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| Preface | |
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| Notation | |
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| A Mathematical Framework for Upscaling Operations | |
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| Representative Elementary Volume (rev) | |
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| Averaging Operations | |
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| Apparent and Intrinsic Averages | |
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| Spatial Derivatives of an Average | |
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| Time Derivative of an Average | |
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| Spatial and Time Derivatives of e | |
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| Application to Balance Laws | |
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| Mass Balance | |
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| Momentum Balance | |
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| The Periodic Cell Assumption | |
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| Introduction | |
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| Spatial and Time Derivative of e in the Periodic Case | |
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| Spatial and Time Derivative of [left angle bracket]e[right angle bracket][subscript Alpha] of in the Periodic Case | |
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| Application: Micro- versus Macroscopic Compatibility | |
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| Modeling of Transport Phenomena | |
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| Micro(fluid)mechanics of Darcy's Law | |
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| Darcy's Law | |
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| Microscopic Derivation of Darcy's Law | |
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| Thought Model: Viscous Flow in a Cylinder | |
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| Homogenization of the Stokes System | |
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| Lower Bound Estimate of the Permeability Tensor | |
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| Upper Bound Estimate of the Permeability Tensor | |
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| Training Set: Upper and Lower Bounds of the Permeability of a 2-D Microstructure | |
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| Lower Bound | |
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| Upper Bound | |
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| Comparison | |
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| Generalization: Periodic Homogenization Based on Double-Scale Expansion | |
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| Double-Scale Expansion Technique | |
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| Extension of Darcy's Law to the Case of Deformable Porous Media | |
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| Interaction Between Fluid and Solid Phase | |
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| Macroscopic Representation of the Solid-Fluid Interaction | |
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| Microscopic Representation of the Solid-Fluid Interaction | |
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| Beyond Darcy's (Linear) Law | |
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| Bingham Fluid | |
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| Power-Law Fluids | |
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| Appendix: Convexity of [Pi](d) | |
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| Micro-to-Macro Diffusive Transport of a Fluid Component | |
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| Fick's Law | |
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| Diffusion without Advection in Steady State Conditions | |
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| Periodic Homogenization of Diffusive Properties | |
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| The Tortuosity Tensor | |
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| Variational Approach to Periodic Homogenization | |
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| The Geometrical Meaning of Tortuosity | |
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| Double-Scale Expansion Technique | |
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| Steady State Diffusion without Advection | |
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| Steady State Diffusion Coupled with Advection | |
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| Transient Conditions | |
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| Training Set: Multilayer Porous Medium | |
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| Concluding Remarks | |
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| Microporoelasticity | |
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| Drained Microelasticity | |
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| The 1-D Thought Model: The Hollow Sphere | |
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| Macroscopic Bulk Modulus and Compressibility | |
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| Model Extension to the Cavity | |
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| Energy Point of View | |
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| Displacement Boundary Conditions | |
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| Generalization | |
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| Macroscopic and Microscopic Scales | |
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| Formulation of the Local Problem on the rev | |
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| Uniform Stress Boundary Condition | |
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| An Instructive Exercise: Capillary Pressure Effect | |
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| Uniform Strain Boundary Condition | |
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| The Hill Lemma | |
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| The Homogenized Compliance Tensor and Stress Concentration | |
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| An Instructive Exercise: Example of an rev for an Isotropic Porous Medium. Hashin's Composite Sphere Assemblage | |
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| The Homogenized Stiffness Tensor and Strain Concentration | |
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| Influence of the Boundary Condition. The Hill-Mandel Theorem | |
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| Estimates of the Homogenized Elasticity Tensor | |
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| The Dilute Scheme | |
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| The Differential Scheme | |
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| Average and Effective Strains in the Solid Phase | |
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| Training Set: Molecular Diffusion in a Saturated Porous Medium | |
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| Definition of a Local Boundary Value Problem | |
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| Estimates of the Effective Diffusion Coefficient | |
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| Linear Microporoelasticity | |
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| Loading Parameters | |
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| The 1-D Thought Model: The Saturated Hollow Sphere Model | |
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| Direct Solution | |
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| Energy Approach | |
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| Generalization | |
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| Definition of a Mechanical Loading on the rev | |
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| Homogenized State Equations | |
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| Symmetry of the Homogenized State Equations | |
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| Energy Approach | |
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| The Macroscopic Variable Set (E, m) | |
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| Application: Estimates of the Poroelastic Constants and Average Strain Level | |
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| Microscopic and Macroscopic Isotropy | |
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| Microscopic and Macroscopic Anisotropy | |
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| Average Strain Level in the Solid Phase | |
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| Levin's Theorem in Linear Microporoelasticity | |
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| An Alternative Route to the Poroelastic State Equations | |
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| An Instructive Exercise: The Prestressed Initial State | |
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| Training Set: The Two-Scale Double-Porosity Material | |
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| Eshelby's Problem in Linear Diffusion and Microporoelasticity | |
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| Eshelby's Problem in Linear Diffusion | |
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| Introduction | |
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| The (Diffusion) Inclusion Problem | |
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| The (Second-Order) P Tensor | |
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| An Alternative Derivation of the P Tensor (Optional) | |
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| The (Diffusion) Inhomogeneity Problem | |
