Nonlinear Continua

Home , Objective stress rate

Eduardo N. Dvorkin Marcela B. Goldschmit

Nonlinear Continua

SPIN 10996775

— Monograph —

May 6, 2005

Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

To the Argentine system of public education

Preface

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 different 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 principles 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- soranalysisisalsopresented. Embedded in the theoretical presentation, application examples are developed to deepen the understanding of the discussed concepts. Even though the mathematical presentation of the different topics is quite rigorous; an effort is made to link formal developments with engineering physical 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 Preface IX

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 eﬀorts 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 eﬀort. 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 ...... 1 1.1 Quantiﬁcationofphysicalphenomena...... 1 1.1.1 Observationofphysicalphenomena...... 1 1.1.2 Mathematicalmodel...... 2 1.1.3 Numericalmodel...... 2 1.1.4 Assessmentofthenumericalresults...... 2 1.2 Linearandnonlinearmathematicalmodels ...... 2 1.3 Theaimsofthisbook...... 4 1.4 Notation...... 5

2 Kinematics of the continuous media ...... 7 2.1 The continuous media and its conﬁgurations ...... 7 2.2 Massofthecontinuousmedia...... 9 2.3 Motionofcontinuousbodies...... 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 Convectedcoordinates ...... 13 2.6 Thedeformationgradienttensor...... 13 2.7 Thepolardecomposition...... 21 2.7.1 TheGreendeformationtensor ...... 21 2.7.2 Therightpolardecomposition...... 22 2.7.3 TheFingerdeformationtensor ...... 25 2.7.4 Theleftpolardecomposition...... 25 2.7.5 Physical interpretation of the tensors t R , t U and t V 26 2.7.6 Numericalalgorithmforthepolardecomposition...... ◦ ◦ ◦ 28 2.8 Strainmeasures...... 33 2.8.1 TheGreendeformationtensor ...... 33 2.8.2 TheFingerdeformationtensor...... 33 2.8.3 TheGreen-Lagrangedeformationtensor...... 34 2.8.4 TheAlmansideformationtensor ...... 35 XII Contents

2.8.5 TheHenckydeformationtensor...... 35 2.9 Representation of spatial tensors in the reference configuration(“pull-back”) ...... 36 2.9.1 Pull-backofvectorcomponents ...... 36 2.9.2 Pull-backoftensorcomponents ...... 40 2.10 Tensors in the spatial configuration from representations in the reference configuration(“push-forward”)...... 42 2.11Pull-back/push-forwardrelationsbetweenstrainmeasures.... 43 2.12Objectivity...... 44 2.12.1Referenceframeandisometrictransformations ...... 45 2.12.2 Objectivity or material-frame indifference ...... 47 2.12.3Covariance ...... 49 2.13Strainrates...... 50 2.13.1Thevelocitygradienttensor...... 50 2.13.2 The Eulerian strain rate tensor and the spin (vorticity) tensor...... 51 2.13.3 Relations between differentratetensors...... 53 2.14TheLiederivative...... 56 2.14.1ObjectiveratesandLiederivatives ...... 58 2.15 Compatibility ...... 61

3StressTensor...... 67 3.1 Externalforces...... 67 3.2 TheCauchystresstensor...... 69 3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem) ...... 71 3.3 Conjugatestress/strainratemeasures...... 72 3.3.1 The Kirchhoff stresstensor...... 74 3.3.2 The first Piola-Kirchhoff stresstensor ...... 74 3.3.3 The second Piola-Kirchhoff stresstensor ...... 76 3.3.4 A stress tensor energy conjugate to the time derivative oftheHenckystraintensor ...... 79 3.4 Objectivestressrates...... 81

4 Balance principles ...... 85 4.1 Reynolds’transporttheorem...... 85 4.1.1 GeneralizedReynolds’transporttheorem...... 88 4.1.2 Thetransporttheoremanddiscontinuitysurfaces ..... 90 4.2 Mass-conservationprinciple...... 93 4.2.1 Eulerian (spatial) formulation of the mass-conservation principle ...... 93 4.2.2 Lagrangian (material) formulation of the mass conservationprinciple...... 95 4.3 Balanceofmomentumprinciple(Equilibrium) ...... 95 Contents XIII

4.3.1 Eulerian (spatial) formulation of the balance of momentumprinciple...... 96 4.3.2 Lagrangian (material) formulation of the balance of momentumprinciple...... 103 4.4 Balanceofmomentofmomentumprinciple(Equilibrium) ....105 4.4.1 Eulerian (spatial) formulation of the balance of momentofmomentumprinciple ...... 105 4.4.2 Symmetry of Eulerian and Lagrangian stress measures 107 4.5 Energybalance(FirstLawofThermodynamics) ...... 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 ...... 115 5.1 Fundamentalsforformulatingconstitutiverelations...... 116 5.1.1 Principleofequipresence...... 116 5.1.2 Principle of material-frame indifference ...... 116 5.1.3 Application to the case of a continuum theory restrictedtomechanicalvariables...... 116 5.2 Constitutive relations in solid mechanics: purely mechanical formulations...... 120 5.2.1 Hyperelasticmaterialmodels ...... 121 5.2.2 Asimplehyperelasticmaterialmodel...... 122 5.2.3 Othersimplehyperelasticmaterialmodels...... 128 5.2.4 Ogdenhyperelasticmaterialmodels...... 129 5.2.5 Elastoplastic material model under infinitesimal strains 135 5.2.6 Elastoplastic material model under finitestrains ...... 155 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations...... 167 5.3.1 Theisotropicthermoelasticconstitutivemodel ...... 167 5.3.2 Athermoelastoplasticconstitutivemodel ...... 170 5.4 Viscoplasticity...... 176 5.5 Newtonian fluids...... 180 5.5.1 Theno-slipcondition...... 181

6 Variational methods ...... 183 6.1 ThePrincipleofVirtualWork ...... 183 6.2 The Principle of Virtual Work in geometrically nonlinear problems ...... 186 6.2.1 IncrementalFormulations ...... 189 6.3 ThePrincipleofVirtualPower...... 194 6.4 ThePrincipleofStationaryPotentialEnergy ...... 195 6.5 Kinematicconstraints...... 207 6.6 Veubeke-Hu-Washizuvariationalprinciples ...... 209 6.6.1 KinematicconstraintsviatheV-H-Wprinciples ...... 209 6.6.2 ConstitutiveconstraintsviatheV-H-Wprinciples ....211 XIV Contents

A Introduction to tensor analysis ...... 213 A.1 Coordinatestransformation...... 213 A.1.1 Contravarianttransformationrule...... 214 A.1.2 Covarianttransformationrule...... 215 A.2 Vectors ...... 215 A.2.1 Basevectors...... 216 A.2.2 Covariantbasevectors...... 216 A.2.3 Contravariantbasevectors...... 218 A.3 Metricofacoordinatessystem...... 219 A.3.1 Cartesiancoordinates...... 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-ordertensors...... 223 A.4.2 n-ordertensors...... 227 A.4.3 Themetrictensor...... 228 A.4.4 TheLevi-Civitatensor ...... 229 A.5 Thequotientrule ...... 232 A.6 Covariantderivatives...... 233 A.6.1 Covariantderivativesofavector ...... 233 A.6.2 Covariantderivativesofageneraltensor ...... 236 A.7 Gradientofatensor...... 237 A.8 Divergenceofatensor...... 238 A.9 Laplacianofatensor...... 239 A.10Rotorofatensor...... 240 A.11 The Riemann-Christoﬀeltensor ...... 240 A.12TheBianchiidentity ...... 243 A.13Physicalcomponents...... 244

References ...... 247

Index ...... 255 1 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 different 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 accuracy, the above-formulated mathematical model. 4. Assessment of the adequacy of the numerical results to describe the phenomenon 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 knowledge 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 diﬀerential 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 stable the analyst has to assess if the unstable 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 1.2 Linear and nonlinear mathematical models 3 their inﬂuence is negligible on the ﬁnal numerical results, is mathematically correct; however, it may not be always practical. The scientist or engineer facing the development of the mathematical model of a physical phenomenon has to decide which nonlinearities have to be kept in the model and which ones can be neglected. This is the main contri- bution of an analyst: formulating a model that is as simple as possible while keeping all the relevant aspects of the problem under analysis (Bathe 1996). In many problems it is not possible to neglect all nonlinearities because the main features of the phenomenon under study lie in their consideration (Hodge, Bathe & Dvorkin 1986); in these cases the analyst must have enough physical insight into the problem so as to incorporate all the fundamental nonlinear aspects but only the fundamental ones. The more nonlinearities are introduced in the mathematical model, the more computational resources will be necessary to solve the numerical model and in many cases it may happen that the necessary computational resources are much larger than the available ones, making the analysis impossible.

Example 1.1. JJJJJ In the analysis of a solid under mechanical and thermal loads some of the nonlinearities that we may encounter when formulating the mathematical model are: Geometrical nonlinearities: they are introduced by the fact that the equi- • librium equations have to be satisfied in the unknown deformed configuration of the solid rather than in the known unloaded configuration. When the analyst expects that for her/his practical purposes the difference between the deformed and unloaded configurations is negligible she/he may neglect this source of nonlinearity obtaining an important simplification in the mathematical model. An intermediate step would be to consider the equilibrium in the deformed configuration but to assume that the strains are very small (infinitesimal strains assumption). This also produces an important simplification in the mathematical model. Of course, all the simplifications introduced in the mathematical model have to be checked for their properness when examining the obtained numerical results. Contact-type boundary conditions: these are unilateral constraints in • which the contact loads are distributed over an area that is a priori unknown to the analyst. Material nonlinearities: elastoplastic materials (e.g. metals); creep behav- • ior of metals in high-temperature environments; nonlinear elastic materials (e.g. polymers); fracturing materials (e.g. concrete); etc.

JJJJJ 4 Nonlinear continua

Example 1.2. JJJJJ In the analysis of a fluid flow under mechanical and thermal loads some of the nonlinearities that we may encounter when formulating the mathematical model are: Non-constant viscosity/compressibility (e.g. rheological materials and tur- • bulent flows modeled using turbulence models). Convective acceleration terms for flows with Re>0 when the mathematical • model is developed using an Eulerian formulation, which is the standard case.

JJJJJ

Example 1.3. JJJJJ In the analysis of a heat transfer problem some of the nonlinearities that we may encounter when formulating the mathematical model are: Temperature dependent thermal properties (e.g. phase changes). • Radiation boundary conditions • JJJJJ There are mathematical models in which the effects (outputs) are proportional to the causes (inputs); these are linear models. Examples of linear models are: linear elasticity problems, • constant viscosity creeping flows, • heat transfer problems in materials in which constant thermal properties • are assumed and radiative boundary conditions are not considered, etc. • Deciding that the model that simulates a physical phenomenon is going to be linear is an analyst decision, after first considering and afterwards carefully neglecting, in the formulation of the mathematical model, all the sources of nonlinearity.

1.3 The aims of this book

This book intends to provide a modern and rigorous exposition of nonlinear continuum mechanics and even though it does not deal with computational implementations it is intended to provide the basis for them. In the second chapter of the book we present a consistent description of the kinematics of the continuous media. In that chapter we introduce the concepts of pull-back, push-forward and Lie derivative requiring only from the reader 1.4 Notation 5 a previous knowledge of tensor analysis. Objective and covariant strain and strain rate measures are derived. In the third chapter we discuss different stress measures that are energy conjugate to the strain rate measures presented in the previous chapter. Ob- jective stress rate measures are derived. In the fourth chapter we present the Reynolds transport theorem and then we use it to develop Eulerian and Lagrangian formulations for expressing the balance (conservation) of mass, momentum, moment of momentum and energy. In the fifth chapter we develop an extensive presentation of constitutive relations for solids and fluids, with special focus on the elastoplasticity of metals. Finally, in the sixth chapter we develop the variational approach to continuum mechanics, centering our presentation on the principle of virtual work and discussing also the principle of stationary potential energy and the Veubeke- Hu-Washizu variational principles. The basic mathematical tool in the book is tensor calculus; in order to assure a common basis for all the readers, in the Appendix we present a review of this topic.

1.4 Notation

Throughout the book we shall use the summation convention; that is to say, in a Cartesian coordinate system

3 aα bα = aα bα α=1 X3 aαβ bβ = aαβ bβ for α =1, 2, 3 , β=1 X and in a general curvilinear system

3 i i ai b = ai b i=1 X3 ij ij a bj = a bj for i =1, 2, 3 . j=1 X Also, our notation is compatible with the notation introduced in continuum mechanics by Bathe (Bathe 1996). We shall deﬁne all notation at the point whereweincorporateit.

2 Kinematics of the continuous media

In this chapter we are going to present a kinematic description of the deformation of continuous media. That is to say, we are going to describe the deformation without considering the loads that cause it and without introducing into the analysis the behavior of the material. Some reference books for this chapter are: (Truesdell & Noll 1965, Truesdell 1966, Malvern 1969, Marsden & Hughes 1983).

2.1 The continuous media and its conﬁgurations

Continuum mechanics is the branch of mechanics that studies the motion of solids, liquids and gases under the hypothesis of continuous media.This hypothesis is an idealization of matter that disregards its atomic or molecular structure. A continuous body is an open subset of the three-dimensional Euclidean space 3 (Oden 1979) 1 .Eachelement“χ” of that subset is called a point or a material< particle. The region of the Euclidean space occupied by the particles χ of the¡ continuous¢ body at time t is called the configuration corresponding to t. B Above, we use the notion of time in a very general sense: as a coordinate that is used to enumerate a series of events.Aninstant t is a particular value of the time coordinate. We can establish a bijective mapping (Oden 1979) between each point of space occupied by a material particle χ at t and an arbitrary curvilinear coordinate system txa,a=1, 2, 3 . Thefactthatateachinstant{ }t the set txa defines one and only one particle χ implies that in a continuum medium,{ } different material particles

1 The requirement of an open subset is introduced in order to eliminate the possible consideration of isolated points, sets in 3 with zero volume, etc. < 8 Nonlinear continua cannot occupy the same space location and that a material particle cannot be subdivided: txa = txa(χ, t); χ = χ(txa) . (2.1) We assume that two arbitrary coordinate systems defined for the configuration at time t are related by continuous and differentiable functions: tx˜b = tx˜b(txa); txa = txa(tx˜b) . (2.2) In a formal way, we say that the configuration of the body corresponding to time t is an homeomorphism of onto a region of the threeB dimensional Euclidean space ( 3) (Truesdell &B Noll 1965). An homeomorphism (Oden 1979) is a bijective< and continuous mapping with its inverse mapping also continuous. The coordinates txa are called the spatial coordinates of the material particle χ in the confi{guration} at time t. We call the motion of the body the evolution from aconfiguration at B an instant t1 to aconfiguration at an instant t2. We select any configuration of the body as the reference configuration (e.g. the undeformed configuration, but not necessarilyB this one); also we can set the time origin so that in the reference configuration t =0.Inthe reference configuration we define an arbitrary curvilinear coordinate system A ◦x ,A =1, 2, 3 :thematerial coordinates. For the reference configuration Eqs.(2.1){ are } A A A ◦x = ◦x (χ); χ = χ(◦x ) . (2.3) From Eqs.(2.1) and (2.3) we obtain the bijective mapping tφ between the configuration at time t and the reference configuration,

t a t a A A t 1 A t a x = φ (◦x ,t); ◦x = φ− ( x ) . (2.4) t 1 t r In a regular motion the inverse mapping φ− £exists¤ and if φ C also t 1 r r ∈ φ− C (Marsden & Hughes 1983), where C is the set of all functions with continuous∈ derivatives up to the order “r”. The formal concept of regular motion agrees with the intuitive concept of a motion without material interpenetration. From Eqs. (2.2) and (2.4) we get

t a t a t b t a t b A t a A x˜ = x˜ ( x )= x˜ φ (◦x ) = φ˜ (◦x ) . (2.5)

The mapping tφ is a function of: h i the reference configuration, • the configuration at t, • the material coordinate system, • the spatial coordinate system. • Since we restrict our presentation to the Euclidean space 3,wecancon- sider tφ as a vector. Hence, in Sect. 2.3.1 we are going to defi

The continuous media have a non-negative scalar property named mass. Our knowledge of Newton laws makes us relate the mass of a body with a measure of its inertia. After (Truesdell 1966) we are going to assume a continuous mass distribu- tion in the body . Concentrated masses do not belong to the field of Con- tinuum MechanicsB (therefore, Rational Mechanics is not part of Continuum Mechanics). We define the density “ tρ ” corresponding to the configuration at time t as

m = tρ tdV, (2.6) t Z V where, m:massofbody , tV :volumeof in the conﬁguration at time t. Equation (2.6) incorporatesB an importantB postulate of Newtonian mechanics: the mass of a body is constant in time.

2.3 Motion of continuous bodies

2.3.1 Displacements

In the mapping tφ ( 3 3) schematized in Fig. 2.1, at a given point < −→ < (particle) χ, the vectors g are the covariant base vectors1 (Green & Zerna ◦ A A 1968) of the material coordinates ◦x (reference configuration; t =0)and the vectors tg are the covariant base{ vectors} of the spatial coordinates txa a (spatial configuration corresponding to time t). { } In the 3D Euclidean space we also define a fixed Cartesian system α t α ◦z z α =1, 2, 3 with a set of orthonormal base vectors eα . { For≡ the Cartesian coordinates} of a particle χ in the reference configuration α we use the triad ◦z and for the Cartesian coordinates of the same particle in the spatial con{figuration} we use the triad tzα . { } In the Cartesian system, the position vector ◦x of a particle χ in the reference configuration is

α ◦x(χ)=◦z (χ) eα , (2.7) and the position vector tx of the particle χ in the spatial conﬁguration is

t t α x(χ, t)= z (χ, t) eα . (2.8) The displacement vector of the particle χ from the reference conﬁguration to the spatial conﬁguration is,

1 See Appendix. 10 Nonlinear continua

Fig. 2.1. Motion of continuous body

t t u(χ, t)= x(χ, t) ◦x(χ) (2.9a) − and the Cartesian components of this vector are,

t α t α α u (χ, t)= z (χ, t) ◦z (χ) . (2.9b) −

2.3.2 Velocities and accelerations

During the motion tφ,thematerial velocity of a particle χ in the t-conﬁguration is

∂tx(χ, t) ∂tu(χ, t) tv(χ, t)= = (2.10) ∂t ∂t assuming that the time derivatives in Eq. (2.10) exist. The material velocity vector is deﬁnedinthespatialconﬁguration (see Fig. 2.2). We can have, alternatively, the following functional dependencies:

t t A v = v(◦x ,t) , (2.11a) tv = tv(txa,t) . (2.11b)

Equation (2.11a) corresponds to a Lagrangian (material) description of motion, while Eq. (2.11b) corresponds to a Eulerian (spatial) description of 2.3 Motion of continuous bodies 11

Fig. 2.2. Material velocity of a particle χ motion. In general, the motion of solids is studied using Lagrangian descriptions while the motion of fluids is studied using Eulerian descriptions; however, this classification is by no means mandatory and combined descriptions have also been used in the literature (Belytschko, Lui & Moran 2000). In any case, we can refer the velocity vector either to the spatial coordinates or to the fixed Cartesian coordinates,

tv = tvatg , (2.12a) a t t α v = v eα . (2.12b)

t t t Assuming an arbitrary tensor field η = η(χ, t)= η( ◦x,t) we define its temporal material derivative (Dtη/Dt) asthetimerateofthetensortη when we keep constant the particle χ or, equivalently, when we keep constant the position vector ◦x in the reference configuration. We call the material acceleration of a particle χ,

Dtv ta = . (2.13) Dt In what follows, we determine the material acceleration vector considering the diﬀerent combinations of: Lagrangian Spatial description + coordinates ⎧ ⎫ ⎧ ⎫ ⎨ Eulerian ⎬ ⎨ FixedCartesian⎬ ⎩ ⎭ ⎩ ⎭ Lagrangian description + spatial coordinates • 12 Nonlinear continua

∂tva ta = + tΓ a tvbtvc tg , (2.14a) ∂t bc a ∙ ¸ t a where the Γbc are the Christoﬀel symbols of the second kind of the spatial coordinates txa 2 . Lagrangian description{ } + ﬁxed Cartesian coordinates •

∂tvα ta = e . (2.14b) ∂t α Eulerian description + spatial coordinates • The material particle that at time t is at the spatial location txa ,at time t + dt will be at txa + tva dt . Hence, { } { } ∂tv ∂tv t+dtv = tv + dt + tvb dt ∂t ∂txb Considering that the base vectors, in a general coordinate system, are functions of the position, we have

∂tva ∂tva ta = + tvb + tΓ a tvbtvc tg . (2.15a) ∂t ∂txb bc a ∙ ¸ Eulerian description + ﬁxed Cartesian coordinates •

∂tvα ∂tvα ta = + tvβ e . (2.15b) ∂t ∂tzβ α ∙ ¸

2.4 Material and spatial derivatives of a tensor ﬁeld

Let tη be an arbitrary tensor ﬁeld, a function of time, and using a Lagrangian description of motion we get,

t t A η = η(◦x ,t) . (2.16a) Now, using an Eulerian description of motion, we get

tη = tη(txa,t) . (2.16b)

We indicate the ﬁrst material time derivative (following the particle) as

t t A t D η ∂ η(◦x ,t) η˙ = = x . (2.17) Dt ∂t |◦ A 2 See Appendix. 2.6 The deformation gradient tensor 13

The first spatial time derivative of a tensor tη defined using a Eulerian ∂tη(txa,t) description is simply indicated as ∂t txa . Let us now calculate the material time| derivative of a tensor tη defined using an Eulerian description,

t t a b t a t t t c t d η = η ··· c d( x ,t) g ... g g ... g (2.18) ··· a b it is easy to derive the following relation (Truesdell & Noll 1965, Slattery 1972), ∂tη ∂tη tη˙ = + tvp . (2.19) ∂t ∂txp Using the spatial gradient of the tensor tη 3 :

∂tη tη = tgp (2.20a) ∇ ∂txp t a b ∂ η ··· c d t k b t a t a k t b =[ t p ··· + η ··· c d Γkp + ...+ η ··· c d Γkp ∂ x ··· ··· t a b t k t a b t k t pt t t c t d η ··· k d Γpc ... η ··· c k Γpd ] g g ... g g ... g − ··· − − ··· a b we can rewrite Eq. (2.19) as,

∂tη tη˙ = + tv ( tη) . (2.20b) ∂t · ∇

2.5 Convected coordinates

Let us consider a body and define in its reference configuration a system of curvilinear coordinates Bθi,i=1, 2, 3 . The curvilinear coordinate{ system} θi is a convected coordinate system (Flügge 1972) if, when the body undergoes{ } a deformation process, for each configuration, the triad θi thatB defines a particle χ,isthesameasinthe reference configuration. { }

2.6 The deformation gradient tensor

Let us consider the motion of the body represented in Fig. 2.1. For the reference conﬁguration (t =0)we can writeB at the point (particle) χ:

dx = dxA g (2.21a) ◦ ◦ ◦ A 3 See Appendix. 14 Nonlinear continua where the vector ◦dx at the point χ in the reference configuration is called a material line element or fiber (Ogden 1984). Due to the motion tφ ,theabovedefined fiber is transformed into a fibre in the spatial configuration,

tdx = tdxatg . (2.21b) a We now deﬁne a second-order tensor: t X ,thedeformation gradient tensor at χ, ◦

t t dx = X ◦dx . (2.22) ◦ · From the above equation,

∂txa t X = tg gA . (2.23) A a ◦ ◦ ∂◦x Using the ﬁrst of Eqs. (2.4) we get the following functional relation,

t a t a B X A = X A(◦x ,t) . (2.24) ◦ ◦ From Eq.(2.23), we see that the tensor t X has one base vector in the reference conﬁguration and the other in the spatial◦ conﬁguration. Hence, it is a two-point tensor, (Marsden & Hughes 1983, Lubliner 1985). t a It is important to note that X A is a function of, ◦ the motion of the body (tφ), • the material coordinate system, • the spatial coordinate system. •

For a regular motion, the tensor that is the inverse of t X at tx is, ◦ t 1 t ◦dx = X− dx (2.25) ◦ · where, A t 1 ∂◦x t a X− = t a ◦g g . (2.26) ◦ ∂ x A Using the second of Eqs. (2.4), we get the following functional relation,

t 1 A t 1 A t b ( X− ) a =(X− ) a ( x ,t) . (2.27) ◦ ◦ We deﬁne the transpose of t X at χ using the following relation (Marsden & Hughes 1983, Strang 1980),◦

t t t T t ( X ◦dx) dx = ◦dx ( X dx) , (2.28a) ◦ · · · ◦ · t T t T A t a if we now deﬁne X =(X ) a ◦g g ,wegetfromEq.(2.28a), ◦ ◦ A 2.6 The deformation gradient tensor 15

t a At b t A t T B t b X A ◦dx d◦x gab = ◦dx ◦gAB ( X ) b dx (2.28b) ◦ ◦ hence, t T B t a t BA ( X ) b = X A gab ◦g . (2.28c) ◦ ◦ and therefore, t T B t B ( X ) b = Xb . (2.28d) ◦ ◦ t t t In the above equations, gab = g g are the covariant components of the a· b t A AB A B metric tensor in the spatial configuration at x and ◦g = ◦g ◦g are the contravariant components of the metric{ tensor} in the reference con· fig- a 4 uration at ◦x . Referring{ the} body to a fixed Cartesian system, and using Eqs. (2.9a- 2.9b), B

t α t ∂ u Xαβ = δαβ + β , (2.29a) ◦ ∂◦z t α t 1 ∂ u ( X− )αβ = δαβ t β , (2.29b) ◦ − ∂ z t T t ( X )αβ = Xβα , (2.29c) ◦ ◦ where δαβ: components of the Kronecker-delta. It is important to remember that when a problem is referred to a Carte- sian system we do not need to make the distinction between covariant and contravariant tensorial components (Green & Zerna 1968).

Example 2.1. JJJJJ In a ﬁxed Cartesian system, a rigid translation is represented by a deformation t gradient tensor with components Xαβ = δαβ. JJJJJ ◦

Example 2.2. JJJJJ In the rigid rotation represented in the following ﬁgure,

4 See Appendix. 16 Nonlinear continua

Rigid rotation

t We can directly calculate the components Xαβ using Eq. (2.29a) but it may be simpler to consider the following sequence:◦ Step 1: Transformation from Cartesian coordinates to cylindrical coordinates in the reference conﬁguration.

1 1 2 2 2 ◦θ = (◦z ) +(◦z ) q 2 l 2 1 ◦z l ∂◦θ ◦θ =tan− ;(X1) = z1 β ∂ zβ µ ◦ ¶ ◦ 3 3 ◦θ = ◦z . Without any motion, the change of the coordinate system in the reference conﬁguration produces a deformation gradient tensor. Step 2: Rigid rotation.

t 1 1 θ = ◦θ t p t 2 2 p ∂ θ θ = ◦θ + γ ;(X2) l = l ∂◦θ t 3 3 θ = ◦θ . Step 3: Transformation from cylindrical coordinates to Cartesian coordinates in the spatial conﬁguration. tz1 = tθ1cos tθ2 ∂tzα tz2 = tθ1 sin tθ2 ;(X )α = 3 p ∂tθp tz3 = tθ3 . Without any further motion, the change of the coordinate system in the spatial conﬁguration produces a deformation gradient tensor. Using the chain rule, 2.6 The deformation gradient tensor 17

t α t α t p l t ∂ z ∂ z ∂ θ ∂◦θ Xαβ = β = t p l β ◦ ∂◦z ∂ θ ∂◦θ ∂◦z t α p l Xαβ =(X3) p (X2) l (X1) β ◦ 1 β For step 1, the derivation of (X1− ) l canbedonebyinspection.Thearray of these components is,

2 1 2 cos ◦θ ◦θ sin ◦θ 0 1 2 − 1 2 X1− = sin ◦θ ◦θ cos ◦θ 0 , ⎡ 001⎤ £ ¤ ⎣ ⎦ inverting 2 2 cos ◦θ sin ◦θ 0 1 2 1 2 [X1]= 1 sin ◦θ 1 cos ◦θ 0 . ◦θ ◦θ ⎡ − 001⎤ For step 2, ⎣ ⎦ 100 [X2]= 010 . ⎡ 001⎤ For step 3, ⎣ ⎦ cos tθ2 tθ1 sin tθ2 0 t 2 −t 1 t 2 [X3]= sin θ θ cos θ 0 . ⎡ 001⎤ Finally, ⎣ ⎦ cos γ sin γ 0 t X = sin γcosγ− 0 . ◦ ⎡ 001⎤ £ ¤ ⎣ ⎦ An important feature of the above matrix is that [t X]T [t X]=[I] ,where ◦t ◦ [I] is the unit matrix. That is to say, the matrix [ X] is orthogonal.JJJJJ ◦

Example 2.3. JJJJJ For the motion represented in the following ﬁgure, we can derive the deformation gradient tensor directly by inspection, 18 Nonlinear continua

Simple deformation process

2.00.00.0 t X = 0.00.50.0 . ◦ ⎡ 0.00.00.6 ⎤ £ ¤ ⎣ ⎦ JJJJJ

Example 2.4. JJJJJ For the motion represented in the following ﬁgure,

Shear deformation

t 1 2 z1 = ◦z + ◦z tan γ t 2 2 z = ◦z t 3 3 z = ◦z , therefore, 1tanγ 0 t X = 010 . ◦ ⎡ 001⎤ £ ¤ ⎣ ⎦ JJJJJ 2.6 The deformation gradient tensor 19

When there is a sequence of motions (some of them can be just a change of coordinate system) like the sequence depicted in Fig. 2.3, we can generalize the result in Example 2.2,

Fig. 2.3. Sequence of motions

n∆t a n∆t a ∆t l n∆t a ∂ x ∂ x ∂ x X P = P = (n 1)∆t b P . (2.30a) ◦ ∂◦x ∂ − x ······ ∂◦x Therefore,

n∆t n∆t (n 1)∆t 2∆t ∆t X = (n 1)∆tX (n−2)∆tX ∆t X X . (2.30b) ◦ − · − ··· · ◦

Example 2.5. JJJJJ It is easy to show that when using a convected coordinate system

t a a X b = δb ◦ a where the δb are the mixed components of the Kronecker delta tensor. JJJJJ

For a motion tφ we deﬁne at a point χ the Jacobian of the transformation (Truesdell & Noll 1965),

tdV tJ(χ, t)= , (2.31) ◦dV t where ◦dV is a differential volume in the reference configuration and dV is the corresponding differential volume in the spatial configuration. Since in a regular motion, a nonzero volume in the reference configuration cannot be 20 Nonlinear continua collapsed into a point in the spatial configuration and vice versa (Aris 1962), t t 1 J and J − cannot be zero. In a fixed Cartesian system we can define, for a motion tφ,

t α t α β z = z (◦z ,t) . (2.32)

The vectors (Hildebrand 1976)

t α t ∂ z t tβ = β eα (β =1, 2, 3) (2.33) ∂◦z are the base vectors, in the spatial conﬁguration, of a convected coordinate α system ◦z and { } 1 2 3 ◦dV = ◦dz ◦dz ◦dz (2.34a) t 1 2 3 t t t dV = ◦dz ◦dz ◦dz [ t ( t t )] . (2.34b) 1 · 2 × 3 Therefore, ∂tz1 ∂tz2 ∂tz3 1 1 1 ∂◦z ∂◦z ∂◦z ⎡ ⎤ tdV =det ∂tz1 ∂tz2 ∂tz3 dV. (2.34c) ∂ z2 ∂ z2 ∂ z2 ◦ ⎢ ◦ ◦ ◦ ⎥ ⎢ ⎥ ⎢ ∂tz1 ∂tz2 ∂tz3 ⎥ ⎢ 3 3 3 ⎥ ⎢ ∂◦z ∂◦z ∂◦z ⎥ Since transpose matrices have⎣ the same determinant,⎦

t t dV = X ◦dV (2.34d) ◦ t t ¯ ¯ where X =det[ X] . ¯ ¯ When|◦ the| motion◦ of a body is referred to a fixed Cartesian coordinate system (Malvern 1969), B tJ(χ, t)= t X . (2.34e) ◦ When in the reference configuration¯ we use¯ a curvilinear system xA ¯ ¯ ◦ and in the spatial configuration a system txa , (Marsden & Hughes{ 1983):} { } ∂tzα ∂tzα ∂txa ∂ xA tJ =det =det ◦ . (2.34f) ∂ zβ ∂txa ∂ xA ∂ zβ ∙ ◦ ¸ " ◦ ◦ # Following the reference (Green & Zerna 1968) and doing some algebra we can show that: ∂tzα 2 tg =det tg = det (2.34g) ab ab ∂txa ∙ ∙ ¸¸ and ¯ ¯ £ ¤ ¯ ¯ 2 A − ∂◦x ◦gAB =det[◦gAB]= det β . (2.34h) | | " " ∂◦z ## 2.7 The polar decomposition 21

Finally,

t gab tJ(χ, t)= t X | | . (2.34i) ◦ s ◦gAB ¯ ¯ | | ¯ ¯

Example 2.6. JJJJJ In a isocoric deformation (without change of volume) tJ =1, hence

t t X gab = ◦gAB . ◦ | | | | When in the problem we¯ use¯ p a ﬁxed Cartesianp coordinate system, we have ◦ ¯ ¯ t X =1. ◦ When we use convective coordinates¯ ¯ ◦ ¯ ¯ t gab = ◦gAB . | | ¯ ¯ ¯ ¯ JJJJJ

2.7 The polar decomposition

The polar decomposition theorem (Truesdell & Noll 1965, Truesdell 1966, Malvern 1969, Marsden & Hughes 1983) is a fundamental step in the development of the kinematic description of continuous body motions. It allows us to locally (at a point χ) decompose any motion into a pure deformation motion followed by a pure rotation motion or vice versa.

2.7.1 The Green deformation tensor

The Green deformation tensor is deﬁned at a point χ as

t C = t XT t X . (2.35) ◦ ◦ · ◦ In some references, e.g. (Truesdell & Noll 1965, Truesdell 1966), the above tensor is referred to as right Cauchy-Green deformation tensor . Using Eq. (2.28c), we get

t t a t AB t b t d t D C = X A gab ◦g ◦g g X D g ◦g , (2.36a) ◦ ◦ B · ◦ d h i h i and therefore, 22 Nonlinear continua

t t a t b t AB D C = X A X D gab ◦g ◦g ◦g . (2.36b) ◦ ◦ ◦ B It is important to note£ that the Green deformation¤ tensor is defined in the reference configuration. Using an equivalent definition of transposed tensor to that given in Eq.(2.28a), we can write

t t T ( C ◦dx1) ◦dx2 = ◦dx1 ( C ◦dx2) (2.37a) ◦ · · · ◦ · where ◦dx1 and ◦dx2 are two arbitrary vectors definedinthereference configuration at the point under study; after some algebra Eq. (2.37a) leads to t A t T A t A C B =( C ) B = CB . (2.37b) ◦ ◦ ◦ The above equation indicates that the Green deformation tensor is symmetric. For an arbitrary vector tdx definedinthespatialconfiguration we can write the following equalities:

t t at t a At dx = dx g = X A ◦dx g (2.38a) a ◦ a t t t b t d BR t t b dx = dxb g = X B ◦g ◦dxR gdb g . (2.38b) ◦ Hence,

t t t a t d t BR A dx dx = X A X B gda ◦g ◦dx ◦dxR . (2.38c) · ◦ ◦ Using Eq. (2.36b), we write

t t t R A t dx dx = C A ◦dx ◦dxR = ◦dx C ◦dx . (2.38d) · ◦ · ◦ · Considering that: tdx tdx 0 . • tdx · tdx =0≥ tdx =0. • t· ⇐⇒ | | t If φ is a regular motion, dx =0 ◦dx =0we • conclude that t C is a positive-de|ﬁnite| tensor (Strang⇐⇒ 1980).| | ◦

2.7.2 The right polar decomposition

We deﬁne, in the reference conﬁguration at the point under study, the right stretch tensor as

1/2 t U = t C (2.39) ◦ ◦ and it follows immediately that the tensor£ t U¤ inherits from t C the properties of symmetry and positive-deﬁniteness (Malvern◦ 1969). ◦ 2.7 The polar decomposition 23

We define the right polar decomposition as a multiplicative decomposition of the tensor t X into a symmetric tensor (t U) premultiplied by a tensor that we will show◦ is orthogonal: the rotation tensor◦ (t R) . Hence, ◦ t X = t R t U . (2.40) ◦ ◦ · ◦ In order to show that t R is an orthogonal tensor we have to show that: ◦ t T t (i) R R = ◦g ◦ · ◦ (ii) t R t RT = tg ◦ · ◦ where g = δA g gB is the unit tensor of the reference configuration ◦ B ◦ A ◦ and tg = δa tg tgb is the unit tensor of the spatial configuration ( δA b a B a 5 and δ b are Kronecker deltas ). To prove the first equality (i), we start from Eq. (2.40) and get

t t t 1 t a t 1 A t B R = X U− = X A ( U − ) B g ◦g . (2.41) ◦ ◦ · ◦ ◦ ◦ a The tensor t R deﬁned by the above equation is, in the same sense as t X ,atwo-point◦ tensor. The components of t R are: ◦ ◦ t a t a t 1 L R A = X L ( U − ) A , (2.42a) ◦ ◦ ◦ and using an similar equation to (2.28c) we have

t T B t a t 1 L t BA ( R ) b = X L ( U − ) A gab ◦g , (2.42b) ◦ ◦ ◦ hence, t T t a t 1 L t BA t b R = X L ( U − ) A gab ◦g ◦g g . (2.42c) ◦ ◦ ◦ B Considering that

t T t d t RB t b X = X R gdb ◦g ◦g g , (2.42d) ◦ ◦ B and t T t 1 L D U− =(U − )D ◦g ◦g , (2.42e) ◦ ◦ L t T t T t 1 L t d R t Dt b U− X =( U − )D X R δL gdb ◦g g , (2.42f) ◦ · ◦ ◦ ◦ and since gD = gDB g we write ◦ ◦ ◦ B

t T t T t 1 L t d t DB t b U− X =( U − )D X L gdb ◦g ◦g g . (2.42g) ◦ · ◦ ◦ ◦ B Comparing Eqs. (2.42g) and (2.42c) it is obvious that:

5 See Appendix. 24 Nonlinear continua

T T T 1 T t R = t U− t X = t U− t X (the last equality follows from the symmetry◦ ◦ of t U·).◦ ◦ · ◦ Hence, ◦ T 1 T 1 t R t R = t U− t X t X t U− . (2.42h) ◦ · ◦ ◦ · ◦ · ◦ · ◦ Using also Eqs. (2.35) and (2.39), we obtain

t T t t 1 t t 1 R R = U− C U− = ◦g (2.42i) ◦ · ◦ ◦ · ◦ · ◦ and the ﬁrst equality (i) is shown to be correct. To prove the second equality (ii), we write (Marsden & Hughes 1983):

2 t R t RT = t R t RT t R t RT = t R t RT , (2.43) ◦ · ◦ ◦ · ◦ · ◦ · ◦ ◦ · ◦ ³ ´ ³ ´ and since t R cannot be a singular matrix, the second equality (ii) is shown to be correct.◦ We will now show that the right polar decomposition is unique. Assuming that it is not unique, we can have, together with Eq. (2.40), another decomposition, for example:

t X = t R˜ t U˜ , (2.44a) ◦ ◦ · ◦ where t R˜ is an orthogonal tensor and t U˜ is a symmetric tensor. We◦ can write ◦ t C = t XT t X = t U˜ t U˜ , (2.44b) ◦ ◦ · ◦ ◦ · ◦ and therefore, 1/2 t U˜ = t C . (2.44c) ◦ ◦ However, comparing the above with£ Eq.¤ (2.39), we conclude that:

t U˜ = t U . (2.44d) ◦ ◦ Then, from Eq. (2.44a),

t t t 1 R˜ = X U− , (2.44e) ◦ ◦ · ◦ and comparing the above with Eq. (2.41), we conclude that:

t R˜ = t R . (2.44f) ◦ ◦ Equations (2.44d) and (2.44f) show that the right polar decomposition is unique. 2.7 The polar decomposition 25

2.7.3 The Finger deformation tensor

The Finger deformation tensor, also known in the literature as the left Cauchy-Green deformation tensor,isdeﬁned at a point χ as: tb = t X t XT . (2.45a) ◦ · ◦ Using Eq. (2.28c), we have

t t d t D t a t BA t b b = X D g ◦g X A gab ◦g ◦g g (2.45b) ◦ d · ◦ B h i h i and therefore, t t d t a t BA t t b b = X B X A gab ◦g g g . (2.45c) ◦ ◦ d It is important to note that the Finger deformation tensor is defined in the spatial configuration. Proceeding in the same way as in Sect. 2.7.1, we can show that: tb is a symmetric tensor. • tb is a positive-definite tensor. • 2.7.4 The left polar decomposition

We define the left polar decomposition as a multiplicative decomposition of the tensor t X into a symmetric tensor (t V) postmultiplied by the orthogonal tensor t◦R. Therefore, ◦ ◦ t X = t V t R . (2.46) ◦ ◦ · ◦ From the above equation, t V = t X t RT = t R t U t RT (2.47a) ◦ ◦ · ◦ ◦ · ◦ · ◦ and taking into account that t U is symmetric, we get ◦ t VT = t R t U t RT = t V . (2.47b) ◦ ◦ · ◦ · ◦ ◦ The above equation shows that the tensor t V ,knownastheleft stretch tensor,issymmetric. ◦ From Eq. (2.45a), we get tb = t X t XT = t V t V (2.48a) ◦ · ◦ ◦ · ◦ and therefore, 1/2 t V = tb . (2.48b) ◦ From the above equation we conclude£ ¤ that the left stretch tensor is defined in the spatial configuration andthatitinheritsfrom tb the positive definiteness. Proceeding in the same way as in Sect. 2.7.2 we can show that the left polar decomposition is unique. 26 Nonlinear continua

2.7.5 Physical interpretation of the tensors t R , t U and t V ◦ ◦ ◦ In this Section, we will discuss a physical interpretation of the second-order tensors introduced by the polar decomposition.

The rotation tensor

t t t Assuming a motion in which U = ◦g and therefore V = g ,we get ◦ ◦ t X = t R (2.49) ◦ ◦ and considering in the reference configuration, at the point under analysis, two arbitrary vectors ◦dx1 and ◦dx2 we have, in the spatial configuration: t t dx1 = R ◦dx1 (2.50a) ◦ · t t dx2 = R ◦dx2 (2.50b) ◦ · hence, in the spatial configuration we can write t t t a t b t A B dx1 dx2 = R A R B gab ◦dx1 ◦dx2 . (2.50c) · ◦ ◦ Using Eqs. (2.42a) and (2.42b) we can rewrite the above equation as t t t a t T C A dx1 dx2 = R A ( R ) a ◦dx1 ◦dx2C , (2.50d) · ◦ ◦ and since t R is an orthogonal tensor, ◦ t t C A dx dx = δ ◦dx ◦dx2 = ◦dx ◦dx . (2.51) 1 · 2 A 1 C 1 · 2 The above equation shows that when t X = t R : ◦ ◦ The corresponding vectors in the spatial and reference configuration have • thesamemodulus. The angle between two vectors in the spatial configuration equals the angle • between the corresponding two vectors in the reference configuration. Hence, the motion can be characterized, at the point under analysis, as a rigid body rotation. We can generalize Eqs.(2.50a-2.50d) for any vector Y that in the reference configuration is associated to the point under analysis. For the particular motion described by t X = t R ,weget ◦ ◦ ty = t R Y , (2.52a) ◦ · and since the rotation tensor is orthogonal, Y = t RT ty . (2.52b) ◦ · If in the reference configuration there is a relation of the form: Y = A W , (2.53a) · where, 2.7 The polar decomposition 27

Y , W : vectors defined in the reference configuration, A : second order tensor defined in the reference configuration, it is easy to show that:

ty = t R A t RT tw . (2.53b) ◦ · · ◦ · h i In the above equation, the term between brackets is the result (in the spatial conﬁguration) of the rotation of the material tensor A. If A is a symmetric second-order tensor we can write it using its eigenvalues and eigenvectors,

AB A = λ ΦA ΦB , (2.54a) where, λAB =0 if A = B, 6 and the set of vectors ΦA form an orthogonal basis in the reference conﬁgu- ration. In the spatial conﬁguration we get, from the rotation of A:

ta = t R A t RT , (2.54b) ◦ · · ◦ and therefore,

t AB t t T a = λ ( R ΦA )(ΦB R ) . (2.54c) ◦ · · ◦ We now deﬁne in the spatial conﬁguration the set of vectors tϕ = t R Φ , a A which obviously constitute an orthogonal basis; using Eq. (2.28a) we obtain◦ ·

t T t t t ΦB R ϕ = R ΦB ϕ , (2.54d) · ◦ · b ◦ · · b ³ ´ and therefore, ¡ ¢ t AB t t a = λ ( R ΦA )( R ΦB ) . (2.54e) ◦ · ◦ · Comparing Eqs. (2.54a) and (2.54e), we conclude that: The material tensor A and the spatial tensor ta have the same eigenval- • ues. The eigenvectors of ta are obtained by rotating with t R the eigenvectors • of A . ◦

Since, according to Eq. (2.47a) t V = t R t U t RT , we can assess that: ◦ ◦ · ◦ · ◦ The material tensor t U and the spatial tensor t V havethesameeigen- • values. ◦ ◦ The eigenvectors of t V (and tb ) are obtained by rotating with t R • the eigenvectors of t U◦ (and t C ). ◦ ◦ ◦ 28 Nonlinear continua

The right stretch tensor

To study the physical interpretation of the right stretch tensor we consider, in the reference conﬁguration, at the point χ under analysis, two vectors ◦dx1 t and ◦dx2 that in the spatial conﬁguration are transformed into dx1 and t dx2 t t dx1 = X ◦dx1 (2.55a) ◦ · t t dx2 = X ◦dx2 . (2.55b) ◦ · After some algebra,

t t t T t dx1 dx2 = ◦dx1 ( X X ) ◦dx2 , (2.56a) · · ◦ · ◦ · and using Eqs. (2.35) and (2.39), we write

t t t t t dx1 dx2 = ◦dx1 C ◦dx2 = ◦dx1 ( U U ) ◦dx2 . (2.56b) · · ◦ · · ◦ · ◦ · It follows from the above equation that the changes in lengths and angles, produced by the motion, are directly associated to the right stretch tensor t U . In the previous subsection we showed that these changes are nil when t◦ U = ◦g . ◦

The left stretch tensor

Starting from Eqs. (2.55a-2.55b) and using a left polar decomposition, we get

t t 1 t 1 t ◦dx1 ◦dx2 = dx1 ( V− V− ) dx2 . (2.57) · · ◦ · ◦ · It is obvious from the above equation that changes in lengths and angles, produced by the motion, are directly associated to the left stretch tensor t V . Above we showed that those changes are nil when t V = tg . ◦ ◦

2.7.6 Numerical algorithm for the polar decomposition

When analyzing ﬁnite element models of nonlinear solid mechanics problems, we usually know the numerical value of the deformation gradient tensor at a point and we need to use a numerical algorithm for performing the polar decomposition. In what follows, we present an algorithm that can be used for the right polar decomposition when we refer the problem to a ﬁxed Cartesian system. √ Starting from the matrix [t X] that is a (3 3)-matrix in the general case, we calculate the symmetric◦ matrix, ×

T t C = t X t X . (2.58a) ◦ ◦ ◦ £ ¤ £ ¤ £ ¤ 2.7 The polar decomposition 29

√ Using a numerical algorithm (Bathe 1996), we calculate the eigenvalues 2 t λA and eigenvectors [ΦA];A =1, 2, 3 of the matrix [ C]. √ From the above step, ◦

t C =[Ψ][Λ][Ψ]T (2.58b) ◦ where £ ¤

[Ψ]=[[Φ1][Φ2][Φ3]] (2.58c) and

2 (λ1) 00 2 [Λ]= 0(λ2) 0 . (2.58d) ⎡ 2 ⎤ 00(λ3) ⎣ ⎦ √ Using Eq. (2.39), we obtain (Strang 1980)

t U =[Ψ][Λ]1/2 [Ψ]T (2.58e) ◦ where £ ¤

λ 00 1 1 [Λ] 2 = 0 λ2 0 . (2.58f) ⎡ ⎤ 00λ3 ⎣ ⎦ √ Finally, using Eq. (2.40) we get

1 t R = t X t U − . (2.58g) ◦ ◦ ◦ And [t R] is now also£ completely¤ £ determined.¤£ ¤ ◦

Example 2.7. JJJJJ For the case analyzed in Example 2.4, and for γ =15◦ we can write

1.00.26795 0.0 t X = 0.01.00.0 . ◦ ⎡ 0.00.01.0 ⎤ £ ¤ ⎣ ⎦ Hence, 30 Nonlinear continua

1.00.26795 0.0 t C = t XT t X = 0.26795 1.0718 0.0 . ◦ ◦ ◦ ⎡ 0.00.01.0 ⎤ £ ¤ £ ¤£ ¤ Solving for the eigenvalues and eigenvectors⎣ of [t C],weget ⎦ ◦ 0.76556 0.00.0 [Λ]= 0.01.30624 0.0 , ⎡ 0.00.01.0 ⎤ ⎣ ⎦ and, 0.75259 0.65849 0.0 [Ψ]= −0.65849 0.75259 0.0 . ⎡ 0.00.01.0 ⎤ Using Eq. (2.58e), we write ⎣ ⎦

0.99114 0.13279 0.0 t U =[Ψ][Λ]1/2 [Ψ]T = 0.13279 1.02672 0.0 , ◦ ⎡ 0.00.01.0 ⎤ £ ¤ ⎣ ⎦ and ﬁnally, using Eq. (2.58g),

0.99114 0.13279 0.0 1 t R = t X t U − = 0.13279 0.99114 0.0 . ◦ ◦ ◦ ⎡ − 0.00.01.0 ⎤ £ ¤ £ ¤£ ¤ ⎣ ⎦ We urge the reader to verify that: [t R]T [t R]=[I] ◦ [t◦X]=[◦ t R][t U] ◦ ◦ ◦ ◦ within the numerical accuracy used in the above calculations. JJJJJ

Example 2.8. JJJJJ For the case analyzed in the previous example, we are now going to describe the deformation of a material fiber that in the reference configuration contains T the (0, 0, 0) point and has the direction [◦n] = 0.01.00.0 . In the reference configuration, £ ¤ ◦dx = ◦dS ◦n 2 ◦dx ◦dx =(◦dS) . · In the spatial configuration,

tdx = tdS tn ; tn =1 k k tdx tdx =(tdS)2 . · Using Eq. (2.56b), 2.7 The polar decomposition 31

t 2 2 t ( dS) =(◦dS) ◦n C ◦n · ◦ · and therefore, in the ﬁxed Cartesian system£ that we are¤ using,

t dS T t 1/2 = [◦n] [ C][◦n] ◦dS ◦ £ ¤ and using the numerical values calculated in the previous example,

tdS =1.03528 . ◦dS The reader can check the above numerical values using very simple geometrical considerations. In the case of ◦n = Φ ,itiseasytoshowthat ◦ i tdS = λi . ◦dS T For [◦m] = 1.00.00.0 , we get ◦ £ ¤tdS =1.0 . ◦dS

The vectors ◦m and ◦n, which are orthogonal in the reference conﬁgu- ration,◦ form an angle tθ in the spatial conﬁguration,

t t t t dx = dSm m ; m =1 m k k t t t t dx = dSn n ; n =1 n k k t t t t t T t dxm dxn =(dSm)(dSn)cosθ =(◦dSm)(◦dSn)[◦m] [ C][◦n] · ◦ t t dS dS t T t cos θ =[◦m] [ C][◦n] dS dS ◦ µ ◦ ¶m µ ◦ ¶n hence, T t t 1 [◦m] [ C][◦n] θ =cos− tdS ◦ tdS " ◦dS m ◦dS n # and using the calculated numerical values,¡ ¢ we get¡ ¢

t θ =75◦ .

