Virtual testing and predictive modeling: for fatigue and fracture mechanics allowables

Virtual Testing and Predictive Modeling Bahram Farahmand Editor Virtual Testing and Predictive Modeling For Fatigue ...

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Virtual Testing and Predictive Modeling

Bahram Farahmand Editor

Virtual Testing and Predictive Modeling For Fatigue and Fracture Mechanics Allowables

123

Editor Bahram Farahmand TASS – Americas, a subsidiary of TASS Inc. 12016 115th Ave NE Suite 100 Kirkland, WA 98034 USA [email protected]

ISBN 978-0-387-95923-8 e-ISBN 978-0-387-95924-5 DOI 10.1007/978-0-387-95924-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009921172 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Acknowledgments

The author is grateful to all co-authors who contributed to this book. Their dedication and effort for submitting their chapters on time are greatly appreciated. This book will be dedicated to my dear mother Gohartaj and my lovely wife Vida. My great appreciation goes to my son, Houman, and my daughter, Roxana, for being extremely helpful with their support during putting sections of this book together.

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Preface

The materials used in manufacturing the aerospace, aircraft, automobile, and nuclear parts have inherent flaws that may grow under fluctuating load environments during the operational phase of the structural hardware. The design philosophy, material selection, analysis approach, testing, quality control, inspection, and manufacturing are key elements that can contribute to failure prevention and assure a trouble-free structure. To have a robust structure, it must be designed to withstand the environmental load throughout its service life, even when the structure has pre-existing flaws or when a part of the structure has already failed. If the design philosophy of the structure is based on the fail-safe requirements, or multiple load path design, partial failure of a structural component due to crack propagation is localized and safely contained or arrested. For that reason, proper inspection technique must be scheduled for reusable parts to detect the amount and rate of crack growth, and the possible need for repairing or replacement of the part. An example of a fail-safedesigned structure with crack-arrest feature, common to all aircraft structural parts, is the skin-stiffened design configuration. However, in other cases, the design philosophy has safe-life or single load path feature, where analysts must demonstrate that parts have adequate life during their service operation and the possibility of catastrophic failure is remote. For example, all pressurized vessels that have single load path feature are classified as high-risk parts. During their service operation, these tanks may develop cracks, which will grow gradually in a stable manner. To avoid catastrophic failure, a thorough nondestructive inspection, a proof test prior to service usage, and a comprehensive fracture mechanics analysis (i.e., safe-life analysis) are requested by the customer. To demonstrate that structural failure of single load path component does not occur and that the part has adequate life during its entire operation, a comprehensive fatigue and fracture mechanics analysis using linear elastic fracture mechanics must be performed. In conducting safe-life analysis, full fracture mechanics data for the material must be available. These data are generated based on the ASTM testing standards. Because fracture toughness is thickness dependent and structures have components that have different sizes and thicknesses, numerous fracture toughness tests must be conducted to include the plane strain, plane stress, and the mixed-mode conditions. In addition to fracture toughness values, the fatigue crack growth rate data must also be available to analysts in order to conduct a meaningful safe-life vii

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analysis. These tests are costly and time consuming, and the cost and time of testing will increase substantially when scatter in fracture allowable, as the result of material variation, need to be considered. Therefore, any method that can reduce the number of tests will be useful to the industry to avoid unnecessary costs when fracture allowables are needed as an input to the life-assessment analysis. The safe-life analysis of high-risk components will demonstrate the ability or tolerance of parts at the presence of existing crack under the load-varying environment. For this reason the fracture mechanics analysis in many cases is called “damage tolerance” analysis. In reality, the total life of structural components is the sum of crack initiation cycles and crack propagation. In aircraft industry, the number of cycles to crack initiation must be accounted for when assessing the total life of the part. Figure 1 shows the process of creating a crack growth/residual strength analysis with emphasis on the comparison (feedback) of analysis with practice. That is, crack growth analysis should be checked against results obtained from the field experience through the interval inspection. Airlines already implemented this approach through their Reliability Centered Maintenance program to adjust their interval inspection period of the entire fleet. As indicated in the figure, the total life analysis is incomplete without having fatigue and fracture allowables through testing.

Fig. 1 Comparison (feedback) of analysis with practice. Analysis is incomplete without fatigue and fracture data

Because the induced stresses in aircraft components must be kept in the elastic range, the high cycle fatigue data generated through the stress to life (S–N) can be useful in the fatigue assessment analysis (also called durability analysis). These data are stress ratio (R) dependent and require considerable time and cost to generate the full range of the S–N curve. Any analytical technique that can be used to generate the S–N data without conducting the traditional laboratory coupon tests will be helpful to the aircraft and aerospace industry to avoid tests.

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The sole purpose of this book is to provide the structural engineers with the present and near-future approaches to the virtual testing techniques, where fatigue and fracture mechanics data can be generated rapidly with minimum amount of tests. As mentioned before, fatigue and fracture allowables are needed as an input to the damage tolerance and durability assessment of fracture critical parts. In many instances, analysts do not have fatigue fracture mechanics values and the budget is not adequate to generate data through testing approach. In other instances, the time does not allow to conduct tests, because of deadlines that designers must meet and laid down by the customers. Therefore, the virtual testing is the right tool to have when both the budget and time do not allow engineers to conduct tests for durability and damage tolerance analysis. The virtual testing technique for generating fatigue and fracture allowables will be presented in this book through two unique techniques. The first approach will use the conventional continuum mechanics approach, which will allow engineers to generate the S–N; fracture toughness, Kc; and fatigue crack growth rate data (da /dN versus ΔK) through analytical approach. The second method of generating these data is based on the fundamental laws of physics (i.e., the ab initio), where material will be assessed from the bottom-up approach. Both approaches to the virtual testing will be presented in this book. The latter utilizes the multiscale modeling and simulation technique to predict the material properties. For this reason the author chooses to use “Virtual Testing and Predictive Modeling” as the title of this book. Chapters 1, 2, 3, 4, 5, and 6 of this book will be dedicated to virtual testing using the continuum mechanics approach. Both metallic and composite materials will be addressed with numerous examples related to the aerospace and aircraft parts. Chapters 7, 8, 9, 10, and 11 will discuss the multiscale modeling and simulation technique. Both quantum mechanics and molecular dynamics approaches will be used to conduct the predictive modeling analysis. Because of outstanding mechanical properties of nanoparticles, there is a strong future demand for their application in aerospace and aircraft structural parts. These particles when combined with polymer matrix will enhance the mechanical properties of polymer, which is an important factor in reducing the weight of the structural parts in modern airplanes. The implementation of multiscale modeling and simulation at the interface region between nanoparticles and matrix is challenging and proper chemistry between nanoparticles and polymer is needed to provide a good bond at the interface region. To make nanoparticles more easily dispersible in polymer, it is necessary to physically or chemically attach certain molecules, or functional groups, to their smooth sidewalls without significantly changing the nanoparticle’s desirable properties. This process is called functionalization. The production of robust composite materials that can allow the transfer of load without causing localized damage requires strong covalent chemical bonding between the filler particles and the polymer matrix that can be achieved through the functionalization process. Chapter 12 will address the most recent approach to the functionalization technique that can be useful to bond dissimilar material with achieving adequate interface strength.

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Finally, Chapter 13 will be allocated to the verification technique using the stateof-the art approach to verify the interface region between nanoparticles and the matrix by applying the transmitted electron microscope (TEM) and atomic force microscope (AFM). The experimental flexibility of these techniques will provide insights into the fundamental structure and deformation processes of nanoscales materials. The in situ measurement of interface region while material under stress will be discussed. Kirkland, Washington

Bahram Farahmand

Contents

1 Virtual Testing and Its Application in Aerospace Structural Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bahram Farahmand 1.1 Introduction to the Virtual Testing . . . . . . . . . . . . . . 1.2 Virtual Testing Theory and Fracture Toughness . . . . . . . 1.3 The Extended Griffith Theory and Fracture Toughness . . . 1.4 Extension of Farahmand’s Theory to Fatigue Crack Growth Rate Data . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Accelerated Region and Fracture Toughness . . 1.4.2 The Paris Constants, C and n . . . . . . . . . . . . 1.4.3 The Threshold Value (Region I) . . . . . . . . . . 1.4.4 The da/dN Versus ΔK from Virtual Testing Against Test Data . . . . . . . . . . . . . . . . . . 1.5 Application of Virtual Testing in Aerospace Industry: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Background . . . . . . . . . . . . . . . . . . . . . 1.5.2 Manufacturing Process and Plastic Deformation of COPV Liner . . . . . . . . . . . . . . . . . . . 1.5.3 Generating Fracture Allowables of Inconel 718 of COPV Liner Through Virtual Testing Technique 1.5.4 Generating Fracture Allowables of 6061-T6 Aluminum Tank Through Virtual Testing Technique 1.6 Summary and Future Work . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Tools for Assessing the Damage Tolerance of Primary Structural Components . . . . . . . . . . . . . . . . . . . . . . . . R. Jones and D. Peng 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 An Equivalent Block Method for Predicting Fatigue Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fatigue Crack Growth under Variable Amplitude Loading . . . 2.3.1 Fatigue Crack Growth in an F/A-18 Aircraft Bulkhead

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Crack Growth in Mil Annealed Ti–6AL–4V under a Fighter Spectrum . . . . . . . . . . . . 2.4 A Virtual Engineering Approach for Predicting the S–N Curves for 7050-T7451 . . . . . . . . . . . . . . . . . . 2.4.1 Computing the Endurance Limit . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Formulae for Computing the Crack Opening Stress . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Cohesive Technology Applied to the Modeling and Simulation of Fatigue Failure . . . . . . . . . . . . . . . . . . . Spandan Maiti 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Models for the Prediction of Threshold Fatigue Crack Behavior . . . . . . . . . . . . . . . . . . . . . 3.2.2 Models for the Prediction of Fatigue Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cohesive Modeling Technique . . . . . . . . . . . . . . . . . . 3.3.1 Reversible Cohesive Model . . . . . . . . . . . . . . . 3.3.2 A Bilinear Cohesive Law . . . . . . . . . . . . . . . . 3.3.3 A Cohesive Model Suitable for Fatigue Failure . . . . 3.3.4 Incorporation of Threshold Behavior . . . . . . . . . . 3.3.5 Finite Element Implementation . . . . . . . . . . . . . 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Paris Curve Simulation . . . . . . . . . . . . . . . . . 3.4.2 Prediction of Threshold Limit of Fatigue Crack Growth 3.4.3 Effect of on the Threshold Limit . . . . . . . . . . . . 3.4.4 Effect of Load Ratio R on Fatigue Crack Threshold . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fatigue Damage Map as a Virtual Tool for Fatigue Damage Tolerance . . . . . . . . . . . . . . . . . . . . . . Chris A. Rodopoulos 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Basic Understanding of Fatigue Damage . . . . . 4.2.1 Development of Fatigue Cracks and Fatigue Damage Stages . . . . . . . . . . . . . . . . 4.2.2 Stage II Fatigue Cracking . . . . . . . . . . . 4.2.3 Stage I Fatigue Cracking . . . . . . . . . . . 4.2.4 Stage III Fatigue Cracks . . . . . . . . . . . . 4.3 Fatigue Damage Map the Basic Rationale – The Navarro–de los Rios Model . . . . . . . . . . . . . . 4.3.1 Fatigue Damage Map – Defining the Stages of Fatigue Damage . . . . . . . . . . . . . . . .

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Fatigue Damage Map – Defining the Propagation Rate of Fatigue Stages . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Predicting Creep and Creep/Fatigue Crack Initiation and Growth for Virtual Testing and Life Assessment of Components K.M. Nikbin 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Background to Life Assessment Codes . . . . . . . 5.1.2 Creep Analysis of Uncracked Bodies . . . . . . . . 5.1.3 Physical Models Describing Creep . . . . . . . . . 5.1.4 Complex Stress Creep . . . . . . . . . . . . . . . . 5.1.5 Influence of Fatigue in Uncracked Bodies . . . . . 5.2 Fracture Mechanics Parameters in Creep and Fatigue . . . . 5.2.1 Creep Parameter C ∗ Integral . . . . . . . . . . . . 5.3 Predictive Models in High-Temperature Fracture Mechanics 5.3.1 Derivation of K and C ∗ . . . . . . . . . . . . . . . 5.3.2 Example of CCG Correlation with K and C∗ . . . . 5.3.3 Modelling Steady-State Creep Crack Growth Rate . 5.3.4 Transient Creep Crack Growth Modelling . . . . . 5.3.5 Predictions of Initiation Times ti Prior Onset of Steady Creep Crack Growth . . . . . . . . . . . . 5.3.6 Consideration of Crack Tip Angle in the NSW Model . . . . . . . . . . . . . . . . . . . . . 5.3.7 The New NSW-MOD Model . . . . . . . . . . . . 5.3.8 Finite Element Framework . . . . . . . . . . . . . 5.3.9 Damage Accumulation at the Crack Tip . . . . . . 5.3.10 Elevated Temperature Cyclic Crack Growth . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Nomenclatures and Abbreviations . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Computational Approach Toward Advanced Composite Material Qualification and Structural Certification . . . Frank Abdi, J. Surdenas, Nasir Munir, Jerry Housner, and Raju Keshavanarayana 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 6.2 Background . . . . . . . . . . . . . . . . . . . . . . 6.2.1 FAA Durability and Damage Tolerance Certification Strategy . . . . . . . . . . . . 6.2.2 Damage Categories and Comparison of Analysis Methods and Test Results . . . . . 6.2.3 FAA Building-Block Approach . . . . . . . 6.2.4 Test Reduction Process . . . . . . . . . . .

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6.3

Computational Process for Implementing Building-Block Verification . . . . . . . . . . . . . . . . . 6.3.1 Multiple Failure Criteria . . . . . . . . . . . . . . 6.3.2 Micro- and Macro-Composite Mechanics Analysis 6.3.3 Progressive Failure Micro-Mechanical Analysis . . 6.3.4 Calibration of Composite Constitutive Properties . 6.3.5 Composite Material Validation . . . . . . . . . . . 6.3.6 Material Uncertainty Analyzer (MUA) . . . . . . . 6.4 Establish A- and B-Basis Allowables . . . . . . . . . . . . 6.4.1 Combining Limited Test Data with Progressive Failure and Probabilistic Analysis . . . . . . . . . 6.4.2 Examples of Allowable Generation for Unnotched and Notched Composite Specimens . . 6.5 Certification by Analysis Example . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Multiscale Modeling of Nanocomposite Materials . . . . . . . . . . Gregory M. Odegard 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Modeling of Multiscale Fatigue Crack Growth: Nano/Micro and Micro/Macro Transitions . . . . . . . . . . . . . . . . . . . G.C. Sih 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Scale Implications Associated with Size Effects . . . . . . . 7.2.1 Physical Laws Change with Size and Time . . . . . 7.2.2 Surface-to-Volume Ratio as a Controlling Parameter 7.2.3 Strength and Toughness: Nano, Micro and Macro . 7.3 Form Invariant of Two-Parameter Crack Growth Relation . . 7.4 Dual-Scale Fatigue Crack Growth Rate Models . . . . . . . 7.4.1 Micro/Macro Formulation . . . . . . . . . . . . . 7.4.2 Nano/Micro Formulation . . . . . . . . . . . . . . 7.5 Micro/Macro Time-Dependent Physical Parameters . . . . . 7.5.1 Macroscopic Material Properties . . . . . . . . . . 7.5.2 Microscopic Material Properties . . . . . . . . . . 7.6 Nano/Micro Time-Dependent Physical Parameters . . . . . 7.6.1 Nanoscopic Material Properties . . . . . . . . . . . 7.6.2 Nanoscopic Fatigue Crack Growth Coefficient . . . 7.7 Fatigue Crack Growth and Velocity Data . . . . . . . . . . 7.7.1 Predicted Micro/Macro Results . . . . . . . . . . . 7.7.2 Predicted Nano/Micro Results . . . . . . . . . . . 7.8 Validation of Nano/Micro/Macro Fatigue Crack Growth Behavior . . . . . . . . . . . . . . . . . . . . . . . 7.9 Implication of Multiscaling and Future Considerations . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Computational Modeling Tools . . . . . . . . . . . . . . Equivalent-Continuum Models . . . . . . . . . . . . . . . 8.3.1 Representative Volume Element . . . . . . . . . 8.3.2 Equivalent Continuum . . . . . . . . . . . . . . 8.3.3 Equivalence of Averaged Scalar Fields . . . . . . 8.3.4 Kinematic Equivalence . . . . . . . . . . . . . . 8.4 Equivalent-Continuum Modeling Strategies . . . . . . . . 8.4.1 Crystalline and Highly Ordered Material Systems 8.4.2 Fluctuation Methods . . . . . . . . . . . . . . . 8.4.3 Static Deformation Methods . . . . . . . . . . . 8.4.4 Dynamic Deformation Methods . . . . . . . . . 8.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Silica Nanoparticle/Polymer Composites . . . . . 8.5.2 Nanotube/Polymer Composites . . . . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Predictive Modeling . . . . . . . . . . . . . . . . . . . . . . Michael Doyle 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Nanocomposites . . . . . . . . . . . . . . . . . . . . . 9.2.1 Nanotechnology and Modeling . . . . . . . . . 9.2.2 Composites . . . . . . . . . . . . . . . . . . . 9.2.3 The Interface Region . . . . . . . . . . . . . . 9.2.4 Functionalization of Interface Region . . . . . 9.2.5 Modeling Approaches . . . . . . . . . . . . . . 9.2.6 Method Developments . . . . . . . . . . . . . 9.3 Multiscale Modeling . . . . . . . . . . . . . . . . . . . 9.4 Continuum Methods . . . . . . . . . . . . . . . . . . . 9.4.1 Predicting Material Properties from the Top-Down Approach . . . . . . . . . . . . . . 9.4.2 Analytical Continuum Modeling . . . . . . . . 9.4.3 Computational Continuum Modeling . . . . . . 9.5 Materials Engineering Simulation Across Multi-Length and Time Scales . . . . . . . . . . . . . . . . . . . . . 9.5.1 Predicting Material Properties from the Bottom-Up Approach . . . . . . . . . . . . . . 9.5.2 Quantum Scale . . . . . . . . . . . . . . . . . 9.5.3 Molecular Scale . . . . . . . . . . . . . . . . . 9.5.4 Molecular Dynamics . . . . . . . . . . . . . . 9.6 Extension of Atomistic Ensemble Methods . . . . . . . 9.6.1 Combining the Top-Down and Bottom-Up Approaches . . . . . . . . . . . . . . . . . . . 9.7 Future Improvement . . . . . . . . . . . . . . . . . . . 9.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Multiscale Approach to Predicting the Mechanical Behavior of Polymeric Melts . . . . . . . . . . . . . . . . . . R.C. Picu 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Single and Multiscale Modeling Methods: Limitations and Tradeoffs . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Atomistic and Atomistic-Like Models . . . . . 10.2.2 Molecular Models . . . . . . . . . . . . . . . . 10.2.3 Continuum Models . . . . . . . . . . . . . . . 10.3 Two Information-Passing Examples . . . . . . . . . . . 10.3.1 General Strategy . . . . . . . . . . . . . . . . . 10.3.2 Calibration of Rheological Constitutive Models 10.3.3 Developing Coarse-Grained Models of Polymeric Melts . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Prediction of Damage Propagation and Failure of Composite Structures (Without Testing) . . . . . . . . . . . . . G. Labeas 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basics of Progressive Damage Modelling methodology . . . 11.2.1 PDM – An Overview . . . . . . . . . . . . . . . . 11.2.2 Multiscale Computational Model . . . . . . . . . . 11.2.3 Prediction of Local Failure at Different Scale Levels 11.2.4 Behaviour of Damaged Material . . . . . . . . . . 11.3 Buckling and Damage Interaction of Open-Hole Composite Plates by PDM . . . . . . . . . . . . . . . . . . 11.3.1 Composite Panel with Circular Cut-Out . . . . . . 11.3.2 Computational Model for the Open-Hole Panel Problem . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Interaction Effects Between Damage Failure and Plate Buckling . . . . . . . . . . . . . . . . . 11.4 Implementation of PDM in Composite Bolted Joints . . . . 11.4.1 Description of Composite Bolted Joint Problem . . 11.4.2 Damage Initiation and Progression Within the Bolted Joint . . . . . . . . . . . . . . . . . . . . . 11.5 Implementation of PDM in Composite Bonded Repairs . . . 11.5.1 Description of the Composite Repair Patch Problem 11.5.2 Details of PDM Model for Composite Repair Patch Analysis . . . . . . . . . . . . . . . . . . . . 11.5.3 Effects of Composite Patch Geometry and Material on the SIF . . . . . . . . . . . . . . . . . 11.6 Multi-Scale Modeling of Tensile Behavior of Carbon Nanotube-Reinforced Composites . . . . . . . . . . . . . . 11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. .

321

. . . . . .

. . . . . .

321 323 323 324 329 331

. . . .

334 334

. .

335

. . . . . .

337 339 339

. . . . . .

342 345 345

. .

346

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348

. . . . . .

350 352 353

Contents

12

13

Functional Nanostructured Polymer–Metal Interfaces Niranjan A. Malvadkar, Michael A. Ulizio, Jill Lowman, and Melik C. Demirel 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . 12.2 Oblique-Angle Polymerization . . . . . . . . . . . 12.2.1 Nanostructured Polymer growth . . . . . 12.2.2 Control of Morphology and Topography . 12.3 Metallization of Nanostructured Polymers . . . . . 12.3.1 Electroless Metal Deposition . . . . . . . 12.3.2 Vapor Phase Metal Deposition . . . . . . 12.3.3 Nanoparticle Assembly . . . . . . . . . . 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

. . . . . . .

357

. . . . . . . . . .

357 358 358 360 361 362 363 364 366 368

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Advanced Experimental Techniques for Multiscale Modeling of Materials . . . . . . . . . . . . . . . . . Reza S. Yassar and Hessam M.S. Ghassemi 13.1 Atomic Force Microscopy (AFM) . . . . . . . . . . . . . . . . 13.1.1 Principles of AFM . . . . . . . . . . . . . . . . . . . 13.1.2 AFM Operation . . . . . . . . . . . . . . . . . . . . . 13.1.3 Application of AFM . . . . . . . . . . . . . . . . . . . 13.1.4 Modeling and Simulation . . . . . . . . . . . . . . . . 13.2 X-Ray Ultra-Microscopy . . . . . . . . . . . . . . . . . . . . . 13.2.1 Principles of XuM . . . . . . . . . . . . . . . . . . . . 13.2.2 Phase Contrast and Absorption Contrast . . . . . . . . 13.2.3 3D Imaging and Multiscale Modeling Applications . . 13.3 In Situ Micro-Electro-Mechanical-Systems (MEMS) Introduction 13.3.1 Principle and Design of MEMS Devices . . . . . . . . 13.3.2 Application of MEMS Devices for Materials Modeling 13.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371 372 372 374 375 379 382 382 384 385 388 389 392 395 396 399

Contributors

Frank Abdi Alpha STAR Corporation, Long Beach, CA, USA, [email protected] Melik C. Demirel The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA, [email protected] Michael Doyle Principal Solution Scientist, Materials Science @ Accelrys, USA, [email protected] Bahram Farahmand Taylor Aerospace (TASS – Americas) Inc., Kirkland, WA, USA, [email protected] Hessam M.S. Ghassemi Mechanical Engineering-Engineering Mechanics Department, Michigan Technological University, 1400 Townsend Dr., Houghton, MI 49931, USA Jerry Housner Analytical Enterprises, Arlington, VA, USA, [email protected] R. Jones DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical and Aerospace Engineering, P.O. Box 31, Monash University, Victoria, 3800, Australia, [email protected] Raju Keshavanarayana Department of Aerospace Engineering/National Institute for Aviation Research (NIAR), Wichita State University, Wichita, KS, USA, [email protected]; [email protected] G. Labeas Laboratory of Technology and Strength of Materials, University of Patras, Panepistimioupolis Rion, 26500 Patras, Greece, [email protected] Jill Lowman The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA Spandan Maiti Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton MI 49931, USA, [email protected] Niranjan Malvadkar The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA, [email protected] xix

xx

Contributors

Nasir Munir North Grumman Corporation, El Segundo, CA, USA, [email protected] K.M. Nikbin Mechanical Engineering Department, Imperial College, London SW7 2AZ, UK, [email protected] Gregory M. Odegard Department of Mechanical Engineering – Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA, [email protected] D. Peng DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical and Aerospace Engineering, P.O. Box 31, Monash University, Victoria, 3800, Australia R.C. Picu Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA, [email protected] Chris A Rodopoulos Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Greece, [email protected] G.C. Sih School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China; Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA, [email protected], [email protected] J. Surdenas Alpha STAR Corporation, Long Beach, CA, USA, [email protected] Michael Ulizio The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA Reza S. Yassar Mechanical Engineering-Engineering Mechanics Department, Michigan Technological University, 1400 Townsend Dr., Houghton, MI 49931, USA, [email protected]

Introduction

Recent failures of several major structures, such as pressure vessels, storage tanks, ships, aircrafts, gas pipe lines, bridges, dams, and many welded parts, have raised concern on issues like loss of life, environmental safety, and high costs associated with repair and replacement of components that have been estimated to be in millions of dollars. Almost in all cases these failures occurred during structural usage where cracks have initiated and advanced in a stable manner to failure under the fatigue load environment well below the material yield allowable. In-depth scientific investigation into the nature of these failures indicated that poor structural design practices, such as the presence of stress concentrations, insufficient material ductility, residual stresses during the fabrication and manufacturing phase of hardware, lack of adequate NDI inspections, corrosive environment, and material degradation in low-temperature environment, each can contribute to an accelerated crack growth, resulting in catastrophic failure and in some cases loss of life. In designing components of commercial aircraft or space vehicles, the induced stresses in the components, as the result of environmental load, must fall below the material yield value. That is, the bulk of the structure must stay elastic and any plastic deformation has to be localized. By enhancing the mechanical properties of material (i.e., design allowables), thickness of structural parts can be reduced and considerable weight saving is possible. The weight saving can be achieved by selecting suitable materials that possess high strength, adequate durability, as well as good ductility. It has been always the goal of the aerospace industry to minimize the weight of components without compromising the structural integrity of the system they include. Over the past few years, more attention has been given to composite materials as a possible solution to the design of aerospace and aircraft parts because of their weight and superior static and fatigue properties over the traditional aluminum alloys. In typical composite materials, high-strength fibers provide reinforcement to the polymer matrix so that they become capable of carrying the high environmental and mechanical loads desired. Recently, reinforcing nano-particles (nano-fillers) in the aerospace polymer resin have shown promise to provide higher mechanical and fatigue properties to composite materials. However, execution of right functionalization at the interface and uniform dispersion of these particles in the polymer matrix are challenging, but both must be implemented correctly in order to have enhancement in mechanical properties. To verify the improvement in xxi

xxii

Introduction

mechanical properties, limited amount of mechanical coupon testing is required. Moreover, to establish material allowables for design purposes, extensive static, fatigue, and fracture mechanics tests must be performed in order to achieve some level of confidence due to material variability. In the case of composite material, it is common to use the “B-basis” allowables for static analysis and average values for fatigue and fracture mechanics allowables. To establish material allowables for both metals and composite parts, it requires extensive tests that carries with it significant cost and time for specimens preparation and data gathering. Therefore, any method that can reduce the number of tests will be extremely useful to the industry to avoid unnecessary costs when fatigue and fracture allowables are required for designing components of structures. The importance of virtual testing and its application in aerospace and aircraft industry was realized by the author since early 1990s while working on the International Space Station (ISS) program. As part of the requirements outlined by customers and written in the ISS fracture control plan, all high-risk parts (fracture critical parts) must shown by analysis to have adequate life during their entire service usage. The safe-life analysis of fracture critical parts requires having fracture allowables (i.e., fracture toughness and fatigue crack growth rate data). On several occasions, the author experienced that these data were not available to analysts for the material under study. Efforts were always made to obtain allowables first through the literature search method and if that failed, the alternative approach was to conduct numerous tests. These tests are labor intense and time consuming and designers had been frustrated because of budgetary constraint to conduct tests. In other instances, the time needed to conduct tests and to establish allowables would exceed the deadline laid down by customers. This dilemma was experienced by the author in several occasions during his involvement with the aerospace industry. The virtual testing methodology discussed in this book is based on the continuum mechanics approach and has been discussed previously in [1, 2, 3]. Data estimated by the virtual-testing technique were in excellent agreement with data generated by the ASTM testing standards. It should be noted that in generating fracture allowables, K c and the da/dN, test coupons were first prepared with a sizable notch machined in the specimen. Subsequently, a sharp natural crack at the tip of the notch is introduced in the specimen through cyclic loading technique. Therefore, all fracture mechanics allowables used in safe-life analysis were based on large crack behavior. In real situation, cracks embedded in material prior to parts service usage are much smaller in length, and linear elastic fracture mechanics concepts and simulative law may not apply to their growth behavior under cyclic loading. With the current approach to the safe-life analysis, the initial flaw size assumption in material is estimated by assuming parts have defects that can be detected through the traditional Non-Destructive Inspections (NDI) techniques. The largest flaw size that may escape from detection can be estimated and used as the initial flaw size in assessing the remaining life of components. The life assessment analysis results based on this approach are too conservative and in some cases lead to redesigning the part and adding unnecessary weight to the structure. Experimental and analytical work showed that the typical crack-like defects found on the part, after machining

Introduction

xxiii

operation, are marks that are much smaller in size than the standard NDI capability. These machining marks on the surface of parts can be as small as a few microns in depth. In contrast to the aerospace industry, the life assessment of aircraft components is divided into two parts: (1) the number of cycles to initiate a crack, where crack reaches to a visible measurable size ∼0.1 in. in length, and (2) the number of cycles from initiation to final failure. The first part of life analysis is called ‘durability’ and the latter part (i.e., the remaining life) is referred to as ‘damage tolerance’ analysis. The durability analysis portion of life assessment uses the traditional high cycle fatigue data (the S–N data) that must be generated through the ASTM testing standards, which require conducting numerous tests to cover all regions of the S–N curve. Just like fracture mechanics allowables, the high cycle fatigue tests are also costly and time consuming to the aerospace and aircraft industry. Therefore, the application of virtual testing technique will be extremely useful to eliminate unnecessary tests and to reduce cost and time associated with generating allowables for structural life analysis.

References 1. B. Farahmand, “Application of Virtual Testing For Obtaining Fracture Allowables of Aerospace And Aircraft Materials,” Book Chapter to “Multiscale Fatigue Crack Initiation and Propagation of Engineering Materials,” by G. Sih, Springer, 2008 2. B. Farahmand, “Predicting Fracture and Fatigue Crack Growth Properties Using Tensile Properties,” Eng. J. Fract. 2007 3. B. Farahmand, “Analytical Development of Fatigue Crack Growth Curve without Conducting ASTM E647 Tests,” AIAA Conference, Houston Texas, May 2006

Chapter 1

Virtual Testing and Its Application in Aerospace Structural Parts Bahram Farahmand

Abstract In many occasions, metallic parts will undergo plastic deformation either during their service usage or prior to their actual operation in the manufacturing and assembly phases. Under these circumstances, material properties may change considerably and must be accounted for when estimating the residual strength capability of parts. The change in material properties will occur when load has been removed and the work-hardening phenomenon has caused increase in material yield value, reduction in percent elongation, and degradation in fracture allowables. For these reasons, new static and fracture data must be generated if safe-life assessment must be performed on fracture critical parts. Fracture data are currently obtained through the ASTM testing standards, which are costly and labor intense. Difficulties associated with preparing the specimen, precracking the notch, recording and monitoring the data, obtaining the variation of fracture toughness versus part thickness, capturing data in the threshold region, and repeating the test in many cases due to invalid results make the virtual testing technique extremely helpful to overcome the time and cost related to the above-mentioned testing difficulties. Farahmand (Fatigue & Fracture Mechanics of High Risk Parts, Chapman & Hall, 1997, Chapter 5; Fracture Mechanics of Metals, Composites, Welds, and Bolted Joints, Kluwer Academic Publisher 2000, Chapter 6) utilized the extended Griffith theory to obtain a relationship between the applied stress and half a crack length by using the full stress– strain curve for the material under consideration. The extension of this work led to estimation of material fracture toughness and construction of full fatigue crack growth rate curve (Farahmand, J. Fract., 2007, Vol. 75, pp 2144–2155). This technique will be very useful to implement when materials undergo plastic deformation (the work-hardening phenomenon) and fracture allowables are needed to conduct a meaningful life analysis assessment of parts. First, Farahmand’s virtual testing theory is described briefly and, subsequently, the application of this methodology to two cases of aerospace pressurized tanks, exposed to plastic deformation, presented in this chapter.

B. Farahmand (B) Taylor Aerospace (TASS – Americas) Inc., Kirkland, WA, USA e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 1,

1

2

B. Farahmand

1.1 Introduction to the Virtual Testing It is estimated that the cost of fully characterizing mechanical properties of a new material, which must be used as part of aerospace or aircraft structures, can be substantial. The static, fatigue, and fracture mechanics data have to be available in order to conduct a meaningful stress, durability, and damage tolerance analysis of highrisk components. The successful prediction of mechanical properties through the virtual testing technique is the goal of aerospace and aircraft industry. The proposed technique utilizes the basic data from a simple static tensile test to estimate the fracture mechanics allowables through the extended Griffith theory [1, 2]. The fracture toughness values can be obtained by taking the energy per unit volume under the uniform and nonuniform portions of the stress–strain curve to calculate the energy consumed at the crack tip up to the failure. Section 1.2 discusses the virtual testing method of obtaining fracture toughness. The fracture toughness value obtained through this approach can be used for the upper portion of the fatigue crack growth rate curve. The upper portion of the da/dN curve is related to the critical value of the stress intensity factor; therefore, the fracture toughness value, estimated through the virtual testing technique, can be used to establish the region III of the da/dN versus ΔK data. Other regions of fatigue crack growth rate can be obtained separately and are presented in Section 1.3 [3–5].

1.2 Virtual Testing Theory and Fracture Toughness Metallic material used for the design of aircraft and aerospace parts exhibits three distinct regions when subjected to static loading. The area under the load versus displacement curve (up to the final failure) will represent the degree of ductility that the material possesses. Under monotonic loading, purely elastic materials (Fig. 1.1a) show no ductility at the onset of failure and are strictly prohibited to be used in the design of aerospace and aircraft parts, where safety of vehicle is of great concern. As part of the aerospace design requirement, material selection must be limited to those alloys that exhibit elongation larger than 4%. Aluminum and titanium alloys are excellent candidates for aircraft and aerospace parts because of their attractive strength-to-weight ratio, as well as their great elongation that almost in all cases exceeds the design requirement of 4%. Since ductility and toughness are directly related to each other, the extended Griffith theory is used to relate the area under the stress–strain curve to the material fracture toughness via Farahmand’s theory. Figure 1.1 shows different stress–strain curves for metallic materials where the area under the curve, WF and WU , can be shown to be related to plastic deformation at the crack tip. The critical value of stress intensity factor, K (i.e., Kc), can be calculated by using the energy under the stress–strain curve and will be used as an input in the safe-life analysis of parts. The results of analysis using Farahmand’s approach indicated that materials with uniform plastic deformation (case b in Fig. 1.1, where WU = 0 and WF = 0) possess lower fracture toughness value when compared with

WF = 0 WU = 0

Stress

Fig. 1.1 Typical stress–strain curves for different materials. The quantity W is the energy per unit volume represented by the area under the curve

3

Stress

1 Virtual Testing and Its Application

(a)

(b) Strain

WF ≠ 0 WU ≈ 0

(c)

Stress

Strain

Stress

WF = 0 WU ≠ 0

WU ≠ 0

WF ≠ 0

(d) Strain

Strain

case (d) where materials display both uniform and nonuniform plastic deformations (WF = 0 and WU = 0). The presence of necking shows materials exhibit considerable plastic deformation and thus have higher resistance to fracture before their final failure. Section 1.3 discusses the methodology and derivation of fracture toughness through the information available via the full stress–strain curve and the extended Griffith theory.

1.3 The Extended Griffith Theory and Fracture Toughness The materials selected by engineers to manufacture fracture-critical hardware must exhibit some amount of plastic deformation and stable slow crack growth at the crack tip region prior to their final failure. The amount of energy consumed at the crack tip for plastically deforming material is largely due to material resistance in that region prior to final failure, which is not properly accounted for in the linear elastic fracture mechanics analysis. Irwin and Orowan [6, 7] independently observed that, for tough material, the amount of energy dissipated at the crack tip for plastic deformation, UP , is much larger than the energy consumed for the formation of two crack surfaces, US . Farahmand showed that the energy consumed at the crack tip for straining material is of two kinds: local nonuniform strainability, UF , and uniform strainability, UU [1]. Thus, fracture behavior can be characterized by two energyreleased terms representing plastic deformation at and near the crack tip. Both terms can be derived by using the areas associated with uniform and nonuniform plastic deformation under the full uniaxial stress–strain curve for the alloy under consideration, Fig. 1.1. The extended Griffith theory in terms of UU and UF , including the two new crack surfaces term, 2T, can be written as

4

B. Farahmand

πσ 2c E

= 2T +

Energy rate associated with final fracture

∂UU ∂c

+

∂UF

(1.1)

∂c

Energy rate associated with uniform straining

Energy rate associated with non-uniform straining

For ductile metals, the two terms associated with uniform and nonuniform plastic deformation are much greater than the two new crack surfaces term, 2T [7]. If quantities to the right of Equation (1.1) are determined, then a relationship between applied stress and half a crack length at the onset of failure can be established. The U term can be derived by assuming that all the crack energy rate associated with ∂U ∂c tip energy due to uniform straining up to the ultimate failure point has been consumed by the amount WU to create permanent slip of height hU : ∂UU = WU h U ∂c

(1.2)

The unrecoverable energy per unit volume can be calculated by integrating the area under the curve from the limit stress up to the ultimate: σT U WU =

σT dεT P

(1.3)

σT L

Equation (1.3) is written in terms of true stress and strain, where for metals the true stress versus true plastic-strain curve can be approximated by the Ramberg– Osgood equation, when fitted at true stress at the limit, σ TL , and ultimate, σ TU . The true stress, σ T , in terms of true plastic strain, εTP , can be written as σT = σT U

εT P εT U

1/n (1.4)

where n is the Ramberg–Osgood exponent. The true plastic strain, εTP εT P = εT U dεT P = nεT U

σT σT U

After integration Equation (1.3) becomes

σT n σT U n−1

(1.5) dσT σT U

(1.6)

1 Virtual Testing and Its Application

5

n σT L n+1 εT U σT U 1 − WU = n+1 σT U

(1.7)

The quantity hU defines the height of the plastic deformation in the uniform strain region near the crack tip. It can be formulated through linear elastic fracture mechanics [1] and in its final format it is n−1 εT U n εT F εT L ∗ n+1 h −1 β hU = n−1 εT U εT εT L

(1.8)

where β for the plane stress and plane strain conditions is 1.3 and 0.127, respecF can be derived by assuming that all the tively. The energy term associated with ∂U ∂c crack tip energy due to uniform straining up to the failure has been consumed by the amount WF to create permanent coarse slip of height hF : ∂U F = WF h F ∂c

(1.9)

where the quantity WF in Equation (1.9) is the area under the stress–strain curve from necking up to fracture (see Fig. 1.1) and is approximately equal to [1]: W F = σ¯U F ε P N

(1.10)

Moreover, the amount of deformation due to slip mechanism, hF , is directly related to the material ability to absorb energy, WF , and can be written as h F = γ (E 2 /3 σU )W F

(1.11)

where E is the modulus of elasticity and σ U is the material ultimate value. The correction factor γ = (8/π )αhmin is a material constant. The minimum value of hF is designated by hmin = 0.000557 in. and α is the material atomic spacing. The constant γ for most metallic material can be approximated as ∼3.5 × 10–8 . Combining Equations (1.2) and (1.9), the energy balance equation for the extended Griffith theory can be written as [1] n σT n+1 E σT U εT U 1 − h min c= 2T + σ¯U F ε P N h F k + π σ 2μ n−1 σT U ⎡ ⎤ ⎫ ⎪ ∗ n − 1 ⎬ εT F εT L ⎢ εT U n − 1⎥ β ⎣ ⎦ ⎪ εT U εT εT L ⎭ (1.12) Equation (1.12) relates half a crack length, c, to the applies stress, σ, in an infinite plate at failure. Other quantities in Equation (1.12) are defined as

6

B. Farahmand

σ¯U F = Average stress between ultimate and final fracture σT U = True stress at ultimate σT F = True stress at failure σT L = True stress at limit εT U = True strain at ultimate εT F = True strain at failure εT L = True strain at limit ε P N = Plastic strain at necking Thickness parameters describing the plane stress and strain conditions for each term of Equation (1.12) are designated by μ, k, and β [1]. As mentioned earlier, by calculating half a crack length, c, for a given applied stress, σ (Equation 1.12), the critical value of the stress intensity factor, Kc , can easily be calculated and will be used for the region III of the da/dN versus ΔK. Detailed discussion related to this topic is presented in Section 1.4.

1.4 Extension of Farahmand’s Theory to Fatigue Crack Growth Rate Data To obtain the da/dN versus ΔK data for all regions of fatigue crack growth rate curve, the virtual testing technique proposed by Farahmand [3] can be extremely useful to apply when there are budget or time limitations to conduct coupon tests. Farahmand’s unique approach is based on establishing each region of the da/dN curve separately and relating them through the Forman–Newman fatigue crack growth equation [8]: th p c(1 − f )n ΔK n (1 − ΔK ) da ΔK = ΔK dN (1 − R)n (1 − (1−R)K c )q

(1.13)

Traditionally, all fatigue crack growth constants (c, n,Kc , and ΔKth ), shown in Equation (1.13), will be provided through the ASTM tests, which are costly and labor intense. However, by using the virtual testing concepts, all constants can be estimated and the da/dN versus ΔK, from Equation (1.13), can be plotted.

1.4.1 The Accelerated Region and Fracture Toughness The region I of the da/dN versus ΔK curve is related to the fracture toughness, Kc , and it is thickness dependent. As mentioned previously, the quantity Kc can be estimated via Equation (1.12) for all ranges of material thickness. Figure 1.2 is the plot of stress–strain curves for 2219-T6 and 6061-T6 aluminums that can be used to estimate fracture toughness [9]. Figure 1.3 is a plot of fracture toughness versus material thickness for the above-mentioned aluminums (2219-T6 and 6061-T6). In all cases the fracture toughness variation with respect to thickness calculated by the virtual testing technique is in good agreement with the NASGRO database generated

1 Virtual Testing and Its Application

7 60

50

50

40

40

30

Stress, ksi

Stress, ksi

60 2219-T6

Longitudinal And Long Transverse

6061-T6

30 20

20

10

10 Typical Thickness: 0.125 –2.00 in.

0.0 0.00

0.02

0.04

0.06

0.08

Typical

0.10

0.12

0.0 0.00

0.02

0.04

Strain, in./in.

0.06

0.08

0.10

0.12

Strain, in./in.

Fig. 1.2 Typical full stress–strain curves for 2219-T6 and 6061-T6 aluminums

Virtual Testing

70 60 50 40 30 20 10 0

NASGRO

KIc

0

0.5

1 1.5 2 2.5 Thickness (Inch)

3

6061-T6 Aluminum Alloy - RT

Fracture Toughness ksi (in.)^0.5

Fracture Toughness ksi (in.)^0.5

2219-T6 Aluminum Alloy - RT

3.5

Virtual Testing

60

NASGRO

50 40

KIc

30 20 10 0 0

0.5

1 1.5 2 2.5 3 Thickness (Inch)

3.5

Fig. 1.3 Virtual testing versus test data for 2219-T6 and 6061-T6 aluminums

by test data [8]. Therefore, by using this technique, the region I of the da/dN versus ΔK curve can be obtained without conducting tests.

1.4.2 The Paris Constants, C and n A significant number of fatigue crack growth rate data for numerous aluminum alloys were extracted from the NASGRO database [8]. The da/dN versus ΔK for all these alloys were plotted, and, as the result, two important observations were obtained that were helpful to establish the Paris region of the da/dN curve. The lower point in the Paris region of the fatigue curves (Fig. 1.4) has a material-independent property so that the ratio of the stress intensity factor at the lower-bound point and the threshold value (ΔK/Kth for R = 0) is ∼1.125 for the crack growth rate per cycle, da/dN∼1.0E-7 in./cycle (∼2.54E-6 mm/cycle). In the upper region of the da/dN

8

B. Farahmand • For da/dN~1.0E–7 inch/cycle, ΔK/Kth~1.125 • For da/dN~0.005 inch/cycle, ΔK/Kc~0.9 2014-T6-L-T

1.00E+00

2014-T651-L-T 2020-T651-L-T 2024-T3-L-T

1.00E–01

2024-T351-L-T 2024-T6-L-T

1.00E–02

2024-T81-L-T 2219-T6

da/dN, in./cycle

2219-T851

1.00E–03

da/dN

2219-T87 6061-T6

Paris Region

7005-T6

1.00E–04

7010-T7365

Accelerated Region

7050-T74

1.00E–05

7050-T745

Threshold Region

7050-T765 7075-T6

1.00E–06

7075-T65

ΔK

7075-T73

1.00E–07

7075-T765 7079-T651 7178-T6

1.00E–08

7178-T765 7475-T61 7475-T651

1.00E–09 1

10

100

7475-T7351

ΔK, ksi-(in.)^0.5

Fig. 1.4 The upper and lower regions of the Paris crack growth have special characteristics common to many alloys

curve (at the end of the Paris region, Fig. 1.4), the ratio of the upper-bound stress intensity factor and its critical value, Kc, (ΔK/Kc for R = 0) is found to be ∼0.9 for da/dN ∼0.005 in./cycle (∼0.127 mm/cycle). These two points are useful to plot the entire region II. The fatigue crack growth curve can then be plotted using Equation (1.13), where the fracture parameters and constants are taken from the estimated Kc , Kth , and the Paris constants C and n values for the case of R∼0. Other ranges of R-ratios can be plotted by using the Newman closure equation, f. Section 1.4.3 discusses the virtual testing approach to obtain the threshold value of region III.

1.4.3 The Threshold Value (Region I) The threshold region of the da/dN versus ΔK curve is difficult to obtain. The complexity is due to plasticity and surface toughness closure phenomenon that makes the actual measurement of the threshold value difficult. The threshold stress intensity factor measurement is time consuming and sometimes may take several days

1 Virtual Testing and Its Application

9

to make a few measurements of the ΔKth value. For these reasons, the approximate value of ΔKth is of interest to engineers when conducting life assessment. Farahmand [10] was able to establish a reasonable relationship between the threshold stress intensity factor ΔKth (for the case of stress ratio, R∼0) and the plane strain fracture toughness, KIc . The result of his observation on the threshold values of more than 100 metallic alloys was such that the quantity ΔKth falls between KIc /4π and KIc /3π (for R = 0), Fig. 1.5. Farahmand argued that materials with high KIc value also possess higher Kth value and, conversely, the lower Kth value belongs to low KIc . Table 1.1 shows the calculated threshold values and the corresponding test data extracted from the NASGRO database. Table 1.1 shows that in most cases the test values fall between the two above-mentioned KIc values. In all cases the estimated Kth values based on KIc /4π are lower than the test values and can be considered as the lower-bound values of Kth (only a few threshold values were equal to KIc /4π , see also Table 1.1). When test data are not available, the lower-bound value of Kth = KIc /4π can be used in life estimation of structural parts. Currently, work is in progress to obtain better fit between the calculated versus test values by incorporating the KIc /Fty ratio to account for material yield value, Fty .

Kth Value ksi-(in.)^0.5

Kth Values For 2000 Series Aluminum (Test Data Versus KIc/3π & KIc/4π) 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

Test KIc/4π KIc/3π 1

4

7

10

13

16

19

22

25

28

31

34

37

40

Number of Data

Fig. 1.5 Bounding threshold values between two values, KIc /4π and KIc /3π

1.4.4 The da/dN Versus ΔK from Virtual Testing Against Test Data As mentioned previously, all regions of da/dN versus ΔK must be estimated separately before being able to plot the fatigue crack growth rate curve. Equation (1.13) (the da/dN versus ΔK) can be plotted by calculating its constants through regions I, II, and III defined in Sections 1.4.1, 1.4.2, and 1.4.3. Figures 1.6, 1.7, and 1.8 are only a few selected cases for aluminum (2014-T3 and 7075-T73), titanium (Ti6Al-4 V, ST(1750) + A(1000F-4 hr) and Ti-6Al-4 V, MA(1350-2 hr) Extruded), and ferrous alloys (Custom 455 H1000 and A286 (AISI 660) 160 UTS), where fatigue crack growth rate data were plotted for stress ratio, R = 0, and checked against

KIc

27 22.5 33 29 33 29 36 30 28 23 35 30 30 26 30 33 30

Material

2014-T6 (L-T) 2020-T651 (L-T) 2024-T3 (L-T) 2024-T3 (T-L) 2024-T3 (L-T) 2024-T351 (T-L) 2024.T62 (L-T) 2024-T62 (T-L) 2024-T852 2024-T861 (L-T) 2048-T851 (L-T) 2048-T851 (T-L) 2124-T851 (L-T) 2124-T851 (T-L) 2219-T62 (-320F) 2219-T851 (L-T) 2219-T87 (L-T)

2.1 1.8 2.6 2.3 2.6 2.3 2.9 2.4 2.2 1.8 2.8 2.4 2.4 2.1 2.4 2.6 2.4

Kth (KIc/4π 2.7 2.2 2.9 2.9 2.9 2.6 2.9 2.9 2.9 2.2 2.7 2.7 2.7 3 2.9 3 2.9

Kth (Test) 3.0 2.4 3.5 3.1 3.5 3.1 3.8 3.2 3.0 2.4 3.7 3.2 3.2 3.2 3.2 3.5 3.2

Kth (KIc/3π ) 2219-T87 (T-L) 7005-T6 & T63 (T-L) 7010-T73651 7050-T736 & T74 (T-L) 7050-T76511 (T-L) 7075-T651 (L-T) 7075-T6510 (L-T) 7075-T6511 (L-T) 7075-T73 (L-T) 7075-T7351 (L-T) 7075-T73510 (L-T) 7075-T73511 (L-T) 7075-T7352 (L-T) 7075-T7651 (T-L) 7079-T651 (L-T) 7149-T73511 (L-T) 7178-T7651 (L-T)

Material 27 40 31 24 24 28 28 28 28 29 31 33 33 23 26 31 28

KIc

Table 1.1 Threshold values for several aluminums (test versus analysis)

2.1 3.2 2.5 1.9 1.9 2.2 2.2 2.2 2.2 2.3 2.5 2.6 2.6 1.8 2.1 2.5 2.23

Kth (KIc/4π 2.9 3.4 2.5 2.3 2 3 3 3 3 3 3 3 3 2.4 2 3 3

Kth (Test)

2.9 4.2 3.3 2.5 2.5 3.0 3.0 3.0 3.0 3.1 3.3 3.5 3.5 2.4 2.8 3.3 2.97

Kth (KIc/3π )

10 B. Farahmand

11

2024-T3 L-T, R = 0

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

7075-T73 Aluminum, R = 0

NASGRO

da/dN –In./Cycle

da/dN – In./Cycle

1 Virtual Testing and Its Application

Virtual Testing

1

10

1.0E+00 1.0E–01 1.0E–02 1.0E–03 1.0E–04 1.0E–05 1.0E–06 1.0E–07 1.0E–08 1.0E–09 1.0E–10

NASGRO Virtual Testing

1

100

10

ΔK–ksi (in.)^0.5

Fig. 1.6

Virtual testing results versus test data (aluminum alloys)

Ti-6Al-4V, MA (1350-2hr) Extruded, R = 0

Ti-6Al-4V, ST (1750) + A (1000F/4hr), R = 0 1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

NASGRO

da/dN –In./Cycle

da/dN- in/cycle

100

ΔK–ksi (in.)^0.5

Virtual Testing

1

10

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

100

NASGRO Virtual Testing

1

10

ΔK-ksi (in)^0.5

100

ΔK–ksi (in.)^0.5

Fig. 1.7 Virtual testing results versus test data (titanium alloys) A-286 (AISI 660) 160 UTS, R = 0

Custom 455 H1000, R = 0

Virtual Testing

da/dN –In./Cycle

da/dN-in./cycle

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1

10

100

1000

NASGRO Virtual Testing

1.00E–00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1

ΔK-ksi (in.)^0.5

10

100

1000

ΔK–ksi (in.)^0.5

Fig. 1.8 Virtual testing results versus test data (ferrous alloys)

available test data [8]. Excellent agreement between test data and virtual testing can be seen. The appendix has collections of more fatigue crack growth curves that have been generated and compared with test data obtained from NASGRO database. In all cases, good agreement between the virtual and test data was obtained. For other stress ratios, R, the fatigue crack growth rate curve can be established by using Equation (1.13), where the quantities R and f are two parameters that will be able to shift the fatigue crack growth curve of R = 0 to the right for R<0 and to the left for R>0. Figure 1.9 shows the fatigue crack growth curve for 2024-T3 and Ti-6Al-4 V, ELI-BA (1900F, 0.5 hr+1325F) alloys with R = 0.5 (R>0). Good

12

B. Farahmand

fit between the virtual testing and test data from NASGRO database can be seen. In addition, the case of R = –1 is shown in Fig. 1.10 for 7075-T73 and 7050-T74511 aluminum alloys. Good correlation between the virtual testing results and test data can be seen.

1.00E+ 00 1.00E– 01 1.00E– 02 1.00E– 03 1.00E– 04 1.00E– 05 1.00E– 06 1.00E– 07 1.00E– 08 1.00E– 09 1.00E– 10

Ti-6Al-4V, ELI-BA (1900F, 0.5hr+1325F) –R = 0.5

NASGRO Virtual Testing

da/dN-In./Cycle

da/dN-In./Cycle

2024-T3 L-T, R = 0.5

1

10

NASGRO

1.00E+ 00 1.00E– 01 1.00E– 02 1.00E– 03 1.00E– 04 1.00E– 05 1.00E– 06 1.00E– 07 1.00E– 08 1.00E– 09 1.00E– 10

100

Virtual Testing

1

10

ΔK–ksi(in.)^0.5

100

ΔK–ksi(in.)^0.5

Fig. 1.9 Virtual testing data versus tests for stress ratio other than R = 0 (R = 0.5)

7050-T74511 Aluminum, R = –1 NASGRO

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11

Virtual Testing

1

10

da/dN-In./Cycle

da/dN-In./Cycle

7075-T73 Aluminum, R = –1

100

ΔK–ksi(in.)^0.5

1000

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

100

1000

ΔK–ksi(in.)^0.5

Fig. 1.10 Virtual testing data versus tests for stress ratio other than R = 0 (R = −1)

1.5 Application of Virtual Testing in Aerospace Industry: Introduction The mechanical properties of the materials selected for manufacturing aerospace and aircraft parts must be fully characterized before their consideration for space application. These properties are needed when assessing the static and fracture mechanics analysis of critical parts. In typical static analysis, stresses throughout the body are estimated by the finite element method and they will be incorporated into the fatigue and fracture mechanics assessment of high-risk parts. These stresses will be used as the far-field stress when conducting the fracture mechanics analysis. To establish the static allowables, numerous dog bone tensile specimens will be prepared, pulled to failure, and, based on statistical analysis, the A basis value for the modulus, yield, and ultimate (E, Fty, Ftu) as well as final elongation will be recorded for the material under consideration. The same is true for fracture allowables. However, because of the cost and time associated with generating these allowables, the

1 Virtual Testing and Its Application

13

number of tests will be limited typically to five specimens per case and the average values from test data are required to be used for safe-life analysis. The average or typical values are acceptable to be used for analysis in order to avoid impacts on structure weight. The lower-bound value of fracture allowables can increase the structural weight and cause economical impact on the program. From a safety perspective, the typical value can be justified because of other factors of safety already embedded in the static and life analysis of parts that are documented in the fracture control plan provided by the customers [11]. Nonetheless, the overall cost and time consumed for establishing these allowables through testing must be avoidable to minimize its effect on the program. The following addresses the application of the virtual testing on one of the space programs where the structure must be launched on time in order to meet the deadline set by the customer. The analyst must demonstrate, through fracture mechanics analysis, that the number of cycles to failure is adequate for the part to survive during its entire service usage. However, the fracture allowable for the material must be first established through testing. The cost of testing in the example shown below may not be an issue, but the time required for completing the test is not tolerable by the customer. For this reason, the application of the virtual testing is of great help to the program, which can be implemented in order to meet the deadline specified by the customer.

1.5.1 Background In many instances, fracture critical parts of aircraft or space structures go through high-stress magnitude cycles during or prior to their service usage. The plastic deformation can occur in localized areas that already have high stresses or the net section area (net section yielding), and in some cases the whole structure can be plastically strained if the load magnitude is above the material yield value. For example, a pressurized tank of a reusable (or expendable) space vehicle can locally undergo severe plastic deformation during the proof test process in the boss area of the doom region. When proof test is completed and pressure is removed, the work hardening causes degradation in material fracture toughness, which shortens the remaining life of the tank during its subsequent operation. In other instances, structural parts are exposed to manufacturing loads that can strain the material above the yield value or, during the operational environment, when the structural components see a few repeated cycles with the stress magnitude exceeding the allowable material yield. In all the above-mentioned cases, both the fracture toughness and the fatigue crack growth rate properties have lower values and must be reevaluated through testing. This is where virtual testing proves to be a valuable tool to implement in order to avoid costly and time-consuming tests. As mentioned previously, the fatigue crack growth rate, as well as the fracture toughness tests, requires specimen preparation and pre-cracking that are time consuming. It can delay life assessment analysis for weeks. The following is an example of a real case in which the virtual testing technique was helpful in calculating the fracture allowables for a safe-life analysis

14

B. Farahmand

of composite over-wrapped pressure vessel. Not using this technique would have delayed the launching of the satellite for weeks.

1.5.2 Manufacturing Process and Plastic Deformation of COPV Liner Composite over-wrapped pressure vessels (COPVs) are used extensively in aerospace industry for their light-weight feature [12]. These tanks have metallic liners that undergo welding process and are wrapped in composite material. In general, metallic pressurized vessels are assembled through the welding process, where the shell (cylindrical) and dome components are joined together at both ends through the circumferential welding technique. In addition, the two halves of cylindrical portion of the tank are also joined together by the longitudinal welding technique (Fig. 1.11). The welding technique can be performed by the traditional fusion welding (the VPPA or GTA techniques). The presence of residual stresses in the heataffected zone (HAZ) and the formation of porosity in the nugget area by the fusion welding are not desirable and therefore must be minimized at all costs to maintain the safety of the structure. The new state-of-the-art welding technique called the friction stir welding (FSW) is recently used throughout the aerospace industry in order to avoid premature failure due to inherent residual stresses embedded in the fusion welding process if post-heat treatment is not feasible. For the COPV metallic liner, the post-heat treatment process is required by the program in order to eliminate the residual stresses. However, the presence of unshaved welds is another main source Outlet tube Dome Vertical weld

Circumferential weld cylinder Circumferential weld Dome

The composite Over-wrapped Pressure vessel

Fig. 1.11 Longitudinal and circumferential welding of a pressurized cylinder [12]

1 Virtual Testing and Its Application Fig. 1.12 Crack initiation and damage on the liner as the result of after the auto-frettage process

15

FM73 adhesive

Bulge inward as the result of compressive residual stresses

Damage on the liner as the result of dis-bond and compressive residual stresses after the auto-frettage process

of failure because of the localized buckling due to the presence of large compressive residual stresses that can occur after depressurization cycles. Under the compressive residual stresses, the occurrence of buckling is assumed to be as a result of (1) the geometrical imperfection due to the presence of a crown at the weld (unshaved weld can act as the source of geometrical imperfection when compared with the parent material) and (2) inadequacy of bond between the liner and the FM73 adhesive material (Fig. 1.12). It should be noted that the problem with shaving the weld is the undercutting issue that may further damage the liner, where the skin thickness can be less than the 0.05 in. It is essential for the COPV tanks to be exposed to several pressurization and depressurization cycles prior to their service usage. These cycles are the autofrettage process cycle, one proof test cycle, and several maximum operating pressure cycles, depending on the program requirement. The auto-frettage process induces global plastic deformation on the liner that can be as high as 5% straining (beyond the yielding) and, upon depressurization part of the cycle, the fracture properties of the liner degrade due to the work-hardening phenomenon. The material degradation, localized buckling along the length of the weld and localized de-bond between the liner and over-wrapped material can reduce the life of the liner considerably. In this work, the remaining safe-life capability of COPV metallic liner is assessed through the fracture mechanics approach. Several assumptions are made regarding the fracture allowables after the auto-frettage process, where the metallic liner material undergoes a severe global plastic deformation. It is assumed that after the auto-frettage process, crack initiation can occur and upon the depressurization phase, when stresses are compressive in the liner, localize buckling will take place (Fig. 1.12). However, for the remaining part of this chapter, discussions are allocated to generating fracture allowables of the liner (made of Inconel 718) after the global plastic deformation (subsequent to the auto-frettage process) by applying the virtual testing technique. As indicated earlier, due to the auto-frettage process, material properties change and there is a need to obtain new allowables for the safelife analysis of COPV liner. Due to intolerable amount of time that is needed to obtain fracture allowables through traditional testing, the virtual testing is a logical approach to implement in order to meet the deadline set forth by the program.

16

B. Farahmand

1.5.3 Generating Fracture Allowables of Inconel 718 of COPV Liner Through Virtual Testing Technique To generate fracture toughness and the fatigue crack growth curve for the Inconel 718 liner, the full stress–strain curve for the material must be available. First, the fracture toughness will be estimated and it will be utilized for generating the da/dN versus ΔK curve. The stress–strain curve prior to plastic deformation for the Inconel 718 is shown in Fig. 1.13, where it is taken from the Metallic Materials handbook [9]. The 5% permanent deformation after the auto-frettage process is also shown in the same figure. Note that the 5% straining is above the yielding point of material (i.e., 2%). After the 5% plastic deformation, when the load has been removed, the Inconel 718 will undergo work-hardening phenomenon. The new stress–strain curve after load removal is plotted as part of Fig. 1.13. Therefore, the new stress–strain curve after load removal will be different and is shown in Fig. 1.14, where the total strain is 0.07 in./in. versus 0.12 prior to the plastic deformation.

180

Stress-strain curve for Inconel 718

Stress, ksi

150

Stress-strain curve after 5% permanent deformation & unloading after autofrettage process

5%

0.12

Strain, in./in. Fig. 1.13 Stress-strain curve before and after the 5% plastic deformation

The area under the stress–strain curve is smaller in Fig. 1.14, which is an indication that the fracture toughness value after the auto-frettage process is lower. For this reason the analyst must use the correct value of Kc in the safe-life analysis assessment. Not considering this fact, the analysis results can be overestimated, which can result in catastrophic failure of liner and loss of the space vehicle during its service usage. Under this situation, the new fracture allowables can best be obtained through the virtual testing technique. Using this approach will help analysts to perform safelife analysis in a short time without compromising the safety of the structure. Based on the extended Griffith theory, values of both the plane strain and plane stress fracture toughness can be calculated from Equation (1.12). This equation can be easily programmed in an Excel sheet where the area under the stress–strain curve must be available as an input to the program.

1 Virtual Testing and Its Application 180 New yield value 170

Stress, ksi

Fig. 1.14 The area under the curve is reduced after the auto-frettage process (5% plastic deformation)

17

Estimated stress-strain curve after 5% plastic deformation

0.035

0.07

Strain, in./in.

Fig. 1.15 The variation of fracture toughness for the Inconel 718 before and after 5% plastic deformation

Fracture Toughness For Inconel 718 (Before & After 5% Plastid Deformation)

Fracture Toughness ksi (in.)^0.5

No Deformation

120

Analysis (5%)

100

KIc = 81 KIc = 70

80 60 40 20 0 0

1

2

3

4

Thickness, in.

Both the fracture toughness and the da/dN versus ΔK data can be generated from Fig. 1.14. Figure 1.15 is the variation of fracture toughness for the Inconel 718 before and after 5% plastic deformation. The plane strain fracture toughnesses before and after 5% straining were calculated by the virtual testing technique and are KIc = 81 and 70 ksi (in.)ˆ0.5, respectively. The “ksi (in.)ˆ0.5” refers to the unit of fracture toughness, KIc. Note that the yield value for the material after the workhardening process is higher than prior to the 5% straining due to the auto-frettage process (170 ksi vs. 150 ksi) and moreover, the total elongation is lower (12% vs. 7%), respectively. Both the increase in the yield value and reduction in percent elongation are main contributors to the drop in the fracture toughness value. The fatigue crack growth rate data must also be available for the case of 5% plastic deformation. In the upper portion of the da/dN versus ΔK data, where crack accelerates, the quantity ΔK approaches the material fracture toughness, Kc. Of course, the value of this quantity depends on the thickness of test specimen. The

18

Inconel 718 (Before & After Plastic Deformation) 0% Deformation

da/dN - In./Cycle

Fig. 1.16 Fatigue crack growth curve for the Inconel 718 before and after 5% Plastic deformation (0% plastic deformation data is provided from NASGRO database)

B. Farahmand

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E+00

5% Deformation

1.00E+01

1.00E+02

1.00E+03

ΔK-ksi (in.)^0.5

variation of Kc versus thickness is already shown in Fig. 1.15. The lower portion of the da/dN versus ΔK curve is related to the threshold value. As indicated previously (see Table 1.1), this value falls between Kth = KIc /4π and Kth = KIc /3π values. Conservatively, the value of Kth = KIc /4π is selected for this analysis. This is the lower-bound value of Kth for the stress ratio of R = 0. Based on the above statement, the threshold value before and after 5% plastic deformation should be 6.4 ksi-(in.)ˆ0.5 and 5.5, respectively. The following fracture mechanics analyses are conducted to check whether the liner is suitable for flight with the assumption that fracture allowables are lower after the 5% plastic deformation. Since the access to the liner is difficult, it is assumed that after the auto-frettage process with 5% plastic deformation, there is a possibility that cracks have been introduced in the material as deep as 95% through the thickness. Figure 1.17 shows the part through crack that is almost becoming a through crack as the result of plastic deformation. In the analysis, it is assumed the crack has initiated in the weld region. The weld will locally bulge when the applied load (causing the 5% plastic deformation) is removed, resulting in the induced residual compressive stresses through the liner. During the operation, when tank is pressurized for the operation, the bulge area may be eliminated and its original geometry Crack is 95% through the thickness

After unloading condition

2c

Fig. 1.17 The part through crack assumption (95% through the thickness) after the plastic deformation

Loading condition

a

0.025”

1 Virtual Testing and Its Application

19 Table 1.2

may be maintained. The safe-life analysis will assume (1) the initial flaw size is 95% through the thickness, (2) fracture allowables are reduced, and (3) the induced hoop stress is 150 ksi. The yield value is higher due to the work-hardening process, and its new value is 170 ksi as shown in Fig. 1.14. Tables 1.2 and 1.3 show results of life analysis conducted by the NASGRO computer code [8], where the abovementioned assumptions were used as the required input in the code. Two extreme cases of crack geometry were considered: (1) a circular crack with the aspect ratio of a/c = 1 and (2) the shallow crack with a/c = 0.2 (Fig. 1.18). The latter crack geometry has shorter life because the depth will advance faster in order to catch up with the crack length (Table 1.3). As shown in Table 1.3, the number of cycles to failure is zero and crack will propagate catastrophically upon the first load cycle with maximum stress of 150 ksi. However, if the crack geometry is a circular crack (a/c = 1) with the same depth, the analysis results show the tank will be able to handle additional 99 cycles prior to becoming a through crack. The following will be an additional example that shows the usefulness of the virtual testing technique when fracture allowable cannot be obtained through the traditional testing because the program is required to meet the customer’s deadline.

20

B. Farahmand Table 1.3

1.5.4 Generating Fracture Allowables of 6061-T6 Aluminum Tank Through Virtual Testing Technique A pressurized container is made of 6061-T6 aluminum alloy and is used as an oxygen tank for a space structure. As part of the requirement, the tank must undergo a proof test and several subsequent pressurization cycles equal to the maximum expected pressure cycle at the start of operation. The proof test is defined as one pressurization and depressurization cycle in excess of maximum expected operating pressure (MEOP). This is done in order to verify the structural integrity of tank [13]. During the proof test operation, the amount of pressure exceeded the required value and caused plastic deformation in most part of the tank. A decision was made by the program manager to use the tank if the analyst could show that it has adequate life during its service operation. Because of induced plastic deformation during the proof test, both the static and fracture properties of 6061-T6 tank are different and must be taken into the consideration when conducting static and fracture analyses. The amount of plastic deformation was estimated to be 5% (above the material yield value, 2%). Figure 1.19 shows the stress–strain curve before and after plastic deformation. Based on our previous discussions related to the COPV liner in

1 Virtual Testing and Its Application

21 The same depth but different length

Fig. 1.18 Two extreme cases of crack geometry were considered (a/c=1 and a/c=0.2) for life assessment analysis

Long Crack, a/c = 0.2

Circular Crack, a/c = 1

t = 0.025 in.

t = 0.025 in.

6061-T6 Stress-Strain Curve -RT (0% & 5% Work Hardening)

Stress, ksi

Fig. 1.19 The stress-strain curve before and after 5% plastic deformation

50 40 30 20 10 0

Longitudinal 5% Plastic Deformation

0

0.05 0.1 Strain, in./in.

0.15

Section 1.5.3, the fracture allowables for the 6061-T6 tank have been degraded due to the work-hardening phenomenon. In Section 1.5.3, the yield value of Inconel 718 has been elevated, but the area under the stress–strain curve was reduced. The same can be said about the above-mentioned tank. The fracture toughness value of 6061-T6 after the 5% plastic deformation can be estimated through the virtual testing technique and, subsequently, the da/dN versus ΔK data can be generated. Figure 1.20 shows the variation of fracture toughness versus material thickness. In part A, the fracture toughness versus thickness for the 0% work-hardening case (no plastic deformation) is compared with the experimental data extracted from the NASGRO database. Excellent agreement between test data and virtual testing can be seen. In part B, the fracture toughness versus thickness for two cases of 5% plastic deformation and prior to 5% plastic deformation (0% work hardening) are also plotted. It can be seen that for the case of 5% plastic deformation, the corresponding KIc value is lower (KIc = 27 versus 22 ksi (in.)ˆ0.5). The fatigue crack growth rate data for the 6061-T6 before and after 5% plastic deformation is shown in Fig. 1.21. The purpose of this plot is to show the capability and accuracy of virtual testing when compared with test data. This is shown in part A where the da/dN versus ΔK data for the material prior to the plastic deformation (0% work hardening) is compared with the NASGRO database. As can be seen from Fig. 1.21, the two data (virtual testing and test data) are almost on top of each other. In addition, the fatigue crack growth rate curves for the 5% plastic deformation and before plastic deformation are plotted in part B. It is clear that the da/dN curve for 5% plastic deformation is shifted to the left, which is an indication that fracture allowable has been reduced. Therefore, in safe-life analysis of pressurized tank after

22

B. Farahmand 6061-T6 Aluminum Alloy -RT

60

0% Work Hardening

0% Work Hardening

50

Fracture Toughness ksi (in.)^0.5

Fracture Toughness ksi (in.)^0.5

6061-T6 Aluminum Alloy -RT

A

40

KIc = 27

30 20

Virtual Testing

10

NASGRO Data

0 0

0.5

1

1.5

2

2.5

3

3.5

Thickness (in.)

60

5% Work Hardening

50 40

B

30 20 KIc = 22

10 0

0 0.5 1 1.5 2 2.5 3 3.5 Thickness (in.)

Fig. 1.20 The variation of fracture toughness versus material thickness before and after 5% plastic deformation

Fig. 1.21 Fatigue crack growth curves before and after 5% plastic deformation

the plastic deformation, the reduced fracture properties must be incorporated in the life analysis computer code.

1.6 Summary and Future Work The virtual testing technique and its application to the aerospace industry are discussed in detail. The method can be used to establish fracture allowables (fracture toughness and da/dN versus ΔK data) when both time and budget do not allow performing tests. These allowables must be available in order to conduct safe-life analysis of fracture critical parts. Two examples were provided by the author that demonstrates the need for this technology when the program must deliver the space hardware reliably to the customer on time. Numerous alloys were used in order to establish the validity of this technique. In all cases, excellent correlation between test data and the virtual testing method were found. Appendix shows several cases where virtual testing technique and test data were compared for randomly selected alloys extracted from NASGRO database. Only in the case of welds a few discrepancies were noticed between the two approaches. This was expected because of errors

1 Virtual Testing and Its Application

23

that can be encountered in testing due to (1) residual stresses and (2) porosity in the weld. Lack of data in the threshold region makes this technique desirable for engineers when designing parts for infinite life.The lack of test data in the threshold region is mainly because of time and cost associated with running the test in that region of the da/dN curve. In most cases, the threshold crack growth reading becomes complicated due to plasticity closure, surface roughness, and environmental effects (oxide particles) at the crack tip. The virtual testing technique utilizes the extended Griffith theory by estimating the amount of energy that has been consumed for straining material at the crack tip. A relationship between the applied stress and critical crack length was formulated, which can be used to calculate the critical value of stress intensity factor, Kc. The proposed technique uses the area under the stress–strain curve and relates the energy per unit volume of a uniaxial tensile test to the energy consumed at the crack tip. Therefore, this technique is required to have only the full stress–strain curve as an input to the theory for the material under study. As already emphasized throughout this chapter, the application of this technique in aerospace industry can be extremely useful. It can significantly reduce the cost and time of testing. It clearly indicates how the virtual testing technique can accurately estimate fracture allowables when they are needed to program for safe-life assessment of structural parts in order to meet the deadline set forth by the customer. Because of the dependency of this method on the stress–strain curve, the author believes there is a need to generate the full stress–strain curve through the multiscale modeling and simulation approach, therefore making the proposed method to become free from ASTM testing. This can be part of the future work in the field of virtual testing. One approach to the multi-scale modeling is the electron density functional theory that imposes deformation constraints to model the energetics of the stress–strain curve in metal and metal alloy systems [14]. In general, multiscale modeling efforts rely on atomistic- or molecular-level information to predict the behavior of the material in response to applied loadings and environmental conditions. The effects of grain boundaries and dislocation are directly incorporated into the prediction of large-scale material behavior using an appropriately chosen atomic potential [15]. Because the prediction of stress–strain behavior of aerospace alloys from molecular-based multi-scale approaches has not been rigorously pursued, future efforts must be focused on developing models for these materials.

Appendix To appreciate the usefulness of the virtual testing technique, numerous fatigue crack growth rate data (da/dN versus ΔK curve) for several aerospace alloys were plotted using the virtual testing approach. In all cases, data generated by this technique were compared with test data extracted from the NASGRO database. Excellent

24

B. Farahmand 2014-T6 ALUMINUM - R=0

2219-T87 Aluminum Alloy, R=0 Virtual Testing

1

da/dN-In./Cycle

da/dN (in./Cycle)

NASGRO Data

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09

10

100

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11

Virtual Testing

1

Δ K-ksi(in.)^0.5

10

100

Δ K -ksi (in.)^0.5

Fig. 1.22 Fatigue crack growth by virtual testing versus NASGRO database (2000 series aluminums) 6082-T651 Aluminum, R = 0

6061-T651 Aluminum, R = 0 NASGRO

Virtual Testing

1.00E–01 1.00E–03 1.00E–05 1.00E–07 1.00E–09 1.00E–11

da/dN -in./cycle

da/dN -in./cycle

NASGRO

1

10 Δ K ksi (in.)^0.5

100

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

100

ΔK -ksi (in.)^0.5

Fig. 1.23 Fatigue crack growth by virtual testing versus NASGRO database (6000 series aluminums)

agreement between the test data and virtual testing technique can be seen. The following is a list of a few alloys that have been used in manufacturing aerospace parts. Figures 1.22, 1.23 and 1.24 are 2000, 6000, and 7000 series aluminum alloys. Figures 1.25, 1.26 and 1.27 are unalloyed and binary, ternary, and quaternary titanium alloys. Figures 1.28, 1.29 and 1.30 are magnesium, copper–bronze, and Russian aluminum alloys. Figures 1.31, 1.32, and 1.33 are miscellaneous super-alloys, miscellaneous corrosion- and heat-resistance steel, and Ni alloys. In all cases, the fracture toughness data from the test data were used for the accelerated region to generate the da/dN versus ΔK curve. If the fracture toughness value is not available for the material, the extended Griffith theory can be used to provide the needed Kc value. However, as mentioned previously, the full stress–strain curve is required for the virtual testing analysis.

1 Virtual Testing and Its Application

25 7175-T7452Aluminum, R = 0 NASGRO

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11

Virtual Testing

da/dN -in./cycle

da/dN-In./Cycle

7010-T7451 T-L Aluminum, R = 0

1

10

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

100

Δ K -ksi (In.)^0.5

10

100

Δ K -ksi (in.)^0.5

Fig. 1.24 Fatigue crack growth by virtual testing versus NASGRO database (2000 series aluminums)

Unalloy Titanium (σ σ ys = 70ksi)

NASGRO

1.00E+00 1.00E–02 1.00E–04 1.00E–06 1.00E–08 1.00E–10

Virtual Testing

1

10 Δ K-ksi (in.)^0.5

da/dN-in/cycle

da/dN-in./cycle

Ti-2.5 Cu Titanium

100

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

100

Δ K-ksi (in)^0.5

Fig. 1.25 Fatigue crack growth by virtual testing versus NASGRO database (binary and unalloyed titanium alloys)

Ti-5Al-2.5Sn Titanium Annealed, R=0

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09

Virtual Testing

1

1

10

Δ K-ksi (In)^0.5

Fig. 1.26 alloys)

NASGRO

Virtual Testing

da/dN-in/cycle

da/dN, In./Cycle

Ti-3Al-2.5V, Titanium, CW +SR (750F)

100

10

100

1000

ΔK-ksi (in)^0.5

Fatigue crack growth by virtual testing versus NASGRO database (ternary titanium

26

B. Farahmand Ti-6Al-6V-2Sn, RA Tiatanium, R = 0

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

100

da/dN -in./cycle

da/dN-in/cycle

Ti-4.5Al-5Mo-1.5Cr Titanium -STA, R = 0

Δ K-ksi (in)^0.5

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09

Virtual Testing

1

10 ΔK -ksi (in)^0.5

100

Fig. 1.27 Fatigue crack growth by virtual testing versus NASGRO database (quaternary titanium alloys)

AZ-31B-H24-Magnesium, R = 0

AM 503 Magnesium as wrought, R = 0

NASGRO

Virtual Testing

da/dN-In./Cycle

da/dN-In/Cycle

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1

10

100

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11

Virtual Testing

1

10

Δ K -ksi(In.)^0.5

100

Δ K -ksi(In.)^0.5

Fig. 1.28 Fatigue crack growth by virtual testing versus NASGRO database (magnesium alloys)

AMg6M Russian Alloy-Annealed, R=0

ZW1 Magnesum as wrought, R=0

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

Δ K -ksi(In.)^0.5

da/dN-In./Cycle

da/dN-In./Cycle

NASGRO

100

NASGRO Virtual Testing

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1

10

100

ΔK -ksi(In.)^0.5

Fig. 1.29 Fatigue crack growth by virtual testing versus NASGRO database (Russian and magnesium alloys)

1 Virtual Testing and Its Application

27 Russian Alloy AMg6H Ann, R = 0

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

NASGRO

Virtual Testing

da/dN -In/Cycle

da/dN-In./Cycle

Al-Bronze CDA 630-R = 0

1.00E+00

Virtual Testing

1.00E–02 1.00E–04 1.00E–06 1.00E–08 1.00E–10 1

1

10

10

100

100

ΔK -ksi (in)^0.5

Δ K -ksi(In.)^0.5

Fig. 1.30 Fatigue crack growth by virtual testing versus NASGRO database (Al-Bronze and Russian zinc alloys)

Nitronic 50 Alloy -Anneal, R = 0 NASGRO

NASGRO Virtual Testing

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1

10

100

Δ K-ksi (in)^0.5

da/dN -in./cycle

da/dN-in/cycle

MP35N STA Super Alloy, R = 0

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09

Virtual Testing

1

10

100

1000

ΔK-ksi (in.)^0.5

Fig. 1.31 Fatigue crack growth by virtual testing versus NASGRO database (super alloy and corrosion resistance Nitronic alloys)

PH13-8Mo H1000 Alloys, R = 0

15-5PH H900 Alloy, R = 0

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09

Virtual Testing

da/dN-in/cycle

da/dN-in/cycle

NASGRO

1

10

100

Δ K-ksi (in)^0.5

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

100

ΔK-ksi (in)^0.5

Fig. 1.32 Fatigue crack growth by virtual testing versus NASGRO database (corrosion resistance alloys)

28

B. Farahmand Rene 41 ST(1950F); A(1400F 16h) Alloy, R = 0 NASGRO

NASGRO

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

Δ K-ksi (in)^0.5

da/dN-in/cycle

da/dN-in/cycle

17-4PH H1025 Alloy, R = 0

100

1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10

Virtual Testing

1

10

100

1000

ΔK-ksi (in)^0.5

Fig. 1.33 Fatigue crack growth by virtual testing versus NASGRO database (corrosion resistance and super alloys)

References 1. B. Farahmand, Fatigue & Fracture Mechanics of High Risk Parts, Chapman & Hall, 1997, Chapter 5 2. B. Farahmand, Fracture Mechanics of Metals, Composites, Welds, and Bolted Joints, Kluwer Academic Publisher, November 2000, Chapter 6 3. B. Farahmand, “Predicting Fracture and Fatigue Crack Growth Properties Using Tensile Properties, Engineering,” J. Fract., Vol.75, 2007, pp. 2144–2155 4. B. Farahmand, De Xie, F. Abdi, “Estimation of Fatigue and Fracture Allowables for Metallic Materials Under Cyclic Loading,” AIAA-2007 5. B. Farahmand, “Multiscaling of Fatigue” Application of virtual testing for obtaining fracture allowables of aerospace and aircraft materials. Springer, 2008 (A Book Chapter by G. Sih) 6. G. Irwin, “Fracture Dynamics,” Fracture of Metals, ASM, 1948, p. 147 7. E. Orowan, “Fracture and Strength of Solids,” Rep. Prog. Phys., Vol. 12, 1949, pp. 185–232 8. Fatigue Crack Growth Computer Program “NASGRO 4.0”, JSC, SRI, ESA, and FAA, 2002 9. MIL-HDBK-5H “Military Handbook Metallic Materials and Elements for Aerospace Vehicle Structure” 10. B. Farahmand, “Virtual Testing for Estimating Material Fracture Properties (Reducing Time & Cost of Testing),” 11th International Conference on Fracture (ICF11), Turin Italy, March 2005 11. NHB 8071 “Fracture Control Requirements for Payloads Using the National Space Transportation System (NSTS),” NASA, Washington, DC, 1985 12. Metal lined Composite Overwrapped Pressure Vessels (COPVs), Arde Inc., 875 Washington Avenue, Carlstadt, New Jersey 07072 13. Military Standard 1522A, “Standard General Requirements for Safe Design Operation of Pressurized Missile and Space System,” USA, May 1984 14. B. Farahmand and M. Doyle, “Obtaining Fracture Properties by Virtual Testing and Multiscale Modeling,” The Aging Aircraft Conference, 2009 15. B. Farahmand and G. Odegard, “Obtaining Fracture Properties by Virtual Testing and Molecular Dynamics Techniques,” The NSTI Conference, 2009

Chapter 2

Tools for Assessing the Damage Tolerance of Primary Structural Components R. Jones and D. Peng

Abstract Fatigue considerations play a major role in the design of optimised flight vehicles, and the ability to accurately design against the possibility of fatigue failure is paramount. However, recent studies have shown that, in the Paris Region, cracking in high-strength aerospace quality steels and Mil Annealed Ti–6AL–4V titanium is essentially R ratio independent. As a result, the crack closure and Willenborg algorithm’s available within commercial crack growth codes are inappropriate for predicting/assessing cracking under operational loading in these materials. To help overcome this shortcoming, this chapter presents an alternative engineering approach that can be used to predict the growth of small near-micron-size defects under representative operational load spectra and reveal how it is linked to a prior law developed by the Boeing Commercial Aircraft Company. A simple method for estimating the S–N response of 7050-T7451 aluminium is then presented. Keywords Fatigue crack growth · Fatigue modelling · Life prediction · Similitude

2.1 Introduction To achieve their design requirements, modern military make extensive use of aluminium, high-strength steels, that is, 4340 and D6ac, and titanium. The Joint Strike Fighter (F-35), the Super Hornet and the F/A-18 make extensive use of 7050-T7451 aluminium. However, there has been an increasing use of titanium in primary structural members due to its high strength, light weight, and good fatigue and fracture toughness properties. As a result, bulkheads in the F-22, the Super Hornet, the Swiss F/A-18, and the Joint Strike Fighter are made of titanium. In the F-22, titanium accounts for ∼36%, by weight, of all structural materials used in the aircraft. R. Jones (B) Department of Mechanical and Aerospace Engineering, DSTO Centre of Expertise for Structural Mechanics, Monash University, Victoria 3800, VIC, Australia e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 2,

29

30

R. Jones and D. Peng

Until recently, it had been thought that fatigue crack growth in 7050-T7451, highstrength aerospace steels and titanium was well understood. However, in his review of fatigue crack growth under variable amplitude loading, Skorupa [1] concluded that, viz: Experimental results also suggest that the underlying causes of load interaction phenomena are not necessarily similar for different groups of metals, e.g. steels of and Al and Ti alloys.

Furthermore, as a result of the Australian Defence Science and Technology Organisation’s Flaw IdentificatioN through the Application of Loads (FINAL) testing program [2] it is now known [3, 4] that similitude-based concepts on which the crack growth programs AFGROW, NASGRO, and FASTRAN are based cannot be used to accurately predict the growth of near-micron-size flaws in 7050-T7451 aluminium alloy under representative in-flight loading. In this context it should also be noted that Forth, James, Johnston, and Newman [5] have reported that crack growth data obtained for D6ac and 4340 steels using compact tension (CT) specimens tested in accordance with the ASTM standards exhibited no R ratio dependency and hence no closure in the Paris region (Region II), see Fig. 2.1. 1.00E–05

Fig. 2.1 Fatigue crack growth data from D6AC steel. Plot reproduced from [5]

R = 0.7 1.00E–06

R = 0.7

da/dN (m/cycle)

R = 0.7 1.00E–07 R = 0.1 1.00E–08

R = 0.1 R = 0.3

1.00E–09 R = 0.3 1.00E–10

1.00E–11 1

10

100

ΔK MPa √ m

This behaviour, that is, the da/dN versus ΔK relationship appearing to be R ratio dependent in Region I but showing no R ratio dependence and hence no closure in the Paris region, is also evident in the work of James and Knott [6] who studied cracking in QIN (HY80) steel, see Fig. 2.2. As such, the various closure-based models and the Willenborg crack growth law, which models load interaction and sequence effects by modifying the effec-

2 Tools for Assessing the Damage Tolerance

31

1.00E–04

Fig. 2.2 Fatigue crack growth data from QIN (HY80) steel. Plot reproduced from [6]

R = 0.7 R = 0.5

da /dN (m/cycle)

1.00E–05

R = 0.35 R = 0.2 1.00E–06

1.00E–07

1.00E–08 1

10

100

Δ K MPa √m

tive R ratio, available within these codes cannot be used to accurately predict crack growth in high-strength steels. Jones, Farahmad, and Rodopoulos [7] have revealed that Mil Annealed Ti–6AL–4V titanium has a similar (near) R ratio independence. As such the various closure-based models and the Willenborg crack growth law cannot be used to accurately predict crack growth in Ti–6AL–4V. (This R ratio independence has also been seen in crack growth in rail steels [8] which have also been found to conform to the generalised Frost–Dugdale crack growth law [8, 9].) When addressing the question of crack growth under representative in-service loading it should also be noted that in the review paper on crack growth and similitude Davidson [10] concluded that similitude was lost during fatigue crack growth under variable amplitude loading and stated that: “Detailed measurements of fatigue cracks undergoing simple load spectra confirm that when ΔKeff is based on Kopen , good correlations are achieved with large crack growth data.This understanding, although useful, does not easily translate to an engineering method for computing crack growth rate under complex variable amplitude loading.” The question thus arises: How can a valid virtual assessment of the performance of an aircraft/rail component under representative operational loading be performed if the fundamental concepts inherent in the existing crack growth codes, viz: AFGROW, FASTRAN, and NASGROW, do not apply to the materials from which the component fabricated, that is, for components made out of 4340 and D6ac steel, QIN (HY80) steel, rail steels, Mil Annealed Ti–6AL–4V, STOA Ti– 6AL–4V, etc.? This chapter presents one possible approach which is based on the equivalent block formulation presented in [8, 11, 18] and reveals how it is linked to spectra where the constant amplitude Region II growth mechanism tends to be suppressed and a single value of C∗ can be used to predict the crack length versus cycles history.

32

R. Jones and D. Peng

2.2 An Equivalent Block Method for Predicting Fatigue Crack Growth It is now known that the mechanisms underpinning crack growth under variable amplitude load differ from those seen under constant amplitude loading [12]. It is also known that many materials either follow a non-similitude-based crack growth law [3, 4, 8, 9], lose similitude as the crack grows [10], or exhibit a near R ratio independence in the Paris Region [5–8]. In these cases, crack growth under representative operational loading cannot be predicted using the concepts inherent in the existing crack growth codes, viz: AFGROW, FASTRAN, and NASGROW, since they do not apply to the materials from which the component is fabricated, and since the data used in these calculations are obtained from constant amplitude tests that may not reflect the mechanisms driving growth under the spectrum of interest [12]. However, many practical engineering problems, that is, cracking in rail and aircraft structures, involve complex load spectra that can be approximated by a number of repeating load blocks. Schijve [13], Gallagher, and Stalnaker [14], Miedlar, Berens, Gunderson, and Gallagher [15], Barsom and Rolfe [16] and Miller, Luthra, and Goranson [17] revealed that these repeated blocks of loads can, in certain circumstances, be treated as equivalent to load cycles. We now show how this concept, that is, an equivalent block approach, can be used to describe crack growth in Mil Annealed Ti–6AL–4V and D6ac steel under complex variable amplitude loading. To this end let us consider the case of block loading, where each block consists of a spectrum with n cycles that have peak stresses of σ i , i = 1. . . n, with the associated cyclic ranges being Δσ i , i = 1. . . n. Let us also assume that: (i) The slope of the a versus block curve has a minimal number of discontinuities. (ii) There are a large number of blocks before failure. With these assumptions, Jones and Pitt [18] derived an “equivalent block” variant of the generalised Frost–Dugdale crack growth law [4, 8] to account for the crack growth per block, da/dB, viz da/dB = C˜ K max γ a 1−γ/2

(2.1)

where C˜ is a spectra-dependent constant and Kmax is the maximum value of the stress intensity factor in the block. (The precise relationship between C˜ and the constant of proportionality in the Paris crack growth law is yet to be determined.) Jones, Molent, and Krishnapillai [11] subsequently extended this “equivalent block” law to have a form consistent with regions I, II, and III, viz ˜ 1−γ/2 K max γ − da/dBo )/(1.0 − K max /K c ) da/dB = (Ca

(2.2)

where a is now the average crack length in the block, and Kc is the apparent cyclic fracture toughness. Here, as described in [11], the term da/dBo reflects the both nature of the discontinuity from which the crack initiates and the apparent fatigue

2 Tools for Assessing the Damage Tolerance

33

threshold for this particular block loading spectra. However, it should (again) be stressed that this variant of the generalised Frost–Dugdale law is only applicable to crack growth data where the slope of the a versus block curve has minimal discontinuities and there are a large number of blocks to failure, see [8, 11, 18]. This formulation is (extensively) validated in [8, 11]. At this stage it is important to note that Miller, Luthra, and Goranson [17], at the Boeing Commercial Aircraft Company, have also developed a related (nonsimilitude) approach whereby instead of Equation (2.2) da/dB was expressed as

da/dB = C(K /g(a/t))m

(2.3)

where the function g(a/t), which is a function of ratio of the crack length (a) to the thickness (t) of the specimen, was experimentally determined and its functional form is presented in [17]. This formulation was necessary to enable the predictions to match the measured crack length histories. However, Jones, Pitt, and Peng have shown [8] that the experimental test data used in [17] to determine the function g(a/t) followed the generalised Frost-Dugdale crack growth law so that the two methodologies essentially coincide. In the next section we present three examples that illustrate how the present nonsimilitude approach, that is, Equation (2.2), can be used to accurately predict crack growth in 7050-T7451, D6ac steel, and Mil Annealed Ti–6AL–4V aluminium specimens subjected to complex variable amplitude load spectra.

