Generalized Models and Non-classical Approaches in Complex Materials 2 (eBook)

The authors of the book dedicated this book to Gérard A. Maugin, an exceptional French engineering scientist and philosopher. Maugin’s achievements in the fields of Physical Sciences and Engineering embrace relativistic continuum mechanics, micro magnetism, electrodynamics of continua, thermo mechanics, surface waves and nonlinear waves in continua, lattice dynamics, material equations and biomechanical applications.

Preface 7
Contents 9
Contributors 11
1 Damping in Materials and Structures: An Overview 14
Abstract 14
1.1 Introduction 14
1.2 Mechanisms of Energy Dissipation 15
1.2.1 Macroscopic Approach 16
1.2.1.1 Viscous Dissipation 17
1.2.1.2 Friction Dissipation 17
1.2.1.3 Magneto-Mechanic Dissipation 17
1.2.1.4 Electro-Mechanic Dissipation 18
1.2.1.5 Plastic Dissipation 18
1.2.2 Microscopic Approach 19
1.2.2.1 Atomic Scale Approach 19
1.2.2.2 Molecular Scale Approach 20
1.2.2.3 Mesoscopic Scale Approach 21
1.3 Modelling Energy Dissipation 22
1.3.1 Internal Forces 22
1.3.2 Work of Internal Forces: Cycling 23
1.3.3 Viscous Dissipation 26
1.3.3.1 Linear Behavior of the Phenomenon 26
Discrete Bi-parametric Model (Like Voigt Model) 28
Continuous Multi-parametric Model (Prony Series) 29
1.3.3.2 Non-linear Behavior of the Phenomenon 29
Schapery Model 30
Valanis-Landel Model 31
Frechet-Volterra Series Model 32
Linearization of the Phenomenon 33
1.3.4 Friction Dissipation 33
1.3.4.1 Coulomb’s Friction Modelling 34
1.3.4.2 Tresca’s Friction Modelling 35
1.3.4.3 Dahl’s Friction Modelling 35
1.3.4.4 Micro-friction 36
1.4 Conclusion 37
References 38
2 The Principle of Virtual Power (PVP): Application to Complex Media, Extension to Gauge and Scale Invariances, and Fundamental Aspects 41
Abstract 41
2.1 First Part 42
2.1.1 Complex Media: Modeling of the Different Continua 42
2.1.2 Thermo-Electro-Magneto-Mechanical Equations 44
2.1.2.1 General Principles in Global Form 44
2.1.2.2 Local Electro-Magneto-Mechanical Balance Equations 50
2.1.2.3 Local Thermodynamical Equations 52
2.1.3 Clausius-Duhem Inequality 52
2.2 Second Part 53
2.2.1 Extension of the PVP to Gauge and Scale Invariances 53
2.2.2 Extended form of d’Alembert’s Principle 54
2.2.3 Unified Global Statement 54
2.2.4 Derivation of Scale, Gauge and Rotational Invariances 56
2.2.5 Local Equations 57
2.2.6 Relativistic Framework 58
2.3 Third Part 59
2.3.1 Foundation of the Principle of Virtual Power (PVP) 59
2.3.2 Main Points of the Leibnizian Dynamical Framework 60
2.3.3 Determination of the Yet Under-Determinate Framework 61
2.3.4 Deduction of the PVP Based on Duality 62
2.3.5 Derivation of Einstein’s Dynamics 62
Acknowledgements 63
References 63
3 The Limitations and Successes of Concurrent Dynamic Multiscale Modeling Methods at the Mesoscale 66
Abstract 66
3.1 Introduction 67
3.2 Review of Dynamic Multiscale Methods 68
3.2.1 Coupled Atomistic and Discrete Dislocation Dynamics 68
3.2.2 Coupled Extended Finite Element Method 70
3.2.3 Concurrent Atomistic Continuum Method 72
3.2.4 The Hot Quasi-Continuum Method 74
3.2.5 The Atomistic to Continuum Method 76
3.3 Analysis 78
3.3.1 Modeling Materials Beyond Monoatomic Crystals 78
3.3.2 Modeling of Defects and Waves 81
3.4 Conclusions 83
Acknowledgements 84
References 85
4 Modeling Semiconductor Crystal Growth Under Electromagnetic Fields 89
