Altenbach / Pouget / Rousseau Generalized Models and Non-classical Approaches in Complex Materials 1

1. Auflage 2018
ISBN: 978-3-319-72440-9
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 89, 799 Seiten

Reihe: Advanced Structured Materials

This book is the first of 2 special volumes dedicated to the memory of Gérard Maugin. Including 40 papers that reflect his vast field of scientific activity, the contributions discuss non-standard methods (generalized model) to demonstrate the wide range of subjects that were covered by this exceptional scientific leader. The topics range from micromechanical basics to engineering applications, focusing on new models and applications of well-known models to new problems. They include micro-macro aspects, computational endeavors, options for identifying constitutive equations, and old problems with incorrect or non-satisfying solutions based on the classical continua assumptions.

The authors have 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, micromagnetism, electrodynamics of continua, thermomechanics, surface waves and nonlinear waves in continua, lattice dynamics, material equations and biomechanical applications.

Autoren/Hrsg.

Altenbach, Holm

Pouget, Joël

Rousseau, Martine

Collet, Bernard

Michelitsch, Thomas

Inhaltsverzeichnis

1;Foreword;7
2;Preface;13
3;Contents;15
4;List of Contributors;32
5;1 Effective Coefficients and Local Fields of Periodic Fibrous Piezocomposites with 622 Hexagonal Constituents;42
5.1;Abstract;42
5.2;1.1 Introduction;43
5.3;1.2 A Boundary Value Problem of the Linear Piezoelectricity Theory;44
5.4;1.3 Homogenization, Local Problems and Effective Coefficients;45
5.4.1;1.3.1 Explicit Form of the Homogenized Problem, Effective Coefficients and Local Problems;46
5.4.2;1.3.2 Local Fields;47
5.5;1.4 Application to a Binary Fibrous Piezocomposite with Perfect Contact Conditions at the Interfaces;48
5.6;1.5 Local Problems for Fibrous Composites with Constituents of 622 Hexagonal Class;50
5.6.1;1.5.1 Local Problems L23 and L1;51
5.6.2;1.5.2 Effective Coefficients Related with the Local Problems L23 and L1;55
5.6.3;1.5.3 Local Problems L13 and L2 and Related Effective Coefficients;55
5.6.4;1.5.4 On the Computation of the Local Fields from the Solutions of the Local Problem L13;58
5.7;1.6 Numerical Examples;59
5.7.1;1.6.1 Square Array Distribution;60
5.7.2;1.6.2 Rectangular Array Distribution;61
5.7.3;1.6.3 Spatial Distribution of Local Fields;62
5.8;1.7 Concluding Remarks;65
5.9;References;65
6;2 High-Frequency Spectra of SH Guided Waves in Continuously Layered Plates;68
6.1;Abstract;68
6.2;2.1 Introduction;68
6.3;2.2 Statement of the Problem and Main Equations;69
6.4;2.3 The Propagator Matrix and Its Adiabatic Approximation;71
6.5;2.4 Boundary Problems and Their General Solutions;73
6.5.1;2.4.1 Spectral Regions Without Division Points;74
6.5.1.1;2.4.1.1 The Range s < min{?(y)}(I1 in Fig. 2.2);74
6.5.1.2;2.4.1.2 The Range s > max{?(y)}(I2 in Fig. 2.2);74
6.5.2;2.4.2 Spectral Regions with one Division Point;74
6.5.2.1;2.4.2.1 The Case ?'(a) > 0 (the Range II in Fig. 2.2);75
6.5.2.2;2.4.2.2 The Case ?'(a) < 0;75
6.5.3;2.4.3 Spectral Regions with two Division Points;76
6.5.3.1;2.4.3.1 The Case ?'(a) > 0, ?'(b) < 0;76
6.5.3.2;2.4.3.2 The Case ?'(a) < 0, ?'(b) > 0 (the Range III in Fig. 2.2);76
6.5.4;2.4.4 Extension for an Arbitrary Number of Division Points;77
6.5.4.1;2.4.4.1 An Odd Number of Division Points N = 2n?1 (n > 1);77
6.5.4.2;2.4.4.2 An Even Number of Division Points N = 2n (n > 1);79
6.6;2.5 The Low-Slowness Approximation and the Cut-Off Frequencies;80
6.7;2.6 Example of Inhomogeneity Admitting an Explicit Analysis;81
6.7.1;2.6.1 The Region 0 < s ? ?0;82
6.7.1.1;2.6.1.1 The Cut-Off Frequencies of the Spectrum;82
