Nonlinear Finite Elements for Continua and Structures

✍ Scribed by Ted Belytschko, Wing Kam Liu, Brian Moran, Khalil Elkhodary

• Publisher: Wiley
• Year: 2014
• Tongue: English
• Leaves: 834
• Edition: 2
• Category: Library

✦ Synopsis

Nonlinear Finite Elements for Continua and Structures p>Nonlinear Finite Elements for Continua and Structures

This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis. New material provides a concise introduction to some of the cutting-edge methods that have evolved in recent years in the field of nonlinear finite element modeling, and includes the eXtended Finite Element Method (XFEM), multiresolution continuum theory for multiscale microstructures, and dislocation- density-based crystalline plasticity.

Nonlinear Finite Elements for Continua and Structures, Second Edition focuses on the formulation and solution of discrete equations for various classes of problems that are of principal interest in applications to solid and structural mechanics. Topics covered include the discretization by finite elements of continua in one dimension and in multi-dimensions; the formulation of constitutive equations for nonlinear materials and large deformations; procedures for the solution of the discrete equations, including considerations of both numerical and multiscale physical instabilities; and the treatment of structural and contact-impact problems.

Key features:

• Presents a detailed and rigorous treatment of nonlinear solid mechanics and how it can be implemented in finite element analysis
• Covers many of the material laws used in today’s software and research
• Introduces advanced topics in nonlinear finite element modelling of continua
• Introduction of multiresolution continuum theory and XFEM
• Accompanied by a website hosting a solution manual and MATLAB^® and FORTRAN code

Nonlinear Finite Elements for Continua and Structures, Second Edition is a must-have textbook for graduate students in mechanical engineering, civil engineering, applied mathematics, engineering mechanics, and materials science, and is also an excellent source of information for researchers and practitioners.

✦ Table of Contents

Nonlinear Finite Elements for Continua and Structures Second Edition
Copyright
Contents
Foreword
Preface
List of Boxes
1 Introduction
1.1 Nonlinear Finite Elements in Design
1.2 Related Books and a Brief History of Nonlinear Finite Elements
1.3 Notation
1.3.1 Indicial Notation
1.3.2 Tensor Notation
1.3.3 Functions
1.3.4 Matrix Notation
1.4 Mesh Descriptions
1.5 Classification of Partial Differential Equations
1.6 Exercises
2 Lagrangian and Eulerian Finite Elements in One Dimension
2.1 Introduction
2.2 Governing Equations for Total Lagrangian Formulation
2.2.1 Nomenclature
2.2.2 Motion and Strain Measure
2.2.3 Stress Measure
2.2.4 Governing Equations
2.2.5 Momentum Equation in Terms of Displacements
2.2.6 Continuity of Functions
2.2.7 Fundamental Theorem of Calculus
2.3 Weak Form for Total Lagrangian Formulation
2.3.1 Strong Form to Weak Form
2.3.2 Weak Form to Strong Form
