Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains

This work describes the finite element implementation of a generalised strain gradient and rate-dependent crystallographic formulation for finite strains and general anisothermal conditions based on a multiplicative decomposition of the deformation gradient. The implementation involved the development of both a novel finite element formulation to determine the spatial slip rate gradients at each material point, and an implicit numerical integration scheme at the constitutive level to update the stresses and solution dependent variables. The time-integration procedure uses a Newton–Raphson scheme with a single level of iteration to solve the incremental non-linear equations associated with the non-local constitutive formulation. Closed-form solutions for the relevant fourth-order Jacobian tensors are given. The proposed numerical scheme is formulated in a general form and hence should be applicable to most existing crystallographic models. The crystallographic formulation is then used to investigate the effect of the morphology and volume fraction of the reinforcing phase of a two-phase single crystal on its macroscopic behaviour.