6.7.1.2;2.6.1.2 Spectrum just Under the Level sl = ?0;83
6.7.2;2.6.2 The Region ?0 < s ? ?m;84
6.7.2.1;2.6.2.1 Spectrum just Over the Level sl = ?0;85
6.7.2.2;2.6.2.2 Spectrum Under the Asymptote s = ?m;85
6.8;2.7 Levels Related to Extreme Points on the Slowness Profile;86
6.8.1;2.7.1 An Absolute Minimum of the Function ?(y);86
6.8.1.1;2.7.1.1 Spectral Features just Under the Level s = ?1;86
6.8.1.2;2.7.1.2 Spectrum Features just Over the Level s = ?1;88
6.8.2;2.7.2 The Level Related to an Inflection Point;90
6.8.3;2.7.3 Asymptote Related to Maximum at the Profile ?(y);91
6.9;2.8 Conclusions;92
6.10;References;93
7;3 Nonlinear Schrödinger and Gross - Pitaevskii Equations in the Bohmian or Quantum Fluid Dynamics (QFD) Representation;94
7.1;Abstract;94
7.2;3.1 Introduction;94
7.3;3.2 Polar Representation of the Wave Function;95
7.4;3.3 Conservation Laws;96
7.4.1;3.3.1 Mass Conservation Equation;96
7.4.2;3.3.2 Energy Conservation Equation;96
7.4.3;3.3.3 The Momentum Equation;97
7.4.4;3.3.4 Pressure Interpretation;97
7.4.5;3.3.5 The Lagrangian Representation;97
7.5;3.4 Adding a Dissipation Term as in Navier - Stokes Equation;98
7.6;3.5 Vorticity;99
7.7;3.6 Closing Remarks;100
7.8;References;101
8;4 The Stability of the Plates with Circular Inclusions under Tension;102
8.1;Abstract;102
8.2;4.1 Introduction;102
8.3;4.2 Problem Statement;104
8.4;4.3 Stability Loss;105
8.4.1;4.3.1 Case with Different Poisson’s Ratio;105
8.4.2;4.3.2 A Plate with a Circular Inclusion under Biaxial Tension;108
8.5;References;109
9;5 Unit Cell Models of Viscoelastic Fibrous Composites for Numerical Computation of Effective Properties;110
9.1;Abstract;110
9.2;5.1 Introduction;111
9.3;5.2 Linear viscoelastic relations;112
9.4;5.3 Numerical Homogenization Model;113
9.5;5.4 Results;116
9.6;5.5 Conclusions;121
9.7;References;121
10;6 Inner Resonance in Media Governed by Hyperbolic and Parabolic Dynamic Equations. Principle and Examples;124
10.1;Abstract;124
10.2;6.1 Introduction;125
10.3;6.2 Dynamic Descriptions of Heterogeneous Linear Elastic Media Without and With Inner Resonance;128
10.3.1;6.2.1 Long Wavelength Descriptions;128
10.3.2;6.2.2 Short Wavelength Descriptions;131
10.4;6.3 Inner Resonance in Elastic Composites;132
10.4.1;6.3.1 Requirements for the Occurrence of Inner Resonance in Elastic Bi-Composites;132
10.4.2;6.3.2 Elastic Bi-Composites: High Contrast of Stiffness, Moderate Contrast of Density;134
10.4.2.1;6.3.2.1 Derivation of the Inner-Resonance Behavior by Homogenization;135
10.4.2.2;6.3.2.2 Comments;139
10.4.3;6.3.3 Elastic Bi-Composites: Significant Contrast of Stiffness and of Density;142
10.4.3.1;6.3.3.1 Co-Dynamics Regime at Anti-Resonance Frequencies;143
10.4.3.2;6.3.3.2 Comments;147
10.4.4;6.3.4 Synthesis on the Resonant and Anti-Resonant Co-Dynamic Regimes;148
10.4.5;6.3.5 Reticulated Media: Inner Resonance by Geometrical Contrast;149
10.5;6.4 Inner Resonance in Poro-Acoustics;153
10.5.1;6.4.1 Double Porosity Media: Inner Resonance by High Permeability Contrast;155
10.5.1.1;6.4.1.1 Homogenized Behavior;157
10.5.1.2;6.4.1.2 Comments and Generalization to Other Diffusion Phenomena;160
10.5.2;6.4.2 Embedded Resonators in Porous Media: Inner Resonance by Geometrical Contrast;161
10.5.2.1;6.4.2.1 Helmholtz Resonator;162
10.5.2.2;6.4.2.2 Homogenized Behavior;163
10.5.2.3;6.4.2.3 Comments;164
10.6;6.5 Inner Resonance in Poroelastic Media: Coupling Effect;166
10.6.1;6.5.1 Double Porosity Poro-Elastic Media - Problem Statement;166
10.6.2;6.5.2 Homogenized Behavior;168
10.6.3;6.5.3 Comments;170
10.7;6.6 Conclusions;171
10.8;Appendix: Elastic Bi-Composites: Moderate Stiffness Contrast and High Density Contrast;172
10.9;References;173
11;7 The Balance of Material Momentum Applied to Water Waves;176
11.1;Abstract;176
