Generalized Models and Non-classical Approaches in Complex Materials 1 (eBook)

Holm Altenbach, Joël Pouget, Martine Rousseau, Bernard Collet, Thomas Michelitsch (Herausgeber)

eBook Download: PDF

2018 | 1st ed. 2018
XLII, 760 Seiten
Springer International Publishing (Verlag)
978-3-319-72440-9 (ISBN)

Infos

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 24.3.2018
• Reihe/Serie: Advanced Structured Materials
• Reihe/Serie: Advanced Structured Materials
• Zusatzinfo: XLII, 760 p.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Complex Materials • continuum approaches • Continuum Mechanics Through the Twentieth Century • Generalized Continua • Gérard MAUGIN • lattice approaches • Non-classical continuum mechanics
• ISBN-10: 3-319-72440-1 / 3319724401
• ISBN-13: 978-3-319-72440-9 / 9783319724409