Computational Methods for Plasticity E-Book

EA de Souza Neto

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 1059

Veröffentlichungsjahr: 2011

Beschreibung

The subject of computational plasticity encapsulates the numericalmethods used for the finite element simulation of the behaviour ofa wide range of engineering materials considered to be plastic- i.e. those that undergo a permanent change of shape inresponse to an applied force. Computational Methods forPlasticity: Theory and Applications describes the theory of theassociated numerical methods for the simulation of a wide range ofplastic engineering materials; from the simplest infinitesimalplasticity theory to more complex damage mechanics and finitestrain crystal plasticity models. It is split into three parts -basic concepts, small strains and large strains. Beginning withelementary theory and progressing to advanced, complex theory andcomputer implementation, it is suitable for use at bothintroductory and advanced levels. The book: * Offers a self-contained text that allows the reader to learncomputational plasticity theory and its implementation from onevolume. * Includes many numerical examples that illustrate theapplication of the methodologies described. * Provides introductory material on related disciplines andprocedures such as tensor analysis, continuum mechanics and finiteelements for non-linear solid mechanics. * Is accompanied by purpose-developed finite element softwarethat illustrates many of the techniques discussed in the text,downloadable from the book's companion website. This comprehensive text will appeal to postgraduate and graduatestudents of civil, mechanical, aerospace and materials engineeringas well as applied mathematics and courses with computationalmechanics components. It will also be of interest to researchengineers, scientists and software developers working in the fieldof computational solid mechanics.

Leseprobe

CONTENTS

Preface

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 Elements of continuum mechanics and 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.2. Isotropic tensor-valued functions

A.3. The two-dimensional case

A.4. The three-dimensional case

A.5. A particular class of isotropic tensor functions

A.6. Alternative procedures

B The tensor exponential

B.1. The tensor exponential function

B.2. The tensor exponential derivative

B.3. Exponential map integrators

C Linearisation of the virtual work

C.1. Infinitesimal deformations

C.2. Finite strains and deformations

D Array notation for computations with tensors

D.1. Second-order tensors

D.2. Fourth-order tensors

References

Index

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Library of Congress Cataloging-in-Publication Data

Neto, Eduardo de Souza.Computational methods for plasticity: theory and applications/Eduardo de Souza Neto, Djordje Peric, DRJ Owen.p. cm.Includes bibliographical references and index.ISBN 978-0-470-69452-7 (cloth)1. Plasticity–Mathematical models. I. Peric, Djordje. II. Owen, DRJ. III. Title.TA418.14.N48 2008531’.385–dc222008033260

A catalogue record for this book is available from the British Library.

ISBN 978-0-470-69452-7

This book is lovingly dedicated to

Deise, Patricia and André;

Mira, Nikola and Nina;

Janet, Kathryn and Lisa.

PREFACE

THE purpose of this text is to describe in detail numerical techniques used in small and large strain finite element analysis of elastic and inelastic solids. Attention is focused on the derivation and description of various constitutive models – based on phenomenological hyperelasticity, elastoplasticity and elasto-viscoplasticity – together with the relevant numerical procedures and the practical issues arising in their computer implementation within a quasi-static finite element scheme. Many of the techniques discussed in the text are incorporated in the FORTRAN program, named HYPLAS, which accompanies this book and can be found at www.wiley.com/go/desouzaneto. This computer program has been specially written to illustrate the practical implementation of such techniques. We make no pretence that the text provides a complete account of the topics considered but rather, we see it as an attempt to present a reasonable balance of theory and numerical procedures used in the finite element simulation of the nonlinear mechanical behaviour of solids.