11.2;7.1 Introduction;176
11.3;7.2 The Balance of Physical Momentum;178
11.4;7.3 The Balance of Material Momentum;180
11.5;7.4 The Energy Balance;185
11.6;7.5 Gerstner’s Wave;186
11.7;7.6 Change of Reference Configuration;191
11.8;7.7 Concluding Remarks;193
11.9;Appendix: Derivatives of the Lagrangian;194
11.10;References;195
12;8 Electromagnetic Fields in Meta-Media with Interfacial Surface Admittance;196
12.1;Abstract;196
12.2;8.1 Introduction;196
12.3;8.2 Mathematical Preliminaries;198
12.4;8.3 General Maxwell Equations and their Fourier Transform;200
12.5;8.4 Maxwell Equations in a Source-Free Domain of an Ohmic, Homogeneous, Isotropic, Dispersive, Linear Medium;202
12.6;8.5 Electromagnetic Fields in a Source-Free Domain of an Ohmic, Homogeneous, Isotropic, Dispersive, Linear Medium;203
12.7;8.6 Plane Wave Solutions in Terms of the Complex Rotation Group;204
12.8;8.7 Interface Conditions for Media Containing Anisotropic, Homogeneous, Planar Interface Consitutive Relations;205
12.9;8.8 Consequences of the Interface Conditions;208
12.10;8.9 Solving the Interface Conditions;213
12.11;8.10 Conclusion;216
12.12;Acknowledgements;218
12.13;References;218
13;9 Evolution Equations for Defects in Finite Elasto-Plasticity;220
13.1;Abstract;220
13.2;9.1 Introduction;220
13.2.1;9.1.1 Defects in Linear eElasticity;221
13.2.2;9.1.2 Defects in Non-Linear Elasticity;221
13.2.3;9.1.3 Defects in Nonlocal Elasticity;222
13.2.4;9.1.4 Elasto-Plastic Models for Defects;222
13.2.5;9.1.5 Aim of this Paper;223
13.2.6;9.1.6 List of Notations;223
13.3;9.2 Elasto-Plastic Materials with Lattice Defects;225
13.3.1;9.2.1 Plastic Connection with Metric Property;227
13.3.2;9.2.2 Measure of Defects;228
13.4;9.3 Free Energy Imbalance Principle Formulated in K;229
13.4.1;9.3.1 Free Energy Function;229
13.4.2;9.3.2 Free Energy Imbalance Principle;232
13.5;9.4 Constitutive Restrictions Imposed by the Imbalance Free Energy Principle;234
13.5.1;9.4.1 Elastic Type Constitutive Equations;234
13.5.2;9.4.2 Dissipation Inequality;235
13.6;9.5 Viscoplastic Type Evolution Equations for Plastic Distortion and Disclination Tensor;236
13.6.1;9.5.1 Quadratic Free Energy;238
13.6.2;9.5.2 Elasto-Plastic Model for Dislocations and Disclinations in the Case of Small Distortions;239
13.7;9.6 Conclusions;241
13.8;References;242
14;10 Viscoelastic effective properties for composites with rectangular cross-section fibers using the asymptotic homogenization method;244
14.1;Abstract;244
14.2;10.1 Introduction;245
14.3;10.2 Statement of the Viscoelastic Heterogeneous Problem;246
14.4;10.3 Two-Scale Asymptotic Homogenization Method to Solve the Heterogeneous Problem;248
14.4.1;10.3.1 Contribution of the Level ??2 Problem;250
14.4.2;10.3.2 Contribution of the Level ??1 Problem;251
14.4.3;10.3.3 Contribution of the Level ? 0 Problem;252
14.5;10.4 Two Phase Viscoelastic Composite;253
14.6;10.5 Numerical Results;255
14.6.1;10.5.1 Model I;255
14.6.2;10.5.2 Model II;257
14.6.3;10.5.3 Viscoelastic Effective Constants for Composites with Rectangular Cross-Section Fibers: Double Homogenization;259
14.7;10.6 Conclusions;261
14.8;Acknowledgements;262
14.9;References;262
15;11 A Single Crystal Beam Bent in Double Slip;264
15.1;Abstract;264
15.2;11.1 Introduction;264
15.3;11.2 3-D Models of Crystal Beam Bent in Double Slip;266
15.4;11.3 Energy Minimization;270
15.5;11.4 Numerical Simulations;276
15.6;11.5 Non-Zero Dissipation;279
15.7;11.6 Numerical Simulation;283
15.8;11.7 Discussion and Outlook;285
15.9;Acknowledgements;286
15.10;References;286
16;12 Acoustic Metamaterials Based on Local Resonances: Homogenization, Optimization and Applications;288
16.1;Abstract;288
16.2;12.1 Introduction;288
16.3;12.2 Locally Resonant Microstructures;291
