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Chapter 8Large-Deformation PlasticitySection 8.1 Large-Deformation Continuum Mechan-ics8.1.1 Continuum DeformationAs a model of mechanical behavior, plasticity theory is applicable primar-ily to those solids that can experience inelastic deformations considerablygreater than elastic ones. But the resulting total deformations, and the rota-tions accompanying them, may still be small enough so that many problemscan be solved with small-displacement kinematics, and this is the situationthat has, with a few exceptions, thus far been addressed in this book. How-ever, when strains or rotations become so large that they cannot be neglectednext to unity, the mechanician must resort to the theory of large or nitedeformations.Since the reference and displaced congurations of a body, as discussedin Section 1.2, may be quite dierent, it is appropriate to use a notation thatmakes the dierence apparent, a notation based on that introduced by Noll[1955] and made current by the monograph of Truesdell and Toupin [1960].A material point is denoted simply X, and its Lagrangian coordinates aredenoted XI (I = 1, 2, 3); a point in the current or displaced conguration isdenoted x, and its Cartesian coordinates (sometimes called Eulerian coordi-nates) are xi (i = 1, 2, 3). Dierent indices are used for the two coordinatesystems because the corresponding bases, the Lagrangian basis (eI) and theEulerian basis (ei), are, in principle, independent of each other.Whenever direct notation is used for tensors in this chapter, vectors aretreated as column matrices and second-rank tensors as square matrices; thusuTv is used for the scalar product u v, while AB denotes the second-ranktensor with components AikBkj. Furthermore, the scalar product of twosecond-rank tensors is A : B = tr (ATB). If is a fourth-rank tensor,465466 Chapter 8 / Large-Deformation Plasticitythen : A denotes the second-rank tensor with components ijklAkl. Inparticular, a fourth-rank tensor with components Uij/Vkl will be denotedU/V, and the second-rank tensor with components Uij/VklAkl will bedenoted (U/V) : A.Deformation GradientThe motion of the body is described by the functional relationx = (X, t).When is continuously dierentiable with respect to X, then the defor-mation gradient at X is the second-rank tensor F(X, t) whose Cartesiancomponents areFiI = iXI.Note that these components are evaluated with respect to both bases simul-taneously; in the terminology of Truesdell and Toupin [1960], the deforma-tion gradient is a two-point tensor.If, in a neighborhood of the material point X, the function (X, t) is in-vertible in other words, if the material points in the neighborhood are inone-to-one correspondence with their displaced positions then, by the im-plicit function theorem of advanced calculus, the matrix of componentsof F(X, t) (the Jacobian matrix) must be nonsingular, that is, J(X, t) = 0,where J(X, t)def= det F(X, t) is the Jacobian determinant. If we consideronly displaced congurations that can evolve continuously from one another,then since J = 1 when the displaced and reference congurations coincide,we obtain the stronger condition J(X, t) > 0.The inverse of F(X, t), denoted F1(X, t), has the components F1Ii (X, t)= XI/xi|x=(X, t). Note that F1is also a two-point tensor, but of a dif-ferent kind from F: while the components of the latter (FiI) are such thatthe rst index refers to an Eulerian and the second to a Lagrangian basis,in the former it is the reverse (F1Ii ).Local DeformationConsider two neighboring material points X and X

= X+u, where u isa small Lagrangian vector emanating from X. If the displaced positionsof X and X

are respectively given by x and x

, then, F being continuous,we have (in the matrix-based direct notation that is used throughout thischapter)x

= x +F(X, t)u +o(|u|) as |u| 0.Since a rigid-body displacement does not change distances between points,Section 8.1 / Large-Deformation Continuum Mechanics 467let us compare the distance between x and x

