Robust integration schemes for generalized viscoplasticity with internal-state variables

This paper is concerned with the development of a general framework for the implicit time-stepping integrators for the flow and evolution equations in complex viscoplastic models. The primary goal is to present a complete theoretical formulation, and to address in detail the algorithmic and numerical analysis aspects involved in its finite element implementation, as well as to critically assess the numerical performance of the developed schemes in a comprehensive set of test cases. On the theoretical side, we use the unconditionally stable, backward Euler difference scheme. Its mathematical structure is of sufficient generality to allow a unified treatment of different classes of viscoplastic models with internal variables. Two specific models of this type, which are representatives of the present state-of-the-art in metal viscoplasticity, are considered in the applications reported here; i.e., generalized viscoplasticity with complete potential structure fully associative (GVIPS) and non-associative (NAV) models. The matrix forms developed for both these models are directly applicable for both initially isotropic and anisotropic materials, in three-dimensions as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). On the computational side, issues related to efficiency and robustness are emphasized in developing the (local) iterative algorithm. In particular, closed-form expressions for residual vectors and (consistent) material tangent stiffness arrays are given explicitly for the GVIPS model, with the maximum matrix sizes ‘optimized’ to depend only on the number of independent stress components (but independent of the number of viscoplastic internal state parameters). Significant robustness of the local iterative solution is provided by complementing the basic Newton–Raphson scheme with a local line search strategy for convergence.