Multi-Scale Modelling of Composite Material Systems: The Art of Predictive Damage Modelling (Woodhead Publishing in Materials)

Multi-scale modelling of composite material systems

i

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Multi-scale modelling of composite material systems The art of predictive damage modelling Edited by

C. Soutis and P. W. R. Beaumont

Woodhead Publishing and Maney Publishing on behalf of The Institute of Materials, Minerals & Mining CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED

Cambridge England iii

Woodhead Publishing Limited and Maney Publishing Limited on behalf of The Institute of Materials, Minerals & Mining Published by Woodhead Publishing Limited, Abington Hall, Abington Cambridge CB1 6AH, England www.woodheadpublishing.com Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2005, Woodhead Publishing Limited and CRC Press LLC © 2005, Woodhead Publishing Limited The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publishers cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from the publishers. The consent of Woodhead Publishing Limited and CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited or CRC Press LLC for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing Limited ISBN-13: Woodhead Publishing Limited ISBN-10: Woodhead Publishing Limited ISBN-13: Woodhead Publishing Limited ISBN-10: CRC Press ISBN-10: 0-8493-3474-8 CRC Press order number: WP3474

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iv

Contents

1

Contributor contact details

xi

Preface

xv

Molecular modelling of composite matrix properties

1

F R JONES, University of Sheffield, UK

1.1 1.2

1.4 1.5 1.6

Introduction Group interaction modelling for the prediction of polymer properties Applying group interaction modelling to polymer matrix composites Conclusions Acknowledgements References

8 31 31 31

2

Interfacial damage modelling of composites

33

1.3

1 3

C GALIOTIS, University of Patras, Greece and A PAIPETIS, Hellenic Naval Academy, Greece

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Introduction: definition of the interface The interface and composite properties Analytical modelling of the shear transfer Interfacial damage modelling Experimental measurement of the stress field at the interface Modelling of the experimentally measured stress transfer Overview and conclusions References

33 35 37 46 48 52 60 62

3

Multi-scale predictive modelling of cracking in laminate composites

65

L N MCCARTNEY, National Physical Laboratory, UK

3.1 3.2

Introduction Predicting undamaged ply properties

65 66 v

vi

Contents

3.3 3.4 3.5 3.6 3.7

Undamaged laminate properties Prediction of ply cracking in laminates Prediction of laminate failure Future trends References

69 73 84 94 96

4

Modelling the strength of fibre-reinforced composites

99

B FIEDLER, Technical University of Hamburg-Harburg, Germany, S OCHIAI, Kyoto University, Japan and K SCHULTE, Technical University of Hamburg-Harburg, Germany

4.1 4.2 4.3

4.5 4.6 4.7 4.8

Introduction Mechanical and thermal response of the polymer matrix Modelling first ply failure by FEA using the partial discretisation approach Stress-strain response and fracture morphology in UD composites Conclusions Future trends Further reading References

110 120 121 121 121

5

Cracking models

124

4.4

99 100 104

P W R BEAUMONT, University of Cambridge, UK and H SEKINE, Tohoku University, Japan

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13

Introduction Empirical and physical modelling Choosing between continuum and physical modelling Combining empirical and physical models Modelling fatigue cracking by delamination Modelling coupled mechanisms in composite cracking Cracking at stress concentrators Bridging cracks: de-bonding’s critical role Modelling stress-corrosion cracking Model implementation Conclusions Acknowledgements References

124 124 128 130 134 138 159 169 179 189 190 191 191

6

Multi-scale modelling of cracking in cross-ply laminates

196

V V SILBERSCHMIDT, Loughborough University, UK

6.1 6.2

Introduction Microstructural randomness of cross-ply laminates

196 197

Contents

vii

6.3 6.4 6.5 6.6 6.7

Damage accumulation Multi-scale modelling Future trends Further information References

205 206 213 213 214

7

Modelling damage in laminate composites

217

M KASHTALYAN, University of Aberdeen, UK and C SOUTIS, The University of Sheffield, UK

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 8

Introduction Stress analysis Predicting stiffness degradation due to intra- and interlaminar damage Predicting onset and growth of intra- and interlaminar damage Conclusions Acknowledgements References Appendices

217 226

240 249 250 250 256

Progressive multi-scale modelling of composite laminates

259

232

C H WANG, Defence Science and Technology Organization, Australia

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Introduction Brief review of failure theories of fibre composites Multi-scale failure theory Phase degradation approach Validation of analysis against experiment Conclusion References Appendices

259 260 261 266 267 273 274 275

9

Predicting fracture of laminated composites

278

I GUZ, University of Aberdeen, UK and C SOUTIS, University of Sheffield, UK

9.1 9.2 9.3 9.4 9.5

Introduction: modelling the compressive response of laminate composites Developing compression models for laminates Identifying critical loads Conclusions References

278 283 287 299 300

viii

Contents

10

Modelling the compressive response behaviour of monolithic and sandwich composite structures

303

C SOUTIS, University of Sheffield, UK, S SPEARING, University of Southampton, UK and P CURTIS, Integrated Systems, UK

10.1 10.2 10.3 10.4 10.5

Introduction Modelling techniques Predicting compressive response Conclusions References

303 304 309 316 317

11

Modelling composite reinforcement by stitching and z-pinning

319

X SUN, H-Y LIU, W YAN, L TONG and Y-W MAI, the University of Sydney, Australia

11.1 11.2 11.3 11.4 11.5 11.6

Introduction Micro-scale models for stitching and z-pinning Assessment of macro-scale delamination toughness of reinforced composites Conclusions Acknowledgements References

319 321 334 350 353 353

12

Finite element modelling of brittle matrix composites

356

V CANNILLO, University of Modena and Reggio Emilia, Italy, and A R BOCCACCINI, Imperial College London, UK

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 13

Introduction Numerical approaches: the finite element method (FEM) Standard FEM analysis for fibre-composite materials Microstructure-based modelling Applications and examples Future developments Acknowledgements References

356 357 359 359 365 369 370 370

Wear modelling of polymer composites

374

K FRIEDRICH, University of Kaiserslautern, Germany, K VÁRADI, Budapest University of Technology and Economics, Hungary, and Z ZHANG, University of Kaiserslautern, Germany

13.1 13.2 13.3

Introduction Rule-of-mixtures-approaches to wear of multi-component materials Wear in relation to other mechanical properties

374 374 382

Contents

ix

13.4 13.5 13.6 13.7

Finite element modelling of composite wear mechanisms Conclusions Acknowledgements References

388 398 398 399

14

Modelling impact damage in composite structural elements

401

A F JOHNSON, German Aerospace Center (DLR), Stuttgart

14.1 14.2 14.3 14.4 14.5 14.6 14.7

Introduction Meso-scale ply damage models Delamination modelling Prediction of impact damage in composite structures Conclusions and future outlook Further information References

401 403 411 415 425 427 428

15

Modelling structural damage using elastic wave-based techniques

430

Z SU and L YE, The University of Sydney, Australia

15.1 15.2 15.3 15.4 15.5 15.6 16

Introduction Fundamentals Models of an active sensor network Lamb wave scattering in defective CF/EP composite laminates Conclusions References Modelling the fatigue behaviour of bonded joints in composite materials

430 433 445 451 465 465 469

M QUARESIMIN, University of Padova, Italy

16.1 16.2 16.3 16.4 16.5 16.6 16.7 Index

Introduction Experimental investigation Finite element analysis of SIF and SERR Fatigue life modelling Conclusions Acknowledgements References

469 471 478 485 489 491 491 495

x

Contributor contact details

(* = main contact)

Chapter 1 Professor F. R. Jones Ceramics and Composites Laboratory Department of Engineering Materials University of Sheffield Mappin Street Sheffield S1 3JD UK E-mail: [email protected]

Chapter 2 Professor C. Galiotis* Institute of Chemical Engineering and High Temperature Chemical Processes Foundation for Research and Technology – Hellas Stadiou Str. Platani PO Box 1414 GR-26504 Patras Greece E-mail: [email protected]

Dr A. Paipetis Hellenic Naval Academy Marine Materials Technology Department Hatzikyriakou Ave., Pireas GR 185 39 Greece Tel: +30210 4581330 Fax: +30210 4181768 E-mail: [email protected]

Chapter 3 Dr L. N. McCartney Division of Engineering and Process Control National Physical Laboratory Hampton Road Teddington TW11 0LW UK E-mail: [email protected]

Chapter 4 Dr B. Fiedler and Professor K. Schulte* Polymer and Composites Section Technical University HamburgHarburg Denickestr. 15, 21073 Hamburg Germany E-mail: [email protected] xi

xii

Contributor contact details

Professor S. Ochiai International Innovation Center Kyoto University Sakyo-ku, Kyoto 606-8501 Japan

Chapter 5 Dr Peter W. R. Beaumont* Department of Engineering University of Cambridge Cambridge CB2 1PZ, UK Professor Hideki Sekine Department of Aeronautics and Space Engineering Tohoku University Aoba-yama 01 Aoba-ku, Sendai 980-8579 Japan

Professor C. Soutis Aerospace Engineering The University of Sheffield Sir Frederick Mappin Building Mappin Street Sheffield S1 3JD UK E-mail: [email protected]

Chapter 8 Dr C. H. Wang Air Vehicles Division Defence Science and Technology Organisation 506 Lorimer Street Fisherman’s Bend VIC 3207 Australia E-mail: [email protected]

Chapter 6 Professor Vadim V. Silberschmidt Wolfson School of Mechanical and Manufacturing Engineering Loughborough University Loughborough Leicestershire LE11 3TU UK E-mail: [email protected]

Chapter 9 Dr I. A. Guz* School of Engineering and Physical Sciences King’s College Fraser Noble Building University of Aberdeen Aberdeen AB24 3UE UK

Chapter 7

E-mail: [email protected]

Dr Maria Kashtalyan* School of Engineering and Physical Sciences University of Aberdeen Fraser Noble Building Aberdeen AB24 3UE UK

Professor C. Soutis Aerospace Engineering The University of Sheffield Sir Frederick Mappin Building Mappin Street Sheffield S1 3JD UK

E-mail: [email protected]

E-mail: [email protected]

Contributor contact details

Chapter 10

Chapter 12

Professor C. Soutis* Aerospace Engineering The University of Sheffield Sir Frederick Mappin Building Mappin Street Sheffield S1 3JD UK

Dr V. Cannillo Dipartimento di Ingegneria dei Materiali e dell’Ambiente University of Modena and Reggio Emilia Via Vignolese 905 41100 Modena Italy

E-mail: [email protected] Professor S. M. Spearing Materials Research Group University of Southampton Southampton SO17 1BJ UK Dr P. T. Curtis DSTL Integrated Systems UK

xiii

Tel: +39 059 2056240 Fax: + 39 059 2056243 E-mail: valeria@ unimore.it Dr A. R. Boccaccini Department of Materials Imperial College London Prince Consort Road London SW7 2BP UK E-mail: [email protected]

Chapter 11 Dr Xiannian Sun, Dr Hong-Yuan Liu, Dr Wenyi Yan, Dr Liyong Tong and Professor Yiu-Wing Mai* School of Aerospace, Mechanical and Mechatronic Engineering (J07) The University of Sydney Sydney NSW 2006 Australia E-mail: [email protected]; [email protected]

Chapter 13 Dr Klaus Friedrich* and Dr Zhong Zhang Institute for Composite Materials (IVW GmbH) University of Kaiserslautern Erwin-Schrödinger Str. Geb. 58 D-67663 Kaiserslautern Germany E-mail: [email protected] [email protected] Dr Károly Váradi Institute of Machine Design Budapest University of Technology and Economics Müegyetem rkp. 3 H-1111 Budapest Hungary

xiv

Contributor contact details

Chapter 14

Chapter 16

Dr Alastair F. Johnson German Aerospace Center (DLR) Institute of Structures and Design Pfaffenwaldring 38-40 70569 Stuttgart Germany

Professor Marino Quaresimin Department of Management and Engineering University of Padova Stradella S.Nicola 3 36100 Vicenza Italy

E-mail: [email protected] E-mail: [email protected]

Chapter 15 Dr Zhongqing Su and Dr Lin Ye* Laboratory of Smart Materials and Structures (LSMS) Centre for Advanced Materials Technology (CAMT) School of Aerospace, Mechanical and Mechatronic Engineering (AMME) The University of Sydney NSW 2006 Australia E-mail: [email protected]

Preface

Structural fibrous composite materials have found applications in aircraft from the first flight of the Wright Brothers’ Flyer 1, in North Carolina on 17 December 1903, to the plethora of uses now enjoyed on both military and civil aircraft, in addition to more exotic applications on unmanned aerial vehicles (UAVs), space launchers and satellites. Their growing use in ground transportation systems, in sporting goods, and power generating systems has arisen from their high specific strength and stiffness, and ease (low energy) fabrication processes when compared to more conventional materials, and our ability to shape and tailor their structure to produce more aerodynamically efficient structural configurations. Polymers containing long strong fibres of carbon, glass or Kevlar, (and combinations of these fibres in GLARE, a fibre/metal laminate), will in a year or two contribute more than 50% of the structural mass of the Airbus A350 and Boeing’s 787. However, affordability is the key to survival in aerospace manufacturing, whether civil or military. Consequently, present effort is devoted to analysis and computational simulation of the manufacturing and assembly process as well as simulation of the performance of the structure, since they are all intimately connected. Prediction of material behaviour and structural performance through modelling is the way forward. Predictive modelling based on the physics of composite material behaviour is wealth generating; by guiding material system selection and process choices, by cutting down on experimentation and associated high costs; and by speeding up the time frame from the research stage to the market place. It is an important tool for industry with its exact role depending on the industrial sector involved (aerospace, automotive, marine, etc.). No longer is computer hardware or software a barrier to the deployment of predictive modelling. Instead, with experience and the building of confidence we can identify the right approach and application of appropriate calculations at the appropriate level of sophistication to solve the correctly identified problem. Optimum material microstructures (and nanostructures) can be forecast and designed rather than found by trial and error, (with the possibility of calamity), whilst maximising structural high performance and sustainable safe life. xv

xvi

Preface

Modelling and optimisation can take many forms, covering a combination of the operating variables, depending on the technological need being addressed. The framework upon which the modelling processes can be placed, and the connections and continuity between them, is illustrated in Fig. P.1. We observe the hierarchy of structural scales from the micron (or less) to the metre (or greater) level of size; from the single fibre to the single ply, and from the laminate coupon to the final structure. Also, the discrete methods of analysis ranging from micro-mechanical (mechanism) modelling to the continuum levels of mathematical prediction of the complete design process (system performance). Unit

Scale

Design process/mechanics

4

10 m System performance (systems engineering)

Complete structure (as-designed structure) 102m Component (e.g. aircraft wing)

Structural interactions (structural mechanics) 100m Element testing (computational mechanics)

Structural element (e.g. stiffened panel) 10–2m

Laminate properties (fracture/damage mechanics)

Coupon (ply thickness, fibre orientation) 10–4m Single ply micro-structure (fibre architecture, layup, etc)

Constituent properties (micro-mechanics) 10–6m Statistical analysis (weakest link statistics, etc)

Constituent behaviour (fibre, whisker, particle, matrix) 10–8m

P.1. Hierarchy of structural scales ranging from the micron to the metre (and greater) level of size, from the single fibre to the fully assembled structure, and discrete methods of analysis in design ranging from micromechanics to the higher structural levels of modelling.

Problem posing and solving are essential components of modelling studies, adding value to our current understanding of the application of predictive modelling of composite material behaviour. The style and level of modelling depends on the problem, and they must all have the right degree of sophistication for the task in hand. Composite materials systems generally exhibit a range of behaviour on different length and different time scales, and any multiscale problems have to be addressed by appropriate methods. Consider, for example, the exploitation of physical modelling: the models suggest forms for constitutive equations, and for the significant groupings of

Preface

xvii

the variables that enter them. Empirical or semi-empirical methods can then be used to establish the precise functional relations between these groups. The end result is a constitutive equation that contains the predictive power of micromechanical modelling with the precision of ordinary curve-fitting of experimental data. In parallel, the broad rules governing material properties can be exploited by creating and checking the database of material properties that enter the equations. This is to follow the path of physical model-informed empiricism or the method of extended empiricism. These chapter contributions are written by leading experts on modelling materials behaviour and structural performance. They focus on the fundamental understanding of the way composite materials fail and address the question of how to model material behaviour from molecular scale to the macroscopic. And they take the next step: how to design the microstructural features of the material, (by tailoring the fibre/matrix interface, for instance), in order to optimise the properties, which can then be built into rules or equations based on the physics of microstructural change over time. The end result is to be able to predict the functional behaviour of the final structure for a specific application that depends critically on strength, toughness, wear resistance, fatigue, or impact damage tolerance, and so forth). To accomplish this requires experience or ‘know how’ of the variety of stress and temperature-related behavioural phenomena of composite materials. In addition, a detailed knowledge is required of those damaging mechanisms in the form of fibre/ matrix debonding, matrix deformation and cracking, splitting and delamination, fibre breakage and fibre microbuckling, and so on that do occur in the structure under static and fatigue loading and environmental attack. All of these things ultimately affect the performance and structural reliability and endurance of the component or structure. Many (perhaps all) of these factors are considered in the modelling techniques (analytical, numerical or theoretical) throughout this book. The advent of powerful computers and software that can be purchased at reasonable cost means that many of these models, that would be cumbersome for design engineers to use, could be implemented as user-friendly computer applications or integrated within commercial finite element design systems. Physical-based damage and failure models can be incorporated into empirical or continuum methods of modelling that would lead to more efficient and reliable experimental programs and the safe design of composite structures. In the development of this book, we have had the enormous pleasure of working and collaborating with talented colleagues from around the world and we would like to thank each of them for their huge contribution to this synergistic effort.

xviii

Preface

The careful text books measure (Let all who build beware!) The load, the shock, the pressure Material can bear. So, when the buckled girder Lets down the griding span, The blame of loss, or murder, Is laid upon the man. Not on the stuff-the Man! Rudyard Kipling’s ‘Hymn of Breaking Strain’ C. Soutis and P.W.R. Beaumont

1 Molecular modelling of composite matrix properties F R J O N E S, University of Sheffield, UK

1.1

Introduction

Fibre composite materials consist of an array of fibres mainly in a polymeric thermosetting matrix. Without support from the resin, the fibres cannot be utilised to their full. In the process of making a composite material, uncured resin is impregnated into the fibres which are maintained in a free defined orientation. After manufacture into a structure, the resin is cured usually in a high temperature stage. The resin used for composite materials depends on the application of the structure that is being manufactured. In fibre reinforced plastics it is common to use unsaturated polyester resins for applications less demanding in mechanical and thermal environments. In advanced composite materials based on high performance fibres such as carbon, most matrices rely heavily on the use of epoxy resins. For high temperature use other thermosetting resins are in use to a limited degree. These include the cyanate esters and bismaleimides. In general, the high performance matrices bring cure issues with them. As a result the epoxy resins in various guises are used extensively for the manufacture of carbon fibre and glass fibre reinforced plastics. These epoxy resins consist of blends of differing functional epoxies with rubber or thermoplastic toughening agents. The thermoplastic modifier also acts as a flow control additive. A variety of hardeners or curing agents are available. With epoxy resins the curing reaction is often a copolymerisation with the curing agent or ring opening initiating polymerisation. This means that the chemical structure of the cured resin varies with the chosen blend of epoxy resins, the related curing agent(s) and the thermoplastic or other toughening agent. As a result the structure of the cured resin can vary significantly from material to material. Furthermore, the detailed curing mechanisms are quite complicated, leading to uncertainty over the actual structure of the resin. This means that modelling procedures to determine the properties of the resin need to take into account a number of factors which are rather uncertain. Any modelling requirement for a thermosetting resin used in this application is determined by the complexity 1

2

Multi-scale modelling of composite material systems

of the chemistry involved. Models which use simplified average structures are therefore preferred if some predictive capability can be achieved. The role of the matrix in determining the performance of a continuous fibre composite is a function of a number of factors depending on the alignment of the fibres. For example, with the fibres at an angle to the load of 10–45∞ the shear properties of the matrix will dominate the failure process of the material. With 0∞ materials, the failure of the composite is dominated by the fibre-breaks, so that the role of the matrix is less clear. However, the nature of the reloading of a broken fibre determines the durability of the composite material under load. It is also a major step in the fracture mechanism. Thus the stress transfer through the matrix back to the fibre at the fibre-break is a very important aspect which determines the strength of a composite material. As a result, the resin needs to withstand high shear stresses. These stresses are concentrated at the fibre interface. Therefore, in a real composite, the interface and/or interphasal region plays a major role. It is normally considered that a small degree of debonding from a random fibre-break in a 0∞ composite has the benefit of reducing the stress concentration which exists on adjacent fibres in the same plane as the initial breaks; thereby reducing the probability of a fracture of the adjacent fibres. Matrix yield at the interface will have the same effect. Therefore with strongly bonded fibres, the yield strength of a resin is an important matrix property which has an impact on the reliability of the structure, by reducing the probability of failure of in-plane adjacent fibres. In this composite configuration, therefore, the yield strength of the matrix is a property worthy of consideration for prediction. For angle ply laminates the micromechanics of failure of each individual lamina is constrained so that the composite material would normally fail by damage accumulation mechanisms. In order to achieve quasi isotropic properties it is normal to use ± 45∞, 90∞ and 0∞ plies in dispersed array to form a structural laminate. Therefore, another factor determining the mechanical properties of a laminate will be the transverse fracture of a 90∞ ply. The transfer strength of an isolated 90∞ ply will be determined by the failure strain of the matrix as well as the strength of the interfacial bond between the fibres and the resin. Apart from the failure strain of the resin matrix the other factor which determines the transfer cracking strain is the residual thermal stress that is inbuilt during the manufacturing process. The residual strain which is present in an angle ply laminate results from the constrained shrinkage of the resin during cure and on cooling from the post-cure temperature. Much of the shrinkage due to cure does not contribute to the residual strain because it occurs before a polymer glass is formed. It is therefore the linear thermal expansion coefficient of the resin and the cooling interval which is responsible for the magnitude of the residual thermal strain and in need of predictive modelling. The other factor which contributes is the strain free temperature. The strain free temperature is the temperature at which the

Molecular modelling of composite matrix properties

3

thermal strains are induced into the material as a result of the solidification of the polymer glass. The appropriate thermal mechanical property of the matrix which determines the strain free temperature is the glass transition temperature. The glass transition temperature is therefore a major property in need of modelling. The glass transition temperature also determines the service temperature of the structure. It is also well known that during the service life, polymeric matrix composites will absorb moisture from the environment. This will lead to a reduction in the glass transition temperature and the strain free temperature. The effect of moisture on these properties is clearly an important issue for consideration. This chapter explores the possibility of predicting relevant matrix properties which can be incorporated into predictive codes for fibre laminates. The most important matrix properties worthy of consideration are: 1. 2. 3. 4.

The The The The

glass transition temperature (Tg) stress or strain free temperature (T1) linear expansion coefficient (am) yield stress of matrix (ty).

Where possible the prediction of a stress strain curve for the resin is ideally needed for incorporation in finite element (FE) modelling. Group interaction modelling (GIM) is a method of predicting the properties of polymers which is derived from classical thermodynamics and atomistic modelling principles and is a generalised type of cell model. In these models, the van der Waals interaction between main groups of elements within a cell consisting of a central unit and six surrounding units in hexagonal structure, forms the basis of the analysis.

1.2

Group interaction modelling for the prediction of polymer properties

Porter [1] has developed the group interaction approach to the prediction of polymer properties. This is a simplified approach which can be applied to a range of polymeric structures. It has been recently extended from linear chains to three-dimensional network polymers [2]. Traditional modelling techniques study either the repeat or ‘mer’ units within the polymer chain or the complete macromolecule. In the latter, the statistical arrangement of the atoms is utilised whereas in the former the interactions between groups of elements are employed. Examples of the differing models described in the literature are: molecular mechanics [3, 4], Monte Carlo methods [5], rotational isomeric state theory [6, 7], group contributions of van Krevelen [8], and the connectivity indices of Bicerano [9]. Group interaction modelling (GIM) relies on additivity principles of the properties of a group of elements and therefore lends itself to the prediction of the properties of the complex resin networks used in composite materials.

4

Multi-scale modelling of composite material systems

Group interaction modelling (GIM) is an example of a generalised cell model which employs five molecular parameters which determine the thermodynamic properties of a polymer. [10, 11] 1. 2. 3. 4. 5.

Strength of the intermolecular interaction potential Range of the interaction potential Number of point centres per molecule Number of intermolecular contacts Number of external degrees of freedom per molecule.

The strength of GIM is in the simple manner in which these aspects are quantified. GIM was developed with the aim of calculating many relevant engineering properties of polymers. These properties arise from the exchange of energy between the material and its environment, which occurs at the molecular level. For polymers, it is the intermolecular bonding between segments in the chain and the conformational energy associated with skeletal rotations, which determine their viscoelastic behaviour. Porter [1] has suggested that these energies can be used to reduce the variables to a common reference frame using the following tools: ∑ a potential function which relates to the separation scale between two adjacent but non-chemically bonded molecules to the total energy of the intermolecular interaction ∑ a thermodynamic balance of the differing energy contributions to the total for an assembly of molecular units. These two alternatives are combined to produce an effective equation of state for polymers which represents a potential functional for the characteristic interaction between adjacent mer units within the cell. The three most important energy components which can be calculated from first principles using quantum mechanics are: 1. Cohesive energy from the polarisability and dipole moments within the molecular structure 2. Thermal energy from vibrational frequencies within the groups of atoms 3. Configurational energy from chemical bond deformations. Quantum mechanics [12] and in particular ab initio calculations are highly computer intensive which limits the number of atoms which can be included in the molecule under study. Therefore the role of GIM is to use a site interaction model which allows much larger groups of fused atoms to be studied rather than the detailed atomistic level calculations. To apply the group interaction model it is necessary to follow this procedure: 1. Identify a characteristic structural group 2. Assign the group parameters 3. Calculate input energies

Molecular modelling of composite matrix properties

5

4. Combine the input energies into a potential function 5. Solve to give an equation of state 6. Calculate the physical properties.

1.2.1

Geometry of the model

An hexagonal cell is chosen which consists of a central polymer chain surrounded by six equidistant polymer chains. The energy of interaction between the chains is considered to be mainly due to near neighbours and has a value of f per interaction. The total interaction energy in a model hexagonal group is 3f because the six interactions occur between 2 ‘mer’ units.

1.2.2

The potential function

The interaction energy, f, can be described as a sum of the following contributions: f = – fo + Hc + HT + HM

1.1

where – fo is the depth of the potential energy-well at a separation distance ro which is the equilibrium position of a pair of interacting molecules at absolute zero. fo is therefore the energy of interaction for the molecular conformation of lowest energy. Hc is the configuration energy. HT is the thermal energy. HM is the mechanical energy. f can also be estimated using a Lennard-Jones potential which describes the decrease in energy resulting from the attraction forces as the segments are brought together. At a separation distance of ro, the segments tend to repel each other giving rise to an increase in energy. È r 12 r 6˘ f = fo Í Ê o ˆ – 2 Ê 0 ˆ ˙ r Ë r ¯ ˚ ÎË ¯

1.2.3

1.2

Cohesive energy

fo is the energy which holds the molecules together and against which the mechanical and thermal energies act. fo can be related to the cohesive energy (Ecoh) which can be estimated from measurements of solubility parameter (do). Thus

do =

E coh Vo

1.3

where Vo is the molar volume of mer units. From the hexagonal geometry, the total interaction energy is 3fo

6

Multi-scale modelling of composite material systems

do =

3 fo 4Vmo

1.4

where Vmo is the volume of one mer unit. By comparison and scaling using the Avogadro number, na we get 3f n 1.5 E coh = o a 4 This provides an expression for the potential energy well of the interaction function in terms of the cohesive energy which is a bulk material property. Cohesive energy is an advantageous input parameter because it has been considered extensively [8, 9]. Furthermore, to a first approximation, it is a molar additive property for segmental groups so that values can be obtained from group contribution methods. Table 1.1 gives a typical example of a group contribution table with values of Ecoh and other parameters. This illustrates the principles behind the method.

1.2.4

The number of skeletal modes of vibration (N)

The other parameter which influences the thermal energy and the interaction is the number of skeletal modes of vibration or degrees of freedom which exist in the structure. In GIM, the number of active skeletal vibrations has to be identified carefully and may not equal the total number because some do not contribute to the interactions. Porter [1] has defined N as the number of skeletal modes which are active in dictating the glass transition temperature (Tg). As a result the Tg can be used as an empirical tool for the estimation of N. N=

0.0513 E coh Tg – 0.224 q 1

1.6

where q1 is The Debye reference temperature. Typical values of N are given in Table 1.1. Typical values of q1 are given in Table 1.2 where the choice of a given value 550 K is shown to be justified.

1.2.5

Model parameters using group contribution

GIM uses the parameters shown in Table 1.3 for the predictions. The relevant variables can be estimated using the group contribution approach, where the parameter for the segments or groups of the polymer can be summed to give a value for a characteristic group of elements which make up a mer unit of the polymer. The group contribution method is described in detail elsewhere [8] and some typical values are given in Table 1.1 The four important parameters M, Vw, Ecoh and N are all molecularly additive, where M is the molecular weight of a mer unit. The choice of

Molecular modelling of composite matrix properties

7

Table 1.1 Typical group contributions to group interaction model parameters [1] Group: Backbone

–CHn– –O– –CO– –CO–O–(ester) –O–CO–O– –CH (OH)– –CO–NH–(amide) –S–

M

Vw (cc/mol)

Ecoh (J/mol)

N

14 16 28 44 60 30 43 32

10.23 5 11.7 17 22 14.8 19 10.8

4,500 6,300 17,500 20,000 19,500 (17,500) 40,000 8,800

2 2 2 4 6 (4) 4 2

76

43.3

25,000

3 or 5

76

43.3

25,000

4 or 6

47,000

(6)

126

(65)

CH3 90

54.5

29,500

4

104

65.5

34,000

4

63,500

8

CH3

CH3 CH3 C

194

119

CH3

values for Ecoh and N have been discussed above. Vw is the van der Waals volume of a mer unit. The group contribution values of van Krevelen [8] are mainly used. The values can be validated by solving the potential function at a temperature for which an experimental value of specific volume exists such as Tg. For polymer glasses Vg = 1.55 Vw

1.7

where Vg is the specific volume of the glassy polymer at Tg. The length of the mer unit (L) in the chain axis, which is required to define the volume of

8

Multi-scale modelling of composite material systems Table 1.2 Experimentally determined values of the Debye temperature q1 [1, 3] Polymer

q1 (K)

Poly(ethylene) Poly(1,4-butadiene) Poly(oxyoctamethylene) Poly(undecanolactone) Nylon-6 Nylon-6,6 Nylon-6,10 Nylon-6,12 Poly(p-phenylene) Poly(oxy-1,4-phenylene) Poly(p-xylylene) Poly(oxy-2,6-dimethyl-1,4-phenylene) Poly(ethylene terephthalate) Poly(butylene terephthalate) Poly(bisphenol-A carbonate)

519 599 480 528 520 450 543 470 544 555 562 564 586 542 569

Table 1.3 Parameters required for group interaction modelling [1] Parameter (units)

Definition

M (g/mol) Vw (cc/mol or m3/mer unit) Ecoh (J/mol) N (mer unit–1)

Molecular weight of mer unit van der Waals volume of mer unit Cohesive energy Skeletal modes of vibration of polymer or degrees of freedom Reference temperature of skeletal modes Length of a mer unit in the chain axis

q1 (K) L (m or Å)

mer units, is generally taken as that of the fully extended chain conformation. This can be calculated from fundamental principles, often as a computer code.

1.3

Applying group interaction modelling to polymer matrix composites

Much of the preceding discussion is directed towards linear polymers. However matrices for composites are usually based on 3D network polymers where the structure combines elements of the monomers and hardeners and/or curing agents. This section demonstrates how GIM can be used to predict relevant matrix properties which can be included in composite mechanics codes and provide understanding of the benefits of optimising the matrix and/or interphase regions in a high performance fibre composite.

Molecular modelling of composite matrix properties

1.3.1

9

Application to service temperature of composite material

Tan d

Storage modulus (log E)

The maximum temperature is determined by the onset of the glass transition. This can be experimentally measured from the reduction in storage modulus (E¢) in a dynamic mechanical thermal analysis experiment (DMTA). This temperature is referred to TgE¢. The traditional definition of Tg is the temperature at which Tan d is maximised. This arises because at the transition the properties of the polymer change from ordinary elastic (glassy) to high elastic (rubbery). Since the principal mechanism of deformation changes from the extension of intermolecular interaction forces to conformational change through rotations about the skeletal bonds, the time-dependence of the latter enables mechanical energy to be converted into heat. This damping behaviour means that the loss modulus and Tan d are maximised at Tg. A typical DMTA spectrum of a resin used in composite fabrication is given in Fig. 1.1. The major problem with polymer matrix composites is the absorption of moisture from the environment which leads to a reduction in Tg and hence TgE¢. Fortunately, the diffusion constants for water into matrix resins are 106 lower than thermal diffusion so that the moisture content of structural composites at typical relative humidities (RH) may not normally reach equilibrium over a lifetime of ten years or more. However, in reality, structures such as aircraft components will experience thermal excursions, sometimes to temperatures as high as 160 ∞C [15]. As with all kinetic processes the rate of diffusion will increase in line with an Arrhenius law, thereby ensuring more rapid equilibration.

Temperature

Temperature

1.1 Schematic of the effect of moisture absorption on the thermomechanical response (tan d and E¢) of a matrix resin (a) dry/as cured (…) (b) after isothermal conditioning in 96% RH at 50 ∞C (–) (c) after isothermal conditioning in 96% RH at 50 ∞C with thermal spikes to an elevated temperature (---) [14].

10

Multi-scale modelling of composite material systems

1.3.2

Group interaction modelling of glass transition temperature of a linear polymer

Equation 1.1 can be solved by including contributions for HC, Hm and HT. At the glass transition temperature, the interaction energy f can be defined fg and is related to fo according to eqn 1.8. fg = 0.787 fo

1.8

Similarly HC + HT = 0.213 fo

1.9

Hm represents the energy stored in the polymer unit and is given by H m = N kDTm 3

1.10

where k is the Boltzmann constant and DTm is a hypothetical temperature increment. HT can be given by the Tarasov approximation of the Debye theory for a one-dimensional chain oscillation between two mer units [16]. Porter [1] has provided a simplified form of the Tarasov approximation by considering only the skeletal vibrations. 6.7 ˆ ˘ q È HT (at Tg) = Hg = N k Í Tg – 1 tan –1 Ê T 3Î 6.7 Ë q 1 g ¯ ˙˚

1.11

Equation 1.11 can be solved by incorporating eqn 1.9 and HC = 0.107 fo. From the relationship between Ecoh and fo in eqn 1.5 a predictive equation for Tg can be obtained. E coh 1.12 N Figure 1.2 shows the correlation of predicted values of Tg for a whole range of linear polymers. Representative examples are given in Table 1.4 which validates the GIM approach.

Tg = 0.224 q1 + 0.0513

1.3.3

Group interaction modelling of glass transition temperature of a network polymer matrix resin

The above analysis has been applied to a crosslinked epoxy resin by Gumen et al. [2]. They chose a system containing two hardeners and two epoxy resins. To solve eqn 1.12 for a cross-linked resin, we need precise values of N and Ecoh. The additive principles have been applied to a model network. N could be obtained by taking into account the number of degree of freedom lost on formation of a crosslink. Equation 1.12 has accordingly been modified;

Molecular modelling of composite matrix properties

11

800

Observed Tg (K)

600

400

200

0

0

200

400 Model Tg (K)

600

800

1.2 The comparison of GIM predictions of the Tg for a range of linear polymers (–) with experimental data (o). [1]. Table 1.4 Model parameters and predictions for linear polymers using group interaction [1] q1 (K)

GIM

Expt.

14

550

599

600

130000

16

550

540

545

146.7

132000

24

550

405

403

302

171.3

150000

31

550

371

388

Poly(m-xylylene adipamide)

246

138.6

136500

28

550

392

363

KAPTONTM

182

190

170300

16

550

669

658

LARCTM

394

197

181500

18

550

640

(538)

Poly(pyromellitimide)

394

197

181500

16

550

705

685

Polymer: aromatic amides

M

Vw (cc/mol)

Ecoh (J/mol)

N

Poly(p-phenylene terephthalamide) KEVLARTM

238

124.6

130000

Poly(m-phenylene isophthalamide) NOMEXTM

238

124.6

Poly(hexamethylene isophthalamide)

246

Poly(p-xylylene sebacamide)

Tg (K)

Tg = 0.224q1 + 0.0513 Ê E coh mon + x st H1 E coh H1 + x st H2 E coh H2 – DE et – w ˆ ¥ Á ˜ (N mon – fN L ) + x st H1 N H1 + x st H2 N H2 Ë ¯

1.13

where xst is the stoichiometric coefficient. The subscripts H1 and H2 refer to the two hardeners. NH1 and NH2 are the average number of degrees of freedom per single mer unit of hardeners H1 and H2. NL is the degree of freedom lost

12

Multi-scale modelling of composite material systems

by each epoxy group on curing. w is cohesive energy conversion coefficient which describes the cohesive energy per epoxy group lost during the formation of a crosslink which is given by eqn 1.14: W = 33000 mcr (C – 0.5)

1.14

where Mcr is the number of monomer crosslinks per single average mer unit (C – 0.5) is the fraction of cure above the critical level of 0.5 which leads to a non-linear chain. The etherification reactions which occur in DDS cured systems [17, 18] were taken into account by including the change in cohesive energy on reaction of an hydroxyl group with an epoxy group. DEet = fmon Cet (E coh OH – E coh et )

1.15

Prediction of Tg for a commercial epoxy resin The Hexcel 924 epoxy resin system was chosen for this study because its approximate structure could be ascertained from safety data sheets. The base epoxy resin was believed to be a 50 mol% blend of tetraglydicyldiaminodiphenyl methane (TGDDM or MY720) in its commercial form (MY721) and the triglycidyl aminophenol (MY0520) which are common monomers in many epoxy resin matrix sytems. The hardeners were 1, 4 diaminodiphenylsulphone (DDS) and dicyandiamide (DICY). Model resins based on single component combinations were also prepared. Figure 1.3 gives the average structure of the cured 924 resin employed in the calculations. Table 1.5 gives the experimental data for the measurement of the degree of cure using differential scanning calorimetry (DSC) and the experimental values Tg from DMTA measurements. Table 1.6 gives the degrees of freedom (N) while Table 1.7 gives the cohesive energy parameters used in the calculations. Initial calculations using eqn 1.13 were found to be in error by 20–30∞ compared to the experimental values. St John and George [19] and GrenierLoustalot et al. [20] reported that MY721 was an impure version of MY720 containing a number of synthesis by-products. The structures of these impurities are given in Fig. 1.4. It is clear that TGDDM is the main component and the others either arise from incomplete substitution or intra cyclisation reactions. St John and George [19] have shown that the concentrations of impurities is 21% by weight. Equation 1.13 neglects the influence of monomer cyclisation reactions and the reactions of non-epoxy groups and was therefore expanded to include the effect of impurities. Tg = 0.224q1 + 0.0513 E coh mon + x st H1 E coh H1 – DE et – w È ˘ ¥ Í ˙ (N – f N ) + x N – w N – N W L st H1 H1 C C ˚ im R Î mon

1.16

Molecular modelling of composite matrix properties

13

MY0510 OH HO

DICY

CH

N

H

CH2

C

N

NH

CH2

O

N

CH2 CH

CH2

CH2 CH

CH2

OH C

N

OH

OH

CH2 CH

CH2

CH2 CH

CH2

N

CH2

N

CH2 CH

CH2

CH2 CH

CH2

OH

NH

OH

O

S

DDS

MY720 O

N

1.3 Approximate chemical structure after a cured epoxy from a blend of two resins (trifunctional, MY0510 and tetrafunctional, MY720) cured with two hardeners (DICY and DDS) used for GIM. [2]. Table 1.5 Experimental degrees of cure (DSC) and Tg (DMTA) of component experimental resins compared to a commercial system [2] Epoxy

Hardener

Cure (%)

Tg (∞C)

MY0510 MY0510 MY721 MY721 924 System

DDS DICY DDS DICY DICY/DDS

95 82 87 85 92

285 218 288 258 234

where NR is number of degrees of freedom lost by non-epoxy group reacting, NC is the degree of cyclisation in the monomers which was taken to be 7%. wim and wc are weight fractions of impurities and the epoxy groups consumed by monomer cyclisation reactions. Table 1.8 shows that the GIM calculations closely predicted the experimental values of the model resins as well as the commercial 924. Gumen and Jones [21] have also investigated the role of these impurities using eqn 1.16. With 21% impurities in TGDDM they were able to show which of the components in Fig. 1.4 was the dominant impurity. Impurities TGDDM-3 and TGDDM5 have uncertain reactability of the pendant hydroxyl groups during cure,

14

Multi-scale modelling of composite material systems

Table 1.6 Calculated values of degrees of freedom (GIM) of base epoxy resins and hardeners with respect to their extent of reaction [2] Chemical name

Chemical formula

Degrees of freedom (N) Unreacted Single Two cross-links Nun cross-link tetrafunctional Trifunctional Nf

Diamino diphenyl sulphone H 2 N (DDS) Dicyan Diamide (DICY)

O S

NH2

18

9

6

9

4

3

46

28

22

34

14

–

O

H 2N C N C N NH2

O TetrafuncCH2 CH tional CH2 glycidyl N amine TGDDM CH2 (M720) CH2 CH O O CH2 CH TrifuncCH2 tional glycidyl N amine CH2 (MY0510) CH2 CH O

O CH

CH2

CH2 N

CH2

CH2 CH

CH2 O

O CH2 CH CH2 O

Table 1.7 Model input parameters for the group interaction modelling of epoxy resins [2] Chemical name

Van der Waals volume (cm3/mol)

Molar volume (cm3/mol)

GIM cohesive energy (J/mol)

Molecular weight (g/mol)

Diaminodiphenylsulphone (DDS)

121.5

176.8

113,000

246.5

Dicyandiamide (DICY)

37.2

50.2

44,000

82.1

Tetrafunctional glycidylamine (M720)

243.4

367.5

204,000

436.5

Trifunctional glycidylamine (MY0510)

164.3

246.0

139,300

297.3

Molecular modelling of composite matrix properties O

O

CH2 CH

CH2

CH

CH2

CH

N

CH2

N

CH2 CH

CH2

CH2 CH

CH2

Tetrafunctional glycidyl amine

O

O

O

O

CH2 CH CH

CH

CH2 CH2

N

CH2

N

CH2 CH

CH2

H

1

O

O

O CH2 CH CH

15

CH

CH2 CH2

N

CH2 2

O

CH2 CH N CH2

CH2

CH2 CH OH

O

O

CH2 CH

CH2

CH

CH2

CH

N

CH2

N

CH2 CH

CH2

CH2 CH

CH2

3

O

OH

Cl

O CH2 CH CH

CH

CH2 CH

CH2 CH2

N

CH2

N CH2 CH

4

O

CH2 CH2

OH

O

O

CH2 CH

CH2

CH

CH2

CH

Cl

O

N

O

CH2

N

5

CH2 CH

CH2

CH

CH2

Cl

CH2

OH

1.4 The structures of TGDDM (MY720) and other synthesis byproducts in MY721 epoxy resin [19, 21].

Table 1.8 GIM predictions of the glass transition temperatures of a range of base epoxy resins in comparison to the prediction of a commercial polymer blend [2] Epoxy resin

Functionality

Hardener

Concentration (%)

Tgexp (∞C)

Tg cal (∞C)

MY0510 MY0510 MY0510 MY721 MY721 924

3 3 3 4 4 3.5

DDS DDS DICY DDS DICY DDS/DICY

36 45 19 26 14 22/8

285 276 218 288 258 234

283 268 213 281 249 241

16

Multi-scale modelling of composite material systems

therefore TGDDM-1, 2 and 4 have been studied in detail. For TGDDM-3 and 5, if the epoxy groups only are considered, a Tg of ⬇270 ∞C was calculated. Table 1.9 shows that these predictions can be used to identify the most important impurity in the commercial TGDDM (i.e. MY721) resin. The experimental value of Tg at 288 ∞C is best predicted when the impurity TGDDM-1 is included in the analysis (Tg = 281 ∞C). TGDDM-1 can be considered to be four-functional whereas TGDDM-2 is three-functional and TGDDM-4 is only two-functional in epoxy groups. Therefore, a lower value of Tg for the latter two is understood. In TGDDM-1, the –NH can react with epoxy groups directly. The close agreement between the resin containing 21% TGDDM-1 and the experimental value of Tg suggests that this byproduct is the most important impurity in MY721 resin. The effect of the degree of cure on Tg It is known that the degree of cure has a major influence on the Tg of a matrix resin. The equations given above include this as a variable. In order to obtain a good prediction, knowledge of the degree of cure is necessary. This can be determined directly by differential scanning calorimetry. However, when it comes to blends of resins including more than one hardener the degree of cure will be averaged over all possible reactions. Examination of Fig. 1.3 indicates that the degree of cure of each of four combinations equates with the degree of incorporation of individual components into the network. Therefore we can use the C parameter as a scaling factor to average the contribution of these segments to the overall network. Figure 1.5 provides the predicted values of Tg up to achievable degrees of cure of the individual resin/hardener pairs. We can therefore use the degree of cure to identify the structure of the 924 blended epoxy through the breadth of the Tan d peak. Figure 1.6 illustrates the component contribution principle to provide an indication of the structure of the network formed when two epoxy resins and two hardeners are incorporated. You can quickly observe that the inclusion of DICY into the structure leads to a lower Tg. This is consistent with the predictions in Table 1.8. Using the degree of cure as a variable enables the individual Gaussian curves to be used for curve fitting of the DMTA curve for the blend. In this way, the onset of the Tg and hence the maximum service temperature can be estimated from a GIM model for Tan d. Equations exist which enable this to be achieved but space does not permit this aspect to be described. The reader is referred to the book by Porter [1].

Composition

DDS (Wt%)

nDDS

xstDDS

C

+

26

0.6

0.44

0.87

0.62

Cmon–DDS

fmon

fDDS–max

EDether (J/mol)

Tg, (∞C) Calc.

4

4

6700

264

MY721 TGDDM

288

TGDDM-1

26

0.58

0.43

TGDDM-2

26

0.60

0.44

TGDDM-4*

26

0.64

0.48

0.65 0.86 0.69

3.79

4

6808

281

3.79

4

3808

268

3.58

4

3597

246

Estimated number of average degrees of freedom for the DDS network unit in this case N = 7.6 Cmon DDS is the degree of cure of DDS-Epoxy Reaction

+

Tg, (∞C) exp.

Molecular modelling of composite matrix properties

Table 1.9 Predictions of the glass transition temperatures for MY721-DDS system assuming that the impurities TGMM-1, 2 and 4 are present at 21% [21]

17

18

Multi-scale modelling of composite material systems 290 280

Temperature, ∞C

270 260 250 240 230 220 210 200

MY 721-DICY MY 721-DDS MY 0510-DDS MY 0510-DICY

190 180 170 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 Degree of cure

0.9 0.92 0.94 0.96

1.5 The effect of cure on the predicted glass transition temperatures (Tg) of component epoxy resins. 0.8 0.7 0.6

tan d

0.5

924-epoxy MY0510-DICY MY721-DICY MY0510-DDS MY721-DDS PES

0.4 0.3 0.2 0.1 0 150

170

190

210 230 Temperature, ∞C

250

270

1.6 The component contribution principle of epoxy resins for the DMTA relaxation curve for 924 resin.

The effect of moisture absorption on Tg It is well established that water diffuses into epoxy resins causing a reduction in the Tg. A typical rule of thumb is that the Tg of an epoxy resin is reduced by 20 ∞C for each 1% moisture [22]. A further response to moisture absorption

Molecular modelling of composite matrix properties

19

is the formation of a shoulder at low temperatures within the tan d loss peak. GIM can be used to understand this phenomenon if we recognise that the absorbed water will not be homogeneously distributed through the resin network but hydrogen bonded to the polar groups within the structure. Thus if we model the effect of water on the Tg on the individual base resins we can perhaps predict the role of water in a blended system such as 924. Moisture absorption can be modelled by considering that the water molecule is a penetrant adding cohesive energy and degrees of freedom. The standard equation for Tg becomes Tg = 0.224q1 + 0.0513 È E coh mon + x st H1 E coh H1 – DE et – w + n H 2 O E coh H 2 O ˘ ¥ Í ˙ (N mon – f N L )+x st H1 N H1 + n H 2 O N H 2 O Î ˚

1.17

where nH2O is the number of moles of water which can be obtained directly from the weight fraction of water within the repeat unit of the polymer. The cohesive energy of water Ecoh = 60 kJ mol–1 and the number of degrees of freedom N H 2 O = 18. Figure 1.7 shows the modelled reduction in Tg through plasticisation of the individual base resins at two hardener concentrations. These follow 260 TGDDM-wt 37% DDS TGDDM-wt 26% DDS MY0510-wt 45% DDS MY0510-wt 35% DDS TGDDM-wt 16% DICY TGDDM-wt 14% DICY MY0510-wt 22% DICY MY0510-wt 19% DICY

250 240 230 220

Tg, ∞C

210 200 190 180 170 160 150 140 130 120 0

1

2 3 Moisture content, (wt%)

4

1.7 The effect of moisture on the predicted glass transition temperatures of a range of base epoxy resins.

20

Multi-scale modelling of composite material systems

approximately the rule of thumb of Wright [22] but the individual combinations of hardener and resin show a range of moisture interaction efficiencies. The implication of this is that the water will be unevenly distributed on the molecular scale but will interact with the sites of highest potential interaction energy. The water affinity constant, Af can be calculated from the relative cohesive energy for hydrogen bonding, for each group within the network structure. Figure 1.8 shows how the GIM methodology is able to predict the broadening of the Tan d peak for the complex 924 system. The individual component groups have differing reductions in average Tg giving rise to a differential shift as illustrated by comparison of Figs 1.6 and 1.8 for the dry and wet 924 resin system. 0.35 0.3

tan d

0.25 0.2

Experimental PES MY0510-DDS MY0510-DICY MY720-DDS MY720-DICY

Tg1

Tg2

0.15 0.1 Tg3 0.05 0 50

100

150 200 Temperature, ∞C

250

300

1.8 Predicted tan d curves for individual group structural components compared to the experimental DMTA curve for 924 resin within a carbon fibre composite containing 3% moistures (6.9% in the resin) [21].

1.3.4

Prediction of the mechanical properties of a composite

To predict the stiffness properties of a composite material we use the HalpinTsai equations which incorporate the law of mixtures estimate for the longitudinal modulus of a unidirectional ply with Halpin-Tsai equation for the transverse modulus. The principal property of the matrix included in these estimates is the resin modulus. For the amorphous glassy polymers used as matrices in most composite laminates it has been established that the intermolecular forces between the macromolecular segments are responsible for the deformation under load. For example, it can be shown using eqn 1.2 that the tensile or compression modulus of a glassy polymer is essentially the

Molecular modelling of composite matrix properties

21

same as a rapidly quenched gas such as methane, at liquid nitrogen temperatures. In this case the gas was not allowed to crystallise and an amorphous glass formed which could be tested under compression. Since the bulk modulus B can be defined as follows:

B = V dP dV

1.18

From eqn 1.18, Porter [1] has derived the following expression:

B = 1.7

E coh Vw

È Ê ro ˆ 6 Ê ro ˆ 12 ˘ Í2 Ë r ¯ – Ë r ¯ ˙ Î ˚ MPa 2 Ê r ˆ –1 Ë ro ¯

1.19

Using appropriate values for the separation distance between segments in an amorphous glass the following generic expression is obtained

B = 6.4

E coh MPa Vw

1.20

From standard mechanics, the tensile modulus (E) is related to its bulk modulus (B) and shear modulus (G). E = 3B(1 – 2n) = 2G (1 + n)

1.21

Porter [1] defines a cumulative loss tangent tan Db Tan Db ⬇ tan db + S tan db

1.22

T1 t

where tan db is the base line value of tan d and tan db is a secondary or b relaxation occurring within the glass. With time (t) and/or temperature (T) the second term in eqn 1.22 becomes more significant. For most glassy polymers, Tb is the temperature at which tan db provides local (or crankshaft type) deformation responsible for shear type mobility. Thus the Poisson ratio n can be defined as 2 n = 0.15 [1 – 0.33 (1 – tan D1/2 b ) ]

1.23

Thus the engineering constants of the matrix resin are predictable. Table 1.10 [1] illustrates the validity of the GIM approach for predicting the modulus of linear polymers. Strength of 0∞ continuous fibre composite In a unidirectional fibre composite in which the failure strain of the matrix is greater than that of the fibres, the first failure involves a fibre fracture. If

22

Parameters Polymer

M

Vw

L

q1

Predictions N

DNs

Ecoh

Bam tan Db (GPa)

Poly(styrene) PMMA PVC PPO Bisphenol-A P Carbonate Poly(sulphone) Poly(ether sulphone) Phenoxy resin PET

104 100 62.5 120 254

66 56 30.5 70 141

2.5 2.5 2.5 4.6 10.8

285 279 368 550 550

6 6 4 6 14

443 232 276 192

236 112 162 98

18.3 10.4 10.7 10.8

550 550 550 550

26 12 22 17

0 0? 1 4 6 20 10 6 5

v

E (GPa)

model

expt.

model

expt.

36000 36000 19500 40300 83000

3.51 4.14 4.12 3.68 3.79

0.008 0.008 0.050 0.078 0.054

0.36 0.36 0.39 1.41 0.40

0.355 0.37 0.39 0.41 0.40

2.96 3.43 2.48 1.95 2.24

3.0 3.2 2.5 2.1 2.3

171100 82300 102000 74000

4.67 4.72 4.05 4.86

0.094 0.096 0.048 0.047

0.42 0.42 0.40 0.40

0.44 0.42 0.40 0.43

2.25 2.25 2.47 2.98

2.2 2.2 2.3 3.1

Multi-scale modelling of composite material systems

Table 1.10 Comparison of predicted moduli for typical glassy linear polymer illustrating the use of GIM parameters [1]

Molecular modelling of composite matrix properties

23

the fibres have a unique failure strength, all of the fibres will fracture at the same time and the load thrown on to the matrix will cause it to fail at the same time. Thus the strength can be predicted simply from the volume fraction of fibres according to equation slu = sfu Vf + s ¢m (1 – Vf)

1.24

where slu is the strength of a 0∞ composite, sfu the strength of the fibres and s ¢m is the stress on the matrix when the fibres break. In reality the fibre strength is not unique and subject to a statistical distribution. Thus the number of fibre-breaks will accumulate in the material before fracture. The factor which determines failure, therefore, is the probability that the adjacent fibres in the same plane, will also fracture to produce a flaw of critical size. Fracture of resin within the multiplet of fibre-breaks will lead the propagation of the failure crack. In this way the fracture toughness of the matrix can play a role in limiting the growth of a flaw. Thus the energy associated with a fibre-break needs to be absorbed by the surrounding material. This can occur by deformation in the matrix or debonding. Traditionally in carbon fibre composites, the fibres are surface oxidised to provide an optimum adhesion to the matrix resin so that a debond will propagate along the interface. The adhesion cannot be so low that composite cannot carry a load, however it should not be so perfect that a crack will propagate throughout the material, from a fibre-break. A small degree of debonding will ensure that the load carried by the fibre prior to fracture is not concentrated on adjacent fibres in the same plane. In this way a higher applied stress will be needed before fracture of the adjacent fibres occurs. This will manifest itself in a composite with a higher failure strain. A similar phenomenon can occur in a composite where either (i) the matrix or (ii) the interphase region yields. In glass fibre composites, the sizing on the fibre which includes a coupling agent and film former often provides an interphase region which is well bonded to the glass fibre surface. In this case the yield strength of the interphase region will determine whether a yield front or debonding occurs at a fibrebreak. Lane et al. [23] examined this phenomenon using finite element modelling. They created a model which enabled the effect of a fibre-break on the stress concentration in neighbouring fibres to be studied. The presence of an interphase of thickness 0.2 mm on a carbon fibre of diameter 7 mm was examined. The full true stress/true strain curves for the matrix and interphase were included in the model. The interphase was given either a higher or lower yield strength than the matrix. The properties of the two resins are given in Table 1.11 [23] The effect of thickness of the interphase and its properties were also studied using a single fibre model [24]. At low applied strains, where the interphase and matrix behave elastically, the effect of a lower modulus interphase was to reduce the rate of stress of stress transfer slightly. With a soft interphase of 11.5 mm thickness the response was similar

24

Multi-scale modelling of composite material systems Table 1.11 Summary of the elastic properties of the two resins [24] Resin system Property

LM

HM

Initial modulus (GPa) Average modulus to yield point (GPa) Yield stress (MPa) Yield strain (%) Assumed value of Poisson’s ratio

1.76 0.79 35.16 4.40 0.36

3.48 0.84 53.50 6.37 0.36

to that for system where the soft interphase material became the matrix. At a higher applied strain of 2% the effect of a soft interphase is more striking when the matrix and interphase behave in an elastic-plastic manner [24]. Figure 1.9 shows how the plasticity within the interphase and/or matrix influences the stress transfer back to fibre at a fibre-break. In the absence of an interphase, the higher modulus matrix (HM) has a significantly higher rate of strain transfer than the lower modulus matrix (LM). The introduction of an interphase of LM into the stiffer HM makes the strain transfer profile similar to that of a composite with a LM matrix alone. Examination of the region over which the strain is transferred elastically, shows that there is a slight dependence on interphase thickness. However, it is clear that the introduction of a soft interphase of ~ 0.2 mm in thickness enables a yield phenomenon to occur within a thin region near to the fibre interface. This, therefore, confirms the long held view that interphases are a design variable for a composite material. Turning to high volume fraction composites, the interphase region will determine the stress concentration which is placed onto neighbouring fibres in the presence of a fibre fracture event. Figure 1.10 illustrates the FE model used in this study. The resins employed had identical properties (Table 1.11) to those used above. Table 1.12 gives the strain concentration factors in composites of fibre volume fraction ranging from 0.38 to 0.58 at two applied strains where (i) elastic phenomena strain transfer operates and (ii) where elastic-plastic strain transfer occurs. In real composites, where fibre failure strain is > 1%, elastic-plastic strain transfer will operate. Table 1.12 shows that the introduction of a soft interphase has a major impact on the strain concentrations in adjacent fibres of a composite. The strength of a 0∞ fibre composite strongly influenced by the probability of fibres fracturing in the vicinity of an initial fibre-break therefore the introduction of a submicron interlayer or interphase can improve the reliability of a composite material.

Molecular modelling of composite matrix properties

25

2.00% 1.80%

Strain attained in the fibre centre

1.60% Extent of plasticity in the pure matrix 1.40% 1.20%

Extent of plasticity in the 1.00% presence of an interphase 0.80% 0.60% 0.40% 0.20% 0.00% 0

50

100

150

200

250

300

–0.20% Distance from fibre end (mm) (a)

Strain attained in the fibre centre

1.80%

1.60%

1.40%

1.20%

1.00% 200

210

220

230 240 250 260 270 Distance from fibre end (mm) (b)

280

290

300

0.2 mm soft interphase, coincident with 2 mm 2 mm soft interphase 11.5 mm soft interphase 6040 (uncoated) 5050 (uncoated)

1.9 Strain development within a carbon fibre fragment in the presence of a soft interphase of differing thickness at 2% applied strain (a) half a fragment (b) between 200–300 mm of the fibre-end. The properties of the resin are given in Table 1.11 [24].

26

Multi-scale modelling of composite material systems Fibre Interphase Broken fibre

Matrix

Y Z X

1.10 Schematic of finite element model indicating the presence of an interphase and location of the fibre-break [23].

The non-linear stress strain behaviour of matrix resins From a modelling point of view we therefore need to be able to predict the complete true stress-true strain curve for a matrix resin. From the foregoing it is clear that this is a complex aim. Figure 1.11 shows the compression stress-strain curves for typical model matrix epoxy resins based on the tetrafunctional TGDDM (MY721) and trifunctional (MY0510) epoxy resins cured with DDS. The highly non-linear behaviour illustrates the complexity of the model required to define the behaviour. In tension, the presence of flaws leads to brittle failure before yield can be achieved so that typical epoxy resin matrices can be difficult to characterise experimentally. It should be recognised that under the complex stress state existing in a ‘thin’ film between fibres in a composite, especially near a fibre-break or in an angle ply laminate where shear stresses dominate, non-linear behaviour is important. It is probably less important under the tensile stresses which develop transverse to fibres especially as a result of the stress concentrations associated with the presence of the high modulus fibres. To model the non-linear behaviour of the glass requires individual models for the low strain and high strain responses. Porter [1] has modelled linear polymers below the yield strain (ey) with some success using the concept of yield, as shown by eqn 1.25:

0.1% Applied strain Matrix

Interphase

Fibre

0.38 Vf

0.48 Vf

1% Applied strain 0.58 Vf

0.38 Vf

0.48 Vf

0.58 Vf

HM

LM

Neighbour

1.10

1.12

1.14

1.05

1.05

1.05

LM

HM

Neighbour

1.08

1.10

1.13

1.05

1.05

1.05

HM

LM

Next-nearest

1.04

1.04

1.05

1.02

1.02

1.02

LM

HM

Next-nearest

1.02

1.03

1.03

1.02

1.02

1.02

Molecular modelling of composite matrix properties

Table 1.12 Maximum strain concentration in the centre of fibres adjacent to a fibre-break showing the effects of an interphase [23]

27

28

Multi-scale modelling of composite material systems

Compressive stress (MPa)

250

200

150

100

MY0510/DDS MY0510/MY721/DDS (50:50) MY721/DDS

50

0 0.00

0.04

0.08

0.12

0.16

0.20 0.24 0.28 Compressive strain

0.32

0.36

0.40

0.44

1.11 Experimental compressive stress-strain curves of DDS cured MY0510, MY721 and a 50:50 blend. They were measured at room temperature at a strain rate of 1.64 10–3 s–1.

Ê ˆ E ( e ) = E o (T) Á 1 – e ˜ e Ë y ¯

1.25

where Eo(T) is the zero-strain modulus at ambient Temperature T, ey is the yield strain, E(e) is the modulus at an applied strain e. Above the yield point, tan Db can be used to introduce the contribution from conformational response. For further details the reader is referred to reference 1. Tripathi et al. [25, 26] demonstrated that the yield stress of a matrix could be used to truncate an elastic shear stress profile at the fibre-matrix interface to provide a good approximation of non-linear behaviour of the matrix polymer. Figure 1.12 compares the tensile stress profiles in a single glass fibre calculated from FE analysis using an elastic-plastic stress-strain curve for the matrix and from a numerical analysis in which the elastic shear stress profile was interrupted for yield using the von Mises criterion (the plasticity effect model of Tripathi and Jones [25]). This demonstrates how shear yield strength can be used as a matrix predictive parameter. Therefore, we seek a means of modelling the shear yield of strength of a matrix resin network and introduce this into a composite code. We are currently predicting stress concentration factors from non-linear stress strain curves using FE analysis which can be included in probalistic models of composite strength. Approximations can be achieved through the use of the experimental and predicted values of shear yield strength.

Molecular modelling of composite matrix properties

29

Tensile stress at the centre of the fibre (MPa)

2000

1500

1000

500 Plasticity effect model Finite element model 0

0

20 40 Distance from the fibre end (x/r)

60

1.12 Comparison of the tensile stress transfer profiles (at an applied strain of 3%) along a glass fibre from its end predicted by FE modelling (䊐) and the plasticity effect model (䊊) [25].

GIM prediction of shear yield strength (sy) of a matrix resin Yield occurs at a point at which significant plastic deformation occurs when brittle fracture is suppressed. Yield in polymers has been reviewed in the books of Kinloch and Young [27] and Bristow and Corneliussen [28]. When the mer units can undergo a large scale translation yield will occur. This is analogous to the molecular response which occurs at the glass transition so that volumetric strain can be used to provide a prediction in GIM. The Lennard-Jones potential can be used to define the volumetric strain Ê DV ˆ required to define the H component. The separation distance between m Ë V ¯ mer units can therefore be used to provide a yield criterion. Ê DV ˆ = 0.02 Ë V ¯y

1.26

from which we obtain sy =

0.02 E 1 – tan D1/2 b

1.27

which is in agreement with the empirical model of Seitz [29] for tensile yield stress of a polymer glass under plane strain.

30

Multi-scale modelling of composite material systems

sy = 0.025 E

1.28

GIM provides the following more general form of the equations for predicting yield phenomena over the range of temperature from T to Tg

e y (T) ª

1.35 N (T – T ) g 1/2 E (1 – tan D b ) coh

1.29

N (T – T ) s y (T) ª 1.35B(1 – tan D1/2 g b ) E coh

1.30

Table 1.13 provides the predictions of compressive yield stress (scy) for model network resins and demonstrates the effectiveness of the GIM methodology for predicting relevant polymer properties. Table 1.13 Comparison of experimental and predicted values of compressive yield strength for DDS cured epoxy resins Resin

Tg (∞C)

C(%)

scy calc (MPa)

scy exp (MPa)

MY0510 MY0510 MY721 MY721

278 296 280 297

93 97 88 94

196 202 191 196

159 168 164 173

1.3.5

Design of composite materials

A polymer matrix composite material relies on combining fibres and resin into different configurations. The configuration of the fibres within a structure is determined by a number of factors which fall into the following criteria: ∑ ∑ ∑ ∑

mechanical performance ease of fabrication complexity of shape cost.

To achieve these aims, the fibres are converted into a range of forms; direct rovings, woven rovings, knitted preforms, non-crimp fabrics, stitched fabrics, chopped fibrous mats, chopped fibres and preimpregnated materials. The fibres need to be sized with polymeric film formers for their conversion in to these useful forms. As a result, fabricated composites will have a complex matrix with the probable formation of an interphase region at the fibre surface. At present, the chemical nature of the sizing or finish is a highly guarded secret, especially for glass fibres. As a result, the predictions of composite performance is in need of understanding of the development of interphase regions. Figure 1.9 demonstrates that with a thin soft interphase or interlayer

Molecular modelling of composite matrix properties

31

the stress transfer at a fibre-break is strongly influenced by the mechanical properties of the region at the fibre interface. Thus for complete design and predictable performance, sizing technologies that enable the properties of the interphase to be matched to those of the matrix are required. Recent work which uses plasma polymers to provide this control is looking promising [30, 31]. The alternative approach is to measure the properties of the interphase for inclusion in predictive models. The obvious approach is to use nanoindentation or the similar technique of AFM to measure the properties of the interphase directly [32]. However, this has proved difficult because of the indentation depth required by the analysis is difficult to achieve in isolation from the fibre at the interface. For example, it has been reported that the modulus of the interphase is higher than can be reasonably expected of a cross-linked network polymer, indicating that the value probably includes a contribution from the fibre. An alternative technique is therefore required to measure directly the relevant properties. Phase stepping photoelasticity is proving to have the power to estimate the shear yield stress of an interphase region from radial birefringent patterns at fibre-breaks [33, 34].

1.4

Conclusions

In this chapter, we have addressed the issue of modelling the properties of a composite material from molecular scale to the macroscopic. The group interaction modelling (GIM) of David Porter [1] has been reviewed and extended to epoxy resins used as a matrices for polymer matrix composite materials. Selected examples of how this modelling approach can be used for providing estimates of relevant properties which can be used in Composite codes for the prediction of micromechanics and strength are discussed. Group interaction modelling appears to have the power to bridge the gap between atomistic modelling of complex resin structures which is highly computer inefficient, and requirements of molecular predictions for inclusions in composite mechanics.

1.5

Acknowledgements

The author acknowledges the contribution of Dr D. Attwood (BAe Systems plc) and Dr D. Porter (Qinetiq Limited) for introducing the Group Interaction Model Methodology; Vadim Gumen (University of Sheffield) for many of the calculations and Shabnam Behzadi for experimental data.

1.6

References

1. Porter D., Group Interaction Modelling of Polymer Properties, Marcel Dekker, New York, 1995.

32 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

Multi-scale modelling of composite material systems Gumen V.R., Jones F.R. and Attwood D., Polymer 42, 5717 (2001). Hutnik H., Argon A.S. and Suter U.W., Macromolecules 26, 1097 (1993). Brown D. and Clarke J.H.R., Macromolecules 24, 2075 (1991). Bicerano J., ed., Computational Modelling of Polymers, Marcel Dekker, New York (1992). Volkenstein M.V., Configurational Statistics of Polymer Chains, Wiley-Interscience, New York (1963). Flory P.J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York (1969). van Krevelen D.W., Properties of Polymers, 3rd edn, Elsevier, Amsterdam (1993). Bicerano J., Prediction of Polymer Properties, Marcel Dekker, New York (1993). Dibenedetto A.T., J. Polymer Sci, Part A 1, 3459 (1963). Paul D.R. and Dibenedetto A.T., J. Polymer Sci, Part C, 16, 1269 (1967). Atkins P.W., Molecular Quantum Mechanics, Oxford University Press, Oxford (1983). Cheng S.Z.D., Lim S., Judovits L.H. and Wunderlich B., Polymer 28, 1165 (1987). Hough J.A., Xiang Z.D. and Jones F.R., Key Engineering Materials, 144, 27 (1998). Whiteside J.B., De Iasi R.J. and Schulte R.L. in O’Brien K., (ed.) ASTM STP813, American Society for Testing Materials pp 192–205 (1983). Tarasov V.V. and Fiz. Khim, ZH., 24, 1430 (1953) Russ. J. Phys. Chem. 39, 1109 (1965). Chino L., Macromolecules 23, 1286 (1990). Cole K., Hechler J. and Noel D., Macromolecules 24, 3106 (1991). St John N.A. and George G.A., Prog Polymer Sci 19, 779 (1994). Grenier-Loustalot M.-F., Metras F., Grenier P., Chenard J.-Y. and Horny P., Eur Polymer J. 26, 83 (1990). Gumen V.R. and Jones F.R., in Proc of 9th Int. Conf on Fibre Reinforced Composites (FRC 2002) Conf. Design Consultants Newcastle, UK, 2002. Wright W.W., Composites, 12, 1981, 201. Lane R., Hayes S.A. and Jones F.R., Comp. Sci. Tech. 61, 565 (2001). Hayes S.A., Lane R. and Jones F.R., Composites pt. A 32, 379 (2001). Tripathi D., Chen F. and Jones F.R., J. Comp. Mater. 27A, 709 (1996). Tripathi D. and Jones F.R., Comp. Sci. Tech. 57, 925 (925). Kinloch A.J. and Young R.J., Fracture Behaviour of Polymers, Elsevier Applied Science, London (1983). Bristow W. and Corneliussen R.D., Failure of Plastics, Hanser, Munich (1986). Seitz J.T., J. Appl. Polymer Sci 49, 1331 (1993). Lopattananon N., Hayes S.A. and Jones F.R., J. Adhesion, 78, 313 (2002). Marks D.J. and Jones F.R., in Silanes and Other Coupling Agents Vol 3, ed. Mittal K., VSP Utrecht, (2004) pp 205–223. Gao S. and Maeder E., Composites Pt A 33, 559 (2002). Zhao F.M., Hayes S.A., Patterson E.A., Young R.J. and Jones F.R., Composites Science and Tech. 63 1783 (2003). Zhao F.M., Martin R.D.S., Hayes S.A., Patterson E.A., Young R.J. and Jones F.R., Composites Pt A. 36, 229 (2005).

2 Interfacial damage modelling of composites C G A L I O T I S, University of Patras, Greece and A P A I P E T I S, Hellenic Naval Academy, Greece

2.1

Introduction: definition of the interface

The inherent notion of a composite material is that it comprises more than one phase. It is desirable to make the optimum use of each distinct phase, something that may only be achieved if the phases are joined together to make a contiguous or integral unit. Upon joining, the phases are separated by an interface which, thus, becomes a dominant feature of the composite. It is the existence of the interface that makes a composite differ physically from its components (Ebert, 1965). The distinct phases in the composite possess different properties, which constitute the reason for joining them. Any loading situation results in an interaction between the two constituents, which plays a crucial role in the composite properties. In the case of fibre reinforced composite materials, the interface is the entire ‘shared’ surface between the fibre and the matrix. There have been a number of basic approaches to studying the nature and the role of the interface: the (global) macroscopic approach, and the microscopic approaches, which in turn, may differ in scale (i.e. microscopic versus atomic). The former is based on classical mechanics and the theory of elasticity. The latter takes into account not only the mechanical but also the chemical and physical nature of the interface. A schematic illustration of the model upon which all various macroscopic approach studies are based is shown in Fig. 2.1(a) (Vlattas 1995). According to this model, there is no difference between the material properties in the vicinity of the interface and those of the bulk material away from the interface. Bonding is assumed to be perfect (Hull 1996). If, on the other hand, we take into account crucial physico-chemical phenomena occurring between the fibre and the matrix in the vicinity of the interface, a more realistic model representing the interface is obtained (Fig. 2.1(b)). Various parameters such as wetting, material absorption and interdiffusion arise during the composite fabrication but they are completely neglected from a macroscopic point of view. Wetting and absorption represent the compatibility between the (solid) 33

34

Multi-scale modelling of composite material systems Matrix Fibre Matrix Macroscopic scale (a)

Amorphous polymer matrix ‘Sizing’ or ‘finish’ Crystalline fibre Microscopic scale (b)

Matrix Chemical van der Waals acid-base hydrogen

Fibre Atomic scale (c)

2.1 The different scales of interfacial adhesion: from the macroscopic to the atomic scale.

fibre and the (liquid) matrix in terms of the thermodynamic work of adhesion when the surfaces are brought close to each other. Interdiffusion is the formation of bonding between two surfaces by the entanglement of the polymer molecules on one surface with the molecular network of the other surface. If we focus even further on the area between fibre and matrix (Fig. 2.1(c)), at the atomic scale level, we may consider the electrostatic attractions and chemical bonding. Electrostatic attractions occur between two surfaces due to opposite charges. The strength of the interface will depend on the charge density. A chemical bond is formed between a chemical grouping of the fibre surface and a compatible chemical group in the matrix. The strength of the bond depends on the number and type of bonds. Bond rupture occurs during interfacial failure. The interface can be visualised as the boundary between two materials with different properties. It is easy to assume that this boundary has no volume for calculation purposes. However, this is not the case on a smaller scale where phenomena such as surface roughness or chemical interaction are of crucial importance. Moreover, it is much easier to attribute a physical meaning to a zone with a gradient of properties between the two constituents. In this case, the zone is termed the interphase and extends in three dimensions. If the mechanical characteristics of the interface in multi-phase materials are

Interfacial damage modelling of composites

35

not affected by the typical length scale of the composite, two-dimensional models can be applied (Herrmann, 1996). However, if the bonding layer is of similar order of magnitude to that of a typical macroscopic size, like the reinforcement diameter, then the interphase has to be included in the mathematical modelling. The recent advent of nanocomposites will certainly require the inclusion of the interphase as a third phase in all calculations. The scale of the interphase may differ dramatically in different composite materials. In the case of fibrous composites it may extend from a few nanometres to several fibre diameters. A typical unsized carbon fibre/epoxy interphase only extends for less than ~10 nm (Guigon, 1994). The interphase of SiC fibres in an Al matrix may extend to hundreds of nanometres (Long, 1996). In thermoplastic matrices, the presence of transcrystallinity changes the matrix properties in the interphase region totally. In this case the interphase may be more than one order of magnitude larger in diameter than the reinforcing fibre (Heppenstall-Butler, 1996).

2.2

The interface and composite properties

The simplest assumption predicting composite properties is the rule of mixtures, by which the properties of the constituent materials are added according to their respective volume fraction. In the case of a continuous fibre unidirectional composite loaded in the fibre direction, the rule of mixtures defines the stress s sustained by the fibre and the matrix, through their respective volume fraction V (Jones, 1975):

s = sf V f + s m V m

2.1

where the subscripts f and m denote the fibre and the matrix, respectively. The rule of mixtures is no longer macroscopically valid when shear forces are present due to the presence of local discontinuities, or if the material is not homogeneous, that is, if the material properties change from point to point (Halpin, 1992). Two simple cases for inhomogeneity may be regarded, where shear stresses are of primary importance: (i) The short fibre composites, where the reinforcement is discontinuous, and (ii) the continuous fibre composites, where inhomogeneity is present at the locus of a discontinuity of the reinforcement. The simplest case is the presence of a fracture in a unidirectional composite loaded in the fibre direction. In that situation, fracture of the fibres is regarded as the primary event which controls the local damage development and accumulation that will lead to the failure of the composite (Reifsnider, 1994). In Fig. 2.2(a), such fractures within the composite zone are shown. The fracture of a fibre causes a local stress perturbation (Fig. 2.2(b)) (Paipetis 2001). The stress is redistributed to the neighbouring fibres through the interface. At some length from the fracture locus, the axial stress is restored

36

Multi-scale modelling of composite material systems

Fibre fractures

(a)

Neighbouring fibres

Stress concentration

1.4

Ineffective length

1.2 1 0.8 0.6 0.4 0.2 0

Fibre fracture Distance along the fibre (b)

2.2 Fibre fractures (a) and shear perturbation around a fibre fracture (b). (Source: Paipetis, 2001.)

on the broken fibre. This length is termed the ineffective (Reifsnider, 1994) or transfer (Galiotis, 1993) length and is dependent on the stress transfer efficiency of the interface. The ineffective length defines the zone of influence of the fracture; small ineffective lengths create large stress concentrations in neighbouring fibres; large ineffective lengths increase the size of the ‘flaw’ within the composite, raising, thus, the possibility of cumulative flaws leading to fracture.

Interfacial damage modelling of composites

37

For a single fibre embedded in an epoxy matrix, Drzal and Madhucar (1993) have identified the failure mechanisms with relation to various levels of interfacial adhesion (Fig. 2.3); as interfacial adhesion increases, the axis of failure propagation changes from a mode II crack which propagates along the fibre axis (Fig. 2.3(a)) to a mixed mode (Fig. 2.3(b)) and finally a mode I crack (Fig. 2.3(c)), which propagates transversely to the loading axis. It is worth noting that increased interfacial adhesion may improve the on-axis properties, such as the tensile strength, by 45% (Drzal, 1993). Matrix Fibre Matrix

Fibre

Fibre

Matrix Fibre Matrix Mode II (a)

Fibre Matrix

Matrix Fibre

Fibre

Matrix

Mixed mode (b)

Mode I (c)

2.3 Modes of interfacial failure.

2.3

Analytical modelling of the shear transfer

2.3.1

The problem statement

The problem of the stress transfer at the locus of a discontinuity is a typical case of torsionless axisymmetric stress state (Timoshenko, 1988). In cylindrical coordinates r, q, z with corresponding displacements components u, v, w, the component q vanishes and u, w, are independent of q. The same is valid for the stress components with trq and tqz being zero (Fig. 2.4). Thus, the strain components are reduced to:

e r = ∂u e q = u e z = ∂w r ∂r ∂z and the equilibrium conditions are:

g rz = ∂u + ∂w ∂z ∂r

2.2

s – sq ∂s r ∂t rz + + r =0 r ∂r ∂z

2.3a

∂t rz ∂s z t + + rz = 0 r ∂r ∂z

2.3b

38

Multi-scale modelling of composite material systems

A

A1

N tr q

tqz

O sz

trz

z sr

dz

r sr +

sz +

∂s r dr ∂r

∂s z dz ∂z

z

tr q

sq

O sr

sq +

∂t rq dr ∂r ∂s r sr + dr ∂r

t rq +

∂s q dq ∂q

2.4 The stress field at the axisymmetric stress state (Timoshenko, 1996).

The solution of eqns 2.3 with the appropriate boundary conditions provides the elastic stress-state. The boundary conditions for a cylinder of length l loaded in shear are (Filon, 1902) for 0 £ z £ l:

sz(z) = sz(–z)

2.4a

sr(z) = sr(–z)

2.4b

trz(z) = –trz(–z)

2.4c

sz = 0, for z = 0, l

2.4d

There is an additional boundary condition where:

trz = 0, for z = 0, l

2.4e

that is, the shear stresses should be zero at the fibre break due to the symmetry of the stress tensor McCartney, 1989; Nairn, 1992, 1997). However, the

Interfacial damage modelling of composites

39

exact solution cannot provide finite stress values at the discontinuity, that is, at the end of the cylinder. Filon (1902) showed that when there is a discontinuous transition from a stressed area to an unstressed one, stresses become infinite. The physical meaning of this finding is that at the discontinuity the stressstate cannot be elastic, and local yielding/damage is always present. The task of solving eqns 2.3 involves defining the appropriate boundary conditions that will provide finite stress-values, by assuming that the effect of the discontinuity is highly localised and does not significantly affect the overall stress-state. It follows that eqn 2.4e should be part of any elasticity analysis in order to provide finite stress values at the discontinuity (Nairn, 1992).

2.3.2

One-dimensional models

The difficulty of providing a closed-form solution for the elastic equilibrium, as well as the statistical nature of interface and/or fibre failure, has led to a variety of analytical treatments. Through the appropriate assumptions, these lead to finite values of the stresses which can be used to model interfacial adhesion. The axial balance of forces at the interface for a cylinder of radius R yields (Fig. 2.5) (Rosen, 1965):

ds z ( z ) t rz = – R 2 dz

2.5

trz(z) 2pRdz

s z (z ) pR 2

[s z (z ) + ds z (z )] pR 2

dz 2R

t rz = – R 2

ds z (z ) dz

2.5 Axial balance of forces for an infinitesimally small fibre element. (Source: Paipetis, 2001.)

Kelly and Tyson (1965) made the simple assumption that trz is constant, which yields:

t rz = –

Rs fu lc

2.6

40

Multi-scale modelling of composite material systems

In this case, sf coincides with the fibre strength which is independent of z and the length lc is the critical length, which is defined as the length where maximum axial stress is reached prior to the final fracture. In this approach, the fibre is perfectly elastic up to fracture, whereas the matrix (or the interface) is perfectly plastic. The stress-state, as postulated in the Kelly and Tyson approach, is shown in Fig. 2.6(b). Although its simplicity makes the constant shear stress approach extremely attractive, the elastoplastic behaviour of the matrix, as well as the lack of definition for radial or hoop stresses (Nairn, 1992) are obvious examples of its limitations. t

sz

Distance z

(a)

t sz

(b)

Distance z

2.6 Axial stress and interfacial shear stress (a) the shear lag model (Cox, 1952) and (b) the constant shear model (Kelly, 1965).

Cox (1952) introduced the shear-lag theory, making the fundamental assumption that the shear force S is proportional to the difference between the axial displacement in the matrix w and the displacement in the matrix w• that would exist if the fibre were absent: S = H(w – w•)

2.7a

where H is a proportionality constant which depends on geometrical and material parameters. Through the axial balance of forces (eqn 2.5), the shearlag assumption yields for the axial stress at the interface: È l Ê ˆ˘ cosh Á b Ê – z ˆ ˜ ˙ Í ¯¯ Ë Ë2 ˙ s z ( z ) = E f e • Í1 – lˆ ˙ Í Ê cosh b Í ˙ 2¯ Ë Î ˚

2.7b

Interfacial damage modelling of composites

41

where:

b=

2 GmR• R R 2 E f ln Ê • ˆ Ë R ¯

2.7c

e• is the far-field strain and GmR• , defines the shear modulus of a matrix cylinder of radius R• within which there is shear perturbation. The interfacial shear stress is:

t rz ( z ) = E f e •

GmR• R 2 E f ln Ê • ˆ Ë R ¯

l sinh b Ê – z ˆ Ë2 ¯ l cosh b 2

2.7d

The shear-lag equation has been derived by different authors and is based on the assumption that, for a given r, all stresses are only a function of z. It is worth mentioning that the shear stress trz is not zero at the vicinity of the discontinuity, but has a finite value (Fig. 2.6(a)). Filon (1902) in a similar solution attributed this to the existence of a determinate system of radial shears over the flat end of the cylinder, which must be self equilibrating due to symmetry. This, however, cannot form part of the shear-lag solution, since radial shearing is ignored. Nayfeh (1977) in a similar approach, proposed a solution for which both the axial stress and the axial displacement were assumed to vary linearly with the radius r. His solution for the case of two concentric cylinders yielded eqn 2.7c but with a different definition of the b parameter. The new definition of b coincides with the definition made by McCartney (1992) and Nairn (1996a). In a thorough analysis of the shear lag approach, Nairn (1997) expressed the approximate one-dimensional shear lag solution and formulated the assumptions that ought to be made. These assumptions are formulated as follows:

∂u << ∂w ∂z ∂r

2.8a

that is, the dependence of the radial displacement u on the symmetry axis z is negligible compared to the axial displacement w on the radius r. The assumption made by Cox (1952) (eqn 2.7a) implies that the first term in eqn 2.8a is zero. The second assumption is formulated as follows:

t rz =

f0 ( z ) r f (z) + 1 r 2

2.8b

that is, the shear stress trz can be expressed as the above sum, where fi (z) are only functions of z. The third assumption is:

42

Multi-scale modelling of composite material systems

·s z Ò nT ·s + s q Ò << + aT T EA r ET

2.8c

where n is the Poisson’s ratio, T is the temperature, a the thermal expansion coefficient and the subscripts A and T denote axial and transverse directions for a transversely isotropic material. This assumption virtually implies that the mean transverse stresses ·sr + sqÒ are negligible compared to the mean axial stress ·szÒ which is fundamental in all shear lag approaches. The final assumption is that sz and w only weakly depend on r. The above assumptions should all be used in order to provide a close form solution to the governing elasticity equations using the shear lag approach. As stated previously, the solutions by Nayfeh (1977), McCartney (1992) and Nairn (1997), define b as follows:

b=

2 R 2 E f Em

È ˘ Í ˙ E f V f + E m Vm Í ˙ Í Vm V Ê ˆ 1 1 1 m ˙ Í 4 G + 2 G ÁË V ln V – 1 – 2 ˜¯ ˙ m m f f Î ˚

2.9a

where Vf and Vm are the fibre and matrix volume fractions: 2 R2 – R2 V f = R 2 and Vm = • 2 2.9b R• R• E and G are the axial and shear moduli respectively, R denotes the radius of the fibre, R• the radius of the matrix cylinder and the subscripts f and m denote the fibre and the matrix respectively. The improved definition of b yields more realistic results compared to finite element analysis (Nairn 1997). However, although shear lag analyses are only approximate, b has been employed to characterise interfacial adhesion (Galiotis and Paipetis 1998) and as such can be used to model the stress transfer as an adjustable parameter. Moreover, as will be shown later, this parameter is the inverse of the length that corresponds to 0.63 of the transfer length, which is the most characteristic property of the interface. If this parameter can be measured or quantified, then the stress transfer characteristics of a composite system may be evaluated by using the aforementioned assumptions. The shear-lag solution (Cox, 1952) can be regarded as an upper bound to the elastic solution as the existence of radial shearing blunts the maximum axial shear. The constant shear model (Kelly, 1965) described previously is a lower bound where the matrix is assumed to be perfectly plastic and flowing freely. However, as should be noted, the constant shear model requires further assumptions for the fibre strength at the critical length, which may introduce additional sources of error.

Interfacial damage modelling of composites

2.3.3

43

Mixed one-dimensional models of interfacial adhesion

Modifications to the above approaches were made in order to account for matrix plasticity, interfacial debonding, as well as for damage, or frictional sliding at different regions of the interface, along the fibre axis. Although these ‘incremental’ approaches are inherently flawed because they introduce unjustifiable discontinuities in the interfacial shear stress, they represent more closely ‘real’ conditions, where complex phenomena coexist and interact to produce a stress field which cannot be uniquely described through a single analysis. Dow (1963) introduced a modified shear-lag approach to account for the non-linear behaviour of the matrix. He introduced an incrementally changing matrix modulus at different levels of strain, simulating, thus, plasticity effects. Piggot (1966, 1980) combined the shear-lag and the constant shear approach to account for matrix yielding and frictional sliding. In the case of reinforced metals (where elastic behaviour for the fibre and elastic-plastic behaviour for the matrix are considered) the interfacial shear stress will be constant up to a certain distance from the fibre ends and equal to the shear yield stress of the matrix:

trz(R, z) = tmu 0 £ z £ l

2.10

where the length l depends on the applied stress and R denotes the radius of the fibre. Near the fibre mid-section the interfacial shear stresses are generated by elastic, Cox-type interaction. A model example for this kind of stress transfer is given in Fig. 2.7(a) for a well-bonded reinforced metal. In the case of reinforced polymers, the stress transfer is controlled by Coulomb’s friction taking place at the interface, between the fibre and the matrix. The frictional stresses are the result of the compressive stresses normal to the fibre-matrix interface, i.e., the radial stresses sr introduced by (i) fibre and matrix contractions during manufacturing (different thermal expansion coefficients) and (ii) Poisson’s contractions of the matrix when a stress is applied to the composite. In this case, the interfacial shear stress near the fibre ends is described by the following equation:

trz(R, z) = msr 0 £ z £ l

2.11

In Fig. 2.7(b) a model example for an imperfectly bonded reinforced polymer is shown. The discontinuity at the interfacial shear stress profile is due to the fact that the adhesion strength is different from the frictional stress. With reinforced cements and similar materials the stress transfer regime is very similar to that of reinforced polymers described above. Since the ultimate tensile strain of the cement is very small, the matrix Poisson’s

44

Multi-scale modelling of composite material systems sz

z

tmy

t

Distance z (a)

sz sz

t

atmy msr

Distance z (b)

t

tmy msr

Distance z (c)

2.7 Mixed models of interfacial adhesion (Kelly, 1965; Piggot, 1966; Piggot, 1980).

shrinkage can be neglected. In the case of reinforced ceramics, the matrix Poisson’s shrinkage cannot be neglected. Also, the moduli of fibres and matrix are not very different and therefore the Poisson’s shrinkage of the fibre must also be included. The interfacial shear stress near the fibre ends is not constant. In fact, it increases towards the fibre ends. A schematic representation of this model is shown in Fig. 2.7(c). It is quite clear from the above models of stress transfer that the maximum interfacial shear stress is not necessarily located at the fibre end as was predicted by the previous approaches (Cox 1952; Dow 1963). In a similar analysis (Piggot 1987), the existence of a thin distinctive layer of interphase around the fibre was introduced. The shear modulus of this layer was assumed to be less than that of the matrix. Analytical solutions were derived, based on the above assumption. The axial and shear stresses were found to be dependent on the Young’s moduli of fibre and matrix, the shear moduli of the matrix and the elastic properties of the interphase layer. These were the volume fraction of the fibre and the interphase, the Poisson’s ratio of the matrix and the fibre packing. Nevertheless, a disagreement with experimental results indicated that a more realistic approach should be taken.

Interfacial damage modelling of composites

45

The interphase may be modelled as a phase with a gradient of mechanical properties in the axial and radial direction rather than uniform properties throughout (Theocharis 1987).

2.3.4

Axisymmetric models

The solution of the equilibrium equation retaining all the components of stress has been attempted by various authors. These solutions aim to satisfy all boundary conditions related to the problem, through an admissible stressstate (Nairn, 1996a), McCartney (1989) managed to provide closed-form solutions to various axisymmetric problems. In order to solve the equations of equilibrium, he introduced average functions f ( z ) for the fibre and the matrix cylinders, over their respective radii: f (z) =

1 pR 2

Ú

R

0

2 prf ( r , z ) dr

2.12

Thus, all the boundary conditions that he defined are satisfied, apart from two stress-strain-temperature conditions, which are satisfied only in an average sense. McCartney (1989) obtained closed-form solutions for all the stresses and extended his analysis to the frictionally bonded case, where Coulomb’s law is valid:

trz(R, z) = msr(R, z)

2.13

Nairn (1996a) managed to derive a stress function
satisfying the equilibrium conditions for all stress components. The derived stress function satisfied all boundary conditions expect one: the stress is zero in an average sense over the surface of the fibre. The stress function has to satisfy (Timoshenko, 1988):

— 12 — 22
= 0

2.14a

where —2 is: 2 2 2 — i2 = ∂ 2 + 1 ∂ + 12 ∂ 2 r ∂r s i ∂r ∂r

2.14b

and si (i = 1, 2) are constants depending on the elastic properties of the material. Through the stress function all components of stress may be defined either for isotropic (where si = 1 and —2 coincides with the Laplace’s operator in cylindrical coordinates (Timoshenko, 1988)), or transversely isotropic materials (Lekhnitski, 1963).

46

2.4

Multi-scale modelling of composite material systems

Interfacial damage modelling

The analyses described in the previous section deal with the problem of stress transfer by shear in axisymmetric systems of finite length. In real composite systems shear-activated mechanisms are intrinsically related to damage mechanisms which stem from the fracture energy that has to be absorbed from the matrix, and secondary mechanisms that include frictional sliding (Piggot, 1980), matrix plasticity (Vlattas and Galiotis, 1992), or matrix cracking (ten Busschen, 1995). These mechanisms are either ignored as in the case of the shear lag analysis (Cox, 1952) or implied as in the case of the constant shear assumption (Kelly, 1965) of the frictional sliding of the interface (Piggot, 1980). However, the intrinsic problem of modelling such phenomena lies with their energy dispersive nature which excludes the possibility of finding exact elasticity solutions to describe them. The modelling approach of the damage process can be performed through the definition of equivalent elastic regimes that can be approached analytically or through energy methods that relate to the fracture toughness of the interface/interphase. The quantification of damage at the interface has been performed though the relaxation of the continuity conditions at the interface. This is better envisaged as an interphase which bridges the properties of the matrix and the fibre which is collapsed in two dimensions (Hashin, 1990). In this case, the damage of the interface is quantified by allowing relative matrix/fibre displacements at the interfacial area, which are proportional to the stress in each direction. Denoting the discontinuity in brackets (Nairn, 1996a), this relative displacement at the interface is expressed for axisymmetric stress state: [u] =

s r , f ( R , z ) s r ,m ( R , z ) = Dn Dn

2.15a

[w] =

t rz , f ( R, z ) t rz ,m ( R, z ) = Ds Ds

2.15b

where Dn and Ds are constants, R is the fibre radius and the subscripts f and m denote the fibre and the matrix respectively. As is obvious, for a perfect interface the relative displacements are zero and the proportionality constants Dn and Ds become infinite. This approach allows for the scaling of the interface stress field in order to provide realistic interfacial stress values, which are expected to be lower than in the ‘perfect’ interface case (Fig. 2.8). Interfacial damage can also be approached through energy methods (Wagner, 1995; Nairn, 1996a). After a fibre break, a considerable amount of energy is released. This energy has to be absorbed by the matrix. Using the classical fracture mechanics approach, this energy will contribute to the formation of new surfaces, that is, will introduce an amount of debonding at the fibre matrix interface:

Interfacial damage modelling of composites

47

1.2 Ds = •

Normalised stress

1.0 0.8

Ds = 100

0.6

Ds = 1000

0.4

Ds = 10 Ds = 1

0.2 0.0

Ds = 0 –0.2 –0.4 0

50

100 Distance (fibre diameters)

150

200

2.8 Normalised axial stress as a function of the interface parameter D(s) (Nairn, 1996a) (Reprinted, with permission, from Fiber, Matrix, and Interface Properties, (STP 1290), Copyright ASTM International, 100 Barr harbour Drive, West Conshocken PA 19428).

eavailable = 2pR LdGi + 2pR2Gf

2.16

where Ld is the debond length and GI and Gf the interfacial and fibre surface energies respectively. Simultaneously, the energy eavailable that will be used for the creation of the debond areas is (Wagner, 1995):

e available = [U prior + U mprior ] + [U post + U mpost ] f f

2.17

The subscripts f and m denote fibre and matrix, the superscripts prior and post refer to the fracture event and U is the strain energy. The terms of eqn 2.17 are defined as follows:

Ê U prior = 1 E f e 2 p R 2 Á Ld + 2 f 2 Ë

Ú

•

0

ˆ dz ˜ ¯

2.18

Ef is the fibre axial modulus, e is the applied axial or far field strain at the moment of the fracture event and R is the fibre radius. This term corresponds to the energy stored in an infinite fibre under axial strain. The matrix term prior to fracture is:

Ê U mprior = 1 E m e 2 p ( R•2 – R 2 ) Á Ld + 2 2 Ë

Ú

•

0

ˆ dz ˜ ¯

2.19

R• is the radius of the matrix cylinder where there is stress perturbation due to the fibre break. The energy terms after the fracture are: U post =2 f

•

Ú Ú 0

R

0

Ê s z2 ( z ) t rz2 ( z , r ) ˆ Á 2 E + 2G ˜ 2 p r drdz Ë ¯ f f

2.20

48

Multi-scale modelling of composite material systems

where Gf is the fibre shear modulus and Ê U mpost = 1 E m e 2 p ( R•2 – R 2 ) Á Ld + 2 2 Ë •

+2

Ú Ú 0

R

0

t rz2 ( z , r ) 2 p rdrdz 2 Gm

Ú

•

0

ˆ dz ˜ ¯

2.21

Gm is the matrix shear modulus. This analysis assumes that the tensile stress in the matrix is uniform. The shear stress is zero everywhere apart form the perturbation area near the fibre break. The analyses by both Hashin (1990) and Wagner et al. (1995) require knowledge of the stress fields around the discontinuity in order to derive values for either the interface parameters in eqns 2.15 or the integrals in eqns 2.18–21. The incorporation of the imperfect interface provides the flexibility needed to simulate any real conditions in the cases where local material variations are not measurable, or propagating interfacial damage creates differences along the axial direction. Nairn (1996b) have extended their elasticity solution to include an imperfect interface. Interfacial damage is quantified through the use of the parameter D (eqns 2.15), by which any displacement discontinuities are proportional to the associated interfacial stress. When combined with the elasticity solution it may provide analytical expressions for the stress field around the discontinuity that parametrically fit to describe ‘real’ interfaces. Wagner (1995) have used the energy equlibrium equation in conjunction with the classical shear lag analysis to yield analytical expressions which relate the debond length Ld to interfacial and fibre surface energies GI and Gf (eqn 2.16). Moreover, they used experimentally measured debond lengths to calculate the interfacial toughness (Fig. 2.9).

2.5

Experimental measurement of the stress field at the interface

The major problem with the assessment of the interface properties in composites lies either with the complexity of the tests that yield interfacial shear strength which is detrimental to their repeatability or the difficulty of interpreting the acquired data due to the complexity of the induced stress field. Round robin tests (Pitkethly, 1993) have shown that interlaboratory data exhibit large discrepancies even with elaborate control of the experimental conditions. Moreover, the task of linking macroscopic mechanical properties of bulk composites to the interfacial shear strength is not straightforward. Several micromechanical testing methods have been developed (Fig. 2.10), in order to provide a quantitative parameter, which is a sole function of interface properties and is free of geometrical or other artefacts (Pitkethly, 1993).

Interfacial damage modelling of composites 100

E-Glass (unsized)

90

Initial debond length (mm)

49

Experimental data Non-linear best fit: G1 = 183 J/m2

80 70 60 50 40 30 20 10 0 0

5

10

15

20 25 30 35 Applied stress (MPa)

40

45

50

2.9 Experimental data for initial debonding length vs. applied strain for glass fibre/epoxy single fibre composite. The data were best fitted to yield the optimum value for the interfacial fracture toughness (Wagner 1995). (Source: Wagner, 1995, Fig. 2 with kind permission of Springer Science and Business Media.)

In general, successful micromechanical methods should (i) simulate the stress state in practical composites, (ii) define interfacial failure as it would appear under service conditions and, most importantly, (iii) link the generated data to composite properties via a preferably simple and reliable model. The stress state imposed on the fibre during the fragmentation (Kelly, 1965) or the pullout tests (Shiryaeva, 1962) does not necessarily correspond to that encountered in bulk composites. The bundle pullout test (Qiu, 1993) is a more promising configuration but prone to the generation of artefacts due to geometrical variations. The indentation test (Mandel, 1986) is the only test that involves bulk composite testing but cannot simulate fibre failure in the bulk composite, since only a single fibre is loaded and the test is performed on a free surface. All aforementioned methods relate measured quantities to interfacial adhesion indirectly, that is, making fundamental assumptions regarding the stress field around the fibre. More elaborate methods provide the possibility of directly measuring the stress field at the vicinity of the interface in conjunction with traditional micromechanical tests or even in practical composites. Standard photoelastic methods have been succesfully applied to yield the nature of the matrix stress field in specially fabricated model composites. The underlying principle of these techniques is based on the ‘optically anisotropic’ (birefringent) behaviour of the matrix material induced by internal

50

Multi-scale modelling of composite material systems

Knife edges

Fibre

Fibre Matrix Matrix Matrix

Fibre Holder (a)

(b)

(c)

Indentor

Microscope

Composite (d) Matrix

Fibre

(e)

2.10 Interfacial test methods; (a) the pull-out test, (b) the microbundle pull-out test, (c) the microbond test, (d) the microindentation test, and (e) the fragmentation test. (Source: Paipetis, 2001.)

stresses arising due to shrinkage or a macroscopic stress applied to the composite. The shear stress at a given point on an arbitrary x axis is uniquely determined from the difference of the principal stresses (s1 – s2) and the angle f which either principal stresses make with the x axis. The shear stress txy is at the same time a function of the photoelastic constant Cs , the thickness t and the birefringence number N: C t xy = 1 (s 1 – s 2 ) sin 2f = s N sin 2f 2 2t

2.22

Stress transfer characterisation involving different combinations of fibre/ matrix adhesions in model composites were carried out by many researchers and are summarised in Table 2.1 (Vlattas, 1995). In one of the first investigations (Schuster, 1964) photoelastic techniques were employed in order to calculate stress concentration factors in a model SFC specimen (sapphire whiskers embedded in a birefringent resin). Three different whisker end conditions were examined: (i) blunt end, (ii) irregular

Interfacial damage modelling of composites

51

Table 2.1 Photoelastic investigations in model composites (Vlattas, 1995) Reference

System

(Schuster, 1964) (Tyson, 1965) (McLaughlin, 1966,1968) (Allison, 1967) (McGarry, 1968) (Hunter, 1973) (Starikovskii, 1984) (Fiedler, 1997) (Ashbee, 1988] (Patterson, 1998) (Zhao 2003)

Sapphire whisker/Photoelastic resin Dural strip/Araldite Strips PLM-4B/Epoxy resin Dural strip/Araldite Glass fibre/Epoxy resin Glass fibre/Epoxy resin Laminated carbon-reinforced plastic Glass fibe/Epoxy resin Carbon fibre/Epoxy resin Glass & carbon fibre/Urethane acrylate matrix Glass fibre/Epoxy resin

end and (iii) fine tapered end. The last condition caused the smallest stress concentration and the first condition the largest. The shear stresses reached their maximum values at approximately half diameter away from the whisker tip. The radial stresses extended into the matrix by approximately four to five whisker diameters. In recent photoelastic studies, phase stepping photoelasticity has been used to obtain a series of matrix and interfacial shear stress contour maps for model composites at various levels of applied matrix stress on a micron scale during fragmentation of an embedded fibre (Zhao, 2003). The improved photoelastic technique uses a phase-stepping automated polariscope, and four CCD cameras in order to capture simultaneously the four images required for isochromatic and isoclinic parameters over a full field of view. Interference fringe patterns from the matrix and the interface are isochromatic and provide contours of the differences in principal stress. The isoclinics are related to principal stress directions. Finally, the shear stress at the interface can be determined experimentally (Patterson, 1998). As is well established by now, laser Raman spectroscopy can be used to study the stress transfer at the vicinity of the interface with less than one micrometre resolution (Paipetis, 1996). In the case of highly orientated crystalline fibres, the anharmonicity of the molecular vibration causing the Raman activity is responsible for a frequency shift in well-characterised Raman bands (Bretzlaff, 1983; Tashiro, 1990). This shift may be calibrated to provide absolute stress and/or strain values at resolutions of the order of a micrometer. The shift has been well documented for carbon (Melanitis, 1993) or Kevlar® (Vlattas, 1992; Bannister, 1995) fibres. Thus, experimental data can be derived which may be used to assess the validity of existing models. A variation of the Laser Raman technique following the same underlying principle regarding the frequency/stress dependence is the fluorescence

52

Multi-scale modelling of composite material systems

technique (Schawlow, 1961). The stress sensitivity of fluorescence lines has been used to calibrate and subsequently locally measure the local interfacial stress filed in ceramic fibre reinforced materials such as alumina/zirconia reinforced glass (Young, 1996). More recently, the European Cynchrotron Radiation Facility in Grenoble has provided the possibility of using high energy intense X-Ray beams to monitor the stress field in the interface of opaque composites with high enough spatial resolution (Preuss, 2002). The X-ray diffraction pattern relates directly to the lattice spacing allowing thus the calculation of the local strain field in the fibre (Young, 2004). Stress built-up profiles using a 50 mm beam diameter have been provided for Ti/SiCf single fibre model composites (Preuss, 2002). The use of even higher resolution and specific energy beams (5 mm beam diameter) has provided the possibility of accurate absolute stress monitoring in PBO and PPTA fibre/epoxy resin single fibre model composites (Young, 2004).

2.6

Modelling of the experimentally measured stress transfer

Knowledge of the stress field in the vicinity of a discontinuity allows the implementation and validation of existing analytical models which may be used parametrically to provide information about stress transfer efficiency and interfacial integrity. The methods presented in the previous section for the experimental determination in the vicinity of the interface have been used in a variety of cases in order to model, validate or parametrically fit the analytical solutions that appear in the literature. Galiotis and Paipetis (1998) have regarded the shear-lag parameter, b, employed in various problems of shear-lag analysis of composites as an unknown parameter which, in certain cases, can be defined experimentally through the fitting of the stress profiles acquired by laser Raman spectroscopy. They argued that, if b is defined as a fitting parameter for the solution of the shear-lag differential equation, then it can effectively serve as a stress-transfer efficiency index. The use of the Cox equation in order to describe the stress transfer is only valid in the absence of damage at the locus of the interface, or what in their work is described as the ‘elastic region’. Therefore, the experimental monitoring was performed in a model system where a fibre of finite length is embedded in a dumb-bell specimen and laser Raman monitoring was performed prior to any fibre failure (Fig. 2.11). Equation 2.7b derived by Cox (1952) can be written as: 1 È ˘ s z ( z ) = E f e • Í1 – cosh ( bz ) + tanh Ê bl ˆ sinh ( bz ) ˙ 2 Ë ¯ ˚ Î

2.23

Interfacial damage modelling of composites

53

A

2~3 mm

A (a)

A

A 0

20

40

Gauge length/mm 0.1 mm

A-A

2 mm

6.6 mm

4 mm (b)

2.11 Specimen geometries used for laser Raman studies: (a) the discontinuous fibre where the fibre ends are included in the gauge length creating a shear perturbation prior to fibre fracture and (b) the continuous fibre coupon where the fibre extends along the gauge length (Galiotis, 1998; Paipetis, 2001).

1 The function tanh Ê bl ˆ takes the value of unity for any value of bl ≥ 10. Ë2 ¯ For fibre composites with finite fibre length for which l ≥ 2000 mm and b ≥ 0.005, then, from the balance of forces eqn 2.5:

sz(z) = sf,•[1 – cosh(bz) + sinh(bz)]

2.24

sz(z) = sf,•[1 – e–bz]

2.25

or where sf,• is the corresponding axial stress in the middle of the fibre. In terms of strain values eqn 2.20 can be written as:

ez(z) = e•[1 – e–bz]

2.26

The significance of eqns 2.25 and 2.26 is that for finite fibres, the stress or strain distributions in the elastic region can be adequately predicted for a number of fibre/matrix combinations by just treating b as an inverse length

54

Multi-scale modelling of composite material systems

fitting parameter rather than attempting to define it analytically. However, as was pointed out, best results are obtained only if bl ≥ 10. Using laser Raman spectroscopy, Galiotis and Paipetis (1998) experimentally measured sz(z) and ez(z) for any z (as well as, sf,• and e•). The value of transfer length, l0, for which x = l0 = 1/b was found from the experimental axial stress or strain profiles at corresponding values of 0.63 sf,• or 0.63 e•. Alternatively, the shear-lag parameter b could also be obtained by fitting an exponential curve of the form of eqns 2.25 or 2.26 to the experimental fibre stress or strain data, respectively (Paipetis, 2001). They confirmed that over a range of far field strains, the shear-lag parameter b remains a system property, as it could satisfactorily describe the stress transfer (Fig. 2.12). Furthermore, the above analysis was employed to define parametrically the modulus at the interface for given perturbation cylinders R • /R. Thus they confirmed the frictional type of the interface in the case of an unsized system and the existence of an interphase in the case of a sized system, as in the former case the predicted shear modulus was considerably less than the resin shear modulus, whereas in the latter case the interfacial shear modulus was almost double the resin shear modulus (Fig. 2.13). In a similar study, Stanford et al. (Stanford, 2000) used the value of R • /R = 3 in order to fit experimentally measured stress data acquired using laser Raman spectroscopy (Fig. 2.14). 1.4

Fibre stress/GPa

1.2

Applied strain: 0.0%

MUS/MY-750 (Short fibre) Applied strain: 0.3% Applied strain: 0.6%

1 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4 0

0.1

0.2 0.3 0.3 0.2 Distance from fibre end/mm

0.1

0

2.12 Experimental data for the axial stress and shear lag fit with a unique parameter b. (Galiotis, 1998). (Source Galiotis, 1998, Fig. 4 with kind permission of Springer Science and Business Media.)

As already mentioned, the aforementioned approach assumes that the interface is intact, or that the stress state is elastic so that it may be described by the shear lag approach. However, during loading there are several

Interfacial damage modelling of composites 2500

55

Matrix shear modulus/MPa

2000

R/r = 5 R/r = 7 R/r = 10

Sized Unsized

1500

Bulk Gm

1000

500

0 0

0.005

0.01

b/mm–1

0.015

0.02

0.025

2.13 Predicted interfacial modulus for two studied systems (Galiotis, 1998). (Source: Galiotis, 1998, Fig. 7 with kind permission of Springer Science and Business Media) 0.5

Fibre strain (%)

0.4

0.3

em = 0.27%

0.2

0.1

em = 0%

0.0

–0.1 –2

0

2

4 6 8 10 12 Distance along the fibre (mm)

14

16

2.14 Tensile strain distribution curve along a copolymer coated glass fibre/epoxy single fibre composite and the respective shear lag fit. The copolymer coating is serving as the Raman sensor (Stanford, 2000) (Reprinted from Stanford, 2000 with permission from Elsevier).

mechanisms that result in the reduction of the stress transfer efficiency of the interface. The shear perturbation near a fibre failure results in the existence of ‘zones’ of variable stress transfer efficiency along the fibre length (Melanitis, 1992). Starting from the locus of fibre fracture, the latter were identified as (i) a zone where the stress transfer is poor, (ii) an intermediate region, where

56

Multi-scale modelling of composite material systems

there is a gradual improvement of the axial stress transfer, and (iii) a ‘well bonded’ region, where stress is transferred efficiently. With increasing applied far field strain, the damaged zone increases to the detriment of the wellbonded zone, whereas the intermediate zone remains constant. Yallee and Young (1998) used a shear-lag approach to model Raman experimental data of single fibre composites, assuming the existence of a debonded zone and an elastic zone to calculate the interface fracture energy. Nairn (1996b) modelled a similar set of data assuming the existence of only ‘poor’ and ‘efficient’ regions (Fig. 2.15). These could be identified through a single interface parameter that corresponds to axial displacements relative to the elastic solution of the problem (Hashin, 1990). By fitting measured axial fibre stress to the Bessel-Fourier series stress analysis in undamaged regions of the fibre/matrix interface, it was possible to measure an interfacial stress transfer property denoted as D(i) s which may then be employed as a material property for predicting the role of the interface in real composite laminates. As the value of D(i) s gets higher, the stress transfer is more efficient. Furthermore, the higher the D(i) s the smaller the amount of energy released when a fibre breaks at low strain (i.e., the strains typically experienced in real laminates) and thus the less interfacial damage that will be caused by those breaks. Fragmentation tests are typically continued to high strain or until the fragmentation process ceases. By analysing LRS experiments as a function of strain Nairn (1996b) managed to follow the length of the interfacial damage zone (ld) and the extent of interfacial damage (characterized by the value of D(d) s) and deduce an interfacial shear strength (ISS). 1.2

Normalised stress

1.0 0.8 0.6 0.4 0.2 0.0 0

100 Distance (fibre diameters)

200

2.15 Two-zone model for interfacial adhesion (Nairn, 1996b) (Reprinted, with permission, from Fiber, Matrix, and Interface Properties (STP 1290), copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428).

Interfacial damage modelling of composites

57

In a more recent paper, Paipetis et al. (1999) proved that high-strain observations usually employed from fragmentation tests are less useful for characterising the fibre/matrix interface, as they are all influenced by the value of D(i) s . A ‘good’ interface, that is one with high D(i) s, leads to the presence of more damage at high strains than that observed in a system with a ‘poor’ interface. By comparing a sized carbon fibre and an unsized carbon fibre epoxy system, they proved that the presence of such damage can mask the difference between the two systems, or that the fragmentation test data can only be unambiguously interpreted if the interfacial damage zone is fully identified and distinguished from the intact interface (low strain data). In summary, the above analysis treats the efficiency of the system as a fraction of the theoretical efficiency that the elasticity solution yields. It defines a stress transfer property D(i) s, which expresses the strain and position dependent property that characterises the damage zone. However, it should be noted that the separation into zones of variable efficiency very much depends on the (i) interpretation of the experimental data and (ii) the implementation of a universal analytical stress transfer model to describe all states of adhesion. In addition, abrupt changes in the ISS distribution that certain models require (Piggott, 1980; Nairn, 1996a; Paipetis, 1999) are not realistic, as they result in shear discontinuities. On the other hand, models that allow continuous changes in the material properties would derive a continuous ISS distribution but they would require an infinite number of damaged zones. Paipetis and Galiotis (2001) adopted an alternative approach that involves the assessment of the energy dissipation during fibre fragmentation by just comparing a given stress transfer state to a well defined reference state. In fact, they define the latter as the state, where no damage is observed and no energy is dissipated (Cox, 1952; Nairn, 1996a; Galiotis, 1998). Fibre fracture releases a considerable amount of energy that has to be dissipated locally. Further loading or increase in the far field strain, triggers secondary failure mechanisms. Therefore, the resulting stress field constantly diverges from the equivalent stress field of the elastic case. On the contrary, at low applied strains in discontinuous fibre systems (Fig. 2.11) the system can be considered as elastic and a single interface parameter may be derived as an index of the fibre/matrix adhesion (Galiotis, 1998). However, when the applied strain approaches the fibre fracture strain, the stress transfer profile starts diverging from its elastic equivalent state. By further extending their study of the undamaged interface Paipetis and Galiotis (2001) used LRS to monitor interfacial efficiency in the presence of damage. If the stress transfer is elastic, that is, if the value of the parameter b remains constant for any applied strain e•, the energy eelastic in the strained fibre can be calculated by integrating equation 2.7b along the length of the fragment:

58

Multi-scale modelling of composite material systems

e elastic =

Ú

l

z =0

È 1 Ê ˆ˘ cosh Á b Ê – z ˆ ˜ ˙ Í ¯¯ Ë Ë2 ˙ dz = E f e • I È1 – 2 tanh b l ˘ E f e • Í1 – ÍÎ 2 ˙˚ bl 1 Í ˙ cosh Ê b ˆ Í ˙ Ë 2¯ Î ˚

2.27 In the case of the measured axial stress profiles, the stored energy on the fibre can be calculated by integrating numerically the measured axial stress values along the fragment length:

e stored =

Ú

l

z =0

s 2 ( z ) dz

2.28

Through equations 2.27 and 2.28, the index of stress transfer efficiency z may be defined, such as:

z=

e stored e elastic

2.29

z provides a measure of the stress transfer efficiency for all systems as a function of applied strain. Although this approach cannot account for the complexity of the stress field along the axis of the fibre, it offers a way of monitoring the energy dissipation mechanisms at the interface without the need of damage length measurements. In Fig. 2.16 (Paipetis, 2001), the axial stress measured by laser Raman spectroscopy is superimposed onto the predicted elastic axial stress distribution for relatively low (Fig. 2.16(a)) and high (Fig. 2.16(b)) applied strains. The prediction was made on the assumption that the interface was intact for a given fragment and the maximum stress in the middle of the fibre is identical to that measured experimentally. The area under both stress vs. length curves corresponds to the elastic stored energy in the fibre per unit fibre crosssectional area. As is evident, the difference in the areas under the two curves corresponds to the energy loss during the fragmentation process. In Fig. 2.17 depicts the parameter z as a function of applied strain. In conjuction with the parametric use of the shear lag parameter b to describe the undamaged state, the parameter z can be used to describe the interfacial damage evolution. Paipetis and Galiotis (2001) showed that the post fibre fracture interface can never be as efficient as in the ideal elastic case and, therefore, energy dissipation is inherent in the fragmentation process. The values of z remained relatively constant for a fairly broad strain region, (Fig. 2.17), where the fracture events are the main contributor to interfacial damage. Secondary damage was significant at high strains (Fig. 2.17), which supersede by far the service strains of ‘practical composites’. As already shown (Melanitis, 1993; Nairn, 1996a; Paipetis, 1999), the useful information

Interfacial damage modelling of composites 5

59

Fibre stress/GPa

4 3 2 1 0 –1 25.8

25.9

26

26.1

26.2

Applied strain: 1.7%

Spline fit

26.3

26.4

Equivalent elastic state

(a) 5 4 3 2 1 0 –1 25.8

25.9

26

Applied strain: 4.8%

26.1

26.2

Spline fit

26.3

26.4

Equivalent elastic state

Position along the fibre/mm (b)

2.16 Axial stress profiles for the intact interface (calculated) and the damaged interface (measured experimentally) for the same far field strain (Paipetis, 2001). (Source: Paipetis, 2001.) 1

z

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

1.5

2

2.5

3

3.5

4

4.5

5

2.17 Index of stress transfer efficiency z as a function of applied strain (Paipetis, 2001). (Source: Paipetis, 2001.)

60

Multi-scale modelling of composite material systems

Tensile stress

Stress concentration Tensile strength

Ineffective length

2.18 Tensile strength of continuous fibre-reinforced laminae as a function of the ineffective length (Paipetis, 2001). (Source: Paipetis, 2001.)

that can be drawn from the fragmentation process is confined to strain ranges concerned with the onset and the bulk of the fragmentation process and not beyond the saturation stage. In other words, the deterioration of stress transfer at the initiation of the fragmentation process, as described by z, corresponds more closely to the conditions that are met in structural composites.

