Albers Continuous Media with Microstructure

1. Auflage 2010
ISBN: 978-3-642-11445-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 389 Seiten

This book discusses the extension of classical continuum models. To the first class addressed belong various thermodynamic models of multicomponent systems, and to the second class belong primarily microstructures created by phase transformations.

Autoren/Hrsg.

Albers, Bettina

Inhaltsverzeichnis

1;Preface;6
2;Contents;11
3;List of Contributors;14
4;Part I SCIENTIFIC LIFE OF PROF. DR. KRZYSZTOFWILMA N´ SKI;18
5;Part II THERMODYNAMIC MODELING;32
5.1;On Pore Fluid Pressure and Effective Solid Stress in the Mixture Theory of Porous Media;33
5.1.1;Introduction;33
5.1.2;Mixture of Elastic Materials;34
5.1.2.1;Summary of Results for Elastic Solid-Fluid Mixtures;36
5.1.2.2;Jump Condition at Fluid-Permeable Surface;37
5.1.3;Saturated Porous Media;37
5.1.3.1;Pore Fluid Pressure;37
5.1.3.2;Equations of Motion;38
5.1.3.3;Linear Theory and Darcy's Law;39
5.1.4;Incompressible Porous Media;40
5.1.4.1;Effective Stress Principle;41
5.1.4.2;An Equilibrium Solution;41
5.1.5;References;42
5.2;An Extrapolation of Thermodynamics to Evolutionary Genetics;43
5.2.1;Introduction;43
5.2.2;Selective Free Energy of a Haploid Population;44
5.2.2.1;Model Population: Number of Realizations and Entropy;44
5.2.2.2;Entropy;45
5.2.2.3;Mutation without Selection. Maximum Entropy;45
5.2.2.4;Selection without Mutation. Minimum of Selective Energy;46
5.2.2.5;Mutation and Selection Together;46
5.2.2.6;Mutational Intensity and Selective “Free Energy”;48
5.2.2.7;Other Forms of the Selective Energy;48
5.2.3;Analogy to Thermodynamics of Binary Chemically Reacting Mixtures;50
5.2.3.1;Chemical Potentials and Law of Mass Action;50
5.2.3.2;“Chemical Equilibrium”;50
5.2.3.3;The Selective Energy Function (5);51
5.2.3.4;The Selective Energy Function (10);51
5.2.4;References;52
5.3;Some Recent Results on Multi-temperature Mixture of Fluids;53
5.3.1;Mixtures in Rational Thermodynamics;53
5.3.2;Euler Fluids and Comparison between $MT$ and $ST$ Models;57
5.3.2.1;Symmetric Hyperbolic System and Principal Subsystems;57
5.3.2.2;Qualitative Analysis;59
5.3.2.3;The K-Condition in the Mixture Theories;60
5.3.3;Average Temperature;60
5.3.4;Examples of Spatially Homogenous Mixture and Static Heat Conduction;62
5.3.4.1;Solution of a Spatially Homogenous Mixture;62
5.3.4.2;Static Heat Conduction Solution;64
5.3.5;Maxwellian Iteration;66
5.3.6;A Classical Approach of Multi-temperature Mixtures;67
5.3.7;Conclusions;69
5.3.8;References;70
6;Part III EXTENSIONS OF CONSTITUTIVE LAWS;72
6.1;Hypocontinua;73
6.1.1;Preamble;73
6.1.2;Essential Ingredients;74
6.1.3;Reminder of Definitions. Comments;76
6.1.4;Balance Laws for Hypocontinua;79
6.1.5;References;82
6.2;On Constitutive Choices for Smectic Elastomers;83
6.2.1;Introduction;83
6.2.2;The Smectic Elastomers;84
6.2.3;Balance Equations for Continua with Vectorial Microstructure;86
6.2.4;Constraints for Smectic Elastomers;87
6.2.5;Restrictions on the Constitutive Equations;90
6.2.6;Final Remarks and Conclusions;91
6.2.7;References;92
6.3;A Note on the Representation of Cosserat Rotation;94
6.3.1;Background: Cosserat Rotations;94
6.3.2;Quaternions as Tensors;95
6.3.3;Application to Cosserat Rotations;97
6.3.4;Conclusions;98
6.3.5;References;99
6.4;Material Uniformity and the Concept of the Stress Space;101
6.4.1;Introduction;101
6.4.2;Hyperelastic Unifomity;102
6.4.2.1;Configurations and the Cauchy Metric;102
6.4.2.2;Material Uniformity;103
6.4.2.3;Material Connections;104