Once again, it is very simple for the reader to check the above numerical result. JJJJJ

In the above examples we have numerically calculated the eigenvalues and eigenvectors of the tensor t C ; however, in some problems it is necessary to ◦ 32 Nonlinear continua diﬀerentiate those eigenvalues and eigenvectors and it is therefore necessary to use an analytical expression of them. As is wellknown the eigenvalues of t C are given by the roots of the following polynomial (Strang 1980), ◦

p(λ2)= λ6 + IC λ4 IC λ2 + IC =0 (i =1, 2, 3) (2.59a) i − i 1 i − 2 i 3 where, (McConnell 1957)

C t I1 = tr( C) (2.59b) ◦ C 1 C 2 t 2 I2 = (I1 ) tr( C ) (2.59c) 2 − ◦ C £t ¤ I3 = det( C) , (2.59d) ◦ and it is easy to verify the Serrin representation (Simo & Taylor 1991):

t C 2 t C 2 t 1 b (I1 λa) g + I3 λa− b− tϕ tϕ = tλ2 − − , (2.60a) a a a 4 C 2 C 2 2 λ I λ + I λ− a − 1 a 3 a t C 2 C 2 t 1 C (I1 λA) ◦g + I3 λA− C− Φ Φ = tλ2 ◦ − − ◦ , (2.60b) A A A 4 C 2 C 2 2 λ I λ + I λ− A − 1 A 3 A with no addition in “a”or “A”in the above equations. According to what we showed in Sect. 2.7.5, λa = λA for a = A .

Example 2.9. JJJJJ To verify Eq. (2.60a)we start from, ◦ tb = λ2 tϕ tϕ i i i t 1 2 t t b− = λ− ϕ ϕ i i i C 2 2 2 I1 = λ1 + λ2 + λ3 C 2 2 2 2 2 2 I2 = λ1 λ2 + λ1 λ3 + λ2 λ3 C 2 2 2 I3 = λ1 λ2 λ3 . For a =1

4 C 2 C 2 t C 2 t C 2t 1 2λ1 I1 λ1+I3 λ1− t t b (I λ ) g + I λ− b− = ϕ ϕ . 1 1 3 1 − 2 1 1 − − λ1 The above veriﬁes Eq. (2.60a) for the case a =1; the demonstrations for a =2, 3 are identical. 2.8 Strain measures 33

To verify Eq. (2.60b) we start from, ◦ t 2 C = λI ΦI ΦI ◦ t 1 2 C− = λI − ΦI ΦI ◦ and proceed as before. JJJJJ

It is important to note that Eqs. (2.60a-2.60b) are only valid if the denominator of the r.h.s. is not zero. The denominator is zero if we have repeated eigenvalues.

2.8 Strain measures

Intheliteraturewecanfind a large number of strain measures that have been proposed to characterize a deformation process. There are different approaches for analyzing the deformation of continuum bodies and we usually find that, for a given approach, one particular strain measure may be more suitable than others. In this section we will present a number of these strain measures without making any claim of completeness.

2.8.1 The Green deformation tensor

We have already presented the Green deformation tensor in Sect. 2.7.1. It is important to remember that this second-order tensor is defined in the reference configuration and that for two vectors ◦dx1 and ◦dx2 defined at a point χ in the reference configuration, the corresponding vectors in the spatial configuration satisfy the relation,

t t t dx1 dx2 = ◦dx1 C ◦dx2 . (2.61) · · ◦ ·

2.8.2 The Finger deformation tensor

We have already presented the Finger deformation tensor in Sect. 2.7.3. It is important to remember that this second-order tensor is deﬁnedinthespatial conﬁguration. Using Eq. (2.45a),

t 1 t T t 1 b− = X− X− , (2.62) ◦ · ◦ t t t and for two vectors dx1 and dx2 deﬁned at a point x in the spatial conﬁguration we can write, 34 Nonlinear continua

t t 1 t t t T dx1 b− dx2 = dx1 X− ◦dx2 (2.63a) · · · ◦ · using Eq. (2.28a), we get

t t 1 t dx b− dx = ◦dx ◦dx . (2.63b) 1 · · 2 1 · 2

2.8.3 The Green-Lagrange deformation tensor

From Eq.(2.61),

t t 1 t dx1 dx2 ◦dx1 ◦dx2 =2◦dx1 ( C ◦g) ◦dx2 . (2.64) · − · · 2 ◦ − · ∙ ¸ The Green-Lagrange strain tensor is defined in the reference configuration as t 1 t ε = ( C ◦g) . (2.65) ◦ 2 ◦ − The second order tensor t ε describes the deformation corresponding to the t-configuration (spatial confi◦guration) referred to the configuration at t =0 (reference configuration).

Example 2.10. JJJJJ Considering a convected coordinate system θi with covariant base vectors t g in the spatial conﬁguration and ◦g in the reference one, we can write, i i © ª t t l m e ε = εlme ◦g ◦g ◦ ◦

e e e and it is easy to show that,

t 1 t t εlm = g g ◦g ◦g . ◦ 2 l · m − l · m h i e e e e e JJJJJ

At the point χ under study, we now evolve from the t-conﬁguration to a t+∆t t + ∆t-conﬁguration by means of a rotation t R. Hence,

t+∆t t+∆t t X = t R X , (2.66a) ◦ · ◦ t+∆t t T t+∆t T t+∆t t C = X t R t R X , (2.66b) ◦ ◦ · · · ◦ and taking into account that the rotation tensor is orthogonal, we get 2.8 Strain measures 35

t+∆tC t C (2.66c) ◦ ≡ ◦ and therefore, t+∆tε t ε . (2.66d) ◦ ≡ ◦ From the above, we conclude that the Green deformation tensor and the Green -Lagrange strain tensor are not aﬀected by rigid body rotations: that istosaytheyareindiﬀerent to rotations.

2.8.4 The Almansi deformation tensor From Eq. (2.63b), we get

t t t 1 t t 1 t dx dx ◦dx ◦dx =2dx ( g b− ) dx . (2.67) 1 · 2 − 1 · 2 1 · 2 − · 2 ∙ ¸ The Almansi strain tensor is deﬁned in the spatial conﬁguration as

t 1 t t 1 e = ( g b− ) . (2.68) 2 − At the point χ under study, we now evolve from the t-conﬁguration to t+∆t the t + ∆t-conﬁguration by means of a rotation t R. Hence, t+∆t 1 t 1 t+∆t T X− = X− t R , (2.69a) ◦ ◦ · and,

t+∆t T t+∆t t T X− = t R X− , (2.69b) ◦ · ◦ using Eq. (2.62), t+∆t 1 t+∆t t 1 t+∆t T b− = R b− R , (2.69c) t · · t and therefore, t+∆te = t+∆tR te t+∆tRT . (2.69d) t · · t Hence, the Finger and Almansi tensors are aﬀectedbyrigid-bodyrotations.

2.8.5 The Hencky deformation tensor The Hencky or logarithmic strain tensor is defined in the reference configuration as t H =lnt U . (2.70) ◦ ◦ When the problem is referred to a fixed Cartesian system using Eq. (2.58e), we get ln λ1 0.00.0 t T [ H]=[Ψ] 0.0lnλ2 0.0 [Ψ] . (2.71) ◦ ⎡ ⎤ 0.00.0lnλ3 Obviously, the Hencky deformation⎣ tensor is indi⎦ fferent to rotations,since from the polar decomposition, we can see that t U does not incorporate the effect of rigid-body rotations. ◦ 36 Nonlinear continua 2.9 Representation of spatial tensors in the reference configuration (“pull-back”)

For the regular motion depicted in Fig. 2.1, we can define: An arbitrary curvilinear coordinate system txa in the spatial configura- • tion. At a point χ (txa,a =1, 2, 3) we can{ determine} the covariant base vectors tg and the contravariant base vectors tga. a A An arbitrary curvilinear coordinate system ◦x in the reference config- • A { } uration. At the point χ (◦x ,A=1, 2, 3) we can determine the covariant base vectors g and the contravariant base vectors gA. ◦ A ◦ A convected curvilinear coordinate system θi .Atthepoint χ in the • spatial configuration we can determine the covariant{ } base vectors tg˜ and a the contravariant base vectors tgã , while in the reference configuration we can determine the covariant base vectors ◦g˜ and the contravariant a a base vectors ◦g˜

2.9.1 Pull-back of vector components

Let us consider in the spatial conﬁguration at the point χ a vector,

tb = tbitg = tb tgi = t˜bitg = t˜b tgi . (2.72) i i i i We deﬁne in the reference conﬁguration the following vectors (Dvorkin, e e Goldschmit, Pantuso & Repetto 1994):

tB = t˜bi g =[tB]A g (2.73a) ◦ i ◦ A tB = t˜b gi =[tB] gA . (2.73b) i ◦ e A ◦ After some algebra, e t A t j t 1 A [ B ] = b ( X− ) j (2.74a) ◦ t t t j [ B ]A = bj X A . (2.74b) ◦ Adopting the notation used in manifolds analysis (Lang 1972, Marsden & Hughes 1983) we deﬁne the pull-back of the contravariant components tbj as

t t j A t A φ∗( b ) =[B ] (2.75a)

t and the pull-back of the covariant£ components¤ bj as:

t t t φ∗( bj) A =[B ]A . (2.75b) We can therefore rewrite Eqs.£ (2.73a-2.73b)¤ and (2.74a-2.74b), 2.9 Representation of spatial tensors in the reference conﬁguration (“pull-back”) 37

t t j t 1 A t j t 1 A C B = b ( X− ) j ◦g = b ( X− ) j ◦gAC ◦g (2.76a) ◦ A ◦ t t t j A t t j AC B = bj X A ◦g = bj X A ◦g ◦g . (2.76b) ◦ ◦ C For two vectors tb and tw ,deﬁned in the spatial conﬁguration at χ , using Eqs. (2.74a-2.74b), it is easy to show that:

tB tW = tB tW = tb tw . (2.77) · · · Also, using Eqs.(2.74a-2.74b) and (2.36a-2.36b), we get

t Bt t [ B ] CAB =[B ]A . (2.78) ◦ Using Eqs. (2.73a-2.73b), we can write

(tg ) = g (2.79a) a ◦ a (tga) = ga . (2.79b) e ◦e Hence, we use the following notation (Moran, Ortiz & Shih 1990): e e t t t φ∗( g )=(g ) = g (2.79c) a a ◦ a tφ (tga)=(tga) = ga . (2.79d) ∗ e e ◦e From the above equations, we can get by inspection the geometrical inter- e e e pretation of the vectors tB and tB : If tb,inthespatialconfiguration, is the tangent to a curve tc(ξ) at a • point χ,then tB is the tangent, in the reference configuration, to the t 1 t curve C(ξ)= φ− [ c(ξ)] ,atthepoint χ. In the transformation tB tb the modulus of the original vector gets • stretched as the material fiber→ to which they are tangent. In convected coordinates we have

tdx =dθi tg (2.80a) i dx =dθi g (2.80b) ◦ ◦e i that is to say, e

t dX = ◦dx . (2.80c) For two vectors tb and tw that are orthogonal in the spatial conﬁguration • it is obvious that: 38 Nonlinear continua

Fig. 2.4. Mappings

tB tW = tB tW = tb tw =0. (2.81) · · · Hence, the orthogonality of tb and tw implies the orthogonality of tB and tW and the orthogonality of tB and tW in the reference conﬁguration. It is important to take into account that in Eqs. (2.74a-2.74b) the terms on the r.h.s. must be written as a function of the coordinates in the reference conﬁguration. We can indicate this using a more formal nomenclature (e.g. Marsden & Hughes 1983),

t A t t a A t 1 A t t a t [ B ] = φ∗( b ) = ( X− ) a φ ( b φ) (2.82a) ◦ ◦ ◦ t £t t ¤ t£ a t t ¤ [ B ]A = φ∗( ba) A = X A ( ba φ) . (2.82b) ◦ ◦ In order to understand£ the¤ above equations we use Fig. 2.4 (Marsden & Hughes 1983). In this ﬁgure we indicate with “f g” the composition of the mapping “g” followed by the mapping “f”. ◦

Example 2.11. JJJJJ For a function f(txa) definedinthespatialconfiguration, we can write: 2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 39 ∂f df = tdxa . ∂txa In the above equation, we use a formal analogy with vector calculus in which ∂f (Marsden & Hughes 1983): df is a vector; ∂txa are its covariant components t a (dfa)andd x are contravariant base vectors. Hence we can do a pull-back operation,

t a t t a ∂ x ∂f ∂f φ∗(dfa) A = X A dfa = A t a = A . ◦ ∂◦x ∂ x ∂◦x £ ¤ Using a more formal nomenclature and the mappings in Fig. 2.7, we get

t t ∂(f φ) φ∗(dfa) A = ◦ . ∂◦xA £ ¤ JJJJJ

Example 2.12. JJJJJ Equation (2.11b) deﬁnes, in the spatial conﬁguration, the velocity of a material point, tv = tvatg . a Using the expressions for the pull-back of contravariant components, we write

t t a A t 1 A t a φ∗( v ) =(X− ) a v . ◦ £ A ¤ Ifthecoordinatesystem ◦x ,defined in the reference configuration, is a convected system with covariant{ } base vectors, tg˜ ,inthespatialconfigura- A tion, we can write dxAtg = tdxatg ◦ A a hence, t e t 1 A t g =(X− ) a g a ◦ A and therefore, t t a t 1 A te v = v ( X− ) a g . ◦ A The components of the material velocity vector in the convected system xA e ◦ are, { } t A t a t 1 A v˜ = v ( X− ) a ◦ and therefore t t a A t A φ∗( v ) = v˜ .

£ ¤ JJJJJ 40 Nonlinear continua

2.9.2 Pull-back of tensor components

Let us consider in the spatial conﬁguration at the point χ (txa,a =1, 2, 3) a second-order tensor,

tt = ttij tg tg = tt tgitgj = tti tg tgj i j ij j i t i . (2.83) = tt˜ij g tg = tt˜ tgi tgj = tt˜ tg tgj i j ij j i We deﬁne in the reference conﬁguration the following second-order tensors (Dvorkin, Goldschmit, Pantusoe e & Repettoe 1994):e e e

tT = tt˜ij g g =[tT]AB g g , (2.84a) ◦ i ◦ j ◦ A ◦ B tT = tt˜i g gj =[tT]A g gB , (2.84b) j ◦ ei ◦ e B ◦ A ◦ tT = tt˜ gi gj =[tT] gA gB . (2.84c) ij ◦ e ◦ e AB ◦ ◦ After some algebra we get e e t AB t ab t 1 A t 1 B [ T ] = t ( X− ) a ( X− ) b , (2.85a) ◦ ◦ t A t a t 1 A t b [ T ] B = t b ( X− ) a ( X) B , (2.85b) ◦ ◦ t t t a t b [ T ]AB = tab ( X) A ( X) B . (2.85c) ◦ ◦ We deﬁne: Pull-back of the contravariant components of tt , •

t t ij AB t AB φ∗( t ) =[T ] . (2.86a) Pull-back of the mixed components£ ¤ of tt , •

t t i A t A φ∗( t ) =[T ] . (2.86b) j B B Pull-back of the covariant£ components¤ of tt , •

t t t φ∗( tij) AB =[T ]AB . (2.86c) From Eqs. (2.84a-2.84c)£ and (2.85a-2.85c)¤ we can write, 2.9 Representation of spatial tensors in the reference conﬁguration (“pull-back”) 41

t t ab t 1 A t 1 B T = t ( X− ) a ( X− ) b ◦g ◦g ◦ ◦ A B t ab t 1 A t 1 B L M = t ( X− ) a ( X− ) b ◦gAL ◦gBM ◦g ◦g ◦ ◦ t ab t 1 A t 1 B L = t ( X− ) a ( X− ) b ◦gAL ◦g ◦g , (2.87a) ◦ ◦ B t t a t 1 A t b BL T = t b ( X− ) a ( X) B ◦g ◦g ◦g ◦ ◦ A L t a t 1 A t b L B = t b ( X− ) a ( X) B ◦gAL ◦g ◦g ◦ ◦ t a t 1 A t b B = t b ( X− ) a ( X) B ◦g ◦g , (2.87b) ◦ ◦ A t t t a t b AL BM T = tab ( X) A ( X) B ◦g ◦g ◦g ◦g ◦ ◦ L M t t a t b A B = tab ( X) A ( X) B ◦g ◦g ◦ ◦ t t a t b BL A = tab ( X) A ( X) B ◦g ◦g ◦g . (2.87c) ◦ ◦ L For two tensors tt and tw deﬁnedinthespatialconﬁguration at χ ( txa,a= 1, 2, 3) , using Eqs. (2.84a-2.84c), it is easy to show that:

tT : tW = tT : tW = tt : tw (2.88) and also, using Eqs.(2.87a-2.87c), we can show that

t t P t [ T ]AB =[T ] B CPA (2.89a) ◦ t t PQ t t [ T ]AB =[T ] CPA CQB . (2.89b) ◦ ◦ If in the spatial conﬁguration, at the point χ (txa,a=1, 2, 3) ,twovectors tb and tw are related by the second order tensor tt, via the equation,

tb = tt tw (2.90) · in the reference conﬁguration, we can easily verify that the following relations are fulﬁlled:

tB = tT tW (2.91a) · tB = tT tW (2.91b) · tB = tT tW . (2.91c) · 42 Nonlinear continua

Example 2.13. JJJJJ t i Instead of using the mixed components t j, it is possible to use the compo- t i nents tj , hence

tt = tt i tgjtg = tt˜ i tg˜j tg˜ j i j i wecanthendeﬁne in the reference conﬁguration the second order tensor,

t t i j t B A T◦ = t˜ g˜ g˜ =[T◦] g g j ◦ ◦ i A ◦ ◦ B after some algebra we then get,

t B t b t 1 B t a [ T◦]A = ta ( X− ) b ( X) A . ◦ ◦ Therefore, t t i B t B [ φ∗ ( tj )]A =[T◦]A . Starting from Eq.(2.90), we can also show that,

t t t B = T◦ W . · JJJJJ

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

In the previous section we obtained representations in the reference configuration of tensors definedinthespatialconfiguration (pull-back). The inverse operation is named, in the manifold analysis literature (Lang 1972, Marsden & Hughes 1983), push-forward. For a second-order tensor, from Eqs.(2.85a-2.85c), we can write:

ab t ab t t AB t a t b t AB t = φ [ T ] = X A X B [ T ] (2.92a) ∗ ◦ ◦ h i t t t t 1 A t 1 B t tab = φ [ T ]AB =(X− ) a ( X− ) b [ T ]AB (2.92b) ∗ ab ◦ ◦ h ia t a t t A t a t 1 B t A t b = φ [ T ] B = X A ( X− ) b [ T ] B . (2.92c) ∗ b ◦ ◦ h i 2.11 Pull-back/push-forward relations between strain measures 43

Example 2.14. JJJJJ Using Eqs. (2.92a-2.92c) we can show that,

t t 1 t A l φ (WA) a =(b− )al φ (W ) . ∗ ∗ £ ¤ £ ¤ JJJJJ

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

In the two previous Sections we defined representations of the types known as pull-back and push-forward. As we will see later on, these kinds of representations are extremely useful in nonlinear continuum mechanics. In Sect. 2.8 we defined a number of strain measures, some of them in the spatial configuration (e.g. the Finger and Almansi tensors) and some of them in the reference configuration (e.g. the Green, Green-Lagrange and Hencky tensors). In the present section we will establish relations of the pull-back/push- forward type between those strain measures. We define, as usual, at the material point χ: In the spatial configuration a coordinate system txa with covariant base◦ vectors tg . { } a A In the reference configuration a coordinate system ◦x with covariant base◦ vectors g . { } ◦ A We can therefore calculate at χ the deformation tensor t X . ◦ The pull-backs of the spatial metric tensor are: •

t t A B t t φ∗( gab) AB ◦g ◦g = g = C , (2.93a) ◦ £t t ab ¤AB t t 1 φ∗( g ) ◦g ◦g = g = C− . (2.93b) A B ◦ The pull-back£ of the Almansi¤ strain tensor is: •

t t A B t t φ∗( eab) AB ◦g ◦g = E = ε . (2.94a) ◦ From Eqs. (2.77)£ ,(2.80a-2.80c)¤ and (2.91a-2.91c), we get

t t t t dx e dx = ◦dx ε ◦dx . (2.94b) · · · ◦ · The pull-back of the left stretch tensor is: • 44 Nonlinear continua

t V = t U . (2.95) ◦ ◦ In many problems related to metallurgy (e.g. metal-forming problems) the • usual practice is to use a logarithmic strain measure (Hill 1978). Therefore, in the spatial conﬁguration the following strain measure is deﬁned

th =lnt V . (2.96a) ◦ Thepull-backofthetensor th is:

t H = tH =lnt V =lnt U . (2.96b) ◦ ◦ ◦ The above tensor is known as the Hencky deformation tensor. In following Chapters we will see the importance of the Hencky strain tensor for the analysis of finite strain problems (Anand 1979). Some recent finite element formulations, developed for the analysis of finite strain elastoplastic problems use this strain measure (e.g. Rolph & Bathe 1984, Weber & Anand 1990, Eterovic & Bathe 1990, Simo 1991, Dvorkin, Pantuso & Repetto 1992 1993 1994 1995, Dvorkin 1995a 1995b 1995c, Pèric, Owen & Honnor 1992).\ \ \ \ \ It is useful to note that we also get Eqs. (2.95) and (2.96a-2.96b) when instead of searching for a representation in the reference configuration, we search for a representation in a configuration rotated by t RT from the spatial configuration: corotational representation. ◦ The push-forward of the reference configuration metric tensor is: •

t t at b t 1 A t 1 B t at b φ ( ◦gAB) ab g g =(X− ) a ( X− ) b ◦gAB g g ∗ ◦ ◦ t 1 £ ¤ = b− (2.97a) t AB ab t t t a t b AB t t φ ( ◦g ) g g = X A X B ◦g g g ∗ a b ◦ ◦ a b t £ ¤ = b . (2.97b) In Chap. 3 we will study relations of the pull-back/push-forward type between stress measures.

2.12 Objectivity

The description of physical phenomena using objective formulations is a cen- tral topic in continuum mechanics. We will ﬁrst present the classical concept of objectivity under rotations and translations (isometries) (Truesdell & Noll 1965). Next, we will present the concept of objectivity under general changes of the reference frame (Marsden & Hughes 1983), we will use the word covariance to refer to this concept. 2.12 Objectivity 45

2.12.1 Reference frame and isometric transformations

We call an event the pair tx,t formed by a vector tx that defines a point in the Euclidean space and{ a time} t. A reference or observation frame is a way of relating the physical world to the points in an 3 Euclidean space and a real axis of time (Truesdell & Noll 1965). < Examples of reference frames: - The system of fixedstarsandaclock. -Thewallsofmyoffice and my watch. - A system of coordinates drawn on a rotating platform and a clock. An isometric transformation of reference frame is a mapping ( 3,t) ( 3,t) in which the distances between spatial points, the time intervals< between→ < events and the time ordering of events are preserved. Obviously, isometric transformations restrict us to Newtonian Mechanics. To describe an isometric transformation of reference frame we define in the spatial configuration:

t A fixed Cartesian system zα . • { t∗ } Another Cartesian system z∗ that rotates and translates. • { α} For simultaneous events, in the fixed system we register a time t,andin the moving system a time t∗. t The base vectors of the fixed and moving Cartesian frames are eα and t∗ eα∗ , respectively; hence, we can write, at an instant t:

t t∗ e = Q(t) e∗ . (2.98) α · α Obviously, the tensor Q(t) is orthogonal. Let us call c(t) the vector that goes from tz =(0, 0, 0) (origin of the t∗ fixed Cartesian frame) to z∗ =(0, 0, 0) (origin of the moving Cartesian frame). t∗ An observer on the moving frame defines an event with the vector z∗ and the time t∗. An observer on the fixed frame defines the same event with the vector tz and the time t. Let us assume that the two observers are observing a lab experiment, e.g. the measurement of the stretching of a spring when it is loaded with a weight of 1 kg. Let us also assume that the lab is moving with the moving frame; for t∗ the observer on this frame z∗ is the position vector of the spring-end. The observer on the fixed frame will see the spring-end moving:

t t∗ z(t)=c(t)+Q(t) z∗ . (2.99a) · Since we are considering now isometric transformations, the following condition holds: 46 Nonlinear continua

t = t∗ a, (2.99b) − where a is a constant. For simplicity, from here on, we will consider a =0(t = t∗). If we include the possibility that Q(t) represents not only rotations but also reflections (transformations between left-handed and right-handed systems), Eqs. (2.99a-2.99b) represent the most general isometric transformation (Truesdell & Noll 1965, Truesdell 1966). We should not confuse the concept of change of reference frame with the concept of change of coordinates system, the latter being an instant concept does not incorporate the frame velocity. An isometric change of reference frame may induce changes in scalars, vectors and tensors. Scalars do not change during an isometric transformation. • Vectors definedinthespatialconfiguration can always be considered as • proportional to the difference between two position vectors in that configuration (Truesdell & Noll 1965); hence, using Eq. (2.99a) we get

t t v = Q(t) v∗ . (2.100) · The above is the transformation law for spatial vectors under an isometric transformation of reference frame. Second-order tensors defined in the spatial configuration relate vectors • defined in the same configuration (quotient rule)6 . In the moving frame, we can write for an arbitrary second-order tensor ts∗

t t t v∗ = s∗ w∗ . (2.101a) · Using Eq.(2.100),

t t T t v = Q(t) s∗ Q (t) w . (2.101b) · · · h i Therefore, the transformation law for spatial second-order tensors under an isometric transformation of reference frame is

t t T s = Q(t) s∗ Q (t) . (2.101c) · · For spatial tensors of higher order, the transformation laws can be derived • in an identical way.

6 See Appendix. 2.12 Objectivity 47

2.12.2 Objectivity or material-frame indiﬀerence

Let us assume a deformation process taking place in the moving reference frame. This deformation process can be referred to a fixed reference configuration. There are general tensors (including scalars and vectors) defined in the reference configuration (e.g. the Green-Lagrange strain tensor), they are called material or Lagrangian tensors. There are general tensors definedinthespatialconfiguration (e.g. the Al- mansistraintensor,thematerialvelocity),theyarecalledspatial or Eulerian tensors. There are general tensors definedinbothconfigurations (e.g. the deformation gradient tensor), they are called two-point tensors. Following (Lubliner 1985) we define the following objectivity criteria under isometric transformations of a reference frame (classical objectivity): A Lagrangian tensor is objective if it is not affected by changes of the ◦ reference frame. An Eulerian tensor is objective if, under a change of reference frame, trans- ◦ forms according to Eqs. (2.100) and (2.101c). A two points second order tensor is objective if, when operating on a Eu- ◦ lerian objective vector, produces a Lagrangian objective vector.

Example 2.15. JJJJJ If we diﬀerentiate Eq. (2.99a) with respect to time we get,

tz˙(t)= tc˙(t)+Q˙ (t) tz + Q(t) tz˙ . · · Hence, the velocity is not an objective vector. JJJJJ

Inthesamewaywecanshowthattheacceleration vector is not objective. Therefore, even though forces are spatial objective tensors, Newton’s second law (tF = m tx¨) is not objective and is only applicable in inertial frames (Truesdell & Noll 1965). In what follows, we will analyze the objectivity of some of the tensors previously deﬁned: The deformation gradient tensor, in the moving frame, satisﬁes the fol- • lowing relation:

t t dx∗ = X∗ ◦dx . (2.102a) ◦ · 48 Nonlinear continua

t The vector ◦dx is an objective Lagrangian vector and dx∗ is an objective • t Eulerian vector, hence X∗ fulfills the objectivity definition. In the fixed frame, ◦

T t t Q (t) dx = X∗ ◦dx , (2.102b) · ◦ · and ﬁnally,

t t X = Q(t) X∗ . (2.102c) ◦ · ◦ Equation (2.102c) is the transformation law for objective two-point second- order tensors. Performing a right polar decomposition on both sides of Eq. (2.102c), we • get

t t t t R U = Q(t) R∗ U∗ . (2.103a) ◦ · ◦ · ◦ · ◦ Taking into account that: ◦ The inner (dot) product of two orthogonal second-order tensors is an orthogonal second order tensor. ◦ The polar decomposition is unique. We get

t t R = Q(t) R∗ (2.103b) ◦ · ◦ t t U = U∗ . (2.103c) ◦ ◦ Hence, the rotation tensor (two-point tensor) and the right stretch tensor (Lagrangian tensor) are objective under isometric transformations. From Eq. (2.47b), we can write •

t V = t R t U t RT . (2.104a) ◦ ◦ · ◦ ·◦ and using Eqs. (2.103a-2.103c), we obtain

t t T V = Q(t) V∗ Q (t) . (2.104b) ◦ · ◦ · The above equation shows that the left stretch tensor (Eulerian tensor) is objective under isometric transformations. The Green-Lagrange strain tensor (Lagrangian tensor) is (Eq. (2.65)) • 2.12 Objectivity 49

t 1 t T t ε = X X ◦g (2.105a) ◦ 2 ◦ · ◦ − h i and using Eq. (2.102c), we get

t ε = t ε (2.105b) ◦ ◦ The above equation depicts an objective behavior under isometric transformations. The Almansi strain tensor (Eulerian tensor) is (Eq. (2.68)): •

t 1 t t T t 1 e = g X− X− . (2.106a) 2 − ◦ · ◦ h i and using Eq. (2.102c), we get

T te = Q(t) te∗ Q (t) . (2.106b) · · The above equation depicts an objective behavior under isometric transformations.

2.12.3 Covariance

Let us assume a body that remains undeformed, referred to a spatial coordinate system that keeps changing (e.g. at a time t we have the spatial system txa and at a time tˆ the system tˆxˆa ). For a spatial second order tensor t{a ,} using the usual tensor transformati{ }ons, we can write (Truesdell & Noll 1965):

∂tˆxâ ∂tˆxˆb tââb = talm , (2.107a) ∂txl ∂txm t l t m tˆ ∂ x ∂ x t aâb = alm , (2.107b) ∂tˆxâ ∂tˆxˆb ∂tˆxâ ∂txm tââ = tal . (2.107c) b ∂txl ∂tˆxˆb m Between the spatial coordinates of the material particles at t and the ˆ tˆ spatial coordinates of the same particles at t we can define a mapping (tφ) tˆ (see Eq. (2.4)). Hence, we can also define a deformation gradient tensor, tX , and we can rewrite Eqs. (2.107a-2.107c) as 50 Nonlinear continua

ab t ab tˆ 1 a tˆ 1 b tˆ lm tˆ tˆ lm a =(tX− ) l (tX− ) m aˆ = tφ∗( aˆ ) , (2.108a) h i t tˆ l tˆ m tˆ tˆ tˆ aab = tX a tX b aˆlm = tφ∗( aˆlm ) , (2.108b) ab h i a t a tˆ 1 a tˆ m tˆ l tˆ tˆ l a b =(tX− ) l tX b aˆ m = tφ∗( aˆ m ) . (2.108c) b h i If we consider now general transformations between the spatial configurations at t and tˆ (not necessarily restricted to simple changes of coordinate system) the spatial tensor ta is defined as covariant or objective (in a more general way than the above defined objectivity under isometric transforma- tˆ tions) if under the mapping (tφ) it transforms following Eqs.(2.108a-2.108c) (Marsden & Hughes 1983). For a two-point tensor, the covariance criterion is only applied to the indices associated to the spatial basis (Lubliner 1985). A Lagrangian tensor is always covariant if it is not affected by changes of the reference frame. The reader can easily check that for the case of isometric transformations the concept of covariance is coincident with the classical concept of objectivity presented in the previous section.

A physical law is objective (either in the classical or in the covariant sense) if all the tensors in its mathematical formulation are objective.

2.13 Strain rates

In the previous sections we presented a static description of the kinematics of continuous media: given a spatial and a reference configuration we developed tools to relate both configurations (deformation gradient tensor, strain measures, etc.) In the present section we will study the kinematic evolution of the spatial configuration. For this purpose we will introduce the time rates of the different tensors described above.

2.13.1 The velocity gradient tensor

ThetimerateofEq.(2.23)is

t a t ˙ ∂ v t l t a t b t A X = A + X A Γlb v g ◦g , (2.109a) ◦ ∂ x ◦ a ∙ ◦ ¸ where tva was deﬁned in Eq. (2.10). 2.13 Strain rates 51

It is important to remember that the functional dependence is

t ˙ a t ˙ a B X A = X A(◦x ,t) . (2.109b) ◦ ◦ Using the chain rule in Eq. (2.109a),

t ˙ a t a t l X A = v l X A . (2.109c) ◦ | ◦ We deﬁne in the spatial conﬁguration the velocity gradient tensor,

t t a t t l l = v l g g , (2.110a) | a we can write the above as

tl = tv , (2.110b) ∇ tlT = tv . (2.110c) ∇ Hence, we can write Eq.(2.109c) as

t ˙ a t a t l X A = l l X A (2.111a) ◦ ◦ and therefore, t X˙ = tl t X . (2.111b) ◦ · ◦ It is important to realize that the above is the material time derivate of D t X t ˙ ◦ the deformation gradient tensor, X = Dt . ◦

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

We can decompose the velocity gradient tensor into its symmetric and skew- symmetric components: tl = td + tω (2.112a) where, 1 T td = tdT = (tl + tl ) (2.112b) 2 is the Eulerian strain rate tensor (deﬁned in the spatial conﬁguration) and,

1 T tω = tωT = (tl tl ) (2.112c) − 2 − is the spin or vorticity tensor,alsodefined in the spatial configuration. Let us assume a deformation process referred to a fixed Cartesian system. The principal directions of t U form, in the reference configuration, a Cartesian system known as Lagrangian◦ system. The principal directions of t V form, in the spatial configuration, a Cartesian system known as a Eulerian system◦ (Hill 1978). 52 Nonlinear continua

Fig. 2.5. Rotations

We can go from one of the above-deﬁned coordinate systems to another one using the rotation tensors sketched in Fig. 2.5. From Fig. 2.5, we get

t t t R = R RL . (2.113) E ◦ · ◦ For two consecutive rotations,

t+∆t t+∆t t R = t R R (2.114a) ◦ · ◦ and therefore,

t+∆t t t R g t R˙ = lim ∆t 0 − t R . (2.114b) ◦ → " ∆t # · ◦ We can deﬁne a rotation rate

t+∆t t t R g tΩ = lim ∆t 0 − (2.114c) R → " ∆t # and using it in Eq. (2.114b) (Hill 1978), we get

t R˙ = tΩ t R (2.115a) ◦ R · ◦ in the same way,

t R˙ = t Ω t R , (2.115b) ◦ L ◦ L · ◦ L tR˙ = tΩ tR . (2.115c) E E · E 2.13 Strain rates 53

Since the rotation tensors are orthogonal we can write

t T t R R = ◦g , (2.116a) ◦ · ◦ taking the time derivative of the above equation and using Eq. (2.115a), we have tΩ + tΩT = 0 (2.116b) R R in the same way,

t Ω + t ΩT = 0 , (2.116c) ◦ L ◦ L tΩ + tΩT = 0 . (2.116d) E E The above equations indicate that tΩ , t Ω and tΩ are skew- R L E symmetric tensors. ◦

2.13.3 Relations between diﬀerent rate tensors

The time derivative of Eq. (2.113) leads to

t t t t t t t t ΩE R = Ω R R + R Ω RL , (2.117a) · E R · ◦ · ◦ L ◦ · ◦ L · ◦ and therefore, t T t t t t R ( ΩE Ω ) R = Ω . (2.117b) ◦ · − R · ◦ ◦ L Using Eqs. (2.111b) and (2.40),

t t t T t t t 1 t T l = R˙ R + R U˙ U− R , (2.118a) ◦ · ◦ ◦ · ◦ · ◦ · ◦ splitting the above equation into its symmetric and skew-symmetric components, we get

t T t t 1 t t 1 t 1 t R d R = ( U˙ U− + U− U˙ ) (2.118b) ◦ · · ◦ 2 ◦ · ◦ ◦ · ◦ and

t T t t t 1 t t 1 t 1 t R ( ω ΩR) R = ( U˙ U− U− U˙ ) . (2.118c) ◦ · − · ◦ 2 ◦ · ◦ − ◦ · ◦ It is very important to recognize that (Hill 1978):

t t t U = ◦g = ω = Ω . (2.118d) ◦ ⇒ R An example of the above situation is the beginning of the deformation process (t =0). In the deformation process depicted in Fig. 2.6 (Truesdell & Noll 1965, Malvern 1969) we can write, using Eqs. (2.30a-2.30b): 54 Nonlinear continua

Fig. 2.6. Relative deformation gradients

τ τ t X = t X X . (2.119a) ◦ · ◦ For a ﬁxed t-conﬁguration, we can write

d τ d τ t X = t X X . (2.119b) dτ ◦ dτ · ◦ Using Eq. (2.111b) and the polar decomposition in the above equation, we get τ τ d τ τ τ d τ t l X =( t R t U + t R t U) X , (2.119c) · ◦ dτ · · dτ · ◦ τ τ for τ = t it is obvious that t U = t R = ◦g , and since the above equation has to hold for any t X, ◦

d τ d τ t R τ=t + U τ=t = l . (2.119d) dτ t | dτ t | It is easy to show that the ﬁrst tensor on the l.h.s. of the above equation is skew-symmetric and the second one is symmetric; hence,

d τ 1 t t T t R τ=t = ( l l )= ω (2.120a) dτ t | 2 −

d τ 1 t t T t U τ=t = ( l + l )= d . (2.120b) dτ t | 2 We can obtain an interesting picture of the deformation process by referring the Lagrangian tensors to the Lagrangian coordinate system (principal directions of t U) and the Eulerian tensors to the Eulerian coordinate system (principal directions◦ of t V). Hence (Hill 1978): ◦ 2.13 Strain rates 55

t t In the Lagrangian system the components of U are λr (weassumethem • t ◦t to be diﬀerent); the components of U˙ are λ˙ rs and the components of t Ω are t ΩL . ◦ L rs ◦ ◦ t t In the Eulerian system the components of V are of course also λr;the • t t ◦ t t components of d are drs; the components of ω are ωrs; the components t t E t t R of ΩE are Ωrs and the components of ΩR are Ωrs. From Eq. (2.117b) we have t E t R t L Ωrs Ωrs = Ωrs , (2.121a) − ◦ from Eq. (2.118b) we get t t t t ˙ λr + λs drs = λrs t t (2.121b) 2 λr λs and from Eq. (2.118c) we get t t t t R t ˙ λr λs ωrs Ωrs = λrs t −t ; (2.121c) − 2 λr λs (in Eqs. (2.121b) and (2.121c) we do not use the summation convention). In the ﬁxed Cartesian system the components of t U form the matrix [t U]; hence, ◦ ◦ t t t t T [ U]=[RL][Λ][RL] , (2.122a) ◦ ◦ ◦ where, t λ1 00 t t [ Λ]= 0 λ2 0 . (2.122b) ⎡ t ⎤ 00λ3 Taking the time derivative of Eq.⎣ (2.122a), we⎦ obtain t t t t T t t t t T [ U˙ ]=[RL][Λ˙][ RL] +[ΩL][ RL][Λ][RL] ◦ ◦ ◦ ◦ ◦ ◦ t t t T t [ RL][Λ][RL] [ ΩL] (2.122c) − ◦ ◦ ◦ hence, t T t t t t T t t t [ RL] [ U˙ ][RL]=[Λ˙]+[RL] [ ΩL][ RL][Λ] ◦ ◦ ◦ ◦ ◦ ◦ t t T t t [ Λ][RL] [ ΩL][ RL] . (2.122d) − ◦ ◦ ◦ The above equation shows that: tλ˙ = tλ˙ (r = s) rs r . (2.123) t ˙ t t t L λrs =( λs λr) Ωrs (r = s) − ◦ 6 From Eq. (2.121b) for the case r = s (diagonal components), we get t ˙ t λr d t drr = t = (ln λr) . (2.124) λr dt 56 Nonlinear continua

Example 2.16. JJJJJ

Using Eqs. (2.65), (2.35), (2.111b) and (2.112b) we can show that: 1 t ε˙ = t C˙ = t XT td t X . ◦ 2 ◦ ◦ · · ◦ JJJJJ

Example 2.17. JJJJJ The Hencky strain tensor components in the ﬁxed Cartesian system are,

t t t t T [ H]=[RL]ln[Λ][RL] , ◦ ◦ ◦ hence,

t t t 1 t t T t t t t T [ H˙ ]=[RL][Λ]− [ Λ˙][RL] +[ ΩL][ RL]ln[Λ][RL] ◦ ◦ t ◦ t t ◦ T t ◦ ◦ . [ RL]ln[Λ][ RL] [ ΩL] − ◦ ◦ ◦ JJJJJ

2.14 The Lie derivative

In the deformation process represented in Fig. 2.2 we can define, for a Eulerian tensor tt,itsLie derivative associated to the flow of the spatial configuration (Simo 1988, Marsden & Hughes 1983):

t t d t t Ltv( t)= φ φ∗( t) . (2.125) ∗ dt ½ ¾ £ ¤ As we already know (see Sect. 2.9) the operation of pull-back is not a tensor operation since it operates on components. Hence, for calculating a Lie derivative using Eq. (2.125) it is important to identify the components of tt that we are using. The Lie derivative of a scalar is

d ∂α ∂α t a Lt α = α = + v . (2.126) v dt ∂t ∂txa The covariant components of the Lie derivative of a spatial vector tw are

t t 1 A d t j t t L v w i =(X− ) i [( X) A wj] , (2.127a) ◦ dt ◦ ½ ¾ ¡ ¢ after some algebra, 2.14 The Lie derivative 57

t t t a t ∂ wi ∂ wi t a ∂ v t Lt w = + v + w . (2.127b) v i ∂t ∂txa ∂txi a Since, ¡ ¢ t it t w wi = α (2.128a) we can write

t i t t i t d t i t t i d t Lt w w + w Lt w = w w + w w (2.128b) v i v i dt i dt i µ ¶ µ ¶ ¡ ¢ ¡ ¢ and from the above we get the contravariant components of the Lie derivative of a spatial vector tw,

t i t i t i t i ∂ w ∂ w t a t a ∂ v Ltv w = + v w . (2.128c) ∂t ∂txa − ∂txa Followingtheaboveprocedurewecanshowthatthemixedcomponents¡ ¢ of the Lie derivative of a general Eulerian tensor tt are

t a...b t a...b t a...b ∂ t c...d ∂ t c...d t p Lt ( t) = + v (2.129) v c...d ∂t ∂txp ¡ ¢ t a t b ∂ v t p...b ∂ v t a...p t c...d t c...d − ∂txp − ··· − ∂txp t p t p ∂ v t a...b ∂ v t a...b + t p...d + + t c...p ∂txc ··· ∂txd

Example 2.18. JJJJJ

To calculate the Lie derivative of the spatial metric tensor tg we can directly use Eq. (2.125)

t t d t t Ltv g = φ φ∗( g) , ij ∗ dt IJ ³ ´ ∙ h i ¸ using now Eq. (2.93a), we get

t t t Ltv g = φ ( C˙ ) . ij ∗ ◦ ij ³ ´ h i Using the result in Example 2.16, we get

t t Ltv g = 2 d . ij ij ³ ´ ¡ ¢ JJJJJ 58 Nonlinear continua

Example 2.19. JJJJJ

To calculate the Lie derivative of the Almansi deformation tensor we use Eq. (2.125) and get

t t d t t Ltv e = φ φ∗( e) ij ∗ dt IJ ∙ ¸ ¡ ¢ £ ¤ and resorting to Eq.(2.94a),

t t t Ltv e = φ ( ε˙) . ij ∗ ◦ ij Taking into account¡ the result¢ obtained£ in¤ Example 2.16 we can ﬁnally write t t Lt e = d . v ij ij ¡ ¢ ¡ ¢ JJJJJ

Example 2.20. JJJJJ

The Lie derivative of the Finger deformation tensor is

t ij t d t t IJ Ltv b = φ φ∗( b) . ∗ dt ∙ ¸ ¡ ¢ £ ¤ Using Eq. (2.97b) we get

IJ t t IJ t IJ φ∗( b) = B = ◦g h i and since £ ¤ ◦g˙ = 0 we get t ij Ltv b =0. ¡ ¢ JJJJJ

2.14.1 Objective rates and Lie derivatives

In this Section we will show that the Lie derivative is the adequate mathematical tool for deriving covariant (objective) rates from covariant (objective) Eulerian tensors. Let us consider the deformation processes schematized in Fig. 2.7. It is obvious that 2.14 The Lie derivative 59

Fig. 2.7. Deformation processes between three conﬁgurations

tˆ tˆ t X = tX X . (2.130a) ◦ · ◦ For a covariant Eulerian tensor tt, that without losing generality we take as a second-order tensor:

tˆ ˆıjˆ tˆ ˆı tˆ jˆ t ab t = tX a tX b t , (2.130b) AB tˆ tˆ ˆıjˆ tˆ 1 A tˆ 1 B tˆ ˆı tˆ jˆ t ab φ∗ t =(X− ) ˆı ( X− ) jˆ tX a tX b t . (2.130c) ◦ ◦ ◦ ¯ ¯ ¯ ¯ Using Eq.¯ (2.130a)¯ in the above, we get

AB tˆ tˆ ˆıjˆ t 1 A t 1 B t lk φ∗ t =(X− ) l ( X− ) k t , (2.130d) ◦ ◦ ◦ ¯ ¯AB ¯tˆ tˆ ˆıjˆ¯ t t lk AB ¯ φ∗ t ¯ = φ∗ t . (2.130e) ◦ ◦ ¯ ¯ ¯ ¯ ¯ ¯ From the above¯ and from¯ Eq. (2.125)¯ it¯ follows that:

ˆ lm lm aˆb tˆ aˆˆb tˆ aˆ tˆ ˆb t tˆ t Ltvˆ ( t) = tX l tX m Ltv ( t) = tφ∗ Ltv ( t) . (2.130f) h i ½ h i ¾

The above equality shows that the Lie derivative of a covariant Eulerian tensor is also a covariant Eulerian tensor. 60 Nonlinear continua

Example 2.21. JJJJJ Considering again the case of a moving Cartesian frame and a ﬁxed one from Example 2.15, we get t α α ˙ α t γ α t γ v = c˙ + Q γ ( z∗) + Q γ ( v∗) , therefore taking into account that c , Q(t) and Q˙ (t), for the case under consideration, are constant in space, ∂ (tz )γ ∂ (tz ) tlα = Q˙ α ∗ + Qα (tl )γ ∗ β γ ∂ tzβ γ ∗ ∂ tzβ hence, t α ˙ α T γ α t γ T l β = Q γ (Q ) β + Q γ ( l∗) (Q ) β . Comparing with Eq. (2.101c) it is obvious that tl is not an objective tensor. Since the velocity gradient tensor is not objective in the classical sense we know that it is not a covariant tensor. JJJJJ

Example 2.22. JJJJJ From Example 2.19, we know that td is the result of a Lie derivative; hence we can assess that the Eulerian strain rate tensor is a covariant (objective) tensor. JJJJJ

Example 2.23. JJJJJ For a Eulerian tensor tt,wedeﬁne in the reference conﬁguration the tensor:

t t 1 A t 1 B t ab T = X− a X− b t ◦g ◦g ◦ ◦ A B which can be written as ¡ ¢ ¡ ¢

t t 1 t t T T = X− t X− . ◦ · · ◦ t 1 t Since, X− X = ◦g we can derive that, ◦ · ◦

d t 1 t 1 t X− = X− l dt ◦ − ◦ · and ¡ ¢ d t T t T t T X− = l X− dt ◦ − · ◦ we can write ³ ´

d t t 1 t t T t 1 t t t T T = X− t˙ X− X− l t X− dt ◦ · · ◦ − ◦ · · · ◦ ³ ´ t 1 t t T t T X− t l X− . − ◦ · · · ◦ 2.15 Compatibility 61

Also, from Eqs. (2.110c)

t T t t p t nt l = v = v n g g . ∇ | p Considering that the time derivative of the reference conﬁguration base vectors is zero and using the above together with the Lie derivative deﬁnition in Eq. (2.125), we get

t ab t ab t nb t a t al t b Ltv ( t) = t˙ t v n t v l . − | − | The above equation£ is¤ going to be used in Sect. 3.4 for deriving objective stress rates. JJJJJ

2.15 Compatibility

In our previous description of the kinematics of continuous media we went through the following path: Assume the existence of a regular mapping tφ

Calculate the⇓ tensorial components of different deformation measures If, instead of the above, we want to start by defining the tensorial components of a given deformation measure, our freedom to define them is limited by the fact that they should guarantee the existence of a regular mapping from which they could be derived. The conditions that the tensorial components of a deformation measure should fulfill in order to assure the existence of a regular mapping are called their compatibility conditions. In what follows we will derive the compatibility conditions for the Green deformation tensor. From Eqs. (2.61), (2.80c) and (2.93a) we have,

Eulerian tensor Spatial conﬁguration Pull-back space

Length diﬀerential tdl tdl

t a t a A A Coordinate diﬀerential dx [( dx ) ] = ◦dx

t t t Metric tensor gab [( gab) ]AB = CAB ◦ Hence, t at t b At B dx gab dx = ◦dx CAB ◦dx . (2.131) ◦ From the above equation it is obvious that the covariant components of the Green deformation tensor are the covariant components of the metric tensor 62 Nonlinear continua of the pull-back space. Note that the pull-back space is by no means coincident with the reference configuration, whose metric tensor has the covariant components ◦gAB. Since we restrict our study of the kinematics of continuous media to the Euclidean space, we can assess that the Riemann-Christoffel tensor is zero in the spatial configuration (McConnell 1957). Hence,

t Rprsq =0. (2.132)

The above equation represents 81 compatibility conditions to be fulfilled in the spatial configuration. However, the covariant components of the Riemann- Christoffel tensor satisfy the following relations (Aris 1962)7 :

t t Rprsq = Rrpsq , (2.133a) − t t Rprsq = Rprqs , (2.133b) − t t Rprsq = Rsqpr . (2.133c)

t t t We must also consider that Riiii =0; Riiij =0; Riijj =0can be easily transformed into a trivial identity of the form 0=0. Hence we are left with only 6 signiﬁcant equations, namely:

t t t R1212 =0 ; R1213 =0 ; R1223 =0; (2.134) t t t R1313 =0 ; R1323 =0 ; R2323 =0.