2.7

Overview and conclusions

The role of the interface in composite materials is crucial to the behaviour of the material in the presence of shear activated mechanisms. The interface may define the mode of failure and play a crucial role in the ultimate tensile strength of unidirectional composites (Drzal, 1993). Although the interface is easily envisaged as the common surface between phases, its complexity from macro to nano scale is rather leading to the more realistic assumption of an ‘interphase’ bridging the properties of the two phases (Theocharis, 1987). This more realistic view of the interface is reflected in the difficulty of modelling the interface as a two-dimensional entity as this approach is bound to lead to stress discontinuities that cannot be justified physically (McCartney, 1992; Nairn, 1996a). Simplifying assumptions are needed in order to find a closed form solution to the problem, which arguably are detrimental to its accuracy. The difficulty of interpretation and the extremely poor repeatability of the various micromechanical tests that relate to interfacial strength do not improve the situation (Pitkethly, 1993). However, the existence of more elaborate methods has provided the possibility of quantitatively measuring the stress at the interface. State of the art photoelastic methods (Zhao, 2003), laser Raman

Interfacial damage modelling of composites

61

spectroscopy (Galiotis, 1993; Bannister, 1995), fluorescence sprectroscopy (Young, 1996), or even high resolution X-ray monitoring at the European cynchrotron facilities (Young, 2004; Preuss, 2002) have provided considerable insight regarding the nature and the magnitude of the stress field near the shear perturbation area. The insight that the aforementioned methods provided opened the way to the validation of the analytical models and/or the understanding of the nature of the stress field. As models like the popular shear lag model (Cox, 1952) or more elaborate ones (McCartney, 1992; Nairn, 1996a) in most cases overestimated the shear stresses, most approaches adopted were led by the need to ‘scale’ the stress field in order to render it realistic. As has been shown in the literature (Galiotis, 1998), the undamaged interface, or the interface that still is in the elastic stress state can be adequately described through the use of the single parameter b. This parameter is independent of the applied far field strain and is easily calculated from the experimental axial stress profiles using the technique of laser Raman spectroscopy. However, the ‘intact interface’ has also been presented as having a measurable amount of damage which is proved by the divergence of the measured stress values from the nearly exact stress solution (Nairn, 1996a). In this case this damage constant coincides with the relative displacement at the interface as defined by Hashin (1990). Apart from this constant that characterises the material system and is independent of strain, other displacement constants are needed in order to describe the evolution of damage at the interface. These constants are no longer far field strain independent, and the length that they describe is increasing with strain. The efficiency of the stress transfer as a function of applied strain depends on the amount of dissipated energy every time a fibre fracture occurs. Throughout this process, the interface is bound to suffer a loss in its stress transfer efficiency. Through an alternative approach (Paipetis, 2001), this is described through a z parameter, which depends on the applied strain and is an index of the energy dissipation resulting from any form of interfacial damage events. The results show that the parameter z remains fairly constant for moderate strains after the first fragmentation event, whereas at higher strains interfacial failure in the form of matrix yielding, fibre/matrix debonding and/or matrix shear cracking reduce its value considerably. The most important conclusions may be summarised as follows: (i) a fibre fracture is bound to damage the interface, (ii) this damage is dominant for a fairly wide range of applied strains which are well above the service strains of a composite and (iii) at fairly large far field strains, secondary mechanisms other than fractures lead to considerable reduction in the stress transfer efficiency. Concluding, it is mostly definite that shear activating mechanisms such as fractures will always be associated with an amount of damage compared to the undamaged state. As the undamaged state can be uniquely described

62

Multi-scale modelling of composite material systems

parametrically, and the amount of primary damage is also measurable, there is the possibility to derive data that will provide the insight for linking the interface to composite properties, which may be performed by the prediction of ineffective length (Reifsnider, 1994) and its correlation to composite properties (Fig. 2.18) (Paipetis, 2001).

2.8

References

Allison I.M. and Hollaway L.C., Brit. J. Appl. Phys., 18, 979 (1967). Ashbee K.H.G. and Ashbee E., J. Comp. Mater., 22, 602 (1988). Bannister D.J., Andrews M.C., Cervenka A.J. and Young R.J., Composites Science and Technology, 53, 411 (1995). Bretzlaff R.F. and Wool J., Macromolecules, 16, 1907 (1983). Cox C.L., The British Journal of Applied Physics, 3, 72 (1952). Dow N.F., General Electric Co., Missiles and Space Div., Report No. R-635D61 (1963). Drzal L.T. and Madhucar M., Journal of Materials Science, 28, 569 (1993). Ebert L.J. and Gadd J.D., in Fiber Composite Materials, American Society for Metals, Ohio, 89 (1965). Fiedler B. and Schulte K., Composites Science and Technology, 57, 859, (1997). Filon L.N.G., Transactions of the Royal Society (London), 14, 147 (1902). Galiotis C., Composites Science and Technology, 48, 15 (1993). Galiotis C. and Paipetis A., J. Mater. Sci., 33, 1137–1143 (1998). Guigon M. and Klinklin E., Composites, 25, 534 (1994). Halpin J.C., Primer on Composite Materials Analysis, 2nd edn, Technomic Publishing Company Inc., Lancaster, Pennsylvania, U.S.A. (1992). Hashin Z., Mechanics of Materials, 8, 333 (1990). Heppenstall-Butler M., Bannister D.J. and Young R.J., Composites: Part A, 27A, 703 (1996). Herrmann K.P., Noe A. and Dong M., Composites: Part A, 27A, 813 (1996). Hull D. and Clyne T.W., An Introduction to Composite Materials, 2nd edn, Cambridge University Press, Cambridge (1996). Hunter R.W., J. Aust. Inst. Met., 18, 173 (1973). Jones R.M., Mechanics of Composite Materials, McGraw-Hill Kogakusha, Ltd., Tokyo (1975). Kelly A. and Tyson W.R., Journal of Mechanical Physics and Solids, 13, 329 (1965). Lekhnitski S.G., Theory of an Anisotropic Body, Holden Day Inc., San Francisco, California (1963). Long S. and Flower H.M., Composites: Part A, 27A, 833 (1996). Mandel J.F., Grande D.H., Tsiang T.H. and McGarry F.J., in Composite materials: testing and design, ASTM STP 893, American Society for Testing and Materials, 87 (1986). McCartney L.N., Proc. R. Soc. Lond., A425, 215 (1989). McCartney L.N., Local Mechanics Concepts for composite material Systems, ed. by Reddy J.N. and Reifsneider K.L., Proc. IUTAM Symposium, Blacksburg, VA, 251, (1992). McGarry F.J. and Fuhwara M., 23rd Annual Technical Conference, Reinforced Plastics Composites Division, The Society of the Plastics Industry, Section 9-B, 1 (1968). McLaughlin T.F., Experimental Mechanics, 6(19), 1 (1966).

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McLaughlin T.F., J. Comp. Mater., 2(1), 44 (1968). Melanitis N. and Galiotis C., Proc. R. Soc. Lond. A, 440, 379 (1993). Nairn J.A., Mechanics of Materials, 13, 131 (1992). Nairn J.A. and Liu Y.C., Composite Interfaces, 4, 241, (1996a). Nairn J.A., Liu Y.C. and Galiotis C., in Fiber Matrix and Interface Properties, ASTM STP 1290, American Society for Testing and Materials, 47 (1996b). Nairn J.A. Mech. Mater. 26, 63, (1997). Nayfeh A.H., Fibre Science and Technology, 10, 195 (1977). Paipetis A. and Galiotis C., Proceedings of the Royal Society of London, 2011, 1555 (2001). Paipetis A., Galiotis C., Liu Y.C. and Nairn J., Journal of Composite Materials, 33(4), 377 (1999). Paipetis A., Vlattas C. and Galiotis C., Journal of Raman Spectroscopy, 27, 519 (1996). Patterson E.A. and Wang Z.F., ‘Simultaneous observation of phase-stepped images for automated photoelasticity’, J Strain Anal, 33 (1998), 1–15. Piggot M.R., Acta Metallurgica, 14, 1429 (1966). Piggot M.R., ‘Load-Bearing Fibres’, Pergamon Press (1980). Piggot M.R., Polymer Composites, 8, 291 (1987). Pitkethly M.J., Favre J.P., Gaur U., Jakubowski J., Mudrich S.F., Caldwell D.L., Drzal L.T., Nardin M., Wagner H.D., Di Landro L., Hampe A., Armistead J.P., Desaeger M. and Verpoest I., Composites Science and Technology, 48, 205 (1993). Preuss M., Rauchs G., Withers P.J., Maire E. and Buffiere J.-Y., Composites A, 33 1381, (2002). Qiu Y. and Schwartz P., Composites Science and Technology, 48, 5 (1993). Reifsnider K.L., Composites, 25, 461 (1994). Rosen B.W., in Fiber Composite Materials, American Society for Metals, Ohio, 37 (1965). Schawlow A.L., in Advances in Quantum Electronics (ed. Senger J.R.), Columbia University Press, New York 1961, 50. Schuster D.M. and Scala E., Trans. Metallurgical Society AIME, 230, 1635 (1964). Shiryaeva G.V. and Andreevska G.D., The Plastics, 4, 40 (1962). Stanford J.L., Lovell P.A., Thongpin C. and Young R.J., Composites Science and Technology, 60, 361–365 (2000). Starikovskii G.P. and Shcherbakov V.T., Mekhanika Kompozitnykh Materialov, 4, 719 (1984). Tashiro K., Wu G. and Kobayashi M., J. Appl. Polym. Sci., 28, 2038 (1990). ten Busschen A. and Selvadurai A.P.S., Journal of Applied Mechanics, 62, 87 (1995). Theocaris P.S., The mesophase concept in composites, ed. by Henrici-Olive G. and Olive S. Springer-Verlag Berlin, Heidelberg (1987). Timoshenko S.P. and Goodier J.N., Theory of Elasticity, McGraw-Hill, New York (1988). Tyson W.R. and Davies G.J., Brit. J. Appl. Phys., 16, 199 (1965). Vlattas C. and Galiotis C., in Interface Micromechanics in Aramid/Epoxy Model Composites in Developments in the Science and Technology of Composite Materials, ECCM 5, eds Bunsell A.R., Jamet J.F. and Massiah A. Bordeaux, France, 415 (1992). Vlattas C., PhD thesis, University of London (1995). Wagner H.D., Nairn J.A. and Detassis M., Applied Composite Materials, 2, 107, (1995). Yallee R.B. and Young R.J., Composites Part A – aApplied Science and Manufacturing, 29, 1353 (1998). Young R.J. and Yang X., Composites part A, 27A, 1996 737–741.

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Young R.J., Eichhorn S.J., Shyng Y.T., Riekel C. and Davies R.J., Macromolecules, 37, 9503–9509 (2004). Zhao F.M., Hayes S.A., Patterson E.A., Young R.J. and Jones F.R., ‘Measurement of micro stress fields in epoxy matrix around a fibre using phase-stepping automated photoelasticity’, Compos Sci Technol, 63 (2003), 1783–1787.

3 Multi-scale predictive modelling of cracking in laminate composites L N M c C A R T N E Y, National Physical Laboratory, UK

3.1

Introduction

Multi-scale modelling is a term of general applicability, even if constrained to the composites field. In its broadest sense multi-scale modelling can be described as traversing all length scales from atoms through to engineering structures of significant size. The predictive tools applicable to multi-scale modelling could in fact range from quantum mechanics, through statistical mechanics, group interaction modelling, to the classical approaches based on linear elastic and non-linear elastic-plastic, not to mention visco-plastic mechanics. The ‘structural’ features determining observed behaviour range from the electrons of quantum mechanics, through atoms and molecules of molecular dynamics, to the fibres and matrix of unidirectional composites, and to plies in laminates having complex lay-ups that are used during the construction of structural elements and engineering structures. This chapter will focus only on a limited range of the modelling methods just described, namely from fibres and matrix through to laminated composites often used in many types of engineering structure. At the fibre and matrix level it is assumed that their behaviour can be characterised by the continuum methods associated with linear elastic behaviour. Section 3.2 will describe the principles involved with one validated model to predict from fibre and matrix properties, and fibre volume fraction, all the thermo-elastic constants of a unidirectional composite that may be used to characterise the properties of the plies in laminates. An elasticity method is used, rather than classical laminate theory that is based on force and moment resultants, as the major objective is to be able to use elasticity methods to predict all the thermoelastic constants of laminates having damage in the form of ply cracks. The elasticity analysis for undamaged laminates, to be considered in section 3.3, is the baseline from which the damage analysis is developed in section 3.4. It is emphasised that the elasticity analysis applied to damaged laminates predicts exactly the same type of stress-strain equations and thermo-elastic constants as those involved when using classical laminate analysis to predict 65

66

Multi-scale modelling of composite material systems

the behaviour of undamaged laminates. The next stage of multi-scale modelling is to describe how first ply cracking, and the subsequent progressive growth of damage, may be predicted using energy methods that are also described in section 3.4. The final stage is to predict the failure of laminates using physically based methods rather than empirical failure criteria that are in widespread use. This important issue will be considered in section 3.5. Three different length scales are thus being considered in the multi-scale models that have just been described, namely, (i) fibre and matrix where fibre diameters are usually measured in microns, (ii) unidirectional plies that usually have thicknesses that range from 0.1 mm to 0.5 mm, and (iii) general symmetric laminates where laminate thicknesses usually range from 1 mm to 10 mm, although thick-section laminates are becoming more common in structures such as aircraft wings and wind turbine blades, not to mention yacht masts. At each length scale, the local thermo-elastic properties can be captured by considering appropriate representative volume elements. Ideally the representative volume element can be embedded in the equivalent homogeneous equivalent without perturbing the stress field. This concept is fundamental to the approaches to be described, and it has been used to test the integrity of the modelling at each of the three length scales.

3.2

Predicting undamaged ply properties

This section is concerned with the deformation processes that occur in continuous fibre unidirectional (UD) composites subject to axial, transverse and shear loading. The results to be presented can be applied to any fibrereinforced composite whose constituents can be regarded as deforming in a linear manner. In general, it is assumed for UD composites that the fibre and matrix are made of different materials (possibly anisotropic having directional properties), and that in the undamaged state they are perfectly bonded everywhere along the length of the fibre/matrix interfaces. The cross-section of the fibres is assumed to be uniform and circular. The fibres are assumed to be uniformly distributed having a hexagonal packing structure leading to transverse isotropic UD composite properties.

3.2.1

Prediction of thermo-elastic constants for undamaged plies

This section presents suitable approximate analytical methods for calculating the thermo-elastic constants of undamaged unidirectional (UD) composites, some of which are almost impossible to measure. Concentric cylinder models of fibre and matrix in conjunction with variational techniques have been used extensively to predict the properties of undamaged UD composites [1]. UD composites are realistically modelled as transverse isotropic solids (i.e.

Multi-scale predictive modelling of cracking

67

axial properties differ from isotropic properties in the plane normal to the fibre direction). UD composites are characterised by seven independent thermoelastic constants (written in italics to distinguish them from corresponding laminate properties): (i) axial Young’s modulus (EA); (ii) axial shear modulus (mA); (iii) transverse Young’s modulus (ET); (iv) axial Poisson’s ratio (nA); (v) transverse Poisson’s ratio (nT); (vi) axial thermal expansion coefficient (aA); and (vii) transverse thermal expansion coefficient (aT). These parameters appear in stress-strain relations for a UD composite of the form:

n n e rr = 1 s rr – T s qq – A s zz + a T DT ET ET EA

3.1

e qq = –

nT n s + 1 s – A s + a T DT E T rr E T qq E A zz

3.2

e zz = –

nA n s – A s + 1 s + a A DT E A rr E A qq E A zz

3.3

e rz =

s rq s s rz , e qz = qz , e rq = 2 m 2mA 2mA T

3.4

The stress s and strain e components refer to cylindrical polar coordinates (r, q, z), with the z-axis orientated parallel to the fibres. The most convenient method for estimating the properties of a unidirectional composite, as a function of fibre properties, matrix properties and fibre volume fraction Vf, uses a concentric two-cylinder model where the inner cylinder of radius R represents the fibre and the outer cylinder, representing the matrix, has an outer radius of a = R/ Vf . For an undamaged UD composite having fibre and matrix volume fractions Vf and Vm (= 1 – Vf) respectively, the axial Young’s modulus, axial Poisson’s ratio and transverse bulk modulus are given by

E A = Vf E fA + Vm E Am + 2 l ( n Am – n fA ) 2 Vf Vm

3.5

Ê ˆ n A = Vf n fA + Vm n Am – l ( n fA – n Am ) Á 1f – 1m ˜ Vf Vm 2 kT ¯ Ë kT

3.6 2

2 1 ∫ 2 (1 – n T ) – 4n A = Vf + Vm – l Ê 1 – 1 ˆ V V f m ˜ ET EA 2 ÁË k fT kT k Tm k Tm ¯ k fT

3.7 where

1 = 1 È 1 + Vf + Vm ˘ . l 2 ÍÎ m Tm k Tm k fT ˙˚

3.8

68

Multi-scale modelling of composite material systems

E, n, k, m and a denote Young’s modulus, Poisson’s ratio, bulk modulus, shear modulus and thermal coefficient of expansion. Suffices f and m denote fibre and matrix respectively. As the values of EA and nA are known, the value of (1 - nT)/ET can be calculated using eqn 3.7. Assuming that the composite is transversely isotropic, then the following relationship holds: ET = 2mT(1 + nT), but EA π 2mA(1 + nA). Clearly, both ET and nT can be calculated once the value of mT has been determined. Christensen and Lo [2] developed a concentric cylinders model (including a surrounding effective composite region) that can be used to estimate transverse shear modulus, which lies between the bounds given by Hashin [1], and is likely therefore to predict more accurate values. The model can be used to calculate the transverse shear modulus mT, thus enabling values of ET and nT to be determined. The following expression given by Hashin [1] (beware a misprint in his paper), leads to either an upper or lower bound value for mT, depending upon the values of mf, mm, kf and km

m T = Vf m fT + Vm m Tm – l 1 ( m fT – m Tm ) 2 Vf Vm where m m 1 = V mf + V mm + kT mT 3.9 m T f T l1 k Tm + 2 m Tm Hashin’s [1] expression for the axial shear modulus mA may be written as:

m A = Vf m fA + Vm m Am – l 2 ( m Am – m fA ) 2 Vf Vm where 1 = m m (1 + V ) + m f V f A A m l2

3.10

The axial and transverse thermal expansion coefficients predicted by the model may be written as: E A a A = Vf E fA a fA + Vm E Am a Am + 2 l ( n Am – n fA ) Vf Vm ¥ [ a Tm + n Am a Am – a fT – n fA a fA ]

3.11

a T = – n A a A + ( a fT + n fA a fA ) Vf + ( a Tm + n Am a Am ) Vm Ê ˆ + l Á 1f – 1m ˜ Vf Vm [( a Tm + n Am a Am ) – ( a fT + n fA a fA )] 3.12 2 Ë kT kT ¯ The relations (3.10–3.12) are equivalent to those given by Hashin [1], and Rosen and Hashin [3], but they are expressed here in a common format consistent with the results (3.5–3.9).

Multi-scale predictive modelling of cracking

3.2.2

69

Characteristics of the two-cylinder fibre/matrix model

The key requirement is to be able to embed a discrete model of a fibre and matrix within an effective homogeneous anisotropic medium such that the thermo-mechanical behaviour of the assembly of the discrete model embedded within the effective medium is identical to that for a homogeneous system having the properties of the effective medium. When the two-cylinder model and the Lame solution are used for the case of biaxial loading, an exact equivalence between the properties of the two-cylinder model and the surrounding effective medium is obtained. Similarly, when using the twocylinder model for the case of axial shear loading, there is again an exact equivalence. However, when the two-cylinder model is subject to transverse shear loading, an exact equivalence is not possible. For this case a residual stress concentration is found to exist in the surrounding effective medium at the location of the interface between the effective medium and the outer cylinder representing the matrix. This issue has been discussed in reference [4] describing a multiple concentric cylinder model.

3.3

Undamaged laminate properties

A composite laminate is a perfectly bonded assembly of individual plies of a unidirectional composite for which the fibres are parallel in a direction that defines the orientation of the ply within the laminate. Before considering damage formation in such laminates, it is necessary to describe the methods for calculating the undamaged properties of the laminate. Classical laminate theory has extensively been used to achieve this objective making use of stress resultants (see for example [5, 6, 7]). Here an elasticity approach is considered, as undamaged laminate theory is the baseline from which damage mechanics will be developed. The assumptions on which the methodology is based are: 1. Laminates are symmetric (i.e. ply orientations are symmetric about midplane of laminate) and constructed of any number of plies that can be made of different anisotropic materials having directional properties, and are perfectly bonded together at all points on the inter-laminar interfaces. 2. Laminates are subject only to loading in the plane of the laminate. Bend deformation of the laminate is not considered. 3. There exists a stress-free temperature To at which laminates, when in an unloaded state, are free of any internal stresses arising from thermal expansion mismatch between plies. The temperature difference DT = T – To, where T is the current temperature, appears in the stress-strain

70

Multi-scale modelling of composite material systems

relations and, when non-zero, leads to thermal residual stresses in the composite. 4. The undamaged plies of laminates have directional properties and linear stress-strain behaviour.

3.3.1

Undamaged general symmetric laminates

In order to characterise the properties of laminates it is useful to introduce orthogonal coordinates to describe the directions of the plies in the laminate. These are the axial y-direction, the transverse z-direction that is normal to the axial direction lying in the plane of the laminate, and the through-thickness x-direction. The stress and strain components referred to these directions are denoted by sxx, exx, sxy, exy, etc. It is essential to be able to calculate the thermo-elastic properties of a general symmetric laminate in an undamaged state referred to a global set of axes. The building block is a set of thermoelastic properties for an undamaged single UD ply using the local coordinate system where the axial direction is aligned with the fibres. The stress-strain relations for any ply in the undamaged laminate whose fibres are parallel to the axial y-direction are of the following form where the superscript i is a label denoting the ply being considered e ixx =

ni ni i s ixx – ia s iyy – it s zz + a it DT i Et EA ET

3.13

e iyy = –

s iyy n iA i n ai i s + – s zz + a iA DT xx E iA E iA E iA

3.14

i e zz = –

n iA i n it i si s – s yy + zzi + a iT DT xx i i ET EA ET

3.15

e ixy =

s ixy , 2 m ai

e ixz =

s ixz , 2 m it

e iyz =

s iyz 2 m iA

3.16

where EA, ET, Et, na, nt, nA, ma, mt, mA, at, aA and aT denote the Young’s moduli (E), Poisson’s ratios (n), shear moduli (m) and thermal expansion coefficients (a) for the undamaged laminate. The upper-case suffices A (axial), T (transverse) indicate that the property to which it is attached is for a direction of the laminate lying within the plane of the laminate. Lower case suffices are attached to properties involving out-of-plane deformation. When the fibres of an individual ply are orientated in the axial direction, and the composite is isotropic in directions normal to the fibres, the stress-strain relations are given in section 3.2, together with formulae that can be used to estimate the thermo-elastic constants of undamaged plies from the fibre and matrix properties.

Multi-scale predictive modelling of cracking

71

For other orientations of the plies, where the fibres in the ith ply are parallel and inclined so that the angle between the fibre direction and the yaxis is fi, the stress-strain relations are of the form [8] i i i i i e ixx = g 11 s ixx + g 12 s iyy + g 13 s zz + g 14 s iyz + a 1i DT

3.17

i i i i i e iyy = g 12 s ixx + g 22 s iyy + g 23 s zz + g 24 s iyz + a 2i DT

3.18

i i i i i i e zz = g 13 s ixx + g 23 s iyy + g 33 s zz + g 34 s iyz + a 3i DT

3.19

i i i i 2 e iyz = g 14 s ixx + g 24 s iyy + g 34 s zz + g i44 s iyz + a i4 DT

3.20

i i i i e ixy = a 11 s ixy + a 12 s ixz, e ixz = a 12 s ixy + a 22 s ixz

3.21

where, in terms of mi = cos fi, ni = sin fi, the coefficients appearing in eqns (3.17–3.21) are: ni ni ni ni i i i = – m 2i it – n 2i ia , = – m 2i ia – n 2i it , g 13 g 11 = 1i , g 12 ET EA EA ET Et Ê ni ni ˆ i g 14 = 2m i n i Á – it + ia ˜ EA ¯ Ë ET

3.22

m4 n4 2 ni ˆ Ê i g 22 = m 2i n 2i Á 1i – i A ˜ + ii + ii , EA ¯ EA ET Ë mA

ni ni Ê ˆ i g 23 = m 2i n 2i Á 1i + 1i – 1i ˜ – m i4 Ai – n i4 Ai mA ¯ ET EA EA Ë EA

3.23

2n 4 2 ni ˆ 2 m 2 Ê i g 24 = m i n i (m 2i – n 2i ) Á 1i – i A ˜ – i i + i i . EA ¯ EA ET Ë mA m4 n4 2 ni ˆ Ê i g 33 = m 2i n 2i Á 1i – i A ˜ + ii + ii EA ¯ ET EA Ë mA

3.24

È 2n 2i ˘ 2 n i ˆ 2 m 2i Ê i g 34 = m i n i Í (m 2i – n 2i ) Á – 1i + i A ˜ + – ˙ EA ¯ E iT E iA ˚ Ë mA Î

3.25

g i44 =

i (m 2i – n 2i ) 2 1 + 2nA ˆ 2 2 Ê 1 + 4 m n + ˜ i i Á i m iA E iT E iA ¯ Ë EA

3.26

a 1i = a it , a 2i = m 2i a iA + n 2i a 1T , a 3i = m 2i a iT + n 2i a 1A , a i4 = 2m i n i ( a iT – a iA )

3.27

72

Multi-scale modelling of composite material systems

n2 ˆ Ê m2 Ê ˆ i i a 11 = 1 Á ii + ii ˜ , a 12 = 1 m i n i Á 1i – 1i ˜ , 2 Ë ma 2 mt ¯ ma ¯ Ë mt n2 ˆ Ê m2 i a 22 = 1 Á ii + ii ˜ 2 Ë mt ma ¯

3.3.2

3.28

Macroscopic effective stress-strain relations for any undamaged symmetric laminate

A symmetric laminate has a ply geometry that is symmetric about the midplane of the laminate. Consider a laminate of total thickness 2h comprising 2N plies. The plies in one half of the laminate are labelled i = 1 … N. The characteristic property of undamaged symmetric laminates subject to inplane loading is that the axial, transverse and in-plane shear strains, e, eT and g respectively, are uniform throughout those parts of the laminate that are not affected by the presence of laminate edges, and that the through-thickness stress st is also uniform in all plies of the laminate. In addition the stress components sxy, sxz are zero everywhere. When the laminate as a whole is considered the stress-strain relations (3.17–3.21) are applied to each ply in i = e T , 2 e iyz = g, sxx = st. The effective the laminate such that e iyy = e , e zz i , s iyz will be applied axial, transverse and in-plane shear stresses s iyy , s zz uniform in each ply but will be discontinuous at ply interfaces. The effective axial stress s, transverse stress sT, and in-plane shear stress t are defined by total normal and shear loads acting on the laminate edges divided by the total area of cross-section of the edges over which the load is acting. The through-thickness displacement on the laminate faces is divided by the half-thickness of the laminate when defining the through-thickness strain et. It is then possible to calculate the effective stresses s, sT and t in each ply together with the effective through-thickness strain et using N

N

N

i s = 1 S h i s iyy , s T = 1 S h i s zz , t = 1 S h i s iyz , h i=1 h i=1 h i=1 N

e t = 1 S h i e it h i=1

3.29

where e it is the uniform through-thickness strain in the ith ply. These values are then used to calculate the effective stresses s, sT and t and throughthickness strain et for the laminate as a whole. Stress-strain relations for the laminate may then be derived [8] which are of the same form as the relations (3.17–3.21) assumed for individual ply properties. The laminate stress-strain relations are written in the following form that generalises the notation given in (3.13–3.16) to include shear-coupling terms

Multi-scale predictive modelling of cracking

et =

st n n l – a s – t s T – t t + a t DT , Et EA ET EA

e= –

3.30

na n l s + s – a s T – A t + a A DT EA t EA EA EA

3.31

nt n s l s – A s + T – T t + a T DT ET t EA ET EA

3.32

lt l l s – A s – T s T + t + a S DT EA t EA EA mA

3.33

eT = – g= –

73

The shear-coupling parameters lA, lT and lt are ratios associated with the effect of shear stress on axial and transverse strain respectively. The parameter aS is the thermal expansion coefficient associated with the effect of temperature changes on the shear strain. This completes the summary of methods to predict undamaged laminate properties from those of the fibre and matrix, and the volume fraction of fibres in individual plies. The remainder of this chapter will be concerned with progressive damage growth in laminates and laminate failure.

3.4

Prediction of ply cracking in laminates

Various methods have been used to develop an understanding of damage growth in laminates. Parvizi et al. [9] developed the first so-called shear lag model of ply cracking in a cross-ply laminate, while Nuismer and Tan [10] adopted the optimum shear lag approach that is recommended for use. More accurate analyses based on the application of variational techniques to ply crack problems have been given by Hashin [11–13], Nairn [14–16] and Nairn and Hu [17]. The most accurate methods consider multi-layer models of laminates where through-thickness stress and deformation variations are modelled by sub-dividing each ply of the laminate into sub-plies. Examples of this approach are the variational approach applied to cross-ply laminates by Pagano [18], Pagano and Schoeppner [19] and Pagano et al. [20], and the technique developed by McCartney [8, 21, 22] for both cross-ply and general symmetric laminates, which is consistent with variational mechanics but provides the complete stress and displacement fields within the laminate. The first damage mode encountered when laminates are subject to multiaxial loading is the formation of ply cracks in plies whose fibre direction is transverse to the principal loading direction. Ply cracks usually form suddenly and completely crack the ply cross-section. Increasing loads lead to the progressive formation of ply crack arrays. When a ply crack forms in an undamaged laminate the load it carried is transferred to neighbouring plies which are intact as the fibres are usually in a different direction to those in

74

Multi-scale modelling of composite material systems

the cracked ply. The stress transfer occurs through the action of shear and normal tractions on the interfaces between the cracked ply and its neighbours. The non-uniform stress field that results has a significant effect on the values of the thermo-elastic constants of the laminate. Ply crack formation in laminates is clearly a complex phenomenon due to the various different locations that are available for cracking, and its dependence on ply properties and geometry such as ply thickness and ply lay-up. Ply cracking is also affected by edge effects where complex stress transfer between plies is occurring at the stress-free edges of laminates introducing further complexity. This problem is dealt with by developing and applying models to the internal regions of the laminate only, for which edge effects have diminished and become negligible. Ply crack formation has been studied in quasi-isotropic laminates, (Crocker et al. [23], McCartney [8, 24] including finite element analysis of cracked laminates for the case of generalised plane strain deformation (Tong et al. [25])). For laminated systems, a methodology has been developed (McCartney [8, 21, 22]) that is capable of predicting accurately the stress and displacement distributions at all points in cross-ply and general symmetric laminates containing arrays of ply cracks having a single orientation. The approach to predicting damage formation in laminates involves three distinct stages. The first stage involves the determination of the local stress field associated with damage, as described in references [21, 22] for crossply laminates and in [8, 21] for general symmetric laminates. The second stage is to use the stress analysis to predict the dependence of the effective thermoelastic constants of the damaged laminate as a function of the damage (ply crack density). The third stage is to use this information to predict the stress-strain curve and dependencies of the effective thermoelastic constants on the applied stress or strain. The analysis to be described considers ply cracking in just one orientation.

3.4.1

Effective laminate strains and associated effective stresses

The development of the model of ply cracking in laminates involves the definition of effective laminate stresses that are applied to the laminate edges. The effective stresses are averages of the actual tractions that would be applied to the laminate boundaries if an exact solution for the stress and displacement distributions were to be available. The deformation applied to the laminate is in fact through external displacements that would produce a homogeneous strain field in an undamaged laminate. It is assumed that inplane biaxial and shear loads are applied to the cracked laminate by imposing displacements on the edges of the laminate that would, in an undamaged laminate, lead to a state of combined uniform biaxial strain and uniform

Multi-scale predictive modelling of cracking

75

shear strain. For a uniformly cracked laminate having length 2L in the ydirection and width 2W in the z-direction, the mathematical description of effective laminate strains is specified by

s xy = 0, n = ± e c L, w = ± g c L + e cT z, on y = ± L, ¸Ô ˝ s xz = 0, n = e c y, w = g c y ± e cT W, on z = ± W, ˛Ô

3.34

where ec, e cT and g c define the imposed in-plane axial, transverse and engineering shear strains for a uniformly cracked laminate. These strains would impose uniform strain distributions in an equivalent homogenised laminate. The external laminate faces are assumed to be subject to a uniform normal traction st, and zero shear tractions. Because applied displacements of the form 3.34 have been imposed on the edges of the cracked laminate, the corresponding stresses syy, szz and syz, resulting from an exact elasticity analysis, will be complex and nonuniform. Effective stresses are defined to be averages of the actual traction distributions over the laminate edges. The mathematical description of the effective applied axial, transverse and shear stresses is given by the following averages ¸ s yy (x, L, z) dx dz, Ô –W –h Ô L h Ô s zz (x, y, W) dx dy, Ô Ô –L –h ˝ W h s yz (x, L, z) dx dz Ô Ô –W –h Ô L h s yz (x, y, W) dx dy, Ô Ô˛ –L –h

Ú s = 1 4 hL Ú t= 1 4 hW Ú = 1 4 hL Ú s=

1 4 hW

T

W

Ú

Ú

Ú

h

Ú

3.35

where 2h is the total thickness of the symmetric laminate. The effective outof-plane transverse strain e ct , corresponding to the uniformly applied throughthickness stress st, is defined for a cracked laminate by e ct =

1 4hLW

Ú Ú L

W

–L

–W

u (h, y, z) dy dz

3.36

The definitions of the effective strains and corresponding effective stresses can be applied to any regions between neighbouring ply cracks in a uniformly cracked laminate where the ply cracks are orientated in a single direction. The concept enables modelling at the ply level to be linked to modelling at the laminate level. An important consequence of defining the effective stresses to be averages of the actual tractions is that the work done by the averaged stresses will be identical to the work done by the actual tractions. This is a key property of the effective stresses that enables the model to develop

76

Multi-scale modelling of composite material systems

accurate relatively simple formulae for the ply cracking stresses, and thermoelastic constants when using the effective stresses in stress-strain relations.

3.4.2

Thermo-elastic constants

In order to understand progressive damage formation and failure in laminates, it is essential to be able to calculate the thermoelastic properties of a general symmetric laminate in a damaged state. When damage exists in the laminate in the form of ply cracks, it has been shown [8] that the effective stress-strain relations for the damaged laminate are of exactly the same form as those for the corresponding undamaged laminate. Such stress-strain relations involve, in addition to E, n and a, the shear moduli (m), the shear coupling ratios (l) and a shear thermal expansion coefficient (aS). For balanced symmetric laminates, l and aS are zero when the laminate is undamaged, but non-zero and small when the laminate is damaged. The stress-strain relations for damaged general symmetric laminates are thus identical in form to those (3.30–3.33) discussed in the previous section. A key issue, when considering the properties of damaged laminates in an engineering application, is how the various thermoelastic constants degrade as the ply cracking damage grows progressively during the loading of the structure. In references [8, 21, 22] it was shown that ply crack development during loading (in the absence of shear deformation) is most easily understood by introducing a macroscopic damage parameter D defined by

D = 1 – 1o EA EA

3.37

The parameter E A is the effective axial modulus of a damaged laminate (constrained so that in-plane shear strain is zero) defined in reference [8], while E Ao ( > E A ) is the corresponding value for the same laminate without ply cracks. Clearly D is zero when the laminate is undamaged. By considering the closure of ply cracks during unloading, it has been shown [8, 21, 22] that the following relations define, in terms of D, the values of the various thermoelastic constants of a damaged laminate which are satisfied for ANY state of cracking in the plies of a general symmetric laminate (see [8] for formal definitions of the various effective thermo-elastic constants) 1 – 1 = (k ¢) 2 D, 1 – 1 = D, 1 – 1 = k 2 D Et E to EA E Ao ET E To

3.38

n Ao n ao n n to nt nA – = kk D, – = kD, – a = k ¢D ¢ o o o ET ET EA EA EA EA

3.39

a t – a to = k ¢k 1 D, a A – a Ao = k 1 D, a T – a To = kk 1 D

3.40

Multi-scale predictive modelling of cracking

77

The three laminate constants k, k¢ and k1 appearing in eqns 3.38–3.40 may be calculated [21, 22] from the properties of the plies, and the reduced thermoelastic properties of the laminate in an undamaged state, so that E [ a + Ba T – C] E k1 = A A ,k= A ET 1 – nA B

k¢ =

E A A – na – nt 1 – nA B

ET EA 1 – nA B

B – nA

EA B ET

3.41

and where A, B and C are constants calculated from unidirectional undamaged ply properties (in italics) defined by A=

nt n n + a A , B = n A, C = aT + n Aa A ET EA

3.42

The parameters k1, k and k¢ are laminate constants that are independent of the state of damage. It should be noted that k and k¢ do not depend on thermal expansion coefficients.

3.4.3

Energy based ply crack formation

While ply failure is often predicted using a critical strength argument (as can be seen from the design criterion known as first ply failure), this approach cannot be justified using physical principles. It is merely a convenient engineering approach that has been used for many years. Our approach is to examine the application of a key physical principle, energy balance, to the ply cracking problem, and to develop relatively simple formulae that can be used to replace the conventional stress-based first ply failure approach. For isothermal conditions, the principle of energy balance equates the increment of work done dW by the externally applied tractions to the various types of energy that can be associated with the system, as follows. dG + dK + dU + dD = dW

3.43

The energy contributions are: dU the increment of stored elastic energy (strictly for isothermal conditions where U is a thermodynamic function known as the Helmholtz free energy), dK the increment of kinetic energy which is never negative, dD an increment of dissipated energy (a positive quantity) if some irreversible processes are occurring such as plastic flow or creep deformation (both relevant mechanisms for polymer composites), and finally dG the energy absorbed in the regions of the ply crack surfaces as the crack forms (related to the fracture energy for ply cracking). Assuming that ply cracks form under fixed applied tractions (rather than fixed applied

78

Multi-scale modelling of composite material systems

displacements), and on making use of the fact that kinetic energy is never negative, the general energy balance equation for isothermal conditions may be recast in a simple inequality involving the increment of Gibbs free energy dG, namely dG + dG < 0

3.44

The Gibbs free energy can be associated with the complementary energy that must be used when state changes occur under fixed loading conditions. The energy changes associated with ply crack formation are easily calculated and applied to the energy balance equation (3.44) leading to a criterion for ply cracking that enables the applied stress conditions for fracture to be determined once the changes in the absorbed energy G and in the Gibbs free energy per unit volume have been calculated. The mean Gibbs free energy per unit volume is obtained (see ref. 8) by converting a volume integral into a surface integral. Use is then made of the definitions of the effective applied stresses, and of the effective thermoelastic constants for a damaged laminate. In order to do this it is also necessary to introduce the crack closure stresses denoted by superscript c, and the result is given by g= 1 V

+

Ú

gdV = –

V

2 2 s 2T s 2t – s – – t 2Et 2EA 2 ET 2mA

na n l n l l s s + A ss T + t s t s + t s t t + A st + T s T t EA t EA ET EA EA EA

– [stat + saA + sTaT + taS] DT + 1 [ s ct a t + s c a A + s cT a T + t c a S ] 2

3.45

While the resulting expression for the Gibbs energy appears complex, it does have a relatively simple structure and is such that the damaged laminate stress-strain relations of the type (3.30–3.33) can be recovered by differentiating the energy with respect to the stress components as indicated in ref. 8. It is possible to eliminate the occurrence of the shear stress in the expression for the Gibbs energy by making use of the shear stress-strain relation, and the definitions of the reduced thermo-elastic constants. The shear strain then appears in the Gibbs energy but the expression is significantly simplified. A remarkable further simplification is possible if the reduced thermoelastic constants are all expressed in terms of the damage parameter D using the various inter-relationships defined by (3.38–3.40). The Gibbs energy per unit volume g may then be expressed as the sum of three terms as follows g = g0 –

where

D [s – s ] 2 – 1 [ m g 2 – m 0 g 2 ] , c A 0 2 A 2 E 0A

s = k¢st + s + ksT,

sc = – k1DT,

3.46

Multi-scale predictive modelling of cracking

79

and where sc is the value of s when the ply cracks just close in the ply. The first is its value g0 when the laminate is undamaged. The second is a term that is proportional to D and the square of a stress term (s – sc), where s is a linear combination of the applied non-shear stresses, and sc is the value of s at the point of crack closure. The third term arises from the shear deformation of the laminate and involves the shear strain g in the damaged and undamaged states (denoted by a subscript 0). For many symmetric laminates, which are balanced and not subject to shear loading, this term is usually small and can be neglected. In practice non-uniform ply cracking in laminates is usually observed where the ply cracks form one at a time. The simulation of such ply cracking is achieved by assuming that in the plies of the laminate there is a set of potential ply crack formation sites where ply cracks may form during loading. In any simulation only a fraction of the sites will actually be cracked. Such behaviour has been modelled by allowing the fracture energy for ply cracking to be statistically distributed. This is achieved by assuming that the fracture energy at each potential ply crack formation site is taken at random from a normal distribution that is characterised by its mean value and standard deviation. Energy methods are used to determine the ply cracking sequence. Assuming that n ply cracks have already formed in the laminate, an energy calculation is performed at each uncracked site to determine the stress needed to form the (n + 1)th crack at that location. Having ‘tested’ all remaining uncracked sites, the site requiring the least stress to form the crack is taken to be the site at which the (n + 1)th crack forms. This procedure is repeated successively as damage progressively increases during loading. The theory to account for non-uniform ply cracking has been described in ref. 26 where an approximate method is used in the model to develop a method of calculating the thermoelastic constants of non-uniformly cracked laminates. The approximation has been compared with accurate numerical solutions (see ref. 26) and found to be excellent for a wide range of conditions. The results of the approximation are then used to generate simple analytical results for first ply failure stress [8, 21, 22]. When applying the analysis, the laminate is rotated so that the plies in which ply cracks are to form are in a 90∞ orientation, and the stress field is correspondingly transformed. Consider progressive cracking between two distinct damage states 1 and 2 having ply crack patterns defined by: State 1 = {L1, L2, . . . Ln}, State 2 = {L1, L2, . . . Ln+1}, and leading to energy absorptions G1 and G2 respectively, involving the fracture energy 2g for ply cracking, where sequences Li denote the separations between neighbouring ply cracks. If the standard deviation of the fracture energy distribution is taken to be zero, then during loading a number of ply cracks will first form simultaneously at a specific characteristic crack density

80

Multi-scale modelling of composite material systems

(depending on laminate lay-up and ply properties), which progressively doubles as the load increases due to the formation of ply cracks at the mid-points between neighbouring cracks. On making use of the simple expression in eqn 3.46 for the Gibbs free energy per unit volume in the energy balance inequality eqn 3.44, the effective stress s, needed to enable the steady state propagation of a ply crack, can be calculated for tri-axial stress states using the following simple formula involving the fracture energy for ply cracking and the reduced axial Young’s modulus (valid in the absence of shear deformation) s>

2[ G 2 – G 1 ] + sc , n ≥ 0. 1 – 1 (2) (1) EA EA

3.47

The absorbed energy function G, appearing in eqn 3.47, is calculated for length 2L of laminate using

G=

h1 m Sdg hL j=1 j j

3.48

where 2h1 is the total thickness of all 90∞ plies in the laminate having total thickness 2h, and where m is the number of potential cracking sites in 90∞ plies which are ordered in a regular way, e.g., from top to bottom in the plies which are taken in order from the centre of the laminate to the outside, symmetry about the mid-plane of the laminate being assumed. The quantity 2gj is the fracture energy for the growth of a ply crack in the jth potential cracking site of the 90∞ plies. The parameters dj describe the crack pattern in the laminate such that

Ï0 dj = Ì Ó1

if j th site of the 90∞ ply is uncracked, if j th site of the 90∞ ply is cracked.

3.49

The energy balance analysis that has been described assumes that the ply crack tip is not influenced by laminate edge effects, and thus it applies only to ply cracks which are sufficiently long so that the stress needed for their growth does not depend on their length c. In practice defects (arising from fibre/matrix debonds) of such a length should not exist in a laminate that has just been manufactured. Smaller defects most certainly will be found, and these defects will initiate the growth of a ply crack if the applied stress is large enough. Two possibilities can arise, (i) the stress for propagation reduces as the defect size increases, or (ii) the stress for propagation increases with defect size. Both possibilities must result in the same steady state propagation value that has been analysed. For case (i), a ply defect will require a larger stress than the steady state limiting value in order to initiate ply cracking. For this case, often observed in practice, ply crack growth is unstable and can

Multi-scale predictive modelling of cracking

81

release a significant amount of energy. As the steady state stress for propagation is less than the initiation stress, designs based on the propagation stress will be conservative but it means that a safe approach is being taken. For case (ii), a ply defect will begin to grow when the applied stress is less than the steady state value. Thus, as the stress is increased the ply crack will grow in a stable manner until the steady state stress is being applied when crack growth can then become unstable. For this case (not usually expected in practice) the use of the steady state stress should be a realistic estimate of the stress needed to cause the growth of a ply crack across the full width of the laminate. The characteristics of the model, which have been described for steady state ply crack growth, can be summarised in a single diagram as shown in Fig. 3.1. From any progressive ply cracking simulation it is possible to calculate, for a given stress state, the difference e – e0 between the axial strain in damaged and undamaged state, and s – sc (where s is the effective stress that is a linear combination of the non-shear applied stresses, and sc is the closure value). It is found that the curve initially rises vertically because the laminate is undamaged and e = e0. When it is energetically favourable for ply cracking to occur, the stress difference s – sc is then a monotonic increasing function of e – e0. If at some point the system is unloaded to the point of ply crack closure, then the unloading curve behaves linearly and approaches the origin as e – e0 tends to zero. The gradient of the unloading curve is in fact the undamaged reduced axial modulus divided by the damage parameter D. The area enclosed by the curve is equal to G that is one half of the energy per unit volume needed to from the ply cracks that are present. A great deal of very useful information can thus be extracted from experimental stressstrain relations when plotted in the manner described. S – Sc

S i – Sc Area G

Gradient =

E0A D

Inelastic strain e – eo

0

3.1 Schematic stress-strain plot indicating a physical interpretation of the damage parameter.

82

3.4.4

Multi-scale modelling of composite material systems

Examples of applications to layered systems

The PREDICT software system [27] has been used to estimate, for both GRP (Silenka/epoxy MY750) and CFRP (AS4/epoxy 35) cross-ply laminates, the effect on first ply failure stress of the thickness of the individual plies. The laminates, all of total thickness 4 mm, that have been considered are [08/ 908]s, [04/904]2s, [02/902]4s and [0/90]8s where both GRP and CFRP systems are assumed to have a ply thickness of 0.125 mm and a fracture energy for ply cracking of 240 J/m2. Thermal residual stresses are included in the model based on a value T – To = –85 ∞C, where T is the temperature of simulation and where To is the stress-free temperature for the laminate. Ply refinement techniques were used to increase the accuracy of the through-thickness stress and displacement distributions by sub-dividing each ply into five sub-elements of equal thickness. The results for first ply failure stress are shown in Fig. 3.2 where it is seen that decreasing the ply thickness while maintaining the total thickness of the laminate at 4 mm leads to dramatic increases in the resistance of the laminates to ply failure. The CFRP laminates perform better, although fibre failure might determine the occurrence of initial damage when the plies are very thin.

First ply failure stress (GPa)

1.2 GRP CFRP

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4 0.6 Ply thickness (mm)

0.8

1

3.2 Effect of ply thickness on first ply failure for GRP and CFRP laminates.

Another example of an application of the energy-based methodology for failure prediction concerns the first cracking of interleaved layers of titanium nitride and steel forming a [TiN/Steel]4s laminate. The layer thickness, laminate thickness and simulation length are all reduced by factors of ten from a thickness of 1 mm down to a thickness of 100 nm. For this example a single model is being used to cover continuously a large range of length scales. The simulations assume a value T - To = –500 ∞C. Layer refinement techniques

Multi-scale predictive modelling of cracking

83

were used to increase the accuracy of the through-thickness stress and displacement distributions by sub-dividing each layer into five sub-elements of equal thickness. The results shown in Fig. 3.3 exhibit a dramatic increase in the first cracking stress as the layer thickness is reduced, illustrating a well-known nano-effect where decreasing length scales lead to improved performance. It must be emphasised that, although strain relaxation cannot be achieved by cracking, other mechanisms such as dislocation formation, will occur invalidating the predictions of Fig. 3.3 when the layer thickness is in the micrometre range. Also, cohesive zones near crack tips, modelling inter-atomic interactions across the fracture surface, may need to be modelled as the layers become very thin. The development of reliable models for laminate strength prediction is in its infancy, although good progress is being made, as demonstrated here by the ability of physically based models to predict laminate thickness effects on ply failure and the cracking of multi-layer coatings. An important feature of both Figs 3.2 and 3.3 is that an energy-based methodology for cracking in layered materials is leading to strong thickness dependencies observed in practice, and such dependencies cannot be predicted using a strength of materials approach that is the basis of widely used conventional failure criteria [31–33] that are often used to predict first ply failure. This issue should be a concern for engineers involved with the design of both laminated structures and other layered systems. 6

Cracking stress (GPa)

5 4 3 2 1 0 –4

–3

–2 log10 (layer thickness (mm))

–1

0

3.3 Size effect on cracking stress for a titanium nitride/steel layered system.

3.4.5

Ply cracking in multiple orientations

Recent work at NPL has focused on further developing the damage model so that ply cracking in more than one orientation can be considered. A

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Multi-scale modelling of composite material systems

homogenisation technique can be used to achieve this approximately, although the solution technique becomes numerically intensive, as many possible ply crack formation sites must be considered [28–30]. The idea is to apply the analysis of sections 3.1–3.4, which can predict ply cracking in a single ply orientation, to a laminate where ply cracks in other orientations have been homogenised. Thus the simulation involves homogenising the properties of a ply having an array of ply cracks, and the converse, together a series of laminate rotations and rotations of the stress field, as described in refs 28– 30. A key result of the homogenisation methodology, is that following homogenisation all the thermoelastic properties of the laminate are identical to those of the laminate where the ply cracks in the ply, where the last ply crack formed, are modelled as discrete entities. This can be achieved only if the homogenised ply properties include a shear-interaction term, as described in ref. 30.

3.5

Prediction of laminate failure

A major concern for engineers is establishing that any component designed using composite materials will perform reliably and safely during service. If an engineering component becomes unserviceable during use, then it has to be regarded as having failed. For example, if a composite component breaks suddenly into two or more distinct pieces then it clearly has failed. However, if instead the component develops microstructural damage such as that associated with localised cracking in the plies of a laminate, delamination of inter-ply interfaces, fibre failure and fibre/matrix interface failure, the stiffness of the component could be degraded to the point at which it becomes unserviceable. For components such as fluid-containing tanks, pipelines and pressure vessels, if leakage occurs then the component must be regarded as having failed. For this case ply cracking in adjacent plies of a laminate is sufficient to enable leakage paths and thus failure, even though the component continues to be capable of supporting mechanical loads safely. It is clear that failure is a concept that cannot be defined uniquely and must often be associated with the engineering purpose for which the component was designed. The longer-term performance of composite components can be also be affected by degradation arising from exposure to aggressive environments (static fatigue), and to cyclic loads (dynamic fatigue), which are topics that will not be considered in this chapter. A large number of failure criteria, designed to ensure that components are not subject to sudden catastrophic failure, have been proposed for composite materials, and some of these are included in commercially available finite element software systems. Invariably the engineer is offered a number of criteria that can be applied to FEA solutions, but which one should he use?

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The wise engineer will try them all and design on the basis of the most pessimistic prediction. This approach is not efficient, and it may not even be safe. To address this very important issue regarding the prediction of the failure of composite materials, an international Failure Prediction exercise has been undertaken during the past ten years or so (see refs 31–33 for Parts A, B and C of the exercise). The organisers identified sets of high quality experimental data that were the results of tests designed to investigate the effects of biaxial loading on the failure of a range of laminates and materials. The laminates, and the associated basic properties used in these tests, were the basis of a variety of test cases that were given to researchers throughout the world, who were asked to use their models of failure to predict how the laminates would behave. The ‘blind’ predictions were then submitted to the organisers who then compared them with the original high quality experimental results. The organisers then commented on the results, and participants were then given the opportunity of commenting on the performance of their own models. The recently published Part C [33] includes contributions from additional participants, who have submitted their model predictions having had sight of the experimental results, and some summary comments and recommendations by the organisers of the exercise. One example of a failure criterion is worth examining in a little more detail, namely the Tsai-Wu criterion [34] 2 s 12 s 22 2 F12 s 1 s 2 Ê tˆ = 1 + + + XY Ë S¯ X2 Y2

3.50

where X and Y are the measured uniaxial tensile strengths in the 1 and 2 directions respectively, and where S is the measured shear strength. The parameter F12 is a normalised interaction term, which is used to optimise the fit of the criterion to the measured test data, and it has a value lying between –1 and 1. The quadratic form would involve linear stress terms for the general case where the axial and transverse uniaxial strengths in tension are different to the compression strengths. If they are equal then the linear terms are removed and the criterion is simplified to the form shown in eqn 3.50. It is emphasised that this criterion, and many other criteria, do not have a physical basis but are phenomenological. Indeed, Lui and Tsai [35], have stated that ‘failure criteria are purely empirical’, being devised to define failure envelopes with the minimum of test data. Even in its simplest form (eqn 3.50) the application of the criterion requires test data from tension, compression and pure shear tests. An additional characteristic of criteria of the quadratic type (eqn 3.50) is that the failure envelopes defined are closed, thus avoiding the possibility of infinite strengths. Quadratic failure criteria are often applied to stress states for structures where extensive damage that occurs before failure, is neglected during engineering assessments. Quadratic failure criteria are often found in commercial FEA codes, thus giving such criteria an apparent

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Multi-scale modelling of composite material systems

and unjustified acceptance of validity. Failure criteria, when applied in such a way, should have associated with them either validation data defining the limits of their validity, or a ‘health’ warning. It is noted that conventional failure criteria cannot easily be applied to environmentally or fatigue damaged composites. Neither do they account for ply thickness (see section 3.4) and ply lay-up effects [30, 40]. The international Failure Prediction Exercise did, however, place the more general version of the Tsai-Wu failure criterion [34] in the leading group of theories, the best being the theory of failure first proposed by Puck [36]. It is emphasised that in Part C of the exercise [33], it is clearly stated that the test cases used did not consider any of the following issues that can be important for practical applications: ∑ ∑ ∑ ∑ ∑

delamination initiation and propagation effects of ply thickness and stacking sequence long term behaviour, creep, fatigue, and degradation environmental effects, including moisture and aggressive substances effects of different temperatures.

Some issues where confidence is lacking were also stated, namely: ∑ ∑ ∑ ∑ ∑

thermal residual curing stress consideration in situ strength of an embedded lamina leakage of pressurised vessels thin and thick laminae effects of lay-up sequence.

The in situ strength of an embedded lamina has been identified as an issue where confidence is lacking. The approach of this section will describe methods that suggest the concept of an in situ strength is in fact ill founded. An additional concern is the lack of test data for multiaxial loading cases where the effects on laminate failure are not known (because of substantial testing difficulties) for in-plane shear loading combined with axial and transverse in-plane loading, and with through-thickness loading. The above discussion emphasises the need to formulate failure prediction methods that are based on physical principles and mechanisms. In section 3.4 it was shown how ply crack formation could be predicted. The objective now is to consider physically based methods of predicting laminate failure. Fibre failure has to be included as a damage mode when 0∞ plies are present, and it is useful to begin by considering the failure of an isolated fibre.

3.5.1

Statistical model of fibre failure

When an isolated fibre is tested its strength at a given temperature can be characterised either by the stress at failure or by the strain at failure. The

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former is a linear function of the latter. For the simplified approaches for a UD composite and a cross-ply laminate based on a parallel bar model, either strength parameter can be used as the fibres are regarded as being a loose bundle when investigating the axial behaviour of the composite. If, however, the fibres in the 0∞ plies are perfectly bonded to the matrix, then a tri-axial state of stress can exist in the fibre. Assuming an axisymmetric state of stress, as would arise when using a concentric cylinder model of a UD composite, it is clear that the transverse stress in the fibre and the thermal residual stresses can affect the axial strain in the fibre. The following question then arises: does the fibre break in a composite when a critical axial stress is reached, or a critical axial strain? It should be noted from the general axial stress-strain relation for the fibre that infinite stresses could in fact be applied while maintaining a bounded axial strain, provided that the ratio of the axial to the transverse fibre stresses is equal to Poisson’s ratio for the fibre. This fact alone suggests that the correct failure criterion for a fibre should be based on axial fibre strain. Such an approach is reinforced if one considers a molecular model of the fibre where atoms interact through central forces that diminish to zero in magnitude as the atoms separate to infinity. While a critical fibre strain criterion has some physical justification, its use requires a careful definition of strain in relation to the temperature. For this reason it is preferable to characterise the failure of isolated fibres using strength, but to impose a critical strain criterion when such strength data are applied to fibres in a composite. A fibre can be thought of as an assembly of elements of the same length joined together, analogous to the links in a chain. Each element contains fibre defects that would initiate fibre failure at a specific applied stress, which will vary from element to element along the fibre length. The fibre will fail during loading when the fibre stress attains the strength of the weakest element. Also the longer the fibre, the weaker is it likely to be. To place these ideas on a sound statistical basis it is useful to let F(s) be the probability that unit length of fibre fails during the stress increase 0 to s. It follows that 1 – F(s) is then the corresponding probability that the fibre survives the stress increase. For a fibre element of length d the survival probability is [1 – F(s)]d and the corresponding failure probability of the fibre must be PF(s) = 1 – [1 – F(s)]d. This is the well-known weakest link failure probability distribution. Fibre strength is well known to be a statistically distributed quantity and its variability of often characterised using the following Weibull distribution function for the cumulative failure probability for unit length of fibre È s ˆm˘ F( s ) = 1 – exp Í – Ê ˙ Î Ë so ¯ ˚

3.51

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Multi-scale modelling of composite material systems

The so-called Weibull modulus m determines the width of the distribution. Small values of m (< 4) indicate a high degree of variability, whereas large values (>12) indicate modest levels of variability. The parameter s0 is a scaling parameter, having the dimensions of stress, that controls the mean strength of the distribution. On using the Weibull distribution (eqn 3.51) to characterise the strength of the fibre elements, the probability of failure PF(s) for the fibre during the fibre stress increase 0 to s is given by the following simple formula that involves the fibre element length d, the fibre stress s, and material parameters m and s0 È s ˆm˘ PF ( s ) = 1 – exp Í – d Ê Ë s o ¯ ˙˚ Î

3.52

The mean value of the strength distribution is given by 1 s 0 d –1/m G Ê 1 + ˆ m¯ Ë

3.53

and the corresponding variance is given by 2 1 ˘ È s 20 d –2/m Í G Ê 1 + ˆ – G 2 Ê 1 + ˆ ˙ m m Ë ¯ Ë ¯˚ Î

3.54

where G(x) is the well known Gamma function.

3.5.2

Composite having weak fibre/matrix interfaces

When applying the statistical approach to a composite material having weak interfaces and considering axial behaviour, the fibres can, to a first approximation, be regarded as being detached from the matrix so that their length is then the length of the composite subjected to a strength test (to be considered later). Alternatively, and more realistically, the composite can be regarded as being a chain of fibre bundles. The bundle length is taken as the stress transfer length d over which the stress, carried by the fibres before they fracture, is transferred to the matrix. If the fibres are used to create a large bundle of loose fibres that are then subject to loading along their length, such that the load carried by failing fibres is shared equally amongst the surviving fibres, then simple expressions can be derived for the mean bundle strength and the fibre stress at the point of bundle failure. The analysis assumes that there are an infinite number of fibres in the bundle so that every fibre strength possible is being sampled during a strength test. If only a finite number of fibres is used then the statistical approach is much more complex, but it is known that, if there are

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more than 200 fibres, the infinite bundle approximation described gives a good estimate of bundle performance [37]. Composite materials use many more fibres than 200 so the formulae given here will be very good estimates of their behaviour if modelled as a loose assembly of fibres. The ultimate strength of a cross-ply laminate having fibres orientated in the principal loading direction is often determined by the strength of the fibres. The total thicknesses of the 0∞ and 90∞ plies in one half of the laminate are denoted respectively by h0 and h90 such that the total thickness of the laminate is 2h where h = h0 + h90. The parallel bar model shown in Fig. 3.4, suitable for the case of weak interfaces between fibres and matrix, and between plies, can be used to develop simple relationships that capture, to a first approximation, the behaviour of the deformation arising from axial loading. The deformation and tensile failure of the laminate is regarded as being equivalent to three distinct mechanical elements, which are loaded in parallel as shown in Fig. 3.4. The fibres of the 0∞ plies are grouped together as a bundle, and the corresponding matrix in the 0∞ plies is grouped together in another bar. The 90∞ plies are represented by the third bar, having homogenised properties to account for any ply crack damage in these plies. In order to estimate the strength of a laminate it is then necessary to be able to relate the stress sf in each of the surviving fibres of the 0∞ plies to the effective applied stress s. F 0∞ plies Fibre bundle

Matrix

90∞ plies

Vf

Vm

h90

sf

sm

s90

Ef

Em

E90

af

am

a90

e

e

e

N0N

F

3.4 Schematic diagram of a parallel bar model of unit width of a cross-ply laminate.

Consider unit width of the cross-ply laminate. The fibres in the 0∞ plies (N0 in number when no fibres have failed) each have cross-sectional area af,

90

Multi-scale modelling of composite material systems

leading to a total cross-sectional area Af = N0af, axial modulus E fA , and axial thermal expansion af. The matrix has total cross-sectional area Am, axial modulus E Am , and axial expansion am. The 90∞ ply has thickness h90, axial modulus E90, and axial expansion a90. The total thickness of the 0∞ plies is denoted by h0. The fibres and matrix are regarded as being fixed to two constrained rigid blocks which slowly move apart defining the axial strain e = (L - L0)/L0 where L is the separation when the load F is applied, and L0 is the separation when the temperature T0 is such that the stress in the fibres and matrix is zero. The temperature T0 is known as the stress-free temperature of the laminate. At this temperature the fibres and matrix are assumed to be stress and strain free if the laminate is in an unloaded state. The parameter DT appearing in the stress-strain relations for the fibre and matrix is defined so that DT = T – T0 where T is the temperature of the composite when loaded. The volume fractions of the fibre and matrix are denoted by Vf and Vm (= 1 – Vf) respectively. The effective axial stress applied to the laminate is denoted by s such that s = F/(h0 + h90) where F is the total axial load applied to unit width of laminate. The axial effective axial modulus EA and the effective axial thermal expansion coefficient aA of the laminate can be estimated using simple rule of mixtures formulae. If the number of surviving fibres N < N0 then the above method can be used to estimate degraded values of modulus and expansion coefficient provided that N0 is replaced by N. The axial strain e is shared by all three elements of the parallel bar model so that e=

s sf s – a fA DT = m – a Am DT = 90 – a 90 DT f E E 90 EA m

3.55

The load-sharing between the three elements when N fibres survive is such that

N V s + V s + h 90 s = h s m m N0 f f h 0 90 h 0

3.56

On making use of (eqn 3.55) the load-sharing rule (eqn 3.56) may be expressed in the form Ê N + a ˆ V s + bDT = h s Ë N0 ¯ f f h0 ,

3.57

where

a= and

h E 90 Vm E m , + 90 Vf E f h 0 Vf E f

b = Vm E Am ( a fA – a Am ) +

3.58 h 90 E ( a f – a 90 ) . h 0 90 A

3.59

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91

If a fibre failure criterion is available, then this is easily applied to the model to predict the failure stress for the laminated composite, in the presence of thermal residual stresses. It is assumed that during loading fibres start to fail so that there are N intact fibres when the effective applied stress is s. If the strengths of all fibres are the same, then the failure of one fibre will lead to the failure of all fibres. Thus the strength of the laminate can easily be predicted once a suitable fibre failure criterion has been imposed. Two parameters a and b control the load sharing that occurs between the fibres and matrix of the 0∞ plies and the 90∞ plies. The parameter a is associated with mechanical loading while b is associated with thermal expansion mismatches between fibres, matrix and the 90∞ plies. It should be noted that b = 0 when the expansion coefficient for the fibre is equal to that of the matrix and the 90∞ plies. To predict the strength of a laminate having statistically distributed fibre strengths, it is necessary to combine the load-sharing rule (eqn 3.57) for fibre fractures together with the Weibull statistical distribution (eqn 3.51) for the fibre strengths of unit lengths of fibre. When the fibre stress is sf the Weibull distribution for the cumulative failure probability implies that the fraction N/ N0 of fibres surviving is known and given by È s m˘ N = N 0 exp Í – Ê f ˆ ˙ Î Ë s0 ¯ ˚

3.60

On insertion into the load-sharing rule (eqn 3.57), the following expression for the strength S of the laminate can be obtained

S=

h0 È ÍV s h Í f max Î

˘ ÏÔ Ê Ê s max ˆ m ˆ ¸Ô Ì a + exp Á – Ë s ¯ ˜ ˝ + bD T ˙ , 0 Ë ¯ Ô˛ ÔÓ ˙˚

3.61

where smax = s0x1/m, mx = 1 + aex.

3.62

It is first necessary to solve the transcendental equation mx = 1 + aex numerically. The solution then defines the parameter smax that appears in eqn 3.61 for the laminate strength S. The parameter smax is in fact the stress in the surviving fibres just prior to the bundle collapsing. This critical point is found by determining the maximum point of the curve defining the relationship between the applied stress s and the fibre stress sf. It is worth noting that the maximum value for the fibre stress smax is independent of the volume fraction Vf of the fibres in the 0∞ plies, and of the temperature difference DT. The new results presented may easily be implemented in software.

92

3.5.3

Multi-scale modelling of composite material systems

Composite having strong fibre/matrix interfaces

When fibre/matrix interfaces are reasonably strong, catastrophic failure is invariably preceded by damage progression where multiple damage modes occur and interact. The objective now is to indicate how various models developed for different length scales may be integrated into a software system that enables the prediction of the formation and progressive growth of various damage modes. Work has already been completed for a multiple-ply crossply laminate subject to biaxial loading and thermal residual stresses. When loaded the first damage forms as ply cracks in the 90∞ plies of the laminate. The load carried by the 90∞ plies is then transferred to the neighbouring 0∞ plies. As loading increases, the stresses in the fibres of the 0∞ plies increases and fibre fracture damage then commences. Eventually the loading of the fibres increases to the point at which they all fail, leading to the failure of the laminate. Accompanying fibre failure will be the debonding of fibre/matrix interfaces that initiate at the fibre fractures. Increasing laminate loading can lead to additional ply cracks, extending of fibre/matrix debonds, and to additional fibre fractures. Clearly, three damage modes are involved, ply cracking, fibre fracture and fibre/matrix debonding, which grow progressively in an interactive manner. To be able to predict laminate failure for such conditions clearly requires an integrated approach involving fibre/matrix models and laminate models, that must be implemented as software in order to enable computer simulations of laminate behaviour during progressive loading and damage development. The behaviour of a cross-ply laminate, where ply cracking and fibre failure are damage modes that can interact, has been modelled using Monte Carlo methods to account for statistically distributed fibre strengths [38]. The approach is to use the methods described in section 3.4 to predict ply crack formation, but taking account of the stiffness loss of the 0∞ plies due to progressive fibre failure. To model the behaviour of the fibres in the 0∞ plies, a representative volume element is selected (see Fig. 3.5) that contains a reasonable number of fibres significantly fewer that those actually present in the 0∞ ply. Such an approach is essential in order to deal with the progressive failure of the fibres, as the computational times associated with Monte Carlo simulations can be excessive and unacceptable if large numbers of fibres are included in the simulation. The problem that arises is that the representative volume element is to be located in a 0∞ ply where the through-thickness stress distribution will be non-uniform due to the presence of ply cracks in the 90∞ plies. Two approaches can be taken where the first is to assume that the stresses applied to the representative volume element correspond to those that would arise in the 0∞ plies of a laminate without any ply cracks. The second is to apply the axial stress averaged over to thickness of the 0∞ ply in the plane ahead of a ply crack. Software has been developed based on the first approach, as described in ref. 38.

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Homogenised 90∞ ply 0∞

90∞

0∞ Static failure of fibres

Fibre/matrix cell

Biaxial loading thermal stresses

Biaxial stresses thermal stresses

3.5 Schematic diagram of multi-scale modelling of a simple cross-ply laminate.

The approach described to predict the failure of a cross-ply laminate could be applied to more general laminates, such as quasi-isotropic laminates, where both 0∞ and 90∞ plies are present. Ply crack formation would be dealt with by using the model developed for general symmetric laminates, while a representative volume element in which a Monte Carlo fibre failure simulation, would be applied to the 0∞ plies in the laminate. It is clear that the statistical modelling of fibre failures in a composite with strongly bonded fibre/matrix interfaces is a complex issue that leads to significant computational requirements if reasonably sized systems are to be simulated.

3.5.4

Ply cracking as an indicator of laminate strength

Consider first of all laminates in which no 0∞ plies are present. For the case of carbon fibre composites, Johnson and Chang [39] have tested many laminate types and provided laminate strength data, and a description of damage modes that were observed during failure. For laminates where the ply angle is greater than or equal to 45∞, significant ply cracking is a dominant damage mode, although edge effects can lead to edge delaminations, which are not considered in this chapter. These observations lead to two possible approaches. First of all, it is pertinent to ask whether the first ply cracking stress is a good indicator of laminate strength for this type of laminate. Results presented in refs 40 indicate that such an approach can be used for CFRP. The second approach makes use of observations [30] from simulations of progressive ply crack formation using the homogenisation methodology

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described in section 3.4.5. When each new ply crack forms the stress is regarded as being increased from zero following the homogenisation of the ply in which the last ply crack formed. A key characteristic of simulations of ply cracking in more than one orientation, when no 0∞ plies are present, is that the stress to form a ply crack first may increase as the number of ply cracks increases. A local maximum ply cracking stress is attained, and thereafter the ply cracking stress decreases as the number of ply cracks increases. The maximum value of the ply cracking stress is regarded as the strength of the laminate, as subsequent damage is interpreted as occurring during the unstable collapse of the laminate. For some laminates, the first ply failure stress can be the maximum stress that occurs during the simulation. When 0∞ plies are present in laminates, as for example in quasi-isotropic laminates, laminate failure can be predicted only if a fibre failure criterion is incorporated into the model. By ignoring the statistical variability of the fibre strengths, this has been carried out for the homogenisation model that deals with ply cracking in multiple ply orientations, and preliminary results for CFRP quasi-isotropic laminates have been obtained [30] where the ply lay-up has been varied. A critical fibre strain criterion has been used in predictions. It is seen from results presented in ref. 39 that the experimental data do lead to an effect of ply thickness and lay-up on laminate strength, and that the model based on homogenisation and energy methods does predict the correct type of trends for laminate strength, e.g., lower strengths result when the 90∞ plies are adjacent to the mid-plane of the laminate so that their thickness is doubled relative to laminates where the 90∞ plies are away from the mid-plane. The development of reliable models for laminate strength prediction is in its infancy, although good progress is being made, as demonstrated here by the ability of physically based models to predict laminate thickness and layup effects. Such effects cannot be predicted by the failure criteria taught in engineering courses, and provided in some FEA packages, and consequently care should be exercised when using such methods in engineering design procedures. It is emphasised that the effects of ply thickness and lay-up were not considered in the International Failure Prediction Exercise [31–33], and that the NPL model described here, and based on physical modelling, is the only one of those included in the exercise that is able to take such effects into account.

3.6

Future trends

The advent of powerful computers that can be purchased at reasonable cost means that complex models, that would be difficult for design engineers to use, can be implemented as user-friendly computer applications. An important future trend will be for systems of this type to become more acceptable for

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use in design offices. Another important development will be to integrate such software systems within finite element design systems so that physicallybased damage and failure models can be applied to composite structures. While some progress has already been made, the ideal design tool will require additional capability so that the effects of additional damage modes, such as delamination, can be integrated into the developing software systems. A severe technical difficulty will be dealing with the excessive computational times that arise when: ∑ ∑ ∑

using homogenisation methods to deal with ply crack formation in multiple ply orientations using Monte Carlo methods to model progressive fibre failures integrating a variety of different types of model within a single userfriendly software system.

One solution will be to develop software systems that can be run on distributed computer systems (e.g. using unused processing capability of the PCs in reasonably sized research organisations), and having access to such system through the Internet so that simulations can be set up, run and analysed remotely. While these difficult concepts clearly define the way forward in the future, an additional hurdle will be developing the acceptance of such an approach by the composite design community. Regulation and certification issues are likely to hinder the rapid development of these ideas. One approach might be to target students (both undergraduates and M Sc and Ph D) who will then import their experiences into companies involved with composite design, leading on to requirements for regulated and certified methods of efficient physically based composite design. The models described in this chapter may be used to perform the following tasks that can be exploited at some stage during the design of composite structural elements, indicating specific ways in which the technology might be exploited in the future: ∑ ∑

∑

Predict undamaged properties of general symmetric laminates from the properties and orientations of the individual plies. Predict the conditions for the initial formation of fully developed ply cracks in a general symmetric laminate subject to general in-plane loading and thermal residual stresses. In components subject to fatigue loading designers will want to avoid any form of microstructural damage. Predict the progressive formation of ply cracks in a single orientation during the monotonic loading of a general symmetric laminate. Multiaxial non-linear stress-strain behaviour is predicted (needed to understand strain softening due to microstructural damage), and the progressive degradation of the thermoelastic constants as a function of applied stress or strain. Such non-linear behaviour leads to load transfer in a structure

96

∑

∑

Multi-scale modelling of composite material systems

from regions of stress concentration, where microstructural damage forms and degrades properties, to other regions in the structure. This phenomenon can lead to composite components performing beyond expectations using designs based on coupon data and the neglect of damage effects. Predict the effects of varying the thicknesses of the plies in a laminate. Ply cracking is well known to become more difficult when the thickness of the plies is reduced. The energy-based approach described in this chapter can model this behaviour, contrasting sharply with stress-based approaches. Doubling the thicknesses and applied loads does lead to a different damage evolution that is not predicted by strength-based methods. Predict the effects of temperature changes on ply crack formation (can be used to investigate thermal cracking following manufacture, or during cryogenic cooling).

To assist in the exploitation of the analysis described in sections 3.2–4, which leads to the capability of operating the above design tasks, a new userfriendly computer application known as PREDICT has recently been released that is a module of the NPL-developed CoDA Composite Design system [27].

3.7

References

1. Hashin Z., ‘Analysis of composite materials – A survey’, J. Appl. Mech. (1983), 50, 481–505. 2. Christensen R.M. and Lo, K.H., ‘Solutions for effective shear properties in three phase sphere and cylinder models’, J. Mech. Phys. Solids, (1979), 27, 315–330. 3. Rosen W.B. and Hashin Z., ‘Effective thermal expansion coefficients and specific heats of composite materials’, Int. J. Engng. Sci., (1970), 8, 157–173. 4. McCartney L.N., Analytical method of calculating the thermo-elastic constants of a multi-phase unidirectional composite, (1992), NPL Report DMM(A)57. 5. Halpin J.C., Primer on composite materials analysis, first published in 1968, second edition 1992, Technomic Publishing Co. Inc., Lancaster, Pa, Basel. 6. Jones R.M., Mechanics of Composite Materials, (1975), McGraw-Hill Book Co., New York, London, Tokyo. 7. Datoo M.H., Mechanics of composite materials, (1991), Elsevier Applied Science, London, New York. 8. McCartney L.N., ‘Model to predict effects of triaxial loading on ply cracking in general symmetric laminates’, Comp. Sci. Tech. (2000), 60, 2255–2279. (See also errata in Comp. Sci. Tech. 62 (2002) 1273–1274. 9. Parvizi A., Garrett K.W. and Bailey J.E., ‘Constrained cracking in glass fibre reinforced epoxy cross-ply laminates’, J. Mater. Sci., (1978), 13, 195–201. 10. Nuismer R.R.J. and Tan S.C., ‘Constitutive relations of a cracked composite lamina’, J. Comp. Mater., (1988), 22, 306–321. 11. Hashin Z., ‘Analysis of cracked laminates: A variational approach’, Mech. of Mater., (1985), 4, 121–36. 12. Hashin Z., ‘Analysis of orthogonally cracked laminates under tension’, J. Appl. Mech., (1987), 54, 872–879.

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13. Hashin Z., ‘Thermal expansion coefficients of cracked laminates’, Comp. Sci. Tech., (1988), 31, 247–260. 14. Nairn J.A., ‘The strain energy release rate of composite microcracking: A variational approach’, J. Comp. Mater., (1989), 23, 1106–1129. 15. Nairn J.A., ‘Some new variational mechanics results on composite microcracking’, Proc 10th Int. Conf. on Composite Materials, Whistler BC Canada, Aug 14–18, (1995), 423–430. 16. Nairn J.A., ‘Fracture mechanics of composites with residual thermal stresses’, J. Appl. Mech., (1997), 64, 805. 17. Nairn J.A. and Hu S., ‘The formation and effect of outer-ply microcracks in crossply laminates: A variational approach’, Eng. Fract. Mech., (1992), 41, 203–221. 18. Pagano N.J., ‘Axisymmetric micromechanical stress fields in composites’, Proc IUTAM Symposium on Local Mechanics Concepts for Composite Material Systems, (1991), 1–26, Springer-Verlag, Berlin-Heidelberg-New York. 19. Pagano N.J. and Schoeppner G.A., ‘Some transverse cracking problems in cross-ply laminates’, Proceedings of AIAA Conference, Orlando, Florida, April 1997. 20. Pagano N.J., Schoeppner G.A. Kim R. and Abrams F., ‘Steady state cracking and edge effects in thermo-mechanical transverse cracking of cross-ply laminates’, Comp. Sci. & Tech., (1998), 58, 1811–1825. 21. McCartney L.N., ‘Physically based damage models for laminated composites’, Proc. Instn. Mech. Engrs., (2003), 217 Part L: J. Materials: Design and Applications, 163– 199. 22. McCartney L.N., ‘Predicting transverse crack formation in cross-ply laminates resulting from micro-cracking’, Comp. Sci. & Tech., 58 (1998) 1069–1081. 23. Crocker L.E., Ogin S.L., Smith P.A. and Hill P.S., ‘Intra-laminar fracture in angleply laminates’, Composites, Part A, (1997), 28A, 839–846. 24. McCartney L.N., ‘An effective stress controlling progressive damage formation in laminates subject to triaxial loading’, Proc. of Conference on Deformation and Fracture in Composites, 18–19 March 1999, IoM Communications, London, pp. 23– 32. 25. Tong J., Guild F.J., Ogin S.L. and Smith P.A., ‘On matrix crack growth in quasiisotropic laminates – I. Experimental investigation’, – II Finite element analysis, Comp. Sci. & Tech., (1997), 57, 1527–1545. 26. McCartney L.N. and Schoeppner G.A., ‘Predicting the effect of non-uniform ply cracking on the thermoelastic properties of cross-ply laminates’, Comp. Sci. Tech., (2002), 62, 1841–1856. 27. Software system ‘PREDICT’ which is a specific module of CoDA. (see http:// www.npl.co.uk/npl/cmmt/cog/coda.html) 28. McCartney L.N., ‘Energy-based prediction of progressive ply cracking and strength of general symmetric laminates using an homogenisation method’, Composites A. (2005), 36, 119–128. 29. McCartney L.N., ‘Prediction of multiple ply cracking in general symmetric laminates’, Proc. ICCM-14, San Diego, Ca , July 14–18 2003. (CD-Rom version only). 30. McCartney L.N., ‘Energy-based prediction of failure in general symmetric laminates’, Engng, Fract. Mech., (2005), 72, 909–930. 31. Part A of the International Failure Prediction Exercise is published in Comp. Sci. Tech., (1998), 58. 32. Part B of the Exercise is published in Comp. Sci. Tech., (2002), 62. 33. Part C of the Exercise is published in Comp. Sci. Tech., (2004), 64.

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34. Tsai S.W. and Wu E.M., ‘A general theory of strength for anisotropic materials’, J. Compos. Mater., (1971), 5, 58–80. 35. Liu K.-S. and Tsai S.W., ‘A progressive quadratic failure criterion for a laminate’, Comp. Sci. Tech., (1998), 58, 1023–1032. 36. Puck A., ‘Calculating the strength of glass fibre/plastic laminates under combined load’, Kunststoffe, German Plastics, (1969), 55, 18–19 (German text pp. 780–787). See also ref. 1 above. 37. McCartney L.N. and Smith R.L., ‘Statistical theory of the strength of fibre bundles’, J. Appl. Mech., 50, 601 (1983). 38. McCartney L.N., Simulation of progressive damage formation and failure during the loading of cross-ply laminates, NPL Report MATC(A)20, July 2001. 39. Johnson P. and Chang F.-K., ‘Characterisation of matrix crack induced laminate failure – Part I: Experiments’. J. Comp. Mater., (2001), 35, 2009–2035. 40. McCartney L.N., ‘Minimising damage and delaying failure of composite laminates’, in Proceedings of Conf. on Advanced Polymer Composites for Structural Applications in Construction (ACIC), Guildford, UK, April 2004, Woodhead Publishing Ltd, Cambridge, UK.

4 Modelling the strength of fibre-reinforced composites B F I E D L E R, Technical University Hamburg-Harburg, Germany, S O C H I A I, Kyoto University, Japan and K S C H U L T E, Technical University Hamburg-Harburg, Germany

4.1

Introduction

The strength of composite materials depends on the strength of the reinforcing fibres, the matrix and the interface. Numerical methods are introduced to estimate the strength of transverse and parallel to the fibre axis loaded unidirectional (UD) composites. The mechanical and thermal properties of the constituents are considered and the interaction of thermal residual stress and strength on the fracture morphology of high performance composites. The evaluation allows determination/prediction of the optimal (necessary) interfacial strength at given matrix and fibre properties. For polymers it is often observed that the strength under tensile and compressive loading is very different. The matrix in a composite is due to micro-residual stresses and the strain magnification due to the presence of the fibres under multi-axial stress state. Modelling the matrix strength requires a description of the dependency of the neat resin matrix strength on the stress state. The parabolic failure criterion used indicates the strength data obtained by tensile, compression and torsion tests of the neat resin. Studies of the fracture surfaces were correlated to the stress-state dependent strength and fracture stress of the neat epoxy resins. FE analysis via a partial discrete model, which consists of a hexagonal unit cell surrounded by an area with orthotropic composite properties, studies the influence of residual stresses on the transverse strength by initial matrix failure and the minimum interfacial strength required for that matrix failure. The local fibre volume fraction Vfl (the hexagonal unit cell) may vary from much less than its average (resin rich areas) to maximum values (close contact) where resin pockets are formed. Therefore, both effects can occur and result in local tensile and compressive stresses. A parametric study by an extended shear lag approach clearly shows the influence of the strength of the constituents (fibre, matrix and interface) on the fracture behaviour and composite strength in fibre direction. The theoretical 99

100

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work is highly validated by the results obtained from tensile tests of UD specimens and the corresponding fracture surfaces.

4.2

Mechanical and thermal response of the polymer matrix

It is often observed that, while the neat resin shows rather brittle fracture behaviour at very low tensile strain, it yields and shows considerable plastic deformation in uni-axial compression or in pure shear [1]. Even composites containing such brittle matrices can deform considerably plastically on the microscopic level [2]. To clear up the role of the matrix in transverse laminates the yielding and fracture behaviour of the neat resin under different kinds of loading, with regard to the thermal dependent properties like Young’s modulus, yield and fracture stress, as well as the non-linear behaviour, have to be considered.

4.2.1

Modelling thermal dependent properties of matrix polymers

The glass transition temperature (TG) of the neat resins and the influence of temperature on stiffness was measured by using dynamic thermal mechanical analysis (DTMA). The experiments were performed at a frequency of 10 Hz in tension mode. For polymers it is well known that the dynamic glass transition temperature is about 30 K higher than the static one [3]. The absolute value of the complex modulus E* is slightly different from the Young’s modulus E. The difference is caused by the viscoelastic behaviour of the resin and its frequency dependence, even at room temperature (RT). From tensile tests carried out in the range between T = –40 ∞C close to TG the dependency of the modulus, ultimate strength and strain to fracture were determined; the details are given elsewhere [4]. The results are a linear relation of stiffness, yield and fracture stress on the temperature. This behaviour as well as the temperature dependence of non-linearity of the stress strain curves are incorporated into the FEA.

4.2.2

Investigation and modelling of polymer properties under different loading conditions

It is often observed that the deformation behaviour of polymers is different in tensile, compressive and shear loading [5, 6]. Ductility, plastic flow and fracture are found to be a function of the state of stress, strain, strain rate, temperature and environment [7]. The initiation of a crack at a certain point in the microstructure generates a local condition and the ductility of the

Modelling the strength of fibre-reinforced composites

101

material governs the rate of crack propagation. For brittle materials like glassy polymers it is characteristic that the strength in compression is greater than in tension. Plastic deformation can be accomplished under compressive states of stress and the scatter in fracture strength is controlled by internal flaws and their distribution. Dog bone tensile specimens, tubes for the torsion test and cubes for the compression tests, were prepared from a single neat resin slab to ensure the same resin and curing conditions for all specimens. The tensile, torsion, and compression tests of the plain resins were carried out by a universal testing machine. To describe and explain the failure of epoxy resins under different kinds of loading conditions, the strength and fracture data used should be independent of geometrical changes of the specimen. To transfer the results of the neat resin to the multi-axial stress-state in the matrix of a fibre reinforced composite makes it especially necessary not to use the engineering strength and fracture data. It is more suitable to use the true stress and true strain to calculate the failure criteria. The individual stress strain curves are given in detail elsewhere [8]. Considering the initial failure of the matrix in composites, the non-linear or viscose deformation of the matrix itself is not initial failure. Hence, void or crack nucleation in the resin or debonding of the interface is initial failure. This can be clearly seen on the micrographs for the different fracture surfaces (Fig. 4.1(a)–(c)). The corresponding mechanical properties are summarised in Table 4.1. The appearance of the tensile fracture surface changes with the crack propagation velocity and leads to the well-known regions mirror, smooth with hyperbolic markings, hackle and severe roughness. The parabolic profiles have been attributed to the generation of secondary cracks in the region of high stress local to the primary crack (Fig. 4.1(b)). In the case of compression tests it is particularly important to reduce the friction between the specimen and the anvils. For this test thin polymer films were taken to act as lubricant, as similarly reported by others [9]. The specimens were completely fractured by longitudinal cracks and no typical pyramidal shaped non-deformed areas were found. This was also found on the microscopic level, i.e., the specimens were saturated with cracks parallel to the applied load (Fig. 4.1(a)). At defects, like micro-voids, the compressive stresses induce tensile stresses perpendicular to the loading direction [10]. The generated tensile stress at the poles is about half the absolute value of the applied compressive stress [11]. The fracture surface should appear similar to that obtained by a tensile test. Actually, the same mechanisms and similar geometric patterns of parabola (triangular) and ellipse, (rhombic) could be found, now generated to 3D geometrical bodies (Fig. 4.1(a)). The stresses and the stressstate are illustrated in Fig. 4.1(e). The fracture surface of the specimen under torsional load shows a pattern

102

(b)

(c) Fracture surface Micro-void

s1

Specimen

s1

s1

(d)

tmax

(e) Normal stress fracture

Shear stress fracture tmax

s3 = – s1 Superposition

tmax

s1

s1

tmax

s3 = – s1

tmax

s1

4.1 Micrographs of the fracture surface of the neat resin #113: (a) V = 0.125 cm3 obtained by the compression test (the arrow indicates the loading direction); (b) typical SEM micrograph of the fracture surfaces obtained by tensile test of the neat resin; (c) SEM picture of the fracture surface of the neat resin #113 obtained by torsion test; (d) crack formation due to compressive induced tensile stress; (e) local stresses and stress-state.

Multi-scale modelling of composite material systems

(a)

Modelling the strength of fibre-reinforced composites

103

Table 4.1 Elastic constants and strength data for the neat resin LY556/HY932 and Toho 113 Material

LY556/ HY932. Tensile Comp. Torsion Toho #113 Tensile Comp. Torsion

Stress s/t [MPa]

Strain e/g [%]

Modulus E/G [GPa]

84 ± 2.7 201 ± 4.6 54.5 ± 8.1

4.7 ± 0.2 40.7 ± 0.6 19.2 ± 3.4

3.7 ± 0.2 3.6 ± 0.3 1.2 ± 0.1

93 ± 3 –243 ± 4 64

4.9 ± 0.6 –36 ± 0.5 58

3.9 3.6 1.35

unusual for a brittle resin. Failure is a combination of plastic deformation and to some extent stable crack growth. In detail, this behaviour must be separated into the acting shear stress and normal stress and the corresponding type of failure or deformation. Macroscopically, the compressive and tensile stresses acting under 45∞ result in a fracture pattern (Fig. 4.1(c)) similar to that of the compression test (Fig. 4.1(a)). The normal stress forms a parabolic fracture pattern, which is superposed on the shear stress induced fracture. The shear stress results in a high plastic deformation. Thus, the shear stress leads to an extended formation of hackles at defects. These hackles are similar to those observed under Mode II crack propagation in carbon fibre reinforced composites. Traces of the parabola and one half of the s-curve shaped hackles are visible. The superposition is indicated in Fig. 4.1(e), the acting stresses and the stress-state forming the ‘brittle’ type fracture surface under torsional loading are shown. The shear stress is responsible for forming the hackle type shape and the normal stress for the parabola shaped cracks. The final macroscopic failure is determined by the acting normal stress. So that fracture, meaning of separation in two or more parts, is caused by tensile stresses. As shown, in Fig. 4.2, the parabolic criterion is applied to the neat resins. The curves are based on the experimental results and the calculated true stresses and strains and fit the experimental results. This criterion includes the dependency on the hydrostatic component of the stress-state, it then has a parabolic shape and is a linear combination of the mean normal stress and the square of the octahedral shear stress [12]. t 20 = a1 – a2 · s0

4.1

The octahedral stresses (s0, t0) in the principal stress space are given by eqns 4.2 and 4.3 [13].

104

Multi-scale modelling of composite material systems 150

Octahedral shear stress (MPa)

100

50

0

–50

–100

–150 –100

–50 0 Octahedral normal stress (MPa)

50

4.2 Parabolic criteria in terms of the octahedral stresses (so, to) applied to characteristic stress data of the neat epoxy resin Toho 113 (dots, solid line) and the Ciba Geigy LY556/HY932 (circle, dashed line).

s0 =

s1 + s 2 + s 3 3

t0 = ±

4.3

(s1 – s 2 ) 2 + (s 2 – s 3) 2 + (s1 – s 3) 2 3

4.2 4.3

Modelling first ply failure by FEA using the partial discretisation approach

The analyses of the thermal residual stresses and the initial matrix failure were carried out using the commercial finite element code MARC/MentatTM. The FE-model consists of two-dimensional plane strain elements and models a microscopic area of fibre and matrix, surrounded by elements with homogeneous properties of a unidirectional composite with transverse orientation (Fig. 4.3). The volume fraction for the composite elements was always constant Vf = 60%. The local fibre volume fraction of the discrete part of the model was again varied from Vfl = 20% to Vfl = 80%. Therefore, hexagon unit cells with a fibre volume fraction of Vfl = 20% to Vfl = 80% were made surrounded by elements with composite properties. The local area is only 1% of the whole FE-model, so that total stiffness is not affected by the variation of the local fibre volume fraction. Fibres and composite

Modelling the strength of fibre-reinforced composites

105

sr s2, e2 10 m

q

T

Composite [90∞] Vf = 60%

Z sz

Fibrepole

m

tqr q 0

n Vfl = 20 to 80% (a)

10 n

Carbon fibre 3mm

trq sq Interface

(b)

4.3 Composite model including hexagonal unit-cell for modelling thermal residual stresses and transverse failure, the hexagonal area is 1% of the total model; stress components acting in the matrix and at the interface.

were taken as orthotropic materials. Tables incorporate the elastic constants and the thermal-mechanical behaviour of the fibre, matrix and composite elements. The epoxy matrix was assumed to be isotropic and the parabolic fracture criterion is taken. The non-linear behaviour and thermal (CTE) and thermal dependent mechanical properties were incorporated according to section 4.1.

4.3.1

Thermal response and residual stresses

The residual stresses are influenced by the distance to the neighbouring fibres and the stress components are not constant along the circumference. More exactly, the values vary over a wide range and can change from positive (tensile) to negative (compression) values depending on the fibre volume fraction. In Fig. 4.4 the results of the stress components for a local fibre volume fraction in the discrete hexagon area of Vfl = 60% are drawn. The identical results obtained by a simple hexagon unit cell model confirms the values of thermal expansion coefficients of the composite obtained by thermal mechanical analysis, which were used for the FE-calculations. As the local fibre volume fraction varies, the thermal expansion coefficient of this discrete area changes as well. The thermal expansion coefficients of the surrounding homogeneous composite elements (Vf = 60%) remain constant. When the local fibre volume fraction Vfl differs from 60%, additional thermal mismatch stresses occur. For fibre volume fractions lower than Vf = 60%, the thermal expansion becomes larger and additional tensile stresses appear due to cooling. A local fibre volume fraction higher than Vf = 60% will result in

106

Multi-scale modelling of composite material systems 40 Longitudinal

Residual stress (MPa)

30 Hoop 20 Composite Hexagon 10 Radial 0 –10

–20 0

60

120 180 240 300 Position along the circumference (deg)

360

4.4 Thermal residual stresses along the circumference obtained by the composite model and the hexagon unit cell model.

additional compressive stresses in the discrete fibre/matrix area. Depending on the volume fraction, the difference between minimum and maximum value of radial stress rises with increasing local fibre fraction. Furthermore, the radial stresses are not always compressive along the circumference. Thermal residual stresses increase with increasing fibre volume fraction [14]. The results of the FE-calculation now show a decrease of the thermal residual stresses with increasing local fibre volume fraction. For the longitudinal stresses, the values change from sZmin = 33.4 MPa and sZmax = 38.7 MPa with a volume fraction of Vfl = 20% to sZmin = 13 MPa and sZmax = 39.6 MPa with a volume fraction of Vfl = 80%, i.e., the maximum value is nearly constant and the minimum value decreases. The radial stresses show a similar pattern. Depending on the volume fraction, the difference between minimum and maximum value of the radial stresses becomes larger with increasing local fibre fraction. Furthermore, the radial stresses are not always compressive along the circumference. The values of stress decrease from sRmin = –1.7 MPa and sRmax = –5.8 MPa with a volume fraction of Vfl = 20% to sRmin = –33.8 MPa and sRmax = 10.9 MPa with a volume fraction of Vfl = 80 %. In case of the hoop stresses, the stress pattern is similar only for the high volume fraction of Vf = 80%. For this volume fraction, the location of the maximum value is again at a = 0∞ and a = 60∞. In the case of lower volume fractions, the location of the maximum value moves to the angle of closest neighbouring fibres (a = 30∞). This is the same behaviour as observed for the calculations of the hexagon unit cell. The values of the stresses decrease from sqmin = 27.8 MPa and sqmax = 30.1 MPa with a volume fraction of Vfl

Modelling the strength of fibre-reinforced composites

107

= 20% to sqmin = 12.5 MPa and sqmax = 25 MPa for a volume fraction of Vfl = 80%. Taking into account the interaction of neighbouring fibres the microresidual stress values are not constant along the circumference of the individual fibres. With increasing fibre volume fraction, the difference between minimum and maximum value is also increasing. Local fibre volume fractions lower than the average lead to higher residual stresses than for constant fibre volume fraction. Hence, for higher local volume fractions (Vfl = 80%), the calculated thermal residual stresses are lower. The reason for this behaviour is the change of the thermal expansion coefficient with the local fibre volume fraction, so that additional thermal mismatch stresses between the composite and the local discrete fibre/matrix areas occur.

4.3.2

First ply failure

After cooling down to room temperature, load in transverse direction is applied. The local octahedral stresses s0, t0 at the location of first failure are plotted for the different fibre volume fractions in Fig. 4.5 together with the thermal residual stresses. In terms of octahedral stresses, the residual stresses are not so much affected by the local fibre volume fraction. With increasing Vfl the stress-state changes only slightly to hydrostatic tension [15]. When stress s2 is applied, locally the matrix gets under a more tri-axial tensile stress-state and for all local volume fractions, the stress-state is more important than the bi-axial tensile stress-state.

Octahedral shear stress to (MPa)

30 Vfl Vfl Vfl Vfl

25 20

= = = =

20% 40% 60% 80% RT, s2 = 0 s2 = 63 MPa

15 10

Cooling DT = – 95 K

s2 = 57 MPa

Vf local

s2 = 57 MPa

5

s2 = 60 MPa 0 0

10

20 30 40 Octahedral normal stress so (MPa)

50

60

4.5 Local octahedral stresses so, to for the location of first matrix failure with thermal effects, local fibre volume fraction Vfl = 20% to Vfl = 80%.

108

Multi-scale modelling of composite material systems

In general, the thermal residual stresses reduce the maximum bearable load. The bearable load is higher only at a high fibre volume fraction of Vfl = 80%. Hence, the location of initial failure is not affected by the thermal residual stresses. In reality this effect will not contribute so much because locally a close contact of the fibres causes a resin rich area elsewhere. Finally, it can be said that the most critical area for transverse strength is the boundary between regions of densely packed fibres and resin rich areas. Hull published typical images of this behaviour [16]. On the discrete microscopic level, the matrix fails at small values of local strain, much smaller than the possible tensile value of the neat resin. Due to the tri-axial stress-state, the matrix fails first and locally at a stress equivalent to about 1/3 of the strain to failure measured in the tensile test of the plain resin. Depending on the local fibre volume fraction, matrix failure occurs at applied global stresses between s2 = 63 MPa (Point A in Fig. 4.6) and s2 = 57 MPa (Point B in Fig. 4.6). The corresponding global strains are between e2 = 0.75% and e2 = 0.67% for A and B, respectively. The corresponding local strains are in the range between e22 = 1.48% for Vfl = 20% (Point A¢ in Fig. 4.6) and e22 = 0.73% for Vfl = 80% (B¢). These results are in good agreement with the experimentally obtained strength of ss = 61 ± 3 MPa at es = 0.78 ± 0.07%. 80 Experiment composite

70 FEM composite (global)

60 50

A B¢

FEM first failure of the matrix (local)

50

Vfl Vfl Vfl Vfl

30

30 20

0

Cooling DT = – 95 K 0

70 60

B

40

10

A¢

0.5 1 Global and local strain e2, elocal (%)

= = = =

20% 40% 60% 80%

40

20 10

Local stress component slocal (MPa)

Applied global stress s2 (MPa)

80

0 1.5

4.6 Transverse tensile test; global experimental stress-strain curve (a); global by FEA calculated stress-strain curve (b); local stress-strain curve in the matrix at the location of initial failure (c).

4.3.3

Interfacial strength for given matrix properties

It is often possible to observe an apparent interfacial failure at the top (a = 0∞) and bottom (a = 180∞) of the fibres, but a cohesive failure along the fibre

Modelling the strength of fibre-reinforced composites

109

midplane. Based on the calculated stress analysis, both tensile and compressive stresses are acting at the interface. In the regions of interfacial compressive stresses (a = 30∞), the failure is shifted to the matrix region [17]. The transverse strength calculated for the case of matrix failure gives the upper bound of the transverse failure caused by interfacial failure. The experimental results show that interfacial failure is the failure-dominating mechanism [15]. In the calculation of the composite transverse failure the interfacial normal (INS) and shear strength (ISS) were varied from ISS = 20 MPa to ISS = 40 MPa and the interfacial normal strength from INS = 30 MPa to INS = 70 MPa. For a local fibre volume fraction of Vfl = 60% the results are given in Fig. 4.7. The composite failure shows a linear dependence on the interfacial strength (INS, ISS). The intersection points with the experimentally observed transverse strength (shaded band) give the upper (ub) and lower (lb) bounds of the interfacial normal strength (INS) and the interfacial shear strength (ISS), respectively.

Composite transverse failure (MPa)

80 Vf local = 60% 70

Transverse tensile test

60

50

Shear failure

lb ub

lb

ub

Normal failure

40

30 10

20

30 40 50 60 70 Interface strength ISS, INS (MPa)

80

4.7 Influence of interfacial normal strength (INS) and interfacial shear strength (ISS) on the composite transverse failure; calculated by FEA with a local fibre volume fraction Vfl = 60%.

For the different local fibre volume fractions (Vfl = 20% to Vfl = 80%) the mean interfacial strength and the upper and lower bounds are summarised in Fig. 4.8. It can be seen that for low local volume fractions (resin rich areas) a high interfacial normal strength is required, but a low value in interfacial shear strength (ISS). The required interfacial shear strength increases with the local fibre volume fraction up to a value of Vfl = 60%; for higher values the strength decreases. The scatter of the interfacial shear strength values is small. In the case of the interfacial normal strength (INS), the required

110

Multi-scale modelling of composite material systems

Interface strength ISS, INS (MPa)

70 Normal strength INS 60

50

40 Shear strength ISS 30

20 10

20 30 40 50 60 70 80 Local fibre volume fraction Vfl (%)

90

4.8 Calculated interfacial strengths (INS, ISS) for the different local fibre volume fractions (Vfl = 20% to Vfl = 80%).

values first decrease and are then nearly constant with increasing Vfl. Only the upper bound of the strength increases. If fibre volume fractions between 20% and 80% are assumed in real composites, the interfacial shear strength ISS is about ISS = 33 MPa and the interfacial normal strength INS = 64 MPa (Fig. 4.8). The calculated interfacial shear strength values are in good agreement with experimentally observed values [18].

4.4

Stress-strain response and fracture morphology in UD composites

4.4.1

Importance of mechanical interaction among the microscopic damages for description of macroscopic behaviour and advantage of the shear lag analysis for description of micro-macro correlation

In unidirectional fibre composites, the strength of components (fibre, matrix) is not scattered, and therefore the strength-value of the components is different from position to position. Thus the fracture of fibres, matrix and interface take place under applied stress at a number of places within the composite. They interact mechanically on each other, determining the species (fibre, matrix and interface) and location of the next fracture. Accordingly, the damage map changes and therefore the mechanical interaction among the damages also changes with progressing fracture. As a result of the consecutive change of the damage-map-dependent interaction, the overall response of

Modelling the strength of fibre-reinforced composites

111

composites such as stress-strain curve, strength and fracture morphology is determined. It is, however, extremely difficult to obtain a rigid solution for the interactions among many spatially distributed damages (so-called multi-body problem). In addition, the damage map changes at every occurrence of new damage. Therefore the new interaction is calculated for the new damage map at each incremental load step. Thus, simplified and approximated methods, which make it possible to solve the change of damage map and the resultant interaction consecutively, shall be developed as a first approximation. In this section, an attempt and some results to solve such a multi-body problem by means of modified shear lag analysis combined with the Monte Carlo technique are presented. Shear lag analysis [19, 28] has been studied as one of the tools to obtain the approximate solution for the problems stated above. Ordinary shear lag analysis has, however, been developed using the approximation that only fibres carry the applied stress and the matrix acts only as a stress-transfer medium. Due to this approximation, it has been applied only to the composites containing soft matrices such as polymers or low yield stress metal matrices [19–26] but not to ceramic matrices. Also, due to this approximation, the residual stresses were not considered. Recently, to overcome these disadvantages, a modified shear lag analysis has been proposed [27, 28], by which the general situation that both fibres and matrix carry the applied stress and both act as stress transfer media in the presence of residual stresses is taken into account. This section describes one of the multi-scale approaches to combine micro and macro behaviour, covering the change in spatial distribution of microscopic fracture of the fibre, matrix and interface with applied strain and its relation to the resultant macroscopic stress-strain behaviour.

4.4.2

Procedure of simulation to correlate microscopic damage evolution to macroscopic behaviour

Details of the mathematical background and the procedure for the modified shear lag-Monte Carlo simulation are shown elsewhere [27, 28]. The outline is briefly described below. Figure 4.9 shows the two-dimensional model composite. Each component (fibre, matrix), numbered as 1,2,…i,... to N from left to right, is regarded to be composed of k + 1 short component elements with a length Dx. The Young’s modulus, shear modulus, width and residual strain of the ‘i’ component are expressed by Ei, Gi, di, and ei,r, respectively, where i = odd and even refers to the matrix and fibre, respectively. The position at x (distance from the top end of the composite) = 0 is numbered as 0 and then 1, 2, 3,...j... k + 1 downward, in step of Dx. The ‘i’ component from x = (j–1) Dx to jDx is named as the (i,j)-component-element, the interface from x = (j–1/2)Dx to (j + 1/2)Dx between ‘i’ and ‘i + 1’ components as the

112

Multi-scale modelling of composite material systems 0 1

1 2 – i

– – N Dx

– – j–1 j j+1 – k+1

Fibre

Displacement ‘j – 1’ ‘i’ ui,j–1 Componentelement (i, j) Interfaceui,j ‘j’ element (i, j) ui,j+1 ‘j + 1’

matrix

4.9 Schematic representation of the two-dimensional model composite for shear lag-Monte Carlo simulation.

(i, j)-interface-element and the displacement of the (i,j)-component-element as x = jDx as ui,j. Introducing the interfacial parameter, ai, j (= 1 and 0 when the (i,j)-interface is bonded and debonded, respectively) and the component parameter, gi,j (= 1 and 0 when the (i,j)-component-element is not fractured and fractured, respectively), the equation for stress balance is given by [27] B1(i, j) ui,j–1 + B2(i, j) ui–1,j + B3(i, j) ui,j + B4(i, j) ui,j+1 + B5(i, j) ui+1,j = B6(i, j)

4.4

where B1(i, j) to B6(i, j) are parameters including the elastic constant, crosssectional area, volume fraction and a residual stress of each component and a damage map that is expressed as the spatial distribution of ai,j and gi,j values. As the values of B1(i, j) to B6(i, j) vary with varying damage map, they are determined at each occurrence of new damage. By using the ui,j values calculated by eqn 4.4, the tensile stress of the (i, j) component si,j and the shear stress at the (i, j) interface ti,j can be calculated. The strength distributions of fibre and matrix are assumed to obey the Weibull distribution function [29]. The strength of each component Si,j is determined by generating a random value based on the Monte Carlo procedure. A fixed shear strength value (tc) is given for all interface-elements for simplicity within the present chapter. The rupture of fibre and matrix elements and interfacial debonding are judged to occur when the exerted tensile stress si,j reaches the strength Si,j and when the exerted shear stress ti,j exceeds the shear strength tc, respectively. The input values for simulation are taken by imaging the polymer-derived SiC fibre-reinforced ceramic matrix composite.

4.4.3

Interfacial debonding process

Figure 4.10 shows the progress of debonding with increasing applied strain ec in the presence of one pre-cut matrix-element and fibre-element each. The thick transverse lines show the pre-cut component-elements components. The thick longitudinal lines show the debonded interface-elements. The rupture

af < am

af > am

2

2 4

1 3

5 1 3 7

6 2 4 8

ec = 0.14%

14 10 6

13 9 5

8 12 16

7 11 15

1

25 21 23

22 18

20 24

17 13 9

18 14 10

11 15 19

12 16 20

18 14 10 5 3 7 12 16 20

45 41 37 33

4628 42 38 32

31 35 39 43 47

34 36 40 44 4830

29

46 42 38 34 30

45 41 37 33 29

32 36 40 44 48

31 35 39 43 47

21 17 26 28

25 27

21

23

19 23

25

ec = 0.30%

22 24 ec = 0.50%

27

26

ec = 0.82%

4.10 Difference in progress of debonding among the cases of af = am, af < am and af > am under the existence of one pre-cut fibre-element and one pre-cut matrix-element. The white and coloured components correspond to the matrix and fibre, respectively. The thick transverse and longitudinal lines indicate the locations of the pre-cut componentelements and debonded interface-elements, respectively. The numbers 1,2,... show the order of the debonding of the interface element.

Modelling the strength of fibre-reinforced composites

af = am

17 13 9 24 6 22 26 4 8 11 15 19

113

114

Multi-scale modelling of composite material systems

of the components during deformation is inhibited in this simulation in order to monitor the interfacial debonding. The numbers 1,2… show the order of the debonding of interface-elements. To a first approximation the residual strains of fibre (ef,r) and matrix (em,r) along the fibre axis were calculated by applying the rule of mixtures for the axial coefficient of thermal expansion (af and am for fibre and matrix, respectively) [30]. It was assumed that no residual stresses are present in the composite at elevated temperature T1 but develop after cooling down to T2 (room temperature). The resulting fibre and matrix strains are as followed: ef,r = (am – af)EmVmDT/(EfVf + EmVm) and em,r = (af – am)EfVfDT/(EfVf + EmVm) where DT is the difference in temperature (T2 – T1). The following three cases were assumed to give different residual stress-state. 1. af = am (no residual stress exists). 2. af < am (am – af = 4 ¥ 10–6/K; corresponding to compressive and tensile residual stresses in fibre and matrix along the fibre axis, respectively). 3. af > am(af – am = 4 ¥ 10–6/K; corresponding to tensile and compressive residual stresses in fibre and matrix along the fibre axis, respectively). Figure 4.10 evidently shows that the existence of residual stresses changes the order and location of occurrence of damage in the damage accumulation process. The following features can be mentioned: (i) When no residual stress exists (af = am) and the fibre volume fraction is high as in the present examples (0.5), the debonding induced from the cut ends of fibre (hereafter noted as cut fibre-induced debonding) occurs prior to the cut matrix-induced one. Also, the former progresses in a narrower strain range (ec = 0.30– 0.53%) than the latter (ec = 0.48–0.78%). (ii) When the residual stresses along the fibre axis are tensile and compressive in the matrix and fibre, respectively (af < am), the cut matrix-induced debonding is encouraged, but the cut fibre-induced one is discouraged in comparison to those without residual stress (af = am). On the other hand, when the residual stresses are reverse (af > am), the cut fibre-induced debonding is encouraged but the cut matrix-induced one is discouraged. (iii) Debonding grows to some extent and stops at a given strain, but then grows again after increment of applied strain. In other words, debonding progresses intermittently.

4.4.4

Influence of residual stress on stress-strain behaviour of composites

In practical composites, ruptures of components and interfacial debonding both occur concurrently. An example showing the influence of residual stresses on stress-strain behaviour is presented in Fig. 4.11. In this example, the matrix failure strain is taken to be lower than that of fibre as in the usual ceramic composites. As the numbers of elements of broken fibres (Nf,e),

Stress, sc (MPa)

af > am

1200 1000 800 600 400 200 00

0.2 0.4 0.6 0.8 1 Strain, ec (%)

1 0.8 0.6 0.4 0.2 0

1.2 1.4

Nm,e Nf,e Nie sc

1 0.8 0.6 0.4 0.2 0

Normalised number of broken elements, Nm, Nf and Ni

Stress, sc (MPa)

0

sc Nm,e Nf,e Nie

1 0.8 0.6 0.4 0.2 0

Normalised number of broken elements, Nm, Nf and Ni

af < am

1200 1000 800 600 400 200 0

Stress, sc (MPa)

af = am

1200 1000 800 600 400 200 00

0.2 0.4 0.6 0.8 1 1.2 1.4 Strain, ec (%) sc Nm,e Nf,e Nie

0.2 0.4 0.6 0.8 1 1.2 1.4 Strain, ec (%) (a)

115

Normalised number of broken elements, Nm, Nf and Ni

Modelling the strength of fibre-reinforced composites

ec = 0.35%

ec = 0.55%

ec = ecu = 0.65%

ec = 0.35%

ec = 0.55%

ec = ecu = 0.8%

ec = 0.35% (b)

ec = 0.55% (c)

ec = ecu = 0.6% (d)

4.11 An example of the simulation result to monitor the influence of residual stresses on the stress-strain curve and damage evolution of the composite. The damage maps at the strains indicated by the arrows in the stress-strain curves are shown on the right-hand side.

116

Multi-scale modelling of composite material systems

matrix (Nm,e) and interface (Ni,e) as a function of applied strain were quite different from each other and could not be shown clearly on the same scale, they were normalised with respect to the final values Nf,f, Nm,f and Ni,f at the fracture of composite, respectively. Such normalised values of Nf = Nf,e/Nf,f, Nm = Nm,e/Nm,f and Ni = Ni,e/Ni,f are superimposed in Fig. 4.11. The following features are mentioned. 1. The stress-strain curves for all samples show serration due to the rupture of the components and interfacial debonding, indicating that the damages are accumulated intermittently. 2. The strains of matrix rupture and interfacial debonding and therefore the composite strength are different among the cases of af = am, af < am and af > am. In the case of af = am, many matrix elements are broken but interface elements are not debonded at ec = 0.35%. In the case of af < am, at the same strain of ec = 0.35%, more matrix elements are broken due to the tensile residual stress in matrix. Also many interface elements are debonded due to the matrix rupture-induced debonding enhanced by the residual stresses. On the contrary, in the case of af > am, no matrix component element is broken at the same strain due to the suppression of matrix rupture by the compressive residual stresses in matrix. Therefore no matrix rupture-induced debonding occurs. 3. At ec = 0.55%, in the cases of af = am and af < am, the number of debonded interface elements increases compared to those at ec = 0.35%, while still no fibre rupture occurs. The number of broken matrix elements increases only slightly, since the growth of debonding reduces the efficiency of stress transfer to the broken matrix. In the case of af > am, matrix rupture, which has been suppressed at ec = 0.35% due to the compressive residual stress of the matrix, occurs at ec = 0.55%. However, the number of broken matrix elements is still small in comparison with those in the case of af = am and af < am. On the contrary, due to the enhancement of fibre rupture by means of tensile residual stresses in the fibres, the weaker fibres are broken. The fibre-rupture-induced interfacial debonding is also enhanced by the tensile residual stress. 4. The strain ecu at ultimate load becomes different under the different residual stress (0.65, 0.8 and 0.6% for af = am, af < am and af > am, respectively). The damage maps at ultimate load at ec = ecu are shown in Fig. 4.11(d). In the case of af = am and af < am, almost all fibre elements carry the applied stress but most matrix elements have lost the stresscarrying capacity due to the rupture, followed by debonding. However, comparing the damage map for af = am with that for af < am in detail, the matrix is more intensively broken in the first case and mainly the interface is debonded in the latter. Thus the strength of the composite for af = am is slightly higher than that for af < am. In the case of af > am,

Modelling the strength of fibre-reinforced composites

117

the matrix-rupture-induced debonding is still suppressed, but the fibre rupture and fibre-rupture-induced debonding are enhanced. However, still many fibres and the matrix can carry the applied stress, resulting in higher strength than the strengths in the other cases.

4.4.5

Influence of fibre strength on composite strength and fracture morphology

To monitor the influence of fibre strength on strength and fracture morphology of the composite, the scale parameter of the fibre was changed. The shape parameter was taken to be 5 for both fibre and matrix. The scale parameter of the matrix was taken to be 500 MPa and that of the fibre (s0,f) was varied from 170 to 1700 MPa. Figure 4.12 shows the simulated relation between the strength of composite and scale parameter of fibre, together with the fracture morphology. Evidently, with decreasing fibre strength, the composite strength decreases and the fracture morphology varies from fibre pullout type to nonfibre pullout type. It is emphasised here that such changes of the strength and fracture morphology occur even though the interfacial bonding strength is not changed. Such a change of morphology is practically observed for the SiC/SiC composite exposed in air at high temperatures, as shown in Fig. 4.13 [31]. In this composite, the fibre strength is reduced by oxidation and accordingly it decreases with increasing exposure temperature and time. As a result, the fracture morphology of the composite varies from fibre pullout type to nonfibre pullout. In this way, the fracture morphology is strongly dependent on fibre strength, indicating that it is not determined solely by the interfacial strength. On this point, further detailed analysis was carried out by the authors [32, 33] based on the non-dimensional approach. The results revealed that the stress-strain behaviour and fracture morphology are determined by the relative strength of the components. Relative strength means the ratio of the scale parameter of components to the interfacial strength. Should fibre strength be far higher than the matrix strength, the relative strength is the ratio of the scale parameter of the fibre to the interfacial strength, as shown in Fig. 4.14. When the relative strength of components is low (relative strength of interface is high), the rupture of the component tends to cause the rupture of the components nearly in the same cross-section and consequently the irregularity of the fracture surface is small. With increasing relative strength of components (with decreasing relative strength of the interface), the rupture of the components in the same cross-section tends not to occur but the debonding tends to occur more and the irregularity of the fracture surface tends to increase. When the relative strength of components is very high (or relative strength of interface is low), the rupture of the components tends to

118

Strength of composite, sc (MPa)

800 600 400 200 0 0

500 1000 1500 Scale parameter of fibre, s0,f (MPa)

2000

4.12 Relation between the strength of composite (sc) and scale parameter of fibre (s0,f) under a fixed interfacial bonding strength (simulation result), together with the change of fracture morphology with fibre strength.

Multi-scale modelling of composite material systems

1000

Modelling the strength of fibre-reinforced composites

119

500 mm (a)

500 mm (b)

500 mm (c)

4.13 Change of fracture morphology of the SiC/SiC composite exposed to air at high temperatures with decreasing fibre strength. (a), (b) and (c) refer to the specimens exposed at 823 K for 3.6 ¥ 105 s, at 1273 K for 3.6 ¥ 104 s and at 1673 K for 3.6 ¥ 104 s, respectively. The strengths of the specimens after exposure were 700, 130 and 65 MPa for (a), (b) and (c), respectively.

Matrix

Fibre

Low

Relative strength of components (strength of components/strength of interface)

High

High

Relative strength of interface (strength of interface/strength of components)

Low

4.14 Change of fracture morphology with relative strength of components.

cause debonding simultaneously over a long distance, due to which the behaviour of the composite tends to be similar to that of the bundles of components and the irregularity of the fracture surface is very large. As the matrix strength is low in comparison with fibre strength, the relative strength

120

Multi-scale modelling of composite material systems

of the interface is practically determined by the ratio of interfacial strength to fibre strength. The result shown in Figs 4.12 and 4.13 correspond to such a case.

4.5

Conclusions

The macroscopic strength values of neat epoxy resins obey the parabolic criterion. The parabolic criterion not only fits the data, it includes the strength dependency of polymers on the hydrostatic pressure. Therefore, the results of the mechanical tests and the application of the parabolic criterion clearly show the strong influence of the stress-state on the final strength and fracture stress. For all three different loading conditions, a parabola, elliptical or similar fracture pattern could be found on the fracture surface. This typical pattern is caused by normal stress fracture. Large plastic deformations of the epoxy resin might have only a small influence on the final value of fracture stress and the fracture behaviour is still brittle. The morphology of the fracture surface is strongly influenced by the previous plastic deformation and the trace of the loading history can be found on the fracture surface. The state of stress controls the plastic flow and the normal tensile stress controls crack propagation. Finally, ductility is a function of the amount of tri-axiality and explains why ductile polymers become brittle when used as a matrix in fibre reinforced composites. Thermal residual stresses and their non-homogeneous distribution along the circumferences for the individual fibres magnify these effects. For the practical use of epoxy resins for composite matrix materials, the use of high-strain matrices increases the toughness and damage tolerance. However, the existence of tri-axial stresses in the matrix limit the ductility of the matrix. Therefore, the improvement in the strain to failure of modified resins or thermoplastic matrices when compared to a conventional thermoset composite is much less than would be expected from the neat resin stress/ strain-behaviour alone. The influence of resin-rich areas and resin pockets between fibres on the initial matrix failure and on the interfacial failure consists of two aspects: (i) with increasing local fibre volume fraction and (ii) the geometrical influence both reduce the stress values of failure. The residual stresses depend on the ratio of local to nominal volume fraction leading to the opposite effect. The calculated strength values show only a slight influence of the thermal residual stresses and the related compensating geometrical change of local fibre volume fraction. In fibre direction the strength of the composite is controlled not only by the strength of the constituents (fibre, matrix, interface). The relative strength which is the ratio of fibre and matrix strength to the interfacial strength controls the fracture morphology and the maximal utilisation of the materials potential.

