Computational Continuum Mechanics.pdf

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COVER

HALF-TITLE

TITLE

CONTENTS

PREFACE

1 INTRODUCTION

1.1 MATRICES

1.2 VECTORS

1.3 SUMMATION CONVENTION

1.4 CARTESIAN TENSORS

1.5 POLAR DECOMPOSITION THEOREM

1.6 D’ALEMBERT’S PRINCIPLE

1.7 VIRTUAL WORK PRINCIPLE

1.8 APPROXIMATION METHODS

1.9 DISCRETE EQUATIONS

1.10 MOMENTUM, WORK, AND ENERGY

1.11 PARAMETER CHANGE AND COORDINATE TRANSFORMATION

PROBLEMS

2 KINEMATICS

2.1 MOTION DESCRIPTION

2.2 STRAIN COMPONENTS

2.3 OTHER DEFORMATION MEASURES

2.4 DECOMPOSITION OF DISPLACEMENT

2.5 VELOCITY AND ACCELERATION

2.6 COORDINATE TRANSFORMATION

2.7 OBJECTIVITY

2.8 CHANGE OF VOLUME AND AREA

2.9 CONTINUITY EQUATION

2.10 REYNOLDS’ TRANSPORT THEOREM

2.11 EXAMPLES OF DEFORMATION

PROBLEMS

3 FORCES AND STRESSES

3.1 EQUILIBRIUM OF FORCES

3.2 TRANSFORMATION OF STRESSES

3.3 EQUATIONS OF EQUILIBRIUM

3.4 SYMMETRY OF THE CAUCHY STRESS TENSOR

3.5 VIRTUAL WORK OF THE FORCES

3.6 DEVIATORIC STRESSES

3.7 STRESS OBJECTIVITY

3.8 ENERGY BALANCE

PROBLEMS

4 CONSTITUTIVE EQUATIONS

4.1 GENERALIZED HOOKE’S LAW

4.2 ANISOTROPIC LINEARLY ELASTIC MATERIALS

4.3 MATERIAL SYMMETRY

4.4 HOMOGENEOUS ISOTROPIC MATERIAL

4.5 PRINCIPAL STRAIN INVARIANTS

4.6 SPECIAL MATERIAL MODELS FOR LARGE DEFORMATIONS

4.7 LINEAR VISCOELASTICITY

4.8 NONLINEAR VISCOELASTICITY

4.9 A SIMPLE VISCOELASTIC MODEL FOR ISOTROPIC MATERIALS

4.10 FLUID CONSTITUTIVE EQUATIONS

4.11 NAVIER–STOKES EQUATIONS

PROBLEMS

5 PLASTICITY FORMULATIONS

5.1 ONE-DIMENSIONAL PROBLEM

5.2 LOADING AND UNLOADING CONDITIONS

5.3 SOLUTION OF THE PLASTICITY EQUATIONS

5.4 GENERALIZATION OF THE PLASTICITY THEORY: SMALL STRAINS

5.5 J2 FLOW THEORY WITH ISOTROPIC/KINEMATIC HARDENING

5.6 NONLINEAR FORMULATION FOR HYPERELASTIC–PLASTIC MATERIALS

5.7 HYPERELASTIC–PLASTIC J2 FLOW THEORY

PROBLEMS

6 FINITE ELEMENT FORMULATION: LARGE-DEFORMATION, LARGE-ROTATION PROBLEM

6.1 DISPLACEMENT FIELD

6.2 ELEMENT CONNECTIVITY

6.3 INERTIA AND ELASTIC FORCES

6.4 EQUATIONS OF MOTION

6.5 NUMERICAL EVALUATION OF THE ELASTIC FORCES

6.6 FINITE ELEMENTS AND GEOMETRY

6.7 TWO-DIMENSIONAL EULER–BERNOULLI BEAM ELEMENT

6.8 TWO-DIMENSIONAL SHEAR DEFORMABLE BEAM ELEMENT

6.9 THREE-DIMENSIONAL CABLE ELEMENT

6.10 THREE-DIMENSIONAL BEAM ELEMENT

6.11 THIN-PLATE ELEMENT

6.12 HIGHER-ORDER PLATE ELEMENT

6.13 ELEMENT PERFORMANCE

6.14 OTHER FINITE ELEMENT FORMULATIONS

6.15 UPDATED LAGRANGIAN AND EULERIAN FORMULATIONS

PROBLEMS

7 FINITE ELEMENT FORMULATION: SMALL-DEFORMATION, LARGE-ROTATION PROBLEM

7.1 BACKGROUND

7.2 ROTATION AND ANGULAR VELOCITY

7.3 FLOATING FRAME OF REFERENCE

7.4 INTERMEDIATE ELEMENT COORDINATE SYSTEM

7.5 CONNECTIVITY AND REFERENCE CONDITIONS

7.6 KINEMATIC EQUATIONS

7.7 FORMULATION OF THE INERTIA FORCES

7.8 ELASTIC FORCES

7.9 EQUATIONS OF MOTION

7.10 COORDINATE REDUCTION

7.11 INTEGRATION OF FINITE ELEMENT AND MULTIBODY SYSTEM ALGORITHMS

PROBLEMS

References

Index

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COMPUTATIONAL CONTINUUM MECHANICS This book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in con- tinuum mechanics are presented and used to develop nonlinear gen- eral ﬁnite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear ﬁnite ele- ment dynamic formulations: a general large-deformation ﬁnite ele- ment formulation and then a formulation that can efﬁciently solve small deformation problems that characterize very and moderately stiff structures. The book presents material clearly and systematically, assuming the reader has only basic knowledge in matrix and vector algebra and dynamics. The book is designed for use by advanced undergraduates and ﬁrst-year graduate students. It