PLASTICITY: FUNDAMENTALS AND APPLICATIONS

Table of Contents

Solid Mechanics and Its Applications

Introduction

Continuum Hypothesis

Elasto-Plastic Solids

Applications of Solid Mechanics

Scope of this Textbook

Review of Algebra and Calculus of Vectors and Tensors

Introduction

Index Notations

Kronecker Delta and Levy-Civita Symbols

Vectors

Transformation Rules for Vector Components under the Rotation of Cartesian Coordinate System

Tensors

Tensors and Vectors in Curvilinear Coordinates

References

Stress

Introduction

Stress at a Point

Surface Forces and Body Forces

Momentum Balance Laws

Theorem of Virtual Work

Cauchy’s Theorem

Transformation of Stress Components

Stresses on an Oblique Plane

Principal Stresses

Maximum Shear Stress

Octahedral Stresses

Hydrostatic and Deviatoric Stresses

Mohr’s Circle

References

Measures of Deformation and Rate of Deformation

Introduction

Deformation

Linear Strain Tensor

Infinitesimal Rotation Tensor

Deformation Gradient

Green Strain Tensor

Almansi Strain Tensor

Logarithmic Strain Tensor

Strain–Displacement Relation in Curvilinear Coordinate

Transformation of Strain Components

Principal Strains

Maximum Shear Strain

Octahedral Strain

Volumetric Strain

Mean and Deviatoric Strain

Mohr’s Circle for Strain

Incremental Strain Tensor

Material and Local Time Derivative

Rate of Deformation Tensor

Spin Tensor

On Relation between Incremental Strain and Strain Rate Tensors

Compatibility Conditions

References

Incremental and Rate Type of Elastic–Plastic Constitutive Relations for Isotropic Materials, Objective Incremental Stress and Stress Rate Measures

Introduction

Elastic Stress–Strain Relations for Small Deformation

Experimental Observations on Elastic–Plastic Behavior

Criteria for Initial Yielding of Isotropic Materials

Modeling of Isotropic Hardening or Criterion for Subsequent Isotropic Yielding

Elastic–Plastic Stress–Strain and Stress–Strain Rate Relations for Isotropic Materials

Objective Incremental Stress and Objective Stress Rate Tensors

Unloading Criterion

References

Eulerian and Updated Lagrangian Formulations

Introduction

Equation of Motion in Terms of Velocity Derivatives

Incremental Equation of Motion

Eulerian Formulation

Example of Eulerian Formulation: A Wire Drawing Problem

Updated Lagrangian Formulation

Example on Updated Lagrangian Formulation: Forging of a Cylindrical Block

References

Calculus of Variations and Extremum Principles

Introduction

Functional

Extremization of a Functional

Solution of Extremization Problems Using δ Operator

Obtaining Variational Form from a Differential Equation

Principle of Virtual Work

Principle of Minimum Potential Energy

Solution of Variational Problems by Ritz Method

References

Two-Dimensional and Axisymmetric Elasto-Plastic Problems

Introduction

Symmetric Beam Bending of a Perfectly Plastic Material (-D Problem)

Hole Expansion in an Infinite Plate (Plane Stress and Axisymmetric Problem)

Analysis of Plastic Deformation in the Flange of Circular Cup during Deep Drawing Process (Plane Stress and Axisymmetric Problem)

Necking of a Cylindrical Rod

References

Appendix A

Appendix B

Contact Mechanics

Introduction

Hertz Theory

Elastic–Plastic Indentation

Cavity Model

Sliding of Elastic–Plastic Solids

Rolling Contact

Principle of Virtual Work and Discretization of Contact Problems

References

Dynamic Elasto-Plastic Problems

Introduction

Longitudinal Stress Wave Propagation in a Rod (-D Problem)

Taylor Rod Problem (Impact of Cylindrical Rod against Flat Rigid Surface, -D Problem)

References

Continuum Damage Mechanics and Ductile Fracture

Introduction

Motivation

Objective and Plan of the Chapter

Classification of Fracture

Global and Local Approaches to Fracture

Limitations of Global and Local Approaches to Fracture

Ductile Fracture

Models of Fracture Initiation

Thermodynamics of Continuum

Continuum Damage Mechanics

Techniques for Damage Measurement

Application of a CDM Model

References

Plastic Anisotropy

Introduction

Normal and Planar Anisotropy

Hill’s Anisotropic Yield Criteria

Plane Stress Anisotropic Yield Criterion of Barlat and Lian

Three-Dimensional Anisotropic Yield Criteria of Barlat and Coworkers

Plane Strain Anisotropic Yield Criterion

Constitution Relations for Anisotropic Materials

Kinematic Hardening

References

Index