Modelling the strength of fibre-reinforced composites

4.6

121

Future trends

Modelling the thermal residual stresses should include the chemical volume shrinkage of the polymer matrix during curing as well as time effects like creep or relaxation of the resin. These effects will reduce the calculated thermal residual stresses during the manufacturing process. In composite materials the structure is, more or less, non-uniform and irregular. For instance, fibre-array is not necessarily uniform; interface is not smooth; the microstructure of fibre and matrix has second phases and/or voids that are not uniformly distributed. Also the strength (and fracture toughness) of fibre and matrix are not uniform, varying from position to position. For a comprehensive description of the relation between micro and macro behaviour, such non-uniformity should be incorporated. The approach presented above is approximate. As the next step, the development of the accurate estimation method of the micro-structural non-uniformity and the methods to describe the damage initiation, extension and accumulation depending on the microstructural non-uniformity, are requested. In such research, the development of the method to solve the interaction among the spatially distributed microstructure and damages, namely, to solve the multibody problem without loss of accuracy, is a key issue. With such an approach, designing of highly efficient utilisation of the constituents and microstructure and therefore the creation of new high-performance composite materials could be realised.

4.7

Further reading

Fibre reinforcements and general theory of composites, ed. by Chou T.-W., (Vol. 1 of Comprehensive composite materials, editors-in-chief, Kelly A. and Zweben C. ), Elsevier. Amsterdam, New York. 2000. Polymer matrix composites, ed. by Talreja R. and Manson J.-A.E., (Vol. 2 of Comprehensive composite materials, Editions-in-chief, Kelly A. and Zweben C.), Elsevier, Amsterdam, New York. 2000. Engineered interfaces in fibre reinforced composites, Kim J.-K. and Mai Y.-W., Elsevier, Amsterdam, Tokyo. 1998.

4.8

References

1. Kinloch A.J. and Young R.J., Fracture Behaviour of Polymers, Elsevier Applied Science, London New York, 1983. 2. Bradley W.L., ‘Matrix toughness and interlaminar fracture toughness’, in Composite Materials Series: Vol. 6: Application of Fracture Mechanics to Composite Materials, edited by Friedrich K., Elsevier, Amsterdam, London, New York, Tokyo, 1989. 3. Young R.J., Introduction to Polymers, Chapman and Hall, London, New York, 1987. 4. Hobbiebrunken T., Hojo M., Fiedler B., Tanaka M., Ochiai S. and Schulte K., ‘Thermomechanical Analysis of the Micromechanical Formation of Residual Stresses and Initial Matrix Failure in CFRP’, JSME, 2004 47 (3) 349–356.

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5. Bowden P.B. and Jukes J.A., ‘The Plastic Flow of Isotropic Polymers’, Journal of Materials Science, 1972 7 52–63. 6. Asp L.E., Berglund L.A. and Talreja R., ‘A Criterion for Crack Initiation in Glassy Polymers Subjected to a Composite Like Stress State’, Composite Science and Technology, 1996 56 1291–1301. 7. Manjoine M.J., ‘Multiaxial Stress and Fracture’, in Fracture, Vol. 3, 265–309 by Liebowitz, Academic Press, New York, London 1971. 8. Fiedler B., Hojo M., Ochiai S., Schulte K. and Ando, M., ‘Failure behaviour of epoxy matrix under different kinds of static loading’, Composites Science and Technology, 2001 61 (11) 1615–1624. 9. Rinne F., ‘Vergleichende Untersuchungen über die Methoden zur Bestimmung der Druckfestigkeit von Gesteinen’, Neues Jahrbuch für Mineralogie, 1909 (12) 121– 128. 10. Sammis C.G. and Ashby M.F., ‘The Failure of Brittle Porous Solids under Compressive Stress States’, Acta metall, 1986 34 (3) 511–526. 11. Goodier J.N., ‘Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws’, Trans. Am. Soc. Mech. Engrs., 1933 55 39–44. 12. Fiedler B., Hojo M. and Ochiai S., ‘The Parabolic Failure Criterion applied to Epoxy Resin’, Proc. of 4th Mesomechanics, Aalborg University, 2002 533–539. 13. Dubbel: Handbook for Mechanical Engineers, edited by Beitz W. and Küttner K.-H., Springer-Verlag: Berlin, Heidelberg, New York, London, Paris, Tokyo, 16th edn, (1987), (in German). 14. Nairn J.A., ‘Thermoelastic Analysis of Residual Stresses in Unidirectional, High Performance Composites’, Polymer Composites, 1985 6 (2) 123–130. 15. Fiedler B., Hojo M. and Ochiai S., ‘The influence of thermal residual stresses on the transverse strength of CFRP using FEM’, Composites Part A, 2002 33 (10) 1323– 1326. 16. Hull D. and Clyne T.W., An introduction to composite materials, 2nd edn, edited by Clarke D.R., Suresh S. and Ward I.M., Cambridge University Press, 1996. 17. Fiedler B., Hojo M., Ochiai S., Schulte K. and Ochi M., ‘Finite-element modelling of the initial matrix failure in CFRP under static transverse tensile load’, Composites Science and Technology, 2001 61 95–105. 18. Ageorges C., Friedrich K. and Ye L., ‘Experiments to relate carbon-fibre surface treatments to composite mechanical properties’, Compos. Sci. Technol., 1999 59 2101–2113. 19. Hedgepeth J.M., ‘Stress Concentrations in Filamentary Structures’, NASA TN D882, Washington, 1961. 20. Oh K.P., ‘A Monte Carlo study of the strength of unidirectional fibre-reinforced composites’, J. Comp. Mater., 1979 13 311–328. 21. Goree J.G. and Gross R.S., ‘Analysis of a unidirectional composite containing broken fibres and matrix damage’, Engng. Fract. Mech., 1980 13 563–578. 22. Dharani L.R., Jones W.F. and Goree J.G., ‘Mathematical modeling of damage in unidirectional composites’, Engng. Fract. Mech., 1983 17 555–573. 23. Nairn J.A., ‘Fracture mechanics of unidirectional composites using shear-lag model I: Theory’, J. Comp. Mater., 1988 22 561–588. 24. Ochiai S., Tokinori K., Osamura K., Nakatani E. and Yamatsuta Y., ‘Stress concentration at a notch-tip in unidirectional metal matrix composites’, Metall. Trans., 1991 22A 2085–2095. 25. Goda K. and Phoenix S.L., ‘Reliability approach to the tensile strength of unidirectional

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26. 27.

28.

29. 30. 31.

32.

33.

123

CFRP composites by Monte Carlo simulation in a shear lag model’, Comp. Sci. Technol., 1994 50 457–468. Curtin W.A. and Takeda N., ‘Tensile strength of fibre-reinforced composites: I. model and effects of local fibre geometry’, J. Comp. Mater., 1998 32 2042–2059. Ochiai S., Okumura I., Tanaka M., Hojo M. and Inoue T., ‘Influences of residual stresses, frictional shear stress at debonded interface and interactions among broken components on interfacial debonding in unidirectional multifilamentary composites’, Comp. Interfaces, 1998 5 363–381. Ochiai S., Tanaka H., Kimura S., Tanaka M., Hojo M. and Okuda H., ‘A Modeling of Residual Stress-Induced Stress-Strain Behaviour of Unidirectional Brittle Fibre/ Brittle Matrix Composite with Weak Interface’, Comp. Sci. Technol., 2003 63 1027– 1040. Weibull W., ‘Statistical distribution function of wide applicability’, J. Appl. Mech., 1951 18 293–297. Hale D.K., ‘Review: The physical properties of composite materials’, J. Mater. Sci., 1976 11 2105–2141. Ochiai S., Kimura S., Tanaka M., Tanaka H., Hojo M., Morishita K., Okuda H., Nakayama H., Tamura M., Shibata K. and Sato M., ‘Residual Strength of PIPprocessed SiC/SiC Single-tow Minicomposite Exposed at High Temperatures in Air as a Function of Exposure Temperature and Time‘, Composites Part A, 2004 35 41– 50. Ochiai S., Hojo M., Schulte K. and Fiedler B., ‘Nondimensional Simulation of Influence of Toughness of Interface on Tensile Stress-Strain Behaviour of Unidirectional Minicomposite’, Composites Part A, 2001 32 749–761. Ochiai S., Hojo M., Tanaka M., Okuda H., Schulte K. and Fiedler B., ‘Nondimensional Simulation of Tensile Behaviour of UD Microcomposite under Energy Release Rateand Shear Stress-Criteria for Interfacial Debonding’, Composite Interfaces, 2004 11 (2) 169–194.

5 Cracking models P W R B E A U M O N T, University of Cambridge, UK and H S E K I N E , Tohoku University, Japan

5.1

Introduction

Oversights in designing with engineering composite materials at the microstructural level of size, have resulted in the evolution of a variety of different cracking mechanisms and the catastrophic fracture of laminated components under stress and environmental attack. The fracture micro-mechanics and underlying physical processes of failure in engineering composite materials are presented. Modelling techniques are described, which quantify this accumulation of damage over time in terms of the important structural features of the composite material, constituent properties, stress-state, temperature, and environment. Particular emphasis is placed on the material internal state variable method of modelling, which relates failure mechanism and material property change. It is important to identify those individual components of damage and to recognise their significance, meaning their special role and dominance in the overall fracture process, of the material on the one hand and component on the other. Proof of identity of individual failure processes based on their direct observation and an understanding of coupling between them are the first steps in the formulation of a complete fracture model. Armed with this information, together with knowledge of the mechanical behaviour of the material, we follow the path of ‘physical modelling’, (sometimes called ‘mechanism modelling’ or ‘micro-mechanical modelling’), or simply ‘micro-mechanics’, which takes us closer to a complete solution to the problems of fracture.

5.2

Empirical and physical modelling

For half a century, the fracture of engineering composite materials that derive stiffness and strength from long strong fibres of glass or carbon has been the subject of extensive experimentation and analytical investigation. Yet despite this acquisition of vast collections of mechanical property data and their plots of dependencies – a phenomenology – and a designer’s intuition based 124

Cracking models

125

on ‘feel’, ‘know-how’ or ‘folklore’– our ability to fully understand this longstanding problem remains restricted. This is because our experience is based (almost) entirely on this store of information being empirical in nature. Our comprehension of the micro-structural changes, which take place in the composite material over time, is somewhat lacking in detail. Oversight in design of microstructure has led to undesirable matrix-dominated load paths. This has resulted in the nucleation and growth of a multiplicity of interacting small cracks in the material over time. Being aware of the accumulation of damage in the component under stress and chemical attack is simply not good enough. Lack of mastery in combining the design of architectural features of the material at the micron level of size with elements of the engineering structure metres in length, has led to the opening of a gap in our knowledge of composite failure, (see, for example, Spearing and Beaumont 1998; Spearing et al. 1998). This weakness can be traced to the changing nature of fracture as size increases: from the fibre to the single ply to the test coupon; and from the component to the sub-assembly to the fully assembled structure (Fig. 5.1). If we consider coming to terms with all sorts of material behavioural complexities at the micron level of size, we might say that we have characterised the properties of the composite at that particular scale by reference to the fibre only. There has been no real consideration of the microscopic architectural features of the composite, or macroscopic geometry of the laminate, or shape of the part. Any notch, hole or cutout is but a geometrical aberration. Coupon level testing Basic material and laminate properties

Design/ analysis

Subelement testing (e.g. stiffener)

Full component testing (e.g. wing)

Structural behaviour of basic forms

Element testing (e.g. stiffened panels)

Behaviour of as-designed structure

Basic structural configuration behaviour

Subcomponent testing (e.g. wing box)

Behaviour of critical component parts

5.1 Flowchart for the ‘building block’ approach for structural composite design/certification.

Conversely, at the level of component design (metre length and above), we have tended to look at the overall geometric shape and thought of the material properties as being set (in a geometric sense) at a global level.

126

Multi-scale modelling of composite material systems

Coming to terms with these differences of scale appears to be a key source of design difficulty because it is precisely at that size, where the material problem becomes a structural one, that this gap in our understanding of composite failure has opened up. This gap has been partially filled in certain engineering material fields using fracture mechanics, where quantitative relationships between atomistic, microscopic, and macroscopic parameters have been developed. This size (or length) scale, which spans several orders of magnitude, provides a framework for understanding the failure characteristics of the material on the one hand and performance limitation of the component on the other. A hierarchy of structural scales and discrete methods of modelling or analysis at each level is illustrated in Fig. 5.2. Almost always, behaviour at one level can be passed to the next level up as one or more parameters or as a simple mathematical function. Unit

Scale

Design process/ mechanics

104 m System performance (systems engineering)

Complete structure (as-designed structure) 102m Component (e.g. aircraft wing)

Structural interactions (structural mechanics) 100m

Structural element (e.g. stiffened panel)

Element testing (computational mechanics) 10–2m Laminate behaviour (fracture/damage mechanics)

Coupon (ply thickness, fibre orientation) 10–4m Single ply microstructure (fibre fraction, layup, etc.

Constitutive properties (micro-mechanics) 10–6m Statistical analysis (weakest link statistics)

Single fibre (individual fibre and matrix) 10–8m

5.2 Hierarchy of structural scales ranging from the micron to the metre (and greater) level of size, and discrete methods of analysis ranging from micro-mechanics to the continuum levels of modelling.

5.2.1

To follow the path of empirical modelling (brick by brick)

Taking this path rewards us with information of the properties of composite materials by a progression of experiments, whereby we see clearly the effects

Cracking models

127

of applied stress (or temperature or time or environment) on the strength, stiffness (or modulus), toughness, or fatigue life of the material. Such experience or ‘know-how’, (the phenomenology), comes from testing in sequence first the test coupon, then the sub-structural element, next the component, and finally, the full-scale structure. This is to follow the ‘building block’ approach (Fig. 5.1). In parallel with experimentation is the design and analysis phase, wherein critical parts of the structure are identified and selected for test verification. Any critical strength features are subsequently isolated in the form of small test articles of progressively increasing geometrical complexity and size (Kedward and Beaumont 1992; de Vouvray et al., 1988). In the aerospace industry, for example, a sequence of component tests might start with a spar shear web and move on to a skin panel rib attachment of an aircraft wing. Successful design by this route relies on transferring empirical information and relating experimental data as a simple mathematical function from one point on the size-scale to the next. Without too much difficulty, a mathematical fit to a new set of experimental measurements can be found. Unfortunately, this empirical equation (representing an empirical model or law) has no power of prediction outside the constraints of the experiment. To attempt to do so could prove fatal. Some have tried; most have failed.

5.2.2

To follow the direction of physical modelling (step by step)

There is available, however, a second direction, an alternative option, which, if we take, we come upon the well-known laws or principles of physics and chemistry. This is to move in the direction of physical modelling, applying the laws of micro-mechanics in the process of formulating a route map called predictive design. This time, such modelling does have powers of prediction derived from those established rules of physical behaviour. But even then, a complete physical treatment is not always possible. For example, a model of a thermally activated chemical reaction, using the law of Arrhenius, has its basis in statistical mechanics. Sometimes the activation energy, which enters that law, can be predicted from molecular models, but the value of the pre-exponential in the equation more often than not eludes current modelling methods; it must be inserted empirically. Interesting material behaviour, (which is frequently dynamic, meaning material properties that are timedependent), originates (usually) from a kinetic process, diffusion or the rate of a chemical reaction, and often contains an empirical component. The dynamic kinking mechanism of a Kevlar fibre in compression is an example (Ganczakowski and Beaumont 1989; Ganczakowski et al. 1989, 1990). Another example is stress corrosion cracking of glass fibre-epoxy composites (Sekine and Beaumont 1998, 2002), of which more is said in section 5.9.

128

5.2.3

Multi-scale modelling of composite material systems

Importance of understanding mechanisms

To understand the consequences of damage in composite material systems requires the design process at each size level of structure to include the dominant (meaning most influential) crack growth mechanism(s). Making links or connections between a material’s fracture resistance and the environment (via its toughness), or the damage tolerance of a component after impact or load cycling, (via its residual compressive strength or stiffness), relies critically on understanding the differences between the structural changes taking place over the entire range of size with time. Predicting material behaviour demands knowledge of these mechanisms. Ultimately, we require answers to the following: how did the cracks get there in the first place; are they dangerous; why did they form; how do they affect material properties; and what can we do to limit their growth?

5.3

Choosing between continuum and physical modelling

Conventional mechanical design is essentially empirical: in perspective, it has been by trial and error. It has either worked or it has not; and if it failed, you simply tried again and learnt by experience. Whilst many old cathedrals still remain standing throughout Europe; many more collapsed prematurely. Consequently, a phenomenology was built up over time. Conventional design has depended explicitly upon the results of that experimentation. In addition to drawing upon ‘know-how’ or ‘feeling’, the modern designer in mechanical engineering relies upon two boxes of tools. The designer’s first box is called mathematics and continuum modelling (sometimes called continuum mechanics). Box one contains those observed rules of familiar material behaviour: the laws of mechanics, of thermodynamics, of rate theory, and so on. From these have evolved the continuum theories of elasticity, plasticity, diffusion, reaction rates, etc. A store of data collected by experimentation can be manipulated using these mathematical methods in a straightforward manner. The second box is called micro-mechanical modelling, (sometimes known as mechanism or physical modelling or simply micro-mechanics). We use these tools to create a picture, a representation or model of the real thing. The model can be compared to a two-dimensional ordnance survey map or a three-dimensional physical relief map of geographical landscape. Whilst a topographical model clearly misrepresents elevations and misleads with distances, it elegantly displays the connectivity with sufficient precision and usefulness. Although the two or three-dimensional model is an idealisation or massive simplification, nevertheless it captures the essential characteristics and features of what truly exists or happens. And, like the ordinance survey

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129

map, the model can be used to display a vast amount of information in a way that is uncomplicated in its interpretation. Some say physical modelling is an art form. The map (or model) of the underground rail network in the City of London hangs in the Tate Gallery. It, too, is a massive simplification or idealisation of the actual system yet it works successfully for tens of thousands of end users every day.

5.3.1

Continuum modelling (via tool box 1)

Closer examination of what has happened shows that continuum theory groups the many variables involved in the engineering design process: working load, displacement, minimum weight, size, etc. It reduces the number of experiments we have to do, and guides us more efficiently to the optimum design. Solutions to engineering problems need not necessarily be complete and, in fact, a total understanding of the problem is rarely required. Mostly a satisfactorily solution requires synthesis, optimization, approximation and ‘feel’, and generally has a time constraint. Unfortunately, the problem that still does not go away is that the constitutive equations of continuum design remain firmly based on experimental evidence. (A constitutive model is a set of mathematical equations that describe the behaviour of a material element when subjected to an external influence: stress, temperature, etc.) Difficulty arises, of course, when experimental conditions become stringent, so that even more properties are involved in the design process. What are needed, of course, are constitutive equations for design that encapsulate all of those intrinsic and extrinsic variables. Intrinsic variables that include the properties of the composite material that in turn depend upon the geometrical considerations of the laminate (fibre orientation, ply thickness, stacking sequence, etc.); and architecture of the microstructure, which includes the shape and size, volume fracture, and dispersion of reinforcement. And the elastic constants of the reinforcing phase and matrix determine the elastic properties of the composite, and so on. Quite obviously, the experimental programme from which these constitutive laws are to be devised becomes formidable. Furthermore, spatial variation appears when stress and temperature or other field variables are non-uniform. While simple geometries can be treated analytically, using, for example, the modelling tools of fracture mechanics, more complex geometries require discrete methods. The finite element method of modelling is an example. Here the material is divided into cells, which respond to temperature, body forces, and stress via constitutive equations, with the constraints of equilibrium, compatibility and continuity imposed at their boundaries. Internal material state variable formulations for constitutive laws are embedded in the finite element computations to give an accurate description of spatially varying behaviour, of which more will be said later with respect to stresses at tips of cracks.

130

5.3.2

Multi-scale modelling of composite material systems

Physical modelling (via tool box 2)

In an engineering context, at first attempt, the physical model could describe concisely a body of fatigue or fracture stress data. A better model, however, would be one that captures the essential physics of the engineering problem of cracking and fracture. By identifying the dominant microscopic process(es) responsible for failure in the first place, we can then model it (them) using the tools of micro-mechanics and our understanding of the theory of defects, of reaction rates, diffusion (and so forth). But most importantly, the physical model would illuminate the basic principles that underline the key elements of the total fracture process. By these means, the micro-mechanical model establishes a physical framework on which empirical descriptions of the behaviour of some of the intrinsic and extrinsic variables could be attached.

5.4

Combining empirical and physical models

The behaviour of composite materials frequently involves four levels of complexity: structural change, multiple mechanisms, linked processes, and spatial variation (when stress, temperature or other field variables are nonuniform). Structural change, (brought about by micro-cracking, for example), is exactly that. The structure within the material under load, and thus the behaviour or performance of the component in service, changes as damage evolves with increasing stress or time. Over time, (or numbers of cycles in fatigue loading), damage accumulates by one (or more) of competing mechanisms, and they more often than not interact. Damage brought about by multiple micro-cracking weakens the material generally by reducing its stiffness (the material becomes more compliant), and lowers its strength. All of this in turn increases further the damage growth-rate and even more cracks accumulate: there is positive feedback (Poursartip et al. 1982a, b, c). Consider the fatigue of a cross-ply glass fibre-epoxy composite (Poursartip and Beaumont 1986; Poursartip et al. 1986). With increasing numbers of cycles (at low applied stress), the modulus falls slowly as the result of progressive transverse ply matrix cracking (Fig. 5.3). (This is called high cycle fatigue). As the stress amplitude, Ds increases, a noticeable change in slope of the modulus degradation curve designates the onset and domination of delamination (inter-laminar) cracking. And if the stress amplitude increases even further, (now called low cycle fatigue), the overwhelming mode of failure becomes fibre fracture. In this example of progressive fatigue failure, the dominant mode of cracking between these competing mechanisms depends upon the independent variable applied stress, s, or stress amplitude, Ds, (temperature constant). For a given composite, with only one such variable, a fairly complete characterisation is practicable. But this does not cater for time-varying stress Ds (t) and temperature DT(t), or for the effect of changes

Cracking models 90∞ 0∞ 90∞

131

Ds

Damage in a composite laminate Transverse ply cracks = D1 Delamination cracks = D2 Fibre breaks = D3 Response = compliance, E–1 (a)

Eo

Modulus, E

Transverse-ply cracks dominant

Delamination dominant

Fibre fracture dominant

Ds5

Ds4

Ds3

Ds2

Ds1

Number of cycles, N (b)

5.3 (a) A model of the composite laminate subjected to repeated load cycling Ds1, Ds2, Ds3 etc., with transverse-ply cracks, delaminations, and fibre fractures – state variables D1, D2, D3 measure the extent of these damaging mechanisms; (b) The response E, with the dominant failure mechanisms identified.

in stress-state, frequency or environment. If we try to include them, we find we are dealing with eight or more independent variables; applied maximum stress smax, and stress amplitude, Ds, the ratios l of stress invariants, the frequency (ns , nT ), temperature change, DT, and so on. It would be possible to set up an experimental programme to characterise in turn the influence of each of these variables on the fracture or fatigue of the composite. But the scope of these test programmes would become immense. Quite simply, this experimental method of extended empiricism would just break down under the unmanageable load of variables. Nevertheless, derived constitutive models would describe this sort of behaviour without too much difficulty.

132

5.4.1

Multi-scale modelling of composite material systems

Constitutive models: internal state variable method

Constitutive models are best derived using the internal state variable method. Briefly, the key ideas are based on the fact that constitutive models have two aspects, response equations and structural evolution equations. The response equation describes the relationship of (say) current modulus, Ec, of the laminate, (a measure of the effect of damage), to the applied stress, s, or stress range, Ds, load cycles, N, and to the current value of the internal state variable, D. We call the internal state variable ‘damage’ because it describes a change in the state of a material, brought about by an applied stress or by load cycling. It (meaning D) uniquely defines the current level of damage in the material, for a given set of test variables. In Fig. 5.3 we observed changes in the composite modulus (stiffness) with the accumulation of fatigue damage. The response equation describes this change of (damaged) modulus, Ec, to the stress magnitude, temperature, time (number of load cycles), and to the current value of the internal state variable D: Ec = f (s, Ds, l, T, DT, t, ns , nT , D, material properties, environment) 5.1 Consider for example, matrix cracking only: D is usually defined as D = 1/s, where s is matrix crack spacing. Damage due to delamination, on the other hand, can be defined as total (meaning actual or measured) delamination crack area normalised with respect to the total area available for delamination (Poursartip et al. 1982a, b, c, 1984, 1986), i.e., D = A/Ao. Or it might be useful to couple matrix crack spacing, s, with delamination crack length ld, (i.e., s/ld ), because, as we shall see later, these two mechanisms, more often than not, are inseparable (Dimant 1994). Since the internal state variable, D, evolves over time with the progressive nature of the damaging processes, its rate of change can be described by: D¢ = g(s, Ds, l , T, DT, t, ns , nT , D, material properties, environment) 5.2

5.4.2

Competing mechanisms

Where several mechanisms contribute simultaneously to the response, (e.g., where modulus degradation is the result of delamination and matrix cracking combined), this time there are two internal state variables, one for each mechanism. Consequently, the model suggests a constitutive equation having a completely different form than before. Instead of trying to characterise the modulus, Ec, as a function of the complete set of independent variables (although we could), we now seek to fit data to a coupled set of differential

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133

equations, one for the modulus E c¢ , and two (or more), depending on the number of damaging mechanisms, for damage propagation, namely D1¢ and D2¢ : E c¢ = f (s, l , T, D1, D2, etc., material properties, environment) 5.3a

D1¢ = g1(s, l, T, D1, D2, etc, material properties, environment)

5.3b

D2¢ = g2(s, l, T, D1, D2, etc., material properties, environment) 5.3c D1 describes the damage due to one mechanism and D2 describes a different damaging mechanism that, when combined with the first, eventually lead to composite failure. E ¢ , D1¢ and D2¢ are their rates of change with time (or numbers of load cycles); f, g1, g2 are simple functions yet to be determined. There are now three independent variables, (s, T, and stress-state, l), whereas before there were eight. These equations can be integrated to track out the change of modulus with the accumulation of damage, and ultimately used to predict fracture of a component or the design life in fatigue (Fig. 5.3). Thus, the modulus-time (cycles) response is found by integrating the equations as a coupled set, starting with E = Eo (the undamaged modulus) and D = 0 (no damage). Step through time (cycles), calculating the increments, and the current values, of Ec and D, and using these to calculate their change in the next step. Equation 5.3a can now be adopted as the constitutive equation for fatigue, and empirical methods can be used to determine the functions f, g1, g2. However, the model points to something else, and it is of the greatest value; it suggests the proper form that the constitutive equation should take. This model-informed empiricism has led to the development of a new branch of mechanics called ‘damage mechanics.’

5.4.3

Physical model informed-empiricism

The point is this: physical models suggest forms for constitutive equations (laws), and for the significant groupings of the variables that enter them. Empirical methods can then be used to establish the precise functional relations between these groups. Finally, we finish up with a constitutive equation that contains the predictive power of micro-mechanical modelling with the precision of ordinary curve fitting of experimental data. In other words, input variables like maximum stress, stress range or stress amplitude, frequency, etc., and temperature, concentration of chemical species, damage-state, are all embedded in the physical model. This is physical model informed-empiricism or the method of extended empiricism.

134

5.4.4

Multi-scale modelling of composite material systems

Single crack mechanics

Finally, the various modes of failure spread throughout the composite and the multiple array of cracks coalesce, until either the net section stress, (there is a loss of section caused by the damage), exceeds the tensile strength, or a single critical crack has evolved, by a concentration of localised fibre breaks in the (0∞) ply, for instance. Having attained a critical size, this crack then propagates catastrophically across the section. Fracture mechanics can be applied to this problem (for example: Beaumont and Harris 1972; Beaumont and Phillips 1972; Beaumont and Tetelman 1972; Beaumont 1974; 1989a,b; Wells and Beaumont 1982, 1987; de Vouvray et al. 1988).

5.5

Modelling fatigue cracking by delamination

During load cycling of a quasi-isotropic carbon fibre-epoxy laminate, damage evolves within it. It consists of several components; matrix cracking, delamination and splitting, and fibre breakage. In high-cycle fatigue, the dominant mode of failure is by delamination. Ignoring other possible forms of damage for the time being, we can define the damage parameter D as the normalised delamination area A/Ao, where A is the actual (meaning measured) delamination area and Ao is the total area available for possible delamination (Poursartip et al. 1982a, b, c, 1984; Poursartip and Beaumont 1982, 1986; Poursartip et al. 1986). At the start of life D is zero, unless, of course, some damage Di has been introduced during fabrication, or by an earlier history of damage to the laminate. Cyclic loading causes the damage, (by delamination), to increase from Di to Df at which point catastrophic failure of the material occurs. This accumulation of damage over time can be monitored non-destructively by measuring the effect it has on one of the properties of the composite; modulus, for instance. Assuming that the fatigue damage accumulation-rate, dD/dN, depends on the cyclic stress range, Ds, the load ratio, R, and on the current level of damage, D, then:

dD = f ( Ds , R, D ) (temperature, environment, etc., constant) 5.4 dN By integrating this equation, we obtain the fatigue life-time, Nf, (meaning the number of cycles to increase D from Di to Df):

Nf =

Ú

Df

Di

dD f ( Ds , R, D )

5.5

The difficulty, of course, is that we do not know the function f. But we can monitor the tensile Young’s modulus, E, of the composite and the accumulated

Cracking models

135

damage, D, with the number of load cycles, N. And then, provided a relation exists between D and E, we can write: E = Eo g (D)

5.6

(Eo is the initial or undamaged modulus and there is a second function g). It follows that:

1 dE = g ¢( D ) 5.7 E o dD (g¢ means the derivative of g with respect to D). Differentiating eqn 5.7 and substituting into eqn 5.4, we get: 1 dE = g ¢ Ê g –1 Ê E ˆ ˆ f Ê Ds , R, g –1 Ê E ˆ ˆ Á Eo d N Ë E o ¯ ˜¯ ÁË Ë E o ¯ ˜¯ Ë

5.8

where g–1 is the inverse of g: E ˆ D = g –1 Ê Ë Eo ¯

5.9

First, we need to establish the function g(D), by experimental means or theoretically, in order that the damage accumulation function f (Ds, R, D) can be determined. This function g (D) depends on the properties and lay-up of the composite laminate, not on how the damage, D, was introduced. There is now a choice of two approaches that can be taken: either a damage accumulation function is proposed and inserted into eqn 5.8, and the result compared with experimental data, or data can be collected of E/E0 as a function of N, the number of load cycles. Then, knowing g(D), the function f (Ds R, D) is determined experimentally using:

f ( Ds , R, D ) =

1 1 Ê dE ˆ E E ˆ ˆ 0 Ë dN ¯ Ê gÁ g –1 Ê Ë E 0 ¯ ˜¯ Ë

5.10

This is a relatively straightforward exercise. First, the right-hand side of the equation is evaluated for different values of Ds at constant E/Eo and constant R; secondly, for different values of R at constant Ds and constant E/Eo; thirdly, for different values of E /Eo at constant Ds and constant R. Finally, at last, the function f (Ds, R, D) can then be determined from a plot of these results.

5.5.1

Modulus reduction with delamination cracking

A simple physical model for the loss of modulus with delamination cracking is: 5.11 E = Eo + ( E* – Eo ) A A0

136

Multi-scale modelling of composite material systems

E* is the modulus corresponding to a completely delaminated composite, (i.e., a totally separated laminate into sub-laminates). Thus, if the laminate completed delaminated into its individual parts, from this simplest rule of mixtures A/Ao = 1 and E/Eo = 0.65. At this point, the function g(D) in eqn 5.6 can be expressed as: g(D) = 1 – 0.35D

5.12

It follows that: E ˆ D = 2.857 Ê 1 – E0 ¯ Ë

5.13

And finally, from eqn 5.10 f ( Ds , R, D ) =

dD dE ˆ = – 2.857Ê 1 dN E d Ë 0 N¯

5.14

With patience, the right-hand side of eqn 5.14 can be evaluated experimentally. Figure 5.4 shows the predicted and measured fatigue damage-rate, dD/dN, as a function of Ds (for R = 0.1). There are three regimes of fatigue damage. At high stress, the damage-rate accelerates leading to catastrophic failure. Essentially, we have a static tensile fracture, not fatigue failure. Below a threshold stress of 250 MPa, no fatigue damage occurs. This critical threshold corresponds to the stress to initiate the first crack (the ‘design ultimate’) observed in a simple tensile test. Between these two extremes, a power fatigue law dominates fracture behaviour.

5.5.2

Fatigue life: the terminal damage state

To predict fatigue life, we need to know the critical or terminal damage state, Df, at ultimate failure. Now the level of reduced, (meaning damaged), modulus at failure depends upon the maximum applied stress, smax. In load-controlled fatigue experiments, the instantaneous strain, e, of the composite increases as the damaged modulus, E, decreases with load cycling, (meaning the material ‘softens’). After some fatigue cycling at smax, the instantaneous strain is given by:

s max 5.15 E Ultimately, the fatigue life is realised when the applied strain, e, attains the critical failure strain, ec, of the composite, i.e., when e = ec. But in a straightforward monotonic tensile test, (assuming no modulus reduction): e=

ec =

s TS E0

5.16

Since fast fracture occurs when the strain, e, during a fatigue cycle equals ec,

Cracking models

10–3

dD = –2.857 Ê 1 dE ˆ Á ˜ dN Ë E 0 dN ¯ CFC (45/90/–45/0)s XAS/914

smax

sTS

R = 0.04–0.12 only f = 0.5 Hz–30 Hz sinewave

10–4

Each point represents a separate test

10–5

dD = 9.189 ¥ 10 –5 Ê Ds ˆ Á ˜ dN Ë s TS ¯

6.393

10–6

10–7

rl

aw

Damage rate dD/dN (cycles–1)

137

Po

we

10–8

10–9 60

120 180 240 300 Stress range Ds (MPa)

600

5.4 Comparison between observed damage growth-rate dD/dN, and cyclic stress range Ds, for a quasi-isotropic carbon fibre-epoxy laminate and prediction of the damage model (after Poursartip et al. 1982 a, b, c, 1984, 1986).

then by equating eqns 5.15 and 5.16 and substituting for Df from eqn 5.13, we have a value of the terminal damage state Df: Ef ˆ s Ê D f = D = 2.857 Á 1 – = 2.857 Ê 1 – max ˆ s TS ¯ E 0 ˜¯ Ë Ë

5.17

For power-law fatigue damage growth only (from Fig. 5.4):

dD Ê Ds ˆ = 9.2 ¥ 10 –5 Á ˜ dN Ë s TS ¯

6.4

(for R = 0.1)

5.18

Quite simply, we determine Nf by integrating eqn 5.18. We assume that at the start of the fatigue test, Di = 0, and at the lifetime Df is given by eqn 5.17. Finally, we are able to construct the S-N curve (Fig. 5.5). Life prediction is shown by the solid curves, which are based on a choice of three possible tensile strengths, sTS, of the laminate. There is consistency with this treatment and the observations shown by the experimental data points. At low stress, high-cycle fatigue, any variations in laminate tensile strength has little effect on life time, whereas in low-cycle fatigue (high cyclic stress), small variations can be significant.

138

Multi-scale modelling of composite material systems 650

sTS = 650 MPa

Maximum stress smax (MPa)

600 sTS = 600 MPa sTS = 550 MPa

CFC (45/90/ –45/0)s XAS/914 R = 0.1 sinewave f = 0.5–10 Hz Indicates run-out

500 450 400 350 300 1

10

102

103 104 105 Cycles to failure Nf

106

107

5.5 Comparison between measured number of cycles, Nf, to complete failure with maximum fatigue stress smax, and prediction of the damage model for a quasi-isotropic carbon fibre-epoxy laminate (after Poursartip et al. 1982 a, b, c, 1984, 1986).

5.6

Modelling coupled mechanisms in composite cracking

5.6.1

Direct observation of matrix and delamination cracking by in-situ SEM

Most, perhaps all, composite systems damage by a number of different failure mechanisms throughout their lifetime and they can be identified by in situ scanning electron microscopy or inferred by post mortem examination using SEM (Kunz-Douglass et al. 1980; Kunz-Douglass and Beaumont 1981; Gilbert et al. 1983, 1984; Mao et al. 1983; Brown et al. 1986; Kortschot et al. 1989; Vekinis et al. (1990; 1991a, b, 1993; Shercliff et al. 1992, 1994a, b; Dimant 1994). For instance, a cross-ply (0∞/90∞)ns glass fibre-epoxy laminate, under monotonic (or cyclic) tensile loading, fails first and foremost by the evolution of a multiplicity of cracks in the matrix of each transverse (90∞) ply (Fig. 5.6). These closely spaced cracks lie roughly parallel to one another, and in a plane that is perpendicular to the direction of applied stress; and they span the thickness and width of every transverse ply. More often than not, a microscopic-sized delamination crack, (sometimes called an interlaminar crack), nucleates by de-cohesive failure of the (0∞/90∞) interface, in front of the tip of an advancing matrix crack (Fig. 5.7). This delamination crack extends by mode II (shear) deformation and stable interfacial fracture. Intersection of the matrix crack tip with a delamination crack surface seems inevitable (Fig. 5.8). Furthermore, some fibres may fracture within the adjacent (0∞) ply at sites close to the plane of the matrix crack. They do so because of

Cracking models

(0/90/0)

(0/90)s

(0/902)s

(0/904)s

139

5.6 Transverse ply cracking patterns in cross-ply glass fibre-epoxy laminates in tensile fatigue loading at 600MPa for 104 cycles, observed by scanning electron microscopy (Dimant 1994).

the local intensification of stress at the matrix crack tip. Damage accumulates as the matrix cracks multiply in number, together with an abundance of these small delamination cracks. As a consequence, the stiffness (modulus) of the laminate decreases, and the material ‘softens’; it becomes more compliant (Dimant et al., 1997, 2002). There are two possible alternative reactions to the localised stress field surrounding the matrix crack tip: either the delamination crack forms, blunting the matrix crack tip and thereby reducing the tip stress intensity, or the interfacial bond remains intact and there is no delamination. If the latter prevails, the magnification of local tensile stress can initiate the breakage of fibres, (in the adjacent load-bearing (0∞) ply), on or close to the matrix crack plane (Figs 5.9, 5.10, 5.11, 5.12).

5.6.2

The physical model of coupled mechanisms

Figure 5.13 shows the physical picture (Dimant 1994). In this model, a matrix crack intersecting a delamination crack is shown circled. Spacing

100 mm (i)

140

(iii)

(a) (ii)

50 mm

(b) (ii)

(i) (ii)

(ii)

(ii)

(i)

100 mm

(iii)

(a)

10 mm (i)

(c)

5.7 (a, b) (i) Two adjacent transverse ply matrix cracks in a (0∞/90∞/0∞) glass fibre-epoxy laminate close to fast fracture, observed by in-situ scanning electron microscopy. (ii) The onset of progressive fibrematrix debonding leading to further de-cohesion (delamination) at the (0∞/90∞) interface in the vicinity of the transverse ply crack tip. (iii) Fibre breaks in the (0∞) ply at the (0∞/90∞) interface, between adjacent transverse ply matrix cracks. (c) A similar in-situ SEM photomicrograph but for a (0∞/90∞4)s laminate (Dimant 1994).

(b)

5.8 (a) Intersection of a transverse ply crack (i) with a delamination crack (iii). A shear band in the epoxy close to the (0∞/90∞) ply interface is evident with the bands at 45∞ to the principal (applied) stress direction. (b) A magnified picture of the shear bands (Mode II deformation). The shear band is a precursor to the formation of the delamination crack (Dimant 1994).

Multi-scale modelling of composite material systems

50 mm

Cracking models

141

200 mm (ii) (iii)

(i)

5.9 Intersection of a transverse ply crack (i) with a longitudinal ply (ii) of a (0∞/90∞)s glass fibre-epoxy laminate in fatigue. Broken longitudinal fibres (iii) in or close to the plane of the transverse ply crack are visible (iii) (Dimant 1994). 50 mm

(ii)

(i)

5.10 Penetration of the longitudinal fibres (ii) by the transverse ply crack (i) of a (0∞/90∞)s glass fibre-epoxy laminate in fatigue (Dimant 1994).

100 mm (i) (ii) (iii)

5.11 Fracture surface of a fatigue delamination crack showing broken (0∞) fibres (i), transverse ply crack (ii) and extensive fibre debonding (iii) (Dimant 1994). 200 mm

(i) (ii)

5.12 Flat (or planar) fracture surface typical of a tensile fracture of a cross-ply glass fibre-epoxy laminate showing four internal transverse plies (i) and two outer surface plies (ii) (Dimant 1994).

142

Multi-scale modelling of composite material systems

between a pair of adjacent matrix cracks is depicted 2s; delamination crack length is ld; d is the thickness of an individual transverse (90∞) ply; and b is the thickness of an individual longitudinal (0∞) ply. The width of the specimen, not shown in this simple edge view, is denoted w. This model of a ‘damage zone’ consists of two parts: that portion of material that is cracked, designated (a) (shown circled in Fig. 5.13), and that portion between two adjacent matrix cracks where the (0∞/90∞) interface remains intact, designated (b). In effect, as damage accumulates the matrix crack spacing gets smaller, while the delamination crack gets longer. Consequently, the distribution of load between portion (a) and portion (b) continuously re-adjusts. It is this re-adjustment of load between adjacent (0∞) and (90∞) plies with damage accumulation that brings about the fracture of (0∞) fibres, or not.

5.6.3

Estimating the modulus

To begin with, let us assume matrix cracking only; ignore for the time being the possibility of microscopic delamination cracking. The crack tip tensile stress in the longitudinal (0∞) ply and the transverse (90∞) ply, with distance x from the crack plane, can be estimated using eqns 5.19 and 5.20, respectively: dE cosh( l x ) ˘ ÈE s1 = Í 1 + Ê 2 ˆ s E bE o ¯ cosh( l s ) ˙˚ a Ë 2 Î

5.19

cosh( l x ) ˘ È 5.20 ÍÎ1 – cosh( l s ) ˙˚ s a E1 and E2 are moduli of the longitudinal and transverse plies, respectively; Eo is the (undamaged) laminate modulus, (when there are no matrix cracks); and sa is the remote applied tensile stress on the laminate. (Depending upon whether the variation in stress with distance obeys a linear or parabolic displacement law, the value of the coefficient l lies somewhere between 1 and 3. Either way, it affects the result little). From knowledge of the elastic strain in the longitudinal ply, and knowing the mean stress in the longitudinal ply, (obtained by integrating eqn 5.19 with respect to distance x), we determine the following expression for reduced modulus, Ec, of a matrix-cracked laminate:

s2 =

E2 Eo

È Ec ˘ = 1 ÍÎ E 0 ˙˚ È dE 2 ˆ tanh( l s ) ˘ Ê Í1 + Ë bE 0 ¯ ˙ ls Î ˚

5.21

It is convenient to make eqn 5.21 dimensionless by normalising Ec with respect to the undamaged modulus Eo. Roughly speaking, the modulus of an

Cracking models

143

undamaged laminate, (meaning there are no matrix cracks), can be determined using a simple rule of mixtures: b E + dE 2 ˘ 5.22 E 0 = ÈÍ 1 Î b + d ˙˚ We extend this model to include microscopic delamination cracking at the matrix crack tip as follows. Begin with the assumption that the reduced or damaged modulus, Ec, of that delaminated portion of laminate, (designated (a) in Fig. 5.13), depends essentially on the modulus of the longitudinal ply only (Dimant 1994): bE1 ˘ È Ec ˘ = È ÍÎ bE1 + dE 2 ˙˚ ÍÎ E 0 ˙˚ a

5.23

(As before, eqn 5.23 is made dimensionless by normalising Ec with respect to the undamaged modulus, Eo, of the laminate (eqn 5.22)). Matrix crack intersects a delamination crack at the interface 2l d

2l d

b

y x

2d

P

b

2s

5.13 This is an example of a physical model of coupled mechanisms. It is a model of a damaged (0∞/90∞)s cross-ply laminate under tensile load P (edge view). The geometry shows two neighbouring transverse ply cracks (of spacing 2s) interacting with local delamination (interlaminar) cracks of length 2ld. It is important to understand the interaction (coupling) between these two failure processes and their influence on fibre fracture and the resulting mechanical properties of the composite. (b is the thickness of the (outside) longitudinal ply; d is the thickness of the transverse ply.)

Next, consider that the damage zone now has an ‘effective’ matrix crack spacing (s – ld). Thus, when we substitute (s – ld) into eqn 5.21, we obtain for the modulus of that portion of (undamaged) laminate (designated (b) in Fig. 5.10):

144

Multi-scale modelling of composite material systems

È Ec ˘ ÎÍ E 0 ˚˙ b

È ˘ Í ˙ 1 = Í ˙ Í 1 + Ê dE 2 ˆ tanh(( s – l d )) ˙ ÍÎ Ë dE1 ¯ l ( s – l d ) ˙˚

5.24

Finally, for a given applied tensile stress, the longitudinal modulus of the damaged laminate is calculated by using a rule of mixtures for (Ec/Eo)a and (Ec /Eo)b:

È Ec ˘ ÍÎ E 0 ˙˚ laminate

5.6.4

È Ec ˘ È Ec ˘ ( s ) ÍÎ E 0 ˙˚ a ÍÎ E 0 ˙˚ b = È È Ec ˘ ˘ È Ec ˘ Í ÍÎ E 0 ˚˙ ( s – l d ) + ÎÍ E 0 ˚˙ ( l d ) ˙ b a Î ˚

5.25

Mapping stiffness change

Consideration of transverse ply cracking only Predicting the damaged modulus (eqn 5.21) of a family of cross-ply glass fibre and carbon fibre-epoxy laminates, showing the effect of matrix cracking only, is shown in Figs 5.14 and 5.15. These are sets of (normalised) modulus curves as a function of matrix crack spacing s, (where s is normalised with respect to the transverse ply thickness, d). Thus, where d /s = 1, crack spacing 1.00

(0/90/0)

0.80

(0/90)s

0.70

(0/902)s

E c / Eo

0.90

0.60 1-D shear-lag theory (0/904)s

Finite element analysis 0.50

0

0.2

0.4

0.6

0.8

1.0

d /s

5.14 Comparison of 1-D shear-lag model with the finite element results (section 5.6.7) for glass/epoxy. The close agreement between the shear-lag model gives credibility to the analytical results of the finite element analysis (Dimant 1994).

Cracking models

145

1.00 (0/90/0) (0/90)s

0.90

(0/902)s

E c /E o

0.95

0.85 (0/904)s 0.80 1-D shear-lag theory Finite element analysis 0.75 0

0.2

0.4

0.6

0.8

1.0

d/ s

5.15 Comparison of 1-D shear-lag model with the finite element results (section 5.6.7) for carbon/epoxy. The close agreement between the shear-lag model gives credibility to the analytical results of the finite element analysis (Dimant 1994).

is equivalent to ply thickness. As d /s Æ 0, the crack spacing tends to infinity. Observe that the decrease in modulus with matrix cracking is greater in laminates constructed from thicker transverse plies, confirmed by experiment. Then, as the matrix crack density goes up, (i.e., s gets smaller), with increasing load, (or load cycling), the transverse ply becomes, (more or less), load-free. At this point, the modulus of the laminate approaches the value predicted by eqn 5.23. Coupling delamination cracking with matrix cracking Adaptation of the transverse ply cracking model to include delamination cracking, (eqn 5.25), is shown in Figs 5.16 and 5.17. Contours of (normalised) damaged modulus, Ec, as a function of delamination crack length, ld, (normalised with respect to matrix crack spacing, s), are computed for the selected spacing of s = 4d. In the absence of delamination cracking, the damaged modulus is indicated on the left axis, which is equal to the reduced modulus indicated on the right axis of the previous two figures (for s = d). Thus, the modulus of a laminate in which the delamination crack has extended completely between two neighbouring matrix crack tips, is equivalent to there effectively being a multiplicity of closely spaced matrix cracks. In this case, the damaged modulus would be given by eqn 5.23. In Figs 5.18 and 5.19, contours of (normalised) modulus Ec as a function of delamination

Multi-scale modelling of composite material systems 1.00

0.90 (0/90/0)

E c /E o

0.80 (0/90)s 0.70

(0/902)s

0.60

0.50 1-D shear-lag theory Finite element analysis 0.40

0

0.2

0.4

0.6

(0/904)s 0.8

1.0

ld/s

5.16 Comparison of 1-D shear-lag model with the finite element results (section 5.6.7) for glass/epoxy. The close agreement between the shear-lag model and finite element analysis indicates that the shear-lag model can accurately predict the change in laminate modulus after delamination (Dimant 1994).

1.00 (0/90/0) 0.95 (0/90)s 0.90

E c / Eo

146

(0/902)s 0.85

0.80 1-D shear-lag theory Finite element analysis 0.75

0

0.2

0.4

0.6

(0/904)s 0.8

1.0

ld /s

5.17 Comparison of 1-D shear-lag model with the finite element results (section 5.6.7) for carbon/epoxy. The close agreement between the shear-lag model and finite element analysis indicates that the shear-lag model can accurately predict the change in laminate modulus after delamination (Dimant 1994).

Cracking models (0/90/0)

(0/90)s

1.00

1.00

s = 8d

0.95

s = 8d

s = 4d

0.90

s = 2d

0.90

s = 4d

E c /E o

E c /E o

147

s = 1d

s = 2d 0.80 s = 1d

0.85 0.80

0.70 0

0.20 0.40

0.60 0.80 ld/s

0

1.00

0.20 0.40

(0/902)s 1.00

0.90

s = 8d

0.80

0.80

s = 8d

0.60

s = 2d

s = 4d

E c /E o

E c /E o

1.00

(0/904)s

1.00

s = 2d 0.70

s = 1d

s = 4d

s = 1d

0.60 0.50

0.60 0.80 l d /s

0.40 0

0.20 0.40

0.60 0.80 ld/s

1.00

0

0.20 0.40

0.60 0.80 ld/s

1.00

5.18 The effect of local delamination cracking, ld (normalised by matrix crack spacing s) at the tip of a transverse ply (matrix) crack tip on the modulus of cross-ply glass fibre-epoxy laminates (Dimant 1994).

crack length, ld, (normalised with respect to matrix crack spacing, s), are computed for selected values of s, equal to 1, 2, 4, and 8 times the transverse ply thickness, d. From these figures, we see that, in general, the damaged modulus is a non-linear function of delamination crack length for all crack geometries. This is pronounced in laminates having ‘thick’ transverse plies, (meaning more than two plies). Similar behaviour exists for both the glass fibre and carbon fibre laminates. The finite element model is discussed in section 5.6.7.

5.6.5

Delamination cracking, compliance change, strain energy release-rate

Consider the following: an incremental increase in tensile load, DP, on the laminate results in a small extension of a delamination crack, dld (Dimant

148

Multi-scale modelling of composite material systems (0/90/0)

(0/90)s

1.00

1.00

s = 8d

0.99

s = 8d

0.98

s = 4d

E c /E o

E c /E o

s = 4d 0.98 s = 2d 0.97 s = 1d

s = 2d

0.94

s = 1d

0.92

0.96 0.95

0.96

0

0.20 0.40

0.60 ld/s

0.90

0.80 1.00

0

0.20 0.40

(0/902)s

0.60 l d /s

0.80 1.00

(0/904)s

1.00

1.00 0.95 s = 8d

s = 8d

E c /E o

E c /E o

0.95 s = 4d

s = 2d 0.90

0.90 s = 4d

s = 2d 0.85

s = 1d

s = 1d

0.80 0.85

0.75 0

0.20 0.40

0.60 ld /s

0.80 1.00

0

0.20 0.40

0.60 l d/s

0.80 1.00

5.19 The effect of local delamination cracking, ld (normalised by matrix crack spacing s) at the tip of a transverse ply (matrix) crack tip on the modulus of cross-ply carbon fibre-epoxy laminates (Dimant 1994).

1994). The compliance of the laminate increases by a corresponding amount, dC, and stored elastic strain energy is dissipated in the process. The elastic strain energy release-rate, DG, (in Joules/m2) for mode II (shear) delamination cracking only, is given by: È D P 2 ˘ È dC ˘ DG = Í 5.26 ˙ Î 2w ˚ ÍÎ dl d ˙˚ (Recall that the width of the specimen is denoted w.) For a small elastic displacement, the compliance, C, and longitudinal modulus, E, are related: C = L /EA

5.27

(L is length and A is the total cross-sectional area of laminate.) We determine compliance, C, of the laminate by evaluating separately the modulus of cracked portion (a) of material (eqn 5.23) and the modulus of uncracked portion (b) of material (eqn 5.24). First, we obtain the compliance of

Cracking models

149

the cracked portion (a) by multiplying eqn 5.23 by eqn 5.21 and substituting into eqn 5.27: l Ca = d 5.28 bwE1 Likewise, the compliance of un-cracked portion (b) is obtained by multiplying eqn 5.23 by eqn 5.21 and substituting into eqn 5.27:

Cb =

dE tanh( l ( s – l d )) ˘ ( s – ld ) È 1+ Ê 2ˆ ( bE1 + dE 2 )w ÍÎ Ë dE1 ¯ l ( s – l d ) ˙˚

5.29

Now, the compliance of the complete portion of material between two transverse ply matrix cracks, (including the two delamination cracks), is given by: Claminate = Ca + Cb

5.30

Thus, by substituting eqns 5.28 and 5.29 into eqn 5.30, we have for the compliance of the damaged laminate (Dimant 1994): Claminate =

dE tanh( l ( s – l d )) ˘ ld ( s – ld ) È + 1+ Ê 2ˆ ( bwE1 ) ( bE1 + dE 2 ) w ÍÎ Ë dE1 ¯ l ( s – l d ) ˙˚

5.31 Next, differentiate eqn 5.31 in order to determine dC/dld for substitution into eqn 5.26. Alternatively, it is possible to calculate compliance change with delamination cracking directly from incremental values of C and ld computed in spreadsheet form. For discrete values of s, (matrix crack spacing), the difference in compliance, dC, can be calculated between successive values of ld separated by a small increment d ld. Then, we assume that dC/dld approximates dC/dld with sufficient accuracy because dld is small. Compliance change with delamination cracking, (normalised with respect to (dC/dld)max, is shown in Figs 5.20 and 5.21, (where (dC/dld)max, corresponds to a completely delaminated (0∞/90∞) interface between two matrix crack tips). In this special case, compliance is determined by considering the stiffness contribution of the longitudinal plies only. Each curve corresponds to a selected matrix crack spacing, s, equal to 1, 2, 4, and 8 times the transverse ply thickness, d. For matrix cracks widely spaced, (s = 8d, for example), the dependence of compliance, C, on delamination crack length, ld, is essentially linear. For values of s/d less than 2, however, the relation between compliance and delamination crack length becomes non-linear. Our explanation for nonlinear behaviour is that the crack tip stress fields of two approaching delamination cracks overlap. Maximum compliance change with delamination In monotonic tensile loading, the characteristic crack spacing at failure is about twice the thickness of the transverse ply. Figures 5.20 and 5.21 show

150

Multi-scale modelling of composite material systems (0/90/0)

1.0

(dC/dl)max = 6.07e–9/N

0.5

s = 1d 0.0 0

1d

s = 4d

s = 2d 2d

3d

s = 8d

4d

5d

6d

7d

8d

(0/90)s

1.0

(dC/dl)max = 10.66e–9/N

0.5

s = 1d

s = 2d

s = 4d

s = 8d

0.0

(dC/dl)/(dC/dl)max

0

1d

2d

3d

5d

6d

7d

8d

(0/902)s

1.0

(dC/dl)max = 17.10e–9/N

0.5

s = 1d 0.0 0

4d

1d

s = 4d

s = 2d 2d

3d

4d

s = 8d 5d

6d

7d

8d

(0/904)s 1.0 (dC/dl)max = 24.51e–9/N

0.5

s = 1d 0.0 0

1d

s = 2d

s = 4d

2d 3d 4d 5d 6d 7d Delamination crack length (d = 0.15 mm)

s = 8d 8d

5.20 The effect of local delamination cracking and matrix crack spacing on compliance change (dl/dC) of cross-ply glass fibre-epoxy laminates (Dimant 1994).

that for s > 2d, the value of dC/dld at crack initiation is equal to the maximum value of dC/dld. It is possible, in this instance, to calculate the maximum value of dC/dld by considering the compliance change when two delamination cracks first initiate at a pair of neighbouring matrix cracks widely separated. The maximum value of dC/dld can be determined numerically by evaluating eqn 5.31 for a large crack spacing, 2s, and ld = 0. Then, allow for a very small extension only of ld, and consider the compliance change when this delamination crack first initiates. The value of dC/dld is simply the difference between the two compliances divided by the increment in delamination length 4ld. Now set arbitrary values of s and ld; e.g., s = nd (where n > 2) and ld = d. Finally, the maximum value of dC/dld is given by Dimant (1994):

Cracking models

151

(0/90/0) 1.0 (dC/dl)max = 0.55e–9/N 0.5

s = 1d 0.0 0

1d

s = 2d

s = 4d

2d

3d

4d

s = 8d 5d

6d

7d

8d

(0/90)s 1.0 (dC/dl)max = 1.05e–9/N

(dC/dl)/(dC/dl)max

0.5

s = 1d

s = 8d

s = 4d

s = 2d

0.0 0

1d

2d

3d

4d

5d

6d

7d

8d

(0/902)s 1.0 (dC/dl)max = 1.97e–9/N 0.5

s = 1d

s = 2d

s = 4d

s = 8d

0.0 0

1d

2d

3d

4d

5d

6d

7d

8d

(0/904)s 1.0 (dC/dl)max = 3.48e–9/N 0.5

s = 1d 0.0

0

1d

s = 2d

s = 4d

2d 3d 4d 5d 6d Delamination crack length (d = 0.125 mm)

s = 8d 7d

8d

5.21 The effect of local delamination cracking and matrix crack spacing on compliance change (dl/dC) of cross-ply carbon fibreepoxy laminates (Dimant 1994).

dE 2 È dC ˘ 1 = ÍÎ dl d ˙˚ bwE (4 1 ) ( bE1 + dE 2 ) max tanh( l ( n – 1) d ) – tanh( l nd ) ˘ È ¥ Í1 – ˙˚ ld Î

5.32

For large values of n, tanh (l(n – 1)d) – tanh (lnd) tends to zero and eqn 5.32 conveniently simplifies to:

152

Multi-scale modelling of composite material systems

d E2 ˘ È È dC ˘ ˘ 1 = ÈÍ 5.33 ÍÎ dl d ˙˚ ˙˚ ÍÎ b E1 + d E 2 ˙˚ bw E 4 1 Î max At last, we obtain the elastic strain energy release-rate, DG, by substituting equation 5.33 into equation 5.19: È D P 2 d E2 ˘ È ˘ 1 DG = Í 5.34 ˙Í 2 ˙ Î 8bw E1 ˚ Î bE1 + d E 2 ˚ Now we are in a position to determine the relation between damage, (as characterised by matrix and delamination cracking), and ply thickness, interlaminar (delamination) toughness, and fracture strength. As the transverse (90∞j) ply thickness increases, ( j goes from 1 to 8), the value of (dC/dld)max goes up. Furthermore, values of (dC/dld)max are larger for glass fibre-epoxy compared to carbon fibre-epoxy by an order of magnitude. (Table 5.1). This indicates that the available stored elastic strain energy for delamination cracking is greater in the case of a glass fibre laminate at any given applied load. Since Table 5.1 Elastic strain energy release-rate DG for localised delamination cracking at the tip of a transverse ply crack Material

Lay-up

Shear-lag (dC/dl )max per Newton

FE analysis (dC/dl )max per Newton

Fracture stress* (0∞) ply (GPa)

ESE releaserate DG (J/m2)

Glass fibreepoxy Glass fibreepoxy Glass fibreepoxy Glass fibreepoxy Carbon fibre-epoxy Carbon fibre-epoxy Carbon fibre-epoxy Carbon fibre-epoxy

(0/90/0)

6.07e–9

5.67e–9

1.24

423

(0/90)s

10.66e–9

10.11e–9

1.20

692

(0/902)s

17.10e–9

16.41e–9

1.13

974

(0/904)s

24.51e–9

23.34e–9

1.06

1249

(0/90/0)

6.07e–9

0.54e–9

1.91

62

(0/90)s

6.07e–9

0.99e–9

1.91

120

(0/902)s

6.07e–9

1.81e–9

2.15

284

(0/904)s

6.07e–9

3.25e–9

2.30

577

*Mean tensile strength obtained from 30 waisted (‘dog-done’) tensile specimens made from 913G-E (glass/epoxy system) and 924C-T300 (carbon/epoxy system). These fracture stresses were calculated using the failure load and area of the longitudinal plies only. This represents the physical damage state at fracture of the laminate where the transverse ply is saturated by a parallel array of matrix cracks and is assumed to carry little load in comparison to the longitudinal (0∞) layers. (For comparative purposes only, the Weibull modulus of the glass/epoxy was determined as 12 and for the carbon/epoxy it was 20, and showed no dependence on lay-up geometry.)

Cracking models

153

dC/dld decreases as the delamination crack length increases, this effectively reduces the elastic strain energy available to sustain delamination cracking. We see that as matrix crack spacing, s, decreases (with respect to d) (Figs 5.20 and 5.21), the compliance curve shifts to the left, (i.e., to a smaller delamination crack size). For a matrix crack spacing s > 2d, dC/dld is already at maximum value when the delamination crack first initiates. For s < 2d, dC/dld decreases as the transverse ply crack spacing, s, decreases. This is determined by the intercept of the curve with the axis of the graph. As an illustration, consider a (0∞/90∞)s glass fibre-epoxy laminate. The variation of dC/dld with matrix crack spacing, s, indicates that as separation between matrix cracks gets smaller, then the value of dC/dld at the initiation of a delamination crack decreases. Now, we know that crack spacing is a function of the geometry of the laminate and the applied tensile stress, furthermore, that the characteristic crack spacing in static loading is approximately s = 2d, (in fatigue loading it is s = d). Since the elastic strain energy available for delamination is a function of the applied load and dC/ dld, it follows that as the crack spacing, s, approaches 2d, the applied load must increase sufficiently to compensate for the decrease in dC/dld if delamination cracking is to be sustained.

5.6.6

Interlaminar toughness (GIIc) and fracture strength

The critical elastic strain energy release-rate, DGc, (equivalent to the interlaminar toughness), can be determined using the values of dC/dld and eqn 5.26, incorporating experimental data into it. Table 5.1 is a summary of such data. We observe in a tensile test coupon made from (0∞/90∞2)s glass fibre-epoxy, the onset of delamination cracking immediately prior to fast fracture only. A measurement of about 1 GPa for the fracture stress of our test specimen is equivalent to an applied failure load of about 600N. By substituting 600N for load, DP, and 17.10e–9/N for (dC/dld)max (see Table 5.1) into eqn 5.26, produces a value for DGc equal to 770 J/m2. Fractographic evidence presented in section 5.6.1, indicates that the delamination propagates by mode II shear cracking. The manufacturer’s reported value for GIIc of about 750 J/m2 is remarkably close to the value arrived at by our theory. Values of critical elastic strain energy release-rate, DGc, for delamination growth for a family of glass fibre-epoxy and carbon fibre-epoxy laminates are listed in Table 5.1. The data shows that the elastic strain energy to drive a delamination crack in glass fibre-epoxy is insufficient if the laminate is of (0∞/90∞/0∞) and (0∞/90∞)s construction since DG < Gc. In contrast, a delamination crack can propagate in the (0∞/90∞4)s laminate since DG > Gc. For the carbon fibre-epoxy laminate of (0∞/90∞4)s construction, the maximum elastic strain energy release-rate, DG, is 580 J/m2, determined by substituting the measured failure load of 1150 N, (corresponding to a fracture stress of 2 GPa), into eqn 5.26.

154

Multi-scale modelling of composite material systems

In the lay-ups we investigated, despite the greater loads carried by the carbon fibre-epoxy specimens, the maximum strain energy release-rate was always below DGc required for delamination crack growth. Assuming GIIc is more or less independent of fibre type, and substituting this and other appropriate material properties into eqn 5.34, yields a value for the minimum ply thickness, d, of about 1.68 mm for delamination to be possible in the carbon fibre laminate system. This would be equivalent to a cross-ply laminate containing 12 or 14 (90∞) plies between single (0∞) outer plies. Jen and Sun (1992) observed the formation of a delamination crack at the tip of a matrix crack but only in a carbon fibre-epoxy laminate of (0∞/90∞6)s construction. Substituting the material properties of the 6-ply material system used by Jen and Sun, into eqn 5.34, the minimum transverse ply thickness for matrix cracking turns out to be 1.6 mm (i.e., 12 layers of the (90∞) ply). Likewise, we never observed a local delamination crack at a matrix crack tip in less than twelve layers of (90∞) plies, which gives confidence in the predictive powers of this model.

5.6.7

Finite element model of residual (damage) strength

The finite element model of residual (damage) laminate strength is based on the residual strength of the longitudinal (0∞) plies (Dimant (1994). Inputs to the model include the elastic constants E1, E2, G12, n12, n21, which can be determined by experiment. The size of matrix crack is determined by thickness of the transverse ply. Other structural variables include matrix crack spacing and delamination crack length. In-situ scanning electron microscopy (SEM) enabled us to measure the delamination crack size and crack spacing in monotonic and cyclic (fatigue) loading. Crack tip damage zone: tensile stress (sx ) in the (0∞) ply The in-plane stresses sx, sy and txy close to the tip of a transverse ply crack are indicated in Fig. 5.22. The localised stress sx can result in fibre fracture or matrix cracking; sy and txy combine to bring about delamination cracking. Idealised meshing of a finite element model of a coupled matrix crack and delamination crack is shown in Fig. 5.23. Under monotonic and cyclic tensile loading, (close to ultimate fracture in both cases), the local stress field around the matrix crack tip looks like this; contours of tensile stress, sx, in the longitudinal (0∞) ply of cross-ply laminates (Fig. 5.24). Whilst there is similarity of shape between the stress field contours, the magnitude of sx is some 20– 30% lower in the fatigue-damaged material. Observe in the simple tensile test, (Fig. 5.24), however, that the 1.2 GPa stress contour around the matrix crack tip encapsulates a volume of material equal to that of the 1 GPa stress

Cracking models

155

Matrix crack

txy sy

sx

Transverse ply: material set 2

Interface

Crack

Longitudinal ply: material set 1

5.22 A schematic showing the in-plane localised tensile stresses, sx, sy, and shear stress, txy, close to the transverse ply (matrix) crack tip. Also, the geometry of a cracked laminate used in a finite element model (Dimant 1994).

Crack

90∞ ply

Delamination

0∞ ply

(a) Deformed mesh Unloaded mesh

(b) Deformation of mesh

5.23 Idealised meshing of a (0∞/90∞)s laminate showing a simple transverse ply (matrix) crack intersecting a delamination crack. The displaced profile of the laminate under tensile load shows that contact elements prevent penetration of the crack surfaces (Dimant 1994).

156

Multi-scale modelling of composite material systems

x=0

Crack tip

Crack tip

x=0

1.8GPa 1.4GPa

1.4GPa 1.1GPa 1.0GPa

1.2GPa

0∞ ply

0∞ ply

1.1GPa

(a)

(b)

5.24 Mapping the local tensile stress sx in the longitudinal (0∞) ply of a (0∞/90∞/0∞) glass fibre-epoxy laminate at the point of fast fracture in (a) monotonic and (b) cyclic loading in tension (Dimant 1994).

contour of the fatigued specimen. Also, note that the strength of the fatiguedamaged material is 20% less than the specimen tested in monotonic loading. The inference is that in fatigue, due to the greater damage by transverse ply (matrix) cracking, more glass fibre has broken in the load bearing (0∞) ply at the point of fast fracture. Quite simply, cyclic loading weakens the (0∞) ply and reduces the strength of the laminate. Strength depends on size (volume) In a laminate of eight (90∞) plies at the point of ultimate tensile fracture, the local tensile stress field of the (0∞) ply, adjacent to the matrix and delamination crack tips, looks quite different (Fig. 5.25). When comparing, however, the patterns of stress contours of a fatigue-damaged laminate with material subjected to monotonic loading, as before we see a certain similarity. Looking no further than the simplest failure criterion, we propose that ultimate fracture occurs when the average localised tensile stress, sx, attains some critical value. Whilst this proposal appears sound for the laminate containing the single (90∞) ply, in the case of the laminate containing eight (90∞) plies, observation shows there is clearly greater volume of (0∞) ply under tensile stress. By experiment, we determined that the strength of the laminate containing eight (90∞) plies is 20% lower than of a laminate construction containing one (90∞) ply. The evidence supports the idea of a size (volume) dependence on strength. In the carbon fibre-epoxy laminate containing a single (90∞) ply, the pattern (shape) of stress (sx) contours in the (0∞) ply is more or less identical to that of the glass fibre-epoxy; however, the localised volume of (0∞) ply close to the matrix crack tip at the point of fracture is smaller for the carbon fibre laminate (Fig. 5.26). Increasing the number of (90∞) plies affects this stress

Cracking models x = 0.78s

Delamination crack tip

157

Crack tip 1.8GPa 1.3GPa 1.2GPa 1.1GPa

0∞ ply 1.0GPa 0.9GPa 0.8GPa

(a) Static failure geometry: s = 1.00 mm, smax = 5.16GPa, sf = 1.06 GPa Delamination crack tip x = 0.64s Crack tip 1.4GPa 1.3GPa 1.2GPa 1.1GPa

0∞ ply 1.0GPa 0.9GPa 0.8GPa

(b) Fatigue failure geometry: s = 0.62 mm, smax = 4.85 GPa, sf = 1.10 GPa

5.25 Mapping the local tensile stress sx in the longitudinal (0∞) ply of a (0∞/90∞4)s glass fibre-epoxy laminate at the point of fast fracture in (a) monotonic and (b) cyclic loading in tension (Dimant 1994). x=0

Crack tip

x=0 1.90GPa 1.85GPa

0∞ ply

(a) Static failure geometry: s = 2d, smax = 11.9GPa, sf = 1.91GPa

Crack tip 2.15GPa 1.95GPa

0∞ ply

(b) Fatigue failure geometry: s = d, smax = 11.1GPa, sf = 2.02GPa

5.26 Mapping the local tensile stress sx in the longitudinal (0∞) ply of a (0∞/90∞/0∞) carbon fibre-epoxy laminate at the point of fast fracture in (a) monotonic and (b) cyclic loading in tension (Dimant 1994). This map is not significantly altered by increasing the thickness of the transverse ply.

158

Multi-scale modelling of composite material systems

field pattern little. The suggestion is this; it is the fracture of individual carbon fibres (or clusters of fibres) within the matrix crack tip zone that produces a single ‘large crack’ that subsequently propagates catastrophically. For glass fibre-epoxy, it is the volume of material within the matrix crack tip zone that affects the fracture stress by degrading the strength of load bearing (0∞) plies.

5.6.8

A failure strain criterion

In constant stress amplitude cyclic loading, the density of matrix cracks increases with time; the crack spacing becomes smaller (Table 5.2). In essence, the material ‘work softens’ and becomes more compliant. The result is that the strain to failure increases with the accumulation of fatigue damage. This is true for laminates containing four transverse (90∞) plies (or more). But the reverse is observed where there is a single (or double) thickness (90∞) ply only. In this case, we postulate that in the vicinity of closely spaced matrix crack tips, more fibres in the (0∞) ply fracture in fatigue than in static loading. Thus, the overall laminate is weakened and the ultimate strain to failure is reduced. In-situ SEM studies of (0∞/90∞/0∞) and (0∞/90∞)s laminates clearly show a concentration of glass fibre and carbon fibre breaks within (0∞) longitudinal plies in front of matrix crack tips. Table 5.2 First ply cracking stress (strain), crack opening and crack spacing in monotonic and cyclic (fatigue) loading for cross-ply glass fibre-epoxy laminates and glass fibreepoxy laminates. Measurements were made in the SEM (0/90/0) First ply cracking stress* (GPa) First ply cracking strain (%) Crack opening (mm) (at first ply cracking) Ultimate fracture stress (GPa) Crack opening (mm) (at ultimate failure) Crack spacing (mm) (0 cycles) Crack spacing (mm) (10K cycles)

(0/90)s

(0/902)s

(0/904)s

1.06

0.86

0.72

0.67

2.42

1.79

1.16

0.76

3

5

1.24

1.20

5

9

15

30

160

250

500

1000

100

180

340

620

11 1.12

25 1.06

*All stresses are expressed as (0∞) ply stresses.

In a tensile test, (for small elastic displacement), the relation between ultimate tensile strength, sf, and ultimate failure strain, ef, is given simply by:

Cracking models

sf Ec (Ec is the Young’s modulus). ef =

159

5.35

Recall eqn 5.25 for the reduced (damage) modulus Ec:

È Ec ˘ È Ec ˘ Í E0 ˙ Í E0 ˙ ( s ) Î ˚a Î ˚b

È Ec ˘ ÍÎ E 0 ˙˚ lam = È E ˘ È c˘ È Ec ˘ Í Í E ˙ ( s – ld ) + Í E ˙ ( ld ) ˙ Î 0 ˚a ÎÎ 0 ˚a ˚ Whilst fibre fracture undoubtedly influences the fracture stress of the laminate, it would be prudent not to hazard a guess as to its effect on modulus. If, for the time being, we ignore any possible effect of broken fibres on modulus, then by estimating the reduced (damage) modulus due to matrix and delamination cracking using eqn 5.25, and substituting into eqn 5.35, we can predict the ultimate failure strain (Table 5.3). Inputs to the model are the experimental measurements of crack spacing and delamination crack length. For the carbon fibre-epoxy laminate we took s = 2d for static loading and s = d for cyclic loading. From Table 5.3, it is reasonable to conclude that increasing the thickness (number) of transverse plies reduces the tensile failure strain, ef, of the glass fibre-epoxy laminate. In contrast, however, the failure strain of carbon fibreepoxy laminates seems to be independent of transverse ply thickness (ef ~ 1.4–1.5%). After cyclic loading, the strain to failure, ef, of glass fibreepoxy laminates containing only one or two transverse plies decreases whilst ef for the composite containing four and eight transverse plies increases. We know that the fatigued material is more compliant, (its crack density is greater), and, therefore, according to eqn 5.35 ef would be expected to increase. This logic seems to be valid, however, only for the laminate containing the thicker (four or more) transverse plies. One possible explanation that is Table 5.3 Ultimate laminate failure strain e f (%) determined by combining eqns 5.25 and 5.35 Static or cyclic loading

Laminate configuration

Glass fibreepoxy (% e f )

Carbon fibreepoxy (% e f )

Static Static Static Static Cyclic Cyclic Cyclic Cyclic

(0/90/0) (0/90)s (0/902)s (0/904)s (0/90/0) (0/90)s (0/902)s (0/904)s

2.95 2.72 2.27 1.88 2.55 2.65 2.52 2.33

1.38 1.36 1.45 1.46 1.48 1.51 1.53 1.48

160

Multi-scale modelling of composite material systems

physically sound is to propose that fibre fracture has occurred in the laminate with the thinner (less than four) transverse plies. The consequence of this would be to simultaneously reduce the fracture stress and failure strain of the laminate. For this suggestion to be consistent with an explanation of the fracture behaviour of the carbon fibre-epoxy laminate, we would have to argue in favour of little or no significant (meaning not detrimental) fibre breakage during fatigue prior to ultimate failure. In which case, our observation of a slight increase in strain to failure with cyclic loading would be due to some additional matrix cracking.

5.7

Cracking at stress concentrators

5.7.1

Damage evolution in cross-ply carbon fibre-epoxy laminates

Under tensile loading, damage is sustained in the vicinity of a notch or hole (Kortschot and Beaumont 1990a, b, 1991, 1993; Kortschot et al. 1991; Spearing and Beaumont 1992a, b; Spearing et al. 1991; 1992a, b, c; Cowley and Beaumont 1997a b, c, d). Examination of this damage process zone, (triangularshaped), (Fig. 5.27), shows: ∑ ∑ ∑ ∑

Splitting in the (0∞) plies Matrix cracking in the (90∞) plies Delamination cracking at the (0∞/90∞) interfaces Fibre fracture in the (0∞) plies.

These are coupled mechanisms. With increasing stress or number of load cycles, the notch tip damage zone simply grows in size. What this means, is that the angle, a, made between the delamination and split cracks remains constant, for a given material system and temperature. This is called ‘selfsimilar’ cracking. It follows that damage growth can be explicitly quantified by the measurement of a single failure parameter only, split length,l, providing a has been previously determined and fixed. Under increasing tensile stress, (or load cycling), damage propagates until a terminal damage state is attained. Catastrophic (unstable) fracture of the composite is triggered by the sequential fracture of carbon fibres in the (0∞) plies, localised failure in the vicinity of the notch tip front (Kortschot and Beaumont 1990a, b, 1991; Kortschot et al. 1991). Two phenomena govern the total failure process and they act in synergy. Firstly, localised interlaminar (delamination) cracking and intralaminar (splitting and transverse ply) cracking occur within the notch tip damage zone. Secondly, these cracks reduce the magnitude of the stress field surrounding the notch tip that leads to notch blunting. Fast (unstable) fracture of the composite occurs when the reduced strength of the (0∞) ply, (by the

Cracking models

161

Transverse ply matrix crack Delamination crack between (90∞/0∞) plies

Notch tip

Ds

Split crack tip in (0∞) ply

Single or cyclic stress

90∞ ply 0∞ ply 0∞ ply 90∞ ply

Split surface

t

a

l

Delamination surface

dl F Split surface Delamination surface

l tan a (a)

dl

l

kl 0∞ ply 90∞ ply

ft

2n–1 delaminations

n

0∞ ply

n–1

90∞ ply

0∞ ply 1 Specimen midplane (b)

5.27 Notch tip damage zone showing two individual modes of cracking, splitting, delamination, in a monotonic or cyclic loading of cross-ply carbon fibre-epoxy laminates. The delamination crack surface is triangular in shape. (b) Shows a blown up version of the delamination cracks and coupled splits. No matrix cracks or fibre breaks are included in this simplified model although they do occur (Kortschot et al. 1990a, b, 1991, 1992a, b, c).

162

Multi-scale modelling of composite material systems

accumulation of fibre breaks close to the notch tip), is exceeded by the local tensile stress. Competition exists, therefore, between the blunting effects of the delamination crack and split, on the one hand, and a weakening of the (0∞) ply by fibre fracture, on the other. Furthermore, each transverse ply matrix crack acts as a local stress concentrator and they can initiate the breaking of fibres in the adjacent (0∞) plies. There are complications in the modelling process. Composite strength is a stochastic quantity that depends on the volume of material under stress. Secondly, composite strength is not constant in fatigue where time-dependent, temperature and environment-sensitive processes act at the fibre level, particularly for glass fibre. Furthermore, whilst a can be measured for a given material system and set of test conditions, it is not straightforward to model.

5.7.2

The model: a Griffith-type energy balance

The model is set up in two parts: 1. Assessment of the damage state, D, in monotonic tensile loading and assessment of fatigue damage growth-rate, dD/dN. 2. Determination of the (0∞) ply fracture stress with increasing D. In monotonic tensile loading of a given composite system, the internal state variable, D, is a function of the applied tensile stress, s, and it can be assessed from successive measurements of split length, l. Similarly, in load cycling, dD/dN, (equivalent to dl/dN) is determined in a like manner. The insitu tensile strength of the (0∞) ply is determined by experiment. This idealised model of a damage zone allows a global analysis based on a global elastic strain energy release-rate, DG, calculated solely on the applied tensile loading DP. Such calculations utilise a crude finite element representation of the triangular-shaped damage zone to determine compliance change of the laminate with cracking, dC/dl. Furthermore, the local notch tip stress field is evaluated by the finite element model (Kortschot and Beaumont 1990a, b, c, 1991, 1993). Begin with a single load cycle. The damage zone grows by an increment of split length, d l, and the energy absorbed in forming new surfaces of delamination and split is given by:

dEab = Gstd l + Gd (l tan a)dl

5.36

(The fracture energies (in J/m2) of splitting and delamination are designated Gs and Gd, respectively; and t is the thickness of a single ply.) Energy is dissipated when the split and (coupled) delamination crack extend, with an increase in specimen compliance, dC;

d Er = 1 D P 2 d C 2

5.37

Cracking models

163

(In the model, DP is the increment of applied load on one quadrant of the tensile specimen (i.e., P = sA/4, where A is the cross-sectional area of the specimen.)) For damage to grow, dEr ≥ dEab, i.e., the energy released must be greater than the energy absorbed: 1 D P 2 d C ≥ G td l + G ldl tan a s d 2

5.38

In the limiting case, Er = Eab: 2 1 D P ∂C = G + G l tan a s d t 2 t ∂l

(i.e., DG = Gc).

5.39

It follows, then, for split (damage) growth under monotonic loading:

l=

D P 2 ( ∂C / ∂l ) Gs Ê t ˆ – Gd Ë tan a ¯ 2Gd tan a

5.40

A non-dimensional ratio of l/a, (where a denotes notch length), is dependent on the square of the applied load (or remote applied stress); other terms can be considered as constants in the model. Essentially, eqn 5.40 predicts the damage zone size as a function of the applied stress to the power of 2:

l = C s2 5.41 1 • a The material constants (at a given temperature) are lumped together to give a single constant C1. In fatigue, the damage growth-rate, dD/dN, (characterised by dl/dN), and the global elastic strain energy release-rate, DG, are related through an empirical damage growth-rate Paris (power) law (Beaumont and Spearing 1990; Spearing and Beaumont 1992a b; Spearing et al. 1991, 1992a, b, c): m dl = l 1 ( DG ) 2 dN

5.42

As the split length, l, increases with load cycling, so does the area of the delamination crack increase accordingly, with a dependence on l 2, implying that DG has to be normalised at any point in time by the current laminate toughness, Gc, given by eqn 5.39 m /2

dl È 1/2D P 2 ( ∂C / ∂l ) ˘ = lÍ 5.43 ˙ dN Î Gs t + Gd l tan a ˚ This conveniently removes the need for a pre-multiplying constant having awkward units. The model identifies that the extent of damage for a given stress range, Ds, (corresponding to DP), scales with specimen size (since damage grows in a self-similar manner). This implies that l/a is the appropriate

164

Multi-scale modelling of composite material systems

normalisation for the extent of notch-tip damage. So, for an initial split length, lo, predicted using eqn 5.40, the split length after N load cycles is given by the integral form of eqn 5.43 l=

m +2 ˆ È 1 l DG m /2 Ê Gd tana ÍÎ 2 Ë 2 ¯ 2/( m+2)

˘ ¥ ( Gd tan a ) N + Gs t + Gd l o tana 5.44 ˙ ˚ It only remains to identify values for l2, m, and (∂ C/∂ l). The exponent for the power law is determined by experiment. Thus, DG is equated, via a Griffith-type energy balance, to the appropriate fracture energies of splitting, Gs, and delamination, Gd; the total toughness, Gc, (or work of fracture in J/m2), is the sum of the energy contributions of these two mechanisms. ∂C/∂ l is determined numerically using a finite element model. Gs can be calculated from the measured stress to initiate splitting, whilst Gd is determined from the measured rate of split growth with increasing stress. In practice, they become empirical parameters used to fine-tune the model. It has been proposed – (Spearing et al., 1992c) that splitting is a mode I cracking process, whilst delamination is by mixed modes I/II. Thus, the inclusion of failure mechanisms into an empirical (Paris) law has made that fatigue law predictive of fatigue damage zone size (eqn 5.44). All that remains is to insert the predicted fatigue damage zone size into a tensile failure criterion to arrive at the model for residual (fatigue) notch strength as a function of lifetime. Alternatively, from a single x-radiograph of a notch tip damage zone of a fatigue-damaged specimen, the residual strength can be calculated. ( m+2)/2

5.7.3

A tensile stress failure criterion

A straightforward tensile stress failure criterion is simply (Kortschot and Beaumont 1990a, b, 1991; Kortschot et al. 1991):

s of 5.45 Kt s •f is the remote fracture stress of the notched composite; s of is the localised tensile strength of the (0∞) ply in the damage zone close to the notch tip; and Kt is the terminal notch tip stress concentration factor dependent on damage zone size, and determined from finite element analysis. Based on an idealised meshing of a finite element model of a split and delamination crack (Kortschot et al. 1991) for a centre-notched (90∞/0∞)s laminate: s •f =

K t = C 4 ÈÍ l ˘˙ Îa ˚

–0.28

5.46

Cracking models

165

where C4 is an empirical constant. This empirical equation from FEA relates Kt to the damage zone size and notch length.

5.7.4

The weakest link

The next part of the model is to relate fracture stress of the laminate to the actual damage-state at the notch tip. From the finite element model, we know that the stress concentration factor at the notch tip decreases with increasing split length due to blunting effects. Also, as a consequence, we know that the localised stress gradient in the vicinity of the notch tip becomes less severe with increasing split length. The net effect is an increase in volume of the (0∞) ply at the notch tip and a corresponding decrease in fracture stress of that (0∞) ply, which is subjected to a peak (localised) stress. Given, therefore, that the tensile strength of the (0∞) ply is dependent on the size of the damage zone and on the equivalent volume of ply under peak stress at the notch tip, then the localised strength of the (0∞) ply, s of , can be characterised using a Weibull weakest link statistics model. Observations show that the net effect of this competition between mechanisms, which can simultaneously blunt notches and reduce the (0∞) ply strength, is initially to increase the residual fracture stress of the damaged laminate. Eventually, however, extensive fibre fracture kicks in and weakens the material locally. To predict this, the model must include the current state of damage, the local notch tip stress concentration factor, and the current volume of (0∞) ply under peak stress at the notch tip. s •f is a material property which can be determined independently. Furthermore, the local strength s of of the (0∞) ply within the damage zone depends on the size of that zone, (just like the strength of any brittle solid depends on its volume). Based on the Weibull expression for the strength of a brittle solid containing a distribution of flaw size, we have (Kortschot and Beaumont 1990a, b, 1991; Kortschot et al. (1991): È ˘ Vo s of = s o Í 2 ˙ Î C2 ( l / a )( a t ) ˚

1/ b

5.47

so and Vo are the mean fracture stress and reference volume of the test piece. The denominator is the equivalent volume of the (0∞) ply at the peak stress in the damage zone; its value depends on zone size (characterised by l), (0∞) ply thickness, t, and notch length, a. b is known as the Weibull modulus. Lumping constants together and re-writing eqn 5.47:

s

o f

ÏÈ ˘ ¸ = C3 Ì Í C 2 Ê l ˆ ˙ a 2 ˝ ÓÎ Ë a ¯ ˚ ˛

–1 b

5.48

166

Multi-scale modelling of composite material systems

For a centre-notched (90∞/0∞)s laminate, recall equation 5.46: K t = C 4 ÈÍ l ˘˙ Îa ˚

–0.28

Thus, the damaged strength of a (90∞/0∞)s laminate as a function of notch size can be quantified by combining eqns [5.45–5.48]. After considerable manipulation, finally we arrive at a prediction of notch strength in terms of damage zone size:

s

5.7.5

• f

È ˘ = C5 Í Ê l ˆ a 2 ˙ a Ë ¯ ˚ Î

–1 m

È 1 Ê l ˆ 0.28 ˘ ÍC Ë a ¯ ˙ Î 4 ˚

5.49

Strength dependence on notch size

This equation can be manipulated to show the strength dependence on notch size:

C1 s •f [1–2(0.28)] Ê1ˆ ˘ È Á ˜ •2 Ë b ¯ ˙ Í C1 C3 C2 ( C1s f ) ˙ Í ˚ Î

= a 2/ b

5.50

Setting b = 20 and combining the constants into a single constant C6:

s •f = C6 a –0.2

5.51

Such coupling between damage growth and the volumetric (size) dependence of the (0∞) ply strength is the origin of the so-called ‘hole size effect’ frequently observed in composite materials (eqn 5.51). It is relatively easy to solve the equation by iterative computational methods. There is only one independent variable, s •f , in these equations. Figure 5.28 shows the comparison between prediction and theory. Any inaccuracy is of the same order as experimental scatter. The model has been applied without prior knowledge of the actual terminal damage state in the material. Furthermore, the overall form of the model is apparently correct. Also, there have been no curve-fitting parameters to improve the fit of the data. All parameters used in deriving eqn 5.51 are physically meaningful and they can be measured independently. The model over estimates tensile strength because it does not include matrix cracking in the transverse plies.

5.7.6

Extension to a fatigue damage model

A model for the residual strength of a fatigue-damaged composite is based on the observation that no further damage, beyond that observed by load

Cracking models

167

800 700

Strength (MPa)

600 500 Eqn 5.51 400 300 200 100 0

0

2

4

6 8 10 Notch size: a (mm)

12

14

5.28 Prediction of notch tensile strength based on a physical model of the damaging processes in the notch tip damage zone of a crossply carbon fibre-epoxy laminate. The model overestimates the strength because it does not include matrix cracking. There are no fitting constants to eqn 5.51.

cycling, occurs in the material that may subsequently be loaded monotonically to catastrophic fracture (Spearing and Beaumont 1992a, b; Spearing et al. (1991, 1992a, b, c). Examination of fatigue-damaged specimens and the measurement of the terminal split length, (equivalent to the damage zone size), enables a prediction of residual or post-fatigue strength (eqn 5.49). It requires calculation of the notch tip stress concentration factor and the (0∞) ply strength, (which are both functions of split length). Comparison between prediction and experimental measurement is shown in Fig. 5.29. If higher values of split length are obtained with prolonged cycling, higher residual strengths result reflecting notch tip blunting effects. When the dominant mode of failure is fibre fracture in the (0∞) ply at the notch tip, the laminate becomes weakened and the residual strength will start to fall. Now the quasi-static fracture stress of a (90∞/0∞)s laminate of carbon fibreepoxy is about 360 MPa which truncates the fatigue strength curve shown in Fig. 5.29. The increase in residual fatigue strength with load cycles reflects the greater size of the damage zone and its notch tip blunting effect. We know that after fatigue loading there is no further damage growth during the residual strength test. A successful strength prediction, therefore, depends on two distinct (uncoupled) steps: (i) modelling damage growth and (ii) equating strength to damage size at the end of the fatigue test. The split length for the (90∞/0∞)s laminate after N cycles is given by (after eqn 5.44:

l = C [ Ds m N + C ] 2/( m+2) 7 1 a

5.52

168

Multi-scale modelling of composite material systems 700 600

Strength (MPa)

500

400 300 – smax smax smax smax R = 0.1

200 100

= = = =

324 208 275 300

MPa MPa MPa MPa

0 0

1

2

3 4 5 6 Normalised split length l /a

7

8

5.29 Prediction of post-fatigue tensile strength based on the physical model of a notch tip damage zone of splitting and delamination in a cross-ply carbo fibre-epoxy laminate. Each datum point refers to a different test specimen cyclically loaded to various maximum tensile stresses and numbers of load cycles.

Substituting into eqn 5.49:

s •f = C8{1+ C9 [( Ds m + C7 ) 2/( m+2) ]a 2}1/ b

[( Ds m N + C7 ) 2/( m+2) ] 0.28 C4 5.53

C8, C9 etc, are combinations of material constants which can be derived from earlier expressions. Equation 5.53 predicts residual strength as a function of specimen geometry, notch size, and fatigue loading. The effectiveness of the model is limited by the accuracy of both the damage growth prediction and the damage-based strength model. The overall accuracy of the model is comparable to the scatter of the experimental data due to inherent material variability (Fig. 5.30). Whilst the central problem of transverse ply (matrix) cracking and delamination (interlaminar) cracking in cross-ply laminates has been confronted, the question of crack propagation on planes normal to the principal fibre reinforcement axis still requires resolution. Resistance to crack propagation means possession of some physical mechanism(s) for rendering the crack non-disastrous or innocuous. Basically, the objective of constructing a tough composite is to build a microscopically weak structure into a macroscopically strong, stiff material. This is discussed in the next section.

Cracking models

169

800

Residual strength (MPa)

700 600 500

smax = 324 MPa

400 300 smax = 208 MPa

200

eqn 5.53 smax = 324 MPa smax = 208 MPa

100 0

0

1

2

3

4 Log (N)

5

6

7

5.30 Comparison between the predicted residual or post-fatigue strength of a damaged (notched) cross-ply carbon fibre laminate with number of load cycles and experimental fatigue data.

5.8

Bridging cracks: de-bonding’s critical role

Crack bridging by fibres requires that de-bonding, (de-cohesion of the fibrematrix interface), must occur in preference to fibre fracture at the matrix crack front (Fig. 5.31) (Wells and Beaumont 1985a, b, 1987; Kaute 1993). In the absence of de-bonding or when the sliding (shear) resistance along the de-bonded interface is high, crack tip stresses are concentrated in the fibres and they decay rapidly with distance from the matrix crack plane. Consequently, fibres are more likely to snap at or near to the crack plane rather than sliding out, thus diminishing their vital role in bridging matrix cracks. Under these conditions, the composite would exhibit brittleness. When the conditions for de-bonding are satisfied, however, sliding (frictional) resistance along the de-bonded interface governs the rate of load transfer onto the fibre (Fig. 5.32). De-bonding, then, reduces (relaxes) the amplitude of this stress concentration on the fibre. When de-bonding is sufficiently extensive, the crack circumvents the fibre, leaving it intact in the matrix crack wake. This sliding or frictional (interfacial) shear resistance restricts the extent of de-bonding and opening of the matrix crack. When fibres fracture in the matrix crack wake, a large fibre de-bond length encourages a statistical mode of failure within this de-bond zone, as governed by the flaw size distribution in the fibre. A statistical mode of fracture leads to a fibre failure probability that increases with distance behind the crack tip (Fig. 5.32). Furthermore, the most probable fibre break location

170

Multi-scale modelling of composite material systems Applied stress

(i)

(iii)

(ii)

5.31 The failure (or damage) process zone, where the matrix crack is bridged by strong fibres partially de-bonded in the crack wake (i). Under monotonic loading, those fibres furthest from the crack tip fracture and pull out against friction, as the crack opens and the two matrix surfaces separate (ii). The mechanisms involved are fibre sliding (i), fibre fracture (ii) followed by pull-out. State variables are fibre volume fraction and fibre diameter and the response is toughness (weak bond) or brittleness (strong bond). The total debond length is (iii).

displaces greater distance from the matrix crack surface with increasing debonding. This leads to an enhancement of the material’s toughness due to these frictional (fibre sliding and pull-out) processes. The critical issue concerning toughening processes centres on the extent of this de-bonding mechanism, its dependence on interface properties, and its effect on crack opening and fibre fracture. Composite toughness is promoted by a high rate of de-bonding (i.e., a low interface fracture resistance, G IIC) and by a low sliding resistance, (frictional shear stress, tf), along the interfacial de-bond crack. In addition, composites that exhibit extensive de-bonding and frictional sliding typically have fibre pull-out contributions to the toughness. These contributions can be derived in terms of the constituent properties of the fibre and the shear resistance (mode II shear stress, t f , or shear toughness, G IIC) of the interface. Thus, the question of toughness enhancement concerns the definition of optimum surface treatment of fibre and optimum properties of any coating or inter-phase between the fibre and matrix. The nature of the bond, and possible thermal stresses and shrinkage effects of the matrix during processing (and hygro-thermal ageing) are important, with an emphasis on the integrity of the interface. At the heart of the matter, all mechanisms that provide appreciable toughening of a fibre composite have a common feature, that material elements at, or near to a crack front exhibit non-linear behaviour within a damage process zone, and show hysteresis. Furthermore, enhancement of toughness

Cracking models

171

Region of maximum stress Matrix crack surface

ld /2 Friction

s

s

b

l/2

Fibre stress

s(x) = sp – (sp – sa) exp (–bx) s sd X Fibre strength

sf = sp – (sp – sd) exp (– bd /2)

A e br s Fi res st

X

sp =

È 8 E 1G i ˘ 4 mn f E m eD E f :b = :s d = Í ˙ E 1d (1 + n m) nf Î d ˚

1/ 2

5.32 The simplest classification of failure mechanisms is of a single, partially de-bonded fibre, bridging a matrix crack. Load builds up in the fibre over its de-bonded length by sliding (frictional) forces until it snaps at a weak point, position A, where the stress in the fibre exceeds the fibre strength sf. The stress in the fibre at the outset of de-bonding (i.e., at the de-bond crack tip) is sd. The fibre-matrix misfit strain is eo which sets up a compressive (or tensile) stress on the fibre-matrix interface. m is the coefficient of friction. Ef, nf and Em, nm are the tensile moduli and Poisson’s ratios of the fibre and matrix, respectively. d is the fibre diameter and Gi is the interfacial fracture resistance. sp is the maximum stress carried by the fibre due to sliding friction at the interface. b is an elastic coefficient that takes into account the Poisson contraction of fibre under load (after Wells and Beaumont 1985a, b).

is explicitly related to this hysteresis (Fig. 5.33). Bridging mechanisms in the crack wake exhibit toughening governed by the hysteresis, such that (Evans et al. 1985, 1986; Ahmad et al. (1986):

DG c = Af

Ú

uf

0

s ( u )du

5.54

172

Multi-scale modelling of composite material systems Energy density, d(y)

sm

sy (yield)

sm

Complete unloading

sm

q

q

q

y

dy h y

dx x

Crack surface Process zone sm

Partial elastic unloading q

Maximum stress

sm

q

5.33 A schematic representation of the process zone and the stress/ strain curve characteristics of an element dx dy as it traverses across a strip dy within the damage zone.

(u is the crack opening displacement, uf is the opening at the edge of the fully developed bridging zone (steady-state cracking), s (u) is the bridging tractions on the crack surfaces exerted by the intact loaded fibres, and Af is the area fraction of fibre reinforcement along the crack plane.) The general philosophy in the modelling of toughening processes includes the classical concept of homogenising the properties of the material around the crack and formulating a constitutive law that characterises the material’s response to a stress. Representation of this deformation behaviour by a constitutive law allows computation of the toughness enhancement without the need for a detailed knowledge of the associated micro-structural changes. Thus, trends in toughening can be deduced from the s (u) function (eqn 5.54). For composites that exhibit extensive fibre-matrix de-bonding, fibre (frictional) sliding, followed by fibre fracture and the subsequent pulling out of broken fibre ends, the associated s (u) functions may simply be integrated to predict the maximum toughness. However, more realistic constitutive laws can be derived and used in a quantitative manner to predict trends in toughness, provided characterisation of this non-linear behaviour in the process zone is based on the actual mechanisms best observed directly.

Cracking models

5.8.1

173

Model of fibre bridging

Under monotonic tensile loading, a small length, xs(u), of a de-bonded fibre bridging a matrix crack, pulls (slides) out of its socket (against friction) as the crack opens (Fig. 5.34). As a result, the tensile strain, e(u), in this bridged fibre, (as yet unbroken), relaxes as the interfacial (de-bond) crack extends (Kaute 1993):

(a) De-bonding & sliding

l o¢ Fibre

u

j

Xd

lo

Xd

Matrix De-bond length s

l o¢

Xs

u

s(u), e(u)

Xd

lo (b)

5.34 A schematic of fibre bridging model (Kaute 1993). This represents the physical picture of a split or delamination crack growing between angled plies (e.g., (0∞) and (90∞)) (Vekinis et al. 1991b, 1993).

174

Multi-scale modelling of composite material systems

e (u) =

l o¢ ( u ) – ( l o + 2 x s ( u )) s ( u ) = Ef ( l o + 2 x s ( u ))

5.55

( l o¢ is distance along the bridging fibre, lo is the initial size (length) of the bridged zone, and Ef is fibre modulus). In the process, the fibre slides (against friction) in its matrix socket over its de-bonded length, xd. As a rough estimation, the interfacial (frictional) shear stress, tf, can be found using:

tf =

ds ( u ) 4x d (u )

5.56

Previously, in a post-mortem examination of a fractured tensile test-piece made of a strongly bonded carbon fibre-epoxy composite, we recorded an average fibre pull-out length of the order of five times the fibre diameter (d = 7 microns) protruding above the matrix crack plane. As a first approximation, if we equate this length to one-half of the total fibre de-bond length, and if we guess that the maximum stress s (u) in the de-bonded fibre was 1 GPa, then the interfacial (frictional) shear stress, t f , is calculated to be 50 MPa. This is a reasonable estimate of the shear strength of a carbon fibre-epoxy interface. More recently, in a dynamic in-situ SEM study on a LAS-SiC fibre ceramic composite, we observed a fibre de-bond length of about 800 microns (Vekinis et al. 1991b, 1993). If the maximum fibre stress s (u) was roughly 1.5 GPa and the fibre diameter, d, is 15 microns, then the interfacial shear strength, t f , is calculated to be 10 MPa or so, which is a reasonable estimate for a ceramic composite. Assuming a constant (post) de-bond frictional shear stress during fibre slippage, and a constant (maximum) fibre stress in the plane of the matrix crack, then the pulled out length of intact fibre, xs(u), is equivalent to the extension of the bridged fibre due to an (average) tensile stress s(u)/2, acting over the de-bond distance xd (u) (Fig. 5.34). This distance over which the intact fibre has pulled out of the matrix is given by:

x s (u) =

s (u) x (u) 2E f d

5.57

Alternatively, by combining eqns 5.56 and 5.57: x s (u) =

d s (u) 2 8t f E f

5.58

Since lo is large compared to xs(u), then by substituting for xs(u): Ê l ¢ ( u ) – lo ˆ d s (u) = E f Á o – s (u) 2 ˜ l 4 t E l Ë ¯ o f f o

5.59

By replacing l o¢ with ( l o2 + u 2 )1/2 yields s (u) in terms of entirely material constituents (apart from u and lo):

Cracking models 2 Ê1 ˆ d s (u) = E f Á Ê u ˆ – s (u) 2 ˜ 4t f E f l o Ë 2 Ë lo ¯ ¯

175

5.60

On the right-hand side of this equation, the first part is equivalent to the stress in a tightly gripped fibre, where de-bonding is absent; the second part is the relaxed stress in a (post) de-bonded sliding fibre. If a de-bonded fibre sliding in its matrix socket dominates the bridging process, eqn 5.60 can be approximated to:

s (u) =

2t f E f l o u d lo

5.61

(The assumption must be, however, that the fibre has a bridging angle greater than 5∞.) The resolved part of the bridging tractions, s (u), exerted by the intact fibres normal to the crack surfaces, leads to the following crack closure force per fibre: f (u) =

pd 2 4

2t f E f l o Ê u ˆ 2 d Ë lo ¯

5.62

Since sin j, (fibre bridging angle, roughly equivalent to tan j), is equal to u/ lo, and combining eqns 5.61 and 5.62, an expression for the crack closure force, f(u), as a function of crack opening displacement, u is given by: 3

1 2 d 2 f max = p Ê ˆ (t f E f ) 2 u3 5.63 Ë2¯ l2 To begin with, the force f (u) on the initial bridged fibre increases with crack opening u, whilst the initial bridging length lo, remains constant. During subsequent crack opening, (the (post) de-bond sliding (frictional) phase), for a maximum fibre force, fmax, and an extended fibre bridging length, l, is the following relation that: 3

1 2 d 2 f max = p Ê ˆ (t f E f ) 2 u3 5.64 Ë2¯ l2 This constitutive equation can be simplified by lumping the materials properties together (Kaute 1993):

2

1 4 4 p ˆ3 l = Ê dˆÊ (t f E f ) 3 u 3 = C1u 3 Ë 2 ¯ Ë f max ¯

5.65

Now the bridging length can be given in terms of the material (constituent) properties, where C1 is the bridge (material) constant. Likewise, the bridging

176

Multi-scale modelling of composite material systems

angle for steady-state crack growth can be given in terms of the constituent properties: tan j = u = 1 u –1/3 5.66 C1 l A measurement of this angle directly from in-situ SEM observations for the LAS-SiC system, is in reasonable agreement with the prediction of a steadystate bridging angle, for a crack opening of 100–300 microns (Vekinis et al. 1991b, 1993). Replacing the initial bridging length lo with the bridging length l, and combining eqns 5.61 and 5.65, the expression for the tensile stress s(u) in the bridged fibre is (Kaute 1993):

t f E f f max 1/3 s (u) = 2 u = C2 u 1/3 5.67 d p Once again, the material (constituent) properties are lumped together to define the fibre stress constant C2 . For reasonable values of these material properties, a value of the tensile fibre stress of the order of 1–3 GPa corresponds to a cracking opening of 100–300 microns. This is in good agreement with the initial estimate of the tensile stress on the bridged fibre. Thus, combining eqns 5.54 and 5.67, toughness enhancement, DGc, scales with the crack opening displacement, u, to the power of one-third. Crack bridging extends over hundreds of fibre diameters behind the crack tip, which leads to Rcurve behaviour (Fig. 5.35). The term R-curve refers to a fracture toughness that increases as the crack grows (Vekinis et al. 1993). For the LAS/SiC ceramic composite, the dependence of the limiting or steady-state toughness (or bridging zone size) on crack length (or crack opening) is shown in Fig. 60

Nominal KIc (MPa m)

50 40 30 20 10 0 0

1

2 3 4 Crack extension (mm)

5

6

5.35 Nominal fracture toughness KC against crack extension for grooved CT specimens in 7-ply (0∞/90∞) LAS-SiC material, showing the strong R-curve behaviour.

Cracking models

177

5.35. Evidence is mounting which points to frictional energy dissipation as the dominant contribution to enhanced toughness with increasing crack length (and crack opening).

5.8.2

Toughness mapping

It is useful to have some way of summarising, for a given fibre composite, information about how microstructure affects each mechanism of failure and the resulting total toughness enhancement DGc. A visual display can help here; a mechanism or fracture map, for instance, a versatile method of presenting the results of the model, together with the experimental toughness data and critical failure parameters to which the model has been tuned (Wells and Beaumont 1981, 1985a, b, 1987; Beaumont and Wells 1983; Beaumont et al. 1982; Anstice and Beaumont 1983a, b, c). Such a map has two axes, labelled using any two of the intrinsic material parameters, fibre strength sf and fibre-matrix interfacial (frictional) shear stress tf, for instance, or matrix toughness Gm and interface shear toughness Gs. (There are about sixteen different material variables to completely define the material system.) The map is divided into fields, each one labelled with the name of the dominant mechanism, (fibre de-bonding or fibre pull-out, for instance) meaning it contributes more than 50% to the total toughness. Each field is separated by a boundary, (the point where two mechanisms contribute equally), and which indicates the point of change in the dominant mode of failure. A series of maps would indicate the influence of the various intrinsic material properties on each of the failure mechanisms displayed and the resulting toughness. For example, Fig. 5.36 shows matrix toughness and interface fracture resistance as the two axes of a fracture map for a unidirectional glass fibre-epoxy composite (Vf = 0.6). In computing a fracture or mechanism map, or toughness map, we require values of the unknown terms in the model, fibre de-bond length and fibre pull-out length, together with known data for material parameters which can be measured reasonably accurately, like fibre strength, fibre modulus, and fibre diameter. The actual construction of a map is done by a computer, which searches incrementally over the field to find the field boundaries. The method is amenable to changes in the labelling of the two axes of the map, and the effect can be seen of altering any two material parameters simultaneously on the dominant fracture mechanism and associated toughness. Superimposed onto the map are contour lines of predicted total toughness enhancement, DGc. Coupled failure processes in hygro-thermal ageing The effects of coupled processes are shown in the fracture map of Fig. 5.37.

178

Multi-scale modelling of composite material systems E-glass/epoxy PO

INT

INT

0.05

0.15

0.25 0.3

0.2

0.1

INT

0.35

0.4

400

40

60 PO

500

0.45

Matrix ‘G’ (G2)(Jm–2)

600

80

100

700

EL

300 EL 30

40 50 60 Interface ‘G’ (G1)(Jm–2)

70

5.36 Matrix toughness and interface fracture resistance as the two axes of a fracture map for a unidirectional glass fibre-epoxy composite (Vf = 0.6). The small circles denote the boundary between one dominant mechanism (the field) and another. The dominant mechanisms (fields) shown are PO = pull-out; INT = interfacial debonding; EL = fibre fracture. The symbol D defines a point on the map indicating the typical properties of the material. Also shown are the contour lines of de-bond and pull-out lengths and total toughness enhancement. The gradient of the toughness contours and their spacing indicates the sensitivity of composite toughness on a particular material property (after Wells and Beaumont 1985a, b).

The axes are labelled fibre-matrix misfit strain eo and fibre strength sf. Water molecules diffuse in the epoxy matrix to the fibre-matrix interface, attacking the surface of the glass and weakening the fibre. Simultaneously, water pickup in epoxy causes it to increase in volume. Compressive forces exerted by the resin onto the fibre during processing (cross-linking) and cooling fall, (i.e., the misfit strain eo between fibre and matrix is reduced). First, calculate fibre de-bond length and fibre pull-out length. Alternatively, observe and measure them. Their values together are then used to calculate the expected total toughness of the given composite system. Contours of theoretical toughness are displayed on the map, together with predictions of fibre pull-out length and de-bond length, all other material parameters being held constant at their default values. In addition, the map displays toughness data, obtained by fracturing samples exposed to hot-wet conditions for up to 100 hours. The data are fitted on the map by comparing measurement and

Cracking models

4

5

10

6 15

7 20

8

E-glass/epoxy 9 30 35

25

179

40

1.5

Misfit strain e0 (%)

0

0.15

6 1

1.0

44

0.20

24

0.25 0.30

100

Fibre fractu re

0.5

Fibre pull-out

1

1.2

1.4 1.6 Fibre strength sf (GPa)

1.8

2

5.37 Map for glass fibres in epoxy (0∞/90∞ cross-ply) with the axes labelled fibre-matrix misfit strain e0 and fibre strength sf. The arrows indicate how the measured toughness changes with exposure to hotwet conditions with time measured in hours. Solid contour lines are toughness (kJ/m2); broken line is de-bond length (mm); dashed line is pull-out length (mm).

prediction of toughness and measurement and prediction of fibre de-bond length and fibre pull-out length. Thus, the model becomes ‘fine-tuned’. The map (Fig. 5.37) indicates that hygro-thermal degradation, (the toughness has fallen from about 30 kJ/m2 to 10 kJ/m2 over 100 hours), is brought about by diffusion of water molecules in the epoxy causing the matrix to swell (the misfit-strain falls). Chemical attack of the glass fibre takes place and weakens it and the fibre pull-out length shortens. The kinetics of the two processes can be modelled separately, but in such a way that their fluxes match. Almost always it is simplest to model one mechanism at a time, ignoring the other, but care must be taken when they are subsequently coupled to ensure continuity and equilibrium. The stress corrosion cracking of a glass fibre-epoxy composite is discussed next.

5.9

Modelling stress-corrosion cracking

The combined effects of a stress and chemical (e.g., acidic) attack weaken glass fibre-epoxy laminates. It begins when a micron-sized crack in a glass fibre initiates at a pre-existing surface flaw. The failure process continues as the crack extends slowly, (meaning it is time-dependent), followed by unstable (fast) crack propagation across the remainder of the fibre’s diameter of 10–

180

Multi-scale modelling of composite material systems

20 microns. The crack continues to propagate by fracture of the surrounding matrix and additional fibre breakage, thereby forming a macroscopic crack that can be seen using an optical microscope. Observation by scanning electron microscopy of typical fracture surfaces of glass fibres, shows that the surface of each fibre is characterised by a smooth or ‘mirror’ region, (which indicates slow cracking), extending into a ‘hackle’ region or area of fast fracture (Fig. 5.38). Similar observations have been made by Hogg and Hull (1980); Noble et al. (1983), Price and Hull (1983, 1987).

5.38 An SEM of a fracture surface resulting from stress corrosion cracking in a glass fibre-epoxy laminate. Fibre pull-out length increases as the crack growth-rate increases.

For a slowly propagating macroscopic crack in an adverse environment, there is little or no fibre pull-out. The fracture planes of the fibre and matrix are coincidental. With increasing crack velocity, however, there is an increasing amount of fibre-matrix interfacial de-bonding indicated by longer fibre pullout lengths. Matrix bridges in the crack wake may be observed when tougher, more ductile matrices are used. Matrix bridging, as well as fibre bridging, that act in the crack wake shield the tip of the crack from the localised tensile stress. However, a complete physical treatment of this fracture problem is difficult. For example, a model of the thermally activated chemical reaction above, using the law of Arrhenius, requires knowledge of the activation energy, which enters that law. The activation energy can sometimes be predicted from molecular models, but the value of the pre-exponential in the Arrhenius equation more often than not eludes current modelling methods; it must be inserted empirically.

5.9.1

The micro-mechanical model: thermally activated chemical kinetics

The micro-mechanical model is set up as follows. In bulk glass, the crack propagation rate, da/dt, due to stress-corrosion cracking can be given by (Wiederhorn and Bolz 1970):

Cracking models

DQ – a K I ˆ da = n exp – Ê dt RT Ë ¯

181

5.68

In this Arrhenius equation, DQ is the activation energy of the chemically activated process, KI is the crack tip stress intensity factor, R is the gas constant, T is absolute temperature, and n and a are empirical constants. In our model of a slowing growing microscopic crack in glass fibre (Fig. 5.39), the shape of the crack front is represented as a circular arc of radius r, equal to the fibre radius rf (Sekine et al. 1995). The average crack propagation rate due to some stress-corrosion process can be written thus (from eqn 5.68): Ê 1 Á 2r q Ë f

ˆ Ê dY ˆ Ê DQ – a K I ˆ ˜ Ë dt ¯ = n exp – Ë RT ¯ ¯

2q

r Y

5.69

rf o

Stress-corrosion crack

5.39 Schematic or model of a crack front as it extends across the diameter of a single fibre. The crack front is of circular profile of radius r. The crack area is denoted Y.

Y is the area of the stress-corrosion crack in the fibre, q is half the angle which is made by two fibre radii on the edges of the stress-corrosion crack, and t is time. In eqn 5.69, the stress intensity factor KI, should be interpreted as the average value of KI along the entire front of the stress-corrosion crack tip. Since the stress intensity factor for the crack opening mode (designated by the subscript I) is constant, more or less, along the large central portion of the circular crack front, KI can be represented, more or less, by the stress intensity factor at the maximum depth of the stress-corrosion crack: K I = F (q )s f ( 2p r f )

5.70

where sf is the tensile stress acting on the fibre and the geometrical function F(q) can be seen in Sekine et al. (1995).

182

5.9.2

Multi-scale modelling of composite material systems

Crack propagation-rate and time to failure

Next, the model considers ideally aligned continuous glass fibres distributed, for convenience, in a doubly periodic square array (Fig. 5.40). In this geometrical arrangement, the spacing D between neighbouring rows of fibres is simply:

p r Vf f

D=c

5.71

Vf is the fibre volume fraction and c is a geometrical constant equal to unity for a square fibre array, (0.7 for a face-centred square array, and 0.9 for a face-centred hexagonal array of fibre). sy

Glass fibre

sy =

K 1* 2p x

2D

D a*

x

D

5.40 Schematic of the distribution of fibres ahead of the crack tip (for a doubly periodic square array). The centre-to-centre fibre spacing is denoted D. The localised tensile stress, sy, acting ahead of the crack tip is carried essentially by the fibres. The size of the macroscopic crack is given by a*.

According to the laws of linear elastic fracture mechanics of an orthotropic elastic solid, the local tensile stress at the tip of a crack is given by:

sy =

K I* 2p x

5.72

K I* is the apparent crack tip stress intensity factor for a mode I crack and x is the rectilinear coordinate axis whose origin is located at the crack tip (Fig. 5.40). We can write for the average local tensile stress over a small distance D in front of the crack tip:

s˜ y = 1 D

Ú

D

0

K I* dx = K I* 2 pD 2p x

5.73

Now, we argue, this average local tensile stress is shared between the fibre

Cracking models

183

and matrix according to a law of mixtures. Hence, the average tensile stress s˜ y is given by: (1 – V f ) E m ¸ Ï s˜ y = ÌV f + 5.74 ˝sf Ef Ó ˛ The Young’s moduli of fibre and matrix are denoted Ef , Em, respectively, and the local stress in front of the crack tip carried by the fibre is sf . Hence, the relationship between this tensile stress carried by the glass fibre and the crack tip stress intensity factor is given by: s f = b K I* where

b=

5.75

Ef V f E f + (1 – V f ) E m

2 . pD

If the matrix is made of a ductile polymer, bridges of the polymer may form in the crack wake (Fig. 5.41). Recall from section 5.8, that a bridged crack has a reduced apparent crack tip stress intensity factor: * * K I* = K Ia + K Ib

5.76 Glass fibre

dc

lc

Matrix bridging

a*

5.41 A bridged crack, consisting of sub-micron sized polymeric fibrils or ligaments in the crack wake of a fibre composite made of a ductile polymer matrix. The crack opening is d; the bridged length is lc.

The first term is due to the remote applied stress and the second term is due to crack bridging given by Sih et al. (1965): * K Ib = –4(1 – V f )s c

lc 2p

5.77

In this bridged zone, we have assumed that the polymeric fibrils or ligaments stretched between the crack surfaces behave according to a cohesive force

184

Multi-scale modelling of composite material systems

model. On these ligaments, there is a constant cohesive stress s = sc for 0 £ d £ dc, where d is the crack opening displacement and dc is the opening at the edge of the fully developed bridging zone of length lc (steady-state cracking), as shown in Fig. 5.41. For a small bridged zone compared to the crack length, the relationship between dc and lc is: lc 2p

d c = 4f K I*

5.78

For the plane problem of a rectilinear anisotropic elastic solid

f = b22 {2( b11 b22 )1/2 + 2b12 + b66 }. These elastic constants can be expressed in terms of the constituent moduli n of the composite laminate (in plane stress): b11 = 1 ; b12 = – LT ; b22 = 1 ; ET EL EL 1 b66 = . E and ET are the Young’s moduli of the laminate in the longitudinal G LT L and transverse directions, respectively; nLT is the Poisson’s ratio for transverse strain; and GLT is the shear modulus. Combining eqns (5.76 and 5.77) and eliminating lc (eqn 5.78), we get: K I* where

=

* K Ia +

*2 K Ia – 4(1 – V f ) s c d c / f 2

5.79

* K Ia ≥ 2 (1 – V f )s c d c / f .

