A finite element formulation to solve a non-local constitutive model with stresses and strains due to slip gradients

Solving constitutive models that incorporate the effects of plasticity and slip gradients is often complicated by the non-local nature of the models. This work presents a finite element solution to a crystal plasticity constitutive model that includes kinematic and stress effects due to slip gradients. The foundation of the model is a three term multiplicative decomposition of the deformation gradient that results in a second order differential equation in terms of the stress that drives slip. Converting the equation into a weak form results in an integral equation that includes first order derivatives of the stress as well as boundary conditions for the stress and gradients of slip rate for each slip system. Using this weak form, an incremental finite element method is developed to solve the constitutive model within a finite element solution to the equilibrium equation. Results for the compression of a two-dimensional plate show the effects of including slip gradient effects in the constitutive model and indicate the tendency for localization of the slip and dislocation density into narrow bands separating regions of nearly constant dislocation density and long range strain.