THE FINITE ELEMENT

METHOD IN Thermomechanics

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... Originally published by, and available in North America from, Prentice-Hall Inc.

THE FINITE ELEMENT

METHOD IN Thermomechanics

Tai-Ran Hsu Department of Mechanical Engineering,

University of Manitoba, Winnipeg, Manitoba, Canada

Boston ALLEN & UNWIN

London Sydney

© T.-R. Hsu, 1986

Softcover reprint of the hardcover 1 st edition 1986

This book is copyright under the Berne Convention. No reproduction without permission. All rights reserved.

Allen & Unwin, Inc., 8 Winchester Place, Winchester, Mass. 01890, USA

Allen & Unwin (Publishers) Ltd, 40 Museum Street, London WCIA ILU, UK

Allen & Unwin (Publishers) Ltd, Park Lane, Hemel Hempstead, Herts HP2 4TE, UK

Allen & Unwin (Australia) Ltd, 8 Napier Street, North Sydney, NSW 2060, Australia

First published in 1986

Library of Congress Cataloging in Publication Data

Hsu, Tai-Ran The Finite Element Method in Thermomechanics.

Bibliography: P. Includes Index. 1. Thermal Stresses--Data Processing. 2. Finite Element Method--Data Processing. 3. Continuum Mechanics--Data Processing. 4. TEPSAC (Computer Program). 5. Thermoelasticity--Data Processing. I. Title. II. Title: Thermomechanics. TA418.58.H78 1986 620.1'121 86-7884 ISBN-13: 978-94-011-6000-1 e-ISBN-13: 978-94-011-5998-2

DOl: 10.1007/978-94-011-5998-2

British Library Cataloguing In Publication Data

Hsu, Tai-Ran The finite element method in thermomechanics.

1. Thermomechanics Mathematics 2. Finite I. Title 621.402'0151 '5353 QC318.F5

Set in 10 on 12 point Times by Blackpool Typesetting Services Ltd, Blackpool

and by Anchor Brendon Ltd, Tiptree, Essex

To my wife Su- Yong and Jean, Euginette, Leigh

PREFACE

The rapid advances in the nuclear and aerospace technologies in the past two decades compounded with the increasing demands for high performance, energy-efficient power plant components and engines have made reliable thermal stress analysis a critical factor in the design and operation of such equipment. Recently, and as experienced by the author, the need for sophis­ticated analyses has been extended to the energy resource industry such as in-situ coal gasification and in-situ oil recovery from oil sands and shales.

The analyses in the above applications are of a multidisciplinary nature, and some involve the additional complexity of multiphase and phase change phenomena. These extremely complicated factors preclude the use of classical methods, and numerical techniques such as the finite element method appear to be the most viable alternative solution.

The development of this technique so far appears to have concentrated in two extremes; one being overly concerned with the accuracy of results and tending to place all effort in the implementation of special purpose element concepts and computational algorithms, the other being for commercial purposes with the ability of solving a wide range of engineering problems. However, to be versatile, users require substantial training and experience in order to use these codes effectively. Above all, no provision for any modifi­cation of these codes by users is possible, as all these codes are proprietary and access to the code is limited only to the owners.

The present approach attempts to strike a compromise between the above two extremes. The theoretical derivations presented in the proposed book, in particular Chapters 2-4, will serve as the basis for a computer code TEPSAC (Thermoelastic-plastic Stress Analysis with Creep). The code, although limited to two-dimensional plane, or three-dimensional axisym­metric structures, can handle large classes of thermomechanical problems using a simplex element algorithm. The thermal and mechanical analyses are quasi-coupled by automatic transferring of data back and forth between these two major components. The full listing of the basic TEPSAC code will be included in Appendix 5 so that the readers may relate the corresponding theory in the text to the program in the Appendix, in addition to being able to use the code for their own research and development work.

