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Mechanics of Solids and Materials

Book Description

Mechanics of Solids and Materials intends to provide a modern and integrated treatment of the foundations of solid mechanics as applied to the mathematical description of material behavior. The 2006 book blends both innovative (large strain, strain rate, temperature, time dependent deformation and localized plastic deformation in crystalline solids, deformation of biological networks) and traditional (elastic theory of torsion, elastic beam and plate theories, contact mechanics) topics in a coherent theoretical framework. The extensive use of transform methods to generate solutions makes the book also of interest to structural, mechanical, and aerospace engineers. Plasticity theories, micromechanics, crystal plasticity, energetics of elastic systems, as well as an overall review of math and thermodynamics are also covered in the book.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Part 1: Mathematical Preliminaries
    1. 1. Vectors and Tensors
      1. 1.1. Vector Algebra
      2. 1.2. Coordinate Transformation: Rotation of Axes
      3. 1.3. Second-Rank Tensors
      4. 1.4. Symmetric and Antisymmetric Tensors
      5. 1.5. Prelude to Invariants of Tensors
      6. 1.6. Inverse of a Tensor
      7. 1.7. Additional Proofs
      8. 1.8. Additional Lemmas for Vectors
      9. 1.9. Coordinate Transformation of Tensors
      10. 1.10. Some Identities with Indices
      11. 1.11. Tensor Product
      12. 1.12. Orthonormal Basis
      13. 1.13. Eigenvectors and Eigenvalues
      14. 1.14. Symmetric Tensors
      15. 1.15. Positive Definiteness of a Tensor
      16. 1.16. Antisymmetric Tensors
        1. 1.16.1. Eigenvectors of W
      17. 1.17. Orthogonal Tensors
      18. 1.18. Polar Decomposition Theorem
      19. 1.19. Polar Decomposition: Physical Approach
        1. 1.19.1. Left and Right Stretch Tensors
        2. 1.19.2. Principal Stretches
      20. 1.20. The Cayley–Hamilton Theorem
      21. 1.21. Additional Lemmas for Tensors
      22. 1.22. Identities and Relations Involving ∇ Operator
      23. 1.23. Suggested Reading
    2. 2. Basic Integral Theorems
      1. 2.1. Gauss and Stokes’s Theorems
        1. 2.1.1. Applications of Divergence Theorem
      2. 2.2. Vector and Tensor Fields: Physical Approach
      3. 2.3. Surface Integrals: Gauss Law
      4. 2.4. Evaluating Surface Integrals
        1. 2.4.1. Application of the Concept of Flux
      5. 2.5. The Divergence
      6. 2.6. Divergence Theorem: Relation of Surface to Volume Integrals
      7. 2.7. More on Divergence Theorem
      8. 2.8. Suggested Reading
    3. 3. Fourier Series and Fourier Integrals
      1. 3.1. Fourier Series
      2. 3.2. Double Fourier Series
        1. 3.2.1. Double Trigonometric Series
      3. 3.3. Integral Transforms
      4. 3.4. Dirichlet’s Conditions
      5. 3.5. Integral Theorems
      6. 3.6. Convolution Integrals
        1. 3.6.1. Evaluation of Integrals by Use of Convolution Theorems
      7. 3.7. Fourier Transforms of Derivatives of f (x)
      8. 3.8. Fourier Integrals as Limiting Cases of Fourier Series
      9. 3.9. Dirac Delta Function
      10. 3.10. Suggested Reading
  8. Part 2: Continuum Mechanics
    1. 4. Kinematics of Continuum
      1. 4.1. Preliminaries
      2. 4.2. Uniaxial Strain
      3. 4.3. Deformation Gradient
      4. 4.4. Strain Tensor
      5. 4.5. Stretch and Normal Strains
      6. 4.6. Angle Change and Shear Strains
      7. 4.7. Infinitesimal Strains
      8. 4.8. Principal Stretches