6.4.3;The Multiplicative Decomposition of the Deformation Gradient;105
6.4.4;The Stress Space;108
6.4.5;Examples;110
6.4.6;References;111
6.5;Coupled Nonlinear Thermoelastic Equations for an Orthotropic Beam with Thermal and Viscous Dissipation;112
6.5.1;Introduction;112
6.5.2;Basic Equations;112
6.5.3;Second Law of Thermodynamics;116
6.5.4;Specific Thermoelastic Constitutive Equations;116
6.5.5;Specification of the Assigned Fields;120
6.5.6;Specific Constitutive Equations for the Viscous Terms;122
6.5.7;Initial and Boundary Conditions;123
6.5.8;References;124
6.6;On the Mathematical Modelling of Functionally Graded Composites with a Determistinic Microstructure;125
6.6.1;Preface;125
6.6.2;Analytical Preliminaries;126
6.6.3;Averaging of Tolerance Periodic Functions;128
6.6.4;Tolerance Averaging of Integral Functionals;129
6.6.5;Model Formulation;131
6.6.6;Example;134
6.6.7;References;136
7;Part IV MICRO- AND NANOSCALE MECHANICS;137
7.1;On the Derivation of Biological Tissue Models from Kinetic Models of Multicellular GrowingSystems;138
7.1.1;Introduction;138
7.1.2;The Mathematical Model and Scaling;140
7.1.3;Asymptotic Analysis;143
7.1.4;Critical Analysis and Perspectives;148
7.1.5;References;150
7.2;Instabilites in Arch Shaped MEMS;153
7.2.1;Introduction;153
7.2.2;Mathematical Model;156
7.2.3;Results for a Sinusoidal Arch MEMS;157
7.2.4;Effect of Viscous Damping;157
7.2.5;Conclusions;159
7.2.6;References;159
7.3;Towards Poroelasticity of Fractal Materials;162
7.3.1;Background;162
7.3.2;Fractal Structures and Product Measures;164
7.3.3;Governing Relations;166
7.3.4;Conclusion;168
7.3.5;References;168
7.4;The Maxwell Problem (Mathematical Aspects);170
7.4.1;Introduction;170
7.4.1.1;The State Equation. Closure;170
7.4.2;Linear Analysis. Reduction to a Quadratic Matrix Equation;172
7.4.2.1;Reduction to a Quadratic Matrix Equation;172
7.4.2.2;Solutions to the Quadratic Matrix Equation in the Case $|.| \neq 0$;174
7.4.2.3;Explicit Formula;176
7.4.2.4;The Number of Solutions;177
7.4.2.5;The Lyapunov Equation. Separation of Dynamics;177
7.4.2.6;Crack Condition and the Existence of an Attracting Manifold;180
7.4.3;Nonlinear Analysis. Chapman Projection;182
7.4.3.1;Statement of the Problem and Auxiliaries;182
7.4.3.2;Method of Successive Approximations;184
7.4.3.3;Construction of a Nonlinear Chapman Projection;186
7.4.3.4;Properties of Nonlinear Projections;190
7.4.4;References;192
7.5;Continuum-Molecular Modeling of Nanostructured Materials;194
7.5.1;Introduction;194
7.5.2;Field Quantities in the Discrete Systems;195
7.5.3;Hyperelastic Nanocontinuum;196
7.5.4;Computational Method;201
7.5.5;Numerical Results;203
7.5.6;Conclusions;205
7.5.7;References;205
8;Part V WAVES;207
8.1;LinearWave Propagation in Unsaturated Rocks and Soils;208
8.1.1;Introduction;208
8.1.2;Microstructural Variables, Microscopic Material Parameters;209
8.1.2.1;Porosity and Mass Densities;209
8.1.2.2;Compressibility of the Skeleton, Poisson’s Number, Shear Modulus;210
8.1.2.3;Saturation and Capillary Pressure;211
8.1.2.4;Compressibilities of Fluid and Gas, Viscosity, Permeability;212
8.1.3;Governing Equations;213
8.1.3.1;Linear Three-Component Model for Elastic Porous and Granular Media;213
8.1.3.2;Micro-macro Transition;216
8.1.4;Wave Propagation in Partially Saturated Rocks;217
8.1.5;Numerical Analysis;218
8.1.5.1;Discussion of Numerical Results;218
8.1.5.2;FinalRemarks;222
8.1.5.3;References;222
8.2;Explicit Solution Formulas for the Acoustic Diffraction Problem with a Slit in a Hard and aSoft Screen;224
8.2.1;Introduction;224
8.2.2;The Acoustic Diffraction Problem;225