16.4;12.3 A Survey of Homogenization Techniques;294
16.4.1;12.3.1 Periodic Homogenization;295
16.4.2;12.3.2 Dynamic Homogenization and Willis-Type Constitutive Relations;296
16.4.3;12.3.3 Homogenization from Scattering Properties.;298
16.5;12.4 Topology Optimization;300
16.5.1;12.4.1 Topology Optimization for Local Resonant Sonic Materials;300
16.5.2;12.4.2 Topology Optimization for Hyperbolic Elastic Metamaterials;301
16.5.3;12.4.3 Topology Optimization for Hyperelastic Plates;303
16.6;12.5 Principal Applications: Phononic Crystals;304
16.7;12.6 Conclusions;309
16.8;References;309
17;13 On NonlinearWaves in Media with Complex Properties;316
17.1;Abstract;316
17.2;13.1 Introduction;316
17.3;13.2 The Governing Equations;317
17.3.1;13.2.1 Boussinesq-Type Models;318
17.3.2;13.2.2 Evolution-Type (KdV-Type) Models;319
17.3.3;13.2.3 Coupled Fields;321
17.4;13.3 Physical Effects;322
17.5;13.4 Discussion;325
17.6;References;326
18;14 The Dual Approach to Smooth Defects;328
18.1;Abstract;328
18.2;14.1 Dedication;328
18.3;14.2 Summary of the Direct Approach;328
18.4;14.3 The Dual Perspective;330
18.5;14.4 A Brief Review of Differential Forms;331
18.5.1;14.4.1 Pictorial Representation of Covectors and 1-Forms;331
18.5.2;14.4.2 Exterior Algebra;332
18.5.3;14.4.3 The Exterior Derivative;333
18.5.4;14.4.4 Integration;334
18.6;14.5 An Application to Smectics;334
18.7;14.6 An Application to Nanotubes;337
18.8;14.7 A Volterra Dislocation;339
18.9;References;341
19;15 A Note on Reduced Strain Gradient Elasticity;342
19.1;Abstract;342
19.2;15.1 Introduction;342
19.3;15.2 Reduced Strain Gradient Elasticity. Examples;344
19.3.1;15.2.1 Structural Mechanics;344
19.3.2;15.2.2 Continual Models for Pantographic Beam Lattices;345
19.3.3;15.2.3 Smectics and Columnar Liquid Crystals;347
19.3.4;15.2.4 Other Spatially Non-Symmetric Models;348
19.4;15.3 Conclusions;349
19.5;References;349
20;16 Use and Abuse of the Method of Virtual Power in Generalized Continuum Mechanics and Thermodynamics;352
20.1;Abstract;352
20.2;16.1 Introduction;352
20.3;16.2 Micromorphic and Gradient Plasticity;354
20.3.1;16.2.1 The Micromorphic Approach to Gradient Plasticity;354
20.3.2;16.2.2 Direct Construction of Gradient Plasticity Theory;358
20.4;16.3 Gradient of Entropy or Temperature Models;359
20.4.1;16.3.1 A Principle of Virtual Power for Entropy;359
20.4.2;16.3.2 Gradient of Entropy or Gradient of Temperature?;362
20.5;16.4 The Method of Virtual Power Applied to Phase Field Modelling;363
20.5.1;State Laws and Dissipation Potential;364
20.6;16.5 On the Construction of the Cahn–Hilliard Diffusion Theory;366
20.6.1;16.5.1 Usual Presentation Based on the Variational Derivative;366
20.6.2;16.5.2 Method of Virtual Power with Additional Balance Equation;368
20.6.3;16.5.3 Second Gradient Diffusion Theory;369
20.6.3.1;16.5.3.1 Variational Formulation of Classical Diffusion;369
20.6.3.2;16.5.3.2 Variational Formulation of Second Gradient Diffusion;369
20.7;16.6 Conclusions;372
20.8;References;373
21;17 Forbidden Strains and Stresses in Mechanochemistry of Chemical Reaction Fronts;376
21.1;Abstract;376
21.2;17.1 Introduction;376
21.3;17.2 Chemical Affinity in the Case of Small Strains;379
21.4;17.3 Forbidden Zones;382
21.5;Acknowledgements;386
21.6;References;387
22;18 Generalized Debye Series Theory for Acoustic Scattering: Some Applications;390
22.1;Abstract;390
22.2;18.1 Introduction;390
22.3;18.2 Generalized Debye Series;392
22.3.1;18.2.1 Formulation of the Problem;392
22.3.2;18.2.2 "Local" Modal Reflection and Refraction Coefficients;395
22.3.2.1;18.2.2.1 Reflection and Refraction of aWave Incident from Medium 1 (Fluid) on Medium 2 (Solid);395
22.3.2.2;18.2.2.2 Reflection and Refraction of Wave Incident from Medium 2 on Medium 1;396
22.4;18.3 Transmitted Waves;397