      9. 4.9. Eigenvectors and Eigenvalues of Deformation Tensors
      10. 4.10. Volume Changes
      11. 4.11. Area Changes
      12. 4.12. Area Changes: Alternative Approach
      13. 4.13. Simple Shear of a Thick Plate with a Central Hole
      14. 4.14. Finite vs. Small Deformations
      15. 4.15. Reference vs. Current Configuration
      16. 4.16. Material Derivatives and Velocity
      17. 4.17. Velocity Gradient
      18. 4.18. Deformation Rate and Spin
      19. 4.19. Rate of Stretching and Shearing
      20. 4.20. Material Derivatives of Strain Tensors: Ė vs. D
      21. 4.21. Rate of F in Terms of Principal Stretches
        1. 4.21.1. Spins of Lagrangian and Eulerian Triads
      22. 4.22. Additional Connections Between Current and Reference State Representations
      23. 4.23. Transport Formulae
      24. 4.24. Material Derivatives of Volume, Area, and Surface Integrals: Transport Formulae Revisited
      25. 4.25. Analysis of Simple Shearing
      26. 4.26. Examples of Particle and Plane Motion
      27. 4.27. Rigid Body Motions
      28. 4.28. Behavior under Superposed Rotation
      29. 4.29. Suggested Reading
    2. 5. Kinetics of Continuum
      1. 5.1. Traction Vector and Stress Tensor
      2. 5.2. Equations of Equilibrium
      3. 5.3. Balance of Angular Momentum: Symmetry of σ
      4. 5.4. Principal Values of Cauchy Stress
      5. 5.5. Maximum Shear Stresses
      6. 5.6. Nominal Stress
      7. 5.7. Equilibrium in the Reference State
      8. 5.8. Work Conjugate Connections
      9. 5.9. Stress Deviator
      10. 5.10. Frame Indifference
      11. 5.11. Continuity Equation and Equations of Motion
      12. 5.12. Stress Power
      13. 5.13. The Principle of Virtual Work
      14. 5.14. Generalized Clapeyron’s Formula
      15. 5.15. Suggested Reading
    3. 6. Thermodynamics of Continuum
      1. 6.1. First Law of Thermodynamics: Energy Equation
      2. 6.2. Second Law of Thermodynamics: Clausius–Duhem Inequality
      3. 6.3. Reversible Thermodynamics
        1. 6.3.1. Thermodynamic Potentials
        2. 6.3.2. Specific and Latent Heats
        3. 6.3.3. Coupled Heat Equation
      4. 6.4. Thermodynamic Relationships with p, V, T, and s
        1. 6.4.1. Specific and Latent Heats
        2. 6.4.2. Coefficients of Thermal Expansion and Compressibility
      5. 6.5. Theoretical Calculations of Heat Capacity
      6. 6.6. Third Law of Thermodynamics
      7. 6.7. Irreversible Thermodynamics
        1. 6.7.1. Evolution of Internal Variables
      8. 6.8. Gibbs Conditions of Thermodynamic Equilibrium
      9. 6.9. Linear Thermoelasticity
      10. 6.10. Thermodynamic Potentials in Linear Thermoelasticity
        1. 6.10.1. Internal Energy
        2. 6.10.2. Helmholtz Free Energy
        3. 6.10.3. Gibbs Energy
        4. 6.10.4. Enthalpy Function
      11. 6.11. Uniaxial Loading and Thermoelastic Effect
      12. 6.12. Thermodynamics of Open Systems: Chemical Potentials
      13. 6.13. Gibbs–Duhem Equation
      14. 6.14. Chemical Potentials for Binary Systems
      15. 6.15. Configurational Entropy
      16. 6.16. Ideal Solutions
      17. 6.17. Regular Solutions for Binary Alloys
      18. 6.18. Suggested Reading
    4. 7. Nonlinear Elasticity
      1. 7.1. Green Elasticity
      2. 7.2. Isotropic Green Elasticity
      3. 7.3. Constitutive Equations in Terms of B
      4. 7.4. Constitutive Equations in Terms of Principal Stretches
      5. 7.5. Incompressible Isotropic Elastic Materials
      6. 7.6. Elastic Moduli Tensors
      7. 7.7. Instantaneous Elastic Moduli
      8. 7.8. Elastic Pseudomoduli
