of core damage is to consider the properties of the crushed foam core. The residual strength prediction models of foam core sandwich beams with impact damage used in Refs. [21–24] are all based on detailed descriptions of the crushed core zone, both in terms of geometry and the properties of the crushed core material. The crushed core properties were obtained from experiments where the core was pre-crushed in compression, allowed to relax and then tested in tension, compression and shear in various directions [22]. It was found that the as-crushed moduli and strengths dropped significantly compared to non-crushed foam and that the crushed foam became highly orthotropic. The geometry of the impact damage requires two different loading conditions to be studied. Since the foam is crushed and has significantly reduced strength, and in some cases there is even a cavity underneath the face, shear loading is relevant, just as for an interface crack. This case was experimentally investigated in Ref. [22]

a

b

Fig. 6 Impact induced core damage in sandwich beams. (a) Crushed core with indent in face sheet and (b) crushed core with cavity. Dashed lines indicate extension of crushed core

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b

Fig. 7 Idealized model with parameters for the modeling of impact damages. (a) Crushed core with indent in face sheet and (b) crushed core with cavity (From Ref. [24])

and modelled in Ref. [23] for a configuration with a core cavity. The experimental set-up used a four-point bending configuration, as depicted in Fig. 4, with the impact damage located between an inner and outer support, i.e., in the shear zone. The modelling approach used FE-analysis with a detailed description of the damage following Fig. 7, where different parameters could be varied depending on damage type and impact energy [23]. As a first conservative approach, the damage was modelled assuming a crack extending from the cavity through the crushed core zone, along the interface, having the crack tip at the position where the crushed core meets the non-crushed at the face/core interface. A second, more elaborate, approach was to model this crack assuming crack bridging stresses. The second approach gave excellent comparisons with the experimental results for a range of impact energies [23]. Actually, the failure mode very much resembles that of an interface crack, as shown in Fig. 4b, hence the assumed crack in the crushed core zone. The second relevant load is in-plane compression, or bending of the beam with the damage located near the compressed face sheet. Compression loading of the face sheet is particularly important due to the features of the damage having a perturbed face sheet due to the residual dent resting on a crushed core with reduced stiffness, or partly unsupported (cavity case). The same FE-model was used as in the shear, however, without the assumed crack along the interface in the crushed core zone. Buckling of the face was now sought after which was calculated using geometrically non-linear FE-analyses using Riks-Wempner arc-length control algorithm. Both bending and in-plane compression load cases were studied numerically and experimentally. The correlation between simulations and experiment were less satisfying for this case, with the numerical predictions being 20% non-conservative. In Ref. [24] a configuration having a more ductile core was used leading to an impact as in Fig. 7a, i.e., without any core cavity. The failure mode of such configurations is characterized by stable dent growth followed by a more buckling type unstable dent growth. In order to capture this properly, one still needs to accurately model both the residual face sheet dent and the geometry and material properties of the underlying crushed core. The fatigue behaviour was also pursued in Ref. [25]. The configuration was an impact damage with a cavity as shown in Fig. 7b. The loading was in shear

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Cavity

Fatigue of crushed core

Initiation and crack kinking

Fatigue crack growth (failure)

Fig. 8 Fatigue failure of impact damaged sandwich beam

(using four-point bending tests). The effect of the crushed core was evident in the tests. A fatigue crack started at the tip of the cavity and grew rapidly along the face/core interface in the crushed core until it was arrested when reaching the undamaged core, see Fig. 8. This process took less than 20% of the fatigue life. During the rest of the fatigue life the crack remained (visually) dormant until the very last stages of the fatigue life when a crack initiated and kinked down into the core material causing failure of the specimen.

5 Interface Disbonds in Sandwich Panels Damage in panels is considerably less studied in the literature. This is much more relevant, however more difficult to pursue, both in terms of modelling effort and costs for experiments. Perhaps not always widely known is that the effect of damage in sandwich panels is often quite different than in beam specimens. The reason for this is mainly that damages tend to cause complete and catastrophic failure in beams, whereas damage progression often is stable in panels. This also implies that the effect of damage usually is much smaller in panels than in beams. The effect of large circular debonds in sandwich panels under hydrostatic loading were studied in Refs. [26, 27]. The rationale for studying hydrostatic pressure stems from the design rules for hull panels which are designed for a load corresponding to a uniform pressure. A face/core debond in a sandwich panel can be positioned in different places in the panel, e.g. in the middle of the panel (Fig. 9a) or across the panel boundary (Fig. 9b). The panel boundaries should be interpreted as the position of underlying stiffeners on which the panel rests. An attempt to analytically model the case in Fig. 9a was performed in Ref. [26] by the same type of approach as for the beam case and assuming circular symmetry. This worked reasonably well but not with great accuracy. Models based on FE-simulations were also performed based again on the same approach used

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a

b

Fig. 9 Schematic of face/core debond in a sandwich panel with (a) debond in the middle of the panel or (b) across the panel boundary

for beams [10]. The failure load predictions were reasonably good. Tests of large sandwich panels (800 by 800 mm) were performed using a water bladder technique [28]. By this procedure, panels can be tested under uniform pressure loading. The debonds were simulated with a thin Teflon film placed at the face/core interface during panel manufacturing. The debonds were reasonably large, with diameters ranging from 200 to 500 mm. Despite having so large debonds, the effect of the damage was surprisingly low and the reduction in load bearing capacity was only about 35% for the cases tested with a more ductile type core (Divinycell). The reason for this is that the transverse force in a simply supported panel subjected to uniform pressure is small in the middle and large at the boundaries. The transverse force creates core shear stresses which drives the debond crack to grow. For panels with more brittle cores (Rohacell) the reduction in load bearing capacity was larger. In a subsequent study [27] the debond was placed at the boundary of the panel, Fig. 9b, since this is the location of maximum transverse force in the panel. As expected, the FE-analysis showed that the stress intensity is much higher for this case and the experiments exhibited a much lower failure load than for a debond in the middle of the panel. There are a few important conclusions from this work; first of all, due to the loading, contact between the face and the core along the debonded area must be considered in the numerical analysis. A second perhaps more important practical issue is that not only the size of the debond influences the severity of the damage, but also position. There is a great different between the two cases depicted in Fig. 9. Furthermore, there is a difference whether the debond is located at the upper or lower interface. The studied cases has the debond in the upper interface (Fig. 9) towards the side of the pressure load where the face laminate is in compression. For this case, the sign of the shear stress is such that debond crack will grow along the interface, similar to the test specimen in Fig. 3b. The mode of failure has a high fracture toughness. If the debond is placed in the lower interface, towards the face laminate in tension, the sign of the shear stress changes. For this case, the debond crack will kink into the core material, a mode with low fracture toughness.

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Fig. 10 Schematic of face/core debond in a sandwich panel subjected to in-plane compression load

The situations are therefore in analogy with the interface debond in a beam according to Fig. 4b, where one crack tip want to grow along the interface and the other kink into the core, due to the different signs of the shear stress (and thus KII ). Jolma et al. [29] performed a similar study with debonds placed in the middle, across a boundary and at a corner. They also used a similar experimental set-up applying uniform pressure to the panels. Their analysis models were also based on FE, but with higher accuracy and a better fracture mechanics model including the effect of mode mixity. Their failure mode predictions were in all cases excellent. In-plane compression is perhaps an even more critical load case for sandwich panels with interface debond damage. The scenario is schematically illustrated in Fig. 10. As the in-plane load is increased, the debond will start to “buckle” out. This deformation creates a non-uniform stress intensity along the debond crack front. At some load, the debond crack will start to propagate, stable or unstable, causing the panel to fail. The kinematics of the problem is geometrically non-linear so that the stress intensity varies non-linearly with the applied load. This problem has been thoroughly investigated by Berggreen [30–32]. He studied sandwich panels with circular interface debonds subjected to various types of in-plane compression loading. The numerical analyses uses FE and with a highly refined mesh towards the debond crack tip circumference which enables not only the calculation of the stress intensity along the crack front, but also the mode mixity. The analysis results were combined with values for the interface fracture toughness, still regarding the current mode mixity, from which failure load predictions were made. These predictions were compared with experimental results obtained from compression tests of fairly large (500 by 500 mm) sandwich panels with simulated interface debonds. The correlation between the numerical exercise and the experiments were overall quite good. There are several reports from various sources that if a debond occurs in a sandwich panel, no matter how, this disbond will be perturbed due to an internal pressure. The pressure build-up is caused by outgasing of the core material [33]. When closed cell foams are manufactured, there will be an entrapped overpressure (above atmospheric) in each cell. It is rationalized that this pressure inside individual cells of the foam core will diffuse to volumes of lower pressure, such as a debond. This

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means that with time, the pressure inside the debond increases to that similar to the cells. Then a blister is formed defined by a debond that due to the internal pressure “bulges” out, see Fig. 10. The blister could potentially propagate in size just from the internal pressure build-up alone, but that would normally be a rather slow and stable process. The main worry is that it can propagate unstably under external loads, particularly in-plane compression that increases the deformation of the debond. The analysis and testing of blisters is somewhat different than analyzing debonds without pressure. It involves applying a certain pressure inside the disbond, keeping the entrapped air volume in the blister while applying the external load. Numerically, this can be done by using FE-analysis. The debond must be modeled with a small gap which is filled with fluid type elements. In stage 1, the pressure in the fluid is increased. This creates a volume change of the entrapped air, causing the debond to buckle out a certain amount. The next load step is to apply the external in-plane compression load while keeping the entrapped air volume constant. Due to the applied load the debond out-of-plane displacement will increase, but the pressure inside the blister will decrease due to the increased volume. In each load step the stress intensity (or energy release rate) must be evaluated along the debond crack front. All this must be performed using geometrically non-linear analysis. In terms of experimental validation the same procedure applies. A technique to apply pressure inside a debond is the following. A sandwich panel is manufactured with a debond simulated using a thin Teflon insert. A small hole is drilled from the back side of the panel to the film insert without penetrating the face sheet. Thereafter a pressure gauge, equipped with an air inlet and a valve, is adhesively bonded over the hole on the backside. This means that the pressure can be applied in the artificial blister, the valve shut (which ascertained constant volume in the blister), and the pressure monitored during the in-plane compression loading. The signal from the pressure gauge is monitored both during the initial pressurizing stage and during the in-plane compression stage. The tests are carried out the same way as numerically analyzed. The panels are mounted in a compression test machine. First, the blister is filled with compressed air to a specified pressure. The air valve is then closed to maintain the given volume of air in the blister. The panel is then subjected to a uni-axial in-plane compressive load. An example of such analyses and tests are shown in Fig. 11. The analysis was performed using ABAQUS with the scheme described above. Figure 11 shows a plot of the pressure drop during in-plane loading. Here the pressure and load are plotted versus time (strain step) from a panel with a 50 mm diameter blister and an initial internal pressure of 0.2 MPa. The panel has glass/vinylester face sheets and a 50 mm thick 100 kg=m3 foam core. The lines are taken from the experiments. The nominal compressive strain is taken from Digital Image Correlation (DIC) measurements and the pressure readings from the pressure gauge. It can clearly be seen that first a quick increase in pressure is obtained. After that the in-plane loading compression loading starts and the pressure starts to drop due to the increase in blister volume. The dots in Fig. 11 are from the ABAQUS simulation of the same case. In order to compare, the pressure inside the blister is read from the ABAQUS output at given load increments and not for given time steps. The markers are thus created by taking

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Fig. 11 Blister internal pressure and applied compressive load versus time (strain step), from experiment (lines) and from ABAQUS simulation (dots)

the nominal compressive strain at a given increment and placing them on top of the experimentally measured points (square markers) and then plotting the blister pressure for that particular load increment (diamond markers).

6 Impact Damage in Sandwich Panels In Ref. [34] the residual strength of impact damaged sandwich panels was studied. The configuration was panels with fairly thin (2.4 mm) glass/vinylester face sheets on a 50 mm Rohacell foam core (50 mm thick). The panels were subjected to low-velocity impact from a 25 mm diameter spherical indentor with various impact energies ranging from 10 to 60 J. The resulting damage was of the type depicted in Figs. 6b and 7b, i.e., core crushing, a core cavity and a remaining dent in the face sheet. For all impact energies the impact damage geometrical parameters were measured (see Fig. 7b) by destructive sectioning. The face sheet laminate remained almost intact and was assumed so in the modelling. In-plane compression tests were performed quasi-statically on panels having different impact energy damages. The panels had strain gauges mounted and the surface with the damage was also monitored using a Digital Image Correlation system set-up to measure also out-of-plane displacement by the use of two digital cameras. From this, the dent displacement as function of applied load could be measured. A finite element model was constructed using the damage parameters (dent depth, crushed core geometry, etc.) as input parameters as shown in Fig. 12a. As for the beam studies with similar damage geometry [23], the core was assigned orthotropic properties for crushed core.

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b

Fig. 12 (a) ABAQUS-model used for sandwich panel with impact damage and (b) measured dent profile in the loading direction. Arrow in upper right corner indicates tensile core stresses (From Ref. [23])

As the loading is applied the face sheet dent will start to increase, i.e., move downwards into the core. At low loads this is a fairly stable process both in the analysis and the experiments. Near some critical load, the dent displacement will increase in speed and the panels eventually failed by the face tearing off the core apparently by tensile failure. The dent profile is shown for a few load steps in Fig. 12b where this is clearly seen. In this case the failure load prediction was performed using two strength criteria; compressive failure of the face and tensile rupture of the core. The rationale was that either the face sheet fails in compression as compressive stresses build up in the face sheet in the middle of the dent due to the compressive load with a superimposed bending stress due to dent growth (which causes the laminate to bend). The tensile failure of the core stems from the fact that tensile stresses build up in the core above and below the dent due to the bending of the dent. This is seen in the dent profile in Fig. 12b and is indicated by an arrow. Using this in the FE-model led to the prediction shown in Fig. 13. The predictions thus indicate tensile core rupture, which also was verified experimentally. The failure load predictions for all the impact energies were fairly accurate. In a similar study, reported in Ref. [2], indentation damage was studied on sandwich panels with carbon/vinylester face sheets on a 50 mm Divinycell core material. The damage geometry in this case has the features of that in Figs. 6a and 7a, i.e. crushed core, residual dent but without any core cavity. The panels were subjected to a transverse load from 150 mm diameter steel indentor to varying indentation depths creating various sizes of crushed core zones and residual dents. The panels were then placed in a testing machine and subjected to an in-plane compression load until failure. Again, the panels had a number of strain gauges mounted and DIC was used to measure out-of-plane displacements in the side of the indentation damage. An FE-model was built up and parameterized using the damage parameters indicated in Fig. 7a. The model is shown in Fig. 14a. The same failure criteria was used as in the previous case; compressive failure of the face sheet in the middle of the dent, and tensile rupture of the core.

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Fig. 13 Tensile stress in the crushed core and compressive stress in the middle of the face sheet in the dent center (From Ref. [23])

a

b

Fig. 14 (a) ABAQUS-model used for sandwich panel with indentation damage and (b) measured dent profile in the loading direction (From Ref. [2])

As the in-plane load was applied, the dent started bending inwards towards the core. The process was fairly stable until the near critical load when the dent growth became unstable. This is shown by various measured dent profiles in Fig. 14b. The DIC-measurements are shown in Fig. 15 for zero-load (only showing the residual dent after indentation) and a number of loads just prior to failure. As seen herein, the dent growth is very rapid just before failure and the panel fail by compressive failure of the face sheet laminate. The same type of panels was used to study two types of impact damages, one resulting from a blunt impactor and one form a sharp [2, 35, 36]. These are shown in Fig. 16. Two different thickness of quasi-isotropic lay-up face sheets were used with thickness 1.8 and 5.4 mm with a 60 mm 80 kg=m3 and 50 mm 200 kg=m3 foam cores, respectively. The blunt impact damage was created using a spherical head

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Fig. 15 Digital image correlation measurements showing out-of-plane displacement (From Ref. [2])

a

b

Fig. 16 Damage types occurring from (a) blunt object impact and (b) sharp object impact (From Ref. [2])

impactor with a diameter of 25 mm. The sharp impact damage was created using a pyramid shaped impactor. The energy levels were chosen so that one level created a barely visible damage (BVID) and a second higher level gave clearly visible damage (VID). The blunt impact damage consisted of a very small dent with some minor surface fiber breakage in the VID case. The underlying core showed a small zone of crushing, but apart from that there was no other visible damage indicated. A cross-section of the impact zone is shown in Fig. 16a. However, ultrasonic C-scan investigations revealed significant overlapping delaminations with a diameter 2R D 15–30 [35]. The sharp type impact created a slit in the face sheet, as shown in Fig. 16b. The projected length of this slit was in the order of 2a D 15 to 25 mm, with the higher value for the higher impact energy.

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The panels were then tested in in-plane compression while using a DIC-system to measure the in-plane strain field. Some tests were interrupted just prior to failure [35], and small pieces of the laminate close to the impact point was cut out, ground and polished stepwise in the thickness direction in order to identify location of failure. Even though the panel had not yet failed, this fractographic investigation revealed numerous local compressive failures manifested by kink bands (microbuckling failures). These appeared mainly in the 0ı plies, appearing to start at the edge of the impact damage extending outwards to the panel edges. Kink-band lengths of over 20 mm were found. The DIC-measurements revealed similar evidence. The surface strain mapping revealed clear indications of stable micro-buckling growth before final failure with extensions similar to what was found through fractography. Two such DSP strain map plots are shown in Fig. 17 for a thin panel with a BVID (30 J) blunt impact damage. The first (Fig. 17a) shows the axial strain at 120 kN load, which corresponds to approximately half the failure load. The second strain map (Fig. 17b) shows the axial strain just prior to failure (at 216 kN, or approximately 200 MPa nominal stress). The localized strain (dark area) corresponds to an axial strain of 1%. As seen, strains localize near the damage and damage progresses in a stable manner during the loading of the panel. No visible damage progression could be seen on the surface of the panel until complete fracture. For the predictions, the code Composite Compressive Strength Modeller – CCSM [37], which implements progressive micro-buckling analysis, was used for the blunt impact cases by assuming an equivalent hole representing the damage. The same code was again used for the sharp impact tests, but then an equivalent crack model was used, since a clear crack was produced by the impact. The crack length used in the model was the projected length of the slit, perpendicular to the load direction. In order to find the necessary input data for the CCSM model, un-notched compression tests were performed on the laminates using the Wyoming modified IITRI test set-up. To find the fracture energy, small panel specimens (150 by 150 mm) were manufactured with central holes cut. These were tested in compression from which the fracture energy can be deducted by the use of CCSM. The predictions obtained

Fig. 17 Axial strain map (vertical D load direction) (a) at 120 kN load and (b) just before failure (at 216 kN) for a thin panel with a blunt impact damage. The dark area corresponds to compressive strains of 1% (From Ref. [2])

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from this gave excellent results in terms of failure loads and reasonable accuracy in the kink band lengths. In a sub-sequent paper, Edgren et al. [38] performed a similar but more refined analysis of the same experiments.

7 Damage Tolerance Scheme for Naval Sandwich Structures From a damage tolerance perspective there are several important aspects to consider. One cannot guarantee that damage or defects can be detected, both since NDI of sandwich structures is difficult due to their multi-material constitution and that ship structures are indeed very large. From a practical point of view one must therefore consider only damages that are barely visible or clearly visible. The question often asked is – “Does all damage have to be repaired?” and “Is this damage critical for the integrity of the structure?”. Decisions must be taken on corrective measures relatively quickly once damage has been found. Large damages will obviously be found and often requires immediate repair, either for structural reasons or for functional reasons (water tightness, etc.). The problem addressed herein concerns barely visible impact damage (BVID) and visible impact damage (VID) with the aim of finding how much such damage will affect the load carrying capacity of a panel or component in the ship. Large damage is thus addressed but the focus is on damage that may exist without being detected, or damage that is small enough not requiring repair for functional reasons. Such damage may, or may not, require repair depending on its size, its type and the type of loading at its location. In this paper, only damage related to the core and face/core interface are investigated. From a practical viewpoint all possible threats and damage types must be included in order to provide a comprehensive damage assessment system. To do this one needs not only the information on the localised strength reduction due to damage for individual panels but a more elaborate scheme for evaluating the influence of the damage on the structural performance and functionality of the ship as a whole. In the SaNDI project such a scheme was developed that is fairly generic in terms of damage types and ship structures, although the designs and design requirements may vary significantly. The damage assessment procedure for cases where strength is the main consideration is basically as follows:

Estimate the strength reduction caused by the damage or defect.

Determine the allowable strength reduction based on the original design assumptions, operational envelope, etc.

Compare these. If the residual strength is smaller than the allowable value, consider the possibilities for restricting the operational envelope and/or accepting a lower safety factor until repair can be effected.

If this is not sufficient, carry out an emergency repair or take other emergency measures as necessary.

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To do this, one may need to consider the damage in the local, panel and global (ship) contexts, as illustrated in Fig. 18. The scheme as a whole is shown schematically in Fig. 19.

Ship

Panel

Local

Fig. 18 Damage illustrated in local, panel and global (ship) contexts (From Refs. [1, 2])

Fig. 19 Residual strength approach for local panel strength and global ship strength (From Refs. [1, 2])

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For damage confined to a single panel, we first consider the influence of the damage on the affected panel or component by estimating a strength reduction factor for the panel, Rp . This is then compared with the allowable strength reduction for the panel, Rpa . If the panel suffers from a reduced load bearing capacity, we need further to consider any implications of this at the global (ship) level and estimate any reductions in the global ship strength. For this purpose it is possible to calculate a ship strength reduction factor Rs , which again must be compared with some allowable global strength reduction factor, Rsa . A division is made considering the level of damage, ranging from small local damage confined to a small part of one panel (level 1) to extensive damage affecting larger areas of two or more panels (level 4). For level 1 damage, which applies to the damage cases considered herein, the size of the damage is small compared to the panel size so that the stress distribution within the panel is unaffected by the presence of the damage (no stiffness changes). The assessment is then performed as follows:

Determination of the local strength reduction factor Rl that quantifies the reduction in far-field stress or strain at failure. The value of Rl should preferably already been calculated a priori, and recorded in tables or graphs.

Determination of a sensitivity factor Sp that accounts for the location of the damage in relation to the stress field in the panel for the real loading case.

Combination of these factors to give the panel strength reduction factor Rp D Rl Sp or 1.0, whichever is smaller. The sensitivity factor Sp , which is more correctly referred to as the local location and load case factor, also provides a means of seeing immediately what are most likely to be the critical locations of damage on a panel. It is defined as follows: Sp D

Pali Papi

(3)

where Pali is the value of load that causes the critical stress or strain component at the damage location to reach its maximum allowable value ignoring the damage, and Papi is the maximum allowable value of load on the intact panel. For face sheet damage, the critical stress or strain component is usually assumed to be the inplane compressive stress or strain at the point in question. Sp can never be smaller than unity. As an example, suppose a panel under uniform lateral pressure is designed so that it reaches the allowable limit for core shear stresses before the allowable stresses for face laminate failure. (This will often be the case for panels designed to classification rules, for example, if the minimum thickness requirements result in an increase of laminate thickness relative to the basic strength requirements.) Then Papi is the pressure load at which the allowable core shear stress is reached. Suppose that there is a local impact damage that has reduced the local in-plane compressive strength of the face laminate so this has to be checked at the damage location. Then Pali is the pressure applied to the panel at which the allowable compressive stress (or strain) in the face laminate at that location reaches its maximum

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allowable value, also calculated for the intact panel. Note that all these quantities are calculated for the intact panel. This means that maps of Sp values can readily be drawn for panels with given dimensions and lay-ups for simple load configurations like uniform lateral pressure. There is normally no need to consider an allowable strength reduction if Rp D 1. If Rp <1 the possibility of accepting a reduction of panel strength must be considered unless the damage can be fully repaired immediately. If Rp < 1, the global strength reduction Rs must also be evaluated. For Level 1 damage, and also for some cases of slightly larger damage to a single panel defined as Level 2 damage, it is possible to neglect redistribution of stresses between panels (and other elements) in the structure when estimating the influence of the damage on the global strength. This enables Rs to be found from Rp by a procedure analogous to that for deriving Rp from Rl . For larger damage cases (Level 3 or 4), alternative procedures must be used as discussed in Ref. [1]. In some cases it is not possible to accept any reduction in the global or local strength of the structure. However, not all parts of the structure of a ship are highly stressed. In many cases a given panel may be exposed to a maximum loading that is lower than the allowable value because the design gave more than the minimum required reserve of strength, i.e. there is a lower utilization of the panel than the maximum that is allowed. This is often due to the fact that a limited number of standard sandwich lay-ups are generally used in any given vessel. In such cases it will normally be acceptable to reduce the panel strength by an amount that reflects this extra reserve of strength in the intact structure. The same may apply at the global ship level if the ship has been generally over-designed against the global loads (e.g. because the local load requirements were more severe). These aspects are dealt with here by using the utilization factors Up and Us . The panel utilization factor, Up , is defined as the ratio between the ratio of the maximum load applied to the damaged panel under extreme load to the maximum allowable value of that load for the undamaged panel. The utilization factors Us is defined the same way but for the global load acting on the whole structure [1, 2]. Normally the allowable strength reductions for panel and global strength will be set equal to the respective utilization factors: Rpa D Up and Rsa D Us There are two main additional considerations that may make a strength reduction acceptable:

Reducing the loads (relative to the original design) by restricting the operation in some way.

Accepting a reduced factor of safety in the interim period until a repair is effected. The first of these leads in effect to a lowering of the utilization level in that the extreme design loads are now decreased. The second leads to a lowering of the utilization level in that the allowable loads are increased.

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Fig. 20 Schematic illustration of assessment process (From Ref. [2])

Establish Rp

Rp<1?

Yes

Yes

No, Rp=1

Possible that Rs<1?

Estimate Rs

Estimate Up, Us, Rpa, Rsa

Rp>Rpa and Rs>Rsa? No

Yes Reduce design loads

No

Yes

No further action

Possible to reduce design loads?

Re-calculate Rpa, Rsa No Rp>Rpa and Rs>Rsa?

Yes

No

Possible to reduce safety factors?

Impose operational restriction. Repair later

Yes Reduce safety factors Re-calculate Rp, Rs, Rpa, Rsa

No

No

Emergency repair or proceed to harbour

Rp>Rpa and Rs>Rsa? Yes Take necessary precautions. Repair later

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Note that the procedure described so far deals only with strength considerations. Considerations of functionality (which may include that of stiffness as embodied in deflection criteria) must be made separately. Functionality requirements, such as









Water tightness Weather tightness Smoke-tightness Fire division effectiveness Equipment functioning (machinery, instrumentation) Signatures and shielding Other functionality

may override any repair decisions based on load carrying capacity. Thus, although a damage does not require repair from a structural point of view, functionality may still require repair. Finally, decisions about corrective measures have to be made on the basis of comparing the strength reductions Rp and Rs with their allowables Rpa and Rsa . Figure 20 shows an example of a flow-chart for such a decision-making process, taking account of only the strength reduction aspects. Acknowledgement The work summarised in this chapter has been ongoing since the 1980s with various support. Thanks are due to The Swedish Defence Materiel Administration and Mr. Anders L¨onn¨o for funding parts of this. The Nordic Industrial Fund (NI) provided support for a long time in the 1990s. A lot of the more recent work was performed within the WEAG-project saNDI in collaboration with Prof. Brian Hayman at DNV. During most of the past 10 years invaluable support has been provided from the Structural Mechanics Programme of ONR through programme officer Dr. Y. Rajapakse.

References 1. Hayman B (2007) Approaches to damage assessment and damage tolerance for FRP sandwich structures. J Sand Struct Mater 9: 571–596 2. Zenkert D, Bull P, Shipsha A, Hayman B (2005) Damage tolerance assessment of composite sandwich panels with localised damage. Compos Sci Technol 65: 2597–2611 3. Zenkert D, B¨acklund J (1989) PVC Sandwich core materials: Mode I fracture toughness. Compos Sci Technol 34: 225–242 4. Zenkert D (1989) PVC Sandwich core materials: fracture behaviour under mode II and mixed mode loading. Mater Sci Eng A108: 233–240 5. Shipsha A, Burman M, Zenkert D (2000) On mode I fatigue crack growth in foam core materials for sandwich structures. J Sandwich Struct 2: 103–116 6. Carlson LA (1991) On the design of the cracked sandwich beam (CSB) specimen. J Reinforced Plast 10: 434–444 7. Shipsha A, Burman M, Zenkert D (1999) Interfacial fatigue crack growth in foam core sandwich structures. Fatigue Fracture Eng Mater Struct 22: 123–131 8. Østergaard RC, Sørensen BF, Brøndsted P (2007) Measurement of interface fracture toughness of sandwich structures under mixed mode loadings. Sandwich Struct Mater 9: 445–466 9. Zenkert D (1990) Strength of sandwich beams with mid-plane debondings in the core. Compos Struct 15: 279–299

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10. Zenkert D (1991) Strength of sandwich beams with interface debondings. Compos Struct 17: 331–350 11. Zenkert D, Groth HL (1989) The influence of flawed butt-joints in foam core sandwich beams. In: Olsson KA, Reichard RP (eds.), First International Conference on Sandwich Constructions, EMAS, UK, pp. 363–381 12. Groth HL, Zenkert D (1990) Fracture of defect foam core sandwich beams. ASTM J Testing Eval 18(6): 390–395 13. Zenkert D (1992) Effect of manufacturing-induced flaws on the strength of foam core sandwich beams. In: Masters JE (ed.), Damage Detection in Composite Materials, ASTM STP 1128, ASTM, Philadelphia, PA, pp. 137–151 14. Zenkert D, Schubert O, Burman M (1997) Fracture initiation in foam core sandwich structures due to singular stresses at corners of flawed butt-joints. Mech Compos Mater Struct 4(1): 1–21 15. Burman M, Zenkert D (1997) Fatigue of foam core sandwich beams. Part I: Undamaged specimens. Int J Fatigue 19(7): 551–561 16. Burman M, Zenkert D (1997) Fatigue of foam core sandwich beams. Part II: Effect of initial damages. Int J Fatigue 19(7): 563–578 17. Burman M, Zenkert D (2000) Fatigue of undamaged and damaged honeycomb sandwich. J Sand Struct 2: 50–74 18. Zenkert D, Shipsha A, Persson K (2004) Static indentation and unloading response of sandwich beam. Compos B 35(6–8): 511–522 19. Rizov V, Shipsha A, Zenkert D (2005) Indentation study of foam core sandwich panels. Compos Struct 69: 95–102 20. Koissin V, Shipsha A (2008) Residual dent in locally loaded foam core sandwich structures – analysis and use for NDI. Compos Sci Technol 68: 57–74 21. Hallstr¨om S, Shipsha A, Zenkert D (2000) Failure of impact damaged foam core sandwich beams. In: Rajapakse YDS (ed.), ASME ICEME’2000, ASME AERO/AMD AD-Vol. 62/AMD Vol. 245, pp. 11–19 22. Shipsha A, Hallstr¨om S, Zenkert D (2003) Failure mechanisms and modelling of impact damage in sandwich beams – a 2D approach: Part I – Experimental investigation. J Sandwich Struct Mater 5(1): 7–31 23. Shipsha A, Hallstr¨om S, Zenkert D (2003) Failure mechanisms and modelling of impact damage in sandwich beams – a 2D approach: Part II – Analysis and modelling. J Sandwich Struct Mater 5(1): 33–51 24. Koissin V, Skortsov V, Shipsha A (2007) Stability of the face layer of sandwich beams with sub-interface damage in the foam core. Compos Struct 78: 507–518 25. Shipsha A, Zenkert D (2003) Fatigue behaviour of foam core sandwich beams with subinterface impact damage. J Sand Struct Mater 5(1): 147–160 26. Zenkert D, Falk F (1991) Interface debondings in foam core sandwich beams and panels. In: Springer G, Tsai S (eds.), Proceedings of the International Conference on Composite Materials, Publ. by SAMPE, (ICCM/VIII), 3-H 27. Falk F (1992) Strength of foam-core sandwich panels with face-to-core debonds. In: Weissman-Berman D, Olsson KA (eds.), Proceedings of the 2nd International Conference on Sandwich Constructions, EMAS Ltd, UK, pp. 645–663 28. Wennhage P, Zenkert D (1998) Testing of sandwich panels under uniform pressure. J Testing Eval 26(2): 101–108 29. Jolma P, Segercrantz S, Berggreen C (2007) Ultimate failure of debond damaged sandwich panels loaded with lateral pressure – an experimental and fracture mechanics study. J Sand Struct Mater 9: 167–196 30. Berggreen C (2004). Damage tolerance of debonded sandwich structures, Ph.D. thesis, Department of Mechanical Engineering, Technical University of Denmark 31. Nokkentved A, Lundsgaard-Larsen C, Berggreen C (2005) Non-uniform compressive strength of debonded sandwich panels – I. Experimental investigation. J Sand Struct Mater 7(6): 461–482 32. Berggreen C, Simonsen BC (2005) Non-uniform compressive strength of debonded sandwich panels – II. Fracture mechanics investigation. J Sand Struct Mater 7(6): 461–482

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33. Causes, mechanisms and preventive actions to avoid blistering, Technical Bulletin, DIAB AB, www.diabgroup.com 34. Shipsha A, Zenkert D (2005) Compression-after-impact strength of sandwich panels with core crushing damage. Appl Compos Mater 12(3–4): 149–164 35. Edgren F, Asp L, Bull P (2004) Compressive failure of impacted NCF composite sandwich panels – characterisation of failure process. J Compos Mater 38(6): 495–514 36. Bull P, Edgren F (2004) Compressive strength after impact of CFRP-foam core sandwich panels. Compos B 35: 535–541 37. Sutcliffe M, Xin A, Fleck NA, Curtis P (1999) Composite compressive strength modeller, Engineering Department, Cambridge University Press, Cambridge 38. Edgren F, Soutis C, Asp L (2008) Damage tolerance analysis of NCF composite sandwich panels. Compos Sci Technol 68(13): 2635–2645

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Size Effect on Fracture of Composite and Sandwich Structures Emmanuel E. Gdoutos and Zdenˇek P. Baˇzant

Abstract The objective of this article is to review the work performed on the scaling and size effect in the failure of composites, foams and laminate–foam sandwiches. These materials exhibit quasibrittle behavior which is characterized by a fracture process zone that is not negligible compared to the characteristic size of the structure. The mean size effect is found to be essentially deterministic, caused by energy release due to stress redistribution. The chapter consists of six sections: After introduction, the second section deals with the size effect on the nominal strength of notched specimens of fiber composite laminates under tension. In the third section, the size effect of fiber–composite laminates on flexural strength is studied. The fourth section studies the effect of structure size on the nominal strength of fiber– polymer composites failing by propagation of a kink band with fiber microbuckling. The fifth section deals with the size effect of fracture of closed-cell polymeric foams. The sixth section analyzes the size effect on the compressive strength of sandwich panels subjected to double eccentric axial load and failing by propagation of a softening fracturing kink band. Finally, the seventh section shows that skin imperfections, considered to be proportional to the first eigenmode of wrinkling, lead to strong size dependence of the nominal strength of sandwich structures failing by skin wrinkling.

1 Introduction The effect of structure size on its nominal strength is of paramount importance in extrapolating small-scale laboratory tests to full-scale structures. The question of size effect was discussed by Leonardo da Vinci (1452–1519) who ran tests to determine E.E. Gdoutos School of Engineering, Democritus University of Thrace, GR-67100 Xanthi, Greece e-mail: [email protected] Z.P. Baˇzant () Department of Civil Engineering and Material Science, Northwestern University, Evanston, IL 60208-3109, USA e-mail: [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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the strength of iron wires [1]. He found an inverse proportionality of the nominal strength to the length of the wire for wires of constant diameter. We quote from an authoritative translation of Leonardo’s sketch book [2]: “Observe what the weight was that broke the wire, and in what part the wire broke. . . Then shorten the wire, at first by half, and see how much more weight it supports; and then make it one quarter of its original length, and so on, making various lengths and noting the weight that breaks each one and the place in which it breaks.” This is of course a strong exaggeration of the actual size effect. The rule of Leonardo was rejected by Galileo in his famous book [3], in which he founded mechanics of materials. Galileo argued that cutting a long cord at various points should not make the remaining part stronger. He pointed out, however, that a size effect is manifested in the fact that large animals have relatively bulkier bones than small ones. Half a century later, Mariotte [4] based on extensive experiments on ropes, paper and tin made the observation that “a long rope and a short rope always support the same weight unless that in a long rope there may happen to be some faulty place in which it will break sooner than in a shorter.” He thus initiated the statistical theory of size effect, two and a half centuries before Weibull. The above studies concern with the structural size effect which is studied in this paper. This differs from the material size effect, which was first systematically studied by Griffith [5, 6], who with his key ideas about the strength of solids laid down the foundation of the present theory of fracture. In experiments performed on cracked circular tubes made of glass, Griffith observed that the maximum tensile stress in the tube was of the magnitude of 344 kip=in:2 (2,372 MPa), while the tensile strength of glass was 24:9 kip=in:2 (172 MPa). These results led him to raise the following questions (we quote from Ref. [5]) “If the strength of this glass, as ordinarily interpreted, is not a constant, on what does it depend? What is the greatest possible strength, and can this strength be made for technical purpose by appropriate treatment of the material?” From Griffith’s experiments it follows that the strength increases as the fiber diameter decreases. The maximum strength of glass was found to be 1;600 kip=in:2 (11,000 MPa), which is two orders of magnitude higher than the ordinary strength of glass of 24:9 kip=in:2 (172 MPa). Stanton and Batson [7] reported the results of tests conducted on notched-bar specimens at the National Physical Laboratory, Teddington, England, after the First World War. From a series of tests it was obtained that the work of fracture per unit volume was decreased as the specimen dimensions were increased. Weibull [8] laid down the basic framework of the statistical theory of size effect. The theory applies to structures that fail at the initiation of macroscopic fracture and have at fracture only a small fracture process zone causing negligible stress redistribution. This is the case of brittle materials. It does not apply to quasibrittle materials, such as concrete and mortar made with various cements and admixtures, polymers, rock, ice, fiber or particulate composites, fiber-reinforced concretes, toughened ceramics, bone, biological shells, stiff clays, cemented sands, grouted soils, coal, paper, wood, wood particle board, various refractories, some special tough metallic alloys, filled elastomers, etc., which are characterized by the existence of a large fracture process zone with distributed cracking damage.

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Weibull’s theory attributing the observed size effect to the randomness of material strength dominated until the early 1980s, when Baˇzant [9, 10] revisited the size effect on structural strength. The classical theories of solid mechanics including elasticity and plasticity do not account for the size effect; in other words, the nominal strength of a structure is independent of its size. This is acceptable only for very small structures. This property is not valid for materials that, instead of plastic yielding, exhibit softening damage, such as distributed cracking. In that case, a strong, non-statistical size effect may be caused by stress redistribution creating energy release from an elastically unloaded region of material at the flanks of a propagating damage or large cohesive fracture, taking place before the maximum load is reached. When this deterministic size effect occurs, it normally prevails over the size effect due to the statistical distribution of strength, described by the Weibull theory. In order to extrapolate to large sizes from material strength values obtained from laboratory tests of relatively small specimens, the size effect should be well understood. Baˇzant laid down the foundation of a new concept with his discovery of a deterministic scaling law based on the release of stored energy due to stable growth of large fractures or large damage zones prior to failure [9, 10]. He derived a simple formula for the size effect law which describes the size effect on nominal strength of structures made of quasibrittle materials, which are characterized by the existence of a sizable fracture process zone at the tip of the macroscopic crack. Quasibrittle materials are brittle materials characterized by a fracture process zone (FPZ) that is not negligible compared to the characteristic size D (or cross-sectional dimension) of the structure. Typically, the size of the FPZ, taken as the material characteristic length lch , is about 5 to 50 times the maximum inhomogeneity size, and quasibrittle behavior is observed only for D= lch  1 to 1,000. For larger D= lch , the FPZ can be regarded as a point and then the behavior is brittle, while for D= lch < 1, and approximately for up to about 5, the behavior can be regarded as quasi-plastic. Baˇzant found that in such materials the size effect is transitional between plasticity (for which there is no size effect) and linear elastic fracture mechanics (for which the size effect is the strongest). The curve of the logarithm of the nominal strength versus the logarithm of the size represents a smooth transition from a horizontal asymptote corresponding to the strength criterion (plastic limit analysis) to an inclined asymptote of slope 0:5, corresponding to linear elastic fracture mechanics. He also formulated the crack band model [11,12] which realistically approximates by simple finite element analysis the size effect observed on concrete specimens and structures. This model is nowadays almost the only concrete fracture or damage model used in inˇ dustry and commercial codes (e.g. code DIANA, Rots 1988; code SBETA, Cervenka and Pukl 1994; or ATENA). A more general nonlocal approach to strain-softening damage, capable of describing the size effect in quasibrittle materials in a more fundamental and realistic manner was published by Baˇzant et al. [13], Baˇzant [14], Pijaudier-Cabot and Baˇzant [15], Baˇzant and Pijaudier-Cabot [16], and Baˇzant and Lin [17, 18]. It is used in commercial code OOFEM and in many research projects. Carpinteri et al. [19] proposed, on the basis of strictly geometrical arguments, that the difference in fractal characteristics of cracks or microcracks at different scales of

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observation is the principal cause of size effect in concrete structures. Carpinteri’s theory was subject to extensive criticism by Baˇzant and Baˇzant and Yavari [20–22]. This article reviews the work performed by Baˇzant and coworkers on the scaling and size effect in the failure of advanced composite, foams and sandwich materials. The size effect of the above materials is described in six sections for fiber–composite laminates subjected to tension, compression and flexural loading, to closed-cell polymeric foams and to sandwich panels under eccentric compression and with skin imperfections. Each section includes an introduction, experimental results, the size effect law for the particular problem and conclusions.

2 Size Effect on the Tensile Strength of Notched Fiber–Composite Laminates [23] 2.1 Introduction Failure of composite materials has been described by conventional failure criteria based on maximum stress, maximum strain, deviatoric strain energy (Tsai-Hill) and tensor polynomial (Tsai-Wu). These criteria are macromechanical and do not account for the various micromechanical failure processes occurring in composites, especially near notches. Damage initiation and development take the form of various interacting failure mechanisms which are sensitive to pre-existing defects and micro-structure (micromechanical) anomalies. The damage processes tend to localize and propagate. Their propagation can be studied by energy release, which accounts for the size effect. Fracture of laminated composites with stress concentrators has been studied by using linear-elastic fracture mechanics and the local stress distributions near the notch end. The first approach based on linear elastic fracture mechanics was used by Waddoups et al. [24] who proposed a theory based on the generalized concept of the process zone. The actual crack length is extended by the length of the process zone which is taken equal to a damage zone at the crack tip. Cruse [25] calculated the fracture energy of a multidirectional laminate as the sum of fracture energies of the individual plies. An equivalent summation of the squares of the stress intensity factors has also been proposed by Mandell et al. [26], who pointed out that microcracking zones play the same role as plastic flow in metals, relieving the high local stress concentrations and absorbing the energy released due to fracture propagation. These authors indicated that the damage zone at the crack tip in fiber composite laminates consists of matrix cracks parallel to the fibers and local delaminations of the cracked plies. Following the second approach of local stress distribution near the notch end, Whitney and Nuismer [27] proposed two simplified fracture criteria based on the actual stress distribution near the notch, the so-called point stress and average stress criteria.

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Daniel [28–32] observed that failure of graphite fiber/epoxy matrix composite laminates involves a combination of several microscopic failure mechanisms, including ply microcracking, delamination, fiber breakage and fiber pullout. He found that there is a critical damage zone at the tip of a notch or at the boundary of a hole, whose size at failure was roughly independent of the notch length. These observations revealed the existence of a characteristic length in composite materials. Daniel introduced an equivalent crack equal to the original crack plus the length of the damage zone and calculated an apparent stress intensity factor which was constant for the range of his data, including mixed mode loading. Daniel used his results to obtain an R-curve (resistance curve), giving the dependence of the apparent stress intensity factor on the crack length. In this work the size effect law of Baˇzant is used to predict the size effect on the nominal strength of single and doubled notched specimens of graphite/epoxy composite laminates.

2.2 Experimental Unidirectional, crossply and quasi-isotropic composite laminates were made of IM7/8551-7A (Hercules) graphite/epoxy unidirectional prepreg. The material was fully characterized by testing Œ06 ; Œ908 , and Œ106  coupons under uniaxial tension. Two sets of specimens were prepared for the fracture tests: crossply of Œ0=902 s layup, and quasi-isotropic of Œ0= ˙ 45=90s layup. Each set consisted of four rectangular specimens of the same thickness but different sizes, geometrically similar in the plane of the laminates, with gage length of 6:4  25; 12:7  51; 25:4  102, and 50:8  203 mm .0:25  1:0; 0:50  2:0; 1:00  4:0 and 2:00  8:0 in:/. The size ratios were 1:2:4:8. The thickness of the crossply and the quasi-isotropic specimens was 0.76 mm (0.030 in.) and 1.02 mm (0.040 in.). Two edge notches of length a D D=16 were machined in the crossply specimens and a single edge notch of length a D D=5 was machined in the quasi-isotropic specimens, where D is the specimen width (Fig. 1). The crack tip radius was 0.1 mm (0.004 in.) in all cases. All specimens were tabbed with 38 mm (1.5 in.) long glass=epoxy plates. The specimens were subjected to a uniaxial tensile loading in an Instron servohydraulic testing machine at a constant crosshead rate for the double edge notched specimens and under crack opening displacement (COD) control for the singleedge-notched specimens. The crosshead rate was adjusted for the different size specimens so as to achieve roughly the same average strain rate of 0.2%/min. in the gage section. With that rate, the peak load was reached within approximately 10 min in all cases. Figure 2 shows typical stress–strain curves for the notched crossply specimens of various sizes. Note that for the largest specimen size, these curves are almost linear up to failure, while for the smallest specimen size there is a significant nonlinear segment before the peak stress. This indicates a pronounced brittle behavior for the large specimens and a hardening inelastic behavior and reduced brittleness

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Fig. 1 Geometry of test specimens: (a) double-edge notched specimens, (b) single-edge notched specimens (units: mm)

(or higher ductility) for the small specimens. The failures of the specimens were catastrophic (dynamic), and occurred shortly after the peak load. Damage consisting of microcracks in layers and delamination between layers before peak load was observed. The nominal strength is defined as the average stress at failure based on the unnotched cross section,  D Pmax = hD where D is the specimen width (characteristic dimension) and h the laminate thickness.

2.3 Size Effect The approximate size effect law of Baˇzant [9, 10, 33] is given by N D Bf u .1 C ˇ/1=2 ; ˇ D D=D 0 where ˇ D relative structure size N D cN P =bD D nominal strength of structure P D maximum load D D characteristic dimension (size) of structure b D width of structure in the third dimension

(1)

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Fig. 2 Typical stress–strain curves of [0=902 s crossply double-edge notched specimens of various sizes, showing increasing nonlinearity with decreasing size

cN D chosen coefficient introduced for convenience (for example to make N coincide with the maximum stress in the specimen calculated by bending theory) D0 D constant depending on both fracture process zone size and specimen geometry B D constant characterizing the solution according to plastic limit analysis based on the strength concept fu D reference strength of the material (laminate), introduced to make constant B dimensionless. Equation (1) is valid not only for the two-dimensional similarity considered here (h D constant) but also for three-dimensional similarity. The parameters of the size effect law were determined by regression analysis of experimental data. It was obtained that D0 D 30:9 mm; Bf u D 892 MPa for the crossply specimens, and D0 D 77:5 mm; Bf u D 611 MPa for the quasi-isotropic specimens. The resulting size effect was represented by plotting log .N =Bf u / versus log .D=D0 / in Figs. 3 and 4 for the two sets of specimens. These size effect plots represent a transition from the strength criterion (plastic limit analysis) characterized by a horizontal asymptote, to an asymptote of slope 0:5, representing linear elastic fracture mechanics (LEFM). The intersection of the two asymptotes corresponds to D D D0 , called the transitional size.

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Fig. 3 Size effect measured for crossply specimens with double-edge notches

Fig. 4 Quasi-isotropic specimens with single-edge notch

2.4 Conclusions From the results, the following conclusions may be drawn: 1. The nominal strength of composite laminate specimens that are similar and have similar notches or initial traction-free cracks exhibits a significant size effect. 2. The size effect observed agrees with the size effect law proposed by Baˇzant, according to which the curve of the logarithm of the nominal strength versus the logarithm of the characteristic dimension (size) exhibits a smooth transition from a horizontal asymptote corresponding to the strength criterion (plastic limit analysis) to an inclined asymptote of slope 0:5, corresponding to linear elastic fracture mechanics. 3. Measurements of the size effect on the nominal strength can be used for determining the fracture characteristics of notched fiber composite laminates, including their fracture energy and the effective length of the fracture process zone. From these characteristics, the R-curve can also be calculated.

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3 Size Effect on the Flexural Strength of Fiber–Composite Laminates [34] 3.1 Introduction The size effect on the flexural strength of fiber–polymer laminates has been studied by many investigators [35–38]. Wisnom [35] conducted four-point bending tests and pin-ended buckling tests on unidirectional XAS/913 carbon fiber–epoxy specimens with 25, 50, and 100 plies. Both types of tests showed a significant decrease in strength with increasing specimen size. Most of the small specimens failed in tension, while most of the large specimens failed in compression. From the experimental data he obtained a Weibull modulus (shape parameter) m D 25:4. Jackson [36] performed tests on beams made of unidirectional, angle-ply, crossply and quasi-isotropic ply-level AS4/3502 carbon fiber–epoxy matrix composites, and found an apparent size effect of the specimen size on the flexural strength. Wisnom and Atkinson [37] performed tests on four-point bend specimens of threedimensional-scaled unidirectional E glass/913 specimens of 16, 32 and 64 plies, and showed a clear size effect. Johnson et al. [38] performed tests in AS4/3502 graphite–epoxy laminate beams in the four-point bending, and found that the flexural strength of ply-level scaled laminates decreased significantly with the specimen size, while the sublaminate-level scaled specimens did not show a pronounced size effect. The Weibull modulus values for the angle-ply and quasi-isotropic specimens were obtained as 50.0 and 26.7, respectively. In the present work the size effect on the flexural strength of fiber–polymer composite laminate beams failing at fracture initiation is analyzed by an energeticstatistical size effect law. The size effect is due to stress redistribution engendered by a boundary layer of cracking in structures that fail at the initiation of fracture from a smooth surface, and also by statistics.

3.2 Size Effect For the analysis of the size effect in fiber composite beams under flexural bending the concept of the boundary layer is introduced. The boundary layer develops at the tensile face and has a finite thickness 2Db that is a property of the fiber composite. The laminate cross section is considered homogeneous, so that the elastic bending stress diagram is linear. The laminate is assumed to fail in tension rather than in compression (although compression failure would lead to a similar formula). The average tensile strength of the boundary layer, f 0 r , which is considered to be a constant, is given by fr D M0 .D  Db /=2I (2)

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where D D beam depth M0 D bending moment I D D 3 =12 D moment of inertia of cross section f 0 r D the average tensile strength of the boundary layer, considered to be constant. The nominal flexural strength N is defined as the maximum stress in the beam N D fr D M0 D=2I . Therefore, N D

fr0



Db 1 D

1 (3)

The above equation gives negative ¢N for small values of the beam depth D. Thus it is used only for large sizes D  Db . This limitation is not surprising because the derivation has been correct only up to the first two terms of the asymptotic series expansion in terms of the powers of 1=D [39]. Through a power series expansion in 1=D one may check that by making the replacement .1  Db =D/1 .1 C rD b =D/1=r (r being any positive constant signifying a transition slowness parameter), the first two terms of the asymptotic expansion in terms of 1=D are not affected while, at the same time, N , becomes positive, finite and monotonically decreasing through the entire range of D. With this replacement, Eq. (3) leads to the size effect formula: N D f r D

fr0 q.D/;



rD b q.D/ D 1 C D

1=r (4)

where q.D/ is a positive dimensionless decreasing function of size D having a finite limit for D ! 1. Equation (4) may be modified as follows: N D

fr0 q

.D/ ;

 q.D/ D 1 C

rD b D C rsD b

1=r (5)

where s is a non-negative constant. In situations where the Weibull statistical size effect is considered to play a role when the local strength of material elements is random and the minimum of random strength encountered in the structure decreases with the structure size, Baˇzant and Novak [40, 41] derived the following equation:

N D

fr0

"

rDb D C rsDb

rnd =m

rDb C D C rsDb

#1=r (6)

where nd D number of spatial dimensions in which the structure is scaled (nd D 1, 2 or 3), m D material constant D Weibull modulus.

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Note that for m ! 1 (and s D 0) Eq. (6) coincides with the foregoing energetic (deterministic) Eq. (4). For D  Db or Db D 0 (and s D 0) Eq. (6) becomes N D fr0 .Db =D/nd =m

(7)

and thus one recovers the classical formula for Weibull size effect as the limit case.

3.3 Experimental Studies The energetic-statistical theory of flexural strength of laminates was checked and calibrated from the existing test data in the literature. Results are shown in Fig. 5 presenting the optimum fit of the existing test data on flexural strength versus relative size, in which ¨ is an unbiased estimate of the coefficient of variation corresponding to the standard error of regression and r appears in Eq. (3).

Fig. 5 Optimum fits of existing test data on modulus of rupture versus relative size, in dimensionless coordinates, by (a) deterministic energetic formula; (b) energetic-statistical formula; (c) Weibull size effect formula with m D 5; and (d) Weibull size effect formula with m D 30

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3.4 Conclusions From the results, the following conclusions may be drawn: 1. The size effect on the flexural strength of laminates appears to be primarily energetic (deterministic) rather than statistical, except possibly for very large thicknesses for which the statistical size effect might also be significant. This further implies that fracture mechanics, rather than some strength criterion (or material failure criterion expressed in terms of stresses and strains), needs to be used for evaluating the strength of laminates. The fracture mechanics approach must take into account the quasibrittle (or cohesive) nature of fracture. 2. The statistical theory is inapplicable for the prediction of the size effect of the flexural strength of composite laminates. 3. The existing experimental data support the applicability of both the energetic theory and the energetic-statistical theory. However, the available data do not suffice to demonstrate that the energetic-statistical theory is better than the energetic theory (which is a special case). Experimental data of a broader size range or lower scatter, or both, would be needed for that purpose. Superiority of the energetic-statistical theory so far relies only on theoretical arguments.

4 Size Effect on the Compression Strength of Fiber–Composite Laminates [42] 4.1 Introduction Compression failure of unidirectional fiber composites is a complex material failure involving formation of a so-called kink band which is a row of parallel axial shear cracks combined with microbuckling of fibers. This type of failure has extensively been studied in the literature. The early studies did not consider the size effect. This seems natural because small-scale laboratory tests indicated no size effect and the maximum load has been thought to occur at the very beginning of microbuckling, before the size or the length of the kink band becomes macroscopically significant. Soutis et al. [43] calculated and experimentally verified that the apparent strength in the vicinity of a hole decreases with an increasing diameter of the hole. Although geometric similarity of the hole with the specimen dimensions was not maintained in these tests, the results nevertheless indicated the likelihood of size effect. Budiansky et al. [44] analyzed the propagation of a semi-infinite out-of-plane kink band, approximating the band with a crack whose face is allowed to overlap in compression. Their analysis clearly suggests the existence of a size effect on the nominal strength of geometrically similar specimens. In the present study the effect of structure size on the nominal strength of unidirectional fiber–polymer composites, failing by propagation of a kink band with fiber microbuckling, is analyzed experimentally and theoretically. Tests of novel

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geometrically similar carbon-PEEK specimens are conducted. They confirm the possibility of stable growth of long kink bands before the peak load, and reveal the existence of a strong deterministic size effect.

4.2 Experimental Failure by kink band propagation is usually combined with axial splitting-shear cracks and delaminations. For the present investigation the shape of the specimen and the type of material should be properly selected to lead to pure kink band failure even for very large sizes. A carbon fiber PEEK (poly-ether-etherketon)/thermoplastic matrix polymer composite was chosen. This material is less brittle than carbon/epoxy composites and leads to more stable failures. Thus, if a size effect is found to exist in this type of composite, it should also exist in a more pronounced form in more brittle composites, such as the carbon fiber–epoxy composites. To exclude the effect of random variation of material strength over the specimen volume, notches were introduced in the specimens to ensure that failure begins in a desired place and not at diverse locations where the material is statistically weakest. One-sided, rather than two-sided, starter notches which cause a stable path of the kink band were introduced in the specimens. These notches were made inclined and not perpendicular to the surface of the specimens. The inclination angle was found by trial tests to be 25:4ı , equal to the angle of the out-of-plane kink band. The in-plane dimensions of the specimens were scaled at the ratio 1:2:4, while the thickness b was kept constant .b D 12:7 mm.0:5 in:// (Fig. 6). The notch length was a0 D 0:3D, where D is the specimen width. The notch was machined with a diamond bladed band saw up to 95% of its length. Then the notch was sharpened at the tip by a cut whose depth is 5% of the notch depth. The cut was machined with a 0.2 mm diameter diamond-studded wire, and thus the crack tip radius was 0.1 mm in all the specimens. The depth a0 of the notch considered for scaling and in the analysis included the depth of the wire saw cut. The specimens were prepared by compression molding of 100 plies of sheets 304:8  304:8 mm, 0:05 mm thick of carbon/polymer prepreg sheets supplied by Fiberite, Inc. under temperature 391ı C (735ı F) and pressure 0.69 MPa, using the manufacturer recommended curing cycle. After the specimens were cut from the sheets at the proper dimensions they were provided with massive end caps made of 1,040 hot rod steel, to which they were glued by epoxy. To ensure proper alignment, the end caps were glued only after the specimen had been installed under the loading platens of the testing machine. The end plates were restrained to prevent any rotations. The specimens were tested under a controlled stroke rate of 1:27104 mm=s. Following initiation at the notch tip, the kink band propagated stably on both the front and back sides of the specimen. At the beginning of propagation the kink band on one side was usually slightly longer than that on the other side. However, during propagation the shorter band on one side would soon catch up with the longer band on the opposite side.

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Fig. 6 (a) Geometrically similar single-edge notched carbon-PEEK (poly-ether-ether-keton) specimens tested, and scheme of loading; (b) specimen with an orthogonal notch exhibiting undesirable failure (splitting shear cracks); and (c) transversely slanted notches achieving pure kink band failure

The nominal strength of the specimen is defined as N D P =bD

(8)

where P D maximum load measured b D specimen thickness D D specimen width (chosen as the characteristic dimension). The specimen width D is chosen as the characteristic dimension. Fig. 7 shows the test results plotted in the form of log N versus log D. The stress–strain diagrams of the small, medium and large specimens are given in Fig. 8, where the average stress is defined over the ligament as L D N D=.D  a0 / and the average axial strain © is determined as the stroke of the piston divided by the length between the platens. Note that the load-deflection diagrams exhibit a post-peak stress drop rather than a horizontal yield plateau at peak load. This fact alone suffices to demonstrate that a fracture-type approach (or a nonlocal damage approach) is required.

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Fig. 7 Results of tests of nominal strength of carbon-PEEK specimens (data points) and their optimum fit by the size effect law of Bazant

Fig. 8 Load-deflection diagrams of carbon-PEEK specimens tested

Furthermore, these diagrams reveal the existence of a terminal yield plateau, which confirms the existence of a finite residual stress across the kink band. The experimental results of Fig. 7 indicate a downward trend of nominal strength versus characteristic dimension of the specimens. The downward slope is quite steep and closer to the slope 1=2 corresponding to LEFM than to the horizontal line corresponding to the strength theory. This indicates a strong size effect, even though the PEEK matrix is relatively ductile. Note in Fig. 7 that one of the small specimens, corresponding to the upper left data point, did not develop a kink band but failed by a vertical shear crack that started from the notch tip and produced axial splitting. This point was not excluded, but retained in the data because it is the highest point in the data set, and the load that would cause the kink band failure of this specimen must have been at least as high as this point. The test results exhibit considerable random scatter. This is, however, typical of compression failure of fiber composites (because of their strong sensitivity to fiber misalignment). For this reason, the size effect would not have

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been revealed clearly if the size range were less than 1:4. The present size range of 1:4 appears to be just about the minimum for being able to clearly demonstrate the size effect. To reduce the ratio of the inevitable scatter band width to the range of sizes, the ratio of the sizes of the smallest and largest specimens should be at least 1:8 in future testing. The rotation restraint boundary conditions imposed at the ends of the specimens lead to the development of an unknown moment causing the compression resultant to shift laterally during kink band propagation. However, the end restraint has the advantage that a stable propagation of a long kink band is possible and the postpeak deformations are well beyond the peak load while the residual stress plateau and stresses can be observed.

4.3 Conclusions From the results, the following conclusions may be drawn: 1. Carbon-PEEK laminate specimens with slanted notches can achieve pure out-ofplane kink band failure without the presence of shear splitting cracks. 2. Restraining specimen ends against rotation helps to stabilize kink band growth and makes it possible to demonstrate the possibility of a stable growth of long kink bands before the peak load. 3. Compression tests of notched carbon-PEEK laminates show that the nominal strength of geometrically similar notched specimens failing purely by kink band propagation exhibits a strong non-statistical size effect. 4. The size effect observed is transitional between the asymptotic case of no size effect and the asymptotic case of size effect of linear elastic fracture mechanics, which is governed by energy release. 5. The results of the present carbon-PEEK tests roughly agree with the approximate general size effect law proposed by Baˇzant. 6. The results of the present study suggest that the current design practice, in which the compression failure is predicted on the basis of strength criteria which miss the size effect, is acceptable only for small specimens or structural parts. However, for large structural parts the size effect should be taken into consideration.

5 Size Effect on Fracture of Polymeric Foams [45] 5.1 Introduction Cellular foams are frequently used as core materials in sandwich construction. They are relatively inexpensive and consist of a vast variety of foamed plastics and metals with varying densities, elastic properties and strengths. Commercially

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available plastic foams are made of polyvinyl chloride (PVC), polyurethane (PUR), and polystyrene, among others. Metallic foams are usually made of aluminum. The properties of foams depend on the structure of the cell, the density and the material of which they are made. A detailed characterization of their mechanical behavior is essential for their efficient use in structural applications. A thorough analysis of the mechanical properties of plastic foams is given in the book by Gibson and Ashby [46]. They presented simple micromechanical equations based on mechanics of materials analysis to relate the properties of the foam to the structure of the cells and the properties of the cell material. Gdoutos et al. [47] studied the mechanical behavior of cellular foams under multiaxial stresses. Furthermore, Gdoutos and Abot [48] analyzed the indentation behavior of foams. The objective of this work is to study the effect of structure size on the nominal strength of closed-cell PVC foam (Divinycell H100). Two types of size effect are considered: Type I, characterizing the failure of structures with large cracks or notches, and Type II, characterizing the failure at crack initiation.

5.2 Experimental The material of the specimens was a PVC closed-cell rigid foam under the commercial name Divinycell H100 with density 100 kg=m3 . The specimens were cut from the same plate and had a thickness b D 25W40 mm (1 in.). To determine the size effect in tensile (model I) fracture, geometrically similar specimens in two dimensions with length-to-width ratio 5:2 were selected. The width of the specimens was D D 6:35, 43.94 and 304.80 mm (Fig. 9). In order to ensure that failure begins at a desired place and not start at diverse locations where the material is statistically the weakest (which could cause a Weibull-type size effect), notches were introduced in the specimens. The width of the notches was 1.00 mm and depth 0:4 D. The tip of the notch was sharpened by a blade having the thickness of 0.25 mm. Compressive tests were also performed using the same specimens but only of the middle size. To avoid the opposite faces of the notch from getting in contact before the maximum compressive load is reached, the notch was widened with a band saw to a wedge shape of width 25 mm at the notch mouth. The notch tip was sharpened by a razor blade. The ends of specimens were glued by epoxy to very stiff steel platens which were gripped in the loading machine, with any rotation of the ends prevented. The specimens were loaded in tension in a servo-hydraulic machine. To minimize the viscoelastic effects due to differences in the loading rate, the displacement rate of the platens was uniform throughout the test and was chosen such that the specimens of any size would reach the maximum load within about 5 min. The displacements were measured by LVDT gages mounted across the notch mouth spanning, in the case of tension tests, a base length of 11.50 mm. A typical load – displacement curve is shown in Fig. 10.

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Fig. 9 Foam specimens geometrically similar in two dimensions for tensile fracture test (Divinycell H100)

Fig. 10 A typical load-•CMOD curve in the tensile fracture test

5.3 Size Effect Cellular foams have micro-structural inhomogeneities, such as cells and finite fracture process zones, and therefore, must exhibit a size effect, unless the structural dimensions are far larger than the characteristic length of the foam. Fig. 11 presents

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Fig. 11 Results of size effect tests of nominal strength of geometrically similar prismatic Divinycell H100 foam specimens with similar one-sided notches subjected to tension

a plot of the nominal strength N D Pmax =bD versus log D, where Pmax is the failure load, and D and b the specimen width and thickness. Note in Fig. 11 that there is a very strong size effect. This size effect cannot be explained by the Weibull statistical theory. In that case the plot of log N versus log D, corresponding to the typical values of Weibull modulus, would have to be a straight line of a slope equal to 2=m where m is the Weibull modulus characterizing the coefficient of variation of local material strength. Since its value is typically between 15 and 30, this straight line would have to have a slope between 0:03 and 0:07. However, the size effect observed in Fig. 11 is much stronger than that, overpowering any possible statistical size effect, and is of deterministic character. According to Baˇzant [49] there are three types of deterministic (energetic) size effect. Type I is caused by relatively large notches or fatigued (stress-free) cracks formed prior to maximum load; type II occurs at crack initiation and is caused by a relatively large fracture process zone (FPZ); and type III is caused by large stable crack growth in structures of initially negative geometry; it is quite similar to Type II and will not be considered here. Note in Fig. 11 that the experimental results are very close to the straight line of downward slope 1=2, which indicates that the material behaves in an almost brittle manner. The term “brittle” is understood as the adherence to LEFM, while the term “quasibrittle” refers to nonlinear cohesive softening (non-ductile) fracture with a large FPZ, deviating from LEFM. According to the size effect method of measuring nonlinear fracture properties [50–52], the location in Fig. 11 of the asymptote of slope 1=2 determines the fracture energy Gf of the material, and the rate at which this asymptote is approached determines the effective size of the FPZ, cf , representing the distance from the actual crack tip to the tip of an equivalent LEFM crack, which lies roughly in the middle of the FPZ and can be precisely defined as the tip location that gives the best LEFM fit of the actual size effect curve. The energy release rate in LEFM may always be expressed as G D K12 =E D N2 g .˛/ D=E

(9)

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where ˛ D a=D D relative crack length, a D actual crack length, g.˛/ D Œk.˛/2 D dimensionless energy release function of relative crack length ˛, k.˛/ D KI =N D D dimensionless stress intensity factor, and KI D actual stress intensity factor (we consider only mode I). In the present case, the fracture geometry is positive (i.e., the derivative g 0 .˛/ > 0/, and in that case the FPZ at maximum load must still be attached to the notch tip and the crack begins to propagate at decreasing load. So the maximum load occurs as soon as the crack propagation condition G D Gf is attained. This yields for the nominal strength N u D N at maximum load the well-known general LEFM expression: Nu D

q

EGf =g .˛/ D

(10)

Since the FPZ at maximum load is still attached to the notch tip, we may use the approximation a D a0 C cf , or ˛ D ˛0 C with ˛0 D a0 =D; D cf =D

(11)

where a0 is the length of notch or preexisting traction-free crack; cf D material constant half-length of the FPZ. The size effect for geometrically similar specimens (i.e., specimens for which ˛0 D constant) could be simply described by substituting ˛ D ˛0 C into Eq. (10). However, such an approximation would be valid only for large sizes D because for small enough D the argument of g.˛/ becomes larger than the range of ˛ for which g.˛/ is defined. To find a size effect law applicable for all sizes, we write an asymptotic expansion in terms of 1=D;   g .˛/ D g ˛0 C cf =D D g .˛0 / C g 0 .˛0 / cf =D C . / =D 2 C

(12)

Truncating it after the second term, we get from Eq. (10) the size effect law proposed by Baˇzant [10] and Baˇzant and Kazemi [47]: s Nu D

E 0 Gf N 0 D p g .˛0 / C g 0 .˛0 / cf =D D 1 C D=D0



in which

s ¢N0 D

EGf ; g 0 .˛0 / cf

D0 D cf

g 0 .˛0 / g .˛0 /

(13)

(14)

D0 represents the transitional size delineating the brittle behavior from nonbrittle (ductile) behavior and corresponds to the intersection of the asymptotes in Fig. 11; D0 and N 0 are constant because, owing to geometric similarity, ’0 is a constant for all the specimens tested. The ratio ˇ D D=D0 is called the brittleness number [50]; ˇ  1 means a very brittle response, close to LEFM, and ˇ 1 means a very ductile response. To be able to identify the material fracture parameters from

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size effect tests, the range of ˇ must be sufficiently broad (in this regard, note that variation of the ratio g 0 .’/=g.’/ due to changes in geometry, e.g., the relative notch depth, helps to increase the range of ˇ. Based on the above analysis and using the data of Fig. 11, the following values were obtained for the foam under study: Gf D 0W49 N=mm; cf D 0W83 mm; Kc D 6W53 N=mm3=2 ; ıCTOD D 0W10 mm Knowledge of these material parameters makes it possible to analyze the size effect due to fracture in the core of a sandwich plate by equivalent LEFM, or cohesive crack model, or crack band model. From compression tests of V-notched specimens it was found that there is no size effect, and therefore, the usual plasticity analysis can be used to describe the compressive failure of PVC cellular foams. Data of Zenkert and B¨acklund [53] on tests of beams made of Divinycell H200 under three-point bending were fitted with the sized effect law of Baˇzant. Results are shown in Fig. 12. The specimens have thickness b D 30 mm, span/depth ratio,

Fig. 12 Fit of Zenkert and B¨acklund’s [53] results for three-point bend beams by Bazant’s size effect law (solid line). Dashed line: LEFM. Dash-dot line: strength criterion. Series A (left) and B (right) pertain to different temperature conditions and material from different blocks. Top: Fits when Gf and cf are optimized separately for each series. Bottom: Fits when cf is forced to be the same for both series while Gf values are allowed to be different

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L=D D 4, and ratio of notch length to specimen depth, a D 0:5, but different beam depths D D 30, 60 and 120 mm. Two series of size effect tests, labeled A and B, were performed by using specimens cut from two different blocks of the foam and subjected to slightly different temperature conditions. There were appreciable differences between these two series, which must be due to differences in temperature conditions as well as notoriously high randomness of foam properties and perhaps uncontrollable differences in specimen manufacture. The randomness is documented by very high scatter of the fracture toughness values, KIc , measured by Zenkert and Backlund on specimens cut from different blocks of material.

5.4 Conclusions 1. Notched closed-cell PVC foam (Divinycell H100) panels exhibit a strong size effect. It agrees well with the size effect law proposed by Baˇzant [10], which describes a smooth transition between the asymptotic case of no size effect and the asymptotic case of size effect of linear elastic fracture mechanics. 2. The size effect law permits the fracture energy and the effective fracture process zone length of foam to be easily identified by measuring only the maximum loads of geometrically similar notched specimens of sufficiently different sizes. 3. The single edge-notched tension specimen fixed at the ends provides a suitable fracture test specimen for very light foams. This specimen fails purely by tensile fracture, while other specimens of light foam, such a three-point bend beams, are plagued by simultaneous compression collapse of foam cells at places of load application, which distorts the results. 4. Evaluation of the energy release function for this test specimen must take into account the effect of the lateral shift of the axial load resultant caused by rotational restraints at specimen ends. This effect is easily captured by LEFM if the end rotations canceled by rotational restraint are calculated from the stress intensity factor expression as a function of crack length. 5. Conclusions 1 and 2 for the size effect also apply for notched three-point bend specimens of a heavier foam (Divinycell H200). Realistic values of the fracture parameters of this foam are obtained by using the size effect law. 6. Foam specimens with V-shaped notches (with an angle wide enough to prevent the notch faces from coming into contact) exhibit no size effect in compression. This implies that the cell collapse at the tip of the notch must be essentially a yielding process rather than a softening damage process. 7. The results demonstrate that the current design practice, in which the tensile failure of foam is generally predicted on the basis of strength criteria or plasticity, is acceptable only for small structural parts. For large structural parts the size effect must be taken into account, especially if the foam has large fatigued cracks or large damage zones prior to critical loading to failure.

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6 Size Effect on Compressive Strength of Sandwich Panels [54] 6.1 Introduction A sandwich structure consists of two external thin, strong, stiff facings bonded to a thick light-weight and weaker core. The core carries the through-the-thickness shear loads, while the facings resist in-plane and bending loads. The high specific strength and specific stiffness of sandwich construction coupled with outstanding thermal and acoustic insulation make it ideal in structural design. Sandwich beams may fail in several ways including tension or compression failure of the facings, shear failure of the core, wrinkling failure of the compression facing, local indentation, debonding of the core/facing interface and global buckling. Following initiation of failure by a specific failure mode, interaction of failure modes may occur and failure could progress by another failure mode. Sandwich materials are quasibrittle and exhibit size effect. While in small structural parts the size effect is negligible, large structures display pronounced size effect that should be taken seriously into account in structural design. Quasibrittle behavior automatically implies a significant deterministic size effect of the energetic type, whose cause is the energy release associated with stress redistribution prior to maximum load, engendered either by a large FPZ (type 1 size effect) or by a large crack (type 2 size effect). It is the objective of this work to study the size effect in sandwich panels subjected to eccentric axial compression. The sandwich panels consist of a closed-cell polyvinyl chloride foam, and woven glass-epoxy laminate facesheets. The panels had small notches which caused the failure to occur in only one place in the specimen, and failed by kink band propagation.

6.2 Experimental The specimens had the form of prismatic sandwich panels of rectangular cross section (Figs. 13 and 14). The core material was a PVC cellular foam under the commercial name Divinycell H250, of density 250 kg=m3 . The facesheets were glass-epoxy laminates (7,781 style satin weave glass and Bryte 250 epoxy resin), consisting of several plies. The prepreg material was manufactured by hot-melt film coating and was oven cured under vacuum consolidation. Two types of skins were used in the tests: porous and nonporous The porosity was 3% to 5%, which is representative of manufacturing defects. The specimens were designed so that failure by skin wrinkling and global buckling could not take place. The modulus of elasticity of the core was Ec D 400 MPa and the axial modulus of the orthotropic skins (nonporous) was Es D 24;300 MPa. The compression strength values were about 390 MPa (nonporous skin) and 320 MPa (porous skin), and 5.7 MPa for the foam core.

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Fig. 13 Test specimen

Fig. 14 Specimens showing size variation

The specimens were geometrically scaled in three dimensions to four different sizes of ratio 1:2:4:8. They were subjected to a doubly eccentric axial load in order to ensure that only one FPZ develops within the cross section. Centric loading might be more difficult to interpret because several interacting FPZs could be developing simultaneously in the cross section prior to the maximum load. The ratios ez and ey of load eccentricities to D and b (the specimen thickness and width) are kept constant. To scale the laminate skins, the number of plies n of the facesheets is increased progressively, and is n D 2, 4, 8, and 16 for the respective four specimen sizes. The load is applied through rigid end plates clamped to the ends of the specimen. To ensure that the specimens fail by deterministic, and not statistical, size

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Fig. 15 Development of fracturing compressive kink band at the notch in the laminate skin

effect, semi-circular notches of radius rn and notch depth dn scaled in proportion to D were introduced at the most highly stressed edge of the facesheets. Thus, failure of the specimens would initiate at the radius of the notches. Failure of sandwich panels consists of horizontal propagation of a softening fracturing kink band. In order to detect this failure mode a photoelastic coating was bonded to the specimen. Fig. 15 presents the photoelastic fringe pattern during loading. Kink band propagation with microbuckling causes a reduction of the normal stress transmitted across the band, which is properly regarded as a phenomenon of cohesive fracture, characterized by a certain kink band fracture energy, implying a certain characteristic length of the kink band FPZ, and a finite residual stress on the softening stress-displacement relation of the kink band. The results show that this mode of failure generally produces a significant energetic size effect, and so it is no surprise that a pronounced size effect is exhibited by the present tests.

6.3 Size Effect According to the deterministic size effect theory developed by Baˇzant [49] the allowable strength limit, ¢N , is given by: N D 1 .1 C rD b =D/1=r .Type 1/ N D 0 .1 C D=D0 /1=2 C r .Type 2/

(15) (16)

where 1 , r, Db , D0 , 0 , and r are constants (related to the geometry and properties of the material). The type 1 size effect law applies to failures at fracture initiation from a smooth surface, and type 2 to failures when a large notch or a large crack is present at maximum load. Parameter r represents the residual nominal strength of the specimen, due to frictional-plastic resistance after the fracture is fully formed. Usually r D 0 for tensile failures, but for compression failure r can be nonzero. The allowable strength limit N represents the maximum stress calculated from the elastic theory of bending, and is defined as: N D cP=bD

(17)

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where P is the maximum load, D D tc C2tf , where tc and tf represent the thickness of the core and facesheet, b is the width of the sandwich panel and c is given by: 6ey 6ez 1 C C 3 3 2tf =D C Ec tc =Ef D 2tf =D C Ec tc =Ef D .1  tc =D / C Ec tc3 =Ef D 3 (18) For the present sandwich geometry, the numerical value of c was 10:92. Because the present specimens have a sizeable notch, the type 2 size effect dominates and Eq. (16) applies. The sandwich specimens were compressed at a constant displacement rate, for each size equal to 0.01 in./min. The viscoelastic effects on N were expected to be unimportant. For each test the maximum load was measured. The values of measured nominal strength N , calculated from the maximum load P with c D 10:92, are shown by the data points in Fig. 16. In the same figure the optimum fits by the type 2 size effect law of Eq. (16) are shown by continuous lines and the asymptotes of this law are also marked. In the plots on top, it is assumed that there is no residual strength (r D 0), while in the plots at the bottom, the residual strength r is finite. While Fig. 16 shows the data in logarithmic scales, Fig. 17 shows the same data in the plots of 1=N2 or 1=.N  r /2 versus D=D0 , which are useful because Eq. (16) gets transformed, in such coordinates, to a linear regression plot. In such a plot, the cD

Fig. 16 Measured values (data points) of nominal strength ¢N of eccentrically compressed sandwich prisms of various sizes (thicknesses) D, plotted as log(N  r ) versus log D, and their fit by type 2 size effect law, Eq. (16). Left: Porous laminate skins from standard manufacture. Right: Nonporous laminate skins

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Fig. 17 The same data and fits as in Fig. 16, but replotted as 02 .N  r /2 versus D=D0 to obtain a linear regression plot

optimum (least-squares) fit of the data by Eq. (16) is easily obtained, along with the coefficient of variation ! of the data deviations from the regression line. The optimum values of D0 , 0 , and r obtained by regression are listed in each figure. The data plots in Figs. 16 and 17 make it clear that the size effect exists and is quite pronounced. Therefore, the current design procedures, which are based on the concept of material strength, are not justified for larger sandwich structures under compression.

6.4 Conclusions From the results, the following conclusions may be drawn: 1. The experimental results indicate that compressive failure of laminate-foam sandwich plates exhibits a significant size effect. 2. The thickness range of the tests performed corresponds to the thicknesses of load-bearing fuselage panels of small aircraft, while application to large ship structures will require extrapolation of the measured size effect.

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3. The observed size effect of the nominal strength of sandwich panels under compression is deterministic, due to the introduction of small notches in the laminate skins. The size effect observed can be explained only energetically, as a consequence of stress redistribution prior to the maximum load. 4. The typical porosity of facesheets is a manufacturing defect that makes no significant difference for the size effect.

7 Size Effect of Cohesive Delamination Fracture Triggered by Sandwich Skin Wrinkling [55] 7.1 Introduction Indentation failure under the load is a predominant failure mode of sandwich construction in cases where the applied load is distributed over a small area. Under such circumstances, significant local deformation of the loaded facing into the core of the sandwich construction takes place causing high local stress concentrations. The problem can be modeled as an elastic beam resting on a Winkler foundation. The stress field is obtained by superimposing the global bending of the sandwich construction and the local bending of the loaded face sheet about its own neutral axis. Buckling of imperfect quasibrittle structures generally leads to snapthrough instability which typically exhibits size effect on the nominal strength. The objective of this section is to show that skin imperfections (considered proportional to the first eigenmode of wrinkling) lead to strong size dependence of the nominal strength. For large imperfections, the strength reduction due to size effect can reach 50%. Dents from impact, though not the same as imperfections, might be expected to cause as a similar size effect. A secondary objective is to assess the size effect on the postpeak energy absorption, important for judging survival under blast or dynamic impact.

7.2 Size Effect The analysis of delamination in sandwich structures subjected to pure bending (Fig. 18a) can be simplified by modeling the skin as an axially compressed beam supported by a softening foundation consisting of independent continuously distributed nonlinear springs. This problem for bilinear elastic-softening response of the foundation was solved by Baˇzant and Grassl [55]. For the case when the wavelength of skin wrinkling Lcr h, where h represents the core thickness, the core may be regarded as an infinite half-space. The reason is that the alternating tractions applied on the core by the periodically wrinkled skin (Fig. 18b) are self-equilibrated over a segment of length 2Lcr where Lcr is the half wavelength of skin buckling

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Fig. 18 (a) The deflection of the top skin; (b) equilibrated stress acting on the foam; (c) equivalent height for shortwave; and (d) long wave wrinkling

(Fig. 18c). Therefore, according to the St. Venant principle, the stresses caused by periodic wrinkling must exponentially decay to nearly zero over a distance from the skin roughly equal to 2Lcr . On the other hand, when the critical wavelength Lcr  h (Fig. 18d) the sandwich beam is subjected to bending moment only (i.e., with no axial force). Then the opposite skin is under tension and may be approximated as a rigid base, with no deflection. The transverse compressive stress in the core is now almost uniform, and the foundation stiffness is constant (independent of the critical wavelength). Consider the load parameter  defined by  xDX

Es I s K

1=4 ;

D

1 P .KEs Is /1=2 2

where: P D axial load in the beam (per unit width) K D spring stiffness of the foundation per unit length

(19)

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Es D Young’s modulus of the skin Is D moment of inertia (per unit width) of the cross section of the skin The effect of the structure size on the relation between the load parameter  and the mid-point displacement wa D w .l=2/ is shown in Fig. 19 for three imperfection amplitudes ı D 0:1, 1, 2 (in which, l D L.Es Is =K/1=4 , where the beam length L is chosen to be 17Lcr . As one can see, the analytical results are in reasonable approximate agreement with the more accurate finite element results. The comparison shows that the size has a strong effect on the postpeak part of the load-displacement relation. The larger the size, the less energy is dissipated in relation to the energy dissipated by delaminating the entire skin. The size effect on the dimensionless nominal strength, N D max , shown in Fig. 20, has a form similar to the size effect law for crack initiation in quasibrittle

Fig. 19 Load  versus the midpoint displacement wa obtained with the softening foundation model and the finite element model for the imperfections: (a)  D 0:1; (b)  D 0:5; and (c)  D 1 for two sizes ( D 1 and  D 0:05)

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Fig. 20 Comparison of the size effect law in Eq. (20) with the nominal strength – size curves obtained from the softening foundation model for different imperfections

structures, which reads N D 1 Œ1 C 1=.k C /], where 1 has the meaning of nominal strength of an infinitely large structure. This law, however, is not directly applicable in this case. Therefore, a generalized law of the form

N .ı; / D 1 .ı/ 1 C

 1 ; k .ı/ D cı 4 k .ı/ C ˛ b

(20)

is proposed here, with constants a, b, c, d and parameters 1 and k, depending on the imperfection amplitude ı. For large sizes ( ! 1), the nominal strength is decided by initiation of cohesive crack (w D 1), and in that case we obtain N .ı; 1/ D 1 D 1=.1 C ı/

(21)

Note that here the large-size limit does not correspond to LEFM, which is the case for type 2 size effect [56], seen in specimens with notches or large stress-free cracks. Rather, in the absence of preexisting delamination crack, we see a particular case of type 1 size effect [56] because the geometry is positive causing failure to occur at crack initiation. For small sizes the nominal strength turns into:

1 N .ı; 0/ D 1 .ı/ 1 C k.ı/

 (22)

Parameters a, b, c, d in Eq. (20) are determined as the optimal fits of numerical results using the Marquardt–Levenberg algorithm for nonlinear least-squares optimization. The size effect law in Eq. (20) using these parameters is compared to the results of the softening foundation model in Fig. 20. The approximation is seen to be satisfactory.

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7.3 Conclusions From the results, the following conclusions may be drawn: 1. The delamination fracture of laminate-foam sandwich structures must be treated as a cohesive crack with a softening stress-separation relation characterized by both fracture energy and tensile strength. In contrast to LEFM, no preexisting interface flaw needs to be considered. 2. The skin of sandwich structures can be treated as a beam on elastic-softening foundation, provided that the equivalent (or effective) core depth heq for which the hypothesis of uniform transverse stress gives the correct foundation stiffness is considered to depend on the critical wavelength Lcr of skin wrinkles; heq D core thickness h for the asymptotic case of long wave wrinkling (Lcr = h ! 1), while (because of St. Venant principle) heq is proportional to Lcr for the asymptotic case of shortwave wrinkling (Lcr = h ! 0). 3. Although the nominal strength of sandwich structures failing by wrinklinginduced delamination fracture is size independent when there is no imperfection, it becomes strongly size dependent with increasing imperfection. A size effect causing strength reduction by 50% is possible for larger imperfections. Dents from impact may be expected to have a similar effect, even though they are not merely geometrical imperfections (because of being usually accompanied by initial delamination). 4. There is also a strong size effect on post peak energy absorption by a sandwich structure, both in the presence and absence of imperfections. This is important for impact and blast resistance. Acknowledgment Most of the work reviewed in this paper was sponsored by ONR from the program directed by Dr. Y.D.S. Rajapakse during the years 1994–2005.

References 1. Timoshenko SP (1953) History of the strength of materials, McGraw Hill, New York 2. Irwin GR, Wells AA (1965) A continuum mechanics view of crack propagation. Metal Rev 10: 223–270 3. Galileo Galilei L (1638) Discorsi e Dimostrazioni Matematiche intorno a` due Nuove Scienze, Elsevirii, Leiden; English transl. by Weston T, London (1730), pp. 178–181 4. Mariotte E (1686) Trait´e du mouvement des eaux, posthumously edited by M de la Hire; transl. by Desvaguliers JT, London (1718), p. 249; also Mariotte’s collected works, 2nd edn., The Hauge, 1740 5. Griffith AA (1921) The phenomena of rupture and flow in solids. Phil Trans Royal Soc Lond A221: 163–198 6. Griffith AA (1924) The theory of rapture. In: Proceedings of the First International Congress of Applied Mechanics, Delft, pp. 55–63 7. Stanton TE, Batson RGC (1921) Proc Inst Civ Eng 211: 67–100 8. Weibull W (1939) Phenomenon of rupture in solids. Proc Roy Swedish Inst Eng Res (Ingenioersvetenskaps Akad Handl) 153: 1–55

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9. Baˇzant ZP (1983) Fracture in concrete and reinforced concrete. In: Baˇzant ZP (ed.), Preprints Prager Symposium on Mechanics of Geomaterials: Rock, Concrete, Soils, Northwestern University Press, Evanston, IL, pp. 281–319 10. Baˇzant ZP (1984) Size effect in blunt fracture: concrete, rock, metal. J Eng Mech ASCE 110: 518–535 11. Baˇzant ZP (1982) Crack band model for fracture of geomaterials. In: Eisenstein, Z (ed.), Proceedings of the 4th International Conference on Numerical Methods and Geomechanics, Edmonton, Alberta, Canada, vol. 3, pp. 1137–1152 12. Baˇzant ZP, Oh BH (1983) Crack band theory for rupture of concrete. Mat Struct (RILEM, Paris) 16: 155–177 13. Baˇzant ZP, Belytschko TB, Chang T-P (1984) Continuum model for strain softening. J Eng Mech ASCE 110: 1666–1692 14. Baˇzant ZP (1984) Imbricate continuum and its variational derivation. J Eng Mech ASCE 110: 1693–1712 15. Pijaudier-Cabot G, Baˇzant ZP (1987) Nonlocal damage theory. J Eng Mech ASCE 113: 1512–1533 16. Baˇzant ZP, Pijaudier-Cabot G (1988) Nonlocal continuum damage localization instability and convergence. ASME J Appl Mech 55: 287–293 17. Baˇzant ZP, Lin F-B (1988) Nonlocal smeared cracking model for concrete fracture. J Eng Mech ASCE 114: 2493–2510 18. Baˇzant ZP, Lin F-B (1988) Nonlocal yield limit degradation. Int J Num Meth Eng 26: 1805–1823 19. Carpinteri A (1994) Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech Mater 18: 89–101 20. Baˇzant ZP (1997) Scaling of quasibrittle fracture: The fractal hypothesis, its critique and Weibull connection. Int J Fract 83: 41–65 21. Baˇzant ZP, Yavari, A (2005) Is the cause of size effect on structural strength fractal or energeticstatistical? Eng Fract Mech 72: 1–31 22. Baˇzant ZP, Yavari A (2007) Response to A. Carpinteri, B. Chiaia, P. Cornetti and S. Puzzi’s comments on “Is the cause of size effect on structural strength fractal or energetic-statistical?” Eng Fract Mech 74: 2897–2910 23. Baˇzant ZP, Daniel IM, Li Z (1996) Size effect and fracture characteristics of composite laminates. ASME Eng Mater Tech 118: 317–324 24. Waddoups ME, Eisenmann JR, Kaminski BE (1971) Microscopic fracture mechanisms of advanced composite materials. J Compos Mater 5: 446–454 25. Cruse TA (1973) Tensile strength of notched composites. J Compos Mater 7: 218–228 26. Mandell JF, Wang S-S, McGarry FJ (1975) The extension of crack tip damage zone in fiber reinforced plastic laminates. J Compos Mater 9: 266–287 27. Whitney JM, Nuismer RJ (1974) Stress fracture criteria for laminated composites containing stress concentrations. J Compos Mater 8: 253–264 28. Daniel IM (1978) Strain and failure analysis of graphite/epoxy plate with cracks. Exp Mech 18: 246–252 29. Daniel IM (1980) Behavior of graphite/epoxy plates with holes under biaxial loading. Exp Mech 20: 1–8 30. Daniel IM (1981) Biaxial testing of graphite/epoxy laminates with cracks. ASTM STP 734 Am Soc Test Mater 734: 109–128 31. Daniel IM (1982) Failure mechanisms and fracture of composite laminates with stress concentrations. In: VIIth International Conference on Experimental Stress Analysis, edited by SEM, Haifa, Israel, Aug. 23–27, pp. 1–20 32. Daniel IM (1985) Mixed-mode failure of composite laminates with cracks. Exp Mech 25: 413–420 33. Baˇzant ZP (1993) Scaling laws in mechanics of failure. ASCE J Eng Mater 119: 1828–1844 34. Baˇzant ZP, Zhou Y, Nov´ak D, Daniel IM (2004) Size effect on flexural strength of fiber composite laminates. ASME J Eng Mater Technol 126: 29–37

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35. Wisnom MR (1991) The effect of specimen size on the bending strength of unidirectional carbon fiber-epoxy. Compos Struct 18: 47–63 36. Jackson KE (1992) Scaling effects in the flexural response and failure of composite beams. AIAA J 30: 2099–2105 37. Wisnom MR, Atkinson JA (1997) Reduction in tensile and flexural strength of unidirectional glass fiber-epoxy with increasing specimen size. Compos Struct 38: 405–412 38. Johnson DP, Morton J, Kellas S, Jackson KE (2000) Size effect in scaled fiber composites under four-point flexural loading. AIAA J 38: 1047–1054 39. Baˇzant ZP, Li Z (1995) Modulus of rupture: Size effect due to fracture initiation in boundary layer. ASCE J Struct Eng 121: 739–746 40. Baˇzant ZP, Novak D (2000) Probabilistic nonlocal theory for quasi-brittle fracture initiation and size effect. I. theory and II. application. ASCE J Eng Mech 126: 166–185 41. Baˇzant ZP, Novak D (2000) Energetic-statistical size effect in quasi-brittle failure at crack initiation. ACI Mater J 97: 381–392 42. Baˇzant ZP, Kim J-JH, Daniel IM, Becq-Giraudon E, Zi G (1999) Size effect on compression strength of fiber composites failing by kink band propagation. Int J Fract 95: 103–141 43. Soutis C, Fleck NA, Smith PA (1991) Failure prediction technique for compression loaded carbon fibre-epoxy laminate with open holes. J Compos Mater 25: 1476–1497 44. Budiansky B, Fleck NA, Amazigo JC (1997) On kink-band propagation in fiber composites. J Mech Phys Solid 46: 1637–1653 45. Baˇzant ZP, Zhou Y, Zi G, Daniel IM (2003) Size effect and asymptotic matching analysis of fracture of closed-cell polymeric foam. Int J Solid Struct 40: 7197–7217 46. Gibson LJ, Ashby MF (1997) Cellular solid-structures and properties (2nd edn.), Cambridge University Press, Cambridge 47. Gdoutos EE, Daniel IM, Wang KA (2002) Failure of cellular foams under multiaxial loading. Compos A 33: 163–176 48. Gdoutos EE, Abot JL (2002) Indentation of cellular foams. In: Gdoutos EE (ed.), Recent Advances in Experimental Mechanics – In Honor of Isaac, M. Daniel, Kluwer, Academic Publishers, Dordrecht, The Netherlands, pp. 55–62 49. Baˇzant ZP (2005) Scaling of structural strength, 2nd edn., Elsevier, London 50. Baˇzant ZP, Pfeiffer PA (1987) Determination of fracture energy from size effect and brittleness number. ACI Mater J 84: 463–480 51. Baˇzant ZP, Kazemi MT (1990) Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. Int J Fract 44: 111–131 52. Baˇzant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials, CRC Press, Boca Raton/London (Sections 9.2 and 9.3) 53. Zenkert D, B¨acklund J (1989) PVC sandwich core materials: Mode I fracture toughness. Compos Sci Technol 34: 225–242 54. Bayldon J, Baˇzant ZP, Daniel IM, Yu Q (2006) Size effect on compressive strength of sandwich panels with fracture of woven laminate facesheet. J Eng Mater Technol 128: 169–174 55. Baˇzant ZP, Grassl P (2007) Size effect of cohesive delamination fracture triggered by sandwich skin wrinkling. ASME J Appl Mech 74: 1134–1141 56. Baˇzant ZP (2004) Scaling theory for quasibrittle structural failure. Proc Nat Acad Sci USA 101: 13400–13407

Elasticity Solutions for the Buckling of Thick Composite and Sandwich Cylindrical Shells Under External Pressure George A. Kardomateas

Abstract Thick composite and sandwich shells are used in many naval submersible structures and in other applications such as space vehicles. Stability under the prevailing high external pressure in deep ocean environments is of primary concern. In many other applications the loading involves a combination of external pressure and axial compression. It is well known that for these structures the simple classical formulas are in much error, due to both the large thickness and the large extensional over shear modulus ratios of modern composite and sandwich materials. Although there exist several advanced theories, such as first order shear and higher order shear theories, each based on a specific set of assumptions, it is not easy to determine the accuracy and range of validity of these advanced models unless an elasticity solution exists. This paper presents the research performed over the last 15 years on benchmark elasticity solutions to the problem of buckling of (i) orthotropic homogeneous cylindrical shells and (ii) sandwich shells with all constituent phases i.e., facings and core assumed to be orthotropic. The paper focuses on uniform external pressure loading. In this context, the structure is considered a three-dimensional body. The results show that the shell theory predictions can produce in many cases highly non-conservative results on the critical loads. A comparison with the corresponding formulas from shell theory with shear included, is also performed. The present solutions provide a means of accurately assessing the limitations of the various shell theories in predicting stability loss.

1 Introduction A class of important structural applications of fiber-reinforced composite materials involves the configuration of monolithic composite or sandwich composite shells. Such designs can be used in naval structures such as submersibles and these are G.A. Kardomateas () School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0150 e-mail: [email protected]

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normally characterized by considerable thickness. In addition to the high specific stiffness and strength, composite and sandwich construction can offer enhanced corrosion resistance, noise suppression and reduction in life-cycle costs. Other applications are as components in the aircraft and automobile industries and in space vehicles in the form of circular cylindrical shells as a primary load carrying structure. In these light-weight shell structures, loss of stability is of primary concern especially in the ocean deep environment of high external pressure. This is particularly important in sandwich construction because of the existence of the low-modulus core, which would be expected to make transverse shear effects even more significant than in homogeneous composites. Shell theory solutions for buckling and even initial post-buckling behavior have been produced by many authors (e.g., from the 1960s, Hutchinson [1]; Budiansky and Amazigo [2], many of these works with elegant variational formulations, and later by Simitses, Shaw and Sheinman [3]). Indeed, the existence of different shell theories underscores the need for benchmark elasticity solutions, in order to compare the accuracy of the predictions from the classical and the improved shell theories. Even from the early work on composites, it was shown that considerable care must be exercised in applying thin shell theory formulations to predict the response of composite cylinders [4, 5] due to the anisotropic and transverse shear effects. In order to more accurately account for the above mentioned effects, various modifications in the classical theory of laminated shells have generally been performed [6–8]. These higher order shell theories can be applied to buckling problems with the potential of improved predictions for the critical load. As far as sandwich shell theory, there are but few studies reported in the literature that deal with sandwich shell analyses [9–11]. This work summarizes the research performed by the author on elasticity solutions to the problem of buckling of cylindrical homogeneous orthotropic shells or sandwich shells with orthotropic phases. In particular, the case of uniform external pressure and orthotropic homogeneous material was formulated and solved in Kardomateas [12]; in this study, a long shell was studied (“ring” assumption). This simplifies the problem considerably, in that the pre-buckling stress and displacement field is axisymmetric, and the buckling modes are two dimensional, i.e., no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. The ring assumption was relaxed in a further study [13], in which a nonzero axial displacement and a full dependence of the buckling modes on the three coordinates was assumed. Other three-dimensional elasticity buckling studies are the buckling of a transversely isotropic homogeneous thick cylindrical shell under axial compression [14] and a generally cylindrically orthotropic homogeneous shell under axial compression [15]. The geometry of a circular cylindrical shell is particularly attractive for constructing elasticity solutions due to the axisymmetry which simplifies the analysis. Furthermore, a pre-requisite to obtaining elasticity solutions for shell buckling such as the one by Kardomateas [12], is the existence of three-dimensional

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elasticity solutions to the pre-buckling problem. Elasticity solutions for monolithic homogeneous orthotropic cylindrical shells have been provided by Lekhnitskii [16]. Regarding sandwich shells, elasticity solutions for sandwich shells were obtained by Kardomateas [17] by properly extending the solutions for monolithic structures. The latter is the pre-buckling solution needed to formulate the bifurcation problem in the elasticity context. Kardomateas and Simitses [18] formulated and solved the three dimensional elasticity problem for the buckling of a sandwich long shell under external pressure. For sandwich shells, the comparison to shell theory predictions will be based on the formulae presented in [19, 20] and specialized to an infinite length cylinder, whose behavior is similar to that of a sandwich ring. It should be mentioned that a few other researchers have addressed this topic, in particular, Soldatos and Ye [21] provided a solution based on a successive approximation method for homogeneous hollow cylinders subjected to combined axial compression and uniform external pressure. The results from the present study can be used to assess the accuracy of the classical shell theory and the existing improved shell theories for thick composite and sandwich shell construction.

2 Formulation At the critical load there are two possible infinitely close positions of equilibrium. Denote by u0 , v0 , w0 the r, and z components of the displacement corresponding to the primary position. A perturbed position is denoted by u D u0 C ˛u1 I

v D v0 C ˛v1 I

w D w0 C ˛w1 ;

(1)

where ˛ is an infinitesimally small quantity. Here, ˛u1 .r; ; z/, ˛v1 .r; ; z/, ˛w1 .r; ; z/ are the displacements to which the points of the body must be subjected to shift them from the initial position of equilibrium to the new equilibrium position. The functions u1 .r; ; z/, v1 .r; ; z/, w1 .r; ; z/ are assumed finite and ˛ is an infinitesimally small quantity independent of r, , z. The nonlinear strain displacement equations are: "

# @w 2 rr ; C C @r "      # 1 @v 1 @v 1 @w 2 1 @u v 2 u 1 u 2 ; C C C C  C 

D r @ r 2 r @ r r @ r r @ "   2  2 # @w @v @w @u 2 1 zz D ; C C C @z 2 @z @z @z 1 @u C D @r 2

@u @r

2



@v @r

2



(2a) (2b) (2c)

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G.A. Kardomateas







1 @u @v v @u 1 @u v C C  C  r @ @r r @r r @ r    1 @w @w u @v 1 @v C ; (2d) C C @r r @ r r @r @   @w @u @u @v @v @w @w @u C C C C ; (2e) rz D @z @r @r @z @r @z @r @z

     @v 1 @w @u 1 @u v @v 1 @v u 1 @w @w  z D C C  C C C : (2f) @z r @ @z r @ r @z r @ r r @ @z

r D

Substituting (1) into (2) we find the strain components in the perturbed configuration: 0 0 00 C ˛rr C ˛ 2 rr rr D rr



D zz D

0 0 00 r D r C ˛r C ˛ 2 r ;

0 0 00 

C ˛

C ˛ 2 

0 0 00 zz C ˛zz C ˛ 2 zz

0 0 00 rz D rz C ˛rz C ˛ 2 rz  z D  0z C ˛ 0 z C ˛ 2  00z ;

;

(3a) (3b) (3c)

0 where rr are the values of the strain components in the initial position of equilib0 00 are the rium, rr are the strain quantities corresponding to the linear terms and rr ones corresponding to the quadratic terms. The explicit expressions for these strains in terms of the displacements u0 , v0 , w0 and u1 , v1 , w1 can be found in Kardomateas (1993a). The stress–strain relations for the orthotropic body are

2

3 2 rr c11 6 7 6c 6

7 6 12 6 7 6 6 zz 7 6c13 6 7D6 6  z 7 6 0 6 7 6 4 rz 5 4 0 0 r

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

3 0 07 7 7 07 7 07 7 05 c66

3 rr 6 7 6

7 6 7 6 zz 7 6 7 ; 6  z 7 6 7 4 rz 5 r 2

(4)

where cij are the stiffness constants (we have used the notation 1 r, 2 , 3 z). Substituting (3) into (4) we get the stresses as: 0 0 00 rr D rr C ˛rr C ˛ 2 rr



D zz D

0 0 00 

C ˛

C ˛ 2 

zz0 C ˛zz0 C ˛ 2 zz00

0 0 00 r D r C ˛r C ˛ 2 r ; 0 0 00 rz D rz C ˛rz C ˛ 2 rz ; 0 0 2 00  z D  z C ˛ z C ˛  z ;

(5a) (5b) (5c)

where ij0 , ij0 , ij00 , are expressed in terms of ij0 , ij0 , ij00 , respectively, in the same manner as equations (4) for ij in terms of ij . In the following we shall keep in (5) and (3) terms up to ˛ i.e., we neglect the terms which contain ˛ 2 .

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Governing Equations The equations of equilibrium are taken in terms of the second Piola–Kirchhoff stress tensor † in the form   (6a) div †:FT D 0 ; where F is the deformation gradient defined by F D I C gradVE ;

(6b)

where VE is the displacement vector and I is the identity tensor. Notice that the strain tensor is defined by ED

 1 T F :F  I : 2

(6c)

More specifically, in terms of the linear strains: @u ; @r 1 @u @v v D C  ; r @ @r r

err D er

1 @v @w u C ; ezz D ; r @ r @z @u @v @w 1 @w erz D C ; e z D C ; @z @r @z r @

e

D

(7a) (7b)

and the linear rotations: 2!r D

1 @w @v  ; r @ @z

2! D

@u @w  ; @z @r

2!z D

v 1 @u @v C  ; @r r r @

(7c)

the deformation gradient F is 2

1 C err

6 F D 4 21 er C !z 1 e 2 rz

 !

1 e 2 r

 !z

1 C e

1 e 2 z

C !r

1 e C 2 rz 1 e  2 z

!

