Atomistic Modeling of Materials Failure

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Author: Markus J. Buehler

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Atomistic Modeling of Materials Failure

Markus J. Buehler


123

Markus J. Buehler Laboratory for Atomistic and Molecular Mechanics Department of Civil and Environmental Engineering Massachusetts Institute of Technology 77 Massachusetts Avenue, Room 1-235A&B Cambridge, MA 02139 USA [email protected]

ISBN 978-0-387-76425-2 e-ISBN 978-0-387-76426-9 DOI: 10.1007/978-0-387-76426-9 Library of Congress Control Number: 2008927204 c 2008 Springer Science+Business Media, LLC 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.com

To my wife, for inspiration and loving support

Preface

This book has evolved from lecture notes of undergraduate and graduate level subjects as well as review articles and journal papers. The book provides a review of atomistic modeling techniques that successfully link atomistic and continuum mechanical methods. It intended to be a reference for engineers, materials scientists, and researchers in academia and industry. The writing of this book was motivated by the desire to develop a coherent set of notes that provides an introduction and an overview into the field of atomistic-based computational solid mechanics, with a focus on fracture and size effects. The book covers computational methods and techniques operating at the atomic scale, and describes how these techniques can be used to model the dynamics of cracks and other deformation mechanisms. A description of molecular dynamics as a numerical modeling tool covers the use of interatomic potentials (pair potentials such as the Lennard-Jones model, embedded atom method (EAM), bond order potentials such as Tersoff’s and Brenner’s force fields, as well as the first principles based ReaxFF Reactive force field) in addition to the general philosophies of model building, simulation, interpretation, and analysis of simulation results. Example applications for specific materials (such as silicon, nickel, copper, carbon nanotubes) are provided as case studies for each of the techniques, areas, and problems discussed. Readers will find a physics-motivated discussion of the numerical techniques along with a review of mathematical concepts and code implementation issues. Using specific examples such as investigations of crack dynamics in brittle materials or deformation mechanics of nanomaterials, this volume conveys how atomistic studies have helped to advance developing new theories, or provided insight into the molecular deformation mechanisms, explaining or supplementing experimental results. Many of the examples are adapted from studies carried out by the author of this book, and some of the discussion should therefore not be considered as a comprehensive and inclusive review with respect to the wider range of available results. Rather, they represent a set of specific examples to illustrate the application of the atomistic simulation techniques reviewed here.

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Preface

Completing this book would not have been possible without the help and support of numerous people. The author is most greatly indebted to all who have contributed to this book in some way. In particular, sincere gratitude goes to those individuals from whom he had the opportunity to learn from over the years, in particular his graduate advisor Huajian Gao and postdoctoral advisor William A. Goddard III. The author is deeply humbled by the many contributions that have pioneered the development of this research field over the past decades. The author would also like to thank the Editor Mrs. Elaine Tham of Springer and her staff for the continuous support for this project. The efforts by the reviewers of the manuscript are greatly acknowledged, as they provided valuable suggestions for revisions in the final manuscript. The study of materials failure using atomistic simulation has been a rewarding journey that continues to bring so much joy, excitement, and inspiration. The author hopes to convey some of the excitement about this research field in this book.

Cambridge, MA July 14, 2008

Markus J. Buehler

Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LXIII Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LXV Part I Introduction 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Materials Deformation and Fracture Phenomena: Why and How Things Break . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Strength of Materials: Flaws, Defects, and a Perfect Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Crystal Structures and Molecular Packing . . . . . . . . . . 1.2.2 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Other Defects in Crystals and Other Structures . . . . . 1.3 Brittle vs. Ductile Material Behavior . . . . . . . . . . . . . . . . . . . . . 1.4 The Need for Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . 1.5 Applications: Experimental and Computational Mechanics . . 1.5.1 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Example Applications: The Significance of Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Outline of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 6 8 10 10 12 12 15 18 18 20 27

Part II Basics of Atomistic, Continuum and Multiscale Methods 2

Basic Atomistic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Model Building and Physical Representation . . . . . . . .

31 31 32 34

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2.2.2 The Concept of Computational Experiments . . . . . . . . 2.3 Basic Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Formulation of Classical Molecular Dynamics . . . . . . . . . . . . . 2.4.1 Integrating the Equations of Motion . . . . . . . . . . . . . . . 2.4.2 Thermodynamic Ensembles and Their Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Energy Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Monte Carlo Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Classes of Chemical Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Interatomic Potential or Force Field: Introduction . . . . . 2.6.1 Pair Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Multibody Potentials: Embedded Atom Method for Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Force Fields for Biological Materials and Polymers . . . 2.6.4 Bond Order and Reactive Potentials . . . . . . . . . . . . . . . 2.6.5 Limitations of Classical Molecular Dynamics . . . . . . . . 2.7 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . 2.7.2 Force Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Neighbor Lists and Bins . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Property Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Temperature Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Pressure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Radial Distribution Function . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Mean Square Displacement Function . . . . . . . . . . . . . . . 2.8.5 Velocity Autocorrelation Function . . . . . . . . . . . . . . . . . 2.8.6 Virial Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Large-Scale Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Historical Development of Computing Power . . . . . . . . 2.9.2 Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Visualization and Analysis Methods . . . . . . . . . . . . . . . . . . . . . 2.10.1 Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Centrosymmetry Parameter . . . . . . . . . . . . . . . . . . . . . . . 2.10.3 Slip Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.4 Measurement of Defect Speed . . . . . . . . . . . . . . . . . . . . . 2.10.5 Visualization Methods for Biological Structures . . . . . 2.10.6 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Distinguishing Modeling and Simulation . . . . . . . . . . . . . . . . . . 2.12 Application of Mechanical Boundary Conditions . . . . . . . . . . . 2.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 37 39 40 43 44 46 48 50 54 56 59 68 69 70 71 72 73 73 74 74 75 76 76 78 79 80 82 83 85 86 88 89 89 90 90 90 93

Contents

XI

3

Basic Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Newton’s Laws of Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Definition of Displacement, Stress, and Strain . . . . . . . . . . . . . 3.2.1 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Energy Approach to Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Isotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Nonlinear Elasticity or Hyperelasticity . . . . . . . . . . . . . . . . . . . 3.6 Elasticity of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Example: Solution of a Simple Beam Problem . . . . . . . 3.6.4 Calculation of Internal Stress Field . . . . . . . . . . . . . . . . 3.6.5 Differential Beam Equations . . . . . . . . . . . . . . . . . . . . . . 3.7 The Need for Atomistic Elasticity: What’s Next . . . . . . . . . . .

95 95 97 99 100 103 105 107 108 110 110 111 112 113 116 119

4

Atomistic Elasticity: Linking Atoms and Continuum . . . . . 4.1 Thermodynamics as Bridge Between Atomistic and Continuum Viewpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Atomic and Molecular Origin of Elasticity: Entropic vs. Energetic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Virial Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Elasticity Due to Energetic Contributions . . . . . . . . . . . . . . . . 4.4.1 Cauchy–Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Elasticity of a One-Dimensional String of Atoms . . . . 4.4.3 Elasticity and Surface Energy of a Two-Dimensional Triangular Lattice . . . . . . . . . . . 4.4.4 Elasticity and Surface Energy of a Three-Dimensional FCC Lattice . . . . . . . . . . . . . . . 4.4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Elasticity Due to Entropic Contributions . . . . . . . . . . . . . . . . . 4.5.1 Elasticity of Single Molecules: Worm-Like-Chain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Elasticity of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

5

Multiscale Modeling and Simulation Methods . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Direct Numerical Simulation vs. Multiscale and Multiparadigm Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Differential Multiscale Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Detailed Description of Selected Multiscale Methods to Span Vast Lengthscales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Examples of Hierarchical Multiscale Coupling . . . . . . .

121 122 123 124 124 126 128 142 149 149 150 152 154 157 157 158 159 160 160

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5.4.2 Concurrent Integration of Tight-Binding, Empirical Force Fields and Continuum Theory . . . . . . 5.4.3 The Quasicontinuum Method and Related Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Continuum Approaches Incorporating Atomistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Hybrid ReaxFF Model: Integration of Chemistry and Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Advanced Molecular Dynamics Techniques to Span Vast Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162 165 168 169 175 180

Part III Material Deformation and Failure 6

Deformation and Dynamical Failure of Brittle Materials . 6.1 The Nature of Brittle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Basics of Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . 6.2.1 Energy Balance Considerations: Griffith’s Model of Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Asymptotic Stress Field and Stress Intensity Factor . . 6.2.3 Crack Limiting Speed in Dynamic Fracture . . . . . . . . . 6.3 Atomistic Modeling of Brittle Materials . . . . . . . . . . . . . . . . . . 6.4 A One-Dimensional Example of Brittle Fracture: Joint Continuum-Atomistic Approach . . . . . . . . . . . . . . . . . . . . 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Linear-Elastic Continuum Model . . . . . . . . . . . . . . . . . . 6.4.3 Hyperelastic Continuum Mechanics Model for Bilinear Stress–Strain Law . . . . . . . . . . . . . . . . . . . . . 6.4.4 Molecular Dynamics Simulations of the One-Dimensional Crack Model: The Harmonic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Molecular Dynamics Simulations of the One-Dimensional Crack Model: The Supersonic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 6.5 Stress and Deformation Field near Rapidly Propagating Mode I Cracks in a Harmonic Lattice . . . . . . . . . . . . . . . . . . . . 6.5.1 Stress and Deformation Fields . . . . . . . . . . . . . . . . . . . . 6.5.2 Energy Flow near the Crack Tip . . . . . . . . . . . . . . . . . . 6.5.3 Limiting Velocities of Mode I Cracks in Harmonic Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 186 189 189 194 196 197 201 202 204 207

211

217 219 223 225 227 229 230

Contents

6.6 Crack Limiting Speeds of Cracks: The Significance of Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Crack Speed and Energy Flow . . . . . . . . . . . . . . . . . . . . 6.6.3 Hyperelastic Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 How Fast can Cracks Propagate? . . . . . . . . . . . . . . . . . . 6.6.5 Characteristic Energy Length Scale in Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Crack Instabilities and Hyperelastic Material Behavior . . . . . 6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Design of Computational Model . . . . . . . . . . . . . . . . . . . 6.7.3 Computational Experiments . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Suddenly Stopping Cracks: Linking Atomistic Modeling, Theory, and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Theoretical Background of Suddenly Stopping Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Atomistic Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . 6.8.4 Atomistic Simulation Results of a Suddenly Stopping Mode I Crack . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.5 Atomistic Simulation Results of a Suddenly Stopping Mode II Crack . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Crack Propagation Along Interfaces of Dissimilar Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.1 Mode I Dominated Cracks at Bimaterial Interfaces . . 6.9.2 Mode II Cracks at Bimaterial Interfaces . . . . . . . . . . . . 6.9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Dynamic Fracture Under Mode III Loading . . . . . . . . . . . . . . . 6.10.1 Atomistic Modeling of Mode III Cracks . . . . . . . . . . . . 6.10.2 Mode III Cracks in a Harmonic Lattice – The Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.3 Mode III Crack Propagation in a Thin Stiff Layer Embedded in a Soft Matrix . . . . . . . . . . . . . . . . . 6.10.4 Suddenly Stopping Mode III Crack . . . . . . . . . . . . . . . . 6.10.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Brittle Fracture of Chemically Complex Materials . . . . . . . . . 6.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.2 Hybrid Atomistic Modeling of Cracking in Silicon: Mixed Hamiltonian Gormulation . . . . . . . . . 6.11.3 Atomistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIII

234 236 238 239 242 244 248 249 251 252 255 258 260 260 262 264 268 278 286 287 289 294 297 299 300 300 301 303 303 304 305 307 307 308

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6.11.5 Dynamical Fracture Mechanisms . . . . . . . . . . . . . . . . . . 6.11.6 Reactive Chemical Processes and Fracture Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Summary: Brittle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12.1 Hyperelasticity can Govern Dynamic Fracture . . . . . . . 6.12.2 Interfaces and Geometric Confinement . . . . . . . . . . . . . 7

8

Deformation and Fracture of Ductile Materials . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Continuum Theoretical Concepts of Dislocations and Their Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Properties of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Forces on Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Rice–Thomson Model for Dislocation Nucleation . . . . 7.2.4 Rice–Peierls Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Link with Atomistic Concepts . . . . . . . . . . . . . . . . . . . . . 7.2.6 Generalized Stacking Fault Curves . . . . . . . . . . . . . . . . . 7.2.7 Linking Atomistic Simulation Results to Continuum Mechanics Theories of Plasticity . . . . . . 7.3 Modeling Plasticity Using Large-Scale Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Case Study: Deformation Mechanics of Model FCC Copper – LJ Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Visualization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Case Study: Deformation Mechanics of a Nickel Nanocrystal – EAM Potential . . . . . . . . . . . . . . . . . 7.6 Case Study: Multi-Paradigm Modeling of Chemical Complexity in Mechanical Deformation of Metals . . . . . . . . . . 7.6.1 Atomistic Model and Validation . . . . . . . . . . . . . . . . . . . 7.6.2 Example Application: Modeling Hybrid Metal–Organic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and Fracture Mechanics of Geometrically Confined Materials . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Thin Metal Films and Nanocrystalline Metals . . . . . . . . . . . . . 8.2.1 Constrained Diffusional Creep in Ultra-Thin Metal Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Single Edge Dislocations in Nanoscale Thin Films . . .

311 316 317 320 323 326 327 327 328 329 331 332 337 338 338 339 341 343 343 345 345 355 357 359 360 364

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8.3

8.4

8.5

8.6

8.7

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8.9

8.2.3 Rice–Thompson Model for Nucleation of Parallel Glide Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . Atomistic Modeling of Constrained Grain Boundary Diffusion in a Bicrystal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Introduction and Modeling Procedure . . . . . . . . . . . . . . 8.3.2 Formation of the Diffusion Wedge . . . . . . . . . . . . . . . . . 8.3.3 Development of the Crack-Like Stress Field and Nucleation of Parallel Glide Dislocations . . . . . . . 8.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dislocation Nucleation from Grain Triple Junction . . . . . . . . . 8.4.1 Atomistic Modeling of the Grain Triple Junction . . . . 8.4.2 Atomistic Simulation Results . . . . . . . . . . . . . . . . . . . . . 8.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomistic Modeling of Plasticity of Polycrystalline Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Atomistic Modeling of Polycrystalline Thin Films . . . 8.5.2 Atomistic Simulation Results . . . . . . . . . . . . . . . . . . . . . 8.5.3 Plasticity of Nanocrystalline Bulk Materials with Twin Lamella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Modeling of Constrained Diffusional Creep in Polycrystalline Films . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.6 Summary: Results of Modeling of Thin Films . . . . . . . Use of Atomistic Simulation Results in Hierarchical Multiscale Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Mechanisms of Plastic Deformation of Ultra-thin Uncapped Copper Films . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Deformation Map of Thin Films . . . . . . . . . . . . . . . . . . . 8.6.3 Yield Stress in Ultra-Thin Copper Films . . . . . . . . . . . 8.6.4 The Role of Interfaces and Geometric Confinement . . Deformation and Fracture Mechanics of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Mesoscale Modeling of CNT Bundles . . . . . . . . . . . . . . . 8.7.2 Mesoscale Simulation Results . . . . . . . . . . . . . . . . . . . . . 8.7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flaw-Tolerant Nanomaterials: Bulk Fracture and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Strength of Brittle Nanoparticles . . . . . . . . . . . . . . . . . . 8.8.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nanoscale Adhesion Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Strength of Fibrillar Adhesion Systems . . . . . . . . . . . . . 8.9.2 Theoretical Considerations of Shape Optimization of Adhesion Elements . . . . . . . . . . . . . . . .

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8.9.3 Atomistic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

456 457 460

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Illustration of how the characteristic material scales of technological eras have been reduced from the scales of meters to the scales of individual molecules and atoms. The current technological frontier is the development of molecular and atomistic structures at the interface of physics, biology and chemistry, leading to a new bottom-up approach in creating and characterizing materials . . . . . . . . . . . The plot shows simple, schematic stress–strain diagrams characteristic for a brittle and a ductile material. Similar curves are found for other materials, including polymers or rubber-like materials. The cross symbol (“x”) indicates the point of material failure [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous material (subplot (a)) and material with elliptical hole (subplot (b), length of elliptical hole is 2a). The presence of the elliptical void leads to a magnification of the stress in the vicinity of the tip of the defect (see schematic illustration of stress profile) . . . . . . . . . . . . . . . . . . . . Schematic illustration of a failure process by crack extension in a brittle material. The inlay in the center shows how chemical bonds rupture continuously, leading to formation of new fracture surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of different crystal structures, showing the SC, FCC, and BCC crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . Dislocations are the discrete entities that carry plastic (permanent) deformation; measured by a “Burgers vector.” The snapshots illustrate the nucleation and propagation of an edge dislocation through a crystal, leading to permanent deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Schematic of brittle (a) vs. ductile (b) materials behavior. In brittle fracture, the crack extends via breaking of atomic bonds. In ductile fracture, the lattice around the crack tip is sheared, leading to nucleation of crystal defects called dislocations. Which one the two mechanisms is more likely to occur determines whether a material is brittle or ductile; this distinction is closely related to the atomic structure and the details of the atomic bonding . . . . . . . . . . . . . . . . . . . . . Brittle (a) vs. ductile (b) materials behavior observed in atomistic computer simulations. In brittle materials failure, thousands of cracks break the material. In ductile failure, material is plastically deformed by motion of dislocations . . . . Overview over timescales and lengthscales associated with various problems and applications of mechanical properties (adapted from [2]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental techniques for conducting mechanical tests in single cell and single molecule biomechanics. Reprinted from Materials Science and Engineering C, Vol. 26, C.T. Lim, E.H. Zhou, A. Li, S.R.K. Vedula and H.X. Fu, Experimental techniques for single cell and single molecule biomechanics, c 2006, with permission from pp. 1278–1288, copyright Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nanomechanical experiments of bending deformation of 200-nm gold nanowires. Subplot (a) depicts a schematic of a fixed wire in a lateral bending test with an AFM tip. Subplots (b–e) depict AFM snapshots of the mechanical deformation the nanowire. Subplot (b) depicts results after elastic deformation, subplots (c) and (d) shows results after successive plastic manipulation, and subplot (e) shows an SEM image following the bending test. The SEM picture agrees in detail with the AFM image shown in subplot (d). All scale bars are 1 µm. Reprinted with permission from c 2005 . . . . . Macmillan Publishers Ltd, Nature Materials [3]

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1.14

Mechanical deformation of a red blood cell (RBC) with optical tweezers. Subplot (a) depicts a schematic of the experimental approach. Subplot (b) depicts optical images of a healthy RBC anda RBC in the schizont stage of malaria, in PBS solution at 25◦ C. The left column depicts results prior to stretching, the middle column depicts results at a constant force of 68 ± 12 pN, and the right column plots results at a constant force of 151 ± 20 pN. The P. falciparum malaria parasite can be seen inside the infected RBCs. Reprinted from Acta Biomaterialia, Vol. 1, S. Suresh, J. Spatz, J.P. Mills, A. Micoulet, M. Dao, C.T. Lim, M. Beil, T. Seufferlein, Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and c 2005, with permission from malaria, pp. 15–30, copyright Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Images of an RBC being stretched from 0 to 193 pN. Subplot (a) shows images obtained from experiment, while subplots (b) and (c) depict a top view and a three-dimensional view of the half-model corresponding to the large deformation finite element simulation of the RBC, respectively. The contours of shades of grey in the middle column shows the distribution of constant maximum principal strains. Reprinted from Materials Science and Engineering C, Vol. 26, C.T. Lim, E.H. Zhou, A. Li, S.R.K. Vedula and H.X. Fu, Experimental techniques for single cell and single molecule c 2006, with biomechanics, pp. 1278–1288, copyright permission from Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of cellular mechanical properties in cancer cells. Subplot (a) depicts an optical image demonstrating the round, balled morphology of visually assigned tumor cells, and the large, flat morphology of presumed benign mesothelial, normal cells. Subplots (b–d) show histograms of the associated Youngs modulus E for cytological samples collected from patients with suspected metastatic cancer. Subplot (b) shows the histogram of E for all data collected from seven different clinical samples, indicating that there exist two peaks in the distribution. Subplot (c) shows a Gaussian fit for all tumor cells, and subplot (d) shows a log-normal fit for all normal cells. The analysis suggests that the presence of tumor cells leads to a sharp peak due to a lower Young’s modulus. This method might be used to diagnose cancer based on a mechanical analysis. Reprinted with permission from Macmillan Publishers Ltd, Nature c 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nanotechnology [4]

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Experimental study of fracture mechanics of bone. Subplot (a) shows the geometry of the three-point bending test with an initial notch as a seed for fracture. Subplots (b) and (c) show scanning electron micrographs of microscopic bone fracture mechanisms, obtained from carrying out fracture experiments (arrangement of figure adapted from original source). Reprinted with permission from Macmillan c 2003 . . . . . . . . . . . . . . . Publishers Ltd, Nature Materials [5] AFM experiments of protein unfolding. Subplot (a): Force peaks corresponding to the sequential unfolding of a immunoglobulin-like domains of human cardiac titin (human cardiac I band titin encompassing the immunoglobulin-like domains I27 – I34). The results show large hump-like deviations from the WLC model of entropic elasticity (continuous lines indicate WLC fits, the arrow illustrates the point of deviation). Subplot (b): Detailed view of the first force peak of a sawtooth pattern. Reprinted with permission from Macmillan Publishers Ltd, c 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature [6] Large deformation of a protein, here an example of unfolding of the enzyme lysozyme, result of a reactive force field simulation. The distance between the ends of the protein (Cα -atom of the terminal residues) is continuously increased by applying a continuously increasing force [7]. As the force is increased, the protein molecule undergoes significant structural changes relative to its initial folded configuration . . Molecular dynamics can be used to study material properties at the intersection of various scientific disciplines. This is because the notion of a “chemical bond” as explicitly considered in molecular dynamics provides a common ground as it enables the cross-interaction between concepts used in different disciplines (here exemplified for the disciplines of biology, mechanics, materials science and physics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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This figure illustrates the concept of model building. Panel (a) on the left shows the physical situation of a map of the subway lines. This representation makes it quite difficult to determine a strategy to use the subway system to travel from the cities of Braintree to Revere, for instance. The model representation depicted in panel (b) on the right enables one to determine quite easily which subway line to take, where to change the subway line, and how many subway stops there are in between. This example illustrates that even though the model representation on the right misrepresents the actual distances and directions, it elegantly displays the connectivity. This figure was created based on a snapshot from the Massachusetts Bay Transportation Authority (MBTA) web site (URL: http://www.mbta.com/), reprinted with permission from the the Massachusetts Bay Transportation Authority . . . . . . . Molecular dynamics generates the dynamical trajectories of a system of N particles by integrating Newton’s equations of motion, with suitable initial and boundary conditions, and proper interatomic potentials, while satisfying macroscopic thermodynamical (ensemble-averaged) constraints, leading to atomic positions ri (t), the atomic velocities vi (t), and their accelerations ai (t), all as a function of time, for all particles i = 1 . . . N , each of which has a specific mass mi . . . Schematic of the atomic displacement field as a function of time. The atomic displacement field consists of a low-frequency (“coarse”) and high frequency part (“fine”) . . . Example of harmonic oscillator with spring constant k = φ (r = r0 ), used to extract information about the time step required for integration of the equations of motion. The dashed line shows the (nonlinear) realistic potential function between a pair of atoms, of which the harmonic oscillator is the second-order approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . Example result of an energy minimization, here an example of minimizing the structure of a solvated protein (lysozyme). As the number of iterations progresses, the total potential energy decreases, until it converges and reaches a constant value (see [8] for further details) . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of generation of random perturbation from an initial state A toward a state B, as typically performed in Monte Carlo schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.11 2.12

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2.14 2.15

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List of Figures

Summary of the Metropolis–Hastings Monte Carlo algorithm. Please see Figure 2.7 for an illustration of how state B is generated based on a random perturbation from state A. The procedure is repeated NA times, the number of iterations. The number of steps is chosen so that convergence of the desired property is achieved . . . . . . . . . . . . . Schematic of the typical characteristic of a chemical bond, showing repulsion at small distances below the equilibrium separation r0 , and attraction at larger distances . . . . . . . . . . . . Atoms are composed of electrons, protons, and neutrons. Electrons and protons are negative and positive charges of the same magnitude. In classical molecular dynamics, the three-dimensional atom structure is replaced by a single mass point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview over different simulation tools and associated lengthscale and timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pair interaction approximation. The upper part shows all pair interactions of atom 1 with its neighbors, atoms 2, 3, 4, and 5. When the bonds to atom 2 are considered, the energy of the bond between atoms 1 and 2 is counted again (bond marked with thicker line). This is accounted for by adding a factor 1/2 in (2.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Replacing a full-electron representation of atom–atom interaction by a potential function that only depends on the distance r between the particles . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the LJ potential and its derivative (for interatomic forces) in a parametrization for copper as reported in [9] . . . . Difference in bond properties at a surface. Pair potentials (left panel) are not able to distinguish bonds in different environments, as all bonds are equal. To accurately represent the change in bond properties at a surface, one needs to adapt a description that considers the environment of an atom to determine the bond strength, as shown in the right panel. The bond energy between two particles is then no longer simply a function of its distance, but instead a function of the positions of all other particles in the vicinity (that way, changes in the bond strength, for instance at surfaces, can be captured). Multibody potentials (e.g., EAM) provide such a description . . . . . . . . . . . . . . . . . . . . . . . . . This plot illustrates how an EAM-type multibody potential can represent different effective pair interactions between bonds at a surface and in the bulk . . . . . . . . . . . . . . . . . . . . . . . .

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XXIII

Chemical complexity in proteins involves a variety of chemical elements and different chemical bonds between them. The snapshot shows a small alpha-helical coiled coil protein domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the contributions of the different terms in the potential expressions given in (2.36), illustrating the contributions of bond stretching, angle bending, bond rotations, electrostatic interactions, and vdW interactions . . . The plot shows the cohesive energy per atom (upper plot, in eV) and the bond length (lower plot, in ˚ A), for several real and hypothetical polytypes of carbon, comparing the predictions from the Tersoff potential [10] for C with experimental and other computational results. The structures include a C2 dimer molecule, graphite, diamond, simple cubic, BCC, and FCC structures. The squares correspond to experimental values for these phases and calculations for hypothetical phases [11]. The circles are the results of Tersoff’s model [10]. The continuous lines are spline fits to guide the eye. Reprinted from: J. Tersoff, Empirical interatomic potentials for carbon, with applications to amorphous carbon, Physical Review Letters, c 1988 by the Vol. 61(25), 1988, pp. 2879–2883. Copyright American Physical Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example to demonstrate the basic concept of the ReaxFF potential. It has been developed to accurately describe transition states in addition to ground states . . . . . . . . . . . . . . . Illustration of basic concept of bond order potentials. Subplot (a) shows how the bond order potential allows for a more general description of chemistry, since all energy terms are expressed dependent on bond order. In contrast, conventional potentials (such as LJ, Morse) express the energy directly as a function of the bond distance as shown in subplot (b). Subplot (c) illustrates the concept for a C–C single, double, and triple bond, showing how the bond distance is used to map to the bond order, serving as the basis for all energy contributions in the potential formulation defined in (2.47) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of a ReaxFF study of water formation, comparing the production rate with and without a Pt catalyst. The presence of the Pt catalyst significantly increases the water production rate (results taken from [12]) . . . . . . . . . . . . . . . . . .

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List of Figures

Water production at varying temperature, for constant pressure. Subplot (a) depicts the water production rate. Subplot (b) shows an Arrhenius analysis, enabling us to extract the activation barrier for the elementary chemical process of 12 kcal/mol. This result is close to DFT level calculations of the energy barrier [12] . . . . . . . . . . . . . . . . . . . . . The ReaxFF force field fills a gap between quantum mechanical methods (e.g., DFT) and empirical molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the numerical scheme in carrying out molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the numerical scheme in carrying out molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of force calculation scheme in molecular dynamics, for a pair potential. To obtain the force vector F one takes projections of the magnitude of the force vector F into the three axial directions xi (this is done for all atoms in the system) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of neighbor lists and bins to achieve linear scaling ∼N in molecular dynamics. Panel (a): Example of how neighbor lists are used. The four neighbors of the central atom (in the circle) are stored in a list so that force calculation can be done directly based on this information. This changes the numerical problem to a linear scaling effort. Panel (b): The computational domain is divided into bins according to the physical position of atoms. Then, atomic interactions must only be considered within the atom’s own bin and atoms in the neighboring bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method to calculate the radial distribution function g(r) . . . . Radial distribution function g(r) for various atomistic configurations, including a solid (crystal), a liquid and a gas . Velocity autocorrelation function (VAF) for a gas, liquid, and solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relating the continuum stress with the atomistic stress. The left shows a continuum system in which σij (r) is defined at any point r. In contrast, in the atomistic system the stress tensor is only defined at discrete points where atoms are located . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of how to calculate the stress tensor in a 1D system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Increase in computer power over the last decades and possible system sizes for classical molecular dynamics modeling. The availability of PFLOPS computers is expected by the end of the current decade, which should enable simulations with hundreds of billions of atoms . . . . . . .

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Summary of top 10 of the TOP500 supercomputer list, as of Spring 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modern parallelization scheme. Subplot (a) depicts the schematic of the tunable hierarchical cellular decomposition scheme (THCD). The physical volume is subdivided into process groups, PGγ , each of which is spatially decomposed into processes, Pγπ . Each process consists of a number of computational cells (e.g., linked-list cells in molecular dynamics). Subplot (b) shows the total execution (circles) and communication (squares) times per molecular dynamics time step as a function of the number of processors for the F-ReaxFF molecular dynamics algorithm with scaled workloads (in a 36,288P atom RDX systems on P processors (P = 1, . . . , 1920) of Columbia [Columbia is a supercomputer at NASA]). Reprinted from Computational Materials Science, Vol 38(4), A. Nakano, R. Kalia, K. Nomura, A. Sharma, P. Vashishta, F. Shimojo, A. van Duin, W.A. Goddard III, R. Biswas and D. Srivastava, A divide-and-conquer/cellular-decomposition framework for million-to-billion atom simulations of chemical c 2007, with permission reactions, pp. 642–652, copyright from Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rendering of a large molecular dynamics simulation on a tiled display at USC, showing hypervelocity impact damage of a ceramic plate with impact velocity 15 kms−1 , where one quarter of the system is cut to show the internal pressure distribution (the projectile is shown in white). This figure illustrates how novel visualization schemes provide analysis methods for ultra large-scale simulations. Reprinted from Computational Materials Science, Vol 38(4), A. Nakano, R. Kalia, K. Nomura, A. Sharma, P. Vashishta, F. Shimojo, A. van Duin, W.A. Goddard III, R. Biswas and D. Srivastava, A divide-and-conquer/cellular-decomposition framework for million-to-billion atom simulations of chemical c 2007, with permission reactions, pp. 642–652, copyright from Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of a dislocation network using the energy filtering method in nickel with 150,000,000 atoms [13, 14]. Subplot (a) shows the whole simulation cell with two cracks at the surfaces serving as sources for dislocations, and subplot (b) shows a zoom into a small subvolume. Partial dislocations appear as wiggly lines, and sessile defects appear as straight lines with slightly higher potential energy . . . . . . . . . . . . . . . . .

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List of Figures

Application of the energy method to visualize fracture surfaces in a computational fracture experiment. Only high energy atoms are shown by filtering them according to their potential energy. This enables an accurate determination of the geometry of cracks, in particular of the crack tip. Typically, the analysis is confined to a search region (shown as a dashed line) to avoid inclusion of effects of free surfaces . The figure shows a close view on the defect structure in a simulation of work-hardening in nickel analyzed using the centrosymmetry technique [13, 14]. The plot shows the same subvolume as in Figure 2.38b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of a dislocation using the slip vector approach. From the result of the numerical analysis, direct information about the Burgers vector can be obtained. The slip vector s is drawn at each atom as a small arrow. The Burgers vector b is drawn at the dislocation (its actual length is exaggerated to make it better visible). The dislocation line is approximated by discrete, straight dislocation segments. A line element between “a” and “b” is considered . . . . . . . . . . Analysis of a simple alpha-helix protein structure, with different visualization options, plotted using VMD [148] . . . . . Simulation method of domain decomposition via the method of virtual atom types. The atoms in region 2 do not move according to the physical equations of motion, but are displaced according to a prescribed displacement history. An initial velocity gradient as shown in the right half of the plot is used to provide smooth initical conditions . . . . . . . . . . . Schematic to illustrate the use of steered molecular dynamics to apply mechanical load to small-scale structures (subplot (a): AFM experiment; subplot (b) Steered Molecular Dynamics model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steered molecular dynamics simulations of I27 extensibility under constant force. Subplot (a) shows snapshots of the structure of the I27 module simulated at a force of 50 pN (I, at 1 ns) and 150 pN (II, at 1 ns). At 50 pN, the hydrogen bonds between strands A and B are maintained, whereas at 150 pN they are broken. Subplot (b) displays the corresponding force–extension relationship obtained from the simulations. The discontinuity observed between 50 and 100 pN corresponds to an abrupt extension of the module by 4–7 ˚ A caused by the rupture of the AB hydrogen bonds, and the subsequent extension of the partially freed polypeptide segment. Reprinted with permission from c 1999 . . . . . . . . . . . . . Macmillan Publishers Ltd., Nature [6]

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XXVII

Axial tensile loading of a beam and schematic force– extension response. Reversible deformation denotes the elastic regime; upon unloading of the sample the displacement returns to the initial point. Irreversible deformation denotes the plastic regime; upon unloading (indicated in the graph) the displacement does not return to the initial point. (It is noted that the specific shape of the force-extension curve may vary significantly depending on the type of material) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example for deformation of a beam due to mechanical loading of a distributed force qz . The structure responds to the mechanical forces by a change in shape. Continuum mechanical theory enables us to derive a relationship between applied forces and displacements, strains, and stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The beam problem as multiscale problem. The goal of solving this problem is to connect the global scale (scale on the order of L where boundary conditions are applied, for instance, load P , N , prescribed displacements) with the local scale (section of the beam, e.g., the stress variation σxx , across the section) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-sectional view of a body. Subplot (a) free body with representative internal forces. Subplot (b) enlarged view with components of the force vector split up . . . . . . . . . . . . . . . The most general state of stress acting on a infinitesimal material element. All stresses shown in the figure have positive sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infinitesimal element with stresses and body forces fi acting as volume forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This schematic explains the condition that σij = σji so that there is no moment on the infinitesimal element, since it cannot rotate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic to illustrate the definition of the deformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic to illustrate the difference between rotational and deformation part of the deformation tensor . . . . . . . . . . . . . . . . Illustration of the concept of nonlinear elasticity or hyperelasticity. Subplot (a) shows the stress–strain relationship, and subplot (b) depicts the tangent modulus as a function of strain. Linear elasticity is based on the assumption that the modulus is independent of strain. However, most real materials do not show this behavior. Instead, they show a stiffening effect (e.g., rubber, polymers, biopolymers) or a softening effect (e.g., metals, ceramics) . . . . Geometry of the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Solution field for a simply supported beam under dead load ρg, showing the shear force Qy , bending moment My , rotation ωy , and the beam axis displacement uz . . . . . . . . . . . . 112 Demonstration of the concept of the Navier–Bernouilli assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Solution field for a simply supported beam under a point load P applied at the end of the beam, showing the shear force Qy , bending moment My , rotation ωy , and the beam axis displacement uz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Example to illustrate Cauchy–Born rule in a one-dimensional geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subplot (a) rectangular cell in a uniformly deformed triangular lattice; subplot (b) the geometrical parameters used to calculate the continuum properties of the lattice . . . . . Elastic properties of the Lennard-Jones solid (continuous line) and elastic properties associated with the harmonic potential (dashed line). The dash-dotted lines in the upper plots show Poisson’s ratio. The lower plots show the tangent modulus for this case. This plot is an actual material law representing the schematic shown in Fig. 3.10 . . . . . . . . . . . . . . Elastic properties associated with the tethered LJ potential, and in comparison, elastic properties associated with the harmonic potential (dashed line). Unlike in the softening case, where Young’s modulus softens with strain (Fig. 4.3), here Young’s modulus stiffens with strain . . . . . . . . . . . . . . . . . . Elastic properties of the triangular lattice with harmonic interactions, stress vs. strain (left ) and tangent moduli Ex and Ey (right ). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the shape of the harmonic potential, comparing the one defined in (2.34) (panel (a)) and the one defined in (4.43) with the bond snapping parameter rbreak (panel (b)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure shows the stretching of the triangular lattice in two different directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure plots the elastic properties under uniaxial loading with Poisson relaxation for the harmonic potential. In the plot, stress vs. strain, Poisson’s ratio as well as the number of nearest neighbors are shown. The lower two subplots show Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XXIX

The figure plots the elastic properties under uniaxial loading without Poisson relaxation for the harmonic potential. In the plot, stress vs. strain, as well as the number of nearest neighbors are shown. The lower two subplots show Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the parameters used in the biharmonic potential defined in (4.44). The plot defines r, k0 , k1 , ron , rbreak , as well as the “atomic” strain . . . . . . . . . . . . . . . . . . . . . . Elastic properties of the triangular lattice with biharmonic interactions, stress vs. strain in the x-direction (a) and in the y-direction (b). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bond breaking process along the fracture plane and calculation of fracture surface energy for (a) direction of high fracture surface energy and (b) direction of low fracture surface energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic properties associated with the harmonic potential, [100] crystal orientation, with Poisson relaxation. Poisson ratio is ν ≈ 0.33 and is approximately independent of the applied strain. The plot shows the elastic properties as a function of strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic properties associated with the harmonic potential, [100] crystal orientation, without Poisson relaxation. The plot shows the elastic properties as a function of strain . . . . . . Elastic properties associated with the harmonic potential, [100] crystal orientation, triaxial loading. The plot shows the elastic properties as a function of strain . . . . . . . . . . . . . . . . Elastic properties associated with the harmonic potential, (a) [110] and (b) [111] crystal orientation, uniaxial loading with Poisson relaxation. The plot shows the elastic properties as a function of strain . . . . . . . . . . . . . . . . . . . . . . . . . Elastic properties associated with (a) LJ potential, and (b) an EAM potential for nickel [15], uniaxial loading in [100], [110] and [111] with Poisson relaxation . . . . . . . . . . . . . . . . . . . This plot depicts a series of snapshots of a single molecule with increasing length L, at constant temperature. The longer the molecule, the more wiggly the geometrical shape . .

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List of Figures

Entropy controlled molecular elasticity. Subplot (a) Coiled, entangled state of a molecule with contour length much larger than the persistence length. The end-to-end distance is measured by the variable x. Subplot (b) Response of the molecule to mechanical loading. As the applied force is increased, the end-to-end distance x increases until the molecule is fully entangled. Clearly, the continued disentanglement leads to a reduction of entropy in the system, which induces a force that can be measured as an elastic spring. Once the molecule is fully extended, the change in entropy due to increased force approaches zero, and the elastic response is controlled by changes in the internal potential energy of the system, corresponding to the energetic elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 This plot depicts the entropic response (F < 14 pN) of a single tropocollagen molecule, obtained by direct molecular dynamics simulation using a multi-scale model [17]. This plot also depicts experimental results [16] obtained for TC molecules with similar contour lengths, as well as the prediction of the WLC model with persistence length of approximately 16 nm [17]. The force-extension curve shows a strong hyperelastic stiffening effect (see also Fig. 3.10) . . . . . 152 The concepts of entropic elasticity of single molecules can be immediately applied to understand two-dimensional and three-dimensional networks of molecules in a polymer. This figure demonstrates how a change in state of deformation poses constraints on the end-to-end distances of molecules, influencing the entropy of the system. Such considerations enable to link the properties of single molecules (their entropy) with the overall macroscopic elastic behavior of the material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A summary of a hierarchical multiscale scheme that can be used to develop an understanding of the behavior of materials across scales in length and time . . . . . . . . . . . . . . . . . 158 Overview over the process of predictive multiscale modeling. Quantitative predictions are enabled via the validation of key properties, which then enables to extrapolate and predict the behavior of systems not included in the initial training set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Example for implementation of a hierarchical multiscale method, where parameters are passed through various lengthscales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

List of Figures

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XXXI

Hierarchical modeling of Cybersteel [18]. Subplot (a) shows quantum mechanical calculations that provide the traction-separation law. Subplot (b) depicts concurrent modeling of the submicron cell based on the tractionseparation law. Subplot (c) illustrates concurrent modeling of the microcell with the embedded constitutive law of the submicron cell. Subplot (d) shows results of modeling the fracture of the Cybersteel with embedded constitutive law of the microcell. Subplot (e) depicts the fracture toughness and the yield strength of the Cybersteel as a function of decohesion energy, determined by geometry of the nanostructures. Subplot (f) shows snap-shots of the localization induced debonding process. Subplot (g) summarizes experimental observations. Reprinted from [18], Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 1529–1578, W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, An introduction to computational c 2004, with nanomechanics and materials, copyright permission from Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 This plot shows a multiscale analysis of a 15-walled CNT by a bridging scale method. Subplot (a) illustrates the multiscale simulation model. It consists of ten rings of carbon atoms (with 49,400 atoms each) and a meshfree continuum approximation of the 15-walled CNT by 27,450 nodes. Subplot (b) shows the global buckling pattern captured by meshfree method, whereas the detailed local buckling of the ten rings of atoms are captured by a concurrent bridging scale molecular dynamic simulation. Reprinted from [18], Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 1529–1578, W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, An introduction to computational nanomechanics and materials, copyright c 2004, with permission from Elsevier . . . . . . . . . . . . . . . . . . . . 165 Results of a simulation of a crack in a thin film constrained by a rigid substrate, exemplifying a study using a concurrent multiscale simulation method, the quasicontinuum approach [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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Application of the quasicontinuum method in the simulation of a nanoindentation experiment. Subplots (a) and (b) depicts a cross-sectional view of the test sample used in the nanoindentation simulations for increasing indenter penetration (part of the indenter is also shown). Subplot (c) plots the dislocation structure at the indenter penetration corresponding to the indentation depth shown in subplot (b). Subplot (d) shows a load vs. displacement curve predicted by full atomistic (LS) and quasicontinuum (QC) simulations, illustrating that the two methods show excellent agreement. Reprinted from Journal of the Mechanics and Physics of Solids, Vol. 49(9), J. Knap and M. Ortiz, An c 2001, analysis of the quasicontinuum method, copyright with permission from Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . The interpolation method for defining a mixed Hamiltonian in the transition region between two different paradigms. As an alternative to the linear interpolation we have also implemented smooth interpolation function based on a sinusoidal function. This enables using slightly smaller handshake regions thus increasing the computational efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of the energy landscape of two force fields, a ReaxFF reactive force field and a nonreactive force field. The plot illustrates that the two models yield a similar energy landscape for small deviations from the minimum potential well, the equilibrium position. An exemplification of this effect specifically for silicon is shown in Fig. 6.108 . . . . Example CMDF script (upper part) and schematic of the structure of CMDF (lower part) . . . . . . . . . . . . . . . . . . . . . . . . . . Study of a nanoscale elliptical penny-shaped crack in nickel, filled with O2 , illustrating the hybrid ReaxFF-EAM approach (crystal is loaded in tension, in the horizontal direction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomistic model to study surface diffusion of a single adatom on a flat [100] copper surface . . . . . . . . . . . . . . . . . . . . . Study of atomic mechanisms near a surface step at a [100]copper surface. The living time (or temporal stability) of states A (perfect step) and B (single atom hopped away from step) as a function of temperature. The higher the temperature, the closer the living times of states A and B get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Snapshots of states A (perfect step) and B (single atom hopped away from step) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hybrid CADD-Parallel Replica study of mechanical twinning of a metal. Subplot (a) shows the simulation domain, illustrating the continuum/discrete dislocation regime and the full atomistic domain (blow-up in right part). Subplot (b) shows a comparison of atomistic simulation results with the predictions of an analytical model. The plot shows the time to nucleation of a trailing or twinning partial versus applied load in Al at a temperature of 300 K. The circles refer to the multiscale simulation results covering many orders of magnitudes in timescales. The dashed lines correspond to the predictions of the analytical. Reprinted with permission from Macmillan Publishers Ltd, Nature c 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Materials [21]

6.1 6.2

Picture of shattered glass, a model for a brittle material . . . . . Characteristic length scales associated with dynamic fracture. Relevant length scales reach from the atomic scale of several ˚ Angstrom to the macroscopic scale of micrometers and more . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using the solution to the beam problem to predict the critical force P at which fracture initiates. Subplot (a) shows the geometry of a crack in a beam-like structure. Subplot (b) shows the representation of the upper and lower part as two cantilever beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thin strip geometry. The gray arrows indicate the mode I (tensile loading), by a stress σ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the basic physical processes involved in brittle fracture, that is, the process of dissipating stored elastic energy toward breaking of chemical bonds . . . . . . . . . . . . . . . . . Schematic of cracks under mode I, mode II, and mode III crack loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closing a crack by negating the tractions at the tip, as used in the derivation of the relation between the stress intensity factor and the energy release rate . . . . . . . . . . . . . . . . . . . . . . . . .

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Mode II loading experimental setup for studies of dynamic fracture in Homalite-100. Subplot (a) depicts the geometry of the experiment, indicating the location of projectile impact to generate rapid mode II loading along a weak plane. The dashed circle displays the view of the circular polariscope for the analysis of the stress field. Subplot (b) displays the evolution of crack speed as the shear crack propagates along the weak plane. The crack tip speed was obtained from crack length history (squares) and from shock wave angles (circles) for a field of view around the notch tip (solid symbols), and for a field of view ahead of the notch (open symbols). The analysis confirms intersonic and supersonic regimes of crack propagation. Reprinted from Science, Vol. 284, A.J. Rosakis, O. Samudrala, D. Coker, Cracks Faster than the Shear Wave Speed, copyright c 1999, with permission from AAAS . . . . . . . . . . . . . . . . . . . . . Enlarged view of the isochromatic fringe pattern around a steady-state mode II intersonic crack along a weak plane in Homalite-100. Subplot (a) shows the experimental pattern, and subplot (b) the theoretical prediction [22]. For both cases, β = 53o and v = 1.47cs . Reprinted from Science, Vol. 284, A.J. Rosakis, O. Samudrala, D. Coker, Cracks c 1999, with Faster than the Shear Wave Speed, copyright permission from AAAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the one-dimensional model of fracture . . . . . . . . . One-dimensional atomistic model of dynamic fracture . . . . . . . Bilinear stress–strain law as a simplistic model of hyperelasticity (mimicking the behavior shown in Fig. 3.10). The parameter εon determines the critical strain where the elastic properties change from local (El ) to global (Eg ) . . . . . . Continuum model for local strain near a supersonic crack. The plot shows a schematic of the two cases 1 (subplot (a)) and case 2 (subplot (b)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnitude of the local stress wave for different crack propagation velocities from atomistic simulations, in comparison with the theory prediction . . . . . . . . . . . . . . . . . . . . Dynamic fracture toughness for different crack propagation velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain field near a suddenly stopping one-dimensional crack. The crack is forced to stop at x ≈ 790. As soon as the crack stops (at x = 550), the strain field of the static solution is spread out with the wave speed . . . . . . . . . . . . . . . . . . . . . . . . . . Prescribed fracture toughness and measured crack velocity as the crack proceeds along x . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XXXV

Strain field of a crack travelling in a material with periodically varying fracture toughness . . . . . . . . . . . . . . . . . . . . Elastic properties associated with the biharmonic interatomic potential, for ron = 1.125 and Eg = 8 = 1/4El . . . Subplot (a) Velocity of the crack for different values of the potential parameter ron . The larger ron , the larger the stiff area around the crack tip. As the hyperelastic area becomes sufficiently large, the crack speed approaches the local wave speed αl = 1 corresponding to αg = 2. Subplot (b) shows a quantitative comparison between theory and computation of the strain field near a supersonic crack as a function of the potential parameter ron . The different regimes corresponding to case 1 and case 2 are indicated. The loading is chosen σ0 = 0.1, with kp /k = 0.1 and rˆ = 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence of strain field near a rapidly propagating supersonic 1D crack moving with Mach 1.85 for ron = 1.124. The primary (1) and secondary wave (2) are indicated in the plot. The wave front (1) propagates supersonically through the material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle velocity field near a supersonic crack, comparison between theory and simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation geometry and coordinate system for studies of rapidly propagating mode I cracks in harmonic lattices . . . . . . Comparison between σxx from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between σyy from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between σxy from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between hoop stress from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the maximum principal stress σ1 from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR . . . . . . . . . . . . . . . . . . . . . . .

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XXXVI

6.29

6.30

6.31

6.32

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6.35

List of Figures

Principal strain field at various crack velocities (a) v/cR ≈ 0, (b) v/cR ≈ 0.5, (c) v/cR ≈ 1. In each of the plots (a)–(c), the upper plot is the simulation result and the lower part is the prediction by continuum mechanics . . . . . . . . . . . . . . . . . . . Stress fields close to the crack tip for a crack propagating close to the Rayleigh velocity v/cR ≈ 1. Plots (a), (b), and (c) show σxx , σyy , and σxy . In each of the plots (a)–(c), the upper plot is the simulation result and the lower part is the prediction by continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . Particle velocity field close to the crack tip for a crack propagating close to the Rayleigh velocity, v/cR ≈ 1. Plots (a) shows u˙ x and plot (b) shows u˙ y . In each of the plots (a) and (b), the upper plot is the simulation result and the lower part is the prediction by continuum mechanics . . . . . . . . Potential energy field and magnitude of the dynamic Poynting vector. (a) Potential energy field near a crack close to the Rayleigh speed. (b) Energy flow near a rapidly propagating crack. This plot shows the magnitude of the dynamic Poynting vector in the vicinity of a crack propagating at a velocity close to the Rayleigh speed . . . . . . . . Energy flow near a rapidly propagating crack. This plot shows (a) the continuum mechanics prediction, and (b) the molecular dynamics simulation result of the dynamic Poynting vector field in the vicinity of the crack tip, for a crack propagating close to the Rayleigh wave speed . . . . . . . . . Crack tip history as well as the crack speed history for a soft as well as a stiff harmonic material (two different choices of spring constants as given in Table 4.1) . . . . . . . . . . . . . . . . . . . . The concept of hyperelasticity in dynamic fracture. Subplot (a) shows the region of large deformation near a moving crack, due to the nonlinear elastic behavior of solids (subplot (b)). The linear elastic approximation is only valid for small deformation. Close to crack tips, material deformation is extremely large, leading to significant changes of local elasticity, referred to as “hyperelasticity” (see also Fig. 3.10 and related discussion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.36

6.37 6.38 6.39

6.40

6.41

6.42

6.43

6.44

6.45

XXXVII

This figure shows a continuously increasing hyperelastic stiffening effect, as observed by measuring the elastic properties of a material (subplot (a)). The increasingly strong hyperelastic effect is modeled by using biharmonic potentials, thereby capturing the essential physics: A small-strain spring constant k0 and a large-strain spring constant k1 (subplot (b)), where the ratio of the two is defined as kratio = k1 /k0 . The bilinear or biharmonic model allows to tune the size of the hyperelastic region near a moving crack, as indicated in subplots (c) and (d). The local increase of elastic modulus and thus wave speeds can be tuned by changing the slope of the large-strain stress–strain curve (“local modulus”) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperelastic region in a (a) softening and (b) stiffening system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperelastic region and enhancement of energy flow in the (a) softening and (b) stiffening system . . . . . . . . . . . . . . . . . . . . J-integral analysis of a crack in a harmonic, softening and stiffening material, for different choices of the integration path Γ . The straight lines are a linear fit to the results based on the calculation of the molecular dynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of the crack speed as a function of εon . The smaller εon , the larger is the hyperelastic region and the larger is the crack speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shape of the hyperelastic regions for different choices of εon (the hyperelastic regions are symmetric with respect to the crack propagation direction). The smaller εon , the larger is the hyperelastic region. The hyperelastic region takes a complex butterfly shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intersonic mode I crack. The plot shows a mode I crack in a strongly stiffening material (k1 = 4k0 ) propagating faster than the shear wave speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supersonic mode II crack. Cracks under mode II loading can propagate faster than all wave speeds in the material if there exists a local stiffening zone near the crack tip . . . . . . . . The plot shows a temporal sequence of supersonic mode II crack propagation. The field is colored according to the σxx stress component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the Broberg problem of a crack propagating in a thin stiff layer embedded in soft matrix . . . . . . . . . . . . . . . . . .

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XXXVIII List of Figures

6.46

6.47

6.48

6.49

6.50

6.51

6.52

6.53

Calculation results of the Broberg problem. The plot shows results of different calculations where the applied stress, elastic properties, and fracture surface energy are independently varied. In accordance with the concept of the characteristic energy length scale, all points fall onto the same curve and the velocity depends only on the ratio h/χ . . The plot shows the potential energy field during intersonic mode I crack propagation in the Broberg problem. Since crack motion is intersonic, there is one Mach cone associated with the shear wave speed of the solid . . . . . . . . . . . . . . . . . . . . . Crack propagation in an LJ system. Subplot (a) shows the σxx -field and indicates the mirror-mist-hackle transition. The crack velocity history (normalized by the Rayleigh wave speed) is shown in subplot (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . The concept of hyperelastic softening close to bond breaking, in comparison to the linear elastic, bond-snapping approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Force vs. atomic separation for various choices of the parameters Ξ and rbreak (these parameters are independent from each other). Whereas rbreak is used to tune the cohesive stress in the material, Ξ is used to control the amount of softening close to bond breaking . . . . . . . . . . . . . . . . . . . . . . . . . . Crack propagation in a homogeneous harmonic solid. When the crack reaches a velocity of about 73% of Rayleigh wave speed, the crack becomes unstable in the forward direction and starts to branch (the dotted line indicates the 60◦ plane of maximum hoop stress) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between hoop stresses calculated from molecular dynamics simulation with harmonic potential and those predicted by linear elastic theory for different reduced crack speeds v/cR . The plot clearly reveals development of a maximum hoop stress at an inclined angle at crack speeds beyond 73% of the Rayleigh wave speed . . . . . . . . . . . . . . . . . . . The critical instability speed as a function of the parameter rbreak , for different choices of Ξ. The results show that the instability speed varies with rbreak and thus with the cohesive stress as suggested in Gao’s model, but the Yoffe speed seems to provide an upper limit for the instability speed. The critical instability speeds are normalized with respect to the local Rayleigh wave speed, accounting for a slight stiffening effect of the moduli as shown in Fig. 4.5 . . . . .

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6.54

6.55

6.56 6.57 6.58

6.59 6.60 6.61 6.62

6.63

XXXIX

Subplot (a) schematic of stiffening materials behavior, illustrating the ratio kratio = k1 /k0 . Subplot (b) extension of the hyperelastic stiffening region. Despite the fact that the stiffening hyperelastic region is highly localized to the crack tip and extends only a few atomic spacings, the crack instability speed is larger than the Rayleigh wave speed . . . . . Molecular dynamics simulation results of instability speed for stiffening materials behavior, showing stable super-Rayleigh crack motion as observed in recent experiment. Such observation is in contrast to any existing theories, but can be explained based on the hyperelastic viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allowed velocities for mode I and mode II crack propagation, linear and nonlinear stiffening case . . . . . . . . . . . . . . . . . . . . . . . Simulation geometry for the stopping crack simulation . . . . . . The asymptotic field of maximum principal stress near a moving crack tip (a), when v = 0, (b) dynamic field for v ≈ cR , (c) dynamic field for super-Rayleigh propagation velocities (v > cR ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack extension history vs. time for the suddenly stopping linear mode I crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum principal stress field for various instants in time, mode I linear crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of principal maximum stress and potential energy along the prospective crack line, for a linear mode I crack . . . Variation of stress at fixed distance ahead of the stopped linear mode I crack. At t ≈ 0, the longitudinal wave arrives at the location where the stress is measured. At t ≈ 8, the shear wave arrives and the stress field behind the crack tip is static. The plot also includes the results of experimental studies [23] of a suddenly stopping mode I crack for qualitative comparison (the time is fitted to the MD result such that the arrival of the shear wave and the minimum at t∗ ≈ 3 match) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of static stress field radiated in front of the crack tip. The measurement at gage 1 is used for comparison with MD results. Reprinted from [23] Engineering Fracture Mechanics, Vol. 15, pp. 107–114, B.Q. Vu and V.K. Kinra, Brittle fracture of plates in tension static field radiated by a suddenly stopping crack, copyright c 1981, with permission from Elsevier . . . . . . . . . . . . . . . . . . . .

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List of Figures

6.64

6.65 6.66 6.67 6.68

6.69

6.70 6.71 6.72

6.73

6.74 6.75 6.76 6.77

Crack tip history a(t) and crack tip velocity v as a function of time, suddenly stopping mode I crack. The limiting speed according to the linear theory is denoted by the black line (Rayleigh velocity), and the super-Rayleigh terminal speed of the crack in the nonlinear material is given by the blueish line. When the crack stops, the crack speed drops to zero . . . . Evolution of principal maximum stress and potential energy along the prospective crack line; for a mode I nonlinear crack Maximum principal stress field for various instants in time, for mode I nonlinear crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of stress at fixed distance ahead of the stopped nonlinear mode I crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of waves emitted at a suddenly stopping mode II crack; (a) stopping of daughter crack, (b) stopping of mother crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of (a) principal maximum stress and (b) potential energy along the prospective crack line; for linear supersonic crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential energy field for various instants in time, mode II linear crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of stress at fixed distance ahead of the stopped intersonic mode II crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack extension history vs. time for the supersonic mode II crack. The dashed line is used to estimate the time when the mother crack comes to rest . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of (a) principal maximum stress and (b) potential energy along the prospective crack line; for nonlinear supersonic crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential energy field around the crack tip for various times, suddenly stopping mode II crack . . . . . . . . . . . . . . . . . . . . . . . . . Normalized stresses σ ∗ vs. time, suddenly stopping supersonic mode II crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the simulations of cracks at bimaterial interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack tip history and crack velocity history for a mode I crack propagating at an interface with Ξ = 10. Subplot (a) shows the crack tip history, and subplot (b) shows the crack tip velocity over time. A secondary daughter crack is born propagating at a supersonic speed with respect to the soft material layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 276 277 277

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6.78

6.79

6.80

6.81

6.82

6.83

6.84

The plot shows the stress fields σxx , σyy , and σxy for a crack at an interface with elastic mismatch Ξ = 10, before a secondary crack is nucleated. In contrast to the homogeneous case, the deformation field is asymmetric. The dark grey shades corresponds to large stresses, and the lighter grey shades to small stresses . . . . . . . . . . . . . . . . . . . . . . . The plot shows the particle velocity field (a) u˙ x and (b) u˙ y for a crack at an interface with elastic mismatch Ξ = 10, before a secondary crack is nucleated. The asymmetry of the particle velocity field is apparent . . . . . . . . . . . . . . . . . . . . . . The plot shows the potential energy field for a crack at an interface with elastic mismatch Ξ = 10. Two Mach cones in the soft solid can clearly be observed. Also, the mother and daughter crack can be seen. In the blow-up on the right, the mother (A) and daughter crack (B) are marked . . . . . . . . . . . . The plot shows the stress fields σxx , σyy , and σxy for a crack at an interface with elastic mismatch Ξ = 10. In all stress fields, the two Mach cones in the soft material are seen. The mother crack appears as surface wave behind the daughter crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The plot shows the particle velocity field (a) u˙ x and (b) u˙ y for a crack at an interface with elastic mismatch Ξ = 10. The shock fronts in the soft solid are obvious . . . . . . . . . . . . . . Atomic details of nucleation of the secondary crack under tensile dominated loading. The plot shows the shear stress field σxy near the crack tip. Atoms with the energy of a free surface are drawn as larger atoms. The plot suggests that a maximum peak of the shear stress ahead of the crack tip leads to breaking of atomic bonds and creation of new crack surfaces. After the secondary crack is nucleated (see snapshots (2) and (3)), it coalesces with the mother crack and moves supersonically through the material (snapshot (4)) Crack tip history for a mode II crack propagating at an interface with Ξ = 3. The plot illustrates the mother– daughter–granddaughter mechanism. After a secondary daughter crack is born travelling at the longitudinal wave speed of the soft material, a granddaughter crack is born at the longitudinal wave speed of the stiff material. The granddaughter crack propagates at a supersonic speed with respect to the soft material layer . . . . . . . . . . . . . . . . . . . . . . . . .

XLI

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XLII

6.85

6.86

6.87

6.88

6.89

6.90

List of Figures

Crack tip velocity history during the mother–daughter– granddaughter mechanism, for elastic mismatch Ξ = 3. The plot is obtained by numerical differentiation of the crack tip history shown in Fig. 6.84. The crack speed changes abruptly at the nucleation of the daughter crack, and rather continuously as the granddaughter crack is nucleated. Characteristic wave speeds for the stiff and soft solid are indicated in the plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supersonic mode II crack motion at a bimaterial interface, stiffness ratio Ξ = 3. Subplot (a) depicts the potential energy field of a mode II crack at a bimaterial interface with Ξ = 3, supersonic crack motion. (A) mother crack, (B) daughter crack, and (C) granddaughter crack. Subplot (b) shows the allowed limiting speeds and the observed jumps in the crack speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The plot shows the potential energy field near a shear loaded interface crack with stiffness ratio Ξ = 3 (different shades of grey are used to indicate different levels of stress). The plot shows a small section around the crack tip. The crack surfaces are highlighted. In the upper left plot, the initial configuration with the starting crack is shown. As the loading is increased, the mother crack starts to propagate, eventually leading to secondary and tertiary cracks. Two Mach cones in the soft solid and one Mach cone in the stiff solid can be observed in the lower right figure, suggesting supersonic crack motion with respect to the soft material and intersonic motion with respect to the stiff material . . . . . . The plot shows the σxx field of a mode II crack at a bimaterial interface with Ξ = 3. Subplots (a) and (b) are consecutive time steps, and subplot (c) is a blowup . . . . . . . . . An interfacial crack rupturing the bond between Homalite and aluminium, experimental results. Subplot (a) shows the loading geometry, illustrating how shear loading is induced by impact loading of the lower, stiffer material. Subplot (b) shows the subsonic growth phase and subplots (c) and (d) display the intersonic crack growth phase. Reprinted from Advances in Physics, Vol. 51(4), A. Rosakis, Intersonic shear cracks and fault ruptures, pp. 1189–1257, copyright c 2002, with permission from Taylor and Francis . . . . . . . . . . Allowed velocities for mode III crack propagation, linear and nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

6.91

6.92

6.93

6.94

6.95

6.96

6.97 6.98

6.99

XLIII

Crack tip velocity history for a mode III crack propagating in a harmonic lattice for two different choices of the spring constant ki . The dotted line shows the limiting speed of the stiff reference system, and the dashed line shows the limiting speed of the soft reference system. Both soft and stiff systems approach the corresponding theoretical limiting speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode III crack propagating in a thin elastic strip that is elastically stiff. The potential energy field is shown while the crack propagates supersonically through the solid. The stiff layer width is h = 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Check of the scaling law of the mode III Broberg problem. The continuous line refers to the analytical continuum mechanics solution [24] of the problem. The parameters γ0 = 0.1029 and τ0 = 0.054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suddenly stopping mode III crack. The static field spreads out behind the shear wave front after the crack is brought to rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry used for simulating mode I fracture in silicon. The systems contain between 13,000 and 113,015 atoms A and Ly ≈ 910 ˚ A. The numerical model is with Lx ≈ 550 ˚ capable of treating up to 3,000 atoms with ReaxFF in Ωrx . . Crack propagation with a pure Tersoff potential (subplot (a)) and the hybrid ReaxFF-Tersoff model (subplot (b)) along the [110] direction (energy minimization scheme). The darker regions are Tersoff atoms, whereas the brighter regions are reactive atoms. The systems contain 28,000 atoms and Lx ≈ 270 ˚ A × Ly ≈ 460 ˚ A. . . . . . . . . . . . . . . . . . . . . . Crack dynamics along the [110] direction at finite temperature (T ≈ 300 K), 10% strain applied . . . . . . . . . . . . . . Crack dynamics along the [100] direction at finite temperature (T ≈ 300 K, 10% strain applied). Shortly after nucleation of the primary crack two major branches develop along [110] directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack speed as a function of load, for the (110) system (subplot (a), and the (111) system (subplot (b)) . . . . . . . . . . .

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XLIV

List of Figures

6.100 Dependence of the average crack velocity in a single crystal of silicon, as a function of the steady-state energy release rate, as obtained in experimental studies. The fracture surface is smooth and mirror-like over the entire crack path for the specimen fractured at the lowest G (open circle). A faceted fracture surface is observed at higher G (triangles). At the highest G, the fracture surface is very rough (squares). The continuous line corresponds to the continuum mechanical solution obtained from an expression similar to the one reviewed in (6.34) [25]. Reprinted from: T. Cramer, A. Wanner, and P. Gumbsch, Physical Review c 2000 Letters, Vol. 85(4), 2000, pp. 788–791. Copyright by the American Physical Society . . . . . . . . . . . . . . . . . . . . . . . . 6.101 This plot shows the dynamic fracture toughness as a function of the energy release rate (subplot (a)) and the average crack velocity (subplot (b)). Reprinted from: T. Cramer, A. Wanner, and P. Gumbsch, Physical Review c 2000 Letters, Vol. 85(4), 2000, pp. 788–791. Copyright by the American Physical Society . . . . . . . . . . . . . . . . . . . . . . . . 6.102 Series of snapshots of fracture mechanics in silicon, for a crack oriented in the (111) plane. The figure shows the dynamics of crack extension, leading to failure of the entire crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.103 Velocity–time history of the crack dynamics shown in Fig. 6.102. Soon after nucleation of the crack, the speeds jumps to values of approximately 2 km s−1 . The crack instability sets in at approximately 69% of cR , the Rayleigh wave speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.104 Analysis of the sequence of atomic events during fracture initiation. This series of snapshots show the formation of the 5–7 ring defect at the tip of the crack (a blow-up of this defect structure is shown in subplot (b)) . . . . . . . . . . . . . . . . . . . 6.105 Comparison of the prediction of LEFM, the prediction of the modified LEFM model (6.129), and molecular dynamics simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.106 Comparison of the crack dynamics under mode I (subplot (a)) and mode II loading (subplot (b)) [26] . . . . . . . . . . . . . . . .

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XLV

6.107 Crack dynamics in silicon without oxygen (O2 molecules) (subplots (a) and (c)) and with oxygen molecules present (subplots (b) and (d)). Subplots (a) and (b) show the results for 5% applied strain, whereas subplots (c) and (d) show the results for 10% applied strain. The darker grey regions are Tersoff atoms, whereas the brighter regions correspond to ReaxFF atoms. The systems contain A × Ly ≈ 310 ˚ A. This about 13,000 atoms, with Lx ≈ 160 ˚ demonstrates the dramatic effect of oxygen in making Si brittle 318 6.108 This plot shows the difference in large-strain elasticity between the Tersoff potential and ReaxFF, while both descriptions coincide at small strain. This result demonstrates the importance of large-strain properties close to breaking of atomic bonds [27–29] . . . . . . . . . . . . . . . . . . . . . . 319 6.109 Different length scales associated with dynamic fracture. Subplot (a) shows the classical picture [22], and subplot (b) shows the picture with the new concept of the characteristic energy length χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 7.1

7.2 7.3 7.4

7.5

7.6 7.7

This plot illustrates the difference between a screw dislocation (marked as “S”) and an edge dislocation (marked as “E”). The graph depicts a crystal with a curved dislocation line (the thick curved line). When bl, the dislocation has screw character, and when b⊥l the dislocation has edge character . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of splitting of complete dislocation into two partial dislocations in an FCC lattice . . . . . . . . . . . . . . . . . . . . . Geometry to explain the Schmid law, illustrating the angles φ and λ as well as the uniaxially applied stress σy . . . . . . . . . . A cracked body, with remotely applied tensile and shear stresses σ∞ and τ∞ . Large resolved shear stresses on specific slip planes are the key drivers for dislocation nucleation . . . . . Balance of forces on a dislocation close to a crack, here illustrating the competition between the image force pulling the dislocation back to the surface (Fim ) and the stress induced force pushing the dislocation away from the crack tip (Fstress ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of a dislocation close to a surface, at distance d . . . Geometry of a cracked specimen with a penny-shaped crack (subplot (a)) and a semi-infinite crack (subplot (b)) . . . . . . . .

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XLVI

7.8

7.9 7.10

7.11 7.12

7.13

7.14

7.15

List of Figures

Concept of stacking fault energy, considering Peierl’s concept of periodic shear stress variation along a slip plane, as originally proposed in [30]. Subplot (a) depicts the geometry, subplot (b) the variation of the shear stress with the distance δ from the crack tip (coordinate system shown in subplot (a)), and subplot (c) depicts the variation of the elastic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balance of energies before (1) and after (2) nucleation of the dislocation, to illustrate the concept of energy release rate . . . Calculation of generalized stacking fault (GSF) curves for different EAM potentials fitted to nickel. We consider potentials by Oh and Johnson [31] (O&J), Angelo et al. [32] (AFB), Mishin et al. [33] (M&F) as well as Voter and Chen [34, 35]. Subplot (a) illustrates the calculation method of the GSF curve by sliding two parts of the crystal along the [112] direction. Subplot (b) shows the GSF curves for the four different potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The multiplicative decomposition F = Fe Fp in continuum theory of plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation geometry, lattice orientation, and time-sequence of the work-hardening simulation. (a) Simulation geometry and (b) lattice orientation, also defining the directions for the x, y, z coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interaction of a dislocation line with an obstacle in the material. Subplot (a) shows a sequence of events that illustrate how the dislocation line becomes bent as it cannot pass through the obstacle. Subplot (b) depicts a schematic that shows an equivalent force acting on the dislocation line . Dislocation–particle interaction in ordered matrix materials. A dissociated complete dislocation interacts individually with the particle (subplot (a) for a schematic). A TEM weak-beam micrograph of the dislocation–particle interaction in Fe–30 at.%Al is shown in subplot (b). Reprinted from Acta Metallurgica, Vol. 46(16), pp. 5611–5626, E. Arzt, Size effects in materials due to microstructural and dimensional constrains: A comparative c 1998, with permission from Elsevier . . . . review, copyright Schematic that illustrates the formation of vacancies while jogs (visible as kinks in the dislocation line) are forced to move through the crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.16

7.17

7.18

7.19

7.20

7.21

XLVII

Schematic of different dislocation cutting processes. Subplot (a) shows two partial dislocations cutting each other. Both dislocations leave a trail of point defects after intersection (circles). The arrows indicate the velocity vectors of the dislocations. Subplot (b) shows a partial dislocation (black line) cutting the stacking fault of another partial dislocation. Dislocation number 1 leaves a trail of point defects (circles) once it hits the stacking fault generated by dislocation number 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomistic simulation results of different types of point defects: (a) Trail of partial point defects, (b) vacancy tube, and (c) trail of interstitials. The inlays provide a detailed atomistic view of the defect structure . . . . . . . . . . . . . . . . . . . . . Generation of point defects due to jogs in screw dislocations. Two representative dislocation-cutting processes are shown, (a) leading to formation of an interstitial, (b) leading to formation of vacancy tubes. In case the edge component of the jog is smaller than that of a partial Burgers vector, trails of partial point defects, characterized by generation of local lattice distortion rather than complete rows of missing or additional atoms, are generated . . . . . . . . . . . . . . . . . . . . . . . . Generation of trails of point defects in early stages of the simulation. Dislocation number 1 and number 2 leave a stacking fault plane, which is subsequently cut by dislocation number 3. Therefore, two trails of partial point defects are generated resulting in bowing of dislocation number 3. Subplot (a) shows a centrosymmetry analysis [36] where the stacking fault planes are drawn yellow; subplot (b) shows an energy analysis of the same region where the stacking fault planes are not shown. Subplot (c) shows a close-up view on the dislocation cutting process . . . . . . . . . . . . This plot shows the reaction of the two dislocation clouds originating from opposing crack tips, causing the generation of numerous point defects. The circles highlight the region of interest in which the dislocation reactions occur . . . . . . . . . Activation of secondary slip systems and generation of Lomer–Cottrell locks. Subplot (a) shows a schematic of the cross-slip mechanism. Subplots (b) and (c) show details of activation of secondary slip systems (subplot (c) represents a magnified view of subplot (b), with the region of interest highlighted by a circle). This mechanism of cross-slip of partial dislocations, here first observed in molecular dynamics simulation, was originally proposed theoretically by Fleischer [37], and contrasts the well-known Friedel–Escaig mechanism [38] . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.22

7.23

7.24

7.25

7.26

7.27

7.28

7.29

7.30

List of Figures

Subplots (a) and (b) show a view from the [110]-direction (a) before nucleation of dislocations on secondary glide systems (therefore only straight lines), and (b) after nucleation of dislocations on secondary glide systems (which appear as curved lines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subplots (a) and (b) show detailed views on the formation of sessile Lomer–Cottrell locks, with its typical shape of a straight sessile arm connected to two partial dislocations . . . . The final network from a distant view, including a blow-up to show the details of the network [39]. The characteristic structure of the network is due to the fact that all sessile defects (both trails of partial and complete point defects) as well as sessile dislocations as part of the Lomer–Cottrell locks assume tetrahedral angles and lie on the edges of Thompson’s tetrahedron. The wiggly lines in the blow-up (see the right half of the figure) show partial dislocations, and the straight lines correspond to sessile defects . . . . . . . . . . Development of the density of different defects during the simulation using a method of separating defects of different energies [39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation geometry, lattice orientation, and time-sequence of the EAM simulation. (a) Simulation geometry and (b) lattice orientation, also defining the directions for the x, y, z coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence of snapshots that illustrate the dislocation nucleation process from the crack tip, for the case of an EAM potential. This snapshot depicts the early stages of dislocation nucleation, depicting how dislocations grow from the tips of the crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closer view on the dislocation structure at a later stage in the simulation with the EAM potential (for a system with a single crack at one surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrosymmetry analysis that shows the activation of secondary slip systems, at later stages in the simulation with the EAM potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrosymmetry analysis of the details of the dislocation structure, for the EAM simulation. In contrast to the LJ simulations, here complete dislocations (that is, leading and trailing partial dislocations) are emitted at the crack tip. As a consequence, the dislocation cutting products (partial point defects) have a finite length. Subplot (a) depicts a sequence of two snapshots, illustrating the growth of the dislocation network at the crack tip. Subplot (b) shows a detailed view of the dislocation structure . . . . . . . . . . . . . . . . . .

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7.31

7.32

7.33

7.34

7.35

XLIX

Centrosymmetry analysis of the details of the dislocation structure, illustrating that the point defect trails are of finite length in the case of the EAM simulation. Subplot (a) shows the network with stacking faults, and subplot (b) shows the network without stacking faults. The thick lines refer to the trails of point defects . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the two loading conditions considered here, shear loading (left ) and shock loading (right ). The figure also depicts which domains are handled by EAM (dark grey) and which domains are handled with ReaxFF (light grey) . . . . Centrosymmetry analysis plot of a complete dislocation emitted as two partials from a crack under shear (mode II) loading moving through EAM and ReaxFF regions (atoms with perfect FCC coordination have been removed). The dashed black circles indicate the region of atoms modeled by ReaxFF potential (the exterior of the circle contains a skin of ghost atoms with zero weight on the forces, while the entire interior is completely reactive). Subplots (a–e) are taken at time interval increments of 500 fs each. The distance between the partials appears unchanged throughout the nucleation and propagation process, and does not change as it passes through the handshaking and ReaxFF regions and back into the EAM region . . . . . . . . . . . . . Velocity profile in x-direction for a uniaxial shock wave in a system with ReaxFF and EAM regions coupled [40, 41]. The shock wave front can be identified as a vertical line of high velocity (darker color) atoms. The dotted circles contain all atoms modeled by ReaxFF potential (the exterior of the circle contains a skin of ghost atoms with zero weight on the forces, while the entire interior is completely reactive). Subplots (a), (b), and (c) depict the velocity profile in the x-direction across the sample as the shock wave passes through. There appears to be no change in shock front profile as it encounters and passes through the reactive regions, indicating a smooth handshaking between the two simulation methods without force discontinuities . . . . . . . . . . . Partial dislocation emission from crack tip for the 110112 crack orientation (only part of the crystal close to the crack tip is shown), for the case when no oxygen atoms present (all-atom EAM) [40, 41]. Defects produced at the crack tip are circled. Subplots (a) and (b) show emission of partials from top and bottom of the crack tip at critical strains of 0.039 and 0.041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

7.36

7.37

7.38

Partial dislocation emission from crack tip for the 110112 crack orientation (only part of the crystal close to the crack tip is shown), under presence of oxygen molecules in the void inclusion (same coordinate system as shown in (Fig. 7.35) [40, 41]. Subplots (a),(b) and (c),(d) depict results for two different starting positions of a single O2 molecule in the crack ((a) and (b): oxygen molecule at the tip of the crack, (c) and (d): oxygen molecule at the side of the crack face). Defects produced at the crack tip are circled and the dotted line indicates the approximate region of atoms modeled by ReaxFF. Subplot (a) shows the structure of the crack tip after 10,000 equilibrium molecular dynamics integration steps, and (b) shows the first partial initiating at a strain of 0.049. Subplot (c) depicts the resulting crack structure after 10,000 molecular dynamics integration steps for a different starting position of the O2 molecule (placed at the side of the crack), and (d) shows the partial initiation for this case at a strain of 0.041. The figures show lattice defects produced in the bulk crystal surrounding the crack as a result of oxidation at crack surface, even before strain is applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 Partial dislocation emission from crack tip for the 111112 case, results for the case without oxygen molecules. Subplot (a) shows brittle crack opening for the no oxygen case at a strain of 0.05, but the results depicted in subplot (b) shows that the crack tip emits dislocations as well at a strain of 0.051 [40, 41] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Partial dislocation emission from crack tip for the 111112 case, results for the case with oxygen molecules. Subplot (a) shows the structure of the crack tip with O2 after 10,000 equilibrium molecular dynamics integration steps. The differently shaded region to the left of the crack represents a grain boundary (GB) that has formed during the oxidation process. In subplot (b) at a strain of 0.03 a void has initiated close to crack tip. The crack tip starts to open up in subplot (c) at a strain of 0.05, indicating a brittle fracture mode [40, 41] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

List of Figures

LI

7.39

Summary of critical failure strains for different cases studied (a schematic of each case is shown below the graph). The results indicate that the failure strain is drastically reduced under presence of oxygen molecules, almost by a factor of two. However, the location of where the oxygen attacks matters: If the attack occurs in the vicinity of the crack tip (such as in the first two cases on the left), the failure strain is reduced (and found to be in the range of 2–3%). If oxygen attack occurs away from the crack tip, the failure strain is not reduced and is then very close to the case without any oxygen present (at approximately 5%) [40, 41] . . . . . . . . . . . . . . 371

8.1

This graph shows a proposed deformation-mechanism map for nanocrystalline materials obtained from molecular dynamics simulation results. The map shows three distinct regions in which either complete extended dislocations (Region I) or partial dislocations (Region II), or no dislocations at all (Region III) exist during the low-temperature deformation of nanocrystalline FCC metals. Reprinted with permission from Macmillan c 2004 . . . . . . . . . . . . . . 374 Publishers Ltd, Nature Materials [42] High resolution TEM images of grain boundaries in electrodeposited nanocrystalline Ni (a) and nanocrystalline Cu (b). The samples were produced by gas-phase condensation. The grain boundaries constitutes itself as a very narrow region between crystals of different orientation. Reprinted from Acta Materialia, Vol. 51, K.S. Kumar, H. Van Swygenhoven and S. Suresh, Mechanical behavior of nanocrystalline metals and alloys, pp. 5743–5774, copyright c 2003, with permission from Elsevier . . . . . . . . . . . . . . . . . . . . 375 Normalized critical stress as a function of grain size, d. The plot illustrates the variation of the flow stress (normalized by the value of the critical flow stress at the maximum strength at the transition grain size). The transition from the hardening regime to the softening regime is associated with an increased role of grain boundary mechanisms (the dashed line was added to the plot to guide the eye). Reprinted from Acta Materialia, Vol. 51, K.S. Kumar, H. Van Swygenhoven and S. Suresh, Mechanical behavior of nanocrystalline metals and alloys, pp. 5743–5774, copyright c 2003, with permission from Elsevier . . . . . . . . . . . . . . . . . . . . 376

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8.5

8.6

8.7

Illustration of the maximum in the strength of nanocrystalline copper, as shown by molecular dynamics simulation of up to 100 million atoms [43]. Panel (a) shows the stress–strain curves for ten simulations with varying grain sizes. Panel (b) depicts the flow stress, defined as the average stress in the strain interval from 7 to 10% deformation. The error bars indicate the fluctuations in this strain interval (1 standard deviation). A maximum in the flow stress is seen for grain sizes of 10–15 nm, caused by a shift from grain boundary mediated to dislocation mediated plasticity. Reprinted from Science, Vol. 301, J. Schiotz and K.W. Jacobsen, A Maximum in the Strength of c 2003, with permission Nanocrystalline Copper, copyright from AAAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Size dependence of a bioinspired metallic nanocomposite (schematic of structure shown in left part) [44, 45], illustrating the existence of “a strongest size” [322]. The increase in strength for building block dimensions larger than approximately 50 nm scales rather well according to the Hall–Petch relationship given in (8.1) . . . . . . . . . . . . . . . . . . . . . Study of size-scale effects in inorganic materials by using a focused ion beam (FIB) microscope machining technique, combined with nanoidentation experiments. Ultra-small pillars of different sizes were created using FIB and subsequently deformed plastically under compression from the top surface. Subplot (a) shows a SEM image of the microsample after testing. The dislocation slip lines are clearly visible at the surface. Subplot (b) shows the dependence of the yield strength on the inverse of the square root of the sample diameter for Ni3 Al-Ta. The linear fit to the data predicts a transition from bulk to size limited behavior at approximately 42 µm. The parameter σys denotes the stress for breakaway flow. Reprinted from Science, M.D. Uchic, D.M. Dimiduk, J.N. Florando and W.D. Nix, Vol. 305(9), pp. 987–989, Sample Dimensions c Influence Strength and Crystal Plasticity, copyright 2004, with permission from AAAS . . . . . . . . . . . . . . . . . . . . . . . . Polycrystalline thin film geometry. A thin polycrystalline copper film is bond to a substrate (e.g., silicon). The grain boundaries are typically predominantly orthogonal to the film surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.8

8.9

8.10

8.11

Experimental results of the thin film strength dependence at room temperature, here for copper thin films plotted as the inverse of the film thickness. With decreasing film thickness, flow stress initially rises, but then exhibits a plateau at approximately 630 MPa for films 400 nm and thinner. Each data point is an average of flow stresses from several thermal cycles, with a scatter of less than 5% in each case. Reprinted from [51] Acta Materialia, Vol. 51, T.J. Balk, G. Dehm and E. Arzt, Parallel glide: Unexpected dislocation motion parallel to the substrate in ultrathin copper films, copyright c 2003, with permission from Elsevier . . . . . . pp. 4471–4485, Discrete dislocation dynamics simulation analysis of the strength dependence of a thin metal film. Subplot (a) shows the discrete dislocation model of the polycrystalline film on a semi-infinite elastic substrate. Subplot (b) shows the average film strength at T = 400 K vs. the film thickness hf . Reprinted from Thin Solid Films, Vol. 479(1–2), L. Nicola, E. Van der Giessen and A. Needleman, Size effects in polycrystalline thin films analyzed by discrete dislocation c 2005, with permission from Elsevier . . plasticity, copyright Distribution of dislocations, result from a discrete dislocation dynamics simulation analysis. The plot depicts the dislocation distribution and the stress field σxx at T = 400 K for films with a grain size of 1 µm and various film thicknesses. The plus and minus symbols denote positive and negative dislocations according to the sign convention defined in Fig. 8.9a (all dimensions are in µm). Reprinted from Thin Solid Films, Vol. 479(1–2), L. Nicola, E. Van der Giessen, A. Needleman, Size effects in polycrystalline thin films analyzed by discrete dislocation plasticity, copyright c 2005, with permission from Elsevier . . . . . . . . . . . . . . . . . . . . Change of maximum shear stress due to formation of the diffusion wedge. In the case of no traction relaxation along the grain boundary, the largest shear stress occur on inclined glide planes relative to the free surface. When tractions are relaxed, the largest shear stresses occur on glide planes parallel to the film surface . . . . . . . . . . . . . . . . . . . .

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8.13 8.14

8.15 8.16

8.17 8.18 8.19

8.20

8.21 8.22

8.23

List of Figures

TEM micrographs of an unpassivated copper film showing parallel glide dislocations (subplot (a), Cu film with film thickness hf = 200 nm), and a passivated copper film showing threading dislocations (subplot (b), self-passivated Cu-1%Al film, film thickness hf = 200 nm). Reprinted from [51] Acta Materialia, Vol. 51, T.J. Balk, G. Dehm and E. Arzt, Parallel glide: unexpected dislocation motion parallel to the substrate in ultrathin copper films, pp. 4471–4485, c 2003, with permission from Elsevier . . . . . . . . . . . copyright Mechanism of constrained diffusional creep in thin films as proposed by Gao et al. [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of grain boundary opening ux normalized by a Burgers vector over time, for the case of a copper film on a rigid substrate. The loading σ0 is chosen such that the opening displacement at the film surface (ζ = 0) at t → ∞ is one Burgers vector [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traction along the grain boundary for various instants in time [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress intensity factor normalized by the corresponding value of a crack over the reduced time t∗ = t/τ for identical elastic properties of substrate and film material (isotropic case), rigid substrate (copper film and rigid substrate), and soft substrate (aluminum film and epoxy substrate) [46–48] . . Geometry and coordinate system of the continuum mechanics model of constrained diffusional creep . . . . . . . . . . . Image stress on a single edge dislocation in nanoscale thin film constrained by a rigid substrate . . . . . . . . . . . . . . . . . . . . . . Critical stress as a function of film thickness for stability of one, two, and three dislocations in a thin film. The critical stress for the stability of one dislocation (continuous line) is taken from the analysis shown in Fig. 8.18. The curves for more dislocations (dashed lines) in the grain boundary are estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rice–Thomson model for nucleation of parallel glide dislocations. Subplot (a) shows the force balance in case of a crack and subplot (b) depicts the force balance in case of a diffusion wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dislocation model for critical stress intensity factor for nucleation of parallel glide dislocations . . . . . . . . . . . . . . . . . . . . Disordered intergranular layer at high-energy grain boundary in copper at elevated temperature (85% of melting temperature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample geometry of the atomistic simulations of constrained diffusional creep in a bicrystal model . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

8.24

8.25 8.26

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8.30

8.31

8.32

Change of displacements in the vicinity of the diffusion wedge over time. The continuous dark line corresponds to the continuum mechanical solution . . . . . . . . . . . . . . . . . . . . . . . Diffusional flow of material into the grain boundary. Atoms that diffused into the grain boundary are highlighted . . . . . . . Nucleation of parallel glide dislocations from a diffusion wedge, showing the dynamical sequence of the process (from top to bottom). The arrows indicate the position of the partial dislocation nucleated from the diffusion wedge, illustrating how the dislocation moves away from the source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleation of parallel glide dislocations from a crack, showing the dynamical sequence of the process (from top to bottom). The arrows indicate the position of the partial dislocation nucleated from the crack, illustrating how the dislocation moves away from the source. Upon nucleation, a surface step is formed due to crack blunting . . . . . . . . . . . . . . . Geometry for studies of plasticity in grain triple junctions. A low-energy grain boundary is located between grains 1 and 2, and two high-energy grain boundaries are found between grains 2 and 3 and between 3 and 1 . . . . . . . . . . . . . . . Nucleation of parallel glide dislocations from a grain triple junction. The plot shows a time sequence based on a centrosymmetry analysis, showing how several dislocation half loops nucleate and grow into the grain interior . . . . . . . . . Schematic of dislocation nucleation from different types of grain boundaries. Individual misfit dislocations at low-energy grain boundaries serve as sources for dislocations. At high-energy grain boundaries, there is no inherent, specific nucleation site so that the point of largest resolved shear stress, the grain triple junction, serves as nucleation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation twinning by repeated nucleation of partial dislocations. Repeated slip of partial dislocations leads to generation of a twin grain boundary . . . . . . . . . . . . . . . . . . . . . . Dislocation junction and bowing of dislocations by jog dragging. A trail of point defects is produced at the jog in the leading dislocation, which is then repaired by the following partial dislocation (this is a similar mechanism as that shown in Figs. 7.16 and 7.17a) . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

Generation of trails of point defects. Subplot (a): Dislocation cutting processes with jog formation and generation of trails of point defects. Both dislocations leave a trail of point defects after intersection. The arrows indicate the velocity vector of the dislocations. Subplot (b): Nucleation of dislocations on different glide planes from grain boundaries generate a jog in the dislocation line that causes generation of trails of point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry for the studies of plasticity in polycrystalline simulation sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomistic model of the polycrystalline thin film. Only surfaces (brighter coloring) and grain boundaries (darker color) are shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleation of threading dislocations in a polycrystalline thin film. Subplot (a) shows a view into the interior, illustrating how threading dislocations glide by leaving an interfacial segment. Subplot (b) shows a top view into the grain where the surface is not shown. The plot reveals that the dislocation density is much higher in grains 3 and 4 . . . . . Surface view of the film for different times. The threading dislocations inside the film leave surface steps that appear as darker lines in the visualization scheme. This plot further illustrates that the dislocation density in grains 3 and 4 is much higher than in the two other grains . . . . . . . . . . . . . . . . . . Detailed view onto the surface (magnified view of snapshot 4 in Fig. 8.37). The plot shows creation of steps due to motion of threading dislocations. The surface steps emanate from the grain boundaries, suggesting that dislocations are nucleated at the grain boundary–surface interface. From the direction of the surface steps it is evident that different glide planes are activated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence of a nucleation of a threading dislocation, view at an inclined angle from the film surface. Threading dislocations are preferably nucleated at the grain boundary–surface interface and half-loops grow into the film until they reach the substrate. Due to the constraint by the substrate, threading dislocations leave an interfacial segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.40

8.41

8.42

8.43

8.44

8.45

Nucleation of parallel glide dislocations, small grain sizes. The plot shows that dislocation activity centers on the grain boundary whose traction is relaxed. Due to the crack-like deformation field, large shear stresses on glide planes parallel to the film surface develop and cause nucleation of parallel glide (PG) dislocations. Subplot (a) shows a top view and subplot (b) shows a perspective view. The plot reveals that there are also threading (T) dislocations nucleated from the grain triple junctions . . . . . . . . . . . . . . . . . . Nucleation of parallel glide dislocations, large grains. The plot shows a top view of two consecutive snapshots. The region “A” is shown as a blow-up in Fig. 8.43 . . . . . . . . . . . . . . Nucleation of parallel glide dislocations, large grains. The plot shows a view of the surface. From the surface view it is evident that threading dislocations are nucleated in addition to the parallel glide dislocations. These emanate preferably from the interface of grain boundaries, traction-free grain boundaries, and the surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleation of parallel glide dislocations. The plot shows an analysis of the complex dislocation network of partial parallel glide dislocations that develops inside the grains (magnified view of the region “A” marked in Fig. 8.41). All defects besides stacking fault planes are shown in this plot . . . Nanostructured material with twin grain boundary nanosubstructure. The light gray lines inside the grains refer to the intragrain twin grain boundaries. The thickness of the twin lamella is denoted by dT . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of nanostructured material with twin lamella substructure under uniaxial loading for two different twin lamella thicknesses. Subplot (a) shows the results for thick twin lamella (dT ≈ 15 nm > d) and subplot (b) for thinner twin lamella (dT ≈ 2.5 nm). Motion of dislocations is effectively hindered at twin grain boundaries . . . . . . . . . . . . .

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List of Figures

Simulation results of nanostructured material with twin lamella substructure under uniaxial loading for two different twin lamella thicknesses, all high-energy grain boundaries. Subplot (a) shows the potential energy field after uniaxial loading was applied. Interesting regions are highlighted by a circle. Unlike in Fig. 8.45, dislocations are now nucleated at all grain boundaries. The nucleation of dislocations is now governed by the resolved shear stress on different glide planes. Subplot (b) highlights an interesting region in the right half where i. cross-slip, ii. stacking fault planes generated by motion of partial dislocations and iii. intersection of stacking fault planes left by dislocations is observed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Modeling of constrained diffusional creep in polycrystalline samples; nucleation of threading vs. parallel glide dislocations [49]. The blowup in panel (c) shows an energy analysis of the dislocation structure and visualizes a parallel glide dislocation nucleated from a grain boundary diffusion wedge. The surface steps indicate that threading dislocations have moved through the grain and no threading dislocations exist in grain 1. The dark lines show the network of parallel glide dislocations in grain 1 (in other grains we also find parallel glide dislocations in snapshot (d) but they are not shown) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Series of TEM micrographs of an unpassivated copper film, film thickness hf = 200 nm, showing the nucleation and propagation of parallel glide dislocations. Dislocations appear as white lines. A total of ten dislocations (numbered in the plots) are emitted sequentially from the source at the lower left triple junction. Dislocations are pushed forward by subsequently emitted dislocations, which in turn are not able to glide as far into the grain as the earlier dislocations (compare subplots (b), (d), (f), (h)). Based on their motion and on the grain geometry, dislocations must have undergone glide on the (111) glide plane parallel to the film–substrate interface. Reprinted from [51] Acta Materialia, Vol. 51, T.J. Balk, G. Dehm and E. Arzt, Parallel glide: unexpected dislocation motion parallel to the substrate in ultrathin copper films, pp. 4471–4485, copyright c 2003, with permission from Elsevier . . . . . . . . . . . . . . . . . . . . 429

List of Figures

8.49

8.50

8.51

8.52 8.53

8.54

8.55

Flow stress σY vs. the film thickness hf , as obtained from mesoscopic simulations of constrained diffusional creep and parallel glide dislocation nucleation (data taken from [50]). The results are shown for two different initiation criteria for diffusion and a film-dependent source. In the case of a local criterion for diffusion initiation, the yield stress is film-thickness independent as also observed in experiment [51] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation map of thin films, different regimes. Thin films with thicknesses in the submicron regime feature several novel mechanisms next to the deformation by threading dislocations (a). Plasticity can be dominated by diffusional creep and parallel glide dislocations (b), purely diffusional creep (c), and no stress relaxation mechanism (d) . . . . . . . . . . Deformation mechanism map of thin copper films, here focused on the yield stress. For films in the submicron regime (thinner than about 400 nm), the yield stress shows a plateau. This is the regime where diffusional creep and parallel glide dislocations dominate (regime (B) in Fig. 8.50) . Geometry of a (15,15) single wall carbon nanotube (SWNT) shown in different views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressive deformation mechanics of a CNT with length L = 6 nm and a diameter of d = 1 nm ((7,7) armchair CNT). The plot illustrates the increase of strain energy as a function of compressive strain (subplot (a)), and shows associated deformation mechanisms (subplot (b)). The analysis revealed that CNTs begin to buckle according to a shell-like behavior. Subplot (c) depicts a similar analysis, showing bending of a (13,0) CNT (length L = 8 nm and a diameter of d = 1 nm). The strain energy density increases according to a harmonic behavior until the buckling point is reached. Reprinted from: B.I. Yakobson, C.J. Brabec, and J. Bernholc, Physical Review Letters, Vol. 76(14), 1996, pp. c 1996 by the American Physical 2511–2514. Copyright Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell-rod wire transition of CNTs under compressive loading (schematically shown in subplot (a)), subplot (b) represents shell-like behavior, subplot (c) shows the behavior of CNTs as a rod, and subplot (d) shows a CNT that behaves similarly as a wire (or a long polymer monomer) [370, 371]. The series of plots illustrates the change in compressive behavior as the length of the CNT increases systematically . . Illustration of the process of coarse graining a CNT, leading to a bead-type representation [52] . . . . . . . . . . . . . . . . . . . . . . . .

