Abstract

A new micromechanical model of probabilistic type describing the plastic behaviour of a polycrystalline material is discussed. The possibility of having relative modes of deformation along non-crystalline planes and directions is here taken into account, so that new constitutive assumptions for the plastic part of the velocity gradient of deformation are considered. The usual assumption on this tensor is that it is equal to the sum of slipping rates on crystalline planes. This hypothesis is reinterpreted here in a probabilistic framework: all planes and directions are possible slip directions, but some of them are more likely than others. To discriminate among slip systems a probability measure, which we call slip probability, is introduced. A quantity related to this probability is assumed to be a material ingredient, and so it is given as a constitutive prescription for the single crystal. Obviously, here instead of having a finite number of slipping rates, i.e. one for each slip system, we have a field of slipping: this depends upon the imposed stretching, the orientation of the crystallite and the orientation of the slip system, which could be any. The plastic part of the velocity gradient is then obtained by integrating the slipping field, together with its slip system, with respect to the slip probability. The field of slipping should be objective. A sufficient condition, adopted by us, is to take the field of slipping linear in the stretching. A motivation for this hypothesis relies upon the fact that the field of slipping can be specified as in rate-type approaches to crystalline plasticity. A standard treatment leads to (i) the evolution equations for grain rotation and elastic stretch in terms of the prescribed velocity gradient, (ii) a constitutive equation for a corotational rate of the Kirchhoff stress, arising from direct time differentiation of the linearized response function assumed for the stress itself.

Properties of the polycrystal are derived here by considering orientational averages of two properties at the submacroscopic level: (a) the corotational rate of the stress mentioned above, (b) the plastic spin. To this end, if a point in the polycrystal and a set of orientation are given, the probability of finding a crystal with an orientation in the given set at the given point is considered: this probability is then an orientation probability measure. The orientational averages mentioned above are evaluated with respect to the orientation distribution function" (ODF). The overall "elastoplastic" moduli is shown to derive from that of the single crystal through the knowledge of the ODF. It will be shown that, unlike standard treatments, texture coefficients may be deduced with no a-priori assumptions on the symmetry of the texture itself. This is one of the various advantages of the new probabilistic micromechanical approach, which includes the explicit derivation of the elastoplastic moduli and the flexibility in the choice of the slip probability measure and of the representation of the slipping field.

This is part of a joint work with C.-S. MAN, Department of Mathematics, University of Kentucky, Lexington-KY-USA. mclxyh@ms.uky.edu and and R.PARONI Dipartimento di Architettura e Pianificazione, Universita di Sassari, Palazzo del Pou Salit, Piazza Duomo, 07041 Alghero, paroni@uniss.it

THURSDAY, September 4, 2003
Time: 1:30 P.M.
Location: PPB 300