3

7 !r 5

(8)

1 C ezz

and the equilibrium equation (6a) gives:

    @ 1 1 rr .1 C err / C r er  !z C rz erz C ! C @r 2 2

    1 @ 1 1 C r .1 C err / C 

er  !z C  z erz C ! C r @ 2 2

    1 1 @ rz .1 C err / C  z C er  !z C zz erz C ! C @z 2 2

  1 1 rr .1 C err /  

.1 C e

/ C rz erz C !  C r 2    1   z e z  !r  2r !z D 0 ; 2

(9a)

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1 @ 1 1 r er C !z C 

.1 C e

/ C  z e z  !r C r @ 2 2

    1 1 @ rz C er C !z C  z .1 C e

/ C zz e z  !r C @z 2 2

    1 1 @ rr er C !z C r .1 C e

/ C rz e z  !r C C @r 2 2

      1 1 1 1 C rr er C !z C 

er  !z C rz e z  !r C r 2 2 2    1 erz C ! C r .2 C err C e

/ D 0 ; C  z 2 

    @ 1 1 rz erz  ! C  z e z C !r C zz .1 C ezz / C @z 2 2 

    1 1 @ rr erz  ! C r e z C !r C rz .1 C ezz / C C @r 2 2 

    1 1 1 @ r erz  ! C 

e z C !r C  z .1 C ezz / C C r @ 2 2 

    1 1 1 C rr erz  ! C r e z C !r C rz .1 C ezz / D 0 : r 2 2

(9b)

(9c)

0 0 Introducing the linear strains and rotations in the form (3), e.g. err = err + ˛err , 0 0 1 !z = !z + ˛!z , as well as the stresses from (5) and keeping up to ˛ terms, we obtain a set of equations for the perturbed state in terms of the eij0 , !j0 and eij0 , !j0 . Notice that in addition to the notations we adopted earlier, eij0 and !j0 are the values of eij and !j for u D u0 , v D v0 and w D w0 and eij0 and !j0 are the values for u D u1 , v D v1 and w D w1 . Since the displacements u0 , v0 , w0 , correspond to positions of equilibrium, there must exist also equations of the form (9) with the zero superscript, which are obtained by referring (6a) to the initial position of equilibrium. Thus, after subtracting the equilibrium equations at the perturbed and initial positions, we arrive at a system of homogeneous differential equations which are linear in the derivatives of u1 , v1 and w1 with respect to r, , z. This follows from the fact that ij0 , eij0 , !j0 appear linearly in the equation, and are themselves, in virtue of (7), linear functions of these derivatives. The system of equations, corresponding to (9), at the initial position of equilibrium, is, on the other hand, nonlinear in the derivatives of u0 , v0 , w0 . However, if we make the additional assumption to neglect the terms that have eij0 and !j0 as coefficients i.e. terms eij0 ij0 and !j0 ij0 , we can use the linear classical equilibrium equations to solve for the initial position of equilibrium. Moreover, if we make the assumption to neglect the terms that have eij0 and !j0 as coefficients i.e. terms eij0 ij0 and !j0 ij0 and furthermore, since a characteristic feature of stability problems is the shift from positions with small rotations to positions with rotations substantially exceeding the strains, if we neglect the terms eij0 ij0 thus keeping only the !j0 ij0 terms, we obtain the following buckling equations:

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345

 1 @  0  @  0 0 0 0 0 0 0 !z0 C rz ! C !z0 C  z ! C rr  r r  

@r r @  1 0  @  0 0 0 0 0 0 0 0 0 0 0 C rz ! C  z !r  2r !z0 D 0 ; rz   z !z C zz ! C rr  

C @z r (10a)    @ 1 @  0 0 0 0 0 0 0 0  C rr  0 C r !z  rz !r C !z0   z !r C @r r r @

 1 0  @  0 0 0 0 0 0 0 0 0 0 C  z C rz 2r C rr !z  zz0 !r0 C !z  

!z0 C  z !  rz !r D 0 ; @z r (10b)    @  0 @ 1 0 0 0 0 0 ! C r !r0 C ! 0 C 

!r0 C   rr  0  r @r rz r @ z  1 0  @  0 0 0 0 0 0 0 0 ! C  z !r C ! C r !r0 D 0 : zz  rz rz  rr C (10c) @z r

Boundary Conditions The boundary conditions associated with (6a) can be expressed as:   (11) F:† T :nO D tE.VE / ; where tE is the traction vector on the surface which has outward unit normal nO = .l; m; n/ before any deformation. The traction vector tE depends on the displacement field VE = .u; v; w/. Indeed, because of the hydrostatic pressure loading, the magnitude of the surface load remains invariant under deformation, but its direction changes (since hydrostatic pressure is always directed along the normal to the surface on which it acts). This gives    1 1 lC er  !z C rz erz C ! rr .1 C err / C r 2 2    

1 1 er  !z C  z erz C ! mC C r .1 C err / C 

2 2

    1 1 C rz .1 C err / C  z er  !z C zz erz C ! n D tr ; 2 2

    1 1 rr er C !z C r .1 C e

/ C rz e z  !r lC 2 2   

 1 1 mC er C !z C 

.1 C e

/ C  z e z  !r C r 2 2   

 1 1 n D t ; er C !z C  z .1 C e

/ C zz e z  !r C rz 2 2



(12a)

(12b)

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1 1 erz  ! C r e z C !r C rz .1 C ezz / lC 2 2    

 1 1 erz  ! C 

e z C !r C  z .1 C ezz / mC C r 2 2   

  1 1 erz  ! C  z e z C !r C zz .1 C ezz / n D tz ; C rz 2 2

rr

(12c)

If we write these equations for the initial and the perturbed equilibrium position and then subtract them and use the previous arguments on the relative magnitudes of the rotations !j0 we obtain:   0  0 0 0 0 0 0 0 rr  r !z0 C rz ! l C r  

!z0 C  z ! mC h i  0  1 0 0 C rz   z !z C zz0 ! 0 n D lim tr .VE0 C ˛ VE1 /  tr .VE0 / ; ˛!0 ˛  0   0  0 0 0 0 0 0 0 r C rr !z  rz !r l C 

C r !z0   z !r mC h i   0 1 0 0 t .VE0 C ˛ VE1 /  t .VE0 / ; C  z C rz !z  zz0 !r0 n D lim ˛!0 ˛   0   0 0 0 0 0 0 0 !r0  r ! 0 mC rz C r !r  rr ! l C  z C 

h i   1 0 0 0 0 C zz0 C  z !r  rz ! n D lim tz .VE0 C ˛ VE1 /  tz .VE0 / ; ˛!0 ˛ 

(13a)

(13b)

(13c)

Let nO 0 and nO 1 denote the normal unit vectors to the bounding surface at the initial and perturbed positions of equilibrium, respectively. Before any deformation, this vector is nO D .l; m; n/. For external pressure p loading at the initial position O I t .VE0 / D p cos.nO 0 ; O / I tz .VE0 / D p cos.nO 0 ; zO/ ; tr .VE0 / D p cos.nO 0 ; r/ (14a) and at the perturbed position tr .VE0 C ˛ VE1 / D p cos.nO 1 ; r/ O I

t .VE0 C ˛ VE1 / D p cos.nO 1 ; O / I

tz .VE0 C ˛ VE1 / D p cos.nO 1 ; zO/ :

(14b)

But in terms of the deformation gradient F0;1 :nO D .1 C En0;1 /nO 0;1 ;

(15)

where En0 , En1 is the relative elongation normal to the bounding surface at the initial and perturbed equilibrium positions, respectively. More explicitly, 

    1 1 0 1 0 0 0 0 O D .1 C err /l C e  !z m C e C ! n ; cos.nO ; r/ 1 C En0 2 r 2 rz 0

(16a)

Elasticity Solutions for the Buckling of Cylindrical Shells

O D cos.nO 0 ; /

cos.nO 0 ; zO/ D

1 1 C En0 1 1 C En0





347



1 0 0 /m C e C !z0 l C .1 C e

2 r 

1 0 e  ! 0 l C 2 rz



1 0 e C !r0 2 z





  1 0 e z  !r0 n ; 2

(16b)  0 m C .1 C ezz /n ; (16c)

Similar expressions hold true for the perturbed state. For example, cos.nO 1 ; r/ O D



   1 1 0 1 0 0 0 0 0 C ˛e /l C  !  ! .1 C e C ˛ mC e e rr rr z z 1 C En1 2 r 2 r      1 0 1 0 n ; (17) C erz C ! 0 C ˛ erz C ! 0 2 2

The assumption of small strains allows neglecting En0 and En1 in comparison with unity. Substituting into the expressions (14) for the tractions in terms of the pressure and subtracting the initial and perturbed state and using the same arguments on the magnitude of rotations to neglect eij0 in comparison with ! 0 , we arrive at the following expressions:   tr .VE0 C ˛ VE1 /  tr .VE0 / D p˛ !z0 m  ! 0 n ;   t .VE0 C ˛ VE1 /  t .VE0 / D p˛ !z0 l  !r0 n ;   tz .VE0 C ˛ VE1 /  tz .VE0 / D p˛ ! 0 l  !r0 m ;

(18a) (18b) (18c)

and in lieu of (13) for the lateral surfaces, i.e. for m D n D 0 and l D 1, 0 0 0 0  r !z0 C rz ! D 0 ; rr

(19a)

0 0 0 0 0 r C rr !z  rz !r D p!z0 ;

(19b)

0 0 0 0 rz C r !r0  rr ! D p! 0 ;

(19c)

3 Pre-buckling State Monolithic Composite Shell The problem at hands is that of a hollow cylinder rigidly fixed at its ends and deformed by uniformly distributed external pressure p (Fig. 1a). The axially symmetric distribution of external forces produces stresses identical at all cross sections and dependent only on the radial coordinate r. In this manner the forces at the ends are distributed identically over both surfaces and

348

G.A. Kardomateas

Fig. 1a A composite cylindrical shell under external pressure

reduce to equal and opposite resultant forces and moments. Let R1 be the internal and R2 the external radius and set c D R1 =R2 . Lekhnitskii [16] gave the stress field as follows:    r k1 pc k1 kC1 R2 kC1 C c ; R2 1  c 2k r  k1   p r pc k1 kC1 R2 kC1 0 D k  kc ; 

1  c 2k R2 1  c 2k r  k1 p r 0 .a13 C a23 k/ zz D .1  c 2k /a33 R2  kC1 pc k1 kC1 R2  .a  a k/c ; 13 23 .1  c 2k /a33 r 0 rr

p D 1  c 2k



0 0 0 r D rz D  z D0:

(20a) (20b)

(20c) (20d)

Equations 4 for the orthotropic constitutive behavior, where cij are the stiffness constants as well as the inverse relationship where aij are the compliance constants have been used, i.e.,

Elasticity Solutions for the Buckling of Cylindrical Shells

2

3

2 rr a11 6 7 6a 6

7 6 12 6 7 6 6 zz 7 6a13 6 7D6 6  z 7 6 0 6 7 6 4 rz 5 4 0 r 0

a12 a22 a23 0 0 0

a13 a23 a33 0 0 0

0 0 0 a44 0 0

0 0 0 0 a55 0

349

3 2

0 07 7 7 07 7 07 7 05 a66

3

rr 6 7 6

7 6 7 6 zz 7 6 7 : 6  z 7 6 7 4 rz 5 r

(20e)

Since the rotations at the initial position of equilibrium are either zero or of the same order as the strains, the classical linear elasticity equilibrium and strain– displacement equations can ordinarily be applied to the initial position of equilibrium. Hence, integration of the above stress field through linear strain–displacement relations (Eqs. 7) gives: u0 .r/ D D1 pr k C D2 pr k ; where

s kD

2 a11 a33  a13 D 2 a22 a33  a23

v0 D w0 D 0 ; r

c22 ; c11



1 a13 C a k  .a C a k/ D a 11 12 13 23 a33 .1  c 2k /kR2k1 1 ; D .c11 k C c12 /.1  c 2k /R2k1 

c k1 R1kC1 a13 D2 D   a k  .a  a k/ D a 11 12 13 23 .1  c 2k /k a33

(21)

(22)

D1 D 

D

c k1 R1kC1 : .c11 k C c12 /.1  c 2k /

(23a)

(23b)

Sandwich Shell In this case, the problem under consideration is that of a sandwich hollow cylinder deformed by uniformly distributed external pressure, p (Fig. 1b) and of infinite length (generalized plane deformation assumption). Then, not only the stresses, but also the displacements do not depend on the axial coordinate. Alternatively, this is the assumption we would make if the cylinder were securely fixed at the ends. An elasticity solution to this problem was provided by Kardomateas [17]. The solution is an extension of the classical one by Lekhnitskii [16] for a homogeneous, orthotropic shell and was provided in closed form. All three phases, i.e., the two face-sheets and the core were assumed to be orthotropic. Moreover, there were no restrictions as far as the individual thicknesses of the face-sheets and the sandwich construction could be asymmetric.

350

G.A. Kardomateas

Fig. 1b A cylindrical sandwich shell under external pressure

In this configuration, the axially symmetric distribution of external forces produces stresses identical at all cross sections and dependent only on the radial coordinate r. We take the axis of the body as the z axis of the cylindrical coordinate system and we denote by R1 and R2 the inner and outer radii. Let us also denote each phase by i where i D f2 for the outer face-sheet, i D c for the core and i D f1 for the inner face-sheet. Then, for each phase i , the orthotropic strain–stress relations are as in Eq. 20e with iji , aiji and iji in place of ij , aij and ij . Let us introduce the following notation for constants which enter into the stress formulas and depend on the elastic properties: i 2 i 2 a13 a23 i i I ˇ D a  .i D f1 ; c; f2 / ; (24a) 22 22 i i a33 a33 s i ˇ11 ai ai i D a12  13i 23 .i D f1 ; c; f2 / I ki D I .i D f1 ; c; f2 / : (24b) i a33 ˇ22

i i ˇ11 D a11 

i ˇ12

Then, the pre-buckling stresses in each of the phases, i.e. for i D f1 ; c; f2 , are:   .i/ .i / 0.i/ rr .r/ D p C1 r ki 1 C C2 r ki 1 ;

(25a)

Elasticity Solutions for the Buckling of Cylindrical Shells



0.i/

.i /

.i /

351





.r/ D p C1 ki r ki 1  C2 ki r ki 1 ; 0.i/

(25b)

0.i/

0.i/  z .r/ D rz .r/ D r .r/ D 0 ;

i i i i .i / .a13 C a23 ki / ki 1 .i / .a13  a23 ki / ki 1 r  C r zz0.i/ .r/ D p C1 : 2 i i a33 a33

(25c) (25d)

Furthermore, the pre-buckling radial displacement is found to be:

 i i i i .i/ .ˇ11 C ki ˇ12 / ki .i / .ˇ11  ki ˇ12 / ki r  C2 r ; u0.i/ .r/ D p C1 ki ki

(25e)

the other displacements being zero, i.e., v0.i/ .r/ D w0.i/ .r/ D 0. .i/ .i/ The constants C1 , C2 are found from the conditions on the cylindrical lateral surfaces (traction-free) and the conditions at the interfaces between the phases of the sandwich structure. Specifically, the traction conditions at the face-sheet/core interfaces give two equations [17]: .f1 /

C1

.f1 /

.R1 C f1 /kf 1 1 C2

.c/

.R1 C f1 /kf 1 1 D C1 .R1 C f1 /kc 1 .c/

CC2 .R1 C f1 /kc 1 ; (26a) and .c/

.c/

.f2 /

C1 .R2  f2 /kc 1 C C2 .R2  f2 /kc 1 D C1

.R2  f2 /kf 2 1

.f2 /

CC2

.R2  f2 /kf 2 1 : (26b)

The displacement continuity at the face-sheet/core interfaces, gives another two equations: f1 .f1 / .ˇ11

C1

f

f

f

1 1 C kf 1 ˇ121 / .f / .ˇ  kf 1 ˇ12 / .R1 C f1 /kf 1  C2 1 11 .R1 C f1 /kf 1 D kf 1 kf 1 c c c c .c/ .ˇ11 C kc ˇ12 / .c/ .ˇ11  kc ˇ12 / D C1 .R1 C f1 /kc  C2 .R1 C f1 /kc ; (27a) kc kc

and c .c/ .ˇ11

C1

c c c C kc ˇ12 / .c/ .ˇ11  kc ˇ12 / .R2  f2 /kc  C2 .R2  f2 /kc D kc kc

f2 .f2 / .ˇ11

D C1

f

f

f

2 2 C kf 2 ˇ122 / .f / .ˇ  kf 2 ˇ12 / .R2  f2 /kf 2  C2 2 11 .R2  f2 /kf 2 : kf 2 kf 2 (27b)

352

G.A. Kardomateas

Finally, the conditions of tractions at the lateral surfaces (traction-free inner surface and pressure, p, at the outer) give: .f1 /

C1

k

R1 f 1

1

.f / k 1 C1 2 R2 f 2

.f1 /

C C2 C

kf 1 1

R1

.f / k 1 C2 2 R2 f 2

D0;

(28a)

D 1 :

(28b) .i /

.i /

The six linear equations (6)-(8)can be solved for the six constants C1 , C2 , .i D f1 ; c; f2 /. Other details of the solution can be found in Kardomateas [17].

4 Perturbed State Monolithic Composite Shell In the perturbed position we seek plane equilibrium modes as follows: u1 .r; / D An .r/ cos n I

v1 .r; / D Bn .r/ sin n I

w1 .r; / D 0 :

(29)

Substituting in (7) we obtain 0 D kp.C1 r k1  C2 r k1 / I rr 0 zz

D

 0z

D

0 rz

D

0 r

  0 

D p C1 r k1 C C2 r k1 ;

D0:

(30a) (30b)

The first order strains are given explicitly in Kardomateas [12]. However, let us 0 examine the expression for rr :   0 0 @u1 D 1 C err rr C @r



1 0 e C !z0 2 r



@v1 C @r



1 0 e  ! 0 2 rz



@w1 : @r

Since all terms multiplied by eij0 or !j0 , can be neglected based on the arguments 0 0 = @u1 =@r = err . It turns out that we can use for the first order made previously, rr strains the very much simpler linear strains eij0 , i.e. ij0 = eij0 . Therefore, 0 0 D err D A0n .r/ cos n ; rr An .r/ C nBn .r/ 0 0 

D e

D cos n ; r

 Bn .r/ C nAn .r/ 0 0 D er D Bn0 .r/  sin n ; r r 0 0 0 D  z D rz D0; zz

(31a) (31b) (31c) (31d)

Elasticity Solutions for the Buckling of Cylindrical Shells

353

and the first order rotations are

 Bn .r/ C nAn .r/ 0 0 sin n ; 2!z D Bn .r/ C r ! 0 D !r0 D 0 :

(31e) (31f)

0 0 0 Substituting in (10) and using (4) and (5), e.g. rr = c11 rr + c12 

, we obtain the following two linear homogeneous ordinary differential equations of the second order for An .r/, Bn .r/:

  0  

c11 0 An 2 C C An  c66 C n C c22 r 2 r2      0 nBn0  0 nBn D0; C c12 C c66 

 c22 C c66 C

2 r 2 r2

c11 A00n

(32a)

and 

 0 0  00  

rr rrr Bn0 00 0 Bn C c66 C rr  C C c66 C 2 2 2 r  

00  0 0  

rr Bn nA0n rrr 2  c C c   c66 C c22 n C C C 12 66 2 2 r2 2 r 

00  0 nAn rrr  c66 C c22 C

C D0: (32b) 2 2 r2

The boundary conditions (19) are written as follows:  c12 D0; j D 1; 2 A0n .Rj /c11 C An .Rj / C nBn .Rj / Rj      0 C pj  0 C pj Bn C nAn D0: Bn0  c66  rr c66 C rr 2 2 Rj

(33a) (33b)

where pj D p for j D 2 i.e. r D R2 (outside boundary), and pj D 0 for j D 1 i.e. r D R1 (inside boundary). Sandwich Shell In the perturbed position we seek plane equilibrium modes for each phase in the same form as eqn (29), i.e. : u1i .r; / D A.i/ n .r/ cos n I i D f1 ; c; f2

v1i .r; / D Bn.i / .r/ sin n I

w1i .r; / D 0 ; (34)

Substituting these in the strain vs displacement Eqs. 31 and then using the stress– strain relations in terms of the stiffness constants, ciji , Eq. 4, with iji , ciji and iji in place of ij , cij and ij the buckling equations (10) result in two linear homoge.i / .i / neous ordinary differential equations of the second order for An .r/, Bn .r/. These

354

G.A. Kardomateas .i /

.i /

.i /

0.i/

0.i/

equations are the same as Eqs. 32 with An , Bn , cij , rr and 

in place of 0 0 An , Bn , cij , rr and 

. Notice that i D f 1 for R1  r  R1 C f1 ; i D c for R1 C f1  r  R2  f2 and i D f 2 for R2  f2  r  R2 . The associated boundary conditions are as follows: (a) At the inner and outer bounding surfaces, we have two traction conditions at .i / .i / .i / 0.i/ each of the surfaces, which are the same as Eqs. (33) with An , Bn , cij , rr 0.i/

0 0 and 

in place of An , Bn , cij , rr and 

, where j D f 1 and pj D 0 at r D R1 (inner bounding surface) and j D f 2 and pj D p at r D R2 (outer bounding surface). (b) At the face-sheet/core interfaces, we have the following four conditions at each of the interfaces:

Traction continuity: .j /

/0 c11 A.j C n

.j / .c/ c12  .j / c12  .c/ .c/ / An C nB.j An C nB.c/ D c11 A.c/0 ; n n C n r r

(35a)

and 0.j /

!

0.j /

!

.j /

.j /

Bn C nAn D r ! ! .c/ 0.c/ 0.c/ .c/ Bn C nAn rr rr .c/ .c/ .c/0 Bn  c66  : D c66 C 2 2 r

.j / c66

rr C 2

Bn.j /0



.j / c66

rr  2

(35b)

Displacement continuity: / .c/ A.j n D An I

Bn.j / D Bn.c/ ;

(35c)

where j D f 1 at r D R1 C f1 (inner face-sheet/core interface) and j D f 2 at r D R2  f2 (outer face-sheet/core interface).

5 Solution of the Eigen-Boundary-Value Problem for Differential Equations Monolithic Composite Shell Equations 32–33 constitute an eigenvalue problem for differential equations, with p the parameter (two point boundary value problem), which is solved by the relaxation method. To show the structure of these differential equations, if we define the constants C1 and C2 as follows: C1 D 

1 .1  c 2k /R2k1

I

C2 D

c 2k R2kC1 ; .1  c 2k /

(36a)

Elasticity Solutions for the Buckling of Cylindrical Shells

355

where c D R1 =R2 and k is defined in (22), then Eq. 32a becomes:   c11 0 c22 C n2 c66 k n2 k n2 k3 k3 C C  An C A  C1 pr C2 pr r n r2 2 2 

kn kn C1 pr k2 C C2 pr k2 Bn0  C n.c12 C c66 /  2 2 

kn kn C1 pr k3  C2 pr k3 Bn D 0 ;  n.c22 C c66 / C (36b) 2 2

c11 A00n

and Eq. 32b becomes:     C1 k2 C2 k2 C1 k1 C2 k1 c66 pr pr C pr pr Bn00 C Bn0  C C c66 C 2 2 r 2 2   C1 k3 C2 k3 c66 C n2 c22 C C pr pr Bn   r2 2 2 

n.c12 C c66 / nC1 k2 nC2 k2 0 An   pr pr   r 2 2 

nC1 k3 nC2 k3 n.c22 C c66 / C C (36c) An D 0 : pr pr  r2 2 2 The minimum eigenvalue is obtained for n D 2: An equally spaced mesh of 241 points was used to derive the results. The procedure is highly efficient with rapid convergence. An investigation of the convergence showed that essentially the same results were produced with even three times as many mesh points. Sandwich Shell Again, we have an eigenvalue problem for differential equations, 0.i/ 0.i/ 0.i/0 with p the parameter. An important point is that rr .r/, 

.r/ and rr .r/ depend linearly on the external pressure, p (the parameter) through expressions in the form of Eqs. 25 and this makes possible the direct application of standard solution techniques. The minimum eigenvalue is again obtained for n D 2: With respect to the method used there is a difference between the sandwich and the homogeneous body. The complication in the sandwich problem is due to the fact that the displacement field is continuous but has a slope discontinuity at the facesheet/core interfaces. This is the reason that the displacement field was not defined as one function but as three distinct functions for i D f 1, c and f 2, i.e. the two face sheets and the core. Our formulation of the problem employs, hence, “internal” boundary conditions at the face-sheet/core interfaces, as outlined above. Due to this complication, the shooting method [22] was deemed to be the best way to solve this eigen-boundary-value problem for differential equations. A special version of the shooting method was formulated and programmed for this problem. In fact, for each of the three constituent phases of the sandwich structure, we have five .i/ .i /0 .i / .i /0 variables: y1 D An , y2 D An , y3 D Bn , y4 D Bn , and y5 D p. The five differential equations are: y10 D y2 , the first equilibrium Eq. 32a, y30 D y4 , the second equilibrium Eq. 32b and y50 D 0.

356

G.A. Kardomateas

The method starts from the inner boundary r D R1 and integrates the five first order differential equations from R1 to the inner face-sheet/core interface R1 C f1 (i.e., through the inner face-sheet). At the inner bounding surface, R1 , we have three conditions, the two traction boundary conditions, Eqs. 33, and a third condition of .f 1/ D 1:0; therefore we have two freely specifiable (arbitrarily) setting y1 D An variables. The freely specifiable starting values at R1 are taken as the y5 (pressure), .f 1/ and the y3 D Bn and these are taken as the values from the shell theory (described later). Then, the three boundary conditions at R1 allow finding the starting values for y1 , y2 and y4 . Once we reach the inner face-sheet/core interface, R1 C f1 , the tractions from the inner face-sheet side are calculated; these should equal the tractions from the core side, according to the boundary conditions on the face-sheet/core interface, .c/0 Equations 35a,b. This allows finding the slopes of the displacements, y2 D An and .c/0 y4 D Bn for starting the shooting into the core (notice that the other three func.c/ .c/ tions, y1 D An , y3 D Bn and y5 D p are continuous according to Eq. 13c, and their values at R1 C f1 have already been found at the end of the integration step through the inner face-sheet). The next step is integrating the five differential equations from R1 Cf1 to R2 f2 , i.e. through the core. In a similar manner, once we reach the outer face-sheet/core interface, R2  f2 , the tractions from the core side are calculated; these should equal the tractions from the outer face-sheet side, per (35a,b), and this allows finding the .f 2/0 .f 2/0 and y4 D Bn for starting the shooting slopes of the displacements, y2 D An into the outer-face sheet (again, the other three functions are continuous and their values at R2  f2 have already been found at the end of the integration step through the core). The third step is the integration through the outer-face sheet. Once the outer bounding surface, R2 , is reached, the traction boundary conditions, Eqs. 33, which ought to be zero, are calculated. Multi-dimensional Newton–Raphson is then used to develop a linear matrix equation for the two increments to the adjustable parameters, y5 and y3 at R1 . These increments are solved for and added and the shooting repeats until convergence. For the integration phase, we used a Runge–Kutta driver with adaptive stepsize control. The method produced results very fast and without any numerical complication.

6 Results and Discussion Monolithic Composite Shell Let us determine the critical pressure for a composite circular cylinder of inner radius R1 D 1m, made of unidirectional carbon/epoxy with zero deg. orientation with respect to the hoop direction. The moduli in GN/m2 and Poisson’s ratios used (typical for a carbon/epoxy material) are listed below, where 1 is the radial (r), 2 is the circumferential ( ), and 3 the axial (z) direction: E1 D E3 D 10:3, E2 D 181:0; G12 D G23 D 7:17, G31 D 5:96; 12 D 0:0159, 23 D 0:280, 31 D 0:490.

Elasticity Solutions for the Buckling of Cylindrical Shells

357

Fig. 2a Critical pressure, pcr for a carbon/epoxy monolithic shell

Figure 2a shows the critical pressure as a function of the ratio of outside vs. inside radius R2 =R1 . The elasticity solution is compared with the predictions of classical shell theory [23]. It is seen that the critical load predicted by shell theory is more than twice the one from the elasticity solution for R2 =R1 D 1:30, and is more than three times the elasticity solution for R2 =R1 D 1:50. The direct expression for the critical pressure from classical shell theory is: pcr;cl D

E2 h3 .n2  1/ ; .1  23 32 / 12R03

(37a)

where R0 D .R1 C R2 /=2 is the mid-surface radius, and h D R2  R1 is the shell thickness. In Fig. 2a we also show the predictions of the shell theory corrected for transverse shear: h2 pcr;cl E2 pcr;shear D I ks D : (37b) 1 C 4ks .1  23 32 / 12G12 R02 The shell theory with shear is between the Elasticity and the Classical shell theory curves. But it offers much better accuracy than the classical shell. At R2 =R1 D 1:30, pcr;shear is 28% higher than the Elasticity and at R2 =R1 D 1:50 it is 40% higher than the Elasticity.

358

G.A. Kardomateas

The shell critical value, Eq. 37a, can be found by using the Donnell nonlinear shell theory equations [24] and seeking the buckled shapes in the form (29) where An .r/ D An , i.e. it is now a constant instead of function of r, and Bn .r/ = Bn C .r  R/ˇ with Bn being a constant, i.e. it admits a linear variation through the thickness. Since ˇ = .v1  u1; /=R, the latter can also be written in the form Bn .r/ = Bn C .r  R/.Bn C n/=R. As a consequence, we obtain the following shell theory buckling equations: u1; C v1;



h2 .u1;

 v1;

/ D 0 12R2

(37b)

pR.1  23 32 / h2 .u1;

 v1;

/ C .u1 C v1; /  .v1;  u1;

/ D 0: (37c) 12R2 E2 h Substituting the displacements from (29) and using the previous expressions for An .r/ and Bn .r/ results in the eigenvalue (37a) and the “eigenvectors” given by 

   h2 2 h2 n = n 1C An D 1I Bn D  1 C : 12R2 12R2

(37d)

A comparison of the An .r/ and Bn .r/, which define the eigenfunctions, as derived from the present elasticity solution and the shell theory assumptions of constant An .r/ and linear Bn .r/ can be found in Kardomateas [12] and a considerable nonlinearity in the functions An .r/ and Bn .r/ can be observed. To illustrate the effect of material constants, Fig. 2b shows the critical pressure for the same geometry and another material: the glass/epoxy material with moduli in GN/m2 and Poisson’s ratios as follows: E1 = E3 = 14.0, E2 = 57.0; G12 = G23 = 5.7, G31 = 5.0; 12 = 0.068, 23 = 0.277, 31 = 0.400. It is seen that the difference between elasticity and shell theory is noticeably smaller for the glass/epoxy case. At R2 =R1 = 1.30 the shell theory is 33% higher than the Elasticity and at R2 =R1 = 1.50 the shell theory is 70% higher than the Elasticity. The shell theory with shear is again between the classical shell and the Elasticity curves. The comparison of our elasticity solution was performed with the Donnell shell theory. It has been known [25] that the Donnell shell theory can produce in some instances inaccurate results (such as for long tude behavior), as opposed to the more elaborate Fl¨ugge theory [26] that provides more accurate predictions. However, for the problem under consideration, due to the assumed two dimensional buckling modes (i.e. no z component of the displacement field, and no z-dependence of the r and displacement components), both the Fl¨ugge and Donnell equations would give the same critical load. Indeed, the buckling equations for the Fl¨ugge shell theory would be [27]: the Eq. 37b without h2 the term 12R 2 .u1;

 v1;

/, and the equation (37a) with the first term being h2 12R2

2

h .u1;

C 2u1;

C u1 / instead of 12R 2 .u1;

 v1;

/. Substitution of the buckling modes (29) gives the same critical load, Eq. 37a, as the Donnell shell theory.

Elasticity Solutions for the Buckling of Cylindrical Shells

359

Fig. 2b Critical pressure, pcr for a glass/epoxy monolithic shell

For a more detailed comparison of the results for a range of radii ratios that would constitute practically moderately thick to thick shell construction, the calculation data are presented in tables in Ref. [12]. From the results presented previously, it can be concluded that predictions of stability loss in composite thick structures can be quite non-conservative if classical approaches are used. The degree of non-conservatism is dependent on the material and thickness, as illustrated in the examples presented above. Sandwich Shell For the sandwich shell, we consider the same geometry as before, i.e. R1 D 1:0 m and a range of ratios R2 =R1 . We further assume that the face sheets have the same thickness, f1 D f2 D 0:10h where h is the total thickness of the shell, therefore c D 0:80 h. The face sheets are assumed to be unidirectional carbon/epoxy with the same properties as above (i.e., with 0o orientation with respect to the hoop direction), and the core is made of alloy foam, which is isotropic with modulus Ec D 0:0459 GPa and Poisson’s ratio c D 0:33. Since the shell is considered to be very long, the buckling analysis reduces to that for a ring [19]. If the transverse shear effect is neglected, the expression for the pressure becomes (classical)

360

G.A. Kardomateas

pcr;cl D 3

.EI /eq : R03

(38a)

where .EI /eq is the equivalent bending rigidity, given in terms of the extensional moduli of the face sheets Ef and the core Ec by Ref. [28]: .EI /eq D



 f2 Ec c3 2 C C .f C c/ : c c f f .1  23 32 / 12 2.1  23 32 / 3 Ef f

(38b)

If the transverse shear effect is accounted for, then pcr;shear D

.EI /eq pcr;cl I ks D ; 1 C 4ks CR02

where

(39a)

Z KGdA;

C D

(39b)

A

K being a shear correction factor taken as equal to one and G is the transverse shear stiffness of the sandwich cross section. Two different expressions for C are employed herein. In the first case, it is asc and sumed that only the core contributes, in which case, C D cG12 ks1 D

.EI /eq ; c cG12 R02

(39c)

where Gc is the shear modulus of the core. N which In the second case, an effective shear modulus for the sandwich section, G, includes the contribution of the facings, is derived based on the compliances of the constituent phases [28]. The expression for GN is given by: c 2f C c 2f D f C c ; N G12 G G12

(39d)

f

where G12 , is the shear modulus of the facings. Therefore, in this case, ks2 D

.EI /eq ; N 2 .2f C c/GR

(39e)

0

In all cases, n D 2 was used in the buckling modes, Eqs. 29. This has been well established for isotropic cylindrical shells under external pressure; however, since we are dealing with a sandwich structure whose core has elastic properties that are orders of magnitude different from those of the face sheets, verification of this postulate was needed. Indeed, in all cases examined, an exhaustive search was made for the n that results in the minimum eigenvalue, and it was indeed found that n D 2 corresponds to the lowest eigenvalue.

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Fig. 3a Critical pressure, pcr , on a linear scale, for the sandwich case. The classical shell formula is not included

Figure 3a shows on a linear plot the variation of the critical pressure from the Elasticity and the two shell with shear formulas. We have intentionally left out the classical shell formula because the classical shell is orders of magnitude above these other solutions and, therefore, it would obscure the details of the shear curves. From this figure we can conclude that the second shear correction formula, the one based on ks2 , Eq. 39e, is more accurate. In general, both shear formulas perform in a satisfactory manner. The first shear formula, the one based on ks1 , is always conservative; the second shear correction formula, the one based on ks2 , may be non-conservative but again, it is the most accurate. To compare with the classical shell formula, we show on Fig. 3b all curves on a logarithmic plot. It is evident that the classical shell formula results in very high critical pressures, it is simply prohibitively in error: for R2 =R1 D 1:30 the pcr;cl is about 80 times the Elasticity value; the pcr;shear1 is almost half the Elasticity value but the pcr;shear2 is only 16% below the Elasticity value. If we compare the corresponding curves for the monolithic composite case, Fig. 2a, and the sandwich composite case, Fig. 3a, b, we conclude that the transverse shear effect is indeed very significant in the latter. In the geometries where the differences from the elasticity values in the monolithic composite case may be modest, even with the classical shell formula, it is seen that these differences are very large in the sandwich construction case. This illustrates the nature of sandwich construc-

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Fig. 3b Critical pressure, pcr , on a logarithmic scale, for the sandwich case which now includes the classical shell formula

tion, in which buckling is a more demanding issue. Detailed calculation data are presented in tables in Kardomateas and Simitses [18]. Acknowledgement The financial support of the Office of Naval Research, Mechanics Division, Grants N00014-91-J-1892, N00014-90-J-1995, N00014-0010323 and N00014-07-10373, and the interest and encouragement of the Project Monitor, Dr. Y. Rajapakse, are both gratefully acknowledged.

References 1. Hutchinson J.W. (1968) Buckling and Initial Postbuckling Behavior of Oval Cylindrical Shells Under Axial Compression, Journal of Applied Mechanics (ASME), vol. 35, pp. 66–72. 2. Budiansky B. and Amazigo J.C. (1968) Initial Post-Buckling Behavior of Cylindrical Shells Under External Pressure, Journal of Mathematics and Physics, vol. 47, no. 3, pp. 223–235. 3. Simitses, G.J., Shaw, D. and Sheinman, I. (1985) Stability of Cylindrical Shells by Various Nonlinear Shell Theories, ZAMM, Z. Angew. Math. u. Mech., vol. 65, no. 3, pp. 159–166. 4. Pagano N.J. and Whitney J.M. (1970) Geometric Design of Composite Cylindrical Characterization Specimens, Journal of Composite Materials, vol. 4, p. 360. 5. Pagano N.J. (1971) Stress Gradients in Laminated Composite Cylinders, Journal of Composite Materials, vol. 4, p. 260.

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6. Whitney, J.M. and Sun C.T. (1974) A Refined Theory for Laminated Anisotropic Cylindrical Shells, Journal of Applied Mechanics (ASME), vol. 41, no. 2, pp. 471–476. 7. Librescu, L. (1975) Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures, Nordhoff International, Leyden. 8. Reddy, J.N. and Liu C.F. (1985) A Higher-Order Shear Deformation Theory of Laminated Elastic Shells, International Journal of Engineering Science, vol. 23, no. 3, pp. 319–330. 9. Birman, V. and Simitses, G.J. (1999) Stability of Long Cylindrical Sandwich Shells with Dissimilar Facings subjected to Lateral Pressure, in Advances in Aerospace Materials and Structures, edited by G. Newaz, ASME AD-58, pp. 41–51. 10. Birman, V. and Simitses, G.J. (2000) Theory of Cylindrical Sandwich Shells with Dissimilar Facings Subjected to Thermo-Mechanical Loads, AIAA Journal, vol. 37, no. 12, pp. 362–367. 11. Birman, V., Simitses, G.J. and Shen, L. (2000) Stability of Short Sandwich Cylindrical Shells with Rib-Reinforced Facings, in Recent Advances in Applied Mechanics, edited by J.T. Katsikadelis, D.E. Beskos and E.E. Gdoutos, National Technical University of Athens, Greece, pp. 11–21, 2000. 12. Kardomateas, G.A. (1993) Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure, Journal of Applied Mechanics (ASME), vol. 60, pp. 195–202. 13. Kardomateas, G.A. and Chung, C.B. (1994) Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure Based on Non-Planar Equilibrium Modes, International Journal of Solids and Structures, vol. 31, no. 16, pp. 2195–2210. 14. Kardomateas, G.A. (1993) Stability Loss in Thick Transversely Isotropic Cylindrical Shells Under Axial Compression, Journal of Applied Mechanics (ASME), vol. 60, 1993, pp. 506–513. 15. Kardomateas, G.A. (1995) Bifurcation of Equilibrium in Thick Orthotropic Cylindrical Shells Under Axial Compression, Journal of Applied Mechanics (ASME), vol. 62, pp. 43–52. 16. Lekhnitskii S.G. (1963) Theory of Elasticity of an Anisotropic Elastic Body, Holden Day, San Francisco, also Mir Publishers, Moscow, 1981. 17. Kardomateas G.A. (2001) Elasticity Solutions for a Sandwich Orthotropic Cylindrical Shell Under External Pressure, Internal Pressure and Axial Force, AIAA Journal, vol. 39, no. 4, pp. 713–719, April 2001. 18. Kardomateas G.A. and Simitses G.J. (2005) Buckling of Long Sandwich Cylindrical Shells Under External Pressure, Journal of Applied Mechanics (ASME), vol. 72, no. 4, pp. 493–499, July 2005. 19. Smith, C.V. Jr. and Simitses, G.J. (1969) Effect of Shear and Load Behavior on Ring Stability, ASCE, Journal of EM Division, vol. 95, mo. EM 3, pp. 559–569, June 1969. 20. Simitses G.J. and Aswani M. (1974) Buckling of Thin Cylinders Under Uniform Lateral Loading, Journal of Applied Mechanics (ASME), vol. 41, no. 3, pp. 827–829. 21. Soldatos, K.P. and Ye, J-Q. (1994) Three-Dimensional Static, Dynamic, Thermoelastic and Buckling Analysis of Homogeneous and Laminated Composite Cylinders, Composite Structures, vol. 29, pp. 131–143. 22. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1989) Numerical Recipes, Cambridge University Press, Cambridge. 23. Ambartsumyan S.A. (1961) Theory of Anisotropic Shells, NASA Technical Translation N6422801-N64-22808. 24. Brush D.O. and Almroth B.O. (1975) Buckling of Bars, Plates, and Shells, McGraw-Hill, New York. 25. Danielson D.A. and Simmonds J.G. (1969) Accurate Buckling Equations for Arbitrary and Cylindrical Elastic Shells, International Journal of Engineering Science, vol. 7, pp. 459–468. 26. Fl¨ugge W. (1960) Stresses in Shells, Springer, pp. 426–432. 27. Simmonds J.G. (1966) A Set of Simple, Accurate Equations for Circular Cylindrical Elastic Shells, International Journal of Solids and Structures, vol. 2, pp. 525–541. 28. Huang H. and Kardomateas G.A. (2002) Buckling and Initial Postbuckling Behavior of Sandwich Beams Including Transverse Shear, AIAA Journal, vol. 40, no. 11, pp. 2331–2335, 2002.

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An Improved Methodology for Measuring the Interfacial Toughness of Sandwich Beams Qida Bing and Barry D. Davidson

Abstract Existing interfacial toughness tests are evaluated for their accuracy and suitability for application to a wide range of sandwich structures and environments. It is shown that geometric nonlinearities and/or axial load coupling can cause errors in the perceived toughness as obtained by many of these tests and their associated data reduction methods. A previously proposed modified peel test is selected to eliminate the effect of axial load and a new, modified beam theory based method of data reduction is developed. Experiments and nonlinear finite element analyses are used to show that this approach produces highly accurate values of toughness, even in the presence of geometrically nonlinear behaviors. Mechanically attached loading tabs are described, allowing the test to be used for a wide variety of structures under various simulated usage environments.

1 Introduction Interfacial crack growth is a critical mode of failure for foam core composite sandwich structures. Correspondingly, a large number of tests have been used to investigate interfacial growth behaviors and to determine the associated fracture toughness. Test methods that have been used or proposed include the cracked sandwich beam (also referred to as the asymmetric double cantilever beam) [1–4], the modified cracked sandwich beam [5–10], the tilted sandwich debond [11–15], modified peel [16,17], center-notched flexure [18–20], single cantilever three-point bend sandwich [21], single cantilever beam sandwich [22] and the shear three-point bend sandwich test [23, 24]. The tilted sandwich debond (TSD) test may be used at various tilt angles; it has the maximum amount of opening mode at a tilt angle of 0ı , and the amount of shear increases as the tilt angle increases. The shear three-point Q. Bing and B.D. Davidson () Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, NY 13244 e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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bend sandwich test is a shear-dominated test. The remaining test methods, including the TSD at 0ı , are aimed at obtaining toughness under primarily opening mode conditions. However, there currently is no consensus on preferred test methods, nor has there been a comprehensive assessment of the accuracies of the various tests. This creates difficulties in choosing a method for material characterization. It also complicates the usage of material data that is presently available, as it is unclear whether results by the various test methods can be directly compared. In view of the above, the goal of this study was to choose or develop an interfacial toughness test for sandwich structures, and an associated data reduction method, that could be used to obtain highly accurate toughness data under primarily opening mode conditions. Further, in support of on-going research, the method was to be applicable to a wide range of materials and environments, including both dry and seawater saturated laminates tested from C20ı C to 40ı C. In addition to accuracy and environmental suitability, a secondary criterion related to required face sheet thickness, i.e., all other things being equal, the ability to test laminates with relatively thin face sheets reduces the overall manufacturing burden of a test program. To address the above, a combined numerical and experimental approach is utilized, where finite element (FE) analyses are used to model a series of tests and perform virtual experiments. The trends and conclusions from this FE study are then validated through physical testing.

2 Test Methods Considered An initial evaluation was first performed on those test methods described above that produce primarily opening mode fracture. The goal of this evaluation was to select a limited number of methods for more in-depth study. The cracked sandwich beam (CSB) test was first considered. Due to its asymmetric geometry, the CSB is an inherently nonlinear test [25]. The amount of nonlinearity depends on the bending stiffness mismatch between the two cracked regions and may lead to large rotations of the specimen. This significantly complicates data reduction, and makes it quite difficult to accurately extract the fracture toughness, Gc . It also often produces large face sheet deformations that result in compression failures. For these reasons, the CSB test was not chosen for consideration. The modified cracked sandwich beam (MCSB) test, shown in Fig. 1a, was developed to overcome the difficulties associated with the CSB. As shown in the figure, in the MCSB the free end of the specimen is constrained from rotation. Although a small amount of nonlinearity was often observed in the load versus deflection data from the MCSB [6, 7], a compliance calibration method of data reduction was judged to provide accurate toughness values [9]. Compliance calibration (CC) is an attractive method of data reduction, as it only assumes linear elastic behavior and self-similar crack advance. The former assumption can be verified through examination of the load–displacement data, and the latter can be validated through crack length measurements at both edges of the specimen. When these conditions apply,

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Fig. 1 (a) MCSB, (b) TSD at 0ı , and (c) MP tests

CC is generally considered to produce highly accurate values of Gc . The MCSB was therefore chosen for additional study. The TSD (hereafter used to refer to the TSD at 0ı ), shown in Fig. 1b, is quite similar to the single cantilever beam sandwich test. The TSD is perhaps preferable, as it eliminates the flexural response of the uncracked region, the specimen is a bit shorter, and the bonded area of the specimen is larger than in the single cantilever beam sandwich test. It also appears that the TSD has been more widely used. For these reasons, the TSD was chosen for additional study. The modified peel (MP) test, shown in Fig. 1c, is similar to the TSD, but a linear bearing is used to prevent any axial force from being developed. As will be shown subsequently, evaluation of this test was necessary due to the problems that are created by the axial force in the MCSB and TSD tests. The two remaining methods, the single cantilever three-point bend sandwich and the center-notched flexure test were not evaluated. The former test has not been used extensively, and the work performed on the tests above indicated that it likely would not produce any significant advantages. The latter test was not chosen for further study due to its complexity, issues associated with having two crack tips, and its relatively limited usage.

3 Geometries Considered The three test methods chosen for in-depth evaluation are presented in Fig. 1. All specimens considered in the finite element (FE) and experimental studies were 25.4 mm wide with 25.4 mm thick DIAB Divinylcell H100 PVC foam core. The experimental study considered 6, 12 and 18 ply thick face sheets comprised of BGF

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7532 plain weave glass fabric with an areal weight of 241 g=m2 , whereas the FE study considered specimens with both woven glass and woven carbon face sheets that were 6 and 12 plies thick. All test specimens were manufactured by vacuum assisted resin transfer molding [5] using Derakane 411-350 vinylester resin. The 6 ply face sheets used a Œ0=90=0s stacking sequence, the 12 ply sequence was Œ0=90=0=0=90=0s , and the 18 ply layup was Œ0=90=0=90=0=90=0=90=0s . Here, 0ı is defined to be the warp direction and corresponds with the direction of crack advance. During manufacture, a 75 mm long, 13 m thick teflon insert was placed at one end of the panel at the interface between the core and one of the face sheets to serve as a starter crack.

4 Finite Element Modeling Linear and nonlinear (NL) FE analyses of the test geometries shown in Fig. 1 were conducted. Although loading hinges are shown in this figure, loading blocks were also considered but were shown to considerably increase the nonlinearities that were observed. Thus, all results that are presented in this work are for loading hinges. The material properties used for the various constituents are presented in Table 1. As indicated, two types of glass face sheet and one type of carbon face sheet were considered, for which the planar modulus and Poisson’s ratio are presented. Various choices of physically realistic through-thickness and shear modulus were evaluated for these materials and were found to have no influence on the conclusions that follow. The core, adhesive and loading hinge materials were all modeled as isotropic. The properties for glass face sheet (2) correspond to the specimens that were tested in this study, and the modulus that is presented corresponds to that obtained from flexural tests. All FE models were constructed and analyzed using ABAQUSTM V6.7. The core, face sheets, adhesive layer, and loading hinges were all discretely represented. The adhesive thickness was always modeled as 0.25 mm thick. This was based on the experiments that were conducted, where glass beads of this diameter were mixed in with the adhesive when bonding. In accordance with the experimental set-up, the loading hinges were 12.7 mm long and 1.52 mm thick. The hinge pin was oriented on the side of the hinge closer to the crack tip and had a diameter of 3.18 mm. Based on the test specimens that were manufactured, the 6 ply specimens were modeled as 1.45 mm thick, and the thickness of the 12 ply specimens was taken as 2.51 mm.

Table 1 Material properties Glass face sheet (1)

Glass face sheet (2)

Carbon face sheet

Modulus (GPa) Poisson’s ratio

25.4 0.11

47.5 0.04

10.8 0.11

Core Adhesive 0:065 2:6 0:337 0:30

Loading Hinges 200.0 0.29

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Finite element meshing was performed primarily using two-dimensional, fournoded, plane stress elements, with a few triangular elements in the region where the hinge flange connected to the pin and in other transitional locations. An extensive mesh convergence study resulted in a mesh with square elements at the crack tip with edge lengths of approximately 0.081 mm. Ten rows of these elements were placed on either side of the crack, i.e., within the face sheet and core. This refined region extended 3 mm in either direction from the crack tip. Away from the crack tip, six elements were used through the thickness of the 6 ply face sheets, 12 were used through the thickness of the 12 ply face sheets, and two elements were used through the thickness of the adhesive. The loading hinges used three elements through their thickness with a length-to-height ratio of approximately 1.0. Boundary conditions were imposed in accordance with the physical tests of Fig. 1. For the MCSB, the support roller at the uncracked end of the specimen was modeled as a rigid body. The contact between the roller and the specimen was frictionless with contact elements that were 1 ply thickness in height and just less than 1 mm long. Because crack advance occurred at a bimaterial interface, only total energy release rate (ERR) was evaluated. Linear mesh refinement studies showed that the mesh described above was sufficient to guarantee convergence in the ERR as determined by the virtual crack closure technique (VCCT) [26]. However, convergence studies with the NL models showed that the four-noded elements evidenced extensive drawing and distortion at the crack tip, and the VCCT did not produce converged results. For this reason, the J-integral approach was used to extract G. Comparisons of J with G as determined from a global energy balance approach indicated that this method gave highly accurate results and remained converged with increasing mesh density. Thus, all results for ERR were determined via J, which was considered the “true ERR.” Similar to the study of Ref. [27], these results were then compared to those from various simulated data reduction techniques for the materials in Table 1. Crack lengths from approximately 15 to 86 mm were considered, with focus on specific crack lengths, ao , of 25.4, 50.8 and 76.2 mm. Simulated CC was always performed by fitting compliance values from five different crack lengths. Specifically, the load versus deflection results were used from FE runs at ao , ao ˙ 5 mm, and ao ˙ 10 mm to obtain compliance, and then the ERR, G, was obtained using the relation [28] GD

P 2 @C 2B @a

(1)

where P is the load, C is the compliance, B is specimen width and a is crack length. For these comparisons, various toughness values between 350 and 1;400 J=m2 were assumed. For example, if Gc D 350 J=m2 , then the load versus deflection results at the five crack lengths considered for CC would be examined up to this ERR. For the specimens with 12 ply face sheets, the load versus deflection results were quite linear over this range at all crack lengths. In these cases, there was no problem defining compliance, C(a) would be curve-fit with an appropriate expression, and the value of load that corresponded to J D 350 J=m2 would be used to evaluate

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the CC method of data reduction using Eq. (1). In these situations, G by CC and J were found to be quite close. A similar approach would be used at higher values of assumed toughness. However, as the assumed value of Gc increased, nonlinearities began to become evident in the load versus deflection results. Unless otherwise stated, in these cases the compliance was obtained from the initial, linear portion of the curve. Here, the correspondence between G by CC and J was often poor. The associated physical problem is that, in practice, when geometric nonlinearities can occur and cause errors in the perceived toughness, then a simple criterion should be available for use in the interpretation of experimental results. This led to another measure by which the various tests and associated data reduction methods were evaluated: whether such a criterion could easily be found. Otherwise, it will be difficult to use the chosen approach with high confidence for a wide variety of geometries and environments.

5 MCSB Evaluation Initially, the MCSB test appeared the most promising. It is a simple test to run and various data reduction methods have been evaluated [9]. However, exploratory experiments on MCSB specimens evidenced a host of problems, including perceived toughnesses that depended strongly on face sheet thickness and crack length, bond line failures for many of the load tabs, and compression readings from the load cell at long crack lengths in the 6 ply specimens. It was hypothesized that these behaviors were due to a combination of the axial constraint and NL effects induced by large deformations. That is, the schematic representation of the MCSB test shown in Fig. 1a represents the typical situation in a standard load frame. Figure 2 shows the forces on the specimen, where the subscript “v” is used for “vertical.” The axial force arises because the lower loading pin wants to move to the right as a result of the horizontal shortening of the compliant (lower) leg as it deflects. However, the stiff upper leg does not deflect appreciably, and rotation of the specimen is constrained.

Fig. 2 Forces on the MCSB specimen

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Fig. 3 Reaction forces for 12 ply MCSB. (a) 25.4 mm, (b) 76.2 mm crack length

Therefore, a tensile force is created in the more compliant leg. By equilibrium, this gives rise to an equal magnitude compression force in the stiff leg. The vertical force in the uncracked region arises to counteract the moment created by the axial loads. For small deformations, linear FE analyses predict these forces reasonably well, but NL effects become significant as the opening displacement reaches only a small percentage of the vertical distance between pins in the undeformed condition. Note also that large values of the axial force are consistent with the observations of load tab debonding. Further, a large axial force will cause a large value of Puncracked . If Puncracked becomes sufficiently large, the direction of the vertical force at the load cell will change, i.e., a compressive load cell reading will be obtained, despite the fact that an opening load is applied at the actuator. This corresponds with the observed behaviors and was confirmed by NL FE analysis. Typical results are presented in Fig. 3, which shows the relationship between reaction forces for an MCSB specimen with 12 ply glass face sheets at different crack lengths. It is observed that the reaction force that will be measured is dramatically different than that applied at the actuator. This can cause the perceived value of toughness to differ appreciably from the true value, and the difference will be a function of specimen thickness, material properties, and crack length. Although this problem could be addressed by mounting a load cell on the actuator, we considered common data reduction techniques [9] applied to the NL FE results and generally found poor agreement with G as determined from J. These difficulties caused us to shift focus to the TSD and MP tests.

6 Preliminary Evaluation of the TSD Test A preliminary evaluation of the TSD test indicated a number of potential problems. First, FE analyses indicated that the magnitude of the axial force developed in the TSD is similar to that which occurs in the MCSB for the same specimen geometry

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and applied load. Despite this, for certain test geometries, such as a relatively stiff cracked leg, a short crack length and a moderate toughness, it was found that the load-deflection response will remain reasonably linear and CC will produce accurate values of Gc . The problem here is that the mode mixity is changing with increasing load. For interfaces that have a higher toughness under shear-dominated loading, this will affect Gc , and the interpretation of this effect will be quite difficult. For test specimens with more compliant cracked regions and/or a high interfacial toughness, nonlinearities were observed in the NL FE load versus deflection results. In these cases, the accuracy of Gc as determined by the CC method could be quite poor. Further, both of the above scenarios are influenced by the introduction of extension rods, which are needed to perform tests within an environmental chamber, and which make the effect of the axial force very difficult to quantify in any meaningful way. Finally, there is the required bonding of the lower portion of specimen to a rigid surface, which may be problematic under certain environmental conditions. That is, as will be described subsequently, methods of mechanically attaching both the hinge and base that are relatively insensitive to environment were developed for the MP test, but the axial force in the TSD may cause the solution for the base region to be inappropriate for this test. Nevertheless, to better understand these issues, a decision was made to move forward with experimental evaluations of the TSD using the recommended method [11] of data reduction as well as a few variations. A modified beam theory based method of data reduction, similar to that described below for the MP test, was also considered for the TSD. However, due to the issues described above, this only met with limited success and is not described.

7 Data Reduction in the MP Test A CC method of data reduction method was originally introduced for the MP that is identical to that used for composite double cantilever beam specimens [16]. Here, it is assumed that C D Ran , where R and n are obtained from a least-squares curve fit of the C(a) data. This expression for C is substituted into Eq. (1) to obtain a “load only” (LO) method of data reduction. A “load-deflection” (LD) method may be obtained by assuming that •c .a/ D C.a/Pc .a/, where •c is the critical deflection and Pc is the critical load. The resulting expressions are given by DCBC C D GLO

Pc2 R nan1 I 2B

DCBC C GLD D

nPc ıc 2Ba

(2)

These methods were evaluated using the NL FE results. Even for geometries where the load-deflection plots were essentially linear, it was found that both expressions gave relatively poor correlation with G from J. This was particularly true at the two shortest and two longest crack lengths used for CC, where the shape of the curve fit has the largest influence on slope. Thus, other curve-fits were considered. These consisted of second, third and fourth order polynomials in crack length and the “CC1” and “CC2” expressions typically used for the end-notched flexure test [27].

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Here, for CC1 it is assumed that C D A C ma3 , and for CC2, C D Co C C1 a C C2 a3 . Overall, the third order polynomial and CC2 gave the best results, but Gc as determined by these methods exhibited errors on the order of 10% at the few shortest and longest crack lengths. When considering specimens where geometric nonlinearities occurred, for example for Gc D 1;100 J=m2 for any of the 6 ply face sheet specimens or for the 12 ply face sheet specimen constructed from glass fabric (1), all of the CC methods were very poor. In view of the above, it was determined that a more robust method of data reduction was required. To this end, the modified beam theory (MBT) based LO and LD methods of data reduction developed for composite double cantilevered beam specimens [29] were modified for use with the MP test. Considering that only one cracked leg of the cracked sandwich beam is displacing in the MP test, the expression for ERR by the two methods are given by MBT GLO D

6Pc2 .a C h/2 F I B 2 h3 E1f

MBT GLD D

F 3Pc ıc 2B.a C h/ N

(3)

where h is the face sheet thickness and E1f is the flexural modulus. The expression for G by the LO method for the MP test differs by a factor of 2 from the expression in Ref. [29] due to the fact that only one cracked leg of the cracked sandwich beam is displacing. However, the displacement of only a single leg enters into the LD expression and this expression is therefore unchanged from that in Ref. [29]. The parameters F and N correct for the reduction in moment at the crack tip due to face sheet shortening and the presence of a load tab, and are given by   6 ı 2 3l1 ı  2 I F D1 5 a a " (4)  3  2 #   l2 l2 36 ı 2 9 ıl1 1   N D1 a 4 a2 a 35 a Here, • is the deflection, l1 is the distance from the center of the loading pin to the mid-plane of the delaminated face sheet, and l2 is half of the height of the end tab. If a hinge is used to introduce the load, then l2 D 0. These expressions are also modified from those presented in Ref. [29] to account for the fact that only a single leg is displacing. The crack length correction factor,  in Eq. (3), accounts for crack tip rotations, shear deformations and the beam on an elastic foundation effect. Both E1f and  are obtained from the slope and interception of the experimentally determined .C=N/1=3 versus a curve following an approach similar to that described in Ref. [29], but the following expression is used  1=3  1=3 C 4 D .a C h/ (5) N Bh3 E1f In the above, C is defined as the slope of the deflection versus load plot from the test at any given crack length.

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The key practical issue that arises when implementing the above approach lies in the determination of E1f and ¦. As described above, to obtain these values, one first determines the compliance of the MP specimen at each crack length, a. However, since there will be some nonlinearity in the load versus deflection plots, one will obtain different values of compliance, C, depending on the load and displacement level used for the linear regression analysis used to obtain C. In theory, the parameter N is intended to account for this, but in practice, different values of E1f and ¦ are obtained over different ranges of curve-fit. This effect is much more pronounced in the MP than in typical DCB testing because of the relatively larger deflections that occur, particularly with glass-reinforced face sheets at large crack lengths. In order to address the above, of all the geometries that were considered, those that showed nonlinearity in the NL FE prediction for load versus deflection (P-•) response were isolated. Various criteria for determining compliance were considered. Each was used to determine E1f and ¦ by Eq. (5). The ERR was then determined by the MBT-LO and MBT-LD methods and compared to J. Methods considered for fitting the compliance included constant load, constant displacement, constant ERR, and various compliance offsets. Of the methods evaluated, the most accurate approach, as well as the one that would be most easily implemented in actual testing, was a “95% compliance offset” method. Here, the compliance from the initial portion of the • versus P curve is computed and used to find the intersection of the loading line with the zero load point; in practice, this accounts for any initial nonlinear portion of the curve that is possibly associated with take-up of free play in the system. Next, a line of slope 0.95C (corresponding to a 5.2% stiffness increase) is projected from this intersection point until it intersects with the • versus P curve. That is, the NL FE analysis predicts that there will be a stiffening effect associated with moment arm shortening. The portion of the • versus P curve above any initial nonlinearities and up to this second intersection point is used to compute C/N for use in Eq. (5). If there is insufficient stiffening for the 0.95C and the loading lines to intersect, then the entire loading line (excluding any initial NL region) is used for the determination of C/N. To illustrate, Table 2 presents the error in the LO and LD methods of data reduction (i.e., in comparison to the results by J-integral) for glass (1) fabric and Table 2 Errors from LO and LD data reduction methods Glass face sheet Method Face sheet thickness LO 6-ply LO 6-ply LO 12-ply LO 12-ply LD LD LD LD

6-ply 6-ply 12-ply 12-ply

Gc .J=m2 /

Carbon face sheet

Max error (%) 3:40 11:47 2:12 6:57

F Avg F Min Mean error (%) 0:812 0:756 0:21 0:581 0:432 3:94 0:934 0:914 1:09 0:809 0:772 2:89

Max error (%) 0:97 5:70 1:82 3:90

F Avg F Min

350 1,100 350 1,100

Mean error (%) 2:84 9:81 1:50 5:83

0:912 0:800 0:969 0:916

0:885 0:703 0:962 0:899

350 1,100 350 1,100

2:56 8:00 1:79 5:20

2:95 10:69 2:15 5:38

0:812 0:581 0:934 0:809

0:73 2:62 1:21 2:72

1:47 3:97 1:78 3:45

0:912 0:800 0:969 0:916

0:885 0:703 0:962 0:899

0:756 0:432 0:914 0:772

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carbon fabric face sheets of 6 and 12 ply thicknesses with assumed values of Gc of 350 and 1;100 J=m2 . The mean and minimum errors refer to the average and minimum values, respectively, from all crack lengths studied. The average and minimum values of F, Eq. (4), are also presented and are discussed subsequently. This table shows that ERRs obtained by LD method are slightly more accurate than by the LO method. Since the carbon face sheets are significantly stiffer, little nonlinearity was displayed in the P–• plots, and ERRs obtained by both methods are fairly accurate for all thicknesses and values of Gc . However, for 6-ply glass face sheets in a material with Gc D 1;100 J=m2 , neither method is capable of extracting Gc with extremely high accuracy. Thus, in this type of a situation, testing specimens with face sheets that are at least 12 plies thick would be recommended. The above discussion brings up the interesting problem that, in practical situations one must know – without the benefit of NL FE analyses – what the accuracy of the method is. This information would be critical to deciding whether or not Gc could be extracted accurately from tests of sandwich structure of a given material and face sheet thickness. Alternatively, one could think of this type of information as being used to design appropriate tests. To address this, various “cut-off” criteria were evaluated. Methods considered included expressions based on F, N, and various nondimensionalized displacements. Once a candidate approach was found, additional geometries were then analyzed to make sure that the chosen method would be widely applicable. The most promising method, and one that could also be easily implemented in practice, was based on the parameter F. This is illustrated in Fig. 4. Figure 4a presents the error in the ERR as predicted by the LD method for all cases considered, and similar results for the LO method are presented in Fig. 4b. In practical applications, if one excludes results from tests where F < 0:73 due to potential data reduction errors, then the worst-case error based on all of our simulations would be 5:4% if the LD method is used and 6:8% if the LO method is used. Referring once again to Table 2, it was found that F is less than 0.73 by both methods for all crack lengths for the specimens with 6 ply glass (1) fabric face sheets and a toughness of 1;100 J=m2 . Thus, if one were to perform an experiment on specimens of this type, the first test (assuming growth over a range of crack lengths from  25 to 75 mm) would yield a mean value of F of around 0.58. Based

Fig. 4 Errors in ERR by (a) LD and (b) LO method for MP test

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on the cut-off criterion, this would provide the feedback that this test geometry is unacceptable. One would also expect to see reasonably large nonlinearities in the load versus deflection plots. For sandwich structure with 6 ply carbon fabric face sheets and a toughness of 1;100 J=m2 , the NL FE results indicate that F will be greater than 0.73 for crack lengths less than approximately 74 mm. Thus, if one was performing tests of this geometry, the cut-off criterion provides a means to know up to what crack length the values of Gc are expected to be reliable.

8 TSD and MP Experiments Prior to testing, marks were place on the edges of all TSD and MP specimens at specific distances ahead of the crack tip. During testing, the specimen was rapidly unloaded when the crack reached one of these marks. The crack length was then measured accurately at each edge, and the average of these measurements, along with the load at the instant that the test was stopped, were used in data reduction. In virtually all cases, the exact point of onset of advance from the teflon insert could not be identified, so this value was not used for the determination of toughness. Rather, when presenting resistance (R) curves, the first crack length corresponds to the first time the test was stopped. To eliminate any effects of panel-to-panel manufacturing variations, all specimens tested at a given thickness were cut from the same panel. Figure 5 presents load versus deflection plots from MP and TSD tests of specimens with 12 and 18 ply thick face sheets. It can be observed that the MP tests (Fig. 5a and d) are quite linear. The 12 ply TSD tests with an extension rod, Fig. 5b, were reasonably linear at short crack lengths, but became increasingly nonlinear as the crack extended. Increasing to an 18 ply thick specimen with an extension rod,

Fig. 5 Load versus defection plots. (a) 12 ply MP, (b) 12 ply TSD with extension rods, (c) 12 ply TSD, (d) 18 ply MP, (e) 18 ply TSD with extension rods, (f) 18 ply TSD

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Fig. 5e, improved the results considerably. As shown in Fig. 5c and f, significant amounts of NL were observed in both the 12 and 18 ply TSD tests without extension rods, and it is likely that the CC method of data reduction is not applicable. In fact, when the CC method was applied, these tests gave a very different toughness and R curve than that obtained from the tests where the extension rods were used. As such, toughness data from the TSD tests without extension rods are not presented. It is pointed out that when testing the TSD specimens with extension rods, an in-plane deflection of the rod tips was quite visible due to the shortening effect and associated axial load. The recommended data reduction method for the TSD test is a second order polynomial [11]. This was based on a goodness of fit assessment, rather than through a supporting analysis that indicated the functional relationship of ERR on crack length. Thus, we considered second, third and fourth order polynomials. The corresponding goodness of fit (r2 ) values were generally 0.998, 0.999 and 1.000, respectively. However, the resulting toughness values from the first 1–2 and last several crack lengths varied dramatically based on the curve fit that was chosen. Any significant NL in the load-deflection plot (e.g., at large a) accentuated these differences. Restricting consideration to only those crack lengths where the P–• plots were essentially linear resulted in different toughness predictions by the various curve fits for only the first 1–2 and last 1–2 crack lengths considered. This is where the different curve fits have the most effect on the slope of the C versus a curve (cf. Eq. (1)). Since it is unclear which curve fit predicts toughness accurately, in what follows TSD toughness data are only presented from those crack lengths where good agreement in Gc was obtained from the various compliance fits. Based on the NL FE evaluation, for the MP test, Gc was determined by the MBT-LD method. These agreed well with the MBT-LO results and, except for the first 1–2 crack lengths, with those from CC using a third order polynomial or the CC2 curve fit. Toughness results are presented in Fig. 6a for specimens with 12 ply thick face sheets and in Fig. 6b for specimens with the 18 ply face sheets. For both geometries, the range of “good data” is much larger from the MP than the TSD test. As is evident from a comparison of Fig. 5b and e, more nonlinearity occurs in the P-• data from the 12 than from the 18 ply TSD. Correspondingly, the range of R curve data by the TSD is longer for the 18 ply specimens.

Fig. 6 Toughness results from MP and TSD tests with extension rods. (a) 12 ply. (b) 18 ply

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Providing that extension rods are used with the TSD test, Fig. 6 shows good correspondence for Gc by the MP and TSD tests of 18 ply specimens. Here, the TSD test produces essentially linear P–• data. For the 12 ply specimens, the P–• data from the TSD test evidence nonlinearities and, despite the measures employed to ensure accuracy, the TSD produces higher toughness values. In addition to the effects of the nonlinearity, it is possible that this is strongly influenced by the different mode mixity in the two tests. That is, the horizontal forces due to face sheet shortening that arise in the TSD causes it to have a greater shear component than the MP. This effect increases with decreasing face sheet thickness due to the associated increase in deflection at fracture. Considering the MP test results only, the mean toughness from all specimens at all crack lengths for the 12 ply specimens is 1;138 J=m2 with a standard deviation of 86:6 J=m2 . The mean from the 18 ply specimens is 21% greater, with mean and standard deviations equal to 1,382 and 103:4 J=m2 , respectively. However, the difference between the 12 ply and 18 ply results do not necessarily reflect a thickness effect, as they are also affected by panel-to-panel variation. For example, in our current testing, we have obtained a mean toughness and standard deviation from specimens of a different 12 ply panel equal to 1,355 and 61:8 J=m2 , respectively. In view of this, it appears that the entire difference between the 12 and 18 ply results could be due to manufacturing variations between panels.

9 Mechanical Attachments As shown in Fig. 7, the MP test may readily be used with mechanical attachments. Figure 7a shows the hinge attachment. Here, a region of the core remote from the crack tip is cut away, and three fasteners and a backing plate are used to attach the hinge to the face sheet. The base plate attachment is shown in Fig. 7b. Wedge grips were machined with a step height just less than the smallest face sheet thickness that was tested. Each wedge is pressed into the core to a depth of approximately

Fig. 7 Mechanical attachments for the MP test. (a) Hinge. (b) Base

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2.5 mm. Plastic shim stock is placed beneath the wedge to account for variations in face sheet thickness among specimens. Tests of specimens with these attachments versus those that had bonded connections showed no difference in toughness.

10 Conclusions The modified peel test with the newly developed modified beam theory based data reduction method has been shown to produce highly accurate values of interfacial toughness for sandwich structures with composite face sheets. The MP test provides a larger range of accurate crack growth data from debonding tests than that which can be obtained using existing methods. It also eliminates the need to interpret whether data from a given increment of crack advance are acceptable due to the possibility of errors in dC/da. These may arise due to the sensitivity of the derivative near the extreme values of crack length used for the C(a) fit and/or due to geometric nonlinearities that effect the load versus deflection data. Considering the associated cut-off criterion that was developed for tests that show nonlinear behavior, the MP eliminates the need for subjective interpretations on whether or not some or all of the data from any particular test is “acceptable.” It also follows that the MP will allow the use of thinner face sheet specimens than existing test methods without a compromise in accuracy. Further, eliminating the axial load prevents toughness variations that may arise due to the associated change in mode mix with crack length. The MP is a predominantly opening mode test and should therefore provide a conservative measure of toughness for use in practical applications. Coupled with the method of mechanical attachment, the advantages of this test are quite significant, and serve to ensure that the MP test may be used for accurate determination of the interfacial toughness of all types of sandwich structures under a wide range of environmental conditions. Acknowledgment This work was supported by the Office of Naval Research under Contract N00014-07-1-0418, Dr. Yapa D.S. Rajapakse, Program Manager.

References 1. Prasad S, Carlsson LA (1994) Debonding and crack kinking in foam core sandwich beams – II Experimental investigations. Eng Fract Mech 47(6): 825–841 2. Carlsson LA, Matteson RC, Aviles F, Loup DC (2005) Crack path in foam cored DCB sandwich fracture specimens. Compos Sci Technol 6: 2612–2621 3. Grau DL, Qiu XS, Sankar BV (2006) Relation between interfacial fracture toughness and mode-mixity in honeycomb core sandwich composites. J Sandwich Struct Mater 8: 187–203 4. Matteson RC, Carlsson LA, Aviles F, Loup DC (2005) On crack extension in foam cored sandwich fracture specimens. In: Thomsen OT et al. (eds.), Sandwich structures 7: Advancing with sandwich structures and materials, 7th edn., Springer, The Netherlands

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5. Smith SA, Emmanwori LL, Sadler RL, Shivakumar KN (2000) Evaluation of composite sandwich panels fabricated using vacuum assisted resin transfer molding. In: Proceedings of the 43rd international society for the advancement of material and process engineering conference, Anaheim, CA 6. Smith SA, Shivakumar KN (2001) Modified mode-I cracked sandwich beam (CSB) fracture test. In: Proceedings of the 42nd AIAA/ASME/ASCE/AHS/ASC Structures structural dynamics and materials conference, Seattle, WA 7. Shivakumar KN, Smith SA (2004) In situ fracture toughness testing of core materials in sandwich panels. J Compos Mater 38(8): 655–668 8. Shivakumar KN, Smith SA (2002) Influence of core density and facesheet properties on debond fracture strength of foam core sandwich composites. In: Sun CT, Kim H (eds.), Proceedings of the American society for composites 17th technical conference, West Lafayette, IN 9. Shivakumar KN, Chen H, Smith SA (2005) An evaluation of data reduction methods for opening mode fracture toughness of sandwich panels. J Sandwich Struct Mater 7(1): 77–90 10. Kolat K, Neser G, Ozes C (2007) The effect of sea water exposure on the interfacial fracture of some sandwich systems in marine use. Compos Struct 78: 11–17 11. Li X, Carlsson LA (1999) The tilted sandwich debond (TSD) specimen for face/core interface fracture characterization. J Sandwich Struct Mater 1(1): 60–75 12. Viana GM, Carlsson LA (2003) Influences of foam density and core thickness on debond toughness of sandwich specimens with PVC foam core. J Sandwich Struct Mater 5: 103–118 13. Majumdar P, Srinivasagupta D, Mahfuz H et al. (2003) Effect of processing conditions and material properties on the debond fracture toughness of foam-core sandwich composites. Compos Part A: Appl Sci Manuf 34: 1097–1104 14. Li X, Weitsman YJ (2004) Sea-water effect on foam-cored composite sandwich lay-ups. Compos Part B: Eng 35: 451–459 15. Truxel A, Aviles F, Carlsson LA et al. (2006) Influence of face/core interface on debond toughness of foam and balsa cored sandwich. J Sandwich Struct Mater 8: 237–258 16. Cantwell WJ, Davies P (1994) A test technique for assessing core-skin adhesion in composite sandwich structures. J Mater Sci Lett 13: 203–205 17. Cantwell WJ, Davies P (1996) A study of skin core adhesion in glass fiber reinforced sandwich materials. Appl Compos Mater 3: 407–420 18. Ratcliffe J, Cantwell WJ (2000) A new test geometry for characterizing skin-core adhesion in thin-skinned sandwich structures. J Mater Sci Lett 19(15): 1365–1367 19. Ratcliffe J, Cantwell WJ (2001) Center notch flexure sandwich geometry for characterizing skin-core adhesion in thin-skinned sandwich structures. J Reinf Plast Compos 20(11): 945–970 20. Gates TS, Su X, Abdi F et al. (2006) Facesheet delamination of composite sandwich materials at cryogenic temperatures. Compos Sci Technol 66: 2423–2435 21. Cantwell WJ, Scudamore R, Ratcliffe J, Davies P (1999) Interfacial fracture in sandwich laminates. Compos Sci Technol 59: 2079–2085 22. Cantwell WJ, Broster G, Davies P (1996) The influence of water immersion on skin-core debonding in GFRP-balsa sandwich structures. J Reinf Plast Compos 15(11): 1161–1172 23. Carlsson LA, Sendlein LS, Merry SL (1991) Characterization of face sheet/core shear fracture of composite sandwich beams. J Compos Mater 25: 101–116 24. Shipsa A, Burman M, Zenkert D (1999) Interfacial fatigue crack growth in foam core sandwich structures. Fatigue Fract Eng Mater Struct 22(2): 123–131 25. Sundararaman V, Davidson BD (1997) An unsymmetric double cantilever beam test for interfacial fracture toughness determination. Int J Solids Struct 34(7): 799–817 26. Rybicki EF, Kanninen MF (1977) A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 9: 931–938 27. Davidson BD, Sun X (2006) Geometry and data reduction recommendations for a standardized end notched flexure test for unidirectional composites. J ASTM Int 3(9): 1–19 28. Broek D (1986) Elementary engineering fracture mechanics. 4th rev. edn., Kluwer, Dordrecht, The Netherlands 29. Hashemi S, Kinloch AJ, Williams JG (1990) The analysis of interlaminar fracture in uniaxial fibre-polymer composites. Proc R Soc Lond 427: 173–199

Structural Performance of Eco-Core Sandwich Panels Kunigal Shivakumar and Huanchun Chen

Abstract Eco-Core, a fire resistant core material for sandwich composite structures developed under the US Navy (ONR) program, was used to study its performance as a sandwich beam with glass/vinyl ester face sheet. Performance of Eco-Core was compared with balsa and PVC core sandwich panels. Test specimens were designed to simulate shear, flexural, and edgewise compression loadings. These tests were conducted on Eco-Core as well as balsa and PVC sandwich composite specimens. Failure loads and modes were compared with each other and the analytical prediction. Both Eco-Core and balsa cored sandwich beams had similar failure modes in all three test conditions. In the case of transversely loaded (four-point) beams Eco-Core specimens failed by core shear for span/depth .S=d / ratio less than 4 and the failure mode changed to core tension for S=d > 4. This is attributed to weak tensile strength of the core material. An expression for core tension failure load based on beam theory was derived. On the other hand, ductile materials like PVC failed by core indentation. Under edgewise compression, face sheet microbuckling and general buckling are the two potential failure modes for Eco-Core and balsa core sandwich composites. For specimen length/depth ratio L=d < 7 the failure is by face sheet microbuckling, for 7  L=d  13 the failure is a combination of face sheet microbuckling, debonding and buckling, and for L=d > 13 the failure is by general buckling. Predictions from the existing equations agreed well with the experiment for both core materials. For PVC core, wrinkling/shear buckling and general buckling are the potential failure modes. For L=d  8:5 the failure is wrinkling and for L=d > 8:5 the failure is general buckling.

K. Shivakumar () and H. Chen Center for Composite Materials Research, Department of Mechanical and Chemical Engineering, North Carolina A&T State University, Greensboro, NC 27411, USA e-mail: [email protected]

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1 Introduction Fire has been a major problem for both mobile (mass transit and marine) and immobile (buildings and civil infrastructure) structures. With the wide use of polymer composites in structural applications, potential for fire hazards has increased. Norwegian composite minesweeper fire [1] demonstrated the vulnerability of composite ships against fire. September 11, 2001 twin tower fire and collapse have demonstrated the vulnerability of our unprotected steel skyscrapers. Although fire cannot be completely eliminated, it can be mitigated to reduce the loss of life and property. Extensive research is being conducted to improve fire safety of composite materials for various applications. Some of these results are summarized by Sorathia and Perez [2] for naval applications. Shivakumar and his co-researchers [3–5] have proposed an alternate method of protection through blocking/containing the fire by using a fire resistant core material in sandwich structures. They also developed a fire resistant core called Eco-Core from a waste product from coal burnt thermal power plants. It is well known that sandwich structures are highly efficient in carrying flexural loads. Sandwich structures with composite face sheet and PVC or balsa core materials are widely used in marine applications. US and European navies are using or considering to use composite sandwich construction for building mine sweepers, coastal protection ships, destroyers, etc. Fiber reinforced plastic (FRP) composite material is structurally highly efficient, fatigue insensitive and corrosion proof but its composition of 50% by weight of resin makes it highly susceptible to fire. Like resin, PVC cores offer no resistance to fire. On the other hand balsa tolerates or inhibits the growth of fire but it suffers from non-uniform density (depends on the source and life time seasonal variation), large moisture ingression and rotting. Recent studies [6] have shown about 800% density change and 35% volume change in balsa in the presence of water that makes it susceptible for self destruction. The novelty of the Eco-Core [3–5] is that it uses little binder (high char yield content) and large volume of ceramic hollow microbubbles (Cenosphere), together press molded to required size and shape. The Eco-Core has superior mechanical properties, excellent fire resistant properties (passed Mil Spec 2031 up to 75 kW=m2 ) and is non toxic [3–5]. The focus of this research is an evaluation of structural performance of Eco-Core sandwich panel and its comparison with PVC and balsa core panels. This paper discusses design of test specimen for various types of failure, namely, shear, flexure and edgewise compression; verification of the design by experiments; identification of failure modes and its comparison with PVC and balsa core sandwich panels.

2 Design of Test Specimens Sandwich beam consists of a lightweight core material which is covered by face sheets on both sides. Figure 1 is the schematic of the sandwich beam cross section. The nomenclature used in the design of test specimens is defined: tc is the core

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Face sheet, sf, Ef

D = 2Df + D0 +Dc

tf d = tc + t f d

Core, sc, Ec, tc, Gc

tc

Df = D0 =

tf

Dc =

1 12

Ef tf 3

Ef tf d2 2 Ectc3 12

Fig. 1 Schematic diagram of the sandwich cross section and nomenclature Table 1 Material properties of face sheet and core materials Tensile Compressive Material Face sheet Eco-Core Balsa PVC

Modulus [GPa] 29.2 2.54 3.5 0.13

Strength [MPa] 512.5 6.46 13 3.5

Modulus [GPa] 31.9 1.14 3.92 0.135

Strength [MPa] 363.4 21.85 12.67 2.0

Shear Modulus [GPa] 4.0 0.97 0.16 0.035

Strength [MPa] 77.1 4.61 2.94 1.6

thickness, tf is the face sheet thickness and d is the sandwich thickness which is the distance between two centroids of the face sheets .d D tc Ctf /. The width of the panel is represented by b. Compression strength, elastic modulus, shear strength and shear modulus of the core are c ; Ec ; c and Gc , respectively. Strength and elastic modulus of the face sheet are f and Ef , respectively. The nomenclature used was the same as in the standard text books on sandwich structure [7, 8]. The face sheet considered is FGI 1854 glass/Derakane 510A-40 vinyl ester composite. The core materials considered are Eco-Core (nominal density of 0.5 g/cc), Balteck SB100 (density of 0.15 g/cc) and PVC foam core Divinycell H100 (density of 0.1 g/cc). Table 1 lists the mechanical properties of the face sheet [9] and core materials [3, 10, 11]. The nominal thicknesses of core and face sheet are 25.4 and 1.27 mm, respectively. The test specimens are designed to simulate typical failures of core shear, face yielding, face wrinkling, shear buckling, face microbuckling and general buckling that occur in sandwich panels as in Refs. [7, 8]. These failures can be simulated by short beam shear, flexure (four-point bend), and edgewise compression tests. Test specimens are designed using the equations in Refs. [7, 8] and the material properties listed in Table 1.

2.1 Short Beam Shear Test Specimen The short beam shear test was designed to measure shear strength of the core material. Four-point and three-point bend loaded testing are commonly used. Here we chose four-point bend with quarter point loading. Figure 2a shows the schematic of

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a P/2

P/2

b P/2

S L Shear test

P/2

Core shear failure mode

Fig. 2 Short beam shear test configuration and failure mode. (a) Shear test, (b) core shear failure mode

the short beam shear test. L is the specimen length, S is the span of the support and S=2 is the load span. The critical load at failure Pf is related to core shear strength and test specimen geometry and is given by: Pf D 2c db

(1)

The typical failure mode in short beam shear test is core shear, a 45ı cracking starting from the mid-thickness as illustrated in Fig. 2b.

2.2 Four-Point Flexure Test Specimen Figure 3a shows the schematic of four-point flexure test with third point loading. Under the flexure loading, sandwich beams are observed to fail by three modes, namely, face yielding, core shear and face wrinkling depending on span to depth ratio and face sheet-core thickness ratio, and properties of face sheet and core materials. Figure 3b illustrates the potential failure modes. The associated loads for all the modes are calculated using the following equations. (a) Face yielding When face yielding occurs, the stress in the face sheet exceeds the compressive yield strength of the face sheet. The failure load Pf can be calculated using the beam equation: 12D0 f b (2) Pf D SdEf where D0 is the flexural rigidity of the face sheet, which is defined in Fig. 1. (b) Core shear The failure load for core shear failure can be calculated from Eq. (1): Pf D 2c db

Structural Performance of Eco-Core Sandwich Panels Fig. 3 Four-point flexure test configuration and potential failure modes. (a) Four-point flexure test configuration, (b) potential failure modes

385

a

S/3 P/2

P/2

P/2

P/2 S L Four-point flexure test configuration

b Face yielding

Core shear

Face wrinkling

Face wrinkling Potential failure modes

(c) Face wrinkling When face wrinkling occurs, the stress in the face sheet f exceeds or equals to the critical wrinkling strength cr of the sandwich, which is given by:  1=3 (3) cr D 0:5 Ef Ec Gc The associated failure load for face wrinkling is:  1=3 6D0 Ef Ec Gc b Pf D SdEf

(4)

The failure load Pf for the three different failure modes were calculated (Eqs. (1), (2) and (4)) for the three core sandwich composites and are plotted as a function of span/depth (S=d ) ratio in Fig. 4. Because the face sheet yielding involves only face sheet properties, there is one curve for all three core materials. Face wrinkling for Eco-Core and balsa core occurs at very high load and large span, before that the specimen fails by face sheet yielding. Therefore, face yielding and core shear are the possible failure modes for

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800

Pf /b N/mm

Eco-Core Blasa PVC

600

400

200

0

Core Shear Eq. 1 0

5

10

15

20

25

30

35

S/d

Fig. 4 Failure load versus S=d design chart for different cores under four-point flexure

Eco-Core and balsa under four-point flexure loading. The failure mode is core shear for S=d less than 12 and 18 for Eco-Core and balsa, respectively. The failure mode is face compressive yielding for spans greater than the above limits. For PVC core, the failure load for face yielding is larger than that for face wrinkling. Therefore core shear (S=d  25) and face wrinkling (S=d > 25) are the possible failure modes for PVC core.

2.3 Edgewise Compression Test Specimen The test specimen and the possible failure modes under edgewise compression are shown in Fig. 5. The failure modes are general buckling, shear buckling, face sheet microbuckling (which is the same as face sheet yielding), face wrinkling and face dimpling. The associated failure loads for all the cases are summarized below. (a) General Buckling It is the buckling of a thin face sheet specimen derived from the Euler and shear bucking of column using a parallel spring model. The buckling load Pbuckling is given by:  2D ob (5) Pbuckling D n 2 c L2 C GcDt 2 d Equation (5) simplifies to Eq. (6) for soft core sandwich column. Psoft D

2 2 Df b L2

(6)

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a

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b

L

General buckling

Shear buckling

Test configuration

Face Face microbuckling wrinkling

Face dimpling

Possible failure modes

Fig. 5 Edgewise compression test configuration and possible failure modes. (a) Test configuration, (b) possible failure modes

An equation can be derived for thick face sheet in the form n Pthick D

2 2 Df D0 L4

n

n2  2 D0 L2

C

 2 DGc d 2 L2 tc

C

Gc d 2 tc

o

o b

(7)

(b) Shear buckling Shear buckling is a failure that occurs in low shear modulus (Gc ) core and/or in short columns. The shear buckling failure load Ps is expressed by: Ps D

Gc d 2 b tc

(8)

For thin face sheets the above equation can be approximated by: Ps D Gc tc b

(9)

(c) Face sheet microbuckling Face sheet microbuckling also referred to as face sheet compressive yielding occurs when the compressive stress within the face sheet exceeds the compressive strength of the face sheet material f c . Therefore, the critical face sheet microbuckling load is expressed by: Pcr D 2tf f c b

(10)

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(d) Wrinkling failure There are two types of wrinkling instability; symmetrical and anti-symmetrical modes of buckling of face sheets. The failure stress for symmetrical wrinkling is:  1=3 for h < tc =2 cr D 0:91 Ef Ec Gc s   Ef Ec tf tc cr D 0:817 for h  tc =2 C 0:166Gc tc tf

(11) (12)

where h is the depth over which the core is affected by the wrinkling of the face sheet. Critical stress for anti-symmetrical mode of face sheet buckling is:    1=3 tc C 0:33Gc for h < tc =2 cr D 0:51 Ef Ec Gc tf s   Ef Ec tf tc C 0:387Gc cr D 0:59 for h  tc =2 tc tf

(13) (14)

The critical stresses for all four possible wrinkling failures are shown in Fig. 6 for FGI 1854/Derakane 510A-40/Eco-Core sandwich panel. For this sandwich configuration, the height of the affected core zone h is less than tc =2, and tc =tf is 20. Therefore, symmetric wrinkling is more likely to occur and Eq. (11) can be used to estimate the critical wrinkling stress and the load. For practical design, the critical 1=3  wrinkling stress is approximated to cr D 0:5 Ef Ec Gc , which is the same as

40 Symmetrical @ h ≥ tc / 2

Eq. 12

30 Antisymmetrical @ h < tc / 2 Eq. 13

σcr, GPa 20

Anti-symmetrical @ h ≥ tc / 2 Eq. 14

10

0

Symmetrical @ h < tc / 2 Eq. 11

0

20

40

60 tc / tf

Fig. 6 Critical wrinkling stress as a function of tc =tf

80

100

Structural Performance of Eco-Core Sandwich Panels

389

the flexural critical wrinkling stress expression (Eq. (3)). The wrinkling failure load is expressed by:  1=3 b (15) Pw D tf Ef Ec Gc For panels, the state of stress is plane strain, therefore Ef may be replaced by Ef =.1  f2 ) and Ec may be replaced by Ec =.1  c2 /. Another expression for predicting wrinkling failure stress was proposed by Heath [12], modified by Gdoutos et al. [13] and is given by:

cr D

Ec Ef 2 tf 3 tc .1  13 31 /

1=2 (16)

where 13 and 31 are the Poisson’s ratios of the face sheet. Validity of this equation is also evaluated in this paper. Figures 7 through 9 show the plots of normalized critical load Pcr =b against specimen length/depth (L=d ) ratio corresponding to different failure modes under edgewise compression loading for the three core materials. For Eco-Core and balsa, face sheet microbuckling and sandwich column buckling are competing failure modes. For short specimens (L=d  15), face sheet microbuckling is the failure mode. For L=d  15, the failure mode is buckling of the sandwich column. In the case of PVC specimens, face sheet wrinkling and sandwich buckling are the failure modes. For L=d  8:5 wrinkling is the failure mode and for L=d  8:5 sandwich buckling is the failure mode.

6000 5000

ch . 5 dwi San ling Eq k Buc

Eco-Core

Wrinkling Eq.15

4000 Pcr /b 3000 N/mm

Wrinkling_Heath Eq.16

2000 Microbuckling Eq. 10 1000 0 0

10

15 L/d

20

Fig. 7 Edgewise compression failure load versus L=d design chart for Eco-Core

25

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K. Shivakumar and H. Chen Balteck SB100 6000 Shear Buckling Eq. 8 5000

Wrinkling_Heath Eq. 16

4000

Wrinkling Eq. 15

Pcr /b 3000 N/mm

Sa Eq ndwi . 5 ch B

2000

uc klin g Microbuckling Eq. 10

1000 0

0

5

10 L /d

15

20

Fig. 8 Edgewise compression failure load versus L=d design chart for balsa

PVC Divinycell H100 1200 Shear Buckling Eq. 8

Microbuckling Eq. 10 1000

Wrinkling_Heath Eq. 16 800 Pcr /b N/mm

San

dwicWrinkling Eq. 15 h Bu cklin g Eq .5

600 400 200 0

0

5

10

15

20

L /d

Fig. 9 Edgewise compression failure load versus L=d design chart for PVC

3 Fabrication of Sandwich Panel and Specimen The sandwich face sheets were unidirectional 3-ply FGI 1854 glass/Derakane 510A40 vinyl ester composite fabricated by Vacuum Assisted Resin Transfer Media (VARTM) process as explained in Ref. [14]. The thickness of face sheet achieved was 1.4 mm. Eco-Core, balsa and PVC core panels were adhesively bonded to the face sheets. The adhesive used was Loctite Hysol E-90FL epoxy adhesive. It is a toughened and medium viscosity adhesive with tensile strength of 13 MPa, lap shear

Structural Performance of Eco-Core Sandwich Panels

391

Core

700 μm

Saturation Layer Adhesive Layer 100 μm Face sheet

Fig. 10 Cross-sectional morphology of the face sheet-Eco-Core bonding region

Table 2 Specimen nominal geometry

Test Specimen geometry Length, L, mm Width, b, mm Thickness, d, mm Span length, S, mm

Short beam shear 152 51 27 102

Four-point flexure 178–610 51 27 127–508

Edgewise compression 127–279 25 14, 27 –

strength of 5.6 MPa and elongation of 64%. Vacuum bag was used for the bonding and the pressure applied was 20 inHg (0.068 MPa). The sandwich panel was kept in vacuum bag at room temperature for 8 h for the adhesive to be cured. Figure 10 shows the bonding interface morphology of the Eco-Core sandwich panel. The adhesive layer was about 100 m thick. Notice a layer of microballoons saturated by the adhesive and its thickness is about 700 m. After curing, the sandwich panels were cut into specimens for short beam shear, four-point flexure and edgewise compression tests. Table 2 lists the specimen nominal geometries for different tests.

4 Tests Short beam shear, four-point flexure and edgewise compression tests were conducted on the sandwich specimens made of FGI 1854 glass/Derakane 510A-40 face sheet and the three cores: Eco-Core, balsa and PVC. The tests were conducted in an Instron 4202 testing machine with a 10 kip load cell. During the tests, load and

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displacement were recorded continuously by data acquisition system and the failures of the specimens were monitored visually, by digital camcorder, and a high speed photography for some cases.

4.1 Short Beam Shear Test The short beam shear test was conducted in accordance with ASTM standard C39300. The specimens were 152 mm long, 51 mm wide and 27 mm thick. Quarter point loading was applied. The supporting span length was 102 mm and the upper loading span was 61 mm. Figure 11 shows the test setup. The specimens were loaded at a cross-head speed of 0.5 mm/min. The test was repeated for three specimens. The specimen number, geometric parameters, and failure loads for the three core materials are listed in Table 3. Eco-Core, balsa and PVC are denoted by “GE”, “GB” and “GP”, respectively, in the table.

4.2 Four-Point Flexure Test The four-point flexure test was conducted in accordance with ASTM standard C39300. The specimens were 51 mm wide and 27 mm thick. Third point loading was applied. The Eco-Core sandwich specimens were tested for five different span (S ) lengths, namely, 127, 191, 254, 394 and 508 mm. The balsa sandwich specimens

Fig. 11 Short beam shear test setup

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Table 3 Short beam shear test specimen number, geometries, and failure loads Failure load, Pf ; N Core

Specimen GE-SBS-1 EcoGE-SBS-2 Core GE-SBS-3 GB-SBS-1 Balsa GB-SBS-2 GB-SBS-3 GP-SBS-1 PVC GP-SBS-2 GP-SBS-3 a

tf S b tc [mm] [mm] [mm] [mm] S/d 102 50:9 25:4 1:4 3:8 102 50:6 25:4 1:4 3:8 102 52:5 25:4 1:4 3:8 102 50:3 25:4 1:4 3:8 102 51:0 25:4 1:4 3:8 102 51:0 25:4 1:4 3:8 102 50:9 25:4 1:4 3:8 102 50:7 25:4 1:4 3:8 102 51:0 25:4 1:4 3:8

Experiment Experiment average 11,085 12,206 12,064 (916a ) 12,900 6,348 7,203 (810) 7,958 7,304 5,209 5,298 5,268 (51) 5,298

Prediction Eq. (1) 12,582 12,511 12,975 8,002 8,119 8,123 Not shear failure

Standard deviation.

were tested for three different span lengths, namely, 203, 254 and 394 mm. The PVC specimens were only tested for span length of 394 mm. The specimens were loaded at a cross-head speed of 1.27 mm/min and load–displacement was continuously recorded. The specimen number, geometric parameters, and failure loads for the three sandwich core materials are listed in Table 4. Figure 12 shows the test setup.

4.3 Edgewise Compression Test Edgewise compression test was conducted in accordance with ASTM standard C364-00. The specimen ends were supported by two lateral support clamps made of rectangular steel bars fastened together to prevent the specimen from slipping from the fixture. Figure 13 shows the edgewise compression test setup. The two ends of the specimens were machined to be flat and parallel to each other and perpendicular to the loading. “L” represents the unsupported specimen length. The test was performed at the cross-head speed of 0.5 mm/min. Load and displacement data were recorded continuously and the specimen was monitored by a digital camcorder. The Eco-Core sandwich specimens were tested for six different unsupported specimen lengths (L) resulting in length/depth (L=d ) ratios of 5, 7, 11, 13, 18 and 20. 25.4 mm thick core was used for L=d ratios of 5 and 7 while 12.7 mm thick core was used to simulate long L=d ratios of 11, 13, 18 and 20. Balsa and PVC sandwich specimens were only tested for L=d ratio of 7. The specimen number, geometric parameters, and failure loads for the three sandwich core materials are listed in Table 5.

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Table 4 Four-point flexure test specimen geometric parameters and failure loads Failure load, Pf , N Core Specimen Eco-Core GE-flex-5in-1 GE-flex-5in-2 GE-flex-7.5in-1 GE-flex-10in-1 GE-flex-10in-2 GE-flex-10in-3 GE-flex-15.5in-1 GE-flex-15.5in-2 GE-flex-15.5in-3 GE-flex-20in-1 GE-flex-20in-2 GE-flex-20in-3 Balsa

PVC

GB-flex-8in-1 GB-flex-8in-2 GB-flex-8in-3 GB-flex-10in-1 GB-flex-10in-2 GB-flex-10in-3 GB-flex-15.5in-1 GB-flex-15.5in-2 GB-flex-15.5in-3 GP-flex-1 GP-flex-2 GP-flex-3

S [mm] 127 127 191 254 254 254 394 394 394 508 508 508

b [mm] 49.0 44.8 51.1 49.0 44.8 49.0 50.7 51.0 50.6 48.9 49.1 49.3

tc [mm] 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4

tf [mm] 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4

S/d 4.7 4.7 7.1 9.5 9.5 9.5 14.7 14.7 14.7 19.0 19.0 19.0

Experiment 10,792 10,212 7,144 4,688 4,893 5,293 3,643 3,452 3,879 1,984 2,260 2,451

Predicted core tension 9,468 8,644 6,529 4,684 4,277 4,687 3,130 3,153 3,124 2,338 2,345 2,354

203 203 203 254 254 254 394 394 394 394 394 394

50.2 50.3 50.3 50.4 50.3 50.2 49.3 51.3 51.1 51.1 50.9 50.7

25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4

1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4

7.6 7.6 7.6 9.5 9.5 9.5 14.7 14.7 14.7 14.7 14.7 14.7

4,742 5,022 5,009 4,902 4,186 6,205 5,476 4,889 4,524 2,985 3,203 3,007

8,981 9,000 8,986 7,203 7,200 7,185 4,553 4,743 4,723 Core indentation failure

Fig. 12 Four-point flexure test setup

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395

L

Fig. 13 Edgewise compression test setup Table 5 Edgewise compression test specimen geometries and failure loads Core Specimen L [mm] b [mm] tc [mm] tf [mm] L/d Failure load, Pcr [N] Eco-Core GE-EC-5in-1 127 26.1 25.4 1.4 4.7 25,577 GE-EC-7in-1 178 25.5 25.4 1.4 6.6 29,265 GE-EC-7in-2 178 24.2 25.4 1.4 6.6 26,013 GE-EC-7in-3 178 25.5 25.4 1.4 6.6 29,545 GE-EC-H-6in-1 157 25.9 12.7 1.4 11.1 23,811 GE-EC-H-6in-2 157 25.7 12.7 1.4 11.1 22,984 GE-EC-H-6in-3 157 26.0 12.7 1.4 11.1 24,029 GE-EC-H-7in-1 178 25.8 12.7 1.4 12.6 23,464 GE-EC-H-7in-2 177 25.8 12.7 1.4 12.6 18,932 GE-EC-H-10in-1 254 24.1 12.7 1.4 18.0 19,946 GE-EC-H-10in-2 256 25.2 12.7 1.4 18.1 21,378 GE-EC-H-10in-3 254 25.1 12.7 1.4 18.0 21,583 GE-EC-H-11in-1 279 22.5 12.7 1.4 19.8 19,577 GE-EC-H-11in-2 279 25.1 12.7 1.4 19.8 18,989 GB-EC-1 178 24.5 25.4 1.4 6.6 18,585 Balsa GB-EC-2 178 23.8 25.4 1.4 6.6 19,252 GB-EC-3 178 23.8 25.4 1.4 6.6 18,144 GP-EC-1 178 23.8 25.4 1.4 6.6 18,620 PVC GP-EC-2 178 23.8 25.4 1.4 6.6 17,949 GP-EC-3 178 24.1 25.4 1.4 6.6 16,049

5 Test Results and Discussion 5.1 Short Beam Shear Test Figure 14 shows the load–displacement responses of three different core sandwich specimens. The load–displacement curves of Eco-Core and balsa specimens show

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K. Shivakumar and H. Chen 20000

15000

Eco-Core

Load, N 10000 Balsa PVC

5000

0

0

0.5

1

1.5 2 Displacement, mm

2.5

3

Fig. 14 Short beam shear load–displacement response of three core materials

a sudden load drop, similar to brittle fracture, at failure while PVC specimens experienced yielding and large local deformation before final failure. Maximum load before the first load drop was considered as the failure load for Eco-Core and balsa. For PVC, failure was by indentation instead of shear hence the maximum load is not compared with the shear failure load. Furthermore, there is no indentation failure equation for four-point short beam shear test. Table 3 summarizes the experimental failure loads, the average value and the standard deviation as well as the predicted failure loads based on Eq. (1). Predicted average shear failure loads from Eq. (1) are 12,689 and 8,081 N for Eco-Core and balsa, respectively. These predictions are about 5% and 12% larger than the average test data for Eco-Core and balsa, respectively. This difference is within the experimental scatter that is expected for these materials. Figure 15 illustrates the failure modes of the three core sandwich specimens. The Eco-Core specimen demonstrated typical shear failure forming a 45ı crack in the core, between the top load point and the bottom support at one end. This was followed by immediate crack at the other end. Balsa core specimen had vertical shear cracks in the core. Figure 15c shows a clear indentation failure of PVC sandwich specimen.

5.2 Four-Point Flexure Test The Eco-Core sandwich specimens were tested for span/depth ratios of 5, 7, 10, 15 and 20. None of the specimens failed by the possible modes illustrated in Fig. 3. The failure was initiated by vertical cracks at the bottom of the core within the two upper loading points as core tension failure. Figure 16 shows the typical

Structural Performance of Eco-Core Sandwich Panels

a

397

b

Eco-Core

Balsa

c

PVC

Fig. 15 Shear failure modes of the three core material sandwich beams. (a) Eco-Core, (b) Balsa, (c) PVC 10000 4 8000 3 6000

Core Tension Failure

Load, N 4000

1

2

2000

0

0

1

2

3

4

Displacement, mm

Fig. 16 Typical four-point flexure load–displacement response of Eco-Core beam

5

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a 2

1

3

Initial cracks in the core

b 4

Ultimate failure

Fig. 17 Crack initiation, propagation and ultimate failure of Eco-Core sandwich panel under fourpoint flexure loading. (a) Initial cracks in the core, (b) ultimate failure

load–displacement plot of Eco-Core specimen. Each load drop in the plot matches the crack formation illustrated in Fig. 17 captured by Photron FASTCAM MC2 high speed camera at 8,000 frames/s. The three cracks shown in Fig. 17a are numbered in the order of appearance during the test. Corresponding load drops are shown in Fig. 16. The maximum load before the first load drop (#1) is considered to be the failure load. Figure 17b shows the ultimate failure which is the shear crack followed by face sheet-core separation. Core tension failure was not reported, heretofore, as a failure mode in sandwich panels. Because the Eco-Core is relatively brittle and is stronger in compression than in tension the core tension failure occurred. The core tension failure load can be derived using the beam theory and it is given by: 12Dct b Pf D (17) S tc Ect where ct and Ect are the tensile strength and modulus, respectively, of the core material. Predicted core tension failure loads and the experimental data are listed in Table 4. Predicted failure loads versus span/depth ratio for all possible failure modes are plotted in Fig. 18 along with the experimental data. The test data agrees very well with the predicted core tension failure loads. Therefore, core tension is the major failure mode for Eco-Core or brittle core sandwich composite panels. The failure is tensile for S=d > 4 and core shear for S=d  4. Typical flexure load-deflection responses of the three cored sandwich composites .S=d D 15/ are shown in Fig. 19. Similar to Eco-Core, core shear and core tension are the possible failure modes for balsa core specimens. The failure is core shear for S=d  7:6 and core tension for S=d > 7:6. Balsa core specimens were tested for S=d D 8, 10 and 15. Figure 20 compares predicted and experimental failure loads. For S=d D 15, the experimental failure load was 98 N/mm which agreed well with the predicted core tension failure load of 92 N/mm. For S=d D 8 and 10, the failure was at the transition from core shear to core tension. The experimental results were lower than the predictions. This is associated with the scatter of the core properties. Figure 21a shows the typical core tension failure of the balsa specimen.

Structural Performance of Eco-Core Sandwich Panels

399 Fa

ce

Eco-Core

ce

rin

kli ng

Eq

.4

om C

ld Yie

o si es pr

ing

400

W

e or C

Fa

500

n

. Eq 2

300

Core Shear Eq. 1

Pcr/b N/mm 200

100 Core Tension Eq. 17 0

0

5

10

15

20

25

30

35

S/d

Fig. 18 Comparison of experiment and theory for four-point flexure loaded Eco-Core sandwich beam

6000 5000 Eco-Core

Balsa

4000 Load, N 3000

PVC

2000 1000 0

0

5

10

15

Displacement, mm

Fig. 19 Load–displacement response of four-point flexure tests

PVC core specimens were only tested for S=d D 15. The PVC did not fail by core tension, it failed by face sheet indentation under the loading point as shown in Fig. 21b. The comparison of predicted and experimental results is illustrated in Fig. 22. The experimental failure load was lower than the core shear and face wrinkling failure loads. Modeling of indentation failure has been studied by a number of researchers [15–17] for three-point bend loading, a similar equation is needed

400

K. Shivakumar and H. Chen Balteck SB100 ce Fa

500

Fa

ce

gE ldin Yie

400

Wr

ink

lin

gE

q.

4

2 q.

300 Pcr / b N/mm 200

Core Shear Eq. 1 100

0

Core Tension

0

5

10

15

20

25

Eq. 17

30

35

S /d

Fig. 20 Comparison of experiment and theory for four-point flexure loaded balsa sandwich beam. (a) Balsa, (b) PVC

a

b

Balsa

PVC

Fig. 21 Flexure failure modes of balsa and PVC core sandwich beams

for four-point bending. A simplified indentation equation which is independent of loading type for elastic-rigid plastic material proposed by Soden [15] is given by: 4 p (18) btf ys c 3 where ys is the yield strength of the core. Using this equation, the predicted indentation failure load for PVC specimen is 52 N/mm while the experimental result is 60 N/mm. The 14% difference may be attributed to limitation of the Soden model. Pcr D

Structural Performance of Eco-Core Sandwich Panels

401

PVC Divinycell H100

Co

re

Fa

Te

sio

nE

ce

300

e Yi

q.

ld

17

in g

C mp ore res sio n

Co

. Eq

200

2

Fa ce W

rin

Pcr /b N/mm

kli n

gE

q.4

100

Core Shear Eq. 1

Indentation Eq. 18 0

0

5

10

15

20

25

30

35

S/d

Fig. 22 Comparison of experiment and theory for four-point flexure loaded PVC sandwich beam 40000

Eco-Core

30000

Load, N 20000 Balsa 10000 PVC 0

0

1

2 3 Displacement, N

4

5

Fig. 23 Load–displacement responses of edgewise compression test

5.3 Edgewise Compression Test Figure 23 shows the typical load–displacement plots of the edgewise compression test for the three core materials. The predicted and experimental failure loads are plotted versus specimen length/depth (L=d ) ratio in Figs. 24 through 26. The symbols represent the experimental data while the lines represent the predicted values. Eco-core specimens were tested for L=d of 5, 7, 11, 13, 18 and 20. The

402

K. Shivakumar and H. Chen h 5 dwic San ling Eq. k Buc

Eco-Core 5000

Wrinkling Eq. 15

4000 Wrinkling_Heath Eq. 16

3000 Pcr /b N/mm 2000

Microbuckling Eq. 10 1000

0

0

5

10 L /d

15

20

Fig. 24 Comparison of experiment and theory for edgewise compression loaded Eco-Core sandwich column

Balteck SB100 6000 Shear Buckling Eq. 8 5000 Wrinkling_Heath Eq. 16 4000

Wrinkling Eq. 15

Pcr /b 3000 N/mm

Sa nd

wic hB uck ling

2000

Eq

.5 Microbuckling Eq. 10

1000 0

0

5

10

15

20

L/d

Fig. 25 Comparison of experiment and theory for edgewise compression loaded balsa sandwich column

experimental failure loads (Pf =b) of Eco-Core for L=d of 5, 7 and 11 were 980, 1,129, and 912 N/mm, respectively. These results agreed well with the predicted face sheet compressive microbuckling failure load (1,015 N/mm) from Eq. (10). The face sheet microbuckling failure is shown in Fig. 27a. For L=d D 5, initial and ultimate

Structural Performance of Eco-Core Sandwich Panels

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PVC Divinycell H100 1200 Shear Buckling Eq. 8

Microbuckling Eq. 10 1000 Wrinkling_Heath Eq. 16 800 Pcr /b N/mm

Wrinkling Eq. 15 dwic h Bu cklin g Eq .5

San

600 400 200 0

0

5

10

15

20

L/d

Fig. 26 Comparison of experiment and theory for edgewise compression loaded PVC sandwich column

a

b

Microbuckling

c

Combination failure General buckling

Fig. 27 Edgewise compression failure modes of Eco-Core sandwich specimens. (a) Microbuckling, (b) combination failure, (c) general buckling

failures were clearly due to compressive microbuckling of face sheet whereas for L=d D 7 and 11, the failure was a combination of face sheet microbuckling, face sheet-core separation/debonding and buckling (see Fig. 27b). For L=d D 13, 18 and 20, failures were a mixture of sandwich general buckling and face sheet separation. One specimen each of L=d D 18 and 20 showed clearly general buckling of the sandwich column (see Fig. 27c). Based on the test results one can conclude that for L=d < 7 the failure is by face sheet microbuckling, for 7  L=d  13 the failure is combination of face sheet microbuckling, debonding and buckling, and for L=d > 13 the failure is by general buckling.

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a

b

Typical failure mode

Face sheet microbuckling

Fig. 28 Edgewise compression failure modes of balsa sandwich specimens. (a) Typical failure mode, (b) face sheet microbuckling

a

b

Typical failure mode

Face sheet failure

Fig. 29 Edgewise compression failure modes of PVC sandwich specimens. (a) Typical failure mode (b) face sheet failure

Balsa and PVC specimens were only tested for L=d D 7. Balsa core showed a similar failure as that of Eco-Core. The failure was a combination of face sheet microbuckling and face sheet-core separation as seen in Fig. 28. For PVC specimens, the test data agreed more closely with wrinkling failure load by Eq. (15). For L=d  8:5 the failure is wrinkling and for L=d > 8:5 the failure is general buckling. Figure 29 shows the failure mode of PVC specimen. Fleck and Sridhar’s experimental data [18] for Divinycell H100 core and glass/epoxy sandwich specimens is shown in Fig. 30 to compare their test data with the present test data. The authors ignored the face sheet thickness and approximated Eq. (8) by Eq. (9) in calculating the shear buckling failure load. Results by both Eqs. (8) and (9) are shown in Fig. 30 and the difference between the two is about 20%. For their specimen configuration, experimental data agreed equally

Structural Performance of Eco-Core Sandwich Panels

405 ckl Bu ing

800

.5 Eq

700

Microbuckling Eq. 10

Shear Buckling Eq. 8

600 500

Wrinkling Eq. 15

Pcr / b 400 N/mm

Shear Buckling Eq. 9

300 200 100 0

0

5

10

15

20

25

30

35

40

L/d

Fig. 30 Comparison of experiment and theory for edgewise compression loaded PVC sandwich columns [18]

well with both wrinkling (as shown above from the present test) and shear buckling (ignoring the face sheet thickness effect) equations. Ignoring the face sheet effect is arbitrary and not justifiable. If the face sheet thickness effect is considered then the wrinkling equation (Eq. (15)) better represents the test data. For large L=d of 35, as expected, the experimental data agreed more closely with general buckling mode Eq. (5).

6 Concluding Remarks Eco-Core, a fire resistant core material for sandwich composite structures developed under the US Navy (ONR) program, was used to study its performance as a sandwich beam with glass/vinyl ester face sheet. Performance of Eco-Core was compared with balsa and PVC core sandwich panels. Test specimens were designed to simulate shear, flexural, and edgewise compression loadings. Tests were conducted on Eco-Core as well as balsa and PVC sandwich composite specimens. Failure loads and modes were compared with each other and the analytical predictions. Both Eco-Core and balsa cored sandwich beams had similar failure modes in all three test conditions. In the case of transversely loaded (four-point) beams Eco-Core specimens failed by core shear for span/depth (S=d ) ratio less than 4 and the failure mode changed to core tension for S=d > 4. This is attributed to weak tensile strength of the core material. An expression for core tension failure load, based ct b . On the other hand, ductile materials on beam theory, is given by Pf D 12D Stc Ect like PVC failed by core indentation. Under edgewise compression, face sheet mi-

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crobuckling and general buckling are the two potential failure modes for Eco-Core and balsa core sandwich composites. For specimen length/depth ratio L=d < 7 the failure is by face sheet microbuckling, for 7  L=d  13 the failure is combination of face sheet microbuckling, debonding and buckling, and for L=d > 13 the failure is by general buckling. Predictions from the existing equations agreed well with the experiment for both core materials. For PVC core, wrinkling/shear buckling and general buckling are the potential failure modes. For L=d  8:5 the failure is wrinkling and for L=d > 8:5 the failure is general buckling. Acknowledgement The authors wish to thank the Office of Naval Research for financial support through grants #N00014-07-1-0465 and #N000140510532. Dr. Yapa Rajapakse was the technical monitor of the grants.

References 1. Olson K (2003) Keynote Lecture, Proceedings of 6th International Conference on Sandwich Structures, March 31–April 2, Fort Lauderdale, FL 2. Sorathia U, Perez I (2004) Improving the fire safety of composite materials for naval applications. SAMPE 2004, May 16–20, Long Beach, CA 3. Shivakumar KN, Argade SD, Sadler RL et al. (2006) Processing and properties of a lightweight fire resistant core material for sandwich structures. J Ad Maters 38(1): 32–38 4. Argade S, Shivakumar K, Sadler R et al. (2004) Mechanical fire resistance properties of a core material. SAMPE, May 16–20, Long Beach Convention Center, Long Beach, CA 5. Shivakumar KN, Sharpe M, Sorathia U (2005) Modification of eco-core to enhance toughness and fire resistance. SAMPE-2005, Long Beach, CA 6. Sadler RL, Sharpe MM, Shivakumar KN (2008) Water immersion of eco-core and two other sandwich core materials. SAMPE 2008, May 18-22, Long Beach, CA 7. Allen HG (1969) Analysis and design of structural sandwich panels. Pergamon, Oxford 8. Zenkert D (1997) An introduction to sandwich construction. Chameleon, London 9. Swaminathan G, Shivakumar KN, Sharpe M (2006) Material property characterization of glass and carbon/vinyl ester composites. Compos Sci Technol 66(10): 1399–1408 10. http://www.baltek.com 11. http://www.diabgroup.com/americas/u products/u divinycell h.html 12. Heath WG (2002) Sandwich construction, Part 2: The optimum design of flat sandwich panels. Aircraft Eng 33: 163–176 13. Gdoutos EE, Daniel IM, Wang KA (2002) Compression facing wrinkling of composite sandwich structures. Mech Mater 35: 511–522 14. Sadler RL, Shivakumar KN, Sharpe MM (2002) Interlaminar fracture properties of split angleply composites. SAMPE 2002, Long Beach, CA 15. Soden PD (1996) Indentation of composite sandwich beams. J Strain Anal 31: 353–360 16. Gdoutos EE, Daniel IM, Wang KA (2002) Indentation failure in composite sandwich structures. Exp Mech 42(4): 426–431 17. Steeves CA, Fleck NA (2004) Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part I: Analytical models and minimum weight design. Int J Mech Sci 46: 561–583 18. Fleck NA, Sridhar I (2002) End compression of sandwich columns. Compos A 33: 353–359

The Use of Neural Networks to Detect Damage in Sandwich Composites David Serrano, Frederick A. Just-Agosto, Basir Shafiq, and Andres Cecchini

Abstract Composite materials fail in complex failure modes that are difficult to detect. No single NDE technique is capable of detecting all damages. The ability to detect and asses the state of the damage is a key issue in order to improve service life of these materials. A Neural Network (NN) was chosen as a means to interpret and classify the information such that the type of damage, severity and location could be identified. The work describes the implementation of a NN based approach which combines thermal damage detection and vibration signatures in order to detect location and extent of damage in sandwich composites consisting of two carbon fiber/epoxy matrix face sheets laminated onto a urethane foam core. The approach analytically characterized and experimentally validated models for both thermal and vibration response. The numerical models were then used to train the neural networks. This approach is significant as it combines two techniques as opposed to just one as generally performed. Results demonstrated that the multi-component neural network approach successfully detected damage in scenarios in which using just a single method would have failed.

1 Introduction Marine, aerospace, ground and civil structures can receive unexpected loading that may compromise integrity during their life span. Therefore, improvements in detecting damage can save money and lives depending upon the application. The prognostic capability is usually a function of the examiner’s experience, background and data collection during the evaluation. Damage detection methods are varied and are in general developed for a particular type of system (material, damage type, loading and environmental scenarios). As a result, one method of damage detection alone D. Serrano (), F.A. Just-Agosto, B. Shafiq, and A. Cecchini College of Engineering, University of Puerto Rico, P.O. Box 9045 Mayaguez PR 00681, USA e-mail: d [email protected], f [email protected], [email protected], [email protected]

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cannot examine all possible conditions and may even give false readings. Furthermore, the probability of obtaining an accurate damage assessment increases when more than one means to detect damage of unknown origin is available. The aim of this work was to develop an intelligent computational system for feature extraction and defect characterization using more than one Non Destructive Evaluation (NDE) technique to detect and identify the location and extent of damages in sandwich composite structures. The research approach followed included: (i) the selection of appropriate NDE techniques in order to investigate critical changes in structural parameters, (ii) determining an adequate Neural Network architecture based on its performance in data classification, (iii) develop appropriate features extraction techniques to preprocess data for NN implementation, (iv) create and validate numerical models to generate different damage scenarios to be used in the NN training (use virtual data in place of experimental data in order to increase flexibility and reduce costs), (v) develop a multi-component NN architecture capable of handling information from different NDE methods, and (vi) perform experiments to test the feasibility of the proposed approach in detecting real damage cases. Vinyl ester sandwich composites consisting of two carbon fiber face sheets laminated onto a urethane foam core with an epoxy matrix were experimentally and analytically characterized using thermal, and vibration response to detect the presence of various types of damage. The sandwich composite material consisted of bi-directional woven [0ı =90ı ] carbon fiber face sheets bonded to polyurethane foam core with epoxy resin. Nominal beam dimensions were 584 51 mm with face sheet thickness of 0.7 mm and core thickness of 6.4 mm. The composite was selected for its relevance to future naval applications.

2 Nondestructive Evaluation Nondestructive inspection techniques are generally used to investigate the critical changes in the structural parameters so that an unexpected failure can be prevented. These methods allow testing or inspection of a system without impairing its future usefulness even without taking it out of service. The most popular NDE techniques currently used are: Ultrasonic testing, Radiography (X-ray, Gamma-ray, etc.), Eddy current testing, Liquid penetrant testing, Magnetic (particle, flux leakage, etc.), Acoustic emission testing, Infrared thermography (images and transient thermal response), Visual testing (optical) and Vibration based testing (matrix updating, modal/curvature response, fatigue behavior, etc.) [1–8]. Ultrasonic testing, radiography, liquid penetrant, magnetic particles and visual testing are not suitable for the proposed approach. Eddy current testing is applicable only to conductive materials while acoustic emission testing is a real time method of damage detection. Other NDE methods that are used in sandwich composites materials include shearography and the electromagnetic infrared method (EMIR)

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[1]. Shearography requires a very complex and expensive equipment while EMIR method is only efficient when the damage leads to water or moisture trapping in the structure. Composite materials present challenges for design, maintenance and repair over metallic parts since they tend to fail by combined failures modes [9]. Furthermore, damage detection in composites is much more difficult due to the anisotropy of the material, the conductivity of the fibers, the insulative properties of the matrix, and the fact that much of the damage often occurs beneath the top surface of the laminate. There are five principal modes of failure of sandwich composites with foam cores: (i) Yielding or Fracture of the tensile face sheet, (ii) Buckling or Wrinkling of the compression face sheet, (iii) Failure of the core in shear although there is also a lesser possibility of tensile or compressive failure of the core, (iv) Failure of the bond between the face sheet and the core and (v) Indentation of the faces and core at the loading points. Many NDE techniques have been used to localize and identify damages in a structure and several algorithms have been implemented to characterize these defects [1–8, 10–17]. However, a combined method of NDE techniques could be more effective in detecting different kinds of damages or a combination of damages. Ideally NDE techniques should complement each other and a particular defect in the structure could be localized by one or more techniques. After careful evaluation infrared thermography and vibrations testing were found to be the most adequate for use with the multi-component neural network approach proposed for damage detection in sandwich composite structures. Together they allow the identification of damages on the surface as well as bellow the surface of the composite structure.

2.1 Thermography Based NDE This method is based on the fact that in presence of a defect; the surface temperature evolution is locally modified producing a thermal contrast whose localization allows identifying the damaged area. The strengths of Passive Infrared Thermography are: fast inspection rate, no contact, safety, ease to interpret results and applicability to many surfaces. The primary weaknesses are: thermal losses, equipment cost and limitations regarding material thickness [16]. Figure 1 shows the experimental setup and sample images (time lapsed) resulting for a test specimen. The setup is used to test and validate the technique. A similar setup may be used to acquire data during the damage detection process. Because the work attempts to use Neural Networks (NN) to perform the damage detection, generating the many cases required for training the NN meant that experimental data would not be practical. Therefore a numerical approach for this purpose was selected. The analysis consists in the acquisition of external surface temperature data from a numerical model in response to a thermal transient excitation induced by external heating and when the model is exposed to free convection condition.

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A mathematical formulation of the problem was developed and a numerical approximation technique is used to solve the governing equations. The thermal distribution on the surface of the model is analyzed when damages are introduced. The numerical simulation was validated experimentally. The validated numerical model was then used to generate the scenarios needed to train the Neural Network Damage Detection System.

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Modeling

The three dimensional transient heat conduction equation that models the temperature distribution is: ¡  cp 

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The terms T , ¡, k and cp in Eq. (1) represent temperature, mass density, thermal conductivity and specific heat capacity respectively, (x1 , x2 , x3 ) are the spatial coordinates and t is the time. This equation is subject to initial and boundary conditions and solved using the finite volume technique [18]. One initial condition in time and two boundary conditions in space are used. The initial conditions are: T .xi ; 0/ DT0 T .xi ; 0/ DT00

xi 2 Q xi 2 

(2)

The Use of Neural Networks to Detect Damage in Sandwich Composites Table 1 Corrected finite volume parameters Parameter Spatial resolution x1, x2, x3 (cm) Time step t (s) Initial temperature Tı .K; ı C) Initial temperature Tı .K; ı C/ Ambient temperature T1 .K; ı C/ Face sheet material thermal diffusivity ˛f (s/cm) Foam core material thermal diffusivity ˛c (s/cm) Damage thermal diffusivity ˛d (s/cm)

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Value 1.58, 1.75, 0.036 0.01 353, 80 298, 25 298, 25 0.062 0.22 0.22

T0 and T00 are initial temperature values for points placed in the regions Q and  of the domain of Eq. (1). The boundary conditions are: ˇ @T ˇˇ D h  .T1  T / k  @xi ˇxi D0Ixi DLxi

(3)

Where h represents the convection heat transfer coefficient, T1 is the ambient temperature and Lxi is the model length in xi direction. The coefficient h for the vertical face was approximated by using the correlation for a vertical plate. This correlation is formulated in terms of the Nusselt and the Raleigh non dimensional numbers [19]. The resulting system of equations was solved using the Conjugated Gradient Stabilized algorithm [20]. Analytical surface temperature images for various given time steps were stored and converted into bitmap files. A sensitivity analysis was performed on the physical properties used in the modeling: diffusivity, convection heat transfer coefficient, conductivity, density and specific heat in order to adjust their values such that the simulation results agreed with the experimental results with a minimum error. Table 1 shows the corrected finite volume parameter values used in the numerical analysis.

2.1.2

Validation

To perform experimental validation, the test specimen were mounted vertically and heated to 80ı C with a blower as illustrated in Fig. 1. The glass transition temperature of the epoxy used in the interface ranged from 130ı C to 250ı C, therefore, a temperature of 80ı C is not expected to significantly alter the bonding/material characteristics. A Thermal Video System with an InSb FPA detector on an infrared imager was used to measure the surface emissions in the 3:5–5:1 m range. The IR camera recorded surface temperature from a 400 mm distance. A field view of 102 by 90 mm was used and a sequence of 12 thermal images with a 256  256 pixels size was recorded using 5 s time step intervals for each sample tested. In order to synchronize the images in different series, the temperature of an arbitrary point was set as a reference to start recording the thermal response of the beam.

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2.1.3

Test Cases and Results

The analytical model was used to generate various surface temperature distributions for a given transient time period. To model damaged scenarios, defects were simulated as reductions in thermal diffusivity thus producing a hot spot as the surface temperature decreases. Experimental transient data was obtained to identify the four types of damage scenarios used in the research. Figure 2 shows the cases studied: (a) crack in the face sheet, (b) de-lamination or bond failure between the face sheet and core, (c) indentation or perforation in the face sheet and (d) core failure. These cases were tested with the thermographic and the curvature based models in various locations and extent of damage. Figure 3 shows the IR images corresponding to a face sheet crack and a hole; experimentally detected damage is spotted exactly by the analytical formulation in both scenarios. The experimental results matched well with the analytical ones. A good qualitative agreement with the literature was also observed [13, 14, 21]. Quantitative comparison was not possible as this technique is sensitive to material used and equipment resolution. Furthermore, no literature is available on the sandwich composites tested. The method successfully identified very small face sheet damage (less than 1 mm) but could not identify low amplitude impact damage that did not physically fracture the face sheet. The foam core was resistive to thermal conduction, thus, the method was not successful in identifying face sheet de-lamination either. Similar results have been reported in the literature [13]. The principal limitation of the transient thermal analysis is the inability to detect subsurface defects because the

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Fig. 3 IR images (a) crack (b) hole, numerical simulation (c) crack (d) hole

thermal distribution only changes in the vicinity of the damage. However, the technique was found to be very suitable for complementing the Curvature Based method because of its ability to detect small face sheet fractures and holes invisible to the vibration technique.

2.2 Vibrations Based NDE Any localized defect in a structure produces variations in its dynamic response characteristics. Localized changes in stiffness result in a mode shape that has a localized change in slope and consequently its curvature, a feature that may be used in damage detection. Because a particular damage may be easier to detect in any one of the fundamental displacement mode shapes, the first three were used to identify the damage sites corresponding to structural changes. The eigen-parameter based technique was applied to detect localized damage in geometrically complex sandwich composite structures using 3D orthotropic finite element analysis. It was shown that differences in the fundamental displacement mode shape curvature of a sandwich structure can be used to identify the unique damage sites corresponding to a reduction in stiffness, which show up as spikes in the curve. Figure 4a shows a bond failure between the face sheet and the core, Fig. 4b shows the FEA model and Fig. 4c the three fundamental mode shapes for the damaged beam. Figure 4d shows the curvature for each mode and Fig. 4e the difference in curvature between the unharmed beam and the damaged beam, where spikes due to the damage are clearly visible.

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The technique was shown to work well for planar geometries. For non-planar or discontinuous structures, such as a ship hull, a segment wise analysis approach was used by the authors. This technique handled material complexities efficiently but with added mesh generation effort.

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Modeling

The analysis was performed on cantilever and simply supported sandwich composite beams modeled using the FEM. The model solved consisted of two layers of eight node “thin composite elements” for the face sheets and six layers of eight node “brick elements” for the foam core. Partial structural damage was modeled as

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localized reduced element stiffness. Gap elements were used to represent complete loss of stiffness as required in the modeling of face sheet de-bond, small perforations and cracks. A sensitivity analysis was performed to determine which FEA parameters had the most influence on the measured mechanical behavior (natural frequency and mode shape). The sensitivity analysis performed for the thin composite and brick elements showed that only six of the ten elastic properties required to model the FEA affected the output significantly. The sensitive parameters were then adjusted by minimizing the mean square error between frequencies obtained numerically, ! n , and experimentally measured frequencies, ! e , using the experimental setup shown in Fig. 5. The resulting object function was minimized: f .f ; EfL ; Gf

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The parameters f ; EfL ; Gf ; c ; Ec ; Gc correspond to face sheet density, face sheet longitudinal modulus of elasticity, face sheet shear modulus of elasticity, core density, core modulus of elasticity and core shear modulus of elasticity. Equation (4) was reduced using the quadratic interpolation method yielding errors between analytical and experimental results in the order of less than 2%. Figure 4c and 4e show the agreement between experimental values and the numerical simulation. The final material properties used in the model to obtain the mode shape curvatures are given in Table 2. LDV

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D. Serrano et al. Table 2 Material properties used in the finite element model Property Face sheet Mass density (kg=m3 ) 3  103 2 Longitudinal modulus of elasticity (MN=m ) 389 Transversal modulus of elasticity (MN=m2 ) 389 Major Poisson’s ratio 0.3 Minor Poisson’s ratio 0.3 Shear modulus of elasticity (MN=m2 ) 4.3

Foam core 2:5  104 4:7  103 n/a 0.3 n/a 0.27

For a beam in bending the curvature (k) can be approximated by the second derivative of the deflection: d 2z (5) D dx 2 In addition, numerical and experimental mode shape data is discrete in space, thus the change in slope at each node can be estimated using finite difference approximations. The central difference equation was used to approximate the second derivative of the displacements at node i : D

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where h is the element length and zi1 , zi and ziC1 are the nodal displacement of the structure vibrating at either the first, second, or third natural frequency. The absolute difference between the undamaged and damaged scenarios can be obtained from, ˇ ˇ ˇ ˇ ˇ  ki jWith ˇˇ : ./ D ˇ ki jNo Damage Damage

(7)

As a result of this analysis, a set of curvature vectors for different damage localizations are obtained. The vectors containing absolute difference between the undamaged and damaged scenarios highlight the differences in curvature among the different scenarios and are therefore suitable for feature extraction and classification. By examining the absolute difference between curvature changes in the mode shapes, damage can show up as spikes as shown in Figs. 4 and 6. Figure 6 shows results for various damage scenarios: (a) bond failure, (b) face sheet crack and (c) face-sheet indentation and (d) core failure. Information on the location of the damage, as well as, type of damage (shape of spike) can be obtained. These are key features in the implementation and enhancement of the NN performance.

2.2.2

Validation

The vibration set up used is illustrated in Fig. 5. Only the translational modes were considered in the experiments. The structural displacements were measured by positioning the Laser Doppler Vibrometer at several locations along the span of the

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Fig. 6 Characteristic curvature difference curves for various damage scenarios

test specimen. Output vectors were analyzed to compute curvature mode shapes. Experimental results agreed well with those obtained through numerical simulation as was shown in Fig. 4c and 4e for the bond damaged beam depicted in Fig. 4a. Various cases were used to validate the approach as well as to produce scenarios to test the neural networks described in the following sections, bond failure, surface indentation, surface crack and core damage examples are shown in Figs. 2 and 6. Shape and location of spikes vary in general according to type of damage, magnitude and location.

2.2.3

Test Cases and Results

Damaged and undamaged sandwich composite cantilever beams were experimentally tested to verify numerical results. Various types of damages, such as, bond failure, face sheet cracks and perforations were introduced into the composite. Multi-defect cases were also tested and the method successfully can be used to identify these. Figure 7 shows the curvature difference as a result of multiple defects. Experimental results agreed well with those obtained through numerical simulation

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especially for large face sheet cracks, however, the results were not so encouraging in the case of small holes, interface or foam core failures. In other words, change may not be observed unless the stiffness of the composite is significantly reduced. The method was somewhat limited in its ability to detect damages that did not actually crack the face sheet (scratches or perforations with dimensions less than the face sheet thickness). In summary, this technique offers a quick and reliable analysis when the damage occurs in the face sheet of planar members. The drawbacks of this technique are mesh and node numbering complexity in the curved members; complex experimentation setup to obtain mode shapes in nonplanar geometries; and de-coupling of rotational and translational eigenvectors.

3 Artificial intelligence (AI) in Damage Detection Algorithms that can recognize patterns in the data are probably the most essential components of an automated damage detection system. They are necessary to decipher and interpret the collected data. Examples of algorithms that have been used successfully in the literature include spectral analysis, Expert Systems (ES), codes that perform wavelet decomposition, Fuzzy Logic and Artificial Neural Networks. Spectral analysis [22, 23] involves the frequency response function (FRF) using the fast Fourier transform algorithm. This technique provides a quick and accurate approach in obtaining structural FRFs that may be used to extract parameters.

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Wavelet analysis may be viewed as an extension of the traditional Fourier transform (FT) with adjustable window location and size [21]. The merits of wavelet analysis [21, 24] lie in its ability to examine local data to provide multiple levels of details and approximations of the original signal. Therefore, transient behavior of the data can be retained. However, other algorithms are more appropriate for implementation in fault diagnosis and detection such as rule based systems [25], fuzzy logic systems [26] and neural networks (NN) [3, 25]. A rule based ES is defined [25] as one which contains information obtained from a human expert and represents that information in the form of rules, such as IFTHEN. The rule can then be used to perform operations on data to inference in order to reach appropriate conclusion. These inferences are essentially a computer code that provides a methodology for reasoning about information in the rule base or knowledge base, and for formulating conclusions. Fuzzy systems are based on membership functions and rules of association that process vague, imprecise or ambiguous information [23, 26]. The proper selection of these functions and rules is essential to successful development of a fuzzy logic based decision system. Known inputs and corresponding correct decisions are used to generate the membership functions used to develop the rule base. Fuzzy sets are commonly predetermined or manually defined. This teaching method is referred to as supervised learning because the fuzzy sets are prescribed. Artificial Neural Networks, ANNs (also known as NN), main advantage over the other methods is that it is capable of using unsupervised learning [25]. An unsupervised system learns from training data that is clustered into patterns to form classes. In recent years, there has been a growing interest in using ANNs, a computing technique that operates in a manner analogous to that of biological nervous systems. This concept is used to implement software simulations for the massively parallel processes that involve processing elements interconnected in network architecture. ANNs are suitable for pattern classification and natural information processing tasks and they are finding applications in almost all branches of science and engineering [3, 27–32].

3.1 Neural Network Based Damage Detection The diagnosis of a structure based on its response is an inverse process; the causes must be discerned from the effects. Interpretation of the changes in the structural response or properties due to damage is a critical task that can be viewed as a classification problem. A unique solution often does not exist for an inverse problem, especially when insufficient data may exist. Algorithms are necessary to decipher, and interpret the collected experimental data. Neural networks have the advantage of using unsupervised learning [25]. The classification and the generalization properties of NN allow the identification of the type of damage, location, and estimate of extent of damage because it may learn from training data which is clustered from patterns. The advantages of using NN are their capacity to diagnose correctly, even when

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trained with partially inaccurate or incomplete data, and their ability to continue learning and improve their performance when presented with new training data. The current work deals with the recognition of features extracted from the NDE damage detection techniques presented above and used concurrently in the NN system. The information is represented in diverse formats (in the form of vectors and bitmaps) and may be contaminated by noise (in experimental data). In addition, discrepancies may occur when matching these features with known (learned) patterns. To solve these issues a statistical based approach was implemented using a Bayesian classifier [27] that stores a single probabilistic summary for each class. Probabilistic neural networks (PNNs) belong to Bayesian classifiers and are used for classification problems, therefore, a Bayesian NN was selected for the implementation. The PNN training data was obtained from numerical simulations rather than from experimental data in order to allow a large number of scenarios to be available for training in a cost effective manner. In order to prepare the data from the chosen NDE techniques, suitable preprocessing algorithms were required. Algorithms from digital signal processing were incorporated to filter, homogenize and perform feature extraction in order to improve the PNN performance. The trained PNN is tested by presenting it with new damage scenarios (not part of the original training set). The PNN responds with the scenario that best fits its knowledge base (corresponding type of damage, location and extent of damage).

3.1.1

Thermographic Based NN Implementation

Figure 8 illustrates the PNN architecture and training process for the implementation of the Thermographic based NN. Multiple damage scenarios generated using the numerical model described above are used to train the PNN. Each scenario contains the image (bit map) and the corresponding state for which the data was obtained (example: location, size, depth, type of failure, etc.). The data is preprocessed to enhance the feature recognition and classified/clustered by the PNN as it learns. Once the training process is complete if the trained PNN is presented with an image it will retrieve from its memory the exact match if present in the knowledge base or the best match possible. The pre-processing method for the thermal images is shown in Fig. 9. Consider the bitmaps used for the thermographic approach, by applying median based filters, noise and optical distortions were removed from the thermal images and the original IR images were converted to Windows Bitmaps Images.

Fig. 8 Thermographic based NN

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Fig. 9 Thermographic image pre-processing

Fig. 10 PNN selects the best match (highest probability)

The thermal images are then converted to grayscale by eliminating the hue and saturation information while retaining the luminance. Then, 2D adaptive noise removal filtering was performed. Later a defect edge extraction procedure based on image gradient computation was applied to the images resulting in a set of binary images. This method allowed the extraction of relevant contour features from the images for damage characterization purposes. A set of images resulting from the numerical simulation were used to train the PNN for damage scenarios including that of an indentation damage of 1.5 mm at different locations. The top four images in Fig. 10 show the simulated damages (as red/dark spots) which were used in the PNN training. The PNN was then tested with a thermal image obtained experimentally. Each single feature of a thermal image was analyzed separately by the PNN Bayesian Classifier and matched to the training data. The bottom row in Fig. 10 contains the test image from the IR Camera used as the test damage scenario and fed to the PNN for damage evaluation. The last image

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Fig. 11 Curvature based NN implementation

corresponds to the case determined by the PNN to be the closest match, i.e., the image in the training set with the highest probability of matching the test damage scenario presented as the input.

3.1.2

Curvature Based NN Implementation

Figure 11 illustrates the architecture and the training process for the Curvature based PNN. The numerical models are used to generate damage scenarios and the curvature difference for each of the three mode shapes is generated for all the nodes in the model. The data is organized and sorted as part of the pre-processor. The data corresponding to for each mode shape is fed into its own PNN which associates the particular features in the data to its corresponding state (type of damage, location and extent). The output of each of the three PNN’s is then sorted, classified and associated to the input scenario by the fourth PNN which uses the “opinion” of the first three to generate the output. Figure 12 shows a beam with an induced damage at node 68. The PNN was trained with data damages corresponding to damages at different locations and then presented with the test scenario (not part of the training set) with damage at node 68. The PNN points out node 69 as the most probable damage scenario. Figure 13 shows a test case with a more complex geometry (a ship hull like structure). This case was performed completely with simulation data.

3.1.3

Multi-component NN Implementation

The complete system implementation requires a total of six PNN subsystems along with the pre and post processing algorithms. Figure 14 shows the overall MultiComponent Neural Network Architecture Implementation. The input data as before is preprocessed according to its NDE technique and processed through the corresponding PNNs as before.

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3.2 Testing and Evaluation The technique has been successfully tested with single damage cases using both curvature based analysis and thermo-graphic analysis, as in the previous examples, Figs. 10, 12, 13. In addition, testing was performed with multiple damage scenarios using both curvature based analysis and thermo-graphic analysis. Figures 15 and 16 show multiple damage scenarios. Training sets are shown in the first row and the test case is shown as the input to the PNN. Notice no multiple damage scenarios are used in the training, yet the result corresponds to two cases that together produce the closest match to the test case (input).

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Fig. 17 Multi-damage combined thermographic and curvature based example

Figure 17 shows a test case in which a de-bonding and a small face sheet perforation were used as the damage scenario. The NN easily detected the de-bonding using the curvature based data that could not be detected with a thermal analysis alone, and it detected the perforation using the IR images that could not be detected with the curvature method alone. It was found that neural networks successfully handled damage under novel conditions not included in the training data in addition to identifying multiple damage scenarios without the need of additional training. The NN approach was capable of identifying damage scenarios from both of the NDE techniques, therefore minimizing false diagnostics as they complement each other to confirm a diagnostic. When presented with a multi-damage case the NN will select from its knowledge base the cases which best match the given input data. The outputs are the corresponding states that produce the selected scenarios (type of damage, location and extent). Although both the thermographic and curvature based techniques are capable of handling multiple damage scenarios it is noteworthy to mention that with the Neural Network implementation single case scenarios are enough for training as the multi-component approach finds the best match for each case using all the information simultaneously. This greatly reduces the training cost and allows for a greater number of possible case scenarios that may be identified. It was found that neural networks successfully identified damage under novel conditions not included in the training data in addition to handling multiple damage scenarios without the need of additional training. The NN approach was capable of identifying damage scenarios from both of the NDE techniques, therefore minimizing false diagnostics as they complement each other to confirm a diagnostic.

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4 Conclusions Various NDE methods were evaluated and two were selected for use in the Neural Network damage detection system. A thermographic technique and a vibration based technique were chosen. In the vibrations based technique modal curvature response and FE modeling were used to locate and determine damage that successfully identified impact damage and face sheet de-lamination. However, the experimental method was not successful in determining small face sheet damage, such as, a small perforation (less than 1/8 in. dia.). In addition to the vibration method, location and severity of the damage was also obtained using changes in thermal properties of the composites. Transient temperature data was obtained to identify the same three types of damage as was tested with the curvature based modal experimentation. The thermal analysis could not identify tool drop impact scenarios that did not physically fracture the face sheet nor face sheet de-lamination. Transient thermal behavior was found to successfully detect damage in face sheet surface cracks, small punctures and other anomalies while the curvature based analysis detected de-laminations and surface damage (greater than 1/8 in. dia.). When both methods were used together with a Bayesian probabilistic neural network, curvature and thermal base analysis complement each other to augment the damage detection capabilities for sandwich composite materials identifying the type, location and extent of the damage. A neural network using a combined vibration technique and a thermographic technique, not previously mentioned in the literature was investigated. When only vibrations based modal curvature response was tested, various face sheet, interface and core damage scenarios were successfully identified. Thermal analysis by itself could not identify tool drop impact scenarios that did not physically fracture the face sheet nor face sheet de-lamination. However, thermal analysis was successful in detecting very small face sheet perforations, small punctures and other anomalies that the curvature based analysis had failed to do. When both methods were used together with a Bayesian probabilistic neural network, curvature and thermal base analysis complemented each other to augment the damage detection capabilities for sandwich composite materials identifying the type, location and extent of the damage. Other methods can be used however the two chosen can be applied to sandwich composite construction configurations usually found in ship hull manufacture. The method successfully identified various types of low amplitude localized damages, and it is speculated that the techniques developed can be extended to study large scale blast impacts by incorporating foam compression and face sheet fragmentation scenarios on a global scale. Literature offers substantial evidence of the validity of each of the chosen damage detection schemes separately. However, this paper shows that these methods can work jointly to complement each other in detecting the state of a composite sandwich structure under conditions that may be experienced by naval applications. Both the classification and the generalization properties of the neural networks are important in order to identify the type of damage scenario, location and estimate extent of damage. These algorithms may eventually be used in adaptive control strategies to help increase the life and serviceability of a given structure.

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Acknowledgments The authors wish to acknowledge the financial support of the ONRComposites for Marine Structures Division. Special thanks are due to Dr. Yapa Rajapakse, ONR program manager for his guidance and support.

References 1. Balageas D, Bourasseau S, Dupont M, Bocherens E, Dewynter-Marty V, Ferdinand P (2000) Comparison between non-destructive evaluation techniques and integrated fiber optic health monitoring systems for composite sandwich structures. J Intel Mater Sys Struct 11: 426–437. 2. Balageas D, Deom A, Boscher C (1987) Characterization of NDT of carbon epoxy composites by a pulsed photothermal method. Mater Eval 45(4): 461–465. 3. Cecchini A (2005) Damage detection and identification in sandwich composites using neural networks. M.S. thesis, University of Puerto Rico, Mayaguez, Puerto Rico. 4. Doebling SW, Farrar CR, Prime MB, Shevitz DW (1996) Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Los Alamos National Laboratory, LA-13070-MS, UC-900 41–49. 5. Pandey AK, Bisws M, Samman MM (1991) Damage detection from changes in curvature mode shapes. J Sound Vibrat 169(1): 3–7. 6. Plotnikov YA, Winfree William P (1988) Thermographic imaging of defects in anisotropic composites. QNDE 17A: 457–464. 7. Ratcliffe CP, Bagaria WJ (1998) A vibration technique for locating delamination in a composite beam. AIAA J 36(6): 1074–1077. 8. Salawu OS, Williams C (1994) Damage location using vibration mode shapes. In: Proceedings of the 12th International Modal Analysis Conference, Honolulu, Hawaii, USA, pp. 933–939. 9. Allen HG (1969) Analysis and design of structural sandwich panels. Pergamon, Oxford/ New York. 10. Just F, Shafiq B, Jairo J, Serrano D (2004) Damage detection using transient temperature response. Int J Nat Disasters, Accidents, Civil Struct 3(3): 129–141. 11. Kessler SS, Spearing SM, Atalla MJ, Cesnik CE (2001) Damage detection in composite materials using frequency response methods. Department of Aeronautics and Aeronautics, Massachusetts Institute of Technology, Cambridge, MA. 12. Kondo I, Hamamoto T (1994) Local damage detection of flexible offshore platforms using ambient vibrations measurements. In: Proceedings of the 4th International Offshore and Polar Engineering Conference, Osaka, Japan, vol. 4, pp. 400–407. 13. Maldague X (2000) Applications of infrared thermography and nondestructive evaluation. In: Rastogi P and Inaudi D (eds.), Trends in optical nondestructive testing, Elsevier Science Amsterdam, pp. 591–601. 14. Maldague X (2002) Introduction to NDT by active infrared thermography. Mater Eval 6(9): 1060–1073. 15. Plotnikov YA, Winfree WP (1988) Advanced image processing for defect visualization in infrared thermography. NASA Langley Research Center, M.S. 231, Hamplton, VA 23681-0001. 16. Shilbayeh N, Iskandarani MZ (2004) Application of new feature extraction technique to PVT images of composite structures. Inform Technol J 3(3): 332–336. 17. Toro C, Shafiq B, Serrano D, Just F (2003) Application of neural networks to eigen-parameter based damage detection in multi-component sandwich ship hull structures. In: Sixth International Conference on Sandwich Structures, Fort.Lauderdale, FL, pp. 25–29. 18. Versteeg H, Malalasekera W (1995) An introduction to computational fluid dynamics the finite volume method. Pearson Prentice Hall, Malaysia. 19. Oosthuizen PH, Naylor D (1999) Introduction to convective heat transfer analysis. McGraw Hill, New York. 20. Quarteroni A, Sacco R, Saleri F (2000) Numerical mathematics. Texts in applied mathematics 37, Springer, New York.

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21. Hou Z, Noori M, Amand RS (2000) Wavelet based approach for structural damage detection. J Eng Mech 126(7): 677–683. 22. Wu S, Ghaboussi J, Garret Jr JH (1992) Use of neural networks in detection of structural damage. Compos Struct 42(4): 649–659. 23. Zang C, Imregun M (2000) Combined neural network and reduced FRF techniques for slight damage detection using measured response data. Arch Appl Mech 71: 525–536. 24. Spanos PD, Rao VRS (2001) Random field representation in a biorthogonal wavelet basis. J Eng Mech ASCE 127(2): 194–205. 25. Liao SH (2004) Expert systems methodologies and applications, a decade review from 1995 to 2004. Expert Syst Appl 28: 93–103. 26. Sawyer JP, Rao SS (2000) Structural damage detection and identification using fuzzy logic. AIAA J 38(12): 2328–2335. 27. Bishop CM (2004) Neural Networks for Pattern Recognition, Oxford University Press, New York. 28. Byon O, Ben G, Nishi Y (1988) Damage identification of CFRP laminated cantilever beam by using neural network. Key Eng Mater 141–143: 55–64. 29. Masri SF, Chassiakos AG, Caughey TK (1992) Identification of nonlinear dynamic systems using neural networks. J Appl Mech Trans ASME 60: 123–133. 30. Sahin M, Shenoi RA (2003) Quantification and localization of damage in beam like structures by using artificial neural networks with experimental validation. Eng Struct 25: 1785–1802. 31. Sahin M, Shenoi RA (2003) Vibration based damage identification in beam like composite laminates by using artificial neural networks. J Mech Eng Sci 217 Part C: 661–676. 32. Zhao J, Ivan JN, DeWolf JT (1998) Structural damage detection using artificial neural networks. J Infr Syst 4(3): 93–101.

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On the Mechanical Behavior of Advanced Composite Material Structures Jack Vinson

Abstract During the period between 1993 and 2004, the author, as well as some colleagues and graduate students, had the honor to be supported by the Office of Naval Research to conduct research in several aspects of the behavior of structures composed of composite materials. The topics involved in this research program were numerous, but all contributed to increasing the understanding of how various structures that are useful for marine applications behaved. More specifically, the research topics focused on the reaction of structures that were made of fiber reinforced polymer matrix composites when subjected to various loads and environmental conditions. This included the behavior of beam, plate/panel and shell structures. It involved studies that are applicable to fiberglass, graphite/carbon and Kevlar fibers imbedded in epoxy, polyester and other polymeric matrices. Unidirectional, crossply, angle ply, and woven composites were involved, both in laminated, monocoque as well as in sandwich constructions. Mid-plane symmetric as well as asymmetric laminates were studied, the latter involving bending-stretching coupling and other couplings that only can be achieved with advanced composite materials. The composite structures studied involved static loads, dynamic loading, shock loading as well as thermal and hygrothermal environments. One major consideration was determining the mechanical properties of composite materials subjected to high strain rates because the mechanical properties vary so significantly as the strain rate increases. A considerable number of references are cited for further reading and study for those interested.

J. Vinson () Department of Mechanical Engineering, University of Delaware, 126 Spencer Lab Newark, DE 19716, USA e-mail: [email protected]

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1 Introduction The ONR sponsored research discussed herein is divided into two areas, each of which comprised a main effort during the 12 years of sponsorship. These are: (1) high strain rate effects on mechanical properties, and (2) behavior of composite sandwich structures.

2 High Strain Rate Effects on Composite Material Properties Starting in 1993, research was conducted at the University of Delaware under ONR sponsorship to investigate the effects of high strain rates on the mechanical properties of various composite materials that are and could be used in marine applications. Because all materials behave significantly different from their response to static loads when the material undergoes high strain rates due to applied dynamic loads, this research program was undertaken to investigate these important phenomena. This program was conducted in three phases: 1. The high strain rate testing of various composite materials involves a Split Hopkinson pressure bar for compression testing, and a High Speed Instron test facility for tensile testing to obtain the ultimate strength, yield strength, modulus of elasticity, strain to failure, elastic strain energy density and the strain energy density to failure for various composite materials at various strain rates up 2,000/s. 2. Close examination of the fracture surfaces of some of the specimens tested to gain insight into the deformation and fracture processes. 3. Evaluation of current material behavior models that describe strength, deformation, and failure of isotropic materials to assess their usefulness for modeling composite materials at high strain rates, and/or developing new behavior models. Numerous papers resulted from the research listed above. Among the materials experimentally investigated (at room temperature) in this research program are the following: Compression Tests: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Unidirectional E-glass/3502 Epoxy Random, Non-Woven Glass/Cycom 4201 Polyester Unidirectional T40 Graphite/ERL 1908 Epoxy Quasi-Isotropic AS4 Graphite Cloth/3501 Epoxy Metton 6061-T6 Aluminum Carbon/Aluminum Metal Matrix Composite (MMC) Silicon carbide/Aluminum MMC Unidirectional Continuous Fiber Carbon/Glass Ceramic Matrix Composite (CMC) 10. Silicon carbide Reinforced 2080 Aluminum MMC

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Cycom 5920/1583 Unidirectional IM7 graphite/8551-7 Epoxy AS4 Graphite/3501 Epoxy AS4 Graphite/K3B Polyimide IM7 Graphite/K3B Polyimide AS4 graphite/PEKK Thermoplastic Unidirectional and Cross-Ply K-49 Kevlar/3501-6 Epoxy Kevlar-29 and Glass Stitched Uniwaeve AS4/3501-6 Graphite/Epoxy IM7 Graphite/E7TI-2 Epoxy Various Seeman Composite Resin Infusion Molded Process (SCRIMP) composites Various Resin Transfer Molding (RTM) composites Unidirectional Glass/Epoxy (3M Scotchply 1003) composites Kevlar-29/Polyethylene IM7/977-3 Composite

Tensile Tests: 25. Dow Derakane 510AVinylesterResin 26. Dow Derakane 8084 Vinyl ester resin with 26 0z. Woiven Roving Glass fabric. Made from both Contact Molding and SCRIMP 27. SP-365/1583 Glass fabric/Epoxy composite The results from thousands of tests on the above materials are: Material property Can change from the static value Yield stress 3.6 Yield strain 3.1 Strain to failure 4.7 Modulus of elasticity 2.4 Elastic strain energy density 6.0 Strain energy to failure 8.1

With factors such as these, it is clear that if any structure is subjected to dynamic loads, then dynamic material properties, not static properties, should be utilized in the design. Otherwise the structure may be considerably overweight or the structure may fail unexpectedly. Even at static loads the strength to density ratio of structural composites is far superior to metal alloys used in marine applications. This can result in significant weight savings in structures employing advanced composite materials. From the above the advantages of using composites for structures subjected to dynamic loads are increased manifold. And of course the corrosion resistance and attendant cost reductions from maintenance such as painting make advanced composites very superior for marine applications. In all of the above only room temperature considerations have been discussed.

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During the last few years, period of 2000–2003, the high strain research has concentrated on the effects of low temperature. Four of the materials for which room temperature data had been obtained were studied. The low temperature effects at high strain rates had not been studied previously even though composite structures are being used increasingly at low temperatures for naval and other applications. The temperatures investigated include the range from room temperature down to liquid nitrogen temperatures (196ı C). In this range, the properties of most structural metals changes very little. However it was found that with the composites tested there were significant changes in the properties of each of the composites. These changes are for the good, and provide another basis for the increased use of composite structures for applications involving cold temperatures. These results for E-glass/Urethane are presented in Fig. 1. From the figure above, it is easily seen that properties along x- and y-directions, i.e. in-plane directions, have much larger increases from room temperature to liquid nitrogen temperature, than z-direction, the through-thickness direction, which is matrix dominated and therefore comparatively weak. At the liquid nitrogen temperature point, the ultimate strength values along the in-plane x- and y-directions are equal to each other and are greater than the values in the thickness direction. X direction: the ultimate compressive strength increases 249.2%. Y direction: the ultimate compressive strength increases 298.2%. Z direction: the ultimate compressive strength increases 121.1%. These increases in strength at low temperatures are truly significant. Over and above the superior values of strength to density of this marine composite material over metal alloys at room temperature, these outstanding increases in strength at low

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temperatures make the material significantly superior to metal alloys for marine applications in Arctic temperatures both above and below the waterline. And in addition there is no corrosion, nor maintenance (painting).

3 Composite Sandwich Structures The use of sandwich structures is growing very rapidly around the world. It is used in many applications. One reason for the increased use of sandwich construction is because it has such a high ratio of flexural stiffness to weight. As a result, sandwich constructions result in lower lateral deformations, greater buckling resistance, and higher natural frequencies than do other types of construction. Thus sandwich constructions quite often provide a lower structural weight than other configurations do for a given set of mechanical and environmental loads. In most sandwich structures there are two faces, identical in material and thickness, which primarily resist the in-plane and lateral (bending) loads. However, in special cases the faces may differ in thickness, materials, or fiber orientation, or any combination of these three. This may be due to the fact that one face is the primary load carrying, low temperature portion of the structure while the other face may have to withstand an elevated temperature, corrosive environment, etc. Assuming a uniform core, the former sandwich is regarded as a mid-plane symmetric sandwich, the latter a mid-plane asymmetric sandwich. Consider the usual mid-plane symmetric construction with two identical faces and a much thicker core of either honeycomb, balsa or foam construction, and compare it with a monocoque construction of the same material as the sandwich faces whose thickness is the sum of the two faces. In this comparison the extensional in-plane stiffness of the two constructions are the same, since the aforementioned cores do not take any appreciable portion of the in-plane loads or bending loads. Hence, the two constructions have the same in-plane stresses when subjected to in-plane loads only (i.e. separating the thickness of the two faces has no effect when considering in-plane loads only). However, for bending loads the sandwich construction is far superior to the monocoque construction, because the two faces are separated by the much thicker core. If the core thickness is 20 times the face thickness, then for bending loads only, the sandwich construction is 300 times as stiff as the monocoque construction of two faces only. In addition the stresses in the thin monocoque faces are one thirtieth (1/30) those of the bending stresses on the face of the monocoque construction discussed above. This is independent of the face material used in the constructions, whether the material is isotropic or anisotropic. Even with these advantages, it is important to develop the means by which to optimize the sandwich construction for minimum weight in order to: 1. Determine the absolute minimum weight for a given geometry, loading, and material system

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2. Rationally compare one type of sandwich construction with other types of sandwich construction 3. Rationally compare the best sandwich construction with alternative structural configurations (monocoque, rib-reinforced, etc.) 4. Rationally select the best face and core materials 5. Select the best stacking sequences for faces that are composed of laminated composite materials 6. Rationally compare the weight of the optimized construction to weights required when there are some restrictions, i.e., cost, minimum gage, manufacturing limitations, material availability, etc. Sandwich construction is relatively new. In England, sandwich construction was first used in the Mosquito night bomber of World War II. During World War II, the concept of sandwich construction in the United States originated with the faces made of reinforced plastic and a lower density core, where in 1943 the Wright Patterson Air Force Base designed and fabricated the Vultee BT-15 fuselage using fiberglassreinforced polyester as the face material and using both glass-fabric honeycomb and balsa wood core. The first research paper concerning sandwich construction was written by Marguerre in 1944 dealing with sandwich panels subjected to in-plane compressive loads. Also in the 1940s, two young World War II veterans formed Hexcel, which is still perhaps the most prestigious firm associated with sandwich construction. In 1951, Bijlaard studied sandwich optimization for the case of a given ratio between core depth and face thickness, as well as for a given total thickness of the isotropic sandwich plate. An abridgement of this research is in the proceedings of the first U.S. National Congress of Applied Mechanics in 1952. Ericksen issued a U.S Forest Products Laboratory (USFPL) Report accounting for the effects of transverse shear deformation on the deflections in isotropic sandwich plates, using a double Fourier series to represent the deflections in the simply-supported plate, i.e. a Navier solution. Also presented were general expressions for the strain components in sandwich panels with orthotropic faces and cores. In 1952, Eringen used the Theorem of Minimum Potential Energy to obtain four partial differential equations for the bending and buckling of rectangular isotropic sandwich plates under various loads and boundary conditions. In 1952, March also published a USFPL report on sandwich panel behavior, as did Ericksen in 1956. During the early post World War II period, the Forest Production Laboratory was the primary group in the development of analysis and design methods for sandwich structures. In 1966, Plantema published the first book on sandwich structures [1], followed by another book on sandwich structures by H.G. Allen in 1969 [2]. These books remained the “bibles” for sandwich structures until the mid-1990s. Also in the mid-1960s, the U.S. Naval Air Engineering Center sponsored research to develop a fiberglass composite sandwich constructions to compete in weight with conventional aluminum aircraft construction. Much of this research effort was in the development of minimum weight optimization methods so that the fiberglass sandwich construction could compete with the established aluminum construction.

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In 1995, a monograph by Zenkert [3] supplemented much of the material contained earlier in the Plantema and Allen texts, which by this time were out of print. In 1999, another textbook was published by Vinson [4]. To date there have been only eight International Conferences on Sandwich Constructions: the first in Stockholm in 1989, the second in Gainesville in 1992, the third was in Southampton in 1995, the fourth in Stockholm in 1998 and the fifth Conference in Zurich in September of 2000. The sixth Conference was held in Ft. Lauderdale in March 2003. The seventh Conference was held in Aalborg, Denmark in 2005, and the eighth conference was held in Porto, Portugal in 2008. The Journal of Sandwich Structures and Materials was first published in 1990 and is the only journal dealing exclusively with this topic. As stated previously in the United States, much of the fundamental research in sandwich structures and materials was sponsored by the Office of Naval Research, under the direction of Dr. Yapa S. Rajapakse. The U.S. navy now employs hexagonally shaped mast of the USS Radford which is 93 ft tall weighing 90 t. Not only is this a foam core sandwich but the use of exterior materials for stealth purposes make this an asymmetric sandwich. The core of a sandwich structure can be of almost any material or architecture, but in general, cores fall into four types: (a) foam or solid core, (b) honeycomb core, (c) truss core, and (d) corrugated or truss core, as shown in Fig. 2. Since World War II, honeycomb cores have been widely used. The two most common types are the hexagonally-shaped cell structure (hexcell) and the square cell (egg-crate). Web core construction is analogous to a group of I-beams with their flanges welded together. The U.S. Navy refers to this web core construction as “double hull” construction. Truss or triangulated core construction can vary from being very simple to being of a complex cross-section, such as in many box materials. In all cases the primary loads, both in-plane and lateral, are carried by the faces, while the core resists the transverse shear loads. In most foam core and honeycomb core sandwich constructions, one can assume for all practical purposes that all of the in-plane loads and lateral bending loads are carried by the faces only. However in truss core and web core constructions, a portion of these loads are carried by the core. Foam or solid cores are relatively inexpensive and can consist of balsa wood or an almost infinite selection of foam/plastic materials with a continuous variety of densities and shear moduli. As in modeling all composite constructions, thermal and hygrothermal considerations must be taken into account. Concerning the thermal effects, with increased temperature, there are three: thermal expansion, degradation of elastic properties; and an increase in non linear creep/viscoelastic effects. For the “hygro” part of the effects, there is moisture expansion in all polymer materials. This is modeled analogous to the thermal expansion. The good news is that if one has the thermal effect solution then one also has the “hygro” solution and they can be superimposed. Moisture also affects the glass transition temperature. One major difference between the thermal and the moisture effects on a polymer matrix structure difference is in the time scales. It takes weeks or months to saturate a specimen or structure with water.

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Fig. 2 Types of sandwich construction

Under ONR sponsorship under the direction of Dr. Rajapakse, minimum weight optimization studies have been performed for the honeycomb core, the foam or solid core, the truss core and the web core sandwich panels subjected to in-plane compressive loads, and in-plane shear loads as well. In the process figures of merit were determined that are most helpful in material selection and comparison. In addition, these methods also provide the optimum stacking sequence for the face plates if a laminated construction is used. This research has appeared in many papers, but one succinct source to find all of the equations by which to analyze, design and optimize these sandwich panels is given in Ref. [4]. Dedication This paper is written to honor Dr. Yapa Rajapakse, who sponsored the research discussed. Dr. Rajapakse has played a uniquely significant role in sponsoring research throughout the world that is useful to the United States Navy. He has been the person responsible for the bulk of research involving composite materials for naval applications. Starting with three noted researchers, Dr. Isaac Daniel, Dr. Jack Weitsman and Dr. Su Su Wang, he built his program to a notable “stable” of the world’s best researchers in this area. It was indeed an honor and privilege to be part of the Yapa team of researchers. Dr. Rajapakse has been uniquely outstanding as a sponsor of basic research over his long and prolific career.

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References 1. Plantema FJ (1966) Sandwich construction: The bending and buckling of sandwich beams, plates and shells, Wiley, New York 2. Allen HG (1969) Analysis and design of structural sandwich panels, Pergamon, Oxford 3. Zenkert D (1995) An Introduction to sandwich construction, EMAS, West Midlands 4. Vinson JR (1999) The behavior of sandwich structures of isotropic and composite materials, Technomic, Lancaster, PA

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Application of Acoustic Emission Technology to the Characterization and Damage Monitoring of Advanced Composites Emmanuel O. Ayorinde

Abstract This chapter gives some highlights of the research done by the author in the area of acoustic emission technology for the Navy under the much valued guidance and support of program manager Dr. Yapa S. Rajapakse over a number of years. Although the author’s ONR research covered a wider base, such as low velocity impact, static and dynamic loadings with damage studies of monolithic and foam-cored sandwich composites, and application of some other NDE methods, the acoustic emission NDE method was widely applied, and it merits a dedicated mention. It is not possible, in such a short paper, to describe every investigation undertaken over the period of time, but an attempt is made to outline a representative number of cases which together present an informed cross-sectional view of work done on this topic. Control samples of monolithics like aluminum and steel were used in some of the studies for the sake of comparisons, but focus was centered on composite materials. It is well known that composite materials are being increasingly utilized in ship structures, and in fact generally in both the civilian and military sectors of the economies across the world. The need for rapid, low-cost, non-destructive and reliable methods for obtaining mechanical property data for materials and evaluating damage in them has boosted this kind of research.

1 Background Advanced composite materials have recently been manifesting increasing presence in the consumer industries, civil construction and military applications, all across the world. The chief reasons are their superior stiffness-to-weight and strength-toweight ratios compared to traditional materials, as well as their better corrosion resistance and potential for tailored and multifunctional construction. Application has been significant in the area of transportation – land, sea and air. It is also established E.O. Ayorinde () Mechanical Engineering Department, Wayne State University, Detroit, Michigan 48202 e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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that the failure modes of composite materials are many more and far more complicated than those of monolithic materials since these are now dependent on the mechanical and chemical properties of the constituent elements – fiber, particle, matrix, and interfaces, as the case may be for each material system being tested. It therefore becomes important to have rapid, low-cost and reliable methods to obtain mechanical property data for these materials, and assessing damage in them. Some standards exist for traditional static tests but the methods are slow, expensive and destructive. It is clear that it is not feasible to always test to destruction. Nondestructive test methods such as acoustic emission, vibration and others are thus preferred to these static tests in efficiency, cost and non-destruction of the test material. Vibration methods are useful in obtaining gross material properties over structures, but acoustic emission methods are by nature earlier and more intensive in responsiveness. AE has been found to be perhaps the technique that best monitors the mechanics of internal damage [1–4], and is recognized as being information laden, particularly in the area of energy and frequency contents [5], thus confirming the appropriateness of AE as in characterizing singularity events. Examples of the AE NDE approach include analyses through several transient and classical AE parameters, such as amplitude, frequency, modal and statistical variables, and transform methods such as FFT and wavelet [6–12]. Acoustic emission (AE) technology is quite proper for testing inhomogeneous materials, as these manifest numerous sites of singularities of geometry, properties, materials, and other variables, and these sites are all potential seeding locations for emissions. This testing approach therefore tends to provide more response information than other procedures, and may sometimes be the only successful method available. Apart from the testing of very different material systems utilized in various marine structures, some part of the ONR programs dealt with low temperature exposure of ship hulls. There is a need to understand material response to extreme conditions, for example the low temperatures to which ships that travel to polar regions are exposed. The study of the behavior of advanced composite materials in appreciably cold conditions thus becomes quite necessary, more so as composite materials are being increasingly deployed into ship structures. It was also found prudent to explore the use of wavelet analysis of captured acoustic emission data. The wavelet transform method basically merges time, frequency and intensity descriptions. Jeong and Jang [13] concluded that wavelets are good for accurate source location in wave propagation in dispersive media, and permit the use of a single frequency component in an array of sensors, which is an improvement over traditional methods. For a function f(t) such that f(t) 2 L2(Re) its continuous wavelet transform is given by   Z t £ 1 f .t /® dt (1) F W .a; £/ D p a a where ’ represents scale level (related to inverse of frequency), £ is the translation parameter and ¥ is the wavelet function. The investigations carried out by this author in the area of marine applications of interest to the ONR covered static testing of tensile, compressive, flexure and

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strain-rate loadings; dynamic testing of impact and fatigue testing; geometries that included laminated composite beams and plates and sandwich structures and isogrid panels; and materials that included glass/epoxy, glass/polyester, carbon/epoxy, nano-sized augmentator materials and a number of foam cores. However, a few representative samples of the work will be described, showing how they have been conducted, and the results that were obtained.

2 Sample Acoustic Emission Applications The particular examples of the author’s work for the ONR (under Program Manager Dr. Y. Rajapakse) that will be discussed involve edgewise compression tests of polycore sandwich material, and test involving the isogrid construction, and also composite sandwich fatigue.

2.1 Edgewise Compression Tests of Polycore Sandwich Material 2.1.1

Background

This work took a more detailed look at the use of acoustic emission NDE in test monitoring and damage characterization of a woven composite sandwich in edgewise compression. Foam-filled fabric composite sandwich panels were fabricated from preform having integrally woven face skins and cross bars that make it essentially three-dimensional. It was found that acoustic emission technology chronicled the failure history of the composite specimens quite well. AE information covered the damage type, timing and sequence. Both the AE parametric approach (based on several parameters, such as amplitude, time, energy, counts, etc.) which can be plotted against one another and the transient (based on the waveform trace and its many transformations) and combinations e.g. [14–22] have been found to be quite useful by many workers. This latter use of transient AE and its transformations, such as FFT (fast Fourier transform), STFT (short time Fourier transform), Gabor, wavelet and EMD (empirical modal decomposition) procedures can be even much more useful in advanced applications for more analysis. General types of service loads experienced include compression, bending, shear, and impacts, and they induce common failure mechanisms in sandwich composites materials like indentation, skin buckling under compression, delamination (in sandwiches this could occur at a skin/core interface, or within the plies of the usually laminated skin), interlaminar matrix cracking, core cracking, core shear, longitudinal matrix splitting, fiber transverse fracture, fiber pull-out, and fiber/matrix debonding [16].

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Testing

Integrally woven 3-D E-glass fabric composite sandwich panels (having the core directly interwoven with skins through vertical pile yarns in the preform) were manufactured using a vacuum assisted resin transfer molding (VARTM) process employing SC-15 resin, and cured for about 24 h. The preform comprised two bi-directional woven E-glass fabric surfaces having mechanical inter-connections with several vertical woven piles. The structure is marketed as “parabeam”. The orthogonal-grid parabeam structure was then filled with a mixed two-part liquid polyurethane foam, and cured for about 5 h. Parabeam sandwich (PS) is lighter than conventional sandwich (CS) structures and has higher bending stiffness and core/skin debonding resistance. The parabeam architecture effectively defeats the major setback of conventional sandwiches, which is the skin/core delamination under loading that normally brings performance degradation or outright failure of the component [23, 24]. Edgewise compression tests as per ASTM C364-94 were conducted with acoustic emission (AE) response being monitored synchronously by a Physical Acoustics LOCAN 320 AE system, using a wideband piezoelectric AE sensor for signal acquisition. Edgewise compression tests were chosen since fiber/core delamination is more effective in edgewise compression as compared to flat wise compression [24]. Figure 1 shows the unfilled and the foam-filled parabeam structures. The edge wise compression experimental test set-up with the attached AE system is shown in Fig. 2. The parabeam sandwich specimen size was code-standardized at 2  5:5  0:75 in. The test specimen ends were securely held in the test fixtures with epoxy adhesive, and crosshead movement rate was fixed at 0.1 in./min. Acoustic emission pre-amplifier gain of 35 dB and a threshold level of 30 dB were utilized, while the PDT, HDT, HLT settings of the AE system were made 35, 150 and 300 respectively as per manufacturer recommendations for composite materials.

2.1.3

Discussion of Results

In Fig. 3, the stress–strain curves show higher compressive strength for PS composites than for CS types, with the maximum compressive strength for the tested PS

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Fig. 1 (a) unfilled, and (b) foam-filled (top); unfilled (middle) and foam-filled (bottom) parabeam sandwich composite samples

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Fig. 2 Experimental set up, indicating sensor location

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composites being about 354 psi. This compressive strength can be further enhanced by reducing the voids content of PS, which tends to be high due to the use of short resin infusion time and open atmospheric curing. The stress–strain curves also show a higher slope for the initial linear portion of the PS indicating a higher compressive modulus for this type. Damage first occurred in the matrix of the PS composite specimens. At a relatively low load value, matrix cracks initiated in the face sheets of the material, and in comparison with CS materials, this indicates comparatively weaker face sheets. This weakness too can be improved or removed with extra resin infusions, such as infusing the resin into the skin in two consolidation processes. Failure occurred at the center of the specimen gage length in the edge-wise compression test of the PS material, accompanied by visible cracks in the face sheets, and with no visible delamination. The absence of delamination has been generally regarded as the reason for the higher compressive strength of PS composites. PS test specimens did not manifest any appreciable core cracking, perhaps due to the presence of piles in the core, which practically serve as reinforcements within the core and thus reduce the shear crimping. To illustrate NDE results, load displacement curves were compared with amplitude–time–frequency, energy–time–amplitude, duration–time–RMS and frequency–amplitude charts of the acoustic emission records. For each particular displacement, the corresponding AE signals can be compared. The AE energy level signals in composite materials testing supplies several response parameter information items including the amount of energy released during failure events such as matrix-fiber debonding, matrix deformation, matrix cracking, fiber pullout, fiber push out, fiber breakage, etc. Traditional AE count indicates the number of these events above a pre-selected amplitude threshold value and event duration and signal frequency are regarded as good characterization variables about failure event characteristics. Since the crosshead movement rate was steady, time synchronization of mechanical and NDE test results can be correlated on a one-to-one basis. Figure 4 shows the load-versus-displacement diagram for the test range while Fig. 5 focuses on the primary failure zone which the AE data covers. In Fig. 5 a small drop of load (from about 560 lb to about 380 lb) at a displacement of 0.028 in. indicates the matrix cracking of the PS composite. A small recovery follows, from about 380 lb to about 430 lb. Figure 5 then shows a sharp drop of load (from about 430 lb to about 100 lb) at a displacement value of about 0.04 in. There then occurs a recovery, to about 170 lb, and then a gradual rise with small slope changes at about 0.054 and 0.06 in. displacements, but with an overall slope that is somewhat less than that of the initial loading rise, up to about 540 lb. Figure 5 also indicates a load drop at a displacement value of about 0.08 in., from about 540 to 350 lb. Figure 6 shows the failure modes of the PS, and shows the fiber fracture band around final failure The levels of AE energy, frequency, amplitude, duration and RMS signals also attest in a detailed manner to each one of these changes as Figs. 7 to 10 clearly show. Indications have been made that low energy AE events in such material systems indicate debonding between individual fibers and surrounding matrix, medium energy events correspond to matrix cracking and deformation, and high energy level

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events to fiber failures [25–27]. Here, the first load drop around 0.028 in. displacement corresponds to debonding failure, the next one at about 0.04 in. displacement corresponds to matrix deformation and cracking failure, while the last one around 0.08 in. displacement corresponds to the onset of fiber fracture. These are the three main failure phenomena encountered by this specimen type in these tests. The stratifications in the AE figures clearly show this. The displacement range 0.1 to 0.4 in. where there is little change in load with displacement is the plastic deformation region, and there was no appreciable count level. It would seem that, the main event occurring in this region is the densification of the materials.

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Fig. 7 Frequency–time– amplitude chart

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Fig. 10 Frequency– amplitude chart

2.1.4

Concluding Remarks

Testing and evaluation of parabeam composite sandwich samples with acoustic emission monitoring was undertaken. This woven sandwich fabric perform with integrally woven fabric composite architecture manifested high skin/core debonding resistance. Edge wise compression test of PS revealed no visible delamination between the core and the face sheets, and these samples also showed higher compressive strength and stiffness than CS materials. Acoustic Emission (AE) analysis provided an accurate representation of damage history of the test specimens. The results indicate that several parameters, including the location, timing, and classification of damages (matrix cracking, delamination, fiber breakage etc.) in composite materials can be inexpensively and accurately obtained from AE measurements for complex composite structures such as the parabeam sandwich. The results also suggest that AE has potential as a reliable and inexpensive method for monitoring and characterizing the structural integrity of composite materials.

2.2 Isogrid Construction 2.2.1

Background

In composite applications involving relatively thin structural sections, it is sometimes necessary to provide stiffening reinforcement. Grid-patterned stiffeners constitute one main way to achieve this objective. Composites have been quite useful as laminates and sandwich panels, but applications as grid stiffened structures are also increasing. In this particular work being reported here, grid stiffened structures were fabricated with filament winding technology in the form of isogrid cylinders made out of advanced composite materials. The mechanical axial compression and simultaneous acoustic emission (AE) testing of such isogrid cylinders having triangular rib geometry were investigated. Analytical, experimental and AE results were found to agree well, and it seems that AE could also characterize isogrid failure history very well, and be used to accurately predict failure. Composites can usually tolerate only slight plastic deformation, but they rather absorb destructive input energy by fracture failure of large areas, undergoing

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strength and stiffness reductions [28]. The isogrid concept which utilizes a repetitive equilateral triangle geometry of stiffening ribs, provides plenty of such areas in the ribs, and finds ready application in composite constructions. These structures are isotropic and can be analytically transformed into effectively homogeneous material layers for further computations [29,30]. The stiffening ribs of isogrid structures may also be used for secondary attachments.

2.2.2

Testing

Isogrid cylinders were fabricated from commercially obtained Hercules IM7/8552 pre-impregnated tapes filament-wound on a Composite Machines Company (CMC) machine, as in Fig. 11. A foam mandrel was made from the Marcore foam system which consisted of an isocyanate A-part and a polyol resin B-part with additives and a blowing agent. A hollow steel mold, approximately 6:2500 diameter and 18 in. long, was used to prepare the foam. The foam components were mixed in a vacuum chamber for about 15 s. The mold was kept between 120ı F and 140ı F. The foam was released after the mandrel cooled down. The mandrel was machined by Computer Numerically Controlled (CNC) filament winder down to 5.45 in. diameter, and the isogrid patterns were machined into the mandrel at 0ı , C60ı and 60ı with a Sears Craftsman 0:2500 router bit. Sparkling putty was spread evenly over the mandrel and the surface was smoothened down on drying. An epoxy resin was applied to seal off any pores so that any mold release agent applied would not possibly be soaked up into such pores. A weight and levers arrangement was used to control the fiber tension in winding. The assembly was bagged and cured in an autoclave. Post-cure hydro-blasting and a minimum of chiseling were utilized to obtain the final form of the foam mandrel. Strain gages were attached to the skin at the cylinder center, on both sides, parallel to the bending direction such that responses from two adjacent 0ı ribs could

Fig. 11 Winding apparatus

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register. A scanner, programmed to record micro-strains at half-second intervals, was connected to the strain gages to check the null balance and the gage calibrations. The cylinders were tested in axial compression in a servo-hydraulically controlled MTS machine, with the program TESTWARE SX to control loading and motion, in displacement control modes at the rate of 0.01 in./min. Simultaneous acoustic emission (AE) readings were taken in-situ with a Physical Acoustics LOCAN 320 system having an auxiliary ATASC software for translating machine language data files into numeric forms to be processed by spreadsheet and graphing application programs. The load–deflection and acoustic emission signals were both automatically recorded during the tests. A single Physical Acoustics WD 709 broadband AE sensor was utilized, thus monitoring only the overall effects in the specimens. The results were processed with mechanical test and AE software which yielded various classical mechanical and AE parameters.

2.2.3

Discussion of Results

The two strain gages initially had similar output indications, but at a particular value of compressive load, the readings began to diverge, with one now showing tensile loading. This was taken as signifying the start of instability, which is due to global buckling of the cylinder. Figure 11 shows in essence the winding apparatus. Figure 12 shows the load–displacement curve. Figures 13 to 18 are different presentations of the AE data. The mechanical records show that critical buckling seemed to have occurred at a displacement of 0.94 in., and a load of about 24,500 lbf. The catastrophic failure load was about 27,000 lbf. The difference between the experimental failure load, analytical (about 40,700 lbf) and the numerical FEA (about 120,000 lbf average) values has been ascribed [31] to the fact that analysis assumed global buckling and higher rib strength than skin strength, while the numerical analysis assumed constant elasticity modulus. Practical conditions normally deviate from these ideals. AE results were similar to the mechanical test records and showed great detail of failure history. The two basic AE plots shown, Figs. 13 and 14, which respectively display the AE counts and energy histories, provide a good measure of time-based

Fig. 12 Load versus displacement

452 Fig. 13 AE counts versus time

Fig. 14 AE energy versus time

Fig. 15 AE frequency –time–amplitude plot

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Fig. 16 AE energy –time–amplitude plot

Fig. 17 AE duration –time–RMS plot

Fig. 18 AE frequency –amplitude plot

failure history of the test. However, as advanced by Ayorinde et al. [21], the presentations of the four-pack Amplitude–Time–Frequency, Energy–Time–Amplitude, RMS–Time–Duration and Frequency–Amplitude presentation is often enough to supply the majority of diagnostic information about the monitored test. Each AE record type makes its own contribution to the unfolding of the progressive failure history. AE counts (Fig. 13) show a general level increase up to a maximum before the collapse, but counts density (counts per unit time) seem to have decreased about 1 min before final failure, and the failure region saw a burst of emissions. This agrees

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with the ramp-like structure of the initial part of the loading diagram. The peaking of AE counts at the end of the increase matches the instant of maximum load. The strain-time record (not shown) shows that the critical buckling load appears to have occurred about 1 min before failure, when the slope of the load–deflection curve may be observed to hit its minimum. These results also agree with other AE records. The critical buckling zone is indicated by increase in the time density (i.e. rapidity of occurrence) of the mid-level failures, and the final failure onset by even denser mid-level as well as high-level failure events. At the same time, there are quite a lot more of low-level failures. This situation suggests massive structural failure, and confirms the actual experience. Acoustic emission response thus appears quite sensitive to the mechanical events in the samples tested. The collapse region is seen to be one of high AE activity, and attests to the audible noises heard during the test. AE energy distribution (Figs. 14, 16 and 18) shows that the events up to rupture are of all types of failures, although up to half of the deformation history, the emissions recorded are low-energy types. It has been suggested [32, 33] that the AE amplitudes increase in order from matrix deformation to fiber/matrix debonding to fiber pullout to fiber breakage. From the literature on AE, it appears that in general, about the first third of the range on an AE energy plot is mainly occupied by interfacial failures, such as matrix deformation, delamination, and fiber/matrix debonding; the middle third mainly by matrix cracking and fiber pullout; and the upper third mainly by fiber breakage, and the same goes for Amplitude and Duration of hits. Figure 15 shows that the amplitude distribution over time synchronizes well with the physical test results, for example as in Fig. 12. Besides the trend similarities, the maximum AE amplitude occurs at about the exact time of catastrophic collapse, and the high frequencies associated with end-period fiber breakages appear to be evident. Near the rupture load, the emissions become more intense, and occur more rapidly. Emissions that are occurring at more energy levels have also increased, showing that various failure phenomena are occurring. These energy levels suggest that matrix micro-cracking, delamination and fiber-matrix de-bonding generally occurred first. Following this, before failure, these phenomena were joined by more severe matrix cracking, some fiber pullout, and a little fiber breakage. In the final failure zone, all these were joined by more fiber breakages at higher energy levels. AE duration (Fig. 17) also indicates some trends. Up to the critical load, the AE events are of low-duration. Near the critical load, mid-level events join in. Final failure is ushered in with high-level events. Notable fiber breakages appear to occur at failure onset and at two other following periods. According to the broad three-zone division scheme, it appears that that response to the loading regime up to critical buckling load is primarily low-level, with a small content of mid-level failure. From the critical buckling load onwards, low-level and mid-level emissions increase significantly and continuously. From the failure initiation onwards, highlevel emissions (fiber breakages) occur in largely sustained fashion till the end of the test. AE amplitude (Figs. 15, 16 and 18) behaves in a generally similar way to the parameters considered earlier. The three levels of failure appear to commence from just a little moment into the testing.

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AE RMS (Fig. 17) is a kind of statistical summary history of overall failure, and is quite useful to obtain a bird’s eye view. RMS trend approximately reveals the gross failure levels in the structure. The AE data suggests that gross damage is relatively small for the first half of the loading history of the specimens. Matrix micro-cracking is suggested by the succession of high and low slopes values in the first half of the record. The first, small RMS jump is probably due to the settlement into true leveling of the loading platens. Debonding of the ribs from the skin apparently started at a displacement of about 0.06 in. The audible pinging sound heard at about the middle load level during the experiment confirms this. Debonding continued in different ribs, until about 0.07 when buckling started to occur in the ribs. The audible noise was noted to have increased with load increase, thus suggesting that more ribs were getting debonded with increasing load, as revealed in both the AE trace and the load-slope data. A continuation of this is seen through the 0.07 to 0.08 displacement region, as more failure types including fiber pullout, delamination in the ribs, matrix cracking, and other low-and medium-energy level failures also appear. In the same way, the Frequency–Amplitude curve of Fig. 18 supplies a signature that quantitatively portrays at a glance distribution of events types involved in the damage process. The significantly inverse frequency–amplitude relationship is clearly depicted. Overall, a slight specimen buckling appears to occur around 0.087 to 0.094 displacement values, when also a major drop in the slope of the load–deflection curve is seen. There being no significant indication on the actual load–deflection diagram, nor any observable physical evidence, such as cylinder barreling, it seems that this effect was quite small in the cylinders tested. It seems that fracture of the ribs, which are the stiffeners or support structure for the skin, appears to directly initiate skin failure, as would be expected. Failure of a specimen thus appears to be a combination of events as indicated clearly by the multi-level points on the AE diagrams. The cylinder failure is well illustrated by a sharp rise in the RMS value, and a sharp drop in the slope values of the load–displacement diagram (signifying drastic loss of stiffness), as well as the loud popping noises audibly registered during the test. Examination of the failed cylinders showed a delamination of the 90ı plies across about half of the circumference.

2.2.4

Concluding Remarks

Load–displacement records of the tests provided faithful mechanical response of the cylinders, as confirmed by the independent acoustic emission and strain-gage records. Different independent acoustic emission parameters all showed agreement with themselves and with the mechanical test data, as to occurrences in the structure each instant of time. The audible noises during the test, strain gage readings, acoustic emission records, the load–displacement diagrams, post-test examination of the cylinders, and fractographs, may be utilized to reconstruct the failure sequence in general terms. It appears that matrix deformation and cracking occurred first, followed by fiber microbuckling and fiber debonding from the matrix, and then

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skin-rib delamination. Rib-buckling and associated skin cracking, deformation and fiber pullout and breakage then ensued. In the rib, ply delaminations and then breakage occurred. Multiple rib failures brought on the final catastrophic collapse of the cylinder.

2.3 Flexural Fatigue of Foam-Cored Composite Sandwich 2.3.1

Background

Acoustic emission (AE) was used in investigation of developing damage as marine composite sandwich samples were loaded in both static and fatigue bending, at room and sub-zero temperatures. The AE test data was collected with wide-band sensors, and analyzed with classical parametric, general joint time–frequency and wavelet transform techniques. The wavelet transform method, which basically combines measures of time with those of frequency and intensity, is regarded as one of the most advanced signal processing methods.

2.3.2

Testing

Four-point bending tests were performed on the composite sandwich (CS) beams having Rohacell (PMI) foam core and unidirectional carbon/epoxy skin material at 22ı C; 0ı C, 30ı C and at 60ı C respectively inside an environmental chamber of an EnduraTec servo-pneumatic testing machine. four-point fatigue bending tests in accordance with ASTM C393-62 were performed at 2 Hz frequency and 0.1 loading ratio (min. load/max. load per cycle) to simulate sea wave effects at the different temperatures of RT, 0ı C and 60ı C with load levels, and in the range from 70% to 90% of static ultimate load. Crack propagation was tracked with a digital camera in the fatigue tests. AE data was processed with a Physical Acoustics AE-Win system. AE data was concurrently acquired with miniature PAC pico sensors. The test setup is shown in Fig. 19.

2.3.3

Results and Discussion

Figure 20 shows the good agreement of AE records with mechanical test results in a comprehensive manner across materials and temperatures used in the tests. At each temperature, the changing slope of the load–displacement curve reveals changing stiffness as damage accumulates. A digital camera aimed at the specimens could not capture any cracking, even up to catastrophic failure, but every specimen was observed to suddenly fail in the core shear mode. However, the AE data and their analysis show a more graphic trending of the damage process. The results also showed that stiffness, strength and elastic limit were enhanced by cooling

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Fig. 19 Experimental setup

while displacement to failure decreased. The corresponding AE energy-release events showed increase in events-density as loading progresses, and also increasing amplitudes. Stiffness reduction at 60ı C was slight, notably increasing useful fatigue life, unlike at RT with significant early stiffness reduction. Events density is seen to be generally proportional to the area under the loading curve showing that AE closely mimics the mechanical test. In general, the catastrophic failures occurred too rapidly for the high-speed camera. On the other hand, he AE approach captured practically everything that happened, in an event-by-event record of the damage progression and ultimate failure. This is a major advantage of AE as an NDE tool in such rapidfailure phenomena. The GF/RC and CF/RC responses show that the glass samples took longer (some four to five times longer) to final failure than the carbon samples. The carbon samples show more intense AE activities than the glass, perhaps reflecting their relatively higher brittleness. It was also observed that the lower the temperature, the longer the sample takes to fail in fracture. The pattern of most energetic terminal failure is also apparent for all the samples and temperatures. It was noted that the bending fatigue loading results showed that at all the temperatures, the number of cycles to failure increased as the load level decreased, and that the GF/RC failed after more cycles than the CF/RC. At each load level, the number of cycles to failure increased with temperature reduction. All these effects were shown very clearly by AE parameters, in both static and fatigue cases. As an example, Fig. 21 shows the variation of Energy with Amplitude for the glass fiber/rohacell sample at the lowest test temperature. The stratification into events types is readily seen, when viewed on the basis of AE energy, as in the literature [16, 18]. Thus it appears that for the zones shown, identifiable failure events include the very low energy core failure events, the slightly higher energy interfacial

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Fig. 20 Static load (lb) versus displacement (in.) curves with superposed AE amplitudes (dBAE) versus time (s) curves for GF/RC (left) and CF/RC (right) for temperatures RT22ı C (top), 0ı C, 30ı C and 60ı C (bottom)

failure events, the moderate energy matrix damage events and the high-energy fiber failures. Although the major failure mode was clearly visible as core shear, accompanying occurrence of other failures even in minute quantities also show up due to the sensitivity of the acoustic emission approach. Each event has a waveform which may be transformed by the Fourier or Wavelet approaches. Waveforms and 2-D and 3-D wavelet transform plots of typical

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Fig. 21 AE energy–time plot for GF/RC at RT

Fig. 22 2D-Wavelet for #3 event of GF/RC at 60ı C, 0.8 load level

breaking events are presented in Figs. 22 and 23. We see a clarifying presentation of the time-evolution of frequency (proportional to inverse of scale) and intensity in the wavelet description, to aid in the failure classification and description process. Modal AE principles may also be applied to the waveform to establish arrival times and relative maximum amplitudes of the extensional and flexural waves in each case to determine which is dominant, and what failure type may have given rise to it.

2.3.4

Concluding Remarks

AE amplitude and several other parameters were found to show very similar behavior to the mechanical test results. Across the samples tested, failure tended to follow the pattern of AE amplitude increases with viscoelastic matrix deformation

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Fig. 23 3D-Wavelet for #3 event of GF/RC at 60ı C, 0.8 load level

and damage reaching climaxing at the break point. The dominant failure mode all across the samples tested was core shear under static and fatigue loading at the different temperatures. AE tends to capture much more details than most other NDE methods, and the damage progression and failure history can be re-constructed to an almost endless degree of precision and detail, and in close agreement with the mechanical test results. For composite sandwich materials at lower temperatures, where very rapid failure mechanisms may manifest, AE appears to be an invaluable acquisition and analysis of the test responses.

3 Conclusion This brief review has been a small extract culled to fit into the context of the publication of which this chapter is a part, from research work done by the author and his colleagues for the ONR under the direction of program manager Dr. Yapa D.S. Rajapakse. The author bears sole responsibility for the NDE part of the investigations. Acknowledgements The author expresses his gratitude to ONR and program manager Dr. Yapa D.S. Rajapakse for several research grants over the years that have made this work possible. He gratefully acknowledges ONR program manager Dr. Kelly Cooper who co-supported the work on flexural testing with grant number N000140810647.

References 1. Mizutani Y, Nagashima K, Takemoto M, Ono K (2000) Fracture mechanism characterization of cross-ply carbon fiber composites using acoustic emission analysis. NDT & E Int 33: 101–110 2. Hamstad MA (1992) An examination of AE evaluation criteria for aerospace type fiber/polymer composites. Proc 4th Int Symp AE Compos Mater (AECM-4) ASNT, Seattle, WA, pp. 436–449

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3. Nixon JA, Phillips MG, Moore DR, Prediger RS (1988) A study of the development of impact damage in cross-ply carbon fibre/PEEK laminates using acoustic emission. Compos Sci Technol 31(1): 1–14 4. Valentin D, Bonniau P, Bunsell AR (1983) Failure mechanism discrimination in carbon fibrereinforced epoxy composites. Composites 14(4): 345–351 5. Hamstad MA, O’Gallagher A, Gary J (2002) Examination of the application of a wavelet transform to acoustic emission signals: Part 2. Source Location. J Acoust Emiss 20: 62–68 6. Ni Q-Q, Iwamoto M (2002) Wavelet transform of acoustic emission signals in failure of model composites Eng Fract Mech 69: 717–728 7. Surgeon M, Wevers M (1998) Modal analysis of acoustic emission signals from CFRP laminates NDT & E Int 32: 311–322 8. Kotsikos G, Evans JT, Gibson AG, Hale JM (2000) Environmentally enhanced fatigue damage in glass fibre reinforced composites characterized by acoustic emission. Compos A: Appl Sci Manuf 31(9): 969–977 9. Qi G (2000) Wavelet-based AE characterization of composite materials. NDT & E Int 33: 133–144 10. Dzenis Y, Saunders I (2002) On the possibility of discrimination of mixed mode fatigue fracture mechanisms in adhesive composite joints by advanced acoustic emission analysis. Int J Fract 117: 123–128 11. Johnson M (2000) Classification of AE transients based on numerical simulations of composite laminates. NDT & E Int 36(5): 133–144 12. Choi N-S, Woo SC, Kim TW, Rhee KY (2003) Monitoring of failure mechanisms in fiber reinforced composites during cryogenic cooling by acoustic emission. Int J Mod Phys B 17(8–9): 1763–1769 13. Jeong H, Jang Y-S (2000) Wavelet analysis of plate wave propagation in composite laminates. Compos Struct 49(4): 443–450 14. Ayorinde E, Islam S, Mahfuz H, Gibson R, Deng F, Jeelani S (2002) Basic NDE of some nano composites. Proc ASME Int Mech Eng Con Exp, New Orleans, LA, vol. 23, pp. 30–44 15. Yamaguchi K, Oyaizu H, Jokkaji I, Kobayashi Y (1991) Acoustic emission technology using multi-parameter analysis of waveform and application of GFRP tensile tests. In: Sachse W, Roget J, Yamaguchi K (eds.), Acoustic emission: current practice and future directions, ASTM STP 1077 American Society for Testing and Materials, Philadelphia, PA, p. 123 16. Cantwell WJ, Morton J (1991) The impact resistance of composite materials – a review. Composites 22(5): 347–362 17. English LK (1987) Listen and learn: AE testing of composites. Mater Eng 104: 38–41 18. Ziola SM, Gorman MR (1999) Source location in thin plates using cross-correlation. J Acoust Soc Am 90(5): 2551–2525 19. Benevolenski OI, Karger-Kocsis J (2001) Comparative study of the fracture behavior of flowmolded GMT-PP with random and chopped-fiber mats. Compos Sci Technol 61: 2413–2423 20. Dzenis YA, Qian J (2001) Analysis of microdamage evolution histories in composites. Int J Solids Struct 38: 1831–1854 21. Ayorinde EO, Deng F, Kulkarni S, Mahfuz H, Jeelani S (2003) Evaluation of AE response of loaded nanophased composites. Paper IMECE2003-43571 Proc. 2003 IMECE, Washington, DC 22. Ayorinde E, Mohammed G, Mahfuz H, Jeelani S (2004) Acoustic emission NDE of some isogrid composite cylinders. 22nd South Eastern Conf Theor Appl Mech, Tuskegee University, Tuskeegee, AL 23. VanVuure AW, Ivens JA, Verpoest I (2000) Mechanical properties of composite panels based on woven sandwich fabric performs. Compos A 31: 671–680 24. Bannister MK, Braemar R, Crothers PJ (1999) The mechanical performance of 3D woven sandwich composites. Compos Struct 47: 687–690 25. Ahmet C, Sabri T, Alptekin A (1999) AE response of 316L SS during SSR test under potentiostatic control. Corros Sci 41: 1175–1183 26. Caprino G, Teti R (1995) Residual strength evaluation of impact GRP laminates with acoustic emission monitoring. Compos Sci Technol 53: 13–19

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27. Ma XQ, Cho S, Takemoto M (2001) Acoustic emission source analysis of plasma sprayed thermal barrier coating during four-point bend tests. Surf Coat Technol 139: 55–62 28. Rutem A (1998) Residual flexural strength of FRP composite specimens subjected to transverse impact loading. SAMPE J 24(2): 19–25 29. Slysh P (1976) Isogrid structural tests and stability analysis. Aircr J 13: 10 30. Jenkins WC (1972) Determination of critical buckling loads for isogrid stiffened cylinders. MDC Report No. 02722 31. Mahfuz H, Dean D, Jeelani S, Baseer M, Mohammed G, Hampton E, Rangari V (2003) Innovative manufacturing and structural analysis of composite isogrid structures for space applications. Report AFRL-SR-AR-TR-04 32. Czsigarny T, Karger-Kocsis J (1993) Comparison of the failure mode in short and longreinforced injection molded polypropylene composites by acoustic emissions. Polym Bull 31: 495–501 33. Wolters J (1985) Description of compound parameters of particle-filled thermoplastic materials by acoustic emission techniques. J Acoust Emiss 3: 51–58

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Ballistic Impacts on Composite and Sandwich Structures Serge Abrate

Abstract Composite and sandwich structures are sometimes subjected to impacts that result in complete perforation. Tests are conducted to determine the velocity required to achieve complete penetration for a given projectile and a model is required for data reduction purposes, to understand the effect of various parameters and to extrapolate for other test conditions. In addition, models capable of predicting the ballistic limit and the extent of damage in composite and sandwich structures are also needed. This chapter presents a comprehensive and critical assessment of the existing literature on this topic.

1 Introduction Extensive studies of the effects of impacts on composite materials in recent years resulted in a large number of publications that deal mostly with low velocity nonpenetrating impacts [1]. The penetration of fabric armor was treated in recent review articles [2, 3] and will not be considered here. This article focuses on penetrating impacts that can occur at both low and high velocities. General experimental observations are included in Section 2. Following Zukas et al. [4], mathematical models are divided into three broad categories: empirical models, approximate engineering models, and numerical models. Engineering models are usually based on basic assumptions regarding the interaction between the projectile and the target. These models capture the main features observed in experimental results while remaining simple. Numerical models attempt to provide a more general capability for analyzing the effect of many factors with only a minimal number of assumptions. This article presents a critical review of the literature concerning ballistic impacts on structural composites and sandwich structures. Section 2 discusses models based on assumptions regarding the penetration resistance which is the resisting S. Abrate () Southern Illinois University, Carbondale, IL 62901-6603, USA e-mail: [email protected]

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force applied on the projectile. This is an approach that has been used for centuries to analyze penetration into soil, rock concrete, and metals. With the correct assumptions one can obtain an accurate approach for analyzing experimental results and predicting the ballistic limit. Section 3 discusses models based on assumptions regarding the stresses acting on the surface of the projectile during the penetration process. In an approach used by many investigators, the normal stress is a function of initial velocity and the shape of the projectile. Here we show that if the normal stress is independent of the projectile’s shape, the penetration energy is the same regardless of that shape. Section 4 discusses the effects of target thickness, projectile diameter, stacking sequence, obliquity and projectile density on the ballistic limit. The approaches discussed in Sections 2 and 3 do not directly account for the energy dissipated during fiber failure or delamination. Section 5 describes models used by several investigators in which force–deflection curves determined during quasi-static tests are used to predict the ballistic limit of the laminate. Other models are based on the conservation of energy and prior knowledge of failure mechanisms for similar types of laminates (Section 6). Section 7 discusses numerical models for the analysis of ballistic impacts and in Section 8 we review the literature related to ballistic impacts on sandwich structures.

2 Models Based on Assumptions Regarding the Penetration Resistance Many experimental studies have been conducted to determine Vb , the ballistic limit of various laminated composites, which is the lowest velocity that results in total penetration of the laminate. To account for the variability in experimental results, some authors also define V50 the ballistic limit as the velocity at which 50% of the specimens will be completely penetrated. In order to determine the ballistic limit for a particular laminate, several tests resulting in complete penetration are conducted. The projectile has a mass M and the initial velocity Vi and the residual velocity VR are measured for each test. Then, a data reduction procedure is needed to estimate Vb . A common approach dating back several hundred years is based on making simple assumptions on the force opposing the motion of the projectile. The projectile is assumed to be rigid and the erosion of its surface is assumed to be negligible. This section shows that five assumptions on the resisting force applied to the projectile lead to three well-known relationships between initial and residual velocities as illustrated in Fig. 1. During the penetration process (Fig. 2b), the motion of the rigid projectile is opposed by a force F called the penetration resistance (Fig. 2d) and, applying Newton’s law, the motion of the projectile is governed by F D M v

dv dx

(1)

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Equation of motion for a rigid projectile

Assumption on penetration resistance

F=co

F=co K.E. of ejecta

F = co + c2 v2

F = c1v + c 2v2

F= co + c1v

(

)1/2

VR = α Vi2 −Vb2

UR = Ui − Up

(

VR= aVi − b

)

UR = α2 Ui −Up

Fig. 1 Assumptions on the penetration resistance and resulting relationships between initial and residual velocities

c

a Vi

VR

M h

d

b M

dv dt

F

x

Fig. 2 Penetration of a target by a rigid projectile: (a) initial condition; (b) penetration; (c) residual velocity of the projectile; (d) free body diagram of the projectile during penetration

Integrating with respect to x gives 1 1 MV2i D MV2R C 2 2

Z Fdx

(2)

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where VR is the residual velocity (Fig. 2c). In Eq. (2), Ui D 12 MV2i is the initial kineticR energy of the projectile, UR D 12 MV2R is its residual kinetic energy and Up D Fdx is the energy required for perforation. Then, Eq. (2) can be written as UR D Ui  Up

(3)

Equation (3) suggests a relationship between the residual kinetic energy and the initial kinetic energy of the projectile. The exact nature of this relationship depends on the penetration resistance. In this section we examine five commonly made assumptions and determine which ones lead to a better fit of the experimental data.

2.1 Constant Penetration Resistance Robins (1742) and Euler (1745) assumed that the penetration resistance is constant .F D co / and is proportional to the cross-section of the projectile [5]. Then, the penetration energy is constant .UP D co h/ and is proportional to h, the thickness of the target. Equation (3) shows that, when plotting UR versus Ui , the experimental data points should plot on a straight line with a unit slope and a horizontal intercept equal to Up . Results from several penetrating impacts can be used to determine Up and the ballistic limit. It is also instructive to examine the variation of the projectile velocity or is kinetic energy during the penetration process since those quantities are often measured during experiments. For a constant penetration resistance, integrating Eq. (1) between x D 0 and an arbitrary position x shows that U, the kinetic energy of the projectile, decreases linearly with x 1 U D Mv2 D Ui  Fx (4) 2 Figure 3 shows that, starting with an initial kinetic energy above the perforation energy, the kinetic energy of the projectile decreases linearly until the target is completely perforated. At that time, the residual kinetic energy is UR . If the initial velocity is the ballistic limit, the residual velocity will be zero (Fig. 3). Often, the velocity of the projectile is plotted versus its displacement. In that case, Eq. (4) indicates that the straight lines shown in Fig. 3 become the lower part of parabolas with a vertices at (0, Vi ) that open towards the negative horizontal axis. Recalling that in Eq. (1), v dv the acceleration of the projectile can also be written as dv , intedx dt grating with respect to time gives v D Vi 

F t m

(5)

Figure 4 shows that, according to this model, the velocity of the projectile decreases linearly with time. When the initial velocity is equal to the ballistic limit, the residual

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Kinetic energy

Ui

F

Ub

UR F

Displacement

Fig. 3 Kinetic energy of the projectile as a function of the displacements assuming a constant penetration resistance

Non-dimensional velocity v/Vb

2.5

2

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

Non-dimensional time t / tb Fig. 4 Projectile velocity as a function of the time assuming a constant penetration resistance. F tb The slope of the straight lines is equal to mV b

velocity is zero and the duration of the penetration process is tb . For higher initial velocities, the duration of the event is tp with 2 3 s  2 tp Vi 4 Vi 5 1 1 (6,7) tb D MVb =F and D tb Vb Vb The dashed line in Fig. 4 represents the end of the perforation process as given by Eq. (7). As the initial velocity increases, the duration of the event becomes shorter and the difference between initial and residual velocities also becomes smaller.

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The assumption of a constant penetration resistance appears to be too restrictive when several factors suggest that this might not be the case. For example, for projectiles penetrating thin laminates, the area of contact between the projectile and the target varies greatly so it is unlikely that the penetration resistance will remain constant. This assumption can be relaxed by assuming that the penetration resistance is independent of velocity. Then, Up , which is the area under the force versus displacement curve, remains constant and the same data reduction procedure applies. Sun and Potti [6] assumed that during ballistic impacts, the penetration energy is the same as the static penetration energy. Several authors [6–14] assumed that the initial and residual velocities are related by Eq. (3). Unfortunately, experimental data often shows that, while UR is a linear function of Ui , the slope of the line is generally less than one so that curve fitting experimental results by Eq. (3) leads to significant errors in estimating the ballistic limit. A number of experimental investigations show that the absorbed energy defined as Uabs D Ui  UR (8) increases with Ui . Roach et al. [15] showed that the dynamic penetration energy can be 2.5 to 3.5 times higher than the static penetration energy. Lee et al. [14] measured the energy required for perforation of 5-ply laminates of Spectra fabric reinforced composites with vinylester and polyurethane resin matrices. Three types of tests were conducted using the same FSP geometry for the penetrator: (1) quasi-static puncture; (2) dropweight impact (3.78 m/s, 12.3 kg); (3) ballistic impact (220–260 m/s, 1.1 g). Results indicate that the energy absorbed for full penetration during ballistic impacts is nearly twice that absorbed during dropweight impact tests. Similarly, in the experiments conducted by Mines et al. [16] on glass– polyester composites, the perforation energy measured during ballistic impacts was significantly larger than the static penetration energy. The elastic moduli of Spectra composites vary significantly with strain rate, which is advanced as an explanation for the increased energy absorption during high velocity impacts [17].

2.2 Assumption 2: Kinetic Energy Absorbed by Ejecta One possible explanation for the fact that experimental results cannot be fitted by Eq. (3) is that material ahead of the projectile is being ejected at high velocities during perforation. The kinetic energy absorbed by this material might be significant. Usually, many particles of varying sizes are ejected at various velocities. Some of the initial kinetic energy of the projectile is then stored into the ejecta in the form of kinetic energy. To account for this effect, consider that, after penetration, the projectile plus an equivalent mass of ejecta m is moving with a residual velocity VR . Using the work-energy principle 1 1 .M C m/ V2R D MV2i  UP 2 2

(9)

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Then, Eq. (1) can be written as   UR D ’2 Ui  Up

(10)

p where ’ D 1= 1 C .m=M/. Eq. (10) shows that, when the residual kinetic energy is plotted versus the initial kinetic energy, the data should plot on a straight line with a slope that is less than or equal to one. In this model, the kinetic energy absorbed by the material removed during perforation is the reason why the slope of the line is less than one in a UR versus Ui plot. Equation (10) is a special case of the well-known equation (see Zukas et al. [4])  p p 1=p VR D ’ Vi  Vb

(11)

that is used to fit experimental data for impacts on metals and other materials. In this form, Eq. (11) is called the Recht–Ipson equation and when p D 2 it is called the Lambert–Jonas equation. The parameter p is usually equal to 2 for rigid projectiles. ’ can be significantly less than one. For example, Larsson [27] reported ’ values of 0.7920, 0.7926, 0.8825, and 0.8427. Equation (11) was mentioned by Lee and Sun [9,10] but ’ was immediately taken to be one and Eq. (1) is recovered. Figure 5 shows the data presented by Lee and Sun [9]. Round symbols represent experimental results for penetrating impacts that were fitted by a straight line in Fig. 5a with a high coefficient of correlation. Square symbols represent results for non-penetrating impacts. In this case ’ D 0:9174 and Vb D 38:55 m=s. Assuming a unit slope as in Ref. [7], leads to a poor fit of the entire data (Fig. 5b) and an over-estimation of the ballistic limit with Vb D 42 m=s. With the present approach, the estimated ballistic limit (38.55 m/s) is higher than the highest velocity for non-penetrating impacts and lower than the lowest velocity for penetrating impacts in the experiments. The ballistic limit obtained by previous authors (42 m/s) is higher than that of two of the penetrating impacts in the experiments. The data of Jenq et al. [18] leads to ’ D 0:9066 and a ballistic limit of 71.82 m/s. In Ref. [18] it was assumed that m D 0 and Eq. (1) was used to estimate the ballistic limit Vb from each data point. This approach leads to the erroneous conclusion that the ballistic limit varies between 68.8 and 91.1 m/s as the impact velocity increases from 68.6 to 172 m/s. Similar errors are made by others. The Lambert–Jonas equation was used in many articles [e.g. [19–27]. Gu and Xu [28] presented experimental results for the ballistic penetration of braided composites. The initial and residual velocities of the projectile (Table 1) were fitted by Eq. (10) with ’ D 1:034 and Vb D 214 m=s. Another set of experimental results on the same type of composites was presented by Gu and Ding [29]. The data presented in that reference (Table 2) was fitted by Eq. (10) with ’ D 1:011 and Vb D 189 m=s. In both cases, the residual velocities calculated using Eq. (10) are in good agreement with the measured values. Values of ’ that are slightly larger than one are often encountered. There does not appear to be much significance to the fact that it is larger than one except that it is obtained from curve fitting experimental results.

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a 70

Residual energy (J)

60 y = 0.8417x - 18.765 R2 = 0.9883

50 40 30 20 10 0 20

0

40

60

80

100

Impact energy (J)

b 70

Residual velocity (m/s)

60 50

Present

40 30 20 Lee and Sun

10 0 0

10

20

30

40 50 60 Initial velocity (m/s)

70

80

90

Fig. 5 Experimental data from Lee and Sun [7,8]: (a) Residual energy versus initial kinetic energy, (b) Residual velocity versus initial kinetic energy of the projectile

Using Eq. (9), the absorbed energy can be expressed as   Uabs D Ui m C Up M = .M C m/

(12)

and it increases linearly with the impact energy. This result has been observed experimentally by several authors. Wonderly et al. [30] presented experimental results for the perforation of glass fiber/vinyl ester and carbon fiber/vinyl ester composites. From that data we extract ’ D 0:9463 and Vb D 108:9 m=s for the glass fiber/vinyl ester composites and, for the carbon fiber/vinyl ester ’ D 0:9543 and Vb D 99:52 m=s. In both cases, the ballistic limits determined by this procedure is

Ballistic Impacts on Composite and Sandwich Structures Table 1 Data from Gu and Xu [28]

Table 2 Data from Gu and Ding [29]

Vi .m=s/ 291 295 298 299 383 386 390 505 531 650 658

Vi .m=s/ 241 253 256 263 279 287 326 328 378 385 391 523 532 661 670

473 VR exp. (m/s) 210 225 204 231 325 316 329 479 502 641 638

VR exp. (m/s) 158 168 170 182 224 206 276 279 326 342 343 486 501 647 648

VR cal. (m=s) 204 210 214 216 328 338 337 473 503 635 643

VR cal. (m/s) 152 170 175 185 208 219 269 271 331 339 346 493 503 641 650

larger that the initial velocities of all the non-penetrating impacts and lower than the initial velocities of all the penetrating impacts. The absorbed energy was shown to be a linear function of the initial kinetic energy Ui . Here Eq. (10) is used to analyze the experimental results given by Shokrieh and Javadpour [31] for the penetration of a projectile in ceramic composite armor and to determine the ballistic limit. The armor consists of one layer of boron carbide ceramic bonded to a composite material with Kevlar 49 fiber reinforcement (Fig. 6). Figure 7 shows the linear relationship between the square of the residual velocity and the square of the initial velocity which indicates that Eq. (10) is also applicable in this case. Using this approach, a ballistic limit of 309 m/s is obtained. This example is a simplified version of the type of composite armor that are considered for military vehicles. Gama et al. [32] and Vaidya et al. [33] describe more complex configurations in with the ceramic layer is covered on the outside by a composite layer and is separated from the structural composite layer by a rubber layer. On the inside, a phenolic layer is added to the structural layer for fire resistance.

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Fig. 6 Impact of a projectile on ceramic composite armor

V Projectile

Plastic zone

Ceramic 65o

Composite

1.6 y = 0.9954x - 95261 R2 = 0.9999

1.4 1.2 1 V2R 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2

Vi

Fig. 7 Square of the residual velocity .x106 m2 =s2 / versus square of the initial velocity of the projectile for ballistic impacts on ceramic-composite armor (Data from Ref. [31])

Senf et al. [34] considered a simpler arrangement with only three layers where the ceramic tile and the glass reinforced panel are separated by a layer of honeycomb.

2.3 Poncelet’s Assumption It is well-known that, with polymer matrix composites, strain rate effects can be significant during dynamic events. Then, it is logical to assume that the penetration resistance will increase with the velocity of the projectile. In 1829, Poncelet assumed that the penetration resistance is a function of velocity in the form [5] F D co C c2 v2

(13)

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Integrating the equation of motion (Eq. (1)) as the projectile moves from zero to x and the velocity decreases from Vi to v gives the evolution of the velocity during the perforation process as

   2c2 x co C c2 v2 D co C c2 V2i exp  M

(14)

Replacing v by VR and x by h gives the relationship between the initial and residual velocities. A simpler but approximate expression is obtained using the Taylor series expansion ex D 1 C x. After some algebraic manipulations, Eq. (14) becomes   U D Ui  co C c2 V2i x

(15)

Equation (15) indicates that, during the perforation process, the kinetic energy of the projectile decreases linearly with x as in Fig. 3 but the slope of the line is steeper and depends on the initial velocity. It is equal to the static component co plus a dynamic component c2 V2i . From Eq. (15) we find that   2c2 h U i  co h UR D 1  M

(16)

Plotting the residual kinetic energy UR versus the initial kinetic energy Ui will result in a straight line with a slope that is less than one. Equation (16) can be written q in the form of the Lambert–Jonas equation (Eq. (10)) but in this case,

’ D 1  2cM2 h . With this model, the energy required to penetrate the target during ballistic penetration   2c2 h (17) Ub D co h= 1  M

is higher than the static penetration energy since the denominator on the right hand side of Eq. (11) is less than one. In addition, Eq. (16) also implies that the energy absorbed during the penetration event increases linearly with the impact energy. There are two possible reasons for the slope of the line in the UR versus Ui plot to be less than one: (1) kinetic energy is absorbed by the target material ejected during perforation as shown in Section 2.1; (2) the penetration resistance contains a term depending on v2 as here.

2.4 Penetration Force Increases Linearly with the Velocity In modeling the perforation of composite structures, it is often assumed that the resisting force increases linearly with the velocity of the projectile F D co C c1 v

(18)

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Integrating the equation of motion (Eq. (1)) as the projectile moves from zero to x and the velocity decreases from Vi to v gives 

    h c0 c1 c1 VR  Vi C 2 ln 1 C Vi  ln 1 C VR D M c1 co co c1

(19)

2

Using the Taylor series expansion ln .1 C x/ D x  x2 which gives less than 3% error when x < 0:3, Eq. (19) reduces to Eq. (1) with UP D co h. Therefore, the effect of the velocity on the penetration force are negligible as long as c1 Vi < 0:3 co .

2.5 Penetration Force Varies with v and v2 In 1895, Resal assumed that the penetration resistance is a function of velocity of the form [5] (20) F D c1 v C c2 v2 Integrating the equation of motion gives

 c2 h c1 C c2 VR D .c1 C c2 Vi / exp  m

(21)

A Taylor series expansion of the exponential function reveals that   c1 h c2 h  VR D Vi c1  m m

(22)

In other words, the residual velocity increases linearly with the initial velocity of the projectile. It is interesting to note that sometimes investigators find that when the residual velocity is plotted versus the initial velocity, experimental results appear to fall on a straight line

2.6 Summary The examination of experimental data presented by numerous authors showed that the simple assumption of a constant penetration resistance is not valid because it leads to an incorrect relationship between the residual and initial kinetic energies of the projectile. It was found that two factors cause the slope of the line representing the relationship between these two kinetic energies to be less than unity: the kinetic energy absorbed by the ejecta and internal dissipation in the target. What is usually called the Lambert–Jonas equation can be used to fit experimental data with great accuracy and produce a close estimate of the ballistic limit.

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3 Projectile-Target Interaction Models In modeling the penetration of projectiles into targets, instead of making assumptions on the resisting force as was discussed before, several authors make similar assumptions on the stresses at the interface between the projectile and the target. First we examine a commonly made assumption made on the normal stress acting on the surface of the projectile that accounts for the initial velocity and the nose shape. This approach is applied to projectile of different shapes and show that the penetration energy is equal to the product of the cross-sectional area of the projectile by the interface stress, a fact that was not brought out in the many articles that have used this approach before. Then, the effects of friction between the projectile and the target are examined.

3.1 Normal Pressure on the Surface of the Projectile Wen [35, 36] assumed that the normal pressure on the surface of the projectile is uniform and is given by   p ¢ D ¢e 1 C “Vi ¡t =¢e

(23)

where ¢e is the static linear elastic compression limit in the through-thickness direction, “ is a constant depending on the shape of the projectile, ¡t is the density of the target and Vi is the initial velocity of the projectile. According to Eq. (23), the normal pressure on the surface of the projectile depends on the initial velocity of the projectile and also on the shape of the projectile since 8 ˆ 1 ˆ ˆ ˆ <sin ™ “D 3 2 ˆ ˆ 4§ ˆ ˆ : 1:5

for a flat end for a conical end for an ogival end

(24)

for a spherical end

Wen [35, 36] developed formulas for calculating the penetration resistance for several types of projectiles. The penetration resistance is calculated as the resultant of this pressure acting on the surface of the projectile that is in contact with the projectile. Using Eq. (23), the penetration resistance depends on the impact velocity but that it remains constant during the penetration process. This may be an acceptable approximation when the initial velocity is much higher than the ballistic limit and the difference between initial and residual velocities is small. However, near the ballistic limit, the velocity of the projectile goes from all the way down to zero so the penetration resistance should not remain the same during the entire penetration process.

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Wen [35] considered cylindrical projectiles with either conical or ogival noses and in Refs. [36, 37] projectiles with truncated, conical, flat, ogival and hemispherical ends were considered. These models were used by Ulven et al. [38]. He et al. [39] used the model for projectiles with conical tips. Cantwell and coworkers [40–42] used Eq. (23) to predict the ballistic limit of fiber-metal laminates (FMLs). The same assumption (Eq. (23)) was made by Ben-Dor et al. [43–45] in their study of the penetration of fiber-reinforced plastics. In the following, this assumption (Eq. (23)) is used to predict the penetration resistance and the penetration energy of blunt-ended, conical, and spherical-tipped projectiles.

3.2 Blunt-Ended Projectile Considering a blunt-ended cylindrical projectile (Fig. 8), the penetration resistance and the penetration energy are given by F D  R2 ¢ and Up D  R2 ¢h

(25, 26)

With this assumption, the penetration energy is proportional to the thickness of the laminate and the square of the radius of the projectile. Furthermore, Up increases linearly with the initial velocity.

3.3 Conical-Tipped Projectile For a cylindrical projectile with a conical nose (Fig. 9) with a short nose (L < h), the penetration process consists of three phases. When x < L, the penetration resistance and the energy dissipated during this phase are  x 2 F D  R2 ¢ and U1 D  R2 ¢h (27, 28) L When L < x < h, the penetration resistance is the same as that for a blunt projectile (Eq. (25)) and the energy dissipated during this phase is U2 D  R2 ¢ .h  L/

(29)

R

x

Fig. 8 Penetration by a blunt projectile

h

Ballistic Impacts on Composite and Sandwich Structures

479

Fig. 9 Penetration by a conical projectile

R

x

h Phase I: x
h

x

Phase II: h<x
h x

Phase III: x>Ln

When h < x < h C L, the penetration resistance is " 2

F D  R ¢

 1

xh L

2 # (30)

and the energy dissipated during this third phase is "

1 U3 D  R2 ¢L 1  3



2 # L tan ’ R

(31)

Up D U1 C U2 C U3 , the total energy dissipated during the penetration process obtained from Eqs. (28), (29), (31), is exactly the same as that obtained for a flatended cylindrical projectile. This result is given by Wen [35] who also shows that this result also holds true for cylindrical projectiles with ogival tips.

480

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Reid et al. [46] used Eq. (27) to model the first phase of the indentation of glass– phenolic laminates by a conical projectile. In their analysis, the dependence of the strength ¢ on the cone half angle was taken to be ¢ D Rip sin2 ™ C Rtt cos2 ™

(32)

The coefficients Rip and Rtt correspond to the inplane and through-thickness resistances of the laminate and were determined from experiments. The improved dynamic penetration resistance of the laminate was shown to be mostly due to an increase of the through-thickness resistance.

3.4 Spherical Tipped Projectile For a cylindrical projectile with a spherical tip (Fig. 10) or for a spherical projectile with R
(33)

  R F D  R ¢ and U2 D   R ¢h 1  h

(34)

When R < x < h, 2

2

When h < x < h C L, 

x h F D  R ¢ 1  C R R 2

2

and U3 D  R2 ¢ h

R 3h

(35)

The total energy dissipated during the penetration process Up D U1 C U2 C U3 D  R2 ¢h is the same as that obtained for a flat ended cylindrical projectile. The general result obtained here for the first time is that, using a uniform normal pressure over the surface of the projectile interacting with the target, the perforation energy is always given by Eq. (26) regardless of the shape of the projectile. It is also worth noting that Up is always proportional to R2 h. In other words, the penetration resistance is proportional to both the cross-sectional area of the projectile and to the thickness of the target. Comparing the different types of projectiles, there are differences in the penetration resistances but the area under each curve, which is the penetration energy, is the same for the same normal pressure ¢. However, ¢ depends on the initial velocity and the shape of the projectile so Up will also depend on the shape of the projectile. To illustrate this point, consider the work of Ulven et al. [38] who used

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481

Fig. 10 Penetration by a spherical projectile

R x

r1

h

Phase I: x
r1 x

h

r2

Phase II: h<x
r r

h

x

Phase III: x>R

the results obtained by Wen [35, 36], for cylindrical projectiles with flat, spherical and conical tips and fragment simulating projectiles. The ballistic limit can be predicted by equating the initial kinetic energy of the projectile to the penetration energy (Eq. (26)) 1 (36) MV2b D  R2 h¢ 2 On the right hand side of Eq. (36), ¢ depends on the initial velocity of the projectile which is Vb in this case. So, Eq. (36) is a quadratic equation to be solved for Vb . In the experiments conducted by Ulven et al. [38] on carbon/epoxy composite panels, the projectiles were cylindrical, they had the same radius R and the same mass M but, because they had a different tips, the ballistic limits were different. For conical tips with ™ D 151ı , “ D 0:968. For flat tips “ D 1 and “ D 1:5 for spherical tips. As “ increases so does the penetration resistance. Therefore, in that example [38], the

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conical projectile will have the highest ballistic velocity followed by the flat-ended projectile and the spherical tipped projectile. Reyes Villanueva and Cantwell [42] used this approach to predict the ballistic limit of FML and FML faced sandwich plates impacted by hemispherical projectiles.