In addition to the transient (or steady state) thermal-elastic-plastic-creep capability that has been included in the basic TEPSAC code, additional topics such as Fourier series approximation for nonaxisymmetric loading, elasto­dynamic stress analysis, thermofracture mechanics, finite strain theory, coupled thermoelastic-plasticity and program organization will also be included. These topics were carefully selected on the basis of industrial needs.

ix

PREFACE

As the intention of this book is to embrace as wide a range of topics in the nonlinear thermo mechanical analysis as possible, it is necessary to cut back on some of the fundamental principles of the finite element method. Topics such as element variety, in-depth study of the variational principle, inte­gration schemes, convergence criteria, etc. will be omitted in the text. However, these are adequately discussed in many published books on the finite element method.

Chapter 1 presents only the bare fundamental principle of the finite element method. The essence of the discretization concept described in this chapter will enable the readers to appreciate the potential of the finite element method and to acquire a correct perception of its nature. The steps of the general finite element method will be presented in a way that includes the formation of some key equations. At the end of Chapter 1, readers are expected to have a firm grasp of this technique and its application in various branches of the engineering science.

Chapter 2 is devoted to the derivation of the finite element formulation of heat conduction analysis. A review of basic mathematical expressions for heat conduction in solids and convection in fluids as well as appropriate initial and boundary conditions will be presented first. It will then be followed by the discretization of axis ymmetrica I solids, the derivations of the element's thermal equilibrium equations and the conductivity and heat capacity matrices. An alternative method of implementing boundary conditions to the discretized models will also be included. Axisymmetric solid geometry was chosen, as planar structures can often be treated as degenerated axisymmetric geometry with a large artificial radius of curvature.

Chapter 3 deals with the finite element formulation of thermoelastic­plastic analysis, and it plays a very important part in the TEPSAC code. A brief review of relevant theories of elasticity and plasticity will be presented first. A comprehensive derivation of the element equations in both elastic and elastic-plastic cases is provided, including both the effects of thermal and the associated material properties variation. Chapter 3 also presents a detailed formulation for both isotropic and kinematic hardening behavior of the structural material. A clear distinction is made between these hardening rules and their respective applications. The solution procedure offers the reader a clear idea of how the respective mathematical expressions are implemented in the TEPSAC code.

Chapter 4 deals with the derivation of the finite element algorithm for structures subject to creep. It will be demonstrated in this chapter that the only major required modification for the creep effect on the main thermo­elastic-plastic formulation is the force matrix. The reader will follow the logical sequence of a solution presented in the book and, it is hoped, will appreciate the relative simplicity of this important aspect of nonlinear thermomechanical analysis contained in the basic TEPSAC code.

x

PREFACE

Many engineering applications may involve axisymmetric structures subject to nonaxisymmetric loadings. Typical are large horizontal pressure vessels or pipelines containing heavy fluids. The analysis is clearly of a three­dimensional nature. However, such analysis in the thermoelastic-plastic regime would require an enormous amount of computer time and storage. An approximation, by means of Fourier series, for the third directional variation of the physical quantities in load description and displacement field in a two-dimensional analysis code, may prove to be a great deal more economical and hence more practical. Chapter 5 presents the theory and algorithm used in conjunction with such a technique. Although this method has been proposed in the past for elastic stress analysis, the present approach extends the use of this technique into the elastic-plastic region. A numerical example on the sag of a heavy pipe illustrates the merit of this method.

Chapter 6 presents the finite element equations for stress wave propagation through structures, caused by dynamic loads such as impact or explosions. Again, the dynamic effect could be treated as a special module to the base TEPSAC code for the cases involving strong thermal effect. A numerical example is included to illustrate the physical significance of the dynamic effect.

The importance of fracture mechanics analysis on large structures has been well recognized by engineers and researchers. While the theory of linear fracture mechanics for brittle materials has been well established, the treat­ment of initially stable, but eventually unstable, cracks commonly occurring in ductile materials is less complete. In view of the many such recent cata­strophic structural failures, emphasis on the fracture analysis of ductile materials is warranted.

Chapter 7 presents specifically some of the unique techniques that could be used to predict the growth of cracks in structures under combined thermomechanical loads. A review of the literature on linear elastic and elastic-plastic fracture mechanics is included with the presentation of brief relevant mathematical expressions, as well as the application of the finite element method. Detailed description is, however, given to the use of the "breakable" element concept which was originally developed at the author's research laboratory and has been incorporated into a special version of the TEPSAC code. This unique technique can provide users with detailed infor­mation associated with the growing crack, which is missing from many other existing methods. The subject of crack growth due to creep is also included. The existing C·-integral method of solution is thoroughly described. Readers will find that the "breakable" element concept can be readily extended to handle creep crack growth, with some unique advantages over the C*-integral approach.