22.5;18.4 Contribution to the Resonance Scattering Theory;398
22.5.1;18.4.1 Case of Solid Submerged Elastic Objects;398
22.5.2;18.4.2 Case of Solid Submerged Lossy Elastic Objects;399
22.5.3;18.4.3 Case of Submerged Elastic Shells;400
22.6;18.5 Non Resonant Background;402
22.7;18.6 Space-Time Dependence of a Bounded Beam Inside an Elastic Cylindrical Guide;405
22.7.1;18.6.1 Propagation Equations;406
22.7.2;18.6.2 Initial Conditions and Limiting Conditions;406
22.7.3;18.6.3 Solution of the Problem: Generalized Debye Series;407
22.7.4;18.6.4 Velocity Fields and Simulation;410
22.8;18.7 Conclusions;412
22.9;References;413
23;19 Simplest Linear Homogeneous Reduced Gyrocontinuum as an Acoustic Metamaterial;416
23.1;Abstract;416
23.2;19.1 Introduction;417
23.3;19.2 Basic Equations for the Linear Reduced Gyrocontinuum;418
23.4;19.3 Special Solution in Case ? = ?0;420
23.5;19.4 LongitudinalWaves and Spectral Problem for the Shear-Rotational Wave;420
23.6;19.5 Shear-Rotational Wave. Reduced Spectral Problem;421
23.7;19.6 Shear-Rotational Wave Propagating Perpendicular to the Rotors’ Axes (k ·m = 0).;422
23.8;19.7 Shear-Rotational Wave Propagating Parallel to the Rotors’ Axes (k×m =0).;423
23.9;19.8 Shear-Rotational Wave Directed in General Way with Respect to the Rotors’ Axes;424
23.10;19.9 Conclusions;426
23.11;Acknowledgements;426
23.12;References;427
24;20 A Mathematical Model of Nucleic Acid Thermodynamics;428
24.1;Abstract;428
24.2;20.1 Introduction;428
24.3;20.2 Denaturation of DNA;429
24.4;20.3 Mathematical Model;430
24.5;20.4 The Role of Parameter b;432
24.6;20.5 Discussion;432
24.7;20.6 Conclusion;434
24.8;References;435
25;21 Bulk Nonlinear Elastic StrainWaves in a Bar with Nanosize Inclusions;436
25.1;Abstract;436
25.2;21.1 Introduction;436
25.3;21.2 Refinement of the Model of a Continuous Microstructured Medium;438
25.4;21.3 Nonlinear Strain Waves in a Bar;442
25.4.1;21.3.1 The Model for Wave Propagation in an Isotropic Bar;442
25.4.2;21.3.2 The Refined Model Application for a Bar with Nanosize Inclusions;446
25.5;21.4 Conclusions;452
25.6;Acknowledgements;453
25.7;21.5 Supplement;453
25.7.1;21.5.1 The Model for Longitudinal Nonlinearly Elastic Damping Waves Propagation in a Microstructured Medium;453
25.7.2;21.5.2 Coefficients of the Coupled Equations in (21.24);455
25.8;References;456
26;22 On the Deformation of Chiral Piezoelectric Plates;458
26.1;Abstract;458
26.2;22.1 Introduction;458
26.3;22.2 Basic Equations;459
26.4;22.3 Chiral Piezoelectric Plates;461
26.5;22.4 General Theorems;464
26.6;22.5 Equilibrium Theory;474
26.7;22.6 Effects of a Concentrated Charge Density;475
26.8;22.7 Conclusions;477
26.9;References;478
27;23 Non-Equilibrium Temperature and Reference Equilibrium Values of Hidden and Internal Variables;480
27.1;Abstract;480
27.2;23.1 Introduction;480
27.3;23.2 Internal Variables and Hidden Variables;481
27.4;23.3 Temperatures in Steady States;484
27.4.1;23.3.1 Asymptotic Equilibrium Expressions for Caloric and Entropic Temperatures;484
27.4.2;23.3.2 Dynamical Steady-State Expressions for Caloric and Entropic Temperatures;485
27.5;23.4 A Model for System’s Aging;488
27.6;23.5 Concluding Remarks;489
27.7;Acknowledgements;490
27.8;References;490
28;24 On the Foundation of a Generalized Nonlocal Extensible Shear Beam Model from Discrete Interactions;492
28.1;Abstract;492
28.2;24.1 Introduction;492
28.3;24.2 The Mechanical Model;494
28.4;24.3 Extensible Engesser Elastica;495
28.4.1;24.3.1 Discrete Extensible Engesser Elastica;495
28.4.1.1;24.3.1.2 Analytical Solution for Short Linkages;499
28.4.1.2;24.3.1.3 Numerical Solution;503
28.4.2;24.3.2 Asymptotic Limit: the Local Extensible Engesser Elastica;503
28.4.3;24.3.3 Continualized Nonlocal Extensible Engesser Elastica;507