From Eqs. (A.79a-A.79e),

2 t 2 t 2 t 2 t t 1 ∂ gpq ∂ grs ∂ gps ∂ grq Rprsq = + 2 ∂txr ∂txs ∂txp ∂txq − ∂txr ∂txq − ∂txp ∂txs µ ¶ t mn t t t t + g Γrsm Γpqn Γrqm Γpsn (2.135a) − t where the Γijk are¡ the Christoﬀel symbols of the¢ﬁrst kind corresponding to the coordinate system txa . Hence, using Eq. (A.79b){ }

2 t 2 t 2 t 2 t t 1 ∂ gpq ∂ grs ∂ gps ∂ grq Rprsq = + 2 ∂txr ∂txs ∂txp ∂txq − ∂txr ∂txq − ∂txp ∂txs µ ¶ (2.135b) 1 ∂ tg ∂ tg ∂ tg ∂ tg ∂ tg ∂ tg + tgmn sm + mr rs qn + np pq 4 ∂txr ∂txs − ∂txm ∂txp ∂txq − ∂txn ∙ µ ¶µ ¶ 1 ∂ tg ∂ tg ∂ tg ∂ tg ∂ tg ∂ tg qm + mr rq sn + np ps =0. −4 ∂txr ∂txq − ∂txm ∂txp ∂txs − ∂txn µ ¶µ ¶¸ 7 See Appendix. 2.15 Compatibility 63

Doing a pull-back operation on Eq. (2.132) we obtain,

t t t p t r t s t q t φ∗ Rprsq PRSQ = X P X R X S X Q Rprsq =0. (2.136a) ◦ ◦ ◦ ◦ £ ¡ ¢¤

Example 2.24. JJJJJ Equation (2.135b) represents the components of the tensorial equation

tR = 0 .

If in the spatial conﬁguration we change from the txi coordinate system to the tx˜i system, we write Eq. (2.135b) using, { } { } ∂txl ∂txm tg˜ = tg pq lm ∂tx˜p ∂tx˜q ∂tx˜p ∂tx˜q tg˜pq = tglm ∂txl ∂txm and the equation would look like

2 t t t 1 ∂ g˜pq t mn 1 ∂ g˜sm R˜prsq = + + g˜ + =0 . 2 ∂tx˜r∂tx˜s ··· 4 ∂tx˜r ··· µ ¶ ∙ µ ¶¸ If we now want to do a pull-back of Eq.(2.135b) the algebra can get quite lengthy, but we can use an analogy with the above tensor transformations:

t l t m t t t ∂ x ∂ x t φ∗( glm)PQ = glm P Q = CPQ ∂◦x ∂◦x ◦ P Q £t t lm PQ¤ t lm ∂◦x ∂◦x t 1 PQ φ∗( g ) = g t l t m = C− . ∂ x ∂ x ◦ £ ¤ ¡ ¢ Using this formal analogy we can easily write:

2 t 2 t 2 t t t 1 ∂ CPQ ∂ CRS ∂ CPS φ∗ ( Rprsq) = ◦ + ◦ ◦ PRSQ 2 ∂ xR∂ xS ∂ xP ∂ xQ − ∂ xR∂ xQ ∙ ◦ ◦ ◦ ◦ ◦ ◦ £ ¤ 2 t ∂ CRQ t 1 MN P◦ S + C− −∂ x ∂ x ◦ ◦ ◦ ¸ t t t t t t 1 ∂ CSM ∂ CMR ∂ CRS ¡∂ CQN¢ ∂ CNP ∂ CPQ ◦ + ◦ ◦ ◦ + ◦ ◦ 4 ∂ xR ∂ xS − ∂ xM ∂ xP ∂ xQ − ∂ xN ∙ µ ◦ ◦ ◦ ¶µ ◦ ◦ ◦ ¶ t t t t t t 1 ∂ CQM ∂ CMR ∂ CRQ ∂ CSN ∂ CNP ∂ CPS ◦ + ◦ ◦ ◦ + ◦ ◦ −4 ∂ xR ∂ xQ − ∂ xM ∂ xP ∂ xS − ∂ xN µ ◦ ◦ ◦ ¶µ ◦ ◦ ◦ ¶¸ JJJJJ 64 Nonlinear continua

Hence,

2 t 2 t 2 t 2 t 1 ∂ CPQ ∂ CRS ∂ CPS ∂ CRQ t 1 MN R◦ S + P◦ Q R◦ Q P◦ S + C− 2 ∂ x ∂ x ∂ x ∂ x − ∂ x ∂ x − ∂ x ∂ x ◦ ∙ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ¸ ¡ (2.136b)¢

t t t t t t 1 ∂ CSM ∂ CMR ∂ CRS ∂ CQN ∂ CNP ∂ CPQ ◦ + ◦ ◦ ◦ + ◦ ◦ 4 ∂ xR ∂ xS − ∂ xM ∂ xP ∂ xQ − ∂ xN ∙ µ ◦ ◦ ◦ ¶µ ◦ ◦ ◦ ¶ t t t t t t 1 ∂ CQM ∂ CMR ∂ CRQ ∂ CSN ∂ CNP ∂ CPS ◦ + ◦ ◦ ◦ + ◦ ◦ =0. −4 ∂ xR ∂ xQ − ∂ xM ∂ xP ∂ xS − ∂ xN µ ◦ ◦ ◦ ¶µ ◦ ◦ ◦ ¶¸ t We can now deﬁne in the pull-back space, with metric CAB ,theChristof- fel symbols of the ﬁrst kind: ◦

t t t 1 ∂ CSM ∂ CMR ∂ CRS ∗ΓRSM = ◦ + ◦ ◦ (2.136c) 2 ∂ xR ∂ xS − ∂ xM µ ◦ ◦ ◦ ¶ therefore we obtain the following 6 compatibility conditions for the covariant components of the Green deformation tensor:

2 t 2 t 2 t 2 t t 1 ∂ CPQ ∂ CRS ∂ CPS ∂ CRQ φ∗( Rprsq)= ◦ + ◦ ◦ ◦ 2 ∂ xR∂ xS ∂ xP ∂ xQ − ∂ xR∂ xQ − ∂ xP ∂ xS µ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ¶ t 1 MN + C− ( ∗ΓRSM ∗ΓPQN ∗ΓRQM ∗ΓPSN )=0. (2.136d) ◦ − t The¡ above¢ equation indicates that CAB is a metric of the Euclidean space. Taking into account the Bianchi identities◦ in Eq. (A.82d) (Synge & Schild 1949) and Eqs.(2.135a), we get

Example 2.25. JJJJJ Using as a metric the Green tensor, we can deﬁne an analog to Eq.(A.59):

t t ∂ AI t D I A GRADt C ( A )= A AD ∗ΓIA ◦g ◦g ◦ ∂ x − ∙ ◦ ¸ where for the Eulerian vector ta,

t t i A = X I ai , I ◦ ³ ´ 2.15 Compatibility 65 and, t t t D 1 t 1 DK ∂ CIK ∂ CAK ∂ CIA ∗ΓIA = C− ◦ A + ◦ I ◦ K . 2 ◦ ∂ x ∂ x − ∂ x ∙ ◦ ◦ ◦ ¸ Taking into account that¡ ¢

t r t s t ∂ x ∂ x t CIK = I K grs , ◦ ∂◦x ∂◦x D K t 1 DK ∂◦x ∂◦x t rs C− = t r t s g , ◦ ∂ x ∂ x it is easy to show that:¡ ¢

t t φ∗ a = GRADt A . ia IA C ∇ ◦ IA £ ¡ ¢ ¤ h ³ ´i JJJJJ

Example 2.26. JJJJJ In a Cartesian coordinate system, from Eqs. (2.9a-2.9b), (2.29a-2.29c), (2.65) and (2.68) we get

t α t β t γ t γ t 1 ∂ u ∂ u ∂ u ∂ u εαβ = β + α + α β , ◦ 2 ∂ z ∂ z ∂ z ∂ z µ ◦ ◦ ◦ ◦ ¶ t α t β t γ t γ t 1 ∂ u ∂ u ∂ u ∂ u eαβ = + . 2 ∂tzβ ∂tzα − ∂tzα ∂tzβ µ ¶ For linear kinematics, ∂tuα β << 1 , ∂◦z and also ∂(.) ∂(.) = . β ∼ t β ∂◦z ∂ z Therefore, for linear kinematics

t α t β t 1 ∂ u ∂ u t εαβ = β + α = εαβ , ◦ ∼ 2 ∂ z ∂ z µ ◦ ◦ ¶ t wherewecall εαβ the components of the inﬁnitesimal strain tensor. From Eq. (2.65) we know that:

t 1 t εαβ = Cαβ δαβ . ◦ 2 ◦ − ¡ ¢ Introducing the above in Eq. (2.135b) and linearizing (neglecting the higher t powers of εαβ ) we get the compatibility equations corresponding to the assumption◦ of linear kinematics: 66 Nonlinear continua

2 t 2 t 2 t 2 t ∂ εαβ ∂ εγδ ∂ εαδ ∂ εγβ γ δ + α β γ β α δ =0. ∂◦z ∂◦z ∂◦z ∂◦z − ∂◦z ∂◦z − ∂◦z ∂◦z The above represents a set of 6 equations, that proceeding as in Eqs. (2.137), can be reduced to 3 independent compatibility conditions. The above result was obtained for a Cartesian system. Generalizing for an arbitrary coordinate system we get

t t t t εAB CD + εCD AB εAD BC εBC AD =0. | | − | − | JJJJJ 3 Stress Tensor

To deform a continuous body the exterior medium has to produce a loading on that body; therefore we get external forces acting on it. Also, during the deformation of a continuous body, neighboring particles exert forces on each other, they are the internal forces in the body. The study of the internal forces in a body leads to the notion of stresses, that we are going to develop in this chapter. Some reference books for this chapter are: (Truesdell & Noll 1965, Truesdell 1966, Malvern 1969, Marsden & Hughes 1983).

3.1 External forces

When studying the deformation of a continuous body, to classify a force as either external or internal, we have to carefully take into account our definition of the body under consideration. For example, in Fig. 3.1 a, we study the spatial configuration at t of the t t t t body ( ΩA ΩB) with the external forces FA and FB andinFig.3.1b,to ∪ t t study the same physical problem, we choose to consider ΩA and ΩB as free bodies. Obviously both analyses will lead to the same conclusions but, it t t is clear that in the first case FAB and FBA have to be considered as internal forces and in the second case as external forces. We can find different types of external forces, for example: External forces acting on the elements of volume or mass inside the body • (they are defined per unit volume or per unit mass). Theses forces are called body forces and some examples are: gravitational forces, inertia forces, elec- tromagnetic forces, etc. (in general they are “action-at-a-distance” forces (Malvern 1969). Let us define in the t-configuration of a body a Cartesian system tzˆα , t { α =1, 2, 3 with base vectors êα and an arbitrary curvilinear system txi ,i=1}, 2, 3 with covariant base vectors tg . We consider the vector { } i 68 Nonlinear continua

Fig. 3.1. External and internal forces in a bar

field tb(tx) to define the forces per unit mass acting on the body at t . We can write tb = tˆbαtê = tbitg . (3.1) α i

The resultant of the forces per unit mass is:

t t t t tˆαt t t t it t ρ b dV = ρ b êα dV = ρ b g dV (3.2) t t t i Z V Z V Z V where, tρ :densityofthebodyint-configuration and tV : volume of the body in the t-configuration. External forces acting on the elements of the body’s surface (they are • defined per unit surface). These forces are called surface forces and some examples are: pressure forces, contact forces, friction forces, etc. We consider the vector field tt(tx) to define the forces per unit surface t acting on the region Sσ, a subset of the surface of the body at t .Wecan write 3.2 The Cauchy stress tensor 69

tt = ttˆαtˆe = ttitg . (3.3) α i The resultant of the forces per unit surface is:

t t tˆαt t t it t t dS = t ˆeα dS = t g dS. (3.4) t t t i Z Sσ Z Sσ Z Sσ It is important to point out that our description of the external forces acting on a body excludes the possibility of considering distributed torques per unit volume, mass or surface. Therefore, the moment with respect to a point P of the considered external forces in the t-conﬁguration is:

t t t t t t t t MP = ρ r b dV + r t dS (3.5) t × t × Z V Z Sσ ¡ ¢ ¡ ¢ where tr is the vector that in the t-conﬁguration goes from the moment center P toapointinsidethebody(ﬁrsttermonther.h.s.)ortoapointonthe body surface (second term on the r.h.s.).

3.2 The Cauchy stress tensor

In Fig. 3.2 a we represent the spatial conﬁguration of a body corresponding toatimet, and we identify a particle P . B

Fig. 3.2. Internal forces at a point inside a continuum

The external forces acting on per unit mass are given by the vector ﬁeld tb and the external forces per unitB surface are given by the vector ﬁeld tt . 70 Nonlinear continua

t We now section the body ,inthet-configuration, with a surface Sc B t t passing through P . The normal to the surface Sc at P is n (see Fig. 3.2 b). t If we now analyze in Fig. 3.2 b the left part, ΩL ,asafreebody,wehave to consider as external forces the internal forces at P in Fig. 3.2 a. t t Considering on the surface Sc an area ∆S around P ,thesetofexternal forces acting on t∆S can be reduced to a force t∆F through P and a moment t ∆MP . When t∆S 0: → t∆F lim = tt (3.6a) t∆S 0 t∆S → t∆M lim P = 0 (3.6b) t∆S 0 t∆S → The vector tt is known in the literature as traction. Equations(3.6a-3.6b) incorporate two fundamental hypotheses: The limit in Eq. (3.6a) exists. Therefore we exclude from the contin- • uum mechanics field the consideration of concentrated forces (concentrated forces are also not physically possible). The condition in Eq. (3.6b) is a strong requirement in the classical for- • mulation of continuum mechanics. There are alternative formulations that do not require the fulfillment of Eq. (3.6b) (e.g. the theory of polar media (Truesdell & Noll 1965, Malvern 1969). In this book we limit our study to the classical case of non-polar media. It is interesting to note that from Eq. (3.6a) we can assess that, if we consider different surfaces through P , which share the external normal tn (tangent surfaces), we will arrive at the same traction vector tt (see Fig. 3.3). We can define, in the t-configuration at the point P, a second-order tensor tσ, the Cauchy stress tensor, via the following equation:

tt = tn tσ . (3.7) · Since tt and tn are vectors, using the quotient rule (Sect. A.5), it is evident that tσ is a second-order tensor. We can consider Eq. (3.7) to be a condition of equivalence between external forces and stresses inside a continuum. 3.2 The Cauchy stress tensor 71

Fig. 3.3. Tangent surfaces at P have the same traction vector

3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem)

In Sect. 4.4.2 we will prove that from the equilibrium equations of a nonpolar continuum we get

tσ = tσT , (3.8) that is to say, the Cauchy stress tensor is symmetric.

Example 3.1. JJJJJ In the following ﬁgure, at the point P , inside the t-conﬁguration of a continuum body, the stress tensor is tσ. t Cutting the continuum with the surface S1,wegetatP the traction vector

tt = tn tσ , 1 1 · t if we cut with S2, the traction vector at P is:

tt = tn tσ . 2 2 · 72 Nonlinear continua

Secant surfaces at a point inside a continuum

t In an arbitrary coordinate system xi we can write { } t t t t ij t t n = n1 σ n2 , 1 · 2 i j t t t t ij t t n = n2 σ n1 , 2 · 1 i j and since from Eq. (3.8) tσij = tσji we get

tt tn = tt tn . 1 · 2 2 · 1 The above result, a direct consequence of the Cauchy Theorem, is known as the projection theorem or reciprocal theorem of Cauchy (Malvern 1969).JJJJJ

3.3 Conjugate stress/strain rate measures

Let us assume, at an instant (load level) t,acontinuumbody in equilibrium under the action of external body forces tb and external surfaceB forces tt. Assuming a velocity ﬁeld tv(tx) on ,thepower provided by the external forces is: B

t t t t t t t t Pext = ρ b v dV + t v dS. (3.9a) t · t · Z V Z S Using Eq. (3.7) we can rewrite Eq.(3.9a) as 3.3 Conjugate stress/strain rate measures 73

t t t t t t t t t Pext = ρ b v dV + n σ v dS. (3.9b) t · t · · Z V Z S From the Divergence Theorem (Hildebrand¡ 1976), ¢

t t t t t t t Pext = ρ b v + σ v dV (3.9c) t · ∇ · · Z V introducing Eq. (2.110a-2.110c)£ and after some¡ algebra,¢¤

t t t t t t t t Pext = σ : l + ρ b + σ v dV. (3.9d) t ∇ · · Z V Since the t-con£figuration is an¡ equilibrium confi¢guration,¤ the following equation (to be proved in Chap. 4, Eq.(4.27b)) holds Dtv tρ tb + tσ = tρ , (3.9e) ∇ · Dt and an obvious result that we also need is: Dtv D 1 tv = tv tv . (3.9f) Dt · Dt 2 · ∙ ¸ The kinetic energy of the body ,attheinstantt,isdefined as B tρ 1 tK = tv tv tdV = tv tv dm. (3.9g) t 2 · 2 · Z V Zm In the second integral of the above equation we integrate over the mass of the body . Since the mass of the body is invariant, B DtK Dtv = tv dm. (3.9h) Dt Dt · Zm Finally, using the decomposition of the velocity gradient tensor in Eq. (2.112a) and considering that since (tσ) is a symmetric tensor and (tω) is a skew-symmetric one, tσ : tω =0 (3.9i) we get t t D K t t t Pext = + σ : d dV. (3.9j) Dt t Z V We define t t t t Pσ = σ : d dV (3.10) tV t Z t as the stresses power. Obviously Pσ is the fraction of Pext that is not transformed into kinetic energy and that is either stored in the body material or dissipated by the body material, depending on its properties (see Chapter 5). From Eq. (3.10) we define the spatial tensors tσ and td to be energy conjugate (Atluri 1984). In what follows we will define other pairs of energy conjugate stress/strain rate measures. 74 Nonlinear continua

3.3.1 The Kirchhoﬀ stress tensor

From Eqs. (3.10) and (2.34d) and the mass-conservation principle (to be discussed in Chapter 4, Eq.(4.20d)) we get

t t t t ◦ρ t t Pσ = σ : d dV = t σ : d ◦dV. (3.11) t ρ Z V Z◦V The Kirchhoff stress tensor is defined as ρ tτ = ◦ tσ (3.12) tρ where, ◦ρ: density in the reference configuration and ◦V : volume of the reference configuration. It is important to note that although the Kirchhoff stress tensor was in- t troduced by calculating Pσ via an integral defined over the reference volume, Eq. (3.12) clearly shows that tτ is definedinthesamespacewhere tσ is defined: the spatial configuration. Hence, using in the t-configuration an arbitrary curvilinear coordinate system txa with covariant base vectors tg we a obtain { }

tτ = tτ ab tg tg (3.13a) a b tσ = tσab tg tg (3.13b) a b and ρ tτ ab = ◦ tσab . (3.13c) tρ

3.3.2 The ﬁrst Piola-Kirchhoﬀ stress tensor

From Eqs. (3.11) and (3.9i) we can write

t t t Pσ = τ : l ◦dV. (3.14) Z◦V In Chap. 2 we learned how to derive representations in the reference configuration of tensors defined in the spatial configuration via pull-back operations. We will now obtain a representation of the Kirchhoff stress tensor in the form of a two-point tensor. In the reference configuration we define an arbitrary coordinate system A ◦x with covariant base vectors ◦g ;andinthespatialconfiguration a { } A system txa with covariant base vectors tg .Wealsodefine a convected a { i } system θ with covariant base vectors in the reference configuration ◦g { } i and covariant base vectors in the spatial configuration tg . i In the spatial configuration we can write the Kirchhoff stress tensor as e e 3.3 Conjugate stress/strain rate measures 75

tτ = tτ˜ij tg tg (3.15a) i j a pull-back of the above tensor to the referencee e conﬁguration is: IJ tT = tτ˜ij g g = tT g g . (3.15b) ◦ i ◦ j ◦ I ◦ J h i We deﬁne a two-point representatione e of tτ as

t t ij t t Ij t P = τ˜ ◦g g = P ◦g g . (3.15c) ◦ i j ◦ I j £ ¤ After some algebra, e e

I t j l m I t j t Ij t lm ∂◦x ∂ x t pq ∂θ ∂θ ∂◦x ∂ x P = τ˜ l m = τ t p t q l m ◦ ∂θ ∂θ " ∂ x ∂ x # ∂θ ∂θ

t pj t 1 I = τ X− p . (3.15d) ◦ The second-order two-point¡ ¢ tensor t P is the first Piola-Kirchhoff stress tensor.ItisapparentfromEq.(3.15d)thatitisa◦ non-symmetric tensor. We can write, due to the symmetry of the Kirchhoff stress tensor:

t ab t t ba t Pσ = τ lab ◦dV = τ lab ◦dV. (3.16a) Z◦V Z◦V Hence, using Eq.(3.15d):

t Ba t b t Pσ = P X B lab ◦dV. (3.16b) ◦ ◦ Z◦V Using Eq. (2.111a) we have

t Ba t t t Pσ = P X˙ aB ◦dV = P X˙ ◦dV. (3.17) ◦ ◦ ◦ ·· ◦ Z◦V Z◦V We can also write the above as (Malvern 1969):

t T t Pσ = P : X˙ ◦dV. (3.18) ◦ ◦ Z◦V The above equation defines the two-point tensors t PT and t X˙ as energy conjugates. ◦ ◦ We will not try to force a so-called “physical interpretation” of the first Piola-Kirchhoff stress tensor; instead we will regard it only as a useful mathematical tool. 76 Nonlinear continua

3.3.3 The second Piola-Kirchhoﬀ stress tensor

The pull-back conﬁguration of tτ to the reference conﬁguration is:

t t ij t 1 I t 1 J T = τ X− i X− j ◦g ◦g . (3.19) ◦ ◦ I J h ¡ ¢ ¡ ¢ i The tensor tT,defined by the above equation, is the second Piola- Kirchhoff stress tensor and it is a symmetric tensor. Using Bathe’s notation (Bathe 1996), we identify the second Piola-Kirchhoff stress tensor, corresponding to the t-configuration and referred to the configuration in t =0as t S. ◦

Example 3.2. JJJJJ From Eqs. (3.15d) and (3.19) we get

t IJ t 1 J t Ij S = X− j P ◦ ◦ ◦ that is to say, ¡ ¢

t IJ t t Ij IJ S = φ∗ P . ◦ ◦ £ ¡ ¢¤ JJJJJ

From Example 2.16:

t t t ε˙ AB = φ∗ dab AB (3.20a) ◦ and since from Eq. (3.19), £ ¡ ¢¤

t AB t t ab AB S = φ∗ τ (3.20b) ◦ using Eqs. (3.11) and (2.88), we get £ ¡ ¢¤

t t t Pσ = S : ε˙ ◦dV. (3.20c) ◦ ◦ Z◦V From Eq. (3.20c) we define the tensors t S and t ε˙ to be energy conjugate (Atluri 1984). ◦ ◦ Here, we will also not try to force a “physical interpretation” of the second Piola-Kirchhoff stress tensor. An important point to be analyzed is the transformation of t S under rigid- body rotations. ◦ Let us first consider the t-configuration of a certain body . At an arbitrary • point P the Cauchy stress tensor is tσ. B 3.3 Conjugate stress/strain rate measures 77

Let us now assume that we evolve from the t-conﬁguration to a (t + ∆t)- • conﬁguration imposing on and on its external loads a rigid body rotation. B At the point P we can write:

t+∆t t+∆t t X = t R , (3.21a) and therefore,

t+∆tdx = t+∆tR tdx . (3.21b) t · For the external loads, ◦

t+∆tt = t+∆tR tt , (3.22a) t · t+∆tb = t+∆tR tb . (3.22b) t · For a velocity vector, ◦

t+∆tv = t+∆tR tv . (3.22c) t · For the external normal vector, ◦

t+∆tn = t+∆tR tn . (3.22d) t ·

Example 3.3. JJJJJ For an arbitrary force vector tf (it can be a force per unit surface, per unit volume, etc.) and considering the evolution described above,

t+∆tf t+∆tv = t+∆tR tf t+∆tv · t · · = tf t+∆tRT t+∆tv · t · = tf t+∆tRT t+∆tR tv · t · t · and since the rotation tensor is orthogonal,

t+∆tf t+∆tv = tf tv . · · The above equation states the intuitive notion that a rigid-body rotation cannot aﬀect the value of the deformation power performed by the external forces. JJJJJ 78 Nonlinear continua

At t we can write tt = tn tσ (3.23a) · and at (t + ∆t), t+∆tt = t+∆tn t+∆tσ . (3.23b) · Introducing Eq. (3.22d) in the above,

t+∆tt = t+∆tR tn t+∆tσ . (3.23c) t · · And with Eq. (2.28a) and (3.22a),¡ we ﬁnally¢ have

tt = tn t+∆tRT t+∆tσ t+∆tR . (3.23d) · t · · t h i For deriving the above equation, we used that t+∆tR tt = tt t+∆tRT . t · · t Hence, t+∆tσ = t+∆tR tσ t+∆tRT . (3.23e) t · · t The above equation indicates that the Cauchy stress tensor fulfills the criterion for objectivity under isometric transformations, established for Eulerian tensors in Sect. 2.12.2. We define an arbitrary system txa0 in the t-configuration and a system t+∆txa in the (t + ∆t)-configuration.{ } Hence, from Eq. (3.23e), { } d t+∆tσa = t+∆tRa tσc0 t+∆tRT 0 (3.24a) b t c0 d0 t b and using Eq. (2.28c), we get ¡ ¢

t+∆tσa = t+∆tRa tσc0 t+∆tRl t+∆tg tgm0d0 (3.24b) b t c0 d0 t m0 lb therefore, t+∆tσal = tσc0m0 t+∆tRa t+∆tRl . (3.24c) t c0 t m0 It is easy to show that for the Kirchhoﬀ stress tensor we can also write

t+∆tτ = t+∆tR tτ t+∆tRT . (3.25) t · · t From Eq. (3.19), we obtain

t+∆t IJ t+∆t ij t+∆t 1 I t+∆t 1 J S = τ X− i X− j (3.26a) ◦ ◦ ◦ but since, ¡ ¢ ¡ ¢

t+∆t a t+∆t a t a0 X A = t R a X A (3.26b) ◦ 0 ◦ t+∆t 1 A t 1 A t+∆t T a0 X− a = X− a t R a (3.26c) ◦ ◦ 0 using Eqs. (3.24c)¡ and (3.26c)¢ in Eq.¡ (3.26a),¢ we¡ get ¢ 3.3 Conjugate stress/strain rate measures 79

t+∆tSIJ = t SIJ (3.27a) ◦ ◦ therefore, t+∆tS = t S . (3.27b) ◦ ◦ The above equation indicates that the second Piola-Kirchhoﬀ stress tensor fulﬁlls the criterion for objectivity under isometric transformations, established for Lagrangian tensors in Sect. 2.12.2.

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

In Sect. 2.8.5 we deﬁned the logarithmic or Hencky strain tensor. Let us now deﬁne, via a pull-back operation, the following stress tensor:

t t t Γ = R∗ ( τ ) . (3.28) ◦ t With the notation R∗( ) we deﬁne the pull-back of the components of the tensor ( ) using the tensor◦ ·t R (Simo & Marsden 1984), that is to say, tΓ is an unrotated· representation◦ of tτ . From the symmetry of tτ ,theabovedeﬁnition implies the symmetry of tΓ . We will now demonstrate, following (Atluri 1984), that for an isotropic material tΓ and t H˙ are energy conjugate. We can write Eq.◦ (3.14) as

t t AB t t Pσ = Γ R∗ dab AB ◦dV (3.29a) ◦ Z◦V £ ¡ ¢¤ therefore, t t AB t a t t b Pσ = Γ R A dab R B ◦dV. (3.29b) ◦ ◦ Z◦V From Eq. (2.28c),

t a t T L t al R A = R l g ◦gAL (3.29c) ◦ ◦ and using the above, the integrand¡ in Eq.¢ (3.29b) is:

t AB t T L t al t t b Γ R l g ◦gAL dab R B . (3.29d) ◦ ◦ It is also easy to show that¡ ¢

t T t t t T L t al t t b A B R d R = R l g ◦gAL dab R B ◦g ◦g . (3.29e) ◦ · · ◦ ◦ ◦ h i Hence, using Eq. (2.118b),¡ ¢

t 1 t t t 1 t 1 t Pσ = Γ : U˙ U− + U− U˙ ◦dV. (3.29f) V 2 ◦ · ◦ ◦ · ◦ Z◦ ³ ´ 80 Nonlinear continua

In order to simplify the algebra, in what follows we will work in a Cartesian system; [A] willbethematrixformedwiththeCartesiancomponentsofa second-order tensor A. From Eqs. (2.122a-2.122d), we get

t t 1 t t t 1 t T t [ U˙ ][ U]− =[RL][Λ˙][Λ]− [ RL] +[ΩL] (3.30a) ◦ ◦ ◦ ◦ ◦ t t t T t t t 1 t T [ RL][Λ][ RL] [ ΩL][RL][Λ]− [ RL] − ◦ ◦ ◦ ◦ ◦ and

t 1 t t t 1 t t T [ U]− [ U˙ ]=[RL][Λ]− [ Λ˙][ RL] (3.30b) ◦ ◦ ◦ ◦ t t 1 t T t t t t T t +[ RL][Λ]− [ RL] [ ΩL][ RL][Λ][RL] [ ΩL] ◦ ◦ ◦ ◦ ◦ − ◦ it follows from the above two equations that

1 t t 1 t 1 t t t 1 t t T [ U˙ ][ U]− +[U]− [ U˙ ] =[RL][Λ]− [ Λ˙][RL] (3.30c) 2 ◦ ◦ ◦ ◦ ◦ ◦ n o 1 t t 1 t T t t t t T + [ RL][Λ]− [ RL] [ ΩL][RL][Λ][ RL] 2 ◦ ◦ ◦ ◦ ◦

1 t t t T t t t 1 t T [ RL][Λ][RL] [ ΩL][RL][Λ]− [ RL] , − 2 ◦ ◦ ◦ ◦ ◦ and using once again Eqs. (2.122a-2.122d), we get

1 t t 1 t 1 t t t 1 t t T [ U˙ ][ U]− +[U]− [ U˙ ] =[RL][Λ]− [ Λ˙][RL] (3.30d) 2 ◦ ◦ ◦ ◦ ◦ ◦ n o 1 t 1 t t 1 t t t 1 + [ U]− [ ΩL][ U] [ U][ΩL][ U]− . 2 ◦ ◦ ◦ − 2 ◦ ◦ ◦ Using the result in Example 2.17, we can write

1 t t 1 t 1 t t t t [ U˙ ][ U]− +[U]− [ U˙ ] =[H˙ ] [ ΩL][ln U] (3.30e) 2 ◦ ◦ ◦ ◦ ◦ − ◦ ◦ n o t t 1 t 1 t t 1 t t t 1 +[ln U][ ΩL]+ [ U]− [ ΩL][U] [ U][ ΩL][U]− . ◦ ◦ 2 ◦ ◦ ◦ − 2 ◦ ◦ ◦ Using the above in Eq. (3.29f) and working with the matrix components,

t t t t t Pσ = [ Γ ]αβ [ H˙ ]αβ [ ΩL]αγ [ln U]γβ (3.31a) V ◦ − ◦ ◦ Z◦ n t t 1 t 1 t t +[ln U]αγ [ ΩL]γβ + [ U − ]αγ [ ΩL]γδ [ U]δβ ◦ ◦ 2 ◦ ◦ ◦

1 t t t 1 [ U]αγ [ ΩL]γδ [ U − ]δβ ◦dV. − 2 ◦ ◦ ◦ ¾ t t t Since [ Γ ] and [ U] are symmetric and [ ΩL] is skew-symmetric, we can rewrite the above equation◦ as 3.4 Objective stress rates 81

t t t Pσ = [ Γ ]αβ [ H˙ ]αβ ◦dV ◦ Z◦V t t t t t [ Γ ][H] βγ [ H][Γ ] βγ [ ΩL]βγ ◦dV − ◦ − ◦ ◦ Z◦V n £ ¤ £ ¤ o 1 t 1 t t t t t 1 t + [ U]− [ Γ ][U] γδ [ U][Γ ][ U]− γδ [ ΩL]γδ ◦dV. 2 ◦ ◦ − ◦ ◦ ◦ Z◦V n £ ¤ £ ¤ o (3.31b)

We will show in Chap. 5 that for isotropic materials the Eulerian tensors tσ (tτ ) and t V have coincident eigenvectors (they are coaxial). Taking into◦ account that

t t t U = R∗( V) (3.32) ◦ ◦ ◦ and the deﬁnition of tΓ in Eq. (3.28) we conclude that the Lagrangian tensors tΓ , t U and t H are also coaxial for isotropic materials. Obviously,◦ the◦ coaxiality of tΓ and t H implies that ◦ [tΓ ][t H] [t H][tΓ ]=[0] (3.33a) ◦ − ◦ and the coaxiality of tΓ and t U implies that ◦ t 1 t t t t t 1 [ U]− [ Γ ][ U] [ U][Γ ][U]− =[0]. (3.33b) ◦ ◦ − ◦ ◦ Finally, for isotropic materials,

t Isot.Mat. t t ˙ Pσ = Γ : H ◦dV (3.34) ◦ Z◦V and therefore in this case tΓ and t H˙ are energy conjugates. ◦

3.4 Objective stress rates

In Sect. 2.14.1 we show that the adequate tool for deriving objective rates of Eulerian tensors is the Lie derivative. The Lie derivative of the Cauchy stress tensor is:

ab t t ab t ab t cb t a t ac t b σ◦ = Ltv( σ) = σ˙ σ l σ l (3.35) − c − c ³ ´ £ ¤ the above stress rate is known as Oldroyd stress rate (Marsden & Hughes 1983). 82 Nonlinear continua

Example 3.4. JJJJJ To derive the expression of Oldroyd’sstressratewe start from Eq. (2.129),

t ab t ab t a t ab ∂ σ ∂ σ t p ∂ v t 1 P t pb t L v σ = + t p v P X− p σ ∂t ∂ x − ∂◦x ◦ t b ¡ ¢ ∂ v t 1 P t ap ¡ ¢ P X− p σ . − ∂◦x ◦ ¡ ¢ From Eq. (2.109a)

t a t ˙ ∂ v t p t a t l t A X = A + X A Γpl v g ◦g , ◦ ∂ x ◦ a ∙ ◦ ¸ and Eq. (2.111b) t X˙ = tl t X , ◦ · ◦ we get, t a ∂ v t a t m t a t st r A = l m X A Γrs v X A . ∂◦x ◦ − ◦ Then, from tσ = tσab tg tg , a b we get,

∂tσab ∂tσab tσ˙ = + tvl + tσmb tΓ a tvl + tσam tΓ b tvl tg tg . ∂t ∂txl ml ml a b ∙ ¸ Replacing in the ﬁrst equation, after some algebra, we ﬁnally get

t ab t ab t a t pb t b t ap Ltv σ = σ˙ l σ l σ . − p − p ¡ ¢ JJJJJ

The Lie derivative of the Kirchhoﬀ stress tensor is: • ab t t ab t ab t cb t a t ac t b τ◦ = Ltv( τ ) = τ˙ τ l τ l (3.36) − c − c the aboveh ratei is known£ as the¤ Truesdel l stress rate (Marsden & Hughes 1983).

Example 3.5. JJJJJ From Eq. (3.19), t ij t IJ t i t j τ = S X I X J , ◦ ◦ ◦ hence, 3.4 Objective stress rates 83

t ij t ij t i ∂ τ ∂ τ t p t ˙ IJ t i t j t IJ ∂ v t j + t p v = S X I XJ + S I X J ∂t ∂ x ◦ ◦ ◦ ◦ ∂◦x ◦ t j t IJ t i ∂ v + S X I J ◦ ◦ ∂◦x but ij t t ˙ IJ t ˙ IJ t i t j φ ( S ) = S X I XJ . ∗ ◦ ◦ ◦ ◦ h i Therefore,

ij t ij t ij t t ˙ IJ ∂ τ ∂ τ t p φ ( S ) = + t p v ∗ ◦ ∂t ∂ x h i t i t lm t 1 I t 1 J ∂ v t j τ ( X− ) l ( X− ) m I X J − ◦ ◦ ∂◦x ◦ t j t lm t 1 I t 1 J ∂ v t i τ ( X− ) l ( X− ) m J X I . − ◦ ◦ ∂◦x ◦

Using algebra along the lines of Example 3.4 and Eq.(3.36) we ﬁnally get,

ij ij tτ◦ = tφ (t S˙ IJ) . ∗ ◦ h i JJJJJ

The above example shows that the necessary and suﬃcient condition for the second Piola-Kirchhoﬀ stress tensor to remain constant is that the Trues- dell stress rate is zero (Eringen 1967). We can also perform pull-back and push-forward operations using the rotation tensor t R (Simo & Marsden 1984). Let us consider an arbitrary Eulerian stress tensor◦ tt (e.g.tσ or tτ ), and perform on its components a t R-pull- back: ◦ t t AB t ab t T A t T B R∗( t) = t ( R ) a ( R ) b . (3.37a) ◦ ◦ ◦ In order to simplify£ our¤ calculations we will now work in a Cartesian system; hence, t t t T t t R∗( t) =[R] [ t][R] . (3.37b) ◦ ◦ ◦ £ d ¤ T d T Taking into account that dt [R] = dt [R] , we get

d t t t T t t © tª T t© t ª t T t t R∗( t) =[R] [ t˙][R]+[R˙ ] [ t][ R]+[R] [ t][ R˙ ] (3.37c) dt ◦ ◦ ◦ ◦ ◦ ◦ ◦ and£ using Eqs.¤ (2.115a) and (2.116a) we get

d t t t T t t t T t t t R∗( t) =[R] [ t˙][ R] [ R] [ ΩR][t][R] dt ◦ ◦ ◦ − ◦ ◦ £ ¤ t T t t t +[ R] [ t][ΩR][R] . (3.37d) ◦ ◦ 84 Nonlinear continua

Since, t t d t t t T Lt R( t) =[R] R∗( t) [ R] (3.37e) ◦ ◦ dt ◦ ◦ we ﬁnally arrive ath i £ ¤

t t t t t t Lt R( t) =[t˙] [ ΩR][t]+[t][ΩR] (3.37f) ◦ − h i that in an arbitrary spatial coordinate system leads to

ab t t ˙ab t a t cb t ac t b Lt R( t) = t ΩRc t + t ΩRc. (3.38) ◦ − h i The above Lie derivative is the well-known Green-Naghdi stress rate (Di- enes 1979, Marsden & Hughes 1983, Pinsky, Ortiz & Pister 1983, Simo & Pister 1984, Cheng & Tsui 1990). From its derivation it is apparent that the Green-Naghdi stress rate is objective under isometric transformations,therefore it is known as a corotational stress rate. In the case of the Kirchhoff stress tensor, its Green-Naghdi rate is the t R-push-forward of the rate of tΓ. ◦ If as a reference configuration we use the t-configuration (Dienes 1979), we will fulfill Eq. (2.118d), that is to say,

tω = tΩ . (3.39) R Using the above in Eq. (3.38) we obtain the Jaumann stress rate (Truesdell & Noll 1965). It is important to realize that (Dienes 1979): The Jaumann stress rate is only coincident with the corotational stress • t rate when U ◦g. ◦ ≈ When formulating a corotational constitutive relation, we will only be able • to use the Jaumann stress rate when the spatial and reference conﬁgura- tions are coincident. 4 Balance principles

In this chapter we are going to present a set of basic equations (balance of mass, momentum, angular momentum and energy) that govern the behavior of the continuous media in the framework of Newtonian mechanics. We are going to present these basic principles in an integral form and also in the form of partial diﬀerential equations (localized form). On presenting the basic principles we are going to use both, the Eulerian (spatial) and Lagrangian (material) descriptions of motion. Some reference books for this chapter are: (Truesdell & Toupin 1960, Fung 1965, Eringen 1967, Malvern 1969, Slaterry 1972, Oden & Reddy 1976, Mars- den & Hughes 1983, Panton 1984, Lubliner 1985, Fung & Tong 2001).

4.1 Reynolds’ transport theorem

We begin this chapter by presenting Reynolds’ transport theorem; which will be used in what follows as a tool for calculating material derivatives of integrals defined in a spatial domain. Let us define in the spatial configuration of a continuum body an arbitrary coordinate system txi ,i =1, 2, 3 and let us assume a continuousB Eulerian tensor field tψ(tx{i ,t) to be a single-valued} function of the coordinates txi and of time t.Also,wedefine in the reference configuration an { } I arbitrary coordinate system ◦x ,I =1, 2, 3 . We define tV as a volume{ in the spatial con}figuration and

D t t t ψ ( xi,t) dV (4.1) Dt t Z V to be the material time derivative of a spatial volume integral;thatistosay, Eq. (4.1) measures the rate of change of the total amount of the tensorial property tψ carried by the particles that at time t areinsidethevolumetV . Using Eq.(2.31) we can write, 86 Nonlinear continua

D t t i t d t I t ψ ( x ,t) dV = ψ (◦x ,t) J ◦dV (4.2a) Dt t dt Z V Z◦V hence,

t I t D t t i t d ψ (◦x ,t) t t I d J ψ ( x ,t) dV = J ◦dV + ψ (◦x ,t) ◦dV. Dt t dt dt V ◦V ◦V Z Z Z (4.2b)

Example 4.1. JJJJJ Working in Cartesian coordinates we can write, from Eq. (2.34e) (Fung 1965):

t t t t J = eαβγ Xα1 Xβ2 Xγ3 . ◦ ◦ ◦

Using Eqs.(2.111a-2.111b) we can calculate the time rate of tJ:

t t t t t t t t t J˙ = eαβγ [ lα X1 Xβ2 Xγ3 + Xα1 lβ X2 Xγ3 t ◦ t ◦ t ◦ t ◦ ◦ ◦ + Xα1 Xβ2 lγ X3] . ◦ ◦ ◦ After some algebra, the reader can easily verify that,

t t t t t J˙ = l eαβγ Xα1 Xβ2 Xγ3 . ◦ ◦ ◦ Generalizing the above for any set of curvilinear coordinates in the Euclidean spacewecanwrite, tJ˙ =( tv) tJ ∇ · where tv is the velocity vector at the t-conﬁguration. A proof of the above result in general curvilinear coordinates can be found in (Marsden & Hughes 1983). JJJJJ

Using the result of Example 4.1 in Eq. (4.2b) we obtain, D tψ (txi,t) tdV = (4.3a) Dt t Z V t I d ψ (◦x ,t) t I t t + ψ (◦x ,t)( v) J ◦dV. dt ∇ · Z◦V " # Returningtothet-conﬁguration we obtain, D tψ (txi,t) tdV = (4.3b) Dt t Z V Dtψ (txi,t) + tψ (txi,t)( tv) tdV. t Dt ∇ · Z V ∙ ¸ 4.1 Reynolds’ transport theorem 87

Equation (4.3b) is one way of expressing Reynolds’ transport theorem. In this Section, we will also discuss other expressions for this theorem. Using Eq. (2.20b) we can rewrite Eq. (4.3b) as,

D ∂tψ (txi,t) tψ (txi,t) tdV = [ (4.3c) Dt t t ∂t Z V Z V + tv tψ (txi,t)+ tψ (txi,t)( tv)] tdV. · ∇ ∇ ·

Example 4.2. JJJJJ Let us assume a general tensor ﬁeld:

tψ = tψa...b tg ...tg tgc...tgd c...d a b where the tg are the covariant base vectors of the coordinate system txi . i The velocity vector ﬁeldcanbewrittenas, { }

tv = tvstg s therefore we can write, using Eq.(A.59),

tv tψ = tvstg tψa...b tgntg ...tg tgc...tgd · ∇ s · c...d |n a b t st a...b t t t c t d ¡ ¢ = v ψ c...d s g ... g g ... g . | a b

Also, using Eq.(A.64),

t t t a...b t s t t t c t d ψ( v)= ψ v s g ... g g ... g . ∇ · c...d | a b Finally, using Eq. (A.62b),

t t t sta...b t t t c t d ( v ψ)= v ψ s g ... g g ... g ∇ · c...d | a b ³t sta...b ´ t a...b t s t t t c t d = v ψ s + ψ v s g ... g g ... g . c...d | c...d | a b ³ ´ From the above equations we get,

tv ( tψ)+tψ ( tv)= (tv tψ) . · ∇ ∇ · ∇ · JJJJJ

Using the above result we can express Reynolds’ transport theorem as, 88 Nonlinear continua

D ∂ tψ (txi,t) tψ (txi,t) tdV = + (tv tψ(txi,t)) tdV. Dt tV tV ∂t ∇ · Z Z ∙ ¸ (4.4) The generalized Gauss’ theorem can be stated as (Malvern 1969):

tψ tdV = tn tψ tdS (4.5) t ∇ · t · Z V Z S where tS is the closed surface that bounds the volume tV and tn is the surface’s outer normal vector. Using Gauss’ theorem in Eq. (4.4) and rearranging terms, we get the following expression of Reynolds’ transport theorem:

∂ tψ D tdV = tψ tdV tn tv tψ tdS. (4.6) t ∂t Dt t − t · Z V Z V Z S Following (Malvern 1969) we can state:

Rate of increase Rate of increase Net rate of outward of the total amount of the total amount t ⎧ t ⎫ ⎧ t ⎫ ﬂux of ψ carried of ψ inside of ψ possessed ⎧ ⎫ ⎪ t ⎪ = ⎪ ⎪ by the material ⎪ avolume V ⎪ ⎪ by the material ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ transport through ⎪ ⎨ in the spatial ⎬ ⎨ instantaneously ⎬ ⎨ t ⎬ t the closed surface S conﬁguration inside the volume V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ The volume tV ⎭is usually⎩ called the control⎭ volume and the surface tS is usually called the control surface.

4.1.1 Generalized Reynolds’ transport theorem

In the previous section, for deriving Reynolds’ transport theorem we considered a material volume tV bounded by a material surface tS.Inthissection, we are going to generalize the previous derivation considering in the spatial t- configuration an arbitrary volume Ω(t) bounded by a surface σ(t) that moves with an arbitrary velocity field tw. We can define inside the t-configuration of the body asurface, B tf (txi,t)=0. (4.7)

If the surface moves with the particles instantaneously on it, we say that it is a material surface. The Lagrange criterion states (Truesdell & Toupin 1960) that the necessary and suﬃcient condition for the above-deﬁned surface to be material is that its material time derivative is zero; using Eq.(2.20b),

∂ tf ∂ tf tf˙ = + tvk =0. (4.8) ∂t ∂ txk 4.1 Reynolds’ transport theorem 89

Fig. 4.1. Body with the surface tf txi,t =0 B ¡ ¢ To prove the Lagrange criterion, let us consider the following general case, In Fig. 4.1, at a point P on tf =0we deﬁne: - The unit vector tn normal to the surface. -Thematerialvelocitytv of the particle instantaneously at P . -Thevelocitytw of the surface. The condition that the point P (not the particle instantaneously at P ) remains on tf =0when the surface moves is given by, ∂ tf ∂ tf + twk =0. (4.9a) ∂t ∂ txk From geometrical considerations,

t ∂ f tgk tn = ∂ txk (4.9b) ∂ tf ∂ tf t lm ∂ txl ∂ txm g and we can write, q tw = twptg . (4.9c) p Hence, t k ∂ tf t t t w ∂ txk wn = w n = (4.9d) · ∂ tf ∂ tf t lm ∂ txl ∂ txm g and using in the above Eq.(4.9a), we getq

∂ tf t ∂t wn = . (4.9e) − ∂ tf ∂ tf t lm ∂ txl ∂ txm g q 90 Nonlinear continua

We can also deﬁne, using the material velocity of the particle instantaneously at P , t k ∂tf t t t v ∂txk vn = v n = . (4.9f) · ∂ tf ∂ tf t lm ∂ txl ∂ txm g Using Eqs. (4.9e), (4.9f) and the ﬁrstq equality in Eq.(4.8), we obtain

∂ tf ∂ tf t ˙ t lm t t f = t l t m g ( vn wn) . (4.9g) r∂ x ∂ x − If the particle instantaneously at P remains on the surface during the motion, from obvious geometrical considerations

t t vn = wn , (4.9h) and using the above in Eq.(4.9g), we have for a material surface

tf˙ =0, (4.10) which demonstrates the Lagrange criterion. Equation (4.9h) indicates that if we consider in the spatial configuration at time t fictitious particles (Truesdell & Toupin 1960) moving with a velocity field tw,thevolumeΩ(t) and the surface σ(t) canbeconsideredmaterialand we can write, using Eq.(4.6) for any Eulerian tensor field tψ,

t Dt ∂ ψ w tψ tdV = tdV + tn tw tψ tdS. (4.11) Dt ∂t · ZΩ(t) ZΩ(t) Zσ(t)

With Dtw ( )/Dt we indicate that when taking the material time derivative, the velocity· ﬁeld tw is considered. Equation (4.11) is the expression of the generalized Reynolds’ transport theorem. An application of this theorem is presented in Example 4.5.

