Qualitative and Numerical Analysis of Quasistatic Problems in Elastoplasticity

The quasistatic 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 nondiierentiable functional and is imposed on a subset of a space. This VI diiers from the standard parabolic VI in that time derivatives of the unknown variable occurs 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 quasistatic problem of elastoplasticity: semi-discrete approximations, in which only the spatial domain is dis-cretised, by nite elements; and fully discrete approximations, in which the spatial domain is again discretised by nite elements and, in addition, the temporal domain is discretised 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 quasistatic problem of elastoplasticity; in particular, these estimates express rates of convergence of successive nite 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 diiculty in solving the problems is caused by the presence of the nondiierentiable terms. We consider some regularization techniques for overcoming the diiculty. 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.