MICROPOROMECHANICS

By MARK ANDERSON

CONTENTS 1 A MATHEMATICAL FRAMEWORK FOR UPSCALING OPERATIONS REPRESENTATIVE ELEMENTARY VOLUME (REV) AVERAGING OPERATIONS APPARENT AND INTRINSIC AVERAGES SPATIAL DERIVATIVES OF AN AVERAGE TIME DERIVATIVE OF AN AVERAGE SPATIAL AND TIME DERIVATIVES OF E APPLICATION TO BALANCE LAWS MASS BALANCE MOMENTUM BALANCE THE PERIODIC CELL ASSUMPTION INTRODUCTION SPATIAL AND TIME DERIVATIVE OF E IN THE PERIODIC CASE SPATIAL AND TIME DERIVATIVE OF ?E?? OF IN THE PERIODIC CASE APPLICATION: MICRO- VERSUS MACROSCOPIC COMPATIBILITY PART I MODELING OF TRANSPORT PHENOMENA 2 MICRO(?UID)MECHANICS OF DARCY’S LAW DARCY’S LAW MICROSCOPIC DERIVATION OF DARCY’S LAW THOUGHT MODEL: VISCOUS FLOW IN A CYLINDER HOMOGENIZATION OF THE STOKES SYSTEM LOWER BOUND ESTIMATE OF THE PERMEABILITY TENSOR UPPER BOUND ESTIMATE OF THE PERMEABILITY TENSOR TRAINING SET: UPPER AND LOWER BOUNDS OF THE PERMEABILITY OF A 2-D MICROSTRUCTURE LOWER BOUND UPPER BOUND COMPARISON GENERALIZATION: PERIODIC HOMOGENIZATION BASED ON DOUBLE-SCALE EXPANSION DOUBLE-SCALE EXPANSION TECHNIQUE EXTENSION OF DARCY’S LAW TO THE CASE OF DEFORMABLE POROUS MEDIA INTERACTION BETWEEN FLUID AND SOLID PHASE MACROSCOPIC REPRESENTATION OF THE SOLID–FLUID INTERACTION MICROSCOPIC REPRESENTATION OF THE SOLID–FLUID INTERACTION BEYOND DARCY’S (LINEAR) LAW BINGHAM FLUID POWER-LAW FLUIDS APPENDIX: CONVEXITY OF ?(D) 3 MICRO-TO-MACRO DIFFUSIVE TRANSPORT OF A FLUID COMPONENT FICK’S LAW DIFFUSION WITHOUT ADVECTION IN STEADY STATE CONDITIONS PERIODIC HOMOGENIZATION OF DIFFUSIVE PROPERTIES THE TORTUOSITY TENSOR VARIATIONAL APPROACH TO PERIODIC HOMOGENIZATION THE GEOMETRICAL MEANING OF TORTUOSITY STEADY STATE DIFFUSION WITHOUT ADVECTION STEADY STATE DIFFUSION COUPLED WITH ADVECTION TRANSIENT CONDITIONS TRAINING SET: MULTILAYER POROUS MEDIUM CONCLUDING REMARKS PART II MICROPOROELASTICITY 4 DRAINED MICROELASTICITY THE 1-D THOUGHT MODEL: THE HOLLOW SPHERE MACROSCOPIC BULK MODULUS AND COMPRESSIBILITY MODEL EXTENSION TO THE CAVITY ENERGY POINT OF VIEW DISPLACEMENT BOUNDARY CONDITIONS GENERALIZATION MACROSCOPIC AND MICROSCOPIC SCALES FORMULATION OF THE LOCAL PROBLEM ON THE REV UNIFORM STRESS BOUNDARY CONDITION AN INSTRUCTIVE EXERCISE: CAPILLARY PRESSURE EFFECT UNIFORM STRAIN BOUNDARY CONDITION THE HILL LEMMA THE HOMOGENIZED COMPLIANCE TENSOR AND STRESS CONCENTRATION AN INSTRUCTIVE EXERCISE: EXAMPLE OF AN REV FOR AN ISOTROPIC POROUS MEDIUM HASHIN’S COMPOSITE SPHERE ASSEMBLAGE THE HOMOGENIZED STIFFNESS TENSOR AND STRAIN CONCENTRATION IN?UENCE OF THE BOUNDARY CONDITION THE HILL–MANDEL THEOREM ESTIMATES OF THE HOMOGENIZED ELASTICITY TENSOR THE DILUTE