السلام عليكم
كتاب اخر في مجال النانو تكنولوجي لعله يكون نافعا لمن يحتاجة
Title of book
Nano Mechanics and Materials-THEORY,MULTISCALE METHODS AND APPLICATIONS
authors:
Wing Kam Liu
Northwestern University, Illinois, USA
Eduard G. Karpov
Northwestern University, Illinois, USA
Harold S. Park
Vanderbilt University, Tennessee, USA
publisher :John Wiley & Sons Ltd
date:2006
ISBN-13: 978-0-470-01851-4
ISBN-10: 0-470-01851-8
no. of pages:320
size: 7,481 KB
كتاب اخر في مجال النانو تكنولوجي لعله يكون نافعا لمن يحتاجة
Title of book
Nano Mechanics and Materials-THEORY,MULTISCALE METHODS AND APPLICATIONS
authors:
Wing Kam Liu
Northwestern University, Illinois, USA
Eduard G. Karpov
Northwestern University, Illinois, USA
Harold S. Park
Vanderbilt University, Tennessee, USA
publisher :John Wiley & Sons Ltd
date:2006
ISBN-13: 978-0-470-01851-4
ISBN-10: 0-470-01851-8
no. of pages:320
size: 7,481 KB
description of the book
This text, written by respected researchers in the field, informs researchers and practitioners about the fundamental concepts in nano mechanics and materials, focusing on their modelling via multiple scale methods and techniques. The book systematically covers the theory behind multi-particle and nanoscale systems, introduces multiple scale methods, and finally looks at contemporary applications in nano-structured and bio-inspired materials.
Nano Mechanics and Materials: Theory, Multiscale Methods and Applications | Nanotechnology is a progressive research and development topic with large amounts of venture capital and government funding being invested worldwide. Nano mechanics, in particular, is the study and characterization of the mechanical behaviour of individual atoms, systems and structures in response to various types of forces and loading conditions.
This text, written by respected researchers in the field, informs researchers and practitioners about the fundamental concepts in nano mechanics and materials, focusing on their modelling via multiple scale methods and techniques. The book systematically covers the theory behind multi-particle and nanoscale systems, introduces multiple scale methods, and finally looks at contemporary applications in nano-structured and bio-inspired materials.
contents
Preface xi
1 Introduction 1
1.1 Potential of Nanoscale Engineering . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation for Multiple Scale Modeling . . . . . . . . . . . . . . . . . . . . 2
1.3 Educational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Classical Molecular Dynamics 7
2.1 Mechanics of a System of Particles . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Mechanical Forces and Potential Energy . . . . . . . . . . . . . . . 8
2.1.3 Lagrange Equations ofMotion . . . . . . . . . . . . . . . . . . . . 10
2.1.4 Integrals of Motion and Symmetric Fields . . . . . . . . . . . . . . 12
2.1.5 Newtonian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Molecular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Pair-Wise Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Multibody Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Molecular Dynamics Applications . . . . . . . . . . . . . . . . . . . . . . . 28
3 Lattice Mechanics 37
3.1 Elements of Lattice Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Basic Symmetry Principles . . . . . . . . . . . . . . . . . . . . . . 40
3.1.3 Crystallographic Directions and Planes . . . . . . . . . . . . . . . . 42
3.2 Equation of Motion of a Regular Lattice . . . . . . . . . . . . . . . . . . . 42
3.2.1 Unit Cell and the Associate Substructure . . . . . . . . . . . . . . . 43
3.2.2 Lattice Lagrangian and Equations of Motion . . . . . . . . . . . . . 45
3.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 53
viii CONTENTS
3.4 StandingWaves in Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 NormalModes and Dispersion Branches . . . . . . . . . . . . . . . 55