2.3.3 Physical Names of Virtual Work Terms
2.3.4 Principle of Virtual Work
2.4 Finite Element Discretization in Total Lagrangian Formulation
2.4.1 Finite Element Approximations
2.4.2 Nodal Forces
2.4.3 Semidiscrete Equations
2.4.4 Initial Conditions
2.4.5 Least-Square Fit to Initial Conditions
2.4.6 Diagonal Mass Matrix
2.5 Element and Global Matrices
2.6 Governing Equations for Updated Lagrangian Formulation
2.6.1 Boundary and Interior Continuity Conditions
2.6.2 Initial Conditions
2.7 Weak Form for Updated Lagrangian Formulation
2.8 Element Equations for Updated Lagrangian Formulation
2.8.1 Finite Element Approximation
2.8.2 Element Coordinates
2.8.3 Internal and External Nodal Forces
2.8.4 Mass Matrix
2.8.5 Equivalence of Updated and Total Lagrangian Formulations
2.8.6 Assembly, Boundary Conditions and Initial Conditions
2.8.7 Mesh Distortion
2.9 Governing Equations for Eulerian Formulation
2.10 Weak Forms for Eulerian Mesh Equations
2.11 Finite Element Equations
2.11.1 Momentum Equation
2.12 Solution Methods
2.13 Summary
2.14 Exercises
3 Continuum Mechanics
3.1 Introduction
3.2 Deformation and Motion
3.2.1 Definitions
3.2.2 Eulerian and Lagrangian Coordinates
3.2.3 Motion
3.2.4 Eulerian and Lagrangian Descriptions
3.2.5 Displacement, Velocity and Acceleration
3.2.6 Deformation Gradient
3.2.7 Conditions on Motion
3.2.8 Rigid Body Rotation and Coordinate Transformations
3.3 Strain Measures
3.3.1 Green Strain Tensor
3.3.2 Rate-of-Deformation
3.3.3 Rate-of-Deformation in Terms of Rate of Green Strain
3.4 Stress Measures
3.4.1 Definitions of Stresses
3.4.2 Transformation between Stresses
3.4.3 Corotational Stress and Rate-of-Deformation
3.5 Conservation Equations
3.5.1 Conservation Laws
3.5.2 Gauss’s Theorem
3.5.3 Material Time Derivative of an Integral and Reynolds’ Transport Theorem
3.5.4 Mass Conservation
3.5.5 Conservation of Linear Momentum
3.5.6 Equilibrium Equation
3.5.7 Reynolds’ Theorem for a Density-Weighted Integrand
3.5.8 Conservation of Angular Momentum
3.5.9 Conservation of Energy
3.6 Lagrangian Conservation Equations
3.6.1 Introduction and Definitions
3.6.2 Conservation of Linear Momentum
3.6.3 Conservation of Angular Momentum
3.6.4 Conservation of Energy in Lagrangian Description
3.6.5 Power of PK2 Stress
3.7 Polar Decomposition and Frame-Invariance
3.7.1 Polar Decomposition Theorem
3.7.2 Objective Rates in Constitutive Equations
3.7.3 Jaumann Rate
3.7.4 Truesdell Rate and Green–Naghdi Rate
3.7.5 Explanation of Objective Rates
3.8 Exercises
4 Lagrangian Meshes
4.1 Introduction
4.2 Governing Equations
4.3 Weak Form: Principle of Virtual Power
4.3.1 Strong Form to Weak Form
4.3.2 Weak Form to Strong Form
4.3.3 Physical Names of Virtual Power Terms
4.4 Updated Lagrangian Finite Element Discretization
4.4.1 Finite Element Approximation
4.4.2 Internal and External Nodal Forces
4.4.3 Mass Matrix and Inertial Forces
4.4.4 Discrete Equations
4.4.5 Element Coordinates
4.4.6 Derivatives of Functions
4.4.7 Integration and Nodal Forces
4.4.8 Conditions on Parent to Current Map
4.4.9 Simplifications of Mass Matrix
4.5 Implementation