4.1.2 The transport theorem and discontinuity surfaces

In Fig. 4.2 we represent a body in its spatial configuration corresponding to time t. We assume that the EulerianB tensor field tψ has a jump discontinuity t t across a surface S12 inside the body and we let the material velocity field v t to be also discontinuous across S12 (Truesdell & Toupin 1960). At any point t on the discontinuity surface we define its normal ( n12) and its displacement velocity (tw), not necessarily coincident with the material velocity of the particle instantaneously at that point (tv). t Considering the region on the negative side of n12,wecandefine fictitious particles with the following velocity field:

t t v on S− • 4.1 Reynolds’ transport theorem 91

Fig. 4.2. Discontinuity surface

t t w on S12 • and using the generalized Reynolds’ transport theorem we obtain,

t Dt ∂ ψ w tψ tdV = tdV + tn tv tψ tdS Dt t t ∂t t · Z V − Z V − Z S− t t t t + n12 w ψ− dS. (4.12a) t · Z S12 Using the generalized Gauss’ theorem (Eq.(4.5)),

(tv tψ) tdV = tn tv tψ tdS t ∇ · t · Z V − Z S− t t t t + n12 v− ψ− dS (4.12b) t · Z S12 hence,

t Dt ∂ ψ w tψ tdV = + (tv tψ) tdV (4.12c) Dt t t ∂t ∇ · Z V − Z V − ∙ ¸ t t t t t ψ− n12 ( v− w) dS. − t · − Z S12 t For the region on the positive side of n12, in the same way, we get t Dt ∂ ψ w tψ tdV = + (tv tψ) tdV (4.12d) Dt t + t + ∂t ∇ · Z V Z V ∙ ¸ t + t t + t t + ψ n12 ( v w) dS. t · − Z S12 92 Nonlinear continua

The velocity field of the fictitious particles is coincident with the velocity t field of the actual particles everywhere except on S12. Therefore:

D Dt Dt tψ tdV = w tψ tdV + w tψ tdV. (4.12e) Dt t Dt t Dt t + Z V Z V − Z V From Eqs.(4.12c) to (4.12e) we get, D ∂tψ tψ tdV = + (tv tψ) tdV Dt t t ∂t ∇ · Z V Z V ∙ ¸ t t t t + [[ ψ ( vn wn)]] dS (4.13) t − Z S12 where, t t t t + t + t t t t [[ ψ ( vn wn)]] = ψ ( vn wn) ψ− ( vn− wn) , t t − t − − − vn = n12 v , t t · t wn = n w . 12 · In order to obtain a localized version of Reynolds’ transport theorem at the discontinuity surface we consider the arbitrary material volume enclosed by the dashed line in Fig. 4.3.

t t + t Fig. 4.3. Derivation of the jump discontinuity condition ν = ν ν− ∪

For the enclosed material volume, using Eq.(4.13) and the generalized Gauss’ theorem, we write: D ∂tψ tψ tdV = tdV + tn tv tψ tdS (4.14) Dt t t ∂t t · Z V Z V Z S− t t t t t t t t + n v ψ dS + [[ ψ ( vn wn)]] dS, t + · t − Z S Z S12 4.2 Mass-conservation principle 93

t + t t t t + t when dS S12 and dS− S12 ; ν 0 and ν− 0 we get from Eq.(4.14): → → → →

t t t t t t [[ ψ vn]] + [[ ψ ( vn wn)]] dS =0. (4.15) t − Z S12 ¡ ¢ Therefore, in order for the above integral equation to be valid for any arbitrary part of the discontinuity surface, we must fulﬁll

t t t t t [[ ψ vn]] + [[ ψ ( vn wn)]] = 0 (4.16a) − t at every point on S12. Equation (4.16a) is known as the jump discontinuity condition. t t t If we call = wn vn the discontinuity’s propagation speed, we can write U − t t t t [[ ψ ]] = [[ ψ vn]] . (4.16b) U The above equation is known as Kotchine’s theorem (Truesdell & Toupin 1960).

4.2 Mass-conservation principle

In Sect. 2.2, Eq.(2.6) introduced the concept of mass of a continuum body . In the study of continuum media, under the assumptions of NewtonianB mechanics, it is postulated that the mass of a continuum is conserved. Hence, D tρ tdV =0 (4.17) Dt t Z V where tρ = tρ (txi,t).

4.2.1 Eulerian (spatial) formulation of the mass-conservation principle Using in Eq.(4.17) the expression of Reynolds’ transport theorem given in Eq.(4.4) we obtain: D ∂tρ tρ tdV = + (tρ tv) tdV =0. (4.18) Dt t t ∂t ∇ · Z V Z V ∙ ¸ Since the above equation has to be fulfilled for any control volume inside the continuum, we can write for any point inside the spatial configuration: ∂tρ + (tρ tv)=0. (4.19) ∂t ∇ · The above partial differential equation is the localized spatial form of the mass-conservation principle in a Eulerian formulation anditiscalledthe continuity equation. 94 Nonlinear continua

Example 4.3. JJJJJ Using components in a general curvilinear spatial coordinate system, the continuity equation is written as

t t ∂ ρ t a ∂ ρ t t a + v + ρ v a =0. ∂t ∂txa | JJJJJ

Example 4.4. JJJJJ D tρ For an incompressible material Dt =0; hence the continuity equation is: tv =0, ∇ · or in components, t a v a =0. | JJJJJ

Example 4.5. JJJJJ

Let a fluid of density tρ = tρ (txi,t) have a velocity field tv. Let us consider in the spatial configuration a volume Ω(t) bounded by a surface σ(t) that moves with an arbitrary velocity field tw. Following (Thorpe 1962), we first calculate the fluid mass instantaneously inside the volume Ω(t): M = tρ tdV ZΩ(t) and using the expression of the generalized Reynolds’ transport theorem in Eq.(4.11), we get

t dM Dt ∂ ρ = w tρ tdV = tdV + tn tw tρ tdS dt Dt ∂t · ZΩ(t) ZΩ(t) Zσ(t) where tn is the external normal of the surface σ(t). Using Eq.(4.5) (generalized Gauss’ theorem), we get

tρ tv tdV = tn (tρ tv) tdS ∇ · · ZΩ(t) Zσ(t) ¡ ¢ and subtracting the above equation from the previous one, dM ∂tρ = + (tρ tv) tdV dt ∂t ∇ · ZΩ(t) ∙ ¸ + tρ tn (tw t v) tdS. · − Zσ(t) 4.3 Balance of momentum principle (Equilibrium) 95

Using Eq.(4.19), we see that the first integral on the r.h.s. is zero; hence, dM = tρ tn (tw t v) tdS. dt · − Zσ(t) The above equation is an integral equation of continuity for a control volume in motion in the fluid velocity field (Thorpe 1962). JJJJJ

4.2.2 Lagrangian (material) formulation of the mass conservation principle Equation (4.17) implies that,

A t t a t ◦ρ (◦x ) ◦dV = ρ ( x ,t) dV (4.20a) t Z◦V Z V t t where (◦ρ, ◦V ) correspond to the reference conﬁguration and ( ρ, V ) to the spatial conﬁguration. Using Eq.(2.31) in the r.h.s. of Eq.(4.20a) and changing variables in the expression of tρ we obtain,

A t A t ◦ρ (◦x ) ◦dV = ρ (◦x ,t) J ◦dV, (4.20b) Z◦V Z◦V hence, t t (◦ρ ρ J) ◦dV =0. (4.20c) − Z◦V Since the above equation has to be fulfilled for any control volume that we define inside the continuum, we can write for any point inside the reference configuration: t t ◦ρ = ρ J, (4.20d) and therefore, D (tρ tJ)=0. (4.20e) Dt The above equation is the localized material form of the continuity equation.

4.3 Balance of momentum principle (Equilibrium)

The principle of balance of momentum is the expression of Newton’s Second Law for continuum bodies. Quoting (Malvern 1969): “The momentum principle for a collection of particles states that the time rate of change of the total momentum of a given set of particles equals the vector sum of all the external forces acting on the particles of the set, provided Newton’s Third Law of action and reaction governs the internal forces. The continuum form of this principle is a basic postulate of continuum mechanics”. 96 Nonlinear continua

4.3.1 Eulerian (spatial) formulation of the balance of momentum principle

For a body in the t-configuration we define its momentum as, B tρ tv tdV (4.21a) t Z V the resultant of the external forces acting on the elements of mass inside the body are, from Eq.(3.2): tρ tb tdV, (4.21b) t Z V and the resultant of the external forces acting on the elements of the body’s surface are, from Eq.(3.4): tt tdS. (4.21c) t Z S Using Eqs.(4.21a-4.21c), we can state Newton’s Second Law for the body B as, D tρ tv tdV = tρ tb tdV + tt tdS. (4.22) Dt t t t Z V Z V Z S Using the condition of equivalence between external forces and Cauchy stresses inside a continuum, defined in Eq.(3.7), we get:

D tρ tv tdV = tρ tb tdV + tn tσ tdS. (4.23) Dt t t t · Z V Z V Z S Using in the above the expression of Reynolds’ transport theorem given in Eq.(4.4), we get

∂(tρ tv) + (tρ tv tv) tdV = tρ tb tdV t ∂t ∇ · t Z V ∙ ¸ Z V + tn tσ tdS.(4.24) t · Z S From Example 4.2, we obtain

(tρ tv tv)=tv (tρ tv) + tρ tv ( tv) (4.25a) ∇ · · ∇ ∇ · also, from Eq.(2.20b), we get £ ¤

D (tρ tv) ∂ (tρ tv) = + tv (tρ tv) , (4.25b) Dt ∂t · ∇ £ ¤ and, from Eq.(4.5) (Generalized Gauss’ Theorem), we get 4.3 Balance of momentum principle (Equilibrium) 97

tn tσ tdS = (tσ) tdV. (4.25c) t · t ∇ · Z S Z V Using Eqs.(4.25a-4.25c) in Eq.(4.24) we arrive at the integral form of the Eulerian formulation of the balance of momentum principle:

D (tρ tv)+tρ tv ( tv) tdV = tρ tb + tσ tdV. t Dt ∇ · t ∇ · Z V ∙ ¸ Z V £ ¤(4.26) Since the above equation has to be fulfilled for any control volume that we define inside the continuum, we can write for any point inside the spatial configuration: D (tρ tv)+tρ tv ( tv)=tρ tb + tσ (4.27a) Dt ∇ · ∇ · and using in the above the continuity equation, we have

D tv tρ = tρ tb + tσ . (4.27b) Dt ∇ · The above equation is the localized form of the balance of momentum principle in an Eulerian formulation and it is known as the equilibrium equation.

Example 4.6. JJJJJ Using Eq.(A.62b), in the general Eulerian curvilinear system txa ,weget, { } t t ab t σ = σ a g ∇ · | b hence, using Eq.(A.55b),

∂tσab tσ = + tσsb tΓ a + tσas tΓ b tg . ∇ · ∂txa sa sa b ∙ ¸ From the result in Example A.10, we can easily get

1 ∂ tg ∂ tg ∂ tg tΓ m = tgmj ij + jl li . il 2 ∂ txl ∂ txi − ∂ txj ∙ ¸ Therefore,

∂tσab 1 tσ = + tσsb tgaj + tσas tgbj ∇ · ∂txa 2 ∙ ∂ tg ∂ t¡g ∂ tg ¢ sj + ja as tg . ∂ txa ∂ txs − ∂ txj b µ ¶¸ JJJJJ 98 Nonlinear continua

Example 4.7. JJJJJ A perfect fluid is defined as a continuum in which, at every point, and for any surface, tn tσ = tγ tn , · where tγ is a scalar (no shear stresses). Since tσ is a symmetric second order tensor (to be shown in Sect. 4.4), its eigenvalues are real and its eigenvectors are orthogonal (Appendix, A.4.1). Referring the problem to the Cartesian system defined by the normalized t eigenvectors, êα we can write, t t t n = nˆα êα 3 t t t t σ = σˆββ êβ êβ . β=1 X Then, for the perfect fluid,

t t t t nˆβ σˆββ = γ nˆβ (β =1, 2, 3)(no addition on β) .

t The above set of equations is fulﬁlled only if the three eigenvalues σˆββ are equal (hydrostatic stress tensor). Hence,

t t t t σ = p ˆeβ ˆeβ . It is easy to show that as the three eigenvalues of tσ are equal, the above equation is valid in any Cartesian system; hence we can write,

t t σij = pδij , where tp is the pressure. Generalizing the above to any arbitrary coordinate system tσ = tp tg . Using Eq. (A.62b) and the result in Example A.11, the equilibrium equation, Eq. (4.27b), can be written as, Dtv ∂p tρ = tρ tb + tgij tg . Dt ∂ txi j From Eq. (A.57) we identify the last term on the r.h.s. of the above equation as tp, hence we can write the equilibrium equation for a perfect fluid as, ∇ D tv tρ = tρ tb + tp. Dt ∇ The above equation is known as the Euler equation for perfect fluids.Many authors get a minus sign for the second term on the r.h.s. because they define

t t σij = pδij . − JJJJJ 4.3 Balance of momentum principle (Equilibrium) 99

Example 4.8. JJJJJ Following with the topic discussed in Example 4.5 we consider a fluid, moving with a velocity field tv, and a moving control volume, moving with a velocity field tw. In this example, following (Thorpe 1962), we are going to analyze the momentum balance inside the moving control volume. Using the generalized Reynolds’ transport theorem (Eq.(4.11)) for the fluid momentum,

t t Dt ∂( ρ v) w tρ tv tdV = tdV Dt ∂t ZΩ(t) ZΩ(t) + tρ tv (tn tw) tdS. · Zσ(t) From the generalized Gauss’ theorem (Eq.(4.5)),

(tρ tv tv) tdV = tn (tρ tv tv) tdS. ∇ · · ZΩ(t) Zσ(t) Subtracting the above equation from the previous one,

t t Dt ∂( ρ v) w tρ tv tdV = + (tρtvtv) tdV Dt ∂t ∇ · ZΩ(t) ZΩ(t) ∙ ¸ + tρ tv tn (tw t v) tdS. · − Zσ(t) £ ¤ Using the result in Example 4.2 and Eq.(4.19) (continuity equation),

t Dt ∂ v w tρ tv tdV = tρ + tv tv tdV Dt ∂t · ∇ ZΩ(t) ZΩ(t) µ ¶ + tρ tv tn (tw t v) tdS. · − Zσ(t) £ ¤ Onther.h.s.oftheaboveequation,thetermbetweenthebracketsinthe ﬁrst integral is the ﬂuid particles material acceleration. We can state, using Newton’s second law, that the external force instantaneously acting on the particles inside Ω(t) is,

tF = tρ ta tdV. ZΩ(t) Hence, D tF = w tρ tv tdV + tρt v tn tv t w tdS. Dt · − ZΩ(t) Zσ(t) £ ¡ ¢¤ JJJJJ 100 Nonlinear continua

Example 4.9. JJJJJ Let us consider the body and the particle P on its external surface. We define at P a convected coordinateB system θi with covariant base vectors tg i in the spatial configuration and g in the material configuration. The con- ◦ i vected system is definedsoastohavetg and tg in the plane tangent to tS 1 2 e at tP ; and therefore g and g define the plane tangent to S at P . ◦ 1 ◦ 2 e ◦ ◦ e e e e

Material and spatial normal vectors (Nanson’s formula)

The external unit normals at P are tg tg tn = 1 × 2 , tg tg 1 2 | e × e | and, e e ◦g ◦g n = 1 × 2 . ◦ g g ◦ 1 ◦ 2 | e × e | Also, the surface-area diﬀerentials are e e tdS tn =(tg tg )dθ1 dθ2 (A) , 1 × 2 1 2 ◦dS ◦n =(◦g ◦g )dθ dθ (B) . e 1 × e 2 e e If we deﬁne,

tt =dθ1 tg , 1 1 t 2 t t2 =dθ g , e2 t =dθ1 g , ◦ 1 ◦ 1 2 e ◦t2 =dθ ◦g , e2 e 4.3 Balance of momentum principle (Equilibrium) 101 it is obvious from the results in Sect. 2.9.1 that

t ◦t1 = T1 t ◦t2 = T2 .

We can now define an arbitrary curvilinear system txi in the spatial config- I { } uration and another one ◦x in the material configuration, with covariant base vectors tg and g {respectively.} i ◦ I tt =(tt )ktg 1 1 k tt =(tt )ktg 2 2 k t t k t 1 K T1 =(t1) ( X− ) k ◦g ◦ K t t k t 1 K T2 =(t2) ( X− ) k ◦g ◦ K t t t i n = ni g I ◦n = ◦nI ◦g where, t i t i ∂ x X I = I . ◦ ∂◦x We write Eqs.(A) and (B) using the above as,

t t t t j t k dS ni = ijk ( t1) ( t2) t j t k t 1 J t 1 K ◦dS ◦nI = ◦IJK ( t1) ( t2) ( X− ) j ( X− ) k . ◦ ◦ t 1 I Multiplying both sides of the above equation by ( X− ) i,weget ◦ t 1 I t 1 I t 1 J t 1 K t j t k ◦dS ◦nI ( X− ) i = ◦IJK ( X− ) i ( X− ) j ( X− ) k ( t1) ( t2) ◦ ◦ ◦ ◦ but, from Eqs.(A.37e) and (2.34g)

◦IJK = eIJK gAB . | ◦ | Hence, using Eq.(A.37c), we get p

t 1 I t 1 t j t k ◦dS ◦nI ( X− ) i = ◦gAB X− eijk ( t1) ( t2) ◦ | ||◦ | and again using Eq.(A.37e), we getp

t t 1 I t 1 ijk t j t k ◦dS ◦nI ( X− ) = gAB X− ( t1) ( t2) . i ◦ t ◦ | ||◦ | gab p | | Therefore, p 102 Nonlinear continua

t 1 I t t t 1 ◦gAB ◦dS ◦nI ( X− ) i = dS ni X− | t | ◦ |◦ | gab s | | and using Eq.(2.34i), we get t t t t 1 n dS = J ◦n X− ◦dS. · ◦

The above equation is called Nanson’s formula (Bathe 1996). JJJJJ

Example 4.10. JJJJJ ForanEulerianvectorta we deﬁne, using Eq.(2.76a),

t t 1 B t b A = X− b a ◦g . ◦ B h¡ ¢ i Using the generalized Gauss’ theorem (Eq.(4.5)),

ta tdV = tn ta tdS. t ∇ · t · Z V Z S In the r.h.s. integral we introduce Nanson’s formula derived in Example 4.9; hence,

t t t t 1 t a dV = J ◦n X− a ◦dS t ∇ · · ◦ · Z V Z◦S t t 1 B t b = J ◦n X− b a ◦g ◦dS · ◦ B Z◦S h¡ ¢ i t t = ◦n J A ◦dS. S · Z◦ ³ ´ Using again the generalized Gauss’ theorem,

t t t t a dV = DIV J A ◦dV. tV ∇ · V Z Z◦ ³ ´ With the notation DIV ( ) we indicate a divergence in the reference conﬁgu- ration. Using in the r.h.s.· integral Eq.(2.31), we get

t t t t J a ◦dV = DIV J A ◦dV. V ∇ · V Z◦ Z◦ ³ ´ The localized form¡ of the above¢ equation is known as the Piola Identity (Mars- den & Hughes 1983),

tJ ta = DIV tJ tA . ∇ · ¡ ¢ ³ ´ JJJJJ 4.3 Balance of momentum principle (Equilibrium) 103

Example 4.11. JJJJJ We can write the Piola Identity, derived in the above example, as:

t t b t t 1 B t c J a b = J ( X− ) c a B . | ◦ | Aftersomealgebra,weget £ ¤

t t b t t 1 B t c t t c J a b = J ( X− ) c B a + J a c . | ◦ | | Hence, £ ¤ t t 1 B J ( X− ) c B =0 ◦ | that is to say £ ¤ t t 1 DIV J X− = 0 . ◦ ¡ ¢ JJJJJ

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

In the previous section we derived, in the spatial configuration, the integral and localized forms of the balance of momentum principle (equilibrium equations). In this section we are going to derive, in the material configuration, the integral and localized forms of the equilibrium equations. We are going to refer Eq.(4.23) to volumes and surfaces definedinthe material configuration, using Eq.(4.20d) and Nanson’s formula (Example 4.9).

D t t ◦ρ t 1 t ◦ρ v ◦dV = ◦ρ b ◦dV + t ◦n X− σ ◦dS. Dt ρ ◦ ◦V ◦V ◦S · · Z Z Z (4.28a) I In the above equation, all magnitudes are written as functions of (◦x ,t). Using the generalized Gauss’ theorem together with the definition of the first Piola-Kirchhoff stress tensor, we obtain

D t t t (◦ρ v) ◦dV = ◦ρ b ◦dV + DIV ( P) ◦dV.(4.28b) Dt ◦ Z◦V Z◦V Z◦V In the above (Malvern 1969),

t t Aa t DIV ( P)= P A g ◦ ◦ | a t Aa ∂ P t Da A t Ad t i t a t = ◦ A + P ◦ΓDA + P XA Γid g . ∂ x ◦ ◦ ◦ a ∙ ◦ ¸ Equation (4.28b) is an integral form of the equilibrium equations. It is important to note that although the integrals are calculated on volumes defined 104 Nonlinear continua in the reference configuration, the equilibrium is established in the spatial configuration. The corresponding localized form is,

t D v t t ◦ρ = ◦ρ b + DIV ( P) . (4.29) Dt ◦

Example 4.12. JJJJJ From Eqs.(4.27b) and (4.29) we get,

◦ρ t t t σ = DIV ( P) . ρ ∇ · ◦ The above equation is a particular application of the Piola Identity. JJJJJ

In order to write the equilibrium equations in terms of fully material tensors we have to pull-back Eq.(4.29). For the material velocity ﬁeld:

t t t a A V = φ∗ ( v ) g . (4.30) ◦ A A“physical interpretation” of£ the pull-back¤ of the material velocity contravariant components was presented in Example 2.12. In the same way, for the material acceleration, Dtv ta = = taatg (4.31) Dt a and

A t t t a A t A A = φ∗ ( a ) = a˜ (4.32) the ta˜A are the componentsh i of the material£ acceleration¤ vector in the convected A system ◦x (convected acceleration (Simo & Marsden 1984)). For the{ external} loads per unit mass, we deﬁne

t t t a A B = φ∗ ( b ) g . (4.33) ◦ A Since DIV (t P) is an Eulerian£ vector, ¤ ◦ t t Ia A t 1 A t Ia φ∗ P I =(X− ) a P I . (4.34) ◦ | ◦ ◦ | Therefore, the pull-back£ ¡ of Eq.(4.29)¢¤ is

t t t 1 t ◦ρ A = ◦ρ B + X− DIV ( P) . (4.35) ◦ · ◦ In a Lagrangian formulation, the above equation is the localized form of the equilibrium equations. 4.4 Balance of moment of momentum principle (Equilibrium) 105 4.4 Balance of moment of momentum principle (Equilibrium)

Quoting (Malvern 1969) again: “In a collection of particles whose interactions are equal, opposite, and collinear forces, the time rate of change of the total moment of momentum for a given collection of particles is equal to the vector sum of the moments of the external forces acting on the system. In the absence of distributed couples, we postulate the same principle for a continuum”. The condition of no distributed couples (nonpolar media) was introduced in Sect. 3.2.

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

For a body in the t-conﬁguration we deﬁne its moment of momentum with respect to aB given point O as,

tρ tr tv tdV, (4.36a) t × Z V where tr = tx tx ; − ◦ and tx ; tx are the position vectors of an arbitrary point P and of the point O, respectively.◦ The resultant moment with respect to O of the external forces acting on the elements of mass inside the body is,

tρ tr tb tdV, (4.36b) t × Z V and the resultant moment with respect to O of the external forces acting on the elements of the body’s surface is,

tr tt tdS. (4.36c) t × Z S Using Eqs.(4.36a-4.36c) we can state the balance of moment of momentum principle for the continuum body : B D tρ tr tv tdV = tρ tr tb tdV + tr tt tdS. (4.37a) Dt t × t × t × Z V Z V Z S With the condition of equivalence between external forces and Cauchy stresses inside a continuum deﬁned in Eq.(3.7), we get 106 Nonlinear continua D tρ tr tv tdV = tρ tr tb tdV + tr (tn tσ) tdS. Dt tV × tV × tS × · Z Z Z (4.37b)

Example 4.13. JJJJJ The last integral on the r.h.s. of Eq.(4.37b) can be written as,

tr tn tσ tdS = tn tσ tr tdS t × · − t · × Z S Z S ¡ ¢ ¡ ¢ = tσ tr tdV − t ∇ · × Z V ¡ ¢ where we used Eq. (4.5) (Generalized Gauss’ Theorem). For any second-order tensor, td , and for any vector, tc ,wecanwrite

t t t pq t rt t t t pq t rt t t o d c = d c g g g = d c qro g g × p q × r p t pq t rt t t o = d c rqo g g . − p Also,

t t T t l t T mn t t t t lt nm t t ot c d = c d g g g = c d lmo g g × l × m n n and ¡ ¢ T t t T t lt nm t t t o c d = c d lmo g g . × n Hence, ³ ´ T td tc = tc tdT . × − × Therefore, ³ ´

t T r tn tσ tdS = tr tσT tdV. t × · t ∇ · × Z S Z V ¡ ¢ ¡ ¢ JJJJJ

Using the above result, D tρ tr tv tdV (4.38a) Dt t × Z V T = tρ tr tb tdV + tr tσT tdV. t × t ∇ · × Z V Z V ¡ ¢ 4.4 Balance of moment of momentum principle (Equilibrium) 107

Using in the above equation the expression of Reynolds’ transport theorem given in Eq.(4.4), we get

∂ (tρ tr tv)+ (tρ tv tr tv) tdV t ∂t × ∇ · × Z V ∙ ¸ T = tρ tr tb tdV + tr tσT tdV. (4.38b) t × t ∇ · × Z V Z V ¡ ¢ With the result in Example 4.2, we get

(tρ tv tr tv)=tv (tρ tr tv) +(tρ tr tv)( tv) (4.39) ∇ · × · ∇ × × ∇ · using Eqs. (4.39) and (2.20b)£ we can write Eq.(4.38b)¤ as,

D (tρ tr tv) × + tρ tr tv tv tdV t Dt × ∇ · Z V ∙ ¸ ¡ ¢¡ ¢ T = tρ tr tb tdV + tr tσT tdV t × t ∇ · × Z V Z V ¡ ¢ and therefore, Dtρ D tr tv + tρ tv + tρ tr tv tdV t × Dt ∇ · Dt × Z V ½ ∙ ¸ ¾ ¡ ¢ ¡ ¢ T ¡ ¢ = tρ tr tb tdV + tr tσT tdV. (4.40a) t × t ∇ · × Z V Z V ¡ ¢ Equation (4.19), the Eulerian continuity equation, can be written as,

Dtρ + tρ tv =0 (4.40b) Dt ∇ · hence, introducing the above into Eq.(4.40a),¡ ¢ we obtain

T tρ tr ta tdV = tρ tr tb tdV + tr tσT tdV. t × t × t ∇ · × Z V Z V Z V ¡ ¢ (4.40c) For the Eulerian formulation, the above is the integral expression of the balanceofmomentofmomentumprinciple. From it, we obtain the localized form:

T tρ tr ta = tρ tr tb + tr tσT . (4.41) × × ∇ · × ¡ ¢ 4.4.2 Symmetry of Eulerian and Lagrangian stress measures

From the Eulerian localized form of the balance of moment of momentum principle, we will ﬁrst derive the symmetry of the Cauchy stress tensor. 108 Nonlinear continua

Using an intermediate result that we got in Example 4.13, we can write the localized form of the balance of moment of momentum as,

D tv tr tρ tr tρ tb = tσ tr . (4.42) × Dt − × − ∇ · × ¡ ¢

Example 4.14. JJJJJ The divergence of the cross product between a second order tensor, tσ ,and a vector, tr ,is

∂ tσ tr = tgn tσ tr ∇ · × ∂txn · × ¡ ¢ £ ¤ = tσ tr + tgn tσ tg ∇ · × · × n ¡ ¢ h i = tr tσ + tgn tσab tg tg tg − × ∇ · · a b × n h i t ¡ t ¢ t n t ab t t t o = r σ + g σ g bno g − × ∇ · · a t h t t ti nb t o ¡ = ¢ r σ + bno σ g . − × ∇ · ¡ ¢ JJJJJ

Using the above result in Eq.(4.42), we get

t t t D v t t t t t nb t o r ρ ρ b σ = bno σ g . (4.43a) × Dt − − ∇ · − ∙ ¸ Using the Eulerian localized equilibrium equation (Eq.(4.27b)),

t t nb t o bno σ g = 0 . (4.43b)

Introducing in the above Eq. (A.37e), we can write

∂tzi te tσnb =0(o =1, 2, 3) (4.43c) ∂txj bno ¯ ¯ ¯ ¯ ¯ ¯ hence, ¯ ¯ (o =1) tσ23 tσ32 =0, − (o =2) tσ31 tσ13 =0, (4.43d) − (o =3) tσ12 tσ21 =0. − Equations (4.43d) show that, 4.5 Energy balance (First Law of Thermodynamics) 109

tσ = tσT , (4.44) that is to say, they show that the Cauchy stress tensor is symmetric. The symmetry of the Cauchy stress tensor implies the symmetry of the Kirchhoff stress tensor, the second Piola-Kirchhoff stress tensor and the stress tensor defined in Eq.(3.28). For the first Piola-Kirchhoff stress tensor, which is not symmetric, we can define the following symmetry condition (see Eq.(3.15d)):

t i t Ij t j t Ii X I P = X I P . (4.45) ◦ ◦ ◦ ◦

4.5 Energy balance (First Law of Thermodynamics)

To study the conservation of energy in our framework of Newtonian continuum mechanics, we will add to our set of variables a new one: the internal energy. Quoting (Marsden & Hughes 1983) we can state that the internal energy “represents energy stored internally in the body, which is a macroscopic reflec- tion of things like chemical binding energy, intermolecular energy and energy of molecular vibrations”. Examples of internal energy are: The energy stored in a deformed spring (elastic energy). • The energy stored in a heated body (thermal energy). • The energy stored in a bottle of oil (chemical energy). • In the next chapter, when we study the different material constitutive relations we will present some phenomenological relations between different forms of internal energy and the continuum state variables (stress, strain, temperature, etc.). In the following subsections, we will derive the Eulerian (spatial) and Lagrangian (material) formulations of the First Law of Thermodynamics (Malvern 1969).

4.5.1 Eulerian (spatial) formulation of the energy balance

For a body in the t-configuration we define tu as its internal energy per unit mass.BeingB tK the kinetic energy of defined by Eq.(3.9g), the total energy tE in the considered body, at the instantB t,is

tE = tK + tρ tu tdV. (4.46) t Z V The external forces acting on the body provide a mechanical power input t Pext. Using Eq.(3.9j), we write 110 Nonlinear continua

t t D K t t t Pext = + σ : d dV. (4.47) Dt t Z V We call “heat”(tQ) the energy that flows due to a temperature gradient. We will consider two types of heat: An outflowing heat flux through the body external surface (tq:heatflux • vector). Examples are radiative and convective heat exchanges between the body and the external medium. An internalB distributed heat source per unit mass (tr). Examples are chem- • ical reactions, phase changes, etc. The total heat input to at the instant t is, B

t t t t t t t Qinput = q n dS + ρ r dV. (4.48) − t · t Z S Z V Using the First Law of Thermodynamics, we can write

DtE = tP + tQ (4.49a) Dt ext input hence, D tρ tu tdV = tσ : td tdV tq tn tdS Dt t t − t · Z V Z V Z S + tρ tr tdV. (4.49b) t Z V Using Gauss’ theorem (Eq.(4.5)), we get

D tρ tu tdV = tσ : td tdV tq tdV Dt t t − t ∇ · Z V Z V Z V + tρ tr tdV. (4.49c) t Z V Using in the above equation the expression of Reynolds’ transport theorem given in Eq.(4.4), we get

∂ (tρ tu) + tv tρ tu tdV (4.49d) t ∂t ∇ · Z V ∙ ¸ ¡ ¢ = tσ : td tdV tq tdV + tρ tr tdV. t − t ∇ · t Z V Z V Z V Now, we introduce in the integral on the l.h.s. of the above equation the result in Example 4.2, the Eulerian continuity equation in Eq.(4.19) and the deﬁnition of material derivative in Eq.(4.49d); hence, 4.5 Energy balance (First Law of Thermodynamics) 111

Dtu tρ tdV = tσ : td tdV tq tdV t Dt t − t ∇ · Z V Z V Z V + tρ tr tdV. (4.49e) t Z V The above equation is the integral form of the Eulerian formulation for the energy conservation principle. Since the above equation has to be fulﬁlled for any control volume that we deﬁne inside the continuum, we obtain the localized form of the energy conservation principle in the Eulerian formulation,

Dtu tρ = tσ : td tq + tρ tr. (4.50) Dt − ∇ ·

Example 4.15. JJJJJ Following from the topic discussed in Examples 4.5 and 4.8, we consider a fluid moving with a velocity field tv and a moving control volume with a velocity field tw. In this example, we are going to analyze the energy balance inside the moving control volume. Using the generalized Reynolds’ transport theorem (Eq. (4.11)) for the fluid internal energy,we get

t t Dt ∂ ( ρ u) w tρ tu tdV = tdV + tρ tu tn tw tdS. Dt ∂t · ZΩ(t) ZΩ(t) Zσ(t) ¡ ¢ From the generalized Gauss’ theorem (Eq.(4.5)),

tρ tu tv tdV = tn tρ tu tv tdS. ∇ · · ZΩ(t) Zσ(t) ¡ ¢ ¡ ¢ Subtracting the above equation from the previous one,

t t Dt ∂ ( ρ u) w tρ tu tdV = + tρ tu tv tdV Dt ∂t ∇ · ZΩ(t) ZΩ(t) ∙ ¸ ¡ ¢ + tρ tu tn tw tv tdS. · − Zσ(t) ¡ ¢ Using the result in Example 4.2 and Eq.(4.19) (continuity equation),

t Dt ∂ u w tρ tu tdV = tρ + tv tu tdV Dt ∂t · ∇ ZΩ(t) ZΩ(t) ∙ ¸ ¡ ¢ + tρ tu tn tw t v tdS. · − Zσ(t) ¡ ¢ On the r.h.s. of the above equation, the term between brackets in the ﬁrst 112 Nonlinear continua integral is the material derivative of the ﬂuid internal energy. Hence, using Eq.(4.50),

tσ : td tq + tρ tr tdV − ∇ · ZΩ(t) Dt £ ¤ = w tρ tu tdV + tρ tu tn tv t w tdS. Dt t · − Z Ω Zσ(t) ¡ ¢ JJJJJ

4.5.2 Lagrangian (material) formulation of the energy balance

Let us call tU the internal energy per unit mass in the t-conﬁguration but referred to the reference conﬁguration. Obviously,

t I t t i U (◦x ,t)= u ( x ,t) . (4.51a) From Eqs.(3.10) and (3.20c),

t t t t t σ : d dV = S : ε˙ ◦dV (4.51b) t ◦ ◦ Z V Z◦V and using Eq.(2.76a), we deﬁne

t t A t 1 A t a Q =(Q ) ◦g =[(X− ) a q ] ◦g . (4.51c) A ◦ A For the heat sources per unit mass we can also deﬁne in the reference conﬁguration, t I t t i R ◦x ,t = r x ,t . (4.51d) Using the Piola Identity³ (Example´ 4.10),¡ we can¢ write

t t t t q dV = DIV J Q ◦dV. (4.51e) tV ∇ · V Z Z◦ ³ ´ Hence, introducing into Eq.(4.49e) the results in Eq. (4.51a-4.51e), we get the integral form of the Lagrangian formulation for the energy conservation principle,

t D U t t t t ◦ρ ◦dV = S : ε˙ ◦dV DIV J Q ◦dV V Dt V ◦ ◦ − V Z◦ Z◦ Z◦ ³ ´ t + ◦ρ R ◦dV. (4.52) Z◦V Since the above equation has to be fulfilled for any volume that we define in the reference configuration, we get the localized form of the energy conservation principle in the Lagrangian formulation, 4.5 Energy balance (First Law of Thermodynamics) 113

t D U t t t t t ◦ρ = S : ε˙ DIV J Q + ◦ρ R. (4.53) Dt ◦ ◦ − ³ ´

Example 4.16. JJJJJ Using Eq.(3.18), we can write an alternative localized form of the energy conservation principle,

t D U t T t t t t ◦ρ = P : X˙ DIV J Q + ◦ρ R. Dt ◦ ◦ − ³ ´ JJJJJ

5 Constitutive relations

Following (Marsden & Hughes 1983) we can state that the constitutive relations in a continuum are the functional forms that adopt the stress tensor, thefreeenergyandtheheatflow as functions of the continuum deformation and temperature. Constitutive relations can be formulated using two different methodologies: (i) Studying the phenomena that takes place on the atomic scale (deformation of the atomic lattices, movement of dislocations, etc. (Dieter 1986)). (ii) Using phenomenological mathematical models that can match laboratory observations at the macroscopic scale. The phenomenological constitutive relations are generally used in continuum mechanics and we will concentrate on this approach in this chapter. This chapter is intended as an introduction to a large number of different constitutive models; hence, the recommended literature has to be classified into different areas:

For studying the fundamentals that have to be considered for formulating • constitutive relations some reference books are: (Truesdell & Noll 1965, Malvern 1969, Marsden & Hughes 1983). For studying hyperelasticity: (Ogden 1984). • For studying plasticity: (Hill 1950, Mendelson 1968, Johnson & Mellor • 1973, Lubliner 1990, Simo & Hughes 1998, Koji´c & Bathe 2005). For studying viscoplasticity: (Perzyna 1966, Koji´c & Bathe 2005). • For studying viscoelasticity: (Pipkin 1972). • For studying damage mechanics: (Lamaitre & Chaboche 1990). • 116 Nonlinear continua 5.1 Fundamentals for formulating constitutive relations

In this section, we will discuss some principles that shall be considered when developing a constitutive relation for modeling the behavior of any material.

5.1.1 Principle of equipresence This principle has been proposed in (Truesdell & Toupin 1960) and very generally states that any independent variable that is included in the formulation of any constitutive relation for a given material has to be included in the formulation of all the other constitutive relations that are developed for the same material unless it is shown that its inclusion is either not necessary or violates some physical law. Often, the constitutive models used by scientists and engineers are derived considering a very simplified material behavior and some variables are not included in the derived relation; e.g. usually when stating the relation between heat flux and temperature the continuum deformations are not considered (Fourier’s Law). The principle of equipresence requires that these simplifications should not be made “by default” and that each of them should be specifically analyzed. These analyses will also be helpful for evaluating and understanding the limitations of the obtained results.

5.1.2 Principle of material-frame indifference This principle states that the continuum constitutive relations shall be formulated using objective physical laws (see Section 2.12). When using Cartesian coordinate systems in the spatial and material configurations, only classical objectivity is required. In more general cases, covariant formulations should be used. Quoting (Ogden 1984), we can describe classical material objectivity as: “An important assumption in continuum mechanics is that two observers in relative motion make equivalent (mathematical and physical) deductions about the macroscopic properties of a material under test. In other words, material properties are unaffected by a superposed rigid motion, and the relation between the stress and the motion has the same form for all observers”.

5.1.3 Application to the case of a continuum theory restricted to mechanical variables This theory considers some measure of the continuum deformations as the only independent variable and some measure of the continuum stresses as the only dependent variable. Since many of the physical problems usually analyzed by scientists and engineers correspond to this category, it is an important case to be considered. In (Truesdell & Noll 1965) the following principles are proposed for the speciﬁc case of a continuum theory restricted to mechanical variables: 5.1 Fundamentals for formulating constitutive relations 117

Stresses are deterministic functions of the continuum deformation history. • Local action: the stresses acting on a material particle (material point) • are only a function of the strains at the same material particle and not of the strains at neighboring particles. It is important to note that nowadays nonlocal continuum theories are used for very speciﬁc problems (Pijaudier- Cabot, Baˇzant & Tabbara 1988). Hence, we consider the principle of local action as a convenient hypothesis that, although it does not represent a physical law, provides a simpliﬁcation to the constitutive relations that agrees with physical observations for many materials.

Example 5.1. JJJJJ Let us assume that for a problem in which only mechanical variables are considered, we formulate the following constitutive relation:

tσ = tm t X ◦ ¡ ¢ where tm is a tensorial function that maps the space of invertible two-point tensors into the space of symmetric Eulerian tensors (Ogden 1984). It is important to realize that tm depends on the selected reference configuration. Since we have restricted ourselves to study a purely mechanical problem we can accept that the proposed constitutive relation fulfills the principle of equipresence. To study the objectivity of the formulation (classical objectivity) we consider in the spatial configuration two Cartesian coordinate systems: a fixed one t t zα and a moving one zα∗ . Since{ } the Cauchy stress is{ an objective} spatial tensor, from Eq. (2.101c),

t t T σ = Q(t) σ∗ Q (t) · · where Q(t) is an orthogonal tensor. And since the deformation gradient tensor is a two-point objective tensor, from Eq. (2.102c): t t X = Q(t) X∗ . ◦ · ◦ Since the above relation is valid for any orthogonal Q,inparticularitisalso valid for t R (from the polar decomposition t X = t R t U ). The observer◦ in the moving frame writes, due◦ to material◦ · ◦ objectivity,

t t t σ∗ = m X∗ ◦ and using variables measured in the stationary¡ ¢ frame,

t t t T t t t T t t σ∗ = m R R U = R σ R . ◦ · ◦ · ◦ ◦ · · ◦ ³ ´ 118 Nonlinear continua

Hence, tσ = Q(t) tm t U QT (t) . · ◦ · The principle of material-frame indiﬀ¡erence¢ imposes tm t U instead of tm t X . ◦ ◦ ¡ ¢ The requirements of determinism and local action are obviously fulﬁlled. JJJJJ ¡ ¢

Example 5.2. JJJJJ As an alternative formulation to the one presented in Example 5.1, we assume that for a problem in which only mechanical variables are considered

t PT = t N t X ◦ ◦ ◦ where t N is a tensorial function that maps¡ the¢ space of invertible two-point tensors◦ into a space of general two-point tensors. The relation between tm (in Example 5.1) and t N can be derived using Eq. (3.15d). For the fulfillment of◦ the principle of equipresence, we make the same comment as in the above example. Again, to study the objectivity of the formulation, we consider in the spatial t configuration two Cartesian systems: a fixed one zα and a moving one t { } zα∗ . Since{ } the first Piola-Kirchhoff stress tensor is an objective two-points tensor, from Eq. ( 2.102c): t T t T P = Q (t) P∗ , ◦ · ◦ where Q (t) is an orthogonal tensor. Inthesameway t t X = Q (t) X∗ . ◦ · ◦ The observer in the moving frame writes, due to material objectivity,

t T t t P∗ = N X∗ , ◦ ◦ ◦ and particularizing for Q (t)=t R¡ (from¢ the polar decomposition ◦ t X = t R t U ), ◦ ◦ · ◦

t T t t T t t t T t T P∗ = N R R U = R P . ◦ ◦ ◦ · ◦ · ◦ ◦ · ◦ ³ ´ Hence, t PT = Q(t) t N t U . ◦ · ◦ ◦ The principle of material-frame indiﬀerence¡ imposes¢ the above form for the constitutive relation. JJJJJ 5.1 Fundamentals for formulating constitutive relations 119

Example 5.3. JJJJJ Another alternative formulation for a pure mechanical problem is,

t S = t M t X ◦ ◦ ◦ where t M is a tensorial function that maps¡ the¢ space of invertible two-point tensors◦ into the space of symmetric Lagrangian tensors. The relation between tm (in Example 5.1) and t M can be derived using Eq. (3.19). For the fulfillment of the◦ principle of equipresence, we make the same comment as in Example 5.1. To study the objectivity of the formulation, as usual, we consider two Cartesian systems in the spatial configuration: a fixed one t t zα and a moving one zα∗ . {Since} the second Piola-Kirchho{ } ff stress tensor is an objective Lagrangian tensor, we have:

t t S = S∗ . ◦ ◦ From the objectivity of the deformation gradient tensor it follows that (Eq. (2.102c)),

t t X = Q (t) X∗ ◦ · ◦ where Q (t) is an orthogonal tensor. Due to material objectivity, the observer in the moving frame writes:

t t t S∗ = M X∗ ◦ ◦ ◦ and particularizing for Q (t)=t R¡ (from¢ the polar decomposition ◦ t X = t R t U ), ◦ ◦ · ◦ t t t T t t t S∗ = M R R U = S. ◦ ◦ ◦ · ◦ · ◦ ◦ Hence, ³ ´ t t t S∗ = M U , ◦ ◦ ◦ and using ¡ ¢ 1/2 t U = 2 t ε + I ◦ ◦ we get, £ ¤ t S = t M t ε . ◦ ◦ ◦ The principle of material-frame indiﬀerence¡ ¢ imposes the above form of the c constitutive relation. JJJJJ 120 Nonlinear continua 5.2 Constitutive relations in solid mechanics: purely mechanical formulations

In this section we will analyze some of the constitutive relations that are used to model the mechanical behavior of solids, neglecting the couplings with other physical phenomena. An elastic material model (also called Cauchy elastic material (Ogden 1984)) predicts a material behavior independent of the material history and time,thatistosay,stressesareunivocally determined by strains and vice versa. Of course, an elastic material model cannot be used to model: permanent deformation phenomena, damage of materials, creep effects, strain-rate effects, etc. In Chap. 3 we presented the definition of conjugate stress and strain rate measures. If we define an arbitrary stress measure T (it can be either a La- grangian, Eulerian or two-point tensor) and its conjugate strain-rate measure E˙ , we can write the stress power per unit volume as,

T Pσ = T : E˙ , (5.1a) for an elastic solid, T = T(E) , (5.1b) hence, T Pσ = T (E):E˙ . (5.1c) In general (Ogden 1984), we cannot assess on the existence of a scalar function, U(E) , such that,

U˙ = TT (E):E˙ , (5.1d) that is to say, in general Pσ is not an exact diﬀerential. When Pσ is an exact diﬀerential, we can write ( 1944) ∂U U˙ = : E˙ = TT (E):E˙ , (5.1e) ∂E and we say that the material model is hyperelastic (also called Green elastic material model (Ogden 1984)). For a hyperelastic material model, from Eq. (5.1e) taking into account that E˙ is arbitrary, ∂U TT (E)= , (5.1f) ∂E and U is called the elastic energy function per unit volume. 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 121

Example 5.4. JJJJJ For a hyperelastic material, ∂U ∂U T ij = ; T kl = ∂Eij ∂Ekl and since ∂2U ∂2U = , ∂Eij ∂Ekl ∂Ekl ∂Eij we must have ∂Tij ∂Tkl = . ∂Ekl ∂Eij JJJJJ

The inelastic mechanical behavior of some materials can be described with equations of the form,

dT = C(T , E):dE , (5.2) which are the hypoelastic material models.