3.5 Effect of Friction Wang and Chou [47] assumed that the surface of the projectile is subjected to a constant normal stress ¢1 and to a tangential frictional stress ¢1 (Coulomb friction). Ben-Dor et al. [48–51] assumed that the normal pressure on the projectile followed a Poncelet-type dependence and neglected frictional effects. Jones and coworkers studied the penetration of rigid projectiles in isotropic targets. Jones and Rule [52] assumed that the normal pressure followed a Poncelet type of dependence and Coulomb type friction. Jones et al. [53] discussed the use of more complex frictional laws. Ben-Dor et al. [54] determined a class of projectile-target interaction laws for which the Lambert–Jonas equation (Eq. (7)) is satisfied. The model developed by Zhu et al. [55, 56] for penetration of Kevlar laminates by conical-tipped indentors also belongs to this category. The penetration process is divided into three stages: indentation, perforation, and exit of the projectile. During indentation motion of the projectile is resisted by uniform pressure pm and the resistive force is obtained by multiplying that pressure pm by Ap , the projected area of the projectile. The means pressure pm is said to be proportional to the yield stress of the material and to be best determined through experiments. For the perforation phase, a damage factor defined as dm D Nb =Nt where Nb is the number of broken fibers and Nt is the total number of unbroken fibers in front of the contact region is introduced P D pm Ap .1  dm / . In the exit stage, a frictional force P determined through static tests provides the only resistance to the motion. The damage factor varies through the penetration process as successive layers fail. Fiber failure is predicted by a maximum strain criterion and the total strain in the fiber depends on the global deformation of the plate and on the local deformation near the indentor. Global deformation is predicted using the first order shear deformation theory and the local damage model includes bulging and delamination.

4 Factors Affecting the Ballistic Limit Extensive experimental studies have been conducted to determine the effect of various factors on the ballistic limit. In this section we will analyze the effects of five factors: (1) target thickness and projectile diameter; (2) laminate stacking sequence; (3) obliquity; (4) projectile density.

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4.1 Effect of Laminate Thickness and Projectile Diameter Assuming that the penetration resistance is constant, the penetration energy should then be proportional to the thickness of the laminate. Similarly, the models discussed in Section 3 predict that the penetration energy should be proportional to the thickness. Gupta and Davids [57] studied the penetration resistance of fiberglass-reinforced plastics against small-caliber projectiles. Their experimental results (Fig. 11) indicate that the energy absorbed increases linearly with the areal density or, equivalently, with the thickness of the laminate. Iremonger and Went [58] studied the penetration of nylon 6,6/ethylene vinyl acetate composite laminates by fragment simulating projectiles (FSPs). In their experiments, the ballistic velocity was proportional to the square root of the thickness which means that Ub is also proportional to h. Bartus and Vaidya [59] performed ballistic impact tests on of polypropylene (PP)/E-glass composites using flat ended and conical-tipped cylindrical projectiles. The penetration energy increased linearly with the areal density of the plate and for each thickness, a significant scatter of the results was shown probably for the first time. The penetration energy for conical tipped projectile was 27% lower than that for flat ended projectile due to different failure modes. Jacobs and Van Dingenen [60] studied the ballistic penetration of composites with Dyneema fiber reinforcement by FSPs. Results indicated that the energy absorbed during perforation is proportional to the area density of the laminate and also to the cross-sectional area of the projectile. In the present study, the values of Ub obtained from the experimental data in Ref. [60] were fitted by the expression Ub D c: .¡h/ :D1:9399 180

Energy absorbed (ft.lb)

160

y = 9494.4x R2 = 0.9893

140

0.22 caliber bullet

120 100 80 60 40 20 0 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Weight / unit area (lb/in2)

Fig. 11 Energy absorbed versus areal density (Data from Ref. [22])

(37)

484 Table 3 Comparison of experimental (from Ref. [60]) and calculated (from Eq. (39)) ballistic limits. Calculated ballistic limits are in parentheses and numbers are the percent errors are in brackets

Table 4 Glare laminates Lin and Hoo Fatt [62]

S. Abrate Diameter of FSPs (in.) 2

Areal density .lb=ft / 0.22 1.4 167.4 (171.9) [2.7] 1.85 217.3 (227.2) [4.6] 2.36 302.9 (289.8) [4.3]

Areal density .N=m2 / 36.65 43.41 61.15 91.92 124.46

Exp. ballistic limit (m/s) 136 150.5 155.5 182 209

0.3 307.9 (313.8) [1.9] 402.9 (414.7) [2.9] 534.3 (529.0) [0.99]

0.5 774.8 (845.4) [9.1] 1032.9 (1117.1) [8.2] 1540.6 (1425.1) [7.5]

Calc. ballistic limit (m/s) 137 144 160 182 207

Table 3 shows that, for the nine cases considered in Ref. [60], agreement between experimental results and values calculated using Eq. (18) are in excellent agreement. Lin and Bhatnagar [61] conducted a series of 76 tests in which Spectra fiber reinforced composites with three areal densities where impacted by three types of fragment simulating projectiles. Hoo Fatt and coworkers [62, 63] provided data on the ballistic limit of glare laminates as a function of thickness or equivalently as a function of the areal density w. A least square fit of the data (Table 3) gave V2b D 273:1 w C 8898. The ballistic limit calculated using this expression is in good agreement with the experimental data (Table 4). Lee et al. [13] showed that for composite materials with Spectra 900, Spectra 1000, Kevlar 29 and S2 glass fiber reinforcement, the ballistic limit increased with ¡h, the areal density of the target, according to Vb D a .¡h/b

(38)

For the examples in Ref. [13], b is a constant between 0.51 and 0.65. All these examples support the thesis that, at the ballistic limit, the penetration energy is proportional to the thickness of the target. However, several examples indicate that in certain experiments the ballistic limit is proportional to the thickness of the target. Bhatnagar et al. [64] and Lin et al. [65] presented results for spectra reinforced composites that showed a linear variation of the ballistic velocity Vb with the areal density of the laminate. The same trend was observed for Kevlar– Polyester [55, 56], for both Kevlar and graphite laminates [66] and by Kasano [19]. The experimental results presented by Kasano [20] for the perforation of CFRP

Ballistic Impacts on Composite and Sandwich Structures

485

laminates by steel balls show that the ballistic limit increases linearly with the thickness of the plate. The same trend was observed for plates made out of glass fiber reinforced plastics, polycarbonate and PMMA in the experiments conducted by Kasano et al. [26]. Vasudev and Meehlman [67] also showed that the ballistic limit for glass reinforced plastics increased linearly with the areal density. Abdullah and Cantwell [41] have performed ballistic impact tests on Fiber–Metal Laminates (FML) and presented results showing that the ballistic limit increases almost linearly with the thickness of the laminate. Justo and Marques [68] considered two types of composite materials: (1) a Kevlar 129 fiber reinforced phenolic matrix composite; (2) a Dyneema SK66 reinforced composite with a thermoplastic matrix. Under ballistic impacts with 1.1 g, 0.22 caliber fragment simulating projectiles, it was shown that for both material systems, the ballistic limit increased linearly with the areal density. In these experiments the ballistic limit varied between 350 and 750 m/s while the areal density was in the 2.5 to 11 kg=m3 . Caprino et al. [69,70] postulated that the penetration energy is related to the laminate thickness t, the fiber volume fraction vf , and the diameter of the projectile D by Up D K .t:vf :D/’

(39)

where K and ’ are constants to be determined from the experimental data. The exponent ’ was found to be 1.40 for CFRP and 1.30 for GFRP. Equation (39) is similar to the formulas proposed by deMarre in 1886 (see Ref. [4]) and the modified deMarre equation t1:4 MV2 MV2 D ’ D ’ .t=D/“ (40) D3 D1:5 D3 Reddy et al. [11] proposed a more complicated expression for the quasi-static penetration energy.

4.2 Effect of Stacking Sequence Bhatnagar et al. [64] studied the ballistic performance of vinyl ester composite with glass and Spectra fibres subjected to 22 caliber FSPs. In one series of experiment, the front layer had glass reinforcement while the back layer had spectra fiber reinforcement. In a second set of experiments, the position of the two layers was reversed. Figure 12 shows that the order in which those layers are placed has a significant effect on the ballistic limit. Similar results were found by Grujicic et al. [71] for alternating layers with Kevlar and carbon fiber reinforcement. Jenq et al. [72] studied the effect of stacking sequence and projectile shape on the ballistic limit. For the same GFRP material system and the same conical-tipped projectile, the ballistic limit was 67.06 m/s for a Œ05 =905 =05  laminate and 80.85 m/s for a Œ.0= ˙ 45=90/2 S laminate. For the same Œ05 =905 =05  layup, the ballistic limit is 67.06 m/s for projectile with a conical tip and 96.60 m/s for a projectile with a blunt end. Therefore, both the stacking sequence and the shape of the projectile have a significant effect on the ballistic limit.

486

S. Abrate 2500 y = – 0.0994x2 – 0.2251x + 2007.9 R2 = 0.9962

Ballistic limit (ft.s)

2000

1500

1000

500

G/S S/G

y = 0.0818x2 – 19.047x + 2045.7 R2 = 0.9942

0 0

20

40 60 80 Percentage of glass reinforcement

100

Fig. 12 The ballistic limit depends on which type of fibers is located on the front face of the laminate (Data from Bhatnagar et al. [29])

4.3 Effect of Obliquity Few investigators have considered the effect of obliquity on the ballistic limit. Fernandez et al. [73] determined the ballistic limit of carbon epoxy laminates using a finite element approach in which three separate criteria were used to predict the failure of the fibers, the matrix, and the delamination of ply interfaces. A simple criterion was used to remove failed elements: if the strain in the fiber direction exceeds the ultimate strain, the element is removed. A number of cases are considered to generate a data set that is used to train an artificial neural network that is then used to predict the occurrence of perforation and the residual velocity of the projectile without having to perform a full numerical analysis. This work is an extension of a previous study on the ballistic impact of steel armor [74] that did not consider the effect of obliquity. The experiments of Hazell et al. [75] showed that the kinetic energy absorbed by the target increased with obliquity. The study also focused on determining the type of failure mode (plugging or petalling) and the extent of damage. Lopez-Puente et al. [76] studied the effect of obliquity on impact damage of carbon–epoxy laminates as the impact velocity increases. It was found that below the ballistic limit the damage area increases rapidly with the velocity of the projectile. Above the ballistic limit the damage are decreases slowly as the velocity increases. The ballistic limit consistently increased with the obliquity angle. Chu et al. [77] have conducted a series of tests on laminates with thicknesses ranging from 5.6 to 16.9 mm at obliquity angles (Fig. 13) from 0ı to 70ı . The results (Table 5) show the strong influence of both variables. A plot of that data shows that, for each value of the obliquity angle ™, the square of the ballistic limit increases linearly with the thickness of the laminate. S, the slope of those lines, increases

Ballistic Impacts on Composite and Sandwich Structures

487

Fig. 13 Oblique impact θ V

h

h cosθ

Table 5 Ballistic limit as a function of thickness and obliquity angle (data from Chu et al. [77]; numbers in parentheses: predictions from Eq. (41)) Obliquity angle (ı ) Thickness (mm) 5.6 9.9 10.9 16.9

0 492.9 (464.1) 615.6 (617.1) 657.2 (647.6) 805.8 (806.3)

30 497.2 (479.4) 636.7 (637.4) 675.5 (668.9) 816.2 832.8)

45 535.3 (504.0) 651.1 (670.1) 708.4 (703.1)

60 567.7 (555.5) 751.2 (738.5) 757.7 (774.9)

70 605.5 (628.1) 812.4 (835.1) 904.5 (876.3)

with the angle ™. Since the distance traveled through the target varies with 1= cos ™ we plot S versus 1= cos ™ which results in another linear plot. As a result of this data reduction procedure, the experimental data is approximated by the following expression   16:623 2 Vb D C 21:849 h (41) cos ™ Table 5 shows that the values of the ballistic limit predicted by Eq. (41) are in very good agreement with the experimental results given in Ref. [77].

4.4 Effect of Projectile Density Czarnecki [78, 79] conducted a series of ballistic tests on graphite–epoxy (AS4/3501-6) laminates with either 32 or 128 plies. The stacking sequences were Œ.0=90= C 45=  45/4 S and Œ.0=90= C 45=  45/16 S respectively. Square 7  7 in. specimens were clamped around the edges and support conditions were assumed to be between ideally simply supported and clamped. These specimens were impacted by 0.5 in. spheres made out of aluminum, steel, or tungsten. The tests on 32-ply laminates showed that the ballistic limits for 1=2 in. diameter spherical projectiles made out of aluminum, steel, and tungsten were 625, 371.3 and 281.9 ft/s

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respectively. If the penetration energy Up is constant, as assumed in Eq. (1), the ballistic limit should be inversely proportional to the square root of the density of the projectile. A least square fit through the experimental data shows that this is p true and that V50 D 1550:4= ¡. Experimental results show that the slope of the line in the UR versus Ui graph is 0.82 which leads to an m/M ratio of 0.2195. With aluminum projectiles, the slope is 0.63 which indicates that there might be another phenomenon involved: the deformation of the projectile.

5 Models Based on Static Test Results One approach to realistically account for interactions between the projectile and the target including the effect of failures modes such as delaminations, fiber tensile failure and shearing is to conduct a quasi-static test of the same laminate with the same boundary conditions. The force–displacement curve recorded during that test is then used to estimate the ballistic limit. Sun and coworkers [9, 10, 80–82] conducted static tests and impact tests on graphite–epoxy laminates with blunt-ended projectiles at velocities up to 100 m/s with 24 mm long, 14.5 mm diameter, 30 g projectiles. The variation of the penetration force with displacement can be idealized as in Fig. 14. The behavior is linear until delaminations are induced (point A). The stiffness drops suddenly from A to B and the load increases again up to point C where shear stresses cause perforation and the formation of a plug. Similar results were described by Ursenbach et al. [83]. The area under the curve in Fig. 14 represents the energy stored into the target during the deformation up to point C. The quasi-static perforation tests conducted by Found et al. [84] on graphite–epoxy panels with blade stiffeners show the same behavior at that shown in Fig. 14 until point C. After that point, the load dropped progressively as the indentor displacements progressed significantly instead of the sudden drop shown in Fig. 14. The results in Ref. [84] confirm those obtained earlier for graphite–epoxy laminates by Sykes and Stoakley [85] who showed that cure temperature has a significant effect on the behavior during static penetration. Higher cure temperatures were shown to result in penetration soon after point B in Fig. 14. This resulted in lower maximum

C

Force

A

B

Fig. 14 Force–displacement behavior in experiments [9, 10]

D Displacement

Ballistic Impacts on Composite and Sandwich Structures Fig. 15 Force–displacement behavior in experiments [8]

489 A

Force

B

C

D

Displacement

force before penetration but also in a much lower penetration energy since the area under the curve is much smaller. Quasi-static tests on glass–epoxy composites with a spherical indentor by Jenq et al. [8] showed a different load displacement behavior (Fig. 15) where the behavior is linear up to point A. At point A, delamination occurs and results in a sudden drop in load from A to B. As the punch moves further, fibers in the contact area start to break and are pushed into the direction of penetrator motion. At C the projectiles exits the back side of the laminate and from C to D the resistance to the motion of the projectile is due to friction. Zee et al. [86] designed a sensor to measure the velocity of the projectile during ballistic impacts. The specimens tested had an epoxy matrix and woven fabric reinforcement with either polyethylene (PE), polyester (PET) or graphite fibers. With graphite fibers, the results are similar to those in Ref. [85]. For PE and PET composites the penetration force increased almost linearly with the displacement of the projectile, reached a maximum and decreased linearly with a similar slope instead of dropping down quickly as in Fig. 15. Several references noted that a significant amount of energy was dissipated by friction between the projectile and the laminate after penetration was complete [6, 8–10, 86]. Experiments conducted by Lee et al. [14] on vinylester or polyurethane resins reinforced by Spectra fabric the static load–deflection curves showed a nonlinear behavior until a maximum load is reached and sudden failure occurs. The nonlinear effect is due to large deflections. Similar behavior was observed by Lee and Sun [9, 10] in some of their tests. The previous examples illustrate some possible failure modes but there are many others. Several authors [e.g. [67,87]] reported that for ballistic impacts the thickness of the laminate can be divided into three regions: I – shear failure, II – tensile failure, III – tensile failure and delamination. The models currently available consider the projectile to be non-deformable even though in some cases projectiles experience significant deformations [88]. In addition, Langlie and Cheng [89] showed that during high velocity impacts the deformation of the target remains localized in a small region surrounding the impacted area and consists mainly of transverse shear deformation. As the deformed zone expands, the effective mass of the target increases and the effective stiffness decreases with time.

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Models developed based on quasi-static perforation results rely on the assumption the deformation of the target during high velocity impact is similar to that produced during quasi-static tests. This is not the case during high velocity impacts when overall bending and shear deformations are limited to a small region near the impact point. The second assumption is that the same failure mechanisms are involved and that the material properties are strain rate independent. Clearly, those assumptions have limited applicability.

6 Energy – Balance Models Another approach consists of accounting for all energy-dissipating mechanisms: bending, shear, local indentation, and delamination. For example, Cantwell and Morton [90] considered that, to perforate a composite beam, energy is dissipated through four mechanisms: bending deformation of the target, contact deformation, delamination, and shear out. The total energy required for perforation is written as U D Uf C Uc C Ud C Us

(42)

where the subscripts f, c, d, and s stand for flexural, contact, delamination and shear respectively. Each term on the right hand side of Eq. (21) is estimated using insights from experiments. With this approach, perforation energies were predicted with good accuracy for a limited number of cases for which experimental results were available. Similar approaches were used in Refs. [91–93]. The energy dissipated by delamination Ud was determined by multiplying the area of delamination by the fracture energy per unit area: 700 J=m2 for the material used. The energy required for shearing out a plug Us as shown in Fig. 16 is obtained by multiplying the shear strength by the area of the conical failure surface. With this approach, perforation energies were predicted with good accuracy for a limited number of cases for which experimental results were available. In Ref. [91], since the affected zone is small, Uf is neglected and the Uc is not considered. The shear

D Delamination

Fig. 16 Failure mechanisms assumed by Caprino et al. [95]

Shear surface

α

Ballistic Impacts on Composite and Sandwich Structures

491

strength for carbon–epoxy materials tested was determined experimentally to be 37:25 kJ=m2 . Cantwell and Morton [92] assumed that for high velocity impacts the delamination zone can be idealized as shown in Fig. 16. The fracture energy for delamination was taken to be 500 J=m2 . Goldsmith et al. [66] used a similar approach to study the perforation of carbon fiber laminates. The penetration energy was estimated by estimating the contributions of several dissipation mechanisms: global plate deflection, fiber breakage, delamination, formation and bending of petals, hole enlargement and friction between the projectile and the target. Each of these mechanisms is shown to make a significant contribution and cannot be neglected. Harel et al. [93] estimated the penetration energy for polyethylene/polyethylene composites perforated by bullets. Expressions were obtained for estimating the energy absorbed by two mechanisms: fiber failure and delamination. Delamination was the dominant energy dissipation mechanism and the predicted penetration energy agreed well with the experimental value. A number of articles propose models that account for the actual failure modes encountered during ballistic impact perforation. Zee and Hsieh [94] identified four mechanisms for energy loss during perforation: matrix cracking, fiber failure, delamination and friction between the projectile and the target. Experiments were conducted to determine the contribution of each mode to the total energy loss. With composites with graphite fiber reinforcement, delamination is the dominant energy absorption process due to the large energy release rate. With Kevlar or-49 or Spectra900 PE fiber reinforcement, the contribution of delaminations plays a lesser role. In all cases, friction accounted for 10–20% of the total energy loss. Caprino et al. [95] estimated the penetration energy using an approach first proposed by Cantwell and Morton [91] which assumes that penetration results in delaminations at each interface and shearing in a conical shape as shown in Fig. 16. The energy required to shear the target at a 45ı angle is estimated using p Ups D ks As D ks : 2 h .h C D/

(43)

where ks is a coefficient and As is the area of the conical surface. The energy absorbed by the creation of delaminations is assumed to be proportional to the delaminated area Ad which in turn is assumed to be proportional to As Upd D kd Ad D kd kAs

(44)

The total penetration energy can be written as Up D .ks C kd k/ As D

p

2  .ks C kd k/ :h .h C D/

(45)

p where the material properties are lumped into the single term 2  .ks C kd k/ that is evaluated from experiments. In other word, in Ref. [91], the penetration energy Up is expected to be proportional to h.h C D/. For a thin laminate .h D/, the penetration energy is proportional to hD.

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Lopez-Puente et al. [96] proposed a simple model for analyzing the perforation of CFRP by spherical projectiles accounting for three energy dissipating mechanisms: laminate crushing under the projectile, momentum transfer to the target, tensile fiber failure. They defined three non-dimensional parameters to estimate the relative importance of the three mechanisms Rc D

2h¢c A ; MV2i

Rm D

hA¡t ; M

Rf D

4h©f Xt r2 MV2i

(46)

With the typical values h D 103 m, ¢c D 108 Pa, A D 105 m2 , M D 103 kg, ¡ D 103 kg=m3 , R D 103 m, Xt D 109 N=m2 , ©f D 102 , these three parameters have the following orders of magnitude Rc D

103 10 ; Rm D 102 ; Rf D 2 2 Vi Vi

(47)

In what is called the zero-order approximation, the ballistic limit is given as s vbl D

2¢c .exp .Rm /  1/ ¡t

(48)

Since Rm is small, using the Taylor series expansion exp.Rm / D 1 C Rm , after some algebraic manipulations we obtain 1 MV2b D ¢c hA 2

(49)

which simply expresses the principle of conservation of energy (Eq. (4)) with a constant penetration resistance due to crushing under the projectile. For penetrating impacts,  2

 VR 2¢c D exp .Rm / 1  .exp .Rm /  1/ (50) Vi ¡t V2i Using the same Taylor series expansion for the exponential functions, we obtain 1 1 MV2R D M .1  Rm / V2i  ¢c hA 2 2

(51)

which in the same form as the Lambert–Jonas Equation (Eq. (11)). These energy approaches are often successful for predicting trends as various parameters are varied but they require prior knowledge of the failure mechanisms involved which are usually determined from experiments. Therefore, those models usually have limited applicability.

Ballistic Impacts on Composite and Sandwich Structures

493

7 Numerical Models Many articles describe numerical models for the analysis of ballistic impacts on composite structures. This section presents an overview of the approaches followed by most investigators without attempting to present an exhaustive account of the literature. To analyze the perforation of composite structures by a projectile a threedimensional finite element model of both the projectile and the target is usually used. Well known finite element codes such as DYNA-3D [97, 98], PAM-CRASH, ABAQUS [e.g. [73, 76, 99], ANSYS [100] and others are used. Some investigators have developed their own codes. The analysis should consider both the projectile and the target, it should include algorithms to account for the dynamic contact interaction between the two bodies, and it should give a detailed and accurate description of the state of stress in the impact region. Adequate failure criteria should be used to determine the onset and the eventual propagation of damage. Then, procedures are to account for the degradation of material properties. These elements of the analysis procedure are discussed in the following. For low velocity impacts, the projectile is considered to be a rigid body and its interaction with the target is accounted for using a statically determined contact law [1]. This approach cannot be used in analyzing penetration problems. The contact between the two bodies can be handled by several approaches. Chan et al. [97] discuss three approaches available with LS-DYNA: the Kinematic Constraint method, the Penalty method, and the Distributed Parameter method. Similar approaches are used by other investigators. One issue is to determine the onset of damage inside each element and at the interface between adjacent plies. Generally, individual failure criteria are used to predict intraply failure rather than interactive failure criteria such as the Tsai-Wu failure criterion. For example, in Refs. [73, 76], fiber failure in plies with woven fabric reinforcement are determined using the following criteria 

¢11 XT



2 C

2 2 ¢12 C ¢13 S2f



 1

¢22 YT



2 C

2 2 ¢12 C ¢23 S2f

 1

(52)

where directions 1 and 2 are the two fiber directions with a 90ı angle between them. XT and YT are the tensile strength in the 1 and 2 directions and Sf is the throughthe-thickness shear strength. Once intralaminar damage has been predicted, some procedure is adopted to account for the degradation in the ability of the element to carry loads. For example, in Ref. [76], when one of the two criteria (Eq. (52)) is satisfied all the stress components that appear in the equation are set to zero. However the element can still carry some stresses. For example if criterion 52.a is satisfied, the element can still carry stresses in the other direction .¢22 / and in the transverse normal direction .¢33 /.

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To predict the onset of delamination, Refs. [73, 76] used the criterion presented by Hou et al. [101, 102] 

¢33 ZT



2 C

¢13 S13



2 C

¢23 S23

2  1 when ¢33 > 0

(53)

that involves all three stress components at the interface and the corresponding strengths. This criterion assumes that no delamination occurs under a compressive normal strength. Other criteria are available and have been used in modeling impacts. In some investigations, interface delamination is predicted using such a criterion throughout the event following an element-by-element, step-by-step approach. This approach has been criticized by several authors (e.g. Li et al. [103]) as being unrealistic. To predict the propagation of an existing interfacial crack, the model can follow two approaches. With the first one, the strain energy release rates can be calculated and a criterion can be used to predict whether or not the crack will extend. If only the total strain energy release rate can be determined, a Griffith type failure criterion is used but, if the individual strain energy release rates are available, one of many mixed-mode fracture criteria can be used. One way to determine the strain energy release rate is to use the Virtual Crack Closure Technique (VCCT). The VCCT is described in details in a recent review article by Krueger [104]. The second approach for modeling delamination growth is to use a cohesive model as a number of investigators have done for low velocity impacts on composite structures [105–110]. A review of methods to predict impact induced delaminations is given by Elder et al. [111]. Chan et al. [97] estimated that strain rate effects on the strength of the composite have negligible effects. Other failure criteria are used in the literature [e.g. xx4] and a complete discussion of this topic will be presented elsewhere. Similarly, many strategies have been employed to account for material degradation once failure onset has been predicted and for element removal once it is unable to carry anymore stresses. Property degradation models are discussed by Iannucci et al. [98] for example.

8 Impact on Sandwich Structures In this section we consider the perforation of sandwich structures by rigid projectiles focusing on sandwich structures with composite facesheets and cores made out of cellular materials. A comprehensive review of the literature on impacts on sandwich structures with laminated facings was presented by Abrate [112] in 1997. Most of the literature on this topic deals with low velocity impacts. In the following, we examine impacts resulting in either partial or complete penetration of sandwich structures. The sequence of events leading to complete perforation of such sandwich structures has been described by several authors. Roach et al. [113] conducted tests on

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laminated plates with a polyester resin and glass fiber reinforcement backed by a closed cell PVC foam layer using a flat-ended solid cylindrical indenter at velocities of 30, 60 and 120 m/s. Under static loading, the penetration load increased linearly with the thickness of the laminate when the facesheets where tested alone and also when tested with the foam backing. Under static loading, the penetration energy was also found to increase linearly with the thickness of the laminate. The penetration energy was found to be significantly higher during impact tests. For plates with foam backing, ratios between the dynamic and static penetration energies were found to vary between 1.72 and 3.02 for sandwich plates impacted at 30 m/s, 2.22 to 3.62 at 60 m/s, and 2.70 to 5.27 at 120 m/s. In the impact tests, the penetration energy was found to increase linearly with the thickness of the laminate. During those tests, laminates absorb a significant amount of energy by deflection. Laminated skins with core did not undergo any significant deflections and required lest energy to penetrate. In the second part of that study, Roach et al. [114] showed that the failure of laminates with a core resulted in a region of localized delaminations. The low velocity experiments of Mines et al. [115] also show that the energy required for penetration increases with the impact velocity. Raju et al. [116] conducted experiments on sandwich specimens with Nomex honeycomb cores with either plain weave carbon fabric/epoxy or fiberglass/epoxy laminated facings. Under dropweight impacts, it was shown that, as suggested by Roach et al. [113, 114], that the core limits the deformation of the top facing and the amount of energy that can be absorbed by such deformation. The thickness of the facings is also important as a stiffer facing spreads the local deformations over a larger area. The initiation of damage in sandwich beams composed of E-glass/Epoxy and PVC foam under three-point bending was brought out by several examples in experiments conducted by Lim et al. [117]. Three failure modes were shown as the thickness of the facings increased: (a) facesheet failure for a thickness t D 0:45 mm; (b) core compressive yield when t D 1:5 mm; and (c) core shear yield when t D 1:95 mm. An analysis produced a failure map that is divided into three regions corresponding to facesheet fracture, core shear, and core compression, when the ratio of facesheet to core thickness is plotted versus the ratio of span length to core thickness. It was concluded that, to enhance impact energy absorption, sandwich beams should be designed to fail in the face failure mode. Reyes Villanueva and Cantwell [118] used the approach described in Section 3 to predict the ballistic limit for sandwich plates with aluminum foam cores with either glass/polypropylene or fiber metal laminate facings. The normal pressure on the impactor is given by Eq. (23) and “ D 1:5 for a hemispherical tip. Predicted and experimental values of the ballistic limit are in excellent agreement. The same approach was used by Abdullah and Cantwell [119] to predict the ballistic limit of sandwich structures with aluminum alloy skins and a polypropylene fibre reinforced polypropylene (PP/PP) composite core. Hoo Fatt and coworkers [120–122] developed an analytical model for the perforation of a sandwich structures.

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9 Conclusions This chapter presents a critical review of the literature dealing with ballistic impacts on composite and sandwich structures. Simple assumptions on the penetration resistance showed that the residual velocity of the projectile increases linearly with the initial kinetic energy. It is important to note that energy is not conserved and that the experimental data can be fitted very well by the well-known Lambert–Jonas equation to obtain a good estimate of the ballistic limit from the results from a set of penetrating impacts. Other engineering models are reviewed and the effects of different parameters are ascertained from experimental results presented by several authors. An overview of numerical models for the analysis of ballistic impacts is given and the literature concerning penetrating impacts on sandwich structures is reviewed.

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Performance of Novel Composites and Sandwich Structures Under Blast Loading Arun Shukla, Srinivasan Arjun Tekalur, Nate Gardner, Matt Jackson, and Erheng Wang

Abstract The current chapter focuses on the experimental observations of the resistance of different composite material systems to air blast loadings. These material systems include traditional two dimensional (2D) woven laminated composites, layered composites and sandwich composite materials. A controlled blast loading of pre-defined pressure magnitude and rise time were obtained using a shock tube apparatus. Rectangular plate elements of the desired material system were subjected to such a controlled blast loading and the effect of the blast loading on these elements were studied using optical and residual strength measurements. A high speed imaging technique was utilized to study the damage modes and mechanisms in real time. It was observed that layering of a conventional composite material with a soft visco-elastic polymer provided better blast resistance and sandwiching the polymer greatly enhanced its survivability under extreme air blast conditions. Aside from layering the conventional composite material with a soft visco-elastic polymer, it was observed that layering or grading the core can successfully mitigate the impact damage and thus improve the overall blast resistance as well. In addition to these, three dimensional (3D) woven skin and core reinforcements were introduced in the conventional sandwich composites and their effects on the blast resistance were studied experimentally. It was observed that these reinforcements also enhance the blast resistance of conventional sandwich composites by changing the mechanism of failure initiation and propagation in these sandwich structures. The energies during the blast loading process were estimated to illustrate the energy absorption and energy redistribution properties of the composite panels. The effect of pre-existing impact damage on the failure mechanisms in sandwich structures was also studied.

A. Shukla (), N. Gardner, M. Jackson, and E. Wang Department of Mechanical Engineering and Applied Mechanics, Dynamic Photo Mechanics Laboratory, University of Rhode Island, Kingston, RI 02881 e-mail: [email protected] S.A. Tekalur Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48864 e-mail: [email protected]

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1 Introduction Composite materials have replaced metals in various engineering applications due to their numerous advantages, such as high strength/weight ratio, low cost, better stealth properties, etc. They are widely used in the defense industry (applications like naval ships, warplanes, armored vehicles), aerospace industry (applications like wing construction, tail construction, fiber composite bodies) and transportation industry (lightweight bodies for buses and trains, composite truck bodies). In recent times, threats to such defense and civilian structures in the form of terrorist attacks have added to the significance of comprehensive understanding of material and structural damage behavior under blast loading, in order to design and fabricate these structures to withstand such loadings. Experimental and instrumental difficulties associated with blast loadings have limited the extent of applied research in this area. But with growing concerns on safety and human lives involved, the significance of such a research cannot be understated. Blast loading is a complex and interesting problem that has been studied from the early twentieth century. When a structure is loaded with a blast wave, there are two main events occurring: 1. Transfer of energy from the medium that the blast wave is traveling to the structure. 2. Internal response of the structural material to the shock wave that is propagating in it. The internal response might lead to blast mitigation and hence good resistance to the blast wave or initiate several different failure mechanisms in the material which might lead to ultimate failure of the structure itself. G I Taylor’s work on the pressure and impulse of underwater explosion waves on freestanding plates is considered as one of the pioneering works on studying impulse transfer to plates subjected to blast loading conditions. His work revealed that the density of the material and medium in which the blast wave propagates has the maximum effect on the impulse transfer. Given this, there are many research efforts that try to use the same for mitigation of blast wave by modifying the material into several different forms like inhomogeneous, layered, anisotropic, etc. Composites are considered to be an effective material for blast mitigation due to their lower bulk densities when compared to metals and many E-Glass and Carbon fiber based composites have found a widespread use in naval structures. Extensive research on the behavior of materials like metals and concrete under extreme loading conditions have been performed and reported in the literature. There are various studies in literature that characterize these materials under different quasi-static and ballistic loadings. The current chapter focuses mainly on composite materials under blast loading conditions. One of the earliest works to study the response of composite materials to blast loading was done by Rajamani and Prabhakaran [1]. The authors studied the strain response of clamped rectangular plates of an isotropic material (aluminum) and a uni-directional composite (E-glass/epoxy) subjected to blast loadings from a rectangular shock tube. The effect of boundary conditions

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and rigid body motion of the panels are two key factors that play a significant role in designing and conducting successful laboratory experiments. The research by Hall [2], addresses some of these issues from an experimental approach. Librescu and Nosier [3] presented a theoretical analysis of the dynamic response of a simply supported, laminated rectangular composite panel exposed to sonic boom and blast loads. Mouritz et al. [4, 6, 8, 14] experimentally studied the damage and failure of composites when subjected to explosive blast and submerged under water. Post impact testing of coupons of the blast loaded composite panels was performed to provide quantitative idea of residual strength in them. Theoretical and computational analysis of blast loaded composite panels are as demanding as experimental analysis. Different approaches have been adopted to model the failure in composite materials due to blast loads. These can be found in literature [5, 7, 9–13]. Modifications to the composite architecture (braided composites and fiber metal laminates) to sustain blast loadings [15, 16] have also been undertaken. In addition to laminated composites, sandwich composites are highly desired for energy absorbing structural elements. The mechanical behavior and structural response of sandwich materials under quasi-static loadings are studied in the literature [17, 18]. While in service, structures constructed of sandwich composites may experience impact events that compromise the integrity of the structure. Such events can include high velocity/low mass impacts from gun fire, fragments from artillery, fragments from IED’s, etc. An understanding of how this type of damage affects the response and failure mechanisms of the sandwich structure during a blast event is vital in order to design robust structures. In recent times, the response of sandwich structures to such dynamic loadings has been an area of interest [19]. The overall dynamic response of the sandwich is dependent, among several other factors, on the construction of the skin, compressive modulus and failure strength of the core, and the interface strength between the skin and core. Due to the inherent nature of its construction, the strength of the sandwich construction is often limited by the strength of the core material. The thickness of the skin increased beyond certain limits, also negates the weight beneficiary advantage of sandwich construction. In lieu of these considerations and to facilitate better performance under transient conditions, several design modifications have been sought after. These include better choice of core material, introduction of soft inter layers (e.g. polyurea layer) between the core and the skin, and discretely layering or grading the core. In recent years, stepwise graded materials, where the material properties vary gradually or layer by layer within the material itself, were utilized as a core material in sandwich composites. Since the properties of graded/layered core structures can be designed and controlled, they show great potential to be an effective core material for absorbing the blast energy and improving the overall blast resistance of sandwich structures. The numerical investigation by Apetre et al. [20] has shown that a reasonable core design can effectively reduce the shear forces and strains within the structures. Consequently, they can mitigate or completely prevent impact damage on sandwich composites. Li et al. [21] examined the impact response of layered and graded metal-ceramic structures numerically. He found that the choice

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of gradation has a great significance on the impact applications and the particular design can exhibit better energy dissipation properties. Also, in recent times, Zdirectional pins have been utilized [22] to modify the core in sandwich material and its high strain rate impact response has been studied. Failure modes of carbon fiber based sandwich beams reinforced with similar Z-directional pins under three-point bending [23] were studied under quasi-static loading conditions.

2 Material Systems The experimental study involved the evaluation of three different major material systems, namely: 1. Laminated composites (a) E-glass vinyl ester (b) Carbon fiber vinyl ester composites 2. Layered composites (a) Polyurea layered materials 3. Sandwich composites (a) (b) (c) (d) (e)

Polyurea sandwich composites Sandwich composites with 3D woven skins Core reinforced sandwich composites Sandwich composites with stepwise graded core Pre-damaged sandwich composites

A brief description on the construction and characteristic properties are given below:

2.1 Laminated Composites 2.1.1

E-Glass Vinyl Ester Composite (EVE)

The composite used in this study was manufactured using glass fibers and vinyl ester matrix. The glass fabric used in this study was woven roving E-glass supplied by Fiber Glass Industries. The areal weight was 610 g=m2 (18 oz=yd2 ) with an unbalanced construction having 59% and 41% of fibers in warp and fill directions respectively. The matrix used was Dow Chemical’s Derakane 510A-40, a brominated vinyl ester, formulated for the VARTM (vacuum assisted resin transfer molding) process. Complete details on the material composition, fabrication, mechanical characterization and quasi-static properties are given in Ref. [24].

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Carbon Fiber Vinyl Ester Composite (CVE)

The fabric used in the carbon fiber vinyl ester composite (CVE), was an equi-biaxial fabric produced using Toray’s Torayca T700 12k carbon fiber tow with a vinyl ester compatible sizing. The areal weight of the fabric was 634 g=m2 with 315 g=m2 of fiber in the 0ı direction and 305 g=m2 in the 90ı direction. Both the directional fibers were stitched with a 14 g=m2 polyester knitting thread. The static mechanical properties of these composite materials are given in Ref. [24].

2.2 Layered Composites 2.2.1

Polyurea Layered Materials

Polyurea is a cross-linked amorphous isocyanate monomer or prepolymer and polyamine curative. Layered materials were fabricated by casting polyurea over E-glass vinyl ester composite (EVE) plates. The thickness of each component of the layered materials was 6.35 mm (0.25 in.). Further details on the properties and fabrication of these layered plates can be found in Ref. [25].

2.3 Sandwich Composites 2.3.1

Polyurea Sandwich Composites

Similar to the fabrication of layered materials, sandwich composites were fabricated out of the E-glass vinyl ester composite and polyurea material combinations. Two different sandwich configurations were manufactured. Whereas one had a soft core (PU) sandwiched between two hard skins (EVE), the other had a hard core (EVE) sandwiched between two soft skins (PU). The sandwich composites along with their dimensions are given below, 1. 3.18 mm (0:12500 ) EVE C 6.35 mm (0:2500 ) PU C 3.18 mm (0:12500 ) EVE 2. 3.18 mm (0:12500 ) PU C 6.35 mm (0:2500 ) EVE C 3.18 mm (0:12500 ) PU Based on the experimental observations, the beneficial effects of the sandwich construction was predominant only in the sandwich with soft core configuration. Hence the results and discussions focus on that particular class of the sandwich alone.

2.3.2

Sandwich Composites with 3D Woven Skin

Three-dimensional orthogonal woven composites, hereafter referred to as 3D composites are different from traditional 2D laminates in that they are able to sustain

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loads in three linearly independent directions. Fiber preforms constructed in a three dimensional fashion not only contain warp and weft fibers in plane of the lamina, but also a certain fraction in the through thickness, or z-direction. The complete details of construction and advantages of utilizing this 3 D woven skin in composite construction can be obtained in Refs. [26–28]. The composite skin materials utilized in this study were made by VARTM method from 3:25 kg=m2 .96 oz=yd2 ) areal density 3WEAVEr E-glass glass roving preforms using Dow Derakane 8084 epoxy–vinyl ester resin. This fabric construction results in 49% warp-, 49% filland 2% Z-fiber content per preform volume. The total fiber volume fraction in a single-layer composite made with such preform was estimated as 49%. The core material used in the sandwich construction was made of Dow Chemical TRYMERTM 200L Polyisocyanurate foam, which is a cellular polymer. From the manufacturer’s data, the density of the foam is 32 kg=m3 . This foam is not an isotropic material; besides, its tensile and compressive properties are markedly different. Specifically, from the foam manufacturer data the following tensile, compressive and shear moduli are given in the direction parallel to rise: 8.27, 5.17 and 1.79 MPa, respectively. For constructing the sandwich composite panels, the first two layers of the front skin (to be exposed to the shock wave) were made of the standard 0:81 kg=m2 .24 oz=yd2 / areal density 2-D E-glass plain weave. Those were followed by three layers of 3WEAVEr fabric and subsequently by a layer of above mentioned foam core. The back skin preform contained (counting from the back surface of the core) two layers of the same 3WEAVEr fabric followed by two layers of the same E-glass plain weave. The front skin was intentionally made one fabric layer thicker than the back one, to enhance its strength. The average front skin thickness was approximately 9.1 mm (0.36 in.) and the average back skin thickness was approximately 6.91 mm (0.27 in.) respectively.

2.3.3

Core Reinforced Sandwich Composites

The work on fabricating the core reinforced sandwich composites, designated as TRANSONITEr , was performed by Martin Marietta Composites.1 This sandwich type is produced in accordance with patented technology and provides an alternative for structural and non-structural flat panel applications. In addition to all the characteristics explained in the previous section on sandwich composites, the TRANSONITE panels have core reinforcement in the form of stitches introduced in their through thickness direction, as shown in Fig. 1 (without the core). To begin the pultrusion process, varying layers of fabrics are pulled together to produce the desired thicknesses of top and bottom skin preforms. Then a foam layer 1

Further information about TRANSONITE can be obtained on company website www. martinmarietta.com.

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Fig. 1 TRANSONITE composite sandwich with removed foam; through thickness “stitches”, impregnated with resin, can be seen [Ref: Martin Marietta Composites website]

is inserted between the top and bottom layers of the fabric forming a sandwich and subsequently the stitching is performed. Once the resin is infused, the sandwich is drawn through a heated die forming a cured composite sandwich. Finally, the panel is cut to the appropriate length.

2.3.4

Sandwich Composite with a Stepwise Graded Core

The sandwich composites were manufactured using E-Glass vinyl ester (EVE) composites and CorecellTM A-series Styrene acrylonitrile (SAN) foams. The woven roving E-glass fibers of the skin material were placed in a quasi-isotropic layout [0=45=90=  45s . The fibers were made of the 610 g=m2 (18 oz=yd2 ) areal density plain weave. The resin system used was Ashland Derakane Momentum 8084 and the front skin and the back skin consisted of identical layup and materials. The core materials were manufactured by Gurit SP Technologies and are specifically designed for marine sandwich composite applications. Three types of CorecellTM A-series foam were used and they were A300, A500, and A800. Table 1 lists important material properties of the three foams from the manufacturer’s data [29]. Based on the density and compression modulus, the wave impedances (WI) of the three types of foams show the following relationship, W IA800 > W IA500 > W IA300 The VARTM-fabricated panels were produced from a plain weave E-glass fabric type. Fabric lay-up was an 8-ply balanced/symmetric quasi-isotropic layout [0=45=90=-45s . For the core, each layer of foam was 12.7 mm (0.5 in.) respectively. The overall dimensions for the samples were 102 mm (4 in.) wide, 254 mm (10 in.) long, and 48 mm (1.9 in.) thick. The foam core itself was 38 mm (1.5 in.) thick, while the skin thickness was 5 mm (0.2 in.). The average areal density of the samples was 19:02 kg=m2 .

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A. Shukla et al. Table 1 Material properties for foam core [29] Nominal density Compressive modulus [lb=f t 3 =kg=m3 ] [psi/MPa] A300 3.6/58.5 4,640/32 A500 5.7/92 9,290/64 A800 9.3/150 16,980/117

2.3.5

Shear elongation [%] 62 69 50

Pre-damaged Sandwich Composite

Sandwich composite specimens that were to be damaged by impact and then subsequently subjected to blast loading were constructed of E-glass, vinyl ester face sheets and a styrene acrylonitrile (SAN) foam core. The face sheets were constructed of 810 g=m2 (24 oz=yd2 ) woven roving in a layup of [0=45=90=  45s . The core of the pre-damaged sandwich specimens was CorcellTM A800. Table 1 lists the manufactures material properties for the core material. Specimen dimensions were 102 mm (4 in.) wide, 254 mm (10 in.) long and had a thickness of 60 mm (2.3 in.). The face sheets, core and complete sandwich had average areal densities of, 9:11 kg=m2 (0:21 oz=in:2 ), 7:75 kg=m2 (0:18 oz=in:2 ) and 27:9 kg=m2 (0:63 oz=in:2 / respectively. Pre-damage was created by impacting the specimens with high velocity projectiles.

3 Experimental Setup An explosion is usually accompanied by rapid expansion of air, in which are entrained constitutive chemicals and adjacent debris. In the present work, attention was focused solely on the damage imparted by gaseous expansion and the associated rapid release of energy on composite targets. Several sources can produce such pressure waves, the first being explosives. Capable of producing pressure waves identical to those of explosives, both from the behavioural standpoint of the gases and from their impact on a structure, are compressed gas sources such as highly pressurized vessels (shock tube). A shock tube can produce the air blast alone while an actual explosive could induce burning in the material in addition to the air blast.

3.1 Shock Tube The current chapter focuses on utilizing a shock tube, shown in Fig. 2, for providing blast loading in a laboratory environment. In its simplest form a shock tube consists of a long rigid cylinder, divided into a high-pressure driver section and a low pressure driven section, which are separated by a diaphragm. The tube is operated by pressurizing the high-pressure section with helium gas until the pressure difference

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Fig. 2 The shock tube facility to produce controlled blast loading

across the diaphragm reaches a critical value and it ruptures. This rapid release of gas creates a shock wave that travels down the tube to impart air blast loading on a specimen. The intensity of the blast loading can be controlled to a repeatable extent by varying the thickness and number of diaphragms used. Initial calibrations for quantification of these input parameters were performed. Shock pressure and velocity measurements were done in real time to provide accurate input parameters for the study. A dynamic pressure sensor is mounted at the muzzle section of the shock tube and graphite rods are used as break circuit initiators to measure the shock velocity. Thin mylar sheets (10 mil) are used as diaphragm material. The driver pressure and hence the shock pressure obtained from the burst of these diaphragms are controlled by the number of plies of sheets used. Complete description of the shock tube and its calibration can be found in Ref. [27]. The shock tube utilized in the present study has an overall length of 8 m and is divided into a 1.82 m driver section, 3.65 m driven section and a final 2.53 m muzzle section. The diameter of the driver and driven section are 0.15 m. The final diameter of the muzzle section that is in contact with the specimen is 0.07 m. Mylar diaphragms are ruptured due to a pressure differential created between the driver and driven section, which develops and drives a shock wave down the length of the shock tube. A typical pressure profile obtained at the sensor location by using 3 plies of 10 mil mylar is shown in Fig. 3.

3.2 Loading and Boundary Conditions The shock loading obtained from the end of the muzzle covered a circular region of 76 mm .300 / diameter. Several different boundary conditions like fixed-free, clamped and simply supported can be configured for the experiments. But each of these conditions imposes a unique failure mode in the specimen (e.g. tearing along the clamped edges, rigid body motion along the simply supported edges) which may not be desirable. The use of large structures compared to the dimensions of the loading

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Reflected Pressure

1.8

Pressure (MPa)

1.6 1.4 Incident / Shock Pressure

1.2 1 0.8 0.6 0.4

Dependent on distance of the sensor from the end of tube

0.2 0 0

2000

4000

6000

8000

Time (Microsec)

Fig. 3 Typical shape of the pressure – time profiles obtained in the shock tube

region is a preferred way of circumventing these failures that are created by boundary conditions. But in practical aspects, such large structures are extremely costly to produce and the facilities for such experiments need to be correspondingly scaled. In small structural elements, maintaining the boundary conditions throughout the loading and response period is an experimental necessity and a challenge. Given that the size of the specimen is small or the response time of the structure is comparable to the duration of the pressure pulse obtained from the shock tube, simply supported conditions as shown in Fig. 4 were used. To achieve this condition, the specimens were held along two knife edges as seen in the figure. This was preferred over other conditions due to the ease of setup and possible quantification and separation of the out of plane displacement and rigid body motion in the plate structures due to the loading. Even though the bulk of these experiments were conducted under simply supported boundary conditions, some were performed under clamped boundary conditions.

3.3 Pre-damage Procedure Experiments involving the blast response of damaged sandwich composites require that the specimens have a measurable amount of impact damage prior to any blast loading experiment. Impact damage was imparted to the specimens in a controlled fashion in order to regulate the level of damage in each specimen. The level of damage was controlled by the number of times that an individual specimen was struck by a projectile. The projectiles used were 10.69 g (165 grain). APM2 armor piercing bullets fired from a 300 Winchester magnum. Bullets were hand loaded in order to

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Fig. 4 Specimen held in simply supported condition at the end of a shock tube

Table 2 Image and dimensions of projectile 0.30 cal. 165 grain Spitzer Copper Jacket Stainless Steel Core

Diameter

Length

Mass

Average velocity

7.89 mm

34.9 mm

10.69 g

320 m/s

control the charge and overall length, therefore obtaining a consistent velocity. The projectile image, type and dimensions are shown in Table 2. The impact velocity and exit velocity of the projectiles were measured with velocity traps constructed of break screens connected to a timing circuit. High speed imaging was utilized to observe in real time the projectile impact and exit from the specimen and to confirm that the bullet exited the specimen undamaged.

3.4 High Speed Imaging In the experiments reported here, one of the free ends of the simply supported plate was viewed from the side using an IMACON high-speed camera. The camera is capable of taking 16 photographs at framing rates as high as 200 million frames/s with exposure times as low as 5 ns. Typical blast loading events are in the order of 2–6 ms and the deformation of the plate is recorded using the camera over the response period of the plate. Post analysis of these images provided the macroscopic deformation mechanisms and quantitative deformation-time history of the center point of the plate during the deformation. High speed imaging, in conjunction with optical techniques, has also been used by others [30] for blast imaging.

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3.5 Blast Energy Calculation Procedure The energy of the incident shock wave (incident energy), the reflected shock wave (reflected energy) and the energy that deforms the specimen (deflection energy) can be calculated based on the shock wave pressure profiles and the high speed deflection images obtained from the shock tube experiment. When a shock wave with pressure profile p(t ) propagates with a particle speed u located behind the shock wave front through a tube with cross section area S and hits a specimen, the work that the shock wave imparts on the specimen during time dt is p(t)Sudt. Thus, the total energy that the shock wave applies on the specimen during the loading process can be obtained by integrating this with respect to time, as follows. Z  ED p.t/ S  u dt Since the incident and reflected pressure profiles can be obtained as shown in Fig. 3 and the particle speed located behind the shock wave front can be calculated by gas dynamic theory, the incident and reflected energies can be calculated. The difference between incident and reflected energies is the total energy lost during the shock wave loading process. This lost energy is transformed into specimen deflection, rigid body motion, and other types of energies including heat, sound, light etc. The deflection energy associated with the specimen can be calculated by integrating the force–deflection curve. The deflection–time curve of every point on the front surface of the composite can be obtained from the high speed images, shown in Figs. 8, 11, 19, 20, 23 and 24. Assuming the pressure on the front surface of the composite is uniform, the reflected pressure profile obtained from Fig. 3 gives the force–time curve of every point on the front surface of the composite. The force– deflection curve then was obtained by combining the deflection–time curve and the force–time curve. The formula to calculate the deflection energy is, I Z Edef lecti on D

 preflected .t /d ldeflection dS

Stube

4 Results and Discussion 4.1 Blast Resistance of Laminated Composites Two different series of experiments were conducted on the laminated composite materials of very similar kind. In the first series, E-glass vinyl ester (EVE) and Carbon fiber vinyl ester (CVE) were subjected to blast loading with a fixed boundary condition. The specimen used was 305 mm .1200 / square plates of 2.5 mm .0:100 / thickness. These plates were fixed on all ends using rigid clamping fixture which

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0.2 MPa

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0.5 MPa

0.8 MPa 0.66 MPa

Fig. 5 Damage progression in E-glass/vinyl ester composites, subjected to shock blast loading. Shown here is the rear face of the panels with shock levels in inset

exposed a square area of 228  228 mm .900  900 / to the blast. The muzzle of the shock tube was flushed against the plate. Damage in EVE composites under dynamic blast conditions were observed and examined. The panels were subjected to increasing level of dynamic loading by varying the driven pressure in the shock tube. The input shock pressure varied from 0.2 MPa (30 psi) to 0.8 MPa (116 psi). Figure 5 shows the damage progression in these plates as they were subjected to increasing shock pressures, quoted in the inset. As seen in the figure, the blast loading induced visual damage in the center and along the boundary regions. The spread and area of these damage regions increased as the input shock pressure was increased. These panels also endured permanent deformation due to the shock. The magnitude of this deformation increases as the input shock level is increased. Panels of CVE composites were subjected to similar shock pressures as used for EVE composites. The modes of damage in these panels were significantly different from those observed before in the E-glass fiber composite panels. Figure 6 shows the damage behavior of these panels. As seen, these panels tend to resist damage

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0.62 MPa

0.43 MPa

0.8 MPa

Fig. 6 Strike face damage progression in carbon fiber/vinyl ester composites subjected to shock blast loading

until a certain level of input shock pressure (0.6 MPa), which can be referred as “Threshold” pressure. After this threshold level, extensive delaminations and fiber breakage were observed in the panel. All the panels below the threshold shock input suffered no external damage and/or permanent deformation. The threshold level in the E-glass composite was comparatively lower at 0.2 MPa (30 psi). While the E-glass composite panels had a slow and progressive damage behavior, failure in Carbon fiber composite was more drastic and rapid. In the second series of experiments, blast loading was applied to rectangular specimens of E-glass fiber composite held in simply supported conditions. Rectangular plates of size 0:23  0:102 m .900 400 / were fabricated for the study. The plates were simply supported over a span of 0:152 m .600 / along two edges (shorter edges) and the other two edges were free. The blast loading covered a circular region of 76 mm .300 / diameter as explained in previous sections. Damage in these panels was concentrated predominantly in the central region as shown in Fig. 7 (photograph taken utilizing a bottom light source). Figure 8 depicts the typical real time deflection in laminated E-glass fiber composite. The macroscopic response of the laminated composite to the applied blast

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Fig. 7 Damage in E-glass/vinyl ester composite plates under varying blast loadings

loading can be seen as the development of flexural bending of the plate in these high speed images. Quantitative values for the deflection were obtained by calculating the deflection of center point of the plate. A detailed comparison of these values is provided in subsequent sections.

4.2 Blast Resistance of Layered Composites The layering of polyurea onto laminated composites provided enhanced blast resistance. This enhancement can be attributed to two main reasons: (1) weight (inertia) addition leading to increased impulse required for damage initiation (2) Energy dissipation in the polyurea material. Since weight addition is not a sought after method to enhance blast resistance, given the requirement of light weight structures, the beneficial factor can be justified only if the energy dissipation mechanisms in the layered structure are superior to laminated composites. It requires a systematic experimental and analytical approach to completely understand these mechanisms. It is equally important to optimize the amount of layering required and the orientation of the layering with respect to blast loading direction. When experiments were performed using Split Hopkinson bar apparatus, it was found that the orientation of the polyurea layer with respect to the applied loading had no significant effect on the dynamic response of the layered sample. But due to the presence of dynamic

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Fig. 8 Typical real-time deformation event of a plain woven composite plate, when subjected to blast load (0.60 MPa)

equilibrium in these experiments, a significant effect is not expected either. So to understand the effect of orientation, two different series of experiments were conducted in the blast loading setup. In the first series, the polyurea side of the layered composite faced the blast loading and this series is referred to as PU/EVE composites. In the second series, the E-glass vinyl ester side of the layered composite faced the blast loading and hence is referred to as EVE/PU composites. A comparative case of damage caused in the laminated composite and a layered composite, PU/EVE, under the same level of blast loading is shown in Fig. 9.

4.2.1

PU/EVE Layered Material

Whereas panels of plain woven composite failed at an incident shock pressure of 0.62 MPa, the PU/EVE layered composite required 0.75 MPa of incident shock pressure to fail. Figure 10 provides a comparison of damage in the laminated and

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PU

519

EVE

EVE

0.62 MPa

0.62 MPa

0.62 MPa

0.62 MPa

Fig. 9 Damage in E-glass/vinyl ester composite and layered plates under blast loading of same intensity

EVE 0.62 MPa

PU/EVE 0.75 MPa

EVE/PU 0.75 MPa

Fig. 10 Post blast view of damage in laminated and layered composites. The EVE composite side is shown in the layered materials

layered composites with reference to the orientation of the polyurea layer. Damage progression in the PU/EVE layered composite was very similar to the damage progression in the EVE but the pressure required to induce the same level of damage was higher in the case of the layered composite. 4.2.2

EVE/PU Layered Material

In the reverse case (EVE/PU), when EVE was on the strike face, the weaker compressive strength (compared to the tensile strength) of EVE attributed to extensive

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t = 0 ms

0.3 ms

0.6 ms

0.75 ms

1.05 ms

Fig. 11 Typical real-time deformation event of a EVE/PU layered composite plate, when subjected to blast load (0.75 MPa)

damages observed in the plate, particularly on the strike face which comprised of glass composite material. The strike face of EVE/PU material had higher delamination area shown as bright white regions in Fig. 10. The qualitative difference of the high speed images of bending in laminated and layered composites due to the applied blast loading is negligible. Quantitatively, the deflection of center point of the plate was calculated from the high-speed images for all the cases of laminated and layered composites. Detailed comparisons of these values are provided in subsequent paragraphs. Figure 11 shows a typical sequence of bending in layered materials when subjected to blast loading.

4.3 Blast Resistance of Sandwich Composites Three broad classifications of sandwich composites were evaluated for blast resistance. The first type was based on glass fiber composite and polyurea material system, the second type was based on 3D composites and polymer foam (Corecell) material system. The third type was based on glass fiber composites and polymer foam (Corecell) material system.

4.3.1

Polyurea Based Sandwich Composites

As in the case of layered construction, two different cases of sandwich construction were evaluated for blast resistance. The construction details of these sandwich materials were discussed in the Materials section. Results are provided only for the case of sandwich composites with a polyurea layer acting as the core with skin materials comprising of E-glass fiber composite material. This sandwich composite

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0.62 MPa

0.75 MPa

0.97 MPa

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1.17 MPa

1 2 Panel 1

Panel 2

Panel 3

Panel 4

3 4

Rear Face View Side View

Fig. 12 Four different EVE/PU/EVE sandwich composite plates under blast loading of varying intensities (Shock pressure shown in inset)

system showed minimal damage as shown in Fig. 12 under increasing blast loading intensities. These minimal damages in the panels were visualized on the strike face predominantly and there is no evident external damage on the rear face. On a macroscopic scale, no damage was observed in the EVE/PU/EVE sandwich system despite the fact that these sandwich panels were subjected to 85% higher pressure than the plain woven composite and 33% higher pressure than the PU/EVE layered material. Figure 12 also shows the side view of the sandwich panels after being subjected to increasing intensities of blast loadings. There was no visible damage or deformation induced in them due to the blast loads. The damage behavior of soft-core (EVE/PU/EVE) composites was different from that of the hard-core sandwich (PU/EVE/PU). Under similar magnitude of loading (1.17 MPa input pressure), PU/EVE/PU panel showed signs of failure as wrinkles on the strike face and shear failure on the composite core. Focusing on the quantitative measure of deflection, the deflection of the center point of the plate specimen in all the above cases was calculated from the high-speed images. Figures 13–15 show the deflection time history of the laminated composite (EVE), layered and sandwich composite materials. The input pressure is quoted on the legend for each material. These plots reveal that the deflections observed in the layered and sandwich constructions were lower than those observed in the laminated composite plates, as expected. The quantitative estimate of reduction in deflections can be observed from these plots. Also to be noted is that the input blast pressure is much lower for the plain composite compared to the layered and sandwich constructions. Observing these deflections and when one calculates the deflection per unit thickness, it can be observed that macroscopic failure in the specimen is observed when

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50

EVE0.62

Deflection (mm)

EVE0.65 40

30

20

10

0 0

500

1000 Tim e (m icrosec)

1500

2000

Fig. 13 Center point deflections of the plain composites (EVE) under different input blast pressures

35

30 EVE/PU 1.15 MPa

Deflection (mm)

25

EVE/PU 0.75 MPa 20

PU/EVE 0.75 MPa

15

10

5

0 0

200

400

600

800

1000

1200

Time (microsec)

Fig. 14 Center point deflections of the layered materials under different input blast pressures

the deflection of the sample is in the range of 2.5–3 times its thickness. This observation is phenomenological. It can further be deduced from the experimental observations that in case of the plain composite materials, the “failure” point (deflections equaling 2.5 times the thickness) is produced at an earlier time compared to the layered system under comparable input blast loadings. In the case of the layered composites with polyurea facing the blast, the failure point is not observed at all.

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20 18

Deflection (mm)

16 14 12

EVEPUEVE0.62

10

EVEPUEVE0.75 EVEPUEVE1.17

8 6 4 2 0 0

500

1000

1500

Time (microsec)

Fig. 15 Center point deflections of the sandwich materials under different input blast pressures

The macroscopic damage in the plate also corroborates with the observed real time trend, vis-`a-vis, the PU/EVE configuration showing lower damage area compared to the EVE/PU configuration. Under the same input blast loadings to the layered and sandwich composites, the sandwich configuration showed normalized deflections less than one, which is well within the elastic limits of the plate. Here again, it was observed that the PU/EVE configuration reached the failure point at a later time stage compared to the EVE/PU configuration. The delay in the attainment of this failure point between the layered configurations can be attributed to the internal strengthening mechanisms that are present in the PU/EVE system.

4.3.2

Sandwich Composites with 3D Skin and Polymer Foam Core

In addition to polyurea based sandwich composites, several design modifications were sought after to reinforce lightweight sandwich composite structures against blast loading. Two major design changes that were incorporated were: (1) use of three dimensional skin materials and (2) use of reinforced core materials; by introducing stitches in the transverse direction of the panel. Such sandwich composites were subjected to identical blast loading of given high intensity, with an input pressure in the range of 1.2–1.4 MPa and reflected pressure in between 5.5–5.8 MPa. Rectangular flat sandwich specimens were utilized for the present experimental study. They were held under simply supported boundary conditions, as shown in Fig. 6. The specimen size was 300  102 mm. The span between the supports was 152 mm. The dynamic loading was applied over a central circular area 76 mm in diameter. The results of post-mortem evaluation of the shock wave tested sandwich

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Skin Fracture in core

Top View

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Fig. 16 Post mortem damage evaluation in Sandwich 1

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Fig. 17 Post mortem damage evaluation in Sandwich 2

composite panels are shown in Figs. 16–18. In case of Sandwich 1, the core disintegrated completely when tested and was lost beyond retrieval. The front skin (top in Fig. 16) shows severely fractured fibers in the central region (where the dynamic

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Fig. 18 Post mortem damage evaluation in Sandwich 3

pressure was applied); it was also fractured into two layers there. The back skin did completely delaminate along the interface between two 3-D woven composite layers and separated into two pieces having thicknesses 2.92 and 3.86 mm respectively. Contrary to the above, post mortem analysis of Sandwich 2 (Fig. 17) shows only localized delaminations in the skins accompanied by the foam damage in the core. The damage in this case is confined to the mid-section of the front skin; minimal visual damage is observed in the back skin. The observed residual deformation of this sandwich sample is relatively small. Separation of the core chunks from the vertical stitch composite bars are seen in this sample. Buckling of the stitch bars themselves was observed under close-up visual examination. The location of such

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local buckling was identified by bright white shear-like spots on the side view of the sandwiches. Such bright white spots were detected closer to either one of the skins rather than in the mid-region of the core. It was also observed that microdelaminations appeared within the skin of this sandwich specimen. Post-mortem analysis of Sandwich 3 (Fig. 18) shows minimal damage in the skins and core and practically no global permanent deformation. Both front and back skins show no signs of fiber breakage or other failure types. Only small cracks parallel to the stitches were found in the core, and small micro-delamination cracks are present in the skins. Overall, it can be concluded that this sandwich had survived the shock loading significantly better than Sandwich 2 and did not lose its overall integrity. This improvement can be attributed to the higher stitching density and to the marginal increase in the thickness of the skin (which replaces the corresponding foam material in the core). The dynamic transient behavior of all the above sandwich specimens under blast loading was recorded using a high speed digital camera and thoroughly analyzed. The real time observation of the shock loading of Sandwich 1 is shown in Fig. 19.

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Fig. 19 Sandwich composite with 3D Woven Skin when subjected to blast load (1.2 MPa Incident)

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The measurements showed that at 500 s time instant, central deflection magnitude was about 5.5 times higher for the front skin than for the back one. It was also noted that initial separation at one of the sandwich edges between the front skin and the core is seen as early as at 200 s time instant, which indicates high stress concentration at the front skin-core interface along the sample edges. The indentation failure of the front skin is observed from 200 s time. The onset of core failure is observed at 400 s time, and complete collapse of the core is seen at 500 s time instant. The conclusion is that for Sandwich 1 all important deformation, damage initiation and total failure events had occurred within 0.5 ms time from the arrival of blast wave. The first principal failure mechanism evident in Sandwich 1 is progressive damage of the core, which starts with some dispersed damage near the front skin and simultaneous formation of a large inclined crack in the central region of the core. These two damage zones gradually extend and coalesce at approximately 400 s time instant which is followed by the formation of a much larger damage zone at 500 s time instant and rapid crush of the core after that time instant. Possibly, around the same time the front skin suffered significant damage. Likely, the back skin fails soon after that, and the whole sandwich is crushed. These images revealed that Sandwich 1 does not provide sufficient transient load transfer from the front skin to the back skin and, as a result, it cannot withstand the applied dynamic pressure. Analysis of such corresponding real time observations for Sandwich 2 and Sandwich 3 revealed that the dynamic pressure is much better transferred from the front to the back of the sandwich, and we see well synchronized local bending of both skins. The core is gradually damaged as the skin bending progresses. Quantitative measurements of deflections showed that the difference between central deflection values of the front and back skins is relatively small. Also, no separation between the front skin and the core is seen in these high-speed camera frames, thus the aforementioned high stress concentration for Sandwich 1 at the skin-core interface has been reduced by core stitching. Failure mode of this stitched sandwich can be defined as a combination of core shear and face failure types. In the case of Sandwich 2, the first two frames (up to 100 s time instant) do not reveal any macroscopic damage to the sandwich. At 200 s time instant initial cracks are seen in the core; they start in the mid-section of the core in the direction preferably parallel to the loading plane. The transition of “hinge point” takes place between time instants 300 and 800 s. Further, at time instant 900 s, the cracks in the core become more pronounced and the “hinging” of the whole sandwich (including the span and overhang) along a center line can be observed. Figure 20 illustrates the deformation process of Sandwich 3. This appears to be very similar to the one seen for Sandwich 2. Though on a closer observation, two significant differences in the real time deformation were observed. The first difference is that the deflection values at 500 s time instant are by 30% smaller at the center of the specimen in this case than for Sandwich 2. This can be attributed to two factors: (a) more dense core stitching and (b) resin layers compensating for reduced thickness of the foam core material in Sandwich 3. Both factors provide additional

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Fig. 20 Sandwich composite with 3D Woven Skin and “stitching” of core, when subjected to blast load (1.2 MPa Incident)

stiffening to the sandwich. The second difference is in the deformation and failure progression. It is observed that at 100 s time instant there are no visible cracks in the core and the overhangs of the simply supported sandwich are straight, whereas the span portion is hinged about the support lines. The hinge point moves from the support line to the center of the sandwich as the time progresses, which is seen between time instants 200 and 400 s. Further on, at 700 s time instant the transition is completed and the whole sandwich is hinged about the center. So, in a contrast to the results shown in Fig. 19 for Sandwich 1, the real time deformation sequences shown in Fig. 20 for Sandwich 3 indicate that the reinforced core damage does not include any visible macro-cracks, although one can see certain micro-damage accumulation and propagation from frame to frame. The principal conclusion made from the presented and discussed experimental results is, that through-thickness stitching of the sandwich before resin infusion makes a remarkable positive effect on the transient deformation, damage initiation and progression processes in the studied sandwich materials. The major failure modes observed in this study under the transient loading conditions are sketched in Fig. 21. These failure modes can be broadly classified as (a) Front Skin Indentation (b) Core Shear Failure and (c) Face Failure. Unlike in respective static loading case, where given load type usually corresponds to a well-defined, dominating single failure mode, in

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Fig. 21 Major failure modes in sandwich composite subjected to dynamic loading

the case of highly transient dynamic loading (as in this study) the initiation and development of more than one different failure modes can be observed simultaneously.

4.3.3

Sandwich Composite with E-Glass Fiber and Polymer Foam Core

Two different types of sandwich composites with E-glass fiber and a polymer foam material system were evaluated for blast resistance. The first type consisted of a sandwich composite with a stepwise graded core, in which the material properties vary layer by layer within the material itself. The second type consisted of a sandwich composite with a homogeneous foam core. Also this composite was predamaged prior to blast loading. The construction details, as well as the material properties of these sandwich composites were discussed in the Materials section.

4.3.4

Sandwich Composites with Stepwise Graded Foam Cores

Again, two different types of graded foam core sandwich specimens were studied (Fig. 22a). Configuration 1 consisted of a sequence of core gradation as A300/A500/A800 (low/medium/high density), and configuration 2 consisted of a sequence of core gradation as A500/A300/A800 (medium/low/high density). With these configurations it should be noted that the first core layer is the one first subjected to the shock wave loading. An actual sample can be seen in Fig. 22b. Both types of sandwich specimens were subjected to an identical blast loading with an incident peak pressure of approximately 1 MPa and a shock wave velocity of approximately 1,030 m/s. Rectangular flat sandwich panels were utilized for the present study. They were held under simply supported boundary conditions, shown in Fig. 4. The span between the supports, as well as the area where the dynamic loading was applied, were identical to those utilized earlier in the study pertaining to sandwich composites with 3D skin and polymer foam core, and they were 152 and 76 mm (diameter) respectively.

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The real time observations of the transient behavior of configuration 1 (A300/A500/A800) and configuration 2 (A500/A300/A800) under shock wave loading are shown in Figs. 23 and 24. The shock wave propagates from the right side of the image to the left side and some detailed deformation mechanisms are pointed out in the figures. For configuration 1, core compression in the A300 foam is very prominent and visible. At the time when this compression is first observed, there is no obvious compression in the other two core layers. Due to the compression of the foam, the high dynamic pressure applied to the front skin is substantially weakened when it reaches the back skin. The onset of core failure, where core cracking begins, is also observed as well as the delamination of the front skin from the core. Even though this occurs, complete core collapse and failure is not observed in this configuration. In configuration 2, A500 is the first core layer subjected to the shock wave and the core gradation begins with the foam of middle density, next the foam of least density, and then the highest density foam. The images in Fig. 24 show that indentation failure occurs at approximately the same time as that of configuration 1 .70 s/. Unlike configuration 1, central core compression is not as prominent in this sandwich and the delamination occurs between the core layers of foam (A500/A300) and not between the front skin and core. This delamination is due to the fact that the A300 is a weaker foam in comparison to the A500 foam. Also note that the onset of core failure does indeed lead to complete core collapse and failure for this configuration. The major failure mechanism in configuration 2 is progressive damage of the core and the sandwich, which starts at the back skin and is evident in Fig. 24. This crack becomes a large inclined crack and propagates through the core from the back skin to the front skin. By t D 490 s the crack has extended completely through the core and delamination between the A300 and A500 foam is very prominent.

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Fig. 23 Real time observation of A300/A500/A800 under shock loading

Also at this time, many cracks are visible in the core which is followed by a rapid crushing of the core after that and catastrophic failure of the sandwich. This shows that configuration 2 cannot withstand the applied shock wave pressure which has a peak value of 4:83 MPa (710 psi). Contrary to the case of configuration 2 the real time deformation sequences seen in Fig. 23 for configuration 1 indicate that the core lay-up improved the overall performance of the structure. The onset of core failure took twice as long to be visible in this configuration as opposed to configuration 2 and no complete core collapse was evident. Even though delamination did occur, it was between the face sheet and foam core only. Overall configuration 1 outperformed configuration 2, and this was directly related to the location of the A300 foam. With its ability to compress and thus absorb blast energy, configuration 1 was able to weaken the shock wave pressure by the time it reached the back skin and thus reduce the overall damage.

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Fig. 24 Real time observation of A500/A300/A800 under shock loading

The mid-point deflections of each graded sandwich panel and all of its constituents were obtained from the high speed images. The deflection of the front face (front skin), interface 1 (between first and second core layer), interface 2 (between second and third core layer), and back face (back skin) for configuration 1 and configuration 2 are plotted in Figs. 25 and 26. It can be seen in Fig. 25 for configuration 1 that the front face deflects to 33 mm at t D 840 s, which is approximately 25% more than the other three constituents. Note that the difference between the front face (skin) and interface 1 is the A300 foam, and almost all compression occurs here (7 mm [0.3 in.]). On the contrary, all of the constituents of configuration 2 deflect in the same manner (Fig. 26). This shows almost no obvious compression, even though the core foams of configuration 1 and configuration 2 are identical, but in a different gradation. Also this graded sandwich panel only deflects to 29 mm at t D 840 s.

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Fig. 25 Deflection of A300/A500/A800

Fig. 26 Deflection of A500/A300/A800

The damage patterns in the graded sandwich composites after the shock event occurred were visually examined and recorded using a high resolution digital camera as shown in Fig. 27. When configuration 1 was subjected to the highly transient loading, the damage was confined to the area where the supports were located in the shock tube and core cracking is visible in these two areas. Delamination is visible between the front skin and the foam core, as well as the back skin and the foam core. The core compression can be seen clearly and distinctively in the A300 foam. Unlike the damage visible in configuration 1, configuration 2 suffered catastrophic damage. The core of the sandwich disintegrated and the front skin (blast

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b

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Configuration 1 A300/A500/A800

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Fig. 27 Visual examination of sandwich composites after being subjected to high intensity blast load

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total energy lost deflection energy 1500 Energy (J)

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Fig. 28 Comparison of the estimated energies

side) of the sandwich fractured into two pieces at the midsection. The back skin showed heavy fiber delamination in the central region as well. Figure 28a shows the energies calculated by the methods described above in the Experimental Setup section as well as compares the total energy lost and deflection energy for the sandwich composites with stepwise (A300/A500/A800) foam core (configuration 1). It can be seen that there is a large energy difference between the incident and reflected energies. From the comparison of the energy lost and the deflection energy of Fig. 28b, it can be seen that only a small amount of the energy lost is absorbed by the specimen and becomes deflection energy, while most of the energy is dissipated into the air.

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4.3.5

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EVE/Corecell sandwich structures were utilized in all experiments focused on blast resistance of damaged sandwich composites. The impact damage was imparted to the specimens as discussed previously in the Experimental Setup section of this chapter. During impact experiments, results showed a reduction in energy absorption for each consecutive impact on any one specimen (Fig. 29). This can be attributed to the close proximity of each impact to each other and the resultant weakening of the back face sheet due to large delamination. Figure 30 shows post impact images of the front face sheet and back face sheet. Damage to the front face sheet is comprised of a puncture with the same diameter as the projectile, 7.62 mm (0.308 in.) and delamination around the puncture with a diameter of 25 mm (1 in.). Fiber pullout and matrix cracking are also present. The back face sheet exhibits much greater damage. The diameter of the back face sheet puncture is slightly enlarged in comparison to the front face sheet. This is due to increased matrix cracking and fiber pullout. The area of delamination on the back face sheet is larger with an average diameter

Fig. 29 Comparison of the average energy absorption for each consecutive impact on any one specimen

Fig. 30 Impact damage on sectioned specimen

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Fig. 31 Maximum deflection of back face sheet of damaged sandwich composite during blast loading

Fig. 32 Pressure profiles for sandwich composites with increasing level of ballistic damage

of 50 mm (2 in.). The large delamination on the back face sheet is due to the fact that the back face sheet is an unsupported free surface normal to the projectile path. The free surface is able to develop larger strains over a larger area as tensile loads are applied by the projectile, leading to a large area of delamination. Post mortem sectioning (Fig. 31) of impact specimens shows impact damage concentrated on the exterior side of the back face sheet. The damage to the interior side of the back face sheet is comparable to the damage on the front face sheet. Delamination is confined to a 25 mm diameter (1 in.) and fiber pullout and matrix cracking are reduced compared to the front face sheet.

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Front face

Side view

Fig. 33 Typical post blast images

Fig. 34 Damage to specimen shot 3 times 102 mm

102 mm

After impact damage was imparted to the specimens the same specimens were subjected to a blast experiment by means of a shock tube. Specimens were subject to input shock pressures of 1.7 MPa (250 psi). Reflected pressures were 8.5 MPa (1,200 psi). Boundary conditions were simply supported as previously discussed. Figure 32 shows the pressure profiles obtained during the blast loading experiments. Plotted data falls into two groups; the first is comprised of a specimen with no damage, and specimens with damage from one and two impacts. The second group is comprised of specimens with damage from three, four and five impacts. This trend is also seen in the time history of the back face deflection (Fig. 33). Deflection data was extracted from high speed images. Typical post blast images of specimens are shown in Fig. 34. The photograph shows that extensive shear cracks in the core have developed. Large delamination between the front face sheet and the core is also evident. The front face sheet shows considerable fiber breakage while the back face sheet shows transverse cracks and some delamination.

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5 Summary The current chapter focused on experimental evidence of damage mechanisms in several polymer based composite systems, namely the laminated composite (E-glass and carbon fiber), layered composite and sandwich composite. Design modifications to strengthen the skin and core of sandwich composite were incorporated based on experimental study on individual components, namely the skin and the core of the sandwich composites. These modifications were observed to be beneficial to the blast resistance but they are obtained at reasonable addition to the bulk weight of the sandwich structure. In quantitative terms, a 300% increase in the input pressure required to damage the structure were observed in modified sandwich materials by adding 1.5–2 times the weight as compared to the laminates. On a qualitative basis, the damage progression in E-Glass fiber composite was observed to be continuous. On the other hand, the Carbon fiber composites showed no signs of external damage until a certain threshold shock pressure beyond which, the panel failed catastrophically. Predominant failure modes in these panels were fiber breakage and delaminations in the strike face. It is observed that, layering of glass fiber composites with a soft layer provides better blast resistance. This enhancement of blast resistance is more pronounced when the softer material faces the blast. It is experimentally observed that, of the different possible material constructions using polyurea and glass fiber composites, sandwich materials made by sandwiching a soft layer (PU) in between woven composite skins (EVE) had the best blast resistant properties. Simultaneously, the weight addition for the layered and sandwich composites are 60% more than the plain composite alone. But the performance enhancement in the layered material is about 25% better (when polyurea faces the blast) and in case of sandwich composite (EVE/PU/EVE), the blast performance is enhanced by more than 100%. Based on the current experimental observations, it can be deduced that laminates and layered composites will be suitable for constructing structures to withstand mild blast loadings. The sandwich composites with through-thickness stitch reinforcement in the core exhibit much lower dynamic deformation, delayed damage initiation and higher dynamic damage tolerance with minimal visual residual damage. Also by grading the core foams from lowest density to highest density, the gradation of the sandwich can help mitigate the blast damage and thus improve the overall performance of the specimen. Hence these materials are highly beneficial in constructing structures that are designed to withstand moderate to severe blast loadings. Acknowledgements The authors kindly acknowledge the financial support and encouragement provided by Dr. Yapa Rajapakse, under Office of Naval Research Grant No. N00014-04-10268. The authors acknowledge the support provided by the Department of Homeland Security (DHS) under Cooperative Agreement No. 2008-ST-061-ED0002: The authors thank Dr. Alex Bogdanovich of 3TEX Inc, for several inputs and discussions during the preparation of the manuscripts on 3D composites. The authors acknowledge Dr. Kunigal Shivakumar of NC A&T University, Materials Science Corporation, and Martin Marietta Composites for supplying composites and sandwich materials used in this study. Authors also thank Dr. Stephen Nolet and TPI Composites for providing the facility for creating the stepwise graded sandwich composites used in the study.

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References 1. Rajamani A, Prabhakaran R (1980) Response of composite plates to blast loading. Exp Mech 20(7): 245–250 2. Hall DJ (1989) Examination of the effects of underwater blasts on sandwich composite structures. Compos Struct 11(2): 101–120 3. Librescu L, Nosier A (1990) Response of laminated composite flat panels to sonic boom and explosive blast loadings. AIAA J 28(2): 345–352 4. Mouritz AP, Saunders DS, Buckley S (1994) The damage and failure of GRP laminates by underwater explosion shock loading. Composites 25(6): 431–437 5. Lam KY, Chun L (1994) Analysis of clamped laminated plates subjected to conventional blast. Compos Struct 29(3): 311–321 6. Mouritz AP (1995) Effect of underwater explosion shock loading on the fatigue behavior of GRP laminates. Composites 26(1): 3–9 7. Singh SK, Singh VP (1995) Mathematical model for damage assessment of composite panels subjected to blast loading from conventional warhead. Def Sci J 45(4): 333–340 8. Mouritz AP (1996) The effect of underwater explosion shock loading on the flexural properties of GRP laminates. Int J Impact Eng 18(2): 129–139 9. Dyka CT, Badaliance R (1998) Damage in marine composites caused by shock loading. Compos Sci Technol 58: 1433–1442 10. Yen CF, Cassin T, Patterson J, Triplett M (1998) Progressive failure analysis of composite sandwich panels under blast loading. Am Soc Mech Eng, Pressure Vessels and Piping Division (Publication) PVP 361: 203–215 11. Sun CT, Luo J, McCoy RW (1994) Analysis of wave propagation in thick section composite laminate using effective moduli. Collection of Technical Papers – AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 3: 1675–1682 12. Turkmen HS, Mecitoglu Z (1999) Nonlinear structural response of laminated composite plates subjected to blast loading. AIAA J 37(12): 1639–1647 13. Turkmen HS, Mecitoglu Z (1999) Dynamic response of a stiffened laminated composite plate subjected to blast load. J Sound Vib 221(3): 371–389 14. Mouritz AP (2001) Ballistic impact and explosive blast resistance of stitched composites, Compos (Part B) 32: 431–439 15. Tang GY, Yan Y, Chen X, Zhang J, Xu B, Feng Z (2001) Dynamic damage and fracture mechanism of three-dimensional braided carbon fiber/epoxy resin composites. Mater Des 22(1): 21–25 16. Langdon GS, Cantwell WJ, Nurick GN (2004) The blast performance of novel fiber-metal laminates. In: Jones N, Brebia CA (eds.), Structures and Materials, Structures Under Shock and Impact VIII, vol. 15, pp. 455–464 17. Tagariell VL, Fleck NA (2005) A comparison of the structural response of clamped and simply supported sandwich beams with aluminium faces and a metal foam core. J Appl Mech Trans ASME 72(3): 408–417 18. Koissin V, Shipsha A, Rizov V (2004) The inelastic quasi-static response of sandwich structures to local loading. Compos Struct 64(2): 129–138 19. Tagarielli VL, Deshpande VS, Fleck NA (2007) The dynamic response of composite sandwich beams to transverse impact. Int J Solids Struct 44(7–8): 2442–2457 20. Apetre NA, Sankar BV, Ambur DR (2006) Low-velocity impact response of sandwich beams with functionally graded core. Int J Solids Struct 43(9): 2479–2496 21. Li Y, Ramesh KT, Chin ESC (2001) Dynamic characterization of layered and graded structures under impulsive loading. Int J Solids Struct 38(34–35): 6045–6061 22. Vaidya UK, Nelson S, Sinn B, Mathew B (2001) Processing and high strain rate impact response of multi-functional sandwich composites. Compos Struct 52: 429–440 23. Rice MC, Fleischer CA, Zupan M (2006) Study on the collapse of pin-reinforced foam sandwich cores. Exp Mech 46(2): 197–204

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24. Tekalur SA, Shivakumar K, Shukla A (2008) Mechanical behavior and damage evolution in E-glass vinyl ester and carbon composites subjected to static and blast loads. Compos Part B: Eng 39(1): 57–65 25. Tekalur SA, Shukla A, Shivakumar K (2008) Blast resistance of polyurea based layered composite materials. Compos Struct 84(3): 271–281 26. Mohamed MH, Bogdanovich AE, Dickinson LC, Singletary JN, Lienhart RB (2001) A new generation of 3D woven fabric preforms an composites. SAMPE J 37(3): 8–17 27. LeBlanc J, Shukla A, Rousseau C, Bogdanovich AE (2007) Shock loading of threedimensional woven composite materials. Compos Struct 79(3): 344–355 28. Tekalur SA, Bogdanovich AE, Shukla A (in Press, Accepted Manuscript, Available online 28 March 2008) Shock loading response of sandwich panels with 3-D woven E-glass composite skins and stitched foam core, Compos Sci Technol 29. Corecell A-Foam - Structural Core Material (v7) SP http://www.gurit.com/core/core picker/ download.asp?documenttable=libraryfiles&id=957. February 16, 2009 30. Espinosa HD, Lee S, Moldovan N (2006) A novel fluid structure interaction experiment to investigate deformation of structural elements subjected to impulsive loading. Exp Mech 46(6): 805–824

Single and Multisite Impact Response of S2-Glass/ Epoxy Balsa Wood Core Sandwich Composites Uday K. Vaidya and Lakshya J. Deka

Abstract Impact damage reduces the structural integrity and load bearing capacity of a composite structure. Most studies on high velocity impact damage have been limited to single-site impacts, with little consideration given to the effect of cumulative damage from multiple impacts. In this study, the impact damage response of S2-glass/epoxy balsa wood core sandwich composite is evaluated experimentally and supported by finite element modeling for single-site and multi-site impacts from 0.30 and 0.50 caliber spherical projectiles. During high velocity impact, a composite laminate undergoes progressive damage; hence a progressive failure model based on Hashin’s criteria is used to predict failure. When subjected to multi-site impact loading, a sandwich composite structure exhibits synergistic and cumulative damage causing extensive fiber breakage, matrix cracking and delamination. An excellent correlation between experimental and numerical results is obtained.

1 Introduction Polymer matrix composites (PMC) have been extensively used in the marine, military and aerospace sector due to their high specific strength, specific modulus and impact energy absorption capability. PMCs and sandwich PMCs are frequently subjected to impact loading from primary and secondary threats such as fragments from blast debris, shrapnel and multiple bullet impacts. Despite extensive research and development of laminated and sandwich composite structures, their response to dynamic loading is less understood [1–4]. When subjected to a transverse impact, a sandwich composite undergoes damage by fiber breakage, matrix cracking and delamination of the strike side facesheet followed by penetration of the impactor into the core [5]. At higher impact velocities

U.K. Vaidya () and L.J. Deka Professor of Material Science and Engineering, University of Alabama at Birmingham, Alabama, USA e-mail: [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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a critical stress condition is reached when the local contact stress exceeds the local strength, which could be laminate bending strength, core compression strength or interface delamination strength. This stress leads to partial or complete penetration of the projectile into the sandwich composite. Numerous researchers [6–24] have investigated single-site impacts to laminated and sandwich composites. Deka et al. [16, 17] investigated high velocity impact of laminated S2-glass/epoxy and E-glass/polypropylene composite using Hashin’s failure criteria [25] combined with the continuous damage mechanics (CDM) approach proposed by Matzenmillar et al. [26]. Balsa wood is a common core material used in the construction of a sandwich composite. Balsa wood is a naturally occurring anisotropic material; it possesses excellent specific strength, stiffness and specific energy dissipation capacity. Wood has been used as a protective material for high velocity impact events for several centuries [27, 28]. There have been only a few systematic studies of the behavior of wood that have investigated high rates of impact loading [27–33]. The damage area in a laminate or a sandwich composite subjected to a multi-site impact is higher than that of a single site impact [20]. A single site impact refers to impact of the composite plate by a single projectile at its geometric center. Nearsimultaneous multisite impact refers to multiple projectiles arriving at the target at approximately same instance of time. Sequential multisite impact refers to sequential arrival of projectiles, each following the previous one. In a multisite impact, the material response is governed by synergistic effects of propagation and interaction of stress waves (additive effects) and dynamic cracks/damage (cumulative effects). Bartus et al. [20, 21, and 23] investigated the multisite impact response of S2-glass/epoxy laminates for near-simultaneous and sequential impacts. Bartus et al. [20, 21, and 23] used projectiles of different masses and sizes to assess synergistic and cumulative effects of energy absorption, delamination and failure mechanisms of the laminate. Deka et al. [16, 18] conducted numerical analysis using LS-DYNA to verify the experimental results of Bartus et al. [24]. This study deals with multisite impact response of a S2-glass/epoxy balsa wood core sandwich composite. The single site and multisite impact response of the sandwich composite to 0.30 and 0.50 caliber spherical steel projectiles is investigated. The effect of various parameters on the impact damage including the caliber of the projectile, impact location and material constituents, namely the facesheet, the core and the assembled sandwich plate has been investigated.

2 Experimental 2.1 Specimen Fabrication The sandwich composite specimens were fabricated using Vacuum Assisted Resin Infusion/Transfer Molding (VARTM) [23]. The facesheets consisted of a 24 oz yd2 , plain woven S-2 glass with 933 sizing. Applied Poleramic Inc. SC-15 rubber

Single and Multisite Impact Response Table 1 S2-glass/epoxy balsa core sandwich characteristics Sandwich Dimensions Number of composite specimen (mm  mm) plys in facesheets S2-glass/epoxy 200  200 3 unscored balsa

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Average sandwich thickness (mm) 27.40

toughened epoxy resin was used as the matrix. Immersion density technique [34] was used to determine the average fiber volume fraction, which was 40:1% ˙ 0:2%. The composite facesheets had an average thickness of 2 ˙ 0:05 mm. Commercial balsa wood (trade name Baltekr SB.100 structural end-grain balsa) with a nominal density of 151 kg m3 was procured in the form of unscored sheet with dimensions 609:6  1219:2  25:4 mm3 . Specimens of dimensions 20:3  20:3 cm2 were cut from the panel and post cured at 82ı CC for 5 h. Details of the sandwich panel constructions and average dimensions are provided in Table 1.

2.2 High Velocity Impact Set Up The apparatus used for impact experiments is a gas gun capable of launching single and up to three projectiles in a near-simultaneous or sequential mode with controlled impact locations [18–23]. The gas gun has three barrels, equally spaced 120ı apart on a 20 mm radius. The firing apparatus consists of a pressure vessel, single to trifire barrel system, a nitrogen/helium tank and a capture chamber. The details of the gas gun setup are reported in [23]. Specimens of dimensions 200  200 mm were clamped on all the four sides and impacted simultaneously with 0.30 and 0.50 caliber spherical steel projectile above the ballistic limit [23].

3 Model Description 3.1 Mesh Generation and Contact Definition Hypermesh v 8.0 and Finite Element Model Builder (FEMB) was used as the preprocessor for developing the model. The sandwich composite plate was modeled as three layer S2-glass/epoxy facesheets and the balsa wood core. The grid geometry of the sandwich plate was designed as three layers of brick elements with one element through the thickness per facesheet layer and ten elements through the thickness of the core. Each plain weave layer and the balsa core had 16,000 and 27,600 brick elements respectively. The 0.30 and 0.50 caliber steel spherical projectile was modeled with 1,300 and 2,500 brick elements respectively. A gradient in mesh density was applied with sufficient detail (0:79  0:79  0:70 mm3 with aspect

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ratio of 1.1) in the impact area and larger elements (6:9  1:8  0:7 mm3 with aspect ratio of 10.74) towards the edges to provide computational efficiency with smooth stress gradient from the impact point to the edge [16]. Hourglassing i.e. mesh distortion is a common numerical instability observed at the impact region [35]. An adequate hourglass energy (HGE) coefficient is an input parameter to the model, because this coefficient inhibits the hourglass mode of the target and projectile. Type 3 and 4 hourglass control with a HGE coefficient .QM/ D 0:01 was used for balsa wood core and composite facesheets respectively. A QM value of 0.01 minimized the HGE at the impact region. The contact between the projectile and the sandwich composite plate was defined using CONTACT ERODING SINGLE SURFACE, which is a penalty method [35]. Eroding contact type is recommended when solid elements in the contact definition are subjected to erosion (element deletion) to avoid numerical disturbance due to large element distortion at the impact region [12–17]. Penetration through the facesheets is addressed using eroding elements with strain based failure criterion, whereas penetration through the balsa wood core is modeled with damage induced stress reduction criteria. If an element undergoes tensile failure and exceeds the axial tensile strain (E LIMIT), then it is automatically eroded. In all the simulations, E LIMIT has been activated using a value of 1.2 [12, 16]. The material properties used for the simulation of the laminate and the balsa wood core are summarized in Table 2 [15, 16] and Table 3 [36, 37], and for the projectile(s) in Table 4 [12, 16], respectively.

Table 2 Material properties of a plain weave S2-glass/epoxy laminates [15, 16] Density, ¡, kg mm3 1:85E-06 Tensile modulus, EA, EB, EC, GPa 27.1, 27.1, 12.0 Poisson’s ratio, v21, v31, v32 0.11, 0.18, 0.18 Shear modulus, GAB, GBC, GCA, GPa 2.9, 2.14, 2.14 Inplane tensile strength, SAT, SBT, GPa 0.604 Out of plane tensile strength, SCT, GPa 0.058 Compressive strength, SAC, SBC, GPa 0.291 Fiber crush, SFC, GPa 0.85 Fiber shear, SFC, GPa 0.30 Matrix mode shear strength, SAB, SBC SCA, GPa 0.075, 0.058, 0.058 Residual compressive scale factor, SFFC 0.30 Friction angle, PHIC 10 Damage parameter, AM1, AM2, AM3, AM4 0.6, 0.6, 0.5, 0.2 Strain rate parameter, C1 0.10 Delamination, S DELM 1, 5 Eroding strain, E LIMIT 1.20 Table 3 Material properties for the tool steel spherical projectile [12, 16]

Density, ¡, kg mm3 7:86E-06 Young’s modulus, E, GPa 210 Poisson’s ratio 0.28 Yield strength, GPa 1.08

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Table 4 Material properties of end-grain balsa wood core [36, 37] Density, ¡, kg mm3 1:55E  07 Moisture content, % 1:20E C 01 Stiffness: Parallel normal modulus, EL, GPa 5.30 Perpendicular normal modulus, ET, GPa 0.20 Parallel shear modulus, GLT, GPa 0.166 Perpendicular shear modulus, GLR, GPa 0.085 Parallel major poisson’s ratio 0.25 Strength: Parallel tensile strength, XT, GPa 0.0135 Perpendicular tensile strength, YT, GPa 0.0004 Parallel compressive strength, XC, GPa 0.0127 Perpendicular compressive strength, YC, GPa 0.0023 Parallel shear strength, SXY, GPa 0.003 Perpendicular shear strength, SYZ, GPa 0.004

3.2 Composite Progressive Failure Model and Strain Softening Characteristics Fiber breakage, fiber crushing, matrix cracking and delamination are the major failure mechanisms in laminated composites subjected to ballistic impact [6]. While a variety of models exist in LS-DYNA for the prediction of failure in composite laminates, recent work [12–17] has shown that the progressive failure model, MAT 162 (MAT COMPOSITE DMG MSC) provides reasonable damage prediction. A discussion on the progressive failure of a composite and strain softening characteristics can be found in references [14–17].

3.2.1

Wood Material Model

Wood is a porous, fibrous, complex anisotropic material. It exhibits different properties as a function of time, temperature, moisture content and loading rate [33, 38]. Under static conditions or low strain rate, wood can be treated as a linear elastic material. With increasing loading rate the cell walls start to buckle locally and the behavior of wood becomes non-linear [38]. Reid and Peng [29] demonstrated localized deformation of different type of woods under quasi-static compression and dynamic loading. They reported that for different loading conditions the damage in wood progresses as a result of compressive (crushing) wave fronts. Vural and Ravichandran [31] investigated the dynamic response and the energy dissipation of balsa wood through experimental and analytical studies. They reported that the initial failure stress is very sensitive to the strain rate. Buckling and kink band formation are two major failure modes in quasi-static and dynamic loading. Wood behaves linearly in longitudinal and transverse tension, while in compression and shear, the behavior is non-linear. For analytical purposes,

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wood can be assumed to be an orthotropic material because it possesses different properties in three directions; the longitudinal, tangential, and radial directions [38]. Very few studies have dealt with non-linearity of wood from a modeling standpoint [32, 33, 38–41]. Tabiei and Wu [38] developed 3D constitutive equations for wood using incremental-loading and curve-fitting technique to trace the non-linear behavior, and Johnson rate dependent model to account for the strain rate. Their study implemented a non-linear model which was incorporated as a user-defined material model in LS-DYNA. Tagarielli et al. [32] developed an elastic-plastic constitutive model for transversely isotropic compressible foams and implemented it to predict the indentation response of balsa wood to a conical indenter. Murray et al. [33] developed a wood model to simulate the deformation and failure of wooden guard rail posts impacted by vehicles. Their material model is currently implemented in LS-DYNA as MAT 143. Material model 143 requires 29 material input parameters to simulate wood behavior. It has inherent damage mechanisms and strain softening characteristics. Murray et al. [33] evaluated quality factor, fracture energy, softening parameters, and hardening parameters of pine wood comparing simulations with experimental quasi-static and dynamic post tests. In our current study we did not include the fracture energy, strain softening parameter, rate effect and hardening effect in the material model. To generate these parameters for end-grain balsa wood would be the scope of a future study. For most practical purposes, balsa wood can be categorized as transversely isotropic with an isotropic plane being perpendicular to the axis of the tree [32, 33]. Material model 143 is based on transverse isotropy and has hardening effect under compression. It has peak-softening characteristics under tension. In material model MAT 143, separate damage parameters are incorporated to account for parallel and perpendicular grains. With progression of damage, the stiffness of the wood reduces along these directions. Strain rate sensitivity and separate yield surfaces for parallel and perpendicular to the grain modes are also incorporated in MAT 143 [33, 35]. The visible structure of wood indicates that the planes perpendicular to the longitudinal (z), radial (r) and tangential .™/ directions, respectively, as shown in Fig. 1 are considered as planes of elastic symmetry (orthotropic). Hence, the element erosion criteria is based on the fact that the elements automatically erode when

z

r

Fig. 1 The three principal directions in balsa wood

θ

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six parallel stress components reach 99% damage via the parallel (longitudinal) damage parameter (DMAXPA [33]). The elements do not erode when the three perpendicular stress components reach 99% damage via the perpendicular damage parameter (DMAXPE) [33]. In our current study we evaluated wood damage and energy dissipation associated with each failure mode using material models MAT 143 and MAT 2. Single site and multisite simultaneous impacts to the sandwich composite is evaluated. Energy absorption and projected delamination area was estimated from experiments and finite element model. The experimental and modeling results are presented for (a) 0.30 caliber and 0.50 caliber single site projectile impact, and (b) 0.30 caliber and 0.50 caliber multisite simultaneous impact to the sandwich plate.

4 Results and Discussion 4.1 Single Site Projectile Impact As a first step, single site impact tests were performed on three-layer S2-glass/epoxy composite facesheets (2 mm thickness) and balsa wood core (25 mm thickness) separately to establish the accuracy of the numerical simulation. Detailed information on the impact response of the composite facesheet can be found in Ref. [16, 23]. Impact testing of the balsa wood core was conducted on 200 mm2 cross section balsa wood specimens and the results are tabulated in Table 5.

4.1.1

Single Project Impact: Balsa Wood Core Only

Experiment The average incident velocity for 0.30 and 0.50 caliber single projectile was 261.6 and 273:8 m s1 respectively. Both impact velocities were above the ballistic limit of the balsa wood and full penetration was observed. The damage in the wood was Table 5 Unscored balsa wood core, single projectile results above the ballistic limit

Specimen Projectile 1 2 3 4 5 6

0.30 Caliber 0.30 Caliber 0.30 Caliber 0.50 Caliber 0.50 Caliber 0.50 Caliber

Average Residual energy energy absorption (J) (J) 51:19 44:13 72:01 66:09 6.55 89:16 82:48 287:90 245:5 303:486 259:78 42.54 352:865 311:36

Incident Residual Impact velocity velocity energy (m s1 ) (m s1 ) (J) 224:02 265:70 295:65 262:12 269:13 290:20

208:00 254:54 284:37 242:62 249:00 272:60

Predicted average energy absorption (J) MAT 143

MAT 2

10.60

5.87

32.40

48.11

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localized and the average kinetic energy absorption was experimentally measured to be 6.55 J and 42.54 J for the 0.30 and 0.50 caliber projectile impact respectively.

Simulation Material model MAT 2 considers wood as an orthotropic-elastic solid [35]. In this study we adopted MAT ADD EROSION in LS-DYNA in which failure criteria is defined by any of the following: (a) principal stress at failure .1  max /, (b) principal strain at failure ."1  "max /, and (c) shear strain at failure .1  max / [35]. .1  max /, (b) principal strain at failure ."1  "max /, and (c) shear strain at failure .1  max / [35]. Material model MAT 143 predicts the wood erosion based on damage (stress reduction) criteria. As element damage (DMAXPA) approaches 99%, the strength and stiffness approach 1% of their original values. Because the damaged elements have nearly zero strength and stiffness, they erode. The impact velocity was assigned to be 295:65 m s1 and 269:13 m s1 for 0.30 and 0.50 caliber projectile. The numerical prediction of energy absorption of the balsa wood for 0.30 caliber and 0.50 caliber projectile is within the 96–98% of corresponding experiment results (Table 5) using both material models 2 and 143. Figure 2a–d illustrate the wood damage comparisons using MAT 2 and MAT 143 for a 0.50 caliber projectile at impact velocity of 269:13 m s1 . The maximum shear stress of 0.005 GPa, max (red area) observed at the impact region in Fig. 2a Shear stress (GPa)

a

Shear stress (GPa)

b MAT 143

MAT 2

vM stress (GPa)

vM stress (GPa)

c

MAT 2

247.4 m s–1

d

MAT 143

254.2 m s–1

Fig. 2 Simulation damage on a balsa wood subjected to a 0.50 caliber single projectile impact at 269:13 m s1 showing (a) maximum shear stress of 0.005 GPa (red area) using MAT 2 (b) maximum shear stress of 0.001 GPa using MAT 143 just prior to element erosion (c) von Mises (vM) stresses with the projectile exit velocity of 247:4 m s1 (MAT 2) (d) vM stresses with projectile exit velocity of 254:2 m s1

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corresponds to 1% shear strain at failure, max using MAT 2 [32]. The simulation using MAT 143 results in a lower max value of 0.001 GPa prior to the element erosion, Fig. 2b. Both material models predict the projectile energy lost due to penetration to be in close agreement with the experimental results for impact on balsa wood. However MAT 2 did not perform well in a sandwich composite configuration, although it predicted the constituent balsa adequately. In simulations on the sandwich composite using MAT 2, excessive element erosion was seen to occur along the facesheet-core interface which resulted in nodal mass loss of up to 40% in the wood, which is unrealistic wood damage along the interface. Therefore, all the single impact and multi-site impact simulations were conducted using MAT 143 for balsa wood. The discussion and presentation of experimental and finite element analysis (FEA) results in later sections will focus on MAT 143.

4.1.2

Single Projectile Impact: Sandwich Composite

0.30 caliber and 0.50 caliber steel spherical projectiles were used to impact the sandwich composite at a single site at velocities of 220–280 m s1 . The impacted specimens were sectioned to study the damage modes. Tables 6 and 7 summarize the experimental results and corresponding numerical predictions respectively. 0.30 Caliber Impact – Experiment The damage section of a representative sandwich plate subjected to impact energies between 49.5 and 66.99 J for a 0.30 caliber single projectile impact is shown in Figs. 3 and 4. These energy levels are below the ballistic limit of the sandwich plate for the 0.30 caliber impact. The sandwich composite plate subjected to a 49.5 J impact shows that the strike side facesheet undergoes complete perforation resulting in fiber fracture and delamination at the facesheet-core interface. The balsa core exhibited localized

Table 6 S2-glass/epoxy balsa core sandwich plate, single projectile impact results New surface creation (cm2 ) Incident Residual Impact Residual Energy velocity velocity energy energy absorption Top Bottom (J) (J) (m s1 ) (m s1 ) (J) Specimen Projectile facesheet facesheet 1 2 3 4 5

0.30 Caliber 0.30 Caliber 0.30 Caliber 0.30 Caliber 0.30 Caliber

220.37 0 254.80 0 256.34 0 266.10 0 307.24 113:69

49:50 66:18 66:99 72:23 96:28

0 0 0 0 13:18

49:50 66:18 66:99 72:23 83:10

13.45 16 17.64 12.50 14.60

15:22 39:64 38:53 35:40 78:65

7 8 9 10 11

0.50 Caliber 0.50 Caliber 0.50 Caliber 0.50 Caliber 0.50 Caliber

220.37 0 235.91 0 268.10 118:20 267.23 103:20 268.20 108:70

203:48 233:18 301:16 299:22 301:39

0 0 58:54 44:62 49:05

203:48 233:18 242:62 254:59 252:34

20.52 21.48 19.12 20.42 21.44

60:20 165:20 154:40 157:80 159:80



D Bottom facesheet delamination with no penetration,  D Full penetration.

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Table 7 Numerical predictions of single projectile impact of S2 glass/epoxy balsa core sandwich plate Energy absorption (J) New surface creation (cm2 ) Incident velocity Specimen (m s1 ) 1 220.37 3 256.34 8 235.91 11 268.2

Residual velocity (m s1 ) 0.00 0.00 0.00 110.20

Top facesheet (J) 32.58 33.59 115.43 119.67

Bottom facesheet (J) 0.00 11.90 75.74 86.77

Wood core (J) 16.92 21.50 42.01 43.09

Total energy absorption (J) 49.50 66.99 233.18 249.53

Top facesheet (cm2 ) 14.20 15.60 20.40 19.40

Bottom facesheet (cm2 ) 0.00 43.50 154.00 163.00

1, 3 D Specimens under 0.30 caliber impact, 8, 11 D Specimens under 0.50 caliber impact,  D Bottom facesheet delamination with no penetration,  D Full penetration.

a

12.6 mm

Wood cells being pushed by the impactor

Impact energy

= 49.5 J

Delamination

b 12.4 mm Composite facesheets Wood (MAT143) damage

Fig. 3 (a) S2-glass/epoxy balsa wood core sandwich panel subjected to impact energy of 49.5 J (b) simulation showing wood damage and delamination

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a

Impact energy

= 66.99 J

b Delamination at the facesheet -core interface 21 mm

Fig. 4 0.30 caliber projectile impact at 66.99 J to S2-glass/epoxy balsa wood core sandwich composite; (a) experimental damage (b) simulation showing delamination (red area) at the top and the bottom facesheet

crushing directly below the point of impact (Fig. 3a) up to a distance of 12.6 mm through the core thickness. The 0.30 caliber projectile remains embedded within the balsa core. Localized compression of the wood cells occurs around the projectile. Kinetic energy is dissipated in deforming the wood cells. Small amount of delamination .Š15:22 cm2 / was observed at the distal side of the specimen (Table 6). The delamination can be attributed to the energy imparted by the wood cells as they get compressed by the projectile towards the non-impact side facesheet shown in Fig. 3a. 0.30 Caliber Impact – Simulation The simulation of the sandwich composite plate subjected to a 49.5 J impact event is shown in Fig. 3b. The predicted energy absorption was 49.5 J which is in close agreement with the experimental result. The composite facesheets were deplied from the balsa core to measure the projected delamination area in the facesheets and the wood core. The strike side facesheet delamination area .Š14:2 cm2 / was adequately predicted, while the non-impact side facesheet delamination .15:2 cm2 / was under predicted. This can be explained as follows. The make up of the balsa core cells is in the form of parallel grains of wood as shown in Fig. 3a. During penetration, the cells are subjected to shear along the faces of the cell walls, which cause debonding of the interface between the core to the non-impact side facesheet. The model does not consider the resolution of the cell walls within the balsa core. The core is modeled simply as a transversely isotropic material.

