On the Finite Element implementation of higher-order gradient plasticity, with focus on theories based on plastic distortion incompatibility

We consider work-conjugate Gradient Plasticity (GP) theories involving both energetic and dissipative higher-order contributions. We show that the conceptually most straightforward Finite Element (FE) implementation, in which the displacement components and the relevant plastic distortion contributions are employed as nodal degrees of freedom, leads to a very efficient Backward-Euler FE algorithm if a proper viscoplastic potential is adopted, the latter in general involving dissipative higher-order terms. We also show that the proposed viscoplastic constitutive law can accurately represent rate-independent behaviour, without losing computational efficiency. To draw our conclusions we consider many benchmarks (simple shear of a constrained strip, bending of thin foils, micro-indentation) and both phenomenological and crystal GP theories whose distinctive feature is a contribution to the free energy, called the defect energy, written in terms of Nye's dislocation density tensor.