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Porous media theories play an important role in many branches of engineering, including material science, the petroleum industry, chemical engineering, and soil mechanics, as well as biomechanics. This book offers a consistent treatment of the material-independent fundamental equations of the theory of porous media, formulates constitutive equations for frictional materials in the elastic and plastic range, and traces the historical development of porous media theory. Thus, for the first time, a unique treatment of fluid-saturated porous solids is presented. The corresponding theory is…mehr
Porous media theories play an important role in many branches of engineering, including material science, the petroleum industry, chemical engineering, and soil mechanics, as well as biomechanics. This book offers a consistent treatment of the material-independent fundamental equations of the theory of porous media, formulates constitutive equations for frictional materials in the elastic and plastic range, and traces the historical development of porous media theory. Thus, for the first time, a unique treatment of fluid-saturated porous solids is presented. The corresponding theory is explained by its historical progression, and its current state is thoroughly described.
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1 Introduction.- 2 The Early Era.- 2.1 The Development of the Principles of Mechanics.- 2.2 The Dynamics of Rigid Bodies.- 2.3 The Theory of Ideal Fluids.- 2.4 Euler's Description of a Porous Body.- 2.5 Coulomb's Earth Pressure Theory.- 2.6 Woltman's Contribution to the Porous Media Theory: The Introduction of the Angle of Internal Friction and the Volume Fraction Concept.- 2.7 Concluding Remarks.- 2.8 Biographical Notes.- 3 The Classical Era.- 3.1 Cauchy's Formulation of the Stress Concept.- 3.1.1 Cauchy's Predecessors.- 3.1.2 The Final Step.- 3.1.3 Biographical notes.- 3.2 The Development of the Linear Elasticity Theory.- 3.2.1 Theoretical Molecular Formulations.- 3.2.2 Continuum Mechanics Approach.- 3.2.3 Completion of the Theory.- 3.2.4 Some Solutions of the Fundamental Equations.- 3.2.5 Final Remarks.- 3.2.6 Biographical Notes.- 3.3 Discovery of Fundamental Laws (Delesse, Fick, Darcy).- 3.3.1 The Delessian Law.- 3.3.2 Fick'sLaw.- 3.3.3 Darcy's Law.- 3.3.4 Biographical Notes.- 3.4 The Development of the Theory of Viscous Fluids.- 3.4.1 Introduction: The Navier-Stokes Equations.- 3.4.2 The Historical Development of the Theory.- 3.4.3 Biographical Notes.- 3.5 The Mohr-Coulomb Failure Condition and other Plasticity Theory Studies.- 3.5.1 W.J. Macquorn Rankine's Fundamental Failure Condition for Granular Material.- 3.5.2 O. Mohr's Contributions to the Determination of the Elasticity and Failure Limits.- 3.5.3 Extension of the Plasticity Theory.- 3.5.4 Biographical Notes.- 3.6 Motion of Liquids in Rigid Porous Solids.- 3.6.1 Motion of Liquids Through Narrow Tubes.- 3.6.2 Flow of a Liquid Through Porous Bodies with Statistically-Distributed Pores.- 3.6.3 Application.- 3.7 Foundation of the Mixture Theory.- 3.7.1 Introduction.- 3.7.2 Stefan's Development of the Mixture Theory.- 3.7.3 Biographical notes.- 3.8 The Foundation of Thermodynamics.- 3.8.1 Development in the Early Days.- 3.8.2 The Achievements of Carnot (1796-1832) and Clapeyron (1799-1864).- 3.8.3 Robert Mayer, the Discoverer of the Mechanical Equivalent of Heat.