When we embarked on the project of writing this text, our initial idea was to produce a rather concise book – based primarily on our own research experience – whose bulk would consist of the description of numerical algorithms required for the finite element implementation of small and large strain plasticity models. As the manuscript began to take shape, it soon became clear that a book designed as such would be most appropriate to those already involved in research on computational plasticity or closely related areas, being of little use to those willing to learn computational methods in plasticity from a fundamental level. A substantial amount of background reading from other sources would be required for readers unfamiliar with topics such as basic elastoplasticity theory, tensor analysis, nonlinear continuum mechanics – particularly nonlinear kinematics – finite hyperelasticity and general dissipative constitutive theory of solids. Our initial plan was then gradually abandoned as we chose to make the text more self-contained by incorporating a considerable amount of basic theory. Also, while writing the manuscript, we decided to add more advanced (and very exciting) topics such as damage mechanics, anisotropic plasticity and the treatment of finite strain single crystal plasticity. Following this route, our task took at least three times as long to complete and the book grew to about twice the size as originally planned. There remains plenty of interesting material we would like to have included but cannot due to constraints of time and space. We are certainly far more satisfied with the text now than with its early versions, but we do not believe our final product to be optimal in any sense. We merely offer it to fill a gap in the existing literature, hoping that the reader will benefit from it in some way.

The text is arranged in three main parts. Part One presents some basic material of relevance to the subject matter of the book. It includes an overview of elementary tensor analysis, continuum mechanics and thermodynamics, the finite element method in quasi-static nonlinear solid mechanics and a brief description of the computer program HYPLAS. Part Two deals with small strain problems. It introduces the mathematical theory of infinitesimal plasticity as well as the relevant numerical procedures for the implementation of plasticity models within a finite element environment. Both rate-independent (elastoplastic) and rate-dependent (elasto-viscoplastic) theories are addressed and some advanced models, including anisotropic plasticity and ductile damage are also covered. Finally, in Part Three we focus on large strain problems. The theory of finite hyperelasticity is reviewed first together with details of its finite element implementation. This is followed by an introduction to large strain plasticity. Hyperelastic-based theories with multiplicative elastoplastic kinematics as well as hypoelastic-based models are discussed, together with relevant numerical procedures for their treatment. The discussion on finite plasticity and its finite element implementation culminates with a description of techniques for single crystal plasticity. Finite element techniques for large-strain near-incompressibility are also addressed.

We are indebted to many people for their direct or indirect contribution to this text. This preface would not be complete without the due acknowledgement of this fact and a record of our sincere gratitude to the following: to J.M.P. Macedo for the numerous valuable suggestions during the design of the program HYPLAS at the very early stages of this project; to R. Billardon for the many enlightening discussions on damage modelling; to R.A. Feijóo and E. Taroco for the fruitful discussions held on many occasions over a long period of time; to M. Dutko for producing some of the numerical results reported; to Y.T. Feng for helpful discussions on the arc-length method; to F.M. Andade Pires for his key contribution to the development of F-bar-Patch elements, for producing the related figures presented and for thoroughly reviewing early versions of the manuscript; to P.H. Saksono for his involvement in the production of isoerror maps; to A. Orlando for literally ‘scanning’ through key parts of the text to find inconsistencies of any kind; to L. Driemeier, W. Dettmer, M. Vaz Jr, M.C. Lobão, M. Partovi, D.C.D. Speirs, D.D. Somer, E. Saavedra, A.J.C. Molina, S. Giusti and P.J. Blanco for carefully reviewing various parts of the manuscript, spotting hard-to-find mistakes and making several important suggestions for improvement. Last, but not least, to our late colleague and friend Mike Crisfield, for the numerous illuminating and passionate discussions (often held on the beach or late in the bar) on many topics addressed in the book.

EA de Souza NetoD PeriDRJ Owen

Swansea

Part One

Basic concepts

2

ELEMENTS OF TENSOR ANALYSIS

THIS chapter introduces the notation and reviews some fundamentals of vector and tensor calculus which are extensively employed in this book. Throughout this text, preference is given to the use of intrinsic (or compact) tensor notation where no indices are used to represent mathematical entities. However, in many of the definitions introduced in this chapter, indicial notation is also used. This will allow readers not yet familiar with compact notation to associate compactly written entities and operations with their indicial forms, which will be expressed exclusively in terms of Cartesian coordinate systems. We note that the use of Cartesian, rather than curvilinear, coordinates for indicial representation is sufficiently general for the applications considered in this book. In the subsequent chapters, the use of indicial notation will be much less frequent. Readers who are familiar with tensor analysis and, in particular, the use of compact notation, may comfortably skip this chapter.