2.3 Fatigue Crack Growth under Variable Amplitude Loading The first problem considered is that of crack growth in the 1969 General Dynamics, now Lockheed Martin Tactical Aircraft Systems (LMTAS), F-111 wing fatigue tested under a representative F-111 usage spectra. (An early F-111 in-flight failure was largely responsible for the USAF adopting a damage tolerance approach.) In this test, cracking was measured at a cut-out location designated as fuel flow hole 58 [19] on the lower (tension) surface of the D6ac steel wing pivot fitting, see Figs. 2.3 and 2.4. Before attempting to predict crack growth in the pivot fitting we first confirmed that growth in D6ac steel conformed to the generalised Frost–Dugdale law. This was done via a collaborative project with Dr. Scott Forth at NASA [20]. As part of this project we examined the results of a detailed NASA study into crack growth in D6ac steel CT specimens. The test matrix evaluated is given in Table 2.1. In this study it was found, √ see Fig. 2.5, that √ if we restrict ourselves to regions where Kmax < 115.0 Ksi in ( = 125 MPa m) then the data conforms to the generalised Frost–Dugdale crack growth law, viz da/dN = 8.12 × 10−9 a (1−γ/2) (Δκ)γ − 2.79 × 10−7

(2.4)

34

R. Jones and D. Peng

Fig. 2.3 Full 3D F-111 model, from DSTO 734247 694766 655285 615804 576323

Mousehole 58

536842 497361 457880 418399 378917 339436 299955 260474 220993 181512 142031 102550 63069

Fig. 2.4 Interior of the DSTO 3D F111 model

where the value of γ = 2.6 was taken from Murtagh and Walker [19] and where as per Walker [21] we have defined the crack driving force as Δκ = K max (1− p) ΔK p

(2.5)

where a value of p = 0.95 was found to best collapse the data. This low value of p confirmed the finding reported in [5] that the crack increment per cycle (da/dN) essentially has no R ratio dependency. Having established that crack growth in D6ac steel conforms to the Generalised Frost–Dugdale law we assumed that in the 1969 wing tests there was, as reported in

2 Tools for Assessing the Damage Tolerance

35

Table 2.1 Test matrix Test frequency Hz Ct3-5-tl Ct3-10b-lt Ct3-12-lt Ct3-25-lt Ct3-27-lt Ct3-29-lt Ct3-46-lt Ct3-47-lt

Constant Kmax = 15 Constant R = 0.3 LI Constant R = 0.9 LI Constant R = 0.7 LI Constant R = 0.9 LI Constant R = 0.3 LI R = 0.1 LI R = 0.8 LI

18 20 20 20 22 10 20 10

LI = Load increasing test, Kmax = constant Kmax test. 1.0E–02

Fig. 2.5 Crack growth in D6ac steel, from [20]

C3-5-lt Ct3-12-lt

1.0E–03

da /dN (mm/cycle)

Ct3-27-lt

1.0E–04

Ct3-25-lt Ct3-47-lt

1.0E–05 Ct3-46-lt Equation (4)

1.0E–06

y = 8.12E–09x – 2.79E–07 R2 = 1.00

Ct3-29-lt

1.0E–07 Ct3-10b-lt

1.0E–08 1

10

(ΔK

100 0.95 K

1000

10000 100000 1E+06

0.05)2.6/a 0.3 MPa2.6 m

max

[19], an initial 0.19-mm semi-circular flaw. At each increment of crack growth, the stress intensity factors were computed using a weight function technique together with the stress field determined from the finite element model shown in Figs. 2.3 and 2.4. Crack √ growth was then predicted using Equation (2.2) with γ = 2.6 and KC = 87 MPa m, as given in [19], and C˜ = 3.0 × 10−6 . The load spectra used in the 1969 test, and in this study, was provided by the Australian Defence Science and Technology Organisation (DSTO) and corresponds to that used in [19]. The resultant predicted crack depth histories are presented in Fig. 2.6 where we see good agreement between the predicted and the measured crack depth histories. In this example, when using Equation (2.2) to compute crack growth at the deepest point of the semi-elliptical surface flaw the quantity ‘a’ on the left- and the righthand sides of Equation (2.2) is the crack depth. Similarly, when using Equation (2.2) to compute crack growth at the surface points, the quantity ‘a’ on the left- and the

36

R. Jones and D. Peng 10 LMTAS Experimental

Crack depth (mm)

Equivalent Block

1

0.1 0

1000 2000 3000 4000 5000 6000 7000 8000 Simulated Flight Hours

Fig. 2.6 Measured and predicted crack growth in the 1969 F-111 wing test

right-hand side of Equation (2.2) is the half crack surface length. In this fashion, we allow for the variation of the crack aspect ratio during crack growth.

2.3.1 Fatigue Crack Growth in an F/A-18 Aircraft Bulkhead The next problem considered involved cracking in an F/A-18 Y488 bulkhead tested as part of the DSTO Flaw IdeNtification through the Application of Loads (FINAL) test program, see Dixon et al. [2]. This test program utilised ex-service Canadian Forces (CFs) and US Navy (USN) wing attachment centre barrel (CB) sections loaded using an industry-standard-modified mini-FALSTAFF spectrum, see [2], which is representative of flight loads seen by fighter aircraft. Since cracking in the bulkhead was three-dimensional, a three-dimensional FE model was required, see Figs. 2.7 and 2.8. The location of the crack is shown in Fig. 2.8, where node 4390 represents the centre of the initial semi-elliptical surface flaw. This problem had previously been studied using a cycle-by-cycle approach [4] and it was known that cracking in 7050-T7451 conformed to the generalised Frost–Dugdale law, [4, 22]. As in [4] we again used a weight function technique together with the stress field as determined from the FE model of the bulkhead to compute the associated stress intensity factors. The crack growth history from initial equivalent pre-crack sizes (EPS) of√0.003 mm was predicted using Equation (2.2) with γ = 3.36 and Kc of 35.4 MPa m as given in [4] and C˜ = 2.25 × 10−10 .

2 Tools for Assessing the Damage Tolerance

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Fig. 2. 7 The bulkhead structure

Node 4390

Fig. 2.8 The local mesh

The predicted crack depth history, allowing for changes in the aspect ratio of the flaw as the crack grows, is shown in Fig. 2.9 together with the associated experimental test result, where we see that there is very good agreement. Figure 2.9 also contains a comparison with predictions, presented in [4], made using FASTRAN II. Here we see that FASTRAN II predicted a very long fatigue life. Furthermore,

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Fig. 2.9 Experimental and predicted crack growth histories

10

Crack Depth (mm)

Y488 Experimental Results 1

FASTRAN Average Block Solution

0.1

0.01

0.001 0

20000

40000

60000

80000

Flight Hours

the shape of the crack depth versus cycles curve predicted by FASTRAN II differed markedly from the test data. In Fig. 2.9 we see that the experimental and predicted (from Equation 2.2) crack depth histories show a behaviour that is consistent across three decades of crack lengths, that is, from 0.003 mm to more than 5 mm.

2.3.2 Crack Growth in Mil Annealed Ti–6AL–4V under a Fighter Spectrum Jones, Farahmad, and Rodopoulos [7], who analysed the data presented in [23, 24], found that crack growth in Mil Annealed Ti–6AL–4V titanium was essentially R ratio independent, see Fig. 2.10. Figure 2.10 shows that cracking in Mil Annealed Ti-6AL-4V also appears to conform to the generalised Frost–Dugdale law, viz da/dN = C ∗ a (1−γ/2) (Δκ)γ − da/dN0

(2.6)

with C∗ ∼ 2.5 10–11 , γ = 2.5, p = 0.08, Kc = 100 MPa √ Δκ as given in Equation (2.5) –9 m and da/dN0 = 4.45 × 10 . As explained in [4, 8, 9] the term da/dN0 reflects both the nature of the discontinuity from which the crack initiates and the apparent fatigue threshold. The small value of p reveals that crack growth in Mil Annealed Ti–6AL–4V titanium has a very weak R ratio dependency. We also see that this relationship, that is, Equation (2.6), holds over 3 orders of magnitude, that is, 2 × 10–9 < da/dN < 2 × 10–6 . It should also be noted that this value of γ compares well with that of γ = 2.6 obtained by Zhuang et al. [25] for Mil Annealed Ti–6AL–4V tested under spectrum loading. With this in mind let us now examine the crack growth data presented by Northrop-Grumman [26] who studied crack growth in 6-inch-wide and 0.289-inchthick centre cracked Mil Annealed Ti-6AL–4V panels subjected to a fighter load

2 Tools for Assessing the Damage Tolerance

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1.00E–05

Fig. 2.10 Crack growth in Mil-Annealed Ti–6AL–4V, from [7]

y = 2.56E–11 x – 4.45E–09 R2 = 0.984

da/dN (m/cycle)

1.00E–06

1.00E–07 R = 0.85 R= 0.66 R = 0.43 R = 0.25

1.00E–08

1.00E–09 100

1000

10000

100000

1000000

(Δ K /(1–R)0.08)2.5/a 0.25 MPa2.5 m

spectrum with a peak remote stress of 103 ksi (710 MPa). The resultant predictions are shown in Fig. 2.11 where we again see an excellent agreement between the measured and the computed crack length histories. In this case, the left hand √ side ˜ = 2.83 × 10−10 , γ = 2.5, and Kc = 150 ksi in of Equation (2.2) is da/dBlock, C √ (163 MPa m). 100

Measured

a (mm)

Fig. 2.11 Grumman centre cracked panel crack growth under a fighter spectra

Predicted

10 240

260

280 Blocks

300

320

The above examples illustrate how the equivalent block method may be used to simulate crack growth under variable amplitude loading both for aluminium alloys and for the materials that exhibit minimal R ratio dependency. However, it must be stressed that this approach has a number of fundamental requirements, viz:

40

R. Jones and D. Peng

(i) There are a large number of blocks before failure. (ii) The slope of the a versus block curve has a minimal number of discontinuities. Applications of this methodology to a range of aluminium alloys as well as to cracking under a Helicopter load spectra and spectra corresponding to several control points in the Joint Strike Fighter are given in [8, 11, 27, 28]. White, Barter, and Molent [12] studied block loading which consisted of a large number of variable amplitude loads interspersed with a single block of constant amplitude loading. They found that at the onset of the constant amplitude loading, the crack changed planes and subsequently reverted back to its original plane after the constant amplitude loading ceased. This indicated that the mechanism’s driving constant amplitude and variable amplitude loading differed and that, in the Paris region, the constant amplitude mechanism was suppressed during variable amplitude loading. This observation explains why, in the examples presented above, only one value of C∗ is needed to represent crack growth. At this stage it should be noted that Liu [29] has shown that the Frost–Dugdale law has different slopes in regions I and II. Tiong and Jones [30] revealed that for aluminium alloys the value of C∗ in Region II is approximately 5 times its Region I value. However, when the Region II growth mechanism is suppressed crack growth can be predicted using the C∗ value associated with Region I.

2.4 A Virtual Engineering Approach for Predicting the S–N Curves for 7050-T7451 Section 2.1 when taken together with the cycle-by-cycle study presented by Jones, Molent, and Pitt [4] illustrates the ability of the Generalised Frost–Dugdale law to simulate the growth of near-micron-size flaws in 7050-T7451 aluminium alloy. As a result, it is possible to use this formulation to derive the S–N curve for 7050T7451. To illustrate this approach let us assume that the material contains a small semi-circular surface initial defect and that it retains this semi-circular shape during growth. Then √ ΔK = FΔσ (aπ)

(2.7)

where Δσ is the remote stress, F is a geometry factor, also termed β, which is also commonly called a boundary correction factor. For a small three-dimensional semielliptical surface flaw we can approximate F as F = 2 × 1.12/π

(2.8)

√ ΔK = (2 × 1.12/π)Δσ (a/π)

(2.9)

so that

2 Tools for Assessing the Damage Tolerance

41

To account for R ratio effects in aluminium alloys under constant amplitude loading, we will adopt Newman’s [31] proposal that ΔK be replaced by ΔKeff , which for the present problem we can express as √ ΔK eff = (1 − σo /σmax )(2 × 1.12 σmax (a/π))

(2.10)

where σ o is the so-called “crack opening stress”, see Appendix. However, when performing crack growth calculations for surface flaws, FASTRAN-II [31] only uses 0.9 of this value, that is, ΔK eff (in calcs.) = 0.9(1 − σo /σmax )(2 × 1.12 σmax

√

(a/π))

(2.11)

so that √ da/dN = C ∗ (0.9(1 − σo /σmax )(2 × 1.1.2 σmax / π)))γ a

(2.12)

Integrating Equation (2.12) gives √ N = ln(af /ai )/C ∗ (2 × 0.91.12 × (1 − σo /σmax )σmax / (π))γ

(2.13)

where ai is the initial defect size, which as shown by Molent et al. [32], for 7050T7451 aluminium has a mean value of ∼10 microns and af is the crack size at failure.

2.4.1 Computing the Endurance Limit If we say that there will be no growth if the computed value of da/dN (at the initial flaw size ai ) is less than a critical value then this will give an endurance stress. In this work we will take this value to be between 1–2 × 10–10 m/cycle. This produces a different endurance limit for each stress. For 7050-T7451 C∗ = 1.21 × 10–12 and γ = 3.36. The resultant predicted S–N curve is plotted in Fig. 2.12 along with the associated Mil Handbook 5 S–N curve. Note that the yield stress for this material in the thick plate condition is in the range 455–496 MPa (66–72 ksi).

2.5 Conclusion The Australian Defence Science and Technology Organisation’s Flaw IdentificatioN through the Application of Loads (FINAL) testing program revealed that the crack growth programs AFGROW, NASGRO, and FASTRAN cannot be used to accurately predict the growth of near-micron-size flaws in 7050-T7451 aluminium alloy under representative in-flight loading. This paper has shown that the Region II crack growth data reveals that cracking in high-strength aerospace quality steels and

42

R. Jones and D. Peng Predicted R = 0 Predicted R = 0.5 Mil Hndbk R = 0 Mil Hndbk R = 0 Fit Mil Hndbk R = 0.5 data Mil Hndbk R = 0.5 Fit Predicted R = –1 Mil Hndbk R = –1 Mil Hndbk R = –1 Fit

S max (ksi)

100

Predicted endurance limit(s) if we impose the requirement that da/dN must be greater than 1- 210– 10

m/cycle

10 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 N, Cycles Fig. 2.12 Measured and predicted S–N curves for 7050-T7451 aluminium alloy

Mil Annealed Ti–6AL–4V titanium is essentially R ratio independent. As a result, the crack closure and Willenborg algorithm’s available within commercial crack growth codes are also inappropriate for predicting/assessing cracking under operational loading in these materials. To help overcome this shortcoming this chapter has presented an alternative engineering approach that is linked to the formulation developed by the Boeing Commercial Aircraft Company, which can be used to predict the growth of small near-micron-size defects under representative operational load spectra. This approach: i. is generally consistent with experimental results, ii. can be used to predict crack growth from near-micron-size initial flaws, and iii. has the potential to accurately predict crack growth in real aircraft structures under complex load spectra. However, it should be stressed that this variant of the Generalised Frost–Dugdale law is only applicable to crack growth data where the slope of a versus block curve has minimal discontinuities and there are a large number of blocks to failure. In such cases the constant amplitude Region II growth mechanism tends to be

2 Tools for Assessing the Damage Tolerance

43

suppressed and a single value of C∗ can be used to predict the crack length versus cycles history. Acknowledgments This work was performed under the auspices of the DSTO Centre of Expertise in Structural Mechanics which is supported by the RAAF Directorate General Technical Airworthiness Air Structural Integrity Section. We specifically acknowledge the support given by Lorrie Molent, Functional Head Combat Aircraft (Structural Integrity), Dr Weiping Hu, Science Team Leader: Structural Lifing Methods and Tools, Dr. Scott Forth at the NASA Johnson Space Center, Prof. Chris Rodopoulos at the University of Patras, Greece, and the Materials and Engineering Research Centre, Sheffield Hallam University, England, and Dr. Bob Farahmand at TASS (Los Angeles).

Appendix: Formulae for Computing the Crack Opening Stress Newman [31] defined an opening load, which he denoted as S0 , as: S0 /S max = A0 + A1 R + A2 R 2 + A3 R 3 for R ≥ 0

(2.14)

S0 /S max = A0 + A1 = R for R < 0

(2.15)

and

for Smax < 0.8σ0 , Smin > −σ0 , where Smax and Smin are the maximum minimum stress in the cycle and σ 0 is the yield stress. If S0 /Smax is less than R then S0 = Smin , whilst if S0 /Smax is negative then S0 /Smax = 0.0. The A j coefficients in Equations (2.14) and (2.15) are functions of α, the constraint factor, and Smax /σ0 and are given in [31] as: Ao = (0.825 − 0.34α + 0.05α 2 )[COS(π Smax F/2σ0 ]1/α

A1 = (0.415 − 0.071α)Smax F/σ0

A2 = 1−A0 − A1 −A3

A3 = 2A0 + A1 − 1

(2.16)

for α = 1 to 3. The boundary correction factor, F, accounts for the influence of finite width on the stresses required to propagate the crack. For 3D small 3D surface cracks we can approximate F as F ∼ 2 × 1.12/π.

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References 1. M. Skorupa, “Load Interaction Effects During Fatigue Crack Growth Under Variable Amplitude Loading—A Literature Review. Part II: Qualitative Interpretation,” Fatigue Fract. Eng. Mater. Struct., Vol. 22, 1999, pp. 905–926. 2. B. Dixon, L. Molent, and S.A. Barter, “The FINAL program of enhanced teardown for agile aircraft structures,” Proceedings of 8th NASA/FAA/DOD Conference on Aging Aircraft, Palm Springs, 31 Jan–3 Feb, 2005. 3. L. Molent, R. Singh, and J. Woolsey, “A method for evaluation of in-service fatigue cracks,” Eng. Fail. Anal., Vol. 12, 2005, pp. 13–24. 4. R. Jones, L. Molent, and S. Pitt, “Crack growth from small flaws,” Int. J. Fatigue, Vol. 29, 2007, pp. 658–1667. 5. S.C. Forth, M.A. James, W.M. Johnston, and J.C. Newman, Jr., “Anomolous Fatigue Crack Growth Phenomena in High-strength Steel,” Proceedings Int. Congress on Fracture, Italy, 2007. 6. M.N. James and J.F. Knott, “An Assessment of Crack Closure and the Extent of the Short Crack Regime in QlN (HY80) Steel,” Fatigue Frac. Eng. Mater. Struc., Vol. 8, No. 2, 1985, pp. 177–191. 7. R. Jones, B. Farahmand, and C. Rodopoulos, “Fatigue crack growth discrepencies with stress ratio,” Theor. Appl. Frac. Mech., doi: 10.1016/tafmec.2009.01.004. 8. R. Jones, S. Pitt, and D. Peng, “The Generalised Frost–Dugdale Approach to Modeling Fatigue Crack Growth,” Eng Fail Anal, 15, 2008, pp. 1130–1149. 9. R. Jones, B. Chen, and S. Pitt, “Similitude: Cracking in Steels,” Theor. Appl. Frac. Mech., Vol. 48, No. 2, pp. 161–168. 10. D.L. Davidson, “How Fatigue Cracks Grow, Interact with Microstructure, and Lose Similitude,” Fatigue and Fracture Mechanics: 27th Volume, ASTM STP 1296, R.S. Piascik, J.C. Newman, and N.E. Dowling, Eds., American Society for Testing and Materials, 1997, pp. 287–300. 11. R. Jones, L. Molent, and K. Krishnapillai, “An Equivalent Block Method for Computing Fatigue Crack Growth,” Int. J. Fatigue, Vol. 30, 2008, pp. 1529–1542. 12. P. White, S.A. Barter, and L. Molent, “Observations of Crack Path Changes Under Simple Variable Amplitude Loading in AA7050-T7451,” Int. J. Fatigue, Vol. 30, 2008, pp. 1267–1278. 13. J. Schijve, “Fatigue Crack Growth Under Variable-Amplitude Loading,” Eng. Frac. Mech., Vol. 11, 1979, pp. 207–221. 14. J.P. Gallagher and H.D. Stalnaker, “Developing Normalised Crack Growth Curves for Tracking Damage in Aircraft, American Institute of Aeronautics and Astronautics,” J. Aircraft, Vol. 15, No. 2, pp. 114–120. 15. P.C. Miedlar, A.P. Berens, A. Gunderson, and J.P. Gallagher, “Analysis and Support Initiative for Structural Technology (ASIST),” AFRL-VA-WP-TR-2003-3002, 2003. 16. J.M. Barsom and S.T. Rolfe, “Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics,” Butterworth-Heinemann Press, 1999. 17. M. Miller, V.K. Luthra, and U.G. Goranson, “Fatigue Crack Growth Characterization of Jet Transport Structures,” Proc. of 14th Symposium of the International Conference on Aeronautical Fatigue (ICAF), Ottawa, Canada, 1987. 18. R. Jones, and S. Pitt, “Crack Patching: Revisited,” Comp. Struct., Vol. 32, 2006, pp. 218–223. 19. B.J. Murtagh and K.F. Walker, “Comparison of Analytical Crack Growth Modelling and the A-4 Wing Test Experimental Results for a Fatigue Crack in an F-111 Wing Pivot Fitting Fuel Flow Hole Number 58”, DSTO-TN-0108, 1997. 20. R. Jones and S.C. Forth, “Cracking In D6ac Steel,” Submitted J. Theor. Appl. Fract. Mech., 2008 (in press).

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21. E.K. Walker, “The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7076-T6 Aluminium.” In: Effect of Environment and Complex Load History on Fatigue Life, ASTM STP 462, American Society for Testing and Materials, Philadelphia, 1970, pp. 1–14. 22. R. Jones, C. Wallbrink, S. Pitt, and L. Molent, “A Multi-Scale Approach to Crack Growth,” Proceedings Mesomechanics 2006: Multiscale Behavior of Materials and Structures: Analytical, Numerical and Experimental Simulation, Porto, Portugal, 2006. 23. C.M. Hudson, “Fatigue-Crack Propagation in Several Titanium and One Superalloy StainlessSteel Alloys, NASA TN D-2331, 1964. 24. T.R. Porter, “Method of Analysis and Prediction for Variable Amplitude Fatigue Crack Growth,” Eng. Fract. Mech., Vol. 4, 1972, pp. 717–736. 25. W. Zhuang, S. Barter, L. Molent, “Flight-By-Flight Fatigue Crack Growth Life Assessment,” Int J Fatigue, Vol. 29, 2007, pp. 1647–165. 26. P.D. Bell and M. Creager, “Crack Growth Analysis For Arbitrary Spectrum Loading,” Volume I – Results and Discussion, Final Report: June 1972 – October 1974, Technical Report AFFDL-TR-74-129, 1974. 27. R. Jones, S. Pitt, and D. Peng, “An Equivalent Block Approach to Crack Growth,” In: Multiscale Fatigue Crack Initiation and Propagation of Engineering Materials: Structural Integrity and Microstructural Worthiness, G.C. Sih, Ed., ISBN 978-1-4020-8519, Springer Press, 2008. 28. L. Molent, S. Barter, and R. Jones, “Some Practical Implications of Exponential Crack Growth,” In: Multiscale Fatigue Crack Initiation and Propagation of Engineering Materials: Structural Integrity and Microstructural Worthiness, G.C. Sih, Ed., ISBN 978-1-4020-8519, Springer Press, 2008. 29. H.W. Liu, Crack Propagation in Thin Metal Sheet Under Repeated Loading, Wright Air Development Center, WADC TN, 1959, pp. 59–383. 30. U.H. Tiong and R. Jones, “Damage Tolerance Analysis of a Helicopter Component,” Int. J. Fatigue, 2008 doi:10.1016/j.ijfatigue.2008.05.012 31. J.C. Newman, Jr., FASTRAN-II- A fatigue Crack Growth Structural Analysis Program, NASA Technical Memorandum 104159, 1992. 32. L. Molent, Q. Sun and A.J. Green, “Characterisation of equivalent initial flaw sizes in 7050 aluminium alloy,” Fatigue Fract. Engng. Mater Struct., Vol. 29, 2006, pp. 916–937.

Chapter 3

Cohesive Technology Applied to the Modeling and Simulation of Fatigue Failure Spandan Maiti

Abstract Estimation of fatigue and fracture properties of materials is essential for the safe life estimation of aging structural components. Standard ASTM testing procedures being time consuming and costly, computational methods that can reliably predict fatigue properties of the material will be very useful. Toward this end, we present a computational model that can capture the entire Paris curve for a material. The model is based on a damage-dependent irreversible cohesive failure formulation. The model relies on a combination of a bilinear cohesive failure law and an evolution law relating the cohesive stiffness, the rate of crack opening displacement, and number of cycles since the onset of failure. Threshold behavior of the fatigue crack propagation is determined by the initial value of the damage parameter of the cohesive failure law, while the accelerated region is the natural outcome of the cohesive formulation. The Paris region can be readily calibrated with the two parameters of the proposed cohesive model. We compare the simulation results with the NASGRO material database, and show that the threshold region is adequately captured by the proposed model. We summarize a semi-implicit implementation of the proposed model into a cohesive-volumetric finite element framework, allowing for the simulation of a wide range of fatigue problems.

3.1 Introduction Fatigue failure of materials has been a topic of investigation for more than 150 years till date, but still complete physical understanding of this failure process is not understood. The critical aspect of fatigue failure is that it occurs at driving force levels less than that required for monotonic failure of the same material. Fatigue crack propagation behavior for materials shows four different fatigue S. Maiti (B) Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 3,

47

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S. Maiti

allowables needed to quantify fatigue crack behavior for any material, that is, threshold limit (ΔKth ), slope (m), intercept (C), and accelerated crack propagation limit (KIC ). Under the defect tolerant design for fatigue failure, all components are assumed to be initially flawed creating stress concentration under loading that might lead to failure. Thus, for designing components for infinite life, stress concentration produced by largest flaw in the component should be kept lower than the threshold limit for fatigue crack propagation. Critical parts in airplanes like fuselage are periodically tested for the largest crack length present in them. Knowledge of the fatigue allowable is necessary for the calculation of critical crack length below which crack growth rate will not be dangerously fast to call the part unusable as well as estimation of remaining life of that part. Currently in the industry, experimental analysis are carried out to determine this fatigue allowable; but experimental procedures are generally expensive and time consuming and are not feasible while doing design revisions or using mathematical techniques like optimization. Computational models for fatigue failure with predictive capabilities would appreciably help in such scenarios. Particularly in case of threshold behavior, which is critical for the life estimation of a component, a quick and robust model will be of immense help. However, numerical methods currently available treat crack initiation and crack propagation regimes under separate mathematical models even though they are the part of the same phenomenon [1]. The main focus of this chapter is to present a unified model for the prediction of stage I crack initiation and stage II and stage III crack propagation under the same numerical scheme. To do so, a fatigue crack growth simulation scheme using cohesive elements for stage II and stage III crack propagation has been modified to be used for the study of fatigue crack propagation in metals, and crack initiation (stage I) prediction capability was added to it [2]. In the current study, we are particularly interested in the predictive modeling of fatigue crack behavior in various materials important to aerospace industry. In organization of this chapter, the second section discusses various modeling efforts undertaken in the literature. Section 3.3 presents our modeling framework with implementation details. Fatigue crack growth simulation results along with a parametric study are presented in Section 3.4. We conclude our discussions in Section 3.5. With help of the developed unified model for fatigue crack propagation, we show that the presented model can predict the complete fatigue crack propagation curve including initiation and propagation regimes for various alloys successfully. We relate the threshold for the fatigue crack propagation with fracture toughness of the material with a new parameterSinit . Comparing the predicted value of Sinit with the experimentally evaluated values of lower threshold and fracture toughness of a number of aerospace alloys, we find that it varies between 0.98 and 1.0 for most of the materials studied.

3 Cohesive Technology Applied to the Modeling and Simulation

49

3.2 Background The steps to fatigue failure can be typically described as (i) accumulation of permanent damage at the microstructure level, (ii) creation of microscopic cracks at the damages sections, (iii) growth and coalescence of these microcracks to form dominant crack, (iv) stable propagation of the dominant crack, and (v) final complete failure [3]. Any numerical model that aims to predict fatigue crack growth behavior should incorporate these steps into the model. One of the oldest and most used numerical techniques for fatigue crack propagation is Paris law [4]. This method is phenomenological in nature and can successfully predict fatigue crack growth in stage II, known also as Paris region. However, the original Paris law is not useful for the prediction of fatigue crack growth in stage I or stage III. Even with subsequent modifications with a number of researchers [5], the inherent problem with this methodology remains in its phenomenological nature and consequent lack of true predictive capability.

3.2.1 Models for the Prediction of Threshold Fatigue Crack Behavior Observation of crack closure at the low-stress-intensity levels has given rise to a number of micromechanical models for the prediction of threshold behavior of metals [6–9]. Basic assumption in these models is that materials do not possess any intrinsic lower threshold: presence of the lower threshold is solely due to crack closure. But with presence of fatigue crack threshold even in vacuum, it has been realized that crack closure may not be the only reason for the presence of threshold. Weertman’s [10] model for prediction of lower threshold was based on the rupture energy of atomic bonds in metal crystals. He argued that the slowest possible crack growth will be of the order of the interatomic distance per cycle and found the following relationship for the threshold: ΔK th ≈ βGb1/2

(3.1)

with ΔK th as the threshold stress intensity range, G as the shear modulus, and b as the interatomic distance. The parameter β in the original model was taken to be the order of unity. To compare this model with experiments, β has been computed for five different aluminum alloys (Table 3.1). Aluminum has a cubic closed-packed structure and the interatomic distance for an aluminum crystal is 404.95 pm. The value of β is calculated as a ratio of experimental Kth and the value predicted from the above model [11]. It can be seen that β is about 5 for all the cases chosen. The application of discrete dislocation dynamics to the problem of fatigue threshold calculation was presented by Deshpande and Needleman [12–15]. Along with discrete dislocation dynamics, reversible and irreversible cohesive laws were used by the authors to simulate fatigue threshold value for metals. They were also

50

S. Maiti

Table 3.1 Evaluation of Weertman’s model for threshold prediction. Experimental values are taken from the NASGRO database [11] Material (Aluminum alloys)

Shear modulus (Pa)

6061-T62 2014T6 6061-T6 2024-T3

2.59E+10 2.72E+10 2.59E+10 2.75E+10

KIc (Pa-m1/2 )

Kth (from NASGRO) (Pa-m1/2 )

Predicted Kth = Gb∧ 0.5 (Pa-m1/2 )

β = Experimental/ predicted threshold

2.55E+007 3.00E+007 2.70E+007 2.90E+007

2.70E+006 2.60E+006 3.00E+006 2.90E+006

5.21E+05 5.48E+05 5.21E+05 5.53E+05

5.18 4.75 5.76 5.24

successful in demonstrating the effect of various parameters: loading parameters such as load ratio, tensile overloads, and microstructural parameters such as obstacle density and slip geometry. Recently, Farkas et al. [16], for the first time, simulated fatigue crack growth using atomistic simulations for nanocrystalline materials. They were successful in simulating the Paris curve and predicting lower threshold value to a good degree of accuracy.

3.2.2 Models for the Prediction of Fatigue Crack Propagation Finite elements codes incorporating fracture mechanics concepts have been a natural choice for the simulation of fatigue crack growth. A number of finite element studies have been performed to simulate the fatigue crack propagation in stages II and III [17]. However, most of these models accomplish the fatigue crack growth in an element-by-element manner by releasing nodes once in each cycle [18–20]. These models do not incorporate the damage event occurring at the crack tip, as described earlier, and thus may not represent the physics involved in a sound manner. Recently, cohesive modeling techniques suitable for fatigue crack propagation have appeared in the literature. These techniques are appealing compared to the earlier techniques due to their capability to incorporate progressive damage of the material naturally into the modeling framework. These models are also capable of predicting arbitrary crack propagation, branching, and coalescence in a solid domain, thus representing the physical phenomenon closely. Cohesive modeling technique has been applied for fatigue crack growth in metals [12–15, 21, 22], along interfaces [23] and in quasi-brittle materials [24]. In one of the early models proposed in [21], no distinction was made between the loading and unloading paths, but a damage parameter was assumed. The evolution of this parameter with the number of load cycles was prescribed explicitly in the model. The presence of plasticity in the bulk material around the crack tip influenced the crack closure and hence the failure of the material. But it was found that the crack ceases to grow after a few cycles due to plastic shakedown [22]. It has since been identified that a distinction needs to be made between the loading and unloading paths allowing for hysteresis so that subcritical crack growth becomes possible. Nguyen

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51

and co-workers have worked out a one-parameter cohesive model for metals [22], which is able to capture experimental Paris curves quite well and, in particular, the slope m of the curves (equal to approximately 3 for most metals). In the irreversible cohesive model mentioned in [12–15], the unloading path is assumed to be parallel to the previous loading path, thus leaving certain amount of residual separation in the cohesive zone after each cycle. It is argued that the environment-assisted oxidation of the crack faces can give rise to this kind of behavior. But this type of crack closure effect is not very common in polymers. The fatigue model for interface cracks presented by Roe and Siegmund [23] is based on damage mechanics, where a history-dependent damage parameter gives rise to irreversibility. In this model also, the unloading path is not toward the origin. Values of the slope m of the Paris curves up to 3.1 have been reported in that study. Fatigue crack growth in quasibrittle materials has also been studied by cohesive techniques in [24], where the irreversibility of the loading and unloading paths is taken into account. A polynomial expression for the cyclic behavior is postulated in that study. A special case of Paris law, where the multiplicative constant is functionally dependent on the maximum loading, has been reported by these authors. Maiti and Geubelle have proposed a two-parameter cohesive model for the simulation of fatigue crack growth in stages II and III for polymers [25]. This particular model has been successfully used to study the crack closure behavior for polymeric materials [26]. A multiscale modeling technique for self-healing polymers has also been suggested by these researchers [27]. It becomes clear from the previous discussion that most of these models, barring a few, treat the threshold regime and fatigue crack propagation regime in separate computational frameworks. Models capable of simulating stage I and stage II in a single framework [12–16] are typically computationally costly, and may not be able to perform simulations up to stage III of fatigue crack propagation. We outline a cohesive methodology-based unified model capable of predicting all the stages of fatigue crack propagation in subsequent sections.

3.3 Cohesive Modeling Technique The first model for the computation of crack extension in solids was postulated by Griffith [28]. He postulated that when a crack passes through a body, due to the effect of applied stresses, it causes decrease in the potential energy of the system. This decrease in the potential energy is influenced by the displacement of the outer boundaries and changes to the stored elastic energy. The decrease in the potential energy should be balanced by the increase in the surface energy released due to crack extension. Later, Dugdale [29] envisioned a narrow strip of plastic region ahead of the crack tip which is of near-zero thickness and extends to a distance rp ahead of the crack tip. The damage resulting into new fracture surface is supposed to be happening exactly in front of the crack tip in this strip-like zone. This particular assumption eliminated the presence of an infinite stress at the crack tip, as predicted

52

S. Maiti

by linear elastic fracture mechanics (LEFM), and provided a structure to the crack process zone. As per his model, the process zone rp ahead of the crack tip can be calculated as rp =

π 8

kI 2 , σy

(3.2)

where KI is the stress intensity factor in the opening mode and σ y is the yield stress of the material. The model also assumes the presence of an opening displacement δ = 2v at the crack tip. Figure 3.1 conceptually describes Dugdale’s strip yield model.

σ farfield

Fig. 3.1 Dugdale’s strip failure model

Crack Tip σy

2v Cohesive Zone Tip

Cohesive Zone

σ farfield

The strip yield model, developed by Barenblatt [30], is analogous to the Dugdale model. Barenblatt added that the crack proceeds when the crack face traction σ y reaches a critical value σ th where σ th is the theoretical bond rupture strength, and the cohesive zone size reaches a critical value rco. The energetic relationship for fracture can be expressed in terms of critical cohesive zone size rco or critical crack opening displacement δc = 2νc :

νc

Gc = 2 0

σ y dv =

8σth 2rc0 = 2γs πE

(3.3)

Idealistic scenario of Barenblatt’s strip yield model can be seen in Fig. 3.2, where the dark strip of material (as shown in the figure) is the strip of material with nearzero thickness undergoing failure process.

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53

Fig. 3.2 Barenblatt’s strip yield model

3.3.1 Reversible Cohesive Model From the models of Dugdale and Barenblatt, a computational scheme for failure of materials has been formulated [31, 32] which is generically known as the cohesive failure methodology. The failure of any material under this theory is governed by two parameters, cohesive traction σ max and the critical crack opening displacement Δnc . Here σ max is analogous to the σ th , that is, the bond rupture strength, and Δnc is analogous to the critical crack opening displacement δc from the Barenblatt model. For explanation purpose consider a cracked body put under loading perpendicular to the crack length as seen in Fig. 3.3. Let the far field stress be σ . Because of the presence of geometric discontinuity in the domain in the form of the crack, there will be a stress concentration K in front of the crack tip. Hence the yield strip (cohesive zone) ahead of crack tip will be experiencing a traction Tn that is much higher that

σ

Crack Tip Cohesive Zone Tip

Δn Δnc

Tn

a Cohesive Zone Length σ

Fig. 3.3 Cohesive failure modeling scheme

54

S. Maiti

far field stress σ . Now as per Dugdale and Barenblatt strip yield model, this traction will create a cohesive crack opening displacement Δn at the crack tip. The cohesive opening Δn is a function of Tn and the material properties for that material. Once the crack opening displacement reaches a critical value Δnc, the traction on the surface vanishes and new cracked surface is created with complete failure of the material in question. It is worthy to note that if crack is unloaded without crack opening displacement reaching Δnc, failure will not occur and the crack will not advance. The crux of the cohesive methodology of crack propagation is the relationship between the traction acting on the cohesive surface and the displacement jump between two opening sides. The traction–separation law, also known as the cohesive law, can be derived from a potential function. A number of cohesive laws have been postulated that differ in the description of the envelope of the traction–separation relationship. For all these cohesive laws, energy U spent for the creation of new fracture surface is given by U=

Δnc

Tn dΔn

(3.4)

0

Where Tn is the traction on the cohesive surface in the cohesive zone and Δn is the crack opening displacement over the cohesive zone and are integrated over cohesive process zone length rp . For implementation into finite element scheme, cohesive elements are introduced at the boundary of volumetric elements where failure is expected. The cohesive elements allow spontaneous initiation and propagation of crack through them when enough amount of energy is supplied to them. The cohesive elements act like nonlinear springs during the failure process, where the springs resist opening under applied load. The resistance of the springs is instantaneous stiffness of that cohesive element. This stiffness is governed by the specific traction–separation relationship. Eventually the opening displacement reaches the critical value and the traction provided by the cohesive element reduces to zero, thus creating two separate bodies on each side of the cohesive element.

3.3.2 A Bilinear Cohesive Law The bilinear rate independent but damage-dependent cohesive model discussed by Geubelle and Baylor [33] is used as the base cohesive model for this study and hence will be explained in details here after. It couples the normal and tangential components of traction and also allows relatively easy implementation. The damage achieved during any phase of simulation is remembered through a special ‘damage parameter’ S defined by ˜ 2 S = 1 − |Δ|

(3.5)

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55

˜ is defined as following Where | |2 denotes Euclidean norm and Δ ˜ = Δ

˜n Δ ˜t Δ

=

Δn /Δnc Δt /Δtc

with subscripts n and t denoting normal and tangential components, respectively. The strength parameter S is initially assigned to an initial value Sinit . As the tractions are applied on the cohesive elements, S monotonically decreases from the initial value to zero. The permanence of damage is achieved by storing the minimum value of S achieved so far in the simulation. The expression for S is as follows: ˜ 2) . S = min S pr ev , max(0, 1 − |Δ|

(3.6)

The cohesive traction–separation law in mode I for this particular model takes the following form: Tn =

S Δn σmax 1 − S Δnc Sinit

(3.7)

where σmax is the failure strength of the cohesive element and Δnc is the critical normal displacement jump. Similar expression can be derived for the tangential traction and separation as well. The traction separation law is depicted in Fig. 3.4.

Fig. 3.4 Bilinear traction–separation law for cohesive failure

The energy expended for failure is area under the traction–separation law envelope curve which can be equated to fracture energy for that material so that GIC =

1 σmax Δnc . 2

(3.8)

Since the damage is stored in parameter S, the path traced on reloading takes into consideration the energy already spent for partial damage achieved during first

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S. Maiti

loading phase. Hence, even though the failure does not occur in one loading phase, the energy used for failure will always be equal to G I C . Generally, KIC , critical stress intensity factor is used for characterization of fracture properties for any material. KIC is related to GIC through following relationship:

GIC =

K I2C . E

(3.9)

Here E is the Young’s modulus of the material. Typically for metals, σmax is taken as the yield strength of the material and Δnc is calculated as follows: Δnc =

2G I c . σmax

(3.10)

3.3.3 A Cohesive Model Suitable for Fatigue Failure Although it prevents healing of the fracture surfaces through the enforcement of monotonic decay of the damage parameter, the cohesive model discussed in the previous section leads to similar unloading and reloading paths in the traction– separation curve. This behavior is illustrated in Fig. 3.4. This characteristic prevents crack growth under subcritical cyclic loading due to the progressive degradation of the cohesive properties in the cohesive failure zone. This limitation suggests the need for an evolution law to describe the changes incurred by the cohesive strength under fatigue. The cohesive model for mode I fatigue, as developed by Maiti and Geubelle [25], is developed with some modification to the monotonic cohesive crack propagation model. The main difference between the monotonic crack propagation and fatigue crack propagation is the method for application of load and total energy supplied to the system. For fatigue crack propagation the load is applied in sequence of loading and unloading patterns. The load applied is not large enough to create stress intensity factor equal to or more than the stress intensity factor KIc required for monotonic crack propagation. In addition, the energy required for fatigue crack propagation is typically less than its quasi-static counterpart. The mechanism for fatigue crack propagation, as explained by Ritchie [34], reveals that the crack propagation below the fracture toughness value for any ductile solid on a simplistic level involves cyclic damage accumulation due to cyclic plastic deformation in the plastic zone ahead of the crack tip. In implementation of cohesive law for fatigue crack propagation, generally few elements ahead of the crack tip as undergoing cyclic fatigue failure which can be related to the plastic zone ahead of the crack. The evolution law for the instantaneous cohesive stiffness kc , that is, the ratio of the cohesive traction Tn to the displacement jump Δn during reloading can be expressed in the general form as [25]

3 Cohesive Technology Applied to the Modeling and Simulation

Kc =

57

dTn = F(N f , Tn ). dΔn

where Nf denotes the number of loading cycles experienced by the material point since the onset of failure, that is, at the time cohesive traction Tn first exceeded the failure strength σ max . Maiti and Geubelle adopted a separable form [25]: dTn = −γ (N f )Tn dΔn

(3.11)

leading to an exponential decay of the cohesive strength, with the rate of decay controlled by the parameter γ . In the present study, we use the following two-parameter power–law relationship [25]: γ =

1 Nf a

−β

(3.12)

where α and β are material parameters describing the degradation of the cohesive failure properties. Here α has the dimension of length and β denotes the history dependence of the failure process. The proposed evolution law for the cohesive model can also be expressed as 1 ˙n >0 ˙ n if Δ K˙ c = − (N f )−β K c Δ a ˙n ≤0 = 0 if Δ ˙ n is rate of change of normal separation. where Δ With these modifications, the traction separation law during cyclic loading can be described with 3.5. The cohesive law during the first loading cycle behaves like monotonic loading. Once the traction goes beyond σmax , the process of damage accumulation begins at that location. In the next loading cycle the evolution law for the cohesive stiffness will take effect and the stiffness at the integration point will reduce as shown in Fig. 3.5. The amount of degradation in the cohesive stiffness is dictated by (3.13), and can be calibrated with the actual physical damage occurring ahead of the crack tip. Observe from this figure that no stiffness degradation occurs during the unloading process. So, the developed model incorporates the physics of the fatigue crack growth into its formulation, and is expected to exhibit good predictive capability. Another important observation is that the cohesive model does not explicitly include any particular set of material properties, thus enabling it to simulate fatigue crack growth in a wide array of materials. Material properties are embedded only in the volumetric finite elements so that the specification of relevant constitutive law is required only for these elements. Cohesive elements include only the failure properties of the material that can be evaluated separately. Though the original model was developed for polymeric materials, we will show that it is equally successful in capturing fatigue crack growth behavior in metals and alloys through a proper calibration of the model parameters. Finally, note that the cohesive

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S. Maiti

Fig. 3.5 Behavior of the traction–separation law for a typical fatigue crack growth simulation scenario

stiffness gradually reduces to zero, at which point the fatigue crack is said to have propagated through the integration point under observation. This event is not dictated by any pre-mediated cycle-by-cycle node release procedure; it rather depends on the loading and geometric conditions as well as the material properties of the physical system under study. Typically, the displacement jump at this event does not reach the critical displacement jump Δnc as to be expected for the crack propagation under quasi-static loading. Also observe that the energy expended in the process, the area under the traction–separation law, is much lesser than that for the monotonic case. This observation is critical for the success of irreversible cohesive models, as this particular property of this class of models enables them to simulate subcritical crack growth.

3.3.4 Incorporation of Threshold Behavior In contradiction to the closure-based theory for lower-threshold prediction, in the current formulation, it is assumed that all materials show an intrinsic lower threshold. This is particularly true as lower threshold for various metals has been even seen in vacuum [35] which cannot be explained with closure-based theory. Stress intensity factor applied below this intrinsic value will not produce crack initiation. As discussed earlier, the cyclic stiffness degradation of cohesive element does not start till the traction on it reaches σmax . Hence, as a consequence, if the loading on the geometry is such that the traction on any cohesive element is less than σmax at any point of time in loading cycle, there will not be any degradation of the cohesive

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59

stiffness. So the loading and unloading path for this type of loading will be following the initial loading path for the cohesive element for infinite number of cycles, that is, the fatigue crack will not initiate under the current scheme of simulation. The maximum value of loading on a given geometry at which the fatigue crack will not grow (traction is less than σmax ) will give us the lower threshold for fatigue crack propagation. For detection of threshold value, crack opening displacement corresponding to the cohesive strength σmax is monitored continuously. Loading at which the crack opening displacement reaches Δnth is the threshold load value for a particular geometry. Threshold stress intensity factor can be calculated from the knowledge of threshold loading and geometry of the specimen. The equation for traction under cohesive scheme is given by Tn =

S Δn σmax 1 − S Δnc Sinit

Now to find out expression forΔnth , we put Tn equal to σmax . Also note that the value S is still Sinit as the traction is less than σmax . Hence, putting these values in above equation we get Δnth = (1 − Sinit )Δnc

(3.14)

Hence to detect the achievement of threshold value, crack opening displacement at all integration points is checked in the simulation code. The simulation starts with a low value of load applied on the DCB specimen, this load is incrementally increased till the threshold is reached. A close examination of (3.14) reveals that the threshold value depends on two parameters: the critical displacement jumpΔnc , which in turn depends on the fracture toughness of the materials, and the parameter Sinit . The last parameter controls the location of the peak of the traction–separation curve. So, it can be said that the peak of the cohesive law envelope is critical for the determination of the threshold limit of fatigue crack propagation. We will show later that the value of Sinit is quite close for a number of materials.

3.3.5 Finite Element Implementation The cohesive model described above is readily implemented in a finite element framework, normally called the cohesive volumetric finite element (CVFE) scheme, using the principle of virtual work:

Ω

S : δEdΩ −

Γex

Tex δudΓex −

Γc

Tn δΔn dΓc = 0

(3.15)

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S. Maiti

where u is the displacement vector, S and E denote the internal stress and strain tensors, respectively, Tex is the externally applied traction, and Ω, Γe , and Γex , respectively, denote volume, cohesive boundary, and exterior boundary of the deformable body. The last term corresponds to the virtual work done by cohesive traction Tn for a virtual separation of δΔn . The expression for internal component of virtual work depends upon the type of volumetric element used in the analysis, which in turn affects the expression used for calculation of internal stress S and virtual strain δE. For general implementation of cohesive elements into the finite element scheme, these elements are modeled as if they are sandwiched between two volumetric elements between which the failure is supposed to occur. In addition, these elements do not have any thickness associated with them to start with. For the cohesive formulation used here, the cohesive elements are inserted before the simulation, pre-specifying the flow of the crack through the material. As discussed earlier, the traction at an integration point is given by (3.7). Value of S does not degrade till the traction value reaches σmax , which is the threshold of fatigue crack propagation as explained in the last section. Crack opening displacement at a particular cohesive integration point is calculated throughout the simulation and is checked against Δnth . If the crack opening displacement is more than Δnth , damage parameter S is evolved using (3.6). Minimum value of S achieved at each integration point is stored and new calculated value at each step in cycle is checked against previously achieved value. The expression for cyclic cohesive stiffness degradation, as given by (3.13), has been discretized for the finite element implementation using two different methods, namely, forward difference method and Tustin transformation. Forward difference method was used in earlier studies by Maiti and Geubelle [25–27]. But this particular method is stable only at very small time-step values. With the required reduction in the time steps, total time required for solution becomes very high. Since the simulations done here are cycleby-cycle simulations for the entire fatigue crack life, increase in the number of load steps per cycle increases the total time considerably. Tustin transformation is known for its stability so that larger time steps can be used. However, too large time steps can compromise the accuracy of the solution considerably. A convergence study was undertaken by us to find the stable yet accurate time step for both the methods. Our study shows a significant increase in the acceptable time step for the Tustin transform. A quasi-implicit load stepping scheme has been used to form equations of equilibrium in this investigation. The time increment at each step is calculated from the frequency of loading and number of steps per cycle. The discretization of the cohesive degradation expression with the forward difference method is as follows: K cj+1

1 j β j+1 1 σmax sj j 1− Nf Δn − Δn . = 1 − s j Δnc sinit α

Δnj+1 ≥ Δnj .

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61

With Tustin transform the expansion is as following K cj+1

=

2α − N j

−β

2α + N f

−β

(Δn+1 − Δn ) (Δn+1 − Δn )

K cj .

Crack location is detected by finding out the coordinate of the newly completely failed cohesive element (S = 0).

3.4 Simulation Results Finally, we are in a position to simulate the fatigue crack growth behavior in different materials by the developed model. We restrict our simulations to a simple double cantilever beam (DCB) arrangement. Due to symmetry of the problem on hand, only the half portion of the DCB specimen is modeled. The calculation of crack opening displacement for cohesive elements is taken care of accordingly. The length of the DCB specimen is taken to be 150 mm with an initial crack of 50 mm. The length of the cohesive elements depends on the materials properties [25], and has been calculated accordingly.

3.4.1 Paris Curve Simulation First we show the capability of the presented cohesive model in simulating all three stages of the Paris curve. For this purpose, the simulations are run for two Aluminum alloys 6061-T62 and 2024 T3 and one Titanium alloy Ti-2.5Cu. The material properties for these alloys are as given in Table 3.2, and are taken from NASGRO database [11]. Table 3.2 Material properties for simulated alloys Material

Young’s Modulus Yield stress (GPa) KIc (MPa-m1/2 ) (MPa)

GIc =

6061 –T62 Al 2024 T3 Al T1-2.5Cu STA

68.9 73.1 105

9437.59071 J 68.4 μm 11504.788 J 69.6 μm 19716.6667 J 59.0 μm

25.5 29.0 45.5

276 331 668

K I cz Z

Δnc =

2G I c σmax

The experimental values for fatigue properties for these three materials taken from NASGRO database [11] are provided in Table 3.3. All the simulations were performed for a frequency of loading of 1 Hz while the load ratio was kept constant at R = 0. Number of steps per cycle was set to be 50 for the load stepping scheme. The output for crack extension versus number of cycles for one of the materials, Al 6061-T62, is shown in Fig. 3.6.

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S. Maiti Table 3.3 Experimental values of fatigue properties of the alloys described in Table 3.2 Material

ΔKth (from NASGRO) (MPa)

Intercept C (m)

Slope n

6061 -T62 Al 2024 T3 Al T1-2.5Cu STA

2.70 2.90 4.60

5.50E-010 8.4E-010 2.5E-10

2.8 2.75 2.6

Fig. 3.6 Crack extension versus number of cycles for Al 6061-T62

Notice that the cohesive model does not possess any in-built assumption about the Paris curve of the material. The crack extension data was post-processed to derive the Paris curve and is presented in Fig. 3.7. The slope of the Paris curve from the simulation is 2.82 and the intercept is 4.4e-10 m, which is very close to the experimental values. In addition, we can note that the threshold limit as well as the accelerated region is also very well captured by the simulation. Two thick vertical lines at the bottom of the plot denote these two limits for this particular alloy. The values of α and β used for this simulation are 8 μm and 0.1, respectively, while Sinit was kept constant at 0.988. Figure 3.8 shows the simulated Paris curve for 2024 T3 aluminum alloy. The Paris curve for the Titanium alloy Ti–2.5Cu is depicted in Fig. 3.9. It can be observed from these plots that the limits are very well captured as was the case with the first material. The simulation values of slope and intercept for various alloys are tabulated in Table 3.4. On comparison with the experimental values from NASGRO database [11] in Table 3.5, we can see that the results of the simulation are fairly accurate even for the Ti–2.5Cu, which has far higher values of fracture toughness, yield strength, and Young’s modulus compared to two other alloys under study. This exercise shows that the model can be used for various types of alloys regardless of their material properties. Our model can predict the fatigue crack growth in all

3 Cohesive Technology Applied to the Modeling and Simulation Fig. 3.7 Paris curve for Al 6061

63

da/dN (mm/cycle)

10–2

10–3

10–4

10–5 0 10

101

102

ΔK (MPa-(m)1/2) Fig. 3.8 Simulated Paris curve for 2024 T3 Al alloys

da/dN (mm/cycle)

10–2

10–3

10–4

10–5

101

102

ΔK (MPa-m1/2)

three stages for a number of materials in a single computational framework. The model calibration parameters α and β are shown in Table 3.6. These parameters can also be evaluated from fatigue crack growth experiments for stage II. A look at the numerical values of these parameters shows that while the parameter α can change appreciably for different material, β keeps an almost constant value. As

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Fig. 3.9 Simulated Paris curve for Ti–2.5Cu STA alloy

da/dN (mm/cycle)

10–2

10–3

10–4

10–5 0 10

Table 3.4 Simulation values of slope and intercept for the three materials described in Table 3.2

Table 3.5 Experimental values of slope and intercept for the three materials described in Table 3.2. [11]

101 Δ K (MPa-m1/2)

102

Material

ΔKth (MPa)

Intercept C (m)

Slope m

6061 -T62 Al 2024 T3 Al T1-2.5Cu STA

2.70 2.90 4.60

5.0E-010 8.3E-010 2.35E-10

2.8 2.75 2.65

Material

ΔKth (MPa)

Intercept C (m)

Slope m

6061 -T62 Al 2024 T3 Al T1-2.5Cu STA

2.70 2.90 4.60

5.0E-010 8.3E-010 2.35E-10

2.8 2.75 2.65

shown by Maiti and Geubelle [25], the first parameter is responsible for the intercept of the Paris curve while the second parameter changes its slope. As the slopes of the experimental Paris curves for all the alloys are very close, the parameter β is also similar in magnitude for all the cases. The wide variation in the intercept of the experimental Paris curves is reflected by the variation in the parameter α in Table 3.6.

3 Cohesive Technology Applied to the Modeling and Simulation Table 3.6 α and β values used for the simulation of three materials listed in Table 3.2

65

Material

α

β

6061 -T62 Al 2024 T3 Al T1–2.5Cu STA

8 μm 13 μm 17 μm

0.1 0.105 0.18

3.4.2 Prediction of Threshold Limit of Fatigue Crack Growth In this section, we turn our attention exclusively to the simulation of threshold limit for different aerospace alloys. As per the procedure explained previously, threshold values for few aluminum alloys are evaluated. The parameter Sinit is chosen to be 0.9828 for all the simulation cases. As we can see from Fig. 3.10, the values predicted by the simulation as very close to the actual experimentally evaluated values except for 6061-T6 GTA weld. For rest of the materials, the simulated lower threshold value is within 10% of the experimental value. Hence with the input values of material stiffness, fracture toughness and yield stress, and one model parameter, Sinit , lower threshold stress intensity factor for most of the materials can be simulated using this model. A proper calibration of this parameter produces simulated threshold limits still closer to the experimental values, as listed in Table 3.7. For the outlying value of 6061-T6 plt:GTA weld, the

Fig. 3.10 Simulated threshold stress intensity range for different aluminum alloys

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S. Maiti Table 3.7 Simulated and experimental lower threshold values with other material properties

Material 6061 -T62 Al 2014T6 6061-T6 plt; T-L 6061-T6 plt;GTA weld 2024-T3clad plt sht t-l DW

Young’s modulus (GPa)

Kic (MPa-m1/2 )

Yield Stress ΔKth (NASGRO) (MPa-m1/2 ) (MPa)

Simulated ΔKth (MPa-m1/2 )

Sinit

68.9 72.4 68.9

25.5 30.0 27.0

276 414 282

2.7 2.6 3

2.46 2.72 2.76

0.989 0.992 0.988

68.9

27.0

158

4.5

3.20

0.972

73.1

29.0

331

2.9

2.79

0.990

difference in the simulated and experimental value could be due to effect of parameters like material microstructure which are explicitly not taken into consideration in this model.

3.4.3 Effect of on the Threshold Limit Fracture energy for the material is embedded into the cohesive law such that GIC =

1 σmax .Δnc 2

We assume that the energy required to start a fatigue crack has a threshold value below which no crack propagation is possible. We further assume that this “minimum energy” G th is an intrinsic property of the material. An expression for this energy can be found from the observation of the cohesive traction–separation curve, and can be expressed as G th =

1 σmax .Δnth 2

But as seen previously from (3.14), Δnth = (1 − Sinit )Δnc Combining these equations, G th = (1 − Sinit ). GIC

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67

But G th = ΔK th 2 /E (at R = 0) and G I C = K I C 2 /E Hence we get ΔK th 2 = (1 − Sinit ). KIC2

(3.16)

Now let us look at the data from NASGRO database [11] for threshold values of different aerospace materials. We can calculate the Sinit value for various materials from the above derived equation. Figure 3.11 shows the plot for ΔKth /KIC for close to 200 aerospace materials taken from the NASGRO database.

Fig. 3.11 ΔKth /KIC for various materials calculated from NASGRO database

It is interesting to observe from Fig. 3.11 that except for just a few materials, most of the materials have Sinit values ranging between 0.98 and 1.0. We observe a very tight bound on the parameter, Sinit . So, this particular parameter can be taken as a material constant, valid at least for aerospace alloys. In the absence of detailed experimental analysis, a choice of Sinit close to 1.0 will yield satisfactory value of the fatigue crack propagation threshold for these materials. This observation points to the ability of our model to predict the fatigue lower threshold stress intensity range for most of the materials relevant for aerospace applications. Equations (3.14) and (3.16) are the key results presented in this discussion; (3.16) relates the experimental fatigue threshold stress intensity range with the model

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parameter Sinit with a very tight bound for this parameter over a range of materials, whereas (3.14) shows that this particular parameter can also be related to another set of failure properties, namely crack opening displacement (COD). There exist a number of analytical approaches to relate the crack tip opening displacement to the geometry of the specimen, and the external loading condition. So, enumeration of COD, and consequently Sinit , can be performed in a computational framework without taking recourse to further experiments.

3.4.4 Effect of Load Ratio R on Fatigue Crack Threshold Experimental results on the effect of load ratio R on the threshold value normally exhibit a large scatter. But the general trend is that the threshold limit decreases linearly with R for most of the materials. Blacktop and Brook [36] noted that threshold value should touch zero when the load ratio assumes a value of 1. On the other hand, other researchers have observed a leveling off of the threshold value at higher load ratio [37]. Mechanistic explanation for this effect is [38] a. Presence of crack closure at low values of stress intensity Kmin, b. Presence of static fracture modes as the maximum stress intensity Kmax approaches fracture toughness for the material. When only the effect of crack closure is taken into consideration, Schmidt and Paris [39] hypothesized that the threshold value will reduce with load ratio till the minimum stress intensity Kmin is less than the crack closure stress intensity Kcl . After some critical value of R (beyond which Kmin > Kcl ) effect of load ratio on the fatigue threshold limit will not be observed any more. Oxide-induced crack closure was cited as one of the mechanisms resulting in the effect of load ratio [40]. Boyce and Ritchie [38] also confirmed these results for Ti–6Al–4 V alloy, where they observed that there is heavy reduction in ΔKth till R = 0.5 but after that point the slope of ΔKth vs. R curve does not become zero. They tried to explain this phenomenon by predicting the presence of sustained load cracking mechanism. They also cited observations of Davidson [41] who reported the presence of closure in the local region of crack tip (within ∼10 μm). Our computational model does not incorporate the effect of crack closure, and hence is not expected to exhibit the leveling off of the ΔKth vs. R curve, as discussed above. We rather choose to compare our simulation results with the experimental trend obtained for a variety of metal alloys [42] that shows a linear reduction for this curve. The results from the simulations for Al 6061–T62 alloy with varying load ratio R when plotted on above scale exactly matches the experimental data (experimental values are shown by filled circles and simulated results are shown by the black line) as seen in Fig. 3.12. So, in the absence of crack closure, our model is capable of reproducing the experimental variation of the fatigue threshold limit with R quite accurately.

Ratio of Threshold Stress Intensity Range at R to that at R = 0.5

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2.50 2.00 1.50 1.00 0.50 0.00

0

0.5 R-Ratio

1

Fig. 3.12 Effect of load ratio R on the threshold value of fatigue crack propagation. Experimental data (filled circles) have been taken from [42]

3.5 Conclusions We have presented a cohesive law-based fatigue crack propagation model that can simulate and predict all the stages of fatigue crack propagation in aerospace alloys. The model was used to simulate fatigue crack growth for five different materials. Except for the threshold limit of 6061-T6 GTA weld, it could successfully predict the fatigue allowables for rest of the materials. We hypothesize that the variation in simulated results for this particular material is due to the microstructural changes caused by the welding process as the model could predict accurate threshold value for 6061-T6 alloy without weld. The scope of future improvement for the presented model lies in the incorporation of microstructural effects. We have also observed that the position of peak of the cohesive law envelope plays an important role in determining the threshold limit. Model parameter Sinit determines the position of this peak, and generally lies between 0.98 and 1.0 for most of the materials studied. The tight bound of this parameter leads us to conclude that Sinit may be an intrinsic material parameter. Finally, we have demonstrated that the model can also predict the effect of load ratio on the fatigue crack threshold behavior quite accurately.

References 1. R. Sunder, “A unified model of fatigue kinetics based on crack driving force and material resistance,” Int. J. Fatigue, Vol. 29, No. 9–11, 2007, pp. 1681–1696. 2. B. Farahmand and S. maiti, “Estimation of threshold of fatigue crack growth curve for aerospace alloys,” Aging Aircraft 2008, Phoenix, 2008. 3. S. Suresh, Fatigue of Materials, 2nd edition, Cambridge University Press, 1991. 4. P. Paris and F. Erdogan, “A Critical Analysis of Crack Propagation,” J. Basic Eng., Vol. 85, 1963, pp. 528–34.

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5. A. Vasudevan, K. Sadananda, and N. Louat, “A Review of Crack Closure, Fatigue Crack Threshold and Related Phenomena,” Int. J. Fatigue, Vol. 18, No. 1, 1996, p. 62. 6. W. Elber, “Fatigue Crack Closure Under Cyclic Tension,” Eng. Frac. Mech., Vol. 2, 1970, pp. 37–45. 7. J.C. Newman, “A Finite Element Analysis of Fatigue Crack Closure,” ASTM STP 590, 1976, pp. 281–301. 8. B. Budiansky and J.W. Hutchinson, “Analysis of Closure in Fatigue Crack Growth,” J. Appl. Mech., Vol. 45, 1978, pp. 267–276. 9. J.T. Gray, J.C. William, and A.W. Thompson, “Roughness Induced Crack Closure; Explaination of Microstructurally Sensitive Fatigue Crack Behavior,” Metallurgical Trans., Vol. 14A, 1983, pp. 421–433. 10. J. Weertman, “The Paris Exponent and Dislocation Crack Tip Shielding,” In: High Cycle Fatigue of Strutcutral Materials, TMS Publication, 1997, pp. 41–48. 11. B. Farahmand, “Fatigue and Fracture Mechanics of High Risk Parts,” Chapman and Hall, 1997. 12. V.S. Deshpande, A. Needleman, and E. Van Der Giessen, “A Discrete Dislocation Analysis of Near Threshold Fatigue Crack Growth,” Acta Materialia, Vol. 49, No. 16, 2001, pp. 3189–3203. 13. V. Deshpande, A. Needleman, and E. Van der Giessen, “Discrete Dislocation Modeling of Fatigue Crack Propagation,” Acta Materialia, Vol. 50, No.4, 2002, pp. 831–846. 14. V. Deshpande, A. Needleman, and E. Van der Giessen, “Discrete Dislocation Plasticity Modeling of Short Cracks in Single Crystals,” Acta Materialia, Vol. 51, No. 1, 2003, pp. 1–15. 15. V. Deshpande, A. Needleman, and E. Van der Giessen, “Scaling of Discrete Dislocation Predictions for Near-Threshold Fatigue Crack Growth,” Acta Materialia, Vol. 51, No. 15, 2003, pp. 4637–4651. 16. D. Farkas, M. Willemann, and B. Hyde, “Atomistic Mechanisms of Fatigue in Nanocrystalline Metals,” 10.1103/PhysRevLett.94.165502, 2005. 17. S. Roychowdhury and R.H. Dodds, Jr., “A Numerical Investigation of 3-D Small-Scale Yielding Fatigue Crack Growth,” Eng. Frac. Mech., Vol. 70, 2003, pp. 2363–2383. 18. S. Roychowdhury and R. H. Dodds, Jr., “Effect of T-stress on Fatigue Crack Closure in 3-D Small-Scale Yielding,” Int. J. Solids Struct., Vol. 41, 2004, pp. 2581–2606. 19. J. Wu and F. Ellyin, “A Study of Fatigue Crack Closure by Elasto-Plastic Finite Element Analysis for Constant-Amplitude Loading,” Int. J. Fract., Vol. 82, 1996, pp. 43–65. 20. A.G. Carlyle and R.H. Dodds, Jr., “Three-Dimensional Effects on Fatigue Crack Closure under Fully-Reversed Loading,” Eng. Frac. Mech., Vol. 74, 2007, pp. 457–466. 21. A. de-Andres, J.L. Perez, ´ and M. Ortiz, “Elastoplastic Finite Element Analysis of ThreeDimensional Fatigue Crack Growth in Aluminium Shafts Subjected to Axial Loading,” Int. J. Solids Struct., Vol. 36, 1999, pp. 2231–2258. 22. O. Nguyen, E.A. Repetto, M. Ortiz, and R.A. Radovitzky, “A Cohesive Model of Fatigue Crack Growth,” Int. J. Frac., Vol. 110, 2001, pp. 351–369. 23. K.L. Roe and T. Siegmund, “An Irreversible Cohesive Zone Model for Fatigue Crack Initiation,” Eng. Frac. Mech., Vol. 70, 2002, pp. 209–232. 24. B. Yang, S. mall, and K. ravi-Chandar, “A Cohesive Zone Model for Fatigue Crack Growth in Quasibrittle Materials,” Int. J. Solids Struct., Vol. 38, 2001, pp. 3927–3944. 25. S. Maiti and P.H. Geubelle, “A Cohesive Model for Fatigue Failure of Polymers,” Eng. Frac. Mech., Vol. 72, 2004, pp. 691–708. 26. S. Maiti and P.H. Geubelle, “Cohesive Modeing of Fatigue Crack Retardationin Polymers: Crack Closure Effect,” Eng. Frac. Mech., Vol. 73, 2006, pp. 22–41. 27. S. Maiti, C. Shankar, P.H. Geubelle, and J. Kieffer, “Continuum and Molecular-Level Modeling of Fatigue Crack Retardation in Self-Healing Polymers,” J. Eng. Mat. Tech., Vol. 128, 2006, pp. 595–602. 28. A.A. Griffith, “The Phenomenon of Rupture and Flow in Solids,” Philos. Trans. R Soc. Lond., Vol. 221, 1920.

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29. D.S. Dugdale, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solids, Vol. 8, No. 2, 1960, pp. 100–104. 30. G.I. Barenblatt, “The Mathematical Theory of Equilibrium of Cracks in Brittle Fracture,” Adv. Appl. Mech., Vol. 7, 1962, pp. 55–129. 31. G.T. Camacho and M. Ortiz, “Computational Modelling of Impact Damage in Brittle Materials,” Int. J. Solids Struc., Vol. 33, 1996, pp. 2899–2938. 32. X. Xu and A. Needleman, “Numerical Simulations of Fast Crack Growth in Brittle Solids,” J. Mech. Phys. Solids., Vol. 42, No. 9, 1994, pp. 1397–1407. 33. P. Geubelle and J. Baylor, “Impact-Induced Delamination of Composites: a 2 D Simulation” Composites Part B, Vol. 29B, 1998, pp. 589–602. 34. R.O. Ritchie, “Mechanisms of Fatigue-Crack Propagation in Ductile,” Int. J. Frac., Vol. 100, 1999, pp. 55–83. 35. R.O. Ritchie, V. Schroeder, and C.J. Gilbert, “Fracture, Fatigue and Environmentally-Assisted Failure of a Zr-based Bulk Amorphous Metal,” Intermetallics, Vol. 8, No. 5–6, 2000, pp. 469–475. 36. J. Blacktop and R. Brook, “Compendium,” Eng. Fract. Mech., Vol. 12, 1979, pp. 619–20. 37. T.M. Ahmed and D. Tromans, “Fatigue Threshold Behavior of Alpha Phase Alloys in Desiccated Air: Modulus Effect,” Int. J. Fatigue, Vol. 21, 2004, pp. 641–649. 38. B.L. Boyce and R.O. Ritchie, “Effect of Load Ratio and Maximum Stress Intensity on the Fatigue Threshold in Ti-6Al-4 V,” Eng. Frac. Mech., Vol. 68, 2000, pp. 129–147. 39. R. Schmidt and P. Paris, “Threshold for Fatigue Crack Propagation and the Effects of Load Ratio and Frequency,” Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, ASTM, Philadelphia, 1973, pp. 79–94. 40. S. Suresh, G.F. Zamiski, and R.O. Ritchie, “Oxide-Induced Crack Closure: An Explanation for Near-Threshold Corrosion Fatigue Crack Growth Behavior,” Metallurgical Mater. Trans. A, Vol. 12, No. 8, 1981, pp. 1435–1443. 41. D. Davidson, Damage Mechanisms in High Cycle Fatigue. AFOSR Final Report, Project 068243. Southwest Research Institute, 1998. 42. L. Lawson, E.Y. Chen and M. Meshii, “Near-Threshold Fatigue: A Review,” Int. J. Fatigue, Vol. 21, 1999, pp. S15–S34.

Chapter 4

Fatigue Damage Map as a Virtual Tool for Fatigue Damage Tolerance Chris A. Rodopoulos

Abstract Using only readily available material properties and the concept of dislocation density evolution ahead of the crack tip, the fatigue damage map attends to develop a virtual tool able to predict the limits and the corresponding crack tip propagation rates characterising each of the fatigue stages, namely crack arrest, microstructurally and physically short crack (Stage I), long crack growth (Stage II), and Stage III growth.

4.1 Introduction In principle, the method of damage tolerance fatigue design is trying to fulfil two needs. The first is the description of “What is happening when I have a crack?” Herein, elastic solutions representing fatigue damage are transformed into the wellknown stress intensity factor, K [1]. In other words, fatigue damage is limited to the single geometric parameter of length and the tensor of stress. The second need is “When this crack will start causing problems?” The latter has been approached with a number of mathematical models [2–5], all of which have been based on the similitude concept. The concept assumes that each material inherently delivers a specific crack growth rate, da/dN. Herein, a is the crack length and N is the number of loading cycles. With the above in mind and considering the particulars emanating from having an elastic solution, it was assumed that once the value of K is known only the material can affect the growth rate. In other words, same material and same K value will always deliver the same crack growth rate. Despite its success, the similitude concept has created a number of additional problems. Large crack growth databases were needed in order to provide the designer with a relationship between K and da/dN. Towards such experimental requirement, especially the aerospace companies have heavily invested in creating C.A. Rodopoulos (B) Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Greece e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 4,

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testing procedures. The issue was further reinforced by ASTM standards to further normalise crack growth testing [6]. Testing revealed a number of unprecedented problems: (a) only a small portion of the testing results was likely to follow the similitude concept and (b) the material seemed to play a more vital role in controlling this portion (see Fig. 4.1).

Fig. 4.1 Typical representation of crack growth stages

All the above forced the industry to seek a solution able to fulfil a number of prerequisites: (a) the effect of the material properties should be acknowledged; (b) testing effort should be kept at minimum; (c) the solution should acknowledge complex conditions, that is, environmental effects, variable amplitude loading, notches, etc.; (d) the scatter emanating from each material should be addressed; and (e) finally the solution should have the capability to operate in a predictive mode. The quest to such approach lasted for over 20 years. This chapter examines the basic assumptions, the steps followed, the problems ahead, the solution, and through that an overall demonstration of the applicability of the fatigue damage map method. In order for the reader to understand the classification of fatigue stages, in order to better acknowledge the necessity of being able to distinguish and classify them, the first section of the chapter provides a brief review.

4.2 The Basic Understanding of Fatigue Damage 4.2.1 Development of Fatigue Cracks and Fatigue Damage Stages Research into the creation of fatigue cracks on free surfaces has confirmed that the damage process is related to the forward and backward motion of dislocations

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along the slip planes of metallic crystals, that is, the reversed plastic flow. A consequence of these repeated dislocation movements are the creation of small-localised deformations called extrusions and intrusions at sites where the persistent slip bands emerge at the material surface [7]. In this respect, a large number of models have been formulated to explain nucleation of microcracks [8], thereby making a distinction between intrusions and microcracks. Furthermore, as a consequence of microscopic material defects, such as second-phase particles, inclusions or precipitates, surface notches, and machining marks; microstructural features, such as grain boundaries, triple points, and twin boundaries [9, 10] and also environmental effects like pitting corrosion [11] a local concentration of stress which may exceed the yield strength of the material is developed. Consequently, cyclic plastic deformation due to the higher stresses of these stress concentration sites and also due to the lower degree of constraint of the near-surface volumes from a cyclically loaded material is developed. It is well known that fatigue damage occurs only when cyclic plastic strains are generated [12]. Therefore, given the intrinsic heterogeneity of polycrystalline metals, the above conditions, separately or in a variety of combinations, can lead to the nucleation of microcracks. The preferred mechanism of initiation will, therefore, depend on the microstructure and manufacturing process of the material, the type of loading, and shape of the component. Once nucleated, a microcrack may be arrested by a microstructural barrier or may propagate until reaching a critical size, causing the final failure [13]. Cyclic crack growth is found to be generally divided into three stages. Stage I fatigue crack growth occurs by a shear mechanism in the direction of the primary slip system over a few grains. The crack propagates on planes oriented at approximately 45◦ to the stress axis1 , that is, the crack follows the best-orientated grain path, for example, a high-angle grain boundary where, due to the high local stress concentration, it will form a new slip band in the next grain giving raise to crack extension [14]. The effect of this crack growth mode is the characteristic zig-zag crack path as defined by Forsyth [15]. At this stage, the crack length may reach a length of a few grains. However, the zone of near-tip plasticity is smaller than the grain dimensions [7]. As such, if the grain is considered as the local self-equilibrating medium, then conditions of large plasticity prevail. Such case negates the application of the stress intensity factor and hence the similitude concept. Favourable conditions and higher stress intensity range values allow the crack to grow longer, as the plastic zone size increases, and also to be able to overcome the resistance offered by successive barriers. After a certain growth, there is a transition from stage I to stage II crack propagation, where the crack overcomes microstructural barriers with ease and simultaneous shear planes develops [15]. At stage II, also known as the steady-state or Paris regime, crack propagation is driven by the stress normal to the crack face and the mode will change to mode I. It is proposed to the Schmid’s law (τ c = σ cos θ cos ϕ, where for θ and ϕ equal to 45◦ , τ c is a maximum), plastic deformation occurs when, the applied tensile stress σ , resolved as τ c on a particular slip plane, exceeds a determined shear stress τ y .τ y = σ y /2 if the material observes the Tresca’s principles, where σ y is the corresponding yield stress.

1 According

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that at the transition from stage I to stage II, the plastic zone will be of the order of the grain size [16]. Fatigue stages are shown in Fig. 4.2. When the crack tip stress intensity reaches values close to those required for unstable crack propagation, stage III begins [17]. tm

ax

t

s

Grains with different crystallographic orientation Plastic Zone

Stage I (shear crack)

Stage II (tensile crack)

Stage III (meandering crack)

s Fig. 4.2 Schematic representation of the fatigue crack growth stages. As indicated (on the left) is a general dislocation model of crack nucleation from the free surface, at the largest and bestorientated grain, as described by Mutoh [18]. The circles in stage III represent voids

The transition from stage II to stage III crack growth has not been widely studied, but it has been observed regularly by E. Hay and M.W. Brown [19]. At this stage, a tearing mechanism is dominant, and the crack branches from the main crack path (crack meandering) [20] as illustrated in Fig. 4.3. In high-cycle fatigue, Stage III represents an insignificant proportion of life, that is, failure follows very rapidly.

Fig. 4.3 Crack meandering from the main crack path in stage III of a shot peened specimen (Al 2024–T351) fatigued under four-point bending constant amplitude loading. The general direction of crack extension is indicated by the arrow

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Despite the fact that short or stage I fatigue cracks will precede to long or stage II cracks, it is imperative to start with the latter in order to highlight differences both in their intrinsic mechanism as well as in the way they can be modelled.

4.2.2 Stage II Fatigue Cracking The theory of fracture mechanics provides the invaluable concept of the stress intensity factor (SIF), denoted by K. As aforementioned, SIF is a measure of the intensity of the near-tip stress fields. However, when such intensity is under linear elastic conditions, the approach used is the so-called linear elastic fracture mechanics (LEFM). This approach can readily be used to analyse and/or to predict fracture, provided the plastic zone surrounding the crack tip is sufficiently small (small-scale yielding) in a way that the K-elastic stress field or K-dominance is not significantly altered. In this respect, and according to Irwin, the singular solutions of the near-tip stress fields, σ ij , are correlated with K as follows: σij = √

1 2πr

K I f ijI (θ ) + K II f ijII (θ ) + K III f ijIII (θ )+ . . . ,

(4.1)

where r is the distance from the crack tip, θ the polar angle measured from the crack plane, fij a dimensionless function of θ at different modes of fracture. KI, II, III are the SIFs for each loading mode, which is generally expressed for mode I as: √ K = Yσ πa,

(4.2)

where Y is a non-dimensional function of the loading and crack geometry, σ is the uniform applied stress remote from the crack and a is the real crack length. Long crack growth rate has been successfully characterised in terms of SIF, K by P.C. Paris & Erdogan [21, 22]. For a cyclic variation of the imposed stress field under LEFM (quasi-elastic) conditions, the empirical relationship between crack growth increment da/dN and K is given by the power law: da = CΔK m dN

(4.3)

where C and m are experimental constants depending on material microstructure, min min = KKmax ; and ΔK is frequency, environment, temperature, and stress ratio R = σσmax the stress intensity factor range, defined as ΔK= Kmax – Kmin . It is important to indicate at this point that in LEFM, the initiation of crack propagation under monotonic, quasi-static loading conditions is characterised by the critical value of the SIF, KC . When the critical value is obtained for mode I stress intensity factor, it is known as fracture toughness and is universally denoted as KIC . In the shearing and tearing modes, fracture toughness is referred to as KIIC and KIIIC , respectively.

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It is well recognised that in Irwin’s analysis the plasticity ahead of the crack tip is assumed negligible. However, when the extent of plasticity is not very small compared to the crack length, the application of LEFM can lead to imprecise predictions. In this respect, views concerning the extent to which the elastic-based theory is applicable have long been documented, among these are: i. The maximum plastic zone at the crack tip taken to be one-fiftieth of the crack length provides a small-scale yielding [23]. ii. The applied stress should be up to two-thirds of the cyclic yield stress [17]. In cases where the plastic zone size is comparable to the crack length, that is, small-scale yielding conditions are not met, the application of elastic–plastic fracture mechanics is the appropriate approach to employ. Discrepancies or offset predictions delivered by Equation 4.3 forced a number of researchers to further analyse the mechanism of crack growth. Perhaps the most known and yet disputable scenario emanates from experimental works indicating that the crack faces do not open immediately with the application of stress. The crack closure effect, first discussed by Elber [24], has increasingly concerned researchers of fatigue crack growth behaviour, particularly in the near-threshold stress intensity levels of long cracks [25, 26]. Herein, it is important to note that the near-threshold behaviour should be considered as being governed by similar to Stage I mechanisms. The differentiation is due to the fact that near-threshold conditions of growth can only be found when the crack emanates from a through thickness notch/slit. This is because the fractured surfaces in the wake of an advancing crack tip close when the far-field load is still tensile before the attainment of the minimum load. Premature contact of the fracture flanks occurs and, as a result, the crack tip does not experience the full range of ΔK, that is, the real driving force or effective stress intensity range, ΔKeff (ΔKeff = Kmax – Kop ) is lower than the nominal ΔK and, therefore, a lower da/dN is expected. Here, Kop is the stress intensity when the crack is fully opened (≥ Kmin ). The most accepted mechanism for such an effect is that of the constraint effect on the residual plastically stretched material which is left on the wake of the crack front by the elastic material which surrounds it, when the crack tip continues advancing through the plastic zone. However, several other mechanisms are envisaged as possible, e.g., due to the presence of corrosion debris within the crack (oxide-induced closure) [27] and due to the contact between rough fracture surfaces (roughnessinduced closure) [28]. Figure 4.4, shows a typical fracture surface corresponding to stage II crack growth.

4.2.3 Stage I Fatigue Cracking It is well documented that short fatigue cracks (SFCs) behave markedly different to long cracks. Their growth could occupy a significant portion of the total fatigue

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Fig. 4.4 Striations found on the fracture surface of 2024-T351 represent clear and welldocumented evidence of steady state crack growth. Note that striations are not always visible or uniform along the propagation plane indicating conditions or strong anisotropy

log da/dN

life of several structures/components. The SFC problem was originally proposed by Pearson [29] with his experimental work performed on aluminium alloys. In this pioneer work it was realised that cracks of the order of the grain size tend to propagate at rates far higher than LEFM predictions suggest. In contrast, experimental evidence on 7075-T6 aluminium alloys [30] indicates that SFCs propagate at rates slower than that of long cracks subjected to the same nominal ΔK. Moreover, some SFCs grow at stress intensities well below ΔK threshold. The crack growth rate decelerates or even arrests in some other SFCs. Others cracks are known to reverse such trends and accelerate as much as two orders of magnitude higher than those of corresponding large cracks. The general ‘anomalous’ behaviour of SFCs is depicted in Fig. 4.5.

Short Crack Growth LEFM Long Crack Growth

Above Yield Stress At Yield Stress

Fig. 4.5 Schematic representation of short crack growth behaviour at different stress levels in 7075 aluminium alloy [30, 31]

At Fatigue Limit Below the Fatigue Limit

log ΔK

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Acceleration, deceleration and crack arrest is commonly attributed as the ‘anomalous’ behaviour of SFCs [32]. Lankford [30] suggested that such behaviour is caused by the difficulty of cracks to nucleate microplasticity in certain crystallographic orientations and/or smaller grains. A comprehensive review on short fatigue crack behaviour has been published by Miller [33, 34]. Other researchers indicated that SFCs may be divided into two primary zones of interest [35]: i. The microstructurally short crack zone/regime (MSC) in which the crack is small in relation to the surrounding microstructural features (e.g. cracks which are comparable to the grain size). Crack growth is strongly influenced by microstructure. The micromechanical description of its propagation is expressed by means of the microstructural fracture mechanics (MFM). Figure 4.6a. ii. The physically small cracks (PSC), which are significantly larger than the microstructural dimension and the scale of local plasticity, but are physically small with length typically smaller than a millimetre or two. Here, the microstructure is not the main parameter affecting their propagation but rather, PSC are strongly dependent on the stress level. PSC are conveniently described in terms of Elasto Plastic Fracture Mechanics (EPFM). Figure 4.6b.

A) MICROSTRUCTURALLY SHORT CRACK

B) PHYSICALLY SMALL CRACK a = 0.1 mm

a

C) LONG CRACK a = 10 mm

a > 100 grains a < 1 grain

FEATURES HIGH STRESS MFM-MODE II AND III STAGE I (SHEAR) CRACK a/rp <1

a < 10 grains

FEATURES HIGH STRESS EPFM-MODES I, II AND III STAGE II CRACK a/rp intermediate values

FEATURES

LOW STRESS LEFM-MODE I(STAGE II) TENSILE CRACK a/rp >>1

Fig. 4.6 Classification of SCs: (a) MSC, (b) PSC and for comparison (c) long crack. A briefly description of their main features according to Miller [14] is given (where, a is the crack length and rp is the plastic zone)

Suresh and Ritchie [36], in turn, suggested a comprehensive classification of SCs which include: microstructurally, mechanically, physically and chemically small cracks. A significant contribution to the understanding of SFCs was put forward by Kitagawa and Takahashi [37] who developed the well known Kitagawa–Takahashi diagram (K-T), shown in Fig. 4.7. This diagram shows, for a number of metals, that there is a dividing line, considered to be the bounding condition between propagation leading to failure and non-propagating cracks or crack arrest.

LOG STRESS LEVEL RANGE, Δσ

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Δσ = Fatigue Limit Range

Microstructurally Short Cracking

LEFM Type Cracking

Physically Short Cracking

Δσ =

MFM

ΔΚ th Υ πa

EPFM

a1

ao

a2

LOG CRACK LENGTH, a

Fig. 4.7 Schematic representation of schematic Kitagawa–Takahashi diagram [37, 32, 38]

In Fig. 4.7, the crack lengths a1 and a2 define the deviation of the constant stress and constant stress intensity behaviour, respectively. Crack lengths between a1 and a2 can be expected to propagate faster and to have lower ΔKth , than cracks larger than a2 . This latter crack length represents the point below which the use of LEFM and the Paris law predictions are non-conservative. On the other hand, the point a1 represents the crack length below which there is no crack length effect on fatigue strength [39]. The K-T may be approximated by two straight asymptotic lines. The line given by ΔKth represents the low-stress threshold condition above which a crack should propagate according to LEFM given by [40, 41]: ΔK th Δσ = √ Y πa

(4.4)

It follows that, log Δσ = log

ΔKth √ Y π

−

1 log (a) 2

(4.5)

where Y is the geometric factor of both the geometry of loading and the geometry of the cracked specimen and a is the crack length. The horizontal line is the fatigue limit itself, i.e., the limiting conditions for the propagation of a crack in a plane specimen. In many works it was suggested that the intersection of the fatigue endurance and the LEFM threshold lines takes place at a critical crack length of ≈ 10 grains (PSC) [38, 39]. This assumption supports the fact that a crack must be very much larger than the microstructural features for a

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LEFM concept to be valid. More accurately, the intersection occurs when the crack length is a2 =

1 π

ΔKth YΔσ o

2 (4.6)

for short cracks, Δσ o = Δσ f l . This critical crack length ‘a2 ’ (transition crack size) has been employed as an empirical parameter to account for the differences in propagation rates between long and short cracks even when they are under the same driving forces from a LEFM standpoint [7]. In this sense, K-T may well illustrate the fracture behaviour under different combinations of stress range levels and crack lengths. The relevant K-T has been further extended by Brown [17]. The interfaces of several crack propagation mechanisms occurring at different stress levels have been eloquently identified in a fatigue damage map as depicted in Fig. 4.8. From the socalled Brown map, it is established that cracks may initiate and propagate at stress levels below the fatigue limit and LEFM threshold. These cracks eventually decelerate until they arrest just below the fatigue limit as a consequence of the existing microstructure. Consequently, the Brown map is useful for applying more accurately the several MFM, EPFM and LEFM models since the dominant mode of growth is correctly established. Δσ (MPa)

0.40 STEEL d = 100μm da/dn = 1000 nm

MCG CRACKS MODE II STAGE I

103

/cycle

10

102

MODE II CRYSTALLOGRAPHIC CRACKING

PSB FORMATION

10 10

MICRO STRUCTURAL BARRIER

FATIGUE LIMIT

102

2 σu EPFM MODE III

EPFM CRACKS MODEI STAGE II

2 σy

0.1

2σ y /3 LEFM MODE I

TH

RE

NON PROPAGATING CRACKS

103

SH

OL

D

104

105

α (μm)

Fig. 4.8 The Brown map showing the boundary conditions between short, long and nonpropagating cracks. The fatigue fracture-mode map encompasses six zones, namely (i) LEFM mode I, (ii) EPFM mode I, (iii) EPFM mode III, (iv) mode II stage I, (v) mode II crystallographic cracks, and (vi) non-propagating modes I/II [17]

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83

The large number of papers dealing with the quantification of short crack propagation rate behaviour reported in the literature indicates the importance of this type of fatigue cracks in metallic materials. In this sense, modelling of SFC propagation rate certainly has contributed to a better understanding of MSC and PSC and subsequently to better fatigue-based designs. Furthermore, attributed to developments of SFCs microstructural analysis, it is now possible to incorporate materials effects in an explicit form within the crack system. The behaviour of SFCs has been formulated as an answer to the belief that LEFM principles are violated due to the relatively large cyclic plasticity at the crack tip, which modifies substantially the strength of the stress field ahead of the crack. In addition to the above it was determined that the problem can lead to significant overestimations with overall effect the premature failure of components. Figure 4.9 shows such a case.

Fig. 4.9 The significance of short crack problem is illustrated for an Astroloy. The plot shows the number of fatigue cycles to failure, estimated using LEFM and small crack growth kinetics, as a function of the initial flaw size. The dashed line comes from experimental results

A typical fracture surface paying significant tribute to the different mechanisms governing the propagation of short cracks is shown in Fig. 4.10. The problem of short cracking was originally approached through several LEFM modifications. For example, a strain intensity factor that took into account cracks propagating in cyclic plastic strain fields was proposed by Boettner et al. [42], which is believed to be the first attempt to model SFC propagation. A model dealing with the blocking action of grain boundaries was analysed by Chang [43], in which critical strain energy must be exceeded at the tip of a crack in order for the crack to propagate. As a result, it is argued that an incubation period arises when cracks encounter grain boundaries. A modified LEFM equation with disorientation between grains analysis was put forward by Chan et al. [43]. Tanaka [44], on the

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Fig. 4.10 Short fatigue cracking in 2024-T351 aluminium alloy. The almost smooth fracture surface (facets) indicates crystallographic crack propagation. Explanation for such behaviour is provided in Section 4.2

other hand, considered the effect of grain boundaries on the development of slip bands and formulated the relationship of SFC propagation behaviour as a function of crack tip displacement and closure. In the same work, SFCs are related to the LEFM threshold stress intensity for long cracks. The application of LEFM principles in the modelling of short cracking was the cause for several concerns regarding the ability of the models to accurately predict such complex physics. Hobson, Brown and de los Rios [45, 46] proposed empirical models to quantify both short and long crack propagation rates, which incorporated the effect of the microstructure in Aluminium 7075-T6 and steel. These empirical relationships were later extended by Angelova and Akid [47] in an attempt to describe more precisely short fatigue crack behaviour not only in air but in an aggressive environment.

4.2.4 Stage III Fatigue Cracks The propagation of stage III or unsteady cracks represents a small portion of the overall fatigue life of components. In general, the stress field ahead of the crack tip is considered as strong (large) enough as to negate the application of LEFM models [7]. In general, it is believed that the high stress field will cause the crack to propagate in an unsteady mode. That is the crack tip plasticity rate exceeds that of the crack. For many [48], the mechanism leading to such behaviour replicates necking in a typical static test. Herein, excess plastic energy, created by the inability of the material to transform all the energy into crack growth, is transported into

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85

the crack tip plastic zone causing the development of voids and subsequent void growth. It is generally believed that when the strain ahead of the crack tip reaches a critical value voids will initiate especially around second-phase particles [48, 49]. The voids being under a tri-axial stress state will expand and coalesce by a simple plastic deformation mechanism (internal necking) [50]. Figure 4.11, shows a typical stage III fracture surface.

Transition from Stage II to Stage III

Vo ids Vo th ow Gr id

Vo ids

Fig. 4.11 Fracture surface exhibiting stage III crack growth in 2024-T351. The arrows indicate voids and void growth

4.3 Fatigue Damage Map the Basic Rationale – The Navarro–de los Rios Model Based on dislocation theory [51], Navarro and de los Rios [52–54] developed a model which characterises short crack behaviour. The Navarro–de los Rios (N-R) model considers three different zones (see Fig. 4.12): (a) the crack itself; (b) the fatigue damage (crack length and crack tip plastic zone, length 2c = i∗ D) and (c) the grain boundary zone (the barrier to further spread of plastic deformation) which represents either the locked source or the boundary itself. A microstructural short crack is assumed to grow along a persistent slip band (PSB) in the most favourable grain in terms of size and crystallographic orientation (<111> in fcc) [55, 56]. The plastic zone (slip band ahead of the crack tip)

86

C.A. Rodopoulos σ

σyc

σ2

σ1

crack zone

Plastic zone

0

a

0

n1

Boundary zone

iD/2 iD/2+ro

n2

1

σ

Fig. 4.12 Three zones of the N-R model. D is the grain diameter, ro is the width of the grain boundary and i is the number of half grains, σ 1 is the crack closure stress, σ y c is the cyclic yield stress and σ 2 is the slip band stress concentration

is blocked by the first barrier (grain boundary) and remains blocked until the crack attains the critical length required to activate slip in the next grain by unpinning dislocations. Typical blocking of crack tip plasticity is shown in Fig. 4.13. Unpinning of the dislocations will cause the plastic zone to extend to the next grain boundary and the crack growth rate increases to a new maximum. The process of crack growth, deceleration, slip spreading to the next grain and acceleration repeats itself for several grains until, at the transition point between the short and long crack, the oscillation of the propagation rate ends and a period of monotonic increase in crack growth rate begins. According to the model, the crack and the associated plastic zone are presented by means of continuous distribution of infinitesimal dislocations. The initiation of a new slip band will occur whenever the stress concentration is sufficiently high to activate an appropriate dislocation source in a neighbouring grain a distance r0 away. The stress acting upon such a source is given approximately as 2 τ0 1 1 S(ξ0 ) τ0 1− =√ √ cos−1 n + τ π τ τ (ξ − 1) 2 0

(4.7)

where ζ = (r0 + c)/c > 1 : c is the extent of the damage zone (crack and plastic zone) whose end coincides at any instant with a grain boundary, i.e. c = iD/2 (D is the grain diameter and i is the number of half grains). If a is the crack length, the parameter

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Fig. 4.13 Early crack growth from a surface notch in aluminium alloy 2024-T351: Initiation of slip band in a next grain

n = a/c defines the position of the crack tip in relation to the grain boundary, τ is the applied stress and τ0 the friction stress which opposes the movement of dislocations (lattice friction). This friction stress was taken to be constant and lower than the applied stress [57]. This is reasonable in the early stages of crack growth. However, as deformation proceeds, the associated rise in dislocation density brought about by the general increase in the number of dislocations, both within the slip band in which the crack grows and some other systems activated in the vicinity required by the compatible deformation between grains must result in an increase in the frictional stress. This agrees with experimental evidence of the Hall–Petch relationship [58]: τ = τ0 + K ε D − 2 1

(4.8)

where τ 0 is the friction stress, D is the grain size, Kε is the strain-dependent slope. The operation of a dislocation source requires the attainment of a critical stress σ c that would allow the operation of critical resolved shear stress, τ c . Hence, the critical condition is given as S(ξ0 ) =

1 ∗ m τc 2

(4.9)

where m∗ is the grain orientation factor representing the difficulty of generating dislocations in different angled grains. Once the crack tip has reached a position close to the grain boundary, i.e. once a critical value of n = a/c is reached: n 1 = n 1 c , n 1 c = cos

π 2

σ − σ Li σcomp

(4.10)

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C.A. Rodopoulos

where

σF L σ Li = √ i

(4.11)

σ comp is an appropriate comparison stress greater than the applied stress σ and σ Li is the stress equivalent to the threshold stress for a non-propagating crack spanning over an arbitrary i number of half grains. At the instance that the critical stress is achieved, a new slip plane (dislocation source) is activated and the plastic zone will extend to the next grain. At this instance, the parameter n decreases to a new value ns n s i+2 = n c i

i for i = 1,3,5, · · · i +2

(4.12)

which is obtained by relating crack length to the two successive values of c, that is, iD/2 before slip extension and (i + 2)D/2 after slip, where i gives the position of the tip of the crack in terms of the number of half grain diameter (D/2). As the crack grows the value of n1 will show an intermittent, oscillating pattern. For a growing crack, the initiation of a new slip band is triggered off by the proximity of the crack tip and so there is a sustained increment in the local dislocation density in the crack tip plastic zone. The distribution of dislocations is i i i i i i −1 1 − n 1 ξ −1 1 + n 1 ξ − cosh σ cosh − σ 2 1 ni − ξ i ni + ξ i π2 A 1 1 i i i i i 1 − n 1 + n ξ ξ i −1 −1 2 2 + σ3 − σ2 cosh n i − ξ i − cosh ni + ξ i 2 2 i i 2 ξ 1 i −1 i i i −1 i i + 2 σ τ − σ sin sin − σ n + σ − σ n + 2 1 1 3 2 2 3 π A 1 − ξ i2 1/2 2

f (ξ i ) =

1

(4.13) where τ is the applied stress, σ1i is the crack closure stress, σ2i is the flow resistance of the material to plastic deformation and σ3i is the stress intensity due to dislocation pile-up at the boundary. A = Gb/2π for screw dislocations, or A = Gb/2π(1 – υ) for edge dislocations. G is the shear modulus, b the Burgers vector and υ the Poisson’s ratio. A graphical representation of bounded dislocations is shown in Fig. 4.14. The process of overcoming the microstructural barrier can be thought of as either pushing dislocations through the barrier zone or of unpinning the dislocation source within that zone. In the first case, and for a given combination of applied stresses and crack lengths, r0 varies in relation to the number of dislocations being pushed through the zone. On further crack growth, a critical situation is reached at n i1 = n ic when σ3i attains the required level to overcome the barrier and the slip spreads right across the next grain. In the second case, r0 is considered constant, but an increasing number of dislocations are compressed into the boundary zone as the crack propagates towards the barrier. When the crack reaches the critical length defined by n i1 = n ic , σ3i attains the level of the unpinning stress, slip spreads into the next grain and the crack is able to propagate across the barrier. This friction stress σ3i

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89

Fig. 4.14 Graphical presentation of dislocation distribution along a crack plane

representing the strength of the barrier, which is required to maintain equilibrium, may be calculated in such a way that no singularity appears into the stress field. Setting the condition so that infinite stresses vanish in Equation 4.13 (by equating to zero the term multiplying the singularity) then i 1 τ − σ3i π = 0 σ2 − σ1i sin−1 n i1 + σ3i − σ2i sin−1 n i2 + 2

(4.14)

and the resulting expression is σ3i =

1 cos−1 n i2

σ2i sin−1 n i1 +

1 τ − σ2i π 2

(4.15)

Equation 4.15 can be simplified as σ3i

=

i −1 i 1 i i −1 i σ2 − σ1 sin n 1 − σ2 sin n 2 + σ π 2 cos−1 n i2 1

(4.16)

The parameter n1 and n2 represent in a dimensionless form the crack length and the fatigue damage size. Equations 4.15 and 4.16 describe the geometrical relationship between crack and plastic zone.

4.3.1 Fatigue Damage Map – Defining the Stages of Fatigue Damage 4.3.1.1 Crack Arrest Within the concepts of dislocation pile-up and slip band nucleation described above, conditions for crack arrest are assumed when the slip band concentration, σ 3 , is unable to attain the level required to initiate a slip band in the next grain before the crack tip reaches the grain boundary. Figure 4.15 shows in a schematic way the

90

C.A. Rodopoulos 1000 R=0 R = 0.5 R = –0.5 R = 0.7 R = –0.7

Δσarrest (MPa)

Plain Fatigue Limit

100

10 10–5

10–4

10–3

10–2

Crack Length (m)

Fig. 4.15 Stress ratio effect on crack arrest stress range for 2024-T351 according to Equation 4.24. The mechanical and physical properties used for the calculations are: σ FL(R = 0) = 200 MPa, mi = 1 + 0.35 ln(2a/D), D = 52 μm, α = 0.5 and σ 1 = 0

conditions for crack arrest. The conditions for arrest can be obtained from Equation 4.16, considering that n1 = n2 = 1 and σ2i = 0 (no crack tip plastic zone): 2 i c σ 3 cos−1 n 2 + σ i 1 = σarr est π

(4.17)

where σ3i c is the stress required for the development of a new slip-band (critical constraint effect) and σ1i is the crack closure stress. By employing the approximation cos–1 n2 ≈(2(1 – n2 ))1/2 and the condition n2 = a/(a + ro ), Equation 4.9 is rewritten as √ ro 2 2 c σ i 3 + σ i 1 = σarr est (4.18) π ro + a where a is the crack length and ro is the width of the grain boundary. In [59], Equation 4.18 was further simplified to yield σarr est =

m i σ F L − σ1 + σ1 , √ m i=1 i

i = 2a/D

(4.19)

mi is the grain orientation factor denoting the difficulty of spreading plasm i=1 ticity along different oriented grains, where σ arrest is the crack arrest or threshold where

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91

stress, a is the crack length, σ FL is the fatigue limit of the material (N > 107 cycles), D is the average transverse grain size and σ 1 is the crack closure stress. It should be noted that mi increases monotonically with crack length from m1 = 1 until it reaches the saturated Taylor value of 3.07 (truly polycrystalline behaviour). The hypothesis m1 =1 is rationalised by the fact that, in many cases, crack nucleation takes place initially in grains that are most favourably oriented so that the resolved shear stress can easily reach the maximum value. The threshold stress defined by Equation 4.19 identifies two controlling parameters: (a) the strength of the grain boundary that is part of the σ FL factor (fatigue limit) and (b) the effect of the grain orientation, mi . These two parameters reflect the microstructural influence on crack arrest; (1) by relating the strength of the boundary to the threshold stress for crack propagation and (2) by incorporating the effect of the increasing number of grains transverses to the crack front as the crack grows. The above reflects the increasing probability of a “hard” grain being included in the plastic zone. As hard grain we denote grains with principle slip orientation which exhibit strong angle difference with those already within the plastic zone. Kujawski [60] proposed that the stress ratio, R, on the threshold stress intensity factor range can be given as ΔK th = (1 − R)α for R ≥ 0 ΔK th,R=0

(4.20)

ΔK th = (1 − R) for R ≤ 0 ΔK th,R=0

(4.21)

and

where ΔKth,R = 0 is the threshold stress intensity factor range corresponding to R = 0 and α is a fitting parameter ranging between 0 and 1. A value of α = 0.5 was suggested for aluminium alloys and martensitic steels [60]. Navarro et al. [61] suggested that the smooth specimen fatigue limit is related to the threshold stress intensity factor through K th = σ F L π

D 2

(4.22)

Using a simple substitution between Equations 4.21 and 4.22, R-ratio effect on the fatigue limit is given by ΔσFL = (1 − R)α for R ≥ 0 σFL(R=0)

(4.23a)

Δσ F L = (1 − R) for R ≤ 0 σ F L(R=0)

(4.23b)

and

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C.A. Rodopoulos

Substitution of the above into Equation 4.19, R-ratio effect on the crack arrest stress can be written as Δσarrest =

m i (1 − R)α σFL(R=0) − σ1 + σ1 for R ≥ 0 √ m1 2a/D

(4.24a)

Δσarrest =

m i (1 − R) σFL(R=0) − σ1 + σ1 for R ≤ 0 √ m1 2a/D

(4.24b)

and

Figure 4.15 shows a typical outcome of Equation 4.24 for an aluminium alloy. It is worth noting that the approach is accurate for opening mode growing cracks. In order to avoid the use of the experimentally determined parameters a, Equation 4.24 can be written according to Goodman as Δσarr est

2σ Fmax L(R=−1) σU T S

1 m i 2σU T S (1 − R) + 2 (1 + R) σ F L(R=−1) ≤ √ Y m1 2a/D

− σ1 +σ1 , -∞ < R1<1

(4.25) Where σ FL(R = –1) is the fatigue limit for fully reverse loading and σ UTS is the ultimate tensile strength. The parameter Y is the crack correction factor and should be used with care in order to apply only to small-yielding (LEFM) conditions. mi , since it can signifiOf outmost importance is the realistic representation of m1 cantly affect the ability of Equations 4.24 or 4.25 in predicting crack arrest. In [62] it was suggested that its effects can be determined from extrapolating experimental long crack threshold data into the short crack region. The method has been criticised for mitigating crack shape errors into the short crack region [63]. To resolve the problem in the same work it was suggested that three rational assumptions have to be made. The first considers that the orientation factor of the first grain is always set at 1, to encapsulate the high probability of finding a favourably oriented grain for slip along the surface while, the crack arrest capacity of that grain equals the smooth fatigue limit, the second that saturation should take place when LEFM conditions apply (the polycrystalline behaviour does not affect the LEFM crack arrest capacity of the material and thus a constant slope gradient can be maintained) and the third that the maximum crack arrest capacity is provided by the first grain. The above can be written as mi = 1 for i = 1, Δσarrest = σ F L m1

(4.26a)

mi σY = 3.07 (FCC) or 2.0 (BCC), Δσarrest ≤ m1 3

(4.26b)

i ≥ 1, Δσarrest ≤ σ F L

(4.26c)

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93

σY , where σ y stands for the monotonic yield stress, 3 implies the truthful application of LEFM to describe the elastic stress field ahead of the crack tip with an error not exceeding 7% [64]. Incorporation of such boundary condition into Equation 4.26 yields The conditionΔσarrest ≤

2 mi 81σFL = 3.07 (FCC) or 2.0 (BCC), i ≥ m1 σY 2

(4.27)

Using a two parameter exponential function to meet the boundary conditions set by Equation (4.27), the following relationship can be established mi 2.31 = 3.07 − for FCC metals m1 (1 + 0.05i)1/0.34

(4.28)

mi 1.2 = 2.0 − for BCC metals m1 (1 + 0.05i)1/0.34

(4.29)

Figure 4.16, shows typical results from the application of Equation 4.25. Herein, it is important to note that when a takes values of a single interatomic spacing – defining a single dislocation – the crack arrest stress tends to the Elastic Modulus 106 105

Elastic modulus

Engineering fatigue limit maximum defect size

Stress range (MPa)

104 Creation of crack tip by scatter of defect size 103 102 101 PSB Creation 100 Creation of crack tip by PSB decohesion 10–1 10–10

10–9

10–8

10–7

10–6

10–5

10–4

10–3

10–2

10–1

100

101

Crack length (m) Interatomic Spacing

Fig. 4.16 Schematic representation of crack arrest for an Aluminium alloy according to Equations 4.25 and 4.28. The parameters used are D = 52 μm, R = 0 and σ FL(R = –1) = 140 MPa

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C.A. Rodopoulos

(E = 72000 MPa for the aluminium alloy used in calculations). Of course, the above represents ideal conditions not applicable to engineering metals where the dominant defect is likely to control the maximum allowable stress for arrest. Hence, the dashed line in Fig. 4.16 stands for the engineering fatigue limit. Of outmost importance is to acknowledge that representation of the engineering fatigue limit by a straight line extending all the way down to sizes defining a single interatomic spacing is under physical terms unrealistic. In reality, the absence of large defects within the matrix of a surface grain are likely to promote cracking by decohesion of a persistent slip band (PSB) at stress level above that defining the plateau of the cyclic strain hardening curve. Typical values are in the range of 25–33 MPa at frequency range at about 1–5 Hz [65]. In Fig. 4.16, the above condition is represented by a straight line owning to lack of information in the region. Yet, the interception at a = 1.5–2.0 μm is realistic considering that such value represents the typical width of a highly plastically strained PSB which is more than likely to overshadow crack tip creation from defects of similar or smaller size. At higher stresses and below the engineering fatigue limit, the scatter of defect size is likely to control crack tip creation and its arrest.

4.3.1.2 Transition from Stage I or Short Crack Growth to Stage II – Steady State Growth In the previous section it was discussed that short crack region is divided into two areas that are of microstructurally and physically short crack. The above division, against popular belief, is not related to length but rather to the degree of their ability to retard the crack tip propagation rate. Of course, both the magnitude and the extent of retardation depend on a number of parameters, including the maximum stress level, average grain size, environmental effects, stress ratio and the cyclic properties of the material. Taira et al. [66], noted that da/dN exhibits minima when the size of crack tip plasticity is approximately equal to that of the grain. In [67], Rodopoulos and Kermanidis proposed that the transition from microstructurally to physically short crack takes place when fatigue damage exceeds in size that of a single grain (c ≥ a + D). Yoder et al. [68] and Curtis et al. [68], on the other hand, successfully related transition from physically short to stage II cracks (end of retardation effect) when the size of crack tip plasticity is between one and two grain sizes (a + D ≤ c ≥ a + 2D). The two transition conditions are shown below Δσ M ≤

1 Δσ P ≤ Y

1− R 1+ R

1− R 1+ R

2 y σ π c

2 y σ π c

D a+D

4D a + 2D

(4.30a) (4.30b)

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95

where σ c y is the cyclic yield stress. The subscripts M and P refer to microstructurally and physically short crack, respectively. It is worth noting that in the case of c ≥ D + a, the shape of the crack is unlikely to affect the boundary condition and hence can be neglected. The above is rationalised by the fact that the size of D would provide geometrical limitations towards the available range of aspect ratios (length over depth). Since transition points towards steady-state crack growth the use of a geometric parameter Y, emanating from LEFM, can be used.

4.3.1.3 Transition from Stage II – Steady-State Growth to Stage III – Unsteady Crack Growth Once the stress range/crack coordinates are outside the boundary defined by Equation 4.30b, stage II crack tip propagation begins. At this point crack tip and crack tip plasticity, propagation rate can be estimated using the Paris–Erdogan relationship as well as adaptations of linear-elastic and elasto-plastic fracture mechanics. Termination of such steady state will take place when the strain field ahead of the crack tip is sufficiently large as to achieve rupture strains which will degrade the flow resistance of the material [69]. In [69], such boundary conditions has been suggested to follow

⎞⎤ Δσ I I →I I I π ⎜ (1 − R) ⎟⎥ 8a y ⎢ ⎟⎥ σc ln ⎢ sec ⎜ y ⎠⎦ ⎣ ⎝ πE 2σc ⎡

⎛

x D/2

= ln 1 − ε f , x > 4

(4.31)

where x is the number of half grains constituting crack tip plasticity (c = α + xD/2), Δσ II→III is the transition from stage II to stage III stress range and f is the elongation to failure. The absence of the parameter Y reflects the independency of the boundary condition due to the type of failure (large-scale yielding). Application of Equations 4.25, 4.30 and 4.31 leads to the creation of the fatigue damage map. A typical example is shown in Fig. 4.17. Herein the reader can see how the stages of the growth can change with the stress ratio and also identify the effect of stress ratio on short crack growth (after a particular stress ratio short crack diminishes it effect). The application of the fatigue damage map also allows the user to see how different materials respond to specific growth stages (Figs. 4.18 and 4.19). A typical example is shown in Fig. 4.20. The reader can easily identify by comparing Figs. 4.17 and 4.20 that the tendency of the Ti-alloy towards short cracking diminishes while other stages of growth increase.

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C.A. Rodopoulos

Stress Range (MPa)

1000.0

100.0

Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III Crack growth Toughness or Testing Failure

10.0 10

100

1000 Crack Length (microns)

10000

100000

10000

100000

Fig. 4.17 Fatigue damage map for 2024-T351 at R = 0

Stress Range (MPa)

1000.0

100.0

Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III crack growth Toughness or Tearing Failure

10.0 10

100

1000 Crack Length (microns)

Fig. 4.18 Fatigue damage map for 2024-T351 at R = 0.3

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97

Stress Range (MPa)

1000.0

100.0

Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III crack growth Toughness or Tearing Failure

10.0 10

100

1000 Crack Length (microns)

10000

100000

10000

100000

Fig. 4.19 Fatigue damage map for 2024-T351 at R = 0.5

Stress Range (MPa)

1000.0

100.0

Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III crack growth Toughness or Tearing Failure

10.0 10

100

1000 Crack Length (microns)

Fig. 4.20 Fatigue damage map for a fine grained Ti-Alloy at R = 0

4.3.2 Fatigue Damage Map – Defining the Propagation Rate of Fatigue Stages In order for the fatigue damage map to provide damage tolerance indications, it is necessary to be able to predict the crack growth rate without the need for expensive testing.

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In [70], de los Rios et al. argued that the elasto-plastic crack tip opening displacement can describe the growth rate of the crack tip through the relationship da = φδti p dN

(4.32)

where da/dN is the crack tip propagation rate ,a is the crack length, N is the number of loading cycles and δ tip is the crack tip opening displacement. The factor φ is a dimensionless parameter representing the fraction of dislocations in the slip band participating in the process of crack extension and takes values between 0 and 1. Herein, the size of the slip band is idealised to that of crack tip plasticity. This assumption has been previously discussed in detail in a number of works [71–73]. The parameter φ should depend on the number of pre-existed dislocations which interact with on-growing dislocations emitted by the crack tip, which in turn depend on the grain size and other strong microstructural barriers able to confine plasticity, like twin boundaries or large unshearable precipitates [70]. In [63], it was reported that φ follows a linear association with the far-field stress range Δσ . Such finding is partially correct since it excludes the crack tip and crack tip plasticity size effects. Of course, Equation 4.32 requires experimental data to incorporate material and crack type variations and thus can be easily subjected to experimental errors during the data input stage. Such problems can be overcome if a pure analytical crack tip prediction rate model is employed. Such model has been presented and compared to numerous stage II experimental results by Nicholls [74]: 1 da = ΔK 4 y dN 4Eσc K I c 2

(4.33)

where σ c y is the cyclic yield stress, KIc is the fracture toughness under plane strain conditions, E√is the modulus of elasticity and ΔK is the stress intensity factor range (ΔK = Δσ πa with Δσ being the far-field stress range). The use of Equation 4.33 does not represent a panacea to prediction error but a rather controllable output through macroscopic and not experimental/fitting parameters. If linear elastic conditions are assumed, Equation 4.32 can be modified following [75] λΔK 2 da =φ y dN Eσc

(4.34)

where λ contains not physical meaning and it is rather a mathematical moderator primarily used to fit experimental results. As opposed to the original estimate from Wells [75] of λ = 4/π, Nicholls proposed a value of unity to best match experimental results. By equating the above formulas (Equations 4.33 and 4.34) and assuming a stress ratio, R = 0, the participating fraction of dislocations can be expressed as φ=

σ 2 πa 4K c2

(4.35)

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99

The introduction of the effect of crack tip plasticity size on crack tip propagation rate should reveal the fraction of the participating dislocations. However, due to the complexity of the issue and the lack of experimental evidence, such association is beyond an analytical solution. Nevertheless, such relationship, with some degree of error, can be sought through the effect of crack tip plasticity size on δ tip . Such, relationship was expressed as φ2 k da = dN S

√

1 − n2 σa n

(4.36)

where n is the dimensionless value of a over the fatigue damage, c (c = a + crack tip plasticity size), k = 1 or 1 – ν (where ν is the Poisson’s ratio) for screw or edge dislocations, respectively, and S is the stiffness modulus with values between the shear modulus G the elastic modulus E. Considering that Equation 4.33 delivers such a wide range of crack tip propagation rates, most likely to include rates corresponding to stage I up to stage III crack growth, the effect of crack tip plasticity size on crack tip propagation rate can be given as da = dN

σ 2 πa

2k S

4K c 2

√

1 − n2 σa n

(4.37)

Equation 4.37 can be used in the equations providing the transition from the fatigue damage stages leading to,

da dN

=

Δσarr est 2 πa

4K I c 2

arr est

& ' ' (1 − 2k S

a a+

a a+

2 D 25

Δσarr est a

(4.38)

D 25

crack growth rate for crack arrest.

da dN

da dN

=

P

4K c 2

M

=

Δσ M 2 πa

Δσ P 2 πa 4K c

2

2k S

*

) 2k S

1−

2 a a+D

a a+D

a 1− a + 2D a a + 2D

Δσ M a

(4.38a)

2 Δσ p a

(4.38b)

crack growth rate for the transition from microstructurally to physically to stage II crack growth and

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C.A. Rodopoulos

da dN

I I →I I I

=

Δσ I I →I I I 2 πa 4K c

2

& ' ' (1 −

a a + x D2 ⎛ ⎞

2k S

⎜ ⎝

2 Δσ I I →I I I a

(4.39)

⎟ D⎠ a+x 2 a

the crack growth rate for the transition from stage II to stage III. Typical examples are shown in Figs. 4.21 and 4.22.

Crack Growth Rate (m/cycle)

1.0E–7

1.0E–8 Crack Arrest Low Threshold

1.0E–9

High Threshold Long Crack

1.0E–10

1.0E–11

1.0E–12 10

100

10000

1000 Crack Length (microns)

100000

Fig. 4.21 Fatigue damage map and corresponding crack growth rates for a fine-grained Ti-Alloy at R = 0

Crack Growth Rate (m/cycle)

1.0E–7

1.0E–8 Crack Arrest Low Threshold

1.0E–9

High Threshold Long Crack

1.0E–10

1.0E–11

1.0E–12 10

100

1000

10000

100000

Crack Length (microns)

Fig. 4.22 Fatigue damage map and corresponding crack growth rates for a fine-grained Ti-Alloy at R = 0.3

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4.4 Conclusions The chapter aims to familiarise the reader with the principles of the fatigue damage map and to draw attention towards the fact that the response of the material is inherently included in the basic material properties. With knowledge of the parameters affecting fatigue damage, the propensity of the material towards fatigue loading is possible using only readily available information. The fatigue damage map can be used as: (a) a virtual tool for examining the response of the material towards fatigue damage and hence provide a measure for material comparison; (b) a virtual tool for predicting the corresponding crack growth rates without expensive and complex testing procedures; and (c) a virtual tool which in comparison with finite element stress analysis can provide a strong damage tolerance indications.

References 1. A.A. Griffith, “The Phenomenon of Rupture and Flow in Solids,” Philos. Trans. R. Soc. Lond. A, Vol. 221, 1921, pp. 163–197. 2. P.C. Paris and F.A. Erdogan, “Critical Analysis of Crack Propagation Laws,” J. Basic Eng. TRANS ASME, Vol. 85(Series D), 1963, pp. 528–534. 3. N.E. Frost and D.S. Dugdale, “The Propagation of Fatigue Cracks in Test Specimens,” J. Mech. Phys. Solids, Vol. 6, 1958, 92–110. 4. J. Schijve, Fatigue of Structures and Materials, Kluwer Academic Publishers, The Netherlands, 2001. 5. Fatigue Design Handbook AE-10, Society of Automotive Engineers, USA, 1988. 6. S.R. Swanson, Handbook of Fatigue Testing, ASTM STOP 566, 1974. 7. S. Suresh, Fatigue of Materials, 2nd edition, Cambridge, University Press, Cambrige, 1998. 8. P. Luk´asˇ, “Fatigue Crack Nucleation and Microstructure,” ASM Handbook Volume 19: Fatigue and Fracture, ASM International, Materials Park, Ohio, 1996, pp. 96–109. 9. J.R. Yates, “Fatigue of Engineering Materials. MSc in Structural Integrity (MPE603), course notes,” Department of Mechanical Engineering, The University of Sheffiel, Sheffield, U.K., 1999. 10. K.J. Miller, “Fundamentals of Deformation and Fracture,” In: Proc. Eshelby Memorial Symposium, Cambridge, U.K., B.A. Bilby, Cambridge University Press, 1985, pp. 477–500. 11. D.W. Hoeppener, “Model for Prediction of Fatigue Lives Based upon a Pitting Corrosion Fatigue Process. Fatigue Mechanisms,” In: Proc. of an ASTM-NBS-NSF Symposium, Kansas, City, Mo., J.T. Fong, Ed., ASTM STP 675, 1978, pp. 841–870. 12. R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd edition New York, USA, John Wiley & Sons, 1989. 13. F. Guiu, R. Dulniak, and B.C. Edwards, “On the Nucleation of Fatigue Cracks in Pure Polycrystalline a-iron,” Fatigue Fract. Eng. Mater. Struct., Vol. 5, 1982, pp. 311–321. 14. K.J. Miller, “Materials Science Perspective of Metals Fatigue Resistance,” Mater. Sci. Technol., Vol. 9, 1993, pp. 453–462. 15. P.J.E. Forsyth, “A two stage process of fatigue crack growth, in crack propagation,” In: Proceedings of Cranfield Symposium, London, Her Majesty’s Stationery Office, 1962, pp. 76–94. 16. V.M. Radhakrishnan and Y. Mutoh, “On Fatigue Crack Growth in Stage I,” In: The Behaviour of Short Fatigue Cracks, EGP Pub. 1. Great Britain, K.J. Miller and E.R. de.los.Rios, Eds., Mechanical Engineering Publications Limited, 1986, pp. 87–99.

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17. M.W. Brown, “Interfaces Between Short, Long and Non-Propagatig Cracks,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 423–439. 18. Y. Mutoh and V.M. Radhakrishnan, “An Analysis of Grain Size and Yield Stress Effects on Stress at Fatigue Limit and Threshold Stress Intensity Factor,” J. Eng. Mater. Technol., Vol. 103, 1986, pp. 229–233. 19. E. Hay and M.W. Brown, “Initiation and Early Growth of Fatigue Cracks from a Circunferential Notch Loaded in Torsion,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 309–321. 20. R.O. Ritchie, F.A. McClintock, H. Nayeb-hashemi, and M.A. Ritter, “Mode III Fatigue Crack Propagation in Low Alloy Steel,” Metall. Trans., Vol. 13A, 1982, pp. 101–110. 21. P.C. Paris and F.J. Erdogan, “A Critical Analysis of Crack Propagation Law,” J. Basic Eng. Trans. ASME, Series D, Vol. 85, No. 4, 1963, pp. 528–535. 22. P.C. Paris, “Fracture Mechanics and Fatigue: A Historical Perspective,” Fatigue Fract. Eng. Mater. Struct., Vol. 21, 1998, pp. 535–540. 23. J.F. Knott, Fundamentals of Fracture Mechanics, Butterworths, London, 1973. 24. W. Elber, “Fatigue Crack Closure Under Cyclic Tension,” Eng. Fract. Mech., Vol. 2, 1970, pp. 37–45. 25. S. Suresh and R.O. Ritchie, “The Propagation of Short Fatigue Cracks,” Int. Met. Rev., Vol. 29, No. 6, 1984, pp. 445–501. 26. N. Louat, K. Sadananda, M. Duesbery, and A.K. Vasudevan, “A Theoretical Evaluation of Crack Closure,” Metall. Trans., Vol. 24-A, 1993, pp. 2225–2232. 27. R.O. Ritchie, S. Suresh, and C.M. Moss, “Near Threshold Fatigue Crack Growth in 21/4 Cr 1Mo Pressure Vessel Steel in Air and Hydrogen,” J. Eng. Mater. Technol. (Trans. ASME), Vol. 102, 1980, pp. 293–299. 28. S. Suresh and R.O. Ritchie, “A Geometric Model for Fatigue Crack Closure Induced by Fracture Surface Roughness,” Metall. Trans. A, Vol. 13A, 1982, pp. 1627–1631. 29. S. Pearson, “Initiation of Fatigue Cracks in Commercial Aluminium Alloys and the Subsequent Propagation of Very Short Cracks,” Eng. Fract. Mech., Vol. 7, 1975, pp. 235–247. 30. J. Lankford, “The Growth of Small Fatigue Cracks in 7075-T6 Aluminium,” Fatigue Fract. Eng. Mater. Struct., Vol. 5, No. 3, 1982, pp. 233–248. 31. D. Kujawski and F. Ellyin, “A Microstructurally Motivated Model for Short Crack Growth Rate,” Short Fatigue Cracks, ESIS 13, K.J. Miller and E.R. de los Rios, Eds., Mechanical Engineering Publications, London, 1992. 32. D. Taylor and J.F. Knott, “Fatigue Crack Propagation Behaviour of Short Cracks: The Effect of Microstructure,” Fatigue Fract. Eng. Mater. Struct., Vol. 4, No. 2, 1981, pp. 147–155. 33. K.J. Miller, “The Behaviour of Short Fatigue Cracks and their Initiation. Part I – A Review of Two Recent Books,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 1, 1987, pp. 75–91. 34. K.J. Miller, “The Behaviour of Short Fatigue Cracks and their Initiation. Part II – A General Summary,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 1, 1987, pp. 75–91. 35. K. Tokaji and T. Ogawa, “The Growth Behaviour of Microstructurally Small Fatigue Cracks in Metals,” Short Fatigue Cracks, ESIS 13, K.J. Miller and E.R. de los Rios, Eds., Mechanical Engineering Publications, London, 1992, pp. 85–99. 36. S. Suresh and R.O. Ritchie, “The Propagation of Short Fatigue Cracks,” Int. Met. Rev., Vol. 29, No. 6, 1984, p. 445. 37. H. Kitagawa and S. Takahashi, “Applicability of Fracture Mechanics to Very Small Cracks or the Cracks in the Early Stage,” In: 2nd International Conference on Mechanical Behaviour of Materials (ICM2), Boston, USA. American Society of Metals, Metal park, Ohio, 1976, pp. 627–631. 38. K.J. Miller, “The Behaviour of Short Fatigue Cracks and their Initiation. Part I – A Review of Two Recent Books,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 1, 1987, pp. 75–91.

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39. D. Taylor, “Fatigue of Short Cracks: The Limitations of Fracture Mechanics,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 479–490. 40. J.M. Kendall, M.N. James, and J.F. Knott, “The Behaviour of Physically Short Fatigue Cracks in Steels,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 241–258. 41. E.R. de.los.Rios, P. Mercier, and B.M. El-Sehily, “Short Crack Growth Behaviour Under Variable Amplitude Loading of Shot Peened Surfaces,” Fatigue Fract. Eng. Mater. Struct., Vol. 19, No. 2/3, 1996, pp. 175–184. 42. R.C. Boettner, C. Laird, and A.J. McEvily, “Crack Nucleation and Growth in High Strain-Low Cycle Fatigue,” Trans. Metall. Soc. AIME, Vol. 233, 1965, pp. 379–385. 43. K.S. Chan and J. Lankford, “A Crack Tip Model for the Growth of Small Fatigue Cracks,” Scr. Metall., Vol. 17, 1983, pp. 529–538. 44. K. Tanaka, “Modelling of Propagation and Non-Propagation of Small Cracks,” In: Small Fatigue Cracks, R.O. Ritchie and J. Lankford, Eds., Metallurgycal Society Inc. 1986, pp. 343–362. 45. P.D. Hobson, “The Formulation of a Crack Growth Equation for Short Cracks,” Fatigue Fract. Eng. Mater. Struct., Vol. 5, No. 4, 1982, pp. 323–327. 46. P.D. Hobson, M.W. Brown, and E.R. de.los.Rios, “Two Phases of Short Crack Growth in a Medium Carbon Steel,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 441–459. 47. D. Angelova and R. Akid, “A Note on Modelling Short Fatigue Crack Behaviour,” Fatigue Fract. Eng. Mater. Struct., Vol. 21, 1998, pp. 771–779. 48. J.R. Rice, The Mechanics of Fracture, ASME AMD, Vol. 19, 1976, pp. 23–53. 49. R.M. McMeeking, “Finite Deformation Analysis of Crack Tip Opening in Elastic-Plastic Materials and Implications for Fracture,” J. Mech. Phys. Solids, Vol. 25, 1977, pp. 357–381. 50. J.F. Knot, “Microscopic Aspects of Crack Extension,” In: Advances in Elasto-Plastic Fracture Mechanics, L.H. Larsson, Ed., Applied Science Publishers, Essex, England, 1980. 51. B.A. Bilby, A.H. Cottrell, and K.H. Swinden, “The Spread of Plastic Yielding from a Notch,” Proc. R. Soc. Lond. A, Vol. 272, 1963, pp. 304–314. 52. A. Navarro and E.R. de.los.Rios, “A Model for Short Fatigue Crack Propagation with an Interpretation of the Short-Long Crack Transition,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 2, 1987, pp. 169–186. 53. A. Navarro and E.R. de.los.Rios, “Compact Solution for a Multizone Bcs Crack Model with Bounded or Unbounded End Conditions,” Philos. Mag. A, Vol. 57, No. 1, 1988, pp. 43–50. 54. A. Navarro and E.R. de.los.Rios, “Fatigue Crack Growth Modelling by Successive Blocking of Dislocations,” Proc. R. Soc. Lond. A, Vol. 437, 1992, pp. 375–390. 55. A.H. Cottrell and D. Hull, “Extrusion and Intrusion by Cyclic Slip in Copper,” Proc. R. Soc. Lond. A, Vol. 242, 1957, pp. 211–213. 56. P.J.E. Forsyth, “Slip Band Damage and Extrusion,” Proc. R. Soc. Lond. A, Vol. 242, 1957, pp. 198–202. 57. E.R. de los Rios, M.W. Brown, K.J. Miller, and H.X. Pei, “Fatigue damage accumulation during cycles of non-proportional straining,” In Basic questions in fatigue, Vol. 1, ASTM STP 924: 1988, pp. 194–213, ASTM, Philadelphia. 58. N.J. Petch, “The Cleavage Strength of Polycrystals,” J. Iron Steel Inst., Vol. 174, 1953, pp. 25–28. 59. E.R. de los Rios, “Dislocation Modelling of Fatigue Crack Growth in Polycrystals,” Eng. Mech., Vol. 5, No. 6, 1998, pp. 363–368. 60. D. Kujawski, “A Fatigue Crack Driving Force Parameter with Load Ratio Effects,” Inter. J. Fatigue, Vol. 23, 2001, pp. S236–S246. 61. A. Navarro, C. Vallellano, E.R. de los Rios, and X.J. Xin, Notch Sensitivity and Size Effects Described by a Short Crack Propagation Model,” In: Engineering Against Fatigue, J.H.

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66. 67.

68.

69.

70. 71. 72.

73. 74. 75.

C.A. Rodopoulos Beynon, M.W. Brown, T.C. Lindley, R.A. Smith and B. Tomkins, Eds., Balkema, Netherlands, 1999. E.R. de los Rios and A. Navarro, “Considerations of Grain Orientation and Work Hardening on Short-Fatigue Crack Modelling,” Phil. Mag. A, Vol. 61, No. 3, 1990, pp. 435–449. C.A. Rodopoulos, “Predicting the Evolution of Fatigue Damage Using the Fatigue Damage Map Method,” Theor. Appl. Fract. Mech., Vol. 45, 2006, pp. 252–265. M.W. Brown, “Interfaces Between Short, Long and Non-Propagating Cracks,” In: The Behaviour of Short Fatigue Cracks, K.J. Miller and E.R. de los Rios, Eds., Mechanical Engineering Publications, London, 1986, pp. 423–439. A.T. Winter, “Cyclic Deformation: the Two Phase Model,” Proceedings of the Eshelby Memorial Symposium, International Union of Theoretical and Applied Mechanics, B.A. Bilby, K.J. Miller and J.R. Willis, Eds., Cambridge University Press, Cambridge, 1984, pp. 573–582. S. Taira, K. Tanaka, and Y. Nakai, “A Model of Crack Tip Slip Band Blocked by Grain Boundary,” Mech. Res. Comm., Vol. 5, 1978, pp. 375–381. C.A. Rodopoulos and Al.Th. Kermanidis, “Understanding the Effect of Block Overloading on the Fatigue Behaviour of 2024-T351 Aluminium Alloy Using the Fatigue Damage Map,” Inter. J. Fatigue, Vol. 29, No. 2, 2006, pp. 276–288. G.R. Yoder, L.A. Cooley, and T.W. Crooker, “On Microstructural Control of Near-Threshold Fatigue Crack Growth in 7000-Series Aluminium Alloys,” Scr. Metall., Vol. 16, 1982, pp. 1021–1025. C.A. Rodopoulos, E.R. de los Rios, J.R. Yates, and A. Levers, “A Fatigue Damage Map for the 2024-T3 Aluminium Alloy,” Fatigue Fract. Eng. Mater. Struc., Vol. 26, No. 7, 2003, pp. 569–576. E.R. de los Rios, Z. Tang, and K.J. Miller, “Short Crack Fatigue Behaviour in a Medium Carbon Steel,” Fatigue Fract. Eng. Mater. Struct., Vol. 7, No. 2, 1984, pp. 97–108. R.M. Pelloux, “Crack Extension by Alternating Shear,” Eng. Fract. Mech., Vol. 1, 1970, pp. 697–704. F.A. McClintock, “Considerations for Fatigue Crack Growth Relative to Crack Tip Displacement,” In: Engineering Against Fatigue, J.H. Beynon, M.W. Brown, T.C. Lindley, R.A. Smith and B. Tomkins, Eds., A.A. Balkema Publishers, Rotterdam, Netherlands, 1999. J.N. Eastbrook, “A Dislocation Model for the Rate of Initial Growth of Stage I Fatigue Cracks,” Inter. J. Fract., Vol. 24, 1984, pp. R43–R49. D.J. Nicholls, “The Relation Between Crack Blunting and Fatigue Crack Growth Rates,” Fatigue Fract. Eng. Mater. Struct., Vol. 17, No. 4, 1994, pp. 459–467. A.A. Wells, “Application of Fracture Mechanics at and Beyond General Yielding,” Br. Weld. J., Vol. 10, 1963, pp. 563–570.

Chapter 5

Predicting Creep and Creep/Fatigue Crack Initiation and Growth for Virtual Testing and Life Assessment of Components K.M. Nikbin

Abstract Predicting creep and creep/fatigue crack initiation and growth, under static and cyclic loading, in engineering materials at high temperatures is an important aspect for improved life assessment in components and development of virtual test methods. In this context, a short overview of the present standards and codes of practice as well as experimental methods and models to predict failure are presented. Following a brief description of engineering creep parameters and basic elastic–plastic fracture mechanics methods, high-temperature fracture mechanics parameters are derived by analogy with plasticity concepts. Techniques are shown for determining the creep and fatigue fracture mechanics parameters K and C∗ to predict experimental crack growth using uniaxial creep data. An analytical ‘failure strain/constraint’ based ductility exhaustion model called the NSW model is presented. The model uses a ductility exhaustion argument and constraint at the crack tip is able to predict, within a range of as much as a factor of ∼30 crack initiation and growth at high temperatures, over the plane stress to plane strain regimes. By taking into account angular damage distribution around the crack tip NSW model is further refined as the NSW-MOD model. This model is able predict a much improved upper/lower bound of a factor of between ∼0.5–7, depending on the creep index n, in steady-state crack initiation and growth over the plane stress/strain region in components containing defects. The presence of cyclic load is assumed to introduce a cycle-dependent fatigue component with a linearly summed cumulative damage effect with the creep response. The prediction for the fatigue component can be handled using either the Paris law or the method proposed by Farahmand in this book. Hence, this chapter does not get into the details of the fatigue response. In conclusion, the creep/fatigue modeling presented can be used as a tool in component design of metallic parts, as well as in life assessment of cracked components at elevated temperatures and in predicting virtual cracking behavior in fracture mechanics specimens.

K.M. Nikbin (B) Mechanical Engineering Department, Imperial College, London SW7 2AZ, UK e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 5,

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5.1 Introduction Engineering life assessment and component lifetime prediction against fracture at high temperatures utilise models based on physical principles but which always need to be validated under practical and operational circumstances [1–9]. The use of computational methods to develop predictive methods in creep crack initiation and growth is important as this will allow to optimise the number of tests that will be needed as well as help improve life prediction methods in components. In this chapter a short description of the background to life assessment methodologies for cracked components which have their roots in creep analysis of uncracked bodies is presented. Following this the engineering creep parameters ranging from uniaxial to multiaxial states of stress are considered. However, by the very nature of the subject’s diversity this chapter cannot go into detail of the derivation of the parameters and models and the reader should follow the appropriate references for further detail [10–21]. The mechanism of time-dependent deformation is shown to be analogous to deformation due to plasticity. Therefore, elastic– plastic fracture mechanics parameters such as K or J are reviewed and linked to high-temperature fracture mechanics parameter C∗ and techniques are shown for determining the creep fracture mechanics parameter C∗ using experimental crack growth data, collapse loads and reference stress methods [3–9]. Finally, models for predicting creep crack initiation and growth in terms of C∗ and the creep uniaxial ductility are developed [22–32]. Cumulative damage concepts are used for predicting crack growth under static and cyclic loading conditions [34–37].

5.1.1 Background to Life Assessment Codes Components in the power generation and petro-chemical industry operating at high temperatures are almost invariably submitted to static and/or combined cycle loading. They may fail by net section rupture, crack growth or a combination of both. The development of codes in different countries has moved in similar direction and in many cases the methodology has been borrowed from a previously available code in another country. The early approaches to high-temperature life assessment used methodologies that were based on defect-free assessment codes. For example, ASME Code Case N-47 [1] and the French RCC-MR [2], which have many similarities, are based on lifetime assessment of uncracked structures. The materials properties data that are used for these codes are usually uniaxial properties and S–N curves for fatigue. However, with the development of non-destructive examination (NDE) methods and improvements in the ability to find smaller and smaller cracks in aging parent and welded components it has become clear that fracture mechanics should be considered in component design and predicting remaining life. More recent methods, therefore, make life assessments based on the presence of defects in the component. The codes dealing with defects [3–9] vary in the extent of the range of failure

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behavior they cover. However, they define a new direction for life assessment as well as component design based on the presence of cracks which should realistically improve predictions. This is especially so where welds or creep-damaged components which substantially aged are concerned so that the presence of a predetermined crack size would need to be assumed even if one is not physically observed due to NDE limitations. Therefore, fracture mechanics solutions and models dealing with creep and creep/fatigue interaction in initiation and growth of defects are treated in the codes in a robust and pragmatic way [3–9] so that a user can relate his/her findings to the plant life cycle. In terms of creep crack growth, all the codes propose similar approaches but use different formulae for the analysis of the crack under linear and non-linear conditions. This difference is most likely to affect the predictive solutions. In such codes, material properties dealing with crack growth data that are needed are more complex compared to uniaxial data both in terms of testing methods and derivation. Consequently, valid methods for material characterization coupled with advanced fracture mechanics modeling is an important part for design and life prediction in components.

5.1.2 Creep Analysis of Uncracked Bodies The time-dependent deformation mechanism occurring at elevated temperature that is generally non-reversible is defined as creep. Creep is most likely to occur in components that are subjected to high loads at elevated temperatures for extended periods of time. Creep may ultimately cause fracture or assist in developing a crack in components subjected to stresses at high temperatures. In the last 30 years, rapid development has taken place in the subject and references at the end of the chapter give an indication of this work. The phenomenon of creep is based on a time-dependent process whereby the material deforms irreversibly. Creep in polycrystalline materials occurs as a result of the motion of dislocations within grains, grain boundary sliding and diffusion processes. A creep curve can simply be split up into three main sections as shown in Fig. 5.1. All the stages of creep are not necessarily exhibited by a particular material for given testing conditions. In Fig. 5.2 the representation of the rupture times versus the applied stress covers the response of the material throughout the three regimes. The primary region is a period of decreasing creep rate where work-hardening processes dominate and cause dislocation motion to be inhibited. The secondary or steady-state region of creep deformation is frequently the longest portion and corresponds with a period of constant creep rate where there is a balance between workhardening and thermally activated recovery (softening) processes. The final stage is termed the tertiary region. This is a period of accelerating creep rate which culminates in fracture. It can be caused by a number of factors which include increase in stress in a constant load test, formation of a neck (which also results in an increase in stress locally), voiding and/or cracking and overageing (metallurgical instability in alloys).

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Fig. 5.1 The creep curve, showing definitions of creep strain rates

× ε&A

Creep Strain εc

Failure strain

ε&s Tertiary

Primary

Secondary

Time, t

tr

Fig. 5.2 Stress/time to rupture relation

Log σ −1/ν

Log tr

5.1.3 Physical Models Describing Creep A number of processes dominate the creep processes [10–21] producing a creep curve of creep strain versus time as schematically shown in Fig. 5.1. When secondary creep dominates, it is often possible to express the minimum secondary creep strain rate ε˙ s or ε˙ sc , in the form ε˙ sc ασ n exp (−Q/RT )

(5.1)

The existence of several creep processes indicates that in general n and Q in Equation (5.1) will change as a mechanism boundary is crossed. In addition, it has been established that a simple power law stress dependence is not always satisfactory. At high stresses an exponential expression of the form ε˙ sc α exp (βσ ) exp (−Q/RT )

(5.2)

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where β is a material-dependent constant is often more adequate. It has shown [11] that Equations (5.1) and (5.2) can both be encompassed by the relation ε˙ sc α (sinh ασ )n exp (−Q/RT )

(5.3)

provided β = αn. When αn < 0.8, Equation (5.3) reduces to Equation (5.1) and when ασ > 1.2 it reverts to Equation (5.2). No satisfactory physical model has yet been developed which produces expressions of the form of Equation (5.2) or (5.3). There are creep laws which deal with the time dependence of creep. The stress and temperature dependence of the material parameters introduced will not be examined. However, for the most part, they can be described by expressions that are similar to those used for secondary creep in the previous section. Model-based laws where creep strain is predicted from motion of dislocations give an understanding of the creep behaviour but are rarely useful for engineering purposes. As a result, empirical laws have been produced [11–12] to give more accurate descriptions of the observed shapes of creep curves. A representative selection is listed below with an indication of their ranges of applicability. Usually for T/Tm < 0.3, work-hardening processes dominate and primary creep is observed which can often be described by a logarithmic expression of the form εc = α ln (1 + βt)

(5.4)

where α and β are parameters which in general are functions of stress and temperature. Within the temperature range 0.3
(5.5)

where m < 1 and takes the value 1/3 in the Andrade [13] expression. In Equation (5.5) the first term describes the primary region and the second term describes secondary creep. For T/Tm > 0.5, Equation (5.5) can still be employed [11], but an alternative expression that has been used is −t + ε˙ sc t εc = εt 1 − exp τ

(5.6)

Other empirical laws have been proposed that have wider applicability than those just presented and which can also accommodate tertiary creep [11–13]. Two representative equations are 1

εc = αt 3 + βt + γ 2

(5.7)

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and εc = θ2 {1 − exp (−θ2 t)} + θ2 {exp (−θ4 t) − 1}

(5.8)

In their most general formulations, each of these parameters consists of a summation of terms involving stress and temperature. The equations describe primary, secondary and tertiary creep. They can correlate a wide spread of behaviour because of the number of disposable parameters that are used. They do however need a large body of data to identify all the terms. They are of most use in extrapolating experimental data to longer times. Details of describing and modeling the creep curve have been presented. However, in most circumstances and especially in the case of the modeling carried out in this chapter, the simple secondary creep strain rate ε˙ sc or the average creep strain rate ε˙ A (shown schematically in Figs. 5.1 and 5.2) are sufficiently accurate to describe the creep response of metals and structures. These are represented by a power law ε˙ sc = As σ n s

(5.9)

and the average creep rate obtained directly from creep rupture data (e.g. in Figs. 5.1 and 5.2) has been proposed to account for all three stages of creep as ε˙ A =

εf = ε˙ 0 tr

σ σ0

n A = A Aσ nA

(5.10)

This is particularly useful in modelling the crack tip where the combinations of the three regimes interact to develop damage, crack initiation and growth within a process zone. The models developed in this chapter make use of Equations (5.9), (5.10) to predict the creep response of structures containing cracks.

5.1.4 Complex Stress Creep Because creep deformation is not linearly dependent on stress, the effects of stresses that are applied in different directions cannot be superimposed linearly. However, it is found experimentally that: (i) hydrostatic stress does not affect creep deformation; (ii) the axes of principal stress and creep strain rate coincide; (iii) and no volume change occurs during creep. These observations are the same as those that are made for plastic deformation [15]. This is not surprising when both processes are controlled by dislocation motion. The observations imply that the definitions of equivalent stress and strain increment used in classical plasticity theory can be applied to creep provided strain rates are written in place of the plastic strain increments. Therefore, for creep, the Levy–Mises flow rule becomes

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ε˙ 3c ε˙ 1c ε˙ 2c ε¯˙ c = = = (5.11) [σ1 − 0.5(σ2 + σ3 )] [σ2 − 0.5(σ1 + σ3 )] [σ3 − 0.5(σ1 + σ2 )] [σ¯ ] Equation (5.11) satisfies the experimental observations (i) to (iii) provided appropriate definitions are chosen for σ¯ and ε˙ c . From (i) and the observation that dislocations are mainly responsible for creep, it may be inferred that shear stresses govern creep deformation so that either the Von-Mises or Tresca criterion can be employed. With the Von-Mises definition being given as 1 σ¯ = √ = [(σ1 − σ2 )2 + (σ2 − σ3 )2 (σ3 − σ1 )2 ]1/2 2

(5.12)

and √ + 2 2 2 ,.1/2 2 c ε˙ 1 − ε˙ 2c + ε˙ 2c − ε˙ 3c + ε˙ 3c − ε˙ 1c ε˙ = 3 c

Assuming the Tresca definition σ¯ = (σ1 − σ2 )

(5.14)

ε¯˙ = 2/3(˙ε1c − ε˙ 3c )

(5.15)

and

The Von-Mises definition can be regarded as a root mean square maximum shear stress criterion and the Tresca definition as a maximum shear stress criterion. Most investigations of equivalent stress criteria were carried out mainly on thin-walled cylinders subjected to different combinations of tension, torsion and internal pressure [16]. The case of internal pressure alone will now be considered as an application of the complex stress creep analysis. It can be shown for any complex stress state that the equivalent stress calculated using the Tresca definition is always greater than, or equal to, that determined from the Von-Mises definition. Use of the Tresca criterion will, therefore, always produce the same, or a higher, creep rate than is obtained from the Von-Mises criterion. As most experimental results usually fall between the two predictions [17–18] assumption of the Tresca criterion is therefore likely to be conservative. The list of creep laws and models presented for describing the time dependence of creep and multiaxiality effects are by no means extensive. For further reading the reader should go to the references which cover the range of models in depth [17–20].

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5.1.5 Influence of Fatigue in Uncracked Bodies Under combined static and cyclic loading at elevated temperatures creep and fatigue processes can take place together [34–36]. It is possible for these processes to occur independently or in conjunction depending upon the controlling mechanisms in each case. In the former case failure will be dominated by the process which occurs most rapidly and in the latter by the slowest process. When creep mechanisms govern, it is likely that the fracture surface will be intergranular and when fatigue processes dominate transgranular. When both mechanisms contribute to failure a mixed mode of fracture may be expected. Cumulative damage laws are available for dealing with superimposed mean and cyclic loading and for assessing the influence of variable amplitude cycling. Most engineering components experience variable amplitude loading during operation. It is usually supposed that fatigue damage incurred under these conditions can be accommodated using Miner’s cumulative damage law. This states that failure occurs when the fractional damage accumulated at each condition sums to one, that is, - i ti n3 n1 + + + . . . = 1 or =1 N1 T1 N3 N

(5.16)

The relation should be used as a general guide only as experimental evidence indicates that the fractional damage suffered at failure can range between about 0.5 and 2.0. This can usually be attributed to sequence of loading effects which are not accommodated in a simple expression like the Miner’s Law. It has been proposed that when creep and fatigue processes occur independently, the fractions of damage incurred by each mechanism can be summed separately to predict failure when -1 - n =1 + tr N

(5.17)

Other approaches have been proposed for dealing with more complicated interaction effects. The two most common are due to Coffin [34] and Manson [35]. In the former, a frequency term is incorporated to allow for time-dependent effects. In the latter, a strain range partitioning (SRP) approach is adopted for separating the total strain range of each cycle into creep, fatigue and mixed creep and fatigue components. With this method, a total of four components can be identified depending upon the type of loading cycle and whether hold times are present. Further reading to gain detailed insight into the fatigue of uncracked bodies can be found in the literature [10–21].

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5.2 Fracture Mechanics Parameters in Creep and Fatigue For situations where linear elastic conditions prevail (short times and/or low loads) the linear elastic stress intensity factor, K, may be used to predict creep crack growth either in creep or fatigue [22–23]. For non-linear conditions the J integral can be used to define a particular contour integral that is equal to the energy release rate in a non-linear elastic material as ∂u i W dy − Ti ds (5.18) J= ∂x Γ

where W is the non-linear elastic strain energy as shown in Fig. 5.3 is given as εi j W =

σi j dεi j

(5.19)

Γ

and Ti is given as Ti = σ ij nj and nj is the normal unit vector outside from the path Γ as shown in Fig. 5.4. This contour integral is a conservation integral and provides the path-independent values when contours are taken around a crack tip [24]. As the elastic energy release rate G is related to K, the J integral, which is also energy release rate in non-linear materials, can also characterise the stress and strain

ni x Γ

Fig. 5.3 Integration path for J integral

y

σ

W

Fig. 5.4 Strain energy W definition

0

ε ij

ε

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fields around a crack tip. When non-linear behaviour of a material is expressed by a power law as

ε εo

=

σ σo

N (5.20)

The stress and strain at a distance r are predicted by the following equations using the J integral. The stress and strain fields are often called the HRR field [25–26]. These are described as 1/(N +1) J σi j = σo σ˜ i j (5.21) In σo εo r εi j = εo

J In σo εo r

N /(N +1)

ε˜ i j

(5.22)

The calculated value of In [25] varies between 2–6 for a range of n between plane stress to plane strain [25]. Because the stress and strain fields given by Equations (5.21) and (5.22) are based on small-deformation theory, the true stress state will be different, close to a crack tip where the assumption of small-deformation theory is not adequate. For this reason, the stress or strain field characterised using the J integral is not the actual state at a crack tip itself. However, the J integral can still be considered as a valid fracture parameter as long as the fracture process zone is related to the surrounding J integral stress and strain fields. This is similar to the concept of applying the stress intensity factor for describing fracture in a material which undergoes small-scale yielding.

5.2.1 Creep Parameter C∗ Integral The arguments for high-temperature fracture mechanics essentially follow those presented above. For creeping situations, where elasticity dominates, the stress intensity factor K may be sufficient to predict crack growth. However, as creep is a non-linear, time-dependent mechanism even in situation where small-scale creep exists, linear elasticity may not be the answer. The behaviour in plasticity can be compared to the creep response of a body by observing that Equations (5.9) and (5.20) are similar. Hence, by analogy, using the J definition estimation procedures the C∗ parameter can be developed. Thus, in the case of large-scale creep where stress and strain rate determine the crack-tip field case, the parameter C∗ analogous to J has been proposed for this purpose. Substantial body of work exists in this respect [22–24] and the C∗ integral has been widely accepted as the fracture mechanics parameter for this purpose both in standards [3–8] as well as in Codes of Practices [4, 7, 9].

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The theoretical definition of the C∗ integral, therefore, is obtained by substituting strain rate and displacement rate for strain and displacement of the J integral defined by equation as . / C ∗ = ∫ Ws∗ dy − Ti (δ u˙ i /δx )ds Γ

(5.23)

and W∗ is given as ε˙i j c

Ws∗ =

σi j d ε˙ icj

(5.24)

o

By analogy between non-linear elasticity and non-linear viscosity, the stress and strain rate in materials where (u˙ i = du i /dt) is the displacement rate. The other notations are the same as in the J integral definition. As the J integral characterises the stress and strain state, the C∗ integral is also expected to characterise the stress and strain rate around a crack. C∗ may also be interpreted as an energy-release rate analogous to the energy definition of J [22], so that C∗ = −

1 dU ∗ B da

(5.25)

This energy estimate of the C∗ integral has been widely used as a parameter for correlating CCG rate under steady-state creep conditions [22–23]. For a non-linear elastic material, it was shown that the asymptotic stress and strain fields are expressed by Equations (5.21) and (5.22). For creep in order to relate the uniaxial properties to the stress and strain rate distributions ahead of the crack tip, the asymptotic stress and strain rate fields in creep are used and they are thus expressed by [25] σi j = σo ε˙ i j = ε˙ o

C∗ In σo ε˙ o r C∗ In σo ε˙ o r

1/(n+1) σ˜ i j (θ, n)

(5.26)

ε˜ i j (θ, n)

(5.27)

n/(n+1)

Therefore, the stress and strain rate fields of non-linear viscous materials are also HRR-type fields. In the next section this description of the stress and strain filed ahead of a creep crack is used to develop a model which can predict crack initiation and growth at elevated temperatures.

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5.3 Predictive Models in High-Temperature Fracture Mechanics At elevated temperatures where creep is dominant, time-dependent crack growth is observed. The rate of this time-dependent crack growth is measurable and parameters are needed to predict it. In order to predict the crack behaviour in such materials, ˙ creep crack growth rate a(= da/dt) must be estimated using appropriate parameters. Several fracture parameters have been applied for this purpose. The most commonly used parameters are stress intensity factor K, the C∗ integral [8, 22–23, 27–33]. Steady-state crack growth rate a˙ s is usually given as follows using these parameters: a˙ s = AK m

(5.28)

a˙ s = DC ∗φ

(5.29)

A suitable parameter to describe crack growth at elevated temperature will depend on material properties, loading condition, size, geometry and the period of time during which crack growth is observed. Any of these variables could affect the stress state at the crack tip of a creeping material, hence making the cracking to behave in a ductile or brittle manner. The testing standard ASTM E1457 [8] goes to a considerable length to quantify these regions with respect to the appropriate parameters.

5.3.1 Derivation of K and C∗ The definitions for the stress intensity factor K are readily available for numerous geometries in the literature [38–41]. For C∗ this analysis is more complicated and the parameter is both non-linear as well as time dependent. There are a number of ways to derive C∗ [22, 23] both numerically and experimentally. However, only the validated methods for laboratory specimens and components adopted in standards and codes of practice are presented here. C∗ is estimated experimentally, using Equation (5.25), from measurements of creep load-line displacement according from tests carried out on compact tension (C(T)) specimens based on the recommendations of ASTM E 1457-07 [8] giving C∗ =

˙c P ·Δ ·F Bn · (W − a)

(5.30)

Values of F as function of n are given for a number of geometries in ASTM E1457-07. The crack initiation and growth rate data obtained from C(T) tests are usually considered as ‘benchmark’ for creep crack growth properties of the materials in the same way as creep strain rate and rupture for uniaxial creep tests. These data can then be used directly in crack initiation and growth prediction models as described in the different codes [3–9] to estimate residual lives in components. For

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components, such as cracked pipes and plated, C∗ must be determined from finite element analysis or reference stress methods described briefly below. In most cases, the reference stress approach of evaluating C∗ is adopted in line with that used in a number of defect assessment codes [3–9]. With this approach C∗ is expressed approximately as Cr∗e f = σr e f .˙εr e f .

K σr e f

2 (5.31)

Where the reference creep strain rate ε˙r e f at an applied stress σr e f can be derived directly from uniaxial data. When described in terms of the Norton creep law in Equations (5.9) and (5.31) can be rewritten as 2 Cr∗e f = A.σrn−1 e f .K

(5.32)

The typical value for n is between 5 and 12 for most metals. In addition, average creep rate, ε˙ A , from Equation (5.10) obtained directly from rupture data, described earlier to account for all three stages of creep as an approximate method can be used for estimating ε˙ A−r e f in conjunction with uniaxial data and σr e f .

5.3.2 Example of CCG Correlation with K and C∗ Figure 5.5a represents a typical relationships for creep crack growth a˙ versus the linear stress intensity factor K in compact tension specimens for an aluminium alloy Al-2519 at 135◦ C. Crack growth might also be characterised to a certain extent by the stress intensity factor K in such ‘creep brittle’ materials using Equation (5.28). It is clear, however, that K does not adequately correlate the data as different test loads 100

Crack Growth Rate, da/dt (mm/h)

100

10–1

10–1

10–2

Al 2519 at 135°C K = 17.07 (B = 22 mm) K = 18.97 (B = 22 mm) K = 18.24 (B = 22 mm) K = 16.37 (B = 22 mm) K = 15.36 (B = 22 mm) K = 21.44 (B = 6.35 mm) K = 20.44 (B = 6.35 mm) K = 19.53 (B = 6.35 mm) K = 18.22 (B = 6.35 mm) K = 18.97 (B = 6.35 mm)

–3

10

10–4

10–5 10

20

30

Stress Intensity Factor, K (MPa

(a)

40

a 10–2 (mm/h)

10–3

10–4 10–3

10–2

10–1

100

101

102

C* (KJ/m2h)

m1/2)

(b)

Fig. 5.5 A typical relationship for creep crack in compact tension specimens growth versus (a) K for an aluminium alloy Al-2519 at 135◦ C and (b) C∗ correlation for a 1CrMoV steel at 550◦ C showing an initial transient ‘Tail’ and a steady crack growth region

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initiate different cracking rates. In Fig. 5.5b, the non-linear C∗ parameter correlating cracking rate in a 1CrMoV steel at 550◦ C is more appropriate. The C∗ correlation in Fig. 5.5b exhibits an initial transient ‘tail’ and a steady crack growth region. If a material is elastic, immediately upon loading, the stress distribution around the crack tip will be elastic. For this situation, the stress and strain at the crack tip are described by linear elastic stresses. The ‘transient’ region will be modelled later in this chapter in order to derive initiation times. Thus, in a material that shows little creep deformation, the stress distribution will remain virtually unchanged by creep and the tail is usually small. On the other hand, in a ‘creep ductile’ material, where large creep strains can be dominant and the ‘tail’ is usually larger. In the extreme where creep deformation is prevalent everywhere in the material, the singularity at the crack tip will be lost and correlation in terms of the reference stress on the uncracked ligament could be obtained. Since the reference stress is a parameter which describes the overall damage across an uncracked ligament and may characterise net section rupture of the ligament rather than creep crack growth. For materials between the two extremes of creep brittle and very creep ductile, substantial creep deformation could accompany fracture, and stress redistribution will occur around a crack tip but a singularity by the crack will still remain. For this situation, creep crack growth will be characterised by C∗ using Equation (5.29). For engineering metals, most experimental evidence suggests that the widest range of correlation is achieved with C∗ [27–33].

5.3.3 Modelling Steady-State Creep Crack Growth Rate Once a steady-state distribution of stress and creep damage has been developed ahead of a crack tip, it is usually found that creep crack growth rate can be described by Equation (5.28). Most often, the constants D and φ are obtained from tests that are carried out on compact tension (C(T)) specimens based on the recommendations of ASTM E 1457-07 and described above. Using the steady-state assumptions at the crack tip model of creep crack growth under steady state conditions, proposed previously [27], is briefly presented in this section. The model called the NSW model is based on stress and strain rate fields characterised by the C∗ integral combined with a creep damage mechanism and ductility exhaustion at the crack tip. For a material which deforms according to the Norton’s creep law described by Equation (5.9), the stress and strain rate distributions ahead of a crack are given by Equations (5.26) and (5.27) respectively. When the creep damage accumulated within the creep zone shown in Fig. 5.6a reaches a critical failure value, the crack is postulated to progress. If the material starts to experience creep damage when it enters the process zone at r = rc and accumulates creep strain εij by the time it reaches a distance r from the crack tip, the condition for crack growth is given using the ductility exhaustion criterion as

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σ Creep Strain Rate

r

.

∫ε dt = εφ

θ .

as

Creep Process Zone rc

Distance r

a)

rc

b)

Fig. 5.6 Concept of the steady-state NSW model, (a) the creep process zone, (b) creep strain rate distribution in the process zone

r εi j =

ε˙ i j dt

(5.33)

r =rc

Using the strain rate distribution in Equation (5.27), the creep strain can be written as

r εi j =

εo r =rc

c∗ In σo ε˙ o r

n/n+1 ε˜ i j

dt dr dr

(5.34)

If a steady (constant) crack growth rate a˙ s is assumed dr/dt = a˙ s

(5.35)

Then Equation (5.34) is analytically integrated as εi j = (n + 1)˙εo

C∗ In σo ε˙ o

n/(n+1)

. 1/(n+1) / rc − r 1/(n+1)

(5.36)

When the creep ductility considering the stress state (multiaxial stress condition) is given by ε∗f , substituting εi j = ε∗f at r = 0 into Equation (5.36) gives (n + 1)˙εo a˙ s = ε∗f

C∗ In σo ε˙ o

n/(n+1)

. / ε˜ i j rc1/(n+1)

(5.37)

The non-dimensional function ε˜ i j is normalised so that its maximum equivalent becomes unity. Hence, assuming this maximum value, the constants D and φ in Equation (5.29) become

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D=

(n + 1)˙εo ε∗f

1 In σo ε˙ o

n/n+1 (5.38)

where n (5.39) n+1 The creep ductility considering the stress state ε∗f is equal to the uniaxial creep ductility ε f in plane stress conditions. This may be estimated from the multiaxial stress factor (MSF), using φ=

ε∗f = MSF. ε f

(5.40)

The value of the MSF may be estimated using appropriate models [42, 43]. In the present study the Cocks and Ashby void growth and coalescence model [40] was used. The calculated MSF using the Cocks and Ashby model is given as 0 n − 1/2 σm 2 n − 1/2 sinh 2 = sinh MSF = εf 3 n + 1/2 n + 1/2 σe ε∗f

(5.41)

This model will make ε∗f dependent on the stress triaxiality, the ratio between the mean (hydrostatic) stress and equivalent (Mises) stress, σ m /σ e , and the creep stress exponent, n. The stress triaxiality ratio, σ m /σ e , may be evaluated using FE for specific geometries or estimated for some geometries. For plane strain conditions, using Equation (5.41), ε∗f could range anywhere between 25 to 80 times smaller than ε f depending on the creep void growth model assumed [44, 45]. The relationships are relatively insensitive to the creep index n. Originally, in the NSW model, ε f /ε∗f = 50 was recommended as a extreme upper-bound value for plane strain conditions [27]. This factor was further refined to ε f /ε∗f = 30 for most relevant engineering materials [46]. Furthermore, if available data exist for a range of sizes and geometries for a specific material, this factor can be further reduced depending on the level of conservatism that is sought. Thus from the relationship between normalised ductility ε f /ε∗f and the ratio of hydrostatic tensile stress over the equivalent stress (σ m /σ e ), it is apparent that an increase in hydrostatic tension causes a large reduction in creep ductility [44, 45]. As an example, experimental data bands in Fig. 5.7 show a range of data for 316H stainless steel. The variation in cracking rate is due to effect of specimen size and geometry and scatter. The cracking rate for the for 316H stainless steel at 550◦ C varies by a factor of ∼7 and in the extreme a factor of 15 for the large specimens. The NSW model from Equation (5.37) predicts a factor of 30 using a failure strain εf = 0.2 using minimum strain rate date to derive the creep index n. The average creep strain rate ε˙ A can also be used from Equation (5.10) to derive n since the material at the crack tip undergoes three stages during its transverse of the creep zone similar to its uniaxial creep behaviour.

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Fig. 5.7 Size and geometry effects on steady-state CCG rate for C(T) and middle tension M(T) ◦ data versus C∗ for 316 H stainless steel at 550 showing the effect of constraint. NSW plane stress/strain prediction (×30) was derived from Equation (5.37). NSW-MOD prediction (grey band x ∼6) Plane stress/strain was derived from Equation (5.54)

To simplify the predictions for crack rate with respect to C∗ for most steels, n > 1, usually 5–10, so that Equation (5.37) becomes relatively insensitive to the value of rc . Considering most engineering materials, Equation (5.37) has been reduced to [32]: a˙ =

3C ∗0.85 ε∗f

(5.42)

The model therefore indicates that crack growth rate should be inversely proportional to the creep ductility ε∗f appropriate to the state of stress at the crack tip. The bounds on this value for plane stress and plane strain are as described above. Using this simplified prediction is particularly useful when the user does not have material cracking data and needs a conservative initial estimate using the plane strain upper-bound lines for prediction component cracking behaviour.

5.3.4 Transient Creep Crack Growth Modelling A transient creep crack growth model has also been proposed as described schematically in Fig. 5.8 [22, 47]. The differences between the steady crack growth model and the transient crack growth model are that transient creep crack growth starts from the undamaged state in the creep damage process zone, whereas the steadystate model assumes a steady-state distribution of damage in the process zone. Also creep damage accumulates in elements with width dr ahead of a crack tip. Steady

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.

∫ε dt = εφ

a)

Distance r

b)

rc

Fig. 5.8 Schematic of the transient NSW model process zone showing (a) the creep zone and (b) the creep strain rate field both showing a schematic view of pseudo-uniaxial specimens operating in the creep zone under local uniaxial conditions

state creep crack growth can be predicted analytically but numerical integration is necessary for the transient model. From the transient creep crack growth model, the initial crack growth rate a˙ can be estimated by the following equation assuming secondary creep behaviour: ε˙ o a˙ i = ∗ εf

C∗ In σo ε˙ o

n/n+1 (dr )1/n+1

(5.43)

This can be further simplified [48] to relate the initial transient crack growth a˙ i to the steady state cracking rate a˙ s as (1/n+1) dr 1 a˙ i = a˙ s n + 1 rc

(5.44)

For the same reason described for the steady state model for the creep zone rc in Equation (5.37), the choice of dr as the size of finite creeping region behaving as a pseudo round uniaxial specimens, as shown schematically in Fig. 5.8 appears to be insensitive for predicting the initial cracking rate because it is raised to a small fractional power when n>>1. For element i, the model predicts a crack growth rate a˙ i of ε˙ o a˙ i = ∗ ε f − ε1.i−1

C∗ In σo ε˙ o

n/n+1 (dr )1/n+1

(5.45)

where εi,i –1 is the creep strain accumulated at r = ri = idr when the crack reaches R = (ri –1) = (i –1)dr The ligament dr can be chosen to be a suitable fraction of rc . However since dr/rc is raised to a small power in Equation (5.45) this will give a˙ i =

1 a˙ s n+1

(5.46)

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For most engineering materials, therefore, the initial crack growth rate is expected to be approximately an order of magnitude less than that predicted from the steady-state analysis. The cracking rate will progressively reach the steady-state cracking rate as damage is accumulated. Thus, using the transient and the steady-state NSW models, the extreme regions of the crack growth behaviour is covered adequately within the bounds of the present model. Figure 5.9 shows the cracking rate bounds covering a wide range of a number of alloys taken from [32]. The lighter shade is the band of data for creep ductile materials (such as 316H, P22 steels, 1CMV). Figure 5.9 acts as a material-independent engineering crack growth assessment diagram where cracking rate is multiplied by the uniaxial ductility of the material. Note that most of the grey data band below the NSW plane stress line is in the initiation and transient region and not steady cracking rates. Therefore, the transient region which is usually below the plane stress line, assuming that the initial cracking rate is given as a˙ i ≈ a˙ s /10, can be introduced as an extreme lower-bound band to estimate the initiation times and the transition ‘tail’ region of damage development observed in many test pieces and components. Fig. 5.9 Material-independent engineering assessment diagram showing CCG rate (da/dt. εf ) versus C∗ over the plane stress/strain and transient bounds using the NSW and NSW-MOD prediction lines

Therefore, the most conservative line to choose for life assessment would be the plane strain NSW line. However, for a narrower range of data sets of creep ductile materials (shown in Fig. 5.9) this may be overly conservative. Furthermore, using known specific material properties with standard size and geometries a much lower acceptable safety factor would be needed to predict cracking behaviour and reduce over-conservatism. To deal with this, a modified NSW-MOD model [44, 45] based on the assumption that damage would occur over a wide range of angles at the crack tip was introduced. NSW-MOD in effect reduced the crack growth bounds relevant to ‘creep ductile’ engineering materials. This is shown in the example in Fig. 5.9 where it is found that an approximate factor of 6 predicts the divergence

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between plane stress and plane strain cracking rates. This modification to NSW will be described in detail later in this chapter.

5.3.5 Predictions of Initiation Times ti Prior Onset of Steady Creep Crack Growth In creep crack growth experiments and in real components, an incubation period when the crack seems to be stationary can often be observed. In cases of sensitive and vital component parts an initiation of a crack could mean the end of life. Also sometimes the incubation time occupies most of the life of a cracked body. Therefore, methods to predict this period are important. In addition, in order to reduce conservatism in predicting crack growth life times it may be important to incorporate the incubation time into creep crack growth predictions in practical applications. There are many ways to deal with this problem and as yet no one methodology can cater for all conditions. In the transient analysis described earlier the initiation time prediction was calculated from the NSW model. Lower and upper bounds can be obtained from the steady creep crack growth model and transient model shown above as follows: ε∗f rc = Lower bound ti := a˙ s (n + 1) εo Upper bound ti :

ε∗f rc = a˙ i εo

= (n + 1)

In σo ε˙ o rc C∗

r 1/n+1 c

dr

In σo ε˙ o rc C∗

n/n+1

n/n+1

rc 1/n+1 dr

ti (lower bound)

(5.47)

(5.48)

(5.49)

Further, the upper-bound incubation can be obtained by taking a˙ = a˙ i ti =

ε∗f o εo

In σo ε˙ o rc C∗

ν/n+1 (5.50)

Alternative estimates of an incubation period can also be obtained from the approximate creep crack growth rate Equation (5.42) assuming steady state. The prediction based on an approximate steady-state cracking rate expression will be the lower bound of the incubation period. ti =

rc ε∗f 3C ∗0.85

(5.51)

When the initial crack growth rate is used instead of the steady-state crack growth rate, the upper bound is determined as

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Fig. 5.10 Experimental incubation periods for different geometries for crack extension Δai = 0.2 mm for Type 316H austenitic stainless steel at 550◦ C compared to NSW crack initiation predictions (using dotted lines) based on Equations (5.51) and (5.52) and on uniaxial creep properties for Type 316H austenitic stainless steel at 550◦ C shown in Table 5.1

ti =

(n + 1) rc ε∗f 3C ∗0.85

(5.52)

Figure 5.10 shows the predicted ti from Equations (5.51) and (5.52) versus experimental incubation times for different geometries for crack extension Δai = 0.2 mm for Type 316H austenitic stainless steel at 550◦ C. The scatter is typical of hightemperature data where 0.2-mm crack extension occurring over long times is difficult to measure accurately. Figure 5.10 does not indicate a clear difference in initiation times due to geometry. Whereas the overall inherent scatter of data shows a much larger difference. Using the steady-state model the prediction lines from Equation (5.52) are conservative but using the initiation line from Equation (5.51) the lower bound covers most of the data.

5.3.6 Consideration of Crack Tip Angle in the NSW Model Based on the form of the crack tip fields in Equations (5.26) and (5.27) and using a ductility exhaustion argument as in the NSW model, it has been shown that the ˙ may be predicted over the bounds of plane strain/stress creep crack growth rate, a, conditions. It was also recommended that under plane stress conditions the multiaxial ductility, εf∗ , be taken as the uniaxial failure strain, εf , and εf /30 under plane strain conditions. This recommendation would cover a wide range of materials and conditions which may make it necessarily over conservative.

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In the NSW model, it is implicitly assumed that fracture occurs first at the value of the crack tip angle, θ , at which the equivalent creep strain, quantified by ε˜ e (θ, n) in Equation (5.27), reaches its maximum value. A more general expression can be obtained, which considers the dependence of ε˜ e (θ, n) and εf∗ on angle, θ . For this condition, the NSW model may be extended to give a modified crack growth rate, (hereafter referred to as the ‘NSW-MOD model’) as described below.

5.3.7 The New NSW-MOD Model In the NSW model the value of the non-dimensional equivalent stress function, used to describe the Mises equivalent stress and strain rate fields in Equations (5.26) and (5.27), was taken to be its maximum value of unity. This is considered to be a conservative measure and implicitly assumes that failure will occur first at the angle, θ , where σ˜ e attains its maximum value [44, 45]. Figure 5.11 shows the dependence of crack-tip stress field and ε f /ε∗f on θ under plane stress and strain. Since both σ m and σ e depend on n and θ , then the MSF is also a function of n and θ which can be then estimated by considering all angles. 1

1

Plane strain n=5

Plane stress 0.1

σ~e (θ , n) 0.1

εf*/εf

n

n = 20

Plane strain

Plane stress 0.01

n=5 n = 10 n = 20

n = 10 0.01

a)

0

45

90

θ

135

180

0.001

b)

0

45

90

θ

135

180

Fig. 5.11 Dependence of (a) σ˜ en on angle θ and n and (b) ε∗f on angle θ and n (note that the lines in this case are relatively insensitive to n)

Figure 5.11 shows the stress distributions σ˜ e (θ, n) and ε f /ε∗f against θ for n = 5, 10 and 20 under plane stress and strain. It is seen that the maximum value of [σ˜ e (θ, n)]n is unity at θ ≈ 0◦ and 90◦ under plane stress and plane strain conditions, respectively. At θ = 0◦ , which is the condition assumed in the NSW model, the difference in the value of [σ˜ e (θ, n)]n under plane stress and plane strain conditions can be up to a factor of 50–100 depending on the value of n. It is also seen in Fig. 5.11(b) that εf∗ increases with angle, both under plane stress and plane strain conditions. At θ = 00 the difference of εf∗ between plane stress and plane strain conditions is well above a factor of 100. This variation will also affect the value of εf∗ in Equation (5.41) which itself will vary for a range of stress states. In the NSW-MOD1model, therefore, failure is considered to occur first where the angular function σ˜ en MSF attains its maximum value. The resulting crack growth rate equations are

5 Predicting Creep and Creep/Fatigue Crack Initiation and Growth

1

a˙ N SW −M O D = (n + 1) (Arc ) n+1

C∗ In

n n+1

127

1 σ˜ en (θ, n) ε f MSF (θ, n) max

(5.53)

Giving a˙ N SW −M O D

ε˙ 0 = (n + 1) ∗ ε f (θ, n)

C∗ ε˙ 0 σ0 In

n/(n+1) rc1/(n+1) ε˜ e (θ, n)

(5.54)

.

.

10

10

8

8

aPE/aPS for θ = 0°

aPEmax / aPSmax

Comparing the maximum value of CCG rate (see Fig. 5.12(a)), the CCG rate for plane strain conditions is about 3–7 times faster than that for plane strain condition, although the ratio depends on the value of n. However, partly because of side grooves, experimentally creep crack growth in C(T) specimens is contained in the direction of θ = 0◦ . Hence, comparing the value of CCG rate at θ = 0◦ , the CCG rate under plane strain conditions is about 0.5–7 times faster than that under plane stress conditions, depending on the values of n (the ratio is decreasing with increasing of the values of n).

6 4

.

2

4

2

0

0

5

a)

6

.

10

15

20

n

5

b)

10

15

20

n

Fig. 5.12 Comparison of (a) the maximum value of CCG rate (a˙ max ) and (b) CCG rate at the angle of 0◦ between plane stress and plane strain conditions

From this analysis the indications are that in Fig. 5.9, as shown in the lighter shade data band, the upper bound can be reduced and modified for most creep ductile alloys such as the 316H stainless steel to a factor of ∼6–7. Clearly, this is an improvement from the factor of 30 predicted in Fig. 5.9 for the NSW model. It should also be noted if pedigree data were available for a particular batch the predicted bounds could be further reduced.

5.3.8 Finite Element Framework The NSW model using the ductility exhaustion and crack-tip constraint effects on damage has been independently verified using a finite element framework.

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Numerical predictions, using basic materials elastic–plastic–creep properties, may be developed to predict CCG rate in fracture mechanics geometries [44]. The elastic, plastic and creep strains are taken to be independent of strain rate giving the total strain as ε = εel + ε p + εc

(5.55)

Power-law creep behavior as in Equation (5.9) is assumed or using the materials average creep strain rate properties as in Equation (5.10). As an example of the modelling procedure the material properties for 316H stainless steel at 550◦ C are shown in Table 5.1. The plastic response of the material is assumed to be governed by a Mises flow rule with isotropic strain hardening, which was obtained by fitting to uniaxial tensile test data at 550◦ C. In the FE model, the yield strength of the material is taken to be its 0.2% proof stress, and the post-yield strain hardening response is treated as piece-wise linear up to the point relating to the ultimate tensile stress, σ UTS , and corresponding strain, εUTS , beyond which no strain hardening occurs. Table 5.1 Uniaxial tensile and creep properties for 316H at 550◦ C used Young’s modulus, E (GPa) σ 0.2 (MPa) σ UTS (MPa) εUTS (%) A (h–1 MPa–n ) n 140

170

415

33

–32

7.24×10

εf (%)

10.6 21

Figure 5.13 gives an example for a C(T) mesh with fine element sized in the expected crack growth line to allow for debonding on damage accumulation.

5.3.9 Damage Accumulation at the Crack Tip For identifying a criterion for damage development leading to a finite crack extension, the principle of the ductility exhaustion NSW model proposed above is used. An uncoupled damage approach is employed, where the rate of damage accumulation, ω, ˙ is controlled by the equivalent creep strain rate, ε˙ c , and determined from the relation ω˙ =

ε˙ c ε˙ c = ε∗f MSF ε f

(5.56)

where ε∗f is calculated from the uniaxial creep failure strain (creep ductility) of the material, εf , using the multiaxial strain factor (MSF) given in Equation (5.41). It should be noted that within an FE analysis the value of ε∗f changes for any fixed material point since it depends on the derived triaxiality through Equation (5.41). Therefore, ε∗f changes with time and crack length and at every angular position ahead of the crack tip as the stress redistributes locally.

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a W Fig. 5.13 Example of finite element mesh for a C(T) specimen with the centre line region refined to allow for virtual crack growth simulation by element debonding at the nodes based on the MSF criterion

(a)

(b)

Fig. 5.14 (a) Damage contour plot for an elastic–plastic–creep analysis of a CT specimen using the node-release technique. (b) Contours of plastic strain at the same amount of crack growth

A creep damage parameter, ω, can be defined such that 0 ≤ ω ≤ 1 is introduced to quantify the extent of creep damage in a body at any instant, which is evaluated from the time integral of ω˙ in Equation (5.56), i.e.

t

ω= 0

t ω˙ dt = 0

ε˙ c dt ε∗f

(5.57)

Initially, at t = 0, it is assumed that there is no damage anywhere in the material, thus ω = 0 and failure occurs at a material point when ω = 1. Note that the evolution of damage is not coupled to creep deformation, such that the creep rate of the material is not enhanced due to the accumulation of damage.

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In a numerical model of, for example, a C(T) specimen, the parameter C∗ can be calculated from the calculated FE creep load line displacement rate as in Equation (5.30). Hence, independently, a virtual crack growth rate, da/dt, can be predicted using Equation (5.57) incrementally allowing crack-tip element debond when ω reaches a value of unity and with the current crack-tip position taken to be at the position of the most recently released node. Figure 5.15 show a graphic example of damage buildup and plastic deformation at the crack tip for the virtual cracking method described. Thus, by modelling the crack tip and using the MSF method used in the NSW and NSW-MOD models it has been shown that the results compare well [44]. Clearly there are possibilities that can be explored to develop robust virtual creep crack growth methods both to reduce and optimize the number of test and to predict component crack growth behaviour at elevated temperatures. Fig. 5.15 Influence of frequency on the crack profile of a corner crack tension (CCT) specimen of AP1 nickel-base superalloy tested ◦ at 700 C [49] showing the specimen cross section and the region of creep and fatigue domination

2 mm x-section

Fatigue creep

Fatigue

5.3.10 Elevated Temperature Cyclic Crack Growth With temperature increase, time-dependent processes become more significant even under fatigue loading but are dependent on the loading frequency. Creep and environmentally assisted crack growth can take place more readily since they are aided by diffusion and rates of diffusion increase with rise in temperature. A graphic example of creep and fatigue mechanisms on crack growth is shown in Fig. 5.15 [49]. The regions of high-frequency loading (10 Hz) on a square section corner crack tension specimen shows no effect of stress state as the crack extension is the same at the centre and at the surface of the specimen. However, when static creep is applied there is a clear state of stress effect. The creep regime in this case has been dealt with the modelling described above. For the fatigue component, the analysis is much simpler and has been described by the Paris Law described below. The summation of the two in the manner that the codes of practice [3, 4] treat creep/fatigue is described below.

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When alternating loads are applied to high-temperature structures, the crack growth in the structures will be subject to both creep and fatigue. The Paris law is usually adequate to describe fatigue at high frequencies given as da/dN = C ΔK c

(5.58)

Some interaction between creep and fatigue is expected under cyclic loading. Some of the causes of creep–fatigue interaction might be the enhancement of fatigue crack growth due to embrittlement of grain boundaries or weakening of the matrix in grains and enhancement of creep crack growth due to acceleration of precipitation or cavitation by cyclic loading. The importance of creep–fatigue interaction effects is largely dependent on the material and loading conditions and detailed analysis for every cycle will reveal a complex interaction of creep and fatigue mechanisms. Nevertheless, for the macro-cracking behaviour of a crack under creep/fatigue, the simple linear summation rule for creep–fatigue crack growth has been successfully applied to predict the crack growth in several engineering metals. This is given as da/dN = (da/dN )c + (da/dN ) f = (1/3600 f )(da/dN )c + (da/dN ) f

(5.59)

where da/dN is crack growth per cycle in mm/cycle, da/dt is crack growth rate in mm/hour, and f is frequency in Hz. There is clear experimental evidence for this general rule [4]. Figure 5.16 shows the fracture mode changes going from transgranular at high frequency to intergranular at lower frequencies. At the intermediate frequencies, there is a mixture of cracking mode indicating that the two mechanisms can run in parallel. Therefore, using the predictive model in creep cracking and fatigue in this chapter, the combined cracking response under creep/fatigue interaction can be predicted in a conservative manner by simple summation depending on the loading history.

5.4 Conclusions This chapter has covered topics relating to the prediction of creep and creep/fatigue crack initiation and growth, under static and cyclic loading, in engineering materials at high temperatures. A short overview of the present standards and codes of practice for fracture-based methods suggest that trends towards a fracture-based design and remaining life prediction is an important methodology for the future. Following a brief description of engineering creep parameters of basic elastic– plastic fracture mechanics methods, high-temperature fracture mechanics parameters are derived from plasticity concepts. Techniques are shown for determining the

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a) 10 Hz

b) 0.1 Hz c) .01 Hz

d) 0.001 Hz Transgranular (Fatigue) >> Intergranular (Creep)

Fig. 5.16 Effects of frequency on mode of failure for AP1 Astroloy Nickel-base superalloy tested at 700◦ C [49]

creep and fatigue fracture mechanics parameters K and C∗ using experimental crack growth data. Models to predict crack initiation, growth and failure under creep conditions are also presented. An analytical failure strain constraint based model called the NSW model has been shown to cover the range of plane stress/strain behaviour found in creep crack growth. The model based on a ductility exhaustion argument and constraint level at the crack tip is able to predict, in a range of as much as a factor of 30 in crack initiation and growth at high temperatures, over the plane stress to plane strain regimes. By taking into account angular damage distribution around the crack tip, the model is further refined and named as the NSW-MOD model is able predict a much improved upper/lower bound of a factor of at most ∼0.5–7 (depending on the value of n) in steady-state crack initiation and growth over the plane stress/strain region in components containing defects. The presence of cyclic load is assumed to introduce a cycle-dependent fatigue component with a linearly summed cumulative effect with the creep. In order to verify the models, a virtual crack extension numerical model was developed which made use of the development of damage at the crack tip based on crack-tip constraint and ductility exhaustion. As advanced computational capabilities hardware and soft-ware improve and new expert systems for software are developed, the task to use predictive methods are made easier. Where there is known to be scatter in the data, probabilistic methods rather than deterministic analysis will need to be used. The models presented in this chapter can be used in predictive techniques in fracture mechanics-based creep crack initiation and growth rate analysis. In conclusion, the creep/fatigue modelling presented can be used in component design of metallic parts, as well as in life assessment of cracked components at elevated temperatures and in predicting

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virtual cracking behaviour in fracture mechanics specimens in order to reduce overconservatism in the analysis.

5.5 Nomenclatures and Abbreviations

˙c Δ θ ε˙ 1c , ε˙ 2c , ε˙ 3c ε˙¯ c ε˙ , ε˙ min , ε˙ A φ i, , Di a˙ i , a˙ s ε˙ , ε˙r e f , ε˙ A−r e f ε˙ A , ε˙ sc , ε f , ε∗f o u˙ i = du i /dt a, ai, af A, n As , ns AA , nA σ˜i j and ε˜ i j B, Bn , W C∗ c’, C’ D, H, α, β, m, φ da/dt or a˙ dr F G, K, ΔK In J N, da/dN N’,n n1 , n2 , n3 N1 , N2 , N3 nj P, Q R rc T, Tm

load line creep displacement rate in a cracked geometry the crack tip angle principal creep strain rates, corresponding equivalent creep strain rate creep strain rate, minimum and average creep strain rates constant for CCI versus C∗ in ti = Di C∗φi initial transient, steady state CCG rate (mm/h) creep rate, creep strain rate at reference stress, average creep strain rate at reference stress average creep rate, secondary creep rate, failure strain, appropriate failure strain relevant to crack-tip constraint displacement rate crack length (mm), initial crack length (mm), final crack length (mm) material constant, Norton’s creep index in ε˙ = Aσ n material constant, creep index value for secondary creep rate material constant, creep index value for average creep rate equivalent stress and strain functions of angle θ and n thickness (mm), net thickness (of a side-grooved geometry) (mm), width (mm) Steady-state creep correlating parameter (MJ/m2 .h) fatigue crack growth (FCG) rate index and material constant in da/dN= C’ΔKc’ material constants dependent on temperature and stress state. creep crack growth rate (mm/h) an increment of distance within creep zone rc factor which depends on geometry and creep index, n the elastic√energy release rate, the stress intensity factor √ (MPa m), stress intensity factor range Δ(MPa m) the non-dimensional factor of n or N’, J-integral Number of loading cycles, FCG/ cycle (mm/cycle) the hardening exponent, the creep index number of cycles spent at each condition the endurance or critical cycle at these conditions. the normal unit vector outside from the path Γ applied load (MN) activation energy Gas constant Creep zone size (mm) Absolute temperature, Absolute melting temperature

134

Ti t, ti , tr U∗ W, W∗ εel , εp , εc σ 1, σ 2 and σ 3 θ 1, θ 2, θ 3 θ 4 Δai , Δaf Γ, ds α, β, γ , φ, D σ, σy σ m , σ e, σ ref σ , ε, r ω, ω˙ MSF CCG CCI FCG C(T), SEN(T), M(T), SEN(B), DEN(T), CS(T)

K.M. Nikbin

the traction force test time (h), time to initiation (h), uniaxial rupture time (h). potential energy rate per crack extension Strain energy density change, strain energy density change rate elastic, plastic and creep strain components principal stresses stress and temperature dependent material parameters. initial crack extension (mm), total crack extension (mm) the line integration path is the length along the path Γ stress and temperature dependent material parameters. power index and material constant in da/dt = DC∗φ stress and normalized yield stress (MPa) hydrostatic stress, equivalent stress, reference stress (MPa) material constants as the stress (MPa) and strain (1/h) at a distance r from crack tip, creep damage, creep damage rate parameters Multiaxial strain factor Creep Crack Growth Creep Crack Initiation Fatigue Crack Growth compact tension, single-edge notch tension, middle tension, single-edge notch bend, c-ring tension specimens

Acknowledgement The author would like to express his gratitude to colleagues at Imperial College, British Energy and VAMAS work group and EU project partners in numerous projects undertaken to develop the findings in this chapter.

