Continuous Media with Microstructure

✍ Scribed by Bettina Albers (editor)

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 384
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Title Page
Preface
Contents
List of Contributors
Part I SCIENTIFIC LIFE OF PROF. DR. KRZYSZTOFWILMA N´ SKI
Part II THERMODYNAMIC MODELING
On Pore Fluid Pressure and Effective Solid Stress in the Mixture Theory of Porous Media
Introduction
Mixture of Elastic Materials
Summary of Results for Elastic Solid-Fluid Mixtures
Jump Condition at Fluid-Permeable Surface
Saturated Porous Media
Pore Fluid Pressure
Equations of Motion
Linear Theory and Darcy's Law
Incompressible Porous Media
Effective Stress Principle
An Equilibrium Solution
References
An Extrapolation of Thermodynamics to Evolutionary Genetics
Introduction
Selective Free Energy of a Haploid Population
Model Population: Number of Realizations and Entropy
Entropy
Mutation without Selection. Maximum Entropy
Selection without Mutation. Minimum of Selective Energy
Mutation and Selection Together
Mutational Intensity and Selective “Free Energy”
Other Forms of the Selective Energy
Analogy to Thermodynamics of Binary Chemically Reacting Mixtures
Chemical Potentials and Law of Mass Action
“Chemical Equilibrium”
The Selective Energy Function (5)
The Selective Energy Function (10)
References
Some Recent Results on Multi-temperature Mixture of Fluids
Mixtures in Rational Thermodynamics
Euler Fluids and Comparison between $MT$ and $ST$ Models
Symmetric Hyperbolic System and Principal Subsystems
Qualitative Analysis
The K-Condition in the Mixture Theories
Average Temperature
Examples of Spatially Homogenous Mixture and Static Heat Conduction
Solution of a Spatially Homogenous Mixture
Static Heat Conduction Solution
Maxwellian Iteration
A Classical Approach of Multi-temperature Mixtures
Conclusions
References
Part III EXTENSIONS OF CONSTITUTIVE LAWS
Hypocontinua
Preamble
Essential Ingredients
Reminder of Definitions. Comments
Balance Laws for Hypocontinua
References
On Constitutive Choices for Smectic Elastomers
Introduction
The Smectic Elastomers
Balance Equations for Continua with Vectorial Microstructure
Constraints for Smectic Elastomers
Restrictions on the Constitutive Equations
Final Remarks and Conclusions
References
A Note on the Representation of Cosserat Rotation
Background: Cosserat Rotations
Quaternions as Tensors
Application to Cosserat Rotations
Conclusions
References
Material Uniformity and the Concept of the Stress Space
Introduction
Hyperelastic Unifomity
Configurations and the Cauchy Metric
Material Uniformity
Material Connections
The Multiplicative Decomposition of the Deformation Gradient
The Stress Space
Examples
References
Coupled Nonlinear Thermoelastic Equations for an Orthotropic Beam with Thermal and Viscous Dissipation
Introduction
Basic Equations
Second Law of Thermodynamics
Specific Thermoelastic Constitutive Equations
Specification of the Assigned Fields
Specific Constitutive Equations for the Viscous Terms
Initial and Boundary Conditions
References
On the Mathematical Modelling of Functionally Graded Composites with a Determistinic Microstructure
Preface
Analytical Preliminaries
Averaging of Tolerance Periodic Functions
Tolerance Averaging of Integral Functionals
Model Formulation
Example
References
Part IV MICRO- AND NANOSCALE MECHANICS
On the Derivation of Biological Tissue Models from Kinetic Models of Multicellular GrowingSystems
Introduction
The Mathematical Model and Scaling
Asymptotic Analysis
Critical Analysis and Perspectives
References
Instabilites in Arch Shaped MEMS
Introduction
Mathematical Model
Results for a Sinusoidal Arch MEMS
Effect of Viscous Damping
Conclusions
References
Towards Poroelasticity of Fractal Materials
Background
Fractal Structures and Product Measures
Governing Relations
Conclusion
References
The Maxwell Problem (Mathematical Aspects)
Introduction
The State Equation. Closure
Linear Analysis. Reduction to a Quadratic Matrix Equation
Reduction to a Quadratic Matrix Equation
Solutions to the Quadratic Matrix Equation in the Case $|Λ| \neq 0$
Explicit Formula
The Number of Solutions
The Lyapunov Equation. Separation of Dynamics
Crack Condition and the Existence of an Attracting Manifold
Nonlinear Analysis. Chapman Projection
Statement of the Problem and Auxiliaries
Method of Successive Approximations
Construction of a Nonlinear Chapman Projection
Properties of Nonlinear Projections
References
Continuum-Molecular Modeling of Nanostructured Materials
Introduction
Field Quantities in the Discrete Systems
Hyperelastic Nanocontinuum
Computational Method
Numerical Results
Conclusions
References
Part V WAVES
LinearWave Propagation in Unsaturated Rocks and Soils
Introduction
Microstructural Variables, Microscopic Material Parameters
Porosity and Mass Densities
Compressibility of the Skeleton, Poisson’s Number, Shear Modulus
Saturation and Capillary Pressure
Compressibilities of Fluid and Gas, Viscosity, Permeability
Governing Equations
Linear Three-Component Model for Elastic Porous and Granular Media
Micro-macro Transition
Wave Propagation in Partially Saturated Rocks
Numerical Analysis
Discussion of Numerical Results
FinalRemarks
References
Explicit Solution Formulas for the Acoustic Diffraction Problem with a Slit in a Hard and aSoft Screen
Introduction
The Acoustic Diffraction Problem
The Soft Screen with a Slit
The Hard Screen with a Slit
Explicit Solution Formulas for the Operators with the Logarithmic Kernel
Diffraction through a Slit in a Soft Screen
Diffraction through a Slit in a Hard Screen
References
On the Stability of the Inversion of Measured SeismicWave Velocities to Estimate Porosity in Fluid-Saturated Porous Media
Introduction
Propagation of SeismicWaves in Fluid-Saturated PorousMedia
Estimate of Porosity from Measured SeismicWave Velocities
Stability of the Inversion Algorithm
Applications
Conclusions
References
On Two Insufficiently Exploited Conservation Laws in Continuum Mechanics: Canonical Momentum and Action
On Two Insufficiently Exploited Conservation Laws in Continuum Mechanics: Canonical Momentum and Action
Introduction
Motivation: Variational Formulation and Noether’s Theorem
General Case of Simple Materials
Nonsimple and Microstructured Materials
The Appearance ofWave Action and Its Conservation
References
Waves and Dislocations
Introduction
Elasticity Theory
Sound Waves and Dislocations
SpinWaves and Dislocations
Matter Waves and Dislocations
References
Part VI PHASE TRANSITIONS
Liquid-Solid Phase Transitions in a Deformable Container
Introduction
The Model
Balance Equations
Energy and Entropy
Equilibria
References
Composite Beams with Embedded Shape Memory Alloy
Introduction
Constitutive Relations for a Shape Memory Alloy
Bending of a SMA Composite Beam
Incremental Problem
Numerical Examples
Conclusion
References
Microstructures in the Ti50Ni50−xPdx Alloys’Cubic-to-Orthorhombic Phase Transformation: A Proposed Energy Landscape
Introduction
Preliminaries
The Twinning Condition
A Basic Free Energy Density
Conclusions
References
Part VII GRANULAR MEDIA
Analysis of Shear Banding with a Hypoplastic Constitutive Model for a Dry and Cohesionless Granular Material
Introduction
Concept of Hypoplasticity
Modelling Inelastic Material Properties
Modelling SOM Properties
Modelling Limit Stress States and Critical States
Specific Hypoplastic Model for Sand by Gudehus and Bauer
Shear Band Bifurcation Analysis
Conclusion
References
Principal Axes and Values of the Dispersion Coefficient in the 2D Axially Symmetric Porous Medium
Introduction
Dispersion in Two-Dimensional Anisotropic Porous Media
References
The Importance of Sand in Earth Sciences
The Perception of Sand Outside the Earth Sciences
Similarities to Other Geomaterials
Similarity to Rock
Similarity to the Earth Mantle
Mechanical Behaviour of Sand
The Physics of the Grain Skeleton
Strain Localisation and Pattern Formation
Experimental Observations—Proportional Loading
Other Experimental Evidence
Mathematical Models
Barodesy
Plasticity Theory without Yield Surfaces
References
Author Index