Review of formulations for elastostatic frictional contact problems

Frictional contact problem formulations have been extensively studied in the literature, both from a rigorous mathematical viewpoint and more recently in the context of finite element analysis. The former primarily dealt with variational inequalities and existence and uniqueness issues using simple problems while the later employed variational equalities with numerical analysis tools to solve large-scale engineering problems. These formulations make use of the standard as well as nonsmooth optimization techniques. In this paper both the Variational inequality and the variational equality formulations are reviewed and some example problems are solved to illustrate them. Usage of standard as well as nonsmooth optimization techniques is explained. Despite extensive research in the fields of mathematics and engineering, frictional contact remains as one of the most challenging problems due to the difficulties involved in its formulation and solution procedures. So far, variational inequality formulation for frictionless contact problems has been established firmly. This is due to the fact that equivalent standard optimization problems can be associated with these inequalities. However, for general frictional contact problems this formulation runs into difficulties due to lack of equivalent optimization problem (more precisely, lack of a genuine minimization functional such as the total potential energy) which has restricted its usage to solve large-scale problems. Variational equality approach on the other hand has been quite successful to solve large-scale frictional contact problems, but it relies heavily on numerical schemes (such as Newton-Raphson method) and user defined numerical parameters. A review of elastostatic frictional contact problems employing variational inequality and equality formulations is provided. Computational procedures are studied and some insights are provided for their numerical aspects. The need for an alternate frictional contact problem formulation that can alleviate the drawbacks in current variational inequality and equality formulations is discussed.