Contact processes between deformable bodies abound in industry and everyday life and, for this reason, considerable efforts have been made in their modelling and analysis. Owing to their inherent complexity, contact phenomena lead to new and interesting mathematical models. Here and everywhere in this book by a mathematical model we mean a system of partial differential equations, associated with boundary conditions and initial conditions, eventually, which describes a specific contact process.
The purpose of this book is to introduce the reader to some representative mathematical models which arise in Contact Mechanics. Our aim is twofold: first, to present a sound and rigorous description of the way in which the mathematical models are constructed; second, to present the mathematical analysis of such models which includes the variational formulation, existence, uniqueness and convergence results. To this end, we use results on various classes of variational inequalities in Hilbert spaces, that we present in an abstract functional framework. Also, we use various functional methods, including monotonicity, compactness, penalization, regularization and duality methods. Moreover, we pay particular attention to the mechanical interpretation of our results and, in this way, we illustrate the cross fertilization between modelling and applications on the one hand, and nonlinear analysis on the other hand.
This book is intended as a unified and readily accessible source for graduate students, as well as mathematicians, engineers and scientists. Its reading requires only basic knowledge of linear algebra, general topology, functional analysis and mechanics of continua.