Continuous Media with Microstructure 2 (eBook)

Preface 6
Contents 8
Contributors 11
Krzysztof Wilmanski (1940--2012) 14
Part I Continuum Mechanics 21
Virtual Power and Pseudobalance Equations for Generalized Continua 22
References 32
Hypocontinua 33
1 Preamble 33
2 Balances 39
3 Hypocontinua 47
4 Extended Mechanics 50
References 53
Some Remarks to Higher Order Frames Occurring in Continuum Mechanics 54
1 Introduction 54
2 The Configuration 55
3 Frame Bundles (of the First Order) 57
4 Jet Groups: Toupin Subgroups 58
5 Frame Bundles (of a General Order) 59
6 Higher Order Deformation Bundles and Reductions of Jet Groups---Interesting Challenges 60
References 61
On the Origin of Balance Equations in Simple and Complex Continua: Unified View 62
1 Preamble 62
2 Reasons for Thinking of the Origins of the Balance Equations 63
3 Remarks on the Geometric Language 66
4 Balance Equations: Ways to Get Them 67
4.1 A Summary of Possibilities 67
4.2 Remarks on the List Above 68
5 Multi-field Setting for Continua with Active Microstructure 70
6 Observers and Their Changes 73
6.1 Synchronous Isometry-Based Changes of Observers in Multi-field Setting 74
6.2 Extending the Setting: Diffeomorphism-Based Changes in Observers 76
7 External Power: Consequence of the Invariance Under Isometry-Based Changes in Observers 76
8 Presence of Macroscopic Material Mutations: The Relative Power 80
9 Covariance of the Second Law 82
9.1 Covariance Principle in Dissipative Setting 82
9.2 An Extended Expression of the Mechanical Dissipation Inequality 82
References 83
A Refined Theory of the Layered Medium with the Slip at the Interface 86
1 Introduction 86
2 Refined Equations 87
2.1 Task in Cell for n=1 90
2.2 Task in Cell for n=2 90
2.3 Task in Cell for n=3 91
3 Variants of Averaged System of Equations 92
4 Wave Properties of Layered Medium with Slippage at Inter-layer Boundaries 95
4.1 Plane Harmonic Waves 95
4.2 Surface Rayleigh Waves 99
5 Conclusion 103
References 103
Affinely Rigid Body and Affine Invariance in Physics 104
1 Introduction 104
2 Lie Groups as Configuration Spaces 105
3 Affinely Rigid Body as a Mechanical System on the Homogeneous Space 113
4 Towards Affine Dynamical Models 118
References 126
Part II Thermodynamics 128
An Alternative to the Allen-Cahn Phase Field Model for Interfaces in Solids---Numerical Efficiency 129
1 Introduction 129
2 The Clausius-Duhem Inequality and the Allen-Cahn Equation 130
3 Formulation of an Alternative Phase Field Equation 132
4 The Allen-Cahn and the Hybrid Models 134
5 Model Error and Asymptotics 135
6 Characteristic Equations 139
7 Comparison of the Models, Numerical Efficiency 141
References 144
Thermoelastic Waves in Microstructured Solids 145
1 Introduction 145
2 Internal Variables Formalism 146
2.1 Single Internal Variable in One Dimension 146
2.2 Dual Internal Variables 149
3 One-dimensional Thermoelasticity in Solids with Microstructure 151
3.1 Reminder: Classical Linear Thermoelasticity 151
3.2 Microstructure Influence: Dual Internal Variables 153
4 Numerical Simulations 155
5 Conclusions 157
References 157
Unconventional Thermodynamical Model of Processes in Material Structures 159
1 Introduction 159
2 General Structure of Thermomechanical Modelling 160
2.1 Jump Conditions Across a Singular Surface for Balances 162
2.2 Jump Conditions for External Fields Influencing the Material System 163
2.3 Jump Conditions for Evolution Equations of Tensor, Vector and Scalar Fields 163
3 Application of the Proposed Model 164
4 Conclusion 167
References 168
A Monatomic Ideal Gas---Prototype of a Continuous Medium with Microstructure 169
1 Introduction 170
2 Microstructural Effects 171
2.1 Coriolis Effect on the Heat Flux 171
2.2 Explicit Form for N = 3. Navier-Stokes-Fourier Equations and Grad 13-Moment Equations 173
2.3 Acceleration Waves 175
2.4 Shock Waves 177
3 Light Scattering 178
References 181
From Second Law Violations to Continuum Mechanics 182
1 Motivation 182
2 Background: Fluctuation Theorem 183
3 Entropy Is a Submartingale 185
4 Violations of Second Law in Diffusion Problems 186
5 Micropolar Fluid Model 188
5.1 Dissipation Functions 188
5.2 Upscaling from Stochastic to Deterministic Media 191
6 Conclusions 192
References 192
Phase Change Materials and Thermochemical Materials for Large-Scale Energy Storage 194
1 Introduction 195
2 Theory 195
3 Experiments 196
3.1 Reactor Design 196
3.2 Microscopy 199
4 Sugar Alcohols as PCMs 200
5 Molecular Modeling Method for PCMs 202
6 Molecular Modeling Methods for TCMs 203
References 203
Distribution of Temperature in Multicomponent Multilayered Composites 205
1 Introduction 205
2 Object of Analysis 206
3 Modelling Concepts 207
4 Modelling Procedure 209
4.1 First Assumption 209
4.2 Second Assumption 209
5 Modelling Equations 210
6 Examples 210
6.1 Periodicity Cell Composed of an Odd Number of Layers 211
6.2 Periodicity Cell Composed of an Even Number of Layers 213
7 Conclusions 217
References 218
Part III Porous Media 221