- 3.8.4 The Contributions of Mohr, Seguin, Colding, Holtzmann, and Helmholtz.- 3.8.5 The Decisive Investigations of Joule.- 3.8.6 The Foundation of Thermodynamics by Clausius, Rankine and Thomson.- 3.8.7 Discussions on the Correct Form of the Mechanical Theory of Heat and Further Developments.- 3.8.8 Biographical Notes.- 4 The Modern Era.- 4.1 Discovery of Fundamental Effects of Liquid-Saturated Rigid Porous Solids.- 4.2 The Treatment of the Liquid-Saturated Deformable Porous Solid by von Terzaghi.- 4.3 The Foundation of Modern Porous Media Theory by Fillunger.- 4.4 The Tragic Controversy Between the Viennese Professors Fillunger and von Terzaghi in 1936/37.- 4.5 The Further Development of the Viennese Affair and in Soil Mechanics.- 4.6 Biographical Notes.- 4.7 The Followers of von Terzaghi and Fillunger: Biot, Heinrich and Frenkel.- 4.7.1 Biot's Theory.- 4.7.2 Heinrich's Theory.- 4.7.3 Frenkel's Description of Moist Soil.- 4.7.4 Further Developments.- 4.7.5 Biographical Notes.- 4.8 Further Development of the Elasticity and Plasticity Theories..- 4.8.1 Elasticity Theory.- 4.8.2 Plasticity Theory.- 4.9 Modern Continuum Mechanics and Mixture Theory.- 4.10 Theories of Immiscible Mixtures.- 5 Current State of Porous Media Theory.- 5.1 Introductory Remarks to Porous Media Theory.- 5.2 The Volume Fraction Concept.- 5.3 Kinematics.- 5.4 Balance Equations.- 5.4.1 Balance of Mass.- 5.4.2 Balance of Momentum and Moment of Momentum.- 5.4.3 Balance of Energy.- 5.5 Entropy Inequality.- 5.6 The Closure Problem and the Saturation Constraint.- 5.7 Principle of Virtual Work.- 5.8 Constitutive Theory.- 5.8.1 Principle of Material Objectivity.- 5.8.2 The Introduction and Evaluation of the Entropy Inequality for a General Binary Porous Medium Model.- a) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Medium Model with Incompressible Constituents.- b) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Model with Compressible Constituents.- c) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Medium Model with Compressible Solid and Incompressible Fluid Constituents (Hybrid Model of First Type).- d) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Medium Model with Incompressible Solid and Compressible Fluid Constituents (Hybrid Model of Second Type).- e) Additional Remarks.- 5.8.3 Thermoelastic Compressible Porous Solid Filled with an Incompressible Viscous Fluid.- 5.8.4 Rigid Ideal-Plastic Porous Solid Filled with an Inviscid Compressible Fluid.- 5.8.5 Elastic-Plastic Behavior of an Incompressible Porous Solid Filled with an Incompressible Inviscid Fluid.- 5.8.6 Constitutive Relations and Transport Phenomena in Fluid-Saturated Rigid Porous Solids.- a) Heat Conduction.- b) Motion of an Incompressible, Viscous Fluid.- c) Further Transport Phenomena: Diffusion, Capillarity, Filtration, and Motion of Moisture.- 5.9 Applications.- 5.9.1 Uplift, Friction and Capillarity: three Fundamental Effects for Liquid-Saturated Porous Solids.- 5.9.2 One-Dimensional Transient Wave Propagation in Fluid-Saturated Incompressible Porous Media.- a) Field Equations.- b) One-Dimensional Transient Wave Propagation Solution.- c) General Properties of the Analytical Solution.- d) An Illustrative Example of a One-Dimensional Soil Column Subject to three Different Surface Loadings.- 5.9.3 One-Dimensional Consolidation (von Terzaghi's Differential Equation).- 5.9.4 The Elastic-Plastic Compaction of Metallic Powders.