Time-Continuous Evolution of Microstructures in Finite Plasticity

Plastic deformation of crystalline solids very often gives rise to the initation of material microstructures experimentally visible as dislocation patterns. These microstructures are not inherent to the material but occur as a result of deformation. Modeling a physically deformed crystal in finite plasticity by means of the displacement field and in terms of a set of internal variables which capture the microstructural characteristics, we employ energy principles to analyze the microstructure formation and evolution as a result of energy minimization. In particular, for non-quasiconvex energy potentials the minimizers are no longer continuous deformation fields but small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. We briefly review the variational concept of the underlying energy principles for inelastic materials. As a first approximation of the relaxed energy density, we assume first-order laminate microstructures, thus approximating the relaxed energy by the rank-one convex envelope. Based on this approach, we present explicit time-evolution equations for the volume fractions and the internal variables, then outline a numerical scheme by means of which the microstructure evolution can be computed and we show numerical results for particular examples in single and double-slip plasticity. In contrast to many approaches before we do not globally minimize a condensed energy functional to determine the microstructure but instead incrementally solve the evolution equations at each time step, in particular accounting for the dissipation required to rearrange the microstructure during a finite time increment with already existing mictrostructure at the beginning of the time step.