References 1. ASME Boiler and pressure vessel code, section XI: Rules for in-service inspection of nuclear power plant components, American Society of Mechanical Engineers, 1998 2. AFCEN, Design and construction rules for mechanical components of FBR nuclear islands, RCC-MR, Appendix A16, AFCEN, Paris, 1985 3. BS7910, Guide on methods for assessing the acceptability of flaws in fusion welded structures, London, BSI, 2007 4. R5, Defect assessment code of practice for high temperature metallic components, British Energy Generation Ltd., 2007 5. R6, Assessment of the integrity of structures containing defects, Revision 3, British Energy Generation Ltd., 2007 6. P.J. Budden, “Validation of the High-Temperature Structural In-tegrity Procedure R5 by Component Testing,” R5, Document, Vol. 8–9, July, 2003 7. B. Drubay, D. Moulin, and S. Chapuliot. A16: Guide for Defect Assessment and Leak Before Break, Third Draft, Commissariat a l’Energie Atomique (CEA), DMT 96.096, 1995 8. ASTM, “E 1457-07: Standard Test Method for Measurement of Creep Crack Growth Rates in Metals,” Annual Book of ASTM Standards, Vol. 3, 2007, pp. 936 –950 9. API RP-579-Recommended Practice (RP) “Standardized Fitness-for-Service Assessment Techniques for Pressurized Equipment Used in the Petrochemical Industry” American Petroleum Inst. 2004

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10. J.E. Dorn (Ed.), Mechanical Behaviour of Materials at Elevated Temperatures, McGraw-Hill, Inc., New York, 1961 11. A.J. Kennedy, Processes of Creep and Fatigue in Metals, Wiley, New York, 1962 12. F. Garofalo, Fundamentals of Creep and Creep-Rupture in Metals, MacMillan, New York, 1965 13. G.A. Webster, “A Widely Applicable Dislocation Model for Creep,” Phil. Mag., Vol. 14, 1966, pp. 775–783 14. J. Gittus, Creep, Viscoelasticity and Creep Fracture in Solids, Applied Science, London, 1975 15. F.A. Mclintock and A.S. Argon, Mechanical Behaviour of Materials, Addison-Wesley, Massachusetts, 1966 16. J.D. Lubahn and R.P. Felgar, Plasticity and Creep of Metals, Wiley, New York, 1961 17. A.E. Johnson, J. Henderson, and B. Khan, Complex Stress Creep, Relaxation and Fracture of Metallic Alloys, HMSO, London, 1962 18. I. Finnie and W.R. Heller, Creep of Engineering Materials, McGraw-Hill, New York, 1959 19. Y.N. Rabotnov, Creep Problems in Structural Members, F.A. Leckie, Ed., North Holland, Amsterdam, 1969 20. R. Viswanathan, Damage Mechanisms and Life Assessment of High-Temperature Components, ASM International, Metals Park, Ohio, 1989 21. L.M. Kachanov, Introduction to Continuum Damage Mechanics, Kluwer Academic Publishers, Dordrecht, 1986 22. G.A. Webster and R.A. Ainsworth, High Temperature Component Life Assessment, Chapman and Hall, London, 1994 23. A. Saxena, “Evaluation of C∗ for the Characterization of Creep Crack Growth Behavior in 304 Stainless Steel,” Fracture Mechanics: Twelfth Conference, 1980 ASTM STP 700, ASTM, pp. 131–151 24. H. Riedel and J.R. Rice, “Tensile Cracks in Creeping Solids,” Fracture Mechanics: Twelfth Conference, ASTM STP 700, ASTM, 1980, pp. 112–130 25. J.W. Hutchinson, “Singular Behavior at the End of a Ten-sile Crack in a Hardening Material,” J. Mech. Phys. Solids, Vol. 16, 1968, pp. 13–31 26. C.F. Shih, “Tables of Hutchinson-Rice-Rosengren Singular Field Quantities,” Brown University Report MRL E-147, Providence, RI. 1983 27. K.M. Nikbin, D.J. Smith and G.A. Webster, “Pre-diction of Creep Crack Growth from Uniaxial Data,” Proc. R. Soc. Lond. A, Vol. 396, 1984, pp. 183–197 28. A. Saxena, “Evaluation of Crack Tip Parameters for Characterizing Crack Growth: Results of the ASTM Round-Robin Program,” Mater. High Temp., Vol. 10, 1992, pp. 79–91 29. K.M. Nikbin, “Foreward: Creep Crack Growth in Components,” Guest Editor,’ Int. J. Pres. Ves. Pip. Elsevier Ltd., Vol. 80, No. 7–8, 2003, pp. 415–595 (July–August 2003) 30. K.H. Schwalbe, R.H. Ainsworth, A. Saxena, and T. Yoko-bori, “Recommendations for Modifications of ASTM E1457 to Include Creep-Brittle Materials,” Eng. Fract. Mech., Vol. 62, 1999, pp. 123–142 31. M. Tabuchi, T. Adachi, A.T. Yokobori, Jr., A. Fuji, J.C. Ha, and T. Yokobori, “Evaluation of Creep Crack Growth Properties Using Circular Notched Specimens,” Int. J. Pres. Ves. Pip., Vol. 80, 2003, PP. 417–425 32. K.M. Nikbin, D.J. Smith, and G.A. Webster, “An Engineering Approach to the Prediction of Creep Crack Growth,” J. Eng. Mater. Technol. Trans. ASME, Vol. 108, 1986, pp. 186–191 33. B. Dogan, K.M. Nikbin, and B. Petrovski, “Development of European Creep Crack Growth Testing Code of Practice for Industrial Specimens,” Proc. EPRI Int. Conf. on Materials and corrosion experience for fossil power plants, Isle of Palm, SC, USA Nov. 18–21, 2003 34. L.F. Coffin, “Fatigue at high temperature” in “Fatigue at elevated temperatures,” ASTM STP 520, 1973, pp. 5–34 35. S.S. Manson, “A Challenge to Unify Treatment of High Temperature Fatigue,” ASTM STP 520, 1973, pp. 744–775 36. J. Bressers, (Ed.) Creep and Fatigue in High Temperature Alloys, Applied Science, Barking, UK, 1981

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37. R.A. Ainsworth, G.C. Chell, M.C. Coleman, I.W. Goodall, D.J. Gooch, J.R. Haigh, S.T. Kimmins, and G.J. Neate, CEGB Assessment Procedure for Defects in Plant Operating in the Creep Range, Fatigue and Fracture of Engineering Materials and Structures, Vol. 10(2), 1987, pp. 115–127 38. H. Tada, Stress Analysis of Cracks Handbook. Paris Productions Incorpo-rated, 2nd edition, 1985 39. J.C. Newman and I.S. Raju, “An Empirical Stress In-tensity Factor Equation for the Surface Crack”. Eng. Fract. Mech., Vol. 15, No. 1–2, 1981, pp. 185–192 40. D.P. Rooke and D.J. Cartwright, Compendium of Stress Inten-sity Factors by. HMSO, London, 1976 41. C. Davies, K.M. Nikbin, and N.P. O’ dowd, “Experimen-tal Evaluation of the J or C∗ Parameter for a Range of Cracked Geometries,” ASTM STP 1480, 2007 42. J.R. Rice and D.M. Tracey, “On the Ductile Enlarge-ment of Voids in Triaxial Stress Fields,” J. Mech. Phys. Solids, Vol. 17, 1969, pp. 201–217 43. A.C.F. Cocks and M.F. Ashby, “Intergranular Fracture During Power-Law Creep Under Multiaxial Stress,” Met. Sci., Vol. 14, 1980, pp. 395–402. 44. M. Yatomi, N.P. O’ Dowd, and K.M. Nikbin, “Computational Modelling of High Temperature Steady State Crack Growth Using a Damage-Based Approach,” in PVP-Vol. 462, Application of Fracture Mechanics in Failure Assessment Computational Fracture Mechanics, ASME 2003, P. -S. Lam, Ed., ASME New York, NY 10016, 2003, pp. 5–12 45. M. Yatomi, K.M. Nikbin, and N.P. O’ Dowd, “Creep Crack Growth Prediction Using a Creep Damage Based Approach,” Int. J. Pres. Ves. Pip., Vol. 80, 2003, pp. 573–583 46. M. Tan, N.J.C. Celard, K.M. Nikbin, and G.A. Webster, “Comparison of Creep Crack Initiation and Growth in Four Steels Tested in HIDA,” Int. J. Pres. Ves. Pip., Vol. 78, 2001, pp. 737–747 47. K.M. Nikbin, “Transient Effects on the Cyclic Crack Growth of Engineering Materials,” Published in ‘3rd Conference on low cycle fatigue and elasto-plastic behaviour of materials III, LCF3’, K.T. Rie, Ed., Elsevier Publishers, 1992, pp. 558–664 48. C.M. Davies, N.P. O’Dowd, K.M. Nikbin, and G.A. Webster, Prediction of Creep Crack Initiation under Transient Stress Conditions, American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP, 2006, Vancouver, BC, Canada, 2006, American Society of Mechanical Engineers, New York, NY 10016-5990, United States, p. 9 49. M.R. Winstone, K.M. Nikbin, and G.A. Webster, “Modes of Failure Under Creep/Fatigue Loading of a Nickel-Based Superalloy,” J. Mater. Sci., Vol. 20, 1985, pp. 2471–2476

Chapter 6

Computational Approach Toward Advanced Composite Material Qualification and Structural Certification Frank Abdi, J. Surdenas, Nasir Munir, Jerry Housner, and Raju Keshavanarayana

Abstract The objective of this chapter is to perform accurate simulation of physical tests using multi-scale progressive failure analysis (PFA) and to simulate the scatter in the physical test results by using probabilistic analysis. The multi-scale analysis is based on a hierarchical analysis, where a combination of macro-mechanics and micro-mechanics is used to analyze material and structures in great detail. To calculate the correct micro-mechanical constituent properties for the multi-scale analysis, a three-step process is used: (1) calibration step, (2) verification step, and (3) probabilistic analysis step. The discussion in this chapter will focus mainly on the use of FAA composite material certification requirements and estimation of mechanical and fracture properties of composites; A-basis and B-basis allowable properties generation that are recognized as statistical in nature; and categories of damage tracking for composite structure under service.

6.1 Overview The use of advanced composites in product design is becoming increasingly more attractive due to advantageous weight-to-stiffness and weight-to-strength ratios. Increasingly, composite structures are being subjected to severe combined environments and are expected to survive for long periods of time. There is neither an adequate test database for composite structures nor significant long-life service experience to aid in risk assessment. To ensure safe designs, it is estimated that aerospace companies spend an estimated $350 million per year on testing. Due to the difficulty and cost in assessing and managing risk for new and untried systems, the general method of risk mitigation consists of applying multiple conservative factors of safety and significant inspection requirements to already conservative designs in lieu of costly full system tests. Unfortunately, this approach can lead to excessively conservative designs and the full potential of composite systems is often not fully realized. F. Abdi (B) Alpha STAR Corporation, Long Beach, CA, USA e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 6,

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This chapter primarily focuses on the use of FAA advanced composite material and structural components certification requirements with a view to obtaining efficient designs. The following items are covered: • Treating FAA categories of damage tracking for composite structures under service. • Estimation of mechanical and fracture properties of composites. • Generation of A-basis and B-basis allowable properties that are statistical in nature. The objective is to perform a virtual testing (VT) process which involves an accurate simulation of physical tests using multi-scale progressive failure analysis (PFA), including the scatter in physical tests by using probabilistic analysis. The multi-scale analysis is based on a hierarchical analysis, where a combination of micro-mechanics and macro-mechanics is used to analyze material and structures in great detail. Certification required predictions are important for reducing risk in structural designs. Moreover, determination of allowable properties is a time-consuming and expensive process, since a large amount of testing is required. In order to reduce costs and product lead time, VT can be used to reduce necessary physical tests both for certification and for determining allowables. In summary, whereas the aerospace industry tends to rely on expensive testintensive empirical methods to establish design allowables for sizing advanced composite structures, the developed VT methodology relies on physics-based failure criteria to reduce its dependence on such empirical-based procedures. This is more than a simple mix of analysis and test because: (1) the root cause of failure at the micro-scale is modeled for accurate failure and life prediction, (2) VT is incorporated into each stage of the FAA building-block process and the FAA categories of damage tolerance, and (3) natural material and manufacturing data scatter is created giving rise to the unique capability to estimate A- and B-basis allowables.

6.2 Background 6.2.1 FAA Durability and Damage Tolerance Certification Strategy The generally accepted strategy for verifying an aircraft structural design for FAA certification is a building-block testing approach consisting of coupon, subelement, and full-scale prototype experimental testing. Building a comprehensive VT database of building blocks that conforms to FAA requirements will put at designers’ disposal a readily available compendium of certified designs that can be beneficially interrogated relative to the FAA certification potential of a newly proposed design.

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To insure advanced composite aircraft flightworthiness, the Federal Aviation Administration (FAA) requires that the aircraft builder/user address the damage levels for primary structures. Categories of damage and defect considerations for primary composite aircraft structures are outlined in Table 6.1. Table 6.1 FAA Categories of damage and defect considerations primary composite aircraft structures (Courtesy of FAA) Category

Examples

Category 1: Damage that may go undetected by field inspection methods (or allowable defects) Category 2: Damage detected by field inspection methods @ specified intervals (repair scenario) Category 3: Obvious damage detected within a few filights by operations focal (repair scenario) Category 4: Discrete source damage known by pilot to limit flight maneuvers (repair scenario) Category 5: Servere damage created by anomalous ground or flight events (repair scenario)

BVID, minor environmental degradation ,scratches, gouges, allowable mfg defects VID (ranging small to large), mfg defects/mistakes, major environmental degradation Damage obvious to operations in a “walk-around” inspection or due to loss of form/fit/function Damage in flight from events that are obvious to pilot (rotor burst, bird-strike, lighithing) Damage occurring due to rare service events or to an extent beyond that considered in design

Safety Considerations (Substantiation, Managment) Demonsrate reliable service life Retain Ultimate Load capability Design-driven safety Demonstrate reliable inspection Retain Limit Load capability Design, maintenance, mfg Demonstrate quick detection Retain Limit Load capability Design, maintenance, operations Defined discrete-source events Retain “Get Home” Capability Design, operations, maintenance Requires new substantiation Requires operations awareness for safety (immediate reporting)

6.2.2 Damage Categories and Comparison of Analysis Methods and Test Results Five damage categories are identified by the FAA, ranging from minor to severe. This section describes the damage and the corresponding analysis methods that can be employed to simulate the damage events of each category. Category 1: Damage That May Go Undetected by Field Inspection Methods Barely visible damage can occur due to matrix transverse cracking and microcrack density formation during manufacturing and service (e.g., static loading, fatigue loading). Quantifying and characterizing the micro-cracking “transverse matrix crack” response during the composite cool down process and subsequent

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in-service fatigue life is important because the micro-cracks can form continuous paths through the thickness of the laminates resulting in lower stiffness, and leakage (Fig. 6.1). Fig. 6.1 Typical micro-cracks in the polymer matrix [7]

Methods for estimating stress fields in cracked laminates are the variational approach [1], the shear lag method [2–3], approximate elasticity [4] and internal variable models [5]. The shear lag method is an efficient and simple means of calculating stresses in laminate fibers and matrix. It has also been extended to micro-crack density prediction in laminates. Zhang et al. [6, 7] developed an equivalent constraint model (ECM), which predicts the reduction in stiffness properties due to transverse ply cracks as well as the initiation and growth of matrix cracking with increasing mechanical load. In their study, an improved two-dimensional (2-D) shear lag analysis is used to determine the stress distribution in the cracked laminates. Though crack density prediction using this method shows consistency with test results, it can only be applied to simple lay-ups subjected to uniaxial tension. In order to predict micro-crack densities in structures, the ECM was incorporated into a VT progressive failure finite element analyzer (GENOA) [8]. Stress and strain fields calculated using the finite element analyzer are transferred into the ECM. Within the ECM, the average stress and strain fields in each constituent are used to calculate the micro-crack formation and growth as described by the magnitudes of the micro-crack density. This method assumes that the micro-crack spacing is uniform in the ECM. Only transverse cracking is considered in the ECM. Longitudinal cracking involves fiber failure, which is the macro-fracture of the entire ECM. The degradation of composite properties due to the existence of cracks in a ply is determined by the crack density, stress level and constraints supplied by the adjacent plies. This may be determined using an iterative process as introduced into the VT ECM method that determines the stress redistribution resulting from matrix damage [7]. In each iteration, crack densities and the corresponding degradation of

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composite properties are re-calculated. The iterative process reaches convergence when crack densities throughout the structure reach saturation under the current load level. New cracks may occur when loads are increased to the next higher load level. The cool-down thermal stresses in the ply transverse direction are often large enough to cause damage in laminate matrices. For such conditions, crack density verification was performed by comparing analytic predictions with the experimental test observations reported in the literature [7, 8] for T300/934 uniaxial laminates under tension (Figs. 6.2a and b). Considering the nature of test data scatter, it is safe to say that the crack density simulation agreed reasonably well with the test results. Crack density development in the angled plies of the laminate in the case of fatigue are illustrated in Fig. 6.3 Micro-cracks in the matrix at cryogenic temperature multiplied more rapidly than those at room temperature as shown in both simulation and test observations. Fig. 6.3b compares simulation and test fatigue lives of the laminate at various stress levels. It can be seen that the numerical results fell within scattered test data at each stress level. Both test and analysis show that the laminate fatigue S–N curve stabilizes at a higher stress level at cryogenic (liquid nitrogen) temperature than at room temperature. As shown in Figs. 6.2 and 6.3 the simulation agrees with test results for most cases. Crack counting in tests is sensitive to sampling locations, mechanical test processes and other factors. Thus the discrepancies shown are acceptable. (b) Laminates are T300/934 [0/902]sand [25/-25/904]s 12 Simulation - [0/90]s Test - [0/90]s Simulation - [25/-25/90/90]s Test - [25/-25/90/90]s

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Fig. 6.2 Comparison of crack density in 90 plies – predicted vs. test [1]

Category 2–3: Damage Detected by Field Inspection Methods Visible damage may be observed during manufacturing such as wrinkling and fiber waviness and void distribution in thick laminates (Category 2). In addition, obvious damage may be detected within a few flights by flight operations and maintenance personnel (Category 3). Low-speed impact, tool drop and part buckling are representative events of these categories. Low-speed impact and static indentation on composite laminates may cause interior damage. As a result, its residual strength during its service loading is reduced: (a) compression after impact (CAI) and (b) tension after impact (TAI).

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Fig. 6.3 Comparison between simulation and test: (a) Crack density development in the laminate under tension, (b) Cyclic loading of the laminate at room and cryogenic temperatures. Laminate lay-up is [0/45/90/-45]s

Impactor Fixture plate

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Panel Size: 5 inch by 5 inch panel Impact Speed: 3.01 ft/sec Impact Energy: 7.58 ft-lbs Total Time: 19.75 msec Test Fixture: Clamped boundary condition on all four sides Impactor: Spherical Steel (diameter 1 in) Material: G30-500/45 R6367: Ply Orientation: /-45/0/90/0/90/0/90/0/90/45/45ly Orientation: /

Fig. 6.4 Schematic of impacting panels and fixtures

Figure 6.4 shows an example of low-velocity impact [9]. Figure 6.5 shows the experimentally observed damaged plies and prediction by the progressive failure dynamic analysis. Figure 6.6 shows the impact of a steel ball on an advanced composite plate, the impact event load versus time, and the TAI comparison between test and simulation [10]. Figure 6.7 shows the impact of a steel ball on a composite foam sandwich plate and CAI [11–13]. Comparisons between test and simulation include: (a) impact energy, (b) impact peak load, (c) CAI residual strength, and (d) failure mechanisms. Analyses such as these may be used to assess damage tolerance of composite components to impact events in Categories 2 and 3 of Table 6.1. Category 4: Discrete Source Damage Discrete source damage (DSD) can limit an aircraft’s flight maneuver envelope. Herein, discrete source damage is defined as a through-penetration of a structure with an area of collateral (non-visible) damage emanating from where high-density (>0.1 lb/in.3 ) projectiles impact the structure at velocities sufficient for penetration. A DSD event that the FAA is concerned with is the disintegration of a mechanical

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a) Experimental

b) Simulated Damaged Area

Fig. 6.5 Comparison of the fracture pattern between test and simulation

(a) Impact of steel ball on composite plate

(b) Load vs. time

(c) Tension after impact (TAI) stress vs. damage size

Fig. 6.6 Tension after impact, residual strength – test vs. simulation

(a) Residual impact foot print (Compression load = 0 lb)

(b) Crack path (Compression load = 24,680 lb)

Fig. 6.7 Compression after impact

(c) Test vs. Simulation

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Shallow Angle Impact Fan Shrapnel + 1/3 Disc Sector 1/3 Disc Sector 3-Blade Sector Small Shrapnel

ACT96-051

Fig. 6.8 Potential engine fragment paths [14]

component, such as an engine failure that ejects items such as a blade disk, blades, and/or blade fragments (Fig. 6.8). The components and fragments (usually made of titanium or steel) can strike and penetrate the surface of the lower wing skin and possibly penetrate through the upper wing skin. The FAA describes such events in AC20-128 [14] to include ejection of: • One-third of a fan disk ejected at the critical rotation speed. (This usually occurs at take-off.) • A turbine blade fragment of 3 pounds at a velocity of 900 ft/s. • Small blade fragments weighing 0.6 lb at velocities up to 400 ft/s Federal Aviation Regulation Part-25 is the principal certification guideline [15]. A discrete source damage event is described in Section 25.571 of this document as one facet of a damage-tolerant design that must be considered for certification. A DSD event is an incident that is immediately apparent to an aircraft’s flight crew or ground personnel at the time of its occurrence, and the post-event residual strength of the structure must be 70% of its design limit load [16]. As stated, this type of damage is often caused by projectiles emanating from disintegrating engine components with variations in projectile mass, velocity, and impact location [14]. These variables create a complex problem for engineers designing structures that are DSD tolerant. To reduce this design complexity, DSD evaluations are typically simplified by evaluating the residual strength of multi-stringer test panels containing a two-bay crack or slot across a principal stiffening element [17, 18]. Certification guidelines also state that any future application of composite materials in primary structures will be required to demonstrate a level of damage tolerance after the occurrence of such an event [19]. Therefore, the successful development and verification of a DSD

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analysis methodology for laminated composites [20] would reduce future certification costs of any advanced composite structures fabricated with these materials. DSD Geometry: An initial demonstration of discrete source damage tolerance of composite materials applicable to primary wing structures used machined slots (saw cuts) to simulate DSD sites [21]. Also, prior discussions with Federal Aviation Administration and B-2 bomber durability and damage tolerance personnel strongly suggested the use of a two-bay slot to demonstrate DSD residual strength of composite structures [17, 22]. This geometry provides a low-cost alternative to ballistic tests and produces a repeatable damage site at a precise location that offers consistent test comparisons between sets of data. All the slots are positioned in the center of each specimen, both in the vertical and horizontal directions. The machined slot geometries used for the compression and tension test specimens are shown in Fig. 6.9. Both slots were 18 cm (7.0 inches) long with a 0.24 cm (0.094 inches) tip radii. The compression slot was modified into a diamond configuration to reduce the possibility of the upper and lower surfaces coming into contact during testing. Straight slots were used in the initial DSD evaluation of composite structures. Post-failure analysis of the compression specimen raised concerns that the slot had closed under load during the test and loaded the machined surfaces. Minimizing the potential for the machined surfaces to contact each other during a test was considered an experimental refinement over the prior DSD compression test approach.

Fig. 6.9 Compression and tension slot geometries simulating DSD

Compression DSD Failure Behavior: Damage development in the compression specimens progressed in a stable manner from the slot radii toward the outer stringer flanges. The development of this type of failure mode from a flaw site is consistent with behavior observed in compression-dominated post-impact fatigue tests performed on composite materials [23]. A close-up view of one edge of a failed compression test article is shown in Fig. 6.10b. This is typical of the failures observed in compression test articles. The compression failures displayed classic transverse shear surfaces typically observed in stitched composites [23]. Another feature of the failure mode of these compression test articles included no observed stringer pull-off problem.

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Shear Cracks

(b) Close-up view of compression test article failure zone

(a) DSD test article

(c) Tension test Article translaminar shear damage zones

Fig. 6.10 A DSD test article – compression and tension

Tension DSD Failure Behavior: Generally, the three-stringer tension specimen failures were characterized by initiation of damage at the radii of the slots. Increasing the applied load on each panel caused the tension failure damage to rapidly propagate transversely to the loading direction until it reached the inner flanges of the outer stringer region of each specimen. Once this tension damage reached the outer stringer flanges, the damage then propagated parallel to the flanges in the loading direction. This vertically oriented translaminar shear failure mode was observed in all four tension test articles. The stable propagation of this shear failure mode continued until the damage neared the loading tabs and caused catastrophic panel failure. A post-test view of a tension panel is shown in Fig. 6.10c [24]. DSD Residual Strengths: Results of the residual strength tests and analytical predictions for the three-stringer stitched resin film infusion (S/RFI) panels containing DSD sites are shown in Table 6.2. Table 6.2 Summary of virtual test predictions with experimental results Compression Results

Tension Results

Sealed Post-test Panel Geometry envelope re-analysis

Experimental Sealed Post-test Experimental value envelope re-analysis value

6-Stack skin 2.3 Inch blades 6-Stack skin 1.8 Inch blades 4-Stack skin 2.3 Inch blades 4-Stack skin 1.8 Inch blades

291

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421

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427

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336

382

238

247

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341

309

199

230

206.8

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The catastrophic failure behavior of the compression test articles allowed the ultimate load a panel carried to be used as its failure load. However, the more complex behavior of the tension panels required closer reviews of strain gage and video data to determine when each panel reached its defined ultimate failure load. Tension panels were considered to have failed when translaminar damage started to propagate along the inner flanges of the outer stringers. In Table 6.2, the columns labeled, “Sealed Envelope” refer to pre-test computer predictions provided independently to the aircraft company overseeing the test program. The predictions provided in Table 6.2 indicate that the analysis methodology shows promise for predicting the residual strength of S/RFI structural panels containing DSD sites. The pretest predictions for the compression panels were within 5% of the experimental failure loads. Analytical predictions of tension residual strengths were also reasonably close to the experimental values. The pretest tension predictions were conservatively below the test results by no more than 5%. After the experimental evaluations were completed, a unified S/RFI material database was developed based on coupon compression, tension, and shear test data. The columns labeled “Post-Test Re-analysis” in Table 6.2 refer to analyses carried out using the unified material database. Using this database, the compression and tension panel failure load predictions were within 11% and 12% of the test results, respectively. Category 5: Severe Damage Created by Anomalous Ground or Flight Events In-flight fire is an example of a scenario that requires a half-hour emergency mission abort. Fire gives rise to high temperatures, which can cause epoxy resins to soften or burn, thus effectively undermining the strength of a composite part. The effects of fire are particularly serious in the confined space of vehicles, aircraft, boats, and trains. Therefore, innovative designs should not only be cost-effective but fire resistant. Developing and using composite life prediction analytical procedures to study any composite experiencing extreme fire conditions is important since the reduced analysis and/or experimental effort can represent a significant cost savings. This applies to both the design and redesign of a composite structure. This composite analytical procedure must predict the response of a composite structure accurately. 6.2.2.1 Example of Simulation of Composite Deck under Fire The fire resistance of simply supported deck sandwich (E-Glass DKNE face sheet, and balsa wood core) panel and deck-bulkhead assembly structures (Fig. 6.11) subject to service mechanical loads of 300 psf and the heat load due to the action of fire with the maximum flame temperature of 2,000◦ F was considered [25]. The mid-span deflection of the deck panel was chosen as a criterion for the fire-resistance assessment of the both structures. Two simulation approaches of coupled and un-coupled thermal structural analysis were employed to determine the mid-span deflections of the deck panel [26–27]. First, the PFA was performed based on the equivalent

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a) quarter model

b) Mid-span deflection of the bottom facesheet of the deck sandwich panel

Fig. 6.11 Deck-bulkhead assembly subject to service mechanical loads and surface heat fluxes

temperature load profile, which was obtained from the given fire profile using the NASA CSTEM computer program. In the second case, the coupled CSTEM– progressive failure simulation was performed where the PFA was invoked immediately after the CSTEM temperature redistribution had been calculated for each time interval. In both cases, the effect of fire on the bottom part of the deck panel was simulated through the application of surface heat fluxes to the bottom of the panel and using CSTEM to calculate the equivalent temperature load required for the PFA [28–31]. The calculated mid-span deflection of the simply supported deck sandwich panel after one-hour exposure to fire correlated well with the experiment. Both the simulation and experimental results show that the mid-span deflection was significantly below the 2-foot deflection limit. In addition, the initiation and propagation of damage and fracture patterns over the deck structures were computed. Figure 6.11a shows the finite element model of the deck-bulkhead assembly. In this case, the equivalent uniform service load on the deck panel was 300 psf. The finite element model of the deck-bulkhead assembly consisted of the 7,761 threedimensional isoparametric solid elements and 16,579 nodes. Figure 6.11b shows the predicted deflection after thirty minutes of exposure was about 1.75 and 2.75 inches according to coupled and uncoupled analysis, respectively. The coupled simulation was much closer to the measured 2.5 inches. Both analyses predicted that the panel design was adequately fire-resistant according to the prescribed limit of 2 feet, which also agreed with the experimental observations. Figure 6.12 shows delamination due to the transverse and normal tensile damages throughout the deck-bulkhead structure after 10 minutes of fire exposure.

6.2.3 FAA Building-Block Approach Within the composite engineering community, the structural substation process, which uses testing and analysis at increasingly complex levels, has become known as the “building-block approach.” Such an approach has traditionally been used to

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Fig. 6.12 Transverse tensile damage in the deck-bulkhead assembly after 10 minutes of exposure to fire in the: (a) balsa wood core; and (b) facesheets (damaged areas are shaded)

address durability and damage tolerance as well as static strength for both metal and composite aircraft structure. The virtual and experimental testing building-block approaches are interactive. Experimental test results are used to validate methods for analytical predictions and reduce uncertainties in VT results. VT provides assistance to planning and reduction of experimental testing at coupon and large component levels. With experimental verification, VT of composite structure can be performed to understand: (1) crack initiation at multiple sites; (2) uncertainties in material properties; (3) effects of barely visible, visible, and discrete source damage; (4) means of predicting damage growth and residual strength; and (5) how to demonstrate durability and robustness to assist in the FAA certification process. Figure 6.13 provides a conceptual schematic of test included in a building-block approach for wing structures. Lower levels of testing are more generic and likely to be applicable to other parts of the airplane and other products. In order to perform these analyses, the material stress–strain curve needs to be established to failure (or

Fig. 6.13 FAA Schematic diagram of building-block tests (Courtesy of FAA)

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a strain cutoff in the test methods) for each composite material used in the design. Analysis has proven reliable to minimize the numbers of tests needed to define this characteristic for laminated composite material forms. There are generally more repetitions at lower levels, such as needed to provide a statistical basis for material performance. Since some lower levels of building-block tests can be considered generic, the concept of databases shared between programs is reasonable. Engineering protocol for base material qualification and the equivalency testing to use shared composite databases has been published previously. Each certification project will have its own certification plan and methods approved by the local aircraft certification office. The integration of the design and manufacturing process becomes evident in larger studies. The larger scales of testing are needed to address the effects of more complex loads and geometry. Fewer tests are performed at larger scales. The relevance of these tests is to address specific structural details. Analysis validation is an important part of the building-block process because it provides a basis to expand beyond the specific tests performed in development and certification. Such validation starts with prediction of the structural stiffness, internal load paths, and stability. Verification of internal load paths may require additional building-block tests, which are designed to evaluate load share between bonded and mechanically attached elements of a design. This is particularly difficult analytically as failure is approached, where some nonlinear behavior can be expected. Combined load effects can further complicate the problem of analytical predictions. 6.2.3.1 Damage, Defects, Repeated Load, and Environmental Effects Prediction of the effect of multiple influences (environment, repeated loads, damage, and manufacturing defects) on the failure modes that affect structural strength traditionally relies on the building-block tests. Often, semi-empirical analyses have been adopted for composite strength. In such analyses, special considerations are given to structural discontinuity (for example, joints, cutouts or other stress risers) and the other design or process-specific details. One of the most important parts of the building-block analysis and test development comes in providing engineering databases to deal with manufacturing defects, field damage, and repairs likely to occur in production and service. Traditionally, not enough attention was given to these issues during composite product development and certification. This has caused significant work slowdowns and increased costs for subsequent product manufacturing and maintenance. Without sufficient analysis and a test database to cover commonly allowed manufacturing defects, damages, and repairs, engineers are often forced to either adopt conservative assumptions (part rejections or expensive repairs) or generate the data as it is needed (leading to down time and associated cost or lost revenue). Unfortunately, production and service experiences with new technologies such as composite materials are often needed to completely define the problems and to plan for unanticipated defects and damage. Nevertheless, an awareness of the likely

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production and service issues will help define practical levels of building-block tests and analyses to be performed as part of the structural substantiation. MIL-HDBK-17 (DoD Composite Material Handbook) provides some detailed background on the engineering practices that have been successfully applied with composite materials used in airplane structures. Chapter 2.1 from Volume 1 – Polymer Matrix Composite (PMC), provides some introduction to this subject, including a synopsis of test levels and the data uses. Many of the engineering practices outlined in MIL-HDBK-17 were derived from composite applications to military and commercial transport structures. The composite material types, structural design details, and associated manufacturing processes selected for such applications may have significant differences from those used for small airplanes.

6.2.4 Test Reduction Process The building-block strategy builds from the lowest configuration level fiber/matrix to the full-scale assembly. To perform test reduction, failure mechanism predictions (such as longitudinal tensile, longitudinal compressive, transverse tensile, transverse compressive, delamination, in-plane shear, out-of-plane shear, peel-off stress, etc.) are determined and then used to identify which tests to perform. The following process (Fig. 6.14) is proposed for characterizing a new material: 1. Material and Manufacturing Development: This step is done by the material vendor where the chemistry, physical form (sheet, plate, cast, etc), and preliminary static and fatigue properties are obtained. 2. Unnotched Static and Fatigue Testing: The basic static material properties and unnotched fatigue material behavior are determined for different stress ratios.

Material and Mfg. Characterization Unnotched Static & Fatigue Testing

Probabilistic Analysis

Deterministic Analysis

Reduced Testing

Fig. 6.14 Test reduction process

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3. Probabilistic Analysis: Treating the material and manufacturing as variables, statistical distributions are applied to simulate test coupons. These simulated test coupons will have different properties, and hence the scatter that is seen in traditional test plans will be reproduced. 4. Deterministic Analysis: Using the test coupons generated in Step 3, a VT matrix in concert with traditional testing is completed to characterize the material completely. To perform the test reduction process, three analytic processes are used. 1. Material Property Characterization Analysis including the effects of manufacturing and service conditions such as hygral (thermal, moisture) effects. Material characterization is used to predict mechanical properties for temperatures where no test data is available including: 1) mechanical properties at the laminate, lamina, and constituent (fiber/matrix) levels under room, cryogenic, and high-temperature testing conditions. Material characterization also validates the simulation results against experimental observation 2. Material Property Uncertainty Analysis of failure mechanisms and the percent contribution of mechanical properties to these failures are used to determine their sensitivity to variations over a temperature range and to generate mechanical properties for temperatures where no test data was available. 3. Probabilistic Analysis. In order to properly characterize a material, several factors must be taken into consideration. These factors, depending on the severity and distribution, have different effects on the fatigue life of the material. The following list provides some of the larger variables: 1. Lot-to-Lot Material Variation (large effect on scatter) a. Porosity (size, density, shape, location) b. Inclusions (same as above) 2. Surface Quality Variation (moderate-to-large effect on scatter) a. Surface finish b. Machining marks, scratches 3. Residual stresses imparted or removed due to cutters, drill bits, etc. 4. Deburring techniques used on edges, holes 5. Loading Variation (moderate effect on scatter) a. Test machine/specimen alignment b. Accuracy of loads/strains applied during test c. Crack measurement/detection capability (for small cracks or to determine crack initiation) 6. Geometry Variation (small effect on scatter) 7. Geometric tolerances causing variation in Kt ∗ from specimen to specimen

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Each these variables will have a statistical distribution depending on how these values change from one specimen to another. Once these distributions have been defined, probabilistic analysis will then come up with a specified number of specimens with a distribution of properties that have the same test scatter.

6.3 Computational Process for Implementing Building-Block Verification The building-block approach focuses on hierarchical progressive failure analyses at each step of the design process to verify basic material constituents, joints, builtup substructures (e.g., spars and bulkheads), and the final product (Fig. 6.15). At each verification stage, materials and structures require evaluation of their mechanical properties and the corresponding uncertainties to determine the adequacy of the structure’s durability and reliability. (PFA) (Fig. 6.16) implements the basic concept that a structure will fail when defects and flaws, that may initially be microscopic, grow and/or coalescence to a critical size at which the structure no longer has an adequate safety margin to avoid catastrophic global fracture (Fig. 6.17). Damage is considered to progress through five stages: (1) initiation, (2) growth, (3) accumulation and coalescence of propagating flaws, (4) stable propagation (up to critical dimensions), and (5) unstable or very rapid propagation to catastrophic failure. Computational PFA involves a formal procedure for identifying the five different stages of damage, quantifying the amount of damage at each stage, and relating the damage to the overall behavior of the deteriorating structure.

Material Constituent Verification 1. 2. 3. 4. 5. 6. 7. 8. 9.

Strength Stiffness Material Uncertainty Fiber/Matrix Interface Micro Mechanics–Failure Void Fiber Volume Fraction Margin of Safety Multi-Factor/Degradation Fracture Toughness

Coupon Verification Strength due to Notch Effects Calibration Basis Waviness Material Uncertainties Load Redistribution ASTM 3039 ASTM 5766 ASTM 3410 ASTM 4255 ASTM 5379

Sub-Element Verification Virtual Testing Ply Failure Waviness Ply Schedule Residual Strength Margin of Safety Residual Strain Load Redistribution Energy Release Rate Design Guidelines Joints

Full Scale Verification Global Multi Site Crack Initiation Multi Site Crack propagation Residual Strength Contribution of Failure Photo Elasticity Deflectometer & Strain Gauge Far Field Strain Inspection Predictions for Non destructive Testing Sensitivity of Failure Modes

Fig. 6.15 Virtual testing building-block verification strategy for FAA FAR-025 certification

Utilization of PFA design has already been demonstrated by successfully predicting damage tolerance limits and failure criteria of significant aerospace structural

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Large Scale Structural Component Multi Site Damage (Initiation, Progression, Residual) Linear/ non linear Structural Analysis Failure Mechanism Contribution Part Inspection Guidance Life Prediction, Final Failure Loads Fracture crack growth Analysis Probabilistic Assessment Reliability Based Optimization Building Block Verification Strategy Certification Process, Conforms to FE Standards

Fig. 6.16 Functionality of virtual testing (GENOA) software modules

Fig. 6.17 Virtual testing multi-scale hierarchical progressive failure analysis process

designs. Results such as these would support satisfying Federal Aviation Regulations for certification of critical aerospace structures.

6.3.1 Multiple Failure Criteria Even though many failure criteria have been developed, there is a lack of evidence to show whether any of the criteria can provide accurate and meaningful predictions of failure for other than a very limited range of circumstances. This conclusion may be surprising to many. After all, there is a large body of composite materials research to draw upon, spanning at least 50 years, along with numerous examples (aircraft, boat hulls, etc.) where composite materials have been used widely and successfully for primary load-bearing structures. One might therefore logically conclude that design procedures (including strength prediction) for composite structures are fully mature. Closer examination reveals that current commercial design practices place little or no reliance on the ability to predict the ultimate strength of the structure with substantial accuracy. Failure theories are often used in the initial calculations to “size” a component (i.e. to establish the approximate dimensions, such as panel thickness, width, etc.). Beyond that, experimental tests of coupons or structural elements are used to determine the global design allowables. These are typically set at levels which are less than 30% of the ultimate load carrying capability, thereby providing

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a wide safety margin to accommodate loss in performance due to fatigue, operating environments, impact and any other possible aspect. The coupon/structural-element testing approach is widespread in the aerospace industry leading to establishment of large databases at great expense. Small-to-medium-sized companies tend to follow a broadly similar path, though on a much smaller scale. A “make and test” approach combined with generous safety factors is commonplace. There are several well-known failure criteria such as Puck, Tsai-Wu, and TsaiHill that are able to predict strength values for some composites extremely well, but differ when the ply lay-up or material system is switched. Using these failure criteria judicially can eliminate the complexity involved, while predicting the failure design envelop without deviating seriously (<10%) from the actual test data. Here we present such a set of carefully selected default multiple failure criteria that can help predict strength parameters of several fiber/matrix systems, which includes complex ply lay-ups, such as unidirectional, biaxial, or triaxial. The PFA process: a) b) c) d) e)

damage initiation, damage propagation, fracture initiation, fracture propagation, and final residual strength [32–35] is validated for both shell and solid elements using several finite element models of coupons (Fig. 6.18). In addition, the linear elastic fracture mechanics (LEFM) formulation [36–39] is used to augment the load displacement curve after the peak.

Fig. 6.18 Fiber matrix, inter-laminar, and interactive failure criteria applied. Damage in subdivided unit cell and delaminations tracked at the micro-scale

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Fig. 6.19 Process of computational micro-mechanics procedure for integration to progressive failure analysis

6.3.2 Micro- and Macro-Composite Mechanics Analysis The micro- and macro-composite mechanics analysis methods used in VT formulate composite properties based on the constituent properties (Fig. 6.19). Then the composite properties are integrated through the entire structure to form the overall structural stiffness for finite element (FE) analysis (Fig. 6.17). This approach considers manufacturing anomalies (e.g., voids, cool down process, fiber architecture/content, and fiber waviness) and environmental effects such as moisture [35]. The composite properties are constituted with the composite configuration information, i.e. fiber and matrix properties, the fiber architecture and content, and the manufacturing defect content using the micromechanics formulations developed at NASA [35]. The overall structural stiffness is formed following classical finite element theory [35]. The (PFA) software, GENOA, augments finite element software by providing progressive failure analysis based on damage tracking and material property degradation at the micro-scale of fiber and matrix, where damage and delamination have their source. The VT software performs multi-scale (full-hierarchical) damage tracking and micro-mechanics material engineering. The software uses micro and macro interaction methods in the composite structural PFA modules shown in Fig. 6.16. Micro-stresses and damages are computed on the constituent level and the corresponding material degradation is reflected in the macroscopic finite element structural stiffness. Displacements, stress, and strains derived from the structural-scale FEA solution at a node/element of the finite element model are passed to the laminate and lamina scales using laminate theory. Unlike the process depicted in Fig. 6.17, most FEA analyses, which are not augmented with GENOA, evaluate failure at the lamina or laminate scale and do not pursue failure beyond this point. Unfortunately, failure does not originate at the lamina and laminate level and, instead, originates at lower

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scales. Hence, GENOA augments FEA analysis, with a full-hierarchical modeling that goes down to the micro-scale of sub-divided unit cells composed of fiber bundles and their surrounding matrix. Stresses and strains at the micro-scale are derived from the lamina scale using micro-stress theory. The sub-divisions of the unit cell (small pieces of fiber and/or matrix), shown in Fig. 6.17a, are then interrogated for damage using a set of failure criteria listed in Fig. 6.18. The relevant formulation can be found elsewhere [34]. Similarly, matrix subdivisions in the unit cell are interrogated for delamination as depicted in Fig. 6.17b. Once damage or delamination occurs, VT determines which fiber and matrix material properties to degrade by applying a set of rules that are based on materials engineering and experience. As damage accumulates in the unit cell, the cell will eventually fracture. This means that a lamina has failed at a node of the finite element model. When all lamina at a node or element fail, the node or element is considered as fractured. Because damage is tracked at the micro-scale, it is quite possible that a node or element may experience two or more types of damage, simultaneously. For example, there may be matrix cracking and fiber breaking in the same unit cell and same lamina of a particular node of the FE mesh. This behavior is especially important when examining damage initiation, accumulation, and growth. It represents a level of detail that provides [40] the foundation for remarkable accuracy.

6.3.3 Progressive Failure Micro-Mechanical Analysis Extensive efforts to identify the various modes of damage in composite materials have been undertaken in recent years. The primary finding of most of these investigations was that macroscopic fracture was usually preceded by an accumulation of the different types of microscopic damage and occurred by the coalescence of this small-scale damage into macroscopic cracks. Additionally, it was generally found that analyses based on classical fracture mechanics did not adequately model the damage effects and did not provide a satisfactory degree of predictive capability [3, 4]. Damage stability is influenced by both local factors, such as constituent material properties at the location of damage, and global factors, such as structural geometry and boundary conditions. The interaction of these factors is further complicated by the numerous possibilities of material combinations, composite geometry, fiber orientations, and loading conditions. Predicting crack propagation in a PMC structure is further complicated by the existence of a multiplicity of design options arising from the availability of numerous choices of fibers, fiber coatings, fiber orientation patterns, matrix materials, constituent material combinations, and hybridizations [15]. The resulting large array of design parameters, presents a logistical problem that complicates and prolongs design optimization and certification processes and adds significantly to the cost of composite parts. Consequently, this complexity makes it difficult to identify and isolate all significant parameters affecting damage propagation and stability without a modelbased computer code capable of incorporating all factors pertinent to determining structural fracture progression, fundamental to evaluating the durability and life of composite structures [5–13].

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6.3.4 Calibration of Composite Constitutive Properties Since it is known that the root cause of failures in composites initiates at the fiber, matrix, and interphase scales, it is critical for successful analysis to ensure that these properties are calibrated to their corresponding physical properties. An approach to accomplishing that is discussed here. Material Characterization Analysis (MCA) predicts the composite lamina, and laminate properties under manufacturing and environmental conditions (e.g. humid environment). MCA is useful during the early phases of concept/product development in evaluating the impact of changes in volume fraction involved in deciding on an appropriate fabrication approval or assessing environmental effect, degradation of material properties to environmental (moisture, thermal), manufacturing (defects, residual strains), etc. The process first requires calibration of constituent material properties using five simple test data obtained from longitudinal tension and compression, transverse tension and compression, and shear tests. Thereafter, the recommended failure criteria are used to predict the strength (design envelop) for higher-order coupon level tests such as open-hole compression and verified against test data. Once confidence in predictions is built using the proposed approach, the predicted failure design envelope could be used in adjusting other failure criteria for improved predictions, verification, and certification of composite parts. To calculate the correct micro-mechanical constituent properties (fiber, matrix, and interphase properties), i.e. to “DETERMINE THE ROOT CAUSE OF FAILURE” properties, for the multi-scale analysis, a three-step process is used: 1. Calibration step is performed using physical un-notched material (lamina) tests. The calibration is achieved through an optimization step, where the stress–strain curves of the ASTM standard tests and the analytic stress–strain curves are compared and the error between these curves is minimized. The calibrated constituent data is then used in the next step, test verification. 2. Verification step is performed, using physical notched laminate and un-notched laminate material tests. Test results are compared to analytic predictions based on the calibrated constituent properties from step 1. 3. Probabilistic analysis step is performed to account for test data scatter and match the probability distributions and standard deviations that are representative for the constituent and manufacturing properties. The probability distribution is then used to obtain the A-basis and B-basis allowable values. Fiber and matrix properties calibration is performed using a reverse-optimization process to determine the matrix-strength/stiffness (stress–strain curves), and the fiber strength/stiffness to match the un-notched (longitudinal/transverse tensile, longitudinal/transverse compression, and shear) composite coupon tests at the lamina and laminate levels (Fig. 6.20a). The simulation uses: Material Characterization Analysis (MCA), and progressive failure methodology (PFA) to verify the test data. VT predicts the stiffness, strength, Poisson’s ratio, and strength of the lamina and

6 Computational Composite Material Qualification and Structural Certification Start Manufacturer fiber and matrix stiffness and strength

GENOA Module Material Constituent Analyzer (MCA) Not Acceptable

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Sensitivity Data

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ASTM D638, D3039

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ASTM Coupon Test Data Lamina or Laminate • Longitudinal Tension • Longitudinal Compression • Transverse Tension • Transverse Compression • Shear

ASTM D3518

a) Calibration

b) Validation Recommendation

Fig. 6.20 ASTM-based static coupon verification process

laminates. Material Uncertainty Analysis (MUA), which is subsequently discussed, is performed to identify the effects of composite fiber/matrix material property and manufacturing uncertainties on laminate response. In order to obtain the in-plane material properties five relatively simple physical coupon tests are required (Fig. 6.20a). These tests include tension and compression tests in the weft and warp direction and an in-plane shear test. The types of tests needed for the calibration processes are not limited to certain ASTM or other standards. The main issue is to create a good virtual counterpart of the physical tests. In other words the material buildup, boundary conditions, loading and test conditions should be included in the model as accurately as possible. For the calibration process stress–strain information is used as comparison parameter between the virtual model and the physical test. To get good results complete stress–strain test data is desired. This means that the stress–strain (load-displacement) has to be recorded and documented through the entire test until the test structure has collapsed. It is recommended to do this for all the verification tests.

6.3.5 Composite Material Validation Static strength and stiffness predictions are validated based on (Fig. 6.20b) ASTM standard test data to qualify a building-block validation strategy. Example of Calibration of lamina properties of IM7/PETI-5 at –423◦ F, –320◦ F, ◦ 75 F, 350◦ F, 400◦ F and 450◦ Fwas conducted using GENOA-PMC module, which is a reverse-calculation process of lamina properties by selecting the correct constituent (fiber/matrix) properties to match the test lamina or ply properties [41–43]. The objective of calibration is to obtain a reliable constituent (fiber/matrix) databank as the base for laminate and structural analysis. The calibration process included matching the test observed lamina mechanical properties: (1) longitudinal modulus, (2) transverse modulus, (3) shear modulus, (4) longitudinal tensile strength, (5) longitudinal compressive strength, (6) transverse tensile strength, (7) transverse compressive strength, (8) shear strength, and (9) coefficients of thermal expansion (Figs. 6.21 and 6.22).

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Calibration Test

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Longitudinal Modulus (msi)

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b) Transverse modulus

Shear Modulus (msi)

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a) Longitudinal modulus 320

Calibration Test

300

280

260

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220 -423F

-320F

75F

350F

400F

450F

Temperature

c) Shear modulus

c) Longitudinal tensile strength

Fig. 6.21 Calibration of IM7/PETI-5 lamina properties – Characterization of material and lamina mechanical properties as a function of temperature (–423◦ F, –320◦ F, 75◦ F, 350◦ F, 400◦ F, 450◦ F)

Validation of Calibrated Mechanical properties of a 13-ply quasi-orthotropic laminate used in NASA’s Reusable Launch Vehicle (RLV) program with lay-up, [45/903/-45/03/-45/903/45] (RLV(0)) of IM7/PETI-5. The fiber volume fraction in each ply was assumed to be 60%. The coupon laminate analyses include unidirectional un-notched tension, and notched tension specimens to simulate the effect of flaws on the laminate strength. The simulation was compared with tests that were conducted by NASA-Langley [43]. Tensile strengths prediction of un-notched RLV(0) at –423◦ F, –320◦ F, 75◦ F, 350◦ F, 400◦ F, and 450◦ F were compared with NASA observed tests (Fig. 6.23). Simulation revealed that: 1. Stiffness of RLV(0) in the simulation is consistent with test data, except at 350◦ F. At 350◦ F, neither the lamina longitudinal modulus nor lamina transverse modulus is higher than those at other temperatures. Therefore, simulated stiffness of RLV(0) at 350◦ F does not show higher magnitudes than those at other temperatures like the test data does. 2. The laminate strengths of both RLV(0) at high temperature (350◦ F–450◦ F) are lower than those at low temperature in the simulation results. Compared to the test results, the deviation of the simulation results is 10%–15%, except at 400F and 350F at which the deviations are 20% and 24%, respectively. 3. Sensitivity analysis shows the upper limits of the tensile strengths of RLV(0) reach the test results, with a 5% variation of constituent properties (moduli, strength and CTE) as standard deviation for normal distribution.

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d) Shear Strength

e) Coefficients of thermal expansion Fig. 6.22 Calibration of lamina properties – Characterization of material and lamina mechanical properties of IM7/PETI-5 vs. test data at six temperatures

6.3.6 Material Uncertainty Analyzer (MUA) MUA predicts the percent contribution of composite mechanical properties at the constituent, lamina, and laminate levels to potential failure mechanisms such as delamination. MUA is useful during the early phases of concept/product development in evaluating the impact of changes in fiber architecture, volume fraction, and material mechanical properties on potential fracture damage.

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b) Tensilestiffness

Fig. 6.23 Comparison of simulating and testing of un-notched laminate tensile strength properties as a function of temperature (–423◦ F, –320◦ F, 75◦ F, 350◦ F, 400◦ F, 450◦ F); Laminate configuration: [45/903/-45/03/-45/903/45], Material: IM7/PETI-5

Sensitivity analysis of the lamina properties of IM7/PETI-5 to constituent properties was performed. The contribution of constituent properties to lamina properties of polymer matrix composites can be usually described as follows: 1. Lamina longitudinal properties are mainly determined by the fiber properties 2. Transverse properties of the lamina depend on the matrix properties.

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

S12 S22C S22T S11C S11T Ef22

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Em

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Sensitivity

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If the fiber properties are not sensitive to temperature change, the dependency of the lamina properties on constituent properties still holds with varying temperatures. Therefore, the sensitivity of IM7/PETI-5 is similar at different temperatures. As a

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Fig. 6.24 Sensitivity of IM7/PETI-5 lamina properties to the constituent properties at room temperature

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representative for those at various temperatures, the sensitivity of IM7/PETI-5 at room temperature is shown in Figs. 6.24 and 6.25. The sensitivity was obtained by GENOA-MUA with a 5% variation in each constituent property used as the standard deviation for a normal distribution. In Fig. 6.24, the definition of each parameter is: Ef22: fiber transverse modulus, SfT: fiber tensile strength, SfC: fiber compressive strength, Em: matrix modulus, SmT: matrix tensile strength, SmC: matrix compressive strength, SmS: matrix shear strength, Vf: fiber volume fraction, S11T: lamina longitudinal tensile strength, S11C: lamina longitudinal compressive strength, S22T: lamina transverse tensile strength, S22C: lamina transverse compressive strength, S12: lamina shear strength. 140

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Fig. 6.25 Variation of the uniaxial tensile strength of RLV(0), with 5% of variation of constituent properties (moduli, strength and CTE) as a function of temperature (–423◦ F, –320◦ F, 75◦ F, 350◦ F, 400◦ F, 450◦ F) – Sensitivity analysis

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400

500

Temperature (F)

6.4 Establish A- and B-Basis Allowables Accurate establishment of the A-basis and B-basis strength values is critical to reducing risk in structural design of composite aircraft structures. The calculation of polymer matrix composite allowables for aerospace applications is governed by FAA and Military Handbook 17-E standards and rules [44–48]. The process is costly and time consuming as large numbers of coupon tests are inevitable. This section describes an innovative computational approach to predict composite material allowables with minimum testing. The methodology combines multi-scale composite modeling with progressive failure and probabilistic analysis and minimum test data to determine A- and B-basis values. The scatter in material strength is determined by iterating on coefficient of variations of random variables from single or multiple sources of uncertainties (i.e. fundamental material properties and fabrication variables). The iterative process is necessary to replicate scatter in the strength value obtained from the test of one coupon of each material batch. The methodology is applicable to notched and un-notched coupons and structures and has the potential of reducing the coupon count for testing by over 60%. The A-basis allowable strength is usually used for single-point catastrophic failure with no chance for load redistribution. For the A-basis, at least 99% of the population of material values is expected to equal or exceed this tolerance bound with 95% confidence. As for B-basis, 90% of the population of material values

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is expected to equal or exceed that strength value. The B-basis strength is generally associated with redundant load path with load redistribution capability. In most cases, strength allowables are defined using ‘design point strategy’, meaning that allowables will be obtained for certain predefined lay-up configurations. After which only these configurations are allowed for use in the design. The generation of material allowables is governed by procedures specified in MIL-HDBK-17E (Military Handbook for Polymer Matrix Composites) and FAA CFR 14 (Aeronautics and Space). The guidelines set by FAA for generation of allowables of polymer matrix composites for aircraft applications [44–46] addresses the following: 1. 2. 3. 4. 5.

Specimen manufacturing Required environmental conditions Non-ambient testing to quantify the effect of temperature Specimen geometry, testing methods, and laminates to be tested Recommended physical and chemical property test for cured and uncured lamina to qualify materials 6. Statistical analysis The specimen manufacturing requirements include recommendations for number of specimens, number of panels, panel sizes, panel manufacturing, tabs, specimen machining, specimen selection, specimen naming, specimen strain gage bonding, and specimen dimensioning and inspection. The required number of specimens is 55 coupons for A-basis (6 batches with two panels in each batch and 6 coupons test per panel) referred to as Robust Sampling. For the B-basis, although Robust Sampling is recommended, Reduced Sampling of 18 tests is accepted (3 batches of two panels per batch and 3 coupons per panel). The FAA recommends testing under the following conditions to capture the environmental effects: 1. 2. 3. 4.

Cold temperature dry (–65F with as-fabricated moisture content) Room temperature dry Elevated temperature dry (180◦ F with as-fabricated moisture content) Elevated temperature wet (180◦ F with an equilibrium moisture weight gain in a 85% relative humidity environment)

6.4.1 Combining Limited Test Data with Progressive Failure and Probabilistic Analysis The computational process that predicts A-basis and B-basis strength values using a combination of test data and PFA [2] coupled with probabilistic analysis [3] is described here. The GENOA software [4] is used for progressive failure and probabilistic analysis. The steps involved in this process, depicted in Fig. 6.26, are as follows:

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Fig. 6.26 Process of generating polymer matrix composites A-basis and B-basis strength values using virtual simulation with limited test data

1. Collect the strength from the test of one coupon from each batch where a minimum of three batches are recommended. 2. Perform PFA to reproduce the coupon average failure strength 2.1 Calibrate/adjust material constituent properties as needed to replicate the average of the test. The process for calibrating the constituent properties is shown in Fig. 6.14. To characterize the fiber and matrix properties, the strength and stiffness needs to be reproduced for 5 ASTM tests: longitudinal tensile/compressive, transverse tensile/compressive and in-plane shear. The objective here is to derive effective fiber/matrix properties and fabrication variables that reproduce mean failure stress for the 5 tests. Generally, some minor adjustments are needed in the constituents (±5–10% variation from vendor-declared properties). This process enables evaluating failure in the composite at the inception source where it naturally occurs in the fiber/matrix and interface [5]. 3. Select random variables and associated coefficient of variations (COV) and distribution types. 3.1 Material properties random variables: fiber longitudinal, transverse and shear stiffness, fiber tensile and compressive strength, matrix stiffness and matrix tensile/compressive and shear strength, and thermal properties of fiber and matrix (for temperatures other than ambient temperatures). 3.2 Fabrication random variables: fiber volume ratio, void volume ratio, ply angle, use temperature, and cure temperature. The relationship of the

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random variables to the failure of the composite is governed by the PFA described in the next section. 3.3 Perform probabilistic PFA to determine the cumulative distribution function of the strength. 4. Compare the cumulative distribution function generated by the analysis to the one fitted to a normal distribution from the limited test of one coupon from each batch. 5. Iterate on the COV until the predicted cumulative distribution matches that of the test. 6. Retrieve the A-basis and B-basis at 1/100 and 1/10 probabilities, respectively, from the cumulative distribution function generated by the analysis.

6.4.2 Examples of Allowable Generation for Unnotched and Notched Composite Specimens 6.4.2.1 Prediction of A- and B-Basis Values for a Fiber-Resin Fabric The composite system is a fiber resin-weave (graphite epoxy material) with the following test conditions: temperature of –65º ± 5ºF with moisture content as fabricated. The test data is published by FAA and can be found in reference [44]. Table 6.3 lists the strength values obtained from the test. Test results were provided for three coupons per panel (with two panels per batch and three batches for the material). A total of 18 coupons were tested making it a reduced sampling test plan. Table 6.3 Compressive strength test data (Cold temperature dry) 1 ksi = 6.8947 MPa Batch

Panel

Strength (ksi)

Batch

Panel

Strength (ksi)

1 1 1 1 1 1 2 2 2

1 1 1 2 2 2 3 3 3

103.26 104.281 102.65 111.336 102.967 108.615 113.21 111.15 102.32

2 2 2 3 3 3 3 3 3

4 4 4 5 5 5 6 6 6

106.58 102.36 101.65 112.15 114.12 104.56 108.29 107.63 100.96

The approach for generating allowables was applied to the fabric coupons of graphite fiber and epoxy matrix. The test failure strength from the first coupon tested in each batch was considered in the validation of this methodology for a total of only three coupon tests used. The test values used to guide the computation are marked in red in Table 6.3. As a first step, the authors applied PFA to reproduce the average failure strength for the given coupon (based on three tests only). According to FAA report [44], the

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Table 6.4 List of property and fabrication random variables (1psi = 0.0068947 MPa) Random variable description

Mean value

Distribution type

EF11:Fiber Longitudinal Modulus (psi) SF11C:Fiber Compressive Strength (psi) EM:Matrix Modulus(psi) SMC:Matrix Compressive Strength (psi) FVR:Fiber Volume Ratio VVR:Void Volume Ratio

30,000,000 273,000 606,000 49,000 0.53 0.02

Normal Normal Normal Normal Normal Normal

coefficient of variation is usually between 4% and 10% for the failure strength of polymer composites. To generate A- and B-basis allowables using computation, as discussed earlier, it is imperative to select random variables thought to possess some scatter naturally. Such random variables are tabulated in Table 6.4 and they are: fiber longitudinal modulus, fiber compressive strength, matrix modulus, matrix compressive strength, fiber volume ratio, and void volume ratio. Probabilistic analysis was then performed using various coefficients of variations for these random variables. As presented in Fig. 6.27, the three test data are fitted to a normal distribution then the cumulative distribution function (CDF) for the strength of the coupon from the analysis is compared to that of the test. Examining the CDFs from the simulation, one concludes that the best fit with the test is the one obtained with a coefficient of variation of 6% for all the random variables. If information from test is unavailable, it is advised to use the same approach but impose a coefficient of variation of 10% on all random variables. This will provide a conservative prediction. The allowables are retrieved from the CDF plot at 1/100 probability for A-basis and 1/10 probability for B-basis.

Fig. 6.27 Predicted A- and B-basis allowables compared to those from reduced sampling test (best solution obtained with 6% COV on random variables)

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Running the 18 test points from the test through the equation that uses the tolerance factors KA and KB as specified in the Military Handbook-17, yields Aand B-basis strength values of 91.8 ksi (633.71 MPa) and 97.87 ksi (674.78), respectively. The allowables predicted analytically compare very well with the ones obtained using 18 test points (the error for A-basis prediction is 0.2%, while the error for B-basis prediction is 1.3%). Setting the coefficients of variations to 6% for all random variables is a way to represent all the possible uncertainties into some equivalent effect. Once the allowables are successfully predicted for a few selected configurations covering the over-all design envelope, the designer and analyst can apply the analytical approach to determine A- and B-basis values for new configurations without resorting to test every time. By then, the engineer would have identified effective values for the coefficients of variations that capture the scatter in the material behavior. 6.4.2.2 Prediction of A- and B-Basis Values for an Open Hole Loaded in Tension The authors applied the same methodology to predict the allowables of a composite open hole loaded in tension [48]. The specimen is fabricated from T300 carbon PPS thermoplastic composite material in a woven configuration with [±45/0/90]3S lay-up. Figure 6.28 shows the dimensions of the tested open-hole specimen loaded in tension. As a first step, PFA was used to calibrate the constituent properties of the composite material using un-notched coupons (longitudinal tensile/compressive, transverse tensile/compressive and in-plane shear). Once PFA reproduced the five tests using unique set of constituent fiber and matrix properties, the authors analyzed the open hole using the calibrated properties. The mean failure load predicted by PFA was 0.2% higher than that of the mean value from the test. The damage initiation caused by matrix cracking around the edge of the hole as predicted by PFA is presented in Fig. 6.29a. Figure 6.29b shows the fracture initiation prior to structural failure started around the hole edge due to fiber failure.

Fig. 6.28 Dimensions of the open-hole tension specimen

Table 6.5 summarizes the normalized failure load from the test for the openhole tension case. The range of the failure load varied from 0.9218 to 1.051 with a standard deviation of 0.028. Table 6.6 lists the assumed random variables for the

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Fig. 6.29 Open-hole failure process – Tension case Table 6.5 Normalized tensile failure load for open hole composite coupon Test #

Failure load

Test #

Failure load

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.9218 0.9507 0.9593 0.9690 0.9709 0.9728 0.9853 0.9853 0.9892 0.9950 0.9950 0.9960 0.9979 0.9979 0.9998

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.0008 1.0027 1.0027 1.0065 1.0065 1.0065 1.0143 1.0143 1.0268 1.0287 1.0335 1.0335 1.0393 1.0470 1.0510

allowables prediction, and they are: fiber stiffness and strength, matrix stiffness and strength, fiber volume ratio, and void volume ratio. For this particular case, the data provided to the authors were not grouped per batch or panel therefore all of the probabilistic distribution of the test data was compared to that from the simulation. PFA was used in conjunction with probabilistic analysis to replicate the scatter in the failure strength of the open-hole coupon under tensile loading condition. The random variables were selectively perturbed by the analysis engine to populate enough data to predict the cumulative distribution of the failure stress. As indicated in Fig. 6.30, the scatter from the test data did not agree with that from the simulation when a coefficient of 5% was applied uniformly to all random variables. However, reducing the coefficient of variation to 1% for the fiber and matrix stiffness and strength and the fabrication variables yielded excellent agreement with test

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Coefficient of variation

Fiber longitudinal modulus Fiber longitudinal tensile strength Matrix modulus Matrix tensile strength Matrix shear strength Fiber volume fraction Void volume fraction

5% 5% 5% 5% 5% 5% 5%

Fig. 6.30 Simulation of test data by PFA and probabilistic analysis (coefficient of variation of 5%)

(as shown in Fig. 6.31). Processing the 30 test data points through the allowables model per Mil Handbook-17 standard resulted in an A-basis value of 0.92 and a B-basis value of 0.9486 with respect to a mean normalized strength of 1.0. The solution from the coupled PFA and probabilistic analysis resulted in A-basis value of 0.9104 and a B-basis value of 0.959 retrieved from 1/100 and 1/10 probabilities, respectively. The maximum error from the prediction with respect to test is 1.1% as shown in Table 6.7. It appears then that a rule of thumb in selecting a viable coefficient of variation to match the calculated CDF to the limited test data CDF is to select a coefficient of variation such that the limited test data CDF and calculated CDF coalesce well about the mean. Then the PFA should provide a good reproduction of the wings of the calculated CDF curve. Indeed, the results presented for the two cases (fabric under compression and open hole under tension) demonstrate that computation can be used to minimize the number of tests needed to certify and qualify aerospace materials.

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Fig. 6.31 Allowables as predicted by the analytical simulation using combined PFA and probabilistic analysis (coefficient of variation of 1%)

Table 6.7 Open-hole tension comparison of A- and B-basis values – test and analysis

A-Basis B-Basis

Test (Mil-HDBK)

Analysis

% Error

0.92 0.9486

0.9104 0.959

1.04% –1.10%

6.5 Certification by Analysis Example The utilization of advanced composite material systems in general aviation and transport aircraft is rapidly growing. This is placing a growing demand on the certification process for composite components, parts, and repairs to meet regulatory requirements and statutes. Presently, certification requires significant testing, which is both expensive and time consuming. Therefore, it is very desirable to establish and demonstrate the technical capability to perform “certification by analysis” (CBA) to reduce test requirements, perform fatigue analysis, and predict residual strength and remaining life after damage is incurred. An example of “certification by analysis” was demonstrated for typical general aviation advanced composite fuselage components. Advanced composite honeycomb sandwich fuselage panel test specimens were built by Adam Aircraft for the “Full-Scale Damage Tolerance of Composite Sandwich Structures” tests to be performed at the “Full-Scale Aircraft Structural Test Evaluation & Research” (FASTER) fixture at the FAA’s William J. Hughes Technical Center in Atlantic City, NJ (Fig. 6.32) [49–52]. The panels were representative of a general aviation/business jet upper fuselage panel (Fig. 6.32a). Geometry and internal panel design are shown in Figs. 6.32b and 6.32c. The sandwich facesheets were made of TORAY COMPOSITES T700SC-12 K-50C/#2510 PWCF plain weave fabric laminate with a (45/0/45) lay-up. The honeycomb core was 0.75inches thick PlascoreNomexPN2-3/16-3.0 honeycomb (0.75 in thickness). The fabricated panel and the FASTER test rig are shown in Figs. 6.32d and 6.32e.

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a) Fuselage panel geometry

b) Fuselage panel geometry

d) Fabricated fuselage panel

c) Fuselage panel details

e) FASTER test rig

Fig. 6.32 Advanced composite honeycomb sandwich fuselage panel test specimens tested at the “Full-Scale Aircraft Structural Test Evaluation & Research” (FASTER) fixture in the FAA’s William J. Hughes Technical Center

A systematic GENOA-based “building-block” CBA process was used for the fuselage panels (Fig. 6.33). First, the advanced composite fiber/matrix constituent stiffness and strength properties were calibrated from standard ASTM unidirectional coupon tests. Next, the calibrated constituent properties were verified using nohole and open-hole coupon tests for different laminate lay-ups. Then the calibrated fiber/matrix properties were verified with uniaxial tension honeycomb element tests. These represented no-damage and damaged states (different size through slots in the sandwich). Finally, the damage initiation, damage propagation, fracture initiation, fracture propagation and final failure in undamaged and damaged fuselage panels (with different size through holes and slots) subjected to internal pressurization, inplane longitudinal tension and combined tension/pressure loadings were predicted with a finite element-based PFA program (GENOA) and the calibrated constituent fiber/matrix properties.

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Calibrate Constituent Fiber/Matrix Properties

Laminate Tests

Laminate Verification with GENOA

Honeycomb Tensile Element Tests

Fuselage Panel Tests

Tensile Element Verification with GENOA

Fuselage Panel Verification with GENOA

Fig. 6.33 A virtual testing building-block “certification by analysis process for general aviation advanced composite fuselage panels

Basic material data for the material systems used in the honeycomb panels were obtained through the National Institute for Aviation Research (NIAR) [53]. These included both coupon test data for unidirectional tape, plain woven fabric, and the honeycomb core. In addition, test data for flat sandwich test specimens were obtained for further correlation of the material models. This material information was used for the Material Characterization Analysis (MCA) and the Material Constituent Optimization (MCO) performed with GENOA. Calibration of the fiber/matrix constituent material properties was excellent. Predicted material stiffnesses properties and strengths were within 10% of test values (Fig. 6.34). In the building-block CBA process, validity of the fiber/matrix constituent calibration needs to be verified before proceeding to the next level of structural complexity. This was accomplished by comparing test and VT-predicted cross-ply laminate coupon responses. Test data was obtained from NIAR’s AGATE database [53]. Strength predictions were generally within 10% of the published AGATE test values (Fig. 6.35). The (40/20/40) fabric no-hole compression specimen shows a large discrepancy (32%) between test and predicted values. Considering that the great majority of predictions are in good agreement with test results, it is desirable to determine the reasons for the outstanding discrepancy. Figure 6.36 shows a comparison on AGATE

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Strengths 350

18 16

300

Fabric - Test Fabric - GENOA Tape - Test Tape - GENOA

14

250 12 10

Fabric - Test

200

Fabric - GENOA

8

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Tape - Test

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Tape - GENOA

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4

50

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0

0 E11(msi)

E22(msi)

G12(msi)

m12

S11T (ksi)

S11C (ksi)

S22T (ksi)

S22C (ksi)

S12 (ksi)

Fig. 6.34 Verification of virtual test fiber/matrix constituent properties calibration Lamina predictions vs. test results 50% 45% 40% 35% 30%

% Error - Predicted vs.Test

25%

Physical Test Anomaly Open–Hole Strength No–Hole Modulus No–Hole Strength

20% 15% 10% 5% 0% –5% –10% –15% –20% –25% –30% –35% –40% –45% –50%

Fig. 6.35 Virtual testing predicted vs. test – Tape and fabric laminates

test data from Toray with the ViVT predictions for no-hole compression for different percentages of ±45◦ layers. VT and Agate results agree except that Agate tests show that 20% and 50% ±45◦ laminates have essentially the same strength while VT predictions show a complete monotonic relationship between compressive strength and percentage of ±45◦ layers. This suggests that there is an anomaly in the reported Agate test data [53]. Next, five sets of honeycomb sandwich tensile elements, with and without through-the-sandwich holes were evaluated. Each had the same fabric laminate in the facesheets and a core height of 0.75 inches. Figure 6.37a shows the finite element model of the tensile specimen with a 2-inch long transverse slot. Damage initiation is depicted in Fig. 6.37b while final failure in this specimen is shown in Fig. 6.37c.VT failure predictions matched the test results closely, as seen in Fig. 6.38 and Table 6.6.

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Longitudinal Compressive Strength

120 Test GENOA

100

80

60

40

TEST DATA TAKEN FROM: TORAY COMPOSITES (AMERICA), INC. AGATE LAMINATE MATERIAL QUALIFICATION REPORT NO: TCQAL-T-1026 DATE : December 4, 2002 Page 50

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0 0

10

20

30

40 50 Percent ± 45 layers

60

70

80

90

Fig. 6.36 Virtual testing predicted vs. agate test data compressive strength – Unnotched fabric laminates

After the initial steps in the “building-block” process, calibration, and verification were completed, GENOA progressive failure analyses (PFA) were performed for representative advanced composite honeycomb curved fuselage panels. Replicates consisted of an undamaged panel and then for panels with a circular hole, a radial slot, an oblique slot, and a longitudinal slot. The goal of the physical tests and the VT analyses was to simulate a fuselage panel under internal pressure, longitudinal and hoop loading. VT results were compared with the physical test results prior to failure; computational strain surveys were compared with test strain gage data up to the limit test loads (no failure). VT evaluations for damage initiation, damage propagation, fracture initiation, fracture propagation, and final failure of the undamaged and the various damaged curved panels were obtained. Three different types of curved panels were simulated with GENOA 1. Baseline panel with no holes, 2. Panel with a central circular hole, and 3. Panels with different size and orientation slots. Table 6.8 shows the GENOA predicted failure loads and pressures for the different NIAR curved honeycomb sandwich panels tested. All GENOA results were obtained using fiber/matrix property inputs. Test failure loads for the panel

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Fig. 6.37 FEM model of sandwich specimen with 2” notch

a) Finite element model

b) Damage Initiation

c) Final Failure

20% 15%

% Difference with Test

10%

7.36%

5% 0% Unnotched –5%

With 1” hole –0.39%

With 2” hole –0.21%

With 1” notch

With 2” notch

–4.49%

–10% –12.33%

–15% –20% Honeycomb Specimen

Fig. 6.38 Virtual test predicted vs. NIAR test data honeycomb tensile specimens

580 lb/in

12.50 psi 747 lb/in NA 8.00 psi 970 lb/in 780 lb/in

8.00 psi 297 lb/in 594 lb/in 8.00 psi 496 lb/in 496 lb/in

Oblique slot Combined loading Oblique slot Longitudinal loading Circumferential slot Combined loading Circumferential slot Longitudinal loading

10.20 psi

5.00 psi

Longitudinal slot

NA

NA NA NA NA NA NA 21.40 psi

TEST

8.00 psi 1231 lb/in 1231 lb/in

14.22 psi 1198 lb/in 1588 lb/in

14.14 psi

1969 lb/in

80.40 psi 5921 lb/in 40.40 psi 3423 lb/in 13.63 psi 803 lb/in 25.48 psi

GENOA

Final fracture

∗Values at fracture initiation are the damage values in the NIAR supplieds spreadsheet.

Slotted panels

Pressure only

Longitudinal loading only

Circular hole panels

33.5. psi 2423 lb/in 23.45 psi 1896 lb/in 5.29 psi 161 lb/in 11.82 psi

Combined pressure and longitudinal loading Pressure only Longitudinal loading Combined loading

Baseline panels

GENOA

Loading condition

Test panel

Damage initiation∗

8.00 psi 1220 lb/in NA

18.30 psi 1153 lb/in NA

13.30 psi

NA

NA NA NA NA NA NA NA

TEST

Preload 8 psi Then apply longltudinal loading to failure Preload 1 psi. Then apply longitudinal loading to failure. Test stopped at 1220 lb/in Machine constraint on deformation

No Data

Unloaded after 18.8 psi and 1160 lb/in. No damage or failure Damage Initiation at 21.4 psi. could not load further Unloaded after 1250 lb/in. No damage or failure Preload longitudinally to 100 lb/in. Then Increase pressure to failure Hoop loading/longitudinal loading=1.2

Elastic loading only

WSU/NIAR Test results

Table 6.8 Predicted failure loads and pressures of NIAR curved panels.

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Test (lb/in)

Virtual test (lb/in)

Un-notched With 1” hole With 2” hole With 1” notch With 2” notch

3,386 1,818 1,435 1,784 1,195

3,234 1,811 1,432 1,564 1,283

Comparison of GENOA FEA and FASTER Test - Strain Gage 25 Baseline Curved Honeycomb NIAR Panel Longitudinal Loading

600

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SG-25-GENOA FEA SG-25-Longitudinal Test

200

100

0 0

200

400

600

800 1000 1200 1400 1600 Longitudinal Strain (microstrain)

1800

2000

2200

Fig. 6.39 Baseline panel – longitudinal loading comparison of virtual test prediction vs. test data strain gage at center of panel – convex side

components were not available. However, failure predictions, locations and load levels can be used for risk mitigation. VT predictions appeared reasonable, except for the curved honeycomb panel with central circular hole loaded with longitudinal edge loads. Examination of the FEM indicated that the applied edge load distributions are questionable. Comparisons of VT predicted strains to test data were performed for the baseline panel and the panel with a radial slot, both under longitudinal loading. Correlation was very good to excellent, indicating that the VT models accurately represent test behavior. Figure 6.39 shows a comparison of the predicted and tested strain gage readings for the baseline panel at the centerpoint of the convex side. Figure 6.40

Longitudinal Load (lbf/in)

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Comparison of GENOA FEA and FASTER Test - Strain Gage 5 Baseline Curved Honeycomb NIAR Panel Longitudinal Loading

2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

Test SG-5-GENOA FEA

0

500

1000

1500

2000 2500 3000 3500 4000 4500 Longitudinal Strain (microstrain)

5000

5500

6000

Fig. 6.40 Curved panel with radial slot – longitudinal loading comparison of virtual test prediction vs. test data

shows a comparison of the predicted and tested strain gage readings for the panel with a radial slot. The strain gage was located on the convex panel side along the centerline, 24 inches from the slot. Both show excellent correlation between test and simulation. Damage initiation, propagation, fracture initiation, fracture propagation, and final failure loads and modes were predicted with VT for all the fuselage panels and loading conditions tested. Figures 6.41 and 6.42 show the damage initiation and propagation for the fuselage panel with a radial slot. Figure 6.43 shows the damage/fracture distributions on a layer-by-layer aspect.

Fig. 6.41 Panel with radial slot longitudinal load only damage initiation: running load:496 lb/in

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Fig. 6.42 Panel with radial slot longitudinal load only

Fig. 6.43 Panel with radial slot – ply damage longitudinal load only

Next the combined loading condition was considered for the circumferential slotted panel. The combined loading condition was applied in the manner shown in Figure 6.44a. First, the pressure was increased in five increments to 8 psi without applying the longitudinal loading. Then the longitudinal loading was incremented in ten steps to the maximum test load while the internal pressure was maintained at

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Curved NIAR Sandwich Panel Center Radial Slot Applied Loading Time History

90000 80000 70000 60000

Normal Pressure (x 10,000)

50000 Loading

Longitudinal Load

40000 30000 20000 10000 0 0

300

600

900

1200

1500

1800

–10000 Test Time in seconds

a) Pressure and longitudinal loading history Curved NIAR Sandwich Panel Circumferential Slot Strain Gage #5 and #17 Time History

6000 5000 Panel Longitudinal Strain in Micro-in./in.

Hand Check Genoa Prediction - Longitudinal Strains in Finite Element 12322

4000

Longitudinal Strain Gage SG5 Longitudinal Strain Gage SG17

3000 2000 1000 0 0

300

600

900

1200

1500

–1000 –2000 –3000 Test Time in Seconds

b) Strain Gage SG17 - Predicted GENOA vs. test data Fig. 6.44 Curved panel with circumferential slot – Combined longitudinal and pressure loading

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8 psi. Figure 6.44b compares the test results for strain gage SG-17 with the GENOA predictions and with a hand stress check. Test strain gage readings (SG-5 and SG-17) revealed an increasing compressive longitudinal strain during pressurization induced by Poisson’s effect. The longitudinal strain transitioned to a tensile strain when the longitudinal loading was applied. GENOA analysis, performed with the same sequential pressure and longitudinal loading, confirmed this. The GENOA predicted strains were verified with a typical hand stress check. All three curves show a good match.

6.6 Summary This chapter has reviewed the significant role of progressive failure analysis (PFA) and virtual testing (VT) in the FAA certification process. The role of PFA and VT in each of the FAA categories of damage and defect considerations in primary aircraft structure was discussed in detail. Special emphasis throughout the chapter was given to the FAA certification building-block process which involves multiple coupon tests, sub-component tests, and full-scale structural tests to verify design failure loads. A unique capability appearing in this chapter is the calculation of A-basis and B-basis allowables. Using probabilistic methods, this chapter demonstrated how PFA can be effectively used to calculate A- and B-basis allowables even when little or even no test data is available. These calculations are critical to reduce risk in structural design of composite aircraft structures and represent significant cost savings because conventional generation of allowables is costly and time consuming as large numbers of coupon tests are inevitable. Consistent with the FAA and MIL-HDBK-17E standards and rules for generating A-basis and B-basis allowables for polymer matrix composites, the following conclusions can be made from the work and research made to this point:

1. Analysis combined with minimum testing can be used to generate material allowables. 2. Analysis can be relied on to generate allowables for configurations (lay-ups) that are not included in the test plan as long as the simulation results are verified for few tests that represent the over-all design envelope. 3. The scatter that naturally occurs in the material can be reproduced analytically by combining progressive failure and probabilistic analysis. 4. Sensitivity analysis can be used to reduce scatter in material response as it identifies influential random variables. 5. Physics-based failure criteria accounting for matrix cracking, delamination, and fiber failure are essential to accurately predicting the failure of composite structures.

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Since aerospace companies spend an estimated $350 million per year on experimental tests alone, use of reliable PFA and VT methods to reduce testing can produce significant cost savings. It is estimated that test costs can be reduced by up to 50% using these methods. It is envisioned that all aerospace structures will be designed and maintained with the help of VT in the future. When VT is fully adopted into the design process, more robust, congruent, and economical composite air vehicle structures result.

References 1. Design Considerations for Minimizing Hazards Caused by Uncontained Turbine Engine and Auxiliary Power Unit Rotor and Fan Blade Failures, Advisory Circular 20–128, Federal Aviation Administration, Washington, DC, 1988 2. Z. Hashin, “Analysis of Stiffness Reduction of Cracked Cross-Ply Laminates,” Eng. Fract. Mech., Vol. 25, 1986, pp. 771–778 3. N. Laws and G.J. Dvorak, “Progressive Transverse Cracking in Composite Laminates,” J. Compos. Mater., Vol. 22, 1988, pp. 900–915 4. Y.M. Han and H.T. Hahn, “Ply Cracking and Property Degradations of Symmetric Balanced Laminates Under General In-plane Loading,” Compos. Sci. Technol., Vol. 35, 1989, pp. 337–397 5. R.J. Nuismer and S.C. Tan, “Constitutive Relations of a Cracked Composite Lamina,” J. Compos. Mater., Vol. 22, 1988, pp. 306–321 6. D.H. Allen, C.E. Harris, and S.E. Groves, “A Thermo-Mechanical Constitutive Theory for Elastic Composites with Distributed Damage,” Int. J. Solids Struct., Vol. 23, 1987, pp. 1301–1338 7. J. Zhang, J. Fan, and C. Soutis, “Analysis of Multiple Matrix Cracking in [±θm/90n]s Composite Laminates – Part 2: Development of Transverse Ply Cracks,” Composites, Vol. 23, No. 5, Sept. 1992, pp. 299–304 8. S.L. Donaldson, R.Y. Kim, and R.E. Trejo, “Damage Development in Laminates Mechanically Cycled at Cryogenic Temperature,” 43rd AIAA Structures, Structural Dynamics, and Materials Conference, Palm Spring, CA, AIAA-2004-1774, April 2004 9. X. Su, F. Abdi, and R.Y. Kim, “Prediction of Micro-Crack Densities in IM7/977-2 Polymer Composite Laminates Under Mechanical Loading at Room and Cryogenic Temperatures,” AIAA/SDM 46, Austin, Texas, 2005 10. F. Abdi, Q. Li, D. Huang, and V.S. Sokolinsky, “Progressive Failure Dynamic Analysis for Composite Structures,” 9th Japan International SAMPE Symposium & Exhibition (JISSE-9), Tokyo, Nov 2005 11. F. Abdi, D. Huang, M. Khatiblou, and C. Chamis, “Impact and Tension After Impact of Composite Launch Space Structure,” Sampe Conference Paper, 2001 12. M. Garg and G. Abumeri, “Assessment of Residual Strength in Impacted Composite Panels,” JEC 2007 Journal Publication 13. D. Abe, J. Ymaki, and Y. Yuroshiyama, “Determination of Bending Characteristics of CFRP Beam Using Progressive Fracture Model,” Japan Society of Automotive Engineering, September 2007 14. D. Abe and Y. Yuroshiyama, “Examination of Dynamic Bending Characteristics of a CFRP Beam Using Progressive Fracture Model,” Japan Society of Automotive Engineering (JSAE) Annual Congress (spring), 2008 15. Federal Aviation Regulations, Part 25, Airworthiness Standards: Transport Category Airplanes, Federal Aviation Administration, Washington, DC, 1993 16. Damage-Tolerance and Fatigue Evaluation of Structure, Advisory Circular 25.571-1A, Federal Aviation Administration, Washington, DC,1986

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17. Personal communication with Mr. Joseph R. Soderquist, FAA damage tolerance contact, 9 August 1996 18. J. Sutton, “A Proposed Method of Compliance to Damage Tolerance Requirements for Commercial Aircraft Composite Primary Wing Structure,” Sixth NASA/DoD Advanced Composite Technology Conference, Anaheim, CA, NASA-CP-3326, Vol. 1, Part 1, 1996, p. 159 19. Composite Aircraft Structure, Advisory Circular 20-107A, Federal Aviation Administration, Washington, DC, 1984 20. P.L.N. Murthy and C.C. Chamis, “Integrated Composite Analyzer (ICAN): Users and Programmers Manual,” NASA-TP-2515, Lewis Research Center, Cleveland, OH, 1986 21. J. Sutton et al., “Design, Analysis, and Tests of Composite Primary Wing Structure Repairs,” Fifth NASA/DoD Advanced Composites Technology Conference, Seattle, WA, NASA-CP3294, Vol. 1, Part 2, 1995, p. 916 22. Personal communication with Mr. Bob Walters at Boeing, B2 Bomber damage tolerance contact, 12 August 1996 23. D.G. Moon and J.M. Kennedy, “Post-Impact Fatigue Response of Stitched Composites,” Composite Materials: Fatigue and Fracture – Fifth Volume, ASTM STP 1230, R.H. Martin, Ed., American Society for Testing and Materials, Philadelphia, PA, 1995., pp. 351–367 24. D. Moon, F. Abdi, and B. Davis, “Discrete Source Damage Tolerance Evaluation of S/RFI Stiffened Panels,” Sampe Conference Paper, 1999 25. F. Abdi, “Fire Resistance Simulation of Horizontal Flat Sandwich Panel and Deck-Bulkhead T-Joint Assembly with Temperature and Pressure Loads,” Contract No: N00014-02-M-0213 April 30, 2001 26. Z. Qian, F. Abdi, R. Miraj, A. Mosallam, R. Iyer, J.-J. Wang, T. Logan “The Post-fire Residual Strength of Composite Army Bridge,” 4th International Conference on Composites in Fire (CIF-4), New Castle England, September, 2005 27. J. Qian and F. Abdi, “The Residual Strength of Composite Army Bridge after Fire Exposure,” AIAA-2006-1842, Newport, RI, May 1–5, 2006 28. R. Miraj, Z. Qian, and F. Abdi, “Fire Resistance Simulation of Loaded Deck Sandwich Panel and Deck–Bulkhead Assembly Structures,” ONR 2006 Journal Publication 29. A. Mossallam, F. Abdi, and J. Qian, “Service Fire Resistance Simulation of Loaded deck Sandwich Panel and Deck Bulk Head Assembly Structure,” http://www.elsevier.com/ copyright, Elsevier Journal Publication, 2007, 1359-836815, 2007.02.2002 30. A. Mossallam, F. Abdi, J. Qian, and R. Miraj, “Residual Strength of Composite Army Bridges Exposed to Fire,” Book Chapter, CDCC 2007, Third International Conference on Durability and Field Applications of Fiber reinforced Polymer (FRP), Composites for Construction 31. A. Mosallam, F. Abdi, and R. Miraj, Post Fire Residual Strength of Composite Army Bridge, JEC, Canada, Feb 2008 32. D. Huang, F. Abdi, and A. Mossallam, Comparison of Failure Mechanisms in Composite Structure, SAMPE 2003 Conference Paper 33. M. Garg, G.H. Abumeri, and D. Huang, “Predicting Failure Design Envelop for Composite Material System Using Finite Element and Progressive Failure Analysis Approach,” Sampe 2008 Conference Paper, Long beach, CA, May 2008 34. GENOA User Manual, http://www. ascgenoa.com 35. F. Abdi, L. Minnetyan, and C. Chamis, “Durability and Damage Tolerance of Composites,” Book Chapter 8- Composites, Welded Joints, and Bolted Joints, Kluwer Academic Publisher, 2000 36. De Xie, M. Garg, D. Huang, and F. Abdi, “Cohesive Zone Model for Surface Cracks using Finite Element Analysis,” AIAA-2008-106742, Chicago, Illinois, April, 2008 37. De Xie, Z. Qian, D. Huang, and F. Abdi, “Crack Growth Strategy in Composites under Static Loading,” 47th AIAA-2006-1842, Newport, RI, May 1–5, 2006

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38. D. Xie, A.G. Salvi, A.M. Waas, and A. Caliskan, “Discrete Cohesive Zone Model to Simulate Static Fracture in 2D Triaxially Braided Carbon Fiber Composites,” J. Compos. Mater., Vol. 40, 2006, pp. 2025–2046 39. D. Xie and A.M. Wass, “Discrete Cohesive Zone Model for Mixed-mode Fracture Using Finite Element Analysis,” Eng. Fract. Mech., Vol. 73, 2006 pp. 1783–1796 40. F. Abdi, T. Castillo, and E. Shroyer, “Risk Management of Composite Structure” Book Chapter 45, CRC Handbook, January 2005 41. F. Abdi and X. Su “Progressive Failure Analysis of RLV Laminates of IM7/PETI-5 – at High, Room, and Cryogenic Temperatures,” 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 7–10, 2003 42. K.S. Whitley and T.S. Gates, “Thermal/Mechanical Response and Damage Growth in Polymeric Composites at Cryogenic Temperatures,” 43rdAIAA/ASME/ASCE/AHS /ASCAIAA-2002-1416 Structural Dynamics, and Materials Conference, Denver, Co, 2001 43. T.F. Johnson. and T.S. Gates, “Temperature Polyimide Materials in Extreme Temperature Environments,” 42ndAIAA/ASME/ASCE/AHS /ASC Structural Dynamics, and Materials Conference, Seattle, WA, 2001 44. R. Rice, R. Randall, J. Bakuckas, and S. Thompson, “Development of MMPDS Handbook Aircraft Design Allowables,” Prepared for the 7th Joint DOD/FAA/NASA Conference on Aging Aircraft, September 8–11, New Orleans, LA, 2003 45. DOT/FAA/AR-03/19, Final Report, “Material Qualification and Equivalency for Polymer Matrix Composite Material System: Updated Procedure,” Office of Aviation Research, Washington, D.C. 20591, U.S. Department of Transportation Federal Aviation Administration, September 2003 46. Department of Defense Handbook-OF Polymer Matrix Composites, “Volume 1. Guideline For Characterization of Structural Materials,” MIL-HDBK-17-1E, Volume 1of 3, 23 January 1997, Superseding MIL-HDBK-17-1D, 25 February, 1994, Approved for Public Release, distribution unlimited 47. G. Abumeri, M. Garg, and M. Taleghani, “A Computational Approach for Predicting A- and B-Basis Allowables for Polymer Composites,” SAMPE -2008, Memphis Tennessee, September, 2008 48. M.R. Talagani, Z. Gurdal, F. Abdi, and S. Verhoef, “Obtaining A-basis and B-basis Allowable Values for Open-Hole Specimens Using Virtual testing,” AIAAC-2007-127, 4. Ankara International Aerospace Conference, 10–12 September, 2007 – METU, Ankara 49. JAMS 2006, “Full-Scale Damage Tolerance of Composite Sandwich Structures,” report, p. 11, K.S. Raju, National Institute for Aviation Research 50. Test Plan – JAMS, “Full-Scale Damage Tolerance of Composite Sandwich Structures,” report p. 15–17, J.S. Tomblin & K.S. Raju, National Institute for Aviation Research, Wichita State University, Wichita, KS 67260-0093, 2007 51. JAMS, “Full-Scale Damage Tolerance of Composite Sandwich Structures,” report, J.S. Tomblin & K.S. Raju, National Institute for Aviation Research, Wichita State University, Wichita, KS 67260-0093, 2007 52. J.S. Tomblin, Damage Tolerance Testing and Analysis Protocols for Full-Scale Composite Airframe Structures Under Repeated Loading, Wichita State University, Wichita 53. AGATE-WP3.3-033051-131, “A – Basis and B – Basis Design Allowables for Epoxy – Based Prepreg, TORAY T700SC-12 K-50C/#2510 Plain Weave Fabric [US Units],” J. Tomblin, J. Sherraden, W. Seneviratne, and K.S. Raju, National Institute for Aviation Research, Wichita State University, Wichita, KS 67260-0093

Chapter 7

Modeling of Multiscale Fatigue Crack Growth: Nano/Micro and Micro/Macro Transitions G.C. Sih

Abstract Nanocracks less than 10–2 mm have been detected to run at about 0.15–9 × 10–6 mm/s in stainless steels that undergo intergranular stress corrosion cracking (IGSCC). Macrocracks propagate in cm/s with more than an order of magnitude difference in velocity. The difference in the rate of energy dissipated due to damage by cracking obviously cannot be overlooked in the development of multiscale crack models, particularly if the design were to depend on virtual testing rather than physical testing. If the loading rate is not low enough to take advantage of the nanostructure, then the added reinforcement may not serve a useful purpose. In other words, micro and/or macro reinforcements should suffice for structures that entail only fast energy release. The consideration of nanoscale in addition to micro and macro is a relatively new development that has received much attention. Although triple scaling appears to be a refinement of dual scaling, the scale range accuracy is actually determined by the two nearest scales such as nano–micro or micro–macro. A triple nano–macro scale model may be too coarse and may result into the insertion of a meso- or the micro-scale. This in effect returns to the treatment of two dual scales consisting of the nano–micro and micro–macro as mentioned earlier. It can thus be said that multiscaling constitutes a series of dual-scale models, each having a well-defined range of accuracy. Hidden subtleties can arise from the underlying physics of damage when the scale range is changed. A case in point is the 1/r0.5 behavior of a classical macrocrack and the double singular behavior of the microcrack having a weak singularity of 1/r0.15 and a strong singularity of 1/r0.75 . The physics of nanocracking can introduce additional alterations. A triple scale model involving nano-, micro-, and macro-cracks will be developed using a new paradigm for evaluating the explicit time history of the internal structure degradation process of solids from a knowledge of three micro/macro parameters ∗ , and d∗micro for connecting the quantities at the micro- and macro-scale μ∗micro , σmicro G.C. Sih (B) School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China; Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA e-mail: [email protected], [email protected]

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∗ and another three nano/micro parameters μ∗nano , σnano , and d∗nano for connecting the quantities at the nano- and micro-scale by using only the properties of the undamaged material at time t = 0. Since there can be no valid counterparts, say micro or nano, to specimen data collected at the macro-scale, to which lower-scale material property data can be related but not determined separately. The present analysis will entail the inverse approach of specifying a unique solution for nonlinear problems and finding the conditions to which the solution will satisfy. This corresponds to specifying the critical crack growth states and determining the evolution of the microstructure and nanostructure material degradation process. Crack growth rate da/dN models are constructed from range of micro/macro micro energy density factor ΔSmacro micro and range of nano/micro energy density factor ΔSnano to reflect the scale transitory behavior as the crack grows through from nano- to micro- to macro-size, that is, anano→ amicro→ amacro. The multiscale scheme is applied to fatigue crack growth data for pre-cracked 2024-T3 and 7075-T6 aluminum panels. Assumptions are made for obtaining the time-dependent character of (μ∗micro , ∗ ∗ , d∗micro ) and (μ∗nano , σnano , d∗nano ) from which the fictitious nano and micro mateσmicro rial properties and their corresponding geometric and restraining characteristics are found. Nanocrack growth rates are found to be in the range of 10–7 to 10–9 mm/s for 2024-T3 and 7075-T6 aluminum with initial crack size of ao = 10–4 mm. They reached their critical conditions about 5–8 years at which the nanocracks become microcracks. The segment transition is assumed to simplify the discussion. The same procedure can be used to predict the transition of microcracks to macrocracks and finally the instability of macrocracking. Starting from the initial defect of ao = 1.8 mm growing at an approximate rate of 10–6 mm/s for 18 years, macrocracking instability can be reached although this terminal condition has been set arbitrarily. It is not a restriction of the approach. The end life can be preset with the appropriate crack growth rate to match the design conditions.

