Table Of ContentComputational Fluid and Solid Mechanics
Series Editor:
Klaus-Jürgen Bathe
Massachusetts Institute of Technology
Cambridge, MA, USA
Advisors:
Franco Brezzi Olivier Pironneau
University of Pavia Université Pierre et Marie Curie
Pavia, Italy 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
Authors:
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
Library of Congress Control Number: 2005929275
ISBN-10 3-540-24985-0 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-24985-6 Springer Berlin Heidelberg New York
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To the Argentine system of public education
Preface
ThisbookdevelopsamodernpresentationofContinuumMechanics,oriented
towards numerical applications in the fields of nonlinear analysis of solids,
structures and fluids.
Kinematicsofthecontinuumdeformation,includingpull-back/push-forward
transformations between di(cid:30)erent configurations; stress and strain measures;
objectivestressrateandstrainratemeasures;balanceprinciples;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.
Embeddedinthetheoreticalpresentation,applicationexamplesaredevel-
oped to deepen the understanding of the discussed concepts.
Eventhoughthemathematicalpresentationofthedi(cid:30)erenttopicsisquite
rigorous;ane(cid:30)ortismadetolinkformaldevelopmentswithengineeringphys-
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 hisenthusiasm,for his encouragementto undertakechallenges
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 e(cid:30)orts 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
introducingmetoComputationalMechanics.Iamalsogratefultomystudents
at the University of Buenos Aires and to my research assistants at the Center
for Industrial Research of FUDETEC for their willingness and e(cid:30)ort.
I want to recognize the permanent support of my mother Esther, of my
sister Mónica and of my friends and colleagues.
Marcela B. Goldschmit
Contents
1 Introduction(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61) 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 (cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61) 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 (cid:119)R (cid:62) (cid:119)U and (cid:119)V 26
(cid:17) (cid:17) (cid:17)
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 indi(cid:30)erence ............ 47
2.12.3 Covariance ....................................... 49
2.13 Strain rates............................................. 50
2.13.1 The velocity gradient tensor ........................ 50
2.13.2 TheEulerianstrainratetensorandthespin(vorticity)
tensor............................................ 51
2.13.3 Relations between di(cid:30)erent 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 (cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61) 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(cid:30) stress tensor ......................... 74
3.3.2 The first Piola-Kirchho(cid:30) stress tensor ............... 74
3.3.3 The second Piola-Kirchho(cid:30) stress tensor ............. 76
3.3.4 Astresstensorenergyconjugatetothetimederivative
of the Hencky strain tensor ........................ 79
3.4 Objective stress rates .................................... 81
4 Balance principles (cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61) 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)formulationofthemass-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 balance112
5 Constitutive relations(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)115
5.1 Fundamentals for formulating constitutive relations ..........116
5.1.1 Principle of equipresence ...........................116
5.1.2 Principle of material-frame indi(cid:30)erence ..............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 (cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)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