3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Green’s Function Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.1 Solution for a Unit Pulse . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.2 Free Lattice with Initial Perturbations . . . . . . . . . . . . . . . . . 61
3.5.3 Solution for Arbitrary Dynamic Loads . . . . . . . . . . . . . . . . 61
3.5.4 General Inhomogeneous Solution . . . . . . . . . . . . . . . . . . . 62
3.5.5 Boundary Value Problems and the Time History Kernel . . . . . . . 62
3.5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Quasi-Static Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.1 Equilibrium State Equation . . . . . . . . . . . . . . . . . . . . . . 66
3.6.2 Quasi-Static Green’s Function . . . . . . . . . . . . . . . . . . . . . 67
3.6.3 Multiscale Boundary Conditions . . . . . . . . . . . . . . . . . . . . 67
4 Methods of Thermodynamics and Statistical Mechanics 79
4.1 Basic Results of the Thermodynamic Method . . . . . . . . . . . . . . . . . 80
4.1.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Energy Conservation Principle . . . . . . . . . . . . . . . . . . . . . 84
4.1.3 Entropy and the Second Law of Thermodynamics . . . . . . . . . . 86
4.1.4 Nernst’s Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.5 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Statistics of Multiparticle Systems in Thermodynamic Equilibrium . . . . . 91
4.2.1 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.2 Statistical Description of Multiparticle Systems . . . . . . . . . . . 93
4.2.3 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.4 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.5 Maxwell–Boltzmann Distribution . . . . . . . . . . . . . . . . . . . 104
4.2.6 Thermal Properties of Periodic Lattices . . . . . . . . . . . . . . . . 107
4.3 Numerical Heat Bath Techniques . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Berendsen Thermostat . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.2 Nos´e–Hoover Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.3 Phonon Method for Solid–Solid Interfaces . . . . . . . . . . . . . . 119
5 Introduction to Multiple Scale Modeling 123
5.1 MAAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Coarse-GrainedMolecular Dynamics . . . . . . . . . . . . . . . . . . . . . 126
5.3 Quasi-Continuum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 CADD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5 Bridging Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Introduction to Bridging Scale 131
6.1 Bridging Scale Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.1 Multiscale Equations of Motion . . . . . . . . . . . . . . . . . . . . 133
6.2 Removing Fine Scale Degrees of Freedom in Coarse Scale Region . . . . . 136
6.2.1 Relationship of Lattice Mechanics to Finite Elements . . . . . . . . 137
6.2.2 LinearizedMD Equation ofMotion . . . . . . . . . . . . . . . . . . 139
CONTENTS ix
6.2.3 Elimination of Fine Scale Degrees of Freedom . . . . . . . . . . . . 141
6.2.4 Commentary on Reduced Multiscale Formulation . . . . . . . . . . 143
6.2.5 Elimination of Fine Scale Degrees of Freedom:
3D Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.6 Numerical Implementation of Impedance Force . . . . . . . . . . . 150
6.2.7 Numerical Implementation of Coupling Force . . . . . . . . . . . . 151
6.3 Discussion on the Damping Kernel Technique . . . . . . . . . . . . . . . . 152
6.3.1 Programming Algorithm for Time History Kernel . . . . . . . . . . 157
6.4 Cauchy–Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.5 Virtual Atom ClusterMethod . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.5.1 Motivations and General Formulation . . . . . . . . . . . . . . . . . 159
6.5.2 General Idea of the VACModel . . . . . . . . . . . . . . . . . . . . 163
6.5.3 Three-Way Concurrent Coupling with QM Method . . . . . . . . . 164