LIX

433

435

436 438

439

440 441

LX

List of Figures

8.56

8.57 8.58

8.59

8.60

8.61

Atomistic analysis of the tensile properties of a (5,5) CNT [52]. Subplot (a): Stress vs. strain behavior during stretching of a (5,5) CNT, result obtained using the Tersoff potential. The straight lines show the mesoscale model that is developed based on the atomistic simulation results. Subplot (b): Fracture mechanism of a (5,5) CNT, modeled using the Tersoff potential (plots show the atomistic mechanics as the lateral tensile strain is increased). Fracture initiates by generation of localized shear defects in the hexagonal lattice of the CNT, reminiscent of 5–7 Stone–Wales defects . . . . . . . . Bundle of several individual SWNTs as obtained from a molecular simulation model [52] . . . . . . . . . . . . . . . . . . . . . . . . . . Response of a CNT bundle to mechanical compressive loading. Even for relatively small strains, the structure starts to buckle, eventually leading to significantly deformed and buckled shapes [52] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Full atomistic calculations of the properties of ultra-long CNTs (subplot (a)) [371, 372] and corresponding results obtained using the mesoscale model (subplot (b)) [52] . . . . . . . Bone-like materials consist of a hierarchical microstructure made of nanoscale constituents [53, 54]. Left : The plot depicts the microstructure of such bone-like biological materials at the smallest scale. Such materials typically consist of fragile, brittle mineral platelets embedded in protein matrix materials as soft as human skin. The combination of these two phases in a nanocomposite results in superior materials properties. In the studies, we focus on the fracture properties of mineral platelets since they play a critical role in determining the strength of these materials. Right : The tension–shear chain model showing the path of load transfer in the mineral–protein composites. The mineral platelets carry tensile load and the protein transfers loads between the platelets via shear. In this section we focus on the strength of the mineral platelets . . . . . . . . . . . . . . The geometry and dimensions of a cracked platelet. This model is used in the continuum and atomistic studies of fracture at small scales. We consider a thin strip of width ξ, in which the crack length extends half way through the length of the slab in the x-direction. The system is under mode I tensile loading as indicated in the plot (mode I loading in the y-direction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

443 444

444

445

447

448

List of Figures

8.62

8.63

8.64

8.65

8.66

Fracture and adhesion strength as a function of the size of the material. The plot depicts the results of bulk fracture as well as surface adhesion. The results are normalized with respect to the theoretical strength and normalized with respect to the critical lengthscale for flaw tolerance. These results suggest that the principle of dimension reduction is valid in a variety of systems, including surface adhesion as well as bulk fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress distribution ahead of the crack in a thin mineral platelet just before failure, for different materials sizes (the x-coordinate is scaled by the characteristic length scale ξcr ). The thinner the slab, the more homogeneous is the stress distribution. When the slab width is smaller than the critical size, the stress distribution becomes homogeneous and does not depend on the size of the platelet any more (see values ξcr /ξ > 2.67 and larger). The normal stress σyy is normalized with respect to the maximum strength at the onset of failure, σth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microscopic view of the contact elements in flies (left ) and geckos (right ). The terminal setae of flies consist of insect cuticle, that is, chitin–fiber reinforced protein, and have typical dimensions of 2 µm. The terminal elements (“spatulae”) of geckos are made of keratin and have typical dimensions of 200-nm diameter [55] (micrographs courtesy of S. Gorb, Max Planck Institute for Metals Research). Reprinted from Materials Science and Engineering: C, Vol. 26, E. Arzt, Biological and artificial attachment devices: Lessons for materials scientists from flies and geckos, pp. c 2006, with permission 1245–1250, copyright from Elsevier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The schematic of the model used for studies of adhesion. The model represents a cylindrical Gecko spatula (as shown in Fig. 8.64) with radius attached to a rigid substrate (left ). A circumferential crack represents flaws for example resulting from surface roughness. The parameter α denotes the dimension of the crack. The regime 0 < r < αR corresponds to an area of perfect adhesion, whereas αR < r < R represents regions of no adhesion. This model resembles the effect of surface roughness as depicted schematically on the right -hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The geometry of the system considered is a periodic array of punches of radius R. The rigid–elastic interface leads to singular stress concentrations for flat punches. We vary the shape of the rigid punch surfaces to avoid these singular stress concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LXI

451

452

453

454

455

LXII

8.67

8.68

8.69

8.70

8.71

8.72

List of Figures

The shape function defining the surface shape change as a function of the shape parameter. For Ψ = 1, the optimal shape is reached and stress concentrations are predicted to disappear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atom rows in the rigid punch are displaced according to the continuum mechanics solution of the optimal surface shape (theoretical solution see Fig. 8.67). This method allows achieving a smoothly varying surface and enables a continuous transition from a flat punch (left ) to the optimal shape (right ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress distribution in the rigid punch slightly before complete detachment (the stress is calculated in a thin strip along the diameter, within the area of contact Rcut = 2αR). The simulations reveal that for large radii, a stress concentration develops at the exterior sides of the cylinder. For small dimensions, this stress distribution starts to vanish. For dimensions smaller than the critical radius for flaw tolerance (large ratios of Rcut /R), the stress distribution becomes homogeneous and does not vary with the cylinder diameter any more . . . . . . . . . . . . . . . . . . . . . . . . . . Stress distribution in the elastic punch slightly before complete detachment (the stress is calculated in a thin strip along the diameter, within the area of contact Rcut = 2αR). Here we keep the dimension fixed and vary the adhesion energy (γ0 corresponds to the surface energy) and the elastic properties (E0 corresponds to the Young’s modulus obtained for k0 = 57.23). We find that thestress distribution becomes homogeneous for large ratios of Rcr /R, in agreement with the other results (see Figs. 8.63 and 8.69) . . . . . . . . . . . . . . . . . . Stress distribution along the diameter of the punch for different choices of the shape parameter describing the punch shape. The results indicate that when the optimal shape is reached (Ψ = 1), the stress distribution is completely flat as in the homogeneous case (λ = 1) without stress magnification. We observe that for Ψ < 1, a stress concentration develops at the boundaries of the punch, whereas for Ψ > 1 the largest stress occurs in the center . . . . . Adhesion strength for different choices of the shape parameter Ψ . The results indicate that although optimal adhesion can be achieved at any lengthscale by changing the shape of the attachment device (by choosing Ψ = 1), robustness with respect to variations in shape while at the same time keeping a strong adhesion force can only be achieved at small lengthscales . . . . . . . . . . . . . . . . . . . . . . . . . . . .

456

457

458

459

460

461

List of Tables

2.1 2.2 2.3

2.4

2.5 4.1

4.2

4.3

4.4

4.5

4.6

Overview of various thermodynamical ensembles (the parameter µ is the chemical potential) . . . . . . . . . . . . . . . . . . . . . Overview over force fields suitable for organic substances . . . . . Explanation of the different measures of computing power, as well as a description when it became available. In comparison a state-of-the art personal computer (PC, laptop) provides a performance of approximately 30 GFLOPS . . . . . . . . . . . . . . . . . Centrosymmetry parameter ci for various types of defects, normalized by the square of the lattice constant a20 . In the visualization scheme, we choose intervals of ci to separate different defects from each other . . . . . . . . . . . . . . . . . . . . . . . . . . . Distinguishing modeling and simulation, for tasks associated with classical molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . Elastic properties and wave speeds associated with the harmonic potential (see (2.34)) in a two-dimensional solid for different choices of the spring constant k . . . . . . . . . . . . . . . . . . . Failure strain of the two-dimensional solid associated with the harmonic potential with snapping bonds under different modes of uniaxial loading for rbreak = 1.17 . . . . . . . . . . . . . . . . . . Elastic properties and wave speeds associated with the harmonic potential (see (2.34)) in a 3D solid for different choices of the spring constant k, cubical crystal orientation . . . Elastic properties associated with the harmonic potential (2.34) in a three-dimensional solid for different choices of the spring constant k and [110] and [111] crystal orientation . . . . . [100] [110] [111] Cohesive strains εcoh , εcoh , and εcoh for the LJ potential and the EAM potential. In all potentials, the weakest pulling direction is the [110] directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of fracture surface energies for a selection of different potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 60

79

87 91

133

136

143

144

146 147

LXIV

List of Tables

4.7

Summary of frequently used units to SI units and/or definition of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.1

Activation energy for different state transitions . . . . . . . . . . . . . . 179

6.1

Overview over wave speeds in a variety of materials, indicating the longitudinal wave speed cl , the shear wave speed cs , and the Rayleigh wave speed cR . . . . . . . . . . . . . . . . . . Critical load R0 for fracture initiation, for different values of the spring constant kp of the pinning potential. The results are in good agreement with the theory prediction when kp becomes much smaller than k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of energy flow to the crack tip, due to a bilinear softening or stiffening interatomic potential . . . . . . . . . . . . . . . . . Griffith analysis of the atomistic models, for mode I and mode II cracks, and different potentials. The predicted values based on continuum calculations agree well with the molecular dynamics simulation results . . . . . . . . . . . . . . . . . . . . . .

6.2

6.3 6.4

8.1 8.2

8.3

198

214 241

268

Material parameters for calculation of stress intensity factor over the reduced time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Critical stress intensity factor K PG for nucleation of parallel glide dislocations under various conditions, for both a diffusion wedge and a crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Summary of mesoscopic parameters derived from atomistic modeling [52] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Nomenclature

Variable U , Utot S T F K H kB Ui φ rij k, ki 0 σ F r Rcut Rtrans r˙ r¨ = a Rbuf wi BOij E µ ν ρ cijkl S, Ψ

Description Total potential energy of a system Entropy of a system Temperature (in Kelvin) Free energy of a system Kinetic energy of a system Total Hamiltonian of a system Boltzmann constant Potential energy of atom i Interatomic potential energy function Distance between atom i and atom j Spring constant of the harmonic potential Energy well depth, Lennard-Jones potential Length parameter, Lennard-Jones potential Atomic force vector Atomic position vector Cutoff radius for interatomic potential Size of transition region in multiparadigm model Atomic velocity Atomic acceleration Size of buffer region in multiparadigm model Relative weight of force field engine in multiparadigm model Bond order of pair of atoms (ReaxFF model) Young’s modulus Shear modulus Poisson’s ratio Material mass density Elasticity tensor coefficients Strain energy density

LXVI

Nomenclature

Pi P ron rbreak kratio εon rH χ t tl tinit H(s) δij a a˙ = v u, ui u˙ i A L c0 cR cs cl Θ r σij εij σi , εi σΘ , εΘ σY σth , τth γs γus γsf ∆γ G K, KI,II,III

Dynamic Poynting vector component Magnitude of dynamic Poynting vector Critical atomic separation for onset of hyperelastic effect (biharmonic potential) Critical atomic separation for bond breaking (biharmonic potential) Ratio of large-strain to small-strain spring constant Critical onset strain for hyperelastic effect (biharmonic potential) (Effective) size of hyperelastic region Characteristic energy lengthscale Time (usually in reduced atomic units) Loading time of a cracked crystal Fracture initiation time Heaviside unit step function Kronecker delta function Crack length Time derivative of crack length (=crack speed) Displacement vector (i stands for the spatial direction component) Particle velocity vector Area (for instance, cross-sectional area of beam) Length (for instance, length of beam) Wave speed in a one-dimensional model string of atoms Rayleigh wave speed Shear wave speed Longitudinal wave speed Angle in cylindrical coordinate system at crack tip Radial direction in cylindrical coordinate system at crack tip Stress tensor Strain tensor Principal stress/strain Hoop stress/hoop strain Yield stress Theoretical maximum stress or shear stress Fracture surface energy Unstable stacking fault energy Stacking fault energy Adhesion energy Energy release rate in fracture problems Stress intensity factor (loading modes I, II, III)

Nomenclature

Σij m, mi Ξ ci sα i bi , b b a0 hf d ξ ξp R τ δz Th δgb Dgb D Ds

Characteristic variation of asymptotic stress field near static or moving crack Atomic mass Elastic mismatch across bimaterial interface Centrosymmetry parameter of atom i Slip vector component i of atom α Burgers vector (components and vector) Magnitude of Burgers vector (=| b |) Lattice constant of cubic metals Film thickness Grain size in a polycrystalline material Characteristic dimension of nanocrystal Persistence length of a polymer chain Characteristic radius of adhesion punch Characteristic time for stress relaxation Diffusive displacement Homologous temperature Grain boundary diffusivity Diffusivity Surface diffusivity

LXVII

Part I

Introduction

1 Introduction

Catastrophic phenomena that afflict millions of lives, ranging from the failure of the Earth’s crust in earthquakes, to the collapse of buildings, to the failure of bones due to injuries, all have one common underlying theme: the breakdown of the basic constituents of any material ultimately leads to the failure of its overall structure and intended function. Understanding how materials fail has always been of great importance to enable and advance technologies. Over several thousands of years, the knowledge about the behavior of materials in extreme conditions has furnished the way for modern technologies, by enabling the use of materials for a variety of applications. At this moment we are at the crossroads to a new era where humans, for the first time, start creating structures and technologies at the scale of single atoms and molecules (see Fig. 1.1). Such nanotechnology could revolutionize the way we live, learn, and organize our lives in the decades to come. Computational modeling, in particular atomistic and molecular simulation, is becoming increasingly important in the development of such new technologies. At nanoscale, the effects of single atoms, individual molecule, or nanostructural features may dominate the material behavior. Thus novel modeling and simulation approaches are a vital component in enabling the engineering design process. Atomistic models, by providing a material description that starts at a fundamental scale, are thus expected to be important not only for scientists but also for engineers. How will engineers of the future treat nanoscale systems? Atomistic, mesoscale and multiscale modeling, and simulation as discussed in this book may provide an important tool for future engineers. How can these techniques be applied to solve problems? What kind of information can be obtained, and how can this be related to existing theories, methods, and concepts, and most importantly, experimental studies? How do existing engineering theories relate to the results obtained from advanced simulation approaches? This book will address some of these questions by providing a fundamental description of the various aspects of atomistic and molecular simulation approaches. In addition to practical questions regarding algorithms and implementation, this book explores the science behind fracture

4


Fig. 1.1 Illustration of how the characteristic material scales of technological eras have been reduced from the scales of meters to the scales of individual molecules and atoms. The current technological frontier is the development of molecular and atomistic structures at the interface of physics, biology and chemistry, leading to a new bottom-up approach in creating and characterizing materials

and deformation of different classes of materials, linking the viewpoints of scientific disciplines such as chemistry and physics with those of engineers. This merger of scientific and engineering disciplines is an exciting opportunity that results from research at this fundamental, atomic scale. This book will provide readers with a basic understanding about the fundamentals, application areas, and potential of classical molecular dynamics for problems in mechanics of materials. Another emphasis is on developing a sensitivity for the significance of mechanics in different areas, and how atomistic and continuum viewpoints can be integrated to build a new platform of control in the analysis and description of the behavior of complex materials under extreme conditions. The focus of this book is on atomistic modeling of small-scale dynamics phenomena with large-scale simulations, combined with theoretical methods such as continuum mechanics. Throughout the text, the reader will find a coherent discussion of theory, simulation results, and experimental studies. The discussion of model building and analysis of simulation results illustrate some of the critical aspects of this computational tool. A core contribution of this book to the available literature in this field is the comparative discussion of atomistic simulation results with continuum theories and other engineering concepts. The main question we are concerned with

1 Introduction

5

is how materials fail under extreme conditions, and how the microscopic or macroscopic failure processes are related to atomistic details.

1.1 Materials Deformation and Fracture Phenomena: Why and How Things Break When materials are deformed, they display a small regime where deformation is reversible, a behavior referred to as the elastic regime. Once the forces on the material are increased, deformation becomes irreversible, and the deformation of a body caused by the applied stress remains after the stress is removed. This behavior is referred to as the plastic regime, and may cause the material to fail [1]. In terms of thermodynamics, elastic deformation is characterized by a reversible process. That is, all mechanical work done on the system is fully recoverable. In contrast, materials failure represents an irreversible process. That is, not all mechanical work done on the system can be recovered, as it is dissipated during processes associated with permanent deformation. Materials can fail in many ways. Brittle materials like glass shatters and quickly breaks into many small pieces. Ductile metals can be deformed permanently without breaking, with moderate resistance against the forces. Many biological tissues such as skin, spider silk, or polymers and rubber are capable to sustain quite large deformation before they break suddenly. Figure 1.2 depicts a schematic stress–strain curve, comparing a brittle and a ductile material, introducing one of the most fundamental distinction of material behavior. The well-known failure mechanisms are macroscopic phenomena and its manifestations are often visible to our eyes. However, the origin of virtually all

Fig. 1.2 The plot shows simple, schematic stress–strain diagrams characteristic for a brittle and a ductile material. Similar curves are found for other materials, including polymers or rubber-like materials. The cross symbol (“x”) indicates the point of material failure [1]

6


deformation and fracture phenomena lies at much smaller, atomistic scales. Whether or not a material behaves brittle or ductile, for instance, depends strongly on how the atoms and molecules are arranged inside the material, and how these structures respond to an applied load. Much of this book is dedicated to elucidate the question of the underlying atomistic and molecular deformation mechanisms and how they can be linked to macroscale phenomena that are measured in engineering laboratories. In many cases, what matters for an engineering application is the behavior at larger macroscopic scales.

1.2 Strength of Materials: Flaws, Defects, and a Perfect Material How strong are materials, and what are the fundamental mechanisms and features in materials that control their deformation behavior and their strength limit? These aspects have been subject to studies in a variety of scientific and engineering disciplines. Materials have been studied by physicists, often considering the smallest lengthscales, featuring a few atoms and below. On the other hand, chemists considered the bonding between different atoms or the interactions of different chemical compounds and molecules. Engineers, on the other hand, have mainly used continuum descriptions of materials, considering the material as matter that can be divided infinitely many times. These traditional engineering theories have their origin in neglecting the discreteness of matter, which is a reasonable assumption when dealing with larger structures that feature characteristic geometric dimensions much larger than the inhomogeneities of the material (e.g., molecular structure, grains, particles). Research carried out over the last few decades revealed that the integration of these different viewpoints, that is, those of physics, chemistry, and engineering is critical to make important breakthroughs to understand and improve the mechanical properties of materials. As we shall discuss in the forthcoming chapters, the mechanical properties of materials are strongly influenced by the presence of defects or imperfections. Nothing is perfect! There are many defects in a material, even though through our eyes it may appear as if no defects are visible. For example a piece of small copper wire will include many millions of crystal imperfections such as voids, cracks, inclusions of foreign materials, and interfaces between crystal grains of different orientations. The list of possible defects in crystalline materials can be extended much further. All these defect structures have in common that they represent a deviation from the ideal, perfect crystal lattice that is considered the reference structure. Why do defects matter? The reason is that upon application of an external load to the material, these imperfections lead to a magnification of the local stresses, and thereby induce failure since the forces between atoms become so large that the chemical bonds are sheared or ruptured. If this occurs collectively over larger lengthscales, the material undergoes permanent

1 Introduction

7

Fig. 1.3 Homogeneous material (subplot (a)) and material with elliptical hole (subplot (b), length of elliptical hole is 2a). The presence of the elliptical void leads to a magnification of the stress in the vicinity of the tip of the defect (see schematic illustration of stress profile)

shape change or begins to fracture, if a large number of interatomic bonds are compromised. The effect of a defect in changing the distribution of forces within the material can be demonstrated mathematically by using a simple example that compares a material with and without an elliptical hole as shown in Fig. 1.3. For the homogeneous material (Fig. 1.3a), the stress inside the material is identical to the applied stress. However, the presence of an elliptical hole leads to a magnification of the stress in the vicinity of the corners of the hole (Fig. 1.3b). The stress magnification was first quantified by Inglis in 1913 [56], and it can be shown as: a σyy = σ 1 + 2 , (1.1) ρ where 2a is the extension of the elliptical inclusion in the x-direction and ρ is the radius of curvature at the sharp corners. This equation includes the special case of a homogeneous material (for a = 0), when σyy = σ. This shows that in this case the stress magnification vanishes. The sharper the radius (that is, for smaller ρ), or the longer the elliptical hole (that is, for larger a), the greater the stress magnification at the tip of the hole. This example illustrates the reason why defects can be sources of failure, since for sufficiently large loads or sufficiently sharp holes, the local stresses are large enough to break or shear the bonds between the atoms. The details of how exactly these bonds respond to extreme loading conditions control the overall material behavior. It all starts at the atomic scale!

8


Fig. 1.4 Schematic illustration of a failure process by crack extension in a brittle material. The inlay in the center shows how chemical bonds rupture continuously, leading to formation of new fracture surfaces

One mode of failure is the repeated rupture of interatomic bonds, leading to spreading of fractures in the material as shown in Fig. 1.4. This example further illustrates that in order to understand materials failure, it is not simply possible to average over microscopic features. The presence of a tiny defect in the material can make a huge difference, as it is the source for steady growth of a fracture that will eventually destroy the entire material or structure. 1.2.1 Crystal Structures and Molecular Packing There exists a large variety of atomic structures with distinct packing arrangements and different chemical bonds, giving rise to the numerous classes of materials. For instance, metals or ceramics typically feature a densely packed, highly organized crystal structure. Often, the bonds between these atoms are quite strong, as it is the case for metallic bonding or in covalent bonding (e.g., forming metals like titanium, copper, nickel, iron, or ceramics such as alumina, quartz, or feldspar). Metals or ceramics without an organized crystal structure are also found, with glass being a prominent example. These materials contain a random, amorphous arrangement of the atoms. Many polymers such as rubber also have a rather amorphous microstructure, where molecules are arranged in a less-ordered fashion. These materials are typically less dense and softer, because fewer or less strong atomic bonds per unit volume resist mechanical deformation. Once stretched significantly, the disordered molecules in amorphous polymers often arrange into regular patterns, and the material stiffens significantly. This phenomenon occurs for instance when a rubber band is stretched further and its resistance increases.

1 Introduction

9

Biological materials and tissues, for instance materials made out of proteins, represent an intermediate case – they can feature highly organized and ordered, quite precise molecular structures in which arrays of intermolecular H-bonds and ionic interactions provide the basis for their properties. In addition to differences in the molecular packing, many biological and natural materials feature hierarchical structures, that is, a different characteristic organization of atoms, molecules, supermolecules at different lengthscales. The particular molecular deformation mechanisms that emerge under large loads depend very strongly on the atomic, molecular, or hierarchical microstructure and the associated chemical bonds. Similarly, the definition of defects depends on the reference geometry of the material. A large part of this book is focused on the deformation and failure of crystalline materials. For this class of crystalline materials, we will explain how some of the basic deformation and fracture mechanisms can be studied using atomistic methods. Many crystalline materials are associated with great technological significance: metals being used as structural materials, as materials for electronic applications, or as decorative components in buildings. Crystalline materials such as silicon or silica play a key role in microelectronics and micromechanical (e.g., MEMS) devices. Silicon, in particular, is of great importance due to its semiconductor properties and due to its availability in a very pure form. Important crystal structures frequently found in metals or ceramics include simple cubic (SC), face centered cubic (FCC), body centered cubic (BCC), and hexagonal closest packing (HCP). A few crystal structures are shown in Fig. 1.5. Variations of these crystal structures exist for two-dimensional geometries. In this book, we will focus on deformation behavior of the simplest crystal structures, primarily studying FCC crystals and triangular lattices.

Fig. 1.5 Overview of different crystal structures, showing the SC, FCC, and BCC crystal structure

As indicated above, imperfections in the material structure may give rise to stress magnifications, leading to breaking of atomic bonds. This breaking of atomic bonds typically induces additional defects within the crystal. Some of these defects play a role as mediators of deformation, with the most prominent

10


example being dislocations. Dislocations are imperfections of a crystal lattice that allow the material to permanently change its shape. We will discuss various defects often found in crystals in the following few sections. 1.2.2 Cracks Cracks or sometimes referred to as fractures are a prominent example of a material defect. Physically, cracks are represented by surface areas inside a solid, in other words, an inclusion of “void” inside a crystal lattice. Cracks also often appear at surfaces of materials. Mathematically, cracks are represented by traction-free boundary conditions at the surface of the crack, that is, no stresses can be transmitted from one surface of the crack to the other one. Similarly as shown in Fig. 1.3, the tip of cracks typically represents locations of extremely large stresses and thus interatomic forces, which are much larger than the background load applied. In fact, an infinitely sharp crack represents a mathematical singularity for the stresses, which scale as 1 σ∼√ . r

(1.2)

Cracks are typically characterized by their geometry, most often by their size, since this parameter controls the influence of a crack on the stresses in the surrounding material (this can also be seen in (1.1)). As will be discussed later, an extensive theoretical framework has been developed that deals with the stress fields around cracks, the geometric effects for various kinds of cracks, the link between the applied load further away from the crack, and the amount of stress magnification observed in the vicinity of the crack. There exist intermediates between cracks and a perfect crystal lattice material. For instance, grain boundaries feature a reduced traction or reduced bonding strength across a geometric interface. 1.2.3 Dislocations Dislocations are crystal defects that enable the deformation of crystals, constituting the fundamental carriers of plasticity or the mediators of plasticity. The existence of dislocations was discovered when it was observed that the predicted strength of materials (based on the strength of individual atomic bonds) was much larger than the strength of materials actually measured. This meant that somehow it was impossible to take advantage of the actual bond strength and that deformation was controlled by mechanisms that operated at much lower critical stress levels. For instance, the theoretical shear strength of a material can be estimated by µ τth ≈ √ , 30

(1.3)

1 Introduction

11

where µ is the shear modulus (often between 25 and 100 GPa for many metals). The predicted shear strength would then be close to several GPa and larger. In contrast, the maximum shear resistance of crystallographic planes measured in real metals is many orders of magnitudes smaller. In 1934, Orowan, Polanyi, and Taylor proposed that this difference is due to existence of crystal defects referred to as dislocations [57–59]. The discrepancy between the theoretical shear strength and the measured values can be explained straightforwardly. Whereas the estimate in (1.3) assumes that all interatomic bonds participate collectively in shear deformation, the concept behind a dislocation is that only a few atomic bonds participate in deformation of a larger piece of a crystal. Instead of homogeneously shearing a crystal, a localized wave of displacements moves through the solid that deforms it step by step. The localization of displacements is energetically more favorable over the homogeneous deformation path, and thus, it is the dominating mechanism in the deformation of metals. Since their discovery in the early 1930s, the concept of dislocations has helped to explain many of the perplexing physical and mechanical properties of metals [57].

Fig. 1.6 Dislocations are the discrete entities that carry plastic (permanent) deformation; measured by a “Burgers vector.” The snapshots illustrate the nucleation and propagation of an edge dislocation through a crystal, leading to permanent deformation

Figure 1.6 shows the concept of a dislocation in a simple, schematic illustration. In order to nucleate a dislocation, the material must be sheared locally. This local shear requires large forces, sufficient to break atomic bonds that have formed across the sheared plane. For instance, the large stresses in the vicinity of a crack can induce nucleation of dislocations (recall that mathematically, cracks represents singularities of stress). In this sense, cracks and dislocations form a symbiotic pair: Cracks lead to large stresses that lead to nucleation of dislocations, which in turn make the material “deformable,” or ductile. The behavior of dislocations in crystals is very complex and involves multiple mechanisms for generation and annihilation, as summarized in [38, 60].

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Collective events may occur through interaction among many dislocations or between dislocations and other defects such as grain boundaries. 1.2.4 Other Defects in Crystals and Other Structures Crystals may feature many other defects, in addition to cracks and dislocations that were introduced above. For instance, if grains of crystals of different orientation interface, defects referred to as grain boundaries are formed. At the atomistic scale, grain boundaries constitute high-energy regions in which the ideal atomic lattice is disturbed in order to provide a bridge between the two crystal orientations. The presence of these high-energy domains can play a critical role in the deformation of materials. Grain boundaries may serve as sources for dislocations and as obstacles for the motion of dislocations. The interplay of these competing effects is controlled by the interaction of elastic energy of dislocations, the size of the crystal grains, and the particular structure of the grain boundary. It leads to intriguing material phenomena, such as the strengthening of materials by reducing the size of crystal grains. Other crystal defects include point defects, that is, missing atoms at lattice sites typically referred to as vacancies, interstitial atoms represented by an additional atom situated between the regular lattice sites, or impurities where foreign atom types are inserted into a crystal lattice, often substituting the atom at an actual lattice position. Each of these defects plays a significant role in the material behavior, as they influence how the carriers of plasticity (dislocations) nucleate and propagate or how other deformation mechanisms such as diffusion or crack extension become activated. We will discuss some of these in later chapters.

1.3 Brittle vs. Ductile Material Behavior As briefly introduced earlier, materials failure is often divided into two generic types, brittle and ductile. In the brittle case, atomic bonds are broken as material separates along a crack front. The type of failure of such materials is often characterized by the simultaneous motion of thousands of small cracks, as it is for example observed when glass shatters. This type of failure usually happens rapidly, as cracks may propagate at velocities close to the speed of sound in materials. An enormous amount of research has been carried out over the last century, which has been summarized in recent books [22, 61]. The origin of fracture research dates back to the early twentieth century in studies by Griffith [62] and Irwin [63]. These studies resulted in theories that enabled a quantitative prediction of the fracture strength of brittle materials. The so-called Griffith criterion provides a quantitative estimate of the condition under which material fails, and is based on simple energetic and thermodynamic arguments. The Griffith criterion is based on the assumption that brittle materials fail when the mechanical elastic energy released by crack

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Fig. 1.7 Schematic of brittle (a) vs. ductile (b) materials behavior. In brittle fracture, the crack extends via breaking of atomic bonds. In ductile fracture, the lattice around the crack tip is sheared, leading to nucleation of crystal defects called dislocations. Which one the two mechanisms is more likely to occur determines whether a material is brittle or ductile; this distinction is closely related to the atomic structure and the details of the atomic bonding

extension, referred to as G, equals the energy required to generate two new surfaces, given by 2γs : (1.4) G = 2γs , where the parameter G is referred to as the mechanical energy release rate. In other words, during brittle fracture, elastic energy is dissipated at a rate G during the fracture process. The dissipated elastic energy is utilized for the creation of new fracture surfaces and, as in the case of dynamic fracture, a continuous rise in the temperature. This thermodynamic view of fracture was the foundation for the field of linear elastic fracture mechanics (often referred to as LEFM). This continuum mechanical theory of fracture is a well-established framework, and has been summarized in several recent textbooks [22, 61, 64, 65]. In ductile failure, a catastrophic event such as rapid propagation of thousands of cracks is prevented by the activation of an alternative deformation mechanism. Tough materials like metals (such as copper, nickel, gold) do not shatter. Instead, they deform and bend easily because plastic deformation occurs by the motion of dislocations inside the material. The competition between brittle and ductile failure depends on how the interatomic bonds close to the tip of a crack respond to the locally large interatomic forces. Thus the tendency of materials to be ductile or brittle depends quite strongly on the atomic microstructure, in particular on the relative ease of either dislocation nucleation vs. the energy required to create new surfaces. The face centered cubic (FCC) packing is known to have a quite strong tendency toward ductility. In contrast, body centered cubic (BCC) crystals are less ductile. Glasses, as they not have extended crystallinity because atoms are randomly packed, have no slip-planes, and therefore dislocation nucleation is

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not possible. Thus they mostly exhibit brittle failure with little ductility. The atomistic mechanisms of brittle fracture or ductile failure is quite different. While atomic bonds are broken by stretching the solid in brittle fracture, the sliding between planes is achieved by shearing the solid in ductile failure. The ductile vs. brittle failure of materials is schematically summarized in Fig. 1.7. Figure 1.7a shows brittle materials failure by propagation of cracks and Fig. 1.7b depicts ductile failure by generation of dislocations at a crack tip. The atomic details of such different behavior is shown in Fig. 1.8, providing a snapshot of an atomistic simulation.

Fig. 1.8 Brittle (a) vs. ductile (b) materials behavior observed in atomistic computer simulations. In brittle materials failure, thousands of cracks break the material. In ductile failure, material is plastically deformed by motion of dislocations

The origin of brittle vs. ductile behavior of materials has been linked to atomic processes in work by Rice and Thomson [66]. They considered quantitatively the energy penalty associated with shearing atomic planes on top of each other vs. the mechanism of creating new surfaces. These studies suggested the concept that there exists a competition between ductile (dislocation emission) and brittle (cleavage) mechanisms at the tip of a crack. The model by Rice and Thomson has been extended further in many following studies, to include additional material parameters that required to characterize this behavior. Most importantly, the introduction of the unstable stacking fault energy γus [30, 67] describes the resistance of the material to the nucleation and motion of dislocations. This provides an corresponding parameter to the fracture surface energy, which describes the resistance of materials to fracture by creating new surfaces. It has been shown that once both of these material parameters are known, it may be possible to quantitatively predict the material behavior, that is, if a material is rather ductile or brittle. Dislocation-based processes and cleavage are not the only mechanism for deformation of materials. Materials under geometric confinement, also referred to as materials in small dimensions or interfacial materials, often show a

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dramatically different behavior. This is caused by the fact that the behavior the carriers of plasticity or fracture changes at very small dimensions, so that they either shut down or other mechanisms are activated. Examples for such materials are nanocrystalline materials [68, 69] or ultra-thin submicron metallic films [51]. It was shown that in materials with ultra-small grain sizes of tens of nanometers and below, deformation can be completely dominated by grain boundary processes such as grain boundary diffusion at high temperatures or by grain boundary rearrangements and grain boundary gliding [68–70]. Due to the small sizes of the grains, dislocations cannot be generated (e.g., sources for dislocations, including Frank–Read sources are too large to fit within a grain, or because dislocations are energetically very expensive under very small geometrical confinement [71–73]). Even though such material behavior is still perceived as ductile (since materials can be bent without fracturing catastrophically), no dislocation motion is required to mediate deformation.

1.4 The Need for Atomistic Simulations The classical theories of continuum mechanics have been the basis for most theoretical and computational tools of engineers, forming the foundation for approaches such as the finite element method (often referred to as FEM), finite volume methods, or finite difference methods. It was not until the 1980s that scientists and engineers began to include atomistic descriptions into models of materials failure and plasticity. For example, in the early stages of computational plasticity, dislocations and cracks were often treated using linear continuum mechanics theory, often relying on phenomenological assumptions. Over the last decades, there has been a new realization that understanding the nanoscale behavior is required for understanding how materials fail, which opens great opportunities to provide new design methods for materials from the bottom up. The significance of atomistic processes to describe deformation and failure of materials is apparent from the discussion in the last sections. The significance of including atomistic or nanoscale mechanisms is also due to the increasing trend toward miniaturization. As shown in Fig. 1.1, relevant lengthscales of materials approach several nanometers in modern technology (e.g., thin films, semiconductors, new composites, carbon nanotubes, new nanotechnologies based on biological concepts). Once the dimensions of materials reach the submicron lengthscale, the continuum description of materials is questionable and the full atomistic information about the material state is often necessary to study relevant materials phenomena. In addition the significance of nanoscopic phenomena for deformation, engineers can now also design materials at the nanoscale. The ability to create nanostructures by design has reached a level of perfection that now enables us to make almost arbitrary structures and shape at ultra-small scales, either through self-assembly or by

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utilizing small-scale material manipulation and cutting techniques (such as focused ion beam, FIB). This ability opens a new era in the design of materials, from bottom up, enabling new functions, properties, and behaviors for a very rich set of applications. However, these advances also require new theoretical concepts to describe the materials behavior, as it can be quite different from what is conventionally known from larger scale materials. Atomistic models are often quite suitable to capture these and other effects. Assuming an atomistic viewpoint has another quite important aspect: It allows for the seamless communication of various scientific disciplines with one another, in particular during the study of deformation and fracture. The reason is that the notion of the “chemical bond” is a concept with which many disciplines can be linked. In this sense, chemistry is the most fundamental language of materials science and related disciplines. Atomistic models typically contain extremely large number of particles, even though the actual physical dimension may be quite small. For example, even a crystal with dimensions below a few micrometers sidelength has several tens of billions of atoms. Predicting the behavior of such large manyparticle systems under explicit consideration of the trajectory of each particle is only possibly by numerical simulation, and must typically involve large computational facilities. In the continuum viewpoint, materials are treated without explicitly considering the underlying inhomogeneous microstructure, that is, the basic assumption is that matter can be divided indefinitely, without a change of the material properties. Only a few special cases of continuum models can be solved analytically and written in closed-form solutions. Many problems require a numerical solution of the governing partial differential equations, for instance by finite difference methods, boundary element or finite element approaches. Many numerical methods used to solve continuum problems thereby require the discretization of the domain into integration points (e.g., mesh generation in the FEM). A higher density of integration points provides a more accurate solution of the problem, and higher densities of integration points must be used in regions of large gradients of stresses and deformation. In atomistic modeling, the discreteness of matter at the ultimate, atomic scale is explicitly considered. For example, the discreteness of an atomic lattice in a metal, where atoms are glued to their equilibrium crystal positions. Therefore in atomistic modeling no spatial discretization is necessary, since this is given by the atomic distances, providing a natural measure for spatial discretization. Each atom is considered to be an individual particle that cannot be divided further. Atomistic models can rarely be solved analytically, so that most models are implemented in numerical simulations that model the motion of each atom in the material over the course of the simulation time span. The simulation thereby carries out a step-by-step integration to successively progress the timescale during a study. The collective behavior of the atoms allows one to understand how the material undergoes deformation, observe phase changes, or investigate many other phenomena.

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The observed phenomena in a larger ensemble of atoms provides the links between the atomic scale to mesoscopic or macroscopic phenomena. However, since the basic informations of an atomistic simulation are only atomic positions, velocities, and forces, the interpretation of these numbers can be quite challenging. Continuum approaches provide very powerful methods to obtain the solution of elastic and sometimes plastic problems. What are their limitations? Clearly, one limitation is the reduction of characteristic material lengthscales to the sub-100 nm regime. In this regime, continuum mechanical methods become questionable, as the material inhomogeneities reach lengthscales comparable to the overall size of the structure. And therefore, the basic assumption of continuum theories that material can be divided without changing the material properties does not hold any more. Another important area is the introduction of complex geometrical features, such as nanostructures, and also the existence of complex chemistry that involves discrete reactions that can severely influence the behavior of a material, making it impossible to neglect the discrete nature of molecules. A fundamental challenge of any model and simulation implementation is the determination of parameters. This is important for continuum approaches, as many atomistic or molecular mechanisms (e.g., the generation of dislocations, diffusion, creep) are not explicitly described at the relevant scale of the underlying physics, but rather treated by empirical relations that mimic the overall collective effect of these processes. Important parameters for continuum approaches include elastic coefficients (typically referred to as constitutive equations), cohesive laws that describe the traction–separation behavior of interfaces, or the inclusion of parameters to describe the conditions at which plastic deformation begins. The determination of these parameters and criteria often involves important challenges. Atomistic modeling, on the other hand, provides a first principles-based approach to this problem, as it is capable of capturing the elementary physical mechanisms of failure. The continuum and atomistic viewpoint provide two fundamentally different approaches in treating materials, with different appeal and significance for specific applications. For many applications, the two views are complementary and the joint use of the approaches can provide much insight into the behavior of materials. The atomistic models provide a fundamental description of material properties and processes, which enables one to communicate data of such studies seamlessly to concepts from other scientific disciplines. Atomistic methods provide a rather general description of matter, since the same atomistic model of a material may be suitable to study elasticity problems as well to study dissipative materials failure such as fracture. Atomistic simulation is capable of solving the dynamical evolution of equilibrium and nonequilibrium processes, providing in particular important detailed insight into the physics of elementary processes.

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1.5 Applications: Experimental and Computational Mechanics A key aspect in the analysis of mechanical properties of materials is the ability to test properties experimentally and to enable proper guidance to interpretation of the results. The combination of experimental testing with computational and theoretical concepts has proven to be a particularly fruitful strategy, for both scientific discovery and for technological innovations and applications. Why do we study mechanical properties of materials? Mechanical properties are of interest throughout a range of lengthscales and timescales, and for a variety of applications. At larger scales, the integrity of airplanes, ships, or space shuttles involves the ability to tolerate large mechanical loads. Bridges, buildings, or dams must withstand enormous pressures and temperature ranges and different loading conditions. At smaller scales, reaching micrometers and scales much below, mechanical properties are quite important for the reliability of devices. For instance, integrated circuits in electronic microdevices must be able to provide electrical conductivity, but also persist large ranges of temperatures that lead to large strains and large stresses. Mechanical properties are also significant for the understanding of biological systems and biological processes. The mechanical properties of biological tissue is critical in several other areas in biology, in particular at the subcellular scale and for the behavior of biological tissue, for example during injuries. Biological systems often use mechanical deformation as signals for cell–cell interaction or cell–tissue interaction as an important part of the physiologic function. Figure 1.9 summarizes specific lengthscales and timescales associated with modeling the mechanical properties of materials for materials and structures of wide interest. This plot illustrates the grand challenges associated with modeling and simulation the complex phenomena of mechanical deformation, spanning many orders of timescales and lengthscales. 1.5.1 Experimental Techniques Experimental techniques that enable one to probe the mechanical properties of materials and structures has significantly developed over the past decades. In particular, a rich set of experimental tools suitable to probe the mechanical behavior at micro- and nanometer lengthscales is now available. Since many fundamental deformation mechanisms operate at that scale, these approaches provide us with the opportunity to understand and test material theories and simulation results at unparalleled scales. Mechanical testing at scales between millimeters and meters is often carried out by using tensile test machines or through indentation experiments and many variations of these approaches. Mechanical tensile test machines provide a means to apply controlled forces to samples, while measuring the elongation. This information is then used to produce force–extension or stress–strain curves as shown in Fig. 1.2. In indentation experiments, the compression of a

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Fig. 1.9 Overview over timescales and lengthscales associated with various problems and applications of mechanical properties (adapted from [2])

hard indenter tip into the material is used to the extract information about the load–displacement relationships, which enables one to characterize the material hardness. The combination of these approaches with temperature control, environmental chambers to control humidity enables the study of material behavior at a diverse range of controlled conditions. For more details, we refer the reader to several textbooks that describe mechanical testing at macroscales and microscales (for example, [1, 65]). Until more than 20 years ago, the scales accessible to mechanical testing of materials was largely limited to millimeter scales. Certainly, material properties of individual molecules or assemblies of molecules could not be determined. The early 1990s marked a decade of many new developments that enabled the study of material properties at ultra-small scales. The experimental analysis of such nanomechanical properties at submicrometer scales is possible with techniques such as the atomic force microscope (AFM), nanoindentation, or optical tweezers. These techniques are based on the development of finer instrumentation and control mechanisms that enable one to characterize forces on the order of hundreds of pN, with displacements on the order of nanometers. Optical tweezers are based on the concept of optical traps, and allow one to probe comparable force and displacement levels. This has led to new applications of the study of mechanical properties in a variety of fields, including materials science, biology, and physics. Figure 1.10 provides a summary of experimental techniques, here focused on applications in

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Fig. 1.10 Experimental techniques for conducting mechanical tests in single cell and single molecule biomechanics. Reprinted from Materials Science and Engineering C, Vol. 26, C.T. Lim, E.H. Zhou, A. Li, S.R.K. Vedula and H.X. Fu, Experimental techniques for single cell and single molecule biomechanics, pp. 1278–1288, copyright c 2006, with permission from Elsevier

biomechanics to probe the mechanical characteristic of DNA, proteins, cells, and biological tissue. Experimental techniques for the analysis of materials is a very wide field. Due to the scope of this book on modeling and simulation we refer the reader to the literature for additional details. Throughout this text, we will make reference to experimental results and experimental techniques that complement the studies of materials failure reviewed here. 1.5.2 Example Applications: The Significance of Mechanics Mechanical properties are important for many areas in science and engineering, ranging through a broad spectrum that includes biology, physics, materials science, and medicine. Here we briefly review the significance of mechanical properties of materials and structures via a brief review of a few studies in different disciplines. Much recent interest in mechanical properties is sparked by the enhanced experimental capabilities that enables a simultaneous comparison between computational and experimental testing, and the ability to probe fundamental deformation mechanisms at the scale of molecular unit events.

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Fig. 1.11 Nanomechanical experiments of bending deformation of 200-nm gold nanowires. Subplot (a) depicts a schematic of a fixed wire in a lateral bending test with an AFM tip. Subplots (b–e) depict AFM snapshots of the mechanical deformation the nanowire. Subplot (b) depicts results after elastic deformation, subplots (c) and (d) shows results after successive plastic manipulation, and subplot (e) shows an SEM image following the bending test. The SEM picture agrees in detail with the AFM image shown in subplot (d). All scale bars are 1 µm. Reprinted with c 2005 permission from Macmillan Publishers Ltd, Nature Materials [3]

Nanostructures and Nanomaterials At nanoscale, significant recent interest has been devoted to the mechanical properties of nanowires and tubular structures such as carbon nanotubes. The miniaturization of materials from macroscales or microscales to the scales of several nanometers leads to severe changes in the mechanical properties. Such observations have been made for many nanomaterials that have been rigorously studied over the past decades. For instance, the strength of nanostructures is often greater than the strength of its macroscopic counterparts made out of the same material. This is believed to be due to geometric confinement, the lack of structural defects, or a combination of these effects. In particular, metallic nanowires have received considerable interest as novel interconnects for electronic devices or as components of both electrical and electromechanical devices. Figure 1.11 depicts results of AFM manipulations of a 200-nm gold nanowire. It was found that whereas Young’s modulus is independent of the diameter of the nanowire, the yield strength is largest for the nanowires with the smallest diameter. Notably, the strength of nanowires reaches up to 100 times that of corresponding bulk materials. This can be explained partly based on the fact that the conventional mechanism of dislocation nucleation and propagation is hindered due to the geometric confinement. Thus the material reaches strengths that are much greater than observed in conventional materials, approaching the strength values comparable to the theoretical strength of materials. The strength of nanowires is

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also substantially larger than the values reported for nanocrystalline metals. Such AFM studies therefore provide important insight into the deformation mechanisms of metals at ultra-small lengthscales. Similar AFM-based testing has been carried for several other nanomaterials and nanostructures, including carbon nanotubes of different geometry, ceramic nanowires, and thin films. In many of these studies, the behavior was found to differ substantially from the properties known from the macroscopic counterparts. Biomechanics and Biomedical Applications The fundamental components of life are structures with characteristic dimensions on the order of nanometers as represented by DNA or proteins. The analysis of their mechanical behavior not only is the key to enable the development of new biomaterials (e.g., prostates, material replacements), but also is crucial for the advancement of the understanding of the physical mechanisms of biological processes. Injuries such as bone fracture or scars represent severe plastic deformations and ruptures of elementary components of organisms. Even though the understanding of associated basic deformation mechanisms is still in its infancy, the ability to probe mechanical properties and processes at ultra-small scales has already contributed significantly to the field. An interesting aspect of mechanical properties is the relation with diseases. Many diseases are associated with a change of mechanical properties of cells or tissues, which renders mechanics quite important for the study and diagnosis of human diseases. For example, at the cellular scales, the change of mechanical properties during disease progression has been studied in the context of malaria, where it was discovered that the disease has a dramatic effect of making red blood cells much stiffer [74, 75]. The change of the mechanical properties of individual cells has thereby major consequences for the spreading of malaria in the human body. Cells infected by malaria show more than ten times increased stiffness, whereas healthy cells are significantly more elastic. Since the cells are so stiff, it is impossible for them to squeeze through very thin blood vessels and thereby adversely affect other biological tissues. Figure 1.12 depicts the results of an optical tweezers experiment, illustrating this effect [75]. Clearly, the cells infected by malaria are much less elastic at the identical force compared with an uninfected cell. Finite element computational mechanics has been used to simulate the micromechanical process of cellular deformation. Figure 1.13 depicts images of a comparison between experimental snapshots and the results of a three-dimensional finite element simulation. Further, a combined optical tweezers – molecular dynamics study of the deformation mechanics of red blood cells has been reported in [76], illustrating the potential of combining molecular modeling with nanomechanical experiments. Similar observations that illustrate the importance of mechanical properties as signature of human diseases have been made in the case of cancer, where

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Fig. 1.12 Mechanical deformation of a red blood cell (RBC) with optical tweezers. Subplot (a) depicts a schematic of the experimental approach. Subplot (b) depicts optical images of a healthy RBC anda RBC in the schizont stage of malaria, in PBS solution at 25◦ C. The left column depicts results prior to stretching, the middle column depicts results at a constant force of 68 ± 12 pN, and the right column plots results at a constant force of 151 ± 20 pN. The P. falciparum malaria parasite can be seen inside the infected RBCs. Reprinted from Acta Biomaterialia, Vol. 1, S. Suresh, J. Spatz, J.P. Mills, A. Micoulet, M. Dao, C.T. Lim, M. Beil, T. Seufferlein, Connections between single-cell biomechanics and human disease states: c 2005, with permission gastrointestinal cancer and malaria, pp. 15–30, copyright from Elsevier

a drastic change of the mechanical properties of cancer cells has been observed by atomic force microscopy analysis of the mechanical stiffness [4]. Figure 1.14 displays some of the images and analyses, in particular illustrating the change of mechanical properties of cancer cells. The analysis suggests that tumor cells feature a lower Young’s modulus. The analysis method reported in [4] might be used to diagnose cancer based on a mechanical analysis, enabling one to develop new inexpensive methods of cancer detection. The medical significance of using mechanical signals as a method to diagnose cancer is based on the fact that the appearance of cancer is biochemically extremely diverse, making it difficult to find unique biochemical signatures for detection. In contrast, the analysis reported in [4] suggests that there exists a signature of the change of mechanical properties that is common to variety of cell types, even for different tumor types and different patient fluids. These effects of a variety of phenomena on the mechanical properties illustrate the possibility to provide new paths toward diagnosis of diseases, and maybe also for the development of treatment methods. The mechanical deformation of cells and tissues plays an integral role in the functioning of organisms. For example, the mechanical deformation of

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Fig. 1.13 Images of an RBC being stretched from 0 to 193 pN. Subplot (a) shows images obtained from experiment, while subplots (b) and (c) depict a top view and a three-dimensional view of the half-model corresponding to the large deformation finite element simulation of the RBC, respectively. The contours of shades of grey in the middle column shows the distribution of constant maximum principal strains. Reprinted from Materials Science and Engineering C, Vol. 26, C.T. Lim, E.H. Zhou, A. Li, S.R.K. Vedula and H.X. Fu, Experimental techniques for single cell and single c 2006, with permission from molecule biomechanics, pp. 1278–1288, copyright Elsevier

tissue by muscle cells is an important part of the physiological function in our body, involving the concerted motion of millions of molecules, involving the continuous rupture and reformation of chemical bonds. Failure of biological tissues is observed for instance in injuries, when large external forces or rapid displacements lead to a disruption of the healthy tissue formation. The fracture of bone is one of the most prominent examples of such an injury. Many studies in the past years, using advanced mechanical testing methods across a wide range of scales, have contributed to significantly improved understanding of how bone breaks. Figure 1.15 illustrates a sequence of microscopic mechanisms that shows how a small crack propagates in bone tissue. The combination of mechanical testing with new imaging techniques

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Fig. 1.14 Change of cellular mechanical properties in cancer cells. Subplot (a) depicts an optical image demonstrating the round, balled morphology of visually assigned tumor cells, and the large, flat morphology of presumed benign mesothelial, normal cells. Subplots (b–d) show histograms of the associated Youngs modulus E for cytological samples collected from patients with suspected metastatic cancer. Subplot (b) shows the histogram of E for all data collected from seven different clinical samples, indicating that there exist two peaks in the distribution. Subplot (c) shows a Gaussian fit for all tumor cells, and subplot (d) shows a log-normal fit for all normal cells. The analysis suggests that the presence of tumor cells leads to a sharp peak due to a lower Young’s modulus. This method might be used to diagnose cancer based on a mechanical analysis. Reprinted with permission from Macmillan c 2007 Publishers Ltd, Nature Nanotechnology [4]

(e.g., SEM, AFM imaging) provides key advances in the understanding of how molecular and multiscale hierarchical features in these materials participate in their deformation. The analysis of mechanical properties is not limited to tissue or cellular scales. Recent experimental and computational progress enabled the study of the deformation and fracture behavior of individual protein molecules. For example Fig. 1.16 depicts the force–extension profile of an immunoglobulin-like domains of human cardiac titin [6]. Fig. 1.16a depicts the large-deformation regime, showing characteristic peaks that correspond to repeated unfolding of protein domains. Fig. 1.16b depicts a close-up view of the first peak.

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Fig. 1.15 Experimental study of fracture mechanics of bone. Subplot (a) shows the geometry of the three-point bending test with an initial notch as a seed for fracture. Subplots (b) and (c) show scanning electron micrographs of microscopic bone fracture mechanisms, obtained from carrying out fracture experiments (arrangement of figure adapted from original source). Reprinted with permission from Macmillan c 2003 Publishers Ltd, Nature Materials [5]

Figure 1.17 depicts the mechanically induced deformation of a protein structure as reported in [7]. This field was pioneered in the early 1990s, when molecular dynamics simulations of protein unfolding were first carried out [6]. The combination of atomistic modeling and experiments with AFM or optical tweezers has been a particularly fruitful combination. For a review of nanomechanical experiments of biological structures please see also [74, 77–80]. The examples reviewed in this section illustrate new frontiers of mechanics, at very small scales, in nanomaterials, or in exceedingly complex biological structures and tissues. Most methods and results reviewed in this book are not yet applicable to these complex materials. However, maybe with future development and research it will be possible to achieve a similar understanding of the fundamental failure mechanisms as it exists for crystalline systems today.

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Fig. 1.16 AFM experiments of protein unfolding. Subplot (a): Force peaks corresponding to the sequential unfolding of a immunoglobulin-like domains of human cardiac titin (human cardiac I band titin encompassing the immunoglobulin-like domains I27–I34). The results show large hump-like deviations from the WLC model of entropic elasticity (continuous lines indicate WLC fits, the arrow illustrates the point of deviation). Subplot (b): Detailed view of the first force peak of a sawtooth pattern. Reprinted with permission from Macmillan Publishers Ltd, Nature [6] c 1999

1.6 Outline of This Book This book is organized into three main parts: An introduction, a part discussing the basic concepts of atomistic, continuum, and multiscale approaches, as well as a large part that reviews examples of modeling of deformation and fracture phenomena in a variety of materials. The discussion begins with the introduction of the conventional concepts of fracture and deformation mechanics, reviewing basic contributions over the past five and more decades. Along with the presentation of the conventional theory, we present molecular simulation studies of these phenomena illustrating how atomistic approaches can be used to model these processes. The book concludes with a focus on small-scale materials phenomena. Along the way we will emphasize on analogies and synergies from the interactions between the different fields of studies. This book reviews the techniques that are introduced in specific examples or case studies. The prime goal of this approach is to convey to the reader the underlying approach, the methods and the strategy used to build, use and analyze the computational and theoretical models. Many examples and case studies are taken from earlier studies of the author of this book. The discussion should therefore not be considered as a

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Fig. 1.17 Large deformation of a protein, here an example of unfolding of the enzyme lysozyme, result of a reactive force field simulation. The distance between the ends of the protein (Cα -atom of the terminal residues) is continuously increased by applying a continuously increasing force [7]. As the force is increased, the protein molecule undergoes significant structural changes relative to its initial folded configuration

comprehensive and inclusive review with respect to the wider range of available results. Rather, the examples used in this book should be regarded as examples to illustrate application of the numerical techniques reviewed in the book.

Part II

Basics of Atomistic, Continuum and Multiscale Methods

2 Basic Atomistic Modeling

Atomistic modeling provides a fundamental description of the materials behavior and materials deformation phenomena. Molecular dynamics simulations represent the numerical implementation to solve the equations of motion of a system of atoms or molecules, resulting in the dynamical trajectories of all particles in the system. The purpose of this chapter is to present an introduction into molecular dynamics modeling and simulation approaches. The discussion includes the physical basics, numerical implementation and examples of atomistic models for specific materials, as well as a brief introduction into multiscale simulation methods. We also review analysis methods, in particular visualization schemes that can be used to extract useful information from large atomistic systems.

2.1 Introduction Molecular dynamics simulation techniques are very widely applicable, for a range of materials and states, including gases, liquids, and solids. The first molecular dynamics studies were focused on modeling thermodynamical behavior of gases and liquids in the 1950s and 1960s [81–84]. It was not until the early 1980s when the first molecular dynamics modeling works were published that were applied to the mechanical behavior of solids. As a consequence of the general applicability of molecular dynamics, many of the methods and approaches described in this book can also be useful for the study of gases and liquids, as well as the interaction of those with solids in systems that contain both solids and liquids. The outline of this chapter is as follows. We begin with a presentation of the basic formulation of molecular dynamics (sometimes also referred to as “MD”). After discussing of the numerical strategies associated with molecular dynamics, we introduce interatomic potentials that mimic the energy landscape predicted by quantum mechanics (we emphasize here that quantum mechanics will not be discussed explicitly in this book, and the reader

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Fig. 2.1 Molecular dynamics can be used to study material properties at the intersection of various scientific disciplines. This is because the notion of a “chemical bond” as explicitly considered in molecular dynamics provides a common ground as it enables the cross-interaction between concepts used in different disciplines (here exemplified for the disciplines of biology, mechanics, materials science and physics)

is kindly referred to other literature). We briefly review statistical mechanics concepts that provide the theoretical and numerical basis for property calculation from the results of molecular dynamics simulations. We discuss the calculation of temperature, measures for the geometry of a particular atomic system, methods to analyze and display crystal defects, and some correlation functions that enable one to predict transport properties from molecular dynamics studies. As illustrated in Fig. 2.1, due to the fundamental nature of the description of the material behavior, molecular dynamics can be used to study material properties at the intersection of different scientific disciplines.

2.2 Modeling and Simulation The significance of the atomistic viewpoint for failure processes and the enormous computational burden associated with such problems makes modeling and simulation of failure a promising and exciting area of research. In this section we discuss some fundamental concepts associated with model building and the solution of the particular numerical problems to be computed in molecular dynamics simulations. The terms modeling and simulation are often used in conjunction with the numerical solution of physical problems. However, it is important to note that the two words have quite distinct meanings. The term modeling refers


33

to the development of a mathematical model of a physical situation, whereas simulation refers to the procedure of solving the equations that resulted from model development. Models are often simplifications or idealizations of rather complex physical systems or phenomena. A key aspect of model development is the ability to map the essential physics features of a system into a description. M.F. Ashby of Cambridge University used the example of a subway map to illustrate the concept of model building [85]: A model is an idealization. Its relationship to the real problem is like that of the map of the London tube trains to the real tube systems: a gross simplification, but one that captures certain essentials. The map misrepresents distances and directions, but it elegantly displays the connectivity. The quality or usefulness in a model may be measured by its ability to capture the governing physical features of the problem. Ashby states that all successful models unashamedly distort the inessentials in order to capture the features that really matter. At worst, a model is a concise description of a body of data. At best, it captures the essential physics of the problem, it illuminates the principles that underline the key observations, and it predicts behavior under conditions which have not yet been studied. The concept of model building is illustrated in Fig. 2.2, here shown for the subway system in the Boston area. The comparison of the left and right panels illustrates how models can facilitate to capture the essential information and features of a physical system. The concepts of modeling and simulation are intimately coupled: Without model development, a simulation can not be carried out. Often, the partial differential equations itself that may be a direct result from model development do not allow to draw significant conclusions for a physical system at hand, until simulations are carried out. This is nicely summarized in a phrase coined by Sidney Yip of MIT [86] who stated that Modeling is the physicalization of a concept, simulation its computational realization. Both modeling and simulation have their specific challenges. The tasks associated with modeling requires insight into the physics of the system, its constituents or the behavior of the particles. The setup of the simulation requires knowledge in the field of numerical techniques that are suitable to solve complicated systems of partial differential equations, or to make computation proceed fast on modern supercomputers. To make efficient use of results from simulation, strategies need to be used to analyze and interpret this data. As indicated earlier, the results of atomistic simulations are merely numbers that represent the position and velocities of atoms at different time steps. Making sense of this huge amount of information can be a daunting task. This becomes more evident as the simulation sizes increase to systems with billions of atoms. Even with today’s largest computers, system sizes with only a few billion atoms can be simulated, whereas a cubic centimeter of material already contains more than 1023 atoms. This book provides illustrates techniques and approaches for both modeling and simulation. The concepts will be applied to specific problems of describing materials failure phenomena. However, the concepts can be

34


Fig. 2.2 This figure illustrates the concept of model building. Panel (a) on the left shows the physical situation of a map of the subway lines. This representation makes it quite difficult to determine a strategy to use the subway system to travel from the cities of Braintree to Revere, for instance. The model representation depicted in panel (b) on the right enables one to determine quite easily which subway line to take, where to change the subway line, and how many subway stops there are in between. This example illustrates that even though the model representation on the right misrepresents the actual distances and directions, it elegantly displays the connectivity. This figure was created based on a snapshot from the Massachusetts Bay Transportation Authority (MBTA) web site (URL: http://www.mbta.com/), reprinted with permission from the the Massachusetts Bay Transportation Authority

transferred to other applications where similar methods could be fruitfully applied, such as self-assembly, diffusion studies, or studies of phase transformation. 2.2.1 Model Building and Physical Representation Together with data analysis, model building and finding an appropriate physical representation is probably the most difficult task in computational materials science. It is imperative that great care must be taken when models are built, and when results of simulations are analyzed. Atomistic simulations have proven to be a powerful way to investigate the complex behavior of dislocations, cracks, and grain boundary processes at a very fundamental level. Atomistic methods have gained an increasingly important role and level of acceptance in modern materials modeling. One of the strengths and the reason for the great success of atomistic methods is its very fundamental viewpoint of materials phenomena. For instance, the only physical law that is used to simulate the dynamical behavior is Newton’s law,


35

along with a definition of how atoms interact with each other (for a discussion of Newton’s law, see Sect. 3.1). Despite this quite simple basis, very complex phenomena can be simulated. Unlike many continuum mechanics approaches, atomistic techniques require no a priori assumption on the defect dynamics or its behavior. Once the atomic interactions are chosen, the entire material behavior is determined. This aspect of atomistic modeling provides a terrific opportunity to build insightful models, that is, models that capture the essentials, elucidate fundamental mechanisms, and thereby provide an elegant representation of the key principles that underline the key observations. While in some cases it is difficult to find an appropriate interatomic potential for a material, atomic interactions can also be chosen such that generic properties common to a large class of materials are incorporated (e.g., describing a general class of brittle or ductile materials). This approach refers to the design of “model materials” to study specific materials phenomena. Despite the fact that such model building has been in practice in fluid mechanics for many years, the concept of “model materials” in materials science is relatively new. On the other hand, atomic interactions can also be chosen very accurately for a specific atomic interaction using quantum mechanical methods such as the density functional theory (DFT) [87], which enables one to approximate the solutions to Schrodinger’s equation for a particular atomistic model. Richard Feynman has also emphasized the importance of the atomistic viewpoint in his famous Feynman’s Lectures in Physics [88], where he stated that the atomic hypothesis (or the atomic fact, whatever you wish to call it) that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another [...] provides an enormous amount of information about the world, if just a little imagination and thinking are applied.” This underlines atomistic simulations as a natural choice to study materials at a fundamental level. This is particularly true for studies of materials failure! The atomistic level provides the most fundamental, sometimes referred to as the ab initio, description of the failure processes. Many materials phenomena are multiscale phenomena. For a fundamental understanding, simulations should ideally capture the elementary physics of single atoms and reach length scales of thousands of atomic layers at the same time. This can be achieved by implementation of numerical models of atomistic models on very large computational facilities. 2.2.2 The Concept of Computational Experiments An increasing number of researchers now consider the computer as a tool to do science, similar as experimentalists use their lab to perform experiments. Computer simulations have thus sometimes been referred to as “computer experiments” or “computational experiments.”

36


The art of a computational experiment is to (1) build an appropriate model of the physical situation, and to (2) construct it in such a way that the results obtained by numerical simulation of this model can be utilized to advance the understanding of a particular phenomenon. The third important step is to (3) analyze and interpret the results computational experiments to advance the understanding of the simulated process. Computational experiments must be set up with care. For instance, if a model is highly accurate but contains a very large number of numerical parameters, it may be impossible to understand how these parameters are related to a phenomenon of interest. Only after reduction of the large number of parameters into a simpler set of reduced variables it is possible to draw significant conclusions about the behavior of the system. Finding these reduced variables is a central aspect in designing clever computational experiments.

2.3 Basic Statistical Mechanics Statistical mechanics provides methods that help us yo analyze molecular dynamics simulation and to interpret results from these simulations. In particular, it leads to a direct link between an ensemble of microscopic states and the corresponding macroscopic thermodynamical properties. The conversion of the microscopic information to macroscopic observables such as pressure, stress tensor, strain tensor, energy, heat capacities, and others requires theories and strategies developed in the realm of statistical mechanics. Atomistic data (e.g., the pressure tensor) is not valid instantaneously but needs to be averaged over multiple configurations of the microscopic system. One of the most central and probably most useful theorems in the practical application of atomistic simulation is the Ergodic hypothesis. The Ergodic hypothesis states that the ensemble average of a property A equals the time average (the symbol . describes an averaged variable): AEnsemble = ATime .

(2.1)

This is a most useful relation that enables the calculation of thermodynamical properties by simply averaging over sufficiently long time trajectories (and thereby measuring the appropriate ensemble properties). The ensemble average of a property A is defined as AEnsemble = A(p, r)ρ(p, r)dpdr, (2.2) p

r

with pi = mi vi as the linear momentum of particle i, and p = {pi } being the set of all linear momentums in the system, for i = 1 . . . N . Similarly, r = {ri } represents the position vector of all particles. The system state is uniquely defined by the combination (p, r) since the Hamiltonian H = H(r, p).


37

In (2.2), the function ρ(p, r) is the probability density distribution, which is defined as H(r, p) 1 (2.3) ρ(p, r) = exp − Q kB T

with Q=

p

H(r, p) dpdr. exp − kB T r

(2.4)

To evaluate these expressions, we would need to known any possible state of the system, characterized by all possible values of p and r. Obtaining this information is very difficult, which immediately shows the significance of the Erdogen hypothesis. The time average from a molecular dynamics simulation can be calculated by M 1 ATime = A(p, r), (2.5) M i=1 where M is the number of measurements taken. The Ergodic hypothesis further lays the foundation for Monte Carlo techniques. Whereas molecular dynamics generates trajectories over time, Monte Carlo generates those within the constraint of an ensemble average. The Ergodic hypothesis states that both viewpoints are equal, allowing one to calculate thermodynamical properties using Monte Carlo. In this sense, Monte Carlo schemes generate a number of possible microscopic states along with specific probability densities ρ, which are then summed up discretely to obtain an estimate for the observed variable.

2.4 Formulation of Classical Molecular Dynamics In atomistic simulations, the goal is to predict the motion of each atom in the material, characterized by the atomic positions ri (t), the atomic velocities vi (t), and their accelerations ai (t) (see Fig. 2.3). Each atom is considered as a classical particle that obeys Newton’s laws of mechanics. The collective behavior of the atoms allows one to understand how the material undergoes deformation, phase changes, or other phenomena by providing links between the atomic scale to meso- or macroscale phenomena. Extraction of information from atomistic dynamics can be challenging and typically involves methods rooted in statistical mechanics. The total energy of such as system is H = K + U,

(2.6)

where K is the kinetic energy of the entire system and U the potential energy. The kinetic energy is a function of the kinetic energy of all N particles,

38


Fig. 2.3 Molecular dynamics generates the dynamical trajectories of a system of N particles by integrating Newton’s equations of motion, with suitable initial and boundary conditions, and proper interatomic potentials, while satisfying macroscopic thermodynamical (ensemble-averaged) constraints, leading to atomic positions ri (t), the atomic velocities vi (t), and their accelerations ai (t), all as a function of time, for all particles i = 1 . . . N , each of which has a specific mass mi

1 mi v2i , 2 i=1 N

K=

(2.7)

and the total potential energy is the sum of the potential energy of all particles: U (r) =

N

Ui (r),

(2.8)

i=1

noting that the potential energy of each particle Ui depends on the position of itself and all other particles in the system, expressed by r = {ri } as defined above. For now we leave this expression as an unknown. The total energy H is also referred to as the Hamiltonian. We note that K = K(p) and U = U (r), that is, the kinetic energy depends only on the velocities or the linear momenta of the particles and the potential energy is a function only of the position vectors. To satisfy Newton’s law Fi = mi ai for each particle i in the system, the following equation governs the dynamics of the system: mi

dU (r) d2 ri =− . 2 dt dri

(2.9)

The right-hand side corresponds to the gradient of the potential energy, which is the force (note that the potential energy of the system depends on the positions of all particles r). Equation (2.9) represents a system of coupled second-order nonlinear partial differential equations, corresponding to a coupled system N -body problem for which no exact solution exists when N > 2.


39

However, the equation can be solved by discretizing the equations in time. We note that the spatial discretization for the problem is given by the atom size, as discussed in Sect. 1.4. 2.4.1 Integrating the Equations of Motion A simple solution strategy is to develop a stepping method that gives new coordinates and velocities from the old ones, for each particle i, such as ri (t0 ) → ri (t0 + ∆t) → ri (t0 + 2∆t) → ri (t0 + 3∆t) . . .

(2.10)

A numerical scheme can be constructed by considering the Taylor expansion of the position vector ri : 1 ri (t0 + ∆t) = ri (t0 ) + vi (t0 )∆t + ai (t)∆t2 + . . . 2

(2.11)

and

1 ri (t0 − ∆t) = ri (t0 ) − vi (t0 )∆t + ai (t)∆t2 + . . . 2 Adding these two equations together yields ri (t0 + ∆t) = −ri (t0 − ∆t) + 2ri (t0 ) + ai (t)∆t2 + . . .

(2.12)

(2.13)

Equation (2.13) provides a direct link between new positions (at t0 + ∆t) and the old positions and accelerations (at t0 ). The accelerations can be obtained from the forces by considering Newton’s law, ai =

Fi . m

(2.14)

This updating scheme is referred to as the Verlet central difference method. There exist many other integration schemes that are frequently used in molecular dynamics implementations. In the following few sections we summarize a few additional popular algorithms. Leap-Frog Algorithm In the Leap-frog algorithm, the positions are updated as

and

1 ri (t + ∆t) = ri (t) + vi (t + ∆t)∆t 2

(2.15)

1 1 vi (t + ∆t) = vi (t − ∆t) + ai (t)∆t. 2 2

(2.16)

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Velocity Verlet Algorithm In the Velocity Verlet algorithm, the positions are updated as 1 ri (t + ∆t) = ri (t) + vi (t)∆t + ai (t)∆t2 2 where vi (t + ∆t) = vi (t) +

1 (ai (t) + ai (t + ∆t)) ∆t. 2

(2.17)

(2.18)

2.4.2 Thermodynamic Ensembles and Their Numerical Implementation When the equations reviewed in the previous section are integrated, the resulting thermodynamical ensemble is N V E, which means that the particle number N , the system volume V , and the total energy of the system E remain constant throughout the simulation. Other thermodynamical ensembles can be realized by modifying the equations of motion in an appropriate way, leading to N V T or N P T ensembles. Table 2.1 shows an overview over various thermodynamical ensembles.

Ensemble NV E NV T NP T µV T

Ensemble name Microcanonical ensemble Canonical ensemble Isobaric–isothermal ensemble Grand canonical ensemble

Table 2.1 Overview of various thermodynamical ensembles (the parameter µ is the chemical potential)

To illustrate the approach of modifying the equations of motion to obtain a specific thermodynamical ensemble, here we briefly review a simple algorithm to enable an N V T ensemble, the Berendsen thermostat. The approach is based on the idea to change the velocities of the atoms so that the temperature (which is a direct function of the atomic velocities, as discussed later on in Sect. 2.8.1) approaches the desired value, mimicking the effect of a heat bath. This is realized by calculating a rescaling parameter λ ∆t T λ= 1+ , (2.19) τ Tset−1 where ∆t is the molecular dynamics time step and τ is a parameter called “rise time” that describes the strength of the coupling to the hypothetical heat bath. The velocities are then rescaled according to this parameter, where the new velocity vectors are given by


vnew,i = λvi

41

(2.20)

for each atom i. Other approaches to enable the N V T ensemble include methods based on Langevin dynamics and the Nose–Hoover scheme. For N P T ensembles, the Parrinello–Rahman approach provides a popular choice for an algorithm. In this method, in addition to adjusting the temperatures to approach the desired control value, the pressure is adjusted by changing the cell size of the simulated system. States with high energy will occur less often, states with low energy more often. Each microscopic state has a certain probability associated with the corresponding energies. During the integration of the equations of motion, molecular dynamics naturally samples these microscopic configurations and provides a collection of snapshots that after averaging correspond to the proper macroscopic state. The obtained trajectories can then be used to calculate thermodynamical properties by simply averaging over the sampled configuration without further weighting. For example, a selection of N particles in a box at pressure P , temperature T , and volume V has many microscopic configurations (p, r) that all correspond to the same thermodynamical macroscopic state. Molecular dynamics can for instance be used to calculate the pressure for a given temperature and system volume, by solving the dynamical evolution of the system over long time scales. In this spirit, molecular dynamics enables one to sample the phase space for admissible configurations, and giving them proper weighting. According to the Ergodic hypothesis (see (2.1)), the molecular dynamics time average is equal to ensemble average, and system properties can be calculated with both methods. The results will be identical.

Fig. 2.4 Schematic of the atomic displacement field as a function of time. The atomic displacement field consists of a low-frequency (“coarse”) and high frequency part (“fine”)

42


A critical step in solving the dynamical equations (for instance, using the Velocity Verlet scheme) is to consider the size of the time step. Figure 2.4 depicts a schematic of the atomic displacement field as a function of time. As can be seen, the displacement history consists of low- and high-frequency contributions, where the total displacements can be written as u(t) = u(t) + u (t),

(2.21)

with u (t) as the fine contribution and u(t) as the coarse part. To solve the equations of motion, the fine part needs to be discretized, which results in a significant computational burden as most systems require time steps of approximately 1 fs or 10−15 s to discretize the rapid oscillations of u (t). Interatomic bonds that involve relatively light hydrogen atoms sometimes require even smaller time steps on the order of 0.1 fs. There are also adaptive techniques that are based on the idea to dynamically change the time step in a simulation, depending on the maximum atomic velocities [18]. Such approaches may help to increase the efficiency of molecular dynamics studies without adversely affecting the results.

Fig. 2.5 Example of harmonic oscillator with spring constant k = φ (r = r0 ), used to extract information about the time step required for integration of the equations of motion. The dashed line shows the (nonlinear) realistic potential function between a pair of atoms, of which the harmonic oscillator is the second-order approximation.

To estimate the time step for a particular system, one can estimate the oscillation frequency of a harmonic oscillator: 1 k (2.22) ν= 2π m


43

where k is the spring constant, given by the second derivative of the potential function with respect to the diameter (k = φ (r = r0 )), evaluated at r = r0 . The function φ describes how the energy of the bond changes as a function of radius r. This is schematically shown in Fig. 2.5. The time step should then be chosen 1 ∆tmin (2.23) ν In summary, the time step ∆t needs to be small enough to model the vibrations of atomic bonds correctly. The vibration frequencies may be extremely high, in particular for light atoms as √ ∆tmin ∼ m (2.24) The stiffer the bond is around its equilibrium position, the lower the critical time step as 1 ∆tmin ∼ √ . (2.25) k The fact that the time step is on the order of several femtoseconds has major implications on the time scale molecular dynamics can reach. For example, approximately 1,000,000 integration steps are needed to calculate a trajectory that covers 1 ns, providing a severe computational burden. Further, the time step typically cannot be varied during simulation. The total time scale that can be reached by molecular dynamics is typically limited to a few nanoseconds. Some exceptionally long simulations have been reported that cover up to a few microseconds. However, such simulations typically take up to several months. This aspect of molecular dynamics is sometimes referred to as the time scale dilemma. Even though the number of atoms in a simulation can be easily increased by adding more processors (e.g., using parallel computing), time cannot easily be parallelized. As can be see in the updating scheme (2.13), an atomistic system is generally not independent in time: The behavior at t0 influences the state at t1 > t0 , and the time stepping cannot be carried out independently, on multiple processors. Several researchers are currently developing techniques such as the temperature accelerated dynamics method, parallel replica method, and many others to overcome this limitation, and make use of massively parallelized computing to expand the accessible time scales [89]. Some of these techniques will be discussed later. 2.4.3 Energy Minimization Energy minimization is an approach during which the potential energy of the system, at zero temperature, is minimized. Energy minimization corresponds to the physical situation of cooling down a material to the absolute zero point. Methods in which the deformation behavior of a material or structure is probed during continuous energy minimization is also referred to as molecular

44


statics. Such approaches have been used to study dislocation nucleation from crack tips or the deformation of carbon nanotubes. It mimics a quasistatic experiment, albeit neglecting the effect of finite temperature. A variety of algorithms exist to perform energy minimization, most notably conjugate gradient methods or steepest descent methods. Figure 2.6 depicts an example result of an energy minimization, showing how the potential energy of the system decreases systematically with the number of iteration steps and finally converges to a finite value.

Fig. 2.6 Example result of an energy minimization, here an example of minimizing the structure of a solvated protein (lysozyme). As the number of iterations progresses, the total potential energy decreases, until it converges and reaches a constant value (see [8] for further details)

2.4.4 Monte Carlo Techniques Statistical mechanics provides a theoretical framework to link a number of microscopic states to macroscopic thermodynamical variables. To achieve this link, one needs to obtain samples of microscopic states, for instance by using a dynamical simulation (e.g., molecular dynamics). An alternative approach to generate this data is to calculate the appropriate ensemble averages directly. This is done in Monte Carlo techniques by sampling phase and state space. Monte Carlo Techniques: Brief Introduction The key task in computing the appropriate ensemble average is achieved by repeated random sampling. This is achieved by generating system states


45

Fig. 2.7 Illustration of generation of random perturbation from an initial state A toward a state B, as typically performed in Monte Carlo schemes

according to suitable Boltzmann probabilities [90, 91]. The procedure can be summarized as follows (Metropolis–Hastings algorithm): 1. Draw random numbers from a random number generator. 2. Advance system according to these random numbers (e.g., for the case of a molecular structure, move atoms accordingly, as illustrated in Fig. 2.7). 3. Accept or reject new configuration, according to an energy criterion. 4. If N < NA , back to 1. Otherwise, continue with 5. 5. The set of configurations obtained based on this scheme is used to calculate ensemble properties. Many Monte Carlo algorithms employ such a procedure to determine a new state for a system from a previous one. Thereby, the specific moves can be chosen arbitrarily, which makes this method very widely applicable. However, a drawback or limitation of this method is that it requires additional knowledge of the system behavior, in particular how the system may evolve as it is required for the generation of new configurations. Figure 2.8 summarizes one of the most popular Monte Carlo schemes, the Metropolis–Hastings algorithm. Comparison of Monte Carlo and Molecular Dynamics In contrast to Monte Carlo, molecular dynamics enables one to obtain actual deterministic trajectories, and thus provides detailed information about the full dynamical trajectories. Molecular dynamics can model processes that are characterized by extreme driving forces and that are nonequilibrium processes. A prominent example for which molecular dynamics is particularly suitable is fracture. Provided expressions exist for the atomic interactions, molecular dynamics modeling provides an excellent physical description of the fracture processes, as it can naturally describe the atomic bond breaking processes. Other modeling approaches, such as the finite element method, are based on empirical relations between load and crack formation and/or crack propagation. In contrast, molecular dynamics does not require such input parameters.

46


Fig. 2.8 Summary of the Metropolis–Hastings Monte Carlo algorithm. Please see Figure 2.7 for an illustration of how state B is generated based on a random perturbation from state A. The procedure is repeated NA times, the number of iterations. The number of steps is chosen so that convergence of the desired property is achieved

In principle, all parameters required for a molecular dynamics simulation can be derived from first principles, or quantum mechanical calculations (see also discussion above). Monte Carlo can typically only be used for equilibrium processes, as it does not provide information about how a system goes from an unstable state A to a stable state B. Many materials deformation processes such as fracture are examples for such phenomena. Thus, in the remainder of this book we focus on molecular dynamics methods, as they are capable of providing insight into the fracture mechanisms, which is an important aspect of modeling and understanding how materials fail under extreme conditions.

2.5 Classes of Chemical Bonding The behavior of molecules is intimately linked to the interactions of atoms, which are fundamentally governed by the laws of quantum chemistry. In metals, for example, bonding is primarily nondirectional, and can be characterized by positive ions embedded in a gas of electrons (this is often referred to as the electron gas model). Other materials show greater chemical complexity – often


47

Fig. 2.9 Schematic of the typical characteristic of a chemical bond, showing repulsion at small distances below the equilibrium separation r0 , and attraction at larger distances

featuring many different chemical bonds with varying strength, such as exemplified in materials including cement, proteins, or ceramics, or at interfaces between metallic systems and organic components. Despite the differences between different chemical bonds, many atom–atom interactions show similar characteristic featyres. Figure 2.9 depicts the typical characteristic of a chemical bond, showing repulsion at small distances below the equilibrium separation r0 of a pair of atoms, and attraction at larger distances. In general, for any material we must consider the interplay of chemical interactions that include, ordered by their approximate strength: • Covalent bonds (due to overlap of electron orbitals, e.g., found in carbon nanotubes, C–C bond, organic molecules such as proteins) • Metallic bonds (found in all metals, e.g., copper, gold, nickel, silver) • Electrostatic (ionic) interactions (Coulombic interactions, e.g., found in ceramics such as Al2 O3 or in SiO2 ) • Hydrogen-bonds (e.g., found in polymers, proteins), as well as • Weak or dispersive van der Waals (vdW) interactions (e.g., found in wax). Electrostatic interactions can be significantly weakened by screening due to electrolytes, which can lead to interactions that are weaker than vdW interactions.

48


For the implementation of molecular dynamics, we must have mathematical expressions available that provide models for the energy landscape of these chemical interactions. In other words, reviewing the formula provided in (2.54), we must know how the potential energy stored in a bond changes based on the geometry or the position of the atoms. These models are referred to as force fields or interatomic potentials. The term “force field” is often used in the chemistry community, whereas the term “potential” is frequently used in the physics community. Here we will use both terms, as it fits for the corresponding force field expressions.

2.6 The Interatomic Potential or Force Field: Introduction Figure 2.10 depicts a fundamental simplification made in classical molecular dynamics to replace the atom as a three-dimensional structure by a single point with a finite mass. This simple picture also illustrates the grand challenge in developing expressions, to somehow and as accurately as possible describe the (often complex) effect of the electrons on the atomic interactions. The goal of interatomic potentials or force fields is to give numerical or analytical expressions that estimate the energy landscape of a large particle system. This energy landscape is therefore the fundamental input into molecular simulations, in addition to structural information (position of atoms, type of atoms, and their velocities and accelerations). In this section, we will review a variety of classical approaches in modeling atomic interactions, ranging through different levels of accuracy and complexity. Numerous potentials with different levels of accuracy have been proposed, each having its disadvantages and strengths. The approaches range from accurate quantum-mechanics-based treatments (e.g., first-principle density functional theory methods [87] or tight-binding potentials [92]), reactive potentials [93] to multibody potentials (e.g., embedded atom approaches as proposed in [94]) to the most simple and computationally least expensive pair potentials (e.g., Lennard-Jones) [9,84]. One of the first molecular dynamics studies was a Lennard-Jones model of argon in 1964 [83]. Previous studies used hard-sphere models to describe phase transformations [81,82]. Many potential expressions are fit carefully so that they closely reproduce the energy landscape predicted from quantum mechanics methods, while retaining computational efficiency. The interatomic potential thereby allows to generate a direct link between the empirical molecular dynamics methods and quantum chemistry.


49

Fig. 2.10 Atoms are composed of electrons, protons, and neutrons. Electrons and protons are negative and positive charges of the same magnitude. In classical molecular dynamics, the three-dimensional atom structure is replaced by a single mass point

Fig. 2.11 Overview over different simulation tools and associated lengthscale and timescale

There is no single approach that is suitable for all materials and for all materials phenomena. The choice of the interatomic potential depends very strongly on both the application and the material. Popular choices in particular for modeling mechanical properties of materials are semiempirical or empirical methods, which typically allow one to simulate large systems with many thousands to billions of particles. However, to address different aspects of the mechanical behavior of a specific material typically requires the application of a range of simulation approaches. An overview over the most prominent materials simulation techniques is shown in Fig. 2.11. In the plot we also indicate which lengthscale and timescale can be reached with the various methods. The methods included in the figure refer to quantum mechanics based methods, classical molecular

50


dynamics methods, as well as numerical continuum mechanics methods. Quantum-mechanical-based treatments are typically limited to very short time- and length scales, on the order of a few nanometers and picoseconds. The assumption of empirical interactions in classical molecular dynamics scheme significantly reduces the computational burden, and the lengthscale and timescale that can be reached are dramatically increased, approaching micrometers and several nanoseconds. For comparison, we include also continuum mechanics-based simulation tools that can treat virtually any length scale, but they may lack a proper description at small scales, and they are therefore often not suitable to describe materials failure processes in full detail (see discussion in Sect. 1.4). Mesoscopic simulation methods such as discrete dislocation dynamics can bridge the gap between molecular dynamics and continuum theories by generating an intermediate scale at which clusters of atoms or small crystals are treated as a single particle [50, 95–101]. The remainder in this section will be focused mostly on empirical potential expressions that are suitable for the study of mechanical properties of materials. How do empirical potentials describe the various chemical interactions? Often, energy contributions from covalent atom–atom interactions, electrostatic interactions, vdW interactions, and others are summed up individually, so that U = UElec + UCovalent + UvdW + UH-bonds + . . . . (2.26) The challenge is how can these individual terms be approximated, most accurately, for a specific material? In the following sections we will describe some of the most common empirical potentials that provide such approximations. 2.6.1 Pair Potentials We begin with the simplest atom–atom interactions for which the potential energy only depends on the distance between two particles. The total energy of the system is given by summing the energy of all atomic bonds over all N particles in the system. The total energy is then given by Utotal =

N N 1 φ(rij ), 2 j=1

(2.27)

i=j=1

where rij is the distance between particles i and j. Note the factor 1/2 to account for the double counting of atomic bonds. The procedure of summing up the energies is shown in a schematic in Fig. 2.12. The term φ(rij ) describes the potential energy of a bond formed between two atoms, as a function of its distance rij . How can one obtain expressions for φ? A possible approach is sketched in Fig. 2.13, showing how a full-electron representation of the pair of atoms is reduced to a case where two point particles interact. The energy–distance relationship must be identical in both cases. An approach often used is to carry out quantum mechanical calculations that provides the relationship between distance and energy of a pair of


51

1 5

2 3 4 1

5

2 3 4

Fig. 2.12 Pair interaction approximation. The upper part shows all pair interactions of atom 1 with its neighbors, atoms 2, 3, 4, and 5. When the bonds to atom 2 are considered, the energy of the bond between atoms 1 and 2 is counted again (bond marked with thicker line). This is accounted for by adding a factor 1/2 in (2.27)

atoms, which is then used to determine the parameters of the pair potential expression. Pair potentials must capture the repulsion at short distances due to the increasing overlap of electrons in the same orbitals, leading to high energies due to the Pauli exclusion principle. At large distances, the potential must capture the effect that atoms attract each other to form a bond. In many pair potentials, two separate terms are used to describe repulsion and attraction, and the sum of these repulsive and attractive interactions yield the total energy dependence on the radius: φ = φRepulsion + φAttraction .

(2.28)

A pair of atoms is in the equilibrium position at a balance between the attractive and repulsive terms. Pair potentials are the simplest choice for describing atomic interactions. Even so, in some materials the interatomic interactions are best described by pair potentials (because the underlying quantum mechanical governing equations actually predict such a behavior). Prominent examples include the noble gases (e.g., argon, neon) [83] as well as Coulomb interactions due to partial charges. Pair potentials have also proven to be a reasonable model for more complex materials such as SiO2 [102]. The potential energy of an individual atom is given by

52


Fig. 2.13 Replacing a full-electron representation of atom–atom interaction by a potential function that only depends on the distance r between the particles

Ui =

Ni

φ(rij )

(2.29)

j=1

where Ni is the number of neighbors of atom i (this expression corresponds to the first summation in (2.27)). Usually, the number of neighbors considered for inclusion in the potential energy calculation is limited to the second or third nearest neighbors. Popular pair potentials for the simulation of metals include the Morse potential [103] and the Lennard-Jones (LJ) potential, which are described, for instance, in [9, 84, 104]. The LJ 12:6 potential is defined as 6 12 σ σ φ(rij ) = 40 . (2.30) − rij rij The LJ potential can be fitted to the elastic constants and lattice spacing of metals (however, this model has some shortcomings with respect to the stacking fault energy and the elasticity of metals). The term with power 12 represents atomic repulsion, and the term with power 6 represents attractive interactions. The parameter σ scales the length and 0 the energy of atomic bonds. Often, pair potentials are cutoff smoothly with a spline cutoff function (see for instance [104] or [105]). For the LJ potential, the equilibrium distance between atoms (denoted as r0 ) is given by √ (2.31) r0 = σ 6 2. The maximum force between two atoms is Fmax,LJ =

2.3940 . σ

(2.32)

Figure 2.14 shows a plot of the LJ potential (dotted line) and its derivative (continuous line, describing the interatomic forces), in a parametrization


53

for copper as reported in [9]. It also illustrates important points in the LJ potential, such as the equilibrium distance r0 and the point of largest force Fmax,LJ .

Fig. 2.14 Plot of the LJ potential and its derivative (for interatomic forces) in a parametrization for copper as reported in [9]

Another popular pair potential is the Morse potential, defined as 2

φ(rij ) = D [1 − exp (−β(rij − r0 ))] .

(2.33)

A fit of this potential to different metals (as well as different forms of the Morse potential) can be found, for instance in [106]. The parameter r0 stands for the nearest neighbor lattice spacing, and D and β are additional fitting parameters. The Morse potential is computationally more expensive than the LJ potential due to the exponential term (however, this is more realistic for many materials). An advantage of using pair potentials is the computational efficiency. Another important advantage is that fewer parameters are involved, which may simplify parameter studies and the fitting process to different materials. For example, the LJ potential has only two parameters, and the Morse potential has only three parameters. The potentials given by (2.30) and (2.33) are strongly nonlinear functions of the radius r. In some cases it is advantageous to linearize the potentials around the equilibrium position and define the so-called harmonic potential (see also Fig. 2.5) 1 φ(rij ) = a0 + k(rij − r0 )2 2

(2.34)

54


where k is the spring constant r0 the equilibrium spacing, and a0 is a constant parameter. An important drawback of pair potentials is that elastic properties of metals cannot be modeled correctly. 2.6.2 Multibody Potentials: Embedded Atom Method for Metals The concept that the total energy of the system is simply a sum over the energy contributions between all pairs of atoms in a system is a great simplification that leads to great challenges. For example, at a surface of a crystal, the atomic bonds may have different properties than in the bulk. Pair potentials are not capable of capturing this effect. The limitation of pair potentials to model more complex situation, in particular the dependence of the properties of chemical bonds between pairs of atoms on the environment, is sketched in Fig. 2.15. This behavior is particularly important for metals, because of quantum mechanical effects that describe the influence of the electron gas.

Fig. 2.15 Difference in bond properties at a surface. Pair potentials (left panel) are not able to distinguish bonds in different environments, as all bonds are equal. To accurately represent the change in bond properties at a surface, one needs to adapt a description that considers the environment of an atom to determine the bond strength, as shown in the right panel. The bond energy between two particles is then no longer simply a function of its distance, but instead a function of the positions of all other particles in the vicinity (that way, changes in the bond strength, for instance at surfaces, can be captured). Multibody potentials (e.g., EAM) provide such a description

To accurately represent the change in bond properties at a surface, a description is needed that considers the environment of an atom to determine the bond strength. Therefore, the bond energy between two particles is no


55

longer simply a function of its distance, but instead a function of the positions of all other particles in the vicinity. This behavior can be captured in multibody potentials. The idea behind multibody potentials is to incorporate more specific information on the bonds between atoms than simply the distance between two neighbors. In such potentials, the energy of bonds therefore depends not only on the distance of atoms, but also on its local environment, that is, on the positions of neighboring atoms. In the case of metals, the interactions of atoms can be quite accurately described using potentials based on the embedded atom method (EAM), or other so-called n-body potentials (e.g., [94, 107]. Several variations of the classical EAM potentials exist [108, 109]). Another, similar approach is based on effective medium theory (EMT) [110, 111]. Particularly appropriate models have been reported for metals such as copper and nickel. Other metals (e.g., aluminum) are more difficult to model with such approaches [112, 113]. An EAM potential for metals is typically given in the form Ui =

Ni

φ(rij ) + f (ρi ),

(2.35)

j=1

where ρi is the local electron density and f is the embedding function. The electron density ρi depends on the local environment of the atom i, and the embedding function f describes how the energy of an atom depends on the local energy density. The electron density itself is typically calculated based on a simple pair potential that maps the distance between atoms to the corresponding contribution to the local electron density. The potential features a contribution by a two-body term φ to capture the basic repulsion and attraction of atoms (just like in LJ or Morse potentials), in conjunction with a multibody term that accounts for the local electronic environment of the atom. Overall multibody potentials allow a much better reproduction of the elastic properties of metals than pair potentials (e.g., [109]). For instance, most real materials violate the Cauchy relation (that is, the condition that c1122 = c1212 ). Any pair potential predicts an agreement with the Cauchy relation [109]. Multibody potentials are capable of reproducing the appropriate elastic behavior. Figure 2.16 illustrates how an EAM-type multibody potential can represent different effective pair interactions between bonds at a surface and in the bulk. However, most conventional multibody potentials are not capable of modeling any effect of directional bonding. Whereas this is not important for metals such as Ni or Cu, this becomes quite significant in materials with a more covalent character of the interatomic bonding. To address these effects, modified embedded atom potentials (MEAM) have been proposed that can be parameterized, for instance, for silicon [114].

56


Fig. 2.16 This plot illustrates how an EAM-type multibody potential can represent different effective pair interactions between bonds at a surface and in the bulk

2.6.3 Force Fields for Biological Materials and Polymers The bases for simulations of polymers, organic substances, or proteins are force fields that describe the various chemical interactions based on a combination of energy terms. This is required since in these materials, the set of characteristic chemical bonds is much more diverse than in metals, for instance, which requires the explicit consideration of ionic, covalent, and vdW interactions. Figure 2.17 illustrates this chemical complexity, exemplified in a small alpha-helical coiled coil protein domain. A prominent example for this approach is the classical force field CHARMM [115]. The CHARMM force field is widely used in the protein and biophysics community, and provides a reasonable description of the behavior of proteins. This force field is based on harmonic and anharmonic terms describing covalent interactions, in addition to long-range contributions describing vdW interactions, ionic (Coulomb) interactions, as well as hydrogen bonding. Since the bonds between atoms are modeled by harmonic springs or its variations, bonds (other than H-bonds) between atoms cannot be broken, and new bonds cannot be formed. Also, the charges are fixed and cannot change, and the equilibrium angles do not change depending on stretch. The CHARMM force field belongs to a class of models with similar descriptions of the interatomic forces; other models include the DREIDING force field [116], the UFF force field [117], the GROMOS force field, or the AMBER model (see [118], for instance, for a review of various force fields for biological systems). In the CHARMM model, the mathematical formulation for the empirical energy function that contains terms for both internal and external interactions


57

Fig. 2.17 Chemical complexity in proteins involves a variety of chemical elements and different chemical bonds between them. The snapshot shows a small alphahelical coiled coil protein domain

Fig. 2.18 Schematic of the contributions of the different terms in the potential expressions given in (2.36), illustrating the contributions of bond stretching, angle bending, bond rotations, electrostatic interactions, and vdW interactions

58


has the form: Usystem = Ubond + Uangle + Utorsion + UCoulomb + UvdW + . . . ,

(2.36)

representing an approach to split up different energy contributions. However, most of the terms, in particular those describing the bond interactions, are harmonic expressions. An expression similar to (2.34) describes the energy contributions to the bond stretching term Ubond (the total energy due to bond stretching is obtained by summing over all pairs of atoms). For bond bending between three atoms i, j, and k, 1 φbend = b0 + kbend (θijk − θ0 )2 2

(2.37)

where θ0 is the equilibrium bond angle (depends on the particular triplet of atoms considered), and θ is the angle between the three atoms i, j, and k. For torsional energies between a group of four atoms, 1 φtorsion = t0 + ktorsion (1 − cos(θ1 )) 2

(2.38)

where θ1 is the torsional angle and ktorsion is the appropriate spring constant that describes the magnitude of the resistance to torsional deformation of a group of atoms. The contributions from all pairs of atoms (for bond stretching), all triplets of atoms (for bond bending), and quadruples of atoms (for bond rotation) are summed up to yield the total potential energy of the system: Ubond = φbond , (2.39) pairs

Ubend =

φbend ,

(2.40)

triplets

and Utorsion =

φtorsion .

(2.41)

quadruples

Due to the harmonic approximation, these expressions are only valid for small deformation from the equilibrium configuration of the bond. Large deformation or fracture of these bonds cannot be described. Figure 2.18 schematically illustrates the energy contributions provided in (2.36). The Coulomb energies are evaluated between pairs of atoms and are described as qi qj (2.42) φCoulomb = 1 rij where qi and qj are the partial charges of atoms i and j, and 1 is the effective dielectric constant (1 = 1.602 × 10−19 C for vacuum). The total Coulomb energy is then given by


UCoulomb

N N qi qj = r j=1 1 ij

59

(2.43)

j=i=1

The calculation of electrostatic interactions provides a significant computational burden, as the particle interactions are long range. In particular when using small, periodic unit cell approaches, the interactions from one cell side to another must be considered very carefully. Several methods have been developed to address this problem. The most prominent technique is the particle-mesh Ewald method (PME), which significantly reduces the computational effort in accurately calculating these interactions. Most modern molecular dynamics codes have implementation of the PME technique or related approaches. The vdW terms are typically modeled using Lennard-Jones 6–12 terms. Both the vdW terms and the Coulomb terms contribute to the external or nonbonded interactions. H-bonds are often included in the vdW terms. Some flavors of CHARMM-type potential provide explicit expressions for Hbonds that involve angular terms to provide a more refined description of the spatial orientation between H-bond acceptor and donor pairs. The parameters in such force fields are often determined from quantum chemical simulation models by using the concept of force field training. Which specific terms of the force field formulation are considered for a particular chemical bond are controlled by atom type names. That is, each atom type is not only specified by its element name but also by a tag that denotes which type of chemical bond is attached to it. For instance, the tag CA refers to an aromatic carbon atom, CC to a carbonyl carbon atom, and C to a polar carbon atom (e.g., in a protein backbone). Thus, a critical step before a molecular simulation with a CHARMM-type force field can be carried out is the assignment of these atom types. The information of element types as provided in the Protein Data Bank, for instance, is insufficient. Automated programs have been developed to carry out this typing tasks by comparing particular molecular structures with templates in a large database, and assigning appropriate atom types according to a best fit comparison. Force fields for protein structures typically also include simulation models to describe water molecules (e.g., TIP3P, TIP4P, SPC and SPC/E, ST2), an essential part of any simulation of protein structures. These water force fields are composed of similar harmonic, bond angle, dispersive, and Coulomb expressions. Table 2.2 summarizes several popular choices for organic force fields. 2.6.4 Bond Order and Reactive Potentials An important class of multibody potentials is based on the concept of bond orders, a model particularly suitable to describe the forces in covalently bonded materials (that is, chemical bonds that have a strong directional dependence). This is important, for instance, in carbon-based materials, such as carbon nanotubes. The key concept is that the bond strength between two

60

Atomistic Modeling of Materials Failure Force field name

Primary application

CHARMM AMBER CVFF CFF GROMOS

Proteins Proteins Small molecules Organic substances Proteins, organic molecules

UFF DREIDING TIP3P, TIP4P SPC, SPC/E

MM3 Organic molecules Organic molecules Water molecules Water molecules

Notes

Implemented in GROMACS code Organic molecules Universal force field

Table 2.2 Overview over force fields suitable for organic substances

atoms is not constant, but depends on the local environment, similar to the EAM approach (please see [119] for a discussion). However, specific terms are included to specify the directional dependence of the bonding. The idea is based on the concept of mapping bond distances to bond orders, which enables one to determine specific quantum chemical states of a molecular structure. The concept of bond order was initially introduced by Pauling [120]. Very well-known models in this class of potentials are the reactive bond order potentials (REBO) [121, 122], the Tersoff potential [10, 123], the Stillinger–Weber potential [124], Brenner’s force fields [125, 126], the Stuart reactive potential [127], and more recently, the ReaxFF reactive force field [93]. Abell–Tersoff Approach The basic concept of bond order potentials is simple to explain. The key idea is to modulate the bond strength based on the atomic environment, taking advantage of some theoretical chemistry principles. Instead of expressing φ(rij ) as a harmonic function or an LJ function (see above), in the Abell–Tersoff approach the interaction between two atoms is expressed as φ(rij ) = φRepulsion (rij ) − Mij φAttraction (rij ),

(2.44)

where φRepulsion (rij ) is a repulsive term and φAttraction (rij ) is an attractive term. The parameter Mij that multiplies the attractive interactions represents a many-body interaction parameter. This parameter describes how strong the attraction is for a particular bond, from atom i to atom j. Most importantly, the parameter Mij can range from zero to one, and describes how strong this particular bond is, depending on the particular chemical environment of atom i. It can thus be considered a normalized bond order, following the concept of


61

Fig. 2.19 The plot shows the cohesive energy per atom (upper plot, in eV) and the bond length (lower plot, in ˚ A), for several real and hypothetical polytypes of carbon, comparing the predictions from the Tersoff potential [10] for C with experimental and other computational results. The structures include a C2 dimer molecule, graphite, diamond, simple cubic, BCC, and FCC structures. The squares correspond to experimental values for these phases and calculations for hypothetical phases [11]. The circles are the results of Tersoff’s model [10]. The continuous lines are spline fits to guide the eye. Reprinted from: J. Tersoff, Empirical interatomic potentials for carbon, with applications to amorphous carbon, Physical Review Letters, Vol. c 1988 by the American Physical Society 61(25), 1988, pp. 2879–2883. Copyright

the Pauling relationship between bond length and bond order. Abell suggested that (2.45) Mij ∼ Z −δ where δ depends on the particular system and Z is the coordination number of atom i that depends on the bond radius. For pair potentials, Mij ∼ Z, which is not true for many real materials. The Abell–Tersoff approach provides a realistic model for these effects. In the Tersoff potential [10, 123], the many-body term depends explicitly on the angles of triplets of atoms, in addition to considering the effect of coordination. Thus, this and related potentials are also referred to as threebody potentials. The explicit angular dependence also illustrates the difference to EAM potentials: Here, the multibody term solely depends on the electron density, and not on any directional information. Tersoff has successfully parametrized his potential approach for carbon, silicon, and other semiconductors. Figure 2.19 shows the cohesive energy per atom for several real and

62


hypothetical polytypes of carbon, comparing the prediction from experiment with the results obtained from the potential. These equations immediately lead to a relationship between bond length, binding energy, and coordination, through the parameter Mij . The modulation of the bond strength effectively leads to a change in spring constant as a function of bond environment, k(r) ∼ k0 Mij (Z, δ).