      9. 7.9. Elastic Moduli of Isotropic Elasticity
      10. 7.10. Elastic Moduli in Terms of Principal Stretches
      11. 7.11. Suggested Reading
  9. Part 3: Linear Elasticity
    1. 8. Governing Equations of Linear Elasticity
      1. 8.1. Elementary Theory of Isotropic Linear Elasticity
      2. 8.2. Elastic Energy in Linear Elasticity
      3. 8.3. Restrictions on the Elastic Constants
        1. 8.3.1. Material Symmetry
        2. 8.3.2. Restrictions on the Elastic Constants
      4. 8.4. Compatibility Relations
      5. 8.5. Compatibility Conditions: Cesàro Integrals
      6. 8.6. Beltrami–Michell Compatibility Equations
      7. 8.7. Navier Equations of Motion
      8. 8.8. Uniqueness of Solution to Linear Elastic Boundary Value Problem
        1. 8.8.1. Statement of the Boundary Value Problem
        2. 8.8.2. Uniqueness of the Solution
      9. 8.9. Potential Energy and Variational Principle
        1. 8.9.1. Uniqueness of the Strain Field
      10. 8.10. Betti’s Theorem of Linear Elasticity
      11. 8.11. Plane Strain
        1. 8.11.1. Plane Stress
      12. 8.12. Governing Equations of Plane Elasticity
      13. 8.13. Thermal Distortion of a Simple Beam
      14. 8.14. Suggested Reading
    2. 9. Elastic Beam Problems
      1. 9.1. A Simple 2D Beam Problem
      2. 9.2. Polynomial Solutions to ∇[sup(4)]ϕ = 0
      3. 9.3. A Simple Beam Problem Continued
        1. 9.3.1. Strains and Displacements for 2D Beams
      4. 9.4. Beam Problems with Body Force Potentials
      5. 9.5. Beam under Fourier Loading
      6. 9.6. Complete Boundary Value Problems for Beams
        1. 9.6.1. Displacement Calculations
      7. 9.7. Suggested Reading
    3. 10. Solutions in Polar Coordinates
      1. 10.1. Polar Components of Stress and Strain
      2. 10.2. Plate with Circular Hole
        1. 10.2.1. Far Field Shear
        2. 10.2.2. Far Field Tension
      3. 10.3. Degenerate Cases of Solution in Polar Coordinates
      4. 10.4. Curved Beams: Plane Stress
        1. 10.4.1. Pressurized Cylinder
        2. 10.4.2. Bending of a Curved Beam
      5. 10.5. Axisymmetric Deformations
      6. 10.6. Suggested Reading
    4. 11. Torsion and Bending of Prismatic Rods
      1. 11.1. Torsion of Prismatic Rods
      2. 11.2. Elastic Energy of Torsion
      3. 11.3. Torsion of a Rod with Rectangular Cross Section
      4. 11.4. Torsion of a Rod with Elliptical Cross Section
      5. 11.5. Torsion of a Rod with Multiply Connected Cross Sections
        1. 11.5.1. Hollow Elliptical Cross Section
      6. 11.6. Bending of a Cantilever
      7. 11.7. Elliptical Cross Section
      8. 11.8. Suggested Reading
    5. 12. Semi-Infinite Media
      1. 12.1. Fourier Transform of Biharmonic Equation
      2. 12.2. Loading on a Half-Plane
      3. 12.3. Half-Plane Loading: Special Case
      4. 12.4. Symmetric Half-Plane Loading
      5. 12.5. Half-Plane Loading: Alternative Approach
      6. 12.6. Additional Half-Plane Solutions
        1. 12.6.1. Displacement Fields in Half-Spaces
        2. 12.6.2. Boundary Value Problem
        3. 12.6.3. Specific Example
      7. 12.7. Infinite Strip
        1. 12.7.1. Uniform Loading: −a ≤ x ≤ a
        2. 12.7.2. Symmetrical Point Loads
      8. 12.8. Suggested Reading
    6. 13. Isotropic 3D Solutions
      1. 13.1. Displacement-Based Equations of Equilibrium
      2. 13.2. Boussinesq–Papkovitch Solutions
      3. 13.3. Spherically Symmetrical Geometries
        1. 13.3.1. Internally Pressurized Sphere
      4. 13.4. Pressurized Sphere: Stress-Based Solution
        1. 13.4.1. Pressurized Rigid Inclusion