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As the impact energy increased, the projectile penetrated to the non-impact side facesheet, which then exhibited small amounts of splitting at the core to facesheet interface as shown in Fig. 4a. The strike face undergoes fiber breakage, fiber-matrix debonding, interlaminar and facesheet to core interface delamination. No fiber breakage was observed at the non-impact side facesheet. A closer examination of the balsa wood core revealed complete crushing of the core cells with indications of significant shear deformation. The projectile was arrested at the non-impact side facesheet. The average delamination (or new surface creation) at the strike side facesheet was 15:7 cm2 (Table 6) for the impact energy range of 49.5–66.99 J, while that at the non-impact side facesheet was 39:1 cm2 for an impact energy of 66.6 J. The predicted energy absorption and delamination area for impact energy 49.5– 66.99 J for 0.30 caliber projectile is summarized in Table 7. Figure 4b shows 180% higher delamination growth (red area) for the non-impact side facesheet compared to the strike side facesheet for an impact energy of 66.99 J. The 0.30 caliber projectile penetrated through the balsa wood until it made contact with the non-impact side facesheet, and rebounded from the non-impact side facesheet with a velocity of 39 m s1 , Fig. 4b. 0.50 Caliber Impact – Experiment and Simulation Figure 5 illustrates a cross section of the sandwich composite impacted at 233.18 and 301.40 J by a 0.50 caliber projectile respectively. The damage of the strike side facesheet and the wood core are clearly observed. The damage is in the form of fiber fracture and matrix damage on the strike side facesheet and crushing of the balsa wood cells in the core (Figs. 5a–d). The 0.50 caliber projectile was arrested within the sample for an impact of 233.18 J, while complete projectile penetration was observed for the higher impact energy, 301.4 J. The facesheets and wood core absorbed energy by indentation, delamination/splitting and progressive collapse for the 233.18 J impact. The non-impact side facesheet suffers fiber breakage in the outer most ply with extensive interlaminar delamination and core-facesheet splitting as shown in Fig. 5a. The specimen subjected to higher impact energy – 301.4 J exhibits more fiber breakage on the strike side as well as the non-impact side facesheet, Fig. 5d. The delamination in the strike side facesheet subjected to impact energies of 233.18 and 301.4 J is shown in Fig. 6. The projected delamination area on the strike side facesheet for both these energy levels is approximately the same .21:46 cm2 , area covered by the dotted line). The delamination grows preferentially along the primary yarns of the strike side facesheet. Representative stereographs of the nonimpact side facesheet are shown in Figs. 7 and 8. These figures show that unlike the strike side facesheet, the delaminations are circular in shape in the non-impact side facesheet. The growth of the delamination hence can be deduced to occur in an equal rate along primary and secondary yarns resulting in a circular shape. The average delaminated region at the non-impact side facesheet is 645% higher than that of the strike side facesheet, Table 7. The projectile penetrates the strike side facesheet by localized shear. Due to pressure exerted by the projectile to the sandwich plate the entire plate goes into flexure. As a result the balsa core debonds at the facesheet to core interface. Due to the action of flexure (non-impact side facesheet is in tension) the non-impact side facesheet has 178% higher damage than the strike side.

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553

a

Interface splitting

Impact energy = 233.18 J

b MAT 143

c

Interface splitting

Impact energy = 301.4 J

d MAT 143

110. 2 m s–1 Fig. 5 0.50 caliber impact damage to S2-glass/epoxy balsa wood core sandwich composite at impact energies between 233.18 and 301.4 J showing damage, experimental result [(a), (c)] and numerical analysis showing maximum principal strain at the eroded regions [(b), (d)].

Energy dissipation in 0.30 and 0.50 caliber impact: Numerical prediction of energy dissipation and delamination at the strike side and non-impact side facesheets of the sandwich composite are tabulated in Table 7 for the specimens numbered 1 and 3 (0.30 caliber impact) and 8 and 11 (0.50 caliber impact).

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a

New surface creation

c

b

18 mm

d

16 mm

= 10.24 cm2

301.4 J impact Top fa-

New surface creation

= 9.56 cm2

Fig. 6 Projected delamination of strike side S2-glass/epoxy composite facesheet for 0.50 caliber projectile impact (a), (c) experimental, (b), (d) corresponding numerical predictions

a Bottom facesheet

Impact energy: 233.18 J Delaminated region

b

35 mm

Fig. 7 Projected delamination at the back side facesheet for a 0.50 caliber projectile, impact energy 233.18 J (a) experimental, (b) corresponding numerical prediction

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a

Bottom facesheet

Impact energy: 301.4 J Delaminated area

b

36 mm

Fig. 8 Projected delamination of the bottom composite facesheet for a 0.50 caliber Projectile, impact energy 301.4 J (a) experimental, (b) corresponding numerical prediction

The results show the energy lost during penetration through each constituent i.e. strike side facesheet, balsa core and non-impact side facesheet of the sandwich plate. The projectile velocity lost during the penetration process at different time states for the sandwich specimens 1, 3, 8 and 11 is shown in Fig. 9. During the 0.30 caliber impact, the front composite facesheet absorbed an average of 36.16 J energy, while 121.3 J impact was absorbed by the front facesheet for the 0.50 projectile impact, Table 7. The average impact energy absorption by a three layer S2-glass/epoxy laminate for a 0.30 and 0.50 caliber projectile impact was 43.9 and 104.5 J, respectively [16, 23]. In our previous work [16] we evaluated the average energy absorption of a three layer composite laminate alone and it was found to be 43 and 108 J for 0.30 caliber and 0.50 caliber single projectile impact respectively, Table 8. In case of the sandwich composite, the laminate constitutes the strike side and non-impact side facesheets. In a sandwich construction, the balsa wood core restricts the flexure of the facesheets to a transverse impact. Hence the energy absorption of the strike side facesheet reduces to 33 J (17% lower than the constituent laminate) for 0.30 caliber impact due to the stiffer response of the facesheets. However, with 0.50 caliber impact projectile the energy absorption of the strike side facesheet was found to be 117 J (16% higher) than that of the constituent laminate. This can be attributed to localized crushing of more wood cell area .Š300%/ in the impact region which contributes to the resistance to penetration.

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Projectile velocity (m s–1)

Specimen # 11 250

Specimen # 8

200

Specimen # 3

Specimen # 1

150

100

50

0 0

0.1

0.2

0.3

0.4

Time (ms)

Fig. 9 Predicted projectile velocity lost versus time history plot for a single projectile impact to different S2-glass/epoxy balsa wood core sandwich specimens Table 8 Three layer laminate, simultaneous impact results for the steel (8.38 g) 0.50 caliber projectile Predicted Predicted average new New average Test surface Incident Residual Impact Energy surface energy config- velocity velocity energy absorpcreation creation absorpSpecimen uration (m s1 ) (m s1 ) (J) (cm2 ) (cm2 ) tion (J) tion (J) 1 A, B, C 222.1 160.7 206.8 295.8 270.6 2 A, B, C 222.1 172.2 206.8 247.4 NA 257 at Sd D 5 3 A, B, C 203 129.9 172.6 305.6 309.7 319 296 at Sd D 7 4 A, B, C 203 125.7 172.6 253.1 418 5 A, B, C 221.5 161 205.6 291.2 265.6

4.2 Simultaneous 0.30 and 0.50 Caliber Three Projectile Impact on the Sandwich Specimens Results from simultaneous three projectile impact to sandwich composite specimens are summarized in Table 9. The average incident velocity for a 0.30 caliber projectile was 392:46 m s1 .SD D 3:03 m s1 / and 327:42 m s1 .SD D 17:78 m s1 / for a 0.50 caliber projectile. The average impact energy absorbed by the sandwich specimens during 0.30 and 0.50 caliber impact events were found to be 350:2 J .SD D 30:45 J/ and 689:13 J .SD D 46:73 J/ respectively. Since the individual projectile velocity at impact locations A, B and C could not be measured experimentally

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Table 9 S2-glass/SC15 epoxy balsa core sandwich, three projectiles simultaneous impact results Predicted Average new Predicted average new surface average surface creation energy Test Incident Residual Impact Energy (cm2 ) creation (cm2 ) absorption config- velocity velocity energy absorpTop Bottom Specimen uration (m s1 ) (m s1 ) (J) tion (J) Top Bottom (J) 1 2 3 4

A, B, C A, B, C A, B, C A, B, C

389.5 391 392.8 396.5

182.2 228.9 174.95 205.4

464.2 467.8 472.13 481.17

362.7 307.5 378.5 352.1

5 6 7 8

A, B, C A, B, C A, B, C A, B, C

310.28 322.17 324.9 352.34

211.22 219.15 231.03 253.89

1210.16 1304.68 1326.89 1560.48

649.36 700.99 655.97 750.22

39

156

363.4

33

144

54

282

569

37

265

1,2,3,4 D specimens impacted by 0.30 caliber. 5,6,7,8 D specimens impacted by 0.50 caliber projectile.

a

b

38 mm

Fig. 10 Delamination interaction in the constituent S2-glass/epoxy laminate for a 0.50 caliber simultaneous impact at impact velocity, 212:3 m s1 (a) experiment (b) simulation damage for Sd D 7

for simultaneous impact (chronograph reads only one projectile speed) the total energy absorption by the plate was calculated by tripling the average energy absorption. The delamination followed a conical progression originating from the strike face for the 0.30 and 0.50 caliber impact study. The delamination zone near the non-impact side facesheet was spherical in shape and the area of the delamination was higher than at the strike side facesheet. In general, as projectile mass increased, the severity of damage increased. This was also noted for both the simultaneous and sequential impact studies [1]. Delamination growth near the strike side and the nonimpact side facesheet of the sandwich plate for a 0.50 caliber impact study was 38% and 81%, respectively. Unlike the 0.30 caliber simultaneous impact, delamination in 0.50 caliber simultaneous impact was found to extend across the clamped region. This was also the case for a 0.50 caliber simultaneous impact to the constituent laminate, Fig. 10.

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Projectile velocity lost (ms–1) Projectile energy lost (J)

a I II III 159 J Total energy absorption = 363.4 J 68.4J 136 J

C

A+B+C

B A

Time (ms)

Projectile velocity lost (ms–1) Projectile energy lost (J)

b

1500

I

II

III

175. 5 J 205 J

1000 188.5 J Total energy absorption = A+B+C 500

0 A

–500

245

327.4 Time (ms)

Fig. 11 Projectile energy and velocity lost versus time history prediction (a) 0.30 caliber simultaneous impact at impact velocity, 293 m s1 (b) 0.50 caliber simultaneous impact at impact velocity, 327:4 m s1 . I, II and II correspond to the energy absorption regime by the strike side facesheet, balsa wood core and the non-impact side facesheet, respectively

Figure 11 depicts the projectile kinetic energy and velocity lost versus time history during the projectile penetration. The average energy prediction for a 0.30 and 0.50 caliber simultaneous impact is 363.4 and 569 J which correspond to 93%

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and 83% of the experiment, respectively, Fig. 11. Three distinct energy absorption regimes, namely I, II and III were observed during projectile perforation for both 0.30 and 0.50 caliber impacts. I, II and III refer to the energy absorbed by the strike side facesheet, balsa wood core and non-impact side facesheet, respectively. It can be noted that balsa core absorbed more amount of energy (205 J) for the 0.50 caliber simultaneous impact compared to 0.30 caliber impact. In the case of 0.50 caliber impact, the three projectiles followed the same velocity profile with a common residual velocity of 245 m s1 , whereas the residual velocities of 0.30 cal projectiles were within 90% of each other. In addition, projectile penetration time in the core and the non-impact side facesheet was found be always higher (>200%) than the strike side facesheet penetration time. The impact event was transient for both 0.30 and 0.50 caliber impact. Multisite 0.30 Caliber Impact: Projected Delamination Area The projected delamination area on the strike side facesheet for the 0.30 caliber event was 39 cm2 .SD D 4 cm2 /, whereas delamination at the non-impact side facesheet (non-impacted face) was found be 156 cm2 .SD D 7 cm2 /. Figure 12 shows the delamination damage on the impacted face for 0.30 caliber simultaneous projectile impact of 464.2–481.2 J. The delamination propagating from each impact point did not completely interact with each other irrespective of the impact energy level. Delaminated area around each impact point was 14 cm2 .SD D 3 cm2 /. Some partial delamination interaction can be observed between A and C, but no interaction between A and B, or B and C. This can be attributed to the balsa core support and

a Top B 38 mm A

C

Impact energy: 464.2 J

b Top B 38 mm

Fig. 12 Delamination damage on the strike side facesheet of the sandwich specimen for 0.30 caliber projectile

A

C

Impact energy: 481.2 J

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the stress wave propagation along primary yarns. The impact points A and C lie on the primary yarns and hence, partial delamination interaction occurred due to longitudinal stress wave propagation along the primary yarns. The impact points A and B and C and B lie on the secondary yarns (along which no longitudinal stress wave propagation occurred) and there was no delamination interaction. Figure 13 illustrates delamination growth on the non-impact side facesheet for impact events of 464.2–481.2 J. The non-impact side facesheet delamination was observed to be 300% higher than that of the strike side facesheet delamination. FEA prediction for delamination at impact energy of 472 J is illustrated in Fig. 14. Although FEA prediction was almost within the 92% of the corresponding exper-

a Bottom

B

A

C

Impact energy: 464.2 J

b Bottom

B

A

Fig. 13 Delamination damage on the non-impact side facesheet of the sandwich specimen for 0.30 caliber projectile impact

C

Impact energy: 481.2 J

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Fig. 14 Delamination prediction showing 311% higher delaminated area on the non-impact side face sheet compared to the strike side facesheet of the sandwich specimen for 0.30 caliber simultaneous impact, impact velocity of 393 m s1

Top facesheet

38 mm

Bottom facesheet

38 mm

imental results, unlike the experiment, the FEA shows delamination interaction on the strike side facesheet for a 0.30 caliber impact. Multisite 0.50 Caliber Impact – Experiment and Simulation Delamination damage for 0.50 caliber simultaneous impact is shown in Figs. 15 and 16 for impact events of 649.36–750.22 J. The average delaminated area was 54 .SD D 8 cm2 / and 282 cm2 .SD D 12 cm2 / for the strike side and the non-impact side facesheet. There was complete delamination interaction between impact points A, B and C on the strike side facesheet for a 0.50 caliber impact events, Fig. 15a, b. This is due to higher momentum transfer to the strike side facesheet which results in higher individual delamination growth from each impact point. The non-impact side facesheet delamination was seen to be 422% higher than that of the strike side facesheet delamination, Fig. 16. Numerical predication of delamination for a 0.50 caliber projectile simultaneous impact is shown in Fig. 17. Delamination at the non-impact side facesheet is seen to propagate to the boundary in the case of the experiment and FEA prediction, Figs. 16–17.

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a Top

B 38 mm

A

C

Impact energy 649.36 J

b Top

B 38 mm

A

C

Impact energy 750.22 J

Fig. 15 Projected delamination in the strike face of the sandwich plate for a 0.50 caliber impact event

4.3 Delamination Factor, Sd for Multisite Impact Prediction The delamination factor Sd is a numerical value which controls (increases or decreases) the delamination area in simulation. The choice of the Sd parameter is iterative and is based on representative experimental measurements of the projected delamination area. It also depends on the mesh pattern (circular or cylindrical), outof-plane tensile strength and element size.

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a Bottom

B

C Fiber breakage

Impact energy 649.36 J

b Bottom

B

Fiber breakage

Impact energy 750.22 J

Fig. 16 Projected delamination damage at the non-impacted side of the sandwich plate for a 0.50 caliber simultaneous impact

Material model MAT 162 estimates the delamination within the elements adjacent to the ply interface if the parallel matrix failure criterion (1) [15–17, 35] is achieved. (     ) 23 2 31 2 h3 i 2 2 fdela min ation D Sd 1D0 (1) C C ZT S23 S31

564 Fig. 17 Projected delamination showing 600% higher delaminated area on bottom face sheet than the top facesheet of the sandwich plate for a 0.50 caliber simultaneous impact at impact velocity of 327:42 m s1

U.K. Vaidya and L.J. Deka Top facesheet

38 mm

Bottom facesheet

where ZT ; S23 and S31 are the failure strength properties and 3 ; 23 and 31 are corresponding stress state. A scale factor Sd is introduced to achieve better correlation of simulated delamination area with experimental values. In the present work, the Sd factor was determined iteratively for the constituents and the sandwich plate studies. In broad terms, a caliber impact events. An average delamination area for these was in the range of 70–300 cm2 . 0.50 Caliber Single Impact on Sandwich Panel For the sandwich composite impact the delamination on the strike side facesheet was smaller .21:46 cm2 / due to the core-to-facesheet interaction compared to that of the constituent laminate (which was 170 cm2 ). Hence a smaller value of Sd D 1 provided reasonable results to simulate the front face delamination. However the non-impact side facesheet delamination was 178% higher in comparison to the strike side facesheet. Hence a higher value of Sd D 5 gave a reasonable prediction for the delamination between the non-impact side facesheet and the core.

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0.30 and 0.50 Caliber Multi Impact Study on the Sandwich Panel Delamination prediction on the non-impact side facesheet is 300% and 600% higher than that of the strike side facesheet for 0.30 and 0.50 caliber impact respectively. Sd D 1 provided delamination prediction on the strike side facesheet reasonably well for 0.30 caliber simultaneous impact, but underestimated by 30% for 0.50 caliber simultaneous impact. This is because the 0.50 caliber imparts higher damage to the strike side facesheet; hence the Sd parameter was iteratively increased to 2 or 3 to match with the experimental results. A value of Sd D 05 resulted in the best match to the experiment (non-impact side face sheet delamination) for 0.30 and 0.50 caliber simultaneous impact on the sandwich specimens.

4.4 Fiber and Wood Damage In addition to delamination damage, there was fiber damage in the strike side facesheet and non-impact side face along with the localized crushing of the balsa core for a 0.30 caliber and 0.50 caliber impact, Figs. 12, 13, 15, 16. Figures 12,

a

b C

C

B

A

B

A

c

d

38 mm

38 mm

Damage fringes

Fig. 18 Localized balsa wood damage for a 0.30 and 0.50 caliber impact showing complete penetration (a), (b) cross-section, experiment (c, d) numerical prediction

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13 and 15 show no fiber breakage along the primary yarns or secondary yarns. Figure 16 reveals significant fiber debonding and pullout along the primary yarns. Impact points A and C experienced high strain to failure for a simultaneous impact. Balsa wood core damage was found to be completely localized at the three impact points for the 0.30 and 0.50 caliber impact events, Fig. 18. A cross-section of the balsa wood core reveals collapsing of the core cell walls along the direction of penetration for a simultaneous impact, Fig. 18a, b. No damage interaction was observed among the impact points. This was verified numerically as well. Figure 18c, d illustrate damage of the wood core for 0.30 and 0.50 caliber impact events. Damage fringes of the wood core as shown in Fig. 18c, d clearly indicate no damage interaction between impact points A, B and C, confirming localized erosion of the wood elements.

5 Summary and Conclusions The response of S2-glass/epoxy balsa core sandwich structures to high velocity impact has been investigated with the aid of experiments and finite element modeling. Progressive damage and delamination of composite facesheet has been modeled using LS-DYNA with material model MAT 162 which incorporates continuum damage mechanics of anisotropic materials. This work extends the study of the author’s previous work on multisite impact of constituent laminate, while the present work dealt with sandwich plates [16]. Therefore, mesh density gradient of the facesheets (the laminate) was kept similar to Ref. [16] to maintain consistency of progressive deformation and strain softening sensitivity of the facesheets during impact. Impact on the balsa wood core was simulated using MAT 2 and MAT 143. Although both material models showed good correlation of the impact damage of balsa wood with experiments, MAT 143 was found to perform better (within 92% of the corresponding experimental result) in the case of impact to both balsa wood and sandwich composite. Microscopic studies confirm a number of failure and energy absorbing mechanisms such as fiber breakage, fiber-matrix delamination, facesheet-core delamination, fiber splitting/debonding along the primary yarns and localized collapse of the balsa wood cells. Delamination at the non-impact side face was found to be 178% and 645% higher than the strike side face for 0.30 and 0.50 caliber impact, respectively. FEA using Sd factor of 1 and 5 for strike side and non-impact side facesheets agreed well with the experiment to predict kinetic energy absorption and delamination (98% and 95%, respectively). The sandwich composite specimens were subjected to a multisite impact using three 0.30 and 0.50 caliber projectiles simultaneously. Significant increase in projected damage ( 300–600%) was observed during simultaneous impact compared to single impact damage. With increasing number of impacts, momentum transfer to the sandwich specimens increases, hence more damage occurs on the strike side and non-impact side facesheets and in the balsa core as well. 0.50 caliber impact on the sandwich plate resulted in 49% higher energy absorption than the 0.30 caliber

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impact. Delamination damage growth on the strike side face sheet was restricted by the wood core in both 0.30 caliber and 0.50 caliber impacts, however non-impact side facesheet delamination extended to the boundary in the case of a 0.50 caliber impact. Hence, boundary effect or edge effect on the delamination growth on the non-impact side facesheet for a 0.50 caliber simultaneous impact was more as compared to 0.30 caliber simultaneous impact. Balsa core damage was observed to be localized. However numerical prediction shows that energy absorption by the balsa wood was effective in absorbing energy ( 200% increase) for a 0.50 caliber impact, compared to a 0.30 caliber impact. The engagement of more number of wood cells by a 0.50 caliber projectile resulted in extensive fiber pull out along the primary yarns on the non-impact side facesheets. No fiber pull out was observed for 0.30 cal impacts. Numerical verification of the multi-site impact study with MAT 162 (laminate) and MAT 143 (balsa core) revealed that FEA captured the main features of the impact phenomenon and predicted the different damage modes, i.e., matrix cracking and delamination, fiber breakage (erosion at the impact points) and wood crushing in close agreement with the experiment. Acknowledgement The support provided by Office of Naval Research (ONR), Program Manager – Dr. Yapa Rajapakse is gratefully acknowledged. We also acknowledge Dr. Frederick Just-Agosto and Dr. Basir Shafiq from the University of Puerto Rico, Mayaguez.

References 1. Arias A, Zaera R, Lopez-Puente J, Navarro C (2003) Numerical modelling of the impact behaviour of new particulate-loaded composite materials. Compos Struct 61: 151–159. 2. Shim VPW, Yap KY (1997) Modelling impact deformation of foam-plate sandwich systems. Int J Impact Eng 19: 615–636. 3. Wada A, Kawasaki T, Minoda Y, Kataoka A, Tashiro S, Fukuda H (2003) A method to measure shearing modulus of the foamed core for sandwich plates. Compos Struct 60: 385–390. 4. Lopatnikov SL, Gama BA, Haque MJ, Krauthauser C, Gillespie JW, G¨uden M, Hall IW (2003) Dynamics of metal foam deformation during Taylor cylinder–Hopkinson bar impact experiment. Compos Struct 61: 61–71. 5. Abrate S (1998) Impact on Composite Structures. Cambridge University Press, Cambridge. 6. Langlie S, Cheng WA (1989) High velocity impact penetration model for thick fiber reinforced composites. ASME, Pressure Vessels and Piping Division, 174, New York, 151–158. 7. Cantwell WJ, Morton J (1990) Impact perforation of carbon fiber reinforced plastic. Compos Sci Technol 38: 119–140. 8. Richardson MOW, Wisheart MJ (1996) Review of low velocity impact properties of composite materials. Compos 27(A): 1123–1131. 9. Villanueva GR, Cantwell WJ (2004) The high velocity impact response of composite and FMLreinforced sandwich structures. Compos Sci Techn 64: 35–54. 10. Abot JL, Yasmin A, Daniel IM (2001) Impact behavior of sandwich beams with various composite facesheets and balsa wood core. Proceedings of 2001 ASME Int Mech Eng Cong and Exp, November, 11–16. 11. Tagarielli VL, Deshpande VS, Fleck NA (2007) The dynamic response of composite sandwich beams to transverse impact. Int J Solids Struct 44: 2442–2457. 12. Yen Chian-Fong (2002) Ballistic impact modeling of composite materials. Proceedings of the 7th International LS-DYNA Users Conference, Detroit, Michigan, pp. 15–25.

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13. Chan S, Fawaz Z, Behdinan K, Amid R (2007) Ballistic limit prediction using a numerical model with progressive damage capability. Compos Struct 77: 466–474. 14. Brown K, Brooks R, Warrior N. (2005) Numerical simulation of damage in thermoplastic composite materials. Proceedings of the 5th European LS-DYNA Users Conference, Birmingham, UK, May 25–26. 15. Xiao JR, Gama BA, Gillespie Jr JW (2007) Progressive damage and delamination in plain weave S-2 glass/SC-15 composites under quasi-static punch-shear loading. Compos Struct 78(2): 82–196. 16. Deka LJ, Bartus SD, Vaidya UK (2009). Multi-site impact response of S2-glass/epoxy composite laminates. Compos Sci Technol 69(6): 725–735. 17. Deka LJ, Bartus SD, Vaidya UK (2008) Damage evolution and energy absorption of E-glass/polypropylene laminates subjected to ballistic impact. J Mater Sci 43: 4399–4410. 18. Deka LJ, Bartus SD, Vaidya UK (2007). Numerical modeling of simultaneous and sequential multi-site impact response of S2-glass/epoxy composite laminates. Composites and Polycon, American Composites Manufacturers Association, October 17–19, Tampa, FL, USA. 19. Bartus SD, Vaidya UK (2005). Performance of long fiber reinforced thermoplastics subjected transverse intermediate velocity blunt object impact. Compos Struct 67(3): 263–277. 20. Bartus SD, Vaidya UK (2007) Near-Simultaneous and Sequential Multi-Site Impact Response of S-2 Glass/Epoxy Laminates. International Conference on Composite Materials 16, July, pp. 8–13, Kyoto, Japan. 21. Bartus SD, Deka LJ, Vaidya UK (2006) Simultaneous and Sequential Multi-Site Impact Response of S-2 Glass/Epoxy Composite Laminates. SAMPE, 30 April–4 May, Long Beach, CA. 22. Bartus SD, Vaidya UK (2004) Fragment Cloud Impact Response of Carbon-Epoxy Plates. 45th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 19–22 April, Palm Springs, CA. 23. Bartus SD (2007) Simultaneous and sequential multi-site impact response of composite laminates. PhD Thesis. The University of Alabama at Birmingham. 24. Bartus SD, Vaidya UK (2007) Impact on composite structures – a review. J Adv Mat 39 (3): 3–21. 25. Hashin Z (1980) Failure criteria for unidirectional fiber composites. J of Appl Mech 47: 329–334. 26. Matzenmillar A, Lubliner J, Taylor RL (1995) A constitutive model for anisotropic damage in fiber composites. Mech of Mater 20: 125–152. 27. Johnson W (1986) Historical and present-day references concerning impact on wood. Int J Impact Eng 4: 161–174. 28. Johnson W (1986) Mostly on oak targets and 19th century naval gunnery. Int J Impact Eng 4: 175–183. 29. Reid SR, Peng C (1997) Dynamic uniaxial crushing of wood. Int J Impact Eng 19 (5–6): 531–570. 30. Buchar J, Rolc S, Lisy J, Schwengmeier J (2001) Model of the wood response to the high velocity of loading. 19th International Symposium of Ballistics, 7–11 May, Interlaken, Switzerland. 31. Vural M, Ravichandran G (2003) Dynamic response and energy dissipation characteristics of balsa wood: experiment and analysis. Int J Solid Struct 40: 2147–2170. 32. Tagarielli VL, Deshpande VS, Fleck NA, Chen C (2005) A constitutive model for transversely isotropic foams, and its application to the indentation of balsa wood. Int J Mech Sci 47: 666–686. 33. Murray YD, Reid JD, Faller RK, Bielenberg BW, Paulsen TJ (2005) Evaluation of LS-DYNA Wood Material Model 143, U.S. Department of Transportation, FHWA-HRT-04–096, August. 34. Standard Test Method for Density Determination for Powder Metallurgy (P/M) Materials containing less than Two Percent Porosity (1997) ASTM Designation B311–93, Annual Book of ASTM Standards, Vo1.2.05, ASTM, West Conshohocken, PA, 80–82. 35. LS-DYNA Theoretical Manual (2003) Version 970, Livermore Software Technology Corporation, April. 36. Baltekr SB Structural end-grain balsa, Data Sheet/Issue (2005) Replaces issue 02/05, Alkan Baltek Corporation, NJ, USA.

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37. Bekisli B, Grenestedt GL (2004) Experimental evaluation of a balsa sandwich core with improved shear properties. J Comp Sci Technol 64(5): 667–674. 38. Tabiei A, Wu J (2000) Three-dimensional non-linear orthotropic finite element material model for wood. Compos Struct 50:143–149. 39. Foschi RO (1974) Load-slip characteristics of nails. Wood Sci l7: 69–77. 40. Davalos-Sotelo R, Pellicane PJ (1992) Bolted connections in wood under bending/tension loading. J Struct Eng ASCE 118, 999–1013. 41. Patton-Mallory M, Smith FW, Pellicane PJ (1998) Modeling bolted connections in wood: a three-dimensional finite-element approach. J Testing Eval JTEVA 26: 115–124.

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Real-Time Experimental Investigation on Dynamic Failure of Sandwich Structures and Layered Materials L. Roy Xu and Ares J. Rosakis

Abstract We present a systematic experimental investigation of the generation and subsequent evolution of dynamic failure modes in sandwich structures and layered materials subjected to out-of-plane low-speed impact. Model sandwich specimens involving a compliant polymer core sandwiched between two metal layers and other model layered materials were designed to simulate failure evolution mechanisms in real sandwich structures and layered materials. High-speed photography and dynamic photoelasticity were utilized to study the nature and sequence of such failure modes. In all cases, inter-layer (interfacial) cracks appeared first. These cracks were shear-dominated and were often intersonic even under moderate impact speeds. The transition from inter-layer crack growth to intra-layer crack formation was also observed. The shear inter-layer cracks kinked into the core layer, propagated as opening-dominated intra-layer cracks and eventually branched as they attained high enough growth speeds causing brittle core fragmentation. In-depth failure mechanics experiments on the dynamic crack branching, crack kinking and penetration at a weak interface, interfacial debonding ahead of a main incident crack were also conducted to understand the physical insight of the dynamic failure modes and their transition observed from sandwich structures and layered materials.

1 Introduction Layered materials and sandwich structures have diverse and technologically interesting applications in many areas of engineering. These include the increased use of composite laminates in aerospace and automotive engineering; the introduction L.R. Xu () Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA e-mail: [email protected] A.J. Rosakis Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA e-mail: [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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of layered concrete pavements in civil engineering; the use of thin films and layered structures in micro-electronic components, and recently, the introduction of sandwich structures in a variety of naval engineering applications (Hutchinson and Suo [1]; Rajapakse [2]). While failure characteristics of layered materials and sandwich structures subjected to static loading have been investigated extensively in the past years, their dynamic counterparts have remained elusive (Sun and Rechak [3]; Cantwell and Morton [4]; Abrate [5]). Indeed, the presence of highly complex and transient dynamic failure modes in such materials and the inaccessibility of internal damage have resulted in experimental studies limited to only the final impact damage and the residual strengths. To begin addressing the need for real-time observations of failure events, the work presented here focuses on the study of such events in model sandwich structures, and in particular, on the identification of their nature, chronological evolution and interaction. To identify the evolution of failure modes, it is convenient to first classify these modes based on the material constitutions of layered/reinforced structures. As shown schematically in Fig. 1, there are two major categories of failure observed in post-mortem studies. The first major failure category is decohesion (or cracking) between bonded layers at an interface. This is often referred to as delamination in composite laminates or interfacial debonding in thin films or sandwich structures. It is also called inter-layer failure. Generally, two distinct inter-layer failure modes are observed. The first one involves opening-dominated inter-layer cracking or “delamination buckling” (Gioia and Ortiz [6]; Kadomateas [7]). The second one involves shear-dominated inter-layer cracks or “shear delaminations,” and often occurs in layered materials subjected to out-of-plane impact (Choi et al. [8]; Lambros and Rosakis [9]). The second major category is referred to as intra-layer failure. There are three possible intra-layer failure modes depending on the material constitution. The first

Inter-layer failure

1. Opening dominated inter-layer crack 2. Shear dominated inter-layer crack 3. Intra-layer crack (matrix crack)

4. Fiber breakage Intra-layer failure

5.fiber/matrix debonding

Fig. 1 Failure modes for layered materials based on material constitution

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Kinked matrix cracks Delamination

56 m/s

Field of view

100 μs

375 μs

492 μs

667 μs

Fig. 2 A series of back lit photos showing the dynamic failure process for a fiberglass face plates/ PVC foam core sandwich structure

one is called intra-layer cracking or matrix cracking as seen in Fig. 2. This type of cracking often occurs inside the matrix of fiber-reinforced composites or within the soft core of sandwich structures [10]. It is also found in the form of tunneling cracks in thin film/substrate structures [1]. Another possible intra-layer failure is the failure of reinforcements such as fiber breakage and fiber kinking within a layer (Oguni et al. [11]). The third intra-layer failure mode is interfacial debonding between the matrix material and the reinforcement such as debonding between particle/fiber and matrix occurring within a constituent layer [12]. Indeed, the sequence, nature and interaction of these dynamic failure modes were never properly clarified. In perhaps the first attempt to visualize impact failure in sandwich structures used in Naval applications, Semenski and Rosakis [13] tested thin sections of such plate structures composed of PVC foam cores, sandwiched between E-glass faceplates. A pulsed laser was used to illuminate the specimens from the back side and a high-speed camera recorded the deformation and failure events. A sequence of photographs corresponding to this process is shown in Fig. 2 together with the postmortem picture of the recorded specimen. As evident from the post-mortem picture, there are, at least, two types of failure present. Inter-layer failure demonstrates itself in the form of delamination between the face plates and the foam core at the vicinity of the impact site and free edges. On the opposite side, delamination is evident only on the top and on the bottom part of the specimen. Intra-layer failure in the form of mode I, opening cracks in the soft core is also observed forming a highly

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symmetric pattern. Because the core is opaque, the high-speed pictures shown below are of limited use. What they show, however, is the emergence and propagation of the opening intra-layer (matrix) cracks inside the foam core. Indeed, these cracks seem to originate at the fiber glass/PVC interface opposite to the side of impact, and to symmetrically propagate towards the impact point. These cracks originate at the same location where the fiberglass/PVC delamination terminates. However, the time sequence and interaction between such inter-layer delamination and the visible intra-layer, opening cracks in the core are not obvious. Indeed, the backlit real-time photographs do not show any evidence of interfacial delamination within the time window of observation. As we will show later, this observation is misleading and is due to the fact that inter-layer fractures are typically shear-dominated. As such they do not allow for light to go through during the recording event because the shear crack faces remain in contact at the early stages of this process. The inability of back-lit photography to visualize the failure process completely, motivates the use of partially transparent model sandwich systems which allow the use of full field optical techniques capable of capturing the nucleation and growth of both opening and shear-dominated cracks and their transition from one mode to the other [14]. For many complex engineering problems, model experiments may prove extremely useful as intermediate steps, which reveal the basic physics of the problem and provide relatively straightforward explanations of the failure patterns observed in post-mortem observations. A successful approach was adopted by Walter and Ravichandran [15], who designed a model aluminum/PMMA/aluminum specimen to simulate and visualize the static debonding and matrix cracking process in ceramic matrix composites. In our experiments, we also adopt the same idea and introduce an appropriate intermediate model configuration. In order to simulate the difficult three-dimensional problem of the out-of-plane impact of real sandwich structures and to simultaneously preserve the essence of the failure phenomena involved, we introduce a two-dimensional, plane stress specimen, which represents a cross-sectional cut of the layered material as illustrated in Fig. 3.

Delamination

Fig. 3 Model layered specimens are idealized cross sections of real structure subjected to out-of-plane impact

Matrix cracking Metal Polymer

Transition points

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For this type of model specimen, the failure process is easy to record, visualize and analyze. It is noted that although the exact impact mechanics involved in two configurations is not identical (the real case is three-dimensional while the model specimen is closer to a plane stress state), the mechanisms of stress wave propagation and failure progression of the real and the model layered materials are quite analogous. In designing these model two-dimensional sandwich specimens, it is important to select model materials whose elastic mismatch is similar to that of materials used in real engineering applications (in our case PVC/composites). Selecting similar Dundurs’ parameters [1] may ensure similarity of the elasto-static response for the interfacial mechanics problem. Meanwhile, selecting model material combinations with similar ratios of wave speeds of two constitution materials to the real structure is perhaps the most important consideration in the dynamic case, where timing of events and stress intensity are governed by the constituent material wave speeds. Also, the ratio of inter-layer and intra-layer strengths (or fracture toughnesses) is important. These three issues provide sets of similarity rules to connect the real structures to our models experiments. As schematically shown in Fig. 3, matrix cracks and delamination are the two major impact failure modes in sandwich structures and layered composites. At some intersection points, matrix cracks and delamination are connected as also seen in the post-impact picture of Fig. 2. One frequently asked question in the literature is whether the matrix cracks lead to the delamination or the delamination happens first and subsequently kinks into the adjacent layer inducing the matrix crack. This is a typical problem of sequence and failure mode transition identification. Since the nature and origin of such failure mechanisms can only be theorized by post-mortem observations, the necessity of full-field real-time, high-speed measurements becomes obvious. To this effect, the objectives of the current work are to conduct systematic experimental studies of the time evolution and nature of different dynamic failure modes and to investigate their interactions. Through these model experiments, we try to identify the basic physical phenomena, and to provide guidance for theoretical models and much needed, real-time, validation of numerical codes. To make this comparison more meaningful, we choose model material combinations that have the ratios of wave speeds very close to those used in real sandwich structures.

2 Experimental Procedure 2.1 Materials and Specimens Two kinds of materials were used in the experiments described below. A 4340carbon steel was employed to simulate the stiff and strong fiberglass faceplates of sandwich structures. The polymeric material, which was used to simulate the weak core layer, such as the PVC foam core or balsa wood in sandwich structures or

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the 90ı plies in cross-ply laminates, is Homalite-100. The adhesive used to bond the metal/polymer interface is Weldon-10. The detailed properties of this adhesive and the effect of interfacial strength variation on dynamic failure mode selection are reported by Xu and Rosakis [16, 17]. The shear wave speed is an important parameter in this investigation. The shear wave speed ratio for the core and faceplate is 3.2 for typical E-glass/PVC sandwich structures of the type that have recently been used in construction of full-scale composite ships. Details on the complete set of physical and constitutive properties for E-glass composite materials have recently been discussed by Oguni et al. [11]. For comparison, the same shear wave speed ratio, for the idealized steel/Homalite model sandwiches, is about 2.7 based on the data from Ref. [14]. Although the absolute values of these constituent properties are very different in the “idealized” versus the “real” solids, the idealized material combinations have been chosen in such a way as to have a shear wave speed ratio that is very similar to its real sandwich counterparts. As shown in Fig. 4, three different types of model sandwich specimen geometries were designed and tested. Type A specimens have equal layer widths and involve two different materials. They contain two metal layers with one polymer layer sandwiched between them. Type B specimens involve two thin metal layers (faceplates) and one polymer layer. This type of specimens is quite similar in geometry (ratio of core to face plate thickness) to realistic sandwich plates used in engineering applications. The only difference between type C and type B specimens is their lengths. Type C specimens are twice as long as type B specimens. The purpose of type C specimens is to explore the impact failure patterns with least edge effect present in the time scale of the failure process. All three types of specimens have the same out-of-plane thickness of 6.35 mm.

a 254 mm

b 254 mm 38mm 38mm 38mm

c 508 mm 6 mm 38 mm 6 mm

Fig. 4 Model specimens simulating sandwich structures (shaded layers – metals; transparent layers – polymers)

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2.2 Experimental Setup The majority of experiments in this investigation were performed using dynamic photoelasticity. This classical method has recently found a lot of new applications such as study of the dynamic fracture processes in functionally gradient materials (FGMs) [18]. The Coherent Gradient Sensing (CGS) method [19] was also used in a small number of cases. A schematic of the dynamic photoelasticity setup used here is given in Fig. 5. Two circular polarizer sheets were placed on either side of the specimen. A coherent, monochromatic, plane polarized laser light output was collimated to a circular beam of 100 mm in diameter. The laser beam was transmitted through the specimen and the resulting fringe pattern was recorded by the highspeed camera. A Cordin model 330A rotating-mirror type high-speed film camera was used to record the images. During the impact test, a projectile was fired by the gas gun and impacted the specimen center. The generation of isochromatic fringe patterns is governed by the stress-optic law. For the case of monochromatic light, the condition for the formation of fringes can be expressed as [20]: O 1  O 2 D

Nf h

(1)

where O 1  O 2 is the principal stress difference of the thickness averaged stress tensor. f is the material fringe value, N is the isochromatic fringe order and h is the specimen thickness. The isochromatic fringe patterns observed are proportional to contours of constant maximum in-plane shear stress, Omax D .O 1  O 2 /=2.

Laser

Gas Gun Lens

Polarizer 1

High Speed Camera Specimen Polarizer 2 X2

X1

X3

Fig. 5 Schematic of the dynamic photoelasticity and high-speed photography setup

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3 Results and Discussion 3.1 Failure Process of Short Model Sandwich Specimens with Equal Layer Widths The diameter of the laser beam was 100 mm; however, the maximum length of the zone that had to be investigated was 254 mm long. In order to observe all possible dynamic failure modes present in each case, the field of view had to be moved from one location to another for each specimen configuration under the same impact condition. Figure 6 presents a series of photoelastic images of the Homalite core layer of a type A specimen. The dark circular spot at the upper right corner is a scaling mark (diameter 6.35 mm) bonded on the specimen. The thin horizontal dark line, seen around the center of every image, is the streak line of the camera. This line provides a stationary reference when the whole specimen moves during the impact process. At first, the field of view was centered on the middle of the specimen because it was close to the impact position and failure was expected to initiate from this zone. As shown in Fig. 6b, about 158 s after impact, two inter-layer cracks at the lower interface entered the field of view and propagated towards the specimen center. Later on (around 182 s), the two inter-layer cracks, identified by the moving concentration of fringes at their tips, are seen to meet each other in Fig. 6d. Similar to shear-dominated interfacial cracks in bimaterials [21], those inter-layer cracks are also shear-dominated. Because the Homalite and steel layers are still in contact up to that time, no visual evidence of decohesion is apparent in the images, although these cracks have already broken the interface in a combination of compression and shear. After these two inter-layer cracks meet at the center, a bright gap between the Homalite and steel layers can be seen to appear in Fig. 6e. Along this clearly opened interface and on the Homalite side, two Rayleigh surface waves are now seen to propagate, originating from the center and moving outwards, towards the specimen edges. Perhaps the most interesting conclusion deriving from the sequence shown in Fig. 6 is the fact that delamination did not initiate in the interface directly above the impact point but did so outside our central field of view at two symmetric, off-axis, locations along the lower interface. In order to discover the location of crack nucleation, we must move our field of view off the specimen center to investigate the origins of these inter-layer cracks. In order to conclusively identify the origins of the upper and lower inter-layer cracks, the field of view was once more moved to the specimen edge as shown in Fig. 7a. After impact at the specimen center, the stress waves in the bottom steel layer propagated towards the edge creating a visible head wave structure on the lower wave speed polymer side (see Fig. 7b). Right after the stress wave reached the free edge, due to the existence of a stress singularity at the bimaterial corner [22,23], an inter-layer crack initiated at the lower interface as seen in Fig. 7c. This crack propagated towards the specimen center. After around 160 s, another inter-layer crack initiated at the upper interface also moving towards the center. This upper inter-layer crack soon kinked into the core layer and branched into a fan of multiple mode-I intra-layer cracks. Crack branching as reported by

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b a Field of view

Steel Polymer Steel

c

V=33 m/s

d

Inter-layer cracks

e

Rayleigh surface wave Fig. 6 Early stages of the failure process of a three-layer specimen with equal layer widths. The central field of view reveals the early occurrence of shear-dominated delamination at the lower interface. At later times (Fig. 6e), the debonding becomes clearly visible at the lower interface

previous researchers often initiates when a crack in a homogeneous solid reaches high fractions of the shear wave speed [24], for example, 30–40% shear wave speed of Homalite-100. Based on experimental observations from different fields of view, the major dynamic failure modes and sequence in model three-layer materials can be summarized in Fig. 8. After the stress wave reaches the free edges, two shear-dominated inter-layer cracks initiate and propagate towards the specimen center. These shear cracks separate the whole lower interface and a Rayleigh surface wave forms on the

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b a Field of view

Homalite Steel V=33 m/s

interfacial crack initiation

c

d

e

f

Fig. 7 Edge view of damage evolution: delaminations are shown to form at the intersection of, first lower, and then upper interfaces with the specimen edge. A fan of kinked intra-layer cracks branches into the core layer from the upper interface

separated free surface. At a later stage, inter-layer cracks also originate from the upper interface at the free edge and travel towards the specimen center. However, these upper inter-layer cracks soon kink into the core layer to form opening-dominated intra-layer cracks. Under certain circumstances (e.g., if the core material is very brittle), such kinked cracks may also branch into a fan of multiple branches fragmenting the core. The model experiments described here seem to capture the basic nature of the post-mortem impact failure modes observed in real sandwich structures. Indeed, the kinked matrix crack of the core layer of the glass fiber/foam core sandwich shown in Fig. 8e seems to follow the same initiation and propagation process as the kinked intra-layer crack in the model three-layer specimens schematically shown in Fig. 8d.

3.2 Failure Process in Long Model Sandwich Specimens In type A specimens, inter-layer cracks always initiated from the specimen free edges due to the bi-material stress concentration at such locations. In order to study

Dynamic Failure of Sandwich Structures and Layered Materials

a

Lower interfacial debonding

b

c

Upper interfacial debonding

d

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Completely debonding

Crack kinking

Crack branching

Rayleigh wave traveling on the opening interface

e

Fig. 8 Conceptual summary of typical failure modes and sequence (a–d) in a short three-layer specimen with equal layer widths. (e) Is a comparison with a post-impact picture of a real sandwich specimen

the impact damage modes and failure sequence in either very large structures or ones that are clamped along the edges, our model specimens featured long specimens. These specimens were long enough such that any damage from the edges, such as inter-layer cracks induced by the edge effect, arrived in the area of observation long after the local damage sequence had been completed near the impact point. To study this effect, we tested long sandwich style specimens of type C (see Fig. 9). As shown in Fig. 9d, at 79 s after impact, an inter-layer crack tip is seen at the lower interface (fringe concentration within the dashed circle). This crack is similar in nature to our previously observed inter-layer fractures but has not originated at the specimen free edge, which for type C specimens is far away from our field of view. Indeed, if this crack originated from the specimen free edge, it would take at least 150 s to enter the field of view. Closer scrutiny reveals that this crack originates from a much closer location to the impact point. This location is marked here by the circle in Fig. 9c within which a concentration of photoelastic fringes points to the concentration of shear stresses that is responsible for its nucleation. Indeed the crack nucleates at a location where the inter-layer shear stress reaches a local maximum, whose value equals the shear strength of the bond. To rationalize this, one should consider the symmetry of our impact configuration and recall the strong wave speed mismatch between the lower faceplate and the core material. The shear stress component ¢12 at the specimen centerline will always vanish because of this symmetry but is expected to anti-symmetrically increase away from the centerline as compressive waves begin to spread along the steel faceplate. As a result of these

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b a Field of view

V=20 m/s

c

d

e

f

Fig. 9 Nucleation of an intersonic inter-layer crack at the vicinity of the impact area of a long model sandwich specimen

observations, the scenario that seems to be emerging is as follows: shear-dominated cracks are generated at two points to the right and left of the center line and more backwards towards the impact point. A series of photographs confirming the existence of an inter-layer crack coming from the right-hand side of the impact point is shown in Fig. 9. Indeed Fig. 9c corresponds to the nucleation of this crack while Fig. 9d–f confirm its high-speed motion towards the impact site. As this inter-layer crack and its symmetric companion from the left meet above the impact point, they create a central shear delamination between the core and the bottom faceplate. The speed of this crack is very high as evident from the shear shock wave that appears as a dark inclined line radiating from its moving tip (Fig. 9e, f). Figure 10 corresponds to another impact experiment which featured the same load condition as shown in Fig. 9. The end point of the central delamination

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a C Propagation direction of the intra-layer crack

Initiation location of the inter-layer crack

O

A

V=19 m/s

B

No debonding

b

Fig. 10 Local view of the post-mortem damage in a long sandwich specimen (a) and high-speed snap shot capturing the formation of intra-layer cracks (b)

described above is denoted by A. Yu et al. [25] show that the inter-layer normal stresses near the impact point are compressive while the inter-layer shear stresses exhibit two clear peak values (same magnitude) moving symmetrically away from the centerline. Point A in Fig. 10 is corresponding to the old location of the maximum inter-layer shear stress. This point can now acts as a stress concentration from which further damage (to the core as well as to the rest of the interface) will subsequently evolve. Indeed as seen in Fig. 10, intra-layer cracks now are generated and propagated into the core (along AC), also accompanied by a new inter-layer debond (along AB) also originating at point A. The high-speed snapshot that appears in the same photograph confirms this scenario. Figure 11 summarizes the proposed failure evolution sequence for the long sandwich style specimens described above. One point that should be made clear here is that following the formation of the central (shear) delamination, the choice of the inclination angle “ and the possibility of further delamination along the bottom interface depend on the impact speed and on the relative values of the matrix material and interfacial bond strengths. The same is true for the exact locations of points A and B. However, we expect that if impact speeds are high enough to promote this localized failure mode, the general features described here will continue appearing even as the projectile speed increases further. For the initiation of intra-layer cracks (matrix cracks), previous researchers theorized that such cracks initiated from the center of the weak layer and propagated toward adjacent interfaces to lead to inter-layer cracks or delaminations [26]. However, no real-time experimental evidence was

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a

b Central shear delamination A

Intra-layer crack formation

B

c

d b

Fig. 11 Failure sequence observed in long sandwich-style specimens with minimal edge effects

ever observed to support such a scenario. Here, we clearly show that the intra-layer cracks always initiate at the interfaces following the shear-dominated delamination, which kinks into the core layer resulting in intra-layer cracking.

3.3 Effect of Impact Speeds In Section 3.1, we investigated the nature and sequence of failure mechanisms in relation to model sandwich specimens subjected to an impact speed of 33 m/s, as seen in Fig. 6. This impact speed situation will be taken as the baseline for our comparisons. Figure 12 describes an experiment of the same geometry that corresponds to an impact speed is 45 m/s [16]. As seen from Fig. 12b, two inter-layer cracks appeared at the lower interface and propagated towards the center, racing towards each other with intersonic speeds. At a later time, inter-layer cracks at the upper interface also appeared propagating towards the center (Fig. 12c). The locations of these four inter-layer cracks (two at the top and two at the lower interfaces) are indicated by the white arrows. As clearly seen from Fig. 12e, intra-layer damage also spreads from the interface in to the core in the form of a periodic series of mode-I cracks inclined at a small angle to the vertical axis. These cracks are nucleated at the upper interface at locations that are behind the horizontally moving inter-layer shear crack. Their nucleation and growth result in the eventual fragmentation of the specimen core. The inter-layer cracks propagating at the lower bimaterial interface and facing towards each other in Fig. 12d–e feature clearly formed shock-like or Mach-like discontinuities (shear shock waves) which are emitted from their crack tips. These discontinuities in photoelastic patterns represent traveling discontinuities

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b a Field of view Homalite Steel

c

V=45 m / s

d

e

Shock waves Fig. 12 Growth of four inter-layer cracks at the center of a three-layer specimen

in maximum shear stress and are clear proofs that crack tips have exceeded the shear wave speed of Homalite [19]. These shock waves formed a clear testimony to the intersonic nature of the inter-layer crack growth even before any detailed crack measurement was ever attempted. The crack speed history for the lower, right interlayer crack is plotted in Fig. 13 as a function of distance from the free edge. The figure shows that the crack speed of the higher impact speed case (45 m/s) is always higher than the baseline equivalent remaining always intersonic within the window of observation.

3.4 Dynamic Failure Mode Transition From the above experimental observation, we find that different dynamic failure modes and their transition are very interesting. As seen in Fig. 11b, an interface

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Crack speed (m/s)

1800 V=45m/s

1600

V=33m/s

1400 1200 Cs 1000 3lshssbwd-8 & 6 Lower inter-layer crack 800 100

110

120

130

Distance from the free edge (mm)

Fig. 13 Comparison of crack speed distributions of two identical specimens subjected to different impact speeds. The dash line is the dynamic shear wave speed of the Homalite-100

a

Crack kink

b

Crack penetration

c

Early debonding

Material 2 Material 1 Incident crack

Incident crack

Incident crack

Fig. 14 Common failure modes when a crack encounters an interface

crack kinks from the interface to form a matrix or intra-layer crack. While this matrix/intra-layer will deflect at the second interface to form a inter-layer crack again. Generally, when a crack propagates in elastic solids and encounters an interface, one of the three situations may occur as seen in Fig. 14: (a) after the crack reaches the interface, it kinks out of its original path and continues to propagate along the interface [27]. This phenomenon is often called “crack kinking or deflection” [28]; (b) the crack penetrates the interface and continues to propagate along its original path, i.e., crack penetration [29, 30]; (c) early interface debonding initiates before the incident crack reaches the interface, or it refers to the “Cook-Gordon mechanism” [31–33]. In the open literature, efforts have been primarily focused on analyzing the first two cases, crack kinking and crack penetration [1]. The energy release rate ratios of the incident and kinked interfacial cracks, and the fracture toughness ratios of the matrix material and the interface are identified as major parameters to govern crack deflection/penetration [34]. Recently, Xu et al. [30] experimentally and analytically studied the dynamic crack deflection/penetration phenomena. They also presented an energy-based

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criterion to investigate the competition between the dynamic crack penetration and deflection. The criterion for interfacial dynamic crack kinking is a ratio comparison between two energy release rates for the deflected/kinked crack along the interface, and the incident crack inside the matrix (driving force) and the fracture toughness values (material resistance): G d .ˇ; v2 / cIT .v2 /  MA P GI .v1 / Id .v1 /

(2)

where ˇ is the crack kinking angle or interface angle, and v1 and v2 are the incident parent crack speed and the kinked daughter crack speed respectively (Freund [40]). MA Id .v1 / is the dynamic fracture toughness of the matrix material and CI T .v2 / is the interfacial fracture toughness. If the left side is less than the right side, a dynamic crack penetration will occur. However, in order to apply the energy-based criterion, putative crack deflection and crack penetration lengths are needed to evaluate the corresponding energy release rates. Several researchers [29, 35] have demonstrated that the two putative lengths have a significant effect on the energy release rate ratios, and sometimes the energy-based criterion fails to predict the crack deflection or crack penetration. For these cases, “Cook-Gordon mechanism” provides an alternative explanation since a crack may not kink right after it reaches the interface as shown in Fig. 14a. The case (interface debonding before kinking) shown in Fig. 14c is quite possible.

3.5 Dynamic Interface Debonding Ahead of a Main Incident Crack In Fig. 14c, correlations of the fracture mechanics parameters of the kinked interfacial crack and the incident crack are not easy to obtain. Therefore, in order to model the “Cook-Gordon mechanism”, we tend to use strength-based criteria to predict interfacial debonding initiation only (rather than crack growth) induced by an incident crack. In terms of the dynamic “Cook-Gordon mechanism”, only Needleman and co-workers [36–38] have simulated this problem using a cohesive element model. In their model, an artificial initial flaw was introduced so they assumed some material properties for predictions. In our investigation, a strength criterion with direct interfacial strength measurements will be used to predict the critical distance rc of the incident crack tip to the intersection point of the incident crack path and the interface. Indeed, our work will be complementary to Needleman’s work, since our work aims to predict interfacial debonding initiation, while their efforts were focused on simulating the late interfacial crack propagation after crack initiation. Figure 15 shows a series of photoelasticity snap shots following impact of a bonded Homalite specimen [33]. In all experiments, the projectile impacted the center of the bottom layer on a steel buffer as shown in Fig. 15a. Figure 15b shows a fan of mode I cracks (symmetric fringe patterns) appearing from the upper free edge

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a

b Field of view Interface

V=33 m/s

c

d

e

f

Fig. 15 Dynamic failure process in a two-layer specimen showing the interaction of a fan of mode I incident cracks and the resulting interfacial crack. The thin horizontal line is the weak interface

at approximately 93:8 s after impact. After impact, the longitudinal compressive stress wave traveled from the lower impact side towards the upper free edge. This compressive stress wave reflected from this edge as a tensile wave and its intensity was sufficient to nucleate a fan of branched cracks from the free edge. As time goes on (Fig. 15c–f), the nucleated fan of cracks widens significantly by producing a multiplicity of both successful and unsuccessful branches, some of which move towards the still coherent interface (for a discussion of crack branching phenomenon in bulk Homalite, see Ravi-Chandar and Knauss [24]. The average speed

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of these locally mode I, branched cracks is 0:41 CS , which is the branching speed in bulk Homalite. Well before the branched cracks reached the interface, a central inter-layer crack was nucleated at the intersection of the specimen center line and the bond line as seen in Fig. 15c. This interfacial crack propagated in both directions off the center as shown in Fig. 15d. At the specimen centerline, the shear stresses vanish because of symmetry. As a result, the nucleated inter-layer crack is initially and, for a very short time, mode I dominated. Its nucleation is induced by the stress field produced by the fan of branched cracks approaching the interface. As this crack spreads symmetrically, opening up the interface (see distinct evidence of decohesion in Fig. 15f, the fan of branched cracks decelerates and arrests just before these cracks reach the decohered interface. The above described scenario is perhaps the first real-time visualization of the dynamic equivalent of the “Cook-Gordon Mechanism” describing the remote decohesion of an interface due to the approach of a matrix (intra-layer) crack. As the interfacial crack spreads away from the specimen centerline, it almost immediately encounters increasing amount of interfacial shear stress, which quickly converts it to a mixed-mode and eventually to a mode II dominated crack. Unlike propagating cracks in bulk Homalite, interfacial cracks are constrained to propagate along the weak interface and, as a result, they can do so under mixed-mode or primarily mode II conditions. They can also propagate at very high (even intersonic) speeds compared to their bulk (intra-layer) counterparts. In order to predict the dynamic crack initiation, Wang and Xu [27] proposed a strength criterion 2  0 2  0 =s D 1 (3) 22 =t C 12 0 0 0 ; 12 ; 22 are stresses where t ; s are the interface tensile and shear strengths, 11 acting on the interface which are induced by an incident dynamic crack close to the interface. The stress field of a steady mode I crack inside a linear elastic solid is given by a well-known form [39, 40]:

KI .t / I †ij . ; v/ C T ıi1 ıj1 C O.1/ ijI D p 2 r

(4)

where KI .t / is the dynamic stress intensity factor of the mode I crack as a function of time t ;T is a non-singular term, which is called “the T-stress” or ¢ox [20]; O.1/ represents higher order terms; the functions †Iij . ; v/ that represent the angular variation of stress components for an instantaneous crack tip speed v. For the kinked/deflected interface crack, it is a mixed-mode crack with a stress field: ij D ijI C ijII KI .t / XI KII .t / XII D p . ; V / C T ıi1 ıj1 C p . ; V / C O.1/ ij ij 2 r 2 r

(5)

where KII .t / is the dynamic stress intensity factor of the mode II crack as a function of time t and the functions †IIij . ; v/ represents the angular variation of stress components for an instantaneous crack tip speed v.

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3.6 New Progress on Dynamic Crack Branching As observed in the above dynamic failure experiments, we find that dynamic crack branching is an important dynamic failure mode for brittle materials, and for ductile materials subjected to certain conditions. This phenomenon has received extensive attention in the past decades. However, previous major efforts have primarily focused on analytical and numerical studies [41–43]. Very few new experimental results were available to verify predictions or to provide guidance for modeling [24, 44]. Some important issues, such as the crack speed change before and after branching, effect of dynamic loading rate on the crack branching, and crack branching induced by stress wave loading are still open. Therefore, we try to focus on how to realize various forms of dynamic crack branching under a variety of conditions [45]. The objectives are to elucidate a series of new phenomena; and to provide guidance for developing theoretical models and validating numerical simulations.

3.6.1

Special Experiments for Dynamic Crack Kinking and Branching

Two kinds of polymeric materials were used in conjunction with two kinds of optical diagnostic techniques. Homalite-100 was chosen for the photoelasticity experiments while PMMA was used in the Coherent Gradient Sensing experiments. Various types of specimens were designed and some of them had pre-notches with different radii. One major specimen used in this investigation was a novel wedge-loaded specimen, which was designed to produce a single, straight dynamic crack as shown in Fig. 16a. An aluminum wedge was inserted into a pre-notch and impacted by a projectile, causing the wedge to open the notch faces thus producing a single mode I crack [30]. The notch tip was cut using a diamond afering blade (Buehler, Series 15 LC). A strain gauge was bonded onto the wedge to trigger the high-speed camera and laser system.

3.6.2

Dynamic Crack Branching and Kinking from a Weak Interface

As shown in Fig. 16, the in-plane Homalite specimen dimensions were 457 mm long, 254 mm wide and the plate thickness was 9.5 mm. In this photoelasticity experiment, the initial notch radius was 0.127 mm (0.005 in.). For a low impact speed, V D 19 m=s, only a pure mode I crack initiated from the notch. As seen in Fig. 16b, we created an artificial interface (an inclined thin line) in front of the horizontally propagating crack. The incident mode I crack approached the interface at about 151 s after impact and transitioned into a mixed-mode interfacial crack. A vertical line appearing in every image is the camera streak line, which was used for positioning and reference purposes. At approximately 177 s, this mixed-mode interfacial crack kinked into the right side of the interface. A significant caustics (or shadow spot) is seen in Fig. 16d to show the mode I nature of the kinked crack. The speed of the kinked crack was

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b a Wedge

V=19m/s

Notch

Field of view

c

d

e

f

Fig. 16 Formation and propagation of branched cracks kinked from an interfacial crack

high enough to induce multiple branches, which are visible in Fig. 16e, f. Crack length and speed records of the main crack and the branched cracks are presented in Fig. 17. Differentiation of the crack length record furnishes the tangential crack tip speed before and after crack branching. Since the differentiation process is based on a three-point-fitting procedure of the crack length history, the exact crack speed at the crack branching could not be obtained. However, it is interesting to notice that

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a Crack length (mm)

30 25 20 15 10 Main crack

5

Lower branched crack 0 170

190

210

230

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Time (microseconds)

b

Crack speed (m/s)

600 500 400 300 Main crack

200

lower brached crack 100 Upper branched crack 0 180

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200 210 220 Time (microsecond)

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Fig. 17 Crack length history and fitted crack tip speeds of the main crack and the branched cracks

the main crack tip speed is almost equal to the branched crack tip speed under the current time resolution (2:6 s=frame). The crack branching speed was about 28% of the shear wave speed of Homalite at the high strain rate.

3.6.3

Dynamic Crack Branching Initiated from a Notch Subjected to High Impact Loading

If we kept all other conditions from the previous case but raised the impact speed to 30 m/s (58% increase), a mode I crack was observed at first as shown in Fig. 18b. However, this main crack soon branched into two cracks as seen in Fig. 18c. At a later time, one branched crack (upper branch) generated two new sub-branching cracks shown in Fig. 18d. Previous experiments reported that the branching angle was less than 45ı [20, 46]. However, in our experiment, the first branching angle is around 45ı and the second branching angle is about 67ı , which is approximately the theoretical branching angle (60ı ) first reported by Yoffe [41]. The mechanism of

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b a Wedge

Notch

V=30m/s Field of view

c

d

67

45

Fig. 18 Crack branching after a mode I crack initiation if the notch radius was small but impact loading was high

crack branching subjected to a high loading (impact) rate can be explained using the dynamic energy release rate (driving force). The availability of kinetic energy due to high impact speed tends to create more fracture surfaces for absorbing energy. Therefore, branched cracks easily occur in a high impact loading case.

3.7 Dynamic Crack Kinking and Penetration at an Interface 3.7.1

Weak Interfaces with Different Interfacial Angles

We define a weak interface if the strength or toughness of this interface is much less than that of the strength or toughness of the bulk material (here is Homalite). The dynamic crack kinking phenomenon as shown in Fig. 16 will be slightly different if the interface crack is changed. For the case of the interfacial angles of 45ı [30], the incident crack reached the interface around 153 s after impact as seen in Fig. 19a. It is observed that the symmetric fringe pattern of the incident mode-I crack disappeared as soon as the crack deflected into the interface. The shape of the

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b a Notch

Interface

Interface V=19m/s

Wedge

b

Field of view 45ⴗ

457 x 254 x 10 (mm)

c

d

Fig. 19 Crack deflection process at a weak interface (interface angle 45ı )

fringe pattern of Fig. 19b suggests that this interfacial crack is shear-dominated at the latter propagation stage. In Fig. 19c, it is interesting to observe that after this shear-dominated crack propagated some distance along the interface, some secondary cracks formed at one side of the interface. These secondary cracks are locally mode-I and form on the tension side of the sheared interface. They form after the dominant crack has propagated along the interface and thus after the interface has already failed in shear. These types of secondary cracks that are a by-product of the shear crack growth along interfaces have already been observed experimentally and are always associated with dynamic shear-dominated crack growth along weak interfaces. As the interfacial angle is changed to 60ı , the dynamic crack deflection behavior is slightly altered, as seen in Fig. 20 on the crack speed records. First, we notice the crack speed jump across the interface at about 150 s. Obviously, the initial interfacial crack speed of 700 m/s is much higher than the incident crack speed which is about 400 m/s for the interfacial angle 60ı case. However, the interfacial crack speed reduced to 350 m/s soon after the interfacial crack kinked into the right side of monolithic Homalite (Fig. 16d). The experiment also suggests that just before the crack kinking, there was a brief crack speed reduction characteristically seen in several failure mode transition experiments [45]. In Fig. 20, the comparison of the crack peed history is made for two different interfacial angle cases. It is noticed that the interfacial crack speed for the interfacial angle 45ı case

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Crack speed (m/s)

900 45 degree

800 700

60 degree

600 500 400 300

Interface

200 100 0 100

120

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180

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Time (microseconds) Fig. 20 Crack speed history before and after crack deflection at a weak interface (for interfacial angles 45ı and 60ı )

is always higher than the crack speed for the interfacial angle 60ı case. This experimental phenomenon was analyzed by the proposed dynamic fracture mechanics model [30].

3.7.2

Modeling of Dynamic Failure Modes Across an Interface

For the incident crack, we may use dynamic fracture mechanics theory [40] to fit the stress intensity factors KId , KIId and the T stress of the incident crack. Then, we can predict the key parameters of the kinked interfacial crack using fracture mechanics theory [47]. Our first step is to obtain the static counterparts of the dynamic stress intensity factors. After crack deflection, the static stress intensity factors of the kinked crack can be calculated using these relationships. Let KIsk , KIIsk denote static mode-I and mode-II stress intensity factors for the deflected (kinked) mixedmode crack, and they are related to the static stress intensity factors of the incident dynamic cracks as a function of the kinking angle “ [1, 48]: KIsk D c11 KIs C c12 KIIs KIIsk D c21 KIs C c22 KIIs where the coefficients are:   3 ˇ 1 3ˇ c11 D cos C cos 4 2 4 2   1 ˇ 1 3ˇ sin C sin c21 D 4 2 4 2

c12 c22

  3 ˇ 3ˇ D sin C sin 4 2 2   1 ˇ 3 3ˇ D cos C cos 4 2 4 2

(6)

(7)

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Let KId k , KIId k be dynamic mode-I and mode-II stress intensity factors for the deflected (kinked) mixed-mode crack. We still assume that the universal relation between the dynamic and static stress intensity factors also holds for the deflected crack tip, i.e., KId k D kI .v2 /KIsk KIId k D kII .v2 /KIIsk

(8)

where kI .v2 / and kII .v2 / are universal functions of the crack tip speed defined by Freund [40]. Based on the above relations, if we know the dynamic stress intensity factors, the crack tip speed of the incident crack, the kinking angle as well as the crack tip speed of the kinked crack, we can get the dynamic stress intensity factors of the kinked crack and hence the crack tip stress fields around the deflected crack using Eq. 7. Then, the fringe patterns of the interfacial crack at the moment of crack deflection can be predicted using Eqs. 1 and 5. Because it is very hard to record the exact moment of crack kinking at the interface in dynamic fracture experiments, average values of the stress intensity factors of the incident crack were used to calculate the stress intensity factors of the kinked interfacial crack. Figure 21 shows the predicted fringe pattern of a kinked interfacial crack (interfacial angle 60ı ). The coordinate origin is located at the intersection point of the incident crack and the kinked crack, and its x-axis is along the interface. For this case, the crack tip speed of the incident crack was around 400 m/s. Right after crack kinking, the interfacial crack tip speed was about 700 m/s [30]. Since the T-stress of the incident crack is around 1 MPa, and there are no accurate results for the T-stress of the kinked crack [49], the T-stress of the kinked crack was assumed to be zero in all our predictions. In order to highlight our comparison of the predicted and experimental fringe patterns, only fringe order 1 was plotted. Figure 21a presents the experimental fringe showing the transition from an incident crack to an interface crack (the horizontal line was the interface). Figure 21b showed the predicted fringe and the two kinds of photoelasticity fringes were very similar so our dynamic fracture mechanics modeling and assumptions were reasonable. However, some discrepancy is also noticed because it is very hard to take one photo at the right time and right position as the theoretically predicted one. One interesting observation is the large concave wedge effect. As seen in Fig. 21a, two caustic spots (one caused by the kinked crack tip and the other due to a large concave wedge) were clearly observed when a mode-I incident crack kinked along a weak interface with a large kinking angle (60ı , see the same photo in Fig. 16c). In most previous crack kinking analyses, researchers only considered the singular stress field due to a kinked daughter crack and ignored the singular stress field of a concave wedge. Interestingly, William’s classical solution of wedge stress singularities [22] is the foundation of the full-field stress field of a traction-free crack in Linear Elastic Fracture Mechanics. Indeed, Cotterell and Rice’s classical work mainly deals with a slightly kinked crack, not a large kinking angle case [50]. To authors’ knowledge, only Azhdari and Nemat-Nasser provided a simple explanation to this phenomenon for a static crack kinking case [51].

Dynamic Failure of Sandwich Structures and Layered Materials Fig. 21 Comparison of (a) experimental fringe (strong interface) and (b) predicted fringe pattern (v1 D 400 m=s, v2 D 700 m=s, N D 1, interfacial angle 60ı )

597

a Caustics caused by the large concave wedge

Caustics caused by the interfacial crack tip

b

y (mm) 2 Incident crack

1 60

Interface

–2

–1

1

x (mm)

–1 –2 –3

3.7.3

Mode Mixity of the Kinked Interfacial Crack

Mode mixity is an important parameter in interface fracture analysis. It is the ratio measure of the mode II stress intensity factor/energy release rate over its mode I counterparts. There were numerous studies on mode mixity in static fracture cases but very few results were reported in dynamic fracture investigation [52, 53]. Indeed, mode mixity is a key parameter in controlling failure mode transitions along interfaces. In this investigation, when the incident mode I crack reached the interface, it kinked along the interface and became one mixed mode crack. Based on Eq. 7, for the kinked interfacial crack, its mode mixity depends on the kinked crack tip speed and the interfacial or kinking angle. Obviously, the dependence on the dynamic mode mixity on the crack tip speed is a special phenomenon in dynamic fracture mechanics. The variations of the mode mixity with the interfacial angle and the kinked crack tip speed are plotted in Fig. 22. It is not surprising to see that when

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a 2.5 V/Cs=0 V/Cs=0.3 V/Cs=0.6 V/Cs=0.9

Mode-Mixity

2 1.5 1 0.5 0 0

10

20

30

40

50

60

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90 100

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b 1.4

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1.2 Mode-Mixity

Interfacial angle=45° 1

Interfacial angle=60°

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

V/Cs

Fig. 22 Dynamic fracture mode mixity as functions of (a) interfacial angle and (b) crack tip speed

the crack tip speed of the interfacial crack remained constant, mode mixity increased with the increase of the kinking angle. In other words, the larger of the interfacial angle, the larger is the mode II component for the mixed mode crack. This result is similar to the common conclusion in static crack kinking analysis [48]. As a special feature of dynamic crack kinking, the mode mixity increases with the increasing kinked crack speed if the interface angle is fixed. In Fig. 22a, for a fixed kinking or interfacial angle 50ı , the mode mixity for a high crack tip speed (90% of the shear wave speed of the matrix material) is almost doubled compared to the mode mixity for a static kinked crack. Here, we should notice that the crack tip speed of the kinked crack is related to the interfacial bonding strength [30]. A weak interface will lead to a fast interfacial crack tip and a high mode-II component as a result. Figure 22b shows the mode mixity dependence on the crack tip speed for different kinking angles. It is interesting to see that each curve has a similar shape and is shifted by some amount for a different kinking angle. In this investigation, the kinking or interfacial angle is limited to 0–90ı . Recently, Rousseau and

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Rosakis [54] examined important crack kinking behavior for very large interfacial angle (greater than 90ı ). One important difference is that their incident crack was an inter-sonic shear crack along a weak path rather than a slow mode-I crack as in our investigation. In order to suppress possible crack branching, our incident crack speed was controlled to be less than the crack branching speed (around 30–40% of the shear wave speed for Homailte-100 [55].

3.8 Two-layer Specimens with Direct Impact on the Brittle Polymeric Layer (Fig. 23) The above discussions are focused on the three-layer specimen or the model sandwich structure. In order to understand the dynamic response and failure of general layered materials or structures, experimental investigation on the two-layer specimen is necessary [56]. In some applications of layered materials, direct impact on the brittle layer such as ceramics layer in composite armor is also very important since their failure behavior might be very different [57]. As shown in Fig. 23 a complicated stress wave propagation, projectile penetration and large deformation process was observed for a two-layer system with a weak and ductile 5083 bond. As soon as the projectile impacted the transparent Homalite layer, a series of fringe patterns related to stress waves developed around the impact site. The projectile head kept moving and complicated stress wave interaction and propagation were observed around 10 s after impact (Fig. 23b, c). It should be noticed that the movement of the projectile was not obvious. In the next stage of projectile penetration, a black contact zone was observed at the impact site and an intra-layer crack initiated and propagated towards the interface. In Fig. 23e, it seemed that this intra-layer crack arrested at the interface since no significant interfacial crack propagation (moving fringe concentration) was observed in the following images. As discussed by Xu and Rosakis [33], the crack arrest mechanism by a ductile and weak adhesive layer is due to the dramatically stress wave gradient change across the adhesive layer. After 300 s of impact (Fig. 23f), a large contact zone appeared in a growing black zone connected to the projectile. Due to large deformation inside the contact zone, those transmitted laser rays were deflected and cannot enter the camera so only a black zone was recorded. Meanwhile, some materials were flying out from the Homalite layer because of the fragments of brittle Homalite subjected to high contact force. Compared to the impact on the strong steel layer of previous cases, direct impact on the brittle Homalite layer has more severe damage in the local impact site.

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b a Field of view

V=44 m/s

c

d

e

f

Fig. 23 Failure sequence of a two-layer specimen (a) impacted directly on the brittle Homalite layer (b–c) show a complicated stress wave process, (d–f) reveal intra-layer crack initiation and propagation and a penetration procedure with large local deformation

4 Conclusions Inter-layer crack growth (delamination) is the dominant dynamic failure mode for layered materials and sandwich structures subjected to out-of-plane impact. These cracks appear to be shear-dominated and proceed with intersonic speeds. Intra-layer cracking always occurs soon after some amount of inter-layer delamination has already happened and proceeds through the spreading and branching of local mode I cracks into the core layer. Intra-layer or core/matrix cracking often initiates at the interface as a result of inter-layer crack kinking into the core. If the speed of the kinked intra-layer crack reaches a critical value, multiple crack branching may also occur inside the brittle core layer. If free-edge effects at the bimaterial corners

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are eliminated (long model sandwich specimens), the failure sequence is slightly modified. Specifically, the inter-layer cracks initiate from positions where the inter-layer shear stress reaches a local maximum equal to the shear strength of the bond. These cracks create a local shear-driven delamination directly above the point of impact. Intra-layer cracks following this process also kink from these positions into the sandwich core. For some individual dynamic failure modes, we find that the dynamic crack branching can exceed the limit angles 45ı (theoretical result) for a second dynamic crack, and the crack tip speeds before and after the crack branching are almost same. Crack kink and penetration at interfaces are mainly governed by the ratio competition of the dynamic energy release rates of the incident crack and the kinked crack, and the dynamic fracture toughness ratio of the matrix (bulk material) and the interface. For the kinked crack speed, it increases if the interfacial angle and interfacial strength decrease. A strength-based criterion is proposed to predict the interfacial debonding ahead of a main crack well. Acknowledgements The authors gratefully acknowledge the support of the Office of Naval Research through a MURI grant to Caltech (N00014–06–1–0730), a research grant to Vanderbilt (N00014–08–1–0137), Dr. Y.D.S Rajapakse, Program Manager of both projects.

References 1. Hutchinson JW, Suo Z (1992) Mixed mode cracking in layered materials. Adv Appl Mech 29: 63–191. 2. Rajapakse YDS (1995) Recent advances in composite research for marine structures. In: Allen, HG (ed) Sandwich Construction 3, Proceedings of the Second International Conference. Chameleon, London, Vol. II, pp 475–486. 3. Sun CT, Rechak S (1988) Effect of adhesive layers on impact damage in composite laminates. In: Whitcomb JD (ed) Composite materials: testing and design (eighth conference). ASTM STP 972, American Society for Testing and Materials, Philadelphia, pp 97–123. 4. Cantwell WJ, Morton J (1991) The impact resistance of composite materials – a review. Comps 22: 347–362. 5. Abrate S (1994) Impact on laminated composites: recent advances. Appl Mech Rev 47: 517–544. 6. Gioia G, Ortiz M (1997) Delamination of compressed thin films. Adv Appl Mech 33: 119–192. 7. Kadomateas GA (1999) Post-buckling and growth behavior of face-sheet delaminations in sandwich composites. In: Rajapakse YDS, Kadomateas GA (eds) Thick Composites for Load Bearing Structures. AMD 235: 51–60. 8. Choi HY, Wu HT, Chang FK (1991) A new approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity impact: part II – analysis. J Comp Mat 25: 1012–1038. 9. Lambros J, Rosakis AJ (1997) An experimental study of the dynamic delamination of thick fiber reinforced polymeric matrix composite laminates. Exp Mech 37: 360–366. 10. Lee JW, Daniel IM (1990) Progressive transverse cracking of crossply composite laminates. J. Comp Mat 24: 1225–1243. 11. Oguni K, Tan CY, Ravichandran G (2000) Failure mode transition in unidirectional E-Glass/Vinylester composites under multiaxial compression. J Comp Mat 34: 2081–2097. 12. Ju JW (1991) A micromechanical damage model for uniaxially reinforced composites weakened by interfacial arc microcracks. J Appl Mech 58: 923–930.

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13. Semenski D, Rosakis, AJ (1999) Dynamic crack initiation and growth in light-core sandwich composite materials. Proceedings of the 17th Danubia-Adria Symposium on Experimental Mechanics in Solid Mechanics, Prague, pp 297–300. 14. Xu LR, Rosakis AJ (2002) Impact failure characteristics of sandwich structures; Part I: Basic Failure Mode Selections. Int J Sol Struct 39:4215–4235. 15. Walter ME, Ravichandran G (1997) Experimental simulation of matrix cracking and debonding in a model brittle matrix composite. Exp Mech 37: 130–135. 16. Xu LR, Rosakis AJ (2002) Impact failure characteristics of sandwich structures; Part II: effects of impact speeds and interfacial bonding strengths. Int J Sol Struct 39: 4237–4248. 17. Xu LR, Sengupta H. Kuai (2004) An experimental and numerical investigation on adhesive bonding strengths of polymer materials. Int J Adh Adhes 24: 455–460. 18. Parameswaran V, Shukla A (1998) Dynamic fracture of a functionally gradient material have discrete property variation. J Mat Sci 33: 3303–3311. 19. Rosakis AJ, Samudrala O, Singh RP, Shukla A (1998) Intersonic crack propagation in bimaterial systems. J Mech Phys Solids 46: 1789–1813. 20. Dally JW (1979) Dynamic photoelastic studies of fracture. Exp Mech 19: 349–61. 21. Singh RP, Shukla A (1996) Subsonic and intersonic crack growth along a bimaterial surface. J App Mech 63: 919–924. 22. Williams ML (1957) Stress singularities resulting from various boundary conditions in angular corners in extension. J Appl Mech19: 526–528. 23. Xu LR, Kuai H, Sengupta S (2004) Dissimilar material joints with and without free-edge stress singularities; Part I: a biologically inspired design. Exp Mech 44: 608–615. 24. Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26: 141–154. 25. Yu C, Ortiz M, Rosakis, A (2003) 3D modelling of impact failure in sandwich structures. Fracture of Polymers, composites and adhesives, II Elsevier and ESIS, pp 527–537. 26. Geubelle PH, Baylor JS (1998) Impact-induced delamination of composites: a 2D simulation. Composites B29B: 589–602. 27. Wang P, LR Xu (2006) Dynamic Interfacial Debonding Initiation Induced by an Incident Crack. Int J Solids Struct 43: 6535–6550. 28. Martinez D, Gupta V (1994) Energy criterion for crack deflection at an interface between two orthotropic media. J Mech Phys Solids 42(8): 1247–1271. 29. He MY, Hsueh CH, Becher PF (2000) Deflection versus penetration of a wedge-loaded crack: effect of branch-crack length and penetrated-layer width. Comps Part B: Engng 31: 299–308. 30. Xu LR, Huang YY, Rosakis AJ (2003) Dynamic crack deflection and penetration at interfaces in homogeneous materials: experimental studies and model predictions. J Mech Phys Solids 51: 461–486 31. Cook J, Gordon JE (1964) A mechanism for the control of crack propagation in all brittle systems. Proc Royal Soc London 282A: 508–520. 32. Martin E, Leguillon D, Lacroix C (2001) A revisited criterion for crack deflection at an interface in a brittle material. Comp Sci Techn 61: 1671–1679. 33. Xu LR, Rosakis AJ (2003) An experimental study of impact-induced failure events in homogeneous layered materials using dynamic photoelasticity and high-speed photography. Optics Lasers Engng 40: 263–288. 34. Evans AG, Zok FW (1994) Review the physics and mechanics of fiber-reinforced brittle matrix composites. J Mat Sci 29: 3857–3896. 35. Ahn BK, Curtin WA, Parthasarathy TA, Dutton RE (1998) Criterion for crack deflection/penetration for fiber-reinforced ceramic matrix composites. Comp Sci Techn 58: 1775–1784. 36. Siegmund T, Fleck NA, Needleman A (1997) Dynamic crack growth across an interface. I J Fract 85: 381–402. 37. Arata JJM, Needleman A, Kumar KS, Curtin WA (2000) Microcrack nucleation and growth in lamellar solids. Int J Fract 105: 321–342. 38. Xuan W, Curtin WA, Needleman A (2003) Stochastic microcrack nucleation in lamellar solids. Engng Fract Mech 70: 1869–1884.

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39. Ramulu M, Kobayashi AS (1985) Mechanics of crack curving and branching – a dynamic fracture analysis. Int J Fract 27:187–201. 40. Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, New York. 41. Yoffe E H (1951) The moving Griffith crack. Phil Mag, Series 7, 42, 739. 42. Gao H (1993) Surface roughness and branching instabilities in dynamic fracture. J Mech Phys Solid 41 (23): 457–486. 43. Seelig Th, Gross D (1999) On the interaction and branching of fast running cracks – a numerical investigation. J Mech Phys Solids 47: 945–952. 44. Sharon E, Fineberg J (1999) Confirming the continuum theory of dynamic brittle fracture for fast cracks. Nature 397: 333–335. 45. Xu LR, Rosakis AJ (2003b) Real-time experimental investigation of dynamic crack branching using high-speed optical diagnostics. Exp Techn 27: 23–26. 46. Shukla A, Nigam H, Zervas H (1990) Effect of stress field parameters on dynamic crack branching. Engng Fract Mech 36: 429–438. 47. Xu LR, Wang P (2006) Dynamic fracture mechanics analysis of failure Mode transitions along weaken interfaces in elastic solids. Engng Fract Mech 73: 1597–1614. 48. Anderson TL (1995) Fracture Mechanics, 2nd edn. CRC, Boca Raton. 49. Li XF, Xu LR (2007) T-stresses across static crack kinking. ASME J Appl Mech 74: 181–190. 50. Cotterell B, Rice JR (1980) Slightly curved or kinked cracks. Int J Fract 16(2): 155–169. 51. Azhdari A, Nemat-Nasser S (1996) Energy-release rate and crack kinking in anisotropic brittle solids. J Mech Phys Solids 44: 929–951. 52. Gupta V, Argon AS, Suo Z (1992) Crack deflection at an interface between two orthotropic materials. J Appl Mech 59: 79–87. 53. Ravi-chandar K, Lu J, Yang B, Zhu Z (2000) Failure mode transitions in polymers under high strain rate loading. Int J Fract 101: 33–72. 54. Rousseau C-E, Rosakis AJ (2003) On the influence of fault bends on the growth of SubRayleigh and intersonic dynamic shear ruptures. J Geophys Res 108: 2411–2431. 55. Broberg KB (1999) Cracks and fracture. Academic, San Diego. 56. Xu LR, Rosakis AJ (2005) Impact damage visualization of heterogeneous two-layer materials subjected to low-speed impact. Int J Dam Mech 14: 215–233. 57. Cheeseman B, Jensen R, Hoppel C (2004) Protecting the future force: advanced materials and analysis enable robust composite armor. AMPTIAC 8(4): 37–43.

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Characterization of Fatigue Behavior of Composite Sandwich Structures at Sub-Zero Temperatures Samirkumar M. Soni, Ronald F. Gibson, and Emmanuel O. Ayorinde

Abstract This chapter summarizes recent studies of the flexural fatigue characteristics of foam core carbon/epoxy and glass/epoxy composite sandwich beams over the temperature range from 22ı C to 60ı C. Core shear was found to be the dominant fatigue failure mode for the test specimens over this temperature range. Significant increases in the useful fatigue life with brittle type core shear failure were observed at low temperatures by comparison with the corresponding room temperature behavior. Fatigue failure at the subzero temperatures was catastrophic and without any significant early warning, but the corresponding failures at room temperature were preceded by relatively slow but steadily increasing losses of stiffness. Two different approaches were used to investigate stiffness reductions during fatigue tests, and both approaches led to the same conclusions. Static finite element analyses confirmed the experimentally observed locations of fatigue crack initiation.

1 Introduction The sub-zero temperature fatigue behavior of composite sandwich structures must be studied in order to evaluate these materials for possible use in the hull structures of ships which operate in cold regions. The analysis of the mechanical behavior of composite sandwich structures is fairly well understood, as shown in the books by Zenkert [1] and Vinson [2]. When such structures are used in the hulls of marine craft, the long-term repetitive loading from wave action on the hulls may lead to fatigue failure. In general, the fatigue characteristics of composite sandwich structures are also reasonably well understood, as summarized by Sharma et al. [3]. However, S.M. Soni and E.O. Ayorinde Mechanical Engineering Department, Advanced Composites Research Laboratory, Wayne State University, Detroit, MI 48202 e-mail: [email protected]; [email protected] R.F. Gibson () Mechanical Engineering Department, University of Nevada-Reno, MS312 Reno, Nevada, USA e-mail: [email protected]

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when ships operate in cold regions such as those near the North and South Poles, the effects of the low temperatures on the hull structural materials must be determined. For example, the mean temperature in Antarctica ranges between 15ı C and 70ı C. The key aspects of static and fatigue behavior of composite sandwich structures have been summarized by Sharma et al. [3]. The failure mechanisms in foam core sandwich beams under static loading have been studied by Daniel et al. [4], and by Steeves and Fleck [5, 6], who found four different modes of failure; core shear, face sheet wrinkling, face sheet micro buckling and face sheet indentation. Harte et al. [7] examined failure modes in aluminum foam core sandwich beam under both static and fatigue loading conditions, while Burman and Zenkert [8, 9] studied fatigue characteristics of foam core sandwich beams experimentally as well as analytically. From these experimental results, it was concluded that the predominant failure mode under cyclic fatigue was core shear, apparently because of reductions in the residual shear strength of the foam core under cyclic loading. Kanny and Mahfuz [10] showed that the stiffness and fatigue strength of sandwich beams increased with increased core density and the number of cycles to failure increased with the frequency, but the time to failure showed the opposite trend. This seems to contradict findings by Burman and Zenkert [8, 9] and by Sharma et al. [11] that increasing frequency reduced the number of cycles to failure due to the corresponding increase in core temperature. It appears that the effects of temperature on fatigue of foam core composite sandwich structures have only been investigated for elevated temperatures, not cold temperatures. For example, Kanny et al. [12] investigated the effects of temperatures up to 80ı C on the flexural fatigue behavior of PVC foam core sandwich structures with glass/vinylester composite face sheets. It was reported that the fatigue lives at given stress levels were significantly reduced by increasing temperatures, and that the failure mode changed from core shear at room temperature to partial plastic yielding of the core and face-core delaminations at elevated temperatures. In conclusion, it appears that the effects of subzero temperatures on the fatigue life and failure modes of foam core composite sandwich beams remains to be investigated, and this provided the motivation for the present work. The objectives of the study were to determine the effects of subzero temperature on the failure modes under static and cyclic loading conditions and on the fatigue lives of the composite sandwich beams. Accordingly, this paper describes the characterization of the effects of temperatures from 22ı C down to 60ı C on low cycle flexural fatigue behavior of unidirectional carbon fiber/Rohacell foam core (CF/RC) sandwich beams and woven glass fiber/Rohacell foam core (GF/RC) sandwich beams by using both experiments and numerical simulations.

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2 Experiments 2.1 Specimens As shown in Fig. 1, two types of specimens were used for the experimental studies of the sandwich beams. Both groups of specimens have polymethacrylimide (PMI) rigid foam (Rohacell 71 IG (Industrial Grade)) as the core. This foam is 100% closed cell, isotropic and has constant through-the-thickness shear strength. Skins for CF/RC beams are unidirectional T300/A534 carbon fiber/epoxy laminates having three layers with fiber direction along the length of the beam. For GF/RC beams, the skins are woven 7781 style E-glass fiber/epoxy fabric laminates with three plies. The weave pattern is 8 HS (Harness Satin). Warp and fill yarns run at 0 and 90 degrees with the length, respectively. Thus, the fabrics are anisotropic, or strong in only two directions. All specimens had dimensions of 8 in. long by 1 in. wide by 0.556 in. thick. Preliminary nondestructive screening tests were conducted by measuring the first three flexural modal vibration frequencies of the beam specimens in free-free impulse response vibration tests at temperatures from 22ı C to 60ı C [13]. These frequencies are directly related to the flexural stiffness of a beam. There were no permanent changes in the frequencies after freezing both types of sandwich beams at 18ı C for 50 days and then warming them back to room temperature, but both GF/RC and CF/RC beams exhibited increases in the average frequencies with decreasing temperature. These changes were found to be reversible, however, as the frequencies returned to their original values upon warming up to room temperature. More details regarding the materials and the preliminary screening tests may be found in the thesis by Soni [13].

2.2 Static Flexure Tests Baseline reference data on the static mechanical properties of the specimens from 22ı C to 60ı C were determined from static 4-point bending tests (Fig. 2) before

Fig. 1 Sandwich beam specimens used for the experimental work

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Fig. 2 Composite sandwich specimen and fixture for 4-point bending test

Fig. 3 Apparatus for 4-point bending tests at low temperatures

performing the fatigue tests. Since the literature survey indicated that core shear is the dominant mode of failure under flexural fatigue loading conditions, static 4-point bending tests were performed in accordance with the ASTM standard C 39300 [14] to check for possible core shear failure under static loading. Tests of three replicate specimens each of CF/RC and GF/RC beams were conducted inside the environmental chamber of an Endura-Tec servo-pneumatic testing machine (Fig. 3). The loading configuration was quarter point loading with a 4.72 in. span and 1.57 in. between the loading points. The support and loading pins each had a diameter of 0.315 in. to avoid local face sheet indentation failure. Experiments were performed under displacement-control at a crosshead speed of 0.0002 in./s until failure. In order to capture the crack initiation and growth with a digital camera, it was found necessary to use a slower loading rate than that recommended by ASTM C 393-00. The loading rate and the response of the system were controlled and recorded with the help of the Endura-Tec Win-Test software. A digital camera and StreamPix software on a personal computer were used to monitor and record the failure events during the test with a scanning rate of 11 frames per second (fps). The temperature of the environmental chamber was controlled by the EnduraTec WinTest software through a PC 200 temperature controller. Evaporated liquid nitrogen was used as the coolant.

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2.3 Flexural Fatigue Tests Using the same 4-point flexural test setup described in Figs. 2 and 3, flexural fatigue tests were performed on CF/RC and GF/RC beams in accordance with ASTM C-393-00 [14] under load control at RT, 0ı C and 60ı C, a frequency of 2 Hz and different load levels. By using the ultimate strength data acquired from the static 4-point bending tests at different temperatures, the maximum flexural load applied per cycle was determined at different load levels, r, as defined by: rD

Pmax Pul

(1)

where, as shown in Fig. 4, Pmax is the maximum load applied per fatigue cycle and Pul is the ultimate static failure load. The minimum flexural load applied per cycle, Pmin , was determined from the loading ratio R, as defined by: RD

Pmin Pmax

(2)

The loading ratio R D 0:1 was kept constant during all the fatigue tests to observe the sole effect of low temperature on the fatigue behavior. Initially, flexural fatigue tests were performed at room temperature, different load levels (from r D 0:9 to r D 0:6) and a frequency of 2 Hz on CF/RC and GF/RC sandwich beams. The maximum test running time at any load level was 48 h (345,600 cycles) by considering the unforeseen fatigue behavior of the beams at low temperatures and limited availability of liquid nitrogen per cylinder. Three replicate specimens of the beams were used for the fatigue tests at each temperature. Two types of approaches have been used to determine the stiffness reduction during flexural fatigue tests performed on both CF/RC and GF/RC beams at different load levels and temperatures. In the first approach, graphs were plotted using the absolute mean crosshead displacement per cycle as ordinate and number of cycles as abscissa, as shown in Fig. 5. In the second type of data acquisition file, 25 points were scanned and stored per fatigue cycle showing load applied at specific points during each fatigue cycle

Fig. 4 Illustration of parameters used to define load level, r, and loading ratio, R

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Fig. 5 Calculation of absolute mean displacement for each cycle

Time (sec) 0.435 0

0.2

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0.395

Displacement of Crosshead (inch)

0

0.39 –250

load vs time curve

0.385

displacement vs time curve

Fig. 6 Data acquisition for fatigue tests

as well as the system response in terms of displacement of crosshead in the vertical direction at those points (Fig. 6). These data points were then used to plot the hysteresis loops of applied load vs. crosshead displacement in a particular fatigue loading cycle, as shown in Fig. 7. The hysteresis loops were then analyzed to determine the dynamic stiffness, S, and damping loss factors, LF, during that cycle using the equations [13]

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Fig. 7 Hysteresis loop for determination of stiffness and damping loss factor

a=2 b=2 c=2 LF D a=2 SD

(3) (4)

These equations were based on the assumption of a perfectly elliptical hysteresis loop.

3 Finite Element Modeling Static finite element (FE) analysis was done for two reasons: (1) to predict the elastic behavior of both CF/RC and GF/RC sandwich beams at room temperature and at temperatures down to 60ı C under the static loading conditions, and (2) to verify the possible differences in the location of fatigue crack initiation and failure modes between CF/RC and GF/RC sandwich beams under fatigue loading conditions. The 2D model of an undamaged CF/RC beam, which was created by Baba et al. [15], was used for the FE analysis of the CF/RC beam, while for modeling GF/RC beam, the same model was used except for different skin properties. Moreover, to predict the behavior of both types of beams at low temperatures, an approximate coefficient of thermal expansion for each component was used in both models. For the static FEA analysis, HyperMesh 7.0 was used as pre-post processor and ABAQUS 6.5 was used as a solver. Both GF/RC and CF/RC beam models were subjected to two type of loading: (1) Mechanical four point bending loading at 22ı C, and (2) thermal and mechanical loading by applying the mechanical four point bending loading at a temperature of 60ı C.

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4 Results 4.1 Static 4-Point Bending Static four point bending tests were performed at 22ı C, 0ı C, 30ı C and at 60ı C to establish baseline reference data on the strength of the CF/RC and GF/RC sandwich beams at the respective temperatures. For each case, during the loading period no cracks were observed by the digital camera up to the final failure point. All specimens failed suddenly due to core shear at all of the temperatures, as seen in Fig. 8. The entire crack formation and growth sequence occurred in a fraction of a second, so it was not possible to follow the sequence exactly with the digital camera. However, the final crack pattern observed under static loading appeared to be quite similar to that reported in the literature for fatigue loading. The effects of temperature on the stiffness, strength and ductility of both the GF/RC and CF/RC sandwich beams are clearly shown in the load-displacement curves in Fig. 9a, b, respectively. As the temperature decreases, stiffness increases for both CF/RC and GF/RC beams. The slope, elastic limit and displacement at failure all varied almost linearly with respect to temperature change. At 22ı C, both the beams exhibited significant yielding and ductile behavior, but a more brittle type sudden failure occurred near the ultimate loading points at 30ı C and at 60ı C, with corresponding reductions in the displacement at failure. As expected, the average flexural strength (as determined by the ultimate load) of the unidirectional CF/RC composite sandwich beams is greater than that of woven GF/RC composite sandwich beams with the same dimensions and geometry at any temperature thus the skin materials influence the strength of any sandwich beam. The average ultimate flexural strength of the beams increased with corresponding reductions in temperature. However, the jump in ultimate strength data is more prominent during temperature reduction from 22ı C to 0ı C than during 0ı C to 60ı C. The corresponding displacement at failure for all specimens decreased with reductions in temperature.

Fig. 8 Modes of failure of CF/RC sandwich beams at different temperatures

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Fig. 9(a) Load-displacement graphs for GF/RC beams under static flexural loading conditions at different temperatures

Fig. 9(b) Load-displacement graphs for CF/RC beams under static flexural loading conditions at different temperatures

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4.2 Flexural Fatigue The main purpose of the fatigue tests on the sandwich beams was to compare the fatigue life, fatigue failure mode, stiffness and damping factors at low temperatures with the corresponding room temperature behavior. The experimental results of flexural fatigue tests performed on both GF/RC and CF/RC beams at different load levels, r, and different temperatures are presented in Table 1 in terms of number of cycles to failure, Nf , where Nf was recorded just before final failure.

4.3 Fatigue Life at Low Temperatures The fatigue life data of GF/RC and CF/RC sandwich beams at different load levels and temperatures were presented by using modified S–N curves. For convenience, the modified S–N curves were plotted with the load level, r, on the vertical axis, and the logarithm of the number of cycles to failure, Nf on the horizontal axis. Figure 10 shows the modified S–N curve data for (a) GF/RC and (b) CF/RC beams at different load levels and temperatures. At all temperatures, the specimens exhibited increases in the number of cycles to failure with reductions in the load level, r. At 22ı C and at 0ı C, the GF/RC beams exhibited greater average number of cycles to failure compared with CF/RC beams

Table 1 Fatigue life data of CF/RC and GF/RC beams at different load level and temperatures Temperature

Load level

22ı C Load applied (lb)

No of cycles to failure

60ı C Load applied (lb)

No of cycles to failure

230 207 195.5 161 – –

0.25 528 650 5580 – –

235 210 199.75 – – –

0.25 24872 33146 – – –

184.5 166 147.6 129.2 –

0.25 953 2227 16335 –

193.5 174.2 154.8 – –

0.25 27516 100000 – –

No of cycles to failure

0ı C Load applied (lb)

0.25 258 290 3328 49622 345600 0.25 1415 3959 23181 345600

CF/RC BEAM 1 0.9 0.85 0.7 0.6 0.5

201 180 170.8 140.7 120.6 100.5

1 0.9 0.8 0.7 0.6

160 144 128 112 96

GF/RC BEAM



Specimen did not fail after this number of cycles and test was stopped.

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Fig. 10 Modified S–N curve data for GF/RC and CF/RC sandwich beams showing effect of low temperatures on fatigue life at different load levels

at each load level but at 60ı C and 0.9 load level both the beams had almost the same average number of cycles to failure. At 60ı C and 0.9 load level, both types of beams showed tremendous increases in the fatigue life compared with RT but CF/RC beams exhibited greater percentage improvement in average fatigue life than GF/RC beams. At 0.8 load level and 60ı C, the GF/RC beams lasted for approximately 105 cycles without any failure or crack initiation. Fatigue lives of CF/RC beams at 60ı C were increased by almost one hundred times compared with those at 22ı C. Thus, the fatigue life of both types of composite sandwich beams was much improved at low levels of loading at 60ı C but this improvement also depends on the skin material properties. Kanny and Mahfuz [10] derived a simple expression for predicting the fatigue life of foam core sandwich beams from experimental data which has been plotted in a log-log fashion. The equation for a straight line on the log-log plot is of the form: log r D a  b log Nf

(5)

Log-log plots for the CF/RC and GF/RC beam fatigue data are shown in Fig. 11, and values of the regression parameters a and b are tabulated in Table 2. These results clearly show that the number of cycles to failure, Nf , at a given stress level increases as the temperature is decreased in both types of sandwich beams. This is consistent with the observation that the slope of the log r vs. log Nf curve, b, decreases with decreasing temperature, and with the results of Kanny et al. [12], which showed that elevated temperatures caused reductions in Nf for similar sandwich structures.

4.4 Stiffness and Damping at Low Temperatures Figures 12 and 13 show the variation in the absolute mean displacement of the CF/RC beams with numbers of fatigue cycles, as calculated according to Fig. 5.

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a

Regression analysis on GF/RC beams 0.1 0.05

Log r

0 –0.05 –0.1

Experimental at 22 C Calculated at 22 C Experimental at 0 C Calculated at 0 C Experimental at –60 C Calculated at –60 C

–0.15 –0.2 –0.25 –1

0

1

2

3

4

5

6

Log Nf

Fig. 11(a) Log-log regression curves for fatigue data of GF/RC beams at different temperatures

b

Regression analysis for CF/RC beam 0.1 0.05 0

Log r

–0.05 –0.1 –0.15

Experimental at 22 C Calculated at 22 C Experimental at 0 C Calculated at 0 C Experimental at –60 C Calculated at –60 C

–0.2 –0.25 –0.3 –0.35 –1

0

1

2

3

4

5

6

Log Nf

Fig. 11(b) Log-log regression curves for fatigue data of CF/RC beams at different temperatures

For each graph, the final data points show the deformation of the beam just before the core shear failure. The final core shear stage is not included in these graphs. These graphs provide important information about the stiffness reduction during the fatigue tests, as each fatigue test was performed under load control and so changes in the absolute mean displacement with increasing number of cycles indirectly provide a measure of stiffness reduction during that period.

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Table 2 Effect of low temperatures on regression parameters a and b in Eq. 5 GF/RC Beam CF/RC Beam Temperature a b a b 22ı C 0.005458 0.03406 0.016915 0.049678 0ı C 0:00558 0.02769 0:00506 0.028918 60ı C 0:00618 0.01405 0:00668 0.011581

Fig. 12 Graphs showing absolute mean displacement vs. number of cycles for both (a) CF/RC and (b) GF/RC beams at 0.9 load level and different temperatures

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Fig. 13 Graphs showing absolute mean displacement vs. number of cycles for both (a) CF/RC and (b) GF/RC beams at 0.85 and 0.8 load levels, respectively at different temperatures

General observations from Figs. 12 and 13 for are: (1) Both CF/RC and GF/RC beams showed the clear effect of temperature reduction on the stiffness degradation during fatigue testing. At 22ı C, the slow but steady decrease in stiffness with increasing number of cycles gave early warning of impending fatigue failure well in advance of the failure. At 0ı C, the slope of the absolute mean displacement as well as the warning of the failure decreased at each load level, (2) at 60ı C, the total ab-

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solute mean displacement was very low up to the failure point with negligible early warning compared with the 22ı C data, and there was no sign of impending failure throughout the fatigue life. Thus, both CF/RC and GF/RC beams became stiffer and stronger against fatigue loading at low temperatures and exhibited tremendous increases in fatigue life, (3) the behavior of GF/RC beams was a bit confusing at 0ı C at each load level, as they showed little reduction in fatigue life compared with the corresponding room temperature results. One possible reason was the negligible change in the stiffness at 0ı C in GF/RC beams compared with 22ı C and the scatter in the fatigue data, (4) the number of cycles to failure was shifted for both type of beams with reduction in temperature, but after a certain deformation (absolute mean displacement D 0:004–0:005 in:) both types of beams acted non-linearly irrespective of temperature. At 22ı C and at 0ı C, both types of beams were ductile and exhibited plastic deformation by exhibiting slow increases in core shear crack growth, but at 60ı C both types of beams became brittle and failed suddenly during the shift from the elastic to the plastic deformation region. In the second approach, dynamic stiffness, S, and damping loss factor, LF, of the specimens were determined from the hysteresis loops at specific numbers of cycles during the fatigue life at 0.9 and 0.85 load levels and at RT, 0ı C and 60ı C as shown in Eqs. 3 and 4, respectively, and the results are shown in Fig. 14. Only the data for CF/RC beams are shown in Fig. 14, as the hysteresis loops were nearly elliptical. GF/RC beams exhibited bilinear hysteresis loops throughout the fatigue life at each load level and temperature, so Eqs. 3 and 4 were not valid. This behavior may have been due to transverse ply (90ı ) failure in the woven 0/90 fiber laminated skins. The following observations were made from the graphs in Fig. 14: (1) at 22ı C, the stiffness curves sloped downwards while damping loss

Fig. 14 Graphs showing changes in stiffness and damping loss factors of CF/RC beams during fatigue loading at different temperatures and load levels

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factor curves sloped upwards with increasing numbers of cycles at both 0.9 and 0.85 load levels. Both curves exhibited significant nonlinearity leading to final failure, (2) at 0ı C, both stiffness and damping factor curves exhibited behavior similar to that at 22ı C, but the amount of nonlinearity in the curves was reduced at both load levels, (3) at 60ı C, the stiffness curves were almost flat throughout the fatigue life at both 0.9 and 0.85 load levels with little nonlinearity in the last few cycles. So at 60ı C, the effect of increasing fatigue cycles on the stiffness was negligible and that was the reason behind much longer fatigue life, (4) the damping loss factor curve sloped downwards a bit with increasing numbers of cycles at 60ı C and at both load levels.

4.5 Fatigue Failure Modes A digital camera and StreamPix software were used to monitor the failure modes and crack progression throughout the fatigue tests at different temperatures on both types of CS beams. At each temperature (22ı C, 0ı C and 60ı C) and load level, both CF/RC and GF/RC beams exhibited core shear failure as shown in Fig. 15. Thus, the primary failure mode was core shear failure for the whole range of temperatures for both types of sandwich beams. However, the crack initiation position was different for CF/RC and GF/RC beams. Figure 15 explains the approximate starting point of minute cracks in the core at different positions, propagation as a single crack and final rupture point in both types of specimens at 22ı C. The different crack initiation sites for CF/RC and GF/RC beams are attributed to the difference in skin stiffnesses, which led to more significant stress concentrations under the loading points for GF/RC beams than for CF/RC beams. At 60ı C, the beams failed with the same sequence but again the whole failure event took place in the final few cycles.

5 Finite Element Analysis Using the model shown in Fig. 16, FE analysis was performed to predict the elastic behavior of composite sandwich beams and for the comparison of the predicted load–displacement curves from static FEA with the experimental static 4-point bending test results for the sandwich beams. At 22ı C, the slope of the linear elastic region of the curves obtained from FEA matched well with the experimental values of slope. However, there was little or no apparent thermal effect on the results at 60ı C for the FEA model, because data on the temperature dependence of skin and core properties were not available. Thus, the present models were only useful for predicting the behavior of sandwich beams at RT. In another analysis on the same model, the von Mises stress contours generated inside the core along the length of the beam between loading and supporting points were investigated for both the GF/RC and CF/RC beams subjected to static load only as shown in Fig. 17.

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Fig. 15 Core shear failure sequence in CF/RC and GF/RC beams under flexural fatigue loading at 22ı C

The mechanical loading caused the maximum von Mises stresses to occur just below the loading points at the skin-core interface in the GF/RC beam model, while in the case of the CF/RC beam model, these stresses were almost uniformly distributed along the region in between the loading points and the supporting points. Thus, these predictions from the static FE analysis threw more light on the experimentally

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Fig. 16 Finite element model for static 4-point flexural loading condition

Fig. 17 Von Mises stress contours from the FE analysis for CF/RC and GF/RC beams

observed difference between the failure initiation phenomena of GF/RC and CF/RC beams. In GF/RC beams, failure started with minute cracks just below the loading point in the core due to skin indentation and local core crushing, while in CF/RC beams failure started with minute shear cracks in the region between the loading points and the supporting points, as the much stiffer carbon fiber skins prevented skin indentation and local core crushing.

6 Conclusions The effects of subzero temperatures on the flexural fatigue behavior of GF/RC and CF/RC sandwich composite beams have been investigated and the following conclusions have been drawn:

Core shear was the only failure mode observed in both types of sandwich beams under both flexural static loading conditions and flexural fatigue loading conditions at different temperatures and there was no effect of subzero temperature on the failure mode of the sandwich beams.

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At 60ı C and high load levels, the fatigue lives of CF/RC beams were increased by almost hundred times compared with 22ı C, while this increment in number of cycles was limited to 20–40 times compared with 22ı C for the case of GF/RC beams. Thus, the percentage improvement in useful fatigue life was much better in CF/RC beams than in GF/RC beams with corresponding reduction in the temperature.

As there was no failure after 105 cycles in GF/RC beams at 60ı C and 80% load level, it was confirmed that the fatigue life was greatly improved at such a low temperature.