Keywords Mesomechanics · Multiscaling · Micro/macro · Nano/micro · Scale transition · Scale segmentation · Uni-modulus · Aspect ratio · Fatigue · Crack growth · Microstructure degradation · Time dependency · Size and time effect · Damage · Cracking

7.1 Introduction This work can also be qualified as multiscaling fracture mechanics for several reasons. First of all, it is concerned with cracking for at least three scales: nano, micro, and macro. The analytical model makes use of the two-parameter crack growth rate da/dN relation [1] that has been in application for more than four decades. The motivation for such an approach arose from the measurements of nanocrack growth data by high-resolution modern electronic transmission microscope using thin-slice metal alloy specimens [2, 3]. What this means is that a line crack model can be used with sufficient accuracy to describe microcrack and nanocrack damage if the

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appropriate physical mechanisms can be modeled [4–7]. Even though the nanoscale results may need additional validation, those found at the micro/macro-scale are encouraging, and they suggest probing deeper down the scale range [8, 9]. The order of magnitude prediction for the nanocrack growth rate also suggests that there are high hopes for developing a workable fracture mechanics model of nanocracking. Already the work has led to valuable information concerning what analytical modeling is capable of achieving and what are the inherent limitations [10, 11]. It is valuable to realize that test specimens apply only to one-scale range which is normally taken at the macroscopic level. There are no physical counterparts at the microscopic scale that can be tested independently. Micro-entities are contained within the macro-portion differing in the system homogeneity. However, multiscale model development necessitates the definition of fictitious microscopic material properties. This is a mathematical requirement because a part becomes the whole upon physical separation. Thus it can be said that the whole is the part and the part is the whole. A continuum element exists only in a fictitious sense with imaginary interface that can be used to calculate the rate change of volume with surface dV/dA [12], a quantity that is not admitted in the classical theories of continuum mechanics that in principle has ignored the size effects of material microstructure. To reiterate, separation of any part from the whole would introduce additional free boundaries of the part, which in itself becomes another whole. The knowledge that high strength can be vulnerable to brittle fracture has led to the concept of fracture toughness where strength trade-off for toughness can be achieved by microstructure processing. This technology was slow to be incorporated into the early model of the two-parameter crack growth rate relation [1] involving da/dN and ΔK. The a = a(N) curve that yields da/dN, however, automatically contains the microscopic-scale effects, manifested by the threshold known as region I. The quantity ΔK being macroscopic is unable to detect microscopic effects. Fatigue in fracture mechanics entails nonvanishing mean stress. This differs from the classical definition of fatigue of smooth specimens where the mean stress σ m is zero. The fracture mechanics model of ΔK considers only the stress amplitude σ a even though σ m may not be zero. On the other hand, constant stress loading defines creep loading for smooth specimens. The discrepancy in definition arose [11] when attention was switched from the testing of smooth specimens to pre-cracked specimens. In retrospect, the fracture mechanics of fatigue crack growth includes creep as well since the mean stress may not be zero. Creep–fatigue crack growth would be a more precise description [10]. Additional inconsistencies prevail when microcracking entered into the da/dN versus ΔK data while the formulation is restricted to macrocracking. In other words, creep loading were ignored in ΔK but the effect appeared in the a = a(N) data. Attempts to correct the inconsistencies were couched in ΔKeff or ΔKeq that offered a variety of short-lived explanations. The fact that the crack growth rate undergoes the transitions damicro nano /dN and macro macro damicro /dN is indicative of the need to seek for the respective ΔKmicro nano and ΔKmicro that will respond to the scale transitions of da/dN. The inclusion of both σ a and σ m macro calls for an energy-based consideration such as ΔSmicro nano and ΔSmicro where ΔS has been known as the energy density factor [13]. The requirement of consistency was

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needed to account for the energy dissipated at both the micro and macro scale. The presence of nanocracking extends the discussion to even smaller-scale damage. In general, the structure life may consist of damage by nanocracking, microcracking, and macrocracking. Should macrocracking occur without substantial damage of the nanostructure, then the life of the structure could have been cut short by the portion that would have sustained by nanocracking. Whether nanoscale damage and/or nanostructure reinforcement needs to be considered or not is a decision that can be made at the design stage. Fatigue crack growth curves for 2024-T3 and 7075-T6 aluminum panels will be predicted to illustrate the proposed methodology for reinforcing the structure at the nano stage. The can be accomplished from a knowledge of the undamaged bulk mechanical properties referred to t = 0. The task involves obtaining the analytical description of the degradation history of the nano- and microstructure. Validation of can be made by comparing the predicted crack growth results with the known test data [14]. In a nut shell, it can be stated that A viable multiscale creep-fatigue crack damage model can be applied to assist the virtual testing of nanomaterials.

Although separate considerations should be given to the use of nanocomposites for large air transport systems and nanoparticle-reinforced metals for highperformance engines, the underlying principle of multiscale damage is the same.

7.2 Scale Implications Associated with Size Effects The strength of seemingly smooth specimen can vary widely for reasons that are not obvious at first sight. The causes may be identified with surface nanodefects or smaller, material microstructure variations arising from grains and subgrains and size of macrocracks relative to the test specimen. Each of the three effects at the different size scale can affect the critical applied stress. Weibull statistics, strain gradient theories, and fracture mechanics have been applied to study the respective cause, although the underlying physical implications depend on the characteristic length parameter that can vary from nano (or smaller) to micro and macro. A common base is needed for weighing their relative effects. This can be accomplished by using dV/dA [12] that possesses three components, one in each of the three orthogonal directions. The precious expressions can be computed for any geometry at any scale [15].

7.2.1 Physical Laws Change with Size and Time Physical laws are not only not immune to size scale, their interpretation is also intertwinned with the dualism of particle and continuum. These complications arose with the advent of nanotechnology during the 1950s. Dependency of physical laws on the particle size were found and referred to the surface A to volume V ratio A/V of the atom. Invoked, however, is that the atom itself has been regarded as a continuum

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with a uniformly distributed mass in contrast to the discrete view of atomic cloud model [16] where the particle is now subatomic in size. The size of the whole that was the atom 10–9 m has been shifted to a subatomic particle 10–13 m or smaller, which now becomes the part. There is no absolute size attached to the whole and part but rather the definition depends on the physics. By in large, A/V has been used in the particle model, while the rate change of volume with surface dV/dA for the continuum model [12]. Here, dV/dA can be a vector with three components. Nonvanishing dV/dA was responsible for explaining size effects associated with nonequilibrium behavior. A relation between uniaxial and three-dimensional stress–strain states were then established in the isoenergy density space [12]. The separate use of V/A and dV/A also reinforces the dualistic usage of the discrete and continuous for interpreting physical data. The difference between the local and global material properties can also use V/A and/or dV/A as a measure. To reiterate, macroscopic physical laws become invalid when the body size becomes vanishingly small such that surface effects start to dominate. The 18th century development of continuum mechanics was based exclusively on large bodies where the volume or bulk controls. The sustaining concept turned into a handicap for computer chip designers. According to bulk theories, dissipative energy would accumulate and computer electronics would overheat if the chip is reduced below the critical size. However, chips can be made smaller until the volume is so tiny compared to the surface that energy would not have a chance to accumulate and overheating can be eliminated in principle. Along the same line, super-miniaturized bearings can run smoothly without lubrication. The law of friction would thus be different for nanowires, nanotubes, and nanoparticles. The same applies to the melting temperature as the dimensions of a material reduces to the atomic scale. The decrease in melting temperature can be of the order of tens to hundreds of degrees for metals with nanometer dimensions [17, 18]. It is now recognized that the thermodynamic and thermal properties for small bodies are different. To support what has been said, the drastic drop of melting temperature of nanoparticles has been shown to occur at the critical diameter of less than 50 nm for common engineering metals [17, 18]. This is relevant in catalyst, sensor, medicinal, optical, magnetic, thermal, electronic, and alternative energy applications at elevated temperatures. It should be further emphasized that the use of composites with nanowires, nanotubes, and nanoparticles as reinforcements are even more critical when they are applied to structures that involve the safety of human lives. Whether nanoparticlereinforced systems would benefit or impede structures will not be known until the size-effect behavior of the parameters governing the system integrity can be assessed quantitatively. The atomic cohesive energy should be better understood.

7.2.2 Surface-to-Volume Ratio as a Controlling Parameter Although the fracture of macro-dimension specimens has been related to the cohesive energy calculated from the interatomic potential of two interacting atoms and

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the use of empirical constants and even though the classical cohesive energy treatment concerns with atomistic dimensions, surface-to-volume effects are still left out. Atoms at or near the surface of the nanoparticle have reduced cohesive energy due to a reduced number of cohesive bonds. The surface-to-volume ratio (A/V) or the rate change of surface with volume (dA/dV) can be used to indicate when the surface starts to dominate. Nanoparticles have a much greater surface-to-volume ratio than bulk materials. The increased A/V ratio means surface atoms have a much greater effect on chemical and physical properties of a nanoparticle. Surface atoms bind in the solid phase with less cohesive energy because they have fewer neighboring atoms in close proximity compared to atoms in the bulk of the solid. Each bond an atom shares with a neighboring atom provides cohesive energy, so atoms with fewer bonds and neighboring atoms have lower cohesive energy. Scale shifting factor [19] has been used to translate the cohesive energy results from the atomic to the macro scale. .Studies in [20–22] have also shown thathttp://en.wikipedia.org/wiki/Image:Lennard-Jones.jpg the cohesive energy of nanocrystals can increase or decrease with A/V depending on different conditions.

7.2.3 Strength and Toughness: Nano, Micro and Macro Strength and toughness are the macroscopic quantities used to characterize the mechanical integrity of specimens and structural components. When referred to damage in fatigue, the crack length versus the number of cycles data are collected as shown in Fig. 7.1. The application started at the macroscopic scale and gradually extended to smaller scales to include microcracks and nanocracks. Cracks 10–2 to 10–4 mm can be measured for thin specimens such that the electron microscopic would have sufficient power to acquired good resolution. Regions I, II, and III in Fig. 7.1 correspond to those of the sigmoidal curve in a log–log da/dN versus ΔK

Micro

Macro

Crack length a

Nano

Region III

Region II Region I

Fig. 7.1 Scaling of crack growth in fatigue: nano, micro, and macro

O

Fast

Number of cycle N

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diagram that is developed from the classical continuum mechanics approach. In this respect, strength and toughness are measured at the macroscopic scale, while those at the micro and nano scale are regarded as fictitious as they cannot be measured separately by using smaller specimens without encountering conceptual difficulties. What this means is that the transition from nano to micro to macro can only be achieved in the fictitious sense if critical nano and micro strength or toughness were to be used as the criterion for scale transition. There is no inconsistency conceptually if the crack length were used for the transition as anano→ amicro →amacro such that the relations danano /dN, damicro /dN and damacro /dN can then be established. The continuum model permits the use of only one real strength and one real toughness. Those at the micro- and nanoscale would be regarded as fictitious.

7.3 Form Invariant of Two-Parameter Crack Growth Relation The two-parameter fatigue crack growth relation [1] da = C(ΔK )n dN

(7.1)

was first used to analyze the fatigue growth data of pre-cracked aluminum panels. Referring to Eq. (7.1), C and n were assumed to be material constants that are determined empirically. The range of macro-stress intensity factor depends on the stress amplitude range and the square root of the half crack length a for a panel having a central crack of length 2a. Taking the log of Eq. (7.1), there results log

da = n log(ΔK ) + log C dN

(7.2)

In Eq. (7.2), n is the slope and logC the y-intercept for the portion of the curve marked region II in Fig. 7.2. To be specific, application of Eq. (7.1) will be made to the pre-crack aluminum panel data in [14]. Note that long cracks refer to a from 10 to 40 mm, while short cracks refer to a from 1 to 10 mm. The portion marked region I has been referred to as the “threshold”. What has remained hidden is the crack mouth opening that changes from invisible and tight to visible and loose. This corresponds to the, respective, change of surface-traction to traction-free boundary as region I switches to II. For lower da/dN, the crack size is even smaller, with dimensions of 10–2 to 10–1 mm. The recent concern is whether the classical ΔK correlation should be replaced by other quantities to account for the smaller microcracks, the behavior of which may not be describable by ΔK. The procedure would presumably be the same for connecting micro- and macrocracking by the application of the dual-scale stress intensity factor range ΔKmacro micro or the dual-scale energy density micro micro factor ΔSmacro micro . That is to introduce ΔKnano or ΔSnano . Even more essential is to connect the da/dN results at the nano, micro, and macro scale. Linear interpolation

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Fig. 7.2 Crack size increases with growth in fatigue

would be an obvious choice if the danano /dN, damicro /dN and damacro /dN data can be transformed onto a straight line. This property is known as form invariant [23]. The idea can be illustrated schematically by Fig. 7.3a and b. Because of the lack of reliable data for nanocrack growth, the discussion will center on the transformation macr o of micro- and macrocrack data by usingΔK micr o . The object is to rotate region I and III in Fig. 7.3a clockwise to arrive at the straight line in Fig. 7.3b assuming that macr o the same can be done for the nanocrack region. The difference of ΔK micr o and ΔK macr o in Eq. (7.1) can be found in [8, 9]. The same can be done by using ΔSmicr o and micro ΔSnano . In what follows, the ΔS will be used such that both the mean stress and stress amplitude effects are included. (a) Before transformation (b) After transformation

7.4 Dual-Scale Fatigue Crack Growth Rate Models micro The construction of ΔSmacro micro or ΔSnano in any dual-scale fatigue crack growth model requires a knowledge of the interconnecting behavior of the crack, say for nano and micro or for micro and macro. To begin with, considerΔSmacro micro where the microcrack behavior needs to be connected with that of the macrocrack. To this end, the 1/r0.5 behavior of the macrocrack falls in between the double-singular behavior of the microcrack [8, 9] which possesses a weak singularity of 1/r0.15 and a strong singularity of 1/r0.75 . Note that the order 0.5 is in between 0.15 and 0.75 as illustrated in Fig. 7.4. The physics of nanocracking can still depend on the application. A triple-scale model involving nano-, micro- and macrocracks will be developed using a new paradigm for evaluating the explicit time history of the internal structure degradation ∗ , process of solids from a knowledge of three micro/macro parameters μ∗micro , σmicro

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(a) Before transformation

(b) After transformation Fig. 7.3 Form invariant of crack growth rate in relation to regions I, II, and III

and d∗micro for connecting the quantities at the micro- and macro-scale and another ∗ , and d∗nano for connecting the quantities at three nano/micro parameters μ∗nano , σnano the nano- and microscale by using only the properties of the undamaged identified with time t = 0 as shown in Table 7.1 [14].

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Macro-crack zone

Macro-micro Micro-crack transition zone zone Distance r –4

10–1

10 Micro-stress (strong field)

Macro-stress (Medium field)

cm

Far-stress (Weak field)

Fig. 7.4 Interconnecting singular behavior of micro- and macrocracks Table 7.1 Undamaged mechanical properties of 2024-T3 and 7075-T6 aluminum alloys at t = 0 Material type

Yield strength σ yd (MPa)

Ultimate strength σ ult (MPa)

Shear modulus μo (GPa)

Poisson’s ratio νo

2024-T3 7075-T6

363.58 463.54

473.34 519.40

27.56 26.87

0.33 0.33

7.4.1 Micro/Macro Formulation Consider a two-parameter micro/macro fatigue crack growth model as presented in [8–10]: da o macr o ψ = Ψmacr micr o [ΔSmicr o ] dt

(7.3)

in which Ψmacro micro accounts for the scale transitional behavior while ψ represents the slope of the da/dt versus ΔSmacro micro curve in a log-log plot. Without going into details, ΔSmacro micro can be found in [8–10] as ΔSmacro micro

2(1 − 2νmicro )(1 − νmacro )2 aσa σm μmicro = μmacro μmacro

2 σomicro dmicro 1 − macro (7.4) σ∞ r

Before putting Eq. (7.4) into (7.3) to yield da/dt, it should be noted that the near-tip radial distance r is a micro/macro transitional parameter. To accentuate this property, the normalized quantity d∗micro will be introduced by using the factor dmicro macro 0.5 such that (dmicro /r)0.5 in Eq. (7.4) can be written as (d∗micro dmicro to bring out the macro /r) scale transitional character of ΔSmacro micro . It follows that ⎡ ⎤ψ * ) micro 2 2(1 − 2ν )(1 − ν ) aσ σ d da micro marco a m ∗ ∗ ⎣ = Ψmacro μmicro (1 − σmicro )2 d∗micro macro ⎦ micro dt μmarco r (7.5)

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The dual-scale physical parameters are μ∗micro =

μmicro ∗ σomicro ∗ dmicro , σmicro = macro , dmicro = micro μmaco σ∞ dmacro

(7.6)

The subscript and superscript notation in Eq. (7.4) is self-explanatory, where m stands for micro while m stands for macro. It should now be clear that r in Eq. (7.3) is a scale transitional parameter. Approximating da/dt by Δa/Δt, the ith increment of crack growth Δai can be found as 2 (1 − 2νmicro ) (1 − νmacro )2 σa σm Δai = Ψmacro micro μmacro ⎤ψ * (7.7) ) micro d 2 ψ macro ∗ ⎦ ai−1 Δt ×μ∗micro 1 − σmicro d∗micro r which can be inserted into ai = ai − 1 + Δai

(7.8)

to obtain ai and hence a = a(t).

7.4.2 Nano/Micro Formulation In a similar fashion, a nano/micro fatigue crack growth model may be used. Let start with the nano/micro dual-scale model da micro ψ = Ψmicro nano [ΔSnano ] dt

(7.9)

Two new parameters Ψmicro nano and ψ are used such that ΔSmicro nano =

2(1 − 2νnano )(1 − νmicro )2 aσa σm μnano σonano 2 ∗ dnano (1 − micro ) dnano micro (7.10) μmicro μmicro σ∗ r

with dnano being the damage size at the nano scale. Because r in Eq. (7.10) is a scale transitional parameter, the quantity d∗nano is introduced to accentuate this property by micro using the normalization factor dnano micro . This exhibits the behavior of ΔSnano where the nanocrack and microcrack may interchange because of the alternating character of fatigue loading and the self-restraining capability of the near-tip material. It follows that ψ 2 2 ∗ dnano da micro micro 2(1 − 2νnano )(1 − νmicro ) aσa σm ∗ ∗ = Ψnano μnano 1 − σnano dnano dt μmicro r (7.11) after applying the definition

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μ∗nano =

μnano σonano dnano ∗ , σnano = micro , d∗nano = nano μmicro σ∞ dmicro

(7.12)

The jth increment of nanocrack growth can thus be found: Δai =

Ψmicro nano

2(1 − 2νnano )(1 − νmicro )2 σa σm ∗ dnano ∗ μnano (1 − σnano )2 d∗nano micro μmicro r

ψ

ψ

ai−1 Δt

(7.13) This determines the crack growth rate in the Eq. (7.11). Keep in mind that r is a scale transitional distance that can in fact adopt the notation rm/m. The approximate relation can be used to find a = a(t): ai = ai − 1 + Δai

(7.14)

7.5 Micro/Macro Time-Dependent Physical Parameters The material would age with time in addition to be damaged by load. This process is irreversible. Under normal circumstances, the material parameters are expected to change monotonically with the time t. Simple tests may be carried out to determine the trend of this time variation.

7.5.1 Macroscopic Material Properties The undamaged macroscopic shear modulus and Poisson’s ratio at t = 0 are given by μo and ν o which appear as coefficients in the time-dependent relations: μmacro (t) = μo Fμ (t), νmacro (t) = νo Fν (t)

(7.15)

For simplicity, assume that Fμ (t) = Fν (t) = F(t)

(7.16)

Note that 0≤F(t)≤1. The progressive damage be aging should be slow at first and increase as time elapses. This can be described as F(t) = 1, t = 0; F(t) = 0.95, t = 10; F(t) = 0.8, t = 20.

(7.17)

A parabolic function can be used: F(t) = −0.0005t2 + 1

(7.18)

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Fig. 7.5 Time variations of Fμ (t), Fν (t) and FΨ (t)

Displayed in Fig. 7.5 is the decay of F(t) in Eq. (7.18) with time over a period of 20 yrs. Equations (7.15) can thus be written as μmacro (t) = μo (−0.0005t2 + 1)

(7.19)

νmacro (t) = νo (−0.0005t2 + 1).

(7.20)

Shown in Fig. 7.6 are the time decay of the macroscopic shear modulus with time with μo = 27.56GPa for 2024-T3 and μo = 26.87GPa for 7075-T6. The two curves

Fig. 7.6 Variation of macroscopic shear μmacro with time t with μo = 27.56GPa for 2024-T3 and μo = 26.87GPa for 7075-T6

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are nearly parallel with 2024-T3 having a higher stiffness than 7075-T6. Variation of macroscopic Poisson’s ratio νmacro with time t for 2024-T3 and 7075-T6 aluminum with νo = 0.33 can be found in Fig. 7.7. Fig. 7.7 Variation of macroscopic Poisson’s ratio νmacro with time t for 2024-T3 and 7075-T6 aluminum with ν o = 0.33

Decreasing of fatigue life with time can be reflected by increase of the coefficient Ψmacro micro with time t. This gives Ψmacro micro (t) = Ψo FΨ (t)

(7.21)

where Ψo is the initial value of Ψmacro micro at the undamaged state. Obviously, the loaddependent function FΨ (t) is different from the functions. Fν (t) and Fμ (t) in Eq. (7.16) or given specifically in Eqs. (7.19) and (7.20). To this end, note that t = 0, FΨ (t) = 1; t = 10, FΨ (t) = 1.05; t = 20, FΨ (t) = 1.2.

(7.22)

such that FΨ (t) = 0.0005t2 + 1

(7.23)

Inserting Eq. (7.23) into Eq. (7.21), the results is 2 Ψmacro micro (t) = Ψo (0.0005t + 1)

(7.24)

Time variation of Ψmacro micro with time t for aluminum 2024-T3 with ψ = 2 and Ψo = 0.54 mm3 /(N2 ·year) for t = 0 can be found in Fig. 7.8. The curve rises slowly at first and then more quickly as time increases. This trend is slightly different for 7075-T6 in Fig. 7.9. The rise in the curve with time is less pronounced.

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Fig. 7.8 Variation of micro/macro crack velocity coefficient Ψmacro micro with time t for aluminum 2024-T3 with ψ = 2 and Ψo = 0.54 mm3 /(N2 year) for t = 0

Fig. 7.9 Variation of micro/macro crack velocity coefficient Ψmacro micro with time t for 7075-T6 with ψ = 2 and Ψmacro = 3.63 mm3 /(N2 year) at t = 0

7.5.2 Microscopic Material Properties Since the macroscopic material properties at t = 0 are found by tests, the corresponding microscopic properties cannot be measured independently but only determined analytically in a fictitious sense for simulating the scale transition behavior. ∗ , and d∗micro will be used for this purpose. They are The three parameters μ∗micro , σmicro also assumed to vary as a convex parabola in time. Their specific forms are shown by Eqs. (7.25), (7.26), and (7.27) as follows:

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μ∗micro (t) =

μmicro = −0.002t2 − 0.01t + 3 μmacro

(7.25)

with μ∗micro = 3 for t = 0, μ∗micro = 2.7 for t = 10 and μ∗micro = 2 for t = 20 yrs. ∗ σmicro (t) =

σo = −0.001t2 + 0.5 σ

(7.26)

∗ ∗ ∗ = 0.5 for t = 0, σmicro = 0.4 for t = 10 and σmicro = 0.1 for t = 20 yrs. with σmicro

d∗micro (t) =

dmicro dmicro macro

= −0.01t2 + 5

(7.27)

with d∗micro = 5 for t = 0, d∗micro = 4 for t = 10 and d∗micro = 1 for t = 20∗ yrs. Figure 7.10 shows that the characteristic length parameter d∗micro decreases more ∗ . It should also be said that the relative change sharply with time than μ∗micro and σmicro of the micro and macro parameters are more easily anticipated than the individual micro and macro quantities. For this reason, they are found prior to the timedependent microscopic material property. The micro-shear modulus μmicro can thus be written as μmicro (t) = μ∗micro (t)μmacro (t)

(7.28)

where μmacro (t) is given by Eq. (7.19). Now Eqs. (7.19) and (7.25) can be inserted into Eq. (7.28). This yields μmicro (t) = μo (−0.002t2 − 0.01t + 3)(−0.0005t2 + 1)

Fig. 7.10 Variations of normalized parameters ∗ μ∗micro , σmicro and d∗micro with time t for 2024 and 7075

(7.29)

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Fig. 7.11 Microscopic shear μmicro with time t for 2024-T3 and 7075-T6

Plots of Eq. (7.29) for 2024-T3 and 7075-T6 are given in Fig. 7.11. The difference is caused from μo , the shear modulus at the undamaged state. The micro-Poisson’s ratio ν micro can be written as νmicro (t) = ν∗micro (t)νmacro (t)

(7.30)

where ν macro (t) has been given by Eq. (7.20). The ratio ν∗micro (t) is defined as ν∗micro (t) = νmicro /νmacro

(7.31)

The ratio ν∗micro should also depend on time. A convex parabola function is assumed. ν∗micro (t) = −0.001t2 − 0.005t + 1.5

(7.32)

with ν∗micro = 1.5 for t = 0, ν∗micro = 1.35 for t = 10 and ν∗micro = 1 for t = 20 yrs. The variations of Eq. (7.32) with time are exhibited in Fig. 7.12. It follows that νmicro (t) = νo (−0.001t2 − 0.005t + 1.5)(−0.0005t2 + 1)

(7.33)

The undamaged macro-Poisson’s ratio is taken as ν o = 0.33 for both 2024-T3 and 7075-T6. A plot of Eq. (7.33) with time can be found in Fig. 7.13.

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Fig. 7.12 Micro/macro time-dependent function ν∗micro (t) for 2024 and 7075

Fig. 7.13 Variation of micro-Poisson’s ratio νmicro with time t for 2024 and 7075

7.6 Nano/Micro Time-Dependent Physical Parameters Determination of nanoscopic material properties are even more involved as they have to be related to those at the microscopic as well as the macroscopic scale. Based on the earlier arguments, the analytical expressions for the nanoscopic material properties are also fictitious in character. In contrast to the micro/macro model in which ψ = 2, the exponent ψ in da/dt for the nano/micro model in Eq. (7.11) is taken as ψ = 1 to distinguish the difference in crack growth rate for the micro/macro and nano/micro models, while Ψo for 2024-T3 and 7075-T6 are assumed to remain unchanged. In the absence of test data for fatigue crack growth at the nanoscale, the following conditions are taken:

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ψ = 1, Ψo = 0.54 mm2 /(N.year) for 2024 − T3

(7.34)

ψ = 1, Ψo = 3.63 mm2 /(N.year) for 7075 − T6

(7.35)

7.6.1 Nanoscopic Material Properties ∗ The nano/micro relative physical parameters μ∗nano (t), d∗nano , and σnano are again nano determined such that the fictitious quantities μnano , dnano, and σo can be found, although only μnano will serve a useful purpose in this work. The nano-Poisson’s ratio taken as νnano = 0.495. As pointed out in [10, 11], for two-dimensional dualscale problems, there is no loss in generality to use the formalism of the theory of elasticity if the range of applicability is sufficiently small. Any nonlinear behavior can be approximated by connecting linear segments. The material properties can be described by two parameters referred to as the uni-modulus and the aspect ratio in replace of the Poisson’s ratio to avoid confusion. Defined is the function

μ∗nano (t) =

μnano μmicro

(7.36)

such that μ∗nano = 21 for t = 0, μ∗nano = 20.496 for t = 4 and μ∗nano = 19.544 for t = 8 yrs. This gives μ∗nano (t) = (−0.012t2 − 0.06t + 18) × 103

(7.37)

the results of which are exhibited graphically in Fig. 7.14 for time up to 8 yrs. It is anticipated that the damage will initially consists of nanocracking. The precise period will be determined in accordance with the prescription of final instability of macrocracking as related to micro- and nanocracking. The numerical results in Fig. 7.14 can now be substituted into the relation μnano (t) = μ∗nano (t)μmicro (t)

(7.38)

together with those in Fig. 7.11 for μmicro (t) as given by Eq. (7.29). This leads to the expression μnano (t) = μo (−0.012t2 − 0.06t + 18)2 (−0.0005t2 + 1) × 103 (GPa)

(7.39)

Refer to Fig. 7.15 for a graphical description of μnano (t) as a function of time for 2024-T3 and 7075-T6. Both curves decrease with time for damage by nanocracking. ∗ are inconsequential for the present As mentioned earlier, the forms of d∗nano and σnano work and they can be taken arbitrary as

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Fig. 7.14 Nano/micro time-dependent shearμ∗nano (t) for 2024-T3 and 7075-T6

Fig. 7.15 Variation of μnano with time t for 2024-T3 and 7075-T6

∗ σnano (t) =

σonano = −0.001t2 + 0.5 micro σ∞

(7.40)

dnano = −0.01t2 + 5 dnano micro

(7.41)

d∗nano (t) =

Equations (7.40) and (7.41) are the same as Eqs. (7.26) and (7.27) for the micro/macro model which have already been plotted in Fig. 7.10.

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7.6.2 Nanoscopic Fatigue Crack Growth Coefficient There remains the determination of Ψmicro nano (t) which is needed for finding the fatigue macro crack growth rate at the nanoscale. In order to connect Ψmicro nano (t) with Ψmicro (t), a ∗ ∗ nano/micro fatigue crack growth coefficient Ψ = Ψ (t) is defined: Ψ∗ (t) =

Ψmicro nano (t) macro Ψmicro (t)

(7.42)

Recall that Ψmacro micro (t) is already known from Eq. (7.24) whose numerical results are shown in Fig. 7.8. The selection of Ψ∗ (t) can be based on making damicro /dN to macro be smaller than damacro /dN such that Ψmicro nano (t) is smaller than Ψmicro (t). In addition, damicro /dN is assumed to increase gradually with the crack size a and merged into damacro /dN. Under these considerations, the function Ψ∗ (t) is taken as 0.5–1. More specifically, let Ψ∗ = 0.5 for t = 0, Ψ∗ = 0.85 for t = 10 and Ψ∗ = 1 for t = 20. A convex parabola function gives Ψ∗ (t) = −0.001t2 + 0.045t + 0.5

(7.43)

as plotted in Fig. 7.16. Eq. (7.42) can be written as ∗ macro Ψmicro nano (t) = Ψ (t)Ψmicro (t)

(7.44)

while Ψmacro micro (t) has been given by Eq. (7.24), the use of which leads to 2 2 Ψmicro nano (t) = Ψo (−0.001t + 0.045t + 0.5)(0.0005t + 1)

Fig. 7.16 Micro/macro to nano/micro crack velocity coefficient Ψ∗ (t) for 2024-T3 and 7075-T6

(7.45)

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Here, Ψo is the initial value of Ψmacro micro (t) at t = 0. Plotted in Figs. 7.17 and 7.18 are, respectively, the curves corresponding to Eq. (7.34) for 7075-T6 and Eq. (7.35) for 7075-T6 of Eq. (7.45). They all rise with time up to 20 yrs although the micro and nano portion is limited to less than 10 yrs. This completes the determination of the nano/micro parameters from which the crack velocity and growth rate can be found. Fig. 7.17 Variation of Ψmicro nano (t) with time for 7075-T6

Fig. 7.18 Variation ofΨmicro nano (t) with time t for 7075-T3

7.7 Fatigue Crack Growth and Velocity Data The final check to decide on the desired time-dependent behavior of the damaged material is based on the satisfaction of the predicted fatigue crack growth rate data that would correspond to the desired life. Both the initial crack size and the fatigue

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crack growth in the micro/macro and nano/micro range would be specified. In other words, the material would be fabricated according to the time-dependent damage specification making sure that the micro- and nanostructure are damaged in the process of absorbing the time varying energy from the load.

7.7.1 Predicted Micro/Macro Results Micro/macro fatigue crack growth life of about 18 yrs can be assured from the timedependent nano–micro–macro material properties derived earlier with ao = 1.8 mm and da/dt = 5–30 mm/yr = 1.6–9.6 × 10–6 mm/s for 2024 and a critical crack length of acr ≈26 mm. This illustrated in Fig. 7.19. The shear modulus of 7075 is 26.87 GPa, being smaller than 2024 which is 27.56GPa, the life of 7075 is about 15.5 yrs with ao = 0.3 mm and a critical crack with crack length of acr ≈12 mm for 7075. Refer to Fig. 7.20. Recall that ψ = 2 and [Ψo ]2024 = 0.54 mm3 /(N2 ·year) for 2024 and ψ = 2 and [Ψo ]7075 ≈3.63 mm3 /(N2 year) for 7075. Since the time-dependency portion of Ψmacro micro (t) for 2024 and 7075 are the same, the foregoing results hold for (t)] /[Ψmacro [Ψmacro 7075 micro micro (t)]2024 = [Ψo ]7075 / [Ψo ]2024 . It can be seen from the results in Figs. 7.19 and 7.20 that the critical crack with crack length of acr ≈26 mm for 7075 in Fig. 7.20 is much more sensitive to the initial crack size ao . More numerical accuracy is needed to pinpoint the precise acr that determines the fatigue life. Such an exercise, however, would not gain additional physical in sight to the problem. Plotted in Figs. 7.21 and 7.22 are the respective crack growth rates of 2024 and 7075. Both Figs. 7.19 and 7.20 show that for time less than 10 yrs, the nano/micro crack growth characteristics would govern. More accuracies are needed to determine the details of the curves.

Fig. 7.19 Crack length a versus time t curve for 2024-T3

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Fig. 7.20 Crack length a versus time t curve for 7075-T6 with acr ≈12 mm

Fig. 7.21 Crack growth rate da/dN versus time t curve for 2024-T3

7.7.2 Predicted Nano/Micro Results Based on the fatigue crack growth rate coefficients in Eqs. (7.34) and (7.35), the crack growth data are found for 2024 and 7075. Figures 7.23 and 7.24 show the details of Fig. 7.19 for the, respective, time range of 6–8 and 1–4 yrs for 2024. The situation for a time range of 2–2.5 yrs for 7075 is different. Figs. 7.25 and 7.26 show the details of Fig. 7.20 or the, respective, time and 0.5–1 yrs for 7075. Details for even smaller crack length can be obtained by modeling crack growth damage below the nano scale.

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Fig. 7.22 Crack growth rate da/dN versus time t curve for 7075-T6

Fig. 7.23 Crack growth in the time range 6–8 yrs for 2024-T3

Using the data in Figs. 7.23 and 7.24, the crack growth rates for 2024 can be found and they are displayed in Figs. 7.27 and 7.28, respectively. Referring to Fig. 7.27, an initial crack length ao = 1.5 × 10–4 mm would correspond to a life of about 8 yrs with da/dt = 2–10 mm/yr being equivalent to 6.4 × 10–7 mm/s to equivalent to 6.4 × 10–7 mm/s to 32 × 10–7 mm/s since 1 yr/hr = 3.2 × 10–7 mm/s. Figure 7.28 shows the crackgrowth rate of 10–2 mm/yr for 1–4 yrs life of 2024-T3. Using the crack growth data of 7075-T6 in Figs. 7.25 and 7.26, da/dt can be computed and plotted in Figs. 7.29 and 7.30. Figure 7.29 shows the crack growth rate data for ao = 0.5 × 10–5 mm and a life of 1.8 yrs with da/dt = 1 mm/yr = 3.2 × 10–7 mm/s. For fatigue life of about 0.6–1 yr with da/dt≈3 × 10–3 mm/hr =

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Fig. 7.24 Crack growth in the time range 1–4 yrs for 2024-T3

Fig. 7.25 Crack growth in the time range of 2–2.5 yrs for 7075-T6

9.6 × 10–10 mm/s, Fig. 7.30 gives different results for different initial crack sizes which vary from 0.1 to 0.5 × 10– 5 mm. This shows that crack growth is sensitive to initial imperfections.

7.8 Validation of Nano/Micro/Macro Fatigue Crack Growth Behavior For a common initial crack size of ao = 2 mm subjected to σm = 44.1 MPa, σa = 31.36 MPa and R = 0.169, the crack growth for 2024-T3 and 7075-T6 can be plotted on the same graph as shown in Fig. 7.31. Converting the time scale into number of

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Fig. 7.26 Crack growth in the time range of 0.5–1 yr for 7075-T6

Fig. 7.27 Crack growth rate of 2–10 mm/yr for 6–8 yrs life of 2024-T3

cycles, the predicted results in Fig. 7.31 can be compared to those found in [14]. To this end, let dN/dt = 0.2 × 105 cycle/yr. This corresponds to 25 yr × 0.2 × 105 cycle/yr = 5 × 105 cycle. The time of 25 yrs can be converted into 5 × 105 cycles. A re-plot of the time scale in Fig. 7.31 to number of cycles N is done in Fig. 7.32 and they agreed with the measure data [14]. The object of the exercise is to derive the available test data and identified them with the time-dependent nano–micro–macro fatigue crack growth behavior with critical crack size at global instability.

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Fig. 7.28 Crack growth rate of 10–2 mm/yr for 1–4 yrs life of 2024-T3

Fig. 7.29 Crack growth rate of 1 mm/yr for 1.8 yrs life with ao = 0.5 × 10–5 mm for 7075-T6

7.9 Implication of Multiscaling and Future Considerations Separation of a part from the whole requires the creation of an interface, an artifact that has acted as the curtain between nature and man-developed theories. Particle physics quantifies such effects by the surface-to-volume ratio A/V [20–22] while continuum mechanics by the rate change of volume with surface dV/dA [12]. The surface-to-volume effects represent an integral part of the interface whether real or imagined. By tradition, the strength of material study is based on testing a representative element, referred to as the specimen having a definite shape and size as endorsed

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Fig. 7.30 Crack growth rate of 3 × 10–3 mm/hr for 0.6–1 yrs of 7075-T6

Fig. 7.31 Comparison of fatigue crack growth for 2024-T3 and 7075-T6 with ao = 2 mm subjected to σ m = 44.1 MPa, σ a = 31.36 MPa, and R = 0.169

by the professional societies for the lack of support from science and engineering. Suppose that the shape and size of the specimen were changed from a rectangular block to that of a solid cylinder. The test results would differ even for the same material micro- and nanostructures. This seemingly trivial problem still remains unresolved if the effect of dV/dA is not taken into account [13]. The discipline of fracture mechanics considers the initial damage of a material by cracking. The presence of a local interface being the crack boundary is introduced. This is in addition to the global interface, the specimen boundary as illustrated in Fig. 7.33(a). As such, the global and local interface can represent any two scales, say structure and macro, macro and micro, micro and nano or nano and atomic. The choice of

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Fig. 7.32 Test results for 2024-T3 and 7075-T6 with ao = 2 mm under σ m = 44.1 MPa, σ a = 31.36 MPa, R = 0.169 using dN/dt = 0.2 × 105 cycle/yr

(a) Global and local interface

(b) Multiscaling

Fig. 7.33 Global/local interface and multiscaling of crack-tip damage

the specimen scale which is an artifact, analogous to the “original sin” committed only once in a lifetime. Should it be the macroscopic scale, then the remaining representative elements micro, nano, atomic, as shown in Fig. 7.33(b) or otherwise would be regarded as fictitious. Each scale is characterized by the strength of the singular field such as r–0.5 for the macro-stresses and r–0.15 and r–0.75 for the dual scale micro-/macro-stresses [8, 9] such that the micro/macro transition is accounted for. The same can be constructed for the nano/atomic crack-tip element by having the appropriate transitional singular stresses used in [24] and dislocation theories [25, 26]. (a) Global and local interface, (b) Multiscaling Referring to Fig. 7.34, the fatigue crack velocity history is segmented into any desired number of dual-scale models such as micro/macro, nano/micro, atomic/nano, etc. Each of the dual-scale models can be quantified by the twoparameter sigmoidal fatigue crack growth relations such as that in Eq. (7.3) or (7.9).

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Fig. 7.34 Dual-scale fatigue crack growth models connected by interface

The interface size can be minimized to obtain the form invariant curve as derived in [23]. Although the choice of numerical approximations is arbitrary, conceptual consistency of the underlying physics should be observed. The combined scheme of multiscaling and mesomechanics [27] has been applied to devise a semi-empirical fatigue crack growth model that can evaluate the effectiveness of structural materials reinforced by nanostructure. Unless the fabrication of the nanostructure is tailormade to counter the time history of the energy dissipated, there is no assurance whether the addition of nanostructure reinforcement would enhance or impede the life of the structure. The nano–micro–macro crack growth model provides the groundwork for the development of a multiscale damage scheme that can be used in virtual testing. The objective has been accomplished by capturing the microscopic and nanoscopic material details in two sets of dual-scale ∗ , and d∗micro , for connecting the quantities at the micro- and parameters, μ∗micro , σmicro ∗ macroscales and another three nano/micro parameters μ∗nano , σnano , and d∗nano for connecting the quantities at the nano- and microscales. The oversimplified assumptions invoked for illustrating the use of the dual-scale parameters need to be scrutinized and examined in connection with realistic physical models. These additional efforts are necessarily use specific and cannot be demonstrated without the support of nanostructure material fabricators.

References 1. P.C. Paris, “The Growth of Cracks Due to Variations in Load,” Ph.D. Dissertation, Department of Mechanics, Lehigh University, 1962 2. S.M. Bruemmer, “Private Communication,” In: Chemistry and Microscopy of Intergranular Stress Corrosion Cracking in LWRs, Pacific Northwest National Laboratories (PNNL), Washington, USA, June, 2005 3. L.E. Thomas, J.S. Vetrano, S.M. Bruemmer, P. Efsing, B. Forssgren, G. Embring, and K. Gott, “High-Resolution Analytical Electron Microscopy Characterization of Environmentally

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G.C. Sih Assisted Cracks in Alloy 182 Weldments,” In: Proc. of the Eleventh International Conference of Environmental Degradation of Materials in Nuclear Power Systems – Water Reactors, American Nuclear Society, 2003 p. 1212 G.C. Sih, “Implication of Scaling Hierarchy Associated with Nonequilibrium: Field and Particulate,” In: G.C. Sih and V.E. Panin, Eds., Prospects of Meso-Mechanics in the 21st Century, Special issue of J. Theor. Appl. Fract. Mech., Vol. 37, 2002, pp. 335–369 G.C. Sih and B. Liu, “Mesofracture Mechanics: A Necessary Link,” J. Theor. Appl. Fract. Mech., Vol. 37, No. 3, 2001, pp. 371–395. G. C. Sih and X. S. Tang, “Scaling of Volume Energy Density Funtion Reflecting Damage by Singularities at Macro-, Meso- and Micro-Scopic Level,” J. Theor. Appl. Fract. Mech., Vol. 43, No. 2, 2005, pp. 211–231 G.C. Sih, “Segmented Multiscale Approach by Microscoping and Telescoping in Material Science,” In: G.C. Sih, Ed., Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, Springer, 2006, pp. 259–289 G.C. Sih and X.S. Tang, “Dual Scaling Damage Model Associated with Weak Singularity for Macroscopic Crack Possessing a Micro/Mesoscopic Notch Tip,” J. Theor. Appl. Fract. Mech., Vol. 42, No. 1, 2004, pp. 1–24 X.S. Tang and G.C. Sih, “Weak and Strong Singularities Reflecting Multiscale Damage: Micro-Boundary Conditions for Free-Free, Fixed-Fixed and Free-Fixed Constraints,” J. Theor. Appl. Fract. Mech., Vol. 43, No. 1, 2005, pp. 1–58 G.C. Sih and X. S. Tang, “Micro/Macro Crack Growth Due to Creep -Fatigue Dependency on Time-Temperature Material Behavior, “J. Theor. Appl. Fract. Mech., Vol. 50, No. 1, (2008), pp. 9–22. G.C. Sih, “Anomalies Concerned with Interpreting Fatigue Data from Two-Parameter Crack Growth Rate Relation in Fracture Mechanics,” J. Theor. Appl. Fract. Mech., Vol. 50, No. 2, 2008, pp. 142–156 G.C. Sih, “Thermomechanics of Solids: Non-equilibrium and Irreversibility,” J. Theor. Appl. Fract. Mech., Vol. 9, No. 3, 1988, pp. 175–198 G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publishers, Boston, 1991. D. Broek and J. Schijve, “The Influence of the Mean Stress on the Propagation of Fatigue Cracks in Aluminum Alloy Sheets,” National Aeronautics and Astronautics Research Institute NLR-TN M. 21111, Amsterdam, 1963, pp. 1–57 G.C. Sih, “Signatures of Rapid Movement of Electrons in Valence Band Region: Interdependence of Position, Time and Temperature,” J. Theor. Appl. Fract. Mech., Vol. 45, No. 1, 2005, pp. 1–12 G.C. Sih, “Electron Cloud Overlap Related to Specific Energy Threshold and Breakdown at High Temperature, Short Time and Nano Distance,” J. Theor. Appl. Fract. Mech., Vol. 50, No. 3, 2008, pp. 173–183 A. Jiang, N. Awasthi, A.N. Kolmogorov, W. Setyawan, A. Borjesson, K. Bolton, A.R. Harutyunyan, and S. Curtarolo, “Theoretical Study of the Thermal Behavior of Free and AluminaSupported Fe-C nanoparticles,” Phys. Rev. B, Vol. 75, 2007, pp. 205–426 J. Sun and S.L, Simon, “The Melting Behavior of Aluminum Nanoparticles,” Thermochim. Acta, Vol. 463, 2007, p. 32 G.C. Sih, “Introduction to a Series on Mechanics of Fracture: Scale Shifting Factor,” In: Methods of Solutions of Crack Problems, Mechanics of Fracture, Vol. I, G.C. Sih, Ed., Noordhoff International Publishing, Leyden, 1973, pp. IX–XII D. Xie, M.P. Wang, and W.H. Qi, “A Simplified Model to Calculate the Surface-to-Volume Atomic Ratio Dependent Cohesive Energy of Nanocrystals,” J. Phys. Condens. Matters, Vol. 16, 2004, pp. L401–L405 K. Ando and Y. Oishi, “Effect of Surface Area to Volume on Oxygen Self-Diffusion Coefficients Determined for Crushed MgO-Al2 O3 Spinels,” J. Am. Ceram. Soc., Vol. 66, No. 8, 1983, pp. C131–C132

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22. M.S. Conradi, M.A. Bruns, A.L. Sukstanskii, S.S. Gross, and J.C. Leawoodsa, “Feasibility of Diffusion-NMR Surface-to-Volume Measurements Tested by Calculations and Computer Simulations,” J. Magn. Reson., Vol. 69, 2004, p. 196 23. G.C. Sih and X.S. Tang, “Form-Invariant Representation of fatigue Crack Growth Rate Enabling Linearization of Multiscale Data,” J. Theor. Appl. Fract. Mech., Vol. 47, No. 1, 2007, pp. 1–14 24. G.C. Sih and X.S. Tang, “Triple Scale Segmentation of Non-equilibrium System Simulated by nano-Micro-Atomic Line Model with mesoscopic Transitions,” J. Theor. Appl. Fract. Mech., Vol. 44, No. 2, 2005, pp. 116–115 25. G.C. Sih and X.S. Tang, “Screw Dislocations Generated for Crack Tip of self-Consistent and self-Equilibrated Systems of Residual Stresses: Atomic, Meso and Micro,” J. Theor. Appl. Fract. Mech., Vol. 43, No. 3, 2005, pp. 261–307 26. J. Friedel, Dislocations, Pergamon Press, Oxford, 1964 27. G.C. Sih, “Birth of Mesomechanics Arising from Segmentation and Multiscaling: NanoMicro-Macro,” J. Phys. Mesomechanics (English-Elsevier), Vol. 11, No. 3–4, 2008, pp. 128–140

Chapter 8

Multiscale Modeling of Nanocomposite Materials Gregory M. Odegard

Abstract Composite and nanocomposite materials have the potential to provide significant increases in specific stiffness and specific strength relative to materials used for many engineering structural applications. To facilitate the design and development of nanocomposite materials, structure–property relationships must be established that predict the bulk mechanical response of these materials as a function of the molecular- and micro-structure. Although many multiscale modeling techniques have been developed to predict the mechanical properties of composite materials based on the molecular structure, all of these techniques are limited in terms of their treatment of amorphous molecular structures, time-dependent deformations, molecular behavior detail, and applicability to large deformations. The proper incorporation of these issues into a multiscale framework may provide efficient and accurate tools for establishing structure–property relationships of composite materials made of combinations of polymers, metals, and ceramics. The objective of this chapter is to describe a general framework for multiscale modeling of composite materials. First, the fundamental aspects of efficient and accurate modeling techniques will be discussed. This will be followed by a review of current state-of-the-art modeling approaches. Finally, a specific example will be presented that describes the application of the approach to a specific nanocomposite material system.

8.1 Introduction While many experimental-based methods for characterizing the mechanical behavior of composite materials have proven to be robust and accurate, they incur high costs due expensive testing equipment, specimen fabrication, and labor. Fortunately, G.M. Odegard (B) Department of Mechanical Engineering – Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 8,

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computational modeling can be used to facilitate the development, characterization, and testing of composite materials by providing reliable and efficient structure–property relationships that are determined via inexpensive computational cycles. In the past two decades, computational molecular modeling approaches [1] have emerged as important tools that can be used to predict atomic structure, vibrational frequencies, binding energies, heats of reaction, electrical properties, and mechanical properties of organic and inorganic materials. These methods are ideal for studying the behavior of atoms on the scale of nanometers or below. While the very largest of these simulations can include up to a billion (1×109 ) atoms, most engineering structures contain atoms numbering on the order of at least Avagadro’s number (1×1023 atoms). Clearly, many orders of magnitude exist between the number of atoms that computational chemistry methods can model and which engineering structures contain. Since the 17th century, mathematicians, scientist, and engineers have continually worked to establish models for the understanding of behavior of bulk quantities of material [2, 3]. These models are important for understanding the motion, deformation, failure, buckling, and vibration of engineering structures and components. Because models that are established for these purposes cannot realistically contain all of the details used in computational chemistry simulations, for the reasons discussed in the preceding paragraph, approximations to the behavior of material must be made for which the number of atoms is minimally on the order of 1×1023 atoms. The assumption in continuum mechanics is that matter is modeled in a three-dimensional Euclidean point space for which points in a body in motion do not represent individual atoms. Instead, a body in motion contains an infinite number of points which form a continuum. For the development of nanostructured materials for engineering applications, it is often necessary to establish structure–property relationships, for which the molecular structure must be related to engineering-scale behavior, which is a feat not commonly performed in classical elasticity analysis of engineering structures. This clearly requires tools to relate the predicted molecular structure and behavior from computational chemistry techniques to the continuous nature of solid and fluid mechanics models. These tools must span the length scale difference between nanometers and meters for computational chemistry and engineering methods, respectively. Basic descriptions of the computational mechanics and computational chemistry modeling tools used in multiscale modeling will be described below. Following this, methods for relating computational chemistry and continuum mechanics will be presented. It is not intended to present an exhaustive review of the effective modeling techniques of nanostructured materials that numerous researchers have established in the last few decades. Instead, this chapter focuses on establishing a mathematical framework that presents a general approach to modeling polymer-based nanostructured materials with effective continuum models.

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8.2 Computational Modeling Tools Molecular dynamics (MD) [1] is a computational technique in which a time evolution of a set of interacting atoms is followed by integrating their equations of motion. The forces between atoms are due to the interactions with the other atoms. A trajectory is calculated in sixN -dimensional phase space (three position and three momentum components for each of the N atoms). Typical MD simulations are performed on molecular systems containing up to millions of atoms and for simulation times up to nanoseconds. The physical quantities of the system are represented by averages over configurations distributed according the chosen statistical ensemble. A trajectory obtained with MD provides such a set of configurations. Therefore, the computation of a physical quantity is obtained as an arithmetic average of the instantaneous values. Statistical mechanics is the link between the nanometer behavior and thermodynamics. Thus the atomic system is expected to behave differently for different pressures and temperatures. The interactions of the particular atom types are described by the total potential energy of the system, U, as a function of the positions of the individual atoms at a particular instant in time. U = U (xi , . . . , x N )

(8.1)

where xi represents the coordinates of atom i in a system of N atoms. The potential equation is invariant to the coordinate transformations, and is expressed in terms of the relative positions of the atoms with respect to each other, rather than from absolute coordinates. Continuum mechanics [2, 3] describes the motion and interaction of a set of 0-dimensional particles that form a mathematical continuum. For a given set of initial and boundary conditions of a particular volume of a continuous medium, the motion and interaction of the particles is governed by a series of fundamental laws. The balance of mass governs the change in mass that occurs for a control volume. The balance of linear momentum and angular momentum for a continuum ultimately leads to equations of motion. The balance of energy forces the continuous system to follow the first law of thermodynamics and includes changes in energy due to damage or other physical phenomena. The Clausius–Duhem inequality ensures that the continuous system obeys the second law of thermodynamics. Finally, the constitutive equations govern the behavior between thermodynamic forces (e.g., temperature, applied deformation, and applied electric and magnetic fields) and thermodynamics fluxes (e.g., stress, entropy, and electric polarization). Perhaps the most important aspect of continuum mechanics is its absolute flexibility for any type of material system. By appropriately modifying the balance laws and constitutive equations, complex material systems such as biological tissue, cellular materials, piezoelectric materials, and composites can be modeled on any length-scale level. Although this flexibility allows the behavior of any material to be modeled on the bulk level, it also renders continuum models to be dependent on

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empirical data obtained through testing for accurate material characterization. This is in contrast with the MD approach in which the atomic behavior is established from basic physical principles of atomic theory. Often, structures must be analyzed that are assumed to be composed of a continuous material that behaves according to a predefined-continuum theory. For many complex structure geometries and loading conditions, closed-form solutions to the governing laws of continuum mechanics are not available. In this case, the finite element analysis (FEA) method [4] is particularly useful in predicting the behavior for a wide range of structural geometries and load conditions. Using FEA, the stress and strain fields in an engineering structure are determined by discretizing the continuum into elements of primitive shapes (e.g., bricks and tetrahedrons). The nodes are at the corners, and sometimes, on the midsides of the element boundaries. As long as the geometry of the elements (mesh) is not too coarse, the overall predicted properties of FEA models can be accurately estimated by solving for the stress and strain fields of all of the elements in the model using standard numerical approaches. Micromechanical methods are used to predict the bulk properties of continuous yet heterogeneous solids. While the assumption of continuity is maintained, it is assumed that different phases of the material exist that interact with each other by transferring load. A large number of micromechanical approaches have been developed [5] with a wide range of assumptions. While many approaches are efficient with less accuracy, others are mathematically complex with tremendous accuracy. It is important to note that FEA can be used as a micromechanical tool. The analysis tools described in this section will be placed into the context of multiscale modeling throughout the rest of this chapter.

8.3 Equivalent-Continuum Models Many material properties are used to describe the behavior of continuum-based systems, such as Young’s modulus, density, and strength. While these types of parameters are indispensable to design engineers, they can only be defined for a mathematical continuum. Therefore, these quantities cannot be used to describe the properties of a discrete structure. For example, a lattice structure (truss or molecular structure) as a whole cannot have a corresponding value of Young’s modulus associated with it directly. However, the lattice structure can be modeled as an equivalent-continuum structure in which the overall behavior is similar to that of the lattice structure for the same loading conditions. The equivalent continuum must have equivalent material properties such as Young’s modulus associated with it in order to describe the mechanical behavior. This brings up the question: How is the properties of an equivalent continuum of a discrete structure determined? This is the topic of this section.

8.3.1 Representative Volume Element Consider Fig. 8.1, which shows three different molecular models with different length-scale levels. The structure of these three molecular models is compared with

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Fig. 8.1 Different length scales of molecular models

the structure of an element of a mathematical continuum. When comparing the ˚ to the continuum model, it is clear molecular model with a side length of 10 A that the points in the continuum do not have a one-to-one correspondence with the atoms in the molecular model. Even if the centroid of each atom in the molecular model were mapped to the corresponding points in the continuum model, there are points in the continuum model that do not correspond to atoms in the molecular model, thus eliminating a one-to-one correspondence of points and atoms. As the ˚ to 50 A, ˚ it is clear that there length scale of the molecular model increases from 10 A are more points in the continuum model that correspond to centroids of atoms in the molecular model, although there is still not a one-to-one correspondence between ˚ it becomes difficult to disthe two. As the length scale continues to grow to 200 A, cern individual atoms in the image, as the atom sizes become very small compared to the molecular model size. In fact, the cubes of the molecular models with side ˚ and 200 A ˚ contain approximately 80 and 640,000 atoms, respeclengths of 10 A ˚ molecular model with the continuum tively. Therefore, comparison of the 200 A ˚ molecular model, model clearly shows the nearly continuous nature of the 200 A as the significance of the behavior of individual atoms significantly decreases with increased molecular model size. In fact, it is generally assumed that molecular mod˚ model behave like the continuous, mathematical els of about the size of the 200 A continuum model with clearly defined elastic constants and smooth fields. However, large molecular models are computationally difficult to work with, so methods are necessary to model smaller molecular models as continuum-based entities. A representative volume element (RVE) is such a model. The morphological structural features of the RVE do not necessarily behave like points in a

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homogeneous continuum when a motion or deformation is applied. Instead, the RVE is defined as the volume element that efficiently and effectively describes the structure of the material in a statistical sense. That is, if a solid is built up of identical RVEs that are placed side by side with a perfect packing, then the resulting macrostructure will effectively describe the structure of the material. In some cases, RVEs can perfectly describe the minimum building block necessary to describe the continuum, such as the unit cell of a single crystal metal (Fig. 8.2) or the fiber composite material with hexagonal packing (Fig. 8.3). These materials are referred to as periodic materials. In other cases, the RVE can only approximately describe the minimum-sized building block of a material, such the RVE of bulk metallic glass (Fig. 8.4). Such amorphous materials have no long-range order, so that an RVE of an amorphous material can never exactly model the structure of the material at any spatial point. Rather, the RVE of an amorphous material only models the structure in a statistically accurate manner.

Fig 8.2 Crystal unit cell

In Figs. 8.2, 8.3, and 8.4, the periodicity vectors are indicated by L(1) ,. . .,L(4) . The periodicity vector components describe the length between opposite sides of a RVE. For the cubic RVEs shown in Figs. 8.2 and 8.4, there are three vectors, L(1) , L(2) , and L(3) , each corresponding to a pair of sides of the RVE. For the larger RVE shown in Fig. 8.3, there are four vectors, L(1) ,. . .,L(4) . The question remains: How is the RVE of a heterogeneous non-continuous structure used to establish the continuum-based balance laws and constitutive equations? As we have seen, the balance laws and constitutive equations of continuum mechanics assume the presence of a mathematical continuum, which clearly does not exist in the RVEs shown in Figs. 8.2, 8.3, and 8.4. The answer to this question will be addressed in the next section.

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Fig. 8.3 Fiber composite microstructure

Fig. 8.4 Bulk metallic glass molecular model

8.3.2 Equivalent Continuum At some level, no material is truly described by a mathematical continuum. All materials contain a heterogeneous or discrete structure on some length-scale level. Some materials contain a definite structure at the micrometer length scale, such as wood and fiber-reinforced composites. Some materials exhibit nearly a homogeneous texture except on the atomic level, such as single-crystal metals. Therefore, it can be argued that any material that is described by a constitutive equation is an equivalent continuum. However, the concept of the equivalent continuum is applied

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to the modeling of materials at length scales in which the characteristic size of the inhomogeneity exists. For example, the modeling of the mechanical behavior of a fiber-reinforced composite material at length scales that are orders of magnitude larger than the fiber diameters does not require an effective continuum, as the material on that level closely approximates a continuum (recall the large molecular model of Fig. 8.1). That is, the fibers are so small with respect to the modeled continuum that they behave similar to a material point in the field equations described above. If the behavior is modeled at length scales that are about the same size of the fiber, then the material will no longer behave in the smooth manner associated with the field equations of a classical continuum. In this case, the equivalent-continuum properties are determined by studying the effective mechanical response of the RVE. Consider the RVE of the LaRC–CP2 polyimide system shown in Fig. 8.5. Suppose that an equivalent volume and shape of a mathematically continuous solid is used to represent the RVE, denoted as region R (with boundary∂R). The equivalent-continuum solid should mimic the behavior of the molecular model as closely as possible under all mechanical loadings. It is important to note that an equivalent-continuum model will ideally represent the behavior of an arbitrary volume of the actual material, not just the shape defined by the RVE. The definition of R is only necessary for relating the mechanical response of the RVE to that of the equivalent material points in the equivalent continuum.

Fig. 8.5 Equivalent continuum model of a molecular RVE

The mechanical response of the RVE can only be studied by applying boundary conditions to the RVE. This is usually performed computationally. For example, displacements can be applied to the surface of a RVE to calculate the resulting loads, and hence mechanical properties, of a solid material. Although electric and magnetic fields can also be applied as boundary conditions, the following discussion will focus on mechanical properties. There are four types of boundary conditions for RVEs in static equilibrium: displacement-controlled boundary conditions (also called kinematic or Dirichlet boundary conditions), traction-controlled boundary conditions (also called static or Neumann boundary conditions), periodic boundary conditions, or a mixture of these three. The boundary conditions are applied at locations in the RVE (such as the centroids of atoms) that have the same locations as material points in the equivalent continuum R.

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For displacement-controlled boundary conditions, the components of the prescribed displacement vector u¯ (boldface indicates a vector quantity) are specified everywhere on ∂R u¯ i = u i

∀X ∈ ∂R

(8.2)

where u is the displacement vector. For traction-controlled boundary conditions, the components of the prescribed traction vector s¯ are specified everywhere on ∂R as s¯i = Si j N j

∀X ∈ ∂R

(8.3)

where Si j are the stress tensor components and N j are the surface unit normal vector components. For periodic boundary conditions, the prescribed displacements and tractions are given by u¯ i (X + L) = u i (X) + u iave

s¯ (X + L) = −s (X)

∀X ∈ ∂R

(8.4)

where uave is the average displacement vector associated with the bulk deformation of the solid material, and L is the periodicity vector of the RVE (shown in Figs. 8.2, 8.3, and 8.4). For the case of mixed boundary conditions, the boundary of region R can be divided into three sub-boundaries, ∂Rd , ∂Rt , and ∂R p , such that ∂R = ∂Rd ∪ ∂Rt ∪ ∂R p

∂R◦d ∩ ∂R◦t ∩ ∂R◦p = ∅

(8.5)

where ∅ is the null set and the superscript ◦ denotes the relative interior. The corresponding boundary conditions are u¯ i = u i ∀X ∈ ∂Rd s¯i = Si j N j ∀X ∈ ∂Rt u¯ i (X + L) = u i (X) + u iave ; s¯ (X + L) = −s (X) ∀X ∈ ∂R p

(8.6)

Although the simplest approach to applying boundary conditions to an RVE (either computationally or analytically) is to use the displacement- or tractioncontrolled boundary conditions, it has been shown [6, 7] that the application of periodic or mixed boundary conditions to the RVE results in a more realistic material response. When determining effective-continuum properties of a material by computationally simulating the deformation of an RVE, the natural question arises: What size do the edges of an RVE need to be? The answer to this question depends mostly on two issues: the periodicity of the specific material and the boundary conditions that will be used to compute the material structure and properties. Specifically, the minimum size of an RVE for a periodic material is the minimum size of a possible repeatable structure, as shown in Fig. 8.3 for the smaller RVE of the fiber-composite material. However, if periodic boundary conditions are applied to the edges of the RVE, then the minimal size of the RVE is the minimum size of necessary to construct

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the material structure without rigid rotations of the RVE, such as the larger RVE in Fig. 8.3 and the RVE in Fig. 8.2. If periodic boundary conditions are not used, then the computational results will depend on the RVE size. This dependence will depend on the morphology and properties of the material. Generally, the larger the RVE, the more accuracy is obtained with the results. For example, consider the elastic properties of aligned-fiber composites established by Jiang et al. [7]. Tractioncontrolled, displacement-controlled, and periodic boundary conditions were applied for bulk-level transverse shear (shear in the plane transverse to the fiber direction). Two cases were simulated, one which contained inclusions that were stiffer than the matrix by an order of magnitude and one in which the matrix was stiffer than the inclusions by an order of magnitude. The Young’s modulus of the matrix was assumed to be 1 GPa. The corresponding transverse shear modulus was determined using all three boundary conditions for the two composite systems for RVE sizes of δ = δ0 ,2δ0 ,3δ0 , and 4δ0 , where δ0 is the RVE size shown in the inset of Figs. 8.6 and 8.7. Figures 8.6 and 8.7 show the calculated shear modulus for the stiff inclusions and matrix, respectively, for the different RVE lengths. The data has been plotted with smooth lines to emphasize the overall trend. Clearly, for the case of stiff inclusions shown in Fig. 8.6, the discrepancy between the predicted modulus from the periodic and displacement boundary conditions is larger than that between the periodic and traction boundary conditions. Both discrepancies decrease as the RVE size increases. For the case of the stiff matrix shown in Fig. 8.7, the predicted shear modulus from the displacement boundary conditions has better agreement with the periodic boundary conditions than does the modulus predicted with the

Fig. 8.6 Transverse shear modulus of fiber composite with stiff inclusions

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Fig. 8.7 Transverse shear modulus of fiber composite with stiff matrix

traction boundary conditions. Again, as the RVE size increases, the different sets of boundary conditions predict a more similar shear modulus. For amorphous materials, such as that shown in Fig. 8.4, the minimum size of the RVE is the size that statistically represents the structure of the material. This is a fairly ambiguous statement; however, there is no general agreement of a minimum necessary size of an RVE that produces more accurately predicted bulk-level properties. Of course, as the RVE size increases, the more likely the predicted properties will agree with those measured at the bulk-length-scale level. Once an RVE of a molecular structure is established, then it must be used to established equivalent-continuum constitutive properties. This process is described in the next section.

8.3.3 Equivalence of Averaged Scalar Fields So far, the necessity of constructing an RVE to mimic the bulk-scale mechanical response (equivalent continuum) of a material has been discussed. However, the determination of the properties of the equivalent continuum based upon the response of the RVE to applied boundary conditions has not been detailed. The establishment of an equivalent continuum model that behaves the same as the RVE under identical boundary conditions is important when attempting to understand molecular structure/bulk-level property relationships. In general, an equivalent continuum must meet two requirements in order to accurately predict the behavior of a particular material:

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1. Under identical applied far-field deformations (or loads), the RVE and the equivalent continuum must have identical (or nearly identical) values of one or more scalar fields that are averaged over the volume of the RVE and volume R. 2. The material points of the equivalent continuum volume R must have the same kinematic motion as material points (atoms in some cases) of the heterogeneous RVE at the same locations relative to some defined basis set and origin. This section describes Requirement #1, the Equivalence of averaged scalar fields, in detail. Requirement #1 expresses the need to have one or more scalar parameters; such as the scalar strain-energy density or the six scalar components of the stress tensor averaged over the RVE; to have equal values under identical loads applied to the RVE and equivalent-continuum models. For example, if periodic boundary conditions are applied to the RVE via Equation (8.4) such that the total potential energy, as calculated with an atomic potential, is equal to the strain energy of the effective continuum if a homogeneous deformation field, which matches the deformations averaged over the RVE, is applied to the effective continuum. In other words, if identical deformations are applied to a very large RVE and the equivalent continuum, the total energies should be the same. Moreover, the material parameters of the equivalent continuum can be adjusted such that the energies of the two models match. It is important to note that many researchers choose to match components of virial and continuum stress tenors (from the RVE and effective continuum models, respectively) under identical conditions. Although this approach makes intuitive sense, matching a single scalar parameter, such as strain energy, is a much more efficient approach than matching six independent components of a symmetric stress tensor. Further details on this requirement can be found elsewhere [8]. Once Requirement #1 is satisfied, then Requirement #2 must be considered.

8.3.4 Kinematic Equivalence Requirement #2 is often referred to as the Cauchy–Born Rule, which requires the kinematic motions of the RVE and the equivalent continuum to match for each atom (in the case of a molecular RVE) or each material point (for a heterogeneous microscale model) of the RVE. For example, in the case of the RVE shown in Fig. 8.2, the equivalent continuum should deform in the same identical manner on both the edges and in the interior as the center of the atoms in the RVE. This requirement often requires higher-order elasticity theories [9–11] to be used to accurately match RVE and effective continuum deformations. Although this requirement can be easily satisfied for the deformations of simple RVEs such as that shown in Fig. 8.2, this rule is usually ignored for more complex RVEs, such as those shown in Figs. 8.3 and 8.4, for one of the following reasons. Either (a) this requirement is over-restrictive and unnecessary given the required predictive accuracy of the effective continuum, or (b) this requirement is extremely

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difficult to impose on very complex RVEs. An example of point (a) is captured with the heterogeneous RVE of Fig. 8.3. In the composites community, it is often unnecessary to have predictive effective continuum models to predict the point-topoint kinematic mechanical behavior of the microstructure. The relaxation of this requirement has presented few difficulties in the successful design and implementation of most fiber-reinforced composite materials in the last several decades. An example of item (b) is for the amorphous RVE shown in Fig. 8.4. Given the complex atomic interactions that occur on this length scale for a set of atoms that have no local geometric order, the kinematic motion of the atoms is not expected to be uniform. Establishing a higher-order effective continuum model to match a highly nonuniform deformation field would be an exhausting and unnecessary task. Therefore, although Requirement #2 is rigorously followed for simple crystalline (or highly ordered) material systems, it is rarely followed for amorphous or structurally complex materials. The followed section describes how these requirements are met for specific material systems.

8.4 Equivalent-Continuum Modeling Strategies This section addresses specific approaches to establishing an equivalent-continuum representation of a molecular representation of a material.

8.4.1 Crystalline and Highly Ordered Material Systems A vast majority of the research conducted into determining the effective properties of engineering materials based upon molecular structure has focused on crystalline material systems and highly ordered systems, such as carbon nanotubes. While the origins of multiscale modeling of simple-structured materials relied on many simplifying assumptions [12, 13], recent efforts have been focused on computational simulation. This section presents a brief review of these efforts. The Macroscopic, Atomistic, ab initio dynamics (MAAD) modeling approach was developed by Abraham et al. [14] to simultaneously simulate tight binding, molecular dynamics, and finite element processes of a material element. In this scheme, the finite element mesh is fine enough such that the element sizes are on the order of the atomic spacing. The Coarse-Grained Molecular Dynamics (CGMD) approach was developed at about the same time [15]. The CGMD approach removes the tight-binding simulation component of the simultaneous simulation methodology such that only molecular dynamics and finite element simulations are used. Perhaps the most well-known effective continuum approaches for crystalline materials is the Quasi-Continuum approach [16]. This approach takes advantage of the Cauchy–Born rule to establish an effective finite element domain that simulates the motion of the crystalline molecules. The effective properties of the finite element mesh are derived from an interatomic potential energy function, and an

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adaptive finite element technique is used to simulate the material response to large deformations. Higher-order elasticity theories have been used in conjunction with the Quasi-Continuum method to model heterogeneous deformations. The Coupled Atomistics and Discrete Dislocation (CADD) approach was developed [17–19] to incorporate dislocation plasticity into the molecular statics and effective continuum models. Therefore, dislocations generated within the atomics simulation region can effectively pass onto the continuum model region, such that the generation and behavior of dislocations can be modeled on separate length scales. Challenges for the CADD approach include the extension to dynamic problems and to three-dimensional simulations. The Bridging Domain Method was developed by Xiao and Belytschko [20]. This approach couples molecular dynamics region with a continuum region that surrounds the molecular dynamics domain. A finite-sized overlap in the two models is called the bridging domain. The kinetic and potential energies associated with the bridging domains are a graded mixture of those associated with the molecular and continuum models. This approach can model the different time steps of the molecular and continuum models, and has been used in wave and crack propagation problems. The Bridging Scale modeling approach was proposed by Wagner and Liu [21] which couples molecular dynamics simulations and continuum mechanics modeling by projecting the MD solution onto finite-element shape functions. As a result, the kinetic energies of the two simulations are decoupled and separate time step sizes can be used. Therefore, the finite element solution is not limited to the time scale of the molecular simulations. The Atomic-Scale Finite Element Method (AFEM) was developed [22] to model crystalline materials and carbon nanotubes-based materials. This method directly incorporates atomic potentials into the finite element method. A simple energy minimization is used to establish the mechanical behavior of the material. Although this approach has been used to analyze a wide range of nanotube-related problems, similar challenges remain as with the other methods discussed in this section. The application of these methods to problems of complex molecular structure with a wide range of chemical bonding types and atomic varieties has yet to be achieved. Yamakov et al. [23] used MD simulations to model the grain–boundary fracture behavior in aluminum. Effective-continuum elastic properties and the effectivecontinuum decohesion law were established using energies associated with atomic forces near the crack tip. Although these approaches are very powerful, the following sections describe alternative approaches for non-crystalline material systems.

8.4.2 Fluctuation Methods A broad class of methods has been developed to predict effective mechanical properties of materials based on the thermal fluctuation behavior of a RVE when no

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loads or deformations are applied. The resulting fluctuation methods are based on the work of Parrinello and Rahman [24]. As the RVE is simulated using MD, the geometric fluctuations of the RVE are correlated with the corresponding potential energy of the system. Because the average stress components in the RVE can be related to the derivative of free energy with respect to deformation, a series of stress–strain behaviors are observed computationally as the molecular system evolves over time. The thermal fluctuations of the RVE include axial deformation and shear deformation modes. Therefore, for a given sampling of simulation time, the six independent components of stress can be correlated with the six independent components of strain, and the corresponding elastic constants are subsequently established via the constitutive equation. A wide range of specific techniques have been utilized for specific materials and simulation ensembles [25–31]. Although the fluctuation methods offer an efficient approach to establishing elastic constants of a wide range of materials, the obvious drawback is the inability of this approach to predict the mechanical response of materials to large deformations. To achieve this, the RVE must be subjected to applied loads or deformations of the appropriate magnitude. This process is described in the next two sections for static and dynamic problems.

8.4.3 Static Deformation Methods Perhaps the most common approach to determining the elastic properties of an effective continuum is with the static deformation method [32–53]. Although many variations exist, this technique involves the application of homogeneous axial, volumetric, and shear deformations/loads to the boundaries of an RVE. Consistent with requirement #1 of an effective continuum, the corresponding strain energy, stress, or strain is calculated. Constitutive equations are used to determine the elastic properties based on the response of the RVE to applied loads. This approach is applicable to amorphous material systems. A reference configuration of the molecular structure of the RVE is determined by subjecting an equilibrated molecular structure to an energy minimization. The reference configuration of the RVE is subsequently exposed to the applied deformations. Initially, the boundaries and the atoms are displaced uniformly and the molecular structure is again subjected to an energy minimization to establish the deformed configuration. The potential energy or the stresses are calculated and compared to the corresponding values in the reference configuration, and elastic properties are determined based on the constitutive equations. Alternatively, loads can be applied to the reference configuration, and the resulting deformations can be determined for an equilibrated deformed state. This method has been extended into a hyperelastic framework to allow for large deformations of materials [49]. The entire static molecular deformation process is usually performed using molecular mechanics or FEA. Therefore, the effects of the thermal motion of atoms on the mechanical behavior of the material are disregarded with this method.

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However, the efficiency and accuracy of this method has made it a popular choice for predicting bulk mechanical properties of finite-sized molecular systems. For modeling time-dependent effects, such as thermal motion, many researchers use the dynamic methods described in the next section.

8.4.4 Dynamic Deformation Methods Similar to the static deformation method, the dynamic deformation method involves deformation of the RVE to determine effective continuum properties [45, 49, 54–60]. However, in the dynamic approach, the motion of the atoms in response to a deformation is determined with MD. Therefore, the time-dependent response of the atoms to applied deformations is computed with the integrated equations of motion, not energy minimizations. This allows the effect of the thermal motion of the atoms on the predicted properties to be determined. Associated with the deformation of the RVE with this approach is a strain rate (or loading rate). The applied deformation is incrementally prescribed onto the RVE to simulate an applied bulk strain rate. However, because such simulated strain rates are orders of magnitude larger than those experienced in the laboratory and in most engineering applications, the resulting calculated effective continuum properties may not match experimentally determined values and they may not be practical for the design of engineering materials. Furthermore, determination of effective continuum properties with the dynamic approach generally requires significantly more computational time than with the static approach. The selection of one of the methods described in this section strongly depends on the material and behavior type that needs to be modeled. Every particular problem has a different approach, as demonstrated in the following section.

8.5 Examples This section presents brief examples of the modeling strategies outlined in the previous section for determining the effective continuum properties of nanostructured materials.

8.5.1 Silica Nanoparticle/Polymer Composites This section presents a brief example of the static modeling strategy outlined in the previous section for determining the effective continuum properties of a silica nanoparticle/polymer composite. The example follows the work of Odegard et al. [40], in which a continuum-based constitutive model was developed for silica nanoparticles/polyimide composites with four different nanoparticle/polyimide interfacial treatments. The model incorporated the molecular structure of the

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nanoparticles, polyimide, and interfacial region. The model was used to examine the elastic properties of the composite as a function of nanoparticles radius. The silica nanoparticles had an α-quartz crystal structure, and the nanoparticles ˚ The polymer were nearly spherical in shape, with an approximate radius of 6 A. modeled was a thermoplastic polyimide with an amorphous molecular structure. The first variation of the composite had a silica nanoparticle without surface treatment that was not bonded to the surrounding polyimide. The second variation had the nanoparticle surface comprised of hydroxyl groups that were bonded to the silicon atoms. In this variation, there were no covalent bonds between the polyimide molecules and the nanoparticle. The third variation had phenoxybenzene groups chemically bonded to the surface of the nanoparticle, and the phenoxybenzene groups were not directly bonded to the polyimide matrix. The fourth variation had a hydroxylated surface with the nanoparticle covalently bonded (functionalized) to the surrounding polyimide molecules. These material variations are shown in Fig. 8.8. In addition to the four composite systems, the pure silica and pure polyimide materials were examined.

Fig. 8.8 Molecular structures of silica nanoparticle/polyimide composites

Molecular modeling techniques were initially used to model the molecular structure of the polymer and the silica nanoparticles. The functionalized structure was constructed by inserting an oxygen atom covalently bonded to a silicon atom in the nanoparticle and a nearby carbon atom in the polyimide. A total of 10 chemical

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bonds were inserted between the silica particle and the polymer matrix. The resulting structures of the four composite systems are shown in Fig. 8.8. It was assumed that the equivalent continuum that represented each of the composite molecular models was homogeneous and isotropic. Hooke’s law was used to modeling both the constituent and bulk composite constitutive behavior. The components of the stiffness tensor were determined using the static method of Theodorou and Suter [48]. As a result, the Young’s moduli of the pure silica, pure polyimide, silica nanocomposite, hydroxylated silica nanocomposite, phenoxybenzene silica nanocomposite, and functionalized silica nanocomposite were 88.7 GPa, 4.2 GPa, 3.4 GPa, 3.3 GPa, 2.2 GPa, and 4.0 GPa, respectively. From these values it is clear that the functionalized composite had a higher Young’s modulus relative to the other composite systems. This is an expected result because the covalent bonds between the nanoparticle and the surrounding polymer increase the load transfer to the stiff silica nanoparticle relative to the non-functionalized systems. Furthermore, it is apparent from this data that the silica composite, the hydroxylated silica composite, and the phenoxybenzene silica composite have bulk Young’s moduli that are less than that of the polyimide alone. Therefore, the non-functionalized surface treatments appear to inhibit the desired reinforcing effect. It is important to note that these results are specific only to this material system. The choice of a different matrix or reinforcement phase can change these trends. Although only elastic properties were determined in this particular study, the multiscale model could also be expanded to predict strength properties as well. While this study offers an excellent example of modeling nanocomposites with spherical reinforcement, the next section will address multiscale modeling of non-spherical reinforcement.

8.5.2 Nanotube/Polymer Composites In 2005 Odegard et al. [41] reported on the constitutive modeling of CNT/polyethylene composites for the two cases in which the CNT reinforcement was functionalized and was not functionalized with the surrounding polyethylene molecules. The purpose of the investigation was to determine if the chemical bonds between the nanotubes and surround polymer matrix transferred more load than the van der Waals forces alone in a non-functionalized composite. A SWNT (10,10) ˚ was modeled as being surrounded by crystalline polyethylene of radius 6.78 A with the polymer chains aligned with the nanotube. Both functionalized and nonfunctionalized systems were modeled. Figure 8.9 shows the modeled RVE for the two systems. Additionally, the pure crystalline polyethylene was also modeled. To establish the equilibrium molecular structures of the three systems, the CNT and polymer chains were replicated across the periodic boundaries of the RVE making them infinitely long in the x1 -direction. In the composite with nanotube functionalization, two polymer chains were attached to six carbons on the nanotube by chemical linkages of two CH2 groups.

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Fig. 8.9 RVEs of non-functionalized and functionalized CNT composite materials

The geometry of the equivalent continuum was assumed to be cylindrical for ease in use in subsequent micromechanical analyses. Thus, the equivalent continuum for these materials is henceforth referred to as the effective fiber. It was assumed that the polymer molecules that were near the polymer/nanotube interface were included in the effective fiber. The effective fiber radius and length were 1.1 nm and 4.3 nm, respectively. The material composing the effective fiber was assumed to exhibit orthotropic symmetry. The elastic properties of the effective fiber were determined by equating the total strain energies of the effective fiber under and molecular model RVE under identical boundary conditions. Further details on this portion of the analysis can be found elsewhere [41]. Once the effective fiber properties were determined, a micromechanics analysis was performed to model the reinforcement of a polyethylene matrix reinforced with effective fibers (Fig. 8.9). It was assumed that the matrix polymer surrounding the effective fiber had mechanical properties equal to those of bulk amorphous polyethylene. All relative movement and interaction of the polymer chains with respect to each other were modeled at the molecular level. This motion and interactions was therefore indirectly considered in the subsequent determination of the properties for the effective fibers, and it is therefore assumed that perfect bonding existed between the nanotube/polymer effective fibers and the surrounding polymer matrix in the micromechanical analysis. The elastic stiffness components, volume fraction, length, and orientation of the effective fiber were used for the fiber properties in the micromechanical analysis. Although the nanotube and effective fiber lengths were equivalent, the nanotube

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volume fraction was determined to be 52.9% of the effective-fiber volume fraction. The overall composite stiffness was calculated for effective fiber lengths up to 450 nm and effective fiber volume fractions up to 90%, which corresponds to the maximum volume fraction for hexagonally packed fibers. The calculations were performed assuming both perfectly aligned and three-dimensional randomly oriented effective fibers in an amorphous polyethylene matrix. The calculated Young’s moduli of the nanotube composites with the amorphous matrix are plotted in Fig. 8.10 as a function of nanotube length, for a 1% nanotube volume fraction. The longitudinal Young’s modulus of the aligned composite and the Young’s modulus of the random composite had a nonlinear dependence on the nanotube length. An increase in the Young’s modulus with respect to an increase in nanotube length is expected to correspond to a relative increase in load-transfer efficiency between the nanotube and surrounding polymer. The data in Fig. 8.10 indicate that at a nanotube length of about 400 nm, the efficiency of load transfer is nearly maximized, as evidenced by the nearly zero slope in the data curve. Further increases in nanotube length beyond 400 nm resulted in relatively small increases in Young’s modulus for a given nanotube volume fraction. As the nanotube length became greater than approximately 100 nm, the Young’s modulus for the composite without nanotube functionalization became larger than that of the composite with nanotube functionalization for the random composite and longitudinal deformation of the aligned composite. At 450 nm, the functionalization reduced the longitudinal Young’s modulus of the aligned composite and the Young’s modulus of the random composite by 11% and 7%, respectively. In contrast, the transverse Young’s modulus

12 Non-functionalized Functionalized

Young's modulus (GPa)

10

8

Aligned nanotubes, Y1

6

4 Randomly-oriented nanotubes, Y 2 Aligned nanotubes, Y2 ~ Y3 0 0

100

200

300

400

Nanotube length Fig. 8.10 Young’s moduli of the composite systems for a 1% nanotube volume fraction

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of the aligned composite had no dependence on nanotube length. Also, there was no effect of functionalization on the transverse deformation of the aligned composite. The longitudinal Young’s moduli of the random and aligned composites are plotted in Fig. 8.11 as a function of nanotube volume fraction, for a constant nanotube length of 400 nm. At this nanotube length, the change in the longitudinal Young’s modulus with respect to nanotube volume fraction of the aligned composite was nearly linear. Over the complete range of nanotube volume fraction, the functionalization of the nanotube reduced the longitudinal Young’s modulus of the two composites. 500 Non-functionalized Functionalized

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Fig. 8.11 Young’s moduli of the composites systems for nanotube lengths of 400 nm

The transverse Young’s moduli of the aligned composite systems with a range of volume fractions and a nanotube length of 400 nm are shown in Fig. 8.12. In contrast to the Young’s moduli in Fig. 8.11, the transverse Young’s moduli of both composites improved when the nanotubes were functionalized. The enhancement is evident for nanotube volume fractions greater than 10% in both composites. The primary implication of these results is that although chemical functionalization of single-walled carbon nanotubes has been considered as a means to increase load-transfer efficiency in nanotube/polymer composites, this functionalization has, in fact, degraded most of the macroscopic elastic stiffness components of the composite materials considered in this study. This is possibly due to the change in chemistry at the functionalization site. The carbon atom on the nanotubes that is functionalized goes from an aromatic state to an aliphatic state upon functionalization. As a result, it is possible that the stiffnesses of the C–C bonds in the nanotube are reduced at the functionalization site, thus reducing the overall composite stiffness in the direction parallel to the nanotubes. However, the increase in C–C bond

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Fig. 8.12 Transverse Young’s moduli of composite systems for nanotube lengths of 400 nm

stiffness perpendicular to the surface of the nanotube after functionalization naturally increases the overall transverse stiffness of the composite. It is important to note that these results are highly dependent on the specific materials that are considered. Different combinations of reinforcement and matrix materials would likely yield different results.

8.6 Summary This chapter describes the fundamental aspects of multiscale modeling, including basic analysis tools and representative element geometries. Examples of the multiscale modeling of nanoparticle/polymer and nanotube/polymer composites have been presented. From the two example cases, it is clear that small changes on the molecular level can have a dramatic influence of macroscale properties. Specifically, for the nanoparticle composites, chemical functionalization of the reinforcement with the polymer matrix improved the overall composite properties. For the nanotube/polymer composite, the functionalization had a detrimental effect on the bulk-level properties. These examples demonstrate the efficient nature in which multiscale modeling can predict the effects of changes in chemistry on the overall material performance before in the design stage of an advanced material. No consistent approach has been taken with multiscale models in the literature since material structures are very different among material classes and because different strategies must be employed for various loading conditions. As a result, these type of tools need to be more rigorously tested and validated for a wider range

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of materials. As research progresses in this field, more refined multiscale modeling approaches may emerge. Future research topics in multiscale modeling include the development of improved methods for modeling amorphous materials; composites with viscoelastic constituents, and polymer/metallic interfaces in nanostructured composites.