5.2.1 Hyperelastic material models

For a hyperelastic material, the elastic energy in the spatial conﬁguration per unit volume of the reference conﬁguration can be written as:

t t IJ t d U = S d εIJ , (5.3a) ◦ ◦ ◦ using the second Piola-Kirchhoﬀ stress tensor and the Green-Lagrange strain tensor. If we use tU as the elastic energy in the spatial conﬁguration per unit mass, we get dm d t U =dtU dm . (5.3b) ◦ ◦ρ Hence, t 1 t IJ t d U = S d εIJ , (5.3c) ◦ρ ◦ ◦ and t t t IJ ∂ U ∂ U S = ◦ρ t =2◦ρ t . (5.3d) ◦ ∂ εIJ ∂ CIJ ◦ ◦ Using the chain rule and considering Eq. (4.51a), we can write (Marsden & Hughes 1983) t t t ∂ u ∂ U ∂ CAB t = t ◦t , (5.4a) ∂ gab ∂ CAB ∂ gab ◦ 122 Nonlinear continua from Eq. (2.93a) t t t a t b CAB = gab X A X B , (5.4b) ◦ ◦ ◦ and therefore

t t t ∂ u ∂ U t a t b t ∂ U t = t X A X B = φ ( t ) . (5.4c) ∂ gab ∂ CAB ◦ ◦ ∗ ∂ CAB ◦ ◦ Pushing-forward Eq. (5.3d) and using the above, we obtain

t t ij ∂ u τ =2◦ρ t , (5.4d) ∂ gij and using Eq. (3.12) we obtain the Doyle-Ericksen formula (Simo & Marsden 1984): t t ij t ∂ u σ =2ρ t . (5.4e) ∂ gij It is important to realize that the above is the correct relation for deriving the Cauchy stress tensor from the elastic energy per unit mass function, and that (Marsden & Hughes 1983):

t t ij t ∂ u σ =2ρ t . (5.5) 6 ∂ eij Using the result in the Example 4.16 and remembering that for a hyperelastic material the stress is only a function of the state variables, we also get

t t B ∂ U Pa = ◦ρ t a . ◦ ∂ X B ◦

5.2.2 A simple hyperelastic material model

In this section we will develop the simplest possible hyperelastic material model: the isotropic, linear hyperelastic material model. To use Eq. (5.3d), in the reference configuration we need to define the strain energy per unit volume of the reference configuration

t t t U = U( εIJ) , (5.6a) ◦ ◦ ◦ the simplest deﬁnition is:

t t t IJ t 1 t ∧ IJKL t t U = A + B εIJ + C εIJ εKL , (5.6b) ◦ ◦ ◦ ◦ ◦ 2 ◦ ◦ ◦ where t A ; t BIJ and t C∧ IJKL are constants (independent of the deformation).◦ ◦ ◦ ◦ 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 123

The value of t A is arbitrary since we are only interested in the derivatives • of t U , therefore◦ ◦ we set ◦ t A =0 . (5.6c) ◦ ◦ Using Eq. (5.1f) with Eq. (5.6b) we get, •

t IJ t IJ 1 t ∧ IJKL t ∧ KLIJ t S = B + C + C εKL . (5.6d) ◦ ◦ 2 ◦ ◦ ◦ µ ¶ Since we are not considering an initial stress/strain state, t t IJ εIJ =0 S =0and therefore, we must have ◦ ⇐⇒ ◦ t BIJ =0. (5.6e) ◦ Hence, for the simplest hyperelastic material model, we have the elastic energy per unit volume of the reference conﬁguration expressed as a quadratic form of the Green-Lagrange strain tensor;

t 1 t ∧ IJKL t t U = C εIJ εKL (5.7a) ◦ 2 ◦ ◦ ◦ and a linear stress/strain relation,

t IJ t IJKL t S = C εKL (5.7b) ◦ ◦ ◦ where (Malvern 1969), 1 t CIJKL = t C∧ IJKL + t C∧ KLIJ . (5.7c) ◦ 2 ◦ ◦ µ ¶ Doing a push-forward of the above equation to the spatial conﬁguration we get, t ij t ijkl t τ = c ekl (5.8a) where the spatial elasticity tensor is deﬁned by

t ijkl t IJKL t i t j t k t l c = C X I X J X K X L . (5.8b) ◦ ◦ ◦ ◦ ◦ Note that the obtained spatial elasticity tensor has components that are a function of the deformation (not constant). The Second Law of Thermodynamics indicates that for deforming a real (stable) material we must spend an amount of work; hence,

t U 0 (5.9) ◦ ≥ and we can only have t U =0when t ε = 0 . Therefore, t U is a positive◦ deﬁnite quadratic◦ form. Using Eqs.◦ (5.7b) and (5.8a) together with the quotient rule (Sect. A.5) we realize that t CIJKL and tcijkl are components of two fourth-order tensors, t C and tc .◦ ◦ 124 Nonlinear continua

t I Symmetries of C ◦ In Eq. (5.7b) t SIJ and t εIJ are components of symmetric tensors; hence, ◦ ◦

t CIJKL = t CJIKL (5.10a) ◦ ◦ t CIJKL = t CIJLK . (5.10b) ◦ ◦ Therefore the original 81 components of t C are reduced to only 36 inde- ◦ pendent ones. Using for this case the result in Example 5.4 we arrive at,

t CIJKL = t CKLIJ . (5.11) ◦ ◦ Therefore, we are left with only 21 independent constants. It is important to note that while Eqs. (5.10a-5.10b) apply to any material, Eq. (5.11) is only valid for a hyperelastic material.

Without introducing any symmetry inherent to a particular material model, for the description of the most general linear hyperelastic material model we have to use 21 material constants.

Now we will consider materials with inherent symmetries, that is to say, with planes of elastic symmetry.Letusﬁrst consider a material with one plane of elastic symmetry; this means that if we deﬁne two coordinate systems I I ◦x and ◦x˜ with: { } 1 { }2 ◦x and ◦x on the symmetry plane, • 3 ◦x normal to the symmetry plane, • 1 1 2 2 3 3 ◦x˜ = ◦x ; ◦x˜ = ◦x and ◦x˜ = ◦x , we must• have − t CIJKL = t C˜IJKL . (5.12) ◦ ◦ Taking into account the tensor transformations

I J K L t ˜IJKL t PQRS ∂◦x˜ ∂◦x˜ ∂◦x˜ ∂◦x˜ C = C P Q R S (5.13) ◦ ◦ ∂◦x ∂◦x ∂◦x ∂◦x it is obvious that the fulﬁllment of Eq. (5.12) imposes that the components with an odd quantity of indices “3” have to be zero. Hence, a plane of elastic symmetry reduces the material constants from 21 to 13. Now we consider a further simpliﬁcation imposed by the consideration of more elastic symmetries: the orthotropic hyperelastic material model. In this case we have to consider three mutually orthogonal planes of elastic symmetry. The intersection of the three orthotropy planes determines a Cartesian α coordinate system ◦z . { } 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 125

Besides the symmetry considerations with respect to the (◦z1 , ◦z2) plane, derived above, we have to consider a symmetry with respect to the (◦z2 , ◦z3) plane (Sokolnikoﬀ 1956). The fulﬁllment of Eqs. (5.12) and (5.13) imposes, for this second symmetry, a further 4 zero material constants; hence, an orthotropic material model has only 9 material constants. Examples of materials that are adequately described using orthotropic material models are wood, rolled steel plates, etc. It is interesting to examine further the orthotropic constitutive relation using the following arrays:

t 11 t 1111 t 1122 t 1133 t S C C C 00 0 ε11 ◦ ◦ ◦ ◦ ◦ ⎡t 22⎤ ⎡ t 2222 t 2233 ⎤ ⎡ t ⎤ S C C 000 ε22 ⎢◦ ⎥ ⎢ ◦ ◦ ⎥ ⎢ ◦ ⎥ ⎢t 33⎥ ⎢ t 3333 ⎥ ⎢ t ⎥ ⎢ S ⎥ ⎢ C 000⎥ ⎢ ε33 ⎥ ⎢◦ ⎥ = ⎢ ◦ ⎥ ⎢ ◦ ⎥ . (5.14) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢t S12⎥ ⎢ t C1212 00⎥ ⎢2 t ε ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 12⎥ ⎢◦ ⎥ ⎢ ◦ ⎥ ⎢ ◦ ⎥ ⎢t 23⎥ ⎢ t 2323 ⎥ ⎢ t ⎥ ⎢ S ⎥ ⎢ C 0 ⎥ ⎢2 ε23⎥ ⎢◦ ⎥ ⎢ ◦ ⎥ ⎢ ◦ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢t S31⎥ ⎢ sym t C3131⎥ ⎢2 t ε ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 31⎥ ⎣◦ ⎦ ⎣ ◦ ⎦ ⎣ ◦ ⎦ It is obvious from the above equation that if the orthotropy axes are coincident with the principal axes of strain, they are also the principal axes of the stress tensor (collinearity between the stress and strain tensor). When for an orthotropic material the constitutive relation is written in a coordinate system that is not coincident with the material orthotropy system, the convenient form of Eq. (5.14) is lost.

I The isotropic hyperelastic material model An important case of materials with inherent symmetry is analyzed in this section: when every plane is a plane of elastic symmetry we have an isotropic material; therefore t C is an isotropic fourth-order tensor (Aris 1962). ◦ It was shown in (Aris 1962) that the most general fourth-order isotropic tensor has Cartesian components of the form, t Cαβγδ = λδαβ δγδ + µ ( δαγ δβδ + δαδ δβγ )+τ ( δαγ δβδ δαδ δβγ ) . ◦ − (5.15) Any of the two Eqs. (5.10a-5.10b), derived from the symmetry of the stress and strain tensors, imposes the condition:

τ =0 . (5.16)

Hence, it is obvious from Eqs. (5.15) and (5.16) that in the case of isotropic materials there are only 2 independent constants; usually (Malvern 1969): 126 Nonlinear continua

E : Young’s modulus • ν : Poisson’s ratio. • For any Cartesian system we can write Eq. (5.14) as, (Bathe 1996):

t 11 ν ν t S 1 1 ν 1 ν 000 ε11 ◦ − − ◦ ⎡t 22⎤ ⎡ ν ⎤ ⎡ t ⎤ S 1 1 ν 000 ε22 ⎢◦ ⎥ ⎢ − ⎥ ⎢ ◦ ⎥ ⎢t 33⎥ ⎢ ⎥ ⎢ t ⎥ ⎢ S ⎥ E (1 ν) ⎢ 1000⎥ ⎢ ε33 ⎥ ⎢◦ ⎥ = − ⎢ ⎥ ⎢ ◦ ⎥ , ⎢ ⎥ (1+ν)(1 2ν) ⎢ ⎥ ⎢ ⎥ ⎢t S12⎥ − ⎢ 1 2ν 00⎥ ⎢2 t ε ⎥ ⎢ ⎥ ⎢ 2(1− ν) ⎥ ⎢ 12⎥ ⎢◦ ⎥ ⎢ − ⎥ ⎢ ◦ ⎥ ⎢t 23⎥ ⎢ 1 2ν ⎥ ⎢ t ⎥ ⎢ S ⎥ ⎢ 2(1− ν) 0 ⎥ ⎢2 ε23⎥ ⎢◦ ⎥ ⎢ − ⎥ ⎢ ◦ ⎥ ⎢t 31⎥ ⎢ 1 2ν ⎥ ⎢ t ⎥ ⎢ S ⎥ ⎢sym 2(1− ν) ⎥ ⎢2 ε31⎥ ⎢◦ ⎥ ⎢ ⎥ ⎢ ◦ ⎥ ⎣ ⎦ ⎣ − ⎦ ⎣ ⎦(5.17) and the stress and strain tensors are always collinear. It is important to realize that the numerical values of E and ν usually found in the engineering literature refer to the relation [Cauchy stresses/infinitesimal strains] rather than to the relation [second Piola-Kirchhoff stresses/Green-Lagrange strains] that we use in this section. We can define K , the volumetric modulus and G , the shear modulus as:

E K = (5.18a) 3(1 2ν) − E G = (5.18b) 2(1+ν) and write for the simplest hyperelastic material model that we consider in this section,

t t t 0 Sαβ = K εV δαβ +2G εαβ (5.19a) ◦ ◦ ◦ t 1 t 2 t t U = K εV + G εαβ0 εαβ0 (5.19b) ◦ 2 ◦ ◦ ◦

t where, εV is the volumetric or hydrostatic component or the Green-Lagrange ◦ t t t strain tensor, εV = εαβ δαβ ,and εαβ0 ] is the deviatoric component of ◦ ◦ t ◦ t 1 t the Green-Lagrange strain tensor, εαβ0 = εαβ 3 δαβ εV . From Eq. (5.19a-5.19b) we conclude◦ that:◦ − ◦

K 0 (5.20a) ≥ G 0 . (5.20b) ≥ 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 127

t It is important to realize that εV is not a measure of the continuum volume change. ◦ The above implies,

E 0 (5.20c) ≥ 1 ν 0.5 . (5.20d) − ≤ ≤ We must remember that the symmetries discussed in the last two Sections are directional properties and not positional properties. Even if a material has a certain elastic symmetry at each point, the properties may vary from point to point in a manner not possessing any symmetry with respect to the shape of the analyzed body (Malvern 1969). When considering an isotropic material t S and t ε are always collinear tensors. ◦ ◦

Example 5.5. JJJJJ In the above equations we expressed the constitutive tensor for orthotropic and isotropic material in Cartesian coordinates. I To express it in a general coordinate system ◦x we use: { } t ˜IJKL t αβγδ I J K L C = C eα ◦g eβ ◦g eγ ◦g eδ ◦g . ◦ ◦ · · · · Even though the material¡ is still¢¡ deﬁned by¢¡ the same number¢¡ of¢ material constants, the convenient form of Eqs. (5.14) and (5.17) is lost. JJJJJ

Example 5.6. JJJJJ A hyperelastic constitutive model cannot be formulated, in the spatial conﬁg- uration, as t ij t ijkl t σ = c ekl (A) where tc is a constant and isotropic fourth order tensor If Eq. (A) is an acceptable constitutive relation, then taking Lie derivatives on both sides of the equal sign, we get for a hyperelastic material,

ij t t ijkl t σ◦ = c dkl . In Example 5.14, we will show that the above equation cannot represent a hyperelastic material; hence, Eq. (A) cannot either. It is important to point out that in (Simo & Pister 1984) it was also demonstrated that the spatial elasticity tensor cannot be an isotropic and constant fourth-order tensor. Simo and Pister developed their demonstration without the need of going through an equivalent hypoelastic material; however, we could not reproduce the demonstration in (Simo & Pister 1984). JJJJJ 128 Nonlinear continua

5.2.3 Other simple hyperelastic material models

In Sect. 5.2.1 we have formulated a simple hyperelastic material model using: 1 t U = t ε : t C : t ε (5.21a) ◦ 2 ◦ ◦ ◦ t t ∂ U S = t◦ . (5.21b) ◦ ∂ ε ◦ We can use other conjugate stress/strain rate measures and get, using for the elastic energy per unit volume in the reference configuration quadratic forms with constant coefficients: 1 t U = t H : t C : t H (5.22a) ◦ 2 ◦ ◦ ◦ t t ∂ U Γ = ◦ . (5.22b) ∂t H ◦ As we saw in Sect. 3.3.4, Eqs. (5.22a-5.22b) can only be used with isotropic materials. Even considering the above limitation, Eqs. (5.22a-5.22b) are very useful for formulating practical constitutive relations because it was experimentally shown by Anand that for metals, the values of E and ν that are used in the usual engineering environment [Cauchy stresses/infinitesimal strains]can be accurately used in Eqs. (5.22a-5.22b) for quite large strain values (Anand 1979). Also, for the case of infinitesimal strains, 1 t U = tε : t C : tε , (5.23a) ◦ 2 ◦ t t ∂ U σ = ◦ , (5.23b) ∂ tε

t t ◦ρ ∂ U τ = ◦ , (5.23c) tρ ∂ tε µ ¶ where tε is the inﬁnitesimal strain tensor deﬁned in Example 2.26. In this t t t case since ρ ◦ρ , σ τ . The above≈ equations≈ are possible because

D tε = td . (5.24) Dt As we have seen above, in Eq. (5.5), it is not possible to use an alternative hyperelastic formulation in terms of the Almansi strain tensor. 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 129

5.2.4 Ogden hyperelastic material models

For an isotropic material we can write,

t t U = U(λ1,λ2,λ3) (5.25) ◦ ◦ t where the λi are the eigenvalues of the second order tensor, U,thatistosay, the principal stretches deﬁned in Eq. (2.58e). ◦ Since C C C λi = λi(I1 ,I2 ,I3 ) (5.26) C C C t where the values (I1 ,I2 ,I3 ) are the invariants of C deﬁned in Eqs. (2.59b- 2.59d); we can write ◦

t t C C C U = U(I1 ,I2 ,I3 ) (5.27a) ◦ ◦ C 2 2 2 I1 =(λ1) +(λ2) +(λ3) (5.27b) C 2 2 2 2 2 2 I2 =(λ2) (λ3) +(λ3) (λ1) +(λ1) (λ2) (5.27c) C 2 2 2 I3 =(λ1) (λ2) (λ3) . (5.27d) t C C C Ogdenproposedtowrite U(I1 ,I2 ,I3 ) as an inﬁnite series in powers of C C C ◦ I1 3 , I2 3 , I3 1 (Ogden 1984). Thus,− − − ¡ ¢ ¡ ¢ ¡ ¢

t ∞ C p C q C r U(λ1,λ2,λ3)= Cpqr I1 3 I2 3 I3 1 (5.28) ◦ p,q,r=0 − − − X ¡ ¢ ¡ ¢ ¡ ¢ where the coeﬃcients Cpqr are independent of the deformation. For the unstrained conﬁguration,

λ1 = λ2 = λ3 =1

C I1 =3 C I2 =3 C I3 =1. Therefore, in the unstrained conﬁguration the strain energy per unit mass is zero provided that C000 =0. Also, the unstrained conﬁguration has to be stress free; hence, ∂t U ◦ IC =IC =3;IC =1 =0 for i =1, 2, 3 . (5.29a) ∂λi | 1 2 3 The above equation leads using Eq. (5.28), to the condition,

C100 +2C010 + C001 =0. (5.29b) 130 Nonlinear continua

Example 5.7. JJJJJ C The third invariant, I3 , has an important physical interpretation. Consid- t ering the eigendirections of U ,adifferential volume ◦dV defined in the reference configuration along◦ those directions is transformed, in the spatial configuration, in a differential volume tdV :

t dV = λ1λ2λ3 ◦dV hence,

t t dV C J = = I3 . ◦dV C q That is to say, the invariant I3 describes the volume change during the continuum deformation. Obviously, for incompressible deformations,

t C J = I3 =1.

JJJJJ

Here Ogden introduces a simpliﬁcative hypothesis: the strain energy is decoupled in two parts, a deviatoric part that is independent of the volume change and a volumetric part that is only a function of the volume change (Ogden 1984). The above hypothesis is introduced in Eq. (5.28) by imposing,

t ∞ C p C q U(λ1,λ2,λ3)= Cpq0 I1 3 I2 3 ◦ p,q=0 − − X ¡ ¢ ¡ ¢ ∞ C r + C00r I3 1 . (5.30) r=1 − X ¡ ¢ We can state,

C ∞ C r g(I3 )= C00r I3 1 (5.31) r=1 − X ¡ ¢ and retain only a few terms in Eq. (5.30); in this way we obtain speciﬁc hyperelastic relations that have been successfully used for particular materials. For example, for incompressible materials:

C I3 =1 (5.32a) C g(I3 )=0 (5.32b)

t ∞ C p C q U = Cpq0 I1 3 I2 3 . (5.32c) ◦ p,q=0 − − X ¡ ¢ ¡ ¢ 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 131

Retaining only two terms in Eq. (5.32c) we obtain the Mooney-Rivlin strain • energy function that has been successfully used for rubber-like materials,

t C C U = C100 I1 3 + C010 I2 3 . (5.33) ◦ − − If C010 =0the above reduces¡ to the¢ neo-Hookean¡ ¢strain energy function • which has played an important role in the development of the theory and applications of nonlinear elasticity,

t C U = C100 I1 3 . (5.34) ◦ − The principal values of the collinear¡ Green-Lagrange¢ and second Piola- Kirchhoﬀ tensors are related by,

t t t ∂ U ∂ U ∂λj Si = ◦ = ◦ (5.35a) ◦ ∂εi ∂λj ∂εi taking into account that the εi are the eigenvalues of the Green-Lagrange strain tensor, we can write

∂λj 1 = δji (no addition on j) . (5.35b) ∂εi λj Hence, t t 1 ∂ U Si = ◦ (no addition on i) . (5.35c) ◦ λi ∂λi With a push-forward of the above, we get the eigenvalues of the Kirchhoﬀ stress tensor, t t ∂ U τ i = λi ◦ (no addition on i) . (5.35d) ∂λi Hence, the eigenvalues of the Cauchy stress tensor are,

t t t ρ ∂ U σ = λ ◦ (no addition on i) . (5.35e) i ρ i ∂λ µ ◦ ¶ i When the material is incompressible the three values of λi are not independent since they have to fulﬁll the relation,

t J 1=λ1λ2λ3 1=0. (5.36a) − − In this case, Eq. (5.35a) determines the eigenvalues of the Cauchy stress tensor except for the value of the hydrostatic pressure, P : 4 t t ∂ U σi = λi ◦ + P (no addition on i) . (5.36b) ∂λi 4 For determining the value of the hydrostatic pressure the equilibrium equations shall be used. 132 Nonlinear continua

Example 5.8. JJJJJ In the case of the biaxial tension of a square incompressible sheet (Ogden 1984),

λ1λ2λ3 =1

λ1 = λ2 = λˆ therefore, 1 λ3 = . (λˆ)2 Also, t t t σ1 = σ2 = σˆ t σ3 =0. Using the constitutive equation of the incompressible neo-Hookean material we get, t 2 2 2 U = C100(λ1 + λ2 + λ3 3) . ◦ − Equations (5.36b) give,

t 2 σˆ =2C100(λˆ) + p 4 1 0=2C100 + p. (λˆ)4 4 Hence,

t 2 1 σˆ =2C100 (λˆ) . " − (λˆ)4 #

In plane Cauchy stresses in the biaxial tension of a square incompressible sheet

JJJJJ 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 133

Example 5.9. JJJJJ For the case of an incompressible cube subjected to a uniform tension on its faces, in (Marsden & Hughes 1983) a neo-Hookean material is considered with a strain energy function of the form,

t 2 2 2 U = C100 λ1 + λ2 + λ3 3 . ◦ − £ ¤

Incompressible cube subjected to a uniform tension on its faces

Using Eqs. (5.36b), we get

t 2 σi =2C100(λi) + p (i =1, 2, 3) . 4

From the equilibrium between internal and external forces, and taking into account that due to incompressibility

λ1λ2λ3 =1 we get,

2 2C100(λ1) + p = tλ1 , 4 2 2C100(λ2) + p = tλ2 , 4 2 2C100(λ3) + p = tλ3 , 4 where t = t . | | 134 Nonlinear continua

Eliminating p gives, 4

[2C100 (λ1 + λ2) t](λ1 λ2)=0 − − [2C100 (λ2 + λ3) t](λ2 λ3)=0 − − [2C100 (λ3 + λ1) t](λ3 λ1)=0. − −

In the intuitive case

λ1 = λ2 = λ3 and the above equations are automatically fulﬁlled. However, we will explore the possibility of other solutions. If we assume • λ1 = λ2 = λ3 6 6 we must have

t =2C100 (λ1 + λ2)=2C100 (λ2 + λ3)=2C100 (λ3 + λ1) .

Hence, we must have λ1 = λ2 = λ3. This contradiction shows that the assumed solution is not possible. If we assume that two λis are equal and the third one is diﬀerent, for example:• λ1 = λ2 = λ3 , 6 we get t =2C100 (λ1 + λ3) and due to incompressibility we must have, 1 2C100 λ1 + 2 t =0. " (λ1) # −

In order to obtain λ1 = λ1(t) we must solve the equation

3 t 2 f(λ)=λ1 λ1 +1=0. − 2C100 Considering that only the positive roots are admisible, it is shown in (Marsden & Hughes 1983) that: 3 ¤ If t<3 √2 C100 = no roots λ1 > 0. 3 ⇒ ¤ If t =3√2 C100 = one root λ1 > 0. 3 ⇒ If t>3 √2 C100 = two positive roots λ1 > 0. ¤ ⇒

3 Hence, t =3√2 C100 is a bifurcation load. JJJJJ 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 135

5.2.5 Elastoplastic material model under inﬁnitesimal strains

In this section we are going to present the elastoplastic material model, considering only a purely mechanical formulation and inﬁnitesimal strains. In forthcoming sections we are going to present a thermoelastoplastic material model and we are going to extend the model to consider ﬁnite strains. The elastoplastic material model has to describe the following experimentally observed phenomena: For loads below a certain limit loading condition, established via a yield • criterion, the material behavior is elastic and can be described using the hyperelastic relations that we have discussed above. When the limit loading condition is reached there is an onset of permanent • or plastic deformations. The plastic deformations produce an evolution of the yield condition that • is described via a hardening law. When the limit loading condition is reached and then an unloading takes • place, elastic deformations are developed. The material behavior is rate independent, that is to say, the material • behavior is not a function of the loading or deformation rate. The material is stable: we have to spend mechanical work in order to • deform it.

I 1D case The above observations easily ﬁt into our experience of the 1D load/displacement curve of a steel sample under tension. In the 1D case it is obvious that (see Fig. 5.1):

Total L = Elastic L + Plastic L. (5.37a) 4 4 4 For the case of inﬁnitesimal strains, dividing the above by ◦L, we have tε = tεE + tεP (5.37b) where tε : total axial strain in the sample, tεE : elastic axial strain in the sample and tεP : plastic axial strain in the sample. During the plastic loading we can also write, following Fig. 5.2, that

ε = εE + εP . (5.37c) 4 4 4 Dividing Eq. (5.37c) by t and taking the limit for t 0 we get for thesimple1Dcase, 4 4 → tε˙ = tε˙ E + tε˙P . (5.37d) For the 3D case we generalize the above equation using Eq. (5.24) and get the additive decomposition of the strain-rate tensor, 136 Nonlinear continua

Fig. 5.1. Tensile test of a steel sample

Fig. 5.2. Plastic loading (zoom)

td = tdE + tdP . (5.38)

It is important to note that, for the moment, we are postulating the above decomposition only for the case of inﬁnitesimal strains, a very important one for standard engineering applications. In the 1D mechanical test described in Fig. 5.1 it is quite obvious that the loading condition is, tP>˙ 0 , (5.39a) and when loading: t if P< ◦Pyield 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 137

E P L = 4 ◦L (5.39b) 4 AE LP =0, (5.39c) 4 t if P > ◦Pyield

E P L = 4 ◦L (5.39d) 4 AE LP > 0 , (5.39e) 4 where A is the sample transversal area, that due to the assumption of in- ﬁnitesimal strains we consider as being constant, and E is the steel Young’s modulus. t As is shown in Fig. 5.1 when we reach a load Pyield and unload

tP<˙ 0 (5.40a)

E P L = 4 ◦L (5.40b) 4 AE LP =0. (5.40c) 4 Let us assume now that, when unloading, we reach a value PUL > 0 and then start loading again: tP>˙ 0 (5.41a) the behavior will be elastic (Eqs. (5.39b) and (5.39c)) for

t t P< Pyield (5.41b) and elastoplastic (Eqs. (5.39d) and (5.39e)) for

t t P > Pyield . (5.41c) Let us divide in Fig. 5.3 the P values by A (assumed constant during − the deformation) and the L values by ◦L . We get the diagram shown in Fig. 5.3 (Cauchy stress vs.4 inﬁ−nitesimal strain).

For a material having a softening behavior, rather than a hardening behavior, we would have the σ ε diagram shown in Fig. 5.4 . It is evident in this ﬁgure that when we go from− point “1” to point “2” : 138 Nonlinear continua

Fig. 5.3. σ-ε for a steel sample

Fig. 5.4. Softening in the σ ε relation −

σ<0 (5.42a) 4 εP > 0 (5.42b) 4 and therefore σ εP < 0 . (5.42c) 4 4 We will now show that the above stress/strain relation is incompatible with the requirement of material stability. 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 139

Let us now assume that the 1D sample is in equilibrium at “0”andan external agent makes it describe the loading-unloading path 0 1 2 3, described in Fig. 5.4. − − − The work performed by the external agent per unit volume of the sample is,

1 2 3 We.a. = σ ε˙ dt + σ ε˙ dt + σ ε˙ dt 0 1 2 Z 1 Z 2 Z 3 σ ε˙ dt σ ε˙ dt σ ε˙ dt. (5.43) − ◦ − ◦ − ◦ Z0 Z1 Z2 For a stable material it must be

We.a. > 0 . (5.44) Taking into account that in the paths 0 1 and 2 3 weareinsidethe elastic range, we can write − −

1 2 E E We.a. = (σ σ ) ε˙ dt + (σ σ ) ε˙ dt + 0 − ◦ 1 − ◦ Z 3 Z 2 (σ σ ) ε˙ E dt + (σ σ ) ε˙P dt. (5.45) − ◦ − ◦ Z2 Z1 The ﬁrst three integrals on the r.h.s. of the above equation add up to zero because they correspond to a loading-unloading elastic cycle and there is neither energy dissipated nor generated in that cycle.

2 P We.a. = (σ σ ) ε˙ dt. (5.46) − ◦ Z1 When “1”and“2”areinﬁnitesimaly close,

2 P d We.a. =dσ dε 6 0 . (5.47) The points on the softening branch are in unstable equilibrium.

The material shown in Fig. 5.4, with a softening behavior is not stable.

However, it is possible to obtain diagrams P with a softening region; for example: − 4 In the 1D test of a steel sample after necking (localization). In this case, the • hypothesis of constant area is not valid and using at every conﬁguration the actual area for constructing the diagram in Fig. 5.4 (true stress), we get a hardening behavior rather than a softening one. 140 Nonlinear continua

In the 1D test of a concrete sample. During the descending branch of the • P ∆ curve there are micro-cracks propagating inside the sample; hence, again− the consideration of a uniform stress across the transversal section is not realistic (Ottosen 1986). In what follows, we will generalize the intuitive concepts of this 1D example to a general 3D formulation.

I The general formulation For the general formulation of an elastoplastic material model we need the following three ingredients: A yield surface that in the 3D stress space describes the locus of the points • where the plastic behavior is initiated. A ﬂow rule that describes the evolution of the plastic deformations. • A hardening law that describes the evolution of the yield surface during • the plastic deformation process.

The yield surface In the stress space, for a given material we can deﬁne a yield surface,

t t t f( σ , qi i =1,n )=0 (5.48)

t t where σ is the Cauchy stress tensor and qi are internal variables to be deﬁned for every particular yield criterion. The elastic range is described by the inequality

tf<0 (5.49a) and the plastic range by the equality

tf =0. (5.49b)

Note that tf>0 is not possible in the elastoplastic framework.

Many different yield functions have been proposed over the years to phenomenologically describe the behavior of different materials (Chen 1982, Lubliner 1990). In this section, we concentrate on the yield function that is generally used to describe the behavior of metals: the von Mises yield function. In his experimental work, developed in the 1950s, Bridgman found that for metals, it can be assumed that the yield function is not affected by the confin- ing hydrostatic pressure - at least for not very extreme hydrostatic pressures (Hill 1950 and Johnson & Mellor 1973). It is important at this point to introduce the following decomposition of the Cauchy stress tensor: 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 141 1 tσ = ts + tσ : tg tg (5.50a) 3 ³ ´ which in Cartesian coordinates is written as, 1 tσ = ts + tσ tδ tδ . (5.50b) αβ αβ 3 γϕ γϕ αβ In the above equation ts is the ¡deviatoric¢ Cauchy stress tensor and 1 t t 3 σ : g is the hydrostatic or spherical component of the Cauchy stress tensorh ³ . í Using the above decomposition together with Bridgman experimental observations for the case of metals, we can write Eq. (5.48) as,

t t t f( s , qi i =1,n )=0. (5.51) Taking into account that the ﬁrst invariant or trace of the deviatoric Cauchy stress tensor is, ts : tg =0 (5.52) we can write Eq. (5.51) for an isotropic material whose behavior is symmetric in the stress space as,

t t t t f J2 , J3 , qi i =1,n =0 (5.53)

t t where J2 and J3 are¡ the second and third invariant,¢ respectively, of the deviatoric Cauchy stress tensor. By isotropic material we mean that the yield condition does not distinguish orientations predeﬁned in the material. Experimental results indicate that when the yield surface is intersected in the stress space with a plane that forms equal angles with the three principal stress axes, a good approximation for the obtained curve is a circle.

Therefore, von Mises proposed

t t t f J2 , qi i =1,n =0. (5.54)

t Hence, the von Mises yield¡ function is also known¢ as the J2 -yield function (Simo & Hughes 1998). More speciﬁcally the actual form of Eq. (5.54) can be written as,

1 (tσ )2 tf = ts tα : ts tα y =0 (5.55) 2 − − − 3 or, ¡ ¢ ¡ ¢

1/2 t 3 t t t t t f = s α : s α σy =0. (5.56) 2 − − − ∙ ¸ In the above we¡ have introduced¢ ¡ the following¢ internal variables: 142 Nonlinear continua

Fig. 5.5. Von Mises yield surface t σy : uniaxial yield stress at the t-configuration; that is to say corresponding t to a given plastic deformation. The evolution of σy is going to be described by the hardening law. tα : back-stress tensor at the t-configuration. In many metals subjected to cyclic loading it is experimentally observed that the center of the yield surface experiences a motion in the direction of the plastic flow; the back- stress tensor describes this behavior (Mc Clintock & Argon 1966). The evolution of tα is going to be described by the hardening law. We will show that tα is a traceless tensor.

In Fig. 5.5 we represent von Mises yield surface in the space of princital (over-)stresses.

The von Mises yield surface in the stress space is a circular cylinder whose axis forms equal angles with each of the coordinates axes. When, later, we discuss Drucker’s deﬁnition of stable materials, we will show that for any stable material whose behavior is modeled using any yield surface, at every time during the yield surface evolution, this surface has to remain as a convex surface in the stress space.

The flow rule As we stated at the beginning of the section, now we are focusing on the case of infinitesimal strains; hence, the starting point for describing the deformation of an elastoplastic solid during plastic loading is Eq. (5.38) which decomposes the material flow into the addition of an elastic flow plus a plastic flow. 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 143

During the material plastic ﬂow there is a plastic dissipation per unit volume of mechanical energy given by:

t = tσij tdP (5.57) D ij (Hill 1950, Lubliner 1990, Simo & Hughes 1998). If tdP = 0 (no plastic loading) t =0, otherwise the Second Law of Thermodynamics imposes that t > 0.D For many materials, such as metals,D the plastic ﬂow is developed so as to maximize the plastic dissipation. In mathematical form we can say that for deﬁning the plastic loading we seek for the maximum of t under the constraint D tf =0. (5.58a)

We deﬁne (Luenberger 1984),

t = t tλ˙ tf (5.58b) D D − where tλ˙ is a Lagrange multiplier used to enforce the constraint in Eq. (5.58a). The conditions for t to attain a maximum under the constraint in Eq. (5.58a) are, D ∂t D =0 (5.59a) ∂tσij ∂t D =0. (5.59b) ∂tλ˙ The ﬁrst of the above equations leads to ∂tf tdP = tλ˙ (5.60) ij ∂tσij and the second one to Eq. (5.58a). Note that

tf<0= tλ˙ =0(elastic behavior) , ⇒ tf =0 = tλ>˙ 0 (plastic loading). ⇒ The above equations can be summarized using the Kuhn-Tucker conditions for constrained optimization (Luenberger 1984),

tλ˙ tf =0, (5.61a) t ˙ λ > 0 , (5.61b) t f 6 0 . (5.61c) The Principle of Maximum Plastic Dissipation states that, for a given plastic strain rate tdP , among the possible stresses tσ satisfying the yield 144 Nonlinear continua criterion, the plastic dissipation attains its maximum for the actual stress tensor (Simo & Hughes 1998). The flow described by Eqs. (5.60-5.61c) is called associative or associated plastic flow and presents many important properties that we will study in what follows. A general or nonassociated plastic flow is described by

Yield function : tf =0 (5.62a) ∂tg Flow rule : tdP = tλ˙ (5.62b) ij ∂tσij where tg is called the plastic potential and is diﬀerent from the yield function.

Example 5.10. JJJJJ For a material model developed using the von Mises yield criterion in Eq. (5.55), we can write in a Cartesian coordinate system:

t t t ∂ f ∂ f ∂ sγδ t t t = t t = sαβ ααβ . ∂ σαβ ∂ sγδ ∂ σαβ − ¡ ¢ Hence, in the case of associated plastic ﬂow

t P t t t d = λ˙ sαβ ααβ . αβ − ¡ ¢ Since ts and tα are traceless tensors (see Eqs. (5.78) and (5.52)), it is obvious that, t P dαα =0, which is the condition of incompressible plastic ﬂow (see Example 4.4). The above is, of course, a direct consequence of the fact that due to Bridg- man experimental observations the yield function does not include the trace t of σ. JJJJJ

Stable materials - Drucker’s postulate The principle of maximum plastic dissipation (associated plasticity) implies that the plastic ﬂow develops in the direction of the yield surface external normal, Eq. (5.60). For an arbitrary yield function we schematize this normality rule in Fig.5.6. Note that in the case of inﬁnitesimal strains, it was shown in . Eq. (5.24) that td = tε ; which is the nomenclature used in Fig. 5.6. Let us assume an elasticplastic material that is initially at a stress state tσ , inside the elastic range. Let us also assume that an “external agent” (one that is independent of whatever has produced the current loads) slowly 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 145 applies an incremental load resulting in a stress/strain increment and then slowly removes it. For a stable material, the work performed by the external agent in the course of the cycle consisting of the application and removal of the external load is non-negative (Lubliner 1990).

Fig. 5.6. Normality rule in associated plasticity (maximum plastic dissipation)

The above is Drucker’s definition of a stable material, also known as Drucker’s postulate. In order to analyze the consequences of the above definition let us schematize the above defined cycle in Fig. 5.7.

The total work per unit volume performed during the cycle 0-1-2-0 is,

1 2 0 ij ij ij WTOTAL = σ dεij + σ dεij + σ dεij (5.63a) Z0 Z1 Z2 hence, the work performed by the external agent is,

1 2 ij t ij ij t ij We.a = σ σ dεij + σ σ dεij − − Z0 Z1 0 ¡ ¢ ¡ ¢ ij t ij + σ σ dεij . (5.63b) − Z2 ¡ ¢

For the small strains case, using Eq. (5.38) we obtain, 146 Nonlinear continua

Fig. 5.7. Drucker’s postulate (work-hardening material)

Tra jectory dεij

E 0-1 dεij

E P 1-2 dεij +dεij

E 2-0 dεij

Taking into account that

ij E σ dεij =0 (5.63c) I t ij E σ dεij =0 (5.63d) I we get,

2 ij t ij P We.a = σ σ dε . (5.63e) − ij Z1 For a stable material ¡ ¢ We.a 0 . (5.63f) ≥ The above requirement implies that the integrand in Eq. (5.63e) has to be non-negative, that is to say 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 147

Fig. 5.8. Convexity of the yield surface as a consequence of Drucker’s postulate

σij tσij dεP 0 . (5.63g) − ij ≥ We now specialize the¡ above equation¢ for the case in which point “2”is inside an inﬁnitesimal neighborhood of “1”; hence,

dσij dεP 0 . (5.63h) ij ≥ The above constrain has already been derived for the 1D case, see Eq. (5.47). Equations (5.63g) and (5.63h) are two mathematical constraints that a stable material has to fulﬁll. We can rewrite Eq. (5.63h) as,

tσ˙ ij tdP 0 . (5.64) ij ≥ The above equation indicates that the plastic strain rate cannot oppose the stress rate (Lubliner 1990). Note that:

t ij t P t ij t P While σ˙ dij > 0 indicates a work-hardening material, σ˙ dij =0 • indicates a perfectly plastic material. Drucker’s postulate excludes from the range of stable materials the possi- • bility of strain-softening materials. However, Drucker’s postulate has been obtained in the environment of stress - space plasticity, i.e. in our plasticity theory the stress is the independent variable. Since the 1960s strain - space plasticity formulations have been proposed even though their application has not been widespread yet (Lubliner 1990). 148 Nonlinear continua

As a consequence of Drucker’s postulate, we can show that for a stable material the yield surface has to be a convex surface in the stress space. It is obvious from the two cases schematized in Fig. 5.8 that the nonconvex one fails to fulﬁll Eq. (5.63g). Therefore, in the environment of stress-space plasticity, a stable material has to have a convex yield surface in the stress space.

Example 5.11. JJJJJ In this example we are going to show that a frictional material cannot be modeled using an associated plasticity formulation (Baˇzant 1979). Let us assume the simplest frictional material represented in the following ﬁgure: two rigid plates slide one on top of the other, and the sliding surface hasaCoulombfrictioncoeﬃcient µ. We formulate a yield function in the stress space using T and N (the modulus of T and N) as independent variables,

tf = T µN , − when tf<0 no sliding and when tf =0 sliding. → →

The simplest frictional material

The sliding is a nonreversible (plastic) deformation ◦ The elastic deformations are zero (rigid-plates hypothesis). In◦ the following ﬁgurewedrawtheyieldsurfaceandthediﬀerential plastic displacement obtained under the assumption of associated plasticity. 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 149

Yield function and plastic deformation predicted using an associated plasticity formulation (simplest frictional material)

The hypothesis of associated plasticity produces a nonphysical plastic displacement component in the N-direction (remember that the plates were assumed to be rigid). Therefore, to model frictional materials, it is necessary to use nonassociated plasticity formulations (Baˇzant 1979, Vermeer & de Borst 1984, Dvorkin, Cuitiño & Gioia 1989). JJJJJ

Stress - strain relation Let us consider an elastoplastic material during the process of plastic loading, to relate the stress and strain increments. For the small-strains case in a Cartesian coordinate system Eq. (5.38) can be written as

E P dεαβ =dεαβ +dεαβ , (5.65a) therefore, for a linear elastic behavior,

E E dσαβ = Cαβγδ dεγδ . (5.65b) The fourth-order tensor CE is the material elastic constitutive tensor, it is constant for linear elastic behavior and it is a function of the total elastic strains for nonlinear elastic behavior. In this book we are not going to discuss the inﬂuence of the plastic deformations on the material elastic properties (damage theory (Lamaitre & Chaboche 1990)). From Eq. (5.60), for an associated plasticity formulation,

t t t P ∂ f ∂ f ∂ sγδ dεαβ =dλ t =dλ t t (5.66a) ∂ σαβ ∂ sγδ ∂ σαβ 150 Nonlinear continua and using Eq. (5.55), the von Mises yield criterion, t ∂ f t t t = sγδ αγδ , (5.66b) ∂ sγδ − ¡ ¢ also, from Eq. (5.50b),

t ∂ sγδ 1 t = δγαδδβ δαβδγδ . (5.66c) ∂ σαβ − 3 Taking into account that ts and tα are traceless tensors,

P t t dε =dλ sαβ ααβ . (5.67a) αβ − Using the above together with Eq.¡ (5.65a) in Eq.¢ (5.65b) we get E t t dσαβ = C dεγδ dλ sγδ αγδ . (5.67b) αβγδ − − During plastic loading dλ>£0 ; hence, using¡ Eq. (5.61a),¢¤ we get the consistency condition df =0 (5.68a) which leads to, ∂tf ∂tf dσ + dεP =0. (5.68b) t αβ t P αβ ∂ σαβ ∂ εαβ Using the above, we get t t E t t sαβ ααβ C dεγδ dλ sγδ αγδ − αβγδ − − ∂tf ¡ ¢ + £ dλ ts ¡ tα =0¢¤. (5.68c) t P αβ αβ ∂ εαβ − ¡ ¢ Hence, t t E ( sαβ ααβ) C dεγδ dλ = − αβγδ . (5.68d) t t E t t ∂tf t t ( sεζ αεζ ) Cεζηϑ ( sηϑ αηϑ) ∂tεP ( sεζ αεζ ) − − − εζ − Replacing in Eq. (5.67b) dλ with the above-derived value, we get

E dσαβ = Cαβγδ (5.69) " − t t E E t t ( sνµ ανµ) Cαβνµ Cϕξγδ ( sϕξ αϕξ) − − dεγδ . t t E t t ∂tf t t ⎤ ( sρπ αρπ) Cρπητ ( sητ αητ ) ∂tεP ( sρπ αρπ) − − − ρπ − t EP ⎦ The term between brackets, C αβγδ, represents the Cartesian components of the continuum tangential elastoplastic constitutive tensor. Therefore, dσ = tCEP :dε .

It is important to note that the following symmetries are present: 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 151

tCEP = tCEP • αβγδ βαγδ tCEP = tCEP • αβγδ αβδγ tCEP = tCEP • αβγδ γδαβ It is important to realize that the last symmetry is lost in the case of non-associated plastic models (see Eq. ( 5.62b)) (Baˇzant 1979, Vermeer & de Borst 1984).

The hardening law Following Hill we can say that “The yield law for a given state of the metal must depend, in some complicated way, on the whole of the previous process of plastic deformation since the last annealing”(Hill 1950). In order to solve the equations that describe the elastoplastic deformation process, we have to describe the yield surface evolution during plastic deformation, that is to say, we have to describe the material hardening. The simplest hardening models are : Isotropic hardening model. • Kinematic hardening model. • While the first one does not describe the Bauschinger effect (Hill 1950), the second one was developed to model the basic features of this effect. In Fig. 5.9 we present a schematic description of the Bauschinger effect. T For an initially isotropic hardening material, after loading in tension to σy , when we unload, the yield stress is

T σy when we reload in tension, • C σy when we load in compression. •

Isotropic hardening We assume that in Eq. (5.55)

tα = 0 (5.70a) and t t t P σy = σy( W ) . (5.70b) Hence, the yield surface remains centered and the yield stress is a function of the irreversible work performed on the solid (work hardening). In the above equation tW P is the plastic work per unit volume performed on the material. We can state that,

DtW P t = . (5.70c) D Dt The total work per unit volume that has to be spent to deform a solid from t its unstrained configuration to a configuration defined by εγδ is, assuming infinitesimal strains, 152 Nonlinear continua

Fig. 5.9. Bauschinger eﬀect

t εγδ t W = σαβ dεαβ . (5.71a) Z0 Using Eq. (5.65a), we get

t E t P εγδ εγδ t E P W = σαβ dεαβ + σαβ dεαβ . (5.71b) Z0 Z0 Note that we have not yet deﬁned the upper limits of the above integrals. The ﬁrst integral on the r.h.s. of Eq. (5.71b) is the elastic (reversible) work per unit volume while the second integral is the plastic (irreversible) work per unit volume. For an isotropic hardening von Mises material during plastic loading

t t t P σ = σy( W ) , (5.72a)

t 3 t t where σ : equivalent stress = 2 s : s . The rate of plastic work perq unit volume is, considering that the plastic ﬂow is incompressible: t ˙ P t t P W = sαβ dαβ . (5.72b) We deﬁne, for the isotropic hardening von Mises material the equivalent plastic strain rate, P 2 td = tdP : tdP . (5.73a) 3 r 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 153

Hence, P tσ td = ts : ts tdP : tdP . (5.73b) r ³ ´ Using Eq. (5.60) we can see that¡ tdP¢ and ts are collinear tensors when isotropic von Mises plasticity is considered; hence, we can write Eq. (5.73b) as, P tσ td = ts : tdP = tW˙ P (5.73c)

P therefore tσ and td are energy conjugate. We deﬁne the equivalent plastic strain as

tεP P tεP = td dt. (5.74a) Z0 Considering Eqs. (5.72a) and (5.73c)

P t t P P dW = σy( W )dε . (5.74b)

Hence,

t t t P t ∂ σy t P ∂ σy ∂ W P ∂ σy P dσy = σy dε = dε = dε . (5.74c) ∂tW P ∂tW P ∂tεP ∂tεP Therefore, we can construct a curve,

t t t P σy = σy ε . (5.75)

“The assumption that one universal¡ stress¢ - strain curve of the form of Eq. (5.75) governs all possible combined - stress loadings of a given material is obviously a very strong one” (Malvern 1969).

Example 5.12. JJJJJ In a uniaxial tensile test, before the necking is localized, with the I-axis the loading direction and the II and III - axes orthogonal ones, we can write 2 tσ = σ , ts = σ , I ∗ I 3 ∗ t t 1 σII =0 , sII = σ∗ , −3 t t 1 σIII =0 , sIII = σ∗ . −3 Hence, σ = σ∗ . Also, due to incompressibility 154 Nonlinear continua

t P εI = εP∗ t P 1 ε = ε∗ II −2 P t P 1 ε = ε∗ . III −2 P Hence, t P ε = εP∗ . Therefore, for an isotropic hardening von Mises material, the complete universal stress - strain curve is determined with only an uniaxial test. JJJJJ

Kinematic hardening In this hardening model, which was developed to simulate the basic features of the Bauschinger eﬀect, we assume:

t σy = ◦σy = const. (5.76)

Kinematic hardening represents a translation of the yield surface in the stress space by shifting its center. This is in fairly good agreement with the Bauschinger eﬀect for those materials whose stress-strain curve in the work- hardening range can be approximated by a straight line (“linear hardening”) (Lubliner 1990). Prager, in his kinematic hardening model, assumes a linear hardening (Malvern 1969): t t P α˙ ij = c dij (5.77) where c is a constant. t P t Since dαα =0(incompressibility), the back-stress tensor α is also traceless; that is to say

tα : tg =0 . (5.78) In the nine-dimensional stress space, the yield surface is displaced in the direction of its external normal at the load point.

Example 5.13. JJJJJ We consider the 1D case of monotonic loading, also considered in the previous example. Considering the yield function in Eq. (5.55) under the constraints in Eqs. (5.76) and (5.77); and imposing that during loading

tf˙ =0, we get, 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 155

σ¯· c = . 3 · 2 ¯ε Therefore, the hardening parameter, c, is constant for linear hardening materials. JJJJJ

When using the kinematic hardening model, as also discussed for the isotropic hardening model, the complete universal stress-strain curve is determined with one axial test. It is important to realize that the results for monotonic loading are exactly coincident when considering either the isotropic or the kinematic hardening models.

5.2.6 Elastoplastic material model under ﬁnite strains

I Hypoelastic models Following what has been done for the case of elastoplastic materials under infinitesimal strains, we start this Section by assuming a hypoelastic description of the deformation process. Referring the problem to the equilibrium configuration at time t, we may again use td = tdE + tdP . (5.79) As we have shown in Example 2.22, td is an objective strain rate; therefore, considering Eq. (5.79), both tdE and tdP have to be considered as objective strain rates too. Hence, we can extend the constitutive laws used for infinitesimal strains and write

tσ◦ = tcE : tdE (5.80a)

tdP = λ· ts . (5.80b)

In the above equation tσ◦ is an objective stress rate such as the Oldroyd stress rate (see Sect. 3.4) and tcE is the spatial elasticity tensor. Equation (5.80a) is the hypoelastic expression of an elastic behavior and we expect it to conserve energy; that is to say, it has to be the hypoelastic expression of a hyperelastic material behavior. 156 Nonlinear continua

In the next example we show that only for the case of inﬁnitesimal strains, it is valid to assume that tcE is a constant and isotropic tensor.

Example 5.14. JJJJJ Following (Simo & Pister 1984) we will show that the hypoelastic equation

ij t t ijkl t σ◦ = c dkl fails to model a hyperelastic behavior when the fourth-order constitutive tensor is constant and isotropic (see also Example 5.6).