        2. 13.4.2. Disk with Circumferential Shear
        3. 13.4.3. Sphere Subject to Temperature Gradients
      5. 13.5. Spherical Indentation
        1. 13.5.1. Displacement-Based Equilibrium
        2. 13.5.2. Strain Potentials
        3. 13.5.3. Point Force on a Half-Plane
        4. 13.5.4. Hemispherical Load Distribution
        5. 13.5.5. Indentation by a Spherical Ball
      6. 13.6. Point Forces on Elastic Half-Space
      7. 13.7. Suggested Reading
    7. 14. Anisotropic 3D Solutions
      1. 14.1. Point Force
      2. 14.2. Green’s Function
      3. 14.3. Isotropic Green’s Function
      4. 14.4. Suggested Reading
    8. 15. Plane Contact Problems
      1. 15.1. Wedge Problem
      2. 15.2. Distributed Contact Forces
        1. 15.2.1. Uniform Contact Pressure
        2. 15.2.2. Uniform Tangential Force
      3. 15.3. Displacement-Based Contact: Rigid Flat Punch
      4. 15.4. Suggested Reading
    9. 16. Deformation of Plates
      1. 16.1. Stresses and Strains of Bent Plates
      2. 16.2. Energy of Bent Plates
      3. 16.3. Equilibrium Equations for a Plate
      4. 16.4. Shear Forces and Bending and Twisting Moments
      5. 16.5. Examples of Plate Deformation
        1. 16.5.1. Clamped Circular Plate
        2. 16.5.2. Circular Plate with Simply Supported Edges
        3. 16.5.3. Circular Plate with Concentrated Force
        4. 16.5.4. Peeled Surface Layer
      6. 16.6. Rectangular Plates
        1. 16.6.1. Uniformly Loaded Rectangular Plate
      7. 16.7. Suggested Reading
  10. Part 4: Micromechanics
    1. 17. Dislocations and Cracks: Elementary Treatment
      1. 17.1. Dislocations
        1. 17.1.1. Derivation of the Displacement Field
      2. 17.2. Tensile Cracks
      3. 17.3. Suggested Reading
    2. 18. Dislocations in Anisotropic Media
      1. 18.1. Dislocation Character and Geometry
      2. 18.2. Dislocations in Isotropic Media
        1. 18.2.1. Infinitely Long Screw Dislocations
        2. 18.2.2. Infinitely Long Edge Dislocations
        3. 18.2.3. Infinitely Long Mixed Segments
      3. 18.3. Planar Geometric Theorem
      4. 18.4. Applications of the Planar Geometric Theorem
        1. 18.4.1. Angular Dislocations
      5. 18.5. A 3D Geometrical Theorem
      6. 18.6. Suggested Reading
    3. 19. Cracks in Anisotropic Media
      1. 19.1. Dislocation Mechanics: Reviewed
      2. 19.2. Freely Slipping Crack
      3. 19.3. Crack Extension Force
      4. 19.4. Crack Faces Loaded by Tractions
      5. 19.5. Stress Intensity Factors and Crack Extension Force
        1. 19.5.1. Computation of the Crack Extension Force
      6. 19.6. Crack Tip Opening Displacement
      7. 19.7. Dislocation Energy Factor Matrix
      8. 19.8. Inversion of a Singular Integral Equation
      9. 19.9. 2D Anisotropic Elasticity – Stroh Formalism
        1. 19.9.1. Barnett–Lothe Tensors
      10. 19.10. Suggested Reading
    4. 20. The Inclusion Problem
      1. 20.1. The Problem
      2. 20.2. Eshelby’s Solution Setup
      3. 20.3. Calculation of the Constrained Fields: u[sup(c)], e[sup(c)], and σ [sup(c)]
      4. 20.4. Components of the Eshelby Tensor for Ellipsoidal Inclusion
      5. 20.5. Elastic Energy of an Inclusion
      6. 20.6. Inhomogeneous Inclusion: Uniform Transformation Strain
      7. 20.7. Nonuniform Transformation Strain Inclusion Problem
        1. 20.7.1. The Cases M = 0, 1
      8. 20.8. Inclusions in Isotropic Media
        1. 20.8.1. Constrained Elastic Field
        2. 20.8.2. Field in the Matrix
        3. 20.8.3. Field at the Interface
        4. 20.8.4. Isotropic Spherical Inclusion
