A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem

Abstract

In this article, we derive a posteriori error estimates for the Hencky plasticity problem. These estimates are formulated in terms of the stresses and present guaranteed and computable bounds of the difference between the exact stress field and any approximation of it from the energy space of the dual variational problem. They consist of quantities that can be considered as penalties for the violations of the equilibrium equations, the yield condition and the constitutive relations that must hold for the exact stresses and strains. It is proved that the upper bound tends to zero for any sequence of stresses that tends to the exact solution of the Haar–Karman variational problem. An important ingredient of our analysis is a collection of Poincaré type inequalities involving the L 1 norms of the tensors of small deformation. Estimates of this form are not new, however we will present computable upper bounds for the constants being involved even for rather complicated domains.