Meanwhile, from a geometrical consideration of the area of the stress-corrosion crack in the single fibre (Sekine and Beaumont 1998):

dY dq 5.80 = 4r f2 sin 2 q dt dt Recall that Y is the area of the stress-corrosion crack in a glass fibre (Fig. 5.39). Substituting eqn 5.80 into eqn 5.69, and combining equations 5.70 and 5.75, we obtain: dt = where

Ê ab F (q ) 2p r * ˆ 2r f sin 2q exp Á – K I ˜ dq RT nk q Ë ¯

5.81

DQ ˆ k = exp Ê – . Ë RT ¯

Now the time, tF, required for the slow crack growth stage of failure of a single glass fibre by stress-corrosion cracking is obtained by integrating eqn 5.81: tF =

2r f nk

Ú

qf

q0

sin 2q exp Ê – a b F (q ) 2p r f K * ˆ dq I ˜ Á RT q ¯ Ë

5.82

Cracking models

185

(q0 is half the angle made by two fibre radii on the edges of the pre-existing (inherent) surface flaw, and qF is the critical value at unstable (fast) fracture of the glass fibre.) Unstable fracture takes place when K I* attains the fracture toughness of glass KIC. By combining eqns 5.70 and 5.75, we obtain the critical angle qF at fast fracture: ˆ Ê K IC q F = F –1 Á ˜ * Ë b K I 2p r f ¯

5.83

F –1 is the inverse function of F given by F(q ) = 1 – cosq {1.12 – 3.4(1 – cos q) + 13.87(1 – cos q)2. The time of the brittle fracture (final) stage of the glass fibre is much shorter than time tF given by eqn 5.82. It follows, therefore, that the macroscopic crack propagation rate, da*/dt, is approximately given by:

da* = D dt tF

5.84

Recall that D is the distance between neighbouring fibres in a doubly periodic square array of fibre (eqn 5.71). By introducing the following quantities:

ab 2p r f Ê n kD ˆ x=Á and m = ˜ RT Ë 2r f ¯

5.85

and by combining eqns 5.82 and 5.84, we obtain:

da* V = dt I

5.86

where I=

Ú

qf

q0

sin 2q exp{– m K * F(q )}dq I q

5.87

Now consider the integrand of the integral above equation. Choose sensible values of a = 0.11 – 0.216 m5/2/mol; Vf = 0.40~0.57; V = 0.5 for a cross-ply laminate (V is that fraction of plies of the laminate in which glass fibre is perpendicular to the macroscopic crack, equal to 1.0 for a unidirectional laminate); R = 8.31 J/(mol.K) and T = 298 K. This gives a value of m estimated within the range m = 97 ~ 295 (MPa m )–1. Generally speaking, the range of experimentally observed values of stress intensity factor KI at the crack tip is between 2 and 15 MPa m . It follows that the integrand tends to zero, except for very small values of q. Since the angle q0 is small and much smaller than qF, it follows that Iª

Ê q ˆ 1 4 + 0 ˜ exp(–0.8 mq 0 K I* ) * Á * 2 ¯ 1.6 m K I Ë 1.6m K I

5.88

186

Multi-scale modelling of composite material systems

Table 5.4 shows the approximate values of I calculated using eqn 5.88 for a unidirectional glass fibre-epoxy composite having a face-centred hexagonal pattern of fibres. Table 5.4 Values of I q0

m = 150 (MPa m1/2)–1 Approximate

0.2∞ 0.4∞ 0.6∞ 0.8∞ 1.0∞ 1.5∞ 2.0∞ 2.5∞ 3.0∞

1.1048 2.3381 4.1456 6.7450 1.0428 8.6259 6.4355 4.5282 3.0681

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

10–6 10–7 10–8 10–9 10–9 10–12 10–14 10–16 10–18

Exact

1.0958 2.3091 4.0783 6.6146 1.0205 8.4477 6.3813 4.6162 3.2768

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

10–6 10–7 10–8 10–9 10–9 10–12 10–14 10–16 10–18

m = 250 (MPa m1/2)–1 Error (%)

Approximate

0.8 1.3 1.7 2.0 2.2 2.1 0.8 –1.9 –6.4

1.4520 8.2069 3.7541 1.5586 6.1195 1.6302 3.8954 8.7588 1.8942

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

10–7 10–9 10–10 10–11 10–13 10–16 10–20 10–24 10–27

Exact

1.4356 8.0521 3.6580 1.5102 5.9056 1.5730 3.8306 9.0002 2.0979

Error (%) ¥ 10–7 ¥ 10–9 ¥ 10–10 ¥ 10–11 ¥ 10–13 ¥ 10–16 ¥ 10–20 ¥ 10–24 ¥ 10–27

1.1 1.9 2.6 3.2 3.6 3.6 1.7 –2.7 –9.7

Typical values taken for the Young’s moduli of fibre and matrix and fibre volume fraction are: 72 GPa, 4 GPa, and 0.5 GPa, respectively. The apparent value of K I* was fixed at 5 MPa m1/2. For the fracture toughness of glass, we used 0.73 MPa m1/2, or for SiO2 glass, a fracture energy gc = 3.7 J/m2. Based on these values, we obtain for the angle qF (using eqn 5.83) 4.3∞ (7.5 ¥ 10–2 rad). Hence, by combining eqns 5.86 and 5.88, the macroscopic crack propagation rate, da/dt, as a function of K I* can be written in the form: Ê ˆ da* 1 = 1.25xm 2 K I*2 Á exp(0.8 mq 0 K I* ) * ˜ dt Ë 2 + 1.6 mq 0 K I ¯

5.89

This equation indicates that the macroscopic crack propagation rate is independent of fibre radius r and fracture toughness of glass fibre. Figure 5.42 shows a logarithmic plot of macroscopic crack propagation rate, da*/dt, versus the crack tip stress intensity factor, K I* , for selected values of q0 and dc. (eqn 5.89). Values of x and m are set at 5 ¥ 10–15 m/s and 115 (MPa m1/2)–1, respectively. The cohesive stress of a polymeric bridged crack is taken as sc. The figure reveals a linear log (da*/dt) – log K I* relationship over an order of magnitude range of da*/dt. In particular, the figure indicates that the larger the inherent flaw size, the faster the crack growth rate. Furthermore, matrix crack bridging effects, which become noticeable at lower values of K I* , shift the threshold value of K I* to higher values. The lowest threshold value of K I* is indicated by the arrow (Fig. 5.42) and is given by: * K Iscc =

(1 – V f )s c d c j

5.90

Cracking models

187

10–3

d c = 0 mm 1 mm 2 mm

10–4

da*/dt m/s

10–5 10–6

q0 = 0.4∞

10–7

0.2∞ 10–8

0∞

10–9 10–10 10–11

1

2

3 4 5 6 7 8 910 K I*a MPa m1/2

20

5.42 Logarithmic plot of macroscopic crack growth-rate versus crack tip stress intensity factor for selected values of q0 and dc (eqn 5.89). Also indicated by Kl* is a threshold level (shown by the arrow), which is shifted to higher values of K with a bridged matrix crack.

5.9.3

Verification by experiment

There is experimental data in the literature on a unidirectional laminate of Eglass fibre (50% by vol.) in an orthophthalic polyester resin matrix. Crack propagation experiments were carried out in 1N sulphuric acid at room temperature. Figure 5.43 shows a logarithmic plot of the experimental values of macroscopic crack propagation rate versus the crack tip stress intensity * factor K Ia . Comparison between experiment and theory can be made by setting values of x = 8.5 ¥ 10–14 m/s, m = 118 (MPa m )–1, q0 = 0.076∞, scdc =1.85 kPa m, and f = 8.6 ¥ 10–2 (GPa)–1. The prediction of macroscopic crack propagation rate is shown by the solid line and is in good agreement with experimental measurement. This gives confidence in the physical model. In the special case of q0 = 0∞ and dc = 0, the relationship between da*/dt * (from eqns 5.79 and 5.89), reduces to: and K Ia da* = 0.625xm 2 K *2 Ia dt

5.91

This is a straightforward Paris law to the power of two.

5.9.4

Failure maps

In a cross-ply (0/90)n composite, resistance to crack propagation is determined essentially by the fracture toughness of the load-bearing (0∞) plies. The physical

188

Multi-scale modelling of composite material systems 10–4 Experimental data by Aveston and Sillwood (1982)

Present theory

da*/dt m/s

10–5

10–6

10–7

10–8

10–9

3

5 K I*scc

10

20

30

40 50

K I*a MPa m1/2

5.43 Experimental evidence of crack velocity, da*/dt, versus stress intensity factor K I*a showing the existence of a stress corrosion cracking threshold, K I*scc .

model is based on the initial phase of stable cracking of fibres followed by fast fracture of the individual (0∞) ply. Macroscopic laminate fracture, then, is brought about by a sequence of failed (0∞) plies. As a first approximation, this ignores any cracking resistance of the transverse (90∞) ply. The total time to failure, tF, of the cross-ply laminate is given by: n tF = 1 S xu i=1

Ú

aF

ai

1.26q o ˆ Ê 1.6 * Á m K *2 + K * ˜ exp (–0.8 mq o K Ii ) da* Ë ¯ Ii Ii

5.92

where K Ii* =

* K Iai +

*2 K Iai – 4(1 – V f )s c d c / f

2

* K Ii* = K Iai – 1.12(1 – V f )s c p a*

(for a* ≥ lc)

(for a* £ lc)

5.93a 5.93b

ai, is the size of stable inherent flaw or initial crack depth at the surface of the (n + 1 – i)th (0∞) ply; and aF is the critical length of unstable crack in the (n + 1 – i)th (0∞) ply. In computing the failure map Figs 5.44 and 5.45), the depth of the pre-existing macroscopic surface crack in the (0∞) ply is assumed to be ai = ao(i =1, 2, 5). Typically, the apparent fracture toughness, KQ, of a cross-ply glass fibre-epoxy laminate is 35 MPa m1/2.

Cracking models 103

189

dc = 0 mm 1 mm

sa MPa

102

a *0 = 20 mm, q0 = 0.3∞ a *0 = 60 mm, q0 = 0.3∞ a *0 = 60 mm, q0 = 0.7∞

10

1 –1 10

1

102

10 t F*

103

104

d

5.44 Failure map: time to failure as a function of applied (working) stress. Crack opening in the crack wake is considered to be zero or 1 micron. The initial crack size has been taken as 20, 60 microns, and q0 is 0.3∞ or 0.7∞. 80 *

tF

dc = 0 mm 1 mm

= Id

2

6

4

8 10 12 40

14

a *0 mm

60

18

20

24 30

0

0∞

0.2∞

0.4∞

q0

0.6∞

0.8∞

1.0∞

5.45 Failure map: the contour lines depict time to failure in days.

5.10

Model implementation

Physical models often rely on numerical methods for their implementation. For example, notch-tip stress distributions and global strain energy releaserates are most easily obtained by relatively crude finite element representations of the damaged zone (Kortschot and Beaumont 1990a b, 1991; Kortschot et al. 1991). In the case of cross-ply laminates, the (0∞) and (90∞) plies are modelled as discrete elastic orthotropic layers in plane stress. Usually it is not necessary to simulate the fully three-dimensional nature of the problem. Furthermore, it makes it possible to perform parametric studies of the effects of varying damage zone shape and size; varying material properties that are affected by temperature or environment; and in processing by changing ply thickness and lay-up geometry, and residual stress-state on the strain energy

190

Multi-scale modelling of composite material systems

release-rates and stress distributions (Spearing and Beaumont 1998). Delaminations are simulated by decoupling nodes between adjacent layers over the delaminated area, and splits are represented by decoupling nodes within the 0∞ ply. For cyclic loading of notched cross-ply carbon fibre-epoxy laminates, there is an intermediate step in the physical model, however, in which damage is monitored up to 106 cycles, or more (Spearing and Beaumont 1992a, b; Spearing et al. 1991, 1992a, b, c). Such observations and measurements are necessary in order to determine the empirical parameters of the fatigue damage power law. The fatigue version of the model becomes capable of predicting fatigue damage growth and post-fatigue residual strength, for a range of hole sizes, ply thickness, stacking sequences, and fatigue history.

5.11

Conclusions

Physical modelling provides the means to assess the relative severity of different loading regimes, as well as load/environment interactions. Furthermore, the economic advantage of reducing the high cost of vast experimental programmes in assorted environments and stress-states having durations of many thousands of hours is potentially huge. The capability of physical modelling is important in the design of experimental test programmes that ensure critical loading regimes are examined. Such models are valuable inasmuch as they draw attention to the precise features of the failure process that have wider significance and implications. Added benefits include more options being made available to the designer, a reduced need for extensive and costly testing and more efficient and shorter design iteration cycles. This last point requires elaboration. Modelling a particular problem is only a sub-element of the overall design process. We believe that the philosophy behind the physical modelling approach has general applicability. In particular, the foundation of physical modelling could be applied to solving a range of problems in composites. Existing design methodologies at the higher structural size scales can be supported and justified by fundamental understanding at lower size scales. In critical fatigue situations, where the objective is to design for longevity, durability, and structural integrity, then the balance is shifted in favour of physical modelling and away from the empirical approach. It is in this area that the application of the physical model is most powerful. Furthermore, the model points to something of the greatest value; it suggests the proper form that the constitutive equation should take. Successful modelling of physical processes can be achieved by following a set of steps: identify the physical mechanisms (preferably by direct observation); construct the model (using previously modelled problems or applying existing modelling tools); test the model (by comparing with data)

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191

and tune the model (lumping together empirical parameters). In other words, determine the dominant mechanisms, simplify it (them), and exploit the modelling successes of others in materials science and engineering. But even now the job is still incomplete; the last word is ‘iterate’.

5.12

Acknowledgements

It is a special pleasure for me (pwrb) to acknowledge the many colleagues at Cambridge with whom I have collaborated for more than three decades, in particular my graduate students who have gone on to greater things. Although their names are given in the list of references, it is a delight to name those who have contributed greatly, in one way or another, to this chapter: Professor S Mark Spearing, Professor Mark T Kortschot, Professor Anoush P Poursartip, Dr Ron A Dimant, and Dr John K Wells, to name but a few. Other colleagues who have played a part include Dr Hugh Shercliff and Professor Paul A Smith. We have all benefited considerably from the input of Professor Mike Ashby. Much of the research on which this chapter is based has been carried out mainly with the support of EPSRC research studentships.

5.13

References

Ahmad Z.B., Ashby M.F. and Beaumont P.W.R., (1986), ‘Contribution of Particle Stretching to the Fracture Toughness of Rubber Modified Polymers’, Scripta Metallurgica, 20 843–848. Anstice P.D. and Beaumont P.W.R., (1982), ‘Hygrothermal Ageing of Fibrous Composites’, 4th International Conference (ICCM-IV) Progress in Science and Engineering of Composites, Tokyo, Japan, 25–28 October, Japan Society for Composite Materials, pp 1001–1008. Anstice P.D. and Beaumont P.W.R., (1983a), ‘Hygrothermal Ageing and Fracture of Glass Fibre-Epoxy Composites’, J. Mater. Sci. 18 3404–3408. Anstice P.D. and Beaumont P.W.R., (1983b), ‘Direct Observations of the Failure and Toughness of Hybrid Composites’, J. Mater. Sci. Letters 2 617–622. Beaumont P.W.R. (1974), ‘A Fracture Mechanics Approach to Failure in Fibrous Composites’, J. Adhesion, 6 107–137. Beaumont P.W.R., (1989a), ‘Failure of Composites’, Stress Concentrations, Cracks and Notches’, Concise Encyclopaedia of Composite Materials, pp 75–77, Pergamon Press, Oxford, Editor A. Kelly. Beaumont, P.W.R. (1989b), ‘Failure of Fibre Composites: An Overview’, Special issue of J. Strain Analysis, Vol 24, No 4, October, pp 189–205. Beaumont, P.W.R. and Harris, B. (1972), ‘The Energy of Crack Propagation in Carbon Fibre-Reinforced Resin Systems’, J. Mater. Sci., 7 1265–1279. Beaumont P.W.R and Phillips D.C., (1972), ‘Tensile Strengths of Notched Composites’, J. Comp. Mater., 6 (1972) 32–46. Beaumont P.W.R. and Spearing S.M., (1990), ‘Development of Fatigue Damage Mechanics for Application to the Design of Structural Composite Components’, EUROMECH 269: Mechanical Identification of Composites, Published by Elsevier Applied Science, Edited by Vantrin and Sol pp 311–326. St-Etienne, France, 3–6 December.

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Beaumont P.W.R. and Tetelman A.S. (1972), ‘The Fracture Strength and Toughness of Fibrous Composites’, The Metallurgical Society, AIME Spring Meeting, Boston, USA: Failure Modes in Composites. Beaumont P.W.R. and Wells J.K., (1983), ‘Origins of the Toughness of Fibrous Composites’ in Polymer Processing and Properties, Astarita and Nicolais (eds), Plenum Press, London, pp. 328–336. Beaumont P.W.R., Wells J.K. and Anstice P.D., (1982), ‘Development of Fracture Maps for Fibrous Composites’, 37th Annual Conf. Reinf. Plastics/Composites Institute. Soc. of the Plastics Industry (SPI), Session 20A, 11–15 January, pp 1–6. Brown D.J., Windle A.H. Gilbert D.G. and Beaumont P.W.R., (1986), ‘Dynamic Studies of Peel Failure in Metallized Polyester Films’, J. Mater. Sci. 21 314–320. Cowley K.D. and Beaumont P.W.R., (1997a), ‘Part 1 – Damage Accumulation at Notches and the Fracture Stress of Carbon Fibre-Polymer Composites: Combined Effects of Stress and Temperature’, Composite Science and Technology 52 295. Cowley K.D. and Beaumont P.W.R., (1997b), ‘Part 2 – Modelling Problems of Damage at Notches and the Fracture Stress of Carbon Fibre-PolymerComposites: Matrix, Temperature and Residual Stress Effects’, Composite Science and Technology 52 310 (1997). Cowley K.D. and Beaumont P.W.R., (1997c), ‘Part 3 – The Interlaminar and Intralaminar Fracture Toughness of Carbon Fibre-Polymer Composites: The Effect of Temperature’, Composite Science and Technology 52 325 (1997). Cowley K.D. and Beaumont P.W.R., (1997d), ‘Part 4 – The Measurement and Prediction of Residual Stresses in Carbon Fibre-Polymer Composites’, Composite Science and Technology 52 334. de Vouvray A., Haug E., Dowlatyari E., Beaumont P.W.R. and Spearing S.M., (1988), ‘Advanced Model Development and Validation for Composite Damage Mechanics’, European Space Agency Conference: Advanced Structural Materials for Design of Space Applications, Noordwijk, The Netherlands, 23–25 March, pp 343–360. Dimant R.A., (1994), ‘Damage Mechanics of Composite Laminates’, Cambridge University Engineering Department PhD Thesis. Dimant R., Beaumont P.W.R. and Shercliff H., (1997), ‘Damage and Fracture of Glass Fibre-Epoxy Laminates; Modelling the Failure Processes’, International Conference on Composite Materials (ICCM-11), Brisbane, Australia, (July). Published by Woodhead Publishers, T W I, Great Abington, Cambridge, UK. Dimant R., Shercliff H. and Beaumont P.W.R., (2002), ‘Evaluation of a damage-mechanics approach to the modelling of notched strength in KFRP and GRP’, Composites Science and Technology 62 255–263. Evans A.G., Williams S. and Beaumont P.W.R., (1985), ‘On the Toughness of Particulatefilled Polymers’, J. Mater. Sci. 20 3668–3674. Evans A.G., Ahmad Z.B. Gilbert D.G. and Beaumont P.W.R., (1986), ‘Mechanisms of Toughening in Rubber Toughened Polymers’, Acta Metallurgica, 34 79–87. Ganczakowski H.L. and Beaumont P.W.R., (1989), ‘Behaviour of Kevlar Fibre-Epoxy Laminates under Static and Fatigue Loadings – Part 1’, Composites Science and Technology 36, 299–319. Ganczakowski H.L. Beaumont P.W.R. and Smith P.A., (1989), ‘On the Modulus of KFRP Laminates in Static and Fatigue Loading’, ICCM-6 and ECCM-2 International Conference: Composite Materials, September, London, Volume 3 pp 3.166–3.175. Elsevier Applied Science, Editors: Matthews, Buskell, Hodgkinson, Morton. Ganczakowski H.L., Ashby M.F., Beaumont P.W.R. and Smith P.A., (1990), ‘Behaviour

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of Kevlar Fibre-Epoxy Laminates under Static and Fatigue Loading – Part 2’, Composites Science and Technology, 37, 371–392. Gilbert D.G., Beaumont P.W.R. and Nixon W.C., (1983), ‘Direct Observations of the Micromechanisms of Fracture in Polymeric Solids using the SEM’, ICCM – IV International Conference: Mechanical Behaviour of Materials, Stockholm, Sweden (15–19 August). Vol. 2 pp 705-7 10. Pergamon Press, Editors: Carisson and Ohison. Gilbert D.G. Beaumont P.W.R. and Nixon W.C., (1984), ‘Direct Observations of Dynamic Fracture Mechanisms in Polymeric Materials’, J. Mater. Sci. Letters 3 961–964. Hogg P.J. and Hull D., (1980), Met. Sci. 14 441. Jen K.C. and Sun C.T., (1992), ‘Matrix Cracking and Delamination Prediction in GraphiteEpoxy Laminates’, Journal of Reinforced Plastics and Composites, 11, pp 1163– 1175. Kaute (1993), PhD Thesis, Cambridge University Engineering Department, Cambridge, UK. Kedward K.T. and Beaumont P.W.R., (1992), ‘The Treatment of Fatigue and Damage Accumulation in Composite Design’, Intl. J. Fatigue 14 (5) 283–294. Kortschot M.T. and Beaumont P.W.R., (1990a), ‘Damage Mechanics of Composite Materials I: Measurement of Damage and Strength’, Composite Science and Technology 39, 289–302. Kortschot M.T. and Beaumont P.W.R., (1990b), ‘Damage Mechanics of Composite Materials II: A Damage-based Notched Strength Model’, Composite Science and Technology 39, 303–326. Kortschot M.T. and Beaumont P.W.R., (1991), ‘Damage Mechanics of Composite Materials IV: Effect of Lay-up on Damage Growth and Notched Strength’, Composite Science and Technology 40, 167–180. Kortschot M.T. and Beaumont P.W.R., (1993), ‘Damage-Based Notched Strength Modeling: A Summary’, in Composite Materials: Fatigue and Fracture, edited by O’Brien T.K., ASTM STP 1110, 55–71. Kortschot M.T., Beaumont P.W.R., Nixon W.C., and Breton B.C. (1989), ‘Crack Advancement in a Carbon Fibre-Epoxy Composite Observed by Dynamic SEM’ ICCM6 and ECCM-2 International Conference: Composite Materials, September, London, Volume 3 pp 3.166–3.175. Published by Elsevier Applied Science, editors: Matthews, Buskell, Hodgkinson, Morton. pp 3.388–3.396. Kortschot M.T., Beaumont P.W.R. and Ashby M.F., (1991), ‘Damage Mechanics of Composite Materials III: Prediction of Damage Growth and Notched Strength’, Composite Science and Technology 40, 147–166. Kunz-Douglass S.C. and Beaumont P.W.R., (1981), ‘Low Temperature Behaviour of Epoxy-Rubber Particulate Composites’, J. Mater. Sci., 16 3141–3152. Kunz-Douglass S.C. Beaumont P.W.R. and Ashby M.F., (1980), ‘A Model for the Toughness of Epoxy-Rubber Particulate Composites’, J. Mater. Sci., 15 1109–1123. Mao T.H., Beaumont P.W.R. and Nixon W.C., (1983), ‘Direct Observation of Crack Propagation in Brittle Materials’, J. Mater. Sci. Letters 2 613–616. Noble B., Harris S.J. and Owen M.J., (1983), J. Mater. Sci. 18 1244. Poursartip A.P. and Beaumont P.W.R., (1982), ‘A Damage Approach to the Fatigue of Composites’, IUTAM Symposium: Mechanics of Composite Materials, Virginia Polytechnic Institute, Blackburg, Virginia, USA (16–19 August) pp 449–456. Poursartip A.P. and Beaumont P.W.R., (1986), Fatigue Damage Mechanics of a Carbon Fibre Composite Laminate: Part 2’, Composites Science and Technology, 25, 283– 299.

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Poursartip A.P., Ashby M.F. and Beaumont P.W.R., (1982a), ‘Damage Accumulation during Fatigue of Composites’, Scripta Metallurgica, 16 601–606. Poursartip A.P., Ashby M.F. and Beaumont P.W.R., (1982b), ‘Damage Accumulation During Fatigue of Composites’, 4th International Conference (ICCM-IV). Progress in Science and Engineering of Composites, Tokyo, Japan (25–28 October). Japan Society for Composite Materials. Vol I pp 693–700. Poursartip A.P., Ashby M.F. and Beaumont P.W.R., (1982c), ‘Damage Accumulation in Composites During Fatigue’, Proc. 3rd RISØ International Symposium: Fatigue and Creep of Composite Materials, 6–10 September, Roskilde, Denmark pp 279–284. Poursartip A.P., Ashby M.F. and Beaumont P.W.R., (1984), ‘The Fatigue Damage Mechanics of Fibrous Composites’, Polymer NDE, Terma do Vimeiro, Portugal (4–5 September) pp 250–260. Technomic Publishers, Basel, Switzerland. Editor: KHG Ashbee. Poursartip A.P., Ashby M.F. and Beaumont P.W.R., (1986), ‘Fatigue Damage Mechanics of a Carbon Fibre Composite Laminate: Part 1’, Composites Science and Technology, 25, 193–218. Price J.N. and Hull D., (1983), J. Material Sci. 18 2798. Price J.N. and Hull D. (1987), Composites Sci. Tech. 28 193. Sekine H. and Beaumont P.W.R., (1998), ‘A physically based micromechanical theory of macroscopic stress-corrosion cracking in aligned continuous glass fibre- reinforced polymer laminates’, Composites Science and Technology, 58, (10) 1659–1665. Sekine H. and Beaumont P.W.R., (2002), ‘Physical Modelling of the Engineering Problems of Composites and Structures’, Key Engineering Materials, 221–222, pp 255–266. Trans. Tech. Publications, Switzerland. Sekine H., Hu N. and Fukunaga H., (1995), Composites Sci. Tech. 53 317. Shercliff H.R., Vekinis G. and Beaumont P.W.R., (1992), ‘Dynamic In-situ Scanning Electron Microscopy of Ceramic Composites’, 1st International Conference: Fatigue and Fracture of Inorganic Composites’, Cambridge, England (April) ButterworthHeinemann. Shercliff H., Vekinis G. and Beaumont P.W.R., (1994a), ‘Direct Observation of the Fracture of CAS-Glass/SiC Composites-Part 1 Delamination’, J. of Materials Science 29 3643– 3652. Shercliff H., Beaumont P.W.R. and Vekinis G., (1994b), Direct Observation of the Fracture of CAS-Glass/SiC Composites – Part 2 Notched Strength’, J. Materials Science 29, 1994, 4184–4190. Sih G.C., Paris P.C. and Irwin G.R., (1965), Int. J Fract. Mech. 1 189. Spearing S.M. and Beaumont P.W.R., (1992a), ‘Fatigue Damage Mechanics of Composite Materials I: Experimental Measurement of Damage and Post-Fatigue Properties’, Comp. Sci. Tech., 44, 159–168. Spearing S.M. and Beaumont P.W.R., (1992b), ‘Fatigue Damage Mechanics of Composite Materials III: Prediction of Post-Fatigue Strength’, Comp. Sci. Tech., 44, 299–307. Spearing S.M. and Beaumont P.W.R., (1998), ‘Towards a Predictive Design Methodology of Fibre Composite Materials’,. Applied Composite Materials, 5 (2) 69–94. Spearing S.M., Beaumont P.W.R. and Ashby M.F., (1991), ‘Fatigue Damage Mechanics of Notched Graphite-Epoxy Laminates’, in Composite Materials: Fatigue and Fracture, edited by T. K. O’Brien, ASTM STP 1110, 596–616. Spearing S.M., Beaumont P.W.R. and Ashby M.F., (1992a), ‘Fatigue Damage Mechanics of Composite Materials II: A Damage Growth Model’, Comp. Sci. Tech., 44, 169–177. Spearing S.M., Beaumont P.W.R. and Smith P.A. (1992b), ‘Fatigue Damage Mechanics of Composite Materials IV: Prediction of Residual Stiffness’, Comp. Sci. Tech., 44, 309–317.

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Spearing S.M., Beaumont P.W.R. and Kortschot, M.T., (1992c), ‘The Fatigue Damage Mechanics of Notched Carbon Fibre/PEEK laminates’, Composites 23, 305–311. Spearing S.M., Lagace P.A. and Mcmanus, H.L.N., (1998), ‘On the Role of Lengthscale in the Prediction of Failure of Composite Structures: Assessment and Needs’, Applied Composite Materials, 5 (3) 139–149. Vekinis G., Ashby M.F. and Beaumont P.W.R., (1990), ‘R-Curve Behaviour of Alumina Ceramics’, Acta Metallurgica, 38 (6) 1151–1162. Vekinis G., Ashby M.F. and Beaumont P.W.R., (1991a), ‘Compressive Failure of Alumina Containing Controlled Distributions of Flaws’, Acta Metallurgica 39 (11) 2583– 2588. Vekinis G., Shercliff H.R. and Beaumont P.W.R., (1991b), ‘Dynamic Testing of Ceramics and Ceramic Composites in the SEM’, Metals and Materials, May, 279–284. Vekinis G., Ashby M.F. and Beaumont P.W.R., (1993), ‘The Micromechanisms of Fracture of Alumina and a Ceramic-based Fibre Composite: Modelling the Failure Processes’, Composite Science and Technology, 48, 325. Wells J.K. and Beaumont P.W.R., (1981), ‘Construction and Use of Toughness Maps in a Fracture Analysis of the Micromechanies of Composite Failure’, ASTM STP 787 Composite Materials: Testing and Design (6th Conference) pp 147–162, Phoenix, USA (12, 13 May). Editor: I M Daniel. Wells J.K. and Beaumont P.W.R., (1982), ‘Correlations for the Fracture of Composite Materials’, Scripta Metallurgica, 16 99–103. Wells J.K. and Beaumont P.W.R. (1985a), ‘Debonding and Pull-out Processes in Fibrous Composites’, J. Mater. Sci. 20 1275–1284. Wells J.K. and Beaumont P.W.R., (1985b), ‘Crack Tip Energy Absorption Processes in Fibre Composites’, J. Mater. Sci. 20 2735–2749. Wells J.K. and Beaumont P.W.R., (1987), ‘Prediction of R-Curves and Notched Tensile Strength for Composite Laminates’, J. Mater. Sci. 22 1457–1468. Wiederhorn S.M. and Bo1z L.B., (1970), J. American Ceram. Soc. 53 543.

6 Multi-scale modelling of cracking in cross-ply laminates V V S I L B E R S C H M I D T, Loughborough University, UK

6.1

Introduction

Cross-ply laminates demonstrate a broad variety of failure mechanisms at different scales (Herakovich, 1998; Reifsnider and Case, 2002). Several of these mechanisms are not linked to an instantaneous fracture event of the entire component/structure but result in deterioration of the load-bearing capacity of composite materials. An obvious example of such failure mechanism is transverse (matrix) cracking – formation of cracks in 90º layers, normally crossing their entire thickness. Application of tensile fatigue to cross-ply composites results in matrix-crack initiation at very early stages of their service (Lafarie-Frenot and Hénaff-Gardin, 1991) and, in many cases, at stress levels below the one necessary for the first-crack formation under static loading conditions. With an increase in the number of cycles, the density of matrix cracks grows up to a certain saturation level, known as the characteristic damage state (Reifsnider and Highsmith, 1981; Stinchcomb, 1986). Matrix cracking in cross-ply laminates under fatigue conditions was extensively studied experimentally (Reifsnider and Talug, 1980; Jamison et al., 1984; Boniface et al., 1987; Lafarie-Frenot and Hénaff-Gardin, 1991; Bergmann and Block, 1992), using mainly the edge replication technique, X-ray radiography or direct observations of cracking in transparent glassepoxy laminates. It was found that matrix cracking in cross-ply laminates is not the immediate cause of the global failure of a laminate. It obviously affects the longitudinal stiffness of composites, to a greater extent – glass/ epoxy laminates as compared to carbon/epoxy ones; it is explained by the higher transverse ply stiffness of the former group (Herakovich, 1998; Berthelot, 2003). In carbon/epoxy composites this stiffness reduction is relatively small even in laminates with a high ratio of weak 90∞ layers: less than 3% in [0/905/0] and some 12% in [0/9010/0] for a crack density 700 m–1 (Lee and Hong, 1993). Alongside stiffness reduction, a more dangerous role of transverse cracking in failure of cross-ply laminates is initiation of delamination due 196

Multi-scale modelling of cracking in cross-ply laminates

197

to stress concentration at crack tips near interfaces between 0∞ and 90∞ layers. The stochastic nature of the matrix-cracking process in fatigue was evident from the initial stage of its research (Manders et al., 1983; Fukunaga et al., 1984; Bergmann and Block, 1992): distances between two neighbouring matrix cracks (known as spacing) demonstrate a considerable – up to hundreds of per cent – scatter. Among the main reasons for such non-uniformity in a distribution of transverse cracks is spatial randomness at the microscopic level in composites due to their manufacturing. Various types of faults (microcracks, voids, fluctuations in reinforcement density, etc.) are present in the as-delivered state of laminates. Comprehension of the stochastic character of matrix cracking was reflected in different models, incorporating various schemes to describe the material’s randomness. Among the suggested approaches are the introduction of the initial distribution of microcracks (flaws) (Wang et al., 1984), the use of spatial strength distributions (Fukunaga et al., 1984; Berthelot and Le Corre, 2000) and a more general phenomenological scheme based on the suggestion of the Itô stochastic differential equation for fatigue-damage accumulation within the concept of continuum damage mechanics (CDM) (Ihara et al., 1988). These schemes were used in a combination with fracture mechanics or stress transfer rules (shear-lag analysis). Analytical schemes can also be successfully used to predict effective properties of laminates with non-uniform crack distributions (McCartney and Schoeppner, 2002). Stress distributions in laminates for specific statistical realisations of crack sets were analysed using either quasiunidirectional shear stress analysis (Berthelot et al., 1996a) or 2d finiteelement simulations (Silberschmidt, 2005). This chapter analyses spatial stochasticity in carbon-fibre reinforced epoxy laminates at the microscopic scale and suggests an approach to incorporate this randomness into a macroscopic model for matrix cracking.

6.2

Microstructural randomness of cross-ply laminates

6.2.1

Material’s randomness

Manufacturing composite materials from (at least) two components – fibres and matrix in case of cross-ply laminates – usually results in considerable heterogeneity of the obtained composite. Microstructural analysis of this system is usually concerned with manufacture-induced defects such as fibre/ matrix debonding, interface cracks, microvoids, fibre ruptures and kinks, etc. Reifsnider and Case (2002) estimated that in any cross-sectional area one-tenth of fibres are broken in the manufacturing process. These defects not only result in deterioration of a material’s properties but also serve as

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stress concentrators and, hence, nuclei of macroscopic fracture initiation. Still, even neglecting their effect on the composites’ randomness, another source of spatial randomness in materials’s properties is important for many types of laminates, namely, a non-uniform distribution of their constituents. Micrographs of transverse cross-sections of unidirectional carbon/epoxy plies vividly demonstrate a considerable extent of non-uniformity in a spatial distribution of fibres. Still, due to a relative small diameter of individual carbon fibres (normally, 5–10 mm), a majority of mechanical approaches uses the average volume fraction of fibres ·v fÒ as a single global parameter. But local fluctuations of v f lead to respective fluctuations in Young’s moduli that mean exposure of various parts of the composite to different levels of stresses even under the macroscopically uniform boundary conditions (forces and/or displacement). For instance, the effective axial modulus of the composite E11 can be presented as RoM E11 = E11 + DE11

6.1

RoM is the axial modulus calculated according to the linear rule of where E11 mixtures, and DE11 is a non-linear term that is negligibly small compared RoM for unidirectional carbon/epoxy lamina. After rearranging the standard to E11 presentation of the linear rule of mixtures to the form RoM E11 = v f ( E Lf – E m ) + E m

6.2

proportionality of the local magnitude of the elasticity modulus to the local volume fraction of fibres becomes obvious. In eqn 6.2 E Lf is the longitudinal module of fibres (transversely isotropic in the case of carbon ones) and Em is the Young’s modulus of the (isotropic) matrix.

6.2.2

Analysis of microstructure

To estimate the extent of non-uniformity in the spatial distribution of constituents in carbon/epoxy laminates, a micrograph of the ply’s crosssectional area, incorporating a sufficiently large number of fibres is analysed. The size of the window is 345 mm ¥ 250 mm, i.e., its dimensions are larger than the thickness of standard plies in cross-ply laminates (normally, 120– 150 mm); it contains (centres of) 603 fibres with diameter d f = 10 mm. Several parameters of the fibres’ set are introduced to quantify its spatial randomness. Figure 6.1 presents a distribution of the distance between the nearest neighbours (i.e., their centres) in the set. It is obvious that though the majority of fibres have the spacing (the distance diminished by d f) to the nearest neighbour less than 1 mm, still a considerable minority (35%) have the respective spacing outside this interval, in some cases up to five times larger. And even within this 1 mm interval some fibres are in the contact with

Multi-scale modelling of cracking in cross-ply laminates

199

0.2 0.18 0.16

Density

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 10.0

11.0

12.0 13.0 Distance, mm

14.0

15.0

6.1 Distribution of distance between the nearest neighbours in the set of carbon fibres in a lamina.

their neighbours while others belong to different bands of spacing. Orientational analysis (Fig. 6.2) shows that this set of non-uniformly distributed fibres could not be considered quasi-isotropic; the maximum density for a 10∞ band is 2.6 times higher than the minimum one. Hence, there is no obvious anisotropy in the transverse distribution of carbon fibres in laminate, but the latter is not fully transversally isotropic at this scale. 0.09 0.08 0.07

Density

0.06 0.05 0.04 0.03 0.02 0.01

70

50

30

10

–10

–30

–50

–70

–90

0.00 Orientation angle, ∞

6.2 Orientational distribution of carbon fibres (pairs of nearest neighbours) in a lamina.

The next step to quantify the local randomness is to study the spatial distribution of constituents (or, rather, of fibres since the matrix is an embedding medium). To implement this study, the analysed window is discretised into

200

Multi-scale modelling of composite material systems

0.84

0.96

0.6

0.72

0.48

0.24

0.36

0

0.12

Vf

0.6

0.72

0.84

0.96

0.72

0.84

0.96

0.48

0.6

0.24

0.36

19 mm

0

Density

0.48

0.24

0.36

6 mm

0

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.12

0.84

31 mm

Vf

Density 0.96

0.84

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.96

0.72

0.6 0.6

0.72

0.36

0.48

0.24

0

Vf

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.12

0.96

0.84

0.6

0.72

0.48

0.36 0.36

Vf

12 mm

0.12

Density

0.24

24 mm

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Vf

0.48

0

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.24

0

Density

49 mm

0.12

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.12

Density

Density

cells with dimensions changing from one discretisation to another, and the volume fraction of fibres is determined for each cell. Respective histograms for various cell dimensions are given in Fig. 6.3. The general trend with the decrease in the length scale is obvious; the distribution width increases alongside respective flattening of histograms. Introduction of two bounds – f f maximal ( v max ) and minimal ( v min ) – for distributions of the volume fraction of fibres confirms their two natural trends with variation in the length scale (Fig. 6.4). Firstly, for sufficiently small scales, these two bonds tend to the f f Æ 0 (i.e. the volume fraction mono-phase asymptotes: v max Æ 1 and v max m of the matrix v Æ 1). A transition to the former limit occurs at the length

Vf

6.3 Effect of the length scale (box size) on distributions of the volume fraction of fibres.

Multi-scale modelling of cracking in cross-ply laminates

201

1 0.9 0.8 0.7

Vf

0.6 0.5 0.4 0.3

Max Min Average

0.2 0.1 0

0

20

40

60 80 Length scale, mm

100

120

6.4 Effect of the length scale on bounds of distributions of the volume fraction of fibres.

scale below the magnitude of the fibre diameter (due to different shape of the cell and reinforcement’s cross-section). The latter limit is attained earlier (i.e. at the higher length scale that is even larger than d f) for the average volume fraction of fibres ·v fÒ = 0.55 of the studied carbon/epoxy composite. Secondly, both bounds should converge at high levels of the length scale to the average value: f v min Ô¸ f ˝ Æ · v Ò. f v max Ô˛

This trend is also distinct (Fig. 6.4) but the full convergence of the bounds is not reached even at the length scale of 115 mm. This relatively slow convergence has severe implications for numerical modelling of laminates. Any attempt to adequately reproduce spatial (nonuniaxial) stress distributions in such composites (especially for the purpose of determining possible places for crack generation) should incorporate statistics of scatter in local properties. For instance, application of 2d finite elements presupposes the use of at least several nodes along the transverse (throughthickness) direction. For a single ply’s thickness 120–150 mm, the maximum element dimension should not exceed 30–50 mm. Considering the microscopic estimate for the scatter in the volume fraction of fibres at these two length scales, one can determine respective bounds for effective elastic moduli. Various schemes can be used for this purpose for carbon/fibre laminates, e.g., the self-consistent approach (Hill, 1965; Budiansky, 1965), Mori-Tanaka method (Mori and Tanaka, 1973; Benveniste, 1987), concentric cylinder assemblage model (Hashin and Rosen, 1964; Hashin, 1983; Christensen, 1980), or method of cells (Aboudi, 1991). The last three approaches give very close results for many fibre-reinforced laminates (Herakovich, 1998).

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Multi-scale modelling of composite material systems

Table 6.1 shows ratios of the maximum values of local effective elastic moduli to the minimum ones obtained using the concentric cylinder assemblage model for the studied case of the distribution of carbon fibres (here E 22 and G12 are the effective transverse modulus and effective axial shear modulus, respectively). Table 6.1 Ratios of local effective elastic moduli of carbon-fibre composite for various length scales

Length scale 30 mm 50 mm

max E 11

max E 22

G 12max

min E 11

min E 22

G 12min

1.41 1.21

2.06 1.53

2.09 1.47

Such estimation demonstrates that the maximum values of the effective axial moduli (both normal and shear) are some 50% higher than the minimum ones for the length scale of 50 mm. A decrease in the length scale to 30 mm results in the twofold difference in these parameters. The scatter in the effective transverse modulus is lower, but still its maximum local values are 20 to 40% higher than the minimum ones.

6.2.3

Multifractal characterisation

The above analysis has shown that there are various parameters that could characterise the extent of the spatial randomness in heterogeneous composite materials at the microscale. A multifractal approach (Chhabra et al., 1989; Harte, 2001) has been found to be a useful tool to analyse spatial scaling of non-uniform distributions. For a stochastic distribution in some area, the local probability Pi introduced for a set of elements (boxes), compactly covering this area and labelled by index i, scales as: Pi ( l ) µ l a i ,

6.3

where l is a length scale (box size) and ai is a respective exponent (known also as singularity strength). The number of elements with probability characterised by the same exponent is linked to the length scale by the fractal (Hausdorff) dimension f (a): N(a) µ l –f(a ).

6.4

Hence, f (a) describes the entire (finite) spectrum of scaling exponents for non-uniform distributions. A direct use of eqns 6.4 and 6.5 results in inaccurate estimates for f (a) due to poor convergence (Grassberger et al., 1988), hence another procedure was suggested (Chhabra and Jensen, 1989; Chhabra et al., 1989) based on the parametric presentation of f and a.

Multi-scale modelling of cracking in cross-ply laminates

203

A modification of this approach for the case of a 2d distribution of fibres in a part of the ply’s transverse cross-section has the following form: N N

x z a = lim 1 S S m ij log nij , lÆ0 log l i=1 j =1

N

6.5

N

x z f = lim 1 S S m ij log m ij , lÆ0 log l i=1 j =1

where

m ij =

Ê Nx Nz nijq Á Ë k =1 m=1

S S

q ˆ n km ˜

¯

6.6

–1

.

Here l is a length scale (box size), M = NxNz = LxLz/l2 is the total number of boxes necessary to cover the entire area Lx ¥ Lz under study, nij = Nij/Nf is a relative number (probability) of (centres of) fibres within the box (i, j), Nf is the total number of fibres in the area. The results of calculations of f (a) based on this approach are given in Fig. 6.5 for the micrographs described in section 6.2.2. Apparently, the nonuniform distribution of fibres in the area under study has a multifractal character since the graph demonstrates the properties of a multifractal spectrum: 1. It is a cup convex which lies under the bisector f = a. 2. It has a single connection point with this bisector where f ¢(a) = 1. The value of f (a) at this point (for q = 1) is called the informational dimension (Evertsz and Mandelbrot, 1992; Falconer, 2003). 3. The maximum value of the f(a) curve is the box-counting dimension of 2 Bisector 1.75

f

1.5 1.25 1 0.75 0.5 1

1.25

1.5

a

1.75

2

2.25

6.5 Multifractal spectrum f (a) for the distribution of carbon fibres in a lamina.

204

Multi-scale modelling of composite material systems

the geometric support of the measure D: D = 2 for a distribution over a 2d region. The width of the multifractal spectrum is linked to the extent of randomness of the distribution, with the increase in its uniformity the spectrum tends to the point f (D) = D, which is characteristic for the total isotropy.

6.2.4

Randomness of matrix cracking

The spatial non-uniformity of a material’s properties at the microscopic level in carbon/epoxy laminates is the main reason for randomness in the matrixcracking process under tensile-fatigue conditions. Various schemes and measures can be introduced to quantify this randomness. The Weibull distribution function (Weibull, 1951) could be used to characterise the set of crack spacing distances for laminates at different stages of their loading history (Bergmann and Block, 1992; Silberschmidt, 2005). Berthelot et al. (1996b) used histograms for numbers of matrix cracks in bands of a constant width, into which the test specimen is divided. The multifractal formalism can be also used to characterise the type of randomness in a matrix-crack set (Silberschmidt, 1995, 1998; Silberschmidt and Hénaff-Gardin, 1996). It was shown that multifractal spectra f (a) for distributions of cracks along the specimen’s length uniquely characterise matrix cracking in 90∞ layers, provided the sets contain a sufficiently large number of cracks (normally more than 20). The high level of closeness of spectra for twin specimens loaded under similar conditions (both the number and positions of single cracks in these specimens vary considerably) justifies the robustness of the multifractal approach to cracking characterisation. Multifractal spectra reflect changes in the level of load, loading history and the structure (stacking order) of carbon/ fibre laminates (see Silberschmidt (1995) for details). For instance, the transition from the initial stage of cracking to the saturated one is manifested by a decrease in the width of multifractal spectra, that practically does not change for a considerably large number of cycles (more than 5 ¥ 105 cycles). Such a decrease in randomness at this stage is naturally explained by a difference in the role of different mechanisms at various stages of cracking in specimens, exposed to a macroscopically uniform axial load. At the initial stage with a vanishing (or low) crack density, generation of a crack is linked to the weakest point, randomly situated along the specimen’s length. An increase in the number of cracks strengthens the effect of another mechanism – stress re-distribution in the direct vicinity of transverse cracks; areas close to the crack are unloaded due to its shielding effect. With initial spacing considerably exceeding zones of this effect, matrix cracking is governed mainly by the spatial randomness in a material’s properties. Increasing crack density results in the growth of the portion of the length occupied by shielding

Multi-scale modelling of cracking in cross-ply laminates

205

zones, limiting the probable nucleation sites mainly to parts of the specimen situated around the middles of spacings between the neighbouring cracks. This ordering of the cracking is reflected in the narrowing of multifractal spectra of their distributions.

6.3

Damage accumulation

Tensile fatigue of cross-ply laminates results in initiation of multiple damage mechanisms, with matrix cracking in 90º layers being the main one at the initial stages of loading, as was discussed above. Matrix cracks are initiated from the microdefects and rapidly – within a single cycle – propagate through the layer’s thickness between two adjoining 0∞ layers of the [0n/90m/0n] laminate. Their propagation through the width of the specimen is slower. In contrast to quasi-static loading with cracks instantaneously propagating from one edge to another, the transverse length of cracks evolves with the loading history in a complex way (Lafarie-Frenot and Hénaff-Gardin, 1991; Silberschmidt and Hénaff-Gardin, 1996). This process is not analysed here; we limit our consideration to a 2d situation neglecting differences in the through-width length of cracks. Due to orientational degeneracy of transverse cracks in relatively thin layers, a scalar damage parameter p can be introduced as an additional variable at the macroscopic level of our description. This parameter reflects the deformational effect of the microscopic evolution of the ensemble of defects. The damage evolution law for fatigue loading can be introduced in the form of a modified Coffin-Manson law that can be written in a general form as (Silberschmidt, 1997)

dp = F (s , p, . . .), dN

6.7

where N is a number of cycles, s is the maximum macroscopic stress in a tensile fatigue. The damage parameter, introduced within the framework of CDM (Lemaitre, 1996), is different from traditional damage metrics used in mechanics of fatigue such as the Palmgreen-Miner rule (Reifsnider and Case, 2002; Suresh, 1998). It was shown (Silberschmidt and Chaboche, 1994a) that the third-order polynomial F(s, p) = –(ap3 + bp2 + cp – ds) used as the right-hand part of the damage accumulation law eqn 6.7 is sufficient to characterise the specific features of damage evolution in (quasi) brittle materials. The introduced parameter p describes the macroscopic evolution of disperse damage; hence an additional condition is necessary to determine generation of a macroscopic matrix crack as a result of this evolution. A standard approach is to use a threshold stress value as the failure criterion alongside the introduction of the spatial distribution of strength. As was shown above, the local stress value is scale-dependent, not to mention its dependence on the loading

206

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conditions and the structure of the composite. Within the suggested CDM approach, it is more natural to use the damage-based criterion. Various studies of microdefect evolution in different materials have shown (Naimark and Silberschmidt, 1991) that the local pre-fracture density of defects is practically independent of the loading conditions. Thus, the transition from the disperse damage accumulation to macroscopic fracture can be linked to overcoming of the threshold damage value pc, and the local failure criterion would have the following form: p = p c.

6.4

Multi-scale modelling

6.4.1

RVE and lattice schemes

6.8

Understanding the necessity to incorporate microscopic features into macroscopic models of composites results in the suggestion of various approaches. The major part of approaches is based on the idea of an effective media which, being spatially uniform, responds to external loading in exactly the same manner as its heterogeneous counterpart. Effective properties of such a media can be obtained by various methods – analytical (mainly incorporating various homogenisation schemes (Bakhvalov and Panasenko, 1989; Milton, 2002)) and numerical. The latter are usually based on the concept of a representative volume element (RVE). Such an element is representative in a sense of reproducibility of the effective macroscopic material’s response to loading, and at the same time it incorporates a direct account for the microstructural features of the composite. In this case, a media is formed by ‘tiling’ of space with RVEs, each behaving in the same way. This approach is useful to analyse a pre-critical material’s behaviour up to initiation of macroscopic fracture in a form of a crack. Crack formation is linked with a transition from a spatially disperse failure process to a highly localised one. At this stage, the concept of representativeness in a general case could be no longer used (an obvious exclusion is a case of periodic crack systems). Sets of matrix cracks in carbon/epoxy cross-ply composites, exposed to tensile fatigue, demonstrate a considerable extent of scatter in inter-crack spacing (Silberschmidt, 1995), hence the application of the RVE concept alongside the assumption of the uniform distribution of transverse cracks is limited. Another way to analyse this problem can be suggested on the basis of a lattice approach (Silberschmidt and Chaboche, 1994b; Silberschmidt, 1997) that was initially developed for problems in statistical physics and later applied to fracture (see, e.g., Hermann and Roux (1990)). In the suggested model the entire specimen is represented as a lattice of elements, each having various properties due to variations in microstructure. The model bridges

Multi-scale modelling of cracking in cross-ply laminates

207

macro and micro levels of description. Microscopic processes are accounted for each element (as in a case of RVEs) in terms of the damage parameter, while the macroscopic processes of stress redistribution – due to the spatial randomness of material’s properties, structure of composites as well as matrix cracking – are considered at the scale of the entire specimen (lattice of elements). The practical implementation of the approach begins with the discretisation of the axial cross-section of the laminate into (in a general case) rectangular elements with dimensions lx ¥ lz (axes x and z are given in Fig. 6.6). Elements are labelled by a pair of indices (i, j) that denotes the element in the ith column and jth row of distribution, with rows being parallel to the specimen’s axis and columns being parallel to z axis (Fig. 6.6). z j

i

x

90∞

(i –1, j –1)

(i , j –1)

(i –1, j )

(i , j )

(i +1)

(i –1, j +1) (i, j +1)

6.6 Scheme of the analysed problem and discretisation of 90∞ layers.

In the case of matrix cracking in carbon/epoxy cross-ply laminates, the spatial non-uniformity in stiffness is considered as the main source of the initial randomness in the material’s properties. To adequately incorporate this feature into the model, histograms of the volume fraction of fibres, obtained from the microstructural analysis for the respective scale length that coincides with the elements’ dimensions, are used to determine the type of randomness in the material’s local stiffness. The values of the local stiffness of 90∞ layers, calculated using the microstructural data, are approximated by the three-parameter Weibull distribution with the cumulative distribution function having the form: È ÊC – g ˆb ˘ F ( C ) = 1 – exp Í – Á ˜ ˙, ÍÎ Ë h ¯ ˙˚

6.9

ij min is the ratio of the local transverse modulus to its minimum where C = E 90 / E 90

208

Multi-scale modelling of composite material systems

value for the respective length scale (elements’ dimensions). The obtained values of three parameters of this distribution are b = 13.0, h = 0.395 and g = 0.76. The graphical presentation of the function is given in Fig. 6.7 together with the microstructure-based data. 1 0.8

CDF

0.6 0.4 0.2

Microstructure Weibull

0 1

1.05

1.1

1.15

1.2

1.25

1.3

ij min E 90 E 90

6.7 Cumulative distribution functions for relative transversal modulus: analytical (Weibull) and microstructural data.

6.4.2

Stress redistribution mechanisms

The spatial distribution of the stiffness in 90º layers should result in variation of local stress levels in elements even in the case of the uniform external loading conditions. This is one of the factors (others are discussed below) resulting in a deviation of the local stress from its average level for the effective, homogeneous (or homogenised) material. To deal with such local fluctuations, stress-renormalising coefficients Kij can be introduced for the set of elements. Then the local axial stress (considered to be constant within elements) is introduced in the following form:

s ij = K i js•

6.10

where s• is the externally applied axial stress. Stress coefficients Kij for the initial (without matrix cracks) state of the lamina can be presented in the form K ij = K c

ij N1 E 90 N1

S E ik k =1 90

6.11

that reflects two factors affecting the local stress level: (i) the global transfer of the external stress, accounted in terms of Kc and (ii) the local variations in the transverse modulus (the second term of the right-hand part). Here N1 is

Multi-scale modelling of cracking in cross-ply laminates

209

a total number of elements in a column covering the entire thickness of the 90∞ layers. Parameter Kc can be introduced using different analytical estimates, the simplest being Kc =

E 90 E

6.12

where E 90 is a transversal Young’s modulus of the ply, E = (2 nE 0 + mE 90 )/(2 n + m ) is an effective axial modulus for the [0n/90m/ 0n] laminate. This stress redistribution process results in a different level of the initial local stress in various parts (elements) of the 90∞ layers that, in its turn, causes a spatial non-uniformity in evolution of damage. Hence, some of the elements, which demonstrate higher damage accumulation rates, will attain the critical damage level pc earlier than others. Fulfilment of the criterion (eqn 6.8) for an element (i, j) (obviously, with a changed notation p Æ pij for a suggested lattice scheme) is considered as the local failure criterion for this element, forming a part of a transverse crack. Depending on the laminate’s structure, various modelling assumptions, regarding evolution of matrix cracking, could be implemented. For composites with thin 90∞ layers an event of the local failure of the element (j, j) could be treated as an instantaneous formation of the transverse crack occupying the entire ith column of elements. This is justified by the experimental observations of matrix cracks growing through the lamina’s thickness within a single cycle. Any matrix crack causes stress redistribution in its direct vicinity due to the formation of its stressfree surfaces. This shielding effect for the axial stress can be treated in different ways, with analytical schemes being mainly based on variants of the shear-lag approach (Berthelot et al., 1996a; McCartney et al., 2000; Vasiliev and Morozov, 2001). The traditionally used assumption of the uniform through-thickness distribution of the axial stress is a fair approximation for a thin layer. To incorporate the shielding effect into the suggested lattice model, an additional multiplier K shi is introduced into the right-hand part of eqn 6.11. Shielding coefficients for elements close to a matrix crack can be obtained by integration (for respective elements) of the known analytical relations. Hence, for a single matrix crack occupying the uth column of the elements, shielding coefficients for the elements in the ith column k shiu have the following form: x =|i – u+1|l x

Ï exp(– k1 x ) ¸ k shiu = 1 Ì [2 k1 k 2 cos k 2 x + ( k12 – k 22 )sin k 2 x ]˝ , l x Ó k 2 ( k12 + k 22 ) ˛ x =|i –u|l x

6.13 where

k1 =

A + B,k = 2 2

B – A, 2

210

Multi-scale modelling of composite material systems

A=

20 [2 E 90 n (1 + n 13 ) + E 0 ( m – 2nn 23 )], E 0 Ch 2 (2n + m )

B = 22 h

15 E , C = 3n2 + 6nm + 2m2 for the [0n/90m/0n] laminate. E 0 Cnm

Here n13 and n23 are Poisson’s numbers of a carbon/epoxy unidirectional ply. The change of the shielding coefficients with the distance from the matrix crack for a case of lx = 50 mm is presented in Fig. 6.8. Generation of the matrix crack disturbs the stress distribution in [02/908/02] laminates as far as 2 mm from it. 1 0.9

Shielding coefficients

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1 1.5 2 Distance from crack, mm

2.5

3

6.8 Shielding coefficients in [02/908/02] carbon/epoxy laminate.

With the saturated magnitude of inter-crack spacing being considerably less than 1 mm, on overlap of shielding zones from two neighbouring transverse cracks significantly change the axial stress along the entire spacing, and even the next-nearest neighbouring cracks at the advanced stage of fatigue have a non-vanishing effect on the stress magnitude. Hence, using the superposition principle the multiplier K shi has the following form: N cr

i K sh = P (1 – k shij n ) n=1

6.14

where Ncr is the total number of matrix cracks in the modelled specimen and jn is the axial position (in numbers of columns) of the nth matrix crack. Obviously, contributions from the cracks situated far away from the ith row to K shi vanish and the account of the shielding zones could be limited to some specified distance, but nowadays introduction of the general form (eqn 6.14) into a numerical model practically does not affect the speed of calculations.

Multi-scale modelling of cracking in cross-ply laminates

211

In the case of laminates with considerably thick 90∞ layers, with matrix cracks growing through their thickness during several loading cycles, matrix cracks disturb the distribution of axial stresses not only in the longitudinal direction but also in the transverse one. For instance, the matrix crack that occupies not the entire thickness causes considerable stress concentration near its tip. This could be accounted by introduction of local stress-concentration coefficients (Silberschmidt and Chaboche, 1994b Silberschmidt, 1997) based on the expansion of fracture mechanics to the case of lattice models. In such a case, the interplay of two mechanisms – the initial spatial randomness and stress concentration – are responsible for the direction of crack propagation. Under conditions of uniaxial fatigue the latter mechanism attempts to straighten the crack in the direction, normal to the loading one. Still, the material’s randomness could be responsible for local crack deviations from the transverse direction in cases when elements in adjoining columns have already accumulated a significant amount of damage and are considerably closer to the threshold pc.

6.4.3

Implementation of the algorithm

In general terms, the suggested algorithm, which is based on the local damage evolution relation, can be considered as a mapping of a dynamic matrix of stress-renormalising coefficients onto the lattice of elements, covering 90∞ layers. Such a matrix incorporates effects of the initial material’s randomness as well as the disperse evolution of damage and its transition to spatially localised matrix cracking. Hence, implementation of this structured algorithm into numerical modelling is a rather unambiguous process. An additional advantage is the possibility for incorporation of additional effects, reflecting various mechanisms of stress redistribution (e.g., the effect of layers adjusting interfaces between 90∞ layers and 0∞ ones (Silberschmidt, 1997)). This scheme could also be expanded to include other failure mechanisms such as delamination, provided that either analytical or numerical estimates of their effects on stress distributions are available. There are obvious limits to the suggested approach; in the case of a complicated type of loading determination of a matrix of renormalising stress coefficients will be infeasible. A principal advantage of the suggested scheme in comparison with the finite-element method is its straightforward analysis of multiple cracking. In finite-element schemes any newly formed transverse crack (or its incremental growth in the case of thick layers) should cause the reformulation of the boundary-value problem due to the formation of new traction-free surfaces. An example of implementation of the suggested lattice scheme is the study of matrix cracking in a specimen of [02/908/02] carbon/epoxy laminate. At the first stage of modelling, the random distribution of stiffness, based on

212

Multi-scale modelling of composite material systems

the Weibull scheme, is introduced, thus resulting in the global spatial stress redistribution for the uniformly applied axial loading. This initial non-uniformity affects damage evolution, causing its accelerated pace in some elements. With the damage level attaining its critical magnitude in an element, this element fails forming the nucleus of the transverse crack. Initiation of the first transverse crack causes the local stress-redistribution process that interferes with the initial (global) randomness. The interplay of these processes governs the local damage accumulation and, consequently, the scenario of matrix cracking. Figure 6.9 demonstrates distribution of matrix cracks in a specimen of T300-934 (axial length 50 mm) loaded by tensile fatigue with the maximum cyclic stress 450 MPa at various stages of the loading history. It is obvious that initial stages of matrix cracking are characterised by a considerable extent of randomness and a large scatter in the spacing length. With a transition to an advanced fatigue stage, newly formed transverse cracks tend to form closer to the mid-spacing. This is obviously due to the mutual action of the shielding zones from neighbouring cracks that at this stage exceed the average spacing. Still, at all stages there are cracks that are formed relatively close to existing ones (see Figs 6.9(b) and 9(d)). In this case, the effect of a large local fluctuation in material’s properties is so strong that even the local stress reduction due to the shielding effect cannot prevent cracking. (a) (b)

(c)

(d)

6.9 Positions of matrix cracks in [02/908/02] laminate at different moments of loading history: (a) 100 cycles, (b) 4 ¥103 cycles, (c) 105 cycles and (d) 2 ¥ 105 cycles. Vertical to horizontal scale 2:1.

The obtained results regarding the number and position of matrix cracks in a studied specimen should be understood in a probabilistic sense; the change in the initial spatial distribution of material’s properties will result in another set of matrix cracks for the same loading conditions and history. Hence, any result is a single statistical realisation. Still, multifractal analysis demonstrates the similarity of these statistical realisations (i.e. sets of transverse cracks in specimens of the same structure) due to the closeness of their multifractal spectra for similar loading conditions. These spectra, obtained from experimental and numerical results, could also be used to validate computational schemes.

Multi-scale modelling of cracking in cross-ply laminates

6.5

213

Future trends

An evident future development in this area is linked with further coupling of lattice schemes and finite elements in order to combine strengths of both approaches; an ease of dealing with multi-scale problems with changing connectivity of the former and detailed field descriptions of stresses and strains of the latter. Here, two main trends are already starting to take shape. The first trend is a broader use of solutions to partial problems, obtained by FEA, in lattice schemes to determine matrixes of stress-redistribution coefficients. Two levels of description are of interest here, macroscopic and microscopic. The macroscopic (global) finite-element analysis of the entire specimen under applied loading conditions provides information on the initial stress field in various parts of the composite that can be used as the input data for a lattice scheme. Simulations of microscopic (local) problems are an invaluable source of information for implementation of local stressredistribution schemes in the lattice model. Obvious examples could be detailed solutions for interface (resin-rich) and edge areas, estimates of stress concentration near delamination zones, etc. The second trend is incorporation of various elements of randomness into FE modelling schemes. Different variants of stochastic finite elements have been suggested recently (Ostoja-Starzewski and Wang, 1999) to introduce microstructural material randomness. Another approach is to incorporate microscopic features/processes, e.g., damage evolution, at the sub-element level alongside introduction of the initial spatial randomness at the global level of the model. The latter can be implemented, for instance, in terms of the initial damage level (linked to manufacturing-induced defects) or local strength based on the experimental statistics. Another line of future developments is linked with enhancement of the CDM approach with regard to composite materials. From the early works on continuous damage in composites (e.g., Talreja (1985), Wnuk and Kriz (1985), Ladevèze (1986)) there has been a strong development in this area. One of the modern trends is an introduction of several damage parameters with respective damage evolution laws to describe different damage mechanisms and their interaction. This is a more promising approach if compared to the introduction of more complicated (usually, of the higher tensorial rank) damage parameters with the associated complication of constitutive relations.

6.6

Further information

A relatively recent review of various sources of information on composite materials and structures has been accomplished by Bogdanovich and Sierakowski (1999).

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The Composite Materials Handbook of the US Department of Defence is a useful source of the data on properties of various composites (including also properties of their constituents). According to the authors, ‘it is the primary and authoritative source for statistically-based characterization data of current and emerging polymer matrix, metal matrix, and ceramic matrix composite materials, reflecting the best available data and technology for testing and analysis, and including data development and usage guidelines’. An on-line access is possible via http://www.mil17.org/ (or the former website http:// mil-17.udel.edu/). Some useful data on properties of matrixes, fibres and composites can be also found in various databases: MatWeb (http://www.matweb.com/), Ecomposites (http://www.e-composites.com/, Material Properties Database (http://composite.about.com/library/data/bldata.htm).

6.7

References

Aboudi J., (1991), Mechanics of Composite Materials: A Unified Micromechanical Approach, Amsterdam, Elsevier. Bakhvalov N. and Panasenko G., (1989), Homogenization: Averaging Processes in Periodic Media – Mathematical Problems in the Mechanics of Composite Materials, Dordrecht, Kluwer Academic Publishers. Bergmann H.W. and Block J., (1992), Fracture/Damage Mechanics of Composites – Static and Fatigue Properties, Braunschweig, Institut für Strukturmechanik DLR (DLRMitteilung 92-03). Benveniste Y., (1987), ‘A new approach to the application of Mori-Tanaka’s theory in composite materials’, Mech. Mater., 6(2), 147–157. Berthelot J-M., (2003), ‘Transverse cracking and delamination in cross-ply glass-fiber and carbon-fiber reinforced plastic laminates: Static and fatigue loading’, Appl. Mech. Rev., 56(1), 111–147. Berthelot J-M. and Le Corre J-F., (2000), ‘Statistical analysis of the progression of transverse cracking and delamination in cross-ply laminates’, Compos Sci. Technol., 60(14), 2659–2669. Berthelot J-M., Leblond P., El Mahi A. and Le Corre J-F., (1996a), ‘Transverse cracking of cross-ply laminates: Part 1. Analysis’, Compos. A, 27A(10), 989–1001. Berthelot J-M., El Mahi A. and Leblond P., (1996b), ‘Transverse cracking of cross-ply laminates: Part 2. Progressive widthwise cracking’, Compos A., 27A(10), 1003–1010. Bogdanovich A.E. and Sierakowski R.L., (1999), ‘Composite materials and structures: Science, technology and applications – A compendium of books, review papers, and other sources of information’, Appl. Mech. Rev., 52(12/1), 351–366. Boniface L., Smith P.A., Ogin S.L. and Bader M.G., (1987), ‘Observations on transverse ply crack growth in (0/902)s CFRP laminate under monotonic and cyclic loading’, in Proc. 2nd Eur. Conf. Composite Materials, London, Vol. 3, 156–165. Budiansky B., (1965), ‘On the elastic moduli of some heterogeneous materials’, J. Mech. Phys. Solids, 13(4), 223–227. Chhabra A.B. and Jensen R.V., (1989), ‘Direct determination of the f(a) singularity spectrum’, Phys. Rev. Lett., 62(12), 1327–1330.

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Chhabra A.B., Meneveau C., Jensen R.V. and Sreenivasan K.R., (1989), ‘Direct determination of the f (a) singularity spectrum and its application to fully developed turbulence’, Phys Rev A, 40(9), 5284–5293. Christensen R.M., (1980), Mechanics of Composite Materials, John Wiley & Sons. Evertsz C.J.G. and Mandelbrot B.B., (1992), ‘Multifractal measures’, in Peitigen H.O., Jürgens H. and Saupe D., Chaos and Fractals. New Frontiers of Science, Berlin e.a., Springer, 921–953. Falconer K.J., (2003), Fractal Geometry. Mathematical Foundations and Applications, Chichester, Wiley. Fukunaga H., Chou T-W., Peters P.W.M. and Schulte K., (1984), ‘Probabilistic failure strength analyses of graphite/epoxy cross-ply laminates’, J. Compos. Mater., 18, 339– 356. Grassberger P., Badii R. and Politi A., (1988), ‘Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors’, J. Stat. Phys., 51(1-2), 135–178. Harte D., (2001), Multifractals: Theory and Applications, Chapman & Hall/CRC. Hashin Z., (1983), ‘Analysis of composite materials – A survey’, Trans ASME. J. Appl. Mech., 50, 481–505. Hashin Z. and Rosen B.W., (1964), ‘The elastic moduli of fiber-reinforced materials’, Trans ASME. J. Appl. Mech., 31, 223–232. Herakovich C.T., (1998), Mechanics of Fibrous Composites, New York e.a., John Wiley & Sons. Hermann H.J. and Roux S., (eds) (1990), Statistical Models for the Fracture of Disordered Media, North-Holland, Elsevier Science Publications. Hill R., (1965), ‘A self-consistent mechanics of composite materials’, J. Mech. Phys. Solids, 13(4), 213–222. Ihara C., Misawa T. and Shigeyama Y., (1988), ‘Stochastic approach to fatigue damage of carbon fiber composites’, Journal of JSMS, 37, 198–203. Jamison R.D., Schulte K., Kenneth L., Reifsnider K.L. and Stinchcomb W.W., (1984), ‘Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/epoxy composite’, in Wilkins J, Proc. 6th Int. Conf. on Composite Materials, ASTM STP 836, 21–55. Ladevèze P., (1986), ‘Sur la Mécanique de l’Endommagement des Composites’, in Bathias C. and Menkès D., Comptes Rendus des Cinquièmes Journées Nationales sur les Composites, Paris, Pluralis, 667–683. Lafarie-Frenot M.C. and Hénaff-Gardin C., (1991), ‘Formation and growth of 90 ply fatigue cracks in carbon/epoxy laminates’, Comp. Sci. Technol., 40(3), 307–324. Lee J.H. and Hong C.S., (1993), ‘Refined two-dimensional analysis of cross-ply laminates with transverse cracks based on the assumed crack opening deformation’, Comp Sci Technol, 46(2), 157–166. Lemaitre J. (1996), A Course on Damage Mechanics, Berlin e.a., Springer. Manders P.W., Chou T-W., Jones F.R. and Rock J.W., (1983), ‘Statistical analysis of multiple fracture in 0∞/90∞/0∞ glass fibre/epoxy resin laminates’, J. Mater. Sci., 18, 2876–2889. McCartney L.N. and Schoeppner G.A., (2002), ‘Predicting the effect of non-uniform ply cracking on the thermoelastic properties of cross-ply laminates’, Compos. Sci. Technol., 62(14), 1841–1856. McCartney L.N., Schoeppner G.A. and Becker W., (2000), ‘Comparison of models for transverse ply cracks in composite laminates’, Compos. Sci. Technol., 60(12/13), 2347– 2359.

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Milton G., (2002), The Theory of Composites, Cambridge, Cambridge University Press. Mori T. and Tanaka K., (1973), ‘Average stress in matrix and average elastic energy of materials with misfitting inclusions’, Acta Metall, 21(5), 571–574. Naimark O.B. and Silberschmidt V.V., (1991), ‘On fracture of solids with microcracks’, Europ J Mech A/Solids, 10 (6), 607–619. Ostoja-Starzewski M. and Wang X., (1999), ‘Stochastic finite elements as a bridge between random material microstructure and global response’, Com. Meth Appl Mech Eng 168(1-4), 35–49. Reifsnider K.L. and Case S.W., (2002), Damage Tolerance and Durability of Material Systems, New York, Wiley-Interscience. Reifsnider K.L. and Highsmith A., (1981), ‘Characteristic damage states: a new approach to representing fatigue damage in composite laminates’, in Sherratt F. and Spargean J. B., Materials Experimentation and Design in Fatigue, Surrey, Butterworth, 246–269. Reifsnider K.L. and Talug A., (1980), ‘Analysis of fatigue damage in composite laminates’, Int J Fatigue, 2(1), 3–11. Silberschmidt V.V., (1995), ‘Scaling and multifractal character of matrix cracking in carbon fibre-reinforced cross-ply laminates’, Mech Compos Mater Structures, 2(3), 243–255. Silberschmidt V.V., (1997), ‘Model of matrix cracking in carbon fiber-reinforced crossply laminates’, Mech Compos Mater Structures, 4 (1), 23–37. Silberschmidt V.V., (1998), ‘Multifractal characteristics of matrix cracking in laminates under T-fatigue’, Comput Mat Sci, 13(1-3) 154–159. Silberschmidt V.V., (2005), ‘Matrix cracking in cross-ply laminates: effect of randomness’, Compos Part A – Appl Sci Manufact, 36(2), 129–135. Silberschmidt V.V. and Chaboche J-L., (1994a), ‘Effect of stochasticity on the damage accumulation in solids’, Int J Damage Mech, 3(1), 57–70. Silberschmidt V.V. and Chaboche J-L., (1994b), ‘The effect of material stochasticity on crack-damage interaction and crack propagation’, Engng Fracture Mech, 48(3), 379– 387. Silberschmidt V.V. and Hénaff-Gardin C., (1996), ‘Multifractality of transverse cracking and cracks’ length distribution in a cross-ply laminate under fatigue’, in Petit J, ECF 11. Mechanisms and Mechanics of Damage and Failure, London, EMAS Ltd, Vol. 3, 1609–1614. Stinchcomb W.W., (1986), ‘Nondestructive evaluation of damage accumulation in composite laminates’, Comp Sci Technol, 25(2), 103–118. Suresh S., (1998), Fatigue of Materials, Cambridge, Cambridge University Press. Talreja R., (1985), ‘A continuum mechanics characterization of damage in composite materials’, Proc R Soc A, 399(1817), 195–216. Vasiliev V.V. and Morozov E.V., (2001), Mechanics and Analysis of Composite Materials, Amsterdam e.a., Elsevier. Wang A.S.D., Chou P. and Lei S.C., (1984), ‘A stochastic model for the growth of matrix cracks in composite materials’, J Compos Mater, 18, 239–254. Weibull W., (1951), ‘A statistical distribution function of wide applicability’, ASME J Appl Mech, 18, 293–297. Wnuk M.P. and Kriz R.D., (1985), ‘CDM model of damage accumulation in laminated composites’, Int J Fract, 28(3), 121–138.

7 Modelling damage in laminate composites M K A S H T A L Y A N, University of Aberdeen, UK and C S O U T I S, The University of Sheffield, UK

7.1

Introduction

The failure of glass- and carbon-fibre reinforced plastic (GFRP and CFRP) laminates subjected to static or cyclic tensile loading acting in the plane of reinforcement, and also under thermal fatigue, is a complex process. It involves sequential accumulation of various types of intra- and interlaminar damage, which gradually lead to the loss of the laminate’s load-carrying capacity. The main damage mechanisms, exhibited in composite laminates, are matrix cracking, delamination, fibre debonding, and fibre breakage. Damage mechanisms in composite laminates can be studied theoretically following two approaches. Using the continuum damage mechanics approach, various types of damage are accounted for via the damage tensor (Talreja, 1985a, b, 1986; Allen et al., 1987a, b; Li et al., 1998). A composite is then described as a continuum with mechanical properties depending on the damage tensor. Using the damage micromechanics approach, stress analysis of the damaged composite is carried out in the explicit presence of damage. Various types of damage are analysed directly with the aim to predict their onset and growth, and also their effect on the properties of the laminate. While for homogeneous isotropic materials it is often possible to obtain exact solutions within the linear elasticity theory, stress analysis of damaged composite laminates is approximate in the majority of cases. If interaction between various types of damage is especially complex, stress field can only be determined by numerical methods such as the finite-element method (Dharani and Tang, 1990; Berthelot et al., 1996).

7.1.1

Intralaminar damage

The first type of damage observed during the initial stages of the failure process is the intralaminar damage in the form of matrix cracks running parallel to the fibres in off-axis plies of the laminate. Matrix cracking initiates long before the laminate loses its load-carrying capacity. It gradually reduces 217

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the stiffness and strength of the laminate (Highsmith and Reifsnider, 1982) and changes its coefficients of thermal expansion (Bowles, 1984), moisture absorption (Lundgren and Gudmundson, 1999) and the natural frequency (Birman and Byrd, 2001). Cracked matrix may cause leaks in laminated composite pressure vessels. Matrix cracking triggers development of other, more harmful damage mechanisms. Stress concentration near the crack tip at the ply interface may cause either delamination (Nairn and Hu, 1992b) or matrix cracking, or sometimes both, in the adjacent ply (Bailey et al. 1979; Jamison et al., 1984; Charewicz and Daniel, 1986). Delaminations may result in fibre breakage in the primary load-bearing plies (Jamison et al., 1984) and lead to the loss of the load-carrying capacity of the whole laminate. Studies of matrix cracking have been focusing predominantly on transverse cracks, i.e., matrix cracks in the 90∞-plies of a laminate. The laws of transverse cracking in GFRP and CFRP laminates are in many respects similar. As a rule, cracks in the matrix occur at equal distances from each other (Garrett and Bailey, 1977) and immediately propagate from edge to edge, cleaving the entire thickness of the damaged ply. Under quasi-static loading, the strain corresponding to the cracking onset decreases with an increased ply thickness (Parvizi et al., 1978). Under cyclic loading, the cycle number corresponding to the beginning of cracking increases with an increased loading amplitude (Daniel and Charewicz, 1986). The degree of transverse cracking is characterised by crack density, i.e., the number of cracks per unit length. After cracking has begun, crack density abruptly increases with the applied load. The rate of crack density increase gradually decreases and the matrix cracking comes to saturation state, which is sometimes called a characteristic damaged state. The features of transverse cracking of the matrix in [90n/0m]s cross-ply CFRP laminates were studied by Highsmith and Reifsnider (1982) and Smith et al. (1998). It was established that transverse cracking in the outer 90∞-plies begins at lower strains than in the inner plies with a double thickness. However, the saturation crack density is lower in this case. Moreover, cracks are staggered rather than aligned in the outer 90∞-plies. The overwhelming majority of studies investigating behaviour and properties of composite laminates with matrix cracks assume that cracks are equally spaced and therefore the analysis can be restricted to a representative segment of the laminate, containing one crack. The analysis of a cross-ply laminate element with a transverse crack is usually reduced to a plane problem in the plane xOz (Fig. 7.1). Shear-lag-based models remain the most commonly used for calculating the reduced stiffness properties of laminates with transverse cracks. To eliminate the dependence on the z-co-ordinate, it is assumed that the stress sz, averaged across the layer thickness, is equal to zero, and variation in the transverse displacement with the longitudinal co-ordinate may be neglected (Nuismer and Tan, 1988). The dependence of in-plane displacements on the transverse

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z x

z

y 0∞ 90∞

x

0∞ Transverse crack

7.1 Cross-ply composite laminate with transverse matrix cracks.

co-ordinate is assumed either linear (Flaggs, 1985; Tan and Nuismer, 1989) or quadratic (Han et al., 1988; Han and Hahn, 1989; Lee and Daniel, 1990; Smith and Wood, 1990; Berthelot et al., 1996). The latter is equivalent to the assumption that the shear stresses depend linearly on the transverse coordinate (Nuismer and Tan, 1988). Some shear-lag models assume that the shear stresses due to transverse cracking act only within a thin resin-rich layer adjacent to a 90∞-ply (Highsmith and Reifsnider, 1982; Fukunaga et al., 1984; Lim and Hong, 1989). Zhang et al. (1992a, b) assumed that shear stresses in [0m/90n]s laminates vary linearly throughout the thickness of the 90∞-ply and one mth the thickness of the 0∞-ply and equal zero in the other parts. For cross-ply laminates with a thick outer 0∞-ply, Berthelot (1997) assumed that the dependence of the longitudinal displacements on the transverse coordinate is quadratic in the 90∞-ply and is linear in the 0∞-ply, but with the coefficient of proportionality being an exponential function of the density of transverse cracks. Hashin (1985) was the first to solve a plane stress problem for a cross-ply laminate with transverse cracks based on the variational principles. He assumed that the stresses sx in each ply depend only on x and do not depend on z. The stresses determined based on this hypothesis satisfy the equilibrium equations, the boundary conditions, and the continuity conditions at interfaces, and the unknown constants can be determined from the principle of minimum additional energy. The tensile stresses sx obtained by the variational and shear-lag methods are qualitatively close. The fundamental difference is observed for the shear stress t at the interface between plies. Within the framework of the shear-lag methods, it turns out to be non-zero for x = s, which contradicts the assumption that the crack faces are free from load. Moreover, the variational methods, in contrast to the shear-lag methods, allow us to determine the transverse stress sz. Hashin’s variational approach was further developed by Nairn (1989), Varna and Berglund (1991, 1992), and Berglund and Varna (1994). The

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model of Varna and Berglund (1991) is a further development of Hashin’s model. The authors assume that in a 90∞-ply the stress sx depends on the transverse co-ordinate z, the shear stress sxz is a linear function of z, and the stress sz is a quadratic function of z. In a 0∞-ply, there may be an inhomogeneous stress distribution described by an exponential function with an unknown shape parameter. The problem is reduced to a differential equation of the fourth order with constant coefficients derived in minimising the additional energy. This equation and its solution contain an unknown shape parameter, which is calculated in the subsequent minimisation of the additional energy. The refined model developed by the same authors (Berglund and Varna, 1994) admits the inhomogeneity of the stress distribution in both plies. McCartney (1992, 1993) assumed that the stresses sx in each ply depend only on x and do not depend on z. However, according to his approach, the problem is reduced to a system of recurrent relations and an ordinary differential equation of the fourth order. Also plies are divided into subplies of smaller thickness, since the elastic relations in the transverse direction are satisfied in the averaged sense (Takeda et al., 2000). McCartney showed that stresses and displacements determined by his method are in agreement with those determined by the variational method based on Reissner’s variational theorem. Schoeppner and Pagano (1998) directly used Reissner’s variational principle. These approaches are compared by McCartney et al., 2000). The variational principles were used by Nairn (1989) and Nairn and Hu (1992a) to study transverse cracking in outer 90∞-plies. Both codirectional and noncodirectional cracks were considered. Viscoelastic analyses of transverse cracking in crossply composite laminates were made in (Zocher et al., 1997; Kumar and Talreja, 2001). In evaluating the effect of transverse cracks on the stiffness of cross-ply laminates, many authors considered only the longitudinal elastic modulus (Ogin et al., 1985; Charewicz and Daniel, 1986; Daniel and Charewicz, 1986; Highsmith and Reifsnider, 1982, 1986; Kobayashi et al., 2001; Ogihara et al., 2001). Hashin (1985) obtained the exact lower bound for the longitudinal elastic modulus. Nairn and Hu (1992a, b) used the variational principles to show that with the same density of transverse cracks the stiffness of [90n / 0m]s laminates decreases more than that of [0m /90n ] s laminates whose 90∞plies are inner and adjacent. Both codirectional and noncodirectional cracks were considered. For cross-ply laminates with both transverse and longitudinal cracks, a representative segment could be defined by intersecting a pair of transverse and longitudinal cracks (Hashin, 1987; Tsai and Daniel, 1992; Henaff-Gardin et al., 1996a, b). Kashtalyan and Soutis (1999b, 2000b, c) suggested analysing cross-ply laminates with damage in both plies using the Equivalent Constraint Model. Instead of the damaged laminate, two ECM laminates are considered and analysed simultaneously. In the first laminate, 0∞ layers contain damage

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explicitly, while 90∞ plies are replaced with equivalent homogeneous ones with reduced stiffness properties. These reduced stiffness properties are assumed to be known from the analysis of the second laminate, in which the 90∞ layer contains damage explicitly, while the damaged 0∞ plies are replaced with equivalent homogeneous ones with reduced properties, assumed to be known from the analysis of the first laminate. Thus, problems for both laminates are inter-related. Angle-ply laminates exhibit much more complex morphologies of intralaminar damage than cross-ply laminates. Comprehensive observations of sequential accumulation of matrix cracks in off-axis plies have been reported for quasi-isotropic [0/45/– 45/90]s carbon/epoxy (Reifsnider and Talug, 1980), [0/90/– 45/45]s carbon/epoxy (Masters and Reifsnider, 1982) and glass/epoxy (Tong et al., 1997a) laminates, balanced [±45/90]s and [90/±45]s glass/epoxy (Marsden et al., 1999) laminates and angle-ply [02/q2/– q2]s carbon/epoxy (O’Brien and Hooper, 1991) and [0/q/0] (Crocker et al., 1997) and [0/ ± q4/01/2]s (Varna et al., 1999) glass/epoxy laminates. It was found that longitudinal strain for matrix cracking initiation decreases with increasing ply orientation angle (Crocker et al., 1997). Also, ply stresses normal to the fibres at crack formation were found to become progressively smaller as the ply orientation angle increased (Crocker et al., 1997). Stress fields in the cracked off-axis plies of angle-ply laminates were examined by means of finite element method (Tong et al., 1997b; Marsden et al., 1999) and analytically (Lapusta and Henaff-Gardin, 2000). Application of the Equivalent Constraint Model (ECM) to quasi-isotropic laminates with matrix cracking in all but 0∞ layers was presented by Zhang and Herrmann (1999). Kashtalyan and Soutis (2001) predicted analytically strain energy release rate associated with matrix cracking in the q-layer of unbalanced symmetric [0/q]s composite laminate using a 2-D shear lag stress analysis and the ECM. Comparison of theoretical predictions with experimental data for glass/epoxy laminates, obtained by Crocker et al., (1997), showed that a quadratic mixed mode fracture criterion can successfully predict the cracking onset strain for ply orientation angles 75∞ £ q £ 90∞.

7.1.2

Interlaminar damage

Matrix cracks, developing in the off-axis plies of the laminate, are either arrested at the interface or cause interlaminar damage leading to delamination due to high interlaminar stresses at the ply interface. Figure 7.2 summarises schematically types of intra- and interlaminar damage observed in cross-ply and angle-ply composite laminates: transverse and longitudinal matrix cracks in cross-ply laminates (Fig. 7.2(a)); transverse and longitudinal crack tip delaminations in cross-ply laminates (Fig. 7.2(b)); off-axis ply cracks in

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Multi-scale modelling of composite material systems Delaminations

Delaminations

Transverse crack Transverse cracks (a)

Longitudinal crack (b)

Delamination

Delamination Off-axis crack

Off-axis crack

(d)

(c) Edge delaminations

Ply cracks (e)

7.2 Types of intra- and interlaminar damage observed in cross-ply and angle-ply composite laminates: (a) transverse and longitudinal matrix cracks in cross-ply laminates; (b) transverse and longitudinal crack tip delaminations in cross-ply laminates; (c) off-axis ply cracks in balanced symmetric laminates and uniform local delaminations; (d) off-axis ply cracks in balanced symmetric laminates and partial local delaminations; (e) matrix-crack induced edge delamination.

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balanced symmetric laminates (Figs 7.2(c–d)); uniform (Fig. 7.2(c)) and partial (Fig. 7.2(d)) local delaminations induced by angle ply matrix cracks; crack induced edge delaminations (Fig. 7.2(e)). Studies of delaminations induced by matrix cracking have been focusing predominantly on delaminations caused by transverse cracks, i.e., matrix cracks in the 90∞-plies of a laminate. Crossman and Wang (1982) made comprehensive observations of transverse cracking and delamination in balanced symmetric [±25/90n]s, n = 0.5; 1; 2; 3; 4; 6; 8 graphite/epoxy laminates. A significant reduction in the delamination onset strain was noted for the laminates with n ≥ 4 A transition from edge delamination to local delaminations growing from the tip of a matrix crack in the 90∞ ply occurred between n = 3 and n = 4. The onset and growth of edge delamination in [(±30)2/90/ 90] s graphite/epoxy laminates under static tension and tensiontension fatigue loading was studied by O’Brien (1982). Stiffness loss was monitored simultaneously with delamination growth and found to decrease linearly with delamination size. Armanios et al. (1991) applied a shear deformation theory and sublaminate approach to analyse local delaminations originating from transverse cracks in CFRP [±25/90n]s laminates. Predictions of their model, which also takes into account hygrothermal effects, are in reasonable agreement with delamination onset strain data by Crossmann and Wang (1982). Nairn and Hu (1992b) used two-dimensional variational approach to analyse crack tip delaminations in [(S)/90n]s laminates, where (S) denotes a balanced sublaminate, e.g., (±qm). They predicted that matrix cracking should reach some critical density before delamination initiates. The critical crack density for delamination initiation is determined by material properties, laminate structure as well as fracture toughnesses for matrix cracking and delamination and is nearly independent of the properties of the supporting sublaminate (S). Zhang et al., (1994a, b) used a 2-D improved shear lag analysis to predict the strain energy release rate for edge and local delaminations in balanced symmetric [±qm /90n ]s laminates. For edge delamination, they were able to capture a zigzag delamination pattern, i.e., edge delamination switching from one (q/90) interface to another through a matrix crack, and improve O’Brien’s formula for strain energy release rate for edge delamination (O’Brien, 1982) incorporating the effect of matrix cracking. For local delaminations, they obtained the strain energy release rate as a function of crack density and delamination area. Their predictions for delamination onset strain agree well with the experimental data of Crossman and Wang (1982) and capture the transition from edge to local delamination quite accurately. Initiation and growth of local delaminations from the tips of transverse cracks in cross-ply [0/90n]s n = 2; 4; 6 carbon/epoxy laminates under static tension was examined by Takeda and Ogihara (1994). Delamination was

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noted to grow more rapidly and extensively in the laminates with thicker 90∞ plies. Ogihara and Takeda (1995) used a modified shear lag method featuring interlaminar shear layer to predict strain energy release rate for transverse crack tip delaminations in cross-ply [0/90n]s laminates and to model interaction between transverse cracking and delamination. However, the effect of cracking/ delamination interaction was found to be negligible in prediction of delamination growth. Henaff-Gardin et al. (1996a, b) and Kobayashi et al. (2001) observed damage development in carbon/epoxy cross-ply laminates under thermal cycling. The first damage mode observed consisted of matrix cracks in 0∞ and 90∞ plies. The first damage mode observed consisted of matrix cracks in 0∞ and 90∞ plies. Most of the matrix cracks spanned the entire width or length of the specimen. Then delaminations initiated between 0∞ and 90∞ plies along the pre-existing cracks in 0∞ ply. More recently, Selvarathinam and Weitsman (1998, 1999) observed and modelled, by means of finite elements and shear lag methods, delaminations induced by matrix cracking in cross-ply laminates under environmental fatigue. By comparing strain energy release rates associated with matrix cracking and delamination, they were able to explain the extensive delaminations and reduced crack densities that arise under immersed fatigue conditions, as compared with fatigue in air. Zhang et al., (1999) studied delaminations induced by transverse cracking at the (f /90) interfaces in […/j i / f m /90n ] s laminates loaded in tension using a sublaminate-wise first-order shear deformation theory. In particular, they were interested in the constraining effect of the immediate neighbouring plies and remote plies on stiffness reduction and strain energy release rate for delaminations. It was found that the strain energy release rate for local delamination and stiffness reduction of the constrained transverse plies largely depends on a local lay-up configuration of a damaged laminate. Kashtalyan and Soutis (1999a, 2000a) examined the effect of crack tip delaminations on stiffness reduction for cross-ply [0m /90n]s laminates with local delaminations along transverse as well as longitudinal cracks. It was established that reduction in the laminate shear modulus and Poisson’s ratio is much more significant than in the axial modulus. For balanced symmetric [±qm /90 n ] s, the effect of constraining ply orientation angle q on reduction of the laminate in-plane stiffness properties was also examined. Delaminations induced by angle ply matrix cracks in carbon/epoxy [02/ q 2 /– q 2 ] s, q = 20∞, 25∞, 30∞ laminates subjected to tension fatigue loading were observed by O’Brien and Hooper (1991) and O’Brien (1991). Matrix cracks formed near the stress free edge and delaminations, bounded by the free edge and the crack, developed in the q/(– q ) interface. They were termed partial local delaminations. Using a quasi-3D finite element (FE) analysis, Salpekar and O’Brien

Modelling damage in laminate composites

225

(1991) found that the strain energy release rate for uniform local delamination calculated from O’Brien (1985) expression matched the value obtained by FE analysis in the laminate interior. O’Brien and Hooper (1991) and O’Brien (1991) observed matrix crack induced delaminations in symmetric angle-ply [02/q2/–q2]s carbon/epoxy laminates under quasi-static and fatigue tensile loading (q = 15∞; 20∞; 25∞; 30∞). Delaminations occurred in the (q /– q) interface, bounded by the cracks in the (– q)-ply and the stress free edge. The laminated plate theory and a quasi-3D finite element analysis were used to examine stresses in the (– q )ply. For the considered range of ply orientations, stresses normal to the fibres were found to be compressive and shear stresses along the fibres to be high in the laminate interior, while near the free edge high tensile stresses normal to the fibres were present. Two closed form expressions for strain energy release rate were derived on the basis of simple load shearing rules, one for a local delamination growing from an angle ply matrix crack with a uniform delamination front across the laminate width, and one for a partial local delamination growing from an angle ply matrix crack and bounded by the free edge. Salpekar and O’Brien (1993) used a 3-D FE analysis to study matrix crack induced delaminations in (0/q/–q)s graphite/epoxy laminates (q = 15∞; 45∞) loaded in tension. For (0/45/– 45)s laminate, the strain energy release rate for local delamination growing uniformly in the (45/– 45) interface from the matrix crack in the (– 45∞)-ply was found to be higher near the laminate edge than in the interior of the laminate. Later, Salpekar et al., (1996) computed strain energy release rates associated with local delamination originating from matrix cracks and bounded by the free edge in (0/q/– q)s and (q/– q/0)s graphite/epoxy laminates using a 3-D FE method. The total strain energy release rate was calculated using three different techniques: the virtual crack closure technique, the equivalent domain integral technique, and a global energy balance technique. Kashtalyan and Soutis (2002) theoretically modelled local delaminations growing uniformly from the tips of matrix cracks in an angle-ply laminate loaded in tension. They obtained closed-form expression for strain energy release rates, associated with these delaminations, as linear functions of the first partial derivatives of the effective elastic properties of the damaged layer with respect to delamination area. Strain energy release rate is independent of delamination area and crack density, thus taking into account the cumulative effect of damage. The total strain energy release rate depends linearly on crack density both in balanced [02/q2/– q2] s and unbalanced [02 / q 2 ] s laminates. The dependence on delamination area is linear in balanced and non-linear in unbalanced laminates. Comparison with results by O’Brien (1991) showed that O’Brien’s closed-form expression for uniform local delamination significantly overestimates the value of strain energy release rate. For the

226

Multi-scale modelling of composite material systems

same ply orientation angle, crack density and delamination area, delaminationinduced changes in stiffness properties are much more significant in unbalanced laminates than in balanced laminates. In all above studies of crack-induced delaminations it was assumed that delamination surfaces, like matrix crack surfaces, are stress free. Besides that, delaminations were assumed to behave in a self-similar manner, i.e., boundary conditions prescribed at the delaminated surfaces were assumed to be the same for small and large delaminations. More recently, Ashkantala and Talreja (1998) and Berthelot and Le Corre (2000) examined transverse crack tips delaminations in cross-ply laminates with shear friction between the delaminated plies. While Berthelot and Le Corre (2000) assumed the magnitude of the interlaminar shear stress at the delaminated interface to be constant, i.e., independent of delamination length, Ashkantala and Talreja (1998) considered both linear and cubic polynomial shear stress distribution at the delamination interface. Selvarathinam and Weitsman (1998, 1999) observed and modelled, by means of finite elements and shear lag methods, delaminations induced by matrix cracking in cross-ply laminates under environmental fatigue, with delamination surface loaded with hydrostatic pressure.

7.2

Stress analysis

Figure 7.3 shows a schematic of a symmetric [(S)/f]s laminate, consisting of the outer sublaminate (S) and the inner f-layer damaged by matrix cracks and local delaminations growing from their tips at the (S)/f interface. The outer sublaminate (S), or layer 1, may consist either of a single layer or a group of layers and can also be damaged (in this case it needs to be replaced in the analysis with an equivalent homogeneous layer with reduced stiffness properties). The laminate is referred to the global Cartesian co-ordinate system xyz and local co-ordinate system x1x2x3, with the axis x1 directed along the fibres in the damaged f-layer, or layer 2. The laminate is subjected to inplane biaxial tension s x and s y . Since the laminate is symmetric, no coupling exists between in-plane loading and out-of-plane deformation. Matrix cracks are assumed to be spaced uniformly, with crack spacing 2s and span the whole width of the laminate. Local delaminations are assumed to be strip shaped, with strip width 2l, (Fig. 7.3). Due to the periodicity of damage, the stress analysis may be carried out over a representative segment containing one matrix crack and two crack tip delaminations. Due to symmetry, it can be further restricted to one quarter of the representative segment, (Fig. 7.4) referred to the local co-ordinate system x 1 x 2x 3. Let {s˜ (1)} and {e˜ (1)} denote the in-plane microstresses and microstrains in the layer 1, and {s˜ (2)} and {e˜ (2)} denote the in-plane microstresses and

Modelling damage in laminate composites y

227

x1

sy x2 f

sx

x 2l

sx

2s

Delamination

Off-axis crack

sy

( S) (f )

h1 2h1

7.3 Front and edge view of a [(S)/f]s laminate subjected to biaxial tensile loading and damaged by matrix cracks and crack-induced delaminations. Local (x1x2x3) and global (xyz) co-ordinate systems for the damaged f-layer (front view in the negative x3 ∫ z direction). x3 Delamination

(S ), or layer 1

f , or layer 2 Off-axis ply crack

l

x2

s–l

7.4 A quarter of the representative segment containing a matrix crack and delamination.

microstrains in the layer 2 (i.e. stresses and strains averaged across the respective layer thickness). Since it is assumed that there is no frictional contact between the layers in the locally delaminated portion of the representative segment (0 < | x2 | < l, | x3 | < h2), the in-plane microstresses

228

Multi-scale modelling of composite material systems

(2) (2) = s˜ 12 = 0, i.e., this region is stress-free. in the delaminated portion are s˜ 22 Assumption of stress-free crack tip delamination surfaces, and the resulting implication that the portion of the damaged ply bounded by matrix crack and delamination surfaces is stress-free, has been widely used in the studies of delaminations. Besides that, delaminations are assumed to behave in a selfsimilar manner, i.e., the boundary conditions prescribed at the delaminated surfaces were assumed to be the same for small and large delaminations. In the perfectly bonded region (l < | x2 | < s) of the representative segment, they are determined from the equilibrium equations

d s˜ (2) – t j = 0 , dx 2 j 2 h2

j = 1, 2

7.1

where tj are the interface shear stresses and h2 is the thickness of the f-layer. By averaging the out-of-plane constitutive equations for both layers across the layer thickness, the interface shear stresses tj can be expressed in terms of the in-plane displacements and shear lag parameters Kij as

t j = K j 1 ( u˜1(1) – u˜1(2) ) + K j 2 ( u˜ 2(1) – u˜ 2(2) )

7.2

The shear lag parameters K11, K22, K12 ∫ K21 are determined assuming that the out-of-plane shear stresses s˜ (j k3 ) vary linearly with x3 (Fig. 7.5), see Appendix I. Substitution of eqn 7.2 into eqn 7.1 and subsequent differentiation yields d 2 s˜ (2) + K (g˜ (1) – g˜ (2) ) + K ( e˜ (1) – e˜ (2) ) = 0 , j1 j2 12 12 22 22 dx 2 j 2

j = 1, 2 7.3

x3 (1) (1) s 13 , s 23

h1

Layer 1

t1 , t2

h2

Layer 2

(2) (2) s 13 , s 23

x2

7.5 Variation of out-of-plane shear stresses. (1) (2) (1) (2) The strain differences ( e˜ 22 – e˜ 22 ) and (g˜ 12 – g˜ 12 ) involved in eqn 7.3 (2) (2) can be expressed in terms of stresses s˜ 12 , s˜ 22 using the constitutive equations for both layers, the laminate equilibrium equations below

c {s˜ (1)} + {s˜ (2)} = [1 + c )[ T ]{s }

7.4a

Modelling damage in laminate composites

È cos 2 f Í [ T ] = Í sin 2 f ÍÎ – sin f cos f

sin 2 f cos 2 f sin f cos f

{s } = {sx, sy, 0}T,

2 sin f cos f ˘ ˙ – 2 sin f cos f ˙ cos 2 f – sin 2 f ˙˚

c = h1/h2

229

7.4b

7.4c

and the assumption of the generalised plane strain condition (1) (2) e˜11 = e˜11

7.5

In the local co-ordinate system x1x2x3, the layer 2 is orthotropic, (2) (2) ¸ È Sˆ11 Ï e˜11 ÔÔ (2) ÔÔ Í (2) Ì e˜ 22 ˝ = Í Sˆ12 Ô ˜ (2) Ô Í ÔÓg 12 Ô˛ ÍÎ 0

(2) Sˆ12 Sˆ (2) 22

0

(2) ¸ 0 ˘ Ïs˜ 11 ˙ ÔÔ (2) ÔÔ 0 ˙ Ìs˜ 22 ˝ (2) ˙ Ô (2) Ô Sˆ66 ˙˚ ÔÓs˜ 12 Ô˛

7.6a

while the layer 1 is anisotropic (1) (1) (1) (1) ˘ (1) ¸ È Sˆ11 ¸ Ï e˜11 Ïs˜ 11 Sˆ12 Sˆ16 ˙ Ô ÔÔ (1) ÔÔ Í (1) Ô Ô (1) Ô (1) (1) Sˆ22 Sˆ26 ˙ Ìs˜ 22 ˝ Ì e˜ 22 ˝ = Í Sˆ12 7.6b ˙ Ô Ô ˜ (1) Ô Í ˆ (1) Ô (1) (2) (1) Sˆ26 Sˆ66 ˙˚ ÔÓs˜ 12 Ô˛ ÔÓg 12 Ô˛ ÍÎ S16 where [ Sˆ ( k ) ] is the compliance matrix for the kth layer. Finally, eqn 7.3 can be reduced to a system of two coupled second order ordinary differential equations (see Appendix II)

(2) d 2 s˜ 12 (2) (2) – N11s˜ 12 – N12 s˜ 12 – P11s x – P12 s y = 0 dx 22

7.7a

(2) d 2 s˜ 22 (2) (2) – N 21s˜ 12 – N 22 s˜ 22 – P21s x – P22 s y = 0 dx 22

7.7b

Here Nij and Pij are laminate constants depending on the layer compliances Sˆij( k ) , layer thickness ratio c, shear lag parameters K11, K22, K12 and angle f (Appendix II). Equations 7.7a and 7.7b can be uncoupled at the expense of increasing the order of differentiation, resulting in a fourth order non-homogeneous ordinary differential equation (2) (2) d 4 s˜ 22 d 2 s˜ 22 (2) – ( + ) – ( N 21 N12 – N11 N 22 ) s˜ 22 N N 11 22 dx 24 dx 22

+ [N11(P21 + a P22) – N21(P11 + a P12)] s x = 0

7.8

230

Multi-scale modelling of composite material systems

Here a = s y / s x is the biaxiality ratio. The boundary conditions for eqn 7.8 are prescribed at the stress-free boundary between locally delaminated and perfectly bonded portions of the representative segment (2) (2) s˜ 22 | x 2 =± l = 0, s˜ 12 | x 2 = ±l = 0

7.9

Finally, the in-plane microstresses can be expressed in the following form (2) (2) (2) s˜ 11 = a 22 s˜ 22 + a12 s˜ 12 + bx s x + bys y

7.10a

cosh l 2 ( x 2 – s ) È cosh l1 ( x 2 – s ) ˘ s˜ (2) + Bj + C j ˙ s x , j = 1, 2 j 2 = Í Aj cosh l1 ( s – l ) cosh l 2 ( s – l ) ˚ Î 7.10b where coefficients a22, a12, bx and by are given in Appendix II, lj are the roots of the characteristic equation and Aj, Bj and Cj are constants depending on Nij and Pij, see Appendix III. In cross-ply and balanced laminates the outer sublaminate (S) is orthotropic, (1) (1) (1) = Sˆ26 = 0 and stiffnesses Qˆ 45 = 0 . In this case with compliances Sˆ16 shear lag coefficients K12 ∫ K21 = 0 vanish, and equilibrium equations are reduced to two uncoupled second-order differential equations. Details of this case are given elsewhere (Zhang et al., (1994b), Kashtalyan and Soutis (1999a, 2000a)). Figure 7.6 shows the in-plane stresses in the q-ply of the [02/q2]s laminate as a function of ply orientation angle q under uniaxial (s y / s x = 0) and biaxial (s y / s x = 0.5; 1; 2) tensile loading as calculated from the laminated plate theory (Daniel and Ishai, 1994). The applied stress is s x = 100 MPa. The material system is AS4/3506-1 graphite/epoxy (O’Brien and Hooper, 1991). Its engineering properties are as follows: axial modulus E11 = 135 GPa, transverse modulus E22 = 11 GPa, in-plane shear modulus G12 = 5.8 GPa, major Poisson’s ratio n12 = 0.301. Nominal ply thickness is t = 0.124 mm. The in-plane stresses were transformed into the local co-ordinate system, Fig. 7.3, in order to determine stresses normal to the fibres, s˜ 22 , and shear stresses along the fibres, s˜ 12 , since these stresses contribute directly to the formation of matrix cracks. Under uniaxial tension, stresses normal to the fibres in the q-ply of the [02/q2]s laminate are compressive for q smaller than 25∞ and tensile for q greater than 25∞, (Fig. 7.6(a)). Under biaxial loading, stresses in the (q) ply normal to the fibres are tensile. For all considered biaxiality ratios a = s y / s x , the highest normal stresses are observed for q = 15∞. They decrease significantly as q increases and at q = 60∞ reach almost a constant value that is almost independent of the biaxiality ratio. For 15∞ £ q £ 60∞, the magnitude of normal stresses increases with increasing biaxiality

Modelling damage in laminate composites

231

Normal stress, MPa

200 Uniaxial 0.5 1 2

150

100 Yt = 57 MPa 50

0

15

30 45 60 75 Ply orientation angle, degrees (a)

90

Uniaxial 0.5 1 2

100

75

Shear stress, MPa

S = 71 MPa 50

25

0

–25 15

30

45 60 75 Ply orientation angle, degrees (b)

90

7.6 Ply stresses in the q-ply of the AS4/3506-1 graphite/epoxy [02/q2]s laminate as a function of ply orientation angle q under uniaxial s y / s x = 0) and biaxial ( s y / s x = 0.5; 1; 2) tensile loading: (a) stresses normal to the fibres; (b) shear stress. The applied stress is s x = 100 MPa.

ratio, sometimes exceeding the value of the transverse tensile strength of the material. Under uniaxial tension, relatively high shear stresses are present along the fibres, (Fig. 7.6(b)), that may contribute to the formation of matrix cracking in the q-ply of the [02/q2]s laminate.

232

7.3

Multi-scale modelling of composite material systems

Predicting stiffness degradation due to intraand interlaminar damage

The reduced stiffness properties of the mth ply can be determined by applying the laminate plate theory to the ECMm laminate after replacing the explicitly damaged mth ply with an equivalent homogeneous one. In the local co-ordinate system x1x2x3, the constitutive equations of the ‘equivalent’ homogeneous layer are

{s (2)} = [ Q (2) ]{e (2)}

7.11

In the local co-ordinates, the modified in-plane stiffness matrix [Q(2)] of the homogeneous layer equivalent to the damaged one is related to the in-plane stiffness matrix [ Qˆ (2) ] of the undamaged layer as (2) 2 ˆ (2) È ( Qˆ 12 ) / Q22 L 22 Í (2) (2) (2) ˆ ˆ [Q ] = [Q ] – Í Q12 L 22 Í 0 Î

(2) Qˆ 12 L 22 (2) ˆ Q12 L 22 0

0 ˘ ˙ 0 ˙ (2) Qˆ 66 L 66 ˙˚

7.12

Here L(22m ) , L(66m ) are the In-situ Damage Effective Functions (IDEFs) (Zhang, et al. 1992a). They can be expressed in terms of lamina macrostresses and macrostrains as L 22 = 1 –

(2) s 22 (2) (2) (2) (2) Qˆ 12 e 11 + Qˆ 22 e 22

, L 66 = 1 –

(2) s 12 (2) (2) Qˆ 66 g 12

7.13

The lamina macrostresses {s (2) } and macrostrains {e (2) } are obtained by averaging respectively microstresses {s˜ (2) } , eqn 7.10, and microstrains {e˜ (2) } , eqn 7.6a, across the length of the representative segment. The lamina macrostresses s ij(2) are (2) (2) (2) s 11 = a 22 s 22 + a12 s 12 + bx s x + bys y

7.14a

l1* (1– Dld ) È Dmc s (2) tanh j 2 = Í Aj 2 Dmc l1* (1 – Dld ) Î + Bj 2

Dmc l 2* (1– Dld )

tanh

l 2* (1 – Dld ) ˘ + C j 2 (1– Dld ) ˙ s x , Dmc ˚

j = 1, 2

7.14b where D = h2/s denotes relative crack density and D = l/s denotes relative delamination area. The macrostrains in the individual homogeneous layers and the laminate are assumed to be equal mc

ld

(1) (2) (1) (2) (1) (2) = e 22 = e 22 , g 12 = g 12 = g 12 e 11 = e 11 = e 11 , e 22

7.15

Modelling damage in laminate composites

233

Using the constitutive equations for the layer 1, eqn 7.6b, and equations of the global equilibrium of the laminate, eqn 7.4, the lamina macrostrains in layer 2 are

{e

(2)

} = [ Sˆ (1) ] c –1 ((1 + c )[ T ]{s } – {s (2)})

7.16

where the transformation matrix [T] is given by eqn 7.4b. Thus, the lamina macrostresses, eqn 7.14, and macrostrains, eqn 7.16, are determined as explicit functions of the damage parameters Dmc, Dld. Finally, the modified stiffness matrix [ Q ] 2 of the ‘equivalent’ homogeneous layer in the global co-ordinates xyz can be obtained from the modified stiffness matrix [Q(2)] in the local co-ordinates, eqn 7.12 as [ Q ] 2 = [T]–1[Q(2)][T]–T

7.17

where the transformation matrix [T] is given by eqn 7.4b. The extension stiffness matrix [ A ] of the ‘equivalent’ laminate in the global co-ordinates xyz can then be determined as [ A ] = S [ Q ] k hk ,

k = 1, 2

7.18

k

where [ Q ]1 is the in-plane stiffness matrix of layer 1, or the outer sublaminate, in the global co-ordinates.

7.3.1

Stiffness degradation due to off-axis ply cracking

In this subsection, predictions of stiffness properties as a function of damage are presented and discussed for balanced [±q]s glass/epoxy laminates and unbalanced [02/q2]s graphite/epoxy laminates. The results are presented in terms of the laminate stiffness properties, that is the axial modulus Ex, transverse modulus Ey, shear modulus Gxy, major Poisson’s ratio nxy, as well as shearextension coupling coefficients hxy,x and hxy,y. The shear-extension coupling coefficients hxy,j = gxy /ej characterise shearing in the xy plane caused by normal stress in the j th direction (j = x, y). To validate the developed approach, a limiting case of a cross-ply [0/90] laminate was considered (Kashtalyan and Soutis, 2000b). Cross-ply laminates cannot be analysed using the developed approach for angle-ply [q1/q2]s laminates directly, because when q1 = 0 and q2 = 90, the system of differential equations, eqn 7.9, becomes uncoupled, and the solution, given by eqn 7.12, is no longer valid. However, it works for any q2 close enough to 90∞. Figure 7.7 shows normalised (i.e. referred to their value in the undamaged state) stiffness properties versus crack density in the inner ply for a [0/89]s glassepoxy laminate, obtained using the current method. They appear to be in a good agreement with results for a cross-ply [0/90]s laminate obtained using

234

Multi-scale modelling of composite material systems Normalised stiffness property

1.0 Transverse modulus 0.9

Axial modulus

0.8 Shear modulus 0.7 Poisson’s ratio

0.6 0.5

0

10 20 Crack density (cracks/cm)

30

7.7 Normalised stiffness properties of [0/90]s (filled symbols) and [0/89]s (open symbols) glass/epoxy laminates as a function of crack density Cmc in the inner ply (cracks/cm).

the ECM/2-D shear lag model (Kashtalyan and Soutis, 1999b, 2000b). The material properties of a unidirectional Silenka E-glass 1200 tex fibre reinforced MY750/HY917/DY063 epoxy composite are as follows: EA = 45.6 GPa, ET = 16.2 GPa, GA = 5.83 GPa, nA = 0.278, GT = 5.79 GPa, single ply thickness t = 0.25 mm. Figure 7.8 shows the normalised stiffness properties of two angle-ply [±q]s glass/epoxy laminates as a function of the crack density in the inner (–q) ply. Crack density C2 = (2s(2))–1 in the inner (–q) layer varies from 0 (no matrix cracking) to 30 cracks/cm, which corresponds to variation in the damage parameter D2mc from 0 to 1.5. In the [30/–30]s laminate, Fig. 7.8(a), the reduction of the axial modulus Ex is bigger than that of the transverse modulus Ey, while for the [55/–55]s laminate the opposite is true (Fig. 7.8(b)). In both laminates, the reduction of the shear modulus Gxy is smaller than that of Ex and Ey, in contrast to the cross-ply [0/90]s laminate, where the reduction in the shear modulus and the Poisson’s ratio is bigger than that of the axial modulus (Fig. 7.7). It is also worth noting that in angle-ply laminates matrix cracking may actually increase the Poisson’s ratio – a phenomenon not observed in cross-ply laminates. In the [30/–30]s laminate the increase in the Poisson’s ratio occurs for all crack densities, while in the [55/–55]s laminate, it is observed only at higher crack densities (Fig. 7.8(b)). In the undamaged state, angle-ply [± q]s laminates are balanced and orthotropic and exhibit no coupling between extension and shear. When matrix cracking occurs, the laminate becomes unbalanced, resulting in coupling between extension and shear, reflected by non-zero shear-extension coupling coefficients hxy,x and hxy,y. Dependence of shear-extension coupling coefficients on the crack density is shown in Fig. 7.9 for [30/–30]s and [55/–55]s glass/

Modelling damage in laminate composites

235

Normalised stiffness property

1.5 1.3 1.1 0.9 0.7 0.5 0

10 20 Crack density (cracks/cm) (a)

30 Axial modulus Transverse modulus Shear modulus Poisson’s ratio

Normalised stiffness property

1.1 1.0 0.9 0.8 0.7 0.6 0

10 20 Crack density (cracks/cm) (b)

30

7.8 Normalised stiffness properties of angle-ply [q/–q]s glass/epoxy laminates as a function of crack density Cmc in the inner (–q) layer (cracks/cm): (a) [30/–30]s laminate; (b) [55/–55]s laminate.

epoxy laminates. When 0∞ < q < 45∞, the hxy,x the coefficient increases more rapidly with the crack density than the hxy,y, while for 45∞ < q < 90∞ the opposite is true. Figure 7.10 shows the normalised stiffness properties for [02/552]s and [02/752]s AS4/3506-1 laminates as a function of crack density Cmc = (2s2)–1. It may be seen that in both angle-ply laminates the most significantly reduced properties are transverse and shear moduli. In a [02/552]s laminate (Fig. 7.10(a)), a slight increase of the Poisson’s ratio with the crack density is observed. Due to their unbalanced configuration, [02/q2]s laminates exhibit shear extension coupling characterised by axial hxy,x = g xy / e x and transverse hxy,y = g xy / e x shear-extension coefficients. Figure 7.11 shows variation of the shear-extension coupling coefficients with the crack density Cmc in a [02/

236

Multi-scale modelling of composite material systems

Shear extension coupling coefficients

0.15 Axial Transverse

0.12 0.09 0.06 0.03 0.00 0

10 20 Crack density (cracks/cm) (a)

30

Shear extension coupling coefficients

0.10 Axial Transverse

0.08 0.06 0.04 0.02 0.00 0

10 20 Crack density (cracks/cm) (b)

30

7.9 Shear extension coupling coefficients of angle-ply [q/–q]s glass/ epoxy laminates as a function of crack density Cmc in the inner (–q) layer (cracks/cm) for: (a) [30/–30]s laminate; (b) [55/–55]s laminate.

752]s laminate. While the axial shear-extension coupling coefficient is almost unaffected by matrix cracking, the transverse one is increased by the absolute value.

7.3.2

Stiffness degradation due to crack-induced delamination

Figure 7.12 shows normalised stiffness properties of T800H/3631 carbon/ epoxy [0/90n]s, n = 2, 4, 6 cross-ply laminates containing transverse cracks and delaminations. The axial modulus Ex, shear modulus Gxy and Poisson’s ratio nxy, normalised by their values in the undamaged state, are plotted against the transverse crack density Cmc = (2s2)–1. The relative delamination area is Dld = 10%, which corresponds to l/s = 0.1. For the axial modulus, predictions are compared to experimental data obtained by Takeda and Ogihara

Modelling damage in laminate composites

237

Normalised stiffness property

1.1

1

0.9 Axial modulus Transverse modulus Shear modulus Poisson’s ratio 0.8 0

1 2 3 4 Crack density (cracks/cm) (a)

5

Normalised stiffness property

1

0.9

Axial modulus Transverse modulus Shear modulus Poisson’s ratio 0.8 0

1 2 3 4 Crack density (cracks/cm) (b)

5

7.10 Normalised stiffness properties of angle-ply [02/q2]s graphite/ epoxy laminates as a function of crack density Cmc in the inner q layer (cracks/cm): (a) [02/552]s laminate; (b) [02/752]s laminate.

(1994) and appear to be in acceptable agreement. However, predictions show that reduction in shear modulus and Poisson’s ratio due to crack tip delamination is more significant. Figure 7.13 shows the variation of the normalised stiffness properties of the [02/302/–302]s AS4/3506-1 graphite/epoxy laminate with the relative delamination area Dld = l/s. Matrix crack density in the inner (–30∞) ply is assumed equal to Cmc = 2 cracks/cm. Values at Dld = 0 indicate stiffness properties of the laminate at this crack density without delaminations. It can be seen that local delaminations further decrease the laminate moduli and,

238

Multi-scale modelling of composite material systems Shear extension coupling coefficients

0.6

Axial Transverse

0

– 0.6

–1.2

–1.8 0

1 2 3 4 Crack density (cracks/cm)

5

7.11 Shear extension coupling coefficients of angle-ply [02/752]s graphite/epoxy laminate as a function of crack density Cmc in the inner layer (cracks/cm).

for the considered lay-up, increase the Poisson’s ratio (Fig. 7.13(a)). In balanced [02/q2/–q2]s laminates uniform local delaminations result in an increase in the absolute value of the axial shear-extension coupling coefficient for q < 45∞ and of the transverse shear-extension coupling coefficient for q > 45∞. However, all shear-extension coupling coefficients are significantly smaller than those for unbalanced laminates (see later and Fig. 7.13(b)). Figure 7.14 shows the normalised stiffness properties in the [02/302]s AS4/3506-1 graphite/epoxy laminate. Axial modulus Ex, transverse modulus Ey, shear modulus Gxy and major Poisson’s ratio nxy normalised by their value for the undamaged laminates are plotted as a function of the relative delamination area Dld in Fig. 7.14(a). The axial/transverse shear-extension coupling coefficients that characterise shearing in the xy plane caused by respectively axial/transverse stress are plotted in Fig. 7.14(b). Matrix crack density in the 30∞ ply is assumed equal to Cmc = 2 cracks/cm, values at Dld = 0 indicate residual stiffness properties of the laminates at this crack density without delaminations. It can be seen that reduction of the laminate moduli and, for the considered lay-up, increase the Poisson’s ratio due to local delaminations are more significant in the unbalanced [02/302]s laminate than in the balanced [02/302/– 302]s laminate with the same orientation as the damaged ply. Matrix cracking and crack tip delaminations are expected to amplify the shear-extension coupling exhibited in the undamaged unbalanced [02/q2]s laminates. As in balanced [02/q2/– q2]s laminates, crack tip uniform local delaminations in unbalanced laminates result in an increase in the absolute value of the axial shear-extension coupling coefficient for q < 45∞ and of the transverse shear-extension coupling coefficient for q > 45∞.

Modelling damage in laminate composites

239

Normalised stiffness property

1

0.8

0.6

Axial modulus (experiment) Axial modulus (prediction) Shear modulus Poisson’s ratio

0.4 0

5 10 Crack density (cracks/cm) (a)

15

Normalised stiffness property

1

0.8 Axial modulus (experiment) Axial modulus (prediction) Shear modulus Poisson’s ratio

0.6

0.4

0.2 0

5 10 Crack density (cracks/cm) (b)

15

Normalised stiffness property

1

0.8

Axial modulus (experiment) Axial modulus (prediction) Shear modulus Poisson’s ratio

0.6

0.4

0.2 0

5 10 Crack density (cracks/cm) (c)

15

7.12 Normalised stiffness properties of T800H/3631 cross-ply laminates as a function of crack density Cmc: (a) [0/902]s; (b) [0/904]s; (c) [0/906]s. Transverse delamination area Dld = 10%.

