Augmented Lagrangian method for Eulerian modeling of viscoplastic crystals

A robust numerical algorithm for an Eulerian rigid-viscoplastic crystal model that accounts for high-strain rates, large strains, and large material and lattice rotations, was developed. The viscoplastic law is obtained from Schmid law by using an overstress approach; the numerical instabilities associated to the classical Norton law are thus eliminated. A time implicit (backward) Euler scheme for time discretization, was used. At each time iteration, a four steps iterative algorithm was proposed. To handle the non-differentiability of the plastic terms an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method was adopted. This formulation was modified to fit to the crystal (non-isotropic) viscoplastic model, for which the stress deviator is not coaxial with the rate of deformation tensor. The proposed algorithm is consistent and permits to solve alternatively, at each iteration, the equations for the velocity field and for the lattice orientation. A mixed finite element-finite volume strategy was adopted: the equation for the velocity field is discretized using the finite element method while a finite volume method, with an upwind choice of the flux, is adopted for the hyperbolic equation related to the lattice orientation. Several two-dimensional boundary value problems are selected to analyze the robustness of the numerical algorithm. The influence of the mesh and of the time step on simulation of the in-plane flow of a fcc crystal in an equal channel angular die extruder was investigated. The transitional flow of a grain embedded in a parent crystal was computed. The grains interaction during channel die compression of a multi-crystal was analyzed using an ALE description.