      9. 20.9. Suggested Reading
    5. 21. Forces and Energy in Elastic Systems
      1. 21.1. Free Energy and Mechanical Potential Energy
      2. 21.2. Forces of Translation
        1. 21.2.1. Force on an Interface
        2. 21.2.2. Finite Deformation Energy Momentum Tensor
      3. 21.3. Interaction Between Defects and Loading Mechanisms
        1. 21.3.1. Interaction Between Dislocations and Inclusions
        2. 21.3.2. Force on a Dislocation Segment
      4. 21.4. Elastic Energy of a Dislocation
      5. 21.5. In-Plane Stresses of Straight Dislocation Lines
      6. 21.6. Chemical Potential
        1. 21.6.1. Force on a Defect due to a Free Surface
      7. 21.7. Applications of the J Integral
        1. 21.7.1. Force on a Clamped Crack
        2. 21.7.2. Application of the Interface Force to Precipitation
      8. 21.8. Suggested Reading
    6. 22. Micropolar Elasticity
      1. 22.1. Introduction
      2. 22.2. Basic Equations of Couple-Stress Elasticity
      3. 22.3. Displacement Equations of Equilibrium
      4. 22.4. Correspondence Theorem of Couple-Stress Elasticity
      5. 22.5. Plane Strain Problems of Couple-Stress Elasticity
        1. 22.5.1. Mindlin’s Stress Functions
      6. 22.6. Edge Dislocation in Couple-Stress Elasticity
        1. 22.6.1. Strain Energy
      7. 22.7. Edge Dislocation in a Hollow Cylinder
      8. 22.8. Governing Equations for Antiplane Strain
        1. 22.8.1. Expressions in Polar Coordinates
        2. 22.8.2. Correspondence Theorem for Antiplane Strain
      9. 22.9. Antiplane Shear of Circular Annulus
      10. 22.10. Screw Dislocation in Couple-Stress Elasticity
        1. 22.10.1 Strain Energy
      11. 22.11. Configurational Forces in Couple-Stress Elasticity
        1. 22.11.1. Reciprocal Properties
        2. 22.11.2. Energy due to Internal Sources of Stress
        3. 22.11.3. Energy due to Internal and External Sources of Stress
        4. 22.11.4. The Force on an Elastic Singularity
      12. 22.12. Energy-Momentum Tensor of a Couple-Stress Field
      13. 22.13. Basic Equations of Micropolar Elasticity
      14. 22.14. Noether’s Theorem of Micropolar Elasticity
      15. 22.15. Conservation Integrals in Micropolar Elasticity
      16. 22.16. Conservation Laws for Plane Strain Micropolar Elasticity
      17. 22.17. M Integral of Micropolar Elasticity
      18. 22.18. Suggested Reading
  11. Part 5: Thin Films and Interfaces
    1. 23. Dislocations in Bimaterials
      1. 23.1. Introduction
      2. 23.2. Screw Dislocation Near a Bimaterial Interface
        1. 23.2.1. Interface Screw Dislocation
        2. 23.2.2. Screw Dislocation in a Homogeneous Medium
        3. 23.2.3. Screw Dislocation Near a Free Surface
        4. 23.2.4. Screw Dislocation Near a Rigid Boundary
      3. 23.3. Edge Dislocation (b[sub(x)]) Near a Bimaterial Interface
        1. 23.3.1. Interface Edge Dislocation
        2. 23.3.2. Edge Dislocation in an Infinite Medium
        3. 23.3.3. Edge Dislocation Near a Free Surface
        4. 23.3.4. Edge Dislocation Near a Rigid Boundary
      4. 23.4. Edge Dislocation (b[sub(y)]) Near a Bimaterial Interface
        1. 23.4.1. Interface Edge Dislocation
        2. 23.4.2. Edge Dislocation in an Infinite Medium
        3. 23.4.3. Edge Dislocation Near a Free Surface
        4. 23.4.4. Edge Dislocation Near a Rigid Boundary
      5. 23.5. Strain Energy of a Dislocation Near a Bimaterial Interface
        1. 23.5.1. Strain Energy of a Dislocation Near a Free Surface
      6. 23.6. Suggested Reading
    2. 24. Strain Relaxation in Thin Films