6.5.4 Tight-Binding Method for Carbon Systems . . . . . . . . . . . . . . 167
6.5.5 Coupling with the VACModel . . . . . . . . . . . . . . . . . . . . 169
6.6 Staggered Time Integration Algorithm . . . . . . . . . . . . . . . . . . . . . 170
6.6.1 MD Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.6.2 FE Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Summary of Bridging Scale Equations . . . . . . . . . . . . . . . . . . . . 172
6.8 Discussion on the Bridging ScaleMethod . . . . . . . . . . . . . . . . . . . 173
7 Bridging Scale Numerical Examples 175
7.1 Comments on Time History Kernel . . . . . . . . . . . . . . . . . . . . . . 175
7.2 1D Bridging Scale Numerical Examples . . . . . . . . . . . . . . . . . . . . 176
7.2.1 Lennard-Jones Numerical Examples . . . . . . . . . . . . . . . . . . 176
7.2.2 Comparison of VAC Method and Cauchy–Born Rule . . . . . . . . 178
7.2.3 Truncation of Time History Kernel . . . . . . . . . . . . . . . . . . 179
7.3 2D/3D Bridging Scale Numerical Examples . . . . . . . . . . . . . . . . . . 182
7.4 Two-DimensionalWave Propagation . . . . . . . . . . . . . . . . . . . . . 184
7.5 Dynamic Crack Propagation in Two Dimensions . . . . . . . . . . . . . . . 187
7.6 Dynamic Crack Propagation in Three Dimensions . . . . . . . . . . . . . . 195
7.7 Virtual Atom Cluster Numerical Examples . . . . . . . . . . . . . . . . . . 200
7.7.1 Bending of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . 200
7.7.2 VAC Coupling with Tight Binding . . . . . . . . . . . . . . . . . . 200
8 Non-Nearest Neighbor MD Boundary Condition 203
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2 Theoretical Formulation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2.1 Force Boundary Condition: 1D Illustration . . . . . . . . . . . . . . 207
8.2.2 Displacement Boundary Condition: 1D Illustration . . . . . . . . . . 210
8.2.3 Comparison to Nearest Neighbors Formulation . . . . . . . . . . . . 211
8.2.4 Advantages of Displacement Formulation . . . . . . . . . . . . . . . 212
8.3 Numerical Examples: 1D Wave Propagation . . . . . . . . . . . . . . . . . 212
8.4 Time-History Kernels for FCC Gold . . . . . . . . . . . . . . . . . . . . . . 213
8.5 Conclusion for the Bridging Scale Method . . . . . . . . . . . . . . . . . . 215
8.5.1 Bridging Scale Perspectives . . . . . . . . . . . . . . . . . . . . . . 220
x CONTENTS
9 Multiscale Methods for Material Design 223
9.1 Multiresolution Continuum Analysis . . . . . . . . . . . . . . . . . . . . . . 225
9.1.1 Generalized Stress and Deformation Measures . . . . . . . . . . . . 227
9.1.2 Interaction between Scales . . . . . . . . . . . . . . . . . . . . . . . 231
9.1.3 Multiscale Materials Modeling . . . . . . . . . . . . . . . . . . . . 232
9.2 Multiscale Constitutive Modeling of Steels . . . . . . . . . . . . . . . . . . 234
9.2.1 Methodology and Approach . . . . . . . . . . . . . . . . . . . . . . 235
9.2.2 First-Principles Calculation . . . . . . . . . . . . . . . . . . . . . . 235
9.2.3 Hierarchical Unit Cell and Constitutive Model . . . . . . . . . . . . 237
9.2.4 Laboratory Specimen Scale: Simulation and Results . . . . . . . . . 239
9.3 Bio-InspiredMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.3.1 Mechanisms of Self-Healing inMaterials . . . . . . . . . . . . . . . 244
9.3.2 Shape-Memory Composites . . . . . . . . . . . . . . . . . . . . . . 246
9.3.3 Multiscale Continuum Modeling of SMA Composites . . . . . . . . 250
9.3.4 Issues ofModeling and Simulation . . . . . . . . . . . . . . . . . . 256
9.4 Summary and Future Research Directions . . . . . . . . . . . . . . . . . . . 260
10 Bio–Nano Interface 263
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.2 Immersed Finite ElementMethod . . . . . . . . . . . . . . . . . . . . . . . 265
10.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.2.2 Computational Algorithm of IFEM . . . . . . . . . . . . . . . . . . 268
10.3 Vascular Flow and Blood Rheology . . . . . . . . . . . . . . . . . . . . . . 269
10.3.1 HeartModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.3.2 Flexible Valve–Viscous Fluid Interaction . . . . . . . . . . . . . . . 270
10.3.3 Angioplasty Stent . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.3.4 Monocyte Deposition . . . . . . . . . . . . . . . . . . . . . . . . . 272
10.3.5 Platelet Adhesion and Blood Clotting . . . . . . . . . . . . . . . . . 272
10.3.6 RBC Aggregation and Interaction . . . . . . . . . . . . . . . . . . . 274
10.4 Electrohydrodynamic Coupling . . . . . . . . . . . . . . . . . . . . . . . . 280
10.4.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
10.4.2 Electro-manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.4.3 Rotation of CNTs Induced by Electroosmotic Flow . . . . . . . . . 285
10.5 CNT/DNA Assembly Simulation . . . . . . . . . . . . . . . . . . . . . . . 287
10.6 CellMigration and Cell–Substrate Adhesion . . . . . . . . . . . . . . . . . 290
10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Appendix A Kernel Matrices for EAM Potential 297
Bibliography 301
Index 315
1 Introduction 1
1.1 Potential of Nanoscale Engineering . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation for Multiple Scale Modeling . . . . . . . . . . . . . . . . . . . . 2
1.3 Educational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Classical Molecular Dynamics 7
2.1 Mechanics of a System of Particles . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Mechanical Forces and Potential Energy . . . . . . . . . . . . . . . 8
2.1.3 Lagrange Equations ofMotion . . . . . . . . . . . . . . . . . . . . 10
2.1.4 Integrals of Motion and Symmetric Fields . . . . . . . . . . . . . . 12
2.1.5 Newtonian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Molecular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Pair-Wise Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Multibody Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Molecular Dynamics Applications . . . . . . . . . . . . . . . . . . . . . . . 28
3 Lattice Mechanics 37
3.1 Elements of Lattice Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Basic Symmetry Principles . . . . . . . . . . . . . . . . . . . . . . 40
3.1.3 Crystallographic Directions and Planes . . . . . . . . . . . . . . . . 42
3.2 Equation of Motion of a Regular Lattice . . . . . . . . . . . . . . . . . . . 42
3.2.1 Unit Cell and the Associate Substructure . . . . . . . . . . . . . . . 43
3.2.2 Lattice Lagrangian and Equations of Motion . . . . . . . . . . . . . 45
3.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 53
viii CONTENTS
3.4 StandingWaves in Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 NormalModes and Dispersion Branches . . . . . . . . . . . . . . . 55
3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Green’s Function Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.1 Solution for a Unit Pulse . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.2 Free Lattice with Initial Perturbations . . . . . . . . . . . . . . . . . 61
3.5.3 Solution for Arbitrary Dynamic Loads . . . . . . . . . . . . . . . . 61
3.5.4 General Inhomogeneous Solution . . . . . . . . . . . . . . . . . . . 62
3.5.5 Boundary Value Problems and the Time History Kernel . . . . . . . 62
3.5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Quasi-Static Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.1 Equilibrium State Equation . . . . . . . . . . . . . . . . . . . . . . 66
3.6.2 Quasi-Static Green’s Function . . . . . . . . . . . . . . . . . . . . . 67
3.6.3 Multiscale Boundary Conditions . . . . . . . . . . . . . . . . . . . . 67
4 Methods of Thermodynamics and Statistical Mechanics 79
4.1 Basic Results of the Thermodynamic Method . . . . . . . . . . . . . . . . . 80
4.1.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Energy Conservation Principle . . . . . . . . . . . . . . . . . . . . . 84