In Chapter 8 is presented the theory and the corresponding fInite element formulation on finite strain plasticity. The theoretical formulation of the problem is again made adaptable to the TEPSAC code as an optional module. It will be demonstrated that finite element analysis for finite strain

xi

PREFACE

plasticity can be achieved by including two additional element stiffness matrices, Kz for the nonlinear terms in the strain-displacement relations and K3 for the necessary correction of loading due to excessive rotational displacement components. Numerical examples on the simple elongation of a prismatic bar serve to illustrate the merit of this approach, as well as to indicate the inadequacy of the usual small strain approach for the instability of metal deformation.

Chapter 9 deals with the very important subject of coupled thermomech­anics. The effect of thermomechanical coupling can be observed in many metal forming and thermal shock processes. This topic, in the author's view, has received far less attention than it deserves, presumably because of its complex nature and ofthe difficulty of its solution by classical methods. This chapter, however, provides the reader with a physical sense of the mathe­matical formulation, which starts with a review of the first and second law of thermodynamics, and the energy equations which link the thermal and mechanical effects for a deforming solid. The finite element discretization of the coupled thermoelastic-plastic equations is carried out, and shown to be related to the existing quasi-coupled approach. Again, a numerical example on the simple tension-compression of prismatic bars illustrates the physical significance of this effect, as well as providing direction for further research on this subject.

The remaining chapter of the book, Chapter 10, is devoted to a descrip­tion of the basic TEPSAC code. It provides the reader with case illustrations on the use of the code. Most of the cases included are excerpts from published work.

The task of preparing a book of this breadth certainly cannot possibly be accomplished by a single person's effort. The author has the fortune and great pleas).1re to have had many young and dedicated research associates and students working for him during the past dozen years. The following are just a few names identifiable with specific chapters in the book: A. W. M. Bertels for his early, but very important, assistance in shaping up the TEPSA code (TEPSAC version without creep), in particular, the theories described in Chapter 3 and the concept of breakable element in Chapter 7; W. K. Tam for his work in improving the kinematic hardening theory, Y. J. Liu for his contribution in creep analysis and the formulation of the generalized creep­fracture theory in Chapter 7; R. Y. Wu for Chapter 5; D. A. Scarth for Chapter 6; Y. J. Kim for the excellent work in the improvement of the breakable element algorithm and significant contribution in Chapter 7; S. Y. Cheng for a very major contribution in formulating the finite strain theory given in Chapter 8; A. Banas and N. S. Sun for their significant work in the coupled thermoelastic-plastic theory in Chapter 9; and, more recently, the valuable assistance by G. G. Chen and Z. Wozniak for the solutions of several important numerical examples in the book. Assistance by other technical and research staff and graduate students at the author's laboratory

xii

PREFACE

is also gratefully acknowledged. Among these individuals are D. Fedorowich, D. Kuss, H. A. Ashour, Z. L. Gong, J. R. Yu, Z. H. Zhai, E. A. Abdel-Hadi and K. S. Bhatia.

A special acknowledgement is due to the author's senior research associate, G. S. Pizey, for his invaluable contribution in housekeeping the TEPSAC code and in continuously upgrading it.

The development of the TEPSAC code and other related topics covered in this book obviously was not a trivial task. It would not have been possible without generous support by several institutions such as the Whiteshell Nuclear Research Establishment of the Atomic Energy of Canada Ltd and the Natural Sciences and Engineering Research Council of Canada. Support of computing facility and time 'llocation by the author's university was essential to the success of this development.

Finally, the author is indebted for the excellent cooperation received from his colleagues and the support staff in the Department of Mechanical Engineering. He also gratefully acknowledges the assistance given by the book's reviewers, Professor R. W. Lewis, Dr C. Patterson and Dr P. Roberts. He is particularly grateful to Mrs Violet Lee for her extraordinary patience in typing a substantial portion of the manuscript.

