Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity

Abstract

The quasi-static problem of elastoplasticity with combined kinematic-isotropic hardening is formulated as a time-dependent variational inequality (VI) of the mixed kind; that is, it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space. This VI differs from the standard parabolic VI in that time derivatives of the unknown variable occur in all of its terms. The problem is shown to possess a unique solution. We consider two types of approximations to the VI corresponding to the quasi-static problem of elastoplasticity: semidiscrete approximations, in which only the spatial domain is discretized, by finite elements; and fully discrete approximations, in which the spatial domain is again discretized by finite elements, and the temporal domain is discretized and the time-derivative appearing in the VI is approximated in an appropriate way. Estimates of the errors inherent in the above two types of approximations, in suitable Sobolev norms, are obtained for the quasi-static problem of elastoplasticity; in particular, these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, of the time step size k. A major difficulty in solving the problems is caused by the presence of the nondifferentiable terms. We consider some regularization techniques for overcoming the difficulty. Besides the usual convergence estimates, we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem.