CRYSTAL HARDENING AND THE ISSUE OF UNIQUENESS

As a contribution to the discussion on uniqueness of the plastic slipping mode of crystals, we consider the elastic plastic uniform response to mixed, monotonic, and uniform loading conditions, of a finitely strained single crystal obeying the Schmid flow criterion; such conditions are dealt with, for instance, by Taylor or other rate independent models for the plasticity of aggregates, as well as by finite elements methods. Once given the constitutive equations of the flow under these conditions, we first recall the assumptions on the usual work-hardening law expression which allow the flow conditions to derive from a potential function. We stress the fact that these assumptions are very questionable from a physical point of view and that, even so, several completely admissible solutions in terms of the active slip systems can still sometimes have to be discriminated. We then investigate, with some support from literature in the field of the linear complementarity theory, the "nonsymmetrical problem" for which such a derivation from a potential function is not permitted. In order to select an optimal solution when several completely admissible slipping modes coexist for the nonnecessarily symmetrical problem, we finally introduce a partly conjectural criterion which generalizes, in the symmetrical case, an extremum principle of the regular plasticity theory. Such an optimization of the flow solution, which is achieved without reference to a potential function, allows us to consider more general single crystal-hardening expressions than the usual strain-hardening law. More physically satisfying descriptions of the hardening mechanisms can therefore be accounted for.