(2.46)

Note that k0 is a reference spring constant, which is modulated by the atomic environment that is essentially dependent on the bond radius. This method has been very successful to describe the interatomic bonding in several covalently bonded materials, for example the C–C bonds in diamond, graphite, and even hydrocarbon molecules [125, 126]. The coordination number is a concept also used in lattice systems, for example crystals. In organic molecules, the coordination number can be thought of as the amount of covalent bonds that an atom has made. Reactive Force Fields Many attempts have failed to accurately describe the transition energies during chemical reactions using more empirical descriptions than relying on purely quantum mechanical (QM) methods. Reactive force fields represent a strategy to overcome some of the limitations classical force fields, in particular the fact that these descriptions are not able to describe chemical reactions. In fact, the behavior of chemical bonds at large stretch has major implications on the mechanical response, as it translates into the properties of molecules at large-strain, a phenomenon referred to nonlinear elasticity or hyperelasticity.

Fig. 2.20 An example to demonstrate the basic concept of the ReaxFF potential. It has been developed to accurately describe transition states in addition to ground states


63

Reactive potentials are based on a more sophisticated formulation than most nonreactive potentials. A bond length to bond order relationship is used to obtain smooth transition between different bond types, including single bonds, double bonds and triple bonds. Typically, all connectivity-dependent interactions (that means, valence and torsion angles) are formulated to be bond-order dependent. This ensures that their energy contributions disappear upon bond dissociation so that no energy discontinuities appear during reactions (see Fig. 2.20). The reactive potential also features nonbonded interactions (shielded van der Waals and shielded Coulomb). Several flavors of reactive potentials have been proposed in recent years. Reactive potentials can overcome the limitations of empirical force fields and enable large-scale simulations of thousands of atoms with quantum mechanics accuracy. The reactive potentials, originally only developed for hydrocarbons, have been extended recently to cover a wide range of materials, including metals, semiconductors, and organic chemistry in biological systems such as proteins. Here we review in particular the ReaxFF formulation [93, 128– 131, 131]. The most important features of the class of ReaxFF reactive force fields are • A bond length to bond order relationship is used to obtain a smooth transition of the energy from a nonbonded to single, double, and triple bonded molecules. • All connectivity-dependent interactions (that is, valence and torsion angles) are made bond-order dependent: Ensures that their energy contributions disappear upon bond dissociation. • Features shielded nonbonded interactions that include van der Waals and Coulomb interactions, without discrete cutoff radius to ensure a continuous energy landscape. • ReaxFF uses a geometry-dependent charge calculation scheme (similar to the Charge Equilibration method, QEq [132]) that accounts for polarization effects and redistribution of partial atomic charges as the molecule or cluster of atoms changes its shape (e.g., determine the partial atom charges qi and qj ). • Most parameters in the formulation have physical meaning, such as corresponding distances for bond order transitions, atomic charges and others. • All interactions in ReaxFF feature a finite cutoff of 10 ˚ A. The total energy of a system in the ReaxFF model is expressed as the sum of different contributions that account for specific chemical interactions. Usystem = Ubond + Uunder + Uover + Uangle + Utors + Uconj + UH-bonds + UCoulomb + UvdW

(2.47)

The term Ubond describes the energy contributions due to covalent bonds. The terms Eunder and Uover describe energy penalties for under- and overcoordination. Angular effects are included in Uangle , and contributions due to

64


torsion are included in Utors . The term Uconj describes energetic contributions of resonance effects. A maximum contribution of the conjugation energy is obtained when successive bonds have bond order values of 1.5, as it is the case in benzene, for instance. H-bonds are treated in the term UH-bonds , and its interactions are calculated between groups X–H and Y, where X and Y are atoms that can form H-bonds (for instance, N, O). Nonbonded two-body interactions are included in UvdW and in UCoulomb . They are included for all atom pairs whether they are bonded or nonbonded. This is important to avoid energy discontinuities when chemical reactions occur. To enable the calculation for these interactions for all atom pairs, a shielded Coulomb potential of the form Ni Ni qi qj (2.48) φi = 3 + γ −3 )1/3 (r 1 ij ij i=1 j=1 is used (the parameter γij is a force field parameter that is adapted to reproduce the orbital overlap contributions). We explain the approach used in ReaxFF for the example of the term Ubond . First the bond order is calculated, according to bππ b b rij σ rij π rij + exp aπ + exp aππ . BOij = exp aσ r0 r0 r0 (2.49) The terms ai and bi are fitting parameters that describe the dependence of the bond order on the bond geometry. The numerical values are adapted for each bond type. The term rij is the distance between atoms i and j. This equation yields the graphs shown in Fig. 2.21. Corrected bond orders BOij are calculated from BOij via correction functions to account for the effect of over- and under-coordination (the correction functions are a function of the degree of deviation of the sum of the uncorrected bond orders around an atomic center from its valency Vali ),

∆i =

n bonds

BOij − Vali .

(2.50)

j=1

This correction refers to the fact that carbon, for instance, cannot have more than four bonds, or hydrogen cannot have more than one bond. This expression illustrates that the bonding term is a multibody expression, as it depends on all j neighbors of an atom i. Once the bond orders are known, the energy contributions can be calculated. For the term considered here,

φbond, ij = −De BOij exp pbe,1 1 − BOpbe,1 ij , (2.51) where De and pbe,i are additional bond parameters. The total bond energy is then given by a summation over all bonds in the system,


65

Fig. 2.21 Illustration of basic concept of bond order potentials. Subplot (a) shows how the bond order potential allows for a more general description of chemistry, since all energy terms are expressed dependent on bond order. In contrast, conventional potentials (such as LJ, Morse) express the energy directly as a function of the bond distance as shown in subplot (b). Subplot (c) illustrates the concept for a C–C single, double, and triple bond, showing how the bond distance is used to map to the bond order, serving as the basis for all energy contributions in the potential formulation defined in (2.47)

Ubond =

φbond,ij

(2.52)

all bonds ij

Equation (2.51) also illustrates that the energy contributions vanish when the bond order goes to zero, which corresponds to a broken chemical bond. All other terms are also expressed as a function of bond orders. For instance, the angle contributions are given as Uangle = f (θijk , BOij , BOjk ).

(2.53)

It is noted that this illustrates a distinction to the Tersoff potential. In Tersoff’s approach, the angular depedence is included in the multibody term for

66


pairs of atoms. In the ReaxFF approach, an explicit angular term is included, similar to the approach used in CHARMM-type force fields. A crucial aspect of the ReaxFF force field is that all parameters are derived from fitting against quantum mechanical data (DFT level) [93]. This process is referred to as force field training. The basic concept is to require a closeas-possible agreement between the quantum mechanical prediction and the force field result, for a wide range of properties. Typically these properties include elastic properties and equation of state, surface energies, dissociation energies and landscapes, or the interaction energies of organic compounds with surfaces. Due to the increased complexities of force field expressions and the charge equilibration step that is carried out at each force calculation, reactive force fields are between 50 and 100 times more expensive than nonreactive force fields, yet several orders of magnitude faster than DFT-level calculations that would be able to describe bond rupture as well.

Fig. 2.22 Results of a ReaxFF study of water formation, comparing the production rate with and without a Pt catalyst. The presence of the Pt catalyst significantly increases the water production rate (results taken from [12])

An example simulation is shown in Fig. 2.22 [12]. This plot depicts the results of a ReaxFF study of water formation, comparing the production rate with and without a Pt catalyst (same pressure and temperature, the only difference is the catalyst). It is evident that the presence of the Pt catalyst significantly increases the water production rate.


67

Fig. 2.23 Water production at varying temperature, for constant pressure. Subplot (a) depicts the water production rate. Subplot (b) shows an Arrhenius analysis, enabling us to extract the activation barrier for the elementary chemical process of 12 kcal/mol. This result is close to DFT level calculations of the energy barrier [12]

68


Figure 2.23 depicts an analysis of the system with Pt catalyst, for variations of the temperature. These simulations involves thousands of reactive atoms, a computational task that cannot be achieved using DFT or similar approaches. Thus, the ReaxFF approach provides a very useful bridge between quantum mechanical methods and empirical potentials, as illustrated in Fig. 2.24. ReaxFF simulations have been reported with system sizes approaching millions of atoms, as recently reported in the ReaxFF parallelized algorithm [133].

Fig. 2.24 The ReaxFF force field fills a gap between quantum mechanical methods (e.g., DFT) and empirical molecular dynamics

2.6.5 Limitations of Classical Molecular Dynamics Atomistic or molecular simulations is a fundamental approach, since it considers the basic building blocks of materials as its smallest entity, atoms. Stress singularities at crack tips are handled naturally, thereby avoiding many challenges associated with continuum methods. Molecular dynamics is also a quite appropriate tool for describing material deformation under extremely high strain rates that are not accessible by other methods (FE, DDD, and other approaches). However, molecular dynamics simulations feature several limitations. It is important to remember these limitations during the course of model development and data analysis and interpretation. For example, molecular dynamics simulation models typically only allow one to study materials with dimensions of several hundred nanometers or below. The time scale limitation is another serious limit of molecular dynamics, which has prevented many researchers from studying phenomena of interest or to make rigorous links to laboratory experiments that are often carried out at much different time scales. Please see [134] for a description of these issues. Further associated aspects and limitations will be discussed in the forthcoming sections. To treat electrons properly one needs to incorporate quantum mechanical methods explicitly into the molecular models, which feature degrees of freedom that describe the structure of electrons. This is particularly significant for


69

electronic properties, optical phenomena, and magnetic properties, which all require such quantum mechanical treatments. The physical reason for this is that these properties are derived from the electronic structure associated with atoms, molecules, or system of atoms rather than being derived only from the positions of atoms or the interatomic forces. Some methods have been developed that utilize simple potential expressions to describe the complex quantum mechanical effects. For instance, the electron force field, or eFF [135], utilizes a single approximate potential between nuclei and electrons, and correctly describes many phases relevant for warm dense hydrogen. The eFF model thereby provides a simplified solution of the time-dependent Schroedinger equation. The potential formulation requires only moderate computational effort, since the computational complexity of the terms is similar to those used in traditional force fields that treat atoms only as particles. Thus it may be possible to use this new force field to simulate large excited systems that are currently beyond the reach of quantum mechanical methods.

2.7 Numerical Implementation Here we summarize a few important numerical and implementation aspects of molecular dynamics simulation methods.

Fig. 2.25 Schematic of the numerical scheme in carrying out molecular dynamics simulations

Figure 2.25 depicts the basic numerical scheme of carrying out a molecular dynamics study. The basic steps of a molecular dynamics simulation are • Define initial conditions and boundary conditions (including positions and velocities at t = t0 ); typically the velocities of particles are drawn from a Maxwell–Boltzmann distribution.

70


• Obtain updating method, e.g., the Verlet scheme as described in (2.13), choose time step and thermodynamical ensemble. • Obtain an expression for forces, by defining an approximation of the potential energy landscape U (r). • Analyze data using statistical methods.

Fig. 2.26 Schematic of the numerical scheme in carrying out molecular dynamics simulations

2.7.1 Periodic Boundary Conditions Periodic boundary conditions (PBCs) is a widely used concept in molecular dynamics. Figure 2.26 depicts a schematic of implementation of periodic boundary conditions. PBCs allows one to study bulk properties (that means, there are no free surfaces) with small number of particles (here N = 3 for three particles). For a variety of thermodynamical properties it has been demonstrated that this approach is very efficient. However, for mechanisms or phenomena that involve inhomogeneous stress and deformation fields, the approach does not give useful results. To represent such spatially varying fields it is vital to make the system larger and model the entire collection of atoms that resemble the physical space. When periodic boundary conditions are used, all particles are “connected” and do not sense the existence of a free surface. For numerical reasons, the original cell is surrounded by 26 image cells (8 in 2D). Image particles in these image cells move in exactly the same way as original particles. These image copies are necessary to carry out force calculations in the system, as discussed in Sect. 2.7.2.


71

2.7.2 Force Calculation We recall that at each integration step, forces are required to obtain accelerations. Forces are calculated from the positions of all atoms, by considering the atomic potential energy surface U (r) that depends on the positions of all atoms. According to (2.9), the force vector is given by taking partial derivatives of the potential energy surface with respect to the atomic coordinates of the atom considered, dU (r) Fi = − . (2.54) dri

Fig. 2.27 Schematic of force calculation scheme in molecular dynamics, for a pair potential. To obtain the force vector F one takes projections of the magnitude of the force vector F into the three axial directions xi (this is done for all atoms in the system)

For the special case where the interatomic forces are described by a potential φ that depends only on the distance between pairs of atoms, here denoted by r, the contributions to the total force vector due this particular interaction can be obtained by taking projections of the magnitude of the force vector F into the three axial projections ri of the vector between the pair of particles. The magnitude of the force vector due to interactions of pairs of atoms is then given by dφ(r) , (2.55) F = dr where F =| F |. The ith component of the force vector is then given by Fi = −F

ri . r

(2.56)

Figure 2.27 depicts this approach schematically. In principle, all atoms in the system interact with all other atoms, which requires a nested loop for calculation of the force vectors of all atoms. This renders the total computational time requirement to solve the problem second order with respect to the number of particles N ,

72


Fig. 2.28 Use of neighbor lists and bins to achieve linear scaling ∼N in molecular dynamics. Panel (a): Example of how neighbor lists are used. The four neighbors of the central atom (in the circle) are stored in a list so that force calculation can be done directly based on this information. This changes the numerical problem to a linear scaling effort. Panel (b): The computational domain is divided into bins according to the physical position of atoms. Then, atomic interactions must only be considered within the atom’s own bin and atoms in the neighboring bins

ttot ∼ N 2 .

(2.57)

The first loop goes over all atoms, and the second loop goes through all possible neighbors of each atom. The following pseudocode illustrates this process: for i=1 to N # loop over all atoms i for j=1 to N (i not equal to j) # second loop over all atoms neighboring atom i r=distance(i,j) # calculate distance between atoms i and j F=f(r) # calculate force depending on radius

Such second-order scaling of the force calculation time is a computational disaster and would prevent us from solving large systems. Several strategies are commonly used to avoid this type of scaling, as discussed in the next few sections. 2.7.3 Neighbor Lists and Bins Avoiding the second-order scaling of force calculation ∼N 2 is important to develop numerically feasible strategies. A bookkeeping scheme that is often used in molecular dynamics simulation is a neighbor list that keeps track of


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that are the nearest, second nearest, and so forth neighbors of each particle. This method is used to save time from checking every particle in the system every time a force calculation is made. The list can be used for several time steps before updating. Each update is expensive since it involves N × N operations for an N -particle system. In low-temperature solids where the particles do not move very much, it is possible to perform an entire simulation without or with only a few updating, whereas in simulation of liquids, updating every 5 or 10 steps is quite common. Figure 2.28a shows a schematic of how neighbor lists are used. We note that neighbor lists can only be implemented if particles interact only up to a certain cutoff radius; for very long range interactions, the definition of neighbor lists may not be feasible. An alternative to generation of neighbor lists is the decomposition of the computational domain into bins. The size of the bin is chosen comparable to the cutoff radius of the potential, so that atomic interactions must only be considered within the atom’s own bin and the neighboring bins (see Fig. 2.28b).

2.8 Property Calculation In this section, we summarize important methods to calculate system properties from molecular dynamics simulations. We introduce definitions and numerical procedures for the following material properties: • • • •

Temperature T (thermodynamic property) Pressure P (thermodynamic property) Radial distribution function g(r) (structural information) Mean square displacement function (measure for mobility of atoms, relates to diffusivity) • Velocity autocorrelation function (transport properties) • Virial stress tensor (link to continuum Cauchy stress) 2.8.1 Temperature Calculation Using the kinetic energy of the system K, the temperature is given by T =

2 K . 3 N kB

(2.58)

Note that the numerical value for the Boltzmann constant kB = 1.3806503 × 10−23 m2 kgs−2 K−1 , relating energy and temperature at the level of individual atoms or particles (its units are energy per absolute temperature). Please see also Table 4.7 for other units and their conversion to SI units.

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2.8.2 Pressure Calculation The pressure P is given by P = N kB T −

N N 11 dφ rij . 3 V i=1 j=1,j 1

Table 2.4 Centrosymmetry parameter ci for various types of defects, normalized by the square of the lattice constant a20 . In the visualization scheme, we choose intervals of ci to separate different defects from each other

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Fig. 2.41 Analysis of a dislocation using the slip vector approach. From the result of the numerical analysis, direct information about the Burgers vector can be obtained. The slip vector s is drawn at each atom as a small arrow. The Burgers vector b is drawn at the dislocation (its actual length is exaggerated to make it better visible). The dislocation line is approximated by discrete, straight dislocation segments. A line element between “a” and “b” is considered

An example using this centrosymmetry technique is shown in Fig. 2.40. This plot shows the same section as in Fig. 2.38b. Unlike in the analysis with the energy method, stacking fault regions can be visualized with the centrosymmetry technique. 2.10.3 Slip Vector Although the centrosymmetry technique can distinguish well between different defects, it does not provide information about the Burgers vector of dislocations. The slip vector approach was first introduced by Zimmerman and coworkers in an application of molecular dynamics studies of nanoindentation [147]. This parameter also contains information about the slip plane and Burgers vector. The slip vector of an atom α is defined as sα i = −

nα 1 αβ xαβ , − X i i ns

(2.72)

α=β

is the vector difference of where ns is the number of slipped atoms, xαβ i atoms α and β at the current configuration, and Xiαβ is the vector difference of atoms α and β at the reference configuration at zero stress and no mechanical deformation. The slip vector approach can be used for any material microstructure, unlike the centrosymmetry parameter which can only be used for centrosymmetric microstructures. Figure 2.41 shows the result of a slip vector analysis of a single dislocation in copper [39]. The slip vector s is drawn at each atom as a small arrow. The Burgers vector b is drawn at the dislocation, where its actual length is exaggerated to make it better visible. The dislocation line can be determined


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from an energy analysis, and the line direction of a segment between point “a” and “b” of the dislocation line is indicated by the vector l. The Burgers vector b is given by the slip vector s directly. The analysis reveals that the dislocation has Burgers vector b = 16 [112]. The unit vector of line direction of the segment is l ≈ [−0.3618 0.8148 − 0.5530]. The length of the line segment is approximately 9 nearest neighbor distances in the [110] direction. The slip plane normal is given by the cross product ns = l × b ∼ [111], and the dislocation thus glides in the (111) plane. 2.10.4 Measurement of Defect Speed Accurate determination of the propagation speed of defects is crucial in the analysis in particular of rapid materials failure phenomena. For instance, the crack tip velocity is an important measure in the analysis of dynamic fracture of brittle materials. The crack tip velocity is determined from finding the crack tip position. The geometry of the crack tip is determined by finding the surface atom at the tip of the crack. This is achieved by considering the surface atom with the highest y position, for instance, for the case of crack propagation in the y-direction. This search is carried out in the interior of a search region inside the slab. This quantity is averaged over a small time interval to eliminate very high frequency fluctuations. To obtain the steady state velocity of the crack, the measurements of the crack speed must be taken within a region of constant stress intensity factor (see, for instance [114] for an additional discussion on this issue). Figure 2.39 illustrates this approach based on the energy filtering method. 2.10.5 Visualization Methods for Biological Structures For the visualization of organic molecules (such as proteins), specific tools have been developed. Many visualization tools exist that are capable of displaying biological protein molecules and molecular clusters. A rather versatile, powerful and widely used visualization tool is the Visual Molecular Dynamics (VMD) program [148]. This software enables one to render complex molecular geometries using particular coloring schemes. It also enables one to highlight important structural features of proteins by using a simple graphical representation, such as alpha-helices, or the protein’s backbone. The simple graphical representation is often referred to as cartoon model. These visualizations are often the key to understand complex dynamical processes and mechanisms in analyzing the motion of protein structures and protein domains, and they represent a filter to make useful information visible and accessible for interpretation. Figure 2.42 shows the visual analysis of a simple alpha-helix protein structure, with different visualization options.

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Fig. 2.42 Analysis of a simple alpha-helix protein structure, with different visualization options, plotted using VMD [148]

2.10.6 Other Methods Other researchers have used a common neighbor analysis to analyze the results of molecular dynamics simulations of crystalline structures [149–151]. In this method, the number of nearest neighbors is calculated, which allows one to distinguish between different defects. Additional analysis to analyze more complex structures such as grain boundaries is possible based on the medium-range-order (MRO) analysis. This method is capable of determining the local crystallinity class. The MRO analysis has been applied successfully in the analysis of simulations of nanocrystalline materials, where an exact characterization of the grain boundary structure is important (e.g., [152–154]).

2.11 Distinguishing Modeling and Simulation The specific meaning of modeling versus simulation has been introduced in Sect. 2.2. Here we briefly explain the application of these terms for the example of a generic molecular dynamics model. Table 2.5 provides a list of tasks associated with molecular dynamics and its classification into modeling and simulation.

2.12 Application of Mechanical Boundary Conditions The application of boundary conditions in atomistic and molecular systems is essential, in particular for studies of the mechanical behavior.


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Table 2.5 Distinguishing modeling and simulation, for tasks associated with classical molecular dynamics Modeling Simulation Mathematical model of physical system Numerical solution of problem Choice of model geometry Choice of numerical integrator (e.g., Verlet) Choice of interatomic potential and parameters Force calculation Choice of boundary conditions Implementation of boundary conditions Choice of system size Choice of thermodynamical ensemble Algorithm to specify the thermodynamic ensemble (e.g., temperature control, pressure control)

× × × × × × ×

× ×

×

×

Fig. 2.43 Simulation method of domain decomposition via the method of virtual atom types. The atoms in region 2 do not move according to the physical equations of motion, but are displaced according to a prescribed displacement history. An initial velocity gradient as shown in the right half of the plot is used to provide smooth initical conditions

Most straightforward are approaches to apply displacement boundary conditions. In these applications, the dynamics of corresponding atoms (or groups of atoms forming a physical domain whose displacement is prescribed) is altered so that this group of atoms follows a prescribed motion rather than following the dynamics according to the equations of motion. To model weak layers or different interatomic interactions in different regions of the simulation domain, one can assign a virtual type to particular groups of atoms. On the basis of the type definition of interacting pairs of atoms, it is possible to calculate different interatomic interactions. For instance, this enables one to model an atomically sharp crack tip by removing any atomic interaction across a material plane (effectively describing this part as a free surface). An example for this type of domain decomposition is shown

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in Fig. 2.43. To apply mechanical load by controlling the displacement, a linear velocity gradient is established prior to simulation to avoid shock wave generation from the boundaries (see also Fig. 2.43). To strain the system, a few layers of atoms at the boundary of the crystal slab are not moved according to the natural equations of motion (e.g., by the Verlet algorithm), but are instead displaced in each step, according to the prescribed displacement, with the velocity that matches the initial velocity field. This procedure has been used in several atomistic studies of dynamic fracture [28, 146, 155, 156]. It is possible to stop the increase of loading after a prescribed loading time, from which on the boundaries are kept fixed. In an alternative method, the system can be strained prior to the beginning of the simulation (according to the particular loading direction), and the outermost material layers are then kept fixed during the simulation (that is, atoms in this group do not move at all). The application of stress or pressure boundary conditions can be more challenging, but it can be achieved by utilizing appropriate ensemble schemes, as for instance the Parinello–Rahman method. This approach enables one to prescribe a particular stress tensor to the system. By changing the prescribed value of the stress tensor one can simulate slow loading conditions. To apply the forces to the molecule that induce deformation, steered molecular dynamics (SMD) has evolved into a useful tool. Steered molecular dynamics is based on the concept of adding a harmonic moving restraint to the center of mass of a group of atoms. This leads to the addition of the following potential to the Hamiltonian of the system: U (r1 , r2 , ..., t) =

1 k (vt − (r(t) − r0 ) · n) , 2

(2.73)

where r(t) is the position of restrained atoms at time t, r0 denotes original coordinates and v and n denote pulling velocity and pulling direction, respectively. The net force applied on the pulled atoms is F (r1 , r2 , ..., t) = k (vt − (r(t) − r0 ) · n) .

(2.74)

By monitoring the applied force (F ) and the position of the atoms that are pulled over the simulation time, it is possible to obtain force-vs.-displacement data that can be used to derive the mechanical properties such as bending stiffness or the Young’s modulus (or other mechanical properties). SMD studies are typically carried out with a spring constant k = 10 kcal mol−1 ˚ A−2 , albeit this value can be varied depending on the particular situation considered. Figure 2.44 depicts a schematic that illustrates how load is applied with SMD, comparing an AFM experiment with the numerical scheme. One of the first applications of the SMD technique was to study protein unfolding. Fig. 2.45 shows steered molecular dynamics simulations of I27 extensibility under constant force, illustrating the molecular geometry under different force levels (compare with Fig. 1.16a) as well as the force–extension curve (compare with Fig. 1.16b).


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Fig. 2.44 Schematic to illustrate the use of steered molecular dynamics to apply mechanical load to small-scale structures (subplot (a): AFM experiment; subplot (b) Steered Molecular Dynamics model)

2.13 Summary We summarize the main points presented in this section. We discussed analysis techniques, to extract useful information from molecular dynamics results, including the velocity autocorrelation function, the atomic stress, and the radial distribution function. These are useful analysis methods since they provide quantitative information about molecular structure in the simulation, for example during phase transformations, to study how atoms diffuse, for elastic (mechanical) properties, and others. We introduced several interatomic potentials that describe the atomic interactions. The basic approach in developing such models is to condense out electronic degrees of freedom and to model atoms as point particles. Properties accessible to molecular dynamics can be classified into these broad categories [86]: • Structural - crystal structure, g(r), defects such as vacancies and interstitials, dislocations, grain boundaries, precipitates • Thermodynamic – equation of state, heat capacities, thermal expansion, free energies • Mechanical – elastic constants, cohesive and shear strength, elastic and plastic deformation, fracture toughness • Vibrational – phonon dispersion curves, vibrational frequency spectrum, molecular spectroscopy • Transport – diffusion, viscous flow, thermal conduction

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Fig. 2.45 Steered molecular dynamics simulations of I27 extensibility under constant force. Subplot (a) shows snapshots of the structure of the I27 module simulated at a force of 50 pN (I, at 1 ns) and 150 pN (II, at 1 ns). At 50 pN, the hydrogen bonds between strands A and B are maintained, whereas at 150 pN they are broken. Subplot (b) displays the corresponding force–extension relationship obtained from the simulations. The discontinuity observed between 50 and 100 pN corresponds to an abrupt extension of the module by 4–7 ˚ A caused by the rupture of the AB hydrogen bonds, and the subsequent extension of the partially freed polypeptide c segment. Reprinted with permission from Macmillan Publishers Ltd., Nature [6] 1999

Molecular dynamics is a useful method to • Perform virtual experiments, that is, computational experiments • Implement a computational microscope to visualize and analyze microscopic processes • Gain fundamental understanding about behavior of materials Further, molecular dynamics: • Has an intrinsic length scale, given by the distance of atoms in the material (typically on the order of the length of a chemical bond), that is, between 1 and 5 ˚ A • Handles stress singularities (e.g., at crack tips) intrinsically • Is ideal for deformation under high strain rate and extreme conditions, which are not accessible by other methods (such as the finite element method, discrete (mesoscale) dislocation dynamics, and other simulation approaches)

3 Basic Continuum Mechanics

The purpose of this chapter is to provide an introduction into basic continuum mechanical concepts, including stress and strain tensor and equilibrium conditions. Using the concepts introduced in this chapter, we will derive simple solutions to elastic beam bending problems to demonstrate how problems can be addressed in the realm of continuum mechanics. The objective of solving such elasticity problems is to provide a link between applied forces, boundary conditions, and the stresses, strains, and displacements inside the material. Such information is critical for instance to find out if a designed structure can operate safely, away from its failure condition. There are several additional applications and significant components of elasticity problems in many other fields, as will be discussed later in this chapter. Topics covered in this chapter include definition of elasticity, elastic response, energetic vs. entropic elasticity, Young’s modulus, stress and strain tensor, as well as the mechanics of a beam. The continuum mechanics concepts introduced in this chapter will be used later on, when a direct comparison between atomistic and continuum methods is carried out.

3.1 Newton’s Laws of Mechanics Sir Isaac Newton (1642–1727) proposed three fundamental laws of mechanics, laying the foundation for the continuum mechanical framework. All continuum mechanical governing partial differential equations can be derived from these three concepts. • First law: An object in a state of rest or uniform motion tends to remain in that state of motion unless an external force is applied to it. That is, as long as the sum of forces acting on it is zero F= Fi = 0, (3.1) the direction and magnitude of velocity of it does not change.

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• Second law: The change of motion is proportional to the applied force to an object. That is, d(mv) = F. (3.2) dt For constant mass m, this law simplifies to m

dv = ma = F, dt

(3.3)

where a is the acceleration (for the application in the molecular dynamics formulation, please see Sect. 2.4). It is noted that the second law as stated above is strictly valid only for a single point mass that cannot undergo rotation. For a collection of N point masses, the equation has to be extended to include d (xi × mi vi ) = (xi × Fi ) = Mi . dt i=1 i=1 i=1 N

N

N

(3.4)

• Third law: To every action there is always a reaction with opposed direction. In other words, the mutual interaction of two bodies are always equal in magnitude, and directed contrary to each other. For two interacting particles i and j, this implies that the mangitude of interaction Fij = Fji , and for the vectors Fij = −Fji .

(3.5)

In the following sections we will demonstrate how these simple laws can be used to post and solve mechanical problems. Due to the scope of this book, this discussion is brief and we refer the reader to further literature for additional details.

Fig. 3.1 Axial tensile loading of a beam and schematic force–extension response. Reversible deformation denotes the elastic regime; upon unloading of the sample the displacement returns to the initial point. Irreversible deformation denotes the plastic regime; upon unloading (indicated in the graph) the displacement does not return to the initial point. (It is noted that the specific shape of the force-extension curve may vary significantly depending on the type of material)


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3.2 Definition of Displacement, Stress, and Strain We begin with a brief introduction into basic concepts. We consider a beam under axial, tensile loading, as shown in Fig. 3.1. The globally applied boundary conditions (shear force P and axial force N ) lead to a local section stress distribution. This problem features several fundamental aspects of mechanics of materials. Considering the force–extension curve shown in Fig. 3.1, several regimes of deformation can be recognized. In the first regime, the F –∆u curve reveals a linear relationship between applied force F and extension ∆u, ∆u ∼ F.

(3.6)

This relationship was first noted by Hooke (1635–1703), who stated ‘ut tensio, sic vis’, which means translated “extension is directly proportional to force.” This discovery is an important foundation for modern elasticity theory. The observation that ∆u ∼ F can be used to propose an expression that links the force to the extension, F = ks ∆u,

(3.7)

where ks is a proportionality factor that can be regarded similar as a spring constant. It describes the stiffness of the mechanical element. The formulation put forward in (3.7) is dependent on the size of the mechanical element considered. For instance, making the beam longer results in a smaller factor ks . To be able to characterize material properties independent of size of the structure, (3.7) can be rewritten in a renormalized way, by dividing by the volume V of the element considered. The volume can be expressed as V = AL, where A is the cross-sectional area and L is the length of the beam. Then, ks ∆u F 1 = . (3.8) AL A L The quantity F σ= (3.9) A is defined as stress, and ∆u ε= (3.10) L as strain. Rewriting (3.8) yields σ = Eε, where

(3.11)

Lks (3.12) A is defined as Young’s modulus. Equation (3.12) is typically referred to as Hooke’s law. E=

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Deformation in the first regime is elastic, that is, the original shape of the beam is assumed after forces are released. However, for large forces, deformation is irreversible and permanent deformation remains even after the load has been released. Permanent deformation is accompanied by dissipative processes, that is, work done to the system by external forces (e.g., F in the example) cannot be recovered. As we shall explain in the forthcoming sections, Newton’s laws can be used to solve more complex mechanical problems, as for instance, a typical beam bending problem as shown schematically in Fig. 3.2.

Fig. 3.2 Example for deformation of a beam due to mechanical loading of a distributed force qz . The structure responds to the mechanical forces by a change in shape. Continuum mechanical theory enables us to derive a relationship between applied forces and displacements, strains, and stresses.

The goal of the continuum mechanical framework is to derive a relationship between applied forces and displacements, strains, and stresses in the structure. In a sense, such solutions represent a simple class of multiscale problems, as they provide a link between local forces and the global structural response. Figure 3.3 illustrates this concept. The goal of solving this type of problem is to connect the global scale (scale on the order of L where boundary conditions are applied that is, the applied load P , N , constraints of displacements at x = 0) with the local scale (that is, the section of the beam). In particular, here the stress state inside the beam σxx and how it varies over the cross-sectional area is of interest.


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Fig. 3.3 The beam problem as multiscale problem. The goal of solving this problem is to connect the global scale (scale on the order of L where boundary conditions are applied, for instance, load P , N , prescribed displacements) with the local scale (section of the beam, e.g., the stress variation σxx , across the section)

3.2.1 Stress Tensor

Fig. 3.4 Cross-sectional view of a body. Subplot (a) free body with representative internal forces. Subplot (b) enlarged view with components of the force vector split up

The concepts introduced in the previous section for a one-dimensional case can easily be generalized for a three-dimensional situation. Figure 3.4 depicts a sectioned body with internal forces (the plot on the right shows a split of the internal force vector into its components ∆Px , ∆Py , and ∆Pz ).

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Fig. 3.5 The most general state of stress acting on a infinitesimal material element. All stresses shown in the figure have positive sense

The idea of normalizing the forces by the area they act on can be generalized from the 1D case to 3D case. Here we explicitly consider the different spatial directions of forces and the plane orientation. The plane cut area and its orientation is characterized by Ai (by the normal vectors ni ), where the index i relates to the normal orientation of described surface). The stress tensor is then defined as ∆Pj . (3.13) σij = lim ∆A→0 ∆Ai The components σii are called normal stresses, and the components σij where i = j are called shear stresses. The stress tensor is a second-order tensor with generally nine unknowns. The most general state of stress acting on a infinitesimal material element is illustrated in Fig. 3.5. This figure provides a visual illustration of the definition introduced in (3.13). 3.2.2 Equilibrium Conditions The expressions for the stress tensor enable us to write differential equilibrium conditions based on Newton’s first law. The integration of the differential equilibrium conditions, by application of boundary conditions to solve for unknown integration constants, will then enable us to predict the behavior of an entire structure.


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Fig. 3.6 Infinitesimal element with stresses and body forces fi acting as volume forces

The force balance on the infinitesimal element is shown in Fig. 3.6, for a two-dimensional example (with an out-of-plane thickness of “1”). This simple schematic is now used to derive the differential equilibrium equations for continuum mechanics problems.

Fig. 3.7 This schematic explains the condition that σij = σji so that there is no moment on the infinitesimal element, since it cannot rotate

First, the moment equilibrium has to be satisfied. The condition that no net torque acts on an infinitesimal element (conservation of the angular momentum) yields the condition that the stress tensor is symmetric (see schematic in Fig. 3.7), σij = σji . (3.14) This condition reduces the nine components of the second-order tensor to six.

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Further, based on Newton’s first law (3.1), a differential material volume dV = dxdydz is in equilibrium if the net force applied to this element is zero, fi = 0. The differential equilibrium equations can be derived by applying Newton’s laws. Carrying out a balance of forces in the x-direction (note that the equation is expressed per unit depth of the infinitesimal cube) yields ∂σxx ∂σyx dx − σxx dy + σyx + dx − σyx dx + fx (dx × dy) = 0, σxx + ∂x ∂y (3.15) resulting in ∂σyx ∂σxx + + fx = 0. (3.16) ∂x ∂y This balance is carried out also for the y-direction and a similar result is obtained, leading to ∂σyy ∂σyx + + fy = 0. (3.17) ∂x ∂y For the three-dimensional case, this approach results in three equations: ∂σxy ∂σxz ∂σxx + + + fx = 0, ∂x ∂y ∂z ∂σyy ∂σyz ∂σyx + + + fy = 0, ∂x ∂y ∂z ∂σzy ∂σzz ∂σxz + + + fz = 0. ∂x ∂y ∂z

(3.18) (3.19) (3.20)

These equations can be written more generally for all directions and in three dimensions as ∂σij + fi = 0. (3.21) ∂xj This equation is sometimes also written as div σ + f = 0.

(3.22)

If inertia terms are included (that is, the time dependence of the solution is considered), the dynamic equilibrium equations are ∂ 2 ui ∂σij + fi = . ∂xj ∂t2

(3.23)

In index notation, the equations are written as σij,j + fi = ρ∂tt ui .

(3.24)

Solving these partial differential equations is generally quite difficult, and it can only be achieved in specific situations. Often, numerical approaches such as the finite element method are used to approximate the solutions.


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Fig. 3.8 Schematic to illustrate the definition of the deformation tensor

3.2.3 Strain Tensor Figure 3.8 illustrates the concept of the deformation tensor. The deformation tensor relates the deformed and undeformed configurations of a material or structure. We consider an undeformed and deformed beam with two points A and B, at a distance of dx. The displacement of point A is given by u. Noting that the term du/dx describes the change of displacement in the x-direction; assuming a linear relationship we can estimate the additional displacement at x + dx (at point B) given by u+

du dx. dx

(3.25)

The initial distance between points A and B is dx; the deformed distance between the two points is du dx + dx. (3.26) dx According to the definition of the strain above (ε = ∆L/L), the strain is defined as du dx − dx dx + du dx ε= = . (3.27) dx dx The displacement of any point can be expressed as u=

du x, dx

(3.28)

e=

du . dx

(3.29)

where

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In general, the displacements ui can be expressed in terms of the deformation tensor eij . The deformation tensor for small deformation is defined as eij =

∂ui . ∂xj

(3.30)

With the deformation tensor, the displacements can be written as ui = eij xj .

(3.31)

The deformation tensor includes both rigid body rotations and shear deformations. The deformation tensor can be rewritten so that the parts describing shear deformation and those describing rotation are separated. The deformation tensor is given by eij =

1 1 (eij + eji ) + (eij − eji ) , 2 2

(3.32)

where 1 (eij + eji ) (3.33) 2 is defined as the strain tensor describing shear deformation, and the second term 1 (eij − eji ) (3.34) 2 describes the rotational component of deformation that does not contribute to a shape change of the material. This is also illustrated in Fig. 3.9. εij =

Fig. 3.9 Schematic to illustrate the difference between rotational and deformation part of the deformation tensor


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3.3 Energy Approach to Elasticity How are stresses and strains related? A thermodynamical view of the physical process of elastic deformation provides insight into this problem. The key physical concept in describing elastic deformation is the fact that it refers to reversible deformation. Thus, here we focus on the first regime of the example shown in Fig. 3.1 (the elastic regime in which deformation is indeed reversible, that is, upon release of the applied force, the beam returns to its undeformed configuration). For the discussion in this section, we recall that the relationship between stretching force and extension is given by F = ks ∆u. The first law of thermodynamics states that ∂U = δW + δQ, ∂t

(3.35)

where δW = xF ˙ e (Fe is the applied force, and δQ is the heat provided to the system. The second law of thermodynamics states that δQ ∂S ≥ , ∂t T

(3.36)

where δQ/T is the change in entropy supplied to the system in the form of heat. This second law basically states that the change in entropy is always greater or equal than the entropy supplied to the system in the form of heat. The difference between the left and right side in (3.36) corresponds to an internal heat source due to dissipation. The dissipation rate is therefore given by ∂D ∂S =T − δQ ≥ 0. (3.37) ∂t ∂t Recalling that δQ = dU/dt − δW (the first law of thermodynamics), we can write ∂ ∂D = δW − (U − T S) . (3.38) ∂t ∂t The quantity F = U − T S is defined as free energy, or Helmholtz energy, describing the maximum internal capacity of the system to do work. During elastic deformation, no dissipation exists by definition (all mechanical work has to be completely recoverable), and therefore ∂D =0 ∂t and

(3.39)

∂F =0 (3.40) ∂t The part δW is known and can be written as δW = xF ˙ e , and therefore δW −

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xF ˙ e− or

∂F ∂x =0 ∂x ∂t

∂F x˙ Fe − ∂x

(3.41)

= 0.

(3.42)

Since this expression has to be valid for any deformation rate x, ˙ the deformation force Fe can be calculated directly from the change of free energy with respect to deformation x, ∂F Fe = . (3.43) ∂x Assuming that only the internal energy U changes during deformation, Fe =

∂U . ∂x

(3.44)

The change of the force Fe with respect to deformation x yields the spring constant ∂2U ks = (3.45) ∂x2 and to satisfy the condition ks > 0 the function U needs to be convex. By normalization of the force with respect to the area it acts on, (3.44) can be rewritten as ∂Ψ σ= , (3.46) ∂ε where Ψ = F/V is the free energy density. Equation (3.46) is easily obtained by dividing (3.44) by AL, noting that dx = Ldε. Further, (3.46) can be generalized to ∂Ψ . ∂εij

(3.47)

∂2Ψ . ∂εij dεkl

(3.48)

σij = The elasticity tensor is then given by cijkl =

This reveals that as soon as an expression for F = U − T S is known for a specific material, a rigorous link between stress and strain is provided within the framework of linear elasticity. This is a core component of the concepts described here, as it provides a link between traditional and new fields of mechanics. The quantity Ψ can be calculated from first principles atomistic calculations, for example. We will utilize this concept later when we make a link between atomistic and continuum mechanical elasticity, or the stress–strain behavior.


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Hooke’s law is an approach to link the measure of deformation with the corresponding stress necessary to induce this deformation. The stress and strain tensor – both second-order field tensors – are related by a fourth-order tensor, the elasticity tensor cijkl via the generalized Hooke’s law: σij = cijkl εkl .

(3.49)

The elasticity tensor cijkl describes the elastic material properties in the most general form. This equation is also referred to as the constitutive law. In tensor notation, (3.49) is written as σ = cε, (3.50) with the stress tensor σ = σij ei ⊗ ej , ε = εkl ek ⊗ el (both second-order tensors) and the fourth-order elasticity tensor, c = cijkl ei ⊗ ej ⊗ ek ⊗ el . The elasticity tensor has 81 entries. However, due to the symmetries of the stress and strain tensor and the independence of the order of differentiation in (3.48), the number of independent entries reduces to 21. These properties are referred to as minor and major symmetries of cijkl . The minor symmetries reflect the fact that cijkl = cijlk = cjikl , (3.51) which is a direct consequence of σij = σji and εlk = εkl . The minor symmetries reduce the number of elastic coefficients to 36. The major symmetry is cijkl = clkij ,

(3.52)

which is a consequence of the independence of the order of differentiation in (3.48), resulting in 21 independent elastic coefficients.

3.4 Isotropic Elasticity In the most general case, 21 independent elastic coefficients cijkl exist, as discussed in the previous section. In isotropic elasticity, the number of independent elastic coefficients is significantly reduced, and there are only two independent coefficients that describe the relation between stress and strain. In isotropic materials, the force–load response, or equivalently, the stress– strain relationship does not depend on direction of applied load. Examples for isotropic materials are polycrystalline metals, in which texture formation leads to anisotropy, or amorphous materials that have no preferred orientation even at atomistic scales. In contrast, in anisotropic elasticity, mechanical properties depend on direction of the applied load. Examples for anisotropic materials include single crystals of materials, since different symmetry directions in crystals lead to different elasticity.

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There are several ways of writing the relationship between stress and strain for an isotropic solid. A common method is to use Lame coefficients λ and µ, so that the elasticity tensor coefficients can be written as: cijkl = µ(δik δjl + δil δjk ) + λδij δkl .

(3.53)

From µ and λ, other important quantities can be calculated, including K= E= ν= µ= λ=

1 3 = 3λ + 2µ B 2µ(1 + ν) E − 2µ 2µ E 2(1 + ν) 3νµ . 1 − 2ν

(3.54) (3.55) (3.56) (3.57) (3.58)

The parameter K is the bulk modulus, ν, the Poisson’s ratio, and µ the shear modulus. The Lame parameters are related to cijkl as µ = c1212 =

1 (c1111 − c1122 ) 2

λ = c1122 λ + 2µ = c1111 .

(3.59) (3.60) (3.61)

Expressions for stress and strain relations can be derived from these equations: σxx = (λ + 2µ) εxx + λεyy + λεzz

(3.62)

σyy = λεxx + (λ + 2µ) εyy + λεzz σzz = λεxx + λεyy + (λ + 2µ) εzz

(3.63) (3.64)

σyz = 2µεyz σzx = 2µεzx

(3.65) (3.66)

σxy = 2µεxy .

(3.67)

The strain energy density Ψ is given by Ψ=

1 (λ + 2µ)(ε11 + ε22 + ε33 )2 + 2µ(ε223 + ε231 + ε212 − ε11 ε22 − ε11 ε33 − ε22 ε33 ). 2 (3.68)

3.5 Nonlinear Elasticity or Hyperelasticity The type of elasticity discussed in the previous sections assumed that the elastic stiffness does not change under deformation. Here we briefly introduce a concept referred to as nonlinear elasticity or hyperelasticity, that


109

is, the phenomenon that the material’s stiffness changes under deformation. Figure 3.10 shows schematic stress–strain plots and the corresponding stiffness–strain plots, for a linear elastic reference case, hyperelastic stiffening and hyperelastic softening. Linear elasticity is based on the assumption that the modulus is independent of strain. The framework of hyperelasticity enables one to describe the behavior of real materials that do not show a linear elastic behavior. Many materials show a stiffening effect (e.g., rubber, polymers, biopolymers) or a softening effect (e.g., metals, ceramics) once the applied strain approaches large magnitudes.

Fig. 3.10 Illustration of the concept of nonlinear elasticity or hyperelasticity. Subplot (a) shows the stress–strain relationship, and subplot (b) depicts the tangent modulus as a function of strain. Linear elasticity is based on the assumption that the modulus is independent of strain. However, most real materials do not show this behavior. Instead, they show a stiffening effect (e.g., rubber, polymers, biopolymers) or a softening effect (e.g., metals, ceramics)

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3.6 Elasticity of a Beam

Fig. 3.11 Geometry of the beam

Here we present one of the simplest systems for which we can develop a closed form, analytical solution using the tools discussed in the last few sections. The purpose of this discussion is to illustrate a simple approach in utilizing the differential equilibrium equations to solve real problems. We consider a beam with length L and cross-sectional dimensions b and h, where b/L 1 and h/L 1, as shown schematically in Fig. 3.11. Due to this particular slender geometry, the beam problem is a quasi-one dimensional problem. The key feature of a beam problem is the fact that it is a geometrically simplified, special case for which the governing partial differential equations take a simpler form. The goal of solving beam problems is to develop solutions that yield expressions for the displacements and stress inside the beam as a function of the applied load. We begin with an explanation of the simplifications that can be done to the governing equations. First, by considering the ratios b/L and h/L one can show that most stress tensor coefficients are zero. This is a direct consequence of the particular geometry of the beam, in which L b, h. The only nonzero components are σxx , σyx , and σzx . Among these, σxx σyx , σzx so that the only remaining stress tensor coefficient is σxx in a first approximation. Thus the entire stress tensor field in a beam is approximated by only one entry, σxx . 3.6.1 Reduction Formulas A key ingredient in exploiting the properties of beam problems is the introduction of reduction formulas, that is, the calculation of equivalent force and moment values per section of the beam. These section quantities are then only a function of the x-coordinate (beam axis), as the variation of the stresses with y and z within a section is lumped into a single value. Thereby the dimensionality of the problem can be reduced from 3D to 1D. By calculating the force vector at a cross-section of the beam, at a given location x,


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Nx =

σxx (y, z)dS,

(3.69)

σyx (y, z)dS,

(3.70)

σzx (y, z)dS.

(3.71)

S

Qy =

S

Qz =

S

The total force vector is FS = Nx ex + Qy ey + Qz ez .

(3.72)

In addition to the force expressions, we can also calculate section moments: (yσzx (y, z) − zσyx (y, z)) dS (3.73) Mx = S

My =

zσxx (y, z)dS

(3.74)

S

Mz = −

yσxx (y, z)dS

(3.75)

S

3.6.2 Equilibrium Equations At each section of the beam, the differential equilibrium conditions in (3.21) have to be satisfied. These equilibrium equations are derived in analogy to the considerations done at the differential element, which lead to the more general form of the governing equations. The equilibrium conditions can be written as (note that the force density qi has units of force per unit length) for the forces, dNx + qx = 0 dx dQy + qy = 0 dx dQz + qz = 0 dx

(3.76) (3.77) (3.78)

and for the moments, dMx =0 dx dMy − Qz = 0 dx dMz + Qy = 0 dx

(3.79) (3.80) (3.81)

112


These equations can be solved by integration while considering proper boundary condition at x = 0 and x = L to solve for the resulting unknown integration constants. For example, moments or forces at the end of the beam may be specified. 3.6.3 Example: Solution of a Simple Beam Problem Here we consider a simply supported beam subject to its own dead weight, q = −ρgAez , as shown in Fig. 3.12.

Fig. 3.12 Solution field for a simply supported beam under dead load ρg, showing the shear force Qy , bending moment My , rotation ωy , and the beam axis displacement uz


113

The equilibrium equations for this particular problem are dNx =0 dx dQy =0 dx dQz − ρAg = 0. dx

(3.82) (3.83) (3.84)

Integration yields Nx = C1 , Qy = C2 , and Qz = C3 + ρAgx. The moment equilibrium conditions yield dMx =0 dx dMy − (C3 + ρAgx) = 0 dx dMz + C2 = 0 dx

(3.85) (3.86) (3.87)

Integration yields Mx = C4 , My = C5 + C3 x + 12 ρAgx2 , and Mz = C6 − C2 x. The application of boundary condition now allows us to find the solution to the problem: The forces F(x = L) = 0 and moments M(x = L) = 0 since there is a free end at x = L. Therefore, C1 = C2 = C4 = C6 = 0 and C3 = −ρgLA. Since M (x = L) = 0, C5 = 12 ρgL2 A. Overall the only nonzero force and moment distributions are given by, Qz (x) = ρgA(x − L) 1 My (x) = ρgA(x − L)2 . 2

(3.88) (3.89)

Figure 3.12 shows the solution field of this problem. 3.6.4 Calculation of Internal Stress Field Even though the distribution of forces and moments are now known, displacements and stresses remain unspecified. To find these quantities, it is necessary to introduce material parameters that provide a link between force and deformation. The strategy to achieve this is to express the section forces as a function of section strains and material properties as well as stresses. Then, by knowing the section force or moment we can easily calculate the corresponding stress or strain state. For the beam geometry, we can express the section displacements and section strains as a function of the variable x: u = u0 (x) + us (y, z)

(3.90)

ε = ε0 (x) + εs (y, z)

(3.91)

114


Fig. 3.13 Demonstration of the concept of the Navier–Bernouilli assumption

According to the Navier–Bernouilli assumption, an initially plane beam section that is perpendicular to the beam’s reference axis remains plain after deformation, and is also perpendicular in the deformed state (see Fig. 3.13). Then, the displacements in the section can be written as us = u0 (x) + ω(x) × xs

(3.92)

where xs = yey + zez , and ω(x) as the infinitesimal rotations: ∂uy,0 , ∂x ∂uz,0 . =− ∂x

ωz,0 =

(3.93)

ωy,0

(3.94)

This leads to the following expressions for the strains εxx = εxx,0 + θy,0 z − θz,0 y.

(3.95)

We note that the parameters θz,0 and θy,0 are defined as the change of the rotations or curvatures: ∂ 2 uy,0 , ∂x2 ∂ 2 uz,0 =− . ∂x2

θz,0 =

(3.96)

θy,0

(3.97)

We reiterate that due to the slender geometry of the beam, the shear stresses are much smaller than the normal stresses, thus σxz = σyx = 0, and therefore we only have one nonzero stress component, σxx . The typical solution of a beam problem results in section quantities Nx (section normal), My , and Mz (both section moments). We recall that these quantities can be obtained by the reduction formulas: Nx = σxx dS, (3.98) S My = zσxx dS, (3.99) S Mz = −yσxx dS. (3.100) S


115

Now we assume an isotropic material behavior (with constant E along the length x), which results in (3.101) σxx = Eεxx , where εzz = εyy = −νεxx . Considering (3.95) and (3.101), we find that σxx = E (εxx,0 + θy,0 z − θz,0 y) .

(3.102)

Then, the normal force and section moments are Nx = E (εxx,0 + θy,0 z − θz,0 y) dS, S zεxx,0 + θy,0 z 2 − θz,0 yz dS, My = E S −yεxx,0 − θy,0 yz − θz,0 y 2 dS. Mz = E

(3.103) (3.104) (3.105)

S

This set of equations can be written in simpler form as a matrix product # # ⎞⎡ ⎤ ⎤ ⎛ # dS ⎡ εxx,0 zdS − S ydS S S Nx # 2 # # ⎟⎢ ⎥ ⎣ My ⎦ = ⎜ z dS − S zydS ⎠ ⎣ θy,0 ⎦ . (3.106) ⎝ S zdS S # # 2 # Mz θ − ydS − zydS − y dS S

S

S

z,0

The terms that appear in the matrix of this equation can be identified as purely geometric properties of the cross-section of the beam. The quantity # # # dS = A is the zero-order area moment, the quantities ydS and S zdS S # S are the first-order area moments, and the quantities Iij = S xi xj dS is referred to as the second-order area moment. The normal force can be expressed as Nx = Eεxx,0 A.

(3.107)

For a rectangular cross section (defined by b × h), the second-order area moments are given by b3 h , (3.108) Iyy = 12 Izz =

bh3 , 12

(3.109)

and Izy = 0.

(3.110)

In fact, it can be shown that for any cross-section with at least one symmetry around one of the section axis Izy = 0.

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In this case, the moments can be expressed as My = EIzz θy,0 ,

(3.111)

Mz = EIyy θz,0 .

(3.112)

These expressions can now be used in (3.102) to find an expression for the stress σxx (x; y, z), in terms of the section quantities Nx (x), My (x), and Mz (x): z Nx (x) y + My (x) − Mz (x). (3.113) σxx (x; y, z) = A Izz Iyy Similar expressions can be developed also for cases when there is no geometric symmetry in the cross–section. Finally, we briefly outline how the stress distribution in the section can be obtained based on these equations. Recall the solution of the problem solved above, 1 ρgA(L − x)2 . 2

My (x) =

(3.114)

The largest section moment appears at x = 0, being My = 12 ρgA. Therefore, σxx (x = 0; y, z) =

z 1 ρgAL. Izz 2

(3.115)

The entire distribution as a function of all variables x, y, and z is given by σxx (x; y, z) =

z 1 ρgA(L − x)2 . Izz 2

(3.116)

Note that due to the particular boundary condition, there is no variation of stresses in the y-direction. Figure 3.3 illustrates schematically the distribution of σxx inside the section. 3.6.5 Differential Beam Equations The beam equilibrium equations can be expressed as d2 Mx = 0, dx2

(3.117)

2

d My + qz = 0, dx2 d2 Mz − qy = 0. dx2

(3.118) (3.119)

Now we take advantage of the result from (3.112), considering (3.97). Therefore,


∂ 4 uz,0 + qz = 0, ∂x4 ∂ 4 uy,0 − qy = 0. EIyy ∂x4

−EIzz

117

(3.120) (3.121)

These are the fourth-order differential equations that govern beam elasticity. These are the equations most frequently used to solve beam problems. Integrating these equations four times, while considering the boundary conditions, leads to solutions for the displacement field in the beam. We note that for a two-dimensional beam geometry (that is displacements only in the z direction) the shear forces are given by Qz (x) = −EIzz

∂ 3 uz,0 , ∂x3

(3.122)

My (x) = −EIzz

∂ 2 uz,0 , ∂x2

(3.123)

the moment is given by

the rotations are given by ωy (x) = −

∂uz,0 , ∂x

(3.124)

and the displacements are uz,0 . Once the solutions for the deformation of a beam are known, the stored elastic strain energy can be calculated by integrating over the strain and curvature fields: 1 2 2 EAε2xx,0 + EIzz θy,0 dx. (3.125) + EIyy θz,0 Ψ= 2 x Using the differential beam equations, the same solution as constructed earlier can be obtained, with the following boundary conditions: The displacement uz (x = 0) = 0 and Qz (x = L) = 0 as well as My (x = L) = 0, noting that qz = −ρAg. Integration of these equations yields for the geometry shown in Fig. 3.12a Qz (x) = ρgA(x − L)

(3.126)

and My (x) =

1 ρgA(x − L)2 . 2

(3.127)

This solution is identical to the one presented in (3.89) and (3.89) and is shown in Fig. 3.12b and c. Further integration of the equations leads to expressions for the rotations and the displacements:

118


ωy (x) =

ρgA 3 x 6EIzz

(3.128)

ρgA 4 x . 24EIzz

(3.129)

and uz (x) = −

These solutions are sketched in Fig. 3.12d and e. Note that whereas forces and moments have a maximum at x = 0, the rotations and displacements are largest at x = L. The largest displacement at the tip of the beam is given by uz (L) = −

ρgA 4 L . 24EIzz

(3.130)

This maximum tip displacement is proportional to the load ρgA, uz (L) ∼ ρgA

(3.131)

and it is inversely proportional to the parameter EIzz (EIzz is also referred to as bending stiffness), uz (L) ∼

1 . EIzz

(3.132)

Note that typically the indexes “0” are dropped, as the solution always refers to the result of the beam reference axis. The rotation and displacement functions are shown in Fig. 3.12c and d. Figure 3.14 depicts the solution of another case, for a beam subject to a point load P applied at the tip of the beam. For this case, qz = 0, the shear force Qz (x) = −P,

(3.133)

My (x) = P (L − x),

(3.134)

the moment

the rotation ωy (x) = −

P EIzz

x2 − xL , 2

(3.135)

and the displacement P uz (x) = EIzz

x2 L x3 − 6 2

.

(3.136)


119

Fig. 3.14 Solution field for a simply supported beam under a point load P applied at the end of the beam, showing the shear force Qy , bending moment My , rotation ωy , and the beam axis displacement uz

3.7 The Need for Atomistic Elasticity: What’s Next Elastic deformation is an important aspect of engineering design. Structures such as buildings, microelectronic components, displays, airplanes, and many others are typically designed for elastic deformation, that is, to specify the amount of deformation under typical loads load. Elastic properties are also important for plastic flow, since the critical conditions for the onset of permanent deformation are related to the material behavior at small bond stretches. Further, as we shall see in Chap. 6, the driving force for brittle fracture is also related to the elastic energy stored in a solid. For both geometries of beam problems reviewed here (Figs. 3.12 and 3.14), the solution of the displacement of the beam reference axis depends on two

120


classes of material parameters. First, geometric parameters that are captured in Izz and A. Second, a material parameter E, Young’s modulus. Continuum mechanics is very powerful in providing us with a relation between displacements, moments, stress, and other quantities, and the applied load, as a function of material parameters. However, continuum mechanics does not provide us with a description of how the material parameters are determined – E remains an unknown parameter. One source of determining this quantity is to measure it by experimental studies. Another possibility is to calculate the value of E from the atomistic structure of the material. This will be done in the forthcoming chapter, when we will explore methods that enable us to use atomistic simulation to predict the values of Young’s modulus.

4 Atomistic Elasticity: Linking Atoms and Continuum

This chapter is dedicated to provide a link between the atomistic models and continuum theoretical concepts of stress and strain. We review the basic thermodynamical concepts that enable a rigorous link between atomistic systems and corresponding continuum theoretical concepts. The discussion includes a review of entropic and energetic contributions to elasticity, free energy models, entropic effects in rubber-like materials as well as the Cauchy–Born rule. We discuss elastic properties of a variety of crystal structures for different interatomic potentials.

4.1 Thermodynamics as Bridge Between Atomistic and Continuum Viewpoints The link between atomistic models and the concept of elasticity used in continuum mechanical theories can be most directly established by considering concepts of thermodynamics: By calculating how the free energy of a system changes due to deformation, one can calculate the stress state. How is this possible? In the spirit of handshaking, thermodynamics provides this link since both a statistical mechanics approach and a continuum theory can be related to thermodynamical concepts. In this sense, thermodynamics is the glue between atomistic methods and continuum theories, linking microscopic states with a macroscale system. In the following sections, we will describe how this concept can be used to link atomic interactions with concepts of stress and strain. Such handshaking concepts also lay the foundation for some multiscale simulation techniques, as they allow to transcend throughout distinct scales. However, it is emphasized that during the transition from microscale to macroscale, information is lost: Whereas the atomistic scale provides a deterministic description of the atomistic state, an averaged, continuum description does not. Some of the most fundamental concepts of thermodynamics tell us that many microscopic states refer to the same macroscopic state. This represents an important issue in the case when a transition is to be made

122


from microscale to macroscale and vice versa. Whereas statistical concepts can be used in the case of equilibrium phenomena (e.g., by assigning random atomic velocities due to a specific temperature, following an appropriate Boltzmann distribution); this is a tremendously challenging task for nonequilibrium systems. The need to advance methods that combine atomistic and continuum analysis is becoming increasingly compelling with rapid advance in computational resources. Many supercomputer facilities now provide peak performances of several TFLOPs per second, and system sizes with billions of particles can readily be simulated. In order to compare atomistic simulations with continuum analysis level, it is necessary to use methods that allow transition between the two levels of descriptions [157–159]. Of particular interest is the relationship between interatomic potentials and associated elastic properties. Here we limit the scope of the discussion primarily to the calculation of properties at thermodynamical equilibrium.

4.2 The Atomic and Molecular Origin of Elasticity: Entropic vs. Energetic Sources Elasticity stems from the change of the free energy due to the collective interactions of atom and their bonds under deformation, and thus it is intimately linked to the chemistry that defines the atomic interaction energies. The elastic properties of materials can be expressed as the partial derivative of the free energy density with respect to the strain tensor that characterizes the resistance to deformation, as discussed in Sect. 3.3. We briefly review the main steps delineated above, based on the free energy F that is composed out of energetic U and entropic contributions S (temperature is denoted by T ). The free energy is F = U − T S,

(4.1)

we can define the free energy density (V denotes the volume), Ψ=

F . V

(4.2)

∂Ψ ∂ε

(4.3)

The (scalar) stress is then is given by σ=

and the elastic modulus can be expressed as E=

∂ 2Ψ , ∂ε2

since the stress and strain are related by Hooke’s law, σ = Eε.

(4.4)


123

This finding is quite significant: If one can calculate the free energy density of a atomistic system for various deformation states, then one can estimate the stress as well as Young’s modulus! Provided that an accurate model for the atomic interactions is used, this provides a theoretical strategy to predict the elastic properties of materials, addressing the point discussed in Sect. 3.7. Equation (4.3) can be written as a tensor equation, considering all 21 independent elastic coefficients cijkl and all independent entries in the stress and strain tensor. In more general terms, the stress tensor σij and the elasticity tensor cijkl can be written as: σij = and cijkl =

∂2Ψ ∂εij

∂2Ψ . ∂εij ∂εkl

(4.5)

(4.6)

These equations inform us that the change of the energy of a system controls the elastic properties, and thus the elastic properties are strongly influenced by the actual contributions to the free energy F . For instance, the free energy density of crystalline materials (e.g., metals, semiconductors) at moderate temperatures is primarily controlled by energetic changes of the internal energy. In natural and biological materials, the free energy density is dominated by entropic contributions. Therefore in many crystalline materials, the entropic term can be neglected, so that in (4.3) and (4.5) can be directly substituted by U/V , the internal energy, and therefore Ψ≈

U . V

(4.7)

However, in biopolymers, entropic contributions can dominate the elasticity, in particular for small deformation, and therefore Ψ can be substituted by −T S/V , so that TS Ψ ≈− . (4.8) V The dominance of entropic behavior is a well-known and well-studied phenomenon in many polymers. The contributions to the entropic term to elasticity can be described in several ways, including classical descriptions such as the WLC or the freely jointed Gaussian chain model. These models will be discussed shortly.

4.3 The Virial Stress and Strain The virial stress and strain provide important concepts to couple deformation behavior of atomistic systems with continuum theories and concepts.

124


The virial stress is defined in Sect. 2.8.6. As most atomistic quantities, the stress tensor coefficients must be averaged in space and time in order to be compared with continuum concepts. Thus, the virial stress needs to be averaged over space and time to converge to the Cauchy stress tensor. The strain field is a measure of geometric deformation of the atomic lattice [112]. The local atomic strain is calculated by comparing the local deviation of the lattice from a reference configuration. Usually, the reference configuration is taken to be the undeformed lattice. In the atomistic simulations, the information about the position of every atom is readily available, either in the current or in the reference configuration and thus calculation of the virial strain is relatively straightforward. We limit the consideration to the two-dimensional case. We define the following tensor for atom l N kl 1 ∆xkl i ∆xj l , (4.9) qij = N r02 k=1

were = and ∆xj = xlj − xkj . The quantity N refers to the number of nearest neighbors considered. The left Cauchy–Green strain tensor is given by N kl N l 1 ∆xkl i ∆xj l bij = qij = , (4.10) λ λ r02 ∆xkl i

xli − xki

k=1

where λ is a prefactor depending on the lattice considered. For a twodimensional triangular lattice with nearest neighbor interaction, λ = 3; λ = 2 for a square lattice with nearest neighbor interaction; and λ = 4/3 for a face-centered cubic lattice. This definition provides an expression for a measure of deformation defined using continuum mechanics and in terms of atomic positions. The Eulerian strain tensor of atom l is obtained from (4.10), elij = 12 δij − blij . One can √ calculate the engineering strain ε = b − 1. Unlike the virial stress, the atomic strain is valid instantaneously in space and time, since it is a purely geometric definition. However, the expression is only strictly applicable away from surfaces and interfaces.

4.4 Elasticity Due to Energetic Contributions A possible method of linking atomistic and continuum concepts is to use the Cauchy–Born rule [160, 161], which provides a relation between the energy created from a macroscopic strain field and the atomistic potential energy found in a stretched crystal lattice [157–159, 162]. 4.4.1 Cauchy–Born Rule The central assumption of the Cauchy–Born rule is that the energy of an atomic system can be expressed as a function of a macroscopically applied


125

continuum strain field. Thereby it is assumed that the continuum fields can immediately be mapped to the atomic scale. This is only possible if the strain field is slowly varying at the atomistic scale. In other words, at the microscopic atomic scale, it is assumed that no gradients in the strain field occur. In this approach, the atomic displacements are calculated directly from the continuum strain fields by imposing the appropriate displacements directly at the atomistic lattice. This provides the positions of all atoms in the atomic system as a function of the deformation field, assuming that a continuous interpolation function can be used. The knowledge of the atomic positions then enables one to calculate the strain energy density (SED) Ψ of an atomic system as a function of the continuum strain field. The first derivatives of the SED with respect to the strain yield the stress, the second derivatives of the SED with respect to the strain give the modulus, as given in (4.5) and (4.6). In general form, the SED can be expressed as follows: 1 U (r)dΩ, (4.11) Ψ (εij ) = Ω Ω where Ω is the volume of the atomic system and U (r) is the potential energy as a function of all atomic coordinates, denoted by r. In order to obtain the desired relationship of the SED as a function of the strain tensor, the atomic coordinates must be expressed as a function of the strain tensor by using a mapping function r = DΩ (εij ). (4.12) The mapping function DΩ maps the strain tensor field to atomic positions, thereby providing a direct relationship of r(ε). By integrating over the volume of the atomic system considered, the total energy is determined directly as a function of the strain tensor field: 1 Ψ (εij ) = U (DΩ (εij ))dΩ. (4.13) Ω Ω This approach is particularly simple for numerical evaluation in periodic systems, in which the atomistic system corresponds to the unit cell of a crystal lattice. In this case, the expression given in (4.13) is typically expressed as energy contributions of individual bonds in all possible directions as a function of the strain tensor. Further, in crystal lattices with pair wise interactions, there is typically a finite number of bonds, so that the integral is replaced by a summation over all bonds. The summation typically involves the pair potential function φ, evaluated at the particular bond length r(εij ), which is then directly summed up to yield the total potential energy. The sum over all these energy contributions resulting from all α bonds in the system yields the SED:

126


Ψ (εij ) =

1 φ(rα (εij )). Ω α

(4.14)

In summary, the typical steps in applying the Cauchy–Born rule involve: • Define lattice and corresponding unit cell, along with cell volume. • Define interatomic interactions (that is, choose a potential function). • Express the energy of the unit cell (denoted by Ψ ) as a function of all atomic coordinates, as a function of the applied strain tensor. In the simple case of pair-wise interactions, the calculation typically reduces to considering the distances between all pairs of atoms, which can then be used to determine the energy contribution of all pairs of atoms. The final result of this step is an expression Ψ (εij ). • Use the expression of Ψ (εij ) to calculate the stress tensor and elasticity coefficients, according to (4.5) and (4.6). 4.4.2 Elasticity of a One-Dimensional String of Atoms As a first example, we consider a one-dimensional string of atoms, as shown in Fig. 4.1.

Fig. 4.1 Example to illustrate Cauchy–Born rule in a one-dimensional geometry

In its undeformed configuration, the distance between pairs of atoms is r0 . This one-dimensional crystal is defined by a unit cell that contains one bond and one atom. The volume of the unit cell is Ω = r0 × 1 × 1 (assuming unit lengths in the out-of-plane directions). Assuming existence of a function describing the relation of bond energy and bond length, the SED is given by Ψ (ε) =

1 φ(r), r0 Ω

(4.15)

where D = 1 × 1 denotes the out-of-plane cross-sectional area. The function φ(r) could, for example be a Lennard-Jones 12 : 6 potential, defined in (2.30). The bond length r in the deformed state, as a function of the strain ε, is given by r = (1 + ε) r0 . (4.16) Therefore, 1 1 φ(r) = φ ((1 + ε) r0 ) . (4.17) r0 Ω r0 Ω Since we are interested in obtaining elastic coefficients within the concept of linear elasticity, the second derivatives of Ψ (ε) with respect to ε should be Ψ (ε) =


127

constant. Thus we consider a second-order Taylor expansion of the interatomic potential around the equilibrium bond length r0 : φ(r) = a0 + a1 (r − r0 ) + a2 (r − r0 )2 + · · · .

(4.18)

Since the force vs. bond stretch is zero at the equilibrium bond length, a1 = φ (r0 ) = 0. The parameter a2 = φ (r0 ) corresponds to the force–bond distance slope around r0 and is now denoted as k. The parameter a0 is a constant, which can be set to zero since we are only interested in the derivatives of Ψ (ε). Thus, we can approximate k 2 (r − r0 ) . 2

(4.19)

r0 1 k 2 ((1 + ε − 1)r0 ) = kε. r0 ΩD 2 2D

(4.20)

φ(r) ≈ We can then rewrite (4.17) as Ψ (ε) =

The first derivative of Ψ (ε) yields the stress for a given strain ε, r0 ∂Ψ (ε) = σ(ε) = kε. ∂ε D

(4.21)

The second derivative yields the modulus, ∂ 2 Ψ 2 (ε) r0 = E = k. 2 ∂ε D

(4.22)

Note that since we only considered a second-order expansion of the interatomic potential, the modulus is independent of strain, as assumed in linear elasticity. Equations (4.21) and (4.22) provide relationships between stress and strain and define Young’s modulus for this system. We can use this expression to calculate other material properties, such as the wave speed. For the one-dimensional system, the (only) wave speed is given by E , (4.23) c= ρ where ρ is the atomic density, given by ρ=

m , r0

(4.24)

with m denoting the atomic mass. These relationships have been established by only considering (1) the atomic microstructure, (2) the interatomic potential, and (3) by assuming that the Cauchy–Born rule can be applied (that is, that one can map the continuum displacement field to the atomic lattice). No other phenomenological assumptions have been made.

128


The Cauchy–Born rule is not applicable to all atomic structures. In the case of molecular crystals, or more complex lattice structures (e.g., in graphene, carbon nanotubes), condition (3) may not be satisfied. This is because in such systems, atomic displacements within the unit cell do not necessarily correspond to the continuum displacement field. Other systems where this is relevant are amorphous materials where the Cauchy–Born rule can only be applied in a statistical sense.

Fig. 4.2 Subplot (a) rectangular cell in a uniformly deformed triangular lattice; subplot (b) the geometrical parameters used to calculate the continuum properties of the lattice

4.4.3 Elasticity and Surface Energy of a Two-Dimensional Triangular Lattice Here we review a case study that illustrates the methods that can be used to assess the elastic properties of a simple two-dimensional triangular lattice (see [27] for further details). We note that here we use large-deformation elasticity theory, a concept that was briefly rigorously introduced in Sect. 3.5, discussing the stress–strain relations in softening and stiffening material behavior (see Fig. 3.10). Large-deformation elasticity theory allows one to describe how the elastic response changes under increasing deformation, as it is for instance the case during stiffening of rubber, or in softening of metals. We refer the reader to the literature for additional details in regards to this approach, in particular to find a more rigorous introduction into this topic (see for instance [163]). Classical hyperelastic continuum theory is based on the existence of a strain energy function [163]. Using the Cauchy–Born rule, applied to the triangular lattice considered in this section (see Fig. 4.2 for the geometry of the unit cell), the strain energy density per unit undeformed area is given by [27, 160, 161]


129

2 Ψ = √ (φ(l1 ) + φ(l2 ) + φ(l3 )) , (4.25) 3 where the values of li are determined from geometric relations of the triangular lattice as shown in Fig. 4.2. The function φ(r) refers to the interatomic potential. The unknowns Eij are the Green–Lagrangian strain components [163, 164], and these can be determined to be Exx = Λ21 − 1 /2, Eyy = Λ22 − 1 /2, Exy = Eyx = Λ1 Λ2 cos (Θ/2) . (4.26) Here, Θ is the shear angle, while Λi describe the elongation of the sides of a lattice unit cell as indicated in Fig. 4.2. From geometric relations, it is found that (4.27) l1 = 1 + 2Exx , ' √ (4.28) l2 = 1 + 1/2Exx + 3/2Eyy − 3/2(Exy + Eyx ), and l3 =

' √ 1 + 1/2Exx + 3/2Eyy + 3/2(Exy + Eyx ).

(4.29)

The symmetric second Piola–Kirchhoff stress tensor is given by Sij =

∂Ψ . ∂Eij

(4.30)

The “slope” of the S−E relationship is often called the material tangent modulus ∂2Ψ Cijkl = . (4.31) ∂Eij ∂Ekl For infinitesimal strains, the Green–Lagrangian strain reduces to the stress tensor of linear elasticity Eij → εij . The same argument can be used for the stresses, and the second symmetric Piola–Kirchoff stress tensor reduces to the linear elasticity stress tensor Sij → σij , as well as Cijkl → cijkl . This scheme is universally applicable, as long as the interatomic potential and thus the strain energy function Ψ is known (it can for instance be applied to pair potentials or the embedded atom method (EAM) for metals). Poisson’s ratio ν is defined as the ratio of transverse strain to longitudinal strain in the direction of stretching force. Poisson’s ratio can be found by choosing a Green–Lagrangian strain Exx and finding a value for Eyy such that Syy assumes zero (and vice versa). We can define ν=−

Eyy Exx

(4.32)

as Poisson’s ratio valid also for large strain. To obtain a linear elasticity formulation with first-order stress–strain law, the strain energy given in (4.25) is expanded up to second-order terms. After some lengthy calculations it can be shown that

130


√ Ψ=

3 φ (r0 ) 3ε2xx + 2εxx εyy + ε2yy + (εxy + εyx )2 , 8

(4.33)

where r0 is the nearest neighbor distance. Using σij =

∂Ψ , ∂εij

(4.34)

one can derive expressions for stress–strain relations, like for instance √ ∂Ψ 3 φ (r0 ) (εxx + 3εyy ) . σyy = = (4.35) ∂εyy 4 As in the case of the one-dimensional example, this equation provides a rigorous link between the elastic properties and the interatomic potential. In the following sections, we review the application of this approach to different interatomic potentials. Lennard-Jones Potential We begin with the elastic properties of a solid in which atoms interact according to a LJ potential as defined in (2.30) (we choose σ = 0 = 1). LJ type potentials have frequently been used in simulating fracture using molecular dynamics [146]. Solids defined by this potential behave as a very brittle material in a two-dimensional triangular lattice. Figure 4.3 shows numerical estimates of the elastic properties of a LJ solid. The systems are loaded uniaxially in the two symmetry directions of the triangular lattice. The plot of the LJ system shows that the y-direction requires a higher breaking strain than in the x-direction (about 18% vs. 12%). The tangent Young’s modulus drops significantly from around 66 for small strain until it reaches zero when the solid fails [163]. Poisson ratio remains around 1/3, but increases slightly when loaded in the x-direction and decreases slightly when loaded in the orthogonal direction. Nonlinear Tethered LJ Potential The objective is to obtain a solid with the property that its tangent moduli stiffen with strain, in contrast to the LJ potential described above [163]. In addition, the small-strain elastic properties should be the same as in the LJ potential. The nonlinear tethered LJ potential is obtained by modifying the well-known LJ 12:6 potential: The potential is mirrored at r = r0 = 21/6 ≈ 1.12246, leading to a strong stiffening effect instead of the normal softening associated with atomic separation. ⎧ 6 12 ⎪ 1 1 ⎪ if rij < r0 , − ⎨40 rij rij φ(rij ) = (4.36) 12 6 ⎪ 1 1 ⎪ ⎩40 if r − ≥ r , ij 0 (2r0 −rij ) (2r0 −rij )


131

Fig. 4.3 Elastic properties of the Lennard-Jones solid (continuous line) and elastic properties associated with the harmonic potential (dashed line). The dash-dotted lines in the upper plots show Poisson’s ratio. The lower plots show the tangent modulus for this case. This plot is an actual material law representing the schematic shown in Fig. 3.10

where the parameter 0 can be chosen to change the small-strain elastic properties (here we assume σ = 0 = 1) [156, 165]. Figure 4.4 plots the elastic properties associated with this potential for a two-dimensional triangular lattice. The upper two subplots show the stress vs. strain behavior under uniaxial stress loading. The left refers to uniaxial stress loading in the x-direction, and the right plot shows the stresses for uniaxial y loading. The Poisson ratio is calculated to be around ν ≈ 0.33. The nonlinear nature of this potential can clearly be identified in these plots. The tangent moduli stiffen strongly with strain, and agree with the small-strain elastic properties of the LJ potential. Therefore, the wave velocities assuming small perturbation from the equilibrium position are the same in both the LJ and the tethered LJ potential.

132


Fig. 4.4 Elastic properties associated with the tethered LJ potential, and in comparison, elastic properties associated with the harmonic potential (dashed line). Unlike in the softening case, where Young’s modulus softens with strain (Fig. 4.3), here Young’s modulus stiffens with strain

Harmonic Potential We introduce a harmonic potential with the objective to mimic linear elastic material behavior, as assumed in most theories of fracture. The linear spring potential given by (2.34) corresponds to the “ball-spring” model of solids and yields a plane-stress elastic sheet for a triangular lattice. Using expressions similar to (4.35), Young’s modulus E and shear modulus µ can be shown to be √ 2 3 (4.37) E = √ k, µ = k. 4 3


133

√ When we assume k = 72 3 2 ≈ 57.14 so as to match the small-strain elastic properties of the LJ potential, E ≈ 66 and µ ≈ 24.8. Equation (4.35) (expression for infinitesimal strains) can be used to show that Poisson’s ratio ν = 1/3. Using the above given values for elastic properties, the wave speeds can be obtained straightforwardly. The longitudinal wave speed can be calculated from the elastic properties to be 3µ (4.38) cl = ρ √ with ρ = 2/21/3/ 3 ≈ 0.9165 (assuming mass m = 1). The shear wave speed is given by the square root of the ratio of the shear modulus µ to the density ρ thus µ . (4.39) cs = ρ Finally, the speed of elastic surface waves, the Rayleigh speed, is given by cR ≈ βcs .

(4.40)

The value of β can be found by solving the following equation Γ = cs /cl : (4.41) β 6 − 8β 4 + 8 3 − 2Γ 2 β 2 − 16 1 − Γ 2 = 0. This solution is found to be β ≈ 0.923. Spring constant, k √ 3 36 √ 2 ≈ 28.57 72 3 2 ≈ 57.14

Young’s modulus, E

Shear modulus, µ

Poisson’s ratio, ν

cl

cs

cR

33 66

12.4 24.8

0.33 0.33

6.36 9

3.67 5.2

3.39 4.8

Table 4.1 Elastic properties and wave speeds associated with the harmonic potential (see (2.34)) in a two-dimensional solid for different choices of the spring constant k

The wave speeds are given by cl = 9, cs = 5.2, and cR ≈ 4.8.

(4.42)

Results for elastic properties and wave speeds are summarized in Table 4.1 for two different choices of the spring constant. To check if the predictions by (6.67) hold even for large strains, we investigate the elastic properties numerically. The numerically estimated elastic properties for uniaxial tension are shown in Fig. 4.5 for the two different crystal orientations in a triangular lattice and k ≈ 28.57. We find reasonable agreement, which could be verified by comparing the values reported in Table 4.1 with the results shown in Fig. 4.5.

134


Fig. 4.5 Elastic properties of the triangular lattice with harmonic interactions, stress vs. strain (left) and tangent moduli Ex and Ey (right). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero

Young’s moduli agree well with the continuum mechanics prediction for small strains. However, we observe a slight stiffening effect for large strains, that is, E is increasing with strain. As predicted, the lattice is found to be isotropic for small deformations, but the results show there exists an anisotropy effect for large deformations. The values of Poisson’s ratio match the linear approximation for small strains, but deviate slightly for large strains. This suggests that even if harmonic potentials are introduced between atoms, the triangular lattice structure yields a slightly nonlinear stress–strain law. We note that the values for Young’s modulus associated with the LJ potential at small strains are in consistency with the results using the harmonic potential with k ≈ 57.14 (see Fig. 4.3). The small-strain elastic properties also agree in the case of the tethered LJ potential (see Fig. 4.4). The comparison with the harmonic potential nicely illustrates the softening and stiffening effect of the LJ and tethered LJ potential. Since the small-strain elastic properties agree with the harmonic potential in both cases, the small-strain wave speed is also identical and thus given by (4.38)–(4.40). Note that there is no unique definition of the wave speed for large strains in the nonlinear potentials. Harmonic Bond Snapping Potential In contrast to the elastic properties of a solid that never breaks as reported in Sect. 4.4.3, here we discuss the elastic properties of a triangular lattice with harmonic interactions where the bonds break upon a critical separation r > rbreak . The interatomic potential is defined as if rij < rbreak , a0 + 12 k(rij − r0 )2 φij (rij ) = (4.43) a0 + 12 k(rbreak − r0 )2 if rij ≥ rbreak .


135

Fig. 4.6 Illustration of the shape of the harmonic potential, comparing the one defined in (2.34) (panel (a)) and the one defined in (4.43) with the bond snapping parameter rbreak (panel (b))

Figure 4.6 shows a graph that displays the shape of the harmonic potential, comparing the potential defined in (2.34) and the one defined in (4.43) with the bond snapping parameter rbreak .

Fig. 4.7 The figure shows the stretching of the triangular lattice in two different directions

The elastic properties for rij < rbreak are identical to those discussed in Sect. 4.4.3 and shown in Fig. 4.5, but for large strains close to the failure of the solid there are strong differences. We focus on the differences in elastic properties due to stretching in the x-direction vs. the y-direction. Figure 4.7

136


shows the crystal orientation for stretching of the triangular lattice in the two different directions. We define two different bond types r1 and r2 : The bonds denoted by r1 have a component only in the x-direction, whereas bonds r2 have a component in the x as well as in the y-direction. Loading direction Poisson relaxation x Yes Yes y x No No y

εbreak 0.08 0.1 0.26 0.09

Table 4.2 Failure strain of the two-dimensional solid associated with the harmonic potential with snapping bonds under different modes of uniaxial loading for rbreak = 1.17

We start with a discussion of stretching in the x-direction (Fig. 4.7a), and consider stretching with and without Poisson relaxation. For uniaxial tension without Poisson contraction, the length of both bonds r1 and r2 increases. In contrast, for uniaxial tension with Poisson contraction, the length of bonds r1 increases, whereas the condition that σyy = 0 requires that |r2 | = r0 . Therefore, with the assumption of Poisson relaxation, for arbitrarily large strains in the x-direction, only two bonds r1 break while the other four bonds r2 never break. However, these bonds do not contribute to the stress. In contrast, if no Poisson relaxation is assumed, these bonds do indeed contribute to the stress and fail at much higher strain than the first two bonds, this increasing the critical strain for bond breaking. Such behavior is indeed observed in the numerical calculation of the elastic properties. Figure 4.8 (left) shows uniaxial tension with Poisson relaxation. Under stretch in the x-direction, the solid fails at about 8% strain. As could be verified in Fig. 4.9 (left), the solid fails at about 26% strain when no Poisson relaxation is assumed. A reduced modulus E ≈ 10 is found between the failure of the first two bonds and the failure of the remaining four bonds. Note that in the figures, the number of bonds is also indicated by the dotted line. The reason for the huge difference in the two cases is, as outlined above by theoretical considerations, the contribution of the four bonds r2 which only break at very high strains. Note that for any stress state not equal to uniaxial tension, there will be a contribution to the stress from the two remaining bonds. We continue with a discussion of stretching in the y-direction (Fig. 4.7b), and also consider stretching with and without Poisson relaxation. For uniaxial tension without Poisson contraction, the length of both bonds r2 increases while the bond length of bonds r2 remains r0 and does not contribute to the stress. Under unixial tension with Poisson relaxation, the length of all bonds is adjusted to satisfy the condition that σxx = 0. In both cases, upon a critical strain the number of bonds drops to two (since the four bonds r2 break) and the remaining bonds r1 do not contribute to the stress in both cases with and without Poisson relaxation. This is a significant difference to the


137

Fig. 4.8 The figure plots the elastic properties under uniaxial loading with Poisson relaxation for the harmonic potential. In the plot, stress vs. strain, Poisson’s ratio as well as the number of nearest neighbors are shown. The lower two subplots show Young’s modulus

behavior in the other loading direction. As a consequence, the critical strain for failure is comparable under both loading conditions in the y-direction. Such behavior is indeed observed in the numerical calculation of the elastic properties. Figure 4.8 (right) shows uniaxial tension with Poisson relaxation. Under stretch in the y-direction, the solid fails at about 10% strain. As could be verified in Fig. 4.9 (right), the solid fails also at about 10% strain when no Poisson relaxation is assumed. In summary, there is a strong dependence of the failure strain on the loading condition. Table 4.2 summarizes the failure strains for different modes of loading in the x- and y-direction. Under large stretching, harmonic lattices behave differently than solids defined by the LJ potential since bonds contribute little to the stress as they weaken strongly with strain. Therefore, the direction with lowest breaking strain is associated with loading in the x-direction, which is the direction with highest fracture surface energy. The consequence of this is that crack propagation is stable along the direction of high fracture energy [146, 166]. Solids associated with harmonic bond snapping potentials show a different behavior: In the harmonic bond snapping systems, the failure strain is larger in

138


Fig. 4.9 The figure plots the elastic properties under uniaxial loading without Poisson relaxation for the harmonic potential. In the plot, stress vs. strain, as well as the number of nearest neighbors are shown. The lower two subplots show Young’s modulus

the x-direction and smaller in the y-direction (see Fig. 4.9). Cracks are therefore expected to propagate stable along the direction of lower fracture surface energy (crack extension along x-direction). We will verify this prediction with molecular dynamics results in a chapter 6. Biharmonic Potential Thus far, we have studied a LJ potential that yields elastic properties that soften strongly with strain, a tethered LJ potential that yields a solid strongly stiffening with strain and a harmonic potential. To be able to smoothly interpolate between harmonic potentials and strongly nonlinear potentials, we adopt a biharmonic, interatomic potential composed of two spring constants k0 and k1 similar to that discussed in Sect. 6.4.5 for the one-dimensional case (all quantities given are in dimensionless units). We consider two “model materials,” one with elastic stiffening and the other with elastic softening behavior. In the elastic stiffening system, the spring constant k0 is associated with small perturbations from the equilibrium distance r0 , and the second spring constant k1 is associated with large


139

bond stretching for atomic separation r > ron . The role of √ k0 and k1 is reversed in the elastic softening system (k0 = 2k1 and k1 = 36/ 3 2). Purely harmonic systems are obtained if ron is chosen to be larger than rbreak . Poisson’s ratio ν is found to be approximately independent of strain and around ν ≈ 0.33 for all potentials. In the stiffening system, the small deformation (up to about 0.5% of strain) Young’s modulus is E ≈ 33 with shear modulus µ ≈ 12.4, and the large deformations tangent Young’s modulus is E ≈ 66 with shear modulus µ ≈ 28.8. The values are reversed for the softening system where the small deformation Young’s modulus is E ≈ 66, and the large deformation tangent Young’s modulus is E ≈ 33. The biharmonic potential is defined as a0 + 12 k0 (rij − r0 )2 if rij < ron , (4.44) φij (rij ) = a1 + 12 k1 (rij − r1 )2 if rij ≥ ron , where ron is the critical atomic separation for onset of the hyperelastic effect, and 1 1 a1 = a0 + k0 (ron − r0 )2 − k1 (ron − r1 )2 2 2

(4.45)

as well as r1 =

1 (ron + r0 ) 2

(4.46)

are found by continuity conditions of the potential at r = ron (note that the expressions (4.45) and (4.46), derived from energy continuity, are only valid for the ratio 2 of the large to small spring constant; similar expressions can be developed for other ratios). The values of k0 and k1 refer to the small-strain and large-strain spring constants. Figure 4.10 shows a graphical explanation of the parameters used in the biharmonic potential defined in (4.44).

Fig. 4.10 Illustration of the parameters used in the biharmonic potential defined in (4.44). The plot defines r, k0 , k1 , ron , rbreak , as well as the “atomic” strain

140


The elastic properties associated with the biharmonic potential are shown in Fig. 4.11. The wave speeds for small and large strains are given by the values of the corresponding harmonic potentials. Therefore, the wave speeds associated with large strains are given by cl,1 = κcl,0 , cs,1 = κcs,0 and cR,1 = κcR,0 where κ = k1 /k0 . Similarly as described in the previous section, a critical bond breaking distance rbreak can be introduced allowing for snapping bonds.

Fig. 4.11 Elastic properties of the triangular lattice with biharmonic interactions, stress vs. strain in the x-direction (a) and in the y-direction (b). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero

Fracture Surface Energy The fracture surface energy γs is an important quantity for the nucleation and propagation of cracks. It is defined as the energy required to generate a unit distance of a pair of new surfaces (cracks can be regarded as sinks for energy, where elastic energy is converted into surface fracture energy). The Griffith criterion predicts that the crack tip begins to propagate when the crack tip energy release rate G reaches the fracture surface energy 2γs , G = 2γs [62]. The fracture surface energy can be expressed as ∆φ , (4.47) d where d is the crack advance and ∆φ is the energy necessary to break atomic bonds as the crack advances a distance d. The bond breaking process is γs = −


141

depicted in Fig. 4.12a for cracks propagating along the direction with highest fracture √ surface energy. In this case, four bonds break while the crack proceeds d = 3r0 . Figure 4.12b shows the bond breaking process for crack orientation along the direction of lowest fracture surface energy. In this direction, two bonds break while the crack advances d = r0 .

Fig. 4.12 Bond breaking process along the fracture plane and calculation of fracture surface energy for (a) direction of high fracture surface energy and (b) direction of low fracture surface energy

For the case considered in the simulations, the fracture surface energy is determined assuming that bonds between nearest neighbors snap during crack propagation. Unlike the wave velocity, the fracture surface energy is well defined for both linear and nonlinear cases. We summarize the results for different potentials described in the earlier sections of this chapter. The fracture surface energy for the harmonic bond snapping model for crack propagation along the direction of high fracture surface energy (as shown in Fig. 4.12) is given by γsbs,h =

k(rbreak − r0 )2 E(rbreak − r0 )2 √ = , 2r0 3r0

(4.48)

which yields γbs = 0.0332 for rbreak = 1.17. For the direction of low fracture surface energy, k(rbreak − r0 )2 , (4.49) γsbs,l = 2r0 which yields γbs = 0.0288 for rbreak = 1.17 and is about 15% smaller than in the other direction. The surface energy for the biharmonic bond snapping model along the direction of high fracture surface energy is given by γsbi,h =

, E0 (ron − r0 )2 − E1 (ron − r1 )2 − (r1 − rbreak − r0 )2 2a1 + k1 (r1 − rbreak )2 √ = . 2r0 3r0

(4.50) For the purely harmonic case, the fracture surface energy reduces to the expression given by (4.48). In the direction of lower fracture surface energy

142 γsbi,l =

Atomistic Modeling of Materials Failure , E0 (ron − r0 )2 − E1 (ron − r1 )2 − (r1 − rbreak − r0 )2 2a1 + k1 (r1 − rbreak )2 √ = . 2r0 4/ 3r0

(4.51) 4.4.4 Elasticity and Surface Energy of a Three-Dimensional FCC Lattice Thus far, we have focused attention to one- and two-dimensional models. We dedicate this section to the discussion of mechanical and physical properties of three-dimensional solids associated with a face-centered cubic lattice. The aim is to determine the elastic properties and the fracture surface energy, to be used for computational experiments of mode III cracks (see section 6.10 and 6.11). We will discuss results for harmonic potentials, as well as LJ and EAM solids. Harmonic Potential Here we focus on harmonic interactions between atoms as defined in (2.34). As in the two-dimensional models, atoms only interact with their nearest neighbors. We begin the discussion with the elastic properties and wave speeds for cubical crystal orientation. We assume that nearest√neighbor distance is r0 = 21/6 ≈ 1.12246, so that √ 1/6 the lattice constant a0 = 2r0 = 2 2 ≈ 1.5874. For mass m = 1, the density is given by ρ ≈ 1, since the volume of one unit cell is V = 4, and there are four atoms per unit cell with mass unity. The atomic volume is Ω0 = 1. In a FCC crystal with pair potential atomic interactions, c1111 = b1111 /Ω0 ,

c1122 = c1212 = b1122 /Ω0 .

(4.52)

The fact that c1122 = c1212 shows that the Cauchy relation holds. For a cubical crystal orientation (that is, x = [100] and y = [010] and z = [001]), the nonzero factors bijkl are given by 4φ (a0 /2) b1111 = √ 2 , a0 / 2 4

2φ (a0 /2) b1122 = b1212 = √ 2 . a0 / 2 4

(4.53)

The second derivative of the potential φ = k, where k is the spring constant associated with the harmonic potential. The shear modulus can be expressed in terms of the spring constant and the nearest neighbor distance as µ=

r02 k. 2

(4.54)

This analysis is based on the material discussed in [109]. For k0 = 28.57 this leads to numerical values c1111 ≈ 36 and µ = c1122 = c1212 ≈ 18. Note that λ = 2µ, and therefore Young’s modulus is

4 Atomistic Elasticity: Linking Atoms and Continuum Crystal orientation [100] [100]

Spring constant k √ 36 3 2 ≈ 28.57 √ 72 3 2 ≈ 57.14

143

E

µ

ν

cl

cs

cR

48 96

18 36

0.33 0.33

8.48 12

4.24 6

3.86 5.56

Table 4.3 Elastic properties and wave speeds associated with the harmonic potential (see (2.34)) in a 3D solid for different choices of the spring constant k, cubical crystal orientation

E=

8 µ (3λ + 2µ) = µ ≈ 48 λ+µ 3

(4.55)

and the shear modulus is µ = c1122 = c1212 ≈ 18. Poisson’s ratio is determined to be ν=

λ = 1/3. 2(λ + µ)

This yields wave velocities (1 − ν) E cl = ≈ 8.48, (1 + ν)(1 − 2ν) ρ

(4.56)

cs =

µ ≈ 4.24, ρ

(4.57)

and finally the Rayleigh wave speed is given by cR ≈ 0.91cs ≈ 3.86.

(4.58)

√ For k1 = 2k0 ≈ 57.14, the wave speeds are a factor of 2 larger. The results are summarized in Table 4.3. Figure 4.13 shows the numerically estimated elastic properties associated with the harmonic potential with k0 = 28.57 in the [100] crystal orientation with Poisson relaxation. The values for the elastic properties show good agreement. Additional results are shown in Fig. 4.14 for uniaxial loading without Poisson relaxation, and in Fig. 4.15 for triaxial loading. Note that under uniaxial loading as shown in Figs. 4.13 and 4.14, Young’s modulus increases slightly with strain, while it decreases with strain under triaxial loading as shown in Fig. 4.15. Now we discuss some additional, numerical results of elastic properties for uniaxial tension with Poisson relaxation in the [110] and the [111] direction. Figure 4.16a plots the results for uniaxial tension with Poisson relaxation in the [110] direction. Young’s modulus is approximately E ≈ 72. It is notable that Poisson’s ratio is different in the y- and the z-direction. The relaxation in the z-direction is νz ≈ 0.5, and in the y-direction there is no relaxation. This result, as well as the values for Young’s modulus can also be obtained from continuum mechanics theories based on generalized Hooke’s law (calculation not shown here). Unlike the two-dimensional triangular lattice, the threedimensional FCC lattice is not isotropic.

