ELASTOPLASTICITY THEORY by V. A. Lubarda

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1 ELASTOPLASTICITY THEORY by V. A. Lubarda

2

3 Contents Preface xiii Part 1. ELEMENTS OF CONTINUUM MECHANICS 1 Chapter 1. TENSOR PRELIMINARIES Vectors Second-Order Tensors Eigenvalues and Eigenvectors Cayley Hamilton Theorem Change of Basis Higher-Order Tensors Traceless Tensors Covariant and Contravariant Components Vectors Second-Order Tensors Higher-Order Tensors Induced Tensors Gradient oftensor Functions Isotropic Tensors Isotropic Functions Isotropic Scalar Functions Isotropic Tensor Functions Rivlin's Identities Matrix Equation A X + X A = B Tensor Fields Differential Operators Integral Transformation Theorems 23 References 25 Chapter 2. KINEMATICS OF DEFORMATION Material and Spatial Description of Motion Deformation Gradient Polar Decomposition Nanson's Relation Simple Shear Strain Tensors 33 iii

4 iv Material Strain Tensors Spatial Strain Tensors Infinitesimal Strain and Rotation Tensors Velocity Gradient, Velocity Strain, and Spin Tensors Convected Derivatives Convected Derivatives of Tensor Products Rates of Strain Rates of Material Strains Rates of Spatial Strains Relationship between Spins W and! Rate of F in Terms of Principal Stretches Spins of Lagrangian and Eulerian Triads Behavior under Superimposed Rotation 50 References 52 Chapter 3. KINETICS OF DEFORMATION Cauchy Stress Continuity Equation Equations of Motion Symmetry of Cauchy Stress Stress Power Conjugate Stress Tensors Material Stress Tensors Spatial Stress Tensors Nominal Stress Piola Kirchhoff Stress Stress Rates Rate of Nominal Stress Stress Rates with Current Configuration as Reference Behavior under Superimposed Rotation Principle of Virtual Velocities Principle of Virtual Work 75 References 76 Chapter 4. THERMODYNAMICS OF DEFORMATION Energy Equation Material Form of Energy Equation Clausius Duhem Inequality Reversible Thermodynamics Thermodynamic Potentials Specific and Latent Heats Irreversible Thermodynamics Evolution of Internal Variables Gibbs Conditions of Thermodynamic Equilibrium Internal Rearrangements without Explicit State Variables 91

5 v 4.6. Relationship between Inelastic Increments 93 References 96 Part 2. THEORY OF ELASTICITY 99 Chapter 5. FINITE STRAIN ELASTICITY Green-Elasticity Cauchy-Elasticity Isotropic Green-Elasticity Further Expressions for Isotropic Green-Elasticity Constitutive Equations in Terms of B Constitutive Equations in Terms of Principal Stretches Incompressible Isotropic Elastic Materials Isotropic Cauchy-Elasticity Transversely Isotropic Materials Transversely Isotropic Cauchy-Elasticity Orthotropic Materials Orthotropic Cauchy-Elasticity Crystal Elasticity Crystal Classes Strain Energy Representation Elastic Constants of Cubic Crystals 122 References 124 Chapter 6. RATE-TYPE ELASTICITY Elastic Moduli Tensors Elastic Moduli for Conjugate Measures with n = ± Instantaneous Elastic Moduli Elastic Pseudomoduli Elastic Moduli of Isotropic Elasticity Components of Elastic Moduli in Terms of C Elastic Moduli in Terms of Principal Stretches Hypoelasticity 140 References 143 Chapter 7. ELASTIC STABILITY Principle of Stationary Potential Energy Uniqueness of Solution Stability of Equilibrium Incremental Uniqueness and Stability Rate-Potentials and Variational Principle Betti's Theorem and Clapeyron's Formula Other Rate-Potentials Current Configuration as Reference Uniqueness of Solution to Rate Problem Bifurcation Analysis 156