(a) Calculation of tσ˙ ij Using Eq. (3.35), we get

tσ◦ = tσ˙ tl tσ tσ tlT . − · − ·

Also, using the deﬁnition of the second Piola-Kirchhoﬀ stress tensor, we can write t t 1 t t t T σ = J − X S X , (A) ◦ · ◦ · ◦ t ◦ρ where J = tρ . Using the result in Example 5.3, we can write

t S = t S t C . ◦ ◦ ◦ From Eq. (A) ¡ ¢

t ˙ t J t t t t T t 1 t t t t t T σ˙ = t 2 X S C X + J − l X S C X − J ◦ · ◦ ◦ · ◦ · ◦ · ◦ ◦ · ◦ t t t 1 t ∂ S¡ C¢ t t T ¡ ¢ ◦ ◦ · + J − X t C X ◦ · ∂ C · ◦ · ◦ ◦¡ ¢ t 1 t t t t T t T + J − X S C X l . ◦ · ◦ ◦ · ◦ · ¡ ¢ Also, t KM t kmpq 1 t k t m t p t q ∂ S c =2J − X K X M X P X Q ◦t . (5.81) ◦ ◦ ◦ ◦ ∂ CPQ ◦ Using Example 4.1 we can write tJ· = tJ ( tv)=tJ td : tg ∇ · t ab and taking into account that g i =0, after same algebra, we³ get, using´ components, the equation (Truesdell| & Noll 1965) 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 157

t km t kmpq t σ˙ = h lpq (B.1)

thkmpq = tσkm tgpq + tgkp tσqm + tckmpq + tσkq tgmp . (B.2) −

In the derivation of the above equation we made use of the symmetry condition tckmpq = tckmqp .

(b) The Bernstein formula Following (Truesdell & Noll 1965) we now present the conditions that thkmpq in Eqs. (B.1-B.2) needs to fulﬁll in order to model a hyperelastic material behavior. For a hyperelastic material behavior, we can write, using Eq. (5.4e),

t t ij t ∂ u σ =2 ρ t , ∂ gij where tu is the elastic energy per unit mass. We can also write the following functional dependence,

t km km t a σ = ϕ XA . (C) ◦ Hence, ¡ ¢ km t km ∂ϕ t a t b σ˙ = t a l b XA . ∂ XA ◦ ◦ Therefore, using the above and Eq. (B.1), we get

km ∂ϕ t a t b t kmpq t t a l b XA = h lpq , ∂ XA ◦ ◦ and after some algebra

km ∂ϕ t km q t 1 A t p = h p X− q . (D) ∂ XA ◦ ◦ ¡ ¢ The above is written in (Truesdell & Noll 1965) as Eq. (100.24). The function ϕkm has to fulﬁll the following condition to be a potential function,

∂2ϕkm ∂2ϕkm t p t r = t r t p . (E) ∂ XA ∂ XB ∂ XB ∂ XA ◦ ◦ ◦ ◦ Using in Eq. (E) the equality (D) and considering that (Truesdell & Toupin 1960) the equation 158 Nonlinear continua

t 1 B t q B X− q X M = δM (5.82) ◦ ◦ leads to ¡ ¢ t 1 B ∂ X− q B A ◦ t 1 t 1 t p = X− p X− q (5.83) ∂¡ X A¢ − ◦ ◦ ◦ we ﬁnally get the Bernstein formula: ¡ ¢ ¡ ¢

∂thkmpq ∂thkmjl thrsjl thrspq thkmpl tgjq + thkmjq tgpl =0. (F) ∂tσrs − ∂tσrs −

(c) The constant and isotropic constitutive tensor Finally, we are going to show that when using a constant and isotropic constitutive tensor in Eq. (5.80a), the Bernstein formula is not fulﬁlled and therefore, a hyperelastic material behavior cannot be modeled. A general constant isotropic tensor is written as (Aris 1962)

tckmjl = λ tgkm tgjl + µ tgkj tgml + tgkl tgmj ¡ ¢ then we replace in Eq. (F) and we ﬁnd that the equality can be satisﬁed only for (λ + µ)=0. Since this constraint on the material properties does not correspond to a physical acceptable material model, we conclude that it is not possible to use a constant and isotropic constitutive tensor in Eq. (5.80a) to model a hyperelastic material behavior. JJJJJ

I The multiplicative decomposition of the deformation gradient The use of hypoelastic models to numerically analyze finite strains elastoplastic problems leads to many difficulties: The strain- and stress-rate measures have to be both objective and incre- • mentally objective (i.e. ∆ε and ∆σ have to be objective tensors). As we have shown in Sect. 3.4 the Jaumann stress-rate, which is an objective stress-rate measure, is not suitable for producing an objective incremental stress measure. The limitation discussed in Example 5.14 to represent an elastic behavior • using a hypoelastic constitutive model. An alternative kinematic formulation to numerically analyze finite-strain elastoplastic problems can be found in (Lee & Liu 1967, Lee 1969). Lee’s kinematic formulation: the multiplicative decomposition of the deformation gradient, is based on a micromechanical model of single-crystal metal plasticity (Simo & Hughes 1998). 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 159

Lee’s multiplicative decomposition of the deformation gradient has been the basis of the hyperelastic model for finite-strain elastoplasticity developed by Simo and Ortiz (Simo & Ortiz 1985, Simo 1988, Simo & Hughes 1998). We will now present an “intuitive description” of Lee’s multiplicative decomposition of the deformation gradient (Lubliner 1990). Let us consider a reference configuration that we assume to be unstressed and unstrained and the spatial configuration corresponding to a certain time “t”. In this spatial configuration, we have reversible (elastic) deformations and permanent (plastic) deformations. If we now cut the spatial configuration into hexahedric infinitesimal volumes and if we assume for each of those volumes an elastic unloading process, we obtain a stress-free configuration that is called the intermediate configuration. Obviously, the infinitesimal hexahedrals will not match together in the stress free or intermediate configuration to form a continuum because compatibility was lost when we divided the spatial continuum into infinitesimal parts. Therefore, the intermediate configuration is not a proper configuration because there is not a bijective mapping between the material particles and 3. < In Fig. 5.10 we schematize the three configurations: the reference one, the intermediate one and the spatial one. We also indicate in this figure the arbitrary coordinates defined on the reference and spatial configurations and the corresponding deformation gradients.

From this ﬁgure

t X = t XE t XP . (5.84) ◦ ◦ · ◦ Notes: Since the intermediate configuration, due to its lack of compatibility is not • aproperconfiguration, the tensors t XE and t XP cannot be calculated using the definition of a deformation◦ gradient tensor◦ (Eq. (2.23)). The mapping represented by t XE is purely elastic and the stresses in the • spatial configuration are developed◦ during this mapping.

The velocity gradient is deﬁned in the spatial conﬁguration and the following relation holds t t t 1 l = X˙ X− . (5.85a) ◦ · ◦ Using the multiplicative decomposition in Eq. (5.84), we obtain

E 1 P 1 1 tl = t X˙ t XE − + t XE t X˙ t XP − t XE − . (5.85b) ◦ · ◦ ◦ · ◦ · ◦ · ◦ ³ ´ ³ ´ ³ ´ 160 Nonlinear continua

Fig. 5.10. Lee’s multiplicative decomposition of the deformation gradient

P P 1 We call t¯l = t X˙ t XP − atensordeﬁned in the intermediate ◦ · ◦ conﬁguration, and we rewrite³ the´ above equation as:

1 t t E t E − t t P l = X˙ X + Xe ¯l . (5.85c) ◦ · ◦ ◦ ³ ´ ³ ´ t In the above equation we used the notation Xe (.) to indicate the push- forward of the components of the tensor (.) from◦ the intermediate configuration to the spatial configuration. At the intermediate configuration, we can make the following additive decomposition P P t¯l = td¯ + tω¯P . (5.85d) A standard hypothesis is that if the material under consideration has isotropic elastic properties, we can impose

tω¯P 0 . (5.86) ≡ The above hypothesis was used in (Weber & Anand 1990, Eterovic & Bathe 1990, Dvorkin, Pantuso & Repetto 1994). By doing the polar decomposition of the elastic and plastic deformation gradients, obtained in Eq. (5.84) via Lee’s multiplicative decomposition, we get 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 161

t XE = t RE t UE = t VE t RE (5.87a) ◦ ◦ · ◦ ◦ · ◦ t XP = t RP t UP = t VP t RP . (5.87b) ◦ ◦ · ◦ ◦ · ◦ We can also deﬁne the elastic Hencky strain tensor as

t HE =ln t UE . (5.88) ◦ ◦ ³ ´ I Stresses and the yield criterion

Considering an elastically isotropic material, the energy conjugate of t HE is (see Sect. 3.3.4), ◦ t t t Γ = Re( τ ) . (5.89) ◦ Following the work by Lee (Lee & Liu 1967, Lee 1969) we formulate the yield criterion in terms of Kirchhoﬀ stresses. Since we are interested in the modeling of elastoplastic deformation processes in metals, we use the von Mises (J2) yield criteria combined with isotropic/kinematic hardening and we have

1 2 t 3 t t t t t f = τ α : τ α σy =0. (5.90) 2 D − D − − ∙ ³ ´ ³ ´¸ In the above, tτ : deviatoric Kirchhoﬀ stress tensor; D tα : back-stress tensor (traceless); t σy: yield stress in the t-conﬁguration.

Example 5.15. JJJJJ In (Lee 1969), Lee noted that the rate of plastic work invested per unit volume of the reference conﬁguration is,

ρ tσ : tdP ◦ (A) tρ µ ¶ t where ◦ρ and ρ are the densities of the reference and spatial configurations. Since the plastic flow is incompressible, then only during the elastic deformation can the densities change. The plastic work in Eq. (A) is related to the material plastic hardening. Equa- tion (A) indicates a decrease in hardening when we increase the hydrostatic t pressure ( ρ>◦ρ) . To avoid this coupling between the elastic and plastic behavior, Lee suggests using, in the finite deformation elastoplastic formulation, the Kirchhoff stress tensor rather than the Cauchy one in the yield criterion. Obviously, for infinitesimal strains, both stress tensors are identical. JJJJJ 162 Nonlinear continua

The tensors deﬁned by

t IJ t t ij B = Re α (5.91a) ◦ t IJ t ¡ t ij¢ ΓD = Re τ D ◦ ³ ´ are traceless. t By doing a Re-pull-back of Eq. (5.90), we get ◦ 1 2 t 3 t t t t t f = Γ B : Γ B σy =0. (5.92) 2 D − D − − ∙ ³ ´ ³ ´¸ We consider the following evolution equations (Eterovic & Bathe 1990, Dvorkin, Pantuso & Repetto 1994):

t t P σ˙ y = βh d¯ (5.93a)

2 P tB˙ = (1 β) h td¯ . (5.93b) 3 − In the above, h = h te¯P is the hardening module, td¯P is deﬁned in (5.73a) and β [0, 1] is the hardening ratio; β =0corresponds to purely kinematic hardening∈ and¡β =1¢ corresponds to purely isotropic hardening. We also deﬁne the equivalent plastic strain as:

t te¯P = td¯P dt. (5.94) Z0

I Energy dissipation We now introduce tψ as the free energy at the spatial conﬁguration per unit volume of the reference conﬁguration. Considering that the mechanical problem is uncoupled from the thermal problem, we can write (Simo & Hughes 1998)

tτ : td tψ˙ 0 . (5.95a) − > The above equation, known as the Clausius-Duhem inequality, is a re- striction imposed by the Second Law of Thermodynamics (Simo & Hughes 1998): For the elastic case the deformation is reversible and the equal sign holds. • When there are plastic deformations the process is irreversible and the • greater-than sign holds. 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 163

Therefore, using Lee’s multiplicative decomposition, we can write t P t t t ˙ Xαβ = δαβ τ : d ψ =0 (5.95b) ◦ −→ − t P t t t ˙ Xαβ = δαβ τ : d ψ>0 . (5.95c) ◦ 6 −→ − For the free energy of the purely mechanical problem, we can state the following functional relation:

t t t E t t ψ = ψ H , σy, B . (5.96a) ◦ ³ ´ Following Simo 1988, we use the following uncoupled expression for the free energy, t t t E t t t ψ = ψe H + ψp σy, B . (5.96b) ◦ t ³ ´ Considering a metal, ψe is the elastic free¡ energy¢ that we identify with t the atomic lattice deformation energy and ψp is the energy associated with atomic lattice defects (e.g. dislocations) (Lubliner 1990). From Eq. (5.85b),

tl = tlE + tlP (5.97a)

E 1 tlE = t X˙ t XE − = tdE + tωE (5.97b) ◦ · ◦ ³ ´ P 1 tlP = t XE t¯l t XE − = tdP + tωP . (5.97c) ◦ · · ◦ ³ ´ Therefore, tτ : td = tτ : tdE + tdP . (5.98) Aftersomealgebra,wecanwrite ³ ´

T T P tτ : tdP = t XE tτ t XE − : t¯l . (5.99) ◦ · · ◦ ∙³ ´ ³ ´ ¸ Since we restrict this formulation to the case of isotropic elastic properties, we can write (see Sect. 3.3.4)

E tτ : tdE = tΓ : t H˙ (5.100) ◦ and the Clausius-Duhem inequality takes the form,

t T T t ∂ ψe t ˙ E t E t t E − t¯P t ˙ Γ E : H + X τ X : l ψp > 0 . − ∂t H ◦ ◦ · · ◦ − Ã ! ∙ ¸ ◦ ³ ´ ³ ´ (5.101) Since the above must also be valid for the case of an elastic deformation, we obtain 164 Nonlinear continua

∂tψ tΓ = e . (5.102) ∂t HE ◦ We deﬁne as dissipation (Simo 1988):

T T t t E t t E − t¯P t ˙ = X τ X : l ψp > 0 . (5.103) D ◦ · · ◦ − ∙³ ´ ³ ´ ¸ Considering that: For elastically isotropic materials the tensors tΓ , t HE and therefore • ◦ t UE are collinear. ◦The contraction of a symmetric tensor with a skew-symmetric one is zero. • We get, P t = tΓ : td¯ tψ˙ 0 . (5.104) D − p > We search for the value of tΓ that maximizes the dissipation under the t unilateral constraint f 6 0 . For the case tf<0 (elastic loading or unloading)

P td¯ = 0 . (5.105)

For the case tf =0 (plastic loading) to solve the minimization problem, we use the Kuhn-Tucker conditions (Luenberger 1984) and we obtain the well-known associated ﬂow rule (see Eqs. (5.57) to (5.61c) for the case of inﬁnitesimal strains):

3 tΓ tB P 2 D td¯ = tλ˙ − (5.106a) ³ ´ 3 tΓ tB : tΓ tB 2 D − D − r ³ ´ ³ ´ where from P 2 P P t¯ε· = td¯ :td¯ . (5.106b) 3 r P we obtain tλ˙ = t¯ε· .

I The incremental formulation If we assume a conservative loading and a fixed intermediate configuration, in order to satisfy the equilibrium equations at time (load level) t + t we have to fulfill the Principle of Minimum Potential Energy (see Chap. 6):4

t+ t t+ t E δ 4 Π 4 H =0 (5.107) ◦ ◦ ³ ´ 5.2 Constitutive relations in solid mechanics: purely mechanical formulations 165

t+ t where 4 Π is the potential energy corresponding to the t + t conﬁgu- ration. ◦ 4 From Chap. 6, Eq. (6.51), we can write

t+ t t+ t t+ t 4 Π = ◦ρ 4 U ◦dV + 4 g (5.108a) ◦ Z◦V t+ t where ◦V : volume of the reference configuration; 4 U : elastic energy per t+ t unit mass stored in the (t + t)-configuration; and, 4 g : potential of the external loads acting on the (4t + t)-configuration. 4 UsingtheelasticfreeenergydefinedinEq.(5.96b),wecanwrite

t+ t t+ t t+ t 4 Π = 4 ψe ◦dV + 4 g. (5.108b) ◦ Z◦V Equation (5.107) together with the above leads to

t+ t t+ t E t+ t t+ t 4 Γ : δ 4 H ◦dV = 4 f b δu 4 dV (5.108c) V ◦ t+ tV · Z◦ ³ ´ Z 4 t+ t t+ t + 4 f s δu 4 dS. t+ t · Z 4 S In the above equation: t+ t 4 V : volume of the spatial configuration, t+ t 4 S : external surface of the spatial configuration, t+ t 4 f b : body forces per unit volume, acting on the (t + t)-configuration, t+ t 4 4 f s : surface forces acting on the (t + t)-configuration, u : displacement from t to (t + t). 4 4 Also,

t+ t E ∂ 4 H t+ t E ◦ t+ t δ 4 H = t+ t : δ 4 H . (5.109a) ◦ ∂ 4 H ◦ ◦ t+ t E ∂ 4 H For the calculation of the fourth-order tensor ◦t+ t , we follow the ∂ 4 H ◦ appendix in (Dvorkin, Pantuso & Repetto 1994) and we get

∂t HE ∂t HE ∂t CE ∂t C IJ = IJ MN RS . (5.110a) ◦t t◦ E ◦ t t◦ ∂ HKL ∂ C MN ∂ CRS ∂ HKL ◦ ◦ ◦ ◦ To calculate the derivative in the ﬁrst factor on the r.h.s. of the above equation, we use t HE =ln t CE (5.110b) ◦ ◦ q 166 Nonlinear continua

T t E being ΦI =[αI ,βI ,γI ](I =1, 2, 3) the 3 eigenvectors of C and λI its 3 eigenvalues ◦ ∂t HE ∂t HE ∂α ∂t HE ∂β IJ = IJ K + IJ K t◦ E ◦ t E ◦ t E ∂ CMN ∂αK ∂ CMN ∂βK ∂ CMN ◦ ◦ ◦ ∂t HE ∂γ ∂t HE ∂λ + IJ K + IJ K . (5.110c) ◦ t E ◦ t E ∂γK ∂ CMN ∂λK ∂ CMN ◦ ◦ To calculate the derivative in the second factor on the r.h.s. of Eq. (5.110a), we use T 1 t CE = t XP − t C t XP − , (5.110d) ◦ ◦ · ◦ · ◦ and to calculate the derivative³ in´ the third factor³ we´ use t H =ln t C , (5.110e) ◦ ◦ and proceed as in Eq. (5.110c) but usingq the eigenvectors and eigenvalues of t C. ◦ For the elastic deformation, we use t+ t E t+ t E 4 Γ = C : 4 H , (5.111) ◦ where CE is a constant and isotropic elasticity tensor (Hooke’s law). The main aspects of the finite strain elastoplastic formulation that we presented in this Section are: We use Lee’s multiplicative decomposition of the deformation gradient. • We use a hyperelastic constitutive relation for the elastic strains/stress • relation; that is to say, we relate total strains with total stresses. We describe the plastic flow using the maximum dissipation principle (as- • sociated plasticity). We use the total Hencky strain as our strain measure. The reason for • this choice is that according to the experimental data reported in (Anand 1979), “the classical strain energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain energy function is replaced by the Hencky or logarithmic measure of finite strain”. From a numerical viewpoint, an additional advantage in using the Hencky strain measure is that the first invariant of the logarithmic strain tensor is the logarithmic volume strain; therefore many techniques developed for handling incompressibility in the infinitesimal strain problem can be carried over for the finite strain problem. We discussed the above formulation, the “total Lagrangian - Hencky formulation”anditsfinite element implementation in (Dvorkin 1995a 1995b 1995c, Dvorkin & Assanelli 2000, Dvorkin, Pantuso & Repetto 1992 1993 \1994 1995).\ \ \ \ 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 167 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations

5.3.1 The isotropic thermoelastic constitutive model

Considering a hyperelastic solid under mechanical loads and thermal evolution, the two point variables that deﬁne the state of the solid at any instant are a strain measure and the temperature. Therefore, the stresses at any point in the solid are a function of the strains and temperature at that point (local action). Using, for example, the Green-Lagrange strain tensor; and tT being the temperature, we can write for any particle in the spatial conﬁguration its internal energy per unit mass (elastic energy + caloric energy) as

tU = tU t ε,t T (5.112a) ◦ and considering a reversible process (Boley¡ &¢ Weiner 1960), we can write

tη = tη t ε,t T (5.112b) ◦ t where η is the spatial entropy per unit¡ mass.¢ The principle of energy conservation (First Law of Thermodynamics) can be written as DtU 1 = t S : t ε˙ + tT t η.˙ (5.113) Dt ◦ρ ◦ ◦ We can deﬁne Helmholtz’s free energy per unit mass, (Boley & Weiner 1960, Malvern 1969) as:

tψ t ε,t T = tU t ε,t T tT tη. (5.114a) ◦ ◦ − Using the above and¡ Eq. (5.113),¢ we¡ get ¢

t t ∂ ψ 1 t IJ t ∂ ψ t t ˙ t S ε˙ IJ + t + η T =0. (5.114b) ∂ εIJ − ◦ρ ◦ ◦ ∂ T µ ◦ ¶ µ ¶ t t Hence considering the isothermal ( T˙ =0)and isometric ( ε˙ IJ =0) cases, we have ◦

t t IJ ∂ ψ S = ◦ρ t (5.114c) ◦ ∂ εIJ ◦ ∂tψ tη = . (5.114d) −∂tT In a hyperelastic material the stresses and entropy are a function of the strain/temperature value at the point and not of its history; then, the above equations are valid for any process. 168 Nonlinear continua

Since the free energy, tψ , is invariant under changes of reference frame, for an isotropic material, it can only depend on the invariants of t ε ,whichfor the t-conﬁguration, and considering the strain tensor Cartesian◦ components, are

t t I1 = εαα (5.115a) ◦ ◦ t 1 t t 1 t 2 I2 = εαβ εαβ I1 (5.115b) ◦ 2 ◦ ◦ − 2 ◦

t 1 t ¡t ¢t I3 = eαβγ eδζ εαδ εβ εγζ . (5.115c) ◦ 6 ◦ ◦ ◦ Hence, t t t t t t ψ = ψ I1, I2, I3, T . (5.116) ◦ ◦ ◦ The most general expression¡ for the free energy¢ is

t t t t t t 2 ψ = a0 + a1 I1 + a2 I2 + a3 I3 + a4 T TR + a5 T TR ◦ ◦ ◦ − − t t t t t t + a6 I1 T TR + a7 I2 T T¡R + a8 ¢I3 T¡ TR ¢ ◦ − ◦ − ◦ − t 2 + a9 I1 ¡ + ... .¢ ¡ ¢ ¡ ¢ (5.117) ◦

In the above¡ ¢ equation TR is the reference temperature, usually the temperature of the undeformed solid. Using Eq. (5.114c), we get

t t t ∂ ψ ∂ Ii Sαβ = ◦ρ t t◦ , (5.118a) ◦ ∂ Ii ∂ εαβ ◦ ◦ where

t ∂ I1 t◦ = δαβ , (5.118b) ∂ εαβ ◦ t ∂ I2 t t t◦ = εαβ I1 δαβ , (5.118c) ∂ εαβ ◦ − ◦ ◦ t ∂ I3 1 t t t◦ = eαγδ eβζ εγ εδζ . (5.118d) ∂ εαβ 6 ◦ ◦ ◦ Physical restrictions used to determine the material constants:

t εαβ =0 t ◦t = Sαβ =0 ; hence, a1 =0. • T = TR ⇒ ◦ For the above condition, we set an arbitrary value for the free energy; • hence, we set a0 =0. Doing the same consideration for the entropy, we set a4 =0. • 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 169

I Specialization for small strains and small temperature increments In this case: t t εαβ εαβ (inﬁnitesimal strain tensor) ◦ t −→ t Sαβ σαβ (Cauchy stress tensor). ◦ −→ t Using Eq. (5.114c) and (5.117) and neglecting higher-order terms in εαβ and t ( T TR) , we get, for an isotropic material (Malvern 1969), − t t t Eα t σαβ = λδαβ εγγ +2G εαβ T TR δαβ (5.119) − (1 2ν) − − ¡ ¢ E where E : Young’s modulus; ν : Poisson’s ratio; G : shear modulus, G = 2(1+ν) ; νE λ = (1 2ν)(1+ν) ;and,α : linear coeﬃcient of thermal expansion. For− an isotropic solid with constant and uniform material properties, in the absence of volumetric heat generation or consumption, the heat conduction equation is (Boley & Weiner 1960):

2 t t t k T = ◦ρ T η˙ (5.120) ∇ where k is the heat conduction coeﬃcient of the material (Fourier’s law). The above equation together with Eq. (5.114d) leads to,

2 t 2 t 2 t t ∂ ψ t ∂ ψ t k T = ◦ρ T ε˙αβ + T˙ . (5.121) ∇ − ∂tT∂tε ∂tT 2 ∙ αβ ¸ The speciﬁc heat of the material is deﬁned as:

∂tU tc = (5.122) ∂tT and after some algebra, ∂tη tc = tT . (5.123) ∂tT Using Eqs. (5.114c) and (5.114d) in (5.121), we get

t t 2t t ∂ σαβ t t ∂ η t k T = ◦ρ T ε˙ αβ + ◦ρ T T.˙ (5.124) ∇ − ∂tT ρ ∂tT µ ◦ ¶ And using Eqs. (5.119) and (5.123) in (5.124), we get

t 2t TEα t t t k T = ε˙αα + ◦ρ c T.˙ (5.125) ∇ (1 2ν) − Equations (5.119) and (5.125) show that in the most general case of thermoelastic materials the heat transfer and stress analysis problems are coupled. However, in most engineering applications the ﬁrst term on the r.h.s. of Eq. (5.125) can be neglected. 170 Nonlinear continua

In (Boley & Weiner 1960) numerical examples in aluminum and steel were considered and it was shown that the coupling is negligible when tε˙ αα 20 . (5.126) 3 α tT˙ ¿

5.3.2 A thermoelastoplastic constitutive model

In Sect. 5.2.5 we developed the constitutive relation for an elastoplastic material under infinitesimal strains; in that section we made the assumption of a purely mechanical formulation, now we shall remove that limitation incor- porating thermal effects. However, we will keep the other assumptions made in that section: infinitesimal strains, limit loading condition (yield criterion), rate-independent behavior, stable material, etc. Sincewehavetotakeintoaccountthethermalstrains,werewriteEq. (5.38) as: td = tdE + tdP + tdTH . (5.127) For the yield condition, we rewrite Eq. (5.48) as:

t t t t f σ , qi i =1,n , T =0. (5.128) Considering that we are¡ focusing on the behavior¢ of metals, we will keep on using the von Mises yield criterion and therefore in the case of isotropic hardening, we can rewrite Eq. (5.55) as:

1 tσ2 tf = ts : ts y =0 (5.129a) 2 − 3 t t t P t where σy = σy ¯ε , T . In the case of kinematic hardening we can rewrite Eq. (5.55) as (Snyder 1980): ¡ ¢ 1 tσ2 tf = ts tα : ts tα y (5.129b) 2 − − − 3 t t t t ¡ t τ ¢ ¡ t ¢ t t t P where σy = σy ( T ); αij = α˙ ij dτ;and, α˙ ij = c ( T ) dij . ◦ For the unstressed state withR no previously accumulated plastic strains (Boley & Weiner 1960) tf(0, 0,t T ) < 0 . (5.130) During plastic loading the consistency equation takes the form (Boley & Weiner 1960), ∂tf ∂tf ∂tf f˙ = s˙ij + tdP + tT˙ =0. (5.131) t ij t P ij t ∂ s ∂ εij ∂ T Following (Boley & Weiner 1960) we will study three possible cases: 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 171

(a)

∂tf ∂tf ts˙ij + tT˙ =0. (5.132a) ∂tsij ∂tT Using the above equation together with Eq. (5.131), we see that it is possible for the stress components and the temperature to change so that the point remains on the yield surface but without further plasticity development. t P Therefore this change is called neutral dij =0 . (b) ¡ ¢

∂tf ∂tf ts˙ij + tT<˙ 0 . (5.132b) ∂tsij ∂tT t P The above equation represents an unloading and therefore dij =0. (c)

∂tf ∂tf ts˙ij + tT>˙ 0 . (5.132c) ∂tsij ∂tT The above equation represents a condition of plastic loading and using t P Eq.(5.131), we get dij.

I Stress - strain - temperature relation As we did above for the case of isothermal plasticity we consider that the plastic ﬂow maximizes the plastic dissipation (Lubliner 1985). We again deﬁne the plastic dissipation via Eq. (5.57) and imposing during plastic loading the condition tf =0 we obtain, in a general curvilinear system, ∂tf ∂tf dσij tdP = tλ˙ dσij + dT . (5.133) ij ∂tσij ∂tT ∙ ¸ t ˙ Considering Eq. (5.132c) and since λ > 0, we get for plastic loading the condition t ij t P σ˙ dij > 0 , (5.134) which is the same stability condition already obtained for the isothermal case (Drucker’s postulate). Again, we obtain the convexity condition for the yield surface in the stress space. From Eq. (5.133),

∂tf ∂tf tdP tλ˙ dσij + dT =0. (5.135) ij − ∂tσij ∂tT ∙ ¸ Hence, the normality rule already developed for the isothermal case applies: 172 Nonlinear continua

∂tf tdP = tλ˙ . (5.136) ij ∂tσij

Stress - strain relations for the case of isotropic hardening Since we are considering the case of inﬁnitesimal strains, we can write in a Cartesian system,

E P TH dεαβ =dεαβ +dεαβ +dεαβ (5.137a) and t t E t E σαβ = Cαβγδ εγδ . (5.137b) We consider an isotropic linear elastic material with elastic constants function of the temperature; therefore (Snyder 1980),

∂tCE dσ = tCE dεE + αβγδ tεE dT. (5.138) αβ αβγδ γδ ∂tT γδ Hence,

t E t E P TH ∂ Cαβγδ t E dσαβ = C dεγδ dε dε + ε dT. (5.139a) αβγδ − γδ − γδ ∂tT γδ For a von Mises£ material, ¤

P t dεγδ =dλ sγδ (5.139b) and for an isotropic thermal expansion,

TH t dεγδ = α dTδγδ . (5.139c) During plastic loading df =0 and using Eq. (5.131),

t t t ∂ f ∂ f P ∂ f t dσαβ + t P d¯ε + t dT =0. (5.140) ∂ σαβ ∂ ¯ε ∂ T Developing each of the terms in the above equation, we obtain

t ∂ f t t E t t t dσαβ = sαβ [ Cαβγδ dεγδ dλ sγδ α dTδγδ ∂ σαβ − − ∂tCE ¡ ¢ + αβγδ tεE dT ] (5.141a) ∂tT γδ ∂tf 4 ∂tσ d¯εP = tσ2 y dλ (5.141b) ∂t¯εP − 9 y ∂t¯εP t t ∂ f 2 t ∂ σy dT = σy dT, (5.141c) ∂tT − 3 ∂tT therefore (Snyder 1980), 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 173

t E t t t E t ∂ Cαβγδ t E 2 t ∂ σy sαβ [ Cαβγδ (dεγδ α dTδγδ)+ ∂tT εγδ dT ] 3 σy ∂tT dT dλ = − t − . t t E t 4 t 2 ∂ σy sηθ Cηθµξ sµξ + 9 σy ∂t¯εP (5.142) Hence, we introduce the above in Eqs. (5.139a-5.139c) and we can inmedi- ately relate increments in strains/temperature with stress increments. In order to be able to evaluate the terms in Eq. (5.142) it is necessary to t t ∂ σy ∂ σy relate ∂t¯εP and ∂tT to the actual material behavior (Snyder 1980). From³ the´ data obtained³ ´ in isothermal tensile tests of virgin samples, we can develop the idealized bilinear stress-strain curves shown in Fig. 5.11.

Fig. 5.11. Stress-strain curves at diﬀerent temperatures, Ti

For a constant temperature curve, we can write

t t ◦σy σy =(◦σy) + ε (Et) (5.143a) T − E T ∙ µ ¶T ¸ tσ tε = tεP + y . (5.143b) (E)T Therefore, t t P (EEt)T σy =(◦σy)T + ε . (5.143c) (E Et) − T 174 Nonlinear continua

Using, as in the isothermal case, the concept of a universal stress-strain curve that is valid for any multiaxial stress-strain state, we can use Eq. (5.143c) for any stress - strain state, provided that tεP is replaced by t¯εP (Eq. (5.74a)). Hence, in Eq. (5.142), we have

t ∂ σy EEt t P = (5.144a) ∂ ¯ε E Et µ − ¶T t ∂ σy ∂◦σy t P ∂ EEt t = t + ¯ε t . (5.144b) ∂ T ∂ T ∂ T E Et µ ¶T ∙ µ − ¶¸T Hence, we can rewrite Eq. (5.142) as:

t E t t E t ∂ Cαβγδ t E sαβ C (dεγδ α dTδγδ)+ t ε dT αβγδ − ∂ T γδ dλ = ∙ ¸ t t E t 4 t 2 EEt sηθ C sµξ + σ ηθµξ 9 y E Et − T

2 t ∂◦σy t P ∂ EE³t ´ σy t + ¯ε t dT 3 ∂ T ∂ T E Et T − T . (5.145) − t h³t E ´ t ³4 t 2³ EEt ´´ i sηθ C sµξ + σ ηθµξ 9 y E Et − T ³ ´ In the above equation, we consider a linear isotropic elastic model; therefore using Eqs. (5.15) and (5.16), we get

t E Cαβγδ =(λ)T δαβδγδ +(G)T (δαγ δβδ + δαδδβγ) (5.146) t and taking into account that sαα =0,weget t t E t sαβ Cαβγδ =2 (G)T sγδ . (5.147) Taking into account that (Snyder 1980)

t E 1 ν 1 C − = δαβ δγδ + (δαγ δβδ + δαδδβγ) (5.148) αβγδ − E T 4(G)T h ¡ ¢ i ³ ´ we can show that ∂tCE 1 ∂G ts αβγδ tεE = ts tσ . (5.149) αβ ∂tT γδ G ∂tT αβ αβ ∙ µ ¶¸T Hence,

t 1 ∂G t t 2(G) sγδ dεγδ + sαβ sαβ dT dλ = T G ∂T T 4 t 2 4 t 2 EEt (G) ( σy) + σ 3 T ¡ 9 ¢y E Et − T ³ ´ 0 2 t ∂ σy t P ∂ EEt σy + ε t dT 3 ∂T ∂ T E Et T − T . (5.150) − 4h³ ´t 2 4³t 2 ³ EEt ´´ i (G) ( σy) + σ 3 T 9 y E Et − T ³ ´ 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 175

Stress - strain relations for the case of kinematic hardening We use the kinematic hardening results in Section 5.2.5 and adapt them for the case of nonisothermal processes,

t 1 t t t t 1 t 2 f = sγδ αγδ sγδ αγδ σ =0 (5.151a) 2 − − − 3 y t t ¡ t ¢¡ ¢ σy = σy T (5.151b)

t ¡ ¢ t t αγδ = α˙ γδ dt (5.151c) Z0

t t t P α˙ γδ = c dγδ (5.151d)

tc = tc tT . (5.151e)

During the plastic¡ loading¢

∂tf ∂tf ∂tf t dσγδ + t dαγδ + t dσy =0. (5.152) ∂ σγδ ∂ αγδ ∂ σy Developing each of the terms in the above equation, we obtain

t ∂ f t t t dσγδ = sγδ αγδ (5.153a) ∂ σγδ − ¡ ¢ t E t E t t t ∂ Cγδϕξ t E Cγδϕξ dεϕξ dλ sϕξ αϕξ α dTδϕξ + εϕξ dT ( − − − ∂T ) £ ¡ ¢ ¤ t ∂ f t t t t t 2 t t 2 t dαγδ = sγδ αγδ c dλ sγδ αγδ = c dλ σy ∂ αγδ − − − −3 ¡ ¢ ¡ ¢ (5.153b)

t t ∂ f 2 t ∂ σy t dσy = σy t dT (5.153c) ∂ σy −3 ∂ T therefore (Snyder 1980),

t E t t t E t ∂ Cγδϕξ t E ( sγδ αγδ) C (dεϕξ α dTδϕξ)+ ε dT − γδϕξ − ∂T ϕξ dλ = ∙ ¸ t t t E t t 2 t t 2 ( sηθ αηθ) C ( sµψ αµψ)+ c σ − ηθµψ − 3 y t 2 tσ ∂ σy dT 3 y ∂tT (5.154) t t t E t t 2 t t 2 − ( sηθ αηθ) C ( sµψ αµψ)+ c σ − ηθµψ − 3 y 176 Nonlinear continua

Again, as in the case of isotropic hardening, we relate the above expression to the actual material behavior using the information contained in the isothermal uniaxial stress-strain curves.

For an isothermal loading in a bi-linear material, we can use the result in Example 5.13 and obtain, 2 EE tc(T )= t . (5.155) 3 E Et µ − ¶T

5.4 Viscoplasticity

In Sects. 5.2 and 5.3, we discussed constitutive relations that have a common feature: the response of the solids is instantaneous;thatistosay,whenaload is applied, either a mechanical or a thermal load, the solid instantaneously develops the corresponding displacements and strains. We know, from our experience, that this is not the case in many situations; e.g. a metallic structure under elevated temperature increases its deformation with time; a concrete structure in the ﬁrst few months after it has been cast increases its deformation with time, etc. There is also an important experimental observation related to the response of materials, in particular metals, to rapid loads: the apparent yield stress increases with the deformation velocity. In the previous sections, when considering instantaneous plasticity, we represented the strain hardening of metals with equations of the form:

σy = σy(ε, T ) . (5.156) To take into account the above commented experimental observation, the yield stress has to present the following functional dependence (Backofen 1972):

σy = σy(ε, ε,˙ T ) . (5.157) We can say that the strain-rate effect shown in Eq.(5.157) is a viscous effect. There are basically two ways in which a viscous effect can enter a solid’s constitutive relation: In the viscoelastic constitutive relations, the elastic part of the solid defor- • mation presents viscous effects. In this book, we are not going to discuss this kind of constitutive relations and we refer the readers to (Pipkin 1972) for a detailed discussion. In the viscoplastic constitutive relations (Perzyna 1966), the permanent • deformation presents viscous effects. The examples we discussed above are described using viscoplastic constitutive relations and also, other important problems like metal-forming processes are very well described using this constitutive theory (Zienkiewicz, Jain & Oñate 1977, Kobayashi, Oh & Altan 1989). 5.4 Viscoplasticity 177

As in the case of elastoplasticity, we can divide the total strain rate into its elastic and viscoplastic parts; hence, we get an equation equivalent to Eq. (5.38), but now for an elastoviscoplastic solid:

td = tdE + tdVP (5.158) where, tdVP is the viscoplastic strain rate tensor. In some cases, for example when modeling bulk metal-forming processes (Zienkiewicz, Jain & Oñate 1977), tdE << tdVP. Therefore, we can set tdE = 0, introducing a very important simpliﬁcation in the model without any signiﬁcant loss in accuracy; these are the rigid-viscoplastic material models.

I The yield surface

As in the case of plasticity, a yield surface is deﬁnedinthestressspacewith an equation identical to Eq. (5.48):

t t t f ( σ, qi i =1,n)=0. (5.159)

t The internal variables qi indicate that in the viscoplastic case, the yield surface is also modiﬁed in its shape and/or position by the hardening phenomenon. In the case of elastoplastic material models, we remember that Eqs. (5.49a- 5.49b) established that in the stress space every point in the solid is either inside the yield surface ( tf<0 and therefore the behavior is elastic and tdP = 0 )oron the yield surface ( tf =0and therefore the behavior is elastoplastic and permanent deformations are generated with tdP = 0). Intheviscoplastictheory,thepointcanbeeitherinside the yield6 surface ( tf<0 and therefore tdVP = 0)oroutside the yield surface ( tf>0 and in this case tdVP = 0 ). 6

I The ﬂow rule

In a Cartesian system, for viscoplastic materials, we use the following ﬂow rule (Perzyna 1966):

t t VP ∂ f t dαβ = γ t φ f . (5.160) ∂ σαβ ¡ ¢® Intheaboveequation,weusetheMacauleybracketsdeﬁned by:

a = aifa>0 (5.161a) h i a =0 if a 0 . (5.161b) h i ≤ 178 Nonlinear continua

An important difference between the flow rate for the viscoplastic constitutive model (Eq. (5.160)) and the flow rate for the plastic constitutive model (Eq. (5.60)) is that in the present case, γ the fluidity parameter is a material constant, while in the plasticity theory tλ˙ is a flow constant, derived by imposing the consistency condition during the plastic loading. Obviously, the correct value of γ and the correct expression for φ (tf) are derived from experimental observations. In what follows we will concentrate on the details of a rigid-viscoplastic relation suited for describing the behavior of metals with isotropic hardening,

1 δ 2 t t 1 t t σy φ( f)= sαβ sαβ (5.162) 2 − √3 · "µ ¶ # IntheaboveequationthetermbetweenbracketsisthevonMisesyield function. Using the definition of the second invariant of the deviatoric Cauchy stresses we get, ∂f 1 t = sαβ (5.163) ∂σ 2√tJ αβ ¯t 2 ¯ hence, using Eq.(5.160), we get ¯ ¯ γ tdVP = ts tf δ (5.164) αβ t αβ 2√ J2 · ® The above equation indicates that with the selected yield function the resulting viscoplastic flow is incompressible; a result that matches the experimental observations performed on the viscoplastic flow of metals. Using the definition of equivalent viscoplastic strain associated to the von Mises yield function, Eq. (5.73a), we have

t γ t δ ε˙ VP = f (5.165) √3 · ® Therefore, for tf 0 ≥ t δ √3 t f = ε˙VP (5.166) γ · Formulating, for a rigid-viscoplastic¡ ¢ material model, the relation among deviatoric stresses and strains as,

t t t VP sαβ =2µ dαβ (5.167) and using the above equations we get, for tf 0 ≥ 1 t δ σy + √3 tε· √3 γ VP tµ = ∙ ¸ (5.168) t √3 ε˙VP · 5.4 Viscoplasticity 179

From Eqs. (5.167) and (5.168) we see that a rigid-viscoplastic material behavesasanon-Newtonianfluid. It comes as no surprise that the solid be- haves in a “fluid way”, since we have neglected the solid elastic behavior and therefore its memory; the material memory is the main difference between the behavior of solids and fluids. In the limit, when γ Eq. (5.168) describes the behavior of a rigid- plastic material (inviscid),→∞ in this case, tσ tµ = y (5.169) t 3 ε˙ VP ·

Example 5.16. JJJJJ An important experimentally observed eﬀect, that the viscoplastic material model explains, is the increase in the apparent yield stress of metals when the strain rate is increased (Malvern 1969) (strain-rate eﬀect). Let us assume a uniaxial test in a rigid-viscoplastic bar,

σ11 = σ

σ22 = σ33 =0 · b Therefore, 2 s = σ 11 3 1 s22 = s33 = σ b −3 ·

Also, for the viscoplastic strain rates we canb write,

VP · d11 = ε 1 dVP = dVP = ε˙ 22 33 −2 · Hence, the equivalent viscoplastic strain rate is,

· εVP = ε.˙ Using Eqs. (5.167) and (5.168) together with the above we get,

1/δ √3 σ = σy + √3 ε˙ . Ã γ !

In the above equation, σby is the bar yield stress obtained with a quasistatic test and σ is the apparent yield stress obtained with a dynamic test. When γ (inviscid plasticity), the strain-rate eﬀect vanishes. →∞ Using otherb functions in Eq. (5.162) more complicated strain-rate dependences can be explained (Backofen 1972). JJJJJ 180 Nonlinear continua

In (Zienkiewicz, Jain & Oñate 1977) a ﬁnite element methodology, based on a rigid-viscoplastic constitutive relation was developed, for analyzing bulk metal-forming processes. This methodology known as the ﬂow formulation has been widely used since then for analyzing many industrial processes (Dvorkin, Cavaliere & Goldschmit 2003, Cavaliere, Goldschmit & Dvorkin 2001a 2001b, Dvorkin 2001, Dvorkin, Cavaliere & Goldschmit 1995 1997 1998, Dvorkin\ & Petöcz 1993). \ \

5.5 Newtonian ﬂuids

We define as an ideal or Newtonian fluid flow a viscous and incompressible one. The first property of a Newtonian fluid is the lack of memory: Newtonian fluids do not present an elastic behavior and they do not store elastic energy. Regarding the incompressible behavior we can write the continuity equation, using the result of Example 4.4 as,

tv =0 (5.170) ∇ · · It is important to remark that even though there are some fluids that can be considered as incompressible, most of the cases of interest in engineering practice are flows where Eq. (5.170) is valid even though the fluids are not necessarily incompressible in all situations (e.g. isothermal air flow at low Mach numbers) (Panton 1984). The constitutive relation for the Newtonian fluids can be written in the spatial configuration as,

tσ = tp tg +2µ td . (5.171)

In the above equation, tσ is the Cauchy stress tensor, tp is its first invariant also called the mechanical pressure, td is the strain-rate tensor and µ is the fluid viscosity that we assume to be constant (it is usually called “molecular viscosity”). t t t D Note that for an incompressible flow dii =0and therefore d = d . Taking into account the incompressibility constraint in Eq. (5.170), it is important to realize that the pressure cannot not be associated to its energy conjugate: the volume strain rate, because it is zero; hence, the pressure will have to be determined from the equilibrium equations on the fluid-flow domain boundaries. Therefore it is not possible to solve an incompressible fluid flow in which all the boundary conditions are imposed velocities, at least at one boundary point we need to prescribe the tractions acting on it. Many industrially important fluids, like polymers, do not obey Newton’s constitutive equation. They are generally called non-Newtonian fluids. When 5.5 Newtonian fluids 181 bulk metal forming processes are described neglecting the material elastic behavior (i.e. neglecting the material memory) the resulting constitutive equation is usually a non-Newtonian one (Zienkiewicz, Jain & Oñate 1977).

5.5.1 The no-slip condition

When solving a fluid flow usually two kinematic assumptions are made: At the interface between the fluid and the surrounding solid walls the • velocity of the fluid normal to the walls is zero. At the interface between the fluid and the surrounding solid walls the • velocity of the fluid tangential to the walls is zero. The first of the above assumptions is quite obvious when referring to non- porous walls: the fluid cannot penetrate the walls. The second of the above assumptions is not so obvious and, as a matter of fact, it has been historically the subject of much controversy; our faith in it is only pragmatic: it seems to work (Panton 1984).

6 Variational methods

In this chapter we will assume that the reader is familiar with the fundamentals of variational calculus. The topic can be studied from a number of references, among them (Fung 1965, Lanczos 1986, Segel 1987, Fung & Tong 2001). The most natural way for starting the presentation of the theory of mechanics is by accepting the Principle of Momentum Conservation as a law of Nature and then stepping forward to demonstrate the Principle of Virtual Work as a consequence of the momentum conservation; this is perhaps the most direct way for developing the mechanical concepts because the Principle of Momentum Conservation is quite intuitive to the reader with a background in basic mechanics. An alternative route for developing the theory of mechanics is by accepting the Principle of Virtual Work as a law of Nature and then stepping forward to demonstrate the Principle of Momentum Conservation. This route is perhaps not as intuitive as the ﬁrst one but equally valid from a formal point of view. However, more important than deciding which formulation is aesthetically more rewarding, an important fact for the scientist or engineer interested in solving advanced problems in mechanics is that the Principle of Virtual Work, and the other variational methods that can be derived from it, are the bases for the development of approximate solutions to problems for which analytical solutions cannot be found (Washizu 1982, Fung & Tong 2001).

6.1 The Principle of Virtual Work

We have represented in Fig. 6.1 the spatial conﬁguration of a continuum body t ; its external surface tS can be subdivided into: tB Su : on this surface the displacements are prescribed as boundary conditions, t Sσ : on this surface the external loads are prescribed as boundary conditions. 184 Nonlinear continua

Fig. 6.1. Spatial conﬁguration of a continuum

t It is important to realize that a given point can pertain to Su in one t direction and to Sσ in another direction, but at one point, the displacement and the external load corresponding to the same direction cannot be simulta- t t neously speciﬁed. Taking this into account we realize that the surfaces S, Su t and Sσ have to be deﬁned as the addition of the surfaces corresponding to each of the three space directions. Also,

t t t S = Su Sσ , t t ∪ Su Sσ = ∅ . ∩ The external forces acting on the body t in the spatial configuration are: t B t t : external loads per unit surface acting on Sσ, tb : external loads per unit mass. We refer the continuum body to a spatial Cartesian coordinate system tzα . { To} each point in the t configuration, which is an equilibrium configuration of the continuum body, we− can associate a displacement vector tu. We can also define an admissible displacements field as (Fung 1965),

tu(tzα)= tu(tzα)+δtu(tzα) . (6.1)

In the above δtu is the variation of the displacements ﬁeld, called the e virtual displacements. The virtual displacements have to satisfy the boundary t t t condition δ u 0 on Su and they are arbitrary on Sσ, (Fung 1965). ≡ 6.1 The Principle of Virtual Work 185

Assuming that when the continuum evolves from tu to tu,theexternal loads remain constant, the work performed by them is the virtual work of the t external loads ( δ Wext), e

t t t t t t t t δ Wext = b δ u ρ dV + t δ u dS. (6.2) t · t · Z V Z Sσ Using in the above Eq.(3.7), which is an equilibrium equation for the particles on the body surface, we get

t t t t t t t t δ u dS = σαβ δ uβ nα dS (6.3a) t · t Z Sσ Z Sσ ¡t t ¢t t = σαβ δ uβ nα dS (6.3b) t Z S ¡ t t ¢ t = σαβ δ uβ ,α dV. (6.3c) t Z V ¡ ¢ In the above, for deriving the last line we have used Gauss’ theorem. Hence,

t t t t t t t t t t δ u dS = σαβ,α δ uβ dV + σαβ δ( uβ,α) dV. (6.4) t · t t Z Sσ Z V Z V In the last integral we have used the equality (Fung 1965)

∂tu ∂ δ β = δtu . (6.5) ∂tz ∂tz β µ α ¶ α ¡ ¢ Using in Eq. (6.4) the momentum conservation equation (Eq.(4.27b)),

t t t t t t t t t t t δ u dS = bα δ uα ρ dV + σαβ δ uβ,α dV.(6.6) t · − t t Z Sσ Z V Z V ¡ ¢ In the derivation of the above equation we have assumed in Eq. (4.27b) Dtv that Dt =0; however, dynamic problems can also be considered by including the inertia forces among the external loads per unit mass (Crandall 1956). It is easy to show that,

t t t t σαβ δ uβ,α = σαβ δ εαβ (6.7a) t t 1 ∂ δ uα ∂ δ uβ δtε = + . (6.7b) αβ 2 ∂tz ∂tz " ¡ β ¢ ¡ α ¢#

t In the above equations the terms δ εαβ are the inﬁnitesimal strain components developed by the virtual displacements; hence, we refer to them as virtual strain components. 186 Nonlinear continua

Note that the actual strains in the t-conﬁguration are arbitrary, only the virtual strains are inﬁnitesimal.