144


Fig. 4.13 Elastic properties associated with the harmonic potential, [100] crystal orientation, with Poisson relaxation. Poisson ratio is ν ≈ 0.33 and is approximately independent of the applied strain. The plot shows the elastic properties as a function of strain

Figure 4.16b plots the results for uniaxial tension with Poisson relaxation in the [111] direction. Young’s modulus is approximately E ≈ 100. Poisson’s ratio is identical in the y- and z-direction and is found to be ν ≈ 0.2. As for the loading in [110], this result can also be obtained from continuum mechanics theories. The elastic properties in the [110] and [111] direction are summarized in Table 4.4.

Loading direction [110] [111]

k √ 3 36 2 ≈ 28.57 √ 36 3 2 ≈ 28.57

E 72 100

νy 0 0.2

νz 0.5 0.2

Table 4.4 Elastic properties associated with the harmonic potential (2.34) in a three-dimensional solid for different choices of the spring constant k and [110] and [111] crystal orientation


145

Fig. 4.14 Elastic properties associated with the harmonic potential, [100] crystal orientation, without Poisson relaxation. The plot shows the elastic properties as a function of strain

Lennard-Jones and EAM Potentials Figure 4.17 shows the stress–strain curves for a pair potential and a multibody potential. Figure 4.17a shows the results for a LJ potential with nearest neighbor interaction, and Fig. 4.17b shows the results for an EAM potential for nickel. In both cases, the [110] direction is very weak and fails at about 12% strain in the case of an LJ potential, and it fails at a strain of only 8% in the case of an EAM potential. In contrast to this, the cohesive strain in the [100] direction is largest in the EAM potential and smaller when the LJ potential is used. The critical cohesive strains are summarized in Table 4.5. An important difference to the harmonic potentials studied in the previous section is that Young’s modulus significantly decreases with strain, leading to a strong softening effect. For the small-strain elastic properties of a LJ potential describing a threedimensional FCC lattice, a direct relation between the LJ potential parameters σ and 0 can be found. For instance, the bulk modulus can be expressed as 0 (4.59) K = 64 3 , σ by assuming only nearest neighbor interactions. There exists a direct relationship between σ and the lattice constant a0 ,

146


Fig. 4.15 Elastic properties associated with the harmonic potential, [100] crystal orientation, triaxial loading. The plot shows the elastic properties as a function of strain

σ=

√ 2a0 √ . 262

(4.60)

These relationships can be used, for instance, to determine the potential parameters for a specific metal. For copper with a bulk modulus K = 140 GPa, one can determine ε0 = 0.161 eV and σ = 2.277 ˚ A. These parameters can be obtained by first determining σ so that the equilibrium lattice constant of copper is reproduced using (4.60), and then by using (4.59) to determine the other unknown to fit the bulk modulus. These potential parameters for LJ copper are similar to those reported in [9]. Potential type Lennard-Jones (LJ) Embedded atom method (EAM) [100]

[110]

[100]

Cohesive strain εcoh 0.25 0.35 [111]

[110]

εcoh 0.17 0.23

[111]

εcoh 0.13 0.08

Table 4.5 Cohesive strains εcoh , εcoh , and εcoh for the LJ potential and the EAM potential. In all potentials, the weakest pulling direction is the [110] directions


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Fig. 4.16 Elastic properties associated with the harmonic potential, (a) [110] and (b) [111] crystal orientation, uniaxial loading with Poisson relaxation. The plot shows the elastic properties as a function of strain

Fracture Surface Energy

Potential Harmonic (with rbreak = 1.17, k ≈ 28.57) Harmonic (with rbreak = 1.17, k ≈ 57.14) Lennard-Jones Tethered LJ (with rbreak = 1.17)

γsh 0.033 0.066 1.029 0.119

γsl 0.029 0.057 0.891 0.103

Table 4.6 Summary of fracture surface energies for a selection of different potentials

For other potentials, we do not give the analytical expression but summarize the results in Table 4.6. The results for harmonic and biharmonic potentials are also included.

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Fig. 4.17 Elastic properties associated with (a) LJ potential, and (b) an EAM potential for nickel [15], uniaxial loading in [100], [110] and [111] with Poisson relaxation

The fracture surface energy of a three-dimensional FCC system can be expressed as 2γ = Nb ρA ∆φ,

(4.61)

where ρA = 2/a20 ≈ 0.794 is the density of surface atoms along the fracture plane and ∆φ denotes the potential energy per bond. The factor Nb = 4 since each atom has 4 bonds across the [100] plane (thus (010) crack faces). The potential energy per bond is given by ∆φ =

1 k0 (rbreak − r0 )2 2

(4.62)

and ∆φ ≈ 2.26 × 10−3 for rbreak = 1.17 and k0 ≈ 57.32. Therefore, the fracture surface energy is 2γ ≈ 0.21. As in the twodimensional case, note that γ ∼ k0 and therefore γ ∼ E.


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Summary The results reviewed in this chapter provide analytical expressions for the elastic properties of three-dimensional solids with harmonic interatomic potentials. The analytical predictions were verified by numerical calculations of the elastic properties. We also report results of elastic properties of FCC solids with LJ interatomic potential and EAM potentials. An interesting observation was that when pulling in the [110] direction, the solid fails at a very low strain (and at very low stress) compared to other pulling directions ([100] and [111]). This phenomenon is likely due to the strong softening of the bonds in the LJ and EAM potential. In contrast, such phenomenon does not appear in the harmonic potential since the bonds do not weaken with stretching (see Figs. 4.13 and Fig. 4.16). In fact, Young’s modulus in the [110] direction is in between the values of the [100] and [111] direction. Similar observations have been made in earlier studies for LJ pair potentials [156]. The results show that this also applies to EAM potentials. Therefore, it is expected that this phenomenon should occur in metals. The observation of this “weak” crystal orientation could potentially have impact on the design of nanowires or electronic interconnects in integrated circuits. 4.4.5 Concluding Remarks The studies shown in this section suggest that by designing the interatomic potential different elastic properties can be obtained. The LJ system shows a strong softening of Young’s moduli with strains (see Fig. 4.3). In contrast, the tethered LJ system yields a solid whose elastic properties stiffen with strain (see Fig. 4.4). The harmonic potential serves as a reference that yields approximately linear elastic properties (see Fig. 4.5). The LJ and tethered LJ potential yields continuously changing Young’s moduli, which may complicate the analysis of materials failure phenomena. Therefore, we propose a simplistic potential to describe these hyperelastic effects, the biharmonic potential. The biharmonic potential is composed of two harmonic potentials and yields bilinear elastic properties (see Fig. 4.11). An important feature of the biharmonic potential is that it allows to define unique wave speeds for small and large strains. The approach described in this chapter exemplifies the development of model materials for computer simulations. The potentials defined in this chapter will be used to study specific aspects of the dynamics of brittle fracture.

4.5 Elasticity Due to Entropic Contributions The dominance of entropic behavior is a well-known phenomenon from many polymers, including biological structures. The contributions to the entropic term to elasticity can be described in several ways, including classical

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descriptions such as the worm-like chain model or the freely jointed Gaussian chain model (for a good introduction into these concepts, please see [167, 168]). Such descriptions are similar to constitutive models in continuum elasticity, and require input parameters that are typically determined empirically. In contrast to these models, molecular dynamics modeling can provide a first principles-based description of entropic elasticity – without any additional fitting parameters beyond the atomic interactions. 4.5.1 Elasticity of Single Molecules: Worm-Like-Chain Model

Fig. 4.18 This plot depicts a series of snapshots of a single molecule with increasing length L, at constant temperature. The longer the molecule, the more wiggly the geometrical shape

We first focus on entropic elasticity of a single molecule, the building block of a polymeric material. The persistence length is defined as the molecular length at which entropic contributions to elasticity become important, as the molecule shows significant bending purely due to its thermal energy. A molecule with length far beyond the persistence length will bend, even without application of forces, and assume a conglomerated, wiggly shape (see Fig. 4.18 for a series of snapshots of a single molecule with increasing length, at constant temperature). With the bending stiffness of a molecule denoted as EI, the persistence length is defined as


ξp =

EI . kB T

151

(4.63)

When the length of molecules, denoted by L is beyond the persistence length, that is, L ξp , thermal energy can bend the molecule, and entropic elasticity typically plays a role. On the other hand, when L ξp , entropic effects play a minor role, and energetic elasticity governs.

Fig. 4.19 Entropy controlled molecular elasticity. Subplot (a) Coiled, entangled state of a molecule with contour length much larger than the persistence length. The end-to-end distance is measured by the variable x. Subplot (b) Response of the molecule to mechanical loading. As the applied force is increased, the end-toend distance x increases until the molecule is fully entangled. Clearly, the continued disentanglement leads to a reduction of entropy in the system, which induces a force that can be measured as an elastic spring. Once the molecule is fully extended, the change in entropy due to increased force approaches zero, and the elastic response is controlled by changes in the internal potential energy of the system, corresponding to the energetic elasticity

A schematic of a convoluted molecule is shown in Fig. 4.19, also showing the change in molecular configuration as the molecule is stretched. Entropic effects become important and appear in measurements, for example when one stretches a convoluted molecule. Assuming that the initial point-to point distance is x < L (expressing the fact that the molecule is convoluted) the force that resists stretching is given by kB T 1 x 1 1 F (x) = + . (4.64) − ξp 4 (1 − x/L)2 4 L This model is called the Worm-Like-Chain Model (WLC) or Marko-Siggia equation (see, for instance [167,169,170]). The molecular properties enter this equation in form of the persistence length, which is a function of the bending

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stiffness. If these properties are known from atomistic calculations, the WLC model provides a quantitative estimate of entropic elasticity. Figure 4.20 depicts the entropic response (F < 14 pN) of a single tropocollagen molecule, illustrating that the WLC model is a good approximation for experimental and numerical results of stretching biopolymers [17]. The plot also shows a comparison with experimental results [16].

Fig. 4.20 This plot depicts the entropic response (F < 14 pN) of a single tropocollagen molecule, obtained by direct molecular dynamics simulation using a multi-scale model [17]. This plot also depicts experimental results [16] obtained for TC molecules with similar contour lengths, as well as the prediction of the WLC model with persistence length of approximately 16 nm [17]. The force-extension curve shows a strong hyperelastic stiffening effect (see also Fig. 3.10)

4.5.2 Elasticity of Polymers Similar considerations as those reviewed above have been used to understand the macroscopic elastic response of rubber – a class of material whose elasticity is largely dominated by entropic effects. Figure 4.21 depicts a sample unit cell of an entropic polymer that undergoes deformation, characterized by the extension ratios λi = rr,new /ri (see Fig. 4.21). We briefly review how to link the molecular properties to the overall elasticity, using an approach that is similar to the Cauchy–Born rule, following an


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Fig. 4.21 The concepts of entropic elasticity of single molecules can be immediately applied to understand two-dimensional and three-dimensional networks of molecules in a polymer. This figure demonstrates how a change in state of deformation poses constraints on the end-to-end distances of molecules, influencing the entropy of the system. Such considerations enable to link the properties of single molecules (their entropy) with the overall macroscopic elastic behavior of the material

example reviewed in [168]. Considering a freely jointed Gaussian chain with links of length each, the entropy of such a chain is S = c − kB b2 r2 ,

(4.65)

where

3 . 2nl2 The change in entropy due to deformation is given by (λ21 − 1)x2 + (λ22 − 1)y 2 + (λ23 − 1)z 2 . ∆S = −kB b2 b2 =

(4.66)

(4.67)

Nb

By averaging over all chains in the system, ∆S = −kB b2 (λ21 − 1) < x2 > + (λ22 − 1) < y 2 > + (λ23 − 1) < z 2 > . (4.68) Assuming an isotropic solid, that is, the end-to-end distances of the Nb chains are directed equally in all directions, we find that < x2 > = < y 2 > = < z 2 > = Since rRMS =

1 < rb2 >. 3

' √ nl = < rb2 >.

(4.69)

(4.70)

and < rb2 > = nl2 , we arrive at < x2 > = < y 2 > = < z 2 > =

1 2 1 nl = 2 . 3 2b

(4.71)

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and finally at ∆S =

−kB Nb 2 (λ1 − 1) + (λ22 − 1) + (λ23 − 1) . 2

(4.72)

The change in free energy is then given by A = −T ∆S =

1 kB N T (λ21 − 1) + (λ22 − 1) + (λ23 − 1) . 2

(4.73)

The free energy density is given by Ψ=

A . V

(4.74)

This expression of the free energy density function can be used to derive expression for Young’s modulus, leading to E = 3N ∗ kB T, linking uniaxial stress σ and uniaxial extension ratio λ through 1 E λ2 − , σ= 3 λ

(4.75)

(4.76)

linking uniaxial stress σ and uniaxial extension ratio λ through E. This provides a link of microscopic parameters (N ∗ = N/V , the density of molecular chains per unit volume) with a macroscopic measurable, E. From this equation, we also note that the stiffness is proportional to the temperature, E ∼ T. Despite the significant progresses made in advancing the understanding of brittle or ductile materials, the atomistic and molecular mechanisms of deformation of natural and biological materials in which entropy governs are often poorly understood. The study of elastic and plastic deformation of such materials using large-scale computer simulations poses great challenges and thus great opportunities.

4.6 Discussion In this chapter, we have summarized the main concepts of how to link atomistic models with continuum concepts, via the utilization of a language common to both approaches: Thermodynamics. Several case studies have been provided, including model materials for two-dimensional and threedimensional crystal lattices. It is noted that these cases will be revisited later when the propagation of fractures in these lattices is discussed. We will provide a few concluding remarks regarding the use of units in molecular dynamics simulations. To interpret the resulting quantities from MD simulation properly, it is important to consider the particular reference


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units used in the code. For instance, many EAM codes use the following reference units: Energies are expressed in E ∗ = 1 eV, distances are expressed in L∗ = 1 ˚ A, and the mass is expressed in M ∗ = 1 amu. In order to concert these units to SI units, appropriate unit versions must be carried out. For example, resulting pressure or stress tensor components will be in the reference stress unit σ ∗ = 160.2 GPa and temperature results will be expressed in 11, 600 K (energy divided by the Boltzmann constant). The reference time unit corresponds to T ∗ = 1.0181 × 10−14 s. Similar conversions can be carried out for other physical quantities such as the density or for other reference units (for example, many chemistry codes use kcal mol−1 as the energy reference unit). Table 4.7 displays a summary of conversions of frequently used units to SI units. The table also shows frequently used conversions between units often found in molecular dynamics codes. Unit/physical constant

Conversion result

1 eV 1 kcal 1 kcal mol−1 1 mol 1 Pa

1.60217646 × 10−19 Nm (1 Nm = 1 J) 4184 Nm (1 Nm = 1 J) 4184 m2 kgs−2 mol−1 6.02214 × 1023 particles (=Avogadro’s number) 1 Nm−2

1 amu kB , Boltzmann constant

1.6605402 × 10−27 kg 1.3806503 × 10−23 m2 kgs−2 K−1

1J 1J 1 kcal mol−1 1 cal

1.439325215 × 1020 kcal mol−1 0.0002390057361 kcal 6.947700141 × 10−21 J 4.184 J

Table 4.7 Summary of frequently used units to SI units and/or definition of constants

5 Multiscale Modeling and Simulation Methods

This chapter is dedicated to a discussion of multiscale modeling and simulation methods. These approaches are aimed to provide a seamless bridge between atomistic and continuum approaches, sometimes by introducing intermediate “mesoscopic” methods of simulation. This chapter gives a brief overview over available models and approaches, along with a discussion of the historical development. The quasicontinuum method and a hybrid ReaxFF method are discussed in more detail. Several case studies are reviewed.

5.1 Introduction The motivation for multiscale simulation methods is that it is not always necessary to calculate the full atomistic information in the whole simulation domain. Based on this insight, several researchers have articulated the need for multiscale methods [20,171–175] by combining atomistic simulations with continuum mechanics methods (for instance, finite element methods). A variety of different methods have to be developed to achieve this. An important motivation for this is to save computational time and by doing that, to extend the lengthscale or timescale accessible to the simulations. It is common to distinguish between hierarchical multiscale methods and on-the-fly concurrent multiscale methods. In hierarchical multiscale methods, a set of different computational tools are used sequentially. First, the most accurate method (e.g., quantum mechanics) is used to determine parameters for the next computational approach (e.g., via force field fitting to generate interatomic potentials). Molecular dynamics simulations with interatomic potentials are then used to determine constitutive equations, or criteria for plasticity, which are utilized as parameters in finite element approaches. These approaches can be carried out for a variety of computational methods. In onthe-fly concurrent multiscale methods, the computational domain is divided into different regions where different simulation methods are applied. A critical issue in both methods is the correct mechanical and thermodynamical

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coupling among different models. This applies to different geometric regions in a concurrent approach and to different simulation methods in a hierarchical approach. There are many computational challenges associated with these schemes, as some of the computational engines may require more computational effort than others, so that load balancing becomes an important issue. There exist several review articles on multiscale modeling of materials in the literature [18, 176, 177], including a mathematical perspective. In addition to methods aimed to extend accessible lengthscales, several methods have been developed to cover larger timescales. In the following sections, we provide a discussion of selected multiscale approaches. First, we focus on methods to span vast lengthscales, and second, we focus on methods to span vast timescales.

5.2 Direct Numerical Simulation vs. Multiscale and Multiparadigm Modeling

Fig. 5.1 A summary of a hierarchical multiscale scheme that can be used to develop an understanding of the behavior of materials across scales in length and time

Material deformation is a phenomenon that cannot be understood at a single scale alone. It requires the consideration of multiple scales to capture the progression of the elementary physical mechanisms. How can one capture multiple scales? One possibility is to simply simulate all particles in a system. Another possibility is to use a combination of methods with different accuracy or resolution. This idea is motivated by the fact that


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in many problems, a high resolution and a high accuracy is only required in regions that are small compared with the overall specimen size. This is for instance the case in the example of a crack-like defect, where only atoms close to the crack tip experience large stresses, whereas atoms further away undergo only small deformation. The connection of multiple simulation techniques, coupled by handshaking or parameter passing, is illustrated in Fig. 5.1. We will discuss this figure later in much more detail and also illustrate in selected case studies how the handshaking between various computational techniques can be achieved. The systematic integration of models that range from the quantum mechanical to macroscopic scales can enable one to make quantitative predictions of complex phenomena with few (or without, in some cases) empirical parameters. Figure 5.2 depicts an overview over the process of predictive multiscale modeling. Quantitative predictions are enabled via the validation of key properties, which then enables to extrapolate and predict the behavior of systems not included in the initial set of parameters used to develop the model.

Fig. 5.2 Overview over the process of predictive multiscale modeling. Quantitative predictions are enabled via the validation of key properties, which then enables to extrapolate and predict the behavior of systems not included in the initial training set

5.3 Differential Multiscale Modeling Computational modeling can be used to predict quantitative numbers of material properties. Since parameter or geometry changes can typically be implemented quite easily, it can also be used to perform a procedure referred to as differential multiscale modeling. Differential multiscale modeling, in contrast to predictive multiscale modeling focuses on the differential aspects of how macroscopic properties change due to variations of microscopic properties and microscopic structures. In this approach, it is not required that each individual simulation provides a quantitative predictive capability of the phenomena. Rather, it is the change of properties that is predicted. It has been argued that this approach, in some ways a weak form of the predictive method, may provide a robust set of results and can thereby provide

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important insight into the physical basis of the simulated phenomena. The use of differential multiscale methods is particularly fruitful in the analysis of very complex processes or phenomena, in which there is no a priori known theoretical guidance about the behavior. It can also be very helpful in developing physics-based theories to describe phenomena, as it is possible to center the attention on the core properties at distinct lengthscales. A combination of differential and predictive tools can be used for the study of problems, providing an advantageous method in the analysis in particular of very complex mechanisms. For example, quantitative methods can be used to help establish an atomistic model in the regime of interest, thereby forming a reference system with proper variables and parameters. Then, differential methods can be used to explore the behavior in the vicinity of this reference system. It is noted that the reference system is sometimes also referred to as the control system in the spirit of the analysis of biological laboratory experiments. Notably, these laboratory methods were developed based on the difficulty of obtaining quantitative results. However, one is able to effectively control the boundary conditions and thereby being able to study the correlations between microscopic parameters and the system behavior.

5.4 Detailed Description of Selected Multiscale Methods to Span Vast Lengthscales In this section, the aim is to provide a brief review of selected multiscale methods used for modeling mechanical deformation in crystalline materials – in particular metals – along with their advantages, potentials, and drawbacks. 5.4.1 Examples of Hierarchical Multiscale Coupling An example for hierarchical multiscale modeling is a study of the shear strength of crystals [178]. The authors investigate the shear strength of crystals based on a multiscale analysis, incorporating molecular dynamics, crystal plasticity, and macroscopic internal state theory applied to the same system. The objective of the studies was to compare different levels of description and to determine coupling parameters. Further studies of climb dislocations in diffusional creep in thin films [50] and mesoscopic treatment of grain boundaries during grain growth processes [179, 180] have also employed hierarchical simulation approaches. The development of hierarchical multiscale approaches has also contributed to new methods and simulation strategies in a variety of fields. For example, mesoscopic dislocations dynamics simulations treat dislocations as particles embedded in a linear elastic continuum (see, for instance [95, 96, 181, 182]). Dislocation particles interact according to their linearelastic fields and move according to empirical laws for dislocation mobility. All


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Fig. 5.3 Example for implementation of a hierarchical multiscale method, where parameters are passed through various lengthscales

nonelastic reactions between dislocations that may occur have to be included in the simulation setup as interaction rules. Thus, an important issue in these approaches is to identify proper coupling variables to transition between the different scales. This is typically achieved by hierarchically coupling to full atomistic methods. Multiscale methods have also been particularly useful for polymers and biological materials. Figure 5.3 depicts a schematic that shows a systematic coarse graining of the molecular structure of a tropocollagen molecule. The concept here is to divide a larger molecular structure into a representation of super-atoms, or beads, which as a whole represent the entire tropocollagen molecule. A similar approach will be discussed in Sect. 8.7.1 for modeling of large carbon nanotube systems. The concept of hierarchical integration of computational approaches has also been used in the design of new materials, from bottom up, as illustrated in the design of Cybersteel [18, 19]. Figure 5.4 illustrates the hierarchical integration of computational tools in the design process. Such new material design methods are based on the bottom-up multiscale design, using multiscale modeling as the basic engineering tool. In the study reviewed in Fig. 5.4, the design process begins at the quantum scale, where the particle–matrix interface decohesion is analyzed by using first principles methods, in order to determine the appropriate traction-separation law. This traction-separation law is then embedded into the simulation of the submicron cell that contains secondary particles. The sequential information flow across multiple hierarchies is repeated several times until the required macroscopic behavior is achieved.

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Fig. 5.4 Hierarchical modeling of Cybersteel [18]. Subplot (a) shows quantum mechanical calculations that provide the traction-separation law. Subplot (b) depicts concurrent modeling of the submicron cell based on the traction-separation law. Subplot (c) illustrates concurrent modeling of the microcell with the embedded constitutive law of the submicron cell. Subplot (d) shows results of modeling the fracture of the Cybersteel with embedded constitutive law of the microcell. Subplot (e) depicts the fracture toughness and the yield strength of the Cybersteel as a function of decohesion energy, determined by geometry of the nanostructures. Subplot (f) shows snap-shots of the localization induced debonding process. Subplot (g) summarizes experimental observations. Reprinted from [18], Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 1529–1578, W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, An introduction to computational nanomechanics c 2004, with permission from Elsevier and materials, copyright

5.4.2 Concurrent Integration of Tight-Binding, Empirical Force Fields and Continuum Theory Rigorous multiscale approaches for mechanical properties of solids were first reported in the 1990s. One of the first approaches in this field included an integration of atomistic simulation with finite element models [183]. The authors discuss simulations performed with a hybrid atomistic-finite element (FEAt) model, and compared the results with the continuum-based Peirls–Nabarro model for different crack orientations in a nickel crystal. The researchers demonstrated the basic assumptions of the continuum model for dislocation nucleation, that is, stable incipient slip configurations are formed prior


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to dislocation nucleation, and found relatively good agreement of the FEAt model with the Peierls model for critical loading associated with dislocation nucleation. In the FEAt model, the region with atomistic detail is determined prior to computation and cannot be updated during the simulation. Another concurrent modeling approaches spanning scales from quantum methods to continuum was developed as a method to model fracture dynamics of brittle materials [144, 174, 184]. Fracture dynamics is governed by processes over a range of interconnected lengthscales, all of which are vital in deciding how a crack propagates in crystals. This multiscale methodology links lengthscales ranging from the atomistic scale, treated with a tight binding (TB) approximation, through microscale, treated by classical molecular dynamics, to the mesoscale, handled by finite element methods in the context of continuum elasticity. The method was applied to study dynamical fracture in silicon, a covalently bonded brittle material whose fracture mechanics is intrinsically complicated and cannot easily be captured by empirical approaches. The formation and breaking of covalent bonds at and in the vicinity of the crack tip is treated by a near quantum mechanical approach, a semiempirical nonorthogonal tight-binding formulation, which describes the bulk, amorphous, and surface properties of Si well. Somewhat further away from the crack tip region, where strains are large, the chemical bonds are not broken but deviate from their ideal bulk bonding arrangements. In this region, the Stillinger–Weber [124] empirical interatomic potential is used to describe the material with classical molecular dynamics. A continuum finite element region is used in the far-field, where the atomic displacements from ideal positions and strain gradients are small. The use of three different regions requires two different hand-shaking zones: FE/MD and MD/TB. The total Hamiltonian for the system is written as Htot = HFE + HTB + HFE/MD + HMD/TB .

(5.1)

The degrees of freedom of the Hamiltonian are the atomic positions r and atomic velocities v for the TB and MD regions, and nodal displacements and their velocities for the FE region. To achieve FE/MD coupling, the FE mesh dimensions are brought down to atomic dimensions at the interface between the two regions. Moving away from the interface and into the continuum the mesh size is expanded. The FE and MD regions share atoms and nodes on either side of the interface where they are given half weights each to the Hamiltonian. Since the FE/MD interface is far from the crack and any plastic deformation zone, the mapping is unambiguous and gives correct forces near the interface. For the MD/TB coupling, the dangling bonds on the TB side are saturated by pseudohydrogen atoms. The Hamiltonian matrix of these pseudohydrogen atoms are carefully constructed to tie off a single Si bond and ensure absence of any charge transfer when the atoms are in a perfect Si lattice position. The TB terminating atoms called “silogens” are fictitious monovalent atoms

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forming covalent bonds with the strength and length of bulk Si bonds. Thus, at the perimeter of the MD/TB region, there are silogens sitting directly on top of the atoms of the MD region. The MD atoms of the interface, on the other hand, have a full complement of neighbors, including neighbors whose positions are determined by the dynamics of atoms in the TB region. Major successes in the description of silicon fracture through this method have been the correct depiction of brittle fracture at low temperatures with experimental agreement with crack propagation speeds, and elucidation of possible mechanisms of brittle-to-ductile transition in fracture at higher temperatures. A general problem with such interface coupling methods is the spurious reflection of elastic waves (phonons) as the boundaries due to changes in system description. In a subsequent paper, the authors reported that there was no visible reflection of phonons at the FE/MD interface and no obvious discontinuities at the MD/TB interface. However, this particular scheme is very confined to covalently bonded crystalline materials. One of the reasons for this limitation is that transition regions for other systems (e.g., metals) are more difficult to implement. More recent ongoing efforts are exploring the possibility of applying it to metals and metallic alloys, where the MD/TB region has to be coupled very differently owing to nondirectional bonding in metals. Also, the use of the TB method at the crack tip to describe bond breaking and making, limits the size of the crack affected zone computationally, since TB is a much costlier method than empirical potentials and FE methods. Thus, in materials where cracks branch off and/or have large plastic zones or voids around them, the computational requirements for the problem can escalate drastically. Other methods in this area coupled DFT level methods with empirical potentials, as for instance done for the case of metals [185,186]. These methods also included an incorporation of the quasicontinuum method (see Sect. 5.4.3 for details). A new development in this field is the bridging scale technique, which enables a seamless integration of atomistic and continuum formulations throughout the entire computational domain [18, 19, 187]. A core feature of this method is that it is assumed that the continuum and atomistic-scale solutions exist simultaneously in the entire computational domain. Molecular dynamics calculations are only performed in the parts of the domain where this level of accuracy is required. This is possible by decomposing the displacement functions into a slowly varying and rapidly varying part (see also the schematic of the displacement time history as shown in Fig. 2.4). By subtracting the bridging scale from the total solution, the authors arrive at a coarse-fine decomposition that decouples the kinetic energy of the two simulations. A major advantage is two different time-step sizes can be used for the two different scales. This method is particularly suitable for finite temperature applications, enabling to connect atomistic and continuum domains seamlessly, even at finite temperature. This is achieved by the definition of


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Fig. 5.5 This plot shows a multiscale analysis of a 15-walled CNT by a bridging scale method. Subplot (a) illustrates the multiscale simulation model. It consists of ten rings of carbon atoms (with 49,400 atoms each) and a meshfree continuum approximation of the 15-walled CNT by 27,450 nodes. Subplot (b) shows the global buckling pattern captured by meshfree method, whereas the detailed local buckling of the ten rings of atoms are captured by a concurrent bridging scale molecular dynamic simulation. Reprinted from [18], Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 1529–1578, W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, An introduction to computational nanomechanics and materials, c 2004, with permission from Elsevier copyright

appropriate boundary conditions for the atomistic simulation domain that mimics the fine atomistic-scale vibrations of atoms. Figure 5.5 depicts a multiscale analysis of a 15-walled CNT by a bridging scale method. The method has found many other applications, for instance to model intersonic crack propagation as reported in [188]. 5.4.3 The Quasicontinuum Method and Related Approaches A quasicontinuum (often also referred to as QC) model for quasistatic atomistic-continuum simulations was developed by Tadmor, Miller, Ortiz, Phillips, Shenoy, and coworkers [20, 175, 189] beginning in the mid 1990s. The quasicontinuum method is based on the observation that in many large-scale atomic simulations a large section of atomic degrees of freedom can be described by effective continuum models and only a small subset of atomic degrees do something different. The chief objective is to systematically coarsen the atomistic description by judicious introduction of kinematic constraints on the full atomistic representation. These kinematic constraints allow full atomic resolution to be preserved where required, for example in the vicinity of defects and interfaces, and to treat large number of atoms collectively in regions where the deformation fields vary slowly on the scale of the lattice. For example, in the simulation of dislocation mechanisms, the method enables to describe the dislocation core region fully atomistically, while most of the bulk region surrounding the dislocation is treated as a continuum.

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The method begins from a conventional atomistic approach that computes the energy as a function of atomic positions. The configuration space of the solid is then reduced to a subset of representative atoms. The position of the remaining atoms are obtained by piecewise linear interpolations of the representative atom positions. Effective equilibrium equations are then obtained by minimizing potential energy of the solid over the reduced configuration space. The energy of the system is defined as the weighted sum of the representative atom energies. N ni U i , (5.2) Utot ≈ i=1

where ni is the quadrature weight signifying how many atoms a given representative atom stands for and Ui is the energy of the ith representative atom. The selection of representative atoms is based on the local variations of the deformation field. For example, near dislocation cores and on planes undergoing slip, full atomistic resolution is attained with adaptive meshing. Far away from defects, the density of representative atoms sharply decreases, and the collective motion of very large number of atoms is described by a small number of representative atoms. The quasicontinuum method has been extended to complex Bravais crystals and polycrystalline materials. It has been applied to many problems, including dislocation structures, interaction of cracks with grain boundaries, dislocation junctions, and other crystal defects. One drawback is that because of the particular expressions for energy in the quasicontinuum method, the actual atomistic methods that can be implemented are limited to ones that can easily be expressed in a suitable form. Also, finite temperature applications remain challenging, although some new developments have been proposed that combine coarse-grained dynamical simulations with the quasicontinuum method (see, for instance [190]). Further, a fully nonlocal three-dimensional version of the method has been introduced and applied to the study of nanoindentation. A recent thrust area in the quasicontinuum field has been on incorporating ab initio methods, such as orbital free DFT, for example, instead of relying entirely on empirical potentials. One potential pitfall of the quasicontinuum method is the so-called “ghost force” at the interface between the coarse-grained representative atoms and the atomically resolved regions [35]. The error arises because of the discontinuity between neighboring cells where the cell sizes are less than the range of the atomistic potential. Care must be taken to correct these “ghost forces.” Quasicontinuum approach also shares some features with hierarchical methods as the constitutive equations for FE nodes are drawn from atomistic calculations, and hence there is a message passing across scales. To exemplify the approach and to illustrate how simulation domains appear in these methods, we review two examples. The quasicontinuum method finds particularly useful applications in studies of fracture and deformation, as it is illustrated here in a simple example of a thin films constrained


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Fig. 5.6 Results of a simulation of a crack in a thin film constrained by a rigid substrate, exemplifying a study using a concurrent multiscale simulation method, the quasicontinuum approach [20]

by a substrate. A set of results for this case is shown in Fig. 5.6. Here we investigate a thin copper film with a (111) surface on a rigid substrate (the film thickness is hf ≈ 30 nm). The interatomic interactions are modeled by Voter and Chen’s EAM potential for copper [34,35]. We consider a crack orthogonal to the surface. Such a crack could for instance be created by grain boundary cracking or constrained grain boundary diffusion [46]. Figure 5.6a shows different snapshots as the lateral mode I opening loading of the film is increased (the black line indicates the interface of substrate and thin film). The atomic region adapts and expands, as dislocations gliding on glide planes parallel to the film surface are nucleated and flow into the film material. Figure 5.6b shows a zoom into the crack tip region. Figure 5.7 shows another example application of the quasicontinuum method, here illustrating the simulation of a nanoindentation experiment [191]. A coupled atomistic and discrete dislocations (CADD) method has been developed [192], exemplifying a multiscale approach aimed at coupling a fully atomistic region to a “defected” dislocation dynamics region. In the CADD method, dislocations in the continuum region are treated with a standard discrete dislocation method, and the atomistic region can have any kind of atomic scale defects. Key strengths are automatic detection and smooth passing of dislocations back and forth in the atomistic and continuum regions. So far, this approach is restricted to two-dimensional, quasistatic problems. Good agreement to fully atomistic simulations has been shown in atomic scale void growth and two-dimensional indentation problems. A major challenge to extension

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Fig. 5.7 Application of the quasicontinuum method in the simulation of a nanoindentation experiment. Subplots (a) and (b) depicts a cross-sectional view of the test sample used in the nanoindentation simulations for increasing indenter penetration (part of the indenter is also shown). Subplot (c) plots the dislocation structure at the indenter penetration corresponding to the indentation depth shown in subplot (b). Subplot (d) shows a load vs. displacement curve predicted by full atomistic (LS) and quasicontinuum (QC) simulations, illustrating that the two methods show excellent agreement. Reprinted from Journal of the Mechanics and Physics of Solids, Vol. 49(9), J. Knap and M. Ortiz, An analysis of the quasicontinuum method, copyright c 2001, with permission from Elsevier

to three-dimensional problems is the representation of three-dimensional dislocation loops that would extend across the atomistic/continuum interface. 5.4.4 Continuum Approaches Incorporating Atomistic Information Recently, a virtual internal bond (VIB) model has been proposed as a bridge of continuum models with cohesive surfaces and atomistic models with interatomic potentials [157]. The VIB method differs from an atomistic model


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in a sense that a phenomenological “cohesive force law” is adapted to act between material particles, which are not necessarily atoms. A randomized network of cohesive bonds is statistically incorporated into the constitutive response of the material based on the Cauchy–Born rule (see Sect. 4.4.1). This is achieved by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. The basic idea of this method is to compute the constitutive equation directly from the interatomic potential. Other features of the VIB model [157] can be found elsewhere. The method has been used to study crack propagation in brittle materials, and is able to reproduce many experimental phenomena such as crack tip instabilities or branching of cracks at low velocities. An important implication of the VIB method is that it provides a direct link between the atomic microstructure and its elastic properties, for any given potential. The method was recently extended to model viscoelastic materials behavior [193]. The fact that this method is able to perform simulations on entirely different lengthscales makes it interesting for numerous applications particularly in engineering, where more complex situations have to be modeled. 5.4.5 Hybrid ReaxFF Model: Integration of Chemistry and Mechanics In this section, we review a method to integrate different force fields within a single computational domain. This method is not a true multiscale method, but it rather is a multiparadigm method. The integration of chemistry and mechanical properties remains a challenging issue, in particular in systems at finite temperature and when thousands and more reactive atoms are present. Chemistry at the atomic scale is well handled by quantum mechanical methods such as QC and DFT methods. However, the number of reactive atoms in these approaches remains limited to a few hundred at most, which constitutes a severe limitations in modeling defect structures and deformation phenomena that emerge at larger scales. The combination of the ReaxFF force field, which is capable of treating chemically complex materials, with nonreactive potentials is a possible strategy to overcome these limitations. Basic Concepts The first principles based reactive force field ReaxFF (see also discussion in Sect. 2.6.4) has been shown to provide the versatility required to predict catalytic processes in complex systems nearly as accurate as QM at computational costs closer to that of simple force fields, including the capability to describe charge transfer during chemical reactions. There have been other attempts of modeling charge transfer in metal/metal-oxide systems, such as the modified charge transfer-embedded atom method potentials [194, 195].

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However, ReaxFF has a wider range of applicability to other types of atomic structures, such as hydrocarbons, proteins, and semiconductors. It has been demonstrated that ReaxFF reproduces quantum mechanical results for both reactive and nonreactive systems, including hydrocarbons, nitramines, ceramics, metal alloys, and metal oxides. Due to the complexity of the underlying mathematical expressions in ReaxFF, and the necessity to perform a charge equilibration (QEq) [132] at each iteration, ReaxFF is approximately 10–50 times more expensive computationally than simple FFs such as CHARMM, DREIDING, or covalent force fields such as Tersoff’s force field. However, ReaxFF is several orders of magnitude faster than quantum mechanics-based ab initio methods. For details about the ReaxFF methodology and development we refer the reader to Sect. 2.6.4, where some of the important aspects are reviewed. EAM-based models, on the other hand, have emerged as a well-established methodology to study mechanical deformation of metals. Fitted to experimental and ab initio data, they reproduce a wide range of properties such as equations of state, surface energies, stacking fault energies, and others quite accurately, which are important in correct representation of deformation behavior. The computational expense of EAM models is on the same order as that of empirical potentials, and systems of sizes of multibillion atoms have been investigated using EAM potentials. Thus a multiparadigm integration of ReaxFF and empirical force fields such as EAM and Tersoff, to capture chemistry at the regions of interest, for example at crack surfaces, using ReaxFF, and bulk elastic and plastic deformation using the EAM or Tersoff model is a quite promising approach to take advantage of the best of both approaches. Formulation of the Hybrid ReaxFF Model The force and energy contribution from different simulation engines is weighted as shown in Fig. 5.8. Every computational engine i has a specific weight wi associated with it that describe how much the energy of this particular computational engine contributes to the total energy. Thus, for two computational engines, the total Hamiltonian of the system is written as (here done for a combination of ReaxFF and EAM, but a similar approach can be used to couple ReaxFF): Htot = HReaxFF + HEAM + HReaxFF−EAM,

(5.3)

where HReaxFF−EAM = wReaxFF (x)HReaxFF + (1 − wReaxFF (x))HEAM

(5.4)

describes the hybrid Hamiltonian formulation in the transition region shown in Fig. 5.8. The assignment of weights is based on the concept to specify a


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Fig. 5.8 The interpolation method for defining a mixed Hamiltonian in the transition region between two different paradigms. As an alternative to the linear interpolation we have also implemented smooth interpolation function based on a sinusoidal function. This enables using slightly smaller handshake regions thus increasing the computational efficiency

relative contribution from the two force fields that are being connected in the transition region. In this equation, the parameter wReaxFF (x) is the weight of the reactive force field in the handshaking region. Forces on individual atoms given by the negative of the partial derivatives of Htot with respect to the atom’s coordinates (see the discussion in Sect. 2.7.2 and (2.54), noting also that the kinetic part of the Hamiltonian does not contribute to the forces as the velocities do not depend on the atomic coordinates). Using (5.3) to calculate the forces in the domains in which only a single force field is used (that is, either the ReaxFF or the EAM domain in this example) is straightforward to implement. The forces are calculated the same way as for the individual potentials, either ReaxFF, Tersoff, or EAM. In the handshaking region, however, due to a gradual change of weights with position, one obtains FReaxFF−EAM = wReaxFF (x)FReaxFF + (1 − wReaxFF (x))FEAM ∂wReaxFF (HReaxFF − HEAM ) , − ∂x which reduces to

(5.5)

FReaxFF−EAM = wReaxFF (x)FReaxFF + (1 − wReaxFF (x))FEAM ∂wReaxFF (UReaxFF − UEAM ) , − (5.6) ∂x since the difference in the kinetic part of the different force fields in the transition region is identical (it only depends on the particle linear momenta).

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Fig. 5.9 Example of the energy landscape of two force fields, a ReaxFF reactive force field and a nonreactive force field. The plot illustrates that the two models yield a similar energy landscape for small deviations from the minimum potential well, the equilibrium position. An exemplification of this effect specifically for silicon is shown in Fig. 6.108

This equation can be simplified quite a bit based on two conditions. First, if wReaxFF (x) varies slowly in the spatial domain from zero to one at the edges of the handshaking regions (that is, if the gradient ∂wReaxFF /∂x is small), the last term in the equation can be neglected. Further, if the potential energies for the reactive force field and the EAM method are almost the same (that is, if the difference UReaxFF −UEAM is negligible in the transition region), the last term in (5.6) can be neglected. Both conditions lead to a simplified expression that only involves the weighting of forces in the transition region and not the considerations of the potential energies. FReaxFF−EAM = wReaxFF (x)FReaxFF + (1 − wReaxFF (x))FEAM .

(5.7)

In summary, this force handshaking algorithm requires a large transition region and therefore a slowly varying interpolation function, and if the handshaking is done in a regime in which both potentials provide an identical or very similar energy landscape. It works well when a transition region of larger than 7 ˚ A width with a linear or sinusoidal interpolation of the force contributions is used. Figure 5.9 depicts the energy landscape of two force fields, illustrated here for interactions around the equilibrium position of atoms. Typically, for small deviations from the equilibrium position both potentials provide a similar description and as long as the handshaking is done in this regime, (5.7) provides a suitable approximation. This hybrid model is in principle not limited to two methods, but it can be generalized to NC different computational methods:


Htot =

NC

wi Hi ,

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(5.8)

i=1

where the weight of NC computational engines add up to unity, NC

wi = 1.

(5.9)

i=0

The forces and energies are weighted accordingly, and the force vector of an atom j is calculated as Fj,i wi , (5.10) Fj = i=0,...,N

where Fj,i is the force contribution on atom j due to computational engine i and Fj is the resulting force vector on atoms j. The width of the transition region Rtrans depends on the nature of the system, but it is generally a few times the typical atomic distance in a lattice or in an organic molecule. The width of the buffer layer Rbuf describing the ghost atoms is about 10% larger than any long-range cutoffs to rule out possible boundary effects. The important point here is that the atoms at the interface to the ghost atoms (which still contribute a small amount to the total force) should not sense the existence of the boundary of the buffer layer at any point, thus Rbuf > Rcut . Numerical Implementation: The Computational Materials Design Facility (CMDF) Multiscale methods often feature great computational complexity, and the development, application, and modification of numerical implementations may require significant effort. Thus several techniques have been developed that aid in making it easier to integrate different computational tools by providing a modular structure. For example, the computational materials design facility (CMDF) is a Python [196]-based simulation framework allowing multiparadigm multiscale simulations of complex materials phenomena operating on disparate lengthscale and timescale [197]. Individual computational engines are wrapped using the “Simplified Wrapper and Interface Generator” (SWIG) for rapid integration of low-level codes with scripting languages [196]. This framework enables complex multiscale simulation tasks encompassing a variety of simulation paradigms, such as quantum mechanics, reactive force fields, nonreactive force fields, coarse grain mesoscale, and continuum descriptions of materials. The CMDF framework enables the combination of ReaxFF to capture the QM description of reactions with classical nonreactive potentials to describe nonreacting regions providing the means for describing many complex materials failure processes as reported herein.

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Since all simulation tools and engines can be called from a Python scripting level, the scale agnostic combinations of various modeling approaches can easily be realized. This strategy enables complex simulations to be simplified to a series of calls to various modules and packages, whereas communication between the packages is realized through the CMDF central data structures that are of no concern to the applications scientist. An excerpt of a CMDF script is shown in Fig. 5.10. CMDF is designed to: • Provide a general, extensible approach of a simulation environment utilizing a library of a variety of computational tools spanning scales from quantum mechanics to continuum theories. • Establish a reusable library of highly complex computational tools that can be used as black boxes for most applications, while being initialized with standard parameters for easy usage in standard cases. • Enable atomistic applications to be used by engineers and experimental scientists, while retaining the possibility of building highly complex simulations and models. • Close the gap in coupling fundamental, quantum mechanical methods such as DFT to the ReaxFF reactive force field, to nonreactive force field descriptions (e.g., DREIDING, UFF). • Provide a test bed for developing new model and algorithms, making it simple to develop new communication channels between computational engines (e.g., developing a new force fields combining distinct methods as QEq, Morse potentials, ReaxFF, or M/EAM). The CMDF approach reviewed here is only one out of many other approaches. Many other software suites such as Konrad Hinsen’s Molecular Modeling Toolkit (MMTK) or the CAMD Open Software project (CAMPOS) of the Center for Atomic-Scale Materials Design at Danmarks Tekniske Universitet provide similar approaches. In addition, codes like NAMD can also be driven by a Python script, providing further opportunity to integrate other codes. Example of CMDF Model of Oxidation Figure 5.11 depicts an example study of a nanoscale elliptical penny-shaped crack in nickel filled with O2 . The system is under 10% tensile strain loading in the x-direction (orthogonal to the long axis of the elliptical defect). Oxidative processes leading to formation of an oxide layer are competing with extension of the crack. In later stages of the simulations, the oxide layer still remains, keeping the Ni half spaces together, indicating that it involves strong Ni–O bonds. The reactive region can expand or shrink during the simulation and is determined by the positions of the oxygen atoms. Failure initiates by formation of nanovoids in the Ni bulk phase. Classical modeling schemes, for example based on the EAM method, cannot describe such complex organic–metallic systems.


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Fig. 5.10 Example CMDF script (upper part) and schematic of the structure of CMDF (lower part)

5.5 Advanced Molecular Dynamics Techniques to Span Vast Timescales Not only multiscale methods are developed to bridge spatial dimensions, but also other methods are focused on bridging across vast timescales. In classical molecular dynamics schemes it is in principle possible to simulate arbitrarily large systems, provided sufficiently large computers are available. However, the timescale remains confined to several nanoseconds. Surprisingly, this is also true for very small systems (independent of how large computers we use). The reason is that very small systems cannot be effectively parallelized. Also, time cannot easily be parallelized. Therefore, surprisingly there exists little trade off

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Fig. 5.11 Study of a nanoscale elliptical penny-shaped crack in nickel, filled with O2 , illustrating the hybrid ReaxFF-EAM approach (crystal is loaded in tension, in the horizontal direction)

between the desired simulation time and desired simulation size. This problem is referred to as the time-scale dilemma of molecular dynamics [89, 134, 198]. Many systems of interest spend a lot of time in local free energy minima before a transition to another state occurs. In such cases, the free energy surface has several local minima separated by large barriers. This is computationally highly inefficient for simulations with classical molecular dynamics methods. An alternative to classical molecular dynamics schemes is using MonteCarlo techniques such as the Metropolis algorithm. In such schemes, all events and their associated energy must be known in advance. Note that in kinetic Monte-Carlo schemes all events and associated activation energy that take place during the simulation should be known in advance. For that purpose, the state space for the atoms has to be discretized on a lattice. Besides having to know all events, another drawback of such methods is that no real dynamics is obtained. To overcome the time-scale dilemma and still obtain real dynamics while not knowing the events prior to the simulation, a number of different advanced simulation techniques have been developed in recent years (for a more


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extensive list of references see [199]). They are based on a variety of ideas, such as flattening the free energy surface, parallel sampling for state transitions, and finding the saddle points or trajectory-based schemes. Such techniques could find useful applications in problems in nanodimensions. Time spans of microseconds, seconds, or even years may be possible with these methods. Examples of such techniques are the parallel-replica (PR) method [200, 201], the hyperdynamics method [202], and the temperature-accelerated dynamics (TAD) method [203]. These methods have been developed by the group around Voter [89] (further references could be found therein) and allow calculating the real time-trajectory of atomistic systems over long time spans. Other methods have been proposed by the group around Parrinello, who for instance developed a Non-Markovian coarse grain dynamics method [199]. The method finds fast ways out of local free energy minima by adding a bias potential wherever the system has been previously, thus quickly “filling up” local minima. The methods discussed in these paragraphs could be useful for modeling deformation of nanosized structures and materials over long time spans, such as biological structures (e.g., mechanical deformation of proteins and properties at surfaces). A drawback in many of these methods is that schemes to detect state transitions need to be known. Also, the methods are often only effective for a particular class of problems and conditions. We give an example of using the TAD method in calculating the surface diffusivity of copper (modeled by an EAM potential [34]). We briefly review the method. The simulation is speed up by simulating the system at a temperature higher than the actual temperature of interest. Therefore, in this method two temperatures are critical: The low temperature at which the dynamics of the system is studied, and a high temperature where the system is sampled for state transitions during a critical sampling time. This critical sampling time can be estimated based on theoretical considerations in transition state theory [89]. For every state transition, the time at low temperature is estimated based on the activation energy of the event. Among all state transitions detected during the critical sampling time, only the state transition that would have occurred at low temperature is selected to evolve the system and the process is repeated. To calculate the surface diffusivity of copper, we consider a single atom on top of a flat [100] surface as shown in Fig. 5.12. The atom is constrained to move at the surface. The total simulation time approaches ∆t = 3 × 10−4 s. This is a very long timescale compared to classical molecular dynamics timescales (see Fig. 2.11). The surface diffusivity is calculated according to | xi (t) − xi (t0 ) |2 Ds = lim . (5.11) t→∞ 6(t − t0 ) The simulation is carried out at a temperature of T ≈ 400 K with N = 385 atoms. The high temperature in the TAD method is chosen to be 950 K. The

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Fig. 5.12 Atomistic model to study surface diffusion of a single adatom on a flat [100] copper surface

Fig. 5.13 Study of atomic mechanisms near a surface step at a [100] copper surface. The living time (or temporal stability) of states A (perfect step) and B (single atom hopped away from step) as a function of temperature. The higher the temperature, the closer the living times of states A and B get

integration time step is δt = 2 × 10−15 s. The diffusivity is then calculated to be (5.12) DsMD = 7.53 × 10−14 m3 /s−1 . This value is comparable to experimental data Dsexp ≈ 11×10−14 m3 /s−1 [204]. The activation energy of all state transitions is found to be 0.57 eV. We further show an example of how the temperature accelerated method could be used. Here we consider the atomic activities near a surface step in a [100] copper surface. We find that atoms at the surface step tend to hop away from the perfect step. This defines two states (A), the perfect step, and (B), when the atom is hopped away from the step. The simulation suggests that over time, the two states A and B interchange. Figure 5.13 shows the time-averaged stability of the two states as a function of temperature. It can be observed that for low temperatures, the living time of state (B) is much smaller compared to that of state (A). State (A) is observed to be

5 Multiscale Modeling and Simulation Methods State transition (from to)

Activation energy (eV)

A→B B→A

0.609 0.217

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Table 5.1 Activation energy for different state transitions

Fig. 5.14 Snapshots of states A (perfect step) and B (single atom hopped away from step)

stable up to several hundred seconds. Figure 5.14 shows the two states in a three-dimensional atomic plot. Table 5.5 summarizes the different activation energies is higher than that of the reverse process. The activation energy to get from state (A) to state (B) is higher than that of the reverse process. This immediately explains why state (B) is not as stable as state (A). Such methods have recently also been applied to better understand rate dependence effects in the deformation of metals [21]. In this work, the authors illustrated the rate dependence of twinning of metals across a large range of timescales, showing how simulation and experimental results can be connected. The authors used a combination of the CADD method to reduce the number of atomic degrees of freedom, together with the parallel replica method. This method enabled them to study dynamical materials failure mechanisms over many orders of magnitudes of timescales (see Fig. 5.15). The brief examples reviewed here illustrate the great appeal of these advanced simulation techniques. Experimental techniques are currently not able to provide the resolution in space and time to track the motion of single atoms. On the other hands, advanced molecular dynamics simulation techniques can track the motion of atoms on a surface on a relatively long timescale, with a very high resolution of time.

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Fig. 5.15 Hybrid CADD-Parallel Replica study of mechanical twinning of a metal. Subplot (a) shows the simulation domain, illustrating the continuum/discrete dislocation regime and the full atomistic domain (blow-up in right part). Subplot (b) shows a comparison of atomistic simulation results with the predictions of an analytical model. The plot shows the time to nucleation of a trailing or twinning partial versus applied load in Al at a temperature of 300 K. The circles refer to the multiscale simulation results covering many orders of magnitudes in timescales. The dashed lines correspond to the predictions of the analytical. Reprinted with permission c 2007 from Macmillan Publishers Ltd, Nature Materials [21]

5.6 Discussion Multiscale simulation methods have developed quite significantly over the past decades. In particular in the past 10 years, many new methods have been developed that contributed to an extensive database of available methods.


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The new methods have contributed both to extend the accessible lengthscales as well as timescales. The methods find useful applications for both scientific applications (see for instance Fig. 5.15) and also for new methods in the design of novel materials (see for instance Fig. 5.4).

Part III

Material Deformation and Failure

6 Deformation and Dynamical Failure of Brittle Materials

In this chapter we review applications of molecular dynamics simulation to study the fracture behavior of brittle materials. Starting with a review of theoretical concepts of dynamic brittle fracture at the continuum scale, we move on to discuss a simple one-dimensional model of the dynamics of brittle fracture. We proceed with a discussion of two-dimensional and three-dimensional models, followed by a case study of multiparadigm modeling of fracture of silicon. This chapter illustrates the use of model potentials in a systematic application to computational experiments to elucidate how the interatomic potential properties and the chemistry is linked with macroscopic observables in dynamic fracture (crack initiation conditions, crack speed, crack surface structure). This chapter deals with pure brittle fracture (for both elastic and hyperelastic conditions). The effects of dislocations and plasticity will be covered in subsequent chapters.

Fig. 6.1 Picture of shattered glass, a model for a brittle material

Figure 6.1 shows a picture of shattered glass. Glass is a very brittle material that typically shatters into many pieces upon fracture.

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6.1 The Nature of Brittle Fracture As schematically visualized in Fig. 6.2, brittle fracture is a complex multiscale process. At the scale of several atomic distances (that is, several ˚ Angstroms), interatomic bonding and the atomic microstructure determine important material properties for fracture, as for instance the fracture surface energy. At this scale, the chemistry of atomic interaction and therefore quantum mechanics can play an important role. Breaking of atomic bonds occurs in the fracture process zone at length scales of several nanometers [22, 61]. In a region around the crack tip extending a few tens of nanometers, the material experiences large deformation and nonlinearities between stress and strain become apparent. The macroscopic fracture process on a scale of several micrometers can only be understood if the mechanisms on smaller length scales are properly taken into account.

Fig. 6.2 Characteristic length scales associated with dynamic fracture. Relevant length scales reach from the atomic scale of several ˚ Angstrom to the macroscopic scale of micrometers and more

Modeling of dynamic fracture can be quite challenging. A variety of numerical tools have been developed over the last decades. Modeling attempts focused on cohesive surface models [205], for instance. However, since these methods are based on continuum mechanics theories, a priori knowledge about the failure path must be known. Continuum models incorporating atomistic information like the VIB method [157, 206] overcome this limitation, as they include information about the underlying atomic lattice (see also discussion in Sect. 5.4.4), for instance. In contrast, atomistic methods require no a priori knowledge about the failure. Studying rapidly propagating cracks using


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atomistic methods is particularly attractive, because cracks propagate at speeds of km s−1 , which corresponds to nm ps−1 . This scale is readily accessible with classical molecular dynamics methods. Much of the research of dynamic fracture focused on understanding the atomic details of crack propagation and its relation to macroscopic theories [22] as well as experiments of fracture [207]. The first part of this review describes simulation work that mostly treats generic “brittle model materials” rather than specific materials. Afterward we will focus on simulations that discuss fracture in specific materials. The studies in the area of dynamic fracture will be focused on the following points: • How can one build appropriate atomistic models of brittle materials to study their fracture behavior? • How do atomistic simulations results compare with continuum mechanics theory predictions? • What is the role of material nonlinearities (that is, hyperelasticity as shown in Fig. 3.10) in dynamic fracture? • What is the effect of geometric confinement and crack propagation along material interfaces? • What is the effect of the details of chemical interactions, bond rearrangements, or chemically aggressive environments on fracture? This chapter begins with a discussion of some theoretical aspects and a review of continuum theory of fracture mechanics, presented in Sect. 6.2. We then proceed with a discussion of a one-dimensional model of fracture in Sect. 6.4. We discuss a linear elastic continuum theory serving as a basis for the extension of the analytical model to the nonlinear case. We report an atomistic model of one-dimensional fracture and show that the continuum theories agree reasonably well with the atomistic simulation results. It is shown that hyperelasticity (see Fig. 3.10) can significantly alter the dynamics of fracture, in agreement with the analytical model. The one-dimensional model allows to study some of the phenomena that also appear in higher dimensional models in a mathematical and numerical simple framework. We recall that Sect. 4.4.3 was devoted to a discussion on mechanical and physical properties of two-dimensional solids. A good understanding of these material properties is critical to compare the atomistic simulation results with continuum mechanics theories. We present methods to calculate elastic properties and wave speeds from the interatomic potential. Several choices of interatomic potentials are discussed. We also address the issue of calculating the fracture surface energy. In Sect. 6.5, we report joint continuum-atomistic studies of the deformation fields near a moving mode I crack in a harmonic lattice. It will be shown that in harmonic lattices corresponding to linear elastic material, continuum mechanics theory is a reasonable model. We compare the stress and strain fields, particle velocity distribution, potential energy field, and energy flow.

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It will be shown that the predicted limiting speed of cracks agrees with the simulation result and the harmonic atomistic model can be used as a reference system. Section 6.6 focuses on the role that material nonlinearities play on the limiting speed of cracks propagating along a prescribed straight fracture path. It will be shown that hyperelasticity can govern dynamic fracture when the size of the nonlinear region around the crack tip approaches a newly discovered length scale associated with energy flux to the crack tip. The characteristic energy length scale helps to explain many experimental and computational results. The analysis illustrates that under certain conditions, cracks can break through the sound barrier and move supersonically through materials. An important aspect of the analysis is the prediction of intersonic mode I cracks. The preceding section focuses on the dynamics of constrained cracks, Sect. 6.7 focuses on the dynamics of unconstrained cracks and the effect of hyperelasticity. The main focus is an investigation of the critical crack speed when straight crack motion becomes unstable. By a systematic study with different model materials representing weak and strong hyperelastic effects we review evidence that suggests that hyperelasticity governs the critical speed of crack tip instabilities. In Sect. 6.8 we discuss inertia properties of cracks by investigating the dynamics of suddenly stopping cracks. We will show good agreement of suddenly stopping mode I cracks with theory and experiment [22,23], and discuss the dynamics of suddenly stopping mode II cracks with respect to continuum mechanics theories [208]. We also address the role of material nonlinearities, and report a Griffith analysis for crack initiation for different interatomic potentials. Section 6.9 discusses several aspects of dynamic fracture along interfaces of dissimilar materials. We will show that mother–daughter mechanisms, formerly believed to exist only under mode II loading, also exist in the dynamics of mode I cracks along interfaces of elastically dissimilar materials. Further, we illustrate that mode II cracks moving along interfaces of dissimilar materials feature a mother–daughter–granddaughter mechanism. The final two sections are devoted to the dynamics of mode III cracks. Since mode III cracks can only be modeled with three-dimensional models, we utilize the results of mechanical and physical properties of three-dimensional solids (as those presented in Sect. 4.4.4). Section 6.10 contains a discussion of the dynamics of mode III cracks. We will study a crack in a stiff material layer embedded in a soft matrix, and confirm the existence of the characteristic length scale for energy flux also for mode III cracks. The results of atomistic simulations are quantitatively compared with continuum mechanics theory. Section 6.11 is dedicated to a case study of studies of brittle fracture of silicon using a multiparadigm approach coupling ReaxFF force fields with empirical Tersoff models.


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6.2 Basics of Linear Elastic Fracture Mechanics The mechanics of fracture of materials has interested scientists and engineers for several hundred years. One of the earliest scientists to work on fracture was Galileo Galilei. He studied the strength of materials approximately 350 years ago, in 1635, when he stated that one cannot reason from the small to the large, because many mechanical devices succeed on a small scale that cannot exist in great size. This statement poses one of the most fundamental questions related to the strength of materials, relating how defects, or flaws, influence the strength of materials, and how the size of materials could influence their strength. It was not until the early twentieth century that scientists had a good physical understanding about the origins of these effects. How strong are materials? How does the strength depend on the size of structures? Such questions have always played a critical role in engineering science. In particular, with the new paradigms of materials science that were developed in the second half of the twentieth century, relating structure– function property, the science of fracture – in particular its theoretical and analytical, and also experimental investigation – has received tremendous attention. Today, the physics of fracture remains a highly active research field. In particular, with the increased importance of new nanoscale materials, or the analysis and synthesis of biological materials, fracture theories have received much attention. Theories of fracture, in particular the continuum mechanics approach, have proven to be very successful and widely applicable. In fact, some of the most powerful predictions have been made by the continuum mechanical treatments of fracture, such as crack limiting speeds or the stress fields surrounding a moving crack. Some of these theories and models will be reviewed in the following sections, focusing on important ideas and concepts of linear elastic fracture mechanics. 6.2.1 Energy Balance Considerations: Griffith’s Model of Fracture Theoretical fracture mechanics, in particular solutions based on linear elasticity theories applied to crack problems has been extremely successful over the past 100–150 years. Here we review some of the basic concepts, in particular the underlying mathematical approaches to derive solutions for crack problems (for a review of fracture mechanics, please see [209]). Griffith (1921) [62] formulated a criterion for unstable crack extension by considering a balance of energies, including mechanical strain energy and surface energy – the energy necessary to create new surfaces. When a crack grows, the potential energy stored in the material – at the atomic level, these energetic contributions come from the distortions of atomic bonds – decreases. This can be visualized straightforwardly by considering that after fracture and crack propagation some piece of the material that was previously strained is now completely relaxed. Its potential energy was transformed into another

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form of energy – it was used to create new surface area of the material. This energy is characterized by γs A = Ws , where A is the newly created surface area (note that when A = 2aB, Ws = 2γs aB, where a is the crack length and B the thickness of a specimen, assuming that the crack extends through the entire thickness). Griffith first postulated that a crack starts to extend when the decrease in potential energy due to crack growth equals the energy necessary to create new material surface. This model can be illustrated in a few simple equations. The total energy in the system is given by U = Wp + Ws ,

(6.1)

where Wp is the potential energy of the cracked material. Thus, crack growth occurs when dWp dWs dU = + = 0, (6.2) dA dA dA where dWp G := − (6.3) dA is defined as the energy release rate, typically denoted by the symbol G. This quantity was first introduced by Irwin [63]. The criterion to describe the onset of fracture is then given by G = 2γs . (6.4) This condition for fracture initiation can be understood from the first law of thermodynamics: When a system goes from nonequilibrium to an equilibrium state there is a net decrease in energy. Before the crack is nucleated, the potential energy Wp (corresponding to the elastic energy) increases, since the applied load increases. Thus, the elastic energy stored in the material increases. The surface energy Ws remains constant, since the crack has not yet nucleated. At the critical point, the change in surface energy and the change in elastic energy with respect to an infinitesimal increase in crack length increases – the crack starts to nucleate. To apply the idea Griffith put forward, we must have an expression of Wp as a function of crack length, load applied, and geometry of the cracked specimen. The quantity Ws is typically found based on the crack geometry and the surface energy (here we will only consider cases in which the crack extends through the entire thickness of a material). This concept can be illustrated by considering the geometry shown in Fig. 6.3. To develop an expression of the potential energy as a function of crack length we consider the beam case shown in Fig. 3.14, noting that the length of each of the two beams is l. Using beam theory, the potential energy Wp is given by Ws = −

P 2 l3 3EI

(6.5)


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Fig. 6.3 Using the solution to the beam problem to predict the critical force P at which fracture initiates. Subplot (a) shows the geometry of a crack in a beam-like structure. Subplot (b) shows the representation of the upper and lower part as two cantilever beams

by utilizing the solution of the beam problem in (3.125) to calculate the potential energy of the beam under load and by considering that there are two beam structures. From this expression the energy release rate is given by 2 3 ∂ P l G= , (6.6) ∂A 3EI where dA = Bdl with B being the out-of-plane thickness of the beam, and therefore P 2 l2 G= , (6.7) BEI and with the Griffith condition G = 2γs , 1 2γs EI . (6.8) Pf = B l2 A similar calculation can be carried out for a thin strip geometry with a semi-infinite crack, as shown in Fig. 6.4. The energy release rate is easily obtained by considering a representative material element ahead of the crack and one far behind the tip of the crack, of width a˜, thickness B, and the height of the strip ξ. The elastic energy in such a representative element far ahead of the crack is given by Wp(1) =

σ 2 (1 − ν 2 ) ξ˜ aB 2E

(6.9)

where the term a ˜B corresponds to the volume of this control element, E is Young’s modulus, ν is Poisson ratio, and σ0 is the applied stress. Behind the crack, the elastic energy is completely relaxed: Wp(2) = 0.

(6.10)

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Fig. 6.4 Thin strip geometry. The gray arrows indicate the mode I (tensile loading), by a stress σ0

Thus, the energy stored in this geometry – as a function of crack length a – is given by σ 2 (1 − ν 2 ) ξ˜ aB (6.11) Wp = Wp(2) − Wp(1) = − 2E with strip width ξ, the energy release rate can be expressed as (note that here we take the partial derivative with respect to A = aB, which is the area of one crack surface) σ 2 (1 − ν 2 )ξ . (6.12) G= 2E From classical fracture mechanics, the critical stress for crack nucleation in this perfectly brittle material is given by the Griffith condition G = 2γs . At the critical point of onset of crack motion, the energy released per unit length of crack growth must equal the energy necessary to create a unit length of two new surfaces. Using the Griffith condition we arrive at 4γs E σf = . (6.13) ξ(1 − ν 2 ) For the case of an elliptical crack in an infinite medium (see Fig. 1.3 for the geometry) – a solution developed by Inglis – the expression for Wp is Wp (a, σ) = −

πa2 σ 2 B , 2E

(6.14)

where E = E/(1 − ν 2 ) for plane strain and E = E for plane stress. The energy release rate is then given by 2 2 ∂ πa σ B G= , (6.15) ∂A 2E


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and therefore

πaσ 2 . (6.16) E At the onset of crack growth, G = 2γs , which leads to a condition for the failure stress as a function of material parameters and crack size a: 2γs E . (6.17) σf = πa G=

Undeformed

Stretching=store elastic energy

Release elastic energy dissipated into breaking chemical bonds

Fig. 6.5 Summary of the basic physical processes involved in brittle fracture, that is, the process of dissipating stored elastic energy toward breaking of chemical bonds

It is noted that the characteristic length dimensions in the various cases considered here (e.g., a finite size crack embedded in a continuum and a thin strip geometry) have a different meaning, referring to the crack size in the first case and to the material dimension in the second case. Notably, in the thin strip case, the fracture stress actually does not depend on the size of the crack (which is infinity due to its semi-infinite nature), but instead on the width of the material strip, given by ξ. Solutions for many other geometries can be derived with similar methods. To account for plasticity and other mechanisms at the crack tip, the surface energy is sometimes replaced with γs + γp , where the second term γp can be much larger than γs . This allows to treat materials that are not perfectly brittle within the same theoretical framework, including metals. Others have generalized this idea to include other dissipative effects.

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Finally, Fig. 6.5 summarizes the basic physical processes involved in brittle fracture, that is, the process of dissipating stored elastic energy toward breaking of chemical bonds. 6.2.2 Asymptotic Stress Field and Stress Intensity Factor

Fig. 6.6 Schematic of cracks under mode I, mode II, and mode III crack loading

It is commonly distinguished between different modes of crack loading, depending on how the load is applied. The geometry of mode I, mode II, and mode III loading is shown schematically in Fig. 6.6. At least two-dimensional atomistic models of dynamic fracture are required to describe the behavior of mode I and mode II cases, and three-dimensional models are required for mode III cracks. The stress intensity factor is an extremely useful concept to calculate the energy release rate G for different geometries and loading conditions. The stress intensity factor describes the impact of the geometry on the stress field in the vicinity of a crack. The stress field in the vicinity of the crack tip is given by an asymptotic solution [22, 210, 211]. With KI as the stress intensity factor for a mode I crack, KI (1) Σij (Θ) + σij + O(1). σij (Θ) = √ 2πr

(6.18)

The functions Σij (Θ) represent the variation of stress components with angle Θ [22]. For mode I loading, the functions Σij are given by Θ 3Θ Θ 1 − sin sin , (6.19) Σxx = cos 2 2 2 3Θ Θ Θ cos sin , (6.20) Σxy = cos 2 2 2 Θ 3Θ Θ Σyy = cos 1 + sin sin . (6.21) 2 2 2 The asymptotic field has a universal character because it is independent of the details of the applied loading, characterized by the stress intensity factor


195

(1)

KI . The values of σij and the first-order contribution O(1) are determined from the boundary conditions, and can be neglected in areas very close to the crack tip. It is important to note that the crack tip represents a singularity for stresses, as 1 (6.22) σij ∼ √ . r

Fig. 6.7 Closing a crack by negating the tractions at the tip, as used in the derivation of the relation between the stress intensity factor and the energy release rate

There exists a unique relationship between the energy release rate G and the stress intensity factor KI,II,III . By considering the scenarios of closing a crack with length a + δa to length a by applying proper negating tractions at the tip of the crack, we can calculate the amount of energy necessary (see Fig. 6.7): 0 1 GδaB = δΦ = −2B σ22 (x1 )u2 (x1 )dx1 (6.23) −δa 2 Some insight can already be gained by studying some scaling relationships. We note that KI σ22 ∼ √ (6.24) r and u2 ∼

KI √ r E

(6.25)

KI2 . E

(6.26)

so that Gδa ∼ Specifically,

KI σ22 = , 2π(δa + x1 )

(6.27)

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so that the singularity of the crack tip is at x1 − δa. The displacements in the y-direction are given by KI −2x1 1 − ν2 , (6.28) u2 = E π noting that the negative sign stems from the fact that for the displacements we integrate from the left to the right, opposing the direction of the polar coordinate system. The displacements are obtained by considering the displacement field behind a crack with tip at a + δa, whereas the tractions are obtained by considering the region ahead of a crack with tip at a. Evaluating the integral (6.23) with the expressions for stress and displacements leads to the relationship between G and KI : G=

KI2 . E/(1 − ν 2 )

(6.29)

This relationship can be used to obtain the stress intensity factor for a specific geometry, such as the thin strip case. For this case, KI = σ0 ξ/2. (6.30) For the case of a crack included in an elastic medium, √ KI = σ0 πa.

(6.31)

For a small crack at a surface of an infinite elastic medium loaded in tension (mode I), √ KI = 1.12σ0 πa. (6.32) Many other expressions for KI , KII , and KIII can be found in the literature (e.g., in stress intensity factor handbooks [61, 64, 212]). 6.2.3 Crack Limiting Speed in Dynamic Fracture After nucleation of a crack in a brittle material, its propagation speed typically increases significantly. The crack propagation speed is defined as the derivative of the crack tip position with respect to time, v=

∂a . ∂t

(6.33)

Larger applied mechanical loads generally lead to larger propagation speeds. However, the maximum speed that a crack can attain is limited by a maximum value that is related to the speed of waves in the elastic media in which the crack propagates. This is similar to the speed of light, which provides the upper bound of the velocity at which light can travel.


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The specific limiting speed of cracks depends on their mode of loading (that is, whether the applied load is mode I, mode II, or mode III). Mode I cracks are limited by the Rayleigh wave speed cR , mode II cracks are limited by the longitudinal wave speed cl , and mode III cracks are limited by the shear wave speed cs . A particular notable feature of mode II cracks is that even though the limiting speed is cl , velocities between cR and cs are not admitted, which leads to a velocity gap. Specific crack mechanisms have been discovered that enable the crack to overcome this gap and “jump” from subRayleigh speeds directly to intersonic crack speeds (this jump occurs via the “Burridge–Andrews mechanism” in mode II interfacial cracks). The physical reason for the limitation of the maximum crack propagation speed is the dependence of the energy release rate G on the crack velocity, as given here for a mode I crack: G(v) ∼ 1 −

v , cR

(6.34)

indicating that the energy release approaches zero when v → cR , and would assume negative values for velocities in excess of cR (this is a result of the fact that the stress intensity factor is a function of the crack speed [22]). In this situation, the crack would represent a source of energy, which is physically impossible. The aspect of the crack limiting speed will be discussed in great detail in the subsequent chapters.

6.3 Atomistic Modeling of Brittle Materials The earliest molecular dynamics simulations of fracture were carried out more than 30 years ago by Ashurst and Hoover [213]. Many features of dynamic fracture were described in that paper, although their simulation size was extremely small (only 64 × 16 atoms with crack lengths around ten atoms). In one of the first papers on large-scale atomistic modeling of fracture [155], the authors reported molecular dynamics simulations of fracture in systems up to 500,000 atoms, which was a significant number at the time. In these atomistic calculations, a Lennard-Jones potential as described in (2.30) was used. The results in [146, 155] were striking because the molecular dynamics simulations reproduced phenomena that were discovered in experiments a few years earlier [207]. The most important observation was the so-called mirror-misthackle transition. It was observed that the crack face morphology changes as the crack speed increases. The phenomenon is also referred to as dynamic instability of cracks. Up to a speed of about one-third of the Rayleigh wave speed, the crack surface is atomically flat (mirror regime). For higher crack speeds the crack starts to roughen (mist regime) and eventually becomes very rough (hackle regime), accompanied by dislocation emission. Such phenomena were observed at similar velocities in experiments [207]. Since the molecular

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dynamics simulations are performed in a perfect lattice, it was concluded that these dynamic instabilities are a universal property of cracks. The instabilities were subject to numerous other studies (e.g., [214]) in the following years. Material cR (in m s−1 ) cs (in m s−1 ) cl (in m s−1 ) Steel Al Glass PMMA

2,940 2,850 3,030

3,200 3,100 3,300

6,000 6,300 5,800

920

1,000

2,400

Table 6.1 Overview over wave speeds in a variety of materials, indicating the longitudinal wave speed cl , the shear wave speed cs , and the Rayleigh wave speed cR

A question that has attracted numerous researchers is that of the limiting speed of cracks [22]. The crack speed is limited by an impenetrable barrier that is related to the speed of sound in the material. The limiting speed for mode I cracks is the Rayleigh wave speed. For mode II cracks, velocities below the Rayleigh speed, and those between the shear wave speed and the longitudinal wave speeds are admissible. Between these two regimes, there is an impenetrable velocity gap, which led to the uncertainty that mode II cracks may also be limited by the Rayleigh wave speed. The review clearly points out that the wave speeds are critically important in the study of cracks in brittle materials. An overview of wave speeds in several materials is provided in Table 6.1. Despite the existence of this velocity gap, experiments have shown that shear-loaded (mode II) cracks can move at intersonic velocities through a mother–daughter mechanism [215, 216]. The experiments reported in [215] provided the first unambiguous evidence that mode II loading drive the the crack to intersonic speeds, even in purely homogeneous systems with only one distinct set of wave speeds. Figure 6.8 depicts a velocity analysis of such an intersonic crack, propagating in Homalite-100. Figure 6.9 shows isochromatic fringe patterns near the intersonic crack, clearly showing the existence of the shock Mach cone. Molecular dynamics simulations reproduced this observation, and provided a quantitative continuum mechanics analysis of this mechanism [165]. A short distance ahead of the crack, a shear stress peak develops that causes nucleation of a daughter crack at a velocity beyond the shear wave speed. This topic is an example where atomistic simulations could immediately be coupled to experiments. This also led to the development of the fundamental solution of intersonic mode II cracks [217]. The fundamental solution was then used to construct the solution describing the dynamics of a suddenly stopping intersonic crack [208].


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Fig. 6.8 Mode II loading experimental setup for studies of dynamic fracture in Homalite-100. Subplot (a) depicts the geometry of the experiment, indicating the location of projectile impact to generate rapid mode II loading along a weak plane. The dashed circle displays the view of the circular polariscope for the analysis of the stress field. Subplot (b) displays the evolution of crack speed as the shear crack propagates along the weak plane. The crack tip speed was obtained from crack length history (squares) and from shock wave angles (circles) for a field of view around the notch tip (solid symbols), and for a field of view ahead of the notch (open symbols). The analysis confirms intersonic and supersonic regimes of crack propagation. Reprinted from Science, Vol. 284, A.J. Rosakis, O. Samudrala, D. Coker, Cracks c 1999, with permission from AAAS Faster than the Shear Wave Speed, copyright

Other researchers [165] reported simultaneous continuum mechanics and atomistic studies of rapidly propagating cracks. The main objective of the studies was to investigate if the linear continuum theory can be applied to describe nanoscale dynamic phenomena. The studies included the limiting speed of cracks and Griffith analysis [165]. The results suggest that continuum mechanics concepts could be applied to describe crack dynamics even at nanoscale, underlining the power of the continuum approach. Materials in small dimensions have also attracted interest in the area of dynamic fracture. Studies of such kind involve crack dynamics at interfaces of different materials (e.g., in composite materials). Since interfaces play an important role in the dynamics of earthquakes, cracks at interfaces have been significantly studied in recent years [218]. Some investigations revealed that shear-dominated cracks at interfaces between dissimilar materials can move at intersonic and even supersonic velocities [219, 220]. If shear-dominated cracks propagate along interfaces between two dissimilar materials, multiple mother–daughter mechanisms have been observed, and they were referred to as mother–daughter–granddaughter mechanisms [219]. Other studies of brittle fracture were based on lattice models of dynamic fracture [214,221]. These models have the advantage that crack dynamics can be solved in closed form for some simplified cases [221]. In contrast to the large-scale molecular dynamics models described above, lattice models are usually small and do not rely on big computers.

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Fig. 6.9 Enlarged view of the isochromatic fringe pattern around a steady-state mode II intersonic crack along a weak plane in Homalite-100. Subplot (a) shows the experimental pattern, and subplot (b) the theoretical prediction [22]. For both cases, β = 53o and v = 1.47cs . Reprinted from Science, Vol. 284, A.J. Rosakis, c O. Samudrala, D. Coker, Cracks Faster than the Shear Wave Speed, copyright 1999, with permission from AAAS

In [222], the authors report an overview over atomistic and continuum mechanics theories of dynamic fracture, emphasizing the importance of the atomic scale in understanding materials phenomena. They discuss scaling arguments allowing to study crack dynamics in small atomic systems and scaling it up to larger length scales comparable to experiment. A study of fracture in tetravalent silicon based on the Stillinger–Weber (SW) potential is discussed. The authors state that the SW potential has problems describing brittle fracture in silicon well, since the experimentally preferred fracture planes (111) and (110) could not be reproduced. The authors further discuss other possible potentials for silicon in terms of their applicability to model fracture of silicon. A velocity gap is discussed implying that at zero temperature there is a minimum speed at which cracks can propagate. Various simulations of fracture of silicon are summarized [223]. In [221], further issues of atomic brittle fracture are discussed, such as lattice trapping. Also, the author showed a relation of crack velocity and loading indicating that there are regimes of forbidden velocities, so that the crack speed increases discretely with increase of loading. In another publication they compared the crack velocity as a function of energy release rate calculated by molecular dynamics to experimental results [224]. Further discussion on the role of the potential in dynamic fracture can be found in a review article [220]. Very large-scale atomistic studies of dynamic fracture involving 10–100 million atoms have been reported in the late 1990s [225]. These researchers studied fracture of silicon nitride, fracture of graphite, and fracture in gallium arsenide. They also report studies of fiber-reinforced ceramic composites (silicon nitride reinforced with silica-coated carbide fibers). More recently, the research group reported molecular dynamics simulations with up to one billion


201

atoms [139,140]. In a review, further approaches of modeling dynamic fracture are summarized [137]. Others reported a series of molecular dynamics simulations to evaluate the influence of several aspects on the dynamic crack tip instability based on various potentials [105]. The authors also report a velocity gap for crack speeds. They use a particular type of boundary conditions leaving the crack in an elliptical-shaped boundary with viscous damping at the outside to avoid reflection of waves from the boundary. Due to its shape similar to a stadium, it was referred to as “stadium damping” [105]. The crack propagates within an N V E ensemble in an elliptical “stadium” that is characterized by center and stadium. Outside this inner ellipse viscous damping or N V T temperature control is applied. This setup is chosen because stress waves reflecting from the boundaries can severely influence the dynamics of cracks, leading to crack arrest. The authors found that the limiting speed of cracks is between 30 and 40% depending on the potential. It was reported that cracks release the excess energy by emitting strong acoustic waves during breaking of every single atomic bond. Further, the authors did not observe crack branching since the velocity was too low for this phenomenon to be observed. Other research focused on mechanical behavior of quasicrystals [226, 227]. Quasicrystals, for the first time observed in 1984, show a symmetry “between crystal and liquids” and cannot be described as a Bravais lattice [228]. They are metallic alloys whose positions of atoms are long range translationary ordered. Research in this field focused on dislocation motion and crack propagation. Unlike in crystals where dislocations leave the lattice undisturbed after they have passed, in quasicrystals they leave a phason-wall that weakens the binding energy and may serve as paths for crack propagation [226, 227, 229]. Atomic studies helped to clarify the fracture mechanism in such materials.

6.4 A One-Dimensional Example of Brittle Fracture: Joint Continuum-Atomistic Approach In the following sections, we will discuss a simple one-dimensional model to illustrate some key aspects of dynamic fracture, in particular focusing on crack limiting speed, inertia properties of cracks, and the effect of material nonlinearities on dynamic fracture. In this first, simple case study it will be illustrated by joint continuumatomistic studies of a one-dimensional model of fracture that hyperelasticity, the elasticity of large strains, plays the governing role in the dynamics of fracture in brittle materials and that linear theory is incapable of capturing all phenomena, such as the speed of crack propagation in real materials. The first part of the section is dedicated to a systematic comparison of the linear elastic continuum model with molecular dynamics simulations featuring harmonic interatomic potentials. The results for wave propagation velocities, the critical condition for fracture, inertia properties of the crack as well as

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Fig. 6.10 Geometry of the one-dimensional model of fracture

stress and deformation fields around the crack tip suggest good agreement of the atomistic model with the continuum theory. In the second part of the section, the one-dimensional model is used to study crack dynamics in nonlinear materials. On the basis of the concept of local elastic properties [27], an analytical model is proposed for the dynamics of the crack and for the prediction of the deformation field. An important prediction of this model is the possibility of supersonic crack propagation if there is a local elastically stiff region close to the crack tip. By atomistic simulations, it is shown that this hypothesis is true and that an elastically stiff zone at the crack tip allows for supersonic crack propagation. This suggests that local elasticity at the crack tip is crucial for the dynamics of fracture. In most classical theories of fracture it is believed that there is a unique definition of how fast waves propagate in solids. Our results prove that this concept cannot capture all phenomena in dynamic fracture, and instead should be replaced by the concept of local wave speeds. 6.4.1 Introduction Most of the theoretical modeling and most computer simulations have been carried out in two or more dimensions (e.g., [22, 61, 155, 219]). One of the important objectives in understanding hyperelasticity in dynamic fracture is to obtain analytical models. However, finding analytical solutions for dynamic fracture in nonlinear materials seems extremely difficult, if not impossible in many cases [230]. To investigate the nonlinear dynamics of fracture at a simple level, we propose a one-dimensional (1D) model of dynamic fracture, as originally reported by Hellan [231] for linear elastic material behavior. The model can be described as a straight, homogeneous bar under lateral loading σ0 . Part of the bar is attached to a rigid substrate, and this attachment can be broken, so that a crack-like front of debonding moves along the bar (in the following, we refer to the front of debonding as crack tip). The model is depicted in Fig. 6.10. A complete analytical solution of this problem is available based on linear elastic continuum mechanics theory [231]. Theory predicts


203

Fig. 6.11 One-dimensional atomistic model of dynamic fracture

that the 1D model has many of the features of higher-dimensional models of dynamic fracture. For instance, there exists a limiting speed for the onedimensional crack associated with the wave velocity, and a critical condition for fracture initiation similar to the Griffith criterion can be formulated. Due to its simplicity, the one-dimensional model of fracture seems an ideal starting point for analyses of the complex dynamics of fracture in nonlinear materials, rather than immediately relying on two-dimensional models. It will be shown that it is possible to extend the linear continuum model to describe the nonlinear dynamics of cracks for a bilinear stress–strain law. This elastic behavior is characterized by two distinct Young’s moduli, one for small strains and one for large strains, and provides the most simple constitutive law of hyperelasticity. The new continuum model predicts that the crack propagates supersonically, if there exists a local zone around the crack tip with stiffer elastic properties than in the rest of the material (which is elastically softer). On the basis of the continuum model, we construct an atomistic model as illustrated in Fig. 6.11. The model features a one-dimensional string of atoms. Part of the atoms are bonded to a rigid substrate by a “weak potential,” whose bonds snap early leading to a finite fracture energy. Bonds between the atoms never break. Using harmonic interatomic potentials, the elasticity of a string of atoms corresponds to a straight linear elastic bar of homogeneous material. Using nonlinear interatomic potentials, the atomistic model is readily able to model a nonlinear material response. A bilinear stress–strain law as assumed in the continuum model can be mimicked at the atomic scale by using a biharmonic potential. The new continuum model of one-dimensional fracture in nonlinear materials in conjunction with the nonlinear atomistic simulations allow to carry out simultaneous atomistic-continuum studies of the nonlinear dynamics of fracture. The plan of this chapter is as follows. After a review of the linear continuum theory of one-dimensional dynamic fracture, we present the continuum model for one-dimensional fracture in the nonlinear case. In joint continuumatomistic studies, we investigate the predictions of both linear and nonlinear continuum theories with atomistic simulation results. We find reasonable

204


agreement at the two scales. The results provide evidence that the predictions of the new continuum model for a bilinear stress–strain law are reasonable. We will show that the crack limiting velocity is indeed associated with the elastic properties localized to the crack tip. 6.4.2 Linear-Elastic Continuum Model The analytical continuum solution is discussed in detail elsewhere [231–233]. We only summarize the main results here. With particle displacement u, particle velocity u˙ = ∂u/∂t, density ρ, coordinate system x, and stress σ, the equation of motion in the absence of body forces is ∂2u ∂σ =ρ 2, ∂x ∂t

(6.35)

where ρ is the material density (this is a simplified version of (3.24)). This equation can be combined with Hooke’s law, given by σ = Eε = E

∂u , ∂x

(6.36)

with E as Young’s modulus and ε as strain. This leads to a partial differential equation to be solved for u(x, t) c20

∂2u ∂u2 = ∂x2 ∂t2

(6.37)

where c0 is the wave velocity. It can be shown that (6.37) has solutions of the form u = f (x ∓ c0 t) = f (ξ), because 2 ∂2f ∂2u ∂2u 2∂ f = , = c . 0 ∂x2 ∂ξ 2 ∂t2 ∂ξ 2

(6.38)

This solution represents a signal travelling in the positive or negative x-direction. Also, it follows that a stress wave σ=E

∂f = E H(ξ) = E H(x ∓ c0 t) ∂ξ

(6.39)

is moving with the sound velocity c0 , and the particle velocity u˙ = ∓c0

c0 ∂f = ∓c0 H(x ∓ c0 t) = ∓ σ. ∂ξ E

(6.40)

In (6.39) and (6.40), the function H(s) is the unit step function (H(s) = 0 for s < 0, and H(s) = 1 for s ≥ 0). In the model of one-dimensional fracture (as shown in Fig. 6.10), we assume that the left part of the string of atoms (which is free and not attached to the substrate) is loaded with stress σ0 . We assume that the crack front moves at propagation velocity a˙ in the positive


205

x-direction. When the crack front has moved by the length da = dta, ˙ a point which has formerly been situated at the crack tip is displaced backward by du = −εda, because the detached part of the string has attained the axial strain ε. A crack represents a signal constrained to be travelling at a lower velocity than a˙ ≤ c0 . According to (6.40), this corresponds to the particle velocity u˙ = −εa˙ = −

a˙ σt , E

(6.41)

where σt is the local stress to the left to the crack tip. Furthermore, we assume that the stress behind the crack tip can be expressed as the sum of the initial stress σ0 , and an emitted stress wave to the separation, σe , so that σt = σ0 + σe .

(6.42)

The emitted stress wave is related to the particle velocity u˙ =

c0 σe . E

(6.43)

Equations (6.41) through (6.43) can be solved for the three unknowns σt , σe and u. ˙ We define α = a/c ˙ 0 as the ratio of crack propagation velocity to the sound velocity. The particle velocity behind the crack tip is given by u˙ = −

a˙ σ0 , 1+α E

(6.44)

and the local stress wave behind the crack tip carries σt =

1 σ0 . 1+α

(6.45)

The emitted stress wave is σe = −

α σ0 . 1+α

(6.46)

The ratio of local to initial strain is εt /ε0 =

1 , 1+α

(6.47)

where ε0 is the initial strain prior to crack propagation. Also, εt =

1 σ0 . 1+α E

(6.48)

In these equations, the crack speed a˙ remains an unknown. However, we can make use of the energy balance G=W −

dT dφ − = R(α), da da

(6.49)

206


where W is the external work, dT is the increment of kinetic energy, dφ is the increment of potential energy, and R(α) is the dynamic fracture resistance. Balancing kinetic and potential energy using (6.43), (6.44), and (6.46) we arrive at G = G0 g(α) = R(α) with G0 = σ02 /(2E), and g(α) =

1−α . 1+α

(6.50)

We emphasize that the crack driving force vanishes for α → 1, independent of how large we may choose G0 , because g(a) ˙ → 0 in this case, and therefore the sound velocity provides an upper bound for the crack propagation velocity. An energy balance for fracture initiation (a˙ = 0) in the spirit of Griffith’s analysis leads to an expression for crack initiation σ02 = R0 2E

(6.51)

where R0 is fracture surface energy defined as the energy required to break atomic bonds per unit crack advance. Since R(a) ˙ is generally not known, it has to be determined from experiments or numerical calculations. To determine the curve R(α), one may apply a stress σ0 , measure the crack limiting speed α, and calculate the value of R(α) as R(α) =

σ02 1 − α . 2E 1 + α

(6.52)

If this curve is known, the crack equation of motion can be solved completely. A simplification to make the one-dimensional problem solvable in closed form is to assume a constant dynamic fracture toughness, thus R(α) = R0 g(α). This assumption is usually a good approximation for low propagation velocities. For higher velocities close to the crack limiting speed a˙ → c0 , however, it is expected that even though the stress is increased significantly, the crack speed will not change much [22, 231]. Equation (6.52) states that the dynamics of the crack responds immediately to a change in loading or fracture energy, implying that the crack carries no inertia. However, the information about the change in loading or fracture resistance is transmitted with the sound velocity, as indicated by (6.39). When the crack suddenly stops from a high propagation velocity, the local strain immediately changes from the magnitude at high propagation velocity to εt = ε0 (static field solution). This can be verified using (6.46). The crack carries no inertia since the crack immediately responds to a change in the boundary conditions. The crack tip velocity responds instantaneously to a change in fracture energy. We summarize the predictions of the continuum model. A critical loading is necessary to initiate fracture (that is, to break the first bond), similar to


207

Fig. 6.12 Bilinear stress–strain law as a simplistic model of hyperelasticity (mimicking the behavior shown in Fig. 3.10). The parameter εon determines the critical strain where the elastic properties change from local (El ) to global (Eg )

the Griffith condition. While the crack propagates, it sends out a stress wave with a magnitude depending on the crack propagation velocity. For α = 0, no stress wave is emitted and the local stress σt = σ0 . For α → 1, the local stress wave has magnitude σt = σ0 /2. For intermediate values of α, the stress wave magnitude decreases monotonically from σ0 to σ0 /2, as α increases from zero to one. The theory predicts that the largest velocity the crack may achieve is the sound velocity c0 , hence αmax = 1. As higher-dimensional cracks, it is predicted that the 1D crack carries no inertia. 6.4.3 Hyperelastic Continuum Mechanics Model for Bilinear Stress–Strain Law If the stress–strain dependence is not linear as assumed in (6.36), the theory discussed in the last paragraph does not hold. However, the linear theory can be extended employing the concept of local elastic properties and local wave velocities, in the spirit of the work discussed in [27]. It was hypothesized that hyperelastic effects become important in the dynamics of cracks because of the strong deformation gradients in the vicinity of the crack [27, 219]. Within a relatively small region, elastic properties may change drastically due to hyperelastic effects. The term “local” is hereby referred to as the region very close to the crack tip, and “global” refers to regions far away from the crack tip. A bilinear stress–strain law serves as a unique tool to study the nonlinear dynamics of cracks: This model features two Young’s moduli, El associated with small perturbations from the equilibrium position (strain smaller than εon ), and Eg associated with large deformations (strain larger or equal than εon ). The parameter εon allows tuning the strength of the hyperelastic effect. The bilinear stress–strain law is shown schematically in Fig. 6.12. There is a conceptual difference between the higher-dimensional models of fracture and the one-dimensional model of fracture. In the higher-dimensional models of fracture, the zone of large deformation is local to the crack tip, with large deformation gradients. In the one-dimensional model of fracture, the

208


zone of large-deformation is found in regions far away from the crack tip, but the zone of small deformation close to the crack tip is associated with large deformation gradients. Considering the stress field in the vicinity of a moving crack based on the continuum model, this can be verified straightforwardly, since 1/2σ0 ≤ σt ≤ σ0 , and the stress ahead of the crack is zero, while it is σ0 far behind the crack tip. Therefore, if the stress–strain law shows softening with increasing strain, the elastic properties at the crack tip tend to be stiffer than in the far-field. Even though there exists this qualitative difference of the elastic fields near a 1D and a higher-dimensional crack, the dynamics of these systems can be compared immediately if proper interpretation of the features of the deformation fields is done. A stiffening potential in higher-dimensional models tends to yield an elastically stiff zone at the crack tip. In the 1D model, a potential softening with strain is required to provide an elastically stiff zone at the crack tip. Here we focus on the case when El > Eg , which implies that there exists a region close to the crack tip where the material is elastically stiffer than in regions far away. In a string of atoms, the stress σ due to strain ε is given by if ε < εon , El ε (6.53) σ= Eg (ε − εon ) + El εon if ε ≥ εon , where εon is the critical onset strain for hyperelasticity. Therefore, the initial equilibrium strain due to an applied stress σ0 is given by if ε0 < εon , σ0 /El (6.54) ε0 = σ0 /Eg − εon El /Eg + εon if ε0 ≥ εon . In the remainder, we confine the investigations to the choice of El /Eg = 4. Equation (6.54) is then simplified to if ε0 < εon , σ0 /El (6.55) ε0 = 4σ0 /El − 3εon if ε0 ≥ εon . The concept of local and global elastic properties enables us to define two reduced crack speeds αg = v/cg and αl = v/cl . We note that El · αl , (6.56) αg = Eg which yields αg = 2αl in the case considered here. In the following, we derive expressions for the local strain field near the crack tip for a crack moving in a hyperelastic material. We distinguish two cases: • Case 1: The local strain near the crack tip εt , is smaller than the onset strain of the hyperelastic effect, εon . The crack dynamics is governed by


209

Fig. 6.13 Continuum model for local strain near a supersonic crack. The plot shows a schematic of the two cases 1 (subplot (a)) and case 2 (subplot (b))

the local elastic properties in this case, and due to the signal travelling to the left with lower strain, the hyperelastic stiff region expands to the left of the crack (see Fig. 6.13a). • Case 2: The local strain near the crack tip εt is larger than the onset strain of the hyperelastic effect εon . Therefore, the region of hyperelastic material response remains confined to the vicinity of the moving crack tip (see Fig. 6.13b). Case 1: Expanding Region of Local Elastic Properties We assume that the crack advance and material detachment occurs in a region with local elastic properties (associated with El ), as shown schematically in Fig. 6.13a. The local emitted strain in the hyperelastic case is therefore predicted to be εt =

1 σ0 . 1 + αl El

(6.57)

This equation is only valid if εt < εon , that is, the local strain wave lays completely within the zone of local (stiff) elastic properties. An important implication of the assumption is that the limiting speed of the crack is determined by the local elastic wave speed. Since αmax = 1, and αg = 2αl , the crack l can propagate supersonically with respect to the global elastic properties. The

210


ratio of local strain to initial strain is given by combining (6.57) and (6.54) ⎧ ⎪ ⎨

1 1 + αl εt /ε0 = 1/(1 + αl )σ0 /El ⎪ ⎩ σ0 /Eg − εon El /Eg + εon

if ε0 < εon , if ε0 ≥ εon .

