Microstructural pattern formation in finite-deformation single-slip crystal plasticity under cyclic loading: Relaxation vs. gradient plasticity

We investigate microstructure formation and evolution during cyclic loading in rate-dependent crystal plasticity at finite strains. The non-quasiconvex free energy density in multiplicative single-slip crystal plasticity leads to fine-scale microstructure whose characteristics and resulting effective stress-strain response are studied by two independent approaches: (i) using an incremental formulation based on variational constitutive updates we approximate the quasiconvex hull by lamination, i.e. by constructing an energy-minimizing first-order laminate microstructure, and (ii) a strain-gradient plasticity model applied to a representative unit cell whose effective properties are obtained from homogenization. In the lamination model, three different formulations for updating the accumulated plastic strains are compared and discussed with a specific focus on identifying a suitable description to account for hardening due to changes of the laminate volume fractions. The gradient-plasticity model also predicts a first-order laminate microstructure to form at a comparable stress level upon microstructure initiation. However, the energy associated with the dislocation network is shown to affect the microstructure evolution, leading to considerably higher strain levels at laminate initiation and a stress overshoot. In both models, cyclic loading leads to a degeneration of the stress-strain hysteresis which ultimately experiences elastic shakedown. The amount of work hardening significantly depends on how fast the degeneration occurs. To allow for a comparison, we consider cyclic loading after pre-deformation in the gradient model which delays the degeneration of the stress-strain hysteresis. For low hardening, the two models predict differences in the stress-strain hysteresis, mainly owing to laminate migration in the gradient-plasticity model. As work hardening increases, this phenomenon is restricted and the agreement of the effective stress-strain response between the two models is excellent. Accounting for the energy stored in the domain walls leads to a delayed lamination which is in agreement with the gradient plasticity model.