14 Conclusions, Remarks and Future Directions

Abstract

This monograph shows clearly that the branch of the Mathematical Theory of Contact Mechanics which deals with quasistatic processes has made an impressive progress in the last decade. Indeed, from a handful of mathematical results, mainly on static problems and very few on quasistatic and dynamic ones, it has developed into a body of results that encompasses many of the fundamental processes present when deformable bodies come in contact. These include friction, wear, adhesion, thermal effects, and material damage among others. Currently, these results mainly concern the modelling of these processes, their weak or variational formulations, the existence and possibly uniqueness of the weak solutions, and, on occasions, the well-posedness of the models. In the course of analysis of the models, new mathematical results have been obtained, extending the Theory of Variational Inequalities, which were directly motivated by the needs of the analysis. This cross fertilization between modelling and applications on the one-hand and mathematical analysis on the other-hand is one of the important aspects of dealing with contact problems which inherently are nonlinear, diverse, and rather complex.