(6.58)

The particle velocity u˙ is given by u˙ = −

αl c0,l σ0 . 1 + αl El

(6.59)

Case 2: Local Hyperelastic Region Here we consider the case when εt as given by (6.57) is larger than the onset strain of the hyperelastic effect, that is εt ≥ εon . The emitted strain wave cannot lay within the soft material since crack motion is supersonic with respect to the global soft elastic properties and no signal faster than the sound speed can be transported through the material. Therefore, a shock wave will be induced when the elastic properties change from stiff to soft. The signal of stress relief is transported through the soft material as a secondary wave and represents a wave travelling at cg , the wave speed of the soft material, independent of how fast the crack propagates. In summary, there are two waves propagating behind the crack tip. The first wave features a magnitude (1)

εt

= εon ,

independent of the crack speed. The second wave has magnitude El σ0 /2 (2) εt = − εon + εon + εon Eg Eg

(6.60)

(6.61)

representing a signal travelling in the soft material at the wave speed of the soft material, also independent on the crack speed. The model is schematically summarized in Fig. 6.13b. The particle velocity u˙ is given by 1 σ0 u˙ (2) = − − εon c0,g . (6.62) 2 Eg Summary of the Predictions of the Hyperelastic Continuum Model We summarize the major predictions of the hyperelastic continuum model for a bilinear stress–strain law. We distinguish two cases, case 1 when the local


211

strain near the moving crack is smaller than the onset strain of the hyperelastic effect and case 2 when it is larger. In case 1, the local elastic properties completely govern the dynamics of the crack. As a consequence, the model predicts that the crack can propagate supersonically. The upper limit of the propagation speed is given by the wave speed associated with the local elastic properties. In case 2, detachment of the material occurs completely in the hyperelastic region and remains confined during crack growth. In this case, two waves with (1) (2) magnitude εt and εt are moving behind the crack tip, one is a shock front associated with the change in elastic properties and the other represents a signal travelling in the elastically soft material carrying the stress relief due to crack propagation at the wave speed of the soft material. As the size of the hyperelastic zone shrinks with decreasing εon , the limiting crack speed is also expected to decrease and approach the limiting value of αg → 1 for εon → a0 . This is because the material detachment eventually occurs completely within the zone of soft elastic properties. On the other hand, if εon is chosen larger, the stiff zone expands and eventually the situation corresponding to case 1 is attained when εt < εon and the dynamics is completely governed by the local elastic properties. In any case, when the crack propagates supersonically, a dramatic reduction in the ratio εt /ε0 is possible due to the local stiffening effect. 6.4.4 Molecular Dynamics Simulations of the One-Dimensional Crack Model: The Harmonic Case According to Fig. 6.11, the atoms are numbered from left to right with increasing index, with a total number of atoms Nt . We assume that atoms with index i > Nf are attached to the substrate, and atoms with i ≤ Nf are free and only interact with other nearest neighbor atoms. The state of an atom i is uniquely defined by a position xi and its velocity x˙ i . The mass of each particle is m = 1. Only nearest neighbor interaction is considered. The systems contain up to 20,000 atoms, which equals a string of atoms of length of about 20 µm in physical dimensions. To study one-dimensional fracture, we have developed a specific molecular dynamics code optimized for one-dimensional analyses. The basis for the atomic interactions is the Lennard-Jones interatomic potential defined in (2.30). We express all quantities in reduced units, so lengths are scaled by the LJ parameter σ which is assumed to be unity in this study, and energies are scaled by the parameter 0 = 1/2, the depth of the minimum of the LJ potential. The reduced temperature is kB T /0 with kB being the Boltzmann constant. To study a harmonic system, we expand the LJ potential around its equilibrium position a0 = 21/6 ≈ 1.12246, and consider only first-order terms yielding harmonic atomic interactions. The mass of each atoms in the models is assumed to be unity, relative to the reference mass m∗ . The reference time unit is then given by t∗ = mσ 2 /. For

212


example, when choosing electron volt as reference energy ( = 1 eV), Angstrom A), and the atomic mass unit as reference mass as reference length (l∗ = 1 ˚ (m∗ = 1 amu), the reference time unit corresponds to t∗ = 1.0181 × 10−14 s. In the simulation procedure, we distinguish an equilibration phase and a fracture simulation phase. In the equilibration phase, we initialize the free part of the bar with a prescribed homogeneous strain, given by ε0 = σ0 /E and let the system equilibrate for a longer time. During that time, we introduce a viscous damping force fd,i = −u˙ i η into the system with η = 0.3 to damp out waves generated during equilibration, so that the particle velocities (and strain gradients) are damped out relatively fast. During equilibration, atomic bonds glued to the substrate can never break, and the total energy of the system is given by 1 1 Utot = k(rij − a0 )2 + H(i − Nf )kp rî2 (6.63) 2 2 i,j i where k is the spring constant for interatomic interaction, and kp is the spring constant of the pinning potential. The variable rî =| x0,i − xi |,

(6.64)

and the variable xi is the current position of the atom i. The variable x0,i stands for the initial position of atom i. We integrate the equations of motion using a velocity verlet algorithm, and choose a time step ∆t = 0.000, 036 in reduced atomic units of σ m/. When all strain is equilibrated in the free standing part of the string, we begin the fracture simulation phase where the bonds to substrate have finite energy. The total energy of the system is then given by 1 1 2 2 Utot = k(rij − a0 ) + H(i − Nf )H(rbreak − rî )kp rî (6.65) 2 2 i,j i where rbreak is the snapping bond distance for the pinning potential. The fracture energy R0 in (6.51) is given by R0 =

1 kp rˆ2 . 2 a0

(6.66)

Assuming a stress–strain law as given by (6.36), we define a Young’s modulus for a one-dimensional string of atoms [114] E = k a0 .

(6.67)


The wave velocity in a string of atoms is given by E c0 = ρ

213

(6.68)

with density ρ = m/a0 for the present one-dimensional lattice. For k = 28.5732, Young’s modulus E = 32.07, and c0 ≈ 6. The elastic properties are determined numerically as a check if the assumptions are valid. We define an atomic strain of atom i which is directly related to the continuum mechanics concept of strain [112], considering only nearest neighbors in a one-dimensional system εi =

xi−1 − xi+1 . 2 a0

(6.69)

In the remainder of this chapter, we preferably use the atomic strain to analyze the simulation results, since it provides a useful way to study the state of deformation in the atomic lattice. We start with a comparison of the theoretical prediction of the elastic properties of the one-dimensional string of atoms with atomistic simulations. The numerically estimated elastic properties agree well with the theory. The measurements of applied stress σ0 vs. strain, and the numerically estimated local modulus in the string of atoms match the theoretical predictions given by (6.67) well. Additional studies of wave propagation velocity show good agreement of the predicted wave velocity with the measured wave velocity. Simulations with other spring constants and consequently other wave velocities provide evidence that the agreement of theory and simulation is generally good. Griffith criterion predicts that fracture initiates when the elastic energy released per unit crack advance equals the energy to create free surface per unit crack advance. The fracture energy is given by (6.66). Setting this quantity equal to the energy release rate allows to determine the critical load to initiate fracture. The computational results are compared to the theory prediction in Table 6.2. A result of the simulations is that the results converge to the theory prediction as kp becomes much smaller than k, but we find larger disagreement with the theory prediction if kp is large. This could be due to the fact that the fracture process zone becomes very small when kp is large, leading to very large strain gradient at the crack. Equation (6.46) predicts that the local stress wave depends on the crack propagation velocity. Figure 6.14 plots the magnitude of the local stress wave for different propagation velocities from atomistic simulations, in comparison with the theory prediction. The dynamic fracture toughness is a function of α and σ0 , and is given by (6.52). Atomistic simulations provide an ideal tool to provide information on this curve. Figure 6.15 plots the dynamic fracture toughness for different crack propagation velocities. As can be verified, the assumption that R0 = const. is reasonable as long as the crack velocity is below 80% of the wave velocity. For

214

Atomistic Modeling of Materials Failure Ratio kp /k Predicted initiation load R0pred Measured initiation load R0 10.0 1.0 0.1 0.01 0.003

0.00014 × 10−4 0.0015 0.0030 0.0040 0.0040

0.0039 0.0039 0.0039 0.0039 0.0039

Table 6.2 Critical load R0 for fracture initiation, for different values of the spring constant kp of the pinning potential. The results are in good agreement with the theory prediction when kp becomes much smaller than k

Fig. 6.14 Magnitude of the local stress wave for different crack propagation velocities from atomistic simulations, in comparison with the theory prediction

larger velocities, the curve deviates significantly from a constant and increases dramatically. This behavior is expected from theory [232] (and also for higher dimensions, as discussed, for instance, in [22]). If α < 1, the crack front propagates slower than the local wave front behind the crack. If the material left to the crack is of finite length, the reflected wave from the left end will eventually hit the crack tip at a time δt =

2L + ∆a , c0

(6.70)

where L denotes the initial free length of the bar, and ∆a is the distance the crack has travelled until it is hit. Once the reflected wave front impinges the crack, the stress will suddenly increase causing a jump in crack propagation speed [232]. In the atomistic simulations, we observe this effect, but note that the crack does not reach a steady-state as predicted by the theory. Instead, the


215

Fig. 6.15 Dynamic fracture toughness for different crack propagation velocities

crack speed seemed to decrease continuously, much below the value predicted by the theory. During this process, the temperature in the system increased continuously and energy seems to be dissipated into heat (“thermalization” process). To investigate the dynamics of a suddenly stopping crack, we let the crack propagate at a high velocity α ≈ 0.9, and then force the crack to stop. This is achieved by setting rˆ to a large number rˆ∞ rˆ for all atoms with identification number greater than istop > Nf . This forces the crack to suddenly stop once the crack tip reaches the atom with index equal istop : rˆ0 if i < istop , (6.71) rˆ(i) = rˆ∞ if i > istop . The simulation results illustrate that the theory prediction is satisfied, and the local strain immediately attains the magnitude ε0 as soon as the crack is stopped. The static field spreads out with the wave velocity. The results are plotted in Fig. 6.16. The discussion of the suddenly stopping one-dimensional crack proves that a one-dimensional crack carries no inertia. According to this observation, the crack tip velocity should immediately respond to a change in the fracture energy. For example, if the crack senses a higher fracture surface energy, the velocity should instantaneously decrease, and if the crack senses a lower fracture surface energy, vice versa. We test this statement by introducing a periodically varying fracture surface energy as the crack propagates along x. The velocity should change in antiphase with the change of fracture surface

216


Fig. 6.16 Strain field near a suddenly stopping one-dimensional crack. The crack is forced to stop at x ≈ 790. As soon as the crack stops (at x = 550), the strain field of the static solution is spread out with the wave speed

energy. A variation in fracture energy is achieved by varying the bond snapping distance rˆ of the pinning potential (see (6.66)) according to r sin(x/p), rˆ = rˆ0 + ∆ˆ

(6.72)

where rˆ0 is the value around the snapping distance. The bond snapping distance oscillates with amplitude ∆ˆ r and period factor p. In Figs. 6.17 and 6.18, results are plotted for rˆ0 = 0.008, ∆ˆ r = 0.003, and p = 30. The velocity oscillates around α ≈ 0.6, which is in agreement with the velocity of a crack under loading σ0 = 0.02 and a fracture toughness of rˆ = 0.008. The same observation applies to the upper and lower limit of the propagation velocity, which correspond to the limiting velocity of the crack if it would be propagating along a path with constant fracture energy of the corresponding magnitude. Therefore,


217

Fig. 6.17 Prescribed fracture toughness and measured crack velocity as the crack proceeds along x

v = vˆ0 + ∆v sin(x/p),

(6.73)

where v0 is the velocity associated with rˆ0 , and ∆v ≈ 1.3 can be approximated r. by the difference of the propagation velocity associated with rˆ0 + ∆ˆ 6.4.5 Molecular Dynamics Simulations of the One-Dimensional Crack Model: The Supersonic Case This section is dedicated to molecular dynamics simulations of supersonic cracks. To achieve a bilinear stress–strain law according to (6.53), the total potential energy of the nonlinear system is given by 1 1 2 2 k(rij − a0 ) + βk H(r − ron )(rij − ron ) Utot = 2 2 i,j (6.74) 1 2 + H(i − Nf )H(rbreak − rî )kp rî , 2 i where ron is a potential parameter allowing for different onset points of the hyperelastic effect (thus controlling the strength of the hyperelastic effect), and εon =

ron − a0 . a0

(6.75)

The choice of β allows for different types of nonlinearities. If −1 < β ≤ 0, the potential softens with strain, and if β = 0, the model reduces to harmonic interactions. The small-perturbation spring constant is always given by k0 = k, and the large-strain spring constant is given by k1 = (1 + β)k. Elastic properties for β = −3/4 are shown in Fig. 6.19, which plots the atomic stress σ vs. atomic separation and the tangent modulus E. The local sound velocity

218


Fig. 6.18 Strain field of a crack travelling in a material with periodically varying fracture toughness

c0,l is readily obtained from E. The figure shows that the tangent modulus softens with strain. We reiterate that if the stress–strain law softens with strain, the elastic properties at the crack tip are stiffer than in the far-field. The simulation procedure when using the bilinear stress–strain law is identical to the previously described procedure. However, the dynamics of the crack with the bilinear stress–strain law is significantly different from the harmonic case. It is observed that the crack can propagate supersonically with respect to the global elastic properties. Figure 6.20a plots the limiting velocity of the crack for different values of the potential parameter ron . For large values of ron , the local hyperelastic zone becomes larger and the limiting velocity approaches Mach 2, or αg ≈ 2. For ron → a0 , the hyperelastic zone shrinks and the velocity of the crack approaches αg ≈ 1. This plot proves that the limiting velocity of the crack is very sensitive with respect to the potential parameter ron . A small change in ron affects the extension of the hyperelastic area and has impact on the limiting velocity. The simulation results prove that supersonic crack propagation is possible even if the hyperelastic zone is very small.


219

Fig. 6.19 Elastic properties associated with the biharmonic interatomic potential, for ron = 1.125 and Eg = 8 = 1/4El

When the crack is propagating at αg > 1, the local strain wave has a magnitude of less than 50% of the equilibrium strain. This is in disagreement with the classical theory stating that the local strain wave is always equal or larger than 50% of the equilibrium strain for a crack propagating at the limiting speed (sound velocity). However, these observations can be explained well by the new continuum model proposed based on the concept of local elastic properties. Figure 6.20b plots a comparison of the continuum model with molecular dynamics simulation results of supersonic crack propagation. The agreement is reasonable. The regimes where case 1 and case 2 are valid is also indicated. Figure 6.21 depicts the strain field in the vicinity of a supersonic crack for ron = 1.124. Figure 6.22 depicts the particle velocity field near the moving supersonic crack. 6.4.6 Discussion and Conclusions We have used a simple one-dimensional model of dynamic fracture to investigate fundamentals of the nonlinear dynamics of fracture. On the basis of the continuum model of one-dimensional dynamic fracture, we have proposed an atomistic model of a string of atoms. We have verified that the continuum model of one-dimensional dynamic fracture can be successfully applied at the atomistic level, if harmonic interactions are assumed between atoms. It was shown that the one-dimensional crack carries no inertia, a phenomenon that is also found in higher dimensions [23,208,234]. The fact that we find good agreement of the one-dimensional atomistic model featuring harmonic interactions

220


Fig. 6.20 Subplot (a) Velocity of the crack for different values of the potential parameter ron . The larger ron , the larger the stiff area around the crack tip. As the hyperelastic area becomes sufficiently large, the crack speed approaches the local wave speed αl = 1 corresponding to αg = 2. Subplot (b) shows a quantitative comparison between theory and computation of the strain field near a supersonic crack as a function of the potential parameter ron . The different regimes corresponding to case 1 and case 2 are indicated. The loading is chosen σ0 = 0.1, with kp /k = 0.1 and rˆ = 0.001

with the continuum theory corresponds to work on the comparison of the atomistic level with continuum theory [165] for mode II cracks. Finding analytical solutions for dynamic fracture in hyperelastic materials in higher dimensions is very difficult, if not impossible in many cases. However, analytical understanding of the nonlinear dynamics becomes possible based on the simple one-dimensional model. We have proposed a continuum model based on the local elastic properties to predict the elastic fields around the crack tip, when a bilinear stress–strain law is assumed. The major prediction of the continuum model is supersonic crack propagation, if there exists a local elastically stiff region confined to the crack tip. By molecular dynamics simulations, it was shown that the local elastic properties at the crack tip indeed govern the dynamics of fracture, in agreement with the predictions of the model. If there is an elastically stiff zone close to the crack tip, the crack can propagate supersonically through the material. We emphasize that this


221

Fig. 6.21 Sequence of strain field near a rapidly propagating supersonic 1D crack moving with Mach 1.85 for ron = 1.124. The primary (1) and secondary wave (2) are indicated in the plot. The wave front (1) propagates supersonically through the material

is true even if the hyperelastic region is highly confined to the crack tip. The observation of supersonic crack motion has been found by other researchers as well in 2D and 3D studies [219]. The finding that the dynamics of the crack is governed by the local elastic properties (the local wave speed) has been predicted theoretically [27] and observed previously [155, 219]. The case of stiffening material response corresponds to materials such as polymers, showing a hyperelastic stiffening effect. Due to the large deformation in the vicinity of the crack, the elastic properties in such materials are stiffer close to the crack than in regions far away from the crack. Laboratory experiments of dynamic fracture in such materials could provide further insight into the nature of hyperelastic stiffening dynamic fracture and associated supersonic crack propagation. In this section, we have concentrated on the case when local elastic properties are stiffer than in the far-field elastic properties, crack propagation is supersonic. In the same sense, if the local elastic properties are softer, crack propagation must be subsonic on a local scale. We have also performed similar

222


Fig. 6.22 Particle velocity field near a supersonic crack, comparison between theory and simulation

one-dimensional molecular dynamics simulations as reviewed here, and found similar results. This study illustrates the potential of an atomistic appraoch in studying brittle fracture. According to conventional continuum-type theories of fracture, it has been widely believed that there is a unique definition of how fast waves propagate in solids. Our results prove that this concept cannot capture all phenomena in dynamic fracture, and instead should be replaced by the concept of local wave speeds. In materials where the large-strain elasticity differs significantly from the small-strain elasticity, the concept of global wave velocities cannot be used any more to describe the dynamics of the crack. Instead, the concept of local elastic properties, and associated local wave velocities govern the dynamics of the crack. Since “real” materials all show strong nonlinear effects, this suggests that hyperelasticity is crucial for dynamic fracture. The one-dimensional model has found useful applications in addressing other fundamental questions of mechanics of materials. A potential application is strain gradient effects in elasticity, and its possible implications on dynamic fracture. The mechanics of one-dimensional structures could also be important in the emerging bio-nanotechnologies, often involving functionalization of single molecules. An objective of future studies could be the development of experimental techniques based on the one-dimensional model of fracture.


223

6.5 Stress and Deformation Field near Rapidly Propagating Mode I Cracks in a Harmonic Lattice In this section, we review large-scale atomistic simulations to study the nearcrack elastic fields, focused on mode I dynamic fracture from both atomistic and continuum mechanics viewpoints. The static solution was discussed in Sect. 6.2.2. Here we review the more general solution that also includes the case of moving cracks. In the continuum theory, the stress field in the vicinity of the crack tip is given by an asymptotic solution [22,210,211]. With KI (t, v) as the dynamic stress intensity factor, KI (t, v) (1) σij (Θ, v) = √ Σij (Θ, v) + σij + O(1). 2πr

(6.76)

The functions Σij (Θ, v) represent the variation of stress components with angle Θ for any value of crack speed v [22]. These functions only depend on the ratio of crack speed to wave speeds. The functions Σij (Θ, v) that appear in (6.76) are defined as follows [22]. . / 1 cos(1/2Θl ) cos(1/2Θs ) Σxx (Θ, v) = (1 + α2s )(1 + 2α2l − α2s ) , − 4α α √ √ s l D γl γs (6.77) . / 2αl (1 + αs )2 sin(1/2Θl ) sin(1/2Θs ) , (6.78) Σxy (Θ, v) = − √ √ D γl γs and 1 Σyy (Θ, v) = − D

. / cos(1/2Θs ) 2 2 cos(1/2Θl ) (1 + αs ) . − 4αd αs √ √ γl γs

Further, γl =

1 − (v sin(Θl /cl )2 ),

(6.79)

(6.80)

tan(Θl ) = αl tan Θ, γs = 1 − (v sin(Θs /cs )2 ),

(6.81)

tan(Θs ) = αs tan Θ.

(6.83)

(6.82)

and The two factors αs and αl are defined as αs = 1 − v 2 /c2s

(6.84)

' 1 − v 2 /c2l .

(6.85)

and αl =

The asymptotic stress field in the vicinity of a dynamic crack depends only on the ratio of crack speed to the wave velocities in the solid. Similar expressions for the asymptotic field have also been derived for mode II cracks [22].

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The asymptotic field strongly depends on the crack velocity, and has universal character because it is independent of the details of applied loading. (1) The values of σij and the first-order contribution O(1) are determined from the boundary conditions, and neglected in the remainder of this work since the first term dominates very close to the crack tip. In the following sections, we review a systematic comparison of atomistic simulations and linear elastic continuum theory of the stress and deformation field near rapidly propagating cracks. Harmonic interatomic potentials are used to model a linear elastic plane-stress sheet. To compare the results for different crack velocities, we report atomistic simulations with different loading rates driving the crack to different terminal velocities. Figure 6.23 shows the slab geometry used in the simulations. The slab size is given by lx and ly . The crack propagates in the y-direction, and its extension is denoted by a. The crack propagates in a triangular hexagonal lattice with nearest neighbor distance along the crystal orientation shown in Fig. 6.23. A weak fracture layer is introduced to avoid crack branching by assuming harmonic bonding in the bulk but an LJ potential across the weak layer (see also [165]).

Fig. 6.23 Simulation geometry and coordinate system for studies of rapidly propagating mode I cracks in harmonic lattices

All simulations presented here are two dimensional. Previous studies have provided evidence that 2D molecular dynamics is a good framework to investigate the dynamics of fracture [165, 219]. This is because the atomistic simulations of a two-dimensional solid and a three-dimensional solid show no difference in the details of the dynamics of the crack. The 2D model captures important features of dynamic fracture such as surface roughening and crack tip instabilities [146, 155]. The outline of this section is as follows. It will be shown that in molecular dynamics simulations of cracks traveling in perfect harmonic lattices the


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prediction of stress and strain fields by continuum mechanics is reproduced quantitatively. An important observation is that the hoop stress becomes bimodal at about 73% of Rayleigh speed, in agreement with the continuum theory. In addition, we report comparison of continuum theory with molecular dynamics simulation of the strain energy field near the crack tip as well as the energy transport field near rapidly moving cracks. 6.5.1 Stress and Deformation Fields In this section, we compare stress and deformation field near a rapidly moving crack tip with continuum mechanics theories. Angular Variation of Stress We analyze the angular variation of the principal stress and hoop stress close to the crack tip and compare the results of the simulation to the continuum mechanics solution given by (6.76). Atomic quantities are evaluated in a small region around a constant radius of r ≈ 11 centered at the crack tip. The continuum theory solution and the simulation results are both normalized with respect to the dynamic stress intensity factor. We find that if the stress field measurements are taken while the crack accelerates too rapidly, the agreement of measured field and continuum theory prediction can be poor. Acceleration effects can severely change the resulting stress fields. Although the crack tip is regarded as inertia-less since it responds immediately to a change in loading or fracture surface energy, it takes time until the elastic fields corresponding to a specific crack speed spread out! In fact, the fields spread out with the Rayleigh velocity behind, and with the shear wave velocity ahead of the crack. In other regions around the crack tip, the fields are reached in the long-time limit (t → ∞) [22, 23]. Therefore, we choose a moderate strain rate ε˙xx = 0.000 01. As a consequence of the relatively low strain rate and the finite slab size, the crack only achieves about 87% of Rayleigh wave speed. We calculate the stress for different crack speeds ranging from 0 to 87% of the Rayleigh speed. Figures 6.24–6.26 show the angular variation of σxx , σyy as well as σxy . Figure 6.27 shows the angular variation of the hoop stress σΘ . Figure 6.28 shows the angular variation of the maximum principal stress σ1 near the crack tip. In all figures, the continuous line is the corresponding analytical continuum mechanics solution [22]. It can be observed from the plots that the hoop stress becomes bimodal at a velocity of about 73% of the Rayleigh wave velocity. This is in agreement with the predictions by continuum mechanics theories [22]. Elastic Fields near the Crack Tip Here we use a higher strain rate of ε˙xx = 0.0005 to drive the crack close to the Rayleigh wave speed.

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Fig. 6.24 Comparison between σxx from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR

Fig. 6.25 Comparison between σyy from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR

The principal strain field is shown in Fig. 6.29 for different velocities of v/cR ≈ 0, v/cR ≈ 0.5, and v/cR ≈ 1. The upper plot is the simulation result, while the lower part is the prediction by continuum mechanics. We note that the principal stress field is in good agreement with the continuum theory. The


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Fig. 6.26 Comparison between σxy from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR

typical trimodal structure of the asymptotic principal strain and principal stress field develops close to the Rayleigh velocity, in contrast to the bimodal structure at low crack speeds. The stress fields σxx , σyy , and σxy for a crack propagating close to the Rayleigh wave velocity are shown in Fig. 6.30(a–c). As before, the upper plot is the simulation result, while the lower part plots the prediction by continuum mechanics. Finally, Fig. 6.29 plots the particle velocity near the crack tip for a crack propagating close to the Rayleigh velocity. Figure 6.31a shows u˙ x , and Fig. 6.31b shows u˙ y . The continuum theory prediction and the atomistic simulation result match well. The particle velocity field behind the crack tip is found to be smeared out more in the simulation results due to thermalization effects not accounted for in the continuum theory. 6.5.2 Energy Flow near the Crack Tip Here we discuss the energy flow near a crack tip in molecular dynamics simulations compared with the continuum theory [22]. A similar study has been reported in [235]. In contrast to the treatment of the dynamic Poynting vector for steady-state cracks at high velocities in analogy to the discussion in [22], the authors in [235] only considered the static Poynting vector to study the energy radiation of rapidly moving cracks. The dynamic Poynting vector for a crack moving at velocity v in the y-direction can be expressed as

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Fig. 6.27 Comparison between hoop stress from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR

Fig. 6.28 Comparison between the maximum principal stress σ1 from molecular dynamics simulation with harmonic potential and the prediction of the continuum mechanics theory for different reduced crack speeds v/cR

Pj = σij u˙ i + (U + T )v δ2j ,

(6.86)

where δij is the Kronecker delta function. The kinetic energy is given by T = 1 ˙ i u˙ i , and the strain energy density for an isotropic medium is given by [236] 2 ρu


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Fig. 6.29 Principal strain field at various crack velocities (a) v/cR ≈ 0, (b) v/cR ≈ 0.5, (c) v/cR ≈ 1. In each of the plots (a)–(c), the upper plot is the simulation result and the lower part is the prediction by continuum mechanics

Ψ=

1 , 2 2 2 σ11 + σ22 . − 2νσ11 σ22 + 2(1 + ν)σ12 2E

(6.87)

The magnitude of the dynamic Poynting vector is calculated as P = P12 + P22 , and can be identified as a measure for the local energy flow. Figure 6.32a shows the strain energy field near the crack tip predicted by both the continuum theory prediction and the molecular dynamics simulation result. Figure 6.32b shows the magnitude of the dynamic Poynting vector field. Figure 6.33 shows in panel (a) the continuum mechanics prediction, and in panel (b) the molecular dynamics simulation result of the dynamic Poynting vector field in the vicinity of the crack tip, for a crack propagating close to the Rayleigh speed. 6.5.3 Limiting Velocities of Mode I Cracks in Harmonic Lattices We also study the dependence of crack dynamics in harmonic materials with different spring constants. Linear elastic fracture mechanics predicts that the limiting crack speed for mode I cracks should only depend on the elastic properties, and therefore, in case of harmonic potentials, on the spring constant k.

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Fig. 6.30 Stress fields close to the crack tip for a crack propagating close to the Rayleigh velocity v/cR ≈ 1. Plots (a), (b), and (c) show σxx , σyy , and σxy . In each of the plots (a)–(c), the upper plot is the simulation result and the lower part is the prediction by continuum mechanics

For mode I cracks considered in this section, the limiting speed is given by the Rayleigh wave speed [22]. Figure 6.34 shows the crack tip position history a(t) as well as the crack speed history a(t) ˙ for a soft as well as a stiff harmonic material, for a mode I crack. The results are in consistency with the predicted limiting speed (see data in Table 4.1). Similar studies have been carried out for mode II cracks, as is discussed in [237]. Additional results for mode III cracks will be discussed in Sect. 6.10. 6.5.4 Summary Simulations of cracks propagating along a confined fracture path in a harmonic lattice show that continuum mechanics theory of fracture can be successfully applied even at the atomistic level. We compared the virial stress and strain from atomistic simulation results with the continuum mechanics solution of the asymptotic field for different propagation velocities.


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Fig. 6.31 Particle velocity field close to the crack tip for a crack propagating close to the Rayleigh velocity, v/cR ≈ 1. Plots (a) shows u˙ x and plot (b) shows u˙ y . In each of the plots (a) and (b), the upper plot is the simulation result and the lower part is the prediction by continuum mechanics

Fig. 6.32 Potential energy field and magnitude of the dynamic Poynting vector. (a) Potential energy field near a crack close to the Rayleigh speed. (b) Energy flow near a rapidly propagating crack. This plot shows the magnitude of the dynamic Poynting vector in the vicinity of a crack propagating at a velocity close to the Rayleigh speed

The results suggest that the agreement of molecular dynamics simulations and continuum mechanics is generally good, as it is shown for the stress tensor components σxx , σyy , and σxy . It is observed that there is some disagreement at larger angles Θ > 150◦ , perhaps due to surface effects in the atomistic simulations. In Fig. 6.24 it is observed that for σxx , the shape of σxx (Θ) is qualitatively reproduced well over the entire velocity regime between 0 and 87% of Rayleigh speed. However, the angles of the local maxima

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Fig. 6.33 Energy flow near a rapidly propagating crack. This plot shows (a) the continuum mechanics prediction, and (b) the molecular dynamics simulation result of the dynamic Poynting vector field in the vicinity of the crack tip, for a crack propagating close to the Rayleigh wave speed

Fig. 6.34 Crack tip history as well as the crack speed history for a soft as well as a stiff harmonic material (two different choices of spring constants as given in Table 4.1)

and minima are shifted to slightly smaller values compared to the theory prediction. Figure 6.25 illustrates that the shift of the maximum in the σyy (Θ) curve from about 60◦ to about 80◦ is reproduced only qualitatively. For low velocities the maximum is found at lower angles around 40◦ , but it approaches the value of the continuum theory at higher velocities. At 87% of Rayleigh speed, the difference is only a few degrees. The shear stress σxy shown in Fig. 6.26 also agrees qualitatively with the continuum theory. As in the previous cases, the angles of local minima and maxima are shifted to lower values in the


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simulation, but the agreement gets better when the crack velocity is faster. Even though we see small deviations in σxx , σyy , and σxy , the hoop stress σΘ agrees quantitatively with the continuum theory as shown in Fig. 6.27. The angles of the maxima and minima during crack acceleration compare well with theory. However, the angles of the maxima and minima of the maximum principle stress shown in Fig. 6.28 are also shifted to slightly lower values. However, it is observed that two local maxima and one local minima develop at a velocity of about 73% in quantitative agreement with continuum theory (“trimodal structure”). The magnitude of the local maxima and minima also agree quantitatively. The analysis of the potential energy field near a crack close to the Rayleigh speed agrees qualitatively with the prediction by the continuum mechanics theory. As Fig. 6.32a shows, in both theory and computation the field clearly shows three local maxima with respect to the angular variation (“trimodal structure”), similar to the principal stress field. At larger distances away from the crack tip, it is observed that other stress terms begin to dominate in the simulation, so the distribution of the potential energy deviates from the prediction by theory. As is expected since only the first term of (6.76) is considered, these contributions are missing in the continuum solution. Similar observations also hold for the magnitude of the dynamic Poynting vector, as it can be verified in Fig. 6.32b. The dynamic Poynting vector field calculated by molecular dynamics is also in reasonable agreement with the continuum mechanics prediction. This could be verified in Fig. 6.33. In both theory (a) and molecular dynamics calculation (b), the orientation of the dynamic Poynting vector is dominated by the direction opposite to crack motion. The vector field seems to bow out around the crack tip, an effect that is more pronounced in the simulation than predicted by theory. Also, the flow ahead of the crack is larger in simulation than predicted by theory. At the free surface of the crack, the measurement from the simulation and the prediction by theory show differences. This could be based on the fact that the continuum theory does not treat surface effects properly, in particular short-wave length Rayleigh waves (see also discussion in [235]). The calculation of the virial stress as shown here does not include the particle velocity contribution, following the suggestion put forward in [238] on the linkage between virial stress and Cauchy stress of continuum mechanics. Finally, it is noted that the virial expression of the stress tensor is classically thought to be only valid under equilibrium conditions [239]. The results reviewed in this section show that it is approximately valid even under dynamic fracture conditions which are far from equilibrium.

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Fig. 6.35 The concept of hyperelasticity in dynamic fracture. Subplot (a) shows the region of large deformation near a moving crack, due to the nonlinear elastic behavior of solids (subplot (b)). The linear elastic approximation is only valid for small deformation. Close to crack tips, material deformation is extremely large, leading to significant changes of local elasticity, referred to as “hyperelasticity” (see also Fig. 3.10 and related discussion)

6.6 Crack Limiting Speeds of Cracks: The Significance of Hyperelasticity The elasticity of a solid clearly depends on its state of deformation. Metals will weaken or soften, and polymers may stiffen as the strain approaches the state of materials failure. It is only for infinitesimal deformation that the elastic moduli can be considered constant and the elasticity of the solid linear. However, many existing theories model fracture using linear elasticity. Certainly, this can be considered questionable since material is failing at the tip of a dynamic crack because of the extreme deformation, as illustrated in Fig. 6.35. We review studies that show by large-scale atomistic simulations that hyperelasticity, the elasticity of large strains, can play a governing role in the dynamics of fracture and that linear theory is incapable of capturing all phenomena. We introduce the concept of a characteristic length scale for the energy flux near the crack tip and demonstrate that the local hyperelastic wave speed governs the crack speed when the hyperelastic zone approaches this energy length scale. Large-scale atomistic simulation studies reviewed here show that hyperelasticity, the elasticity of large strains, can play a governing role in the dynamics of brittle fracture. This is in contrast to many existing theories of dynamic fracture where the linear elastic behavior of solids is assumed sufficient to predict materials failure [22, 61, 240]. Some experimental work [241, 242] as well as many computer simulations [146, 155, 205] have shown a significantly reduced crack propagation speed in comparison with the predictions by the theory. In contrast, other experiments indicated that over 90% of the Rayleigh wave speed can be


235

achieved [25, 224, 243–247]. Such discrepancies between theories, experiment, and simulations cannot always be attributed to the fact that real solids feature a variety of imperfections such as grain boundaries and microcracks (either preexisting or created during the crack propagation), as similar discrepancies also appear in molecular dynamics simulations of cracks traveling in perfect atomic lattices. Earlier studies have independently led to the conclusion that hyperelastic effects at the crack tip may play an important role in the dynamics of fracture [27, 146, 155, 248]. Their suggestions have been used to help explaining phenomena related to crack branching and dynamic crack tip instability, as well as explaining the significantly lower maximum crack propagation speed observed in some experiments and many computer simulations. However, it is not generally accepted that hyperelasticity should play a significant role in dynamic fracture. One reason for this belief stems from the fact that the zone of large deformation in a loaded body with a crack is highly confined to the crack tip, so that the region where linear elastic theory does not hold is extremely small compared to the extensions of the specimen [22, 61]. In this study, we use large-scale molecular dynamics simulations [219] in conjunction with continuum mechanics concepts [22, 61] to prove that hyperelasticity can be crucial for understanding dynamic fracture. The study reviewed here shows that local hyperelasticity around the crack tip can significantly influence the limiting speed of cracks by enhancing or reducing local energy flow. This is true even if the zone of hyperelasticity is small compared to the specimen dimensions. The hyperelastic theory completely changes the concept of the maximum crack velocity in the classical theories. For example, the classical theories clearly predict that mode I cracks are limited by Rayleigh wave speed and mode II cracks are limited by longitudinal wave speed. In contrast, both super-Rayleigh mode I and supersonic mode II cracks are allowed by hyperelasticity and have been seen in computer simulations [156, 219]. In the simulations, it is found that there exists a characteristic length scale associated with energy flow near the crack tip such that hyperelasticity completely dominates crack dynamics if the size of hyperelastic region approaches this characteristic length. In earlier simulations [156, 219], a nonlinear interatomic stiffening was assumed, and there was no sharp distinction between the linear and nonlinear elastic regimes for the stretched solid. In contrast, the model is based on a biharmonic potential composed of two spring constants, one associated with small deformations and the other with large deformations (see discussion in Sect. 4.4.3). This serves as a simplistic model material for hyperelasticity, allowing us to investigate the generic features of hyperelasticity common to a large class of real materials.

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6.6.1 Modeling We consider propagation of a crack in a two-dimensional simulation geometry shown in Fig. 6.23. The slab size is given by lx and ly . The crack propagates in the y-direction, and its extension is denoted by a. The crack propagates in a triangular hexagonal lattice with nearest neighbor distance along the crystal orientation shown in Fig. 6.23. To avoid crack branching, a weak fracture layer is introduced by assuming that atomic bonds across the prospective crack path snap at a critical atomic snapping distance rbreak while those in the rest of the slab never break. As outlined in Sect. 4.4.3, the snapping distance can be used to adjust the fracture surface energy 2γ.

Fig. 6.36 This figure shows a continuously increasing hyperelastic stiffening effect, as observed by measuring the elastic properties of a material (subplot (a)). The increasingly strong hyperelastic effect is modeled by using biharmonic potentials, thereby capturing the essential physics: A small-strain spring constant k0 and a large-strain spring constant k1 (subplot (b)), where the ratio of the two is defined as kratio = k1 /k0 . The bilinear or biharmonic model allows to tune the size of the hyperelastic region near a moving crack, as indicated in subplots (c) and (d). The local increase of elastic modulus and thus wave speeds can be tuned by changing the slope of the large-strain stress–strain curve (“local modulus”)

For a systematic study of hyperelastic effects in dynamic fracture, we adopt the biharmonic potential defined in (6.53). This potential is composed of two spring constants k0 and k1 . Here we consider two “model materials,” one with elastic stiffening and the other with elastic softening behavior. In the elastic


237

stiffening system, the spring constant k0 is associated with small perturbations from the equilibrium distance r0 , and the second spring constant k1 is associated with large bond stretching for r > ron . The role of√k0 and k1 is reversed in the elastic softening system (k0 = 2k1 and k1 = 36/ 3 2). The elastic properties associated with this potential are shown in Fig. 4.11. Figure 6.36 shows how the biharmonic potential is used to model a continuous stiffening effect, and how this model can be used to control the size of the hyperelastic region near the crack. A similar approach is used to model hyperelatic softening material behavior. To strain the system, we use two approaches. The first is using a constant strain rate applied over a loading time by displacing the outermost rows of atoms. After the loading time, the boundaries are kept fixed. In the second method, we strain the system prior to simulation in the loading direction, and keep the boundary fixed during simulation. In either way, the crack starts to move once a critical strain is applied. It can be shown that the stress intensity factor remains constant in a strip geometry inside a region of [114] 3/4lx < a < (ly − 3/4lx).

(6.88)

This ensures that the crack achieves a steady-state during propagation through the slab. The slab is initialized at zero temperature prior to simulation. The length ly is several times larger than lx , with the ratio ranging from two to five. The slab width lx considered ranges from 1,150 (smallest) up to 4,597 (largest, corresponding to micrometer length scale in physical dimensions). The largest model contains over 70 million atoms. All quantities in this section are given in reduced units. The condition for small-scale yielding is satisfied in all cases (with harmonic, stiffening and softening potentials), since breaking of atomic bonds occurs over a region involving only a few atoms along the weak layer (that is, very small fracture process zone with a size on the order of a few atomic distances). There is no dislocation processes and the system is perfectly brittle. The slab is loaded with a maximum of a few percent strain, according to the crack loading mode. The loading is significantly lower than other studies [114]. A slit of length a is cut midway through the slab as an initial, atomically sharp crack. Accurate determination of crack tip velocity is important because we need to be able to measure even smallest changes in the propagation speed. The crack tip position is determined by finding the surface atom with maximum y position in the interior of a search region inside the slab. This quantity is averaged over a small time interval to eliminate very high frequency fluctuations. To obtain the steady-state velocity of the crack, the measurements are taken within a region of constant stress intensity factor [114]. In addition to checking the velocity history, steady state is verified by path-independency of the energy flux integral [22].

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6.6.2 Crack Speed and Energy Flow Molecular dynamics simulations suggest that a localized, small hyperelastic region around the crack tip can have significant effects on the dynamics of crack propagation. In all simulations, the slab is statically loaded with 0.32% strain in mode I. The strain energy density far ahead of the crack tip is given by ε2 lx E , (6.89) S = xx 2(1 − ν)2 where E is the Young’s modulus at small strain. The linear elastic expression of strain energy density is valid because material far ahead of the crack is strained always below the onset threshold of the bilinear law, that is it remains in the linear elastic regime of material response. The strain and strain energy density both vanish far behind of the crack. For a unit distance of crack propagation, a strip of material with energy density S ahead of the crack is replaced by an identical strip with zero strain energy behind the crack.

Fig. 6.37 Hyperelastic region in a (a) softening and (b) stiffening system

According to the linear elastodynamic theory of fracture [22], the crack speed should satisfy the dynamic energy release rate equation A(v/cR ) =

2γ S

(6.90)

where the function A(v/cR ) is a universal function of crack velocity v for a given material. Assuming that the small-strain elasticity completely governs the dynamics of fracture, the linear theory predicts that crack velocity should depend only on the ratio S/γ. During crack propagation, the energy stored ahead of the crack tip is partly converted by the bond breaking process into fracture surface energy, and partly dissipated into atomic motion. In the purely harmonic case, the fracture surface energy γ depends on rbreak and E. In the biharmonic case, the fracture surface energy depends on rbreak , ron , E0 , and E1 . The strategy is to focus on the prediction from linear theory that crack velocity depends only on


239

the ratio S/γ. To achieve this objective, we keep the ratio S/γ constant in all of the simulations. In the harmonic systems, as S ∼ E and γ ∼ E, we choose the parameter rbreak to be identical in all cases. In the biharmonic systems, we adjust the parameter rbreak , at given values of ron , E0 , and E1 , to always keep S/γ constant. We choose rbreak = 1.17 for the harmonic systems. The failure strain at the crack tip can reach a magnitude of several percent, which is comparable to many “real materials.” In the harmonic systems (with Young’s modulus equal to E0 or E1 ), the crack achieves the same propagation velocity around 80% of the Rayleigh wave speed. This is consistent with the linear theory. For the biharmonic systems, we choose ron = 1.1275 and rbreak = 1.1558 in the stiffening system and rbreak = 1.1919 in the softening system to keep S/γ constant. In contrast to the linear theory prediction, we find that the crack propagation velocity is about 20% larger in the stiffening system and 30% smaller in the softening system. These deviations cannot be explained by the linear theory. The fact that we change the large-strain elasticity while keeping the small-strain elasticity constant indicates that hyperelasticity is affecting crack dynamics. 6.6.3 Hyperelastic Area A geometric criterion based on the principal strain is used to characterize the area with hyperelastic material response close to the crack tip. The region occupied by atoms having a local maximum principal strain ε1 ≥ εon =

ron − r0 r0

(6.91)

defines the hyperelastic area AH by an integral over the whole simulation domain Ω AH = H(ε1 − εon ) dΩ. (6.92) Ω

Figure 6.37a shows the hyperelastic area in the case of a stiffening material, and Fig. 6.37b shows the hyperelastic area in the case of an elastically softening material, indicating that the hyperelastic effect is highly localized to the crack tip (these pictures show a portion of the simulation slab near the crack tip). However, the effect of hyperelasticity on crack velocity is significant, independent of the slab size. Enhancement or Reduction of Energy Flow A measure for the direction and magnitude of energy flow in the vicinity of the crack tip is the dynamic Poynting vector [22, 235]. The magnitude of the dynamic Poynting vector ' P =

P12 + P22

(6.93)

240


Fig. 6.38 Hyperelastic region and enhancement of energy flow in the (a) softening and (b) stiffening system

may be identified as a measure for the local energy flow. A measure for the enhancement or reduction of energy flow is obtained by subtracting the magnitude of the dynamic Poynting vector in the harmonic case from that in the biharmonic case at every point in the slab ∆P = Pbiharm − Pharm .

(6.94)

If the difference is negative, energy flow is reduced, and if the difference is positive, energy flow is enhanced. The steady-state fields are averaged over space as well as time to obtain good statistics. Figure 6.38 shows the energy flow enhancement and reduction in the vicinity of the crack tip for the elastically stiffening bilinear system (a) and for the elastically softening system (b). In each plot, the local hyperelastic zone is indicated by a dotted line. The energy flow in the vicinity of the crack tip is enhanced in the bilinear stiffening case and reduced in the softening case. In these plots, we also indicate the direction of energy flow with arrows and note that in the softening case, the energy flow ahead of the crack almost vanishes. The plots show that the local hyperelastic effect leads to an enhancement (stiffening system) or reduction (softening system) in energy flow. The small hyperelastic regions enhance the energy flow around the crack tip. The higher crack velocity in the stiffening system and the lower velocity in the softening system are due to enhancement or reduction of the energy flow in the vicinity of the crack tip. Table 6.3 summarizes change of net energy flow, as well as change of energy flow toward and away from the crack tip, in comparison to the harmonic system. The results quantify those depicted in Fig. 6.38a, b and show that the net energy flow as well as the flow of energy toward and away from the crack tip are all enhanced in the stiffening case, and reduced in the softening case.


241

Change of net Change of Change of Change of energy flow to energy flow energy flow away limiting crack tip to crack tip from crack tip speed Stiffening Softening

+19% −32%

+20% −32%

+25% −35%

+20% −30%

Table 6.3 Change of energy flow to the crack tip, due to a bilinear softening or stiffening interatomic potential

Fig. 6.39 J-integral analysis of a crack in a harmonic, softening and stiffening material, for different choices of the integration path Γ . The straight lines are a linear fit to the results based on the calculation of the molecular dynamics simulation studies

J-Integral Analysis The integral of energy flux, or path-independent dynamic J-integral is defined as F (Γ ) = (σij nj u˙ i + (U + T )v n2 ) dΓ. (6.95) Γ

It can be shown that its value is path-independent for steady-state crack motion [22]. We find that F (Γ ) around the crack tip increases by 19% in comparison with the harmonic case for the stiffening system, while decreasing by 32% for the softening system. The results are shown in Fig. 6.39. To calculate the integral, we choose a circular shape of Γ centered around the crack tip. Atomic quantities like stress and particle velocity are averaged spatially and over time and then the line integral is computed. The plot shows the value of the dynamic J-integral is independent of the shape of Γ which proves that crack motion is in steady state.

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6.6.4 How Fast can Cracks Propagate?

Fig. 6.40 Change of the crack speed as a function of εon . The smaller εon , the larger is the hyperelastic region and the larger is the crack speed

It has been shown that a local hyperelastic zone around the crack tip can have significant effect on the velocity of the crack. For a mode I tensile crack, linear theory predicts that the energy release rate vanishes for all velocities in excess of the Rayleigh wave speed [233], implying that a mode I crack cannot move faster than the Rayleigh wave speed. This prediction is indeed confirmed in systems with the harmonic potential where crack velocity approaches the Rayleigh wave speed independent of the slab size, provided that the applied strain is larger than 1.08% and the slab width is sufficiently large (lx > 1,000). The systems are loaded dynamically in this case. The strain levels are about ten times lower than in many other studies [114]. We consider hyperelastic effects of different strengths by using a biharmonic potential with different onset strains governed by the parameter ron . The parameter governs the onset strain of the hyperelastic effect εon =

ron − r0 . r0

(6.96)

The simulations reveal crack propagation at super-Rayleigh velocities in steady state with a local stiffening zone around the crack tip. Intersonic Mode I Cracks Figure 6.40 plots the crack velocity as a function of the hyperelasticity onset strain εon . The crack speeds shown in Fig. 6.40 are determined during


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steady-state propagation. It is observed that the earlier the hyperelastic effect is turned on, the larger the limiting velocity. Measuring the hyperelastic area AH using the principal strain criterion, we find that AH grows as εon becomes smaller. A correlation of the square root of the hyperelastic area with the achieved limiting speed of the crack is shown in Fig. 6.40. In Fig. 6.41, we depict the shape of the hyperelastic area near the crack tip for different choices of εon . The shape and size of the hyperelastic region is found to be independent of the slab width lx . In all cases, the hyperelastic area remains confined to the crack tip and does not extend to the boundary of the simulation.

Fig. 6.41 Shape of the hyperelastic regions for different choices of εon (the hyperelastic regions are symmetric with respect to the crack propagation direction). The smaller εon , the larger is the hyperelastic region. The hyperelastic region takes a complex butterfly shape

Figure 6.40 shows that the hyperelastic effect is very sensitive to the potential parameter and the extension of the local hyperelastic zone. Mode I cracks can travel at steady-state intersonic velocities if there exists a locally stiffening hyperelastic zone. For example, when the large-strain spring constant is chosen to be k1 = 4k0 , with ron = 1.1375 and rbreak = 1.1483 (that is, “stronger” stiffening and thus larger local wave velocity than before), the mode I crack propagates 21% faster than the Rayleigh speed of the soft material, and becomes intersonic, as shown by the Mach cone of shear wave front depicted in Fig. 6.42. Supersonic Mode II Cracks We have also simulated a shear-dominated mode II crack using the biharmonic stiffening potential. We define rbreak = 1.17, and ron is chosen slightly below to keep the hyperelastic region small. The dynamic loading is stopped soon after the daughter crack is nucleated [156, 165, 219]. The result is shown in Fig. 6.43. The daughter crack nucleated from the mother crack propagates supersonically through the material, although the hyperelastic zone remains localized to the crack tip region.

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Fig. 6.42 Intersonic mode I crack. The plot shows a mode I crack in a strongly stiffening material (k1 = 4k0 ) propagating faster than the shear wave speed

Supersonic mode II crack propagation as shown in Fig. 6.44 has been observed in other molecular dynamics studies [219] using an anharmonic stiffening potential. However, a clearly defined hyperelastic zone could not be specified in these simulations. Our result proves that a local hyperelastic stiffening effect at the crack tip causes supersonic crack propagation, in clear contrast to the linear continuum theory. The observation of super-Rayleigh and intersonic mode I cracks, as well as supersonic mode II cracks, clearly contradicts the prediction by the classical theories of fracture. 6.6.5 Characteristic Energy Length Scale in Dynamic Fracture The problem of a super-Rayleigh mode I crack in an elastically stiffening material is somewhat analogous to Broberg’s [249] problem of a mode I crack propagating in a stiff elastic strip embedded in a soft matrix. The geometry of this problem is shown in Fig. 6.45. Broberg [249] has shown that, when such a crack propagates supersonically with respect to the wave speeds of the surrounding matrix, the energy release rate can be expressed in the form G=

σ2 h f (v, c0 , c1 ) E

(6.97)


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Fig. 6.43 Supersonic mode II crack. Cracks under mode II loading can propagate faster than all wave speeds in the material if there exists a local stiffening zone near the crack tip

where σ is the applied stress, E the local Young’s modulus of the strip material, h is the half width of the stiff layer, and f is a nondimensional function of crack velocity v and wave speeds in the strip (c0 ) and the surrounding matrix (c1 ). The dynamic Griffith energy balance requires G = 2γ, indicating that crack propagation velocity is a function of the ratio h/χ where χ∼

γE σ2

(6.98)

can be defined as a characteristic length scale for local energy flux. By dimensional analysis, the energy release rate of our hyperelastic stiffening material is expected to have similar features except that Broberg’s strip width h should be replaced by a characteristic size of the hyperelastic region rH (note that rH could, √ for instance, be defined as the square root of the hyperelastic area, rH = AH ). Therefore, we introduce the concept of a characteristic length χ=β

γE σ2

(6.99)

for local energy flux near a crack tip. The coefficient β may depend on the ratio between hyperelastic and linear elastic properties as well as on the dynamic

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Fig. 6.44 The plot shows a temporal sequence of supersonic mode II crack propagation. The field is colored according to the σxx stress component

Fig. 6.45 Geometry of the Broberg problem of a crack propagating in a thin stiff layer embedded in soft matrix

energy balance. The characteristic energy length scale is defined such that h/χ equals 1 when the increase in crack speed is 50% of the difference between the shear wave speed of soft and stiff material. We have simulated the Broberg problem and found that the mode I crack speed approaches the local Rayleigh wave speed as soon as h/χ reaches values around 20. Numerous simulations verify that the scaling law in (6.99) holds when γ, E, and σ are changed independently. The results are shown in Fig. 6.46. From the simulations, we estimate numerically β ≈ 4.4 and therefore χ ≈ 38. The potential energy field near a crack propagating at an intersonic speed is shown in Fig. 6.47.


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The transition from the limiting speed of the soft material to the limiting speed of the stiff material depicted in Fig. 6.46 is reminiscent of the observations in the one-dimensional model of dynamic fracture (see Fig. 6.20a showing the dependence of the crack speed as a function of the potential parameter ron ).

Fig. 6.46 Calculation results of the Broberg problem. The plot shows results of different calculations where the applied stress, elastic properties, and fracture surface energy are independently varied. In accordance with the concept of the characteristic energy length scale, all points fall onto the same curve and the velocity depends only on the ratio h/χ

The existence of a characteristic length χ for local energy flux near the crack tip has not been discussed in the literature and plays the central role in understanding the effect of hyperelasticity. Under a particular experimental or simulation condition, the relative importance of hyperelasticity is determined by the ratio rH /χ. For small rH /χ, the crack dynamics is dominated by the global linear elastic properties since much of the energy transport necessary to sustain crack motion occurs in the linear elastic region. However, when rH /χ approaches unity, as is the case in some of the molecular dynamics simulations, the dynamics of the crack is dominated by local elastic properties because the energy transport required for crack motion occurs within the hyperelastic region. The concept of energy characteristic length χ immediately provides an explanation how the classical barrier for transport of energy over large distances can be undone by rapid transport near the tip.

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Fig. 6.47 The plot shows the potential energy field during intersonic mode I crack propagation in the Broberg problem. Since crack motion is intersonic, there is one Mach cone associated with the shear wave speed of the solid

6.6.6 Summary It has been shown that local hyperelasticity can have a significant effect on the dynamics of brittle crack speeds and have discovered a characteristic length associated with energy transport near a crack tip. The assumption of linear elasticity fails if there is a hyperelastic zone in the vicinity of the crack tip comparable to the energy characteristic length. Therefore, we conclude that hyperelasticity is crucial for understanding and predicting the dynamics of brittle fracture. The simulations prove that even if the hyperelastic zone extends only a small area around the crack tip, there may be crucial effects on the limiting speed and the energy flow toward the crack tip, as illustrated in Fig. 6.40. If there is a local softening effect, we find that the limiting crack speed is lower than in the case of harmonic solid. The study has shown that hyperelasticity dominates the energy transport process when the size of hyperelastic zone becomes comparable to the characteristic length χ ∼ γE/σ 2 . (6.100) Under typical experimental conditions (that is, relatively small stresses), the magnitude of stress may be one or two orders smaller than that under molecular dynamics simulations. In such cases, the characteristic length χ is relatively large and the effect of hyperelasticity on effective velocity of energy transport is relatively small. However, χ decreases with the square of the applied stress. At about 1% of elastic strain as in the simulations, this zone is already on the order of a few hundred atomic spacings and significant hyperelastic effects are observed.


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The simulations indicate that the universal function A(v/cR ) in the classical theory of dynamic fracture is no longer valid once the hyperelastic zone size rH becomes comparable to the energy characteristic length χ. Linear elastic fracture mechanics predicts that the energy release rate of a mode I crack vanishes for all velocities in excess of the Rayleigh wave speed. However, this is only true if rH /χ 1. A hyperelastic theory of dynamic fracture should incorporate this ratio into the universal function so that the function should be generalized as A(v/cR , rH /χ). (6.101) The local hyperelastic zone changes not only the near-tip stress field within the hyperelastic region, but also induces a finite change in the integral of energy flux around the crack tip. We find that the dynamic J-integral around a super-Rayleigh mode I crack is still path-independent but no longer vanishes in the presence of hyperelasticity. Similarly, the supersonic mode II crack motion as shown in Fig. 6.44 can only be understood from the point of view of hyperelasticity. A single set of global wave speeds is not capable of capturing all phenomena observed in dynamic fracture. We believe that the length scale χ, heretofore missing in the existing theories of dynamic fracture, will prove to be helpful in forming a comprehensive picture of crack dynamics. In most engineering and geological applications, typical values of stress are much smaller than those in molecular dynamics simulations. In such cases, the ratio rH /χ is small and effective speed of energy transport is close to predictions by linear elastic theory. However, the effect of hyperelasticity will be important for nanoscale materials, such as highly strained thin films or nanostructured materials, as well as high speed impact phenomena. The prediction of intersonic mode I cracks (see Fig. 6.42) has also been verified in experimental studies of dynamic fracture of rubber [250].

6.7 Crack Instabilities and Hyperelastic Material Behavior Cracks propagating in homogeneous materials show a very interesting dynamics: From experiment [207] and computer simulation [146, 155, 157, 206] it is known that cracks propagate straight for low speeds with perfect cleavage (“mirror”), and become unstable at higher speeds. The onset of instability results in an increasingly rough crack surface (‘mist”), which becomes more intense when the crack speed increases further (“hackle”). This phenomenon was referred to as the “mirror-mist-hackle” transition. Computer simulation played an important role in this area [155], since it showed that the crack tip instability also occurs in perfect atomic lattices and is therefore not due to material imperfections. It was proposed [27, 155, 207] that the mirror-misthackle transition is due to an intrinsic dynamic crack tip instability.

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Fig. 6.48 Crack propagation in an LJ system. Subplot (a) shows the σxx -field and indicates the mirror-mist-hackle transition. The crack velocity history (normalized by the Rayleigh wave speed) is shown in subplot (b)

The dynamic crack tip instability can nicely be observed in LJ systems [146, 155]. A simulation result of such a study is shown in Fig. 6.48. After an initial phase where cleavage is mirror-like, the crack surface starts to roughen at about 30% of the Rayleigh wave speed. Eventually, the crack surface turns into a hackle region accompanied by emission of dislocations. The final speed of the crack is around 50% of the Rayleigh speed. These observations are in agreement with the results discussed in [146, 155]. Fracture instabilities have already been predicted in the classical literature based on linear elastic fracture mechanics theories, where it has been proposed that the governing stress measure determining the direction of crack propagation is the circumferential or hoop stress [22, 210]. If this view of crack tip instabilities is adopted, the instability should occur at speeds around 73% of


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Rayleigh wave speed. In contrast to this, in many experiments [207] as well as several computer simulations [146, 155] the crack tip instability was observed at speeds as low as 1/3 of the theoretical limiting speed. In this section, we will show that this discrepancy may also be due to hyperelasticity. The outline of this section is as follows. We begin with a discussion of the direction of stable crack propagation in harmonic lattices. In agreement with the analysis described in Sect. 4.4.3, crack propagation is stable along the direction of lowest fracture surface energy for low velocities. We then continue with a discussion of the crack-speed dependent tip instability in harmonic solids and show that in agreement with Yoffe’s analysis [210], cracks become unstable at a speed of about 73% of Rayleigh wave speed. In contrast, by using a softening LJ potential it will be shown that the instability occurs at crack velocities around 1/3 of Rayleigh wave speed [29]. We then continue with results using stiffening and softening potentials and show that hyperelasticity, the elasticity of large strains, plays a governing role in the stability of cracks. 6.7.1 Introduction There are several models for the instability problem proposed in the literature. Some theories assume that the stress distribution ahead of the crack determines the onset of instability [22, 155, 210], while others are based on energy flow in the vicinity of the crack tip [27, 248]. In the classical literature based on linear elastic fracture mechanics, the instability was explained by the fact that the circumferential or hoop stress σθ [22, 210] has a maximum straight ahead of the crack at low speeds, but features two maxima in directions inclined to the crack at high crack speeds (see Fig. 6.27). According to this criterion, the instability should occur at speeds around 73% of Rayleigh wave speed. Other suggestions were based on a perturbation analysis of the asymptotic stress field [251] that predicted unstable crack motion at 65% of Rayleigh wave speed, thus at a comparable speed as given by the Yoffe criterion. Both criteria predict that the crack changes to another cleavage plane inclined about 60◦ to the initial crack plane. There are two experimental and computational observations that disagree with the Yoffe criterion. Firstly, in most experimental and computational investigations, the instability establishes as wiggly crack path with crack branches inclined 30◦ to the initial crack plane. This is in contrast to Yoffe’s prediction of an angle of 60◦ relative to the initial crack plane. In addition, in many experiments [207] as well as in computer simulations [146, 155] the crack tip instability was observed at speeds as low as 30% of the theoretical limiting speed thus much lower than the theoretical prediction of 73%. In the literature it has been suggested that this lower critical speed for the instability may be due to hyperelastic softening around the crack tip [27, 146, 155, 248]. One attempt of explanation was a nonlinear continuum analysis carried out by Gao [27,248] focusing on the energy transport near the

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moving crack. The model, for the first time, allowed quantitative estimates for the instability speed. The main idea of the hyperelastic continuum mechanics analysis was that once the crack speed exceeds the speed of local energy transport near the crack tip (the local wave speed), the crack becomes unstable. Due to the strong softening of many materials, the speed of energy transport is significantly reduced in the vicinity of the crack tip. The theoretical analysis of the critical instability speed [27] was in consistency with the value observed in molecular dynamics simulation for cracks propagating in LJ solids. In contrast to Gao’s analysis, Abraham and coworkers [146, 155] proposed that due to the local softening around the crack tip, the hoop stress becomes flattened at much lower speed than predicted by the linear elastic continuum theories. It was also suggested that the instability could be a consequence of lower lattice vibration frequencies in the soft region near the crack tip. It was argued that once the crack starts to see local fluctuations of the atoms ahead of the crack tip, the crack becomes unstable. This was assumed to occur when the time for the crack to traverse one lattice distance becomes comparable to the lattice vibration period. By systematically changing the large-strain elastic properties while keeping the small-strain elastic properties constant and thus tuning the strength of the hyperelastic effect, we will show that the elasticity of large strains governs the instability dynamics of cracks. Linear elastic materials serve as reference systems for the studies, where we find that the instability speed agrees well with the predicted value from Yoffe’s linear analysis [210]. Changing the strength and type of hyperelastic effect (stiffening vs. softening) allows tuning the instability speed. 6.7.2 Design of Computational Model Figure 6.49 illustrates the concept of hyperelastic softening in contrast to linear elastic behavior. In the spirit of “model materials” as introduced earlier, we develop a new, simple material model which allows a systematic transition from linear elastic to strongly nonlinear material behaviors, with the objective to bridge different existing theories and determine the conditions of their validity. By systematically changing the large-strain elastic properties while keeping the small-strain elastic properties constant, the model allows us to tune the size of hyperelastic zone and to probe the conditions under which the elasticity of large strains governs the instability dynamics of cracks. In the case of linear elastic model with bond snapping, we find that the instability speed agrees well with the predicted value from Yoffe’s model. We then gradually tune up the hyperelastic effects and find that the instability speed increasingly agrees with Gao’s model. In this way, we achieve, for the first time, a unified treatment of the instability problem leading to a generalized model that bridges Yoffe’s linear elastic branching model to Gao’s hyperelastic model.


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Fig. 6.49 The concept of hyperelastic softening close to bond breaking, in comparison to the linear elastic, bond-snapping approximation

Although simple pair potentials do not allow drawing conclusions for unique phenomena pertaining to specific materials, they enable us to understand universal, generic relationships between potential shape and fracture dynamics in brittle materials; in the present study we use a simple pair potential that allows the hyperelastic zone size and cohesive stress to be tuned. The potential is composed of a harmonic function in combination with a smooth cut-off of the force based on the Fermi–Dirac (F–D) distribution function to describe smooth bond breaking. We do not include any dissipative terms. The force vs. atomic separation is expressed as 1−1 0 dφ Ξ (r) = k0 (r − r0 ) exp r +1 . (6.102) dr rbreak − Ξ Assuming that the spring constant k0 is fixed, the potential has two additional parameters, Ξ and rbreak . The parameter rbreak (corresponding to the Fermi energy in the F–D function) denotes the critical separation for breaking of the atomic bonds and allows tuning the breaking strain as well as the cohesive stress at bond breaking. It is further noted that σcoh ∼ rbreak .

(6.103)

The parameter Ξ (corresponding to the temperature in the F–D function) describes the amount of smoothing at the breaking point. In addition to defining the small-strain elastic properties (by changing the parameter k0 ), the present model allows one to control the two most critical physical parameters describing hyperelasticity, (1) cohesive stress (by changing the parameter rbreak ), and (2) the strength of softening close to the crack tip (by changing the parameter Ξ). Figure 6.50 depicts force vs. atomic separation of the interatomic potential used in the study reviewed here. The upper part shows the force vs. separation

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Fig. 6.50 Force vs. atomic separation for various choices of the parameters Ξ and rbreak (these parameters are independent from each other). Whereas rbreak is used to tune the cohesive stress in the material, Ξ is used to control the amount of softening close to bond breaking

curve with respect to changes of rbreak , and the lower part shows the variation in shape when Ξ is varied. For small values of Ξ (around 50), the softening effect is quite large. For large values of Ξ (beyond 1,000), the amount of softening close to bond breaking becomes very small, and the solid behaves like one with snapping bonds. The parameter rbreak allows the cohesive stress σcoh to be varied independently. This model potential also describes the limiting cases of material behavior corresponding to Yoffe’s model (linear elasticity with snapping bonds) and Gao’s model (strongly nonlinear behavior near the crack tip). Yoffe’s model predicts that the instability speed only depends on the smallstrain elasticity. Therefore, the instability speed should remain constant at 73% of the Rayleigh wave speed, regardless of the choices of the parameters rbreak and Ξ. On the other hand, Gao’s model predicts that the instability speed is only dependent on the cohesive stress σcoh (and thus rbreak ):


vinst =

σcoh ∼ ρ

rbreak ρ

255

(6.104)

(note that ρ denotes the mass density, as defined above). According to Gao’s model, variations in the softening parameter Ξ should not influence the crack instability speed. 6.7.3 Computational Experiments We carry out systematic numerical studies based on continuously varying potential parameter. The investigations will focus on the predictions of Gao’s model vs. those of Yoffe’s model. We begin with harmonic systems serving as the reference and then increase the strength of the hyperelastic effect to study the dynamics of crack tip instability in hyperelastic materials.

Fig. 6.51 Crack propagation in a homogeneous harmonic solid. When the crack reaches a velocity of about 73% of Rayleigh wave speed, the crack becomes unstable in the forward direction and starts to branch (the dotted line indicates the 60◦ plane of maximum hoop stress)

Harmonic Potential – The Linear Elastic Reference System We find that cracks in homogeneous materials with linear elastic properties (harmonic potential, achieved by setting Ξ to infinity) show a critical instability speed of about 73% of the Rayleigh wave speed, independent of the choice of rbreak . The crack surface morphology is shown in Fig. 6.51. This observation is in quantitative agreement with the key predictions of Yoffe’s model. It is observed that the occurrence of the instability can be correlated with the development of a bimodal hoop stress as proposed by Yoffe, as is shown in the sequence of hoop stress snapshots as a function of increasing crack speed depicted in Fig. 6.52. We conclude that in agreement with Yoffe’s prediction, the change in deformation field governs the instability dynamics in the harmonic systems.

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Fig. 6.52 Comparison between hoop stresses calculated from molecular dynamics simulation with harmonic potential and those predicted by linear elastic theory for different reduced crack speeds v/cR . The plot clearly reveals development of a maximum hoop stress at an inclined angle at crack speeds beyond 73% of the Rayleigh wave speed

Hyperelastic Materials Behavior in Real Materials It is observed that crack dynamics changes drastically once increasingly stronger softening is introduced at the crack tip and linear elastic Yoffe model fails to describe the instability dynamics. The predictions by Yoffe’s model are included in Fig. 6.53 as the red line, and the predictions by Gao’s model are plotted as the blue points. As seen in the plots, it can be observed that for any choice of rbreak and Ξ, the instability speed lies in between the prediction by Gao’s model and that by Yoffe’s model. Whether it is closer to Gao’s model or to Yoffe’s model depends on the choice of rbreak and Ξ. For small values of rbreak and Ξ, we find that the instability speed depends on the cohesive stress, which is a feature predicted by Gao’s model. The instability speed seems to be limited by the Yoffe speed (as can be confirmed for large values of rbreak and Ξ). Whereas the observed limiting speeds increase with rbreak , they saturate at the Yoffe speed of 73% of Rayleigh wave speed for larger values (see Fig. 6.53, bottom curve for Ξ = 300). In this case, the instability speed is independent of rbreak and Ξ. This behavior is reminiscent of Yoffe’s deformation controlled instability mechanism and suggests a change in governing mechanism for the instability speed. Overall, the results indicate that the instability speed depends on the strength of softening (parameter Ξ) and on the cohesive stress close to bond breaking (parameter rbreak ). Stiffening Materials Behavior: Stable Intersonic Mode I Crack Propagation A generic behavior of many rubber-like polymeric materials is that they stiffen with strain. What happens to crack instability dynamics in such materials? Recent experiments have shown intersonic mode I crack propagation in rubber-like materials with elastic stiffening characteristics. According to the existing theories, such high-speed crack propagation should not be possible in


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Fig. 6.53 The critical instability speed as a function of the parameter rbreak , for different choices of Ξ. The results show that the instability speed varies with rbreak and thus with the cohesive stress as suggested in Gao’s model, but the Yoffe speed seems to provide an upper limit for the instability speed. The critical instability speeds are normalized with respect to the local Rayleigh wave speed, accounting for a slight stiffening effect of the moduli as shown in Fig. 4.5

homogeneous materials. We hypothesize that local stiffening near the crack tip may lead to a locally enhanced Yoffe speed and enhanced energy flow, so that the onset of crack tip instability is shifted to higher velocities. The simulations are based on a simple model in which we change the large-strain spring constant and small-strain spring constant, similar to studies described in the previous section except for the smoothing part near bond snapping. The model is depicted schematically in Fig. 6.54a. Upon a

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Fig. 6.54 Subplot (a) schematic of stiffening materials behavior, illustrating the ratio kratio = k1 /k0 . Subplot (b) extension of the hyperelastic stiffening region. Despite the fact that the stiffening hyperelastic region is highly localized to the crack tip and extends only a few atomic spacings, the crack instability speed is larger than the Rayleigh wave speed

critical atomic separation ron , the spring constant of the harmonic potential is changed and switched to a new “local” large-strain value. The knowledge of ron allows for a clear definition of the extension of the hyperelastic zone near the crack tip, as can be verified in Fig. 6.54b. Here the F–D function is also used to smoothly cut off the potential at rbreak . We discuss two different choices of the ratio k1 /k0 = 2 and k1 /k0 = 4. It is observed that if there exists a hyperelastic stiffening zone, the Yoffe speed is no longer a barrier for the instability speed and stable crack motion beyond the Yoffe speed can be observed. This behavior is shown in Fig. 6.55. We find that the stronger the stiffening effect, the more rapid the increase of instability speed with increasing value of the ratio of rbreak . Note that all points fall together once rbreak becomes comparable to ron , corresponding to the case when no hyperelastic zone is present and the potential is harmonic with smooth F–D bond breaking (here the potential shape is identical because bonds rupture before onset of the hyperelastic effect). As can be clearly seen in Fig. 6.55 for k1 /k0 = 4, the instability speed can even be super-Rayleigh and approach intersonic speeds. This is inconsistent with the classical, linear elastic theories but can be understood from a hyperelasticity point of view. This observation suggests that the stiffening materials behavior tends to have a stabilizing effect on straight crack motion. 6.7.4 Discussion and Conclusion In this section we reviewed studies of the dynamics of crack tip instabilities. We find that the onset of the instability is governed by a critical crack speed,


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Fig. 6.55 Molecular dynamics simulation results of instability speed for stiffening materials behavior, showing stable super-Rayleigh crack motion as observed in recent experiment. Such observation is in contrast to any existing theories, but can be explained based on the hyperelastic viewpoint

the instability speed. The most important result of this section is that hyperelasticity, the elasticity of large strains (extending only a small domain from the crack tip) governs the instability speed. By keeping the small-strain elastic properties constant and systematically changing the large-strain elasticity, we demonstrated that the instability speed can be tuned to higher and lower values. An important consequence of the results is that linear elastic theory cannot be applied to describe the instability dynamics in nonlinear materials. Since most real materials show nonlinearities at large strains, linear elastic theory cannot be applied to describe crack dynamics in real materials. The results conform that the local wave speed near the crack tip governs the dynamics [27]. This also explains experimental [207] and other computational results [146, 155]. We summarize the main findings. • Cracks in purely harmonic lattices with the harmonic bond snapping potential move straight as long as the propagation speed is below about 73% of the Rayleigh wave speed. This finding is in agreement with the classical theory proposed by Yoffe [210] (since the hoop stress develops a bimodal structure). • Large-strain elastic properties and therefore the local wave speeds can dominate the instability dynamics. This result was verified in Fig. 6.53 where a correlation of the instability speed to the local wave speed is shown. • In softening materials, the instability is reduced to much lower values than the 73% mark of the Rayleigh wave speed.

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• In stiffening materials, the instability is increased to much higher values than the 73% mark of the Rayleigh wave speed, even reaching superRayleigh velocities. Hyperelasticity, the elasticity of large strains, can control dynamic crack tip instabilities. This explains the discrepancy of measured instability speeds in “real materials” [207] and predicted instability speeds by linear elastodynamic theory [22]. The reason is that virtually all real materials show a strong softening close to materials failure.

6.8 Suddenly Stopping Cracks: Linking Atomistic Modeling, Theory, and Experiment This section addresses the following question: “What happens if a crack propagating at very high velocities suddenly comes to rest?” We all know that if we try to stop a heavy object like a car, we feel considerable resistance due to its inertia. How does a crack stop? Does the crack carry properties like “inertia” or mass? The research in the last decades has shed light on these fundamental questions about the nature of fracture. It was found that the stress field at the crack immediately responds to changes in loading condition or fracture surface energy for velocities lower than or equal to the Rayleigh wave speed. This result led to the terminology of the crack being “mass less,” because it responds instantaneously to a change in crack driving force. If we turn back to the analogy of stopping a heavy object, this implies that we could stop it instantly from any velocity without feeling any resistance, and the object would react to any applied force immediately without delay. In this section, we carry out large-scale atomistic simulations to focus on the atomic details of the dynamics of suddenly stopping cracks. 6.8.1 Introduction Large-scale atomistic simulations are used to study suddenly stopping cracks under mode I (tensile) as well as mode II (shear) loading conditions. The crack velocity, denoted as v, is related to the crack tip position a by v = a˙ = da/dt. The time history of the crack tip velocity is arranged to be v = v ∗ − v ∗ H (t − tstop ) ,

(6.105)

where t denotes the time and H(s) is the unit step function. The variable v ∗ stands for the constant propagation velocity which corresponds to the limiting velocity of cracks in our simulations. As shown by (6.105), we study a crack that propagates at its limiting speed up to time tstop and then suddenly stops. The reason for crack stopping could, for example, be that the resistance of the material to fracture increases dramatically. This problem is


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important for constructing solutions for nonuniform crack growth [22,234,252], and for understanding crack propagation in materials with changing resistance to fracture. The limiting speed of a mode I crack is the Rayleigh wave velocity, therefore (6.106) v ≤ cR . In mode II, the allowed velocities are sub-Rayleigh (6.106) as well as intersonic crack propagation speeds [165, 216, 219, 253] c s < v ≤ cl

(6.107)

There is a forbidden velocity regime c R < v ≤ cs

(6.108)

which can be overcome by a mother–daughter mechanism involving nucleation of a secondary crack (daughter crack) at some distance away from the primary crack (mother crack) [165,237,254]. Intersonic crack propagation has also been reported in earthquakes since 1982 [255] and has led to active research in this field. The discovery of intersonic crack propagation has almost doubled the limiting crack velocity from Rayleigh to longitudinal wave speed. In the case of nonlinear materials, the limiting velocities can be lifted to even higher speeds! This allows for supersonic crack propagation as discussed in Sect. 6.6. The allowed velocity regimes are depicted in Fig. 6.56.

Fig. 6.56 Allowed velocities for mode I and mode II crack propagation, linear and nonlinear stiffening case

The stress field around a crack propagating rapidly (also referred to as the dynamic field) is very different from the field around a static crack, as discussed in Sect. 6.5. In mode I, the static field is expected to arise as soon as the crack stops and is emitted with the shear wave speed. For sub-Rayleigh mode II cracks, the situation is the same. For intersonic crack propagation in mode II, it has been shown [208] that the static field spreads out with the shear wave speed as soon as the mother crack has “reached” the daughter crack tip. In any loading case, the corresponding static field is established instantaneously on a line ahead and behind the crack propagation direction

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(“prospective crack plane”) [234], while in other areas around the crack tip this is achieved only in the long-time limit (after a number of elastic eaves have been emitted) [22, 23]. Although we do not study a mode III crack in this chapter, we would like to note that the static field is radiated out behind a circular wave front and the region of instantaneous “switch” to the static field is not confined to the prospective crack line [252]. First experimental observations on nonuniform crack growth in the sub-Rayleigh regime were published in [256]. Here we review molecular dynamics model suitable to simulate the dynamical processes of a suddenly stopping crack. Explaining this phenomenon at the atomistic scale will help forming a more complete picture of dynamic fracture. We will review a study of suddenly stopping cracks using a combination of continuum theory, laboratory experiments, and computer simulations. Important references for the study will be [22] and [23], where analytical and experimental results for the mode I case are described. For the mode II case, we will compare the findings to the analytical work in [208, 217], where the fundamental solution for an intersonic mode II crack and the solution for a suddenly stopping crack were derived. No laboratory experiments are available up to date for the suddenly stopping intersonic mode II crack. The outline of this section is as follows: For both mode I and mode II loading conditions, linear system solutions are established by assuming the interaction between atoms to be a central pair potential similar to a harmonic ball-spring model. It will be shown that these simulations reproduce the continuum mechanics solution for the plane stress case. Subsequently, we use the linear study as a reference to probe crack dynamics in nonlinear materials characterized by an “anharmonic” tethered LJ potential [165, 219, 237]. The harmonic potential is the first-order approximation of the anharmonic potential. It will be shown, for the first time at the atomistic scale, that the sub-Rayleigh crack indeed behaves like a massless particle and that this feature does not hold for the intersonic case. It will also be demonstrated that the massless feature of cracks does not hold for nonlinear materials: The crack does not behave like a massless particle in the nonlinear case. Cracks being strictly massless is therefore confined to sub-Rayleigh cracks. 6.8.2 Theoretical Background of Suddenly Stopping Cracks The suddenly stopping crack is important for studies related to nonuniform crack growth. Solutions to this problem are often denoted as fundamental solutions for crack growth. The core idea is to construct the solution of nonuniform crack growth from the solution for uniform crack growth at constant velocity [22,234]. This becomes possible because the dynamic stress intensity factor can be written as a product of the static stress intensity factor for given geometry and a universal function which depends only on the propagation velocity. The dynamic stress intensity factor can be expressed by [22] ˙ = kI,II (a)K ˙ I,II (a, a˙ = 0). KI,II (a, a)

(6.109)


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The universal functions kI and kII may be approximated by [22]

and

1 − a/c ˙ R k(a) ˙ I≈ 1 − a/c ˙ l

(6.110)

1 − a/c ˙ R k(a) ˙ II ≈ 1 − a/c ˙ s

(6.111)

for most practical purposes. The stress intensity factor responds instantaneously to a change in propagation velocity. In fracture mechanics, one often writes the so-called equation of motion of a crack as [22] G(a, a, ˙ loading , . . . ) = Γ (a), ˙

(6.112)

where G denotes the dynamic energy release rate and Γ (a) ˙ represents the dynamic fracture toughness, a material property measuring the fracture resistance. For mode I, one can write more precisely, EΓ (a) ˙ ≈ g(a). ˙ (1 − ν 2 )KI (a, 0)2

(6.113)

The right-hand side can be shown to be [22, 114, 221] g(a) ˙ = 1 − a/c ˙ R.

(6.114)

In linear elastic fracture mechanics, the bulk elastic properties consist of elastic constants, while the effects of loading and geometry are included in the expression for KI (a, 0). Equations of this type can be integrated to obtain a solution for a(t), if Γ (a) ˙ and KI (a, 0) are both known. We would like to remark that the crack propagation history a(t) could in principle be solved using molecular dynamics simulations. In contrast, the dynamic fracture resistance Γ (a) ˙ cannot be determined from continuum mechanics theory and is also difficult to be measured by experiments. The massless behavior of the crack is also reflected by the fact that only the first derivative of the crack tip position appears in the equation of motion (6.112). This is different from a moving dislocation: It takes an infinite time for the static field to establish itself around a suddenly stopping dislocation [38]. Equations (6.109)–(6.114) are only valid for sub-Rayleigh crack growth [22,23,208,217]. In the intersonic case, it will take some time after the crack has completely stopped before the shear and Rayleigh waves reach the tip, as will be discussed shortly. The analytical solution for the suddenly stopping crack can be derived using the superposition principle: First, the solution for a crack propagating at a constant velocity v is determined. Subsequently, the solution for a moving dislocation is superposed to negate the crack opening displacement ahead of the crack tip where the crack has stopped. The solution for mode I loading was derived in [234], and that for mode II sub-Rayleigh cracks was

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first considered in [257]. The fundamental solution for the intersonic crack was recently determined in [208, 217]. In these papers, the Wiener–Hopf technique [22] was used to address this problem, by transforming the problem into the complex space and then applying the theory of complex functions. A scalar Wiener–Hopf problem is derived which can be solved by transforming the complex functions back to real space employing the deHoop method of integral inversion. 6.8.3 Atomistic Simulation Setup Figure 6.57 shows the simulation geometry which consists of a 2D atomic lattice with dimensions lx and ly . The suddenly stopping crack is modeled by a finite length weak layer (similar as in Sect. 6.5, however, here with a finite length to enforce that the crack stops to propagate at a critical point). Once the crack tip reaches the end of the weak layer, it cannot propagate any further and is forced to stop. The crack tip does not sense the existence of the barrier before it actually reaches it because the material is elastically homogeneous.

Fig. 6.57 Simulation geometry for the stopping crack simulation

The simulations are performed using a microcanonical N V E ensemble (constant number of particles N , constant volume V , and constant energy E), an appropriate choice for nonequilibrium phenomena such as dynamic fracture. The slab is initialized with very low temperature, T ≈ 0, which increases during the simulation to slightly higher temperatures. The loading starts when the outermost rows are displaced according to a given strain rate. To avoid wave emission from the boundaries, an initial velocity field according to the prescribed strain rate is established prior to simulation. The simulations are done with a slab size of 1512 × 3024 atom rows, and the system contains about 4,500,000 particles.


265

Suddenly Stopping Mode I Crack: Simulation Setup The crack is stopped after it reaches its limiting speed (the Rayleigh wave speed, or higher for nonlinear simulations) and has traveled at this velocity for some time. The value of ystop denotes the position at which the crack stops, corresponding to the coordinate of the end of the weak layer (Fig. 6.57). The system is loaded until time tl , after which the boundaries are no longer displaced but held fixed. Somewhat different loading histories are chosen for different simulations and will be indicated in the corresponding sections. Suddenly Stopping Mode II Crack: Simulation Setup Similar to the mode I case, we also perform mode II simulations. For the mode II simulations, the loading has to be significantly larger to achieve nucleation of the daughter crack and the limiting speed. The large deformation around the crack tip leads to large local dilatations soon after the crack has been stopped. In the simulation, this could cause bonds to break as they are driven out of the cutoff radius. Nearest neighbors are searched only within a cutoff radius (rcut = 2 in reduced atomic units). This leads to finite values of rbreak instead of the theoretical, continuum mechanics assumption rbreak → ∞. The variable rbreak stands for the atomic separation when atomic bonds snap. Breaking of atomic bonds in the bulk is avoided by increasing the potential energy barrier for higher strains by introducing a fourth-order term in the potential. Modifying the potential gives additional barrier for bond breaking without affecting the rest of the slab, where the strains are much lower. Only very localized to the crack tip, and only for a very short time after the crack is stopped, this modification of the potential marginally affects the dynamics. This procedure is not applied with the tethered LJ potential (see Sect. 4.4.3), because the barrier for bond breaking has proven to be high enough due to its natural stiffening. In addition to a slight opening displacement loading, we impose a strong shear loading on the outermost rows, displacing the upper border atoms to the left and the lower border atoms to the right during loading. We quickly note here that without the additional potential barrier, dislocations would be observed to emit when the crack is stopped. This phenomenon shows the competing mechanisms of atom separation and atom sliding in nature [66]; the former yielding brittle fracture and the latter giving ductile response. We deliberately avoid such effects because we wish to focus on the crack dynamics. Interatomic Potentials We briefly present an analysis of the interatomic pair potentials used for the simulations. The choice of simple interatomic force laws is consistent with our objective to study generic properties of a many-body problem common

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to a large class of real physical systems. We deliberately avoid the specific complexities of a particular atomic force law. The simple interatomic force laws can be regarded as providing model materials for computer experiments. In this study, we use two model materials to study brittle fracture: 1. Linear elastic material with bond snapping across the weak layer 2. Hyperelastic stiffening material, also with bond snapping across the weak layer As in the studies reviewed above, here we use pair potentials (for instance, harmonic potentials or modified Lennard-Jones potentials), rather than multibody EAM potentials [94], to model a generic brittle material. A horizontal slit of 400 atoms distance is cut midway along the left-hand vertical slab boundary. The crack is oriented orthogonal to the close-packed direction of the the triangular lattice. For positions y < ystop , atomic bonds are assumed to snap at rbreak = 1.1625 across the weak layer. The quantity rbreak can be used to control the fracture surface energy distribution. This confines the crack to propagate along prescribed weak layer without branching. With this approach, we deliberately try to suppress branching and dislocation emission in the current work by introducing a weak interface. For the linear spring potential, the value for k is assumed to be k = 140/r02 . Wave velocities are cs ≈ 7.10, cl ≈ 12.29, and cR ≈ 6.55 [258]. The nonlinear tethered LJ potential is described in detail in Sect. 4.4.3, and we choose 0 = 1.9444 to match the small-strain elastic properties with the elastic properties of the harmonic potential associated with the spring constant k = 140/r02 . Griffith Analysis The question why, how, and under which conditions cracks initiate can be investigated by comparing atomistic and continuum predictions. We assume that the onset of crack motion is governed by the Griffith criterion. The Griffith criterion predicts that the crack tip begins to propagate when the crack tip energy release rate G reaches the fracture surface energy 2γ: G = 2γ.

(6.115)

The energy release rate G can be universally expressed as G=

2 KI2 + KII , E

(6.116)

where KI,II are the mode I and mode II stress intensity factors. In both cases, bonds across the weak layer breaks at rbreak = 1.1625. For the triangular lattice and the given crack orientation, the fracture surface energy is γ harm = 0.0914 for the harmonic system. For the tethered LJ potential, the fracture surface energy is determined to be γ LJ = 0.1186 following the same approach described in Sect. 4.4.3.


267

Crack Initiation Time in an Infinite Solid with a Semi-Infinite Crack This analysis follows the considerations in [165] for a mode II crack using atomistic and continuum methods. To estimate the crack initiation time due to the applied loading, an infinite plane stress solid containing a semi-infinite crack is subject to far field tensile (mode I) and shear (mode II) loading. Initially assuming a perfect solid without a crack, the background stress rate σ˙ 11 is given by 2λG + 2G ε˙11 (6.117) σ˙ 11 = λ + 2G with λ=

νE (1 + ν)(1 − 2ν)

(6.118)

E . 2(1 + ν)

(6.119)

and G=

The background shear rate is given by σ˙ 12 = Gε˙12 . The stress intensity factor KI can be determined as [22] √ 4 2 − 2ν(1 + ν)cs t3 KI (t) = σ˙ 11 3 π

(6.120)

(6.121)

and KII is found to be [22] 4 KII (t) = σ˙ 12 3

2(1 + ν)cs t3 . π

(6.122)

Equations (6.121), (6.122), and (6.116) can be used to derive an expression for the initiation time of crack motion: 2 3 9πµγ pred ' tinit = 3 (6.123) . 3 4 1−ν 2 2 + 8cs σ˙ 12 σ ˙ 11 2 In the case of pure mode I cracks, the shear rate σ˙ 12 is set to zero in (6.123). Crack Initiation Time Predictions The loading strain rate for the mode I simulations is ε˙xx = 0.000 05. The predicted crack initiation time for the mode I linear crack is tpred init = 41.51. Assuming small perturbations, the crack initiation time for the nonlinear mode I crack is predicted to be tpred init = 45.19. We may assume that the nonlinearity

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Atomistic Modeling of Materials Failure Theory prediction

Simulation result

Fracture loading mode Linear tpred Nonlinear tpred Linear tinit Nonlinear tinit init init I II

41.51 28.26

45.19 30.76

42.12 32.04

46.26 35.82

Table 6.4 Griffith analysis of the atomistic models, for mode I and mode II cracks, and different potentials. The predicted values based on continuum calculations agree well with the molecular dynamics simulation results

is localized to the crack tip, and the slab region can be described by small perturbation elastic properties. The values are summarized in Table 6.4. The initiation time decreases with stiffer systems (larger linear spring constants), faster loading, and smaller values of the fracture surface energy. For mode II, the loading rates are ε˙xx = 0.000 015 and ε˙xy = 0.000 2. We predict an initiation time for the crack in an harmonic solid tpred init = 28.26. For the nonlinear solid, we predict a slightly higher value tpred init = 30.76 because of the higher fracture energy. As before, we assume that the nonlinearity is localized to the crack tip, and the slab region can be described by small perturbation elastic properties. The values are summarized in Table 6.4. 6.8.4 Atomistic Simulation Results of a Suddenly Stopping Mode I Crack In the following we present the results for a suddenly stopping mode I crack. The plan is to start with the linear system, and subsequently move on to the nonlinear system. Harmonic Systems The crack propagating close to the Rayleigh velocity displays a distinct signature from cracks at lower speeds. We use the maximum principal stress field to analyze the simulation results. We find this field to be a simple and powerful measure to be compared with continuum solutions, because it displays a significant dependence on the propagation speed (and can therefore distinguish a static field from a dynamic field). The stress field close to the crack tip is best described by the asymptotic solution of continuum mechanics [22]. The field shows only one maximum for low speeds, and exhibits another maximum for sufficiently high velocities. The stress state ahead of the crack at high velocities is more complicated than at low velocities. The asymptotic field obtained by atomistic simulation is shown in Fig. 6.58a for the quasistatic case (v = 0, and low velocities), and for the case v = cR in Fig. 6.58b. Crack initiation time is determined as tinit = 42.12, in good agreements with the continuum theory prediction, 41.51. The loading is stopped at


269

Fig. 6.58 The asymptotic field of maximum principal stress near a moving crack tip (a), when v = 0, (b) dynamic field for v ≈ cR , (c) dynamic field for super-Rayleigh propagation velocities (v > cR )

tl = 72.8 by setting the strain rate to zero. The crack has a velocity v ≈ 6.55 before stopping, close to its limiting speed. The crack speed does not increase significantly even if the loading is kept for longer time. The maximum strain is εxx = 0.0073. The stress field, as well as numerical estimation of the crack velocity, clearly identifies a crack propagating close to Rayleigh velocity. In Fig. 6.59, the history of the crack length a(t) is shown.

Fig. 6.59 Crack extension history vs. time for the suddenly stopping linear mode I crack

Once the mode I crack is stopped, two circular waves are emitted from the crack tip. The first wave front corresponds to the longitudinal wave front, while the second one is the shear wave front. The Rayleigh surface wave can

270


be seen on the plane behind the crack. The static field was measured to spread out with a velocity v ≈ 7.05, and the longitudinal wave emitted by the stopped crack is propagating at v ≈ 12.2. Both values are, taking into account measurement errors, reasonably close to the continuum mechanics prediction. In the prospective crack plane, the stress field takes on its static counterpart immediately after the shear wave has passed. Behind the crack tip, the static field is established after the Rayleigh wave has passed. In other areas, the static field is only reached in the long-time limit. Continuous wave emission and rapid attenuation in regions surrounding the crack tip is observed. The frequency of these waves increases with time. The wave period attains atomic distance rapidly and elastic energy is dissipated as heat (thermalization). This is visualized in Fig. 6.60.

Fig. 6.60 Maximum principal stress field for various instants in time, mode I linear crack

It can be verified that at late stages (after the waves have attenuated), the static field inside the shear wave front remains constant, and no additional wave emission is identified (see lower right snapshot of Fig. 6.60). The results confirm the experimental observations in [23]: In Fig. 6.61a, the evolution of the maximum principal stress along the prospective crack line is shown. The evolution of potential energy is shown in Fig. 6.61b. The first kink in the plots refers to the longitudinal wave front and a second kink corresponds to the shear wave front at which the static field is radiated. Additional evidence is provided by different snapshots of the stress field after the crack has been stopped. These results are depicted in Fig. 6.60.


271

Fig. 6.61 Evolution of principal maximum stress and potential energy along the prospective crack line, for a linear mode I crack

The stresses ahead of the crack tip are closely related to the stress intensity factor KI . We choose a fixed location to measure the stress over time. A similar approach was used in experiment [23]. The result is depicted in Fig. 6.62. The plot also shows the results of experimental studies of a suddenly stopping mode I crack [23]. In both experiment and simulation, the stress decays slightly after the longitudinal wave has reached the measurement location, and increase again soon afterward. This decrease in stress is related to the arrival of longitudinal wave and persists even when we change the measurement location to

272


different y positions. The stresses continuously change until they reach the corresponding static solution. The result of Fig. 6.62 agrees qualitatively with experimental data, as can be verified in the plot (see also Figure 2 in [23]). In particular, we note that the minimum at time step t∗ ≈ 3 and the smaller, local minimum at time step t∗ ≈ 7 are qualitatively reproduced.

Fig. 6.62 Variation of stress at fixed distance ahead of the stopped linear mode I crack. At t ≈ 0, the longitudinal wave arrives at the location where the stress is measured. At t ≈ 8, the shear wave arrives and the stress field behind the crack tip is static. The plot also includes the results of experimental studies [23] of a suddenly stopping mode I crack for qualitative comparison (the time is fitted to the MD result such that the arrival of the shear wave and the minimum at t∗ ≈ 3 match)

These results show good agreement among atomistic simulations, continuum theories and experiments. The atomistic simulation demonstrated that the rapid thermalization of elastic waves near the crack tip did not change the basic nature of crack tip stress fields predicted by continuum mechanics. This result would not have been possible by continuum mechanics alone. Figure 6.63 displays the experimental results that have been used for comparison with the MD simulations. The data obtained from gage 1 (closest to the crack tip) have been used for comparison with MD results, as shown in Fig. 6.62. Anharmonic Systems It was shown that the harmonic solid reproduces continuum mechanics solutions, and may serve as a reference system when we further probe into nonlinear material behaviors. Atomistic simulations provide an extremely helpful tool to investigate the nonlinear case – a situation which usually cannot


273

Fig. 6.63 Experimental results of static stress field radiated in front of the crack tip. The measurement at gage 1 is used for comparison with MD results. Reprinted from [23] Engineering Fracture Mechanics, Vol. 15, pp. 107–114, B.Q. Vu and V.K. Kinra, Brittle fracture of plates in tension static field radiated by a suddenly stopping c 1981, with permission from Elsevier crack, copyright

be solved in closed form. We present simulations to address the following questions: • Does the result agree qualitatively and quantitatively with the linear solution? • What is the “wave” speed in the nonlinear case, that is, how fast can the static field be established? We start with a simple Griffith analysis to calculate the time for the onset of fracture due to the applied loading. Crack initiation time is found to be tinit ≈ 46.26. The initiation time agrees well with the prediction tpred init = 45.19. The loading is kept up to tl = 144. The maximum strain we achieve is εxx = 0.0144. The limiting speed observed is v ≈ 7.5. As soon as loading is stopped,

274


the velocity remains at the value it has at the moment where the strain rate is set to zero. We make a few remarks at this point: 1. The crack speed is significantly higher than in the linear case (7.5 vs. 6.55; this is about 16% higher limiting speed than predicted by the linear theory). 2. The crack speed is also larger than the corresponding wave velocity in the far-field. This second finding is in good agreement with other results we have obtained with bilinear hyperelastic potentials (see Chap. 6.6), as well as previous studies on the topic of hyperelastic brittle fracture [27, 248]. These results indicate that the local stress state at bond breaking is important [27, 219, 248]. The higher local wave speed leads to a higher limiting velocity. The crack can funnel energy faster than the far-field wave speeds would allow. 3. The dynamic maximum principal stress field provides signatures of nonlinear material response. For a super-Rayleigh crack, this field is shown in Fig. 6.58c. The stresses are higher compared to the harmonic case, and the angular variation of the asymptotic field is different. The histories of crack tip position and the velocity for the super-Rayleigh crack can be found in Fig. 6.64 which plots the limiting speed calculated from the atomistic simulations and visualizes how the crack accelerates and approaches its limiting speed. The history of maximum principal stress in the line ahead of the crack tip is shown in Fig. 6.65a, and the potential energy field is shown in Fig. 6.65b. Even in the nonlinear case, we can identify “bulk wave fronts” associated with a localized group of nonlinear waves. The distributions of stress and energy along y-direction are different in the linear and nonlinear cases. In very early stages, the shear wave front propagates with vy ≈ 8.6. Later, when the stress in y-direction is reduced at the crack tip, we measure vy ≈ 7.3. The bulk of longitudinal waves emitted from the stopped crack is moving at vy ≈ 15.3 in early stages, and at vy ≈ 12.3 later, approaching the linear sound velocity. This can be attributed to the fact that the material ahead of the stopped crack is not strained as severely in the y-direction. The propagation speed in the x-direction (orthogonal to the crack) remains higher than that in the linear case. The longitudinal wave front orthogonal to propagating direction is moving faster than ahead of the crack at late stages. Similar finding applies qualitatively to the shear wave front. The results show that the wave velocity depends on the stress state, and is significantly affected by the loading condition. The fact that the wave fronts propagate faster orthogonal to the propagation direction leads to elliptical wave fronts. In particular, the local wave speeds differ significantly from the linear elastic wave speeds. This observation is found in all of the nonlinear simulations. The discontinuities of the longitudinal and shear wave front are smeared out compared to the corresponding harmonic simulation. This observation is again consistent


275

Fig. 6.64 Crack tip history a(t) and crack tip velocity v as a function of time, suddenly stopping mode I crack. The limiting speed according to the linear theory is denoted by the black line (Rayleigh velocity), and the super-Rayleigh terminal speed of the crack in the nonlinear material is given by the blueish line. When the crack stops, the crack speed drops to zero

with the idea that there is no unique wave speeds near the crack tip. There exists a train of “longitudinal” and “shear” waves associated with the rapidly changing stress state near the crack tip. Consequently, the static field is not established as soon as the crack has stopped behind the shear wave. In one of the simulations, it takes δt ≈ 61 since stopping of the crack for the stresses to reach a static, constant value at δy = 15 ahead of the crack tip. This time is found to be shorter if the stresses are measured closer to the crack tip. For example, at δy = 5 ahead of the crack tip, the time to establish the static field is determined to be δt ≈ 30. The closer to the crack tip, the less the time required to establish the static field. The reason could be nonlinear wave dispersion. We emphasize that such large changes in the stress after the shear wave has passed are not observed in the linear case. The time it takes until the stresses do not change any more depends on the strength of the nonlinearity and on the amount of lateral loading. For the tethered LJ potential, it is observed that for more compliant systems and longer loading time this effect becomes more severe, presumably due to larger displacements and more nonlinear dispersion. As in the harmonic case, emission of elastic waves occurs

276


Fig. 6.65 Evolution of principal maximum stress and potential energy along the prospective crack line; for a mode I nonlinear crack

soon after crack arrest, and subsequent thermalization suppresses additional wave emission. The maximum principal stress field is shown in Fig. 6.66 for various instants in time. The discontinuities are smeared out, and it becomes evident that the definition of a unique wave front can be difficult. In the nonlinear case the shape of the wave fronts is different (elliptical vs. circular shape). The normalized maximum principal stress over time, recorded at a constant distance ahead of the crack tip, is plotted in Fig. 6.67 for an anharmonic


277

Fig. 6.66 Maximum principal stress field for various instants in time, for mode I nonlinear crack

simulation. One can observe the difference in shape compared to the linear case shown in Fig. 6.62.