Tai-Ran Hsu Winnipeg, Manitoba, Canada

xiii

ACKNOWLEDGEMENTS

R. Y. Wu, D. A. Scarth, Y. J. Kim and A. Banas kindly allowed the use of ideas and extracts from their theses. Extracts from Volume 2 of the Proceedings of the 4th International Conference on Pressure Vessel Tech­nology, held in London on May 19-23, 1980, are reproduced by permission of the Council of the Institution of Mechanical Engineers. Extracts from" A numerical analysis of stable crack growth under increasing load" (Journal oj Fracture 20, 17-32) are reproduced by permission of Martinus Nijhoff. Extracts from the Journal oj Engineering Fracture Mechanics are reproduced by permission of Pergamon Press. Parts of Chapter 10 are reproduced by permission of the American Society of Mechanical Engineers.

xiv

CONTENTS

PREFACE

ACKNOWLEDGEMENTS

1 FUNDAMENTALS OF THE FINITE ELEMENT METHOD 1.1 Introduction 1.2 The concept of discretization 1.3 Steps in the finite element method

References

2 FINITE ELEMENT ANALYSIS IN HEAT CONDUCTION

page ix

xiv

1 2 6

21

2.1 Introduction 23 2.2 Review of basic formulations 24 2.3 Finite element formulation of transient heat conduction in solids 30 2.4 Transient heat conduction in axisymmetric solids 34 2.5 Computation of the thermal conductivity matrix 38 2.6 Computation of the heat capacitance matrix 40 2.7 Computation of thermal force matrix 41 2.8 Transient heat conduction in the time domain 41 2.9 Boundary conditions 45

2.10 Solution procedures for axisymmetric structures 51 References 52

3 THERMOELASTIC-PLASTIC STRESS ANALYSIS 3.1 Introduction 53 3.2 Mechanical behavior of materials 54 3.3 Review of basic formulations in linear elasticity theory 59 3.4 Basic formulations in nonlinear elasticity 65 3.5 Elements of plasticity theory 68 3.6 Strain hardening 75 3.7 Plastic potential (yield) function 76 3.8 Prandtl-Reuss relation 77 3.9 Derivation of plastic stress-strain relations 78

3.10 Constitutive equations for thermoelastic-plastic stress analysis 79 3.11 Derivation of the [Cep] matrix 83 3.12 Determination of material stiffness (H') 87 3.13 Thermoelastic-plastic stress analysis with kinematic hardening

rule 91 3.14 Finite element formulation of thermoelastic-plastic stress analysis 98 3.15 Finite element formulation for the base TEPSAC code 102 3.16 Solution procedure for the base TEPSA code 110

References III

xv

CONTENTS

4 CREEP DEFORMATION OF SOLIDS BY FINITE ELEMENT ANALYSIS page

4.1 Introduction 113 4.2 Theoretical background 113 4.3 Constitutive equations for thermoelastic-plastic creep stress

analysis 117 4.4 Finite element formulation of thermoelastic-plastic creep stress

analysis 118 4.5 Integration schemes 119 4.6 Solution algorithm 120 4.7 Code verification 121 4.8 Closing remarks 123

References 124

5 ELASTIC-PLASTIC STRESS ANALYSIS WITH FOURIER SERIES

5.1 Introduction 125 5.2 Element equation for elastic axisymmetric solids subject to

nonaxisymmetric loadings 126 5.3 Stiffness matrix for elastic solids subject to nonaxisymmetric

loadings 132 5.4 Elastic-plastic stress analysis of axisymmetric solids subject to

nonaxisymmetric loadings 135 5.5 Derivation of element equation 137 5.6 Mode mixing stiffness equations 138 5.7 Circumferential integration scheme 142 5.8 Numerical example 143 5.9 Discussion of the numerical example 149

5.10 Summary 150 References 151

6 ELASTODYNAMIC STRESS ANALYSIS WITH THERMAL EFFECTS 6.1 Introduction 152 6.2 Theoretical background 153 6.3 Hamilton's variational principle 154 6.4 Finite element formulation 154 6.5 Direct time integration scheme 157 6.6 Solution algorithm 159 6.7 Numerical illustration 159

