Rate-Independent versus Viscous Evolution of Laminate Microstructures in Finite Crystal Plasticity

In this chapter we investigate the variational modeling of the evolution of inelastic microstructures by the example of finite crystal plasticity with one active slip system. For this purpose we describe the microstructures by laminates of first order.We propose an analytical partial relaxation of an incompressible neo-Hookean energy formulation, keeping the internal variables and geometric microstructure parameters fixed, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. Based on the minimization of a Lagrange functional, consisting of the sum of rate of energy and dissipation potential, we derive an incremental strategy to model the time-continuous evolution of the laminate microstructure. Special attention is given to the three distinct cases of microstructure evolution, initiation, rotation, and continuous change. We compare a rate-independent approach with another one that employs viscous regularization which has certain advantages concerning the numerical implementation. Simple shear and tension/compression tests will be shown to demonstrate the differences between both approaches and to show the physical implications of the models introduced