with that between X and X


x| =_(x


x)=_uTFT(X, t)F(X, t)u +o(|u|2)=_uTC(X, t)u +o(|u|),where Cdef= FTF is known, in the NollTruesdell terminology, as the rightCauchyGreen tensor. The notation will henceforth be simplied by writingF for F(X, t) and so on. The components of C are given byCIJ = i,I i,J ,where (),I = ()/XI. C is a Lagrangian tensor eld which, moreover, issymmetric (C = CT) and positive denite (uTCu > 0 for any u = 0).Stretch and StrainThe stretch at (X, t) along a direction u is dened byu = limh0|(X+hu, t) (X, t)|h|u| =uTCuuTu .We have u = 1 for every u if and only if C = I; then the displacement islocally a rigid-body displacement. If u()( = 1, 2, 3) are the eigenvectorsof C and if def= u() , then the 2 are the eigenvalues of C, and the (the principal stretches) are the eigenvalues of Udef= C12.A strain is a measure of how much a given displacement diers locallyfrom a rigid-body displacement. In particular, a Lagrangian strain is ameasure of how much C diers from I. The following Lagrangian straintensors are commonly used: (a) the GreenSaint-Venant strain tensor1al-ready mentioned in Section 1.2, E = 12(C I), with eigenvalues 12(2 1);(b) the conventional strain tensor Ee = U I, with eigenvalues 1(the principal conventional strains); and (c) the logarithmic strain tensorEl = lnU, with eigenvalues ln (the principal logarithmic strains). Notethat all three strains may be regarded as special cases of (1/k)(UkI), with(a), (b), and (c) corresponding respectively to k = 2, k = 1 and the limit ask 0. Furthermore,1k(k1) = 1k[(1 + 1)k1] = 1 +o(| 1|),so thatEeEl_ = E+o(||E||),1Often called simply the Lagrangian strain tensor.468 Chapter 8 / Large-Deformation Plasticitywhere ||E|| denotes some measure of the magnitude (a norm) of E. In otherwords, the dierent Lagrangian strain tensors are approximately equal whenthey are suciently small. For large strains the GreenSaint-Venant straintensor is analytically the most convenient, except in cases where the principaldirections u()are known a priori.Since U is symmetric and positive denite, we can form the two-pointtensor R = FU1. Note thatRTR = U1FTFU1= U1U2U1= I,that is, R is orthogonal. Also, det U = J, so that det R = J/J = 1, andconsequently, R is a proper orthogonal tensor, or a rotation. The decompo-sition F = RU is the right polar decomposition of F, and U is called theright stretch tensor; this is why C is called the right CauchyGreen tensor.R is usually called simply the rotation tensor.If the displacement is locally a rigid-body one, then we simply haveF = R. To study the general case, let us consider points near X given byX()= X+hu(),where u()is, as above, an eigenvector of C. The displaced images of thesepoints arex()= x +hv()+o(h),wherev()= Fu()= RUu().Since, however, u()is also an eigenvector of U, it follows that Uu()=u(), and thereforev()= Ru(),so that R represents the rotation of the eigenvectors of C.For an arbitrary vector u, Uu is not in general parallel to u. In fact, ifwe consider the ellipsoid centered at X, given byuTCu = r2,with principal semiaxes r/, we see that its displaced image is approx-imately the sphere of radius r centered at x. The ellipsoid given by thepreceding equation is called the reciprocal strain ellipsoid.We may also ask what is the eect of the displacement on the sphereof radius r, centered at X, in the reference conguration. If x + v is thedisplaced image of X + u, then v = dotFu, so that uTu = dotvTB1v(where B = FFT), and vTB1v = r2denes an ellipsoid centered at x withprincipal semiaxes r; this ellipsoid is called simply the strain ellipsoid.The tensor B1is called the Finger deformation tensor, while B is calledSection 8.1 / Large-Deformation Continuum Mechanics 469(again in the terminology of Truesdell et al.) the left CauchyGreen tensor.It is an Eulerian tensor, and its eigenvalues are also 2. Its name derives,as may be surmised, from the left polar decomposition of F (see Exercise1). It can easily be shown that B = RCRT.The most commonly used Eulerian strain tensor is the Almansi straintensor Ea = 12(I B1). It is easy to show that E = FTEaF, so thatEa = RU1EU1RT. Hence, in order for Ea to be approximately equal toE (i.e. to have Ea = E+o(||E||)), it is necessary not only for ||E|| but alsofor ||RI|| to be small.In many treatments the right CauchyGreen tensor C is given anotherdenition, namely, as a metric tensor. If the Lagrangian coordinates XIare used to describe points in the displaced body, then they are no longerCartesian coordinates, since the surfaces XI = constant are not necessarilyplanes. An innitesimal vector dx in the displaced body has the square ofits magnitude given bydx dx = CIJdXIdXJ.Hence C is often called the metric tensor in the displaced (strained) body.The identity I is accordingly regarded as the metric tensor in the unstrainedbody. The analysis of continuum deformation based on metric tensors isusually carried out by using curvilinear coordinates to begin with (see, e.g.,Green and Zerna [1968], Eringen [1962], Marsden and Hughes [1983]). Forour purposes, its main usefulness is in the derivation of compatibility con-ditions, that is, the nite-deformation analogue of Equation (1.2.4), and tothat end Cartesian coordinates are adequate. The result is usually given asthe vanishing of a fourth-rank tensor c