4.1.3 Entropy and the Second Law of Thermodynamics . . . . . . . . . . 86
4.1.4 Nernst’s Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.5 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Statistics of Multiparticle Systems in Thermodynamic Equilibrium . . . . . 91
4.2.1 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.2 Statistical Description of Multiparticle Systems . . . . . . . . . . . 93
4.2.3 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.4 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.5 Maxwell–Boltzmann Distribution . . . . . . . . . . . . . . . . . . . 104
4.2.6 Thermal Properties of Periodic Lattices . . . . . . . . . . . . . . . . 107
4.3 Numerical Heat Bath Techniques . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Berendsen Thermostat . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.2 Nos´e–Hoover Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.3 Phonon Method for Solid–Solid Interfaces . . . . . . . . . . . . . . 119
5 Introduction to Multiple Scale Modeling 123
5.1 MAAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Coarse-GrainedMolecular Dynamics . . . . . . . . . . . . . . . . . . . . . 126
5.3 Quasi-Continuum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 CADD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5 Bridging Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Introduction to Bridging Scale 131
6.1 Bridging Scale Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.1 Multiscale Equations of Motion . . . . . . . . . . . . . . . . . . . . 133
6.2 Removing Fine Scale Degrees of Freedom in Coarse Scale Region . . . . . 136
6.2.1 Relationship of Lattice Mechanics to Finite Elements . . . . . . . . 137
6.2.2 LinearizedMD Equation ofMotion . . . . . . . . . . . . . . . . . . 139
CONTENTS ix
6.2.3 Elimination of Fine Scale Degrees of Freedom . . . . . . . . . . . . 141
6.2.4 Commentary on Reduced Multiscale Formulation . . . . . . . . . . 143
6.2.5 Elimination of Fine Scale Degrees of Freedom:
3D Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.6 Numerical Implementation of Impedance Force . . . . . . . . . . . 150
6.2.7 Numerical Implementation of Coupling Force . . . . . . . . . . . . 151
6.3 Discussion on the Damping Kernel Technique . . . . . . . . . . . . . . . . 152
6.3.1 Programming Algorithm for Time History Kernel . . . . . . . . . . 157
6.4 Cauchy–Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.5 Virtual Atom ClusterMethod . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.5.1 Motivations and General Formulation . . . . . . . . . . . . . . . . . 159
6.5.2 General Idea of the VACModel . . . . . . . . . . . . . . . . . . . . 163
6.5.3 Three-Way Concurrent Coupling with QM Method . . . . . . . . . 164
6.5.4 Tight-Binding Method for Carbon Systems . . . . . . . . . . . . . . 167
6.5.5 Coupling with the VACModel . . . . . . . . . . . . . . . . . . . . 169
6.6 Staggered Time Integration Algorithm . . . . . . . . . . . . . . . . . . . . . 170
6.6.1 MD Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.6.2 FE Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Summary of Bridging Scale Equations . . . . . . . . . . . . . . . . . . . . 172
6.8 Discussion on the Bridging ScaleMethod . . . . . . . . . . . . . . . . . . . 173
7 Bridging Scale Numerical Examples 175
7.1 Comments on Time History Kernel . . . . . . . . . . . . . . . . . . . . . . 175
7.2 1D Bridging Scale Numerical Examples . . . . . . . . . . . . . . . . . . . . 176
7.2.1 Lennard-Jones Numerical Examples . . . . . . . . . . . . . . . . . . 176
7.2.2 Comparison of VAC Method and Cauchy–Born Rule . . . . . . . . 178
7.2.3 Truncation of Time History Kernel . . . . . . . . . . . . . . . . . . 179
7.3 2D/3D Bridging Scale Numerical Examples . . . . . . . . . . . . . . . . . . 182
7.4 Two-DimensionalWave Propagation . . . . . . . . . . . . . . . . . . . . . 184
7.5 Dynamic Crack Propagation in Two Dimensions . . . . . . . . . . . . . . . 187