The stiffness reductions during the fatigue tests were measured by two different methods which gave results that were in good agreement. At 22ı C and at 0ı C, both types of beams showed significant failure warnings by the slow but steady increase in the rate of stiffness reduction and the appearance of minute visible cracks during fatigue loading after almost 85% of the useful life. However, at 0ı C, the amount of early warning and the slope of the absolute mean displacement vs. number of cycles were reduced, which was a clear indication of failure transition from ductile type to brittle type behavior. Finally, at 60ı C, both types of beam became very stiff and exhibited very little reduction in stiffness during the whole fatigue life. The result was a significant increase in the fatigue life for both beams at low temperatures, but a lack of early warning of impending fatigue failure.

At 60ı C, both beams failed with sudden brittle type core shear failure without any failure warning in the form of minute cracks or stiffness reduction. Thus, at a low temperature working condition, it is difficult to monitor failure events in composite sandwich structures and to take any safety measures similar to those used in their metal counterparts.

It was confirmed from the FE analysis that the failure initiation location depends on the stiffness of the skin materials. As the unidirectional carbon/epoxy skin is stiffer than the woven glass/epoxy skin with the same dimensions, in CF/RC beams the core shear cracks starts somewhere in the core while in GF/RC beams, they start just below the loading point due to core crushing. Acknowledgement This work was performed while all of the authors were located at Wayne State University. The authors gratefully acknowledge the financial support of the United States. Office of Naval Research, and the guidance of ONR Program Officers Dr. Kelly Cooper and Dr. Yapa Rajapakse.

References 1. Zenkert D (1995) Introduction to sandwich construction. Chameleon, London 2. Vinson JR (1999) The behavior of sandwich structures of isotropic and composite materials. Technomic, Lancaster, PA. 3. Sharma N, Gibson RF, Ayorinde EO (2006) Fatigue of foam and honeycomb core composite sandwich structures: a tutorial. J. Sandw. Struct. Mater. 8(4):263–319

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4. Daniel IM, Gdoutos EE, Wang KA, Abbot JL (2002) Failure modes of composite sandwich beams. Int. J. Damage Mech. 11:309–334. 5. Steeves CA, Fleck NA (2004) Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part I: analytical models and minimum weight design. Int. J. Mech. Sci. 46:561–583. 6. Steeves CA, Fleck NA (2004) Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part II: experimental investigation and numerical modeling. Int. J. Mech. Sci.46:585–608. 7. Harte AM, Fleck NA, Ashby MF (2001) The fatigue strength of sandwich beams with an aluminum alloy foam core. Int. J. Fatigue 23:499–507. 8. Burman M, Zenkert D (1997) Fatigue of foam core sandwich beams – 1: undamaged specimens. Int. J. Fatigue 19(7):551–561. 9. Burman M, Zenkert D (1997) Fatigue of foam core sandwich beams – 2: effect of initial damage. Int. J. Fatigue 19(8):563–578. 10. Kanny K, Mahfuz H (2005) Flexural fatigue characteristics of sandwich structures at different loading frequencies. Compos. Struct. 67:403–410. 11. Sharma SC, Murthy HN, Krishna M (2004) Interfacial studies in fatigue behavior of polyurethane sandwich structures. J. Reinf. Plast. Compos. 23(8):893–903. 12. Kanny K, Mahfuz H, Thomas T, Jeelani S (2004) Temperature effects on the fatigue behavior of foam core sandwich structures. Polym. Polym. Compos. 12(7):551–559. 13. Soni S (2006) Characterization of fatigue damage in composite sandwich hull materials at low temperatures. M.Sc. Thesis, Wayne State University, Department of Mechanical Engineering, Detroit, MI. 14. Annual Book of ASTM Standards (2000). C 393–00: Standard Method of Flexure Test of Flat Sandwich Constructions, 15(03):23–24. American Society for Testing and Materials, Philadelphia, PA. 15. Baba BO, Gibson RF, Soni S (2006) The influence of fatigue cracks on the modal vibration response of composite sandwich beams. Proc. 21st Annu. Tech. Conf. American Soc. Compos. Dearborn, Michigan, paper No. 101.

Impact and Blast Resistance of Sandwich Plates George J. Dvorak, Yehia A. Bahei-El-Din, and Alexander P. Suvorov

Abstract Response of conventional and modified sandwich plate designs is examined under static load, impact by a rigid cylindrical or flat indenter, and during and after an exponential pressure impulse lasting for 0.05 ms, at peak pressure of 100 MPa, simulating a nearby explosion. The conventional sandwich design consists of thin outer (loaded side) and inner facesheets made of carbon/epoxy fibrous laminates, separated by a thick layer of structural foam core. In the three modified designs, one or two thin ductile interlayers are inserted between the outer facesheet and the foam core. Materials selected for the interlayers are a hyperelastic rate-independent polyurethane; a compression strain and strain rate dependent, elastic–plastic polyurea; and an elastomeric foam. ABAQUS and LS-Dyna software were used in various response simulations. Performance comparisons between the enhanced and conventional designs show that the modified designs provide much better protection against different damage modes under both load regimes. After impact, local facesheet deflection, core compression, and energy release rate of delamination cracks, which may extend on hidden interfaces between facesheet and core, are all reduced. Under blast or impulse loads, reductions have been observed in the extent of core crushing, facesheet delaminations and vibration amplitudes, and in overall deflections. Similar reductions were found in the kinetic energy and in the stored and dissipated strain energy. Although strain rates as high as 104 =s1 are produced by the blast pressure, peak strains in the interlayers were too low to

G.J. Dvorak () Northwestern University, Evanston, Illinois, USA e-mail: [email protected] Y.A. Bahei-El-Din Center for Advanced Materials, The British University in Egypt, El Shorouk City, Egypt e-mail: [email protected] A.P. Suvorov McGill University, Montreal, Quebec, Canada e-mail: [email protected] 

This research was conducted at Rensselaer Polytechnic Institute in Troy, NY.

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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raise the flow stress in the polyurea to that in the polyurethane, where a possible rate-dependent response was neglected. Therefore, stiff polyurethane or hard rubber interlayers materials should be used for protection of sandwich plate foam cores against both impact and blast-induced damage.

1 Introduction Sandwich plates of the conventional kind consist of stiff and strong face sheets, separated by a thick layer of structural foam, honeycomb, or another light material with reasonably high shear stiffness. Vinyl ester matrices reinforced by glass or carbon fibers or fabrics, and PVC foam cores are the commonly used materials in shipbuilding applications, both to cover parts of the upper structure and as hull surfaces. In both cases, and especially in the latter, the sandwich plate has to withstand impact loads, caused for example, by contact with floating objects, moorings and tools. More severe exposure awaits in combat, where blast and shrapnel resistance are required. Both visible and hidden forms of damage can be expected to occur under any such loads, and then grow under subsequent service loads. The hidden damage, in the form of internal delamination of large interface areas between the face sheet and foam core, can jeopardize structural integrity of the sandwich plate and cause catastrophic failure. Earlier experimental and analytical studies of such delaminations in conventionally designed sandwich plates were conducted by Wu and Sun [1], Lee and Sun [2], Wu et al. [3], Lindholm and Abrahamsson [4], Vadakke and Carlsson [5], and others. Our work on impact introduces and evaluates modified sandwich plate configurations, designed to reduce the risk of crack nucleation and extension. The mechanism of such crack nucleation is well known, caused by generally reversible deflection of the face sheet under the point of contact with a concentrated force, and irreversible crushing of an underlying volume of the structural foam core. Upon unloading, the face sheet tends to spring back, but the core cannot follow, leaving a delamination crack that can be extended by the residual stress field for some distance from the nucleation site, along the interface between the outer facesheet and structural foam core. Subsequent cyclic loads in service can extend such delaminations over large interface areas. The modified sandwich plate designs introduce ductile interlayers, such as polyurethane or hard rubber, as well as elastomeric foam, inserted and bonded between the exposed outer face sheet and the core. These can reduce the risk of delamination cracking in different ways. Under a given contact force, the stiff polyurethane or PUR interlayer limits deflection of the outer face sheet, and it also helps to shield the structural foam core from crushing, by absorbing a part of the face sheet deflection. The very compliant elastomeric foam or EF interlayer allows an elevated deflection of the outer face sheet, but it better protects the structural foam core from crushing. Combining both PUR and EF sheets into a double interlayer is also explored for certain applications. Similar results could be achieved using

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suitable substitutions for the interlayer materials selected here as representative of a wider class. They may include various hard rubbers, polyurethanes or silicones. Our initial analysis of both conventional and modified sandwich plate designs focused on quasi-static loads and on low to medium velocity (20–40 knots) impact by a rigid indenter. The results show that the interlayers reduce face sheet deflections and/or the interface displacements leading to core crushing. An interlayer can deform with the face sheet and core after unloading, preventing delamination at both interfaces. For nucleated cracks, it reduces the crack driving forces. It can also seal the core from water penetration in the case of matrix cracking in the face sheet. Next, we examine response of four different designs to blast loads, and as a possible substitute for the polyurethane, add polyurea interlayers which exhibit substantial strain hardening under high strains and strain rates. Both interlayer materials reduce the extent of face sheet delamination, of core crushing and thickness reduction, longitudinal or membrane strains in the face sheets, and plate deflections. They also reduce the imparted kinetic energy, and absorbed strain energy in the blast-deformed plate. However, the strain magnitudes found in our blast response simulations are too low to elevate the flow stress of the polyurea to that of the polyurethane, which also offers better protection from impact damage. Chapter 2 describes response of both conventional and modified sandwich plate configurations to uniform static load. The third chapter is concerned with their response under low and medium velocity impact by rigid indenters with pointed or flat contact surfaces. Local and overall deflections, strain and stress distributions are found under typical loading conditions that may be encountered in service. Moreover, magnitudes of the strain energy release rate of cracks extending along different interfaces are computed after unloading or loss of contact. Chapter 4 is concerned with response of four sandwich plate designs to uniformly applied impulse or blast loads. Extensive finite element simulations with the LS-Dyna software monitor the dynamic response of different parts of a single span of a continuously supported sandwich plate during the time period following arrival of a pressure wave. The Appendix presents a listing of additional ONR-sponsored publication by the authors since 1998.

2 Response to Uniform Pressure The sandwich plate under consideration is continuous, supported by periodically spaced hull frames, with typical span length selected here as L D 800 mm. A representative span of one modified sandwich plate, loaded by a uniform pressure is shown schematically in Fig. 1. The top or outer and bottom or inner face sheets are made of a 3.6 mm thick carbon/epoxy laminate. The outer face sheet is supported by a 5 mm thick ductile interlayer, selected either as a relatively stiff polyurethane (PUR), or as a very compliant elastomeric foam (EF). Both can undergo large, fully reversible deformations. The balance of the cross section thickness is filled with a PVC H100 structural foam which has limited ductility, except in crushing under

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t Fig. 1 Sandwich plate with an interlayer, subjected to a uniform load q

compression. In a conventional design, the sandwich plate has the same outer dimensions, but does not have an interlayer; instead, the structural foam fills the space between face sheets. Response of the selected sandwich plate configurations was examined under constant pressure applied to the entire span, and also under local tractions applied by cylindrical or flat indenters, that make contact at different distances from the supports at several relative velocities. Large plate width is assumed in the y-direction, implying a state of plane strain. Standard stress and displacement and traction continuity conditions are enforced at layer interfaces and periodic boundary conditions are applied at supports. In analysis of overall deflections, each layer is regarded as a linear elastic solid. The interlayers are isotropic, but orthotropic material symmetry is admitted in the face sheets. Material properties appear in Table 1 and Fig. 2. The polyurethane or PUR layer is assumed to be nearly incompressible, with the Poisson’s ratio  D 0:49, while the EF interlayer is regarded as compressible, with  ! 0. The face sheets are quasi-isotropic, .0=˙45=90/s symmetric laminates made of AS4/3501-6 graphite/epoxy plies. The stiff interlayer properties are those of the polyurethane Isoplast 101 resin manufactured by Dow Plastics. The elastomeric foam moduli were selected as typical of the actual range for closed cell polyethylene foam. A finite element model of the plate half-span, created using ABAQUS [6] software, was used to evaluate deflections and local stresses. A low Poisson’s ratio was selected to improve convergence of the solution in the inelastic range. Strength of the foam in hydrostatic compression was taken equal to that in uniaxial compression ¢c0 D 1:7 MPa [4]. The polyurethane interlayer was modeled as an incompressible Neo-Hookean material using hyperelastic material model of ABAQUS. Stress analysis was also performed by expressing the displacements in Fourier sine series [7]. Figure 3 compares the bottom surface deflections caused in the single span by the uniform pressure q D 0:1 MPa as shown in Fig. 1. As expected, the stiff PUR interlayer .E D 1500 MPa/ causes a small reduction in overall deflection in

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Table 1 Dimensions and elastic properties Foam PVC H100 Ec D 111 MPa; c D 0:1, c0 D 1:7 MPa; c D 100 kg=m3 Polyurethane (PUR) Ee D 1;500 MPa; e D 0:49, e D 1;200 kg=m3 Elastomeric foam (EF) Ee D 10 MPa; e D 0 E1 D E2 D 55;022 MPa, Face sheet .0=˙45=90/s E3 D 10;792 MPa, AS4/3501-6 carbon/epoxy laminate G12 D 21;319 MPa, G13 D G23 D 4953 MPa, 12 D 0:29, 13 D 23 D 0:248, f D 1;500 kg=m3 Aluminum honeycomb (AlH) Ec D 2;411 MPa; c D 0:22 c0 D 9:65 MPa; c D 130 kg=m3 Aluminum honeycomb with Ec D 2;411 MPa, c D 0:22 c0 D 19:3 MPa, PVC H100 foam (AlH/H100) c D 200 kg=m3 Cylindrical indenter i D 7;800 kg=m3

hc C te D 50 mm te D 5 mm te D 5 mm tf D 3:6 mm

hc C te D 50 mm hc C te D 50 mm

Radius Ri D 25 mm h D 57:2 mm, L D 800 mm

Sandwich plate, Fig. 1

1.1

5.5

Stress (MPa)

1.0

Polyurethane (PUR)

4.5

0.9

4.0

0.8

3.5

0.7

3.0

0.6

2.5

0.5

2.0

0.4

H100 Foam

1.5

Elastomeric Foam (EF)

1.0

0.3 0.2

0.5

0.1

0.0

0.0

–0.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Stress (GPa)

5.0

–0.1 0.9

Strain (m/m)

Fig. 2 Stress-strain curves of elastomeric foam and H100 closed cell foam under compression, and of polyurethane under uniaxial tension or compression. Notice that the right-hand stress scale applies to the polyurethane

comparison with the conventionally designed plate having a homogeneous foam core .E D 111 MPa/. A stiffer EF interlayer with .E D 45 MPa/ causes a small elevation of the displacement, but another EF interlayer with .E D 10 MPa/ elevates the displacement to almost twice of that of the conventionally designed plate.

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Fig. 3 Deflection of bottom surface of sandwich plates subjected to distributed load q D 0:1 MPa; plate dimensions and material properties are listed in Table 1.

3 Response to Impact 3.1 Medium Velocity Impact at Different Contact Locations When the sandwich plate is subjected to low or moderate velocity impact (10–20 m/s or 19–38 knots), both local and overall deformation of the sandwich plate must be evaluated as a function of location of the impact point relative to the supports of the beam, and of the kinetic energy imparted by an indenter, Fig. 4. The supports are selected here as 2d D 60 mm wide, similar in size to the total height h D 57:2 mm of the plate. Dynamic loading is applied at a variable distance x D s from the left support, by an indenter which has an initial velocity V and mass M . The relative distance of the impact point from the support is defined as q D s=L. The finite element package ABAQUS EXPLICIT [8] was used to model the sandwich plate with inelastic constituents at relatively short dynamic response times. Most exposed to impact damage are sections at and close to the supports, where overall deflections are limited, but local deflections are maximized. Large indentations lead to the extensive core crushing and high crack driving forces upon rebound of the indenter. To reduce indentations, stiffer core materials, such as balsa wood, aluminum honeycomb, or aluminum honeycomb filled with PVC foam should replace the PVC foam above and next to supports. Elastic properties of the aluminum honeycomb, and of honeycomb filled with foam, are identical, Table 1, and correspond to those of the PLASCORE Aluminum Military Grade Honeycomb with cell size 1/8 inches, and foil thickness equal to 0.002 in.

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Fig. 4 Continuous sandwich plate in contact with a rigid cylindrical indenter with 25 mm radius

Behavior of the aluminum honeycomb in inelastic region is qualitatively similar to that of the PVC foam. However, the compression strength is much larger; for aluminum honeycomb it is equal to 9.65 MPa, and for the honeycomb filled with PVC foam it is taken as 19.3 MPa. Increase of the compression strength of the honeycomb core by filling its cells with PVC foam was described by Wu et al. [3]. The deceleration of the indenter by the plate was converted into a time-dependent contact force d 2w P D maz D m 2 (1) dt where az is the acceleration and w is the displacement of the center of mass of the indenter. Since the indentation problem utilizes symmetry with respect to both the x D 0 and y D 0 planes, the az is the only nonzero acceleration component. The force P is the resultant of all forces imposed by the decelerating indenter on the top face sheet. Core indentation under the point of impact, as a function of the relative distance q from the support, are shown in Fig. 5a for the initial velocities of the indenter, V D 20 m=s. Gray lines in the figure illustrate the response of the sandwich plate without the honeycomb reinforcement at the supported sections. Black lines denote the response of beams with the reinforcing aluminum honeycomb or aluminum honeycomb filled with PVC H100 foam inserts at the supported sections. Results are plotted for sandwich plates with or without the PUR interlayer. The total length of the support region is 60 mm, but due to symmetry with respect to x D 0, only 30 mm of the total appears in the figure. The PVC foam core indentation remains small and almost constant if the impact takes place away from the support, at a distance larger than about 0.15 L, or 12 cm, comparable to twice

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a

b

Fig. 5 (a) Maximum core indentation and (b) maximum contact force for sandwich plate with or without PUR interlayer subject to impact at distance s D qL from support; indenter initial velocity V D 20 m=s; length L D 800 mm

the total plate thickness. Reducing this distance below 0.15 L leads to a dramatic increase of foam indentation. Inserting the PUR interlayer causes reduction of the core indentation up to 25% if the impact point is at 0:05  q  0:15. On the other

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hand, adding the aluminum honeycomb or one filled with the foam at the supported sections of the beam, greatly reduces maximum indentation of the core. Reduction can be as large as 70%. Figure 5b shows the maximum contact force P , applied by the indenter to the top face sheet of the plate. Corresponding core indentations are shown in Fig. 5a. The contact force remains at an almost constant low value if the impact occurs at a distance q > 0:1 L, or less than two plate thicknesses away from the support. If the soft PVC foam is the core material (gray line), then the contact force at the supported section increases to only twice the lowest value. With the stiff aluminum honeycomb, or foam filled honeycomb in the supported region, the maximum contact force increases to about three or five times the value at the center of the beam. The unfilled honeycomb seems to offer a reasonable compromise between reduction of core indentation and face sheet strain elevation. Although the reduction of the core deflections by the PUR interlayer is relatively small, about 10%, the energy release rates of interfacial cracks are substantially lower when the interlayer is present, as shown below.

3.2 Response to Impact at Support To illustrate the effect of indenter impact on deflection and contact force magnitudes and rates at lower relative velocities of 1–3 m/s, a cylindrical (R D 25 mm) indenter, with mass m D 0:06 kg=mm per unit y-direction length was applied to different sandwich plate configurations. In all cases, contact took place at the support, where the contact force reaches a maximum, at q=L D 0 in Fig. 4. In this case, dynamic response of the sandwich plate was evaluated with an implicit direct-integration dynamic operator, or the ˛-method of Hughes et al. [9], Chung and Hulbert [10], implemented in the ABAQUS software. Figure 6a shows deflections ˛ h=2 of the top face sheet, and ˛ i nt of the foam core interface with either the top face sheet F, or the PUR or EF interlayers. The interfaces between two adjoining materials are denoted by letter symbols, for example, F/C means the interface between the face sheet and foam core. External surface of the face sheet is also denoted by letter F. Indenter velocity is 3 m/s. Deflection rates are comparable to the impact velocity, of the order of 1 m/s. Dashed lines show the surface and F/C interface deflections in a conventionally designed plate. The two adjacent solid lines indicate that the stiff PUR interlayer causes a small reduction of both these deflections. As expected, the compressible EF interlayer allows much larger deflection of the face sheet, while the foam core deflection has been much reduced. Figure 6b shows magnitudes of contact forces P generated in different plate configurations by the indenter mass used in Fig. 6a, approaching the top face sheet at several impact velocities. Again, dashed lines apply to the conventionally designed plate. Increasing the impact velocity V from 1 to 3 m/s has a significant effect, raising the contact force maximum from 90 to 190 N/mm, or by more than 100%. The

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a

b

Fig. 6 (a) Deflection at interface of foam core ˛ i nt and outer face sheet indentation ˛ h=2 ; (b) contact force P for a sandwich plate with or without interlayer impacted by a cylindrical indenter of radius Ri D 25 mm

two solid lines were both found at impact velocity V D 3 m=s, for plates with either PUR or EF interlayers. The stiff interlayer tends to elevate both the contact force maximum and the loading rate. On the other hand, both the load and its rate are reduced in plates with the compressible EF interlayer.

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Approximate values of strain rate magnitudes in the foam or PUR and EF interlayers can be deduced from the results in Fig. 6b. For example, at V D 3 m=s, the time to reach P D 200 N=mm is approximately equal to 1:2  103 s. Stress caused by this load at the foam core interface turns out to be zi nt  2:5 MPa. This suggests the stress rate of about 2083 MPa/s. Dividing this by the foam core modulus EC D 111 MPa gives the strain rate of 18.7/s. For the PUR interlayer, with Ee D 1500 MPa, the strain rate is equal to 1.38/s. Therefore, the local strain rates expected at impact velocities of 1–30 m/s are of the order of 1–200/s, depending on the local material stiffness. For the PUR interlayer at temperatures below the glass transition temperature T < Tg 100ı C, the effect of the strain rate on the elastic moduli is very small and can be neglected. The same is true for the PVC structural foam core which inherits the strain-rate and temperature dependence of the solid of the cell walls. However, the compressive strength C0 of the foam depends on the strain-rate, even at ambient temperatures. In particular, according to Gibson and Ashby [11] T 104 C0 =N C0 D 1  0:1 ln (2) Tg "P where N C0 is the strength at 0ı K. For example, assuming "P D 18:7 and T D 20ı C, gives C0 D 0:874N C0 . At "P D 102 , there is C0 D 0:723N C0 , showing that the compressive strength is elevated at higher strain rates. This can be explained, in part, by the tighter compression of the cell walls near the impact zone which elevates densification of the foam and the crushing stress. In the EF interlayer, with Tg  70ı C, below the room temperature, both the elastic moduli Ee and the compressive strength of the foam are raised by an increase in strain rate, at least at low densities of  D 10  50 kg=m3 . These foams have low Young’s modulus Ee of the order of 0.1–1 MPa, hence an addition of 0.1 MPa caused by the atmospheric pressure of gas inside the cells of the foam may not be neglected. The EF foam used in the present study has a much higher Young’s modulus, Ee D 10 MPa. Therefore, the effect of strain rate on response of the constituent layers of the modified sandwich plate design can be neglected at the low and medium velocities of 1–30 m/s and ordinary operating conditions. However, such effects may become significant in other interlayer and core materials, or under higher temperatures and loading rates. Of course, inertia effects on overall deflections need to be taken into account.

3.3 Effect of Indenter Shape, Interlayer Moduli and Thickness In the absence of strain rate effects, local deformation of the sandwich plate caused by contact with a foreign object can be modeled by applying to the top face sheet a quasi-static traction distribution, generated by a vertical dynamic force at a cylindrical or flat rigid indenter. The plane strain deformation state is retained. Analytical solutions of such problems are available for elastic layers on both rigid and elastic

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substrates [12–14]. According to the above results in Fig. 5, indentation of the face sheet caused by a given magnitude of applied force, and subsequent damage to the interfaces, reach a maximum when the indenter makes contact at one of the supports of the plate. The solution domain and boundary conditions are similar to those used in Fig. 4, but q=L D 0, hence the indenter is applied at a support. Two indenter contours were used, the cylinder shown in Fig. 4 and a flat indenter with a 40 mm wide contact area. Elastic properties of the face sheets (F) in Table 1 are retained, but those of the interlayers and foam core are expanded beyond the elastic range, Fig. 2. In particular, the structural foam core PVC H100 material (C) is assumed to be elastic in its initial response, up to its compression strength and then undergo inelastic deformation at higher compressive stresses. Plastic strain in uniaxial compression test is related to the applied compressive stress by a power-law strain hardening equation which was implemented in the crushable foam model provided by ABAQUS. A low Poisson’s ratio was selected to improve convergence of the solution in the inelastic range. Compression strength of the foam in hydrostatic compression was taken equal to that in uniaxial compression C0 D 1:7 MPa. Tensile strength was selected as t0 D 0:1C0 (ABAQUS 6.2 Standard User’s Manual) and t0 D 0:8C0 for evaluation of energy release rates. The polyurethane Isoplast 101 (PUR) interlayer was modeled as an incompressible Neo-Hookean material using hyperelastic material model of ABAQUS. Deflections of face sheets and foam core were computed for the applied force P per unit width of the plate in the y-direction, selected in the range of 0–200 N/mm for the cylindrical indenter and up to 300 N/mm for the flat indenter. These selections allowed inelastic deformation or crushing of the foam core, but kept the deflections and the membrane stresses in the face sheet within allowable strength limits. Results that follow show the effects of the two selected interlayer materials in reducing the magnitude of foam core deformation relative to that caused in conventionally designed sandwich plates. Figure 7a, b present a comparison of the indentation deflections caused at the center of the contact zone in different sandwich plate designs by the cylindrical and flat indenters, loaded by P . As expected, deflection of the face sheet supported by the stiff and incompressible PUR interlayer is smaller than that of the face sheet in the conventionally designed plate. The plate with a compliant elastomeric foam (EF) interlayer experiences the largest surface deflection. Examination of the force-deflection diagrams reveals that both the conventionally designed and modified plate with the PUR interlayer exhibit deviations from linear response at P D 100–150 N=mm. This is associated with onset of crushing of the underlying foam core, which allows the face sheet deflections to increase at a higher rate with the applied force. In contrast, the deviation from linearity in the plate with the compliant but compressible EF interlayer starts at much higher applied force of P D 160–200 N=mm. Figure 8a, b show the deflection profiles of the interface of the foam core for the two indenters and loading conditions used in Fig. 7a, b. For a given load, the cylindrical indenter causes higher local deflections, and the PUR interlayer is seen

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a

b

Fig. 7 Force – skin indentation relationship for sandwich plates with or without an interlayer, impacted by a cylindrical or a flat indenter. Notice the different vertical scales

to reduce their magnitude at the foam core surface. Both indenters appear to affect a relatively large area of the plate. Of interest in design is the relative effect of interlayer thickness and elastic modulus on deflection of the foam core, and on normal membrane stress in the face

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a

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Fig. 8 Deflection distribution at the foam core interface ˛ i nt in a sandwich plate with or without an interlayer. Forces applied to the cylindrical or flat indenters are P D 200 N=mm or P D 300 N=mm, respectively

sheet. Figure 9a shows the magnitude of the maximum deflection (at x D 0 in Fig. 4), caused by the cylindrical indenter at the foam core interface with an incompressible interlayer. The selected ranges of elastic moduli and thicknesses of the interlayer should accommodate most candidate materials and sandwich plate thicknesses. The

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a

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Fig. 9 Deflection of the foam core interface ˛ i nt ; and maximum tensile stress xi nt in the face sheet supported by an incompressible interlayer with Young’s modulus Ee and thickness te ; Poisson’s ratio e D 0:49. Load applied to the cylindrical indenter is Pmax D 200 N=mm

implication is that both high interlayer thickness and modulus reduce interface deflections, and crushing of the foam core. Interlayer thickness has a significant effect on deflection magnitude even at high modulus values, where the influence of interlayer stiffness is much reduced. The

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Fig. 10 Distribution of the stress xi nt in the top face sheet, at the interface with the foam core C, or PUR interlayer, for a sandwich plate impacted by a cylindrical or a flat indenter; applied force 200 N/mm

relative effect of interlayer thickness and elastic modulus on the face sheet normal stress at the interface with the interlayer is illustrated in Fig. 9b. The overall trend is similar to that shown in Fig. 9a, but an increase in interlayer stiffness appears to have a stronger influence on face sheet stress reduction, and on its persistence at high stiffness values. Figure 10 shows the effect of indenter shape on local distribution of the maximum tensile stress xi nt in the face sheet. For the cylindrical indenter, the maximum values are found at the center x D 0 of the contact zone. Maxima caused by the flat indenter are found in the vicinity of the corner at x D 20 mm, in the coordinates of Fig. 4. For both indenter shapes, the stiff PUR interface offers a significant reduction of the face sheet stress.

3.4 Energy Released by Interfacial Cracks Crushing of the foam core and the resulting residual stresses at interfaces with the foam core and face sheet can impair the strength of the interfacial bond. It can initiate cracks that may then extend under cyclic service loads, and cause extensive but hidden internal delamination of the sandwich plate. The propensity to interfacial cracking is evaluated here in terms of the energy rate G released by interfacial

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Fig. 11 Refinement of the finite element mesh at the crack tip

cracks. These cracks are driven only by the residual stresses during the last stage of unloading, when the resolved normal stress across the particular interface changes from compression to tension. Two well known methods are used in the evaluation of G; the crack closure integral and the J -integral. Both are computed using the crack tips fields obtained from finite element analysis. Figure 11 shows the refined mesh used for cracks at the interfaces PUR/C or EF/C between the two interlayers and foam core C. Similar refinements were used for cracks at F/C and F/PUR, F/EF interfaces between face sheet F and foam core C, or face sheet and interlayers. The crack closure integral evaluation of G assumes purely elastic response of the materials adjoining the crack tip. That was enforced by elevating the tensile strength of the foam core in tension and thus avoiding inelastic deformation during unloading. As noted by Moran and Shih [15], the presence of thermal or residual stresses renders the J -integral globally path dependent. However, path-independence is restored if the -contour is sufficiently small. In the present evaluation, coincides with the outer boundary of the second ring of elements surrounding the crack tip, which has the radius of 0.25 mm. Typical results of the strain energy release rate calculations for cracks driven by the residual stresses at interfaces inside the sandwich plate are shown in Fig. 12. As noted earlier, cracking follows unloading from the selected value of Pmax , and the energy release rate is monitored here at load levels P D Pu , when G reaches reasonably large magnitudes. Dashed lines in Fig. 12 indicate magnitudes of energy release rate G for cracks extending along the F/C interface between the face sheet and foam core in a conventionally designed plate, without the interlayer. As the applied load is reduced from Pu D 30 N=mm to Pu D 0, the magnitude of G grows substantially. Solid lines indicate the much lower magnitudes of G computed for cracks extending at the interface PUR/C between the stiff polyurethane interlayer and the foam core. Even lower magnitudes of the strain energy release rate were computed at the interface F/PUR between the face sheet and polyurethane interlayer. Energy release

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Fig. 12 Energy release rate G versus half crack length a, for Pmax D 200 N=mm applied to cylindrical indenter. Crack extends at interface with foam core. Points and lines represent G found from crack closure and J integrals, respectively. Dashed (solid) lines correspond to the energy release rate for the plate without (with) interlayer

Table 2 Energy release rates G (N/m) at the interfaces with the foam core C and the face sheet F, for various plate designs loaded by a cylindrical indenter Pmax D 200 N=mm Pmax D 150 N=mm Crack length F/C PUR/C EF/C F/C F/PUR F/EF 2a (mm) 16 508 330 245 203 32 25 20 583 376 249 222 48 25 36 188 86 11

rates at the interface between the PUR interlayer and face sheet F are similar to those found at the PUR/C interface. Similar results computed after unloading from different impact force magnitudes appear in Bahei-El-Din et al. [14]. In summary, values of G found at PUR/C and F/PUR interfaces are much lower than those at the F/C interface. Of course, the maximum applied force has a strong effect on the value of G after unloading, as shown in Table 2 below. This table compares values of energy release rates G and the J -integrals evaluated from finite element solution, for different designs of the sandwich plate without interlayer, and with PUR or EF interlayers, loaded by a cylindrical indenter. G values are given at interface with the foam core, at final stage of unloading, i.e. Pu D 0, from the maximum load Pmax D 200 N=mm, and at interface with the face sheet at Pu D 0 when Pmax D 150 N=mm.

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Table 3 Values of debond fracture toughness Gc (N/m), for the face sheet – foam core interface in a conventionally designed sandwich plates Foam core Face sheet Gc (N/m) Reference H100 PVC composite 500 Li and Weitsman [16] H130 PVC E-glass/Vinyl ester 900 Veazie et al. [17] Nomex honeycomb graphite/epoxy 1,177 Berkowitz et al. [18] H100 PVC carbon-fabric/epoxy 00 Shivakumar et al. [19]

The results clearly illustrate the principal benefit provided by presence of the interlayer, which is reduction of the interfacial crack driving forces derived from residual stresses generated by inelastic deformation caused by low-velocity impact. While the elastomeric foam (EF) interlayer offers greater reduction of G, it lacks the stiffness sufficient for maintaining small plate and face sheet deflections under distributed and concentrated loads. Comparison of the computed values with experimentally measured interface fracture toughness, Table 3, indicates that in the conventionally designed plates, the energy release rates G may exceed the toughness Gc . When the sandwich plate is impacted by an indenter, the value of toughness Gc are expected to be lower than that indicated in Table 3 due to crushing of the foam core. Also, the values of energy release rates G can be higher than those found if the foam core with lower hardening is selected in the sandwich plate design. However, interface cracking can often be prevented if the PUR interlayer is inserted between the facesheet and the foam core.

4 Response to Impulse or Blast Loads This part of our research examines the effect of design modifications on response of sandwich plates to impulse or blast loads [14, 20–23]. The objective is to limit damage by delamination of the facesheets and by crushing of the structural foam core. To this end, we introduced structural elements for energy storage, and for reduction of core crushing that dominates response of conventionally designed sandwich plates. In particular, ductile interlayers inserted between the outer facesheet and the foam core, can absorb a significant part of the incident energy, and protect the foam core from excessive deformation. These modifications are similar to those proposed above to mitigate the effect of low and medium velocity impact on sandwich plates, where they enhanced resistance to local deflections of the facesheet, foam crushing and interface delaminations [24–26]. At the high strains and strain rates generated by impulse loads, polyurea interlayers, which exhibit elevated flow stress under such loading conditions, appear to be a suitable replacement for the polyurethane. However, as discussed in Section 4.6 below, both materials respond in a similar manner to blast loads, and the polyurethane offers better protection against impact-induced damage.

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Explicit dynamic finite element solutions were developed with the LS-Dyna software. Contact algorithms were invoked to model intermittent separation and rejoining of the facesheets and the inner core following delamination. Both impulse and explosive pressure loads were applied to the outer facesheet of a section of a continuously supported plate.

4.1 Geometry and Material Properties The sandwich panel considered in this is continuously supported by equally spaced rigid stiffeners, as shown in Figs. 13a, b. Total width, measured in the X2 -direction of Fig. 13a, is assumed to be sufficiently large, so that the plate can be analyzed in plane strain, with displacements u2 D 0 everywhere. Figures 13c, d show shapes of loading pulses applied to the plate in Fig. 13b. The four designs of plate cross sections shown in Fig. 14 are as follows: Design (1) is the conventional arrangement, where laminated composite facesheets are bonded directly to a structural foam core, to form a symmetric sandwich cross section. Designs (2) is modified by inserting a polyurea interlayer between the outer facesheet and the foam core. Performance comparisons with a similar design that uses the polyurethane Isoplast 101, Table 1, are described in Section 4.6. Design (3) is modified by inserting a fairly compliant elastomeric foam (EF) interlayer between the outer facesheet and the foam core. Design (4) is modified by inserting both polyurea and EF interlayers between the outer facesheet and the foam core.

Fig. 13 Geometry and loading of a continuous sandwich plate

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1

2

3

4

Fig. 14 Cross sections of standard and modified designs of sandwich plates

1.8 1.6

Polyurethane (rate-independent) Polyurea (rate-dependent)

1.4 Stress (GPa)

1.2 1.0

8000 s–1

0.8 0.6

5000

0.4 2000

0.2 0.0 0.0

0.2

0.4

0.6

0.8 1.0 Strain (m/m)

1.2

1.4

1.6

1.8

Fig. 15 Stress–strain diagram of polyurea and polyurethane Isoplast 101 under compression

In Designs (2) and (4), the polyurea interlayer exhibits substantial increase in stiffness under large shock compression, Fig. 15, to better support the outer facesheet. Design (3) employs a fairly compliant elastomeric foam (EF) interlayer, which protects the foam core, albeit at the expense of higher facesheet and overall deflections. Design (4) utilizes both polyurea and EF interlayers, each 5.0 mm thick, in an attempt to combine the described benefits. Facesheets are made of a AS4/3501-6 carbon/epoxy fibrous composite laminate, and each consists of eight plies arranged in a quasi-isotropic .0=˙45=90/s

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symmetric layup. Their properties appear in Table 1. While the carbon fiber is elastic, the epoxy matrix may exhibit time dependent deformations. However, elastic response is expected to dominate the behavior of the laminate under high strain rates. Therefore, the facesheets are assumed to remain linearly elastic during the loading cycle and are modeled as homogeneous orthotropic material layers using LS-Dyna Material Type 2. Polyurea interlayer exhibits pressure and rate sensitivity under shock compression [27, 28], and was used in analysis of Design (2) by Bahei-El-Din et al. [14, 21–23]. Experiments on polyurea under high velocity impact suggest that the material can be modeled as elastic-plastic-hydrodynamic, and plastically incompressible with a pressure dependent yield strength, Fig. 15. The constitutive relation corresponding to the polyurea response shown in Fig. 15 is (3) y D y0 C Eh "Np C .a1 C a2 p/ maxŒp; 0 where y0 is yield stress of the virgin material, Eh is elastic-plastic tangent modulus of the stress-true strain curve under compression, and "Np is effective plastic strain. Isotropic hardening due to induced pressure p is given by the last term in Eq. 3, where a1 ; a2 are material parameters in Table 4. Material behavior under hydrostatic load, or dilatation caused by shock compression, is described by Gruneisen’s equation of state  0 C 2 1 C .1  0 =2/  a 2 =2 i C .0 C a / e (4) pD h 2 3 1  .S1  1/  S2 .1C /  S3 .1C /2 Here, e is the internal energy per current specific volume and D .=0 /  1 D .V0 =V /  1 is a measure of dilatation, with 0 and V0 denoting the initial mass density and volume, and  and V their current counterparts. Constants C , S1 , S2 , and S3 are fitting parameters for the shock velocity - particle velocity curve, 0 is the Gruneisen gamma, and a is the first order volume correction to 0 . In the finite element analysis, the polyurea interlayer was represented by LS-Dyna Material Type 10 and Equation of State Type 4. Table 5 presents the material parameters utilized in (3) and (4) for polyurea to qualitatively correlate the computed response with the experimental results [28]. Predictions of this material model under strain-controlled compressive strain, applied at various rates in Fig. 15, exhibit sensitivity to both strain rate, strain and pressure. Reversing applied strain causes a sudden reduction in the stress due to unloading of the induced shock pressure, followed by recovery of elastic strains.

Table 4 Plasticity and equation of state (EOS) parameters for polyurea Model Reference equation Parameters Plasticity y0 D 10 MPa; Eh D 10 MPa Eq. 3 a1 D 2:0; a2 D 0 Gruneisen’s EOS C D 25 m=s; o D 1:55; a D 1:0 Eq. 4 S1 D 2:0; S2 D S3 D 0

Impact and Blast Resistance of Sandwich Plates Table 5 Elastic properties and dimensions of sandwich plate constituents H100 .0= ˙ 45=90/s AS4/3501-6 Divinycell Property Carbon/epoxy foam (units) Polyurea Isotropic, elastic–plastic Orthotropic, Isotropic, hydrodynamic elastic crushable Material type LS-Dyna material 2 63 10 number 55022.0 E1 D E2 (MPa) 111.0 2500.0 E3 (MPa) 10792.0 21319.0 G12 (MPa) 50.45 860.0 G13 D G23 (MPa) 4953.0 0.29 12 0.1 0.4653 13 D 23 0.248 3 .kg=m / 1,580 100 1,070 Compressive yield – 1.7 10.0 strength (MPa) Tensile strength – 0.3 – (MPa) Maximum tensile – 0.28 – strain (%) Maximum shear – 3.5 – strain (%) Thickness (mm) Design (1) 50.8 – Design (2) 3.6 45.8 5.0 Design (3) 45.8 – Design (4) 40.8 5.0

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Elastomeric Foam (EF) Isotropic, crushable 63 10.0 5.0 0.0 148 0.0264 – – –

– – 5.0 5.0

Elastomeric (EF) foam is closed-cell polyethylene, with variable properties. LS-Dyna Material Type 63 represented the elastomeric foam interlayer. Structural foam core material is H100 Divinycell, an isotropic, closed cell foam. Under uniaxial compression, it deforms as shown in Fig. 2 [29]. Essentially incompressible response is reached at the densification strain of approximately 80%. At a small Poisson’s ratio, this is essentially a uniaxial crushing model. Unloading is elastic up to a small tensile cutoff stress as shown by the dashed line. Reloading is elastic up to a stress and strain pair, which falls on the curve of Fig. 2. In the finite element calculations, the foam was also modeled by LS-Dyna Material Type 63 with properties given in Table 5. Interfaces are assumed to be well bonded. Delamination cracks are expected to occur in the Divinycell structural foam, along a path adjacent to its interface with either the facesheets or interlayers. To this end, the foam core was subdivided by a thin layer of elements at such interfaces. Delamination is modeled by removing from the mesh thin foam interface elements, using material erosion capability of LS-Dyna. Maximum strain failure criteria are utilized to initiate failure of the foam

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interface elements and their elimination from the mesh, when element principal normal strain, "1 , and maximum shear strain, max , satisfy "1  "ult or max  ult

(5)

where "ult D 0:28% and ult D 3:5% are the ultimate normal and shear strains. They were estimated by assuming elastic behavior of the foam under tensile and shear loads, with ultimate strength magnitudes of 0.3 MPa and 1.4 MPa, respectively. All interfaces between the foam core and facesheets, or polyurea interlayer, were kept in contact without cohesion in stress free state, and allowed to open under tensile traction, and slide relative to each other under shear, while interpenetration was prevented. Blast loads were idealized by either uniform pressure impulse, p.t/ D p0 ı.t/, where ı.t/ is the Dirac delta function, Fig. 14c, or an exponentially decaying function of time t [30, 31], Fig. 14d   t p.t/ D pmax 1  e bt=td td

(6)

where pmax is the peak reflected positive pressure and td is the time for pressure reversal. Pressure decay rate is given by the ratio b=td . Negative pressure phase of the blast load diminishes for b  1:0. Since negative pressure phase is usually neglected in the analysis of structures subjected to explosives, the magnitude of b was assumed equal to 2.0. In the present study, it was assumed that the peak pressure is 100 MPa, and pressure reversal time is 0.05 ms. A separate estimate shows that this peak blast pressure is equivalent to that caused by a spherical, 6.7 kg of TNT explosive charge, placed at a distance of 35 cm from the outer surface of the sandwich plate, measured from the center of the charge.

4.2 Finite Element Models Response of the four sandwich plate designs to blast loading was examined using the finite element method. The LS-Dyna software [32] was used. It performs a Lagrangian dynamic analysis using an explicit, central difference integration scheme. In the integration, the nodal accelerations are used to advance the velocity solution to time t C t=2, which in turn is used to calculate the displacements at time t C t . The central difference operator is conditionally stable for time increments that are smaller than Courant limit, t  `=c, where ` is the smallest element dimension, c D Œ. C 2 /=1=2 is the speed of sound waves in the medium, ; are Lame’s constants, and  is the density of the material. Inserts in Fig. 16 show details of the mesh for the four Designs (1)–(4). A uniform mesh was utilized in the longitudinal, X1 -direction in all material parts. Element dimensions in the thickness direction X3 varied among the material parts. Thickness

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Fig. 16 Finite element solution domain and mesh. PUR denotes polyurea

of the interlayers was divided into two elements; however a finer mesh with ten elements across the thickness was used with the polyurea interlayer in Design (2), to provide a refined simulation of wave dispersion in this rate sensitive material.

4.3 Response to a Full Span Pressure Impulse When the pressure impulse of Fig. 13c is applied to the entire span of the sandwich plate, Fig. 13b, each of the four designs of Fig. 14 undergoes a particular deformation history, which determines distribution of the kinetic and strain energy absorbed by different layers of the sandwich structure. Response can be divided into two time periods. In the first period of approximately 0–0.2 ms, the plate undergoes thickness reduction and core crushing. The second period, lasting several ms, is dominated by top facesheet delamination and overall bending. Deformations and facesheet delaminations in standard sandwich plate Design (1) and the modified Design (4), are illustrated in Fig. 17 at t D 0:1, 0.2 ms, beyond the transient response loading period of the impulse load of Fig. 13c. The foam core undergoes large compressive or crushing deformation in the top half of the core layer. Under the uniformly applied pressure, the foam compression is largest in center of the span, and then decreases in sections that are located closer to supports.

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(t = 0.1ms)

Design (1)

(4)

(t = 0.2 ms)

(1)

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Fig. 17 Deformed geometry of a sandwich plate with a standard Design (1) and modified Design (4) with two interlayers

Delamination occurs in all designs, but the opening displacement is lowest in Design (4), moderate in Design (2), and very extensive in Design (1). Large displacement gradients are present in both the outer and inner facesheets. In the outer facesheet, these are associated with the significant deflections of its surface under the applied pressure, while in the inner facesheet gradients are caused by deflections constraints imposed by the supports. Figure 18 compares the deflection histories for the four designs at midspan. In all cases, progressive crushing of the foam core slows down the incident compression wave and delays the onset of deflection by approximately 0.1 ms. Results show that a single EF interlayer (Design (3)) enhances response of the plate to impulse loading from that of the standard Design (1) only marginally. Also, the EF interlayer used in combination with a polyurea interlayer in Design (4) shows only a small reduction in deflection compared to that obtained with a single polyurea interlayer (Design (2)). Consequently, our analysis focused on the behavior of sandwich plates in Design (2) with polyurea interlayer, to take advantage of its enhanced performance under shock compression. Averaged core compressions at midspan, computed as the ratio hc = hc , where hc is thickness of the foam core and hc is the change in its magnitude, for Designs (1) and (2) are compared in Fig. 19. A steady state is reached within about 0.3 ms, after a rapid rise to average strains of 0.35–0.65. The stiffening interlayer in Design (2) appears to absorb the induced shockwave, and thus better protect the inner foam core from crushing, which is reduced to about 50% of that in Design (1).

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Fig. 18 Comparison of mid-span deflection of a sandwich plate subjected to impulse load, with a standard Design (1) and new Designs (2)–(4).

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0.4 0.3 0.2 0.1 0.0 0.0

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Fig. 19 Average core compression of a sandwich plate subjected to blast load, for a standard Design (1) and modified Design (2) with a polyurea interlayer

While these averages indicate the magnitude of thinning of the sandwich plate cross section, they are caused by much larger compressive or crushing strains, which are not uniform across the thickness. Figure 20 shows distribution of the lateral strain "33 within the foam core thickness at mid-span for Designs (1) and (2) at 0.5 ms. At a densification strain of about 0.8, it is evident from the stress– strain response of the H100 foam, Fig. 2, that the foam in Design (1) reaches full

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Fig. 20 Crushing of the foam core under blast load. In Design (2), PUR is polyurea interlayer

densification for the outer 60% of the foam cross section. On the other hand, only the outer 20% of the core cross section in Design (2) is fully compressed.

4.4 Energy Absorption Total energy imparted by the applied pressure impulse to the sandwich plate layers is converted to kinetic and strain energy. The latter is either dissipated by inelastic deformation and damage, or it is stored in the individual layers after unloading. Elastic strain energy is stored primarily in the outer and inner composite face-sheets. Energy dissipation occurs in the crushable foam core, and in the elastic–plastic polyurea interlayer in Design (2). The reported energy magnitudes refer to those in the total volume of each layer. We note that the volume ratio of the outer facesheet to the foam core is 7% in Design (1) and 8% in Design (2). The volume ratio of the polyurea interlayer to the foam core in this design is 11%. Figure 21 shows time histories of kinetic energy of individual layers of the sandwich plate Designs (1) and (2). Kinetic energy is initially imparted by applied load to outer composite facesheet. The induced compression wave propagates into the underlying materials, with particle velocity affected by the different material properties. In Design (1), the facesheet is supported by the foam core, which exhibits permanent crushing at low stress. Therefore, total kinetic energy in this design is mostly imparted to the outer facesheet at a peak value of 60 J.

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Fig. 21 Distribution of kinetic energy

Fig. 22 Distribution of stored strain energy

Distribution of strain energy among the sandwich plate material parts is shown in Fig. 22. In both designs, the energy absorbed by the elastic composite facesheets is small, and is eventually recovered. Most of the strain energy of the sandwich plates is dissipated by permanent deformations of the foam core in both designs, and of the polyurea interlayer in Design (2). In Design (1), the energy dissipated by crushing of the entire volume of the

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foam core is 40 J. In the enhanced Design (2), both crushing of the foam core and plastic deformation of the polyurea account for the total dissipated energy of 25 J; a reduction of about 38%.

4.5 Effect of Change in Total Mass Since the overall dimensions of all plate designs were kept constant, replacing the top 5 mm of the inner foam core with a polyurea interlayer of equal thickness in Design (2), increases the total mass by 30% in over that in Design (1), Table 5. Whether the improvements found in response of the sandwich Design (2) under blast loads can be realized while the total mass of the plates is kept constant, was examined by modifying the thickness of the foam core. Since it is not possible to achieve equal mass in the two designs examined here by reducing the thickness of Design (2), the thickness of the foam core in the conventional Design (1) was increased from 50.0 to 98.5 mm. In this case, the total thickness of the plate in Design (1) is 105.7 mm, compared to 57.2 mm in Design (2), and the mass per unit surface area for each plate is 21:23 kg=m2 . The enhanced Design (2) with a polyurea interlayer continues to show a significant improvement in response to blast load compared to the conventional sandwich Design (1) of equal mass. The total, peak kinetic energy was reduced by 43%, and energy dissipation in the entire volume of the foam core was reduced by 38%. These reductions are similar to those found when the standard and modified sandwich plates have equal thickness. The reason is that a smaller volume of the foam core is permanently compressed when a polyurea interlayer is used. However, the larger thickness of the conventional sandwich plate with mass equal to that in the modified design leads to smaller bending deformations. Of course, differences in total mass and thickness have no effect on response to impact, which affects, at least prior to large scale delamination, only a small material volume surrounding the point of contact.

4.6 Performance Comparison of Polyurea and Polyurethane In a more recent paper [22], we have compared the dynamic response of Design (2), where the interlayer was made either of the polyurea (Table 5), or Isoplast 101 polyurethane (Table 1). Material properties, geometry, as well as loading and boundary conditions were identical to those described herein. As shown in Figs. 2 and 15, the compressive flow stress of polyurea is much lower that that of the polyurethane at strains as high as 1.0, or (100%). Therefore, the polyurea may not be fully utilized at lower compressive strains. Moreover, since dynamic response of polyurethane was not known, it was neglected in the simulation. Flow stress elevation at high strain rates would enhance utility of the polyurethane under blast loads.

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Overall dimensions of all plates, including the total thickness were not altered, resulting in slight increase in mass of the polyurea interlayer. When compared to conventional sandwich plate designs, both interlayers provided significant protection to the foam core and reduced the kinetic energy imparted to it by the pressure impulse. The underlying mechanism leading to this effect is that the stiff interlayers deliver a lower amplitude of the compression wave to the foam core. Crushing of the core, as well as the dissipated energy, deflections, and facesheet strains were significantly reduced. However, the plate with the hyperelastic polyurethane interlayer showed a slightly better energy absorption than that with a rate-dependent, elastic-plastic polyurea interlayer. In all other respects, the benefits offered by both types of interlayer materials are about equal. The main reason for this similar behavior, despite the distinct response to compressive loading, are the relatively small strains produced through the thickness of the interlayers. The maximum compressive strain computed in the polyurethane and polyurea interlayers was 0.60%, and 35.0%, respectively at the interface with the outer facesheet, and 0.25% and 10.0% at the interface with the foam core. However, due to the large flow stress difference, the corresponding stresses are much closer, and therefore, a compression wave of similar intensity passes from the interlayer to the foam core. The distinct, stiffening behavior of polyurea under shock compression observed in high velocity impact experiments is caused by strains that are larger by an order of magnitude than those computed here under blast loading. Under static or impact load, the initially soft behavior of polyurea interlayer is expected to cause large deflections in the sandwich plate, perhaps comparable to the softer EF interlayer .E D 10 MPa/ in Fig. 2. Therefore, the Isoplast 101 polyurethane, or a similar material, should be preferred in protecting sandwich plate against both impact and blast loads.

5 Conclusions The results demonstrate the role of the interlayers in enhancing structural performance of sandwich plates under both impact and blast loads. Compared to conventional sandwich plate designs, the benefits gained from the modified designs can be enumerated as follows: Under impact loads, there is a substantial of reduction in the depth of the outer facesheet indentation and in the magnitude in-plane normal tension stress. While local delamination of the foam core is not prevented at all load magnitudes, the inserted polyurethane interlayer substantially reduces the magnitude of the energy release rate that may drive interfacial cracking. Under blast loads, facesheet delamination remains a major damage mechanism in both standard and modified sandwich plate designs. Permanent crushing of the foam core occurs in all designs. However, the proposed Design (2) provides enhanced performance, both during the initial blast response and during the bending and stretching phase. Compression of the crushable core is reduced by more than

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50%. Longitudinal strain in the facesheets was reduced by about 25%, and plate deflection was reduced by 15%. Substantially lower curvature of the inner facesheet was found at the supports. The much more graceful response found in this design is attributed to the stiffness increase caused by straining a polyurea interlayer. Both foam core crushing, vibrations of the face sheet, and overall deflection of the plate are reduced. As a result, the total kinetic energy of the modified sandwich plate is much lower than that of a conventionally designed plate. Similar reductions are found in the stored and dissipated strain energy. A large part of the reduced strain energy is absorbed by the polyurea interlayer, where it is dissipated by inelastic deformation. As discussed in Section 4.6, under the strain and strain rate magnitudes computed in the blast load simulations, the polyurethane described in Table 1 has a much higher flow stress than the polyurea in Table 5 and in Fig. 15. The polyurethane was assumed rate independent, but its actual flow stress may be further enhanced under high rates. Therefore, it appears to offer superior protection under both impact and blast loads. Acknowledgment The authors gratefully acknowledge sustained support provided to us by the Office of Naval Research and Rensselaer Polytechnic Institute for more than 20 years. Dr. Yapa D.S. Rajapakse served as program monitor.

References 1. Wu C.L., Sun C.T. (1996). Low velocity impact damage in composite sandwich beams. Composite Structures, 34(1), 21–27. 2. Lee, S.W.R., Sun, C.T. (1993). A quasi-static penetration model for composite laminates. Journal of Composite Materials, 27(3), 251–271. 3. Wu, C.L., Weeks, C.A., Sun, C.T. (1995). Improving honeycomb-core sandwich structures for impact resistance. Journal of Advanced Materials, July, 41–47. 4. Lindholm, C.J., Abrahamsson, P. (2003). Modelling and testing of the indentation behavior of sandwich panels subject to localized load. Proceedings of the 6th International Conference on Sandwich Structures, 279–291. 5. Vadakke, V., Carlsson, L.A. (2004). Experimental investigation of compression failure of sandwich specimens with face/core debond. Composites Part B (Engineering), 35B (6–8), 583–590. 6. ABAQUS, Version 6.2. (2001). Hibbit, Karlsson and Sorenson. Pawtucket, RI. 7. Suvorov, A., Dvorak, G. J. (2005). Cylindrical bending of continuous sandwich plates with arbitrary number of anisotropic layers. Mechanics of Advanced Materials and Structures, 12, 247–263. 8. ABAQUS EXPLICIT (2001). Hibbit, Karlsson and Sorenson. Pawtucket, RI. 9. Hughes, T.J.R., Hilber, H.M., Taylor, R.L. (1976). A reduction scheme for problems of structural dynamics. International Journal of Solids and Structures, 12(11), 749–767. 10. Chung, J., Hulbert, G. (1994). A family of single-step time integration algorithms for structural mechanics. Computer Methods in Applied Mechanics and Engineering, 118(1–2), 1–11. 11. Gibson, L.J., Ashby, M.F. (1997). Cellular solids: structure and properties. New York, Cambridge University Press. 12. Johnson, K.L. (1987). Contact mechanics. New York, Cambridge University Press.

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13. Barovich, D., Kingsley, S.C., Ku, T.C. (1964). Stresses on a thin strip or slab with different elastic properties from that of the substrate due to elliptically distributed load. International Journal of Engineering Sciences, 2, 253–270. 14. Bahei-El-Din, Y.A., Dvorak, G.J., Fredricksen. O.J. (2006). A blast-tolerant sandwich plate design with a polyurea interlayer. International Journal of Solids and Structures, 43, 766–744. 15. Moran, B., and Shih, C.F. (1987). Crack tip and associated domain integrals from momentum and energy balance. Engineering Fracture Mechanics, 27(6), 615–642. 16. Li, X., Weitsman, J. (2003). Sea water effects on foam-cored composite sandwich layups. Proceedings of the 6th International Conference on Sandwich Structures, 443–455. 17. Veazie, D.R., Robinson, K.R., Shivakumar, K. (2003). Marine environmental effects on the interfacial fracture toughness of PVC core sandwich composites. Proceedings of the 6th International Conference on Sandwich Structures, 485–495. 18. Berkowitz, C.K., Johnson, W.S., Makeev, A. (2003). Environmental effects on the fatigue and fracture of Nomex honeycomb sandwich structure. Proceedings of the 6th International Conference on Sandwich Structures, 496–510. 19. Shivakumar, K., Chen, H., Smith S. (2003). An evaluation of data reduction methods for opening mode fracture toughness of sandwich plates. Proceedings of the 6th International Conference on Sandwich Structures, 770–783. 20. Dvorak, G.J., Bahei-El-Din, Y.A. (2005). Enhancement of blast resistance of sandwich plates. Proceedings of the 7th International Conference on Sandwich Structures, 29–31 August 2005, Aalborg University, Aalborg, Denmark. 21. Bahei-El-Din, Y.A., Dvorak, G.J. (2007). Wave propagation and dispersion in sandwich plates subjected to blast loads. Mechanics of Advanced Materials and Structures, 14, 465–475. 22. Bahei-El-Din, Y.A., Dvorak, G.J. (2007). Behavior of sandwich plates reinforced with polyurethane/polyurea interlayers under blast loads. Journal of Sandwich Materials and Structures, 9, 261–282. 23. Bahei-El-Din, Y.A., Dvorak, G.J. (2008). Enhancement of blast resistance of sandwich plates. Composites Part B, 39, 120–127. 24. Dvorak, G.J., Suvorov, A.P. (2006). Protection of sandwich plates from low-velocity impact. Journal of Composite Materials, 40, 1317–1332. 25. Suvorov, A.P., Dvorak, G.J. (2005). Enhancement of low-velocity impact resistance of sandwich plates. International Journal of Solids and Structures, 42, 2323–2344. 26. Suvorov, A.P., Dvorak, G.J. (2005). Dynamic response of sandwich plates to medium-velocity impact. Journal of Sandwich Structures and Materials, 7, 395–412. 27. Boyce, M. (2004). Experimental characterization of polyurea with constitutive modeling and simulation. Proceedings of ONR, ERC, ACTD Workshop, MIT, November 2004, Cambridge, Massachusetts. 28. Amirkhizi, A.V., Isaacs, J., Mc Gee, J., Nemat-Nasser, S. (2006). An experimentally-based constitutive model for polyurea, including pressure and temperature effects. Philosophical Magazine, 86(36), 5847–5866. 29. Fleck N.A. (2004). Collapse mechanisms of sandwich beams with composite faces and foam core, loaded in three-point bending. Part II: experimental investigation and numerical modeling. International Journal of Mechanical Sciences, 46, 585–608. 30. Gantes, C.J., Pnevmatikos, N.G. (2004). Elastic-plastic response spectra for exponential blast loading. International Journal of Impact Engineering 30, 323–343. 31. Low, H.Y, Hao, H. (2001). Reliability analysis of reinforced concrete slabs under explosive loading. Structural Safety 23, 157–178. 32. LSTC, (2003), LS-Dyna 970, Livermore Software Technology Corporation, Livermore, CA 94550.

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Appendix Here we present a chronological list of additional publications that had appeared in 1998–2003, all completed with the support of the Office of Naval Research; Dr. Yapa D.S. Rajapakse served as program monitor.

1. Reiter T. and Dvorak G.J. (1998a). Micromechanical models for graded composite materials IIThermomechanical loading. Journal of the Mechanics and Physics of Solids 46(9): 1655–1674. 2. Reiter, T. and Dvorak, G.J. (1998b). Micromechanical modeling of functionally graded materials. Transformation Problems in Composite and Active Materials, edited by Y.A. Bahei-El-Din and G. J. Dvorak. Proceedings of the IUTAM Symposium, in Cairo, Egypt, March 1997, Kluwer, pp. 173–184. 3. Srinivas, M.V. and Dvorak, G.J. (1998). Design of composite cylinder fabrication process; ibid, pp. 209–220. 4. Dvorak, G.J. and Srinivas, M.V. (1999). New estimates of overall properties of heterogeneous solids. Journal of the Mechanics and Physics of Solids, 47(4), 899–920. 5. Dvorak, G.J., Srinivas, M.V. and Prochazka, P. (1999). Design and fabrication of submerged cylindrical laminates – I. International Journal of Solids and Structures, 36(26), 3917–3944. 6. Srinivas, M.V., Dvorak, G.J. and Prochazka, P. (1999). Design and fabrication of submerged cylindrical laminates II. Effect of fiber prestress. International Journal of Solids and Structures, 36, 3945–3976. 7. Dvorak, G.J. and Suvorov, A.P. (1999). Size effect in fracture of unidirectional composite plates. International Journal of Fracture, 95, 89–101. Also published in Fracture Scaling, Zdenek P. Bazant and Yapa. D.S. Rajapakse (Eds.) Kluwer, Dordrecht. 8. Bahei-El-Din, Y.A. and Dvorak, G.J. (2000). Micromechanics of inelastic composite materials, Chapter 15 in Volume I: Fiber Reinforcement and General Theory of Composites (ed. by T.-W. Chou), of Comprehensive Composite Materials (ed. by A. Kelly and C. Zweben) Elsevier. 9. Dvorak, G. J. (2000) Composite materials: inelastic behavior, damage, fatigue and fracture. International Journal of Solids and Structures, 37, 155–170. Also published in Research Trends in Solid Mechanics, A report form the U.S. National Committee on Theoretical and Applied Mechanics, Elsevier, Oxford, U.K. (ISBN 0-08-043572-6). 10. Dvorak, G.J. and Suvorov, A.P. (2000). Effect of fiber prestress on residual stresses and onset of damage in symmetric laminates. Composite Science and Technology, 60, 1129–1139. 11. Dvorak, G.J., Zhang, J. and Canyurt, O. (2000). Prestressed adhesive strap joints for composite laminates. Proceedings of the 23rd Annual Meeting of the Adhesion Society, edited by G.L. Anderson. The Adhesion Society (ISSN 1086-9506), 186–187. 12. Dvorak, G.J., Zhang, J. and Canyurt, O. (2000). Adhesive joints for thick composite laminates. Proceedings of the 23rd Annual Meeting of the Adhesion Society, edited by G.L. Anderson. The Adhesion Society (ISSN 1086-9506), 389–391. 13. Dvorak, G.J., Zhang, J. and Canyurt, O. (2001). Adhesive tongue and groove joints for thick composite laminates. Composites Science and Technology, 61, 1123–1142. 14. Bahei-El-Din, Y.A. and Dvorak, G.J. (2001). New adhesive joints for thick composite laminates. Composites Science and Technology, 61, 19–41. 15. Dvorak, G.J. (2001). Transformation field analysis of composite materials. Handbook of Material Behavior Models, edited by Jean Lemaitre, Section 10.5. Academic, 996–1003. 16. Suvorov, A.P. and Dvorak, G.J. (2001). Optimized fiber prestress for reduction of free edge stresses in composite laminates. International Journal of Solids and Structures, 38, 6751–6786. 17. Suvorov, A.P. and Dvorak, G.J. (2001). Damage control in composite laminates. ThreeDimensional Effects in Composite and Sandwich Structures, edited by Y.D.S. Rajapakse, AMD vol. 248. ASME, New York, 107–116.

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18. Matous, K. and G.J. Dvorak (2002). Stress redistribution in skin-flange assemblies. Mechanics of Advanced Materials and Structures, 9(3), 257–272. 19. Matous, K. and Dvorak, G.J. (2002). Design of prestressed skin-flange assembly. Journal of Sandwich Structures and Materials, 4, 367–387. 20. Suvorov, A.P., and Dvorak, G.J. (2002). Stress relaxation in prestressed composite laminates. Journal of Applied Mechanics, 69, 459–469. 21. Suvorov, A.P. and Dvorak, G.J. (2002). Rate forms of the Eshelby and Hill Tensors. International Journal of Solids and Structures, 39 (22), 5659–5678. 22. Matous, K. and Dvorak, G.J. (2003). Optimization of electomagnetic absorption in laminated composite plates. IEEE Transactions of Magnetics, 39,1827–1835. 23. Suvorov, P. and Dvorak, G.J. (2003). Analysis of elastic and viscoelastic prestressed composite laminates. Proceedings of 6th International Conference on Sandwich Structures, 842–850.

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Modeling Blast and High-Velocity Impact of Composite Sandwich Panels Michelle S. Hoo Fatt, Leelaprasad Palla, and Dushyanth Sirivolu

Abstract Analytical models for predicting the deformation and failure of composite sandwich panels subjected to blast and projectile impact loading are presented in this paper. The analytical predictions of the transient deformations and damage initiation in the composite sandwich panels were compared with finite element solutions using ABAQUS Explicit. For the blast model, the predicted transient deformation of the sandwich panel was within 7% of FEA results, while the predicted damage initiation using Hashin’s composite failure criteria was about 15% higher than FEA results in most cases. For the high velocity impact model, the predicted transient deformations were within 20% of FEA results.

1 Introduction Lightweight polymer composite sandwich panels are becoming more widely used in military and civilian transport vehicles because they offer greater load-bearing capabilities per unit weight and easier maintenance. In some instances, these composite sandwich panels may be subjected to blast and/or high velocity impact from flying debris of a nearby explosion. During blast and high velocity impact loading of the composite sandwich panel, loads are characterized by a very short time duration compared to panel structural response time. The sharply applied loads initially involve the transmission or propagation of stress waves, which after multiple reflections from the panel boundaries result in structural vibration. If the transient loads are of high intensity, however, they may cause fracture or perforation of the structure during the initial transient phase. This paper provides analytical techniques that can be used to obtain the transient response and determine the damage initiation of composite sandwich panels subjected to blast and high velocity impact.

M.S. Hoo Fatt (), L. Palla, and D. Sirivolu Department of Mechanical Engineering, The University of Akron, Akron, OH 44325–3903, USA e-mail: fhoofatt;lp21;[email protected]

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There is a paucity of research articles addressing analytical methods for determining the behavior and survivability of composite sandwich panels under dynamic loading [1]. Analytical models are essential for elucidating physical mechanisms that control the deformation and survivability of composite panels under blast and impact. Analytical solutions are used by many researchers as a starting point for setting up test and parametric studies. Analytical solutions are also useful in numerical studies because they can be used to benchmark more refined finite element analysis. Analytical models are developed separately for a composite sandwich panel under blast and high velocity projectile impact in the following sections.

2 Impulsively-Loaded Sandwich Panels Consider the fully clamped, composite sandwich panel of radius a, as shown in Fig. 1. The facesheets consist of orthotropic composite plates of thickness h, and the core is crushable polymeric foam of thickness H. Assume for simplicity that the panel is subjected to a uniformly distributed pressure pulse (   po 1  t ; 0  t   (1) p .t/ D 0; t > where po is the peak pressure and  is the load duration. Other more complicated pressure transients can be used to more accurately simulate underwater and air explosions [2, 3], and it will be found later in this study that it is not the actual function used to describe the pressure transient but rather the impulse (integrated area under the pressure pulse diagram) that governs the blast response and ultimate failure of the panel. Provided no failure has occurred to the panel during the blast, the response of the composite sandwich panel may be described by the three phases of motion depicted in Fig. 2a–c. In Phase I, a through-thickness, compressive stress wave propagates from the incident facesheet to the distal facesheet. In this phase the sandwich panel experiences primarily core crushing, while impulsive transverse shear reaction forces are induced around the clamped boundaries. As indicated in Fig. 2a, there is no global deflection in Phase I.

Fig. 1 Composite sandwich panel subjected to uniformly distributed, pressure pulse

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Fig. 2 Three phases of blast response: (a) Phase I: through-thickness wave propagation, (b) Phase II: transverse wave propagation and (c) Phase III: vibration

At the end of Phase I, momentum and kinetic energy are transferred globally to the panel and the now established impulsive transverse shear reaction force just begins to propagate from the clamped boundaries towards the panel center. The pressure pulse resulting from the blast would have either ended or decayed to almost negligible amplitude by the start of Phase II. Momentum, equivalent to the impulse from the blast, would be transferred to the sandwich panel with a reduced core thickness from Phase I. In Phase II, the transverse shear stress wave due to the reaction forces at the clamped boundary propagates from the clamped boundary towards the center of the panel. This transverse stress wave is an unloading wave, causing bending and shear deformations to develop behind its front, as shown in Fig. 2b. The elastic unloading transverse shear wave brings the panel to maximum deflection when it reaches the panel center. At the end of Phase II, the unloading transverse shear wave reverses sign and direction of travel thereby causing the panel to rebound. The transverse shear wave reflects back and forth from the boundary to the panel center in Phase III. As depicted in Fig. 2c, elastic vibrations take place in Phase III. During Phase I, high intensity transverse shear stresses are developed at the clamped boundary and these may cause transverse shear fracture at the clamped boundaries of the panel. Transverse shear fracture can be avoided by using reinforcements at boundaries. The second mode of failure that can occur during blasts is tensile fracture in the center of the panel where the bending strains are at a maximum at the end of Phase II. These two failure modes in addition to permanent

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deformation were first observed on impulsively-loaded aluminum beams by Menkes and Opat [4] and later on aluminum plates by Teeling-Smith and Nurick [5]. They have also been experimentally observed on composite plates by Franz et al. [6]. Analytical methods for predicting the panel response in Phases I and II are given in the following sections. These analytical solutions will then be used to establish damage initiation criteria for the sandwich panel.

2.1 Phase I – Through-Thickness Wave Propagation The wave speed in polymeric foam is low compared to the wave speed in fiberreinforced laminates or metals. A thick composite sandwich panel with a polymeric foam core is likely to undergo transient local facesheet indentation and core crushing while the pressure pulse is still acting. Take for example, H100 PVC foam core with a density of 100 kg=m3 and a compressive elastic modulus of 35 MPa. Elastic uniaxial stress waves propagate through a 25 mm thick core made of H100 PVC foam in about 0.04 ms. An initial pressure pulse duration of this magnitude is common for naval composite sandwich ships subjected to underwater and air blast explosions [2]. Thus one can assume that permanent plastic deformations of the core will take place from a transient event, i.e., during the load application. Phase I response is described by stress waves propagating through the thickness of the facesheets and core.

2.1.1

Transmission and Reflection at Interfaces

The transmission and reflection of a stress waves through the multi-layered composite sandwich panel is shown in Fig. 3. Stress waves are transmitted from the incident facesheet to the foam at Interface 1 and from the foam to the distal facesheet at Interface 2. When the incident stress I first reaches Interface 1, the transmitted stress T1 and the reflected stress R1 are given as follows (see Ref. [7]): 2c Cc  f Cf C c Cc

T1 D kT1 p.t  t1 /u ht  t1 i ; kT1 D  and

 R1 D kR1 p.t  t1 /u ht  t1 i ; kR1 D 

 c Cc   f Cf

f Cf C c Cc

(2)

 

(3)

where t1 D h=Cf is the wave travel time through the facesheet, Cf and Cc are the wave speeds in the facesheet and core, respectively; f and c are the density of the facesheet and core, respectively; and u h i is the unit step function.

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels sT1 -sR1

sR1 sI

Incident Facesheet

sR2

sT1

sR2

sT1

sR2

sT2

-sT2 Distal Facesheet

Foam

Interface 1

665

Interface 2

Fig. 3 Transmission of stress waves through facesheets and foam in sandwich panel

The wave speed in an orthotropic plate in uniaxial strain is derived in Appendix A as s .1  12 / E33 (4) Cf D Œ1  12  32 .13 C 23 / f where Eij and ij are elastic modulus and Poisson’s ratio of the orthotropic facesheet. This wave speed is usually higher q than the more commonly used uniaxial stress wave speed, which is equal to Ef33 . The wave speed in the foam will be discussed in the following section. The reflected wave in the incident facesheet is tensile because f Cf  c Cc . This reflected wave is again reflected, but as a compressive stress wave, when it reaches the outer surface of the incident facesheet. The process of reflection and transmission of waves at Interface 1 repeats itself over and over again at intervals 2t1 . Thus the transmitted stress in the foam at Interface 1 is given as T1 D kT1 p.t  t1 /u ht  t1 i C kT1 kR1 p.t  3t1 /u ht  3t1 i kT1 kR2 1 p.t  5t1 /u ht  5t1 i : : : C .1/nC1 kT1 kRn 1 p.t  .2n C 1/ t1 /u ht  .2n C 1/ t1 i

(5)

where n is the number of reflections up to that time. The transmitted stress wave in the foam T1 will reflect back as a compressive wave into the foam and be transmitted as a compressive stress wave in the distal facesheet when it first reaches Interface 2. The transmitted stress in the distal facesheet T2 is further reflected as a tensile stress wave from the outer surface of the distal facesheet. This reflected stress waves will then be transmitted as tensile stress wave in the foam and reflected back as compressive stress wave into the facesheet. The part that is transmitted to the foam will add to the reflected stress waves in the foam R2 . This process repeats itself indefinitely so that the reflected stress wave at any time is given by

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R2 D  kR2 T1 .t  t2 /u ht  t2 i C kT2 kT1 T1 .t  2t1  t2 /u ht  2t1  t2 i C  kT2 kT1 kR1 T1 .t  4t1  t2 /u ht  4t1  t2 i C kT2 kT1 kR2 1 T1 .t  6t1  t2 /u ht  6t1  t2 i : : : C .1/nC1 kT2 kT1 kRn 1 T1 .t  2nt1  t2 /u ht  2nt1  t2 i

(6)

2f Cf . C  C / and kR2 D  f C f Cc Cc : The reflected stress .f Cf Cc Cc / . f f c c/ is a tensile unloading elastic stress wave. Permanent plastic strains or local indentation of the foam results after elastic unloading.

where t2 D H=Cc ; kT2 D

2.1.2

Elastic and Plastic Waves in Foam

The facesheets are very stiff and remain elastic during wave transmissions, but the polymeric foam core is elastic–plastic with a compressive stress–strain characteristic as shown in Fig. 4 [8]. The foam is linear elastic with a compressive modulus of Ec until yielding at a flow stress q. Rapid compaction of cells causes the density to change during the plateau region until full densification has occurred at "D . The stress rises to a maximum plastic stress p at the densification strain. The maximum plastic stress at the densification strain depends on the load intensity. If the pressure pulse amplitude is high enough to yield the core, elastic and plastic waves would be generated in the foam during Phase I.pIn Ashby et al. [9], the elastic uniaxial stress wave speed q in the foam is given by Ec =c and the plastic wave  q

p , where p is the stress in the densification region speed is given by Cp D  c "D (see Fig. 4). Elastic waves propagated first in the core and are later followed by plastic waves, as shown in Fig. 5. By substituting isotropic properties for foam in Eq. 4, one can derive the following expression for elastic wave speed in the foam in a state of uniaxial strain: s .1  c / Ec Ce D  (7) 1  c  2c2 c

s

sp Plateau

Densification

q Ec

Fig. 4 Compressive stress–strain curve of polymeric foam

Elastic eD

e

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels Fig. 5 Elastic and plastic wave fronts in foam

Facesheet

sp

q

Vp

Ve

rD

667

Foam

rc Cp

Plastic wave front

Ce

Elastic wave front

The densification strain is related to particle velocities in the elastic and plastic zone gage length is Cp t regions, Ve and Vp , respectively. After time  t, the plastic  and the compression in the plastic zone is Vp  Ve t . Therefore, the densification strain is Vp  Ve (8) "D D Cp The particle velocity in the plastic region is in turn related to the plastic stress p and density of foam after densification D [7]: Vp D

p D Cp

(9)

c where D D .1" . Similarly, Ve D cqCe . Combining Eqs. 8 and 9 to eliminate Vp D/ and expressing Cp in terms of p give the following quadratic equation that can be solved for p :



c  D c

2.1.3

2

p2 C



2D q c



c  D c

 

2  Ve2 D 2 q 2 V 2  2 q p C D2 C e D D 0 (10) c "D c c "D

Local Indentation

Permanent plastic strains arise when the elastic unloading wave reaches the plastic wave front. The local indentation is confined in the plastic zone and may be calculated from the densification strain and the characteristic gage length of the plastic zone. This characteristic gage length is Cp #T , where #T is the time from the start of transmission of p to the time when the elastic unloading wave reaches the plastic wave front. Thus the local indentation is ı D "D Cp #T

(11)

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Let tp be the start time of plastic wave transmission of p at Interface 1. Then,   H  Cp #T H C  tp (12) #T D 3t1 C Ce Ce Solving for #T , one gets 

2H=Ce C 3t1  tp   #T D 1 C Cp =Ce

 (13)

The start time of plastic wave transmission tp is determined from the transmitted core stress given in Eq. 5.

2.2 Phase II – Global Bending/Shear The global sandwich deformation and velocity fields in Phase II are shown in Fig. 6a, b. Subsequent to Phase I, the load has ended and the core has crushed permanently to a height of H 0 D H  ı. Momentum is transferred to the sandwich panel, which has become impulsively loaded with a uniformly distributed velocity field (see top diagram in Fig. 6b). Conservation of momentum gives the initial velocity of the panel as po   vi D  (14) 2 c H C 2f h Denote the distance from the center of the panel to the wave front of the transverse shear wave as . A transverse shear elastic unloading wave propagates from the P This unloading wave instantaneously brings clamped boundaries with velocity . the plate to rest behind the wave front. As the plate is brought to rest, it undergoes shear and bending deformations as exemplified in Fig. 6a, b.

Fig. 6 Global panel bending/shear response: (a) deformation profiles and (b) velocity fields

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

2.2.1

669

System Lagrangian

Dynamic equilibrium of the complete sandwich can be expressed in terms of the maximum deflection at the center, #, and an equivalent shear angle, ˛o . These two degrees of freedom have associated velocities, vi and ˝, respectively. The kinetic energy for the sandwich is thus T D 12 meff v2i C 12 Ieff ˝ 2 , where meff D  2 2f h C c H is the effective sandwich mass and Ieff is the effective sandwich rotary inertia. Assume the rate of angular rotation is a maximum at the wave front and decreases linearly to zero at the boundary: ( ˛P .r/ D

0;

0r 

˝ .ar/ .a/ ;


(15)

Then, the effective rotary inertia for the sandwich is Ieff D 6 IQ .a  / .a C 3/, 3    c 3 P   k z3k  z3k1 D 12f 3hH 2 C 3h2 H C h3 C 12 H : where IQ D kD1

The elastic potential energy of the system is equivalent to the bending/shear strain energy of the sandwich, ˘ D U . The Lagrangian for the whole model is L D T  ˘ . For dynamic equilibrium, @ @t



and @ @t

2.2.2

@L @vi 



@L @˝



 @L @U D 2 P 2f h C c H vi C D0 @# @#

(16)

@Ieff @L @U @˝ D˝ D0 C Ieff C @˛o @t @t @˛o

(17)

 

Bending/Shear Strain Energy Potential

Assume in-plane deformations are negligible compared to the transverse deformation. The elastic strain energy of the symmetric sandwich panel with orthotropic facesheet is then given as 8 Z < s  2 D11 @˛N s U D C D12 : 2 @x

!2 "   # s D22 @ˇN @w 1 @w 2 ˛N 2 C ˛N C As55 2 @y 2 @x 2 @x S 2 39 " # ! 2    2 ˇN 2 @˛N @ˇN @˛N @w 1 @w 2 1 @ˇN 5= s s 41 N dS (18) CD66 C CA44 Cˇ C C ; 2 @y 2 @y 2 @y @y @x 2 @x @ˇN @y

!

@˛N @x



C

where w is the transverse deflections, ˛N and ˇN are shear angles associated with the x- and y-directions, respectively, Dijs is the sandwich bending stiffness matrix, As44 and As55 are the transverse shear stiffnesses, and S is the panel surface area. The

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superscript “s” is used to denote the sandwich properties. Equation 18 is a special case of a more general expression for the elastic strain energy of a symmetric sandwich panel with orthotropic facesheet [10]. Finite element analysis using ABAQUS Explicit indicates that the transverse deformation w and the shear rotations with respect to the radial direction ˛N are of the following forms: 8 ˆ 0r  <#;  2 2  w .r/ D r ˆ ; 
8 <0;

0r 

:4’o .r/.ar/ ; .a/2


(19)

(20)

where # is the global deflection and ˛o is the rotation at r D .a C / =2. The deflection profile described by Eq. 19 was found by fitting functions to the transient deflection profiles in the region. The transverse shear angle function was derived from the transverse shear strain, rz , and slope of the deflection profile: ˛N D rz  @w=@r. To evaluate the integral expression in Eq. 18 to polar coordinates, set dS D rdrd @ @ @ and derivatives with respect to x and y as @x D Cos @r@  Sin and @y D Sin @r@ C r @ Cos @ , respectively. r @ As55 , Eq. 18 becomes

U D

2.2.3

s s Furthermore, for the special case of D22 D D11 and As44 D

2 8 .a C /  s s s ˛o D11 C 2D12 C .2 C / D66 3 .a  /   2 As55  C 28 a3  a 2  a2  C  3 ˛o2 105 .a  /   C 176a2 C 16a C 160 2 ˛o # C .29 C 35a/ #2

(21)

Equations of Motion

The propagation speed of the unloading elastic wave P is assumed constant and and # D vi t , defined as a negative quantity in Figs. 6a and b. Denoting P D .a/ t one gets the following coupled equations of motion from Eqs. (16) and (17):  .a  / 2 As55 vi C Œ16 .a  / .11a C 10/ ˛o 2 2f h C c H  t 105 .a  / C2 .29 C 35a/ vi t  D 0 (22)

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

671

and d 2 ˛o  Q .a  / .a  3/ d˛o  Q  I I .a  / .a C 3/ 6 dt 2 3 t dt  16 .a C / s s s C 2D12 C .2 C / D66 C ˛o D11 3 .a  / C

  2 s  A55 56 a2   2 ˛o  16 .11a C 10/ vi t D 0 105

(23)

where ˛o .0/ D 0 and ˛P o .0/ D 0.

2.3 Transient Deformations As an example consider a fully clamped, sandwich panel made of E-glass vinyl ester facesheets and Divinycell H100 foam core, with a radius 250 mm, facesheet thickness 2 mm, and core thickness 25 mm. Material properties for the E-glass vinyl ester and Divinycell H100 foam are given in Table 1. The materials properties for the

Table 1 Facesheet and foam material properties E-Glass/Vinyl Ester Density .kg=m3 / 1391.3 Thickness (mm) 2 17 E11 .C/ (GPa) 17 E22 .C/ (GPa) E33 .C/ (GPa) 7.48 E11 ./ (GPa) 19 19 E22 ./ (GPa) E33 ./ (GPa) – 0.13 12 D 21 13 D 23 0.28 31 D 32 0.12 4.0 G12 D G21 (GPa) G23 D G32 (GPa) 1.73 1.73 G13 D G31 (GPa) q (MPa) – ©D – 270 ¢1f .C/ (MPa) ¢1f ./ (MPa) 200 270 ¢2f .C/ (MPa) ¢2f ./ (MPa) 200 40 £12f .C/ D £21f .C/ (MPa) £13f .C/ D £31f .C/ (MPa) 31.6 £23f .C/ D £32f .C/ (MPa) 31.6

Divinycell H100 100 25 0.126 0.126 0.126 0.035 0.035 0.035 0.31 0.31 0.31 0.01335 0.01335 0.01335 1.4 0.76 3.2 1.53 3.5 1.53 1.47 1.47 1.47

Divinycell H200 200 25 0.170 0.170 0.170 0.105 0.105 0.105 0.3 0.3 0.3 0.0403 0.0403 0.0403 4.35 0.7 6.4 4.36 6.4 4.36 3.86 3.86 3.86

Klegecell R300 300 25 – – – 0.263 0.263 0.263 0.234 0.234 0.234 0.106 0.106 0.106 7.8 0.285 – – – – – – –

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Divinycell H100 and H200 foams in Table 1 were taken from Mines and Alias [11]. Let the sandwich panel be subject to a uniformly distributed pressure pulse of the form given in Eq. 1, where po D 10 MPa and  D 0:05 ms. This problem was modeled in 2D assuming axi-symmetric conditions for the Phase I response and in full 3D for both Phase I and Phase II responses using ABAQUS Explicit Version 6.7. In the 2D model, continuum axi-symmetric, 4 node reduced integration, CAX4R, were chosen for both facesheets and foam. In the 3D model, four node, reduced integration shell elements S4R and continuum 3D, eight node reduced integration elements C3D8R were chosen for the facesheets and core, respectively. The E-glass vinyl ester was considered as orthotropic elastic material, and the Divinycell H100 foam was modeled as crushable foam with isotropic hardening. The plastic hardening curves were taken from Ref. [11] for the Divinycell H100 and H200 foams and Ref. [12] for the Klegecell R300 foam. Although full integration elements could have been used for the FEA analysis, it was found that there would be very minor differences in the solution with full and reduced integration elements. Use of full integration elements simply did not warrant the very long computational run time and huge data files associated with them.

2.3.1

Local Core Crushing: Phase I Response

The FEA predicted distributions of the through-thickness particle velocity and transverse compressive stress at various times during Phase I are shown in Fig. 7a, b. It is clear to see that elastic waves propagated at a faster speed than the plastic waves. The elastic wave front is marked by a jump in the stress amplitude of q D 1:4 MPa. Particles behind the elastic wave front would have a particle velocity of Ve D cqCe D 20:1 m=s. The analytically predicted particle velocity in the elastic region compares very well to the FEA values shown in Fig. 7a, b. According to Eqs. 4 and 7, the elastic wave speed in the E-glass vinyl ester facesheets and the Divinycell H100 foam would be 2,414 m/s and 696.5 m/s, respectively. For the 2 mm thick facesheets and the 25 mm core, the elastic compressive stress wave reaches the distal facesheet at about 0.0375 ms, which corroborates with the FEA results at 0.0377 ms shown in Fig. 7a, b. Behind the elastic stress waves are the plastic stress waves. The amplitude of the plastic stress wave exceeds the flow stress of 1.4 MPa and increases up to a peak value, which can be determined from Eq. 10. The highest transmitted stress was calculated from Eq. 10 as p D 2:1 MPa. At this plastic stress value the plastic wave speed was calculated as Cp D 95:97 m=s. The transmitted stress at Interface 1, which is described in Eq. 5, must be calculated separately for elastic and plastic responses because the transmission and reflection factors, kT1 and kR1 , respectively, depend on the density and wave speed in the foam. A FORTRAN program was written to evaluate the time variation of the transmitted stress at Interface 1 in two parts: an initial elastic response

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

673

Fig. 7 Distribution of through-thickness particle velocity and stress at the panel center during and just after Phase I: (a) particle transverse velocity and (b) transverse stress

whereby T1 < q and c Cc D c Ce followed by a plastic response whereby T1 > q and c Cc D D Cp . The FORTRAN results are shown by the solid line in Fig. 8. There is a jump in the transmitted stress when the core changes from linear elastic response to plastic response at the densification strain. This follows from the approximate compressive stress–strain relation for foams [8, 9]. The peak plastic stress p D 2:1 MPa occurs at tp D 0:039 ms. The transmitted stress at Interface 1 from FEA is also shown in Fig. 8 for comparison. The predicted peak plastic stress was about 4.5% lower than the maximum compressive stress of 2.2 MPa found at 0.0396 ms from FEA. The transmitted stress from FEA is smooth and shows no jump discontinuity when the core begins to plastically flow because the plastic hardening curve is more gradual and only approximates the ideal case of a plateau and an infinite gradient at the densification strain shown in Fig. 4. From the calculated values of Cp and tp , local core crushing was estimated at 2.3 mm from Eqs. 11 and 13.

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Transmitted Stress (MPa)

2.5 2 1.5

Analytical FEA

1 0.5 0 0

0.01

0.02

0.03

0.04

0.05

Time (ms)

Fig. 8 Transmitted stress at Interface 1 up to peak stress

2.3.2

Global Bending/Shear: Phase II Response

The initial global panel velocity was determined from Eq. 14 as vi D 31 m=s. The sandwich bending and shear stiffness were evaluated with a reduced core thickness H 0 D 22:7 mm. Note that a 2.3 mm indentation changes the core height from 25 to 22.7 mm. Since the bending stiffness is proportional to the cube of core height, this reduces the bending stiffness by 25%. A MATLAB program was written to solve Eqs. 22 and 23 for  and ˛o . First, an expression for  in terms of ˛o was derived from Eq. 22 and then substituted into Eq. 23 to eliminate . Then, the resulting second order non-linear differential equation in ˛o was solved using a Runge Kutta ordinary differential equation solver (ode45) in MATLAB. Eq. 22 is cubic in , but only one of the three roots for  gives physically realistic solutions for ˛o . The transient deflection profile is fully determined from Eq. 19, knowledge of .t/ and the fact that # D vi t. As shown in Fig. 9, the predicted transient deformation profiles compared very well to FEA results; the predicted value for  was within 7% of FEA. The solution for ˛o .t / will be used to predict strains and damage initiation in the panel in the following section.

2.4 Damage Initiation An important reason for developing analytical models is to provide simple design tools for determining the survivability of the panel when it is subjected to an intense pressure pulse load. There are critical impulses, combinations of peak pressures and pulse durations, which would just cause damage to initiate in the panel. Recall from the transient deformation analysis in the previous section that the maximum bending strains occur either in the center or clamped edges of the sandwich panel. It is assumed that the clamped edges would be protected from damage so that the panel center is the most critical area for damage initiation.

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels 25

Deflection (mm)

Analytical FEA

0.7ms

20

675

0.5ms 15 10 0.3ms

5 0 0

50

100

150

200

250

Half Span (mm)

Fig. 9 Transient deflection profiles of composite sandwich

So that the analytical predictions could be compared to ABAQUS Explicit results, Hashin’s failure criteria [13] was chosen to predict damage initiation. In Hashin’s theory, the following four damage initiation mechanisms are considered for a unidirectional laminate: fiber tension, matrix tension, fiber compression, and matrix compression. These are expressed in terms of principal stresses ij , material strengths, and the following failure parameters, Fiber tension   2   2 11 12 C (24) Fft D XT SL Matrix tension   2   2 22 12 Fmt D C (25) YT SL Fiber compression   2 11 (26) Ffc D XC Matrix compression Fmc

" # 2   2   2 YC 22 22 12 D C C 1 2S T SL 2S T YC

(27)

where X T and Y T are the longitudinal and transverse tensile strengths, X C and Y C are the longitudinal and transverse compressive strengths, S L is the longitudinal shear strength and S T is the transverse shear strength. When Fft D 1; Fmt D 1; Ffc D 1, or Fmc D 1, the corresponding damage mode initiates. For the 0/90 degree orthotropic laminate facesheets, 11 D 22 and X T D Y T , so that fiber tension and matrix tension failure conditions in Eqs. 24 and 25 would be identical. The fiber and matrix compression criteria, Eqs. 26 and 27, would also apply in both principal directions.

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For the orthotropic facesheet with fibers in 0 and 90 degrees parallel to the x- and y-axes, one gets that the following relationship between principal stresses and strains 8 9 2 38 9 QN 11 QN 12 0 < "x = <11 = D 4QN 12 QN 11 0 5 "y (28)  : 22 ; : ; 0 0 QN 66 12 xy where QN ij is the transformed stiffness matrix. The strains in rectangular coordinates are evaluated in transformed polar coordinates as follows: @˛N @˛N D zCos @x @r @ˇN @˛N "y D z D zSin @y @r   @˛N @˛N @˛N xy D z D z .Cos C Sin / C @x @y @r "x D z

(29) (30) (31)

Thus, the principal stresses are expressed in polar coordinates as  4˛o .a C   2r/  N Q11 Cos C QN 12 Sin 2 .a  /  4˛o .a C   2r/  N Dz Q12 Cos C QN 22 Sin 2 .a  / 4˛o .a C   2r/ N Dz Q66 .Cos C Sin / .a  /2

11 D z

(32)

12

(33)

12

(34)

The principal stress components are greatest at the top or bottom of the outer facesheets, z D ˙ .H 0 =2 C h/, and for QN 11 D QN 22 , each stress components will be maximum at D 45 degrees. Furthermore, the impulse that would just cause damage relate to either tensile fiber or matrix conditions (Eqs. 24 or 25) and first occurs at r D 0 and when  D 0. A criterion for damage initiation following tensile fiber or matrix failure is given by 2   !2 2˛o2 .H 0 C 2h/2 4 QN 11 C QN 12 C 1D a2 XT

2QN 66 SL

!2 3 5

(35)

The above failure criterion gives a combination of permanent core height H 0 and shear angle ˛o for damage initiation. A critical impulse would be responsible for this combination of H 0 and shear angle ˛o . For the sandwich panel with H100 PVC foam core in the example problem, it was predicted that a critical impulse Icr D 54:83 Pa-s (po D 2:2 MPa and  D 0:05 ms) would just cause damage initiation in the panel center. At this value of the pressure pulse the core had almost negligible permanent deformation at the end of

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

677

Fig. 10 Variation of critical impulse to failure with core types

Phase I, H 0 D 24 mm. The ABAQUS Explicit finite element program was run using Hashin’s failure criteria for damage initiation. It was found that damage initiated near the panel center when the pressure pulse was adjusted to po D 3 MPa and  D 0:05 ms or an impulse of 75 Pa-s. Below this peak load and duration, no damage occurred in the FEA. Thus the critical impulse to failure is Icr D 75 Pa-s as predicted by FEA. The analytical critical impulse to failure is about 27% less than FEA predictions. The discrepancy between analytical and FEA predictions was attributed to the fact that in the FEA, tensile fiber and/or matrix damage took place near but not exactly in the panel center as was assumed in the analytical model. Damage initiation of sandwich panels with the same 2 mm-thick, E-glass vinyl ester facesheets and two other cores, namely Divinycell H200 and Klegecell R300 foams, were also considered. Material properties for the Divinycell H200 and Klegecell R300 foams are listed in Table 1. The analytical predictions for the critical impulse to failure using the wave propagation model and Hashin’s tensile fiber and/or matrix failure criteria compared better to FEA predictions than with the Divinycell H100 foam core, as indicated in Fig. 10. The analytical predictions for the Divinycell H200 and Klegecell R300 foams were only about 13% higher than FEA results. The actual failure site for damage initiation in the sandwich panels with the Divinycell H200 and Klegecell R300 foams were exactly in the center of the distal facesheet, as assumed in the analytical model.

3 High-Velocity Impact of Sandwich Panels Consider the composite sandwich panel, as shown in Fig. 11. The facesheets are orthotropic plates of thickness h, and the core is crushable polymeric foam of thickness H . The blunt cylindrical projectile has a radius rp , a mass Mo and a velocity Vo . The projectile is assumed rigid compared to the sandwich panel. Upon impact, the panel experiences local indentation as well as global bending/shear deformation. At the early stages of impact, compressive stress waves are generated under the projectile. These stress waves must travel through the incident facesheet, core and distal facesheet before global transverse shear and bending waves can be transmitted laterally in the sandwich panel. During this phase, the

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Fig. 11 Projectile impact of composite sandwich panel

Fig. 12 Local and global sandwich deformations: (a) local indentation, (b) global bending/shear and (c) simultaneous local and global deformations

problem becomes one of local indentation only, i.e., the incident facesheet deflects under the projectile and the core crushes as if the distal facesheet of the sandwich panel were rigidly-supported. In Fig. 12a, the local indentation under the projectile is denoted ı and the lateral extent of deformation is denoted . Once the through-thickness compressive stress waves have reached the distal side of the sandwich panel, global panel bending/shear deformation can initiate. These global bending/shear deformations are also shown Fig. 12b, where the maximum global deflection under the projectile is denoted # and the lateral extent of global deformation is denoted $ . Simultaneous local indentation and global deformation actually occurs until the core can no longer crush because of material densification or failure has occurred (see Fig. 12c). Analytical solutions for the local and local/global transient response of the panel will be given in the following section.

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3.1 Phase I: Local Indentation The local indentation is found by considering projectile impact of a facesheet resting on a foam foundation. The duration of Phase I is determined from the wave travel time through the thickness of the panel.

3.1.1

Through-Thickness Wave Propagation

Compressive stress waves must pass through the full thickness of the sandwich, i.e., two facesheets and core, before any response can be characterized as global bending/shear deformation. The through-thickness wave travel time is given by tI D

2h H C Cf Cd

(36)

where Cf is the wave speed in the transverse direction of the facesheet and Cd is the elastic dilatational wave speed in the core. The transverse wave speed in an orthotropic plate is given by s Cf D

E33 .1  12 21 / f .1  12 21  23 32  13 31  221 32 13 /

(37)

where Eij ; ij and f are the elastic modulus, Poisson’s ratio and density of the orthotropic facesheet, respectively [14]. The dilatational wave speed in the core is given in terms of Lame’s constant,  and , as follows: s Cd D where  D

3.1.2

Ec c .1Cc /.12c / ,

D

Ec 2.1Cc /

. C 2 / c

(38)

and c is Poisson’s ratio of the foam.

Local Indentation

Local indentation is found by considering the projectile presses onto the incident facesheet resting on a foam foundation. A single degree-of-freedom equation of motion governing the dynamics of the projectile and effective facesheet mass can be written considering the system Lagrangian. The Lagrangian L for a system is defined as L D T  ˘ , where T and ˘ are the kinetic energy and potential energy of the system, respectively.

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Kinetic Energy The kinetic energy of the system is given by T D

1 1 1 Mo V12 C mf V12 C ma V12 2 2 2

(39)

where V1 is the velocity under the projectile, mf is the effective mass of the facesheet and ma is the added mass of core. The last two terms in Eq. 39 increase over time as waves propagate laterally across the panel and through the thickness of the panel. The effective mass of the facesheet is found by assuming that the projectile induces the following linear velocity field in the facesheet: 

r w P D V1 1  

(40)

The spread of the deformation zone  in the above equation varies with time, i.e., the velocity field has a moving boundary and the effective facesheet mass grows as the velocity field spreads away from the impact site. The total kinetic energy of the facesheet is thus given by 1 Tf D .2/ 2

Z

f hV12



r 1 

2 rdr

(41)

0

Integrating the kinetic energy of the facesheet and setting it equal to the equivalent kinetic energy produced by an effective facesheet mass mf give mf D

f h 2  6

(42)

The added mass of core is more complicated to evaluate because core particle velocities are governed by elastic and plastic response. Elastic and plastic regions in the foam are shown in Fig. 13a. Under the projectile, the elastic wave front has advanced a distance Cd t , while the plastic wave front has advanced a distance Cp t . As the deformation spreads laterally, the distances from the incident facesheet to the elastic and plastic wave fronts decrease in an approximately linear fashion. The volume of plastic and elastic foam is described by a cone and a truncated cone, respectively. The particle velocity in the plastic zone is controlled by the projectile impact speed and the velocity filed of incident facesheet, which is described in Eq. 40. The particle velocity in the elastic zone depends on the magnitude of the elastic stress V D cCd . Thus the maximum particle velocity in the elastic zone is Ve D cqCd . It is assumed that the elastic particle velocity decreases linearly through the thickness of the foam. The variation of the particle velocities through the foam thickness under the projectile and half-way across the local deformation zone is shown in Fig. 13b. Following

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

681

Fig. 13 Propagation of elastic and plastic waves in the core during Phase I (particle velocity fields shown right)

these velocity distributions, the kinetic energy of the elastic and plastic parts of the foam is derived in Appendix B and is taken as   c ı c ı 2 2 Ta D  V1 C (43) Cd t   2 Ve2 20 .1  "D / "D 18 "D The first and second terms on the right-hand side of Eq. 43 are the kinetic energy of the plastically and elastically deformed core, respectively. The added mass of the core is therefore   V2 c ı c ı ma D 2 C (44) Cd t   2 e2 10 .1  "D / "D 9 "D V1

Potential Energy The total potential energy of the system, …, consists of the elastic strain energy of the facesheet, Ul , and the work dissipated in crushing the core, D: … D Ul C D

(45)

Assuming in-plane deformations are negligibly small compared to transverse deflections, w, the elastic strain energy due to bending and membrane stretching in an orthotropic facesheet is 1 Ul D 2

Z "

 D11

S

1 C 8

@2 w @x 2

Z "

 A11

2

@w @x

@2 w @2 w C2D12 2 2 C D22 @x @y

4

 C A22

@w @y

4



@2 w @y 2

2

 C 4D66 

@w C .2A12 C 4A66 / @x

@2 w @x@y

2 

@w @y

2 # dS

2 # dS

S

(46)

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M.S. Hoo Fatt et al.

where Dij and Aij are the bending and membrane stiffness of the facesheet and S is the surface area associated with indentation. When deflections are less than 0.5 h, the bending energy, which is calculated from the first integral in Eq. 46, dominates in the expression for Ul . The membrane energy, which is represented by the second integral in Eq. 46, becomes greater than the bending energy when deflections are greater than h. Finite element analysis has shown that the transverse deflection of the facesheet may be assumed as follows: "  2 #3 r (47) w.r/ D ı 1   where r 2 D x 2 C y 2 . To evaluate the integral expression in Eq. 46 in polar co@ D ordinates, set dS D rdrd and derivatives with respect to x and y as @x @ Sin @ @ @ Cos @ Cos @r  r @ and @y D Sin @r C r @ , respectively. The strain energy due to facesheet bending and membrane stretching is ı2 ı4 Ul D DN 2 C AN 2  

(48)

 Œ3 .D11 C D22 / C 2D12 C 4D66  and AN D 220 Œ27 .A11 C A22 / where DN D 6 5 C18A12 C 36A66 . The elastic strain energy and plastic work in the foam are derived separately in Appendix C, giving a total energy consumed in the foam as   q 2 qı 2 ı C Cd t  2 DD (49) 3 18Ec "D

The first and second terms on the right-hand side of Eq. 49 are the plastic work and elastic strain energy, respectively.

Equation of Motion Lagrange’s equation of motion for the projectile and effective facesheet mass is 

@ @t where L D T  ˘ and V1 D  Mo C mf C C

dı . dt

 

@L D0 @ı

(50)

Substituting T and ˘ into Eq. 50 gives

  f h 2 c ı 2 dı d 2ı C C 2 dt 3t 5 .1  "D / "D t dt (51) 2 2 3 N N c Ve  2Dı 4Aı q 2 q 2  2 C C 2 C 2 C D0  18"D   3 18Ec "D

c ı 2 10 .1  "D / "D

c  2 V12 20 .1  "D / "D

@L @V1



Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

683

where it is assumed that P D =t . Conservation of linear momentum relates the projectile and effective facesheet mass velocity, V1 , with : Z Mo Vo D Mo V1 C 2

  r f hV1 1  rdr C Lp C Le 

(52)

0

where Lp and Le are the momentum associated with the plastic and elastic regions of the foam, respectively. The following expressions were derived for them in  c ıV1  2 c Ve  2 Cd t  "ıD . Momentum balance Appendix B: Lp D 6"D .1"D / and Le D 6 thus gives Mo Vo D Mo V1 C

f hV1  2 c ıV1  2 c Ve  2 C C 6 6"D .1  "D / 6

 Cd t 

ı "D

 (53)

Note that the momentum of the facesheet and core has been integrated using the velocity and mass field described earlier. Equation 51 becomes a nonlinear second order differential equation in ı when Equation 53 is used to eliminate . The initial Mo Vo , where to D Chf . conditions for Eq. 51 are ı .to / D 0 and ddtı .to / D M C . o rp2 f h/

3.2 Phase II: Global Bending/Shear At the end of Phase I, transverse shear waves propagate from the point of impact across the sandwich panel. The global panel deflection is denoted # and the lateral extent of global deformation is denoted $ , as shown in Fig. 12c. A two degreeof-freedom model shown in Fig. 14 is considered for combined local indentation and global panel bending/shear deformations. Degrees of freedom, X1 and X2 , are related to the position of the projectile and incident facesheet, and the deflection of the sandwich panel under the projectile if there were no local indentation or core crushing, respectively. Therefore, ı D X1  X2 and # D X2 . Applying Lagrange’s equations of motion results in two equations of motion:

Fig. 14 Two degree-of-freedom model for coupled local and global deformation

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M.S. Hoo Fatt et al.





@ @L  @t @V1   @ @L  @t @V2 where V1 D

3.2.1

dX1 dt

and V2 D

@L D0 @X1 @L D0 @X2

(54) (55)

dX2 . dt

Kinetic Energy

Assume the mass of the core is small compared to the mass of the facesheet and neglect rotary inertia of the sandwich panel. Then the kinetic energy of the system is given by 1 1 1 (56) T D Mo V12 C mf V12 C ms V22 2 2 2 where ms is the effective mass of the sandwich and V2 is the velocity of the sandwich under the projectile. The added mass of the core is taken into account in the effective mass of the sandwich. The effective mass of the sandwich is calculated by again assuming a linear velocity field for the sandwich:  r (57) w P D V2 1  $ From the above velocity field, one finds that the effective sandwich mass is ms D

3.2.2

  2f h C c H $ 2 6

(58)

Global Bending/Shear Energy

The elastic strain energy due to bending and shear of a symmetric sandwich panel with orthotropic facesheet is !   @˛N 2 @˛N @ˇN s Ug D C D12 @x @y @x S 2 !2 !2 3  2 s N N N D @˛N @ˇ @˛N @ˇ 1 @ˇ s 41 5 C 22 C D66 C C 2 @y 2 @y @y @x 2 @x " "  2 #  2 #) 2 N2 ˇ ˛ N @w 1 @w @w 1 @w C As55 dS C ˇN C C ˛N C CAs44 2 @y 2 @y 2 @x 2 @x (59) Z (

s D11 2



where w is again used to express transverse deflections, ˛N and ˇN are shear angles associated with the x- and y-directions, respectively, Dijs is the sandwich bending

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

685

stiffness matrix, and As44 and As55 are the transverse shear stiffnesses. The superscript “s” is used to denote sandwich properties. Assume the following global transverse deflection and shear rotations: 8   <# 1   r 2 3 ; 0  r  $ $ w .r/ D :0; r >$ and

( ˛N .r/ D ˇN .r/ D

27˛o r 4 $



1

 r 2 $

0;

(60)

; 0r $

(61)

r >$

where # is the global deflection under the projectile and ˛o is the shear rotation at the center of the panel. Substituting derivatives of the expressions in Eqs. 60 and 61 into Eq. 59 gives the following expression for the strain energy:   1  s A C As55 5472$ ˛o # C 729$ 2 ˛o2 C 1344#2 (62) 4480 44   s  s s s where DN s D 243 C 2 . C 2/ D66  D11 C D22 C 4D12 . 320 Ug D DN s ˛o2 C

3.2.3

Equations of Motion

Satisfying Lagrange’s equations of motion stated in Eqs. 54 and 55 one gets  d 2 X1 f h 2 V1 2DN .X1  X2 / 4AN .X1  X2 /3 q 2  C D0 Mo C m f C C C dt 2 3t 2 2 3 (63) and     S  2f h C c H $ 2 dX2 A44 C AS55 d 2 X2 ms C .171$ ˛o C 84X2 / C dt 2 3t dt 140 2DN .X1  X2 / 4AN .X1  X2 /3 q 2    D0 2 2 3 (64) where it is assumed that $P D $=t . Rotary inertia is neglected so that   2736 As44 C As55 $ X2   ˛o D 4480DN s C 729 As C As $ 2 44

@˘ @˛o

D 0 gives (65)

55

The extent of local indentation and global deflections,  and $ , are evaluated from conservation of momentum and energy. Conservation of linear momentum is given as

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M.S. Hoo Fatt et al.





Mo Vo D Mo V1 C

 2f h C c H $2 3



V1 C V2 2

 (66)

Conservation of energy stipulates that   1 1 ı2 ı4 qı 2 V1 C V2 2 1 C DN 2 C AN 2 C Mo Vo2 D Mo V12 C ms C DN S ˛o2 2 2 2 2   3   1  S C A44 C AS55 5472˛o #$ C 1344#2 C 729 ˛o2 $ 2 4480 (67) The momentum and kinetic energy of the panel in Eqs. 66 and 67 are given in average forms in order to simplify the solution for  and $ . Equations 63 and 64 are solved with the following initial conditions: X1 .tI / D X10 ; XP 1 .tI / D XP10 ; X2 .tI / D 0; and XP 2 .tI / D 0, where tI is defined in Eq. 36 and X10 and XP10 are taken at the end of Phase I.

