Geometrical material structure of elastoplasticity

Pergamon International Journal of Plasticity, Vol. 14, Nos. 1-3, pp. 109-115, 1998

© 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0749-6419/98 $19.00+0.00

PII: S0749-6419(97)00043-0

GEOMETRICAL MATERIAL STRUCTURE OF ELASTOPLASTICITY

G. A. Maugin 1. and M. Epstein 2

ILaboratoire de Modrlisation en Mrcanique, associ6 au C.N.R.S. (URA 229), Universit~ Pierre et Marie Curie, Tour 66, 4 place Jussieu, Case 162, 75252 Paris Crdex 05, France

2Department of Mechanical Engineering, University of Calgary, Calgary, Alberta, B.C., Canada, T2N 1 N4

(Received in final revised form 12 July 1996)

Abstract--G-structures are the geometric backbone of the theory of material uniformity in continuum mechanics. Within this geometrical framework, plasticity is seen as a result of evolving distributions of inhomogeneity with driving force provided by the material Eshelby stress tensor. Constitutive principles governing the time evolution of the G-structure underlying the finite-strain theory of elastoplasticity are proposed together with a thermodynamically admissible formulation. © 1998 Elsevier Science Ltd. All rights reserved

Key words: B. finite strain.

I. INTRODUCTION

An appealing feature of the geometrical approach is that its terminology has been often matched to that of the theory of continuous distributions of dislocations, e.g. in Bilby et al. (1957) and Kr rne r (1958, 1960). Since plasticity is a process of rearrangement of dislocation patterns, it seems natural to at tempt to cast it within his geometrical mold. This is achieved by introducing the essential notion of G-structure (Section III), and identifying the relevant thermodynamical driving force as the Eshelby stress tensor on the material manifold (Section IV). Contact with more traditional presentations involving a multiplicative decomposition of the deformation gradient and an intermediate configuration is then established (Section V). We owe an immense debt to the fundamental experimental and theoretical works of James F. Bell--synthesised in the encyclopedic work of Bell (1968). This short essay is our contribution to J. F. Bell's memorial.

II. PRELIMINARY: G-STRUCTURE

In pure elasticity with an internal energy WR = W(F; X) per unit reference v o l u m e - - where F is the deformation gradient from that configuration KR and X denotes a material point, an element of the material manifold M3--we say that the body B is materially uniform whenever there exists a (uniformity) map P(X) from a reference crystal to the reference configuration of B such that (Epstein and Maugin, 1990; Maugin, 1993)

*Corresponding author.

109

110 G.A. Maugin and M. Epstein

WR = d~l W(FP(X)) (1)

when Jr :=- det P. If G is the material symmetry group of the reference crystal and G a (density preserving) member of G, then P(X)G is also a uniformity map, P(X) = P(X)G is the totality of uniformity fields, and Gx = P(X)Gp-I(x) is the symmetry group of the material at X. In terms of crystallographic bases, we see that a fixed basis in the reference crystal induces at each X, by means of the set P(X), a subset of all the permissible bases at that point. Technically speaking, we have obtained a so-called G-structure, i.e. a reduction of the principal bundle of frames FB at that point (Elzanowski et al., 1990). This sub- bundle has a reduced structural group G C GL (3). Prescribing one particular uniformity field implies choosing one element from each fibre, that is, taking a section of the G- structure. Such sections may not be integrable. In other words, the material G-structure expresses geometrically the distributions of inhomogeneities.

lII. PRINCIPLES OF EVOLUTION

In pure elasticity (e.g. Elzanowski et al., 1990) the material G-structure remains unchanged. Plasticity involves a mechanism which modifies the distribution of material inhomogeneities (dislocation patterns), i.e. the material G-structure. The problem consists then in formulating, possibly on a thermodynamical basis, the law of evolution from one G-structure to another one. This belongs to constitutive modelling but we note the following two physically meaningful principles of formulation (Epstein and Maugin, 1996):

(a) G-covariance. A law of evolution of a G-structure must be form-invariant under parallel changes of sections within the structural group.

(b) Principle of actual evolution. For a law of evolution of a G-structure to prescribe at all times an actual change of inhomogeneity distribution, the inhomogenity velocity gradient

Lr = pp-1 (2)

must not belong to the Lie algebra of the instantaneous symmetry group G.