4.5.1 Indicial to Matrix Notation Translation
4.5.2 Voigt Notation
4.5.3 Numerical Quadrature
4.5.4 Selective-Reduced Integration
4.5.5 Element Force and Matrix Transformations
4.6 Corotational Formulations
4.7 Total Lagrangian Formulation
4.7.1 Governing Equations
4.7.2 Total Lagrangian Finite Element Equations by Transformation
4.8 Total Lagrangian Weak Form
4.8.1 Strong Form to Weak Form
4.8.2 Weak Form to Strong Form
4.9 Finite Element Semidiscretization
4.9.1 Discrete Equations
4.9.2 Implementation
4.9.3 Variational Principle for Large Deformation Statics
4.10 Exercises
5 Constitutive Models
5.1 Introduction
5.2 The Stress–Strain Curve
5.2.1 The Tensile Test
5.3 One-Dimensional Elasticity
5.3.1 Small Strains
5.3.2 Large Strains
5.4 Nonlinear Elasticity
5.4.1 Kirchhoff Material
5.4.2 Incompressibility
5.4.3 Kirchhoff Stress
5.4.4 Hypoelasticity
5.4.5 Relations between Tangent Moduli
5.4.6 Cauchy Elastic Material
5.4.7 Hyperelastic Materials
5.4.8 Elasticity Tensors
5.4.9 Isotropic Hyperelastic Materials
5.4.10 Neo-Hookean Material
5.4.11 Modified Mooney–Rivlin Material
5.5 One-Dimensional Plasticity
5.5.1 Rate-Independent Plasticity in One Dimension
5.5.2 Extension to Kinematic Hardening
5.5.3 Rate-Dependent Plasticity in One Dimension
5.6 Multiaxial Plasticity
5.6.1 Hypoelastic-Plastic Materials
5.6.2 J2 Flow Theory Plasticity
5.6.3 Extension to Kinematic Hardening
5.6.4 Mohr–Coulomb Constitutive Model
5.6.5 Drucker–Prager Constitutive Model
5.6.6 Porous Elastic–Plastic Solids: Gurson Model
5.6.7 Corotational Stress Formulation
5.6.8 Small-Strain Formulation
5.6.9 Large-Strain Viscoplasticity
5.7 Hyperelastic–Plastic Models
5.7.1 Multiplicative Decomposition of Deformation Gradient
5.7.2 Hyperelastic Potential and Stress
5.7.3 Decomposition of Rates of Deformation
5.7.4 Flow Rule
5.7.5 Tangent Moduli
5.7.6 J2 Flow Theory
5.7.7 Implications for Numerical Treatment of Large Rotations
5.7.8 Single-Crystal Plasticity
5.8 Viscoelasticity
5.8.1 Small Strains
5.8.2 Finite Strain Viscoelasticity
5.9 Stress Update Algorithms
5.9.1 Return Mapping Algorithms for Rate-Independent Plasticity
5.9.2 Fully Implicit Backward Euler Scheme
5.9.3 Application to J 2 Flow Theory – Radial Return Algorithm
5.9.4 Algorithmic Moduli
5.9.5 Algorithmic Moduli: J2 Flow and Radial Return
5.9.6 Semi-Implicit Backward Euler Scheme
5.9.7 Algorithmic Moduli – Semi-Implicit Scheme
5.9.8 Return Mapping Algorithms for Rate-Dependent Plasticity
5.9.9 Rate Tangent Modulus Method
5.9.10 Incrementally Objective Integration Schemes for Large Deformations
5.9.11 Semi-Implicit Scheme for Hyperelastic–Plastic Constitutive Models
5.10 Continuum Mechanics and Constitutive Models
5.10.1 Eulerian, Lagrangian and Two-Point Tensors
5.10.2 Pull-Back, Push-Forward and the Lie Derivative
5.10.3 Material Frame Indifference
5.10.4 Implications for Constitutive Relations
5.10.5 Objective Scalar Functions
5.10.6 Restrictions on Elastic Moduli
5.10.7 Material Symmetry
5.10.8 Frame Invariance in Hyperelastic–Plastic Models
5.10.9 Clausius–Duhem Inequality and Stability Postulates