240

Multi-scale modelling of composite material systems

Normalised stiffness property

1.3

Axial modulus Transverse modulus Shear modulus Poisson’s ratio

1.1

0.9

0.7

Shear extension coupling coefficients

0

30 60 Relative delamination area (%) (a)

90

0.07 Axial Transverse 0.05

0.03

0.01

– 0.01 0

30 60 Relative delamination area (%) (b)

90

7.13 Stiffness properties of AS4/3506-1 [02/302/–302]s laminate as a function of relative delamination area Dld: (a) normalised moduli and Poisson’s ratio; (b) shear-extension coupling coefficients. Matrix crack density Cmc = 2 cracks/cm.

7.4

Predicting onset and growth of intra- and interlaminar damage

7.4.1

Strain energy release rate for off-axis ply cracking and crack-induced delamination

The concept of the ‘equivalent’ laminate can be used to calculate strain energy release rates for intra- and interlaminar damage modes. The total strain energy release rate G associated with a particular damage mechanism is equal to the first partial derivative of the total strain energy U stored in the

Modelling damage in laminate composites

Normalised stiffness property

1.5

241

Axial modulus Transverse modulus Shear modulus Poisson’s ratio

1.3

1.1

0.9

0.7

0.5

Shear extension coupling coeffcients

0

30 60 Relative delamination area (%) (a)

90

–0.2

–0.4

–0.6

– 0.8 Axial Transverse –1

–1.2 0

30 60 Relative delamination area (%) (b)

90

7.14 Stiffness properties of AS4/3506-1 [02/302]s laminate as a function of relative delamination area Dld: (a) normalised moduli and Poisson’s ratio; (b) shear-extension coupling coefficients. Matrix crack density Cmc =2 cracks/cm.

damaged laminate with respect to the total damage area for this damage mode provided the applied strains {e } are fixed and the areas covered by other damage modes remain unchanged G = – ∂U ∂A

{e }

7.19

The strain energy release rates Gmc and Gld associated respectively with matrix cracks and local delaminations growing from the tips of matrix cracks can be effectively calculated using the ‘equivalent’ laminate, in which the

242

Multi-scale modelling of composite material systems

damaged ply is replaced with an equivalent constraint layer with degraded stiffness properties. In the global co-ordinates, the total strain energy stored in the laminate element with a finite gauge length L and width w is U = wL S ( z k – z k –1 )({e } + {e kthermal} + {e khygro}) T [ Q ] k ({e } 2 k + {e kthermal} + {e khygro})

7.20

where {e kthermal} and {e khygro} are respectively residual thermal and residual hygroscopic strains in the laminate due to the temperature and moisture difference between the stress-free and actual state, and [ Q ] k is the in-plane reduced stiffness matrix of layer k in the global co-ordinates. Since the area of a single crack is equal to amc = 2h2w/| sin f |, the total area covered by all cracks is equal to Amc = amcCmcL = LwDmc/| sin f |

7.21

Likewise, since the area of a single crack tip delamination is equal to ald = 2lw/| sin f | (Fig. 7.3) the total delamination area is equal to Ald = 2ald CL = 2LwDld/| sin f |

7.22

If hygrothermal effects are neglected, the strain energy release rate for matrix cracking and crack-tip delaminations, calculated from eqns 7.19–22, are G mc ( e , D mc ) = – h2 {e }T G ld ( e , D mc , D ld ) = –

∂[ Q ] 2 {e }| sin f | ∂D mc

h2 ∂[ Q ] 2 {e }T {e }| sin f | 2 ∂D ld

7.23a

7.23b

Under uniaxial strain, eqns 7.23 simplify to 2 G mc ( e xx , D mc ) = – h2 e xx

G ld ( e xx , D mc , D ld ) = –

∂Qxx ,2 | sin f | ∂D mc

h2 2 ∂Qxx ,2 e | sin f | 2 xx ∂D ld

7.24a

7.24b

Calculation of the in-plane axial stiffness Qxx ,2 using eqn 7.16 and the transformation formulae given by eqn 7.18, yields the strain energy release rates in terms of the in-situ damage effective functions (IDEFs) L22, L66 and stiffness properties of the undamaged material Qˆ ij(2) as follows: for off-axis ply cracking

Modelling damage in laminate composites

243

È Ê Qˆ (2)2 (2) 2 G mc ( e xx , D mc ) = h2 e xx sin 2 f cos 2 f Í Á 12(2) cos 4 f + 2 Qˆ 12 ˆ ÍÎ Ë Q22

7.25a

(2) + Qˆ 22 sin 4 f

) ∂∂DL

22 mc

(2) + 4 Qˆ 66 sin 2 f cos 2 f

∂L 66 ∂D mc

˘ ˙˚ |sin f |

for crack-induced delamination G ld ( e xx , D mc , D ld ) =

(2)2 h2 2 È Ê Qˆ 12 (2) e xx Í Á (2) cos 4 f + 2 Qˆ 12 sin 2 f cos 2 f 2 ˆ ÍÎ Ë Q22

)

∂L 66 ∂L 22 (2) (2) + Qˆ 22 sin 4 f + 4 Qˆ 66 sin 2 f cos 2 f ld ∂D ∂D ld

˘ ˙˚ |sin f |

7.25b

The first partial derivatives of IDEFs that appear in eqns 7.25 are explicit functions of the damage parameters Dmc and Dld and can be calculated analytically. For local delaminations growing uniformly from the tips of matrix cracks, O’Brien (1985) suggested a simple closed-form expression for the strainenergy release rate, based on simple load shearing rules and classical laminated plate theory. It gives the strain energy release rate that depends only on the laminate lay-up and thickness, the location of the cracked ply and subsequent delaminations, the applied load and the laminate width, and is independent of delamination size and matrix crack density. In the nomenclature of this chapter it is given by 2 ˆ 1 G ld = NEˆ x h Ê – 1 ˜ Á 2 2 m Ë ( N – n ) Eˆ ld e xx NEˆ x ¯

7.26

where h is the laminate thickness, N is the number of plies, n is the number of cracked plies, Eˆ x and Eˆ ld are respectively the laminate modulus and the modulus of the locally delaminated sublaminate as calculated from the laminated plate theory. Parameter m has a value of 2 if the cracked ply is in the interior of the laminate, corresponding to local delamination on either side of matrix crack and that of 1 if the cracked ply is a surface ply. Later, O’Brien (1991) showed that this simple closed-form expression is valid for the total strain energy release rate associated with uniform local delamination growing from an angle ply matrix crack. It is worth noticing that the strain energy release rate given by eqn 7.26 is independent of the delamination size. Also, the effect of matrix cracking is not taken into account when calculating the laminate modulus Eˆ x . Nairn and Hu (1992b) established that O’Brien’s expression for strain energy release rate applies only to

244

Multi-scale modelling of composite material systems

12

4

3

2 45 60 75 90 1 0

1 2 3 4 5 Crack density (cracks/cm) (a)

Normalised strain energy release rate (MJ/m2)

Normalised strain energy release rate (MJ/m2)

delaminations induced by isolated matrix cracks, i.e., when crack density is very small and the influence of neighbouring cracks is negligible. For crack densities, at which delaminations are observed to initiate, strain energy release rate depends both on delamination area and crack density. It is worth noting that the expressions derived in this chapter, eqns 7.25, give strain energy release rates that depend on crack density and, for delaminations, also on delamination area. 2 for offFigure 7.15 shows normalised strain energy release rate Gmc/ e xx axis ply cracking, calculated using eqn 7.23b, as a function of crack density Cmc for respectively carbon/epoxy and glass/epoxy [0/q]s laminates. Results are presented for four ply orientation angles: 45∞, 60∞, 75∞ and 90∞. It may 2 decreases faster be seen that in carbon/epoxy angle-ply laminates Gmc/ e xx with the crack density than in a cross-ply [0/90]s laminate. For smaller crack 2 in a [0/75]s carbon/epoxy laminate are higher densities, the values of Gmc/ e xx than in a cross-ply laminate (Fig. 7.15(a)). It is also worth noticing that the glass/epoxy angle-ply laminates exhibit significantly higher levels of normalised strain energy release rates for matrix cracking than carbon/epoxy ones (Fig. 7.15(b)).

8

4 45 60 75 90 0

0

1 2 3 4 5 Crack density (cracks/cm) (b)

2 7.15 Normalised strain energy release rate Gmc/ e xx for matrix cracking in a [0/q]s laminate as a function of crack density Cmc in the q layer: (a) carbon/epoxy laminate; (b) glass/epoxy laminate.

2 Figure 7.16 shows the normalised strain energy release rate Gld/ e xx , calculated from eqn 7.23b, as a function of the delamination length normalised by the single ply thickness l/t. The laminate lay-up is [02/252/–252]s, and crack half-spacings are s = 40t and s = 20t. This is equivalent to the crack

Modelling damage in laminate composites

Normalised strain energy release rate (MJ/m2)

0.38

245

s = 40 t s = 20 t

0.36

0.34

0.32

0.3

0

0.8 1.6 2.4 3.2 Normalised delamination width

4

2 7.16 Normalised strain energy release rate Gld/ e xx for uniform local delamination in a cracked [02/252/–252]s AS4/3506-1 laminate as a function of normalised delamination length l/t. Matrix crack spacing s = 20t and s = 40t.

densities of approximately Cmc = 1 cm–1 and Cmc = 2 cm–1 respectively. It can be seen that the present approach gives the strain energy release rate for uniform local delamination that depends both on crack density and delamination length. The result of eqn 7.26 for the same lay-up is found equal to 12.7 MJ/ m2 provided shear-extension coupling and bending-extension coupling are taken into account (O’Brien, 1991). Still, it is much higher than our predictions, since the model of eqn 7.26 is for a single isolated matrix crack and associated local delamination and does not account for the cumulative effect of multiple cracking and local delaminations as illustrated in Fig. 7.2. 2 Figure 7.17 shows the normalised strain energy release rate Gld/ e xx associated with uniform local delaminations induced by off-axis ply cracking in a graphite/epoxy [02/q2]s laminate. Dependence on the relative delamination area Dld is shown in Fig. 7.17(a), and on the crack density in Fig. 7.17(b). Results are presented for four different ply orientations angles: 45∞, 60∞, 75∞ and 90∞. It may be seen that strain energy release rate non-linearly depends on delamination area and almost linearly on the crack density. While in a 2 cross-ply [02/902] laminate the value of G ld / e xx | D ld =0 at the delamination onset is almost independent of crack density, in [02/q2]s it strongly depends on it. Also, in [02/602]s and [02/752] laminates it is significantly higher than in a cross-ply [02/902] laminate, suggesting lower delamination onset strains.

Multi-scale modelling of composite material systems 2.5

45 60 75 90

2

1.5

1

0.5

0

0 20 40 60 80 100 Relative delamination area (%) (a)

Normalised strain energy release rate (MJ/m2)

Normalised strain energy release rate (MJ/m2)

246

2.5 45 60 75 90 2

1.5

0 0

1 2 3 4 Crack density (cracks/cm) (b)

5

2 7.17 Normalised strain energy release rate Gld/ e xx for uniform local delamination in a cracked AS4/3506-1 [02/q2]s laminate: (a) as a function of relative delamination area Dld (crack density 1 crack/cm); (b) as a function of crack density Cmc in the q-ply (relative delamination area Dld = 0, i.e. onset of delamination).

7.4.2

Predicting onset and growth of off-axis ply cracking

Even under the uniaxial loading, damage development in the off-axis plies of general symmetric laminates always occurs under mixed mode conditions due to shear-extension coupling. It is therefore important in the calculation of the total strain energy release rate to be able to separate Mode I and Mode II contributions. For a [(S)/f]s laminate with damaged f-layer modelled by an ‘equivalent’ laminate, the total strain energy release rate for off-axis ply cracks and crack-induced local delaminations is equal to the first partial derivative of the portion of the total strain energy stored in the ‘equivalent’ homogeneous layer with respect to damage area

G mc = – ∂U mc ∂A

(2)

G ld = – ∂U ld ∂A

7.27a {e }

(2)

7.27b {e }, C

In the local co-ordinates (Fig. 7.3), this portion of the total strain energy can be separated into extensional and shear parts

Modelling damage in laminate composites

247

(2) (2) (2) 2 (2) 2 U (2) = U I(2) + U II(2) = Lwh2 (s 11 e 11 + s 22 e 22 ) + Lwh2 s 12 g 12 7.28

Under uniaxial strain e xx , strains and stresses in the ‘equivalent’ homogeneous layer are {e (2)} = {cos2 f, sin2 f, 2 cos f sin f}T e xx , {s (2)} = [Q(2)]{cos2 f, sin2 f, 2 cos f sin f}T e xx

7.29

where the modified stiffness matrix [Q(2)] of the ‘equivalent’ homogeneous layer in the local co-ordinates is given by eqn 7.12. Substitution of eqns 7.22, 28 and 29 into eqn 7.27 gives Mode I and Mode II contributions into the total strain energy release rate as follows: For off-axis ply cracking GImc = –

∂U I(2) 2 f1mc ( D mc ) = e xx ∂A mc

7.30a

ˆ ∂L(2) Ê Qˆ (2)2 (2) (2) 22 f Imc ( D mc ) = h2 Á 12(2) cos 4 f + 2 Qˆ 12 sin 2 f cos 2 f + Qˆ 22 sin 4 f ˜ |sin f | mc ˆ ∂ D ¯ Ë Q22

7.30b GIImc = –

∂U II(2) 2 f 2mc ( D mc ) = e xx ∂A mc

(2) ∂L 66 f 2mc ( D mc ) = 4 h2 Qˆ 66 cos 2 f |sin 3 f | ∂D mc for crack-induced delaminations

7.31a

(2)

GIld = –

∂U I(2) 2 = e xx f1 ( D ld ) ∂A ld

7.31b

7.32a

f1(Dld)

=

(2)2 ˆ (2) h2 Ê Qˆ 12 4 ˆ (2) sin 2 f cos 2 f + Qˆ (2) sin 4 f ∂L 22 |sin f | cos f + 2 Q 12 22 ˜ ∂D ld 2 ÁË Qˆ (2) ¯ 22

7.32b GIIld = –

∂U II(2) 2 = e xx f 2 ( D ld ) ∂A ld

7.33a

(2) ∂L 66 f 2 ( D ld ) = 2 h2 Qˆ 66 cos 2 f |sin 3 f | 7.33b ∂D ld These expressions can be used with appropriate fracture criteria to estimate (2)

248

Multi-scale modelling of composite material systems

the onset of local delamination in an already cracked laminate. The resulting total strain energy release rate G ld = GIld + GIIld coincides with eqn 7.25. To predict the development of the off-axis ply cracks in the q-layer of a [0/q]s laminate a mixed mode fracture criterion is suggested M N Ê G1 ˆ + Ê GII ˆ = 1 Ë GIC ¯ Ë GIIC ¯

7.34

where GIC and GIIC are respectively Mode I and Mode II interlaminar fracture toughnesses. The exponents M and N depend on the material system. Following Rikards et al. (1998), for a glass/epoxy system they can be taken as M = 1, N = 2. Figure 7.18 shows predicted and experimentally observed cracking onset strains for [0/q]s glass/epoxy laminates. Crocker et al. (1997) measured cracking onset strains in specimens with as-cut, polished and notched edges. At that, an independence of strain at onset of crack propagation in notched samples on the notch depth was observed. To predict cracking onset strains, GImc and GIImc values are calculated from eqns 7.30 and 7.31, and cracking onset strain e xx is found as a root of the following equation Ê e xx4 Á

f 2mc ( D mc ) ˆ ˜ Ë GIIC ¯

2

mc mc mc 2 Ê f1 ( D ) ˆ + e xx Á ˜ = 1 when D Æ 0 7.35 GIC Ë ¯

3

Cracking onset strain (%)

As-cut edges Polished edges Notched edges Prediction 2

1

0 45

60 75 Ply orientation angle (degrees)

90

7.18 Off-axis ply cracking onset strain as a function of ply orientation angle q in glass/epoxy [0/q]s laminates.

Modelling damage in laminate composites

249

Since the exact GIC and GIIC critical values for the considered glass/epoxy system are not known, predictions are made using typical for glass/epoxy systems values of GIC = 200 J/m2 and GIIC = 1500 J/m2. Comparison with limited experimental data shows that the mixed mode fracture criterion, eqn 7.9, can successfully predict the initiation of matrix cracking for ply orientation angles 75∞ £ q £ 90∞. For 45∞ £ q £ 75∞, measured strains are much higher than predictions. Also, they increase steeply as q decreases. Further work is required to develop an appropriate fracture or failure criterion that captures initiation and development of matrix cracks in off-axis plies of composite laminates reinforced by glass or carbon fibres, especially for q < 75∞. Further work is required to validate theoretical predictions. For the layups, damage modes and loading conditions examined in this chapter the experimental data are currently not available.

7.5

Conclusions

The fracture process of composite laminates subjected to static or fatigue tensile loading involves a sequential accumulation of intra- and interlaminar damage, in the form of transverse cracking, splitting and delamination, prior to catastrophic failure. Matrix cracking parallel to the fibres in the off-axis plies is the first damage mode observed. It triggers development of other harmful resin-dominated modes such as delaminations. Since a damaged lamina within the laminate retains a certain amount of its load-carrying capacity, it is important to predict accurately the stiffness properties of the laminate as a function of damage as well as progression of damage with the strain state. Incorporation of analytical models of stiffness degradation and damage progression into a finite element code will constitute the most effective tool for progressive failure modelling of composite plates with more complex configurations, e.g., holes, notches and other stress concentrators. In multidirectional laminates subjected to in-plane tensile or thermal loading matrix cracks parallel to the fibres develop in several off-axis plies. Comprehensive observations of sequential accumulation of matrix cracks in quasi-isotropic and balanced carbon/epoxy and glass/epoxy laminates under quasi-static and fatigue tensile loading have been extensively reported in the literature. Concurrent matrix cracking in the adjacent off-axis plies is an extremely complex problem to model and has been analysed in the literature mostly using finite elements method. A theoretical model that describes damage development under complex loading conditions does not yet exist. Here, analytical modelling of off-axis ply cracking and crack tip delaminations in balanced and unbalanced angle-ply composite laminates subjected to in-plane tensile loading is presented and discussed. A 2-D shearlag analysis is used to determine ply stresses in a representative segment and the equivalent laminate concept is applied to derive expressions for Mode I,

250

Multi-scale modelling of composite material systems

Mode II and the total strain energy release rate associated with uniform local delaminations. These expressions can be used with appropriate fracture criteria to estimate the onset and growth of damage in off-axis plies. To calculate strain energy release rates for off-axis ply cracking and uniform local delaminations growing along matrix cracks in balanced and unbalanced angle-ply laminates, the damaged layer of the laminate is replaced with an equivalent homogeneous one with effective elastic properties. Closed form expressions for strain energy release rate associated with matrix cracking and crack induced uniform local delaminations have been derived, representing them as linear functions of the first partial derivatives of the effective elastic properties of the damaged layer with respect to appropriate damage parameters. Dependence of strain energy release rates and the laminate stiffness properties on delamination area, crack density and ply orientation angle has been examined. It appears that matrix cracking and delamination area influence the strain energy release rate value significantly. Comparison with results obtained by O’Brien (1991) shows that O’Brien’s closed-form expression for uniform local delamination significantly overestimates the value of the total strain energy release rate leading to lower theoretical strains for the initiation of local delamination and therefore overconservative designs. Also, it gives the total strain energy release rate as independent of delamination area and does not take into account the cumulative effect of damage. It is found, in particular, that the reduction due to matrix cracking of the laminate axial and transverse moduli is more significant in angle-ply than in cross-ply laminates, while for the shear modulus, the opposite is true. Matrix cracking in angle-ply CFRP and GFRP laminates can result in an increase in the Poisson’s ratio – a phenomenon not observed in cross-ply laminates. Also, matrix cracking in angle-ply laminates introduces coupling between extension and shear. In near future work, the analytical predictions will be compared to numerical (finite element) and experimental data, for which the lay-ups, damage modes and loading conditions examined in this study are currently not available.

7.6

Acknowledgements

Financial support for this research by the Engineering and Physical Sciences Research Council (EPSRC/GR/L51348 and EPSRC/GR/A31001/02) and the British Ministry of Defence is gratefully acknowledged.

7.7

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7.8

Multi-scale modelling of composite material systems

Appendices

Appendix I Variation of the out-of-plane shear stresses has the form

s (2) j3 =

tj x , 9 £ | x3 | £ h2, j = 1, 2 h2 3

7.A1a

s (1) j3 =

tj ( h – x 3 ) , h 2 £ | x3 | £ h h1

7.A1b

Constitutive equations for the out-of-plane shear stresses (k) (k) Ïs 13 ¸ È Q55 Ì (k) ˝ ª Í (k) Ós 23 ˛ Î Q45

(k) ˘ ∂ Ïu1( k ) ¸ Q45 , i = 1, 2 ( k ) ˙ ∂x Ì ( k ) ˝ Q44 ˚ 3 Óu2 ˛

7.A2

After substituting eqn 7.A2 into eqns 7.A1, multiplying them by x3 and by h – x3 respectively and integrating with respect to x3 we get h1 3

(1) Ï t 1 ¸ È Qˆ 55 = Ì ˝ Í ˆ (1) Ót 2 ˛ ÍÎ Q45

h2 3

(2) Ï t 1 ¸ È Qˆ 55 Ì ˝= Í Ót 2 ˛ ÍÎ 0

(1) ˘ Ê Ïu˜1(1) ¸ Ï V1 ¸ˆ Qˆ 45 – Ì ˝˜ (1) ˙ Á Ì (1) ˝ Qˆ 44 ˚˙ Ë Óu˜ 2 ˛ ÓV2 ˛¯

0 ˘ ˙ (2) Qˆ 44 ˙˚

7.A3a

Ê Ï V1 ¸ ÏÔu˜1(2) ¸Ôˆ Á Ì ˝ – Ì (2) ˝˜ Ë ÓV2 ˛ ÔÓu˜ 2 Ô˛¯

7.A3b

Here {V } = {u (1) }| x 3 = h2 = {u (2)}| x 3 =h2 are the in-plane displacements at the interface. After rearranging eqns 7.A3 become

Ïu˜1(1) ¸ Ïu˜1(2) ¸ Ê h1 Ì (1) ˝ – Ì (2) ˝ = ÁÁ 3 Óu˜ 2 ˛ Óu˜ 2 ˛ Ë

(1) (1) ˘ È Qˆ 55 Qˆ 45 Í ˆ (1) ˆ (1) ˙ ÍÎ Q45 Q44 ˙˚

–1

h È Qˆ (2) 0 ˘ + 2 Í 55 3 Í 0 Qˆ (2) ˙˙ 44 ˚ Î

–1

ˆ Ït ¸ 1 ˜Ì ˝ ˜ Ót 2 ˛ ¯ 7.A4

Inversion of eqn 7.A4 leads to

Ï t 1 ¸ È K11 Ì ˝= Í Ót 2 ˛ Î K 21

K12 ˘ ˙ K 22 ˚

Ê ÏÔu˜1(1) ¸Ô ÏÔu˜1(2) ¸Ôˆ Á Ì (1) ˝ – Ì (2) ˝˜ Ë ÓÔu˜ 2 ˛Ô ÓÔu˜ 2 ˛Ô¯

7.A5

with Êh [K] = Á 1 Á 3 Ë

(1) (1) ˘ È Qˆ 55 Qˆ 45 Í ˆ (1) ˆ (1) ˙ ÍÎ Q45 Q44 ˙˚

–1

h + 2 3

(2) È Qˆ 55 0 ˘ Í (2) ˙ ÍÎ 0 Qˆ 44 ˙˚

–1

ˆ ˜ ˜ ¯

–1

7.A6

Modelling damage in laminate composites

257

Appendix II On referring to the constitutive equations 7.A6, the generalised plane strain condition, eqn 7.5, becomes (1) (1) (1) (1) (1) (1) (2) (2) (2) (2) s˜ 11 + Sˆ12 s˜ 22 + Sˆ16 s˜ 12 = Sˆ11 s˜ 11 + Sˆ12 s˜ 22 Sˆ11

7.B1

Using the laminate equilibrium equations 7.4, stresses in the constraining (2) layer (layer 1) can be excluded, so that the microstress component s˜ 11 is given by (2) (2) (2) s˜ 11 = a 22 s˜ 22 + a12 s˜ 12 + bx s x + bys y

a 22 = –

bx =

(1) (1) (2) Sˆ16 Sˆ12 + cSˆ12 , a = – 12 (1) (2) (1) (2) Sˆ11 + cSˆ11 Sˆ11 + cSˆ11

(1) (1) (1) (1 + c )( Sˆ11 cos 2 f + Sˆ12 sin 2 f – Sˆ16 sin f cos f ) (1) (2) ˆ ˆ S + cS 11

by =

7.B2

11

(1) (1) (1) (1 + c )( Sˆ11 sin 2 f + Sˆ12 cos 2 f – Sˆ16 sin f cos f ) (1) (2) Sˆ + cSˆ 11

11

Strain differences are expressed in terms of stresses as (1) (2) ÏÔg˜ 12 ¸Ô – g˜ 12 1 È L11 = – Ì (1) ˝ (2) c ÍÎ L21 ÔÓ e˜ 22 – e˜ 22 Ô˛

L12 ˘ ˙ L22 ˚

(2) ÏÔs˜ 12 ¸Ô 1 È M11 Ì (2) ˝ + c Í ÔÓs˜ 22 Ô˛ Î M 21

M12 ˘ Ïs x ¸ ˙Ì ˝ M 22 ˚ Ós y ˛

7.B3 Here (1) (1) (1) L11 = Sˆ66 + a12 Sˆ16 + cSˆ66 (1) (1) L12 = Sˆ26 + a 22 Sˆ16

(1) (1) (2) L21 = Sˆ26 + a12 Sˆ12 + ca12 Sˆ12 (1) (1) (2) (2) L22 = Sˆ22 + a 22 Sˆ12 + c ( Sˆ22 + a 22 Sˆ12 ) (1) (1) (1) (2) M11 = (1 + c )[( Sˆ16 + a12 Sˆ11 ) cos 2 f + ( Sˆ26 + a12 Sˆ12 )sin 2 f

(1) (1) – ( Sˆ66 + a12 Sˆ16 ) sin f cos f ]

7.B4a

258

Multi-scale modelling of composite material systems (1) (1) (1) (1) M 21 = (1 + c )[( Sˆ12 + a 22 Sˆ11 ) cos 2 f + ( Sˆ22 + a 22 Sˆ12 )sin 2 f (1) (1) – ( Sˆ26 + a 22 Sˆ16 ) sin f cos f ] (1) (1) (1) (2) M12 = (1 + c )[( Sˆ16 + a12 Sˆ11 ) sin 2 f + ( Sˆ26 + a12 Sˆ12 ) cos 2 f

(1) (1) + ( Sˆ66 + a12 Sˆ16 ) sin f cos f ] (1) (1) (1) (1) M 22 = (1 + c )[( Sˆ12 + a 22 Sˆ11 ) sin 2 f + ( Sˆ22 + a 22 Sˆ12 ) cos 2 f (1) (1) + ( Sˆ26 + a 22 Sˆ16 ) sin f cos f ]

7.B4b

Substitution into the equilibrium equations 7.B3, yields the following coupled second-order differential equations (2) ÏÔs˜ 12 ¸Ô Ì (2) ˝ ÓÔs˜ 22 ˛Ô

d2 dx 2

È K11 – 1 Í h1 Î K 21

K12 ˘ Ê È L11 ˙Á Í K 22 ˚ Ë Î L21

(2) L12 ˘ ÏÔs˜ 12 ¸Ô È M11 ˙ Ì (2) ˝ + Í L22 ˚ ÓÔs˜ 22 ˛Ô Î M 21

M12 ˘ Ïs x ¸ˆ ˙ Ì ˝˜ = 0 M 22 ˚ Ós y ˛¯

7.B5 or (2) ÏÔs˜ 12 ¸Ô È N11 Ì (2) ˝ – Í ÓÔs˜ 22 ˛Ô Î N 21

d2 dx 2

(2) N12 ˘ ÏÔs˜ 12 ¸Ô È P11 ˙ Ì (2) ˝ + Í N 22 ˚ ÓÔs˜ 22 ˛Ô Î P21

P12 ˘ Ïs x ¸ ˙ Ì ˝ = 0 7.B6 P22 ˚ Ós y ˛

where [N] = h1–1 [K][L] and [P] = h1–1 [K][M], with matrices [K], [L] and [M] defined by eqns 7.A6, 7.B4a and 7.B4b respectively.

Appendix III. l12 – N 22 l 2 – N 22 C N + P21 + aP22 A2 , B1 = 2 B2 , C1 = – 2 22 N 21 N 21 N 21 7.C1

A1 =

A2 = –

( P21 + aP22 )( N 21 N12 – N11 N 22 ) + Rl 22 ( l 22 – l12 )( N12 N12 – N11 N 22 )

B2 =

( P21 + aP22 )( N 21 N12 – N11 N 22 ) + Rl12 ( l 22 – l12 )( N 21 N12 – N11 N 22 )

C2 =

R , R = N11(P21 + aP22) N 21 N12 – N11 N 22

– N21(P11 + aP12)

7.C2 7.C3

7.C4

8 Progressive multi-scale modelling of composite laminates C H W A N G, Defence Science and Technology Organisation, Australia

8.1

Introduction

Because composite structures offer a number of advantages over conventional metallic materials, including lightweight and better corrosion resistance, there is an increasing tendency in the development of new aircraft towards an even more widespread use of composites. For instance, Boeing’s 7E7 aircraft will contain 50% by weight of carbon fibre reinforced composites, while the US Air Force’s F-35 Joint Strike Fighter (JSF) stealth aircraft will contain 40% by weight of composites with most of its exterior functioning as load-bearing structures. Given the fact that there are numerous examples where composite materials are being successfully used in primary load-bearing structures, one might logically conclude that design procedures (including strength prediction) for fibre-reinforced composites are fully mature. To the contrary, even at the lamina or laminate level, there is a lack of evidence to show that any of the failure criteria developed so far could provide accurate and meaningful predictions of failure beyond a very limited range of circumstances [1]. Therefore it remains a persistent difficulty to accurately predict laminate failure under combined loading by using either unidirectional composite (ply) data or with basic constituent material properties. This has been the main motivation for a major world-wide failure exercise over the past decade [2]. To accelerate the insertion of composite materials/structures into application and to reduce the associated certification cost, aircraft manufacturers are moving towards certification by analysis supported by test and demonstration. To this end, an advanced predictive capability is required that is able accurately to assess the damage tolerance and durability of fibre-reinforced composites. In the context of through-life support and management of composite structures, a predictive capability would also improve the design and qualification of repairs. Because the failure process of composite structures involves failures at multiple length scales, including the microscopic scale (fibre, matrix and fibre/matrix interface), the mesoscopic scale (lamina, laminate), the macroscopic scale (structure), a predictive capability would need to deal 259

260

Multi-scale modelling of composite material systems

with the onset of microscopic failures, their progression to the mesoscopic scale, and the ultimate coalescence into macroscopic damage causing structural failure. For failures dominated by the initiation and growth of cracks and delaminations, a computational-based predictive method, GENOA-PFA [3], has been developed on the basis of a progressive-fracture methodology for composites [4]. Recently the strain-invariant failure theory (SIFT) [5] has emerged as a promising predictive method for composite laminates. Compared to the traditional composite failure theories, including the Tsai-Wu interactive failure theory [6], and the physically based failure criteria by Hashin [7] and Rotem [8], the SIFT method represents a first step towards multi-scale modelling that attempts to link structural failure to events at the fibre-resin level. In this context, SIFT is similar to the multi-continuum theory [9], although SIFT does appear to have the advantage that the critical strain invariants determined from lamina data were reported to correlate well with net resin properties [10]. This important feature of the SIFT provides a basis for an accelerated durability assessment methodology. In the strain-invariant failure theory, the matrix and fibre phases are characterised by separate failure criteria. Matrix is considered to fail either under dilatation mode or shear mode, while fibre is assumed to fail by shear distortion. Consequently, composite failure is completely characterised by three critical invariants. However, such an approach does not capture the micro-buckling failure mode under compression and therefore would overpredict the strength of compression failure which is often dominated by micro-buckling. Furthermore, SIFT was essentially developed to determine the onset of failures in composites, rather than the ultimate load-carrying capacity. Nevertheless, attempts have been made to extend SIFT to predict the maximum load-carrying capacity by using a maximum energy retention (MER) method [5]. The purpose of this chapter is to present a multi-scale failure theory that accounts for the pressure-sensitivity of polymer matrix [11, 12], fibre fracture under tension, and fibre micro-buckling under compression. Similar to the SIFT, the present approach treats a laminate as a stack of transversely isotropic plies. At the ply level, matrix failure is due to the combination of dilatation and distortion, while the fibres can fail either by micro-buckling or by distortion. Damage progression through plies is simulated by judiciously degrading the constituent stiffness, hence the ply stiffness.

8.2

Brief review of failure theories of fibre composites

Due to the complex structure of composite laminates, consisting of plies made of anisotropic fibres embedded in a polymeric resin, there are a number

Progressive multi-scale modelling of composite laminates

261

of complex failure modes, rendering the characterisation very difficult. For instance, a unidirectional laminate subjected to compressive loading may fail by fibre breakage, fibre micro-buckling, or transverse delamination. In this regard, extensive research has been devoted to developing predictive failure theories [1]. Early failure theories, such as the Tsai-Wu theory, treated the lamina as the basic building block. Consequently, a large number of mechanical properties are required to determine the coefficients of empirical failure models. Because a ply consists of two phases (fibres and resin) and can fail in several modes, the laminate properties are strongly dependent on stacking sequence, thickness, and operating temperature. Furthermore, it is extremely difficult to extend the conventional laminate failure theories to account for time-dependent failure mechanisms, such as creep and exposure to hot/wet environment. Recognising the limitations of the conventional laminate failure theories, a number of researchers have attempted to develop failure criteria that separately model the failure modes of the matrix and the fibres, including the Hashin criterion [7] and the Rotem criterion [8]. Both these models use the average stresses in the fibre phase and the matrix phase in determining the onset of initial failure. Stiffness degradation is then required to simulate the fibrebridging effect in cross-ply laminate. Instead of the average stresses as in the Hashin and Rotem criteria, the multi-continuum theory [9, 13] employs the phase-averaged stresses, which take into account the stress amplification effect due to fibre reinforcement in a lamina. By contrast, the strain-invariant failure theory (SIFT) developed by HartSmith [14] and Gosse [5] proposed that fibre failure and matrix failure are characterised by the maximum strain invariants, not the average stresses or strains as in the multi-continuum theory. In SIFT, fibres are assumed to fail by shear, while the matrix fails by dilatation or shear distortion. Consequently the failure of a lamina is completely determined by three material constants. It is worth noting that micro-buckling of fibres under compression is not considered. Furthermore, the use of strains instead of stresses makes the analysis of thermal loading rather complicated, due to the need to determine the so called ‘effective mechanical strain’ and magnification factors relating to thermal expansion.

8.3

Multi-scale failure theory

8.3.1

Stress-invariant failure criteria

One common feature between the multi-continuum theory and the straininvariant failure theory is the use of micro-mechanics in determining the stresses within a lamina or ply, which is modelled by a periodic fibre/matrix structure, either of diamond pattern or a hexagonal pattern, as shown in Fig.

262

Multi-scale modelling of composite material systems

8.1. The failure of a fibre-composite material lamina will occur either in the fibres or in the matrix, each having two failure modes.

(b)

(a)

8.1 Three-dimensional unit cell for fibre reinforced composites (60% volume fraction): (a) diamond configuration; and (b) hexagonal configuration.

Fibre failure Under compressive loading, fibres would normally fail by micro-buckling, which can be described as

s 1( f ) £ s c( f ) , s 1( f ) £ 0

8.1

where s c( f ) denotes the critical micro-buckling strength of fibres embedded in a matrix subjected to compression. Theoretical estimates of the buckling strength corresponding to plastic kinking can be found in ref. 15. For practical purposes, the critical buckling strength can be measured experimentally by using a standard test configuration [16]. When the fibres are under tensile loading, the conventional approach is to assume that the failure is dictated by the tensile stress along the fibre direction. However, due to the strong anisotropy of carbon fibres, such an approach would neglect the effects of lateral stresses. In the absence of a validated anisotropic failure theory for carbon fibres, the shear distortion criterion [14] provides a good alternative. The relevant failure criterion can be expressed as, (f) e vM ≥ e c( f ) , with

(f) e vM = 1 2

(e1 – e 2 ) 2 + (e 2 – e 3 ) 2 + (e 3 – e1 ) 2

where e c( f ) denotes the critical von Mises strain for the fibres.

8.2

Progressive multi-scale modelling of composite laminates

263

Matrix failure For matrix made of polymeric resin, there are two possible failure modes: dilatational and distortional. The dilatational failure mode can be simply expressed as

p ≥ pc( m ) , p ≥ 0,

8.3

where p = (s1 + s2 + s3)/3 and pc( m ) denotes the material’s critical hydrostatic stress. To account for the pressure sensitivity of polymeric matrix in the shear distortion failure mode [12], the modified von Mises yield criterion will be adopted [11, 12] (as illustrated in Fig. 8.2),

s vm + m p ≥ s c( m )

8.4

where denotes the critical von Mises stress for the matrix, m the pressure sensitivity of the matrix. In the special case of equal triaxial tension (s1 = s2 = s3 > 0), consistency between eqns 8.3 and 8.4 implies that (m) s vmc

m = s c( m ) / pc( m ) .

8.5

Therefore only four material properties (s c( f ) , e c( f ) , pc( m ) , and s c( m ) ) are required to completely describe fibre failure and matrix failure. To determine the stresses in the fibres and in the matrix, a two-step analysis is required. Firstly the stresses (or strains) for a lamina are determined by using the laminate theory. Then a micro-mechanics analysis is performed by using the finite element method to solve for the fibre and matrix stresses where the strains determined in the first step serve as the boundary conditions. Von Mises shear stress svM Shear distortion

s c(m ) Pressure sensitive svm + mcp = s c(m )

Pressure sensitive svm + mtp = s c(m )

Dilatation

Hydrostatic stress sm

8.2 Matrix failure criterion.

p c(m )

264

8.3.2

Multi-scale modelling of composite material systems

Laminate analysis

Consider a laminate, which has undergone a heating-curing-cooling process, is operating at temperature To, the thermal strains in a ply orientated at angle q to the x-axis of the global coordinate are, denoting the room temperature and the curing temperature respectively as Tr and Tc, T {e }12 = {a P }( Tc – Tr ) + [q ] – T {a L }( To – Tc ),

8.6

where the subscripts 12 denote quantities in the local material coordinate (fibre direction = 1, lateral direction = 2), the parameter a the coefficient of thermal expansion. The subscripts P and L signify quantities pertinent to the ply and laminate, and [q]–T the transfer matrix given by

[q ]

–T

È m2 Í = Í n2 Í Î –2mn

n2 m

2

2mn

˘ ˙ – mn ˙ , m = cos(q ), n = sin(q ). ˙ m2 – n2 ˚ mn

8.7 •

Assuming the applied stresses are {s} , the total strains in a ply are given by T {e }12 = {e }12 + [q ] – T [ S L ]{s }•

8.8

Then the stresses in the ply are {s}12 = [QP] ({e}12 – {aP}(To – Tr)) = [QP][q]–T [SL]{s}• + [QP]([q]–T{aL} – {aP}) (T0 – Tc) 8.9 where [QP] and [SL] denote respectively the ply stiffness and the laminate compliance.

8.3.3

Micro-mechanics solutions

For either of the unit cells shown in Fig. 8.1, there is a substantial nonuniformity in the matrix stress, due to the disparity in stiffness. An example is shown in Fig. 8.3, which depicts the stress distribution due to an applied stress s2 = 1. The maximum fibre stresses {sf} and the maximum matrix stresses {sm} are related to the ply stresses, {sm} = [Am] {s}12

8.10

{sf} = [Af]{s}12

8.11

where matrices [Am] and [Af] denote the amplification factors, which can be determined by using the finite element method for a given fibre volume fraction and fibre-matrix combination. The micromechanics model is based

Progressive multi-scale modelling of composite laminates

265

1.25+00 1.19+00 1.14+00 1.08+00 1.02+00 9.66–01 9.09–01 8.52–01 7.95–01 7.38–01 6.81–01 6.24–01 5.67–01 5.11–01

Y Z

4.54–01 X

3.97–01 7.00–01 6.27–01 5.53–01 4.80–01 4.07–01 3.33–01 2.60–01 1.87–01 1.13–01 4.00 –02 –3.33–02 –1.07–01 –1.80–01 –2.53–01

Y Z

–3.27–01 X

–4.00–01

8.3 Stress distributions in a carbon/epoxy lamina subjected to a unit applied x-stress: (a) sxx and (b) syy.

on an assumed periodic fibre configuration within the lamina’s matrix, either in the form of repeating diamond or hexagonal unit cell shown in Fig. 8.1. To enforce compatibility of the unit cell boundaries, i.e., boundaries remain plane after deformation, the nodes along the unit cell boundaries are (i)

266

Multi-scale modelling of composite material systems

constrained to undergo the same displacement normal to the boundary and (ii) subjected to zero shear tractions. An alternative method to obtain the stress amplification factors is to determine the strain amplification factors defined as {em} = [Mm]{e}12

8.12

{ef} = [Mf]{e}12

8.13

where only displacement boundary conditions are required at the unit cell boundaries. It can be readily shown that the stress magnification factors are related to the strain magnification factors via the following relationship, [Am] = [Qm][Mm][SP]

8.14

[Af] = [Qf][Mf][SP]

8.15

where [Qm] and [Qf] denote the stiffness matrix of the matrix and the fibres, while [SP] denotes the compliance matrix of the pertinent unit cell. Values of the strain magnification factor matrices for the hexagonal unit cell configuration are given in Appendix II.

8.4

Phase degradation approach

The failure criteria presented in section 8.3.1 are valid only for a lamina. For a composite laminate, the failure of a single lamina may not cause the rupture of the laminate, as the failed lamina may be bridged by other plies. Therefore the ultimate failure strength of a laminate may be significantly higher than the initial failure strength of the most highly stressed lamina in the laminate. One efficient approach to account for this damage progression through the laminate is to adopt a phase degradation approach in which the fibre stiffness or the matrix stiffness is reduced in accordance with failure modes; longitudinal fibre stiffness is reduced to zero should fibre fracture occur, whereas the matrix shear stiffness is reduced if matrix failure is detected. Ply stiffness is then re-calculated by using the modified rule of mixture [17]; details will be presented in the following. As the damage progresses through the laminate, eventual failure will occur due to either excessive deformation or insufficient load-carrying capacity (i.e., failures have completely progressed through all the laminae). In the case of fibre failure, resulting from either shear distortion or compressional micro-buckling, the longitudinal stiffness of the fibre phase is assumed to degrade to zero. In other words, the effective longitudinal stiffness can be expressed as ÔÏ E f 1 E *f 1 = Ì ÔÓ 0

(f) < e c( f ) or s 1( f ) > s c( f ) e vM

otherwise

8.16

Progressive multi-scale modelling of composite laminates

267

For the matrix, it is necessary to account for the relatively high degree of plastic deformation. In this case, the effective shear modulus is taken to be equal to the secant shear stiffness before the shear strain reaches a critical value, as described by the following relation,

Ï Gm Ô (m) (m) Gm* = Ìt oct / g oct Ô 0 Ó

s vm + m p < s c( m ) (m) £ g c( m ) s vm + m p ≥ s c( m ) and g oct

8.17

otherwise

(m) (m) where t oct and g oct denote the octahedral shear stress and shear strain in the matrix. The value of g c( m ) for a polymeric matrix material can be readily obtained through the in-plane shear stress/strain data. As an example, for the four fibre/epoxy laminates employed in the world-wide failure exercise study, the values of g c( m ) range between 2% to 4%, as listed in Table 8.5. Now the ply properties can be expressed in terms of the fibre and matrix properties via the modified rule of mixture [17], * E11 = V f E *f 1 + (1 – V f ) E m* * * E 22 = E33 = 2(1 + n m ) Gm*

8.18

1 + x1h1V f E f 2 / Em – 1 , with h1 = 1 – h1V f E f 2 / E m + x1

8.19 * G12 = Gm*

1 + x2h2 V f G f 12 / Gm – 1 , with h 2 = 1 – h2 V f G f 12 / Gm + x 2

* n 12 = V f n f 12 + (1 – V f )n m

* n 23 = nm

n f 23 / n m – 1 1 + x 3h 3 V f , with h 3 = 1 – h3 V f n f 23 / n m + x 3

8.20 8.21 8.22

where x1, x2, and x3 need to be determined so that predictions of the modified rule of mixture match the ply properties in the absence of damages. It is interesting to note that both the major Poisson’s ratio and the through-thickness Poisson’s ratio are not affected by damage, while the longitudinal, transverse, and in-plane shear stiffness are strongly dependent on the degree of damage in the fibre and the matrix.

8.5

Validation of analysis against experiment

To verify the accuracy and robustness of the proposed model, a critical comparison will be made between the model prediction and the extensive experimental data reported in the world-wide failure exercise [18]. Based on the properties of the fibres, matrices, and the laminae [19], which are summarised in Tables 8.1–8.3 for completeness, the strength properties of

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Multi-scale modelling of composite material systems

Table 8.1 Fibre properties [19] Fibre type

Carbon fibre AS4

Carbon fibre T300

E-Glass fibre 21 ¥ K43

E-Glass fibre 1200 tex

Longitudinal modulus Ef 1 (GPa)

225

230

80

74

Transverse modulus Ef 2, Ef 3 (GPa)

15

15

Ef 1

Ef 1

In-plane shear modulus Gf 12, G f 13 (GPa)

15

15

Ef1 2(1 + n f 12 )

Ef1 2(1 + n f 12 )

Major Poisson’s ratio nf 12 = nf 13

0.2

0.2

0.2

0.2

Transverse Poisson’s ratio n f 23

0.07

0.07

0.2

0.2

Longitudinal thermal coefficient af1 (10–6/∞C)

–0.5

–0.7

4.9

4.9

Transverse thermal coefficient a f 2 (10–6/∞C)

15

12

4.9

4.9

Note: transverse shear modulus G f 23 = E f 2/ 2(1 + n f 23 )

Table 8.2 Resin properties [19] Resin type

3501-6 epoxy

BSL914C epoxy

LY556 epoxy

MY750 epoxy

Modulus Em (GPa) Poisson’s ratio nm Thermal coefficient am (10–6/∞C)

4.2 0.34 45

4.0 0.35 55

3.35 0.35 58

3.35 0.35 58

AS4

T300

3501-6

BSL914C

E-Glass 21 ¥ K43 LY556

E-Glass 1200 tex MY750

1950

1500

1140

1280

1480

900

570

800

48

27

35

40

79

80

72

73

Table 8.3 Laminar strengths [19] Laminar type

Longitudinal tensile strength XT (MPa) Longitudinal compressive strength XC (MPa) Transverse tensile strength YT (MPa) In-plane shear strength S12 (MPa)

Progressive multi-scale modelling of composite laminates

269

the fibres and matrices are determined using the analysis presented in sections 8.3.1 and 8.3.3; the results are given in Tables 8.4–8.5. Table 8.4 Strength properties of fibres Fibre type

Carbon fibre AS4

Carbon fibre T300

E-Glass fibre 21 ¥ K43

E-Glass fibre 1200 tex

Critical distortional strain e (cf ) Micro-buckling strength s (cf ) (MPa)

0.019 –2640

0.0133 –1466

0.0266 –831

0.0349 –1369

Table 8.5 Strength properties of matrix Matrix type

3501–6 epoxy

BSL914C epoxy

LY556 epoxy

MY750 epoxy

Pressure sensitivity m+ Pressure sensitivity m– Distortional strength s (cm ) (MPa) Critical shear strain g (cm )

2.08 0.327 95 0.02

5.8 0.143 130 0.04

12.26 0 46.55 0.04

10 –1.17 43.5 0.037

While the initial failure strength of a composite laminate can be readily determined, determination of the ultimate failure strength requires an iterative process in which the effective ply properties are gradually modified as the applied stress is increased, using the analysis presented in section 8.4. In the following the comparisons between model predictions and experimental data reported in [18] for four different fibre/matrix composites will be presented. AS4/3506-1 laminate The biaxial failure stress data for quasi-isotropic (90∞/±45∞/0∞) laminate of AS4/3506-1 were generated by subjecting tubular specimens, of three different diameters, to internal pressure and axial loads. Since large diameter tubular specimens tended to suffer global buckling failure, those data points were not included in Fig. 8.4, which shows the predicted initial and final failure strengths. It can be seen that very good agreement has been achieved in both the tension-tension quadrant and the tension-compression quadrant. It is also important to note that the initial failure strength is much lower than the final laminate strength, indicating the importance of accounting for damage progression. Due to the reported problem of tube buckling for test configurations employed in these experiments, no valid compression-compression data are currently available.

270

Multi-scale modelling of composite material systems 1000 Experimental data Initial failure Final failure

syy (MPa)

500

0

–500

AS4/3501-6 (0/90/45–45) –1000 –1000

–500

0 sxx (MPa)

500

1000

8.4 Biaxial failure stress envelope for (90∞/ ± 45∞/0∞) laminate of AS4/ 3506-1 under combined sy and sx stresses.

T300/BSL914C laminate The experimental data were obtained from experiments on axially wound tubes made from prepreg T300/BSL914C carbon/epoxy. The tubes were tested under combined axial tension or compression and torsion. The correlation between predictions and experimental data shown in Fig. 8.5 suggests a good agreement for combined compression and torsion loading, as well as combined tension and torsion, as long as the shear stress is below the shear strength of the laminate. Clearly the high shear strength at axial stress between 700 and 1000 MPa is not predicted. Since the shear strength is actually higher than the lamina shear strength, these high values are unlikely to be physically possible, but further study is required. E-Glass/LY556 laminate Circumferentially filament wound tubes, made from E-Glass/LY556 epoxy, were tested under torsion combined with axial tension or compression. The experimental data are shown in Fig. 8.6 together with the model prediction. It is clear that the model correctly predicts the interaction between the transverse stress and the shear strength, except for a few data points at negative transverse stress. It is not clear what contributed to these puzzling situations, but it is likely to be caused by the pressure sensitivity of the matrix. Filament wound tubes made from (±90∞/±30∞) E-Glass/LY556 were tested

Progressive multi-scale modelling of composite laminates

271

140 DFVLR tension DFVLR compression MBB test Prediction

In-plane shear stress (MPa)

120 100 80 60 40 20 0

T300/BSL914C (0∞ unidirectional lamina)

–1500

–1000

–500

0 500 1000 Longitudinal stress (MPa)

1500

2000

8.5 Biaxial failure stress envelope for 0∞ unidirectional laminate of T300/BSL914C under longitudinal and shear loading (sx versus tx).

120 E-Glass/LY556 (0∞ unidirectional lamina) 100

Experimental data Prediction

txy (MPa)

80

60

40

20

0 –200

–150

–100

syy (MPa)

–50

0

50

8.6 Biaxial failure stress envelope for 0∞ unidirectional laminate of E-Glass/LY556 epoxy under transverse and shear loading (sy versus txy).

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Multi-scale modelling of composite material systems

under combined direct stresses and under combined tension and axial load. The failure envelope data are shown in Fig. 8.7 and 8.8. These specimens 800 600

Experimental data Prediction

400

syy (MPa)

200 0

–200 –400 –600 –800 –800

–600

–400

–200

0 200 sxx (MPa)

400

600

800

1000

8.7 Biaxial failure stress envelope for (90∞/ ± 30∞/90∞) laminate of EGlass/LY556 epoxy under combined sy and sx stresses. 800 Experimental data Prediction

E-Glass LY556 (90∞/±30∞/90∞)

txy (MPa)

600

400

200

0 –800

– 600

– 400

–200

0 sxx (MPa)

200

400

600

800

8.8 Biaxial failure stress envelope for (90∞/ ± 30∞/90∞) laminate of EGlass/LY556 epoxy under combined sx and txy stresses.

Progressive multi-scale modelling of composite laminates

273

contained 82.8% circumferential layers and 17.2% of ± 30∞ layers. Therefore the laminate is not quasi-isotropic and the strength depends on the loading direction. It can be seen that the model predictions correlate well with the experimental data. E-Glass/MY750 laminate Angle ply E-Glass/MY750 epoxy tube specimens were made through filament winding at a ±55∞ helical pattern (the winding angle is measured between the fibre direction and the tube axis). Specimens of two different diameters (100 mm and 51 mm inner diameter) with various wall thicknesses were tested under combined internal pressure and axial load. The experimental data shown in Fig. 8.9 were from tubes that failed by rupture and not by buckling. Overall the model predictions are in reasonable agreement with the experimental data, except that there is a minor over-prediction of axial strength.

600 400

Thin tube without liner Thick tube Thick tube with liner Prediction

sxx (MPa)

200 0 –200 –400 E-Glass/MY750 (±55∞ angle ply laminate)

–600 –1000

– 500

0 syy (MPa)

500

1000

8.9 Biaxial failure stress envelope for (±55∞) angle ply of E-Glass/ MY750 epoxy under combined sy and sx stresses.

8.6

Conclusion

A multi-scale failure theory that is based on constituent failure modes and micro-mechanics analysis has been developed to predict the initial and ultimate failure loads of composite laminates. In this approach, the ultimate failure of a laminate is the result of damage progression through lamina, a process involving failures in the fibre or in the matrix. Comparison with experimental

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Multi-scale modelling of composite material systems

data reported in the world-wide failure exercise confirms the good predictive capability of the proposed approach.

8.7

References

1. Hinton M.J. and Soden P.D., ‘Predicting failure in composite laminates: the background to the exercise’. Composite Science and Technology 1998. 58: p. 1001–1010. 2. Hinton M.J., Kaddour A.S. and Soden P.D., ‘A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence’. Composite Science and Technology 2004. 64: p. 549–588. 3. NASA, GENOA-PEF: Progressive Fracture in Composites Simulated Computationally. 1999, NASA. 4. Chamis C.C., Murthy P.L.N. and Minnetyan L., ‘Progressive fracture of polymer matrix composite structures’. Theoretical and Applied Fracture Mechanics 1996. 25(1): p. 1–15. 5. Gosse J., ‘Strain invariant failure criteria for fiber reinforced polymeric composite materials’. in Proceedings of 13th ICCM conference. 2001. Beijing, China. 6. Tsai S.W. and Wu E.M., ‘A general theory of strength for anisotropic materials.’ J. Composite Materials, 1971. 5: p. 58–80. 7. Hashin Z., ‘Failure criteria for unidirectional composites’. J. Appl. Mech., 1980. 47: p. 329–335. 8. Rotem A., ‘The Rotem failure criterion: theory and practice’. Composite Science and Technology, 2002. 62: p. 1663–1671. 9. Mayes J.S. and Hansen A.C., ‘A comparison of multicontinuum theory based failure simulation with experimental results’. Mechanics of Composite Materials and Structures, 2003. 10. Kuraishi A., ‘Accelerated durability assessment of composite structures’. in Proceedings of ICCM-14. 2003. San Diego. 11. Wang C.H. and Chalkley P.D., ‘Plastic yielding of a film adhesive under multiaxial stresses’. International Journal of Adhesion and Adhesives, 2000. 20: p. 155–164. 12. Haward R.N., The physics of glassy polymers. 1973, London: Applied Science. 13. Mayes J.S. and Hansen A.C., ‘Multicontinuum failure analysis of composite structural laminates’. Mechanics of Composite Materials and Structures, 2001. 8(4): 249–262. 14. Hart-Smith L.J., ‘Mechanistic failure criteria for carbon and glass fibres embedded in polymer matrices’. in AIAA-2001-1184. 2002. Seattle, WA, USA: AIAA. 15. Budiansky B. and Fleck N.A., ‘Compressive failure of fibre composites’. J. Mech. Phys. Solids, 1993. 41(1): p. 183–211. 16. ASTM, Standard test method for compressive properties of rigid plastics, D695–96. 1996. 17. Tsai S.W., ‘Theory of Composites Design’. 1992: Think Composites. 18. Soden P.D., Hinton M.J. and Kaddour A.S., ‘Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: failure exercise benchmark data’. Composite Science and Technology, 2002. 62: p. 1489–1514. 19. Soden P.D. Hinton M.J. and Kaddour A.S., ‘Lamina properties, lay-up onfigurations and loading conditions for a range of fibre-reinforced composite laminates’. Composite Science and Technology, 1998. 58: p. 1011–1022.

Progressive multi-scale modelling of composite laminates

8.8

Appendices

Appendix I: Nomenclature Symbol

Meaning

T A aL aP af am q Q S M E

Temperature Stress amplification factor Laminate’s coefficient of thermal expansion Ply’s coefficient of thermal expansion Fibre coefficient of thermal expansion Matrix’s coefficient of thermal expansion Ply angle Stiffness matrix Compliance matrix Strain magnification matrix Young’s modulus

Subscript or superscript P L f m a c

Ply Laminate Fibre Matrix Ambient Cure

Appendix II: Strain amplification matrix for diamond unit cell Carbon/epoxy diamond unit cell (Vf = 0.6) È 1 Í 0.0036 Í Í 0.0036 Mm = Í Í 0 Í 0 Í Î 0

0 1.98 –0.32 0 0 0

0 –0.32 1.98 0 0 0

0 0 0 1.86 0 0

0 0 0 0 2.4 0

0 ˘ 0 ˙ ˙ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ 2.4 ˚

275

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Multi-scale modelling of composite material systems

È 1 Í –0.0017 Í Í –0.0017 Mf = Í Í 0 Í 0 Í Î 0

0 0.642 0.12 0 0 0

0 0.12 0.641 0 0 0

0 0 0 0.63 0 0

0 0 0 0 0.43 0

0 0 0 0 1.95 0

0 ˘ 0 ˙ ˙ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ 1.95 ˚

0 ˘ 0 ˙ ˙ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ 0.43 ˚

E-Glass/epoxy diamond unit cell (Vf = 0.6) È 1 Í –0.3 Í Í –0.3 Mm = Í Í 0 Í 0 Í Î 0

0 3.33 –0.3 0 0 0

0 –0.3 3.33 0 0 0

0 0 0 2.9 0 0

È 1 Í –0.2 Í Í –0.2 Mf = Í Í 0 Í 0 Í Î 0

0 0.2 0.06 0 0 0

0 0.06 0.2 0 0 0

0 0 0 0.16 0 0

0 0 0 0 0.13 0

0 ˘ 0 ˙ ˙ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ 0.13 ˚

Appendix III: Strain amplification matrix for hexagonal unit cell Carbon/epoxy hexagonal unit cell (Vf = 0.6) È 1 Í –0.022 Í Í –0.0019 Mm = Í Í 0 Í –0.0345 Í Î 0

0 1.76 –0.188 0 0.66 0

0 –0.29 1.22 0 0.091 0

0 0 0 1.06 0 0.15

0 0.44 0.12 0 1.8 0

0 ˘ 0 ˙ ˙ 0 ˙ ˙ 0.15 ˙ 0 ˙ ˙ 1.25 ˚

Progressive multi-scale modelling of composite laminates

È 1 Í 0.0076 Í Í –0.0076 Mf = Í Í 0 Í 0 Í Î 0

0 0.7 0.13 0 0 0

0 0.13 0.7 0 0 0

0 0 0 1 0 0

0 0 0 0 0.54 0

0˘ 0˙ ˙ 0˙ ˙ 0˙ 0˙ ˙ 1˚

E-Glass/epoxy diamond unit cell (Vf = 0.6) È 1 Í 0.41 Í Í 0.03 Mm = Í Í 0 Í 0.66 Í Î 0 È 1 Í –0.14 Í Í –0.14 Mf = Í Í 0 Í 0 Í Î 0

0 3.18 –0.31 0 2.47 0 0 0.22 0.022 0 0 0

0 0.117 1.48 0 1.26 0 0 0.017 0.17 0 0 0

0 0 0 1.17 0 0.31 0 0 0 1 0 0

0 0.76 0.27 0 2.5 0 0 0 0 0 0.17 0

0 ˘ 0 ˙ ˙ 0 ˙ ˙ 0.3 ˙ 0 ˙ ˙ 1.5 ˚ 0˘ 0˙ ˙ 0˙ ˙ 0˙ 0˙ ˙ 1˚

277

9 Predicting fracture of laminated composites I A G U Z, University of Aberdeen, UK and C S O U T I S, University of Sheffield, UK

9.1

Introduction: modelling the compressive response of laminate composites

The compressive strength of currently used carbon fibre reinforced plastics is generally 30–40% lower than the tensile strength due to fibre microbuckling (Soutis, 1996), thus it is recognised that the compressive strength is often a design-limiting consideration. It should be underlined that zones of compressive stresses can appear in composite structures even under tensile loads. They could be due to the presence of holes, cut-outs and cracks, or generated by impact. Previous experimental studies by Dow and Grunfest (1960), Rosen (1965), Guz (1990), Guynn et al. (1992), Soutis and Turkmen (1995) have revealed that a possible mechanism of failure initiation is fibre or layer microinstability (microbuckling) that may usually occur in regions where high stress gradients exist, for instance, on the edge of a hole or near free edges (Berbinau et al., 1999; Zhuk et al., 2001). At that, various cases of interfacial adhesion breakdown may occur in real laminated composites during the fabrication process or in service. In the case of compression of composites along layers and, therefore, along the mentioned interlaminar defects the classical Griffith-Irwin criterion of fracture or its generalisations are inapplicable. Actually, in such situations all stress intensity factors and crack opening displacements are equal to zero. This fact emphasises the importance and the necessity of a most careful (possibly exact) investigation of fracture due to specific mechanisms inherent to heterogeneous (piecewisehomogeneous) materials. The loss of stability in the heterogeneous structure of composites is one of such mechanisms; the moment of stability loss in the microstructure of the material – internal instability according to Biot (1965) – is associated with the onset of fracture. This assumption was suggested for the first time by Dow and Grunfest (1960) and later used in numerous publications on the subject, see the reviews by Guz (1990, 1992), Budiansky and Fleck (1994), Schultheisz and Waas (1996), Soutis (1996), Babich et al. (2001). 278

Predicting fracture of laminated composites

279

A better understanding of the compression failure mechanisms, specific only to heterogeneous materials, is therefore crucial to the development of improved composite materials. The task of deriving three-dimensional (3-D) analytical solutions to describe the compressive response is considered as one of great importance. Such solutions, if obtained, enable us to analyse the behaviour of a structure on the wide range of material properties, and kinematic and loading boundary conditions, without the restrictions imposed by simplified approximate methods. The schematic of different approaches to the problem of instability (microbuckling) of composite materials in compression is shown in Fig. 9.1. The use of 3-D stability theory, which was expounded, for instance, in (Guz, 1999), places the methods into the category of ‘exact’ approaches, as opposed to approximate models based on certain simplifications when describing the stress-strain state. Instability (microbuckling) of composites in compression (and fracture, if the onset of fracture process is associated with moment of stability loss)

Exact approach (based on the 3-D stability theory)

Model of a piecewisehomogeneous medium

Advantage: • the most accurate approach in solid mechanics Drawback: • due to its complexity restricted to a very small group of problems

Continuum theory

Advantage: • simple • wide range of problems can be solved Drawback: • accuracy is not known

Approximate models (Rosen and many others)

Give significant discrepancy in comparison with the exact approach and experimental data

Too simplistic

9.1 Classification of the models.

Generally speaking, in mechanics of heterogeneous (piecewisehomogeneous) media, there are two major distinctive approaches to describe the behaviour of solids. One of them is based on the model of piecewisehomogeneous medium, when the behaviour of each material constituent is described by 3-D equations of solid mechanics provided certain boundary conditions are satisfied at the interfaces. This approach enables the investigation

280

Multi-scale modelling of composite material systems

of phenomena occurring in the internal structure of solids (e.g. in the composite microstructure) in the most rigorous way. However, due to its complexity the method is restricted to a small group of problems. The other approach, or continuum theory, involves significant simplifications. Within the continuum theory, the composite is simulated by a homogeneous anisotropic material with effective constants, by means of which physical properties of the original material, shape and volume fraction of the constituents are taken into account. The continuum theory may be applied when the scale of the investigated phenomena is considerably larger than the scale of material internal structure. The approach, based on the model of the piecewise homogeneous medium, is free from such restrictions and is, therefore, the most accurate one. As applied to the internal instability of fibrous or laminated composites, the exact approach was utilised for the first time within the continuum theory and the model of a piecewise-homogeneous medium in Guz (1967) and (1969) respectively. The continuum theory, due to its simplicity, is widely used to characterise the mechanical behaviour of composites, but questions on its accuracy and domain of applicability always arise (Fig. 9.1). This problem was successfully resolved in the recent works by Guz (1991b), Guz and Soutis (2001a, b), Soutis and Guz (2001). It was shown for various types of composites (including the non-linear ones) that the continuum theory of brittle fracture is asymptotically accurate. Apart from the exact approaches to the considered problem, which are based on the 3-D stability theory, there are also approximate models proposed by Rosen (1965) and by many other authors, see for example (Schuerch, 1966; Sadovsky et al., 1967). The detailed reviews of the later approximate models are given by Soutis (1991, 1996), Budiansky and Fleck (1994), Soutis and Turkmen (1995), Schultheisz and Waas (1996). However, the approximate approaches proved to be not worth applying (Fig. 9.1). This and the application of kink-band models are discussed in some detail in the next sub-section. However, all works mentioned above considered perfectly bonded layers only. Since in practical cases the assumption of perfect bonding between neighbouring layers in composites does not correspond to reality, this chapter attempts to fill the gap. It focuses on investigation of composite materials with specific kinds of interlaminar defects. The analysis establishes the upper and the lower bounds for the critical loads for layered composites under equi-biaxial and uniaxial compression along such defects. In order to calculate the bounds, the non-axisymmetrical problem of internal instability (microbuckling) is considered within the scope of the most accurate (exact) statement based on the application of the piecewise-homogeneous medium model and the equations of the 3-D stability theory. The 3-D approach presented in this chapter allows us to take into account large deformations, geometrical and physical non-linearities and load biaxiality that the simplified methods

Predicting fracture of laminated composites

281

cannot consider. Part of the analysis is based on previous works by the authors (Guz and Soutis, 2000; Soutis and Guz, 2001) and on the general approaches developed by Guz (1991a, 1998), so the equations derived earlier are presented only for clarification purposes without much detail. Since practical composite materials contain not only interlaminar, but also various sorts of intralaminar defects, which further complicate the problem, the effect of intralaminar damage can be studied in the next stage. The presence of damage can be accounted for by considering layers with reduced stiffness properties – see, for example, Kashtalyan and Soutis (2000a, b, 2001, 2002a) and the review by Kashtalyan and Soutis (2002b).

9.1.1

On the applicability of approximate models

As mentioned in the previous subsection, the most accurate (‘exact’) approach to studying internal instability is based on the piecewise-homogeneous medium model, when the behaviour of each component of the material is described by the 3-D equations of solid mechanics provided certain boundary conditions are satisfied at the interfaces. This approach that enables us to investigate in the most rigorous way any phenomena occurring in the composite microstructure is implemented throughout this chapter. Probably the first solutions to the problem of internal instability for a layered material obtained within the most accurate (exact) approach were reported by Guz (1969), Babich and Guz (1969, 1972), where the problem for linear-elastic layers under uniaxial compression was solved. This solution was included in numerous books, for example, Guz (1990), and comprehensive reviews on the topic (Guz, 1992; Babich et al., 2001). This problem seems to have remained topical for more than thirty years and is still being ‘reexamined’. A recent paper by Parnes and Chiskis (2002) reports the solution (by a very approximate method, based on modelling rigid layers as 2-D beams embedded in the matrix) of precisely the same problem that was solved more than thirty (!) years ago by Guz (1969) and Babich and Guz (1969, 1972) using the exact approach. Later the exact solutions were derived also for more complex problems: for orthotropic, non-linear elastic and elastic-plastic, compressible and incompressible layers including the case of large (finite) deformations – see, for example, Guz (1989a, b) and the reviews by Guz (1990, 1992, 1999), and Babich et al. (2001). These publications contain many examples of calculation of critical stresses/strains for particular composites as well as analyses of different buckling modes. The importance and the complexity of the considered phenomena prompted a large number of publications which put forward various approximate methods aimed at tackling the problems with different levels of accuracy – see, for example, Rosen (1965), Schuerch (1966), Sadovsky et al. (1967) and the

282

Multi-scale modelling of composite material systems

reviews by Soutis (1991, 1996), Guz (1992), Budiansky and Fleck (1994), Soutis and Turkmen (1995), Schultheisz and Waas (1996), Niu and Talreja (2000). It was concluded after the detailed analyses (Guz, 1990, 1992, 1999; Soutis and Turkmen, 1995; Niu and Talreja, 2000) that the approximate methods are not very accurate when compared to experimental measurements and observations. For instance, the model suggested by Rosen (1965) involves considerable simplifications, modelling the reinforcement layers by the thin beam theory and the matrix as an elastic material using one-dimensional stress analysis. It makes the results of this method inaccurate even for simple cases. It was shown by Guz (1990, 1992), Guz and Soutis (2000), Soutis and Guz (2001) that the approximate model can give a significant discrepancy in comparison with the exact approach and with experimental data even for the simplest case of a composite with linear elastic compressible layers undergoing small pre-critical deformations and considered within the scope of geometrically linear theory. For small fibre volume fractions the approximate approach gives physically unrealistic critical strains. It does not describe the phenomenon under consideration even on the qualitative level, since it predicts a different mode of stability loss from that obtained by the 3-D exact analysis. Figure 9.2 gives an example of the critical strain plotted against the fibre volume ecr 1 1 2

0.5

3 1 2 3

0.1

0.05 Exact solution (the piecewisehomogeneous medium model within the 3-D stability theory) Rosen model 0.01

Vr 0

0.2

0.4

0.6

0.8

9.2 Critical strain plotted against fibre volume fraction for the extension mode (the second mode); logarithmic scale (Guz, 1990).

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283

fraction (logarithmic scale) for the extension mode (the second mode) calculated using the exact solution and the Rosen model. The plot was taken from (Guz, 1990) for the case of a composite consisting of two alternating linear-elastic compressible layers. Lines 1, 2, and 3 in the plot correspond to the ratios of Young’s modulus of the fibres to the shear modulus of the matrix equal to 50, 100 and 200, respectively; the Poisson ratios for both layers were always 0.25. For more complex models, which take into account large deformations and geometrical and physical non-linearity (e.g. those considered in this chapter), the approximate theories are definitely inapplicable and one can expect an even bigger difference between the exact and approximate approaches. The exact approach presented in this chapter allows us to take into account large deformations, geometrical and physical non-linearities and load biaxiality that the simplified methods cannot consider. Another approach, which is commonly utilised, is based on the investigation of fibre kinking. From the literature on compressive fracture it is easy to get the impression that fibre instability (microbuckling) and kinking are competing mechanisms. In fact, a kink band is an outcome of the microbuckling failure of actual fibres, as observed experimentally by Guynn et al. (1992). Fibre microbuckling occurs first, followed by propagation of this local damage to form a kink band. A comprehensive comparative analysis of the Rosen model, Argon-Budiansky (kinking) model, and Batdorf-Ko model was presented by Soutis and Turkmen (1995). Studies of the kinking phenomenon were also reviewed by Soutis (1996) Budiansky and Fleck (1994) and Fleck (1997). It was shown by Soutis and Turkmen (1995) that the existing kinking analyses are able to account for some, but not all, of the experimental observations. They correctly predict that shear strength and fibre imperfections are important parameters affecting the compressive strength of the composite. However, within this model it is not possible to say exactly how the strength will vary with fibre content, and the value of misalignment is chosen arbitrarily. This model requires knowledge of the shear strength properties, the initial fibre misalignment and, most importantly, the kink-band orientation angle which is a post-failure geometric parameter. All works mentioned in this sub-section considered perfectly bonded layers only. Moreover, the approaches based on the Rosen model and kink-band model cannot be altogether applied in the case of large pre-critical deformations.

9.2

Developing compression models for laminates

Let us consider the statement of the static non-axisymmetrical problem of stability for layered composites. For this special attention will be paid to accounting for the biaxiality of compressive loads. The composite consists of alternating layers with thicknesses 2hr and 2hm, Fig. 9.3, which are simulated

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Multi-scale modelling of composite material systems

X3

X1

O 2h r

2h m

X2

9.3 Equi-biaxial compression of a laminated material.

by compressible non-elastic transversally isotropic solids with a general form of the constitutive equations. Henceforth all values referring to these layers will be labelled by indices r (reinforcement) and m (matrix). The values of displacement, stress and strain corresponding to the precritical state will be marked by the superscript ‘0’ to distinguish them from perturbations of the same values (ui0 and ui , e ij0 and eij, Sij0 and Sij respectively). Suppose also that the material is compressed in planes of the layers along the axis OX1 and OX2 by static ‘dead’ loads applied at infinity in such a manner that equal deformations along all layers are provided. Within the scope of the most accurate approach, i.e., using the piecewisehomogeneous medium model and the equations of the 3-D stability theory (Guz, 1999), the following eigen-value problem is solved. The equations of stability for the individual incompressible layers are (Guz, 1999) ∂ t r = 0, ∂ tm = 0 9.1 ∂ x i ij ∂ x i ij where t ij = w ija b

∂ua ∂x b

9.2

Here tij is the non-symmetrical Piola-Kirchhoff stress tensor. This is the nominal stress tensor according to Hill (1958). Further, we shall consider also the symmetrical stress tensor Sij which reduces to sij for the case of small precritical deformations. The components of the tensor wijab depend on the material properties and on the loads (i.e. on the precritical state). For the orthotropic case, w is given by Guz (1999) as

wijab = l 0 l a0 [d ij d ab Ab i +(1 – d ij )(d ia d jb m ij + d ib d ja m ji )] j 0 + d ib d ja Sbb

9.3

Predicting fracture of laminated composites

285

where Aij and mij are the quantities which characterise the axial and shear stiffness, and lj are the shortening factors along the OXj axis. The quantity 0 or strain characterising the precritical state, i.e., a stress component S11 0 component e 11 , is the parameter in respect to which the eigen-value problem is solved. The components of tensor w can be obtained for the specific constitutive equations. For instance, in the simplest case of a linear elastic solid: A i b = l + 2 d i b m, 0 m ij = m , Sbb = 1 l ( l12 + l 22 + l 32 – 3) + m ( l b3 – 1) 9.4 2 where the parameters l and m are the Lamé constants.

9.2.1

Different cases of interlaminar adhesion

To complete the problem statement, boundary conditions should be written for each interface. In the most general case, the layer interfaces T consist of zones of perfectly connected (bonded) layers Tp–c and defects of interlaminar adhesion of various types. Defects can be represented either by cracks (Tcr) or by cleavage-type delaminations (Tsl), i.e., T = Tp–c » Tcr » Tsl

9.5

For the perfectly bonded layers (or for the remaining parts of a perfect interlaminar adhesion) we have the continuity conditions for the stresses and for the displacements r m r m r m t 31 = t 31 , t 32 = t 32 , t 33 = t 33 , u3r = u3m , u 2r = u 2m , u1r = u1m

if (x1, x2) Œ Tp–c

9.6

For the case of stress-free crack surfaces (the classical model), when the mathematical sections simulate cracks regardless of the reasons of their occurrence, the following relations hold true r r m r m m t 31 = 0 , t 31 = 0 , t 32 = 0 , t 33 = 0 , t 33 = 0 = 0 , t 32

if (x1, x2) ΠTcr

9.7

Equation 9.7 is valid since the initial moment of the stability loss, not the postcritical behaviour, is considered in the present chapter. In a certain sense this model is an idealisation, and in practical composites other kinds of interlaminar adhesion breakdown may also occur. For example, a change in the nature of the interlaminar contact is possible, when an interaction of the layers is implemented so that infinitesimal sliding is allowed but still there are no gaps between the layers (e.g., molecular chains in some kinds of glue connection, Fig. 9.4). This kind of cleavage-type delamination is called ‘defects

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Multi-scale modelling of composite material systems

Matrix, 2hm

Rigid layer, 2hr

Defects of interlaminar adhesion

Rigid layer

Matrix Interface

9.4 A cleavage-type delamination (defect with connected edges).

with connected edges’. For defects with connected edges, the continuity at the interface is retained for normal components only, with boundary conditions for the perturbations of stresses and displacements in the form of r m r m r m t 31 = 0 , t 31 = 0, t 32 = 0 , t 32 = 0, t 33 = t 33 , u3r = u3m

if (x1, x2) ΠTsl

9.8

Also, for this case Aboudi (1987) and Librescu and Schmidt (2001) use the terminology ‘perfectly lubricated interfaces’.

9.2.2

The bounds for critical loads

Let us now consider a layered composite with an unidentified set of defects with connected edges. The following estimation can be suggested to find the lower and the upper bounds for the critical loads. If in a material with cleavage-type delaminations all remaining zones of a perfect bonding are removed, the material of the same type with an ideal slippage (sliding without friction) between all the layers is obtained. In other words, all interfaces become perfectly lubricated. The substantiation of the lower bound is based on a general principle of mechanics, which states that release from a part of the connections inside the mechanical system cannot increase the value of the critical load. Therefore, the critical load, Pcrsl , for a material with sliding layers must be smaller than the critical load, Pcrimp , for a material with the imperfections of interlaminar adhesion, i.e.,

Predicting fracture of laminated composites

Pcrsl £ Pcrimp

287

9.9

which, if written in terms of strains, takes the following form

e crsl £ e crimp

9.10

The critical shortening strain e crsl is determined by solving eqn 9.1 for T = Tsl, Tcr = 0, Tp–c = 0, i.e, when the boundary conditions on all interfaces have the form of eqn 9.8. If e crsl is found, the bound required follows from eqn 9.9 or 9.10. This is true for an arbitrary set of defects with connected edges, i.e., for an arbitrary number, size and disposition of the defects. Applying the same principle, it can be also said that the critical load for a material without imperfections (i.e. with perfectly bonded layers), must be larger than the critical load, Pcrpb , for a material with the same internal structure containing cleavage-type delaminations, i.e. Pcrpb ≥ Pcrimp

9.11

which in terms of strains means

e crpb ≥ e crimp

9.12

Hence, from eqns 9.9–12. imp Pcrpb ≥ Pcrimp ≥ Pcrsl and e crpb ≥ e cr ≥ e crsl

9.13

The above-mentioned bounds were considered earlier for other material properties by Guz (1991a, 1998), and also used later by Guz and Soutis (2000) and Soutis and Guz (2001), where the investigation was restricted by the case of small deformations only, and by Guz and Herrmann (2003) for incompressible hyperelastic materials. Of course, practical composite materials contain not only interlaminar, but also various sorts of intralaminar defects, cracks, etc. The latter further complicate the problem. However, the suggested method allows accounting for the presence of intralaminar damage by considering layers with reduced stiffness properties. The reduction in the axial material stiffness as well as in the Poisson’s ratio due to matrix cracking was recently extensively investigated (see the review by Kashtalyan and Soutis (2002b)). Based on these results, the effective properties of materials can be calculated and incorporated into the analysis described above. Also, internal instability does not always cause material fracture. For studying the problem on fracture of composites in compression completely, it is also necessary to analyse postcritical behaviour. However, it will not change the lower bounds for the critical loads, which are found in the present chapter.

9.3

Identifying critical loads

The exact solutions of the 3-D non-axisymmetrical problems of internal instability for compressible elastic and elastic-plastic layers were found by

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Multi-scale modelling of composite material systems

Guz (1989a, b) within the scope of the piecewise-homogeneous medium model. At that, only perfectly bonded layers with boundary conditions given by eqn 9.6 were investigated (i.e. T = Tp–c, Tcr = 0, Tsl = 0). The characteristic determinants were derived for the case of equi-biaxial compression as applied to four modes of stability loss, Fig. 9.5. The plane problem (uniaxial compression) for such materials was studied by Guz (1969, 1990, 1992) and Babich et al. (2001).

First (shear) mode

Third mode

Second (extension) mode

Fourth mode

9.5 Modes of microbuckling.

In this subsection, the most general case of the 3-D non-axisymmeterical modes of stability loss is considered for the case of sliding without friction along all interfaces, i.e., for T = Tsl, Tcr = 0, Tp–c = 0 – perfectly lubricated interfaces following Aboudi (1987) and Librescu and Schmidt (2001).

Predicting fracture of laminated composites

289

According to the previous section, it will yield the lower bound for the critical load in composites with interlaminar defects. Solutions of eqn 9.1 (i.e. perturbations of stresses and displacements) can be expressed through the functions C and Y which, in their turn, are the solutions of the following equations (Guz, 1999) 2 Ê 2 ∂ D + x Á 1 1 ∂ x 32 Ë

2 2 ˆÊ ˆ Ê 2 ∂ 2 ∂ Y = 0, D + x D + x 1 ˜ Á 1 3 2 ∂ x 32 ˜¯ ÁË ∂ x 32 ¯ Ë

ˆ ˜ X = 0 9.14 ¯

where

x12 =

w 3113 , x 22 = C + w 1221

x 32 = C –

C2 –

C2 –

w 3333w 3113 , w 1111w 1331

w 3333w 3113 , w 1111w 1331

2 2 D1 = ∂ 2 + ∂ 2 , ∂ x1 ∂x 2

C=

w 1111w 3333 + w 1331w 3113 – (w 1313 + w 1133 ) 2 w 1111w 1331

9.15

The parameters x rj and x jm , depend on the components of the tensor wijab and, therefore, on the properties of the layers and on the loads. It was proved 2 2 by Guz (1989a) that x rj and x jm are always real and positive. Before proceeding with the construction of solutions for the same four modes of stability loss, Fig. 9.5, as those considered in (Guz, 1989a), we introduce the notations, which will be useful later

ar = p h r l –1, am = p h m l –1,

l –1 =

l1–2 + l 2–2

9.16

Here li is the half-wavelength of the modes of stability loss along the OXi axis, and a is the normalised wavelength. 2 2 2 2 For the case of x 2r π x 3r and x 2m π x 3m the potentials C and Y can be set up for the first (shear) mode (Fig. 9.5) as follows (subscript j denotes the number of the layer): X rj = c r (cosh x 3 )

Ê ˆ p ∫ Á A r cosh r x 3 + B r cosh pr x 3 ˜ sin p x1 sin p x 2 , l1 l2 lx 2 lx 3 ¯ Ë Y jr = y r (sinh x 3 ) ∫ C r sinh p r x 3 cos p x1 cos p x 2 , l1 l2 l x1

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Multi-scale modelling of composite material systems

X mj = c m (cosh x 3 ) Ê ˆ ∫ Á A m cosh pm x 3 + B m cosh pm x 3 ˜ sin p x1 sin p x 2 , l1 l2 l x lx 2 Ë ¯ 3 Y jm = y m (sinh x 3 ) ∫ C m sinh pm x 3 cos p x1 cos p x 2 l1 l2 lx1

9.17

For the second (extensional) mode (Fig. 9.5): X rj = c r (cosh x 3 ), Y jr = y r (sinh x 3 ), X mj = c m (sinh x 3 ), Y jm = y m (cosh x 3 ), X rj +1 = – c r (cosh x 3 ), Y jr+1 = – y r (sinh x 3 ), X mj +1 = – c m (sinh x 3 ), Y jm+1 = –y m (cosh x 3 )

9.18

For the third mode, (Fig. 9.5):

X rj = c r (sinh x 3 ), Y jr = y r (cosh x 3 ), X mj = c m (sinh x 3 ), Y jm = y m (cosh x 3 )

9.19

For the fourth mode (Fig. 9.5):

X rj = c r (sinh x 3 ), Y jr = y r (cosh x 3 ), X mj = c m (cosh x 3 ), Y jm = y m (sinh x 3 ), X rj +1 = – c r (sinh x 3 ), Y jr+1 = –y r (cosh x 3 ), X mj +1 = – c m (cosh x 3 ), Y jm+1 = –y m (sinh x 3 )

9.20

The components of tij and ui can be expressed (Guz, 1999) through the potentials C and Y. Substituting them into the boundary conditions, eqn 9.8 which due to the periodicity of both, the material and the solution, are to be satisfied on one interface only, we get the (6 ¥ 6) characteristic determinant. This determinant can be analytically reduced to the (4 ¥ 4) determinant in the following form for the first, second, third and fourth modes respectively: 2

Ê l 22 ˆ –1 –1 r r m m Á 1 + l 2 ˜ (w 1133 + w 1313 ) (w 1133 + w 1313 ) Ë 1 ¯ ¥

r m w 3113 w 3113 a a coth rr coth mm det ||g ij || = 0 r m x1 x1 x1 x1

9.21

Predicting fracture of laminated composites

291

2

Ê l 22 ˆ –1 –1 r r m m Á 1 + l 2 ˜ (w 1133 + w 1313 ) (w 1133 + w 1313 ) Ë 1 ¯

¥

r m w 3113 w 3113 a a coth rr sinh mm det ||g ij || = 0 r x1 x1m x1 x1

9.22

2

Ê l 22 ˆ r r m m 1 + (w 1133 + w 1313 ) –1 (w 1133 + w 1313 ) –1 Á l12 ˜¯ Ë

¥

r m w 3113 w 3113 a a sinh rr sinh mm det ||g ij || = 0 r m x1 x1 x1 x1

9.23

2

Ê l 22 ˆ r r m m 1 + (w 1133 + w 1313 ) –1 (w 1133 + w 1313 ) –1 Á l12 ˜¯ Ë r m ¥ w 3113 w 3113 sinh a r coth a m det ||g ij || = 0 x1r x1m x1r x1m

9.24

It was proved by Guz (1989a, b) that the inequalities

w3113 > 0, x1 > 0

9.25

always hold for any material. It is obvious from eqn 9.14 that ar > 0 and am > 0. Hence, the roots of eqns 9.21–24 will be defined by the (4 ¥ 4) characteristic determinant

det ||g rs || =

g 11

g 12

0

0

g 21

g 22

g 23

g 24

0

0

g 33

g 34

g 41

g 42

g 43

g 44

9.26

=0

The elements of the determinant for the first (shear) mode of stability loss (Fig. 9.5) are given below: r r r r g 11 = (w 1111 w 1313 + x 2r –2 w 3113 w 1133 )cosh a r x 2r , –1

r r r r g 12 = (w 1111 w 1313 + x 3r –2 w 3113 w 1133 )cosh a r x 3r , –1

r r r r r r r r g 21 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2r w 3113 w 3333 ) –2

¥ x 2r –1 sinh a r x 2r

–1

292

Multi-scale modelling of composite material systems r r r r r r r r g 22 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3r w 3113 w 3333 ) –2

¥ x 3r –1 sinh a r x 3r

–1

m m m m m m m m g 23 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2m w 3113 w 3333 ) –2

¥ x 2m –1 sinh a m x 2m

–1

m m m m m m m m g 24 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3m w 3113 w 3333 ) –2

¥ x 3m –1 sinh a m x 3m

–1

m m m m g 33 = –(w 1111 w 1313 + x 2m w 3113 w 1133 ) cosha m x 2m , –2

–1

m m m m g 34 = –(w 1111 w 1313 + x 3m w 3113 w 1133 ) cosha m x 3m , –2

–1

r r g 41 = (– w 1111 + x 2r w 3113 ) cosha r x 2r , –2

–1

r r g 42 = (–w 1111 + x 3r w 3113 ) cosh a r x 3r , –2

–1

r m g 43 = (w 1111 – x 2m w 3113 ) cosha m x 2m , –2

–1

m m g 44 = (w 1111 – x 3m w 3113 ) cosh a m x 3m –2

–1

9.27

For the second (extension) mode of stability loss (Fig. 9.5): r r r r g 11 = (w 1111 w 1313 + x 2r w 3113 w 1133 ) cosh a r x 2r , –2

–1

r r r r g 12 = (w 1111 w 1313 + x 3r w 3113 w 1133 ) cosh a r x 3r , –2

–1

r r r r r r r r g 21 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2r w 3113 w 3333 ) –2

¥ x 2r sinh a r x 2r , –1

–1

r r r r r r r r g 22 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3r w 3113 w 3333 ) –2

¥ x 3r sinh a r x 3r , –1

–1

m m m m m m r m g 24 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3m w 3113 w 3333 ) –2

¥ x 3m cosh a m x 3m , –1

–1

Predicting fracture of laminated composites

293

m m m m m m m m g 23 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2m w 3113 w 3333 ) –2

¥ x 2m cosh a m x 2m , –1

–1

m m m m g 33 = – (w 1111 w 1313 + x 2m w 3113 w 1133 )sinh a m x 2m , –2

–1

m m m m g 34 = – (w 1111 w 1313 + x 3m w 3113 w 1133 )sinh a m x 3m , –2

–1

r r g 41 = (– w 1111 + x 2r w 3113 )cosh a r x 2r , –2

–1

r r g 42 = (– w 1111 + x 3r w 3113 )cosh a r x 3r , –2

–1

m m g 43 = (w 1111 – x 2m w 3113 )sinh a m x 2m , –2

–1

m m g 44 = (w 1111 – x 3m w 3113 )sinh a m x 3m

–1

–2

9.28

For the third mode of stability loss (Fig. 9.5): r r r r g 11 = (w 1111 w 1313 + x 2r w 3113 w 1133 )sinh a r x 2r , –2

–1

r r r r g 12 = (w 1111 w 1313 + x 3r w 3113 w 1133 )sinh a r x 3r , –2

–1

r r r r r r r r g 21 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2r w 3113 w 3333 ) –2

¥ x 2r cosh a r x 2r , –1

–1

r r r r r r r r g 22 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3r w 3113 w 3333 ) –2

¥ x 3r cosh a r x 3r , –1

–1

m m m m m m m m g 24 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3m w 3113 w 3333 ) –2

¥ x 3m cosh a m x 3m , –1

–1

m m m m m m m m g 23 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2m w 3113 w 3333 ) –2

¥ x 2m cosh a m x 2m , –1

–1

m m m m g 33 = –(w 1111 w 1313 + x 2m w 3113 w 1133 )sinh a m x 2m , –2

–1

m m m m g 34 = –(w 1111 w 1313 + x 3m w 3113 w 1133 )sinh a m x 3m , –2

–1

r r g 41 = (– w 1111 + x 2r w 3113 )sinh a r x 2r , –2

–1

294

Multi-scale modelling of composite material systems r r g 42 = (– w 1111 + x 3r w 3113 )sinh a r x 3r , –2

–1

m r g 43 = (w 1111 – x 2m w 3113 )sinh a m x 2m , –2

–1

m r g 44 = (w 1111 – x 3m w 3113 )sinh a m x 3m , –2

–1

9.29

For the fourth mode of stability loss (Fig. 9.5): r r r r g 11 = (w 1111 w 1313 + x 2r w 3113 w 1133 )sinh a r x 2r , –2

–1

r r r r g 12 = (w 1111 w 1313 + x 3r w 3113 w 1133 )sinh a r x 3r , –2

–1

r r r r r r r r g 21 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2r w 3113 w 3333 ) –2

¥ x 2r cosh a r x 2r –1 , –1

r r r r r r r r g 22 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3r w 3113 w 3333 ) –2

¥ x 3r cosh a r x 3r –1 , –1

m m m m m m m m g 23 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 2m w 3113 w 3333 ) –2

¥ x 2m sinh a m x 2m –1 , –1

m m m m m m m m g 24 = (w 1133 w 1133 + w 1133 w 1313 – w 1111 w 3333 + x 3m w 3113 w 3333 ) –2

¥ x 3m sinh a m x 3m –1 , –1

m m m m g 33 = – (w 1111 w 1313 + x 2m w 3113 w 1133 ) cosh a m x 2m –1 , –2

m m m m g 34 = – (w 1111 w 1313 + x 3m w 3113 w 1133 ) cosh a m x 3m –1 , –2

r r g 41 = (– w 1111 + x 2r w 3113 )sinh a r x 2r –1 , –2

r r g 42 = (– w 1111 + x 3r w 3113 )sinh a r x 3r –1 , –2

m m g 43 = (w 1111 – x 2m w 3113 ) cosh a m x 2m –1 , –2

m m g 44 = (w 1111 – x 3m w 3113 ) cosh a m x 3m –1 –2

9.30

Similarly, the characteristic equations can be derived for the case of uniaxial compression or other modes of stability loss. The proposed method can also give the solutions for modes with periods, which are equal to 3, 4, 5, ….

Predicting fracture of laminated composites

295

periods of the internal structure. However, based on the experience of solving similar problems (Guz, 1991a, 1998; Guz and Soutis, 2000; Babich et al., 2001; Soutis and Guz, 2001, Guz and Herrmann, 2003), the modes with the larger periods are not of practical interest. Other modes with periods, which are not multiples of the period of the internal structure, can also be examined. The solution for them would be based either on the Floquet theorem for ordinary differential equations with periodic coefficients (Brillouin, 1953), or on reducing the problem to an infinite set of equations with the consequent solution by a numerical method (Shul’ga, 1981).

9.3.1

Results for elastic layers

As an example, let us consider a composite consisting of alternating linearelastic isotropic compressible layers with different properties (Young’s moduli E and Poisson’s ratios n). Then for the reinforcement layer we have

(s ij0 ) r = d ij

Er n r Er e0 + e0 (1 + n r )(1 – 2n r ) nn 1 + n r ij

9.31

Emn m Em e0 + e0 (1 + n m )(1 – 2n m ) nn 1 + n m ij

9.32

and for the matrix

(s ij0 ) m = d ij

In the case of equi-biaxial compression the components of tensor w for this model are expressed as r w 1111 =

È 2n r2 Er 0 Ê 1 – n r + e 11 1– Á Í (1 + n r )(1 – 2n r ) Î 1 – nr Ë

ˆ˘ ˜ ˙, ¯˚

r w 3333 =

E r (1 – n r ) Er r r , w 1313 = w 3113 = , (1 + n r )(1 – 2n r ) 2(1 + n r )

r w 1133 =

1 + nr ˆ Er n r Er Ê r 0 , w 1331 = 1 + 2 e 11 , (1 + n r )(1 – 2n r ) 2(1 + n r ) ÁË 1 – n r ˜¯

m w 1111 = m w 3333 =

m w 1133 =

È 2n m2 Em 0 Ê 1 – n + e 1 – m 11 Á (1 + n m )(1 – 2n m ) ÍÎ 1 – nm Ë

ˆ˘ ˜ ˙, ¯˚

E m (1 – n m ) Em m m , w 1313 = w 3113 = , (1 + n m )(1 – 2n m ) 2(1 + n m )

1 + nm ˆ Emn m Em Ê 0 m , w 1331 = 1 + 2e 11 (1 + n m )(1 – 2n m ) 2(1 + n m ) ÁË 1 – n m ˜¯ 9.32

296

Multi-scale modelling of composite material systems

Substituting eqn [9.32] into the characteristic equation derived for the particular mode of stability loss, a transcendental equation is deduced. For each of the modes we have a different characteristic equation in terms of two 0 (applied strain) and ar (normalised half-wavelength). As a variables, e 11 result of solving the characteristic equations for different modes of stability (N) loss, the dependencies e 11 (a r ) are obtained, where N is the number of the mode – in this case N = 1, 2, 3, 4. The critical value for the particular mode, e cr(N ) , can be found as a minimum of the corresponding dependence. The minimal of these four values will be the critical strain of internal instability for the considered layered material determined by the most accurate approach: e crsl , if eqn [9.26] is used, and e crpb , if the corresponding characteristic equation for the case of perfectly bonded layers is used, i.e., (N) ˆ e crsl/ pb = min e cr( N ) = min ÊË min e 11 ¯ N ar N

9.33

Values of e crsl and e crpb , which represent, respectively, the lower and the upper bounds respectively of the composites with interfacial defects with connected edges, are given in Fig. 9.6 and 9.7 for the first two modes of stability loss. The results for equi-biaxial compression, Fig. 9.6, are given in comparison with uniaxial compression, Fig. 9.7. The latter case was considered in detail in by Guz (1990, 1992) for various models of the layers. All results in the section, which correspond to uniaxial compression, are calculated following Guz (1991a, 1998), Guz and Soutis (2001a). According to the suggested estimation, eqn 9.13, critical strains for materials with unidentified set of cleavage-type delaminations lie between curves representing e crsl and e crpb . One can see that the larger the ratio E r /Em, the lower is the critical strain. To the contrary, the ratio does not affect the accuracy of the estimation. Another important observation is that for the considered layer properties (i.e. linear-elastic compressible layers) the critical values for the first mode are significantly lower than for the second mode. At that, the accuracy of the estimation for the second mode is much better. It differs from the case of non-linear incompressible layers where both modes (the first and the second) were able to give the lowest critical strain depending on the layer properties and the accuracy of estimation did not vary too much for different modes. The type of loading (i.e. equi-biaxial or uniaxial compression) does not affect the accuracy of the estimation very much. Of course, critical strains depend on the type of loading, being in the case of equi-biaxial compression smaller than in the case of uniaxial compression. The latter can be explained from the physical point of view. Indeed, in the case of equi-biaxial compression, the composite experiences more loading than in the case of uniaxial compression of the same intensity.

Predicting fracture of laminated composites

297

Upper bound Lower bound

5%

Second mode

Critical strains

4%

3%

2%

First mode

1%

0% 100

200 300 Young’s moduli ratio

400

9.6 Values of critical strains under equi-biaxial compression of linear elastic layers plotted against E r /Em for the case of h r /hm = 0.4, nr = 0.28, nm = 0.35.

9.3.2

Results for elastic-plastic layers

Now, let us consider a layered composite where the reinforcement behaves as a linear-elastic isotropic compressible material, eqn 9.30, while the matrix response is elastic-plastic incompressible described by the following relationship for equivalent stress (s I0 ) and strain ( e I0 ):

s I0 = Am ( e I0 ) k m

9.34

The constitutive equation, eqn 9.34, is typical for metal matrix composites (Honeycombe, 1968; Pinnel and Lawley, 1970). Problems of internal stability of such materials were investigated within the most accurate approach in previous studies. Characteristic determinants (4 ¥ 4) were derived for perfectly bonded (Guz, 1989b, 1998) or sliding without friction layers (Guz, 1991a, 1998). They have the form of eqn 9.13, but the expressions for grs are different. Applying the above method to this kind of material, the upper and the lower bounds for the critical strain, ecr, in

298

Multi-scale modelling of composite material systems 7% Upper bound Lower bound 6% Second mode

Critical strains

5%

4%

3% First mode 2%

1%

0% 100

200 300 Young’s moduli ratio

400

9.7 Values of critical strains under uniaxial compression of linear elastic layers plotted against E r /Em for the case of h r /hm = 0.4, nr = 0.28, nm = 0.35.

composites with an unidentified set of cleavage-type delaminations can be estimated. The bounds, eqn 9.33, are presented in Figs 9.8 and 9.9 for composites with typical values of constants vr and km. In the case of elastic-plastic matrix both modes (the first and the second) can give the lowest critical strain depending on the layer properties. The accuracy of estimation did not vary too much for different modes – see also Guz (1998). This conclusion differs from the case of linear elastic compressible layers considered in the previous subsection. It should also be noticed that critical strains increase with increasing relative stiffness of fibres or with decreasing value of parameter km. An increase in the fibre Poisson’s ratio, nr, affects critical strains causing them to decrease slightly. Besides that, if the fibre volume fraction is very small (h r /hm < 0.02), the critical strains do not depend on the layer thickness ratio. In this case, the investigation of instability of the considered composite laminates is reduced to investigation of instability of an elastic layer between two half-spaces (or half-planes, if a plane problem is considered) with properties

Predicting fracture of laminated composites

299

3.6%

3.2%

Critical strains

2.8%

2.4%

2.0% Upper bound Lower bound 1.6% 0.0005 0.0015 0.0025 0.0035 Ratio of parameter A and Young’s modulus E

9.8 Bounds for critical strain for composites with elastic-plastic matrix under equi-biaxial compression (h r /hm = 0.11, nr = 0.32, km = 0.2).

of the matrix. Numerical results for composites undergoing equi-biaxial compression or uniaxial compression show that the strain bounds present a reasonable estimation for certain fibre volume fractions and mechanical properties, see Figs 9.8 and 9.9. Therefore, they may be considered as the first approximation on the way to the exact solution of the stability problem in compression along interfacial defects.

9.4

Conclusions

The present analysis finds the upper and the lower bounds for critical compressive loads acting on layered materials with imperfect interlaminar adhesion. The substantiation of the bound is based on a general principle of mechanics, which reads that the release from a part of connections inside of the mechanical system cannot increase the value of the critical load. In order to calculate the bounds for critical loads the problem of internal instability is solved within the scope of the exact statement using the model of a piecewisehomogeneous medium and the equations of 3-D stability theory. This allows

300

Multi-scale modelling of composite material systems 3.8%

3.4%

Critical strains

3.0%

2.6%

2.2%

Upper bound Lower bound 1.8% 0.0005 0.0015 0.0025 0.0035 Ratio of parameter A and Young’s modulus E

9.9 Bounds for critical strain for composites with elastic-plastic matrix under uniaxial compression (h r /hm = 0.11, nr = 0.32, km = 0.2).

us to eliminate the restrictions imposed by using approximate theories as well as the inaccuracies they involve. Special attention was paid to accounting for elastic-plastic deformations and the biaxiality of the compressive loads. The obtained results for practical composites show that the bounds present a good estimation for particular modes and material properties. Therefore, the bounds may be considered as the first step on the way to the exact solution of the problem of stability in compression of composites with particular interlaminar defects. It should be stressed that the most practical composite materials contain not only interlaminar, but also various sorts of intralaminar defects. The effect of intralaminar damage can be accounted for by considering layers with reduced stiffness properties following Kashtalyan and Soutis (2000a, b, 2001, 2002a, b).

9.5

References

Aboudi J., (1987), Damage in composites – modelling of imperfect bonding, Compos. Sci. Technol. 28(2), 103–128.

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Babich I. Yu. and Guz A.N., (1969), Deformation instability of laminated materials, Soviet Appl. Mech., 5(3), 53–57. Babich I. Yu. and Guz A.N., (1972), On the theory of elastic stability of compressible and incompressible composite media, Polymer Mech., 5. Babich I. Yu., Guz A.N. and Chekhov V.N., (2001), The three-dimensional theory of stability of fibrous and laminated materials, Int. Appl. Mech., 37(9), 1103–1141. Berbinau P., Soutis C. and Guz I.A., (1999), On the failure criteria for unidirectional carbon fibres composite materials under compression. Int. Appl. Mech., 35(5), 462– 468. Biot M.A., (1965), Mechanics of incremental deformations, New York, Wiley. Brillouin L., (1953), Wave propagation in periodic structures, New York London, Dover. Budiansky B. and Fleck N.A., (1994), Compressive kinking of fibre composites: a topical review, Appl. Mech. Rev., 47(6), S246-S270. Dow N.F. and Grunfest I.J., (1960), Determination of most needed potentially possible improvements in materials for ballistic and space vehicles. General Electric Co., Space Sci. Lab, TISR 60 SD 389. Fleck N.A., (1997), Compressive failure of fiber composites, Adv. Appl. Mech., 33, 43– 117. Guynn E.G., Bradley W.L. and Ochoa O., (1992), A parametric study of variables that affect fibre microbuckling initiation in composite laminates: part 1: analyses, part 2: experiments, J. Compos. Mater., 26(11), 1594–1627. Guz A.N., (1967), The stability of orthotropic bodies, Soviet Appl. Mech., 3(5), 17–22. Guz A.N., (1969), On setting up a stability theory of unidirectional fibrous materials, Soviet Appl. Mech., 5(2), 156–162. Guz A.N., (1990), Mechanics of fracture of composite materials in compression, Kiev, Naukova Dumka (in Russian). Guz A.N., ed. (1992), Micromechanics of composite materials: focus on Ukrainian research, Appl. Mech. Rev. 45(2), 15–101. Guz A.N., (1999), Fundamentals of the three-dimensional theory of stability of deformable bodies, Berlin Heidelberg, Springer-Verlag. Guz I.A., (1989a), Spatial nonaxisymmetric problems of the theory of stability of laminar highly elastic composite materials, Soviet Appl. Mech., 25(11), 1080–1085. Guz I.A., (1989b), Three-dimensional nonaxisymmetric problems of the theory of stability of composite materials with a metallic matrix, Soviet Appl. Mech., 25(12), 1196–1201. Guz I.A., (1991a), Plane problem of the stability of composites with slipping layers, Mech. Compos. Mater., 27(5), 547–551. Guz I.A., (1991b), Asymptotic accuracy of the continual theory of the internal instability of laminar composites with an incompressible matrix, Soviet Appl. Mech., 27(7), 680– 685. Guz I.A., (1998), Composites with interlaminar imperfections: substantiation of the bounds for failure parameters in compression, Compos Part B, 29(4), 343–350. Guz I.A. and Herrmann K.P., (2003), On the lower bounds for critical loads under large deformations in non-linear hyperelastic composites with imperfect interlaminar adhesion, Eur. J. of Mechanics – A/Solids, 22(6), 837–849. Guz I.A. and Soutis C., (2000), Critical strains in layered composites with interfacial defects loaded in uniaxial or biaxial compression, Plast Rubber Compos, 29(9), 489– 495. Guz I.A. and Soutis C., (2001a), A 3-D stability theory applied to layered rocks undergoing finite deformations in biaxial compression, Eur. J. of Mechanics – A/Solids, 20(1), 139–153.

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Guz I.A. and Soutis C., (2001b), Compressive fracture of non-linear composites undergoing large deformations, Int. J. Solids & Structures, 38(21), 3759–3770. Hill R., (1958), A general theory of uniqueness and stability of elastic-plastic solids, J. Mech. Phys. Solids, 6(3), 236–249. Honeycombe R.W.K., (1968), The plastic deformation of metals, London, Edward Arnold. Kashtalyan M. and Soutis C., (2000a), Modelling stiffness degradation due to matrix cracking in angle-ply composite laminates, Plast Rubber Compos., 29(9), 482–488. Kashtalyan M. and Soutis C., (2000b), Stiffness degradation in cross-ply laminates damaged by transverse cracking and splitting, Compos Part A, 31(4), 335–351. Kashtalyan M. and Soutis C., (2001), Strain energy release rate for off-axis ply cracking in laminated composites, Int. J. Fracture, 112(2), L3-L8. Kashtalyan M. and Soutis C., (2002a), Analysis of local delaminations in composite laminates with angle ply matrix cracks, Int. J. Solids & Structures, 39(6), 1515–1537. Kashtalyan M. and Soutis C., (2002b), Mechanisms of internal damage and their effect on the behavior and properties of cross-ply composite laminates, Int. Appl. Mech., 38(5), 641–657. Librescu L. and Schmidt R., (2001), A general linear theory of laminated composite shells featuring interlaminar bonding imperfections, Int. J. Solids & Structures, 38(19), 3355–3375. Niu K. and Talreja R., (2000), Modelling of compressive failure in fiber reinforced composites, Int. J. Solids & Structures, 37(17), 2405–2428. Parnes R. and Chiskis A., (2002), Buckling of nano-fibre reinforced composites: a reexamination of elastic buckling, J. Mech. Phys. Solids, 50(4), 855–879. Pinnel M.R. and Lawley A., (1970), Correlation of yielding and structure in aluminiumstainless steel composites, Metal. Trans., 1(5), 1337–1348. Rosen B.W., (1965), Mechanics of composite strengthening, in Fiber Composite Materials, American Society of Metals, Metals Park, Ch.3, 37–75. Sadovsky M.A., Pu S.L. and Hussain M.A., (1967), Buckling of microfibers, J. Appl. Mech., 34(12), 1011–1016. Schuerch H., (1966), Prediction of compressive strength in uniaxial boron fibre-metal matrix composite materials, AIAA Journal, 4(1), 102–106. Schultheisz C. and Waas A., (1996), Compressive failure of composites, parts I and II, Progress in Aerospace Science, 32(1), 1–78. Shul’ga N.A., (1981), Fundamentals of mechanics of layered media with periodic structure, Kiev, Naukova Dumka (in Russian). Soutis C., (1991), Measurement of the static compressive strength of carbon fibre/epoxy laminates, Composite Science and Technology, 42(4), 373–392. Soutis C., (1996), Failure of notched CFRP laminates due to fibre microbuckling: a topical review, J. Mech. Behav. Mater., 6(4), 309–330. Soutis C. and Guz I.A., (2001), Predicting fracture of layered composites caused by internal instability, Compos. Part A., 32(9), 1243–1253. Soutis C. and Turkmen D., (1995), Influence of shear properties and fibre imperfections on the compressive behaviour of CFRP laminates, Appl. Compos. Mater., 2(6), 327– 342. Zhuk Y., Soutis C. and Guz I.A., (2001), Behaviour of thin-skin stiffened CFRP panels with a stress concentrator under in-plane compression, Compos. Part B, 32(8), 697– 709.

10 Modelling the compressive response behaviour of monolithic and sandwich composite structures C S O U T I S, University of Sheffield, UK, S M S P E A R I N G, University of Southampton, UK and P T C U R T I S, Integrated Systems, UK

10.1

Introduction

Failure in compression of fibre composite laminates with an open hole is by the initiation and growth of a microbuckle from the edge of the hole.1, 2 They find that failure is governed by microbuckling in the 0∞ plies. The geometric inhomogeneity induces fibre rotation under increasing applied load; deformation localises within a band and a microbuckle is initiated (Fig. 10.1). The microbuckle then propagates in a stable manner for 2–3 mm and the component fails at a higher load than the initiation load. Soutis et al.1, 2 compared this process in carbon-epoxy and carbon-PEEK laminates to an equivalent crack containing cohesive stresses. It is a crack bridging analysis that predicts the size of the buckled region as a function of the applied load, with the local s• Particular fibres Initial geometry

Deformed geometry

d j

w

b

s•

10.1 Fibre kink band geometry.

303

304

Multi-scale modelling of composite material systems

stress supported by the buckled fibres decreasing linearly with the closing displacement of the microbuckle (linear softening cohesive zone law). The model is able to predict successfully the effects of hole size and lay-up upon the compressive strength and has been incorporated into a user-friendly computer program by Xin et al.3 In the past, Soutis et al.2 performed tests to obtain both the laminate unnotched strength and the compressive energy release rate associated with fibre microbuckling, which are required as the model’s input. From a design point of view it is desirable to predict these laminate properties from the mechanical properties of the fibres and the matrix and from the lay-up geometry. In this work, the unnotched strength and fracture energy is estimated from a micromechanics model for fibre microbuckling. Predictions of this approach are compared to measured values and the open hole compressive strengths of laminates made from three different commercially available CFRP pre-pregs are discussed. Strength results for honeycomb-cored composite sandwich panels loaded in compression are also presented.

10.2

Modelling techniques

The compressive strength of long, aligned carbon fibre-reinforced plastics (CFRP) is significantly lower (30–40%) than the tensile strength of the material due to kink band formation introduced by fibre instability (microbuckling). The earliest attempt to model this behaviour was given by Rosen4 who predicted compressive strength based on an elastic fibre microbuckling and related buckling strength to the in-plane shear modulus of the composite (G12). However, failure of modern filamentary composites occurs because of local non-linear matrix deformation at composite strains well below the yield strain of the matrix and elastic analysis substantially over predicts compressive strength. Current models attribute the low compression strength and the mechanism of kink band formation to initial fibre misalignment (waviness) but fibre and fibre-matrix interface properties may also play an important role. The compressive strength is further reduced by the presence of fastener holes and access cut-outs. In this section a brief description of the Budiansky-Fleck and Soutis (BFS) analytical model that appears in a composites design tool3 is presented; later, the predictions of this approach are compared to experimental data.

10.2.1 Unnotched compressive strength It is now well established that the unnotched strength, sc, of unidirectional carbon fibre-epoxy laminates is governed by fibre microbuckling which is associated with non-linear shear of the polymer matrix initiating from regions of pre-existing fibre waviness (of magnitude only few degrees). For an elastic-

Modelling compressive response behaviour

305

perfectly plastic body Budiansky5 showed that 1

2 È ˘2 Ê s Ty ˆ 2 t y Í1 + Á tan b ˙ ˜ Ë ty ¯ ÍÎ ˙˚ s= f0 + f

10.1

where ty and sTy are the in-plane shear and transverse yield stresses of the composite, respectively. f0 is the assumed fibre misalignment angle in the kink band, f is the additional fibre rotation in the kink band under a remote stress s, and b is the band orientation angle, as shown in Fig. 10.1. The critical stress s = sc is achieved at f = 0 in eqn 10.1. By using the above kinking theory, the unnotched strength of the unidirectional laminate can be obtained in terms of the shear properties of the composite and the initial fibre misalignment. Once the failure stress of the 0∞ ply is known, the compressive strength of any multidirectional 0∞dominated lay-up, sun, can be estimated simply by the stiffness ratio method,

s un =

s c N (k) (k) S n E xq NE1 k =1

10.2

where sun is the unnotched laminate strength, sc is the strength of the 0∞ lamina, N is the total number of the laminae in the laminate, E1 is the 0∞ ply stiffness in the fibre direction, n is the number of plies of a given orientation q, and Exq is the modulus of a ply of orientation q in the loading direction (x). Alternatively, a ply-by-ply failure analysis using the classical laminate theory and the maximum stress criterion could be performed for more accurate predictions. However, previous studies6 have indicated that eqn 10.2 results in a very good agreement with measured strengths of multi-directional laminates that fail by 0∞ fibre microbuckling.

10.2.2 Open hole compressive strength Holes are commonly required for fastening the laminate to the substructure and become critical regions under compressive loading. Previous work by Soutis and co-workers1, 2 has found that open holes cause more than 40% reduction in the strength of carbon fibre-epoxy and carbon fibre-PEEK laminates and that damage was initiated by fibre microbuckling in the 0∞ plies at the edge of the hole. This process has been modelled with varying degrees of sophistication. Early models assumed that failure occurred when the maximum stress in the structure equals the unnotched strength of the material (maximum stress criterion) underestimating considerably the residual strength of the composite. To account for the local ‘ductility’ of the material, researchers applied the average stress or point stress failure criteria. They introduced a characteristic length by assuming that fracture depends on attaining

306

Multi-scale modelling of composite material systems

a critical stress (= unnotched strength) at a characteristic distance d0 ahead of the notch or a critical average stress along a characteristic length a0 ahead of the cutout. The characteristic distance is used as a free parameter to be fixed by best fitting the experimental data. Soutis et al.1 compared the damage zone (microbuckling surrounded by delamination) at the edge of the hole to a through-thickness line crack containing cohesive stresses.

10.2.3 Linear softening cohesive zone model Consider compressive failure of a finite width, multi-directional composite panel, which contains a central circular hole. It is assumed that microbuckling initiates when the local compressive stress parallel to the 0∞ fibres at the hole edge equals the unnotched strength of the laminate sun, that is

kts • = sun

10.3 •

where kt is the stress concentration factor (SCF) and s is the remote axial stress. For an orthotropic laminate kt is given by Lekhnitskii’s analysis7

kt = 1 +

Ê 2Á Ë

ˆ Ex E – n xy ˜ + x Ey G xy ¯

10.4

where E, G and n are the in-plane laminate elastic constants (x is the loading axis). Damage development by microbuckling of the 0∞ plies is represented by replacing the damage zone by an equivalent crack, loaded on its faces by a normal traction, T, Fig. 10.2, which decreases linearly with the crack closing s• s un

T

2V Hole Equivalent crack

s un

l s

•

10.2 Damage zone is modelled as a line crack, loaded on its faces by a normal traction, T.

Modelling compressive response behaviour

307

displacement (CCD), 2v. It is assumed that the length of the equivalent crack l represents the length of the microbuckle. When the remote load is increased the equivalent crack grows in length, thus representing microbuckle growth. The evolution of microbuckling is determined by requiring that the total stress intensity factor at the tip of the equivalent crack equals zero, Ktot = K• + KT = 0 •

10.5 •

where K is the Mode I stress intensity factor due to the remote stress s and KT is the stress intensity factor due to the local bridging traction T across the faces of the equivalent crack. When this condition is satisfied, stresses remain finite everywhere.8 The equivalent crack length from the circular hole is deduced as a function of remote stress s • using the following algorithm. For an assumed length of equivalent crack l, we solve for s • and for the crack bridging tractions by matching the crack displacement profile from the crack bridging law to the crack profile deduced from the elastic solution for a cracked body.2 The cracked body is subjected to a remote stress s • and crack face tractions T. The crack is discretised into a number n of elements and the loading T on the crack flanks is represented by piecewise constant loading packets, each of magnitude Ti. As the crack advances, the number of elements increases. The linear stress-displacement relationship in the crush zone, Fig. 10.2, allows direct calculation of the local tractions Ti from the local crack surface displacements vi using the expression, T vi = vc Ê I i – i ˆ s un ¯ Ë

10.6

where Ii ∫ 1 for i = 1, 2,…n, and n is the number of segments into which the cohesive zone is divided and vc is the critical crack displacement. The normal displacement vi of an element of crack surface is calculated by adding the • displacement v is due to remote stress s • and vT due to local stress acting over the buckled zone,

v i = v is

•

+ v iT

10.7

Equation 10.7 is combined with eqns 10.5 and 10.6 to give an expression for the applied compressive stress as a function of microbuckling length, l, unnotched strength, sun, critical crack closing displacement, vc, laminate elastic properties, E and geometry (plate width, W and hole radius, R), i.e., n

s • = S b i Ti = f ( l , s un , v c , E , R, W ) i=1

10.8

Detailed expressions for the functions b i and Ti are given in ref. 2. At a critical length of equivalent crack, lcr, the remote stress s • attains a maximum

308

Multi-scale modelling of composite material systems

value, designated scr, and catastrophic failure occurs. The model contains two unknown parameters, which can be measured independently or predicted analytically; the unnotched strength sun and the critical CCD vc, which is related to the area Gc (fracture energy) under the assumed linear traction – crack displacement curve.

10.2.4 Fracture energy associated with fibre microbuckling Rice9 has shown that the work done to advance the crack by unit area, Gc’ equals the area under the crack traction versus crack displacement curve (s – v) and for a linear softening cohesive zone law, Fig. 10.2, GC is given by

Gc = 2

Ú

vc

0

s ( v )dv = s un v c

10.9

where vc is the critical crack closing displacement on the crack tractioncrack displacement curve, which is analogous to the crack opening displacement in tension. It is assumed that the fracture energy Gc represents the total energy per unit projected area dissipated by fibre microbuckling, matrix plasticity in the off-axis plies and delamination. In previous work,2 Gc was obtained from a separate compressive kink propagation test, wherein the fracture toughness ( K c = Ys p a ) of a laminate containing a sharpened long slit (= 2a) is measured. The alternative method is to estimate the critical crack closing displacement vc, that appears in eqn 10.9. The crack overlap displacement d (= 2v) represents the end-shortening associated with fibre rotation in the microbuckled band and can be estimated by the kinematic expression

d = 2v = w(1 – cos f)

10.10

where f is the fibre rotation given by f = 2b, and b is the orientation of the microbuckle band as shown in Fig. 10.1. The condition f = 2b is a statement that fibres rotate until the volumetric strain in the band vanishes and then ‘lock-up’, see Fleck et al.10 However, once fibre fracture has occurred, the inclination angle f in the crush zone can increase rapidly (f Æ 90∞) if the load is not removed. The broken fibre segments do not lock-up but are free to move into the bore of the hole or delaminations that have been created between neighbouring plies in order to permit the buckled fibres to undergo both compressive and shear deformation. Following this assumption, the critical crack overlap displacement dc from eqn 10.10 is found equal to the kink band width w, which Budiansky5 related in his microbuckling analysis explicitly to fibre diameter and fibre volume fraction by

Modelling compressive response behaviour

309

1

p d f Ê Vf E f ˆ 3 2v c = 4 ÁË 2t y ˜¯

10.11

where df is the fibre diameter, Ef is the fibre elastic modulus and ty is the inplane shear yield stress of the composite. Once the CCD and unnotched strength are known the fracture energy associated with fibre microbuckling can be obtained from eqn 10.9.

