An Algorithm for Determining the Active Slip Systems in Crystal Plasticity Based on the Principle of Maximum Dissipation

In continuum mechanics inelastic deformation is attributed to the aggregate effect of irreversible gliding of atoms along well defined planes whose number and orientation depends on the crystallographic structure of the material. Such a gliding plane and its pertaining gliding direction are termed “slip system”. It is activated by mechanical stresses expressed in terms of the resolved shear stress acting at a material point evaluated for each slip system [1]. The overall rate of inelastic deformation is constituted by the superposition of the velocities of the shear deformation of each single slip system, i.e. the slip rates. The overall constitutive material response can be worked out once these slip rates are determined. The general mathematical framework for small strain theory has been reported e.g. in [2], [3]. The question whether a slip system is activated is decided by a yield criterion mathematically expressed in terms of one inequality per slip system. It essentially states that plastic yielding occurs if the resolved shear stress reaches a critical value subject to the additional constraint that only positive slip rates are permitted. The solution to the problem is generally non-unique, i.e. the solution set contains a discrete number of feasible sets of slip rates, so an additional condition has to be formulated to be able to select the set of slip rates which actually appears in reality. According to fundamental physical laws every process obeys the principle of the maximization of the dissipation rate. Thus, from all feasible sets the one which maximizes the dissipation rate has to be selected. This requires sophisticated search strategies since standard optimization routines fail due to the discrete nature of the search-space.