5.11 Exercises
6 Solution Methods and Stability
6.1 Introduction
6.2 Explicit Methods
6.2.1 Central Difference Method
6.2.2 Implementation
6.2.3 Energy Balance
6.2.4 Accuracy
6.2.5 Mass Scaling, Subcycling and Dynamic Relaxation
6.3 Equilibrium Solutions and Implicit Time Integration
6.3.1 Equilibrium and Transient Problems
6.3.2 Equilibrium Solutions and Equilibrium Points
6.3.3 Newmark β -Equations
6.3.4 Newton’s Method
6.3.5 Newton’s Method for n Unknowns
6.3.6 Conservative Problems
6.3.7 Implementation of Newton’s Method
6.3.8 Constraints
6.3.9 Convergence Criteria
6.3.10 Line Search
6.3.11 The a -Method
6.3.12 Accuracy and Stability of Implicit Methods
6.3.13 Convergence and Robustness of Newton Iteration
6.3.14 Selection of Integration Method
6.4 Linearization
6.4.1 Linearization of the Internal Nodal Forces
6.4.2 Material Tangent Stiffness
6.4.3 Geometric Stiffness
6.4.4 Alternative Derivations of Tangent Stiffness
6.4.5 External Load Stiffness
6.4.6 Directional Derivatives
6.4.7 Algorithmically Consistent Tangent Stiffness
6.5 Stability and Continuation Methods
6.5.1 Stability
6.5.2 Branches of Equilibrium Solutions
6.5.3 Methods of Continuation and Arc Length Methods
6.5.4 Linear Stability
6.5.5 Symmetric Systems
6.5.6 Conservative Systems
6.5.7 Remarks on Linear Stability Analysis
6.5.8 Estimates of Critical Points
6.5.9 Initial Estimates of Critical Points
6.6 Numerical Stability
6.6.1 Definition and Discussion
6.6.2 Stability of a Model Linear System: Heat Conduction
6.6.3 Amplification Matrices
6.6.4 Amplification Matrix for Generalized Trapezoidal Rule
6.6.5 The z -Transform
6.6.6 Stability of Damped Central Difference Method
6.6.7 Linearized Stability Analysis of Newmark b -Method
6.6.8 Eigenvalue Inequality and Time Step Estimates
6.6.9 Element Eigenvalues
6.6.10 Stability in Energy
6.7 Material Stability
6.7.1 Description and Early Work
6.7.2 Material Stability Analysis
6.7.3 Material Instability and Change of Type of PDEs in 1D
6.7.4 Regularization
6.8 Exercises
7 Arbitrary Lagrangian Eulerian Formulations
7.1 Introduction
7.2 ALE Continuum Mechanics
7.2.1 Material Motion, Mesh Displacement, Mesh Velocity, and Mesh Acceleration
7.2.2 Material Time Derivative and Convective Velocity
7.2.3 Relationship of ALE Description to Eulerian and Lagrangian Descriptions
7.3 Conservation Laws in ALE Description
7.3.1 Conservation of Mass (Equation of Continuity)
7.3.2 Conservation of Linear and Angular Momenta
7.3.3 Conservation of Energy
7.4 ALE Governing Equations
7.5 Weak Forms
7.5.1 Continuity Equation – Weak Form
7.5.2 Momentum Equation – Weak Form
7.5.3 Finite Element Approximations
7.5.4 The Finite Element Matrix Equations
7.6 Introduction to the Petrov–Galerkin Method
7.6.1 Galerkin Discretization of the Advection–Diffusion Equation
7.6.2 Petrov–Galerkin Stabilization
7.6.3 Alternative Derivation of the SUPG
7.6.4 Parameter Determination
7.6.5 SUPG Multiple Dimensions
7.7 Petrov–Galerkin Formulation of Momentum Equation
7.7.1 Alternative Stabilization Formulation
7.7.2 The Test Function
7.7.3 Finite Element Equation