Abstract 89
4.1 Introduction 89
4.1.1 Liquid Phase Electroepitaxy 90
4.1.2 Traveling Heater Method 92
4.2 Basic Equations of an Electromagnetic Liquid Continuum 94
4.2.1 Basic Equations 94
4.2.2 Constitutive Equations 96
4.3 Liquid Phase Electroepitaxial Growth of Binary Systems Under Magnetic Field 99
4.3.1 Electromagnetic Mobility 101
4.4 Growth of Binary Systems by the Traveling Heater Method Under Magnetic Fields 102
4.4.1 Growth by the Traveling Heater Method Under Static Magnetic Field 103
4.4.2 Growth by the Traveling Heater Method Under Rotating Magnetic Field 105
4.5 Conclusions 106
Acknowledgements 107
References 107
5 Dispersion Properties of a Closed-Packed Lattice Consisting of Round Particles 111
5.1 Introduction 112
5.2 Discrete Model for a Hexagonal Lattice Consisting of Round Particles 114
5.3 Derivation of the Dispersion Equation 119
5.4 Dispersion Properties of Normal Waves 121
5.5 Conclusions 124
References 126
6 Emulating the Raman Physics in the Spatial Domain with the Help of the Zakharov’s Systems 128
Abstract 128
6.1 Introduction 128
6.2 Soliton Dynamics in an Extended Nonlinear Schrödinger Equation with a Pseudo-Raman Effect and Inhomogeneous Dispersion 131
6.3 Damped Solitons in an Extended Nonlinear Schrödinger Equation with a Pseudo-Raman Effect and Exponentially Decreasing Dispersion 134
6.4 Soliton in a Higher-Order Nonlinear Schrödinger Equation with Pseudo-Raman Effect and Inhomogeneous Second-Order Diffraction 138
6.5 Vector Solitons in Coupled Nonlinear Equations with the Pseudo-Raman Effect and Inhomogeneous Dispersion 140
6.5.1 Analytical Results 141
6.5.2 Numerical Results 144
6.6 Solitons in a Forced Nonlinear Schrödinger Equation with the Pseudo-Raman Effect 147
6.7 Conclusion 151
Acknowledgements 151
References 151
7 Generalized Differential Effective Medium Method for Simulating Effective Physical Properties of 2D Percolating Composites 154
Abstract 154
7.1 Introduction 154
7.2 Generalized Differential Effective Medium Method for Elastic Moduli and Conductivity Prediction 156
7.3 Elastic Properties Calculations 159
7.4 Effective Conductivity Calculations 163
7.5 Concluding Remarks 166
Acknowledgements 167
References 167
8 Nonlinear Acoustic Wedge Waves 169
Abstract 169
8.1 Introduction 170
8.2 Evolution Equation with Second-Order Nonlinearity Only 174
8.3 Nonlinear Evolution of Acoustic Wedge Pulses 179
8.4 Evolution Equation with Second- and Third-Order Nonlinearity 182
8.5 Conclusions 187
Acknowledgements 187
Appendix A 188
Appendix B 189
References 190
9 Analysis of Nonlinear Wave Propagation in Hyperelastic Network Materials 193
Abstract 193
9.1 Introduction 194
9.2 Incremental Scheme for the Computation of the Effective Hyperelastic Effective Models 196
9.3 Identification of a Hyperelastic Strain Energy Density for the Hexagonal Lattice, the Re-entrant Lattice and Plain Weave Textile 197
9.4 Analysis of Nonlinear Wave Propagation in the Homogenized Hyperelastic Continua 201
9.4.1 Wave Propagation Analysis for the Form 1 of the Hyperelastic Effective Medium Energy 201
9.4.2 Wave Propagation Analysis for Form 2 of the Hyperelastic Energy 203
9.5 Conclusion 206
References 207
10 Multiscale Modeling of 2D Material MoS2 from Molecular Dynamics to Continuum Mechanics 209
Abstract 209
10.1 Introduction 209