Replacing with Eq. (6.7a) in Eq. (6.6),

tb δtu tρ tdV + tt δtu tdS = tσ : δtε tdV. (6.8) t · t · t Z V Z Sσ Z V The above equation is the mathematical statement of the Principle of Virtual Work and it states that for a continuum body in equilibrium, the virtual work of the external loads equals the virtual work of the stresses.

Notes: No assumption was made on the material, i.e. on its stress - strain relation; • hence, the Principle of Virtual Work holds for any constitutive relation. No assumption was made on the actual strains in the spatial configuration; • hence, the Principle of Virtual Work holds for finite or infinitesimal strains. No assumption was made on the external loads; hence, the Principle of • Virtual Work holds for conservative and nonconservative loads (Crandall 1956). The integrals in Eq. (6.8) are calculated on the spatial configuration of the • body. The Principle of Virtual Work was derived from momentum conservation • and not from energy conservation. As we see the Principle of Virtual Work is a very general statement, holding for any type of nonlinearities that may be present in the spatial configuration (material and geometrical nonlinearities). It is also possible to go through the inverse route, that is to say, starting from the Principle of Virtual Work to demonstrate the equations of momentum conservation. For this demonstration we refer the reader to (Fung 1965, Fung & Tong 2001).

6.2 The Principle of Virtual Work in geometrically nonlinear problems

In Eq. (6.8) the integrals are calculated in the spatial configuration of the continuum, which is normally one of the problem unknowns; however, for ge- t ometrically linear problems ( uα,β << 1) the difference between the spatial configuration and the unloaded configuration, normally the reference configuration, is negligible; hence

t V ∼= ◦V (6.9a) t S ∼= ◦S. (6.9b) 6.2 The Principle of Virtual Work in geometrically nonlinear problems 187

In the case of geometrically nonlinear problems it is convenient to calculate the integrals in Eq. (6.8) using the reference configuration. t The coordinates zα remain constant during the virtural displacement; hence, dδtε 1 ∂δtv ∂δtv αβ = α + β (6.10) dt 2 ∂tz ∂tz µ β α ¶ where δtv is the virtual velocity vector. Equation (3.11) is valid for any velocity field, in particular when we use the virtual velocity field, we get using Eq. (6.10),

t t t t t σ : δ ε dV = τ : δ ε ◦dV. (6.11) t Z V Z◦V Replacing in Eq. (6.8),

t t t t t t t t t b δ u ρ dV + t δ u dS = τ : δ ε ◦dV. (6.12) t · t · Z V Z Sσ Z◦V Also, using another pair of energy conjugate measures,

t t t t t t t t t b δ u ρ dV + t δ u dS = oS : δ oε ◦dV. (6.13) t · t · Z V Z Sσ Z◦V In the above, t oS: second Piola-Kirchhoﬀ stress tensor, t oε: Green-Lagrange strain tensor. We can deﬁne a general load, either a load per unit mass or per unit surface as: tf = lλ mϕ . (6.14) In the above, lλ is a scalar proportional to the load modulus and (Schweiz- erhof & Ramm 1984): l =0implies that the load modulus is a function of the reference coordi- • nates (body-attached loads); l = t implies that the load modulus is a function of the spatial coordinates • (space-attached loads). The unitary vector mϕ is a direction and (Schweizerhof & Ramm 1984): m =0implies a constant direction load; • m = t implies a follower load (the direction is a function of the body • displacements). 188 Nonlinear continua

Example 6.1. JJJJJ Buckling of a circular ring. In (Brush & Almroth 1975) we ﬁnd that the elastic buckling pressure acting on a circular ring depends on the type of load that we consider:

Load Buckling pressure

EI Hydrostatic pressure loading pcr =3a3

EI Centrally directed pressure loading pcr =4.5 a3

Both are cases of follower loads but, the description of the load as a function of the displacements is diﬀerent. JJJJJ

Using the mass conservation principle in Eq. (4.20d) we can write,

t t t t t t o b δ u ρ dV = b δ u ρ ◦dV. (6.15) t · · Z V Z◦V At each point on the surface bounding the continua we can calculate,

t t dS = JS ◦dS, (6.16) hence, t t t t t t t δ u dS = t δ u JS ◦dS. (6.17) t · · Z Sσ Z◦Sσ Therefore, we can write the principle of Virtual Work calculating the integrals over the reference conﬁguration as,

t t o t t t t t b δ u ρ ◦dV + t δ u JS ◦dS = S : δ ε ◦dV. (6.18) · · o o Z◦V Z◦Sσ Z◦V

Example 6.2. JJJJJ Let us consider the work of the external loads per unit surface of the spatial conﬁguration for the case of a typical follower load: the hydrostatic ﬂuid pressure. In this case, tt = tp tn where tn is the surface external normal. For this case,

t S t t t t δ Wext = pδu n dS. t · Z Sσ 6.2 The Principle of Virtual Work in geometrically nonlinear problems 189

Using Nanson’s formula (Example 4.9) we get,

t S t t t t 1 δ W = pδu J ◦n X− ◦dS. ext · · o Z◦Sσ For a case with inﬁnitesimal strains,

t J ≈ 1 t 1 t T oX− ≈ oR and therefore,

S t T t t δW = R δ u p ◦n ◦dS. ext o · · Z◦Sσ h i £ ¤ In the above equation, the first bracket inside the integral is the displacements variation rotated back from the spatial configuration to the material one; and the second bracket is the load per unit surface, but in the reference configuration. Hence, it is very important to realize that for the case of fluid-pressure loads and infinitesimal strains we can easily calculate the external surface loads virtual work in the reference configuration. JJJJJ

Using other alternative energy conjugate stress/strain measures,

t t o t t t t T t b δ u ρ ◦dV + t δ u JS ◦dS = P : δ X ◦dV,(6.19) · · o o Z◦V Z◦Sσ Z◦V and, for an isotropic constitutive relation

t t o t t t t t b δ u ρ ◦dV + t δ u JS ◦dS = Γ : δ H ◦dV.(6.20) · · o Z◦V Z◦Sσ Z◦V

6.2.1 Incremental Formulations

We have represented in Fig. 6.2 a typical Lagrangian analysis where the configuration at t =0is known and the configuration at t = t1 is sought. Normally, we perform an incremental analysis; that is to say, we determine the sequence of equilibrium configurations at t =0, ..., t, t + ∆t, ...t1. In this incremental analysis, the basic link to be analyzed is the generic step t t + ∆t; that is to say, knowing the equilibrium configuration at t we seek the→ one at t + ∆t. Of course, once this generic step can be solved, then the complete incremental analysis can be performed. In what follows, to describe the step t t + ∆t we follow the presentation in (Bathe 1996). → 190 Nonlinear continua

Fig. 6.2. Lagragian incremental analysis

First, we have to recognize that for describing the t + ∆t-configuration we can use as a reference configuration either the one at t =0or any of the intermediate ones, already known. In what follows we will specifically analyze two particular cases:

The total Lagrangian formulation, whereweuseasthereferenceconfigu- • ration the one at t =0(usually the undeformed configuration) The updated Lagrangian formulation, where we use as the reference con- • figuration the previous one (t).

The total Lagrangian formulation Using the principle of Virtual Work to deﬁne the t + ∆t equilibrium conﬁgu- ration, we write:

t+∆t IJ t+∆t t+∆t o S δo εIJ ◦dV = δ Wext . (6.21) Z◦V In the above equation, t+∆t IJ o S : components of the second Piola-Kirchhoff stress tensor, correspond- ingtothet + ∆t configuration and referred to the configuration at t =0. t+∆t o εIJ: components of the Green-Lagrange strain tensor, corresponding to the t + ∆t configuration and referred to the configuration at t =0.

I A general curvilinear coordinate system ◦x ,I =1, 2, 3 is used in the reference configuration, with covariant base vectors g (L =1, 2, 3). n ◦ L o Thevolumeofthereferenceconfiguration is ◦V and the virtual work of t+∆t the external loads acting at time t + ∆t is δ Wext (see the previous section for a discussion on the calculation of this term). We can write, t+∆t IJ t IJ IJ o S = oS + oS (6.22) IJ where oS are the components of the incremental second Piola-Kirchhoff stress tensor. It is important to recognize that the three tensors in Eq. (6.22) 6.2 The Principle of Virtual Work in geometrically nonlinear problems 191

Fig. 6.3. Total Lagrangian formulation are referred to the same reference conﬁguration (t =0). For the Green- Lagrange strain tensor we can also write an incremental equation,

t+∆t t o εIJ = oεIJ + oεIJ , (6.23) again, in the above equation the three tensors are referred to the same reference conﬁguration (t =0). Replacing with Eqs. (6.22) and (6.23) in Eq. (6.21) and taking into account t t that for the step that we are investigating, oεIJ is data and therefore, δoεIJ = 0,weget t IJ IJ t+∆t (oS + oS ) δoεIJ ◦dV = δ Wext . (6.24) Z◦V We write an incremental constitutive equation of the form,

IJ IJKL oS = oC oεKL (6.25) and get,

t IJ IJKL t+∆t (oS + oC oεKL) δoεIJ ◦dV = δ Wext . (6.26) Z◦V Refering the problem to a ﬁxed Cartesian system, we can write for a generic particle P , as shown in Fig. 6.3:

t P t P P u = x ◦x (6.27a) − uP = t+∆tuP tuP . (6.27b) −

We can show that (Bathe 1996), 192 Nonlinear continua 1 ε = u + u + t u u + t u u + u u . o αβ 2 o α,β o β,α o γ,α o γ,β o γ,β o γ,α o γ,α o γ,β (6.28) ¡ t ¢ ∂uα t ∂ uα In the above equation, ouα,β = β and uα,β = β . ∂◦z o ∂◦z Hence, it is possible to rewrite Eq. (6.28) as,

oεαβ = oeαβ + oηαβ (6.29a) 1 e = u + u + t u u + t u u (6.29b) o αβ 2 o α,β o β,α o γ,α o γ,β o γ,β o γ,α 1 η = ¡u u . ¢ (6.29c) o αβ 2 o γ,α o γ,β The term in Eq. (6.29b) is linear in the unknown incremental displacements, u, while the term in Eq. (6.29c) is nonlinear. We introduce Eq. (6.29a) in Eq. (6.26) and obtain,

t t+∆t [oSαβ + oCαβγδ (oeγδ + oηγδ)] δ(oeαβ +o ηαβ) ◦dV = δ Wext . (6.30) Z◦V The above is the momentum balance equation at time t + ∆t;whichisa nonlinear equation in the incremental displacement vector. In order to solve it we use an iterative technique, in which the ﬁrst step is the linearization of Eq. (6.30) (Bathe 1996). Keeping only up to the linear terms in u,weobtain the linearized momentum balance equation:

t [oCαβγδ oeγδ δoeαβ + oSαβ δoηαβ ] ◦dV (6.31) Z◦V t+∆t t = δ Wext Sαβ δoeαβ ◦dV. − o Z◦V

The updated Lagrangian formulation Using the principle of Virtual Work to deﬁne the t + ∆t equilibrium conﬁgu- ration, we write:

t+∆t IJ t+∆t t t+∆t t S δt εIJ dV = δ Wext . (6.32) t Z V In the above equation, t+∆t IJ t S : components of the second Piola-Kirchhoff stress tensor, correspond- ingtothet + ∆t configuration and referred to the configuration at t. t+∆t t εIJ: components of the Green-Lagrange strain tensor, corresponding to the t + ∆t configuration and referred to the configuration at t. We can write, t+∆t IJ t IJ IJ t S = tS + tS (6.33) 6.2 The Principle of Virtual Work in geometrically nonlinear problems 193

t IJ t IJ IJ where tS = σ and tS are the components of the incremental second Piola-Kirchhoﬀ stress tensor; it is important to recognize that the three tensors in Eq. (6.33) are referred to the spatial conﬁguration at time t. Also, t+∆t t εIJ = tεIJ . (6.34) t because tεIJ =0. Replacing with Eqs. (6.33) and (6.34) in Eq. (6.32), we get

t IJ IJ t t+∆t σ + tS δ (tεIJ) dV = δ Wext , (6.35) t Z V we can write an incremental¡ constitutive¢ equation referred to the t conﬁguration, − IJ IJKL tS = tC tεKL , (6.36) and get,

t IJ IJKL t t+∆t σ + tC tεKL δ (tεIJ) dV = δ Wext . (6.37) t Z V ¡ ¢ In a ﬁxed Cartesian system we can show that (Bathe 1996), 1 ε = ( u + u + u u ) (6.38) t αβ 2 t α,β t β,α t γ,α t γ,β

∂uα where tuα,β = t . ∂ zβ We can decompose the strain increment into a linear and a nonlinear increment in the unknown incremental displacement; that is to say,

tεαβ = teαβ + tηαβ 1 e = ( u + u ) (6.39) t αβ 2 t α,β t β,α 1 η = ( u u ) . t αβ 2 t γ,α t γ,β Hence we can write Eq. (6.37) as,

t t t+∆t σαβ + tCαβγδ teγδ + tηγδ δ teαβ + tηαβ dV = δ Wext . t Z V £ ¡ ¢¤ ¡ ¢ (6.40) The above is the momentum balance equation at time t + ∆t;whichis a nonlinear equation in the incremental displacement vector. Proceeding in the same way as in the total Lagrangian formulation we obtain the linearized momentum balance equation (Bathe 1996):

t t t tCαβγδ teγδ δteαβ dV + σαβ δtηαβ dV (6.41) t t Z V Z V t+∆t t t = δ Wext σαβ δteαβ dV. − t Z V 194 Nonlinear continua

It is easy to show that

o ρ I J SIJ = Sij t X 1 t X 1 (6.42) o tρ t o − i o − j

¡ t i ¢ t ¡j ¢ oεIJ = tεij oX I oX J (6.43) and therefore if the same material is considered in both formulations the incremental constitutive tensors should be related,

o IJKL ρ mnpq t 1 I t 1 J t 1 K t 1 L C = tC oX− oX− oX− oX− . (6.44) ◦ tρ m n p q Any problem can be alternatively¡ ¢ ¡ solved¢ using¡ either¢ the¡ total¢ or the updated Lagrangian formulations and the results should be identical (Bathe 1996). For solving ﬁnite-strain elastoplastic problems, in Sect. 5.2.6 we introduced an adhoc incremental formulation, the total Lagrangian-Hencky formulation.

6.3 The Principle of Virtual Power

There are formulations where the primary unknowns are the material velocities rather than the material displacements (e.g. ﬂuid problems, metal-forming Eulerian (Dvorkin, Cavaliere & Goldschmit 1995, Dvorkin & Petöcz 1993) or ALE formulations (Belytschko, Liu & Moran 2000), etc.). For these cases the momentum conservation leads to,

tb δtv tρ tdV + tt δtv tdS = tσ : δtd tdV. (6.45) t · t · t Z V Z Sσ Z V In the above equation tv is the material velocity at a point and td is the strain-rate tensor. Of course, we can use, for formulating the Principle of Virtual Power, other energy conjugated stress/strain rate measures, for example:

t t o t t t t t b δ v ρ ◦dV + t δ v JS ◦dS = τ : δ d ◦dV, (6.46a) · · Z◦V Z◦Sσ Z◦V t t t o t t t t t b δ v ρ ◦dV + t δ v JS ◦dS = S : δ ε· ◦dV, (6.46b) · · o o Z◦V Z◦Sσ Z◦V t t o t t t t T t · b δ v ρ ◦dV + t δ v JS ◦dS = P : δ X ◦dV, · · o o Z◦V Z◦Sσ Z◦V (6.46c)

t t o t t t t t · b δ v ρ ◦dV + t δ v JS ◦dS = Γ : δ H ◦dV, (6.46d) · · o Z◦V Z◦Sσ Z◦V 6.4 The Principle of Stationary Potential Energy 195 the last one only being valid for isotropic constitutive relations.

6.4 The Principle of Stationary Potential Energy

As we remarked above, the Principle of Virtual Work can be used for any material constitutive relation, for any type of loading and for any nonlinearity inthecasetobeanalyzed. In the present section we will specialize the Principle of Virtual Work for: Hyperelastic materials. • Conservative external loads. • For a hyperelastic material we have seen in Chap. 5 (Eq. (5.3d) that,

t t t IJ o ∂ U( oε) oS = ρ t . (6.47) ∂ oεIJ

The external conservative loads are the external loads that can be derived from a potential. Hence, a load ﬁeld is said to be conservative in a region if the net work done around any closed path in that region is zero (Crandall 1956). A typical conservative load system can be represented as,

t f = ◦λ ◦ϕ . (6.48)

Following the deﬁnitions introduced above, the load system in Eq. (6.48) is a body attached load system with constant direction. For conservative loads per unit mass, we write

∂tG (tu) tb = (6.49) − ∂u and for conservative surface loads ∂tg (tu) tt = . (6.50) − ∂u Note that the above-defined surface loads are defined as loads per unit t reference surface; therefore, its resultant at time t is t ◦dS . ◦Sσ We now define a functional of the function tu called the potential energy R functional:

t t t t oΠ = ◦ρ U + G ◦dV + g ◦dS (6.51) Z◦V Z◦Sσ ¡ ¢ 196 Nonlinear continua

Therefore,

t t t t ∂ U t ∂ G t ∂ g t δ Π = ◦ρ t : δ ε + ◦ρ δ u ◦dV + δ u ◦dS. (6.52) ◦ V ∂ ε ◦ ∂u · Sσ ∂u · Z◦ " ◦ # Z◦ t t In the above, δ u are admissible variations δ u = 0 on ◦Su see Fig. 6.1 and δt ε is derived from the displacement variations. Therefore, ◦ ¡ ¢

t t t t t t t δ Π = S: δ ε ◦ρ b δ u ◦dV t δ u ◦dS. (6.53) ◦ V ◦ ◦ − · − Sσ · Z◦ h i Z◦ Hence, for a hyperelastic material under a conservative load system, the principle of virtual work, in Eq. (6.18), can be written as

δt Π =0. (6.54) ◦ The above equation states that when the t configuration is in equilibrium the potential energy functional reaches a stationary− value; i.e. it fulfills the necessary requirements for being an extreme (Fung 1965). In what follows we show that in the case of infinitesimal strains the potential energy not only is stationary at the equilibrium configuration but it actually attains there a minimum. Using the nomenclature introduced in Eq. (6.1) we write the potential energy functional for an admissible configuration close to the equilibrium one as

t 0 t t t t t t oΠ = ◦ρ U( ε + δ ε) b u + δ u ◦dV (6.55) ◦ ◦ − · Z◦V £ ¡ ¢¤ t t t t u + δ u ◦dS. − · Z◦Sσ ¡ ¢ Using a Taylor expansion,

t 2t t t t t t ∂ U t 1 t ∂ U t U( ε + δ ε)= U( ε)+ t : δ ε + δ ε : t t : δ ε + . ◦ ◦ ◦ ∂ ε ◦ 2 ◦ ∂ ε ∂ ε ◦ ¯t ¯t ··· ◦ ¯ ε ◦ ◦ ¯ ε ¯◦ ¯◦ ¯ ¯ (6.56) Hence, ¯ ¯

2t t 0 t t 1 t ∂ U t oΠ oΠ = δ Π + ◦ρδ ε : t t : δ ε ◦dV + . (6.57) ◦ 2 ◦ ∂ ε ∂ ε ◦ − ◦V ¯t ··· Z ◦ ◦ ¯ ε ¯◦ ¯ Since at equilibrium δt Π =0, the sign of¯ the l.h.s. is the sign of the integrand on the r.h.s.. ◦ 6.4 The Principle of Stationary Potential Energy 197

In the case of infinitesimal strains case we can assume that t ε 0 and we t ◦ ≈ have tU(0)=0(convention) and t S = ρ ∂ U = 0; hence, from Eq. (6.56) 0 ◦ ∂t ε ◦ ◦ 0 ¯ ¯ ¯ ¯ 2t ¯ t t 1 t ∂ U t U(δ ε)= δ ε : t t : δ ε + . (6.58) ◦ 2 ◦ ∂ ε ∂ ε¯ ◦ ··· ◦ ◦ ¯0 ¯ ¯ Since, in a stable material the value of the¯ elastic strain energy is positive for any strain tensor (the elastic strain energy is a positive-definite function) we conclude that, t Π 0 t Π>0 , (6.59) o − o and the potential energy is a local minimum at the equilibrium configuration. In the infinitesimal strains case we call it the minimun potential energy principle (Washizu 1982).

Example 6.3. JJJJJ Conservative and nonconservative loading. (a) Conservative loading

Let us consider a linear elastic, cantilever beam under the conservative end- load shown in the ﬁgure,

Conservative load

The elastic energy stored in the beam is,

L EI d2 tu 2 tU = 2 dtz 2 dtz 2 1 Z0 µ 1 ¶ where E is Young’s modulus and I is the beam cross section moment of inertia. The Principle of Virtual Work states, 198 Nonlinear continua

t t δ U = Pδu2 t t t δ U P u2 =0 − t t t t where Π = U P u2 is¡ the potential¢ energy of the system. −

(b) Nonconservative loading We now consider the same linear elastic cantilever beam but under a follower load, as shown in the ﬁgure

Body-attached follower load

The principle of virtual work states,

t t t t t δ U = P sin θ δu1 + P cos θ δu2 . − For small displacement derivatives¡ ¢ we can approximate¡ ¢

dtu sin tθ tθ 2 ≈ ≈ dtz µ 1 ¶L cos ¡tθ¢ 1 ≈ hence, ¡ ¢ t t t d u2 t t δ U = P δ u1 + δ u2 . − dtz ∙ µ 1 ¶L ¸ Since t t t d u2 t t ∂ G t P δ u1 + δ u2 = δ u − dtz 6 − ∂u · ∙ µ 1 ¶L ¸ the load is nonconservative and the principle of stationary potential energy cannot be used. JJJJJ 6.4 The Principle of Stationary Potential Energy 199

Example 6.4. JJJJJ Stability of the equilibrium configuration (buckling) (Hoff 1956). Let us consider the system shown in the following figure, in equilibrium at time t,inthestraightconfiguration:

tP : axial conservative load ; L : length of the rigid bar ; k : stiﬀness of the linear spring; kT : stiﬀness of the torsional spring.

Assume that the equilibrium conﬁguration is perturbed with a rotation θ<<1. The axial load displacement is obtained from the following scheme: 200 Nonlinear continua

Lθ2 For θ<<1 ∆P and L sin(θ) Lθ. ⇒ ≈− 2 ≈ The potential of the external load is,

t 2 t t PLθ G = P∆P = . − 2 The only deformable bodies are the springs; hence 1 1 tU = k (Lθ)2 + k θ2 . 2 2 T Therefore the potential energy functional of the system is

1 1 tPLθ2 t Π = kL2 θ2 + k θ2 + o 2 2 T 2 and the equilibrium conﬁguration is deﬁned by

δt Π =0 ◦ which leads to, 2 t kL θ + kT θ + PLθ δθ =0. Since δθ is arbitrary£ the bracket has to be zero.¤ Two solutions are possible: (i) θ =0;thatistosay,thestraight(undeformed)conﬁguration. (ii) tP = kL+ kT . − L For the second¡ solution θ¢ is undeﬁned. We call this load value the critical value, Pcr, because at this load there are two branching solutions ( θ =0and θ =0). 6 6.4 The Principle of Stationary Potential Energy 201

Pcr deﬁnes the bifurcation or buckling load. The equilibrium path is

Since in the above derivation the terms higher than θ2 were neglected, we cannot assess anything about the branching equilibrium path. JJJJJ

Example 6.5. JJJJJ Postbuckling behavior. We repeat the previous example derivation keeping terms higher than θ2.By doing this, we get

θ2 θ4 ∆P = L (1 cos θ) L − − ≈− 2 − 4! µ ¶ 2 4 t t t θ θ G = P∆P PL − ≈ 2 − 4! µ ¶ 3 2 t 1 2 1 2 θ Uk = k (L sin θ) kL θ 2 ≈ 2 − 3! µ ¶ 1 tU = k θ2 . T 2 T Hence

4 6 2 4 t 1 2 2 θ θ 1 2 t θ t θ Π = kL θ + + kT θ + PL PL . o 2 − 3 36 2 2 − 24 µ ¶ For the equilibrium conﬁguration δt Π =0and therefore, ◦ 3 5 3 2 2 θ θ t t θ kL θ + + kT θ + PLθ PL δθ =0. − 3 12 − 6 ∙ µ ¶ ¸ 202 Nonlinear continua

Since δθ is arbitrary, we get, neglecting terms higher than θ3,

2 2 2 2 θ t θ kL 1 + kT + PL 1 θ =0 − 3 − 6 ∙ µ ¶ µ ¶¸ which has again two possible solutions: (i) θ =0the straight solution 2 kL 1 2 θ + kT t − 3 L (ii) P = 2 ³ 1 θ ´ − − 6 t kT In the second solution, for θ =0, P = Pcr = kL+ L . The bifurcation point is the same as the one calculated− in the previous example; however, now θ is deﬁned. ¡ ¢ We see that for θ>0 Eq. (ii) provides tP = tP (θ). If we examine the case with kT =0,weget

2 θ2 kL 1 3 tP = − − 1³ θ2 ´ − 6 and we can represent

If we examine the case with k =0,weget

k tP = T − L 1 θ2 − 6 ³ ´ and we can represent 6.4 The Principle of Stationary Potential Energy 203

It is clear that the above cases represent two very diﬀerent behaviors from a structural point of view. For the case kT =0, the buckling is catastrophic because for θ>0 the load-carrying capacity of the structure keeps dimishing: unstable postbuckling behavior. For the case k =0, the load-carrying capacity of the structure increases after buckling: stable postbuckling behavior. JJJJJ

Example 6.6. JJJJJ Natural boundary conditions. Let us study the following linear elastic cantilever beam under conservative loads:

E : Young’s modulus; I : moment of inertia of the beam cross section. t t t Assume u2 = u2( z1) is the beam transversal displacement and using linear beam theory (Hoﬀ 1956) 204 Nonlinear continua

L 2 t 2 L t EI d u2 t t t t Π = d z1 q u2 d z1 2 dtz 2 − 0 µ 1 ¶ 0 Z t Z t t t d u2 PL u2 ML . − L − dtz µ 1 ¶L ¡ ¢ For the equilibrium conﬁguration δtΠ =0; hence

L 2 t 2 t L d u2 d u2 t t t t EI δ d z1 qδ u2 d z1 ((A)) dtz 2 dtz 2 − Z0 µ 1 ¶ µ 1 ¶ Z0 t ¡ ¢ t t t d u2 PL δ u2 ML δ =0. − L − dtz ∙µ 1 ¶L¸ ¡ ¢ 2 t 2 d u2 d t In the ﬁrst integral we use (Fung 1965) δ t 2 = t 2 δ u and inte- d z1 d z1 2 grating by parts twice, we get ³ ´ ¡ ¢

L d2 tu d2 d2 tu d L EI 2 δtu dtz = EI 2 δtu dtz 2 dtz 2 2 1 dtz 2 dtz 2 Z0 µ 1 ¶ 1 ∙ 1 1 ¸0 3 t ¡ L ¢ L 4 t ¡ ¢ d u2 t d u2 t t EI δ u2 + EI δ u2 d z1 . − dtz 3 dtz 4 ∙ 1 ¸0 Z0 1

t t t d u2 At z1 =0we have as boundary conditions u2 = t =0, hence, d z1 t t d u2 δ u2 t = δ t =0. z1=0 d z1 t z1=0 ¡Replacing¢ in (A),h ³ we get´i

L 4 t 3 t d u2 t t t d u2 t t EI q δ u2 d z1 EI + PL δ u2 dtz 4 − − dtz 3 L Z0 ∙ 1 ¸ ∙µ 1 ¶L ¸ 2 t t ¡ ¢ d u2 t d u2 + EI ML δ =0. dtz 2 − dtz ∙µ 1 ¶L ¸ µ 1 ¶L

t t Since δ u2 is arbitrary at every point 0 6 z1 6 L we must fulﬁll the following diﬀerential equation

4 t t d u2 q = EI t 4 d z1 which is the well-known equation of beam theory. t At z1 = L we get 6.4 The Principle of Stationary Potential Energy 205

Essential (rigid) boundary conditions or Natural boundary condition

3 t t t t d u2 Either ( u2)L is ﬁxed and δ ( u2)L =0 or PL = EI dtz 3 − 1 L ³ ´ t t 2 t d u2 d u2 t d u2 Either dtz is ﬁxed and δ dtz =0 or ML = EI dtz 2 1 L 1 L 1 L ³ ´ ³ ´ ³ ´ . JJJJJ

Example 6.7. JJJJJ The Rayleigh-Ritz method. In the previous example, from the principle of stationary potential energy we derived the differential equation that governs the deformation of a cantilever beam. Usually, we need to work in the opposite direction: we know the differential equations that govern the deformation of a continuum but we cannot integrate them and we resort to the principle of stationary potential energy to derive an approximate solution. One method for deriving approximate solutions is the Rayleigh-Ritz method (Hoff 1956). Let us consider again the linear case analyzed in the previous example and letusassumethatwewanttoderiveanapproximatesolutionforthecase t t q = PL =0.Forthiscase

L 2 t 2 t t EI d u2 t t d u2 Π = d z1 ML . 2 dtz 2 − dtz Z0 µ 1 ¶ µ 1 ¶L To derive an approximate solution we consider trial functions that fulﬁll the geometrical or essential boundary conditions,

dtu tu (0) = 2 =0. 2 dtz µ 1 ¶0 For example t t t π z1 u ª( z1)=a 1 cos 2 − 2L µ ¶ where the parameter a will be determined by imposing the minimization condition on t t Πª = Πª (a) . Using the adopted trial function, we get

4 2 t t EIπ a ML aπ Πª = . 64 L3 − 2 L 206 Nonlinear continua

The minimum value that can attain the above functional is, within the considered set of trial functions, our best approximation to the equilibrium con- ﬁguration. Imposing ∂tΠ ª =0, ∂a we get 16 tM L2 a = L . EIπ3 Therefore our approximate solution is

t 2 t t t 16 ML L π z1 u ª( z1)= 1 cos 2 EIπ3 − 2L µ ¶ t 2 t 4 ML L Πª = . − EIπ2 For the case we are analyzing the exact solution is

tM (tz )2 tu exact tz = L 1 2 1 2 EI ¡ ¢ tM 2 L tΠexact = L . − 2 EI t t exact It is obvious that Πª > Π . If we want to improve our approximate solution we enrich the trial function set using, for example

t t t t π z1 π z1 u ⊕ z1 = a 1 cos + b 1 cos . 2 − 2L − L µ ¶ µ ¶ ¡ ¢ It is important to note that the above deﬁned trial function: Fulﬁlls the essential (rigid) boundary conditions. ◦ t t Contains the previous one, u2 ª ( z1),asaparticularcase(b =0). ◦ t Since we will determine the values of both constants by imposing on Π⊕ the necessary conditions for attaining a minimum, it is obvious that

t t Π⊕ 6 Πª . That is to say, we will either ﬁnd the same solution as before (b =0and t t t t Π⊕ = Πª)orabetterone(b =0and Π⊕ < Πª). We cannot deteriorate the solution by adding more6 terms in the trial function. t t Using u2 ⊕ ( z1) , we get

3 t t EIπ ab π 2 π 2 ML aπ Π⊕ = + a + b . 2 L3 3 32 2 − 2 L µ ¶ t t ∂ Π⊕ ∂ Π⊕ Imposing ∂a = ∂b =0,weget 6.5 Kinematic constraints 207

tM L2 a =0.6294 L EI tM L2 b = 0.0668 L − EI t 2 t ML L Π⊕ = 0.4940 . − EI t exact t t It is clear from the derived values that, as expected: Π < Π⊕ < Πª, t t t t and therefore u2 ⊕ ( z1) isa“better”approximationthan u2 ª ( z1). JJJJJ

In the previous example we have introduced three relevant topics that we want to highlight: 1. When obtaining approximate solutions using the Rayleigh-Ritz method, based on the minimum potential energy principle (infinitesimal strains), we can only rank the merit of different solution using their potential energy value; that is to say, if tΠA < tΠB then the A-solution is a better approximation than the B-solution. 2. The trial functions have to exactly satisfy the rigid boundary conditions (admissible functions) but not the natural boundary conditions. 3. Approximate solutions do not need to fulfill exactly either the equilibrium equation inside the dominium or the natural boundary conditions (equilibrium equations on the boundary).

6.5 Kinematic constraints

IntheexampleshowninFig.6.4,wheretP is a conservative load, the potential energy is, 1 tΠ = k tu2 tP tv. (6.60) 2 − For the inextensible string there is a kinematic constraint given by,

2 tv tu =0. (6.61) − Hence, we have to minimize the functional of the potential energy given in Eq.(6.60) under the constraint expressed in Eq. (6.61). Using the Lagrange multipliers technique (Fung 1965, Fung & Tong 2001), we deﬁne a new func- t tional ( Π∗) and we perform on it an unconstrained minimization:

t 1 t 2 t t t t t Π∗ = k u P v + λ 2 v u (6.62) 2 − − ¡ ¢ where tλ is the Lagrange multiplier. 208 Nonlinear continua

Fig. 6.4. Kinematics constraints

t t t t We need to determine the set ( λ, u, v) that satisﬁes δ Π∗ =0.The necessary conditions are,

∂ tΠ ∗ =0 (6.63a) ∂ tλ ∂ tΠ ∗ =0 (6.63b) ∂ tu ∂ tΠ ∗ =0. (6.63c) ∂ tv From the above we get,

tP tu = (6.64a) 2 k tP tv = (6.64b) 4 k tP tλ = k tu = . (6.64c) 2 Example 6.8. JJJJJ Physical interpretation of the Lagrange multiplier in the above example (Cran- dall 1956). Let us assume that the string, instead of being inextensible has a stiﬀness ks; hence, with tλ the tensile load on the spring we have,

t t t λ = ks 2 v u . − ¡ ¢ 6.6 Veubeke-Hu-Washizu variational principles 209

For this system with no constraints we can deﬁne the potential energy

t 1 t 2 1 t t 2 t t Πk = k u + ks 2 v u P v s 2 2 − − t ¡ ¢ and δ Πks =0leads to, t t t k u ks 2 v u =0 − − t t t 2 ks 2 v u P =0. −¡ − ¢ Hence, ¡ ¢ tP tP tu = ; tλ = . 2 k 2 Also, tλ 2 tv tu = 0 . − k ks→ s →∞ Therefore, tλ is independent of the displacement (conservative load) and we can use Eq. (6.62) for writing the potential energy, and identify the Lagrangian multiplier with the string tensile load. JJJJJ

In the above example, it is important to realize that the Lagrangian multiplier is the energy conjugate of the physical magnitude that represents the constraint equation: the string elongation.

6.6 Veubeke-Hu-Washizu variational principles

In the previous section we analyzed a simple mechanical system and we developed the imposition, using the Lagrange multipliers technique, of a kinematic constraint on the stationarization of the potential energy functional. In the present section we are going to generalize the above technique to impose diﬀerent constraints on the potential energy functional (Fraeijs de Veubeke 1965).

6.6.1 Kinematic constraints via the V-H-W principles Let us assume that we relax the compatibility conditions (see Sect. 2.15) when formulating the potential energy functional; that is to say we consider t in Eq. (6.51) a strain tensor oε that can not necessarily be derived from the displacements ﬁeld. We now deﬁne the functional:

t o t t t t t t oΠ∗ = ρ U oε + G( u) ◦dV + g u ◦dS (6.65) Z◦V Z◦Sσ £ ¡ ¢ t t t¤ t ¡ ¢ + λ :( ε u ε) ◦dV o −o Z◦V ¡ ¢ 210 Nonlinear continua and we search for the equilibrium conﬁguration imposing

t δ Π∗ =0 (6.66) ◦ t t t t o ∂tU we have as independent variables: u, oε and λ;withoS = ρ t we get ∂oε

t t o t t t t S : δ ε ◦dV ρ b δ u ◦dV t δ u ◦dS (6.67) o o − · − · Z◦V Z◦V Z◦Sσ t t t t t t + δ λ :(ε ε) ◦dV + λ :(δ ε δ ε) ◦dV =0. o − o o − o Z◦V Z◦V Considering that the variations are arbitrary,

t t o t t t t λ : δ ε ◦dV = ρ b δ u ◦dV + t δ u ◦dS (6.68) o · · Z◦V Z◦V Z◦Sσ t t t S λ : δ ε ◦dV =0 (6.69) o − o Z◦V ¡ t t ¢ t δ λ :(ε ε) ◦dV =0 . (6.70) o − o Z◦V The last equations impose the conditions

t t oS = λ (6.71) t t oε = oε (6.72) which are always fulﬁlled for the continuum problem but not necessarily in ﬁnite element approximations. Adding Eqs. (6.68) and (6.69), we get

t t t t t S : δ ε ◦dV + λ :(δ ε δ ε) ◦dV (6.73) o o o − o Z◦V Z◦V o t t t t = ρ b δ u ◦dV + t δ u ◦dS. · · Z◦V Z◦Sσ From the above, we can state the Principle of Virtual Work as,

t t o t t t t S : δ ε ◦dV = ρ b δ u ◦dV + t δ u ◦dS (6.74) o o · · Z◦V Z◦V Z◦Sσ as long as we fulﬁll the condition of variational consistency (Simo & Hughes 1986), t t t λ :(δ ε δ ε) ◦dV =0, (6.75) o − o Z◦V which is obviously fulﬁlled for the continuum problem. 6.6 Veubeke-Hu-Washizu variational principles 211

6.6.2 Constitutive constraints via the V-H-W principles

t Let us now assume that we consider in Eq. (6.51) a stress tensor oS that is t not necessarily derived from the kinematically consistent strain ﬁeld oε. We can write t t o ∂ U oS = ρ t (6.76a) ∂oε t t o ∂ U oS = ρ t . (6.76b) ∂oε Therefore

t o t t t t t t oΠ∗ = ρ U oε + G( u) ◦dV + g u ◦dS (6.77) Z◦V Z◦Sσ £ ¡ ¢ t t ¤ t ¡ ¢ + λ :( S S) ◦dV. o − o Z◦V We search for the equilibrium conﬁguration imposing

t δ Π∗ =0, (6.78) ◦

t t t t and considering the independent variables u, λ and oε , oS : ¡ ¢ t t t t S : δ ε ◦dV λ : δ S ◦dV =0 (6.79) o o − o Z◦V Z◦V t t o t t t t λ : δ S ◦dV ρ b δ u ◦dV t δ u ◦dS =0 (6.80) o − · − · Z◦V Z◦V Z◦Sσ t t t δ λ :(S S) ◦dV =0 . (6.81) o − o Z◦V The last equation imposes the condition

t t oS = oS (6.82) whichisalwaysfulﬁlled for the continuum problem but not necessarilly in the ﬁnite element approximations. Adding Eqs. (6.79) and (6.80), we get

t t t t t S : δ ε ◦dV + λ :(δ S δ S) ◦dV o o o − o Z◦V Z◦V o t t t t = ρ b δ u ◦dV + t δ u ◦dS. (6.83) · · Z◦V Z◦Sσ From the above, we can state the Principle of Virtual Work as, 212 Nonlinear continua

t t o t t t t S : δ ε ◦dV = ρ b δ u ◦dV + t δ u ◦dS (6.84) o o · · Z◦V Z◦V Z◦Sσ as long as we fulﬁll the condition,

t t t λ :(δ S δ S) ◦dV =0, (6.85) o − o Z◦V which is obviously fulfilled for the continuum problem. Other constraints can also be considered and they constitute the basis of different finite element applications. When using variational principles of the Veubeke-Hu-Washizu type for developing finite element formulations, the interpolation functions selected for the different interpolated fields should fulfill orthogonality conditions of the form of Eqs. (6.75) or (6.85) (Simo & Hughes 1986). Different forms of the Veubeke-Hu-Washizu variational principles have been used to develop mixed and hybrid finite element formulation some of them can be read from the classical paper (Pian & Tong 1969) and also from (Dvorkin & Bathe 1984, Bathe & Dvorkin 1985 1986, Dvorkin & Vassolo 1989, Fung & Tong 2001), etc. \ A Introduction to tensor analysis

In this Appendix, assuming that the reader is acquainted with vector analysis, we present a short introduction to tensor analysis. However, since tensor analysis is a fundamental tool for understanding continuum mechanics, we strongly recommend a deeper study of this subject. Some of the books that can be used for that purpose are: (Synge & Schild 1949, McConnell 1957, Santaló 1961, Aris 1962, Sokolnikoﬀ 1964, Fung 1965, Green & Zerna 1968, Flügge 1972, Chapelle & Bathe 2003).

A.1 Coordinates transformation

Let us assume that in a three-dimensional space ( 3) we can deﬁne a system of Cartesian coordinates: we call this space the Euclidean< space. In this Appendix we will restrict our presentation to the case of the Eu- clidean space. In the 3 spacewedeﬁne a system of Cartesian coordinates zα ,α= 1, 2, 3 , and< an arbitrary system of curvilinear coordinates θi ,i{=1, 2, 3 . The following} relations hold: { }

θi = θi(zα ,α=1, 2, 3) ,i=1, 2, 3 . (A.1)

The above functions are single-valued, continuous and with continuous ﬁrst derivatives. We call J the Jacobian of the coordinates transformation deﬁned by Eq. (A.1). Hence ∂θi J = α . (A.2) "∂z # An admissible transformation is one in which det J =0,thatistosay, a transformation where a region of nonzero volume in one6 system does not collapse into a point in the other system and vice versa. 214 Nonlinear continua

A proper transformation is an admissible transformation in which det J> 0.

A.1.1 Contravariant transformation rule

From Eq. (A.1) we obtain ∂θi dθi = dzα . (A.3) ∂zα When the coordinates system is changed, the mathematical entities ai at a certain point of 3 that transform following the same rule as does the coordinate diﬀerentials< (Eq. (A.3)) are said to transform according to a contravariant transformation rule. We indicate these mathematical entities using upper indices. i Now we consider two systems of curvilinear coordinates θi and θˆ , related by the following equations: { } © ª i i θˆ = θˆ (θj ,j=1, 2, 3) ,i=1, 2, 3 (A.4a) and l θk = θk(θˆ ,l=1, 2, 3) ,k=1, 2, 3 . (A.4b) We can write the coordinate diﬀerentials as:

i i ∂θ ˆj dθ = j dθ (A.4c) ∂θˆ i i ∂ˆθ dθˆ = dθj . (A.4d) ∂θj In the same way, a contravariant mathematical entity can be deﬁned in either of the two systems

ai = ai(θj ,j=1, 2, 3) ,i=1, 2, 3 (A.4e) j aî =âi(θˆ ,j=1, 2, 3) ,i=1, 2, 3 (A.4f) and we transform it from one curvilinear system to the other following a transformation rule similar to the transformation rule followed by the coordinate differentials:

i i ∂θ j a = j aˆ ,i=1, 2, 3 (A.4g) ∂θˆ i ∂θˆ aˆi = aj ,i=1, 2, 3 . (A.4h) ∂θj A.2 Vectors 215

Although the contravariant transformation rule applies to dθi and not to θi, using a notation abuse, we follow the convention of using upper indices for the coordinates.

A.1.2 Covariant transformation rule Given an arbitrary continuous and differentiable function f(θ1,θ2,θ3) and using the chain rule, we write ∂f ∂f ∂θj i = j i ,i=1, 2, 3 . (A.5a) ∂ˆθ ∂θ ∂θˆ We define ∂f aj = ,j=1, 2, 3 . (A.5b) ∂θj i In the ˆθ coordinate system we define { } ∂f aˆj = j ,j=1, 2, 3 (A.5c) ∂θˆ ∂θj aî = aj i ,i=1, 2, 3 (A.5d) ∂θˆ j ∂θˆ ai =âj ,i=1, 2, 3 . (A.5e) ∂θi

When the coordinates system is changed, the mathematical entities ai at a certain point of 3that transform following the same rule as does the derivatives of a scalar< function (Eqs. (A.5d) and (A.5e)) are said to transform accordingtoacovariant transformation rule. We indicate those mathematical entities using lower indices.

A.2 Vectors

There are some physical properties like mass, temperature, concentration of a given substance, etc., whose values do not change when the coordinate system used to describe the problem is changed. These variables are referred to as scalars. On the other hand, there are other physical variables like velocity, acceleration, force, etc. that do not change their intensity and direction when the coordinate system used to describe the problem is changed. They are called vectors. In what follows, we will make use of the above intuitive deﬁnition of scalars and vectors. However, in Sect. A.4 we will see that they represent two particular kinds of tensors (order 0 and 1, respectively). 216 Nonlinear continua

A.2.1 Base vectors

Asetofn linearly independent vectors is a basis of the space n and any othervectorin n can be constructed as a linear combination of< those base vectors. < Let us consider the three linearly independent vectors g (i =1, 2, 3) in i 3. Any vector v inthesamespacecanbewrittenas: < v = vi g . (A.6) i The mathematical entities vi (i =1, 2, 3) are the components of v in the basis g (i =1, 2, 3). i

Example A.1. JJJJJ In a Cartesian system zα ,α=1, 2, 3 thebasevectorsare { }

e1 =(1, 0, 0) ,

e2 =(0, 1, 0) ,

e3 =(0, 0, 1) , wherewehaveindicatedtheprojectionofthebasevectorsontheCartesian axes. The position vector r of a point P in 3 is < α r = z eα . Hence, α dr = dz eα , but also, ∂r dr = dzα . ∂zα Therefore, we get ∂r e = ,α=1, 2, 3 . α ∂zα JJJJJ

A.2.2 Covariant base vectors

In the arbitrary curvilinear system θi ,i=1, 2, 3 we can write, at any point P of the space, { } ∂r dr = dθi . (A.7a) ∂θi Since A.2 Vectors 217

Fig. A.1. Covariant base vectors at a point P

dr = dθig (A.7b) i we obtain ∂r g = ,α=1, 2, 3 . (A.7c) i ∂θi The vectors g ,deﬁned with the above equation, are the covariant base i vectors of the curvilinear coordinate system θi at the point P . From its deﬁnition, the vector g is tangent{ to} the line, θ = θ (P ) and 1 2 2 θ3 = θ3 (P ) . Similar conclusions can be reached for the covariant base vectors g and 2 g . 3 In a Cartesian system, we can write Eq. (A.7c) as:

∂zα g = e ,i=1, 2, 3 . (A.8) i ∂θi α

i In a second curvilinear system θˆ ,i=1, 2, 3 , { } 218 Nonlinear continua

j i ∂θˆ dr = dθˆ gˆ = dθi gˆ . (A.9a) i i ∂θ j Hence, we have j ∂θˆ g = gˆ ,i=1, 2, 3 . (A.9b) i ∂θi j Due to the similarity between Eqs. (A.9b) and Eq. (A.5e) the base vectors g are called covariant base vectors. i

A.2.3 Contravariant base vectors

In an arbitrary curvilinear coordinate system θi ,i=1, 2, 3 we deﬁne the contravariant base vectors (dual basis) (gi ,i{=1, 2, 3) with the} equation

gi g = δi , (A.10) · j j where the dot indicates a scalar product (“dot product”) between two vectors i i i and δj is the Kronecker delta (δj =1for i = j and δj =0for i = j ). i 6 Defining in 3 two curvilinear systems θi and ˆθ and using Eq. (A.9b), we obtain < { } { } ∂θm gî gˆ = gî g = δi . (A.11a) j j m j · · ∂ˆθ Hence, using Eq. (A.10), we obtain

∂θm gi g = gî g . (A.11b) j j m · · ∂ˆθ If we define i ∂θˆ gî = gl ,i=1, 2, 3 (A.11c) ∂θl from Eq. (A.11b), we obtain

i ∂θˆ ∂θm gi g = gl g (A.11d) j l j m · ∂θ ∂ˆθ · and i i ∂θˆ ∂θm ∂ˆθ gi g = δl = = δi , (A.11e) j l j m j j · ∂θ ∂θˆ ∂θˆ where we can see that the relation (A.11a) is satisﬁed. Therefore, Eq.(A.11c) can be considered the transformation rule for the contravariant base vectors. Due to the similarity between Eqs. (A.11c) and (A.4h) the base vectors gi are called contravariant base vectors. A.3 Metric of a coordinates system 219

Fig. A.2. Covariant and contravariant base vectors

InFig.A.2werepresentinaspace 2,atapointP ,thecovariantand contravariant base vectors, in order to provide< the reader with a useful geometrical insight.

It is important to note that in a Cartesian system the covariant and contravariant base vectors are coincident.

A.3 Metric of a coordinates system

If a position vector r deﬁnes a point P in 3 and a position vector (r +dr) < deﬁnes a neighboring point P 0, the distance between these points is given by

ds = dr dr . (A.12) · p A.3.1 Cartesian coordinates

In a Cartesian system in 3,wehave < ds2 =dzα dzβ (e e ) . (A.13a) α · β 220 Nonlinear continua

If we call e e = δαβ , then α · β 2 α β ds =dz dz δαβ . (A.13b)

We call the nine number δαβ the Cartesian components of the metric tensor at P (this notation will be clariﬁed in Sect. A.4.3). From its deﬁnition, it is obvious that δαβ =1if α = β and δαβ =0 if α = β . 6

A.3.2 Curvilinear coordinates. Covariant metric components

In an arbitrary curvilinear system θi ,i=1, 2, 3 the distance between P { } and P 0 is given by

ds2 =dr dr =dθi dθj (g g ) . (A.14a) · i · j We call gij = gji = g g (A.14b) i · j the covariant components of the metric tensor at P in the curvilinear system θi (this notation will be clariﬁed in Section A.4.3.). { }Using Eq. (A.8), we get

∂zα ∂zβ gij = δαβ . (A.15a) ∂θi ∂θj

Defining a second curvilinear system θî ,i=1, 2, 3 and using Eq. (A.9b), we get { } l m ∂ˆθ ∂θˆ gij = gˆlm . (A.15b) ∂θi ∂θj Equations (A.15a-A.15b) are the reason for using the name “covariant” for the metric tensor components defined in Eq. (A.14b).