8.2.2.1;The Soft Screen with a Slit;225
8.2.2.2;The Hard Screen with a Slit;226
8.2.3;Explicit Solution Formulas for the Operators with the Logarithmic Kernel;226
8.2.3.1;Diffraction through a Slit in a Soft Screen;226
8.2.3.2;Diffraction through a Slit in a Hard Screen;231
8.2.3.3;References;233
8.3;On the Stability of the Inversion of Measured SeismicWave Velocities to Estimate Porosity in Fluid-Saturated Porous Media;235
8.3.1;Introduction;235
8.3.2;Propagation of SeismicWaves in Fluid-Saturated PorousMedia;237
8.3.3;Estimate of Porosity from Measured SeismicWave Velocities;240
8.3.4;Stability of the Inversion Algorithm;241
8.3.5;Applications;245
8.3.6;Conclusions;249
8.3.7;References;249
8.4;On Two Insufficiently Exploited Conservation Laws in Continuum Mechanics: Canonical Momentum and Action;252
8.5;On Two Insufficiently Exploited Conservation Laws in Continuum Mechanics: Canonical Momentum and Action;252
8.5.1;Introduction;252
8.5.2;Motivation: Variational Formulation and Noether’s Theorem;254
8.5.3;General Case of Simple Materials;258
8.5.4;Nonsimple and Microstructured Materials;264
8.5.5;The Appearance ofWave Action and Its Conservation;264
8.5.6;References;267
8.6;Waves and Dislocations;270
8.6.1;Introduction;271
8.6.2;Elasticity Theory;273
8.6.3;Sound Waves and Dislocations;275
8.6.4;SpinWaves and Dislocations;277
8.6.5;Matter Waves and Dislocations;280
8.6.6;References;283
9;Part VI PHASE TRANSITIONS;284
9.1;Liquid-Solid Phase Transitions in a Deformable Container;285
9.1.1;Introduction;285
9.1.2;The Model;287
9.1.3;Balance Equations;289
9.1.4;Energy and Entropy;292
9.1.5;Equilibria;294
9.1.6;References;299
9.2;Composite Beams with Embedded Shape Memory Alloy;301
9.2.1;Introduction;301
9.2.2;Constitutive Relations for a Shape Memory Alloy;302
9.2.3;Bending of a SMA Composite Beam;305
9.2.4;Incremental Problem;308
9.2.5;Numerical Examples;309
9.2.6;Conclusion;314
9.2.7;References;314
9.3;Microstructures in the Ti50Ni50 xPdx Alloys’Cubic-to-Orthorhombic Phase Transformation: A Proposed Energy Landscape;316
9.3.1;Introduction;316
9.3.2;Preliminaries;317
9.3.3;The Twinning Condition;320
9.3.4;A Basic Free Energy Density;322
9.3.5;Conclusions;328
9.3.6;References;331
10;Part VII GRANULAR MEDIA;332
10.1;Analysis of Shear Banding with a Hypoplastic Constitutive Model for a Dry and Cohesionless Granular Material;333
10.1.1;Introduction;333
10.1.2;Concept of Hypoplasticity;335
10.1.2.1;Modelling Inelastic Material Properties;336
10.1.2.2;Modelling SOM Properties;337
10.1.2.3;Modelling Limit Stress States and Critical States;338
10.1.3;Specific Hypoplastic Model for Sand by Gudehus and Bauer;340
10.1.4;Shear Band Bifurcation Analysis;344
10.1.5;Conclusion;346
10.1.6;References;346
10.2;Principal Axes and Values of the Dispersion Coefficient in the 2D Axially Symmetric Porous Medium;349
10.2.1;Introduction;349
10.2.2;Dispersion in Two-Dimensional Anisotropic Porous Media;351
10.2.3;References;353
10.3;The Importance of Sand in Earth Sciences;354
10.3.1;The Perception of Sand Outside the Earth Sciences;354
10.3.2;Similarities to Other Geomaterials;356
10.3.2.1;Similarity to Rock;356
10.3.2.2;Similarity to the Earth Mantle;361
10.3.3;Mechanical Behaviour of Sand;365
10.3.3.1;The Physics of the Grain Skeleton;365
10.3.4;Strain Localisation and Pattern Formation;366
10.3.4.1;Experimental Observations—Proportional Loading;368
10.3.4.2;Other Experimental Evidence;369
10.3.5;Mathematical Models;371
10.3.5.1;Barodesy;372
10.3.5.2;Plasticity Theory without Yield Surfaces;376
10.3.5.3;References;377
11;Author Index;379