10.2 Crystal Structure and Interatomic Potential of MoS2 210
10.3 Molecular Dynamics 212
10.4 Thermoelasticity and Sequential Multiscale Modeling 214
10.4.1 Governing Equations of Thermoelasticity 214
10.4.2 Elastic Constants 216
10.4.3 Thermal Conductivity 217
10.4.4 Specific Heat and Thermal Expansion Coefficient 218
10.5 Concurrent Multiscale Modeling from Atoms to Genuine Continuum 219
10.5.1 Interfacial Conditions 221
10.5.2 Multiple Time Scale Algorithm 222
10.5.3 Sample Problems and Numerical Results 223
10.5.3.1 Material Constants Obtained from MD Simulations 223
10.5.3.2 Case Study 224
10.6 Conclusion and Future Work 226
References 227
11 Gradient Elasticity Effects on the Two-Phase Lithiation of LIB Anodes 228
Abstract 228
11.1 Introduction 228
11.2 Theoretical Framework of Gradient Chemoelasticity 230
11.3 Modeling Lithiation of a Spherical Silicon Particle 232
11.3.1 Governing Equations 232
11.3.2 Material and Model Parameters 234
11.3.3 Initial and Boundary Conditions 235
11.3.4 Numerical Solution 236
11.3.5 Stress and Strain Radial Profiles 238
11.4 Conclusions 240
Acknowledgements 240
References 240
12 Generalized Continua Concepts in Coarse-Graining Atomistic Simulations 243
Abstract 243
12.1 Generalized Continuum Mechanics (GCM) 244
12.2 Atomistic Field Theory (AFT) 245
12.3 The Concurrent Atomistic-Continuum (CAC) Method 249
12.3.1 A Comparison Between CAC and Other Multiscale Methods 249
12.3.2 Code Development 251
12.3.3 Numerical Implementations in PyCAC 252
12.4 Applications of the CAC Method to Metal Plasticity 253
12.4.1 Static Dislocation Properties 254
12.4.2 Fast Moving Dislocations and Phonons 256
12.4.3 Dislocation/GB Interactions 258
12.5 Conclusions 260
Acknowledgements 261
References 261
13 Bending of a Cantilever Piezoelectric Semiconductor Fiber Under an End Force 267
Abstract 267
13.1 Introduction 268
13.2 Three-Dimensional Equations 268
13.3 One-Dimensional Equations 270
13.4 A Cantilever Under a Transverse End Force 274
13.5 Numerical Results and Discussion 276
13.6 Conclusions 281
Acknowledgements 282
References 282
14 Contact Mechanics in the Framework of Couple Stress Elasticity 285
Abstract 285
14.1 Introduction 286
14.2 Basic Equations in Plane-Strain 288
14.3 Green’s Functions 291
14.4 Formulation of Contact Problems 298
14.5 Singular Integral Equation Approach 301
14.5.1 Indentation by a Flat Punch 301
14.5.1.1 Complete Contact 302
14.5.1.2 Receding Contact 303
14.5.2 Indentation by a Cylindrical Indenter 304
14.5.3 Indentation by a Wedge Indenter 304
14.6 Results and Discussion 305
14.7 Conclusions 310
References 310
15 Radiation from Equivalent Body Forces for Scattering of Surface Waves by a Near-Surface Cylindrical Cavity 313
Abstract 313
15.1 Introduction 313
15.2 Formulation 315
15.3 Equivalent Body Forces 318
15.4 Surface Waves Generated by the Equivalent Body Forces 322
15.4.1 Surface Waves Generated by the Equivalent Body Forces Due to u_{x} 322
15.4.2 Surface Waves Generated by the Equivalent Body Forces Due to u_{z} 329
15.5 Conclusions 332
Acknowledgements 333
References 333
16 Correction to: Generalized Models and Non-classical Approaches in Complex Materials 2 335
Correction to: H. Altenbach et al. (eds.), Generalized Models and Non-classical Approaches in Complex Materials 2, Advanced Structured Materials 90, https://doi.org/10.1007/978-3-319-77504-3 335