28.4.3.1;24.3.3.1 Numerical Solution: Discrete Versus Nonlocal Extensible Engesser Elastica;508
28.5;24.4 Extensible Haringx Elastica;508
28.5.1;24.4.1 Discrete Extensible Haringx Elastica;510
28.5.1.1;24.4.1.1 Buckling Loads;511
28.5.1.2;24.4.1.2 Analytical Solution for Short Linkages;513
28.5.1.3;24.4.1.3 Numerical Solution;516
28.5.2;24.4.2 Asymptotic Limit: the Local Extensible Haringx Elastica;516
28.5.3;24.4.3 Continualized Nonlocal Extensible Haringx Elastica;520
28.5.3.1;24.4.3.1 Numerical Solution: Discrete Versus Nonlocal Extensible Haringx Elastica;521
28.6;24.5 Conclusions;521
28.7;Appendix A;521
28.8;Appendix B;524
28.9;References;525
29;25 A Consistent Dynamic Finite-Strain Plate Theory for Incompressible Hyperelastic Materials;528
29.1;Abstract;528
29.2;25.1 Introduction;528
29.3;25.2 The 3D Governing Equations;530
29.4;25.3 The 2D Dynamic Plate Theory;532
29.4.1;25.3.1 Dynamic 2D Vector Plate Equation;533
29.4.2;25.3.2 Edge Boundary Conditions;537
29.4.2.1;25.3.2.1 Case 1. Prescribed Position in the 3D Formulation;537
29.4.2.2;25.3.2.2 Case 2. Prescribed traction in the 3D formulation;538
29.4.3;25.3.3 Examination of the Consistency;538
29.5;25.4 The Associated Weak Formulations;540
29.5.1;25.4.0.1 Case 1. Edge position and traction in the 3D formulation are known;542
29.5.2;25.4.0.2 Case 2. Edge position and traction in the 3D formulation are unknown;542
29.6;25.5 Conclusions;543
29.7;References;544
30;26 A One-Dimensional Problem of Nonlinear Thermo-Electroelasticity with Thermal Relaxation;546
30.1;Abstract;546
30.2;26.1 Introduction;546
30.3;26.2 The Nonlinear Equations;548
30.4;26.3 The Associated System of Linear Equations;549
30.5;26.4 Numerical Scheme;552
30.6;26.5 Conclusions;554
30.7;References;556
31;27 Analysis of Mechanical Response of Random Skeletal Structure;558
31.1;Abstract;558
31.2;27.1 Introduction;558
31.3;27.2 Material Characterization;560
31.3.1;27.2.1 Extended Voronoi Tessellation;560
31.3.2;27.2.2 Basic Equations;562
31.3.3;27.2.3 Finite Element Discretization;563
31.4;27.3 Analyses and Discussions;565
31.4.1;27.3.1 Preparation of Skeletal Model;565
31.4.2;27.3.2 Static Tension;566
31.4.3;27.3.3 Dynamic Loading;568
31.5;27.4 Concluding Remarks;570
31.6;Acknowledgements;571
31.7;References;571
32;28 On the Influence of the Coupled Invariant in Thermo-Electro-Elasticity;573
32.1;Abstract;573
32.2;28.1 Introduction;574
32.2.1;28.1.1 Kinematics;575
32.2.2;28.1.2 Balance Laws in Electrostatics;576
32.2.2.1;28.1.2.1 Spatial Configuration;576
32.2.2.2;28.1.2.2 Material Configuration;578
32.2.3;28.1.3 Heat Equation;579
32.2.4;28.1.4 Energy Function;581
32.3;28.2 Non-Homogeneous Boundary Value Problems;583
32.3.1;28.2.1 Deformation of a Cube with a Uniaxially Applied Electric Field;584
32.3.2;28.2.2 Extension and Torsion of a Cylindrical Tube;586
32.4;28.3 Conclusions;591
32.5;Acknowledgements;592
32.6;References;592
33;29 On Recurrence and Transience of Fractional Random Walks in Lattices;595
33.1;Abstract;595
33.2;29.1 Introduction;596
33.3;29.2 Time Discrete Markovian Random Walks on Undirected Networks;598
33.4;29.3 Probability Generating Functions - Green’s Functions;601
33.5;29.4 The Fractional Random Walk;605
33.6;29.5 Universality of Fractional Random Walks;607
33.6.1;29.5.1 Universal Behavior in the Limit ? ?0;607
33.6.2;29.5.2 Recurrence Theorem for the Fractional Random Walk on Infinite Simple Cubic Lattices;608
33.6.3;29.5.3 Universal Asymptotic Scaling: Emergence of Lévy Flights;611
33.7;29.6 Transient Regime 0 < ? < 1 for the InifiniteOne-Dimensional Chain;613
33.8;29.7 Conclusions;617
33.9;References;619
34;30 Micropolar Theory with Production of Rotational Inertia: A Rational Mechanics Approach;621
34.1;Abstract;621
34.2;30.1 Review of the Current State-of-the-Art;622