- 5.9.5 Further Solutions.- 6 Conclusions and Outlook.- A Evaluation of the Entropy Inequality.- B Introduction to the Vector- and Tensor Calculus for Engineers.- B 1. Introduction.- B 2. Basic Concepts.- B 2.1 Symbols.- B 2.2 Einstein's Summation Convention.- B 2.3 Kronecker Symbol.- B 3. Vector Algebra.- B 3.1 Vector Notion and Vector Operations.- B 3.2 Base System.- B 3.3 Reciprocal Base System.- B 3.4 Covariant and Contravariant Coefficients of the Vector Components.- B 3.5 Physical Coefficients of a Vector.- B 4. Tensor Algebra.- B 4.1 Tensor Notion (Linear Mapping).- B 4.2 Algebra in Base Systems.- B 4.3 Scalar Product of Tensors.- B 4.4 Tensor Product.- B 4.5 Special Tensors and Operations.- B 4.5.1 Inverse Tensor..- B 4.5.2 Transposed Tensor.- B 4.5.3 Symmetrical and Skew-Symmetrical Tensors..- B 4.5.4 Orthogonal Tensor.- B 4.5.5 Trace of the Tensor.- B 4.6 Decomposition of the Tensor.- B 4.6.1 Additive Decomposition.- B 4.6.2 Multiplicative Decomposition (Polar Decomposition).- B 4.7 Change of the Base.- B 4.8 Higher-Order Tensors.- B 4.8.1 Introduction of Higher-Order Tensors.....- B 4.8.2 Special Operations and Tensors.- B 4.8.3 Algebra in Base Systems.- B 4.9 Cross Product.- B 4.9.1 Cross Product of Vectors.- B 4.9.2 Cross Tensor Product of Vector and Tensor..- B 4.9.3 Cross Tensor Product of Tensors.- B 4.9.4 Cross Vector Product of Tensors.- B 4.9.5 Special Tensors and Operations.- B 4.10 Fundamental Tensors.- B 5. Vector and Tensor Analysis.- B 5.1 Functions of Scalar Parameters.- B 5.2 Field Theory.- B 5.2.1 Gradient.- B 5.2.2 Derivatives of Higher-Order.- B 5.2.3 Special Operations (Divergence, Rotation, and the Laplace-Operator).- B 5.3 Functions of Vector and Tensor Variables.- B 5.4 Integral Theorems.- B 5.4.1 Transformation of Surface Integrals into Volume Integrals.- B 5.4.2 Transformation of Line Integrals into Surface Integrals.- C Geometric Representation of the Principal Stresses, the Stress Invariants and the Mohr-Coulomb Theory.- C 1. Preliminaries.- C 2. Triaxial Plane.- C 3. Octahedral Plane.- C 4. Geometric Representation of Yield Functions.- C 5. Subspace of the Stress State.- C 6. Mohr-Coulomb Theory.- C 6.1 Coulomb's Strength Hypothesis for Arbitrarily Oriented Surfaces.- C 6.2 Failure Surfaces.- C 7. Failure Condition in Invariant Formulation.- References.- Author Index.
1 Introduction.- 2 The Early Era.- 2.1 The Development of the Principles of Mechanics.- 2.2 The Dynamics of Rigid Bodies.- 2.3 The Theory of Ideal Fluids.- 2.4 Euler's Description of a Porous Body.- 2.5 Coulomb's Earth Pressure Theory.- 2.6 Woltman's Contribution to the Porous Media Theory: The Introduction of the Angle of Internal Friction and the Volume Fraction Concept.- 2.7 Concluding Remarks.- 2.8 Biographical Notes.- 3 The Classical Era.- 3.1 Cauchy's Formulation of the Stress Concept.- 3.1.1 Cauchy's Predecessors.- 3.1.2 The Final Step.- 3.1.3 Biographical notes.- 3.2 The Development of the Linear Elasticity Theory.- 3.2.1 Theoretical Molecular Formulations.- 3.2.2 Continuum Mechanics Approach.- 3.2.3 Completion of the Theory.- 3.2.4 Some Solutions of the Fundamental Equations.- 3.2.5 Final Remarks.- 3.2.6 Biographical Notes.- 3.3 Discovery of Fundamental Laws (Delesse, Fick, Darcy).- 3.3.1 The Delessian Law.- 3.3.2 Fick'sLaw.- 3.3.3 Darcy's Law.- 3.3.4 Biographical Notes.- 3.4 The Development of the Theory of Viscous Fluids.- 3.4.1 Introduction: The Navier-Stokes Equations.- 3.4.2 The Historical Development of the Theory.- 3.4.3 Biographical Notes.- 3.5 The Mohr-Coulomb Failure Condition and other Plasticity Theory Studies.- 3.5.1 W.J. Macquorn Rankine's Fundamental Failure Condition for Granular Material.- 3.5.2 O. Mohr's Contributions to the Determination of the Elasticity and Failure Limits.- 3.5.3 Extension of the Plasticity Theory.- 3.5.4 Biographical Notes.- 3.6 Motion of Liquids in Rigid Porous Solids.- 3.6.1 Motion of Liquids Through Narrow Tubes.- 3.6.2 Flow of a Liquid Through Porous Bodies with Statistically-Distributed Pores.- 3.6.3 Application.- 3.7 Foundation of the Mixture Theory.- 3.7.1 Introduction.- 3.7.2 Stefan's Development of the Mixture Theory.- 3.7.3 Biographical notes.- 3.8 The Foundation of Thermodynamics.- 3.8.1 Development in the Early Days.- 3.8.2 The Achievements of Carnot (1796-1832) and Clapeyron (1799-1864).- 3.8.3 Robert Mayer, the Discoverer of the Mechanical Equivalent of Heat.- 3.8.4 The Contributions of Mohr, Seguin, Colding, Holtzmann, and Helmholtz.- 3.8.5 The Decisive Investigations of Joule.- 3.8.6 The Foundation of Thermodynamics by Clausius, Rankine and Thomson.- 3.8.7 Discussions on the Correct Form of the Mechanical Theory of Heat and Further Developments.- 3.8.8 Biographical Notes.- 4 The Modern Era.- 4.1 Discovery of Fundamental Effects of Liquid-Saturated Rigid Porous Solids.- 4.2 The Treatment of the Liquid-Saturated Deformable Porous Solid by von Terzaghi.- 4.3 The Foundation of Modern Porous Media Theory by Fillunger.- 4.4 The Tragic Controversy Between the Viennese Professors Fillunger and von Terzaghi in 1936/37.- 4.5 The Further Development of the Viennese Affair and in Soil Mechanics.- 4.6 Biographical Notes.- 4.7 The Followers of von Terzaghi and Fillunger: Biot, Heinrich and Frenkel.- 4.7.1 Biot's Theory.- 4.7.2 Heinrich's Theory.- 4.7.3 Frenkel's Description of Moist Soil.- 4.7.4 Further Developments.- 4.7.5 Biographical Notes.- 4.8 Further Development of the Elasticity and Plasticity Theories..- 4.8.1 Elasticity Theory.- 4.8.2 Plasticity Theory.- 4.9 Modern Continuum Mechanics and Mixture Theory.- 4.10 Theories of Immiscible Mixtures.- 5 Current State of Porous Media Theory.- 5.1 Introductory Remarks to Porous Media Theory.- 5.2 The Volume Fraction Concept.- 5.3 Kinematics.- 5.4 Balance Equations.- 5.4.1 Balance of Mass.- 5.4.2 Balance of Momentum and Moment of Momentum.- 5.4.3 Balance of Energy.- 5.5 Entropy Inequality.- 5.6 The Closure Problem and the Saturation Constraint.- 5.7 Principle of Virtual Work.- 5.8 Constitutive Theory.- 5.8.1 Principle of Material Objectivity.- 5.8.2 The Introduction and Evaluation of the Entropy Inequality for a General Binary Porous Medium Model.- a) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Medium Model with Incompressible Constituents.- b) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Model with Compressible Constituents.- c) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Medium Model with Compressible Solid and Incompressible Fluid Constituents (Hybrid Model of First Type).- d) The Introduction and Evaluation of the Entropy Inequality for a Binary Porous Medium Model with Incompressible Solid and Compressible Fluid Constituents (Hybrid Model of Second Type).- e) Additional Remarks.- 5.8.3 Thermoelastic Compressible Porous Solid Filled with an Incompressible Viscous Fluid.