Total Lagrange implementation of a finite-deformation continuum dislocation dynamics model of mesoscale plasticity

Abstract

We present a computational algorithm for solving the recently developed finite-deformation continuum dislocation dynamics theory of mesoscale plastic deformation of single crystals (Starkey et al., 2020). This CDD theory is based on a vector density representation of dislocations governed by curl-type transport-reaction equations subjected to the divergence-free constraint of the appropriate dislocation density. These density evolution equations are to be solved simultaneously with the finite-deformation crystal mechanics. Specifically, our algorithm aims to solve the referential form of the governing equations for a representative volume element (RVE) subject to remote uniform loading. The mechanical fields at the mesoscale are thus split into RVE-averages plus fluctuating components and treated using a strain-driven homogenization scheme. A virtual work-based total Lagrange formulation was used to discretize the governing mechanics equations. A first-order system least squares finite element formulation was used to solve the transport equations. The two schemes are coupled in a staggered fashion. As a part of the crystal mechanics discretization, we derive a consistent tangent modulus and show that the stress update for this model is both linear and global. This linear stress update comes at the cost of solving the dislocation transport equations at every time step to update the plastic distortion caused by dislocation motion. Several test problems are given, demonstrating the ability of the discretization scheme to solve the problem, including the expansion of dislocation loop-like bundles under constant velocity and driven by a mean deformation gradient, dynamic recovery of two oppositely oriented tilt boundaries in a single crystal, and a uniaxial tension test of a single crystal with one slip system activated. In most of these examples, the evolution behavior of the dislocations in the finite deformation regime is demonstrated.