An implicit formulation of the elasto-plastic self-consistent polycrystal plasticity model and its implementation in implicit finite elements

Elasto-plastic self-consistent (EPSC) polycrystal plasticity theory has been used extensively in understanding and predicting anisotropic thermo-mechanical response and underlying microstructure evolution of polycrystalline metals. This paper describes the first implicit formulation of the EPSC model and its implementation in implicit finite elements. To this end, a suitably defined system of non-linear equations at the single crystal level and that at the polycrystal level homogenizing the single crystal solutions in terms of the rotation-neutralized increments in Cauchy stress and strain are formulated and numerically solved. The implicit EPSC model is first validated using the original explicit EPSC model. Subsequently, the implicit EPSC model is coupled with implicit finite elements (FE) through the use of the user material subroutine in Abaqus. To facilitate the efficient coupling, a stress update algorithm is developed and the consistent tangent stiffness operator is analytically derived. Here, every FE integration point embeds the implicit EPSC constitutive law taking into account microstructure evolution and the directionality of deformation mechanisms acting at the single crystal level. The multi-level FE-EPSC model is benchmarked using the single crystal data for copper and then applied to simulate drawing of a cup from an AA6022-T4 sheet. The implementation and insights from these predictions are presented and discussed in this paper.