6 vi Exclusion Functional Localization Bifurcation Acoustic Tensor Strong Ellipticity Condition Constitutive Inequalities 164 References 167 Part 3. THEORY OF PLASTICITY 171 Chapter 8. ELASTOPLASTIC CONSTITUTIVE FRAMEWORK Elastic and Plastic Increments Plastic Stress Increment Plastic Strain Increment Relationship between Plastic Increments Yield Surface for Rate-Independent Materials Yield Surface in Strain Space Yield Surface in Stress Space Normality Rules Invariance of Normality Rules Flow Potential for Rate-Dependent Materials Ilyushin's Postulate Normality Rule in Strain Space Convexity of the Yield Surface in Strain Space Normality Rule in Stress Space Additional Inequalities for Strain Cycles Drucker's Postulate Normality Rule in Stress Space Convexity of the Yield Surface in Stress Space Normality Rule in Strain Space Additional Inequalities for Stress Cycles Infinitesimal Strain Formulation Relationship between Work in Stress and Strain Cycles Further Inequalities Inequalities with Current State as Reference Related Postulates 210 References 211 Chapter 9. PHENOMENOLOGICAL PLASTICITY Formulation in Strain Space Translation and Expansion of the Yield Surface Formulation in Stress Space Yield Surface in Cauchy Stress Space Nonuniqueness of the Rate of Deformation Partition Hardening Models in Stress Space Isotropic Hardening 227

7 vii Kinematic Hardening Combined Isotropic Kinematic Hardening Mróz Multisurface Model Two-Surface Model Yield Surface with Vertex in Strain Space Yield Surface with Vertex in Stress Space Pressure-Dependent Plasticity Drucker Prager Condition for Geomaterials Gurson Yield Condition for Porous Metals Constitutive Equations Nonassociative Plasticity Plastic Potential for Geomaterials Yield Vertex Model for Fissured Rocks Thermoplasticity Isotropic and Kinematic Hardening Rate-Dependent Plasticity Power-Law and Johnson Cook Models Viscoplasticity Models Deformation Theory of Plasticity Deformation vs. Flow Theory of Plasticity Application beyond Proportional Loading J 2 Corner Theory Pressure-Dependent Deformation Theory 282 References 286 Chapter 10. PLASTIC STABILITY Elastoplastic Rate-Potentials Current Configuration as Reference Reciprocal Relations Clapeyron's Formula Variational Principle Homogeneous Data Uniqueness of Solution Homogeneous Boundary Value Problem Incrementally Linear Comparison Material Comparison Material for Elastoplastic Response Minimum Principle Stability of Equilibrium Relationship between Uniqueness and Stability Criteria Uniqueness and Stability for Rigid-Plastic Materials Uniaxial Tension Compression of Column Eigenmodal Deformations Eigenstates and Eigenmodes Eigenmodal Spin 332

8 viii Eigenmodal Rate of Deformation Uniaxial Tension of Elastic-Plastic Material Triaxial Tension of Incompressible Material Triaxial Tension of Rigid-Plastic Material Acceleration Waves in Elastoplastic Solids Jump Conditions for Shock Waves Jump Conditions for Acceleration Waves Propagation Condition Stationary Discontinuity Analysis of Plastic Flow Localization Elastic-Plastic Materials Localization in Pressure-Sensitive Materials Rigid-Plastic Materials Yield Vertex Effects on Localization 360 References 365 Chapter 11. MULTIPLICATIVE DECOMPOSITION Multiplicative Decomposition F = F e F p Nonuniqueness of Decomposition Decomposition of Strain Tensors Velocity Gradient and Strain Rates Objectivity Requirements Jaumann Derivative of Elastic Deformation Gradient Partition of Elastoplastic Rate of Deformation Analysis of Elastic Rate of Deformation Analysis of Spin Ω p Analysis of Plastic Rate of Deformation Relationship between D p and D p Expression for D e in Terms of F e, F p, and Their Rates Intermediate Configuration with! p = Isotropic Hardening Kinematic Hardening Rates of Deformation Due to Convected Stress Rate Partition of the Rate of Lagrangian Strain Partition of the Rate of Deformation Gradient Relationship between ( P _ ) p and ( T _ ) p Normality Properties Elastoplastic Deformation of Orthotropic Materials Principal Axes of Orthotropy Partition of the Rate of Deformation Isoclinic Intermediate Configuration Orthotropic Yield Criterion Damage-Elastoplasticity Damage Variables Inelastic and Damage Rates of Deformation 413

