Continuous Media with Microstructure (eBook)

Preface 6
Contents 11
List of Contributors 14
Part I SCIENTIFIC LIFE OF PROF. DR. KRZYSZTOFWILMA N´ SKI 18
Part II THERMODYNAMIC MODELING 32
On Pore Fluid Pressure and Effective Solid Stress in the Mixture Theory of Porous Media 33
Introduction 33
Mixture of Elastic Materials 34
Summary of Results for Elastic Solid-Fluid Mixtures 36
Jump Condition at Fluid-Permeable Surface 37
Saturated Porous Media 37
Pore Fluid Pressure 37
Equations of Motion 38
Linear Theory and Darcy's Law 39
Incompressible Porous Media 40
Effective Stress Principle 41
An Equilibrium Solution 41
References 42
An Extrapolation of Thermodynamics to Evolutionary Genetics 43
Introduction 43
Selective Free Energy of a Haploid Population 44
Model Population: Number of Realizations and Entropy 44
Entropy 45
Mutation without Selection. Maximum Entropy 45
Selection without Mutation. Minimum of Selective Energy 46
Mutation and Selection Together 46
Mutational Intensity and Selective “Free Energy” 48
Other Forms of the Selective Energy 48
Analogy to Thermodynamics of Binary Chemically Reacting Mixtures 50
Chemical Potentials and Law of Mass Action 50
“Chemical Equilibrium” 50
The Selective Energy Function (5) 51
The Selective Energy Function (10) 51
References 52
Some Recent Results on Multi-temperature Mixture of Fluids 53
Mixtures in Rational Thermodynamics 53
Euler Fluids and Comparison between $MT$ and $ST$ Models 57
Symmetric Hyperbolic System and Principal Subsystems 57
Qualitative Analysis 59
The K-Condition in the Mixture Theories 60
Average Temperature 60
Examples of Spatially Homogenous Mixture and Static Heat Conduction 62
Solution of a Spatially Homogenous Mixture 62
Static Heat Conduction Solution 64
Maxwellian Iteration 66
A Classical Approach of Multi-temperature Mixtures 67
Conclusions 69
References 70
Part III EXTENSIONS OF CONSTITUTIVE LAWS 72
Hypocontinua 73
Preamble 73
Essential Ingredients 74
Reminder of Definitions. Comments 76
Balance Laws for Hypocontinua 79
References 82
On Constitutive Choices for Smectic Elastomers 83
Introduction 83
The Smectic Elastomers 84
Balance Equations for Continua with Vectorial Microstructure 86
Constraints for Smectic Elastomers 87
Restrictions on the Constitutive Equations 90
Final Remarks and Conclusions 91
References 92
A Note on the Representation of Cosserat Rotation 94
Background: Cosserat Rotations 94
Quaternions as Tensors 95
Application to Cosserat Rotations 97
Conclusions 98
References 99
Material Uniformity and the Concept of the Stress Space 101
Introduction 101
Hyperelastic Unifomity 102
Configurations and the Cauchy Metric 102
Material Uniformity 103
Material Connections 104
The Multiplicative Decomposition of the Deformation Gradient 105
The Stress Space 108
Examples 110
References 111
Coupled Nonlinear Thermoelastic Equations for an Orthotropic Beam with Thermal and Viscous Dissipation 112
Introduction 112
Basic Equations 112
Second Law of Thermodynamics 116
Specific Thermoelastic Constitutive Equations 116
Specification of the Assigned Fields 120
Specific Constitutive Equations for the Viscous Terms 122
Initial and Boundary Conditions 123
References 124
On the Mathematical Modelling of Functionally Graded Composites with a Determistinic Microstructure 125
Preface 125
Analytical Preliminaries 126
Averaging of Tolerance Periodic Functions 128
Tolerance Averaging of Integral Functionals 129
Model Formulation 131
Example 134
References 136
Part IV MICRO- AND NANOSCALE MECHANICS 137
On the Derivation of Biological Tissue Models from Kinetic Models of Multicellular GrowingSystems 138
Introduction 138
The Mathematical Model and Scaling 140
Asymptotic Analysis 143
Critical Analysis and Perspectives 148
References 150
Instabilites in Arch Shaped MEMS 153
Introduction 153
Mathematical Model 156
Results for a Sinusoidal Arch MEMS 157
Effect of Viscous Damping 157
Conclusions 159
References 159
Towards Poroelasticity of Fractal Materials 162
Background 162
Fractal Structures and Product Measures 164
Governing Relations 166
Conclusion 168
References 168
The Maxwell Problem (Mathematical Aspects) 170
Introduction 170
The State Equation. Closure 170
Linear Analysis. Reduction to a Quadratic Matrix Equation 172
Reduction to a Quadratic Matrix Equation 172
Solutions to the Quadratic Matrix Equation in the Case $|.| /neq 0$ 174
Explicit Formula 176
The Number of Solutions 177