A.3.3 Curvilinear coordinates. Contravariant metric components

The scalars deﬁned in Eq. (A.14b) by the dot product of the covariant base vectors were named covariant components of the metric tensor. In the same way, we deﬁne gij = gji = gi gj (A.16a) · the contravariant components of the metric tensor at P . Using Eq. (A.11c), we get

i j ij ∂θ ∂θ lm g = l m gˆ . (A.16b) ∂ˆθ ∂θˆ A.3 Metric of a coordinates system 221

The above equation is the reason for using the name “contravariant” for the metric tensor components deﬁned in Eq. (A.16a). It is obvious that in the Cartesian system, we have

αβ δ = δαβ . (A.17a)

i When θˆ is a Cartesian system, Eq. (A.16b) is { } ∂θi ∂θj gij = δαβ . (A.17b) ∂zα ∂zβ

A.3.4 Curvilinear coordinates. Mixed metric components

In any curvilinear coordinate system θi ,i=1, 2, 3 we can deﬁne the mixed components of the metric tensor{ as }

gi = gi g = δi (A.18a) j · j j g i = g gi = δi . (A.18b) j j · j

Example A.2. JJJJJ Any vector in 3 can be written as a linear combination of the covariant base vectors; hence,< we can write

gi = αij g . j

When we postmultiply by gk on both sides, we obtain

αik = gi gk = gik . ·

Thus, we have gi = gij g . j JJJJJ

Example A.3. JJJJJ Proceeding as we did in the previous examples, the reader can easily show that: g = g gj . i ij JJJJJ 222 Nonlinear continua A.4 Tensors

We show in Sect. A.2 that in the space 3 we can deﬁne two sets of linearly independent vectors: the covariant and the< contravariant base vectors. Hence, any arbitrary vector in 3 can be written as: < v = vi g = v gi . (A.19) i i We are now going to show that for the vector v to remain invariant under coordinate changes, the components vi should transform following a contravariant rule and the components vi should transform following a covariant rule. i When we go from the system θi ,i=1, 2, 3 to the system ˆθ ,i= 1, 2, 3 , using Eqs. (A.9b), we obtain{ } { } j ∂θˆ v = vi g = vi gˆ =ˆvj gˆ . (A.20a) i ∂θi j j Hence, j ∂θˆ vˆj = vi ; j =1, 2, 3 . (A.20b) ∂θi We see from the above that when the coordinate system is changed, the components vi transform following a contravariant rule. Using Eq. (A.11c), we can write

ˆj j ∂θ i i v =ˆvj gˆ =ˆvj g = vi g . (A.21a) ∂θi Hence, j ∂θˆ vi =ˆvj ; i =1, 2, 3 . (A.21b) ∂θi We see from the above that when the coordinate system is changed, the components vi transform following a covariant rule. As a conclusion to this section, we can state that the invariance of v under coordinate transformation requires the use of: covariant components + contravariant base vectors • or contravariant components + covariant base vectors. • A.4 Tensors 223

A.4.1 Second-order tensors

Generalizing the concept of vectors that we presented above, we deﬁne as second-order tensors the following mathematical entities:

a = a gi gj = aij g g = ai g gj = a j gi g (A.22) ij i j j i i j that remain invariant under coordinate transformations. In the above equation, we used tensorial or dyadic products between vectors (g g ; gi gj ; gi g ; etc.) that we are going to formally deﬁne in this Section. i j j i For the transformation θi ˆθ using Eq. (A.9b), we get { } → { } k l ∂θˆ ∂θˆ a = aij g g = aij gˆ gˆ =ˆakl gˆ gˆ . (A.23a) i j ∂θi ∂θj k l k l Thus, we have

k l ∂ˆθ ∂θˆ aˆkl = aij ; k, l =1, 2, 3 . (A.23b) ∂θi ∂θj That is to say, the components aij transform following a double contravariant rule. In the same way, we can show that

∂θi ∂θj aˆkl = aij k l ; k, l =1, 2, 3 . (A.23c) ∂ˆθ ∂θˆ

That is to say, the components aij transform following a double covariant rule. We can also show that:

ˆk j k i ∂θ ∂θ aˆ l = a j i l ; k, l =1, 2, 3 (A.23d) ∂θ ∂θˆ j ˆk k i ∂θ ∂θ aˆl = aj l i ; k, l =1, 2, 3 . (A.23e) ∂θˆ ∂θ

i j That is to say, the components a j and ai transform following mixed rules. From Eq. (A.22), we have

a gk gl = aij g g . (A.24a) kl i j

When we postmultiply by g on both sides (the formal deﬁnition of this r operation is given in what follows), we get

a gk δl = aij g g (A.24b) kl r jr i 224 Nonlinear continua and if we now postmultiply (inner product) both sides by g ,weobtain s

l k ij akl δr δs = a gis gjr . (A.24c)

Finally, ij asr = a gis gjr ; r, s =1, 2, 3 . (A.24d) In the same way, we can show the following relations for k, l =1, 2, 3:

j akl = ak gjl (A.24e) kl ik jl k jl a = aij g g = a j g (A.24f) k kj jk a l = a gjl = ajl g . (A.24g)

It is evident that we can raise and lower indices using the proper components of the metric tensor. In Eq. (A.22) and the ones that followed, we wrote dyads of the type g g or gi gj or gi g or g gj that define an operation known as i j j i the tensorial product of two vectors. To define the tensorial product of two vectors a and b (ab in our notation or a b in the notation used by other authors), we will define the properties of⊗ this new operation: Given a scalar α, •

α (ab)=(α a)b = a(α b)=αab . (A.25a) Givenathirdvectorc, •

(ab)c = a(bc)=abc (A.25b) and

a(b + c)=ab + ac = ba + ca =(b + c) a . (A.25c) 6 In general, •

ab = ba . (A.25d) 6 The scalar product of a vector c with the dyad ab is a vector, •

c (ab)=(c a)b , (A.25e) · · where (c a) is a scalar. Also, · A.4 Tensors 225

(ab) c = a(b c) = c (ab)=(c a)b . (A.25f) · · 6 · · It should be notd that (ab) c is a vector with the direction of the vector a , while c (ab) is a vector· with the direction of the vector b . The scalar or· inner product between two dyads is another dyad: •

(ab) (cd)=(b c) ad . (A.25g) · · The double scalar or inner product between two dyads is a scalar: •

(ab): (cd)=(a c)(b d) . (A.25h) · · Besides (Malvern 1969),

(ab) (cd)=(a d)(b c) (A.25i) ·· · · which is a scalar too. Using the above definition, we can perform the scalar product of the second-order tensor a defined by Eq. (A.22) and the vector v defined by Eq. (A.19),

k i ik i k k i a v = aik v g = a vk g = a v g = a vk g (A.26a) · i k i i then, we obtain a vector b = a v with: · k k covariant components: bi = aik v = ai vk ◦ i ik i k contravariant components: b = a vk = a k v . It◦ is easy to show that (v a) is also a vector and that in general · v a = a v . (A.26b) · 6 ·

Eigenvalues and eigenvectors of second-order tensors

We say that a vector v is an eigenvector of a second-order tensor a if

a v = λ v (A.27) · and we call λ the eigenvalue associated to the eigendirection v. It is easy to show that the following relation holds

j ( aij λgij ) v =0. (A.28) − Equation (A.28) represents a system of 3 homogeneous equations (j = 1, 2, 3) with 3 unknowns (v1,v2,v3) . To obtain a solution diﬀerent from the trivial one, we must have 226 Nonlinear continua

aij λgij =0. (A.29) | − | The above is a cubic equation in λ that leads to 3 eigenvalues and therefore 3 associated eigendirections. It is obvious that if a pair (λ, v) satisﬁes Eq. (A.27), the pair (λ, α v) will also do so. Hence, the modulus of the eigenvectors remains undeﬁned. The following properties can be derived: If a is symmetric, the eigenvalues and eigenvectors are real. •

Proof. (Green & Zerna 1968) Assume λ is not real, then

λ = α + iβ (A.30a) vj = ηj + iµj . (A.30b)

From Eq. (A.28), equating real and imaginary parts,

j j (aij αgij) η + βgij µ =0 (A.30c) − j j (aij αgij) µ βgij η =0. (A.30d) − − After some algebra, from the two above equations, we get

i j i j β (gij η η + gij µ µ )=0 (A.30e) for aij = aji. i i i j Since all the η and µ cannot be zero and the terms (gij η η ) and i j (gij µ µ ) are always positive (see Eq. (A.14a)), then

β =0. (A.30f)

Therefore, the eigenvalues are always real and to satisfy Eq. (A.28) the eigenvectors shall also be real.

If a is symmetric, the eigenvectors form an orthogonal set. • A.4 Tensors 227

Proof. (Crandall 1956) For two pairs (λ1, v1) and (λ2, v2) from Eq. (A.27)

v a v v a v =(λ2 λ1) v v . (A.31) 1 · · 2 − 2 · · 1 − 1 · 2 For a symmetric tensor, the l.h.s. of the above equation is zero. Hence, If λ1 = λ2 then v v =0.Thatistosay,v and v are orthogonal. ◦ 6 1 · 2 1 2 If λ1 = λ2 there are inﬁnite vectors v1, v2 that satisfy the above equation. Among◦ them we can select a pair of orthogonal vectors. Hence, in general we assess that for symmetric second-order tensors, the eigenvectors are orthogonal.

Example A.4. JJJJJ As a is a symmetric second order tensor, with eigenvalues λI and eigenvectors v (I =1, 2, 3) with v =1,thecanonical form of a is: I | I |

a = λ1 v1 v1 + λ2 v2 v2 + λ3 v3 v3 also known as the diagonalized form. JJJJJ

A.4.2 n-order tensors

Inthesamewaythatwedefined the tensorial products of two vectors (dyad), we can define the tensorial product of n vectors (n-poliad). Therefore, we can define mathematical entities of the type:

ij...n i j n t = t g g g = tij...n g g g (A.32) i j ··· n ··· ij...k l m n = t lm...n g g g g g g i j ··· k ··· which we call tensors of order n and we associate to them the property of remaining invariant when coordinate transformations are performed. When we go from the curvilinear system θi ,i=1, 2, 3 to the curvi- i { } linear system ˆθ ,i=1, 2, 3 , due to the invariance property,weget { } ij...k l m n t = t lm...n g g g g g g (A.33a) i j ··· k ··· = tˆab...c gˆ gˆ gˆ gˆd gˆe gˆf . de...f a b ··· c ··· Hence, the following relations can be derived: 228 Nonlinear continua

ab...c ij...k a b c tˆ = t lm...n (gˆ g )(gˆ g ) (gˆ g ) de...f · i · j ··· · k (gˆ gl)(gˆ gm) (gˆ gn) (A.33b) d · e · ··· f · and using Eqs. (A.9b) and (A.11c), we obtain

ˆa ˆb ˆc l m n ˆab...c ij...k ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ t de...f = t lm...n i j k d e f . (A.33c) ∂θ ∂θ ··· ∂θ ∂θˆ ∂θˆ ··· ∂ˆθ

A.4.3 The metric tensor

As a particular but important example of second-order tensors, we will refer in this section to the metric tensor, g,

g = g gi gj = gij g g = δi g gj = δj gi g . (A.34) ij i j j i i j In Sects. A.3.1 - A.3.3, we introduced the covariant, contravariant and mixed components of this tensor. We can rewrite Eq.(A.14a) as:

2 i k l j ds = dr g dr = dθ g gkl g g dθ g (A.35a) · · i · · j ³ ´ ³ ´ and therefore, ¡ ¢

2 i j k l i j ds = dθ dθ gkl δi δj = dθ dθ gij . (A.35b) Going back to Eq. (A.34), we post-multiply both sides by the vector gp and i j p kl p gij g g g = g g g g . (A.36a) · k l · Operating, we get g gjp gi = gkl δp g . (A.36b) ij l k Using Eqs. (A.15a) and (A.17b), we arrive at

α β j p α α p jp ∂z ∂z ∂θ ∂θ γδ ∂z ∂z ∂θ gij g = δαβ δ = . (A.36c) ∂θi ∂θj ∂zγ ∂zδ ∂θi ∂zγ ∂zγ Rearranging, p α p jp ∂θ ∂z ∂θ p gij g = = = δ . (A.36d) ∂zα ∂θi ∂θi i Using the above in Eq. (A.36b), we ﬁnally obtain

gp = gpk g . (A.36e) k The above equality was also derived in Example A.2. In an identical way, we can also derive the result of the Example A.3. A.4 Tensors 229

A.4.4 The Levi-Civita tensor

The Cartesian components of the Levi-Civita or permutation tensor are de- ﬁned as:

0 when two of the indices are equal αβγ eαβγ = e = 1 when the indices are arranged as 1,2,3 . ⎧ 1 when the indices are arranged as 1,3,2 ⎫ ⎨ − ⎬ By using the tensorial⎩ components transformation rules in an arbitrary⎭ curvilinear system θi ,i =1, 2, 3 and for the covariant components, we get { } ∂zα ∂zβ ∂zγ ijk = eαβγ . (A.37a) ∂θi ∂θj ∂θk Taking into account that the determinant of a (3 3) matrix can be written as: × i r s t a j = erst a 1 a 2 a 3 (A.37b) it is easy to show that ¯ ¯ ¯ ¯ m r s t eijk a = erst a a a . (A.37c) | n| i j k Another important relation is

αδ δ δ eαβγ e = δ δ δ δ . (A.37d) β γ − β γ Introducing the above relation in Eq. (A.37a) and also using Eq. (A.2), we get ∂zm = e . (A.37e) ijk ijk ∂θn ¯ ¯ ¯ ¯ In the same way, we can show that ¯ ¯ ¯ ¯ ∂θn ijk = eijk . (A.38) ∂zm ¯ ¯ ¯ ¯ ¯ ¯ Some authors define the components¯ of the¯ permutation tensor in any curvilinear system using the same definition that we just used for the Cartesian components. In this way, the tensorial components transformation rules are not fulfilled (invariance is lost) and the permutation tensor in this case is called a pseudotensor. In a Cartesian system, we define the cross product between two vectors as

γ e e = eαβγ e . (A.39) α × β Taking into account that 230 Nonlinear continua

∂θl e = g (A.40a) α ∂zα l ∂zγ eγ = gn (A.40b) ∂θn and using the above-derived curvilinear components of the Levi-Civita tensor, we get l m ∂θ ∂θ n g g lmn g = 0 . (A.40c) ∂zα ∂zβ l × m − Considering that the aboveh expression is valid ini any curvilinear system, we obtain n g g = lmn g . (A.41) l × m

Example A.5. JJJJJ For a second-order tensor we can write Eq. (A.29) using mixed components as: ai λδi =0. | j − j | Hence, using Eq. (A.37b)

r r s s t t erst (a λδ )(a λδ )(a λδ ) =0. 1 − 1 2 − 2 3 − 3 After some algebra£ (Flügge 1972) we get the characteristic¤ equation of the second-order tensor a.

1 λ3 ai λ2 + (ai aj ai aj ) λ ai =0. − i 2 i j − j i − | j| Since the eigenvectors of a areindependentofthecoordinatesystemweuse to describe the tensor, the coeﬃcients of the above equation are invariant against coordinate transformations. We deﬁne the invariants as:

i ij ij II = a i = a gij = aij g = a : g ,

1 i j i j 1 2 III = a a a a = a a I , 2 j i − i j 2 ·· − I i h i IIII = a , ¡ ¢ | j| Finally, the characteristic equation can be written as:

3 2 λ II λ III λ IIII =0. − − − JJJJJ A.4 Tensors 231

Example A.6. JJJJJ In a plane normal to the axis z3,wedefine an arbitrary curvilinear system θi ,i=1, 2 with θ3 = z3. The base vectors of the curvilinear system are { } ∂z1 ∂z2 g = e + e , 1 ∂θ1 1 ∂θ1 2 ∂z1 ∂z2 g = e + e , 2 ∂θ2 1 ∂θ2 2 g = e . 3 3 We define the area differential in the plane normal to z3 as: dA =dθ1 dθ2 g g 3 1 × 2 and after some algebra, we get

1 2 dA = J3 dθ dθ e 3 | | 3 ∂z1 ∂z2 ∂z2 ∂z1 where J3 = 1 2 1 2 . In the particular case when the | | ∂θ ∂θ − ∂θ ∂θ curvilinear system³ θi is in fact a Cartesian´ system zˆi , it is easy to { } { } show that J3 =1. Therefore, | | 1 2 dA3 =dˆz dˆz e3 .

JJJJJ

Example A.7. JJJJJ We deﬁne in the 3 space an arbitrary curvilinear system θi ,i=1, 2, 3 with the following< covariant base vectors { } ∂zα g = e . i ∂θi α We also deﬁne dV =dθ1 dθ2 dθ3 g (g g ) 1 · 2 × 3 h i and after some algebra, we get ∂zm dV = dθ1 dθ2 dθ3 . ∂θn ¯ ¯ ¯ ¯ In the particular case when the curvilinear¯ ¯ system θi is in fact a Cartesian ¯ ¯ m i ∂z { } system zˆ , it is easy to show that, n =1. Therefore, { } ∂θ 1 ¯ 2 ¯ 3 dV =dˆz ¯dˆz dˆ¯z . JJJJJ 232 Nonlinear continua A.5 The quotient rule

rs Let A(ijkrs) be a set of 243 quantities, B the contravariant components of an arbitrary second-order tensor (independent of A(ijkrs))andDijk the covariant components of a third-order tensor. If, in any coordinate system, the relation

rs Dijk = A(ijkrs) B (A.42) is satisﬁed, then we are going to prove that the A(ijkrs) are the covariant components of a ﬁfth order tensor.

Proof. Since D and B are tensors, when we change the coordinate system i from θi ,i=1, 2, 3 to θˆ ,i=1, 2, 3 ,weget { } { } i j k ˆ ∂θ ∂θ ∂θ Dlmn = l m n Dijk (A.43a) ∂θˆ ∂θˆ ∂θˆ r s rs ∂θ ∂θ ˆpq B = p q B . (A.43b) ∂θˆ ∂θˆ

Using Eqs. (A.42) and (A.43b) in Eq. (A.43a), we get

i j k r s ˆ ∂θ ∂θ ∂θ ∂θ ∂θ ˆpq Dlmn = l m n A(ijkrs) p q B . (A.44a) ∂θˆ ∂ˆθ ∂θˆ ∂θˆ ∂θˆ We can write ˆ ˆ ˆpq Dlmn = A(lmnpq) B (A.44b) substracting Eq. (A.44a) from Eq. (A.44b), we get

i j k r s ˆ ∂θ ∂θ ∂θ ∂θ ∂θ ˆpq A(lmnpq) l m n p q A(ijkrs) B =0. (A.44c) " − ∂ˆθ ∂θˆ ∂θˆ ∂ˆθ ∂ˆθ # Since B is an arbitrary second order tensor, from the above equation, we obtain the following relation:

i j k r s ˆ ∂θ ∂θ ∂θ ∂θ ∂θ A(lmnpq) = l m n p q A(ijkrs) . (A.44d) ∂ˆθ ∂θˆ ∂θˆ ∂ˆθ ∂ˆθ

The above equation shows that the A(ijkrs) transform according to a covariant transformation rule that shows they are the covariant components of a ﬁfth- order tensor. A.6 Covariant derivatives 233

The generalization of the case that we analyzed, the quotient rule, is a tool for identifying general tensors.

Example A.8. JJJJJ Let us consider the vectors (ﬁrst-order tensors)

x = xr g r y = yr g r r z = zr g .

If we know that r s t α = A st x y zr is invariant under coordinate transformations (a scalar), then the quotient r rule indicates that the A st are the mixed components of the following tensor

A = Ar g gs gt . st r

JJJJJ

A.6 Covariant derivatives

A.6.1 Covariant derivatives of a vector

Contravariant components

Given a vector v,wecandeﬁne it using its Cartesian components as

α v = v eα , (A.45a) and since the base vectors of a Cartesian system do not change with the coordinates, we get ∂v ∂vα = e . (A.45b) ∂zβ ∂zβ α Using, in the Euclidean space, a system of arbitrary curvilinear coordinates θi,i=1, 2, 3, , we get { } v = vs g (A.46a) s ∂v ∂vs ∂g = g + vs s . (A.46b) ∂θn ∂θn s ∂θn 234 Nonlinear continua

Using Eq. (A.8), we obtain

∂g ∂2zα s = e (A.46c) ∂θn ∂θs ∂θn α and using it once more,

∂g ∂2zα ∂θp s = g = Γ p g . (A.46d) ∂θn ∂θs ∂θn ∂zα p sn p p Γsn is defined as the Christoffel symbol of the second kind in the Euclidean space: ∂2zα ∂θp Γ p = . (A.47) sn ∂θs ∂θn ∂zα It should be noted that: The Christoffel symbol of the second kind is a function of the coordinate • system under consideration θi and of the coordinates of the point where the calculations are performed.{ } The Christoffel symbols of the second kind are not tensorial components • and therefore do not transform as such,

2 α â â ∂ z ∂θ Γbc = b c α . (A.48a) ∂θˆ ∂θˆ ∂z In general, â s n â ∂θ ∂θ ∂θ p Γbc = p b c Γsn . (A.48b) 6 ∂θ ∂ˆθ ∂θˆ It is obvious from Eq. (A.47) that

p p Γsn = Γns . (A.49) It is important to note that in general Eq. (A.49) is not necessarily valid in a non-Euclidean space. In the Cartesian coordinate system Γ α =0. • βγ

From Eqs. (A.46b), (A.46d) and (A.47), we get

∂v ∂vp = + Γ p vs g . (A.50a) ∂θn ∂θn sn p ∙ ¸ Deﬁning p p ∂v p s v n = + Γ v (A.50b) | ∂θn sn we can write A.6 Covariant derivatives 235

∂v p = v n g . (A.50c) ∂θn | p p We call v n the covariant derivative of the contravariant components of v. | p We are going to show in Sect. A.7. that the v n are mixed components of a second-order tensor and that the subindex n,| associated to the variable θn, transforms in a covariant way.

Example A.9. JJJJJ Since gij = g g i · j and using Eq. (A.46d), we get

∂gij p p = Γ gpj + Γ gip . ∂θl il jl JJJJJ

Example A.10. JJJJJ From the above result, we get

∂gij ∂gjl ∂gli p p p + = Γ gpj + Γ gpi + Γ gpl ∂θl ∂θi − ∂θj il jl ji p p p + Γ gpj Γ gpi Γ gpl li − lj − ij p p p p =(Γ + Γ ) gpj + Γ Γ gpi il li jl − lj p p ³ ´ + Γ Γ gpl . ji − ij In the Euclidean space, ¡ ¢

p 1 ∂gij ∂gjl ∂gli gpj Γ = + . il 2 ∂θl ∂θi − ∂θj µ ¶ JJJJJ

Covariant components

We are now going to perform the derivations of the previous Section but, in the present case, for a vector deﬁned using its covariant components and contravariant base vectors, the following results 236 Nonlinear continua

∂v ∂v ∂gs = s gs + v . (A.51) ∂θn ∂θn s ∂θn Taking into account that gs g = δs, we get · t t ∂gs ∂g g + gs t =0. (A.52a) ∂θn · t · ∂θn Using Eq. (A.46d) in the above, ∂gs g + Γ p gs g =0 (A.52b) ∂θn · t tn · p and after some algebra, we have ∂gs = Γ s gt . (A.52c) ∂θn − tn Therefore, ∂v ∂vp s p = vs Γ g . (A.53a) ∂θn ∂θn − pn ∙ ¸ We now call ∂vp s vp n = vs Γ . (A.53b) | ∂θn − pn Hence, ∂v p = vp n g . (A.53c) ∂θn | We call vp n the covariant derivatives of the covariant components of v. | We are going to show in Sect. A.7 that the vp n are covariant components of a second-order tensor. |

A.6.2 Covariant derivatives of a general tensor

Given an arbitrary n-order tensor,

ij k p q r t = t ··· pq r g g g g g g (A.54) ··· i j ··· k ··· we can generalize the previous derivations,

∂ t ij...k p q r = t pq...r n g g g g g g (A.55a) ∂θn | i j ··· k ··· where ij...k ij...k ∂t pq...r sj...k i is...k j t pq...r n = + t pq...r Γ + t pq...r Γ + | ∂θn sn sn ij...s k ij...k s ij...k s + t pq...r Γ t sq...r Γ t ps...r Γ ··· sn − pn − qn ij...k s t pq...s Γ (A.55b) − ··· − rn A.7 Gradient of a tensor 237 is the covariant derivative of the mixed components of the tensor t. ij k We are going to show in Sect. A.7 that the t ··· pq r n are mixed components of a (n +1)-order tensor. ··· |

Example A.11. JJJJJ Using Eq.(A.55b), we get

∂gij p p gij m = gpj Γ gip Γ | ∂θm − im − jm and taking into account Example A.9, we get

gij m =0 . | JJJJJ

A.7 Gradient of a tensor

Let t be a general n-order tensor,

ij...k p q r t = t pq...r g g g g g g . (A.56) i j ··· k ··· We deﬁne the gradient of the tensor t as:

n ∂ ij...k p q r t = g t pq...r g g g g g g . (A.57) ∇ ∂θn i j ··· k ··· h i Using the quotient rule and taking into account that due to the deﬁnition of gradient, d t = dr t (A.58) · ∇ and that dt is an n-order tensor while dr = dθn g is a vector, we n conclude that t is a (n +1)-order tensor. Using Eq. (A.55a),∇ we can rewrite Eq. (A.57) as:

ij...k n p q r t = t pq...r n g g g g g g g . (A.59) ∇ | i j ··· k ··· ij...k Therefore, the t pq...r n are mixed components of a (n +1)-order tensor. | p In the particular case of t being a vector, it is now evident that v n are | mixed components and the vp n are covariant components of the second-order tensor, then | p n n p v = v n g g = vp n g g . (A.60) ∇ | p | 238 Nonlinear continua

Example A.12. JJJJJ We are going to show that if the components of a given tensor t are constant in a Cartesian system, then in any curvilinear coordinate system in the Euclidean space, the covariant derivatives of the components of t are zero. In a Cartesian system zˆα ,usingEq.(A.59),weget { } ∂tˆαβ...γ t = πσ...τ eθ e e e eπ eσ eτ ∇ ∂zˆθ α β ··· γ ··· α ( e = eα in a Cartesian system). If the Cartesian components of t are constant, ∂tˆαβ...γ πσ...τ =0. ∂zˆθ Hence, we get t = 0 . ∇ Since the above is a tensorial equation, it has to be fulﬁlledinanycoordinate system. In particular, in a system θi { } ij...k t pq...r n =0. | JJJJJ

A.8 Divergence of a tensor Let t be a general n-order tensor, ij...k p q r t = t pq...r g g g g g g , (A.61) i j ··· k ··· we deﬁne the divergence of the tensor t as:

n ∂ ij...k p q r t = g t pq...r g g g g g g . (A.62a) ∇ · ∂θn · i j ··· k ··· Aftersomealgebra,wegeth i ij...k p q r t = t pq...r i g g g g g . (A.62b) ∇ · | j ··· k ··· When we write t as t = t j...k gi g g gp gq gr (A.63a) i pq...r j ··· k ··· its divergence is ni j...k p q r t = g t n g g g g g . (A.63b) ∇ · i pq...r| j ··· k ··· Thedivergenceofann-order tensor is a (n 1)-order tensor. In the particular case of a vector, − n ni v = v n = g vi n , (A.64) ∇ · | | the divergence of a vector is a scalar. A.9 Laplacian of a tensor 239 A.9 Laplacian of a tensor

Let t be a general n-order tensor,

ij...k p q r t = t pq...r g g g g g g (A.65) i j ··· k ··· we deﬁne the Laplacian of the tensor t as

2 t = t . (A.66) ∇ ∇ · ∇ Using Eqs. (A.59) and (A.62a-A.62b) and after lengthy algebra, we obtain

2 ij...k nl p q r t = t pq...r nl g g g g g g g (A.67a) ∇ | i j ··· k ··· where 2 ij...k sj...k i ij...k ∂ t pq...r ∂t pq...r i sj...k ∂Γsn t pq...r nl = + Γ + t pq...r | ∂θn ∂θl ∂θl sn ∂θl is...k j ∂t pq...r j is...k ∂Γsn + Γ + t pq...r + (A.67b) ∂θl sn ∂θl ··· ij...s k ij...k ∂t pq...r k ij...s ∂Γsn ∂t sq...r s + Γ + t pq...r Γ ∂θl sn ∂θl − ∂θl pn s ij...k s ij...k ∂Γpn ∂t ps...r s ij...k ∂Γqn t sq...r Γ t ps...r − ∂θl − ∂θl qn − ∂θl ij...k s ∂t pq...s s ij...k ∂Γrn ij...k s Γ t pq...s t pq...r s Γ − ··· − ∂θl rn − ∂θl − | nl sj...k i is...k j ij...s k + t pq...r n Γ + t pq...r n Γ + + t pq...r n Γ | sl | sl ··· | sl ij...k s ij...k s ij...k s t sq...r n Γ t ps...r n Γ t pq...s n Γ . − | pl − | ql − ··· − | rl The Laplacian of a n-order tensor is another n-order tensor.

Example A.13. JJJJJ In the same way we proved the lemma in Example A.12 we can show that if the components of a given tensor t in a Cartesian zˆα system have zero second derivatives, i.e. { } ∂2tˆαβ...γ πσ...τ =0 ∂zν ∂zµ then in any curvilinear coordinate system θi in the Euclidean space, we get { } ij...k t pq...r nl =0. | JJJJJ 240 Nonlinear continua A.10 Rotor of a tensor

Let t be a general n-order tensor,

ij...k p q r t = t pq...r g g g g g g , (A.68) i j ··· k ··· we deﬁne the rotor of the tensor t as:

n ∂ ij...k p q r t = g t pq...r g g g g g g (A.69a) ∇ × ∂θn × i j ··· k ··· n ij...kh p q r i = g t pq...r g g g g g g . × |n i j ··· k ··· Using Eq. (A.41),

ij k nl m p q r t = εnim t · pq r l g g g g g g g . (A.69b) ∇ × · | j ··· k ··· The rotor of a n ordertensorisanothern order tensor. In the particular− case of a vector −

j ni k ijk v = εijk v n g g = ε vj i g , (A.70) ∇ × | | k the rotor of a vector is a vector.

A.11 The Riemann-Christoﬀel tensor

Using Eqs. (A.67a-A.67b) to calculate the Laplacian of an arbitrary vector v, we obtain: 2 i nl v = v nl g g (A.71a) ∇ | i where

2 i s i i ∂ v ∂v i ∂v s v nl = + Γ Γ (A.71b) | ∂θn ∂θl ∂θl sn − ∂θs nl ∂vs ∂Γi + Γ i + sn Γ i Γ p + Γ p Γ i vs . ∂θn sl ∂θl − sp nl sn pl ∙ ¸ i Using the quotient rule it is easy to show that the v nl are the mixed components of a third-order tensor. | s s Since we are working in the Euclidean space where Γnl = Γln (Eq.(A.49)), we write

i i i i ∂Γsn ∂Γsl t i t i s v nl v ln = + Γ Γ Γ Γ v . (A.72a) | − | ∂θl − ∂θn sn tl − sl tn ∙ ¸ Using again the quotient rule, we realize that the term between brackets on the r.h.s. of the above equation contains the mixed components (one A.11 The Riemann-Christoﬀel tensor 241 contravariant index and three covariant ones) of a fourth-order tensor: the Riemmann-Christoﬀel tensor (R). Hence,

i i i s v nl v ln = R v . (A.72b) | − | sln In any Cartesian system, we have

2 α α ∂ v v νµ = (A.73a) | ∂zν ∂zµ 2 α α ∂ v α v µν = = v νµ (A.73b) | ∂zµ ∂zν | and therefore, using the result in Example A.13, in any curvilinear system in the Euclidean space,wehave

i i v nl v ln =0. (A.73c) | − | Therefore, i Rsln =0, (A.73d) that is to say, in the Euclidean space,

R = 0 . (A.74)

In the Euclidean space, we can also prove that the following relation holds

s vi nl vi ln = R vs (A.75a) | − | inl where ∂Γs ∂Γs Rs = il in + Γ s Γ t Γ s Γ t . (A.75b) inl ∂θn − ∂θl tn il − tl in We can use the metric tensor components to lower the contravariant index; hence, s Rijkl = gsi R jkl . (A.76a) Therefore,

s s ∂Γjl ∂Γjk s t s t Rijkl = gsi + Γ Γ Γ Γ . (A.76b) ∂θk − ∂θl tk jl − tl jk ∙ ¸

We now define the Christoffel symbol of the first kind, Γijk,as:

s Γijk = gsk Γij (A.77a) s sk Γij = g Γijk (A.77b) using the above in Eq. (A.76b) we get, 242 Nonlinear continua

∂Γjli ∂Γjki s ∂gsi s ∂gsi Rijkl = + Γ Γski + Γ Γsli . ∂θk − ∂θl jl − ∂θk jk ∂θl − ∙ ¸ ∙ (A.77c)¸ It is very important to realize that Eqs.(A.77a) and (A.77b) are not standard operations to go from contravariant tensorial components to covariant tensorial components and vice versa because we have already established that the Christoﬀel symbols are not tensorial components. The result in Example A.9 can now be rewritten as:

∂gsi = Γsli + Γils . (A.77d) ∂θl Using the above in Eq. (A.77c) and taking into account that in the Euclidean space Γils = Γlis,weget

∂Γjli ∂Γjki s s Rijkl = Γ Γkis + Γ Γlis . (A.77e) ∂θk − ∂θl − jl jk In what follows, we will prove the identities:

(i) Rijkl = Rijlk , (A.78a) − (ii) Rijkl = Rjikl , (A.78b) − (iii) Rijkl = Rklij . (A.78c)

(i) Rijkl = Rijlk Using Eq. (A.77e),− we write

∂Γjli ∂Γjki s s Rijkl = Γ Γkis + Γ Γlis ∂θk − ∂θl − jl jk

∂Γjki ∂Γjli s s = Γ Γlis + Γ Γkis − ∂θl − ∂θk − jk jl ∙ ¸ = Rijlk . (A.79a) −

(ii) Rijkl = Rjikl − s s Since we are working in the Euclidean space, Γij = Γji and Γijk = Γjik. Also, we can rewrite the result of Example A.10 as:

1 ∂gac ∂gbc ∂gab Γabc = + . (A.79b) 2 ∂θb ∂θa − ∂θc µ ¶ Using the above in Eq.(A.77e), we obtain, after some algebra: A.12 The Bianchi identity 243

2 2 2 2 1 ∂ gli ∂ gjk ∂ gjl ∂ gki Rijkl = + 2 ∂θj ∂θk ∂θi ∂θl − ∂θi ∂θk − ∂θj ∂θl ∙ ¸ sa + g [ Γjka Γlis Γjla Γkis ] . (A.79c) − Changing the order of the indices, we obtain

2 2 2 2 1 ∂ glj ∂ gik ∂ gil ∂ gkj Rjikl = + 2 ∂θi ∂θk ∂θj ∂θl − ∂θj ∂θk − ∂θi ∂θl ∙ ¸ sa + g [ Γika Γljs Γila Γkjs ] − 1 ∂2g ∂2g ∂2g ∂2g = li + jk jl ki − 2 ∂θj ∂θk ∂θi ∂θl − ∂θi ∂θk − ∂θj ∂θl ∙ ¸ sa g [ Γlia Γjks Γkia Γjls ] − − = Rijkl . (A.79d) −

(iii) Rijkl = Rklij Using Eq. (A.79c), we can write

2 2 2 2 1 ∂ gjk ∂ gli ∂ glj ∂ gik Rklij = + 2 ∂θl ∂θi ∂θk ∂θj − ∂θk ∂θi − ∂θl ∂θj ∙ ¸ sa + g [ Γlia Γjks Γlja Γiks ] − 1 ∂2g ∂2g ∂2g ∂2g = li + jk jl ki 2 ∂θj ∂θk ∂θi ∂θl − ∂θi ∂θk − ∂θj ∂θl ∙ ¸ sa + g [ Γjks Γlia Γjla Γkis ] − = Rijkl . (A.79e)

A.12 The Bianchi identity

A second-order tensor g canbeconsideredametrictensorinaEuclidean space if it fulﬁlls the set of equations Rijkl =0, derived from Eq.(A.74). However, between those equations, certain relations exist that we are going to demonstrate in this Section. Using Eq. (A.55b), we can write

i i ∂R jkl s i i s R m = + R Γ R Γ jkl| ∂θm jkl sm − skl jm Ri Γ s Ri Γ s , (A.80a) − jsl km − jks lm and with the help of Eq. (A.75b), we get, in the Euclidean space, 244 Nonlinear continua

2 i 2 i i s i ∂ Γjl ∂ Γjk ∂Γsk s i ∂Γjl R m = + Γ + Γ jkl| ∂θk ∂θm − ∂θl ∂θm ∂θm jl sk ∂θm i s s s ∂ m s i ∂Γjk ∂Γjl i ∂Γjk i ∂θ Γjk Γsl m + k Γsm l Γsm − Γ sl − ∂θ ∂θ − ∂θ ∂Γi ∂Γi + Γ s Γ t Γ i Γ s Γ t Γ i sl Γ s + sk Γ s tk jl sm − tl jk sm − ∂θk jm ∂θl jm ∂Γi ∂Γi Γ i Γ t Γ s + Γ i Γ t Γ s jl Γ s + js Γ s − tk sl jm tl sk jm − ∂θs km ∂θl km ∂Γi ∂Γi Γ i Γ t Γ s + Γ i Γ t Γ s js Γ s + jk Γ s − ts jl km tl js km − ∂θk lm ∂θs lm Γ i Γ t Γ s + Γ i Γ t Γ s . (A.80b) − tk js lm ts jk lm i i We can develop similar expressions for R jlm k and R jmk l and remem- a a | | bering that in the Euclidean space Γbc = Γcb , we ﬁnally obtain the Bianchi identity: i i i R m + R k + R l =0. (A.81) jkl| jlm| jmk| Starting from, p Rijkl = gpi R jkl (A.82a) and using the result in Example A.11, we have

p Rijkl m = gpi R m . (A.82b) | jkl| Hence,

p p p Rijkl m + Rijlm k + Rijmk l = gpi R m + R k + R l | | | jkl| jlm| jmk| ³ (A.82c)´ and using Eq.(A.81), we get

Rijkl m + Rijlm k + Rijmk l =0. (A.82d) | | | It is worth noting that the Bianchi identities are not restricted to Euclidean spaces and can be demonstrated in other spaces in which the second Christoﬀel symbol is also symmetric (e.g. Riemmanian spaces (McConnell 1957)).

A.13 Physical components

In an arbitrary curvilinear system θi , we can write the n-order tensor t , using its contravariant components{ and} the covariant base vectors,

t = tij...k g g g . (A.83) i j ··· k A.13 Physical components 245

In general, the covariant base vectors: (i) Do not have a unitary modulus. (ii) Are not dimensionally homogeneous.

Example A.14. JJJJJ In a cylindrical coordinate system where θ1 is the radius, θ2 the polar angle and θ3 z3,we can write ≡ g =cosθ2 e +sinθ2 e 1 1 2 g = θ1 sin θ2 e + θ1 cos θ2 e 2 − 1 2 g = e . 3 3 Therefore,

g =1 1 ¯ ¯ ¯g ¯ = θ1 ¯ 2¯ ¯ ¯ ¯g ¯ =1 ¯ 3¯ ¯ ¯ ¯ ¯ which are obviously not dimensionally¯ ¯ homogeneous. JJJJJ

We can rewrite Eq. (A.83) as:

3 3 3 g g g ij...k i j k t = t √gii √gjj √gkk . ··· ··· gii gjj ··· gkk i=1 j=1 k=1 √ √ √ X X X (A.84a) In the above equation, we did not use the summation convention to avoid misinterpretations. Obviously, g i =1, (A.84b) g ¯ √ ii ¯ ¯ ¯ and therefore the terms ¯ ¯ ¯ ¯ ij...k t = t √gii √gjj √gkk ( noadditiononi,j...k) (A.84c) ··· are the projections of the tensor t on base vectors of unitary modulus. The terms t are known as the physical components of the tensor t. It is obvious that the above-deﬁned physical components are not tensorial components and therefore, when the coordinate system is changed the physical components cannot be transformed using either a covariant or a contravariant transformation rule.

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Index

acceleration, 10 Bianchi identity, 243 convected, 104 body-attached loads, 187 material, 11 Bridgman experimental observations, admissible displacements, 184 140, 144 admissible transformation, 213 buckling, 188, 199 Almansi deformation tensor, 35, 49, 58 pull-back, 43 Cartesian coordinates, 5, 213 associated plastic flow, see associative Cauchy elastic material, see elastic plastic flow material model associative plastic flow, 144 Cauchy stress tensor, 69—72, 78 symmetry, 107 back-stress tensor, 142, 161 Cauchy Theorem, 71 balance of energy, 109 chemical energy, 109 Eulerian (spatial) formulation, 109 classical objectivity, see objectivity Lagrangian (material) formulation, Clausius-Duhem inequality, 162 112 compatibility, 61 localized material form, 112 configuration localized spatial form, 111 reference, 8, 10, 11, 22, 33—35 balance of moment of momentum, 105 spatial, 10, 11, 25, 33, 35 Eulerial (spatial) formulation, 105 configurations, 7 localized spatial form, 107 conjugate stress/strain rate measures, symmetry of stress measures, 107 72 balance of momentum consecutive rotations, 52 Eulerian (spatial) formulation, 96 conservative loads, 195, 197 Lagrangian (material) formulation, consistency equation, 150, 170 103 constant direction load, 187 localized material form, 104 constitutive relations, 115 localized spatial form, 97 continuity equation, 93 balance of momentum principle, 95 continuous body, 7 balance principles, 85 continuous media base vectors, 216 hypothesis, 7 contravariant, 218 kinematics, 7 covariant, 216 contravariant transformation rule, 214 Bernstein formula, 157 controlsurface,88 256 Index control volume, 88 elastoplastic material model convex yield surface, 142 1D case, 135 coordinates finite strains, 155 convected, 13, 34 general formulation, 140 Eulerian, 52 infinitesimal strains, 135 fixed Cartesian, 11 thermal effects, 170 Lagrangian, 52 energy conjugate, 73, 75, 76, 79 material, 8, 14 energy dissipation, 162 spatial, 8, 11, 14 entropy, 167 coordinates transformation, 213 equilibrium, see balance of momentum corotational stress rate, 84 principle covariance, 44, 49 equipresence, 116 covariant derivative, 233 equivalent plastic strain, 153 of a general tensor, 236 equivalent plastic strain rate, 152 of a vector, 233 equivalent stress, 152 covariant rates, 58 essential boundary conditions, 205 covariant transformation rule, 215 Euclidean space, 62, 213 cross product between vectors, 229 event, 45 curvilinear coordinates, 5, 213 fiber, 14 damage mechanics, 115 Finger deformation tensor, 25, 33, 58 decomposition first law of Thermodynamics, see left polar, 25 balance of energy multiplicative, 23, 25 flow rule, 140, 142, 144, 177 polar, 21 follower load, 187 numerical algorithm, 28 forces right polar, 22 concentrated, 70 deformation gradient tensor, 13, 47 external, 67 inverse, 14 internal, 67 multiplicative decomposition, 158 per unit mass, 68 transpose, 14 surface, 68 density, 9 Fourier’s law, 169 deviatoric components free energy, 162, 165 Cauchy stress tensor, 141 frictional material, 148 Green-Lagrange strain tensor, 126 generalized Gauss’ theorem, 88 displacement vector, 9 gradient of a tensor, 237 distribuited torques, 69 Green deformation tensor, 21, 33 divergence of a tensor, 238 Green elastic material, see hyperelastic Doyle-Ericksen formula, 122 material model Drucker’s postulate, 142, 144, 145, 147 Green-Lagrange strain tensor, 34, 48 dyads, 224 Green-Naghdi stress rate, 84 eigenvalues, 28—33 hardening law, 135, 140, 151, 162, 172, eigenvectors, 28—33, 230 175 elastic energy, 109, 120 heat flux, 110 elastic material model, 120 heat source, 110 elasticity tensor, 123 Helmholtz’s free energy, 167 spatial, 123 Hencky strain tensor, 35, 56, 161 symmetries, 124 time derivative, 79 Index 257 homeomorphism, 8 mapping, 8 hydrostatic component mass, 9 Cauchy stress tensor, 141 mass-conservation principle, 74, 93 hyperelastic material model, 120 Eulerian (spatial) formulation, 93 hyperelasticity, 115, 120 Lagrangian (material) formulation, hypoelastic material model, 121, 155 95 localized material form, 95 incompressible flow, 180 localized spatial form, 93 incrementalformulation,164,189 material infinitesimal strain tensor, 65 isotropic, 79, 81 instant, 7 material derivative, 12 internal energy, 109 material particle, 7, 14 isocoric deformation, 21 material surface, 88 isometric transformation, 45 material time derivative, 11, 12 isotropic hardening, 151, 161, 172 material-frame indifference, 116 isotropic materials, 125 mathematical model, 2 linear, 2 J2-yield function, see von Mises yield function nonlinear, 2 Jacobian, 213 metric, 219 of the transformation, 19 Cartesian coordinates, 220 time rate, 86 curvilinear coordinates, 220 Jaumann stress rate, 84 metric tensor, 15, 57, 220, 228 jump discontinuity, 90 contravariant components, 220 condition, 93 covariant components, 220 mixed components, 221 kinematic constraints, 207, 209 pull-backs of the spatial, 43 kinematic evolution, 50 push-forward of the reference, 44 kinematic hardening, 151, 154, 161, 170, minimum potential energy principle, 175 197 kinetic energy, 73 momentum conservation principle, see Kirchhoff stress tensor, 74, 82, 84 balance of momentum principle Kotchine’s theorem, 93 Mooney-Rivlin material model, 131 motion,8,14 Lagrange criterion, 88 continuous body, 9, 10 Lagrange multipliers, 207 Eulerian description, 11, 12 physical interpretation, 208 Lagrangian description, 10—12 Lagrangian system, 51 regular, 8 Laplacian of a tensor, 239 moving control volume Lee’s multiplicative decomposition, 158 Left Cauchy-Green deformation tensor, energy conservation, 111 see Finger deformation tensor mass conservation, 94 left stretch tensor, 25, 28, 48 momentum conservation, 99 physical interpretation, 26 multiplicative decomposition of the pull-back, 43 deformation gradient, 158 Levi-Civita tensor, 229 Lie derivative, 56, 58, 82, 84 n-poliad, 227 local action, 117 Nanson formula, 100 Logarithmic strain tensor, see Hencky natural boundary conditions, 203, 205, strain tensor 207 258 Index neo-Hookean material model, 131—133 polar media, 70 Newtonian fluids, 180 postbuckling, 201 no-slip condition, 181 potential energy, 195 nonassociated plasticity, 144, 149, 151 power, 72 nonconservative loads, 198 principle of maximum plastic dissipa- nonconvex yield surface, 147 tion, 143 nonpolar media, 70 principle of stationary potential energy, notation, 5 195 numerical model, 2 principle of virtual power, 194 principle of virtual work, 183 objective geometrically nonlinear problems, 186 physical law, 50 projection theorem, see reciprocal rates, 58 theorem of Cauchy stress rate, 61, 81 proper transformation, 214 objectivity, 44 pull-back, 36, 75, 79 classical, 47 strain measures, 43 criteria, 47 tensor components, 40 observation frame, see reference frame vector components, 36 Ogden hyperelastic material model, 129 push-forward, 42 Oldroyd stress rate, 81, 82 strain measures, 43 orthotropic material, 125 quotient rule, 232 perfect fluid, 98 Rayleigh-Ritz method, 205 Euler equation, 98 reciprocal theorem of Cauchy, 72 perfectly plastic material, 147 reference frame, 45 permanent deformations, see plastic Reynolds’ transport theorem, 85 deformations discontinuity surface, 90 permutation tensor, see Levi-Civita generalized, 88 tensor Riemann-Christoffel tensor, 62, 240 physical components, 244 right Cauchy-Green deformation tensor, physical phenomena see Green deformation tensor observation, 1 right stretch tensor, 22, 28, 48 quantification, 1 physical interpretation, 26 Piola identity, 102 rigid boundary conditions , see essential Piola Kirchhoff stress tensor bounday conditions first rigid rotation, 15 symmetry, 107 rigid translation, 15 second rotation tensor, 23, 26, 48 symmetry, 107 physical interpretation, 26 Piola-Kirchhoff stress tensor rotor of a tensor, 240 first, 74 second, 76, 83 Serrin representation, 32 plastic deformation, 135 shear modulus, 126 plastic dissipation, 143, 164 softening material, 137, 147 maximization, 143, 164 space-attached loads, 187 Kuhn-Tucker conditions, 143, 164 spatial derivative, 12 plasticity, 115 spin tensor, see vorticity tensor point, see material particle stable materials, 144 polar decomposition, see decomposition strain measures, 33 Index 259 strain rate effect, 176 Truesdell stress rate, 82 strain rate tensor, 51 strain rates, 50 updated Lagrangian formulation, 190, stress tensor, 79 192 stresses, 67 stresses power, 73 variational calculus, 183 symmetry of stress measures, 107 variational consistency, 210 variational methods, 183 variations, 184 tangential constitutive tensor, 150 vector analysis, 213 tensor analysis, 213 vector components, 216 tensors vectors, 215 covariant, 59 velocity, 10 Eulerian, 47, 58 material, 10 isotropic, 125, 156 velocity gradient tensor, 50 Lagrangian, 47, 50 Veubeke-Hu-Washizu variational n-order, 227 principles, 209 orthogonal, 23 constitutives constraints, 211 physical components, 245 kinematic constraints, 209 second-order, 223 virtual displacements, 184 eigenvalues and eigenvectors, 225 virtual strains, 185 symmetric, 22 virtual work, 185 two-point, 14, 23, 47, 50, 74 viscoelasticity, 115 thermal energy, 109 viscoplasticity, 115, 176 thermo-elastoplastic constitutive model, volumetric component 170 Green-Lagrange strain tensor, 126 thermoelastic constitutive model, 167 volumetric modulus, 126 time, 7 von Mises yield function, 140, 141 time rates, 50 vorticity tensor, 51 total Lagrangian formulation, 190 total-Lagrangian Hencky material work hardening, 147 model, 166 traction, 70 yield criterion, 135, 144, 161, 170 transformation yield surface, 140, 144, 148, 171, 177 isometric, 45 Young’s modulus, 126