9 ix Rates of Damage Tensors Reversed Decomposition F = F p F e Elastic Unloading Elastic and Plastic Rates of Deformation 420 References 422 Chapter 12. CRYSTAL PLASTICITY Kinematics of Crystal Deformation Kinetic Preliminaries Lattice Response Elastoplastic Constitutive Framework Partition of Stress and Strain Rates Partition of Rate of Deformation Gradient Generalized Schmid Stress and Normality Rate of Plastic Work Hardening Rules and Slip Rates Uniqueness of Slip Rates for Prescribed Strain Rate Further Analysis of Constitutive Equations Uniqueness of Slip Rates for Prescribed Stress Rate Fully Active ortotal Loading Range Constitutive Inequalities Implications of Ilyushin's Postulate Lower Bound on Second-Order Work Rigid-Plastic Behavior Geometric Softening Minimum Shear and Maximum Work Principle Modeling of Latent Hardening Rate-Dependent Models Flow Potential and Normality Rule 486 References 488 Chapter 13. MICRO-TO-MACRO TRANSITION Representative Macroelement Averages over a Macroelement Theorem on Product Averages Macroscopic Measures of Stress and Strain Influence Tensors of Elastic Heterogeneity Macroscopic Free and Complementary Energy Macroscopic Elastic Pseudomoduli Macroscopic Elastic Pseudocompliances Macroscopic Elastic Moduli Plastic Increment of Macroscopic Nominal Stress Plastic Potential and Normality Rule Local Residual Increment of Nominal Stress Plastic Increment of Macroscopic Deformation Gradient 512

10 x Plastic Potential and Normality Rule Local Residual Increment of Deformation Gradient Plastic Increment of Macroscopic Piola Kirchhoff Stress Plastic Increment of Macroscopic Lagrangian Strain Macroscopic Increment of Plastic Work Nontransmissibility of Basic Crystal Inequality Analysis of Second-Order Work Quantities General Analysis of Macroscopic Plastic Potentials Deformation Space Formulation Stress Space Formulation Transmissibility of Ilyushin's Postulate Aggregate Minimum Shear and Maximum Work Principle Macroscopic Flow Potential for Rate-Dependent Plasticity 537 References 538 Chapter 14. POLYCRYSTALLINE MODELS Taylor-Bishop-Hill Analysis Polycrystalline Axial Stress-Strain Curve Stresses in Grain Calculation of Polycrystalline Yield Surface Eshelby's Inclusion Problem of Linear Elasticity Inclusion Problem Inhomogeneity Problem Inclusion Problem for Incrementally Linear Material Dual Formulation Analysis of Concentration Tensors Finite Deformation Formulation Self-Consistent Method Polarization Tensors Alternative Expressions for Polycrystalline Moduli Nonaligned Crystals Polycrystalline Pseudomoduli Isotropic Aggregates of Cubic Crystals Voigt and Reuss Estimates Elastoplastic Crystal Embedded in Elastic Matrix Concentration Tensor Dual-Concentration Tensor Locally Smooth Yield Surface Rigid-Plastic Crystal in Elastic Matrix Elastoplastic Crystal Embedded in Elastoplastic Matrix Locally Smooth Yield Surface Rigid-Plastic Crystal in Rigid-Plastic Matrix Self-Consistent Determination of Elastoplastic Moduli Kröner-Budiansky-Wu Method Hutchinson's Calculations 599

11 xi Berveiller and Zaoui Accommodation Function Lin's Model Rigid-Plastic Moduli Development of Crystallographic Texture Grain Size Effects 608 References 612 Author Index 623 Subject Index 629

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