3.3 Comparison with Finite Element Analysis As an example, consider a fully clamped sandwich panel made of E-glass vinyl ester facesheets and PVC H200 foam core, with a panel radius of 250 mm, facesheet thickness of 2 mm, and core thickness of 25 mm. Material properties for the E-glass vinyl ester and the PVC H200 foam are given in Table 1. Let the sandwich panel undergo impact by a rigid cylindrical rod of radius 2.5 mm, mass 0.5 kg, and velocity 40 m/s. This impact problem was modeled using ABAQUS Explicit using continuum C3D8R elements for both the facesheets and the foam. The PVC H200 foam was modeled as crushable foam with isotropic hardening. Additional foam properties, such as the plastic hardening curve were taken from Reference [11]. The analytical solution for the incident facesheet under the projectile and global panel deflections are compared to FEA results in Fig. 15. The deflection of the distal facesheet

Analytical, X1

12

FEA, X1

Deflection (mm)

10

Analytical, X2 FEA, X2

8 6 4 2 0 0

0.05

0.1

0.15 0.2 Time (ms)

0.25

0.3

0.35

Fig. 15 Transient deflections of incident and distal facesheets under the projectile

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

687

under the projectile was taken as the global panel deflection in FEA. The analytical deflections are about 15% lower than FEA predictions.

4 Conclusions Wave propagation models were developed to examine the damage resistance of composite sandwich panels subjected to blast and high velocity projectile impact. Analytical solutions were derived for the transient response and damage initiation of a foam-core composite sandwich panel subjected to uniformly-distributed, pressure pulse loading. The panel response was modeled in two consecutive phases: (1) a through-thickness wave propagation phase leading to permanent core crushing deformations and (2) a transverse shear wave propagation phase resulting in global panel deflections. The predicted transient deformation of a sandwich panel consisting of E-glass vinyl ester facesheets and PVC foam core was found to be within 7% of ABAQUS predictions. Analytical predictions of the critical impulse for damage initiation also compared fairly well with ABAQUS predictions. The second wave propagation model was developed for the deformation response of a composite sandwich panel subjected to high velocity impact by a rigid blunt, cylindrical projectile. Waves travelling through the sandwich core thickness and laterally across the panel were incorporated in a previously developed two degreesof-freedom model for the sandwich panel. Modelling the sandwich with two degrees of freedom allowed local indentation and core crushing to be coupled with global bending/shear deformations of the sandwich. An example was given for a composite sandwich panel consisting of orthotropic E-glass vinyl ester facesheets and PVC foam core and subjected to high velocity impact of a blunt cylindrical projectile. The analytical solution for the local indentation and global deflection under the projectile was found to be within 20% of FEA results. Acknowldgement This work was supported by Dr. Yapa Rajapakse at the Office of Naval Research under grant N00014–07–1–0423.

Appendix A Uniaxial Strain Wave Speed in an Orthotropic Plate Stress waves propagating through the thickness of the orthotropic facesheets travel in material that is constrained laterally in the x- and y-directions. To evaluate this wave speed, set "x D "y D 0. For the special case of a 0ı =90ı laminate, x D y , E11 D E22 ; %12 D %21 , and %31 D %32 , two of the three-dimensional stress–strain relations become "y D

.1  %12 / %32 x  z D 0 E11 E33

(A1)

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M.S. Hoo Fatt et al.

and "z D

 .%13 C %23 / 1 x  z E11 E33

(A2)

Solving for x in Eq. A1, one gets x D

%32 E11 z E33 .1  %12 /

(A3)

Substituting the above expression for x in Eq. A2 gives z D

.1  %12 / E33 "z Œ1  %12  %32 .%13 C %23 /

(A4)

Equation A4 represents a stress-strain relation for an orthotropic material in onedimensional or uniaxial strain. The wave speed for the material in this state is given by s .1  %12 / E33 (A5) Cf D Œ1  %12  %32 .%13 C %23 / f where f is the mass density.

Appendix B Momentum and Kinetic Energy of Core During Phase I Both elastic and plastic compressive stress waves propagate in the core with speeds Ce and Cp . Given that Ce > Cp , the plastic zone in the foam is confined to a boundary layer just beneath the incident facesheet. The particle velocity in the foam under the projectile is specified by the projectile velocity V1 , while the particle velocity in the foam away from the projectile is specified by the assumed velocity field of the  incident  facesheet given by Eq. 40. The plastic zone extends a distance hp D "ıD 1  r as shown in Fig. 13 Ahead of the plastic zone is the elastic zone    he D Cd t  "ıD 1  r . The distribution of the particle velocities under the projectile, at r D 0, and at r D =2 are indicated in Fig. 13. The momentum of the plastically deformed portion of the core is Z Lp D 2D 0

2 3  Zhp  r 6 7 d z5 rdr 4 V1 1  

(B1)

0

where densification of the core has caused the density to become D D momentum in the plastic part of the core is

c .1"D / .

The

Modeling Blast and High-Velocity Impact of Composite Sandwich Panels

Lp D

c ı  2 V1 6"D .1  "D /

689

(B2)

The kinetic energy of the plastically deformed portion of the core is 2 3 2 Zhp  r 6 7 2 d z5 rdr 4 V1 1  

Z Tp D D 0

D

0

(B3)

c ı  2 V12 20"D .1  "D /

The momentum of the elastic portion of the core is 3  z d z5 rdr Le D 2c 4 Ve 1  he 0 0   c ı Cd t   2 Ve D 6 "D Z

where Ve D

q c Cd

2



Zhe

(B4)

. The kinetic energy of the elastic portion of the core is 2h 3  2 Ze z d z5 rdr Te D c 4 Ve2 1  he 0 0   ı c D Cd t   2 Ve2 18 "D Z

(B5)

Appendix C Elastic Strain Energy and Plastic Work in Core The core absorbs both elastic strain energy and plastic work in Phase I. The plastic work dissipated in the core plastic region shown in Fig. 13 is Z Dp D 2



r qı 1  

 rdr D

qı 2  3

(C1)

0

The elastic strain energy in the core is Z Zhe De D 2 0

o

1 Ec "2 dzrdr 2

(C2)

690

M.S. Hoo Fatt et al.

   where Ec is the core compressive modulus, he D Cd t  "ıD 1  r and the   elastic strain vary through the thickness as " D Eqc 1  hze . After integration, the elastic strain energy in the core becomes De D

q 2 18Ec

 Cd t 

ı "D



2

(C3)

References 1. Hampson PR, Moayamedi M (2007) A review of composite structures subjected to dynamic loading. Int J Crashworthiness 12: 411–428. 2. Cole RH (1984) Underwater explosions. Princeton University Press, New Jersey. 3. Smith PD and Hetherington JG (1994) Blast and ballistic loading of structures. Butterworth Heinemann, Oxford. 4. Menkes SB, Opat HJ (1974) Broken beams. Exp Mech 13: 480–486. 5. Teeling-Smith, RG, Nurick GN (1991) The deformation and tearing of thin circular plates subjected to impulsive loads. Int J Impact Eng 11: 77–91. 6. Franz T, Nurick GN, Perry MJ (2002) Experimental investigation into the response of choppedstrand mat glass fibre laminates to blast loading. Int J Impact Eng 27: 639–667. 7. Graff KF (1975) Wave motion in elastic solids. Oxford University Press, London. 8. Gibson LJ, Ashby MF (1999) Cellular solids structures and properties. Cambridge University Press, Cambridge. 9. Ashby, MF, Evans AG, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG (2000) Metal foams: a design guide. Butterworth Heinemann, London. 10. Vinson JR (1999) The behavior of sandwich structures of isotropic and composite materials. Technomic Publishing Company, Lancaster. 11. Mines RAW, Alias A (2002) Numerical simulation of the progressive collapse of polymer composite sandwich beams under static loading. Composites Part A 33: 11–26. 12. Rizov V, Mladensky A (2007) Influence of the foam core material on the indentation behaviour of sandwich composite panels. Cellular Polymers 26: 117–131. 13. Hashin Z (1980) Failure criteria for unidirectional fiber composites. J Appl Mech 47: 329–334. 14. Sierakowski RL, Chaturvedi SK (1997) Dynamic loading and characterization of fiberreinforced composites. John Wiley and Sons, New York.

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Effect of Nanoparticle Dispersion on Polymer Matrix and their Fiber Nanocomposites Mohammed F. Uddin and Chin-Teh Sun

Abstract Dispersion of nanoparticles and its effect on mechanical properties were investigated by fabricating nanocomposites via conventional sonication, sol-gel, and modified sonication method. Silica nanoparticles dispersed in epoxy and MEK produced via sol-gel method were procured as Nanopox F 400 and MEK-ST-MS, respectively, to produce silica/epoxy nanocomposite whereas the conventional sonication method was followed to produce alumina/epoxy and carbon nanofibers (CNF)/epoxy nanocomposites. The conventional sonication method was modified by combining it with sol-gel method to improve the dispersion quality as well as to increase the particle loading. The as-prepared nanocomposites were morphologically and mechanically characterized to investigate the effect of dispersion of nanoparticles on polymer matrix nanocomposites. The nanocomposites fabricated via sol-gel method revealed the most improved and consistent properties among all nanocomposites which showed almost proportional properties improvement with particle loading in contrast to conventional nanocomposites. Subsequently, the modified matrix (silica/epoxy) was used to make fiber reinforced nanocomposites via the VARTM process. The effect of improved matrix properties was reflected in the properties of fiber composites which showed significant improvements in compressive strength, tensile strength and modulus, fracture toughness and impact resistance.

1 Introduction There has been a growing use of polymer matrix composites (PMC) materials in a wide spectrum of applications ranging from aircraft structures to sporting goods due to their high strength/weight ratio compared to conventional metal. However, these benefits of PMCs could not be fully utilized as their matrix-dominant properties are much weaker than their fiber-dominated properties. The recent advent of M.F. Uddin and C.-T. Sun () School of Aeronautics and Astronautics Purdue University, West Lafayette, IN 47906, USA e-mail: [email protected]; [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

693

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M.F. Uddin and C.-T. Sun

nanoparticles has attracted much attention in improving the matrix properties by using various nanoparticles as reinforcements. However, experimental results on polymeric nanocomposites so far have revealed a wide range of variations in their mechanical properties. Some researchers reported enhancement in matrix properties [1–3] (e.g. modulus, strength) while others reported adverse effects [4–6] due to nanoparticle inclusions. The large surface area of nanoparticles enhances interactions among particles causing agglomerations of particles. Several processing methods including shear mixing [2], mechanical mixing [7], in-situ polymerization [4] and sonication [8] have been used for dispersing nanoparticles into polymers. Researchers have been able to produce nanocomposites with improved properties only with small particle loadings (less than 5 wt%), even though some conflicting results were also reported using same processing methods. The fact is unanimously agreed that the quality of dispersion of nanoparticle in the polymer matrix is to be held responsible for these anomalies in nanocomposite properties. While a good-dispersion of nanoparticles is the key issue to fully utilize the advantage of nanoparticle reinforcement, the term “well-dispersion” is very loosely defined. It is, therefore, of great importance to develop a processing method to disperse nanoparticles uniformly in the polymer with high particle loadings to maximize their effects. While developing the processing method, it is equally important to understand the mechanisms of property enhancements with the nanoparticle reinforcement. The objective of the present article is to summarize the effect of nanoparticles dispersion on the mechanical properties of fiber nanocomposites with a nanoparticle-enhanced matrix.

2 Effect of Dispersion on Polymer Matrix For decades, organic or inorganic nanofillers have been used as reinforcements in polymers to improve their structural properties. However, delivering a quality nanocomposite consistently is still far from maturity. The challenge primarily comes from the dispersion of nanoparticles, specially at higher particle loading. Hence, a few issues regarding dispersion of nanoparticles need to be addressed: (i) techniques for uniform dispersion of high nanoparticle loading, (ii) dispersion of different types of nanoparticle for their versatile use, and (iii) how dispersion of nanoparticles affects the polymer matrix properties. Although the positive effect of nano-fillers is well known, a unique, consistent method of dispersing nanoparticles is still lacking. Several processing methods have been used to produce improved polymeric nanocomposites, but only with small particle loadings (usually less than 5 wt%). For instance, Subramaniyan and Sun [9] employed both shear mixing and sonication to disperse montmorillonite clay nanoparticle into vinyl ester resin. The clay nanoparticle loading was 3–8% by weight. The compressive results of clay modified resin showed that the resin with 5% nanoclay exhibited a reasonable improvement in compressive strength and a considerable (20%) improvement in modulus. However, the mechanical properties tended to go down when nanoclay

Effect of Nanoparticle Dispersion on Polymer Matrix and their Fiber Nanocomposites

695

t Fig. 1 TEM images of nanoclay/vinyl ester resin (a) 3 wt%, (b) 5 wt% and (c) 8 wt% [9]

was loaded beyond 5 wt%, especially strength. Poor dispersion of nanoclay platelets as shown by TEM images in Fig. 1 was suggested as the reason for the property deterioration at higher particle loading. Even though shear mixing and sonication were able to create a reasonable degree of intercalation and exfoliation of clay nanoparticles, the TEM images show the presence of double platelet at higher particle loading. At lower concentration, shear mixing was somewhat able to separate the double platelets. However, these closely spaced double platelets reduced the load transfer efficiency between polymer and nanoclay which was reflected in the mechanical properties of 8 wt% loaded nanocomposite. Similar results were also reported by Yasmin et al. [2] who used a three-roll mill to disperse clay nanoparticles into a liquid epoxy. In addition to the dispersion problem, they mentioned the processing difficulties due to the increased viscosity with high nanoclay loading. Moreover, the authors reported breakage of nanoclay due to shear force which would reduce the aspect ratio and load transfer efficiency as well. Choi et al. [10] dispersed various loadings of vapor grown carbon nanofibers (VGCF) in two epoxy resins of different viscosities. They also concluded that non-uniform dispersion of VGCFs at higher particle loadings caused decrease in nanocomposite properties. However, 16% improvement in tensile strength and 90% improvement in tensile modulus were reported at 5 wt% loading of VGCFs. On the other hand, Cho et al. [11] concluded that tensile modulus of vinyl ester resin can be improved as high as 8–15% by infusing 3–10 wt% alumina nanoparticles via sonication. However, they also reported agglomeration of nanoparticles in their nanocomposites as shown in Fig. 2 which was more severe at high particle loading and was reflected in their tensile strength results. In general, dispersion plays a more vital role in affecting the strength than the modulus of nanocomposites. From the discussion above, it is obvious that dispersing nanoparticles uniformly is a major challenge to fabricate quality nanocomposites. Feeling the need of an alternative dispersion method, some researchers modified the chemical surface of fillers to enhance the interfacial compatibility with the polymer – a process known as sol-gel [1, 12]. Using this method, a novel epoxy system containing 40 wt% uniformly dispersed nanosilica was developed by a German Company (nanoresins, AG [13]). In this method, organosol (colloidal fillers in organic solvent) is prepared with a required concentration, and then in the subsequent step, the solvent is replaced with a polymer to form a well dispersed agglomerate-free nanoparticle/polymer

696

M.F. Uddin and C.-T. Sun

Fig. 2 TEM images of alumina/vinyl ester resin (a) 3 wt%, (b) 10 wt% [11]

composite. Several studies have been reported on silica/epoxy nanocomposites fabricated from organosilicasol [1, 3, 14–20]. Rosso et al. [15] reported improved mechanical properties such as tensile modulus and fracture toughness with the inclusion of 5 vol% silica nanoparticles. Kinloch et al. reported the fracture behavior of an epoxy adhesive with the inclusion of both silica nanoparticles and rubber particles [3, 14]. They demonstrated that the addition of silica nanoparticles to a rubber-toughened epoxy led to a very significant increase in the toughness as well as the glass transition temperature and single-overlap shear strength. Zhang et al. [19] concluded that epoxy matrix modified with silica nanoparticles (up to 14 vol%) via sol-gel method enhanced the mechanical properties of polymer nanocomposites such as modulus, microhardness, and fracture toughness. Only recently, a few companies have emerged to produce and market epoxy resins containing carbon nanofibers and carbon nanotubes. For instance, a company, named Nanosperse LLC [21] in Akron, Ohio produces epoxy with carbon nanofibers whereas epoxy with carbon nanotubes is produced by Zyvex, Columbus, Ohio [22]. However, the respective dispersion techniques for these two products are not readily available. Beside these two exceptions, the success story of high loading nanocomposites has so far been limited to organosilicasol (colloidal silica in organic solvent) via the sol-gel technique. Similar approaches have not been developed for the use of other nanoparticles which are essential if one desires nanocomposites with additional functions. For example, indium tin oxide would be a good candidate for transparent and electrically conductive nanocomposite. Other than carbon nanofiber and carbon nanotubes, silver nanoparticles can also be used to incorporate electrical conductivity in nanocomposites. In this study, we modify the sonication method by combining it with sol-gel for the possibility in adding different types of nanoparticles to the organosilicasol. During sonication, cavitations occur in the surrounding wall [23] and/or near the tip of the probe creating a localized disturbance (cavitation) in the polymer precursor. Due to this localized effect of sonication, a very good distribution of nanoparticles cannot be assured and agglomeration of particles occurs. Our motivation came from the hypothesis that the presence of well spaced nanoparticles (as in the case of polymers infused with colloidal fillers) which act as nucleation sites for cavitation [24–26]

Effect of Nanoparticle Dispersion on Polymer Matrix and their Fiber Nanocomposites

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may have a positive effect in helping to disperse the added dry nanoparticles through sonication. Moreover, using different types of nanoparticles would facilitate to introduce multifunctional properties in the nanocomposite.

2.1 Materials The epoxy resin used in this study was diglycidylether of bisphenol A (DGEBA) with epoxy equivalent weight of 162 g/mole (SC 79 supplied by Applied Poleramic Inc. [27]) and the hardener was cycloaliphatic amine. The silica nanoparticles were procured from two different sources – one was 40 wt% silica nanoparticles dispersed in DGEBA epoxy resin (Nanopox F 400) from nanoresins AG and the other one was 36 wt% silica nanoparticles dispersed in methyl ethyl ketone (MEK-ST-MS) from Nissan Chemical [28]. The silica nanoparticles were synthesized from aqueous sodium silicate solutions by sol-gel technique [29]. After undergoing a process of surface modification and solvent exchange, the epoxy resin and MEK contain surface modified silica nanoparticles with a very narrow particle size distribution. The average particle sizes of Nanopox F 400 and MEK-ST-MS are 20 and 15–23 nm in diameter, respectively. Alumina nanoparticles and carbon nanofiber (Pyrograf IIITM ) were selected as the hybridizing particles which were procured from Nanostrutured and Materials Inc. [30] and Applied Sciences Inc. [31], respectively. The average particle size of alumina particles was 150 nm with a nearly spherical shape whereas CNF has a diameter of 60–150 nm and length upto 30–100 m.

2.2 Fabrication The required amount of Nanopox F400 was diluted with standard DGEBA to get the desired percentage of silica particles in the end product. In the subsequent step, the hardener was added and stirred with a mechanical stirrer for about 5 min. After degassing, the nano-resin was cured and post cured. For making the silica nanocomposite from MEK-ST-MS, the required amount of MEK-ST-MS and DGEBA was heated (before adding the hardener) at 93ı C for about 8 h to evaporate MEK. After complete evaporation of MEK, the hardener was added and then, the nano-resin was cured and postcured. Two different types of hybrid nanocomposites were made using alumina particles and CNFs as second particles. At first (step 1), a required amount of Nanopox F 400 was diluted with DGEBA to get the desired percentage of silica particles in the end products. In the next step (step 2), alumina particles or CNFs were added to the above mixture and sonicated with high intensity ultrasound (Misonix 3000, 600 W, 20 KHz) at amplitude of 12–15 m with 2.54 cm diameter probe for 30 min in pulse mode (15 s on and 15 s off). The temperature was kept just under the room temperature by circulating the cool water around the beaker throughout the whole sonication

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process. Once the epoxy resin (DGEBA) was loaded with nanoparticles, the rest of the procedure of adding hardener and curing were followed as before. A conventional sonication method was followed to make alumina and CNF nanocomposites for comparison.

2.3 Microstructural and Mechanical Characterization Techniques All samples were imaged using TEM to examine the dispersion of nanoparticles. The TEM images were taken using a Phillips CM-100 TEM operated at 100 kV, 200 m condenser aperture, 70 m objective aperture. Three-point-bending tests were conducted using an MTS machine according to the ASTM standard D790 with a cross-head speed of 1 mm/min. The specimen was 100  13  5 mm with an 80 mm span length. Flexural stress and flexural strain were determined from simple beam theory whereas the flexural modulus is determined from the initial slope of the stress–strain curves.

2.4 Morphological Characterization Figure 3 shows the micrographs of 10 wt% silica nanocomposites fabricated from Nanopox and MEK-ST-MS. The particles are agglomerate-free and the individual particle can be identified very clearly from these images. It is now evident that using organosilicasol (Nanopox and MEK-ST-MS) as starting material, it is possible to fabricate a very well dispersed agglomerate-free polymeric nanocomposite. Micrographs of alumina nanocomposites, CNF nanocomposites, and hybrid nanocomposites are shown in Figs. 4–6 where black dots indicate alumina particles and CNFs, and dark grey color indicates silica nanoparticles. It is evident that alumina and CNF nanocomposites fabricated using the conventional sonication

Fig. 3 TEM images of 10 wt% silica nanocomposites from (a) Nanopox (b) MEK-ST-MS

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b

699

c

2μm

2μm

2μm

Fig. 4 TEM images of (a) 1.5 wt% alumina (b) 3 wt% alumina (c) 7 wt% silica C 3 wt% alumina hybrid nanocomposites

a

b

c

1 μm

500 nm

Fig. 5 TEM images of (a) 1.5 wt% CNF (b) 3.5 wt% Silica C 1.5 wt% CNF (c) 8.5 wt% Silica C 1.5 wt% CNF hybrid nanocomposites

a

b

200 nm

500 nm

Fig. 6 TEM Images showing silica nanoparticles surrounding (a) alumina nanoparticles (b) CNFs

method showed agglomerations of nanoparticles with an average size of 3–5 m. On the other hand, both alumina and CNF hybrid nanocomposites do not show such big agglomerations. Individual alumina particle and CNF can be identified in several cases. Apparently, the manufacturing procedures for both conventional (alumina or CNF) nanocomposites and hybrid nanocomposites seem to be same as long as sonication is concerned. But there is a basic difference in the environment during

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the sonication of second particles and that is the presence of well dispersed silica nanoparticles. As seen in micrographs of the hybrid system, the second nanoparticles (alumina or CNF) are distributed throughout the matrix with fewer or no clusters. This change in the quality of dispersion of alumina particles is attributed to the presence of well dispersed silica nanoparticles. It is well known that, under the influence of sonication, cavitation occurs when stress in the liquid produced during the expansion cycle of a sound wave exceeds the tensile strength of the liquid. If the tensile strength of a liquid can be reduced or the stress in the liquid can be increased, a more favorable condition will be created for cavitation. Collings claimed in his patent that the cavitation was facilitated by the presence of solid particles [25]. Due to ultrasonic irradiation, the stress at the solid-liquid boundary is much greater than that in the bulk liquid itself. It was concluded that the solid particles acted as a surface for ‘seeding’ the cavitation bubbles before it got separated from solid particles whereupon the bubbles implode in the liquid medium. On the other hand, Marschall et al. experimentally proved that the tensile strength of a liquid (water) decreased with the presence of solid spherical particles [26] which, in turn, would create a favorable condition for cavitation. In the hybrid system, the presence of spherical, solid silica nanoparticles might be seeding the cavitation process (during sonication) same way as does the solid particles in a liquid medium. In that way, the cavitation would occur all over the liquid epoxy rather than be localized around the probe or wall as what would happen in the conventional system. Besides influencing the sonication, the presence of very well dispersed silica nanoparticles might also physically affect the movement of second nanoparticles by offering very evenly spaced obstacles. It should also be noted that individual alumina nanoparticles or carbon nanofibers are surrounded by silica nanoparticles (Fig. 6) and thus, preventing the agglomeration of these second particles.

2.5 Mechanical Characterization Figure 7 shows the typical stress–strain curves under flexural loading where all the nanocomposites, as expected, show higher flexural strength, modulus and strain at break than the neat epoxy. Conventionally, inclusion of spherical dry nanoparticles incorporates brittleness into the polymer resulting in a decrease in strain at rupture, even though the overall modulus and strength may improve. In contrast, silica and hybrid nanocomposites show a different behavior introducing more nonlinearity into the system and yielding a higher strain at break. Figure 8 shows the flexural moduli and flexural strengths for all nanocomposites where all the properties are normalized by the corresponding properties of the neat epoxy. Silica nanocomposite fabricated from Nanopox exhibits a gradual increment of flexural modulus, strength with particle loading [32]. Silica nanocomposite (10 wt%) fabricated from MEK-ST-MS shows the similar improvement in modulus and strength. It should be noted that unlike conventional nanocomposites, the properties of silica nanocomposites continue to increase with increasing particle

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Fig. 7 Stress–strain curves for neat epoxy, silica, and hybrid nanocomposites

loading. This is due to the fact that using sol-gel, and solvent exchange methods, it is possible to make well dispersed nanocomposites even for very high particle loadings. On the other hand, hybrid nanocomposites show some interesting results. For instance, flexural modulus of alumina hybrid is same as that of the silica nanocomposite. Since the moduli of both particles are of the same order, one would expect a similar increment in their composite moduli as long as their total particle loadings are same. However, effect of stiffer second particles was reflected in CNF hybrid nanocomposite which shows the largest modulus improvement among all nanocomposites. On the other hand, both alumina and CNF hybrid show significant improvement in strength over silica nanocomposite. It should be noted that the properties of hybrid nanocomposite is better than the combined effect of both individual nanocomposites. This may be attributed to the fact that the agglomeration of second particles was reduced in hybrid nanocomposite due to the present of well-dispersed silica nanoparticles.

3 Mechanical Behavior of FRP Nanocomposites In the second phase, we intended to characterize the fiber reinforced composite with a matrix modified with uniformly dispersed high loading nanoparticles as discussed in previous section. It is highly desirable to make the conventional fiber-reinforced composite materials to possess improved strength, toughness, stiffness, and impact resistance.

702 1.40 Modulus (Enano/Eneat)

Fig. 8 Flexural moduli and strengths of different nanocomposites

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1.30

1.20

1.10

1.00 10% MEK-ST-MS

10% Nanopox

7% Silica + 3% Alumina

8.5% Silica + 1.5%CNF

15% Silica

Flexural Strength (σnano/σneat)

1.50 1.40 1.30 1.20 1.10 1.00 10% MEK-ST-MS

10% Nanopox

7% Silica + 3% Alumina

8.5% Silica + 1.5%CNF

15% Silica

It is well known that most advanced fiber composites are stronger in tension (in the fiber direction) than in compression. This behavior is due to fact that the compressive strength of unidirectional composites is governed by microbuckling of fibers embedded in the matrix [33, 34]. It has been shown that the composite compressive strength is proportional to the elastic–plastic tangent shear modulus of the matrix [35]. Thus, raising the elastic–plastic tangent shear modulus leads to a higher compressive strength of the composite.

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A few attempts have been made to take this advantage of nanoparticle infused matrix to enhance the compressive strength of composite [9, 36, 37]. Subramaniyan and Sun [9] reported about 17–19% gain in compressive elastic modulus with 3–5 wt% nanoclay loadings in vinyl ester resin. Using these stiffened matrices, they found about 22–36% improvement in the longitudinal compressive strength of glass fiber reinforced composite with 36% fiber volume fraction. Iwahori et al. [36] used 5–10 wt% cup-stacked type carbon nanofiber as reinforcing particles to stiffen epoxy matrix and reported 15% or less improvement in compressive strength of a carbon/epoxy composite. On the other hand, Cho et al. [37] infused 3 wt% and 5 wt% disk-like graphite nanoparticles in epoxy via sonication and reported 10% and 16% gain, respectively, in the longitudinal compressive strength of a carbon/epoxy composite with 55% fiber volume fraction. Tsai and Wu [38] investigated the effect of nano-sized organoclay on tensile properties of glass fiber reinforced composites with particles loadings of 2.5–7.5 wt%. They reported 74% and 67% improvement in transverse tensile strength and modulus whereas the properties deteriorated in the longitudinal direction. In all these studies, hand layup was used to fabricate the laminated composites and sonication was used to disperse nanoparticles in the resin (except [36]). These are the two major steps employed to produce fiber nanocomposites which are time and labor intensive and, thus, prevent bulk productions for real field applications. On the other hand, owing to their high crosslink density, unfortunately, epoxies are inherently brittle materials. Most of epoxy systems show low fracture toughness, poor resistance to crack initiation and propagation, and inferior impact strength. Because of the brittle nature of the epoxy matrix, fiber reinforced epoxy composites show delamination and poor impact resistance. The growth of delamination results in progressive stiffness degradation and eventual failure of the composite structure. So the resistance to delamination is an important property of a composite and is of great interest to structural designers [39]. Many different types of additives have been used over the years to toughen the epoxy resin, which include rubbers [40], inorganic glasses [41], acrylates [42]. Whilst many of these additives could toughen epoxy resins effectively, their incorporation could also result in reductions in basic mechanical properties, such as modulus. In general, the addition of nanoparticles induces brittleness in polymer matrix. But, as described in the previous section, we found that infusion of silica nanoparticles increased the strain at break of epoxy nanocomposites as well as their strength and stiffness. The advantage of a stiffened matrix can be used to improve the compressive strength of fiber composites whereas the effect of toughened epoxy matrix should be reflected in interlaminar fracture toughness and impact resistance of fiber reinforced composites. Kinloch et al. reported a moderate improvement in mode I critical strain energy release rate with addition of 12 wt% silica nanoparticles in linen-weave carbon fiber reinforced composites [43–45]. The main purpose of the present work is to verify that an epoxy containing well dispersed nanoparticles can be used to form fiber composites with enhanced mechanical properties.

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3.1 Materials and Fabrication 15 wt% silica nanoparticles (from Nanopox) infused epoxy was used as matrix whereas unidirectional E-glass fibers were used as reinforcements. Unidirectional R [46]. E-glass fiber (E-LR-0908–14) was procured from Vectoreply Fiber reinforced composites were fabricated using the VARTM process with a minor adjustment to make a continuous flow of nanophased epoxy. Conventional VARTM procedures were followed for lay-up and arrangement of 16 layers of glass fibers with a vacuum pressure of 75 KPa. Usually, the air bubbles trapped in the resin and lay-ups are removed during the resin flow under vacuum. In case of the nanophased resin, it was not easy to completely take out the air bubbles due to the increased viscosity. Hence, the nanophased resin was degassed under vacuum for an hour before it was infused into the VARTM mold. For consistency, this step was also followed for neat resin. The laminate was cured in hot press with a uniform pressure of 130 KPa under vacuum for about 4 h at 60ı C and then post-cured. Similar procedures were followed for tensile specimen by arranging 6 layers of glass fibers. For compressive strength tests, composites with two fiber volume fractions of 42% and 50% were fabricated whereas it was 47% for tension test specimens. For DCB and ENF specimens, 12 layers of unidirectional glass fibers were used in VARTM to give a laminate thickness of about 4.5 mm with a 50% fiber volume fraction. A Teflon film 51 m thick was placed in the mid-plane of the laminate to create the pre-crack during VARTM. On the other hand, in order to promote the delamination failure during low velocity impact, symmetric cross-ply laminates [0ı2 =90ı2 s were made with 8 layers resulting a laminate thickness of about 3.5 mm and fiber volume fraction of about 44%.

3.2 Mechanical Characterization Techniques 3.2.1

Compression

For unidirectional fiber reinforced composites, compressive loading along the fiber direction (0ı ) often causes failure in the form of fiber brooming which is an end failure and not a true representation of compressive strength of the composite. In this study, compression tests on off-axis composite specimens with angles of 5ı , 10ı and 15ı were conducted from which the longitudinal compressive strength (0ı ) was extracted. The off-axis block specimen of 6  6  10 mm were cut using a water jet cutting machine at these three angles from a unidirectional composite panel. All the specimens were then lapped with 6 and 15 m diamond slurry to ensure parallel and smooth loading surfaces. In order to avoid bending and ensure uniform compressive pressure, a self adjusting support was used during the test. Lubricant was used on the contact surfaces between the specimens and the supports to minimize the contact friction. Compression tests were performed with a nominal strain rate of 104 =s.

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705

Tension

Tension tests were carried out on an MTS machine in both longitudinal and transverse directions at a cross head speed of 1 mm/min. The longitudinal (0ı ) and transverse (90ı ) coupon specimen dimensions were 230  12  2:2 mm and 170  25  2:2 mm, respectively. Glass/epoxy was used as end-tabs bonded to the coupon specimen using a high strength epoxy adhesive to prevent grip failure or slippage of the specimen during the test. 3.2.3

DCB

Mode I interlaminar fracture tests of double cantilever beam (DCB) specimens with piano hinges were conducted on an MTS machine. The specimen dimension was 125  21  4:5 mm with an initial crack length of 55 mm. The specimens were loaded at a rate of 2 mm/min and the crack extension was monitored using a telemicroscope and PixeLINK camera. The specimen edges were painted with thin, white, brittle correction fluid to ease the crack monitoring. 3.2.4

ENF

Mode II interlaminar fracture tests of End-Notched Flexure (ENF) specimens were performed on an MTS testing machine. The specimen of 120  21  4:5 mm was loaded at a rate of 2 mm/min. The specimen is loaded on a three-point bend fixture with a span (2L) of 100 mm. The initial crack length was 25 mm which gave a/L ratio of 0.5. 3.2.5

Low Velocity Impact

Low velocity impact tests were conducted using DYNATUP 9250 drop tower with 12.7 mm diameter impactor having a hemispherical end. Composite Œ02 =902 S specimens of 97  97  3:5 mm were impacted with 5, 10 and 15 joule energy levels. The drop weight was kept constant at 5.77 Kg while the drop heights were adjusted to give the desired energy level. The impact velocity ranged from 1.32 m/s (5 J) to 2.28 m/s (15 J).

3.3 Compressive Properties 3.3.1

Off-Axis Compressive Strength

The results of compressive strength are shown in Fig. 9 for three different off-axis angles of 5ı , 10ı and 15ı . The figure shows the average values of at least four samples. Figure 9 also shows the comparison of neat and nanophased composites with

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Fig. 9 Off-axis compressive strengths of neat and nanophased fiber composites

two fiber volume fractions (Cf ) of about 42% and 50%, respectively. At all three off-axis angles, composites with the higher Cf show higher compressive strengths than those of composites with the lower Cf for both neat and nanophased composites. The low Cf nanophased composites show 10–30% improvement in strength than the neat composite at corresponding angles. However, the improvement is much pronounced (20–40%) in high Cf composites than the low Cf composites.

3.3.2

Longitudinal Compressive Strength

Theoretically, the longitudinal compressive strength of a unidirectional fiber composite can be measured by conducting a compression test on the 0ı specimen. However, such a test often produces end (brooming) failure rather than microbuckling failure. Lee and Soutis [47] conducted compression tests in combined shear loading and end loading by employing an anti-buckling device. Their test results showed a decreased compressive strength of unidirectional composites as specimen thickness increased, and observed end failures in thick-section composite specimens. In most of the studies in compressive strength of composites, shear loading through the use of end tabs or wedge grips was utilized partially or fully in conducting compression tests. Thus, the compressive strengths measured were influenced by the presence of shear stresses. In this study, the longitudinal compressive strength was extracted from offaxis strength results by extrapolation and by an elastic–plastic fiber microbuckling model. The applied compressive strength on the off-axis specimen is first transformed into longitudinal stress ¢11 and shear stress ¢12 (according to the coordinate transformation law) and then plotted for fiber composites containing neat and nanophased matrices with fiber volume fractions of 42% and 50%, respectively, as

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Fig. 10 (a) Extraction of longitudinal compressive strength (b) prediction of compressive strength

shown in Fig. 10a. The longitudinal compressive strength corresponds to the value of ¢11 at ¢12 D 0, i.e., the 0 degree specimen. According to the fiber microbuckling model to be presented later, the compressive strength is not linearly dependent on shear stress. However, from Fig. 10a, for each composite system a straight line seems to give a good fit of the off-axis strength data. Thus, the longitudinal strength for each composite system is extracted based on this linear projection. This projection technique has been used elsewhere with fairly good results as compared with experimental data or model predictions [9, 48]. The composite compressive strengths obtained in this manner indicate 81% and 62% improvements in longitudinal compressive strength for composites with 15 wt% silica particles for 42% and 50% fiber volume fractions, respectively. An existing fiber microbuckling model developed by Sun and Jun [35] is then used to validate the longitudinal compressive strength of both neat and nanophased unidirectional composites. Sun and Jun [35] developed this model

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based on Rosen’s [34] elastic microbuckling model and obtained the longitudinal compressive strength for unidirectional fiber composites as ep

c D

Gm 1  Cf

(1)

ep

where Cf is the fiber volume fraction and Gm is the matrix elastic–plastic tangent shear modulus. This compression failure model was later extended by Subramaniyan and Sun [9] for large off-axis angles. By using the incremental theory of plasticity, Gm was evaluated as " ep Gm

D

9 sin2 1  C Gm ˇ 2 cos2 C 3 sin2 Hp

#1 (2)

E

f , Hp D dd"NNp is the plastic tangent modulus, Cm where ˇ D ˛Cf 1CCm , ˛ D Ems is the matrix volume fraction, Ef is the fiber modulus, is the off-axis angle, Ems is the secant modulus of the matrix calculated from the uniaxial stress–strain curve of the matrix, d N is the effective stress increment and d "Np is the effective plastic strain increment. As the matrix elastic–plastic tangent shear modulus depends on stresses at the time of microbuckling, Eq. 1 was solved numerically to find out the composite failure stress. The experimental compressive failure stresses for different off-axis angles are shown in Fig. 10b. The model predictions are also presented as solid lines. The initial fiber misalignment angles are chosen to fit the experimental data. The misalignment angles vary between 2ı and 3:6ı for different composite panels which are within the experimental measurements of fiber misalignment of composites [49]. The model predictions agree quite well with the experimental results.

3.4 Tensile Properties The average transverse tensile strengths and moduli of composites with/without silica nanoparticles are shown in Fig. 11. It was found that the nanocomposite shows 32% and 41% increase in tensile strength and modulus in the transverse direction, respectively. For transversely loaded tension specimens, matrix cracking is the dominant failure mode for unidirectional composites. However, it was surprising to find that silica nanoparticles yielded 10% improvement in the longitudinal tensile strength in the fiber composite whereas there was no effect on the modulus (not shown in figure).

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Fig. 11 Tensile strength and modulus in transverse direction

3.5 Fracture Toughness Mode I interlaminar fracture toughness GIC was calculated based on the corrected beam theory proposed by Hashemi et al. [50] as given by Eq. 3 GI D

3P ı 2w.a C j j

(3)

where P is load, ı is load point displacement, w is width of the specimen, a is the crack length and j j is the x axis intercept from a vs. C 1=3 curve. Calculated critical mode I energy release rate GIC was plotted against crack extension to construct the resistance (R) curves as shown in Fig. 12. It can be seen that the energy release rate GI increases initially till the crack extension of 15 mm. After the initial rise, the energy release rate reaches a stable plateau where standard deviation does not exceed 2–5%. As the pre-crack was created artificially, there was no fiber bridging observed during the crack initiation. At this point, the fracture behavior of composites is dominated by the matrix [51]. After the crack initiation, the crack was arrested creating sufficient fiber bridging to stabilize the crack growth producing a plateau region of GI in the resistance curve. In the present study, the critical strain energy release rate at crack initiation, GIC -i nitiation and during crack propagation, GIC were evaluated. GIC -i nitiation corresponds to the first point in the resistance curve. On the other hand, GIC was evaluated by taking the average of those at the plateau region. The nanophased fiber composite shows 34% and 27% increase in GIC -i nitiation and GIC , respectively, over the neat resin fiber composite. Similar improvement in GIC was reported by Kinloch et al. [47]. GIC indicates the energy absorbed by fiber bridging mechanisms of debonding, fiber pull-out, fiber fracture as well as matrix deformation and fracture [51]. GIC -i nitiation represents only the matrix dominated failure. The more significant improvement in GIC -i nitiation is therefore reasonable as only the matrix was toughened by the addition of nanoparticles. Compston et al. [51] concluded that in toughened composites, matrix toughness transfers partially at crack initiation to composite toughness and

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GI (J/m2 )

2500 2000

Neat Nano

1500 1000 500 0 0

5 10 15 20 25 30 35 40 45 50 55 60 Crack Extension (mm)

Fig. 12 Resistance curves of neat and nano composites Neat

Nano

Fig. 13 Fiber bridging at 30 mm crack extension

a

b

Fig. 14 Fracture surfaces of DCB specimen (a) neat (b) nano

fiber bridging effect is more significant during crack propagation. Figure 13 shows the microscopic images during crack propagation and Fig. 14 shows the SEM image of fracture surfaces. It is evident that the nanophased fiber composite shows more fiber bridging than the neat resin fiber composite.

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2500

GIIC (J/m2)

2000 1500 1000 500 0 Neat

Fig. 15 Comparison of GIIC

Nano

Mode II fracture toughness of nanocomposites was evaluated using end-notched flexure tests. In this case, the crack became unstable once the load reached the maximum. The maximum load was used to calculate the mode II critical energy release rate GIIC by using the following equation [52] GII D

9P 2 C a2 2w.2L3 C 3a3 /

(4)

where C is compliance, and 2L is span. The mode II critical energy release rates GIIC of both neat and nanophased fiber composites are shown in Fig. 15. It can be seen that the nanophased fiber composite shows about 55% increase in mode II fracture toughness over the neat resin fiber composite.

3.6 Impact Resistance The impact resistance of Œ02 =902 S composite laminates was evaluated by low velocity impact tests. Impact resistance was characterized by considering the peak load, deflection at the peak load, and the inflicted damage area after impact. The composite laminates were impacted at energy levels ranged from 5 to 15 J. At these energy levels, no penetration would occur during impact. This was chosen to promote the delamination failure rather than perforation. The neat resin fiber composite laminate shows a considerable amount of oscillation in the load-time curve (not shown) as it rises to the peak load. The oscillation in load history indicates the progressive local failure of the composite which is more pronounced in the neat resin fiber composite. The magnitude of the peak load is higher in the nanophased fiber composite laminate for all impact energy levels. Moreover, the deflection at peak loads decreased with nanophased composites.

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10J

15J

Neat

Nano

Damage Area (cm2)

Fig. 16 Comparison of impact damage areas at various energy levels 18 16 14 12 10 8 6 4 2 0

Neat

Nano 5J

Neat

Nano

Neat

10J

Nano 15J

Fig. 17 Comparison of delamination area at various energy levels

X-ray radiography was employed to evaluate delamination failure after impact. Figure 16 shows the x-ray radiographs of neat and nanophased fiber composite specimens after impacts at three energy levels of 5, 10 and 15 J. The damage areas in the nanophased fiber composites show 24–47% decreases as compared with the corresponding damage areas in the neat resin fiber composites for the energy levels considered, see Fig. 17.

4 Conclusion Dispersion of nanoparticles plays the dominant role in improving the mechanical properties of nanocomposites. Very uniform dispersion of nanoparticles in epoxies can not be achieved with commonly used methods, such as melt mixing,

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shear mixing, sonication, etc., specifically at high particle contents. However, nanocomposites made via the sol-gel process showed very promising results. Silica/ epoxy nanocomposite fabricated from Nanopox and MEK-ST-MS showed very uniform dispersion of nanoparticles even with high particle loadings. Due to the uniform dispersion of particles, silica nanocomposites showed improvement in mechanical properties nearly proportional to particle loading. The modified sonication method revealed that agglomeration of dry nanoparticle (dispersed via sonication) can be reduced by the presence of well dispersed silica nanoparticle in the polymer precursor during sonication. The mechanical properties enhancement of all nanocomposites followed the same direction as the dispersion of nanoparticles – greater improvement with better dispersion. Using the modified resin with silica nanoparticle, it was possible to make fiber reinforced composite with improved compressive strength, tensile strength and modulus, mode I and mode II fracture toughness as well as impact resistance. It has been also demonstrated that the VARTM process with a minor adjustment can be used to manufacture quality nanocomposites even with high particle loadings and make it feasible for applications to large composite structures. Acknowledgement This work was supported by ONR through grant N00014–05–1–0552. Dr. Yapa D.S. Rajapakse was the technical monitor.

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Experimental and Analytical Analysis of Mechanical Response and Deformation Mode Selection in Balsa Wood Murat Vural and Guruswami Ravichandran

Abstract This study investigates the influence of relative density and strain rate on the compressive response of balsa wood as a sandwich core material commonly used in naval structures. Compressive strength, plateau stress and densification strain of balsa wood along the grain direction is investigated over its entire density spectrum ranging from 55 to 380 kg=m3 at both quasi-static .103 s1 / and dynamic .103 s1 / strain rates using a modified Kolsky (split Hopkinson) bar. Scanning electron microscopy is used on recovered specimens subjected to controlled loading histories to identify the failure mode selection as a function of density and strain rate. The results indicate that compressive strength of balsa wood increases with relative density though the rate of increase is significantly larger at high strain rates. The failure of low-density specimens is governed by elastic and/or plastic buckling, while kink band formation and end-cap collapse dominate in higher density balsa specimens. Based on the experimental results and observations, several analytical models are proposed to predict the quasi-static compressive strength of balsa wood under uniaxial loading conditions. Results also show that the initial failure stress is very sensitive to the rate of loading, and the degree of dynamic strength enhancement is different for buckling and kink band modes. Kinematics of deformation of the observed failure modes and associated micro-inertial effects are modeled to explain this different behavior. Specific energy dissipation capacity of balsa wood was computed and is found to be comparable with those of fiber-reinforced polymer composites.

M. Vural () Mechanical, Materials and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, IL 60616, USA e-mail: [email protected] G. Ravichandran Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA e-mail: [email protected]

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1 Introduction In general, cellular solids with stochastic or periodic topologies are often utilized as core materials in sandwich structures to increase specific stiffness as well as impact damage protection [1, 2]. Balsa wood is a naturally occurring light-weight cellular material and is one of the most commonly used core material in composite sandwich structures in marine industry because of its salient mechanical and physical properties. As its cellular/porous microstructure allows the application of large deformations, fine composite nano-architecture of wood cell material increases its specific strength and stiffness, giving rise to a high specific energy dissipation capacity. Moreover the fact that balsa wood can be found in a wide range of densities from 40 to 380 kg=m3 , depending on the average size and the wall thickness of cells, provides the flexibility in design since the strength is a monotonic function of its density [3]. Of particular interest for balsa wood are the mechanical properties in longitudinal (along the grain) direction such as its peak strength, plateau stress and corresponding failure mechanisms under compressive loading conditions. Several quasi-static experimental investigations have been carried out in uniaxial compression/tension as well as shear modes to determine the mechanical properties and deformation mechanisms of balsa wood in longitudinal and/or transverse directions [3–7]. Knoell [4] investigated the effects of environmental and physical variables (temperature, moisture content and ambient pressure) on the mechanical response of balsa wood. Soden and McLeish [5] carried out an extensive investigation, which mainly concentrated on the variation of tensile strength with fiber alignment. They also reported compressive strength data. Easterling et al. [6] paid particular attention to the micromechanics of deformation in their experiments, during which they performed in-situ scanning electron microscopy (SEM) observations and defined the end-cap collapse of grains as the dominant compressive failure mechanism in longitudinal direction. Vural and Ravichandran [3] documented the compressive strength, plateau stress and densification strain of balsa wood in its entire density range, identified the variations in failure mechanisms with density and described simple analytical models to represent the observed experimental strength data. Most recently, Silva and Kyriakides [7] performed an extensive experimental study on the compressive and shear behavior of balsa wood and reported both elastic and subsequent nonlinear crushing response of balsa wood along its all three orthotropic axes. However, the dynamic compressive behavior of balsa wood (and in general wood) has received limited attention. Daigle and Lonborg [8] performed dynamic compression experiments on balsa wood and several other crushable materials using drop towers and compared their specific energy absorption capacities. Reid and Peng [9] investigated the dynamic crushing behavior of several wood species, including balsa wood, through the impact of free flying specimens on a rigid wall. While their tests covered a wide range of impact velocities up to approximately 300 m/s, they tested the specimens at a certain density for each species. More recently, Vural and Ravichandran [10, 11] systematically investigated the dynamic response and energy mitigation capacity of balsa wood at high strain rates for its entire density range.

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In this work, we present a comprehensive review of Vural and Ravichandran’s [3, 10, 11] experimental and analytical work to understand the mechanical behavior of balsa wood over its entire density range in both quasi-static and dynamic deformation regimes with particular emphasis on its relation to microstructural deformation modes. The paper is organized as follows. The next section describes the microstructural features of balsa wood, specimen preparation, and experimental methods used. In Section 3, experimental results are discussed, and analytical models based on the kinematics of deformation modes are introduced and compared with experiments, which is followed by conclusions.

2 Experimental 2.1 Microstructural Features of Balsa Wood As shown in Fig. 1, the cellular microstructure of balsa (Ochroma) is mainly composed of long prismatic cells (fibers, grains) of nearly honeycomb shape and the blocks of these cells are separated by narrower rays in which the cells are smaller and of a different shape. Balsa wood has three orthogonal axes in the longitudinal (L, along the grain), radial (R, across the grain and along the rays) and tangential (T, across the grain and transverse to rays) directions forming a heterogeneous porous composite. This composite is highly anisotropic with a high ratio of longitudinal to transverse properties, the latter having also difference in itself due to so-called ray cells oriented in radial direction. It should be noted that, for applications where balsa wood is considered as a sandwich core material, its mechanical response in longitudinal direction as well as corresponding failure mechanisms are of particular interest.

Fig. 1 SEM micrographs showing the typical (a) across the grain and (b) along the grain crosssections of balsa wood

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Balsa, as a diffuse-porous hardwood, has a uniform distribution in type and size of cells throughout the grain cross section (see Fig. 1a). Cells of nearly hexagonal shape (grains) have diameters ranging from 30 to 70 m, with an average of 45 m. These long, hexagonal-prismatic cells are pointed at the ends (Fig. 1b) and have an average length of 650 m, giving the cell an aspect ratio of about 16:1, as also reported by Easterling et al. [6]. Blocks of these large cells are separated by narrower rays in which the cells are smaller and of a different shape. Large sap channels running parallel to the axis of the tree penetrate the entire structure. Average diameter of sap channels is around 350 m. A closer look into the ultrastructure of cell wall material reveals a unique hierarchical architecture at sub-micron scale. Figure 2 schematically shows the details of microstructural arrangement through a cell wall. Wood fibers (cells) normally possess three distinct layers designated as S1, S2 and S3 layers, which is the main source of their high specific strength. It is self-evident from Fig. 2 that secondary wall layering is the result of differences in the angle at which microfibrils are oriented within each layer. Microfibrils are the reinforcing phase of wood cell structure since they contain highly crystalline high strength cellulosic chains. Because of its relative thickness, the S2 layer has a great influence on most wood properties. The steeply inclined wrapping of the microfibrils in the S1 and S3 layers would have the mechanical advantage of stiffening the S2 layer under longitudinal compressive forces. A more detailed discussion of wood cell wall chemistry and architecture can be found in Vural and Ravichandran [11]. It must be noted that even though the grains in balsa wood are highly oriented in longitudinal direction this alignment is far from being perfect, as would be expected in a naturally occurring material. Experimental observations reveal that rays are the major source of grain misalignment in LT plane. Figure 3 shows a typical

Microfibrils S3 Secondary Wall

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Fig. 2 Magnified view of a balsa cell (left, from Ref. [11]) and a schematic of its nanostructured hierarchical infrastructure (right, after Ref. [12]). The typical diameter of microfibrils ranges between 10 and 30 nm, which possibly consist of even smaller sub-units called protofibrils with a few nanometers diameter. The steeply inclined wrapping of the microfibrils in the S1 and S3 layers can stiffen the S2 layer under longitudinal compressive forces

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Fig. 3 SEM micrograph illustrating typical fiber misalignment around ray cells θo

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misalignment caused by the existence of ray cells penetrating the complete structure in radial direction. Even though there exist longitudinal misalignments in LR plane as well, they are both smaller in magnitude and not considered as important as LT misalignments in terms of their effect on the failure mode/strength of balsa wood. This issue will be revisited in detail and its relevance to kink band formation will be given in the discussion of failure models. Scanning electron microscopy (SEM) examinations performed on the LT planes of balsa specimens with various densities show that the maximum fiber misalignment angles are highly scattered in the range between 7ı and 11ı . Apart from this measurable imperfection at the length scale of the cell height, individual cell walls possess further imperfections at smaller length scales that are rather difficult to quantify such as the waviness or non-uniform thickness distribution along the cell walls.

2.2 Specimen Preparation and Geometry Compressive experiments were performed on balsa wood specimens along the grain direction over a wide range of densities between 55 and 380 kg=m3 . Cylindrical specimens were machined from big blocks of balsa wood at five different nominal densities provided by BALTEK Corporation (Northvale, NJ) and polished using 320 grit sand papers. In order to avoid any possible end effects on the mechanical behavior during testing, planar surfaces of specimens were machined and inspected with extreme care to assure good specimen end-surface quality. Densities were measured in a consistent way, paying considerable attention particularly to the moisture content since it has significant effect on both density and properties [13]. Hence, density of each specimen was calculated with an accuracy of 1:5% prior to testing by measuring its weight (within 2 mg) and using its nominal dimensions given below. Due to the variation of density within each block of a certain nominal density, specimens were prepared with densities varying continuously between the range

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specified above. In order to find moisture contents of balsa specimens a series of specimens with different densities were heat-treated in furnace at 110ı C and intermittent weight measurements were performed until they reached a constant weight. Measured moisture content ranged from 8% to 11%, which is the typical value for well-seasoned dry woods. Since the determination of moisture content for each specimen prior to mechanical testing is inapplicable due to the irreversible nature of process, they are assumed to have moisture content within this range. The specimens used in quasi-static testing were typically 19:05 ˙ 0:12 mm in diameter and 25:4 ˙ 0:12 mm in length, corresponding to a length to diameter ratio of 1:33. The dimensions of the specimens used in high-strain-rate testing were 9:5 ˙ 0:12 mm in diameter and 5:08 ˙ 0:12 mm in length, corresponding to a length to diameter ratio of 0.53. It must be noted that the length of all specimens is along the grain direction.

2.3 Quasi-Static Testing Method Figure 4 shows the compression fixture used in the quasi-static compression tests with and without confining sleeve. It ensured that two loading rods were perfectly aligned with each other so that any unwanted shear forces on the specimen are minimized. The specimen is sandwiched in between the loading rods and the compression was applied by a screw-driven materials testing system (Instron, model 4204) at a constant displacement rate of 2 mm/min, which corresponds to a nominal strain rate of 3  103 s1 . The quasi-static compression tests in longitudinal direction (along the grain) were performed with and without lateral confining sleeve to investigate the behavior of balsa wood under compression with proportional confinement and uniaxial compression, respectively. The loading rods are made of high

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Fig. 4 Compression fixture for (a) unconfined and (b) confined experiments of balsa wood specimens

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strength maraging steel (C-350, Rockwell hardness, RC D 60) and the confinement sleeve and end caps are machined from aluminum alloy (Al 2024-T3, Rockwell hardness, RB D 75). The use of bonded resistance strain gages on highly compliant balsa specimens was not considered satisfactory because the reinforcing effect of bonding these gages with adhesive might lead to large errors in the measurement of strains. Therefore, deformation data of both confined and unconfined specimens were obtained from the displacement transducer, which was calibrated to eliminate the machine and fixture compliance. The confining sleeve shown in Fig. 4b is a hollow cylinder, which resists the lateral expansion of the specimen during the axial compression. Thus, lateral confinement, which is proportional to axial compression, is applied on the specimen. The experimental setup for the confined tests consists of a cylindrical specimen placed in a hollow cylinder with a sliding fit and the specimen is axially compressed using loading sleeves [14]. Since the strength and elastic modulus of the confining sleeve are greater by two orders of magnitude than that of the balsa specimens in lateral direction, the stresses within confining sleeve remain elastic at all times during the tests. Moreover, elastic analysis of the confined configuration for balsa wood as the specimen shows that radial expansion of the specimen-sleeve interface remains practically zero. Therefore, the boundary condition at the lateral surface of the specimen can be considered as zero displacement, i.e., rigid wall confinement.

2.4 Dynamic Testing Method The specimens were dynamically loaded using a modified compression Kolsky (split Hopkinson) pressure bar as shown in Fig. 5. Kolsky pressure bar, originally developed by Kolsky [15], has been widely used and modified to measure the dynamic compressive behavior of engineering materials. The present modification through the use of a quartz single crystal, introduced by Chen et al. [16], enables the stress measurements three orders of magnitude more sensitive than achievable in a

Fig. 5 Schematic of modified Kolsky (split Hopkinson) pressure bar used in the dynamic compression of balsa specimens

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conventional Kolsky (split Hopkinson) pressure bar. Hence, stress profiles of low amplitudes experienced by balsa specimens could be measured with high precision by adopting this modification. The lengths of the bars in Kolsky pressure bar setup used in this study are 1219 mm for the incident bar, and, 584 and 610 mm for the two sections of the transmission bars, with a common diameter of 12.7 mm. The striker bars of 12.7 mm diameter were used with varying lengths to achieve desired pulse duration. All the bars are made of precision ground 7075-T651 aluminum alloy. Depending on the striker velocity, either a thin, annealed copper (C11000) disc of 4.8 mm in diameter and 0.81 mm in thickness or a layer of tissue paper was typically used as pulse shaper to control the rise time of the incident pulse and to ensure stress equilibration within the specimen [17]. A high resistance (1000 ) strain gage (Micro-measurements, WK-06–250BF-10C) with excitation voltage of 30 V was used to measure transient surface strain on the incident bar. The raw strain gage signal was recorded using a high-speed 12-bit digital oscilloscope (Nicolet 440). A Valpey-Fisher X-cut quartz single crystal of 12:7 ˙ 0:025 mm in diameter and 0:254 ˙ 0:025 mm in thickness, coated with a 30 nm thick chrome/gold layer on both of the flat surfaces, was glued to the end of the transmission bar in contact with the specimen using a conductive epoxy (CW2400, Chemtronics). Other surface remained in contact with the other half of the transmission bar without bonding, allowing the compressive transmitted pulse to pass through the quartz disk without reflection. Before each experiment, a thin layer of conductive grease (CW7100, Chemtronics) was applied at the contact surface to ensure electrical conductivity. Thin, silver coated copper wires were wrapped and glued (using the same conductive epoxy) to the bar surface near the quartz crystal. The wires were connected to a charge amplifier (Kistler 5010B1) whose output is recorded using the digital oscilloscope for data processing. Using the modified Kolsky pressure bar system described above, balsa wood specimens were uniaxially compressed in longitudinal (L) direction (i.e., along the grain) at a nominal strain rate of 3  103 s1 .

3 Results and Discussion 3.1 Stress–Strain Response Typical engineering stress–strain curves obtained from experiments at both quasistatic and dynamic loading rates are shown in Fig. 6 as a function of specimen density. For both cases the stress–strain curve is almost linear up to the maximum stress (hereafter referred to as failure strength, f ) beyond which, as the deformation is increased, the stress level either remains nearly constant or experience a sudden drop depending mainly on the density of specimen. After this sudden drop, if there is, the specimen continues to deform at a lower level of stress (hereafter referred to as plateau stress, p ). Eventually, at high strains, the cells collapse sufficiently so

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that opposing cell walls touch (or their broken fragments pack together) and further deformation compresses the cell wall material itself. This gives the final, steeply rising portion of the stress–strain curve called densification regime. Figure 6 clearly shows that, while the compressive deformation in plateau region is fairly smooth and occurs almost at a constant stress level for low-density balsa specimens, it is significantly irregular and of a fluctuating nature for high-density specimens. These two different characters of stress–strain response reflect the fact that there occurs a transition both in failure mode and in progressive deformation mechanism as the specimen density increases, which will be discussed in the context of the following paragraphs. It is also immediately observed from Fig. 6 that the failure strength increases at high strain rates while there is an apparent decrease in densification strain.

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End Effects

It should be emphasized that the quality of the loading surfaces of specimens becomes an important issue in compressive testing of cellular solids due to the socalled end effects, which may alter certain features of the mechanical behavior. Due to both frictional effects and the discrete nature of cellular topology, specimens have a tendency to premature localized deformation at the loading surfaces when the surface preparation is of poor quality. For instance, experiments on specimens with roughly cut end-surfaces may result in a lower compressive modulus when using end plate displacements than when using an extensometer because of the end effects [2]. It should be stated at this point that neither the elastic modulus nor its variation with density or strain rate has been the subject of this study. End effects are also important in that they may affect the magnitude of compressive failure strength. If the cellular solid under consideration is a type-II structure (i.e., has a significant strain softening at the end of homogeneous elastic deformation and with the beginning of initial failure localization) such as the honeycombs loaded in longitudinal direction, the geometric imperfections at the specimen ends may obscure the initial stress peak and result in monotonous stress–strain curves. In the present study, planar surfaces of cylindrical specimens have been machined and inspected with extreme care to assure good specimen-end quality and to avoid any possible end effects on the mechanical behavior during testing. Experimental stress–strain curves further verify that the end effects of this type have been avoided (Fig. 6). Finally, as long as the large strain inelastic behavior of cellular solids is concerned (such as the plateau stress and densification strain) end effects are negligible for all practical purposes. Large-strain deformation of balsa wood involves the regions of intense localization which eventually act as favored sites for progressive deformation, replacing any possible degenerative role of end effects.

3.1.2

Initial Failure and Progressive Deformation

The variation of compressive failure strength over the entire density range is shown in Fig. 7a for both quasi-static and dynamic loading rates, where solid lines are the best linear fits to experimental data. It is obvious that both the density and the strain rate have considerable effects on the initial failure strength of balsa wood. Higher density means increased ratio of cell wall thickness to cell diameter, and thus less porosity, therefore its effect towards fortifying the structure is expected and experimentally observed in both failure strength and plateau stress (Fig. 7b). However, the effect of strain rate seems to be more complicated due to the fact that the plateau stress is practically insensitive to the strain rate whereas the failure strength is significantly affected by the increase in strain rate. In other words,experimental evidence indicates that, while the critical stress for failure initiation has a strong dependence on the strain rate, driving force for progressive deformation remains unaffected by the dynamics of deformation. This difference in response to increasing strain rate can simply be explained in terms of the differences in the kinematics

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of deformation, associated micro-inertia and initiation mechanism at micro scale prior to initial failure and during progressive deformation. Initial failure at the peak stress level is preceded by elastic loading. Micromechanics of deformation in this regime is characterized by the uniform axial compression of wood cells in longitudinal direction. Initial failure is mostly triggered by an elastic instability in the form of local micro buckling or kink band formation depending mainly on the density of balsa wood. In dynamic loading, the onset of both these kinds of instabilities are intimately related to the lateral inertia in deforming elements, which in turn enhances the critical stress for failure initiation in the longitudinal direction. Calculations based on the kinematics of micro

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buckling/bending geometries show that even though the initial effect of this inertial strain rate hardening is very strong it rapidly falls off and becomes negligible during the evolution of deformation in a localized zone. However, progressive deformation phase is also characterized, in addition to the continuation of deformation in initially localized regions, by the creation of new localization zones. Following the initial failure/localization, the state of stress experienced by neighboring cells is perturbed and far from being uniaxial due to the kinematics of deformation at the micro scale. This new local situation (state of stress) makes the subsequent progressive deformation (localizations) easier and at a lower nominal stress level by taking and exaggerating, in a sense, the role of local geometric and material imperfections that trigger the initial failure. Within this framework, regarding the formation of new strain localization sites, it appears that the softening effect due to the perturbations in stress field created by preceding localization suppresses the effect of inertial hardening. The effect of strain rate, and hence micro-inertia, is thus restricted to the initiation phase of first instability due to the kinematics of failure (buckling/kink banding). During the following progressive deformation phase the effect of inertia is suppressed either by the kinematical considerations, as in the evolution of deformation within localized region, or by the perturbations in stress field as in the creation of new localization sites. Figure 7b also shows an increased scatter in plateau stress when the specimen density exceeds 170–200 kg=m3 interval. This is attributed to the transition in deformation mode from micro buckling to kink band formation and resulting increased susceptibility of plateau stress to the random nature of increased perturbations in stress field. As the length scale of the deformation geometry increases from a fraction of cell diameter in buckling mode to as much as 10–15 times the cell diameter in kink band formation mode, an increased level of perturbation is induced in the stress field and it generates the increased scatter in plateau stress for denser specimens. As opposed to many other materials, compressive strength of balsa wood in longitudinal direction is insensitive to lateral confinement, and any significant variation in compressive strength, plateau stress or densification strain was not observed in experiments with lateral confinement. This is attributed to the combined effect of high porosity, nature of cellular microstructure and high compliance of cell walls in lateral (across the grain) directions, which altogether give the material flexibility of accommodating the would-be-induced lateral strains within the initial lateral dimensions without exposing itself to significant lateral stresses.

3.1.3

Densification

At the end of plateau region after all cells have collapsed, densification strain is reached and continued compression results in the compaction of deformed and ruptured cell walls against each other, leading to rapid increase in crushing stress since internal voids are almost eliminated. For design purpose, densification strain can be interpreted as the maximum range of safe loading in the sense that beyond this range further compression generates substantial amplification in the stress level over

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350

Density (kg/m3) Fig. 8 Decreasing trend of densification strain in balsa wood with density and strain rate

the plateau stress as well as initial failure strength. Therefore, in applications where shock attenuation and impact damage protection are considered to be of significant importance, densification strain is one of the important material properties because, together with plateau stress, they directly determine the energy absorption capacity of material. Figure 8 illustrates the effects of initial density and strain rate on the densification strain of balsa wood, which is defined in this study as the strain at the last local minimum before the stress starts rising steeply (see also Fig. 6). For quasistatic loading rates, densification strain decreases as a function of specimen density and varies between 0.68 and 0.86. As expected, the denser is the balsa wood the lower is the densification strain due to its close correlation with the level of initial porosity. In dynamic loading regime, the same declining linear trend in densification strain is preserved. However, it is shifted downwards and the bounds decrease to the range between 0.58 and 0.74, suggesting that cellular packing efficiency at the microstructural level degrades by the increasing rate of strain. This could be attributed to the micro-inertial effect associated with the cell wall collapse. Densification strain can alternately be defined as the strain at which the tangent modulus of the stress– strain curve attains the value that corresponds to 3% of the elastic modulus and monotonously increases from that point onwards with further deformation. The data extracted from the original stress–strain curves according to this definition differ from the previous densification strain data by less than 1%. Maiti et al. [18] have a theoretical prediction for densification strain, "d , by assuming that it is reached when all the voids within cellular material are eliminated and the relative density of deformed wood, ."/=s , attains the value of 1, i.e., "d D 1  =s . However, this approach overestimates the present quasi-static

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densification strain data by as much as 15%. Since this was also the case for their experimental data they suggest that the densification strain is reached when the relative density of wood attains the value of 0.5, i.e., "d D 1–2=s . Even though this last relation gives a good correlation with their wide range of data for various species of wood, it still does not correlate well with the quasi-static experimental data presented in Fig. 8. The best fit to the present quasi-static data for balsa wood is given by, (1) "d D 0:87 .1  =s / where  is the initial density of balsa wood and s is the density of wood cell substance. The last linear relation implies that, for the quasi-static data, densification strain occurs 13% before the theoretical prediction [18]. The packing/compaction involves the random folding and crushing of cellular elements and therefore occurs in a stochastic manner, which leads to inefficiency in packing and, thus, to the fact that its prediction is not straightforward. For dynamic data, cellular packing efficiency degrades further due to the effect of increased strain rate and for dynamic data in Fig. 8 the densification strain is given by, "d D 0:75 .1  =s /

(2)

which implies a degradation of 25% over its theoretical value represented by the terms within the brackets. The difference in the degradation of packing efficiency between quasi-static and dynamic loading conditions may, of course, be qualitatively attributed to the effects of micro-inertia experienced by the cell walls during dynamic deformation. However, to quantify its contribution to the packing inefficiency again is not easy. Having discussed these observations, it is still interesting to note that, independent of the initial density of balsa wood, experimental densification strain data correspond to a constant factor (0.87 for quasi-static and 0.75 for dynamic loading) of the theoretical prediction. It may be argued that the transverse deformation effects should also be considered in the discussion presented above since the experimental data comes from the specimens compressed under uniaxial stress conditions. Let’s assume that the densification starts when the initial volume of balsa specimen .Vo / is compressed by V, which is a certain factor (k) of the initial void volume .Vv /. Thus, volumetric strain at this point is given by, V Vv Vo  Vs Dk Dk D k .1  =s / Vo Vo Vo and also

(3a)

3

X V D "k D "d .1  AT  AR / Vo

(3b)

kD1

where  is the Poisson’s ratio and the subscripts A, T and R stand for natural orthogonal directions in wood as described in Section 2.1. Thus, "d D

k .1  =s / 1  AT  AR

(4)

Mechanical Response and Deformation Mode Selection in Balsa Wood

731

in which the last term in parenthesis represents the theoretical densification strain under uniaxial strain conditions, k can be defined as the packing factor, and the denominator of the first term stands for the transverse deformation effects. The comparison of the last relation (Eq. 4) with the Eqs. 1 and 2 justifies the usage of the term “packing efficiency” for the experimental constants 0.87 and 0.75, which correspond to the first term in Eq. 4, because any possible effect of transverse deformation should be the same for both quasi-static and dynamic loading conditions.

3.1.4

Energy Dissipation Capacity

Energy is dissipated through the progressive inelastic deformation of cellular structure where the plateau stress .p / and densification strain ."d / are two major quantities that control its magnitude. Therefore, energy dissipated per unit mass is approximately given by, 1 E D p "d (5)  Figure 9 shows the overall trend for the specific energy dissipation capacity of balsa wood per unit mass basis as a function of its density at both quasi-static and dynamic loading rates. The data presented represent the area under stress–strain curves up to densification strain divided by the density. The S-shaped character of the data is a consequence of failure mode transition from buckling to kink band formation. The first rising part of data at low densities corresponds to the specimens that undergo buckling mode of deformation as shown in Fig. 10a. In this regime, the rate of increase in plateau stress with respect to density is much higher in comparison with the rate of decrease in densification strain and the resulting net effect is an increasing trend in specific energy dissipation as function of density. However, when the

100

Em (kJ/kg)

80 60 40 3000 S–1

20

0.0013 S–1

(a)

0 0

50

100

150

200

250

300

350

Density (kg/m3)

Fig. 9 Variation in the specific energy absorption capacity of balsa wood with density and strain rate

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M. Vural and G. Ravichandran

a

b 3.5 r = 55 kg/m3

. e = 1.310–3 s–1

3.0

Stress (MPa)

2.5 2.0 1.5 1.0 0.5 0.0 20 μm 750 X

0

0.01 0.02 0.03 0.04 0.05 0.06 Strain

Fig. 10 Failure mechanism illustrating plastic buckling (a) and associated stress–strain curve (b) for low-density balsa wood

a

b 30

r = 236 Kg/m3

. e =1.3x10–3 s–1

Stress (MPa)

25 20 15 10 5 0 800 μm 30X

0

0.01 0.02 0.03 0.04 0.05 0.06 Strain

Fig. 11 Failure mechanism illustrating kink band (a) and associated stress–strain curve (b) for high-density balsa wood

density of balsa exceeds approximately 170 kg=m3 deformation mode changes to kink band formation (see Fig. 11a). Even though the transition in failure mode does not affect on the linear decreasing trend of densification strain, it slows down the rate of increase in plateau stress and results in the transient drop of specific energy dissipation observed in Fig. 9. When the density increases further the rate of increase in plateau stress is recovered as is evident from the final rising part of data in

Mechanical Response and Deformation Mode Selection in Balsa Wood

733

Fig. 9. The overall picture of specific energy dissipation in balsa wood as a function of density can be summarized by taking the logarithmic derivation of Eq. 5, dp =d d "d =d 1 dE=d C  D E p "d 

(6)

where densification strain ."d / is known to be a decreasing linear function of density as shown in previous section. Thus, the nonlinear relationship between the plateau stress and density, i.e., the first term on the r.h.s. of Eq. 6, controls the rate of specific energy dissipation and directly responsible for the S-shaped character of curve in Fig. 9. At this point, one should note that the specific energy dissipation capacity of balsa wood lies between 30 and 90 kJ/kg, which is comparable or better than the most synthetic fiber reinforced composite tubes specifically designed for energy dissipation purposes. It is reported by Mamalis et al. [19] in a review study for the crashworthiness capability of composite material structures that specific energy dissipation/absorption capacities of fiber reinforced hallow tubes made of Carbon/Epoxy, Aramid/Epoxy, Glass/Epoxy and Carbon/PEEK lie between 50–99, 9–60, 30–53, 127–180 kJ/kg respectively, variations arising mostly from the fiber orientation angle and stacking sequence. When compared with these highly engineered advanced composite structures it is interesting to see that balsa wood, as a product of nature, outperforms the majority of man-made composite systems in energy dissipation. In fact, balsa wood owes its remarkable energy dissipation capacity to its naturally tailored microstructural features such as the tubular honeycomb geometry of its cells and the fine nano-architecture and composite nature of its cell walls as depicted in Fig. 2. A more detailed discussion of energy dissipation in balsa wood in relation to hierarchical architecture of its fibrous cell wall and its broader implications can be found in Vural and Ravichandran [11].

3.2 Failure Modes Post-test examinations on the failure surfaces of specimens sectioned by a sharp razor blade were performed using scanning electron microscopy (SEM). Both visual observations during compression tests and SEM observations on a series of deformed balsa wood specimens at various densities suggest that the failure mode undergoes transition from elastic/plastic buckling of cell walls to kink band formation as the density of the specimen tested increases. Figure 10a shows a typical SEM micrograph of the failure surface of a low-density balsa wood specimen. This micrograph was taken from a specimen compressed up to a total strain of 3.2%, its stress-strain history is shown in Fig. 10b. It is seen that the deformation is localized into a narrow region in the form of plastic buckling waves and that the region outside of the localized zone is characterized by small wrinkles, which would eventually grow into buckling waves if the deformation were increased

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M. Vural and G. Ravichandran

further. For this particular specimen maximum stress occurs at 2.3% strain and the subsequent stress drop is negligible as usually observed in low-density specimens (Fig. 6a). Figure 11 shows the failure surface and corresponding stress-strain curve of a high-density balsa specimen. SEM micrograph clearly shows that in this case failure occurs by kink band formation. The maximum stress occurs at 2.8% strain and the subsequent deformation is associated with a large stress drop that is characteristic of kinking. The kink band formation represents a compressive instability in highly anisotropic materials. The fibers that deform into the kink band geometry involve a primary axial shear deformation. The initiating mechanism is that of the imperfections in the material, usually expressed through a fiber misalignment angle for the nominally aligned state of fibers. Balsa wood, as shown above (see Fig. 3), has favored sites for misaligned fibers, which in return trigger the observed kink band formation.

3.3 Strength Models Based on Failure Modes Post-mortem observations on both unconfined and confined specimens reveal that elastic/plastic buckling and kink band formation are two major compressive failure mechanisms in balsa wood. Easterling et al. [6] report in their study on the mechanics of balsa wood that compressive failure in axial direction is dominated by the plastic collapse of end-caps (pointed end of long hexagonal cell). Even though end-cap collapse is not observed as the sole mechanism for failure in the present study, pyramidal end-cap collapse and/or fracture is observed to have occurred at some portions of the hinges of the kink bands. Therefore, elastic buckling, plastic buckling, end-cap collapse and kink band formation can be considered to be potential failure mechanisms in balsa wood. It has been shown in the following that simple analytical models based on these failure mechanisms have a strong potential to explain the observed trends and the model predictions correlate well with the experimental data for a natural porous composite such as balsa wood.

3.3.1

Elastic Buckling

The elastic buckling of a thin plate (the cell wall) which is constrained along the two edges that lie parallel to the loading direction is a standard problem in structural mechanics (see, for example, Ref. [20]). The buckling load per unit length which is determined by the flexural rigidity of the wall (D) and by the width (l), not the height (a) of the panel (see Fig. 12a for notation), is given by: Ncrit D k

 2D l2

(7)

Mechanical Response and Deformation Mode Selection in Balsa Wood

a

l

735

b

P

t

dz

l

a r

z

Fig. 12 Schematic of (a) the axial elastic buckling of a hexagonal cell, (b) the collapse of an idealized pyramidal end-cap

If the vertical edges in Fig. 12a are simply supported (i.e., they are free to rotate), and the height a is large as compared to l (a > 3l), then k D 4. If, instead they are clamped, k D 6:97. In the honeycomb structure, the cell wall is neither completely free nor rigidly clamped, therefore, to take a value just in between them appears reasonable. However, in a natural material like wood, the deviations from perfect geometry such as imperfections in cell wall planarity and spatial variation of cell wall thickness are considered highly effective in reducing the critical load for elastic instability. Therefore, the use of a much lower value like, for example, k D 2 is proposed arbitrarily. For simple hexagonal cells, the relative density =s is related to the dimensions of the cell wall by (see Ref. [1]), 2 t  Dp s 3 l



t 1 p 2 3l

2 t Šp 3 l

(8)

for low-density regular hexagons; where  is the density of entire cellular structure (balsa wood) and s is the density of wood cell substance which is generally taken to be approximately 1;500 kg=m3 [21, 22]. Then, by using Eq. 8 and hexagonal configuration of cell walls, Eq. 7 gives the critical stress for elastic buckling .eb / as a function of density,  2 Es  3 =s eb D k =4 (9) 1  s2 where Es and s are the axial Young’s modulus (35 GPa, from Cave [23]) and Poisson’s ratio of the cell wall (assumed to be 0.3), respectively. Zhang and Ashby [24] used the similar approach to predict the out-of-plane compressive strength of Nomex honeycombs and showed that the failure stress for this material could be represented by elastic buckling analysis for relative densities, =s , lower than 0.1. As will be discussed later, the comparison of Eq. 9 with experimental strength data for balsa wood also shows that elastic buckling can be considered as one of the failure mechanisms only for very low densities.

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3.3.2

M. Vural and G. Ravichandran

Plastic Buckling

The axial collapse of hexagonal honeycombs has been treated by McFarland [25] and, more recently improved by Wierzbicki [26]. Wierzbicki identifies a compatible collapse mode which requires plastic hinges both on the cell walls and at the corners, and a limited amount of cell wall extension also at the corners. By minimizing the collapse load with respect to buckling wavelength Wierzbicki’s method, together with Eq. 8, gives the stress for collapse by plastic buckling as  5=3 pb D m ys =s

(10)

where ys is the yield stress for wood cell wall substance (350 MPa, from Cave [27]) and m D 4:4 for regular hexagons with uniform wall thickness t . However, due to the same reasoning discussed above regarding the imperfections in wood cell geometry, a lower arbitrary value of m D 2 has been chosen to show its potential to represent the experimental data. Actually, the expression originally derived by Wierzbicki [26] is for the prediction of progressive crushing strength (plateau stress). However, the fact that that the stress ratio (f =P ) is very close to 1 for lowdensity balsa wood justifies the use of Eq. 10 to predict the failure strength, at least, for low density specimens.

3.3.3

End-Cap Collapse

Figure 12b shows an idealized pyramidal end-cap which represents the tapered ends of long hexagonal cells in balsa wood (see Fig. 1b). When the cell is loaded axially these end-caps collapse by the stretching of the triangular faces, i.e., by the plastic extension of elements like the one shaded in Fig. 12b. The collapse strength .pc / is calculated by equating the work done by a small displacement •u due to the load P to the plastic work required to stretch a triangular face plastically: Zr 0 P ıu D 6

y ıu r t z dz r 02

r and

0

r D

r2 C

3l 2 4

(11)

0

or pc D

1  r    =s y 2 l

(12)

where y is the yield strength of the wood cell wall material in the direction of the length of the shaded element (assumed to be 350 MPa), and the dimensions r, l, z and t are indicated in Fig. 12b. The SEM observations show that r= l ratio varies roughly in the range from 1 to 3 for balsa wood. Therefore, the minimum value of r= l D 1, representing the weakest sites, is adopted for modeling the behavior of balsa wood.

Mechanical Response and Deformation Mode Selection in Balsa Wood

3.3.4

737

Kink Band Formation

Kink band formation has long been recognized as a major failure mechanism limiting the compressive strength in unidirectional fiber reinforced composites. Therefore, this problem has received wide attention. Following the elastic fiber microbuckling concept originally proposed by Rosen [28] to model the compressive instability in composites, Argon [29] took the alternative view that long-fiber composites undergo plastic kinking and recognized that the initial fiber misalignment angle, o , would have a large degrading effect on the compressive strength. Budiansky [30] extended Argon’s formula for an elastic-perfectly plastic composite to a more general expression for stress .kb / required for kink band formation, kb D

G13 1 C o =y

(13)

where G13 is the axial shear modulus of the composite and y D y13 =G13 is the yield strain in longitudinal shear. The fibrous and anisotropic structure combined with the favored sites for large fiber misalignment angles (see Fig. 3) makes the kink band formation a major failure mode in balsa wood, which is also evidenced by SEM observations (Fig. 11a). Therefore, Budiansky’s formula was used to see its correlation with experimental data. Out of plane shear modulus, G13 , in Eq. 13 is replaced by GLT for balsa wood, because the majority of fiber misalignments are observed to exist in LT plane due to the ray cells lying along the radial direction. Furthermore, the fact that ray cells act as a reinforcing phase to increase the shear modulus in LR plane is also supported by the experimental evidence that the kink band failure always occurs in the LT plane. Based on the cellular mechanics calculations of Gibson and Ashby [1] for out of plane shear properties of regular hexagonal honeycombs, and in conjunction with Eq. 8, the shear modulus and shear strength in LT plane for balsa wood can be approximated by  1  (14) GLT D Gs =s 2 and 1   LT D ys =s (15) 2 where Gs and ys are the shear modulus and yield strength in shear of the wood cell wall material, respectively. The values Gs D 2:6 GPa and ys D 30 MPa will be used here, which have been determined to be the relevant values by Gibson and Ashby [1] through the extrapolation of the experimental data for shear moduli and shear strength of a number of woods. The use of Eqs. 14 and 15 give a yield strain .LT =GLT / of y D 0:023 in longitudinal shear, which is independent of the relative density because of their individual dependence on density is the same (linear). The last term that remains to be determined in Eq. 13 is the fiber misalignment angle, o . The SEM examinations (e.g., Fig. 3) performed on the LT planes of balsa specimens with various densities show that the maximum fiber misalignment angles

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M. Vural and G. Ravichandran

are highly scattered in the range between 7ı and 11ı . Hence, an average value of

o D 9ı is used to predict the critical stress .kb / for kink band formation. It should be noted that the two parameters in Eq. 13, G13 and y , are predicted by using Eqs. 14 and 15. Even tough these equations are tuned according to the experimental data in literature they reflect an averaging process over different species of wood, thereby introducing a certain amount of error for the balsa wood. However, the experimental determination of these values for the complete density range of balsa wood would exceed the scope of present paper. Moreover, it is considered that the large scatter observed for the third parameter of Eq. 13, i.e. fiber misalignment angle o , and the selection of an arbitrary average value for it justifies the use of this approximation.

3.4 Comparison with Quasi-Static Experiments The predictions of the four separate strength models described above are plotted in Fig. 13 along with the experimental strength data for balsa wood obtained from compression tests in longitudinal (grain) direction. It is obvious that none of these models is capable of representing the entire data by itself. However, as also suggested by the experimental evidence of failure mode transition, the combination of models successfully correlates with the present experimental data. It should be noted

60

50

Strength (MPa)

40

30

20 Quasi-static data Elastic buckling Plastic buckling End-cap collapse Kink band formation

10

0 50

70

90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 Density (kg/m3)

Fig. 13 Correlation of various failure model predictions with the experimental compressive strength for balsa wood along the grain direction

Mechanical Response and Deformation Mode Selection in Balsa Wood

739

that the plots of the first two models, namely elastic and plastic buckling models, are based on arbitrary assumptions concerning the geometric imperfections such as the spatial variations of thickness and the deviations from planarity in cell walls. Basically, the two failure models based on plastic buckling and kink band formation mechanisms correlate well with the experimental data. The transition region predicted by these models lies between ¡ D 170 kg=m3 and ¡ D 200 kg=m3 and is compatible with experimental observations. At very low densities, elastic buckling seems to have the potential as a competing failure mechanism. In this region progressive failure occurs by plastic folding of cell walls into buckles comparable in length to the width of cell walls. However, to identify the initiation mechanism for failure is extremely difficult. One can only speculate that depending on the type and degree of geometric imperfections at micro scale failure initiates either by elastic or by plastic buckling of the cell walls. At high densities above transition region, the predictions of pyramidal end-cap collapse mechanism correlate with data, interestingly, equally well as those of kink band formation do, hence making these two mechanisms compete at high densities. Actually, geometric perturbations induced by end-cap collapse at favored sites may also trigger kink band formation. Therefore, the last two mechanisms are open to both competition and interaction.

3.5 Models for Inertial Stress Enhancement 3.5.1

Background

It is well known that the strength enhancement observed in materials/structures under dynamic loading conditions is attributed to one or both of the two factors which are not present in quasi-static loading: (i) strain-rate sensitivity of material properties and (ii) inertial forces. The inelastic behavior of cellular solids is governed mainly by the progressive deformation mechanisms which are strongly controlled by the cellular geometry of the structure. Studies on the dynamic response of cellular solids that are made of strain-rate insensitive materials show that, while the materials with periodic topology (e.g., aluminum honeycombs) are prone to dynamic strength enhancement in longitudinal direction [31–35], those with stochastic topology (e.g., aluminum foams) are insensitive to the rate of loading [2, 36], suggesting that there exists a close relationship between the cellular topology and inertial dynamic strength enhancement. It should be noted that due to the anisotropy of materials with periodic topology their response to dynamic loading might vary depending on the loading direction. In fact, an argument of this type can also be related to a series of studies on what is called type I and type II structures [37–42], which have succeeded in explaining certain anomalies regarding their energy absorbing capabilities under dynamic loading conditions. Those studies, in the simplest terms, discuss that the structures with an “unstably softening” quasi-static load-deflection curve (type II) are much more susceptible to inertia based dynamic strength enhancement than the structures with a “flat topped” load-deflection curve (type I). Within

740

M. Vural and G. Ravichandran

this terminology, the mechanical response of type I structures are characterized by that of circular tubes under transverse loading in which the bending mode of collapse defines the deformation for both quasi-static and dynamic loading conditions. However, type II structures are represented by longitudinal struts/plates that undergo bending deformation around localized hinges and, as explained theoretically by Su et al. [40, 41], under dynamic loading conditions lateral inertia forces induce an initial phase of axial compression in the struts which results in dynamic stress enhancement before the bending mechanism is recovered. Thus, within this framework, mechanical response of honeycomb materials in longitudinal direction is reminiscent of type II structures while the response of foams and honeycombs in transverse direction falls into the category of type I structures. This analogy is also supported by the experimental data in the references given above. However, there are two important points that need to be discussed. First, the deformation in cellular materials mostly occurs in localized bands and, therefore, actual strain rates experienced by the localized regions may be an order of magnitude higher than the nominal strain rate applied over the cellular material. This situation requires defining a realistic characteristic length scale to account for localization effects. Second, the term dynamic covers a wide range of impact velocities (or strain rates) and depending on its magnitude the role of inertia should be evaluated in two subcategories: (i) up to a critical impact velocity, lateral inertia (or micro-inertia) is effective in stress enhancement and its effect on dynamic strength enhancement is limited by the constitutive response of the material that make up the cellular solid, it is this inertia that is responsible for the dynamic strength enhancement observed in type II structures, (ii) for higher impact velocities, axial inertia that is associated with the propagation of shock waves through the cellular structure is effective in stress enhancement and it may dominate the dynamic strength enhancement well above the constitutive capacity of material, this type of inertia may result in dynamic strength enhancement in both type I and type II structures. Reid and Peng [9] developed a structural shock wave model and compared its strength predictions with experimental data for wood, which is impacted at velocities in the range from 30 to 300 m/s. The strength enhancement in transversely loaded specimens follows the predictions of their shock model, both increasing substantially with increasing impact velocity. However, the predictions of shock model underestimate the dynamic strength of balsa wood in longitudinal direction for the range of impact velocities they used. They also calculate a critical impact velocity for longitudinal loading above which the deformation will proceed in a shock-like manner and give its value as 360 m/s. Thus, their test velocities lie in a range where the shock wave effects are not expected. Almost all of their data show a dynamic strength enhancement factor in the range 1.5 to 2.3, the latter factor corresponding to the yield stress of wood cell material. They attribute this strength enhancement to micro-inertial effects at the cellular scale by noting the upper bound of 2.3. Even though the literature on the dynamic behavior of cellular materials often refers to the micro-inertia for qualitative explanations of the rate effects observed, an attempt to quantify its effects has not been undertaken yet, which will be the subject of following sections.

Mechanical Response and Deformation Mode Selection in Balsa Wood

741

The post-mortem SEM investigations of the present study on compression loaded balsa wood specimens reveal that the deformation mechanisms operating at the microstructural level are either buckling or kink band formation, depending mainly on the density of wood. Increasing the strain rate does not change these primary deformation mechanisms. However, compressive strength of balsa wood experiences a substantial increase when loaded at high strain-rates (see Fig. 7a). Since the deformation in cellular structures is associated with the large displacements of microstructural elements that make up the cellular solid, these elements experience rapid acceleration under dynamic loading. Therefore, it is not unreasonable to expect that the inertial forces predominate the dynamic response of balsa wood. With this reasoning, and also due to the lack of experimental data for the strain-rate sensitivity of wood cell material (not wood as a whole), the effect of micro-inertia is investigated to quantify its contribution to the dynamic strength enhancement. Based on the kinematics of deformation mode, simple models for buckling and kink band formation are discussed in the following sections.

3.5.2

Buckling

The present analysis is essentially based on the recognition of the subsequent deformation phases that are shown schematically in Fig. 14. The initial configuration of balsa wood is illustrated in Fig. 14a in which the vertical lines represent the longitudinal cell walls. The cell walls are assumed to have an initial imperfection in the form of a misalignment distribution though it is not demonstrated in the figure explicitly. The first phase (I) of dynamic loading is governed by the homogeneous elastic deformation of the cell walls until a critical stress level, which is the quasistatic strength of material, is reached. The second phase (II) represents the initiation of buckling instability and elasto-plastic hinge formation within localized regions. In this phase, lateral inertia comes into play and, contrary to quasi-static process, raises the stress level within cell walls by resisting the transverse motion and, thus,

Fig. 14 Subsequent phases of deformation: (a) initial configuration of cellular structure, (b) homogeneous elastic loading, (c) initiation of localized buckling deformation, (d) progressive deformation through localized regions

742

M. Vural and G. Ravichandran

Fig. 15 Idealization of half-buckle geometry to bars and lumped masses: (a) original configuration with initial cell wall imperfection and (b) deformed configuration showing the definitions of angles (™), transverse displacement (w), vertical shortening (y) and axial shortening (u)

inducing further compression along the cell walls. Therefore, it is this phase that contributes to the dynamic strength enhancement. The third phase (III), which corresponds mainly to the plateau region in a typical stress–strain curve (see Fig. 6), is characterized by further deformation within localized regions by bending at plastic hinges and the formation of new localization sites. As it is evident from the preceding discussion, the effect of transverse inertia on the dynamic strength enhancement is analyzed by modeling the initiation of buckling instability in phase II shown in Fig. 14. Figure 15 shows the idealized geometry of a half-buckle, (a) in its initial condition and (b) at a typical stage after the initiation of the buckling instability. For the sake of simplicity, the segments of cell walls that buckle are idealized as two axially deformable light rods with lumped masses at each end of the rods. It is further assumed that the two rods are freely hinged to each other at the central node. In Fig. 15a the initial misalignment angle is o and the overall length of the initial buckle is l, while in Fig. 15b the current angle is and the overall current length is l  u, where u is the shortening of the length of the cell wall due to elastic/plastic deformation. Between Fig. 15a and 15b, the vertical height of the half-buckle is correspondingly reduced by y, which is computed geometrically, as follows: y D l cos o  .l  u/ cos

(16)

By assuming that the angle remains small during phase II, the truncated Taylor series expansion gives, ı cos D 1  2 2 C    (17) and the transverse displacement w and angle can be related by, w D l=2

(18)

Mechanical Response and Deformation Mode Selection in Balsa Wood

743

Hence, after neglecting the higher-order terms, the vertical shortening of the half-buckle becomes,   y D .2=l/ w2  w2o C u (19) The rate of vertical shortening of the cell wall segment can be obtained by differentiating Eq. 19 with respect to time, yP D .4=l/ w wP C uP D V  C uP

(20)

V  D .4=l/ w w P

(21)

where, The first term on the right hand side of Eq. 20 gives the vertical shortening rate, V , which the half-buckle would have in terms of w and w, P if the cell walls were inextensional, while the second term is the additive contribution to the total shortening rate, y, P due to the lengthwise elastic/plastic compression of cell walls. At the beginning of phase II, the cell walls are already subject to a critical stress, qs , which is the quasi-static strength of the cellular material and a known quantity due to the availability of experimental data as well as analytical predictions described in Section 3.3. Therefore, in phase II, the stress in the cell walls is, ( D

qs C

Eu l

; u < u ; u  u

y

) (22)

ı where u D l.y  qs / E, y and E are the yield stress and Young’s modulus of the wood cell material respectively (y D 350 MPa from Ref. [27], E D 35 GPa from Ref. [23]). One must note that strain-rate sensitivity of cell wall material is ignored in Eq. 22, mainly due to lack of experimental data, and focus is placed solely on inertial effects. There is an axial force of A  on each of the inclined cell walls and these exert a lateral force on the lumped mass at the central node, which in turn provides the lateral acceleration. Hence, the equation of transverse motion for the half-buckle can be written as, wR D

2A w lm

(23)

where A D b tc (cross sectional area of the cell wall or in this case of the rod), m D .1=4/s l b tc , s is the density of solid wood cell substance which is generally taken to be approximately 1;500 kg=m3 [21, 22], b and tc are the width and the thickness of the cell wall respectively. Now, assuming that the vertical shortening occurs at a constant rate yP D Vo , y D Vo t

(24)

744

M. Vural and G. Ravichandran

When coupled with Eqs. 19, 22 and 24, Eq. 23 takes the form, ( wR D

r1 wt C r2 w C r3 w3

; u < u

r4 w

; u  u

) (25)

where  ı  r2 D 2w2o C qs l 2 E k r3 D 2k 8y 8E kD r4 D 4 s l s l 2

r1 D lVo k

(26)

The first part of Eq. 25 corresponds to elastic compression of cell walls. This is a second-order nonlinear ordinary differential equation and can be solved numerically to yield w and wP with the initial conditions, w D wo ; wP D 0

at

t D0

(27)

If the condition u D u is reached at a time t D t  during the phase II, the second part of Eq. 25 corresponds to plastic yielding of cell walls and can be solved analytically with the initial conditions, w D w ; wP D wP  to give, w D w cosh

at

t D t

p  p  r4 .t  t  / C wP  sinh r4 .t  t  /

(28)

(29)



Figure 16 shows the typical plots of y; P V and  as a function of time, t . The upward concave curve is for V  , and the horizontal straight line is for y. P The vertical intercept between the two curves is equal to uP , by virtue of Eq. 20. Furthermore, since uP  0 by hypothesis (as the cell walls are assumed to be deforming

Fig. 16 Typical plots for the evolution of vertical shortening rate y, P V  D .4= l/ww, P axial shortening rate uP , and the cell wall stress  D qs C Eu= l with time during the phase II. (a) When the axial shortening occurs only by elastic compression, and (b) when the yield stress is reached

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in compression), the solution ceases to be valid beyond the time t D t1 at which the two curves intersect. Thus, the intersection of the two curves marks the end of the phase II and the beginning of the phase III, i.e., the phase in which further deformation occurs only by the rotation of the elasto-plastic hinges and the formation of subsequent buckling (localization) sites. Figure 16 also shows the evolution of the stress experienced by the cell walls. As can be seen, under dynamic loading conditions the stress may be significantly raised over its quasi-static critical value due to the lateral resistance offered to the cell walls by transverse inertia. At the end of phase II, as the transverse displacement rate reaches a critical value, extensional shortening rate in the cell walls becomes zero so that further vertical shortening can be accommodated only by the rotation of hinges. This situation results in a peak in the stress–strain response of the material, which is followed by a more complex behavior. The present analysis assumes, for the sake of simplicity and to focus on the effect of inertia, that the hinges are free to rotate. However, actual situation involves the formation of an elasto-plastic hinge that resists the rotation. The complexity arises from the fact that, beyond the critical point, as the inertia does not support the development of further compression and hence the stress enhancement along the cell walls but, on the contrary, favors the partial unloading, the moment at the hinges takes over the role to support the cell wall stress. Depending on the competition between these two opposing effects the overall flow stress of the material may experience a sudden drop or mildly level off. The peak stress, transverse displacement and the cell wall thickness are among the important factors that affect the evolution of this competition. The evolution of stress response beyond the peak stress, i.e., after phase II, is not pursued in the present analysis.

Model Parameters In order to predict the dynamic strength of balsa wood, the model developed above needs several inputs regarding the material properties, initial cell wall imperfections and characteristic length of localization, which will be discussed next. A linear fit to the quasi-static strength data (in MPa) shown in Fig. 7a gives 1 D 0:131   5:23 qs

(30)

where  is the density of the cellular material, i.e., balsa wood, in kg=m3 . Note that Eq. 30 is valid for densities in the range of 55 to 380 kg=m3 . Thus, the corresponding stress experienced by the individual cell walls at the time of quasi-static collapse can be calculated as 1 qs D .s =/ qs (31) The SEM observations on deformed low-density balsa specimens suggest that the half-buckle length of cell walls (l) is roughly equal to the cell width (b) (see Fig. 10a). This is also further supported by the elastic buckling analysis of thin plates. Therefore, half-buckle length (l) that is shown in Fig. 15 is taken as 30 m

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and the misalignment angle o has been chosen as 1ı . Finally, the vertical shortening rate of half-buckles (Vo ) should be provided to the model as the boundary condition. Given the specimen length .Ls / and the strain rate ("P), the overall shortening rate of specimen .Vo / can be easily calculated by Vo D "P Ls

(32)

However, a characteristic length scale .lc / for the localization is still needed in order to have a reasonable vertical shortening rate for half-buckles. The SEM observations on buckled specimens suggest that the characteristic length of localization region is on the order of 200 m. This magnitude for the length scale is also supported by the fact that hexagonal balsa cells are generally divided into smaller prismatic boxes by one or two lateral support membranes along their length. Easterling et al. [6] and Gibson and Ashby [1] report the length of these smaller cellular elements as being in the range 175–285 m, which will be used as the lower and upper bounds for the characteristic localization length, lc . Thus, the vertical shortening rate for half-buckles can be determined as yP D Vo D "P Ls l= lc

(33)

Since the present experiments are performed at a nominal strain rate of "P D 3000 s 1 , and with a specimen length .Ls / of 5 mm, the range of 1.6–2.6 m/s will be used for the vertical shortening rate of half-buckles. Based on the model parameters introduced above for balsa wood, the dynamic strength predictions of the model are discussed and compared with the experimental results in the next section.

3.5.3

Kink Band Formation

As already has been discussed in previous sections, kink band formation is the major failure mode for high-density balsa wood. The basic approach adopted for the above analysis will be preserved also here, which states that the initiation of localized deformation, i.e., kink band formation, occurs after the homogeneous elastic deformation when the quasi-static strength of material is reached. Therefore, the analysis will focus on the phase II where the inertial forces are dominant. Based on experimental observations, Fig. 17 schematically shows the various combinations of kink band formation within cellular composites, e.g., balsa wood. Basically they can be divided into two subcategories: single kink band formation and conjugate kink band formation. Figure 18 shows the idealized geometry of these kink bands that will be employed for the analysis of inertial effects. Equations (16–22, 24) pertaining the kinematics are also valid for kink band analysis if the definitions of Fig. 18 are adopted regarding the geometry and mass for the kink band. The equations of motion in the transverse direction are given by, w R D

2A w l .m C M 0 /

(34a)

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Fig. 17 Various combinations of kink band formation: (a) single kink band, (b) conjugate kink bands, (c) and (d) conjugate kink band formation with added mass

Fig. 18 Details of idealized kink band geometries: (a) single kink band and (b) conjugate kink bands with added mass

for a single kink band, and by, w R D

A w l .2m C m0 /

(34b)

for conjugate kink band geometries where M 0 D .1=2/.Ls  l/b tc s stands for half the mass of specimen outside the single kink band, and m0 D .1=2/l 0 b tc s for the added mass that is accelerated between the conjugate kinks. Using Eqs. (19), (22) and (24), the equations of motion for the single kink and conjugate kink band configurations take exactly the same form as Eqs. 25 and 29 with the coefficients ri (i D 1; 4) replaced by ri0 and ri00 , respectively, which is given by,   k0 l 0 for single kink band (35a) .iD1; 4/ and k D k D ri k 2Ls  l   k 00 l ri00 D ri for conjugate kink band (35b) .iD1; 4/ and k 00 Dk k 4.lCl 0 / ri0

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where ri and k are given by Eq. 26. Thus, the differential equations governing the transverse motion of system are of the same character for both buckling and single/conjugate kink band formation. Therefore, the previous discussion on Fig. 16 regarding the evolution of velocity components and the cell wall stress during phase II is also valid for kink band formation mode.

Model Parameters The critical level of nominal far field stress at the beginning of kink band instability is given by Eq. 30. Therefore, it will be used in conjunction with Eq. 31 to obtain the critical cell wall stress .qs /. Experimental evidence shows that the width of kink band .l/ may vary from 100 m to as much as cell length, which is approximately 650 m. As will be discussed in detail in the next section, the dynamic strength prediction of the model is independent of the kink band width .l/ when the added masses (m0 and M 0 ) are ignored or when they are a certain fraction of kink band mass .4m/. However, when the added mass is kept constant, the smaller the kink band width, the higher is the dynamic stress enhancement. The initial misalignment angle for the kink band formation is much larger than that is assumed for buckling because in kink band mode it represents the misalignment of longitudinal cells as a whole while in buckling mode initial misalignment is a measure of waviness within the cell walls. Based on the SEM observations (see Fig. 3), average misalignment angle is taken to be o D 9ı . The characteristic length of localization is the same as kink band width for single kink mode .lc D l/ and twice the width of kink band for conjugate kink mode .lc D 2l/. Therefore, in conjunction with Eqs. 32 and 33, the vertical shortening rate .y/ P for single kink and conjugate kink band configurations shown in Fig. 18 is computed to be 15 and 7.5 m/s, respectively.

3.6 Comparison of Inertia-Based Models with Dynamic Data Before making the direct comparison between the model predictions and the experimental results, it is helpful to discuss the essential features pertaining to the buckling and the kink band models. The basic parameter common to both the models is the term qs , which represents the state of stress within the individual cell walls at the initiation phase of buckling or kink band instability. Independent of the failure mode, this critical stress level is obtained using the Eqs. 30 and (31). However, it should be noted that it is not a constant but varies as a function of the density of entire cellular structure, i.e., balsa wood. In a naturally occurring material such as balsa wood it is inevitable to have geometric material imperfections at different length scales within the cellular structure. Thus, the second parameter, initial misalignment angle o , represents the measure

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of this geometric imperfection. For buckling model, it defines the extent of waviness along the cell walls and is assumed to be 1ı for present predictions. For kink band model, misalignment angle stands for the geometric imperfection at a larger length scale that is comparable with the cell length. Experimental evidence shows that there exist fiber (cell) misalignments in balsa wood and they vary in the range 7–11ı , particularly on LT plane due to the presence of ray cells. Therefore, an average value of 9ı is adopted for the kink band model, which is also consistent with the value used in quasi-static kink band model for balsa wood (Vural and Ravichandran [3, 10]). In buckling model, the dynamic strength prediction depends on the buckling wavelength only through the characteristic length of localization .lc /. These two quantities together determine, through Eq. 33, the vertical shortening rate of halfbuckles, which in turn forms the boundary condition for half-buckle configuration. If this velocity boundary condition is kept constant, even though the coefficients in the equation of motion, Eq. 25, are functions of the half-buckle length, the strength prediction of the model becomes independent of the buckling wavelength. Since the half-buckle length is chosen to be 30 m, which is both supported by SEM observations and consistent with the theoretical elastic buckling analysis of thin plates, lower and upper bounds for the characteristic length of localization .175–285 m/ set the upper and lower limits, respectively, for the dynamic strength prediction in buckling mode. As shown in Fig. 19, experimental data are in quite good agreement with these limits. In the kink band model, it is recognized that the part of material outside the localization band may also be accelerated and the lateral inertia of this added mass has a contribution to the dynamic stress enhancement. In single kink band configuration this is accounted by the term M 0 , which is half the mass of specimen outside the kink band. This mechanism involves the acceleration of large specimen masses below and above the kink band in opposite directions and, therefore, the stress exerted to cell walls due to lateral inertia reaches the yield stress of the wood substance well within the phase II (Fig. 16b), resulting in a dynamic strength enhancement restricted by the plastic collapse of cell walls. This situation occurs irrespective of the choice of kink band width .100 m < l < 650 m/. The acceleration of the entire specimen mass may be thought to be irrelevant in actual single kink band formation process and instead a deformation geometry that accounts for the bending of cell walls (see Fig. 20) may be considered more appropriate. In this case, the total effective mass accelerated at the tips of kink band will be restricted to a certain factor of kink band mass, which can be as small as the kink band mass itself under the most conservative assumption. Even in this situation that corresponds to single buckle formation, the lateral inertia is found strong enough to induce the yield stress along the cell walls. Therefore, for the current material and model parameters, single kink model produces an upper bound for the compressive strength prediction, which is given by the plastic collapse stress of entire cellular structure. This upper bound is plotted in Fig. 19. On the other hand, the conjugate kink band model represents a unique deformation mode in which the material finds a way out to avoid the large accelerations. When the added mass accelerated between the conjugate kinks .m0 / l 0 / is assumed

750

M. Vural and G. Ravichandran 80 Dynamic Quasi-static Buckling (upper bound) Buckling (lower bound) Kink band (upper bound) Kink band (lower bound) Best linear fit

70

Strength (MPa)

60 50 40 30 20 10 0 50 4

Dynamic/Quasi-Static Stress Ratio

3.5 3

80

110

140

170

200

230

260

290

320

350

380

Experimental data Buckling (upper bound) Buckling (lower bound) Kink band (upper bound) Kink band (lower bound)

2.5 2 1.5 1 0.5 0 50

100

150

200

250

300

350

Density (kg/m3)

Fig. 19 Variation of dynamic and quasi-static compressive strength with density for balsa wood: experimental data versus inertia-based model predictions

not to exist, the model gives the lower bound for the dynamic strength of balsa wood as is shown in Fig. 19. If the length of accelerated region is chosen to be 70% of the kink band width, i.e, if l 0 D 0:7 l, the lateral inertia becomes strong enough to exert the yield stress along the cell walls so that the upper bound of plastic collapse is attained. Thus, depending on the added mass, the predictions of the conjugate kink model covers the entire range between lower and upper bounds plotted in Fig. 19. Even though the buckling and kink band models developed in the preceding section are physically based on experimental observations, significant

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Fig. 20 Schematic geometry of deformation that accounts for cell wall bending

simplifications such as the lumped masses and the neglect of cell wall bending and plastic hinge formation are introduced in order to keep the evolution of inertial forces tractable. However, it should be noted that the present analysis attempts to cover only the initiation phase of buckling and kink band instability, where the inertial forces are the most effective and there is very limited hinge rotation and cell wall bending, rather than the entire progressive deformation process which involves substantial hinge rotation and subsequent localizations in the neighboring sites. As shown in Fig. 19, the upper and lower bound predictions of these simplified models are in quite good agreement with the experimental dynamic strength data. This agreement suggests that the dynamic strength enhancement in balsa wood is dominated by the transverse inertia of the elements that makes up the cellular structure. Even though the present study on balsa specimens shows that the significant amount of dynamic strength enhancement can be well explained by cellular micro-inertia, the question on the possible role of viscous effects still needs to be addressed. One way to do this is to conduct high-strain-rate experiments on isolated cell wall material, which is extremely challenging. Another way may be to test the strain-rate sensitivity of specimens that will be machined out of balsa specimens compressed up to their densification strain or even further. However, the latter method will bring about new questions arising from the anisotropy of cell wall material because the direction of loading for densified balsa specimens will not be along the cell walls. The investigation of viscous effects and its contribution to the dynamic strength enhancement in balsa wood, and more generally in wood cell wall, deserves further study. On the other hand, for the range of strain rates investigated (103 to 103 s1 ), the present experimental data (Fig. 7) shows that the plateau stress is relatively insensitive to the variation in strain rate while the compressive strength (peak stress) exhibits significant sensitivity. Since the progressive deformation in the plateau region is mainly characterized by the intensive localized plastic deformation at around the hinges and the transverse inertia is negligible at this phase of deformation, experimental data indirectly suggest that the contribution of viscous effects to the dynamic strength enhancement should be insignificant as compared to that of inertial effects. It should be noted that the deformation mechanisms and the interactions in the plateau region is quite complex and, therefore, this indirect suggestion needs to be further investigated.