Comments. The first principle essentially expresses the fact that if a G-structure is given in terms of sections, then the law of evolution is independent of the particular section chosen. Here two sections P(X,t) and Q(X,t) belonging to the same G-structure evolve in parallel if and only if Q(X, t) = P(X, t)G for a certain G E G and for all times if this is true at t= 0. The second means that the proposed evolution law actually prescribes an evolution to a truly different G-structure and not a mere change of section within the same G-structure. A consequence of this in the fully isotropic case (G = 0(3)) is that the evolution must allow for a part of Lr that is not skewsymmetric with respect to the metric

C~I := ppr .

For a discrete symmetry group, any nontrivial law in terms of L r is evolutive.

(3)

Geometrical material structure of elastoplasticity 111

IV. DRIVING FORCE: ESHELBY STRESS

The natural logic whereby Eshelby's tensor emerges in the context of uniformity theory as the driving force behind inhomogeneities has been demonstrated in previous works (e.g. Epstein and Maugin, 1990). With

wR = wR(r ;x ) = j 7 l wco~p). (4)

a change of section that implies a change of G-structure brings about a change of energy evaluated as

OWg/OP = - b P - r (5)

where Tn and b defined in quasi-statics by

Tn = OWn~OF

and

(6)

b - OWR p r = WR1R -- FrT~ (7) 0P

are the first Piola-Kirchhoff stress and material Eshelby stress, respectively (see, e.g. Maugin, 1!993, 1995, for this general notion of Eshelby stress). Noting that b is independent of the P-section chosen within the same G-structure, it appears reasonable to demand that the evolution law for Le be driven by the present value of b. This is indeed the case since, with W replaced by the free-energy density

*R = q'R(F, 0, a; X) (8)

where 0 is the thermodynamical temperature, a is an n-vector of internal variables (see Lubliner, Zl990; Naghdi, 1990 or Maugin, 1992 for this general notion), and we have set

TR = oqJnlOF, S R = --OkIIR/O0, A R = --OqIR/OOl (9)

where SR is the entropy density and An the thermodynamic force conjugate to or, it is found from the Clausius-Duhem inequality that there remains the following dissipation inequality:

¢~PR ~- - t r ( b r L e ) + An& + Oqg.VnO -1 > O. (10)

In this inequality which constrains the material heat flux qR and the evolution laws for Le and & the Eshelby stress b has two symmetries. One is the C-symmetry (C = FrF) so as to satisfy the balance of angular momentum. It reads

bC = Cb r or (bC)A = 0 (11)

where subscript A designates the operation of antisymmetrization.


The second symmetry of b is of material origin and consists of having a zero inner product (mechanical power) with the Lie algebra of Gx (2nd principle above). For instance, in the case of full isotropy (with respect to the reference configuration), this symmetry reads (compare to (11)):

bC? = C?b r or (bCp)A = 0 (12)

As a consequence (see below), thermodynamics imposes a restriction only on that part of LF----cf. eqn (10)--which is orthogonal to two subspaces, defined by (11) and (12), of the space of all tensors.

If

V. COMPARISON WITH OTHER FORMULATIONS

F = F e F p (13)

is the Bilby-Kr6ner-Lee multiplicative decomposition of F (after Bilby et al., 1957; Kr6ner, 1958, 1960; Lee, 1969; see also Clifton, 1972 for a critical look at (13)), then the roles of H = FP and p-1 are played by ~ and ~-¢, respectively, and the "classical" plastic distortion rate

Ee = F ( W ) -1

and the inhomogeneity velocity gradient (2) are related by

~? = _ p - l L?P

(14)

(15)

and (10) is replaced by the dissipation per unit volume of the intermediate configuration Kr~ (see Sidoroff, 1973; Teodosiu, 1975; Teodosiu and Sidoroff, 1976 for this concept):

q~R = t r ( S ~ e L P ) s ) + A~dt -F J pIOqR.VRO-I > 0 (16)

where

q/R = ~I/7~CCe, o~, 0), AT~ = -0~I/7?./~t, S~ = 20tl/7~/0Ce, Ce : = (-F-e) T~-e. (17)

But this last formulation found in several books (Lubliner, 1990; Maugin, 1992 and several other works of great interest in finite-strain elastoplasticity (Lubliner, 1986; Armero and Simo, 1993; Miehe and Stein, 1993; Simo and Miehe, 1993; Tsakmakis, 1996) is not G-covariant. Note that the Eshelby stress tensor was previously introduced in the intermediate configuration at the Fourth International Symposium on Plasticity (Baltimore, 1993; cf. Maugin, 1994).