      1. 24.1. Dislocation Array Beneath the Free Surface
      2. 24.2. Energy of a Dislocation Array
      3. 24.3. Strained-Layer Epitaxy
      4. 24.4. Conditions for Dislocation Array Formation
      5. 24.5. Frank and van der Merwe Energy Criterion
      6. 24.6. Gradual Strain Relaxation
      7. 24.7. Stability of Array Configurations
      8. 24.8. Stronger Stability Criteria
      9. 24.9. Further Stability Bounds
        1. 24.9.1. Lower Bound
        2. 24.9.2. Upper Bound
      10. 24.10. Suggested Reading
    3. 25. Stability of Planar Interfaces
      1. 25.1. Stressed Surface Problem
      2. 25.2. Chemical Potential
      3. 25.3. Surface Diffusion and Interface Stability
      4. 25.4. Volume Diffusion and Interface Stability
      5. 25.5. 2D Surface Profiles and Surface Stability
      6. 25.6. Asymptotic Stresses for 1D Surface Profiles
      7. 25.7. Suggested Reading
  12. Part 6: Plasticity and Viscoplasticity
    1. 26. Phenomenological Plasticity
      1. 26.1. Yield Criteria for Multiaxial Stress States
      2. 26.2. Von Mises Yield Criterion
      3. 26.3. Tresca Yield Criterion
      4. 26.4. Mohr–Coulomb Yield Criterion
        1. 26.4.1. Drucker–Prager Yield Criterion
      5. 26.5. Gurson Yield Criterion for Porous Metals
      6. 26.6. Anisotropic Yield Criteria
      7. 26.7. Elastic-Plastic Constitutive Equations
      8. 26.8. Isotropic Hardening
        1. 26.8.1. J[sub(2)] Flow Theory of Plasticity
      9. 26.9. Kinematic Hardening
        1. 26.9.1. Linear and Nonlinear Kinematic Hardening
      10. 26.10. Constitutive Equations for Pressure-Dependent Plasticity
      11. 26.11. Nonassociative Plasticity
      12. 26.12. Plastic Potential for Geomaterials
      13. 26.13. Rate-Dependent Plasticity
      14. 26.14. Deformation Theory of Plasticity
        1. 26.14.1. Rate-Type Formulation of Deformation Theory
        2. 26.14.2. Application beyond Proportional Loading
      15. 26.15. J[sub(2)] Corner Theory
      16. 26.16. Rate-Dependent Flow Theory
        1. 26.16.1. Multiplicative Decomposition F = F[sup(e)] · F[sup(p)]
      17. 26.17. Elastic and Plastic Constitutive Contributions
        1. 26.17.1. Rate-Dependent J[sub(2)] Flow Theory
      18. 26.18. A Rate Tangent Integration
      19. 26.19. Plastic Void Growth
        1. 26.19.1. Ideally Plastic Material
        2. 26.19.2. Incompressible Linearly Hardening Material
      20. 26.20. Suggested Reading
    2. 27. Micromechanics of Crystallographic Slip
      1. 27.1. Early Observations
      2. 27.2. Dislocations
        1. 27.2.1. Some Basic Properties of Dislocations in Crystals
        2. 27.2.2. Strain Hardening, Dislocation Interactions, and Dislocation Multiplication
      3. 27.3. Other Strengthening Mechanisms
      4. 27.4. Measurements of Latent Hardening
      5. 27.5. Observations of Slip in Single Crystals and Polycrystals at Modest Strains
      6. 27.6. Deformation Mechanisms in Nanocrystalline Grains
        1. 27.6.1. Background: AKK Model
        2. 27.6.2. Perspective on Discreteness
        3. 27.6.3. Dislocation and Partial Dislocation Slip Systems
      7. 27.7. Suggested Reading
    3. 28. Crystal Plasticity
      1. 28.1. Basic Kinematics
      2. 28.2. Stress and Stress Rates
        1. 28.2.1. Resolved Shear Stress
        2. 28.2.2. Rate-Independent Strain Hardening
      3. 28.3. Convected Elasticity
      4. 28.4. Rate-Dependent Slip
        1. 28.4.1. A Rate Tangent Modulus
      5. 28.5. Crystalline Component Forms
        1. 28.5.1. Additional Crystalline Forms
        2. 28.5.2. Component Forms on Laboratory Axes
