# Computational Methods for Plasticity: Theory and Applications

# Computational Methods for Plasticity: Theory and Applications

ISBN: 978-0-470-69452-7

Dec 2008

814 pages

In Stock

$202.00

## Description

The subject of computational plasticity encapsulates the numerical methods used for the finite element simulation of the behaviour of a wide range of engineering materials considered to be plastic – i.e. those that undergo a permanent change of shape in response to an applied force.*Computational Methods for Plasticity: Theory and Applications*describes the theory of the associated numerical methods for the simulation of a wide range of plastic engineering materials; from the simplest infinitesimal plasticity theory to more complex damage mechanics and finite strain crystal plasticity models. It is split into three parts - basic concepts, small strains and large strains. Beginning with elementary theory and progressing to advanced, complex theory and computer implementation, it is suitable for use at both introductory and advanced levels. The book:

- Offers a self-contained text that allows the reader to learn computational plasticity theory and its implementation from one volume.
- Includes many numerical examples that illustrate the application of the methodologies described.
- Provides introductory material on related disciplines and procedures such as tensor analysis, continuum mechanics and finite elements for non-linear solid mechanics.
- Is accompanied by purpose-developed finite element software that illustrates many of the techniques discussed in the text, downloadable from the book’s companion website.

This comprehensive text will appeal to postgraduate and graduate students of civil, mechanical, aerospace and materials engineering as well as applied mathematics and courses with computational mechanics components. It will also be of interest to research engineers, scientists and software developers working in the field of computational solid mechanics.

**Part One Basic concepts**

1 Introduction

1.1 Aims and scope

1.2 Layout

1.3 General scheme of notation

**2 ELEMENTS OF TENSOR ANALYSIS**

2.1 Vectors

2.2 Second-order tensors

2.3 Higher-order tensors

2.4 Isotropic tensors

2.5 Differentiation

2.6 Linearisation of nonlinear problems

**3 THERMODYNAMICS**

3.1 Kinematics of deformation

3.2 Infinitesimal deformations

3.3 Forces. Stress Measures

3.4 Fundamental laws of thermodynamics

3.5 Constitutive theory

3.6 Weak equilibrium. The principle of virtual work

3.7 The quasi-static initial boundary value problem

**4 The finite element method in quasi-static nonlinear solid mechanics**

4.1 Displacement-based finite elements

4.2 Path-dependent materials. The incremental finite element procedure

4.3 Large strain formulation

4.4 Unstable equilibrium. The arc-length method

**5 Overview of the program structure**

5.1 Introduction

5.2 The main program

5.3 Data input and initialisation

5.4 The load incrementation loop. Overview

5.5 Material and element modularity

5.6 Elements. Implementation and management

5.7 Material models: implementation and management

**Part Two Small strains 6 The mathematical theory of plasticity **6.1 Phenomenological aspects

6.2 One-dimensional constitutive model

6.3 General elastoplastic constitutive model

6.4 Classical yield criteria

6.5 Plastic flow rules

6.6 Hardening laws

**7 Finite elements in small-strain plasticity problems **7.1 Preliminary implementation aspects

7.2 General numerical integration algorithm for elastoplastic constitutive equations

7.3 Application: integration algorithm for the isotropically hardening von Mises model

7.4 The consistent tangent modulus

7.5 Numerical examples with the von Mises model

7.6 Further application: the von Mises model with nonlinear mixed hardening

**8 Computations with other basic plasticity models**

8.1 The Tresca model

8.2 The Mohr-Coulomb model

8.3 The Drucker-Prager model

8.4 Examples

**9 Plane stress plasticity**

9.1 The basic plane stress plasticity problem

9.2 Plane stress constraint at the Gauss point level

9.3 Plane stress constraint at the structural level

9.4 Plane stress-projected plasticity models

9.5 Numerical examples

9.6 Other stress-constrained states

**10 Advanced plasticity models**

10.1 A modified Cam-Clay model for soils

10.2 A capped Drucker-Prager model for geomaterials

10.3 Anisotropic plasticity: the Hill, Hoffman and Barlat-Lian models

**11 Viscoplasticity**

11.1 Viscoplasticity: phenomenological aspects

11.2 One-dimensional viscoplasticity model

11.3 A von Mises-based multidimensional model

11.4 General viscoplastic constitutive model

11.5 General numerical framework

11.6 Application: computational implementation of a von Mises-based model

11.7 Examples

**12 Damage mechanics**

12.1 Physical aspects of internal damage in solids

12.2 Continuum damage mechanics

12.3 Lemaitre's elastoplastic damage theory

12.4 A simplified version of Lemaitre's model

12.5 Gurson's void growth model

12.6 Further issues in damage modelling

**Part Three Large strains 13 Finite strain hyperelasticity **13.1 Hyperelasticity: basic concepts

13.2 Some particular models

13.3 Isotropic finite hyperelasticity in plane stress

13.4 Tangent moduli: the elasticity tensors

13.5 Application: Ogden material implementation

13.6 Numerical examples

13.7 Hyperelasticity with damage: the Mullins effect

**14 Finite strain elastoplasticity **14.1 Finite strain elastoplasticity: a brief review

14.2 One-dimensional finite plasticity model

14.3 General hyperelastic-based multiplicative plasticity model

14.4 The general elastic predictor/return-mapping algorithm

14.5 The consistent spatial tangent modulus

14.6 Principal stress space-based implementation

14.7 Finite plasticity in plane stress

14.8 Finite viscoplasticity

14.9 Examples

14.10 Rate forms: hypoelastic-based plasticity models

14.11 Finite plasticity with kinematic hardening

**15 Finite elements for large-strain incompressibility**

15.1 The F-bar methodology

15.2 Enhanced assumed strain methods

15.3 Mixed u/p formulations

**16 Anisotropic finite plasticity: Single crystals**

16.1 Physical aspects

16.2 Plastic slip and the Schmid resolved shear stress

16.3 Single crystal simulation: a brief review

16.4 A general continuum model of single crystals

16.5 A general integration algorithm

16.6 An algorithm for a planar double-slip model

16.7 The consistent spatial tangent modulus

16.8 Numerical examples

16.9 Viscoplastic single crystals

Appendices

A Isotropic functions of a symmetric tensor

A.1 Isotropic scalar-valued functions

A.1.1 Representation

A.1.2 The derivative of anisotropic scalar function

A.2 Isotropic tensor-valued functions

A.2.1 Representation

A.2.2 The derivative of anisotropic tensor function

A.3 The two-dimensional case

A.3.1 Tensor function derivative

A.3.2 Plane strain and axisymmetric problems

A.4 The three-dimensional case

A.4.1 Function computation

A.4.2 Computation of the function derivative

A.5 A particular class of isotropic tensor functions

A.5.1 Two dimensions

A.5.2 Three dimensions

A.6 Alternative procedures

B The tensor exponential

B.1 The tensor exponential function

B.1.1 Some properties of the tensor exponential function

B.1.2 Computation of the tensor exponential function

B.2 The tensor exponential derivative

B.2.1 Computer implementation

B.3 Exponential map integrators

B.3.1 The generalised exponential map midpoint rule

C Linearisation of the virtual work

C.1 Infinitesimal deformations

C.2 Finite strains and deformations

C.2.1 Material description

C.2.2 Spatial description

D Array notation for computations with tensors

D.1 Second-order tensors

D.2 Fourth-order tensors

D.2.1 Operations with non-symmetric tensors

References

Index