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Chapter 9

Predictive Modeling Michael Doyle

Abstract It is the goal of the aerospace industry to minimize the weight of structural parts without compromising the structural integrity and failure resistance of the systems they comprise. Over the past few years, more attention has been paid to composite materials as a possible solution to the challenges and tradeoffs inherent in aerospace design. In these complex composite materials, high-strength fibers provide reinforcement to polymers so that they become capable of carrying the high environmental and mechanical loads desired. Recently, reinforcing nanoparticles (nanofillers) have shown promise in the aerospace resin materials sector due to their superior mechanical and fatigue properties. In order to improve the interaction of these particles with the “host” matrix, there is a need to functionalize them for better interaction with and within the matrix. This functionalization can be guided and ultimately achieved through the use of multiscale modeling techniques and simulation, as part of a true four dimensional design space (width, height, length and material). These multiscale approaches and simulations necessitate significant computational capability. But, not only do they address the limits of conventional approaches on the number of atoms that can be simulated, but also they serve to address the time and length scales intrinsic in the atomistic approach and bring them more in line with those of the “meso-scopic regime” or real performance space. These hybrid approaches have recently shown success in solving these classes of systems. In the hybrid approach, multiple regions are defined within the configuration; some with direct atomistic interactions; some with detailed interactions defined by quantum mechanical density functional theory or semi-empirical or tight binding theory; still others defined by electrostatic or mechanical linkages; and yet others treated by a bulk or continuum representation. The objective of these methods is to predict material properties of the modified parent material when reinforced with nanoparticles using an aggregating approach, in which there are multiple connect domains of simulation. Algorithmic improvements to all of these approaches, coupled with the increasing speed of computational hardware, are making it possible to perform M. Doyle (B) Principal Solution Scientist, Materials Science @ Accelrys, USA e-mail: [email protected]

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predictive modeling on ever larger systems. A number of methods are now available that are capable of modeling hundreds of thousands of atoms, and these results can have a significant impact on real-world engineering and failure analysis problems. This work reviews some of the modeling methods currently in use, provides illustrative examples on obtaining mechanical properties through fundamental laws, and discusses multidimensional material and device design. In this section, computational tools are illustrated that are able to bridge the gap between the characteristic time and spatial scales of the nanochemistry and the macro-scale engineering or physics of the aerospace system. Finally, discussions on the prospects for future modeling approaches will be included.

9.1 Introduction As society advances, so does the demand and need for advanced or specialty materials. Whether this is in the area of medical substances or devices, the construction and civil engineering field, the chemical products field or the transportation or military field, the requirements that drive this technological imperative are consistent. Those are improved performance, ease or control of processing, optimization of manufacturing, greater sustainability, and increased feature or function density. As Pliny the Elder, a studied metallurgist, observed [1], composites or mixtures and alloys can have properties that are significantly altered or enhanced over their constituents. For example, gold, an economically important metal in ancient Rome due to its characteristics of ductility and malleability, had its properties “strangely altered” upon alloying with other metals such as silver in electrum. This possibility of counterfeiting was so unsettling and potentially disruptive to the Roman economy and its financial structure that, he advised, it necessitated a “detailed observation and understanding”[2] of the conditions, properties, and characteristics of its occurrence. Clearly this example shows the societal and cultural impact that advanced materials can have. And, as history shows, from the bronze through the iron ages into the nuclear and space age, new materials and mastery of their properties and applications are critical to the growth and survival of society and culture. It is in this role of interpretation, investigation, and design of new materials at the functional, device, or atomic level that simulation and modeling provide unique insights and enrich materials knowledge. It is also true that as industrialization and global competition have increased over the last two centuries, every country has had to cope with constantly changing and expanding demands for new products, as well as requirements for speedier design, better quality, and lower cost systems. A direct consequence of this increased economic and business competition is the obsolescence of traditional trial-and-error experimental approach. In this new situation, the traditional “single variable” approach to material system optimization [3], such as the COST (or change one variable singly at a time), or the conventional ingredient or process single component swap out approach have been shown to be an inefficient way to produce these required materials in a timely way, and so new research

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strategies have been invented. Computational science, combined with experiments, can play a significant role in these new approaches. For example, recent developments in catalysis [4], polymer composites [5], alloys [6], carbon nanotubes (CNTs) [7], and electronic materials [8] have all been aided by computational approaches. The papers cited here also identify how beneficial such an approach is in terms of intellectual property identification, knowledge building, and skills enhancement. The high-performance computing techniques used are based on fundamental equations of quantum mechanics for solids and molecules [9]. These methods of computer modeling and simulation enable detailed examination of materials behavior at the atomic and electronic levels. However, in some areas such as large numbers of molecules, larger dimensions, longer time, and greater structural complexity, these powerful methods still fall somewhat short of representing a realistic system configuration with millions or billions of atoms. This is the key challenge in modern computation and simulation, namely bridging the length and time domains to yield practical engineering and design results (Fig. 9.1). The complexity of the challenge is highlighted since, paradoxically, computational thermo-chemistry [10] has been used in chemistry and chemical engineering research for many decades [11] as a continuum method for “master data” definition, creation, and hence engineering design. Recently, data-mining and artificial intelligence techniques have also been applied in combinatorial chemistry [12] and chemical engineering research [13] with the aim of aggregating and coordinating the information in both the public and private literature alongside simulation results, as an aid to materials selection and the rapid introduction of new materials into the development process. This approach, which shows much promise in both the government [14] and industrial

Fig. 9.1 Predictive modeling provides a framework for experimentalists and theoreticians to work in parallel and streamline the creation of metrology and standards

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research field, has been the subject of recent DARPA-funded activities [15] and is discussed later in the chapter. From the engineering perspective, structural and engineering mechanics provide good theoretical descriptions for the rational design of materials and accurate lifetime prediction of mechanical structures. Typical examples are the design of aircraft to reduce drag, or the testing of large-scale structures like oil rigs and bridges for stability, safety, and performance. These approaches deal with continuous quantities such as strain field that are functions of geometry, space, and time. Constitutive relations such as Hooke’s law for deformation and Coulomb’s law for friction describe the relationships between these macroscopic fields. These constitutive equations contain material-specific parameters such as elastic moduli and friction coefficients, which are often size dependent, and in multiscale simulation approaches have their fundamental constants derived from computational or information mining methods. For example, the mechanical strength of materials is inversely proportional to the square root of the grain size, according to the Hall [16]–Petch [17] relationship [18]. Such scaling laws are usually validated experimentally at length scales above a micron, but interest is growing in extending these relations and performance scaling laws down to a few nanometers. The reason is because it is believed that by reducing the structural scale (such as grain sizes) to the nanometer range, material performance properties such as strength and toughness can be extended beyond the current engineering-design materials limit. This has been the rationale behind the recent DARPA [19] initiative in the rapid introduction of new materials into the aerospace field. Because of the large surface-to-volume ratios in these nanoscale systems, new engineering and mechanics concepts will be required which reflect the enhanced role of atomic structure and interfacial processes [20]. Atomistic simulations will play an important role [21] in scaling down design, these engineering, and mechanics concepts to nanometer scales [22] (Fig. 9.2). To achieve accurate parameter prediction and constitutive relationship derivation at these short scales, a combination of continuum mechanics and atomistic simulations will be needed with the ultimate goal of a single simulation environment which couples the diverse length scales. In order to characterize the nature of composites and nanoscale materials, and to understand these effects, it has become increasingly important to be able to reliably and accurately measure materials distributions, morphologies, interactions, structures, and compositions. At present, the tools for accomplishing these require measurements at the nanoscale using tools like TEM, high-resolution SEM, crystallography, and NMR spectroscopy. In parallel with the gathering of data from these measurement techniques, a further strand of multiscale modeling is pursuing the development of physics-based data correlation models to better understand and explain the effects on properties that result from the observed material morphologies, interactions, and distributions. This chapter addresses an overview of the methods in such calculations, quantum mechanics, molecular mechanics, and meso-scale modeling. It also addresses processes such as crack propagation and fracture in real materials involving structures on many different length scales. They occur on a macroscopic scale but require atomic-level resolution in highly nonlinear regions. To study such multiscale

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Fig. 9.2 Virtual material characterization is a critical component of simulations since it provides the linkage between physical materials testing and experimental testing

materials processes, we need a multiscale simulation approach that can describe physical and mechanical processes over several decades of length scales. This chapter also addresses how they can be integrated and how the information obtained from such simulations may be used to generate parameters, or limiting values for engineering-scale calculations, better mechanistic understanding of chemical and materials transforms, clearer interpretation and understanding of experiments, and as such provide hard-to-identify or test answers that can be used in the search for new and novel materials.

9.2 Nanocomposites “For the next several decades, there will be very few cases in which an entire product is the result of nanotechnology, but more and more we will find that the crucial or enabling component of a system is engineered at the nanometer scale. . .nanotechnology has the potential to greatly improve the properties of nearly everything that humans currently make”, said R. Stanley Williams, Senior HP Fellow and director of quantum science research at HP Labs, during his testimony before the Senate Subcommittee on Science, Technology and Space on September 17, 2002. There is a common definition of nanotechnology as pertaining to phenomena and new properties arising below the 100-nm level. The fact that there are three orders of magnitude in length scale between the 100-nm level and the size of an atom is

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illustrated in Feynman’s famous lecture “there is plenty of room at the bottom” [23], and provides an indication of the challenge faced by modeling and simulation in this field. A description of nanoscale phenomena will require methods that stretch across many scales, and hence many different types of modeling will play a role. Furthermore, the eventual product or ‘system’ enhanced by this nanotechnology will in general at least be a micro-sized system, that is, many orders of magnitude larger. Three factors define nanotechnology: small size, new properties, and the integration of the technology into materials and devices. Nanotechnology covers a broad range of science, drawing concepts, knowledge and expertise, skills, and materials from all the three classical sciences: physics, chemistry, and biology. A nanocomposite is a multi-component material system including at least one type of nanoscale particulate or additive that is compounded or mixed with an appropriate matrix to produce a well-dispersed material with new properties such as lighter aircraft, moreimpact-resistant automotive bumpers, or modified lumber with resistance to environmental elements. A nanomaterial is a single-constituent material form introduced into a nanocomposite to increase at least one desired property, such as conductivity, mechanical strength, and impact resistance. Nanomanufacturing or nanochemistry is best defined as the technology and science to produce the nanomaterials and can be extended to include the incorporation of nanomaterials into nanocomposites. The chief perceived advantage of nanotechnology in composites has been that, when effective, the incorporation of nanoparticles/nanoconstituents into a composite can produce benefits that are significantly beyond what a standard rule-of-mixtures calculation would predict. These larger-than-expected benefits generally arise from effects that occur at length scales below those of continuum mechanics where principles such as the rule of mixtures apply. In order to achieve the full benefits of nanotechnology in composite materials, it will be necessary to link the constituent material properties, the composite-material morphologies, etc., and the nanocomposite properties to parameters that lead to the performance levels desired. Providing these links will require advanced metrology techniques and improved materials models. Data will be gathered for key end-use properties (e.g., strength, stiffness, toughness) and related to the observed nanoscale properties and input into improved models. Ultimately, it should be possible to identify which nanoscale properties lead to improved macro-scale properties and to build models to apply the specific knowledge gathered from a particular materials system more generally. Despite the difficulties, some nanocomposite-enabled products have made it through to the marketplace: reduced-weight and increased-strength automotive bumpers and bed liners, high-strength nanocomposites for sporting goods, and layered barrier materials, to name but a few. One of the oldest success stories for a man-made nanocomposite is the use of alumina–silicate clays that are exfoliated to individual sheets within polymeric matrices in, for example, sound-deadening materials in the automotive industry. In general, the clay/polymer systems are easier and more reliable because the material itself is cheap, quality controls are adequate, processing conditions necessary for exfoliation are well known, and there are reliable

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methods based on X-ray scattering to monitor dispersion. However, these systems have only been used where high mechanical performance and properties are not very important. Other nanoparticle additives are more problematic. Intense research into the use of carbon nanotubes as additives has only resulted in a few notable nanocomposite successes (conductive polymers for fuel lines in automotive applications, high-strength, and high-modulus materials for sporting goods). High costs and low availability of raw materials, variability in the carbon nanotube manufacturing processes, difficulties in purification and de-agglomeration, and challenges in producing strong carbon nanotube/matrix interactions are some of the challenges in this area. The tendency of carbon nano-tubes to agglomerate is particularly a serious problem for nanocomposites manufacturing, because it significantly decreases the aspect ratio and mechanical properties of the added nanocomponent. The agglomeration also leads to higher percolation thresholds for both electrical and thermal conductivity – properties which are both critical to their use in advanced aerospace applications. Therefore, good dispersion of carbon nano-tubes (CNT) in polymer matrix is essential for effectively enhancing the performance of polymer/carbon nanotube nanocomposites, as advanced composite aerospace materials. Various processing measures and fabrication methods have been employed toward this goal. Today, the most common method for achieving good dispersion of carbon nanotubes in polymer matrices is either through chemical functionalization of carbon nanotubes to improve their intrinsic solubility in the polymer or by surrounding them with dispersing agents such as co-polymers and surfactants. The presence of dispersant or wrapped polymer provides a means to improve the initial carbon nanotube dispersion and prevent the re-aggregation during composite processing. In addition to improving the dispersion, chemical functionalization also allows carbon nanotubes to be more readily wetted by, and form covalent bonds with, the matrices. This results in increased interfacial strengths and improved stress-transfer efficiencies in nanocomposites. One important note of the chemical functionalization is the degradation of properties, particularly the electrical and mechanical properties, due to introduction of defects and modification of the electronic structure in the CNT. Multiscale and atomistic-scale simulations have been used [24] in this area of matrix [25], nanotube compatibilization [26], and in the area of fiber compatibilization [27].

9.2.1 Nanotechnology and Modeling The emergence of nanotechnology as a key new field in materials science has led to many reports [28] and roadmaps [29]. Most of these reports have concluded that modeling and simulation tools will play a key role in the development of the fundamental science, and transitioning the science from an academic endeavor to a commercial reality. As a result, there has been an increased attention to modeling approaches and software tools [30], with a number of workshops and white papers asking for further development of multiscale tools for nanotechnology research.

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Computational models for nanocomposites serve as tools for quantitative analysis of nanocomposite bulk material properties versus nanoparticle synthesis/structure, interfacial properties, inclusion methods, matrix properties, and other relevant variables. Simulation, modeling, and multiscale methods have the potential to provide predictive capability for correlating electrical, thermal, mechanical, and acoustic properties to the structural, synthetic, manufacturing processes as well asand the quality-control (QC) metrics for nanocomposites. One of the key requirements in this area is a modeling framework which enables the integration of models at various length and time scales. Such a framework would provide integration to the end bulk material design tools and the ability to model the interaction of nanoparticles in a composite system. So the focus of modeling is also to predict the end state of performance parameters such as electrical, thermal, mechanical, and acoustic properties of complex systems involving composites (Fig. 9.3). For many systems, it is the surface or interfacial properties that are key to the systems performance and provide significant value to products, increasing resistance to environmental effects, affecting the way in which a surface can be bonded, or cleaned, and of course enhancing appearance. It is well known that many of these properties relate to phenomena at the molecular, meso-, and nanoscale, and so the coatings area has been one of the foremost applications of nanotechnology. A common way of modifying the surface properties of a material is by adding nanoparticles (metal oxides, nanoclays [31], carbon, silica) in the layer which will be applied to the corresponding substrate (ceramics, metals, plastics, papers). The design of such a coating will therefore include a nanocomposite structural aspect as well as a bonding aspect. In the case of automotive or multi-layer surface coatings, the nanostructure can be investigated with codes such as dissipative particle dynamics (DPD) to explore

Fig. 9.3 (a) Molecular dynamics simulation of a silicate surfactant–polystyrene nanocomposite. (b) The corresponding ensemble averaged, number density of carbon atoms as a function of distance

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how altering the short-range interactions between components (an acrylic continuous phase, additive, and air) could be used to achieve variations from ideal mixing that would drive select additives to the surface in a controlled manner. DPD provides a fast way to screen new formulation components using accessible variables that synthesis chemists can vary such as polymer composition and molecular weight, and additive properties. An example of such a coating is CeramiClear clearcoat, an award-wining product from PPG, used on several Mercedes models.

9.2.2 Composites A nanocomposite can be defined as an engineering material made of distinctly dissimilar components mixed at the nanometer scale. Research on those hybrid organic/inorganic materials is an exciting field of nanotechnology which delivers materials with interesting novel properties. The inorganic components in nanocomposites can be zeolites [32], clays [33], metal oxides [34], metal phosphates, and chalcogenides [35]. Those molecular frames can host foreign species as polymers, yielding the new composite material (Fig. 9.4). In 1993, Toyota commercialized a nylon-6/clay nanocomposite [36], out of which it produced the timing belt cover for the Camry. These hybrid nanocomposites show significant improvements in their mechanical response characteristics of

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Fig. 9.5 Equilibrium morphologies of (15, 15) CNTPMMA composites at ambient temperature and pressure as modeled by DPD

the native polymer; the improvements were in the areas of yield strength and heat distortion temperature. Considering clay–polymer nanocomposites, a key feature to be understood and optimized is whether the structure consists of intercalated or exfoliated clay sheets. In the first case, the material has a defined number of polymer chains between the inorganic layers, while in the second case the number of polymer chains between layers is variable. While processing parameters play a role, the strength of the interactions and the elucidation of the structure of those materials are fundamental [37]. The General Utility Lattice Program (GULP) [38] has particular strengths in forcefield for mineral and other inorganic systems. In addition, a consortium of academic groups and commercial interests has extended the forcefield coverage in GULP as well as developed the tools to fit, extend, and combine forcefields to facilitate the study of such complex hybrid systems. The end-use properties of the nanocomposite material depend not only on the properties of the parent materials but also on their morphology and interfacial characteristics (Fig. 9.5). To study the morphological characteristics of the mixture of both materials, coarse-grained codes can be used. For example, for the dispersion of carbon nanotubes in a polymer matrix, the DPD code in the Materials Studio can be used to give the possible resulting equilibrium morphologies [39]. It is also important, when working with nanocomposites, to understand the synthetic routes used to insert the polymer chains into the frame structure. DFT codes available in, for example, the Accelrys Materials Studio (MS) can be used in those studies, ONETEP [40] being especially useful if the size of the systems under consideration exceeds a thousand atoms. In fact, theoretical studies of the formation of clay–polymer nanocomposites by self-catalyzed in situ intercalative polymerization have been performed in the past using the DFT code CASTEP [41]. For polymer–metal nanocomposites, the current QMERA module in MS can also be used to understand the polymerization mechanism.

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Current advanced engineered materials systems, such as organic-matrix composites, have a myriad of applications, including aerospace structures, sporting goods, high-performance automobiles, and boats. Composite aerospace structures often have extreme property demands that make the adoption of higher-performance materials systems such as nanocomposites inviting. However, there is also a strong need to balance the multiple demands of performance, weight, processability, risk, and/or life-cycle cost in selecting new structural materials. As air travel continues to grow, lightweight, multifunctional, and easily manufactured structural materials such as polymer matrix composites (PMCs) are being increasingly used. PMCs combat increased fuel and maintenance costs, which account for roughly 50% and 20%, respectively, of the operation costs of a commercial airplane. A particular challenge for advanced composites is the integration, exploitation, and optimization of nanoparticle-enabled properties within a structured composite made with commercially viable processing methods (Fig. 9.6). Although industrial efforts continue to pursue low cost processing methods for composite fabrication, the current large capital investment of composite processing equipment (automated fiber coating and positioning systems, equipment for infusing uncured resin into fiber pre-forms, high-temperature and high-pressure vessels for void-free composite curing, etc.) makes it initially preferable to use traditional composite processing schemes for integrating nanocomposites. This then dictates and governs the scope of initial nanoparticle deployment in aircraft-grade composite systems. Nanoparticles can be incorporated in a number of different ways within the traditional composite material forms, including within a fiber, as a thin coating on a fiber, in place of a bundle of fibers (i.e., a fiber tow), as an inner layer, as a coating, or as a part of the polymer resin system. Mechanical improvements in traditional composite materials through the use of nanocomposites can be targeted toward improvement of resin-dominated properties or, eventually, toward fiber-dominated properties, the latter being a more difficult and challenging goal. Resin-dominated complex properties include inter-laminar shear strength (ILSS),

Fig. 9.6 Multiple scales of modeling used from left to right, quantum (insert), atomistic composite simulation, and finally meso-scale simulation. After work by Maiti et al. on carbon nanotube polymethylmethacryclic acid interactions

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compression strength, and fatigue properties. Fiber-dominated properties include modulus and ultimate tensile strengths in advanced design. The theoretically predicted high modulus and strength of nanoparticles such as single-wall carbon nanotubes (SWNTs) are of interest. For resin-dominated properties such as ILSS and toughness, a detailed understanding of the fracture mechanics and the propagation of crack tips and crack fronts within the nanocomposite morphology is needed to fully exploit the advanced properties of nanocomposites. Characterization of the deformation and failure mechanisms in a manner similar to that used for traditional composites (pull out, bridging, interfacial cracking, matrix cracking, etc.) are needed to enable the construction of models which address this aspect and exploit the potential improvements made with nanocomposites.

9.2.3 The Interface Region Cohesively bonded interfaces play a significant and important role in a variety of products, commonly based on thin, film-coated substrates or layered thin film structures, and adhered layer systems, such as composites used in aircraft manufacturing, or composite material systems. The reliability of such products and devices is often compromised by the occurrence of interfacial separation processes (or delamination). Dedicated experiments to measure interface performance are being developed in industry and academia, but the results clearly reveal that a macroscopic approach to this problem does not enable the development of truly predictive insights under different loading conditions or classes of interfacial functionalization. The key problem resides in the fact that the total fracture energy encompasses contributions from physical de-bonding processes and micro-scale dissipation processes in the bonded material, which cannot be separated without a multiscale approach. The cohesively bonded systems of interest typically involve two different polymers, or a polymer and a substrate of a nonpolymeric material, for example, a metal, glass, or silicon, in the form of a surface fiber (tow) or particles. The adhesion between the layers in such systems has been the subject of great scientific and industrial interest over the past decades [42, 43]. Due to the complexity and limitations of computing power, commercial ventures are at this time still largely relying on trial-and-error procedures for each interfacial system to be designed or optimized. The thickness of the films or layers often is in the range of 30 nm up to 100 μm and more in complex layered materials. Most work, in this area, has been focused on brittle interface failure, for example, in metal/metal and metal/ceramic interfaces. These studies have focused on atomistic degradation [44] and de-lamination and separation studies of interfaces [45]. Ductile separation, that is, failure, has been studied by taking into account the (visco)plastic dissipation mechanisms in the adjacent sections of material near the interface in question [46]. Such adhesive failure has been investigated using classical fracture mechanics with smallscale separations [47, 48]. The interaction between adhesive and cohesive failure has only recently become an area of scientific interest [49]. It is however a key

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area where meso-scale and grouped atom simulations can connect with larger element simulations such as finite element methods and boundary element approaches. The classical reasoning in addressing the ductile separation problem is to study the adhesion on the basis of the physicochemical bonding of the layers at the separating interface. Although this provides clear answers in brittle interfaces, it constitutes a major problem for cohesive interfaces, because during interfacial separation (or de-lamination), a considerable amount of energy is dissipated in breaking chemical bonds, disrupting, rearranging, and deforming the adjacent material. As a result, different experimental configurations or tests do not provide consistent measurements of the work of adhesion at an interface, since the measured fracture energy consists of a contribution from (1) actual de-bonding at the surface, (2) dissipative deformation in the bonded material adjacent to the interface, (3) the interaction between both previous mechanisms, and (4) stress- or energy-induced chemical changes in the materials at the interface. A typical example of this is the measured work of separation in, for example, a polymer-coated metal, where values of 2 J/m2 [50], 30 J/m2 [51], and 194 J/m2 [52] have been reported by different techniques on chemically similar systems. In some systems the fine scale dissipation mechanism near or at the interface was fibrillation [53]. Delaminating comprises various failure processes, including extending fibrils, de-adhesion from the surface, and even breakdown of fibrils. The latter mechanism typically involves the pull-out and scission of polymer or molecular chains. These are processes which can be accurately studied using quantum mechanical methods, molecular dynamics, and meso-scale methods, since quantum mechanics allows the investigation of the bond scission energies at the interface or crack tip, molecular dynamics its atomic motion and meso-scale simulation allows the investigation of the relaxation of the surrounding regions. At the continuum, all these mechanisms result in an overall work of separation that is larger than the classical adhesion energy by an amount equal to the mechanically induced dissipation near the interface. Figure 9.7 shows a clear separation of scales and how the various parts of the system, the chain scission, the work separation, and longer-range distortion/relaxation forces, can be modeled [54]. It is, however, a clear limitation of the coarse segmentation approach that such a description of the model is intrinsically case specific.

9.2.4 Functionalization of Interface Region The literature and history of interface modification of composite systems is extensive [55]. It is important to note that in the field of functionalization, multi-scale modeling applies to the direct functionalization, the reactions produced, and the interactions between the host and the functionalized moiety, and also the nature and composition of the functionalized interfaces. There are many different classes and types of chemical and mechanical treatments; however, many of these methods have difficulties when applied to nanocomposite and complex adhesion systems. In these

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classes of systems, there are three different types of functionalization, namely that of the interface of the nanoparticle and of the host matrix [56]. The latter case is more commonly considered as a solution of modified polymer in the host matrix. There are several reasons why most researchers have chosen to modify the nanotube or polymer matrix as the route to enhanced composite performance. These have been extensively studied with chemical reaction modeling methods and approaches [57]. As noted by Chandra and Ghonem [58], the thermo-mechanical load transfer

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between fibers and matrix in conventional composites is generally affected by both chemical and mechanical bonding between the fiber and the functionalized moiety. Also, under fatigue loading, the final failure will be a result of either complete fiber fracture (fiber dominate) or buckling (matrix dominate) [59]. In woven carbon-fiber reinforced polymers [60], (CFRP), a number of damage modes occur before this final failure. Previous investigations have shown debonding of fiber–matrix interface, fiber breakage, normal and longitudinal matrix cracking under tension–tension and tension–compression loading (Fig. 9.8). While chemical bonding arises from the formation of new phases, mechanical bonding occurs by interlocking of constituent fibers and chains of functionalized nanotubes fibers in composites. Chemical modification of either phase in the system could enhance the cross-linking (chemical bond) and interfacial strength between matrix and fiber. These chemical attachments are also expected to act as “tethers” (mechanical bond) and help in load transfer when the composite is deformed either slowly or rapidly [61]. Again, these chemical modifications, the synthetic reactions that constitute them, as well as the strength of the interlocking and association, interactions are able to be studied at the atomistic and molecular level. Researchers have attached different functional groups to the walls of nanotubes using various experimental techniques. For example, Michelson and co-workers have fluorinated CNTs using alcohol solvents, while others attached hydrogen using ammonia, hydrogenated nanotubes using electric discharge, and attached alkyl chains using amidization. Apart from the possible applications in nanocomposites, functionalization of CNTs is proposed as a useful technique for altering electronic properties (fluorination) and for fabrication of sensors [62] and in the development of coated composite systems of interest in the areas of lightning strike protection [63] and gas storage in reusable space launch vehicles [64]. Such electronic functionalization is one of the clear goals and requirements for the adoption of nanocomposites into

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the aerospace industry, where lightning-strike resistance and static-charge dissipation alongside mechanical and thermal performance are key design requirements. Many questions still remain to be answered in such systems, and it is these complex questions which multiscale modeling attempts to answer.

9.2.5 Modeling Approaches From a modeling point of view, it is useful to distinguish and characterize the often-quoted top-down and bottom-up approaches [65] in engineering and molecular design, fabrication, and assembly of nanomaterials and systems. At a large scale, modeling methods which describe properties of materials and devices have been established on the basis of the laws of chemistry and physics, together with data- or observation-based empirical laws and constitutive equations. These methods have usually been implemented numerically by means of methods such as finite elements, field methods, and computational fluid dynamics methods. However, as shown earlier in this chapter, the atomistic and molecular interactions that govern interfacial and other composite system interactions are key to the accurate understanding of complex composite performance. The bonding at interfaces, whether particle, fiber, or substrate, is also very important in the area of deposit rather than directly chemically modified composite systems. Tools such as molecular dynamics for substrates and surface layers can be used (Fig. 9.9) alongside more computationally expensive methods such as Carr–Parinello quantum dynamics [66] to understand the surface interaction phenomena and chemical changes which may occur at the interface under the influence of high stress and shear fields. For a given atomistic substrate structure, be it a periodic surface or a cluster, atomistic mesoscale and quantum approaches have been used to screen the additives [67] and to evaluate the energetic, atomistic orientation, chemical structure when they physi- or Materials Modeling Approaches and Methods

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chemi-sorbe or dock onto the surface. This information can then be combined with a structure activity model (QSPR or QSAR) or a standard engineering model to predict the overall performance of the system. One commonly used technique in this area is that of algorithms such as the Monte Carlo search approach. This works by scanning the configurational space of a substrate and chemical reactant to identify low energy or relevant configurations using a Boltzmann configuration acceptance criterion. In detail, this functions by slowly decreasing the simulation temperature during a dynamical simulation (Simulated Annealing, in which kinetic energy is added to the atoms in the system and Newton’s Laws of motion are solved) to find the most stable configurations for the additive on the surface, from a number of starting configurations. One can, for example, use two different glass surfaces as substrates and study the adsorption of the same additive on those surfaces, or simply study the adsorption of different additives on the same glass surface. These descriptions typically assume that materials and properties can be described by macroscopic, thermodynamically defined quantities, and that there is generally continuous behavior from the micro- to the macro-scale of the system. It has proven to be possible, in at least a number of cases, to refine these multiscale simulation approaches quite close to the atomistic level in fact. An example would be finite element descriptions of nanotubes themselves [68]. While nanotubes are atomistic rather than continuum entities, finite element models have been shown to give accurate description of their mechanical, that is, bending behavior. The advantage of such an approach is that, within a single methodological approach (in this case, FEM), the range from the nano to the macro-scale can be described (Fig. 9.10). On the other hand, this approach has some difficulties when fundamentally new nanoscale phenomena and associated properties due to electronic or quantum chemical interactions have to be considered. One area of active research is the parameterization of these calculations directly from the atomistic and quantum data warehouses, which are generated by large high-throughput computation farms or engines [69]. The bottom-up approach, on the other hand, starts with the methods of electronic structure, atomistic and molecular simulations, which have become well established in areas such as chemistry and drug discovery in the last 20 years. The routine methods in today’s arsenal of computational chemistry and materials science include density functional quantum methods, semi-empirical methods, classical forcefield simulations, and coarse-grained, or mesoscale methods [70]. Since these methods handle length scales up to about 100 nm, they are a natural fit to the demands of nanotechnology and can probe the all-important new phenomena and desired new properties arising at that scale (Fig. 9.11). The integration of these methods into one coherent user environment, easily accessible with standard personal computers, has further added to the impact of modeling and simulation, making them accessible to the nonexpert simulator, that is, the materials engineer or materials scientist. It is important that realistic complex atomistic structures can be constructed with ease using the graphical interfaces available, as, for example, the nanotube shown in Fig. 9.9.

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It is important to realize the tremendous impact which Internet or web based methods has had on the field of multiscale simulations. As such, scientist and

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engineers think nothing of using the NIST handbook [71] to identify spectral frequencies and characteristics of interest to their physical system. In this sense, spectral multiscale simulation has a key role in the characterization and understanding of the system, as well as in the parameterization, description, and analysis of the state and dynamical change of state the system undergoes over time. A recent paper from Intel Research has identified a number of key industrial application areas, issues, and requirements for theoretical computation and materials design (TCAD) [72]. The paper highlights the need for a hierarchical approach connecting the quantum/atomistic and the device simulation approaches. The challenges and opportunities are summarized as follows: “Moving into the nanotechnology era, the demands on model-based understanding will only increase. Meeting the rapidly advancing challenge of new materials and device structures requires a physically-based, hierarchical modeling approach.” This analysis shows that building from atoms to materials to devices can deliver the understanding that technology development requires and has the potential to reach even further up the hierarchy to enable material-, design-, and system-level tradeoffs. It is this three-way method of action of materials simulation in the nanotechnology space that makes it a complex field to cover, but points to the eventual role multiscale simulation will play. These modes of action are: support for experimental characterization, that is, understanding of what constitutes a system, increased understanding of reactions and processes, that is, insight in reactions and chemistry and property prediction and analysis, and determining the behavior of the system and its response to external fields or forces.

9.2.6 Method Developments Although the range of methods currently available covers the important ‘nano’ length and time scales, in many important and particularly interesting cases there is such a strong coupling of scales in a system that no single method is applicable or has the required range. Potential solutions to this issue are to (a) extend the time and scale range of current methods, the so-called single formalism approach and (b) to construct the so-called hybrid (multiscale) simulations which integrate, for example, quantum and classical methods in one coherent approach, where there is a tight and bi-directional coupling of parameters between the regions and methods. In the hybrid approach, multiple regions are defined within the configuration; some with direct atomistic interactions, some with detail interactions defined by quantum mechanical density functional theory or semi-empirical or tight binding theory, still others defined by electrostatic or mechanical linkages, and yet others treated by a bulk or continuum representation. The objective of these methods is to predict material properties of the modified parent material when reinforced with nanoparticles using an aggregating approach, in which there are multiple connect domains of simulation. Algorithmic improvements of these approaches, coupled with the increasing speed of computational hardware and greater memory availability, are making it possible to perform predictive modeling on ever-larger systems.

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One approach to the development of tools in this area is that of collaborative and shared research. A number of commercial organizations have participated in various consortia. One of these, headed by Accelrys Inc., has been focusing on the development of such tools to a commercial standard since 2004. The consortium has led to the development of a new toolset, which includes the linear-scaling ONETEP method [73], and a hybrid QM/MM method called QMERA [74]. The ONETEP method, for example, enables high-precision total energy calculations for systems of several thousand atoms using modified planewave potentials. Previously, calculations using this method were limited to the order of 100 atoms or more. In particular, QMERA enables any changes in bonding at the interface to be studied while retaining the atomistic information of the surrounding matrix. Finally, the results of the atomistic study can be fed back into meso-scale simulations using a quantum mechanical parameterized DPD approach, completing the cycle of ‘bulk’ and interfacial design of nanoscale coatings (Fig. 9.12). Fig. 9.12 The process of developing theory and the validation of experimental data

9.3 Multiscale Modeling The earliest approach to understanding material behavior is through observation via experiments. Careful measurements of observed data are subsequently used for the development of models that predict the observed behavior under the corresponding conditions. The models are necessary to develop the theory. The theory is then used to compare predicted behavior to experiments via simulation. This comparison serves to either validate the theory, or to provide a feedback

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loop to improve the theory using modeling data. Therefore, the development of a realistic theory of describing the structure and behavior of materials is highly dependent on accurate modeling and simulation techniques. Mechanical properties of nanostructured materials can be determined by a select set of computational methods. These modeling methods span a wide range of length and time scales. For the smallest length and time scales, computational chemistry techniques are primarily used to predict atomic structure using first-principles theory. For the largest length and time scales, computational mechanics is used to predict the mechanical behavior of materials and engineering structures. Computational chemistry and computational mechanics modeling methods are based on thoroughly established principles that have been developed in science and engineering. However, the intermediate length and time scales do not have general modeling methods that are as well developed as those on the smallest and largest time and length scales. Therefore, multiscale modeling techniques are employed, which take advantage of computational chemistry and computational mechanics methods simultaneously for the prediction of the structure and properties of materials. Each modeling method has broad classes of relevant modeling tools. Creation of these tools falls mostly in the academic domain; however, the interfacing and the provision of usable connected networks of these simulations lie somewhat at the interface between commercial and academic interests. The quantum mechanical and nanomechanical modeling tools assume the presence of a discrete molecular structure of matter. Micromechanical and structural mechanics assume the presence of a continuous material structure. The continuum-based methods primarily include techniques such as the finite element method (FEM), the boundary element method (BEM), and the micromechanics approach developed for composite materials. Specific micromechanical techniques include the Eshelby [75] approach, Mori–Tanaka method [76], and Halpin–Tsai method [77] (Fig. 9.9).

9.4 Continuum Methods 9.4.1 Predicting Material Properties from the Top-Down Approach These modeling methods assume the existence of continuum for all calculations and generally do not include the chemical interactions between the constituent phases of the composite. These methods can be classified as either analytical or computational. The so-called correlative or structure, formulation or process correlative methods also fall into this category. It should be noted that the detailed atomistic methods can function as descriptor generators or inputs to these approaches so that a data-based method can be coupled both from a validation and a design standpoint.

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9.4.2 Analytical Continuum Modeling The overall properties of composites can be estimated by a volume-average stress and strain fields of the individual constituents. The self-consistent scheme utilizes an iterative technique to evaluate the modulus for a section of composite material, by averaging the forces over the constituent resin and fiber parts, and then extending it to the whole phase. In one of the best examples of this approach, Pipes et al. used an anisotropic elasticity approach to study the behavior of a layered cylinder with layers of discontinuous CNT [78] following a helical path in each layer [79]. In this simulation, one layer represented the polymer matrix and the other represented the carbon fiber tow. Odegard et al. [80] used the Mori–Tanaka method to predict elastic properties of polyimide/CNT composites at various lengths, orientations, and volume fractions. A similar micromechanics-based approach was used by Odegard et al. to predict the properties of CNT/polyethelene composites. This study also examined the effects of CNT functionalization in CNT/polyethylene composites by adjusting and scaling the particle–particle interaction parameters based on detailed atomistic and quantum simulations. In another study, MWNT/polystyrene composite elastic properties were shown to be sensitive to nanotube diameter by an approach based on Halpin–Tsai micromechanical method [81], and Lagoudas et al. predicted elastic properties of CNT/epoxy composites using a variety of analytical micromechanics approaches [82]. So these methods show that analytical or mathematical continuum modeling is a viable and reasonable approach for large-scale and large focus systems.

9.4.3 Computational Continuum Modeling Continuum-based computational modeling techniques include finite element method (FEM) and the boundary element method (BEM). While these approaches do not always supply exact solutions to complex material challenges, they can provide very accurate estimates for a wide range of systems based on an initial state or a set of assumptions. These can be derived from commercial sources of software, commercial data or information, and property feeds such as those from the NIST [83], MAT WEB [84], ACS [85], and Granta Inc. [86]. Finite element method (FEM) can be used for numerical computation of bulk properties based on the geometry, properties, and volume fraction of constituent phases [87]. FEM involves discretization, that is, segmenting into mathematic localized functions, of a material representative volume elements (RVEs) for which the Hook’s law elastic solutions lead to the determination of the stress and strain field. These fields are then solved using an iterative mechanism leading to a steady state, or in some codes (such as LSDYNA/DYNA 3D), a dynamical or deforming condition. [88]

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The coarseness of the discretization determines the accuracy of the solution and, in many cases, the rate of convergence. Nanoscale RVEs of different geometric shapes can be chosen for simulation of mechanical properties [89]; the shape and extent of the RVEs can be determined from atomistic and sometimes embedded shell codes, such as the previously discussed QMERA code. In this approach, a connection is again made where meso- or intermediate length and time-scale particle dynamics code results can be linked or used as inputs to FEA methods. FEM-based micromechanics have been used extensively for the prediction of mechanical properties of nanostructured composites. Li et al. used an FEM-based approach to investigate the stress concentration at the end of CNTs and the effects of nanotube aspect ratio on the load transfer between nanotubes and matrix [90]. Bradshaw et al. [91] used FEM to evaluate the strain concentration tensor in a composite consisting of wavy CNTs. Fisher et al. [92] used FEM to determine the effect of waviness on effective moduli of CNT composites. Chen et al. [93] used different shapes of RVEs to understand the dependence of predicted properties on the element shape. Boundary element method (BEM) is also a continuum mechanics approach which involves solving surface or boundary equations for the stress and strain fields in a system. This method uses elements only along the boundary layer of a material, unlike FEM, which involves elements throughout the volume. This approximation makes BEM less computationally exhaustive than FEM [94] and enables BEM to be applied from micro- to macro-scale modeling [95], due to the compact nature of the calculations and their short time scale. In BEM, it is assumed that the material exists as in a continuum, and therefore, the details of molecular structure and atomic interactions are absorbed into the elements and their interaction terms. The rigid fiber model has been shown to be very effective in estimation of fiber composites, for example, Ingber [96] et al. have shown agreement in a predicted modulus using BEM for fiber composites. The estimated modulus was found to be very close to that predicted by MD simulations. They concluded that BEM can be very useful for first-order approximation of mechanical properties in large-scale modeling of CNT composites. This approach again forms a phenomenological connection between atomistic and meso-scale.

9.5 Materials Engineering Simulation Across Multi-Length and Time Scales 9.5.1 Predicting Material Properties from the Bottom-Up Approach Over the last two decades, molecular modeling, [97] atomistic [98], quantum [99], and meso-scale [100] have emerged as an important tool in the prediction of physical material properties [101] such as elastic response, atomic structure [102], vibrational frequencies [103], heats of reaction [104], electric permittivity [105], and binding

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energies [106]. Molecular modeling assumes a noncontinuous composition of the material, which makes it a powerful tool for studying interactions of components of composite systems at the nanometer length scale. Due to the discrete nature of these techniques, they are limited by the time scales that can be achieved in the simulations, and thus the techniques can be computationally exhaustive when applied to real-world problems. However, as discussed previously, judicious use and choice of information abstraction can make these methods applicable to intermediate length and time scales. It is traditional to consider simulation science to be grouped into four broad categories: quantum mechanical, molecular mechanical, meso-scale, and bulk scale. This segmentation does, however, ignore the area of statistical and correlative modeling, which, although often focused on bulk-level properties, has been applied at the nanometer scale [107] (Fig. 9.13) and has been one of the most beneficial uses of simulation methods in connection to finite element analysis. Fig. 9.13 A meso-scale representation of a lipid bi-layer interface with solvent molecules

Many papers have been written on the size of systems that are modeled in each domain and the length of time for which a materials system can be simulated, but generally speaking, as the size of the system increases, more approximations are employed and the methods are less numerically precise, but increase in the scope and context of what they are simulating (Fig. 9.14). Three widely used and validated molecular modeling techniques for the prediction of mechanical properties of nanostructured materials are molecular dynamics (MD), Monte Carlo (MC), and ab initio simulation (QM). The time and resource cost of performing a simulation with quantum mechanical accuracy increases rapidly with the number of atoms in the system, anywhere between N1 and N8 , where N is the number of atoms; various seminal studies have focused on approaches to reduce the time of key mathematic steps in these calculations [108], usually the complex matrix inversion [109] and diagonalization steps [110]. The best-optimized algorithms scale in the neighborhood of N2 –N3 , but even so, to keep the calculations manageable, modelers introduce successive levels of approximations into the methods [111], such as the united atom approaches and agglomerating groups of atoms into beads or particles in the particle dynamics approach. While these reduce the computational cost, they also potentially compromise the accuracy of the results [112]. The modeling of systems at the nanoscale, say 10–100 nm, poses special challenges [113]. Because of the size of these systems, modelers typically would use

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Fig. 9.14 Schematic of the interlinked simulations methods, where one approaches output is both a direct value and the input for the next stage in the simulation hierarchy

approximate methods in order to keep the computational requirements (CPU time, disk space, memory) reasonable. Calculations on nanomaterials, however, require a high degree of accuracy, and electronic and quantum mechanical effects play an important role even at the scale of ˜100 nm. Consequently, computationally expensive quantum-mechanical-based calculations or approximate methods that can reproduce quantum mechanical accuracy are often necessary to study these materials [114].

9.5.2 Quantum Scale Unlike most materials simulation methods that are based on classical potentials, the main advantages of ab initio methods, which are based on first-principles quantum mechanics or density functional theory, are the generality, reliability, and accuracy. These methods involve the solution of Schr¨odinger’s [115] equation for each electron in the self-consistent potential created by the other electrons and the nuclei. Methods based on quantum mechanics are the most accurate, and they can predict

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a large number of electronic, optical, and structural properties such as crystal structure [116, 117, 118], optical spectra [119, 120], and catalytic performance [121]. Ab initio methods can be applied to a wide range of systems and properties; however, these techniques are computationally exhaustive, making them difficult for simulations involving large numbers of atoms. In the last decade or so, density functional theory (DFT) [122] has emerged as the method of choice in industrial modeling of nonbiological materials. This method offers an excellent compromise between cost and accuracy and time-number scaling [123] on the order of N2 –N3 [124]. There are three widely used procedures in quantum first principles or ab initio simulation. These procedures are single-point calculations, geometry optimization, and frequency calculation. Another less widely used but insightful approach is that of first-principles molecular quantum mechanics, in which a Newtonian dynamical calculation is performed from the system over time, but the forces are computed from first-principles calculations. Single-point calculations involve the determination of energy and wave functions for a given geometry. Geometry calculations are used to determine energy and wave functions for an initial geometry, and also subsequent geometries with lower energy levels as the system is subject to optimization to its lowest, or in the case of a transition state, its highest energy. Frequency calculations are used to predict Infrared and Raman intensities of a molecular system; this is where the linkage to multiscale simulations and experimental characterization of composites also occurs. Ab initio techniques have been used on a limited basis for the prediction of mechanical properties of polymer-based nanostructured composites. Ab initio simulations are restricted to small numbers of atoms because of the intense computational resources that are required. A study conducted by Mylvaganam et al. demonstrated that nanotubes of smaller diameters have higher binding energies in a polyethylene matrix [125]. Bauschlicher studied the bonding of fluorine and hydrogen atoms to nanotubes [126]. He showed that fluorine atoms favored to bond to existing fluorine atoms. The so-called semi-empirical quantum mechanical methods [127] work by replacing the most computationally demanding portions of a quantum mechanical calculation with empirical parameters [128]. Using this approach, calculations on complex polymer fibers enzymes with thousands of atoms have been performed [129]. The methods are limited in their accuracy, however, and more importantly do not work with all elements. Another approximate quantum mechanical approach is DFT tight-binding (DFTTB) [130, 131]. Here, the parameters are determined by fitting to DFT calculations rather than to a range of empirical data. The method has proved its strengths in a number of biomedical applications, but also works for a number of transition metals, and can be applied to solid materials [132].

9.5.3 Molecular Scale Molecular modeling is generally applied to systems one to two orders of magnitude larger than the quantum scale. Molecular mechanics approximates the inter-atomic

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interactions with empirically derived analytical expressions and parameters, usually based on variants of Hook’s law or spring constants. Owing to the mathematical simplicity of these functional forms, calculations are very fast and can be performed on systems containing from thousands to hundreds of thousands of atoms, for significant periods of time, up to nanoseconds. The atomic interactions can be considered in two groups, that is, long-range and short-range terms; also, various grouping approximations can be made to allow even larger-scale systems to be studied. The behavior of covalently bonded atomistic systems is mainly governed by short-range interactions due to the nature of chemical bonds (Fig. 9.15). This bond energy is represented as a polynomial function, usually of second or higher order based on the bond distances, bond angles, torsional rotations, and out-of-plane bends of the atoms in the constituent species. In many cases, coupling terms among bonds, angles, and torsions are included in order to improve the reproduction of experimental data by the results; this, however, can lead to a change or degree of over parameterization of the system. This constant duality of trying to build more accurate simulations as well as more general simulations is one of the challenges in this field as we seek to extend the scale and level of these modeling tools in the multiscale area. Fig. 9.15 The Morse potential compared to the standard harmonic oscillator

A grouping or a given set of functions and parameters used to reproduce the geometry and physical properties of an experimental system is termed a force field. These force fields are derived by fitting to experiment or to accurate quantum mechanical calculations [133, 134]. Materials that are not dominated by covalent interactions frequently require more complicated short-range potentials, such as Morse, Brenner–Tersoff [135], Buckingham [136], Lennard–Jones, or Axilrod– Teller [137] potentials. It is important to note that a force field will be applicable

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only to systems similar to those used in its derivation and fitting. Hence, many different force fields have been developed and published in the literature. Force fields have been developed for organic molecules (Amber [138], CVFF [139]), Zeolites, [140] microporous materials (CVFF-AUG [141]) organic–inorganic interactions COMPASS [142, 143], OPLS [144, 145], CHARMM [146], Amber [147], and metal oxides GULP [148]. In ionic materials, unlike pure organic systems, the Coulomb interaction is by far the dominant term and can typically represent up to 90% of the total energy. One technical advance that has enabled simulations of ionic materials and that has led to the extension of the type of systems that can be treated with this method, that is, from clusters to slabs and surfaces, is the Ewald approximation. In the Ewald [149] method, the long-range cumulative ionic or electrostatic terms of the ionic material interactions is efficiently achieved through a series approximation method. The potential energy of the system, a single slowly and conditionally convergent series, is recast into the sum of two rapidly converging series plus a constant term. Dispersion energy, or London dissipation forces, constitutes the next largest long-range interaction, arising from dipole–dipole and higher-order terms. These long-range Coulomb and dispersion terms are not exclusive to ionic materials, but must also be accounted for in covalent systems, particularly when considering inter-molecular interactions. A recent review covers these various potentials in more detail [150]. One serious limitation of force fields is their inability to study chemical reactions. The functional forms employed in classical force fields are not able to describe the formation or breaking of chemical bonds. Recently, however, force fields have emerged, such as the Tersoff [151] series of potentials [152], REBO [153] and ReaxFF [154], that are capable of determining material shapes when bond breaking and bond scission is occurring. ReaxFF, for example, has been used to model transition-metal-catalyzed nanotube formation [155]. Finally, a series of organometallic force fields has been developed that allow the analysis and determination of complex coordinated metal polymer [156] and metal organic systems [157].

9.5.4 Molecular Dynamics Molecular dynamics (MD) [158] is one of the most widely used modeling techniques for the simulation of nanostructured materials since it allows the incorporation of time, energy, and external stress/strain into the simulation space. MD allows accurate predictions of interactions between constituent phases at the atomic scale [159]. It involves the determination of the time evolution of a set of interacting atoms, followed by integration of the corresponding equations of motion. The equations of motion are given by Newton’s second law. MD is a statistical mechanics method, where a set of configurations is distributed according to a statistical ensemble or statistical distribution function. The trajectories of the motion of the atoms are calculated under the influence of interaction forces of the atoms in a phase space with 6 N dimensions; three degrees of variation for position and three degrees

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of variation of the momenta for each atom. Calculation of physical quantities by MD simulation is obtained by arithmetic averages of instantaneous energy values from individual simulation steps. MD simulations, if run for a sufficiently long time, can completely sample the phase space; however, in practice, simulation times are limited. Physical quantities are sampled after the molecular system reaches a thermodynamic equilibrium. Examples of the sort of calculations done with molecular dynamics include understanding the deformation of substrate structures under the dynamic loading of atomic force microscope tips. [160]

9.6 Extension of Atomistic Ensemble Methods 9.6.1 Combining the Top-Down and Bottom-Up Approaches 9.6.1.1 Monte Carlo Approaches Monte Carlo (MC) approaches are a class of probabilistic mathematical models for the prediction of the behavior and outcome of a system. The outcomes of MC are statistical in nature, subject to laws of probability and in most cases involve multidimensional integration over the system space. Different MC techniques can be used for determination of material properties; they include classical MC, quantum MC, volumetric MC, and kinetic MC. Classical MC involves drawing samples from a probability distribution, often generated using the classical Boltzmann distribution, to obtain thermodynamic properties or minimum-energy structures. Quantum MC utilizes random walks to compute system configurations that are then energetically evaluated using quantum-mechanical energies and wave functions. Quantum MC is generally used to solve electronic structure problems. Volumetric MC generates random numbers to determine volumes per atom or to perform geometrical analysis. Kinetic MC simulates process by the use of scaling arguments to establish time scales; this has been extensively used in the reaction kinetics field as a method to link the quantum heats of formation studies with surface diffusion, reaction, and poisoning effects [161]. Kinetic MC also includes MD simulations to account for stochastic effects. Based on the dependence of time, MC simulations can be classified as either metropolis MC or kinetic MC. Metropolis MC applies to systems under equilibrium and is independent of time. This method generates configurations according to a statistical-mechanics distribution. Whereas kinetic MC deals with systems under nonequilibrium and uses transition rates that depend on the energy barrier between the states, with time increments formulated so that they relate to the microscopic kinetics of the system [162]. MC techniques have been used to study the mechanical and phase behavior of quartz, cristobalite, coesite, and zeolite structures. The bulk modulus predicted from their model was found to be in good agreement with experimental values. They concluded that the model can be used to determine properties of silica nanostructures with atomistic detail. Chui et al. used a MC-based modeling approach to study

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deformation, rate of deformation, and temperature dependence of large strain deformation in amorphous polymeric materials [163]. 9.6.1.2 Mesoscale Approaches The meso-scale approaches introduce an additional level of approximation on top of those used in molecular mechanics. Dissipative particle dynamics (DPD) is a coarse-grained molecular dynamics method in which the fundamental variables are the positions and momenta of fluid droplets rather than individual atoms. Droplets interact via three forces: a conservative force, a dissipative force, and a random force. The latter two forces are generated to satisfy a state of detailed balance, which conserves the temperature of the system. All forces between beads are pairwise, implying linear and angular momentum conservation, and therefore implicit inclusion of hydrodynamic forces. This approach [164] is derived from molecular dynamics simulations [165] and lattice gas automata (Fig. 9.16). Because of the dimension reduction such an approach has on the entities within the simulation, it enables length and time regimes in complex fluids to be simulated. With this approach, systems with upwards of 1,000,000 atoms can be analyzed [166]. Fig. 9.16 An atomistic representation of a polystyrene polymer showing regions that could potentially be abstracted into beads for a meso-scale level calculation

Also, this approach extends the effective time step of the simulation by over an order of magnitude, so that simulations can last from several microseconds to even a millisecond of real time. An alternative meso-scale approach is based on selfconsistent field theory. In this field approach, a material is not seen as a collection of atoms or particles, but rather as density fields of the various components. The reason for this is that the mathematics of pairwise interactions, which scales exponentially with the number of particles, can be exactly replaced with individual interactions with a mean field continuum [167] (Fig. 9.17). Hence, DPD is the explicit particle representation of the system, and the field methods are the statistical mechanics representation expressed in terms of free R have been used successfully, energy equations. Field methods such as MesoDyn

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Fig. 9.17 Schematic of the relationship between the atomistic and meso-scale particles

in particular in the area of block-copolymer phase behavior and specifically in the kinetics of phase transitions in nanostructured fluids [168]. 9.6.1.3 Equivalent Continuum Approaches The equivalent-continuum method (ECM) [169] is used to determine the bulklevel mechanical properties of a material from the molecular model. ECM is a methodology [170] for linking computational chemistry and solid mechanics [171]. An equivalent continuum, identical to the MD model in geometry, is developed and a constitutive law is used to describe the mechanical behavior of the continuum. The energies of deformation of the molecular and equivalent continuum models are derived for identical loading conditions, and then the continuum model is used for the extended calculations [172]. The unknown mechanical properties of the equivalent continuum are determined by equating the energies of deformation of the two models under these loading conditions. The properties of a larger-scale material are then determined using the equivalent-continuum volume element properties. Odegard et al. [173] have used the ECM/MD to predict the properties of various CNT-based composite systems, and they successfully predicted the elastic properties of PmPV CNT/polyimide composite for a wide range of nanotube lengths, orientations, and volume fractions. They also predicted the performance of functionalized and nonfunctionalized CNT/polyethylene composites. Frankland et al. [174] used MD to study the influence of chemical functionalization on the CNT/polyethylene composites, and studied the critical nanotube length required for effective load transfer. Frankland qualitatively predicted the nanocomposite system’s stress–strain curves from MD and then compared them to those obtained from micromechanical models for CNT/polyethylene composites. This showed that the models formed part of an accurate simulation continuum, since both theories, from different size-scale and time-scale domains, gave similar results. Hu et al. [175] used MD to understand the effect of chemical functionalization on toughness of CNT/polystyrene composites. MD has also been used for the prediction and investigation through simulation of other physical properties of nanocomposites. Wei et al. showed that addition of CNTs to polyethylene resulted in an increase of thermal expansion, glass transition temperature, and diffusion coefficients of the polymer

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[176]. Lordi and Yao calculated sliding frictional stresses between CNT and various polymer substrates based on molecular mechanics simulations [177]. Liang et al. showed the presence of an attractive interaction between SWNTs and epoxy polymer matrix [178]. Frankland et al. characterized the interfacial friction model for the pull-out of SWNTs from a polyethylene matrix [179]. 9.6.1.4 The Bulk-Scale Approach Bulk-scale methods like CAD/CAM are most familiar to engineers. These are entirely phenomenologically based and use well-understood engineering approaches to model problems such as fluid flow, structural stability, or the efficiency of chemical manufacturing plants. In some recent examples, engineering parameters were computed from the results of meso-scale simulations [180], establishing a connection all the way from the quantum level to the bulk level. In the above simulation, atomistic parameters were used to configure a meso-scale simulation. The resulting distributions and phase descriptions were then loaded into a finite element code and the results examined in terms of their electrical and thermal conductivity. Other approaches, mentioned in the introduction, have also been used to link the atomic and engineering scales such as linking continuum and meso-scale simulations using parameterized state equations. In yet other studies, major new contribution to carbon nanotube mechanics have been identified by employing engineering approaches, where nanotube mechanics and MM simulations are linked together. This allows virtual experiments to be performed [181] in which discretely modeled or simulated nanotube are deformed in a manner similar to those performed by testing machines at the macro level. The output (energy and deflection rather than load and deflection) is then used to develop a fully parameterized material response. As indicated in the review above, numerous attempts have been made to study the mechanical behavior of polymer nanocomposites using modeling techniques. From the general results from these studies, several conclusions can be drawn. First, there is a strong effect of the interfacial nature between the nanoreinforcement and matrix on the mechanical properties. The interfacial conditions can improve the load transfer via bonded (functionalization) or nonbonded means (dative or dispersive interactions). Second, there is a measurable influence of nanotube length and diameter on the overall composite properties, and the maintenance of length is key to high tensile load performance. Third, use of traditional element or mechanical theories to predict overall composite properties without the aid of atomistic simulation do not always result in good or reproducible predicted mechanical properties.

9.7 Future Improvement This review has shown that there are many different threads and strands to the multiscale simulation challenge. It can be argued that the current field of materials dynamical simulations began in the early 1960s when Rahman [182] simulated

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864 argon atoms on a CDC 3600 computer. Assuming a simple exponential growth [183], the number of atoms that can be simulated in classical atomistic simulations has doubled approximately every 19 months to reach 6.44 billion atoms in a recent study. Similarly, the number of atoms in DFT-based ab initio MD simulations (started by Roberto Car and Michelle Parrinello in 1985 for eight Si atoms69 ) has doubled every 12 months to 111,000 atoms. Petaflop computers, or simulation engines based on custom-programmed graphical processing units [184], for example, the Sony PS3 system, will maintain the growth rates in these “Molecular Simulation Moore’s Laws”, and it can be expected that we will be able to perform 1012-atom classical and 107-atom quantum molecular dynamical simulations on such computers. However, multi-resolution approaches combined with cache-conscious techniques will be essential to achieve scalability on such petaflop architectures (Fig. 9.18).

Fig. 9.18 Time evolution of finite element (FE) nodes and molecular dynamics (MD) atoms in a hybrid FE–MD simulation of a projectile impact on a silicon crystal. Absolute displacement of each particle from its equilibrium position is color-coded. The figure shows a thin slice of the crystal for clarity of presentation

9.8 Summary This review has identified a number of key aspects of multiscale modeling in nanomaterials and noncomplexes. These include the use of parameterized meso-scale calculations as the input into finite element and dynamical flow codes, as well as the use of engineering models in conjunction with experimental data and quantum level insights as to the functional elements or regions of interest in the system. It has also shown that shell or coupled models such as the QMERA can enable both detailed and wide scope to be satisfied within the same simulation scheme. The convergence of these parameterized solutions from the atomistic to the meso-scale and from the engineering to the parameterized-continuum scale is a clear indication of where the point of connection is between the engineering and design domain and the chemistry, molecular and polymer science domain. It is inherent in all simulations that

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some tradeoff is performed between accuracy and scope. This is perhaps best illustrated when considering Heisenberg’s uncertainty theory, where it is impossible to know both the location and the momentum of a given particle with any certainty, so for all simulations beyond a single electron a degree of approximation is required, thus creating the iterative self-consistent field theory approach, which is how we see the cyclical interaction of quantum, atomistic, meso, and continuum models being used. This need for approximation is not necessarily a poor substitute for a rigorous theory. In fact, it allows different degrees of models to be used that focus on the salient aspects of the systems performance, that is, for the structural, mechanical, electrical, and thermal aspects. Another part of the picture is the growing body of computerized information sources. These allow and enable the derivation of simulation parameters for representative volume elements or models using physical properties or the results of simulations that generate accurate physical and chemical properties. The generation of these large compendiums, automatically, has only recently become a computational reality, and so their impact on the field is yet to be fully determined. However, it already has made significant progress, as has been seen with the Materials Grid project in the UK [185]. In this project a large compendium of design and performance characteristics for a range of different materials are stored, and if not available, computed on demand for users. This then could form part of a data feed that could be used by engineers and scientists in the development of the next generation of multiscale simulation systems. Also in the area of Kinetic Monte Carlo codes, used for the simulation of steady-state or surface-mediated reaction kinetics, such database of thermodynamic parameters is essential. In these approaches a master equation is parameterized with quantum kinetic information about reactions that occur on the surface and the evolution of grid-based autonoma are tracked over time. The large body of work reviewed in this chapter shows that the goals of designing lighter, stronger, higher-performance composite systems, using multiscale approaches, for aircraft manufacturing are achievable. Further, it shows that there are a number of different approaches, for example, loose coupling and tight integrated computational approaches, where this has been shown to be an accurate and beneficial method. The aims of using successive approximation and knowledge distillation approaches, from detailed to broader scope, but less high-resolution methods, has been shown to work, and to provide benefit in the conceptual and capability assessment phases of systems design. Also, it has become obvious that some commonality of access to these tools is desirable, so that the connection of these various computational engines can be performed in a facile and reasonable manner. It is to be hoped that with the rise of methods such as cloud-based computing for computational resources and serviceoriented architectures for software component design and architecture will lead to easy and common interoperability of the various tools used throughout this chapter. It is recently heartening to see that many new tools have been developed which are data or compute source agnostic and allow the easy and visual connection of the different methods and approaches described [186].

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150. W.A. Goddard et al., “Simulation and Design of Materials – Applications to Polymers, Ceramics, Semiconductors, and Catalysis,” Abstracts of Papers of the American Chemical Society 208, 391-PHYS, 1994 151. J. Tersoff, “Empirical Interatomic Potential for Carbon, with Applications to Amorphous Carbon,” Phys. Rev. Lett., Vol. 61, 1988, p. 2879 152. F. de Brito Mota, J.F. Justo, and A. Fazzio, “Hydrogen Role on the Properties of Amorphous Silicon Nitride,” J. Appl. Phys., Vol. 86, 1999, p. 1843 153. D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, and S.B Sinnott,., “Second Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons,” J. Phys. Condens. Matter., Vol. 14, 2002, pp. 783–802 154. A.C.T. van Duin, S. Dasgupta, F. Lorant, and W.A. Goddard III., “ReaxFF: A Reactive Force Field for Hydrocarbons,” J. Phys. Chem. A, Vol. 105, 2001, pp. 9396–9409 155. M.J. Cheng, K. Chenoweth, J. Oxgaard, A.C.T. van Duin, and W.A. Goddard, ”The SingleSite Vanadyl Activation, Functionalization, and Re-oxidation Reaction Mechanism for Propane Oxidative Dehydrogenation on the Cubic V4O10 Cluster,” J. Phys. Chem. B., Vol. 111, 2007, pp. 14440–14440 156. A.K. Rappe and C.J. Casewit, “Molecular Mechanics Across Chemistry,” University Science Books, California, 1997 157. D.M. Root, C.R. Landis, and T. Cleveland, “Valence Bond Concepts Applied to the Molecular Mechanics Description of Molecular Shapes. 1. Application to Nonhypervalent Molecules of the P-Block,” J. Am. Chem. Soc., Vol. 115, 1993, pp. 4201–4209 158. J.L. Suter, P.V. Coveney, H.C. Greenwell, and M. Thyveetil, “Large-Scale Molecular Dynamics Study of Montmorillonite Clay: Emergence of Undulatory Fluctuations and Determination of Material Properties,” J. Phys. Chem. C, Vol. 111, 2007, pp. 8248–8259 159. C. Moon, P.C. Taylor, and P.M. Rodger, “Molecular Dynamics Study of Gas Hydrate formation,” J. Am. Chem. Soc., Vol. 125, 2003, pp. 4706–4707 160. J. Ma, Y. Liu, H. Lu, and R. Komanduri, “Multiscale Simulation of Nanoindentation Using the Generalized Interpolation Material Point (GIMP) Method, Dislocation Dynamics (DD) and Molecular Dynamics (MD),” CMES, Vol. 16, 2006, pp. 41–56 161. A.F. Voter, “Introduction to the Kinetic Monte Carlo Method,” In Radiation Effects in Solids, edited by K.E. Sickafus and E.A. Kotomin, Springer, NATO Publishing Unit, Dordrecht, The Netherlands, 2005 162. A. Chatterjee and D.G. Vlachos, “An Overview of Spatial Microscopic and Accelerated Kinetic Monte Carlo Methods,” J. Computer-Aided Mater. Des., Vol. 14, 2007, p. 253 163. C. Chui and M.C. Boyce, ”Monte Carlo Modelling of Amorphous Polymer Deformation: Evolution of Stress with Strain,” Macromol., Vol. 32, 1999, p. 3795 164. P. Hoogerbrugge and J. Koelman, “Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics,” Europhys. Lett., Vol. 19, 1992, pp. 155–160 165. R. Groot and P. Warren, ”Dissipative Particle Dynamics: Bridging the Gap Between Atomistic and Mesoscopic Simulation,” J. Chem. Phys., Vol. 107, No. 11, 1997, pp. 4423–4435 166. D. Broseta and G.H. Fredrickson, “Phase Equilibria in Copolymer/Homopolymer Ternary Blends: Molecular Weight Effects,” J. Chem. Phys., Vol. 93, 1990, p. 2927 167. G. Fredrisckson, “The Equilibrium Theory of Inhomoge-neous Polymers,” Oxford University Press, ISBN-13: 978-0-19-856729-5, 2005 168. A. Knoll, K. Lyakhova, A. Horvat, G. Krausch, G. Sevink, A. Zvelindovsky, and R. Magerle, “Direct Imaging and Mesoscale Modelling of Phase Transitions in a Nanostructured Fluid,” Nature Mater., Vol. 3, 2004, pp. 886–890 169. G.M. Odegard, T.S. Gates, K.E. Wise, C. Park, and E.J. Siochi, “Constitutive Modeling of Nanotube-Reinforced Polymer Composites,” Compos. Sci. Tech-nol. Elsevier, Vol. 63, 2003, pp. 1671–1687 170. W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park,“An Introduction to Computational Nano Mechanics and Materials, Computer Methods in Applied Mechanics for Nanoscale Mechanics and Materials), 2004

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Chapter 10

Multiscale Approach to Predicting the Mechanical Behavior of Polymeric Melts R.C. Picu

Abstract Modeling the mechanical behavior of polymers and polymer-based materials is notoriously difficult, primarily due to the need to integrate physics taking place on multiple scales. In this chapter we review single-scale models used to represent polymers and their composites, including atomistic, coarse–grained, and continuum models. Each of these has limitations associated with either accuracy or efficiency. To combine their advantages while reducing the associated drawbacks, multiscale methods are desirable. Two strategies are presented, both belonging to the class of “information-passing” methods. In the first, the physics of (dielectric and stress) relaxation is studied on the molecular scale and relevant parameters are calibrated using single-scale molecular dynamics and Monte Carlo techniques. These parameters are then incorporated in constitutive laws whose functional form is physically motivated. Such constitutive laws can then be used in continuum models on larger scales. This strategy is exemplified for a class of model polymer nanocomposites. In the second approach, a system reduction technology is developed to coarse grain the structure and dynamics of atomistic models of dense polymers. The coarse-grained models are calibrated using an equilibrium fine-scale model of a monodisperse system and then are used to predict the behavior of other systems in equilibrium and nonequilibrium.

10.1 Introduction Today, polymeric materials are ubiquitous. They are used in the solid and liquid phase and as additives in almost all engineering applications, performing functions that cannot be performed by other types of material.

R.C. Picu (B) Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 10,

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Viewed from the molecular scale, dense polymers, whether melts or solids, are multibody systems of polymeric chains. The chains may be monodisperse, that is, all of the same length, or polydisperse. In practice, it is difficult to ensure that a given system is monodisperse, so the vast majority of materials in use are polydisperse. The simplest systems contain linear chains, but many polymers of practical importance are branched or combs. Star and ring polymers can also be synthesized. A given chain can be made from a single or from multiple chemical compounds (copolymers). In order to preserve focus, here we will refer to linear homopolymers. Problems of practical interest involve both dense (melts and solids) and dilute (solutions) systems. In some applications, polymers are added in trace amounts as, for example, to reduce drag in water cannons. The mechanical and transport properties of dense polymeric systems are directly and significantly influenced by the chain architecture and chain length, N. Here, N represents the number of repeat units (monomers or, in some coarse models, Kuhn segments) along the chain. The chain length sets an internal length scale for the material. Mechanical relaxation, dielectric relaxation, and self-diffusion are slowed down as N increases, typically following a power of N. Hence, dynamics takes place at different rates on different scales. On the atomic scale, dynamics is on the time scale of atomic vibrations, while on the chain scale and for long chains, relaxation may take seconds or minutes in the melts state. Of course, dynamics is significantly affected by temperature, and relaxation modes with long wavelength are frozen in the solid state. An additional complicating factor, which also leads to slower dynamics, is the fact that interactions are nonlocal, and the nonlocality range is large. In fact, interactions are nonlocal in all discrete systems, but in most situations, as for example, in monatomic crystalline solids, the range of nonlocality is on the order of the range of interatomic potentials, that is, below 1 nm. In polymers, the nonlocality range is significantly larger due to chain connectivity. It can be as large as the chain dimensions in moderately dilute solutions. In dense melts, the entanglement mesh reduces the effect of nonlocal interactions by modulating the nonlocality kernel rather than by reducing its range. These two aspects render multiscale modeling of polymers difficult. Large models are required in order to capture the long-range nonlocal interactions, and these models must be evolved over long time intervals in order to represent all relevant relaxation modes. If chemical specificity is required, the smallest entity present in the model is an atom, which mandates an integration time step on the order of the period of atomic vibrations. These generic requirements clearly indicate that modeling complex polymeric systems with atomistic resolution using the current technology is not feasible or extremely expensive, even with the fastest computers. Multiscale technologies are necessary. On the other hand, predicting the mechanical response of polymeric fluids is technologically important. Melts are non-Newtonian fluids, they swell upon exiting an injection channel and may break when loaded at high rates. They exhibit both shear thinning (reduction of the stress required for flow as the deformation rate increases) and shear thickening (the reverse trend) and this response depends not only on the

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nature of the polymer, but also on the type of loading applied. In complex fluids, in which polymers are mixed with other objects, such as nanoparticles, nanotubes, the situation is even more complicated and the complexity of the multiscale physics is higher. This chapter begins with a brief review of single-scale models emphasizing their limitations and range of applicability. These models are defined on three scales: atomistic models that have atomic-scale resolution and can be chemically specific, mesoscale models defined on the scale of the entire chain, which are too coarse to explicitly represent the chemistry, and continuum models that disregard the discrete nature of the material and use “effective” constitutive equations to represent its behavior. Two examples of multiscale modeling of such systems, both belonging to the “information-passing” class of models, are presented.

10.2 Single and Multiscale Modeling Methods: Limitations and Tradeoffs 10.2.1 Atomistic and Atomistic-Like Models Atomistic models can be divided based on their degree of coarsening. These are usually composed of objects, beads that stand for individual atoms or for a group of atoms. The bead–bead interaction is given by “interatomic potentials.” These provide the energy of a bead function of the position relative to its neighbors. When a bead stands for a single atom, it has chemical specificity and the model is the most refined atomistic model possible. In a sense, this is already a coarse model since all finer-scale details (nucleus and electrons) are represented by the potential. Therefore, atomistic models have poorer accuracy compared with ab-initio representations. However, when modeling polymers, the essential physics is usually controlled by larger-scale phenomena related to the motion of the chain, and quantum mechanics interactions are rarely important. A detailed discussion on this issue is beyond the scope of this chapter. In polymeric models, the energy is written as a sum of bonded and nonbonded interactions. The bonded interactions are those taking place between the beads along the chains, while nonbonded interactions are between beads not immediately connected along the given chain. This last category includes interactions between nonconnected beads belonging to the same and to different chains. This separation of the energy in bonded and nonbonded interactions is an approximation rooted in heuristic considerations. Models in which a bead stands for larger physical entities are also used. These have the advantage of a smaller number of degrees of freedom and can be advanced in time with larger time steps. The beads interact also through effective “interatomic” potentials that represent the net effect of smaller-scale interactions (usually in the mean field sense, see Section 10.3.3.2). Two such examples are given

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below: the bead-spring model and the coarse-grained Monte Carlo model based on the second nearest neighbor diamond lattice (SNND) [1, 2]. In general, spatial coarse graining leads to temporal coarse graining. Models in which a bead stands for entities containing more than an atom can be advanced in time with larger time steps because the relaxation modes on spatial scales smaller than the bead are not represented. Coarse graining also leads to softer potentials, which is in line with the slower dynamics. It also leads to memory (nonlocality in time), as discussed in Section 10.3.3.3. Atomistic models are typically advanced in time with molecular dynamics (MD), which is a standard technique today. It treats the beads as classical objects with constant mass. The beads’ motion obeys classical dynamics. Several techniques exist to control the temperature or the energy of the system, a function of the thermodynamic ensemble in which the simulation is performed. Various algorithms for integration have been developed. Their review is beyond the scope of the present chapter; the reader may consult specialized texts for this purpose [3]. An alternative technique is Monte Carlo. The method is used to obtain the stationary point of a certain functional. In this method, beads are selected at random and moved prespecified distances. The energy of the new state is computed and compared with the original energy. If the energy decreases, the move is accepted. Otherwise, the move is accepted with a probability that depends (through the Arrhenius relation) on the height of the barrier separating the two states. The method has the advantage that the system can overcome multiple energy barriers and reach equilibrium, or some approximation of it, very fast (in computational time) relative to molecular dynamics. The disadvantage is that the system does not follow realistic trajectories toward equilibrium and time is not explicit. To mitigate these deficiencies, kinetic Monte Carlo (kMC) was developed. In this technique a fictitious time is associated with each successful move. MC and kMC can be performed on lattices (the beads can occupy only lattice positions) or in the continuum space. Lattice simulations are by far the most popular and the easiest, but may lead to spurious effects due to the finite spacing of the lattice and to lattice periodicity. However, these errors are expected only on scales comparable with the lattice spacing, while on larger scales lattice and off-lattice simulations lead to similar results. It should be also mentioned that while MD is clearly appropriate for nonequilibrium simulations, MC techniques are not adequate or tedious to use for this purpose. Further information on these single-scale technologies can be obtained from [4, 5]. In the context of multiscale modeling and the coupling of models defined on various scales, it is useful to comment briefly on the two atomistic-like models mentioned above: the bead-spring model and SNND MC. The bead-spring model is a central model in polymer physics. It has been used extensively over the last 20 years to test various theoretical considerations [e.g., 6]. Coarse graining from the finest possible atomistic model (chemically specific model in which each bead represents an atom) to the bead-spring model is entirely heuristic. Each bead stands for an entire segment of a chain, usually a Kuhn segment. The distance between two beads is equal to the chain-persistence length, so the beads are freely rotating relative to each other (this is also called the “pearl necklace” model).

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The nonbonded interactions are described by simple Lennard–Jones potentials. In many instances, the potential is truncated at the position of its minimum and hence is purely repulsive; this ensures that beads do not penetrate each other and hence the chains do not cross. Bonded interactions are represented by linear or nonlinear springs. The nonlinear version mimics the Langevin function and insures the finite extensibility of the chain (FENE springs). These potentials are defined heuristically and are not derived based on finer-scale models (e.g., ab-initio). However, it is possible to calibrate the few parameters of the two potentials so as to reproduce the density and cohesive energy of a certain polymeric system. The bead-spring model captures the essential physics of mechanical and dielectric relaxation in polymeric systems. The SNND Monte Carlo models were developed by Mattice and co-workers [1, 2, 7] and were initially targeted to represent simple chains such as polyethylene and polypropylene. The diamond lattice is a natural choice for the simulation of polyethylene and other vinyl polymers since the bond angle between three successive carbon atoms is approximately tetrahedral and the torsional angles (trans and gauche states) are separated by approximately 120◦ . In the finest model (chemically specific) the beads are placed on the diamond cubic lattice. The coarse-graining step is the mapping of this structure to the SNND lattice which is derived from the diamond (tetrahedral) lattice by eliminating every other site on the lattice as shown in Fig. 10.1. The resulting coarse-grained lattice is identical to the closest packing of uniform hard spheres. The name “second nearest neighbor diamond lattice” is chosen because it conveys a crucial relationship to the underlying diamond lattice that the “uniform packing of hard spheres” does not. The SNND lattice has a coordination number of 12. In the case of n-alkane simulations, each bead at a site on the SNND lattice represents two backbone carbon atoms and the hydrogen atoms bonded to them, that is, a (-CH2 -CH2 -) unit. The occupancy of the SNND lattice is quite low even at bulk densities since each lattice site represents two backbone carbon atoms.

Fig. 10.1 Mapping (coarse graining) from diamond lattice to SNND lattice. The eliminated lattice sites are shown as open circles. The resulting SNND lattice (filled circles on the right-hand side image) is a distorted cubic lattice

One of the advantages of the SNND simulations is that it is possible to reverse map and recover the fully atomistic detail at any time during the simulation. Reverse mapping is accomplished by adding the eliminated lattice sites onto the SNND lattice to recover the underlying diamond lattice. The energetic interactions of SNND

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simulations are defined by short- and long-range interactions. Short-range interactions are derived from the rotational isomeric state (RIS) theory and long-range interactions are derived from the second virial coefficient with a Lennard–Jones potential. The SNND lattice in combination with the short- and long-range interactions is used to simulate high-molecular-weight polymers at their bulk densities. Further details of the fine-to-coarse (and reverse) mapping can be found in the cited articles of the group who originally developed the method.

10.2.2 Molecular Models Another family of models for the dynamics of polymeric systems is based on various physical insight and heuristics and is defined on the much coarser scale of the entire macromolecule. A classical example is the one proposed by Edwards, DeGennes, and later fully developed by Doi and Edwards [8]. In this model, the chain is viewed as a 1D object, not necessarily composed from beads. In dense systems, the chain motion is highly constrained by its neighbors. The only allowed moves are along the chain contour (or the so-called “primitive path”). This type of 1D random walk was named reptation. It was envisioned that the chain relaxes fully when the tube is entirely renewed, that is, when the chain snakes out completely from the original “tube.” The diameter of the tube is a characteristic length scale of the model, a, and is known as the entanglement length. Other works that take related views are [9–12]. A molecular rheological model was developed by Doi and Edwards [9] based on this physical picture (the “tube” model). This model eventually became the dominant model of polymer dynamics. The tube model is of mean field type. It does not capture several multibody effects that appear to be important, especially when polymers are deformed at high rates. For example, phenomena such as contour length fluctuations (fluctuations of the tube length), constrain release (CR), and convective constrain release (CCR) and chain stretch control chain dynamics [13, 14, 15]. It is important to note that accounting for these nonlinear relaxation mechanisms within the Doi–Edwards model is difficult (and inherently ad hoc). This difficulty is due to the mean field nature of the basic representation which comes in contradiction with the multibody nature of the interactions leading to this phenomenology. Corrections to the Doi– Edwards tube model accounting for these effects have been proposed and are briefly reviewed in the next section. Discrete representations of the tube model have been developed. An important example is the stochastic single-chain representations known as CONFFESSIT [16, 17]. This is a numerical model in which a representative chain is modeled in the mean field sense, but the formulation is free of part of the approximations made in molecular rheology, approximations required in order to render the analytic form tractable. Here, an ensemble of single chains is explicitly simulated in real time and the effective behavior is obtained by averaging over the entire population. This effective response can be input in continuum models (e.g., finite element models) in place of the constitutive law.

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Coarse graining to degrees larger than in the models discussed in Section 10.2.2 appears to be a promising approach to the temporal multiscale problem. Masubuchi et al. [18] developed a dynamic network model in which coarse graining is performed to the scale of the entanglement mesh. The chains are “ropes” that perform a random walk on this network with a characteristic length scale equal to the tube diameter, a. A thorough analysis of coarse graining in polymeric systems was performed by Briels et al. [19, 20, 21]. They used a bead-spring-like model with soft potentials, in which chain noncrossability is imposed explicitly. The contact points between neighboring chains are traced in time, and bead dynamics is constrained by these “entanglements.” The model reproduces most features observed experimentally but is rather computationally expensive, the overhead added by tracing the chain contact points being important. In Section 10.3.3 we present a similar model which is a hybrid between a mean field and a multibody representation and in which the chain noncrossability is imposed in a simple manner, without the need of tracing all chain–chain contact points [22].

10.2.3 Continuum Models Continuum models require the input of a constitutive law. A large number of viscoelastic and viscoplastic empirical models have been developed for polymeric fluids and solids [23, 24]. These models are typically calibrated based on the output of macroscopic mechanical tests. Their accuracy is rather limited as no information about the deformation mechanisms or the underlying physics is used to guide model development. An alternative is provided by the much more complex molecular rheology models which are based on some aspects of the physics and various heuristic considerations. For dense polymeric systems, they come in few flavors, but all incorporate some type of constraint to the motion of the chain, akin to the constraint imposed by the tube in the Doi–Edwards theory. For example, one may view the constrains (entanglements) as individual objects rather than as a continuous tube, as “sliplinks” [25], or as a limitation imposed artificially on the Rouse modes of the chain in the direction normal to its contour [26]. Irrespective of how these are imposed, the overall concept is the same: the system of chains is represented by a “representative chain” which moves in a “mean field.” The multibody interactions are imposed (if at all) in a rather ad hoc manner. Attempts to correct these deficiencies have been made over the last 10–20 years. For example, the equations of the Doi–Edwards model have been amended by Marrucci and Grizutti to lead to the DEMG model [15, 16], which, in turn, has been repeatedly amended by a number of researchers [27–31]. The most complete molecular rheological models at this time provide a reasonable correlation with experimental data while preserving their analytic form. However, the large majority are developed for monodisperse linear polymeric melts and dense solutions. Addressing more complex systems such as melts of star polymers, branched polymers, and their mixtures is way behind and requires the use of additional approximations [32, 33].

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A separate more recent line of thought is rooted in the sequential multiscale concept by which one solves a continuum boundary value problem whose constitutive input is provided at each quadrature point by an underlying discrete model [e.g., 34]. The discrete model follows the deformation history of the continuum and returns to the coarse scale the stress state, a generalized force, or the energy state. The inherent problem with these formulations in the context of polymer dynamics is that the many-chain system dynamics cannot be solved throughout the entire coarse-scale deformation history. A molecular dynamics representation of the system at each quadrature point cannot be advanced in time long and fast enough to become really useful for the continuum scale. The characteristic time scales of the two models are too different and hence are difficult to couple them directly. The method may be used for very short chains or simple fluids with short relaxation times, which are however the least interesting in practice.

10.3 Two Information-Passing Examples In this section we present two examples of approaching the problem of deformation of polymer-based materials in a multiscale setting. Both belong to the informationpassing class of multiscale methods. In such models, the physics is integrated over short periods of time using the model with higher accuracy (atomistic, in this case) and parameters relevant for the immediately next coarser-grained model are computed. These parameters are then input into the coarser model which is used to evolve the system over a longer time interval. The parameter calibration step can be repeated after some amount of deformation if it turns out that important structural changes, which would potentially lead to significant parameter variation, may have occurred on the fine scales. The first example refers to the calibration of physics-based constitutive equations using atomistic models. Here, the method is presented in the context of polymer nanocomposites. This is an example of how material parameters relevant to larger scales can be derived from finer-scale models. The second example refers to the development of coarse-grained models for unfilled polymers. As discussed in Section 10.2, this technology is not entirely new. However, several items of novelty are presented, which significantly improve the accuracy and the range of applicability of these models. For example, we calibrate the reduced formulation using a monodisperse system of entangled polymeric chains and then use the resulting model to predict the response of polydisperse melts and monodisperse melts in nonequilibrium. This level of transferability is possible only after accounting for constraint release in the coarse representation.

10.3.1 General Strategy The general approach is shown in Fig. 10.2. The reference model, considered the most accurate in this context, is atomistic. For simplicity, we select a bead-spring

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Fig. 10.2 Schematic of the two approaches proposed for bridging the time-scale gap between atomistic and continuum models of flow

representation for the fully refined model. However, any atomistic representation can be used identically. The objective is to bridge the gap between this formulation and the continuum. The resulting continuum incorporates information pertaining to the physics taking place on smaller scales, which increases its accuracy. Continuum models can be used further to represent the flow of molten polymers and polymer nanocomposites on scales relevant to process design. The linkage is performed either by developing rheological constitutive equations that incorporate parameters calibrated using atomistic models (the right branch in Fig. 10.2) or via a set of coarse-grained models whose temporal time scale is somewhere between those of continuum and atomistics (left branch). The second approach is also of information-passing type because, on the one hand, coarse graining implies parameter calibration based on finer models, and on the other hand, coarse-grained models are supposed to be used in conjunction with continuum models, as representative volume elements associated with each Gauss point of the continuum mesh. The details of the two methods are discussed in the following sections.

10.3.2 Calibration of Rheological Constitutive Models 10.3.2.1 The System This technology is exemplified in the context of polymer nanocomposites. Let us consider a dense system of monodisperse polymeric chains mixed with nanoparticles. The nanoparticles are impenetrable for the chains and are spherical. Their

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volume fraction is kept small (below 15%) in order to mimic conditions relevant for a series of experiments on silica-filled PEO [35]. The fillers are randomly distributed in the matrix. As the filler size decreases at constant volume fraction, f, the mean wall-to-wall distance between them, d, decreases as d/R = (4π/3 f )1/3 − 2, where R is the filler radius. For f = 10%, the wall-to-wall distance is d = 1.5R. Experimental data collected over the last decade suggest that the composite has properties significantly different from those of the matrix polymer only when R is sufficiently small and comparable with the polymer chain gyration radius Rg . Note that classical homogenization predicts properties that scale with the filler volume fraction. For small f, these predictions are in strong disagreement with the experiments. The reduction of the wall-to-wall distance has at least two consequences. The probability that a given chain comes in contact with multiple fillers becomes nonzero. On the other hand, the small polymer volumes in-between fillers are strongly confined and their mechanics is expected to be different than that of bulk polymer. Moreover, if the polymer-filler interaction is strong, the chains that come in contact with given nanoparticle have a very long residence time. The particle is then covered with a dense polydisperse brush which, primarily because the curvature of the respective grafting surface, is entangled with the free chains. This provides an effective load transfer mechanism between fillers and matrix. In this work we develop a set of constitutive equations that take into account the formation of a network of chains connecting fillers [36]. To test the validity of this reinforcement mechanism, we investigate whether this network may lead

F Bridging segment

G

Dangling end

H Loop I J

K Polymer-filler junction (A-point)

Fig. 10.3 Schematic representation of the molecular scale structure. The macromolecules are divided into bridging segments, loops, and dangling ends

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to features akin to those observed in the experiments. The situation envisioned is shown schematically in Fig. 10.3. Here, a representative chain forms bridges between fillers, loops (segments that start and end on the surface of the same filler), and dangling ends. The objective is to homogenize this system, that is, to find constitutive equations describing the response of the ensemble. This way, the inferred macroscopic properties would be informed by the fine scales. In absence of topological entanglements, the molecular structure is modeled using a combination of the classical network theory and the elastic dumbbell model for polydispersed polymer segments and localized junctions. A junction represents an adsorbed polymer-filler attachment (A-point). A-points are reversible junctions and their time evolution is due primarily to the applied deformation, but fluctuations are also possible in equilibrium. Hence, the destruction and creation of segments is a stochastic reversible process. The internal structure has a transient topology and its dynamics is controlled by the rate of the attachment and detachment process and depends on the applied deformation. 10.3.2.2 Formulation of the Constitutive Model Let us consider an arbitrary representative volume element. The end-to-end vector j for a typical bridging or dangling segment is denoted by R. Ψi (R, t) is the distribution function of chain segments of type j. Here j = B or j = D, where B stands for bridging segments and D refers to the dangling ends and loops. A loop of 2n monomers is visualized as two separate pseudo-dangling segments each of length n. i represents the number of “beads” (Kuhn segments) in the respective chain segment. j Hence, Ψi (R, t) dR represents the number density of strands of type j composed of i beads, having end-to-end vectors in the range dR about 2R. The total number j j density of this type of segments at time t is given by Ni (t) = Ψi (R, t) dR, where the integration is performed over the end-to-end vector configuration space. Using the principle of local action, the time evolution of the distribution function for bridging segments is represented by the convection equation [37]: ∂ΨiB ∂ ˙ + G i (R, t) − Di (R, t), =− · (ΨiB R) ∂t ∂R

(10.1)

where G i (R, t) and Di (R, t) represent the rate of generation and destruction of bridging segments per unit volume, respectively. In general, there is no explicit form for the rate of generation and destruction functions in Equation (10.1). These are described by phenomenological relations based on the linear response theory. For example, the bridge destruction function is often taken to be proportional to the current distribution, that is, [38] Di (R, t) = di (R)ΨiB (R, t)

(10.2)

where di (R) is the detachment probability for A-points (Fig. 10.3), and therefore the probability of failure of bridging segments per unit time.

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The physical picture of the bridge formation rate is less obvious. The effect of flow is accounted for by including a term proportional to ΨiB in the equation for the rate of bridge formation: G i (R, t) = gi (R) λiB Ψi,B eq (R) − ΨiB (R, t) ,

(10.3)

B (R) = ΨiB (R, 0). λiB is a where gi (R) is the rate of A-point formation and Ψi,eq dimensionless constant to be determined, which is included in order to keep the formulation consistent at time t = 0 (in equilibrium). The functional forms of di (R) and gi (R) is Arrhenius. Detachment depends on an activation energy which is proportional to the energy of interaction between polymers and fillers and on the force pulling the respective strand from the surface (in nonequilibrium). Attachment occurs at all times when an empty spot exists on the surface, since for sticky fillers surfaces are always saturated. Equations 10.1, 10.2, and 10.3 define the evolution of the distribution function ΨiB (R, t) of bridging segments. A similar equation can be written for dangling ends. Its derivation is not presented here, but is given in full in [36]. Neglecting the contribution of entanglements, the total stress may be expressed as a superposition of contributions from bridging and dangling segments as

T=

-

(TiB + TiD ),

(10.4)

i

where TiB and TiD represent the stress contribution of bridging and dangling strands of length i Kuhn units, respectively. These quantities are evaluated using the virial 3 4 3 4 equation as TiB = FiB R B and TiD = FiD R D . The brackets denote averaging over the respective configuration space. Substituting the expression of the entropic force using the Warner approximation to the inverse Langevin function, the stress tensor reads T=

- 3k B T i

il 2 1 −

Bi 1 T r Bi (il)2 NiB

+

Di 3k B T 1 il 2 1 − (il) 2

T r Di NiD

,

(10.5)

where Bi (t) = ∫ ΨiB RR dR,

Di (t) = ∫ ΨiD RR dR.

(10.6)

and the end-to-end length of a Kuhn segment is denoted by l. Equations (10.1), (10.2), (10.3), (10.4), (10.5) and (10.6) form a full set defining the constitutive model. The model requires the integration of the evolution Equation (10.1) for the distribution functions of bridging and dangling segments of all lengths (all i), ΨiB (t)andΨiD (t). These are functions of time and depend on the motion of ˙ = fillers. The fillers are assumed to move affinely with the macro-deformation, R L · R, where L(t) is the macroscopic velocity gradient tensor (the tensor satisfies the incompressibility condition, tr L = 0). Hence, knowing L and with appropriate

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initial conditions for the distribution functions, one computes ΨiB (t) and ΨiD (t), and hence the total stress tensor T(t). 10.3.2.3 Importing Information from Atomistic Models The unknown quantities that need to be evaluated using discrete models for this structure are the number of bridging, dangling, and loop segments and the details of the chain-filler attachment/detachment process. The equilibrium structure of this system was determined by lattice Monte Carlo simulations [39–41]. Here, we review only those details of the simulation procedure and results that are relevant for the present discussion. The simulation cell is a cube of side length L, which contains spherical impenetrable fillers of radius R arranged in a simple cubic pattern. Periodic boundary conditions are imposed on all faces of the simulation cell. The volume surrounding the fillers is occupied by a monodisperse population of Nc chains of length N links. The simulation is performed for filler volume fractions ϕ = 6% and 12%. The particle diameter is about twice the chain radius of gyration, while the minimum wall-towall distance w equals 1Rg and 2Rg for the two filler volume fractions, respectively. The chains are represented as “pearl necklaces” of spherical beads; each bead representing a Kuhn segment and occupying a lattice site. The chains do not overlap (excluded volume) and interact with the fillers through a potential that has a stiff repulsive term (required in order to impose the noncrossability condition), and a longer-ranged attractive term that models the polymer-filler affinity. The system was equilibrated for several thousand Monte Carlo steps per bead and then chain statistics for the equilibrium structure was collected. The distribution functions of bridging, dangling, and loop segments were determined, as well as the total number of such chain segments per filler. The main conclusions of this study are [39–41]: B (R) exists. The total • A wide distribution of polydisperse bridging segments Ψi,eq number of bridging segments per filler decreases dramatically with the wall-towall distance, essentially vanishing when this parameter equals 2Rg . The bridges are not stretched compared to the chains in the neat bulk. D (R), including long and short seg• Dangling ends form a wide distribution Ψi,eq ments. The dangling segments follow approximately a Gaussian distribution of their end-to-end vectors at given i. The distribution depends on the wall-to-wall distance since as the fillers approach each other, part of the long dangling segments become bridges. • A large number of loop segments form on the surface of each filler. Their distribution is rather narrow and the loops are short (relative to the other segment types). Their number per filler is independent of the filler wall-to-wall distance and is largely independent of the intensity and range of the attraction between polymers and fillers. The dense population of loops increases the effective filler radius, hence enhancing the hydrodynamic interaction between fillers and matrix at large deformations rates.