      6. 28.6. Suggested Reading
    4. 29. The Nature of Crystalline Deformation: Localized Plastic Deformation
      1. 29.1. Perspectives on Nonuniform and Localized Plastic Flow
        1. 29.1.1. Coarse Slip Bands and Macroscopic Shear Bands in Simple Crystals
        2. 29.1.2. Coarse Slip Bands and Macroscopic Shear Bands in Ordered Crystals
      2. 29.2. Localized Deformation in Single Slip
        1. 29.2.1. Constitutive Law for the Single Slip Crystal
        2. 29.2.2. Plastic Shearing with Non-Schmid Effects
        3. 29.2.3. Conditions for Localization
        4. 29.2.4. Expansion to the Order of σ
        5. 29.2.5. Perturbations about the Slip and Kink Plane Orientations
        6. 29.2.6. Isotropic Elastic Moduli
        7. 29.2.7. Particular Cases for Localization
      3. 29.3. Localization in Multiple Slip
        1. 29.3.1. Double Slip Model
        2. 29.3.2. Constitutive Law for the Double Slip Crystal
      4. 29.4. Numerical Results for Crystalline Deformation
        1. 29.4.1. Additional Experimental Observations
        2. 29.4.2. Numerical Observations
      5. 29.5. Suggested Reading
    5. 30. Polycrystal Plasticity
      1. 30.1. Perspectives on Polycrystalline Modeling and Texture Development
      2. 30.2. Polycrystal Model
      3. 30.3. Extended Taylor Model
      4. 30.4. Model Calculational Procedure
        1. 30.4.1. Texture Determinations
        2. 30.4.2. Yield Surface Determination
      5. 30.5. Deformation Theories and Path-Dependent Response
        1. 30.5.1. Specific Model Forms
        2. 30.5.2. Alternative Approach to a Deformation Theory
        3. 30.5.3. Nonproportional Loading
      6. 30.6. Suggested Reading
    6. 31. Laminate Plasticity
      1. 31.1. Laminate Model
      2. 31.2. Additional Kinematical Perspective
      3. 31.3. Final Constitutive Forms
        1. 31.3.1. Rigid-Plastic Laminate in Single Slip
      4. 31.4. Suggested Reading
  13. Part 7: Biomechanics
    1. 32. Mechanics of a Growing Mass
      1. 32.1. Introduction
      2. 32.2. Continuity Equation
        1. 32.2.1. Material Form of Continuity Equation
        2. 32.2.2. Quantities per Unit Initial and Current Mass
      3. 32.3. Reynolds Transport Theorem
      4. 32.4. Momentum Principles
        1. 32.4.1. Rate-Type Equations of Motion
      5. 32.5. Energy Equation
        1. 32.5.1. Material Form of Energy Equation
      6. 32.6. Entropy Equation
        1. 32.6.1. Material Form of Entropy Equation
        2. 32.6.2. Combined Energy and Entropy Equations
      7. 32.7. General Constitutive Framework
        1. 32.7.1. Thermodynamic Potentials per Unit Initial Mass
        2. 32.7.2. Equivalence of the Constitutive Structures
      8. 32.8. Multiplicative Decomposition of Deformation Gradient
        1. 32.8.1. Strain and Strain-Rate Measures
      9. 32.9. Density Expressions
      10. 32.10. Elastic Stress Response
      11. 32.11. Partition of the Rate of Deformation
      12. 32.12. Elastic Moduli Tensor
        1. 32.12.1. Elastic Moduli Coefficients
      13. 32.13. Elastic Strain Energy Representation
      14. 32.14. Evolution Equation for Stretch Ratio
      15. 32.15. Suggested Reading
    2. 33. Constitutive Relations for Membranes
      1. 33.1. Biological Membranes
      2. 33.2. Membrane Kinematics
      3. 33.3. Constitutive Laws for Membranes
      4. 33.4. Limited Area Compressibility
      5. 33.5. Simple Triangular Networks
      6. 33.6. Suggested Reading
  14. Part 8: Solved Problems
    1. 34. Solved Problems for Chapters 1–33
  15. Bibliography
  16. Index