7.8 Path-Dependent Materials
7.8.1 Strong Form of Stress Update
7.8.2 Weak Form of Stress Update
7.8.3 Finite Element Discretization
7.8.4 Stress Update Procedures
7.8.5 Finite Element Implementation of Stress Update Procedures in 1D
7.8.6 Explicit Time Integration Algorithm
7.9 Linearization of the Discrete Equations
7.9.1 Internal Nodal Forces
7.9.2 External Nodal Forces
7.10 Mesh Update Equations
7.10.1 Introduction
7.10.2 Mesh Motion Prescribed A Priori
7.10.3 Lagrange–Euler Matrix Method
7.10.4 Deformation Gradient Formulations
7.10.5 Automatic Mesh Generation
7.10.6 Mesh Update using a Modified Elasticity Equation
7.10.7 Mesh Update Example
7.11 Numerical Example: An Elastic–Plastic Wave Propagation Problem
7.12 Total ALE Formulations
7.12.1 Total ALE Conservation Laws
7.12.2 Reduction to Updated ALE Conservation Laws
7.13 Exercises
8 Element Technology
8.1 Introduction
8.2 Element Performance
8.2.1 Overview
8.2.2 Completeness, Consistency, and Reproducing Conditions
8.2.3 Convergence Results for Linear Problems
8.2.4 Convergence in Nonlinear Problems
8.3 Element Properties and Patch Tests
8.3.1 Patch Tests
8.3.2 Standard Patch Test
8.3.3 Patch Test in Nonlinear Programs
8.3.4 Patch Test in Explicit Programs
8.3.5 Patch Tests for Stability
8.3.6 Linear Reproducing Conditions of Isoparametric Elements
8.3.7 Completeness of Subparametric and Superparametric Elements
8.3.8 Element Rank and Rank Deficiency
8.3.9 Rank of Numerically Integrated Elements
8.4 Q4 and Volumetric Locking
8.4.1 Element Description
8.4.2 Basis Form of Q4 Approximation
8.4.3 Locking in Q4
8.5 Multi-Field Weak Forms and Elements
8.5.1 Nomenclature
8.5.2 Hu–Washizu Weak Form
8.5.3 Alternative Multi-Field Weak Forms
8.5.4 Total Lagrangian Form of the Hu–Washizu
8.5.5 Pressure–Velocity (p–v) Implementation
8.5.6 Element Specific Pressure
8.5.7 Finite Element Implementation of Hu–Washizu
8.5.8 Simo–Hughes B-Bar Method
8.5.9 Simo–Rifai Formulation
8.6 Multi-Field Quadrilaterals
8.6.1 Assumed Velocity Strain to Avoid Volumetric Locking
8.6.2 Shear Locking and its Elimination
8.6.3 Stiffness Matrices for Assumed Strain Elements
8.6.4 Other Techniques in Quadrilaterals
8.7 One-Point Quadrature Elements
8.7.1 Nodal Forces and B-Matrix
8.7.2 Spurious Singular Modes (Hourglass)
8.7.3 Perturbation Hourglass Stabilization
8.7.4 Stabilization Procedure
8.7.5 Scaling and Remarks
8.7.6 Physical Stabilization
8.7.7 Assumed Strain with Multiple Integration Points
8.7.8 Three-Dimensional Elements
8.8 Examples
8.8.1 Static Problems
8.8.2 Dynamic Cantilever Beam
8.8.3 Cylindrical Stress Wave
8.9 Stability
8.10 Exercises
9 Beams and Shells
9.1 Introduction
9.2 Beam Theories
9.2.1 Assumptions of Beam Theories
9.2.2 Timoshenko (Shear Beam) Theory
9.2.3 Euler–Bernoulli Theory
9.2.4 Discrete Kirchhoff and Mindlin–Reissner Theories
9.3 Continuum-Based Beam
9.3.1 Definitions and Nomenclature
9.3.2 Assumptions
9.3.3 Motion
9.3.4 Nodal Forces
9.3.5 Constitutive Update
9.3.6 Continuum Nodal Internal Forces
9.3.7 Mass Matrix
9.3.8 Equations of Motion
9.3.9 Tangent Stiffness