10.3

Predicting compressive response

In the following sections the linear softening cohesive zone model is applied to predict the compressive strength and critical microbuckling length of monolithic carbon fibre-epoxy laminates with an open hole and notched glass fibreepoxy honeycomb sandwich panels loaded statically in uniaxial compression.

10.3.1 Compressive strength of laminates with a hole Experimental work1, 2 has shown that the failure strength of T800/924C carbon fibre-epoxy laminates with an open hole is approximately half that of the unnotched material. However, the experimental data for the six lay-ups examined in ref. 2 lie above the ideally brittle behaviour (maximum stress criterion), due to the development of sub-critical damage in the form of fibre microbuckling accompanied by matrix cracking of the off-axis plies, matrix plasticity and delamination between the plies. This reduces the stress concentration at the edge of the hole and delays final failure at higher applied stresses. Microbuckling of the 0∞ plies nucleates at the sides of the hole at ª 80% of the failure load and propagates like a line-crack into the interior of the specimen. The length of the buckled zone increases with increasing applied load, propagating stably until it reaches a critical length. Then unstable growth begins and the microbuckle traverses the specimen completely. The linear softening model described above predicts that the applied stress attains a maximum at a microbuckle length of approximately 2 mm, depending on lay-up and a softening behaviour thereafter. For the T800/924C carbon fibre/epoxy system, using eqn 10.11 and material data (vf = 0.65, Ef = 294 GPa, ty = 60 MPa) a kink band width of about 12 fibre diameters (ª70 mm) is obtained, which is representative of observed kink band widths, 60–100 mm, see Soutis et al.2, 11 and Berbinau et al.6 In Table 10.1, the theoretical notched strength (sn) and critical microbuckling length (lc) at the hole edges of a laminate fabricated from three currently used carbon fibre/epoxy composite systems are simulated by the cohesive zone model. Using a value of vc = 40 mm appears to predict the notched strength of all three systems very accurately. The difference between theory and experiment is less than 5%, giving confidence to this fracture approach.

310

Multi-scale modelling of composite material systems Table 10.1 Notched compressive strength of a [0/ ± 45/0/90/0/ ±45/0/90/0/ ±45/0]s laminate (hole diameter = 6mm, d/w = 0.167, f 0 = 3∞, b =15∞, vc = 40 mm) Composite system

sunth MPa

Gc KJ/m2 (eqn 10.9)

snth MPa

lc mm

snexp MPa

T800/922

728 (728 ± 37) 743.8 (743.8 ± 23) 743.8 (661 ± 55)

29.1

368.1

1.95

370.4 ± 8

29.8

371.3

1.86

390 ± 4

29.8

371.2

1.86

377 ± 6

T800/924 T800/927

Note: ( ) experimental data

The fracture energy Gc for the three composite systems, using the theoretical unnotched strength and the linear softening crack bridging curve with vc = 40 mm, is found equal to ª29 kJ/m2, which is in good agreement with experimental values (23–29 kJ/m2) obtained in earlier work.11 The 15%– 20% discrepancy observed between predictions and the measured unnotched compressive strength in Table 10.1, is attributed to Euler bending that occurred during testing and thickness variation across the width; most of the specimens failed prematurely near the end-tab. Unnotched compressive strength data is more difficult to generate than tensile data because compression testing is sensitive to factors such as Euler buckling, specimen geometric imperfections, specimen misalignment in the test fixture, fibre misalignment in the specimen and bending/stretching in the laminate.12

10.3.2 Compressive response of composite sandwich panels The honeycomb-cored composite sandwich panel is an efficient structural configuration that is employed in aerospace applications ranging from helicopter rotor blades to secondary structures in commercial and military aircraft.13 One of the factors limiting the wider usage of sandwich panels, particularly for commercial aircraft primary structures, is concern regarding their damage tolerance.14 The combination of thin face-sheets and the low transverse strength of the core material make them susceptible to impact damage.15 Therefore, in keeping with the damage tolerant design approach for flight critical structures, appropriate procedures must be developed to predict the effect of such damage on the load-carrying capability. In this section, the experimental strength data obtained by Toribio and Spearing16 for E-glass/epoxy-NomexTM sandwich panels containing throughholes loaded in compression are analysed using the cohesive zone model.

Modelling compressive response behaviour

311

The work was conducted to address one, idealised, aspect of this wider problem, namely the effect of through-thickness open holes on the compressive strength of honeycomb sandwich panels with composite facesheets.

10.3.3 Materials and specimen geometry The specimens tested in the study16 were fabricated from sandwich panels consisting of woven E-glass/epoxy face-sheets and Nomex™ honeycomb cores. In all cases the core material was nominally 25.4 mm thick with a density of 48 kg/m3. Two face-sheet fibre architectures were investigated. ‘A-type’ face-sheets consisted of two plies of (0/90) eight-harness satin weave with approximately 800 fibres per bundle, giving a nominal face-sheet thickness of 0.49 mm. ‘B-type’ face-sheets consisted of three plies of (0/90) fourharness satin weave with approximately 200 fibres per bundle, giving a nominal face-sheet thickness of 0.28 mm. Specimens of three sizes were cut from the panels, of nominal dimensions: 51 mm ¥ 152 mm, 102 mm ¥ 305 mm, and 203 mm ¥ 406 mm, respectively. The longer dimension corresponds to the 0∞ fibre direction and the loading axis. Circular holes with diameters of 12.7 mm, 25.4 mm, and 50.8 mm were then drilled using diamond tooling at the centre of the specimens through both face-sheets and the core to give a constant ratio of hole diameter to specimen width of 0.25 for all the specimens; a schematic of the specimen geometry is shown in Fig. 10.3. w

Thru notch

h

Nomex 25 mm

10.3 Test specimen configuration.

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Multi-scale modelling of composite material systems

10.3.4 Mechanical testing Compression tests were performed on a digitally controlled servo-hydraulic load frame equipped with a 500 kN load cell. Monitoring of load, stroke and strain data was performed via A/D boards into a desktop computer running a commercially available data acquisition software package. The data acquisition rate was 2 Hz. Tests were performed in displacement control. Compressive loads were first applied at a stroke rate of 0.25 mm/min to approximately 80% of the specimen’s predicted ultimate failure load, and then slowed to 0.013 mm/min in order to be able to interrupt the test and monitor initiation and propagation of damage near the edge of the hole. The alignment of the specimen was checked by comparing the strain readings from two far field back-to-back strain gauges.

10.3.5 Damage characterisation Linear damage zones (LDZs), resembling a line crack, were the only macroscopically visible damage mechanisms. They initiated in the facesheets at the horizontal edges of the hole, shown schematically in Fig. 10.4(a), and were easily distinguishable as sharply defined light-coloured areas against the background of the beige face-sheets. Cross-sectioning studies were carried out in order to determine the damage mechanisms present in the LDZs, as

LDZ

View direction Kinking Loading direction (a)

0.284 mm (b)

10.4 (a) A schematic showing the development of LDZ at the edge of the hole. (b) Cross-section of a material B face-sheet specimen immediately behind the LDZ tip. Warp tow kink bands are clearly visible.17

Modelling compressive response behaviour

313

well as the sequence of damage mechanism development. Specimens were cross-sectioned using diamond tooling and then polished down to 1 mm diamond grit. Successive cross-sections were used to generate a complete picture of the damage evolution employing optical and scanning electron microscopy. The key compressive micro-mechanisms were found to be warp fibre and warp tow micro-buckling, (Fig. 10.4(b)) and weft tow transverse cracking in both A and B materials. By inspection of the material immediately in the wake of the LDZ tip it was found that out-of-plane microbuckling of individual warp tow fibres led to warp tow kinking immediately behind the LDZ tip. Figure 10.5 shows an SEM micro-graph of the micro-mechanisms in a type A face-sheet in the region near the edge of the hole. The warp tows can be seen to have failed by kink band formation, which is accommodated by transverse cracking of the weft tows and delamination. The width of the kink band is 70–80 mm, which is very similar to the size measured for the carbon fibre-epoxy systems discussed earlier. Post-mortem X-ray and cross-sectioning and microscopy indicated no detectable damage away from the linear damage zone. Strain gauges were used to measure local deformation as the LDZ propagated across the width of the specimen; a strain-softening behaviour was observed in the LDZ wake, justifying the application of the cohesive zone model. Out-of-plane kinking

Splitting

Kinking

200 mm

10.5 SEM micrograph of material A face sheet cross-section several millimetres behind the LDZ tip. Warp tow kinking and weft tow cracking are clearly visible.

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Multi-scale modelling of composite material systems

10.3.6 Strength data and predictions Unnotched compressive strength The measured unnotched compressive strengths of the woven E-glass/epoxy face-sheets for material A and B are 162 MPa and 126 MPa, respectively. These data, may be used to infer the degree of fibre misalignment in the woven fabric. Assume that at the instant when compressive failure occurs, the matrix in the off-axis tows is at yield whilst the axially aligned warp tows support the stress necessary to trigger plastic fibre microbuckling. The strength of the warp tows can then be deduced from the rule of mixtures expression

sc = sfVfA + smy(1 – VfA)

10.12

where sc is the compressive strength of the composite, VfA (ª 0.3) is the volume fraction of axial tows, sf is the stress supported by the axial tows and smy (ª 60 MPa) is the matrix yield stress. The deduced value of axial tow compressive strength sf is substituted in the Budiansky expression, eqn 10.1, and fibre misalignment angles of f0 = 4.5∞ (ª 0.0795 radians) for material A and f0 = 6.3∞ (ª 0.109 radians) for material B are obtained; the shear yield stress for the epoxy resin is taken equal to ty = 35 MPa and the kink band orientation angle b = 15∞. Polished axial sections of the woven composites and optical microscopy could confirm the warp tows misalignment.18 These fibre misalignment angles are almost twice those observed for unidirectional carbon fibre-epoxy laminates, but this is due to the woven nature of the glass-epoxy composite causing substantial reduction in the axial compressive strength of the material. The simple fibre microbuckling model, eqn 10.1, together with eqn 10.12 predicts quite reasonably the fibre misalignment of the composite. Of course, the method can be applied to predict the compressive strength of the composite for a given fibre waviness, measured experimentally. However, it does not consider the different fibre architecture of the face-sheet, eight-harness satin weave for A-type versus four-harness satin weave for B-type material. The volume fraction of the fibres in the axially aligned warp tows VfA has a dominant influence on the axial properties of the two-dimensional (2-D) composite. For material A and B, VfA ª 0.3 and this is consistent with the measured value of the axial modulus of the composite (Ex = 22.1 GPa), Table 10.2. A crude rule-of-mixtures estimate gives Ex ª VfAEf, where Ef is the axial Table 10.2 Face sheet engineering elastic constants and fracture toughness Material

E1 GPa

E2 GPa

G12 GPa

n12

sunexp MPa

Gc KJ/m2

nc mm

A B

22.1 22.1

22.1 22.1

3.8 3.8

0.11 0.11

162 126

16 16

98.7 126.9

Modelling compressive response behaviour

315

modulus of the E-glass fibres. For the 2-D weave, Ef = 70 GPa and the estimated axial modulus Ex =21 GPa, which is in good agreement with the measured value of 22.1 GPa. Notched compressive strength The measured notched strength for both material types is more than 60% higher than the value predicted by the maximum stress failure criterion (ideally brittle response), eqn 10.3, due to the development of damage near the edge of the hole in the form of matrix cracking, delamination and warp tow fibre microbuckling, Fig. 10.3. This local damage reduces the stress concentration (elastic SCF, kt = 4.077) at the edge of the hole and delays final failure to higher applied stresses. In Table 10.2 the face sheet engineering elastic constants together with an estimated fracture energy GC are presented. They were used in the cohesive zone model, eqn 10.8, to predict the effect of the open hole on the compressive strength of the sandwich panel and also to estimate the critical microbuckling length (allowable flaw size) that developed at the edge of the hole before catastrophic failure occurred. The theoretical notched strength results are shown in Table 10.3 and compared to experimental measurements for three different hole sizes. The difference between theory and experiment is less than 10%. The specimens exhibit a ‘hole size effect’ widely reported for tests on notched laminates and successfully captured by the cohesive zone model; the ultimate failure stress decreases as the hole size increases. The critical buckling length of 7–9 mm corresponds to the value at which the ultimate stress is reached, after this catastrophic failure occurs. However, in the experiment larger damage lengths were measured, 10–30 mm, because the core supports the damaged material, which is not the case for the monolithic laminates. Also, in the experiment LDZs mainly initiated in only one face-sheet leading to larger damage zones. Table 10.3 Open hole compressive strength of sandwich panels d mm

W mm

sn MPa

l crA mm

sn MPa

l crA mm

12.5

51

5.9

102

50.8

203

77.8 (85) 66.7 (70) 55.1 (53)

8.9

25.4

90.6 (92) 74.7 (71) 63.4 (69)

th, A

5.8 4.6

Note:( ) experimental data, c.o.v ª 5%

th,B

9.3 9.1

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Multi-scale modelling of composite material systems

The fracture energy value of 16 KJ/m2 and eqn 10.9 would result to a crack closing displacement vc of 98–127 mm (Table 10.2), which is much larger than the value of about 40 mm observed experimentally (Figs 10.4 and 10.5). This suggests that in the case of the sandwich panel, although fibre microbuckling is still the critical failure mechanism, a significant amount of energy might be dissipated in other failure modes (‘energy sinks’), such as matrix cracking and delamination. However, in the authors’ opinion the most likely explanation is that the woven nature of the face-sheet and the honeycomb core largely contribute by giving lateral support to the crushed material in the linear damage zone allowing larger local displacement and absorbing more energy, resulting in a more damage-tolerant structure. Taking vc = 40 mm the fracture energy Gc that results from eqn 10.9 would be 6.5 KJ/m2 for A-type and 5.05 KJ/m2 for B-type material, leading to more conservative values for the notched strength (15%–20% lower than the measured strengths). It is important to note that in earlier work17 an attempt was made also to describe the local strain distributions and the LDZ propagation with a consistent traction law. The present approach is focused on optimising the description of the notched compressive strength as a function of hole size and does not attempt to capture these other effects. It is still an open question as to how best to model all three behaviours (i.e., damage growth, strain redistribution and notched strength) in a consistent analytical framework. This is potentially a fruitful topic for future work.

10.4

Conclusions

The compressive failure of current unidirectional and multidirectional carbon fibre-epoxy laminates is controlled by fibre microbuckling. Its initiation depends on material imperfection, such as resin rich regions, voids and fibre misalignment (waviness). The fibres break at two points to create a kink band inclined at an angle b = 5∞–25∞ to the transverse direction. The kink band width w is approximately equal to 60–80 mm (12–14 fibre diameters). In multidirectional laminates, 0∞ fibre microbuckling is accompanied by delamination between the off-axis and 0∞ layers, and by plastic deformation in the off-axis plies. The Budiansky plastic buckling analysis5 accounts for most of the experimental observations and correctly predicts that the shear properties and fibre waviness are the most important parameters affecting the compressive strength of the composite. Although the initial fibre misalignment can be quite arbitrary, for current CFRP systems values of f0 = 2∞–3∞ produce strength predictions for unidirectional and multidirectional laminates within 10–15% of the measured data. The linear softening cohesive zone model of Soutis et al.1, 2 successfully predicts the effects of hole size upon the compressive strength and buckle zone size at failure of monolithic laminates. In the analysis, the inelastic

Modelling compressive response behaviour

317

deformation associated with fibre microbuckling and matrix plasticity is mathematically replaced with a line-crack loaded across its faces by a bridging normal traction that decreases linearly with the overlap displacement of the microbuckle. The model takes as its input the laminate unnotched compressive strength sun and the critical crack overlap displacement, vc or fracture energy Gc. In this work, vc is related to the kink band width w and the fracture energy associated with the crush zone predicted by the new method is in good agreement with experimental Gc values obtained in previous work2 from a separate compressive kink band propagation test. The theoretical notched strengths for three different carbon fibre-epoxy prepregs are in good agreement with experimental measurements. The model applied to E-glass/ epoxy-Nomex TM sandwich panels containing through-holes also gives acceptable strength predictions. Specimens with four and eight harness satin weave fabric face-sheets were studied.16, 17 It was observed that in these materials the principal failure mechanism consists of linear damage zones (LDZs) emanating from the notch tip. LDZs are macroscopically similar to fibre-bridged cracks in tension, and propagate in a stable manner. Cross-sectioning indicated that the LDZ wake is characterised by kinking in all warp tows and weft tow cracking.16, 17 Strain gauges were used to measure local deformation as the LDZ propagated across the width of the specimen; a strain-softening behaviour was observed in the LDZ wake, justifying the application of the cohesive zone. The difference between prediction and measurement for the compressive strength of these sandwich panels is found to be less than 10% (Table 10.3). The maximum stress failure criterion would underestimate the failure stress by more than 60% for both material types leading to a more conservative design and therefore a heavier structure.

10.5

References

1. Soutis C., Fleck N.A. and Smith P.A., (1991), ‘Failure prediction technique for compression loaded carbon fibre-epoxy laminate with an open hole’. J. Comp. Mat., 25, 1476–1498. 2. Soutis C., Curtis P.T. and Fleck N.A., (1993), ‘Compressive failure of notched carbon fibre composites’. Proc. R. Soc. Lond. A (1993), 440, 241–256. 3. Xin X.J., Sutcliffe P.P.F., Fleck N.A. and Curtis P.T., (1995), ‘Composites Compressive Strength Modeller’. Cambridge University Engineering Department, CUED MAT/ TR139. 4. Rosen B.W., (1965), ‘Mechanics of composites strengthening’. In: Fiber Composite Materials. American Society of Metals Seminar, Ohio: ASM, 1965, 37–75 (Chapter 3). 5. Budiansky B., (1983), ‘Micromechanics’. Computers and Structures, 16(1), 3–12. 6. Berbinau P., Soutis C., Goutas P. and Curtis P.T., (1999), ‘Effect of off-axis ply orientation on 0∞ fibre microbuckling’. Composites A, 30(10), 1197–1207. 7. Lekhnitskii S.G., (1968), ‘Anisotropic Plates’. Cheron, Gordon and Breach Science Publishers Inc., NY, 1968.

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8. Dugdale D.S., (1960), ‘Yielding of steel sheets containing slits’, J. Mech. Phys. Solids, 8, 100–104. 9. Rice J.R., (1968), ‘Mathematical analysis in the mechanics of fracture’. Ed. H. Liebowitz, Academic Press, Chapter 3 of Fracture, vol 2. 10. Sivashanker S., Fleck N.A. and Sutcliffe M.P.F., (1996), ‘Microbuckle propagation in a unidirectional carbon fibre-epoxy matrix composite’. Acta. Mater., 44(7), pp. 2581–2590. 11. Soutis C. and Curtis P.T., (2000), ‘A method for predicting the fracture toughness of CFRP laminates failing by fire microbuckling’. Composites: Part A, 31(7), 733–740. 12. Soutis C., Smith F.C. and Matthews F.L., (2000), ‘Predicting the compressive engineering peformance of carbon fibre-reinforced plastics’. Composites: Part A, 31(6), 531–536. 13. Smith B.A., (1995), ‘Airframes stress durability in composites’, Aviation Week and Space Technology, 142, 60–61. 14. Abbot R., (1998), ‘Damage tolerance evaluation of composite honeycomb structures’, International SAMPE Symposium and Exhibition (Proceedings), 43, SAMPE, Covina, CA, USA, 1998, 376–386. 15. Lie S.C., (1989), ‘Damage resistance and damage tolerance of thin composite facesheet honeycomb panels’, Massachusetts Institute of Technology, TELAC Report 89–3, 1989. 16. Toribio M.G., Mirazo J.M. and Spearing S.M., (1999), On the compressive response of notched composite-honeycomb sandwich panels AIAA paper 99-1415, Presented at AIAA SDM meeting, St. Louis, April 1999. 17. Toribio M.G. and Spearing S.M., (2001), ‘Compressive response of notched glassfiber epoxy/honeycomb sandwich panels’. Composites: Part A, 32, 859–870. 18. Soutis C. and Turkmen D., (1995), ‘Influence of shear properties and fibre imperfections on the compressive behaviour of GFRP laminates’. Applied Composite Materials, 6(2), pp. 327–342.

11 Modelling composite reinforcement by stitching and z-pinning X S U N, H - Y L I U, W Y A N, L T O N G and Y - W M A I, The University of Sydney, Australia

11.1

Introduction

Fiber-reinforced composite laminates have been widely used in weight-critical applications, especially in aircraft structures, due to their significant advantages over traditional engineering materials. However, a critical drawback to this traditional layered 2D architecture is its relatively low interlaminar fracture toughness, which makes the laminates susceptible to delamination when subjected to interlaminar stress concentrations. For aircraft structures made from CFRP composites, delamination is inevitable as a result of low energy impact, such as hail impact, accidental impacts from dropped tools during manufacturing, maintenance and servicing, and impacts from stones on the tarmac during take-offs and landings. To retain adequate strength, stiffness and fatigue performance of the delaminated composite structures, thick laminates or improved interlaminar properties are often employed. Apparently, an increase in laminate thickness leads to higher structural weight and material costs. Consequently, a considerable amount of research has been devoted to improving the damage tolerance of a composite structure, either by selecting better materials or by utilizing stitching, knitting, z-pinning, weaving or braiding in order to introduce through-thickness reinforcement into the laminate. As only minor improvements are achieved in delamination toughness by using tougher matrices, there have been more recent efforts devoted to throughthickness reinforcement by stitching and z-pinning [1–4]. Figure 11.1 shows the three most common types of stitches used: the lock stitch, modified lock stitch and chain stitch. In this technology, a high tensile strength yarn, such as carbon, glass or Kevlar fiber, etc., is inserted into a prepreg laminate or fabric using a sewing needle, which may lead to localized damage to the prepreg laminate in the penetration process of the sewing needle and yarn. This damage can be fiber breakage at the stitch hole and misalignment of the fibers in the laminate as they are forced to spread around the stitches. As a result, stitching can cause considerable damage by breaking, spreading and kinking the fibers, the formation of resin-rich regions, porosity 319

320

Multi-scale modelling of composite material systems Needle thread

(a)

Bobbin thread Needle thread

(b)

Bobbin thread

(c)

11.1 Schematic diagram of (a) lock stitches, (b) modified lock stitches and (c) chain stitches [2].

and resin cracks as well as by stress gradient effects [2]. According to Freitas et al. [5], z-pinning converts a 2D prepreg lay-up to 3D on-tool with little or no change to standard cure cycles. As shown in Fig. 11.2, a foam preform containing small diameter rigid fibers is placed on top of the prepreg lay-up on-tool. During cure or debulk, a combination of heat and pressure compacts the foam, which in turn pushes the fibers into the composite. The through-thickness fibers are elastically supported by the foam to prevent buckling during insertion. After cure or debulk, the foam residue and excess fiber is removed along with the bleeder and release ply. The z-fibers or z-pins used in this technology can have a diameter ranging from 0.2 to 0.6 mm, using either composite pins or metallic pins in any desired pattern and spacing. Apparently, z-pinning may introduce damage to the laminates similar to stitching. A review conducted by Mouritz et al. [2] shows that stitching can affect a wide variety of in-plane mechanical properties. It may improve, degrade or retain the in-plane mechanical properties depending on a large number of interacting factors, including the types of laminate (e.g. prepreg tape versus woven fabric), lamination technique (e.g. prepreg lay-up versus liquid moulding), stitching condition (that is, stitch density, type, orientation, yarn diameter, thread tension) and loading configuration. Conversely, Freitas et al. [5] reported that z-pinning results in a 50% reduction in impact delamination

Modelling composite reinforcement by stitching and z-pinning Fibers 1 Place release film and z-preform on top of prepreg lay-up, then bag

Backing

321

Vacuum bag

Foam Prepreg Tool

Release film Pressure 2 During standard cure or debulk cycle, heat and pressure compact foam, forcing the zfibers through the laminate

3 Remove compacted foam and discard along with bagging materials

Elevated temperature or debulk

Z-fibers

Remove and discard foam

Cured laminate

11.2 Schematic of z-fiber insertion process [5].

area, ability to retain co-cured stiffeners on skins subjected to ballistic impact, retention of 91 to 98% of in-plane tensile properties and no reduction in inplane compressive properties. However, due to the complicated damage mechanisms of stitching and z-pinning, it is suggested that each type of laminate must be thoroughly tested and evaluated after stitching or z-pinning to determine any change to the in-plane properties. Despite likely reductions in in-plane properties, the out-of-plane properties of the laminates are significantly improved by stitching and z-pinning, especially for delamination toughness [1, 4–6]. Many theoretical models were proposed to evaluate the enhancement of stitching and z-pinning on delamination toughness in the past two decades. In the meantime, some experiments have been performed to test the improvement in damage resistance and tolerance arising from stitching and z-pinning. In this chapter, these models and experiments on the delamination toughness of laminated composites with through-thickness reinforcements by stitching and z-pinning are reviewed.

11.2

Micro-scale models for stitching and z-pinning

The bridging law of stitching and z-pinning is essential to assess the role of through-thickness reinforcement on the delamination toughness of composite laminates. Although some single fiber pullout tests reveal this complex

322

Multi-scale modelling of composite material systems

relationship between the bridging traction and the pull-out length, current micro-scale models for stitching and z-pinning, which are widely used, are mostly based on Jain and Mai’s frictional stitching model or Cox’s fibrous tow model considering the lateral ploughing effect. In this section, these two models are introduced together with some other bridging laws for z-pins presented by Liu et al. Another metallic z-pin model developed by Tong and Sun, in which the effect of bending resistance capacity on the bridging law is taken into account, is also presented.

11.2.1 Jain and Mai’s stitching model (independent and interconnected) [7–11] Two micromechanics-based models have been proposed by Jain and Mai to study the effect of through-thickness reinforcement (stitching) on improving delamination crack growth resistance. In the first model, it is assumed that the stitches are not interconnected, (referred to as independent stitches) as in most cases the top and the bottom surfaces of the stitched laminates are ground off to remove surface in-plane waviness caused by the stitching loops. In the second model, the interconnected stitches are considered. Force-displacement relationship for an independent stitch Consider a single thread embedded in a matrix with embedded length H as shown in Fig. 11.3. The slip length (or slip zone) Y is defined as the portion of the thread carrying the load before pull-out of the thread begins and the slippage distance S is the slippage of the embedded end after pull-out of the thread begins. As the load F increases, the thread is elastically stretched and the slip length increases from 0 to H. During this process, the load carried by the thread increases from zero to its maximum. So long as the slip length

S

H Y d

F (a)

F (b)

11.3 Pull-out of an independent stitch (a) during elastic stretching (b) after slippage of embedded end begins.

Modelling composite reinforcement by stitching and z-pinning

323

reaches the value H without thread rupture, the thread pulls out steadily and the load borne by the thread decreases to zero. Accordingly, Jain and Mai [6] gave the force in the stitching thread as: F = 2tpdf [YU(x1)U(x3) + (H – S) (1 – U(x1)) U(x2)U(x4)]

11.1

where

[

]

x1 = 2 H – H ln (1 + r ) (1 + r ) – d r x2 = H – d

11.2

x3 = LC – Y x4 = LC – H and t is shear stress at the matrix-stitch thread interface. The diameter of the stitching thread is df · d is the pull-out length of a stitch. LC is the critical embedded length and is defined by: LC =

d f s fu 4t

11.3

The step functions U(xi) (i = 1, º , 4) follow the usual definition of the Heaviside function. Accordingly, the equation to solve for the slip length Y is:

Yr Yr ˆ È ˘ f ( Y ) = 2 Í Y – H ln Ê + 1ˆ ˙ Ê 1 + –d=0 r H H¯ Ë ¯ ˚Ë Î

11.4

and the slippage of embedded end S is obtained from: ( H – S)r ( H – S )r ˆ È ˘ f ( S ) = Í H – H ln Ê + 1ˆ ˙ Ê 1 + r H H Ë ¯ Ë ¯ Î ˚ ( H – S ) r ˆ ˘ Ê ( H – S )r ˆ È + Í ( H – S ) – H ln Ê +1 ˙ 1 + –d =0 r H H Ë ¯ ˚Ë ¯ Î

11.5 where the extensibility ratio r is given by:

r=

tp d f H . Af E f

11.6

Here, Af and Ef are cross-sectional area and Young’s modulus of the stitching thread, respectively. It should be pointed out that the bridging thread always pulls out from one side of the crack surface only.

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Multi-scale modelling of composite material systems

Force-displacement relationship for the interconnected stitch In this case, the stitches are assumed to be interconnected and the embedded end remains intact even after the slip length Y increases from 0 to H provided that the thread does not rupture. Therefore, with further increase in load F, both the crack opening displacement and the load at the embedded end F0 increases. When the load in the stitch thread at the loading end approaches that of the thread strength, the thread breaks. The pre-tension in the stitches is also considered in this model, wherein it is assumed that the pre-tension, Ft, remains constant along the whole length of the embedded thread. Then, the load in the thread is given by:

ÈÊ Ft ˆ F = 2tp d f Í Á Y + U ( x 3 )U ( x 5 ) tp d f ˜¯ Ë Î ˘ F + Ft ˆ Ê +Á H + 0 (1 – U ( x 5 ))U ( x 4 )U ( x 6 ) ˙ tp d f ˜¯ Ë ˚

11.7

where È Ê Í F Á Ê ˆ t r x 5 = 2 Í H – H Á 1+ ˜ ln Á 1+ r A E Ft Ë f f ¯ Í Á 1+ A E Ë ÍÎ f f

ˆ ˘È ˜ ˙Í r ˜ ˙ Í1+ Ft ˙ ˜ Í 1+ A E ¯ ˙˚ ÍÎ f f

˘ ˙ ˙–d ˙ ˙˚

x6 = sfu Af – (tp d f H + F0 + Ft) and x3 and x4 are already given in eqn 11.2. The slip length Y is obtained by solving the equation below: È Ï ¸˘ Í Ô Ô˙ Ft ˆ Ô Yr Ê Ô˙ H Í f (Y ) = 2 Y – 1+ ln 1 + ˝ r ÁË A f E f ˜¯ Ì Í Ft ˆ Ô ˙ Ê Ô HÁ1 + Í A f E f ˜¯ Ô˛ ˙˙ ÔÓ Ë ÍÎ ˚

È ˘ Í ˙ Yr ˙ – d = 0. ¥ Í1 + Í Ft ˆ ˙ Ê HÁ1 + Í A f E f ˜¯ ˙˚ Ë Î Similarly, embedded end, F0, is obtained from:

11.8

Modelling composite reinforcement by stitching and z-pinning

È Ï Í Ft ˆ ÔÔ Ê H r f ( F0 ) = 2 Í H – Á 1 + ˜ ln Ì1 + r A E F + F0 Ë ¯ f f Í Ô 1+ t A Ô ÍÎ f Ef Ó

F0 È r+ Í Af E f ¥ Í1 + Ft Í 1+ A ÍÎ f Ef

325

¸˘ ÔÔ ˙ ˝˙ Ô˙ Ô˛ ˙˚

˘ ˙ ˙ –d=0 ˙ ˙˚

11.9

Equations 11.4, 5, 8 and 9 can be solved by using numerical methods for any given crack opening displacement, d.

11.2.2 Liu et al.’s z-pin bridging law [12–15] Tri-linear z-pin bridging law for mode I delamination [12–14] When a delamination crack opens up, a z-pin experiences an axial tension. As a reaction, the z-pin provides a closure bridging force to the laminates against delamination. This process can be described by the load-displacement curve of z-pin pullout. The functional relationship between the bridging force and associated crack opening during pullout is called the bridging law, which should reflect the bridging mechanisms of z-pinning in composite delamination. A test method to measure the z-pin bridging law was introduced [12]. Load-displacement curves were directly measured by pullout of z-pins and from which a tri-linear bridging law was suggested [13]. In this model, a typical z-pin pullout curve can be simplified by a tri-linear function, which is:

Ïd Ô d 1 Pd ÔÔ d – d d – d1 Ps = Ì 2 Pd + , Pf – d d d 2 1 2 – d1 Ô ÔP + d 2 – d P ÔÓ f h – d2 f

(0 £ d £ d 1 ) (d 1 £ d £ d 2 )

11.10

(d 2 < d £ h )

where, Ps is the bridging force of a single pin and d is the pullout displacement. The whole process consists of three stages as shown in Fig. 11.4, in which h is half-length of a z-pin and is equal to the thickness of one cantilever beam. In the first stage, 0 £ d £ d1, the interface between the z-pin and the laminates is perfectly bonded. The bridging force is caused by the elastic deformation of the z-pin. With increasing load, the interfacial shear stress between z-pin and laminates exceeds the interfacial shear strength. Debonding

326

Multi-scale modelling of composite material systems

Pull-out force Ps

(d 1 , P d )

(d 2, Pf )

0

Displacement d

h

11.4 A simplified tri-linear z-pin bridging law [13].

starts and propagates. Hence, in the second stage, d1 £ d £ d2, the bridging force is caused by both the elastic deformation (in the bonded region) and interfacial friction (in the debonded region). After the interface has fully debonded, the z-pin is pulled out from the laminates. The bridging force in this stage, d2 < d £ h, is completely caused by interfacial friction. The four parameters, maximum debonding force Pd, maximum frictional pull-out force Pf and their corresponding displacements d1 and d2 can be determined by the pullout test. In some cases, if there is no (or very weak) initial bonding between the pin and the laminates, the bridging law can be simplified to a bi-linear function determined by two parameters: maximum load and corresponding displacement. Hence, eqn 11.10 is reduced to the first and third expressions with Pd (= Pf) and d1 (= d2), which is the case of frictional pullout so that [14]: Ï Pa d , Ôd P= Ì a Pa Ô Pa – (d – d a ) h – da Ó

0 £ d £ da da £ d £ h

11.11

This pullout model is totally determined by the peak bridging force, Pa, its corresponding pullout displacement,da, and the ultimate pullout displacement, h, equal to half-thickness of the DCB. Z-pin bridging law for mode II delamination [15] Figure 11.5 is a schematic of the pullout process of a z-pin caused by mode II delamination. During delamination the crack faces move relative to each other along the crack growth direction with a displacement 2d at the location

Modelling composite reinforcement by stitching and z-pinning

327

Pin

Composite

T

Crack

2d Pin

11.5 Schematic of the pullout of a z-pin under mode II delamination.

of the pin. Consequently, the pin is forced to pull out of the laminate. We assume a predominant tensile force T exists in the pin, which is the total axial force of a pin against the pin/laminates interfacial shearing and bending. Experiments on ENF by Cartie [16] indicated that the delaminated crack faces were kept open during mode II crack growth. Thus, the friction effect between the crack faces is not included in the model. This crack opening displacement originates from the bending of the pins at the ends that are being pulled out as shown in Fig. 11.5. Since the opening is very small and crack growth is predominantly mode II, bending of the z-pins is neglected in this model. Under mode II shearing, T acts roughly along the crack surfaces in x-direction and its amplitude varies with relative pullout displacement 2d during the pullout process. Hence, the pullout model can be described by the functional relation between force, T, and half pullout displacement, d. The effect of different pullout models on mode I delamination in a DCB has been investigated by Liu et al. [13]. For mode II sliding, a simple bi-linear law is adopted for z-pin pullout, which is:

Ï Ta d , Ôd T= Ì a Ta Ô Ta – (d – d a ), – da h Ó

0 £ d £ da

da £ d £ h

11.12

In the FE model, the pullout process of a pin from the composite laminates is simulated by a non-linear spring whose properties are described by eqn 11.12.

328

Multi-scale modelling of composite material systems

11.2.3 Cox’s fibrous tow model with lateral ploughing [17–19] The bridging law defined by the stitching models proposed by Jain and Mai is related only to axial deformation and pull-out length. Based on observations and measurements on laminates reinforced through-the-thickness by stitches or short rods, Cox presented a fibrous tow model for mode II delamination cracks that is applicable to general mixed-mode cracks, in which the effect of transverse deformation of the bridging tow and its sideway displacement through the surrounding laminates on the bridging law is considered. In Cox’s model, the bridging tow is represented as a beam that can shear and extend axially. Its axial displacement relative to the laminate is opposed by friction over its debonded periphery. The forces associated with the sideway displacement are estimated by contact mechanics of a punch pressed against a plastic medium. The bridging law is thence reduced to a set of 1D equations. The following assumptions are made for Cox’s model: 1. The fibrous tow deforms by shear as a rigid/perfectly plastic material and the axial shear strength is ~70 MPa. 2. Axial sliding of the tow relative to the laminate is opposed by friction characterized by a uniform shear traction. 3. Elastic deformation of the laminate in the out-of-plane direction has negligible contribution to the displacement of the tow at the fracture plane. 4. All strains in the tow are either uniform axial extension or uniform axial shear, that is, planes initially normal to the out-of-plane direction remain so. 5. Lateral ploughing of the tow through the laminate, Fig. 11.6, is governed by the contact mechanics of a punch pressed against a rigid/perfectly plastic medium in plane strain.

z t

x

Delamination fracture surface

Ls

T

11.6 Nomenclature for the analytical model.

Modelling composite reinforcement by stitching and z-pinning

329

The two displacement variables, which describe the state of the stitch, are denoted by x(z) and z(z) in the x and z direction between z and z + dz, respectively (see Fig. 11.7). The axial strain, e, and the rotation of the fibers, q, are given by,

È Ê ∂x ˆ 2 e = ÍÁ ˜ + ÍÎ Ë ∂z ¯ È ∂x q = tan –1 Í Î ∂z

2 ∂z ˆ ˘ Ê 1 + Á ˜ ˙ ∂z ¯ ˙ Ë ˚ ∂z ˆ ˘ Ê Á1 + ˜ ∂z ¯ ˙˚ Ë

1/2

–1 11.13

Ê ˆ ∂x ∂z dz , z + dz ˜ Áx + ∂z ∂z Ë ¯

q

z (x, z)

x

11.7 Deformation of a slice of the stitch.

The axial stress in the stitch, ss, i.e., the stress component in the local stitch fiber direction is:

s s = eE s

11.14

where Es is the axial Young’s modulus of the stitch. The axial force equilibrium equation can be written as: 2t ∂s s = R ∂z

11.15

whence

s s = s 0 – 2t z ( z < l s ) R ls =

Rs 0 2t

where t is interface friction between stitch and surrounding laminates and R is stitch radius. The transverse force equilibrium equation can be given by: Ê s – 2t z ˆ ∂q = – P x R ¯ ∂z Ë 0 where Px is a component of the laminate resistance in x-direction.

11.16

330

Multi-scale modelling of composite material systems

The boundary conditions at the delamination fracture surface are: T = t0 + s0 sin q

z(0) = 0

11.17

where T is bridging shear traction on the stitch and t0 is shear flow inside the stitch. Equations 11.15 and 11.16 with boundary conditions eqn 11.17 are nonlinear and must be solved by iteration. More boundary conditions and mixed mode traction law for an inclined fibrous tow have been discussed in ref. 19.

11.2.4 Tong and Sun’s metallic z-rod model with bending resistance [20–23] Through-thickness reinforcement in the form of short rods can be either a fibrous z-pin or a metallic one. Compared to a fibrous short rod, which is often referred to as z-fiber, a metallic z-pin or z-rod has reasonably high capability to carry transverse loading. That is, a z-rod can provide both axial and transverse bridging tractions to a delamination crack. Thus, the effect of bending resistance of metallic z-rods on delamination toughness of laminated composites should be included in assessing the mechanical properties of zpinned laminates (see Fig. 11.8). A transverse bending resistance and deflection relationship for a metallic z-rod is derived by modeling the pin embedded in a continuous matrix using classic beam theory.

d/2

l q q

d /2

du dl

H

(a)

(b)

11.8 Transverse deformation of a z-rod (a) mode I (b) mode II.

Mode I transverse bending resistance-displacement relationship for a metallic z-rod [20–22] To develop the mode I transverse bending moment-displacement relation (Fig. 11.8(a)), the following assumptions are made. (i) The z-rod is supported by springs that are attached to two rigid foundations each modeling a substrate

Modelling composite reinforcement by stitching and z-pinning

331

beam. (ii) The movements of each rigid foundation are identical to deformations on the neutral plane of the relevant substrate beam at the point of the z-rod. (iii) Support of substrate laminate to the z-rod is linear elastic, so that the support springs are linearly elastic. (iv) The strength of the z-rod is sufficiently high so that it does not break in the whole bending process. (v) No plastic deformation in the z-rod caused by the bending moment is allowed. Based on the above assumptions, the embedded z-rod in the two delaminated substrate beams is modelled as a beam non-uniformly supported with an infinite number of linearly elastic springs, which are attached to two moving rigid foundations. Since a linear elastic spring model is adopted, there can be no plastic deformation in the substrate beams. Hence, any ploughing of the z-rod through the substrate beams cannot happen. The consequence is that this model will exaggerate the role of bending traction provided by a z-rod. For the segment of the z-rod embedded in a substrate beam shown in Fig. 11.9, the governing equation for the transverse deflection of the z-rod resting on an elastic foundation is: Ef If

d 4 wz È ˘ + k Íwz + q Ê x – 1 H ˆ ˙ = 0 2 Ë ¯ dx 4 Î ˚

(0 £ z £ l)

11.18

where wZ is lateral deflection of z-rod, k is stiffness coefficient of elastic x q

H

z-rod

l

A Ou

MA , qA

h

du

M = MA, q = 0

11.9 Mode I transverse deformation of a z-rod embedded in a substrate beam.

332

Multi-scale modelling of composite material systems

springs for the sub-strate beams. Ef and If are Young’s modulus and moment of inertia of the cross-sectional area of z-rod, respectively. Accordingly, the relative contact displacement between a z-rod and its surrounding matrix is defined by: w R = w Z + q ( x – 12 H ). For the free segment of the z-rod sandwiched between the two substrates, the governing equation can be simplified to: Ef If

d 4 wz =0 dx 4

Ê d ˆ Á – 2 £ x £ 0˜ Ë ¯

11.19

The boundary conditions at the free end of the z-rod x = l are: w Z¢¢ | x =l = 0,

w Z¢¢¢ | x =l = 0

11.20a

The symmetrical boundary conditions at x = – du w Z¢ | x =– d u = 0,

w Z¢¢¢ | x =– d u = 0

11.20b

Then, the following expression for the bending moment acting on the z-rod at point A is obtained from: MA =

where

a=

4

Ef I fq d + sinh a l cosh a l + cos a l sin a l 2 a (sinh 2 a l – sin 2 a l )

11.21

k 4E f I f

An asymmetrical model is also developed in ref. 20 for the bending moment carried by a z-rod in a mode I delamination crack. Mode II transverse bending resistance-displacement relationship for a metallic z-rod [23] To develop the mode II transverse shear force-displacement relation of a zrod (Fig. 11.8(b)), we adopt the similar assumptions as for the mode I case. (i) The z-rod is supported by springs, which are attached to two rigid foundations each modeling a substrate beam. (ii) Movements of each rigid foundation are identical to the deformations on the neutral plane of the relevant substrate beam at the location of the z-rod. (iii) The support between the substrate laminate and the z-rod is linearly elastic, namely, the support springs are linearly elastic. (iv) No plastic deformation in the z-rod caused by the transverse bending moment and shear force due to the fact that a metallic z-rod is stiffer and stronger than the substrate resin. (v) Pullout of a z-rod is ignored in derivation of the transverse shear force-displacement relation because the pull-out length is found to be small. It is assumed that the deformations of

Modelling composite reinforcement by stitching and z-pinning

333

the two substrate beams are anti-symmetric. Based on these assumptions, the embedded z-rod in the two delaminated substrate beams is again modeled as a beam non-uniformly supported with an infinite number of linearly elastic springs attached to two moving rigid foundations. Plastic deformation in the substrate beams, hence ploughing of z-rods through the substrate beams, is ignored. It is apparent that only the anti-symmetric in-plane displacement at the point of the z-rods creates the transverse bending moment and shear force in the rods. Thus, the above assumptions will be used to establish the relation between transverse shear force and in-plane displacement of the substrate beam. As shown in Fig. 11.10, equation 11.18 can be used as the governing equation for the z-rod with 0 £ x £ H. The boundary conditions can be written as: w R¢¢ | x = H = 0

11.22a

w R¢¢¢ | x = H = 0

11.22b

x

MB = 0 q

B

z-rod

h

H A MA = 0

11.10 Mode II transverse deformation of a z-rod embedded in a substrate beam.

and the anti-symmetry conditions in the middle of a z-rod are given by: wR |x=0 = 0

11.22c

w R¢¢ | x =0 = 0

11.22d

Then the transverse shear bridging force carried by a z-rod is:

334

Multi-scale modelling of composite material systems

Q ( x ) = E f I f a 3 d ◊ f (a H ) =

dfd 8

4

p E f k 3 ◊ f (a H )

11.23

where f (a H ) =

11.3

(sin 2 a H – sinh 2 a H ) sin a H cos a H – sinh a H cosh a H

11.24

Assessment of macro-scale delamination toughness of reinforced composites

11.3.1 Experiments on mode I, mode II and mixed mode toughness of z-pin reinforced composite laminates To evaluate the improvement of delamination toughness of composite laminates by using the stitching and z-pinning technologies, DCB and ENF specimens have been used to measure mode I and mode II crack growth resistance, respectively [6, 8, 11, 24, 25]. However, due to the much increased delamination toughness caused by the through-thickness reinforcement, i.e., stitches and z-pins, the laminates may break before the delamination crack propagates to the desired length. Therefore, some modifications to these specimens have been proposed to enable a longer delamination crack growth without laminate failure, such as attaching a pair of aluminum sheets to the top and bottom surface of the DCB specimen [8], a new test setup suggested by Chen et al. [24], and a mixed-mode apparatus proposed by Rugg et al. [25]. The schematics of these three testing configurations are given in Figs 11.11 to 11.13, respectively.

11.3.2 Theoretical and computational analyses Analytical solution of stitched DCB specimen (mode I) [7, 8] A semi-infinite double-cantilever-beam (DCB) specimen with throughthickness reinforcement is shown in Fig. 11.14. This test geometry is most suitable for obtaining crack growth resistance curves during mode I delamination. a0 is initial crack length, Da is crack growth characterized by the stitch bridging zone length, H is half-beam thickness, P is applied load per unit width of the beam, d(t) is crack opening displacement and t is measured from the crack-tip. The crack growth condition is governed by the equality of the net stress intensity factor (Kc) to the intrinsic fracture toughness of the composite (KIC). There are two contributions to Kc, one is the stress intensity factor due to the applied load, Ka, and the other, Kr, is due to the closure traction, p(t), acting across the crack faces due to the bridging zone. Thus, the crack growth criterion becomes:

Modelling composite reinforcement by stitching and z-pinning 15 m

25 mm

m

25 mm

25

15 m

mm

142.5 mm (a)

25 mm

m

25 mm

335

25

mm

142.5 mm (b)

11.11 Schematic representation of (a) un-tabbed specimen, and (b) tabbed specimen for mode I indicating specimen and tab dimensions (after [8]). P

L1

L2

L3

q 10

L4

L1

q 10

L4

L2

L3

11.12 Novel DCB test setup proposed by Chen et al. [24].

336

Multi-scale modelling of composite material systems To instron Load saddle 27 mm c

P

18 mm

Lever arm Kapton release ply

7.3 mm

z x

Z-fibers

Hinge Base

L = 66 mm

L = 66 mm

Width approx, 25 mm

11.13 Schematic of mixed mode bending test specimen and apparatus (not to scale) [25].

L

P

Pitch distance Lp Stitches

2H

2 1 3

z x

Da

a0 P

11.14 Semi-infinite double-cantilever beams with through-thickness reinforcement (crack extension starts from the stitch bridging zone).

Kc = Ka + Kr = KIC

11.25

Equation 11.25 may be rearranged and the crack growth resistance KR is defined as: KR(Da) = KIC – Kr(Da)

11.26

According to Foote and Buchwald [26], the stress intensity factor KI at the crack tip of a DCB specimen due to an applied load P (per unit width) at a distance t from the crack tip can be approximately calculated from:

Modelling composite reinforcement by stitching and z-pinning

t KI = C P f Ê ˆ H ËH¯

337

11.27

where ˘ t 2H È t 0.619 f Ê ˆ = 12 Ê t + 0.673 ˆ + – Í 0.815Ê ˆ + 0.429 ˙ pt Î ËH¯ ËH ¯ Ë H¯ ˚

–1

11.28 and C is a correction factor, which depends on the anisotropic elastic material properties. For an orthotropic material, C is calculated from [27]: C=

Eo E

11.29

where Eo is the orthotropic modulus as shown in ref. 7. The formula for f Ê t ˆ is within 1.1% of the exact values for all values of t with a large H ËH¯ un-cracked ligament. The stress intensity factor, Kr, due to the closure traction may be computed from: Kr (Da) = – C

Ú

Da

0

t p( t ) 1 f Ê ˆ dt H Ë ¯ H

11.30

and the equilibrium crack growth condition then becomes

K R ( Da ) = K IC + C

Ú

Da

0

t p( t ) 1 f Ê ˆ dt H ËH¯

11.31

To obtain the crack growth resistance, KR, the applied load corresponding to the critical stress intensity factor at the tip of the crack should firstly be iteratively determined. Then by removing all the stitch elements, the stress intensity factor can be re-calculated for the same geometry and applied load, and this new value of stress intensity factor is KR. Figures 11.15 and 11.16 show typical KR – Da curves for DCBs reinforced by independent stitches and interconnected stitches. Clearly, through-thickness reinforcement significantly improves mode I crack growth resistance and hence inhibits or delays delamination extension. A large interfacial shear stress and high stitch density together with a small stitch thread diameter are desirable to maximize the mode I crack growth resistance. Smaller stitch thread diameter is also desirable from the point of view of reducing in-plane fiber damage caused by stitching.

338

Multi-scale modelling of composite material systems 3.5

KR (MPam1/2)

3

2.5

2

1.5

1

0

6

12

18 24 Da (mm)

30

36

42

11.15 KR – Da curve for the DCB geometry with independent stitches. 3.5

KR (MPam1/2)

3

2.5

2

1.5

1

0

2

4

Da (mm)

6

8

10

11.16 KR – Da curve for the DCB geometry with interconnected stitches.

Analytical solution of z-pinned DCB mode I delamination [13] A theoretical model based on beam theory was developed by Liu, Yan and Mai to study the delamination growth in z-pinned composite laminates [13]. In this model, the z-pin bridging law given in an earlier section was adopted to simulate the effect of z-pinning on delamination resistance. Based on the experimental observation during delamination of the laminates the bending resistance from z-pins is very small compared to their pullout resistance,

Modelling composite reinforcement by stitching and z-pinning

339

only the bridging force of z-pins was incorporated in the model. A DCB specimen is shown in Fig. 11.17, in which the beams are reinforced by Nc columns and Nr rows. z Nc P Pi d

z o

x

Lc

11.17 Mechanics model of a delamination crack bridged by z-pins. Lc is delamination length; P is load corresponding to applied displacement, d.

Figure 11.17 shows the mechanics model of a delamination bridged by zpins. Due to symmetry only one single beam is considered. The bridging force caused by the ith column of z-pins (i = 1, 2 … Nc) at location xi, is added to the beam as an external force, Pi. From a generalized beam theory, the differential equation of the deflection curve is given by: Elz¢¢ = M(x)

11.32

where EI is the flexural rigidity of the laminated beam and M is the bending moment. Before the delamination tip reaches the first column of pins, 0 £ Lc £ x1, there is no bridging force on the beam and the total bending moment is: (0 £ x £ Lc)

M(x) = Px

11.33

The solution of eqn 11.32 for this initial stage becomes:

z = d 3 ( x 3 – 3 xL2c + 2 L3c ) 2 Lc

(0 £ x £ Lc)

11.34

where d is the applied displacement at the loading end, d = z(0). The fracture energy method is used as the delamination criterion [28]. The strain energy release rate is calculated by: ∂U GI = 1 w ∂ Lc

11.35

in which w is the width of the laminated beam; U is total strain energy in the bent beam and is:

U= 1 EI

Ú

Lc

0

M 2 ( x ) dx

11.36

340

Multi-scale modelling of composite material systems

From equations 11.33, 11.35 and 11.36, we obtain GI =

P 2 L2c wEI

11.37

If the energy release rate of the bent beam is greater than a critical intrinsic toughness of the unpinned DCB, GIC, the delamination will propagate. After the crack has passed the first column of pins, Lc > x1, the pins start to provide closure forces to the opening crack. The differential equation of the deflection curve in each interval is given by: Ï Px EIz ¢¢ = Ì Ó Px – P1 ( z )( x – x1 )

(0 £ x £ x1 )

11.38

( x 1 £ x £ Lc )

The boundary conditions in each interval are: z(x1)+ = z(x1)–, z¢(x1)+ = z¢(x1)–, z¢(Lc) = z(Lc) = 0,

11.39

In eqn 11.38, P1 is the total bridging force from the first column of z-pins at the location x = x1, that is, P1 = Nr ¥ Ps. Here Ps is the bridging force from a single pin, which is a function of the flexural displacement z(x1) given by the bridging law, eqn 11.10 in section 11.2.4. Thus, it is mathematically difficult to obtain a closed-form solution of eqn 11.38. Instead, an iteration method is used to obtain a numerical solution. In the iterative calculation, the displacement, d, and the crack length, Lc, are added step by step by a tiny increment. In the first step, a tiny increase in the crack length is given by, Lc = x1 + DLc

11.40

Since the increment DLc is very small, the displacement of the first pin is very small. Thus, the bridging force of the z-pin can be neglected in eqn 11.38. According to eqn 11.34, the displacement of the first pin can be approximately calculated from:

z ( x1 ) = d 3 ( x13 – 3L2c x1 + 2 L3c ) 2 Lc

11.41

Substituting the obtained displacement, z(x1), into the pin bridging law, the bridging force, P1, can be obtained. Increasing the applied displacement, d = d + Dd, and applying the bridging force, P1 from the previous step to eqns 11.38 and 11.39, the deflection of the beam under this applied displacement, d = d + Dd, can be approximately calculated by: 1 Px 3 + Cx + EId Ï Ô 6 EIz = Ì 1 Px 3 – 1 P ( x – x ) 3 + Cx + EId Ô6 1 6 1 Ó

(0 £ x £ x1 ) ( x 1 £ x £ Lc )

11.42

Modelling composite reinforcement by stitching and z-pinning

341

in which P=

C1 =

3EId P1 ( Lc – x1 ) 2 (2 Lc + x1 ) + 3 2 L3c Lc

11.43

PL2c P1 ( Lc – x 1 ) 2 – 2 2

11.44

The new displacement of the first column of pins, z(x1), can be obtained from eqn 11.42. Substituting z(x1) into the bridging law, a new bridging force at this step can be obtained. This bridging force is subsequently applied to calculate the deflection of the beam in the next step. If the number of columns of the bridging z-pins is Nc when the delamination crack length is Lc, the differential equation of the deflection curve in each interval becomes: Ï Ô Px i ÔÔ E Iz ¢¢ = Ì Px – S Pj ( z )( x – x j ) j =1 Ô Nc Ô Px – S Pj ( z )( x – x j ) ÔÓ j =1

(0 £ x £ x1 ) ( x i £ x £ x i+1, i = 1, 2... N c –1) ( x N c £ x £ Lc ) 11.45

Following the above iteration method, the solution at this step can be solved as:

1 Px 3 + Cx + EId Ï 6 Ô Ô i Ô E I z = Ì 1 Px 3 – S 1 Pj ( x – x j ) 3 + Cx + EId j =1 6 6 Ô Nc Ô1 1 P ( x – x ) 3 + Cx + EId Ô Px 3 – jS j j =1 6 Ó6

(0 £ x £ x1 ) ( x i £ x £ x i+1, i = 1, 2... N c –1) ( x N c £ x £ Lc ) 11.46

in which, N

c C = – 1 PL2c + S 1 Pj ( Lc – x j ) 2 j =1 2 2

P=

1 2 L3c

È Nc ˘ Pj ( Lc – x j ) 2 (2 Lc + x j ) + 6EId ˙ Í jS =1 Î ˚

Then, the displacement of the ith column of pins is:

11.47

11.48

342

Multi-scale modelling of composite material systems

E Iz ( x i ) =

i –1 P Px i3 j – S ( x i – x j ) 3 + Cx i + EId i = 1, 2, . . . N c j =1 6 6

11.49 Adding the applied displacement step by step and using the solved displacement to calculate the current bridging force, Pi in eqn 11.48, a new set of displacement z(xi) can be obtained. Combining eqns 11.32, 11.35, 11.36 and 11.45, the strain energy release rate at the current applied displacement can also be obtained. The above process is repeated until the energy release rate is large enough to cause the delamination crack to grow. Analytical solution of stitched ENF specimen (mode II) [9–11] Figure 11.18 shows a typical stitched or z-pinned ENF specimen with a delamination starting from one end in the mid-plane of a symmetric laminated a0

Da

z-rods

P

P/2 P/2 End-notched flexure (ENF) specimen, mid-plane symmetric laminates (a)

L P

P/2 P/2 Beam without delamination (b)

z

t

tR

x

Problem used for approximate stress analysis of ENF specimen (c)

11.18 End-notched flexure (ENF) specimen and its superposition of two sub-problems.

Modelling composite reinforcement by stitching and z-pinning

343

beam under three-point bending. The specimen has a span of 2L and thickness 2H. The initial delamination crack length is a0 and the delamination crack growth is represented by Da. It should be noted that the through-thickness bridging zone starts right behind the initial crack. P is applied load per unit width of the ENF specimen. Following the general approach, this problem can be analyzed as a superposition of two sub-problems, namely, 1. a beam without delamination (Fig. 11.18(b)), and 2. a beam with surface traction t equal in magnitude but opposite in sign to the tractions at the same location in sub-problem 1 with the force in the bridging entities replaced by closure surface traction tR, acting in or on the delamination surface, shown in Fig. 11.18(c). It is easy to find a solution for sub-problem 1 using the classic beam theory. For sub-problem 2, the laminate may be treated as an assemblage of two sub-laminates adhesively bonded to each other at the mid-plane of the whole laminate except at the delamination zone. The in-plane deformations in the top and the bottom sub-laminates is assumed to be equal in magnitude but opposite in sign. Then considering only the top or bottom sub-laminate is sufficient to solve sub-problem 2 due to symmetry. Using the first-order shear deformation laminated plate theory, the governing equilibrium equations are given by: Ï d N x – t + t U( x – a ) = 0 R 0 Ô dx ÔÔ dM x 1 Ì dx + 2 H[t – t R U ( x – a 0 )] = 0 Ô dQ Ô x =0 ÔÓ dx

11.50

where Nx is in-plane stress resultant, Mx is bending moment and Qx is transverse shear force. The constitutive laws are: Ï Nx ¸ ÈA Ô Ô Í ÌMx ˝ = Í B Ô Ô Í Ó Qx ˛ Î 0

B D 0

0 ˘ Ï u , 0x ¸ Ô ˙Ô 0 ˙ Ì q,x ˝ Ô Ô K ˙˚ Óq + w, x ˛

11.51

where u0 is mid-plane displacement, q is bending rotation, and w is out-ofplane displacement. A, B, D, and K are axial, extension-bending coupling, bending and shear stiffness of the sub-laminates, respectively. The function U(x – a0) is a Heaviside step function, which equals unity when x > a0. The shear traction t in eqn 11.50 can be obtained from the solution of the first sub-problem and is approximated by:

344

Multi-scale modelling of composite material systems

Ê ÊHˆ ˆ Á B¢ + Ë 2 ¯ D¢˜ Ë ¯ t = 1P 4 A*

11.52

where 2

A* = A ¢ + B ¢ H + D ¢ H 4 A¢ = D/(AD – B2)

11.53

2

B¢ = B/(AD – B ) D¢ = A/(AD – B2) Accordingly, the closure traction tR representing the equivalent traction obtained from the load carried by an individual z-rod is given in the micro-scale models. Based on the assumed displacement field, the relative slip between top and bottom sub-laminates at the crack plane becomes:

d = 2Ê u 0 – H q ˆ 2 ¯ Ë

11.54

Using the general theory for this type of modeling of laminated composite structures as an assemblage of sub-laminates, Jain and Mai presented a solution to sub-problem 2, including the closure surface traction tR. For the interconnected stitches, the energy release rate GII available for crack propagation can be given by: GIIR =

Ï Ê sin( l D a ) ˆ A* + a˜ Ìt Á l ¯ cosh 2 ( l ¢Da ) Ó Ë

a ¸ – l Ê 1 ˆ sinh( l D a ) ˝ A* Ë a 2 ¯ ˛ where

l=

2

11.55

Ft ˆ Ê Af E f ˆ Ê A*a 2 , a 1 = 2S D Ft , a 2 = 2 S D Á ˜ Á1 + A E ˜ H Ë ¯Ë f f ¯

where SD is stitch density (number of stitches per unit area) and Ft is pretension in the stitch thread. For independent stitches, the energy release rate GII for crack growth is obtained from: GIIR =

Ï Ê sin( l ¢ D a ) ¸ ˆ Ê a 1¢ ˆ A* + a˜ – l ¢ sin( l ¢Da ) ˝ Ìt Á A* Ë a 2¢ ¯ l¢ ¯ cos 2 ( l ¢Da ) Ó Ë ˛

2

11.56

Modelling composite reinforcement by stitching and z-pinning

where

l¢ =

345

A*a 2¢ , a 1¢ = 2S D tp d f H , a 2¢ = 4S D tp d f

Numerical results show that through-thickness reinforcement improves mode II delamination toughness slightly in the case of continuous stitches. In the case of independent stitches, the improvement in the mode II delamination toughness could be negligible. An increase in stitch thread strength, axial modulus of the beam, beam thickness and stitch density and a decrease in the stitch thread Young’s modulus can lead to significantly higher steady-state potential energy release rate in the case of continuous stitches. For discontinuous stitches, an increase in stitch thread-matrix interfacial shear stress, stitch thread diameter, beam thickness and stitch density and a decrease in beam axial modulus lead to higher potential energy release rates. Numerical analysis Finite element (FE) method is also employed to study the effect of throughthickness reinforcement on the mode I and mode II delamination toughness with through-thickness z-pinning/stitching. A stitched or z-pinned DCB or ENF specimen has been simulated by a beam or plate element [29, 30] or a four-node bi-linear plane stress element [14, 15] due to their geometrical characteristics. With the presence of stitches and z-pins in the crack plane, the traditional algorithm of toughness calculation at the crack tip has to be altered accordingly. Finite element analysis of stitched DCB specimen (mode I) [29, 30] Among those complicated parameters of material properties and stitching conditions, stitch density (number of stitches per unit area) and stitch volume fraction are often used to indicate continuously spatially distributed or spatially averaged stitch traction distribution (for plain stitch pattern) along the delamination crack plane. In many cases, the bridging tractions in discrete bridging entities can be conveniently and accurately replaced by continuously spatially distributed or spatially averaged tractions, acting continuously over the fracture surface, which greatly simplifies subsequent bridged crack analysis. Therefore, the stitch distribution is ignored in most predictive analysis. However, stitching pattern may affect the improvement of stitch, especially for those having longer pitch distance and spacing, and it is therefore important to study the effects of stitching distribution. To investigate the effect of stitch distribution on the improved fracture toughness in a stitched DCB specimen, Sun et al. [29] and Wood et al. [30, 31] employed a beam element or a plate element to model the composite laminates in the DCB specimen, a stitch element was also constructed to model the bridging tractions provided by the discrete stitches. The developed

346

Multi-scale modelling of composite material systems

stitch element is a tension-only nonlinear spring element with the axial force equal to the bridging force carried by each stitch under different axial displacement. The stitch distribution and stitch pattern (i.e. plain stitch pattern versus zigzag stitch pattern) were studied by the proposed FE model. Figure 11.19 shows the KR – Da curves for DCBs with independent and interconnected stitches, predicted by continuous tractions (analytical solutions) and discrete tractions (FEA). It is evident that replacing the discrete bridging tractions by the spatially averaged tractions may lead to some error in the prediction of the reinforced delamination toughness of a DCB specimen, which indicates that pitch distance and stitch line spacing should be used as two independent stitching parameters rather than one parameter – stitch density [28]. Stitch pattern plays no significant role in the improvement in crack growth resistance for stitched DCB specimens specifically when the pitch distances and stitch line spacing are similar. However, when stitch density is maintained constant and stitch pitch distance and stitch line spacing are altered, the results are distinctly different. This implies that stitch distribution is a critical parameter in determining the improvement in crack growth resistance of stitched laminated composites [30]. Finite element analysis of z-pinned DCB and ENF specimen (mode I and II) [14, 15] In this section, a contour integral scheme proposed by Yan et al. [14, 15] is firstly introduced and followed by parametric analysis on mode I and mode II delamination with through-thickness z-pins. According to linear elastic fracture mechanics theory, the toughness of a material or structure can be quantified by the energy release rate, G, which is defined as [28] dU e dU s ˆ G= 1Ê – w Ë da da ¯

11.57

where w is width of the crack front equal to the DCB width, a is crack length, Ue is external work performed and Us is stored elastic energy. The energy release rate G represents the energy available for creation of a unit new crack area. For unpinned DCB specimens, the dissipated energy is completely consumed by the surface energy of the newly created crack surface, which is the composite’s intrinsic toughness GIC. Hence, G = GIC

11.58

For z-pin reinforced DCB specimens, the dissipated energy includes not only the crack surface energy but also the energy dissipated during the z-pin pullout process, which includes the elastic energy of the pins, the surface

Modelling composite reinforcement by stitching and z-pinning

347

3.5

KR (MPam1/2)

3

2.5

2 Continuous tractions Lp = 18.0 mm, Ls = 1.67 mm Lp = 15.0 mm, Ls = 2.00 mm Lp = 12.0 mm, Ls = 2.50 mm Lp = 6.0 mm, Ls = 5.00 mm Lp = 3.0 mm, Ls = 10.00 mm

1.5

1

0

5

10

15 Da (mm) (a)

20

25

30

10 Continuous tractions Lp = 15.0 mm, Ls = 2.0 mm Lp = 12.0 mm, Ls = 2.5 mm Lp = 6.0 mm, Ls = 5.0 mm Lp = 3.0 mm, Ls = 10.0 mm

9 8

KR (MPam1/2)

7 6 5 4 3 2 1 0

5

10 Da (mm) (b)

15

20

11.19 Effect of stitch distribution on the delamination toughness of stitched DCB specimen (a) independent stitches (b) interconnected stitches [29].

348

Multi-scale modelling of composite material systems

debonding energy between pins and the laminates and the friction energy consumed during pullout. Therefore, the total energy release rate of a zpinned DCB specimen, G, consists of two parts: the energy release rate for the new crack surface, GIC, and the energy release rate due to z-pin pullout, Gp, which depends on the extent of delamination. That is, G = GIC + Gp(Da)

11.59

The delamination toughness of z-pinned laminates can be completely described by the total energy release rate, G, which is commonly called the crackresistance GR. The FE method is applied to analyze G or GR of z-pin reinforced DCB specimens. For simplicity, the pullout process of a z-pin from the composite laminate is not explicitly simulated in our FE analysis. Instead, the pin effect is simulated by distributed springs along the thickness of the beam at the same location. The properties of these non-linear springs are given by a simplified z-pin bridging law, i.e., eqn 11.11 in section 11.2.2. The energy release rate is calculated by the contour integral method. According to linear elastic fracture mechanics [28], the energy release rate, G, is equal to a contour integral with the integrating path starting from the lower crack surface and ending at the upper crack surface, i.e., G=

Ú W dz – Ë T s

Ê

x

du x du + Tz z ˆ dS , dx dx ¯

11.60

where Ws is strain energy density of the composite, Tx and Tz are components of the traction vector at the section dS of the contour G. ux and uz are displacement components, see Fig. 11.20. Two contours, G1 and G2, are shown in Fig. 11.20. Contour G1 includes only the composite around the crack-tip without the springs, i.e., z-pins. The calculated energy release rate G based on contour G1 corresponds to GIC, the intrinsic toughness of the unpinned composite, i.e., G = GIC. Contour G2 includes all the springs, that is, all the effects of the z-pins. The calculated energy release rate, G, now represents the total energy release rate, which includes the energy dissipation due to the creation of new crack surfaces GIC and the energy dissipation due to the z-pins Gp. According to fracture mechanics theory, the energy release rate criterion is equivalent to the crack opening displacement criterion. The relation between the energy release rate, G, and the COD is (see ref. 32) COD = 4(2)1/4

( a11 a 22 )1/4 Ê 2a12 + a 66 + Á 2a 11 p Ë

a 22 ˆ a11 ˜¯

1/4

G r , 11.61

where a11, a22, a12 and a66 are determined by material elastic constants, which can be found in refs. 14 or 32. The r in eqn 11.61 represents the distance from the crack-tip.

Modelling composite reinforcement by stitching and z-pinning

349

dS Springs

z

COD

x

r

Crack tip

G1

G2

11.20 Integral contours for calculating energy release rates: G1: contour excluding springs (z-pins); G2: contour including all the springs (z-pins).

Crack growth in z-pinned DCB is fairly complex. Many parameters contribute to the failure process. Here, the effects of some major parameters were studied. Following the dimensional analysis in ref. 14, the function of the crack-resistance of the DCB is simplified as:

P d GR Ê Da d ˆ = gÁ , a , a , c , nc ˜ G IC h h G h h Ë ¯ IC

11.62

Thus, the normalized crack-resistance or ‘apparent’ crack toughness of the zpinned DCB, GR /GIC, is completely determined by the dimensionless crack advance, Da/h, normalized location of peak force in the pullout model, da/h, normalized peak force in pullout model, Pa /(GIC h), normalized pin column spacing, d c /h, and number of columns of pins, nc. Based on eqn 11.62, the effects of these parameters on the delamination toughness of DCB were discussed in detail by Yan, Liu and Mai in ref. 14. The fracture resistance can be greatly increased by increasing the normalized peak value, Pa /(GIC h) in the bridging model. However, the results show that the effect of the pullout parameter da on toughness enhancement is not significant. The number of zpin columns represents the size of the z-pinned zone in the crack growth direction for a given column spacing, dc. Therefore, it is clear that the enhanced toughness GR /GIC covers a longer delaminated distance for higher number of z-pin columns. Their results also indicate that interaction between pin columns can also enhance the delamination toughness of the composite laminate. With nc

350

Multi-scale modelling of composite material systems

increasing continuously from 4 to 8, this effect becomes less efficient. For example, GR /GIC are almost identical for nc = 6 and 8. Hence, it is expected that further increasing nc beyond 8 would not lead to any improvement in GR /GIC. The influence of the normalized column spacing, d c /h, on the normalized energy release rate, GR /GIC, was also studied. By keeping the same number of z-pin columns, the delamination toughness was increased with decreasing column spacing. This confirms the interactive effect between z-pin columns. That is, smaller column spacing provides stronger interaction between z-pin columns. This prediction is consistent with reported experimental results [16]. All calculations were performed by a FE software ABAQUS [33]. Similar to mode I, a parametric study on mode II delamination resistance of z-pinned ENF specimens has also been performed via dimensionless analysis. The dimensionless function of the crack-resistance is given by: Ta d GR Ê Da d ˆ = gÁ , a , , c,n ˜ G IIC Ë h h G IIC h h c ¯

11.63

Thus, the normalized potential energy release rate of a z-pinned ENF specimen, GR/GIIC, is completely determined by the dimensionless crack growth, Da/h; normalized location of the peak force in the z-pin pullout model, da /h; normalized peak force, Ta/GIICh; normalized column spacing, d c /h; and number of columns of pins, nc. The effects of all the parameters on mode II delamination toughness of zpinned ENF specimens were studied by Yan, Liu and Mai [15] and the results are shown in Fig. 11.21. Comparison between Fig. 11.21(a) and 11.21(b) showed that both values of da /h and Ta /GIIC h can affect the fracture resistance of ENF. However, the effect of the dimensionless parameter, Ta / GIIC h, is more dramatic than d a /h. It is suggested that smaller da and higher Ta can lead to a higher fracture resistance of ENF against delamination. Furthermore, as shown in Figs 11.21(c) and 11.21(d), smaller column spacing, dc /h, and longer reinforcing zone (larger nc) give higher normalized toughness, GR /GIIC.

11.4

Conclusions

In this chapter, experiments and theoretical analyses of two through-thickness reinforcing technologies to laminated composites, i.e., stitching and z-pinning, have been reviewed. Despite the possible damage to the in-plane properties by stitching and z-pinning, the coupon test by DCB and ENF specimens shows that the poor interlaminar fracture toughness of laminated composites can be significantly and effectively improved by the through-thickness reinforcements.

Modelling composite reinforcement by stitching and z-pinning

351

On the micro-scale level, the typical bridging law for a single interconnected stitch is mainly related to its Young’s modulus and ultimate strength, while the interfacial friction (between a z-pin and its surrounding materials) is the dominant factor for the bridging law of a z-pin. Current proposed bridging laws for stitches and z-pins have been justified by the z-pin pull-out tests and their applications in coupons, i.e., DCB and ENF tests. Further verification of the metallic z-pin model is required to justify the assumptions made. 3.2 da /h = 3.33E-3 da /h = 6.67E-3 da /h = 1.33E-2

2.8

GR / GIIC

2.4 2.0 1.6 1.2 0.8

0

4.0

4

8 D a /h (a)

12

16

8 D a /h (b)

12

16

Ta /GIICh = 95.24 Ta /GIICh = 190.5 Ta /GIICh = 285.7

3.6 3.2

GR / GIIC

2.8 2.4 2.0 1.6 1.2 0.8

0

4

11.21 Numerical results of delamination toughness, GR /G IIC , affected by (a) normalized pullout model parameter, da /h, with nc = 4, Ta /GIIC h = 190.5 and dc / h = 2.33; (b) normalized pullout model parameter, Ta /GIIC h, with nc = 4, da /h = 0.00667 and dc / h = 2.33; (c) z-pin column number, nc, with Ta /G IIC h = 190.5, da /h = 0.00667 and dc / h = 2.33; (d) column spacing, dc / h, with Ta /GIICh = 190.5, da /h = 0.00667 and nc = 4.

352

Multi-scale modelling of composite material systems

nc nc nc nc

2.8

GR / GIIC

2.4

= = = =

1 2 3 4

2.0

1.6

1.2

0.8

0

4

8 Da / h (c)

12

16

3.2

2.8

GR /GIIC

2.4

2.0

1.6 dc /h dc /h dc /h dc /h

1.2

0.8

0

4

8 Da/h (d)

12

= = = =

0.667 1.333 2.333 3.333

16

11.21 Contd.

On the macro-scale level, DCB and ENF specimens reinforced by stitches and z-pins have been comprehensively studied. On one hand, analytical solutions and numerical modelling have both shown that through-thickness reinforcing technologies of stitching and z-pinning, can effectively fulfil their objective, that is, improve the interlaminar toughness of laminated composites. On the other hand, to validate the enhancement on the interlaminar toughness by stitching and z-pinning, some modifications have been made to the standard DCB and ENF specimen geometries and loading configurations. This has been subsequently experimentally substantiated. The major challenge now is to take the micro-scale modelling of crack-

Modelling composite reinforcement by stitching and z-pinning

353

bridging constitutive laws, the material properties determined from macroscale modes I, II and I/II delamintation tests, and apply them to the structuralscale analysis and design for engineering components. This aspect of research work is currently under way.

11.5

Acknowledgements

We would like to thank the Australian Research Council (ARC) and the Cooperative Research Centre for Advanced Composite Structures (CRCACS) for supporting our research on stitching and z-pinning over a number of years. In particular, we are grateful to Dr Lalit K Jain who has made significant contributions to this field but who has now taken up other nonengineering pursuits. YWM and HYL are, respectively, Australian Federation Fellow and Australian Research Fellow supported by the ARC and tenable at the University of Sydney.

11.6

References

1. Dransfield K.A., Baillie C. and Mai Y.-W., Improving the delamination resistance of CFRP by stitching – a review. Composites Science and Technology, 1994. 50: 305– 317. 2. Mouritz A.P., Leong K.H. and Herszberg I., A review of the effect of stitching on the in-plane mechanical properties of fibre-reinforced polymer composites, Composites, Part A, 1997, 28A, 979–991. 3. Emile G. and Matthew H., The assessment of novel materials and processes for the impact tolerant design of stiffened composite aerospace structures. Composites Part A: Applied Science and Manufacturing, 2003. 34(2): 151–161. 4. Tong L., Mouritz, A.P. and Bannister M.K., 3D fibre reinforced polymer composites, Elsevier, 2002. 5. Freitas G., Magee C., Dardzinski P. and Fusco T., Fiber insertion process for improved damage tolerance in aircraft laminates, Journal of Advanced Materials, 1994, 25(4), 36–43. 6. Jain L.K. and Mai Y.-W., Recent work on stitching of laminated composites – theoretical analysis and experiments. Proceedings of the 11st International Conference on Composite Materials, Gold Coast, Australia, 1997, July. 7. Jain L.K. and Mai Y.-W., On the effect of stitching on Mode I delamination toughness of laminated composites. Composites Science and Technology, 1994, 51, 331–345. 8. Dransfield K.A., Jain L.K. and Mai Y.-W., On the effects of stitching in CFRPs – I. Mode I delamination toughness. Composite Science and Technology, 1998, 58, 815– 827. 9. Jain L.K. and Mai Y.-W., Analysis of stitched laminated ENF specimens for interlaminar Mode II Fracture Toughness, International Journal of Fracture, 1994, 68, 219–244. 10. Jain L.K. and Mai Y.-W., Determination of Mode II delamination toughness of stitched laminated composites, Composites Science and Technology, 1995, 55, 241– 253. 11. Jain L.K., Dransfield K.A. and Mai Y.-W., On the effects of stitching in CFRPs – II.

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15. 16. 17. 18. 19. 20.

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Multi-scale modelling of composite material systems Mode II delamination toughness, Composites Science and Technology, 1998, 58, 829–837. Dai S.-C., Yan W., Liu H.-Y. and Mai, Y.-W., Experimental study on z-pin bridging law by pullout test, Composites Science and Technology, 2004, 64, 2451–2457. Liu H.-Y., Yan W. and Mai Y.-W., Z-pin bridging force in composite delamination, In: Blackman B.R.K., Pavan A. and Williams J.G., editors. Fracture of Polymers, Composites and Adhesives II, ESIS Publication 32, 491–502, Amsterdam: Elsevier, 491–502, 2003 Yan W., Liu H.-Y. and Mai Y.-W., Numerical study on the Mode I delamination toughness of Z-pinned laminates. Composites Science and Technology, 2003, 63:1481– 1493. Yan W., Liu H.-Y. and Mai Y.-W., Mode II delamination toughness of z-pinned laminates, Composites Science and Technology, 2004, 64, 1937–1945. Cartie D.D.R., Effect of z-fibres on the delamination behaviour of carbon fibre/ epoxy laminates. PhD thesis, Cranfield Univeristy, U.K. 2000. Cox B.N., A constitutive model for through-thickness reinforcement bridging a delamination crack. Advanced Composite Letters, 1999, 8(5): 249–256. Cox B.N., Constitutive model for a fiber tow bridging a delamination crack, Mechanics of Composite Materials and Structures, 1999, 6, 117–138. Cox B.N. and Sridhar N., A traction law for inclined fibre tows bridging mixed Mode Cracks, Mechanics of Advanced Materials and Structures, 2002, 9, 229–331. Tong L. and Sun X., Bending effect of through-thickness reinforcement rods on Mode I delamination toughness of DCB specimen I: Linearly elastic and rigidperfectly plastic models, International Journal of Solids and Structures, 2004, 41(24– 25): 6831–6852. Tong L. and Sun X., Effect of through-thickness reinforcement bending on delamination toughness of composite laminates. Proceedings of the 35th International SAMPE Technical Conference, Dayton, Ohio, USA, September 28–October 2, 2003. Sun X., Tong L. and Rispler A., Design curves for metallic z-pinned composite laminated double cantilever beams, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, California, 19–22 April, 2004. Tong L. and Sun X., Bending effect of through-thickness reinforcement rods on Mode II delamination toughness of ENF Specimen I: Linearly elastic and rigidperfectly plastic analyses. Composite, Part A, submitted. Chen L.S., Sankar B.V. and Ifju P.G., A new Mode I fracture test for composites with translaminar reinforcements. Composites Science and Technology, 2002. 62(10-11): 1407–1414. Rugg K.L., Cox B.N. and Massabo R., Mixed mode delamination of polymer composite laminates reinforced through the thickness by z-fibers. Composites Part A, 2002. 33: 177–190. Foote, R.M.L. and Buchwald V.T., An exact solution for the stress intensity factor for a double cantilever beam, International Journal of Fracture, 1985; 29: 125–34. Ye L., Evaluation of Mode I interlaminar fracture toughness for fiber-reinforced composite materials, Composites Science and Technology, 1992; 43: 49–54. Williams J.G., Fracture mechanics of anisotropic materials. In: Friedrich K. editor. Application of fracture mechanics to composite materials. Amsterdam: Elsevier, 1989. Sun X., Tong L., Wood M.D.K. and Mai Y.-W., Effect of stitch distribution on Mode

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30.

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I delamination toughness of laminated DCB specimens. Composites Science and Technology, 2004. 64(7–8): 967–981. Wood M.D.K., Sun X., Tong L. and Mai Y.-W., The effect of stitch pattern on Mode I delamination toughness of laminated DCB specimens, The 14th International Conference of Composite Materials, San Diego, USA, July 14–18, 2003. Wood M.D.K., Sun X., Tong L., Katzos A. and Rispler A., The effect of stitch distribution and stitch pattern on Mode I delamination toughness of stitched laminated composites, The 4th Asian-Australasian Conference on Composite Materials, Sydney, Australia, 6–9 July 2004. Suo Z., Delamination specimens for orthotropic materials. ASME Journal of Applied Mechanics, 1990; (57): 627–634. ABAQUS 2001 Version 6.2. Providence, RI: HKS Inc.

12 Finite element modelling of brittle matrix composites V C A N N I L L O, University of Modena and Reggio Emilia, Italy and A R B O C C A C C I N I, Imperial College London, UK

12.1

Introduction

Composites are a class of materials which include several types and combinations of different constituents. The behaviour of such materials is strongly affected by the properties of the two or more phases as well as by the morphology and spatial arrangement of the materials which constitute the microstructure. Since it is well known that the overall effective mechanical behaviour of composites is greatly affected by microstructural features, such as second phase shape, orientation, dimensions and volume fraction, it is crucial to set up models which are able to relate microstructure parameters with the resulting performance. Regarding the mechanical properties of composite materials, in particular, several different strengthening and toughening mechanisms may arise, depending on the constituents’ characteristics as well as on the conditions at the interfaces between constituents. Several analytical and computational approaches have been proposed in order accurately to describe the behaviour of composite structures. The various classes of composites require different modelling strategies. For example, classical composites containing long unidirectional fibres can be thoroughly described by means of micromechanics equations. Since these composites have a microstructure which is reproducible over the whole composite length, it is relatively straightforward to set up analytical relationships which correlate the composite constituents’ properties and their relative amount with the overall performance. On the other hand, other composite typologies, such as particulate composites containing irregularly shaped inclusions, are not easily described by means of simplified analytical approaches. Due to the processing technologies employed, these materials frequently have a random distribution of irregularly shaped particles dispersed in the volume and thus a more complex modelling strategy is usually required. Therefore, due to the heterogeneity of the microstructure, numerical techniques are often exploited. Among others, the finite element method (FEM) has proven to be an effective technique. This 356

Finite element modelling of brittle matrix composites

357

approach, originally developed for studying structural problems, has now been adapted to analyse long fibre composite laminates. The significant characteristic of the method is its versatility, since it can be applied to model materials with arbitrary geometry and constitutive properties. In recent years, the finite element technique has been extended to investigate heterogeneous microstructures of complex materials. The advantage is that random-particulate composites of inhomogeneous microstructure can be characterised taking into account actual microstructural features. In this chapter, we describe the finite element method with regard to its application to model mechanical properties of particulate composite materials. In particular, the focus of the chapter is the discussion of microstructurebased approaches for the determination of the elastic properties and fracture propagation behaviour of brittle-matrix composites. The chapter is organised in the following manner. In sections 12.2 and 12.3, the basics of FEM are briefly summarised and relevant references for further reading are suggested. The fundamentals of standard FE analysis applied to long fibre composites are given. In section 12.4, microstructure-based approaches suitable for the investigation of particulate composites are presented, while section 12.5 introduces a series of examples of applications of the numerical models on glass matrix composites and on porous materials, considered a limiting case of composites. Finally, in section 12.6 the scope for future developments is outlined.

12.2

Numerical approaches: the finite element method (FEM)

This section is intended to review briefly the fundamentals of the finite element method. A complete and exhaustive treatment of this topic is however beyond the scope of this chapter. The interested reader should consult FEM dedicated textbooks (see, for example, refs 1 and 2). The aim of the section is to illustrate the basics of the method and to show how this approach is useful for the study of complex materials. Initially, the finite element method was created for analysing macroscopic structures and continuous media. Subsequently, the approach was extended to cover problems of magnetic and electric fields, fluid flow and heat transfer. The original core of the method, however, was developed for stress analysis, which is a subject related to the structural performance of engineering materials. The finite element method is a numerical procedure for solving a complex problem that cannot be addressed with an analytical approach. The basic concept is that of discretisation: a continuum medium is subdivided into discrete elements, called finite elements. Each element is of simple geometry compared to the whole structure and therefore much more easily treated. Each element has nodes on its vertices and edges. The displacements of the

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nodes are usually assumed as the degrees of freedoms (dofs) of the problem in structural mechanics. For example, for plane elements such as triangular or quadrangular elements, each node possesses two dofs, i.e., the displacements in the X and in the Y directions. Then, the displacement of every point inside the element is defined as a function of the nodal displacements via particular functions called shape functions. The choice of such functions determines the degree of complexity of the element. In fact the displacement field is interpolated over the elements from the nodal values by using the shape functions. For each element an equilibrium equation can be written: {F}e = [K]e {D}e

12.1

where{F}e is the vector of the forces applied on the nodes, {D}e is the vector containing nodal dofs and [K]e is the so-called element stiffness matrix, which relates nodal displacements to nodal forces and can be determined either by a direct method or by a variational formulation. Finally, all the elements equations such as eqn 12.1 are assembled, thus obtaining the global equilibrium equation {F} = [K]{D}

12.2

where [K] is the global stiffness matrix. From eqn 12.2 the nodal dofs {D} are determined. It should be noted that eqn 12.2 is a system of algebraic equations. Therefore a problem stated by differential equations, such as the elasticity problem, has been reduced to a set of linear algebraic equations, that can be easily solved on a computer. The problem unknowns are thus determined and by differentiation the strains are obtained from the displacements. Finally stresses can be calculated from strains. Obviously, the choice of the element affects the problem complexity. Various types of elements are used, e.g., planar triangular or quadrangular elements, solid brick elements, etc. Elements may have a different number of nodes, for example, quadrangular elements are commonly available with 4, 8 or 9 nodes depending on the particular formulation. Consequently, the number of dofs will vary accordingly and also the dimension of the system of eqn 12.2. It is straightforward that the discretisation, including the choice of the element type, is the crucial step of the procedure. The mesh should be refined enough depending on the particular problem investigated; in fact it has to be accurately refined in order to capture with accuracy fluctuations in the stress and strain fields. The proper degrees of refinement of a finite element mesh as well as the choice of the element formulation are relevant topics which are thoroughly treated in specialist textbooks [1, 2]. The user of a commercial FEM code should be aware of the implications of the assumptions selected. It is worth noting that the FEM is a really versatile tool. The potential of the

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method is that it can be implemented in order to account for non-homogeneous materials, non-linear behaviour as well as complex load situations and boundary conditions.

12.3

Standard FEM analysis for fibre-composite materials

Fibre-composites are commonly made up of a sequence of laminae. Each ply is usually considered to behave as an orthotropic material. Long fibres or short-aligned fibrous plies can be easily represented by assuming average properties that account both for the matrix and for the fibre contributions: usually these properties are calculated by using analytical equations, such as the rule of mixtures or the Halpin and Tsai relations [3]. Due to the orientated reinforcements, the overall material behaviour is usually orthotropic, as mentioned. In fact, each ply has directional properties with a higher stiffness in the direction of the fibres. Moreover, if a composite material has a layered structure, in general properties will vary from ply to ply. When representing plies, generally 2D shell elements or 3D brick elements with such behaviour are adopted. Engineering judgement is needed in order to select the most appropriate type of element. It is possible to stack 3D brick elements, each ply of the composites being represented by one layer of bricks. However, this solution can be very cumbersome in terms of computational time and numerical problems may arise [4]. Often, 2D elements are employed for modelling composite plates. A layered composite is usually represented by using shell elements. Such elements represent the mid-surface plane of the structure, and the stacking sequence of the plies is taken into account for the determination of the element characteristics. Either classical lamination theory or Mindlin theory can be used to formulate shell elements which allow for laminated materials [4]. Many commercial codes that include special elements for composite structures are currently available. These packages allow the prediction of properties of laminates and multi-layered materials. It should be kept in mind that the accuracy of the finite element solutions is strongly affected by the element choice and the mesh refinement.

12.4

Microstructure-based modelling

Conventional FEM analysis for composite materials as illustrated in the previous section is suitable in general for long-aligned fibres, i.e., for a regular arrangement of the reinforcement in the matrix. This analysis is convenient for structures whose length scale is much larger than the characteristic dimension of the single fibre. In this case, the effect of different lay ups can be included directly into the FE model. However, for more

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complex composites, i.e., when the fibres are not aligned or the reinforcement is in the form of randomly distributed irregularly shaped particulate inclusions, the above approach may not suffice for an accurate description of the micromechanic behaviour of the material. Such composite systems, which are not characterized by a regular and periodic arrangement of the two phases, require specific microstructure-based modelling in order accurately to describe the effective material behaviour. It is well known that microstructural features affect the global elastic and fracture properties of the composite materials. However as regards complex microstructures, as mentioned in the introduction, analytical approaches or theoretical equations for the description of such materials are available only for simplified situations (e.g. spherical particles) or particular cases (e.g. orientated microstructures). Therefore, in such composites computed numerical modelling is required in order to correlate the actual microstructural characteristics with their effective mechanical behaviour. In the following subsections, methods for the treatment of heterogeneous materials are illustrated, namely the representative volume element (RVE) approach, a modelling technique based on FEM representation of microstructures, and other related approaches that have been put forward in the literature.

12.4.1 The representative volume element (RVE) Since particulate composites often have a random distribution of an irregularly shaped second phase in a matrix, i.e., the material is inhomogeneous at the microscale, it is convenient to define the smallest portion of material that contains all the relevant microstructural features. This becomes the smallest repetitive periodic unit; which is considered to be representative of the whole microstructure, and is known as the representative volume element (RVE). The local properties are not homogeneous inside the RVE, i.e., in general, thermoelastic properties fluctuate from point to point. However, average homogenised properties can be defined by analysis of the RVE and these properties may be used to model the global composite structure at the macroscale. When the second phase has a regular arrangement inside the matrix, such as for aligned long fibre composites, the RVE can be easily defined. However, for particulate composites, the definition may not be straightforward. In most cases, an idealised distribution of particles is hypothesised, assuming an average size of particles and a repetitive pattern of spatial disposition. Figure 12.1 illustrates two examples. It should be noted that this assumption is a simplification of the real complex microstructure; yet, when global average properties only are sought, this schematisation may suffice. In a more complicated case, when, for example, the size distribution of the

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Fibres

(a)

(b)

(c)

12.1 Various types of representative volume elements (RVE): (a) long fibre composites; (b) idealised particulate composite with a regular spatial disposition of second phase of equal shape and average (constant) particle size; (c) ‘random’ particulate composite, with inclusions of equal shape but varying particle size.

reinforcement is relevant, more complex RVEs may be defined, as also illustrated in Fig. 12.1.

12.4.2 Finite element modelling of complex microstructures The analysis illustrated in the previous subsection is adequate only when average homogenised properties are required. However, many authors pointed out the need for a modelling approach closely related to the actual material microstructure [5, 6], which should allow the analysis of phenomena occurring at the microscale. Microstructural features strongly affect the overall mechanical response. For example, in brittle matrix composites, flaws and microcracks are usually responsible for fracture initiation. Other peculiarities, such as pores or interfaces between materials with different elastic properties, or mismatch of thermal expansion coefficients may cause local stress concentrations. Therefore, if all characteristics of the microstructure should be fully modelled, it is necessary to carry out microstructure-based analyses. In recent years, a numerical method to investigate complex and heterogeneous microstructures has been proposed and developed in the U.S. by W. C. Carter, E. Fuller and S.A. Langer, namely the OOF code [7, 8]. This approach consists in direct modelling of 2D microstructural images of the material under investigation. The innovative idea is that the digitised image of the real microstructure, such as an optical microscopy image or a SEM micrograph of a material cross-section, is directly acquired in the code and then transformed into a finite element mesh. It is possible to attribute to the different parts of the image the corresponding thermo-elastic properties of the phases involved. In this way the computational mesh displays the exact 2-D arrangement of the composite phases and features such as pores and microcracks can be modelled as well. Moreover, the finite element grid may be refined at the interface between the different phases or around defects

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where the stresses are expected to concentrate. On such a mesh, virtual (numerical) tests can be performed in order to assess the mechanical response and fracture behaviour as well as the interactions between the constituents materials. OOF has been increasingly used to analyse microstructure-property correlations in a wide range of materials, especially for complex and heterogeneous materials. For example, Vedula et al. investigated residual stresses and spontaneous microcracking upon cooling in polycrystalline alumina [9]. Saigal et al. examined microcracking of iron titanate [10] and the internal stresses originated in aluminum-silicon alloys [11]. A. Zimmermann et al. determined residual stress distributions in ceramics caused by thermal expansion anisotropy [12], M.H. Zimmermann et al. examined the fracture mechanism of textured anisotropic ceramics [13] and A. Zimmermann et al. investigated damage evolution during microcracking of brittle polycrystalline materials [14]. Hsueh et al. analysed the stress transfer mechanism in platelet-reinforced composites [15], as well as residual stresses and interface asperity effects in thermal barrier coatings [16, 17]. Cannillo and Carter [18] employed OOF to determine the mechanical reliability of composite microstructures. Moreover, Cannillo et al. used the OOF code to predict the elastic and fracture properties of a porous glass [19, 20], investigated the formation of thermal residual stresses and the fracture behaviour of composites with glass matrix and alumina platelets [21–23] and with glass matrix and molybdenum particles [24, 25]. A review of the great variety of applications of the OOF numerical approach for the simulation of thermomechanical behaviour of materials has been published recently [26]. It is worth noting that, in order to prepare reliable microstructure-based finite element models, it is necessary for the computational grid to be a relevant representation of the whole material. This means that, if the mesh is prepared for example from a SEM micrograph, the image should be large enough in relation to the scale of the microstructure to contain all relevant microstructural features. Due to the stochastic variation of randomly dispersed particulate composites, if the image is not properly selected, the results will not be valid for the whole material. Therefore a few different microstructural images must be considered in order to capture all possible situations or microstructural variations. On the other hand, the mesh should be accurate enough to reveal all relevant microstructural details, i.e., small cracks, pores or flaws, in order to reflect the real nature of the material. The increase of the computational power in the last decade makes it feasible to develop meshes of larger regions of material microstructures with high degree of refinement. As a further observation, it should be noted that in any case this kind of analysis is twodimensional. Therefore, 3D mechanisms are represented with the corresponding 2D model. Even if for several problems this is not a serious limitation, it

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should be kept in mind when considering complex phenomena that are threedimensional in nature, such as crack-reinforcement interactions and the effect of internal stresses on crack propagation in composite materials. By using the finite element solver, stresses and strains can be determined. Besides the classical stress analysis, OOF can tackle fracture problems. Specifically, selected elements can be implemented and added to the main code in order to analyse crack propagation in a composite microstructure under loading. In particular, two elements can be used for the investigation of brittle materials, namely the so called Griffith and the Weibull elements. The Griffith element is an element available in the OOF finite element library. The microstructural stresses as calculated by the FEM solver are used in conjunction with the adopted failure criterion in order to analyse fracture propagation. When the prescribed solicitation is applied, local stresses are determined for each element of the mesh. Then, for each element an energy balance is computed. It is assumed that the element fails if the following condition holds [7]:

2lg £ 1 2

Ú se dV

12.3

where l is the crack length (specified by the element size), g is the surface energy of the cracked interface and the volume of integration is the element area (per unit depth). This means that an element fails when the total surface energy required to propagate the crack can be supplied by the elastic energy stored. If the energy balance is favourable for micro-cracking, the element is damaged, i.e., elastic properties of the element are significantly reduced. The released stresses upon element cracking are redistributed among the neighbouring elements; due to this redistribution other elements may fail. In this way it is possible to track crack propagation inside the composite material. It is worth noting that the effect of damage accumulation due to microcracking of the material and the interaction between the propagating crack and the constituent phases can be taken into account. This element has been used in different works, for example, to describe fracture behaviour of polycrystalline alumina [14], and of brittle-matrix particulate composites [23]. The second element useful for the description of brittle materials is the Weibull element. This element was recently implemented and tested by Cannillo and Carter [18, 27] and subsequently added to the main code. A probabilistic failure criterion is recommended in order to account for stochastic distribution of strength. Therefore this element was designed to fail under a Weibull-like criterion. This widely adopted statistical model relates the failure probability to the stress state in the specimen by means of two parameters, the Weibull modulus m and a scale parameter s0 [28], i.e., Pfail = 1 – exp[–(s/s0)m(A/A0)]

12.4

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where the failure probability Pfail is related to the stress s acting on the sample, the normalised area of the sample A/A0 and the aforementioned Weibull parameters m and s0. Therefore, once stresses are calculated by the FEM solver, the local failure probabilities can be calculated for all elements. It should be noted that in eqn 12.4 the principal tensile stress is used. A Monte Carlo process is adopted to simulate fracture of elements: if the computed failure probability exceeds a random generated number in the range 0–1 the element breaks. Further details of the procedure are fully described in ref. 27. Following the probabilistic law, elements undergo damage by losing stiffness as described above for the Griffith element. Damage evolution can be tracked as a function of applied stress. Thus, this stochastic approach offers an alternative to the classic fracture mechanics methods for the study of brittle fracture. It is worth noting that the underlying assumption is that the Weibull statistical analysis, usually adopted (and validated) at the macro scale, can be applied at the microscopic level.

12.4.3 Other approaches Different numerical methods have been proposed in the literature to account for microstructural heterogeneity. Ghosh, Lee and co-workers published for example several papers regarding multi-level computational modelling strategies for complex materials such as composites and porous structures [29–38]. The authors developed a multiple scale analysis protocol for the investigation of heterogeneous materials undergoing microstructural damage. The basic idea is to investigate both macroscopic response with a conventional displacement based FEM code and microscopic behaviour by means of the Voronoi Cell Finite Element method (VCFEM), which enables the modelling of arbitrary microstructural representative material elements [29]. Critical and non-critical regions in the structure can be differentiated and microscopic simulations can be performed where necessary [32]. The approach is able to track the incidence and propagation of microstructural damage in composites and has been tested for different heterogeneous microstructures. Other authors used experimental techniques such as small-angle neutron scattering (SANS) to quantify microstructures [39]. The approach was able to provide average microstructural parameters such as porosity amount and distribution; this information was used to prepare a ‘model microstructure’, that was therefore statistically significant for the whole material. After the reconstruction of the porous structure, a finite element mesh was generated and the effective properties were estimated. Gusev and co-workers proposed an approach based on direct finite element modelling of representative unit cells of short fibre composites [40–43]. Such cells are produced using a Monte Carlo procedure which generates a

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random microstructure based on the measured fibre orientation distribution functions. The study also led to the commercial implementation of a code (called Palmyra) [43].

12.5

Applications and examples

This section illustrates some applications of the microstructure-based finite element modelling of brittle matrix particulate composites and other complex and heterogeneous systems. In particular mechanical properties, fracture behaviour and mechanical reliability are investigated. The goal is to illustrate with examples how it is possible to establish quantitative microstructureproperty correlations by using the numerical approach described above. Such correlations have the final objective of aiding the materials scientist in the optimal design of innovative composite systems.

12.5.1 Particulate composites The example illustrated in this paragraph is referred to as a brittle glassmatrix composite reinforced with ceramic particles, modelled with OOF. The 2-D representation of the microstructure of the two-phase composite considered is illustrated in Fig. 12.2a. The method can be applied to a multiphase composite, with the same procedure described below. The microstructural image is converted via the OOF pre-processor into a finite element mesh, as illustrated in Fig. 12. 2(b). The relevant properties of the composite constituents are attributed to the corresponding phase, since each pixel of the image is associated with a material (in this case either the glass matrix or a ceramic inclusion). The mesh is conveniently refined at the interface between the two constituents, where stresses may concentrate due to elastic properties or thermal expansion coefficient mismatch. In this way the computational mesh accurately reproduces the real material. Obviously, the portion of material considered should be representative of the whole material microstructure. If the mesh is obtained from a microscopy image, this image should be properly selected. Relevant information regarding composite behaviour can be obtained from the model. As illustrated in refs 21–23 for glass matrix composites reinforced with alumina particles, materials can be designed to achieve a particular property by varying the reinforcement content. For example, using the finite element solver and imposing a temperature gradient in the calculations, it is possible to obtain the 2-D distribution of thermal residual stresses which develop in the composite upon cooling from the processing temperature due to different thermal expansion coefficients (CTE) of the two phases. Choosing the two phases with proper CTE, this phenomenon can be exploited to obtain a favourable effect. If the reinforcement is selected with a CTE greater than

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(a)

(b)

12.2 (a) Example of particulate composite with particles of arbitrary shape and random position; (b) detail of the OOF finite element mesh based on the microstructure. The mesh has been refined at the matrix-particle interface.

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that of the matrix, compressive stresses arise in the brittle matrix. This was shown to have a beneficial effect on the global toughness of the composite [21]. An example of the calculation of thermal residual stresses in two-phase composites is illustrated in Fig. 12.3. 0 MPa

–100 MPa

12.3 Thermal residual stresses in a glass-matrix composite with ceramic inclusions. The tangential stresses are compressive in the brittle glass matrix if the CTE of the inclusions is higher than the CTE of the matrix (considering isotropic CTEs of the two phases).

The microstructure-based OOF analysis is useful to understand fully the composite performance. Additional detailed information can be gained if the elastic solution of the finite element solver is used in conjunction with selected failure criteria suitable for brittle fracture. Brittle matrix composites were investigated both by using the Griffith’s approach and the Weibull-based statistical failure criterion. As described in section 12.4.2, this enables the tracking of crack propagation in two-phase brittle microstructures. Damage accumulation with increasing mechanical or thermal loading can thus be investigated. In this way, it is also possible to assess crack-particle interaction and the effect of inclusions on the global fracture toughness of the composite. For example, the investigation on the glass-alumina composites pointed out crack deflection as the main toughening mechanism, which was confirmed by experimental observations [23]. In Fig. 12.4 an example of crack propagation under tensile loads for the mesh of Fig. 12.2 is illustrated. It is must be highlighted that computational simulations should be validated against experimental data at least for a selected volume fraction of reinforcement. Once the model has been validated, it can be used to predict the behaviour of similar composites, e.g., with a different volume fraction of second phase.

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12.4 Example of fracture propagation in the glass matrix composite of Fig. 12.2(a) under tensile load in vertical direction, (in black fractured elements). It is seen that the crack propagates from the matrix through the interface and cuts the inclusion.

The experimental investigation of the alumina-glass system gave results in good qualitative agreement with the computational predictions of the OOF method [21–23].

12.5.2 Porous materials Porous materials can be considered as the limiting case of composites in which the second phase are the pores. In the finite element mesh, pores can be represented by using elements with no elastic stiffness. Pores can have a beneficial effect for the design of materials with special properties, for example, in materials for thermal barrier coatings and in low dielectric constant substrates. However, porosity usually has a detrimental effect on the mechanical properties [44] therefore when designing such materials, the effect of pore size, shape and amount on overall material properties must be known. The microstructurebased numerical modelling approach is useful to find direct correlation between the microstructural porosity parameters and the resulting elastic and fracture behaviour of porous materials. In order to reliably obtain the effective properties of the porous material, several local regions should be modelled separately and then average values should be considered. However, since it is impractical to characterise several possible distributions of pores, it is convenient to generate a ‘random’ pore microstructure which is statistically significant, i.e., such a microstructure should have pores randomly placed in the matrix, but with a prescribed size

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and shape (for example a diameter ranging in a selected interval or a varying axial ratio) and realistic pore volume fraction, related to the actual material under consideration. For example, Fig. 12.5 shows different cases of the same ‘average’ microstructure. The three samples displayed contain pores with the same average size, aspect ratio and volume fraction but the pores occupy different random positions relative to each other. All three microstructures are suitable to represent on average the same porous material. However, the role of pores at the microscale level and the effect on the macroscopic properties can be assessed for each sample using OOF.

12.5 Examples of statistical microstructures of porous materials generated by random positioning pores of prescribed size, shape (circular in this case) and volume fraction.

Once the average characteristics of the porous microstructure have been established, the OOF approach with direct microstructural meshing is useful to determine fracture propagation behaviour. For example, the studies in refs 19 and 20 illustrate the application of the approach to glass samples containing closed, controlled porosity of spheroidal shape [44]. Besides the determination of the Young’s modulus as a function of porosity content and pore shape, pore-crack interactions can be investigated.

12.6

Future developments

The microstructure-based numerical modelling concept presented in this chapter constitutes a promising method for the design of new composites. The potential of the approach is its suitability to be applicable to any complex microstructure. A microstructure-based model is much more versatile and universal than an analytical approach, since arbitrary microstructures with any number of phases, morphology and properties can be modelled. The approach presented here is expected to be applied to complex materials systems. For example, spatially varying composites, such as functionally graded materials (FGMs), could be investigated in a similar fashion. Since in this class of materials the microstructure is tailored to change along one

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direction, it is straightforward that a model which reproduces the exact microstructural gradient is to be preferred. It should be noted that the approach presented here is inherently twodimensional, since the computational model is obtained directly from planar images of the microstructure such as SEM micrographs. However, it would be desirable to extend the approach to fully 3D models, especially for complex problems such as fracture propagation and damage accumulation. It would be possible, in theory, to reconstruct 3D microstructures from 2-D images coupled with stereological analysis and to convert the image to a 3D solid FEM mesh. Although much more expensive and demanding from a computational point of view, this approach will enable more realistic information about material behaviour in 3D to be obtained. In fact, increasing computational power makes it feasible to investigate systems that were considered overwhelming a few years ago. Therefore, the authors believe that computer aided design of advanced composites based on advanced 3D microstructure-based numerical models will become a possible aim in the near future.

12.7

Acknowledgements

The NIST ‘OOF team’ is gratefully acknowledged for discussions and support. VC wishes to thank Craig Carter for his advice, guidance and encouragement.

12.8

References

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10. Saigal A., Jr. Fuller E.R., Langer S.A. Carter W.C., Zimmermann M.H. and Faber K.T., Effect of interface properties on microcracking of iron titanate. Scripta Materialia, 38, 1998, 1449–1453. 11. Saigal A., Jr. Fuller E.R., Analysis of stresses in aluminum-silicon alloys. Computational Materials Science, 21, 2001, 149–158. 12. Zimmermann A., Fuller E.R. and Rödel J., Residual stress distributions in ceramics. J. Am. Ceram. Soc. 82 [11] (1999) 3155–3160. 13. Zimmermann M.H., Baskin D.M., Faber K.T., Jr. Fuller E.R., Allen A.J. and Keane D.T., Fracture of a textured anisotropic ceramic. Acta Materialia, 49 (2001), 3231– 3242. 14. Zimmermann A., Carter W.C. and Fuller E.R., Damage evolution during microcracking of brittle solids. Acta Mater., 2001, 49, 127–137. 15. Hsueh C.H., Fuller E.R., Langer S.A. and Carter W.C., Analytical and numerical analyses for two-dimensional stress transfer. Mater. Sci. Eng. A 268 (1999) 1–7. 16. Hsueh C.H., Haynes J.A., Lance M.J. Becher P.F., Ferber M.K., Fuller E.R., Langer S.A., Carter W.C. and Cannon W.R., Effects of interface roughness on residual stresses in thermal barrier coatings. J. Am. Ceram. Soc. 82 (1999) 1073–1075. 17. Hsueh C-H and Jr, Fuller E.R., Residual stresses in thermal barrier coatings: effects of interface asperity curvature/height and oxide thickness. Materials Science and Engineering A, 283,2000, 46–55. 18. Cannillo V. and Carter W.C., Computation and simulation of reliability parameters and their variations in heterogeneous materials. Acta Materialia, 48, (2000), 3593– 3605. 19. Cannillo V., Leonelli C., Manfredini T., Montorsi M. and Boccaccini A.R., Computational simulations for the assessment of the mechanical properties of glass with controlled porosity. J. Porous Materials, 10, 189–200, 2003. 20. Cannillo V., Manfredini T., Montorsi M. and Boccaccini A.R., Use of numerical approaches to predict mechanical properties of brittle bodies containing controlled porosity. J. Materials Science, 39, 4335–4337, 2004. 21. Cannillo V., Leonelli C. and Boccaccini A.R., Numerical models for thermal residual stresses in Al2O3 platelets/borosilicate glass matrix composites. Materials Science and Engineering, A323: 246–250, 2002. 22. Cannillo V., Corradi A., Leonelli C. and Boccaccini A.R., A simple approach for determining the in-situ fracture toughness of ceramic platelets used in composite materials by numerical simulations. Journal of Materials Science Letters, 20, 1889– 1891, 2001. 23. Cannillo V., Pellacani G.C., Leonelli C. and Boccaccini A.R., Numerical modelling of the fracture behaviour of a glass matrix composite reinforced with alumina platelets. Composites A 34, 43–51, 2003. 24. Minay E., Boccaccini A.R., Veronesi P., Cannillo V. and Leonelli C., Processing of Novel Glass Matrix Composites by Microwave Heating. Journal of Materials Processing Technology, 155–156, 1749–1755, 2004. 25. Cannillo V., Manfredini T., Montorsi M., Boccaccini A.R., Investigation of the mechanical properties of Mo reinforced glass matrix composites. Journal of NonCrystalline Solids, 344, 88–93, 2004. 26. Chawla N., Patel B.V., Koopman M., Chawla K.K., Saha R., Patterson B.R., Fuller E.R. and Langer S.A., Microstructure-based simulation of thermomechanical behavior of composite materials by object-oriented finite element analysis. Mater. Charact. 49 (2002) 395–407.

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27. Cannillo V. and Carter W.C., A stochastic model of damage accumulation in complex microstructures. Journal of Materials Science, 2005, in press. 28. Weibull W., A statistical distribution function of wide applicability. J. Appl. Mech., 1951, 18, 293–297. 29. Lee K., Moorthy S., and Ghosh S. Multiple scale computational model in composite materials. Computer Methods in Applied Mechanics and Engineering, 172, 175– 201, 1999. 30. Lee K. and Ghosh S., A microstructure based numerical method for constitutive modeling of composite and porous materials. Materials Science and Engineering A, 272, 120–133, 1999. 31. Lee K. and Ghosh S., Small deformation multi-scale analysis of heterogeneous materials with the Voronoi cell finite element model and homogenization theory. Computational Materials Science, 7, 131–146, 1996. 32. Ghosh S., Lee K. and Raghavan P., A multi-level computational model for multiscale damage analysis in composite and porous materials. International Journal of Solids and Structures, 38, 2335–2385, 2001. 33. Moorthy S. and Ghosh S., Particle cracking in discretely reinforced materials with the Voronoi cell finite element model. International Journal of Plasticity, 14, 805– 827, 1998. 34. Ghosh, S. and Moorthy S., Particle fracture simulation in non-uniform microstructures of metal matrix composites. Acta Materialia, 46, 965–982, 1998. 35. Ghosh S., Li M., Moorthy S. and Lee K., Microstructural characterization, mesoscale modeling and multiple-scale analysis of discretely reinforced materials. Materials Science and Engineering A, 249, 62–70, 1998. 36. Li M., Ghosh S. and Richmond O., An experimental-computational approach to the investigation of damage evolution in discountinuously reinforced aluminum matrix composite. Acta Materialia, 47, 3515–3532, 1999. 37. Li, M., Ghosh S., Richmond O., Weiland H. and Rouns T.N., Three dimensional characterization and modeling of particle reinforced metal matrix composites: part I Quantitative description of microstructural morphology. Materials Science and Engineering A, 265, 153–173, 1999. 38. Li M., Ghosh S, Richmond O., Weiland H. and Rouns T.N., Three dimensional characterization and modeling of particle reinforced metal matrix composites: part II Damage characterization. Materials Science and Engineering A, 266, 221–240, 1999. 39. Wang Z., Kulkarni A., Deshpande S., Nakamura, T. and Herman H., Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings. Acta Materialia, 51, 5319–5334, 2003. 40. Hine P.J., Lusti H.R. and Gusev A.A., Numerical simulation of the effects of volume fraction, aspect ratio and fibre length distribution on the elastic and thermoelastic properties of short fibre composite, Composites Science and Technology, 62, 1445– 1453, 2002. 41. Hine P.J., Lusti H.R. and Gusev A.A. On the possibility of reduced variable predictions for the thermoelastic properties of short fibre composite. Composites Science and Technology, 64, 1081–1088, 2004. 42. Gusev A.A., Representative volume element size for elastic composites: a numerical study. J. Mech. Phys. Solids, 45, 1449–1459, 1997. 43. Hine P.J., Lusti H.R. and Gusev A.A., Direct numerical predictions for the elastic

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and thermoelastic properties of short fibre composite, Composites Science and Technology, 62, 1927–1934, 2002. 44. Boccaccini A.R., Fabrication, microstructural characterisation and mechanical properties of glass compacts containing controlled porosity of spheroidal shape, Journal of Porous Materials, 6(4): 369–379, 1999.

13 Wear modelling of polymer composites* K F R I E D R I C H, University of Kaiserslautern, Germany, K V Á R A D I, Budapest University of Technology and Economics, Hungary and Z Z H A N G, University of Kaiserslautern, Germany

13.1

Introduction

Polymer composites belong to the group of multiphase materials. They are characterised by a softer matrix containing harder and stronger particles or fibres. The latter act as the reinforcing phase, which sometimes also causes a certain degree of anisotropy. Under tribological loading conditions, the corresponding properties (coefficient of friction; specific wear rate) depend on the system in which these materials must function, including effects of the type and size of the counterpart asperities (e.g. abrasive particles; roughness of the mating steel partner). For given system conditions, the wear behaviour can be modelled using various approaches, of which the most important of them will be discussed in the following parts of this chapter.

13.2

Rule-of-mixtures-approaches to wear of multi-component materials

13.2.1 General remarks A quantitative approach to the wear behaviour of multiphase systems has been reported in the literature by Simms and Freti [1]. In particular, these authors discussed models describing the wear rate as a function of the volume fraction of a second phase. Phase interaction effects and interface phenomena have also been observed. Under defined conditions, a given phase shows a specific wear mode and wear rate, which depends on its individual properties. Consequently, when various phases are combined, forming a multi-phase material, it is expected that the overall behaviour will be a function of the respective contribution of each phase. Based on this approach, the wear

* Partly presented at the 6th International Tribology Conference, AUSTRIB 2002, Perth, Australia, December 2002.

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375

resistance has been mathematically described as a linear function of the volume fraction of the phases present [2]. In terms of the total wear rate, i.e., the inverse of the wear resistance, the model is called the inverse rule of mixtures (IROM):

W –1 = S Vi Wi –1

13.1 –1

where W is the total volume wear rate, W is the total wear resistance, Wi is the wear rate of the ith phase and Vi is the volume fraction of the i th phase. The other existing model is the linear rule of mixtures (LROM): W = ÂViWi

13.2

The relationship gives a balanced value of the wear rate without a predominant effect of any one phase. In general, it was pointed out that LROM is suitable to describe a combination of very similar phases, while IROM better describes structures which consist of phases with very different properties [3]. Another important point is the particle/matrix interfacial bond strength. It can have the most significant effect in multiphase systems. The above models are based on no phase interaction. In fact, important micro-mechanisms, like cracking, wear debris formation, decohesion and pulling out of the reinforcing phase, can occur at interfaces. The relationship between wear rate and second-phase volume fraction is summarised in Fig. 13.1, where the above-described models are compared with previously reported experimental results [3–5]. The LROM was confirmed to describe ferrite-pearlite behaviour in a two-body pin-on-paper abrasion test. In the same work, it was found that a ferrite-martensite system conforms

Wear rate W

Wa

Wb

a

Volume fraction of b phase

b

13.1 Qualitative wear behaviour in various two-phase structures (schematic representation): ferrite-pearlite, LROM(–––-); ferrite) [3]; martensite, IROM (----); chromium cast iron-M7C3 carbides ( chromium cast iron-M7C3 carbides ( ) [4]; NICrBSi W2C ( ) [5].

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better to IROM. All these results are mainly correlated with the secondphase volume fraction. Although abrasive size effects and microstructure size effects were observed [6], they have not been specifically reported. In order to better understand the mechanisms involved in the total tribological system, the interacting effects between the geometrical factors must also be analysed. The total abrasive action on the multiphase system has to be considered as the macroscopic sum of all the microscopic effects produced by the individual abrasive grains. Each of these wear micro-events generates a groove in the material. It is expected that the reinforcing role of a dispersed phase may change since the groove size is smaller or bigger than the microstructural size [7].

13.2.2 Abrasion of particulate composites Hard-particle reinforcement of glassy or crystalline polymers gives an increase in modulus but no improvement or a reduction in tensile strength and impact strength. Particle-filled polymer composites are thus suited to uses where a high modulus is needed, where loads are mainly compressive or where hightemperature creep resistance is important. Dental filling materials are one such application. Wear processes take many forms, but the processes most important for dental fillings presumably involve combinations of indentations and ploughing. This means that one is concerned with abrasive wear in which a hard particle ploughs or cuts a surface. In the work by Prasad and Calvert [8] a simple equation was developed to describe abrasive wear of composite materials in terms of the wear properties of the individual components. This expression is equivalent to a series model of wear (IROM) in which the two components are exposed successively to abrasion. This expression is not expected to be a complete description of composite wear particularly if one must interpret the experimental observations in terms of particle pullout and enhanced interfacial wear. Wear rates for composites against quartz abrasive are shown in Fig. 13.2. In this case the wear rate drops markedly as silane-treated quartz particles are introduced into the resin. Wear rates of PMMA and quartz against the same quartz abrasive were 19.9 and 1.6 ¥ 10–3 mm3/(Nm), respectively. Using these values one can predict an even more marked decrease in wear rate. Non-silanated quartz fillers give little improvement in wear rate and glass beads (diameter of glass-bead filler between 4 –44 mm) give an increased rate. The wear rate for glass sheet against abrasive was 25 ¥ 10-3 mm3/(N m), so that the IROM predicts a slight increase in wear rate with filler content, but not as great as is observed. In ref. 7, other examples of particulate filled polymers were discussed in which some results are better described by the LROM.

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377

Ê mm 3 ˆ Wear rate Á 10 3 Nm ¯˜ Ë

24

18

12

Dental composites vs. quarz abrasive

6

0

0

0.1 0.2 Filler volume fraction

0.3

13.2 Wear rates of composites by 10 mm quartz abrasive. (
) Silantreated quartz filler, () untreated quartz, and () untreated glassbead filler. Full lines fitted using IROM [8].