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• It was observed that varying the affinity between polymers and fillers has no qualitative and little quantitative effect on the previous conclusions. Figure 10.4 shows the distributions of bridging and dangling segments in thermodynamic equilibrium obtained from the numerical model for two filling volume fractions. This information is used directly to calibrate the analytic model.

Number of bridging segments

30 25 20 15

Volume fraction = 12% Volume fraction = 6%

10 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Number of beads in segments

Number of dangling ends

60 50 40 30

Volume fraction = 12% Volume fraction = 6%

20 10 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Number of beads in segments

Fig. 10.4 The number of bridging segments and dangling ends in the simulation box, as determined from discrete simulations [39, 36]

The numerical model can be also used to elucidate the details of the polymer– filler attachment–detachment process [42]. It was observed that attachment time (the time a certain chain segment remains in contact with given filler) scales with the number of attached beads following a power law. The physical explanation for this behavior resides in the Rouse motion of the chains. This scaling holds for many decades in time. An interesting question is how the attachment time depends on the magnitude of the polymer–filler affinity (relative to the polymer–polymer affinity). Intuitively, this should follow an Arrhenius relation which depends on the solvation energy of a bead initially in contact with the filler. In fact, the problem is more complicated since, due to chain connectivity, detachment involves correlated motion in an entire vicinity of the point of attachment. Nevertheless, the numerical results confirm the Arrhenius dependence of the system average attachment time on the

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Fig. 10.5 System average attachment/detachment time, τad , as a function of the polymer-particle energetic interaction parameter, w. The normalization is made with the Rouse time of the respective chain (of N = 80 Kuhn segments) in the unfilled polymer [42]

magnitude of the energetic interaction between polymers and filler. This relation is shown in Fig. 10.5. Some results of this atomistically informed constitutive model are shown in Fig. 10.6. The figure shows the storage and loss moduli of the composite (in fact, these are just the contribution from the transient network) for systems with f = 6% and 12% filling volume fraction. It is interesting to see that the model predicts a feature usually seen in experiments performed with filled polymers: the appearance of a secondary quasi-plateau in G’ at low frequencies. This is the effect of the network of

Fig. 10.6 Storage modulus of nanocomposites with filling volume fraction 6% and 12%

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chains connecting fillers. The curve is continuously decreasing (not perfect horizontal) because the network is transient. G’ is higher in the system with higher filling fraction due to the fact that the network is better defined when filler wall-to-wall distance is smaller. This discussion exemplifies how physical insight into the behavior of the system on the molecular scale, as well as information obtained directly from fine-scale models, can be used to develop and calibrate constitutive laws and hence infer macroscopic properties of the material.

10.3.3 Developing Coarse-Grained Models of Polymeric Melts As discussed in the Introduction, molecular systems have characteristic times that span many orders of magnitudes, from atomic vibrations to hydrodynamic modes. The macroscopic evolution of the system depends on the dynamics on all scales, but, as the scale of observation becomes coarser, the contribution of finer and finer scales is important only in average. This contribution can be described in terms of a much smaller number of “independent” variables than the actual number of degrees of freedom of the system, which scales with the number of particles. The objective of any system reduction technique is to identify the “macroscopically relevant” modes or degrees of freedom and to predict their evolution without the need to fully represent the dynamics of the fine system. This mesoscopic behavior is derived from the finer scales by averaging or projection [43]. For this procedure to be applicable, it is optimal if scale separation exists. The mapping from the fine to the coarse models is not governed by fundamental principles and must be made based on physical intuition about the process to be modeled. Once the coarse-scale variables are defined, their evolution may be described using projection operator techniques. The projection leads to a reduction of the phase space dimensionality and has several important consequences. Although the fine-scale model is deterministic and conservative, the coarse-model dynamics is stochastic, includes dissipation, and has memory. Both the dissipative nature and the memory are associated with the reduction of the phase space dimensionality and the smoothing performed in phase space upon coarse graining. The dissipation represents the energy exchange with the thermodynamic “bath”, that is, with all the degrees of freedom that have been eliminated, and the memory is associated with the crossing of trajectories from the fine phase space upon projection in the coarse phase space. 10.3.3.1 The System and the Coarse Graining Concept Coarse graining is most desirable in systems with a very broad range of characteristic time scales. In the context of polymeric melts, these are systems made from long molecules. In this Section we address this challenge by developing a method to coarse grain a heavily entangled polymeric system. A mesoscopic model which is a hybrid between a multibody and a mean-field representation results [44].

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The reference fine-scale system is a bead-spring representation of a monodisperse polymeric melt. The density corresponds to physical melts and the chain length considered is large enough for the polymer to be entangled (N = 200 Kuhn segments). This model is advanced in time with classical molecular dynamics. The use of a monodisperse system as reference is not a limitation; we calibrate the coarse model using this system and then test its transferability to polydisperse models. Employing an already coarse-grained model as the reference is not a limitation either; a chemistry-specific atomistic model can be used instead. The chain is coarse grained such that each segment of length equal to that of an entanglement segment (Ne consecutive beads) is lumped into a blob. The blobs interact through potentials of mean force. The chain inner blobs are constrained to move along the respective chain coarse backbone path, which insures that chains do not cross. The chain end blobs are free to move in 3D, therefore dynamically redefining the coarse backbone path of the respective chain. The system is evolved with Brownian dynamics. The fine to coarse mapping, the definition of the backbone paths at the initial time, the potentials and the friction coefficient used in Brownian dynamics are all calibrated from the equivalent fine-scale system. The concept is shown schematically in Fig. 10.7. The highlights of this procedure are described next.

Fig. 10.7 Schematic representation of the coarse graining concept

10.3.3.2 Mapping to the Coarse Phase Space and Definition of the Interaction Potentials Coarse graining begins by defining the mapping from the fine scale to the coarse scale.5Generically, if the 6 state of the fine system is described by a set of variables X = Xi , i = 1 . . . N , where N is the total number of particles in the fine system, 5 6 one needs to define a new set of variables Xc = Xci , i = 1...N c as Xc = Xc (X), N c < N .

(10.7)

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This mapping is not unique and needs to be performed based on physical considerations. Here we consider

Xc =

l2 1 - k x , l2 = l1 + Ne − 1, Ne k=l1

(10.8)

that is, Ne consecutive beads along a chain are grouped in a blob. The centers of mass of all groups of Ne beads represent the positions of the blobs and the variables of the coarse system. Since the beads are connected along the chain, they must remain so during system evolution and this facilitates the definition of the coarsegrained objects. The quantity Ne defines the degree of coarse graining, that is, Nc = N/Ne , and, in the first approximation, defines the degree of system reduction and the phase space dimensionality loss. Here Ne is taken to be the number of beads in an entanglement segment of the fine-scale chain at the given density. This quantity can be inferred by simulating the fine-scale system or inferred from macroscopic tests on the respective monodisperse melt. In the fine-scale system the beads interact through interatomic potentials and the total energy is evaluated as the sum over all beads. Let us denote this quantity by E(x). The blobs interact through effective coarse-grained potentials which lead to the system energy, E c (Xc ). It is further required that the probability to find the coarse system in a configuration defined by Xc should be equal to the sum of probabilities to find the fine system in configurations compatible with Xc : exp −β E c (Xc ) = ∫ exp (−β E(x)) dx,

(10.9)

V (X c )

where β = 1/k B T and V (Xc ) is the volume of the fine system phase space containing configurations compatible with Xc . This expression is used to define E c (Xc ). It is noted that no functional form is prescribed for the coarse-grained potentials. Most of the times, for simplicity, one chooses pair potentials. In this work, the same approximation usually made when writing the fine-scale energy is made on the coarse scale: the energy is partitioned as a sum of energies associated with bonded and nonbonded interactions. This is equivalent to a multiplicative decomposition of the phase space probability distribution function. The coarse-grained potentials are determined using an iterative Boltzmann inversion procedure as described by Reith et al. [45]. The procedure starts with the pair distribution function on the coarse scale computed from the fine system. This is also the target distribution. A first guess for the respective potential is constructed as E c = −(1/β) log g (r ). Then, an MD simulation is performed with this potential on the blob scale and the respective pair distribution function is evaluated. Its value is usually different than the target. The potential is updated and the procedure is repeated. Denoting by gi (r) the pair distribution computed in iteration i, the potential for iteration i + 1 is evaluated as

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4

(a)

1.5

(b)

E b (r)/εLJ

2

c

1

c

E nb (r)/εLJ

3

0.5

1

0

2

4

6

8

0

10

5

r/σLJ

10

15

r/σLJ

Fig. 10.8 Coarse-grained bonded and nonbonded potentials calibrated from the fine-scale system (an atomistic model) [23]

gi (r ) , + (1/β) log g(r )

=

c E i+1

E ic

(10.10)

The bonded and nonbonded coarse grained potentials are shown in Fig. 10.8. The nonbonded potential has a repulsive branch only and finite value at r = 0. The potential is less stiff than the equivalent potential of the fine-scale system; the more one coarse grains, the softer the interactions become. Since the system is dense, the blobs must overlap in the coarse system in order to reproduce the density of the reference system. The finite force at r = 0 does not prevent the blobs from fully overlapping and the chains from crossing. Hence, additional constrains need to be imposed to insure noncrossing. Furthermore, the weakly repulsive potentials provide less caging to the blobs and hence, if the coarse-grained system is advanced in time by MD with no additional constrains to the motion of the blobs, their diffusion is observed to be faster than in the reference system [46, 47]. 10.3.3.3 Dynamics in the Coarse-Grained Model The theoretical basis for deriving the equations of motion of the reduced system starting from those of the fine system is provided by the Projection Operators technique developed by Zwanzig [48] and Mori [49]. The equation of motion is put in the form of the generalized Langevin equation: c

t

c

¨ = − ∫ G(t − τ )X ˙ (τ )dτ + F R (t) + Fd (t), M cX

(10.11)

0

where Mc is the mass of the blob (Mc = Ne m0 ), FR and Fd are the random and deterministic forces, respectively. The two forces represent interactions with the degrees of freedom preserved in the model (deterministic force) and those eliminated, or

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R.C. Picu

“the bath” (random force). The deterministic force is obtained from the gradient of the blob-interaction potentials. The projected force is not known exactly, but it must have zero mean and no correlation with the velocities of the coarse-grained degrees of freedom. The fluctuation dissipation theorem provides an expression for the kernel G [50]:

G(t) =

4 β3 R F (t)F R (0) . 3

(10.12)

Working with the generalized Langevin equation is difficult because of the (usually) long range of the kernel. To simplify the calculations, one can make the Markovian approximation that G(t) decays much faster than the characteristic time of vari˙ c (t), a case in which the velocity can be factored out from the integral and ation of X the Langevin equation results: ¨ c = −ξ X ˙ c + F R + Fd , M cX where the friction tensor is given by ξ (t) =

β 3

(10.13)

2t0 3

4 F R (τ )F R (0) dτ . In this expression,

0

t0 is usually taken to be infinity. The calibration of the friction coefficient from atomistic simulations has been the subject of many works. The theory requires that the friction be computed from the autocorrelation function of the projected force FR . However, this quantity cannot be directly evaluated in an MD simulation. The only available quantity is the total c force Fc = P˙ acting on a probe (e.g., a blob defined fictitiously in the reference system). It has been shown by a number of authors [51, 52] that the total force becomes equal to the projected force when the probe becomes infinitely heavy. Moreover, as t0 increases, in order to insure the convergence of the integral to a plateau, the thermodynamic limit must be considered. This implies that the friction computed using this procedure depends on the size of the fine system considered in simulations. This issue is fairly well understood for simple fluids, but less so for polymeric fluids in which correlations are longer and have multiple physical origins (hydrodynamics, chain connectivity, etc.). From Equation (10.13) one can compute the autocorrelation function of the blob momentum (VACF). The friction coefficient can be also extracted from the decay of this function. The VACF was computed from the fine-scale model (with momenta evaluated on the blob scale) and is shown in Fig. 10.9. It exhibits a fast, exponential relaxation, followed by a long tail which is related to hydrodynamic interactions. The VACF was fitted with a sum of an exponential and a stretched exponential (to represent the tail). The friction coefficient per blob estimated through this procedure is 6.5τ L−1J . If one takes t0 → ∞ in Equation (10.13), the friction results 15.1τ L−1J , which shows that the contribution of the VACF tail to the apparent friction is important. A value of the friction of ∼15τ L−1J also results from the fine-scale system Rouse modes and by using the mean square displacement technique.

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1 0.02 0.01

0.2

0

vacf(t)

0.6

–0.01

1/e

–0.02 0

10

20

30

0.4

0.2

0

10–2

100

102 t/τLJ

Fig. 10.9 Velocity autocorrelation function computed on the coarse scale using the fine-scale model. This function is used to compute the friction coefficient

In general, the presence of a tail in VACF precludes the use of the Markov approximation for the equation of motion. However, once the approximation (10.13) is made, the smaller friction coefficient (6.5τ L−1J ) must be used for internal consistency. This is because the coarse model with the Markovian approximation would lead to an exponential decay of the function in Fig. 10.9, with no tail. The hydrodynamics of the coarse model is “poorer” since the soft repulsive potentials used on this scale do not promote strong long-range correlations of this type. Some degree of hydrodynamics, with modes having wavelength much larger than the coarse graining scale is still present. The crucial issue of the selection of the friction coefficient is further discussed in [23] and in the works by Briels and co-workers [20, 21, 22]. 10.3.3.4 Constraints and Constraint Release Evolving the system in the coarse-grained phase space would not lead to a behavior compatible with that of an entangled melt, since the soft repulsive potentials of the coarse model will not prevent chain crossing. A number of solutions that include various degrees of heuristics have been developed for this problem [9, 26, 53]. The approach adopted in this work is in-between the mean field view and a fully discrete (multibody) MD model. Specifically, we use the tube concept for the chain inner blobs, while the chain end blobs are allowed to move freely in space. The middle blobs are constrained to move along the coarse-grained chain backbone (CB).

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The end blobs are not constrained. The path (CB) is being continuously redefined by the trajectory of the end blobs (Fig. 10.7). The advantage of such view is that most relevant many-body interactions are taken into account, while the chain noncrossing condition is fulfilled at all times. It is noted that this procedure leads to an additional reduction of the number of degrees of freedom in the coarse model by 2Nc (N /Ne − 2), where Nc is the number of chains in the model. It should also be emphasized that, following the spirit of the tube model, the level of coarse graining must be taken equal to Ne . If a smaller number is selected, the type of constraints used here cannot be applied explicitly, while if a larger number is used, the physics of topological interactions between chains is violated. The CB of given chain is mapped from the reference system at the initial time as the path joining the blob centers of mass. The path is a random walk in 3D with the mean step length equal to the mean of the entanglement segment end-to-end distribution. The fact that the CB should not be a smooth contour is important. Constraint release is an important component of the physics of these systems. As discussed in the Introduction, this notion stands for the process leading to the release of the constraints imposed by surrounding chains on the motion of a representative chain, constraints that leads to reptation. This release occurs by chain retraction and “dilution” of the neighborhood of the representative chain in time, that is, effectively, due to the reptation of neighboring chains. This process is essential in nonequilibrium, but has a small influence on the average response in equilibrium. In molecular rheology models, constraint release is usually represented by allowing for Rouse-like motion of the primitive path. The frequency and amplitude of the jumps are kept as heuristic parameters. This uncertainty is due to the fact that most models are of mean field type, while constraint release is a multibody process. In the coarse-grained model discussed here, the discrepancy is eliminated since the entire multibody information is available. We developed a method to account for constraint release which is shown schematically in Fig. 10.10. Let us consider that at the current time t, the CB of chain A loses the leftmost segment A0–A1. Note that the figure does not represent the actual chain and the points Ai (i = 0,1. . .5) do not stand for the positions of the blobs. Rather, Ai are the corners of the CB of chain A and the blobs (not shown) are located anywhere between these points; the CB is the support on which the chain middle blobs perform their 1D random walk. The segment A0–A1 is removed when all blobs of chain A retract to the right of point A1. Following such redefinition of the CB of chain A, all CB corners (of chain A and any other chain, say, chains B, C, and D) located within a sphere of radius a are selected and their location is adjusted based on a parameter-free algorithm aimed at homogenizing the distribution of volumes in the region centered at A0. The volume within the sphere is tessellated in 3D using a Delaunay algorithm [54], for which the CB corners at time t are used as nodes. The volumes of the tetrahedra of the tessellation are evaluated. These values define samples from a probability distribution function, which is compared to a reference distribution. A Monte Carlo procedure is then used to adjust the CB corner positions within the respective sphere so as to bring these samples in agreement with

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Chain B

Chain C

A3

Chain A A4

M

Chain D A2

A0 N

A1

Fig. 10.10 Schematic representation of the constraint release algorithm

the target PDF. The reference PDF of tetrahedra volumes is determined in preliminary runs by tessellating the entire model after equilibration. Modifying the position of the CB corners implies shifting the blobs along with the CB segments on which they reside. The chain noncrossing condition is enforced during these moves. This algorithm enforces that chain retraction does not perturb the homogeneity of the CB corner distribution in space, a mandatory condition for the melt. It involves no ad-hoc parameters while making use of the entire multibody information available in the model. 10.3.3.5 Results The coarse-grained model is developed by calibrating the potentials and the friction using a dense system of N = 200 beads per chain. This is moderately entangled, as the entanglement segment length is about Ne ∼ 40 [6]. The system is first considered in equilibrium and we check that the dynamics of the coarse model matches that of the reference, fine system. The Rouse modes of the coarse model, defined by the autocorrelation function 4 3 Xp (t) · Xp (0) 4, gp (t) = 3 Xp (0) · Xp (0)

(10.14)

are shown in Fig. 10.11 [23]. Here, Xp is the normal coordinate for mode p. The curves, labeled “fine scale (MD),” are obtained from the reference system. Only the first four modes of the fine system can be compared directly with the coarse model. The agreement between the coarse and fine models is very good. The only difference

314

R.C. Picu 1 0.9 0.8 0.7 0.6

Fig. 10.11 Rouse modes computed in the reference and coarse-grained systems

0.5

gP(t)

0.4

1/e p=3

p=4

0.3

p=1

p=2

0.2

coarse scale(BD) fine scale(MD)

0.1 102

103

104 t/τLJ

105

is related to the fact that the coarse model leads to more exponential relaxation than the fine model. Another confirmation of adequate calibration is provided by the comparison of the mean square displacement of the blobs, g1c , and the chain centers of mass, g3c and g3 [23]. This agreement shows that the friction coefficient was properly calibrated and the constraints perform properly. The proper verification of the procedure has to involve the use of the model in contexts other than that used for calibration of model parameters. To this end, we considered monodisperse systems of chain length larger than N = 200, up to N = 800. We studied the scaling of the ultimate relaxation time, τd , and that of the diffusion coefficient with N. The relaxation time scales as N 3.4 and the diffusion coefficient decreases with N as N −2.1 [23]. This scaling compares well with the experimental observations and differs from the prediction of the tube model (which leads to exponents 3 and –2, respectively). Further testing of the coarse model was performed by comparing results obtained with the fine and the coarse models for a bi-disperse polymer melt without further parameter calibration [55]. This system contains chains of two different lengths (long and short), NS and NL , which are present with fractions, ϕ S and ϕ L = 1 − ϕ S . A special type of such system is that in which one of the species is present in very small numbers, and one of the two fractions is close to zero (tracer conditions). Figure 10.12 shows curves for NL = 320 in the mixture with NS = 200 at φs = 0.62, and for the monodisperse system with N = 320. Although the trajectory obtained from the fine model is of limited length, one can conclude convergence at times larger than ∼105 τ L J . A speed-up of the dynamics of the long chain is observed in the mixture, in agreement with previous numerical and experimental observations. The agreement between the fine and coarse mean square displacements cannot be expected at early times, as these are below the time scale corresponding to the coarse graining level.

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315

350 250

NL = 320 φsw = 0.62

g3(t), g1(t)

150

50

neat (N = 320, F) Ns = 200 (F) neat (N = 320, CG) Ns = 200 (CG) 100000

700000

t/τLJ Fig. 10.12 Mean square displacement of coarse-grained blobs, g1F−C G (t), and chain centers of mass, g3F (t) (lines with symbols) along with their CG counterparts, g1C G (t) and g3C G (t) (lines without symbols), for the chain of length NL = 320 in the mixture with NS = 200 at φs = 0.62 and in the monodisperse N = 320 system [56]

It is noted that obtaining the fine-scale trajectories shown in Fig. 10.12 is extremely time consuming and advancing the model to times on the order of t = 106 τ L J is very expensive with molecular dynamics. The trajectories shown with continuous lines obtained with the coarse model result in just few hours of running on a single processor. The scaling of the diffusion coefficient of chains of length NS = 200 and 400 with the length of the matrix chains is shown in Fig. 10.13 for systems with φs = 0.62. The results are obtained with the CG model. The diffusion coefficient for several systems with NS = 200 was evaluated using atomistic models (F) and is shown in the figure (filled symbols). The diffusion coefficient reaches a plateau once the matrix chains become long enough. The plateau is broadly documented in the experimental literature [56, 57]. This is the first time the plateau is obtained by means of simulations. Under these conditions, the matrix chains are long enough for the shorter chains to see an environment that does not evolve significantly in time on the time scale of the relaxation of the short chains. Hence, making the matrix chains even longer has no effect on the dynamics of the short chains. This demonstrates that material properties (self-diffusion coefficient in this case) can be obtained by a proper use of atomistic and coarse-grained models, even for systems with very long chains. Finally, it is interesting to look at the behavior of the coarse-grained model in nonequilibrium. All essential features of the rheology functions observed in

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0.14

DτLJ /σ 2LJ × 103

0.12 0.1 0.08

Ns = 200 (F)

0.06

Ns = 200 (CG) Ns = 400 (CG)

0.04 0.02 0

400

200

800

600

1000

NL Fig. 10.13 Variation of the diffusion coefficient of chains of length NS = 200 and 400, with the length of the second type chains, at φs = 0.62. Data from coarse-grained (CG) and finescale (F) models are shown. F model data are available only for NS = 200 in mixtures with itself (monodisperse), with NL = 320 and NL = 400 [56]

0.14 0.12

Wi = 0.93

0.12

Wi = 0.93

0.08

0.08 Wi = 1.875 0.06 0.04

Wi = 3.75

ψ1+(t)

η+(t)

0.1 Wi = 1.875

0.04 Wi = 3.75

0.02 0

0 0

200000

400000

t/τLJ

600000

0

200000

400000

600000

t/τLJ

Fig. 10.14 Transient shear viscosity η+ (t) = σ¯ 12 (t)/W i and transient first normal stress coefficient ψ1+ (t) = (σ¯ 11 − σ¯ 22 )/W i 2 after start-up of shear flow at t = 0, for several values of the Wi numbers [59]

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experiments on monodisperse melts were recovered using the coarse model [58]. The model was used to study start-up and step strain shear flows and reproduce the overshoot during start-up shear flow, the Lodge–Meissner law, the monotonicity of the steady-state shear stress with the strain rate, and shear thinning at large strain rates γ˙ . No comparison is made with predictions of the fine-scale model because data under such small strain rates and with this reduced level of noise cannot be obtained using a molecular dynamics fine-scale model. We reproduce here (Fig. 10.14) only the transient shear viscosity, η+ (t) = σ¯ 12 (t)/W i, and first normal stress coefficient ψ1+ (t) = (σ¯ 11 − σ¯ 22 )/W i 2 following a start-up shear flow. The stress values correspond to the normalization of the actual stress tensor with the stress-optical coefficient, C: σ¯ = σ /C. Wi stands for the Weissenberg number: W i = γ˙ τd , where τd is the ultimate relaxation time of the system. Acknowledgment Several people contributed to the material presented here: Dr. Abhik Rakshit, developed the coarse graining technology, Prof. Alireza Sarvestani, Dr. Murat Ozmusul, Dr. Peter Dionne and Prof. Rahmi Ozisik contributed to the various aspects of the nanocomposite study.

References 1. R.F. Rapold and W.L. Mattice, “Introduction of Short and Long Range Energies to Simulate Real Chains on the 2nnd Lattice,” Macromolecules, Vol. 29, 1996, p. 2457 2. J. Cho and W.L. Mattice, “Estimation of Long-Range Interaction in Coarse-Grained Rotational Isomeric State Polyethylene Chains on a High Coordination Lattice,” Macromolecules, Vol. 30, 1997, p. 637 3. M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, 1987 4. V. Galiatsatos, Molecular Simulation Methods for Predicting Polymer Properties, Wiley, Hoboken, 2005 5. D. Frenkel and B. Smit, “Understanding Molecular Simulation: From Algorithms to Applications,” Academic Press, San Diego, 2002 6. K. Kremer and G.S. Grest, “Dynamics of Entangled Linear Polymer Melts: A MolecularDynamics Simulation,” J. Chem. Phys., Vol. 92, 1990, p. 5057 7. R. Ozisik, P. Doruker, et al., “Translational Diffusion in Monte Carlo Simulations of Polymer Melts: Center of Mass Displacement vs. Integrated Velocity Autocorrelation Function,” Comput. Theor. Polym. Sci., Vol. 10, 2000, p. 411 8. M. Doi and S.F. Edwards, Theory of Polymer Dynamics, Clarendon, Oxford, 1986 9. G. Ronca and G. Allegra, “An Approach to Rubber Elasticity with Internal Constrains,” J. Chem. Phys., Vol. 63, 1975, p. 4990 10. W. Hess, “Generalized Rouse Theory for Entangled Polymeric Liquids,” Macromolecules, Vol. 21, 1988, p. 2620 11. T.A. Kavassalis and J. Noolandi, “A New Theory of Entanglements and Dynamics in Dense Polymer Systems,” Macromolecules, Vol. 21, 1988, p. 2869 12. K.S. Schweitzer, “Microscopic Theory of the Dynamics of Polymeric Liquids: General Formulation of a Mode-Mode-Coupling Approach,” J. Chem. Phys, Vol. 91, 1989, p. 5802 13. R.G. Larson, “A Constitutive Equation for Polymer Melts Based on Partially Extended Strand Convection,” J. Rheol., Vol. 28, 1984, p. 545 14. G. Marrucci and N. Grizzuti, “Fast Flows of Concentrated Polymers: Prediction of the Tube Model on Chain Stretching,” Gazz. Chim. Ital., Vol. 118, 1988, p. 179 15. G. Marrucci and N. Grizzuti, Topics in Molecular Modeling of Entangled Polymer Rheology, Proc. 10 Int. Congr. Rheol., Sydney, 1988

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16. K. Feigl, M. Lasso and C.H. Ottinger, “CONNFFESSIT Approach for Solving a TwoDimensional Viscoelastic Fluid Problem,” Macromolecules, Vol. 28, 1995, p. 3261 17. M. Laso and H.C. Ottinger, “Calculation of Viscoelastic Flow Using Molecular Models: The CONNFFESSIT Approach,” J. Non-Newtonian Fluid Mech., Vol. 47, 1993, p. 1 18. Y. Masubuchi, J. Takimoto, et al., “Brownian Simulations of a Network of Reptating Primitive Chains,” J. Chem. Phys., Vol. 115, 2001, p. 4387 19. R.L.C. Akkermans and W.J. Briels, “Coarse Grained Interactions in Polymer Melts: A Variational Principle,” J. Chem. Phys., Vol. 115, 2001, p. 6210 20. J.T. Padding and W.J. Briels, “Uncrossability Constrains in Mesoscopic Polymer Melts Simulations: Non-Rouse Behavior of C120 H242 ,” J. Chem. Phys., Vol. 115, 2001, p. 2846 21. J.T. Padding and W.J. Briels, “Time and Length Scales of Polymer Melts Studied by Coarse Grained Molecular Dynamics Simulations,” J. Chem. Phys., Vol. 117, 2002, p. 925 22. A. Rakshit and R.C. Picu, “Coarse Grained Model of Entangled Polymer Melts,” J. Chem. Phys., Vol. 125, 2006, p. 164907 23. I.M. Ward and J. Sweeney, The Mechanical Properties of Solid Polymers, Wiley, Chichester, 2004 24. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1987 25. J.D. Schieber, J. Neegaard, S. Gupta, “A Full Chain, Temporary Network Model with Sliplinks, Chain Length Fluctuations, Chain Connectivity and Chain Stretching,” J. Rheol., Vol. 47, 2003, p. 213 26. T.A. Kavassalis and J. Noolandi, “New Theory of Entanglements and Dynamics in Dense Polymer Systems,” Macromolecules, Vol. 21, 1988, p. 2869 27. D.W. Mead, R G. Larson, M. Doi, “A Molecular Theory for Fast Flows of Entangled Polymers,” Macromolecules, Vol. 31, 1998, p. 7895 28. G. Ianniruberto and G. Marrucci, “Convective Orientational Renewal in Entangled Polymers,” J. Non-newtonian Fluids Mech., Vol. 95, 2000, p. 363 29. G. Ianniruberto and G. Marrucci, “A Simple Constitutive Equation for Entangled Polymers with Chain Stretch,” J. Rheol., Vol. 45, 2001, p. 1305 30. R.S. Graham, A.E. Likhtman, T.C.B. McLeish, S.T. Milner, “Microscopic Theory of Linear, Entangled Polymer Chains Under Rapid Deformation Including Chain Stretch and Convective Constrain Release,” J. Rheol., Vol. 47, 2003, p. 1171 31. G. Marrucci and G. Ianniruberto, “Flow-Induced Orientation and Stretching of Entangled Polymers,” Phil. Trans. R. Soc. Lond. A, Vol. 361, 2003, p. 677 32. S.T. Milner and T.C.B. McLeish, “Arm-Length Dependence of Stress Relaxation in Star Polymer Melts,” Macromolecules, Vol. 31, 1998, p. 7479 33. S.T. Milner, T.C.B. McLeish, R.N. Young, A. Hakiki, J.M. Johnson, “Dynamic Dilution, Constrain Release and Star-Linear Blends,” Macromolecules, Vol. 31, 1998, p. 9345 34. E. Weinan, B. Engquist, X. Li, W. Ren, E. Vanden-Eijden, “The Heterogeneous Multiscale Method: A Review,” Comm. Math. Sci., Vol. 23, 2003, p. 432 35. Q. Zhang and L.A. Archer, “Optical Polarimetry and Mechanical Rheometry of PEO-Silica Dispersions,” Macromolecules, Vol. 37, 2004, p. 1928 36. A.S. Sarvestani and R.C. Picu, “Network Model for Viscoelastic Behavior of Polymer Nanocomposites,” Polymer, Vol. 45, 2004, p. 7779 37. R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Wiley, New York, 1987 38. N. Phan Thien and R.I. Tanner, “New Constitutive Equation Derived from Network Theory,” J Non-Newtonian Fluid Mech., Vol. 2, 1977, p. 353 39. M.S. Ozmusul and R.C. Picu, “Structure of Polymers in the Vicinity of Curved Impenetrable Surfaces – The Athermal Case,” Polymer, Vol. 43, 2002, p. 4657 40. P.J. Dionne, R. Ozisik, R.C. Picu, ) “Structure and Dynamics of Polyethylene Nanocomposites,” Macromolecules, Vol. 38, 2005, p. 9351 41. M.S. Ozmusul, R.C. Picu, S.S. Sternstein, S. Kumar, “Lattice Monte Carlo Simulations of Chain Conformations in Polymer Nanocomposites,” Macromolecules, Vol. 38, 2005, p. 4495

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Chapter 11

Prediction of Damage Propagation and Failure of Composite Structures (Without Testing) G. Labeas

Abstract The increasing application of advanced fibrous composites as primary structural materials in aerospace, marine and automotive applications has resulted in the need of developing reliable methods to predict the performance of composite structures beyond initial local failure. The partial replacement of testing in the prediction of damage propagation and failure mechanisms in composite structures beyond the first failure has been motivated by the high cost and difficulty of extensive experiments on composite structures, especially at high- or full-scale level. In the current chapter, the basics of the progressive damage modeling (PDM) methodology, comprising three major components, that is, computational model, failure analysis and representation of damage evolution, are implemented in the prediction of structural response and demonstrated in the case of different composite structures, at various scale levels, including the multiscale. PDM demonstrations include composite open-hole panels, bolted joints, bonded repairs and carbon nanotube reinforced structures.

11.1 Introduction There is a continuously increasing application of advanced fibrous composites as primary structural materials in aerospace, marine and automotive applications. The percentage of composite components for the next generation of airlines will reach or exceed 50%, while in non-aerospace products there are similar trends. The driving force for the widespread use of composite materials mainly arises from their superior specific strength and stiffness and the ability of tailoring them to meet the design requirements, which result in significant improvements in fuel consumption and cost-efficiency. G. Labeas (B) Laboratory of Technology and Strength of Materials, University of Patras, Panepistimioupolis Rion, 26500 Patras, Greece e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 11,

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Advancement of composite structures became possible by the successful development of suitable material systems, comprising fibres and polymeric matrices in numerous forms, the development of reliable and cost-effective production techniques, as well as progression in the design and analysis methods of anisotropic material structures at different scale levels. The demand for continuous performance improvement of composite systems, which arises from the increased exploitation of composites, continuously generates new requirements in the previously mentioned scientific and technological research sectors. With respect to the analysis–design–testing procedures of advanced composite structures, a methodology for the reliable performance prediction of composite structures beyond initial local failure is of major importance. Many composite materials have brittle behaviour, with limited margin of safety as compared to their typical metallic counterparts. Therefore, the replacement of testing in the prediction of damage propagation and failure mechanisms evolution in composite structures beyond the first failure has attracted considerable attention by many researches in recent years, motivated by the high cost and difficulty of extensive experiments on composite structures, especially at high- or full-scale level. Local failure in composite materials usually initiates in the form of matrix cracks, fibre breakage, fibre-matrix shear-out and delamination. These localized damage types usually interact between themselves, propagate in the structure with different rates, depending mainly on the material type, the operating conditions and the load redistribution capability as this is affected by the locality of the damage. Hence the load-carrying capability of structural components can be found below the desired limit load with operational time. To achieve the goal of predicting the structural behaviour at various scales, it is necessary to implement three basic analysis steps The first step includes the descriptive analysis of the geometrical factors, loading factors, boundary conditions, material properties, and so on, a process which is usually performed using computational analysis scheme in order to provide stress, strain and displacement fields within the structure with sufficient accuracy. The second step encapsulates all possible failure modes within a composite system. Each of the failure modes is independently modelled according to appropriate failure theories. The third step, fuelled by the previous two, provides descriptive equations used to reflect the conditions and level of damage. Finally, using an iterative process, the designer can achieve a virtual representation of structural behaviour with time. The progressive damage modelling (PDM) methodology is a logical chain of all the above steps. It aims to practically realise the ambitious goal of prediction the entire response of the composite structure, starting from the very first localized damage generation, up to ultimate failure, by implementing only a minimum number of unavoidable experimental tests. The basic theories on which PDM methodology at different scale levels is based are discussed in Section 11.2. A demonstration of PDM methodology in critical structural elements containing stress concentrators, such as open-hole panels, bolted joints and bonded composite patch repairs, are presented in Sections 11.3, 11.4 and 11.5, respectively. In Section 11.6, carbon nanotubes and nanotube reinforced polymeric matrix composites are investigated

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by PDM principles. Major conclusions about the PDM approach and recommendations with respect to limitations, application guidelines and suggestions for further developments may be found at the end of this chapter.

11.2 Basics of Progressive Damage Modelling methodology 11.2.1 PDM – An Overview The overview of a typical analysis methodology based on the principles of progressive damage modelling (PDM) is presented in Fig. 11.1.

Structure geometry

Boundary conditions

Macro Mechanics Validated computational structural numerical model

Micro Mechanics Nano Mechanics

Nonlinearities (material/geometrical)

Multi-scale mechanics

Material Properties

Applied loads (static, fatigue, environmental Micro Mechanics Failure theories at various scale levels

Damage mechanics Fracture mechanics (cohesive elements)

Prediction of failure sites

Nano Mechanics Multi-scale Mechanics

Response of damaged material (property degradation rules, delamination evolution)

Prediction of progressive failure pattern, mode, size and ultimate failure load

Fig. 11.1 An overview of typical progressive damage modelling (PDM) approach

PDM is essentially based on an iterative methodology comprising three main modules. The first module is a validated multiscale computational model of the structure, which serves the purpose of determining stresses and strains at an

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appropriate scale level for the subsequent analysis steps; in the macromechanics level, lamina stress state is required, while in the micromechanics level, fibre and matrix stress fields should be predicted. The specified operational load is divided into an appropriate number of load steps. Each step is engaging a series of numerical calculations assuming that no changes in the material stress–strain behaviour will take place. The second major module of the methodology deals with the prediction of localized failure initiation based on the various failure theories, which also depend on the scale level. In the macro- and micro-mechanics level, computed lamina or matrix and fibre stresses/strains are compared to material allowables or their combination in order to predict initial failure generation and type. The third basic module, completing PDM methodology, is the process of describing the response of the damaged material volumes due to the progressive damage evolution within the structure. Degrading material properties is the most usual way to match the response of the failed structure to that observed in the experiments, while for the case of delamination, crack interfaces are appropriately introduced. Any degradation should be performed on the same scale level at which local failure has been predicted. For example, in the macromechanics level, when a lamina failure is detected, its properties at certain material axes are modified according to suitable degradation models. Due to the material property reduction, the stiffness of the structure changes locally, therefore, the current stress solution of the computational model does not represent an equilibrium state any more. The structure equilibrium at the current load step should be established again, taking into account the modified material properties of the ‘failed’ material volumes, until no additional failures are detected. Only then the load may be further increased to the next loading level and the previously described main modules of PDM method re-executed iteratively, until catastrophic failure of the structure is detected. The intermediate results of the PDM approach comprise the history of failure patterns and failure modes evolution within the structure, as well as the damage extent and ultimate load at final failure.

11.2.2 Multiscale Computational Model The analysis method used to calculate the stress, strain and displacement fields depends on the scale level the analyst will select to perform failure prediction and material property degradation. Furthermore, the analysis level strongly depends on the type of structure under consideration. For example, in the case of composite structures made of conventional unidirectional laminas, stacked in a desired lamination to form a laminate, the macromechanics approach is well established since many years and can lead to results, which match the experimental observations, not only qualitatively but also quantitatively. However, more modern composite material systems have been recently under development. Fabric composites reinforced not only in-plane but also in the third dimension have been developed in order to meet the need for enhanced out-of-plane properties and higher impact resistance.

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Non-crimp fabric composites have been proposed in order to eliminate the problem of in-plane properties reduction that occurs in three-dimensional fabrics. Carbon nanotube-reinforced polymeric composites are under investigation, aiming to dramatically increase both stiffness and strength characteristics at minimum weight increase. However, the effect of micro- or nanostructural parameters, such as type or density of nanotubes, tow shape or stitching, on the macroscopic response of the material system is not yet satisfactorily established, and failure theories or even damage mechanics of such systems are far from having been understood. Therefore, any PDM approach of these modern composite systems has to include stress analysis up to a level which goes down to the micro- or nanomechanics level. However, as such a detailed computational model is not possible for the entire volume of the structure, it is usual practice to combine solutions at the various scale levels. At macromechanical level, the lamina is the lowest scale element modelled, in which homogenized properties are appropriately assigned, such that the real macroscopic behaviour is achieved. At this level, computational model difficulties originate by the need of accurately modelling the special characteristics of the original undamaged structure, which may have irregular geometry due to geometrical discontinuities, such as holes, cut-outs, thickness reduction in ply drop-off areas, presence of reinforcing elements, high anisotropy; the complexity further increases by the inhomogeneous nature of the locally failed structure. Because of these characteristics, most of the other solution methods that are not numerical are capable of solving the problem only partly, therefore, are incompatible to PDM requirements. One of the most powerful numerical computational methods for performing structural analysis in different scale levels is the finite element (FE) method. For this reason most researchers have based their PDM developments on a validated FE model capable of computing stresses, strains and displacements which match correctly the corresponding experimentally derived magnitudes of the real structure. Modelling the structure by FE should ideally be based on three-dimensional finite elements in order that interlaminar stresses and strains are computed with sufficient accuracy. Interlaminar stress magnitudes are essential for the prediction of delamination, which is strongly a three-dimensional failure mode, as it usually initiates between plies and may propagate in the same ply interface, and also can interact with other in-plane failure modes and kink into other plies interfaces. However, due to the extensive amount of computational effort required for threedimensional analysis, PDM has also been based on either ‘smearing’ the properties of certain number of plies within one element through the thickness or on twodimensional application (using plate and shell finite element models), or on using global–local approaches by dividing critical areas (modelled by 3-D elements) and non-critical areas (modelled by 2-D elements). It is out of the scope of this text to provide an extensive literature review on the different computational schemes already applied to different PDM models, which can be found, for example, in [14] and [52]. At the micromechanics level, fibre and matrix materials are explicitly modelled and their individual elastic properties are introduced. Upon the numerical solution of the respective model, computed fibre and matrix stress and strain results are used to

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perform failure analysis and property degradation based on the individual strength properties of fibre, matrix and their interfaces. As mentioned above, a computational model of such a level is only required when homogenised properties at macroscopic level are not available, or in case that failure criteria at the macroscopic level are not established; otherwise, implementation of such a detailed computational model can be considered as an unnecessary extra effort, with very low additional value. An example of a computational model at the micromechanics level, which refers to two different forms of non-crimp fabric composite system, is presented in Fig. 11.2, from [55]. Three-dimensional solid elements with orthotropic material properties are used to model the matrix material volumes, while stitching simulation is performed through superposition of elastic-beam elements following the stitching patterns onto the matrix FE mesh. It becomes apparent from Fig. 11.2, that such a detailed computational model cannot be efficiently created for the entire volume of a realistic large-scale structure, not even for a structural element, for example. a bolted joint, due to the increased needs for computational power. Therefore, global–local approaches based on representative volume element (RVE) principles are applied; they comprise full characterisation of the RVE using computational models and failure theories at detailed micro-mechanic level and then introduction of the derived behaviour in a global macro-mechanical model. The RVE of the fabric composite material of Fig. 11.2 is appropriately selected, in a way that if adequately repeated in all three space dimensions, the global structure will arise. After proper periodic boundary conditions are applied to the RVE, it is loaded in all possible directions, including multiaxial load combinations, and its response is numerically computed. In such a way, it is then possible to efficiently transfer the mechanical behavior of the micro-structure into the homogenised macro-level global model. Specific attention must be paid to the modules of failure analysis and material property degradation of the RVE in order to achieve accurate stress–strain diagrams for each material direction.

Fig. 11.2 Typical FE computational models of NCF composites, from [55]

At the nanomechanics level, carbon nanotubes (CNTs) and CNT-reinforced polymers are of increasing interest. CNTs are fullerene-related structures discovered by Iijima in 1991 [23] that can be visualised as graphene sheets rolled into hollow cylinders composed of hexagonal carbon rings, as shown in Fig. 11.3 (a) and (b). Each carbon atom is connected to three neighbouring atoms via covalent bonds, one of the strongest chemical bond types in nature.

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Fig. 11.3 Schematic representation of (a) grapheme and (b) cylindrical carbon nanotube, after [55]

(a) Hexagonal carbon rings (b) Cylindrical nanotube

FE analysis is a capable computational method of modelling the exact geometric hexagonal pattern of the nanotube structure; the atomic bond between two carbon atoms is represented by load-carrying members (FE elements) and carbon atoms are joint points of these elements [58]. The element type is 3-D elastic beam with solid circular cross-section and appropriate material properties, which represent the linear and non-linear behaviour of the atomic bond. As CNTs alone cannot be practically used as structural materials, they are usually introduced in polymeric matrices, either as replacements of conventional fibres or as fillers to enhance the properties of advanced composite structures. Due to the impressive mechanical properties of CNTs, which reach to elastic modulus values higher than 1 TPa, tensile strengths in the range of 150 GPa and failure strain up to 20%, the mechanical properties of CNT-reinforced polymeric matrices with only 1% by weight of CNTs can result in a stiffness increase between 36% and 42% and the a tensile strength increase of about 25% for the resulting composite film. It becomes apparent that between nanomechanical computational models (length of some nanometers) and the macroscopic structure (length between few millimetres to few meters), enormous scale differences are involved. Thus, multi-scale computational approaches, combining atomistic and/or micromechanical with continuum mechanics computational approaches should be implemented. Despite the scale level of the developed computational model, a non-linear analysis needs to be performed in order to compute the non-linear response of the structure. Depending on the type of problem analysed, non-linearity may be introduced in the form of geometrical or material non-linearity. An example of geometrical nonlinearity is when the computational model includes contact interfaces in the form of contact elements, for example, to represent the physical contact of bolt-to-hole in the case of bolted joints analysis. Another typical situation of geometrical non-linearity is when the structure undergoes large displacements (e.g. in the case of predicting post-buckling behaviour). Material non-linearity is not usually the case inside one

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load step of the PDM stress solution, as conventional composite laminates typically behave linearly elastic until local structural failures start developing, and after that point any material non-linear behaviour is dealt in a piecewise linear manner by degrading elastic properties; therefore, in each loading step of a macromechanics analysis, the material is always assumed to behave linearly, while at the nanomechanics level non-linear material behaviour may need to be taken into account. The assembled FE non-linear equation system takes the form [K T ] {u} = {Ψ}

(11.1)

In Equation (11.1), [K T ] is the tangent stiffness matrix, {Ψ} is the vector indicating the divergence between externally applied and internal forces and {u} is the displacement vector. The tangent stiffness matrix [K T ] consists of three parts: [K T ] = [K 0 ] + [K σ ] + [K L ]

(11.2)

In Equation (11.2), [K 0 ] is the linear part of the stiffness matrix, [K σ ] is the stress stiffness matrix due to prestress effects and [K L ] is the non-linear portion of the stiffness matrix due to the higher-order terms of the strain-displacement equations. Equation (11.2) may be solved by an iterative Newton–Raphson scheme combined to the frontal method for inverting the stiffness matrix. Although the classic Newton–Raphson scheme is widely used due to its rapid convergence, its drawback of requiring assembly of the tangent stiffness matrix in every numerical iteration becomes significant in the case of PDM, which requires many individual loading steps. To reduce the computational effort, the modified Newton–Raphson procedure is better applicable in PDM analysis; it differs from the classic Newton–Raphson method in that the tangent stiffness matrix is only updated when material properties are degraded due to local failures, leading to a significant change of the tangent stiffness matrix. The updated tangent stiffness matrix takes into account both local stiffness reduction due to material failure and large deformation effects associated with geometric non-linearity. In the deep non-linear post-buckling regime, the arc-length method is a more suitable non-linear solution technique, as it causes the Newton–Raphson equilibrium iteration to converge along an arc, thereby often preventing divergence, even when the slope of the load versus deflection curve becomes zero or negative. After a localised failure has occurred, it is necessary to determine the effect of the size of analysis load step. In the case that the load step size is large, the non-linear solution should be repeated at the current load level using the degraded material properties to assemble the tangent stiffness matrix, until equilibrium is re-established, before any additional load is added to the structure. This process should be repeated until no additional failures are detected. Otherwise, the load should be applied in very small load increments, such that any errors introduced by omitting equilibrium re-establishment at the same load level become insignificant. The incremental iterative PDM analysis is performed up to the load level that convergence of the non-linear solution is no more possible. If this situation does not arise from any other numerical convergence problems of the solution, it usually

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indicates that the structure is no more capable of redistributing load to establish equilibrium; thus the load level reached may be considered the ultimate load-carrying capability of the structure.

11.2.3 Prediction of Local Failure at Different Scale Levels Whatever the type of FE computational model, the outcome is always the same, that is, nodal displacements, element strains usually at the Gaussian integration points of the element and element stresses determined by the strains via the material constitutive equations. It is evident that the theories applied in the prediction of failure strongly depend on the scale of the analysis, implying that different criteria are used in the macro-, micro- or the nano-scale level. Concerning the macromechanical level, the initial failure of a lamina can be predicted by applying one of the numerous failure theories which are available in the literature. The advantage of availability of many choices is partly cancelled by the difficulty of selecting the most appropriate theory, which can determine failure at maximum accuracy. It is the purpose of this section rather to provide guidelines for the selection of failure rules appropriate for PDM analysis than discuss the high number of existing failure criteria that have been proposed in the literature, for example, [19], [20] and [54]. Towards this target and from PDM viewpoint, one may categorise the failure theories into two big groups, namely the polynomial and direct-mode determining criteria. Most of the polynomial failure criteria are polynomials based on curve-fitting suitable composite material tests; they involve interactions between stress and strain components, that is, interactive failure criteria. The most well-known polynomial failure criterion is the Tsai–Wu [54] tensor polynomial criterion, which has been widely used due to its simplicity and easy implementation. The quadratic form of Tsai–Wu criterion may be described as F1 σx + F2 σ y + F3 σz + 2F12 σx σ y + 2F13 σx σz + 2F23 σ y σz 2 + F11 σx2 + F22 σ y2 + F33 σz2 + F44 τx2y + F55 τx2z + F66 τ yz =1

(11.3)

In Equation (11.3), σx , σy , σz , τxy , τyz and τxz are the components of the stress tensor at the principal material directions and the 12 parameters F1 –F66 are experimentally derived material constants. Various other quadratic criteria differ from the Tsai–Wu failure equation in a way the different tensor stress components are introduced; however, they can be represented by a generalised Tsai–Wu quadratic criterion. The essential disadvantage of these polynomial failure criteria with respect to PDM analysis is that they predict the initiation of failure without providing any information about the lamina failure mode. However, when selecting appropriate failure criteria for PDM analysis, it is important to keep in mind that apart from being accurate and compatible with FE computational models, it is of prime importance that these criteria are able to distinguish between the different failure modes.

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It is well known that the basic failure modes that can occur in a unidirectional lamina comprise a fibre mode and a matrix mode subdivided into either tension or compression failure. In the fibre mode, the lamina fails due to fibre breakage in tension or fibre buckling in compression, while in the matrix mode, failure is due to matrix cracking in tension or compression. Furthermore, lamina’s interfaces may fail due to the various types of delamination [40]. The mode of failure basically depends upon the load type, stacking sequence, specimen geometry, stress concentrators, initial imperfection or pre-existing manufacturing damage. In a structural component level, the above ‘material’ failure modes result in characteristic ‘structure’ failure evolution patterns; for the case of bolted joints and composite repair patches, these patterns are discussed in more detail in Sections 11.4 and 11.5, respectively. The direct-mode determining failure theories comprise sets of separate polynomial equations based on the material strengths, each one describing one failure mode. Under this viewpoint, direct-mode determining failure criteria are not only very useful, but necessary in progressive failure analysis. The maximum stress and maximum strain criteria, although representing the simplest type of such direct mode determining failure criteria, usually fail to provide the required accuracy in complex loading problems, as they are non-interactive. Hashin [19] and [20] proposed a set of quadratic failure criteria, based on material strengths, where each equation predicts one distinguished failure type. Hashin criteria are usually applied in PDM for the reasons explained above and have been used in all the macro-level PDM applications presented in the next sections of this chapter. The mathematical form of Hashin’s criteria are presented in Table 11.1 (σ ij are the calculated lamina stress components in the (ij) direction and the denominators are the ultimate strengths in the corresponding direction). As can be seen from Table 11.1, Hashin type failure equations are capable of distinguishing between matrix tensile and compressive cracking, fibre tensile and compressive failure, fibre matrix shear out and delamination in tension and compression.

Table 11.1 Hashin direct-mode distinguishing criteria Failure Mode Matrix tensile and compressive cracking (σ yy > 0) Fibre tensile failure (σx x > 0) Alternative form of fibre tensile failure criterion (shear terms are excluded) Fibre compressive failure (σx x < 0) Fibre-matrix shear-out (σx x < 0) Delamination in tension and compression (σzz < 0)

Failure Criterion 2 2 σ σ + Sx y + Syz ≥ 1 xy yz 2 2 2 σx y σx z σx x + S + S ≥1 XT xy xz σx x ≥1 XT

σ yy YT,C

σx x XC

2

≥1 2 2 2 σ σx x + Sx y + σSx z ≥ 1 XC xy xz 2 2 2 σ yz σzz + S + σSx z ≥ 1 Z

T,C

yz

xz

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In the nano-mechanics level, in order to describe the non-linear ‘damage’ behavior of the carbon nanotube atomic bonds (see Fig. 11.3), the modified Morse interatomic potential [4] is used. According to the modified Morse potential, the interatomic force F between carbon atoms as a function of the interatomic distance r is given by F = 2βDe 1 − e−β(r −r0 ) e−β(r −r0 )

(11.4)

In Equation (11.4), r is the carbon atoms distance, r0 is the equilibrium interatomic distance equal to 0.1421 nm and β and De are parameters of the Morse potential, taking values of 2.625 × 1010 m −1 and 6.03105 × 10−19 N m, respectively. The schematic representation of the interaction between carbon atoms, that is, force F versus bond strain εb of the carbon atomic bond is presented in Fig. 11.4, where the bond strain is defined as εb = (r − r0 )/r0 . As can be seen from Fig. 11.4, for negative values of εb , the force is highly repulsive, while at positive values it turns into attractive. The force–strain diagram of Fig. 11.4 can be used for derivation of the bond stiffness and indicates the strain value at which the bond is no more capable of carrying load, that is, the bond inflection strain. Specifically, for strain εb of about 19% the bond is practically released and its stiffness should be gradually degraded to an almost zero value. 12

Fig. 11.4 Force versus strain diagram of the modified Morse potential, from [58] Force, F (nN)

8 4 0

–4 –8 –12 –16 –20 –20

0

20 40 Bond strain,

b

60 (%)

80

100

11.2.4 Behaviour of Damaged Material In Section 11.2.3 the criteria for detecting any localised failure at certain locations of the composite structure have been discussed. Occurrence of localised material failure implies that the local load-carrying capability of the damaged material volumes is not anymore at their initial undamaged level. To account for this effect, the local load-carrying capability of the structure must be adjusted according to suitable degradation rules. The modelling of damage evolution from the initial to the ultimate

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Stiffness

failure is much more complicated compared to the prediction of the localised failure. The two main routes towards this goal are based on damage mechanics, which describes the evolution of damage in terms of material constitutive equations, as well as fracture mechanics, which considers damage to be represented by propagating flaws or cracks. The degradation of in-plane properties is usually dealt through damage mechanics, while for the out-of-plane damage (i.e. delamination) the fracture mechanics approaches seem more appropriate. In the principles of damage mechanics, progression of damage in the FE model is accomplished by an appropriate degradation of the material stiffness properties. On the fracture mechanics basis, cracks are assumed in proper locations of the structure and modelled by cohesive elements of increased complexity, which are activated to model damage progression. A fracture mechanics approach seems to be more appropriate to predict delamination evolution, despite its generic limitations in describing interaction of delamination with matrix cracks or kinking between different ply interfaces. In both approaches the capability of predicting damage progression patterns which are insensitive to numerical modelling parameters, such as mesh density, degradation factors, and at the same time provide satisfactory agreement to the experimentally achieved damage patterns is a major prerequisite. At the macromechanics level, most of the material property degradation rules existing in the literature [37] may be categorised into two main groups, namely sudden degradation and gradual degradation. A third less usual approach is to keep the stress of the failed ply at the constant level it had by the time that failure occurred, until the laminate ultimate failure occurs, for example, [18]. This approach implies prevention of any further increase in the load-carrying capability of the failed ply [36] as it cannot maintain any additional load. In Fig. 11.5, the most important property degradation schemes are schematically illustrated. In the gradual property degradation method, e.g. [9], [44], [48] and [49], the material elastic properties relative to the failure mode are gradually reduced up to the level at which the failure criterion is no longer satisfied. The computational

Gradual reduction

Fig. 11.5 Sudden and the gradual property degradation schemes

Instantaneous reduction

Load

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model is solved iteratively at the same load level, resulting in a partial unloading of elements. Upon increase of load, repeated failures for the same element will occur, representing damage evolution within the element, until its complete failure occurs. The unloading can follow a linear or exponential curve (see Fig. 11.5). The application of gradual property degradation schemes in PDM results in increased computational effort and thus is more complicated compared to the sudden degradation approach. The sudden degradation method is realised by the instantaneous degradation of the stiffness of the failed material volumes, according to the mode of failure. The most conservative version of sudden degradation is the ply discount method, [12], [41] and [45], according to which, stiffness properties of a failed ply are set to zero values. A version of the ply-discount method assumes that only selected material properties of a failed ply are reduced, depending on the failure mechanisms responsible for the ply failure, [10] and [33]. It is generally observed that the ply-discount method underestimates laminate strength and stiffness because it does not account for the residual load-carrying capability that a failed ply has in reality, as well as in some cases may lead to the computational problem of unconvergent solution that arises from the aero-stiffness terms in the tangent stiffness matrix. To overcome these drawbacks, sudden degradation schemes are applied, in which degradation factors different from zero are assigned to respective stiffness properties, according to the different failure modes. Constant degradation factors of around 10% applied on both elasticity modulus and Poisson’s ratio are quite usual, because of their simplicity and their relatively good predictions in the description of damage propagation they offer. A variation of the simple sudden degradation method assumes that degradation of stiffness properties corresponds to the respective failure mode only, e.g. a fibre failure mode will lead to reduction of elasticity modulus in the fibre direction only; other variations include coupling to other stiffness properties; therefore, they also reduce shear elasticity modulus. Variations also exist with respect to degrading or not degrading Poisson’s ratios of the failed elements. Of course, degradation factors values are usually adjusted (deviating from their usual 10% value) with respect to the specific problem under consideration, aiming to achieve the better possible predictions as compared to the respective experimental results. In the macromechanics applications presented in Sections 11.3, 11.4 and 11.5, the sudden material property degradation rules presented in Table 11.2 have been applied to represent the response of the damaged material volumes. The sudden material property degradation rules of Table 11.2 is a mixed set, comprising the rules of Tan [53] as they were extended by Camanho and Matthews [6] for matrix tensile and compressive cracking and for fibre tensile and compressive failure. The degradation in Poisson’s ratio are restricted by the relationships between the engineering properties of the anisotropic materials [26]. To allow practical application of the degradation rules, the degraded material properties are assumed to have a very small value instead of zero, in order to avoid numerical problems in the FE solution. The in-plane damage material property degradation method has been modified and extended to the prediction of delamination progression.

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Stiffness partially reduced

Matrix tensile cracking Matrix Compressive cracking Fibre Tensile failure Fibre compressive failure Fibre matrix shear-out Delamination in tension Delamination in compression

Eyy = 0.2Eyy , Gxy = 0.2Gxy , Gyz = 0.2Gyz Eyy = 0.4Eyy , Gxy = 0.4Gxy , Gyz = 0.4Gyz Exx = 0.07Exx Exx = 0.14Exx Gxy = νxy = 0 Ezz = Gyz = Gxz = νyz = νxz = 0 Ezz = Gyz = Gxz = νyz = νxz = 0

In the nanomechanics level, as it may be observed from Fig. 11.4, the maximum (positive) force of the interatomic bond occurs at an inflection point of strain value. It is apparent that after the infection point, bond force is exponentially decreased towards zero value, which means that the respective material properties of the beam elements representing the bond should follow the same behaviour in order to properly represent the material property degradation of the nanotube.

11.3 Buckling and Damage Interaction of Open-Hole Composite Plates by PDM 11.3.1 Composite Panel with Circular Cut-Out The PDA methodology described in Section 11.2 is initially demonstrated for the relatively simple case of predicting the interaction effects between post-buckling behaviour and material failure modes of a compressively loaded layered composite plate with a circular cut-out. The investigated problem belongs to the applications of macro-mechanical scale level with the aim to predict the effect of damage propagation on the post-buckling behaviour and to identify the conditions under which this interaction becomes significant. Various investigations on the buckling and post-buckling behaviour of panels with holes, without considering the influence of the various composite material failure types on buckling, may be found in refs. [38] and [39]. The buckling behaviour of damaged composite plates by PDM has also been considered in [51] and [60], by applying the simple ply-discount method and 2-D stress analysis. Buckling analyses of several types of composite plates have been also performed in [61], [21] and [34], in order to examine buckling resistance and post-buckling behaviour, without modelling damage evolution in the plates or discussing the different damage modes. A compressively loaded rectangular composite plate with a circular cut-out having the geometry, boundary conditions and loading shown in Fig. 11.6 is currently considered [31]. An incremental compressive load step Pi is applied to one end of the plate, while the opposite end is fully clamped. The other two edges are constrained with respect to z-axis, that is, preventing the out-of-plane displacement.

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y

W1 D

L1

Pi

W

x t

L z Fig. 11.6 Open-hole panel geometry and loading, after [31]

The three main parts of the progressive failure analysis, that is, computational model, prediction of failure sites and material property degradation, have been programmed as macr- routines under the finite element code Ansys [3]. The current PDM algorithm has been structured in a modular way such that the modules of buckling, stress and failure analysis, as well as the degradation of the material properties, are separate subroutines. Therefore, they can be modified accordingly in order to perform parametric studies by changing the geometry, loading, boundary conditions, parameters of failure criteria, degradation rules, and so on. However, the numerical results presented hereafter refer to L = 508 mm, W = 139.7 mm, D = 19.05 mm, L1 = 317.5 mm, W1 = 69.85 mm and thickness varying from t = 1 mm to t = 6 mm. The composite specimen lamination is [(45/–45/0/90)3 ]s , made of graphite/epoxy layers having the material properties given in Table 11.3. Table 11.3 Material Properties of graphite/epoxy material (stiffness and strength values in GPa) EX X

EY Y

EZ Z

G XY

GXZ

GY Z

νX Y

νX Z

νY Z

130,4 XT 1,38

12,97 Xc 1,14

12,97 YT 0,081

6,38 YC 0,189

6,38 ZT 0,081

4,69 ZC 0,189

0,3 SX Y 0,069

0,3 SY Z 0,021

0,45 SX Z 0,069

11.3.2 Computational Model for the Open-Hole Panel Problem The computational model developed is 3-D, as it is well known that interlaminar effects play an important role in the out-of-plane damage evolution around free edges, such as the open hole in the problem under investigation. If the plate thickness is low, interlaminar effects may not be significant in the pre-buckling regime; however, they influence the post-buckling response due to the interlaminar failures arising from plate bending. Moreover, interlaminar effects become more important in thicker plates in both pre- and post-buckling regimes and in any case determine the fatigue behaviour of plates with cut-outs. The 3-D computational model uses

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the Ansys layered element type ‘Solid46’, which has eight nodes, three displacement degrees of freedom per node and is defined by the ply thickness, ply material direction angles and orthotropic layer material properties. It is typical in such type of FE analysis that the accuracy of the interlaminar stress/strain results increases by increasing the number of elements through the thickness, although the respective analysis becomes extremely time consuming. Different meshes from coarse to very dense have been trialled in order to determine the optimum mesh size. In all cases the in-plane discertisation is denser around the cut-out, as shown in Fig. 11.7 in order to achieve a better accuracy of the stress solution. Referring to the throughthe-thickness dimension, a reference solid model is developed, having one element per layer; that is, it is a complete solid model at ply level. Parametric mesh studies on ‘smearing’ the properties of certain number of plies into one finite element through the thickness have shown that coarser meshes cannot provide good estimations of both the first ply failure and the final failure observed in the respective experimental test; therefore, current results refer to a heavy computational model requiring increased computer capacities. Fig. 11.7 (a) Typical FE mesh of the composite plate and (b) first buckling mode shape as computed by the computational model, [31]

(a)

(b)

For the specific case only, in which the analysed structure has no geometrical imperfections as it is completely flat with symmetric lamination under pure compressive loading, an additional step is required compared to the computational analysis described in Section 11.2. It comprises an initial eigenvalue ‘linear’ buckling analysis of the flat plate before the non-linear static analysis is carried out, which is performed using the FE computational model of Fig. 11.7(a) to solve the equation: ([K0 ] − λ [Kσ ]){u} = 0

(11.5)

In Equation (11.5), [K 0 ] is the linear part of the stiffness matrix, [K σ ] is the stress stiffness matrix due to pre-stress effects, λ is the critical load multiplier and {u} is the eigenvector, which describes the buckled mode shape. The amplitudes of the first buckling mode shape are scaled accordingly to calculate offsets, which are added to the original nodal coordinates, that is, the maximum imperfection displacement applied is equal to the plate thickness times a geometrical perturbation factor. Therefore, in the non-linear static analysis which follows, the in-plane compressive

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loads introduce bending moments which lead to out-of-plane deformations of the panel from the onset of the load application. In Fig. 11.7(b) the calculated first buckling mode of a 3.5-mm composite plate is presented; it can be successfully compared to the respective experimentally derived photograph of Moir´e–Fringe pattern from ref. [51]. A parametric study of the critical buckling load with respect to the panel thickness has indicated that the critical buckling load value increases exponentially for thicker plates. Therefore, it is expected that, as the plate thickness increases, the plate enters the post-buckling regime at higher loading levels and its pre-buckling response is more significantly affected by the various material damages before the geometrical non-linearity becomes important and buckling takes place.

11.3.3 Interaction Effects Between Damage Failure and Plate Buckling The Hashin-type failure criteria and property degradation rules, described in Tables 11.1 and 11.2, respectively, are applied to predict damage initiation and progression, based on the approach that damage within an element has an effect on the elastic properties of that element only. For the elements in which ply failures are detected, the algorithm automatically executes the corresponding properties degradation. Therefore, degradation is done on an element-by-element basis, assuming that the stiffness reduction at a point is confined to the neighbourhood of that point only. After initial buckling occurs, the structural stiffness changes even if no material damage exists, therefore the FE solution is repeated under the same load level to establish equilibrium and the stress field is recalculated. The incremental load is then increasing until the final collapse of the structure is reached. Once all the model numerical parameters, that is, mesh density, initial perturbation factor, failure criteria and degradation rules, have been set, the interaction effect between damage failure and plate buckling is investigated. To this purpose, simulations with and without PDM, as well as including and excluding geometrical non-linear terms, are executed. Hence, for four different plate thicknesses, 1 mm, 2 mm, 3.5 mm and 6 mm, the following analysis types were performed: (a) linear analysis with PDM, (b) non-linear analysis without PDM and (c) non-linear analysis with PDM. In Figs. 11.8 and 11.9, the 1-mm and 6-mm plate thickness responses are presented, as calculated by the three different analysis options (in Figs. 11.8 and 11.9, CBL means Critical Buckling Load, FPF means First Ply Failure, FF means Final Ply Failure) It can be observed from Fig. 11.8 that the initial buckling of the 1-mm plate occurs at the early stages of the analysis as expected since the plate is very thin, leading to large out-of-plane displacements and bending. As a consequence, damage is observed in the post-buckling stages, initiating with fibre–matrix shear-out. Due to the fact that this type of failure is not catastrophic, high stiffness changes

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FPF (Linear)

50000

FF (Linear Analysis PDM)

45000

Non Linear Analysis without PDM Linear Analysis PDM Non Linear Analysis with PDM

Force, P (N)

40000 35000 30000 25000 20000

FF (Non Linear Analysis with PDM)

15000

FPF (Non Linear)

10000 CBL

5000 0 0,000

CBL

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

Displacement, u (m) Fig. 11.8 Force versus edge displacement for the three different analysis options (t = 1 mm), after [31]

CBL (non Linear without PDM)

400000 350000

Force, P (N)

ith ar w

PDM

ne

n Li

o F (N

300000 250000

)

CBL (Non linear Analysis) F

FPF (Non Linear with PDM) FF (Linear with PDM)

200000 FPF (Linear with PDM)

150000 100000

Non Linear Analysis, without PDM Linear Analysis with PDM Non Linear Analysis, with PDM

50000 0 0,000

0,001

0,002 0,003 0,004 Displacement, u (m)

0,005

Fig. 11.9 Force versus edge displacement for the three different analysis options (t = 6 mm), after [31]

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are observed at the latest stages of the analysis, close to the final failure, which is caused by fibre and matrix (compression mainly) failures in the vicinity of the hole. Similarly, the failure process of the 3.5-mm plate also initiates with the evolution of fibre–matrix shear-out failures, propagating from the plate edges towards the open-hole area, followed by matrix-compressive failures and few delamination initiations, which result in initial plate buckling. The out-of-plane deformation during the last loading steps leads to extensive fibre failures accumulation and significant stiffness changes. The final failure, though, is characterised by matrix compression and fibre–matrix shear-out in a large area connecting the outer edge of the panel and the vicinity of the hole. In Fig. 11.9, which refers to the 6-mm plate, the first ply failure (fibre matrix shear-out) happens much earlier before any buckling occurs. It may be observed that linear analysis with PDM is more conservative than the corresponding nonlinear analysis. This is due to the fact that in the non-linear analysis, large out-of plane deformations occur at the vicinity of the hole (see Fig. 11.7(b)) together with in-plane stress concentration and many delamination sites generation at the openhole areas. The in-plane stresses are reduced in the post-buckling region, therefore, less fibre failures are observed in the non-linear analysis compared to the linear one. A main conclusion arising from the open-hole problem investigation is that the critical buckling load is seriously affected by the damage of the composite plate, especially when the thickness is high, implying that a large amount of damage has accumulated in the structure before initial buckling takes place, as may also be observed from the change in the slope of the force–displacement curves. Furthermore, it can also be stated that for low-thickness plates, initial buckling occurs before any serious damage is accumulated in the structure; therefore, prediction of the critical buckling load does not require the consideration of the damage evolution in the analysis, which becomes necessary only in the post-buckling regime.

11.4 Implementation of PDM in Composite Bolted Joints 11.4.1 Description of Composite Bolted Joint Problem The design optimisation of composite bolted joints is affected by numerous parameters, such as geometry, combination of material types of the joined parts, stiffness properties, joint configuration, friction, clamping force and loading type, which significantly influence the many interacting and complicated failure mechanisms taking place during the bolted joint operation. Despite the substantial amount of investigations performed in the area of composite bolted joints analysis, a PDM approach capable of addressing all damage phenomena and operating in affordable computing times is one of the most suitable approaches for analysing the problem. In this context, a 3-D PDM, capable of predicting damage initiation and progression in a composite bolted joint and assessing its residual strength under quasi-static loading is presented, [56] and [57]. The 3-D nature of the computational model is chosen,

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so as to address the through-the-thickness phenomena of the laminate, which play an important role in the joint behaviour. The current model comprises the typical PDM components described in Section 11.2 and programmed into a parametric Ansys macro-routine, as mentioned in Section 11.3. The computational model of Section 11.2 has been replaced by a composite bolted joint FE model subjected to incremental in-plane tensile loading, while the remaining modules of failure analysis and damage description remain identical to those of Section 11.2. The joint configuration, taken from [24] for enabling comparisons to experimental tests, consists of an upper plate made of composite laminate, a lower plate made of aluminium and a titanium bolt with protruding head. The joint geometry is schematically shown in Fig. 11.10.

Fig. 11.10 Geometry of the composite-to-metal bolted single-lap joint, after [56]

The 3-D typical FE mesh of the joint is shown in Fig. 11.11. Perfect-fit is assumed between the bolt and the two plates. Modelling of the composite plate was done using the 8-noded Solid46 Ansys layered element, while the metallic plate and bolt modelled using the Solid45 element type. Eight-layered elements stacked together with one element per ply are used through-thickness to model the [0◦ /90◦ / ± 45◦ ]S laminate. The 3-D type of analysis aims to an accurate representation of the interlaminar stress field developing around the bolt, due to bending effects, clamping force of the washer and through-thickness pressure in the bolt-hole area. Thus, the out-of-plane effects, such as delamination damage, out-of-plane bending and stacking sequence effects, in the area close to the bolt are taken into account in the model. To simulate contact between the two plates, the washer and the plates, as well as between the bolt and the hole surface, the node-to-surface 3-D Contac49 element type is used. The bolt, washer and nut are considered as one unit to limit the number

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(b)

(a)

(c) Fig. 11.11 Finite element model of the bolted single-lap joint (a), detailed mesh of the area around the bolt (b) and model of the protruding head bolt (c), [56]

of contact elements in the model. For all contact surfaces a friction coefficient of 0.3 is assumed. To reduce the size of the problem, symmetry conditions are implemented. As a full and half model solution with the same mesh density did not affect significantly the predicted damage accumulation results, while the computation time was reduced by one order of magnitude, the symmetric model shown in Fig. 11.13 is used for the analysis. The composite plate is loaded with an in-plane uniform tensile pressure at its end, while the aluminium plate is fully built-in. In order to prevent secondary bending of the joint, lateral support is applied in the model. Since the two plates and the bolt are assembled by washers, the clamping torque on the laminate may be simulated by a uniformly distributed pressure. In order to apply the required pressure, first thermal expansion properties are assigned to the material of the bolt; then an appropriate temperature reduction leads to bolt contraction and, hence, to the required pressure. Since the procedure is linear, the appropriate temperature reduction can be found easily for each required torque. Prediction of initial failure for the current bolted joint problem is performed using the same Hashin-type stress-based failure criteria used in the open-hole compression panel, described in Section 11.2. Once failure is detected in a ply, its material properties are degraded by implementing the same sudden property degradation rules used in the compression panel problem of Section 11.2. Using the PDM macro-routine, the bolted joint response is simulating by gradually increasing the applied load. This

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procedure continues until excessive damage is reached, which means that failure has propagated to a specimen edge and the applied load can no longer be redistributed in the damaged areas. As a result, the specimen cannot carry any additional load and catastrophic failure is reached. At this point the residual strength is evaluated and the maximum load, which is the last applied load, is determined.

11.4.2 Damage Initiation and Progression Within the Bolted Joint It has been observed experimentally that there are three basic macroscopic failure modes in composite bolted joints, namely tension, shear-out and bearing, as schematically shown in Fig. 11.12. Which failure mode will initiate first and how failure will propagate depends on the bolt position. The geometries examined were chosen such as the three possible macroscopic failure modes could be observed.

Fig. 11.12 Macroscopic failure modes observed in composite bolted joints

The results presented hereafter refer to basic dimensions of a bolted joint of L = 150 mm, W = 60 mm, D = 17.8 mm, d = 10 mm, e = 30 mm, h = 30 mm, H = 60 mm, t = 4.16 mm (see Fig. 11.10). The laminated plate has been made of composite material system HTA/6376, the full material properties of which can be found in [57] and are similar to those in Table 11.3. In Fig. 11.13 the initial stage of the different damage modes of a geometry failed at 9.9 kN is presented. In the early stages of loading the composite behaves elastically with no sign of damage in the material, while as the load increases, the first ply failures are detected. Matrix cracking initiated at the outer 0◦ ply at the bearing plane in the material located under the washer area due to normal compressive

0°

Fig. 11.13 Prediction of initial stage of the different damage modes at the [0◦ /90◦ /±45◦ ]S plate (d = 10 mm, w/d = 3 and e/d = 4)

Matrix cracking

90°

Fibre-matrix shear-out

0°

Fibre failure

90°/0°

Delamination

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stresses acting on the laminate from the washer mainly during pre-tension. As the load increases, fibre–matrix shear-out initiates at a 45◦ to the load direction along the shear-out plane (90◦ ply) due to high shear stresses. Fibre failure (compressive and tensile) initiated at 68% of the failure load across the hole boundary and at the 0◦ ply due to high normal stresses in the fibre direction. Delamination onset is predicted to occur on the hole edge at the 0◦ /90◦ interface at 75% of the failure load, which is in agreement with the predictions and experiments presented in [5]. In Fig. 11.14 the damage prediction at the –45◦ ply for the fully shear-out failed joint is shown. Because of the small-edge distance e/d, a shear-out failure is predicted as expected, at a failure load of 9984 N. Fig. 11.14 Damage prediction at the –45◦ ply at the final failure load of the [0◦ /90◦ /±45◦ ]S specimen that failed in shear-out (d = 10 mm, w/d = 3 and e/d = 4)

Matrix cracking

Fibre-matrix shear-out

Fibre failure

Delamination

In Fig. 11.15, damage prediction at the 45◦ ply at final failure of the tension-failed joint is shown. The specific type of failure is characterised by the early fibre failure progression from the hole edge towards the specimen free edge at the direction normal to the loading due to the small w/d ratio. As a consequence, the residual strength of the joints with such geometry is very small.

Matrix cracking

Fibre-matrix shear-out

Fibre failure

Delamination

Fig. 11.15 Damage prediction at the 45◦ ply at the final failure load (6822 N) of the ◦ ◦ ◦ [0 /90 /±45 ]S specimen that failed in tension (d = 10 mm, w/d = 2 and e/d = 6)

Figure 11.16 illustrates the prediction of damage at the 0◦ and 90◦ plies at the final failure stage of the specimen failed in bearing due to the large edge distance

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Fig. 11.16 Damage prediction at the 0◦ and 90◦ plies at the final failure load (13470 N) of the [0◦ /90◦ / ±45◦ ]S specimen that failed in bearing (d = 10 mm, e/d = 6 and w/d = 6)

0°

Matrix cracking

Fibre-matrix shear-out

90°

0°

90°

Fibre failure

Delamination

ratio e/d. Damage initiated from the contact surface as a result of the compression failure due to a concentrated bearing load. Occurrence of both fibre–matrix shearout and delamination damages at the 90◦ ply in the bearing plane confirm the experimental observations of [22] that the formation of shear cracks and delaminations are the major characteristics of bearing damage. The shear cracks either propagated towards the free surfaces of the laminate or merged to initiate delaminations, resulting in a significant reduction of laminate stiffness and fibre failure. As a result, the joint could no longer sustain any additional load, and the failure was catastrophic. The final failure criteria used for the definition of catastrophic failure and determination of failure load for the cases presented above are based on damage accumulation. For tension and shear failure modes, the criterion is the propagation of fibre failure until the free edge of the specimen at the directions of 90◦ and −45◦ with regard to loading direction. In Fig. 11.17 the predicted deformed shape of the joint failed in bearing is compared to the corresponding experimental results, as taken from [25]. Secondary bending of the area around the bolt leaned the bolt and led to penetration of the washer in the laminated plate. As can be seen, this phenomenon is accurately captured by the PDM model by predicting both deformation of the joint and damage in the laminated plate resulted from the penetration.

Fig. 11.17 Prediction of damage in the laminated plate due to penetration of washer

The stiffness and strength of bolted joint configurations for which extensive experimental data existed in [25] have been predicted and presented hereafter. In Fig. 11.18 load-displacement curve, predicted by PDM analysis, is compared to the

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15

Fig. 11.18 Comparison of load-displacement curve predicted by PDM analysis to experimental curve, [56]

Load (kN)

12

9

6

PDA Analysis

3

Bounds of experimental error

0 0

0,3 0,6 0,9 1,2 Displacement (mm)

1,5

experimental curve from [25]. Stiffness of the joint has been accurately predicted since the numerical curve falls inside the bounds of experimental error. The successful strength prediction, which occurs due to fibre failure, indicates the accurate detection by the respective fibre failure criterion, implemented in the PDM.

11.5 Implementation of PDM in Composite Bonded Repairs 11.5.1 Description of the Composite Repair Patch Problem Taking into account that damage during service life is a natural characteristic of any airframe, performing repairs to damaged airframe components is virtually inevitable mainly for economic reasons. Bonded repairs offer many advantages over the mechanically fastened doublers, for example, improved fatigue life, reduced corrosion, in situ application, easy conformance to complex aerodynamic contours and high level of bond durability under operating conditions. Design of adhesively bonded repairs using composite patches on cracked metallic aircraft structures is of similar complexity to that of the composite bolted joints, as several parameters affect the progression of damage and final failure. Although implementation of composite bonded repairs started more than 30 years ago, at present, such repairs are certified for use only on secondary structural parts of the airframe. In order to expand their use on primary structural parts also, a detailed knowledge of their mechanical performance under several loading conditions is required. Most of the numerical models existing in the literature used for calculating the stress field of the repairs and optimising the patch geometry with regard to the reduction of stress intensity factor (SIF) have not taken into account the failure analysis

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of the repairs. Designs that ignore certain critical failure modes may cause greater damage than the original damage they are intended to repair. In Fig. 11.19 the main failure modes that might occur in a bonded repair of a cracked fuselage skin, as presented [15], are illustrated. They comprise crack growth, patch debonding and nucleation of new fatigue cracks in the skin at the patch edges. The foremost failure mode is crack growth and almost of the same importance is patch debonding. Fig. 11.19 Illustration of the main macroscopic failure modes in a bonded repair of a cracked fuselage skin, after [15]

11.5.2 Details of PDM Model for Composite Repair Patch Analysis In [42] and [43], PDM has been implemented for assessing the mechanical performance of composite bonded repairs in cracked metallic sheets. The main results obtained are the assessment of repair effectiveness measured in terms of Stress Intensity Factor (SIF) reduction with regard to the SIF of the respective un-patched configuration, the effects of composite patch material and geometry, as well as the initiation and progression of patch debonding. Under the basis of the developed Ansys PDM macro-routine, already described in Sections 11.3 and 11.4, a 3-D computational model for the repair patch configuration has been built. The model is fully parametric with regard to repair geometry, material properties and mesh density. Due to symmetry of geometry and loading, only one-quarter of the repair has been modelled, as shown in Fig. 11.20. A very fine mesh has been adopted in the area around the crack tip to achieve better accuracy in the calculation of the SIF. Away from the crack tip, a coarser mesh has been adopted in order to reduce the total number of elements, and therefore, the computational time. The metallic plate and adhesive material have been modelled as different bodies. Eight-layered elements have been used through the thickness of the composite patch, keeping the rule of using one element per layer. For the adhesive

11 Prediction of Damage Propagation and Failure of Composite Structures

tapered composite patch (height h, width w)

adhesive (height h, width w)

cracked metallic sheet

347

composite repair

Fig. 11.20 Typical FE mesh of the composite patch repair (one-quarter model)

and the metallic plate, eight and two elements have been used, respectively. The use of many elements through the thickness provides greater accuracy to the calculation of through-thickness stresses, which play a significant role on the prediction of patch debonding and the other failures of the adhesive and patch. Additional to the failure modes considered for the composite material, debonding has been also included in the failure analysis. For the detection of the composite patch failure modes, the interactive Hashintype failure criteria, used for the open hole and the bolted joint problems and presented in Table 11.1, are applied. For the adhesive and the adhesive/metal or adhesive/patch interface, three major failure modes are considered, namely adhesive shearing, adhesive/metal peeling and adhesive/patch peeling. The failure crite3 ≥ ps , where σ1 and σ3 are rion used for the detection of adhesive shearing is σ1 −σ 2 the maximum and minimum principal stresses in the adhesive and ps is the shear strength of the adhesive. Adhesive/metal peeling is predicted when: σza ≥ pam , while adhesive/composite patch peeling is predicted when σza ≥ pap , where σza is the stress in the normal direction and pam , pap are the peeling strengths of the adhesive/metal and adhesive/patch interfaces, respectively. To account for the shearing failure of the two interfaces, ps can be considered as the minimum of the adhesive shear strength and the interfaces shear strength values. From the modelling point of view, this consideration seems satisfactory, since the breakage of a single element of the adhesive is considered equivalent to the debonding of the either of the interfaces in this element. When a mode of failure is detected in the composite patch, the degradation rules of Table 11.2 are applied in order to disable the failed material region from

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carrying a specific load. For the adhesive material, which has isotropic properties, the degradation is always achieved by setting the Young’s modulus equal to an almost-zero value, in order to avoid numerical problems that would be caused if zero values were used. The iterative PDM procedure runs until either the crackedplate SIF reaches its critical value or when net-section yielding occurs. More details about the failure criteria used to predict debonding may be found in [42] and [43].

11.5.3 Effects of Composite Patch Geometry and Material on the SIF The configuration of the repair considered comprises a metallic sheet with a central through-thickness crack loaded in tension and repaired using a double-sided rectangular (symmetric) carbon–fibre-reinforced plastic (CFRP) patch with tapered edges. The dimensions of the plate considered were 240 mm × 240 mm × 3 mm. The metallic plate is made of an aluminium alloy with E = 71709 MPa and ν = 0.33. The thickness of the patch is 2.25 mm. The patch height 2 h and width 2w, as well as the crack length 2a are parameters of study. The thickness of the adhesive is 0.15 mm. In Fig. 11.21 the variation of the SIF with regard to half patch width w (length parallel to crack line) is presented, as obtained by the present FE model and compared to the results of ref. [42]. The results refer to half crack length of a = 12 mm and half patch height of h = 24 mm. From Fig. 11.21, it can be observed that both SIF variations indicate a continuous decrease of the SIF with the patch width, as well as a very good agreement between the maximum SIF variation obtained by the present FE model and ref. [42].

Influence of width and height on SIF 6

SIF [MPa-m1/2]

5 4 3 width effect (PDA)

2

width effect (from ref. [xx])

1

Fig. 11.21 SIF variation with regard to patch width and patch height

heigth effect (PDA)

0 0

20 40 60 Geometric Parameter [mm]

80

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In Fig. 11.21 the variation of the SIF with regard to half patch height h (length perpendicular to crack line) is also presented, referring to half crack length of a = 12 mm and half patch width of w = 36 mm. As may be observed in Fig. 11.21, there is an optimum patch height for which the SIF becomes minimum, which can be explained by the relation of SIF value more to the entire repair geometry rather than the patch geometry itself. To assess the effect of patch lay-up on the repair effectiveness, four different lay-ups with eight layers have been considered. The results, shown in Fig. 11.22, suggest that the SIF variation with the applied load is linear in all cases, until the load at which patch debonding causes sudden increase of the SIF, as explained in the previous section. Since the aircraft fuselage, in which the bonded composite repairs are mainly implemented, is subjected to fatigue loading conditions, the SIF thresh√ old (4.4MPa m for the specific material) has been used as criterion of the repair effectiveness. According to these conditions, the most effective lay-up is the [0◦ ]8 , as expected, since at the 0◦ layers the fibres coincide with the direction of the load application.

Fig. 11.22 Effect of patch lay-up on SIF variation, after [42]

However, the performance of the patch lay-ups with regard the SIF reduction cannot be the only criterion for their use to bonded repairs. Other critical parameters, such as thermal expansion and resistance to fatigue, must be also taken into account. For example, although the quasi-isotropic [±45◦ /0◦ /90◦ ]S has proved the less effective concerning the SIF reduction due to the small number of 0◦ layers that it contains, it is well known that it shows better resistance to fatigue and has smaller thermal expansion than the other lay-ups. Therefore, the final selection of the lay-up must be based on a general performance, in which all the specific performances will be taken into account. By investigating the failure progression within the repair patch structure as predicted by the PDM model, it is observed that debonding initiation at the crack-tip region is due to crack opening, but it is not catastrophic for the repair, in the sense that the applied load can be further increased significantly. In Fig. 11.23 the vari-

350 100 Percent debonded area

Fig. 11.23 Increase of the debonded area with the applied load in the composite repair patch

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3% deb. area 70% deb. area 10% deb. area

60 40 20

Debonding initiates at the crack region

0 200

250 300 350 Applied load (MPa)

Debonding initiates at the upper patch edge

400

ation of the percent debonded area with regard to the applied load is presented. It may be observed that debonding initiated at about 220 MPa, while applied stress can be further increased up to 350 MPa, before debonding is also initiated at the upper patch edge and propagate very suddenly leading to complete dissociation of the patch from the metallic sheet. In Fig. 11.23 the debonding progression is schematically presented, indicating that debonding initiated at the crack tip is confined to the area around it, while the later initiated debonding at the upper patch edge propagates very fast, leading to the catastrophic failure of the repair.

11.6 Multi-Scale Modeling of Tensile Behavior of Carbon Nanotube-Reinforced Composites A typical nanotube structure is described in terms of a chiral vector defined by the pair of indices (n,m), according to which, three nanotube types may arise, namely the armchair (n,n), the zigzag (n,0) and the chiral (n,m) CNTs. Recently, the tensile behaviour of such unidirectional nanotubes has been numerically predicted [58], based on a nanomechanics computational FE model of the nanotube structure presented in Fig. 11.3 and a modified Morse potential (Fig. 11.4) for calculating the stiffness properties of the beam finite element representing the interatomic bond between carbon atoms. The results of the local nanoscale model are stress–strain equations of zigzag- and armchair-type nanotubes, which can then be used in the prediction of the behaviour of CNT-reinforced matrices. As explained in Section 11.2, such an analysis bridging different scale levels is only efficient if based on RVE or similar approaches. Having this in mind, the Representative Volume Element (RVE) of a CNT composite comprising a rectangular solid volume representing the polymeric matrix and an embedded nanotube is created, as can be seen in Fig. 11.24. Three-dimensional isotropic solid elements are used for modelling the matrix material, while the nanotube is represented by 3-D elastic beam elements. The nanotube/matrix debonding failure mode is incorporated in the model, by

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Fig. 11.24 (a) Schematic representation of an RVE comprising a reinforcing CNT embedded in polymeric matrix and (b) FE mesh of the RVE, [58]

nanotube (a)

(b)

assuming different nodes for the matrix and for the CNT at their interface, the connection of which may be released if a failure condition is realised. The RVE of the CNT-reinforced composite is subjected to axial tension by fully constraining the nodes of the one end and applying an incremental ‘axial’ displacement at the nodes of the other end. Solution of the RVE computational model has indicated that the reinforcing effectiveness of CNTs is significantly influenced by the degree of interface bonding between the nanotube and the matrix. In the RVE of Fig. 11.24, the nanotube/matrix debonding mode is incorporated. The interface shear stress is calculated by a shearlag analysis and compared to the interfacial shear strength; upon identification of a failed interface, the respective nodes are released and the particular finite elements of both the matrix and the CNT are assumed to have reduced load-carrying capability, which according to sudden property degradation principles are assigned very small stiffness values. Parametric analyses have indicated that although the interfacial shear strength (ISS) value of the armchair-type CNT-reinforced composite system does not affect the resulting stiffness, the tensile strength decreases significantly with decreasing interfacial shear strength, as may be observed in Fig. 11.25. The effect of CNT volume fraction on the resulting tensile strength of the CNTreinforced composite system is presented in Fig. 11.26 for ISS assumed to be 20 MPa. It can be observed in Fig. 11.26 that increasing the volume fraction Vfnt results in increasing strength, which may be explained by the higher load-transfer capability of the matrix/CNT interface, indicated by the fact that for the same applied stress, the shear stress is larger for lower values of Vfnt .

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Vfnt 8% - ISS = 25 MPa Vfnt 8% - ISS = 20 MPa

10

Vfnt 4% - ISS = 30 MPa

Stress [GPa]

Vfnt 4% - ISS = 20 MPa

8

Vfnt 1% - ISS = 100 MPa Vfnt 1% - ISS = 20 MPa

6 4 2 0 0

5

10 Strain [%]

15

20

Fig. 11.25 Predicted tensile stress–strain curves of armchair-type CNT-reinforced composite with various volume fractions Vfnt and different interfacial shear strengths (ISS)

12 Strength (GPa)

Fig. 11.26 Effect of nanotube volume fraction on tensile strength of CNTreinforced system, [58]

Armchair

10 8

Zigzag

6 4 2 0 0

2 4 6 8 10 Nanotube Volume Fraction (%)

11.7 Conclusions The basics of the progressive damage modelling methodology have been presented and the implementation of its three major components, that is, computational model, failure analysis and representation of damage evolution, have been demonstrated for different composite structures, at various scale levels, including the multi-scale. The computational modelling of structures under PDM analysis should ideally be based on 3-D finite elements, despite any computational cost, in order to compute out-of-plane effects with sufficient accuracy. Three-dimensional computational models including all kinds of geometrical and structural details capable of solving for the non-linear behaviour of large scale of even full-scale structures are foreseen in future PDM analyses.

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In the area of failure analysis and material property degradation, there is a lot of research already performed, especially for the conventional composite systems; however, more research is required with respect to the investigation of pre-existing damage or the effect of defects on the behaviour of advanced composites. The prediction and modelling of debonding or delaminations in the frame of PDM is also an open research topic, especially from the viewpoint of fracture mechanics, where the inherent difficulty that a pre-existing delamination must be assumed in order to apply fracture mechanics principles to predict delamination propagation. In addition, reliable criteria to predict the final, catastrophic failure of structural components and accurately evaluate ultimate failure loads are still not well defined. Bridging the analyses of different length scales or of different type, for example, global, local, RVE approaches needs more effort. For example, methods for a reliable transfer or homogenisation of damage states predicted in a very detailed way from local analysis towards a global analysis of higher levels have to be established. Finally, extension of PDM principles to include the time scale in the analysis of composite structures would be valuable as it would enable the application of a damage tolerance design philosophy, in a fashion similar to that currently used for metallic structures, to composite structures too; this would enable assessment of fatigue damage and material property degradation using results from non-destructive evaluation data that would make possible the application of structural health monitoring techniques on smart future composite material structures.

