Nonlinear Continua Part 1 pps

Kinematics of the continuum deformation, including pull-back/push-forward transformations between dierent configurations; stress and strain measures; objective stress rate and strain rat

Computational Fluid and Solid Mechanics

Series Editor:

Klaus-Jürgen Bathe

Massachusetts Institute of Technology

Cambridge, MA, USA

Advisors:

Franco Brezzi

University of Pavia

Pavia, Italy

Olivier Pironneau Université Pierre et Marie Curie Paris, France

Available Volumes

D Chapelle, K.J Bathe

The Finite Element Analysis of Shells - Fundamentals,

2003

D Drikakis, W Rider

High-Resolution Methods for Incompressible and Low-Speed Flows 2005

M Kojic, K.J Bathe

Inelastic Analysis of Solids and Structures

2005

E.N Dvorkin, M.B Goldschmit

Nonlinear Continua

2005

Eduardo N Dvorkin · Marcela B Goldschmit

Nonlinear Continua

With 30 Figures

Eduardo N Dvorkin, Ph.D

Marcela B Goldschmit, Dr Eng

Engineering School

University of Buenos Aires and

Center for Industrial Research

TENARIS Dr Simini 250

B2804MHA

Campana

Argentina

ISBN-10 3-540-24985-0 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-24985-6 Springer Berlin Heidelberg New York

This work is subject to copyright All rights are reserved, whether the whole or part of the material

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To the Argentine system of public education

This book develops a modern presentation of Continuum Mechanics, oriented towards numerical applications in the fields of nonlinear analysis of solids, structures and fluids

Kinematics of the continuum deformation, including pull-back/push-forward transformations between dierent configurations; stress and strain measures; objective stress rate and strain rate measures; balance principles; constitutive relations, with emphasis on elasto-plasticity of metals and variational princi-ples are developed using general curvilinear coordinates

Being tensor analysis the indispensable tool for the development of the continuum theory in general coordinates, in the appendix an overview of ten-sor analysis is also presented

Embedded in the theoretical presentation, application examples are devel-oped to deepen the understanding of the discussed concepts

Even though the mathematical presentation of the dierent topics is quite rigorous; an eort is made to link formal developments with engineering phys-ical intuition

This book is based on two graduate courses that the authors teach at the Engineering School of the University of Buenos Aires and it is intended for graduate engineering students majoring in mechanics and for researchers in the fields of applied mechanics and numerical methods

VIII Preface

I am grateful to Klaus-Jürgen Bathe for introducing me to Computational Mechanics, for his enthusiasm, for his encouragement to undertake challenges and for his friendship

I am also grateful to my colleagues, to my past and present students at the University of Buenos Aires and to my past and present research assistants at the Center for Industrial Research of FUDETEC because I have always learnt from them

I want to thank Dr Manuel Sadosky for inspiring many generations of Argentine scientists

I am very grateful to my late father Israel and to my mother Raquel for their eorts and support

Last but not least I want to thank my dear daughters Cora and Julia, my wife Elena and my friends (the best) for their continuous support

Eduardo N Dvorkin

I would like to thank Professors Eduardo Dvorkin and Sergio Idelsohn for introducing me to Computational Mechanics I am also grateful to my students

at the University of Buenos Aires and to my research assistants at the Center for Industrial Research of FUDETEC for their willingness and eort

I want to recognize the permanent support of my mother Esther, of my sister Mónica and of my friends and colleagues

Marcela B Goldschmit

1 Introduction= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 1

1.1 Quantification of physical phenomena 1

1.1.1 Observation of physical phenomena 1

1.1.2 Mathematical model 2

1.1.3 Numerical model 2

1.1.4 Assessment of the numerical results 2

1.2 Linear and nonlinear mathematical models 2

1.3 The aims of this book 4

1.4 Notation 5

2 Kinematics of the continuous media = = = = = = = = = = = = = = = = = = = = = = = 7 2.1 The continuous media and its configurations 7