9.4 Analysis of the CB Beam
9.4.1 Motion
9.4.2 Velocity Strains
9.4.3 Resultant Stresses and Internal Power
9.4.4 Resultant External Forces
9.4.5 Boundary Conditions
9.4.6 Weak Form
9.4.7 Strong Form
9.4.8 Finite Element Approximation
9.5 Continuum-Based Shell Implementation
9.5.1 Assumptions in Classical Shell Theories
9.5.2 Coordinates and Definitions
9.5.3 Assumptions
9.5.4 Coordinate Systems
9.5.5 Finite Element Approximation of Motion
9.5.6 Local Coordinates
9.5.7 Constitutive Equation
9.5.8 Thickness
9.5.9 Master Nodal Forces
9.5.10 Mass Matrix
9.5.11 Discrete Momentum Equation
9.5.12 Tangent Stiffness
9.5.13 Five Degree-of-Freedom Formulation
9.5.14 Large Rotations
9.5.15 Euler’s Theorem
9.5.16 Exponential Map
9.5.17 First- and Second-Order Updates
9.5.18 Hughes–Winget Update
9.5.19 Quaternions
9.5.20 Implementation
9.6 CB Shell Theory
9.6.1 Motion
9.6.2 Velocity Strains
9.6.3 Resultant Stresses
9.6.4 Boundary Conditions
9.6.5 Inconsistencies and Idiosyncrasies of Structural Theories
9.7 Shear and Membrane Locking
9.7.1 Description and Definitions
9.7.2 Shear Locking
9.7.3 Membrane Locking
9.7.4 Elimination of Locking
9.8 Assumed Strain Elements
9.8.1 Assumed Strain 4-Node Quadrilateral
9.8.2 Rank of Element
9.8.3 Nine-Node Quadrilateral
9.9 One-Point Quadrature Elements
9.10 Exercises
10 Contact-Impact
10.1 Introduction
10.2 Contact Interface Equations
10.2.1 Notation and Preliminaries
10.2.2 Impenetrability Condition
10.2.3 Traction Conditions
10.2.4 Unitary Contact Condition
10.2.5 Surface Description
10.2.6 Interpenetration Measure
10.2.7 Path-Independent Interpenetration Rate
10.2.8 Tangential Relative Velocity for Interpenetrated Bodies
10.3 Friction Models
10.3.1 Classification
10.3.2 Coulomb Friction
10.3.3 Interface Constitutive Equations
10.4 Weak Forms
10.4.1 Notation and Preliminaries
10.4.2 Lagrange Multiplier Weak Form
10.4.3 Contribution of Virtual Power to Contact Surface
10.4.4 Rate-Dependent Penalty
10.4.5 Interpenetration-Dependent Penalty
10.4.6 Perturbed Lagrangian Weak Form
10.4.7 Augmented Lagrangian
10.4.8 Tangential Tractions by Lagrange Multipliers
10.5 Finite Element Discretization
10.5.1 Overview
10.5.2 Lagrange Multiplier Method
10.5.3 Assembly of Interface Matrix
10.5.4 Lagrange Multipliers for Small-Displacement Elastostatics
10.5.5 Penalty Method for Nonlinear Frictionless Contact
10.5.6 Penalty Method for Small-Displacement Elastostatics
10.5.7 Augmented Lagrangian
10.5.8 Perturbed Lagrangian
10.5.9 Regularization
10.6 On Explicit Methods
10.6.1 Explicit Methods
10.6.2 Contact in One Dimension
10.6.3 Penalty Method
10.6.4 Explicit Algorithm
11 EXtended Finite Element Method (XFEM)
11.1 Introduction
11.1.1 Strong Discontinuity
11.1.2 Weak Discontinuity
11.1.3 XFEM for Discontinuities
11.2 Partition of Unity and Enrichments
11.3 One-Dimensional  XFEM
11.3.1 Strong Discontinuity
11.3.2 Weak Discontinuity
11.3.3 Mass Matrix
11.4 Multi-Dimension  XFEM
11.4.1 Crack Modeling
11.4.2 Tip Enrichment
11.4.3 Enrichment in a Local Coordinate System
11.5 Weak and Strong Forms
11.6 Discrete Equations
