A comparative study of integration algorithms for finite single crystal (visco-)plasticity

Computational aspects of finite crystal plasticity have been discussed in the literature for some time, yet open challenges regarding the robustness and uniqueness of predictions for different algorithms remain. In this contribution, a careful comparison between different formulations of single crystal plasticity is therefore presented, covering both rate-dependent and rate-independent behavior in the finite deformation setting. To this end, algorithms are developed for an augmented-Lagrangian formulation and a formulation based on nonlinear complementary functions in the rate-independent case. For the rate-dependent case, algorithmic treatments are considered for two commonly employed viscoplastic formulations. The robustness of the different algorithms is assessed by means of simulations at the material point level, involving proportional fully deformation-controlled tests and non-proportional loading cycles in stress space. The augmented-Lagrangian formulation and a particular viscoplastic formulation are found to be superior in terms of robustness compared to the other algorithm considered in their respective category. It is then further observed that these algorithms of choice, which perform well for the ideally-plastic case and for Taylor-type hardening, induce a significant reduction in acceptable deformation increment size for an anisotropic, phenomenological hardening law, due to the rather complex determination of the active slip systems. Additionally, it is found from a sensitivity analysis that increasing the degree of anisotropy in the hardening law amplifies the sensitivity of the model's response to small initial misorientations, both in the rate-dependent and the rate-independent case. A sensitivity analysis is also applied in the ideally-plastic case to demonstrate that for common choices of viscous parameters, the response of a particular rate-dependent formulation of the crystal plasticity model not necessarily possesses the same characteristic features as the response obtained by the corresponding rate-independent augmented-Lagrangian formulation. Therefore, it is recommended to apply the augmented-Lagrangian formulation to determine the rate-independent model response, rather than employing a rate-dependent formulation with limiting values of the viscous parameters due to the superior robustness of the former. The application of the augmented-Lagrangian formulation in the solution of an inhomogeneous fixed-end torsion problem by means of finite element analysis demonstrates the beneficial numerical properties of this formulation and the ability of the material model to reproduce characteristic features observed in experiments on single crystal copper samples.