2.2 Mass of the continuous media 9

2.3 Motion of continuous bodies 9

2.3.1 Displacements 9

2.3.2 Velocities and accelerations 10

2.4 Material and spatial derivatives of a tensor field 12

2.5 Convected coordinates 13

2.6 The deformation gradient tensor 13

2.7 The polar decomposition 21

2.7.1 The Green deformation tensor 21

2.7.2 The right polar decomposition 22

2.7.3 The Finger deformation tensor 25

2.7.4 The left polar decomposition 25

2.7.5 Physical interpretation of the tensors wR> w Uand wV 26 2.7.6 Numerical algorithm for the polar decomposition 28

2.8 Strain measures 33

2.8.1 The Green deformation tensor 33

2.8.2 The Finger deformation tensor 33

2.8.3 The Green-Lagrange deformation tensor 34

2.8.4 The Almansi deformation tensor 35

X Contents

2.8.5 The Hencky deformation tensor 35

2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 36

2.9.1 Pull-back of vector components 36

2.9.2 Pull-back of tensor components 40

2.10 Tensors in the spatial configuration from representations in the reference configuration (“push-forward”) 42

2.11 Pull-back/push-forward relations between strain measures 43

2.12 Objectivity 44

2.12.1 Reference frame and isometric transformations 45

2.12.2 Objectivity or material-frame indierence 47

2.12.3 Covariance 49

2.13 Strain rates 50

2.13.1 The velocity gradient tensor 50

2.13.2 The Eulerian strain rate tensor and the spin (vorticity) tensor 51

2.13.3 Relations between dierent rate tensors 53

2.14 The Lie derivative 56

2.14.1 Objective rates and Lie derivatives 58

2.15 Compatibility 61

3 Stress Tensor= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 67 3.1 External forces 67

3.2 The Cauchy stress tensor 69

3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem) 71

3.3 Conjugate stress/strain rate measures 72

3.3.1 The Kirchho stress tensor 74

3.3.2 The first Piola-Kirchho stress tensor 74

3.3.3 The second Piola-Kirchho stress tensor 76

3.3.4 A stress tensor energy conjugate to the time derivative of the Hencky strain tensor 79

3.4 Objective stress rates 81

4 Balance principles = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 85 4.1 Reynolds’ transport theorem 85

4.1.1 Generalized Reynolds’ transport theorem 88

4.1.2 The transport theorem and discontinuity surfaces 90

4.2 Mass-conservation principle 93

4.2.1 Eulerian (spatial) formulation of the mass-conservation principle 93

4.2.2 Lagrangian (material) formulation of the mass conservation principle 95

4.3 Balance of momentum principle (Equilibrium) 95

Contents XI

4.3.1 Eulerian (spatial) formulation of the balance of

momentum principle 96

4.3.2 Lagrangian (material) formulation of the balance of momentum principle 103

4.4 Balance of moment of momentum principle (Equilibrium) 105

4.4.1 Eulerian (spatial) formulation of the balance of moment of momentum principle 105

4.4.2 Symmetry of Eulerian and Lagrangian stress measures 107 4.5 Energy balance (First Law of Thermodynamics) 109

4.5.1 Eulerian (spatial) formulation of the energy balance 109

4.5.2 Lagrangian (material) formulation of the energy balance 112 5 Constitutive relations= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 115 5.1 Fundamentals for formulating constitutive relations 116

5.1.1 Principle of equipresence 116

5.1.2 Principle of material-frame indierence 116

5.1.3 Application to the case of a continuum theory restricted to mechanical variables 116

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 120

5.2.1 Hyperelastic material models 121

5.2.2 A simple hyperelastic material model 122

5.2.3 Other simple hyperelastic material models 128

5.2.4 Ogden hyperelastic material models 129

5.2.5 Elastoplastic material model under infinitesimal strains 135 5.2.6 Elastoplastic material model under finite strains 155

5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 167

5.3.1 The isotropic thermoelastic constitutive model 167

5.3.2 A thermoelastoplastic constitutive model 170

5.4 Viscoplasticity 176

5.5 Newtonian fluids 180

5.5.1 The no-slip condition 181

6 Variational methods = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 183 6.1 The Principle of Virtual Work 183

6.2 The Principle of Virtual Work in geometrically nonlinear problems 186

6.2.1 Incremental Formulations 189

6.3 The Principle of Virtual Power 194

6.4 The Principle of Stationary Potential Energy 195

6.5 Kinematic constraints 207

6.6 Veubeke-Hu-Washizu variational principles 209

6.6.1 Kinematic constraints via the V-H-W principles 209

6.6.2 Constitutive constraints via the V-H-W principles 211

XII Contents

A Introduction to tensor analysis = = = = = = = = = = = = = = = = = = = = = = = = = = = = 213