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38. M.P. Nemeth, “Buckling Behaviour of Compression Loaded Symmetrically Laminated Angle-Ply Plates with Holes,” AIAA J., Vol. 26, 1988, pp. 330–336 39. M.P. Nemeth, M. Stein, and E. Johnson, “An approximate buckling analysis for rectangular orthotropic composite plates with centrally located cut-out,” NASA TP 2528, 1986 40. O. Ochoa and J.N. Reddy, Finite Element Analysis of Composite Laminates. Kluwer Academic Publishers, The Netherlands, 1992 41. P. Pal and C. Ray, “Progressive Failure Analysis of Laminated Composite Plates by Finite Element Method,” J. Reinf. Plast. Compos., Vol. 21, 2002, pp. 1505–1513 42. P. Papanikos, K.I. Tserpes, and G.Labeas et al., “Progressive Damage Modelling of Bonded Composite Repairs,” Theor. Appl. Fract. Mech., Vol. 63, 2004, pp. 219–230 43. P. Papanikos, K.I. Tserpes, and Sp. Pantelakis, “Initiation and Progression of Composite Patch Debonding in Adhesively Repaired Cracked Metallic Sheets,” Compos. Struct., Vol. 81, 2007, pp. 303–311 44. P.H. Petit and M.E. Waddoups, “A Method of Predicting the Nonlinear Behaviour of Laminated Composites,” J. Compos. Mater., Vol. 3, 1969, pp. 2–19 45. B.G. Prusty, “Progressive Failure Analysis of Laminated Unstiffened and Stiffened Composite Panels,” J. Reinf. Plast. Compos., Vol. 24, 2005, pp. 633–642 46. X. Qing, H.T. Sun, L. Dagba et al., “Damage-Tolerance-Based Design of Bolted Composite Joints,” Compos. Struct.: Theory and Practice, ASTM PA Vol. 1383, 2000, pp. 243–272 47. D. Qian, E. Dickey, R. Andrews et al., “Load Transfer and Deformation Mechanisms in Carbon Nanotube-Polystyrene Composites,” Appl. Phys. Lett., Vol. 76, 2000, pp. 2868–2870 48. N. Reddy, M.D. Moorthy, and J.N. Reddy, “Non-Linear Progressive Failure Analysis of Laminated Composite Plates,” Int. J. Non-Linear Mech., Vol. 30, 1995, pp. 629–649 49. R.S. Sandhu, “Nonlinear Behavior of Unidirectional and Angle Ply Laminates,” J. Aircr., Vol. 13, 1974, pp. 104–111 50. R.S. Sandhu, G.P. Sendeckyj, and R.L. Gallo, “Modeling of the Failure Process in Notched Laminates,” Mech. Compos. Mater. Recent Adv. 1983, pp. 179–189 51. R.S. Sandhu, G.P. Sendeckyj, and R.L. Gallo, “Modeling of the Failure Process in Notched Laminates.” In: Z. Hashin and C.T. Herakovich, Eds., Mechanics of Composite Materials. Recent Advances, pp. 179–189, Pergamon Press, Oxford 52. T.E. Tay, G. Liu, B.C. Tan et al., ) “Progressive Failure Analysis of Composites,” J. Compos. Mater., Vol. 42, 2008, pp. 1921–1966 53. S.C. Tan, “A Progressive Failure Model for Composite Laminates Containing Openings,” J. Comp. Mater., Vol. 25, 1991, pp. 556–577 54. S.W. Tsai and E.M. Wu, “A General Theory of Strength for Anisotropic Materials,” J. Compos. Mater., Vol. 5, 1971, pp. 58–80 55. K.I. Tserpes and G.N. Labeas, “Mesomechanical Analysis of Non-Crimp Fabric Composite Structural Parts,” Compos. Struct., Vol. 87, 2009, pp. 358–369 56. K.I. Tserpes, G. Labeas, P. Papanikos et al., “Strength Prediction of Bolted Joints in Graphite/Epoxy Composite Laminates,” Composites Part B, Vol. 33, 2002, pp. 521–529. 57. K.I. Tserpes, P. Papanikos, and Th. Kermanidis, “A Three-Dimensional Progressive Damage Model for Bolted Joints in Composite Laminates Subjected to Tensile Loading,” Fatigue Fract. Engng. Mater. Struct., Vol. 24, 2001, 10, 673–686 58. K.I. Tserpes, P. Papanikos, G. Labeas et al., “Multi-Scale Modeling of Tensile Behavior of Carbon Nanotube-Reinforced Composites,” Theor. Appl. Fract. Mech., Vol. 49, 2008, pp. 51–60 59. H.D. Wagner, O. Lourie, Y. Feldman et al., “Stress-Induced Fragmentation of Multiwall Carbon Nanotubes in a Polymer Matrix,” Appl. Phys. Lett., Vol. 72, 1998, pp. 188–190 60. D. Xie, B. Sherill, and Jr. Biggers, “Post Buckling Analysis with Progressive Damage Modelling in Tailored Laminated Plate and Shells with a Cut-Out,” Compos. Struct., Vol. 59, 2003, pp. 199–216 61. J. Yap, R.S. Thomson, M.L. Scott et al., Influence of Post-Buckling Behaviour of Composite Stiffened Panels on the Damage Criticality,” Compos. Struct., Vol. 66, 2004, pp. 197–206

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Chapter 12

Functional Nanostructured Polymer– Metal Interfaces Niranjan A. Malvadkar, Michael A. Ulizio, Jill Lowman, and Melik C. Demirel

Abstract The study of polymer–metal surfaces is important for basic scientific research as well as many practical applications in aircraft, automobile, biomedical, and electronics industries. The possibility of controlling particle size and particle surface chemistry of metals would help us to understand the fundamental mechanism of polymer–metal adhesion in general. We have recently demonstrated that nanostructured polymers can be fabricated by an oblique-angle polymerization method. These structures have a high aspect ratio and the production technique does not require any template or lithography method or a surfactant for deposition. We studied influences of the chemical functionality, morphology, and topology of the nanostructured films on the physical properties of metallic–polymer interfaces. Based on the nanostructured polymer mediated metal technology, we can develop novel polymer–metal interfaces with the following attributes: (1) high surface area materials with controlled roughness, (2) light weight and high adhesion strength of polymer to metal, and (3) industrial-scale deposition.

12.1 Introduction Physical properties of polymers can be improved with micron or nano-size inclusions (i.e., fibers, platelets, or particles of nanocomposites). Polymer nanocomposite materials have shown the potential to have improved physical properties with respect to traditional polymer composites used in engineering applications [1–13]. Because of these improved properties and their relatively low density, polymer–metal surfaces are widely used in many practical applications in aircraft, automobile, biomedical, and electronics industries. The preparation of nanocomposites has been described already 100 years ago [14] but nanocomposites have M.C. Demirel (B) The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA e-mail: [email protected]

B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 12,

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attracted wide attention in science only for the last two decades [1, 15]. A large number of studies have been conducted to develop, model, and test polymer nanocomposite materials [7, 16]. In this chapter, a novel two-step technique to prepare polymer–metal surfaces is discussed. First, polymeric nanostructures have been prepared without any use of lithography, patterning, or template. Second, conformal metal layers are deposited either by electroless or vapor deposition. These films have a variety of applications such as surface-enhanced Raman spectroscopy (SERS) substrate [17], metal catalyst membrane [18], and polymer-mediated nanoparticle assembly [19].

12.2 Oblique-Angle Polymerization Oblique-angle polymerization (OAP) is a novel technique to prepare tunable nanostructures in polymeric film with unique physicochemical properties using a complete bottom-up approach. The method is a one-step growth technique to prepare polymer nanostructures without any post-processing step. In other words, there is no template, lithography, or patterning involved in preparing these nanostructures. The tunable morphological properties of polymer films grown by the OAP method in combination with the ease of chemical functionalization can be applied to prepare metal or ceramic membranes on these films with controllable nanostructure and strong interface strength.

12.2.1 Nanostructured Polymer growth We extensively studied the physicochemical properties and the growth mechanism of nanostructured poly(p-xylylene) (PPX) films and its derivatives (e.g., poly(chloro-p-xylylene), poly(bromo-p-xylylene) and poly(trifluoroacetyl-pxylylene)) deposited via OAP method. We modified the method employed by Gorham that involves the deposition of PPX films using a low-pressure, vapor deposition technique (Fig. 12.1) [20]. The [2.2] paracyclophane and its derivatives are the precursor to deposit PPX films. The precursor is first sublimed at 175◦ C under a pressure of 0.1–10 torr. Afterwards, the precursor vapor is pyrolyzed at 650–700◦ C. The pyrolysis of [2.2] paracyclophane results in the cleavage of the alkyl bridges to form quinodimethane radicals. These radicals undergo polymerization on any substrate (e.g., silicon, polymer, or glass), which is kept at ambient temperature. When the radical flux is directed at an angle of lower than 25◦ to the substrate, the resulting film is a porous low-density polymer with unique nanostructured morphology. The nanostructure consists of parallel assemblies of PPX nanowires (Fig. 12.1) each having a diameter of approximately 150 nm. On the other hand, conventionally deposited PPX films (i.e., without directed flux or directed flux at an angle of 90◦ ) do not possess such morphology.

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Fig. 12.1 Schematic showing the preparation process for OAP-PPX films (left). Figure also shows the picture of the deposition system and cross-sectional scanning electron microscope (SEM) image of nanostructured polymer film deposited by OAP method [21] (right)

PPX film adhesion to the substrate surface is enhanced by an allyl functionalized self-assembled monolayer. Silicon wafers were first sonicated in acetone. Afterwards, the wafers were washed in water and dried under nitrogen gas flow. The wafers were then transferred to a 1/1 (v/v) solution of HCl (36.5–38% w/w) and methanol (98.5% w/w). After 30 min, the wafers were removed, washed with water, and dried with nitrogen gas. The wafers were then kept in concentrated H2 SO4 (95–98% w/w) for another 30 min, after which they were washed and sonicated in water for 10 min. The wafers were thoroughly dried under nitrogen gas. The self-assembled monolayer (SAM) solution was prepared by adding 1% (v/v) allyltrimethoxysilane (Gelest, PA) in toluene containing 0.1% (v/v) acetic acid. The cleaned wafers were transferred to this solution and left for SAM formation for 60 min at 25◦ C. The wafers were removed after 60 min and sonicated in anhydrous toluene for 10 min. The wafers were then dried on a hot plate at 140◦ C for 5 min. Nanostructured films of poly(chloro-p-xylylene) were deposited on these allyl functionalized silicon wafers using oblique-angle vapor deposition at low-vacuum conditions. There are three stages in the growth of oblique-angle polymerized PPX (OAP-PPX) films. First, a thin film of PPX is formed on the substrate with an RMS roughness of 1–5 nm. Second, the surface instabilities are further enhanced due to the self-shadowing effect leading to the evolution of the columnar structure. The growth of the column follows a scaling law (i.e., the diameter of the column scales as a power function of the height) [22]. Finally, the columns grow almost uniformly without any change in the size of the columns beyond a critical height of the columns (∼1 μm) leading to the final structure. Figure 12.1 inset shows cross-sectional scanning electron microscope (SEM) image of nanostructured polymer film deposited by OAP method.

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We showed that OAP-PPX film growth follows a power-law scaling, that is, the size or the diameter of the individual columns is a power function of the height of the column [22]. Mathematically, this can be expressed as d = ch p where d is the diameter and h is the height of the column. This implies that the column size and spatial distribution can be controlled by simply varying the thickness of the OAP-PPX film. The single exponent, p, signifying isotropic growth of columns and the multiplying factor, c depends on the deposition parameters and the side-group chemistry of PPX. For example, the exponent, p, values of PPX-Cl, PPX-Br, and PPX-COCF3 for a 10◦ deposition angle are estimated to be 0.11 ± 0.01, 0.18 ± 0.01, and 0.15 ± 0.01, respectively [23]. A smaller exponent for PPXCl film implies that the columns grow quicker than for the other two films. It is observed that the power law is followed only till a critical thickness is reached, hc , after which the column size remains constant when the deposition continues.

12.2.2 Control of Morphology and Topography During the deposition of OAP-PPX films using oblique-angle polymerization, the substrate can be axially rotated to create additional morphologies. Various configurations of the substrate rotation and the nozzle tilt angle create lateral morphologies, such as columnar deposition (substrate tilt is applied at an angle of lower than 25◦ with respect to the flux), helical deposition (substrate is continuously rotated while the substrate tilt lower than 25◦ ), and chevron deposition (substrate is rotated at a specific angle similar to a pendulum motion). The tilt angle has the greatest effect on the surface roughness of the film. For example, a planar PPX-Cl film has an RMS roughness of 7.9 ± 0.8 nm while an OAP-PPX-Cl film with a tilt angle, α = 10◦ , has an RMS roughness of 46.3 ± 5.0 nm [23]. Topology can also be altered by using microscale lithographically patterned substrates (Fig. 12.2). To prepare the patterned silicon, wafers are first cleaned with EKC 830 for 10–15 min. They are then rinsed with acetone, isopropyl alcohol, and DI water for 1–2 min each and dehydrated at 100◦ C for 1 min. Wafers are then spin coated with PR S1837 and baked at 110◦ C for 1 min and air cooled. Wafers are then exposed for 10.5 s using MA6 mass aligner and developed in CD-26 for 2 min and rinsed with DI water. Patterns are prepared on the silicon substrate using dry reactive ion etching (DRIE). To remove the photoresist, wafers are treated with EKC 830 for 30 min and rinsed in acetone, isopropyl alcohol, and DI water for 1–2 min each. The porosity, which is a function of the surface roughness and the morphology, can therefore be controlled. The simultaneous control of the morphology and topology by simply varying the deposition parameters or the choice of the monomer or substrate can create a new generation of polymer films with optimal physicochemical properties.

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Fig. 12.2 SEM image showing (A) helical OAP-PPX film deposited on patterned silicon substrate. (B) Patterned substrate before film deposition. (C) OAP-PPX film with columnar morphology deposited on patterned PDMS substrate. (D) Patterned PDMS substrate before film deposition

12.3 Metallization of Nanostructured Polymers There are two methods for metallization of OAP-polymer film. The first method is the metallization via electroless deposition which is carried out by functionalization of the OAP-polymer film surface using a metal-binding ligand such as pyridine. The films are then treated with colloidal palladium that bind to the ligand and seeds the metallization. Metal ions are reduced on the catalytic palladium sites to form metal clusters on the colloids. Further increase in the metallization time leads to the formation of porous metal membranes. The second method is the vapor phase deposition which is carried out by direct deposition of metals on the OAP-polymer surface by a vapor phase. In this method metal clusters are directly formed on the surface of the OAP-polymer film. The topology of such a film mimics the surface of the OAP-polymer film, leading to a similar porous structure. Figure 12.3 explains the various parameters, growth phases and applications of the two metallization routes. Metal binding via electroless deposition or vapor phase deposition on OAP-PPX films show unusually high polymer–metal adhesion compared to planar PPX films which shows non-continuous metal deposition with poor adhesive properties. This is due to increases in surface area in nanostructured polymer surfaces [21]. The adhesion on OAP-polymer films is enhanced by adsorption of ligand molecules on the high-energy surfaces in OAP-polymer films in addition to the mechanical anchoring of the metal layer due to the higher surface roughness in such films [21]. Although, a detailed study on the evolution of crystallinity

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Obligue angle deposited PPX films

Electroless Deposition (Liquid Phase Metallization)

Parameters Crystallinity Nanowetting

Ions →Metal clusters→ Porous membrane

Vapor Phase Deposition

Parameters Confinement Surface crystallinity

Metal clusters→ Porous membrane

Applications Catalysis Polymer meciated nanoparticle assembly SERS substrate for biosensing

Fig. 12.3 Overview of the metallization techniques, control parameters, and applications

differences in planar and nanostructured PPX films has not been done, the likely reason for preferential ligand binding is due to the presence of amorphous regions induced by OAP. In addition, the nanowetting due to the columnar morphology also contributes to the enhanced ligand adsorption. In the case of vapor phase deposition of the metal layer, the important property that determines the adhesion is the surface crystallinity of the polymer, the reactivity of the metal species and the confinement effect due to the polymer surface roughness. There are several applications of the porous metal structure. The uniformity and controllability of the underlying polymer nanostructure can be utilized to prepare nanostructured gold films with controlled nanostructural properties which can be used as SERS substrate for critical application such as biosensing [17]. The highly porous structure can be applied to build heterogeneous catalyst [18]. The ability of OAP-polymer film to undergo facile chemical functionalization in addition to the tunable lateral morphologies can be used to prepare nanoparticle assemblies on OAP polymer nanowires. Such engineered materials can be applied to versatile applications such as chemical sensing and catalysis [19].

12.3.1 Electroless Metal Deposition The electroless metal deposition is carried out by a three-step process. OAP-PPX film is first incubated in aqueous pyridine solution (1 M) for 40–48 h. The hydrophobic PPX surface is covered with hydrophilic N sites on the pyridine molecules. Essentially, these films are chemically functionalized using ligands such as pyridine. Instead of aqueous pyridine solution, pyridine vapor can be used to functionalize the PPX surface. PPX samples are suspended on top of a pyridine-containing vial which is heated to a temperature of 110◦ C (pyridine b.p. 115.2◦ C) for 40–48 h.

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Pyridine-functionalized films are then transferred to a Pd(II)-based solution, the preparation of which is described in detail elsewhere [24]. Briefly, the Pd(II)-based colloidal solution is prepared by hydrolysis of Na2 PdCl4 at pH 5 in a 0.01 M NaCl solution. The solution is aged for approximately 20 h before use. The films are kept in the aged Pd(II) solution for 45 min at 25 ± 0.5◦ C, after which they are washed in water and dried under N2 gas. These N sites on the OAP-PPX surfaces can covalently bind the Pd(II) colloids that seeds the metallization process. The nanoparticles enter the pores and binds to the walls of the nanocolumns. Electroless metallization of nickel is carried out at 25◦ C to characterize the effective binding of Pd(II) colloids on the OAP-PPX surface. For nickel, 10% (v/v) NIPOSITTM 468B was used as the electroless bath. The bath is stirred continuously throughout the deposition process to remove the trapped hydrogen bubbles on the surface. The samples are then removed from the bath, washed in water, and dried using nitrogen gas. The metalized OAP-PPX films show a high level of adhesion with <5% delamination in the R test. Thus, metallization is carried out without any roughening of the Scotch Tape polymer surface using a tin-free activation process. The final structure consists of porous membrane that mimics the columnar topology of the underlying OAP-PPX film (Fig. 12.4). In comparison, planar PPX surfaces show lower metal deposition with poor adhesion under identical experimental conditions (not shown). This is due to the poor binding of the pyridine molecules on the planar PPX surface. The enhanced surface disorder typically observed for structured (high curvature) films (∼100 nm structures) show enhanced and rapid ligand binding [21]. A critical palladium concentration of 1015 atoms/cm2 is required for electroless metallization [25]. However, given a longer ligand-binding time, planar films also show a continuous metallization similar to nanostructured PPX films. OAP-PPX films deposited on patterned silicon substrate can create polymer surface with dual-layer roughness. Depositing metal layer on such a surface can result in metal membranes with multi-scale roughness (Fig. 12.5).

12.3.2 Vapor Phase Metal Deposition Silver and gold films are deposited using thermal evaporation from resistively heated tungsten and tantalum boats onto OAP-PPX surface (Fig. 12.6). The cryogenically pumped deposition chamber is maintained at a base pressure of ∼1×10–8 torr. The thicknesses of the gold and silver films are monitored using a parallel quartz crystal microbalance (QCM). We studied the structural properties of gold membrane deposited on OAP-PPX films using AFM and SEM (Fig. 12.6B and C). From the images it can be seen that the gold forms a continuous porous layer on the OAP-PPX film even for a ˚ The topology of the gold layer is replicated from thickness of gold as low as 60 A. the underlying polymer. It can be concluded that the confinement effect created by the roughness of the underlying OAP-PPX film is responsible for the uniform gold deposition.

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Fig. 12.4 (A) Contact mode AFM image showing nickel deposited on OAP-PPX film functionalized by aqueous pyridine solution. (B) SEM image showing nickel deposited on OAP-PPX film functionalized by aqueous pyridine solution. In both cases the ligand incubation time was 48 h, palladium reaction time was 45 min at 25◦ C and nickel bath time was 1 h

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12.3.3 Nanoparticle Assembly Assembly of nanoparticles on the varied tunable nanostructure morphologies can be useful to control the optical, magnetic, electrical and biological properties [26–30]. We utilize the non-covalent ligand-binding property of OAP-polymer films to assemble metal nanoparticles on the surface of the nanocolumns. PPX is used as the polymer, while thiophenol dissolved in ethyl alcohol is used as the ligand to bind copper nanoparticles. OAP-PPX films are incubated in 0.5 M ligand solution

12 Functional Nanostructured Polymer–Metal Interfaces Fig. 12.5 (A) Nickel membrane possessing dual-layered roughness prepared by electroless deposition on OAP-PPX films on both patterned and non-patterned silicon substrate. (B) SEM image showing nickel membrane deposited on OAP-PPX films on patterned silicon substrate

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of thiophenol in ethyl alcohol for 48 h. The films are then washed with the solvent to remove any non-bonded ligand molecules. Copper nanoparticles are prepared by wet-chemistry route as explained elsewhere [31]. Copper nanoparticles show an absorption band at 562 nm, indicating the nanoparticle size of 10–20 nm. Films are kept in copper nanoparticles at 25◦ C. After 45 min, the films are washed in water thoroughly and dried. Assembly of copper nanoparticles is studied using Olympus Fluoview 1000 confocal fluorescence microscope with an excitation wavelength of 488 nm. Figures 12.7B and C show fluorescence microscope image of nanoparticle assembly on OAP-PPX and planar PPX, respectively. Figure 12.7A shows fluorescence microscope image of OAP-PPX without nanoparticles for comparison. From the data, it can be concluded that nanoparticle assemble on OAP-PPX films on the surface of the columns thereby taking its structure. On the other hand, planar-PPX film shows low and patchy deposition of nanoparticles. This confirms that continuous metal

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Au, Ag, Co, Cu, etc…

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Fig. 12.6 (A) An illustration showing deposition of metal membrane on OAP-PPX film using vapor phase metal deposition. (B) SEM image of 60 nm gold membrane deposited on OAP-PPX film. (C) Corresponding AFM image of the gold membrane

layer on OAP-PPX film is a result of uniform and high degree of ligand adsorption leading to uniformly distributed seed layer. In planar-PPX films the patchy deposition of metal layer is due to the insufficient binding of ligand molecules and the seed layer. Thus nanoparticle assembly on OAP-polymer films using non-covalent ligand binding is a novel and a facile method to immobilize and pattern nanoparticles on tunable polymer morphologies. The study of nanoparticle distribution on the various lateral morphologies is left for future studies.

12.4 Conclusions Oblique-angle polymerization is a novel technique to prepare tunable nanostructures on polymer films without any use of lithography, patterning, or template. The column size, periodicity, column density, and the surface roughness of OAP films exhibit a power-law scaling to the height of the columns. Conformal metal layers

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Fig. 12.7 Fluorescence microscope image showing (A) control sample of OAP-PPX film without nanoparticles. (B) nanoparticle assembly on OAP-PPX film using thiophenol as ligand. (C) nanoparticle assembly on planar-PPX film using thiophenol as ligand. Scale bar is 10 μm

can be deposited either by electroless or vapor deposition on OAP films. The morphology and topology of this metal layer can be tuned by varying the deposition parameters for the underlying polymer. The ability to control the chemical and physical properties of nanostructured films has scientific and technological importance in many medical areas including biomedical coatings, biosensing, tissue culture and growth, and (A) biocatalyst supports. PPX is an FDA-approved material and the process of depositing nanostructured PPX film is relatively inexpensive. Based on the nanostructured PPX technology, we can develop a novel surface with the following attributes: (1) high surface area materials with control roughness, (2) light weight and high adhesion strength of polymer to metal, and (3) industrial-scale deposition for medical and engineering applications.

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The combined properties of controlled surface roughness, porosity, and morphology in addition to the ability for chemical functionalization make such films unique for a variety of applications such as surface-enhanced Raman spectroscopy (SERS) substrate [17], metal catalyst membrane [18], and polymer-mediated nanoparticle assembly [19]. Acknowledgments This research is supported by a Young Investigator Program Award from the Office of Naval Research (N000140710801), Research Experience for Undergraduates in Nanoscale Science, Engineering and Technology (to J.L and M.U.) from the Penn State National Nanotechnology Infrastructure Network (National Science Foundation), and Penn State Biomaterials and Biotechnology Summer Institute (National Institutes of Health). We thank Dr. Aman Haque (Penn State), Dr. Metin Sitti (CMU), and Mr. David Welch (summer student) for providing patterned surfaces.

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17. P. Kao, N. Malvadkar, D. Allara, and M.C. Demirel, “Surface enhanced Raman Detection of Bacteria on Metalized Nanostructured Poly(p-xylylene) Films,” Adv. Mater., Vol. 20, 2008, pp. 3562–3565 18. N. Malvadkar, S. Park, H. Wang, M. Macdonald, and M.C. Demirel, “Catalytic Activity of Cobalt Deposited on Nanostructured Poly(p-xylylene) Films,” J. Power Sources, Vol. 182, 2008, pp. 323–328 19. R. Shenhar, T.B. Norsten, and V.M. Rotello, “Polymer-Mediated Nanoparticle Assembly: Structural Control and Applications,” Adv. Mater., Vol. 17, 2005, pp. 657–669 20. W.F. Gorham, “A New General Synthetic Method for Preparation of Linear Poly-PXylylenes,” J. Polym. Sci. Part A-1-Polym. Chem., Vol. 4, 1966, p. 3027 21. M.C. Demirel, M. Cetinkaya, A. Singh, and W.J. Dressick, “Noncovalent Deposition of Nanoporous ni Membranes on Spatially Organized Poly(p-xylylene) Film Templates,” Adv. Mat., Vol. 19, 2007, pp. 4495–4499 22. M. Cetinkaya, N. Malvadkar, and M.C. Demirel, “Power-Law Scaling of Structured Poly(PXylylene) Films Deposited by Oblique Angle,” J. Polym. Sci. Part B: Polym. Phys., Vol. 46, 2008, pp. 640–648 23. A. Cetinkaya, S. Boduroglu, and M.C. Demirel, “Growth of Nanostructured Thin Films of Poly (p-xytylene) Derivatives by Vapor Deposition,” Polymer, Vol. 48, 2007, pp. 4130–4134 24. S.L. Brandow, et al. “Size-Controlled Colloidal Pd(II) Catalysts for Electroless Ni Deposition in Nanolithography Applications,” J. Electrochem. Soc., Vol. 144, 1997, pp. 3425–3434 25. W.J. Dressick, C.S. Dulcey, J.H. Georger, G.S. Calabrese, and J.M. Calvert, “Covalent Binding of Pd Catalysts to Ligating Self-Assembled Monolayer Films for Selective Electroless Metal-Deposition,” J. Electrochem. Soc., Vol. 141, 1994, pp. 210–220 26. A. Rogach, et al. “Nano- and Microengineering: Three-Dimensional Colloidal Photonic Crystals Prepared from Submicrometer-Sized Polystyrene Latex Spheres Pre-Coated with Luminescent Polyelectrolyte/Nanocrystal Shells,” Adv. Mater., Vol. 12, 2000, p. 333 27. F.J. Castano, et al. “Magnetization Reversal in Sub-100 nm Pseudo-Spin-Valve Element Arrays,” Appl. Phys. Lett., Vol. 79, pp. 2001, 1504–1506 28. M. Brust, D. Bethell, C.J. Kiely, and D.J. Schiffrin, “Self-Assembled Gold Nanoparticle Thin Films with Nonmetallic Optical and Electronic Properties,” Langmuir, Vol. 14, 1998, pp. 5425–5429 29. S.H. Sun, et al. “Controlled Synthesis and Assembly of FePt Nanoparticles,” J. Phys. Chem. B, Vol. 107, 2003, pp. 5419–5425 30. C.A. Mirkin, R.L. Letsinger, R.C. Mucic, and J.J. Storhoff, “A DNA-Based Method for Rationally Assembling nanoparticles into macroscopic materials,” Nature, Vol. 382, 1996, pp. 607–609 31. C.M. Coyle, G. Chumanov, and P.W. Jagodzinski, “Surface-Enhanced Raman Spectra of the Reduction Product of 4-Cyanopyridine on Copper Colloids,” J. Raman Spectrosc., Vol. 29, 1998, pp. 757–762

Chapter 13

Advanced Experimental Techniques for Multiscale Modeling of Materials Reza S. Yassar and Hessam M.S. Ghassemi

Abstract From a scientific viewpoint, direct comparison between mechanical tests and computational simulations on a one-to-one basis has the potential to lead to substantial development in the concept of virtual testing of materials. Successful application of virtual testing methodology in our daily life basis requires the use of high-fidelity computational models that are being validated through accurate characterization techniques. The content of this chapter is prepared to cover some of the most recent developments in the area of materials characterizations with great potential for virtual testing and modeling applications. During the last decade, atomic force microscopy (AFM) has evolved into an essential tool for direct measurements of intermolecular forces that can be employed for verification of first-principle and molecular dynamic models. Novel techniques in the area of in situ electron microscopy have been developed in the last decade for investigating the structure–mechanical property relationship of advanced materials. X-ray ultramicroscopy (XuM) and microelectromechanical systems (MEMS) are among the two newest in situ microscopy developments. These techniques provide an excellent platform for direct correlation between structure and properties of nanoscale materials. These systems contain a limited number of atoms and possible equilibrium configurations, which can be identified in real time by means of in situ electron microscopy techniques. In addition, because of the limited number of atoms, these systems can be atomistically modeled within the reach of currently available computational power. This chapter provides a comprehensive review on the abovementioned characterization techniques that can be used to validate computational models at nanometer length scales.

R.S. Yassar (B) Mechanical Engineering-Engineering Mechanics Department, Michigan Technological University, 1400 Townsend Dr., Houghton, MI 49931 USA e-mail: [email protected]

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13.1 Atomic Force Microscopy (AFM) Atomic force microscopy (AFM) belongs to a series of scanning probe microscopes (SPM) invented in the 1980s. This series started with the scanning tunneling microscope (STM), which allowed the imaging of surfaces and single atoms in conducting and semiconducting materials [1, 2]. In parallel, the scanning near-field optical microscope (SNOM) was invented which allowed microscopy with light below the optical resolution limit [3, 4]. The last one of the series is the AFM, invented by Binnig et al. [5]. Since its invention, AFM system has evolved into a useful tool for direct measurements of intermolecular forces with atomic-resolution characterization that can be employed in a broad spectrum of applications such as electronics, semi-conductors, materials and manufacturing, polymers, biology, and biomaterials. AFM provides additional capabilities and advantages relative to other microscopic methods (e.g., scanning electron microscopy (SEM) and transmission electron microscopy (TEM)) in studies of metallic surfaces and micro-structures by providing reliable measurements at the nanometer scale [6–9]. Image resolution in AFM, which can be in the order of sub-nanometer, depends on a variety of factors, including tip sharpness, acoustic isolation of the instrument, sampling medium, and AFM controller precision.

13.1.1 Principles of AFM A typical AFM system consists of a micro-machined cantilever probe and a sharp tip mounted to a PZT actuator and a position-sensitive photo detector for receiving a laser beam reflected off the end-point of the beam to provide cantilever deflection feedback. The resolution of an AFM depends strongly on the shape of the tip [10]. The sharper the tip is, the smaller is the surface area sampled by this tip. An AFM tip consists of a microfabricated pyramidal Si or Si3 N4 tip attached to a flexible cantilever of a specific spring constant. From the deflection of the cantilever, the tip-sample force data can be calculated using Hooke’s law: F = kc z

(13.1)

where F is the magnitude of the force acting between the tip and sample, z is the cantilever deflection at its free end, and kc , the cantilever spring constant, is given by kc =

Ewt 3 4 l3

(13.2)

In the above equation, w is the cantilever’s width, l is its length, t is its thickness, and E is the elastic modulus. If the cantilever spring constant is known, the cantilever deflections may be converted to quantitative force data.

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The tip scans over the sample surface with feedback mechanisms that enable the PZT scanners to maintain the tip at a constant force, or constant height above the sample surface. The cantilever deflects in z-direction due to the surface topography during tip scanning over the sample surface. As the tip scans the surface of the sample, moving up and down with the contour of the surface, the laser beam deflected from the cantilever provides measurements of the difference in light intensities between the upper and lower photo detectors. Feedback from the photodiode difference signal enables the tip to maintain either a constant force or constant height above the sample. In the constant force mode, the PZT transducer monitors real-time height deviation. In the constant height mode, the deflection force on the sample is recorded. Figure 13.1 shows the principle set-up of an AFM.

Fig. 13.1 Schematic representation of an AFM. The cantilever-tip system is deflected by the surface topography of the sample. Cantilever deflections are detected with a laser-optical set-up. The four-segment photodiode detects normal forces (normal force microscopy (NFM)) and frictional forces (FFM) affecting the tip. In most AFM systems the sample rests on a piezotube scanner (not shown in this figure) which allows a scanning motion in x- and y-directions as well as movement in z-direction [25]

A four-segment photodiode detects the deflection of the cantilever through a laser beam focused on and reflected from the rear of the cantilever. A computer processes the electrical differential signal of the photodiode obtained from each point of the surface and generates a feedback signal for the piezoscanner to maintain a constant force on the tip. This information is transferred into a photographic image of the surface. The main forces contributing to the cantilever deflection are electrostatic (Coulomb) repulsive forces and attractive van der Waals forces between the atom within the tip and the atoms of the sample surface. Hence, the AFM is also called scanning force microscope (SFM). Assuming bond energies for ionic bonded samples of U • 10 eV and 10 meV for van der Waals bonded samples (such as organic crystals) and taking the repulsive force as acting through a distance x ≈ 0.02 nm, the interatomic force

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F=

−U x

(13.3)

would be ≤ 10−7 NN for ionic bonds and ≤ 10−11 N N for van der Waals bonds [11]. These values define the requirements for the force applied by an AFM probe, that is, the force constant of the cantilever. AFM eliminates the need for a vacuum that is required by most electron microscopy techniques and therefore has the potential to image samples in ambient air or liquid.

13.1.2 AFM Operation Typical AFMs operate in three different modes: (a) contact mode, (b) non-contact mode, and (c) tapping mode. In the contact mode, the tip is in permanent contact with the sample surface, while in the tapping mode, the tip oscillates (frequencies in air 50–500 kHz, in the fluids approximately 10 kHz) at a tip amplitude of several tens ◦

of nanometers. In the non-contact the cantilever tip hovers about 50–150 A above the sample surface to detect the attractive van der Waals forces acting between the tip and the sample. The interaction forces between the tip and sample in all of the three open-loop modes can be distinctly identified on a force–displacement curve as shown in Fig. 13.2. Fig. 13.2 Interatomic force variation versus distance between AFM tip and sample

When the interatomic distance is quite large, weak attractive forces are generated between the tip and the sample. As the atoms are gradually brought closer to each other, the attractive forces increase until the atoms become so close that the electron clouds begin to repel each other electrostatically. This repulsive force between the atoms progressively weakens the attractive forces as the interatomic distance decreases. The interaction force becomes zero when the distance between the atoms

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reaches a couple of angstroms and becomes fully repulsive when the atoms are in contact (see Fig. 13.2). It can be seen that the tapping and non-contact modes reveal the surface topography of the sample to a greater extent when compared to the contact mode. Also, the operation of the AFM in contact mode damages the sample surface due to the lateral dragging forces exerted by the probe tip. The detailed description and operational characteristics of each mode is described next.

13.1.3 Application of AFM Due to its inherent advantages, AFMs find a variety of applications that range from materials and mechanical sciences to life sciences [12, 13]. In this section we briefly review some of the vital applications of AFM that provide insight to our understanding about materials behavior and subsequent modeling and simulation applications. 13.1.3.1 Characterization of Surface Microstructures Recently, AFM has been recognized as a powerful surface characterization technique and has been widely used to study surface morphology of materials. In addition, TM-AFM has been employed to determine the bulk morphology. Feng [14] used TM-AFM to detect nano-silica particles underneath an ultra-thin polymer film. Figure 13.3 displays the TM-AFM height and phase images of the PVP/silica thinfilm surface, using a set-point amplitude ratio of 0.49. In the phase image, the bright dots are identified as nano-silica particles because silica particles are much harder than PVP. The average diameter of these silica particles that ranges from 10 to 20 nm is in total agreement with their TEM results.

Fig. 13.3 TM-AFM images of the PVP/silica thin-film surface using a set-point amplitude ratio of 0.49. (a) The TM-AFM height image and (b) the TM-AFM phase image [14]

Another application of AFM is imaging of defects like cracks, dislocations, and pores. The high-resolution capability of AFM permits study of the processes of brittle and ductile crack growth in detail. Lu et al. [15] used in situ technique under AFM to observe the nucleation and propagation behaviors of cracks in CuNiAl

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Fig. 13.4 Propagation process of a crack upon loading in AFM, (a) the martensites A ahead of a propagated crack tip, (b) the propagation of the crack along the martensite interface B, (c) the formation of slip bands at positions C and D after significant crack propagation, (d) nucleation and propagation of microcrack along slip bands [15]

shape memory alloy. Figure 13.4 illustrates the propagation of a crack upon loading in AFM. In Fig. 13.4a, there is a propagating crack tip, around which exist different martensite variants. Upon further loading, the crack tip propagated from position A to position B along the martensite interface, as shown in Fig. 13.4b. Upon further loading, the main crack propagated to point C, and many ripple slip bands appeared at the positions D and E, as shown in Fig. 13.4c. Upon further loading, a new microcrack nucleated and propagated along the slip bands at position E, resulting in bifurcation of the main crack, as shown in Fig. 13.4d. 13.1.3.2 Nanomechanical and Nanoindentation Testing Nanomechanical studies in an AFM allow local measurements of the mechanical properties on a nanometer scale. Using a diamond tip mounted to a metal-foil cantilever, a surface can be indented and the indentation can be imaged. Indentation sizes can be reduced to 10 nm or less, which makes it possible for local studies of the mechanical properties of microstructures [16]. Goken [17] determined, for the first time, hardness on the superalloys CMSX-6 and Waspaloy for the γ’ and matrix

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Fig. 13.5 Nano-indentations in the superalloy CMSX-6 obtained with a load force of 500 mN. The larger lateral size of the indents in the matrix phase indicates a lower hardness of matrix in comparison with the γ’ phase [17]

Region B

F

Cantilevrer

Y

Y

F

phases, separately. Figure 13.5 shows different indents in the γ’ and matrix phases of CMSX-6. These indents were performed at large loads of 500 mN. Obviously, the remaining plastic indentation size is larger in the matrix phase. This indicates a higher hardness for the γ’ phase [17]. AFM has been used widely to measure the elastic properties of nanomaterials. The amplitude of cantilever modulation that results from an applied signal varies according to the elastic properties of the sample, as shown in Fig. 13.6.

Region A Sample

Fig. 13.6 Variation of the AFM cantilever oscillation amplitude as function of the mechanical properties of the sample

The system generates a force modulation image, which is a map of the sample’s elastic properties, from the changes in the amplitude of cantilever modulation, which is based on the hardness or softness of the regions. In Fig. 13.6, oscillation of the cantilever on region A, corresponds to soft region, is less than that of region B which corresponds to a hard region. Thus, topographic information can be separated from local variations in the sample’s elastic properties, and the two types of images can be collected simultaneously. AFM tip-based manipulation technique is one of the most powerful approaches for probing the elastic properties of nanotubes [18], nanobelt [19], and nanowire [20]. Ni et al. [21] measured elastic modulus of the amorphous SiO2 nanowires, performing three-point bending tests by directly indenting the center of a suspended nanowire which bridged the channel with a tapping-mode silicon AFM tip. Elastic modulus of the amorphous SiO2 nanowires was measured to be 76.6 ± 7.2 GPa, as shown in Fig. 13.7. Zhang et al. [20] measured Young’s modulus of

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Fig. 13.7 Bending elastic modulus of the amorphous SiO2 nanowire versus nanowire diameters measured by AFM [21]

Fig. 13.8 Young’s modulus as a function of LaB6 nanowire thickness measured by AFM technique [21]

LaB6 nanobelt in six different diameters, based on three-point bending measurement. As shown in Fig. 13.8, all six E values are evenly distributed around an average value of 467.1 ± 15.8 GPa, which equals to that of reported. This result is also in agreement with data measured using nanoindentation. 13.1.3.3 Characterization of Electrical Properties Electrical properties of nanostructures can be directly achieved using conductiveAFM. This not only allows the investigation of nanostructure electrical characteristics but also the nanooxidation of metallic and semiconductor materials [22]. Electrons can flow between the sample and the tip when a bias is applied. The conductance between the tip at each surface point and the substrate is determined by the random configuration of conducting paths (via metallic connecting or tunneling) in materials. Figure 13.9 [23] shows a typical electric current image for x = 0.55 at a bias of 8.2 mV. Here x is the metallic concentration. 13.1.3.4 Characterization of Biological Structures In contrast to conventional biological imaging methods, specimens can be investigated by AFM in their native environment for several hours without damages, which

Fig. 13.9 Electric current image obtained on a sample with x = 0.55 at a bias of 8.2 mV [23]

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0.2 0.4 0.6 0.8

μm

results in the unsurpassed advantage of real-time imaging of live biological samples. The AFM technique acts as a nanoscale finger that probes the local surface topography through a direct mechanical contact. Muller [24] showed that this method can be used for imaging of single native protein, structural identification of individual proteins and observing proteins at work. Jandt et al. [25] applied load into a previously implanted high-density polyethylene hip implant with a vertical transducer nanoidentation AFM leading to highly reproducible force-displacement curves such as shown in Fig. 13.10. These curves in combination with the indentation area allow calculation of sample hardness and the reduced elastic modulus. 800 Force (mN)

Fig. 13.10 Force– displacement curve of an indentation into a previously implanted high-density polyethylene hip implant with a vertical transducer nanoidentation AFM [25]

600 400 200 0 0

200

400 600 Displacement (nm)

800

13.1.4 Modeling and Simulation AFM has proven to be a valuable tool for validation of computational results and for suggesting new insights on materials modeling. In this way, simulations can help establish the connection between experimentation and simpler models. Combined

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AFM and molecular dynamic (MD) simulation is a very useful tool in studying indentation, adhesion, friction, fracture, and surface defects on the atomic scale. Mulliah et al. [26] used MD simulation to investigate on wear and friction behavior of silver (100) surface by nanoscratching in three different orientations. Their simulation consists of three stages, namely: the indentation of the substrate; the relaxation of the system; followed by the scratching of the substrate. Figures 13.11 (a–f) are the top view of the silver substrate showing the dislocations during the initial stage of the scratching simulation, for three orientations of the indenter at ˚ Figs. 13.11(b), (d), and (f) are the same pictures as an indentation depth of 10 A. in Fig. 13.11(a), (c), and (e), respectively, except that atoms in Fig. 13.11(b), (d), and (f) are shaded with regard to depth and energy filtered, whereas the atoms in Fig. 13.11(a), (c), and (e) are shaded according to the slip vector modulus.

Fig. 13.11 Molecular dynamic simulations show top view of the silver substrate at t = 21 ps of ◦

the scratching simulation and at an indentation depth of 10 A for the three indenter orientations. Figures (a), (c), and (e) are shaded according to the slip vector modulus, whereas (b), (d), and (f) are shaded with regard to depth and the atoms are energy filtered [26]

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Yan et al. [27] investigated the effects of the feed in AFM nano-scratching process using the MD simulation approach. Interactive forces between a rigid pyramidal tool simulating an AFM diamond tip and a copper sample were modeled by the Morse function [28]. In another research work [29], they utilized a 3D-MD model to investigate the effect of tool geometry on the deformation process of the workpiece and the nature of deformation process at the atomic scale. The tool model is a cone-shaped tool with a hemisphere at the end. Part of their simulation was focused on the elastically deformed region of the workpiece during the scratching process, which is shown in Fig. 13.12.

Fig. 13.12 Potential energy variation of an atom in elastic deformation region [29]

An atom, the blue atom, is selected from region A, which experiences three kinds of processes, namely: the process before deformation, the deformation process, and the process after deformation. The potential energy of this atom in the whole process is shown in Fig. 13.12(d) as points (a), (b), and (c). Its potential energy is about –6 eV before deformation (point (a)). It increases to –5.7 eV during deformation (point (b)). After the recovery stage, it reduces to the original value, –6 eV (point (c)). Therefore, the features of atoms in the elastically deformed region are: (1) original potential energy remains constant (before deformation), (2) the potential energy increases to a value depending on the extent of deformation (during deformation), and (3) the potential energy reduces to its original value (after deformation). Walter et al. [30] measured surface topography incorporating into an FE model, which allowed simulating the indentation of CrN thin films with a spherical indenter in the elastic deformation range. A comparison to the Hertzian solution of the contact problem was used to check the validity of the FE model. Figure 13.14 shows

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Fig. 13.13 Comparison of simulated FE modeling load–displacement curves with analytical Hertz load–displacement curves of the sample consisting of a 3-μm thick CrN coating on a Si substrate for an AFM experiment [30]

that there is very good agreement between the simulated and the Hertzian load– displacement curves for these two cases. Therefore, the model is assumed to be a valid representation of the indentation process. Furthermore, Fig. 13.13 contains the load–displacement curve for the sample consisting of a 3-μm thick CrN coating on a Si substrate. The deviation of the curve for the thin film from the bulk CrN indicates the effect of the substrate on the indentation behavior.

13.2 X-Ray Ultra-Microscopy The X-ray ultra microscopy (XuM) is one of the latest techniques for revealing the internal structure of materials and has great potential to be used for validation of materials modeling and computer simulations. It is a nondestructive technique that can provide information about the internal structure of optically opaque objects. The XuM has proven to be a versatile and useful instrument, with greatly enhanced visibility of weakly absorbing and fine-scale features. Some of its advantages include revealing the internal structure (simple 2D and stereo images and 3D micro-tomography), submicron resolution (200–400 nm typical, <100 nm possible), exploiting different contrast mechanisms (absorption contrast, phase contrast, and allthe SEM contrast mechanisms) and being SEM-hosted where SEM electron beam used to generate point-source of X-rays and SEM stage used to move sample.

13.2.1 Principles of XuM The XuM setup is hosted by SEM, and originally was developed by Mayo et al. [31, 32]. It has four main components, which are integrated into the host SEM to provide the unique XuM facilities: target positioner, specialized sample holders, X-ray detector, and image acquisition and processing software for stereo imaging and microtomography. The arrangement of the components is shown in Fig. 13.14.

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Fig. 13.14 Diagram showing the main components of the XuM system [32]

The target positioner enables the target – typically a metal foil – to be positioned a few millimeters below the pole piece where the electron beam is focused onto it to produce a microfocus X-ray source. The sample is mounted on the SEM sample stage which enables x, y, and z translation and rotation about the vertical axis. The large depth of field and high spatial resolution of X-ray images is a significant advantage of the X-ray microscopy. SEM can be utilized for the purpose of producing a fine X-ray source suitable for X-ray microscopy. XuM exploits X-ray phase contrast to boost the quality and information content of images. Figure 13.15 compares the results obtained by Wu et al. [33] from SEM and XuM, respectively, on the nylon 6 composite specimens. In contrast to the SEM results, the XuM images provide very clear and spatially resolved filler dispersion patterns for all the composites samples investigated. Unlike most imaging and analytical techniques available on the SEM, the XuM allows the user to look inside the sample structure rather than just examine the surface or near-surface structure. This ability to image internal structure means that many samples can be analyzed completely intact without the need for crosssectioning which would both destroy the sample and would raise the possibility of introducing sample preparation artifacts. It can also reveal features down to less than 0.1 micron, 100 times smaller than conventional X-ray imaging. The XuM has already been used to provide new information about a wide range of sample types from both the biological sciences and materials science. Such applications include the study of electromigration, delamination, and other defects in semiconductors and micro-electronic device, corrosion of aluminum alloys, internal structure of wood and paper products, fish embryos, perforations in polymers, defects in diamonds and minerals, to name but a few.

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Fig. 13.15 Characterization of CaCO3 particle dispersion in a nylon 6 matrix by (a) SEM and (b) XuM [33]

13.2.2 Phase Contrast and Absorption Contrast The imaging in XuM is mostly based on two mechanisms: phase contrast and absorption contrast. Phase contrast is a phenomenon that exploits the wave properties of X-rays. It arises from the refraction (rather than absorption) of X-rays by the sample and is therefore sensitive to changes in refractive index which typically occur at boundaries and edges. As well as providing this edge enhancement, phase contrast allows us to see objects which are either too thin or are insufficiently dense to produce any significant absorption contrast. Under these conditions, the phase contrast component is much stronger than the absorption component so even biological materials can be imaged without the need for heavy metal staining. Absorption contrast is based on the attenuation of X-rays. Figure 13.16 shows the phase and absorption contrast functions represented in terms of the dimensionless coordinate u•(zλ), where u is the spatial frequency, z the propagation distance (for a parallel beam), and λ is wavelength. This coordinate effectively represents the imaging regime, with the near-field region being near the origin. The images in the lower part of the figure show how the appearance of the propagated image changes as this coordinate increases for the peak spatial frequency of the original image. In physical terms, this implies either increasing propagation distance or increasing the wavelength [32]. Figure 13.17 [31] shows examples where the raw phase contrast image gives useful information about the sample. The first is of a dust mite, and clearly shows the grooves on its back despite the small size of these features compared to the overall thickness of the mite. These features would be very difficult to see in an image relying on conventional absorption contrast. The second image is of a 1 mm diameter multilayer sphere composed of concentric shells of different materials. Phase contrast makes a crack in the outermost 30 μm thick shell clearly visible.

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Fig. 13.16 Absorption and phase contrast transfer functions (CTF) represented as a function of the dimensionless coordinate u•(zλ). The images of the face show the effect on a pure phase object of imaging conditions at the corresponding position on the x axis u•(λz) of the CTF, where in this case, u is the peak spatial frequency in the image [32]

As shown, XuM images are particularly useful for observing cracks, voids, boundaries, and surface textures, even when these are small features in a much larger structure.

13.2.3 3D Imaging and Multiscale Modeling Applications An exciting application of XuM is on 3D reconstruction of materials internal structure. Adding a third dimension to 2D images allows material modelers to develop models that can predict materials behavior – including anisotropic properties – in all crystallographic directions. The geometry of the XuM is well suited to tomography, particularly since the SEM sample stage on which the sample is mounted incorporates a vertical rotation axis. This enables the acquisition of a tomographic dataset

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Fig. 13.17 (a) Phase contrast images of a dust mite. The grooved texture of the mite’s back is clearly visible due to phase contrast, despite the small size of these grooves compared to the overall thickness of the mite. (b) Part of a 1 mm diameter multilayer composed of concentric shells of different thicknesses showing a fine crack in the outer shell [31]

by acquiring a series of images with the sample rotated by a small angle between each successive image. A 3D micro-tomography provides the ultimate information by creating 3D solid models of the sample reconstructed from rotational datasets. A wide range of visualization and analysis tools can then be used to digitally “slice open” the sample to reveal the internal structure. Wu et al. [33] modeled the distributions of 1 wt% and 5 wt% calcium carbonate particles in the PP matrix displayed in 3D images as shown in Fig. 13.18. The 3D images

Fig. 13.18 (a) Reconstructed 3D image of the PP sample containing 1 wt% CaCO3 ; (b) Reconstructed 3D image of the PP sample containing 5 wt% CaCO3 [33]

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confirm the existence of aggregates of different sizes in the bulk of the composite samples as well as near the surface region as shown by the 2D inspection. Another application of XuM would be in analysis of damage and fracture in a wide range of materials. Ductile fracture that occurs through the nucleation, growth, and coalescence of voids is a primary mode of material failure, so it is important to predict the condition of void growth and coalescence when a material is subjected to large plastic deformation. In general, void growth and coalescence effects were studied by employing a unit cell including a void or voids. Scientific studies show that void growth direction and shape are quite different for varying crystallographic orientation. In some directions the void growth velocity is larger than other directions. However, the 2D unit cell renders these calculations of more quantitative value in assessing the influence of microstructure. Actually the voids which nucleate from second-phase particles are approximately spherical, so it is necessary to address the 3D model with sphere voids and associated slip laws, and computational evaluations of this nature are valuable because they provide a means to quantitatively assess micro–macro relations, to understand stress-state dependence, and to illustrate the relation of void growth rate with crystallographic orientation directly. Liu et al. [34] concluded that due to plastic flow localization and anisotropic behavior, void which has a spherical shape initially, develops an irregular shape, and some corners are induced. Potirniche et al. [35] showed that void growth was much more dominant than void coalescence. They used molecular dynamic simulation to study the evolution of void growth and coalescence under very high strain rates. The dimensions for the specimens tested with this method were situated in the range of nanometers. The stress–strain curves of the one-void and two-void specimens are presented in Fig. 13.19 indicating the stages of the specimens’ internal evolution of damage.

Fig. 13.19 Average stress versus true strain curve at 1010 /s strain rate for the one-void (left) and two-void (right) nickel specimens with N = 4,408 and N = 5,052 atoms, respectively [35]

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Zabler et al. [36] used 3D image analysis for direct interpretation of fracturing, and quantitative analysis of the fractures in rocks. Because of the limited vertical dimension of the X-ray beam, three tomograms were recorded at different heights to cover the entire sample volume. Figures 13.20 a and b depict how the different parts of sample are aligned. In Fig. 13.20c, the different micro-structural components are illustrated.

Fig. 13.20 Three-dimensional XuM images of the sample: (a) virtual assembly of three tomograms of sample. (b) 2 × 2 × 2 binning is applied to the complete dataset. (c) Reconstruction of linear absorption coefficients allows grains and phases in the graywacke to be distinguished. In red, porosity and a macro-crack forming after strong deformation; in yellow, dense mineral particles reveal a “healed” crack ca. 30◦ inclined to the shortening direction [36]

Their method generates a 3D image of the sample before compression which can be compared with the same sample after brittle failure and allows the sample to be viewed from all perspectives and as many cross-sections as wished may be analyzed and compared with the results from other methods, for example, thin or polished sections.

13.3 In Situ Micro-Electro-Mechanical-Systems (MEMS) Introduction Nanostructures encompass excellent properties such as small size, low density, and high ultimate strength. It is therefore of particular relevance to accurately evaluate the mechanical properties of such nanostructures. In the testing of nanostructures, load–displacement signatures represent the materials mechanical properties, which is a major challenge to characterize in nanoscale materials. MEMS-based testing devices are one of the latest developments in quantitative measurement of nanomaterials behavior inside SEM and TEM. Irrespective of the type of application,

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force (F) is measured as the product of the stiffness (K) of the probe (usually a mechanical structure) and the measured displacement (x) of the probe (F = K∗ x). To achieve a very high force sensing resolution, one can either fabricate sensor structures with very low stiffness or employ very-high-resolution displacement sensors. However, further improvement in resolution using conventional sensing principle is limited by the intrinsic thermal noise and structural instability. In this chapter, we have reviewed some of the MEMS-based testing devices that have been employed to perform mechanical tests on nanomaterials.

13.3.1 Principle and Design of MEMS Devices Saif and Haque [37] developed one of the first-generation MEMS devices for the mechanical characterization of freestanding thin films inside a TEM. An advantage of this design was that the total force and resolution requirements could be met by proper choice of width and depth of the force sensor beam. In this design the central part of the testing chip is the freestanding thin-film specimen, which is held at one end by a force sensor beam, which allowed in-plane deflections only. The other end of the specimen is held by a set of supporting springs. Gripping of the specimen is due to the adhesion between the silicon substrate and the specimen material. Tensile force was applied on the specimen by imposing a displacement on one end of the chip, while the other end was held fixed. This displacement is transmitted to the force sensor beam by the specimen itself, causing a deflection in the beam. When the actuation takes place there is a deflection in the specimen, which in turn causes a deflection force sensor beam. Let this deflection be δ, the force on the specimen is given by F= k∗ δ. Here, k is the stiffness and is given by k=

24E I L3

(13.4)

where L is half of the total length of the force sensor beam, E is the elastic modulus of the material, and I is the moment of inertia obtained from the beam cross section. An accurate value of k can be found by using a nano-indenter. The force resolution thus depends on the spring constant of the force sensor beam, and the resolution of the measurement of δ. The displacement in D and F (Fig. 13.21) can be measured directly from the SEM or TEM observation. Readings of marker F then gives the displacement δ of the force sensor beam. The relative displacements between markers D and F gives the elongation of the specimen, which is subsequently used to compute the strain, associated with the stress value. One disadvantage of this design was related to the requirement of imaging of displacement and load sensor displacement. This could result in losing some important nanomechanism phenomena that happens while the electron beam is switched between sample and load sensor. Another disadvantage of this device was that it

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Fig. 13.21 SEM image of the device by Saif and Haque showing the moving end of the device, the U-spring for the structural stability, support beams, the force sensor beam near the fixed end, the displacement sensor D and F and the magnified image of the freestanding specimen, showing clearly the displacement markers [37]

could perform only mechanical characterizations, and no electrical characterizations could be performed. Recently, the above-mentioned device was redesigned to allow for mechanical and electrical characterizations [38]. The obtained data for a 100-nm-thick freestanding Al film with 65 nm grain size in TEM is shown in Fig. 13.22. The test revealed that nano-grained metal conductivity may increase due to thermal annealing without grain growth, possibly by reorganization of grain boundaries. Similar to

Fig. 13.22 The MEMS testing device by Han and Saif (a) Optical micrograph of the whole device, where the fixed and the pulled ends are clearly visible and the loading direction is shown. (b) Magnified view near the displacement gauges. (c) Components of the stage subjected to possible drag force due to gas flows in the inductively coupled plasma (ICP) chamber shown by the dotted arrows [38]

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Fig. 13.23 Result of uniaxial tensile test on the Al specimen that is 810 μm long, 13 μm wide, and 100 nm thick, having an average grain size of 65 nm and measured with MEMS device shown in Fig. 13.23. Initially the material is being loaded, then it is unloaded, and reloaded. The points 1, 2, and 3 show the states where the resistivity at room temperature prior to annealing, resistivity decreases, resistivity increases with cooling, respectively [38]

the previous design, the requirement of displacement imaging is the major drawback for such a design. Figure 13.23 shows the stress–strain curve which is plotted during the experiment. Initially the material is being loaded, and then it is unloaded, and reloaded. The points 1, 2, and 3 show the states where the resistivity at room temperature prior to annealing, resistivity decreases, resistivity increases with cooling, respectively. To overcome the force–displacement imaging requirements, Espinosa and Zhu [39] described the design, fabrication, and operation of a MEMS-based material testing system that was capable of electronic force measurements. This device offered the possibility of continuous observation of the specimen deformation and fracture with sub-nanometer resolution, while simultaneously measuring the applied load electronically with nano-Newton resolution. Figure 13.24 shows the device structure where the presence of the backside window is very important as it makes the specimen freestanding and also allows the electron beam to pass. The device is designed based on electrothermal actuation (ETA), which means the thermal expansion of freestanding beams when subjected to Joule heating. As a result of the voltage applied across the inclined beams (V-shaped beams), the actuator shuttle moves forward. The overall device performance was demonstrated by testing multi-walled carbon nanotubes (MWCNTs) (Fig. 13.25a). Specimens were loaded incrementally until failure; the failure regions were then further analyzed at high magnifications. Direct observation of the TEM image does not reveal the number of failed shells, but this can be obtained using image-processing techniques (Figs. 13.25b and c). This device achieved resolutions of 0.05 fF in capacitance, 1 nm in displacement, and 12 nN in load.

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Specimen Folded beams

Thermal actuator

Load sensor

200 µm

Fig. 13.24 The MEMS-based testing device developed by Zhu and Espinosa for mechanical testing of nanoscale materials. The device has three main parts: capacitive load sensor, thermal actuator, and a gap for the placement of the specimen. The specimen is kept freestanding with the help of a backside window. The thermal actuator also has the heat-sink beams which restrict the heat from reaching the specimen [39]

a

b

A

Fracture

B

Distance along path A (nm)

Distance along path B (nm) Fig. 13.25 (a) TEM image of MWCNT after fracture. (b) Paths A and B were used to create intensity profiles on either side of the fracture to verify that only a single shell failed [39]

13.3.2 Application of MEMS Devices for Materials Modeling Lu et al. [40] developed a new analytical model for interpreting tensile loading data on “templated carbon nanotubes” (T-CNTs) obtained with a MEMS-based mechanical testing stage. It has been found that the force output from the actuation unit

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depends on the stiffness of the force sensing beam and the nanostructure specimen, as well as the power input. The force used to define the stiffness of the actuation unit (Kd ) is its output force applied to the measurement unit, while the corresponding displacement is the displacement of this unit at point A (Fig. 13.26 (a)) caused by this force only (along the axis of the opposing platforms). Ks is stiffness of the specimen (T-CNT).

Fig. 13.26 (a) Schematics of the MEMS device developed by Lu et al. [57] and (b) an equivalent spring system. Kd is the stiffness of the actuation unit, and K∗ = Kb (without T-CNT attached) or K∗ = Kb + Ks (with T-CNT attached). In illustration (a), the nanostructure is not loaded [40]

The displacement x, used for defining the stiffness of the measurement unit, is the horizontal displacement at the midpoint of the force sensing beam (on the moving platform) which is equivalent to the actual displacement x of point A that they measured. Obviously, the corresponding force for the measurement unit is equal to the value of the output force of the actuation unit. If the two springs are hypothetically separated, the distance between the two resulting relaxed springs is δ0 , which depends on the magnitude of the input power. When the device is actuated and no nanostructures are loaded, as in Fig. 13.26b, the actuation unit is loaded with a force from the force sensing beam only. The force output from the actuation device F1 is F1 = K d (δ0 − x1 ) = K b x1

(13.5)

where x1 is the deformation of the force-sensing beam. The stiffness of the actuation device Kd and the force sensing beam Kb can be calculated by using the finite element analysis software ABAQUS, with the actual dimensions of the device as measured in the SEM. The experimentally measured displacement of the moving platform with and without the T-CNT attached is shown in Fig. 13.27.

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Fig. 13.27 Displacement of the moving platform when a T-CNT is, and is not, loaded [40]

Aebersold et al. [41] presented the testing of a MEMS-based capacitive bending strain sensor utilizing a comb drive. A commercial finite element software package was used to predict sensor actuation, capacitance output, and avoid material failure. Actuation or displacement of the interdigitated fingers was used to calculate capacitance. While undergoing bending, the sensor will exhibit a parabolic increase in capacitance due to its transverse actuation. The capacitance relationship for a parallel-plate system is given by C=

ε0 εr A d

(13.6)

where C is the generated capacitance in farads (F), ε0 is the dielectric constant of free space equal to 8.85 × 10−14 F cm−1 , εr is second dielectric constant and A is overlapping area between the two plates. Figure 13.28 shows the change in capacitance from the FE model and experimental data. Luo et al. [42] used a mechanical FEA program to determine the Poisson’s ratio by knowing Young’s modulus and measuring the deflection at an arbitrary point on a thin film. By simplifying the differential equation of the deflection surface, the Poisson’s ratio ν can be expressed as * v=

1−

4 ∂ w(x, y) Eh 3 ∂ 4 w(x, y) ∂ 4 w(x, y) + 2 + 12q(x, y) ∂x4 ∂ x 2∂ y2 ∂ y4

(13.7)

where x and y are the coordinates of any arbitrary point on the plate, w(x, y) represents the deflection at the point, q(x, y) is the applied pressure at the selected point, h denotes the plate thickness, and E and ν represent Young’s modulus and Poisson’s ratio, respectively. Using this method, Poisson’s ratio can be determined by various experiments. One experimental example is shown in Fig. 13.29. A thin film was

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18.00

Capacitance (pF)

16.00

14.00

12.00

10.00 interdigitated fingers touch 8.00

6.00 0

200

400

600

800 1000 Strain (μe)

FEA Model

1200

1400

1600

1800

Experimental

Fig. 13.28 Graphical results of the entire capacitance range from the FE model and experimental testing [41]

Fig. 13.29 Schematic of an experimental example which can be used to determine Poisson’s ratio with the aid of the method [42]

deposited on the front side of a (100) silicon wafer. A cavity was etched through the wafer from the backside such that the film above the cavity suffered no constraint from underneath. A small force or pressure was used to deflect the film. Also, in the case that Poisson’s ratio is known and Young’s modulus is unknown, the method may be applied to find the Young’s modulus.

13.4 Concluding Remarks The new advances in experimental techniques have enabled scientists to explore materials behavior from a new prospective. In this chapter a number of widely used

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and newly developed in situ techniques were explained from the fundamental basics up to applied sciences. Atomic force microscopy, micro-electro-mechanical devices for in situ microscopy, and X-ray ultra microscopy show great potential for future applications in materials modeling. It is crucial for material modelers and simulation experts to become familiar with the new advances in experimental techniques. Combination of multiscale modeling and experimental validations in subatomic levels up to macro is the future roadmap for virtual testing paradigm.

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20. H. Zhang, J. Tang, L. Zhang, B. An, and L.-C. Qin, “Atomic Force Microscopy Measurement of the Young’s Modulus and Hardness of Single LaB6 Nanowires”, Appl. Phy. Lett., Vol. 92, 2008, p. 173 21. H. Ni, X. Li, and H. Gao, “Elastic Modulus of Amorphous SiO2 Nanowires”, Appl. Phy. Lett., Vol. 88, 2006, p. 43 22. P. Guaino, M. Gillet, R. Delamare, and E. Gillet, “Modification of Electrical Properties of Tungsten Oxide Nanorods Using Conductive Atomic Force Microscopy”, Sur. Sci., Vol. 601, 2007, p. 2684 23. E.Z. Luo, A.B. Pakhomov, Z.-Q. Zhang, M.-C. Chan, I.H. Wilson, J.B. Xu, and X. Yan, “Conductance Distribution in Granular Metal Films: A Combined Study by Conducting Atomic Force Microscopy and Computer Simulation”, Phys., Vol. 279, 2000, p. 98 24. D.J. M¨uller and K. Anderson, “Biomolecular Imaging Using Atomic Force Microscopy”, Tren. Biotech., Vol. 20, 2002, p. 8 25. K.D. Jandt, “Atomic Force Microscopy of Biomaterials Surfaces and Interfaces”, Surf. Sci., Vol. 491, 2001, p. 303 26. D. Mulliah, S.D. Kenny, R. Smith, and C.F. Sanz-Navarro, “Molecular Dynamic Simulations of Nanoscratching of Silver (100)”, Nanotechnology Vol. 15, 2004, p. 243 27. Y. Yan, T. Sun, S. Dong, and Y. Liang, “Study on Effects of the Feed on AFM-Based NanoScratching Process Using MD Simulation”, Comp. Mat. Sci., Vol. 40, 2007, p. 1 28. Y. Isono and T. Tanaka, “Molecular Dynamics Simulations of Atomic Scale Indentation and Cutting Process with Atomic Force Microscope”, JSME Int. J. A, Vol. 40, 1997, p. 211 29. Y.D. Yan, T. Sun, S. Dong, X.C. Luo, and Y.C. Liang, “Molecular Dynamics Simulation of Processing Using AFM Pin Tool”, App. Sur. Sci., Vol. 252, 2006, p. 7523 30. C. Walter, T. Antretter, R. Daniel, and C. Mitterer, “Finite Element Simulation of the Effect of Surface Roughness on Nanoindentaion of Thin Films with Spherical Indenters”, Sur. Coat. Tech., Vol. 202, 2007, p. 1103 31. S.C. Mayo, P.R. Miller, S.W. Wilkins, T.J. Davis, D. Gao, and T.E. Gureyev, “Quantitative X-ray Projection Microscopy: Phase-Contrast and Multi-spectral Imaging”, J. Micr., Vol. 207, 2002, p. 79 32. S.C. Mayo, T.J. Davis, T.E. Gureyev, P.R. Miller, D. Paganin, and A. Pogany, “X-ray PhaseContrast Microscopy and Microtomography”, Opt. Exp., Vol. 11, 2003, p. 2289 33. D. Wu, D. Gao, S.C. Mayo, J. Gotama, and C. Way, “X-ray Ultramicroscopy: A New Method for Observation and Measurement of Filler Dispersion in Thermoplastic Composites”, Comp. Sci. Tech., Vol. 68, 2008, p. 178 34. W.H. Liu, X.M. Zhang, J.G. Tanga, and Y.X. Du, “Simulation of Void Growth and Coalescence Behavior with 3D Crystal Plasticity Theory”, Com. Mat. Sci., Vol. 130, 2007 35. G.P. Potirniche, M.F. Horstemeyer, G.J. Wagner, and P.M. Gullett, “A Molecular Dynamics Study of Volid Growth and Coalescence in Single Crystal Nickel”, Int. J. Pla., Vol. 22, 2006, p. 257 36. S. Zablera, A. Racka, I. Mankea, K. Thermannb, J. Tiedemannb, N. Harthillc, and H. Riesemeier, “High-Resolution Tomography of Cracks, Porosity and Microstructure in Greywacke and Limestone”, J. Str. Geo., Vol. 30, 2008, p. 876 37. M.A. Haque and T. Saif, “A Novel Technique for Tensile Testing of Submicron Scale Freestanding Specimens in SEM and TEM”, Exp. Mech., Vol. 42, 2002, p. 123 38. J.H. Han and M.T.A. Saif, “In Situ Microtensile Stage for Electromechanical Characterization of Nanoscale Freestanding films”, Rev. Sci. Ins., Vol. 77, 2006, p. 045102 39. B. Peng, M. Locascio, P. Zapol, S. Li, S.L. Mielke, G.C. Schatz, and H.D. Espinosa “Measurements of Near-Ultimate Strength for Multiwalled Carbon Nanotubes and Irradiation-Induced Cross Linking Improvements”, Nat. Nanotech., Vol. 3, 2008, pp. 626–631 40. S. Lu, Z. Guo, W. Ding, and R.S. Ruoff, “Analysis of a Microelectromechanical System Testing Stage for Tensile Loading of Nanostructures”, Rev. Sci. Ins., Vol. 77, 2006, p. 056103

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Index

A Absorbed energy, 5 Accelerated region, 6–7, 24, 47, 62 Aerospace industry, virtual testing in, 12–13 fracture allowables of Inconel 718 of COPV liner, 16–20 generating fracture allowables of 6061-T6 aluminum tank, 20–22 manufacturing process and plastic deformation of COPV liner, 14–15 Aerospace steels, 29–30 AFGROW, 30–32, 41 AFM, see Atomic force microscopy (AFM) Aluminums, threshold values, 10 Analytical continuum modeling, 268 Atomic force microscopy (AFM), 372 application, 375 biological structures, characterization, 378–379 electrical properties, characterization, 378 nanomechanical and nanoindentation testing, 376–378 surface microstructures, characterization, 375–376 modeling and simulation, 379–382 operation, 374–375 principles of, 372–374 Atomistic and atomistic-like models, 292–296 Atomistic ensemble methods, 275–278 equivalent-continuum method (ECM), 277–278 mesoscale approaches, 276–277 Monte Carlo (MC) approaches, 275–276 Atomistic modeling, 293–296, 299, 303–306 Auto-frettage process, 15, 16–17

B Bead-spring model, 294–295, 297, 298–299, 307 Bi-disperse melt, 314 Bilinear cohesive law, 54–56 Blobs, 307–314 Bolted joint, 322, 326, 327–328, 330, 339–345, 345–350 Boundary condition, 223, 228–232, 239 Boundary correction factor, 42–43 Bridge formation, 302 Bridging domain, 234 Brown map, 82 Building-block approach, 148–149 Building-block verification, computational process for, 153–154 calibration of composite constitutive properties, 158–159 composite material validation, 159–161 micro- and macro-composite mechanics analysis, 155–157 MUA, 161–163 multiple failure criteria, 154–155 progressive failure micro-mechanical analysis, 157 C Calibration of composite constitutive properties, 158–159 of rheological constitutive models, 299–306 constitutive model, formulation, 301–303 importing information from atomistic models, 303–306 system, 299–301 Carbon nanotube, 322–323, 325, 326, 327, 331, 350–352

399

400 Certification by analysis, 171–182 C∗ integral, 115 Clausius–Duhem inequality, 223 Coarse grained model, 293–295, 297, 298–299, 306–317 Cohesive failure methodology, 53 Cohesive law, 54 Cohesive modeling technique, 51–53 bilinear cohesive law, 54–56 cohesive model, 56–58 finite element implementation, 59–61 reversible cohesive model, 53–54 See also Cohesive models threshold behavior, incorporation of, 58–59 Cohesive models, 47, 50–51, 56–58 Cohesive volumetric finite element (CVFE), 59–60 Composite, 321–353 deck under fire, simulation of, 147–148 Composite material qualification certification by analysis example, 171–179 computational process for implementing building-block verification, 153–154 calibration of composite constitutive properties, 158–159 composite material validation, 159–161 micro- and macro-composite mechanics analysis, 155–157 MUA, 161–163 multiple failure criteria, 154–155 progressive failure micro-mechanical analysis, 157 damage categories composite deck under fire, simulation of, 147–148 damage detected by field inspection methods, 141–142 DSD, 142–147 severe damage created by anomalous ground, 147 undetected by field inspection methods, 139–141 establish A- and B-basis allowables, 163–164 allowable generation for unnotched and notched composite specimens, 166–171 combining limited test data with progressive failure and probabilistic analysis, 164–166 FAA building-block approach, 148–151

Index FAA durability and damage tolerance certification strategy, 138–139 test reduction process, 151 deterministic analysis, 152 material and manufacturing development, 151 probabilistic analysis, 152 unnotched static and fatigue testing, 151 Composite material validation, 159–161 Composite over-wrapped pressure vessels (COPV), 14–15 liner Inconel 718 of, fracture allowables, 16–20 manufacturing process and plastic deformation of, 14–15 Composites, 255–258 Composite structures virtual testing, 155–157 Computational continuum modeling, 268–269 Computational model, 324–329, 335–337, 339–340, 351 nanocomposite materials, 221–222 Computational process for implementing building-block verification, 153–154 calibration of composite constitutive properties, 158–159 composite material validation, 159–161 micro- and macro-composite mechanics analysis, 155–157 MUA, 161–163 multiple failure criteria, 154–155 progressive failure micro-mechanical analysis, 157 CONFFESSIT, 296 Constitutive law, 297 Constraint, 75 and constraint release, 311–313 crack-tip constraint effects, 127–128, 132 critical constraint effect, 90 deformation, 23 ECM, 140 factor, 43 release Rouse modes, 297, 309, 310, 311–313 Consumed energy, 2–3 Continuous medium, 189 Continuum mechanics, 223–224, 226, 234 Continuum methods analytical continuum modeling, 268 computational continuum modeling, 268–269

Index predicting material properties from the top-down approach, 267 Continuum models, 297–298 COPV, see Composite over-wrapped pressure vessels (COPV) Coulomb’s law for friction, 250 Crack growth, 32, 80, 117, 187 D6ac steel, 35 Mil-annealed Ti–6AL–4V, 39 rate, 187, 210–215 See also Fatigue crack growth Crack initiation, 15, 48, 58, 110, 116, 132–133, 149 creep, 106, 132–133 Crack opening stress, 41 Crack tip angle in NSW model, 125–126 Creep complex stress, 110–111 and fatigue creep parameter C∗ integral, 114–115 fracture mechanics parameters in, 113–114 physical models describing, 108–110 uncracked bodies analysis of, 107–108 influence of fatigue in, 112 Crystalline and highly ordered material systems, 233–234 Crystal structure, 233–234, 237 Cyclic crack growth, 75 See also Crack growth D D6ac steel, 30–35 Damage accumulation at the crack tip, 128–130 categories composite deck under fire, simulation of, 147–148 damage detected by field inspection methods, 141–142 DSD, 142–147 severe damage created by anomalous ground, 147 undetected by field inspection methods, 139–141 detected by field inspection methods, 141–142 mechanics, 51, 328, 332 propagation, 321–353 undetected by field inspection methods, 139–141 Damage tolerance, 29–31

401 equivalent block method for predicting fatigue crack growth, 32–33 fatigue crack growth under variable amplitude loading, 33–36 in an F/A-18 aircraft bulkhead, 36–38 Mil Annealed Ti–6AL–4V under a fighter spectrum, 38–40 virtual engineering approach for predicting the S–N curves for 7050-T7451, 40–41 endurance limit, 41 Defence Science and Technology Organisation (DSTO), 35 Depressurization, 15 3D F-111 model, 34 Diffusion coefficient, 314, 315, 316 Discrete, 49–50, 142–147, 149, 191, 224, 227, 234, 296, 303, 304, 311–312 Discrete source damage (DSD), 142–147 residual strength, 146–147 Dissipated energy, 3, 187, 190, 191, 217, 259 Dissipation, 3, 190, 191, 217, 258–259, 262, 274, 276, 306, 310 DSD, see Discrete source damage (DSD) Dynamic deformation methods, 236 E ECM, see Equivalent-continuum method (ECM) Elasto plastic fracture mechanics (EPFM), 80 Elevated temperature cyclic crack growth, 130–131 Elongation, 2, 12, 17, 95, 389 Energy balance equation, Griffith theory, 5 Energy density, 134, 188, 189–190, 193, 232 factor, 188–189 Energy transformation, 84–85, 194, 195, 223 Entanglement length, 296 Equation of motion, 311 Equivalent block method for predicting fatigue crack growth, 32–33 Equivalent continuum, 224–226, 227–230 Equivalent-continuum method (ECM), 277–278 Equivalent-continuum models, 224 equivalence of averaged scalar fields, 231–232 equivalent continuum, 227–231 kinematic equivalence, 232–233 representative volume element, 224–227 strategies, 233 crystalline and highly ordered material systems, 233–234

402 Equivalent-continuum models (cont.) dynamic deformation methods, 236 fluctuation methods, 234–235 static deformation methods, 235–236 Extrusions, 75 F F-111, 33, 34, 36 F/A-18, 29, 36–38 FAA building-block approach, 148–151 FAA durability and damage tolerance certification strategy, 138–139 Failure, 321–353 Farahmand’s theory to fatigue crack growth rate data, 6 accelerated region and fracture toughness, 6–7 da/dN Versus ΔK, 9–12 Paris constants, 7–8 threshold value, 8–9 FASTRAN, 30–32, 37–38, 41 FASTRAN II, 37–38, 41 Fatigue crack propagation, prediction of, 50–51 stage I, 78–84 stage II, 77–78 stage III, 84–85 threshold, effect of load ratio R on, 68–69 Fatigue crack growth equivalent block method for predicting, 32–33 rate, 2, 6–12, 13, 17, 21–22, 194–198, 208, 210 under variable amplitude loading, 33–36 in an F/A-18 aircraft bulkhead, 36–38 Mil Annealed Ti–6AL–4V under a fighter spectrum, 38–40 Fatigue damage, 73–74 cracks and damage stages, 74–77 Navarro–de los Rios (N-R) model, 85–89 fatigue damage map, 89–100 stage I fatigue cracking, 78–84 stage II fatigue cracking, 77–78 stage III fatigue cracking, 84–85 See also Fatigue crack Fatigue failure, 47–49 cohesive modeling technique, 51–53 bilinear cohesive law, 54–56 cohesive model, 56–58 finite element implementation, 59–61 reversible cohesive model, 53–54 threshold behavior, incorporation of, 58–59

Index fatigue crack propagation, prediction of, 50–51 simulation fatigue crack threshold, effect of load ratio R on, 68–69 Paris curve simulation, 61–65 threshold limit, effect on, 66–67 threshold limit of fatigue crack growth, prediction of, 65–66 steps to, 49 threshold fatigue crack behavior, prediction of, 49–50 Fatigue modelling, 51, 105, 132 FE non-linear equation system, 328 Finite elements, 12, 35, 50, 54, 59–61, 127–128, 148, 155–156, 173, 175, 224, 233–234, 262, 263, 270, 394 Finite element implementation, 59–61 First law of thermodynamics, 223 Flaw IdentificatioN through the Application of Loads (FINAL), 30, 36 Fluctuation dissipation theorem, 310 Fluctuation methods, 234–235 Force field, 274 Forman–Newman fatigue crack growth equation, 6 Form invariant, 193 Fracture-critical hardware, 3 Fracture mechanics, 2, 3, 5, 13, 15, 18, 77, 106, 107 Fracture toughness, 2, 13, 77–78 accelerated region and, 6–7 Griffith theory and, 3–6 virtual testing theory and, 2–3 Friction stir welding (FSW), 14 Frost–Dugdale crack growth law, 32, 33, 34–35, 38 FSW, see Friction stir welding (FSW) Functionalization, 358, 361, 362, 368 G Griffith theory, extended, 2, 16, 23 fracture toughness and, 3–6 H High-temperature fracture mechanics, 116 crack tip angle in NSW model, 125–126 damage accumulation at the crack tip, 128–130 elevated temperature cyclic crack growth, 130–131 finite element framework, 127–128

Index initiation times prior onset of steady creep crack growth, 124–125 K and C∗ , 116–117 CCG correlation with, 117–118 NSW-MOD Model, 126–127 steady-state creep crack growth rate, 118–121 transient creep crack growth model, 121–124 Hook’s law, 238, 250, 268, 273, 372 Hybrid (multiscale) simulations, 265 I Inconel 718 of COPV liner, fracture allowables of, 16–19 Informatics, 249, 250, 263, 266, 268, 270 Information passing models, 293, 298–317 Initiation times prior onset of steady creep crack growth, 124–125 In situ micro-electro-mechanical-systems (MEMS), 388–389 application of, 392–395 principle and design, 389–392 In situ microscopy, 388–395 Interface region, 258–259 functionalization of, 259–262 Intrusions, 75 J Joint Strike Fighter (F-35), 29 K K and C∗ , 116–117 CCG correlation with, 117–118 Kinematic equivalence, 232–233 Kinemeticor Dirichlet boundary conditions, 228–229 Kk, 49 L Langevin equation, 309–310 Law of friction, 191 Life assessment methodologies for cracked components, 105–106 codes, 106–107 creep analysis of uncracked bodies, 107–108 Life prediction, 106, 107, 131, 138, 147 Linear elastic fracture mechanics (LEFM), 51–52, 77, 77, 78 equation, 83 Local nonuniform strainability, 3 Lodge–Meissner law, 317

403 M Macro-scale, 248, 263, 269 Macro (transitions), 187–217 Material A-B base allowable and qualification, 163–171, 182 Material degradation, 15 Materials science, 253, 263–264 Material uncertainty analyzer (MUA), 161–163 Maximum expected operating pressure (MEOP), 20 Mean square displacement, 310, 314, 315 Mechanical, 188, 189, 192, 196, 214–215 MEMS, 388–395 Meso-scale, 250, 257, 259, 266, 269, 270, 276–277 Mesoscale approaches, 276–277 Micro- and macro-composite mechanics analysis, 155–157 Micromechanics, 224, 239 Microstructural fracture mechanics (MFM), 80 Microstructure degradation, 190, 194–195 Micro (transitions), 187–217 Mil-Annealed Ti–6AL–4V, 39 Miner’s cumulative damage law, 112 Modeling, predictive, 247–280 Model validation, 379–382 Molecular dynamics (MD), 223, 233–234, 254, 259–260, 262, 274–275, 294, 298, 307, 315 Molecular models, 296–297 Molecular scale, 272–274 Monte Carlo (MC) approaches, 275–276 MUA, see Material uncertainty analyzer (MUA) Multi-length and time scales bottom-up approach, 269–271 molecular dynamics (MD), 274–275 molecular scale, 272–274 quantum scale, 271–272 Multiple failure criteria, 154–155 Multiscale, 23, 51, 187–217, 221–243, 253, 258, 262, 264, 266–267, 291–317, 324–329, 371–396 modeling, 221–243 Multiscale (computational models), 324–329 Multiscale fatigue crack growth, 187–190 dual-scale fatigue crack growth rate models, 194–196 micro/macro formulation, 196–197 nano/micro formulation, 197–198

404 Multiscale fatigue crack growth (cont.) fatigue crack growth and velocity data, 208–209 predicted micro/macro results, 209–210 predicted nano/micro results, 210–212 invariant of two-parameter crack growth relation, 193–194 micro/macro time-dependent physical parameters, 198 macroscopic material properties, 198–201 microscopic material properties, 201–204 multiscaling and future considerations, implication of, 214–217 nano/micro/macro fatigue crack growth behavior, 212–214 nano/micro time-dependent physical parameters nanoscopic fatigue crack growth coefficient, 207–208 nanoscopic material properties, 205–206 scale implications, 190 physical laws change with size and time, 190–191 strength and toughness, 192–193 surface-to-volume ratio as a controlling parameter, 191–192 Multiscale modeling, 23, 51, 266–267, 371 atomic force microscopy (AFM), 372 application of, 375–379 modeling and simulation, 379–382 operation, 374–375 principles of, 372–374 in situ micro-electro-mechanical-systems (MEMS), 388–389 application of, 392–395 principle and design, 389–392 X-ray ultra microscopy (XuM) 3D imaging and multiscale modeling applications, 385–388 phase contrast and absorption contrast, 384–385 principles of, 382–384 Multi-scale progressive failure analysis, 138, 154–155, 158 Multiscale simulation, 250–251, 253, 263, 264–265, 272, 278 N Nanocomposite materials, 221–243, 251–253, 325, 350–352

Index composites, 255–258 computational modeling tools, 223–224 equivalent-continuum modeling strategies, 233 crystalline and highly ordered material systems, 233–234 dynamic deformation methods, 236 fluctuation methods, 234–235 static deformation methods, 235–236 equivalent-continuum models, 224 equivalence of averaged scalar fields, 231–232 equivalent continuum, 227–231 kinematic equivalence, 232–233 representative volume element, 224–227 functionalization of interface region, 258–262 interface region, 258–259 method developments, 265–266 modeling approaches, 262–265 nanotechnology and modeling, 253–255 nanotube/polymer composites, 238–242 silica nanoparticle/polymer composites, 236–238 Nanoparticles, 236–238 Nano (transitions), 187–217 Nanotube/polymer composites, 238–242 Nanotubes, 233–234, 238–242 NASGRO, 6, 7, 9, 22, 30, 41, 62, 67 Navarro–de los Rios (N-R) model, 85–89 fatigue damage map, 89–100 Newman closure equation, 8 Newton’s second law, 274 Nonequilibrium, 191, 275, 294, 298, 302, 312, 315 Non-similitude-based crack growth law, 32 Non-uniform plastic deformation, 3–4 Normal stress coefficient, 316–317 Norton creep law, 117–118 NSW model, 118–119, 120, 123, 127, 128, 132 crack tip angle in, 125–126 NSW-MOD model, 123, 126–127 O Oblique angle, 358–361 ONETEP method, 266 P Paris constants, 7–8 Paris curve simulation, 61–65 Paris law, 49, 51, 130–131

Index Paris regime, 75 Paris region, 7–8, 30, 32, 40, 47, 49 Parylene, 358–359 Pearl necklace model, 294 Performance, 248–250, 252–254, 257, 260, 262, 263, 271, 278 Plane stress and strain fracture toughness, 8–9, 16–17 Plastic deformation, 2–4, 13–17, 18, 20–22, 85, 110, 130 crack tip for, energy consumed at, 3 cyclic, 56, 75 height of, 5 Plastic strain, true, 4 Polymer dynamics, 296, 298 Polymer-filler attachment, 301, 304–305 interaction, 300 Polymeric melts, 291–293 coarse-grained models of, 313–316 constraints and constraint release, 311–313 dynamics, 309–311 mapping to coarse phase space, 307–309 system, 306–307 modeling, 306–317 single and multiscale modeling methods atomistic and atomistic-like models, 293–296 continuum models, 297–298 molecular models, 296–297 two information-passing examples, 298 calibration of rheological constitutive models, 299–306 coarse-grained models, 306–317 general strategy, 298–299 Polymer-metal interface, 356–368 Polymers, 236–242 Power law, 77, 304, 360, 366–368 Power-law scaling, 360 Predicting material properties from the top-down approach, 267 Predictive modeling, 247–251 atomistic ensemble methods bulk-scale approach, 278 ECM, 277–278 mesoscale approaches, 276–277 Monte Carlo (MC) approaches, 275–276 continuum methods analytical continuum modeling, 268

405 computational continuum modeling, 268–269 predicting material properties from the top-down approach, 267 multi-length and time scales bottom-up approach, 269–271 MD, 274–275 molecular scale, 272–274 quantum scale, 271–272 multiscale modeling, 266–268 nanocomposites, 251–253 composites, 255–258 functionalization of interface region, 259–262 interface region, 258–259 method developments, 265–266 modeling approaches, 262–265 nanotechnology and modeling, 253–255 Pressure vessel, 14 Primitive path, 296 Probabilistic analysis, 138, 152–153, 158, 164–166, 170, 171 Progressive damage model, 322–324 Progressive failure micro-mechanical analysis, 157 Properties, 247–248, 250, 251–254, 255, 257–258, 261–263, 268, 269–271, 273, 277, 280 Q QIN (HY80) steel, 30–31 QMERA, 266 Quantum scale, 271–272 Quantum simulation, 257, 264, 268, 269, 271–272 R Ramberg–Osgood equation, 4 Release Rouse modes, 297, 310, 311–313 Repair patch, 330, 345–348 Representative volume element, 224–227 Residual strength, 141–143 DSD, 146–147 Reversible cohesive model, 53–54 Rheological constitutive equation, 297 Risk assessment, 137–138, 139 Rouse modes, 297, 309, 310, 311–313 R ratio, 31, 34–35, 38, 41, 91 S Scalar fields, equivalence of averaged, 231–232 Scanning force microscope (SFM), 373 Schr¨odinger’s equation, 271 Second law of thermodynamics, 223

406 Segmentation, 188, 205, 216–217, 259, 268, 270, 292, 294–295, 300, 301–305, 307–308, 312–313, 373 Semi-empirical quantum mechanical methods, 272 Severe damage created by anomalous ground, 147 Shear viscosity, 292–293, 316–317 Short fatigue cracks (SFC), 78–84 SIF, see Stress intensity factor (SIF) Silica nanoparticle/polymer composites, 236–238 Similitude, 30, 31, 32, 33 Single and multiscale modeling methods atomistic and atomistic-like models, 293–296 continuum models, 297–298 molecular models, 296–297 Single formalism approach, 265 Size, 189, 190–193 SNND lattice Monte Carlo, 294, 295–296 Spectra-dependent constant, 32 Spring constants, 273 Static deformation methods, 235–236 Steady-state creep crack growth rate, 118–121 Storage and loss moduli (dynamic moduli), 305 Stress analysis, 324–329, 335 Stress intensity factor (SIF), 2, 6, 7–9, 23, 32, 35, 36, 52, 56, 58, 59, 65, 73, 75, 77 Stress state, 111, 114, 116, 119, 120, 126, 130, 298, 324, 387 tri-axial, 85 Stress-strain curve, 2–3, 5, 6, 7, 16–17, 20, 21, 23, 149, 387, 391 of armchair-type CNT-reinforced composite, 352 ASTM standard tests, 158 nanocomposite system, 277 one/two-void specimens, 352 Structural analysis, 325 Structural behaviour prediction, 322 Structure, 248, 250–251, 253–257, 263–264, 267–269, 270, 272, 275 Super Hornet, 29 T 7050-T7451 aluminium, 29, 40–42 2219-T6 aluminum, 7 6061-T6 aluminum, 7 tank, generating fracture allowables of, 19–22 Tertiary region, 107

Index Test reduction process, 151 deterministic analysis, 152 material and manufacturing development, 151 material property characterization analysis, 152 material property uncertainty analysis, 152 probabilistic analysis, 152–153 unnotched static and fatigue testing, 151 Theory, 250, 257, 258, 264–267, 272, 277–278, 279 Thermal fluctuation, 191, 234–235 Thermal motion, 235–236 Thin films, 359 Threshold behavior, incorporation of, 58–59 Threshold fatigue crack behavior, prediction of, 49–50 Threshold limit effect on, 66–67 of fatigue crack growth, prediction of, 65–66 Threshold region, 8, 22 Threshold values, aluminums, 10 Ti–6AL–4V steel, 32 Ti–6AL–4V titanium, 42 Time, 190–191, 198–208 Time Boltzmann inversion, 308 Titanium, 30 Traction–separation law, see Cohesive law Traction under cohesive scheme, 59 Transient creep crack growth model, 121–124 True plastic strain, 4 Tsai–Wu failure equation, 329 Two information-passings calibration of rheological constitutive models, 299–306 developing coarse-grained models of polymeric melts, 306–317 general strategy, 298–299 U Uniform plastic deformation, 2 Uniform strainability, 3 Unrecoverable energy, 4 V Vapor deposition, 358, 359, 367 Virtual cracking, 130 Virtual engineering approach for predicting the S–N curves for 7050-T7451, 40–41 endurance limit, 41 Virtual testing theory and fracture toughness, 2–3 Volume/surface ratio, 191–192

Index W Welding technique, 14 Willenborg crack growth law, 30–31 Work hardening, 13, 15, 16–18, 20–22, 107, 109 X X-ray microscopy, 382–388 X-ray ultra microscopy (XuM)

407 3D imaging and multiscale modeling applications, 385–388 phase contrast and absorption contrast, 384–385 principles of, 382–384 Y Young’s modulus, 56, 62, 128, 224, 230, 238, 240–241, 248, 377–378, 394, 395