- 5.8.4 Rigid Ideal-Plastic Porous Solid Filled with an Inviscid Compressible Fluid.- 5.8.5 Elastic-Plastic Behavior of an Incompressible Porous Solid Filled with an Incompressible Inviscid Fluid.- 5.8.6 Constitutive Relations and Transport Phenomena in Fluid-Saturated Rigid Porous Solids.- a) Heat Conduction.- b) Motion of an Incompressible, Viscous Fluid.- c) Further Transport Phenomena: Diffusion, Capillarity, Filtration, and Motion of Moisture.- 5.9 Applications.- 5.9.1 Uplift, Friction and Capillarity: three Fundamental Effects for Liquid-Saturated Porous Solids.- 5.9.2 One-Dimensional Transient Wave Propagation in Fluid-Saturated Incompressible Porous Media.- a) Field Equations.- b) One-Dimensional Transient Wave Propagation Solution.- c) General Properties of the Analytical Solution.- d) An Illustrative Example of a One-Dimensional Soil Column Subject to three Different Surface Loadings.- 5.9.3 One-Dimensional Consolidation (von Terzaghi's Differential Equation).- 5.9.4 The Elastic-Plastic Compaction of Metallic Powders.- 5.9.5 Further Solutions.- 6 Conclusions and Outlook.- A Evaluation of the Entropy Inequality.- B Introduction to the Vector- and Tensor Calculus for Engineers.- B 1. Introduction.- B 2. Basic Concepts.- B 2.1 Symbols.- B 2.2 Einstein's Summation Convention.- B 2.3 Kronecker Symbol.- B 3. Vector Algebra.- B 3.1 Vector Notion and Vector Operations.- B 3.2 Base System.- B 3.3 Reciprocal Base System.- B 3.4 Covariant and Contravariant Coefficients of the Vector Components.- B 3.5 Physical Coefficients of a Vector.- B 4. Tensor Algebra.- B 4.1 Tensor Notion (Linear Mapping).- B 4.2 Algebra in Base Systems.- B 4.3 Scalar Product of Tensors.- B 4.4 Tensor Product.- B 4.5 Special Tensors and Operations.- B 4.5.1 Inverse Tensor..- B 4.5.2 Transposed Tensor.- B 4.5.3 Symmetrical and Skew-Symmetrical Tensors..- B 4.5.4 Orthogonal Tensor.- B 4.5.5 Trace of the Tensor.- B 4.6 Decomposition of the Tensor.- B 4.6.1 Additive Decomposition.- B 4.6.2 Multiplicative Decomposition (Polar Decomposition).- B 4.7 Change of the Base.- B 4.8 Higher-Order Tensors.- B 4.8.1 Introduction of Higher-Order Tensors.....- B 4.8.2 Special Operations and Tensors.- B 4.8.3 Algebra in Base Systems.- B 4.9 Cross Product.- B 4.9.1 Cross Product of Vectors.- B 4.9.2 Cross Tensor Product of Vector and Tensor..- B 4.9.3 Cross Tensor Product of Tensors.- B 4.9.4 Cross Vector Product of Tensors.- B 4.9.5 Special Tensors and Operations.- B 4.10 Fundamental Tensors.- B 5. Vector and Tensor Analysis.- B 5.1 Functions of Scalar Parameters.- B 5.2 Field Theory.- B 5.2.1 Gradient.- B 5.2.2 Derivatives of Higher-Order.- B 5.2.3 Special Operations (Divergence, Rotation, and the Laplace-Operator).- B 5.3 Functions of Vector and Tensor Variables.- B 5.4 Integral Theorems.- B 5.4.1 Transformation of Surface Integrals into Volume Integrals.- B 5.4.2 Transformation of Line Integrals into Surface Integrals.- C Geometric Representation of the Principal Stresses, the Stress Invariants and the Mohr-Coulomb Theory.- C 1. Preliminaries.- C 2. Triaxial Plane.- C 3. Octahedral Plane.- C 4. Geometric Representation of Yield Functions.- C 5. Subspace of the Stress State.- C 6. Mohr-Coulomb Theory.- C 6.1 Coulomb's Strength Hypothesis for Arbitrarily Oriented Surfaces.- C 6.2 Failure Surfaces.- C 7. Failure Condition in Invariant Formulation.- References.- Author Index.
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