A.1 Coordinates transformation 213

A.1.1 Contravariant transformation rule 214

A.1.2 Covariant transformation rule 215

A.2 Vectors 215

A.2.1 Base vectors 216

A.2.2 Covariant base vectors 216

A.2.3 Contravariant base vectors 218

A.3 Metric of a coordinates system 219

A.3.1 Cartesian coordinates 219

A.3.2 Curvilinear coordinates Covariant metric components 220 A.3.3 Curvilinear coordinates Contravariant metric components 220

A.3.4 Curvilinear coordinates Mixed metric components 221

A.4 Tensors 222

A.4.1 Second-order tensors 223

A.4.2 n-order tensors 227

A.4.3 The metric tensor 228

A.4.4 The Levi-Civita tensor 229

A.5 The quotient rule 232

A.6 Covariant derivatives 233

A.6.1 Covariant derivatives of a vector 233

A.6.2 Covariant derivatives of a general tensor 236

A.7 Gradient of a tensor 237

A.8 Divergence of a tensor 238

A.9 Laplacian of a tensor 239

A.10 Rotor of a tensor 240

A.11 The Riemann-Christoel tensor 240

A.12 The Bianchi identity 243

A.13 Physical components 244 References= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 247 Index= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 255

Introduction

The quantitative description of the deformation of continuum bodies, either solids or fluids subjected to mechanical and thermal loadings, is a challenging scientific field with very relevant technological applications

1.1 Quantification of physical phenomena

The quantification of a physical phenomenon is performed through four dif-ferent consecutive steps:

1 Observation of the physical phenomenon under study Identification of its most relevant variables

2 Formulation of a mathematical model that describes, in the framework of the assumptions derived from the previous step, the physical phenomenon

3 Formulation of the numerical model that solves, within the required ac-curacy, the above-formulated mathematical model

4 Assessment of the adequacy of the numerical results to describe the phe-nomenon under study

1.1.1 Observation of physical phenomena

This is a crucial step that conditions the next three Making an educated observation of a physical phenomenon means establishing a set of concepts and relations that will govern the further development of the mathematical model

At this stage we also need to decide on the quantitative output that we shall require from the model

2 Nonlinear continua

1.1.2 Mathematical model

Considering the assumptions derived from the previous step and our knowl-edge on the physics of the phenomenon under study, we can establish the mathematical model that simulates it This mathematical model, at least for the cases that fall within the field that this book intends to cover, is normally

a system of partial dierential equations (PDE) with established boundary and initial conditions

1.1.3 Numerical model

Usually the PDE system that constitutes the mathematical model cannot be solved in closed form and the analyst needs to resort to a numerical model in order to arrive at the actual quantification of the phenomenon under study

1.1.4 Assessment of the numerical results

The analyst has to judge if the numerical results are acceptable This is a very important step and it involves:

• Verification of the mathematical model, that is to say, checking that the numerical results do not contradict any of the assumptions introduced for the formulation of the mathematical model and verification that the numerical results “make sense” by comparing them with the results of a

“back-of-an-envelope” calculation (here, of course, we only compare orders

of magnitude)

• Verification of the numerical model, the analyst has to assess if the numer-ical model can assure convergence to the unknown exact solution of the mathematical model when the numerical degrees of freedom are increased The analyst must also check the stability of the numerical results when small perturbations are introduced in the data If the results are not sta-ble the analyst has to assess if the unstasta-ble numerical results represent an unstable physical phenomenon or if they are the result of an unacceptable numerical deficiency

• Validation of the mathematical/numerical model comparing its predictions with experimental observations

1.2 Linear and nonlinear mathematical models

When deriving the PDE system that constitutes the mathematical model of

a physical phenomenon there are normally a number of nonlinear terms that appear in those equations Considering always all the nonlinear terms, even if