SCHEME THE DIFFERENTIAL SCHEME AVERAGE AND EFFECTIVE STRAINS IN THE SOLID PHASE TRAINING SET: MOLECULAR DIFFUSION IN A SATURATED POROUS MEDIUM DE?NITION OF A LOCAL BOUNDARY VALUE PROBLEM ESTIMATES OF THE EFFECTIVE DIFFUSION COEF?CIENT 5 LINEAR MICROPOROELASTICITY LOADING PARAMETERS THE 1-D THOUGHT MODEL: THE SATURATED HOLLOW SPHERE MODEL DIRECT SOLUTION ENERGY APPROACH DE?NITION OF A MECHANICAL LOADING ON THE REV HOMOGENIZED STATE EQUATIONS SYMMETRY OF THE HOMOGENIZED STATE EQUATIONS THE MACROSCOPIC VARIABLE SET (E, M) APPLICATION: ESTIMATES OF THE POROELASTIC CONSTANTS AND AVERAGE STRAIN LEVEL MICROSCOPIC AND MACROSCOPIC ISOTROPY MICROSCOPIC AND MACROSCOPIC ANISOTROPY AVERAGE STRAIN LEVEL IN THE SOLID PHASE LEVIN’S THEOREM IN LINEAR MICROPOROELASTICITY AN ALTERNATIVE ROUTE TO THE POROELASTIC STATE EQUATIONS AN INSTRUCTIVE EXERCISE: THE PRESTRESSED INITIAL STATE TRAINING SET: THE TWO-SCALE DOUBLE-POROSITY MATERIAL 6 ESHELBY’S PROBLEM IN LINEAR DIFFUSION AND MICROPOROELASTICITY ESHELBY’S PROBLEM IN LINEAR DIFFUSION THE (DIFFUSION) INCLUSION PROBLEM THE (SECOND-ORDER) P TENSOR AN ALTERNATIVE DERIVATION OF THE P TENSOR (OPTIONAL) THE (DIFFUSION) INHOMOGENEITY PROBLEM ESHELBY-BASED ESTIMATES OF THE HOMOGENIZED DIFFUSION TENSOR ESHELBY’S PROBLEM IN LINEAR MICROELASTICITY THE (ELASTIC) INCLUSION PROBLEM THE GREEN TENSOR G AND THE (FOURTH-ORDER) P TENSOR G AND P IN THE ISOTROPIC CASE THE (ELASTIC) INHOMOGENEITY PROBLEM AN INSTRUCTIVE EXERCISE: GEOMETRY CHANGE OF SPHERICAL PORES IN A POROUS MEDIUM SUBJECTED TO COMPACTION IMPLEMENTATION OF ESHELBY’S SOLUTION IN LINEAR MICROPOROELASTICITY IMPLEMENTATION OF ESHELBY’S SOLUTION IN THE DILUTE SCHEME IMPLEMENTATION OF THE DILUTE SCHEME WITH DIFFERENT PORE FAMILIES AN ALTERNATIVE ESHELBY-BASED DERIVATION OF THE POROELASTIC MODEL MECHANICAL INTERACTION BETWEEN PORES: THE MORI–TANAKA SCHEME THE SELF-CONSISTENT APPROACH INSTRUCTIVE EXERCISE: ANISOTROPY OF POROELASTIC PROPERTIES INDUCED BY FLAT PORES COEF?CIENTS OF THE ESHELBY TENSOR APPLICATION OF THE DILUTE SCHEME IN?UENCE OF THE MECHANICAL INTERACTION TRAINING SET: NEW ESTIMATES OF THE HOMOGENIZED DIFFUSION TENSOR THE MORI–TANAKA ESTIMATE OF THE DIFFUSION COEF?CIENT THE SELF-CONSISTENT ESTIMATE OF THE DIFFUSION COEF?CIENT APPENDIX: CYLINDRICAL INCLUSION IN AN ISOTROPIC MATRIX PART III MICROPOROINELASTICITY 7 STRENGTH HOMOGENIZATION THE 1-D THOUGHT MODEL: STRENGTH LIMITS OF THE SATURATED HOLLOW SPHERE MACROSCOPIC STRENGTH OF AN EMPTY POROUS MATERIAL MICROSCOPIC STRENGTH OF THE SOLID PHASE STRENGTH-COMPATIBLE MACROSCOPIC STRESS STATES DETERMINATION OF ?GHOM SOLID STRENGTH DEPENDING ON THE FIRST TWO STRESS