34.3;30.2 Productions of Microinertia and the Coupling Tensor for Transversally Isotropic Media;626
34.4;30.3 Discussion of Special Cases for the Production Term for the Moment of Inertia, ?J;628
34.4.1;30.3.1 Examples for the Isotropic Case;629
34.4.2;30.3.2 Structural Change I: Purely Deviatoric Production;630
34.4.3;30.3.3 Structural Change II: Purely Axial Production;633
34.5;30.4 Dynamics of Micropolar Media with Time-Varying Micro-Inertia;639
34.5.1;30.4.1 General Remarks;639
34.5.2;30.4.2 Axial Elongation and Shrinkage;640
34.6;30.5 Conclusions and Outlook;641
34.7;Acknowledgements;642
34.8;Appendices;642
34.8.1;Representation of the Production of Moment of Inertia;643
34.8.2;Restrictions on the Production of Moment of Inertia by the Second Law;644
34.9;References;645
35;31 Contact Temperature as an Internal Variable of Discrete Systems in Non-Equilibrium;647
35.1;Abstract;647
35.2;31.1 Introduction;647
35.3;31.2 Contact Temperature;648
35.3.1;31.2.1 Definition;648
35.3.2;31.2.2 Contact Temperature and Internal Energy;649
35.4;31.3 State Space and Entropy Rate;650
35.5;31.4 Equilibrium and Reversible "Processes";651
35.6;31.5 Brief Overview of Internal Variables;653
35.7;31.6 Contact Temperature as an Internal Variable;655
35.8;Appendices;656
35.8.1;Heat Exchange and Contact Temperature;656
35.8.2;Contact Temperature and Efficiency;658
35.9;References;660
36;32 Angular Velocities, Twirls, Spins and Rotation Tensors in the Continuum Mechanics Revisited;661
36.1;Abstract;661
36.2;32.1 Introduction;661
36.3;32.2 Rotation Tensor and Angular Velocity Vector;662
36.4;32.3 Rotation Tensors and Spins in the Classical Continuum Mechanics;663
36.4.1;32.3.1 Rotations of Principal Directions and Twirls;664
36.4.2;32.3.2 Logarithmic Spin;667
36.5;32.4 Conclusions;670
36.6;Appendix: Some Operations with Second Rank Tensors;670
36.6.1;Dot Products of a Second Rank Tensor and a Vector;670
36.6.2;Cross Products of a Second Rank Tensor and a Vector;670
36.6.3;Vector Invariant;671
36.7;References;672
37;33 Towards Continuum Mechanics with Spontaneous Violations of the Second Law of Thermodynamics;673
37.1;Abstract;673
37.2;33.1 Dissipation Function in Thermomechanics within Second Law;673
37.3;33.2 Dissipation Function in Statistical Physics beyond Second Law;675
37.4;33.3 Stochastic Dissipation Function;677
37.4.1;33.3.1 Basics;677
37.4.2;33.3.2 Atomic Fluid in Couette Flow;678
37.5;33.4 Closure;679
37.6;References;680
38;34 Nonlocal Approach to Square Lattice Dynamics;681
38.1;Abstract;681
38.2;34.1 Introduction;681
38.3;34.2 Linear Local Model;684
38.4;34.3 Nonlocal Linear Model;686
38.5;34.4 Dispersion Relations Analysis;688
38.6;34.5 Continuum Equations;689
38.6.1;34.5.1 Local Model;690
38.6.2;34.5.2 Nonlocal Model;690
38.6.3;34.5.3 Nonlinear Interaction;692
38.7;34.6 Conclusion;693
38.8;Acknowledgements;693
38.9;References;693
39;35 A New Class of Models to Describe the Response of Electrorheological and Other Field Dependent Fluids;695
39.1;Abstract;695
39.2;35.1 Introduction;695
39.3;35.2 Preliminaries;697
39.4;35.3 Constitutive Relation;699
39.5;35.4 Simple Shear Flow;702
39.5.1;35.4.1 Extra Stress Tensor S is a Linear Function of the Symmetric Part of the Velocity Gradient D;706
39.5.2;35.4.2 Symmetric Part of the Velocity Gradient D is a Linear Function of the Extra Stress Tensor S;707
39.5.3;35.4.3 Extra stress tensor S is a function of the symmetric part ofthe velocity gradient;707
39.5.4;35.4.4 Fully Implicit Constitutive Relation – Constitutive Relation with Bilinear Tensorial Terms;708
39.6;35.5 Conclusion;710
39.7;References;711
40;36 Second Gradient Continuum: Role of Electromagnetism Interacting with the Gravitation on the Presence of Torsion and Curvature;714
40.1;Abstract;714