We refer to (16)-(17) as the "classical" formulation of finite-strain elastoplasticity. In view of the stress contribution to eqn (16), it clearly is tantamount to saying that only the symmetric part of Lp with respect to Ce is thermodynamically constrained by the second law, a statement with which a number of serious authors (Moran et al., 1990; Maugin, 1992; Miehe and Stein, 1993; Van Der Giessen, 1989; Besseling and Van Der Giessen,


1994) agree. In other words, whereas the "plastic strain rate" (CeLe)s--and not (Le)s, an expression without real tensorial meaning--may, for instance, be derived from a pseudo- dissipation ]potential that is positively homogeneous of degree one for rate-independent plasticity (cf. Lubliner, 1986, 1990; Maugin, 1992), the skewsymmetric nice tensorial quantity (C,,Le)A--and not only (Le)a, again an expression without real tensorial meaning--is left indeterminate by thermodynamics for it has no thermodynamically conjugate force. As a :matter of fact, this quantity would not show up at all in the absence of hardening. In still other words, apart from the obvious remark that (Le)A is not a good tensorial quantity, the problem of the "'plastic spin", here (CeLe)A, would not arise, though this has been the object of many studies and judicious remarks (e.g. in Mandel, 1973, 1978; Dafalias, 1985; Cleja-Tigoiu and Soos, 1990, 1996) unfortunately based on the consideration of the ill-fated quantity (Le)A. The question was essentially raised because if the internal variable is chosen to be a true tensor or, e.g. the backstress, then the question is raised of the objectivity of the time derivative &. The solution to this may stem from the introduction of a co-rotational time derivative of ~x, based either on the plastic spin or another suitably defined kinematical quantity such as the rotational velocity of a director triad attached to preferred crystallographic directions as in Mandel (1971, 1973, 1978). But, as already noticed, the formulation (16)-(17) is not G-covariant and, therefore, not fully satisfactory from our viewpoint. What about the G-covariant formulation based on the Clausius-Duhem inequality (10)?

The formulation of finite-strain elastoplasticity that may be derived from the thermodynamical constraint (10) is intrinsically G-covariant and thermodynamically admissible. Furthermore, on account of the general symmetry condition (11), only the symmetric part of Le with respect to C -t is thermodynamically constrained as it is readily shown that

tr(bTLe) =_ tr{(bC)(LeC-l)s }. (18)

Let

B := bC = ~r. (19)

If ot is none other than a material tensor or, then & simply is the material time derivative Oo~/Otlx. Let further ~D --- :D(I~, A) designate a pseudopotential of dissipation that is positively homogeneous of degree n in the generalized (material tensorial) forces ~ and A. Then a G-covariant thermodynamically admissible model is obtained as (disregarding heat condition effects)

(LeC- ' ) s= -i~/~/8~,, a = 0"/~/0A. (20)

A model of rate-independent finite-strain elastoplasticity follows for n = 1 and the pair (~, A) restrained to a closed convex set in the appropriate space. The "plastic-spin-problem" does not show up in this formulation which is completed by noting that, from the definition of b in eqn (7)2

l~ = qJRC + ~ (21)


with

g = CSC = ~r = OqJR/OE (22)

of the free energy q.I R that replaces W (cf. Epstein and Maugin, 1995), is taken as a func- tion qJR(E, or, 0) for a truly materially homogeneous body, and

1 S = T R F - I = S r, E = ~ ( 1 R - C -1) (23)

denote, respectively, the second Piola-Kirchhoff--or thermodynamical--stress and the Piola finite-strain tensor. The latter naturally appears in the formulation of elasticity and derived theories in which the so-called inverse motion X = ×-l(x, t) plays a foremost role (compare to Maugin and Trimarco, 1992). Accordingly, the material Eshelby stress plays a direct and essential role in the accompanying thermomechanical construct. We can claim, by way of conclusion, that as in other thermomechanical theories dealing with the evolution of material inhomogeneities or quasi-inhomogeneities (i.e. phenomena manifest- ing themselves just as material inhomogeneities; these include thermoelastic effects, mag- netostrictive ones, dislocations, cracks, progressing phase-transition fronts)--cf. Maugin (1995)--the Eshelby stress b or its fully covariant counterpart • is the driving force of plasticity, viewed as an evolving inhomogeneity.

Acknowledgements--This research was carried within the framework of NATO CRG 950833. M.E. gratefully acknowledges the support of the Natural Science and Engineering Research Council of Canada. The writing of the paper was completed in the friendly atmosphere of the Dipartimento di Matematica Applicata, "U.Dini" Universit6 di Pisa, while G.A.M. held a CNR Visiting Professorship.

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Geometrical material structure of elastoplasticity