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Deformation Theory of Plasticity Revisited CRNOGORSKA AKADEMIJA NAUKA I UMJETNOSTI GLASNIK ODJELJENJA PRIRODNIH NAUKA, 13, 1999. QERNOGORSKAYA AKADEMIYANAUK I ISSKUSTV GLASNIK OTDELENIYA ESTESTVENNYH NAUK, 13, 1999. THE MONTENEGRIN ACADEMY OF SCIENCES AND ARTS GLASNIK OF THE SECTION OF NATURAL SCIENCES, 13, 1999 UDK 539.319 Vlado A. Lubarda Deformation Theory of Plasticity Revisited A b s t r a c t Deformation theory of plasticity, originally intro duced for in nitesi- mal strains, is extended to encompass the regime of nite deforma- tions. The framework of nonlinear continuum mechanics with loga- rithmic strain and its conjugate stress tensor is used to cast the formu- lation. A connection between deformation and ow theory of metal plasticity is discussed. Extension of theory to pressure-dep endent plasticity is constructed, with an application to geomechanics. Deriva- tions based on strain and stress decomp ositions are b oth given. Dual- ity in constitutive structures of rate-typ e deformation and ow theory for ssured ro cks is demonstrated. Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA Deformation Theory of Plasticity Revisited 2 OSVRTNADEFORMACIONU TEORIJU PLASTICNOSTI I z v o d U radu je data formulacija deformacione teorije plasticnosti ko ja obuh- vata oblast konacnih deformacija. Meto di nelinearne mehanika kon- tinuuma, logaritamska mjera deformacije i njen konjugovani tenzor nap ona su adekvatno up otrebljeni u formulaciji teorije. Veza izmedju deformacione i inkrementalne teorije plasticnosti je diskutovana na primjeru p olikristalnih metala. Teorija je zatim prosirena na oblast plasticnosti koja zavisi o d pritiska, sa primjenom u geomehanici. For- mulacije na bazi dekomp ozicija tenzora deformacije i nap ona su p osebno date. Dualnost konstitutivnih struktura deformacione i inkrementalne teorije je demonstrirana na mo delu stijenskih masa. INTRODUCTION Commonly accepted theory used in most analytical and computa- tional studies of plastic deformation of metals and geomaterials is the so-called ow theory of plasticity (e.g., Hill, 1950,1978; Lubliner, 1992; Havner, 1992). Plastic deformation is a history dep endent phe- nomenon, characterized by nonlinearity and irreversibility of under- lining physical pro cesses (Bell, 1968). Consequently,in ow theory of plasticity the rate of strain is expressed in terms of the rate of stress and the variables describing the current state of material. The over- all resp onse is determined incrementally byintegrating the rate-typ e constitutive and eld equations along given path of loading or defor- mation (Lubarda and Lee, 1981; Lubarda and Shih, 1994; Lubarda and Kra jcinovic, 1995). There has b een an early theory of plasticity suggested by Hencky (1924) and Ilyushin (1947,1963), known as deformation theory of plas- ticity,inwhich total strain is given as a function of total stress. Such constitutive structure, typical for nonlinear elastic deformation, is in Deformation Theory of Plasticity Revisited 3 general inappropriate for plastic deformation, since strain there de- p ends on both stress and stress history, and is a functional rather than a function of stress. However, deformation theory of plasticity found its application in problems of prop ortional or simple loading, in which all stress comp onents increase prop ortionally, or nearly so, without elastic unloading ever o ccurring (Budiansky, 1959; Kachanov, 1971). The theory was particularly successful in bifurcation studies and determination of necking and buckling loads (Hutchinson, 1974). Deformation theory of plasticitywas originally prop osed for non- linear but in nitesimally small plastic deformation. An extension to nite strain range was discussed by Storen and Rice (1975). The purp ose of this pap er is to provide a formulation of the rate-typ e deformation theory for pressure-dep endent and pressure-indep endent plasticity at arbitrary strains. After needed kinematic and kinetic background is intro duced, the logarithmic strain and its conjugate stress are conveniently utilized to cast the formulation. Relationship between the rate-typ e deformation and ow theory of metal plasticity is discussed. A pressure-dep endent deformation theory of plasticity is constructed and compared with a non-asso ciative ow theory of plasticity corresp onding to the Drucker-Prager yield criterion. Devel- opments based on strain and stress decomp ositions are b oth given. Duality in the constitutive structures of deformation and ow theory for ssured ro cks is demonstrated. 