13.2.3 Wear of short fibre reinforced thermoplastics To illustrate the effects of various short-fibre reinforcements on the sliding wear characteristics of various polymers, Fig. 13.3 summarises the results for many different fibre matrix systems. One should look more at the general trends here rather than at the individual data. For those polymers that possess high specific wear rates in the unreinforced condition, almost any type of reinforcing fibre results in both significant reductions in wear and further improvements of the mechanical properties. In addition, carbon fibres often improve the wear resistance by a factor of five or more compared to the same volume fraction of glass fibres [9]. From recent studies carried out with different types of carbon fibre (PAN-based CF vs. Idemitsu high-temperature pitch-based CF) it can be concluded that the composite wear resistance will be better, the less sensitive the fibres are to frictional loading and early breakage. In addition, strong bonding to the matrix helps to maintain broken fibre pieces in the composite surface, thus preventing early formation of third-body abrasives and so increased wear. The corresponding wear mechanisms possess similar features, which are schematically illustrated (for the case of an in-plane fibre arrangement) in Fig. 13.4. It can be assumed that, at the beginning of the wear process, matrix and fibres are worn simultaneously (each component being worn to an amount that depends on its individual wear resistance). After wear thinning of the fibres, fibre cracking starts which eventually ends in the removal of worn fibre pieces from their fibre beds. These broken fibre pieces can temporarily act as third-body abrasives while being entrapped in the form

378

Multi-scale modelling of composite material systems 10–2 p ◊ v = 3 MPa ◊ m sec p ◊ v = 1.7 MPa◊ m sec

GF/CF-PPS 10–3 Ê mm 3 Ws Á Á N◊ m Ë

ˆ ˜˜ ¯

GF-PES GF/CF-ETFE

10–4 GF-LCP (AP/P)

10–5 GF-PET GF-PA6.6 10–6 CF-LCP

GF-LCP (N) 10–7

0

MGF-LCP

CF-PA6.6

10

20

30

40

Vf (%)

13.3 Influence of fibre weight fraction on the specific wear rate of various short-fibre reinforced thermoplastics. Fibre cracking

Matrix wear (plowing, cutting, microcracking)

Pulverised wear debris

Fibre/matrix interfacial separation

Fibre sliding wear

13.4 Schematic of various wear mechanisms in fibre reinforced polymer composite.

of consolidated wear-debris layers on the two surfaces in sliding contact [10]. Recent papers by Voss and Friedrich [11,12] have shown that one can assign partial rates to all of the wear processes described in Fig. 13.4. It is assumed that eqns 13.3 and 13.4 occur sequentially and can therefore be considered as a combined process (Ws,Fci). The removal of fibres also caused an additional matrix wear process because these fibre particles can act as

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379

third body abrasives, as can be deduced from the wear surface micrographs. All these mechanisms, which are not due to pure sliding, are included in Ws,Fci. Hence, the wear rate of the composite (Ws,C) is the sum of wear rates, which accounts for the sliding processes (Ws,s) and one which accounts for the additional wear mechanisms (Ws, Fci). Each partial wear rate is assumed to be a function of the fibre volume fraction, VF. When considering the sequences of the individual wear mechanisms, i.e., at first a parallel sliding wear of the fibre and the matrix, followed by fibre cracking and interfacial removal, a complex rule of mixtures equation results: –1

È ˘ Ws ,C = Í (1 – VF ) 1 + aVF 1 ˙ + bVF Ws ,Fci Ws,M Ws,Fs ˚ Î

13.3

The factors a and b are, in general, also functions of VF. For simplicity, a and b can be assumed to be equal to 0.5. The only partial wear rate which contributes to the overall wear rate of the composite and which is not directly accessible, is Ws,Fci accounting for the post-sliding wear process of the mechanisms in eqns 13.3 and 13.4. However, Ws,Fci can be estimated by adjusting eqn 13.3 to the experimental data. Figure 13.5 gives an example of the results predicted for polyethernitrile (PEN) composites, reinforced with different short carbon fibres [13].

˙ (mm3/Nm) W s

10–4

Matrix

Sliding wear vs. steel (i) CF-PEN composites p · v = 1.7…3 (MPa · m/s)

10–5

Experimental Theoretical PAN-based CF

10–6 Pitch-based ICF 10–7

0

5

10

15

20 25 Vf (vol. %)

30

35

40

13.5 Correlation between experimental and predicted specific wear rate of carbon fibre (CF) reinforced polyethernitrile (PEN) [13].

13.2.4 Abrasive and sliding wear of continuous fibre reinforced polymer composites Polymer composites with continuous-fibre reinforcement of high volume fraction and perfect alignment are known to have very high values of their specific strength and stiffness (when measured in the direction of the fibres).

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Multi-scale modelling of composite material systems

The wear of some unidirectional carbon fibre composites sliding on steel was found to be also (more or less) affected by the fibre orientation relative to the sliding direction, e.g., refs. 14 and 15. In addition, the type of fibres and their volume fraction play an important role in the tribological properties of these composites. Zum Gahr [6] studied the wear behaviour of a steel fibre reinforced polyester composite, where the use of fibres led to an anisotropic microstructure. The modified rule of mixtures equation which described the composite’s abrasive wear resistance best, was found to be: W –1 = [ VSt WSt–1 + VP WP–1 – VSt Vp ( WSt–1 + Wp–1 )]

13.4

The objective of the study by Hawthorne [16] was to determine if hybridisation could provide benefits for composite tribological properties. Consequently, investigations have been made of the friction and wear behaviour of unidirectional single-fibre and mixed carbon and glass fibre/epoxy materials when sliding, with various fibre orientations, against a steel counterface. The results in Fig. 13.6 indicate that all the hybrid carbon/glass fibre composites exhibit lower wear rates than those expected from a linear interpolation (LROM) between the single fibre composite values. For example, substituting 33% of the glass fibres in GFRP by carbon fibres reduces the wear rates between 55–80%, depending upon fibre orientations. Similar comments apply to the initial wear rates, suggesting that such large reductions are determined more by the type of composite than by the sliding-generated topography on the counterface. This greater reduction of wear rates could thus be regarded as a positive effect in the wear behaviour of these materials Sliding distance 15 km Load = 93 N Speed = 0.5 m/s Nominal VF = 57%

Specific wear rate (10–6 mm3/Nm)

8

6

Parallel Antiparallel Normal

4 LROM 2

0

IROM 100% glass

100% carbon Specimen fibre composition

13.6 Variation of the specific wear rate with glass/carbon fibre content [16].

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381

(although the reduction is not as high as one would expect from the inverse rule of mixtures IROM, i.e., the fully drawn line in Fig. 13.6). Another approach to the hybrid model composite is to combine layers of different fibre types and orientations with one laminate. This can result in either a sandwich or a layer structure [7]. Figure 13.7 shows the specific wear rates of hybrid composites consisting of normally orientated aramid fibre and parallel carbon fibre in an amorphous PA matrix. The arrangement of the aramid fibre in the sandwich core resulted in a positive hybrid effect, i.e., the wear rates for the hybrid were lower than those expected from a simple rule of mixtures (IROM) based on the wear rates of the two components when being tested separately. The best results were achieved when more that 50% of the fibres were aramid. For the 50/50 mixture it became evident that a change in the stacking sequence, i.e., the aramid fibre now positioned in the surface layers, does not result in the same positive hybrid effect. Instead, the wear data were found to be on the linear connection between the two limits. 5

˙ (10–7 mm3/N · m) W s

4 Sandwich structure (AF-core)

3

2

1

0

Sliding wear AF/CF-PA66-hybrids vs. steel 0 20 100% AF (N)

Layer structure

40

60 Vf (CF)

80

100 100% CF (P)

13.7 Specific wear rates of aramid-fibre/carbon-fibre/PA 6.6 hybrids.

The hybrid model developed in this study, therefore, contains not only the wear contributions of the two components (when measured separately), but also a wear reduction term, which accounts for possible changes due to hybridisation: WH = WAB – Wred

13.5

where WH is the specific wear rate of the hybrid composite, i.e., the specific wear rate due to simultaneous sliding of the two components A and B against the same counterpart, and Wred is the wear reduction due to interactive protection of one component by the other (to be arranged sequentially with WAB because interaction between A and B builds up with time).

382

Multi-scale modelling of composite material systems

The main term in eqn 13.5 WAB is calculated from the wear resistances of the two components A and B, since the wear of A and B starts simultaneously: WAB = [(1 – VB )WA–1 + VB WB–1 ] –1

13.6

where VB is the relative contribution of component B to the hybrid (VB + VA = 1). The reduction term, Wred, is a more complex quantity because it has to account for different factors: (i) the absolute amount of reduction of the wear rate as a function of hybrid composition and (ii) the relative dominance of one or other fibre in the hybrid effect. A general expression taking these factors into consideration is: Wred = XWAf(VA, VB)

13.7

where X is the hybrid efficiency factor. Further details can be found in ref. 7. Using the latter equations, the experimental data given for the aramidfibre (N)/carbon-fibre (P)/PA6.6 hybrid with sandwich structure, were approached iteratively. Figure 13.8 shows that this can be done with quite good agreement. 5 Rule of mixtures approach

˙ (10–7 mm3/N · m) W s

4 Experimental data 3 Hybrid model approach 2 Sliding wear AF/CF-PA66-hybrids vs. steel

1

00

20 100% AF (N) (core)

40

60 Vf (CF)

80

100 100% CF (P) (surface)

13.8 Comparison of experimental data and theoretical approach of hybrid wear effects in carbon-fibre/PA 6.6 composites [7].

13.3

Wear in relation to other mechanical properties

The correlations between wear resistance and characteristic properties of polymers and composites are another route for wear modelling that has been discussed in terms of various semi-empirical equations by some pioneers. These include, e.g., the Ratner-Lancaster equation [17,18], i.e., the relationship of the single pass abrasion rate with the reciprocal of the product of ultimate tensile stress and strain, or an equation used by Friedrich [19] to correlate the

Wear modelling of polymer composites

383

erosive wear rate of polymers with the quotient of their hardness to fracture energy. Although these equations are quite helpful to estimate the wear behaviour of polymers in some special cases, wear normally is very complicated, and it therefore depends on much more mechanical and other parameters. This means that simple functions cannot always cover all the prevailing mechanisms under wear. A new mathematical approach, artificial neural networks (ANN), has been introduced recently into this field as well.

13.3.1 Various traditional attempts In general, for the wear behaviour prediction for a special pair of material the following can be distinguished: (i) empirical models, (ii) models on the basis of contact mechanics, and (iii) models including additional failure mechanisms. Normally, models based on (ii) and (iii) are customised to the respective system by empirically obtained constants [20, 21]. Most of the empirical models present the wear process of a tribological pair only as a time function. In some cases, these equations consider strain parameters. A well-known empirical model is from Rhee [22]: DW µ Fa · vb · tc

13.8

with DW = mass lost (g), F = load (N), v = sliding velocity (m/s) and t = testing time (s). The exponents a, b, and c are determined experimentally. Newly developed empirical models make it possible to predict the wear behaviour qualitatively. These are pictured in so-called ‘wear maps’, in which the transitions between different wear mechanisms are shown, either depending on p and v (generally) or with respect to the special mechanical properties of the material pair as well as the structure of the tribological system. These wear maps are valid under strictly defined conditions and normally nontransferable. They are mainly applied in ceramics and metals. Wear models based on contact-mechanics establish a relationship between the wear and the mechanical/thermal properties of the paired materials in connection with the surface topography. The main focus lies on the correlation between the mechanical properties and the wear behaviour. New models were developed on this basis, which also consider the failure mechanisms, as it was recognised that the complex process leading to the material loss could not be described completely by specific mechanical properties (e.g. for the contact-mechanical calculation of the real contact surface) [20, 21]. The following paragraphs of this section describe some important models for the contact-mechanics and for the failure mechanisms. These can, in general, be distinguished regarding the dominating wear mechanisms. Archard [23] drafts a very simple basic approach to estimate the adhesive wear. Thus, the adhesive wear depends on the real contact surface. The real

384

Multi-scale modelling of composite material systems

contact surface should be inversely proportional to the hardness H of the softer wear partner. Bowden and Tabor [24] confirmed this simple correlation. Wad µ H–1

13.9

with Wad = specific wear rate (mm3/Nm) and H = hardness (HV). Bartenev [25] found the wear rate to be proportional to the frictional rate as well as the applied tensile energy absorption. The adhesive wear rate is inversely proportional to the tensile energy absorption (approximation by the strength and ultimate strain) of the material:

m se E

Wad µ

13.10

with m = coefficient of friction [1], s = stress at break (MPa), e = ultimate strain (%) and E = Young’s modulus (MPa). Archard assumes that the material is deformed elastically. Bartenev distinguishes elastic from plastic deformations. For the elastic deformation, the wear rate is inversely proportional to the Young’s modulus of the material. For polymeric materials (as shown in Fig. 13.9), Ratner and Lancaster [17, 18] found an additional correlation between the tensile energy absorption (sg e g ) and the abrasive wear behaviour. By a combination of all these approaches one can conclude that the abrasive wear rate Wabr can be considered to be inversely proportional to the hardness and the fracture energy:

Wabr µ

mp Hse

13.11

PS PMMA PAC PP PTEE PE PA 6.6

3 ˙ Ê mm ˆ W s Á ˜ Ë N ◊m ¯

100

10–1

10–2

10–3 10–4 –3 10

10–2

10–1

100 Ê mm 2 ˆ 1 s B ◊ e F ÁË N ˜¯

101

13.9 Ratner-Lancaster correlation of wear rate vs. fracture energy (counterface roughness 1.2 mm).

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385

with p = pressure in [MPa], s = stress at break [MPa] and e = ultimate strain in [%]. The coefficient of friction (m) and the applied pressure (p) also contribute to the wear rate. k is the probability of formation of one wearing particle per unit of the gliding path. This approach was frequently seized upon and enhanced especially by Briscoe, Evans and Lancaster [26]. Lancaster assumes that the simple approach by Ratner is valid only if the polymer wears against a steady counterpart. For real systems (e.g. slide bearings), which always contact the very same surface position of the counterpart, the validity of the relations mentioned above is restricted, due to a change in the properties of the counterpart (roughness, transfer film) [27]. Hornbogen [28] and Zum Gahr [29] relied on the linear fracture mechanics approach to interpret the abrasive wear. The breaking process during the invasion of a rough spike is expressed by the energy release rate GIc. In connection with the approaches mentioned above this results in the following equation for the abrasive wear: Wabr µ

m mE = 2 3/2 3/2 G IC H K IC H

13.12

with GIC = energy release rate (kJ/m2), KIC = fracture toughness (MPa · m0.5).

13.3.2 Artificial neural network approach For predictive purposes, an artificial neural network (ANN) approach has, therefore, been introduced recently into the field of wear of polymers and composites by Velten et al. [30] and Zhang et al. [31]. An ANN is a computational system that simulates the microstructure (neurons) of a biological nervous system. The most basic components of ANN are modelled after the structure of the brain. Inspired by these biological neurons, ANN is composed of simple elements operating in parallel. ANN is the simple clustering of the primitive artificial neurons. This clustering occurs by creating layers, which are then connected to one another. How these layers connect may also vary. Basically, all ANN have a similar structure of topology. Some of the neurons interface the real world to receive its input, and other neurons provide the real world with the network’s output. All the rest of the neurons are hidden from view. As in nature, the network function is determined largely by the interconnections between neurons, which are not simple connections, but some non-linear functions. Each input to a neuron has a weight factor of the function that determines the strength of the interconnection and thus the contribution of that interconnection to the following neurons. ANN can be trained to perform a particular function by adjusting the values of these

386

Multi-scale modelling of composite material systems

weight factors between the neurons, either from the information of outside the network or by the neurons themselves in response to the input. This is the key to the ability of ANN to achieve learning and memory. The multi-layered neural network is the most widely applied neural network, which has been utilised in most of the research work for materials science, reviewed by Zhang and Friedrich [32]. Back propagation algorithms can be used to train these multi-layer feed-forward networks with differentiable transfer functions to perform function approximation, pattern association, and pattern classification. The term ‘backpropagation’ refers to the process by which derivatives of network error, with respect to network weights and biases, can be computed. The training of an ANN by backpropagation involves three stages: (i) the feedforward of the input training pattern, (ii) the calculation and back propagation of the associated error, and (iii) the adjustment of the weights. This process can be used with a number of different optimisation strategies. For materials research, a certain numbers of experimental results are always needed first for developing an artificial neural network that performs well. The coefficient of determination B has been introduced to the ANN quality evaluation, which is defined by M

B=1–

S ( O( p ( i ) ) – O ( i ) ) 2 i=1 M

S (O

i=1

(i)

– O)

13.13

2

where O(p(i)) is the i th predicted property characteristic, O(i) is the i th measured value, O is the mean value of O(i), and M is the number of test data. The coefficient B describes the fit of the ANN’s output variable approximation curve with the actual test data output variable curve. Higher B coefficients indicate an ANN with better output approximation capabilities. To avoid any artificial influence in selecting the test data, a random technique could be applied in the selection, and the entire process will be repeated independently several times (e.g. 50 times). Afterwards the distribution of B values is recorded and the percentage of B ≥ 0.9 is calculated, since this value is identified as a high predictive quality, i.e., less than 15% of the RMSE are between the predicted values and the measured ones. It is clear that the higher the percentage of B ≥ 0.9 the better is the quality. Recently, Zhang, Barkoula, Karger-Kocsis, and Friedrich [33] applied this approach to deal with the erosive wear data of three polymers, i.e., polyethylene (PE), polyurethane (PUR), and an epoxy modified by hygrothermally decomposed polyurethane (EP-PUR). The impact angle of solid particle erosion and some characteristic properties (material compositions for the case of EP-PUR as well) were selected as ANN input variables for predicting the erosive wear rate. It seems that the ranking of the importance

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387

of characteristic properties to the erosive wear rate could offer some information about which property has a stronger relationship to wear of polymers. As an example, for the EP-PUR polymer system, the weight amount of PUR varies from 0%, 20%, 40%, 60%, to 80%. The characteristic properties considered include density, mean molecular mass between crosslinks, glass transition temperature (Tg), rubbery plateau modulus and onset temperature, crosslink density, and fracture energy. Four erosive impact angles, i.e., 30∞, 45∞, 60∞ and 90∞, were applied, and four types of erodents recognised by various mass flow rate and velocity were employed. Therefore, the whole dataset of EP-PUR contains 80 independent groups of data. The duration was 60 seconds for all the erosive measurements. In order to investigate the correlations between erosive wear rate and characteristic properties of these polymers, each characteristic property was used only with the necessary erosive conditions together as input variables for training the ANN. The qualities were analysed by the percentage of B ≥ 0.9, which was used to rank the importance of these characteristic properties to erosive wear as summarised in Table 13.1. It is clear that material compositions, i.e., epoxy and PUR weight content, present the strongest correlation with wear performance, in which a similar effect was also found by Zhang et al. [31] for the ANN prediction of polymer composites. Density holds the second important position due to its strong relation to compositions. Mean molecular mass between crosslinks (Mc) and fracture energy (Gc) exhibits a similar quality, which may be explained by the linear dependence of Gc to Mc1/2.

Table 13.1 Ranking of importance of input variables to erosive wear of polyurethane (PUR) predicted by ANN* [33] Ranking Input variables

Percentage of B ≥ 0.9 (%)

Percentage of B ≥ 0.8 (%)

1 2 3 4 5 6 7 8 9

23 15 13 12 9 7 7 6 5

43 36 34 31 22 28 28 28 19

Loss factor at Tg Failure strain (%) Failure stress (MPa) Hardness (Shore A) Stress s at e = 300% (MPa) Density (g/cm3) Stress s at e = 100% (MPa) Thermal expansion coefficient (¥ 10–1/K) Glass transition temperature Tg (∞C)

* Erosive impact angle and one of the input variables in this table were applied as ANN input to predict the erosive wear rate using a 2-[25]-1 structured neural network, which contains 25 neurons in its hidden layer. The coefficient of determination was calculated according to eqn 13.13, and the percentage of B ≥ 0.9 (additionally with B ≥ 0.8) was applied for ranking.

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Multi-scale modelling of composite material systems

It is ideal when only material compositions and testing conditions serve as ANN input data. Figure 13.10 shows, as an example, a 3D-plane of the predictive results of the erosive rate as a function of epoxy weight content and impact angle of solid particle for one type of erodent. Compared to the real test results (dots in Fig. 13.10), the predictive results are very acceptable.

(mg/kg) wear rate

2500

2000

Erosive

1500 1000 100

w

en

co

40

y

60 70 act a ngle (∞ )

80

90

20

ox

50 Imp

Ep

40

nt

0

t(

80 60

t. %

)

500

13.10 Erosive wear rate of an epoxy modified by hygrothermally decomposed polyurethane (EP-PUR) as a function of epoxy weight content and impact angle of solid particle. Dots are experimental data, whereas the rest of the 3D-plane was calculated by an artificial neural network approach. (Erodent: corundum, size = 60~120 mm, mass flow rate = 0.015 kg/s, velocity = 70 m/s) [33].

Based on this work devoted to predict the wear performance of the indicated polymeric systems by adopting the method of artificial neural networks, the following conclusions can be drawn. (i) Ranking of the importance of characteristic properties to the erosive wear rate could offer some information about which property has a stronger relationship to wear of polymers; (ii) ANN is a helpful mathematical tool in the property analysis and prediction of polymers, being directly based on a limited number of measurement results.

13.4

Finite element modelling of composite wear mechanisms

In addition to the previous approach for the description and understanding of the wear preformances of composite materials against various counterparts, it has been attempted in recent years to analyse the contact, stress and

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389

deformation characteristics during sliding contact by the use of finite element methods. Most of these evaluations traditionally followed a macroscopic approach, assuming homogeneous, anistropic material properties as derived by rule of mixture type relationships. A disavantage of this macroscopic approach is that it is not suitable for modelling the actual interaction of the fibres and matrix of the composite with the asperities of the counterpart. In ref. 34, Ovaert analysed P-fibre orientation (relative to the sliding direction) by using anisotropic half-space models. The individual fibre was modelled as an infinite beam on an elastic foundation, with the foundation of stiffness approximated from the results of the contact simulation. From these results, a fibre stress due to deformation and sliding is estimated. The normal and tangential forces from the rough surface will induce tensile stresses in the fibres at the surface, which are a function of the counterface asperity geometry and the asperity load. These tensile stresses play an important role in the deterioration of the surface fibres, which leads to subsequent fibrematrix separation and enhanced wear. Later Ovaert [35] extended this model to AP-fibre orientation. According to his conclusion, a rough correlation exists between normalised wear rates and calculated stress – deformation parameters for several polymer composites. A FE micro-model was used [36] to determine contact and stress states produced by a steel ball pressed into a fibre-reinforced composite. Location and distribution of sub-surface stresses and strains were studied for N- as well as P-fibre orientation. It was established that in the case of N-fibre orientation there is a high shear and compression to the surface. In the case of P-fibre orientation, the matrix is subjected to both shear and compression type straining, yielding and local plastic deformation, while the characteristic deformations of the fibres are bending and compression. The present study aims at the development of FE micro-models that allow the provision of explanations for different failure mechanisms, in the case of different fibre orientations, based on the evaluation of contact and stress conditions produced by a sliding hemispheric steel asperity (Fig. 13.11). In particular, it is expected that this approach will provide information about the actual fibre stresses, matrix strains, events of fibre/matrix debonding, etc. The composite material studied here is a CF (carbon fibre) reinforced PEEK (polyether-etherketone) with a fibre volume fraction of 0.61. The radius of the steel asperity is R = 0.45 mm for all cases. The normal load is FN = 1N, and the friction coefficients are m = 0.45 (N), m = 0.28 (P) and m = 0.3 (AP), respectively. In the models, a larger radius was required than that of the real average asperity (some ten microns) in order to be able to compare our results with experimental test results.

390

Multi-scale modelling of composite material systems FN

Sliding direction

FR

FN

FR

Sliding direction

FT

FT

R

R

x

y

y

2a

2b 2a

x 2b

z

(a)

z

(b) FR

FN Sliding direction

FT R y

2a

x 2b

z

(c)

13.11 The modelled sliding asperity with normal and tangential loading; (a) N-fibre orientation, (b) P-fibre orientation, and (c) APfibre orientation.

13.4.1 FE macro- and micro-contact models To evaluate the stresses in a fibre/matrix micro-system an FE micro-model was first created. This micro-model is ‘built into’ a larger (homogeneous and anisotropic) macro-model in order to represent a larger segment of the original body (Fig. 13.12), and to achieve higher accuracy. Due to the symmetry condition, half of the structure is modelled, as illustrated in Fig. 13.12(b). The contact elements are orientated under an angle, relative to the perpendicular direction, representing the friction cone. In other words the direction of the contact elements is the same as the direction of the resultant force FR, representing the normal force FN and the friction force FT, shown in Fig. 13.11 for the cases studied.

13.4.2 FE-results Contact results The contact pressure distributions for N-fibre orientation are shown in Fig. 13.13 without and with friction. In both cases mostly the fibres carry the load. The frictional force produces an asymmetric pressure distribution on

Wear modelling of polymer composites

391 Macro

Steel asperity x y Micro z

Macro Composite material

Micro

(a)

(b)

13.12 (a) The global model, and (b) the macro/micro-models with contact elements (the arrows describe the actual loading conditions of the steel asperity).

Sliding

(a)

directio

n

(b)

13.13 Contact pressure distribution (a) without friction, and (b) with friction in the case of N-fibre orientation.

each fibre loaded. The highest contact pressure appears at the rear edge of each fibre within the contact area. Figure 13.14 illustrates the contact pressure distribution in the case of Pand AP-fibre orientations. In both cases the normal and friction forces are practically transferred by fibres, due to their much higher stiffness (compared to the polymer matrix). Displacement and stress results Figure 13.15 shows the deformed shape of the normally orientated composite micro-structure. It is clearly visible that the side edges of the micro-model are not vertical any more, due to the results of the displacement coupling technique. The deformed shape of the fibres is primarily caused by compression and bending/shear type loadings. The Mises equivalent stress distribution in the composite is shown in Fig.

392

Multi-scale modelling of composite material systems

Sliding direction (a)

Sliding direction (b)

13.14 Contact pressure distribution in the case of (a) P-, and (b) APfibre orientations.

13.16. Considering the fibres, high compression stresses appear on top of them. In the middle of the contact area the rear edge of each fibre is ‘overloaded’. The same ‘asymmetric behavior’ is shown in Fig. 13.13(b) for the contact pressure distribution. This high stress may cause fibre cracking at these locations. According to the equivalent stress distribution in the matrix, it can be established that stresses, exceeding the yield strength of the matrix material, are produced near the surface and along the fibre/matrix interfacial regions. As a result, the matrix material becomes deformed and eventually shearedoff in the form of thin wear debris layers [37]. Considering the fibre/matrix interface, debonding phenomena may also occur, due to a repeated compression-tension loading along the interfaces on both sides of the loaded fibres (in sliding direction). The debonding starts within small environments of the stressed surface region and can then propagate into the subsurface. Figure. 13.17 shows the horizontal stress component sy at the fibre/matrix interface as a function of depth at the rear edge of the fibre/matrix contact region.

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x y z

13.15 Deformed shape of the ‘micro-environment’ in the case of Nfibre orientation (deformation scale is 5:1).

Mises 2478 2168 1859 1549 1239 929 619 310 0.00

x y z

13.16 The equivalent stress distribution (in MPa) in the composite for N-fibre orientation.

394

Multi-scale modelling of composite material systems sy (MPa) 0

100

200

300

400

0

Depth (mm)

10 20 30 40 50

13.17 The horizontal stress components sy producing fibre debonding as a function of depth in the case of N-fibre orientation.

In Fig. 13.18, the dominant deformation is caused by compression of the composite system and corresponding bending type loading of the fibres. Figure 13.19 illustrates the equivalent stress distribution in the composite. The highest stresses are built-up within the fibres in the y direction. They are subjected to bending, which causes high compressive stresses in the middle of the contact area. This stress component may finally cause multiple cracking of wear-thinned fibres in the surface region. Behind the contact area, tension also appears in most of the loaded fibres. The most dominant stress component in the matrix is the shear stress in the y direction, located just below the mostly loaded half-fibres. This can

x

y z

13.18 Deformed shape of the ‘micro-environment’ in the case of Pfibre orientation (deformation scale 5:1).

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395

Mises 1167 1021 875 729 583 437 292 146 0.00

x

y z

13.19 The equivalent stress distribution (in MPa) in the composite for P-fibre orientation.

locally produce a shear type fibre/matrix debonding. During the repeated sliding motion this debonding can also propagate to the surface. The dominant deformation in Fig. 13.20 (AP-fibre orientation) is caused by compression of the composite system, associated with bending and torsion type loading of the fibres. The highest stresses (Fig. 13.21) occur within the fibres in the x direction. They are subjected to transverse compression, bending and torsion. The highest equivalent stresses are in the fibres at their rear

x

y z

13.20 Deformed shape of the ‘micro-environment’ in the case of APfibre orientation (deformation scale 5:1).

396

Multi-scale modelling of composite material systems Mises 959.6 839.6 719.7 599.7 479.8 359.8 239.9 119.9 0.000

x

y z

13.21 The equivalent stress distribution (in MPa) in the composite for AP-fibre orientation.

edges, similar to those observed for the N-fibre orientation. Here, however, the maximum stress values are lower than in N-orientation, thus yielding a lower probability for fibre lengths cracking events at these locations. The most dominant stress components are the shear stresses in the x and y directions, parallel to the surface plane. From this it can be concluded that matrix shear failure and tension/compression type fibre/matrix debonding will be the dominant matrix failure events in the case of AP-fibre orientation.

13.4.3 Experimental verification To simulate the wear process, single scratches were produced by the indentor on the composite surfaces. The diamond Rockwell test indentor had a tip angle of 120∞ and a tip radius of 100 mm. In the testing series, normal loads, varying between 1 and 3N, were chosen. Figure 13.22 shows a higher magnification SEM-micrograph of a diamond tip scratched region under 3 N load in the case of N-fibre orientation. The typical wear mechanisms are as follows: (i) enhanced push-up of matrix between the fibres, and (ii) cracking at the rear edges of the fibres. Figures 13.23 and 13.24 show the scratched surfaces of the composite specimens in the case of parallel and anti-parallel fibre orientations. In these figures, shear failure events of the polymer matrix, fibre/matrix debonding and fibre cracking can be identified [38].

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Direction of scratching

13.22 Higher magnification SEM-micrograph of a diamond scratch region under a load of 3 N.

Direction of scratching

13.23 The typical failure mechanisms on a P-orientated CF/PEEKsurface, as produced by a diamond indentor under a load of 3 N.

Direction of scratching

13.24 The typical failure mechanisms on a AP-orientated CF/PEEKsurface, as produced by a diamond indentor under a load of 3 N.

398

13.5

Multi-scale modelling of composite material systems

Conclusions

This chapter shows that a lot of possiblities exist in order to estimate or predict the wear properties of composites materials. The rule-of-mixtures (ROM) approaches are useful if one knows about the individual wear rate of the components and their relative volume fractions. It was, however, shown that the very simple IROM or LROM equations are often not sufficient for a simple description of the trends in the experimental results. In most of the cases, the ROM approaches must be modified so that more complex, but therefore closer to reality, relationships are the result. Wear in relation to other mechanical properties is a traditional attempt to estimate the wear properties from easily measurable mechanical characteristics. It is a quite helpful approach as long as the wear mechanisms are well defined and the mechanical properties chosen are to some degree related to the occuring mechanisms. If a good correlation is found, this helps to reduce the time for usually long lasting wear experiments. The same is true for the relatively new artifical neural network (ANN) approach, which, however, needs for its training a large number of datasets. Once the ANN works, it can be used for (i) predicting the wear rate of new compositions of the composite system under consideration that have not been tested experimentally before, (ii) for parameter studies with regard to external testing conditions, and (iii) for estimations about which input parameter is of more or less important for the output parameter ‘wear rate’. Finite element modelling on a micro-mechanical scale is another tool to predict the interaction between counterpart asperities and the individual compoments of the composite materials studied. It allows the calculation of the critical conditions under which certain types of wear mechanisms are initiated. Local effects of fibre orientation can also be considered. In addition, temperature developments due to the sliding motion of the mating surfaces are predictable with this method. Nevertheless, more studies have to be done in this field in order to be able to conclude from the predicted wear mechanisms the resulting wear rate of the particular material studied. Good examples on how to illustrate the wear mechanisms of different materails schematically are given in reference 39.

13.6

Acknowledgements

The authors would like to acknowledge the support of some of this work by the Deutsche Forschungsgemeinschaft (DFG FR675/45-1). Parts of these studies were also carried out under the German-Hungarian partnership programme of the BMBF (WTZ-HUN 020/99). Z. Zhang is grateful to the Alexander von Humboldt Foundation for his Sofja Kovalevskaja Award, financed by the German Federal Ministry of Education and Research within

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the German Government’s ‘ZIP’ programme for investment in the future. The authors wish to thank Mr. T. Goda for his contribution on the FEM analysis.

13.7

References

1. Simms W. and Freti S., Abrasive wear of multiphase materials, Wear 129 (1989) 105–121. 2. Garrison W.M., Jr. Wear 82 (1982) 213–220. 3. Zum Gahr K.-H., in: Proc. Int. Conf. on Wear of Materials, Vancouver (American Society of Mechanical Engineers, New York, 1985) 45–58. 4. Sandt A. and Krey J., Metall 39(3) (1985) 233–237. 5. De Mello J.D.B., Durand-Charre M. and Mathia T., Colloque International sur les Matériaux résistant à l’usure, St.-Etienne, November 23–25 (1983) 21.4–21.11. 6. Zum Gahr K.-H., Microstructure and Wear of Materials, Tribology Series, Vol. 10, Elsevier, Amsterdam, 1987. 7. Friedrich K., Wear models for multiphase materials and synergistic effects in polymeric hybrid composites, in: K. Friedrich (ed.), Advances in Composites Tribology, Composite Materials Series, Vol. 8 (series editor R.B. Pipes), Elsevier, Amsterdam, 1993, 209– 273. 8. Prasad S.V., and Calvert P.D., Abrasive wear of particle-filled polymers, J. Mater. Sci. 15 (1980) 1746–1754. 9. Voss H., Aufbau, Bruchverhalten und Verschleißeigenschaften kurzfaserverstärkter Hochleistungsthermoplaste, VDI, Reihe 5, Grund- und Werkstoffe, No. 116, VDIVerlag, Düsseldorf, 1987. 10. Williams J.A., Morris J.H. and Ball A., The effect of transfer of layers on the surface contact and wear of carbon-graphite materials, Tribology Int., 30 (1997) 663–676. 11. Voss H. and Friedrich K., On the wear behaviour of short fibre reinforced PEEKcomposites, Wear 116 (1987) 1–18. 12. Voss H. and Friedrich K., Wear performance of a bulk liquid crystal polymer and its short fibre composites, Tribology Int. 19 (1986) 145–156. 13. Friedrich K., Karger-Kocsis J., Sugioka T. and Yoshida M., On sliding wear performance of polyethernitrile (PEN)-composites, Wear 158 (1992) 157–170. 14. Giltrow J.P. and Lancaster J.K., Properties of CFRP relevant to applications in tribology, in: Proc. Int. Conf. on Carbon Fibers, Plastics Institute, UK, 1971, 251– 257. 15. Sung N.-H. and Suh N.P., Friction and wear of fibre reinforced polymeric composites: effect of fibre orientation on wear, in: Proc. 35th Ann. Tech. Conf. Society of Plastics Engineering (1977) 311–314. 16. Hawthorne H.M., Wear in hybrid carbon/glass fiber epoxy composite materials, in: Proc. Int. Conf. on Wear of Materials, Reston, VA, April 11–14, 1983, ed. K.C. Ludema (ASME, New York, 1983) 576–582. 17. Ratner S.N., Farberoua I.I., Radyukeuich O.V., Lure E.G., Correlation between wear resistance of plastics and other mechanical properties, in: D.I. James (ed.), Abrasion of Rubber, MacLaren, London, 1967, 145–154. 18. Lancaster J.K., Friction and wear, Chapter 14, in: A.D. Jenkins (ed.), Polymer Science and Material Science Handbook, North Holland Publishing Co., London, 1972. 19. Friedrich K., Erosive wear of polymer surfaces by steel ball blasting, Journal of Materials Science, 21 (1986) 3317–3332.

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20. Meng H.C. and Ludema K.C., Wear models and predictive equations: their form and content, Wear 181 (1995) 443–457. 21. Williams J.A., Wear modelling: analytical, computational and mapping: a continuum mechanics approach, Wear 225 (1999) 1–17. 22. Rhee S.K., Wear equation for polymers sliding against metal surfaces, Wear 16 (1970) 431–445. 23. Archard J.F., Contact of rubbing surfaces, J. Applied Physics 24 (1953) 981–988. 24. Bowden F.P. and Tabor D., The friction and lubrication of solids (Part II), Oxford: Clarendon Press 1964. 25. Bartenev G.M. and Lavrentev V.V., Friction and wear of polymers, Tribology Series 6. Amsterdam: Elsevier Science Publishers 1981. 26. Briscoe B.J., Evans P.D. and Lancaster J.K., The influence of debris inclusion on abrasive wear relationships of PTFE, Wear 124 (1998) 177–194. 27. Lancaster J.K., Relationship between the wear of polymers and their mechanical properties, Proc. Instn. Mech. Engrs. 183 (1968) 98–106. 28. Hornbogen, E. The role of fracture toughness in the wear of metals, Wear 33 (1975) 251–259. 29. Zum Gahr K.-H., Abrasiver Verschleiß metallischer Werkstoffe, VDI-Fortschr.-Bericht Reihe 5. Nr. 57. Düsseldorf: VDI Verlag 1981. 30. Velten K., Reinicke R. and Friedrich K., Wear volume prediction with artificial neural networks, Tribology International, 33 (2000) 731–736. 31. Zhang Z., Friedrich K. and Velten K., Prediction on tribological properties of short fibre composites using artificial neural networks, Wear, 252 (2002) 668–675. 32. Zhang Z. and Friedrich K., Artificial neural network applied to polymer composites: a review, Composite Science and Technology, 63[14] (2003) 2029–2044. 33. Zhang Z., Barkoula N.M., Karger-Kocsis J. and Friedrich K., Artificial neural network predictions on erosive wear of polymers, Wear 255[1-6] (2003) 709–714. 34. Ovaert T.C. and Wu J.P., Tribology Transactions 36(1) (1993) 120–126. 35. Idem, ibid. 37(1) (1994) 23–32. 36. Váradi K., Néder Z., Friedrich K. and Flöck J., Composites Science and Technology 59 (1999) 271–281. 37. Goda T., Váradi K., Friedrich K. and Giertzsch H., Finite element analysis of a polymer composite subjected to a sliding steel asperity, Part I: Normal fibre orientation, J. Mater. Sci. 37 (2002) 1575–1583. 38. Friedrich K., Váradi K., Goda T. and Giertzsch H., Finite element analysis of a polymer composite subjected to a sliding steel asperity, Part II: Parallel and antiparallel fibre orientations, J. Mater. Sci. 37 (2002) 3497–3507. 39. Stachowiak G.W. and Batchelor A.W., Engineering Tribology, 2nd edn, Butterworth Heinemann, Boston, 2001.

14 Modelling impact damage in composite structural elements A F J O H N S O N, German Aerospace Center (DLR), Stuttgart

14.1

Introduction

This chapter reviews recent progress on composites materials modelling and numerical simulation of impact on fibre reinforced composite structures from both hard and soft projectiles. To reduce certification and development costs, computational methods are required by the aircraft industry to be able to predict structural integrity of composite structures under high velocity impacts from hard objects such as metal fragments, stone debris and from soft or deformable bodies such as birds, hailstones and tyre rubber. Key issues are the development of suitable constitutive laws for modelling composites in-ply and delamination failures, determination of composites parameters from dynamic materials tests, materials laws for deformable impactors, and the efficient implementation of the materials models into finite element (FE) codes. These problems involve both multi-scale and multi-physics modelling techniques which are discussed in the chapter. The multi-scale aspects arise because impact damage is localised and requires fine-scale modelling of delamination and ply damage at the micromechanics level, whilst the structural length scale is much larger. The multi-physics aspects arise in fluid-structure interactions when soft bodies such as gelatine (substitute bird) or ice (hailstones) flow extensively on impact with the structure and require fluid modelling techniques. Polymer composites exhibit a range of failure modes: matrix cracking, transverse ply cracks, fibre fracture, fibre pull-out, microbuckling, interply delamination, etc., which are initiated at the micro level and can be modelled by micromechanics techniques at length scales governed by fibre diameters. The length scale for aircraft structural analysis is in metres, with shell element size for crash or impact simulations in FE analyses measured in cms. The challenge for composites research is to develop appropriate materials models at the structural macro level which embody the salient micromechanics failure behaviour. Impact modelling and simulation of impact damage in composite structures have been extensively studied in recent years, since composite 401

402

Multi-scale modelling of composite material systems

structures are susceptible to impact damage due to low fibre failure strains which lead to brittle failure modes with low energy absorption. The comprehensive review by Abrate [1] discusses impact failure mechanisms in composite structures and summarises impact modelling approaches, based mainly on analytical models. The emphasis in this chapter is on development and validation of numerical methods with finite element (FE) techniques which are suitable for use in assessment of impact damage in composite aircraft structures. The solution proposed is to use meso-scale models based on continuum damage mechanics (CDM) as the link between composites micro- and macroscales. CDM provides a framework for modelling in-ply and delamination failures which is suitable for implementation into explicit FE codes. Damage mechanics for composite materials is an extensive subject, with different philosophies and schools of thought, and is well documented in the monograph by Talreja [2]. Several authors have been concerned with the development of CDM materials laws for composites failure within FE codes, see for example the works of Williams et al. [3], [4], Iannucci et al. [5], Ladevèze [6], and Ladevèze and Le Dantec [7]. The work described here is based mainly on the CDM formulations of Ladevèze and his co-workers. Most reported research on impact in composite structures [1], [3], [5] concentrates on impact damage and modelling from rigid body impactors. However, for aircraft structures soft body impactors such as gelatine (substitute bird) or ice (hailstones) are highly deformable on impact and flow over the structure, spreading the impact load. For reliable damage prediction in composite structures it is thus necessary to develop modelling techniques and appropriate data for highly deformable impactors. Ice impact in composite plates has been studied in some detail both experimentally and theoretically by Kim and Kedward [8]. They used an elastic-plastic ice model with solid elements for the ice projectiles. However, this method was found to be less suitable for gelatine impactors due to the excessive element distortion on impact. Iannucci [9] discusses suitable materials laws for gelatine as it flows during impact and Johnson [10] describes how these soft impactors are modelled by a smooth particle hydrodynamic (SPH) method, in which the FE mesh is replaced by interacting discrete particles. Data from pressure pulses measured during impact of gelatine and ice on rigid plates are used to determine parameters for a soft body equation of state for use with the SPH method. In this chapter the meso-scale ply damage models for composites are described in section 14.2 for both fabric and unidirectional (UD) composite plies. Ply damage is described by three scalar damage parameters representing modulus reductions under different loading conditions due to microdamage in the ply. Damage evolution equations are introduced relating the damage parameters to damage energy release rates in the ply. The formulation of the damage evolution equations in the Ladevèze CDM models is more physically

Modelling impact damage in composite structural elements

403

based than in other work [3]–[5], in which convenient mathematical functions are chosen rather than materials test data. It also allows generalisations to include features such as shear plasticity and delamination modelling. In the present work test data on carbon and glass fabric/epoxy materials have been used to determine the damage evolution equations and lead to a simplified model in which fibre tension/compression behaviour is elastic damaging and may be decoupled from the elastic-plastic ply shear damage. The delamination model for interply failure in laminates is described in section 14.3 and requires two further interface damage parameters which may be determined from measurement of the delamination fracture energy at the interface in modes I and II. The CDM ply and delamination models have been implemented by ESI GmbH in their explicit FE crash and impact code PAM-CRASH/SHOCK [11]. The ply damage model is implemented in layered composite shell elements and the delamination model is introduced using stacked shell elements with a contact interface failure condition. Section 14.4 summarises recent work on FE code validation in which impact simulations are compared with test data from a series of drop tower and gas gun tests of composite plate and shell structures subjected to impact from hard and soft body impactors at velocities in the range 2–200 m/s. These represent impact conditions relevant to civil aircraft structures ranging from tool drop up to foreign object damage (FOD) and bird strike. Test conditions and impact energies were chosen to give both delamination failures, fibre damage and impactor flow in soft bodies. Results are presented which demonstrate the simulation of composites failure modes and failure progression during impact from hard and soft bodies in composite structural elements. Section 14.5 completes the chapter with the current status and future outlook for FE code developments for predicting impact damage in composite structures.

14.2

Meso-scale ply damage models

The development of numerical design tools for predicting impact damage in composite aircraft structures was the main objective of the EU funded HICAS project [12]. During impact loading of composite structures failure may occur by delamination, which is important in lower energy impacts and in failure initiation, and by in-plane ply failure which controls ultimate fracture and penetration in the structure. Attention was thus given in the HICAS project to the development and improvement of composites failure models suitable for implementation into explicit FE codes. The approach adopted was to use continuum damage mechanics (CDM) for composites as developed by Ladevèze [6] as a framework within which in-ply and delamination failure may be modelled. The mathematical models developed and FE code implementations are summarised here.

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Multi-scale modelling of composite material systems

14.2.1 CDM formulation The composite laminate is modelled by layered shell elements or stacked shells with a tied interface which may fail by delamination. The shells are composed of composite plies which are modelled as homogeneous orthotropic elastic or elastic-plastic damaging materials whose properties are degraded on loading by microcracking prior to ultimate failure. A CDM formulation is used in which ply degradation parameters are internal state variables which are governed by damage evolution equations. For shell elements a plane stress formulation with orthotropic symmetry axes (x1, x2) is required. The in-plane stress and strain components are e e e T
= (s 11 , s 22 , s 12 ) T  e = ( e 11 , e 22 , 2e 12 ) .

14.1

Constitutive laws for orthotropic elastic materials with internal damage parameters are described in refs 6 and 7, and take the general form

 e = S


14.2

where
and  are vectors of stress and elastic strain, and S is the elastic compliance matrix. Using a strain equivalent damage mechanics formulation, the elastic compliance matrix S may then be written: e

Ê 1/ E1 (1 – d1 ) S = Á –n 12 / E1 Á Á 0 Ë

–n 12 / E1 1/ E 2 (1 – d 2 ) 0

ˆ ˜ 0 ˜ ˜ 1/ G12 (1 – d12 )¯ 0

14.3

where E1, E2, are the initial (undamaged) Young’s moduli in the fibre and transverse fibre directions, and G12 is the (undamaged) in-plane shear modulus. The principal Poisson’s ratio n12 is assumed here not to be independently degraded. The ply model introduces three scalar damage parameters d1, d2, d12 which have values 0 £ di < 1 and represent modulus reductions under different loading conditions due to microdamage in the material. For UD plies d1 controls damage in the fibre direction, d2 transverse to the fibres and d12 controls in-plane shear failure. For fabric plies d1 and d2 are associated with damage or failure in the principal fibre directions. The general damage mechanics formalism [6] is based on a postulated internal energy function j for the material, j = 1/2sTSs

14.4

with the compliance matrix S defined in eqn 14.3 above. The damage parameters are now assumed to be internal state variables (ISV) in the mathematical model so that additional equations are required in the constitutive model to connect the ISV damage parameters to other known state variables. In the

Modelling impact damage in composite structural elements

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general formulation eqn 14.6 damage energy release rates Y1, Y2, Y12 or ‘conjugate forces’ are introduced corresponding to ‘driving’ mechanisms for materials damage. On introducing vectors of damage parameters d = ( d1, d2, d12 )T and associated conjugate forces Y = ( Y1, Y2, Y12 )T, it can be shown that the elastic strains e e and conjugate damage forces Y are defined from the strain energy function by:
e =

∂j = S ∂

Y=

∂j ∂d

14.5

With the compliance matrix (eqn 14.2), differentiation of the strain energy function leads to the following definitions for the Y components: 2 2 Y1 = s 11 / (2E1(1 – d1)2), Y2 = s 22 / (2E2(1 – d2)2),

2 Y12 = s 12 / (2G12(1 – d12)2)

14.6

The general theory of materials with ISVs is completed by assuming that the damage parameters are functions of these conjugate force functions. In terms of the damage evolution functions f1, f2, f12 they have the general form: d1 = f1(Y1, Y2, Y12), d2 = f2(Y1, Y2, Y12), d12 = f12(Y1, Y2, Y12). 14.7 The complexity of the failure mechanisms and possible interaction between failure modes is controlled by the forms assumed for these damage evolution functions. In practice very simple forms are assumed which are based on physical observations and specimen test data.

14.2.2 Damage evolution equations Fabric ply model The elastic damage mechanics ply fabric model is based on the following assumptions: ∑ Fibre and shear damage modes are decoupled, with fibre damage determined by Y1 and Y2, and shear damage by Y12. ∑ Fibre damage development may be different in tension and compression. ∑ For balanced fabrics (E1 = E2) damage development in the two fibre directions may be different, thus d1 π d2. However, it is assumed that f1 and f2 will have the same functional form (f1 = f2). ∑ The ply material is ‘non-healing’; therefore damage during unloading is held constant until positive loading is applied which causes further damage accumulation. ∑ Damage development does not necessarily lead to ultimate failure of the ply and a global failure criterion should also be applied.

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Due to the non-healing condition, the damage evolution equations are based on the maximum value of the damage forces reached during the previous loading history. The quantities Y 1 , Y 2 , Y 12 are thus introduced which are defined in terms of the maxima of ÷Yi. Test data on unidirectional (UD) composites [7] has shown that the square root of the damage forces is the quantity which arises more naturally, therefore: Y 1 (t) = max {÷Y1 ( t )}, Y 2 ( t ) = max {÷Y2 ( t )}, Y 12 (t) = max {÷Y12 ( t )}, t £ t

14.8

The assumed forms for the damage parameter functions fi allow for an elastic region without damage at the onset of loading, leading to lower damage energy thresholds (Y10, Y120), and cut off at upper damage energy thresholds (Y1f, Y12f),:

d1 = 0, Y 1 £ Y10

d1 = a 1 ( Y 1 – Y10 ), Y10 < Y 1 < Y1 f

d1 = 1, Y 1 ≥ Y1 f d 2 = 0, Y 2 £ Y10

d 2 = 1, Y 2 ≥ Y1 f

d 2 = a 1 ( Y 2 – Y10 ), Y10 < Y 2 < Y1 f 14.9

d12 = 0, Y 12 £ Y120 d12 = a 12 (ln Y 12 – ln Y120 ), Y120 < Y 12 < Y12 f

d12 = 1, Y 12 ≥ Y12 f Study of ply stress-strain data for glass fabric/epoxy in ref. 13 showed that linear forms for d1 and d2, were good approximations for fabric plies, and an equation linear in ln ( Y 12 ) was found to be suitable for modelling the shear behaviour at larger strains. Thus in the model the evolution equations for a balanced fabric ply require the determination of two slope parameters a1, a12 and four damage threshold parameters Y10, Y120, Y1f, Y12f. Further refinements to allow different fibre damage behaviour in tension and compression, may also be introduced in the model, by having two sets of constants a1, Y10 , Y1f, corresponding to data from tensile and compression tests in the fibre direction. Ultimate failure is controlled by setting the damage parameters di = 1, at threshold energy values Y1f ,Y12f. From the definitions for the Yi in eqn 14.6 these ultimate failure conditions may be simply related to ply failure stresses or strains. Additionally the model allows recognised multiaxial ply failure criteria such as maximum strain, modified Puck, etc., to be applied. In this case the CDM model is used to describe ply degradation such as matrix cracking, whilst ply ultimate failure may be controlled by fibre fracture. This is achieved by setting all the di = 1 when the multiaxial failure envelope is reached. For in-plane shear, ply deformations are controlled by matrix behaviour

Modelling impact damage in composite structural elements

407

which may be inelastic, or irreversible, due to the presence of extensive matrix cracking or plasticity. On unloading this can lead to permanent deformations in the ply. The extension of the fabric model to include these irreversible damage effects is now considered, based on the following additional assumptions: ∑ The total strain in the ply is split into the sum of elastic and plastic (or inelastic) parts. ∑ Plastic strains are associated only with the matrix dominated in-plane shear response. ∑ A classical plasticity model is used with an elastic domain function and hardening law applied to the ‘effective’ stresses in the damaged material. ∑ Inelastic or plastic strain increments are assumed to be normal to the elastic domain function. The total strain
=
e +
p is written as the sum of elastic
e and plastic strains
p. The elastic strain component is given by eqns 14.2 and 14.3. A plane stress model for a thin ply is assumed and only shear strains contribute p to plasticity ( e 11p = e 22 = 0, e 12p π 0). Following eqn. 14.6, an elastic domain function is introduced F (s˜ 12 , R ) where s˜ 12 is the ‘effective’ shear stress s˜ 12 = s 12 /(1 – d12 ) and R is an isotropic hardening function. R(p) is a function of an inelastic strain variable p. The elastic domain function has a simple form here since only the effective shear stress leads to plastic deformation: F = | s12 | /(1 – d12) – R(p) – Ro

14.10

where it is assumed that R(0) = 0 and that Ro is the initial threshold value for inelastic strain behaviour. The condition F < 0 corresponds to a stress state inside the elastic domain where the material may be elastic damaging. It follows from the normality requirement that F = 0, F˙ = 0, hence from eqn 14.10 it can be deduced that the plastic strain p is defined by

e˙12p = p˙ /(1 – d12 ) or p =

Ú

p e 12

0

(1 – d12 ) de 12p

14.11

showing that p is the accumulated effective plastic strain over the complete loading cycle. The model is completed by specifying the hardening function R(p). This is determined from cyclic loading tests in which both the elastic and irreversible plastic strains are measured. A typical form assumed for the hardening function (eqn 14.13) which models test data fairly well is an index function, which leads here to the general equation: R(p) = b pm

14.12

The shear plasticity model depends on the parameters b, the power index m and the yield stress Ro.

408

Multi-scale modelling of composite material systems

UD ply model The theoretical framework of the UD ply model follows closely Ladevèze [7]. Here it is sufficient to explain only the differences to the ply fabric model above. In the UD ply it is assumed that there is no degradation in the fibre direction so that that d1 = 0 in eqn 14.3 and the ply is assumed to be linear elastic in the fibre direction with constant modulus E1. Ply failure occurs when an additional criterion such as fibre failure strain or a multiaxial failure condition is imposed. There are now two damage parameters in the model d2 and d12 corresponding to transverse and shear damage respectively, which are matrix dominated. The model has the refinement that transverse and shear effects may be coupled, which introduces additional coupling parameters. It follows that in the notation used here eqn 14.8. is replaced by: Y 2 (t) = max {÷Y2 ( t )}, Y (t) = max {÷[ Y12 ( t ) + bY2 ( t )]}, t £ t 14.13 where b is a tension-shear damage coupling parameter. With these definitions the damage evolution equations for UD plies are assumed to take the form: d 2 = 0, Y < Y20

d 2 = a 1 ( Y – Y20 ), Y20 < Y < Y2 f ,

d12 = 0, Y < Y120

d12 = a 12 ( Y – Y120 ), Y120 < Y < Y12 f , 14.14

with the additional conditions d2 = d12 = 1,

Y 2 ≥ Y2 f

or

Y 12 ≥ Y12 f

14.15

The choice of evolution equations is based on test data on UD composite plies and is explained as follows. The model has damage initiation and threshold (i.e. maximum) energies Y2o, Y2f in transverse tension, and Y12o, Y12f in ply in-plane shear. Below these initiation values the ply is linear elastic (d2 = d12 = 0 ). Above these values the damage parameter is linear in the coupled energy function Y ( t ) defined in eqn 14.13 until the threshold values are reached, when damage is set to its cut-off values. However, if the threshold energies are first reached for brittle failure in transverse tension (Y2f) or shear (Y12f) then the ply fails and the damage parameters take their cut-off values. The inclusion of ply plasticity follows the formulation given above for fabric plies, with the refinement of a coupling between matrix plasticity due to transverse tension and shear. It is now assumed that the plastic strains take p the form e 11p = 0, e 22 π 0, e 12p π 0 and the elastic domain function eqn 14.10 is generalised to: F = ÷{[s12/(1 – d12)]2 + A[s22/(1 – d2)]2} – R(p) – Ro

14.16

where A is the tension-shear plasticity coupling factor. The effective accumulated inelastic strain p is now given by

Modelling impact damage in composite structural elements p (1 – d 2 )] 2 } p˙ = ÷{[2e 12p (1 – d12 )] 2 + (1/ A )[ e 22

409

14.17

which replaces eqn 14.11. Test data on UD carbon/epoxy indicate that the power law relation eqn 14.12 is valid so that ply plasticity is again characterised by the parameters b, the power index m and the yield stress Ro, with the additional coupling parameter A.

14.2.3 Determination of ply damage model parameters The parameters in the ply elastic damaging model are derived from stressstrain curves on standard composite test specimens loaded monotonically to failure. The plastic hardening law parameters require cyclic load tests to identify the irreversible (plastic) strain components in the specimens. Test programme strategies are discussed in detail for the UD ply model in ref. 7 and for fabric plies in ref. 13. To illustrate the methods a cyclic shear test carried out at the DLR in the HICAS project [12] on a glass fabric (GF)/ epoxy test specimen is discussed. A test programme was carried out with cyclic tensile tests on thin GF/epoxy test specimens loaded at ± 45∞ to the principal fibre directions. The tests were quasi-static under displacement control using specimen geometries and test procedures defined in ISO 14129 for shear properties of composites from ± 45∞ tension tests. Figure 14.1 is a typical shear stress-strain curve obtained from the test with five load-unload cycles, which shows extensive inelastic or ‘plastic’ deformation at larger

Shear stress s 12 MPa

Glass fabric/epoxy. Cyclic shear stress-strain curve

0

0.02 e12pl

0.04

0.06 e12el

0.08

0.1 0.12 Total shear strain e12

14.1 Measured cyclic shear stress-strain curve for GF/epoxy.

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Multi-scale modelling of composite material systems

strains. The following data are recorded for each load cycle: shear stress s12, p e and the the total strain e12, the elastic and plastic strain components e 12 , e 12 secant modulus 2Gsec, shown by the dotted lines in Fig. 14.1. The elastic shear strain component in this test is given from eqns 14.2 and 14.3 by e e 12 = s 12 /(2 G12 (1 – d12 ))

14.18

from which it follows that the measured secant shear modulus Gsec is e Gsec = s 12 /2e 12 = G12 (1 – d12 )

14.19

Thus the shear damage parameter d12 may be determined from the test data as a function of the applied shear stress and the elastic shear strain. It follows that the shear damage force Y12 may be computed using eqn 14.6 and hence the elastic shear damage evolution equation d12 – Y 12 plotted. From the resulting evolution curve Fig. 14.2 the measured damage parameter d12 reaches values as high as 0.75. It is apparent that a linear model would be valid only for small values of ÷Y12 and would not be a good basis for modelling damage over the complete strain range. A simple function which gives a reasonable fit to the test data is linear in ln(÷Y12 ) and this is also marked as the broken curve on Fig. 14.2 for comparison. The constants a12, Y120, and threshold parameter Y12f defined in eqn 14.9 are readily obtained from the fitted log function. Evolution eqn – elastic shear 0.9 0.8 0.7 0.6

d12

0.5 0.4 0.3 0.2 0.1 d12 Logarithmic (d12)

0 –0.1

sq rt Y12

14.2 Elastic damage evolution equation in shear for GF/epoxy.

The plastic hardening function requires the determination of the accumulated plastic strain p from (12)2. With the data on the plastic strain e 12p and the damage parameter d12 taken from the cyclic test curve a plot is made of (1 – d12) against plastic strain e 12p and its integration gives the accumulated plastic

Modelling impact damage in composite structural elements

411

strain p. The effective shear stress s12 /(1 – d12) is then plotted against p. On the elastic domain surface it follows from eqn 14.10 with F = 0 that this gives a graph of R(p) + Ro against p, from which Ro follows from the intercept at p = 0. On subtracting the inelastic threshold stress Ro a graph is obtained of the hardening function R(p) against p, as shown in Fig. 14.3. This curve is well fitted by the power law function (eqn 14.12), whence the plasticity parameters b and m are determined. R(p): comparison test data and power law model 700 600 500

R(p)

400 300 200 s12/(1–d12)–Q0 R(p)

100 9 –0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

–100 p

14.3 Plastic hardening function R(p) in shear for GF/epoxy.

14.3

Delamination modelling

The essence of delamination failure is that it occurs over a thin interfacial region in the composite. It follows that models for delamination in FE codes are not usually treated as a materials failure law since they are strongly dependent on the numerical modelling of the interface. The main numerical problem in explicit FE codes arises from modelling very thin interfaces with solid elements, which leads to extremely small time steps. One approach is to use gap or interface elements to replace the thin solid interface so that the delamination model is implemented as a form of interface contact law, which may include interface failure conditions.

14.3.1 Fracture mechanics ply interface model Delamination failures occur in composite structures due to local contact forces in critical regions of load introduction and at free edges. They are

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Multi-scale modelling of composite material systems

caused by the low, resin dominated, through-thickness shear and tensile properties found in laminated structures. Because delamination failure causes rapid interface crack propagation, failure models are generally based on fracture mechanics ideas rather than conventional stress based failure models. A general framework for composites delamination models is described in ref. 14, in which the thin solid interface is modelled as a sheet of zero thickness across which there is continuity of surface tractions but jumps in displacements. General interface constitutive laws are then presented within a CDM formulation. The approach used here follows more closely ref. 15 where specific forms of the interface stress-displacement laws are assumed. The equations of the model are given for the case of mode I tensile failure at an interface. The terminology of section 14.2 is used in which (x1, x2) is in the laminate plane and x3 is the coordinate through the thickness of the thin laminate. Let s33 be the tensile stress applied at the interface, u3 the displacement jump across the interface, and k3 the interface tensile stiffness. An elastic damaging interface stress-displacement model is assumed:

s33 = k3 (1 – d3) u3

14.20

with through-thickness tensile damage parameter d3, 0 £ d3 £ 1. The damage evolution equation is assumed to have the particular form d3 = 0, 0 £ u3 < u30, d3 = c1(1 – u30/u3), u30 £ u3 £ u3m, with

c1 = u3m/(u3m – u30)

14.21

It can be verified that with this particular choice of damage function d3, the stress-displacement function has the triangular form shown in Fig. 14.4, and u30 , u3m correspond to the displacement at the peak stress s33m and at ultimate failure respectively. The tensile failure model requires the two constants u30, u3m. Furthermore these damage evolution constants may be defined in terms of s33m and GIC, the critical fracture energy under mode I interface fracture by: GIC s33 m Stress

Unload/reload

u3O

Displacement

u3 m

14.4 Idealised mode I interface stress-displacement function.

Modelling impact damage in composite structural elements

u30 = s33m / k3

u3m = 2GIC / s33m.

413

14.22

In this case from these expressions it can be shown that the area under the curve in Fig. 14.4 is equal to the fracture energy GIc. This interface model therefore represents an initially elastic interface, which is progressively degraded after reaching a maximum tensile failure stress s33m so that the mode I fracture energy is fully absorbed at separation. Following ref. 15 for mode I interply failure the interface energy GI displacement u3 is defined as GI =

Ú

u3

0

s 33 du3

14.23

When GI exceeds the critical fracture energy value GIc, then the mode I fracture energy is absorbed and the delamination crack is advanced. These equations may be used to define conditions for interface elements in an implicit code as in ref. 15, or applied to an interface contact law in an explicit FE code as described in ref. 16 for PAM-CRASH. For mode II interface shear fracture a similar damage interface law eqn. 14.20 is assumed, with an equivalent set of damage constants u130 , u13m and critical shear mode II fracture energy GIIC. In general loading of an interface there will be some form of mixed mode delamination failure involving both shear and tensile failure. This is incorporated in the model by assuming a mixed mode failure condition, which for mode I/mode II coupling could be represented by an interface failure envelope such as that proposed in ref. 15: n n Ê GI ˆ Ê GII ˆ Á G ˜ + Á G ˜ = eD £ 1 Ë IC ¯ Ë IIC ¯

14.24

Here GI and GII are the monitored interface strain energy in modes 1 and 2 respectively, GIC and GIIC are the corresponding critical fracture energies and the constant n is chosen to fit the mixed mode fracture test data. Typically n is found to be between 1 and 2. Failure at the interface is imposed by degrading stresses when e D < 1 using eqns 14.20 and 14.21 and the corresponding shear relation. When eD ≥ 1 there is delamination and the interface separates. Experimental testing work has been undertaken, using a mixed mode fracture test, which indicates that the interaction of energies is approximately linear in typical composites materials. Thus the interface traction failure law for typical composites could be based on eqn 14.24 with n = 1 for mixed mode loading.

14.3.2 Implementation in FE codes In order to apply the composites failure models developed above in the analysis of composite structures it is necessary to implement and validate the

414

Multi-scale modelling of composite material systems

models in a suitable FE code. In recent years explicit FE methods have proved successful for the analysis of dynamic, highly non-linear problems, particularly where contact plays an important role. In collaboration with the software company Engineering Systems International (ESI) the CDM ply and delamination models were implemented in the commercial explicit crash and impact code PAM-CRASH [11] and described in more detail in ref. 16. The code uses a bilinear four-node quadrilateral isoparametric shell element due to Belytschko [17] with uniform reduced integration in bending and shear. Hour-glass control is applied to compensate for the under integration. A central difference explicit integration scheme is used in time with geometrical nonlinearities accounted for in an updated Lagrangian scheme with co-rotational description. A Mindlin-Reissner shell formulation is used with a layered shell description to model a composite ply, a sublaminate or the complete laminate, depending on the detail required. The layered shells contain one integration point per ply, so that at least four plies are required in a layered shell for the correct bending stiffness. A novel approach has been developed to implement the delamination model of section 14.3.1 into the PAM-CRASH code, in which the laminate is treated as a stack of shell elements. Each ply or sublaminate ply group is represented by one layer of shell elements and the individual ply layers are tied together using a ‘sliding interface’ with an interface traction-displacement law. This approach gives a good approximation for delamination stresses and failure, with the advantage that the critical integration timestep is larger since it depends on the area size of the shell elements. Thus large composite structures may be modelled efficiently with shells, or stacked shells, requiring fewer elements than solid models, and computationally expensive interface solid elements are eliminated. In this stacked shell laminate model the interface failure law is applied to determine tractions and displacement discontinuities at the interface under mixed mode tension/shear loading. Full details of the implementation of the delamination model as a sliding interface with failure between stacked shell elements is given in the paper by Greve and Pickett [18]. In the HICAS project [12] an extensive materials test programme was carried out on carbon and glass fibre reinforced epoxy materials including in-plane and through-thickness tension, compression and shear tests. Cyclic shear tests were also conducted which showed the importance of the plastic strain contribution. As discussed in section 14.2.3 test data were used to obtain the required materials parameters for the fabric ply models. The materials dataset obtained was then used as the basis for code validation on single elements and materials test specimens, see ref. 16. Specimen tests show significant material nonlinearity due to elastic damage and plastic shear strain on unloading, which were successfully modelled in PAM-CRASH and which verified the damage/plasticity model and choice of evolution curves

Modelling impact damage in composite structural elements

415

and hardening function. The delamination model requires interface fracture energy data under mode I (GIC), mode II (GIIC) and the fracture response under mixed mode loading. Pure modes I and II data are obtained from double cantilever beam (DCB) and end notched flexure (ENF) tests respectively, whilst mixed mode data requires a specialised test which can propagate an interface crack under a predetermined ratio of GI and GII as discussed in ref. 18. This paper also contains validation studies in which mixed mode delamination tests were simulated by stacked shell elements giving very good agreement with test data.

14.4

Prediction of impact damage in composite structures

14.4.1 Plates and shells under rigid body impact Encouraged by the code validations on single elements and test specimens, trial impact simulations with the damage/delamination model in PAM-CRASH are now considered on idealised composite structures in the form of CF/ epoxy plates. The plates had nominal dimensions 300 ¥ 300 ¥ 4.5 mm and were made up of 16 plies of carbon fabric/epoxy with quasi-isotropic layup. In the DLR drop tower impact tests the plates were simply supported on a 250 ¥ 250 mm square steel frame. The impactor head was a 50 mm diameter steel sphere with an added mass of 21 kg, and various impact velocities were used in the range 2.33–6.28 m/s, to give different impact energies 57–414 J and failure modes from rebound to full penetration. For the simulation a shell laminate model is used with 16 plies. In order to reduce CPU time the laminate is modelled as four sublaminates of shell elements which are connected by three delamination interfaces. Each sublaminate is a layered shell element with four plies to represent the correct quasi-isotropic laminate layup. This model allows delamination at three interfaces in the plate, which is in agreement with typical observed failure behaviour. The composite plate is simply supported over the frame and impacted at its centre with the sphere, which is modelled here as a rigid impactor. Numerical results are presented for the case of an impact velocity of 6.28 m/s when the impactor penetrated the plate. A lower velocity case of 2.44 m/s impact causing delamination and rebound is described in ref. 16. Post-processing allows information on both the ply damage and ply interface delamination to be assessed. Figure 14.5 shows typical penetration of the plate and fibre fracture. For code validation it is important to compare quantitative predictions of impact loads, plate deflections and energy absorbed with measured test data. These data have been measured in the low-velocity plate impact test programme. The load cell on the impactor head records the contact load-time pulse during impact. Test data for the impactor load in the

416

Multi-scale modelling of composite material systems

14.5 CF/epoxy plate under impact by steel impactor (M = 21 kg, V0 = 6.28 m/s), damaged plate after impact.

6.28 m/s impact tests are plotted in Fig. 14.6. These are compared with simulations using the ply damage laminate model with a single laminated shell (without delamination), and with the four sublaminate stacked shells 12.0 Experiment Sim: 1 shell Sim: 4 shell

10.0

Load (kN)

8.0

6.0

4.0

2.0

0.0 0

1

2

3

4 Time (ms)

5

6

7

8

14.6 Simulation of CF/epoxy plate under impact by steel impactor (M = 21 kg, V0 = 6.28 m/s): Comparison of impactor load-time pulses from test, with simulations using single shell and 4 ply delamination models.

Modelling impact damage in composite structural elements

417

(with delamination). The figure shows that the single laminated shell model which has ply damage and plasticity overpredicts the measured peak failure loads by a factor of two or higher. The simulation with the four-sublaminate delamination model has significantly lower peak loads than the single shell result and is much closer to the test data. The simulations show that delamination in the plate reduces the plate bending stiffness near the impactor which leads to lower peak loads, greater deflections and reduced energy absorption. It follows that the simulation with only laminate ply failure and no delamination gives poor agreement with test data, with predicted peak loads and energy absorption too high. The results for the delamination model are encouraging and show that delamination effects are significant in plate impact and that these can be successfully modelled with the improved code. Attention is next turned to code validations on composite shell structures under high-velocity impacts. In the EU HICAS project [12] gas gun impact tests at velocities in the range 40–150 m/s were carried out at the DLR on cylindrical shell structures with steel balls of nominal diameter 30 mm with masses between 0.11–0.134 kg. Six shells were impacted at the middle point at the top of the shell, each with different impact energies chosen to give a range of damage conditions. Three shells were impacted on the concave face and three on the convex face, with quite different failure modes. Although the structures are idealised, the convex impact case could represent impact on a wing leading edge structure, whilst the concave impact case could arise in aero engine containment or in a nacelle structure impacted by a broken fan blade. Delamination damage in all the shells was determined from C-scan tests using a hand-held ultrasonic probe. The glass fabric GF/epoxy shells were half circular cylinders, with internal diameter 193 mm, length 200 mm and 20 mm flanges at the straight edges. The composite shells were fabricated from an R-glass fabric prepreg and toughened epoxy resin, with 24 plies in a quasi-isotropic lay-up (0∞/45∞)6S and nominal thickness 6 mm. For the gas gun tests the shell flanges were bonded into grooves in a steel backing plate, approximating a fixed edge condition, with the curved edges free. Simulation results are presented here on one convex impact case (m = 0.134 kg, V0 = 107.5 m/s, E = 774 J). Figure 14.7 shows the observed damage states after impact with a steel ball. The silver lines marked on the shell outer surface show the extent of the delamination region as determined by C-scan tests. In the impact test the steel ball rebounded after impact, damaging the convex face of the GF/epoxy shell with a single transverse crack and causing extensive delamination damage over a large area. This was accompanied by large shell deformations so that part of the impactor kinetic energy is stored as elastic strain energy, which reduces local damage at the impactor. The straight edges at the base of the shell were modelled as fixed with rotations

418

Multi-scale modelling of composite material systems

14.7 GF/epoxy shell after gas gun impact test with steel ball (M = 0.134 kg, V0 = 107.5 m/s).

and displacements constrained, the curved edges were free. The impactor was modelled as a rigid body. Since delamination is important in this case, the laminated structure was modelled with four stacked shells, each representing a quasi-isotropic sublaminate with six GF/epoxy plies. Observation of the shell edge in the delaminated region indicated a single main centre plane delamination crack. Hence it was decided that a model with four stacked shells and hence three damaging interfaces should be adequate to include the observed delamination damage. Note that the CPU time for the simulations is critically dependent on the number of stacked shells in the model, so that full models with 24 stacked shells are CPU intensive. The orthotropic elastic damage model was used to model each ply, with materials parameters determined from the HICAS materials test programme. Fracture toughness data for composite laminates based on the same resin system were used to determine suitable values of GIC and GIIC, with interaction parameter n = 1. In the test the impactor caused ply damage and delamination in the shell as seen in Fig. 14.7, but there was not sufficient kinetic energy to pass through and it rebounded. Simulation results for ply damage and delamination agreed well with the observed shell failure behaviour. The high damage values were at the position of the crack in the shell, and the size of the delamination was similar to that measured by C-scan. For the concave face impact tests a different failure mode was observed, in which failure was localised at the impact point and the steel ball penetrated through the shell. This was also well predicted by the FE simulation, as described further in ref. 19. Thus there is encouraging agreement between the impact test results and the FE simulation.

Modelling impact damage in composite structural elements

419

14.4.2 Soft body impact on composite shells In this section attention is turned to validation studies on impact simulations of composite shell structures, which have been impacted with deformable impactors in high velocity impact tests. There are many critical impact situations in aircraft structures such as bird strike and foreign object damage (FOD) from tyre rubber fragments, hailstones, etc., where the impactor is deformable or fragments on impact, so that the assumption of a rigid impactor is not valid. In this case part of the impactor kinetic energy is absorbed by failure of the impactor and the contact surface is modified by impactor flow which can reduce local structural damage. In order to predict damage and failure in composite shell structures in these situations it is necessary to give attention to suitable models for the soft body impactors, since they strongly influence structural failure modes. The two main problems in simulating soft body impacts are defining a suitable materials model for the soft material, and overcoming the high mesh distortion in the impactor which causes numerical problems with the time step in explicit codes. The approach being adopted by the DLR is to use the smooth particle hydrodynamic (SPH) method to model the flow and large deformations in the impactor, in which the FE mesh for the impactor is replaced by interacting particles. This is combined with a materials law for a ‘hydrodynamic solid’ in which the pressure–volume relation is modelled by an equation of state (EOS). Data from pressure pulses measured during soft body impact on rigid plates are used to calibrate material parameters for the EOS for use with the SPH method. The SPH method and EOS used have been implemented in PAM-SHOCK [9], which was used for the simulations presented here. Soft body impactor model Simulation of impacts in which the impactor is highly deformed or fragmented into a debris cloud on impact is a major challenge for FE codes. The main problems in using a conventional FE model with Lagrangian mesh are mesh distortion in the high deformation or flow regions which causes time step and stability problems and the difficulty in modelling disintegration of the impactor. To overcome these a gridless Lagrangian SPH technique has been developed in which the solid FE mesh is replaced by a set of discrete interacting particles. The method was developed originally [20] for problems in astrophysics, has been applied to hypervelocity impacts (~ 10 km/s) in the design of protection shields for space structures [21] and more recently in bird strike on aircraft structures [10], where the impact speeds are much lower (100–300 m/s). SPH is a gridless computational method whose foundations are in

420

Multi-scale modelling of composite material systems

interpolation theory. The material is represented as a set of discrete particles (interpolation points) which are topologically independent from each other. The two main features of the model are the ‘kernel’ approximation and the ‘particle’ approximation. The kernel function defines a range of influence of a point in the continuum. The particle approximation is applied to replace the continuous domain of influence at a point by a set of discrete neighbouring particles. The transformed integral field equations are replaced by summations over discrete particles within the region of influence of a point. Each particle has an associated mass, velocity and stress state which will evolve according to the discretised field equations, as described in more detail in ref. 18 along with discussions of suitable kernel functions. In PAM-SHOCK the initial particle distribution is generated automatically from a conventional solid FE mesh, with mass and particle density determined from those of the original solid element and standard features such as penalty contact laws, boundary conditions, etc., are available with the SPH solver. Contact laws are also valid between particles and conventional finite elements, so that it is possible to combine an SPH impactor model with an FE structural model. The method used here is to model the deformable impactor by SPH particles and the composite structure by finite elements using layered shells or stacked shells with delamination interfaces. The soft body impactor model is completed by assigning a relevant constitutive law to the SPH particles. The material model currently available in PAM-SHOCK is referred to as an ‘elastic-plastic hydrodynamic solid’, which was originally developed for ballistic impact in metals and describes an isotropic elastic-plastic material at low pressures, with an equation of state for the ‘hydrodynamic’ pressure-volume behaviour at high pressures. In this chapter attention is restricted to gelatine impactors used as substitute birds for testing aircraft structures. In this case the elastic-plastic contribution to the materials behaviour may be neglected and the model reduces to the EOS for the pressure p, which is assumed in this case to have the polynomial form: p = C 0 + C 1 m + C 2 m 2 + C 3m 3, m = r / r 0 – 1

14.25

where C0 , C1 , C2 and C3 are materials constants and m is a dimensionless parameter defined in terms of the ratio of current density r to initial density r0 . Suitable values are required for the constants Ci in the polynomial EOS. Since these constants refer to the dynamic behaviour of gelatine at impact pressures they are difficult to measure directly and have to be determined indirectly. Wilbeck [22] has measured high-velocity impact pressures of several materials, including rubber, gelatine and chickens and found that pressure pulses have a characteristic form. This consists of a high peak pressure caused by shock wave propagation in the impactor, followed by a

Modelling impact damage in composite structural elements

421

lower fairly constant pressure due to steady flow of the impactor onto the target. Furthermore, for bird impact the EOS for water could be used as a basis for predicting peak pressures behind a 1-D shock front which were similar to those measured in the test programme. In this case it can be shown [10] that the polynomial EOS constants are approximated by C0 = 0, C1 = r0 c 02 ,

C2 = (2k – 1)C1, C2 = (k – 1)(3k – 1)C1 14.26

where k is the Hugoniot constant and c0 the sound speed in the material. Wilbeck showed that with values for the Hugoniot constant k and sound speed c0 for water the predicted pressure peak across the shock front for gelatine impactors fitted test data fairly well. However, for real chickens the pressure pulse had lower peak pressures and was wider. Since the aim of the simulation tools is to simulate real bird impacts on structures, it was recommended to use materials constants in the EOS which represent a mixture of water with about 10% air. The air content has the effect of reducing density, and lowering the bulk modulus and sound speed. Thus it is possible to determine the EOS constants using a rule of mixtures model, as proposed by Wilbeck [22] and Iannucci [9]. This is the procedure used here to calibrate the SPH model by finding suitable values of r 0 , c0 and k to represent the EOS of a water/air mixture which fit available impact test data. Impact pressure pulse data are available for a 1.82 kg synthetic bird which was modelled as a cylinder length 114 mm, diameter 114 mm, with two hemispherical end caps. A fine FE mesh was adopted for the impactor which was converted to an SPH mesh. Figure 14.8 shows the particle deformation of the 1.82 kg impactor impacting a rigid plate at 225 m/s. The SPH method captures very well the flow of the gelatine onto the target as an expanding disc shaped region. A more critical test of the simulations is to compare the pressure pulses at the centre of the

14.8 SPH simulation of bird impact on rigid plate (M = 1.82 kg, V0 = 225 m/s).

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plate during impact with test data as shown in Fig. 14.9. The figure shows a typical simulated pressure pulse at the centre of the target plate for a porous gelatine impactor model for impact velocity of 116 m/s. The test data were taken from Wilbeck [22] and are presented as normalised pressure-time curves, based on a normalised time tN = tV0/l and normalised pressure p N = p / 1 / 2 r0 V02 , where V0 is the impact velocity and l the impactor length. Porous gelatine 8

Normalised pressure

Exp: Wilbeck (porous gelatine) Sim: calcul. pressure (1 Element) Sim: calcul. pressure (4 Elements) 6

4

2

0 0.0

0.2

0.4

0.6 0.8 Normalised time

1.0

1.2

14.9 Normalised pressure in gelatine impact at 116 m/s: comparison with test data from ref. 22.

The results confirm that the impactor pulse predicted by the SPH simulation has a steady flow region preceded by the short duration pressure peak at the Hugoniot shock, which has an overpressure ratio compared with the steady flow. The shape of the simulated pressure pulse in Fig. 14.9 for porous gelatine agrees well with test data from Wilbeck at 116 m/s impact velocity. Peak pressures occurred at about tN = 0.05, with peak normalised pressures pN of about 5.5–6.0 and steady normalised flow pressures of about 1.5. At 116 m/s impact the predicted pressures at the centre of the rigid target plate are calculated to be steady flow pressures of 9.5 MPa and a peak shock pressure of 31.7 MPa, with a pulse length of 1.96 ms. The good general agreement with measured pressure pulse data from the literature gives confidence in the SPH model for the impactor and the values of the constants assumed in the EOS. Note that for this test case the flow pressure is well above the expected compression strength of gelatine, which justifies the neglect of the elastic-plastic contribution to the impactor material law, and the peak pressure is high enough to cause local damage in composite structures.

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Gelatine impact on leading edge shells An impact test programme was carried out at the DLR in which composite shells, typical of a light aircraft wing leading edge profile, were tested with soft body gelatine cylinder projectiles. The composite shells were fabricated in glass fabric (GF)/epoxy with a quasi-isotropic lay-up [0/45]2S and nominal thickness 2 mm. The shells had a length 200 mm, with a sharp nose profile having radius 15 mm. For the test the shells were bonded into grooves in a steel backing plate, approximating a fixed edge condition, with the curved edges left free. The gelatine cylinder projectiles had diameter 30 mm and length 40 mm with masses in the range 30–34 g. The gelatine used was bovine hide gelatine (260 bloom). After fabrication the projectiles have a soft rubbery consistency with a coarse granular texture, and on impact they fragment into smaller granules which flow over the target before dispersal in the impact chamber. The projectiles were supported by PU foam sabots during acceleration in the gas gun. The shells were impacted at the centre of the leading edge on the convex face at different impact velocities chosen to give different levels of impact damage. Impact angles were normal to the leading edge with an inclination of 0∞ to the plane face of the shells. After impact testing C-scan tests were carried out on the shells with a hand held probe to determine the extent of delamination damage and the delamination boundary marked. Three leading edge shells were tested at impact velocities 132.5, 142.0 and 198 m/s chosen to give a range of damage conditions. In all the tests the gelatine cylinders disintegrated indicating that they had ‘flowed’ over the shell nose as observed in high-speed film sequences, so that the impact load is spread over a wide area and there is very little penetration of the shells, compared with steel ball impact tests. At the lowest impact velocity 132.5 m/s there was surface scratching, but no measurable delamination found by C-scan. At 142.0 m/s damage was observed at the leading edge in the impact region. On increasing the impact velocity to 198 m/s there was extensive delamination over a wide region and significant fibre cracking, as shown in the impacted specimen (Fig. 14.10). In addition there was also extensive cracking at the middle region of the curved shell wall which was well away from the impact point. This shows that there was extensive shell bending which extended beyond the nose region. For the impact simulations with PAM-CRASH a stacked shell FE model was developed for the leading edge structure, based on eight stacked shells for the laminate which corresponds to the eight-ply quasi-isotropic layup. The straight edges at the rear of the shell are fixed, as in the test, with the curved edges left free. The SPH model described above was used for the gelatine cylinders, with GIC and GIIC data for the delamination model and the materials parameters for the GF/epoxy ply fabric damage and plasticity model as discussed in section 14.4.1. Simulation of impacts at 130 m/s led to

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14.10 Geometry of leading edge shell showing impact damage (M = 0.033 kg, V0 = 200 m/s).

mainly elastic deformation in the shell, which returned to its original shape after impact with little significant delamination or fibre damage, in line with the test at this velocity. Computed results for normal impact at the centre of the leading edge at 200 m/s are shown in Fig. 14.11, and it is seen that the SPH gelatine model flows over the leading edge in a similar manner to that observed in tests. Figure 14.11(a) shows the shell deformation at 0.4 ms, with highly deformed gelatine projectile and delamination damage, and Fig. 14.11(b) shows the middle ply damage contours at 0.4 ms.

(a)

(b)

14.11 Simulation of the GF/epoxy leading edge under gelatine impact (M = 0.033 kg, V0 = 200 m/s): (a) Delamination contours at 0.4 ms; (b) Ply damage contours at 0.4 ms.

Figure 14.10 is a photograph of the damaged leading edge after impact at 198 m/s. It shows clearly fibre cracks at the impact contact point, corresponding to the maximum ply damage values, and the marked delamination region obtained from the C-scan tests (marked by silver boundary) is similar to the size of the computed delaminations. For gas gun impact tests quantitative load pulse data to compare with simulated results are difficult to obtain,

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however qualitatively there is good agreement with observed deformations and damage conditions. They indicate that a simulation technique with an SPH impactor model and a stacked shell structural model is very promising for simulating soft body impacts in composite structures.

14.5

Conclusions and future outlook

This chapter has described the current status of materials failure models for composites with UD and fabric reinforcement. Emphasis is on failure models suitable for use in explicit FE codes to predict damage arising in composite structures under crash and impact loads. This is a multiscale problem since composites damage is at the microscale level, while crash and impact loads are applied at the structural level. Composites failure models are separated into models for in-plane failure and interlaminar failure. The in-plane composites models are generally based on laminated shell elements made up of several plies, with associated ply materials models. Ply damage is homogenised and described by damage parameters within a CDM framework which is suitable for implementation in FE structural analysis codes. Failure modelling of delamination is based on laminate interfaces which are introduced in the FE codes as a failure condition between plies or laminate shell elements. In each case the background theory has been briefly reviewed, the model parameters defined, and test programmes for determining the parameters discussed. Model implementation in the explicit FE code PAM-CRASH has been discussed in some detail, since the author was involved with ESI in these developments. However, similar damage mechanics based on ply failure models and delamination models are being implemented in other commercial and research FE codes see, for example, refs 3, 5, 7, 9 and 14. Code validation studies are discussed on ply and fracture mechanics materials test specimens, and on composite plates and shells under low- and high-velocity impact. Structural simulations of gas gun impact tests with steel balls on half cylinders are described, which give good correlation with observed failure modes and test data. An important feature of the model is that it distinguishes clearly between different failure modes in the structure. In post-processing it is possible to follow the progression during impact of the fibre and shear damage parameters in the shell, the fibre strains and the extent of delamination. Thus it is possible to simulate both the impact failure modes and failure progression during impact loading in composite structures. Where a quantitative numerical comparison was possible with measured and computed load pulses from low-velocity impact tests and strain gauge measurements from high-velocity tests there was good agreement in peak loads, peak strains and pulse widths. This is sufficient to show the general validity of the damage mechanics and delamination models, and the code implementations.

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Multi-scale modelling of composite material systems

This was followed by a review of recent progress on numerical simulation of soft body impact on fibre reinforced composite structures, which is an important problem in aircraft structures subject to bird strike or tyre rubber impact. Here FE methods were developed to simulate damage in composite shell structures under impact from highly deformable soft impactors such as gelatine or ice, which flow over the structure spreading the impact load. These soft impactors are modelled by the SPH method, in which the FE mesh is replaced by interacting particles. It is difficult to measure the soft body impactor properties under relevant dynamic load conditions for use in the SPH model. The method adopted was to calibrate the parameters required by simulating soft body impacts on a rigid target and comparing geometrical flow characteristics and pressure pulses observed with simulation results. Numerical simulations using the SPH impactor model with the composites failure model were compared with high-velocity impact test data for gelatine projectiles impacting glass fabric/epoxy cylinders and gave encouraging results. The chapter has concentrated on the composites modelling techniques and FE code validations on idealised composite plate and shell structures. Ongoing work at the DLR and other centres is now concerned with applying these FE modelling techniques to composite aircraft structures under crash and impact loads within the design and development process and for certification studies. This was a main theme in the recently completed EU CRAHVI project [23] in which simulation methods were validated on aircraft structures by full scale tests such as bird strike, stone impact and tyre fragment impact. The reader is referred to Aktay et al. [24] for FE simulation of stone impact damage in sandwich panels, Ubels et al. [25] for bird strike simulation of novel composite wing leading edge structures, and to McCarthy [26] for a detailed study of bird impact on a hybrid fibre composite/metal wing leading edge fabricated from GLARE. Problems being addressed in current research programmes include introducing rate dependent effects into the composites materials models, the development of failure models for projectiles composed of fragmenting materials like ice and stone, and the extension of discrete element methods for modelling fracture of materials on impact. Developments in FE codes which are expected to be of future importance for safety studies in aircraft structures include multiscale FE modelling for studying damaged regions in large structures, and the application of stochastic methods for defining failure envelopes in structures under a range of crash and impact conditions. Although considerable progress has been made in the last decade, it should nevertheless be pointed out that it will take many years for composites modelling in dynamic FE codes to reach the same level as that currently achieved in crash and impact analysis of metallic structures. The reasons are historical. Metal plasticity was understood and established mathematical models date from the middle of the last century. Furthermore, the automotive industry

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was the ‘driver’ for the development of FE modelling particularly in crash codes, so that considerable resources have been invested in modelling, development of test methods, code implementations, validations, etc., for metal structures. Similar resources have not yet been made available for composite materials. In addition the diversity of composite materials, the complex anisotropic failure behaviour, the lack of standardised materials and dynamic test procedures, and the smaller more specialist market for composites all contribute to the delay in establishing good, reliable composites models in FE crash and impact codes.

14.6

Further information

Modelling impact damage in composite structures is a multidisciplinary subject which requires a detailed understanding of composites damage and failure mechanics, impact mechanics, dynamic test methods for composite materials and structures, and recent developments in explicit FE codes. The scientific aspects are covered by the literature cited in the chapter. An excellent overview of different approaches to modelling damage and failure mechanics of composites is given in the book edited by Talreja [2]. In this chapter particular emphasis is placed on the CDM approach based on meso-scale modelling which has been developed over many years by Ladevèze and Allix at ENS de Cachan, France. Recent developments in CDM for composites were discussed in two workshops held at Cachan. Information is available in the proceedings published as journal special issues. These were: ‘Recent Advances in Continuum Damage Mechanics for Composites’, Cachan, 20-22 September 2000, Composites Science and Technology, 61/15, 2147–2344, 2001; and ‘Advances in the Statics and Dynamics of Delamination’, Cachan, 15–17 September 2003, Composites Science and Technology, 65, 2005, in press. A good introduction and review of composites impact mechanics can be found in the book by Abrate [1]. Dynamic test methods for composite materials and structures are actively being developed by scientific societies through pre-normative studies and special conferences. The reader is referred to ASTM (www.astm.org ) and CEN Standards ( www.cenorm.be ) for current information on composites test methods. Study groups in Europe on dynamic test methods are being organised by EURODYMAT (www.dymat.org ) and ESIS ‘European Structural Integrity Society’ ( www.esisweb.org ) where ESIS TC4 is concerned with delamination testing and ESIS TC5 with dynamic materials tests. Code developments for impact simulation and composites failure models are actively pursued by explicit FE code suppliers and general information is available at the code websites: PAM-CRASH/PAM-SHOCK (www.esi-group.com); LSDYNA ( www.lstc.com ); RADIOSS (www.radioss.com). Validation of code developments for impact damage predictions requires impact test data on

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structures, which is usually obtained from instrumented drop tower tests at low velocities and gas gun tests at high velocities. The author has been involved in such validation studies with industry in EU funded research projects HICAS [12] and CRAHVI [23] in which composite structures were impacted in gas gun tests with birds, synthetic birds (gelatine), tyre rubber, ice and stone. A summary of the HICAS project by the author was presented at the EU Aeronautics Days 2001 (Air & Space Europe, 3/4, vol. 3, 2001), and more information on the CRAHVI project is at www.crahvi.net. Finally, there are ongoing aerospace industry GARTEUR working groups studying impact damage and modelling in composite structures. These are: GARTEUR SM/AG 24 ‘Modelling bird strike on aircraft structures’ and GARTEUR SM/ AG 28 ‘Impact damage and repair of composite structures’ ( see www.onera.fr/ garteur/sm-ag28).

14.7

References

1. Abrate S., Impact on Composite Structures, Cambridge University Press, Cambridge, UK, 1998. 2. Talreja R., (ed.), Damage Mechanics of Composite Materials, Composite Materials Series 9, Elsevier, Amsterdam, 1994. 3. Williams K.V., Floyd A.M., Vaziri R. and Poursartip A., Numerical simulation of inplane damage progression in laminated composite plates. 12th International Conference on Composite Materials (ICCM-12), Paris, France, 1999. 4. Williams K.V. and Vaziri R., Finite element analysis of the impact response of CFRP composite plates. 10th International Conference on Composite Materials (ICCM10), Whistler, Canada, 1995. 5. Iannucci L., Dechaene R., Willows M. and Degrieck J., A failure model for the analysis of thin woven glass composite structures under impact loadings. Computers and Structures, 2001 79 785–799. 6. Ladevèze P. and Inelastic strains and damage, Chapt. 4 in Damage mechanics of composite materials, (ed.) Talreja R, Composite Materials Series 9, Elsevier, Amsterdam, 1994. 7. Ladevèze P. and Le Dantec E., Damage modelling of the elementary ply for laminated composites. Composites Science and Technology, 1992 43 257–267. 8. Kim H. and Kedward K.T., Experimental and numerical analysis correlation of hail ice impacting composite structures, 40th AIAA Structures, Structural Dynamics and Materials Conference, St Louis, AIAA-99-1366, 1999. 9. Iannucci L., Bird strike impact modelling. I. Mech. E. Seminar, Foreign Object Impact and Energy Absorbing Structures, Inst. of Mech. Engineers, London, 1998. 10. Johnson A.F. and Holzapfel M., Modelling soft body impact on composite structures, Composite Structures, 2003 61 103–113. 11. PAM-SHOCK™/PAM-CRASH™ FE Code. Engineering Systems International, F94578 Rungis Cedex, France. 12. HICAS: High Velocity Impact of Composite Aircraft Structures. CEC DG XII BRITEEURAM Project BE 96–4238, 1998. 13. Johnson A.F., Modelling fabric reinforced composites under impact loads. Composites A 2001, 32 1197–1206.

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14. Allix O. and Ladevèze P., Interlaminar interface modelling for the prediction of delamination. Composites Structures, 1992 22 235–242. 15. Crisfield M.A., Mi Y., Davies G.A.O. and Hellweg H.B., Finite Element Methods and the Progressive Failure Modelling of Composite Structures. CIMNE Barcelona 1997, D.R.J. Owen et al. eds, Computational Plasticity – Fundamentals and Applications, Part 1; 239–254. 16. Johnson A.F., Pickett A.K. and Rozycki P., Computational methods for predicting impact damage in composite structures. Composites Science and Technology, 2001 61 2183–2192. 17. Belytschko T., Lin J.L. and Tsay C., Explicit algorithms for the nonlinear dynamics of shells. Computer Methods for Applied Mechanics and Engineering, 1984 42 225– 251. 18. Greve L. and Pickett A.K., Delamination testing and modelling for composite crash simulation. Comp. Sci. Tech., 2005, to appear. 19. Johnson A.F. and Holzapfel M., Computational methods for predicting impact damage in engineering composite structures, 5th World Congress on Computational Mechanics, WCCM V, Vienna, Austria, 2002. 20. Monaghan J.J., Smoothed particle hydrodynamics. Ann. Rev. Astro. Astrophys, 1992 30 543–574. 21. Groenenboom P.H.L., Numerical simulation of hypervelocity impact using the SPH option in PAM-SHOCK. Int. J. Impact Engng, 1997 20 309–323. 22. Wilbeck J.S., Impact behaviour of low strength projectiles. Air Force Materials Laboratory, Technical Report AFML-TR-77-134, 1977. 23. CRAHVI: Crashworthiness of Aircraft for High velocity Impact, CEC Project: G4RDCT-2000-00395, 2001–2004. 24. Aktay L., Johnson A.F. and Holzapfel M., Prediction of impact damage on sandwich composite panels. Computational Materials Science, 2005 32 252–260. 25. Ubels R., Johnson A.F., Gallard J-P. and Sunaric M., Design and testing of a composite bird strike resistant leading edge, SAMPE Europe: 24th International Conference and Forum, Paris, 1–3 April 2003. 26. McCarthy M.A., Xiao J.R., McCarthy C.T., Kamoulakos A., Ramos J., Gallard J-P. and Melito V., Modelling of bird strike on an aircraft wing leading edge made from fibre metal laminates: Part 2 Modelling of impact with SPH bird model. Applied Composite Materials, 2004 11 317–340.

15 Modelling structural damage using elastic wave-based techniques Z S U A N D L Y E, The University of Sydney, Australia

15.1

Introduction

Though serving as competitive candidates to meet current and future challenges imposed on airframes, carbon fibre/epoxy (CF/EP) composite structures still run a considerable risk of losing efficiency as a result of structural damage, which can potentially lead to catastrophic failure of the whole system if without timely detection. Cost-effective and reliable nondestructive evaluation (NDE) techniques are therefore critical for the confident acceptance of composite materials. Recent studies (Wang and Chang, 1999, 2000; Lemistre and Balageas, 2001; Kessler et al, 2002a, b; Su and Ye, 2004, 2005) have concluded that detection approaches taking advantage of elastic waves are capable of acting as promising substitutes to conventional NDE tools, due to their uncontroversial capability in linking detection precision with universality of application.

15.1.1 Modelling technique for elastic wave-based damage identification To develop an elastic wave-based damage identification technique for laminated composite structures, it is essential to understand wave propagation behaviour in multilayered anisotropic media. In this respect, finite element simulation is well acknowledged as an indispensable tool, by which characteristics of elastic waves in various structures have been rigorously investigated over the years. For generation of elastic waves, Guo and Cawley (1992, 1993) presented a FEM approach, in which uniform in-plane displacement was applied across the thickness of a laminate beam end to excite a symmetric plate wave, known as a Lamb wave (Bindal, 1999). For wave collection, the propagation of waves was monitored by individually measuring vertical displacement (Alleyne and Cawley, 1991; Guo and Cawley, 1992; Koh et al., 2002) or horizontal displacement (Guo and Cawley, 1993) in the area concerned, as 430

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shown in Fig. 15.1. Such a method has been demonstrated to be particularly effective for one-dimensional structures. Vertical displacement Individual layer

Forced oscillation applied on beam end

Horizontal displacement

15.1 Lamb wave generation and collection (Guo and Cawley, 1992, 1993).

Percival and Birt (1997) and Birt (1998) developed a FEM model to elaborate the dispersion nature of plate wave in carbon fibre (T800)/resin (924) composite laminates. Their model allows particle motion in any direction and hence successfully simulated the horizontal shear wave modes. The influence of different laminate layouts on plate wave dispersion was also evaluated in their work. Giurgiutiu et al. (2001) evaluated the propagation of flexure and pressure waves in aluminium beams and plates under different excitation and boundary conditions, using a four-node shell element on the ANSYS platform, where a flexural wave was excited by transient nodal rotations, while the pressure mode was generated by nodal translations. Moulin et al. (2000) developed a coupled finite element–normal mode expansion approach, considering both bonded and embedded transducers. This method is able to deal with cases of multi-element transducers integrated into a composite structure. The influence of PZT element dimension, position and excitation delay on wave propagation was also assessed. Issa et al. (1994) proposed a self-adaptive finite element modelling technique for oblique ultrasonic waves, while You and Lord (1991) analysed ultrasonic wave propagation using their 3-D finite formulation. On the other hand, to study the scattering of ultrasonic waves in defective composite structures, particularly the interaction with delamination in composite structures, a diversity of damage modelling techniques have also been developed for one-dimensional structural composite beams (Majumdar and Suryanarayan, 1988; Tracy and Pardoen, 1989; Guo and Cawley, 1992, 1993; Islam and Craig 1994; Ostachowicz et al., 2002), for two- and three-dimensional composite plates (Razi and Kobayashi, 1993; Kishore et al., 2000; Wisnom and Chang, 2000). In these approaches, different damage models were proposed using relevant damage assessment criteria. Particularly, a two-dimensional model for infinite isotropic solid with absorbing boundary conditions was

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developed by Kishore et al. (2000), able to simulate complex mode-conversion when pulse interacts with flaw. It highlights the fact that if the scattering object is a void, such as a delamination, part of the incident energy will be converted to radiate energy in all directions and surface wave ‘creep’ around the cavity. Modelling of wave propagation is usually associated with the modelling techniques for piezoelectrics-coupled structures, where the piezoelectrics are used to generate and collected elastic waves (Ha et al., 1991; Chee et al., 1999; Chattopadhyay et al., 1999; Fukunaga et al., 2002; Perel and Palazotto, 2002). Most representatively, Shah et al. (1994) established a quasi-threedimensional model for a piezo layer embedded in a composite laminate containing a delamination, using eight-noded iso-parametric elements as shown in Fig. 15.2(a). Koh et al. (2002) developed a FEM model for a platelike aluminium structure with surface-mounted piezoelectric transducers and an impact damage, as sketched in Fig. 15.2(b), where the impact damage was simulated by a local change in the stiffness of the material. The effects of the size, properties and orientation of the impact damage upon a propagating Lamb wave were evaluated. Their FEM results show that elastic waves excited by surface mounted piezoelectric transducers are suitable for the quantification of impact damage, and the delaminated ply can be located. It was also shown

45∞ Gr/EP –45∞ Gr/EP Piezo layer 0∞ Gr/EP Delamination 90∞ Gr/EP (a) Piezo layer

Vacuum (b)

Aluminium plate

15.2 FEM model for PZT-embedded defective composite structures by (a) Shah et al., (1994); (b) Koh et al (2002).

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that with PZT elements mounted on only the top surface, both symmetric and asymmetric modes could be excited.

15.1.2 About this chapter In this chapter, the dynamic propagation characteristics of guided elastic waves, particularly Lamb waves in quasi-isotropic CF/EP composite laminates containing a delamination, are elaborated via theoretical analyses, finite element simulation (FES) and experimental verification. To generate and acquire Lamb waves, an active sensor network using distributed piezoelectric disks is configured based upon a comparison with other currently available Lamb wave generation/collection methods. A novel concept of ‘Standard Sensor Unit’ (SSU) is introduced in such an approach, by which various active networks can be flexibly and expeditiously customised to accommodate different applications. For simulation, a FEM modelling technique for piezoelectric actuator and sensor coupled with quasi-isotropic composite laminate is developed, to excite and monitor the fundamental symmetric and anti-symmetric Lamb modes, respectively. Delamination in quasi-isotropic CF/EP composite plate is modelled by using contact surface algorithms. Lamb wave signals are notoriously complicated in composite materials due to their unique dispersive features in viscoelastic substances. Traditional signal processing approaches cannot offer efficient identification for them. A signal processing and identification technique using wavelet transform algorithm is established to interrogate Lamb wave signals in the time–frequency domain synchronously. Finally, experimental validation is also undertaken using a signal generation and acquisition system.

15.2

Fundamentals

15.2.1 Lamb waves in composite structures Guided elastic waves Elastic waves in anisotropic multilayered media, e.g., composite laminates, are highly concerned in a wide range of applications, from coating problems of plasma spray on turbine blade, diffusion/adhesively bonded structures, to ice detection for aircraft windshields. Their multilayered structural features introduce many interesting but complex phenomena in elastic wave propagation, such as direction-dependent speed, different phase and group velocities, wave skewing, and others which are somewhat more unusual. More specifically, guided waves are those that compulsorily require boundaries to maintain propagation (Achenbach, 1973), such as waves in a rod, plate, tube, or platelike laminates. Antecedents of this speciality are not hard to identify. It was Lord Rayleigh

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in 1889 who first explained wave propagation along a guided surface, and the wave discovered was then named after him (Rayleigh wave). Following Rayleigh’s work, Love in 1911, Lamb in 1917 and Stoneley in 1924 individually discovered other modes of guided waves in plate-like structures, which are well acknowledged nowadays as Love wave, Lamb wave, and Stoneley wave, respectively (Chimenti, 1997). All these pioneering studies have consolidated the basis for the theory of elasticity. A revival of this topic occurred in the 1970s with the exponential introduction of composite materials, whose direction-dependent mechanical properties classify them as typical anisotropic multilayered media. Generally, guided waves in a plate-like laminated structure can manifest themselves as longitudinal, transverse (shear), Rayleigh (surface), Lamb (plate), Stoneley and creep modes, according to the particle motion style induced (Bindal, 1999). In most cases, guided waves in laminates are made up of a superposition of longitudinal and shear waves, but, because of boundaries or discontinuity, interaction between these waves and boundaries or discontinuity is often caused by reflection and refraction simultaneously, resulting in mode conversion. The guided waves in a plate with thickness of 2h (see Fig. 15.3), regardless of mode, can generally be described in a form of Cartesian tensor notation in the theory of elasticity (Rose, 1999), as

m · ui, jj + (l + m) · u j, ji + r · fi = rüi

15.1

where u is the displacement field tensor, l and m are Lamé constants. Equation 15.1 governs the wave motion, which is sufficient to describe wave propagation in an infinite domain. But for a finite elastic medium, boundary conditions are compulsory. For a plate with boundary conditions, s31 = s33 = 0, at x3 = ± d/2 = ± h (for plane strain), a general solution can be achieved (Rose, 1999)

4k 2 q p m tan( qh ) = tan( ph ) ( l k 2 + l p 2 + 2m p 2 )( k 2 – q 2 )

15.2

x3

h

x1

h

15.3 A plate (2h in thickness) in referred coordinate system.

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where 2 2 p 2 = w 2 – k 2 , q 2 = w 2 – k 2 , and k = w / c p cL cT

where k, cL, cT, cp, w are the wavenumber, velocities of longitudinal and transverse modes, phase velocity, and wave circular frequency, respectively. It is the introduction of boundary conditions and multiple modes that make the guided wave problem difficult to handle. Lamb waves in laminated composite structures Lamb waves, an elastic disturbance existing in a thin plate with parallel free boundaries, were firstly recorded by Horace Lamb in 1917 in his milestone publication, On Waves in an Elastic Plate (Proc. R. Soc. London Ser. A. 93:114-128, 1917) (Viktorov, 1967). Lamb’s discovery did not gain wide attention, due to those extremely complicated equations, until 1945 when Osborne and Hart revisited this topic to analyse the interaction between steel plates and waves generated by underwater explosions. Then this intriguing topic was espoused by some great mathematicians such as Cauchy, Lamé, Green Thomson and Poisson (Lowe, 1995). Such a wave can also be defined with the general motion formula (eqn 15.2). Mathematically comprised of the trigonometrics function sine and cosine, the equation determines that two Lamb modes are synchronously available, i.e., symmetric and anti-symmetric modes, individually simplified as

tan ( qh ) 4k 2 q p , = – 2 tan( ph ) (k – q 2 )2

for symmetric modes,

tan( qh ) (k 2 – q 2 )2 = – , for anti-symmetric modes. tan( ph ) 4k 2 q p

15.3a

15.3b

Equation 15.3, often referred to as the Rayleigh-Lamb frequency equation, associates frequency w with wavenumber k and phase velocity cp for Lamb waves, whose graphic descriptions are frequency spectrum and dispersion curves, respectively. Theoretically, for a given frequency, there are infinite wavenumbers satisfying eqn 15.3, either real or purely imaginary. In spite of its simplicity in appearance, the equation can be solved only by numerical simulation under most circumstances. Physically, a symmetric mode across the entire plate thickness is symmetric for u (horizontal), and anti-symmetric for w (vertical); and an anti-symmetric mode is symmetric for w but anti-symmetric for u, as illustrated in Fig. 15.4. Equation 15.3 also implies that Lamb waves, regardless of mode, are dispersive (velocity depends on propagation frequency).

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Compressional wave (symmetric Lamb mode)

Flexural wave (anti-symmetric Lamb mode)

15.4 Symmetric and anti-symmetric Lamb wave modes.

In addition to Lamb waves, transverse (shear) motion was also observed in layers of finite thickness by Love in 1911 (Rose, 1999). FEM simulations, based on models allowing particle motion in any direction (Percival and Birt, 1997; Birt, 1998), have also demonstrated the existence of horizontal shear waves in the direction perpendicular to the plane of wave travel, besides the normal shear waves (vertical shear mode) and Lamb waves. This wave is therefore named the shear horizontal (SH) or Love wave, as shown in Fig. 15.5. In the figure, waves propagate along the x1 direction, and the particle x2 Wave propagation

SH x1

2h

x3

15.5 Horizontal shear (SH) mode in laminated structure.

Modelling structural damage using elastic wave-based techniques

437

vibrates in the x3 direction, which is caused by SH modes in a plane parallel to the surface of layers. Any mode in the SH family can be considered as a superposition of up- and down-reflecting bulk shear waves, polarised along x3, with wavevectors lying in the x1 – x2 plane and inclined at such an angle that the system of waves satisfies traction-free boundary conditions on surfaces (Rose, 1999). Generation of Lamb waves In spite of many attractive traits, Lamb waves in anisotropic viscoelastic media are notoriously complicated due to multiple modes and dispersive properties. Two technologies are therefore motivated for utility of Lamb waves: (i) active generation of controllable Lamb waves, and (ii) effective signal processing and interpretation (to be addressed in section 15.2.3). Lamb waves can be excited by a variety of means. With each possessing unique intrinsic advantages and disadvantages, these methods can be roughly summarised into four major categories, and compared in Table 15.1 (Su, 2004).

15.2.2 Active sensor network Sensor technology for NDE Analogous to the nervous system of human beings, a sensing network is essential for the practical implementation of online damage identification. As a basic unit in the network, a sensor converts physical responses into electrical or optical signal identifiable for data acquisition hardware. Sensor technology, an interdisciplinary technology, is seen as an integral element in the overall development of NDE technique, spanning areas of physics, chemistry, materials, molecular biology, manufacturing, electronics and signal processing. Basically, sensing devices for NDE purposes require (i) a certain tolerance to general environment, (ii) a service life of at least 5–10 years, (iii) simple and easy handling and attachment (Mathworks, 2001a). For higher performance, they should feature smaller size, lighter mass, higher sensitivity, lower cost/ power, more reliable and quick response to sudden changes, as well as easy integration. To cater for future trends, preferences also include remote control and data transmission, tolerance for vibration and noise, little ageing deterioration and less wire or even wireless. Various sensing/actuation devices are currently available for different purposes, including piezoelectrics, optical fibres, electro-rheological fluids magnetostrictive/electrostrictive, and shape memory alloy/polymer, etc. There is no single principle in selecting sensors for a practical application.

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Multi-scale modelling of composite material systems

Table 15.1 Comparison of major generation approaches for Lamb waves Generation tools

Ultrasonic probes (coupled with angle-adjustable perspex wedges)

Air/fluidcoupled Ultrasonic probes

Merits

Excellent precision and controllability; ability to generate pure Lamb mode; easy and explicit signal interpretation

Advoidance of contact and couplant

Demerits

Time, money & labour consuming; out-of-service of structure, couplant and contact required; inefficient for near-surface damage; nonneglectable mass/volume; not cheap

Application & limitation

For both large and small damage, large thickness; not suitable for online NDE; for flat surface

Inefficient when large differences in mechanical impedance between air/fluid and structure under inspection

Electromagnetic acoustic transducer (EMAT)

Narrow band, avoidance of physical contact

Influence by electrical conductivity of structure under inspection; effective in certain frequency ranges only

Laser interferometer

Avoidance of contact and couplant; exact and explicit signal; flexibly controllable; bandwidth adjustable; remote control; easy scanning, exact calibration and applicability to curved surfaces; possible to generate selective mode

Expensive, bulky volume and heavy mass; affecting structural integrity

Small defect or non-flat surface; Not suitable for cost-effective online NDE

Interdigital transducers (e.g. PVDF)

Higher internal damping; good ability to produce waves with controllable wavelength; soft and flexible, easily

Insensitivity to structural dynamic responses but highly susceptible to environmental factors; weak driving forces

Small defect non-flat surface

Modelling structural damage using elastic wave-based techniques

439

Table 15.1 Continued Generation tools

Merits

Demerits

Application & limitation

Multiple wave modes if no control; non-linear behaviour/ hysteresis under large strains/ voltage/hightemperature; small force/ displacement; brittle, low fatigue life; sophisticated signal processing required

Suitable for cost-effective online NDE

shaped; cheap; adaptive to curved surfaces or complicated geometry.

Piezoelectric elements

Easily shaped, mass/volume-less, low-power consumption; easy integration and activation; low acoustic impedance; wide frequency responses; low cost; possible to be used for network

Typically, strain gauges are able to give localised measurement of deformation economically, but are applicable only for static or dynamic changes at a low rate. Acoustic emission (AE) sensors can be efficient in triangulating damage and monitoring propagation of a crack, but do not sense damage unless it is propagating. Moreover, AE sensors are highly susceptible to noise interference during measurement. Fibre-optic sensors can sense strain and vibration but are difficult to fit into structures with complex geometry. In contrast, self-sensing devices, represented by piezoceramic materials, can be used as active sensors. Compared with passive sensors, they are more flexible and controllable. But used locally, a single sensor is unable to offer enough information for complex damage, and a large number of sensors is often required to configure an active sensor network, ensuring completeness of signal acquisition. Amongst them, the piezoelectric actuator/sensor shows an excellent capacity to satisfy the exigency of online NDE techniques. They are able cost-effectively to generate and collect elastic wave signals without inducing noticeable integrity degradation on host structures. In particular, piezoelectrics can be fabricated to function as both actuator and sensor, permitting significant reduction in transducer number, electrical wiring and associated hardware. They are also known for excellent mechanical strength, wide frequency response ranges, generation of relatively high voltage with a low current and reasonable cost. The insignificant mass and volume make it possible for piezoelectrics

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Multi-scale modelling of composite material systems

to be distributed in a structure with only minor effects on structural mechanical properties. Crystals, ceramics and polymers are the major available forms of piezoelectrics. Modern manufacturing techniques allow piezoceramics and piezo-polymer in any geometric configurations, e.g., plate, disk, ring, tube, etc. These sensors can be flexibly bonded onto or embedded into a structure. Sensor network According to the excitation source, an actuator/sensor network can be either passive or active. In passive sensing, structural responses, naturally induced due to controllable or uncontrollable sources, are collected. Such a sensing scenario can be fulfilled inexpensively and easily. In contrast, an active actuating/sensing approach measures structural responses for a particular purpose, using controllably actuator-generated excitation. Obviously, active sensing is able to offer more flexible and exact monitoring than the passive approach. Two commercial sensor network products, SMART Layer® and HELP layer®, are now available. SMART Layer® (also known as the Stanford Multi-Actuator-Receiver Transduction Layer), jointly produced by Stanford University and Acellent® Technologies, Ltd., integrates a certain number of distributed piezoceramics into a dielectric film (Boller, 2001). Unlike SMART Layer® that can generate and measures stress waves, HELP layer® (Hybrid Electromagnetic Performing Layer), developed by ONERA, is based on the interaction between electromagnetic field and structural abnormality. Mechanical damage, as well as thermal defects, can be diagnosed by their influence on electromagnetic properties (Lemistre et al., 2003). Standard sensor unit (SSU) In practice, structures to be inspected are geometrically diverse. Traditional transducer allocation, where the distribution of actuators/sensors and signal excitation/acquisition must be individually configured for each application, is unlikely to provide an expedient and cost-effective solution. Motivated by methods adopted in the automobile industry for standardised accessories and parts, a sensor network approach based on a concept of standard sensor unit (SSU) was developed in the Laboratory of Smart Materials and Structures (LSMS) at the University of Sydney. In this approach, 4 PZT disks are collocated to encircle a square area of 172.5 mm ¥ 172.5 mm (the value is determined by most applications in their studies) and controlled by a signal excitation/acquisition circuit. Such a configuration is nominated as standard sensor unit (SSU), illustrated in Fig. 15.6(a). In SSU, two-way switches are attached to each PZT wafer, enabling every PZT transducer to act as both the

Modelling structural damage using elastic wave-based techniques

441

To central control unit

172.5 mm

Two-way switcher

Control circuit

PZT sensor

Two-way switcher

To central control/ analysis subsystem

172.5 mm To central control unit Valid coverage area (a)

(b)

15.6 Active actuator/sensor network: (a) a standard sensor unit (SSU); (b) customised network for a geometrically non-regular structure.

actuator and sensor. Theoretically, a total of six actuator-sensor pairs, viz., 12 actuator-sensor paths, can be offered by one SSU. Paradigmatically, an active sensor network involving several SSUs, customised for a non-regular geometric entity, is schematically configured in Fig. 15.6(b).

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Multi-scale modelling of composite material systems

Each PZT disk in the SSU is chosen following an optimal criterion (Kessler et al., 2002a), to minimise geometric effect and consequently avoid uneven wave propagation, 2R =

v wave Ê ◊ n + 1 ˆ = l wave ◊ Ê n + 1 ˆ , ( n = 0, 1, 2. . .) f 2¯ 2¯ Ë Ë

15.4

where R is the radius of the PZT disck; vwave, f and lwave are the wave velocity (circa. 6000 m/s in quasi-isotropic CF/EP laminates for the fundamental symmetric Lamb mode), frequency and wavelength of the relevant Lamb mode, respectively. Based on eqn 15.4 and also considering the relation between PZT disk size and detectable damage area (Su, 2004), PZT disks are shaped with a diameter of 6.9 mm and a thickness of 0.5 mm to trade off the desired excitation force and unpolarised voltage limit. Using this concept, diverse transducer networks can be conveniently customised by appropriately assembling SSUs, flexibly to accommodate different geometries and boundary conditions. The distributed multi-point architecture also contributes to the enhancement of signal-to-noise ratio (SNR). Meanwhile, since each SSU is independent from the others, such a network is robust for signal excitation and acquisition. In a network, several SSUs can be employed, which may own conterminous actuator-sensor paths. Under these circumstances, the number of PZT transducers is minimised by keeping only one transducer at the joint of SSUs, as in the example in Fig. 15.7, where 9 SSUs (16 instead of 4 ¥ 9 = 36 transducers) are used. By way of elucidation, a PZT transducer located in the second row is chosen as the actuator. Synchronously belonging to SSUs A, B, C and D, the selected transducer performs standard excitation/acquisition for each SSU (denoted by solid lines with double arrowheads), and also additional excitation/acquisition for neighbouring SSUs (denoted by dotted lines with double arrowheads). The additional signal acquisition is aimed at improving efficiency with a limited number of PZT transducers. All the standard and additional excitation/acquisition is conducted in sequence mastered by a central control circuit.