The Lyapunov Equation. Separation of Dynamics 177
Crack Condition and the Existence of an Attracting Manifold 180
Nonlinear Analysis. Chapman Projection 182
Statement of the Problem and Auxiliaries 182
Method of Successive Approximations 184
Construction of a Nonlinear Chapman Projection 186
Properties of Nonlinear Projections 190
References 192
Continuum-Molecular Modeling of Nanostructured Materials 194
Introduction 194
Field Quantities in the Discrete Systems 195
Hyperelastic Nanocontinuum 196
Computational Method 201
Numerical Results 203
Conclusions 205
References 205
Part V WAVES 207
LinearWave Propagation in Unsaturated Rocks and Soils 208
Introduction 208
Microstructural Variables, Microscopic Material Parameters 209
Porosity and Mass Densities 209
Compressibility of the Skeleton, Poisson’s Number, Shear Modulus 210
Saturation and Capillary Pressure 211
Compressibilities of Fluid and Gas, Viscosity, Permeability 212
Governing Equations 213
Linear Three-Component Model for Elastic Porous and Granular Media 213
Micro-macro Transition 216
Wave Propagation in Partially Saturated Rocks 217
Numerical Analysis 218
Discussion of Numerical Results 218
FinalRemarks 222
References 222
Explicit Solution Formulas for the Acoustic Diffraction Problem with a Slit in a Hard and aSoft Screen 224
Introduction 224
The Acoustic Diffraction Problem 225
The Soft Screen with a Slit 225
The Hard Screen with a Slit 226
Explicit Solution Formulas for the Operators with the Logarithmic Kernel 226
Diffraction through a Slit in a Soft Screen 226
Diffraction through a Slit in a Hard Screen 231
References 233
On the Stability of the Inversion of Measured SeismicWave Velocities to Estimate Porosity in Fluid-Saturated Porous Media 235
Introduction 235
Propagation of SeismicWaves in Fluid-Saturated PorousMedia 237
Estimate of Porosity from Measured SeismicWave Velocities 240
Stability of the Inversion Algorithm 241
Applications 245
Conclusions 249
References 249
On Two Insufficiently Exploited Conservation Laws in Continuum Mechanics: Canonical Momentum and Action 252
On Two Insufficiently Exploited Conservation Laws in Continuum Mechanics: Canonical Momentum and Action 252
Introduction 252
Motivation: Variational Formulation and Noether’s Theorem 254
General Case of Simple Materials 258
Nonsimple and Microstructured Materials 264
The Appearance ofWave Action and Its Conservation 264
References 267
Waves and Dislocations 270
Introduction 271
Elasticity Theory 273
Sound Waves and Dislocations 275
SpinWaves and Dislocations 277
Matter Waves and Dislocations 280
References 283
Part VI PHASE TRANSITIONS 284
Liquid-Solid Phase Transitions in a Deformable Container 285
Introduction 285
The Model 287
Balance Equations 289
Energy and Entropy 292
Equilibria 294
References 299
Composite Beams with Embedded Shape Memory Alloy 301
Introduction 301
Constitutive Relations for a Shape Memory Alloy 302
Bending of a SMA Composite Beam 305
Incremental Problem 308
Numerical Examples 309
Conclusion 314
References 314
Microstructures in the Ti50Ni50 xPdx Alloys’Cubic-to-Orthorhombic Phase Transformation: A Proposed Energy Landscape 316
Introduction 316
Preliminaries 317
The Twinning Condition 320
A Basic Free Energy Density 322
Conclusions 328
References 331
Part VII GRANULAR MEDIA 332
Analysis of Shear Banding with a Hypoplastic Constitutive Model for a Dry and Cohesionless Granular Material 333
Introduction 333
Concept of Hypoplasticity 335
Modelling Inelastic Material Properties 336
Modelling SOM Properties 337
Modelling Limit Stress States and Critical States 338
Specific Hypoplastic Model for Sand by Gudehus and Bauer 340
Shear Band Bifurcation Analysis 344
Conclusion 346
References 346
Principal Axes and Values of the Dispersion Coefficient in the 2D Axially Symmetric Porous Medium 349
Introduction 349
Dispersion in Two-Dimensional Anisotropic Porous Media 351
References 353
The Importance of Sand in Earth Sciences 354
The Perception of Sand Outside the Earth Sciences 354
Similarities to Other Geomaterials 356
Similarity to Rock 356
Similarity to the Earth Mantle 361
Mechanical Behaviour of Sand 365
The Physics of the Grain Skeleton 365
Strain Localisation and Pattern Formation 366
Experimental Observations—Proportional Loading 368
Other Experimental Evidence 369
Mathematical Models 371
Barodesy 372
Plasticity Theory without Yield Surfaces 376
References 377
Author Index 379