7.6 Dynamic Crack Propagation in Three Dimensions . . . . . . . . . . . . . . 195
7.7 Virtual Atom Cluster Numerical Examples . . . . . . . . . . . . . . . . . . 200
7.7.1 Bending of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . 200
7.7.2 VAC Coupling with Tight Binding . . . . . . . . . . . . . . . . . . 200
8 Non-Nearest Neighbor MD Boundary Condition 203
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2 Theoretical Formulation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2.1 Force Boundary Condition: 1D Illustration . . . . . . . . . . . . . . 207
8.2.2 Displacement Boundary Condition: 1D Illustration . . . . . . . . . . 210
8.2.3 Comparison to Nearest Neighbors Formulation . . . . . . . . . . . . 211
8.2.4 Advantages of Displacement Formulation . . . . . . . . . . . . . . . 212
8.3 Numerical Examples: 1D Wave Propagation . . . . . . . . . . . . . . . . . 212
8.4 Time-History Kernels for FCC Gold . . . . . . . . . . . . . . . . . . . . . . 213
8.5 Conclusion for the Bridging Scale Method . . . . . . . . . . . . . . . . . . 215
8.5.1 Bridging Scale Perspectives . . . . . . . . . . . . . . . . . . . . . . 220
x CONTENTS
9 Multiscale Methods for Material Design 223
9.1 Multiresolution Continuum Analysis . . . . . . . . . . . . . . . . . . . . . . 225
9.1.1 Generalized Stress and Deformation Measures . . . . . . . . . . . . 227
9.1.2 Interaction between Scales . . . . . . . . . . . . . . . . . . . . . . . 231
9.1.3 Multiscale Materials Modeling . . . . . . . . . . . . . . . . . . . . 232
9.2 Multiscale Constitutive Modeling of Steels . . . . . . . . . . . . . . . . . . 234
9.2.1 Methodology and Approach . . . . . . . . . . . . . . . . . . . . . . 235
9.2.2 First-Principles Calculation . . . . . . . . . . . . . . . . . . . . . . 235
9.2.3 Hierarchical Unit Cell and Constitutive Model . . . . . . . . . . . . 237
9.2.4 Laboratory Specimen Scale: Simulation and Results . . . . . . . . . 239
9.3 Bio-InspiredMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.3.1 Mechanisms of Self-Healing inMaterials . . . . . . . . . . . . . . . 244
9.3.2 Shape-Memory Composites . . . . . . . . . . . . . . . . . . . . . . 246
9.3.3 Multiscale Continuum Modeling of SMA Composites . . . . . . . . 250
9.3.4 Issues ofModeling and Simulation . . . . . . . . . . . . . . . . . . 256
9.4 Summary and Future Research Directions . . . . . . . . . . . . . . . . . . . 260
10 Bio–Nano Interface 263
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.2 Immersed Finite ElementMethod . . . . . . . . . . . . . . . . . . . . . . . 265
10.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.2.2 Computational Algorithm of IFEM . . . . . . . . . . . . . . . . . . 268
10.3 Vascular Flow and Blood Rheology . . . . . . . . . . . . . . . . . . . . . . 269
10.3.1 HeartModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.3.2 Flexible Valve–Viscous Fluid Interaction . . . . . . . . . . . . . . . 270
10.3.3 Angioplasty Stent . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.3.4 Monocyte Deposition . . . . . . . . . . . . . . . . . . . . . . . . . 272
10.3.5 Platelet Adhesion and Blood Clotting . . . . . . . . . . . . . . . . . 272
10.3.6 RBC Aggregation and Interaction . . . . . . . . . . . . . . . . . . . 274
10.4 Electrohydrodynamic Coupling . . . . . . . . . . . . . . . . . . . . . . . . 280
10.4.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
10.4.2 Electro-manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.4.3 Rotation of CNTs Induced by Electroosmotic Flow . . . . . . . . . 285
10.5 CNT/DNA Assembly Simulation . . . . . . . . . . . . . . . . . . . . . . . 287
10.6 CellMigration and Cell–Substrate Adhesion . . . . . . . . . . . . . . . . . 290
10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Appendix A Kernel Matrices for EAM Potential 297
Bibliography 301
Index 315
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