Fig. 6.67 Variation of stress at fixed distance ahead of the stopped nonlinear mode I crack

278


Discussion – Mode I One important observation in the simulation is that a large number of waves are generated after crack stopping. This effect is less significant in the prospective propagation direction and becomes more pronounced in other directions. The waves attenuate quickly after the crack is stopped, and in the long-time limit become dissipated as heat (“thermalizing”). In the nonlinear case, we summarize the following findings: 1. There is no unique wave velocity, and the static field does not spread out behind the shear wave front. We find that there exists a train of “longitudinal” and “shear” waves associated with the rapidly changing stress state near the crack tip. The static field is not established until all waves have passed. 2. There is an anisotropy effect. Ahead of the crack, the wave speed approaches the linear limit, and orthogonal to the crack, the wave speed is significantly larger. We observe slightly elliptical wave fronts instead of circular wave fronts. 6.8.5 Atomistic Simulation Results of a Suddenly Stopping Mode II Crack We consider crack propagation under in-plane shear dominated loading, starting with the linear case and moving subsequently to the nonlinear case. For all mode II simulations, a mother–daughter mechanism to overcome the forbidden velocity zone is observed. This mechanisms is assumed to be governed by a Burridge–Andrew mechanism [254, 259]. A peak shear stress ahead of the crack continuously increases as the mother crack propagates through the material. Once this peak of shear stress reaches the cohesive strength of the interface, the daughter crack nucleates at some distance ahead of the mother crack and starts to propagate at an intersonic speed. The observation regarding the mother–daughter mechanism and the limiting speed of cl is consistent with the discussion in [165, 237]. Harmonic Systems The objective is to validate the theoretical results derived in [208, 217] using computer experiments. In particular, we will show that the static field does not spread out until the Rayleigh wave carrying the mother crack reaches the stopped daughter crack. We will determine the stresses slightly ahead of the crack tip from the molecular dynamics data and show similarity to the continuum solution. The linear solution will serve as the reference system when we probe into nonlinear material behaviors. The time to crack initiation is found to be tinit = 32.04, in good agreement to the continuum theory prediction tpred init = 28.26. The daughter crack nucleates at t = 82, this is, δt ≈ 50 later than the initiation of the mother crack.


279

Fig. 6.68 Schematic of waves emitted at a suddenly stopping mode II crack; (a) stopping of daughter crack, (b) stopping of mother crack

The loading is stopped at tl ≈ 84 soon after the daughter crack is nucleated. The mother crack hits the stopped daughter crack at t = 105. The mother crack propagates at v ≈ 6.5, and the daughter crack quickly attains a velocity v ≈ 12.3. The suddenly stopping intersonic crack shows the following sequence of events. The daughter crack is stopped, and the mother crack continues until it reaches the end of the weak layer. For each crack stopping event, two wave fronts are emitted yielding a total number of four wave fronts. The mechanism of the suddenly stopping intersonic crack is visualized schematically in Fig. 6.68a, b [208]. The stresses continuously change after the daughter crack is stopped [208]. Once the mother crack hits the daughter crack, stresses begin to increase dramatically. The static field radiates out from the crack tip with a velocity v ≈ 7.4. This velocity is the shear wave velocity and the observation provides good agreement to the prediction by continuum theory. Other propagation velocities measured from the data also agree with the continuum mechanics predictions. In Fig. 6.69a, the maximum principal stress is shown, and in Fig. 6.69b, the potential energy field is depicted some distance ahead of the crack tip. The potential energy field is shown in Fig. 6.70 at several instants in time. Figure 6.71 shows the normalized maximum principal stress at a fixed measurement location some distance ahead of the crack. When comparing this quantity to the stress intensity factor, care must be taken because the singularity changes continuously as the crack passes through the distinct intersonic velocity phases. The arrival of the shear wave front is characterized by a strong discontinuity induced by the Rayleigh wave [165, 208, 217]. As soon as the shear wave reaches the measurement location, the static field is established and the stress no longer changes afterwards. The simulation results are consistent with the analysis in [208].

280


Fig. 6.69 Evolution of (a) principal maximum stress and (b) potential energy along the prospective crack line; for linear supersonic crack

Anharmonic Systems The linear solution has reproduced results similar to the continuum mechanics solution of the problem. We further consider the dynamics of a suddenly stopping crack in a nonlinear material.


281

Fig. 6.70 Potential energy field for various instants in time, mode II linear crack

Fig. 6.71 Variation of stress at fixed distance ahead of the stopped intersonic mode II crack

The time to crack initiation is determined to be tinit = 35.82, which is somewhat larger than the prediction tpred init = 30.76. The loading is stopped at tl = 129.6, after nucleation of the daughter crack at t ≈ 112. This leads to a far-field strain of εxx = 0.0039 and εxy = 0.052. The crack propagation speeds we measure are v ≈ 8 for the mother crack, and v ≈ 16.8 for the daughter crack. Both velocities are higher than the corresponding velocities of the linear case (22% higher, and 36% higher for mother and daughter crack, respectively) due to material nonlinearities. The daughter crack propagates supersonically through the material. Figure 6.72 plots the crack extension

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Fig. 6.72 Crack extension history vs. time for the supersonic mode II crack. The dashed line is used to estimate the time when the mother crack comes to rest

history a(t). In this figure, the mother–daughter mechanism can be identified straightforwardly. The mother crack hits the daughter crack at t ≈ 197, δt ≈ 77 after the nucleation of the daughter crack. This can also be estimated from Fig. 6.72. Like in the simulations of a mode I crack in nonlinear material, the wave fronts are smeared out, but we can still identify a localized group of nonlinear waves which may be interpreted as a “nonlinear wave front.” The first waves emitted after stopping of the crack can be regarded as the longitudinal wave moving with a velocity v ≈ 16.4 through the solid in the prospective crack line. The strong spike ahead of the crack corresponds to the shear wave induced by the stopped daughter crack and is propagating with a velocity v ≈ 10.5. Both values are higher than the corresponding linear shear wave velocity. The propagation speed of the longitudinal wave is close to the velocity of the daughter crack before stopping. The stopping mother crack leads to nucleation of additional waves. A discontinuity propagating at v ≈ 10.5 is observed, which is emitted when the mother crack is stopped. We associate this with the shear wave front. It is difficult to observe the longitudinal wave front from the molecular dynamics data in this case. Plots of the maximum principal stress field and the potential energy field ahead of the crack tip in the prospective crack line are shown in Figs. 6.73a, b. The plots look quite different from the harmonic case. The shape we have observed in the harmonic counterpart, in particular the strong discontinuity before arrival of shear wave front seems to have disappeared. The discontinuity associated with the shear wave front of the daughter crack is more pronounced than in the linear case. The fact that the discontinuities are smeared out can lead to stresses changing significantly even after the bulk of the shear waves of the mother crack has passed. This effect is reminiscent of the phenomenon


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Fig. 6.73 Evolution of (a) principal maximum stress and (b) potential energy along the prospective crack line; for nonlinear supersonic crack

observed in the nonlinear mode I crack. In this simulation, the static stresses are reached soon after the bulk of shear waves has passed. The delay required to establish the static field after the shear waves have passed, which we have observed in the nonlinear mode I case, is not observed in this simulation, but

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appears in different simulations when the impact of nonlinearities is stronger due to higher lateral strains (see further discussion below).

Fig. 6.74 Potential energy field around the crack tip for various times, suddenly stopping mode II crack

The potential energy field around the crack tip is shown in Fig. 6.74 for various times. The velocity of the wave fronts depends on the angle. This leads to elliptical shapes of the wave front, as also observed in the mode I case. Like in previous simulations, a thermalization effect is observed. In particular, after stopping of the daughter crack, energy is found to be dissipated as heat, as can be verified from Fig. 6.74 (right top). The normalized stresses σ ∗ vs. time are shown in Fig. 6.75 for an anharmonic simulation. Comparing this result to the linear case, we find similarity until the shear wave arrives. Unlike the linear case in which stresses decrease strongly after the mother crack arrives, this stress remains constant until the mother crack arrives and then quickly increase to the steady state value. To investigate the nonlinear dynamics further, we will briefly discuss a second simulation. To obtain softer elastic properties, we alter the tethered LJ potential and choose 0 = 1. The corresponding harmonic wave velocities are cR = 4.8, cs = 5.2 and cl = 9. As we mention in Sect. 6.8.4, this softening in combination with higher lateral strain allows us to study the nonlinear effect better. The loading is kept for a longer time than in previous simulations. All other parameters are kept constant. The daughter crack achieves a speed of v ≈ 12 and is truly supersonic. The mother crack propagates at v ≈ 5.8, which is super-Rayleigh. The wave fronts become very difficult to identify, because they are smeared out more than in the previous simulation. The shape of the discontinuities is clearly elliptical. The longitudinal wave front


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Fig. 6.75 Normalized stresses σ ∗ vs. time, suddenly stopping supersonic mode II crack

of the stopped daughter crack is propagating with v ≈ 11.95, and the shear wave front is associated with a strong discontinuity propagating at v ≈ 7.80 through the solid. We remark that the velocity of the longitudinal wave front is almost identical to the propagation velocity of the daughter crack. Once the mother crack reaches the stopped daughter crack, there is no more significant discontinuity. This is very different from the harmonic case where each wave front is clearly identified by a distinct discontinuity. We find that the mother crack, represented by a surface wave after the secondary crack is nucleated, is smeared out in the nonlinear case. Therefore the arrival of the mother crack is not the arrival of a singularity, but of a distributed stress concentration. The static field is not instantaneously established. The stress intensity continuously increases from the point where the daughter crack has been stopped until it becomes fully steady state. A slight increase of stresses until all waves have passed is observed, just as in the nonlinear mode I case. This also helps to explain the difference in shape of energy and stress distribution in the prospective crack line, when we compare the harmonic with the anharmonic simulations. Moreover, the static field is not established until all waves have passed. Discussion – Mode II We start with the results for the linear case: 1. It was shown that the static field does not establish until the mother crack reaches the stopped daughter crack for the linear reference system.

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2. After the mother crack has reached the stopped crack tip, the stress field (as well as the energy field) ahead of the crack tip is static behind the shear wave front in, and is static behind the Rayleigh surface wave behind the crack. Both observations match the prediction by continuum mechanics. In the nonlinear case, the definition and observation of the longitudinal wave front is difficult. The wave fronts are not as sharp and discontinuities not as strong, which may cause the field behind the shear wave front of the mother crack to change continuously during some transition time until all waves have passed. 6.8.6 Discussion We have studied suddenly stopping cracks by atomistic simulations. We considered a plane-stress elastic solid consisting of a two-dimensional, triangular atomic lattice. For the interatomic interactions, we assumed a tethered Lennard-Jones potential, as well as a harmonic potential. We presented four simulations for mode I and mode II loading conditions, and linear and nonlinear simulation potentials. In addition to the atomistic simulations, we have done continuum analyses to determine the crack initiation time and wave velocities associated with the interatomic potentials. The harmonic atomistic simulations have shown good agreement with the continuum mechanics and experimental results. We summarize the main results below. 1. In mode I, the static solution spreads out as soon as the crack is stopped. This is in agreement to the continuum theory [234]. 2. In mode I, experimental results and atomistic studies show similar results on very different scales [23] (see Fig. 6.62). 3. In mode II, the static solution spreads out as soon as the mother crack has reached the stopped daughter crack. The nature of the mode II intersonic crack is very different from the sub-Rayleigh crack [208]. 4. In both mode I and mode II, we observe emission of waves from the stopped crack. These waves attenuate quickly, in which process energy is dissipated as heat. This does not change the basic nature of stress fields near the crack tip as predicted by continuum mechanics. The anharmonic simulations give somewhat different results, which cannot be explained by linear elastic fracture mechanics only. For the nonlinear case, we would like to summarize the main findings as follows. 1. In both mode I and mode II, the wave fronts and discontinuities are not as sharp as in the linear case. This nonlinear dispersion effect could cause the field to change continuously even after the main wave discontinuities have passed. 2. For supersonic mode II cracks, the mother crack could transform from a stress singularity into a less localized stress distribution. The static field


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is not instantaneously established but stresses increase continuously, until they eventually reach steady state. 3. The nonlinearity can lead to an anisotropy effect of wave propagation. Instead of circular wave fronts, we find elliptical wave fronts. The simulations show that the linear model is a reasonable approximation even when moderate nonlinearities are present. The simulations provide evidence that the crack behaves like a massless particle only in the sub-Rayleigh regime. The Griffith criterion works well in all simulations we presented.

6.9 Crack Propagation Along Interfaces of Dissimilar Materials In this section, we review studies of cracks propagating along interfaces between two dissimilar materials, as schematically shown in Fig. 6.76. Cracks at interfaces are technologically important, since the bonding between two dissimilar materials as for instance between epoxy and aluminum is usually weak and serves as a potential failure initiation point of a structure. The atomistic model featuring the weak layer could be regarded as an idealization of such cases. Another important field where interfaces between dissimilar materials play an important role is the dynamics of earthquakes.

Fig. 6.76 Geometry of the simulations of cracks at bimaterial interfaces (note that in many of the subsequent plots the crystal slab is rotated by 90 degrees)

Several theoretical studies were carried out on cracks in dissimilar media [260–262]. Most of the early investigations focused on static cracks. One of the interesting features of the elastic interfacial crack problem is the characteristic oscillating stress singularity that was determined by Williams [263]. This theoretical finding is incompatible with real materials since the crack faces would penetrate each other at the crack tip. The stress intensity factor is complex-valued for interfacial cracks and it is generally difficult to define a crack nucleation criterion based on the Griffith condition [260]. In recent years, the dynamics of cracks along dissimilar interfaces was increasingly researched. For instance, the asymptotic stress field near dynamic cracks at bimaterial interfaces was studied [264, 265]. The analysis discussed

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in [264] assumed steady-state crack propagation and provided the spatial structure of square-root singular stress field very close to the dynamic crack tip. The analysis led to definition of a complex dynamic stress intensity factor. Later, this analysis was refined relaxing the steady-state assumption and including higher order terms [265]. There are also experimental results available on interfacial cracks, as for instance studies reported in [216, 266, 267] of cracks propagating along interfaces of PMMA and metals. In [266], the researchers focused on the development of a crack growth criterion along interfaces. They also compared the experimental results with theoretical predictions of the stress field near the crack tip. Crack speeds that exceeded that of the shear wave speed of the soft PMMA material were observed. Rather few molecular dynamics analyses of dynamic fracture along bimaterial interfaces have been reported. One example is recent molecular dynamics simulations of mode II cracks along a weakly bonded interface of harmonic– anharmonic materials (material defined by a harmonic potential neighboring a material defined by a tethered LJ potential) [219]. In summary, for cracks at interfaces, existing theory and experiment predicts that • The limiting speed of mode I cracks at bimaterial interfaces can exceed the Rayleigh wave speed of the soft material [265, 268]. However, intersonic or supersonic crack propagation with respect to the soft material layer is not predicted by theory and has not been observed in experiment. • The limiting speed of mode II cracks is given by the longitudinal wave speed of the stiff material and the crack speed can thus be truly supersonic with respect to the soft material [267]. The most important research objective of the following studies is the limiting speed of cracks: The fact that the wave speed changes discontinuously across the interface makes it difficult to define a unique wave speed near the interface, and thus difficult to predict the limiting speed of the crack. Using molecular dynamics, can we determine what is the limiting speed of a crack along dissimilar materials? For mode II cracks in an earlier study a mother–daughter–granddaughter mechanism was observed through which the crack finally approached a velocity faster than the longitudinal wave speed of one of the layers [219]. In this setup, however, one of the half spaces was modeled by harmonic interactions, and the other was modeled by a tethered LJ potential. Although this setup constitutes an interface of different materials, the wave speeds associated with each half space could not be clearly defined since one of the material was hyperelastic. To obtain a more clean model of cracks at interfaces, we propose to study two half spaces with harmonic interatomic potentials, but with different spring constants k0 < k1 . The ratio k1 Ξ= (6.124) k0


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Fig. 6.77 Crack tip history and crack velocity history for a mode I crack propagating at an interface with Ξ = 10. Subplot (a) shows the crack tip history, and subplot (b) shows the crack tip velocity over time. A secondary daughter crack is born propagating at a supersonic speed with respect to the soft material layer

measures the elastic mismatch of the two materials, and the wave speeds are √ thus different by a factor Ξ. The plan of this section is as follows. We start with simulations of mode I cracks along interfaces and show that under sufficiently large loading, the crack approaches the Rayleigh wave speed of the stiffer of the two materials via a mother–daughter mechanism. We continue with a study of mode II cracks along interfaces. It will be shown that a mother–daughter–granddaughter mechanism, in agreement with previous analyses [219], exists and allows the crack to approach the longitudinal wave speed of the stiffer of the two materials. We finally discuss the simulation results and compare the elastic fields of the mode I crack with the solution of continuum mechanics results and the results of atomistic modeling as reported earlier. 6.9.1 Mode I Dominated Cracks at Bimaterial Interfaces In the simulations, the left (upper) part of the slab is the stiff solid, while the right part has lower Young’s modulus and is soft. We consider the case

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Fig. 6.78 The plot shows the stress fields σxx , σyy , and σxy for a crack at an interface with elastic mismatch Ξ = 10, before a secondary crack is nucleated. In contrast to the homogeneous case, the deformation field is asymmetric. The dark grey shades corresponds to large stresses, and the lighter grey shades to small stresses

when the elastic mismatch Ξ = 10. For comparison, the elastic mismatch as between PMMA and aluminum is about 15. Figure 6.77a shows the crack tip history, and b shows the crack tip velocity over time. The crack nucleates at time t ≈ 35, and quickly approaches the Rayleigh speed of the soft material v → cr,0 ≈ 3.4. As loading is increased, the crack speed increases slightly and becomes super-Rayleigh. We observe a large jump in the crack velocity at t ≈ 110, when a secondary crack is nucleated which quickly approaches the √ Rayleigh speed of the stiff material v → Ξcr,0 ≈ 10.7517 > cl,0 ≈ 6.36. The secondary crack is nucleated approximately at a distance ∆a = 11 ahead of the mother crack and propagates with Mach 1.7 through the material! Nucleation of secondary cracks under mode I loading is only found under high-strain rate loading (ε˙xx = 0.000 05). If the strain rate is too low, the crack moves at a super-Rayleigh speed until the solid has separated. The mechanism of nucleation of a secondary crack is reminiscent of the mother–daughter mechanism, a phenomenon so far only observed in cracks under mode II loading. The result suggests that at a bimaterial interface, mode I cracks under very large loading can propagate with the Rayleigh speed of the stiffer materials, and cracks can reach speeds beyond the fastest wave speeds in the soft material. This observation is surprising and has not been reported in experiment so far [216]. In experimental studies of mode I cracks along interfaces, the crack slightly exceeds the Rayleigh speed of the soft material but is never observed to become intersonic or supersonic. Figure 6.78 shows the stress field, and Fig. 6.79 the particle velocity field before the secondary crack is nucleated. At the time the snapshots are taken,


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Fig. 6.79 The plot shows the particle velocity field (a) u˙ x and (b) u˙ y for a crack at an interface with elastic mismatch Ξ = 10, before a secondary crack is nucleated. The asymmetry of the particle velocity field is apparent

Fig. 6.80 The plot shows the potential energy field for a crack at an interface with elastic mismatch Ξ = 10. Two Mach cones in the soft solid can clearly be observed. Also, the mother and daughter crack can be seen. In the blow-up on the right, the mother (A) and daughter crack (B) are marked

the crack propagates at a super-Rayleigh speed through the material. Since crack motion is subsonic, no shock front is established. Figure 6.80 shows the potential energy field for a crack after the secondary crack is nucleated and crack motion of the daughter crack is supersonic. Figs. 6.81 and 6.82 show the stress field and the particle velocity field. The

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Fig. 6.81 The plot shows the stress fields σxx , σyy , and σxy for a crack at an interface with elastic mismatch Ξ = 10. In all stress fields, the two Mach cones in the soft material are seen. The mother crack appears as surface wave behind the daughter crack

secondary crack propagates supersonically through the material and the Mach cones in the right half space (soft material) is clearly visible.

Fig. 6.82 The plot shows the particle velocity field (a) u˙ x and (b) u˙ y for a crack at an interface with elastic mismatch Ξ = 10. The shock fronts in the soft solid are obvious

The mother–daughter mechanism in mode I cracks at interfaces is also observed for elastic mismatch Ξ = 2, 5, 7, and 10 (note that not in all cases crack motion is supersonic with respect to the soft material since the Rayleigh


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wave speed of the stiff material is smaller than the longitudinal wave speed of the soft material).

Fig. 6.83 Atomic details of nucleation of the secondary crack under tensile dominated loading. The plot shows the shear stress field σxy near the crack tip. Atoms with the energy of a free surface are drawn as larger atoms. The plot suggests that a maximum peak of the shear stress ahead of the crack tip leads to breaking of atomic bonds and creation of new crack surfaces. After the secondary crack is nucleated (see snapshots (2) and (3)), it coalesces with the mother crack and moves supersonically through the material (snapshot (4))

The secondary crack is nucleated approximately at a distance 11 atomic distances ahead of the mother crack and rapidly propagates at Mach 1.7 with respect to the soft material. If the strain rate is too low, the crack moves at a sub-Rayleigh speed (soft material) until the solid has separated, without nucleation of secondary cracking. In Fig. 6.83, we plot the shear stress field σxy for several instants in time during nucleation of the secondary crack (note we chose quite small time intervals between the snapshots). Atoms with the energy of a free surface are colored blue and highlighted by larger spheres. The plot suggests that a peak of shear stress ahead of the crack tip may have caused breaking of atomic bonds. After the secondary crack is nucleated, it coalesces with the mother crack and moves supersonically with respect to the soft material. The mechanism of nucleation of a secondary crack is reminiscent of the mother–daughter mechanism predicted by the Burridge– Andrew mechanism and observed in studies of intersonic mode II cracks in homogeneous solids (see discussion in previous sections). It is noted that the

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location of the maximum tensile stress σyy does not coincide with the location of nucleation of the secondary crack; rather, the latter was found to correlate with a peak in shear stress ahead of the mother crack. We therefore conclude that there exists a peak in shear stress ahead of a mode I dominated interfacial crack moving at the Rayleigh wave speed of the soft material and this peak shear stress leads to subsequent nucleation of a secondary crack which breaks the sound barrier at the soft Rayleigh-wave speed. 6.9.2 Mode II Cracks at Bimaterial Interfaces

Fig. 6.84 Crack tip history for a mode II crack propagating at an interface with Ξ = 3. The plot illustrates the mother–daughter–granddaughter mechanism. After a secondary daughter crack is born travelling at the longitudinal wave speed of the soft material, a granddaughter crack is born at the longitudinal wave speed of the stiff material. The granddaughter crack propagates at a supersonic speed with respect to the soft material layer

For the studies of mode II cracks along interfaces we choose Ξ = 3. The crack tip history is depicted in Fig. 6.84. The loading rates are ε˙xx = 0.000 03 for slight mode I opening loading and for the shear loading ε˙xy = 0.000 125. Fig. 6.85 shows the crack tip velocity history during the mother–daughter– granddaughter mechanism. The plot is obtained by numerical differentiation of the crack tip history shown in Fig. 6.84. The crack speed changes abruptly at the nucleation of the daughter crack, and rather continuously as the granddaughter crack is nucleated. Initially, the (mother) crack propagates close to the Rayleigh velocity of the soft slab part (v ≈ 4.8). After a secondary daughter crack is born that travels at the longitudinal wave speed of the soft material, a granddaughter crack is


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Fig. 6.85 Crack tip velocity history during the mother–daughter–granddaughter mechanism, for elastic mismatch Ξ = 3. The plot is obtained by numerical differentiation of the crack tip history shown in Fig. 6.84. The crack speed changes abruptly at the nucleation of the daughter crack, and rather continuously as the granddaughter crack is nucleated. Characteristic wave speeds for the stiff and soft solid are indicated in the plot

Fig. 6.86 Supersonic mode II crack motion at a bimaterial interface, stiffness ratio Ξ = 3. Subplot (a) depicts the potential energy field of a mode II crack at a bimaterial interface with Ξ = 3, supersonic crack motion. (A) mother crack, (B) daughter crack, and (C) granddaughter crack. Subplot (b) shows the allowed limiting speeds and the observed jumps in the crack speed

born at the longitudinal wave speed of the stiff material. The granddaughter crack propagates at a supersonic speed with respect to the soft material layer. If the loading is stopped after the granddaughter crack has nucleated, this velocity is maintained until the whole slab is cracked. For a choice of Ξ = 2,

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Fig. 6.87 The plot shows the potential energy field near a shear loaded interface crack with stiffness ratio Ξ = 3 (different shades of grey are used to indicate different levels of stress). The plot shows a small section around the crack tip. The crack surfaces are highlighted. In the upper left plot, the initial configuration with the starting crack is shown. As the loading is increased, the mother crack starts to propagate, eventually leading to secondary and tertiary cracks. Two Mach cones in the soft solid and one Mach cone in the stiff solid can be observed in the lower right figure, suggesting supersonic crack motion with respect to the soft material and intersonic motion with respect to the stiff material

the qualitative behavior is the same. In Fig. 6.86a we depict the potential energy field near a supersonic mode II crack along a bimaterial interface. We mark the different cracks: (A) is the mother crack, (B) is the daughter crack, and (C) refers to the granddaughter crack. Figure 6.86b shows a schematic of the allowed limiting speeds and the observed jumps in crack speed. Figure 6.87 shows a few snapshots of the potential energy field from the initial configuration until the birth of the granddaughter crack, illustrating the dynamics of the crack propagation mechanism. The stress fields for two different instants in time are shown in Fig. 6.88. Figure 6.88a shows the stress field before nucleation of the daughter crack, and Fig. 6.88b shows the stress field after nucleation of the granddaughter crack. Figure 6.88c shows a magnified view into the crack tip region. Intersonic cracks along bimaterials interfaces have also been observed in the experiment [267]. Figure 6.89 depicts snapshots from such experimental


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Fig. 6.88 The plot shows the σxx field of a mode II crack at a bimaterial interface with Ξ = 3. Subplots (a) and (b) are consecutive time steps, and subplot (c) is a blowup

studies, here carried out for mode II dominated cracks at Al–Homalite interface. The Mach cone are clearly visible in the soft material. 6.9.3 Summary The studies reported in this section show that cracks at interfaces show a very different dynamics than cracks in homogeneous materials. At the interface, the limiting crack speed is not well defined any more since the wave velocities change discontinuously across the interface. Both mode I and mode II cracks can propagate supersonically with respect to the wave speeds in the soft material. We summarize the main findings. • In mode I, it is observed that the limiting speed of cracks at bimaterial interfaces is the Rayleigh wave speed of the stiff material. The nucleation of a secondary daughter crack from the primary mother crack is observed. Supersonic crack motion with respect to the soft layer is possible, and the mother–daughter crack mechanism is reminiscent of the observations in mode II cracks in the homogeneous case. This is a new phenomenon in dynamic fracture not observed in experimental studies so far. It is also in contrast to published experimental results [216]. Preliminary continuum mechanics analysis stimulated by the atomistic simulation results provides theoretical evidence that this dynamical phenomena is possible. The analysis revealed that the energy release rate is positive for crack motion close to the Rayleigh speed of the stiff material [269]. • In mode II, it is observed that the limiting speed is the longitudinal wave speed of the stiff material. Supersonic crack motion with respect to the soft layer is possible, and then we observe a mother–daughter–granddaughter

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Fig. 6.89 An interfacial crack rupturing the bond between Homalite and aluminium, experimental results. Subplot (a) shows the loading geometry, illustrating how shear loading is induced by impact loading of the lower, stiffer material. Subplot (b) shows the subsonic growth phase and subplots (c) and (d) display the intersonic crack growth phase. Reprinted from Advances in Physics, Vol. 51(4), A. Rosakis, c 2002, with Intersonic shear cracks and fault ruptures, pp. 1189–1257, copyright permission from Taylor and Francis

mechanism [219]. This agrees with results of earlier computer simulations. The results also confirm theoretical [265] as well as experimental results [267]. • The elastic fields in mode I and mode II cracks are very different from homogeneous materials. If crack propagation is supersonic with respect to one of the half spaces, multiple shock fronts are observed as shown in Fig. 6.88. If the elastic mismatch is small or nucleation of daughter cracks is suppressed in mode I cracks, the elastic fields in the left and right half are asymmetric. The asymmetric shape of the asymptotic deformation fields matches the predictions by continuum mechanics theories [265].


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Atomistic simulations are a feasible approach to study the dynamics of cracks at interfaces. Future investigations could focus on the comparison of the asymptotic stress field in simulation and theory, as well as on a more detailed and theoretical analysis of the observed mother–daughter and mother–daughter–granddaughter mechanisms, particularly focusing on the nucleation process of secondary cracks.

6.10 Dynamic Fracture Under Mode III Loading In this section, we study three-dimensional models of mode III cracks. A schematic of the mode III antiplane shear crack loading is shown in Fig. 6.6. The study of mode III cracks is motivated by the fact that in mode III, there exists only one wave speed associated with crack dynamics, the shear wave speed cs . This simplifies the theoretical continuum mechanics analysis of the crack dynamics. Recently, a closed form solution for the crack speed of a crack propagating in a stiff material layer embedded in a soft matrix was derived [24]. The analysis revealed that the same concept of a characteristic energy length scale χ also holds for mode III cracks. The most important objective of this chapter is therefore to validate this finding using atomistic simulations similar to those presented in Sect. 6.6.5. Mode III cracks have rarely been studied with molecular dynamics methods before. Some simulations were reported in the literature, but these focused on cracks in ductile materials under mode III loading [270, 271]. According to classical linear elastic theories [22], for mode III cracks all velocities below the shear wave speed are admitted, thus v ≤ cs .

(6.125)

The allowed crack propagation speeds for mode III cracks in linear and nonlinear solids are shown in Fig. 6.90. Similar to the results of mode I and mode II cracks where cracks can move faster than the wave speeds in the material, mode III cracks can also move faster than the shear wave speed and thus become supersonic once the material stiffens with strain. This phenomenon was verified using a tethered LJ potential (results not shown here).

Fig. 6.90 Allowed velocities for mode III crack propagation, linear and nonlinear case

The objectives of the studies in this section are summarized as follows. First, we verify that the limiting speed associated with a crack propagating

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in a harmonic lattice agrees with the theoretical prediction. We then discuss simulation results of crack motion in a thin stiff layer embedded in a soft matrix, also yielding supersonic crack motion (similarly to the Broberg problem discussed in Chap. 6.6). The recently derived analytical continuum mechanics solution of the problem is quantitatively compared with the molecular dynamics results [24]. We find that the energy length scale described in Chap. 6.6 also applies to mode III cracks. 6.10.1 Atomistic Modeling of Mode III Cracks Previous studies have provided evidence that 3D molecular dynamics is a good framework to investigate the dynamics of fracture. For instance, Abraham and coworkers [156] studied dynamic fracture in a three-dimensional solid with LJ interactions. They showed that unlike in two dimensions where the LJ potential yields a very brittle solid (see Fig. 6.48), in three dimensions the LJ potentials leads to a very ductile solid [138]. The researchers studied the dynamics of fracture in different crystal orientation and provided a Schmidt factor analysis [272]. Later, a three-dimensional model using harmonic interactions in the bulk, and using the concept of a weak fracture path was adopted in simulations of dynamic crack propagation [219]. This model corresponds to a perfectly brittle system which allows to study the dynamics of fracture in a clean environment. Here we adopt a similar approach and confine crack motion along a weak layer, which is characterized by a fracture surface energy much smaller than in the bulk. This confined fracture path helps to avoid crack branching and allows to purely focus on the dynamics of cracks. In previous studies, a weak LJ layer was used to model the weak fracture layer [219]. Here we assume a homogeneous material with harmonic interactions. The interactions are defined according to (4.4.3) in the bulk, and according to (4.43) across the weak fracture layer. The slab is initialized at zero temperature and loaded according to mode III, and we also give a slight mode I loading. The loading rates are ε˙xx = 0.000 1 for mode I and the (engineering) shear rate γ˙ xz = 0.000 2. The loading is kept up during a loading time tl , and then the boundaries are held fixed. 6.10.2 Mode III Cracks in a Harmonic Lattice – The Reference Systems The results show that the limiting speed of mode III cracks in a harmonic lattice is given by the shear wave speed. This was verified for two choices of the spring constant k1 ≈ 57.32 and k1 = k0 /2. This observation is in agreement with the predictions by continuum mechanics theories [22]. The crack tip history for the soft and stiff reference system is shown in Fig. 6.91. In both systems, the loading is stopped at tl = 135. Both soft and stiff systems approach the theoretical limiting speeds. Fracture initiation times are tsoft init ≈ 47 for the soft system and tstiff ≈ 41 for stiff system. init


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Fig. 6.91 Crack tip velocity history for a mode III crack propagating in a harmonic lattice for two different choices of the spring constant ki . The dotted line shows the limiting speed of the stiff reference system, and the dashed line shows the limiting speed of the soft reference system. Both soft and stiff systems approach the corresponding theoretical limiting speeds

6.10.3 Mode III Crack Propagation in a Thin Stiff Layer Embedded in a Soft Matrix Here we use the same geometry as shown in Fig. 6.45, with the difference that the slab is predominantly under mode III loading. The main objective is to compare the molecular dynamics simulation results of the curve v(h/χ) with the theoretical prediction. According to theory [24], the energy release rate for a crack propagating in a stiff layer with width h is given by G=

2 hσxz f (v, c0 , c1 ) µ

(6.126)

where f is a function only of the elastic properties of the layer and matrix material as well as the crack propagation velocity. Using the Griffith condition G = 2γ, (6.126) can be numerically solved for v. Therefore, the crack velocity can be expressed as v = f˜(c0 , c1 , h/χ)

(6.127)

where χ=β

γµ 2 σxz

(6.128)

denotes the characteristic energy length scale. The characteristic energy length scale is defined such that h/χ equals 1 when the increase in crack speed is 50% of the difference between the shear wave speed of soft and stiff material.

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Fig. 6.92 Mode III crack propagating in a thin elastic strip that is elastically stiff. The potential energy field is shown while the crack propagates supersonically through the solid. The stiff layer width is h = 50

Most importantly, the crack speed should only depend on the ratio of the layer width h to the characteristic energy length scale χ. According to the values of γ, µ, and the applied shear stress σxz for loading 2 time of tl = 135, γµ/σxz ≈ 1. Figure 6.92 shows a mode III crack propagating in a thin elastic strip which is elastically stiff. The crack propagates supersonically through the solid, and the stiff layer width is h = 50. Figure 6.93 depicts the results of a set of calculations to check of the scaling law for mode III dynamic fracture. The continuous line corresponds to the analytical continuum mechanics solution, and the data points are obtained for different simulation conditions. In the molecular dynamics simulations, the loading σxz , the fracture surface energy γ as well the elastic properties E are changed independently. The results show that all velocities fall on the same curve. From comparison of molecular dynamics results to the continuum solution, the parameter β ≈ 11 and therefore χ ≈ 11. When the inner layer width h approaches this length scale, the crack speed has increased 50% of the difference between soft and stiff shear wave speed. For realistic experimental conditions under 0.1% shear strain and a crack propagating within a thin steel layer, χ is on the order of millimeters. Further details will be included in a forthcoming publication [273].


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Fig. 6.93 Check of the scaling law of the mode III Broberg problem. The continuous line refers to the analytical continuum mechanics solution [24] of the problem. The parameters γ0 = 0.1029 and τ0 = 0.054

6.10.4 Suddenly Stopping Mode III Crack In Chap. 6.8, we discussed suddenly stopping mode I and mode II cracks in linear and nonlinear materials. We have conducted similar studies for a suddenly stopping mode III crack. Theory predicts that the dynamics of the suddenly stopping mode III crack is very similar to the mode I crack [22]. An important difference of the suddenly stopping mode III crack to the mode I case is that the static field spreads out in the whole area around the crack tip, and not only in the line ahead of the crack tip as in mode I [22]. Figure 6.94 shows the potential energy field close to a suddenly stopping mode III crack. The simulation technique is the same as described in Chap. 6.8 with the only difference that a three-dimensional model is used. The result is very reminiscent of the mode I simulation results discussed in Chap. 6.8. The static field spreads out with the shear wave speed as soon as the crack is stopped. In snapshot “1” of Fig. 6.94, the crack propagates at a velocity close to the shear wave speed prior to stopping. Behind the crack, the static field is transported with the Rayleigh wave speed. The Rayleigh surface wave can clearly be observed in Fig. 6.94, snapshots “3” and “4.” The static stress field spreads out in the whole area around the crack tip, and not only in the line ahead of the crack tip as in mode I [22]. 6.10.5 Discussion The limiting speeds of mode III cracks are found to be in agreement with the predicted velocities from the continuum mechanics analysis. With the results discussed in this chapter, it is concluded that for all three modes of loading,

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Fig. 6.94 Suddenly stopping mode III crack. The static field spreads out behind the shear wave front after the crack is brought to rest

the predicted limiting speeds agree well with the observation in atomistic simulations. The most important result of this chapter is that the scaling law found for mode I cracks also holds for the mode III case. A quantitative comparison with the theory provided good agreement. This result strongly corroborates the concept of the energy length scale proposed earlier. The results also suggest, in accordance with the continuum analysis of the problem, that supersonic mode III crack motion is possible [24]. The results are also in agreement with recent theoretical analysis of supersonic mode III crack propagation in nonlinear stiffening materials [230]. Preliminary molecular dynamics simulations of crack motion in a material defined by the tethered LJ potentials have also shown supersonic mode III crack propagation. Further results of a suddenly stopping mode III crack agree qualitatively with the continuum mechanics prediction of a suddenly stopping mode III crack. The static field is found to spread out behind the shear wave front nucleated by the stopping crack. In harmonic lattices, the mode III carries no inertia, just like the mode I crack and the sub-Rayleigh mode II crack. These results are in agreement with continuum theory of dynamic fracture [22].

6.11 Brittle Fracture of Chemically Complex Materials In the previous sections we have reviewed a series of studies, primarily using a model potential approach. These studies provided a comparison between atomistic approaches and continuum theory, and led insight into the significance of the large-deformation interatomic bonding properties or hyperelasticity for the dynamics of cracks in brittle materials. Here we extend these studies to materials in which the atomic bonds forming the lattice are


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more complex, focusing on a case study of a brittle material whose fracture mechanics has received considerable attention, silicon. We will review a study of dynamic fracture in a silicon single crystal in which the ReaxFF reactive force field is used for several thousand atoms near the crack tip, while more than 100,000 atoms of the model system are described with a computationally less expensive Tersoff force field. The hybrid simulation method introduced in Sect. 5.4.5 is used for the simulation studies reviewed here [274, 275]. Since the ReaxFF force field is completely derived from quantum mechanical calculations of simple silicon systems without any empirical parameters, this model provides a fundamental, quantum mechanics based description of fracture of silicon. It will be shown that the results reproduce experimental observations of fracture in silicon including differences in crack dynamics for loading in the [110] or [100] orientations, as well as dynamical instabilities with increasing crack velocity. Further, a correlation of the atomistic simulation results with single silicon crystal fracture experiments will be presented. This study reveals that after the critical fracture load is reached, the crack speed instantaneously jumps from zero to approximately 2 km s−1 . The simulation results provide a mechanistic explanation for these observations, illustrating that chemical rearrangement effects of the atomic lattice control these phenomena. This shows that for many materials, the properties of an ensemble of chemical bonds under large load can control the fracture behavior, and that a simple Griffith condition of crack initiation is insufficient to capture these effects. The experimental validation of simulation results is a crucial aspect in the studies of complex fracture phenomena. 6.11.1 Introduction The basic notion of brittle fracture is that continuous breaking of atomic bonds lead to the formation of two new materials surfaces. Is this simple picture also true for chemically complex materials like silicon? The atomistic models of fracture discussed earlier in this book assume an empirical relationship between bond stretch and force, typically described using a simple pair potential. However, breaking of bonds in real materials can be an extremely complicated process. For instance, it can be captured with sufficient accuracy by using quantum mechanical (QM) methods. However, such methods are limited to approximately 100 atoms, which limits the applicability of these methods to model dynamic fracture, since here thousands of atoms participate in the bond breaking mechanisms. Fracture of silicon has received tremendous attention due to its complexity of bond breaking and due to interesting failure dynamics observed experimentally [25, 222–224, 276–278]. These experimental efforts led to critical insight into deformation modes, such as the mirror-mist-hackle transition and orientation dependence of crack dynamics in silicon single crystals [25].

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Atomistic modeling fracture of silicon has been the subject of several studies using empirical force fields [114, 279]. In contrast to many metals, where fracture and deformation can be described reasonably well using embedded atom (EAM) potentials [31, 33, 35, 106, 109, 280], a proper description of fracture in silicon has proved to be far more difficult, as many models did not agree with experimental observations. This suggested that silicon requires a different, more accurate treatment of the atomic interactions. There have been several earlier attempts to describe fracture of silicon using atomistic methods (see, for example [114, 174, 184, 224]). Early attempts to model fracture in silicon used Tersoff’s classical potential [10] (in the following referred to as “Tersoff potential”) and similar formulations such as the Stillinger–Weber potential [124]) or the EDIP potential [281]. Simulations carried out with those potentials have not been able to reproduce experimentally observed brittle fracture of silicon [222]. It has been suggested that the reason for these discrepancies between computation and experiment is that the description of the atomic bonding at large stretch obtained by empirical potentials deviates significantly from the more accurate, quantum mechanical solution [222]. It is thus believed that to develop models of crack dynamics in silicon that agree with experimental observations, the accuracy of QM for atoms near the propagating crack tip is necessary. Baskes and coworkers used their modified EAM formulation (MEAM) to describe crack motion in silicon [114] and to investigate the critical load for fracture initiation [279]. Even though this model leads to improved results compared to Tersoff-type potentials, the MEAM formulation cannot describe bond formation and breaking of silicon with other elements such as oxygen. Here we review the results for an alternative approach utilizing the ReaxFF reactive force field developed to reproduce the barriers and structures for reactive processes from QM, but at a computational cost many orders of magnitude smaller. Here we use a hybrid simulation technique in which the ReaxFF reactive potential for silicon [93, 131] is used for a modest region of a few thousand atoms close to the crack tip while a computationally inexpensive but nonreactive Tersoff potential [10] is used to describe the other 100,000 more distant atoms needed to include their elastic constraints on the propagating tip. The Tersoff potential and ReaxFF lead to similar materials behavior or equation of state for small strains, but deviate strongly at large strains as demonstrated previously [93]. The fact that both descriptions overlap for small strains enable a smooth handshake between the two methods. As reviewed earlier (in chapter 2), the ReaxFF reactive potentials [93, 131] have been developed to describe combinations of many different elements across the periodic table, including first row elements (C, O, H, N), metals (Cu, Al, Mg, Ni, Pt and others), and semiconductors (Si and others) [93, 128–131, 282]. Thus, the methodology described here could potentially be a valuable tool for describing plasticity and fracture for some materials, where certain details of the bond breaking process are necessary to model crack propagation.


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Here we focus on modeling fracture in pure silicon and the interactions of silicon with oxygen competing with crack extension. 6.11.2 Hybrid Atomistic Modeling of Cracking in Silicon: Mixed Hamiltonian Gormulation We employ the hybrid multiparadigm approach as described in Sect. 5.4.5. As described in [131], the ReaxFF potential for silicon has been tested against quantum mechanical data for a wide range of processes [283], including Si– Si bond breaking in H3 Si–SiH3 and Si=Si bond breaking in H2 Si=SiH2 , equations of state for 4-coordinate silicon (diamond-configurations) and 6coordinate silicon phases (β-tin), and simple cubic crystal. This force field is also capable of treating interactions of Si with O and H [131]. It is emphasized that all parameters are completely derived from QM calculations. The simple approach described in Sect. 5.4.5 is used to describe the transition region between two paradigms, using two parameters, Rtrans for the size of the transition region, and Rbuf for the size of the ghost atom region. A schematic of this approach is shown in Fig. 5.8. In the examples discussed here, we model a reactive region Ωrx embedded into a nonreactive domain Ωnr . The reactive region is updated every Nu = 10 steps during the inteA and Rbuf = 5 ˚ A. These parameters have gration. We choose Rtrans = 6 ˚ been chosen by trial and error to make sure that the crack dynamics is not affected by changing these parameters. The shape and size of the reactive region surrounding the moving crack is based on the strain energy density of each atom. All atoms with a strain energy larger than Ecrit = −3.5 eV are embedded in a cylindrical reactive region of R = 10 ˚ A. The critical strain filters atoms at the tip of cracks and atoms in the vicinity of cracks whose bonds are stretched significantly. The union of all cylindrical regions yields the total reactive region, allowing representation of arbitrary shapes. The final reactive region is typically not circular. Initially, the systems contain several hundred reactive atoms, which corresponds to a small reactive region at the crack tip. This initial size of the reactive region may increase during the simulations approaching several thousand atoms because of crack branching or due to formation of microcracks. When oxygen atoms are present in the system, a similar procedure is applied and in addition to the criterion based on the strain energy density, each oxygen atom is embedded in a cylindrical reactive region of R = 10 ˚ A. 6.11.3 Atomistic Model Figure 6.95 depicts the atomistic model. A perfect crystal with an initial crack of length a serving as the failure initiation point is considered. The thickness of the system is one unit cell in the z-direction with periodic boundary conditions, corresponding to a plane strain case (size in z-direction Lz = 5.43 ˚ A).

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Fig. 6.95 Geometry used for simulating mode I fracture in silicon. The systems A and Ly ≈ 910 ˚ A. The contain between 13,000 and 113,015 atoms with Lx ≈ 550 ˚ numerical model is capable of treating up to 3,000 atoms with ReaxFF in Ωrx

This model is chosen due to computational limitations for keeping the reactive region small. It is noted that this choice of geometry may impose some constraints on the system, which may effect the possibility of dislocation nucleation. The slab is strained at a strain εxx in mode I prior to simulation. The boundaries are held fixed during the simulation so that the stress in the material can only be relieved by crack propagation. The crack starts to nucleate shortly after the simulation is started. 6.11.4 Simulation Results First we compare the dynamics of crack propagation described using pure Tersoff [10] to the results incorporating a region treated by ReaxFF close to the crack tip to demonstrate the importance of QM level accuracy at the crack tip to describe the bond breaking process. The crack direction is [110] with a (110) crack surface. We strain the system by 20% in mode I loading and then minimize the potential energy. We consider two cases, one in which a small region around the crack tip is treated using ReaxFF embedded in a Tersoff region, and the other case in which all atoms are described using Tersoff. The results of these computational experiments are shown in Fig. 6.96. It is observed that with pure Tersoff, the crack does not propagate. Instead the crack becomes blunt (Fig. 6.96a) and eventually amorphizes as the loading is increased sufficiently. This incorrect description of brittle fracture has been observed in previous studies with Tersoff-type potentials [224, 279], and is in sharp contrast to experimental results [25]. Experiment clearly suggest a highly brittle behavior of silicon, in particular when initial cracks have (110) faces. In agreement with experiment, the hybrid treatment leads directly to a correct description of the fracture process, showing a dominance of brittle fracture (Fig. 6.96b) [25,222,224,276]. For the [100] crack direction with (100)


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crack surfaces, similar behavior is observed: The pure Tersoff model leads to amorphization at the crack tip, in contrast to the hybrid model that leads to initiation of brittle fracture.

Fig. 6.96 Crack propagation with a pure Tersoff potential (subplot (a)) and the hybrid ReaxFF-Tersoff model (subplot (b)) along the [110] direction (energy minimization scheme). The darker regions are Tersoff atoms, whereas the brighter regions are reactive atoms. The systems contain 28,000 atoms and Lx ≈ 270 ˚ A × Ly ≈ 460 ˚ A

Fig. 6.97 Crack dynamics along the [110] direction at finite temperature (T ≈ 300 K), 10% strain applied

Figure 6.97 shows various snapshots of a crack propagating in a silicon crystal strained by 10% with the temperature controlled to be around T ≈ 300 K. The crack propagates through the material in a perfectly brittle manner. The crack approaches a speed of 3.41 km s−1 , which is approximately 75% of the Rayleigh wave speed, the limiting speed predicted by continuum theory [22] (cR ≈ 4.5 km s−1 [284]). The observations from the

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hybrid simulations, in particular the onset of the rough surface agree with experimental studies of cracks propagating along the same crystallographic planes [25].

Fig. 6.98 Crack dynamics along the [100] direction at finite temperature (T ≈ 300 K, 10% strain applied). Shortly after nucleation of the primary crack two major branches develop along [110] directions

Figure 6.98 shows crack dynamics for an initial crack oriented into the [100] direction. Experiment shows that the crack branches into multiple (110) surface directions for such a system [224]. Indeed, it is observed that the crack branches off into two [110] directions – forming (110) crack surfaces – shortly after nucleation. This result is consistent with the notion that crack dynamics is most stable along this direction. This observation suggests that proper description of the chemistry of bond breaking (with ReaxFF or QM) agrees with experiment [224]. Branching into [110] directions from an initial [100] crack is not observed with the pure Tersoff model. This model also enables to calculate the crack speed as a function of the applied load. Figure 6.99 plots the crack speed as a function of G/G0 (for two crystallographic orientations). It is observed that the crack speed remains zero for G < G0 , as it is expected. However, the crack speed discontinuously jumps to approximately 3 km s−1 when G = G9 , approaching approximately 4.5 km s−1 for higher loads. This crack dynamics is reminiscent of the latticetrapping effect, which has also been observed in other studies. Fig. 6.99b contains a comparison of the results with experimental and earlier computational results, showing that the model quantitatively reproduces experimental results of crack speeds. We reiterate that other computational attempts, for example those using empirical potentials (Stillinger-Weber, Tersoff, EDIP) have failed to reproduce the experimental results qualitatively, as they predict a slower increase of crack speed with increasing crack driving force or fail to reproduce the brittle character of silicon. Fig. 6.100 shows the dependence of the average crack velocity as a function of the steady-state energy release rate, as obtained in experimental studies [25]. This plot also confirms that the steady-state crack propagation speed


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Fig. 6.99 Crack speed as a function of load, for the (110) system (subplot (a), and the (111) system (subplot (b))

features a minimum crack speed, in agreement with the molecular dynamics simulations. Figure 6.101 shows the dynamic fracture toughness as a function of the energy release rate (Fig. 6.101a) and the average crack velocity (Fig. 6.101b). This experimental result agrees nicely with the molecular dynamics simulations of the 1D fracture model, as shown in Fig. 6.15. 6.11.5 Dynamical Fracture Mechanisms To investigate the atomistic details of the fracture dynamics in silicon, we analyze a computational experiment with a slowly increasing tensile mode I load, for the (111) oriented crystal (at a strain rate of 0.0005% strain increment per integration step). Figure 6.102 depicts a series of snapshots of fracture mechanics in silicon, for a crack oriented in the (111) plane. The initial, static regime is followed by a short period of crack growth during which a perfectly flat, mirror-like surface is generated. Crack propagation becomes increasingly erratic until the entire crystal is fractured.

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Fig. 6.100 Dependence of the average crack velocity in a single crystal of silicon, as a function of the steady-state energy release rate, as obtained in experimental studies. The fracture surface is smooth and mirror-like over the entire crack path for the specimen fractured at the lowest G (open circle). A faceted fracture surface is observed at higher G (triangles). At the highest G, the fracture surface is very rough (squares). The continuous line corresponds to the continuum mechanical solution obtained from an expression similar to the one reviewed in (6.34) [25]. Reprinted from: T. Cramer, A. Wanner, and P. Gumbsch, Physical Review Letters, Vol. 85(4), c 2000 by the American Physical Society 2000, pp. 788–791. Copyright

Fig. 6.101 This plot shows the dynamic fracture toughness as a function of the energy release rate (subplot (a)) and the average crack velocity (subplot (b)). Reprinted from: T. Cramer, A. Wanner, and P. Gumbsch, Physical Review Letc 2000 by the American Physical ters, Vol. 85(4), 2000, pp. 788–791. Copyright Society

Figure 6.103 depicts the crack tip velocity history and the onset of the crack tip instability. Figure 6.104 depicts an analysis of the sequence of atomic events. The first single bond rupture event is due to a local rearrangement of the atoms. After this initial event, further crack extension does not occur even though the load is increased. Instead of crack extension, a change in the local crystal structure at the tip of the crack is observed. Two of the 6-membered silicon rings transform into a 5–7 double ring combination, where the 7-membered


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Fig. 6.102 Series of snapshots of fracture mechanics in silicon, for a crack oriented in the (111) plane. The figure shows the dynamics of crack extension, leading to failure of the entire crystal

Fig. 6.103 Velocity–time history of the crack dynamics shown in Fig. 6.102. Soon after nucleation of the crack, the speeds jumps to values of approximately 2 km s−1 . The crack instability sets in at approximately 69% of cR , the Rayleigh wave speed

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Fig. 6.104 Analysis of the sequence of atomic events during fracture initiation. This series of snapshots show the formation of the 5–7 ring defect at the tip of the crack (a blow-up of this defect structure is shown in subplot (b))

ring is closer to the tip of the crack (see Fig. 6.104b). The creation of this crystal defect appears to be induced by the increased stresses in the vicinity of the crack tip. The eventual crack nucleation after the 5–7 double ring has been formed occurs not at the primary crack tip. Instead, a small secondary microcrack forms ahead of the primary crack (Fig. 6.104c), which eventually reunites with the primary crack (Fig. 6.104d). Subsequently, the crack begins to propagate at a speed close to 2 km s−1 . This minimum crack speed coincides with the smallest admissible speed observed under constant load (see Fig. 6.104c)


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and experimental results. This behavior has been confirmed in simulations of various pulling rates. It is believed that the threshold crack speed is related to formation of the 5–7 double ring. Two observations support this hypothesis. First, there exists a geometric effect due to crack blunting, effectively leading to a reduction of the stress concentration at the tip of the crack (compare Fig. 6.104a, b). In this spirit, the 5–7 defect corresponds to the fracture process zone; since it is only a few atomic distances wide and much smaller than the specimen dimension, the small-scale yielding condition is satisfied.

Fig. 6.105 Comparison of the prediction of LEFM, the prediction of the modified LEFM model (6.129), and molecular dynamics simulation results

Secondly, the 5–7 double ring effectively leads to an increased energy barrier for crack nucleation. The apparent fracture surface energy under presence of the 5–7 double ring is increased to γ5−7 > γ0 . After the critical load is reached sufficient to break the 5–7 double ring, the crack propagates continuously, without formation of the 5–7 double ring, while the crack senses a fracture resistance according to the surface energy of the perfect crystal, γ0 . The jump of crack speed to a finite speed immediately after nucleation can therefore be explained based on a simple consideration of the dynamical energy balance. This behavior can be expressed in this simple model, where the crack speed as a function of the crack driving force G is given by v(G) 0 if G < G0,5−7 , (6.129) = cR 1 − G0 /G if G ≥ G0,5−7 ,

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where G0 is the critical energy release rate according to the (111) fracture surface energy γ5−7 , and G0,5−7 is the critical energy release at which fracture actually occurs (due to the fracture surface energy γs ). The crack speed at the critical fracture load G = G0,5−7 is cR (1 − G0 /G0,5−7 ) > 0,

(6.130)

G0 /G0,5−7 < 1.

(6.131)

while

Thus the admissible crack speeds v are cR (1 − G0 /G0,5−7 ) ≤ v < cR .

(6.132)

This is in contrast to the prediction by classical LEFM that predicts that 0 ≤ v < cR

(6.133)

under identical boundary conditions. Figure 6.105 shows a quantitative comparison of the LEFM model, the modified LEFM model and molecular dynamics simulation. The predictions of theoretical model and the computational results are in good agreement. The modified LEFM model predicts larger crack speeds for larger crack driving forces. A possible reason for this disagreement could be elastic softening at large strains that lead to a reduction of cR . Even though this analysis was focused on silicon, similar mechanisms may occur in other covalently bonded materials. A wider implication of this study is that properties such as the fracture surface energy are not material parameters alone, but may depend on the physical state of the system, as exemplified by generation of the 5–7 double ring that is formed under large stresses near the tip of a crack. Finally, it is noted that the mode of loading can also strongly influence the crack direction [26]. Figure 6.106 depicts a comparison of the crack dynamics under mode I (Fig. 6.106a) and mode II (Fig. 6.106b) loading. Under mode I loading, the crack begins to move straight after the critical load is reached. Under mode II loading, the crack begins to move at an angle of approximately 30◦ to the left. 6.11.6 Reactive Chemical Processes and Fracture Initiation The potential inherent in the hybrid scheme becomes particularly apparent when reactive chemical processes such as oxidation are allowed to compete with brittle crack extension. Figure 6.107 depicts a series of studies in which we investigate crack dynamics in a small penny-shaped crack containing oxygen molecules O2 . As previously, the slab is under mode I tensile loading. The oxygen atoms react strongly with the silicon surface (see Fig. 6.107b, d).


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Fig. 6.106 Comparison of the crack dynamics under mode I (subplot (a)) and mode II loading (subplot (b)) [26]

It is noticed that 5% prestrain is sufficient to nucleate a crack for pure silicon, but when oxidative processes are present such a strain is not sufficient to nucleate a crack. The oxidation leads to crack blunting effectively reducing the stress intensity factor at the crack tip, while leading to formation of strong Si–O bonds, making this Si–O layer harder to break than pure silicon (see Fig. 6.107a vs. Fig. 6.107b). 6.11.7 Summary The hybrid ReaxFF–Tersoff model reproduces experimental results quantitatively, with modest computational cost, thus providing some advantage over other more expensive computational models. The ReaxFF reactive force field was developed solely from first-principles quantum mechanical calculations of

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Fig. 6.107 Crack dynamics in silicon without oxygen (O2 molecules) (subplots (a) and (c)) and with oxygen molecules present (subplots (b) and (d)). Subplots (a) and (b) show the results for 5% applied strain, whereas subplots (c) and (d) show the results for 10% applied strain. The darker grey regions are Tersoff atoms, whereas the brighter regions correspond to ReaxFF atoms. The systems contain about 13,000 atoms, with Lx ≈ 160 ˚ A × Ly ≈ 310 ˚ A. This demonstrates the dramatic effect of oxygen in making Si brittle

equations of state and various reactions in clusters and was not modified to improve the agreement with experimental results. The results show that in addition to instability driving forces such as energy flow [28] and change of asymptotic stress field [27, 29], chemical rearrangements may contribute to fracture instabilities, rendering the situation more complex for many materials, suggesting an intimate connection between fracture mechanics and chemistry [274, 275]. Even though this analysis was focused on silicon, similar mechanisms may occur in other covalently bonded materials. A wider implication of this study is that properties such as the fracture surface energy are not material parameters alone, but may depend on the physical state of the system, as exemplified by generation of the 5–7 double ring that is formed under large stresses near the tip of a crack.


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In this section we reviewed the application of a hybrid numerical method integrating the ReaxFF and Tersoff force fields to allow a physics-based description of the fracture mechanics and fracture dynamics of silicon. The scheme represents a new multiscale approach of coupling the QM scale of chemistry and bond breaking and formation with the scale of mechanics of materials. Unlike other attempts in which empirical potentials were modified heuristically to yield brittle fracture of silicon, our method is completely based on first-principles, with no empirical parameters used for fitting of ReaxFF for silicon.

Fig. 6.108 This plot shows the difference in large-strain elasticity between the Tersoff potential and ReaxFF, while both descriptions coincide at small strain. This result demonstrates the importance of large-strain properties close to breaking of atomic bonds [27–29]

The results (see Fig. 6.96a, b) underline the importance of large-strain properties at bond breaking for the dynamics of fracture, as suggested earlier by Gao [27] and in [28, 29]. Figure 6.108 shows the large deviation between Tersoff and ReaxFF at large strains close to bond breaking by stretching the crystal in the [110] direction. The Tersoff description leads to a sharp rise of the force close to rupture of bonds, which deviates significantly from the more accurate solution by ReaxFF. The model is further capable of reproducing key experimental observations including crack limiting speed, crack instabilities, and directional dependence on crystal orientation (see Fig. 6.97a–c). The studies lead to insight into the atomistic details of the fracture processes, suggesting continuous formation of microcracks ahead of a propagating mother crack (see Fig. 6.97b). This has been debated in the literature and the simulations seem to corroborate this concept. We find that ReaxFF reactive force field [93, 131] successfully models chemomechanical properties of materials as crack propagation. Since

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ReaxFF is capable of describing a wide heterogeneous range of materials [93,128–131,282], our approach may be a practical means to studying the coupling of complex chemical reactions to mechanical properties. This is shown in the studies depicted in Fig. 6.107. Here we have demonstrated the effect of oxygen in changing the fracture behavior of silicon by effectively blunting the crack tip and thus making silicon more brittle, in agreement with experiment. Our hybrid method could enable studies of stress corrosion processes and other degradation and aging mechanisms. A few thousand atoms in the reactive region at the crack tip are sufficient to describe the crack dynamics correctly. Although such calculations are not practical with pure QM, ReaxFF provides QM accuracy for the reacting part of the system, while retaining speeds comparable to that of simple force fields. As demonstrated in the literature and by the simulations reviewed here, the Tersoff potential cannot describe fracture accurately [114, 279], but it is adequate for describing the elastic regions.

6.12 Summary: Brittle Fracture Brittle fracture is an important material phenomenon with great scientific and technological impact. Atomistic simulations have been particularly helpful in understanding the science of fracture, as numerous contributions by many researchers over the past decades illustrated. In the previous sections we have only reviewed a small part of the literature. Particular emphasis of these studies was on the joint investigation of fracture phenomena from a continuum theoretical perspective and by using atomistic simulation. This side-by-side comparison provides an illustration of how a complex topic such as fracture can be tackled by joint analysis from different perspectives. The chapter included both studies with a simple model potential (harmonic potential, biharmonic potential) as well as studies with quantum mechanics based ReaxFF reactive force fields. The investigations and reviews in the area of dynamic fracture focused on the following points. • Comparison of atomistic results with continuum mechanics theory predictions (in particular crack limiting speed, crack tip instability speed, and deformation fields). • Investigation of hyperelastic effects in dynamic fracture (in particular regarding the crack limiting speed and the crack tip instability speed). • Effect of geometric confinement (cracks in thin elastically inhomogeneous layers) and crack propagation along interfaces of elastically dissimilar materials. • Fracture mechanisms in chemically complex materials, here illustrated for the case of silicon.


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The first study discussed in Sect. 6.4 centered on a one-dimensional model of dynamic fracture. The appeal of the one-dimensional model is that many of the physical phenomena of dynamic fracture, such as maximum crack speed and a condition for crack initiation similar to the Griffith theory, also appear in this simple model. An important aspect was that analytical expressions for the nonlinear dynamics of fracture could be derived (see Sect. 6.4.3). The analytical model predicted crack motion faster than the speed of sound, if there is an elastically stiff zone near the crack tip. This illustrates the significance of the details of bond breaking on understanding dynamic fracture. The atomistic model with harmonic interactions is found to reproduce the predictions of the linear elastic continuum theory well. The simulations carried out with nonlinear interatomic potentials revealed that a small zone with stiff elastic properties at the crack tip significantly changes the dynamics and allows the crack to break through the sound barrier. This observation is in agreement with the theoretical predictions of the model described in Sect. 6.4.3. As shown in Fig. 6.20a, the crack propagation speed depends critically on the onset strain of the hyperelastic effect and therefore, the crack speed is highly sensitive to the size of the hyperelastic region. The deformation fields near the one-dimensional crack in nonlinear materials agrees reasonably well with the predictions by continuum theory, as shown in Fig. 6.20b. After studying the one-dimensional crack, we moved on to two-dimensional models. We utilized the elastic properties of two-dimensional solids, as reviewed in Sect. 4.4.3. It was demonstrated that the choice of the potential allows to construct model materials with the objective to probe the effect of specific material properties on the dynamics of fracture. One of the examples of such model materials is the biharmonic potential. This potential yields a solid composed of two linear elastic materials, with one Young’s modulus associated with small strains and one with large strains, representing a simplistic model material for hyperelasticity. Further, the fracture energy was calculated for different choices of the interatomic potential serving as input parameter for the prediction of crack initiation time by the Griffith model [22]. A quantitative comparison of the deformation fields near a rapidly propagating mode I crack in a harmonic lattice revealed that the continuum theory predictions of angular variation of stress and strain agree well with the results of atomistic simulations (Sect. 6.5). It was found that the prediction that the hoop stress becomes bimodal [210] at a critical crack propagation speed is reproduced in atomistic calculations. The occurrence of the bimodal hoop stress is an important aspect in the theories of crack tip instabilities. In summary, the studies in Sect. 6.5 (together with the results of the one-dimensional crack discussed in Sect. 6.4) reveal that atomistic simulations with harmonic potentials are a good model for the linear elastic continuum theory. The results in this section and the results of the one-dimensional model with harmonic interactions both show reasonable agreement with the linear elastic continuum theory [22].

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In Sect. 6.6, the large-strain elastic properties were changed, while keeping the small-strain elastic properties constant to systematically investigate the effect on crack dynamics. The main finding is that the elasticity of large strains can dominate the dynamics of fracture, in contrast to the predictions by many existing theories [22]. With a new concept of the characteristic energy length scale χ in dynamic fracture the experimental and computational results can be explained. This length scale immediately explains under which conditions hyperelasticity is important and when it can be neglected. Cracks moving in solids absorb and dissipate energy from the surrounding material. The new length scale characterizes the zone near the crack tip from which the crack draws energy to sustain its motion. When materials are under extreme stress, this length scale extends only a few dozens nanometers. One of the important consequences of this is that cracks can move supersonically in contrast to existing theories. The finding that the crack speed increases continuously as the size of the hyperelastic region expands (shown in Figs. 6.40 and 6.41) can be explained by the interplay of the hyperelastic region size and the characteristic energy length scale, and is in qualitative agreement with the findings in the one-dimensional model depicted in Fig. 6.20a. Stimulated by the results discussed in Sect. 6.6 [28], intersonic mode I cracks as shown in Fig. 6.42 have recently been verified in the laboratory [250]. In the following section, we investigated the effect of hyperelasticity on the stability of cracks. It is known that cracks propagate straight at low velocities, but start to wiggle when the crack speed gets larger [155, 241]. One of the theoretical explanations [22, 210] is that the hoop stress becomes bimodal at a speed of 73% of Rayleigh wave speed. Indeed, we find in the atomistic simulation that a crack in a harmonic lattice becomes unstable at a speed of about 73% of Rayleigh wave speed, in good agreement with the continuum theory. However, if a softening potential is used, the instability occurs at lower speeds! In contrast, it was demonstrated that if material stiffens with strain, the instability occurs at higher speeds than predicted by linear elastic theory. We therefore conclude that hyperelasticity governs the dynamic crack tip instability. Additional studies focused on cracks at interfaces. We reviewed studies of mode I and mode II cracks along interfaces of elastically dissimilar materials. In mode I, it is observed that cracks are limited by the Rayleigh wave speed of the stiffer of the two materials provided that sufficient loading is applied. A mother–daughter mechanism, similar as known to exist at interfaces of identical materials under mode II loading [165], is observed that allows the crack to break through the sound barrier. In mode II, we find that the crack speed is limited by the longitudinal wave speed of the stiff material and observe a mother–daughter–granddaughter crack mechanism. Whereas a mother–daughter mechanism has not been observed in mode I cracks, the mother–daughter–granddaughter mechanism has been observed in mode II cracks along interfaces of elastically harmonic and anharmonic materials. Most importantly, experimental evidence was reported for the existence of


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such mechanisms for crack propagation along interfaces of aluminum (stiff) and PMMA (soft) [267] corresponding to our atomistic model. The molecular dynamics simulations reproduce some of the experimental findings. The next section was devoted to a discussion of suddenly stopping cracks. The main result was that mode I cracks in harmonic lattices carry no inertia, and the static field spreads out behind the shear wave front immediately after the crack is stopped. This result matches the prediction by continuum theory [22]. A comparison of the suddenly stopping mode I cracks with experimental results [23] also reveals good agreement. The results in discussed in this section are in accordance with the results of a suddenly stopping one-dimensional crack shown in Fig. 6.16. As soon as the crack stops, the strain field of the solution corresponding to zero crack velocity is spread out. We have then shown that mode II cracks behave differently than mode I cracks: In agreement with the predictions by continuum mechanics theories of suddenly stopping intersonic cracks [208], an intersonic mode II crack does carry inertia and the static field does not spread out until the mother crack has reached the stopped daughter crack. In the nonlinear cases of mode I and mode II cracks, the wave fronts are smeared out and the static field is not instantaneously reached but after all trails of waves have passed. In the last two remaining sections we focused on the dynamics of mode III cracks by using three-dimensional atomistic simulations. Firstly, we considered the mechanical and physical properties of three-dimensional solids. We discussed the elastic properties and compared theoretical predictions with numerical estimates for various crystal orientations in an FCC crystal. As in the two-dimensional case, we also calculated the fracture surface energy. The results of the section on the mechanical and physical properties of threedimensional solids provide important information for the studies discussed in Sect. 6.10. After studying cracks in harmonic lattices and showing agreement of the corresponding limiting speed (the shear wave speed), we focused on the critical energy length scale χ. We find that the same concept as that discovered in Sect. 6.6 also holds for mode III cracks. A quantitative comparison with an analytical continuum theory solution [24] showed good agreement. For realistic experimental conditions and cracks propagation within a thin steel layer, the characteristic energy length scale was estimated to be on the order of millimeters. Further studies of mode III cracks included suddenly stopping cracks. It was shown that, as for sub-Rayleigh mode I and mode II cracks, mode III cracks in harmonic lattices carry no inertia [22]. 6.12.1 Hyperelasticity can Govern Dynamic Fracture In this section, the effect of the elasticity of large strains on the dynamics of fracture was was one of the main points of interest. Here we discuss the role of hyperelasticity in more general terms. The results suggest that hyperelasticity has (1) an effect on the crack speed as well as (2) on the instability dynamics of cracks. Unlike in some previous

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studies (e.g. [155]), we used the concept of the weak fracture layer to separate the two problems of limiting speed and instability from one another to obtain clean simulation and analysis conditions. This allowed us to investigate the conditions under which hyperelasticity governs the dynamics of fracture. The approach of defining model materials is a reasonable method to investigate some of the fundamentals of dynamic fracture, and may be considered advantageous over methods where the peculiarities of a specific material are accounted for. In some previous studies, due to the complexities of the potential it was difficult to draw general conclusions about crack dynamics in brittle solids. Limiting Speed of Cracks It was shown that the key to understand the dynamics of cracks in hyperelastic materials is a new length scale that characterizes the zone near the crack tip from which the crack draws energy to sustain its motion.

Fig. 6.109 Different length scales associated with dynamic fracture. Subplot (a) shows the classical picture [22], and subplot (b) shows the picture with the new concept of the characteristic energy length χ

This characteristic length scale is found to be proportional to the fracture surface energy and elastic modulus, and inversely proportional to the square of the applied stress, γE χ∼ 2. (6.134) σ Contrary to the common belief, the crack does not need to transport energy from regions far away from its tip, rather only from a small local region described by the characteristic length scale. The assumption of linear elasticity, and hence the classical theories, fails if the hyperelastic zone becomes comparable to the local energy flux zone. This is because in soft materials


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energy is transported slower, in stiff materials faster. Correspondingly, the crack velocity becomes slower or larger once the hyperelastic region is sufficiently large. If the region around the crack tip becomes stiff due to hyperelasticity, more energy can flow to the crack tip in shorter time. In the opposite, energy transport gets slower when there is a local softening zone around the crack tip. Therefore, hyperelasticity is crucial for understanding and predicting the dynamics of brittle fracture. When hyperelasticity dominates, cracks can move faster than all elastic waves as shown in Fig. 6.44. This is in clear contrast to the classical theories in which it is believed that the longitudinal elastic wave speed is an impenetrable upper limit of crack speed. Such phenomenon can only be understood from the viewpoint of hyperelasticity. Hyperelasticity dominates fracture energy transport when the size of the hyperelastic zone approaches the energy characteristic length. Under normal experimental conditions, the magnitude of stress may be one or two orders of magnitude smaller than that under atomic simulations. In such cases, the characteristic length is relatively large and the effect of hyperelasticity on effective velocity of energy transport is relatively small. At about 1% of elastic strain, the energy characteristic length is on the order of a few hundred atomic spacing and significant hyperelastic effects are observed. It seems that hyperelasticity can play the governing role especially in nanostructured materials such as thin films, or under high-impact conditions where huge stresses occur, so that the region from which the crack needs to draw energy is small. In the conventional picture of dynamic fracture, there exist three important length scales near the crack tip, as shown in Fig. 6.109a. The fracture process zone in which atomic bonds are broken is usually very small and extends only a few angstroms in perfectly brittle systems. Another important length scale is the K-dominance zone, which is relatively large. In between the fracture process zone and the K-dominance zone is the region where material response is hyperelastic. We proposed that there exists an additional length scale near the dynamic crack, the characteristic energy length scale. This new energy length scale is shown in Fig. 6.109b, and it is in between the K-dominance zone and the hyperelastic region. If the size of the hyperelastic region becomes comparable to the energy length scale, hyperelasticity governs dynamic fracture. If it is much smaller, hyperelasticity can be neglected. Crack Tip Instabilities We find that the large-strain elastic properties have a strong impact on the stability of dynamic cracks. Therefore, the dynamics of fracture is predominantly governed by the large-strain elastic properties of the interatomic potential. This was exemplified in a study of a harmonic vs. softening and stiffening potentials. Whereas the crack becomes unstable at 73% of the Rayleigh wave speed in materials with harmonic interactions, the crack becomes unstable at speeds much smaller than the Rayleigh wave speed in softening materials. In contrast, stiffening material behavior allowed cracks close to Rayleigh wave

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speed to propagate stable. On the basis of systematically varying the ratio of large-strain elastic properties while keeping the small-strain elastic properties constant, it was shown that the instability speed depends on the local wave speed. A generalized Yoffe criterion [210] and Gao’s analysis [27] of local limiting speed helped to explain some of the simulation results. With respect to the governing mechanism of the dynamic crack tip instability, the stiffening and softening case need to be distinguished. We illustrated that in softening systems, the reduction in local energy flow governs the instability, and in stiffening systems, the change in deformation field near the crack tip is responsible for the crack to leave its straightforward motion and branch. The analysis of the stress and strain field support these assumptions. As discussed in Sect. 6.11, in chemically complex materials additional atomistic mechanisms may play a role in initiating fracture instabilities. Here we have discussed the formation of 5-7 defects during crack initiation in silicon, as shown in Fig. 6.104b. 6.12.2 Interfaces and Geometric Confinement Interfaces and geometric confinement play an important role in the dynamics of cracks. Crack propagation constrained along interfaces can significantly change the associated maximum speeds of crack motion. This is illustrated for instance by the studies using the concept of a weak fracture layer where the Rayleigh wave speed of cracks can be attained by cracks (see Sect. 6.5), vs. the studies of cracks in homogeneous materials where the crack starts to become unstable at 73% of the theoretical limiting speed. If cracks propagate along interfaces of elastically dissimilar materials, the maximum crack speed can significantly change and new mechanisms of crack propagation such as daughter and granddaughter cracks appear. Geometric confinement as cracks moving inside thin strips (the Broberg problem) has proven to provide strong impact on the dynamics of cracks. If the crack propagates in a small strip with different elastic properties, a significant effect on the propagation speed of the crack is observed as illustrated in Fig. 6.46 for mode I and in Fig. 6.93 for mode III cracks. An implication of crack motion within a thin stiff layer is that mode I cracks can break through the shear wave speed barrier and propagate at intersonic velocities as shown in Fig. 6.47, and that mode III cracks become supersonic as shown in Fig. 6.92. In summary, the atomistic studies suggest that geometric confinement has strong impact on how cracks propagate. This is potentially important in composite materials where understanding of crack dynamics may be critical in designing robust and reliable devices. The results reviewed here suggest that the definition of wave speeds according to the small-strain elastic properties is questionable in many cases and should be replaced by a local wave speed. Similar thoughts apply to the definition of wave speeds across interfaces: When the elastic properties are discontinuous, no unique definition of the wave speed and therefore the crack limiting speed is possible.

7 Deformation and Fracture of Ductile Materials

This chapter is dedicated to the study of the deformation and fracture behavior of ductile materials. Ductile materials are characterized by their capacity to withstand large deformation and to be able to deform permanently (see also the comparison shown in Fig. 1.2). Plastic deformation of metals is often described using continuum mechanics techniques, such as crystal plasticity theories [285–287] or strain gradient formulations [288, 289]. Significant research effort has also been put into the development of mesoscopic discrete dislocation dynamics techniques [50, 95–101]. Yet another approach is to study plasticity using large-scale atomistic simulations. The basic carriers of plastic deformation in crystals are dislocations. Therefore, most of the discussion in this chapter will be focused on the behavior of these basic elements of plasticity. In this chapter, we will review a continuum theoretical and atomistic approach in treating the nucleation, propagation, and interaction of dislocations. The ductile character of the material is captured by the specific interatomic potential. As discussed earlier, for many metals, appropriate EAM-type potentials have been developed. This discussion will be limited to FCC crystals (since several well tested interatomic potentials exist for this class of metals).

7.1 Introduction The understanding of how materials deform and break is often limited to phenomenological engineering theories that describe the macroscale materials behavior, neglecting the underlying atomistic microstructure. However, deformation and fracture of materials is controlled by atom-by-atom processes that are essentially governed by quantum mechanics. These quantum mechanical effects that control chemical bonds are neglected in most existing theories. To include these effects, atomistic models have proven to be a promising approach that are capable to simulate the motion of all atoms in the material, with systems comprising up to several billion particles, thus reaching

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macroscopic scales of material behavior that can be directly observed in experiment. The goal of such atomistic models is to understand the macroscopic response of materials, for example to mechanical stimulation, based on their fundamental, atomistic ultrastructure. Modern multiscale modeling techniques use a sequence of overlapping hierarchies encompassing various simulation tools to bridge the scales from nano to macro. These modeling techniques allow a rigorous linking of material properties from quantum mechanics to mesoscale and macroscale. This fundamental viewpoint could revolutionize the engineering approach to use and create materials, by incorporating the atomistic scale into materials analysis and synthesis. Experimental techniques such as TEM have been very fruitful in the analysis of dislocation structures and dislocation networks. HRTEM can even resolve individual atoms, which has proven to be very useful in the analysis of the structure of grain boundaries, (see, for instance, the plot shown in Fig. 8.2 for an example of such an analysis). For mechanical testing, in addition to macroscopic tensile tests, the use of nanoindentation has led to a novel tool for the investigation of ultra-small-scale plasticity.

7.2 Continuum Theoretical Concepts of Dislocations and Their Interactions

Fig. 7.1 This plot illustrates the difference between a screw dislocation (marked as “S”) and an edge dislocation (marked as “E”). The graph depicts a crystal with a curved dislocation line (the thick curved line). When bl, the dislocation has screw character, and when b⊥l the dislocation has edge character

This section is dedicated to a brief review of continuum mechanical approaches in treating dislocations. Dislocations are the fundamental carrier


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of plasticity, as discussed in Sect. 1.2.3. Figure 1.6 displays the process of nucleating and propagating a dislocation through a crystal, leading to permanent plastic deformation. The geometry of a dislocation is characterized by its Burgers vector b, defining the amount of plastic deformation due to a single dislocation. The direction of the dislocation is referred to as the dislocation line, typically referred to as l. There are two types of dislocations, edge and screw dislocations. Edge dislocations feature a Burgers vector that is perpendicular to the line direction. Screw dislocations feature a Burgers vector that is parallel to the line direction. Figure 7.1 illustrates the difference between a screw dislocation (marked as “S”) and an edge dislocation (marked as “E”). A dislocation can move in a crystal, changing its screw or edge character as the line direction can change. However, the Burgers vector of a dislocation always remains the same. Two dislocations of opposite Burgers vector can annihilate each other when they come sufficiently close. 7.2.1 Properties of Dislocations A dislocation in a solid induces a stress field that scales as 1/r, thus showing a different singularity order as a crack. For an edge dislocation, the stress field under plane strain condition is given by µb y(3x2 + y 2 ) 2π(1 − ν) (x2 + y 2 )2 µb y(x2 − y 2 ) 2π(1 − ν) (x2 + y 2 )2 µb x(x2 − y 2 ) 2π(1 − ν) (x2 + y 2 )2 ν(σxx + σyy ) σyz = 0,

σxx = −

(7.1)

σyy =

(7.2)

σxy = σzz = σxz =

(7.3) (7.4) (7.5)

where µ is the shear modulus and ν is Poisson’s ratio, and x and y are coordinates. The pressure field around an edge dislocation is p=

2 µb(1 + ν) y 2 3 2π(1 − ν) x + y 2

(7.6)

illustrating that the pressure is compressive above the slip plane (due to the additional inserted volume) and tensile below the slip plane. The stress field around a screw dislocation has a simpler form, involving only shear stress components. The reader is referred to the literature for further details [38]. Since dislocations induce a distortion of the atomic lattice, a crystal that contains a dislocation has a higher energy state than a crystal without a dislocation. The elastic strain energy stored per unit length of dislocation is

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EL = Cµb2 ,

(7.7)

where C is a constant that depends on the structure of the core of the dislocation (C typically ranges from 0.5 to 1.0), µ is the shear modulus, and b is the magnitude of the Burgers vector. Dislocations with smaller Burgers vectors are energetically favored over those with large Burgers vectors. This equation can also be used to determine if dislocation reactions are energetically favorable, a method referred to as “Frank’s rule.” Dislocations move through a crystal lattice through either glide or climb. In dislocation glide, the Burgers vector and the glide direction lie in the same plane, and during its motion no volume is generated or taken away. In dislocation climb, motion of the dislocation involves generation or dissipation of volume, which may lead to generation of vacancy defects or interstitials. The repeated slip of dislocations in the same direction leads to shearing of the material, resulting in macroscopic permanent plastic deformation. In dislocation climb, the Burgers vector and the glide planes lie in different crystal planes. Climb of dislocations is typically diffusion controlled (to ensure mass transport to sustain vacancies or interstitials).

Fig. 7.2 Illustration of splitting of complete dislocation into two partial dislocations in an FCC lattice

In FCC crystals, slip is limited almost exclusively to {111} planes in 110 directions. This results in 12 independent slip systems of an FCC lattice. Dislocations in FCC crystals are typically split up into partial dislocations. A complete dislocation with Burgers vector 12 110 decomposes into two partial dislocations (see Fig. 7.2) 1 1 1 110 → 211 + 121. 2 6 6

(7.8)

√ The magnitude of the Burgers vector√of the complete dislocation is a0 / 2, and that of a partial dislocation is a0 / 6. Frank’s rule can be used to illustrate that this split into partial dislocations is energetically favorable. This can be done by considering the square of the Burgers vector for both cases, showing that it is a20 /2 for a full dislocation and a20 /3 for the two partial dislocations. Thus, the energy of the partial dislocations is smaller than that of a full dislocation. The nucleation of the first partial dislocation leads to a stacking fault in the FCC lattice, which is repaired by the second partial dislocations.


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The two partial dislocations repel each other. However, the separation of the partial dislocations is associated with an energy penalty due to creation of a stacking fault. The competition between these two effects leads to an equilibrium separation d of the partial dislocations which scales as d∼

b2 . γsf

(7.9)

7.2.2 Forces on Dislocations

Fig. 7.3 Geometry to explain the Schmid law, illustrating the angles φ and λ as well as the uniaxially applied stress σy

The local stress tensor in a material leads to forces on dislocations. These forces control the activation of specific slip systems. The particular activated slip system is controlled by the resolved shear stress that acts on this particular plane; among all possible planes the one with the largest resolved shear stress is most likely to be activated. The activation of particular slip systems is described based on Schmid’s law: τc = σy cos(φ) cos(λ) (7.10) where σ0 is a uniaxially applied stress, and φ and λ are the angles of the slip plane normal and the slip direction with respect to the axis in which the tensile load is applied. A dislocation is nucleated when τc reaches a critical value (please see Fig. 7.3). These considerations are used to develop criteria that describe the critical conditions for nucleation of dislocations.

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7.2.3 Rice–Thomson Model for Dislocation Nucleation

Fig. 7.4 A cracked body, with remotely applied tensile and shear stresses σ∞ and τ∞ . Large resolved shear stresses on specific slip planes are the key drivers for dislocation nucleation

Fig. 7.5 Balance of forces on a dislocation close to a crack, here illustrating the competition between the image force pulling the dislocation back to the surface (Fim ) and the stress induced force pushing the dislocation away from the crack tip (Fstress )

The Rice–Thomson model is a simple model to characterize the critical conditions for nucleation of a dislocation from a crack tip. It is also a very educational model to illustrate the physical process of nucleation of a dislocation, which motivates us to review it here. In the following paragraph, we will summarize the basic steps to obtain the solution for the critical condition for dislocation nucleation from the tip of a crack, as shown in Fig. 7.4. The basis for this model is to consider the balance of competing forces acting on a dislocation, as shown in Fig. 7.5. In that sense, the dislocation is


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treated as a particle, which moves in the direction of the resulting force vector Ftot . It remains in equilibrium if the resulting force vector is zero Ftot = 0. This idea was initially proposed by Rice and Thomson [66]. Two major forces act on a dislocation that is nucleated at a tip of a crack: First, the image force that drives the dislocation back into the surface. Second, a force that results from the asymptotic stress field at the tip of the crack.

Fig. 7.6 Geometry of a dislocation close to a surface, at distance d

The image force of an edge dislocation at a surface is given by Fim = −

µb2 L0 4π(1 − ν) d

(7.11)

where µ is the shear modulus, b Burgers vector, d the distance of the dislocation to the surface (Fig. 7.6). The parameter L0 refers to the line length of the dislocation (which will cancel out later). The physical origin of the image force is the fact that adding an additional half plane of atoms close to a surface is energetically expensive, and therefore, the dislocation is being pulled back into the surface. This model can also be envisioned by considering a dislocation with opposite sign at the “other” side of the surface, which leads to an attraction toward the surface to annihilate itself (see Fig. 7.6). Note that this result can be derived from considering the change in elastic energy of a dislocation with respect to its position d, F = −∂E/∂d, or by considering (7.7) (during annihilation b changes from a finite value to zero). To quantify the force due to the external stress field, we consider the Peach–Koehler equation F = P × l, (7.12) where Pi = σij bj

(7.13)

and li is the direction of the dislocation line. Now we consider the second force acting on the dislocation, due to the crack tip stress field. Considering the case shown in Fig. 7.4 or Fig. 7.5,

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Fstress = τ (d)bL0 ,

(7.14)

where τ is the local stress at the crack tip, which can be written as KI τ (d) = √ fτ (θ), 2πd

(7.15)

with KI being the stress intensity factor that relates to the overall geometry of the specimen, noting that KI ∼ σ∞ (for mode II, KII ∼ τ∞ ). The function fτ (θ) is the characteristic angular variation of the particular stress component considered. Therefore, the force acting on the dislocation due to the crack tip stress field is KI Fstress (d) = √ (7.16) fτ (θ)bL0 . 2πd The forces Fstress and Fim act in opposite directions. We assume that at the critical moment of dislocation nucleation, these two forces must be equal so that the dislocation starts to move to the right. Since Fim ∼ and

1 d

1 Fstress ∼ √ , d

(7.17)

(7.18)

the dislocation will continue to move away from the crack tip once the critical load is reached. For a geometry as shown in Fig. 7.7a, √ (7.19) KI = σ∞ πa. Then, the critical condition is given by 2d µ b σ∞ (d) = a fτ (θ)4π(1 − ν) d

(7.20)

We note that the critical stress σ∞ is still a function of d, which is defined as the critical distance at which the dislocation is considered to nucleate. It remains unknown, but can be estimated to be one to three Burgers vectors. For d = b (assuming that nucleation occurs at a distance of one Burgers vector), 2b µ σ∞ = (7.21) a fτ (θ)4π(1 − ν)

and therefore σ∞ ∼

b µ. a

(7.22)


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Fig. 7.7 Geometry of a cracked specimen with a penny-shaped crack (subplot (a)) and a semi-infinite crack (subplot (b))

Alternatively, for the geometry shown in Fig. 7.7b and mode II shear loading (τ∞ ), H 2(1 + ν)(1 − 2ν) . (7.23) KII = τ∞ 2 ν(1 − ν) Then,

τ∞ ≈ 2

πb µ , H 4π(1 − ν)

and

(7.24)

b . (7.25) H A drawback of the Rice–Thomson model is that the nucleation condition depends on the critical dislocation distance d at the moment of nucleation, which is a priori an unknown. τ∞ ∼

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Fig. 7.8 Concept of stacking fault energy, considering Peierl’s concept of periodic shear stress variation along a slip plane, as originally proposed in [30]. Subplot (a) depicts the geometry, subplot (b) the variation of the shear stress with the distance δ from the crack tip (coordinate system shown in subplot (a)), and subplot (c) depicts the variation of the elastic energy

Fig. 7.9 Balance of energies before (1) and after (2) nucleation of the dislocation, to illustrate the concept of energy release rate


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7.2.4 Rice–Peierls Model A few years after his original model, Rice introduced another model, referred to as the Rice–Peierls model. This model is not based on a force balance but rather on an energy balance [30]. Rice argued that at the critical point of nucleation of a dislocation, the energy released due to slip equals the unstable stacking fault energy, γus . The concept of the unstable stacking fault energy is depicted in Fig. 7.8. Considering Fig. 7.9, the change in potential energy due to slip of a distance ∆x (this parameter could be one Burgers vector) is given by ∆W = W(2) − W(1) = −

2 τ∞ ∆xHW, 2µ

(7.26)

where H is the height of the system (see Fig. 7.7b), and W is the out of plane thickness. Further, µ µ = (7.27) (1 + ν)(1 − 2ν) for plane strain. Equation (7.26) can be rewritten as ∆W τ2 ∆W = = − ∞ H, ∆xW ∆(xW ) 2µ

(7.28)

corresponding to the energy released per unit crack area advance (note that ∆(xW ) = ∆xW ). The critical condition for dislocation nucleation is then given when the energy released per unit crack advance equals the energy to overcome the energy barrier for dislocation nucleation, γus (energy per unit area): 2 τ∞ H = γus . (7.29) 2µ Solving (7.29) for the critical remote shear stress τ∞ yields 2γus µ . τ∞,disl = H

(7.30)

An alternative mechanism for dislocation nucleation is generation of two new surfaces (that is, brittle fracture). The energy necessary to create two new surfaces is 2γs and thus the critical stress is 4γs µ . (7.31) τ∞,surf = H It is evident that the ratio α=

γus 2γs

(7.32)

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determines which of the two mechanisms sets in first, that is, if the material is ductile (α < 1) or if it is brittle (α > 1). We note that for the general case of mode I loading and more general slip systems, the expressions take a somewhat more complex form. However, the basic scaling relationship of critical energies remains identical. 7.2.5 Link with Atomistic Concepts Dislocation mechanics concepts can be quite conveniently linked with atomistic concepts, and vice versa. Here we briefly discuss two aspects of how such a link can be achieved. 7.2.6 Generalized Stacking Fault Curves The Rice–Peierls model provides a direct link with atomistic concepts. As we discussed, both the unstable stacking fault energy as well as the surface energy can be calculated directly from atomistic calculation. Therefore, for a given potential the parameters are defined uniquely. In [113], the generalized stacking fault energy curves are calculated for different EAM potentials of FCC metals. The authors show that the resulting curves show similar characteristics but vary with respect to their agreement with the experimental estimates of the intrinsic stacking fault energy. The curves have been used to obtain estimates of the unstable stacking fault energy γus [30]. Figure 7.10 shows such curves calculated from atomistic simulation, and indicates also how γus and γsf are calculated from such curves. Figure 7.10a shows the calculation method of the GSF curve by sliding two parts of the crystal along the [112] direction, and Fig. 7.10b shows the GSF curves for four different potentials [31–35]. Note that the best agreement of γsf to first principle calculation results are obtained from the potential proposed by Mishin et al. [33] and Angelo et al. [32]. This finding is in agreement with the results reported in [113]. Because of the nature of the stacking fault, the generalized stacking fault energy curve depends on nonnearest neighbor interactions. This is important to consider when models based on pair potentials are developed. In [113], it is further shown that the EAM models produce the same value for the dimensionless constant γus /(γslip bp ) as the simpler Lennard-Jones interatomic potential. This suggests that the LJ pair potential should produce the qualitatively same behavior as EAM models for a given type of plastic deformation. It is noted that the link between atomistic models and the Rice–Thomson model is much more difficult to make. In that case, a possible strategy is to calculate the stress intensity factor from molecular dynamics and use this information to provide a nucleation criterion.


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Fig. 7.10 Calculation of generalized stacking fault (GSF) curves for different EAM potentials fitted to nickel. We consider potentials by Oh and Johnson [31] (O&J), Angelo et al. [32] (AFB), Mishin et al. [33] (M&F) as well as Voter and Chen [34,35]. Subplot (a) illustrates the calculation method of the GSF curve by sliding two parts of the crystal along the [112] direction. Subplot (b) shows the GSF curves for the four different potentials

7.2.7 Linking Atomistic Simulation Results to Continuum Mechanics Theories of Plasticity Atomic measures for quantities like stress or elastic strain are well described in the literature and it has been shown in several cases that good agreement of continuum mechanics theories and atomistic simulation results can be obtained, even in the dynamic cases as shown in this book (see, for instance, the studies discussed in Chap. 6.5). However, no direct link between continuum mechanics concepts of plasticity such as strain gradient theories of plasticity has been established so far. In this section, we discuss how such coupling could in principle be achieved (based on the discussion reported in [39]). We assume that the deformation gradient is multiplicatively decomposed, thus F = Fe Fp , where the lattice distortion is assumed to be contained in the elastic part Fe , and the plastic slip is contained in Fp [290]. Such deformation

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Fig. 7.11 The multiplicative decomposition F = Fe Fp in continuum theory of plasticity

mapping is illustrated in Fig. 7.11. In the continuum theory of plasticity, the geometrically necessary dislocation density tensor A is defined as [287, 290, 291] −1 T A = det(Fe )F−1 e (curlFe ) =

1 T Fp (CurlF−1 p ) . det(Fe )

(7.33)

Note that Curl is the curl differential operator with respect to the material point in the reference configuration, while curl is the curl operator with respect to a material point in the current configuration. From an atomistic point of view, the dislocation density can be expressed as [287, 290, 291] ∆l (7.34) A=l⊗b ∆v where l denotes the unit tangent vector along the dislocation line segment, ∆l is the element of the dislocation line, and ∆v is the elementary volume. The operator ⊗ denotes a dyad product. Under the assumption of infinitesimal deformation and negligible elastic strain, the dislocation density tensor is directly linked to the plastic distortion [287]. In the case of multiple dislocation segments within a representative volume element, the dislocation density tensor is defined by a linear combination of dislocations A= ηk lk ⊗ bk (7.35) k

where

dlk b. dv An integral formulation of (7.35) is given by 1 A= dl ⊗ b. ∆v ⊥ in ∆v ηk =

(7.36)

(7.37)

Note that statistically stored dislocations do not contribute to A in crystal plasticity since dislocation dipoles cancel. Curved dislocation lines can be approximated by straight dislocation segments.


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Equation (7.35) could be used to calculate the dislocation density tensor from atomistic data. The slip vector approach, as described in Sect. 2.10.3, is a possible candidate for this purpose. As discussed earlier, Fig. 2.41 shows the result of a slip vector analysis of a single dislocation in copper. The quantitative information obtained from atomistic results described in Sect. 2.10.3 can be used to calculate the dislocation density tensor.

7.3 Modeling Plasticity Using Large-Scale Atomistic Simulations Unlike continuum mechanics approaches, atomistic techniques require no a priori assumptions and no formulation of constitutive laws to model the behavior of dislocations and thus describe mechanical properties of materials. “Everything,” that is the complete material behavior, is determined once the atomic interactions are chosen. Atomic interactions can be defined for a specific material such as copper based on quantum mechanics calculations. Alternatively, they can also be chosen such that generic properties common to a large class of materials are incorporated. This allows to develop “model materials” to study specific materials phenomena. Models for ductile materials, for example, thus allow studying the generic features of ductile material behavior. The length and time range accessible to molecular dynamics is suitable for studying dislocation nucleation from defects such as cracks, as well as complicated dislocation reactions. The method also intrinsically captures dynamics of other topological defects, such as vacancies or grain boundaries and its interaction with dislocations. This is an advantage over mesoscopic methods that require picking parameters and rules for defect interaction. Also, using multibody EAM potentials (e.g., [94]), reasonably good models for some metals can be obtained. With sufficient computer resources it is possible to study the collective behavior of a large number of dislocations in systems with high dislocation density. Systems under large strain rates can be readily simulated. In discrete dislocation dynamics methods, such conditions are difficult to achieve. Two distinct length scales are involved in the mechanics of networks of crystal defects. The micrometer length scale is characteristic of the mutual elastic interaction among dislocations, but dislocation cores and formation of junctions and other reaction products is characterized by the length scale of several Burgers vectors and occur at the atomic length scale [38]. The two length scales span over several orders of magnitude, indicating the computational challenge associated with modeling. The rapidly advancing computing capabilities of supercomputers approaching TFLOPs and beyond now allow simulations ranging from nanoscale to microscale within a single simulation. The state-of-the-art of of ultra-large-scale simulations can model billion atom systems [138, 140, 219, 292].