References 162

7 THERMOFRACTURE MECHANICS

Part 1: Review of fracture mechanics concept 7.1 Introduction 164 7.2 Linear elastic fracture mechanics 165 7.3 Elastic-plastic fracture mechanics 167 7.4 Application of the finite element method to fracture mechanics 173

xvi

CONTENTS

Part 2: Thermoelastic-plastic fracture analysis 7.5 Introduction 7.6 Fracture criteria 7.7 J integral with thermal effect 7.8 Numerical illustrations of J integrals with thermal effect 7.9 The "breakable element"

7.10 Numerical illustrations of stable crack growth

Part 3: Thermoelastic-plastic creep fracture analysis

page

175 176 177 182 189 195

7.11 Literature review 203 7.12 Generalized creep fracture model 204 7.13 Path dependence of the C: integral 209 7.14 Creep crack growth simulated by "breakable element" algorithm 211

References 216

8 THERMOELASTIC-PLASTIC STRESS ANALYSIS BY FINITE STRAIN THEORY

8.1 Introduction 222 8.2 Lagrangian and Eulerian coordinate systems 225 8.3 Green and Almansi strain tensors 225 8.4 Lagrangian and Kirchhoff stress tensors 228 8.5 Equilibrium in the large 229 8.6 Equilibrium in the small 231 8.7 The boundary conditions 233 8.8 The constitutive equation 234 8.9 Equations of equilibrium by the principle of virtual work 235

8.10 Finite element formulation 236 8.11 Stiffness matrix [K2] 238 8.12 Stiffness matrix [K3] 239 8.13 Constitutive equations for thermoelastic-plastic stress analysis 242 8.14 The finite element formulation 244 8.15 The computer program 244 8.16 Numerical examples 244

References 252

9 COUPLED THERMOELASTIC-PLASTIC STRESS ANALYSIS

9.1 Introduction 9.2 The energy balance concept 9.3 Derivation of the coupled heat conduction equation 9.4 Coupled thermoelastic-plastic stress analysis 9.5 Finite element formulation 9.6 The (yl matrix 9.7 The thermal moduli matrix (PI 9.8 The internal dissipation factor 9.9 Computation algorithm

9.10 Numerical illustration 9.11 Concluding remarks

References

xvii

253 254 257 259 261 263 264 265 266 267 271 273

CONTENTS

10 APPLICA nON OF THERMOMECHANICAL ANALYSES IN INDUSTRY page

10.1 Introduction 274 10.2 Thermal analysis involving phase change 275 10.3 Thermoelastic-plastic stress analysis 277 10.4 Thermoelastic-plastic stress analysis by TEPSAC code 285 10.5 Simulation of thermo mechanical behavior of nuclear reactor fuel

elements 289 References 294

APPENDIX 1 Area coordinate system for triangular simplex elements 296

APPENDIX 2 Numerical illustration on the implementation of thermal boundary conditions 298

APPENDIX 3 Integrands of the mode-mixing stiffness matrix 308

APPENDIX 4 User's guide for TEPSAC

APPENDIX 5 Listing of TEPSAC code

AUTHOR INDEX

SUBJECT INDEX

314

329

385

389

1.1 5.1

5.2 5.3 5.4 7.1 7.2

7.3

9.1 10.1 10.2

A.2.1 A.2.2 A.2.3 A.2.4

LIST OF TABLES

Actions and reactions Accuracy of Gaussian quadrature as a function of Gauss points for

the stiffness matrix of modes 0 and I Convergence of results by element number Code verification on the sag of a heavy rod by its own weight Computational effort for the sag of a heavy pipe J integral in a center-cracked plate with uniform temperature field J integral in a center-cracked plate with nonuniform temperature

field but with constant material properties J integral in a center-cracked plate with nonuniform temperature

and temperature-dependent properties Properties of annealed copper Versions of TEPSA-based codes TEPSA code predictions Thermophysical properties Matrix elements of the interpolation functions Calculating the heat capacitance matrix Summary of the numerical results

xix

page 16

143 144 149 150 183

186

187 268 274 285 299 300 301 307