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Another issue is that the balsa wood is composed of different kinds of cells and, therefore, the applicability of the present models should be addressed under the concerns about the homogeneity of cell types. Apart from the homogeneously distributed diffuse sap channels, balsa wood has two structurally important cell types: (i) long prismatic hexagonal cells (grains) and (ii) shorter brick-shaped ray cells (see Fig. 1). The groups of ray cells are continuous along the radial direction but they are discontinuous and have a finite depth in the longitudinal direction. The models developed to quantify micro-inertia induced strength enhancement for buckling and kink band formation modes are essentially based on the kinematics of deformation during localized failure initiation. Therefore, the issue of homogeneity in cell types should be considered in terms of its effect on the kinematics of deformation at micro-scale. The SEM studies on recovered specimens reveal that the buckling wavelength in the walls of ray cells tends to be the same as that in grains. In the case of kink band formation, the length scale of deformation is much larger and the groups of ray cells displace exactly in the same manner as the grains. Thus, independent of the mass ratio of ray cells to grains in a balsa specimen, the contribution of ray cells to micro-inertial hardening should be expected to be the same as grains. In other words, as long as the kinematics of deformation is not altered due to the inhomogeneous distribution of cell types, which is the experimentally observed situation for balsa wood, the models should be considered applicable for wood structure.

4 Conclusions The mechanical behavior of a naturally occurring cellular material, balsa wood, has been investigated experimentally over its entire density spectrum from 55 to 380 kg=m3 with particular emphasis on the microstructural characterization and the analytical modeling of compressive failure mechanisms. Results from unconfined and confined experiments show that neither compressive strength nor plateau stress is significantly affected by confinement. Independence of strength from confinement is attributed to the highly porous cellular microstructure of balsa. Compressive strength and plateau stress of balsa wood increases linearly as a function of its density while the densification strain decreases. The post-mortem SEM examinations reveal that failure mode transition from elastic/plastic buckling to kink band formation occurs as the density increases. Based on the experimental results and observations, several analytical models were described and their relevance to represent the failure behavior of a natural material like balsa wood was discussed. Results show that these analytical failure models describing the elastic/plastic buckling, endcap collapse and kink band formation mechanisms have a strong potential to predict compressive strength of balsa wood in a range of densities that differ by an order of magnitude. The dynamic results regarding the compressive strength, plateau stress, densification strain and energy dissipation capacity have been discussed and documented, and compared with the quasi-static results. It has been demonstrated that initial failure

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strength of balsa wood is very sensitive to the rate of loading (50–130% increase over corresponding quasi-static values) while plateau stress remains unaffected by the strain rate. This difference in response to increasing strain rate is explained in terms of the differences in the kinematics of deformation, associated micro-inertia and the level of stress perturbations during initial failure and progressive deformation. The packing efficiency of balsa wood, as manifested by the densification strain, has been found to degrade with the rate of deformation. As in quasi-static loading, buckling and kink band formation were identified to be two major failure modes in dynamic loading as well. Kinematics of deformation of the observed failure modes and associated micro-inertial effects has been modeled and discussed to explain measured dynamic strength enhancement. The predictions of model show that the micro-inertia of cellular elements may result in strength enhancement up to the constitutive bounds of cell wall material. Specific energy dissipation capacity of balsa wood was analyzed and determined to be comparable with those of fiber-reinforced polymer composites. As opposed to the bulk materials, the localized deformation in cellular solids is associated with large local displacements mainly due to the discrete nature of cellular topology, which leads to large accelerations in cellular elements under dynamic loading conditions. The modeling effort introduced in the present study seeks to understand and quantify the role of these cellular accelerations (micro-inertia) on experimentally observed dynamic strength enhancement. As has been elaborated in the preceding sections the governing physics for micro-inertial strength enhancement leads to further compression of cell walls due to the non-zero component of transverse inertial forces along the cell walls. Therefore, the dynamic strength enhancement models introduced in the present study focus on the quantification of these resolved inertial forces irrespective of the complexity of kinematics associated with failure. It should be underlined that the kinematics of failure is 3D in buckling mode, while it is 2D in kink band formation mode mainly because high anisotropy of balsa wood in its transverse plane forces the failure to occur in LT plane. However, the 3D or 2D character of failure kinematics loses its importance in this particular problem since the dynamic strength enhancement is essentially driven by transverse inertia. Thus, the distinction between transverse directions is not critical and a realistic representation of 2D failure kinematics as applied here is sufficient as long as one aims at capturing the governing physics with reasonable simplicity. From this perspective, the present models are based on 2D representation of experimentally observed failure kinematics and keep track of resolved inertial forces along the cell walls. With these features the models attempts to provide a tool to understand and quantify the effect of cellular micro-inertia at intermediate strain rates that correspond to the regime between the quasi-static loading and shock loading. Acknowledgement The support of the Office of Naval Research (Dr. Y. D. S. Rajapakse, program Manager) is gratefully acknowledged. GR acknowledges the support of the DoD MURI at the California Institute of Technology on Mechanics and Mechanisms of Impulse Loading, Damage and Failure of Marine Structures and Materials through the Office of Naval Research.

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References 1. Gibson LJ, Ashby MF (1997) Cellular Solids: Structure and Properties. Cambridge University Press, Cambridge 2. Ashby MF, Evans A, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG (2000) Metal Foams: A Design Guide. Butterworth-Heinemann, Oxford 3. Vural M, Ravichandran G (2003) Microstructural aspects and modeling of failure in naturally occurring porous composites. Mech Mater 35: 523–536 4. Knoell AC (1966) Environmental and physical effects of the response of balsa wood as an energy dissipator. JPL Technical Report No. 32–944, California Institute of Technology, Pasadena, CA 5. Soden PD, McLeish RD (1976) Variables affecting the strength of balsa wood. J Strain Anal 11(4): 225–234 6. Easterling KE, Harrysson R, Gibson LJ, Ashby MF (1982) On the mechanics of balsa and other woods. Proc Roy Soc London A383: 31–41 7. Silva AD, Kyriakides S (2007) Compressive response and failure of balsa wood. Int J Solids Struct 44: 8685–8717 8. Daigle DL, Lonborg JO (1961) Evaluation of certain crushable materials. JPL Technical Report No. 32–120, California Institute of Technology, Pasadena, CA 9. Reid SR, Peng C (1997) Dynamic uniaxial crushing of wood. Int J Impact Eng 19(5–6): 531–570 10. Vural M, Ravichandran G (2003) Dynamic response and energy dissipation characteristics of balsa wood: experiment and analysis. Int J Solids Struct 40(9): 2147–2170 11. Vural M, Ravichandran G (2004) Failure mode transition and energy dissipation in naturally occurring composites. Composites Part B – Eng 35(6–8): 639–646 12. Hahn JJ (1981) Wood-based composites. In: Wangaard FF (Ed.) Wood: Its Structure and Properties. Pennsylvania State University, University Park, PA, pp. 401–427 13. Dinwoodie JM (1975) Timber – a review of the structure-mechanical property relationship. J Microscopy 4(1): 3–32 14. Oguni K, Tan CY, Ravichandran G (2000) Failure mode transition in unidirectional E-glass/ vinylester composites under multiaxial compression. J Compos Mater 34(24): 2081–2097 15. Kolsky H (1949) An investigation of mechanical properties of materials at very high rates of loading. Proc Roy Soc London B62: 676–700 16. Chen W, Lu F, Zhou B (2000) A quartz-crystal-embedded split Hopkinson pressure bar for soft materials. Exp Mech 40(1): 1–6 17. Ravichandran G, Subhash G (1994) Critical appraisal of limiting strain rates for compression testing of ceramics in split Hopkinson pressure bar. J Am Ceram Soc 77(1): 263–267 18. Maiti SK, Gibson LJ, Ashby MF (1984) Deformation and energy absorption diagrams for cellular solids. Acta Metal 32(11): 1963–1975 19. Mamalis AG, Robinson M, Manolakos DE, Demosthenous GA, Ioannidis MB, Carruthers J (1997) Review: Crashworthy capability of composite material structures. Compos Struct 37: 109–134 20. Timoshenko S (1936) Theory of Elastic Stability. McGraw-Hill, New York, pp. 324–418 21. Wangaard FF, (1950) The Mechanical Properties of Wood. Wiley, New York 22. Mark RE (1967) Cell Wall Mechanics of Tracheids. Yale University Press, New Haven 23. Cave ID (1968) The anisotropic elasticity of the plant cell wall. Wood Sci Tech 2(4): 268–278 24. Zhang J, Ashby MF (1992) The out-of-plane properties of honeycombs. Int J Mech Sci 34(6): 475–489 25. McFarland RK (1963). Hexagonal cell structures under post-buckling axial load. AIAA J 1(6): 1380–1385 26. Wierzbicki T (1983) Crushing analysis of metal honeycombs. Int J Impact Eng 1(2): 157–174 27. Cave ID (1969) The longitudinal Young’s modulus of Pinus radiata. Wood Sci Tech 3(1): 40–48 28. Rosen VW (1965) Mechanics of composite strengthening. In: Fiber Composite Materials, American Society of Metals, Metals Park, Ohio, pp. 37–75

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Mechanics of PAN Nanofibers Mohammad Naraghi and Ioannis Chasiotis

Abstract Novel and scalable fabrication methods, such as electrospinning and pulsed laser deposition, provide low-cost polymeric nanofibers for structural applications. However, there is insufficient knowledge about the structural and mechanical behavior of polymeric nanostructures, which, in turn, limits their potential as enabling materials. The knowledge void in the systematic mechanical characterization of polymeric nanofibers is addressed in this chapter with focus on the room-temperature temporal and scale-dependent mechanical response of polyacrylonitrile (PAN) nanofibers. Their molecular structure and deformation processes are intimately related to their fabrication conditions, and this chapter describes the first effort in literature to establish fabrication–structure-properties interrelations. The nanoscale experiments presented here demonstrate strong diameter dependence of the elastic modulus and tensile strength of PAN nanofibers for a variety of electrospinning conditions, while for particular fabrication conditions, the applied strain rate is shown to result in non-monotonic mechanical behaviors and very unusual deformation profiles during cold drawing.

1 Introduction Polymeric nanofibers have large aspect ratios and consequently large free surface per unit mass. All other parameters considered equal, polymer molecules near the free surface are typically associated with lower molecular entanglement and higher molecular mobility compared to those at the core of the fiber. They may also be partially aligned with the free surface as inferred experimentally [1–4] and in simulations [5–8]. Therefore, the mechanical response of polymeric nanofibers could be influenced by surface molecules and hence be quite different from the bulk.

M. Naraghi and I. Chasiotis () Aerospace Engineering, University of Illinois at Urbana Champaign, Urbana Illinois 61801, USA e-mail: [email protected]

I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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In addition to the role of surface molecules, nanoscale heterogeneities with dimensions smaller than the nanofiber diameter, which are formed during fabrication, could modify and bias not only the fiber strength but also the evolution of its mechanical deformation [9, 10]. Nanocrystalinity, partial molecular alignment, and core-skin structure [11–14] are among the structural features that can affect the deformation mechanics of nanofibers. A commonly used amorphous polymer to fabricate nanofibers is polyacrylonitrile (PAN) [15–18]. The mechanical behavior of PAN as a function of processing conditions has been investigated in connection with macroscale PAN fibers and films [19–23]. A detailed report on macroscale PAN fibers by Rosenbaum [19] discusses their mechanical response to axial loading and their strain recovery. In the same work, the mechanical behavior of PAN fibers was divided into three regions: (a) elastic response, in which the free volume of the material increases until yield at 2% strain, (b) post-yield hardening caused by short range straightening of polymeric chains, (c) rubber-like extension at large strains with insignificant or no hardening at all. Yield was observed only below Tg 90–100ı C, while the last region was characteristic for fiber stretching above Tg . It was also reported that, provided the proper kinetic conditions are present, PAN macrofibers can recover to almost zero strain after being stretched to engineering strains as high as 100%. Tsai [20] measured the effect of drawing on the strength and the elastic modulus of macroscale PAN fibers at relatively small drawing ratios, less than 20. The author’s X-ray diffraction studies on drawn PAN fibers revealed an increase in the molecular orientation and packing with increasing drawing ratios, which led to stiffer and stronger fibers. As the drawing ratio increased from 6 to 12, the elastic modulus and the fiber strength increased between 10–15 GPa and 480–700 MPa, respectively. Sawai et al [21] further extended the fiber drawing ratio and established experimental limits to improve on the mechanical behavior of drawn fibers. Their fibers were extruded from PAN films, which, in turn, were formed by gelation of PAN in dimethylformamide (DMF). As the drawing ratio was increased, the PAN density increased too, reaching a plateau at drawing ratios of about 150, suggesting that this was the most compact arrangement of PAN chains. Wide-Angle X-Ray Diffraction (WAXD) patterns confirmed the orientation of the chains along the fiber axis and the formation of crystals at high drawing ratios. The elastic modulus and the tensile strength followed a trend similar to that of the fiber density, increasing as a function of drawing ratio between 1–30 GPa and 0.2–1.5 GPa, respectively. At extreme drawing ratios, the tensile strength was reduced, possibly due to the introduction of defects. Yamane et al [22] investigated the conditions for stable fiber drawing to high ratios, i.e. 100 and above, in order to obtain macroscale fibers with high strength and stiffness from PAN films cast similarly to the previous study. They concluded that high drawing ratios were not achievable in a single stage, but at least one initial drawing to a relatively low ratio (<16) was necessary to change the morphology of the fibers from a randomly oriented low entanglement mix of chains to better oriented chains, which could then sustain further stretching. It was demonstrated that higher, first stage, drawing ratios and proper thermal treatment prevented the localization of strain during the second phase of drawing. In addition, the maximum

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achievable total drawing ratio increased with temperature, with an accelerated trend above 150ı C, and it reached an optimum at 190ı C. The rapid increase of fiber ductility above 150ı C was attributed to a reversible change in the crystalline structure of PAN fibers from orthorhombic to hexagonal, which enhanced the molecular mobility. On the other hand, Bashir et al [23] demonstrated the existence of hexagonally packed (with high disorder along the c-axis) structure of 70 m thick PAN films at lower temperatures, 115ı C, which were stretched to a drawing ratio of 20. Fourier Transform Infrared spectroscopy (FTIR) and X-diffraction verified the molecular alignment and packing of the particular films. While PAN films and macroscale fibers have been studied extensively over the past decades, PAN nanofibers have only recently attracted attention both in terms of academic research and industrial applications. This is owed to the development of methods for the mass production of aligned electrospun nanofibers. PAN nanofibers are now used for large scale filters and as potential precursors to obtain pyrolyzed carbon nanofibers for applications in nanocomposites [24]. Furthermore, as-fabricated PAN nanofibers may be employed in muscle tissue engineering as fast actuating artificial muscles in response to PH changes in their environment [16]. A roadblock in the aforementioned developments is the lack of experimental studies even on the cold drawing response of PAN nanofibers. This void in quantitative information is attributed to several reasons, such as:  Their small dimensions that do not allow for conventional sample manipulation

methods, which are required for single nanofiber mechanical testing  The high resolution in force and fiber extension measurements that are inherently

required at the scale of the nanofibers  The fiber vulnerability to electron beam radiation inside scanning electron mi-

croscopes (SEM), which are often used to test in situ nanostructures so that high resolution measurements are achieved [25] In addition to the aforementioned challenges, one should also consider the fragility and ductility of polymeric nanofibers, and their strain rate dependent mechanical behavior, which place additional requirements for their mechanical characterization. These challenges and requirements underscore the necessity for a universal method to conduct mechanics experiments with single polymeric nanofibers at different strain rates, deformation regimes, and temperatures.

2 Experimental Methods and Materials 2.1 Nanofiber Fabrication Polymeric nanofibers are fabricated by different processes including drawing, template synthesis, and electrospinning. Drawing, which is similar to dry spinning of macrofibers, produces relatively long individual nanofibers in a batch process [26] which is relatively slow. In template synthesis, nanofibers are fabricated in

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nanoporous media resulting in nanofibers that are tens of microns in length [27]. A better alternative method is electrospinning [28], which allows for mass production of long PAN nanofibers in a continuous form. Patented in 1932 [29], this method is cost effective providing nanofibers that can be meters long. The method of electrospinning was employed in this study to fabricate nanofibers from polymeric solutions which were ejected from a capillary toward a metal target with the aid of a high voltage between the capillary and the target. As the polymer jet traveled towards the target, it underwent three consecutive instabilities and it was subjected to fast elongation flows (10–20 m/s). The resulting nanofibers ranged in diameters between a few tens to a few hundreds of nanometers after their complete solidification. The high elongation strains during fabrication resulted in nanofibers with mechanical behaviors which varied with the fabrication parameters. Furthermore, the electrospinning parameters, such as the acceleration voltage and the distance from the collector/target, affected the fiber elastic modulus, ductility and tensile strength, significantly. For brevity, we will limit our discussion on the fabrication procedures and details to allow for a thorough presentation of the mechanical property investigations conducted by this group.

2.2 Mechanical Experiments with Single Polymeric Nanofibers 2.2.1

Background

Several methods have been developed to quantify the mechanical behavior of polymeric nanofibers, such as nanoindentation, bending, and microscale tension tests. Nanoindentation has been used to measure the elastic modulus of polymeric nanofibers [30,31] because it is convenient to perform, and it does not require special sample preparation. It is however, not suitable for nanostructures as the theoretical background developed to analyze nanoindentation data does not account for finite size effects, let alone the uncertainties in the geometry of the contact and the associated adhesion force between the sample and the indenter [32]. Moreover, bending tests conducted with the aid of a nanoindenter or AFM probes, are subject to boundary condition uncertainties, e.g. built-in, sliding or deformable support, which limit the degree of confidence in the results. Uniaxial tension experiments, on the other hand, are suitable to evaluate the properties of nanofibers as the latter are loaded in the mode they naturally carry forces in their service environment. In addition, compared to the aforementioned methods, tension tests involve the least number of assumptions necessary to extract material properties. Different approaches have been developed to perform microscale and nanoscale tension experiments with nanofibers. A common approach is the use of AFM cantilevers as force sensors [12,33]: The nanofiber is fixed at one end to a substrate and at its other end is attached onto an AFM cantilever with known bending stiffness. Unfortunately, this approach results in off-axis loading of the fiber (and the cantilever) when relatively large cantilever deflections occur (i.e. microns). In an

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effort to simplify the experiments with long nanofibers, commercial apparatuses have been employed too [32, 34, 35]. While convenient, the force range of commercial equipment is prohibitive for nanofibers with diameters of a few hundreds or a few tens of nanometers [32]. An approach that has gained ground in the recent years is the use of Microelectromechanical System (MEMS) force sensors and actuators [36–38]. MEMS mechanical testing platforms are comparable in size to the test samples and their planarity facilitates sample alignment and unidirectional sample loading. Such micro-apparatuses incorporate leaf-spring type loadcells, which, unlike AFM cantilevers, allow for a range of forces without off-axis loading. On the other hand, the small range of force output and/or translations generated by typical MEMS actuators impedes their application to mechanical testing of ductile nanofibers [39, 40]. To overcome this limitation, Naraghi, Chasiotis et al combined their MEMS testing platform with an external picomotor actuator which allows for unlimited nanofiber extension and applied force [41]. The authors achieved very good resolution in fiber force and extension by applying Digital Image Correlation (DIC) [42–44] on optical images of the MEMS device during fiber testing in order to resolve the fiber extension and the load cell spring opening. This approach allowed for 25 nm resolution in rigid body displacements, which, for all practical purposes, eliminates the need for SEM imaging. Moreover, in this method, experiments could be conducted without exposing the soft nanofibers to the degrading e-beam radiation of an SEM [25]. Optical imaging allowed for experiments at a wide range of strain (loading) rates, which could not be accomplished inside an SEM. The application of this experimental method and the results are presented in the following sections.

2.2.2

Nanoscale Tension Experiments with Individual Polymeric Nanofibers

The experimental apparatus developed by Naraghi, Chasiotis et al [41] employed a surface micromachined platform for nanofiber gripping, and the measurement of force by an integrated leaf-spring loadcell. One end of the nanofiber was attached to the leaf-spring supported grip, while the other end was held fixed on a stationary grip during the process of fiber drawing. For this purpose, a tipless AFM cantilever attached to a three-axis stage was mounted on a grip (Fig. 1) by using an epoxy adhesive. This cantilever was used to transmit the force from the external piezoelectric (PZT) actuator to the nanofiber. Each chip, hosted approximately 100 MEMS platforms similar to that in the schematic in Fig. 1 [41]. The first class of PAN nanofibers tested in this study were fabricated from solution of PAN in DMF, with average molecular weight of 150,000. Their diameters were in the range of 300–600 nm, while two fiber lengths of 25 and 50 m were used in the experiments. The nanofibers were isolated from a deposition grid and were cut in sections of 200–300 m long. The test specimens were substantially longer than the 25 or 50 m grip spacing to allow for adhesive gripping and for a free fiber segment, not involved in the experiment, which served as reference for post-mortem SEM imaging in order to determine the undeformed fiber diameter. This approach eliminated the need for exposure of the polymeric nanofibers to an SEM before the

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Fig. 1 MEMS test platform for nanofiber mechanical property experiments. The fiber length in the schematic is 50 m, which provides a relative length scale

mechanical testing. Benchmark tests showed that the exposure of PAN nanofibers to an SEM caused significant reduction in their ductility by as much as 90%.

2.2.3

Resolution in Force and Nanofiber Extension Measurements

Using the MEMS platform described in the previous section, microscale tension experiments were performed at optical magnification of 500 by recording digital images with 1280  1024 pixel resolution. The nanofiber diameter was prohibitively small (200–600 nm) in order to measure the nanofiber elongation directly with optics. Thus, optical images of the MEMS test platform were used to measure the cross-head motion by resolving the displacements of the fiber grips and the deflection of the center of the doubly supported column loadcell by DIC [43]. Each image of the test platform contained both force and displacement records and, therefore, the fiber force and stretch ratio could be acquired independently but also synchronously. During an experiment, the sample grips were subjected to rigid body motions that were resolved via DIC with resolution significantly better that the pixel size. The accuracy of rigid body motion calculations by DIC is 1=8th of a datum pixel, or better [43]. In our experiments, the rigid body resolution was shown to be better than 25 nm, as shown in Fig. 2. Unlikely previously published experimental methods, strain rate experiments can be conducted now with this method and the rate of loading is only limited by the frame rate of the CCD camera. The accuracy of the stress–strain curves obtained by this MEMS platform depends on the accuracy in sample force and extension. As far as the displacement measurement is concerned, DIC has been shown to resolve displacements on the order of a few tens of nanometers by using optical imaging [45, 46]. In order to demonstrate the resolution limits of this method, the PZT transducer was actuated at constant speed with a nominal step size of 23 nm. The rigid body motion of the attached MEMS chip was recorded optically and was calculated by DIC. When plotted as a function of the

Fig. 2 Rigid body displacement imposed by a PZT stepper motor (x-axis) compared to displacement measurements by means of DIC (y-axis). Each step of the PZT stepper motor (actuator) is 23 nm

763 Experimental Displacement (nm)

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Fig. 3 Contours of rigid body displacements superposed into the MEMS test platform

PZT step number, the true displacement of the PZT actuator agreed very well with the actuator position as calculated by DIC, starting from the very first datum point (23 nm) in Fig. 2. A simple uncertainty analysis shows that the accuracy in measuring p the relative displacement between two points on the test platform is at most 23 2 33 nm. Since this uncertainty is not cumulative, the measurement of large rigid body motions minimizes its relative contribution to the calculated property values. Thus, longer fibers improve the accuracy of the determined stress–strain curves. In order to extract the force–elongation curves of PAN nanofibers, images of the MEMS testing platform were recorded during each experiment and were compared with images of the unloaded configuration. DIC calculations from optical images require a speckle pattern. The polysilicon devices used here had rough surfaces that scattered light, which generated very good quality, natural, speckle patterns. During an experiment, the rigid body motions of three areas on the MEMS platform were monitored: U1 (region 1 in Fig. 3) where the AFM cantilever was attached, U2 at the freestanding load cell (region 2 in Fig. 3), and U3 at the fixed grip (region 3 in Fig. 3), which served as a reference. The applied force on the fiber was calculated

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from the deflection of the loadcell, which was u1 D U2  U1 , and the loadcell stiffness (see next section on load cell calibration). The elongation of the nanofiber was uf D U3  U2 . A snapshot of the computed displacement contours is shown in Fig. 3. The undeformed fiber diameter and length were measured by an SEM. These quantities were used to extract the engineering stress ¢ D kl ul= A0 and strain © D uf = l0 , where kl was the loadcell stiffness, l0 , was the initial length of the fiber, and A0 was the initial cross-sectional area of the fiber.

2.2.4

Loadcell Calibration

The calibration of the MEMS loadcell was conducted by two approaches. In the first, the stiffness, kc , of an AFM cantilever was determined a priori by analyzing the frequency spectrum of its thermal fluctuations [47], as shown in Fig. 4a. The nominal stiffness of the AFM cantilever was selected to be close to the estimated stiffness of the loadcell in order to minimize the uncertainties in the loadcell calibration [48]. During calibration, the AFM cantilever was pressed against the MEMS loadcell and optical images of the loadcell were used to determine the deflection of the loadcell .ul / and the AFM cantilever .uc / by means of DIC. For example, the three calibration curves in Fig. 4b provided an average stiffness for the particular loadcell a

b

Fig. 4 (a) Frequency spectrum of an AFM cantilever. (b) Three calibration curves of a MEMS loadcell using a pre-calibrated AFM cantilever

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Fig. 5 (a) MEMS folded beam loadcell calibrated with glass beads of known weight, (b) the force-deflection curve of the folded beam loadcell in (a)

of 1:33 ˙ 0:03 N=m, which was close to 1.38 N/m calculated by finite element analysis. The difference between the two values is usually attributed to the uncertainty in the device dimensions and the rigidity of the supports (boundary conditions) of the loadcell. The uncertainty in loadcell calibration using its nominal dimensions can be significantly larger, and, thus, an experimental measurement of the kl is recommended [47]. The advantage of this method is its simplicity and its disadvantage is its dependence on the stiffness of the AFM cantilever. In order to overcome this limitation, an alternative method was also employed to further improve on the accuracy of our force measurements. An optical image of the implementation of the second method is shown in Fig. 5a. In this calibration method, the loadcells were detached from the silicon die and were mounted vertically as shown in Fig. 5a. Next, 10–15 glass beads with known density were attached at the end of the loadcell tip. The loadcell deflection was measured optically while the glass beads were being attached. The calibration curve for a loadcell is shown in Fig. 5b. This loadcell calibration is more accurate compared to the first one in Fig. 4, since it is based on the weight of the beads, which bears fewer uncertainties.

3 Fabrication vs Mechanical Behavior of PAN Nanofibers As will be shown in the next sections, one can hardly speak about mechanical properties of PAN with universal applicability, mainly because modifications in the electrospinning parameters result in significantly different nanofiber properties. The

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fabrication parameters that are expected to affect the mechanical behavior of electrospun PAN can be classified in three groups:  Solution parameters, such as concentration, density, viscosity, charge, molecular

weight and composition  Flow parameters, such as acceleration distance and acceleration voltage  Ambient conditions, such as humidity and temperature

The effects of some fabrication conditions, such as acceleration voltage, on the morphology and size of polymer nanofibers have already been investigated. For instance, an increase in the concentration of the polymer solution has been reported to result in smaller bead size and density [49, 50] and thicker nanofibers [50–52]. The formation of thicker nanofibers has been attributed to solution viscosity which reduces the stretching of the polymer jet [50]. In addition, higher acceleration voltages may increase (or decrease) the bead density [49, 53, 54]. Along the same lines, the reduction in bead size and density with increasing acceleration voltage can be explained in terms of higher electric force on the polymer jet, which stretches the beads into fibers [49]. On the other hand, high voltages may result in neutralizing the jet ions, and, therefore, higher bead density [53]. Further details about the effects of electrospinning parameters on nanofiber morphology can be found in [28,55–61]. While substantial research has been focused on the effect of electrospinning conditions on the morphology of the resulting nanofibers, the effect of fabrication conditions on their mechanical behavior has been rarely addressed due to the inherent difficulties in conducting experiments with the delicate polymeric nanofibers. The universal method for nanofiber testing described in the previous sections [41] served as infrastructure to investigate the mechanical aspects of electrospun PAN fibers. Ongoing research at the authors’ laboratory has been investigating the effects of acceleration voltage and collector distance on the mechanical behavior of PAN nanofibers. As part of an extensive experimental investigation by the authors of this chapter, Naraghi and Chasiotis varied the acceleration voltage and the collector distance as shown in Fig. 6. This spinning parameter space was discretized to limit the number

Fig. 6 Parameter space to study the effect of electrospinning controls on the mechanical properties of PAN

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of combinations of applied voltage and distance to be explored experimentally. PAN nanofibers were fabricated from 9% wt. solution for all the combinations shown in Fig. 6. The thus fabricated PAN nanofibers had uniform circular cross-sections and smooth surfaces. Their diameters varied mostly between 200 and 300 nm. However, nanofibers as slender as 80 nm and as thick as 800 nm were also observed. Individual nanofibers from each fabrication condition in Fig. 6 and with different diameters were tested with the MEMS nanoscale tension tester described in the previous sections. In all cases, it was clearly found that the nanofiber properties, such as the fiber strength and the elastic modulus, depended on the nanofiber diameter, especially for nanofibers thinner than 300 nm, as shown in Fig. 7. On the other hand, the fiber ductility did not change significantly with its diameter. Therefore, the elastic modulus and fiber strength were selected as the metrics to compare the impact of electrospinning conditions. To account for variations in mechanical properties with fiber diameter, the fiber property data were divided into three subsets: 200–300 nm, 300–400 nm, and 400–900 nm. In the first two subsets, the mechanical properties varied significantly with fiber diameter, while in the last one the fiber strength and modulus nearly reached a plateau. This classification is seen in Fig. 7a and b, where the modulus and strength for nanofibers fabricated at 1 kV/cm are plotted as a function of acceleration (collector) distance. In the same plots, it is evident that the change in the fiber properties for acceleration distances 15–20 cm is rather marginal. However, increasing the acceleration distance to 25 cm substantially changed the mechanical behavior of the thinner .200 nm < d < 300 nm/ nanofibers. The origins of this improvement in the elastic and failure response were further investigated by FTIR spectroscopy, which provided information about the molecular alignment in the nanofibers. The absorption spectrum of polarized IR with two planes of polarization, one parallel and one perpendicular to aligned nanofibers, were compared and the degree of alignment of the nitrile side group in the PAN macromolecule with respect to the nanofiber direction was quantified in terms of the orientation factor [23].

Fig. 7 (a) Elastic modulus and (b) tensile strength of electrospun PAN nanofibers as function of their diameter and the collector distance for an average electric field of 1 kV/cm

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Based on our FTIR measurements, the orientation factor of the nanofibers fabricated at acceleration distance 25 cm was larger than 50%, while the orientation factor for nanofibers fabricated at smaller acceleration distances was 21%. An orientation factor of 50% or more has also been achieved in thin films of PAN cast from a dimethyl sulphoxide (DMSO) solution and drawn to 300% at 115ı C [23]. Therefore, the higher strength and modulus of PAN nanofibers at acceleration distance of 25 cm was due to the higher molecular alignment. This improved molecular alignment for large extensions of the polymer solution is due to the longer travel times of the jet, during which solvent evaporation occurred while the jet was stretched under the electric field. Therefore, the acceleration distance plays a key role in improving the mechanical behavior of the final electrospun nanofibers.

4 Mechanical Instabilities During Cold Drawing of PAN Nanofibers SEM images of the morphology of the as-spun PAN nanofibers by the conditions described in Fig. 6 showed smooth fiber surface with a regular diameter along each nanofiber, as shown in Fig. 8a. However, the process of fiber deformation varied according to the electrospinning conditions. SEM images of deformed nanofibers

a

b

c

Fig. 8 (a) Undeformed electrospun PAN nanofiber, (b) tested nanofiber that deformed homogeneously, (c) nanofiber with multiple ripples formed in response to tensile loading

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Fig. 9 (a) Undeformed PAN nanofiber, (b–d) cold drawn nanofibers with different initial fiber diameters. The scale bars correspond to 500 nm. The fibers in (b–d) were tested at a nominal strain rate 0:0025 s1

fabricated at some combinations of acceleration voltage and collector distance showed homogenous fiber deformation at up to 300% engineering strain, Fig. 8b. On the other hand, nanofibers obtained at other fabrication conditions did not deform homogeneously. Their extension was accompanied by the formation of multiple surface ripples as shown in Fig. 8c. Several experimental observations distinguish the classical necking in cold drawn polymeric microfibers from the cascade of fiber ripples observed in the present PAN nanofibers [62] in Fig. 8c. In microfibers, a single neck initiates, stabilizes, and then propagates throughout the length of the fiber. In contrast, in the PAN nanofibers presented here multiple ripples formed during mechanical drawing without subsequent propagation. Secondly, stretching of microscale fibers with given crystallinity, results in a neck whose diameter is a fraction of the original fiber diameter. This means thicker fibers form proportionally thicker necks. On the contrary, as evidenced in Figs. 9b–d, the surface ripples formed in the present PAN nanofibers had the same spatial frequency and amplitude, independently of the initial diameter of the nanofiber. These surface ripples did not correlate with any surface feature on the originally smooth surface of the nanofibers shown in Fig. 9a.

5 Effect of Strain Rate on the Mechanical Deformation of Nanofibers Nanoscale tension experiments at different strain rates [63] showed that the aforementioned process of nanofiber rippling is strain rate dependent. Three nominal strain rates, 2:5  104 s1 , 2:5  103 s1 , and 2:5  102 s1 , were applied by the MEMS-based nanofiber testing platform described in the previous sections [41, 63]. The test specimens were fabricated from PAN with average molecular weight of 150,000. The acceleration voltage and distance were 12.5 kV and 20 cm,

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50 75 100 Engineering Strain (%)

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Fig. 10 Engineering stress–strain curves of PAN nanofibers at three strain rates [63]

respectively, and the feed rate was 0.2–0.5 mL/h. The engineering stress–strain curves of seventeen nanofibers are shown in Fig. 10. These nanofibers were quite short with gage length of 12 m. Their ultimate strain varied between 60% and 120%, monotonically decreasing with strain rate. Their tensile strength was in the range of 30–130 MPa and was in good agreement with results reported by Fennessey and Farris for twisted yarns of PAN [64]. Contrary to the consistent trend in their ultimate strain, their tensile strength did not vary monotonically with the strain rate. Instead, the highest strength occurred at the slowest strain rate .2:5  104 s1 /, while the lowest strength was recorded at the medium strain rate .2:5  103 s1 /. Such behavior was certainly unexpected for homogenous material deformations, and was shown to be due to structural fiber deformations occurring as a result of the competition between the external loading rate and the material stress relaxation rates. As shown in Fig. 11a, the undeformed nanofibers had a uniform cross-section and a smooth surface. When loaded at the fastest strain rate .2:5  102 s1 /, a cascade of periodic surface ripples, Fig. 11c, formed to accommodate the displacements applied to the fiber. As a consequence, these fibers were drawn at smaller applied forces (engineering stresses), although the local stress (true stress) at each ripple was considerably higher than the corresponding engineering stress plotted in Fig. 10. On the other hand, when the nanofibers were drawn at the slowest strain rate .2:5  104 s1 /, they deformed rather homogeneously, Fig. 11b. At slow loading, local defects that otherwise resulted in deformation instabilities and surface ripples were less severe due to local stress relaxations. At the faster two strain rates, on the other hand, local defects resulted in stress concentrations and caused the fiber deformation to localize.

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a

b

Undeformed fiber c

0.00025 s–1

0.0025 s–1 and 0.025 s–1

Fig. 11 (a) Undeformed PAN nanofiber, (b, c) surface morphology of deformed fibers corresponding to the three strain rates

The periodicity of surface ripples in Fig. 11c requires periodic sites for local instabilities to initiate. Experiments have shown surface cracks, which formed at strains larger than 20%. These cracks imply different surface (skin) vs fiber core properties, which is explained in the following section. The same surface cracks are present in the fibers loaded at the slowest strain rate, but stress relaxations caused by macromolecular rearrangements at the fiber surface reduced the severity of these surface cracks and the propensity for subsequent localization of strain. Therefore, nanofibers drawn at 2:5  104 s1 deformed rather uniformly (homogeneously) with small fluctuations in their diameter, Fig. 11b. The lack of local structural deformations resulted in increased engineering stress during fiber drawing and larger axial forces as opposed to the fibers subjected to faster loading rates. The failure modes of the PAN fibers subjected to rippling are also noteworthy. Contrary to macroscopic neck propagation and failure by reduction of the neck diameter, the fracture of PAN nanofibers was owed to core “extrusion” in the form of a cone (wedge) as shown in Fig. 12a and b. Despite their small diameter, fracture due to formation of nanoscale voids in the fiber core, Fig. 12c, was also observed, which would be expected only in thicker polymeric fibers.

6 Origins of Surface Rippling in Electrospun PAN Nanofibers In order to identify the defects which initiate surface rippling, the authors carried out a series of interrupted tension experiments, in which several fibers were drawn to strains from 20–200% at strain rate 0:0025 s1 . These fibers were removed from the

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a

b

c

Fig. 12 (a, b) Matching surfaces of a fractured PAN nanofiber, (c) nanofiber failure due to void formation

a

b ε = 60%

Undeformed fiber

400 nm

c

d

ε = 135%

Post-failure

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Fig. 13 (a) Undeformed PAN fiber, (b) nanofiber loaded to 60% strain showing cracks on its surface, (c) fiber loaded at 135% strain with pronounced localization, and (d) broken fiber after loaded at >200% strain, with morphology similar to (c)

loading apparatus without further change in their loading history and were imaged by an SEM as shown in Fig. 13. At engineering strains larger than 20%, several cracks were evidenced on the fiber surface. Increasing the strain to up to 60% resulted in widespread surface cracks, Fig. 13b, similarly to the process of fragmentation of a brittle layer on a ductile substrate subjected to in-plane loading [65]. The relative brittleness of the fiber skin, compared to fiber core, could be the result of solvent entrapment in the fiber core which acts as a plasticizer. The similarities

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Absorption intensity (Arbit. Unit)

0.08 0.07 N–C=O

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Fig. 14 FTIR spectrum of PAN nanofibers. The peak at 2240 cm1 corresponds to the nitrile group in PAN. The peak at 1630 cm1 corresponds to molecules of DMF, indicating the presence of solvent molecules in the final nanofibers

between the surface cracks in Fig. 13(b–d), suggest that the fragmentation sites in Fig. 13b triggered the formation of ripples in Figs. 13c and d by inducing localization sites on the fiber surface. The formation of skin-core structure in PAN has been introduced in the past and explained by Guenthner et al [14] in terms of the competition between solvent evaporation from the fiber surface and its diffusion to the surface of the fiber from the fiber core. Fast solvent evaporation may result in a dense fiber skin, which significantly limits further solvent evaporation and transport from the fiber core. Such hypothesis is supported by evidence obtained via FTIR, which confirms the presence of DMF in our PAN nanofibers as shown in Fig. 14.

7 Molecular Alignment in Electrospun PAN Nanofibers During electrospinning, the fiber elongation strain rate is on the order of 1000 s1 [66], which is relatively higher than the time scales for conformational relaxation in polymer solutions, i.e. 10–100 s1 [67]. Therefore, following some of the work by de Gennes [68], electrospinning could impose “stretching” on the solution molecules and reconfiguration from random coils to stretched macromolecules, i.e. molecular alignment. Some times this high strain rate may also result in flowinduced crystallinity [69]. Molecular alignment in PAN nanofibers can be detected in the absorption IR spectra of the nitrile side group, as explained in a previous section. Figure 15 shows

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Fig. 15 FTIR spectrum of PAN nanofibers with two normal planes of polarization

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Fig. 16 Elastic modulus of PAN nanofibers as a function of nanofiber diameter

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such a spectrum for fabrication conditions #5 in Fig. 6. The absorption peak at wave number 2240 cm1 belongs to the nitrile group. The amplitude of the IR absorption peak, for IR polarization parallel to fiber direction, is more than 20% higher than that in the normal direction. The ratio of the absorption peak amplitudes corresponds to an orientation factor of 21% [23], with orientation factor zero corresponding to random molecular orientation and 100% to fully aligned molecules. While FTIR spectroscopy pointed out to potential molecular alignment, it is rather inconvenient to be used to detect molecular alignment in single nanofibers due to the very low-signal-to-noise ratio. An alternative (indirect) measure of molecular alignment in individual nanofibers is their elastic modulus [10]. An example is shown in Fig. 16, where the elastic modulus of PAN nanofibers fabricated at conditions #1 in Fig. 6 is plotted as a function of fiber diameter. The nearly sixfold increase in the elastic stiffness indicates that thinner fibers exhibit higher level

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of molecular alignment. Although still an indirect measure, it points to the strong scale dependence of the mechanical behavior of soft nanofibers. Analogous is the variation in fiber strength with respect to the fiber diameters, which further enforces the aforementioned argument for enhanced molecular alignment in thinner nanofibers.

8 Conclusions The work presented in this chapter summarized the state-of-the-art in experimental nanoscale mechanics of soft nanofibers with the goal to quantify and improve on the mechanical performance of nanostructures and the resulting nanostructured materials. It was shown that PAN nanostructures are strongly susceptible to surface processes that control their load bearing capability and mechanics, leading to behaviors that are not encountered in macroscale fibers. Their mechanical response is a strong function of time and strain rate. When surface phenomena dominate, the nanofiber mechanical properties vary non-monotonically with respect to the applied loading rate. On the other hand, there is direct coupling between electrospinning conditions, fiber nanostructure and its mechanical properties, which introduces a pronounced size dependence of key mechanical properties. “Smaller is stronger” but not by virtue of length scale but due to fabrication-induced variation in nanofiber structure and potentially, surface effects. The work presented in this chapter is currently further extended to novel MEMS-based devices for nanoscale experimentation [70], and to their application on problems concerning the failure of carbon nanofibers [71]. Acknowledgement The authors acknowledge the support by the Solid Mechanics Program on Composites for Marine Structures under ONR-YIP grant #N00014–07–1–0888 with Dr. Y. D. S. Rajapakse as the program manager, and the support by the National Science Foundation (NSF) under NSF-NIRT grant DMI-0532320.

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Characterization of Deformation and Failure Modes of Ordinary and Auxetic Foams at Different Length Scales Fu-pen Chiang

Abstract Sandwich panels with foam core have gained substantial importance in marine structures for the past several decades. However, designers of ships still lack the confidence in composites when compared to traditional structural materials such as aluminum or steel. As a result, composite structures tend to be overdesigned to provide added safety. While there have been numerous studies, most investigators treat the foam cores as made of homogeneous and isotropic materials. But at the length scale of the order of millimeter or smaller, foam is neither homogeneous nor isotropic. In this paper, we present some results of the characteristics of deformation and failure mechanism of polymer foam composites at different length scales. Central to this investigation is a multiscale digital speckle photography technique whereby we can measure detailed full deformation with spatial resolution ranging from centimeters to micrometers. We first investigate the size effect on the mechanical properties of polyurethane foams with and without nanoparticles, crack tip deformation field at different length scales, and the crack propagation characteristics in a foam. Then we present results for a newly created auxetic PVC foam composite. Auxetic materials have a negative Poisson’s ratio rendering them to be more resistant to shear failure, indentation, and impact damages. We describe the manufacturing process of this material and demonstrate its advantageous properties as compared to the original foam.

1 Introduction Sandwich panels with foam core have gained substantial importance in marine applications for over 60 years to lighten, stiffen, and strengthen the boat structures [1]. The decreased weight helps to increase top speed and acceleration, increase cargo capacity, and reduce operating and maintenance cost. The increased F. Chiang () Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794-2300 e-mail: [email protected]

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stiffness and strength allow builders to use less skin material resulting in even lower weight structure. The sandwich structure consisting of a foam core and two skins has many advantages including better thermal insulation, better impact/damage resistance, longer fatigue life, and sound attenuation. Foam core materials have been studied by many investigators [2–8]. In most cases, the foam cores were treated as homogeneous and isotropic materials. This assumption does not have a physical base because of the fact that core materials are in general not homogeneous nor isotropic in the macro-domain of millimeters. Experimental testings are mostly done with macro-specimen and by analyzing global responses. In this paper we present some results of mapping a foam core’s deformation at various length scales. We show that many conventional concepts of the deformation and failure modes are not applicable at certain scales. Another investigation presented in this paper is the study of auxetic foams. An auxetic material has a negative Poisson’s ratio. Thus, when loaded in uniaxial tension, the specimen expands in all directions. This rather unusual behavior gives rise to a variety of properties that are more advantageous than ordinary foam as a core material for sandwich constructs. Essential to all these studies is a unique experimental mechanics technique called multi-scale digital speckle photography. In the following this special technique will be presented first before the studies of foam composites are presented.

2 The Multi-scale Speckle Photography Technique Using a random pattern for quantitative displacement/strain measurement is a major milestone in the history of experimental stress/strain analysis. Heretofore, measurement of deformation is done by using a regular geometric pattern (grating, grid, optical markers, fringes, etc.) whose deviation from the norm is used as a gauge of deformation. The speckle methods [9–13], on the other hand, employ a random dot pattern (i.e., speckles) as a measuring gauge. Quantitative measurement is not done by direct comparison; rather it is obtained by maximizing the cross correlation between the random pattern before and after displacement/deformation. In the following we described a multi-scale digital speckle photography technique whereby the displacement sensitivity and spatial resolution can be varied by judiciously selecting the proper speckle size and recording magnification [9, 13]. A random light intensity distribution of any sort can be considered as a speckle pattern, which may be naturally present or artificially created on the surface (or interior) of a specimen. The speckle pattern is digitally recorded sequentially before and after deformation and processed using a specially designed algorithm called Computer Aided Speckle Interferometry (CASI) [12]. The essence of CASI is described in the following. Two speckle patterns, one before and one after deformation, are first segmented into subimages of 32  32 pixels, for example, and the displacement of all points within a subimage is assumed to be constant.

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Fig. 1 Schematic of CASI for calculating displacement vectors

The corresponding subimages of the two recordings are “compared” via a twostep FFT (fast Fourier transform) process to find the displacement vector. Figure 1 is the schematic of the CASI process. In Fig. 1, h1 .x; y/ is the complex amplitudes of the light disturbance of a generic speckle subimages before deformation and h2 .x; y/ is nothing but the original speckle pattern with displacement components added, i.e., h2 .x; y/ D h1 Œx  u.x; y/; y  v.x; y/

(1)

where u and v are the displacement components along the x and y directions, respectively, of the subimage “point”. First a FFT is applied to both h1 and h2 . Then, a numerical “interference” between the two speckle patterns is performed on the spectral domain as follows, H1 .!x ; !y /H2 .!x ; !y / F .!x ; !y / D qˇ ˇ ˇH1 .!x ; !y /H2 .!x ; !y /ˇ qˇ ˇ D ˇH1 .!x ; !y /H2 .!x ; !y /ˇ expfj Œ 1 .!x ; !y /  2 .!x ; !y /g (2) where 1 .!x ; !y / and 2 .!x ; !y / are the phases of H1 .!x ; !y / and H2 .!x ; !y /, respectively, and  denotes complex conjugate. Finally, a halo function is obtained by a second FFT, i.e., G.; / D =fF .!x ; !y /g D G.  u;  v/

(3)

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which is an expanded impulse function located at (u, v) of the Ÿ and ˜ plane. Thus, by detecting the crest of this impulse function, the displacement vector represented by the cluster of speckles within the subimage is uniquely determined. Strains are then calculated using an appropriate strain–displacement relation. By recording the speckle image at incremental loads, strain of almost any finite magnitude can be obtained. And this is the technique we use in all the subsequent discussions on displacement and strain measurement. At macro scales the texture of the foam cells themselves services as the speckle pattern. At microscales, SiC particles of micron size are spread onto the specimen surface to serve as speckle patterns. The speckle patterns at different stages of specimen deformation are recorded digitally via a CCD camera, an optical microscope, or a scanning electron microscope (SEM). Processing is performed using the CASI algorithm.

3 Studies of Ordinary Foams 3.1 Size Effect on Mechanical Properties of Foam Composites Whenever a new material is developed, the first order of business is to determine its mechanical properties. For the determination of Young’s modulus and Poisson’s ratio, the usual procedure is to use a simple coupon specimen mounted with strain gauges in both the longitudinal and transverse directions to measure its strain under load. Sometimes clip gauges or LVDTs are used to measure displacements which are then converted to strain. Little attention is paid to the size of the specimen or the size of the strain gauge relative to the size of the specimen. We shall show that in foam composites not only does the size of the specimen effect the result the relative sizes of the specimen and the strain gauge also affects the result. Measuring strain in a foam material is not necessarily a straight forward matter. Polymeric foams such as polyurethane or PVC tend to be relatively soft as compared to most engineering materials. Using strain gauges is not the best because the cement employed to glue the strain gauges tend to be stronger than that of the foam, thus gives rise to erroneous results. Clip gauges or other similar techniques have a fairly large gauge length which tends to mask the details of foam deformation at critical regions. Thus, there is a need to use a technique that is noncontact and has a high spatial resolution. The white light speckle photography technique is such a technique [9,13]. It was developed by the author over the years with support, in part, by grants from the Office of Naval Research, managed by Dr. Yapa D.S. Rajapakse. Uniaxial tension specimens of the dog-bone type were machined from neat polyurethane foam panels. Figure 2 shows two types of specimens that were used in the experiments. The dimensions of the uniform section of the larger ones are 80  16  8 mm whereas those of the smaller ones are 25  4:8  3:6 mm. The larger specimens were tested using a tabletop InstronTM 1011 uniaxial testing machine, and the smaller ones were tested inside the vacuum chamber of a Hitachi S-2460N

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Fig. 2 Dimensions of the large and small polyurethane foam specimens

contour interval is 18 µm

Fig. 3 Displacement fields of a macro specimen under uniaxial tension

SEM at 30x, 100x and 300x magnifications, respectively. A thin coating of silicon carbide powder (with average powder size of about 20 m for the larger specimens and 1 m for the smaller specimens) was spread on the foam surface to serve as speckles. The specimen was loaded slowly under displacement control. After every incremental loading, the load was stopped and the speckle image was recorded after the force reached steady state (the force drops slightly due to material’s viscoelastic effect). The force was read from a load cell and strain was calculated using CASI. Figure 3 depicts a typical deformation pattern of a larger specimen as calculated by CASI. The lines represent displacement contours along x- (u field) and

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At 30× (contour interval is 8 μm)

At 100× (contour interval is 1 µm)

At 300× (contour interval is 0.7 µm)

Specimen Image

u Field

v Field

Fig. 4 Heterogeneous deformation as the observation length scales is reduced

y-directions (v field), respectively. The contour interval is 18 m. The distribution of the contours indicates that the deformation is fairly uniform, as one would expect from a uniaxially loaded tensile specimen. Thus at this scale, the material behaves as if it were rather homogeneous and isotropic. However, when the observation scale is reduced, interesting deformation patterns are observed. Figure 4 shows the deformation pattern of a small specimen tested inside the SEM. The contour intervals are 8 m for 30x, 1 m for 100x, and 0:7 m for 300x images. The deformation pattern becomes progressively heterogeneous as the length scale is further reduced. A more illustrative way of showing the deformation pattern is given in Fig. 5 wherein the total displacement vector distributions of the cells at different magnification are depicted. It is seen from the pictures that individual foam cells deform differently depending on its spatial orientation in the foam as well as the stress field that surrounds it at the time. We find that a single cell may be in tension in one region and in compression in another. We have even seen a situation in which a single cell is shown to expand in all directions while under a globally applied uniaxial tension as shown in Fig. 6. Figure 7 shows the stress strain relationships of all neat polyurethane foam specimens tested. The stress is the average stress obtained by dividing the load by the

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At 30×

At 100×

At 300×

Fig. 5 Deformation of the foam specimens in terms of total displacement vector

initial cross-sectional area of the specimen and the strain is the average value of strains in all subimages. As can be seen from the figure the material is highly nonlinear and the stress–strain curve is size dependent. And for a given size the result also depends on the dimension of the area from which the strain is measured. The general trend is that the larger specimen shows a higher stiffness. This may be attributed to the fact that the material is a porous one consisting of interconnected foam cells. When the specimen is large enough the mutual constraints of the cells

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Fig. 6 Deformation of a foam cell showing expansion in all directions under uniaxial tension load

give rise to the stiffness of the foam. When the specimen size is reduced, the ratio of cell size to the specimen cross-section is increased. As a result, the material effectively becomes more “porous”, hence the reduced stiffness.

3.2 Crack Tip Deformation in Foam at Different Length Scales Knowing the crack tip deformation field is of course essential to understanding the crack growth behavior and the resulting fracture of a material. Foam is neither homogeneous nor isotropic at a length scale of the order of its unit cell. Unlike common structural materials such as aluminum or steel in which the size of a crystal is many orders of magnitude smaller than the macro length scale used in the structural, the cell size in a foam is only about two to three orders of magnitude smaller than the macro length used in a sandwich structural construct. Thus, it is important that its behavior at different length scales be examined before a realistic predictive model can be constructed. In the following we examine the crack tip deformation of a beam made of PVC foam under a three-point bending load. Figure 8 depicts some of the results. On the left is a pair of micrographs showing the region in which we performed the CASI analysis. The upper picture, taken at 50x magnification shows the tip of a crack created by a sharp razor blade cut which is visible near the bottom of the picture. The region just above the crack tip, marked by a black rectangle is further enlarged to 100x. CASI was used to map the region’s deformation pattern

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Large Specimen #1 Large Specimen #2 Large Specimen #3 Large Specimen #4 Small Specimen #1 (@ Small Specimen #2 (@ Small Specimen #3 (@ Small Specimen #4 (@ Small Specimen #5 (@

2.5

2

30x) 100x) 300x) 30x) 100x)

stress (MPa)

1.5

1

0.5

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

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0.1

strain Fig. 7 Stress–strain curves of various specimens

and the results are shown in the two sets of u- and v-field displacement contours depicted. The displacement sensitivity of the contours are 2 and 1 m, respectively, for the 50x and 10x pictures. At the 50x magnification, the crack tip deformation patterns show only a small resemblance to that of the classical deformation pattern at the crack tip. At 100x magnification, the formation pattern bears no similarity to that of the classical pattern at all. Thus, if one is to construct a realistic analytical or numerical model to predict the deformation of a foam beam or plate under load, these types of deformation must be taken into consideration. Crack propagation in a foam material is also quite different from that of a traditional engineering material. Figure 9 depicts the characteristics of crack propagation in foam at a micro-scale. The specimen is a coupon with a single edge crack under tension. When the load is increased, voids spontaneously appear in front of the crack. Further loading results in blunting of the original crack as well as the enlargement of the void. Eventually, the main crack and the void are linked together resulting in failure of the specimen. The last picture in the figure depicts

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Region of interest with speckles at 50x and 100x

u and v displacement field at 50 x. Contour constant = 2μm

u and v displacement field at 100 x. Contour constant = 1μm

Fig. 8 Crack tip deformation field of a nano-phased PVC foam specimen with an edge crack under three-point bending Void generated

Linking void

Displacement Vectors Surrounding a Crack Tip

Failure

Fig. 9 Crack propagation characteristics in a foam at a micro scale

the deformation of a step loading after the void has been generated. The displacement vector field demonstrates an interesting deformation pattern: all the vectors are pointing towards the crack tip region.

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4 Studies of Auxetic Foams 4.1 Introduction An auxetic material has the unusual property of having a negative Poisson’s ratio. When loaded in uniaxial tension it expands rather than shrinks, laterally. This particular property gives rise to a variety of unusual mechanical behaviors when the material is subjected to load. Manmade auxetic material was first manufactured in 1987 by Lakes [14]. Subsequently, many investigators have ventured into the field [4–23]. An auxetic thin foam has several advantages over the conventional ones as a ship-building material [24]. An auxetic thin plate deflects much less than a conventional plate for a given load [23]. It reduces acoustic noise due to its lower cut-off frequency [25]. It resists indentation and has a lower velocity impact damage [26,27]. When bent it deforms synclastically rather than anticlastically thus rendering it ideally suitable for forming into convex–convex surfaces [28]. Furthermore, it resists shear failure due to the resulting large shear modulus. In this paper, we present the process of manufacturing an auxetic composite from an ordinary polyurethane foam. We characterize the mechanical properties of the resulting material and exploit its use as a core material for sandwich panels for marine construction. There are different mechanisms that would result in auxetic behavior. One of which is the so called “re-entrant structure” whereby when loaded in tension the reentrant structure expands in transverse directions. An example is shown in Fig. 10 which is a microscopic picture of the auxetic polyurethane foam that we manufactured. Note the unit cell within the marked circle in the viewed area. The

Mechanism of auxetic behavior in foam. (Uniaxial tension resulting in expansion in all directions.)

Fig. 10 Auxetic behavior in a foam due to “re-entrant structure”

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images depict a re-entrant structure. When loaded in tension vertically the re-entrant structure expands in both the vertical and horizontal directions progressively as the load is incrementally increased from picture #1 to picture #4.

4.2 Auxetic Polyurethane Foam We selected a soft polyurethane packaging foam for this initial study. We followed the manufacturing process proposed by Lakes [14]. A 90  30  30 mm block of the ordinary polyurethane foam was compressed into a rigid frame made of aluminum of much smaller size. The assembly was then heated (at 160ı C) slightly above its transition temperature for about 17 min. Then it was allowed to cool naturally to the room temperature. The resulting specimen has a dimension of 68  19  10 mm. The fact that the resulting material is auxetic is demonstrated in the global deformation of two thin strips of specimen made of the original conventional foam and the converted auxetic foam, respectively, as shown in Fig. 11. When loaded in uniaxial tension, the conventional foam specimen changes its length from 43 to 60 mm whereas its lateral dimension shrinks from 7.2 to 5.2 mm. On the other hand, the manufactured auxetic foam changes its length in the longitudinal direction from 38 to 54.6 mm and its transverse direction also expands from 8.6 to 10.4 mm.

4.3 Auxetic PVC (H45) Foam 4.3.1

Manufacturing the Auxetic PVC Foam

We manufactured an auxetic foam by compressing ordinary PVC foam (H45-Blue) in three dimensions. The PVC foam has the following properties: size of the foam cell: 50 m; tensile strength: 0:6 MPa; Young’s modulus: 90 MPa; foam density:

Fig. 11 Mechanical response of conventional and auxetic polyurethane foams under uniaxial tension

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Fig. 12 Optical images of foam material (a) before processing, (b) after processing and the manufacturing scheme

44:3 kg=m3 . A 50  85  95 mm PVC foam block was first cut and then compressed in all three dimensions sequentially using an aluminum mold and a hydraulic press. After that, the aluminum mold was heated to about 135ı C and kept at that temperature for about 30 min. Finally, the aluminum mold was allowed to cool down to room temperature and the resulting foam became auxetic. The global volume reduction of the fabricated foam was about 50% from the original but the local volume reduction varies. After compression and heat treatment cells become smaller and crushed inward. The schematic of the manufacturing process and micrographs of the original and auxetic foams are shown in Fig. 12.

4.3.2

Mechanical Properties of Auxetic PVC Foam

Uniaxial Test Specimens were tested under uniaxial loading. A VHX-100 digital optical microscope was used to record the speckle pattern as the load was being applied quasi-statically by a servo controlled stage at a constant rate. Coupon specimens were cut from a processed foam plate of 5 mm in thickness about 5  7  30 mm in size and tested under uniaxial tensile loads. A typical example of the specimen

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Fig. 13 Uniaxial test setup and an image of the foam specimen

dimensions and its corresponding optical micrograph are shown in Fig. 13. The surface roughness of the foam served as an ideal speckle pattern. Speckle pattern under different loads were digitized and analyzed using CASI software. Figure 14 shows a typical specimen and resulting displacement fields. It can be seen clearly from the vector field that with increasing load, the specimen expands in both directions without much rotation and assumes auxetic behavior. The contour lines are not straight as an ideal case would have been. However, the error is relatively minor, in fact, this type of non-uniformity, in practice, is unavoidable because of the local variation of the foam’s properties resulting from the manufacturing process. Total displacement change measured using clip gages or LVDT will not reveal whether or not the specimen is uniformly deformed. Figure 15 shows the resulting stress–strain curve of a coupon specimen under uniaxial load and its Poisson’s ratio as a function of the tensile strain applied. It is noted that in Fig. 15a the LVDT result for the specimen appears to be much softer than the results of the CASI. The reason is that the LVDT measures the total displacement experienced by the loading grips. The end effect is such that the strains near the grips are much larger. The results shown in Fig. 15a are the average of the entire gauge length. In the case of LVDT measurement, the gauge length is the entire length between the grips. Defining Poisson’s ratio as simply the ratio of longitudinal strain over the transverse strain the result is as shown in Fig. 15b. It is seen that as the tensile strain increases the absolute value of Poisson’s ratio decreases. We also employed tests under cyclic loading. The results obtained using CASI and LVDT are shown in Fig. 16. It is noted that while using CASI, the deformation remains essentially elastic with little plastic deformation (within the experimental errors), the results obtained by LVDT show a large amount of plastic deformation.

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Fig. 14 Example of displacement fields obtained by CASI. v field, u field and total displacement vector. Contour Constant is 31 m

Fig. 15 Deformation of an auxetic foam. (a) A typical stress–strain plots. (b) Negative Poisson’s ratio as a function of tensile strain

The degree of auxeticity of the foam thus manufactured shows a dependence on the volume reduction during the manufacturing process. The larger the amount of volumetric change, the larger the negative Poisson’s ratio. The relationship is shown in Fig. 17.

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Fig. 16 Deformation of the auxetic PVC foam under cyclic loading

Fig. 17 Poisson’s ratio as a function of volumetric change

Shear Test Assuming elastic deformation an auxetic material should have a high shear modulus. We prepared a shear test setup with the schematics as shown in Fig. 18. Two identical strips of foam blocks are cemented onto aluminum plates as shown. When the

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Fig. 18 Schematic of shear test setup

central plate is extended the two foam blocks are loaded in simple shear. A typical deformation pattern as obtained by CASI is shown in Fig. 19. The original PVC foam (H45 Blue) has the following mechanical properties: Young’s Modulus E D 90 MPa, Poisson’s Ratio  0.3. Using the relation G D E , we obtain a shear modulus of 28 MPa. Using the displacement fields as 2C.1Cv/ shown in Fig. 19 the average shear modulus is 38 MPa, which is substantially greater than the original one.

Impact Test We also conducted impact resistance tests to check whether or not the auxetic foam is better than the original in resisting low velocity impact. We used a toy BB gun (Daisy 105B) with 4.5 mm steel bullets to shoot the foam target. We employed a grid on the surface to indicate the volume change, thus the Poisson’s ratio. Shots were fired from a very close range to ensure precise positioning of the shots. The penetration values of these shots were measured. The results are presented in Fig. 20. Figure 20a shows the auxetic foam sample with location of the shots, whereas Fig. 20b depicts the penetration depth as a function of Poisson’s ratio. The penetration depth of the original foam is much larger than that of the auxetic foam. Within the latter the result seems to indicate a trend that the larger the negative Poisson, the less the penetration depth.

Indentation Tests Because auxetic material is said to have better indentation resistance, we set out to show whether this is true with the auxetic PVC foam we manufactured. The

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Fig. 19 Deformation of the shear test specimen

Fig. 20 Penetration values of the metal bullets into the foam block as a function of Poisson’s ratio

specimens were 60 mm cubes. Both normal and auxetic foam specimens were tested. Indentation test was done using a loading frame with the maximum load of 500 N (TIRA Test). The resulting load–displacement curve was plotted as shown in Fig. 21. In this figure the red line represents the auxetic foam whereas the green

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Fig. 21 Indentation load and indentation depth relation of auxetic and normal PVC foams

line represents the ordinary foam. As can be seen, the auxetic foam becomes stiffer with increasing load indicating that the auxetic foam does provide better indentation resistance. We believe this is largely due to the fact that under indentation load, the negative Poisson’s ratio causes the foam to shrink, thus providing more resistance. Acknowledgment The author gratefully acknowledges the support provide by the Office of Naval Research’s Solid Mechanical Program under the leadership of Dr. Yapa D.S. Rajapakse. The development of the speckle technique was supported by earlier grants from ONR. The foam composite work was supported by the grant # N000140410357. The encouragement of Dr. Rajapakse is indispensible for the success of the work.

References 1. Vinson JR, Rajapakse YDS, Carlsson L (eds.) (2003) 6th International Conference on Sandwich Structures. CRC press, Boca Raton, FL 2. Prasad S, Carlsson L (1994) Debonding and crack kinking in foam core sandwich beams – I. Analysis of fracture specimens. Eng Fract Mech 47(6): 813–824 3. Prasad S, Carlsson L (1994) Debonding and crack kinking in foam core sandwich beams – II. Experimental investigation. Eng Fract Mech 47(6): 825–841 4. Gibson LJ, Ashby MF, Schajer GS, Robertsson CI (1982) The mechanics of two-dimensional cellular materials. Proc R Soc Lond A382: 25–42 5. Gibson LJ, Ashby MF (1982) The mechanics of three-dimensional cellular materials. Proc R Soc Lond A382: 43–59 6. Zenkert D, B¨acklund J (1989) PVC sandwich core materials: Mode I fracture toughness. Compos Sci Technol 34: 225–242

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7. Zenkert D (1989) PVC sandwich core materials: fracture behaviour under mode II and mixed mode conditions. Mater Sci Eng A108: 233–240 8. Hallstr¨om S, Grenestedt JL (1997) Mixed mode fracture of cracks and wedge shaped notches in expanded PVC foam. Int J Fract 88(4): 343–358 9. Chiang FP (2003) (http://www.engineeringvillage2.org/engresources/images/s.gif) isn’t in document Evolution of white light speckle method and its application to micro/nanotechnology and heart mechanics. Opt Eng 42(5): 1288–1292 10. Chiang FP (1978) A family of 2D and 3D experimental stress analysis techniques using laser speckles. Solids Mech Arch 3(1): 1–32 11. Chen DJ, Chiang FP (1993) Computer aided speckle interferometry using spectral amplitude fringes. Appl Opt 3(2): 225–236 12. Chen DJ, Chiang FP, Tan YS, Don HS (1993) Digital speckle-displacement measurement using a complex spectrum method. Appl Opt 32(11): 1839–1849 13. Chiang FP, Wang Q, Lehman F (1997) New developments in full field strain measurements using speckles. In: Lucas GF, Stubbs DA (eds.), Non traditional methods of sensing stress, strain and damage in materials and structures. ASTM STP 1318: 156–169 14. Lakes RS (1987) Foam structures with a negative Poisson’s ratio. Science 235: 1038–1040 15. Lakes RS, Elm K (1993) Indentability of conventional and negative Poisson’s ratio foams. J Compos Mater 27: 1193–1202 16. Chiang FP (2006) Macro to nanomechanics studies of foam material. Proc ONR Prog Rev ’06 (Rajapakse YDS ed.), University of Maryland, College Park, MD 17. Ting TCT, Barnett DM (2005) Negative Poisson’s ratios in anisotropic linear elastic media. J Appl Mech Trans ASME 72(6): 929–931 18. Ting TCT, Chen T (2005) Poisson’s ratio for anisotropic elastic materials can have no bounds. Q J Mech Appl Math 58(1): 73–82 19. Alderson KL, Webber RS, Kettle AP, et al. (2005) Novel fabrication route for auxetic polyethylene. Part 1. Processing and microstructure. Polym Eng Sci 45(4): 568–578 20. Alderson A, Evans KE (2002) Molecular origin of auxetic behavior in tetrahedral framework silicates. Phys Rev Lett 89(22): 225503-1 21. Alderson KL, Fitzgerald A, Evans KE (2000) Strain dependent indentation resilience of auxetic microporous polyethylene. J Mater Sci 35(16): 4039–4047 22. Alderson KL, Alderson A, Evans KE (1997) Interpretation of the strain-dependent Poisson’s ratio in auxetic polyethylene. J Strain Anal Eng Des 32(3): 201–212 23. Baughman RH, Shacklette JM, et al. (1998) Negative Poisson’s ratios as a common feature of cubic metals. Nature 392(6674): 362–365 24. Whitty JPM, Alderson A, Myler P, Kandola B (2003) Towards the design of sandwich panel composites with enhanced mechanical and thermal properties by variation of the in-plane Poisson’s ratios. Compos Part A: Appl Sci Manuf 34(6) SPEC: 525–534 25. Evans KE (1991) Auxetic polymers: a new range of materials. Endeavour, New Series 15(4): 170–174 26. Chen CP, Lakes RS (1989) Dynamic wave dispersion and loss properties of conventional and negative Poisson’s ratio polymeric cellular materials. Cell Polym 8(5): 343–359 27. Scarpa F, Yates JR, et al. (2002) Dynamic crushing of auxetic open-cell polyurethane foam. Proc Inst Mech Eng Part C: J Mech Eng Sci 216(12): 1153–1156 28. Scarpa F, Ciffo LG, et al. (2004) Dynamic properties of high structural integrity auxetic open cell foam. Smart Mater Struct 13(1): 49–56

Fracture of Brittle Lattice Materials: A Review Ignacio Quintana-Alonso and Norman A. Fleck

Abstract The mechanics of failure for elastic-brittle lattice materials is reviewed. Closed-form expressions are summarized for fracture toughness as a function of relative density for a wide range of periodic lattices. A variety of theoretical and numerical approaches has been developed in the literature and in the main the predictions coincide for any given topology. However, there are discrepancies and the underlying reasons for these are highlighted. The role of imperfections at the cell wall level can be accounted for by Weibull analysis. Nevertheless, defects can also arise on the meso-scale in the form of misplaced joints, wavy cell walls and randomly distributed missing cell walls. These degrade the macroscopic fracture toughness of the lattice.

1 Introduction Lattice materials are enjoying increasing use in engineering applications such as the core of sandwich panels. Extensive research has been conducted into the prediction of stiffness and strength for a wide range of 2D and 3D lattices. For example, the in-plane elastic properties of various lattice topologies are now well understood [1–3]. In contrast, little work has been done on their fracture properties and damage tolerance. The response of lattice materials to several types of defects has been investigated by numerical, analytical and experimental methods. For instance, the mechanical properties of regular hexagonal lattices with defects consisting of missing cells were analysed by Guo and Gibson [4] using the finite element (FE) method. The effect of holes and rigid inclusions on the elastic modulus and yield strength are studied in Chen et al. [5]. Defects in the form of randomly fractured cell-walls have been examined numerically [6] and experimentally [7]. The influence of defect size and cell size on the tensile strength of notched lattices was studied in Andrews and I.Quintana-Alonso and N.A. Fleck () Cambridge University Engineering Department, Trumpington St, Cambridge CB2 1PZ, UK e-mail: [email protected], [email protected] I.M. Daniel et al. (eds.), Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, c Springer Science+Business Media B.V. 2009 

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Gibson [8] by means of FE simulations. Chen et al. [9] provide a comprehensive study of this wide range of geometrical imperfections. They found that fractured cell-walls produce the largest knock-down effect on the yield strength of hexagonal lattices. Damage tolerance is a broad subject, as illustrated by the variety of defects examined in the above studies. The purpose of this review is not to provide an exhaustive list. Rather, we focus on the response of 2D brittle lattices to the presence of a crack or sharp notch. The dependence of fracture toughness upon microstructure is reviewed and the applicability of conventional fracture mechanics concepts to the discrete lattice is assessed.

1.1 Fracture Mechanics Concepts For a linear-elastic material, the stress intensity factor K defines the magnitude of the dominant component of the local stresses near the crack tip. It accounts for the geometry of the body containing the crack and the type of loading to which it is subjected. The power and utility of K is that this macroscopic parameter overcomes our lack of knowledge of the microscopic fracture events at the crack tip. Rapid, unstable crack advance will occur when the local stresses reach some critical value, which will be constant for a given material. It immediately follows that these critical local conditions will correspond to a critical value of the stress intensity factor, K D KC , which we can easily quantify and then use as a measure of a material’s resistance to rapid crack advance; we call this resistance the material’s fracture toughness, KC .1 There are three types of loading that a crack can experience. Mode I loading, where the principal load is applied normal to the crack plane, tends to open the crack. Mode II corresponds to in-plane shear loading and tends to slide one crack face with respect to the other. Mode III refers to out-of-plane shear. A cracked body can be loaded in any one of these modes, or a combination of two or three modes giving rise to mixed-mode fracture. Mode I is technically the most important; the discussions in this review are limited to modes I and II. Conventional linear elastic fracture mechanics (LEFM) was developed for a continuous medium. Continuum theory can predict the stresses near the crack tip, but it is the material’s microstructure that determines the critical conditions for fracture. It is well known that the length scales associated with the microscopic events that lead to failure of the material at the crack tip must be sufficiently small compared with the size of the K-field. In a lattice material, this means that for K to define

1

It is worth emphasising the physical origin of each side of the fracture criterion K D KC . The stress intensity factor on the left hand side is a mechanics parameter, its value determined by the geometry, stress level and crack length in the component, and is a measure of the cracking effort being applied to the component; whilst the right hand side is a material constant, a measure of the particular material’s ability to resist rapid crack advance.

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uniquely the crack-tip stress conditions and be a valid failure criterion, the cell size must be small compared to the size of the singularity zone governed by the K-field. Otherwise, the continuum assumption is invalid and linear elastic fracture mechanics (LEFM) becomes inapplicable.

1.2 Outline of this Review This review is arranged thematically rather than chronologically. It is organised according to the different methods and models used to determine the fracture properties of lattices. First, analytical estimates of fracture toughness obtained by classical beam theory are presented. These studies are based on the assumption that the cell-walls are adequately modelled by a lattice composed of rigidly connected Euler beams. Second, we consider generalised continuum theories that apply homogenisation techniques to the lattice. Third, major contributions obtained by the finite element method are reviewed in detail. We then briefly mention atomic lattice models. Next, the main results obtained by the representative cell method are summarised. Finally, we point out the few experimental studies dealing with the determination of the fracture toughness of lattices.

2 Classical Beam Theory Ashby [10] and Maiti et al. [11] is the starting point for all subsequent investigations of the fracture toughness of lattice materials. The concept of effective fracture toughness KC is based upon the existence of a K-field on a length scale much larger than the cell size of the lattice. Ashby [10] made use of LEFM concepts to estimate the fracture toughness of a hexagonal lattice (Fig. 1a). The stress field of an equivalent linear-elastic continuum was used to calculate the stresses on the discrete cell walls of the lattice directly ahead of the crack tip. The macroscopic fracture toughness was then estimated by assuming that the critical strut directly ahead of the crack tip fails when the maximum tensile stress within it attains the tensile fracture strength T S of the parent solid material.2

2

Tensile fracture strength T S and modulus of rupture f are used indistinctively in this review. The modulus of rupture f is the maximum surface stress in a bent beam at the instant at which it fractures. If the beam is made of a brittle solid, like the cell wall discussed here, the fracture initiates at a microcrack (usually a surface microcrack) in the wall and propagates catastrophically. On average, the modulus of rupture is a little larger than the tensile strength T S because, in bending, only one surface of the beam sees the maximum tensile stress; in simple tension the entire beam is stressed uniformly, so a given microcrack is less likely to be stressed in bending than in simple tension. The statistics of the problem (see, for example, Davidge [12]) show that the modulus of rupture is typically about 1.1 times larger than the tensile strength.

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Fig. 1 Planar lattice topologies: (a) hexagonal, (b) triangular, (c) Kagome, (d) square, and (e) diamond-celled

2.1 The Hexagonal Lattice These analytical models predict that the mode I fracture toughness KIC of the regular hexagonal lattice scales linearly with T S , quadratically with stockiness t = l, and with the square root of the cell size l as p (1) KIC D 0:53 .t = l/2 T S l This approach is described in detail in the fundamental monograph by Gibson and Ashby [1]. In-plane shear stress is the primary loading in sandwich panels, producing a mode II or mixed-mode fracture. The mixed-mode fracture criteria for solid materials are invalid for lattice materials because failure mechanisms are different due to the discreteness of the lattice. Accordingly, Huang and Lin [13] analysed the mixed-mode fracture for hexagonal lattices under a combined loading of uniform tensile and in-plane shear stresses. As a first step they derived the expression for mode II fracture toughness KIIC using dimensional arguments in cooperation with the near tip singular stress field of a continuum. It was concluded that the mode I and II fracture toughness have the same dependence on cell size, stockiness and fracture strength exhibited by Eq. (1), with the coefficient to be determined empirically or numerically. Huang and Lin [13] considered only cell-wall bending (acceptable for the hexagonal topology) to obtain a mixed-mode fracture criterion which is a linear combination of KI =KIC and KII =KIIC . Fleck and Qiu [14] showed that it is in fact piece-wise linear. The mixed-mode fracture criterion for brittle lattice materials contrasts with that of solid materials, for which usually a smooth convex envelope can be described

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by an ellipse. Huang and Lin [13] compared the theoretical mixed-mode fracture criterion with limited experimental data for PVC foams, claiming agreement between theory and experiment. Nevertheless, their conclusion should be regarded with caution, given the scarcity and scatter of the data.

2.2 Other 2D Lattices Fleck and Qiu [14] have evaluated KIC and KIIC for the isotropic triangular and Kagome lattices (Fig. 1b, c). Similar expressions to Eq. (1) are derived. They find that KIC scales linearly with stockiness t = l for the triangular lattice, and scales as .t = l/1=2 for the Kagome topology. More recently, Quintana-Alonso and Fleck [15] have considered the fracture toughness of the orthotropic, diamond-celled lattice, see Fig. 1e. Their model is in the spirit of the analyses of Gibson and Ashby [1] for the hexagonal lattice, and of Fleck and Qiu [14] for the triangular and Kagome lattices. Quintana-Alonso and Fleck [15] find that KIC scales linearly with t = l for the diamond-celled topology.

2.3 Statistics of Brittle Failure The first studies on the fracture properties of brittle lattice materials assumed that the tensile strength of a single cell wall is constant. In practice this is unrealistic. Brittle fracture is a stochastic process, dependent on the presence and distribution of defects in the struts. Depending on the width of this distribution, the failure of the first strut may or may not trigger the failure of the whole. The tensile and compressive strengths of ceramic lattice materials are significantly influenced by the flaw size distribution within them. As a result, variability in the strength of brittle lattice materials is expected. Weibull [16] proposed an empirical formulation with a simple statistic distribution to describe the strength variability in brittle materials such as concrete, wood and glass fibre. The Weibull modulus m describes the flaw size distribution such that the material becomes less brittle as m increases. Huang and Gibson [17] have made use of Weibull statistics to include probabilistic effects in the fracture toughness of hexagonal and square lattices. They concluded that the fracture toughness increases with increasing cell size for a Weibull modulus m > 4, is insensitive to cell size for m D 4, and decreases with increasing cell size for m < 4. This result was later used by Huang and Chou [18] to modify the model for failure envelopes of lattice materials under in-plane biaxial loading [19]. The validity of Huang and Gibson’s [17] result is limited by the assumption that failure always occurs near the crack tip and, thus, only the critical strut directly ahead of the crack tip needs to be considered. This assumption holds true for large values of Weibull modulus m. But for small m, the variability in strength may be

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sufficiently high for struts remote from the crack tip to fail. Quintana-Alonso and Fleck [20] prove that for m < 4, struts remote from the crack tip fail: the effective stressed volume becomes sufficiently large for the predicted fracture toughness to drop to zero according to Weibull theory.

3 Generalised Continuum Theories An alternative analytical method for fracture toughness evaluation is presented by Chen et al. [21]. The authors determined the fracture toughness of lattices with hexagonal, triangular and square cells (Fig. 1a, b, d) by replacing the discrete lattice with a generalised continuum described within Cosserat theory.3 The model is based on equating the continuum approximation of the strain energy of the discrete lattice to the strain energy of an equivalent micro-polar continuum. For the prototypical regular hexagonal lattice they obtained KIC 1:8 .t = l/ T S

p

l

(2)

This result gives a much higher toughness than Eq. (1), particularly for low-density materials, due to the linear rather than quadratic dependence upon t = l. The experimental data quoted by Gibson and Ashby [1] support the quadratic dependence of KIC upon t = l. As indicated by Fleck and Qiu [14], the discrepancy can be traced to the fact that Chen et al. [21] assumed affine deformation in their calculation of the effective properties of the Cosserat medium: this gives too stiff a response as it neglects the dominant contribution of cell wall bending under uniform loading. The analysis of Maiti et al. [11] correctly includes the effect of cell wall bending and thereby gives the correct functional dependence of fracture toughness upon t = l, as given by Eq. (1). It is interesting to note that the possibility of applying generalised continuum theories to lattice structures and frames was already discussed in the pioneering works of Banks and Sokolowski [22] and Bazant and Christensen [23], who developed an analogy between a micro-polar medium and an uncracked rectangular lattice. They showed that to obtain accurate results, the continuum approximation of potential energy of individual struts must be expressed exactly up to terms with second order derivatives of joint rotation. Specifically, this approach is necessary to derive the correct micro-polar constants relating the in-plane moments and bending curvature.

3

Classical continuum theory is well-suited for situations where the variations in stresses and strains are smooth. However, in many circumstances this is not necessarily the case (e.g. near crack tips or boundary layers) and one resorts to enhanced continuum theories (also called enriched or generalised). Such theories include information on the microstructure and take into account the non-uniformity of stresses and strains at the micro-scale. One of the simplest generalised theories is Cosserat (or micro-polar) theory, in which the interaction between neighbouring material points is governed by a moment vector in addition to the force vector from classical continuum theory.

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The continuum description of lattice materials enables the use of powerful analytical methods, such as asymptotic analysis, Fourier transform and Wiener-Hopf technique, to study the fracture of lattice materials. For instance, homogenisation models for cracks in lattice materials with triangular, square and hexagonal cells were studied by Antipov et al. [24, 25] with emphasis on crack propagation. They derive explicit asymptotic formulae for the stress intensity factors, and in addition analyse interaction of cracks in the homogenised lattice. This procedure leads to similar results to those of Chen et al. [21]: for the hexagonal lattice the inferred fracture toughness is too large. A significant body of research has evolved on generalised continuum modelling of 2D periodic lattices. This approach has an advantage over discrete modelling of the lattice especially in design selection exercises involving various cell topologies and complex domains. For example, various 2D lattices can be rank-ordered with respect to fracture toughness or notch resistance [26]. However, the generalised continuum approach has disadvantages: homogenisation techniques are unable to capture microscopic instabilities such as local buckling of individual struts in the discrete lattice.

4 Finite Element Modelling 4.1 Stress Analysis Lattice materials can be viewed as rigid-jointed interconnected simple beams. Consequently, the methods of analysis developed for 2D beams and frames in structural engineering are used to analyse their stiffness and strength. One direct approach is to treat the material as a latticework and determine stresses in various members by a numerical method such as finite element (FE) analysis. This approach, in general, leads to the prediction of strength for a lattice material rather than its fracture toughness. Within the framework of this numerical approach, the fracture toughness is calculated by the FE simulation of a finite lattice plate with several missing struts. Suppose the characteristic size of the plate is equal to L, the crack-like flaw modelled by contiguous removed struts has length 2a, and the cell size is l, see Fig. 2. The requirement for the correct evaluation of the fracture toughness is expressed by the inequality Lal (3) Huang and Gibson [17] used the FE model to examine a rectangular lattice plate subjected to uniaxial tension with a central finite length crack (Fig. 2). They found that for a mode I crack in a diamond-celled lattice the fracture toughness is approximately given by p KIC D 1:7.t= l/2 TS l (4)

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Fig. 2 Centre-cracked plate under uniaxial tension

Previous attempts to model the fracture toughness of brittle lattices and foams applied well-known results for a continuum by assuming that the crack length is large relative to the cell size. In practice, this may be unrealistic, since short cracks of roughly three to four cells are common in brittle lattices and foams. Huang and Gibson [17] detected that Eq. (4) is valid when the stockiness t = l is lower than 0.2 and the ratio of crack length to cell size a= l is larger than 7. For lattices with a= l < 7, the effective fracture toughness is reduced by a factor which is insensitive to changes in the cell geometry. They attributed the discrepancy at ratios of t = l higher than 0.2 to the fact that axial stresses in the cell walls, neglected in the analytic model, become significant. Quintana-Alonso and Fleck [15] demonstrate that the crack length required for the correct fracture toughness evaluation scales with the stockiness of the diamondcelled lattice as a= l / .t = l/2 and, consequently, exceeds the above value for small stockiness. The KIC formulations in LEFM are applicable for solid materials but have some restrictions on specimen geometry [27]. Similarly, the validity of the KIC formulations and their corresponding specimen geometry restrictions must be assessed before they are employed to compute fracture toughness of lattice materials. The single-edge notched beam in three-point bend shown in Fig. 3 is the easy and common test for measuring KIC of brittle lattices. Hence, Huang and Chiang [28] carried out a numerical examination of the restrictions on specimen geometry of the threepoint bend test, including span S, height H and crack depth ratio a=B. These authors suggest that the fracture toughness measured from a three-point bend test is higher than that measured from a uniaxial test, and that the difference between the two measures is reduced as stockiness increases. This conclusion is inconsistent with the definition of fracture toughness – a material property independent of test geometry. Quintana-Alonso et al. [29] have recently reassessed the threepoint bend problem and show that a K-dominated regime only exists for sufficiently large at = l 2 . At low at = l 2 a crack-insensitive regime prevails.

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Fig. 3 Single-edge notched beam in three-point bending

Fig. 4 Boundary layer analysis: lattice material with a macroscopic crack under K-control. The joint displacements and rotations associated to a K-field are prescribed on the outer boundary

4.2 Boundary Layer Analysis The numerical approach to the fracture toughness evaluation can be improved by the use of continuum fracture mechanics. A finite element problem is formulated for the finite lattice domain surrounding the tip of a macroscopic crack. A semiinfinite crack is introduced by breaking a series of discrete cell walls (see Fig. 4). The displacements ui and rotations of the beam ends are prescribed on the outer boundary according to the asymptotic K-field of a crack in an equivalent homogenous material possessing effective elastic properties. The fracture toughness of the lattice is calculated by relating the stresses and strains generated in the cell walls to the magnitude of the applied K-field. This formulation, referred to as a boundary layer analysis, was first employed by Schmidt and Fleck [30] to investigate crack growth initiation and subsequent propagation for regular and irregular elastic-plastic hexagonal lattices. A seminal work on the fracture behaviour of 2D lattice materials is that of Fleck and Qiu [14]. These authors determined the fracture response of three isotropic lattices: hexagonal, triangular, and Kagome (Fig. 1). They found that the fracture toughness KC of these lattices scales with stockiness t = l according to KC p DD T S l

 d t l

(5)

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I. Quintana-Alonso and N.A. Fleck Table 1 Fracture toughness of various lattice topologies. Values of pre-exponent D and exponent d , for scaling law (5) Mode I Mode II Topology D d D d References Hexagonal 1.20 2 0.54 2 Fleck and Qiu [14] 1.80 1 0.77 1 Chen et al. [21] 0.53 2 – – Gibson and Ashby [1] 0.43 2 – Huang and Chiang [28] Triangular 2.10 1 1.40 1 Fleck and Qiu [14] 4.60 1 1.50 1 Chen et al. [21] Kagome 0.27 1/2 0.15 1/2 Fleck and Qiu [14] Square 0.56 1 0.34 3/2 Romijn and Fleck [36] 0.71 1 0.20 1 Lipperman et al. [44] 1.4 1 0.07 1 Chen et al. [21] Diamond 0.44 1 0.45 1 Romijn and Fleck [36] 0.25 1 0.50 1 Lipperman et al. [44] 1.28 2 – – Huang and Chiang [28] 1.66 2 – – Huang and Gibson [17]

where the exponent d equals 2 for the hexagonal lattice, equals unity for the triangular lattice, and equals 1/2 for the Kagome lattice. For each topology, the coefficient D is slightly less for mode II loading than for mode I loading, see Table 1. It is emphasised that the value of the exponent d has a dominant influence on the magnitude of the fracture toughness. For example, at a relative density of 10%, the fracture toughness of the Kagome lattice is ten times greater than that of the hexagonal lattice (see Fig. 5). The unusually high fracture toughness of the Kagome lattice is attributed to the presence of an elastic zone of bending emanating from the crack tip into a remote stretching field. The Kagome lattice exhibits pronounced crack tip blunting due to the deformation band at the crack tip; this elastic blunting phenomenon reduces the stress levels at the crack tip and thereby increases the fracture toughness. Quintana-Alonso and Fleck [31] found a similar effect for the diamond-celled lattice. A particular feature of this lattice topology is a low resistance to shear along the ˙45ı directions; the elastic bending zones emanate from the crack tip along these directions. It has already been noted that Huang and Lin [13] proposed a mixed-mode fracture criterion for lattice materials, which is a linear combination of KI =KIC and KII =KIIC . Fleck and Qiu [14] follow a more systematic approach by making use of the boundary layer analysis to determine the mixed-mode fracture envelope for the three isotropic lattices. They find that the fracture locus in .KI ; KII / space comprises the inner convex envelope of a series of straight lines, corresponding to a particular failure site (Fig. 6). They conclude that the triangular lattice is the most resistant to mode II loading, and the failure envelope is approximately circular in shape. In contrast, the failure envelope for the hexagonal lattice is the most eccentric in shape, with a relatively low value of mode II toughness.

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Fig. 5 The predicted mode I fracture toughness KIC plotted as a function of relative density N / t = l, for the three isotropic lattices: hexagonal, triangular and Kagome

Fig. 6 The failure envelope for mixed-mode loading. The segments of failure surface correspond to the critical sites of failure shown in the insets

Fleck and Qiu [14] also considered the tensile and shear strengths of centrecracked plates (CCP) made from each of the three isotropic microstructures. The hexagonal and triangular lattices are flaw-sensitive, with the strength adequately predicted by LEFM for cracks spanning more than a few cells. In contrast, the Kagome microstructure is damage tolerant: for cracks shorter than a transition length its strength is independent of crack length but somewhat below the unnotched strength u of the lattice. At crack lengths exceeding the transition value,

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the strength decreases with increasing crack length, in accordance with the LEFM 2 =u2 , in estimate. They showed that this transition crack length is given by KIC agreement with previous studies on fully dense materials [32]. Finally, the authors point out the need to include T -stress effects4 in order to explain the predicted strengths of the centre-cracked plates at sufficiently low stockiness, particularly in the hexagonal lattice.

4.2.1

Extrapolation of 2D Results to 3D Lattices

The boundary layer analysis of the fracture toughness problem was also used in the numerical study of Choi and Sankar [35], who investigated mode I and mode II cracks in a three-dimensional (3D) cubic lattice of side length l and composed of square struts of cross-section t  t. It is of concern that Choi and Sankar [35] found values of the coefficient D in Eq. (5) which varied by an order of magnitude depending upon whether they calibrated the equation by varying the cell-wall thickness t or the cell size l. Recently, Romijn and Fleck [36] have reconsidered the problem of the cubic lattice made from elastic-brittle struts. These authors regard the 3D lattice as a separated stack of 2D square grids each of thickness t. One grid is fastened to the next layer at its joints by out-of-plane bars of length l. This allows them to use toughness calculations for a 2D lattice in order to make predictions for the fracture toughness .3D/ of the 3D cubic lattice. The fracture toughness of the 3D lattice KC is then related .2D/ .3D/ .2D/ to the fracture toughness of the 2D lattice KC by KC D .t = l/ KC , leading to .3D/

KIC

.3D/ KIIC

4.2.2

D 0:56.t= l/2 T S 5=2

D 0:35.t= l/

p

T S

l p

(6) l

Sensitivity of Fracture Toughness to Imperfections

Fracture of lattice materials involves multiple length scales: the crack length at the macroscopic level, the dimensions of the lattice (cell size and strut thickness) at the microstructural level, and even lower length scales such as imperfections at the cell wall level. All the above studies on fracture toughness are for perfect lattices. A detailed treatment of the role of various microstructural imperfections (missing

4 The second term in the series expansion of the mode I crack tip field is the so-called T -stress parallel to the crack plane. The T -stress scales linearly with the remote applied stress, but its magnitude depends upon the specimen configuration. The T -stress vanishes for mode II loading, by a symmetry argument. For conventional, fully dense, elastic-brittle solids the T -stress plays a relatively minor role in influencing the fracture process at the crack tip. In ductile fracture and fatigue, however, the T -stress becomes important at short crack lengths and reveals itself in a dependence of inferred fracture toughness upon specimen geometry, see for example [33, 34].

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struts, misplaced joints, and wavy cell walls) upon the in-plane effective properties is given in the recent publication by Symons and Fleck [37] for three isotropic lattices: hexagonal, triangular and Kagome. The knockdown in fracture toughness due to misplaced joints is numerically explored in a parallel study by Romijn and Fleck [36] for the three isotropic lattices, as well as two orthotropic topologies: square and diamond-celled. Both studies conclude that the imperfection sensitivity of modulus and fracture toughness for the lattices examined can be catalogued in terms of the nodal connectivity of each lattice, consistent with the arguments of Deshpande et al. [38]. A connectivity value of three struts per joint for the hexagonal lattice is sufficiently low for the struts to deform by bending, and the random misplacement of joints has little effect upon the stress state in the lattice. The high connectivity value of six for the triangular lattice causes it to deform by cell-wall stretching. Again, it is insensitive to imperfections: misplaced joints have a negligible effect upon the mechanical properties of the lattice. But the square-celled lattices, like the Kagome, have a transition value of connectivity equal to four struts per joint. The response of these lattices can be bending or stretching dominated, depending on the level of imperfection. For instance, under uniform loading the square lattice deforms by strut stretching, but upon introduction of a defect such as a macroscopic crack the struts deform by a combination of strut stretching and strut bending. Thus, the moduli and fracture toughness of these topologies is highly sensitive to imperfections.

5 Atomic Lattice Models for Crack Dynamics Analytical solutions for crack dynamics in lattices obtained over the last 2 decades are based on the model where point masses are connected by massless bonds [39]. The first works along these lines were published by Slepyan [40] and by Kulakhmetova et al. [41], who considered the dynamic problem of propagation of a rectilinear crack in periodic lattices with rectangular and triangular cells, respectively (Fig. 7). These models of the lattice structure can be considered as related to a regular atomic lattice or as a discrete model of a continuous material with periodic structure. With the latter application in mind, a lattice with material bonds of nonzero density has been investigated. This limiting structure can be interpreted as a simplest model of porous material or a material becoming porous in a vicinity of the

Fig. 7 Atomic lattice models with massless bonds for (a) square and (b) triangular lattices

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growing crack tip. It can also be considered as a model of a fibre-reinforced composite or a fabric. Detailed examination of the relation between joint mass and bond mass is undertaken in Ref. [42] for the square-celled lattice. This approach assumes that the bonds behave like trusses rather than beams, i.e. they transmit axial forces but do not resist bending moments. A comprehensive review of these lattice models for crack dynamics can be found in the recent monograph by Slepyan [39], where the author discusses models, phenomena, methods, and his overall view on the subject. The approach has inspired the use of techniques such as the representative cell method described in the following section. In addition, it has resulted in a wealth of research on crack kinetics in atomic lattices.

6 Representative Cell Method A difficulty in the investigation of fracture properties of lattice materials is the need to introduce a large number of degrees of freedom in order to model the material microstructure with a flaw. To overcome this, Lipperman et al. [43, 44] have recently developed an analytical method that enables to determine the stress state of an unbounded infinite lattice with an embedded finite-length crack. The analysis technique hinges on the combined use of the structural variation method and the representative cell method [45]. While the latter allows for the analysis of periodic structures under arbitrary loads, by means of the discrete Fourier transform, the former analyses modified structures (the cracked lattice) on the basis of the analysis of the uncracked periodic structure. The material is viewed as an assemblage of identical parallelogram cells defined by the two vectors of the lattice translational symmetry. It was noted by Renton [46] that such a representation is always possible. In a first study, Lipperman et al. [43] numerically emulate the way in which cracks may nucleate and propagate from a single imperfection until a long and stable crack path is achieved. For the triangular and Kagome lattices, the sequence of broken beams produces straight linear cracks propagating perpendicular to the loading direction. For the hexagonal lattice, however, the crack kinks to a straight line inclined at 60ı with respect to the loading direction. A similar deviation was reported by Schmidt and Fleck [30]. For the square lattice two cracks running parallel to the loading direction emerge. Indeed, the stress state at the crack tips of hexagonal and square lattices is characterised by significant mode II deformations. This phenomenon is explained by the relatively small resistance of the lattices to shear deformation. In a second paper, Lipperman et al. [44] proceed to investigate the fracture toughness of several lattice topologies: triangular, hexagonal, Kagome, square and diamond-celled. Their results for the triangular lattice were found to be in agreement with the exact analytical solution for a semi-infinite crack derived by Slepyan and Ayzenberg-Stepanenko [47] within the context of the atomic lattice models mentioned above. The values of fracture toughness for the hexagonal and Kagome

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Fig. 8 Macroscopic crack in a porous material with square cells

lattices are also in agreement with those obtained by Fleck and Qiu [14] by the boundary layer analysis. For the diamond-celled lattice, they find that the dependence of fracture toughness on stockiness is close to linear, in agreement with Quintana-Alonso and Fleck [15]. The representative cell method is a convenient tool for investigating periodic lattice materials with flaws. It can be applied not only to isolated straight linear cracks but also to more general crack interaction or flaws that are not straight. Lipperman et al. [48] have recently extended their analysis technique based on the discrete Fourier transform, to investigate the fracture properties of 2D periodic porous material (Fig. 8). They considered materials with periodic voids where the porosity is neither high enough to regard them as lattice materials modelled by systems of beams and plates, nor low enough to model them as an elastic plane with dilute voids. The traditional FE approach for determining the fracture toughness of these materials requires a huge computational effort. The technique presented by Lipperman et al. [48] overcomes this difficulty. They perform a parametric study in order to find the optimal voids shape for maximum fracture toughness at a given porosity. For the specific case of large voids (i.e. high porosity) when the model of a material consisting of beam elements becomes valid the results are found to be in agreement with the FE data of Fleck and Qiu [14] for 2D hexagonal lattices. The method employed by Lipperman et al. [48] can be used to evaluate the fracture toughness of more complicated microstructures. For example, their parametric study of void shape could serve to investigate the effect upon fracture toughness of varying the core angle ! of the diamond-celled lattice (Fig. 8). The analytical and numerical work reviewed thus far provides a useful foundation for subsequent investigations of the fracture properties of lattice materials. The next section highlights the few experimental studies identified in the literature.

7 Experimental Studies on Fracture Toughness To date, experimental studies on brittle lattice materials have focused on determining the strength of intact lattices made from glasses or ceramics. Scheffler and Colombo [49] provide a comprehensive summary. But experimental research into

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the fracture toughness of brittle lattice materials is scarce. Huang and Gibson [17] have determined the fracture toughness of a cordierite .2MgO-2Al2 O3 –5SiO2 / lattice with square cells by testing single-edge notched beams in three-point bending, as shown in Fig. 3. The fracture toughness of the diamond-celled lattice was calculated from the measured failure load Pcr , using the K-calibration given in Ref. [50] for an isotropic continuum. Recently, Quintana-Alonso et al. [29] have performed tests over a much wider range of t = l and found that KIC scales linearly with relative density.

8 Concluding Remarks Since the pioneering work of Ashby [10], who provided the first analytical estimate of the fracture toughness of a planar lattice, only a limited amount of research has been conducted on the fracture properties of lattice materials. Attempts to model the fracture behaviour of lattice materials, making use of generalised continuum theory, finite element analysis, and novel techniques such as the representative cell method, have met with various degrees of success. The study of orthotropic lattices, however, has led to contradictory results in the literature. For example, the correct functional dependence of fracture toughness upon stockiness has been debated. The experimental data available is too limited and inconclusive. All the studies reviewed herein have focused on determining the effective fracture properties of brittle lattices. In practice, lattice materials are commonly loaded in a sandwich panel configuration with stiff and strong face-sheets. The flaw sensitivity of these lattice structures has not yet been explored. In view of the above, the following two directions of research are expected to be fruitful: (i) the fracture toughness of orthotropic lattices, and (ii) the cracksensitivity of a sandwich panel containing a defective lattice core.

References 1. Gibson LJ, Ashby MF (1997) Cellular solids: structure and properties, 2nd edition, Pergamon, Oxford. 2. Christensen RM (2000) Mechanics of cellular and other low-density materials. Int J Solids Struct 37(1–2): 93–104. 3. Wang A-J, McDowell DL (2004) In-plane stiffness and yield strength of periodic metal honeycombs. J Eng Mater Technol 126(2): 137–156. 4. Guo XE, Gibson LJ (1999) Behavior of intact and damaged honeycombs: a finite element study. Int J Mech Sci 41(1): 85–105. 5. Chen C, Lu TJ, Fleck NA (2001) Effect of inclusions and holes on the stiffness and strength of honeycombs. Int J Mech Sci 43(2): 487–504. 6. Silva MJ, Gibson LJ (1997) The effects of non-periodic microstructure and defects on the compressive strength of two-dimensional cellular solids. Int J Mech Sci 39(5): 549–563.

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7. Alburquerque JM, F´atima Vaz M, Fortes MA (1999) Effect of missing walls on the compression behaviour of honeycombs. Scripta Mater 41(2): 167–174. 8. Andrews EW, Gibson LJ (2001) The influence of cracks, notches and holes on the tensile strength of cellular solids. Acta Mater 49(15): 2975–2979. 9. Chen C, Lu TJ, Fleck NA (1999) Effect of imperfections on the yielding of two-dimensional foams. J Mech Phys Solids 47(11): 2235–2272. 10. Ashby MF (1983) The mechanical properties of cellular solids. Metall Trans 14(9): 1755–1769. 11. Maiti SK, Ashby MF, Gibson LJ (1984) Fracture toughness of brittle cellular solids. Scripta Metall 18(3): 213–217. 12. Davidge RW (1979) Mechanical behaviour of ceramics, Cambridge University Press, Cambridge. 13. Huang JS, Lin JY (1996) Mixed-mode fracture of brittle cellular materials. J Mat Sci 31(10): 2647–2652. 14. Fleck NA, Qiu X (2007) The damage tolerance of elastic-brittle, two dimensional isotropic lattices. J Mech Phys Solids 55(3): 562–588. 15. Quintana-Alonso I, Fleck NA (2007) Damage tolerance of an elastic-brittle diamond-celled honeycomb. Scripta Mater 56(8): 693–696. 16. Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18: 293–297. 17. Huang JS, Gibson LJ (1991) Fracture toughness of brittle honeycombs. Acta Metall Mater 39(7): 1617–1626. 18. Huang JS, Chou CY (1999) Survival probability for brittle honeycombs under in-plane biaxial loading. J Mat Sci 34(20): 4945–4954. 19. Gibson LJ, Ashby MF, Zhang J, Triantafillou TC (1989) Failure surfaces for cellular materials under multiaxial loads – I. Modelling. Int J Mech Sci 31(9): 635–663. 20. Quintana-Alonso I, Fleck NA (2008) The damage tolerance of a sandwich panel containing a cracked honeycomb core. J Appl Mech in press. 21. Chen JY, Huang Y, Ortiz M (1998) Fracture analysis of cellular materials: a strain gradient model. J Mech Phys Solids 46(5): 789–828. 22. Banks CB, Sokolowski M (1968) On certain two-dimensional applications of the couple stress theory. Int J Solids Struct 4: 15–29. 23. Bazant ZP, Christensen M (1972) Analogy between micropolar continuum and grid frameworks under initial stress. Int J Solids Struct 8(3): 327–346. 24. Antipov YA, Kolaczkowski ST, Movchan AB, Spence A (2000) Asymptotic analysis for cracks in a catalytic monolith combustor. Int J Solids Struct 37(13): 1899–1930. 25. Antipov YA, Movchan AB, Kolaczkowski ST (2001) Models of fracture of cellular monolith structures. Int J Solids Struct 38(10–13): 1659–1668. 26. Kumar RS, McDowell DL (2004) Generalized continuum modeling of 2-D periodic cellular solids. Int J Solids Struct 41(26): 7399–7422. 27. ASTM E399 – (2006) Standard test method for linear-elastic plane-strain fracture toughness KIC of metallic materials. 28. Huang JS, Chiang MS (1996) Effects of microstructure, specimen and loading geometries on KIC of brittle honeycombs. Eng Fract Mech 54(6): 813–821. 29. Quintana-Alonso I, Mai SP, Fleck NA, Oakes DCH, Twigg MV (2009) Fracture toughness measurements of a cordierite lattice. Submitted to Acta Mater. 30. Schmidt I, Fleck NA (2001) Ductile fracture of two-dimensional cellular structures. Int J Fract 111(4): 327–342. 31. Quintana-Alonso I, Fleck NA (2008) Compressive response of a sandwich plate containing a cracked diamond lattice. Submitted to J Mech Phys Solids. 32. Fleck NA, Kang KJ, Ashby MF (1994) The cyclic properties of engineering materials. Acta Metall Mater 42(2): 365–381. 33. Fleck NA (1986) Finite element analysis of plasticity-induced crack closure under plane strain conditions. Eng Fract Mech 25(4): 441–449. 34. Shercliff HR, Fleck NA (1990) Effect of specimen geometry on fatigue crack growth in plane strain: II – overload response. Fatigue Fract Eng Mater Struct 13(3): 287–296.

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35. Choi S, Sankar B (2005) A micromechanical method to predict the fracture toughness of cellular materials. Int J Solids Struct 42(5–6): 1797–1817. 36. Romijn NE, Fleck NA (2007) The fracture toughness of planar lattices: imperfection sensitivity. J Mech Phys Solids 55(12): 2538–2564. 37. Symons DD, Fleck NA (2008) The imperfection sensitivity of isotropic two-dimensional elastic lattices. J Appl Mech 75: 051011. 38. Deshpande VS, Fleck NA, Ashby MF (2001) Foam topology: bending versus stretching dominated architectures. Acta Mater 49(6): 1035–1040. 39. Slepyan LI (2002) Models and phenomena in fracture mechanics, Springer, Berlin. 40. Slepyan LI (1981) Dynamics of a crack in a lattice. Soviet Phys Doklady 26(5): 538–540. 41. Kulakhmetova SA, Saraikin VA, Slepyan LI (1984) Plane problem of a crack in a lattice. Mech Solids 19(3): 102–108. 42. Slepyan LI (2005) Crack in a material-bond lattice. J Mech Phys Solids 53(6): 1295–1313. 43. Lipperman F, Ryvkin M, Fuchs MB (2007) Nucleation of cracks in two-dimensional periodic cellular materials. Comput Mech 39(2): 127–139. 44. Lipperman F, Ryvkin M, Fuchs MB (2007) Fracture toughness of two-dimensional cellular material with periodic microstructure. Int J Fract 146(4): 279–290. 45. Ryvkin M, Fuchs MB, Lipperman F, Kucherov L (2004) Fracture analysis of materials with periodic microstructure by the representative cell method. Int J Fract 128(1–4): 215–221. 46. Renton J (1966) On the analysis of triangular mesh grillages. Int J Solids Struct 2: 307–318. 47. Slepyan LI, Ayzenberg-Stepanenko MV (2002) Some surprising phenomena in weak-bond fracture of a triangular lattice. J Mech Phys Solids 50(8): 1591–1625. 48. Lipperman F, Ryvkin M, Fuchs MB (2008) Crack arresting low-density porous materials with periodic microstructure. Int J. Eng Sci 46(6): 572–584. 49. Scheffler M, Colombo P (eds.) (2005) Cellular ceramics: structure, manufacturing, properties and applications, Wiley, Weinheim. 50. Strawley JE (1976) Wide range stress intensity factor expressions for ASTM E399 standard fracture toughness specimens. Int J Fract 12(6): 475–476.

Author Index

A Abrate, S., 97, 98, 465–501, 572 Ayorinde, E.O., 441–462, 605–624

H Hassan, N.M., 89–111 Hoo Fatt, M.S., 484, 495, 661–690

B Bahei-El-Din, Y.A., 625–659 Batra, R.C., 89–111 Bazant, Z.P., 305–338, 319, 325, 804 Bing, Q., 365–380

J Jackson, M., 503–540 Johannes, M., 229–277 Just-Agosto, F.A., 407–429

C Carlsson, L.A., 67–85, 626 Cecchini, A., 407–429 Chasiotis, I., 757–778 Chen, H., 381–406 Chiang, F.-P., 779–798 Christensen, R.M., 8, 51–65, 232, 233, 266–269, 804 D Daniel, I.M., 197–227, 309, 406, 606 Davidson, B.D., 365–380 Davies, P., 67–85 Deka, L.J., 541–569 Drzal, L.T., 27–50 Dvorak, G.J., 625–659 F Fleck, N.A, 231, 404, 606, 799–816 G Gardner, N., 503–540 Gdoutos, E.E., 197–227, 232, 233, 305–338, 389, 406 Gibson, R.F., 605–624 Gupta, N., 169–194

K Kardomateas, G.A, 75, 339–363 Kimpara, I., 113–132 M Massab`o, R., 133–168 Miyano, Y., 3–25, 52, 53, 55–57, 61 N Nakada, M., 3–25 Naraghi, M., 757–778 P Palla, L., 661–690 Porfiri, M., 169–194 Q Quintana-Alonso, I., 799–816 R Ravichandran, G., 545, 574, 717–755 Rosakis, A.J., 571–603 S Saito, H., 113–132 Serrano, D., 407–429

817

818

Author Index

Shafiq, B., 407–429 Shivakumar, K., 381–406 Shukla, A., 503–540 Sirivolu, D., 661–690 Soni, S.M., 605–624 Sun, C.T., 99, 100, 470–472, 488, 489, 572, 626, 693–715 Suvorov, A.P., 625–659

V Vaidya, U.K., 473, 483, 541–569 Vautard, F., 27–50 Vinson, J.R., 431–439, 605 Vural, M., 545, 717–755

T Tekalur, S.A., 503–540 Thomsen, O.T., 216, 229–277

X Xu, L., 27–50 Xu, L.R., 571–603

U Uddin, M. F., 693–715

Z Zenkert, D., 235, 270, 279–303, 325, 326, 437, 605, 606

W Wang, E., 503–540