Slip systems and flow patterns in viscoplastic metallic sheets with dislocations

•A rate-type viscoplastic model with non-local evolution equations for dislocations and multislip flow rules is considered.•Conditions to ensure the in-plane stress for material with cubic elastic symmetry.•The variational equality for the velocity field is associated with the incremental boundary value problem at time t.•The finite element method for solving the variational problem, coupled with an update algorithms.•Tests accounting for eight activated slip systems, the influence of the diffusion equation for dislocations are analyzed.

The work addresses the rate dependent crystal plasticity with hardening dependent on the dislocation density, when the local and non-local (diffusion-like) evolution equations for the dislocation densities are considered. A new procedure is proposed for solving initial and boundary value problems based on the incremental equilibrium equations in terms of variational results, coupled with a rate-type constitutive model. The problems concerning the deformation of a sheet composed of a single fcc-crystal, generated by different slip systems simultaneously activated, are solved numerically for an in-plane stress state. Necessary and sufficient conditions for obtaining the in-plane stress state are proved under the assumptions of multislip flow rule, and cubic elastic symmetry. The variational problem which defines the velocity field in the actual configuration at time t is solved by the finite element method (FEM) and the current state in the sheet is defined via an update algorithm when either local (differential type) or non-local (diffusion-like) evolution equations describe the dislocation densities. Boundary value problems resulting from the simulation of compressive and tensile tests are solved by considering all eight activated slip systems together with an activation condition. Finally, a comparison between the numerical results obtained by using the non-local (diffusion-like) evolution equations and local (differential type) equations which describe the dislocation densities is carried out.