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We will continue with a review of some of the activities and the historical development of atomistic simulations of dislocations and dislocation interactions in metals, and illustrate that progress in this field was highly coupled to advances in computer resources. Early studies by Hoagland et al. [293] and deCelis et al. [294] treated only a few hundred atoms. The researchers studied the competition of ductile vs. brittle behaviors of solids using quasistatic methods and investigated how and under which conditions dislocations are generated at a crack tip. Such microcracks can be found in virtually any real materials (referred to as material flaws), and serve as seeds for defect generation (see also Fig. 1.7 and the associated discussion). The studies were small in size, and only a few dislocations could be simulated. Due to the lack of dynamic response and the system size limitations, the treatments were valid only until the first dislocation moved a small fraction of the sample size away from the crack tip. Computational resources rapidly developed during the 1990s (see Fig. 2.34). Cleri et al. [9] studied the atomic-scale mechanism of crack-tip plasticity using around 80, 000 atoms. They investigated dislocation emission from a crack tip by extracting the atomic-level displacement and stress fields, so as to link the molecular dynamics results to continuum mechanics descriptions of brittle vs. ductile behavior in crack propagation [30, 66, 67]. Zhou and coworkers [295] performed large-scale molecular dynamics simulations and carried out simulations of up to 35 million atoms to study ductile failure. In these simulations, the atoms interact with Morse pair potentials as well as more realistic EAM potentials. They observed emission of dislocation loops from the crack front, and found that the sequence of dislocation emission events strongly depends on the crystallographic orientation of the crack front. They assumed that systems comprising of 3.5 million atoms are sufficient to study the early stages of dislocation nucleation (since they observed the same feature independent of the system size). Later, simulations with more than 100 million atoms showed generation of “flower-of-loop” dislocations at a moving crack tip [146]. It was observed that the generation of dislocation loops in a rapidly propagating crack occurs above a critical crack speed, suggesting a dynamic brittle-to-ductile transition. Other studies focused on the creation, motion, and reaction of very few dislocations in an FCC lattice, with the objective to understand the fundamental principles. Research activity was centered on atomistic details of the dislocation core making use of the EAM method [296, 297]. Zhou and Hoolian [298] performed molecular dynamics simulations of up to 3.5 million atoms interacting with EAM potentials (they used up to 35 million atoms with pair potentials). They studied the intersection of extended dislocations in copper and observed that the intersection process begins with junction formation, followed by an unzipping event and partial dislocation bowing and cutting. These are unique studies, whose results can be immediately applied in mesoscopic simulations. Additional research was carried out to investigate the screw dislocation structure and interaction in a nickel FCC lattice by Qi


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et al. [299], using a QM-Sutton-Chen many body potential. The researchers studied the core geometry of partial dislocations, as well as the motion and annihilation of oppositely signed dislocations, and discussed cross-slip and associated energy barriers. Atomistic simulations have also been applied to study the interaction of dislocations with other defects. Further studies focused on the ductility of quasicrystals (see also discussion in Sect. 6.3) [226, 227, 229]. Atomistic simulation particularly helped to explain the mechanism of dislocation motion in quasicrystals. An important contribution was the observation of phason-walls that are attached to each moving dislocation. These phason-walls helped to clarify some of the perplexing properties of quasicrystals found in experiments, such as a brittleto-ductile transition at about 80% of the melting temperature. Most recently, three-dimensional atomistic simulations at elevated temperatures were carried out [300] where it was found that dislocation climb processes play an increasingly important role. In summary, molecular dynamics simulations of plasticity have advanced to a quite sophisticated level. Atomistic simulations with multibody EAM potentials can also be applied to describe mechanical properties of thin metal films, thereby providing a path to investigate size effects of materials behavior.

7.4 Case Study: Deformation Mechanics of Model FCC Copper – LJ Potential To illustrate the analysis of an atomistic simulation and the interpretation of the deformation mechanisms in light of theoretical predictions, we review the analysis of a large-scale molecular dynamics simulation of work hardening in a model system of a ductile solid of a simulation originally reported in [138]. Two systems will be considered, first a simple model with an LJ potential (this section). Then, a model simulation with an EAM potential (next section). The comparison between the two results illustrates some of the differences between both atomistic models of metals. 7.4.1 Model Setup Both systems feature a small single nanocrystal with small surface cracks, under mode I tensile loading. In the first case study, two opposing surface cracks on opposite faces of a three-dimensional FCC solid cube are considered. Figure 7.12a and b) shows the geometry and the crystallographic orientation [39, 301]. With tensile loading, the emission of thousands of dislocations from two sharp cracks is observed. The dislocations interact in a complex way, revealing three fundamental mechanisms of work-hardening in this ductile material. These are (1) dislocation cutting processes, jog formation, and generation of trails of point defects; (2) activation of secondary slip systems by Frank–Read

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Fig. 7.12 Simulation geometry, lattice orientation, and time-sequence of the workhardening simulation. (a) Simulation geometry and (b) lattice orientation, also defining the directions for the x, y, z coordinate system

and cross-slip mechanisms, and (3) formation of sessile dislocations such as Lomer–Cottrell locks. On the typical time scale of molecular dynamics simulations, the dislocations self-organize into a complex sessile defect topology. The analysis illustrates numerous mechanisms formerly only conjectured in textbooks and observed indirectly in experiments. The plastic or nonreversible deformation of materials occurs immediately after a regime of recoverable elastic deformations and is governed by the nucleation and motion of defects in the crystal lattice [1, 38, 60]. In experiment, researchers often rely on indirect techniques to investigate the creation and interaction of defects. In theory, predictions are primarily based on continuum theory with phenomenological assumptions. While the continuum description has been very successful in the past, some of the key features of plasticity can only be understood when the atomistic viewpoint is taken into account, and this may be achieved using atomistic simulation methods, like molecular dynamics [28, 84, 138, 149, 152–154, 302–305]. However, most molecular dynamics studies consider a small number of dislocations and address specific dislocation mechanisms; for instance, dislocation nucleation from cracks or dislocation reactions [173, 298]. In the present simulations, a simple interatomic force law is adopted. This choice is motivated by the goal to investigate the generic features of a particular many-body problem common to a large class of real physical systems and not governed by the particular complexities of a unique molecular interaction. The model potential for the present study is the Lennard-Jones potential [84]. With its well-known shortcomings, it nevertheless provides a fundamental description of the generic features of interatomic interaction: Atomic repulsion at close distances, and attraction at large radii of separation.


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The slab (Fig. 7.12a) is initialized to zero temperature, and a mode I tensile strain of 4% is imposed on the outer most columns of atoms defining the opposing vertical y − z-faces of the slab, and kept constant throughout the simulation. The initial purely elastic strain is relaxed into plastic strain during the course of the simulation. Since an N V E ensemble is used, the relief of the potential energy causes an increase of temperature in the crystal. 7.4.2 Visualization Procedure To “see” into the interior of the solid, only those atoms are shown that have potential energy greater than or equal to −6.1, where the ideal bulk value is −6.3. This approach was used very effectively in several atomistic studies using a single crack for displaying dislocations, microcracks, and other imperfections in crystal packing. This filtering scheme reduces the number of atoms seen by approximately two orders of magnitude in 3D; the visible atoms are associated with faces of the slab and initial notch, surfaces created by crack motion, local interplanar separation associated with the material’s dynamic failure at the tip, and topological defects created in the otherwise perfect crystal. Because of periodic boundary conditions, the vertical faces are not exterior surfaces and therefore transparent. For the analysis, the centrosymmetry technique [36] is also used, which is particularly helpful in distinguishing stacking fault regions and partial dislocations as well as point defects. 7.4.3 Simulation Results Upon application of loading to the system, the cracks serve as fertile sources for dislocations. Within a few picoseconds, thousands of dislocations appear and flow into the interior of the solid. The dislocations from the cracks glide on two primary glide planes (111) and (111). This is in agreement with the fact that dislocations are nucleated in the direction of largest shear stress and thus largest Schmid factor in the K-field of the crack [38, 60]. In this early stage, the dislocations glide through the initially perfect crystal without sensing any obstacles. Dislocations with the same Burgers vector and line orientation repel each other and therefore push those previously created rapidly through the crystal [38, 60]. The nucleation of partial dislocations is observed because of the very small stacking fault energy in the LJ model material. The Burgers vectors of the emitted partials are predominantly [211] and [121] on the (111) glide plane, as well as [211] and [121] on the (111) glide plane. Similar cutting mechanisms are also observed for dislocations with the other two possible Burgers vectors. Assuming a positive line direction of the dislocation in the negative zdirection, the Burgers vectors all have a component in the negative x-direction (which is [110)) in the upper cloud. In the lower cloud, the Burgers vectors have the opposite sign.

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Dislocations on the (111) and (111) glide planes in regions very close to the crack tip are also observed. However, these dislocations sense no strong elastic driving force that would allow them to flow further away from the crack tip. The dislocations on the primary glide systems (111) and (111) out run dislocations on the (111) and (111) glide planes, and in the center of the simulation sample, only dislocations on primary glide systems occur.

Fig. 7.13 Interaction of a dislocation line with an obstacle in the material. Subplot (a) shows a sequence of events that illustrate how the dislocation line becomes bent as it cannot pass through the obstacle. Subplot (b) depicts a schematic that shows an equivalent force acting on the dislocation line

Dislocation Cutting Processes The process of work-hardening or strengthening of materials is often due to an additional constraint on the mobility of dislocations. For instance, once a moving dislocation line hits an obstacle, it bends backwards as it senses the retracting force of the obstacle. This general process is depicted in Fig. 7.13. Figure 7.14 shows the dislocation–particle interaction in ordered matrix materials [306]. A dissociated complete dislocation interacts individually with the particle (Fig. 7.14a). A TEM weak-beam micrograph of the dislocation– particle interaction is shown in Fig. 7.14b. During work-hardening, dislocation cutting processes lead to similar mechanisms that hinder the free motion of dislocations.


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Fig. 7.14 Dislocation–particle interaction in ordered matrix materials. A dissociated complete dislocation interacts individually with the particle (subplot (a) for a schematic). A TEM weak-beam micrograph of the dislocation–particle interaction in Fe–30 at.%Al is shown in subplot (b). Reprinted from Acta Metallurgica, Vol. 46(16), pp. 5611–5626, E. Arzt, Size effects in materials due to microstructural and c 1998, with permission dimensional constrains: A comparative review, copyright from Elsevier

Fig. 7.15 Schematic that illustrates the formation of vacancies while jogs (visible as kinks in the dislocation line) are forced to move through the crystal

It is known from the literature [38, 60] that, when two screw dislocations intersect, each acquires a jog with a direction and length equal to the Burgers vector of the other dislocation. Upon intersection, the dislocations cannot glide conservatively since each jog has a sessile edge segment. However, if the applied stress is sufficient large, the dislocations will glide, and the moving jogs will leave a trail of vacancies (see schematic in Fig. 7.15), or a trail of interstitials depending on the line orientation and the Burgers vector of the reacting dislocations.

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Fig. 7.16 Schematic of different dislocation cutting processes. Subplot (a) shows two partial dislocations cutting each other. Both dislocations leave a trail of point defects after intersection (circles). The arrows indicate the velocity vectors of the dislocations. Subplot (b) shows a partial dislocation (black line) cutting the stacking fault of another partial dislocation. Dislocation number 1 leaves a trail of point defects (circles) once it hits the stacking fault generated by dislocation number 2

Generation of interstitials is energetically expensive and thus not observed very frequently. The mechanism of dislocation cutting is shown in Fig. 7.16a. Another possible mechanism of dislocation cutting is when a partial dislocation moves through the stacking fault generated by another partial dislocation as shown in Fig. 7.16b.

Fig. 7.17 Atomistic simulation results of different types of point defects: (a) Trail of partial point defects, (b) vacancy tube, and (c) trail of interstitials. The inlays provide a detailed atomistic view of the defect structure

Intersection of a partial dislocation with the stacking fault of another dislocation is observed in early stages when dislocations within the same cloud intersect the stacking fault generated by other dislocations. The dislocation line forms a jog that features a sessile edge component. Creation of a trail of point defect causes a resulting drag force on the dislocation which results in a bowing out of the segment. Because of the specific Burgers vectors of the


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Fig. 7.18 Generation of point defects due to jogs in screw dislocations. Two representative dislocation-cutting processes are shown, (a) leading to formation of an interstitial, (b) leading to formation of vacancy tubes. In case the edge component of the jog is smaller than that of a partial Burgers vector, trails of partial point defects, characterized by generation of local lattice distortion rather than complete rows of missing or additional atoms, are generated

intersecting dislocations and the line orientation, the cutting dislocations only √ have a small screw component equal to 6/12a0 (half the length of a partial dislocation). Therefore, not a complete point defect is generated but rather a trail of local lattice distortion, a phenomenon that will be referred to as a trail of partial point defects. The defect has a dipole structure, and is depicted in Fig. 7.17a. The dragging force of the trail of partial point defects was estimated by calculating the energy per length for a trail of partial point defects and that of a vacancy tube, and it was calculated to be about 20% of that of a complete vacancy tube. This defect thus causes a significant dragging force on the dislocations, in particular, in a situation where the dislocation density is extremely high as in the present simulation. Two representative dislocationcutting processes are schematically shown in Fig. 7.18. The forces on the jog segment are a combination of tensile and compressive forces in the two 112 directions. This immediately explains the dipole structure seen in Fig. 7.19c. The bowing effect on the dislocations is shown in Fig. 7.19a and b, based on a centrosymmetry analysis in Fig. 7.19a, and on an energy analysis in Fig. 7.19b. So far we have only discussed the dislocation reactions that take place when dislocations of the same clouds react. Numerous dislocation reactions occur when dislocations of the two different clouds start to interact, as shown in Fig. 7.20. Such reactions primarily involve the mechanism is dislocationcutting processes as depicted schematically in Fig. 7.16(a). Due to the Burgers vectors and the dislocation line orientation, when the dislocation clouds meet straight ahead of the crack, the reactions are very similar to those observed in the previous stages when dislocations of the same clouds cut each other’s stacking fault and thus, trails of partial point defects are generated. In Fig. 7.19c, the significant effect of jog dragging on the motion of the dislocations is clearly observed. The elastic interaction of dislocations repelling each other causes a decrease in the dislocation velocity.

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Fig. 7.19 Generation of trails of point defects in early stages of the simulation. Dislocation number 1 and number 2 leave a stacking fault plane, which is subsequently cut by dislocation number 3. Therefore, two trails of partial point defects are generated resulting in bowing of dislocation number 3. Subplot (a) shows a centrosymmetry analysis [36] where the stacking fault planes are drawn yellow; subplot (b) shows an energy analysis of the same region where the stacking fault planes are not shown. Subplot (c) shows a close-up view on the dislocation cutting process

Fig. 7.20 This plot shows the reaction of the two dislocation clouds originating from opposing crack tips, causing the generation of numerous point defects. The circles highlight the region of interest in which the dislocation reactions occur

Due to the crystal orientation and the Burgers vector of the dislocations, only trails of partial point defects as well as interstitials can be generated from the cutting processes on the primary glide systems.


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Fig. 7.21 Activation of secondary slip systems and generation of Lomer–Cottrell locks. Subplot (a) shows a schematic of the cross-slip mechanism. Subplots (b) and (c) show details of activation of secondary slip systems (subplot (c) represents a magnified view of subplot (b), with the region of interest highlighted by a circle). This mechanism of cross-slip of partial dislocations, here first observed in molecular dynamics simulation, was originally proposed theoretically by Fleischer [37], and contrasts the well-known Friedel–Escaig mechanism [38]

Complete vacancy tubes [149, 305] are not generated until later stages of the simulation when the dislocation density becomes very large and secondary slip systems are activated. A large number of such defects are created and appear as straight, thick lines in the plot of the potential energy of atoms. The geometry of complete vacancy tubes is shown in Fig. 7.17b. The vacancy tubes observed in the simulation are only several nanometers long. The formation of some trails of interstitials is also observed. The number of such processes is rather small since the energy to create such defects is extremely large. A trail of interstitials is depicted in Fig. 7.17c. The excess of vacancy concentration is also found in experiments of materials under heavy plastic deformation [307–309]. More recently, there are also several observations of vacancy generation in computer simulations [149, 309, 310]. In particular, the study reported in [309] shows formation of single vacancy tube due to dislocation cutting processes.

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Fig. 7.22 Subplots (a) and (b) show a view from the [110]-direction (a) before nucleation of dislocations on secondary glide systems (therefore only straight lines), and (b) after nucleation of dislocations on secondary glide systems (which appear as curved lines)

Activation of Secondary Slip Planes It is found that secondary slip systems are activated once the dislocation density is above a critical value. Figures 7.21a–c and Fig. 7.22 illustrate the activation of secondary slip systems.

Fig. 7.23 Subplots (a) and (b) show detailed views on the formation of sessile Lomer–Cottrell locks, with its typical shape of a straight sessile arm connected to two partial dislocations

In the simulation, it is found that dislocations on secondary slip systems are generated by cross slip and Frank–Read mechanisms. This is an unexpected observation because cross slip is only possible, according to the classical dislocation mechanics, along at least locally constricted screw segments (Friedel–Escaig’s mechanism) [38]. Due to the low stacking fault energy of the model material used here, only partial dislocations exist in the simulation. Each occurring cross-slip event leaves behind a straight, sessile stair-rod


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dislocation to conserve the Burgers vector. These sessile segments can clearly be observed in Fig. 7.22. The main result is that by this cross-slip mechanism it is possible to observe cross-slip in a situation when only partial dislocations, together with very high stresses are present. Even in systems with only partial dislocations, nature finds a way to relieve elastic energy into secondary slip systems! It is noted that this mechanism of cross-slip of partial dislocations was proposed in 1959 by Fleischer [37]. This simulation confirms the existence of such mechanism. Formation of sessile locks has recently also been found in deformation of nanocrystalline materials [311]. The activation of secondary slip systems is important for the hardening process because dislocation cutting or the formation of sessile locks generates a large number of additional defects. Dislocations on secondary slip systems cannot move easily in the beginning. However, soon afterward, numerous new defects are generated. In particular, it is found that the formation of complete vacancy tubes at this stage dramatically increases the concentration of vacancies. It is noted that cross-slip mechanisms have been studied in other molecular simulations previously [299, 312–314], and we refer the reader to these articles for further reference. Formation of Sessile Locks Another mechanism of dislocation interactions is the formation of sessile locks. An analysis of the Burgers vectors of the primary dislocations reveals that formation of sessile locks is not observed until dislocations on secondary glide planes are activated. Sessile dislocation locks can be formed depending on the Burgers vector whenever two partial dislocations on different glide planes get close together. Some combinations of partial dislocations are attractive and form a dislocation line with Burgers vector of the type 16 [110]. These dislocations are not glissile on any glide system of the FCC lattice, and therefore it is sessile and cannot move. Such defects provide a serious burden for further dislocation glide through the material, since other dislocations approaching the sessile dislocations cannot easily glide through this defect agglomerate, the so-called Lomer–Cottrell lock. One of the possibilities how such locks may be circumvented is by Orowan–mechanisms [1, 38]. Formation of sessile dislocations is also observed in the simulation. It is found that the sessile dislocations severely hinder further dislocation motion, as it is assumed in the classical theories of work hardening [38]. In Fig. 7.23 we show some snapshots when such defects are generated. The typical structure of the Lomer–Cottrell lock is characterized by a straight sessile arm connected with two partials on different glide planes. This can be clearly seen in the figure. Formation of Lomer–Cottrell locks has been studied in several molecular dynamics simulations, as for instance in [315].

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The Work-Hardening Regime The simulation reveals the final sessile structure of a large number of dislocations in the late stages of the plastic deformation. When a situation is reached where the plastic deformation of a solid has generated such a high dislocation density that dislocation motion is hindered by their mutual interactions, one generally speaks of the work-hardening regime of deformation. In the molecular dynamics simulation, the deformation is large enough that this work-hardening regime is reached quickly. In this final stage, a structure composed of point defects, sessile dislocations, and partial dislocations is observed. This geometric arrangement explains the particular structure dominated by the straight defect segments in the final stages of simulation.

Fig. 7.24 The final network from a distant view, including a blow-up to show the details of the network [39]. The characteristic structure of the network is due to the fact that all sessile defects (both trails of partial and complete point defects) as well as sessile dislocations as part of the Lomer–Cottrell locks assume tetrahedral angles and lie on the edges of Thompson’s tetrahedron. The wiggly lines in the blow-up (see the right half of the figure) show partial dislocations, and the straight lines correspond to sessile defects

Fig. 7.24 reveals the final network from a distant view. The blow-up shows a more detailed energy analysis of the network. The characteristic structure of the network is due to the fact that all sessile defects (trails of point defects as well as 16 [110]-sessile dislocations as part of the Lomer–Cottrell locks) assume tetrahedral angles and lie on the edges of Thompson’s tetrahedron. This immediately explains the particular structure of the observed sessile network. In contrast to the initial stages of plastic deformation, where dislocation glide occurred easily, dislocation motion is essentially stopped due to the work hardening and plastic relaxation.


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7.4.4 Summary It is observed that dislocation-cutting processes generate a large number of trails of point defects. An interesting aspect is the observation of generation of trails of partial point defects by dislocation cutting processes with an orientation such that the sessile edge component of the jog is relatively small. This does not cause generation of a complete row of vacancies or interstitials, but rather a local lattice distortion (see Fig. 7.17c). As it was shown in the simulations reviewed here, it has an important effect on hindering free dislocation motion. Such deformation mode could become important in nanostructured materials, where it is now understood that partial dislocations can dominate plastic deformation [153, 154, 303, 304]. Indeed, in some molecular dynamics simulations of nanocrystalline materials, similar cutting mechanisms as described in the present work have been observed [43, 153, 154, 304, 311]. Once the dislocation density in the center of the solid grain exceeds a critical value, dislocations on secondary slip systems are activated. Since only partial dislocations are present in the current study, the dislocations must leave a sessile dislocation segment to conserve the Burgers vector (see Fig. 7.21b). Such mechanism of partial dislocation cross-slip was actually proposed theoretically almost 50 years ago [37], and directly simulated by the molecular dynamics simulation. Another important dislocation reaction mechanism is the formation of sessile Lomer–Cottrell locks (see Fig. 7.21). Formation of sessile locks is not observed until dislocations on secondary glide planes are activated. The formation of complete vacancy tubes is noticed once secondary slip systems are activated (see Fig. 7.16). The observation of generation of vacancies is in agreement with the understanding that heavy plastic deformation causes generation of vacancies. Mott (1960) [316] was the first to predict the vacancy formation by motion of jogged screw dislocations. The production of vacancies [307] and their influence on plasticity has been studied in experiment and theory, [149,305,310,317,318] and has also been included in discrete dislocation modeling recently [317]. Figure 7.25 plots the development of the density of different defects during the simulation using a method of separating defects of different energies. The potential energy of atoms allows discrimination between different types of defects. This figure summarizes the hardening process in the simulation. At the first stage, partial dislocations are emitted from cracks with a high rate. As a consequence of cutting processes trails of point defects are produced. Finally, the activation of secondary glide systems results in additional dislocation cutting processes including the generation of complete vacancy tubes. Another important hardening mechanism is formation of sessile locks. It is interesting to note that the rapid production of partial dislocation ceases, with the creation of cutting debris (and the plastic relaxation), and the density of partial dislocation finally goes into an equilibrium.

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Fig. 7.25 Development of the density of different defects during the simulation using a method of separating defects of different energies [39]

The model simulation reviewed here has revealed some of the fundamental, classical mechanisms of work hardening, [1, 38, 60] in a single computer simulation, including: (1) dislocation cutting processes, (2) cross-slip, and (3) formation of sessile dislocation locks. The study reviewed here exemplifies methods to analyze ultra-large-scale simulations. Similar techniques, based on energy filtering, geometrical analysis, and centrosymmetry parameter studies may be helpful for other future investigations. The collective operation of the basic hardening mechanisms apparently constricts the mobility of dislocations. A large ensemble of defects self-organizes into a complex defect network with a regular structure leading to a final defect network composed of trails of partial point defects and complete vacancy tubes as well as some trails of interstitials, sessile dislocations, and partial dislocations. The characteristic structure of the final network is given by the geometrical condition that the sessile defects appear as straight lines lying at the intersection of stacking faults, thus along the sides of Thompson’s tetrahedron. The results illustrate that even though the LJ potential is a simplistic model for interatomic bonding, it is nevertheless capable of capturing most of the predicted hardening mechanisms [1, 38, 60]. A computer simulation using the LJ potential under extreme conditions of very large strains seems to be able to reproduce the essential deformation mechanisms of natural crystalline materials such as metals. However, in molecular dynamics simulations, systems are generally under relatively “harsh” conditions, such as high stress and strain rates. This may


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lead to activation of all of the possible dislocation mechanisms concurrently, whereas under more realistic conditions, the system discriminates between the different mechanisms, then favoring the lowest-energy path. Therefore, although MD simulation can elucidate the possible mechanisms, experimental verification of the rate limiting processes remains a critical issue. In this spirit, the molecular dynamics experiments could thus be viewed as informative, but not representative of the hardening mechanisms under more normal conditions. To investigate the hardening mechanisms with more realistic potentials for metals, we review additional simulations of work-hardening using potentials developed within the EAM framework in the next section. As expected, similar hardening mechanisms are discussed in this chapter are observed. In particular, dislocation cutting processes, cross-slip, and formation of sessile locks play a dominating role. We emphasize that such detailed atomistic views on fundamental aspects of plasticity, as shown in the study reviewed here, can neither be obtained from experimental techniques, nor can they be calculated with continuum mechanics methods, especially in view of extremely high dislocation densities and strain rates. In this respect, with its limitations understood, molecular dynamics simulations represent a convenient, unique and straightforward way to obtain such information.

7.5 Case Study: Deformation Mechanics of a Nickel Nanocrystal – EAM Potential The objective of this section is to review a computational experiment as discussed in the previous section, this time using a different interatomic potential. Here an EAM potential, parameterized for nickel, is used to carry out a simulation of large-deformation plastic behavior. Figure 7.26 depicts the geometry and the setup of this experiment. Figure 7.27 shows a sequence of snapshots that illustrate the dislocation nucleation process from the crack tip. This snapshot depicts the early stage of dislocation nucleation. Figure 7.28 depicts a closer view on the dislocation structure at a later stage in the simulation. At later stages, secondary slip systems are activated, as shown in a centrosymmetry analysis depicted in Fig. 7.29. A few notable differences to the LJ case are: • Full dislocations are nucleated from the crack tip, due to the finite stacking fault energy (in contrast to the zero stacking fault energy in the LJ case). Full dislocations are visible due to the leading and trailing partial, enclosing the stacking fault region (see also Fig. 7.30 for a direct visualization).

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Fig. 7.26 Simulation geometry, lattice orientation, and time-sequence of the EAM simulation. (a) Simulation geometry and (b) lattice orientation, also defining the directions for the x, y, z coordinate system

Fig. 7.27 Sequence of snapshots that illustrate the dislocation nucleation process from the crack tip, for the case of an EAM potential. This snapshot depicts the early stages of dislocation nucleation, depicting how dislocations grow from the tips of the crack

• Different dislocation interactions take place, in particular, the sessile segments that are generated due to dislocation cutting mechanisms are of finite length (see also Fig. 7.31). This is because the stacking fault regions are also of finite length, and because the sessile defect only appears in regions where two stacking fault regions intersect.


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Fig. 7.28 Closer view on the dislocation structure at a later stage in the simulation with the EAM potential (for a system with a single crack at one surface)

7.6 Case Study: Multi-Paradigm Modeling of Chemical Complexity in Mechanical Deformation of Metals The prediction of the deformation behavior of metals in the presence of environmentally embrittling species like water or hydrogen, or under presence of organic oxidative chemicals presents a critical challenge in materials modeling. This exceeds the capability of EAM-type potentials, since chemical reactivity and a variety of chemical bonds are the key to describe the behavior of these materials appropriately. This task can be accomplished by a combination of the first principles based reactive force field ReaxFF and the embedded atom method (EAM) in a generic multiscale modeling framework, as reported in [41]. This hybrid method enables one to treat large reactive metallic systems within a classical molecular dynamics framework. The hybrid method is based on coupling multiple Hamiltonians by weighting functions that allows accurate modeling of chemically active sites with the reactive force field, while other parts of the system are described with the computationally less expensive EAM potential. In this section, a brief review of a case study is provided that illustrates the significance and the potential of this approach. Here we review the application of a hybrid modeling scheme in the study of fracture of a nickel single crystal under the presence of oxygen molecules, where thousands of reactive atoms are used to model the chemical reactions that occur under the presence of oxygen. This hybrid method constitutes an alternative to existing methods that are based on coupling quantum mechanical methods such as DFT to empirical potentials. The goal of the studies

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Fig. 7.29 Centrosymmetry analysis that shows the activation of secondary slip systems, at later stages in the simulation with the EAM potential

reviewed here is to simulate the effect of organic molecules interacting with a crack opening. It is observed that the oxide formed on the crack surface produces numerous defects surrounding the crack, including dislocations, grain boundaries, and point defects. It is shown that the mode of crack propagation changes from brittle crack opening at crack tip to void formation ahead of crack and void coalescence for {111}112 orientation of the crack. The results illustrate the significance of considering oxidative processes in studying deformation of metals. The solution reviewed here is to take advantage of the best qualities of the ReaxFF force field (that is, its ability to model metal–organic interactions and charge transfer) together with the best qualities of EAM potentials (that is, in its computational speed and accuracy of descriptions of metals and defects in metals) combined in a hybrid scheme, the ReaxFF-EAM coupling as discussed in Sect. 5.4.5 (for a schematic, see Fig. 5.8). This will enable us, for the first time, to carry out finite temperature calculations of deformation of large reactive metal systems that include metal atoms, oxygen molecules, hydrogen molecules, or water molecules, covering a wide range of the periodic table. 7.6.1 Atomistic Model and Validation The successful integration of ReaxFF and Tersoff force fields within the framework of CMDF has already been established for the description of fracture in silicon, as discussed in Sect. 6.11. Here the validity of coupling of reactive force fields and embedded atom EAM potentials within CMDF particularly suitable


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Fig. 7.30 Centrosymmetry analysis of the details of the dislocation structure, for the EAM simulation. In contrast to the LJ simulations, here complete dislocations (that is, leading and trailing partial dislocations) are emitted at the crack tip. As a consequence, the dislocation cutting products (partial point defects) have a finite length. Subplot (a) depicts a sequence of two snapshots, illustrating the growth of the dislocation network at the crack tip. Subplot (b) shows a detailed view of the dislocation structure

Fig. 7.31 Centrosymmetry analysis of the details of the dislocation structure, illustrating that the point defect trails are of finite length in the case of the EAM simulation. Subplot (a) shows the network with stacking faults, and subplot (b) shows the network without stacking faults. The thick lines refer to the trails of point defects

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for modeling metals is discussed. Specifically, the focus of the studies reviewed here is on combining the Ni/Ni–O ReaxFF potential and Baskes-Daw’s Ni EAM potential. Shear Loading (Mode II) of an Initial Crack in a Single Crystal of Nickel A single nickel crystal with an elliptical crack in the 110112 orientation is shear loaded in its x-direction faces, so that a pair of partial dislocations is emitted from the crack tip. As the partials move in the EAM region, the separation distance between them is found to be approximately 17 ˚ A (see Fig. 7.32 for the geometry and Fig. 7.33 for simulation results).

Fig. 7.32 Summary of the two loading conditions considered here, shear loading (left) and shock loading (right). The figure also depicts which domains are handled by EAM (dark grey) and which domains are handled with ReaxFF (light grey)

The partial dislocations are allowed to pass through a ReaxFF region with a suitable transition region from EAM to ReaxFF. A distance of 7 ˚ A is chosen for the transition region width and 7 ˚ A for the buffer region width. The reactive region spans approximately 1,000 atoms and lies in the path of the partials slip direction. The attempt is to see any change in behavior as the partials pass into the reactive region through the EAM potential region. As results in Fig. 7.33 show, the partials maintain the same separation as they pass through the reactive region. The speed and direction of slip are also not affected. Further simulations with varying values of the transition region width down to 1 ˚ A, while keeping the buffer region width fixed at 7 ˚ A and the total number of nickel atoms in the reactive region about the same, show no change in the behavior of the system of two partials as it moves across the boundary of the two force fields. This can be attributed to the close match between the ReaxFF and EAM force fields equations of state over a large range of strain values. This ensures that the (HReaxFF − HEAM ) term in (5.6) is small for this system, and hence, (5.7) is still valid for this system with a small transition


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Fig. 7.33 Centrosymmetry analysis plot of a complete dislocation emitted as two partials from a crack under shear (mode II) loading moving through EAM and ReaxFF regions (atoms with perfect FCC coordination have been removed). The dashed black circles indicate the region of atoms modeled by ReaxFF potential (the exterior of the circle contains a skin of ghost atoms with zero weight on the forces, while the entire interior is completely reactive). Subplots (a–e) are taken at time interval increments of 500 fs each. The distance between the partials appears unchanged throughout the nucleation and propagation process, and does not change as it passes through the handshaking and ReaxFF regions and back into the EAM region

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˚ region. However, as a conservative estimate the transition width is left at 7 A for all simulations reviewed in the remainder of this section. This test calculation proves that the hybrid scheme does not influence dislocation propagation dynamics. In particular, it shows that a dislocation can easily enter and leave a reactive region inside a crystal. Figure 7.33 depicts snapshots of this simulation. Uniaxial Shock Loading of a Perfect Nickel Crystal A single nickel crystal is subjected to uniaxial shock load in the [110] direction with a piston speed of 4 km/s (schematic of the loading conditions see Fig. 7.32, right part). The elastic shock front that moves out in the EAM modeled region is allowed to pass through a ReaxFF region with the same handshaking conditions as discussed earlier. The shape of the shock front and velocity are observed as it passes through different atomistic potential regions [40]. As seen in Fig. 7.34, the shock front maintains its shape as it passes through the reactive region. This shows that the coupling region between the two potentials does not spuriously reflect elastic waves or change the straight geometry of the shock front. The behavior of dislocations and elastic waves as they pass through the coupling region from one atomistic potential region to another shows no anomalies. In particular, the shock front line maintains its straight geometry, which proves that the wave speeds for both methods are similar. 7.6.2 Example Application: Modeling Hybrid Metal–Organic Systems The study of effect of environmental effects of O2 , H2 , and H2 O on crack propagation in metal crystals provides important insight into the embrittling mechanisms that are at play in changing the fracture strength of the material. The effect of formation of a layer of oxide at the surface of cracks in the metal can play an important role in modifying its fracture properties. Here we particularly pay attention to the effect of atmospheric oxygen on single crack propagation in a single crystal nickel under a tensile load in mode I loading. We are interested in early fracture events and in particular, in the first deformation event near the crack tip. This problem has been intractable to empirical potentials atomistic modeling because of the unavailability of a proper empirical potential that describes the nickel oxide correctly. The use of ab initio methods to model the entire problem is prohibitively large. This constitutes a perfect problem for multiscale modeling in CMDF with the oxidation regions, where nickel–oxygen chemistry can be handled by the ReaxFF


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Fig. 7.34 Velocity profile in x-direction for a uniaxial shock wave in a system with ReaxFF and EAM regions coupled [40, 41]. The shock wave front can be identified as a vertical line of high velocity (darker color) atoms. The dotted circles contain all atoms modeled by ReaxFF potential (the exterior of the circle contains a skin of ghost atoms with zero weight on the forces, while the entire interior is completely reactive). Subplots (a), (b), and (c) depict the velocity profile in the x-direction across the sample as the shock wave passes through. There appears to be no change in shock front profile as it encounters and passes through the reactive regions, indicating a smooth handshaking between the two simulation methods without force discontinuities

potential, and the unoxidised bulk Ni can be modeled by the computationally more efficient EAM potentials [40, 41]. Deformation Dynamics of a Single Nickel Crystal with an Elliptical Crack Under Presence of Oxygen Molecules A perfect crystal with an initial elliptical crack to serve as failure initiation point is considered here. Different orientations of a single Ni crystal with different crack orientations were considered in this study to investigate the effect

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of different slips systems being activated first. We focus on two cases to show the variety in deformation mechanisms depending upon crack orientation: • Ni crystal with crack in 110112 orientation with tensile loading in 110 direction • Ni crystal with crack in 111112 orientation with tensile loading in 111 direction The simulation cell size is chosen 150 ˚ A × 150 ˚ A × 8˚ A with free surfaces in the x and y-directions, and periodic boundary conditions (PBC) in the zdirection. The cracks have an elliptical shape, and are inserted in the middle of the samples in the x − y plane, and run through the entire sample thickness in the z-direction. The crack propagation is studied under tensile mode I loading with and without the presence of a single oxygen molecule in the crack region to discern changes in mechanism upon presence of oxygen molecules. The reactive regions in the multiscale modeling framework are defined by a radius of reactive region around each oxygen atom in the system. A region of 5˚ A radius is chosen in the nickel system around an oxygen atom within which the influence of the oxygen atom can be felt and these atoms are described by a reactive Ni–O force field. The handshaking transition region is taken as 7˚ A in width and the buffer region with ghost atoms is also 7 ˚ A in thickness, keeping the same values as in the test cases discussed above. The union of all reactive regions triggered by the oxygen atoms constitutes the entire reactive domain in the system. This method ensures that ReaxFF is applied solely to primarily describe metal-organic chemistry, while the EAM method handles dislocation propagation and reactions in the bulk [40, 41]. Crack in 110112 Direction The system is first set up with no oxygen atoms inside the crack region, and tensile loading at a constant strain rate is applied. Due to the time scales accessible to molecular dynamics, this strain rate is rather high, approaching 1010 s−1 . In this system, all atoms are modeled by an EAM potential as the absence of oxygen atoms triggers no reactive regions. This system is thus a reference case for the nickel–oxygen system. As seen in Fig. 7.35, the crack tip emits partial dislocation on one of the inclined 111 planes at the crack tip, at a strain of 0.039, and emits a second partial at the other crack tip end at a slightly larger strain of 0.041. Now we consider the behavior of this system under presence of an oxygen molecule (Fig. 7.36). A single oxygen molecule is randomly placed inside the crack and the system is allowed to equilibrate chemically over 10,000 molecular dynamics integration steps at a time-step of 0.5 fs. During this relaxation phase, the system includes approximately 750 atoms that are modeled by a reactive potential. Different starting positions are used for the oxygen molecule to investigate if there is a stochastic effect of the initial position of O2 . An NVT


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Fig. 7.35 Partial dislocation emission from crack tip for the 110 112 crack orientation (only part of the crystal close to the crack tip is shown), for the case when no oxygen atoms present (all-atom EAM) [40,41]. Defects produced at the crack tip are circled. Subplots (a) and (b) show emission of partials from top and bottom of the crack tip at critical strains of 0.039 and 0.041

ensemble is used at 400 K to equilibrate the system to allow faster reaction of the nickel and oxygen. After 10,000 molecular dynamics integration steps, the temperature is slowly quenched down close to 0 K, allowing the system to settle into a local energy minimum. Over equilibration, the oxygen molecule immediately binds to the nickel surface, breaking apart into two oxygen atoms in the process. The oxygen atoms at the surface pull nickel atoms from the bulk to create a cluster around them, leading to formation of geometric misfit dislocations inside the bulk nickel, away from the initial oxidized site. This is seen for all cases where oxygen is added, showing the presence of additional defects around the crack tip owing to the presence of oxygen. The shape of the crack changes and volume of empty space inside the crack is decreased. Upon straining the crystal, it is observed that the crack tip emits the first partial dislocation at about the same applied tensile strain as in the case without oxygen. Since the critical nucleation strain does not change in these two cases, the modification in shape and reduction of crack volume thus does not seem to affect the strain at which the first defect is nucleated. The dislocation is emitted from the crack tip end away from the oxide layer and it moves into bulk nickel without encountering the oxide. This hints that a regularly distributed, thick oxide layer all around the entire crack surface may be required for a proper study of crack propagation through the oxide layer. However, a much larger number of O2 molecules are required to successfully cover the entire crack surface with oxide, with a much large reactive region that encompass more than 10,000 reactive atoms. This large number of reactive atoms is beyond the capabilities of the current numerical implementation of ReaxFF.

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Fig. 7.36 Partial dislocation emission from crack tip for the 110 112 crack orientation (only part of the crystal close to the crack tip is shown), under presence of oxygen molecules in the void inclusion (same coordinate system as shown in (Fig. 7.35) [40,41]. Subplots (a),(b) and (c),(d) depict results for two different starting positions of a single O2 molecule in the crack ((a) and (b): oxygen molecule at the tip of the crack, (c) and (d): oxygen molecule at the side of the crack face). Defects produced at the crack tip are circled and the dotted line indicates the approximate region of atoms modeled by ReaxFF. Subplot (a) shows the structure of the crack tip after 10,000 equilibrium molecular dynamics integration steps, and (b) shows the first partial initiating at a strain of 0.049. Subplot (c) depicts the resulting crack structure after 10,000 molecular dynamics integration steps for a different starting position of the O2 molecule (placed at the side of the crack), and (d) shows the partial initiation for this case at a strain of 0.041. The figures show lattice defects produced in the bulk crystal surrounding the crack as a result of oxidation at crack surface, even before strain is applied


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Crack in 111112 Direction The simulation cell is set up identically as described in the previous section, but with 111 being the straining direction and the elliptical crack aligned with major and minor axes along 110 and 111 directions, respectively. As before, we first investigate the reference case with no oxygen present in crack region and all atoms modeled by EAM potentials. The crack tip starts to fracture in a brittle fashion at a strain xx = 0.05. We also observe emission of partial dislocations at the other crack tip end at a strain of xx = 0.051. The placement of a random oxygen molecule inside the crack region leads to rapid oxide formation at the surface of the crack. The system is equilibrated at a temperature of 400 K, using an NVT ensemble at 400 K for 10,000 molecular dynamics steps at a time-step of 0.5 fs, as for the case described above.

Fig. 7.37 Partial dislocation emission from crack tip for the 111 112 case, results for the case without oxygen molecules. Subplot (a) shows brittle crack opening for the no oxygen case at a strain of 0.05, but the results depicted in subplot (b) shows that the crack tip emits dislocations as well at a strain of 0.051 [40, 41]

The equilibration creates a geometric dislocation close to the crack tip as seen in Figs. 7.37 and 7.38. The differently shaded region to the left of the crack represents a grain boundary that has formed during the oxidation process (dashed line indicates grain boundary). The tensile straining of the cell, however, shows a different mechanism of fracture propagation (Fig. 7.37): Upon straining to the critical strain of 0.028 – 0.03, void formation takes place near the crack tip close to the location of the dislocation, and the void enlarges until it joins the main crack at a strain of 0.034. Subsequent propagation is through the emission of partial dislocation at the crack tip at strains of 0.05. A similar behavior is found for a different initial placement of the oxygen molecule near the crack tip, and thus different areas of oxide formation. The

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Fig. 7.38 Partial dislocation emission from crack tip for the 111 112 case, results for the case with oxygen molecules. Subplot (a) shows the structure of the crack tip with O2 after 10,000 equilibrium molecular dynamics integration steps. The differently shaded region to the left of the crack represents a grain boundary (GB) that has formed during the oxidation process. In subplot (b) at a strain of 0.03 a void has initiated close to crack tip. The crack tip starts to open up in subplot (c) at a strain of 0.05, indicating a brittle fracture mode [40, 41]

oxygen attaches at the bottom crack tip end in this case, and upon tensile loading, void formation starts at a strain of 0.018 and the void joins the crack tip at a strain of 0.023. Further opening of the crack is through opening of the crack tip in a brittle manner at a strain of 0.049. On the other hand, placing the O2 molecule away from the crack tip so that the oxygen reacts away from the tip leads to defect formation which is not in vicinity of the tip. This leads to similar crack propagation behavior as for the no oxygen case, with brittle fracture at xx = 0.054. The crack propagation takes place in regions away from the oxide clusters in all cases. These studies show that that oxidation can induce generation of numerous defects including dislocations, even under conditions when no strain is applied. These processes can significantly change the dominating fracture mechanisms. The local chemical reactions lead to crystal defects that can spread far away from the tip of the crack. Figure 7.39 summarizes the critical failure strains for different cases studied. The results indicate that the failure strain is drastically reduced under presence of oxygen molecules, almost by a factor of two. However, the location of where the oxygen attacks matters: If attachment occurs in the vicinity of the crack tip (first two cases on the left), the failure strain is reduced. If attack occurs away from the crack tip, failure strain is not reduced and is very close to the case without any oxygen present.


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Fig. 7.39 Summary of critical failure strains for different cases studied (a schematic of each case is shown below the graph). The results indicate that the failure strain is drastically reduced under presence of oxygen molecules, almost by a factor of two. However, the location of where the oxygen attacks matters: If the attack occurs in the vicinity of the crack tip (such as in the first two cases on the left), the failure strain is reduced (and found to be in the range of 2–3%). If oxygen attack occurs away from the crack tip, the failure strain is not reduced and is then very close to the case without any oxygen present (at approximately 5%) [40, 41]

8 Deformation and Fracture Mechanics of Geometrically Confined Materials

This chapter is dedicated to a discussion of atomistic simulation of size effects in the deformation and fracture mechanics of geometrically confined materials, including nanocrystalline materials, thin films, as well as biological nanomaterials. The discussion is focused on both brittle and ductile nanomaterials. Particular emphasis is given to studies of thin metal films and their plasticity behavior. A direct comparison of simulation results is provided for several cases, illustrating how a direct link can be made between both approaches.

8.1 Introduction The strength of materials depends on the geometry of their microstructure [306]. For instance, it has been established that in most metals, by decreasing the grain size, the strength of the material can be increased. Therefore, finegrained materials are typically stronger than coarse-grained materials. The yield strength increases according to ky σY = σ0 + √ , d

(8.1)

where d is the grain size, σ0 is the yield stress of a single crystal, and ky is a material constant. This leads to the following scaling relation 1 σY ∼ √ . d

(8.2)

This is referred to as the Hall–Petch behavior, and can be derived based on considerations of dislocation pileups in the grains [1,306]. It is a prominent example of a geometric confinement effect. However, materials cannot get infinitely strong as suggested by (8.1). For instance, at elevated temperatures [1, 306, 319, 320] deformation by creep or other grain boundary mediated mechanisms plays an important role in materials with small grain sizes. Fine-grained materials thus tend to fail rapidly

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Fig. 8.1 This graph shows a proposed deformation-mechanism map for nanocrystalline materials obtained from molecular dynamics simulation results. The map shows three distinct regions in which either complete extended dislocations (Region I) or partial dislocations (Region II), or no dislocations at all (Region III) exist during the low-temperature deformation of nanocrystalline FCC metals. c Reprinted with permission from Macmillan Publishers Ltd, Nature Materials [42] 2004

under loading, and cannot get infinitely strong contradicting the prediction of (8.1). Recent research results suggest that even at low temperatures, materials with ultra-fine grain sizes behave quite differently from coarse-grained materials. For instance, in nanostructered materials, the role of grain boundaries becomes increasingly important leading to previously unknown deformation mechanisms. Even though it is generally accepted that grain boundaries provide sources and sinks for dislocations, its role in doing so is still not well understood. Molecular dynamics simulations have provided valuable insight into the deformation mechanisms of nanocrystalline materials as shown in Fig. 8.1. This plot depicts a deformation-mechanism map for nancrystalline FCC metals as reported in [42]. Experimental techniques are capable of studying the details of the structure of grain boundaries, reaching a resolution down to individual atoms. Figure 8.2 depicts high resolution TEM images of grain boundaries in electrodeposited nanocrystalline Ni (a), and nanocrystalline Cu (b). The capability to analyze the structure of materials at nanoscale, combined with the analysis of mechanical properties by tensile tests or nanoindentation enables one to make structure–property links for nanoscale materials features.


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Fig. 8.2 High resolution TEM images of grain boundaries in electrodeposited nanocrystalline Ni (a) and nanocrystalline Cu (b). The samples were produced by gas-phase condensation. The grain boundaries constitutes itself as a very narrow region between crystals of different orientation. Reprinted from Acta Materialia, Vol. 51, K.S. Kumar, H. Van Swygenhoven and S. Suresh, Mechanical behavior of c 2003, with permission nanocrystalline metals and alloys, pp. 5743–5774, copyright from Elsevier

Due to the increased volume fraction of grain boundary regions, the behavior of grain boundaries is critically important in nanocrystalline materials. One of the reasons for the increasing importance of grain boundaries is that classical mechanisms of dislocation generation (e.g., Frank–Read-sources) cannot operate in nanocrystals, because they would not fit within the grain. In addition, defects such as grain boundaries interact in complicated ways with other defects like dislocations. An important consequence of this is that despite the prediction by (8.1), the strength of nanomaterials does not increase continuously with decreasing grain size. Below a critical grain size, experiments have shown that the strength decreases again [321]. This is referred to as the inverse Hall–Petch effect [69, 306, 321]. In this regime, it was proposed that the yield stress scales as √ σY ∼ d, (8.3) although physical foundation of such material behavior is yet to be explored [303]. Such behavior indicates that there may exist a maximum of strength for a certain grain size, described as “the strongest size” by Sidney Yip in 1998 [322]. One of the major objectives of recent research is to quantify this

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Fig. 8.3 Normalized critical stress as a function of grain size, d. The plot illustrates the variation of the flow stress (normalized by the value of the critical flow stress at the maximum strength at the transition grain size). The transition from the hardening regime to the softening regime is associated with an increased role of grain boundary mechanisms (the dashed line was added to the plot to guide the eye). Reprinted from Acta Materialia, Vol. 51, K.S. Kumar, H. Van Swygenhoven and S. Suresh, Mechanical behavior of nanocrystalline metals and alloys, pp. 5743–5774, c 2003, with permission from Elsevier copyright

critical condition and understand the underlying principles, for a variety of materials and a variety of microstructures. This concept has been confirmed in several studies and in different geometries. Figure 8.3 depicts an analysis of the normalized flow stress obtained from various sources, including bubble raft experiments, experiments with electrodeposited nanocrystalline metals (see [323] for references), as well as molecular dynamics simulations [324]. Figure 8.4 depicts the results of a systematic investigation of size effects in nanocrystalline copper, carried out using molecular dynamics simulations [43]. These plots illustrate the existence of a peak at a critical, strongest grain size. A similar behavior is found in other geometries. Figure 8.5 depicts results of molecular dynamics simulations that illustrate the size dependence of a bioinspired metallic nanocompsite, showing the existence of “a strongest size” [44]. The increase in strength scales very well according to the Hall–Petch relationship given in (8.1), with parameters kY = 0.11 MPa m1/2 . and σ0 = 1.2 GPa. The physical reason for the departure from the Hall–Petch behavior is the onset of interfacial sliding mechanism (at the interfaces between the Ni and Al particles) and the associated breakdown of dislocation-based plasticity. Applying classical molecular dynamics to investigations of nanostructured materials is particularly attractive because of the fact that the lengthscale of several tens of nanometers fall well within the range accessible to molecular dynamics simulation. Indeed, classical molecular dynamics methods have proven to be a very powerful tool for these materials. Studying deformation


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Fig. 8.4 Illustration of the maximum in the strength of nanocrystalline copper, as shown by molecular dynamics simulation of up to 100 million atoms [43]. Panel (a) shows the stress–strain curves for ten simulations with varying grain sizes. Panel (b) depicts the flow stress, defined as the average stress in the strain interval from 7 to 10% deformation. The error bars indicate the fluctuations in this strain interval (1 standard deviation). A maximum in the flow stress is seen for grain sizes of 10–15 nm, caused by a shift from grain boundary mediated to dislocation mediated plasticity. Reprinted from Science, Vol. 301, J. Schiotz and K.W. Jacobsen, A Maximum in c 2003, with permission from the Strength of Nanocrystalline Copper, copyright AAAS

Fig. 8.5 Size dependence of a bioinspired metallic nanocomposite (schematic of structure shown in left part) [44, 45], illustrating the existence of “a strongest size” [322]. The increase in strength for building block dimensions larger than approximately 50 nm scales rather well according to the Hall–Petch relationship given in (8.1)

of nanocrystalline materials with molecular dynamics still requires significant computer power. Atomistic studies of nanostructured materials and associated size effects were reported by several groups (see, for example [44, 45, 68–70, 153, 154, 305, 325]). In most of the molecular dynamics studies,

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polycrystalline samples at nanoscale were created (for example by a Voronoi construction), annealed, relaxed, and then exposed to tensile loading. In the following paragraphs, we summarize the main results in this field obtained for different geometries, materials, and simulation conditions (e.g., variation of temperature and loading conditions). Coble creep is a well-known mechanism for creep of polycrystalline materials [320]. The characteristic time for exponential stress relaxation scales as τ ∼ d3 ,

(8.4)

where d is the grain diameter. Appreciating that the grain size in nanostructured materials is on the order of tens of nanometers (in contrast to micrometer grain sizes in coarse-grained materials), this scaling suggests that at very small grain size, diffusive mechanisms at grain boundaries may play a dominating role in nanomaterials even at moderate temperatures. In recent publications, this was investigated using molecular dynamics simulations at elevated temperature [68, 303, 326]. The temperature was increased to render the process of diffusion accessible to the molecular dynamics timescale. The authors [68, 303] used a fully three-dimensional model of palladium with 16 grains having a truncated-octahedral shape arranged on a three-dimensionally periodic BCC lattice. Grain sizes range from d ≈ 3.8 to 15 nm, and the grain boundary misorientations are chosen such that only high-energy grain boundaries are present in the model. A multibody EAM potential was used to model the atomic interactions. They find that grain boundary processes indeed play a dominating role and conclude that grain boundary diffusion fully accounts for plasticity. Under lower strain rates than in molecular dynamics simulation, this result could be valid even at room temperature, once microcracking and dislocation nucleation are suppressed. Dislocation mechanisms are shut down due to the small grain size and moderate loading of the sample! The authors derive a generalized Coble-creep equation and show that the grain-size dependence of the strain rate decreases from the 1/d3 scaling law appropriate for large grain size toward a 1/d2 scaling law as expected in the limit of a very small grain size (critical grain size d ≈ 7 nm in palladium). The grain size scaling observed in molecular dynamics simulations indeed agrees with this prediction [68]. It is also concluded that grain boundary diffusion creep must be accommodated by grain boundary sliding (also referred to as Lifshitz sliding) to avoid microcracking. Experimental reports of the inverse Hall–Petch behavior inspired numerous simulation studies by Schiotz and coworkers [324, 327] and Swygenhoven and coworkers [70, 152, 328–330]. In contrast to the above research of Coble creep, these simulations are all performed at low temperatures making it basically impossible to observe any Coble creep at the molecular dynamics timescale. In these studies, very large stresses in the range of 1–3 GPa were applied. Schiotz et al. [324, 327] determined the yield stress σY as a function of the grain size d. In contrast, the group around van Swygenhoven focused attention on the strain rate. Both groups concluded that the deformation


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mechanism is controlled by grain boundary processes and that the material softens with decreasing grain size (inverse Hall–Petch effect). Nucleation of numerous partial dislocations was observed in their simulations. Schiotz and coworkers [324, 327] considered nanocrystalline copper with grain sizes from 3.3 to 6.6 nm and showed that grain boundary sliding occurs together with grain rotation. When the grain size was larger than about 5 nm, nucleation of partial dislocation was identified under very large stresses. Similar observations were also reported by van Swygenhoven and coworkers [329, 330] in simulations of nickel at average grain sizes of about 5 nm at a temperature of 70 K. The results were confirmed with simulations at higher temperature and for larger grain sizes [70]. The authors suggested that grain boundary sliding occurs through atom shuffling and stress-induced athermal grain boundary diffusion. In a later paper by Wolf et al. [303], the missing issue of the rate-limiting deformation mechanism was addressed. The authors suggest that the accommodation mechanism in the simulations described by van Swygenhoven’s and Schiotz’s group is the same as that in Coble creep, with the difference that there is no activation energy for this athermal process. Therefore, the Coble creep equation should apply. They verified this proposal by an analysis of the data in [330], proving that the data points for the three smallest grain sizes fall on a straight line with a slope 2.73 in a log–log plot of the /σ ˙ vs. the grain size d (Fig. 4 in [303]). It was concluded that the athermal mode of Coble creep is due to the fact that the simulations are carried out in a regime where molecular dynamics can not be used. The fact that Coble creep still dominates may indicate that grain boundary diffusion is a very robust mechanism for stress relaxation [303]. Recent work by Hasnaoui et al. [331] discussed the influence of the grain boundary misorientation on the ductility of nanocrystalline materials. It was shown that at specific low-energy grain boundaries (e.g., twins), several neighboring grains can be effectively immobilized, creating structures that offer significant resistance to plastic deformation. The authors finally discuss the possibility to design more ductile nanostructured materials that feature less low-energy grain boundaries and therefore lead to a more homogeneous deformation. Other studies were carried out on dislocation processes of nanocrystalline aluminum [304]. The authors demonstrate that deformation twinning may play a very important role in the deformation of nanocrystalline aluminum. The simulations demonstrate that molecular dynamics simulations have advanced to predict deformation mechanisms of materials at a level of detail not yet accessible to experimental techniques. Observation of twinning is quite surprising because of the small grain size and the high stacking fault energy of aluminum [112]. The predictions by these simulations have recently been verified experimentally [332]. Experimentalists conclude that twinning in aluminum only occurs in nanocrystalline materials, while it is not observed in coarse-grained aluminum. The findings support the hypothesis that in the nanograin-regime, a transition occurs from normal slip of complete

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dislocations to activities dominated by partial dislocations. The critical stress for nucleation of dislocations in nanocrystalline aluminum was estimated to be 2.3 GPa. Research has focused not only on the polycrystalline nanoscale materials, but also on the mechanics of single crystals with nanometer extension, so-called nanowires [333–338]. Such structures may become increasingly important for example as interconnects in complex integrated circuits or bioelectrical devices. Studies of defect-free single nanocrystals under tension (the crystals had dimensions of several nanometers) have been carried out by Komanduri et al. [106]. Due to the small structural size of the nanocrystals, the dislocations glide quickly through the specimen leaving surface steps, and repeated glide admits plastic deformation. Similar research of mechanical properties of copper were carried out by Heino and coworkers [339].

Fig. 8.6 Study of size-scale effects in inorganic materials by using a focused ion beam (FIB) microscope machining technique, combined with nanoidentation experiments. Ultra-small pillars of different sizes were created using FIB and subsequently deformed plastically under compression from the top surface. Subplot (a) shows a SEM image of the microsample after testing. The dislocation slip lines are clearly visible at the surface. Subplot (b) shows the dependence of the yield strength on the inverse of the square root of the sample diameter for Ni3 Al-Ta. The linear fit to the data predicts a transition from bulk to size limited behavior at approximately 42 µm. The parameter σys denotes the stress for breakaway flow. Reprinted from Science, M.D. Uchic, D.M. Dimiduk, J.N. Florando and W.D. Nix, Vol. 305(9), pp. c 987–989, Sample Dimensions Influence Strength and Crystal Plasticity, copyright 2004, with permission from AAAS

The combination of new micromachining techniques and studies of sizedependent plasticity is a particularly interesting approach. In a recent study, researchers developed a test methodology that allows the exploration of size-scale effects in inorganic materials by using a focused ion beam (FIB)


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microscope for sample preparation. This enables one to create small-scale structures in a controlled way, whose mechanical properties can be tested using a modified nanoindentation technique. In a study reported in [340], the researchers created pillars made out of nickel, consisting of different sizes, and then measured the associated size-dependent plasticity. Figure 8.6 shows results of this experimental study. The measurements of plastic yielding for single crystals of micrometer-sized dimensions for three different types of metals showed that the characteristic size of the sample can limit the lengthscales available for plastic deformation mechanisms. The results reported in [340] show significant size effects even at sample sizes beyond a few micrometers. At micrometer lengthscales, it is crucial to consider both the overall sample geometry and the internal structure to determine the strength of the sample.

8.2 Thin Metal Films and Nanocrystalline Metals The study of the mechanical properties of materials at nanoscales and submicrometer scales is motivated by increasing need for such materials due to miniaturization of engineering and electronic components, development of nanostructured materials, thin film technology, and surface science. When the material volume is lowered, characteristic dimensions are reduced that control the material properties and this often results in deviation from the behavior of bulk materials. Small-scale materials are often referred to as materials in small dimensions, and they are defined as materials where at least one dimension is reduced. For instance, thin films bond to substrates are a relevant example of materials in small dimensions since the film thickness hf is small compared to the planar extension of the film and the thickness of the substrate. Thin films bond to substrates have become an increasingly active area of research in the last decades. This can partly be attributed to the fact that these materials are becoming critically important in today’s technologies, whereas changes in material behavior due to the effects of surfaces, interfaces, and constraints are not completely understood. The focus of this part is on mechanical properties of ultra-thin submicron copper films on substrates. We will show that in such materials, important effects of the film surface and grain boundaries are observed and that the constraint by the film–substrate interface governs the mechanical behavior [46, 51, 96, 341–344]. Polycrystalline thin copper films as shown schematically in Fig. 8.7 are frequently deposited on substrate materials to build complex microelectronic devices. In many applications and during the manufacturing process, thin films are subjected to stresses arising from thermal mismatch between the film material and the substrate. This can have a significant effect on the production yield as well as on the performance and reliability of devices in service. In past years, an ever increasing trend of miniaturization in semiconductor and integrated circuit technologies has been observed, stimulating a growing

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Fig. 8.7 Polycrystalline thin film geometry. A thin polycrystalline copper film is bond to a substrate (e.g., silicon). The grain boundaries are typically predominantly orthogonal to the film surface

interest to investigate the deformation behavior of such ultra-thin films with film thicknesses well below 1 µm. Different inelastic deformation mechanisms are known to operate relaxing the internal and external stresses in a thin film. Experiment shows that for films of thicknesses between approximately 2 and 0.5 µm, the flow stress increases in inverse proportion to the film thickness (see for example [341–343]), that is, 1 σY ∼ . (8.5) hf Figure 8.8 depicts experimental results of the thin film strength dependence, here for a copper thin film [51]. This has been attributed to dislocation channelling through the film [71, 72, 345], where a moving threading dislocation leaves behind an interfacial segment. The relative energetic effort to generate these interfacial dislocations increases with decreasing film thickness, which explains the higher strength of thinner films. This model, however, could not completely explain the high strength of thin films found in experiments [343]. More recent theoretical and experimental work [51, 96, 346–348] indicates that the strength of thin metal


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Fig. 8.8 Experimental results of the thin film strength dependence at room temperature, here for copper thin films plotted as the inverse of the film thickness. With decreasing film thickness, flow stress initially rises, but then exhibits a plateau at approximately 630 MPa for films 400 nm and thinner. Each data point is an average of flow stresses from several thermal cycles, with a scatter of less than 5% in each case. Reprinted from [51] Acta Materialia, Vol. 51, T.J. Balk, G. Dehm and E. Arzt, Parallel glide: Unexpected dislocation motion parallel to the substrate in ultrathin c 2003, with permission from Elsevier copper films, copyright pp. 4471–4485,

films often results from a lack of active dislocation sources rather than from the energetic effort associated with dislocation motion. Discrete dislocation dynamics simulations have directly confirmed the increase of the strength of thin films [349]. Figure 8.9 depicts the result of this analysis. It was found that for most cases, the thin film strength increase does not obey a simple scaling of power-law type, depending on the particular grain size. The strengthening exponent ranges from 0.5 (Hall–Petch behavior) to values greater than 1 (Freund–Nix behavior of thin films). Figure 8.10 shows the distribution of dislocations for different film thicknesses. In copper films, the regime where plastic relaxation is limited by dislocation nucleation and carried by glide of threading dislocations reaches down to film thicknesses of about hf ≈ 400 nm [51]. For yet thinner films experiments reveal a film-thickness-independent flow stress [51, 346, 350], as also shown in

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Fig. 8.9 Discrete dislocation dynamics simulation analysis of the strength dependence of a thin metal film. Subplot (a) shows the discrete dislocation model of the polycrystalline film on a semi-infinite elastic substrate. Subplot (b) shows the average film strength at T = 400 K vs. the film thickness hf . Reprinted from Thin Solid Films, Vol. 479(1–2), L. Nicola, E. Van der Giessen and A. Needleman, Size effects c in polycrystalline thin films analyzed by discrete dislocation plasticity, copyright 2005, with permission from Elsevier

Fig. 8.8. This transition of the deformation mechanism will be discussed in more detail in the forthcoming sections. In-situ transmission electron microscopy observations of the deformation of such ultra-thin films reveal dislocation motion parallel to the film–substrate interface [51, 346]. This glide mechanism is unexpected, because in the global biaxial stress field there is no resolved shear stress on parallel glide planes. This indicates that there must be a mechanism involving long-range internal stresses that decay only slowly on the lengthscale of the film thickness. For sufficiently thin films these internal stresses have a pronounced effect on the mechanical behavior. It has been proposed that constrained diffusional creep may be the origin of this novel deformation mechanism [46]. This deformation mechanism by parallel glide dislocations is not well understood as of today. In this book, we therefore propose atomistic and continuum studies to investigate the behavior of such thin films below 400 nm. The final objective is to draw


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Fig. 8.10 Distribution of dislocations, result from a discrete dislocation dynamics simulation analysis. The plot depicts the dislocation distribution and the stress field σxx at T = 400 K for films with a grain size of 1 µm and various film thicknesses. The plus and minus symbols denote positive and negative dislocations according to the sign convention defined in Fig. 8.9a (all dimensions are in µm). Reprinted from Thin Solid Films, Vol. 479(1–2), L. Nicola, E. Van der Giessen, A. Needleman, Size effects c in polycrystalline thin films analyzed by discrete dislocation plasticity, copyright 2005, with permission from Elsevier

a deformation map that summarizes all relevant deformation mechanisms in submicron thin films. 8.2.1 Constrained Diffusional Creep in Ultra-Thin Metal Films Similarly as in bulk nanostructured materials, it has also been hypothesized that grain boundary processes dominate the mechanical properties in ultrathin films [46, 51, 346, 350, 351]. In recent theoretical studies of diffusional creep in polycrystalline thin films deposited on substrates, a new class of defects called the grain boundary diffusion wedges was predicted [46]. These diffusion wedges are formed by stress driven mass transport between the free surface of the film and the grain boundaries during the process of substrate-constrained grain boundary diffusion. The diffusion wedges feature a crack-like opening displacement, and due to the strong bonding between film and substrate, a stress concentration at the root of the grain boundary builds up. This leads to a singular, crack-like stress field in the film as the grain boundary tractions are relaxed. Because the material inserted into the grain boundary by diffusion takes the shape of a wedge, this new class of defects has been referred to as a diffusion wedge

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[46,48]. An important implication of the crack-like stress field at the diffusion wedges is that dislocations with Burgers vector parallel to the interface may be nucleated at the root of the grain boundary, at the location with highest shear stress. This is a new dislocation mechanism in thin films that contrasts to the well-known Mathews–Freund–Nix mechanism of threading dislocation propagation [71, 72, 345]. For films without relaxation due to diffusional creep, the maximum shear stress is found on inclined planes, leading to strong driving forces for Mathews– Freund–Nix threading dislocations. If diffusional creep is active, the largest shear stresses occur on planes parallel to the film surface. The two modes of deformation are illustrated in Fig. 8.11.

Fig. 8.11 Change of maximum shear stress due to formation of the diffusion wedge. In the case of no traction relaxation along the grain boundary, the largest shear stress occur on inclined glide planes relative to the free surface. When tractions are relaxed, the largest shear stresses occur on glide planes parallel to the film surface

Indeed, results of TEM experiments show that, while threading dislocations dominate in passivated metal films, parallel glide dislocations begin to dominate in unpassivated copper films with thickness below 400 nm. Figure 8.12 depicts TEM micrographs of an unpassivated copper film showing parallel glide dislocations and a passivated copper film showing threading dislocations [51]. The two deformation modes can be distinguished clearly from the shape of the dislocation networks. Parallel glide dislocations are curved lines, since the glide in a plane parallel to the film surface. Threading dislocations are visible as straight lines, as they leave surface steps in these orientations. The discovery of parallel glide dislocations [350] provided experimental support for the constrained diffusional creep model [46]. In turn, constrained diffusional creep provided the basis for interpretation of certain experimental results, especially in regard to the mechanisms for the creation and emission of parallel glide dislocations. Figure 8.13 summarizes this model including the occurrence of parallel glide dislocations in three stages: In stage one, material is transported from the surface into the grain boundary. In stage two, mass transport leads to the formation of a diffusion wedge, as more and more material flows into and accumulates in the grain boundary. The continuum model predicts that the


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Fig. 8.12 TEM micrographs of an unpassivated copper film showing parallel glide dislocations (subplot (a), Cu film with film thickness hf = 200 nm), and a passivated copper film showing threading dislocations (subplot (b), self-passivated Cu-1%Al film, film thickness hf = 200 nm). Reprinted from [51] Acta Materialia, Vol. 51, T.J. Balk, G. Dehm and E. Arzt, Parallel glide: unexpected dislocation motion parallel c 2003, with to the substrate in ultrathin copper films, pp. 4471–4485, copyright permission from Elsevier

Fig. 8.13 Mechanism of constrained diffusional creep in thin films as proposed by Gao et al. [46]

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traction along the grain boundary diffusion wedge becomes fully relaxed and crack-like on the scale of a characteristic time τ . The timescale at that diffusion takes place is usually much larger than that of dislocation glide. However, in the nanoscaled structures investigated here, the model can explain the observed deformation rates even at room temperature, because the timescale of diffusional creep is inversely proportional to the cube of the characteristic structural length (similarly to Coble creep, see (8.4)). In the continuum model [46], diffusion is modeled as dislocation climb in the grain boundary. The solution for a single edge dislocation near a surface is used as the Green’s function to construct a solution with infinitesimal Volterra edge dislocations [38, 352, 353]. The basis for the continuum modeling is the solution for the normal traction σxx along the grain boundary due to insertion of a single dislocation (material layer of thickness b) along (0, ζ) (corresponding to a climb edge dislocation). The coordinate system is given in Fig. 8.17.

Fig. 8.14 Development of grain boundary opening ux normalized by a Burgers vector over time, for the case of a copper film on a rigid substrate. The loading σ0 is chosen such that the opening displacement at the film surface (ζ = 0) at t → ∞ is one Burgers vector [46]

The dislocations “stored” in the grain boundary are a measure of additional material in the grain boundary. With respect to the lattice distortion around the diffusion wedge, the dislocations in the grain boundary exemplify a type of geometrically necessary dislocations [289] that cause nonuniform plastic deformation in the thin film. The eigenvalues measure the rate of decay of each eigenmode. The results show that the higher eigenmodes decay much


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faster than the first eigenmode, so that the diffusion process is dominated by the first eigenmode. The continuum mechanics model was further advanced to capture the effect of surface diffusion [48]. No difference in the qualitative behavior was found, and stress decay in the film is still exponential with a characteristic time proportional to the cube of the film thickness. Further details could be found in [48]. In all cases considered in the literature [46, 48], with the proper definition of the characteristic time τ , stress decay could be described by an exponential law of the form t (8.6) σgb (t) = σ0 exp −λ0 τ with a geometry-dependent constant λ0 = 8.10 + 30.65hf /d.

(8.7)

Note that d characterizes the grain size and σ0 stands for the laterally applied stress as discussed above. Equation (8.7) is an empirical formula and is valid for 0.2 ≤ hf /d ≤ 10.

Fig. 8.15 Traction along the grain boundary for various instants in time [46]

Figures 8.14–8.16 show several numerical examples. Figure 8.14 shows the opening displacement along the grain boundary for several instants in time. Figure 8.15 shows the traction along the grain boundary for various instants in time. These examples show that in the long time limit t → ∞, the solution approaches the displacement of a crack.

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Fig. 8.16 Stress intensity factor normalized by the corresponding value of a crack over the reduced time t∗ = t/τ for identical elastic properties of substrate and film material (isotropic case), rigid substrate (copper film and rigid substrate), and soft substrate (aluminum film and epoxy substrate) [46–48]

Figure 8.16 shows the stress intensity factor normalized by the corresponding value of a crack over the reduced time t∗ = t/τ for identical elastic properties of substrate and film material (isotropic case), rigid substrate (copper film and rigid substrate), and soft substrate (aluminum film and epoxy substrate). The results indicate that in a film on a stiff substrate, the stress intensity factor of a crack is reached faster compared to the homogeneous case. Similarly, the stress intensity factor of a crack is reached slower in the case of a film on a soft substrate compared to the homogeneous case. Table 8.1 summarizes the material parameters used for the calculation.

Cu/rigid Al/epoxy isotropic

νfilm 0.32 0.3 –

νsubs – 0.35 –

µfilm /µsubs 0 23.08 1

Table 8.1 Material parameters for calculation of stress intensity factor over the reduced time

8.2.2 Single Edge Dislocations in Nanoscale Thin Films Mass transport from the surface into the grain boundary toward the substrate is modeled as climb of edge dislocations [46]. At nanoscale, the fact that


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Fig. 8.17 Geometry and coordinate system of the continuum mechanics model of constrained diffusional creep

dislocation climb in the grain boundary is a discrete process becomes more evident. Grain boundary diffusion requires insertion of climb dislocations into the grain boundary one by one.

Fig. 8.18 Image stress on a single edge dislocation in nanoscale thin film constrained by a rigid substrate

To investigate the effect, we consider a single edge dislocation climbing in a grain boundary in an elastic film of thickness hf on a rigid substrate. The elastic solution of edge dislocations in such a film can be obtained using the methods described in [47,352]. The geometry, as well as the coordinate system is shown in Fig. 8.17. In such a geometry, a dislocation placed inside the film is subject to image forces due to the surface and the film–substrate interface.

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The image stress on the dislocation for different film thicknesses is shown in Fig. 8.18. The thinner the films, the stronger gets the effect of the geometric confinement. Between the film surface and the film–substrate interface, the image force is found to attain a minimum value at ζEQ ≈ 0.4hf . Therefore, from the energetic point of view, a minimum critical stress is required to allow even a single climb edge dislocation to exist in the grain boundary. The thicker the film, the smaller the critical stress. This analysis suggests that consideration of single, discrete dislocations can become very important for the nanoscale thin films. The requirement that an edge dislocation in the film is in a stable configuration could be regarded as a necessary condition for constrained grain boundary diffusion to initiate and proceed. If more than one dislocations are stored in the grain boundary, even stronger image forces are expected since different dislocations repel each other.

Fig. 8.19 Critical stress as a function of film thickness for stability of one, two, and three dislocations in a thin film. The critical stress for the stability of one dislocation (continuous line) is taken from the analysis shown in Fig. 8.18. The curves for more dislocations (dashed lines) in the grain boundary are estimates

Another consequence of the geometric confinement of dislocations is that discrete dislocation effects can lead to quantization of stresses in nanostructured devices. Figure 8.19 shows the critical stress as a function of film thickness for stability of one, two, and three dislocations in a thin film. The critical stress for the stability of one dislocation is taken from the analysis shown in Fig. 8.18. The curves for two and more dislocations in the grain boundary are estimates. For a given film thickness, for at least one dislocation to be stable inside the film, the critical stress needs to exceed a critical


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value. For two dislocations to be stable inside the film, the critical stress is even higher. Consequently, the stress relief due to insertion of dislocations will also be quantized. Most importantly, as the film thickness increases the critical stresses for stable dislocations in the film get smaller and smaller, eventually approaching the limit when the role of single dislocations can be neglected and the quantization is negligible. Similar observations have been made in discrete dislocation modeling of constrained diffusional creep in thin films [50]. 8.2.3 Rice–Thompson Model for Nucleation of Parallel Glide Dislocations To characterize the nucleation condition of parallel glide dislocations, a criterion based on a critical stress intensity factor K PG is used. The motivation is that the concept of stress intensity factor is commonly used in the mechanics of materials community and provides a possible link between atomistic and mesoscopic simulation methods. The critical value for nucleation of parallel glide dislocations from a diffusion wedge could be thought of as a new material parameter. The stress intensity factor is defined as K = lim {[2π(ζ − hf )]s σxx (0, z)} ,

(8.8)

ζ→hf

where s refers to the stress singularity exponent determined by [354] cos(sπ) − 2

α−β α − β2 (1 − s)2 + = 0. 1−β 1 − β2

(8.9)

It is assumed that the diffusion wedge is located close to a rigid substrate and the corresponding Dundurs parameters for this case are α = −1 and β = −0.2647. The Dundurs parameter measures the elastic mismatch of film and substrate material [47]. The singularity exponent is found to be s ≈ 0.31 for the material combination considered in the simulations (comparing to s = 0.5 in the case of a homogeneous material). Close to the bimaterial interface, the stress intensity factor is ∂ux (ζ) 2 s K = A × lim (1 − (ζ/hf ) ) (πhf )s , (8.10) ζ→hf ∂ζ where A=

E (1 − α) 2 1 − ν 4 sin(πs)

3 − 2s 1 − 2s − 1+β 1−β

.

(8.11)

The stress intensity factor provides an important link between the atomistic results and continuum mechanics. To calculate the stress intensity factor from atomistic data, the atomic displacements of the lattice close to the diffusion wedge are calculated and the stress intensity factor is then determined using (8.10).

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The Peach–Koehler force on a dislocation can be written as Fd = (σ · b) × dl, where dl is a dislocation element and σ is the local stress [38]. The variable bx stands for the√magnitude of the Burgers vector in the x = [110] direction and bx ≈ 3.615/ 2 × 10−10 m for copper at 0 K. A dislocation is assumed to be in an equilibrium position when Fd = 0. Following the approach of the Rice– Thomson model [66], we consider the force balance on a probing dislocation in the vicinity of a dislocation source to define the nucleation criterion. The probing dislocation is usually subject to an image force attracting it toward the source and a force due to applied stress driving it away from the source. The image force dominates at small distances and the driving force due to applied stress dominates at large distances. There is thus a critical distance between the dislocation and the source at which the dislocation is at unstable equilibrium. Spontaneous nucleation of a dislocation can be assumed to occur when the unstable equilibrium position is within one Burgers vector of the source. Nucleation Mechanism of Parallel Glide Dislocations

Fig. 8.20 Rice–Thomson model for nucleation of parallel glide dislocations. Subplot (a) shows the force balance in case of a crack and subplot (b) depicts the force balance in case of a diffusion wedge

Nucleation of parallel glide dislocations from a crack in comparison to that from a diffusion wedge is shown in Fig. 8.20. The crack is treated as by [66], and forces involved are Fc due to the crack tip stress field, Fimage because


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of the free surface (image dislocation), and Fstep due to creation of a surface step (in the following, we assume Fstep Fimage ). Close to a diffusion wedge, Fstep = 0 since no surface step is involved and a dipole must be created in order to nucleate a parallel glide dislocation from the wedge. This leads to a dipole interaction force Fdipole . The dipole consists of a pair of dislocations of opposite signs, one pinned at the source and the other trying to emerge and escape from the source. The pinned end of the dipole has the opposite sign to the climb dislocations in the diffusion wedge and can be annihilated via further climb within the grain boundary. The annihilation breaks the dipole free and eliminates the dipole interaction force so that the emergent end of the dipole moves away to complete the nucleation process. Therefore, it seems that there could be two possible scenarios for dislocation nucleation at a diffusion wedge. In the first scenario, the nucleation condition is controlled by a critical stress required to overcome the dipole interaction force. In the second scenario, the nucleation criterion is controlled by the kinetics of climb annihilation within the grain boundary which breaks the dipole interaction by removing its pinned end and setting the other end free. No matter which scenario controls the nucleation process, the climb annihilation of edge dislocations in the grain boundary must be completed and will be the rate limiting process. The force balance on the dislocation is illustrated in Fig. 8.20b for two different, subsequent instants in time. Critical Stress Intensity Factor for Dislocation Nucleation in Homogeneous Material

Fig. 8.21 Dislocation model for critical stress intensity factor for nucleation of parallel glide dislocations

It is now assumed that dislocation nucleation at a diffusion wedge is stress controlled (rather than kinetics controlled) and adopt the first scenario of dislocation nucleation as described above. This assumption will later be verified by molecular dynamics simulation results. With this assumption, it is possible to define a nucleation criterion in terms of a critical stress intensity factor for both cracks and diffusion wedges. We illustrate the critical condition for

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dislocation nucleation in Fig. 8.21. A force balance on a dislocation near a crack tip leads to the critical stress intensity factor for dislocation nucleation from a crack PG = Kcr

E(2πbx )s . 8π(1 − ν 2 )

(8.12)

In comparison, a balance of critical stress required to break the dipole interaction in front of a diffusion wedge yields a similar nucleation criterion PG Kdw =

E(2πbx )s . 4π(1 − ν 2 )

(8.13)

For copper with E = 150 GPa, s = 0.31, and ν = 0.33 the predicted values PG PG are Kcr ≈ 12.5 MPa ms and Kdw ≈ 25 MPa ms , and we note a factor of 2 PG PG = 2, for dislocation nucleation at a difference in critical K-values, Kdw /Kcr diffusion wedge and at a crack tip. 8.2.4 Discussion and Summary The study of single edge dislocation showed that in film thicknesses of several nanometers, image stresses on climb edge dislocations can be as large as 1 GPa. This further supports the hypothesis that single dislocations become important in small dimensions and that the discrete viewpoint of dislocation climb needs to be adapted. It also supports the view of a critical stress for diffusion initiation described above. A criterion in the spirit of the Rice–Thomson model was proposed to describe the conditions under which parallel glide dislocations are nucleated from diffusion wedges and cracks. The most important prediction of this model is that the critical stress intensity factor for parallel glide dislocation nucleation from a diffusion wedge is twice as large compared to the case of a crack.

8.3 Atomistic Modeling of Constrained Grain Boundary Diffusion in a Bicrystal Model The continuum model [46, 48] of constrained diffusional creep has been very successful in explaining the origin of the internal stresses in thin films. However, a continuum viewpoint alone can neither yield a description of the nucleation process of parallel glide dislocations from the diffusion wedge, nor incorporate the parallel glide mechanism into the prediction of residual stresses in a film. Under the guidance of the continuum model [46–48], atomistic simulations are an appropriate tool to provide a detailed description of how parallel glide dislocations are nucleated near a diffusion wedge. In this chapter, we review large-scale atomistic simulations to study plastic deformation in submicron thin films on substrates. The simulations reveal that


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Fig. 8.22 Disordered intergranular layer at high-energy grain boundary in copper at elevated temperature (85% of melting temperature)

stresses in the film are relaxed by mass diffusion from the surface into the grain boundary. This leads to formation of a novel material defect referred to as the diffusion wedge. A crack-like stress field is found to develop around the diffusion wedge as the traction along the grain boundary is relaxed and the adhesion between the film and the substrate prohibits strain relaxation close to the interface. The diffusion wedge causes nucleation of dislocations on slip planes parallel to the plane of the film. It is found that nucleation of such parallel glide dislocations from a diffusion wedge can be described by a critical stress intensity factor similar to the case of a crack. 8.3.1 Introduction and Modeling Procedure Atomistic modeling of thin film mechanics becomes feasible with the advent of massively parallel computers on timescales and lengthscales comparable with those usually attained in experimental investigations. Due to the time limitation of the classical molecular dynamics method (time intervals are typically FCC lattice, rotating one-half counterclockwise and cutting out two rectangular pieces. This procedure leads to an asymmetric < 111 > tilt grain boundary with approximately α ≈ 7◦ mismatch. Note that the choice of this geometry and the tilt angle is motivated by recent experimental results [51]. The simulation geometry is depicted in Fig. 8.23. In some simulations, we further rotate the tilted grain counterclockwise around its [112] axis, creating a higher energy grain boundary. The structure is periodic in the y-direction. We impose a homogeneous strain throughout the sample to account for thermal expansion [107]. The boundary and substrate atoms are chosen such that atoms inside the film do not sense the existence of the surface. The atoms of the boundary and the substrate are held pinned at a prescribed location by introducing an additional term in the potential energy of the form 1 φp = k0 rx2 d −xc , (8.14) 2 where xd is the prescribed location of an atom and xc is its current position. The expression rxd −xc =| xd − xc | stands for the radius of separation of the desired and present location, while k0 is a harmonic spring constant which is chosen k0 = 20 in reduced atomic units. The locations of atoms in the substrate are prescribed only in the x- and z-direction, and the y-direction is left unconstrained. The atoms in the boundary are only constrained in the x-direction, so that the boundary is allowed to relax in the y- and z-direction. In terms of a continuum mechanics interpretation of the simulation cell, this resembles a plane strain case with a thin compliant (copper) film on a rigid substrate, with no sliding and no diffusion at the film–substrate interface. The choice of the rigid substrate is partly motivated by the finding that the stress intensity factor of a crack is reached faster in the case of a rigid substrate than in the isotropic case or when the film is attached to a compliant substrate (see Fig. 8.16 and associated discussion). After the “raw” sample is created, a global energy minimization scheme is applied to relax the structure. Subsequently, the sample is heated up to an elevated temperature and annealed for a longer time so that the grain boundary structure relaxes and takes its equilibrium configuration. The virial stresses [239] is relaxed to zero in this initial configuration before loading is applied. To make the diffusive processes accessible to the molecular dynamics timescale, the simulations are carried out at elevated temperatures. The simulations to investigate dislocation nucleation in conjunction with diffusional

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creep are performed at a homologous temperature of Th ≈ 0.8 and Th ≈ 0.9. At elevated temperature, the grain boundary exhibits a highly confined glassy intergranular phase of less than 1-nm width, in accordance with [356]. After annealing we proceed with applying a lateral strain. The prescribed positions xd are calculated according to a homogeneous strain throughout the simulation sample. We use a time step of ∆t ≈ 3 × 10−15 s for integration. The strain rate is on the order of 107 s−1 corresponding to approximately 1% strain per nanosecond. The strain rate is adjusted during the simulation such that the stress in the film remains low to avoid nucleation of dislocations on inclined slip planes, similar to the procedure adopted by [68]. The only deformation mechanism allowed is diffusional creep in the grain boundary. Whenever activation of a different mechanism such as threading dislocations is observed, the simulation is restarted at a lower stress and the strain rate is lowered. This procedure has proven to allow more time for diffusive processes and effectively shut down competing mechanisms. In the simulation with Th ≈ 0.8, the grain boundary width remains less than 1 nm and increases slightly at higher temperatures when the sample is loaded. In the simulation with Th ≈ 0.9 the grain boundary width is further increased and is found to be around 1–2 nm. The systems contain more than 1,000,000 particles, which is a significant number since simulations are carried out over several nanoseconds. The film thickness ranges from 4.5 to 35 nm, the latter value becoming comparable to experimental investigations where films between 35 and 50 nm were investigated [346, 350]. 8.3.2 Formation of the Diffusion Wedge In this section, we discuss the change of the displacement field as the diffusion wedge builds up and show that the displacement field becomes crack-like. Further, we show the diffusive displacement of atoms and hence prove that diffusional mass transport from the surface along the grain boundary leads to formation of a diffusion wedge. Crack-Like Displacement Near a Diffusion Wedge

Fig. 8.24 Change of displacements in the vicinity of the diffusion wedge over time. The continuous dark line corresponds to the continuum mechanical solution


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The snapshots in Fig. 8.24 show how the displacement changes as material diffuses into the grain boundary. The horizontal coordinates have been stretched by a factor of 10 in the x-direction to make the crystal lines clearly visible [x, y, z]new = [10 · x, y, z]orig .

(8.15)

This visualization technique is applied throughout this section. We highlight the additional half planes of atoms close to the grain boundary. The continuous dark line corresponds to the continuum mechanics solution in the long-time limit t → ∞ discussed earlier. The results suggest that the displacement field near the diffusion wedge approaches the continuum mechanics solution. Diffusive Displacement of Atoms in the Grain Boundary

Fig. 8.25 Diffusional flow of material into the grain boundary. Atoms that diffused into the grain boundary are highlighted

To illustrate diffusional motion of atoms in the grain boundary, each atom with diffusive displacement δz larger than a few Burgers vectors is colored according to its displacement. Figure 8.25 plots these fields for several instants in time. Diffusion leads to significant surface grooving, with groove depths up to several nm. One can clearly identify the wedge-shape of the diffused atoms. The atomistic simulations show that atoms inserted into the grain boundary instantaneously crystallize, rendering the structure of the grain boundary invariant (this was observed for temperatures below 1,150 K; at higher temperatures, the width of the grain boundary increases slightly).

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The atoms transported along the grain boundary add to either one of the two grains. This result illustrates that the continuum mechanics assumptions [46–48] are valid also on the atomistic level. It is observed that “classical” threading dislocations which become operative when stresses in the film are high enough to allow nucleation of dislocations [68,154]. A frequently observed phenomenon is the emission of dislocations from the grain boundary on inclined < 111 > glide planes [154,345]. In the sample with hf ≈ 30 nm, such threading dislocations are nucleated at a stress level of σxx ≈ 2.4 GPa. We observe that thinner films require a higher critical stress for dislocation nucleation from the grain boundary, in qualitative agreement with the prediction by the 1/hf scaling law for the yield stress. In films thinner than 10 nm it requires extremely high stresses to nucleate inclined dislocations, which renders this mechanism almost impossible. The studies show that grain boundaries are, as proposed in the literature, fertile sources for dislocations in small-grained materials [68, 154]. Another issue in terms of dislocation nucleation is the stability and mobility of the grain boundary. It is observed that the grain boundary forms jogs at elevated temperatures and relatively low stresses (contrary to intuition, at high stresses the grain boundary remains straight). The diffusion path can be severely suppressed and the local stress concentration at the kink serves as a ready source for dislocations. The grain boundary does not remain straight and oscillates around the initial, straight position. 8.3.3 Development of the Crack-Like Stress Field and Nucleation of Parallel Glide Dislocations Continuum theory assumes that dislocations are nucleated when the stress field around the diffusion wedge becomes crack-like. Critical stress intensity factors for dislocation nucleation measured from the atomistic simulations are shown in Table 8.2 for different simulations. We use (8.10) to determine the stress intensity factor. The stress intensity factor is found independent of geometry (film thickness) and also has similar values at Th = 0.8 and Th = 0.9. Temperature T (in K) hf (nm) Stress intensity factor K PG (in MPa × ms ) Crack 300 27.2 4.95 Diffusion wedge 1150 27.2 11.91 27.2 11.35 1250 1250 34.2 11.23 Table 8.2 Critical stress intensity factor K PG for nucleation of parallel glide dislocations under various conditions, for both a diffusion wedge and a crack


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It is observed that nucleation of parallel glide dislocations depends on the film thickness. In the present quasi-two-dimensional setup with rigid boundaries, it is found that dislocations from the boundaries are nucleated when very large strains are applied, thus imposing a condition on the minimum thickness for nucleation of parallel glide dislocations.

Fig. 8.26 Nucleation of parallel glide dislocations from a diffusion wedge, showing the dynamical sequence of the process (from top to bottom). The arrows indicate the position of the partial dislocation nucleated from the diffusion wedge, illustrating how the dislocation moves away from the source

Dislocation nucleation at a diffusion wedge can be divided into different stages shown in Fig. 8.26. After the critical stress intensity factor is achieved, a dislocation dipole is formed. One end of the dipole is pinned in the grain boundary, while the dislocation at the other end of the dipole slides away from the grain boundary. Subsequently, the pinned dislocation is annihilated or “dissolves into” the grain boundary, while the dislocation at the right end of the dipole begins to move away from the nucleation site. As usually found in FCC metals, the dislocation is decomposed into two Shockley partials. The parallel glide dislocation glides on a slip plane parallel to the plane of the film at a distance of a few Burgers vectors above the film–substrate interface (and

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is therefore completely inside the film material). The core width of the partials extends to about six Burgers vectors around 1.6 nm. The dislocation moves a small distance away from the grain boundary to its equilibrium position. When stresses in the film become larger, it responds by moving further away from the grain boundary. The nucleation process is highly repeatable. Every time one parallel glide dislocation is nucleated, one climb edge dislocation is annihilated, leading to a decay in stress intensity. The additional time required to nucleate another parallel glide dislocation is determined by the time required for diffusion to recover the critical stress intensity. This time is much less than the initial time required to form the diffusion wedge. After the first dislocation is nucleated, more and more parallel glide dislocations are observed. In the confined, finite simulation geometry, the emitted parallel glide dislocations form a “secondary pileup” close to the boundary of the simulation cell. In simulations at lower temperatures (T ≈ 800 K), we also observe constrained grain boundary diffusion and the formation of a diffusion wedge with a lattice displacement field similar to that of a crack. However, due to the time constraints of molecular dynamics, nucleation of parallel glide dislocation is not observed. The nucleation of parallel glide dislocations from a crack in a bimaterial layer is shown in Fig. 8.27. For numerical reasons, the loading rate is chosen higher than in the previous case and the temperature in the simulations is about 300 K. After an incipient dislocation is formed, a dislocation nucleates and moves away from the crack tip. The crack tip is blunted, and each time a parallel glide dislocation is nucleated, one surface step is formed. This process is also highly repeatable, as lateral strain is increased. The nucleation of parallel glide dislocations from a crack tip is observed at loading rate a few orders of magnitude higher than in the case of a diffusion wedge and there seems to be no rate limitation in the case of a crack. As in the case of a diffusion wedge, the dislocation glides on a parallel glide plane a few Burgers vectors above the film–substrate interface. For a crack, nucleation of parallel glide dislocations is observed in films as thin as 5 nm. This may be because the critical stress intensity factor is smaller than that for a diffusion wedge as indicated in Table 8.2. The critical stress intensity for parallel glide dislocation nucleation from a diffusion wedge is about 2.3 times larger than that for a crack. This value is in good agreement with the estimate based on the Rice–Thomson model. 8.3.4 Discussion When classical mechanisms of plastic deformation based on the creation and motion of dislocations are severely hindered in thin films on substrates, constrained diffusional creep becomes a major mechanism for stress relaxation, leading to the formation of a new class of defects called the grain boundary diffusion wedge. Large-scale atomistic simulations are performed to investigate the properties of such diffusion wedges. It was shown by atomistic simulations that material is indeed transported from the surface into grain boundaries


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Fig. 8.27 Nucleation of parallel glide dislocations from a crack, showing the dynamical sequence of the process (from top to bottom). The arrows indicate the position of the partial dislocation nucleated from the crack, illustrating how the dislocation moves away from the source. Upon nucleation, a surface step is formed due to crack blunting

and that such transport leads to a crack-like stress field causing nucleation of a novel dislocation mechanism of parallel glide dislocations near the film– substrate interface. The atomistic simulations of parallel glide dislocations being emitted near the root of the grain boundary have further clarified the mechanism of constrained grain boundary diffusion in thin films and provided an important link between theory and experiments. Theoretical, Experimental, and Simulation Results The experimental data suggests that nucleation occurs only at specific grain boundaries. This can partially be explained by the strong dependence of diffusion coefficients on the structure of the grain boundary [68, 356]. Using different types of grain boundaries, it was verified that high-energy grain boundaries exhibit faster diffusivities than low-energy grain boundaries. The viewpoint proposed in [356] is thus consistent with the simulation results. The fact that this concept was shown to hold in covalently bonded system,

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palladium as well as copper (the present study) shows that the transformation of grain boundaries into liquid-like structures may be a more general concept independent of the details of the atomic bonding. Experimental results [346, 350] indicate that the nucleation of parallel glide dislocation occurs repeatedly from grain boundary sources near the film–substrate interface while strain is increased during thermal cycling. The same phenomenon is observed in the atomistic simulations reported in this chapter, although the conditions are quite different. Repeatedly emitted parallel glide dislocations form a pileup when they move toward an obstacle, which can be other grain boundaries (e.g., twins) in the experiments or boundary atoms in the simulations. Repeated nucleation is possible because by each parallel glide nucleation, only one climb edge dislocation in the grain boundary is annihilated while many of them remain “stored” in the grain boundary. The total Burgers vector stored in the grain boundary is found to remain constant. As discussed in Sect. 8.2.2, in films thinner than 10 nm, image stresses on climb dislocations can be as large as 1 GPa. This can severely hinder climb mediated diffusional creep, suggesting that the behavior of discrete dislocations needs to be considered for the nanoscale thin films. This is also supported by the atomistic results showing that stress cannot be relaxed completely in extremely thin films. In addition to the theoretical and computational evidence, the results are not contradicting experimental results which often show large residual stresses in extremely thin films [342]. Employing the molecular dynamics results that nucleation occurs at a critical stress intensity factor K PG , we estimate the necessary lateral stress σ0 in order to achieve this stress intensity factor at t → ∞. In films thinner than a critical thickness between 10 and 20 nm, the analysis predicts stresses reaching the cohesive strength of the material. Hence, before nucleation of parallel glide dislocations the simulation sample will be destroyed by homogeneous decohesion. In the present quasi-two-dimensional setup with rigid boundaries, an additional issue is that dislocations from the boundaries are nucleated when very large strains are applied. These considerations suggest a minimum thickness for parallel glide dislocation nucleation. The critical stress intensity factor for dislocation nucleation from a crack and a diffusion wedge at 0 K is about three times larger than the values calculated from atomistic simulation results at elevated temperature. This can be explained by the finite temperature in the simulations. Yet it is PG PG /Kcr ≈ 2. important that both approaches suggest that Kdw The discussion reveals that the diffusion wedge has similar properties as a crack, but requires a larger stress intensity factor to nucleate a dislocation. The reason for this is that in the case of a diffusion wedge, a dislocation dipole needs to be formed and the dipole interaction force is twice as strong as the image force on an emergent dislocation near a crack tip. This is an important result of atomistic modeling that corroborates the assertion made in the development of the Rice–Thomson model in Sect. 8.2.3.


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Diffusion Wedge vs. Crack We discuss some of the common and distinct properties observed in atomistic simulations for the two kinds of defects (crack vs. diffusion wedge), both are assumed to lie along the grain boundary under elevated temperatures. For a crack, it is observed: • As the applied stress σ0 is increased, the normal stress σxx along the grain boundary remains zero throughout the film thickness, in consistency with the traction-free crack condition. • The loading rate for dislocation nucleation can be much higher than in the case of diffusional creep; there is no rate limiting factor. • Dislocation nucleation occurs at relatively small stress intensity factor. • Dislocation nucleation starts with an incipient dislocation close to the crack tip. In contrast, for a diffusion wedge, it is observed: • The loading rate must be slow enough to allow for diffusion as a dominant relaxation mechanism. Otherwise dislocation activities on inclined planes are observed instead of grain boundary diffusion. • The stresses in the film are determined by the competition of processes causing stresses to be generated and inelastic deformation mechanisms that allow the stresses to be relaxed. • The nucleation process proceeds much slower, because in order to nucleate a parallel glide dislocation, dislocation climb in the grain boundary has to take place to annihilate part of the newly created dipole. On atomistic timescales, nucleation is an extremely slow process (for a crack, nucleation is very fast). • Dislocation nucleation starts when the stress intensity factor is sufficiently large to create a dislocation dipole near the diffusion wedge. • There exists a minimal thickness for parallel glide dislocation nucleation. If the film is very thin, the applied stress reaches the cohesive strength of the material before the critical stress intensity factor K PG for dislocation nucleation is reached. The two defects have major differences in the timescale associated with creation of dislocations. A crack is a ready source for dislocations, while a diffusion wedge has an intrinsic characteristic time associated with dislocation climb. We finally note that no difference in the mechanism of parallel glide dislocation nucleation is observed at different temperatures. We propose further investigations on discrete dislocation effects at the nanoscale. In particular, it is important to develop continuum level solutions for dislocation nucleation in the spirit of Rice’s analysis based on Peierls concept [30, 360]. It should also be interesting to study thin films creep using mesoscopic methods such as discrete dislocation dynamics. Results of the molecular dynamics simulations can be used as input parameter in a multiscale modeling procedure.