11.6.1 Strain–Displacement Matrix for Weak Discontinuity
11.7 Level Set Method
11.7.1 Level Set in 1D
11.7.2 Level Set in 2D
11.7.3 Dynamic Fracture Growth Using Level Set Updates
11.8 The Phantom Node Method
11.8.1 Element Decomposition in 1D
11.8.2 Element Decomposition in Multi-Dimensions
11.9 Integration
11.9.1 Integration for Discontinuous Enrichments
11.9.2 Integration for Singular Enrichments
11.10 An Example of XFEM Simulation
11.11 Exercise
12 Introduction to Multiresolution Theory
12.1 Motivation: Materials are Structured Continua
12.2 Bulk Deformation of Microstructured Continua
12.3 Generalizing Mechanics to Bulk Microstructured Continua
12.3.1 The Need for a Generalized Mechanics
12.3.2 Major Ideas for a Generalized Mechanics
12.3.3 Higher-Order Approach
12.3.4 Higher-Grade Approach
12.3.5 Reinterpretation of Micromorphism for Bulk Microstructured Materials
12.4 Multiscale Microstructures and the Multiresolution Continuum Theory
12.5 Governing Equations for MCT
12.5.1 Virtual Internal Power
12.5.2 Virtual External Power
12.5.3 Virtual Kinetic Power
12.5.4 Strong Form of MCT Equations
12.6 Constructing MCT Constitutive Relationships
12.7 Basic Guidelines for RVE Modeling
12.7.1 Determining RVE Cell Size
12.7.2 RVE Boundary Conditions
12.8 Finite Element Implementation of  MCT
12.9 Numerical Example
12.9.1 Void-Sheet Mechanism in High-Strength Alloy
12.9.2 MCT Multiscale Constitutive Modeling Outline
12.9.3 Finite Element Problem Setup for a Two-Dimensional Tensile Specimen
12.9.4 Results
12.10 Future Research Directions of MCT Modeling
12.11 Exercises
13 Single-Crystal Plasticity
13.1 Introduction
13.2 Crystallographic Description of Cubic and Non-Cubic Crystals
13.2.1 Specifying Directions
13.2.2 Specifying Planes
13.3 Atomic Origins of Plasticity and the Burgers Vector in Single Crystals
13.4 Defining Slip Planes and Directions in General Single Crystals
13.5 Kinematics of Single Crystal Plasticity
13.5.1 Relating the Intermediate Configuration to Crystalline Mechanics
13.5.2 Constitutive Definitions of the Plastic Parts of Deformation Rate and Spin
13.5.3 Simplification of the Kinematics by Restriction to Small Elastic Strain
13.5.4 Final Remarks
13.6 Dislocation Density Evolution
13.7 Stress Required for Dislocation Motion
13.8 Stress Update in Rate-Dependent Single-Crystal Plasticity
13.8.1 The Resolved Shear Stress
13.8.2 The Resolved Shear Stress Rate
13.8.3 Updating Resolved Shear Stress in Rate-Dependent Materials
13.8.4 Updating the Cauchy Stress
13.8.5 Adiabatic Temperature Update
13.9 Algorithm for Rate-Dependent Dislocation-Density Based Crystal Plasticity
13.10 Numerical Example: Localized Shear and Inhomogeneous Deformation
13.11 Exercises
Appendix 1 Voigt Notation
Appendix 2 Norms
Appendix 3 Element Shape Functions
Appendix 4 Euler Angles from Pole Figures
Appendix 5 Example of Dislocation-Density Evolutionary Equations
A.5.1 Dislocation Mobility
A.5.2 Dislocation Generation
A.5.3 Dislocation Annihilation
A.5.4 General Dislocation Interaction
A.5.5 Final Remarks
Glossary
References
Index