15.2.3 Signal processing and identification Appropriately exploring captured signals, extracting and identifying essential information from them is a prime concern for signal-based NDE, which determines the feasibility and precision that a damage detection technique can offer. Though promising for NDE, elastic waves, particularly Lamb waves, also present several intricate features, limiting their practical applications: (i) in practice, sensors and structures under surveillance may operate in noisy or fluctuating environments, where the acquired wave signals can be severely impaired by diverse disturbances, such as random electrical

Modelling structural damage using elastic wave-based techniques

SSU A

SSU B

SSU C

SSU D

443

15.7 Actuator/sensor network using nine SSUs.

or mechanical noise; (ii) in both experimental tests and numerical simulations, interference from natural structural vibrations, especially low-frequency modes, may obscure the damage-induced wave components in signals; (iii) compared with waves excited by angle-variable ultrasonic probes coupled with angleadjustable perspex wedge, multiple modes of wave exist simultaneously when a piezoelectric element is used directly as actuator for wave generation, and their dispersion behaviour is notoriously complicated; (iv) adoption of high-resolution data acquisition may guarantee precision of identification, but it unavoidably leads to a bulky amount of data to serialise the sampling, burdening signal processing. All these factors weaken the sensitivity of elastic waves to structural damage, and lower detection precision. For this reason, efficient signal processing techniques are of vital importance and necessity. A wide variety of signal processing methods are currently available. Timeseries analysis (Thwaites and Clark, 1995; Sohn and Farrar, 2001), frequency analysis (Wu et al., 1992; Keilers, 1997, 1998; Kawiecki, 1998; Blanas and Das-Gupta, 1999) and time-frequency/time-scale analysis (Abbate et al., 1997; Samuel and Pines, 1997; Staszewski 1998; Hou et al., 2000; Okafor and Dutta, 2000; Kim and Kim, 2000, 2001) are the most common. However, except for some successful applications in location detection, time-series analysis cannot separate defect-scattered composition appropriately from a

444

Multi-scale modelling of composite material systems

raw signal containing all parts with different frequencies. The frequency domain analysis is conducted with the loss of temporal information. Although signal compositions with different eigen-frequencies can be separated, their history is hidden. Neither of them are straightforward for exact quantitative identification. These obstacles have accordingly necessitated a combination of the time domain with the frequency domain. In this category, Short-Time Fourier Transform (STFT), Winger-Ville distribution (WVD), and wavelet transform (WT) analysis are several important examples. Of these the wavelet transform has been demonstrated to be most effective for a non-static signal (Daubechies, 1990; Newland, 1994). Nowadays, wavelet transform-based signal processing enjoys a burgeoning popularity in the NDE approach, covering signal purification, spectrographic analysis, and signal/image compression. There are two types of wavelet transform in implementation, namely continuous wavelet transform (CWT) and discrete wavelet transform (DWT). Generally, CWT is particularly effective for signal analysis and visualisation, while DWT is more effective in signal denoising, filtration, compression and extraction. Fundamentally, with the basic orthogonal wavelet transform function, Y(t), a time-dependent wave signal, f (t), acquired from sensors, is converted into a quadratic expression using the dual parameters scale a, and time b (Hou et al., 2000): W ( a, b ) = 1 a

Ú

+•

–•

t – bˆ f ( t ) ◊ Y*Ê ◊ dt. Ë a ¯

15.5

Operation by eqn 15.5 is CWT and W(a, b) is the CWT coefficient. Y*(t) denotes the complex conjugate of Y(t). Conversely, the signal can also be reconstructed via inverse continuous wavelet transform: f (t ) = 1 CY

+•

+•

–•

a>0

Ú Ú

W ( a , b ) ◊ 1 ◊ Y Ê t – b ˆ ◊ 12 da ◊ db 15.6 Ë a ¯ a a

where CY is a constant depending on Y(t). In terms of eqns 15.5 and 15.6, the energy allocation for the signal f (t) in the time-scale domain is then derived as (Picinbono, 1988)

E=

+•

+•

–•

a≥ 0

Ú Ú

| W ( a , b )| 2 da ◊ db

15.7

where | CWT (a, b) |2, referred to as a scalogram, permits the description of the signal in the form of wave energy over the time-scale space, i.e., spectrographic analysis. Though wavelet transform does not produce a direct time-frequency view, it is not a weakness but a strength of the time-frequency analysis (Mathworks, 2001b).

Modelling structural damage using elastic wave-based techniques

445

To calculate wavelet coefficients at every scale point is computationally expensive. For simplification, eqn 15.5 can be executed only at discretised scale and time using dyadic variables m and n, i.e., DWT (Chui, 1997): a = a 0m and b = na 0m b0 , m , n ΠZ

DWT ( m , n ) =

–m a0 2

Ú f (t ) ◊ Y( a

–m 0 t

15.8a – nb0 ) ◊ dt

15.8b

where a0 and b0 are constants determining sampling intervals along the time and scale axes, respectively. Equation 15.8 decomposes signals into associated scopes with relatively higher and lower frequencies. Inversely, reconstruction of the wave signal can be performed by f ( t ) = c S S C mn ( t ) ◊ DWT ( m , n )

15.9a

m n

–m

C mn ( t ) = a 0 2 Y ( a 0– m t – nb0 )

15.9b

where c is a constant in correlation with Y(t). The time-dependent function Cmn(t) is the DWT coefficient. The more densely the decomposition is conducted, the higher the resolution that is achieved.

15.3

Models of an active sensor network

For evaluation and application of an active sensor network using the concept of SSU (addressed in section 15.2.2), models for both distributed piezoelectric actuator and sensor in the active sensor network are developed.

15.3.1 Model of PZT actuator Single actuator activation Considering a PZT-coupled quasi-isotropic composite laminate, a perfect bonding condition between them is applied to ensure strain continuity at interfaces. The electro-mechanical constitutive mutuality, with regard to direct and converse piezoelectric effects, for a single piezoelectric disk in free status, is (Lin and Yuan, 2001), Ï Q1 ¸ È p1 ÔQ Ô Í 0 Ô 2Ô Í Ô Q3 Ô Í 0 Ô e 11 Ô Í 0 Ô Í Ô Ì e 22 ˝ = Í 0 Ô e 33 Ô Í 0 Ô Í Ô Ô e 23 Ô Í 0 Ô e 13 Ô Í d15 Ô Í Ô Ó e 12 ˛ Î 0

0 p1 0 0 0 0 d15 0 0

0 0 p3 d 31 d 31 d 33 0 0 0

0 0 d 31 c11 c12 c13 0 0 0

0 0 d 31 c12 c11 c13 0 0 0

0 0 d 33 c13 c13 c 33 0 0 0

0 d15 0 0 0 0 c 55 0 0

d15 0 0 0 0 0 0 c 55 0

0 ˘ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ 0 ˙ 0 ˙ ˙ c 66 ˚

Ï E1 ¸ ÔE Ô Ô 2 Ô Ô E3 Ô Ô s 11 Ô Ô Ô ◊ Ìs 22 ˝ Ôs 33 Ô Ô Ô Ôs 23 Ô Ô s 13 Ô Ô Ô Ós 12 ˛

15.10

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Multi-scale modelling of composite material systems

where orthogonal components Qi and Ei (i = 1, 2, 3) are the electric displacement (charge/area) and electric field (voltage/length), respectively. sij and eij are stresses and strains in the PZT disk. Constants dij, pi and cij represent the piezoelectric strain constants, dielectric permittivity and compliance constants, respectively. Provided the poling direction is perpendicular to the surface of the PZT disk, eqn 15.10 in the absence of in-plane external electric fields (E1 = E2 = 0), is simplified in the polar coordinate system, as seen in Fig. 15.8(a), Q3 = p3E3 + d31(sr + sq)

15.11a

Laminate

Z¢

r R

Z

V

q

hPZT

PZT disk

Y

X

hLMT (a)

Z Polarisation

R Interface of coupled laminate

hPZT

X hneu

hLMT

(b)

15.8 Model of PZT actuator/sensor: (a) PZT disk in the polar coordinate system; (b) single-PZT-coupled laminate system; (c) stress/strain distribution throughout thickness for single-PZT-coupled laminate system; (d) stress/strain distribution throughout thickness for double-PZT-coupled laminate system.

Modelling structural damage using elastic wave-based techniques

447

Z

sr–PZT

er–PZT er–LMT

sr–LMT

hPZT X hneu hLMT

(c)

Z

er–PZT

sr–PZT

sr–LMT

hPZT

hLMT

X

er–LMT

(d)

15.8 Continued

sr =

E PZT [( e r + n PZT e q ) – (1 + n PZT ) ◊ d 31 E3 ] 2 1 – n PZT

15.11b

sq =

E PZT [( e q + n PZT e r ) – (1 + n PZT ) ◊ d 31 E3 ] 2 1 – n PZT

15.11c

where sr /e r and sq /eq are radial and tangential stress/strain components in the polar coordinate system, respectively. EPZT and nPZT denote Young’s modulus and Poisson’s ratio of the PZT disk, respectively.

448

Multi-scale modelling of composite material systems

For a free thin piezoelectric disk of hPZT in thickness, in-plane strains, er–PZT and eq–PZT, are generated when an external voltage V is applied, d 31 ◊ V = P. h PZT

e r – PZT =˙ e q – PZT = d 31 E3 =

15.12

Introducing the strain continuity condition at the interface ( e ri – PZT = e ri – LMT = e ri and e qi – PZT = e qi – LMT = e qi , while superscript i denotes variables at the interface between the PZT disk and composite laminate), eqn 15.11b can be rewritten as

s ri – PZT =

E PZT [( e ri + n PZT e qi ) – (1 + n PZT ) ◊ d 31 E3 ] 2 1 – n PZT

15.13

Laterally depicted in Fig. 15.8(b), the z-axis in the referred polar coordinate system originates from the interface, and hneu denotes the distance between the abscissa plane and neutral plane of the PZT-coupled laminate. Considering that both the PZT disk and the laminate are thin, classic lamination theory is applicable and the strain distribution in the laminate and PZT disk can be assumed to be linear across their own thickness, as

s r – LMT =

s ri – LMT ◊ ( hneu + z ) (– h LMT £ z £ 0) hneu

15.14a

sr–PZT =

1–n LMT E PZT s ri – LMT E PZT ◊ ◊ ◊( hneu + z ) – ◊ P (0 £ z £ h PZT ) E 1–n PZT LMT 1 – n PZT hneu 15.14b

as graphically explained in Fig. 15.8(c), where subscript LMT represents corresponding variables and constants for the laminate. Applying equilibrium of moments and force regarding the neutral plane,

Ú

0

Ú

0

– h LMT

– h LMT

2p rs r – LMT ◊ z ◊ dz + 2p rs r – LMT ◊ dz +

Ú

Ú

h PZT

0

h PZT

0

2p rs r – PZT ◊ z ◊ dz = 0

2p rs r – PZT ◊ dz = 0

15.15a

15.15b

yields stresses of the laminate and PZT actuator at the interface, E E PZT s ri – PZT = A˜ ◊ PZT D˜ ◊ P – ◊ P = E˜ ◊ P E LMT 1 – n PZT

15.16a

s ri – LMT = D˜ ◊ P

15.16b

Modelling structural damage using elastic wave-based techniques

449

where

1 – n LMT A˜ = 1 – n PZT

B˜ = 2 E LMT E PZT h LMT h PZT ◊ (1 – n LMT ) 2 2 ) ◊ (2h LMT + 3h LMT h PZT + 2h PZT 2 4 E 2 h 4 (1 – n PZT ) 2 + E PZT h PZT (1 – n LMT ) 2 C˜ = LMT LMT (1 – n PZT ) 3 2 3 ˜ ˜ = E LMT E PZT h PZT (4 E LMT h LMT + 3E LMT h LMT h PZT + AE PZT h PZT ) D B˜ + C˜

E Ê ˜ – E PZT ˆ˜ E˜ = Á A˜ ◊ PZT D E LMT 1 – n PZT ¯ Ë Substituting eqn 15.16 into eqn 15.11 leads to the strain of the PZT actuator at the interface. For a small PZT disk ( e ri ª e qi = e i ) equivalent radial elastic deformation, d r , along the disk circumference, is achieved by integrating the interface strain dr =

Ú

R

0

e i dr = R ◊

˘ d 31 È E˜ (1 – n PZT ) + 1˙ ◊ V . Í h PZT E PZT ˚ Î

15.17

It is implied that the resultant equivalent radial displacement along the circumference of a PZT disk actuator is proportional to the applied external voltage, whose scale factor is determined by eqn 15.17. Dual actuator activation A similar principle is applied for the case of dual PZT activation (see Fig. 15.8(d)), where the resulting neutral plane for the PZT-coupled laminate is consistent with the neutral plane of the laminate, and the stress symmetrically distributes regarding the neutral plane as (Lin and Yuan, 2001)

s r – LMT =

2s ri – LMT ◊z h LMT

(0 £ z £ hLMT/2)

2s ri – LMT E E PZT s r – PZT = A˜ ◊ PZT ◊ ◊z – ◊P E LMT h LMT 1 – n PZT

15.18a (0 £ z £ hPZT) 15.18b

as described in Fig. 15.8(d). The equilibrium of moments regarding the neutral plane can therefore be simplified as

450

Multi-scale modelling of composite material systems

Ú

h LMT /2

2p rs r – LMT ◊ z ◊ dz +

0

Ú

h LMT /2 z +h PZT

h LMT /2

2p rs r – PZT ◊ z ◊ dz = 0 15.19

which gives the stress of the laminate at the interface si = F˜ ◊ V

15.20

r – LMT

F˜ =

6d 31 h LMT E PZT ◊ ( h LMT + h PZT ) E È 3 2 2 (1 – n PZT ) ◊ Í h LMT + 2 A˜ ◊ PZT ◊ h PZT (3h LMT + 4h PZT + 6h LMT h PZT ) ˘˙ E LMT Î ˚

Substituting Eqn 15.19 into eqn 15.18a and integrating the strain at the interface results in

dr =

Ú

R

0

R ◊(1 – n LMT ) e i dr = F˜ ◊ ◊V E LMT

15.21

Serving a similar function to eqn 15.17, eqn 15.21 determines the scale proportional factor for activation under dual PZT disk actuators. Based on eqns 15.17 and 15.21, actuator models are established for finite element analyses in Fig. 15.9, with which various Lamb wave modes can be generated by applying uniform radial displacement constraints in the x-y plane to stimulate desirable Lamb modes in the composite laminate. Imposed displacement

Z

X

Y

Diameter of PZT disk Interface between PZT and laminate

15.9 Actuator model with applied displacement constraints.

15.3.2 Model of PZT sensor Considering deformation in the x¢-y¢ plane only, the constitutive relation, eqn 15.10, for a PZT disk sensor of R in radius and hPZT in thickness, can be simplified in its local polar coordinate system r¢ – q ¢ – z¢, in the absence of an external electric field (Lin and Yuan, 2001), Q = d 31 (s r ¢ + s q ¢ ) =

d 31 E PZT (e + eq ¢ ) 1 – n PZT r ¢

15.22

Modelling structural damage using elastic wave-based techniques

451

where r¢ and q ¢ are the local polar coordinates. Electric charges, Q, accumulated on both surfaces of a PZT disk sensor, can be defined in terms of applied external voltage, Q= 1 4p

Ú Ú Ú — ◊ Q ◊ dV

15.23

Substituting eqns 15.22 into 15.23 and applying Gauss’ theorem leads to Q=

d 31 E PZT 4p (1 – n PZT )

Ú Ú (e

r¢

+ e q ¢ ) ◊ r ¢ ◊ dr ¢ ◊ dq ¢

15.24

Regarded as a capacitor with a capacitance of C, a PZT disk sensor induces the output voltage Votp under the deformation Q ◊ h PZT Votp = Q = C p K 3 p0 R 2

15.25

where K3 and p0 represent the relative dielectric constant and the dielectric permittivity of the free PZT disk, respectively. Combining eqns 15.24 and 15.25 yields

Votp =

d 31 E PZT h PZT 4p K 3 p 0 R 2 (1 – n PZT )

Ú Ú (e

r¢

+ e q ¢ ) ◊ r ¢ ◊ dr ¢ ◊ dq ¢ 15.26

Compared with the laminate, the PZT disk sensor is geometrically small (e r¢ ª eq ¢ ª ecen, and ecen is the strain at the centre of PZT disk). Equation 15.26 then becomes

Votp = G˜ ◊ e cen , G˜ =

d 31 E PZT h PZT 4p K 3 e 0 (1 – n PZT )

15.27

This indicates that for a PZT disk sensor, the output voltage induced by geometrical deformation is directly proportional to the central strain of the ˜ Based on eqn 15.27, the PZT sensor output PZT sensor with the factor of G. responses, in the form of induced output voltage, can be defined using the sensor model by calculating the strain at the centre of PZT sensor.

15.4

Lamb wave scattering in defective CF/EP composite laminates

The scattering phenomena of Lamb waves, generated by a PZT actuator model and collected by a PZT sensor model, in a defective quasi-isotropic CF/EP composite laminate with delamination are then evaluated using dynamic FEM simulation and experimental verification.

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Multi-scale modelling of composite material systems

15.4.1 Model of delaminated composite laminates An 8-ply quasi-isotropic CF/EP (T650/F584) composite laminate in a stacking sequence of [45/–45/0/90]s, measuring 475 mm ¥ 475 mm ¥ 1.275 mm in geometry and having a pair of clamped opposite edges, is considered in finite element modelling, schematically depicted in Fig. 15.10. Four standard sensor units (SSUs), involving nine distributed circular piezoelectric wafers, are used to customise an active sensor network, which is surface-bonded on the laminate. The sensors are 65 mm away from the neighbouring edge(s) or at the plate centre, numbered with Pi (i = 1, 2, . .., 9). With dual piezoelectric effects (i.e. actuating and sensing), a total of 36 actuator-sensor pairs, i.e., 72 actuator-sensor paths, are provided by such a network. For convenience of approach, the whole laminate is factitiously quartered and denoted counterclockwise by Zone-1, Zone-2, Zone-3 and Zone-4 from the lower left quadrant. A delamination occurs in Zone-3 of the laminate. Considering reality, the delamination is assumed to be elliptical, but allowing for different axis length, orientation and interlaminar position. In the delamination region, a volume-split is shaped and enveloped by two isolated surfaces sharing the same boundary, on which FEM nodes are symmetrically allocated, as depicted in Fig. 15.11. Y

Z

P7

P8 Zone-4

P5

P4

P9 Zone-3

Invisible delamination

P6 475 mm

Zone-1

P1

Zone-2

P2

P3

X 0∞ fibre direction

Fixed boundary

475 mm

15.10 Quasi-isotropic CF/EP composite laminate containing an invisible delamination.

For comparison, two quasi-isotropic CF/EP composite laminates with the above-mentioned geometry and layout are modelled without and with the delamination, serving as benchmark and defective laminates, respectively. In the latter, an elliptic delamination (semi-major axis: 15 mm, semi-minor axis: 10 mm, angle between major axis and 0∞ fibre direction: 45∞, centre position: 276 mm to left and 299 mm to bottom edges of laminate, delaminated area: 0.25% of total surface area of laminate) between the first and second layers is introduced, referring to Fig. 15.10.

Modelling structural damage using elastic wave-based techniques

453

Delamination surfaces

Volume split

Single lamina

(a)

Neighbouring area

Upper delaminated surface Lower delaminated surface (b)

15.11 A delamination model: (a) profile diagram, (b) threedimensional view.

Full-scale FEM models for the two laminates are created using eight-node three-dimensional brick elements on the PATRAN® platform, symbolised by FEM-UD1# and FEM-UD2# for the benchmark and defective laminates, respectively. It is clear that the delamination is a complicated 3-D pattern, and highly refined finite element description with laminar-thick elements is adopted to capture all the stress/strain components. To accommodate the actual laminate configuration, eight layers are divided along the laminate thickness to characterise each unidirectional lamina individually, leading to approximately 100,000 solid brick elements for the simulation. Ichnography of the FEM model for vicinity of delaminated area is displayed in Fig. 15.12, where more than ten nodes exist within one wavelength to ensure simulation precision. A surface contact algorithm provided by the ABAQUS/EXPLICIT® FEM package (Hibbitt, Karlsson & Sorensen Inc., 2003) is adopted to process the contact problem arising from delamination, primarily relaxing restrictions on surfaces which may come into contact. The contact algorithm permits a

454

Multi-scale modelling of composite material systems Delamination

Vicinity

15.12 FEM Model for delaminated area in an eight-layer [45/–45/0/90]s quasi-isotropic CF/EP composite laminate.

small relative sliding displacement and arbitrary rotation of two delaminated surfaces. Both the upper and lower surfaces are defined using an elementbased deformable surface, allowing interaction between two surfaces in the normal direction without mutual penetration. PZT actuator/sensor models described in the previous section are invoked, where a uniform radial displacement in the x-y plane, following a five-cycle sinusoidal toneburst windowed with the Hanning function at a central frequency of 0.5 MHz, is applied on the actuator model to generate Lamb waves. Wave propagation is simultaneously monitored at a sample rate of 20.48 MHz by the sensor model.

15.4.2 Elastic properties A three-dimensional fibre/matrix model based on micromechanics (Gommers et al., 1996) is employed to derive the elastic properties for chosen CF/EP composite materials (F584, Hexcel®). In the model, it is presumed E11 = W f Ef11 + WmEm;

E 22 = E33 = 1–

G12 = G13 = 1–

Gm Gm Ê W f Á1 – G f 12 Ë

ˆ ˜ ¯

;

Em Em ˆ Ê W f Á1 – ˜ E Ë f 22 ¯

Modelling structural damage using elastic wave-based techniques

G23 = 1–

455

Gm Gm ˆ Ê W f Á1 – G f 23 ˜¯ Ë

15.28

n 23 =

n12 = n13 = Wfnf12 + Wmnm;

E 22 –1 2G23

where subscripts f and m denote fibre and matrix, respectively. W, G, E and n are volume percentage, shear modulus, Young’s modulus and Poisson’s ratio, respectively. Subscript 11 is the fibre direction. In general, these formulae predict the in-plane elastic constants of unidirectional composites quite well, except for axial shear modulus which is normally underestimated (Gommers et al., 1996). Assuming there are no voids in the composite materials, the density is defined using

r = Wf · rf + Wm · rm

15.29

Based on the elastic properties of individual fibre and matrix (Hexcel Co., 2001) detailed in Table 15.2, effective elastic properties for single unidirectional lamina are determined using eqns 15.28 and 15.29, summarised in Table 15.3. Table 15.2 Elastic properties for composite material (F584, Hexcel®) used in this study (Hexcel Co., 2001) (a) carbon fibre; (b) epoxy matrix (a) Carbon fibre Volume fraction (%)

E f 11 (GPa)

Ef 22 (GPa)

Gf 12 (GPa)

Gf 23 (GPa)

nf 23 (GPa)

rf (g/m3)

0.555

276

17.3

11.24

6.056

0.25

1835

(b) Epoxy matrix Volume fraction (%)

Em (GPa)

Gm (GPa)

nm (GPa)

rm (G/m3)

0.445

4.14

1.532

0.35

1219

Table 15.3 Estimated effective elastic properties for single CF/EP lamina E11 (GPa)

E22 (GPa)

E33 (GPa)

G12 (GPa)

G13 (GPa)

G23 (GPa)

n12

n13

n23

rm (g/m3)

153.67

9.49

9.49

4.26

4.26

3.44

0.295

0.295

0.381

1528

456

Multi-scale modelling of composite material systems

15.4.3 Dynamic simulation Dynamic simulation is accomplished on a supercomputer (physically installed at the Australian National University) using the ABAQUS/EXPLICIT® FEM code. The step of calculation time is controlled to be less than the ratio of the minimum distance of any two adjoining nodes to the maximum wave velocity. Actively generated by the actuator model, visualised wave propagation in FEM-UD1# at several representative moments is captured in Fig. 15.13. Meanwhile, a typical moment of the Lamb wave produced by the PZT actuator model in FEM-UD2# is displayed in Fig. 15.14 for comparison (note that the vertical displacement in the figures is significantly magnified for illustration), where delamination-imposed disturbance on wave propagation can be phenomenally observed.

(a)

(b)

(c)

(d)

15.13 Lamb wave generated by PZT actuator model at (a) 3.5th; (b) 6.0th; (c) 8.5th; (d) 12.0th microsecond.

15.4.4 Experimental validation Two quasi-isotropic CF/EP composite laminates made from unidirectional (UD) prepreg with the same materials (T650/F584), configuration [45/–45/ 0/90]s and geometry (475 mm ¥ 475 mm ¥ 1.275 mm), were manufactured. For the purpose of comparison, a delamination of the same shape and size in

Modelling structural damage using elastic wave-based techniques

457

Delaminated area

15.14 Lamb wave propagation in delaminated area.

FEM simulation is introduced in one of the two laminates to serve as the defective specimen, using two pieces of UPLEX©-R-25 film of 25 mm in thickness during fabrication, while the other is kept intact, symbolised by EXP-UD1#/UD2#, respectively, as summarised in Table 15.4. Table 15.4 Configuration of specimens Specimen no.

Geometric dimension (mm)

Simulated delamination

a

b

f

x

z

q

EXP-UD1# EXP-UD2#

475 ¥ 475 ¥ 1.275 475 ¥ 475 ¥ 1.275

N/A 15

N/A 10

N/A 45

N/A 276

N/A 299

N/A 0.25

a: semi-major axis b: semi-minor axis f: angle between the major axis and 0∞ fibre direction x/z: vertical distance to left/bottom edges from the centre of delamination q: percentage of delaminated area (%) in laminate

An active sensor network with four SSUs is tailored, including nine distributed commercial PZT disks (PI© PIC151, referring to Table 15.5), and instrumented with an in-house signal generation and acquisition setup (Su et al., 2002). Amplified and modulated excitation, in accordance with the FEM simulation, is applied on each PZT wafer in turn to generate Lamb waves. Wave signals are then acquired via each actuator-sensor path at the sample rate of 20.48 MHz.

Multi-scale modelling of composite material systems Table15.5 Material properties of PZT transducer (PI Ceramic, 1994) Product name

PI 151

Geometry Density Poisson’s ratio Charge constant d31 Charge constant d33 Relative dielectric constant K3 Dielectric permittivity P0 Elastic constant E

f : 6.9 mm, hPZT : 0.5 mm 7.80 g/cm3 0.31 –170 ¥ 10–12m/ V 450 ¥ 10–12 m/ V 1280 8.85 ¥ 10–12Fm–1 72.5 GPa

Normalised amplitude

1.0

0.5

0.0

–0.5

–1.0 0 200

220

240

260 Time (ms) (a)

280

300

320

0 200

220

240

260 Time (ms) (b)

280

300

320

1.0

Normalised amplitude

458

0.5

0.0

–0.5

–1.0

15.15 Raw Lamb wave signals acquired via P1-P3 for (a) benchmark FEM-UD1#; (b) defective FEM-UD2#; (c) benchmark EXP-UD1#; (d) defective EXP-UD2#.

Modelling structural damage using elastic wave-based techniques

459

Normalised amplitude

1.0

0.5

0.0

–0.5

–1.0 0 200

220

240

260 280 Time (ms) (c)

300

320

0 200

220

240

260 280 Time (ms) (d)

300

320

Normalised amplitude

1.0

0.5

0.0

–0.5

–1.0

15.15 Continued

15.4.5 Signal processing and identification Signal acquisition is well replicable in both the simulations and experiments. As representative paradigms, captured signals for FEM-UD1#/UD2# and EXPUD1#/UD2#, via actuator-sensor path P1–P3, are displayed in Fig. 15.15. Their frequency spectra via FFT are compared in Fig. 15.16. Certain nuances can be observed between the signals of the benchmark and defective laminates, either in the time or the frequency domain. However, none of them is sufficient for quantitative damage identification. Aided with wavelet transform-based signal processing, digitally filtered

460

Multi-scale modelling of composite material systems

Normalised amplitude

1.0

0.8

0.6

0.4

0.2

0.0 0.0

200.0k

400.0k 600.0k Frequency (Hz) (a)

800.0k

1.0M

200.0k

400.0k 600.0k Frequency (Hz) (b)

800.0k

1.0M

Normalised amplitude

1.0

0.8

0.6

0.4

0.2

0.0 0.0

15.16 FFT spectra for signals in Fig. 15.15: (a) benchmark EXP-UD1#; (b) defective EXP-UD2#.

signals corresponding to active excitation (500 kHZ) are separated from raw signal (Fig. 15.17), where the fundamental symmetric Lamb mode, S0, and the basic horizontal shear (SH) mode, S0¢ , together with the reflected components from laminate edges, denoted by S0 ref , can be observed in sequence. For identification only the first few wave components are useful because the rest may be corrupted by multiple boundary reflections. Mode conversion caused by the existence of delamination is noticeable in the signals for the defective laminate. Compared with the benchmark laminate,

Modelling structural damage using elastic wave-based techniques

461

extra wave component between S0 and S0ref is detected in Figs 15.17(b) and (d), which is recognised as the delamination-induced basic shear mode, S0¢ Delam . The appearance order of wave components in a signal is dependent on the relative locations among defect, boundaries and sensors. Other damage-induced modes are unexploitable because their propagation velocities are either too high to be identified or too low to be sampled in the specific time period. 1.0

S0ref

Normalised amplitude

S0

S 0¢

0.5

0.0

–0.5

–1.0 4500

5000

5500 6000 Sampling points (a)

6500

1.0

S0ref S 0¢

Normalised amplitude

S0 0.5

0.0

–0.5 S 0¢ Delam –1.0 4500

5000

5500 6000 Sampling points (b)

6500

15.17 DWT-filtered signals in the time domain for (a) benchmark FEM-UD1#; (b) defective FEM-UD2#; (c) benchmark EXP-UD1#; (d) defective EXP-UD2#.

462

Multi-scale modelling of composite material systems 1.0

S0 ref

Normalised amplitude

S0

S 0¢

0.5

0.0

–0.5

–1.0 4500

5000

5500 6000 Sampling points (c)

6500

1.0

S0 ref

Normalised amplitude

S0

S 0¢

0.5

0.0

–0.5 S 0¢ Delam –1.0 4500

5000

5500 6000 Sampling points (d)

6500

15.17 Continued

Further, spectrographic analyses based on CWT are also performed. Results for selected signals are compared in Fig. 15.18. In the 3-D energy spectra, a defect-caused shear wave, S0¢ Delam , is detected. During comparison between the benchmark and defective laminates, or between the FEM simulation and experiment, certain distinctions or incomplete agreement can be observed in addition to the delamination-induced extra wave components, which are attributed to the differences of boundary conditions, imperfect similarity of

Modelling structural damage using elastic wave-based techniques

463

S0

S0 ref 6

Wavelet coefficient

4 2 0 –2 –4

S 0¢ Delam

100

–6 4500

5000

50 5500

25

6000

Sc

75 ale

6500

Sampling points (a)

S0 6

S0 ref

Wavelet coefficient

4 2 0 –2 –4 S 0¢ Delam

100

–6 4500

75 5000

50 5500

6000

6500

7000

25

Sc

ale

Sampling points (b)

15.18 CWT spectrographic analysis in the time-scale domain: (a) spectrum for FEM-UD2#; (b) 3-D spectrum for EXP-UD2#.

464

Multi-scale modelling of composite material systems 9

Phase velocity (Km/s)

S0

S1 A1

6 S ¢0

3

A0

FEM model Effective elastic constant model Experiment

0

0.0

0.5

1.0 1.5 2.0 Frequency thickness (MHz mm)

2.5

3.0

15.19 Dispersion curves for Lamb waves in an 8-ply quasi-isotropic CF/EP (T650/F584) composite laminate.

two individual laminates and discrepancy of theoretical simulation with laboratory measurement.

15.4.6 Dispersion nature Conducting the above experiments in a sweep frequency range from 0.05 MHz to 2.0 MHz, the dispersion characteristics for Lamb waves in quasiisotropic [0/–45/45/90]s CF/EP (T650/F584) composite laminates are evaluated. Wave velocities via different actuator-sensor paths are individually measured and averaged upon process using wavelet transform. The dispersion relation for the chosen laminates is diagrammed in Fig. 15.19. For comparison, the dispersion curves simulated using FEM approach and analytically calculated using the effective elastic constant method (Percival and Birt, 1997) are also contrasted, denoted by different symbols. Good correlation between FEM and experiments has been observed. Both reveal that symmetrical and anti-symmetrical Lamb modes are synchronously available, and their phase/group velocities are dependent on the algebraic product of the laminate thickness and the central frequency. A low-frequency region (<1.0 MHz·mm) exists for the present structure, in which the fundamental anti-symmetric Lamb mode, A0, symmetric Lamb mode, S0, and the lowest order shearing mode, S0¢ , exhibit reasonably non-dispersive behaviour. Higher order modes appear gradually with an increase in frequency.

Modelling structural damage using elastic wave-based techniques

15.5

465

Conclusions

Following a brief review of representative modelling techniques of wave scattering in damaged composite laminates for the purpose of damage identification, a FEM modelling approach for distributed piezoelectric devices (actuator/sensor) and delamination is specifically developed, and elastic wave propagation in quasi-isotropic CF/EP (T650/F584) composite structures containing delamination is evaluated. The surface contact algorithm adopted in the simulation exhibits considerable effectiveness in handling problems arising from delamination. The FEM results are then validated by experiments, and good correlation between them is achieved. A wavelet transform-based signal processing and identification approach is employed to characterise the complex Lamb wave signals in composite structures. It has been demonstrated that spectrographic characteristics of Lamb waves in the time–frequency space are quantitatively sensitive to structural damage, offering the capability to detect damage accurately. The sensor technique for online damage identification is addressed, and in particularly, an active sensor network technique using a concept of ‘Standard Sensor Unit (SSU)’ is proposed. Diverse transducer networks can conveniently be customised by appropriately assembling SSUs, flexibly to accommodate different geometries and boundary conditions.

15.6

References

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16 Modelling the fatigue behaviour of bonded joints in composite materials M Q U A R E S I M I N, University of Padova, Italy

16.1

Introduction

A typical problem the designers have to face during the development of a new product in composite material (or in repairing an existing structure) is the presence of joints, metal inserts and openings for the connection of the composite parts between them and the remaining part of the structure. A solution frequently adopted is to bond together the different parts to be joined. Due to the increasing importance of, and interest in, bonded joints in load-carrying structural applications, the behaviour of bonded connections, either in isotropic or composite materials, has been extensively investigated by the scientific community in the last decades, analytically, numerically and experimentally. This activity is testified by the endless literature available on the subject and some good examples are reported in references 1 to 43. Interesting readings on the structural joints in composite materials are also provided by the books of Adams et al.44 and the more recent of Tong and Soutis.45 Although the greater part of the published material concerns the development of data and design tools for the prediction of strength and life of bonded joints, the problem of the fatigue design is still far from being closed due to the significant influence of several design parameters on joint behaviour, as well illustrated in the review presented by De Goeij et al.46 The overall geometry of the joint, the property and thickness of laminates and adhesive, the overlap length and the presence of a spews fillet were reported to have the greatest influence on joint strength. Several textbooks and the literature discuss in detail the influence of each single parameter and therefore it is unnecessary and out of the scope of this chapter to further describe this subject here. Instead it is important to remember the typical feature of a bonded joint. Due to the mismatch of the elastic properties of the connected materials, stress singularities can arise even in the absence of any geometrical discontinuity.2 The stress distributions at the critical location of the joint are 469

470

Multi-scale modelling of composite material systems

therefore characterised by the combined influence of a constitutive singularity due to the differences in the elastic properties of adherends and adhesive and a geometric singularity due to the corner frequently present at the overlap edge. On this basis, a generalised stress intensity factor (SIF) approach was successfully used in the past to predict static strength in bonded joints4,7–9 and it proved to work better than the other possible stress, strain or energy based strength criteria particularly in the case of joints bonded with brittle adhesive. As far as fatigue behaviour is concerned, the simplest, traditional approach to the fatigue design based on nominal stresses and fatigue curves turned out to be unsuitable to represent together data referred to joints of different geometry and to provide a model of general validity. Moreover, the use of fatigue curves, usually made on the basis of the number of cycles to failure, makes it difficult to describe and account for damage evolution. The available experimental results indicate, in fact, the actual mechanics of fatigue damage evolution in the nucleation of one or more cracks at the critical locations in the joint and their subsequent propagation to the point of joint failure.16, 18–20, 26, 29, 47 The fatigue cracks grow at the laminate/adhesive interface or, less frequently, inside the adhesive layer. However, the most recent literature is still debating the relative contribution of crack initiation and crack propagation phase to the total fatigue life of the joint and no definitive indication can be provided, since the relative fraction of life spent in the second phase depends on several design parameters like joint materials, overall and corner geometry, load level and possible presence of defects in the bondline. The majority of the methodologies available for the fatigue life prediction of bonded joints both in metallic or composite materials are based on a fracture mechanics approach, considering the fatigue life of the joint entirely spent in the crack propagation phase.12, 23, 36, 41, 43 On the assumption that this choice is on the conservative side, in some models the assessment of the nucleation phase is explicitly neglected,41 in others it is simply not considered. The fatigue life of the joint is then obtained by the integration of a Paris-like curve relating a fracture parameter like the strain energy release rate (SERR) or, less frequently, the SIF to the crack growth rate (CGR). Although some of the available literature20, 27, 28 discusses the assessment of the fatigue threshold level, as the load level below which the cracks do not generate in a large number of cycles, no models seem to be available for the explicit assessment of the life spent for initiating a crack in a joint subjected to fatigue loading. Instead as shown later, the nucleation phase represents an important part of the fatigue life of the joint and therefore a model not incorporating and describing this phase risks underestimating significantly the joint lifetime. Although on the conservative side, the neglect of the nucleation phase may

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lead to over-designed joints with unjustified increases in weight and cost and the final results of a loss in lightness and specific strength of the whole composite structure the joints are part of. In an attempt to overcome this limitation, this chapter presents the development and validation of a life prediction model based on the results of an extensive experimental and numerical investigation. According to the actual mechanics of the fatigue damage evolution, the model describes explicitly the two phases of the joint life. The first phase of the fatigue life, spent in nucleating one or more cracks, is estimated by using fracture mechanics, extending to composite joints the stress intensity factor approach, recently suggested by Lefebvre and Dillard 21,22 and Lazzarin, Quaresimin and Ferro34, to predict the life to crack initiation in bonded joints made of isotropic materials. Under the assumption that the crack nucleation process, either in the adhesive or at the adhesive/adherend interface, is controlled by the intensity of the singular stress field at the critical location in the joint, it is possible to draw scatter bands suitable for the assessment of the life to crack initiation. The life spent in the propagation phase is then calculated, more conventionally, by the integration of a Parislike curve relating the SERR to the CGR. A schematic representation of the model and the parameters involved is presented in Fig. 16.1. DHo vs. Ni Life to crack initiation

Ni

SERR vs. da/dNp

Nf

Life for crack propagation

16.1 Schematic of the life prediction model.

16.2

Experimental investigation

Fatigue behaviour under tension loading of single lap bonded joints in composite materials was investigated with the aim to obtain a fatigue database for the model validation and information about the actual mechanics of damage evolution. The single lap joint is not a very efficient geometry in terms of load bearing capability, due to the out-of-plane bending induced by the adherend misalignment. Nevertheless, it was chosen because it is frequently present in practical applications for its relative ease of manufacture; sometimes, for instance, when only one side of the structure is accessible, it is instead imposed by the structure geometry as the only possible design solution. The typical geometry of the joints under investigation is presented in Fig. 16.2. The joints were manufactured from autoclave moulded carbon/epoxy laminates (SEAL T300 twill 2 ¥ 2 fabric/toughened ET442 matrix) bonded with 3M 9323 B/A epoxy adhesive. Laminates and adhesive layer are 1.65

472

Multi-scale modelling of composite material systems F 1.65

24

w

SE 260

20

16.2 Geometry of single lap bonded joint; w = 20, 30 and 40 mm F = spew fillet joints (fillet angle = 45∞), SE = square edge joints.

mm and 0.15 mm thick, respectively. Three overlap lengths (20, 30 and 40 mm) and two corner geometries (square edge and spew fillet) were investigated by keeping constant specimen width (24 mm) and overall length (260 mm), being the laminate lay-up [0]6. After some preliminary tests, the laminates were laid up including an external layer of peel-ply for improving adhesion quality and the strength of the joints. After peel-ply removal, the panels were degreased by using methyl-ethyl-ketone (MEK), bonded and then cured in an oven for two hours at 65 ∞C under a pressure of about 0.1 MPa; 150 mm diameter glass spheres were mixed with the adhesive to maintain the required thickness. The joints were cut from the bonded panels and tested at room temperature on a servo-hydraulic MTS 809 machine with 10–100 kN load cells. The static behaviour of the joints was preliminarily investigated by means of tensile tests under displacement control with a crosshead speed equal to 2 mm/min. The fatigue tests were carried out under load control, nominal load ratio R = 0.05 and a test frequency variable in the range of 10–15 Hz depending on the stress level. Significant efforts were dedicated to the analysis of fatigue damage evolution, with particular reference to the assessment of fatigue crack initiation and evaluation of the rate of crack propagation. For doing this, some of the joints were subjected to repeated blocks of loading at constant amplitude, up to failure. At the end of each block, the damage patterns were analysed by means of microscopic observation of the polished edges of the joints to identify the onset of a fatigue crack or to measure the length of the cracks, when present. It was decided to use optical microscopy for crack detection because, although very time consuming, it is far more accurate than the compliance-based methods which are more sensitive, instead, to the overall response of the joint. Only a synthesis of the experimental results is presented here, for a more extensive and exhaustive description the reader is referred to ref. 47, where the complete results are presented and discussed.

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16.2.1 Static and fatigue strength of single lap joints A comparison of static properties of the joints is presented in Table 16.1, with reference to the influence of overlap length and corner geometry. For providing information on both load-carrying capability of the joint and adhesive property, tensile and shear stress at failure are reported together. The stresses are calculated here dividing the load at failure by the transverse section of one adherend and by the joined area, respectively. The results clearly indicate improvement in the static strength of the joint with the increase in the overlap length as well as the beneficial effect of the presence of a spew fillet at the overlap edge. The typical failure mode was the tensile failure of the adherend at the end of the overlap, due to the very high stress concentration; failures of the adhesive layer were also recognised in some cases, for the shortest overlaps. Table 16.1 Static strength of the joints Corner geometry

Overlap (mm)

sUTS (MPa)

c.o.v. (%)

tUTS (MPa)

c.o.v. (%)

Square edge

20 30

347.9 407.9

4.0 10.8

29.2 22.6

4.9 11.5

Spew fillet

30 40

501.7 551.5

2.6 4.2

27.6 22.7

2.9 4.3

The increase in overlap length and the presence of the fillet provides a significant improvement also on the fatigue strength of the joints, as shown by the fatigue curves plotted in Figs 16.3 and 16.4 and the statistical analysis of all the fatigue data, presented in Table 16.2. The fatigue data were analysed assuming log-normal distribution of cycles to failure and a fatigue curve in the form of k Ns max = cost

or

k Nt max = cost

16.1

was calculated for each series. Table 16.2 lists the reference stress at 2·106 cycles for a probability of survival of 50% and 90%, the values of the inverse slope k and scatter index Ts(sA,10%/sA,90%). It is interesting to note the limited variation of the inverse slopes, which seems to indicate a shift of the fatigue curves as a function mainly of the overlap length.

16.2.2 Analysis of crack nucleation and propagation During fatigue loading the cracks were found to nucleate, usually, in the central zone of the bondline at the overlap edge and then to propagate very quickly to the joint edges before steadily growing towards the centre of the

474

Multi-scale modelling of composite material systems R 0.05 0.05

300

sMAX, 50%

k

108.5 77.5

6.07 6.50

smax (MPa)

200

100 Overlap length 40 mm Overlap length 20 mm 1E3

1E4

1E5 Cycles to failure

1E6

5E6

16.3 Influence of the overlap length on the fatigue strength of spew fillet joints.

R 0.05 0.05

400

sMAX, 50% 108.5 85.7

k 6.07 5.84

smax (MPa)

300

200

100 Overlap length 40 mm Spew fillet joints Square edge joints

1E3

1E4

1E5 Cycles to failure

1E6

5E6

16.4 Influence of the corner geometry on the fatigue strength of the joints.

joints. Examples of crack nucleation sites are presented in Figs 16.5a and b, where the cracks are clearly evidenced by the crazing and whitening of the epoxy adhesive. Figure 16.6 shows instead the onset of a crack as it appears from the side view of the polished edges of a square edge joint. Also visible is the small radius sometimes present at the end of the bondline due to the bonding procedure adopted. In very few cases, the sites of crack nucleation were identified also in different locations like in the upper part of the fillet.

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Table 16.2 Fatigue strength data evaluated at 2 · 106 cycles Corner geometry

Overlap sMAX,50% sMAX,90% k (mm) (MPa) (MPa)

Ts

tMAX,50% tMAX,90% No. (MPa) (MPa) of data

Square edge

20 40

53.9 85.7

46.0 70.0

6.13 5.84

1.372 1.498

4.4 3.6

3.6 2.8

4 12

Spew fillet

20 30 40

77.5 106.2 108.5

69.2 98.3 87.6

6.50 7.08 6.07

1.253 1.166 1.534

6.3 5.9 4.4

5.6 5.3 3.5

14 8 13

(a)

(b)

16.5 a,b Examples of sites of crack nucleation.

16.6 Side view of the adhesive layer with a nucleated crack.

The life to crack initiation was defined as the number of cycles at which a crack could be practically observed on the polished edges of the joint; a reference threshold was arbitrarily fixed at 0.3 mm. Examples of the results obtained from damage analysis are reported in Fig. 16.7 which compares the fatigue life to crack initiation and to complete failure for square edge joints, and Table 16.3 which lists life data for spew

476

Multi-scale modelling of composite material systems w = 40 mm – crack initiation w = 40 mm – final failure

300

smax (MPa)

200

100

1E3

Nucleation

Propagation

1E4

1E5 Number of cycles

1E6

5E6

16.7 Fatigue curves to crack nucleation and final failure (from block loading tests on square edge joints). Table 16.3 Fatigue life data for spew fillet joints from block loading fatigue tests Overlap (mm)

smax (MPa)

Cycles to crack initiation Ni (cycles)

Cycles for crack propagation Np (cycles)

20 20 20 20 30 30 30 40 40 40 40

188 168 126 113 211 168 126 253 200 168 126

4500 8700 42000 77000 6800 37000 205000 4800 48000 26000 475000

2080 10955 64603 75000 8428 52802 623018 16628 16235 104180 1726105

Fatigue life Nf (cycles)

6580 19695 106603 152000 15228 89802 828018 21428 64235 130180 2201105

Ni/Nf

0.68 0.44 0.39 0.51 0.45 0.41 0.25 0.23 0.75 0.20 0.21

fillet joints. It is clearly evident that a large fraction of fatigue life is spent in the crack nucleation phase and this corroborates and justifies the choice to develop a model incorporating the description of this phase. After the identification of crack onset, crack evolution during the propagation phase was monitored for the subsequent evaluation of the crack propagation rate; the length of the cracks running from each of the four corners of the bonded area were measured. Two symmetric crack fronts growing uniformly from both sides of the overlap length were usually observed; only in a few cases and mainly for spew fillet joints, the crack front grew in a non-uniform or non-symmetric way. Figures 16.8 and 16.9 show typical crack growth in the two extreme conditions.

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20 Overlap length w: 40 mm smax = 80 MPa Nf = 1352934 cycles

Crack length (mm)

17.5 15 12.5

A

C

B

D

A C

B D

10 7.5 5 2.5 N/Nf (%)

0 20

40

60

80

100

16.8 Symmetric fatigue crack growth in a square edge joint. 17.5

Crack length (mm)

15 12.5

Overlap length w: 20 mm smax = 113 MPa Nf = 200100 cycles A

C

B

D

A C

B D

10 7.5 5 2.5 0 20

N/Nf (%) 40

60

80

100

16.9 Non-symmetric fatigue crack growth in a spew fillet joint.

In view of the applicability of the model under development, a uniform and symmetric crack front was always assumed for the evaluation of the CGR, calculated by using the incremental polynomial method50 on the crack length averaged from the four measurements taken on each joint. The alternative possibility, to use the longest crack, turned out to provide small variations both in the calculated rate of the crack growth47 and, eventually, in the estimated life.49 Trends of the CGRs measured on square edge joints during their fatigue life, i.e., for increasing crack length, are presented in Fig. 16.10. The CGR data are plotted against the crack length a normalised with respect to the overlap length w to allow a direct comparison to be made among data related to joints with different overlap; high and low stress refer in both cases to a medium (about 50,000 cycles) and long (about 1 million cycles) fatigue life.

478

Multi-scale modelling of composite material systems 1.0E-01 w w w w

da/dN (mm/cycle)

1.0E-02

= = = =

20 mm; 20 mm; 40 mm; 40 mm;

high stress level low stress level high stress level low stress level

1.0E-03

1.0E-04

1.0E-05

1.0E-06 0

0.1

0.2

0.3

0.4

0.5

a/w

16.10 Crack growth rates measured on square edge joints.

Since the normalised trends are similar, the joint overlap seems not to have any influence, whereas the CGR is significantly affected by the applied stress level. Comparable behaviour was observed also for spew fillet joints47 with higher values of the rate of crack growth.

16.3

Finite element analysis of SIF and SERR

The prediction of the fatigue life of a bonded joint according to the model presented here requires knowledge of the stress intensity factor (SIF), related to the un-cracked geometry of the joint, for the assessment of the time spent in the crack nucleation phase and the strain energy release rate (SERR) of the propagating crack for the estimation of the propagation phase. The presence of the stress singularity and the anisotropic properties of the composite adherends suggested the use of finite element analysis as an accurate and, most of all, general tool for the calculation of these two parameters. Linear elastic analysis, both for material and geometry, were adopted to evaluate the stress distributions and the associated stress singularity and SIF on un-cracked FE models of the joints. Parallel investigations on the influence of design parameters like overlap length, corner geometry, adhesive properties and thickness were also carried out and the results are discussed in ref. 48. For the evaluation of the SERR, crack propagation was assumed to be at the adherend-adhesive interface, according to the experimental observations. The virtual crack closure technique (VCCT),51–52 modified to consider the actual deformed shape of the joints,53 was applied for the calculation of the SERR components from the results of geometric non-linear analyses which turned out to be those more suitable to describe the out-of-plane displacement of the cracked joints. However, it is important now to remember that the

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evaluation of the SERR components via FE analysis and VCCT for a crack at the interface of a bi-material body can be strongly influenced by the meshsensitivity of the problem. In fact, as pointed out by Sun and Jih,54 recalling the work of Williams,55 due to the oscillatory nature of the stress fields near the crack tip at a bi-material interface, the mode I and mode II components of the SERR, GI and GII, do not converge in terms of crack closure integrals, whereas their sum, i.e., the total SERR GTOT, is well defined. To overcome the problem, some authors23 suggested basing the prediction of the propagation phase on the total strain energy release rate GTOT. However, the use of the total SERR does not keep any information about the possible mode-mixity due to joint geometry and loading conditions, information helpful to explain fatigue damage evolution and useful to define a proper failure model. A possible alternative, as suggested by Wahab and co-authors,36 to maintain the information related to the mode-mixity, is to perform a preliminary analysis of mesh sensitivity and identify the size of the elements near the crack tip suitable to provide constant values for the mode I and mode II components calculated by VCCT analysis. An element-size-to-adhesivethickness ratio larger than 1/8 was suggested36 to obtain SERR components almost insensitive to the element size. Both approaches are considered later and, after the introduction of an equivalent formulation for the SERR, the results are compared in terms of estimated fatigue life.

16.3.1 Finite element modelling Different strategies were considered for the evaluation of the stress intensity factors and strain energy release rates. For the analysis of the stress fields and the associated SIFs, bi-dimensional finite element models of the uncracked joint geometry were developed by using the Ansys 7.0® code and PLANE82 elements under plane strain conditions. The models are representative of the central section of the joint, a location where, according to damage analysis, the majority of cracks originated. For a better description of the asymptotic stress distributions near the singularity, very fine meshes were adopted, by using the KSCON command of the ANSYS software, useful for creating a fracture model around a singularity point. The number of elements in the models, ranging from 5,000 to 28,000, depended on overlap length and presence of fillet, the smallest element size being equal to 10–5 mm. Figure 16.11 shows the overall geometry of the FE model, the applied boundary conditions and some details of the mesh near the singularity; the input material properties are listed in Table 16.4. Geometric linear behaviour was assumed according to the results reported by Pradhan et al.,15 who observed a near-linear dependence between stress intensity factors and applied stress levels. Moreover, preliminary analyses demonstrated that the solution of the model including geometrical non-linearity

480

Multi-scale modelling of composite material systems (a)

(b)

Y snom

X

16.11 Boundary conditions applied to the finite element model and details of the mesh near the singular zone for (a) square edge and (b) spew fillet corner (un-cracked joint geometry). Table 16.4 Elastic properties of adherends and adhesive

Adherends* Adhesive**

EX (MPa)

EY(MPa)

GXY(MPa)

vXY

58050 2870

6000 2870

500 1050

0.27 0.37

Orthotropic Isotropic

*from ref. 56, **from ref. 57

may lead to some changes in stress fields but has only a reduced influence on the generalised SIF values. The neglect of possible material non-linearity of the adhesive and the consequent assumption of elastic behaviour for both adhesive and adherends was also based on dedicated experimental tests and microscopic observations which proved an elastic, brittle behaviour of the adhesive up to failure. The finite element models for the evaluation of the SERR were developed again by using PLANE82 elements under plane strain conditions. In this case, however, a different meshing strategy was adopted, as required by VCCT technique and the region near the crack tip was modelled with elements of regular shape and uniform size, as shown in Fig. 16.12. Since the experimental a Adherend Adherend

Adhesive a

16.12 Details of the mesh at the adhesive-adherend interface for the model of a joint in the presence of a crack.

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481

tests showed a significant increase in the out-of-plane displacements in the presence of cracks, the geometric non-linearity was included in these analyses; the behaviour of the material was, instead, assumed again to be linear elastic. The choice of the smallest element size was based on the results of a preliminary sensitivity analysis, presented in Fig. 16.13. As expected, the individual Mode I and Mode II components seem to converge for larger element size, whereas the total SERR turns out to be almost insensitive to the mesh size. Interestingly, the same insensitivity is exhibited by a new equivalent formulation for the SERR which will discussed later in the chapter. An element size of 30 mm was then chosen for all the analyses, as it represents a good compromise between the need to have large elements to avoid mesh sensitivity and, on the other hand, the limitation of the element size due to the adequate modelling of the adhesive layer, which is only 150 mm thick. Strain energy release rate (J/m2)

500 Gtot 400 Geqv GI

300

200 GII 100

0 0

20

40

60 80 100 120 Element size e (mm)

140

160

16.13 Influence of the element size in the calculation of the SERR components (Square edge joint, overlap length 20 mm, crack length 5 mm, tensile stress 110 MPa).

The last consideration is about location and configuration of the cracks in the model. As already said, the experimental observations suggested modelling the crack at the adherend-adhesive interface. Furthermore, on the basis of the assumption of uniform and symmetric crack front, two symmetric cracks of equal length were introduced in the 2D plane strain model (see again Fig. 16.12). This condition turned out also to be more critical, in terms of SERR, than a single crack running from one side only.48

16.3.2 Stress distributions, SIF and SERR An example of the stress distributions obtained from the FE analyses on the uncracked models of the joints is reported in Fig. 16.14. The stress distributions

482

Multi-scale modelling of composite material systems 100 0∞

0∞

srr-adherend srr-adhesive sqq srq

s ij /snom

10

1

Overlap = 30 mm Empty symbols = square edge joints Filled symbols = spew fillet joints

0.1 0.0001

0.001 r (mm)

0.01

16.14 Normalised stress distributions at the adhesive-adherend interface for square edge and spew fillet joints under tensile loading.

are plotted in polar co-ordinates (in a r, q system centred at the corner), as a function of the distance from the singularity and normalised with respect to the external tensile loading (snom). Similar trends were obtained for other angular positions and overlap lengths. It can be easily observed that the square edge joints are characterised by stress fields more critical in terms of both intensity and degree of singularity and this provides a first explanation of their lower static and fatigue strength with respect to the spew fillet joints. The overall stress distributions near the singularity of a bonded joint could be described by a two-term stress expansion.34 However, as can be seen in Fig. 16.14, in the close neighbourhood of the singularity the stress fields can be described, in a simplified way, by the following relationship:

s ij ( r , q ) = H 0 r s f ij(0) (q )

16.2

representing only the first term of the distribution, given by a radial component raised to an exponent (the eigenvalue) multiplied by a constant parameter H0 and an angular function. The parameter H0 represents the generalised stress intensity factor for the first-order term of the expansion. The values of generalised SIF H0 as well as those of the strength of the stress singularity can be obtained from the best fit of the stress distributions and their values are summarised in Table 16.5 for square edge and spew fillet joints and for the overlap lengths of interest. For the same geometry corner, the normalised values of H0 decrease as the overlap length increases whereas the strength of the singularity remains constant, depending on corner geometry and elastic properties of the materials only. Evident again is the beneficial

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contribution of the spew fillet presence which reduces the values of both stress intensity factor and strength of the singularity. The FE analyses made assuming a linear elastic behaviour, the absolute values of the SIF for a specific stress level can be calculated by simply multiplying the proper normalised value in Table 16.5 for the stress level of interest. Table 16.5 Normalised values of generalised stress intensity factors H0 and degree of singularity Square edge

Spew fillet

–s

w (mm)

H0 /s0 (mm )

s

H0 /s0(mm–s)

s

20 30 40

0.1713 0.1660 0.1577

–0.4285 –0.4285 –0.4285

0.1683 0.1592 0.1493

–0.1987 –0.1987 –0.1987

On the other hand, the evaluation of the SERR trends as a function of the crack length required an independent set of analyses for each level of the applied tensile stress, being in this case the geometric nonlinearities included. All the geometries were investigated; for each of them the crack propagation process was simulated by applying the same stress levels applied during the block loading fatigue test. The data thus obtained, combined with the crack growth rates experimentally measured on the joints allowed the Paris curves to be drawn. Figure 16.15 shows the trends of Mode I and Mode II SERR components, calculated for joints with overlap 20 mm at low and high stress levels. The plots are quite representative also of the results obtained for joints with

Strain energy release rate (J/m2)

1000 GI w GII w GI w GII w

750

= = = =

20 20 20 20

mm mm mm mm

snom snom snom snom

= = = =

110 MPa 110 MPa 60 MPa 60 MPa

500

250

0 0

2

4

6

8

10

a (mm)

16.15 Mode I and Mode II components of the SERR for square edge joints with 20 mm overlap length at different stress levels.

484

Multi-scale modelling of composite material systems

longer overlap length, for which similar trends were found. The analysis of the results obtained for all the conditions investigated indicates that crack propagation occurs under mixed-mode loading, although with a predominance of the Mode I component, more evident for the higher levels of the applied stress. The predominance of the Mode I component can be observed even better from analysis of the mode-mixity ratios represented in Fig. 16.16. The trends presented for the two overlap lengths are clearly different and it is also evident that the mode-mixity ratio changes significantly with the crack length, i.e., during the fatigue life of the joint. This means that the relative contribution of Mode I and Mode II components to crack propagation during the fatigue life of the joint is continuously changing, according also to the results reported by Wahab et al.43 4 w = 20 mm snom = 60 MPa w = 40 mm snom = 80 MPa

Mode-mixity ratio (GI/GII)

3.5 3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

a/w

16.16 Comparison of the mode-mixity ratios at low stress level.

Although the propagation scenario is dominated by the Mode I component, it could be useful for the development of a more general and more reliable life prediction model to define an equivalent energy parameter suitable to incorporate both the Mode I and Mode II components GI and GII as well as the continuous variation of their ratio with the crack length. The total SERR Gtot, in fact, being simply defined as the sum of the two components does not explicitly account for this variation. As a possible alternative to the total SERR, the following formulation for a new equivalent parameter was introduced:48 G(a) eqv = G I (a) +

G II (a) ◊ G II (a) G I (a) + G II (a)

16.3

Modelling the fatigue behaviour of bonded joints

485

The contribution of the two components of SERR is incorporated and the variability of the mode-mixity is also considered. Moreover, as previously shown in Fig. 16.13, it maintains the mesh-insensitivity typical of the total SERR, at least in the range of element size of practical interest. This new parameter could therefore be used in the definition of the model for crack propagation under fatigue loading.

16.4

Fatigue life modelling

The experimental and numerical results presented in the previous paragraphs are summarised and integrated here for the development of the submodels describing the two phases of the fatigue life of a bonded joint, the nucleation and the propagation, according to the schematic already shown in Fig. 16.1.

16.4.1 Crack nucleation phase As already said in the introduction, the model for the nucleation phase is based on the assumption that the onset of a fatigue crack at the corner of a bonded joint is controlled by the intensity of the local stress field. According to this hypothesis, all the fatigue data to crack initiation were recalculated in terms of the range of generalised SIF H0 and analysed together. It was thus possible to summarise all the data available into two well defined scatter bands, one for the square edge joints and another for the spew fillet joint. The data could not be compared directly in one single scatter band due to the different strength of the stress singularity induced by the different corner geometry (see Table 16.5). Figure 16.17 shows the scatter band at 10–90% of probability of survival obtained for the spew fillet joints. It can be seen that, despite the different overlap length, the data can be summarised together with a reasonably reduced variation. The life to crack initiation of a joint can therefore be predicted on the basis of applied stress and corner geometry only.

16.4.2 Crack propagation phase The description and modelling of the crack propagation phase is based, conventionally, on the correlation between an energetic parameter associated to the propagating crack and the rate of crack growth, with the definition of a Paris-like curve. Different proposals are available in the literature for the energetic parameter to be used. Curley et al.23 suggested the use of the total SERR, which has the advantage of being mesh-insensitive and it was proved to be suitable for tough adhesives.10 In other cases the mode I14, 43 or an equivalent formulation37 of the SERR were considered.

486

Multi-scale modelling of composite material systems R DH0.50% 0.05 11.82

40

k 5.86

DH0 (MPa mm0.1987)

30

10% P.S.

20

50% P.S. 90% P.S. 10 Overlap length 40 mm Overlap length 30 mm Overlap length 20 mm 1E3

1E4

1E5 Cycles to crack initiation

1E6

5E6

16.17 10–90% scatter band to crack initiation for fillet joints.

As said above, crack propagation occurs in general under mixed-mode conditions also with a variable mixity-ratio during the fatigue life. Neither the total strain energy release rate nor its Mode I component can account for this situation and therefore an equivalent SERR could be the most effective energy parameter. Xu et al.14 discussed the influence of the mode-mixity on fatigue crack propagation in steel bonded joints and concluded that the use of a Geqv instead of GI does not improve significantly the correlation with the experimental data, in the cases when the Mode I component is predominant. However, the Mode I calculation is strongly mesh-sensitive. To avoid this problem but to consider the mode-mixity and its variability, it was decided to correlate the experimental rates of crack growth alternatively to the total SERR Gtot or to the new equivalent parameter Geqv, and to compare the accuracy of the choice in terms of predicted lives. From the experimental point of view, the influence of the mode-mixity on crack propagation was implicitly considered, since the crack growth rates were measured on the joints under mixed-mode loading conditions. The data were described by using the classical Paris-like, power law formulation:

da = D[ DG( a )] n dN

16.4

Figure 16.18 shows the 10–90% scatter band obtained for all the data available, expressed in terms of equivalent SERR range, which refers to joints with different overlap length, corner geometry and applied stress level. In spite of the different conditions, a very reduced variation of the data is obtained. The characteristic parameters of the Paris curves for the different cases are

Modelling the fatigue behaviour of bonded joints

487

1.E-01

da/dN (mm/cycle)

1.E-02

50%

1.E-03

1.E-04

1.E-05 90% 1.E-06

– w = 20, 30 40 mm – Low and high fatigue stress level

10% 1.E-07 10

100

1000

DGeqv (J/m2)

16.18 Scatter band of the equivalent SERR range vs. CGR data.

summarised in Table 16.6. It is worth noting that the scatter bands calculated for the two alternative combinations of SERR and CGR are quite similar, indicating their substantial equivalence, at least from an engineering point of view. The number of cycles spent in the propagation phase can then be calculated by the integration of eqn 16.4: Np =

Ú

NP

dN =

0

Ú

af

a0

da D[ D G ( a )] n

16.5

where DG(a) can be either DGtot(a) or DGeqv(a). Table 16.6 Paris curve data SERR parameter

50% probability

90% probability

DGtot (J/m2)

D = 1.354◊10–11 n = 2.729

D = 5.235◊10–11 n = 2.729

DGeqv (J/m2)

D = 2.421◊10–11 n = 2.723

D = 9.658–11 n = 2.723

The lower limit of integration ao, which can be assumed as the length of the crack existing at the beginning of the propagation, depends on the assumptions made during the experimental analysis and is fixed at 0.3 mm, as discussed above. The definition of crack length at failure af, can be made by using a static fracture criterion: it is assumed that fatigue failure occurs

488

Multi-scale modelling of composite material systems

when the total SERR Gtot or its equivalent value Geqv reaches the Mode I adhesive fracture toughness GIC. The comparison between the experimental values of the cracks at failure measured on the joints and the predictions made by using, alternatively, one of the two energy parameters, proved their accuracy and substantial equivalence.47 Moreover, apart from a very few cases, the predictions were on the conservative side.

16.4.3 Procedure for fatigue life estimation and validation of the model The procedure for estimation of the fatigue lifetime of bonded joints is presented in Fig. 16.19. The estimation is made on the basis of geometry of joint, elastic properties of the adherends, elastic and toughness properties of adhesive and applied stress level. The life to crack initiation can be simply obtained from the reference scatter band (DH0–Ni) after the evaluation of the generalised SIF H0 associated to the nominal stress of interest. The SIF is provided by the linear elastic FE analysis on the joint and therefore only one analysis is required for each geometry, independently of Input data: material properties, geometry of the joint, applied stress level

H0 from FE analysis

GI (a) and GII (a) from FE analysis calculation of DGTOT (a) and DGeqv (a)

DH0 – Ni scatter band

af from GTOT (a) or Geqv (a) = GIC

Integration of Paris curve

Number of cycles to crack onset, Ni

Number of cycles for propagation, Np

Cycles to failure, Nf = Ni + Np

16.19 Procedure for the assessment of the joint lifetime.

Modelling the fatigue behaviour of bonded joints

489

the applied load. However, it is important to remember that a new reference scatter band is needed when material properties or corner geometry of the joint change, due to the possible variations in the singularity of the local stress field. The assessment of the life spent in propagating the crack up to joint failure is made by the integration of the Paris curve describing the crack grow rate data. The components of the SERR as a function of crack length are provided again by FE analyses and VCCT. In this case, due to inclusion of the geometric non-linearity, one set of analyses for each level of applied stress is required. After the application of the static criterion for the definition of crack length at failure, the Paris curve can be integrated, thus obtaining the cycles spent in the propagation phase. The fatigue life of the joint is then calculated as the sum of the life to crack initiation Ni and life for crack propagation Np. For producing more safe and reliable predictions, instead of using the average curves, the 90% bound of the relevant scatter band can be used for the estimation of both the initiation and the propagation fatigue life. For a preliminary evaluation of the model accuracy, the fatigue lives of the joints tested under constant amplitude loading were compared with the life predictions calculated by using the procedure just described. The global analysis of the results49 indicated a slight tendency to overestimate the nucleation phase only for the joint with shorter overlaps, as one could also argue from Fig. 16.17; the predictions, however, were usually inside the 10– 90% scatter bands of the experimental data. The propagation phase was always predicted on the conservative side, accurately and independently of the overlap. The choice of the total strain energy release rate instead of its equivalent value turned out to have only a negligible influence on the estimated lives, the more conservative predictions being, however, provided by the total SERR. As representative examples of the predictions provided by the model, the best and the worst results obtained during the validation are presented in Figs 16.20 and 16.21. In the worst case, the square edge joints with 20 mm overlap length, the average life estimated by the model is slightly nonconservative, even though inside the 10–90% scatter band of the experimental results. On the other hand, the life predicted by the model at 90% of probability of survival remains on the conservative side. For the spew fillet joints with 40 mm overlap length, the prediction is very accurate and again conservative, with the estimated average life well inside the experimental scatter band. Intermediate results were obtained for the other series of data.49

16.5

Conclusions

The results of extensive experimental and numerical investigations aimed at studying and characterising the behaviour of bonded joints in composite

490

Multi-scale modelling of composite material systems 300 Experimental data Estimated life (50% P.S.) Estimated life (90% P.S.)

smax (MPa)

200

100

Overlap length: 20 mm

1E3

1E4

1E5 Cycles to failure

1E6

1E7

16.20 Comparison between 10–90% P.S. experimental scatter band for square edge joints (overlap 20 mm) and model predictions at 50% and 90% probability of survival (based on DGeqv). 300 Experimental data Estimated life (50% P.S.) Estimated life (90% P.S.)

smax (MPa)

200

100

Overlap length: 40 mm 1E3

1E4

1E5 Cycles to failure

1E6

1E7

16.21 Comparison between 10–90% P.S. experimental scatter band for spew fillet joints (overlap 40 mm) and model predictions at 50% and 90% probability of survival (based on DGeqv).

materials was presented and discussed. These results and in particular the accurate analysis of fatigue damage evolution served as a basis for the development of a physically based model for fatigue life prediction which describes the life of the joint as the sequence of a nucleation phase, up to the onset of cracking, followed by a propagation phase up to final failure. The

Modelling the fatigue behaviour of bonded joints

491

description and modelling of the nucleation phase is usually neglected in the models available in the literature. Therefore, the application of the model proposed here can improve the accuracy and the reliability of fatigue life prediction of bonded joints and, as a result, the overall efficiency of the composite structures using these joints. The model can also represent the first step towards a methodology entirely based on a single energetic parameter for the description of the nucleation and the propagation phase.

16.6

Acknowledgements

The results and the activities presented in this chapter were developed as part of the research program ‘Fatigue design methodologies of joints in composite material’ (2001092449) financially supported by the Italian Ministry of Research and University. The author is grateful to Dr Mauro Ricotta for his important contribution in all phases of the research and to Mr Luca Bortolussi and Mr Luca Martin for their help in the experiments.

16.7

References

1. Goland M. and Reissner E., ‘The stresses in cemented joints’, J. Appl. Mechanics, 1944, 11, A17–A27. 2. Bogy D.B., ‘Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading’, J. Appl. Mechanics, 1968, 35, 460–466. 3. Renton W.J. and Vinson J.R., ‘Fatigue behavior of bonded joints in composite material structures’, J. Aircraft., 1974, 12, 442–447. 4. Gradin P.A., ‘A fracture criterion for edge-bonded bi-material bodies’, J. Compos. Mater, 1982, 16, 448–456. 5. Johnson W.S. and Mall S., ‘A fracture mechanics approach for designing adhesively bonded joints’ in ASTM STP 876, 189–199, American Society for Testing and Materials, Philadelphia, 1985. 6. Matthews F.L. and Tester T.T., ‘The influence of the stacking sequence on the strength of bonded CFRP single lap joints’, Int. J. Adhes Adhes, 1985, 5, 13–18. 7. Adams R.D. and Harris J.A., ‘The influence of local geometry on the strength of adhesive joints’, Int. J. Adhes Adhes, 1987, 7, 69–80. 8. Groth H.L., ‘Stress singularities and fracture at interface corners in bonded joints’, Int. J. Adhes Adhes, 1988, 8, 107–113. 9. Hattory T., Sakata S., Hatsuda T. and Murakami G., ‘A stress singularity parameter approach for evaluating adhesives strength’, JSME Int. J., Series I, 1988, 31, 718– 723. 10. Mall S. and Kochhar N.K., ‘Characterization of debond growth mechanism in adhesively bonded composites under mode II static and fatigue loadings’, Eng. Fract. Mech., 1988, 31, 747–758. 11. Mall S. and Ramamurthy G., ‘Effect of bond thickness on fracture and fatigue strength of adhesively bonded composite joints’, Int. J. Adhes Adhes, 1989, 9, 33– 37. 12. Kinloch A.J. and Osiyemi S.O., ‘Predicting the fatigue life of adhesively-bonded joints,’ J. Adhesion, 1993, 43, 79–90.

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13. Gilmore R.B. and Shaw S.J., ‘The effect of temperature and humidity on the fatigue behaviour of composite bonded joints’, in ASTM STP 1227 – Composite Bonding, 82-95, American Society for Testing and Materials, Philadelphia, 1994. 14. Xu X.X., Crocombe A.D. and Smith P.A., ‘Mixed-mode fatigue and fracture behaviour of joints bonded with either filled or filled and toughened adhesive’, Int. J. Fatigue, 1995, 17, 279–286. 15. Pradhan S.C., Iyengar N.G.R and Kishore N.N., ‘Finite element analysis of crack growth in adhesively bonded joints’, Int. J. Adhes Adhes, 1995, 15, 33–41. 16. Zhang Z., Shang J.K. and Lawrence F.V., ‘Backface strain technique for detecting fatigue crack initiation in adhesive joints’, J. Adhesion 1995, 49, 23–36. 17. Roy A., Mabru C., Gacougnolle J.L. and Davies P., ‘Damage mechanisms in composite/ composite bonded joints under static tensile loading’, Appl. Compos Mater, 1997, 4, 95–119. 18. Dessureault M. and Spelt J.K., ‘Observations of fatigue crack initiation and propagation in an epoxy adhesive’, Int. J. Adhes Adhes, 1997, 17, 183–195. 19. Crocombe A.D. and Richardson G., ‘Assessing stress state and mean load effects on the fatigue response of adhesively bonded joints’, Int. J. Adhes Adhes 1999, 19, 19– 27. 20. Ishii K., Imanaka M., Nakayama H. and Kodama H., ‘Evaluation of the fatigue strength of adhesively bonded CFRP/metal single and single-step double-lap joints’, Compos Sci. Technol., 1999, 59, 1675–1683. 21. Lefebvre D.R. and Dillard D.A., ‘A stress singularity approach for the prediction of fatigue crack initiation in adhesive bonds. Part 1: theory’, J. Adhesion, 1999, 70, 119–138. 22. Lefebvre D.R. and Dillard D.A., ‘A stress singularity approach for the prediction of fatigue crack initiation in adhesive bonds. Part 2: Experimental’, J. Adhesion, 1999, 70, 139–154. 23. Curley A.J., Hadavinia H., Kinloch A.J. and Taylor A.C., ‘Predicting the service-life of adhesively-bonded joints’, Int. J. Fracture, 2000, 103, 41–69. 24. Knox E.M., Bowling M.J. and Hashim S.A., ‘Fatigue performance of adhesively bonded connections in GRE pipes’, Int. J. Fatigue, 2000, 22, 513–519. 25. Briskham P. and Smith G., ‘Cyclic stress durability testing of lap shear joints exposed to hot-wet conditions’, Int. J. Adhes Adhes, 2000, 20, 33–38. 26. Ashcroft I.A., Abdel Wahab M.M., Crocombe A.D., Hughes D.J. and Shaw S.J., ‘The effect of environment on the fatigue of bonded composite joints. Part 1: testing and fractography’, Compos Part A-Appl S, 2001, 32, 45–58. 27. Abdel Wahab M.M., Ashcroft I.A., Crocombe A.D., Hughes D.J. and Shaw S.J., ‘The effect of environment on the fatigue of bonded composite joints. Part 2: fatigue threshold prediction’, Compos Part A – Appl S, 2001, 32, 59–69. 28. Abdel Wahab M.M., Ashcroft I.A., Crocombe A.D. and Shaw S.J., ‘Prediction of fatigue thresholds in adhesively bonded joints using damage mechanics and fracture mechanics’, J. Adhesion Sci. Technol, 2001, 15, 763–781. 29. Potter K.D., Guild F.J., Harvey H.J., Wisnom M.R. and Adams R.D., ‘Understanding and control of adhesive crack propagation in bonded joints between carbon fibre composite adherends I. Experimental’, Int. J. Adhes Adhes, 2001, 21, 435–443. 30. Guild F.J., Potter K.D., Heinrich J., Adams R.D. and Wisnom M.R., ‘Understanding and control of adhesive crack propagation in bonded joints between carbon fibre composite adherends II. Finite element analysis’, Int. J. Adhes Adhes, 2001, 21, 445–453.

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31. Kayupov M. and Dzenis Y.A., ‘Stress concentrations caused by bond cracks in single-lap adhesive composite joints’, Compos Struct., 2001, 54, 215–220. 32. Ashcroft I.A., Hughes D.J. and Shaw S.J., ‘Mode I fracture of epoxy bonded composite joints 1. Quasi-static loading’, Int. J. Adhes Adhes, 2001, 21, 87–99. 33. Ashcroft I.A. and Shaw S.J., ‘Mode I fracture of epoxy bonded composite joints 2. Fatigue loading’, Int. J. Adhes Adhes, 2002, 22, 151–167. 34. Lazzarin P., Quaresimin M. and Ferro P., ‘A two-term stress function approach to evaluate stress distributions in bonded joints of different geometries’, J. Strain Anal. Eng. Des., 2002, 37, 385–398. 35. Mortensen F. and Thomsen O.T., ‘Analysis of adhesive bonded joints: a unified approach’, Compos Sci. Technol., 2002, 62, 1011–1031. 36. Abdel Wahab M.M., Ashcroft I.A., Crocombe A.D. and Smith P.A., ‘Numerical prediction of fatigue crack propagation lifetime in adhesively bonded structures’, Int. J. Fatigue 2002, 24, 705–709. 37. Cheuk P.T., Tong L., Wang C.H., Baker A. and Chalkley P., ‘Fatigue crack growth in adhesively bonded composite-metal double-lap joints’, Compos Struct., 2002, 57, 109–115. 38. Cheuk P.T. and Tong L., ‘Failure of adhesive bonded composite lap shear joints with embedded precrack’, Compos Sci. Technol, 2002, 62, 1079–1095. 39. Barroso A., Manti
V. and Paris F., ‘General solution for anisotropic multimaterial corners’, Int. J. Fracture, 2003, 119, 1–23. 40. Hadavinia H., Kinloch A.J., Little M.S.G., and Taylor A.C., ‘The prediction of crack growth in bonded joints under cyclic-fatigue loading I. Experimental studies’, Int. J. Adhes Adhes, 2003, 23, 449–461. 41. Hadavinia H., Kinloch A.J., Little M.S.G. and Taylor A.C., ‘The prediction of crack growth in bonded joints under cyclic-fatigue loading II. Analytical and finite element studies’, Int. J. Adhes Adhes, 2003, 23, 463–471. 42. Ferreira J.A.M., Reis P.N., Costa J.D.M. and Richardson M.O.W., ‘Fatigue behaviour of composite adhesive lap joints’, Compos Sci. Technol., 2002, 62, 1373–1379. 43. Abdel Wahab M.M., Ashcroft I.A., Crocombe A.D. and Smith P.A., ‘Finite element prediction of fatigue crack propagation lifetime in composite bonded joints’, Compos Part A-Appl. S, 2004, 35, 213–222. 44. Adams R.D., Comyn A. and Wake W.C., Structural Adhesive Joints in Engineering, 2nd edn, London, Chapman and Hall, 1997. 45. Tong L. and Soutis C., Recent Advances in Structural Joints and Repairs for Composite Materials, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. 46. De Goeij W.C., van Tooren M.J.L. and Beukers A., ‘Composite adhesive joints under cyclic loading’, Mater Design, 1999, 20, 213–221. 47. Quaresimin M. and Ricotta M., ‘Fatigue behaviour and damage evolution of single lap bonded joints in composite material’, Compos Sci. Technol., in press. 48. Quaresimin M. and Ricotta M., ‘Stress Intensity Factors and Strain Energy Release Rates in single lap bonded joints in composite materials’, Compos Sci. Technol., accepted for publication. 49. Quaresimin M. and Ricotta M., ‘Life prediction of bonded joints in composite materials’, Int. J. Fatigue, submitted. 50. ASTM E 647-00, Standard test method for measurement of fatigue crack growth rates, American Society for Testing and Materials, 2000. 51. Rybicki E.F. and Kanninem M.F., ‘A finite element calculation of stress intensity factors by a modified crack closure integral’, Eng. Fract. Mech., 1977, 9, 931–938.

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52. Raju I.S., ‘Calculation of strain energy release rates with higher order and singular finite elements’, Eng. Fract. Mech., 1987, 28, 251–274. 53. Krueger R., Minguet P.J. and O’Brien T.K., ‘Implementation of interlaminar fracture mechanics in design: an overview’, 14th International Conference on Composite Materials, San Diego, Society of Manufacturing Engineers, 2003. 54. Sun C.T. and Jih C.J., ‘On strain energy release rates for interfacial cracks in bimaterial media’, Eng. Fract. Mech., 1987, 28, 13–20. 55. Williams M.L., ‘The stresses around a fault or a crack in dissimilar material’ Bull. Seism Soc. Am., 1959, 49, 199–204. 56. Quaresimin M., ‘Fatigue of woven composite laminates under tensile and compressive loading’ 10th European Conference on Composite Materials, Brugge, ESCM, 2002. 57. 3M – Aerospace Technical Data Sheet Nb13 (1998)-Scotch-Weld 9323 B/A.

Index

abrasion continuous fibre reinforced composites 379–82 particulate composites 376–7 see also wear modelling absorbed energy 78, 80 acoustic emission (AE) sensors 439 activation energy 127, 180, 181 active sensor network 433, 437–42, 457 models 445–51 PZT actuator 445–50 PZT sensor 450–1 sensor network 440, 441 sensor technology 437–40 standard sensor unit 433, 440–2, 443 aircraft 259, 401, 402, 403, 426 gelatine impact on leading edge shells 423–5 see also impact damage angle-ply laminates 2 matrix cracking 221, 234, 250 anti-parallel (AP) fibre orientation 389, 390, 391, 392, 395–6, 397 anti-symmetric Lamb wave mode 435–6, 464 approximate models 279, 280 applicability 281–3 aramid fibre/carbon fibre hybrid 381–2 Arrhenius equation 180, 181 artificial neural network (ANN) approach 385– 8, 398 AS4/3506–1 laminate 268, 269–70 atomic force microscopy (AFM) 31 atomic scale 33–4 axial balance of forces 39 axial modulus 198, 202 stiffness degradation due to intra- and interlaminar damage 233–40, 241 axial shear modulus 68, 202 axisymmetric models 45 backpropagation 386 bending 394, 395

bending resistance 330–4 b shear-lag parameter 40–1, 42, 52–4, 61 bird impact 421 bismaleimide 1 Boeing 7E7 aircraft 259 bonded joints see joints boundary conditions critical loads for laminates 286–7 elastic layers 296, 297, 298 elastic-plastic layers 298–9, 300 guided waves 434–5 shear transfer in interfacial damage modelling 38–9 brick elements 359 bridging angle 175–6 bridging cracks see crack bridging bridging laws 321–34 Cox’s fibrous tow model 328–30 Jain and Mai 322–5 Liu et al’s z-pin law 325–7 Tong and Sun’s 330–4 bridging length 172, 174–5 bridging tractions 171, 174–6 brittle matrix composites 356–73 finite element method 357–9 microstructure-based modelling 359–65 applications and examples 365–9 standard FEM analysis for fibre-composite materials 359 brittleness 169 Budiansky-Fleck and Soutis (BFS) analytical model 304–6 ‘building block’ approach 127 bulk modulus 21, 22 bundle pull-out test 49, 50 carbon fibre/epoxy plates 415–17 carbon fibre hybrid composites 380–2 chain stitch 319, 320 characteristic damage state 196, 218 chemical bonding 34 chemical reactions 127

495

496

Index

cleavage-type delaminations 285–7 closed-form expression for uniform local delamination 243–4, 250 closure surface traction 343–4 coefficient of determination 386 cohesive energy 4, 5–8 column spacing 350, 351–2 commercial epoxy resin 12–20 competing mechanisms 130–1, 132–3 complex modulus 100 compliance 148 delamination cracking and compliance change 148–51 compliance matrix 404 compression loading 100–4 compressive response behaviour laminate composites 278–302 applicability of approximate models 281–3 developing compression models 283–7 identifying critical loads 287–99, 300 modelling 278–83 results for elastic layers 295–7, 298 results for elastic-plastic layers 297–9, 300 monolithic and sandwich composite structures 303–18 damage characterisation 312–13 modelling techniques 304–9 predicting compressive response 309– 16 strength data and predictions 314–16 concentric cylinder models 67–8 characteristics 69 configurational energy 4, 5, 10 conjugate forces 404–5 constant shear model 39–40, 42 constitutive models 129, 132–8 contact mechanics 383–5 contact pressure distributions 390–1, 392 continuous fibre composites abrasive and sliding wear 379–82 strength of 21–6 continuous wavelet transform (CWT) 444, 462–4 continuum damage mechanics (CDM) 217 cracking in cross-ply laminates 197, 205– 6, 213 impact damage 402 delamination modelling 403, 411–15, 425 meso-scale ply damage models 402–11, 425 continuum theory/modelling 279, 280 choosing between physical modelling and 128–30 contour integral method 348–9 corner geometries see spew fillet joints; square edge joints

coupled mechanisms 130–1, 132–3, 138–59 coupled failure processes and bridging cracks 178–9 delamination cracking, compliance change and strain energy release rate 148– 51, 152 direct observation of matrix and delamination cracking by in-situ SEM 138–41, 142 estimating the modulus 142–4 failure strain criterion 157–9 finite element model of residual strength 153–7 interlaminar toughness and fracture strength 151–3 mapping stiffness change 142–7 physical model 141–2 Cox’s fibrous tow model 328–30 crack bridging 169–79 model of fibre bridging 172–6 toughness mapping 176–9 crack closing displacement (CCD) 306–7, 308 critical 307–8, 316 crack closure force 175 crack density 218 predicting onset and growth of intra- and interlaminar damage 244, 245–6 predicting stiffness degradation 234–9 crack growth rates 477–8 crack growth resistance 336–8, 346, 347 crack-induced delamination 218, 221–6, 249– 50 stiffness degradation 236–40, 241 strain energy release rate 240–6 stress analysis 226–31 crack nucleation 470–1, 473–8, 485, 486, 489 crack opening displacement (COD) 348, 349 crack propagation bonded joints 470, 471, 473–8, 485–8, 489 brittle matrix composites 363–4, 367, 368 crack propagation rate 181, 188–9 and time to failure 182–8 crack resistance 348–50 crack spacing 148–51 crack tip damage zone 153–4, 155, 156–7 cracking models 124–95 bridging cracks 169–79 choosing between continuum and physical modelling 128–30 combining empirical and physical models 130–8 cracking at stress concentrators 159–69 cross-ply laminates 196–216 damage accumulation 205–6 microstructural randomness 197–205 multi-scale modelling 206–12 empirical and physical modelling 124–8 model implementation 189–90 modelling coupled mechanisms 138–59

Index predictive modelling of cracking see predictive modelling of cracking stress-corrosion cracking 179–89 cracking onset strain 248–9 CRAHVI project 426 critical angle at fast fracture 185 critical crack closing displacement 307–8, 316 critical damage state 136–8 critical elastic strain energy release rate see delamination toughness critical embedded length 323 critical failure strain 136–7 critical length 40 critical loads bounds for 286–7 identifying 287–99 critical shear strain 267, 269 critical shortening strain 287 critical strain 282–3, 296, 297, 298–9, 300 critical threshold stress 136, 187 crosslinked epoxy resin 10–20 cross-ply laminates 189–90, 196–216 damage accumulation 205–6 damage evolution 159–61 future trends 213 microstructural randomness 197–205 analysis of microstructure 198–202 material’s randomness 197–8 multifractal characterisation 202–4 randomness of matrix cracking 204–5 multi-scale modelling 206–12 implementation of the algorithm 211– 12 RVE and lattice schemes 206–8 stress redistribution mechanisms 208– 11 cumulative loss tangent 21, 22 curing agents/hardeners 1 cyanate esters 1 cyclic loading 218 cracking models 130–8, 190 damage accumulation 205–6 damage accumulation function 135 damage evolution equations 405–9, 410 damage evolution law 205 damage growth rate 136, 137, 161–3 damage maps 110–11, 115, 116 damage mechanics 133 damage micromechanics approach 217 damage modelling 217–58 interlaminar damage 221–6 intralaminar damage 217–21, 222 predicting onset and growth of damage 240–9 off-axis ply cracking 246–9 strain energy release rate 240–6 predicting stiffness degradation 232–40 stress analysis 226–31 damage parameters 205

497

macroscopic damage parameter for ply cracking in laminates 76, 81 ply damage models 404–5, 406, 408 determination 409–11 damage process zone model 159–69 damage evolution in cross-ply laminates 159–61 extension to a fatigue damage model 167– 9 Griffith-type energy balance 161–4 tensile stress failure criterion 164–5 weakest link 165–7 damage strength see residual (damage) strength damage tensor 217 debonding 23, 92 bridging cracks 169–79 modelling strength of fibre-reinforced composites 112–14, 116–19 wear modelling 392, 394, 395, 396 debris layers 377–8, 392 Debye reference temperature 6, 8 defects with connected edges 285–7 degree of cure 12, 13 effect on glass transition temperature 16– 18 degrees of freedom 4, 6–8 commercial epoxy resin 12, 14 finite element modelling 357–8 delamination 196–7, 319 coupled mechanisms 139–59 compliance change and strain energy release rate 148–51, 152 direct observation by in-situ SEM 138– 41, 142 maximum compliance change with delamination 149–51 cracking models 130–1, 132, 190 cracking at stress concentrators 159–69 modelling fatigue cracking by 134–5 modulus reduction with delamination cracking 135–6 impact damage modelling 403, 411–15, 425 fracture mechanics ply interface model 411–13 implementation in FE codes 413–15 Lamb wave scattering in composite laminates 451–64 model of delamination 452–4 Liu et al’s z-pin bridging law 325–7 matrix crack-induced see crack-induced delamination Tong and Sun’s metallic z-rod model with bending resistance 330–4 delamination area 242, 244 relative delamination area 237–8, 240, 241, 245–6 delamination-induced shear wave 461, 462, 463

498

Index

delamination toughness (interlaminar toughness) and fracture strength 151–3 stitched and z-pinned reinforced composites 334–50, 351–2 density 455 design of composite materials 30–1 differences of scale and 125–6, 127 dental fillings 376–7 diamond indentor 396–7 diamond unit cell 261–2, 264, 275–6 differential equation of the deflection curve 339–42 dilatational failure 260, 261, 263 dimensionless analysis 350 discrete wavelet transform (DWT) 444–5, 459–62 discretisation 357–8 partial 104–10 dispersion curves for Lamb waves 464 displacement relative displacement 46, 61 transverse bending resistance-displacement relationship for a metallic z-rod 330–4 wear modelling 388–9, 391–6 distortional failure 260, 261, 262, 263, 269 distributed computer systems 95 double-cantilever beam (DCB) specimens 334–42 stitched 334–8 finite element analysis 345–6, 347 z-pinned 338–42 finite element analysis 346–50 drop tower impact tests 415–17 dual actuator activation 449–50 ductility 120 dynamic mechanical thermal analysis (DMTA) 9, 100 E-Glass/LY556 laminate 268, 269, 270–3 E-Glass/MY750 laminate 268, 269, 273 edge delaminations 221–3 edge effects 74 effective laminate strains 74–6 effective laminate stresses 74–6 eigen-value problem 284–5 elastic domain function 407, 408 elastic layers 295–7, 298 elastic modulus see modulus/stiffness elastic-plastic hydrodynamic solid 420 elastic-plastic layers 297–9, 300 elastic-plastic strain transfer 24 elastic strain energy release rate see strain energy release rate elastic wave-based techniques 430–68 active sensor network 437–42 models 445–51 Lamb wave scattering in defective

composite laminates 451–64 delaminated composite laminates 452–4 dispersion curve 464 dynamic simulation 456 elastic properties 454–5 experimental validation 456–8 Lamb waves in composite structures 433– 7, 438–9 modelling technique for damage identification 430–3 signal processing and identification 442–5, 458–64 electromagnetic acoustic transducer (EMAT) 438 electrostatic attractions 34 element size 479, 480–1 empirical modelling 383 approach 126–7 choosing between continuum and physical modelling 128–30 combining with physical modelling 130–8 physical modelling and 124–8 end-notched flexure (ENF) specimens stitched 342–5 z-pinned 346–50, 351–2 energy balance 363 ply crack formation 77–81 application to layered systems 82–3 energy components 4 energy methods for interfacial damage 46–8 energy release rate 346–50, 351–2, 385 see also strain energy release rate epoxy modified by hygrothermally decomposed polyurethane (EPPUR) 386–8 epoxy resins 1 GIM and prediction of glass transition temperature for a commercial epoxy resin 12–20 equation of state (EOS) 419, 420–1 equi-biaxial compression 296, 297, 299 Equivalent Constraint Model (ECM) 220–1 equivalent crack 306–7 equivalent laminate 240–9 equivalent SERR parameter 484, 486–8 erosive wear rate 386–8 see also wear modelling EU CRAHVI project 426 EU HICAS project 403, 414, 417–18 exact approaches 279–81 extended empiricism, method of 133 fabric ply damage model 405–7 failure crack length at failure 487–8 prediction of laminate failure 84–94 time to failure 182–9 failure criteria 205–6, 261 failure strain criterion 157–9 fibre failure criteria 91, 94

Index matrix failure criteria 266–7 prediction of laminate failure 84–6 stress-invariant 261–3 tensile stress failure criterion 164–5 failure maps 189 failure mechanisms 383–5 Failure Prediction Exercise 85, 86, 94 failure strain criterion 157–9 failure theories 259–77 multi-scale failure theory 261–5 phase degradation approach 266–7 review of 260–1 validation of analysis against experiment 267–73 fatigue 196–7 fatigue damage model 167–9, 190 fatigue life 134–5 joints 470–1, 485–8 crack nucleation 475–7, 485, 486 crack propagation 475–7, 485–8 fatigue life modelling 485–9 procedure for fatigue life estimation and validation of the model 488–9, 490 terminal damage state 136–8 fatigue strength 473, 474, 475 fibre failure microbuckling see microbuckling phase degradation approach 266–7 statistical model of 86–8 fibre failure criteria 91, 94 fibre fractures 21–3, 35–6, 218 cracking models 130–1, 134 coupled mechanisms 139, 141, 142 cracking at stress concentrators 159–69 modelling strength of fibre-reinforced composites 112, 114–17 wear of short fibre reinforced thermoplastics 377–9 fibre misalignment 314–15, 316 fibre-optic sensors 439 fibre properties 268, 269 fibre rupture-induced debonding 114–17 fibre sliding wear 377–9 fibre spacing 198–9 fibre strength 117–20 fibre volume fraction critical strain and 282–3 microstructural randomness of cross-ply laminates 198, 199–201 modelling first ply failure 104, 105–10 fibrous tow model 328–30 finite element method (FEM) 3, 129 bonded joints 478–85 stress distributions, SIF and SERR 481–5 brittle matrix composites 356–73 particulate composites 365–8 porous materials 368–9 composite wear mechanisms 388–97, 398

499

contact pressure distributions 390–1, 392 displacement and stress results 391–6 experimental verification 396–7 FE macro- and micro-contact models 390, 391 cracking in cross-ply laminates 197, 206– 8, 211, 213 elastic wave-based techniques 430–1 experimental validation 456–64 Lamb wave scattering 451–6 FE codes impact damage prediction 403, 415–25, 426–7 implementation of ply damage and delamination models in 413–15 fundamentals 357–9 microstructure-based modelling 359–65 finite element modelling of complex microstructures 361–4 other approaches 364–5 RVE 360–1 model of residual strength 153–7 modelling first ply failure using partial discretisation 104–10 standard FEM analysis for fibre-composite materials 359 stitching and z-pinning and delamination toughness 345–50, 351–2 strength of unidirectional fibre composite 23–6 first ply cracking stress 93–4 first ply failure 77 modelling by FEM using partial discretisation 104–10 fluorescence technique 51–2 fractal dimension 202–4 fracture criteria mixed mode criterion 248–9, 413 static fracture criterion 487–8 fracture energy 412–13 associated with microbuckling 308–9 wear rate and 384–5 fracture map 176–9 fracture mechanics 126, 134, 470 interfacial damage modelling 46–8 ply interface model 411–13 fracture morphology influence of fibre strength on composite strength and 117–20 stress-strain response and in UD composite 110–20 fracture strength 151–3 fragmentation test 49, 50 frequency domain analysis 443–4 frequency spectra 459, 460 friction 43 wear modelling 384–5 contact pressure distributions 390–1, 392

500

Index

Friedrich equation 382–3 functionally graded materials (FGMs) 369–70 gas gun impact tests 417–18 gelatine 402, 422 impact on leading edge shells 423–5 generalised plane strain condition 229, 257 Gibbs free energy 78–9 glass fibre 380–1 E-Glass/LY556 laminate 268, 269, 270–3 E-Glass/MY750 laminate 268, 269, 273 glass fibre/epoxy (GF/epoxy) shells 417–18 leading edge shells 423–5 glass transition temperature 3, 9, 100 group interaction modelling 10–20 effect of degree of cure 16–18 effect of moisture absorption 18–20 linear polymers 10, 11 network polymer matrix resin 10–20 prediction for a commercial epoxy resin 12–20 global stiffness matrix 358 Griffith element 363 Griffith-type energy balance 161–4 group contribution approach 6–8 group interaction modelling (GIM) 3–31 application to polymer matrix composites 8–31 design of composite materials 30–1 glass transition temperature of a linear polymer 10 glass transition temperature of a network polymer matrix resin 10– 20 prediction of mechanical properties 20– 30 service temperature of composite material 9 for prediction of polymer properties 3–8 guided elastic waves 433–5 see also elastic wave-based techniques; Lamb waves hackles 103, 180 Halpin-Tsai equations 20–1 hardeners/curing agents 1 hardening function 407, 410–11 hardness 384–5 Hashin criterion 261 Hausdorff (fractal) dimension 202–4 HELP Layer 440 hexagonal unit cell 261–2, 264 strain amplification matrix 276–7 Hexcel 924 epoxy resin system 12–20 HICAS project 403, 414, 417–18 hierarchy of structural scales 126 high-cycle fatigue 130, 134, 138 high-velocity impacts 417–18 hole size effect 166, 315 holes, open see open holes

homogenisation technique 83–4, 93–4, 95 honeycomb sandwich panels 310–16 horizontal shear (SH) mode 434, 436–7, 460–1 Hugoniot constant 421 hybrid composites 380–2 hygro-thermal ageing 178–9 ice impact 402 see also impact damage 402 impact damage 401–29 delamination modelling 403, 411–15, 425 meso-scale ply damage models 402–11, 425 prediction 415–25 rigid body impact 415–18 soft body impact 419–25 impurities 12–16, 17 in-plane microstresses 230–1 in-situ damage effective functions (IDEFs) 232, 242–3 in-situ scanning electron microscopy (SEM) 138–41, 142 indentation test 49, 50 independent stitches 322–3, 337, 338, 345, 346, 347 index of stress transfer efficiency 57–60, 61 ineffective (transfer) length 36, 60, 62 interaction energy 5, 10 interaction potential 4 interconnected stitches 324–5, 337, 338, 345, 346, 347 interdiffusion 34 interdigital transducers 438–9 interface and composite properties 35–7 definition 33–5 prediction of laminate failure strong interfaces 92–3 weak interfaces 88–91 wear modelling and interfacial phenomena 374–6 interface stress-displacement models 412–13 interfacial adhesion, mixed one-dimensional models of 43–5 interfacial damage modelling 33–64 analytical modelling of shear transfer 37– 45 approaches 46–8 experimental measurement of stress field at the interface 48–52 modelling of experimentally measured stress transfer 52–60 modes of interfacial failure 37 interfacial debonding see debonding interfacial normal strength (INS) 109–10 interfacial separation 377–9 interfacial shear modulus 54, 55 interfacial shear strength (ISS) 108–10 interfacial shear stresses 172, 228 interfacial test methods 48–52

Index interlaminar adhesion 285–6 interlaminar damage 221–6 see also damage modelling interlaminar defects 278–302 interlaminar toughness see delamination toughness intermolecular contacts, number of 4 intermolecular interaction potential 4 internal instability see microbuckling internal state variables 404–5 cracking models 129, 132–8 International Failure Prediction Exercise 85, 86, 94 interphase 34–5, 44–5, 60 group interaction modelling 23–6, 27, 30–1 intralaminar damage 217–21, 222 see also damage modelling intralaminar defects 287, 300 inverse rule of mixtures (IROM) 375–6 Jain and Mai’s stitching model 322–5 independent stitches 322–3 interconnected stitches 324–5 joints 469–94 experimental investigation 471–8 analysis of crack nucleation and propagation 473–8 static and fatigue strength of single lap joints 473, 474, 475 fatigue life modelling 485–9 crack nucleation phase 485, 486 crack propagation phase 485–8 fatigue life estimation and validation of the model 488–9, 490 finite element analysis of SIF and SERR 478–85 kernel approximation 420 kink bands 283, 303–6, 313, 316–17 see also microbuckling Lamb waves 430–1, 465 in composite structures 433–7 generation 437, 438–9 scattering in defective CF/EP composite laminates 451–64 see also elastic wave-based techniques laminate composites 2 compressive response see compressive response behaviour modelling damage 217–58 interlaminar damage 221–6 intralaminar damage 217–21, 222 predicting onset and growth of intraand interlaminar damage 240–9 predicting stiffness degradation 232–40 stress analysis 226–31 predictive modelling of cracking see predictive modelling of cracking progressive multi-scale modelling 259–77

501

multi-scale failure theory 261–5 phase degradation approach 266–7 review of failure theories 260–1 validation of analysis against experiment 267–73 laminate failure 84–94 composite with strong fibre/matrix interfaces 92–3 composite with weak fibre/matrix interfaces 88–91 ply cracking as indicator of laminate strength 93–4 statistical model of fibre failure 86–8 laser interferometer 438 laser Raman spectroscopy 51, 52, 53, 54, 55 lateral ploughing 328–30 lattice schemes 206–8, 209–10, 211–12, 213 layer thickness 82–3 leading edge shells 423–5 length of a mer unit in the chain axis 7–8 linear damage zones (LDZs) 312–13, 315, 316, 317 linear-elastic layers 295–7, 298 linear expansion coefficient 2–3 linear polymers 10, 11 linear rule of mixtures 375 linear softening cohesive zone model 306–17 fracture energy associated with microbuckling 308–9 predicting compressive response 309–16 composite sandwich panels 310–11 damage characterisation 312–13 laminates with a hole 309–10 materials and specimen geometry 311 mechanical testing 312 strength data and predictions 314–16 linked processes see coupled mechanisms Liu et al’s z-pin bridging law 325–7 load cycles, number of 135, 137–8 load-sharing rule 90, 91 loading conditions 100–4 local fibre volume fraction 104, 105–10 lock stitch 319, 320 longitudinal cracks 221–3 longitudinal elastic modulus 220 longitudinal ply finite element model of residual laminate strength 153–7 tensile stress 153–4, 155 localised tensile strength 164, 165–6 Love waves 434, 436–7, 460–1 low-cycle fatigue 130, 138 macroscopic damage parameter (D) 76, 81 macroscopic scale 33–4 matrix 1–3 mechanical and thermal response 100–4 molecular modelling of matrix properties see group interaction modelling properties 268, 269

502

Index

see also epoxy resins matrix bridging 180 matrix cracking 130–1, 132, 134 coupled mechanisms 138–59 direct observation by in-situ SEM 138– 41, 142 mapping stiffness change 144–7 cracking at stress concentrators 159–69 cross-ply laminates 196–216 damage accumulation 205–6 implementation of the algorithm 211–12 randomness 204–5 RVE and lattice schemes 206–8 stress redistribution mechanisms 208– 11 delamination induced by see crack-induced delamination laminate composites 217–23, 249–50 predicting onset and growth 240–9 predicting stiffness degradation 232–6 stress analysis 226–31 matrix failure modes 263 dilatational 260, 261, 263 distortional 260, 261, 262, 263, 269 matrix failure criterion 266–7 matrix rupture-induced debonding 114–17 matrix wear 377–9 maximum energy retention (MER) 260 mechanical energy 5, 10 mechanical properties 454–5 prediction using GIM 20–30 wear in relation to other properties 382–8 see also under individual properties mechanism map 176–9 mechanisms, importance of understanding 128 metallic z-rod model 330–4 microbuckling 260, 261, 262, 269 fracture energy associated with 308–9 monolithic and sandwich composite structures 303–18 modelling techniques 304–9 predicting compressive response 309– 16 predicting fracture of laminated composites 278–302 modes of microbuckling 288–95 micromechanical modelling see physical modelling micromechanical testing methods 48–9, 50 microscopic damage mechanical interaction among microscopic damages 110–11 shear-lag Monte Carlo simulation to correlate macroscopic behaviour with 111–20 microscopic scale 33–4 microstructure-based modelling 359–65, 369– 70 applications and examples 365–9 FEM of complex microstructures 361–4

other approaches 364–5 RVE 360–1 ‘mirror’ region 180 mixed mode bending apparatus 334, 336 mixed mode fracture criterion 248–9, 413 mixed one-dimensional models of interfacial adhesion 43–5 mode-mixity 479, 486 modified lock stitch 319, 320 modulus/stiffness 21, 100, 134–5 coupled mechanisms 139 estimating the modulus 142–4 failure strain criterion 157–8 mapping stiffness change 144–7 effect of moisture absorption 9 fibre volume fraction and 198 GIM approach 21, 22 Lamb wave scattering 454–5 length scale and 202 stiffness degradation 196, 287 crack-induced delamination 236–40, 241 delamination cracking 135–6 off-axis ply cracking 233–6, 237, 238 predicting due to intra- and interlaminar damage 232–40 moisture absorption 9 effect on glass transition temperature 18– 20 molecular weight of a mer unit 6–8 Monte Carlo simulation 92, 95 shear-lag Monte Carlo simulation 111–20 multi-body problem 110–11 multi-continuum theory 261 multifractal analysis 202–4, 212 multiphase materials 374–6 see also wear modelling multiple mechanisms 130–1 multi-scale failure theory 261–74 laminate analysis 263–4 micromechanics solutions 264–5 phase degradation approach 266–7 stress-invariant failure criteria 261–3 validation of analysis against experiment 267–73 MY721 12–20 N-fibre orientation 390–4, 396, 397 nanoindentation 31 network polymer matrix resin 10–20 nondestructive evaluation (NDE) 430 sensor technology 437–40 see also elastic wave-based techniques non-symmetrical Piola-Kirchhoff stress tensor 284 non-uniformity see randomness notch size, strength dependence on 166–7 notch tip stress distribution 161, 165, 189 notched compressive strength 315–16 nucleation phase 470–1, 473–8, 485, 486, 489

Index off-axis ply cracking 221–3, 249–50 predicting onset and growth 246–9 strain energy release rate 240–6 stiffness degradation 233–6, 237, 238 one-dimensional models 39–45 mixed one-dimensional models of interfacial adhesion 43–5 OOF analysis 361–4, 365–9 open holes, laminates with 303–18 modelling open hole compressive strength 305–6 modelling techniques 304–9 predicting compressive response 309–16 orientational analysis 199 out-of-plane shear stresses 228, 256 overlap length 472, 473, 474, 475, 482–3 PAM-CRASH/SHOCK 403, 413, 414, 415, 419, 420, 423, 425 parabolic criterion 101–4, 120 parallel bar model 89–91 parallel fibre (P-fibre) orientation 389, 390, 391, 392, 394–5, 396, 397 Paris laws 162–4, 189, 486–7 partial discretisation 104–10 partial local delaminations 224 particle approximation 420 particulate composite materials 356–73 abrasion 376–7 application of FEM 365–8 microstructure-based modelling 359–65 passive sensing 440 phase degradation approach 266–7 phase interaction 374–6 phase stepping photoelasticity 31, 51 photoelastic testing methods 49–51 physical modelling 190–1 approach 127 choosing between continuum modelling and 128–30 combining with empirical modelling 130–8 empirical modelling and 124–8 piecewise-homogeneous media model 279–80, 281 piezoelectric (PZT) transducers 432–3, 439– 42, 457, 458 actuator model 445–50 dual actuator activation 449–50 single actuator activation 445–9 sensor model 450–1 Piola-Kirchhoff stress tensor 284 plasma polymers 31 plastic strain 407, 408–9, 410–11 plasticity 43 ply cracking as indicator of laminate strength 93–4 prediction in laminates 73–84 application to layered systems 82–3 effective laminate strains and stresses 74–6

503

energy balance 77–81 in multiple orientations 83–4 thermo-elastic constants 76–7 ply damage models 402–11, 425 CDM formulation 404–5 damage evolution equations 405–9 determination of ply damage model parameters 409–11 ply properties, undamaged 69–73 predicting 66–9 ply thickness 82, 151 point centres per molecule, number of 4 Poisson’s ratio 21, 22, 455 predicting stiffness degradation due to intra- and interlaminar damage 233–40, 241 Poisson’s shrinkage 43–4 polyethernitrile (PEN) composites 379 polymer matrix see matrix polyurethane (PUR) 386–8 porous materials 368–9 potential function 4, 5 power fatigue law 136, 137 predictive capability 259–60 predictive design 127 predictive modelling of cracking 65–98 predicting undamaged ply properties 66–9 prediction of laminate failure 84–94 prediction of ply cracking 73–84 undamaged ply properties 69–73 pressure pulses 420–2 pressure sensitivity 263, 269 propagation, crack see crack propagation pull-out test 49, 50 pulled out length of fibre 172, 174 pulverised wear debris 377–9, 392 quartz abrasive 376–7 quasi-static loading 218 R-curve 176 randomness, microstructural 197–205 incorporation into FE modelling 213 material’s randomness 197–8 matrix cracking 204–5 Ratner-Lancaster equation 382, 384–5 Rayleigh-Lamb frequency equation 435 Rayleigh wave 433–4 reduced (damaged) modulus 142–4 failure strain criterion 157–8 relative delamination area 237–8, 240, 241, 245–6 relative displacement 46, 61 relative strength 117–20 relative transversal modulus 207–8 representative volume element (RVE) 360–1 modelling of cracking in cross-ply laminates 206–8 prediction of laminate failure 92–3 see also finite element method

504

Index

residual (damage) strength fatigue-damaged composite 167–9 finite element model 153–7 residual stresses 2 brittle matrix particulate composites 365–7 modelling first ply failure by FEM 105–7 shear-lag Monte Carlo simulation 112–14 influence on stress-strain behaviour 114–17 resins 1 non-linear stress-strain behaviour 26–9 see also epoxy resins; matrix response equations 132 Rhee empirical model 383 rigid body impact 415–18 Rosen model 282–3 Rotem criterion 261 rule of mixtures (ROM) 35, 421 approaches to wear 374–82, 398 abrasion of particulate composites 376– 7 continuous fibre reinforced composites 379–82 short fibre reinforced composites 377–9 coupled mechanisms and estimating the modulus 143–4 phase degradation approach 267 unnotched compressive strength 314–15 S-N curve (stress-cycles to failure) 138 scanning electron microscopy (SEM) 138–41, 142 self-similar cracking 160 sensors see active sensor network service temperature 9 shape functions 358 shear distortion 260, 261, 262, 263, 269 shear extension coupling coefficients 234–6, 238, 240, 241 shear horizontal (Love) wave mode 434, 436– 7, 460–1 shear-lag models 40–2, 218–19 modified shear-lag Monte Carlo simulation 111–20 shear-lag parameters b 40–1, 42, 52–4, 61 stress analysis 228, 256 shear modulus 21, 454–5 stiffness degradation due to intra- and interlaminar damage 223–40, 241 shear stresses 2 interfacial damage modelling 35–6 analytical modelling of shear transfer 37–45 modelling of polymer properties under different loading conditions 100–4 shear yield strength 29–30 shell elements 359 shielding zones 204–5, 209–10, 212 short fibre composites 364–5

wear 377–9 signal processing and identification 442–5, 458–64 single actuator activation 445–9 single crack mechanics 134 single lap bonded joints see joints size, dependence of strength on 154–7 skeletal modes of vibration see degrees of freedom sliding interface 414 sliding wear continous fibre reinforced composites 379– 82 short fibre reinforced composites 377–9 see also wear modelling slip length (slip zone) 322–3, 324 slippage distance 322–3 small-angle neutron scattering (SANS) 364 SMART Layer 440 smooth particle hydrodynamic (SPH) method 402, 419–25, 426 soft body impact 402, 419–25 gelatine impact on leading edge shells 423–5 soft body impactor model 419–22 solubility parameter 5–6 spacing of fibres 198–9 spatial variation 130–1 spew fillet joints 472–8, 480, 482–3, 485, 486, 489, 490 splitting 190 cracking at stress concentrators 159–69 square edge joints 472–8, 480, 482–3, 485, 489, 490 stacked shell laminate model 414 rigid body impact 415–18 soft body impact 423–5 standard sensor units (SSUs) 433, 440–2, 443 static fracture criterion 487–8 static strength, of single lap joints 473 stiffness see modulus/stiffness stitch distribution 345–6, 347 stitching 319–55 delamination toughness of reinforced composites 334–50 finite element analyses 345–6, 347 stitched DCB specimen 334–8 stitched ENF specimen 342–5 micro-scale models 321–34 Cox’s fibrous tow model 328–30 Jain and Mai’s model 322–5 Stoneley waves 434 storage modulus see modulus/stiffness strain amplification factors 264–5 strain amplification matrices 275–7 strain energy release rate 189 bonded joints 478–9, 480–5, 489 crack-induced delamination 225–6, 240–6, 250 cracking at stress concentrators 161, 162–3

Index delamination toughness of reinforced composites 339–40, 344–5 modelling coupled mechanisms 148–51 delamination toughness and fracture strength 151–3 off-axis ply cracking 240–6, 250 predicting onset and growth of off-axis ply cracking 246–9 strain free temperature 2–3, 90 strain gauges 439 strain-invariant failure theory (SIFT) 260, 261 strength 99–123 dependence on notch size 166–7 dependence on size 154–7 finite element modelling of first ply failure using partial discretisation 104–10 fracture strength 151–3 honeycomb sandwich laminates with holes 309–16 strength data and predictions 314–16 influence of fibre strength on composite strength and fracture morphology 117–20 laminate 91 ply cracking as an indicator of 93–4 strong matrix-fibre interfaces 92–3 of longitudinal continuous fibre composite 21–6, 27 mechanical and thermal response of polymer matrix 100–4 properties of fibres and matrix 267–9 static and fatigue strength of joints 473, 474, 475 stress-strain response and fracture morphology in UD composite 110– 20 stress analysis 357 modelling damage in laminate composites 217, 226–31, 256–8 stress concentration factor (SCF) 164–5, 306 stress concentrators 197–8 cracking at 159–69 damage evolution in cross-ply laminates 159–61 extension to a fatigue damage model 167–9 model 161–4 strength dependence on notch size 166– 7 tensile stress failure criterion 164–5 weakest link 165–6 stress-corrosion cracking 179–89 crack propagation rate and time to failure 182–8 failure maps 189 micromechanical model 181–2 verification by experiment 188–9 stress distributions bonded joints 469–70, 481–5

505

finite element modelling of wear mechanisms 391–6 notch-tip 161, 165, 189 stress field at the interface 48–52 stress-free temperature 2–3, 90 stress function 45 stress intensity factor (SIF) bonded joints 470, 471, 478, 479–80, 481– 5, 488–9 modelling stress-corrosion cracking 181–9 stitched reinforced composites 334–7 stress-invariant failure criteria 261–3 stress redistribution 204, 211–12, 363 mechanisms 208–11 stress-strain behaviour energy based ply crack formation 81 and fracture morphology in UD composite 110–20 influence of residual stress on 114–17 non-linear behaviour of matrix resins 26–9 ply damage models 409–10 undamaged symmetric laminates 70–3 stress transfer analytical modelling 37–45 axisymmetric models 45 mixed one-dimensional models of interfacial adhesion 43–5 one-dimensional models 39–42 problem statement 37–9 modelling of experimentally measured stress transfer 52–60 stress transfer property 56–7 strong fibre/matrix interfaces 92–3 structural change 130–1 structural evolution equations 132 symmetric Lamb wave mode 435–6, 460–1 symmetric undamaged laminates 70–3 T300/BSL914C laminate 268, 269, 270, 271 tabbed specimens 334, 335 tan d 9 effect of degree of cure on glass transition temperature 16, 18 effect of moisture absorption on glass transition temperature 20 Tarasov approximation 10 temperature glass transition temperature see glass transition temperature modelling thermal dependent properties of matrix polymers 100 service temperature 9 tensile loading 100–4 tensile modulus see modulus/stiffness tensile stress failure criterion 164–5 terminal damage state 136–8 tetraglydicyldiaminodiphenyl methane (TGDDM) 12–20 thermal energy 4, 5, 10

506

Index

thermal residual stresses see residual stresses thermally activated chemical kinetics 181–2 thermo-elastic constants prediction of ply cracking in laminates 76– 7 prediction for undamaged plies 66–8 thermosetting matrix see matrix three-dimensional microstructure-based models 370 three-dimensional (3–D) stability theory 279– 81 threshold damage value 206 time to failure 182–8 time-frequency/time-scale analysis 443 time-series analysis 443–4 Tong and Sun’s metallic z-rod model 330–4 torsion testing 100–4 total strain energy release rate 479, 484, 485, 486–8 toughening agents 1 toughness, delamination see delamination toughness toughness enhancement 171–2, 176, 177 toughness mapping 176–9 transfer (ineffective) length 36, 60, 62 transverse bending resistance 330–4 transverse modulus 202 predicting stiffness degradation due to intra- and interlaminar damage 233–40, 241 transverse ply cracking see matrix cracking transverse shear modulus 68, 202 tri-linear z-pin bridging law 325–6 Tsai-Wu failure criterion 85, 86 Tsai-Wu theory 261 ultimate laminate failure strain 157–9 ultrasonic probes 438 undamaged general symmetric laminates 70–2 macroscopic effective stress-strain relations 72–3 undamaged ply properties 69–73 predicting 66–9 uniaxial compression 296, 298, 299, 300 unidirectional (UD) composites 2, 356 ply damage model 408–9 strength of continuous fibre composites 21–6, 27 stress-strain response and fracture morphology 110–20 undamaged ply properties 66–73 unit cells 261–2, 264 strain amplification matrices 275–7 see also representative volume element (RVE) United States Air Force F-35 Joint Strike Fighter (JSF) stealth aircraft 259

unnotched compressive strength 304–5, 306, 307–8, 314–15 unsaturated polymer resins 1 van der Waals volume of a mer unit 6–8 variational methods 219–20 virtual crack closure technique (VCCT) 478– 9 volume fraction of fibres see fibre volume fraction von Mises yield criterion 263 Voronoi Cell Finite Element Method (VCFEM) 364 warp tow microbuckling 312, 313 water affinity constant 20 wave-based techniques see elastic wave-based techniques wavelet transform-based signal processing 444–5, 459–64 weak fibre/matrix interfaces 88–91 weakest link failure probability distribution 87 weakest link statistics model 165–7 wear maps 383 wear modelling 374–400 finite element modelling of wear mechanisms 388–97 rule-of-mixtures approaches 374–82 wear in relation to other mechanical properties 382–8 ANN approach 385–8 traditional attempts 383–5 weft tow cracking 313 Weibull distribution function 87–8, 91, 112, 204, 207–8 Weibull element 363–4 Weibull weakest link statistics model 165–7 X-ray diffraction 52 yield 2–3, 26–30 compressive yield strength 30 shear yield strength 29–30 Young’s modulus see modulus/stiffness z-pin columns, number of 349–50, 351–2 z-pinning 319–55 delamination toughness of reinforced composites 334–50 DCB specimen 338–42 finite element analysis 346–50, 351–2 micro-scale models 321–34 metallic z-rod model with bending resistance 330–4 z-pin bridging law 325–7 zones of stress transfer 55–6