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8.3.5 Summary This review illustrates how large-scale atomistic simulations can be used to study constrained diffusional creep in thin films deposited on substrates. The formation of diffusion wedges has been confirmed using atomistic simulations. In agreement with theory and experiment, this illustrates that the flow of matter from the film surface into grain boundaries represents an important mechanism of plasticity in submicron thin films. It has been verified that the diffusion wedge exhibits crack-like stress field at the atomistic level, and this mechanism occurs even if the background stress in the film is insufficient to create dislocations. The results of molecular dynamics simulations enabled us to calculate a critical stress intensity factor for nucleation of parallel glide dislocations from the diffusion wedge. The critical stress intensity is found to be independent of the film thickness and does not significantly change in the temperature range of the investigation (from Th = 0.9 to Th = 0.8). The most important result of these simulations is that when grain boundary diffusion is active, the grain boundary can be treated as a crack in a first approximation. In the present section, we have only studied a two-dimensional geometry. In the following chapters, we will show that the condition of traction relaxation along grain boundaries also has dramatic consequences on the dislocation mechanisms in polycrystalline thin films.

8.4 Dislocation Nucleation from Grain Triple Junction In this chapter, we focus on the details of dislocation nucleation close to a triple junction between three grains misoriented with respect to one another. The simulation geometry is shown in Fig. 8.28.

Fig. 8.28 Geometry for studies of plasticity in grain triple junctions. A low-energy grain boundary is located between grains 1 and 2, and two high-energy grain boundaries are found between grains 2 and 3 and between 3 and 1


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In contrast to the simple bicrystal geometry used in numerical studies discussed in Sect. 8.3, experiments are carried out in polycrystalline thin films [51, 346]. The first goal of this section and Sect. 8.5 is hence to extend the quasi-two-dimensional studies of constrained diffusional creep to more realistic polycrystalline microstructures. We will investigate plasticity of thin films with traction relaxation along the grain boundaries due to constrained diffusional creep. The continuum model and the quasi-two-dimensional geometry of previous atomistic simulation of plasticity in thin films could not provide a clear understanding of dislocation nucleation processes from different types of grain boundaries. Yet, experiments show a high selectivity of dislocation nucleation from different grain boundaries [51, 350]. Understanding the features of the grain boundaries as sources for dislocation nucleation is critically important to form a clear picture of thin film plasticity. Therefore, an important objective of this section is to study the details of the dislocation nucleation process near the grain boundary–substrate interface. We show that the grain boundary structure indeed has a significant influence on the dislocation nucleation process. Another interesting result is that the role of partial dislocations is important in very thin films with very small grain diameters. 8.4.1 Atomistic Modeling of the Grain Triple Junction Here we summarize the details of the atomistic modeling procedure for this case. To focus on the nucleation process of dislocations and the effect of different types of grain boundaries in detail, we consider a tricrystal model with a triple junction between three grains. The model is constructed such that it features two high-energy and one low-energy grain boundary. The schematic geometry is shown in Fig. 8.28. As indicated in Fig. 8.28, cracked grain boundaries with traction-free surfaces along zc < z < hf are used to mimic the existence of diffusion wedges in all of the grain boundaries. We choose zc ≈ 1.5 nm so that the crack does not reach the substrate (z = 0 at the substrate). This is motivated by the results of molecular dynamics simulations showing that the glide plane of dislocations is not directly at the substrate but a few atomic layers above. Loading is applied by prescribing a displacement to the outermost rows at the boundary of the quadratic slab. Grain 1 has the reference configuration ([110] in the x-direction, [112] in the y-direction). Grain 2 is rotated counterclockwise by 7.4◦ , and grain 3 is rotated by 35◦ with respect to grain 1. The low-energy grain boundary is situated between grains 1 and 2, and the two high-energy grain boundaries are between grains 2 and 3 and between grains 3 and 1. The structure of the low-energy grain boundary is significantly different from that of the high-energy ones. The former is essentially composed of a periodic array of misfit dislocations, with a strongly inhomogeneous distribution of strain energy along the grain boundary. In contrast, the strain energy along

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the high-energy grain boundaries is more homogeneously distributed. After creation of the sample, the structure is annealed for a few picoseconds and then relaxed for a few thousand integration steps using an energy minimization scheme. Boundary Conditions and Integration Scheme The boundary conditions of all models are chosen such that atoms close to the film–substrate interface are pinned to their initial locations (and also moved according to the applied strain field), mimicking perfect adhesion of a film on a stiff substrate. After the initial atomic configuration is created, a global energy minimization scheme is applied to relax the structure. The studies are carried out using a microcanonical N V E ensemble with a quasistatic energy minimization scheme. The biaxial strain in the samples is gradually increased up to 2.5%. The multibody embedded atom potential (EAM) potential for copper developed by Mishin and coworkers [107] is used for these studies. Analysis Techniques The simulation results are analyzed with the centrosymmetry technique [36], which is a convenient way to discriminate between different defects such as partial dislocations, stacking faults, grain boundaries, surfaces, and surface steps. In some cases, we will also use the slip vector technique proposed recently [147]. This method allows us to extract quantitative information about the Burgers vector and slip plane of dislocations immediately from the simulation data. 8.4.2 Atomistic Simulation Results We will investigate the details of parallel glide dislocation nucleation process near the grain boundary–substrate interface. Dislocation mechanisms associated with grain boundary cracks will be compared and related to experimental results. Nucleation of Parallel Glide Dislocations from a Grain Triple Junction In this section, we focus on the details of dislocation nucleation close to a triple junction between three grains misoriented with respect to one another. As soon as a threshold stress is overcome during loading, the generation of parallel glide dislocations from the grain boundaries is initiated. Snapshots of this processes are shown in Fig. 8.29.


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Fig. 8.29 Nucleation of parallel glide dislocations from a grain triple junction. The plot shows a time sequence based on a centrosymmetry analysis, showing how several dislocation half loops nucleate and grow into the grain interior

Fig. 8.30 Schematic of dislocation nucleation from different types of grain boundaries. Individual misfit dislocations at low-energy grain boundaries serve as sources for dislocations. At high-energy grain boundaries, there is no inherent, specific nucleation site so that the point of largest resolved shear stress, the grain triple junction, serves as nucleation point

We begin with a description of the nucleation process from the low-energy grain boundary between grains 1 and 2. This boundary is composed of an array of misfit grain boundary dislocations which serve simultaneously as multiple nucleation sites for new dislocations. The nucleation sites are therefore not necessarily located close to the triple junction, the region of largest shear stresses. This observation could be reproduced in different geometries. A small number of incipient dislocations grow along the low-energy grain boundary and coalesce to form dislocation half-loops. This mechanism is also visualized schematically in Fig. 8.30. In Fig. 8.31, we show that deformation twinning

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occurs due to repeated nucleation of partial dislocations with the same Burgers vector.

Fig. 8.31 Deformation twinning by repeated nucleation of partial dislocations. Repeated slip of partial dislocations leads to generation of a twin grain boundary

Jog Dragging An interesting observation in the simulations is that some dislocations are strongly bowed at defect junctions. Dislocation junctions obstructing further glide motion are highlighted in Fig. 8.32. The reason for this effect is that the glide planes of the incipient half-loops of partial dislocations are different, but have the same Burgers vector. Using the slip vector approach proposed by Zimmerman and coworkers [147], we have verified that the Burgers vector of the dislocations nucleated in each grain are indeed identical.

Fig. 8.32 Dislocation junction and bowing of dislocations by jog dragging. A trail of point defects is produced at the jog in the leading dislocation, which is then repaired by the following partial dislocation (this is a similar mechanism as that shown in Figs. 7.16 and 7.17a)

Once different half-loops grow, they combine with each other while forming jogs since they glide on different glide planes. The jog has a nonglissile component and cannot move conservatively [38] thus causing generation of


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point defects. This, in turn, exerts a drag force on dislocations causing the dislocation lines to bow. A similar mechanism of jog dragging due to point defect generation is known from dislocation cutting processes as depicted in Fig. 8.33. As discussed in the literature on dislocation mechanics [38], when two dislocations intersect each acquires a jog equal in direction and length to the Burgers vector of the other dislocation. If two screw dislocations intersect, this jog cannot glide conservatively since it features a sessile edge segment. However, if the applied stress is large enough, the dislocation with the jog starts to glide and the jog leaves a trail of vacancies or a trail of interstitials depending on the line orientation and the Burgers vector of the reacting dislocations.

Fig. 8.33 Generation of trails of point defects. Subplot (a): Dislocation cutting processes with jog formation and generation of trails of point defects. Both dislocations leave a trail of point defects after intersection. The arrows indicate the velocity vector of the dislocations. Subplot (b): Nucleation of dislocations on different glide planes from grain boundaries generate a jog in the dislocation line that causes generation of trails of point defects

The mechanism observed in the simulations is similar. The difference is that no dislocation cutting process occurs, but instead the jog in the dislocation line develops due to nucleation of incipient dislocations on different glide planes. It is observed that the sessile component of the jog is rather small and is only a fraction of the partial Burgers vector. Therefore, not a complete point defect is generated but only a trail of “partial point defects.” Trails of “partial point defects” have recently also been observed in large-scale computer simulations of work-hardening in ductile materials [14]. As shown by calculations in [14], the dragging force of the partial point defects is estimated to be approximately 20% of the dragging force exerted by generation of complete vacancy tubes. If these defects appear in large numbers, the effect on dislocation motion can be significant.

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High-Energy vs. Low-Energy Grain Boundaries In the case of high-energy grain boundaries (as between grains 2 and 3) with a more homogeneous structure, nucleation of parallel glide dislocations is found to occur preferably at the triple junction. The process proceeds with an incipient dislocation growing until the second partial is emitted. The parallel glide dislocations often have semi-circular shapes as observed in early stages of dislocation nucleation in experiment [346, 350]. In contrast to the low-energy grain boundary where misfit dislocations serve as nucleation sites for new dislocations, the triple junction acts as the main nucleation source at high-energy grain boundaries. 8.4.3 Discussion Dislocation nucleation depends on the grain boundary structure: We observe that low-energy grain boundaries composed of a periodic array of misfit dislocations provide more fertile sources for threading dislocation nucleation. At low-energy grain boundaries, dislocations are often observed to nucleate close to grain boundary misfit dislocations. This can be referred to as an intrinsic condition, because the concentration of internal grain boundary stresses serves as nucleation site for dislocations. Since the incipient dislocations are often nucleated at different glide planes, complex dislocation reactions take place when several of them combine to form a single dislocation line. Such mechanisms can hinder dislocation motion and cause bowing of the dislocation line. The observation of such nucleation-induced jogs with subsequent generation of trails of point defects has not been described in the literature. In other computer simulation of ductile materials [14], similar mechanisms have been observed, suggesting that this mechanism may play a role in hardening of materials. In the more homogeneous high-energy grain boundaries, there is inherently no preferred nucleation site. Therefore, triple junctions of grain boundaries are preferred as nucleation sites. The overall stress field governs dislocation nucleation, since such a triple junction provides a location with highest stress concentration. Different parallel glide dislocations can interact in a complex way to form networks of dislocations as shown in Fig. 8.32. Another finding is that partial dislocations dominate plasticity in the simulations, as can be verified in Fig. 8.31, where deformation twinning is depicted. This indicates that partial dislocations dominate plasticity at nanoscale. Similar observations have been reported by other groups [154, 303].

8.5 Atomistic Modeling of Plasticity of Polycrystalline Thin Films Atomistic modeling of plasticity in polycrystalline thin films is just at its beginning. Few studies of such systems have been reported in the literature.


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Recently, atomistic simulations of two-dimensional systems were reported by Shen [361]. In this study, the increase in yield strength was investigated and nucleation and motion of threading dislocations was in the focus. However, the model did not contain grain boundaries despite the fact that grain boundaries can serve as fertile sources for dislocations. In contrast to this simplistic model, we propose a three-dimensional model of thin films with a more realistic microstructure. The model studied in this section is a polycrystalline thin film consisting of hexagonal shaped grains as shown in Fig. 8.34. The choice of this geometry is motivated by the grain microstructure found in experiments [346, 350]. An advantage of this model over the geometry studies in Sect. 8.4 is that fully periodic boundary conditions in the x- and y-direction can be assumed. Here we will focus on dislocation nucleation and motion from grain boundaries and a crack–grain boundary interface. One of the important objectives will be to study the effect of grain boundary traction relaxation by diffusional creep on the dislocation mechanism that operate in the film. Further studies will be focused on the details of dislocation nucleation from different type of grain boundaries. As known from Sect. 8.4, the structure of the grain boundaries has strong influence on the nucleation of dislocations. The plan of this section is summarized as follows. After presenting details about the atomistic modeling procedure, we will continue with a discussion of the results of several large-scale atomistic studies comprising of up to 35 million particles. Even for today’s supercomputers, this represents a significant system size. We will show that grain boundary relaxation by diffusional creep gives rise to dominance of parallel glide dislocations, in accordance with experiment. In contrast, if grain boundary tractions are not relaxed, threading dislocations dominate plasticity. We show that low-energy grain boundaries are more fertile sources for plasticity than more homogeneous high-energy grain boundaries. This hypothesis is further supported by a set of atomistic simulations of bulk nanocrystalline copper. 8.5.1 Atomistic Modeling of Polycrystalline Thin Films As indicated in Fig. 8.34, cracked grain boundaries with traction-free surfaces (along zc < z < hf , where zc ≈ 1.5 nm) are used to mimic the existence of diffusion wedges in some of the grain boundaries. Grain 1 is in the reference configuration ([110] in the x-direction, [112] in the y-direction). Grain 2 is rotated counterclockwise by 7.4◦ , grain 3 is rotated by 35◦ , and grain 4 is rotated by 21.8◦ with respect to grain 1. The model contains up to 35 million particles. With this procedure, a low-energy grain boundary is constructed between grains 3 and 4. After creation of the sample, the structure is relaxed for a few thousand integration steps using an energy minimization scheme. Figure 8.35 shows the atomistic model of the polycrystalline thin film. Only surfaces and grain boundaries are shown (explanation of visualization in figure caption).

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Fig. 8.34 Geometry for the studies of plasticity in polycrystalline simulation sample

Fig. 8.35 Atomistic model of the polycrystalline thin film. Only surfaces (brighter coloring) and grain boundaries (darker color) are shown

In contrast to the modeling with the tricrystal model, the simulation cell is fully periodic in the x- and y-direction. Loading is applied by homogeneously straining the sample in the desired direction. Further details on the modeling technique can be found in Sect. 8.4.1. 8.5.2 Atomistic Simulation Results The plasticity of thin films will be investigated with and without traction relaxation at the grain boundaries. This enables one to study the details of the dislocation nucleation process near the grain boundary–substrate interface. Dislocation mechanisms associated with grain boundary cracks will be compared and related to experimental results. The model of constrained diffusional creep [46] predicts that due to mass transport from the surface into the grain boundary, the tractions along the grain boundary are relaxed, and thus a crack-like stress field develops. This


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change should significantly alters the dislocation microstructure that develops in the film under mechanical deformation. While threading dislocations dominate plasticity in films where grain boundary diffusion is shut down, in films where grain boundary diffusion is active parallel glide dislocations are expected to dominate. Indeed, since the continuum model was proposed [46], an experimental group has reported the observation of parallel glide dislocations in copper films with thicknesses below 400 nm [51]. The researchers concluded that grain boundary traction relaxation by diffusional creep leads to change in the deformation field near the crack tip. Experimental results suggest that threading and parallel glide dislocations are competing mechanisms [51, 346, 350] In submicron, uncapped thin films on substrates. In this section, we want to probe this hypothesis by large-scale atomistic simulations. Atomistic modeling of thin film plasticity at the nanoscale provides an ideal tool to study such competing mechanisms and to determine conditions under which they are active. It is known that grain boundaries are important sources for dislocations in nanostructured materials. We illustrate that the structure of the grain boundaries has significant influence on the motion of dislocations into the grain interior. Furthermore, we find that the role of partial dislocations seems to be increasingly important as the grain size approaches nanoscale. Threading Dislocations We start with the polycrystalline sample without relaxation of tractions along the grain boundaries, corresponding to the case when grain boundary diffusion is not active. The simulation results are depicted in Figs. 8.36 and 8.37. In this case, the dominating inelastic deformation mechanism is clearly glide of threading dislocations on inclined glide planes. No dislocations on glide planes parallel to the film surface are observed as expected, because there is no resolved shear stress and thus no driving force for dislocation nucleation on parallel glide planes. Figure 8.36a reveals a complex dislocation structure in the interior of the film. The dislocation structure is analyzed with the centrosymmetry technique. Figure 8.36b shows a more detailed magnified view of a section of the film. Threading dislocations are observed to leave behind interfacial dislocation segments at the film–substrate interface and atomic steps at the film surface. Figure 8.37 shows snapshots of a top view of the film surface at different times, including a magnified view of the surface at snapshot 4 in Fig. 8.38. The surface steps emanate from the grain boundaries, suggesting that dislocations are nucleated at the grain boundary–surface interface. From the direction of the surface steps it is evident that different glide planes are activated. From the number of surface steps created during plastic deformation, it seems that dislocation motion concentrates in grains adjacent to low-energy grain boundaries which apparently provide more fertile sources for dislocation nucleation (within grains 1 and 2). This can also be verified in Fig. 8.36b. The

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Fig. 8.36 Nucleation of threading dislocations in a polycrystalline thin film. Subplot (a) shows a view into the interior, illustrating how threading dislocations glide by leaving an interfacial segment. Subplot (b) shows a top view into the grain where the surface is not shown. The plot reveals that the dislocation density is much higher in grains 3 and 4

Fig. 8.37 Surface view of the film for different times. The threading dislocations inside the film leave surface steps that appear as darker lines in the visualization scheme. This plot further illustrates that the dislocation density in grains 3 and 4 is much higher than in the two other grains

dislocation density in grains 3 and 4 is several times higher than that in grains 1 and 2. Figure 8.39 shows a sequence of a nucleation of a threading dislocation from the grain boundary–surface interface. The plot indeed shows that threading dislocations are nucleated at the grain boundary–surface interface and then the half-loops grow into the film until they reach the substrate. Due to the constraint by the substrate, threading dislocations leave an interfacial segment. This observation is in agreement with the classical understanding of threading


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Fig. 8.38 Detailed view onto the surface (magnified view of snapshot 4 in Fig. 8.37). The plot shows creation of steps due to motion of threading dislocations. The surface steps emanate from the grain boundaries, suggesting that dislocations are nucleated at the grain boundary–surface interface. From the direction of the surface steps it is evident that different glide planes are activated

Fig. 8.39 Sequence of a nucleation of a threading dislocation, view at an inclined angle from the film surface. Threading dislocations are preferably nucleated at the grain boundary–surface interface and half-loops grow into the film until they reach the substrate. Due to the constraint by the substrate, threading dislocations leave an interfacial segment

dislocation nucleation and with experimental results [341–343]. The threading dislocations intersect the surface at an angle of 90◦ [38]. Parallel Glide Dislocations In the following, some of the grain boundaries are treated as traction-free cracks as shown in Fig. 8.34. In Fig. 8.40, we show several snapshots of the dislocation structure. Parallel glide dislocations are generated close to the

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Fig. 8.40 Nucleation of parallel glide dislocations, small grain sizes. The plot shows that dislocation activity centers on the grain boundary whose traction is relaxed. Due to the crack-like deformation field, large shear stresses on glide planes parallel to the film surface develop and cause nucleation of parallel glide (PG) dislocations. Subplot (a) shows a top view and subplot (b) shows a perspective view. The plot reveals that there are also threading (T) dislocations nucleated from the grain triple junctions

Fig. 8.41 Nucleation of parallel glide dislocations, large grains. The plot shows a top view of two consecutive snapshots. The region “A” is shown as a blow-up in Fig. 8.43

film–substrate interface. Figure 8.40a shows a top view while Fig. 8.40b shows a perspective side view of the interior of the film. The section shown has dimensions of approximately 120 nm × 150 nm, and the film thickness is hf ≈ 15 nm. The grain diameter in the x-direction is approximately dx ≈ 40 nm. This plot reveals that not only parallel glide but also some threading dislocations are generated at the grain boundary– surface interface. The plot shows that dislocation activity centers on the grain boundary whose traction is relaxed. Due to the crack-like deformation field, large shear stresses on glide planes parallel to the film surface develop and cause nucleation of parallel glide dislocations. A complex dislocation network


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Fig. 8.42 Nucleation of parallel glide dislocations, large grains. The plot shows a view of the surface. From the surface view it is evident that threading dislocations are nucleated in addition to the parallel glide dislocations. These emanate preferably from the interface of grain boundaries, traction-free grain boundaries, and the surface

develops on a timescale of several picoseconds after the first dislocation nucleation. Figure 8.41 shows the simulation results for a larger grain size. The section shown has dimensions of approximately 300 nm × 400 nm; the film thickness is hf ≈ 15 nm. The grain diameter in the x-direction is approximately dx ≈ 180 nm, about four times larger than in Fig. 8.40 while the film thickness is kept constant at hf ≈ 15 nm. More dislocations are observed to nucleate than in Fig. 8.40, indicating that more dislocations “fit” into the larger grain, and consequently, a more complex dislocation microstructure develops. As the laterally applied strain is continuously increased, the first dislocations to be nucleated are occasionally complete dislocations, while the following dislocations are often pure partial dislocations. Figure 8.42 shows a view of the surface of the results shown in snapshot 2 of Fig. 8.41, revealing surface steps generated from the motion of threading dislocations. Even when the traction of some of the grain boundaries are relaxed, threading dislocations occur. The figure shows that threading dislocations are predominantly nucleated at the junction between traction-free grain boundaries and normal grain boundaries where traction is not relaxed. It is observed that dislocations cannot glide as easily along the low-energy grain boundaries as along the more homogeneous high-energy grain boundaries between grains 1 and 2. This can be verified in Fig. 8.40a. While an extended dislocation (marked by “PG”) in grain 1 is almost a straight line, all dislocations in grains 3 and 4 are strongly curved.

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Fig. 8.43 Nucleation of parallel glide dislocations. The plot shows an analysis of the complex dislocation network of partial parallel glide dislocations that develops inside the grains (magnified view of the region “A” marked in Fig. 8.41). All defects besides stacking fault planes are shown in this plot

Figure 8.43 shows the complex dislocation network of partial parallel glide dislocations that develops inside the grains. In this plot, the stacking fault planes are not shown. The bowing of the dislocations indicates that their motion is hindered by mutual interaction. We observe that formation of jogs and creation of trails of point defects play a very important role as already discussed in Sect. 8.4.2. 8.5.3 Plasticity of Nanocrystalline Bulk Materials with Twin Lamella In the preceding sections, the role of low-energy vs. high-energy grain boundary was discussed. The importance of this concept is further underlined by the studies reported in this section. Here we focus on polycrystalline bulk copper, where the grain size is on the order of several nanometers to tens of nanometers. We consider a polycrystalline microstructure with hexagonal grains, but with different grain orientations as in the previous case (see Fig. 8.44 for details). To further study the effect of geometric confinement on plasticity, we introduce a subnanostructure in the grains. This subnanostructure is established by assuming twin grain boundaries forming very thin twin lamella. Such microstructure can be produced experimentally in copper [362]. With this model, two main objectives are pursued: 1. We show that in bulk nanostructured materials, the type of the grain boundary plays a very important role for dislocation nucleation, as it was found for thin films. 2. We show that the subnanostructure composed of twin grain boundaries provides a very effective barrier for dislocation motion and therefore leads to a very “strong” nanostructured material.


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Modeling To underline the first point regarding dislocation nucleation, we consider two samples in our simulation study. The first sample (i) has the same grain misorientations as in Sect. 8.5.2 (and therefore features a low-energy grain boundary between grains 3 and 4) and we construct a second sample (ii) where all grain boundaries are of the same type. If the proposed concept is correct that dislocation nucleation occurs predominantly from low-energy grain boundaries, the dislocation density in sample (i) should be higher in grains 3 and 4, and should be comparable in all grains in sample (ii).

Fig. 8.44 Nanostructured material with twin grain boundary nanosubstructure. The light gray lines inside the grains refer to the intragrain twin grain boundaries. The thickness of the twin lamella is denoted by dT

The simulation geometry is depicted in Fig. 8.44. The light grey lines inside the grains refer to the intragrain twin grain boundaries. The thickness of the twin lamella is denoted by dT . Simulation Results The material is loaded uniaxially in the x-direction. We start with sample (i), and we consider is a grain size of 12.5 nm × 16.5 nm. The grains have the same

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misorientation as in the study described above. We perform the simulation for two different lamella sizes dT . The results are shown in Fig. 8.45. It is observed that dislocations are generated exclusively from the lowenergy grain boundaries between grains 3 and 4. This is in agreement with the results of the polycrystalline thin films. The fact that we use a different grain orientation in this study with different boundary conditions suggests that the nucleation conditions discussed previously is a more general concept. The results indicate that the twin grain boundaries are an effective barrier for further dislocation motion, since it is found that dislocation pileups at the twin grain boundaries. An important consequence is that the thinner the lamella structure (that is, a small dT ), the less plasticity can be transmitted via the motion of dislocations. This indicates that grains with a nanosubstructure of twin grain boundaries could serve as an effective hardening mechanism for materials.

Fig. 8.45 Simulation results of nanostructured material with twin lamella substructure under uniaxial loading for two different twin lamella thicknesses. Subplot (a) shows the results for thick twin lamella (dT ≈ 15 nm > d) and subplot (b) for thinner twin lamella (dT ≈ 2.5 nm). Motion of dislocations is effectively hindered at twin grain boundaries

We report another study with the same microstructure, but with different grain misorientation angles, sample (ii). In this case, we choose the grain boundary misorientation identical in all grains. Grain 1 is in its reference configuration, grain 2 is rotated by 30◦ , grain 3 by 60◦ , and grain 4 is misoriented by 90◦ . All grain boundaries are now high-angle grain boundaries. The results of this calculation are shown in Fig. 8.46a. Unlike in Fig. 8.45, dislocations are now nucleated at all grain boundaries and the nucleation of dislocations is governed by the resolved shear stress on different glide planes. It is observed that dislocations can easily penetrate through the stacking fault planes generated by motion of other partial dislocations, but build pileups at the twin grain boundaries. We also observe that dislocations with opposite


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Burgers vector annihilate. Further, dislocations cross-slip (see highlighted region in the center of Fig. 8.46b, region i.) in regions with high dislocation densities. The activated primary and secondary glide planes are highlighted in the plot. The primary glide planes are parallel to the twin grain boundaries so that dislocation glide is not restricted. In contrast, once dislocations have cross-slipped to the secondary glide plane their motion is restricted due to the twin grain boundaries (see Fig. 8.46b, ii.). Figure 8.46b, iii. shows intersection of dislocations. A defect is left at the intersection of the stacking fault planes. As in the previous studies of nanostructured materials [303], we also observe that partial dislocations dominate plasticity. Dominance of partial dislocations is verified by the fact that dislocations leave behind a stacking fault.

Fig. 8.46 Simulation results of nanostructured material with twin lamella substructure under uniaxial loading for two different twin lamella thicknesses, all high-energy grain boundaries. Subplot (a) shows the potential energy field after uniaxial loading was applied. Interesting regions are highlighted by a circle. Unlike in Fig. 8.45, dislocations are now nucleated at all grain boundaries. The nucleation of dislocations is now governed by the resolved shear stress on different glide planes. Subplot (b) highlights an interesting region in the right half where i. cross-slip, ii. stacking fault planes generated by motion of partial dislocations and iii. intersection of stacking fault planes left by dislocations is observed

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8.5.4 Modeling of Constrained Diffusional Creep in Polycrystalline Films We have also modeled constrained grain boundary diffusion in polycrystalline thin films, thus extending the two-dimensional studies discussed in Chap. 8.3 to the three-dimensional case. We apply biaxial loading rate on the order of 1% strain per nanosecond. The temperature is, as in the two-dimensional studies, chosen around 90% of the melting temperature. High-energy grain boundaries transform into liquid-like intergranular layers, while low-energy grain boundaries establish as arrays of misfit dislocations. In the following sections we will show that the basic mechanism of parallel glide dislocation nucleation is identical to the results observed in the twodimensional case. The simulations provide direct evidence that the diffusivities depend on the grain boundary structure. Constrained Grain Boundary Diffusion and Dependence on Grain Boundary Structure We model a film of thickness hf ≈ 11 nm with a grain diameter of about 22 nm in the x-direction. The simulation sample is constructed such that we have high-energy as well as low-energy grain boundaries. This is motivated by the goal to investigate the effect of grain boundary structure on the deformation mechanisms. Grain 1 is completely surrounded by high-energy grain boundaries and the other grains feature low-energy grain boundaries (grain 1 is in its reference configuration, grain 2 is rotated by 35.4◦ , grain 3 by 44.7◦ , and grain 4 by 53.4◦ ). It is observed that, in agreement with the predictions in the literature [303], high-energy grain boundaries provide rather fast diffusion paths in contrast to low-energy grain boundaries. This strongly underlines the notion that the grain boundary structure plays a major role in determining the resulting deformation mechanism, and that it therefore needs to be taken into account when diffusivities are determined. Formation of grain boundary diffusion wedges is accompanied by surface grooving at the grain boundary interface. Therefore, the surface height profile provides a reliable indication of diffusive activities in the grain. Fig. 8.47a plots the surface profile of a polycrystalline sample in early stages of the simulation. The observation of surface grooves is in agreement with recent experimental reports [351]. Compared with all other diffusion paths, grain triple junctions provide the fastest paths for diffusion. This is verified in the simulation results since at grain triple junctions, the surface grooves are deepest. Similar as in the two-dimensional case, it is observed that the grain boundaries become curved along the z-direction. Nucleation of Parallel Glide Dislocations According to the hypothesis by continuum theory [46], parallel glide dislocations should only be nucleated along grain boundaries whose tractions are


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Fig. 8.47 Modeling of constrained diffusional creep in polycrystalline samples; nucleation of threading vs. parallel glide dislocations [49]. The blowup in panel (c) shows an energy analysis of the dislocation structure and visualizes a parallel glide dislocation nucleated from a grain boundary diffusion wedge. The surface steps indicate that threading dislocations have moved through the grain and no threading dislocations exist in grain 1. The dark lines show the network of parallel glide dislocations in grain 1 (in other grains we also find parallel glide dislocations in snapshot (d) but they are not shown)

relaxed by diffusional creep. Since high-energy grain boundaries are predominant paths for diffusion, in grains neighboring high-angle grain boundaries parallel glide dislocations should dominate plasticity. In other grains, where tractions along the grain boundaries are not relaxed threading dislocations should dominate. This prediction is verified by the atomistic simulations. Nucleation of parallel glide dislocation only occurs in grains that are completely surrounded by high-energy grain boundaries. In the computational sample, there is no other dislocation activity than parallel glide dislocations after diffusion has proceeded sufficiently long. In other grains where little grain boundary relaxation is possible by diffusional creep, threading dislocations are easily nucleated. Predominant nucleation site are, in agreement with previous results, misfit

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dislocations at the grain boundary. The observation of threading dislocations in other grains is consistent with the studies where some grain boundaries were assumed traction free (see Fig. 8.41). This result of the study of constrained grain boundary diffusion in polycrystalline films is documented in Fig. 8.47, where the first two snapshots in Fig. 8.47a, b show the initial stages of diffusive deformation. The nucleated parallel glide dislocation is shown in Fig. 8.47c as a black line, and additional parallel glide dislocations are nucleated subsequently as shown in Fig. 8.47d. Its shape was determined using the energy filterning method shown in the blow-up of Fig. 8.47c. Additional analysis was performed based on geometrical methods identical to those applied in Sect. 8.3. After the first parallel glide dislocation is nucleated, additional dislocations appear as the loading is increased and the dislocations form a network that is similar to the results shown in Fig. 8.40. It is noted that even at higher applied loads, there are exclusively parallel glide dislocations in grain 1 as shown in Fig. 8.40d. In other grains, we observe parallel glide dislocations at later stages in addition to threading dislocations. The most important result of this section is that constrained diffusional creep and nucleation of parallel glide dislocations can also be modeled in a polycrystalline model. Therefore, this result enables one to perform a direct comparison with experimental results. Figure 8.12a (discussed earlier) shows the structure of parallel glide dislocations. Figure 8.48 depicts a time sequence of experimental snapshots that illustrates the subsequent emission of parallel glide dislocations. 8.5.5 Discussion The results reviewed in this section can be summarized as follows: 1. Threading dislocations dominate deformation when tractions along the grain boundaries are not relaxed. However, if the grain boundary tractions are relaxed, parallel glide dislocations dominate the plasticity of ultrathin films (hf ≈ 15 nm). Almost all plasticity is carried on glide planes that are very close to each other. This transition of the maximum shear stress from inclined planes to planes parallel to the film surface is illustrated in Fig. 8.11. 2. Dislocation nucleation depends on the grain boundary structure. Lowenergy grain boundaries composed of an array of misfit dislocations provide more fertile sources of dislocations than high-energy grain boundaries with a more homogeneous structure. We find that the dislocation density is a few times higher in grains connected by low-angle grain boundaries. This assertion is further supported by the studies of nanostructured bulk material described in Sect. 8.5.3. 3. Different parallel glide dislocations can interact in a complex way to form networks of dislocations as shown in Fig. 8.43 (a blow-up picture, showing only partial dislocations and grain boundaries while filtering out the stacking faults).


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Fig. 8.48 Series of TEM micrographs of an unpassivated copper film, film thickness hf = 200 nm, showing the nucleation and propagation of parallel glide dislocations. Dislocations appear as white lines. A total of ten dislocations (numbered in the plots) are emitted sequentially from the source at the lower left triple junction. Dislocations are pushed forward by subsequently emitted dislocations, which in turn are not able to glide as far into the grain as the earlier dislocations (compare subplots (b), (d), (f), (h)). Based on their motion and on the grain geometry, dislocations must have undergone glide on the (111) glide plane parallel to the film–substrate interface. Reprinted from [51] Acta Materialia, Vol. 51, T.J. Balk, G. Dehm and E. Arzt, Parallel glide: unexpected dislocation motion parallel to the substrate in ultrathin c 2003, with permission from Elsevier copper films, pp. 4471–4485, copyright

The simulations show that relaxation of grain boundary tractions changes the dislocation microstructure and triggers completely different stress relaxation mechanisms in thin films. We have also investigated nucleation of parallel glide dislocations from diffusion wedges using the quasicontinuum method. The results of this simulation are discussed in Sect. 5.1 and results were shown

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in Fig. 5.6, for instance. The same behavior was observed in these simulations as with purely atomistic methods. Without relaxation of grain boundary tractions, threading dislocations dominate thin film plasticity, while under grain boundary diffusion, dislocations on parallel glide planes dominate. Threading dislocations are found to be mostly complete dislocations, while we see a strong tendency to nucleate partial dislocations in the case of parallel glide dislocations in the nanometersized grains investigated here. This is qualitatively consistent with results of atomistic modeling of deformation of nanocrystalline materials [154,303,304]. At nanoscale, the role of partial dislocations becomes increasingly important. Twinning along parallel planes might become an important deformation mechanism at high strain rates as shown in Fig. 8.31. The transition of the deformation mechanism from threading dislocations to parallel glide dislocations is also observed in recent experimental investigations [346, 350]. Experiment has clearly confirmed that once grain boundary diffusion is shut down, for example in very thick films or due to the existence of a capping layer, threading dislocations dominate plasticity [51, 346, 350]. When grain boundary diffusion is active, either because there is no capping layer or because the film thickness is sufficiently small, parallel glide dislocations dominate. These observations indicates that mechanisms relaxing the grain boundary tractions are active during the deformation of ultra-thin films. Experimental results are in good qualitative agreement with the molecular dynamics results reviewed here. Another important feature is that parallel glide dislocations do not glide as easily along inhomogeneous low-angle grain boundaries as they do along homogeneous high-energy grain boundaries as shown in Fig. 8.41. This is explained by the fact that the low-energy grain boundaries are composed of an array of misfit dislocations and thus rather inhomogeneous. Similar mechanism has been observed in experiment. In [51], it was reported that dislocations are effectively repelled from certain type of grain boundaries causing significant bowing. Figure 8.40b reveals that not only parallel glide but also some threading dislocations are generated at the grain boundary–surface interface. This observation is in qualitative agreement with experiment [51]. The studies of constrained grain boundary diffusion in polycrystalline samples led to similar results and are also in qualitative agreement with experiment. 8.5.6 Summary: Results of Modeling of Thin Films We have reviewed deformation mechanisms in submicron thin copper films, focusing on the competition between constrained diffusional creep with subsequent parallel glide dislocation nucleation and the nucleation of threading dislocations. Together with experimental observations [51], the results of atomistic modeling of constrained grain boundary diffusion provide evidence that it is an important deformation mechanism in very thin uncapped copper films.


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The observation of parallel glide dislocations nucleated at grain boundary diffusion wedges (shown in Fig. 8.26) agrees with experimental investigations of deformation of ultra-thin copper films [51,346,350]. The results also agree well with the predictions by recently developed continuum mechanics theory [46, 48], thus closing the experiment–theory–simulation linkage. The most important result is that once other stress relaxation mechanisms (e.g., by dislocation motion) are shut down, diffusional flow of matter along grain boundaries dominate the mechanical properties of thin metal films on substrates. An important contribution of this work is the atomistic study of constrained diffusional creep and subsequent parallel glide dislocation nucleation. We find that grain boundary diffusion could be modeled by classical molecular dynamics, in agreement with reports in previous studies [68, 303]. However, modeling of grain boundary diffusion is difficult with classical molecular dynamics and is only possible under large stress and high temperatures. In the simulations, stress of magnitude of several GPa is applied and diffusion is studied at elevated temperatures on the order of 90% of the melting temperature of copper. Modeling at the atomic scale helped to identify the key mechanism of dislocation nucleation from a diffusion wedge. The simulations allowed to establish a detailed understanding of the dislocation nucleation process close to a diffusion wedge. This led to the definition of a critical stress intensity factor for parallel glide dislocation nucleation. The critical value for dislocation nucleation from a diffusion wedge is twice as large compared to a crack. This was explained by the difference in force balance on the incipient dislocation: For dislocation nucleation near a diffusion wedge, a dislocation dipole needs to be generated where the dislocations are one Burgers vectors apart. In the case of a crack, the incipient dislocation senses the image force of the surface corresponding to a “virtual” dislocation dipole where the dislocations are two Burgers vectors apart (see Fig. 8.21 for a schematic visualization of these considerations). The atomic simulation results support this theoretical model since the ratio of critical stress intensity factor of a diffusion wedge to a crack is found to be around two. As discussed in Sect. 8.3.4, a crack and a diffusion wedge have major differences in the timescale associated with creation of dislocations. A crack is a ready source for dislocations, while a diffusion wedge has an intrinsic characteristic time associated with dislocation climb. In the long-time limit on the order of a characteristic time τ , the diffusion wedge behaves as a crack in agreement with theoretical considerations [46]. The change of maximum resolved shear stress due to climb of edge dislocations into the grain boundary is schematically visualized in Fig. 8.11. Atomistic studies of polycrystalline thin films helped to clarify the nucleation mechanisms of dislocations from different types of grain boundaries. We find that low-energy grain boundaries provide more fertile sources for dislocations than high-energy grain boundaries. This concept helps to explain

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why the dislocation density is several times higher in grains neighboring to low-energy grain boundaries in the simulations. We observe that mostly partial dislocations are nucleated from the grain boundaries in nanostructured thin films. This contradicts the classical theories of deformation [38] where it is predicted that complete dislocations dominate, but it is in agreement with other studies of deformation of nanostructured bulk materials [153, 154, 304, 328, 332]. A detailed investigation of the dislocation nucleation process from low-energy and high-energy grain boundaries provided insight into the atomic mechanisms of this process. One of the important observations is that dislocation half-loops are generated on different glide planes causing formation of sessile jogs in the dislocation line that generate point defects as they move. Additional simulations of constrained grain boundary diffusion in polycrystalline films suggested that there is a strong dependence of the diffusivities on the grain boundary structure. We could also show that parallel glide dislocations are nucleated along grain boundaries with highest diffusivities. This result shows that diffusion and nucleation of parallel glide dislocations are highly coupled, thus supporting the theoretical understanding of the process [46].

8.6 Use of Atomistic Simulation Results in Hierarchical Multiscale Modeling Classical molecular dynamics is rather constrained with respect to the accessible timescale. One of the drawbacks is that parameter studies with varying film thickness are difficult, if not impossible to carry out with today’s computers (maximum film thicknesses that can be reached are on the order of 50–100 nm). Therefore, hierarchical multiscale modeling may serve as a useful tool to reach higher lengthscales and timescales. Such studies are now being frequently applied in materials modeling [179, 180, 348]. Here we briefly describe how a hierarchical coupling of molecular dynamics with mesoscopic simulations was achieved and review the results of a mesoscopic study that was carried out based on the molecular dynamics results discussed in this book. Mesoscopic methods must rely on phenomenological input parameters or rules. The most important contribution by the molecular dynamics simulations was the concept of a critical stress intensity factor which could be translated into a discrete dislocation formulation of diffusional creep [50]. The mesoscopic model reported in [50] follows the well-known discrete dislocation models in two and three dimension described in the literature (see for example [96, 181, 182, 348] for thin film plasticity). In such models, dislocations are considered sources of stress and strain in a linear elastic continuum. The proposed discrete dislocation model for diffusional creep in ultra-thin films proved to be capable of predicting experimentally measurable quantities like the flow stress [50]. The two-dimensional model suggests the existence of


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Fig. 8.49 Flow stress σY vs. the film thickness hf , as obtained from mesoscopic simulations of constrained diffusional creep and parallel glide dislocation nucleation (data taken from [50]). The results are shown for two different initiation criteria for diffusion and a film-dependent source. In the case of a local criterion for diffusion initiation, the yield stress is film-thickness independent as also observed in experiment [51]

a threshold stress for grain boundary diffusion [50]. The investigation of different conditions for nucleation of climb and glide dislocations, as well as their interaction with grain boundaries, suggests that the diffusion threshold stress should only depend on the strain in the top layer of the film, and thus be independent of the film thickness. This gives rise to a thickness-independent flow stress for ultra-thin films, in good agreement with the relevant experimental results [51]. An important point is that only a local criterion for initiation of diffusion leads to a film-thickness-independent yield stress [50, 51]. If the source for diffusion initiation is chosen film-thickness dependent, the yield stress is also a function of the film thickness. Since the yield stress is found independent of the film thickness in experiment [51], there is reason to believe that the assertion of a local criterion for diffusion initiation is correct. This is in contrast to the nucleation criterion for parallel glide dislocations, which is a global criterion dependent on the film thickness. The results for the yield stress obtained from mesoscopic simulation are summarized in Fig. 8.49 [50]. The yield stress as a function of film thickness is shown for two different initiation criteria for diffusion (constant source and therefore local criterion, and film-thicknessdependent source). In the case of a local criterion for diffusion initiation, the yield stress is film-thickness independent as observed in experiment [51]. The yield stress increases slightly for very thin films. The results obtained by this hierarchical multiscale simulation method illustrate the usefulness of the atomistic approach and its possible transferability

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to other materials phenomena for which fully atomistic simulations are not yet feasible. 8.6.1 Mechanisms of Plastic Deformation of Ultra-thin Uncapped Copper Films The results of atomistic simulations, experiments, continuum modeling as well as mesoscopic modeling have advanced to a level that they allow drawing general conclusions about the deformation mechanism in ultra-thin films. Here we summarize the main results in a deformation map of thin films. The objective of this is to provide a clear overview over the different deformation mechanisms in ultra-thin films. 8.6.2 Deformation Map of Thin Films The results from the numerical modeling reviewed here and in [50] together with experimental findings reported by different authors [51] allow us to qualitatively describe different deformation mechanisms that occur in thin films in the submicron regime. We propose that there exist four different deformation regimes. These are: • • • •

Regime A: Deformation with threading dislocations Regime B: Constrained diffusional creep with subsequent parallel glide Regime C: Constrained diffusional creep without parallel glide Regime D: No stress relaxation mechanism with no diffusion and no dislocation motion

A schematic “deformation map” is plotted in Fig. 8.50. This plot shows the critical applied stress to initiate different mechanisms of deformation as a function of the film thickness. We assume that the loading is applied very slowly and the temperature is sufficiently high such that diffusive processes are generally admitted. The critical applied stress to nucleate threading dislocations scales with 1/hf [71–73]. We note that the 1/hf -scaling has been found in two-dimensional molecular dynamics simulations [361] recently. Two-dimensional mesoscopic studies [182] revealed qualitatively that the flow stress increases with decreasing film thickness. For films thicker than a material-dependent value, regime (A) is the dominating deformation mechanism. For thinner films, the stress necessary to nucleate threading dislocations must be assumed larger than the stress to initiate grain boundary diffusion. In this regime (B), diffusion dominates stress relaxation and causes a plateau in the flow stress as shown by the discrete dislocation modeling. Parallel glide helps to maintain grain boundary diffusion until the overall stress level is below the diffusion threshold which is independent of the film


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Fig. 8.50 Deformation map of thin films, different regimes. Thin films with thicknesses in the submicron regime feature several novel mechanisms next to the deformation by threading dislocations (a). Plasticity can be dominated by diffusional creep and parallel glide dislocations (b), purely diffusional creep (c), and no stress relaxation mechanism (d)

thickness. For yet thinner films, grain boundary diffusion stops before a sufficient stress concentration to trigger parallel slip is obtained as suggested by our molecular dynamics simulations. The onset of regime (C) can be described with the scaling of the critical nucleation stress for parallel glide with 1/hsf (s ≈ 0.5). In this regime, the flow stress increases again for smaller films, due to the back stress of the climb dislocations in the grain boundary, effectively stopping further grain boundary diffusion. If the applied stress is lower than the critical stress for diffusion, no stress relaxation mechanism is possible. This is referred to as regime (D). The critical film thickness of 25 nm is estimated based on the PG from molecular dynamics simulations. result for the critical Kdw The investigations of ultra-thin films show the richness of phenomena that occur, as the dimensions of materials are shrunk to nanometer scale. For tomorrow’s engineers, such knowledge may be the key to successful design. 8.6.3 Yield Stress in Ultra-Thin Copper Films The yield stress of thin films resulting from these considerations is summarized in Fig. 8.51 for different film thicknesses. For thicker films, the strength increases inversely proportional to the film thickness as has been shown in many theoretical and experimental studies [51,72,96,343]. If the films thickness is small enough such that grain boundary diffusion and parallel glide are the prevailing deformation mechanisms, the film strength is essentially independent of hf , as shown by the discrete dislocation model (reviewed in Sect. 8.6)

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Fig. yield yield glide


8.51 Deformation mechanism map of thin copper films, here focused on the stress. For films in the submicron regime (thinner than about 400 nm), the stress shows a plateau. This is the regime where diffusional creep and parallel dislocations dominate (regime (B) in Fig. 8.50)

and seen in experiment [51]. However, for films thinner than hf ≈ 25 nm, the modeling predicts an increase in strength with decreasing film thickness (see also Fig. 8.49). In Fig. 8.51, the film thickness of hf ≈ 400 nm below which the yield stress remains constant, as well as the plateau yield stress of 0.64 GPa are taken from experimental results of copper thin films [51, 346]. 8.6.4 The Role of Interfaces and Geometric Confinement The studies show that interface properties and geometric confinement can govern the deformation mechanisms in thin films. Important interfaces in thin films are: • The film surface • The grain boundary between two neighboring grains • The interface of film and substrate (geometrical constraint) In the following paragraphs, we will discuss the role of these different interfaces and constraints on the mechanical behavior. Film Surface The film surface is important since it allows that atoms diffuse along the surface into the grain boundary. As shown in [48], the slower of the processes surface or grain boundary diffusion controls the dynamics of constrained diffusional creep.


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Grain Boundary Structure The governing character of the grain boundary structure in thin films is found either when deformation is mediated by diffusional creep or by dislocation motion: As discussed in [68], the grain boundary structure has a significant influence on the diffusivities, and therefore determines how fast the tractions along the grain boundaries are relaxed and a singular stress field develops. This is also shown in Fig. 8.47 where the depth of grain boundary grooves is deepest at high-energy grain boundaries corresponding to the fastest diffusion paths. Another indication of this is that high-energy grain boundaries lead to more pronounced surface grooves than low-energy grain boundaries when grain boundary diffusion is active. If deformation is carried by nucleation and motion of dislocations, the structure of the grain boundaries also has a significant influence on the details of deformation: Low-energy grain boundaries composed of arrays of misfit dislocations are more fertile sources for dislocation nucleation than homogeneous high-energy grain boundaries. On the other hand, motion of parallel glide dislocations through grains may be hindered due to pinning of dislocations when such a inhomogeneous grain boundary structure is present (see discussion in Sect. 8.5.2). Geometrical Constraints The geometrical constraint of no sliding at the interface of film and surface is the reason for the singular stress field to develop around the diffusion wedge and is therefore responsible for the occurrence of parallel glide dislocations [46]. The geometrical constraint imposed by the grain size strongly influences the dislocation network that develops inside the grain. In very small grains of a few tens of nanometers, only one or two dislocations fit into a grain. In larger grains of several hundred nanometers, a much larger number of dislocations fit into each grain and may form a more complicated network (see Fig. 8.43). Similar considerations apply to the Hall–Petch hardening [1]. Deformation Mechanisms of Small-Scale Materials The dominance of grain boundary processes during deformation of ultra-thin films is in qualitative agreement with recent investigations of other small-scale materials, such as nanostructured materials [303]. The preliminary study on nanostructured materials discussed in Sect. 8.5.3 showed that an intergranular nanosubstructure constituted by twin lamellas could play an important role in effectively strengthening materials. Since twin grain boundaries are relatively poor diffusion paths (since they are low-energy grain boundaries), such materials could potentially be successfully employed at elevated temperatures where “usual” materials with ultra-fine grains cannot be utilized since creep becomes the dominant deformation mechanism.

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The study supports the notion that geometric confinement has strong impact on the deformation, and could potentially be utilized to create materials with superior mechanical properties.

8.7 Deformation and Fracture Mechanics of Carbon Nanotubes Carbon nanotubes (CNTs) constitute a prominent example of nanomaterials, with many potential applications that could take advantage of their unique mechanical, electrical, and optical properties [363–368]. A fundamental understanding of the properties of individual CNTs, or assemblies of CNTs in bundles or nanopillars, or in conjunction with biological molecules such as DNA may be critical to enable new technologies and to engineer CNT-based devices. A series of different views of a single wall carbon nanotube (SWNT) is shown in Fig. 8.52.

Fig. 8.52 Geometry of a (15,15) single wall carbon nanotube (SWNT) shown in different views

The mechanical properties of CNTs are particularly important in many technological applications. This includes cases in which the primary role of CNTs is not related to their mechanical properties. Nevertheless, a thorough understanding of the mechanical properties is essential to design manufacturing processes or to ensure reliability during operation of devices. The interactions of individual CNTs can play a critical role in application and during fabrication processes, and may pose significant challenges compared with macroscopic classical engineering applications. This is because at nanoscale,


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Fig. 8.53 Compressive deformation mechanics of a CNT with length L = 6 nm and a diameter of d = 1 nm ((7,7) armchair CNT). The plot illustrates the increase of strain energy as a function of compressive strain (subplot (a)), and shows associated deformation mechanisms (subplot (b)). The analysis revealed that CNTs begin to buckle according to a shell-like behavior. Subplot (c) depicts a similar analysis, showing bending of a (13,0) CNT (length L = 8 nm and a diameter of d = 1 nm). The strain energy density increases according to a harmonic behavior until the buckling point is reached. Reprinted from: B.I. Yakobson, C.J. Brabec, and J. Bernholc, Physical Review Letters, Vol. 76(14), 1996, pp. 2511–2514. Copyright c 1996 by the American Physical Society

weak dispersive van der Waals interactions (vdW) play a more prominent role, and often govern the mechanics or self-assembly dynamics of those materials. The interplay of such adhesive forces with covalent bonding within CNTs is not well understood for many CNT systems. The mechanics of carbon nanotubes has been discussed in various articles published over the last decade, both from a continuum and atomistic perspective [369]. In a classical article by Yakobsen and coworkers [369], the behavior of single, free-standing SWNTs under compressive loading was investigated using classical, molecular dynamics with an empirical Tersoff–Brenner potential. Yakobson and his co-authors developed a continuum shell model to describe the buckling modes of the CNTs. Figure 8.53 plots some of the results. Figure 8.53a shows the strain energy as a function of compressive strain. Associated snapshots of the CNT geometry are shown in Fig. 8.53b, illustrating a link between the kinks in the energy plot and atomistic deformation mechanisms. Figure 8.53c shows a similar analysis for bending deformation of a CNT. Atomistic simulations of carbon nanotubes have led to significant insight into their mechanical behavior, as they have confirmed a very high Young’s

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modulus (approximately 1 TPa) and a very high strength (exceeding 40– 50 GPa).

Fig. 8.54 Shell-rod wire transition of CNTs under compressive loading (schematically shown in subplot (a)), subplot (b) represents shell-like behavior, subplot (c) shows the behavior of CNTs as a rod, and subplot (d) shows a CNT that behaves similarly as a wire (or a long polymer monomer) [370, 371]. The series of plots illustrates the change in compressive behavior as the length of the CNT increases systematically

This section is kept very brief and we refer the reader to the literature for additional information. We only highlight a few results of simulation studies of CNTs and assemblies of CNTs, in particular focused on assembly and mechanical properties. Figure 8.54 depicts a shell-rod wire transition of CNTs under compressive loading. Figure 8.54a represents the shell-like behavior, Fig. 8.54b shows the behavior of CNTs as a rod, and Fig. 8.54c shows a CNT that behaves similarly as a wire [370, 371]. The series of plots illustrates the change in compressive behavior as the length of the CNT increases systematically. This result


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illustrates that the mechanical properties of CNTs are strongly lengthscale dependent.

Fig. 8.55 Illustration of the process of coarse graining a CNT, leading to a bead-type representation [52]

8.7.1 Mesoscale Modeling of CNT Bundles Atomistic modeling is typically limited to one or few CNTs with moderate lengths. In order to reach longer timescale and lengthscale in simulation, mesoscale methods have been developed that describe the behavior of CNTs at a larger, coarse-grained scale [52]. Figure 8.55 reviews the process of coarse-graining a CNT, leading to a bead-type representation. Many different approaches in coarse-graining CNTs have been described, and the bead-type model is only one of many possible approaches. We briefly review the details of this molecular mesocopic formulation. The total energy of the system is expressed as U = UT + UB + Uweak ,

(8.16)

where UT is the energy stored in chemical bonds due to stretching along the axial direction, UB is the energy due to bending of the CNT, and Uweak constitute weak (vdW) interactions. The total energy contribution of each

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part is given by the sum over all pair-wise and triple (angular) interactions in the system, thus φI (r) (8.17) UT;weak = pairs

for the tensile and weak interactions (both summed pair-wise), and φbend (ϕ), UB =

(8.18)

angles

summed over all triples of particles in the systems. The bending energy of one triplet is given by a simple harmonic potential φbend (ϕ) =

1 kbend(ϕijk − ϕ0 )2 , 2

(8.19)

with kbend as the spring constant relating to the bending stiffness and ϕijk as the bending angle between three particles i, j, and k (the angle ϕ0 is the equilibrium angle). This expression is based on the harmonic angle term given in (2.37), as it is used in the CHARMM potential, for instance. Calculation of the bending angle requires consideration of the position of three atoms and the molecular potential is thus a three-body potential. The nonlinear stress–strain behavior under tensile loading with a bilinear model (similar to the one that has been used successfully in the studies of fracture as discussed above): The biharmonic potential is defined as (0) k (r − r0 ) if r < r1 , dφT (r) = H(rbreak − r) T (8.20) (1) dr kT (r − r˜1 ) if r ≥ r1 . (0)

(1)

In this equation, kT and kT stand for the small- and large-deformation (0) (1) spring constants. The parameter r˜1 = rq − kT /kT (r1 − r0 ) is obtained from force continuity conditions. This model is chosen to reproduce the nonlinear elastic and fracture behavior of carbon nanotubes. The availability of two spring constants enables modeling changes in the elastic properties due to increasing deformation. The Heaviside function enables us to describe the drop of forces to zero at the initiation of fracture of the carbon nanotube. We assume weak, dispersive interactions between either different parts of each molecule or different molecules, defined by a Lennard-Jones 12-6 (LJ) potential. The LJ potential has been shown to be a good model for the dispersive interactions between CNTs. With these three potentials, all terms in (8.16) are defined. The mass of each bead is determined by assuming a homogeneous distribution of mass in the molecular model. The equilibrium distance between mesoscale particles is chosen to be r0 = 10 ˚ A, based on the requirement that it is much smaller than the persistence length. All other parameters are chosen based on the results of full atomistic simulations (see, for instance Fig. 8.56 for


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Fig. 8.56 Atomistic analysis of the tensile properties of a (5,5) CNT [52]. Subplot (a): Stress vs. strain behavior during stretching of a (5,5) CNT, result obtained using the Tersoff potential. The straight lines show the mesoscale model that is developed based on the atomistic simulation results. Subplot (b): Fracture mechanism of a (5,5) CNT, modeled using the Tersoff potential (plots show the atomistic mechanics as the lateral tensile strain is increased). Fracture initiates by generation of localized shear defects in the hexagonal lattice of the CNT, reminiscent of 5–7 Stone–Wales defects Variable Equilibrium bead distance r0 (˚ A) Bead particle mass m0 (amu) (0) A−2 ) Tensile stiffness parameter kT (kcal mol−1 ˚ (1) −1 ˚−2 Tensile stiffness parameter kT (kcal mol A ) Hyperelastic parameter (˚ A) Fracture parameter (bond rupture distance) (˚ A) Equilibrium angle (degrees) A−2 ) Bending stiffness parameter kbend (kcal mol−1 ˚ Dispersive parameter (kcal mol−1 ) Dispersive parameter (˚ A)

Numerical value 10 1,953 1,000 700 10.5 13.2 180 14,300 15.1 9.35

Table 8.3 Summary of mesoscopic parameters derived from atomistic modeling [52]

an analysis of tensile deformation). For instance, the dispersive interactions between CNTs are determined to reproduce the adhesion energy of CNTs as obtained in a full atomistic representation. The tensile spring constant is determined from atomistic calculations of stretch vs. deformation, consider(0) (1) ing the small- and large-deformation regime to determine kT and kT . The bending spring parameter is determined based on the results of the bending stiffness of a single CNT. Table 8.3 summarizes all parameters used in the mesoscopic model.

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Fig. 8.57 Bundle of several individual SWNTs as obtained from a molecular simulation model [52]

8.7.2 Mesoscale Simulation Results CNTs usually assembly in bundles as shown in Fig. 8.57. Large-aspect ratio CNTs are extremely flexible, and can be deformed into almost arbitrary shapes with relatively small energetic effort. Figure 8.58 depicts the response of a CNT bundle to compressive mechanical load. The mesoscale model enables the simulation of such large assemblies of CNTs, which would otherwise not be accessible to atomistic simulation.

Fig. 8.58 Response of a CNT bundle to mechanical compressive loading. Even for relatively small strains, the structure starts to buckle, eventually leading to significantly deformed and buckled shapes [52]

If different parts of the tube come sufficiently close, these attractive forces should also be present and can form self-folded structures of CNTs. Such self-folded structures of CNTs with extremely large aspect ratio were first observed in molecular dynamics simulations of (5,5) CNTs using a combined Tersoff–LJ model. The results from such atomistic and corresponding mesoscale simulations are shown in Fig. 8.59 [370–372]. The stability of the folded structure is governed by the balance of energy required to bend the CNT and energy gained by formation of weak vdW


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“bonds.” Therefore there exists a minimum critical length of the CNT (dependent on the bending stiffness and the adhesion parameter) at which such a structure becomes energetically stable. It was proposed that the critical CNT length (denoted by Lχ ) scales as Lχ ∼

EI . γ

(8.21)

The critical length scale was estimated to approximately 79 nm for a (5,5) CNT in vacuum.

Fig. 8.59 Full atomistic calculations of the properties of ultra-long CNTs (subplot (a)) [371, 372] and corresponding results obtained using the mesoscale model (subplot (b)) [52]

8.7.3 Discussion The presentation of atomistic simulation CNTs was extremely brief and by no means comprehensive. However, it provides a small glimpse into this exciting new field. Further, the development of the mesoscale model has illustrated how atomistic models (in particular the “molecular dynamics” algorithm) can be immediately used to define coarse grained models, as shown here for a bead model [52]. This simple mesoscopic model is particularly suited to describe the mechanical properties and self-assembly mechanisms of CNTs with ultra-large aspect ratio. The mesoscopic model can reach timescales and lengthscales not accessible to the full atomistic model, but still includes information about the fracture mechanics of individual CNTs via the inclusion of appropriate terms in the mesoscale model formulation. The mesoscopic model discussed

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here can be straightforwardly implemented for other CNTs than those studied here, including multiwall CNTs.

8.8 Flaw-Tolerant Nanomaterials: Bulk Fracture and Deformation The material discussed in the previous sections have clearly shown that once the characteristic size of materials reaches nanoscale, the mechanical properties may change drastically and classical mechanisms of materials failure cease to hold. In the following two sections, we focus on joint atomisticcontinuum studies of failure and deformation of nanoscale materials, focusing on a particular phenomenon referred to as “flaw-tolerance” [54, 373, 374]. In particular, we discuss the size dependence of brittle fracture and the effect of characteristic dimensions on adhesion. It will be illustrated that if the characteristic dimension of a material is below a critical lengthscale that can be on the order of several nanometers, the classical Griffith theory of fracture no longer holds. An important consequence of this finding is that materials with nanosubstructures may become flaw-tolerant, as the stress concentration at crack tips disappears and failure always occurs at the theoretical strength of materials, regardless of defects. The atomistic simulations reviewed here complement recent continuum analyses [54]. Both approaches reveal a smooth transition between Griffith modes of failure via crack propagation to uniform bond rupture at theoretical strength below a nanometer critical length. The results may be important for understanding failure of many brittle nanoscale materials, in particular biological structures. Biological materials typically feature nanoscopic features as a most fundamental building block. It has been suggested that in light of the flaw-tolerance concept, scale reduction may be a common design principle found in biological materials to create structural links leading to robust architectures. The analysis reviewed here is also interesting as another illustration of how continuum mechanics theory (here linear elastic fracture mechanics) can be coupled with atomistic simulation approaches to study fracture and adhesion of nanostructures [375]. 8.8.1 Strength of Brittle Nanoparticles In this section, we focus on fracture properties of ultra-small brittle particles and the impact of size variation on fracture properties. The goal of these studies is to understand the limiting cases for the validity of Griffith’s theory of fracture. In addition to general interest to understand if Griffith’s theory can be applied to brittle fracture at ultra-small scales, this study is motivated by the fact that in bone, mineral platelets appearing at ultra-small nanoscale dimensions seem to play an integral role in the load transfer process. Thus, their properties may have implications on the strength


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Fig. 8.60 Bone-like materials consist of a hierarchical microstructure made of nanoscale constituents [53, 54]. Left: The plot depicts the microstructure of such bone-like biological materials at the smallest scale. Such materials typically consist of fragile, brittle mineral platelets embedded in protein matrix materials as soft as human skin. The combination of these two phases in a nanocomposite results in superior materials properties. In the studies, we focus on the fracture properties of mineral platelets since they play a critical role in determining the strength of these materials. Right: The tension–shear chain model showing the path of load transfer in the mineral–protein composites. The mineral platelets carry tensile load and the protein transfers loads between the platelets via shear. In this section we focus on the strength of the mineral platelets

of bone. As illustrated in Fig. 8.60 (right), the mineral platelets are critical for the integrity of the material since they carry the tensile load. We thus focus on the fracture strength of the brittle platelets. The strength of a small mineral particle with a crack under mode I tensile loading as shown schematically in Fig. 8.61 is considered here. We assume that classical continuum theory of fracture can be applied to describe this material. From classical fracture mechanics, the critical stress for crack nucleation in this perfectly brittle material is given by the Griffith condition G = 2γ, where γ is the fracture surface energy and G is the energy release rate. For the strip geometry as shown in Fig. 8.61, with strip width ξ, the energy release rate can be expressed as σ 2 ξ(1 − ν 2 ) G= , (8.22) 2E where E is Young’s modulus, ν is Poisson ratio, and σ is the applied stress. At the critical point of onset of crack motion, the energy released per unit length of crack growth must equal the energy necessary to create a unit length of new surface. Using the Griffith condition, (8.22) can be solved for the critical

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Fig. 8.61 The geometry and dimensions of a cracked platelet. This model is used in the continuum and atomistic studies of fracture at small scales. We consider a thin strip of width ξ, in which the crack length extends half way through the length of the slab in the x-direction. The system is under mode I tensile loading as indicated in the plot (mode I loading in the y-direction)

applied stress

σ=

4γE ξ(1 − ν 2 )

(8.23)

for spontaneous onset of failure. Theoretical Considerations: Breakdown of Griffith Continuum Theory For decreasing layer width ξ, (8.23) predicts an increasing stress for nucleation of the crack, approaching infinity as ξ goes to zero. This, however, cannot be literally accepted, since the stress cannot exceed the theoretical strength of the material, which is denoted by σth . This immediately yields a critical layer width ξcr below which fracture cannot be described by the Griffith theories any more. Instead, the strength of the cracked slab is given by the theoretical strength of the material, regardless of the presence of a crack. This critical length can be calculated to be ξcr =

4γE 2 (1 − ν 2 ) . σth

(8.24)

Similar expressions for critical lengthscales can be derived for a variety of different geometries, and also for the case of ductile material behavior [45]. These considerations have led to the question if the continuum theory based on the Griffith concept is still applicable at very small scales. Now we use atomistic simulation as a tool to gain further insight.


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Atomistic Modeling Atomistic modeling of the strip crack problem is conducted by classical molecular dynamics simulations, utilizing a global energy minimization scheme. A three-dimensional model is used to study crack nucleation and propagation. Consider the geometry depicted in Fig. 8.61. The initial crack extends over half of the slab in the x-direction. The slab size in the x-direction is several times larger than that in the y-direction. The model is built using a FCC crystal oriented in cubic orientations, with x = [100], y = [010], and z = [001]. The crystal is periodic in the z-direction with crack faces along (010)-planes. We use the concept of virtual atom types to distinguish various atomic interactions and to allow application of boundary conditions. Atoms in the boundary region (as shown in Fig. 8.61) are assigned a specific virtual atom type and are displaced according to a prescribed displacement field. We use an energy minimization scheme to relax the crystal after each increment of loading. An increment of strain of magnitude ∆εyy = 0.001 is applied every 3,000 integration steps. Different loading rates are chosen to assure that the results have reached equilibrium before the next loading increment is applied. The loading is constant along the x-direction. As reviewed in Sect. 2.6, interatomic potentials for a variety of different brittle materials exist, many of which are derived from first principles. However, it is difficult to identify generic relationships between potential parameters and macroscopic observables when using such complicated potentials. Furthermore, in many cases the potential parameters do not have immediate physical meaning for the bonding between atoms. Here we use an alternative approach based on model potentials describing the behavior of model materials. To investigate universal scaling behavior between microscopic and macroscopic variables, this has been shown to be very fruitful and allows fundamental insight, in particular into the fracture mechanisms, as illustrated in Chap. 6. By using simple model potentials, the complexities with sophisticated potential formulations can be deliberately avoided. Based on this concept, a simple pair potential based on a harmonic interatomic potential with spring constant k0 in combination with a Lennard-Jones potential is used to describe smooth bond breaking. The harmonic potential is chosen to model linear elastic material behavior as assumed in Griffith’s fracture theory. This allows one to define a well-described reference system. It is important to note that although simple pair potentials do not allow drawing conclusions for unique phenomena pertaining to specific materials, they enable one to understand universal, generic relationships between potential shape and brittle fracture mechanics and adhesion properties of materials, and help elucidate universal scaling laws of fracture mechanics. To model a perfectly brittle solid, harmonic interactions are assumed in the bulk of the strip. Atoms in the bulk only interact with their nearest neighbor, and bonds never break. Crack propagation is constrained along a weak fracture

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layer in the center of the strip. To model bond breaking, we assume that atoms across this weak layer interact according to the 12-6 Lennard-Jones (LJ) potential (2.30). In the simulations, it is assumed that σ = = 1 and r0 = 1.12246 (FCC lattice constant a ≈ 1.587). Interactions across the weak fracture layer (LJ potential) are cut off at a critical distance rcut = 2.5. Bonds only break between atoms that are located at different sides of the weak layer. This procedure ensures that the crack can only extend along a predefined direction. For the analysis of the critical length for flaw tolerance, exact knowledge of elastic properties and fracture surface energy is needed. It can be shown that Young’s modulus is 4r2 E = 0 k0 (8.25) 3 and Poisson’s ratio is given by ν = 1/3. The surface energy is given by the expression reviewed in Sect. 4.4.4. It is emphasized that this setup of bulk material and a weak layer is particular convenient for the studies because Young’s modulus E can be easily varied independent of the other variables (γ and σth ), allowing the critical lengthscale ξcr to be tuned in a range easily accessible to the molecular dynamics simulations. All simulation results are expressed in reduced units: Energies are scaled by the depth of the LJ potential and lengths are scaled by σ. In these reduced units, the critical length scale is ξcr ≈ 115 for the material parameters chosen in the simulation (this corresponds to a few tens of nanometers in real materials). A critical element of the studies is calculation of stresses from atomistic simulations. The atomic stress is calculated based on the virial theorem as reviewed in Sect. 2.8.6. As illustrated in the studies reviewed in Sect. 6.5, the atomistic definitions of stress near a moving crack tip show reasonable agreement with continuum mechanics predictions. Thus this approach is reasonable in performing a direct comparison with continuum mechanical theories. 8.8.2 Simulation Results The study begins with carrying out computational experiments of studies of the fracture strength of thin layers depending on their size ξ. In a series of studies, we calculate the critical fracture stress σ as a function of the material size (layer width ξ). Figure 8.62 shows the results of large-scale atomistic simulations of fracture strength of small perfectly brittle platelet as a function of the inverse of the square root of the size of the material characterized by ξcr /ξ. In the plot, the predictions of Griffith’s theory and the theoretical strength of the material are both included. The fracture stress σ is normalized with respect to the maximum strength at the onset of failure, σth .


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Fig. 8.62 Fracture and adhesion strength as a function of the size of the material. The plot depicts the results of bulk fracture as well as surface adhesion. The results are normalized with respect to the theoretical strength and normalized with respect to the critical lengthscale for flaw tolerance. These results suggest that the principle of dimension reduction is valid in a variety of systems, including surface adhesion as well as bulk fracture

Clearly, the brittle fracture mechanism shows a strong size dependence. Whereas the strength of the materials is predicted well based on Griffith’s theories for large dimensions ( ξcr /ξ < 1), reduction of dimension results in deviation from this prediction, and eventually failure of the material at σcr regardless of the presence of flaws (for ξcr /ξ > 1). This observation suggests a change in behavior once the dimensions of the solid are below the critical length scale: The Griffith theory is not valid for extremely thin layer widths. Now the focus is on the stress distribution ahead of the crack slightly before failure occurs. In macroscopic systems and according to the classical understanding, it is expected that the crack tip always constitutes a location of high stress concentration, according to the inverse square root scaling discussed in Sect. 6.2.2. Is this also true when the characteristic dimensions of materials reach nanoscale? Calculation of the stress distribution ahead of the crack reveals that the stress becomes increasingly homogeneous as the lengthscale of the structure reduces. In fact, it is observed that the distribution becomes completely uniform for structural dimensions well below the critical lengthscale. This is in strong contrast to what we would expect based on a completely classical picture. Further, we observe that the stress distribution does not change with structural size any more once the size of the material is below a critical value (this is observed for ξcr /ξ > 2.67 in the simulations). The results are shown in Fig. 8.63. This size-independence of the fracture strength of materials is also in clear contrast to the conventional knowledge. The results suggest that at nanoscale, materials may behave dramatically differently.

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Fig. 8.63 Stress distribution ahead of the crack in a thin mineral platelet just before failure, for different materials sizes (the x-coordinate is scaled by the characteristic length scale ξcr ). The thinner the slab, the more homogeneous is the stress distribution. When the slab width is smaller than the critical size, the stress distribution becomes homogeneous and does not depend on the size of the platelet any more (see values ξcr /ξ > 2.67 and larger). The normal stress σyy is normalized with respect to the maximum strength at the onset of failure, σth

It is noted that a similar behavior as observed here for tensile loading is also expected for shear loading, as the underlying equations take a quite similar form.

8.9 Nanoscale Adhesion Systems Similar concepts as discussed in the last section also apply to adhesion systems, as reported first in [376]. This application provides another opportunities to link atomistic methods with theories developed within the continuum assumption of engineering mechanics [375, 377]. The focus of the discussion presented here is on the size dependence of cylindrical adhesion systems. It will be demonstrated that optimal adhesion can be achieved by either lengthscale reduction or by optimization of the shape of the surface of the adhesion element. This illustrates two paths to increase the efficacy of adhesion systems. It is found that change in shape can lead to optimal adhesion strength; those systems are not robust against small deviations from the optimal shape. In contrast, reducing the dimensions of the adhesion system results in robust adhesion devices that fail at their theoretical strength, regardless of the presence of flaws. An important consequence of this finding is that even under presence of surface roughness, optimal adhesion is possible provided the size of contact elements is sufficiently small. The


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Fig. 8.64 Microscopic view of the contact elements in flies (left) and geckos (right). The terminal setae of flies consist of insect cuticle, that is, chitin–fiber reinforced protein, and have typical dimensions of 2 µm. The terminal elements (“spatulae”) of geckos are made of keratin and have typical dimensions of 200-nm diameter [55] (micrographs courtesy of S. Gorb, Max Planck Institute for Metals Research). Reprinted from Materials Science and Engineering: C, Vol. 26, E. Arzt, Biological and artificial attachment devices: Lessons for materials scientists from flies and c 2006, with permission from Elsevier geckos, pp. 1245–1250, copyright

atomistic results corroborate earlier theoretical modeling at the continuum scale [376]. The section concludes with a discussion of the relevance of the studies with respect to the biological architecture of bone nanostructures and nanoscale adhesion elements in Geckos. 8.9.1 Strength of Fibrillar Adhesion Systems In this section, the focus is on fibrillar adhesion structures as they appear in many biological systems such as geckos. Figure 8.64 depicts a microscopic view of flies and gecko attachment systems, illustrating that they are made up of a large number of ultra-small fibrillar adhesion elements [55, 378]. To understand adhesion properties at small scales, we have modeled an elastic flat-ended cylindrical hair in adhesive contact with a rigid substrate. The radius of the cylinder is R. To test the ability of the flat cylinder to adhere in the presence of adhesive flaws, imperfect contact between the spatula and substrate is assumed such that the radius of the actual contact area is a = αR, where 0 < α < 1, as shown in Fig. 8.65(left), and the outer rim αR < r < R represents flaws or regions of poor adhesion. In the present model, which is similar to continuum models of this situation [376, 378], there is no adhesion in the region αR < r < R, and so the model resembles a cylinder attached to the substrate with a circumferential crack. The adhesive strength of such an adhesive joint can be calculated by treating the contact problem as a circumferentially cracked cylinder, in which case

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Fig. 8.65 The schematic of the model used for studies of adhesion. The model represents a cylindrical Gecko spatula (as shown in Fig. 8.64) with radius attached to a rigid substrate (left). A circumferential crack represents flaws for example resulting from surface roughness. The parameter α denotes the dimension of the crack. The regime 0 < r < αR corresponds to an area of perfect adhesion, whereas αR < r < R represents regions of no adhesion. This model resembles the effect of surface roughness as depicted schematically on the right-hand side

the stress field near the edge of the contact area has a square-root singularity with stress intensity factor KI =

P √ πaF1 (α), πa2

(8.26)

where F1 (α) varies narrowly between 0.4 and 0.5 for 0 < α < 0.8 (α = 1 corresponds to a perfect, defect-free contact). Equation (8.26) is substituted into the Griffith condition: KI2 = ∆γ (8.27) 2E ∗ where the factor two is due to the rigid substrate and ∆γ is the adhesion energy (corresponding to two times the surface energy). The apparent adhesive strength normalized by the theoretical strength for adhesion, Pc πσth R2

(8.28)

σ ˆc = βα2 ψ,

(8.29)

σ ˆc = is obtained as where

ψ=

∆γE ∗ 2 Rσth

(8.30)


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Fig. 8.66 The geometry of the system considered is a periodic array of punches of radius R. The rigid–elastic interface leads to singular stress concentrations for flat punches. We vary the shape of the rigid punch surfaces to avoid these singular stress concentrations

and

β=

2 παF12 (α)

(8.31)

E 1 − ν2

(8.32)

as well as E∗ =

where E is Young’s modulus. The adhesive strength is a linear function of the dimensionless variable ψ with slope βα2 . The maximum adhesion strength is achieved when the pull-off ˆc = α2 , in which case the traction within the force reaches Pc = σth a2 , or σ contact area uniformly reaches the theoretical strength σth . This saturation in strength occurs at a critical size of the contact area Rcr = β 2

∆γE ∗ . 2 σth

(8.33)

This lengthscale corresponds to ξcr described above. 8.9.2 Theoretical Considerations of Shape Optimization of Adhesion Elements The results in the previous sections indicate that optimal adhesion and optimal fracture strength can be achieved by reducing the dimension of the structure.

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Fig. 8.67 The shape function defining the surface shape change as a function of the shape parameter. For Ψ = 1, the optimal shape is reached and stress concentrations are predicted to disappear

In this section, we focus on the question: Can optimal adhesion be achieved at any size of the punch? The system of interest is a periodic array of rigid punches attached to an elastic substrate as schematically shown in Fig. 8.66 in a quasi-two-dimensional geometry with periodic boundary conditions. The reader is referred to the primary literature [376] for discussions on the concepts of optimal and singular shapes in adhesive contact mechanics. For a punch array, the optimal shape is given by a series expression as described in detail in [376]. The theoretical prediction of the optimal shape as a function of a shape parameter Ψ is shown in Fig. 8.67 (where Ψ = 1 is the optimal shape and Ψ = 0 corresponds to the singular, straight surface shape). The critical lengthscale for a single fiber on a substrate (in analogy the earlier derivations) is given by Rcr =

8 E ∗ ∆γ . 2 π σth

(8.34)

8.9.3 Atomistic Modeling The simulation geometry of the atomistic studies discussed in this section is shown in Fig. 8.65 (studies of size reduction only) and Fig. 8.66 (studies of shape variation). In the first case, the punch is elastic and the substrate is rigid, and in the second case, the punch is rigid and the substrate is elastic. This is in accordance with the continuum mechanics models described above. In both cases, we model the elastic part using a harmonic potential, and we treat the interface between the two parts using a Lennard-Jones potential to model the vdW adhesion interactions. The expressions for the interatomic


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Fig. 8.68 Atom rows in the rigid punch are displaced according to the continuum mechanics solution of the optimal surface shape (theoretical solution see Fig. 8.67). This method allows achieving a smoothly varying surface and enables a continuous transition from a flat punch (left) to the optimal shape (right)

potentials, as well as the overall simulation method, including the expressions for elastic properties and fracture surface energy are identical to the procedure described above. It is noted that this Griffith model of adhesion corresponds to the JKR model [379]. Studies of variations of surface shape require a method to achieve small and smooth variations of surface shape in atomistic models. Such a change in surface shape is achieved by displacing the rows of atoms as shown in Fig. 8.68. This method allows one to achieve a smoothly varying surface in the molecular dynamics simulations. We note that the alternative approach in cutting the optimal shape out of an atomic lattice does not work well because of the discreteness of the lattice and the resulting steps on the order of a Burgers vector. 8.9.4 Simulation Results Here we report the results of a series of computer experiments with the models described in the previous section. The first question we address is how the adhesion strength varies with the diameter of the spatula. Figure 8.62 shows the results of atomistic simulations of the adhesion strength as a function of the size of the material R/Rcr (corresponding to the geometry shown in Fig. 8.65). Whereas the strength of the materials interfaces is predicted reasonably well based on Griffith’s theories for large dimensions, reduction of dimension results in deviation from this prediction, and eventually failure of the material at its theoretical strength σth regardless of the presence of the flaw. In practical terms, that means even if there is surface roughness present, the roughness does not lead to stress concentration, and therefore the adhesion structure adheres robustly to the substrate.

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Fig. 8.69 Stress distribution in the rigid punch slightly before complete detachment (the stress is calculated in a thin strip along the diameter, within the area of contact Rcut = 2αR). The simulations reveal that for large radii, a stress concentration develops at the exterior sides of the cylinder. For small dimensions, this stress distribution starts to vanish. For dimensions smaller than the critical radius for flaw tolerance (large ratios of Rcut /R), the stress distribution becomes homogeneous and does not vary with the cylinder diameter any more

Now we focus on the stress distribution across the adhesion element slightly before detachment occurs. This study includes variation of (1) the dimensions of the adhesion element, (2) the adhesion energy, and (3) the elastic properties of the substrate. A rigid punch on an elastic substrate is considered. Figure 8.69 shows the stress distribution at the punch–substrate interface close to detachment for various choices of the ratio R/Rcr . The simulations reveal that for large radii, a stress concentration develops at the exterior sides of the cylinder. This result is expected from the classical understanding of fracture mechanics and corresponds to the regime where Griffith’s theory holds to describe onset of detachment. For small dimensions, the stress distribution starts to become uniform. For dimensions smaller than the critical radius for flaw tolerance, the stress distribution becomes homogeneous and does not vary with the cylinder diameter any more. Figure 8.70 shows the variations of the stress distribution close to detachment for changes in adhesion energy and elastic properties of the substrate. These results further support the notion that Rcr ∼ E

(8.35)

Rcr ∼ γ.

(8.36)

and


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Fig. 8.70 Stress distribution in the elastic punch slightly before complete detachment (the stress is calculated in a thin strip along the diameter, within the area of contact Rcut = 2αR). Here we keep the dimension fixed and vary the adhesion energy (γ0 corresponds to the surface energy) and the elastic properties (E0 corresponds to the Young’s modulus obtained for k0 = 57.23). We find that the stress distribution becomes homogeneous for large ratios of Rcr /R, in agreement with the other results (see Figs. 8.63 and 8.69)

Finally we discuss the change in the adhesion strength due to variations of the surface shape. The atomistic studies are based on the continuum considerations reviewed above where it was stated that the choice of a specific, optimal surface shape that would always lead to optimal adhesion, at any lengthscale. Figure 8.71 shows the stress distribution along the diameter of the punch for different choices of the shape parameter describing the punch shape. The results indicate that when the optimal shape is reached (Ψ = 1), the stress distribution is completely flat as in the homogeneous case (λ = 1) without stress magnification. It is observed that for Ψ < 1, a stress concentration develops at the boundaries of the punch, whereas for Ψ > 1 the largest stress occurs in the center. Figure 8.72 shows the maximum adhesion strength as a function of the shape parameter Ψ for different sizes of the punch. These observations allow drawing conclusions about the robustness: The results indicate that although optimal adhesion can be achieved at any lengthscale by changing the shape of the attachment device (by choosing Ψ = 1), robustness with respect to variations in shape, while at the same time keeping a strong adhesion force can only be achieved at small lengthscales. Robustness thus seems to be closely coupled to nanodimensioned adhesion systems. We believe that only reduction of lengthscale results in (1) robustness and (2) disappearance of the stress concentration. This may be an explanation

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Fig. 8.71 Stress distribution along the diameter of the punch for different choices of the shape parameter describing the punch shape. The results indicate that when the optimal shape is reached (Ψ = 1), the stress distribution is completely flat as in the homogeneous case (λ = 1) without stress magnification. We observe that for Ψ < 1, a stress concentration develops at the boundaries of the punch, whereas for Ψ > 1 the largest stress occurs in the center

why nature does not primarily optimize shape but instead focuses on reduction of dimension as a design strategy. 8.9.5 Summary We have reviewed a series of continuum and atomistic studies to investigate how the fracture and failure behavior of materials changes as a function of material size. The atomistic simulations reveal a smooth transition between Griffith mode of failure via crack propagation to uniform bond rupture at theoretical strength below a nanometer critical length as shown in Fig. 8.62 for both fracture of nanoparticle and surface adhesion of an elastic punch on a rigid substrate. It was found that below a critical length, the stress distribution becomes increasingly homogeneous and eventually uniform near the crack tip. The atomistic simulations fully support the hypothesis that materials become insensitive to flaws below a critical nanometer lengthscale. The results corroborate earlier suggestions made at the continuum level that the concept of nanoscale flaw tolerance may play a critical role in developing structural links in biological materials, as many biological materials feature small nanoscale dimensions. Small nanosubstructures lead to robust, flaw-tolerant materials.


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Fig. 8.72 Adhesion strength for different choices of the shape parameter Ψ . The results indicate that although optimal adhesion can be achieved at any lengthscale by changing the shape of the attachment device (by choosing Ψ = 1), robustness with respect to variations in shape while at the same time keeping a strong adhesion force can only be achieved at small lengthscales

Surprisingly, this principle is found in different geometries and different situations, including bulk and surface materials. The studies on adhesion systems indicate that optimal adhesion can be achieved at any scale if the adhesion surface shape is adapted to eliminate locations of stress concentration (see Fig. 8.71). However, this design approach does not lead to robust adhesion elements as the smallest deviations from the optimal shape lead to catastrophic failure (see Fig. 8.72). It is found that the reduction of the diameter of cylindrical adhesion systems leads to optimal adhesion including robustness; therefore, “nano is also robust.” We note that the actual displacements of atoms necessary to realize the optimal shape are on the order of a few Angstroms. Due to the discrete nature of atoms with typical interatomic bond distances between 1 and 3 ˚ A, the realization of the optimal shape at such small scale may become difficult. This may be another, alternative reason why shape optimization is not a preferred strategy to achieve optimal adhesion. We refer the reader to other articles for a more detailed discussion of linking the flaw-tolerance concept to biological materials [54, 376, 378]. Atomistic modeling represents a powerful tool in designing virtual experiments to demonstrate the concept and effect of scale reduction and shape changes. Unlike purely continuum mechanics theories, molecular dynamics simulations can intrinsically handle stress concentrations (singularities) well and provide accurate descriptions of bond breaking. Also, it can be rather straightforwardly used to study ultra-small scale systems, thus probing the limiting cases where continuum theories start to break down. The approach of using simplistic model potentials rather than utilizing highly

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Index

N 2 scaling, 72 µVT, 40 Abell-Tersoff approach, 60 Adhesion, 439, 452, 453 Adhesion strength, 459 Advanced molecular dynamics methods, 177 Aluminum nanocrystalline, 379 AMBER, 56 Analysis techniques, 410 Asymptotic stress field, 194 Atomic hypothesis, 35 Atomic interactions, 35 Atomistic simulations, 33 Atomistic theory, 16, 121 Averaging, 36 Avogadro’s number, 79, 155 Barrier dislocation motion, 424 BCC, 9 bcc packing, 13 Beam elasticity, 110 Berendsen thermostat, 40 Billion-atom simulation, 341 Bimaterial interface, 287 Biological materials, 56 Biomechanics, 9, 452 Bond order potentials, 59, 63 Bookkeeping, 73 Boundary conditions, 90, 399 displacement, 90

mechanical, 90 steered molecular dynamics, 92 Brittle failure, 12 Brittle versus ductile, 12 Brittle-to-ductile transition, 342 Buffer region, 171 CADD, 167 Canonical ensemble, 40 Carbon nanotubes, 438 Catalysis, 66 Cauchy relation, 55 Centrosymmetry parameter, 86 Centrosymmetry technique, 410 Charge equilibration, 63 CHARMM, 56 Chemical bonding, 46 Chemical complexity, 56, 62, 63, 304, 359 Classical molecular dynamics, 37 CMDF, 173 Coble creep, 378 Cohesive zone, 446 Common neighbor analysis, 90 Computational efficiency, 54 Computational Materials Design Facility CMDF, 173 Computer experiments, 35 Computer power, 79 Computing power, historical development, 79 Concurrent multiscale modeling, 162

484

Index

Concurrent multiscale simulation tools, 157 Confinement, 244, 301, 436 Constrained grain boundary diffusion, 385 atomistic simulations, 396 bicrystal model, 396 continuum model, 388 experimental evidence, 386 Continuum mechanics, 15, 95 Continuum theory, 16, 121 Copper nanocrystalline, 379 nanostructured, 422 Coulomb potential, 64 Coupling atomistic-continuum, 393 atomistic-continuum theories of plasticity, 339 atomistic-experiment, 249, 271 atomistic-mesoscopic scale, 432 strain, 124 stress, 123 Coupling constant heat bath, 41 Crack, 401, 419 diffusion wedge, 386 hyperelasticity, 207 initiation time, 267 parallel glide dislocations, 404 versus diffusion wedge, 406 Cracks, 10 Cross-slip, 352 Crystal structure, 9 Deformation diffusive, 378 elastic, 5 nanocrystalline materials, 378 plastic, 5, 373 thin films, 430 Deformation map, 434 Deformation mechanisms, 373 Deformation tensor, 104 Deformation-mechanism map, 374 Density functional theory, 48 DFT, 164 Differential beam equations, 116 Differential multiscale modeling, 159

Diffusion, 15, 378 Diffusion wedges, 385, 395, 397, 409 crack like, 401 dislocation glide, 409 formation, 400 versus crack, 406, 408 Diffusive displacement, 401 Diffusivity copper, 178 dependence, 436 surface, 178 Discretization, 76 Dislocation cross slip, 425 pileup, 404 pilups, 425 Dislocation bowing, 412 Dislocation channelling, 382 Dislocation climb, 388 Dislocation cutting, 346 Dislocation density, 418 tensor, 341 Dislocation dipole, 396, 403 Dislocation dragging force, 349 Dislocation motion grain boundaries, 421 Dislocation network, 421 Dislocation pinning, 346 Dislocations, 10, 11, 341 interaction, 424 Displacement, 97 DREIDING, 56 Ductile failure, 13 Dundur’s parameter, 393 Dynamic materials failure, 5 EAM, 54, 166, 357, 359 Edge dislocation, 329, 390 Elastic regime, 5 Electron gas, 55 Electron volt, 155 Electronic properties, 68 Embedded atom potential, 48 Embedded-atom method, 54 Empirical potentials, 50 Energetic elasticity, 122 Energy approach to elasticity, 105 Energy length scale characteristic, 188, 234, 244, 299

Index Energy method, 85 Energy minimization, 43 Energy release rate, 190, 196, 447, 453 Entropic elasticity, 122 Equations of motion, 39 Ergodic hypothesis, 36, 41 Experiments polycrystalline films, 409 Failure, 5 Failure processes, 32 FCC, 9 FCC lattice, 142 FCC packing, 13 FEAt, 163 Flaw tolerance, 446, 457 Flaws, 10 Fleischer mechanism, 351 Force calculation, 71 Force field, 48 Fracture, 12 Fracture instability, 197 Fracture surface energy, 140 Fracture surface energy 3D, 148 Free energy minima, 176 Friedel-Escaig mechanism, 351 Gecko, 452, 453 Geometric analysis, 85 Geometric confinement, 14, 373, 381 Geometrically necessary dislocations, 389 Glassy phase, 398 Glide parallel glide dislocations, 409 Grain boundary, 401 dislocation source, 414, 415 jogs, 402 stability, 402 Grain boundary processes, 378 Grain boundary structure elevated temperature, 400 Grain boundary traction relaxation, 410, 415 Grain boundary tractions, 430 Grain triple junction, 408 Grand canonical ensemble ensemble, 40 Green-Kubo relations, 76 Griffith condition, 13, 266

485

Hall-Petch, 373 Hamiltonian, 37 Hardening, 424 Harmonic potential, 54, 128, 142 HCP, 9 Heat bath, 41 Hierarchical multiscale methods, 157 High-energy grain boundary, 410, 414 Homologous temperature, 400 Hooke’s law, 97 Hybrid models, 169, 304, 359 Hyperelasticity, 62, 108, 260 Image force, 392 Image stress, 406 Insects, 452 Interface crack-grain boundary, 419 Interface effect, 381 Interfaces, 287, 436 dissimilar materials, 287 Interfaces and geometric confinement, 436 Interfacial dislocations, 382, 418 Intersonic mode I cracks, 242 Interstitial tubes, 351 Inverse Hall-Petch effect, 375, 376 Isobaric-isothermal ensemble, 40 Jog dragging, 412 Jogs, 413 kcal/mole, 155 Langevin dynamics, 41 Large-scale computing, 78 Leap-frog algorithm, 39 Length-and time scale Classical molecular dynamics, 341 Lennard-Jones, 48, 52 Limitations, classical molecular dynamics, 68 LINUX supercomputers, 82 Liquid-like grain boundary, 398 Loading strain field, 399 Lomer-Cottrell locks, 355 Long-time limit, 179 Low-energy grain boundary, 410

486

Index

MAAD, 163 Materials failure ductile, 85 nickel, 85 Materials in small dimensions, 381 Mathews-Freund-Nix mechanism, 386 Mean square displacement function, 75 Measurement, 37 Mechanical properties, 142 Medium-range-order analysis, 90 Melting temperature copper, 399 Mesoscopic simulations, 50, 434 Message passing, 80 Metropolis-Hastings algorithm, 45 Microcanonical ensemble, 40 Microelectronic devices, 381 Microscopic configurations, 41 Microstructure, 9, 76 Miniaturization, 381 Mirror-mist-hackle, 197 Mode I fracture Mother-daughter mechanism, 289 Mode II fracture, 243, 294 Mode III fracture, 142, 299 Model materials, 35, 341 Modeling, 32 Modeling and simulation, 32, 33, 90 Molecular dynamics, 37 Molecular statics, 44 Monte Carlo, 37, 45 Morse potential, 53 Mother-daughter mechanism, 289 Mother-daughter-granddaughter mechanism, 294 MPI, 80 Multi-body potential, 54 Multi-scale phenomena, 35 Multi-scale simulations hierarchical, 432 Multiparadigm modeling, 158, 170, 357 Multiscale, 157 Multiscale modeling and simulation, 157 Nanocrystalline copper, 376 Nanocrystalline materials, 15, 353 Nanoindentation, 167 Nanomaterials, 26, 438, 446

Nanoscale, 381 confinement, 391 deformation phenomena, 435 Nanoscale adhesion, 453 Nanostructured materials, 376 strain rate, 379 yield stress, 379 Nanostructures, 446 Nanotechnology, 3, 381 Navier-Bernouilli, 114 Newton, 155 Newton’s laws, 96 Nickel, 85 nanocrystalline, 379 Nonlinear elasticity, 108 NPT, 40 NVE, 40 NVT, 40 On-the-fly concurrent multiscale methods, 158 Organic materials, 56 Oxidation, 357 Pair potential, 48, 50 Parallel glide dislocations, 167, 384, 419 experimental evidence, 386 minimum film thickness, 403 nucleation, 393, 396, 402 nucleation mechanism, 394 Parallel molecular dynamics, 80 Parrinello-Rahman, 41 Partial dislocations, 348, 350, 422 Partial point defects, 413 Pascal, 155 PBC, 70 Peach-Koehler force, 392 Periodic boundary conditions, 70 Petaflop computers, 80 Phonons, 164 Pinning potential, 399 Plane strain, 399 Plastic deformation, 5 Plasticity, 341 atomistic modeling, 414 nanocrystalline materials, 422 polycrystalline thin films, 414 thin films, 409 Point defect generation, 413

Index Polycrystalline films, 409 Polycrystalline thin films atomistic modeling, 415 Polycrystalline thin metal films, 381 Polymers, 9, 56 Post-processing, 85 Potential, 48 Property calculation, 73, 93 Protein unfolding, 92 Proteins, 9, 56 Quantization stress, 392 Quantum mechanics, 32, 35, 68 Quasicontinuum method, QC, 165 Quasicrystals, 201, 343 Brittle fracture, 201 Dislocations, 343 Ductile failure, 343 Radial distribution function, 74 Reactive potentials, 59 ReaxFF, 59, 62, 169, 304, 359 Reduced units, 155 Reference units, 155 Relaxation, 399 Relaxation mechanisms, 430 Rice-Thomson model, 404 Richard Feynman, 35 Rigid boundaries, 403 Rise time, 41 Robustness, 459 SC, 9 Screw dislocation, 329 Self-folding, 439 Sessile locks, 353 Shielded Coulomb potential, 64 Shock loading, 364 Silicon, 55 Simulation, 32 Simulation techniques, 49 Single atoms, 179 Single edge dislocations, 390 Size effects, 373, 381, 438, 452 Slip planes, 353 Slip vector, 88, 341, 410, 412 Small-scale materials, 199 Speedup, 81

487

Spring constant ratio, 236 Stacking fault, 358 State transition, 179 Statistical mechanics, 36 Strain, 97 Strain rate, 266, 400 Stress, 97 Stress intensity, 393 Stress intensity factor, 194, 266, 390, 396 parallel glide dislocations, 402 Stress tensor, 99 Sub-micron scale, 381 Sub-nano structure, 422 Submicron thin films, 385 Suddenly stopping crack, 260 mode I, 268 mode II, 278, 280 mode III, 303 Super-Rayleigh fracture, 242, 274 Supercomputers, 79 Supercomputing, 78, 80, 201, 342 Supersonic fracture, 207, 210, 280, 281, 289 Supersonic mode I cracks, 289 Supersonic mode II cracks, 243 Surface diffusion, 177 Surface diffusivity, 177 Surface effects, 381, 436 Surface steps, 178, 417 Temperature, 73 Temperature accelerated method, 177 Tersoff potential, 61 The strongest size, 376 Thermodynamics, 41, 105, 121 Thin films, 167, 381 deformation map, 434 yield stress, 384, 433, 435 Threading dislocations, 382, 383, 386, 402, 417 versus parallel glide dislocations, 416, 430 Three-dimensional molecular dynamics simulations, 142 Threshold stress, 433 Tight-Binding approach, 162 Tight-binding potential, 48 Tilt grain boundary, 399

488

Index

Time scale, 407 Time scale dilemma, 175, 176 Time scale methods, 175 Time step, 43 top500.org, 80 Transition region, 171 Transport properties, 76 Triangular lattice, 9 Triple junction, 410 Twin grain boundary, 424 Twin lamella, 422 Two-dimensional lattice, 128 UFF, 56 Unit conversion, 155 Unstable stacking fault energy, 14 Vacancy tubes, 351 Velocity autocorrelation function, 76 Velocity verlet, 398 Velocity Verlet algorithm, 40

Verlet algorithm, 39 Virial strain, 124 Virial stress, 76, 77, 123 Virtual internal bond method, VIB, 168 Viscoelasticity, 169 Visualization, 83, 85, 345, 401 distributed computing, 83 movies, 83 Richard Hamming, 84 virtual reality, 83 Volterra edge dislocations, 388 Water formation, 66 Work-hardening, 354 Yield stress, 435 Young’s modulus, 97, 120, 133, 142, 204 bilinear, 138, 207 FCC, 142, 144 Zhou’s virial stress, 77

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