Computational Methods for Plasticity

Eduardo de Souza Neto is a senior lecturer at the School of Engineering, University of Wales, Swansea, where he teaches a postgraduate course on the finite element method, and undergraduate courses on structural mechanics and soil mechanics. He also currently teaches external courses on computational plasticity; and his research interests include, amongst others, damage mechanics, computational plasticity, contact with friction and finite element technology. He is an international advisory board member for the Latin American Journal of Solids and Structures, and has authored 30 papers in refereed research journals as well as many conference papers, and 4 book contributions. David Owen is Professor in Civil Engineering at the University of Wales, Swansea, and chairman of Rockfield Software Ltd. He is an international authority on finite element and discrete element techniques, and is the author of seven textbooks and over three hundred and fifty scientific publications. In addition to being the editor of over thirty monographs and conference proceedings, Professor Owen is also the editor of the International Journal for Engineering Computations and is a member of several Editorial Boards. His involvement in academic research has lead to the supervision of over sixty Ph.D. students. Professor Owen is a fellow of the RAE and ICE. Djordje Peric is Professor in the Department of Civil Engineering, University of Wales, Swansea. He has an established reputation in the field of non-linear computational mechanics and is the author of over 150 research publications. He has also edited two special journal issues, and serves as an editorial board member of five international academic journals. Over the last decade Professor Peric has attracted approximately £2.5 million of research grants and funding from the UK Engineering and Physical Sciences Research Council, and various industries including Unilever, British Steel, Rolls Royce, MIC and Rockfield Software.

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