Hysteresis in Unsaturated Porous Media---Two Models for Wave Propagation and Engineering Applications 222
1 Introduction 222
2 Continuum Model for Partially Saturated Soils 224
3 Model with Two Types of Hysteresis 226
3.1 Constitutive Equations 227
3.2 Mass Balance 227
3.3 Momentum, Energy, and Entropy Balance 228
3.4 Hysteresis Operators 231
4 Final Remarks 233
References 233
Simulation of the Influence of Grain Damage on the Evolution of Shear Strain Localization 235
1 Introduction 235
2 Micro-Polar Hypoplastic Model 237
3 Numerical Investigation of the Influence of Grain Damage on Plane Shearing 241
4 Conclusions 246
References 246
Non-hydrostatic Free Surface Flows: Saint Venant Versus Boussinesq Depth Integrated Dynamic Equations for River and Granular Flows 249
1 Introduction 249
2 Vertically Integrated Balance Laws of Mass and Horizontal Momentum 251
2.1 Governing Equations 251
2.2 Depth Integrated Balance Law of Mass 253
2.3 Depth Integrated Momentum Equations in the Horizontal Plane 253
3 Depth Integrated Momentum Equation in the z-Direction 254
4 Boussinesq-Type Development for Water Flows 257
4.1 Water Waves on a Horizontal Bottom Surface Treated as an Ideal Fluid 257
4.2 Water Waves on a Horizontal Bottom Treated as a Newtonian Fluid 259
5 Boussinesq-Type Extension of the Savage-Hutter Equations 262
5.1 The Savage-Hutter (S-H) Theory 262
5.2 One-Dimensional Non-hydrostatic Granular Flow Model 263
6 Discussion and Conclusion 267
References 268
A Mixture Theory of Porous Media and Some Problems of Poroelasticity 270
1 An Elastic Solid-Fluid Mixture 270
2 Saturated Porous Media 273
2.1 Pore Fluid Pressure 273
2.2 Equations of Motion 274
2.3 Linear Theory 274
3 Problems in Poroelasticity 276
3.1 Some Models of Porous Media 276
3.2 Boundary Conditions 278
4 Acceleration Waves 280
4.1 Wave Front Propagation 280
4.2 Porous Media with Incompressible Solid Constituent 281
4.3 Porous Media with Incompressible Fluid Constituent 285
4.4 Incompressible Porous Media 286
References 287
Plane Waves, Uniqueness Theorems and Existence of Eigenfrequencies in the Theory of Rigid Bodies with a Double Porosity Structure 289
1 Introduction 289
2 Basic Equations 291
3 Plane Harmonic Waves 292
4 Internal Boundary Value Problems of Steady Vibrations 299
5 Uniqueness Theorems 301
6 Connection Between Plane Waves and Uniqueness of Solutions 305
References 306
Seismic Response of Poroelastic Graded Geological Region with Underground Structures by BIEM 309
1 Introduction 310
2 Problem Statement 311
3 BEM Solution and Numerical Results 315
4 Conclusion 320
References 322
Part IV Engineering Applications 324
Determination of Foundation Coefficients for a 2-Parameter Model on the Basis of Railway Sleeper Deflection 325
1 Introduction 326
2 Types of Railway Sleepers 326
3 Influence of Dynamical Factors Due to Train Velocity and Track State (kvJ Coefficient) 329
4 Beam on Elastic Foundation 330
4.1 Introduction 330
4.2 A BE-Beam on a Two-Parameter Vlasov Foundation (Finite Element Formulation) 331
4.3 Identification of the Parameters for Theoretical Calculations 333
4.4 Numerical Investigations 335
4.5 Experimental Investigations for INBK-7 Sleeper 339
5 Final Remarks 340
References 341
Orthotropic Parameters of PU Foam Used in Sandwich Panels 342
1 Introduction 342
2 Standard Approach and Testing Procedure 343
3 Orthotropic Analysis---Experimental Setup 345
4 Results and Discussion 347
4.1 Tension Test 347
4.2 Compression Test 349
4.3 Sandwich Plates---Approaches 351
5 Conclusions 351
References 352
Numerical Elastic-Plastic Model of RPC in the Plane Stress State 353
1 Introduction 353
2 Two-Scale Elastic-Plastic Model of RPC 355
3 Experimental Tests 357
4 Numerical Example 359
4.1 Mixture I 361
4.2 Mixture II 362
5 Conclusions 363
References 364
Causal Damping Ratio Spectra and Dispersion Functions in Geomaterials from the Exact Solution of Kramers-Kronig Equations of Viscoelasticity 365
1 Introduction 365
2 Some Relevant Facts on Viscoelasticity Theory 367
2.1 Review of Constitutive Equations 367
2.2 Velocity and Attenuation of Viscoelastic Waves 369
2.3 Kramers-Kronig Relations 370
3 Examples of Causal Dispersion Functions and Damping Ratio Spectra Pairs 373
4 Conclusions 378
References 379
A Study of Deformation and Failure of Unidirectional Fiber-Reinforced Polymers Under Transverse Loading by Means of Computational Micromechanics 381
1 Introduction 381
2 Micromechanical Models 382
2.1 Numerical Homogenization Technique 383
2.2 Constitutive Equations of Matrix and Interface 384
2.3 Calibration of the Drucker-Prager Plasticity Model 385
2.4 Calibration of the Interface Model 386
3 Results 387
4 Conclusions 392
References 393
The Dynamic Modelling of Thin Skeletonal Annular Plates 395
1 Introduction 395
2 Direct Description 397
3 Modelling Technique 398
4 Averaging Description 401
5 Applications 402
5.1 Tolerance Model 402
5.2 Shape Functions 405
5.3 Shape Functions 405
5.4 Conclusions 415
References 416