1 KINEMATIC PRELIMINARIES The lo cations of material points of a three-dimensional body in its undeformed con guration are sp eci ed by vectors X. Their lo ca- tions in deformed con guration at time t are sp eci ed by x, such that x = x(X;t) is one-to-one deformation mapping, assumed twice continuously di erentiable. The comp onents of X and x are mate- rial and spatial co ordinates of the particle. An in nitesimal material Deformation Theory of Plasticity Revisited 4 elementdX in the undeformed con guration b ecomes @ x dx = F dX; F = (1.1) @ X in the deformed con guration at time t. Physically p ossible deforma- tion mappings have p ositivedetF,henceF is an invertible tensor; dX 1 can b e recovered from dx byinverse op eration dX = F dx. By p olar decomp osition theorem, F is decomp osed into the pro d- uct of a prop er orthogonal tensor and a p ositive-de nite symmetric tensor, such that (Truesdell and Noll, 1965) F = R U = V R: (1.2) Here, U is the right stretch tensor, V is the left stretch tensor, and T , so that U R is the rotation tensor. Evidently, V = R U R and V share the same eigenvalues (principal stretches ), while their i eigenvectors are related by n = R N . The right and left Cauchy- i i Green deformation tensors are T 2 T 2 C = F F = U ; B = F F = V : (1.3) If there are three distinct principal stretches, C and B have their sp ectral representations (Marsden and Hughes, 1983) 3 3 X X 2 2 n n : (1.4) N N ; B = C = i i i i i i i=1 i=1 2 STRAIN TENSORS Various tensor measures of strain can b e intro duced. A fairly general de nition of material strain measures is (Hill, 1978) 3 X 1 1 2n 0 2n E = U I = 1 N N ; (2.1) i i (n) i 2n 2n i=1 where 2n is a p ositive or negative integer, and and N are the i i principal values and directions of U. The unit tensor in the initial Deformation Theory of Plasticity Revisited 5 0 con guration is I . For n = 1, Eq. (2.1) gives the Lagrangian or 2 0 Green strain E = (U I )=2, for n = 1 the Almansi strain (1) 0 2 E = (I U )=2, and for n = 1=2 the Biot strain E = (1) (1=2) 0 (U I ). The logarithmic or Hencky strain is 3 X E =lnU = ln N N : (2.2) i i i (0) i=1 A family of spatial strain measures, corresp onding to material strain measures of Eqs. (2.1) and (2.2), are 3 X 1 1 2n 2n E E = V I = 1 n n ; (2.3) i i (n) i 2n 2n i=1 3 X E E =lnV = ln n n : (2.4) i i i (0) i=1 The unit tensor in the deformed con guration is I, and n are the i 2 E E principal directions of V . For example, E =(V I)=2, and E = (1) (1) 2 (I V )=2, the latter b eing known as the Eulerian strain tensor. 2n T 2n Since U = R V R, and n = R N , the material and i i spatial strain measures are related by T T E E E = R EE R; E = R EE R; (2.5) (n) (n) (0) (0) i.e., the former are induced from the latter by the rotation R. Consider a material line element dx in the deformed con guration at time t. If the velo city eldisv = v (x;t), the velo cities of the end p oints of dx di er by 1 _ dv = L dx; F F : (2.6) The tensor L is called the velo city gradient. Its symmetric and an- tisymmetric parts are the rate of deformation tensor and the spin tensor 1 1 T T D = L + L ; W = L L : (2.7) 2 2 Deformation Theory of Plasticity Revisited 6 3 CONJUGATE STRESS TENSORS For any material strain E of Eq. (2.1), its work conjugate stress (n) T is de ned such that the stress p ower p er unit initial volume is (n) _ T : E = : D; (3.1) (n) (n) where = (det F) is the Kirchho stress. The Cauchy stress is denoted by . For n =1,Eq. (3.1) gives 1 T 1 1 T = F F = U ^ U : (3.2) (1) T The stress ^ = R R is induced from by the rotation R. Similarly, T T = F F = U ^ U: (3.3) (1) More involved is an expression for the stress conjugate to logarithmic strain, although the approximation 2 T =^ + O E ^ (3.4) (0) (n) may be acceptable at mo derate strains. If deformation is such that principal directions of V and are parallel, the matrices E and T (n) (n) commute, and in that case T =^ exactly (Hill, 1978). If principal (0) directions of U remain xed during deformation, 1 _ _ ^ E = U U = D; T =^ : (3.5) (0) (0) E The spatial strain tensors E in general do not have their conju- (n) _ _ T T E gate stress tensors T suchthat T : E = T : E .
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