In this chapter we consider numerical approximations for the primal vari-ational problem of elastoplasticity. We start with the derivation of error estimates for various numerical schemes approximating the solution of the primal variational problem by applying the results for the abstract varia-tional problem proved in the last chapter. We also discuss the convergence property for various schemes under the basic solution regularity condition.

Then we consider the practically important issue of the implementation of numerical schemes and, in particular, the algorithms that are employed in such schemes. The algorithms considered here are of predictor-corrector type. Detailed derivation of the solution algorithms is given in Section 12.2. Convergence of the solution algorithms is discussed in Section 12.3.

A major difficulty in solving the primal variational problem numerically (and similarly, the inequality problem in a corrector step in the solution algorithms discussed in Section 12.2) is the treatment of the nondifferentiable functional j( ). Several approaches can be used to circumvent the difficulty in practice. One approach is the regularization method, where the nondifferentiable term is approximated by a sequence of differentiable ones. Convergence and error estimations for the regularization method are the main topics of Section 12.4. A practically efficient approach is discretizing the inequality for the continuous variables involving the nondifferentiable term to give a set of uncoupled inequalities at integration points. We give a detailed error analysis for one such method in Section 12.5.