Non‐linear B‐stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity

A class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward-Euler return maps as a particular case. More importantly, it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary Helmholz free energy function (B-stability). Since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm, the requirement of B-stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity. The analysis that follows depends critically on the assumption of convexity. In particular, the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation. The proposed algorithms obey the discrete counterpart of this classical principle.