is also a reference for researchers, practicing engineers, and scientists working in compu- tational mechanics, bio-mechanics, computational biology, multibody system dynamics, and other ﬁelds of science and engineering using the general continuum mechanics theory. Ahmed A. Shabana is the Richard and Loan Hill Professor of Engi- neering at the University of Illinois, Chicago. Dr. Shabana received his PhD in mechanical engineering from the University of Iowa. His active areas of research interest are in dynamics, vibration, and control of mechanical systems. He is also the author of other books, including Theory of Vibration: An Introduction, Vibration of Discrete and Continuous Systems, Computational Dynamics, Railroad Vehicle Dynamics: A Computational Approach, and Dynamics of Multibody Systems, Third Edition.

Computational Continuum Mechanics Ahmed A. Shabana Richard and Loan Hill Professor of Engineering University of Illinois at Chicago

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521885690 © Ahmed Shabana 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-38640-4 eBook (EBL) ISBN-13 978-0-521-88569-0 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents Preface page ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Matrices 1.2 Vectors 1.3 Summation Convention 1.4 Cartesian Tensors 1.5 Polar Decomposition Theorem 1.6 D’Alembert’s Principle 1.7 Virtual Work Principle 1.8 Approximation Methods 1.9 Discrete Equations 1.10 Momentum, Work, and Energy 1.11 Parameter Change and Coordinate Transformation PROBLEMS 2 6 12 13 25 27 34 37 40 43 45 48 2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1 Motion Description 2.2 Strain Components 2.3 Other Deformation Measures 2.4 Decomposition of Displacement 2.5 Velocity and Acceleration 2.6 Coordinate Transformation 2.7 Objectivity 2.8 Change of Volume and Area 2.9 Continuity Equation 2.10 Reynolds’ Transport Theorem 2.11 Examples of Deformation PROBLEMS 52 60 67 69 71 75 82 85 89 90 92 100 v

vi Contents 3 Forces and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1 Equilibrium of Forces 3.2 Transformation of Stresses 3.3 Equations of Equilibrium 3.4 Symmetry of the Cauchy Stress Tensor 3.5 Virtual Work of the Forces 3.6 Deviatoric Stresses 3.7 Stress Objectivity 3.8 Energy Balance PROBLEMS 103 106 107 109 111 120 123 127 129 4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . 131 4.1 Generalized Hooke’s Law 4.2 Anisotropic Linearly Elastic Materials 4.3 Material Symmetry 4.4 Homogeneous Isotropic Material 4.5 Principal Strain Invariants 4.6 Special Material Models for Large Deformations 4.7 Linear Viscoelasticity 4.8 Nonlinear Viscoelasticity 4.9 A Simple Viscoelastic Model for Isotropic Materials 4.10 Fluid Constitutive Equations 4.11 Navier-Stokes Equations PROBLEMS 132 134 135 137 144 146 150 164 171 173 174 175 5 Plasticity Formulations . . . . . . . . . . . . . . . . . . . . . . . 177 5.1 One-Dimensional Problem 5.2 Loading and Unloading Conditions 5.3 Solution of the Plasticity Equations 5.4 Generalization of the Plasticity Theory: Small Strains 5.5 J2 Flow Theory with Isotropic/Kinematic Hardening 5.6 Nonlinear Formulation for Hyperelastic–Plastic Materials 5.7 Hyperelastic–Plastic J2 Flow Theory PROBLEMS 179 180 181 190 197 214 225 230 6 Finite Element Formulation: Large-Deformation, . Large-Rotation Problem . . . . . . . . . . . . . . . . . . . . . 231 6.1 Displacement Field 6.2 Element Connectivity 6.3 Inertia and Elastic Forces 6.4 Equations of Motion 6.5 Numerical Evaluation of the Elastic Forces 6.6 Finite Elements and Geometry 6.7 Two-Dimensional Euler–Bernoulli Beam Element 233 240 243 246 250 256 263

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