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| Eshelby Based Estimates of the Homogenized Diffusion Tensor | |
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| Eshelby's Problem in Linear Microelasticity | |
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| Introduction | |
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| The (Elastic) Inclusion Problem | |
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| The Green Tensor G and the (Fourth-Order) P Tensor | |
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| G and P in the Isotropic Case | |
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| The (Elastic) Inhomogeneity Problem | |
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| An Instructive Exercise: Geometry Change of Spherical Pores in a Porous Medium Subjected to Compaction | |
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| Implementation of Eshelby's Solution in Linear Microporoelasticity | |
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| Implementation of Eshelby's Solution in the Dilute Scheme | |
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| Implementation of the Dilute Scheme with Different Pore Families | |
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| An Alternative Eshelby-Based Derivation of the Poroelastic Model | |
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| Mechanical Interaction Between Pores: The Mori-Tanaka Scheme | |
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| The Self-Consistent Approach | |
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| Instructive Exercise: Anisotropy of Poroelastic Properties Induced by Flat Pores | |
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| Coefficients of the Eshelby Tensor | |
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| Application of the Dilute Scheme | |
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| Influence of the Mechanical Interaction | |
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| Training Set: New Estimates of the Homogenized Diffusion Tensor | |
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| The Mori-Tanaka Estimate of the Diffusion Coefficient | |
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| The Self-Consistent Estimate of the Diffusion Coefficient | |
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| Appendix: Cylindrical Inclusion in an Isotropic Matrix | |
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| Microporoinelasticity | |
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| Strength Homogenization | |
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| The 1-D Thought Model: Strength Limits of the Saturated Hollow Sphere | |
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| Macroscopic Strength of an Empty Porous Material | |
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| Microscopic Strength of the Solid Phase | |
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| Strength-Compatible Macroscopic Stress States | |
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| Determination of [Part]G[superscript hom] | |
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| Solid Strength Depending on the First Two Stress Invariants | |
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| Principle of Nonlinear Homogenization | |
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| Von Mises Behavior of the Solid Phase | |
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| The Equivalent Viscous Behavior | |
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| Homogenization of the Fictitious Viscous Behavior | |
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| Validation | |
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| The Role of Pore Pressure in the Macroscopic Strength Criterion | |
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| Von Mises or Tresca Solid | |
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| Drucker-Prager Solid | |
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| Nonlinear Microporoelasticity | |
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| Non-Pressurized Pore Space | |
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| Pressurized Pore Space | |
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| An Alternative Approach to Strength Homogenization | |
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| Non-Saturated Microporomechanics | |
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| The Effect of Surface Tension at the Solid-Fluid Interface | |
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| Representation of Internal Forces at the Solid-Fluid Interface | |
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| Principle of Virtual Work and the Hill Lemma with Surface Tension Effects | |
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| State Equation with Surface Tension Effects | |
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| Macroscopic Strain Related to Surface Tension Effects | |
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| Microporoelasticity in Unsaturated Conditions | |
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| The Bishop Effective Stress in Unsaturated Porous Media | |
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| Surface Tension Effects in Unsaturated Porous Media | |
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| Training Set: Drying Shrinkage in a Cylindrical Pore Material System | |
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| The Capillary Pressure Curve | |
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| The State Equation | |
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| Strains Induced by Drying | |
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| Strength Domain of Non-Saturated Porous Media | |
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| Average Strain Level in a Linear Elastic Solid Phase | |
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| Strength in Partially Saturated Conditions | |
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| Microporoplasticity | |
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| The 1-D Thought Model: The Saturated Hollow Sphere | |
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| Elastic Response | |
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| Elastoplastic Response | |
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| The Concept of Residual Stresses | |
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| Energy Aspects | |
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| State Equations of Microporoplasticity | |
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| First Approach to the Macroscopic Stress-Strain Relationship | |
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| Macroscopic Plastic and Elastic Strain Tensors | |
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| Macroscopic State Equations in Poroplasticity | |
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| Macroscopic Plasticity Criterion | |
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| Link Between the Microscopic and the Macroscopic Plasticity Criterion | |
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| Arguments of the Macroscopic Yield Criterion | |
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| Dissipation Analysis | |
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| Macroscopic Flow Rule | |
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| Energy Analysis | |
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| Effective Stress in Poroplasticity | |
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| On the "Effective Plastic Stress" [Sigma] + [Beta]P1 | |
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| Mrcroporofracture and Damage Mechanics | |
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| Elements of Linear Fracture Mechanics | |
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| Dilute Estimates of Linear Poroelastic Properties of Cracked Media | |
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| Open Parallel Cracks | |
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| Randomly (Isotropic) Oriented Open Cracks | |
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| Anisotropic Distribution of Open Cracks | |
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| Effect of Total Crack Closure on the Overall Stiffness | |
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| Mori-Tanaka Estimates of Linear Poroelastic Properties of Cracked Media | |
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| Open Parallel Cracks | |
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| Closed Parallel Cracks | |
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| Randomly Oriented Interacting Cracks | |
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| Double-Porosity Model of Cracked Porous Media | |
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| Micromechanics of Damage Propagation in Saturated Media | |
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| LEFM-Damage Analogy | |
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| Extension to Multiple Cracks | |
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| The Role of the Homogenization Scheme in the Damage Criterion | |
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| Training Set: Damage Propagation in Undrained Conditions | |
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| The Case of an Incompressible Fluid | |
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| The Case of a Compressible Fluid | |
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| Appendix: Algebra for Transverse Isotropy and Applications | |
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| References | |
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| Index | |