INVARIANTS PRINCIPLE OF NONLINEAR HOMOGENIZATION VON MISES BEHAVIOR OF THE SOLID PHASE THE EQUIVALENT VISCOUS BEHAVIOR HOMOGENIZATION OF THE FICTITIOUS VISCOUS BEHAVIOR VALIDATION THE ROLE OF PORE PRESSURE IN THE MACROSCOPIC STRENGTH CRITERION VON MISES OR TRESCA SOLID DRUCKER–PRAGER SOLID NONLINEAR MICROPOROELASTICITY NON–PRESSURIZED PORE SPACE PRESSURIZED PORE SPACE AN ALTERNATIVE APPROACH TO STRENGTH HOMOGENIZATION 8 NON-SATURATED MICROPOROMECHANICS THE EFFECT OF SURFACE TENSION AT THE SOLID–FLUID INTERFACE REPRESENTATION OF INTERNAL FORCES AT THE SOLID–FLUID INTERFACE PRINCIPLE OF VIRTUAL WORK AND THE HILL LEMMA WITH SURFACE TENSION EFFECTS STATE EQUATION WITH SURFACE TENSION EFFECTS MACROSCOPIC STRAIN RELATED TO SURFACE TENSION EFFECTS MICROPOROELASTICITY IN UNSATURATED CONDITIONS THE BISHOP EFFECTIVE STRESS IN UNSATURATED POROUS MEDIA SURFACE TENSION EFFECTS IN UNSATURATED POROUS MEDIA TRAINING SET: DRYING SHRINKAGE IN A CYLINDRICAL PORE MATERIAL SYSTEM THE CAPILLARY PRESSURE CURVE THE STATE EQUATION STRAINS INDUCED BY DRYING STRENGTH DOMAIN OF NON-SATURATED POROUS MEDIA AVERAGE STRAIN LEVEL IN A LINEAR ELASTIC SOLID PHASE STRENGTH IN PARTIALLY SATURATED CONDITIONS 9 MICROPOROPLASTICITY THE 1-D THOUGHT MODEL: THE SATURATED HOLLOW SPHERE ELASTIC RESPONSE ELASTOPLASTIC RESPONSE THE CONCEPT OF RESIDUAL STRESSES ENERGY ASPECTS STATE EQUATIONS OF MICROPOROPLASTICITY FIRST APPROACH TO THE MACROSCOPIC STRESS–STRAIN RELATIONSHIP MACROSCOPIC PLASTIC AND ELASTIC STRAIN TENSORS MACROSCOPIC STATE EQUATIONS IN POROPLASTICITY MACROSCOPIC PLASTICITY CRITERION LINK BETWEEN THE MICROSCOPIC AND THE MACROSCOPIC PLASTICITY CRITERION ARGUMENTS OF THE MACROSCOPIC YIELD CRITERION DISSIPATION ANALYSIS MACROSCOPIC FLOW RULE ENERGY ANALYSIS EFFECTIVE STRESS IN POROPLASTICITY 10 MICROPOROFRACTURE AND DAMAGE MECHANICS ELEMENTS OF LINEAR FRACTURE MECHANICS DILUTE ESTIMATES OF LINEAR POROELASTIC PROPERTIES OF CRACKED MEDIA OPEN PARALLEL CRACKS RANDOMLY (ISOTROPIC) ORIENTED OPEN CRACKS ANISOTROPIC DISTRIBUTION OF OPEN CRACKS EFFECT OF TOTAL CRACK CLOSURE ON THE OVERALL STIFFNESS MORI–TANAKA ESTIMATES OF LINEAR POROELASTIC PROPERTIES OF CRACKED MEDIA CLOSED PARALLEL CRACKS RANDOMLY ORIENTED INTERACTING CRACKS DOUBLE-POROSITY MODEL OF CRACKED POROUS MEDIA MICROMECHANICS OF DAMAGE PROPAGATION IN SATURATED MEDIA LEFM–DAMAGE ANALOGY EXTENSION TO MULTIPLE CRACKS THE ROLE OF THE HOMOGENIZATION SCHEME IN THE DAMAGE CRITERION TRAINING SET: DAMAGE PROPAGATION IN UNDRAINED CONDITIONS THE CASE OF AN INCOMPRESSIBLE FLUID THE CASE OF A COMPRESSIBLE FLUID APPENDIX: ALGEBRA FOR TRANSVERSE ISOTROPY AND APPLICATIONS REFERENCES