40.2;36.1 Introduction;714
40.3;36.2 Electromagnetism in Minkowski Spacetime;715
40.3.1;36.2.1 Maxwell’s 3D Equations in Vacuum;715
40.3.2;36.2.2 Covariant Formulation of Maxwell’s Equations;717
40.4;36.3 Electromagnetism in Curved Continuum;718
40.4.1;36.3.1 Variational Method and Covariant Maxwell’s Equations;718
40.4.2;36.3.2 Field Equations and Conservation Laws;720
40.5;36.4 Electromagnetism in Twisted and Curved Continuum;723
40.5.1;36.4.1 Faraday Tensor in Twisted Continuum;723
40.5.2;36.4.2 Field Equations, Wave Equations;724
40.5.3;36.4.3 Electromagnetism and Continuum Defects;727
40.6;36.5 Concluding Remarks;730
40.7;References;732
41;37 Optimal Calculation of Solid-Body Deformations with Prescribed Degrees of Freedom over Smooth Boundaries;734
41.1;Abstract;734
41.2;37.1 Introduction;734
41.3;37.2 Method Description;736
41.4;37.3 Method Experimentation;738
41.4.1;37.3.1 Deflections of an Elastic Membrane;738
41.4.2;37.3.2 Torsion of an Elastic Annulus;740
41.5;37.4 Final Comments;742
41.6;Acknowledgements;742
41.7;References;742
42;38 Toward a Nonlinear Asymptotic Model for Thin Magnetoelastic Plates;744
42.1;Abstract;744
42.2;38.1 Introduction;744
42.3;38.2 Summary of the Three-Dimensional Theory for Conservative Problems;745
42.4;38.3 Reformulation;747
42.5;38.4 Legendre-Hadamard Conditions;748
42.6;38.5 Equations Holding on the Midplane and Small-Thickness Estimates;750
42.7;38.6 Potential Energy of a Thin Plate;751
42.8;38.7 Reduction of the Plate Energy;753
42.9;Acknowledgements;755
42.10;References;755
43;39 Modelling of an Ionic Electroactive Polymer by the Thermodynamics of Linear Irreversible Processes;756
43.1;Abstract;756
43.2;39.1 Introduction;756
43.3;39.2 Description and Modelling of the Material;759
43.3.1;39.2.1 Average Process;760
43.3.2;39.2.2 Interface Modelling;760
43.3.3;39.2.3 Partial Derivatives and Material Derivative;761
43.3.4;39.2.4 Balance Laws;762
43.4;39.3 Conservation Laws;763
43.4.1;39.3.1 Conservation of the Mass;763
43.4.2;39.3.2 Electric Equations;763
43.4.3;39.3.3 Linear Momentum Conservation Law;764
43.4.4;39.3.4 Energy Balance Laws;765
43.4.4.1;39.3.4.1 Potential Energy Balance Equation;765
43.4.4.2;39.3.4.2 Kinetic Energy Balance Equation;765
43.4.4.3;39.3.4.3 Total Energy Balance Equation;766
43.4.4.4;39.3.4.4 Internal Energy Balance Equation;766
43.4.4.5;39.3.4.5 Interpretation of the Equations;766
43.5;39.4 Entropy Production;767
43.5.1;39.4.1 Entropy Balance Law;767
43.5.2;39.4.2 Fundamental Thermodynamic Relations;767
43.5.3;39.4.3 Entropy Production;768
43.5.4;39.4.4 Generalized Forces and Fluxes;769
43.6;39.5 Constitutive Equations;770
43.6.1;39.5.1 Rheological Equation;770
43.6.2;39.5.2 Nafion Physicochemical Properties;771
43.6.3;39.5.3 Nernst-Planck Equation;772
43.6.4;39.5.4 Generalized Darcy’s Law;773
43.7;39.6 Validation of the Model: Application to a Cantilevered Strip;774
43.7.1;39.6.1 Static Equations;774
43.7.2;39.6.2 Beam Model on Large Displacements;776
43.7.3;39.6.3 Simulations Results;778
43.8;39.7 Conclusion;779
43.9;39.8 Notations;780
43.10;References;781
44;40 Weakly Nonlocal Non-Equilibrium Thermodynamics: the Cahn-Hilliard Equation;784
44.1;Abstract;784
44.2;40.1 Introduction;784
44.3;40.2 Variational derivation of Ginzburg-Landau and Cahn-Hilliard equations;787
44.4;40.3 The Thermodynamic Origin of the Ginzburg-Landau (Allen-Cahn) Equation;789
44.4.1;40.3.1 Separation of Full Divergences;789
44.4.2;40.3.2 Ginzburg-Landau Equation: a More Rigorous Derivation;790
44.5;40.4 The Thermodynamic Origin of the Cahn–Hilliard Equation;792
44.5.1;40.4.1 Separation of Full Divergences;792
44.5.2;40.4.2 Cahn-Hilliard Equation: a More Rigorous Derivation;793
44.6;40.5 Discussion;796
44.7;Acknowledgements;797
44.8;References;797