Inequality problems in mechanics can be divided into two main categories: that of variational inequalities which is concerned with nonsmooth and convex energy functionals (potentials), and that of hemivariational inequalities which is concerned with nonsmooth and nonconvex energy functionals (superpotentials).  Mathematical studies of variational inequalities started in the sixties, whereas that of hemivariational inequalities are more recent.  Through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth and multivalued constitutive laws, forces, and boundary conditions can be treated successfully.

In this talk, sample variational and hemivariational inequalities in contact mechanics are introduced and studied. The focus will be on recent and new results on the numerical solution of hemivariational inequalities. Numerical examples will be reported to illustrate numerical evidence of the theoretically predicted optimal convergence orders.

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