Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics 270
Contact problems (when two bodies touch each other) arise naturally in many areas of mechanical engineering, machine dynamics and manufacturing, and when predicting earthquakes. In this book, they are studied by mathematical methods. Despite the fact that the notion of contact is quite natural, the corresponding frictional contact problems are rather delicate. The problem has a variational formulation in terms of variational inequalities. The frictional functional turns out to be neither monotone nor compact. Moreover, the Euclidean norm makes the friction non-smooth. The basic method to overcome these difficulties is to approximate the given problem by another problem with a simpler structure. In most cases, the Coulomb law of friction is substituted by a given friction. A solution of the original problem is then obtained by the fixed point theorem. The second method is to penalize Signorini contact conditions and to get a solution of the original problem by taking a limit of the penalty parameter. In both these methods, additional a priori estimates of solutions to approximated problems are needed. Typically, some estimates in Besov spaces are derived by the method of tangential translation.
The necessary background from the theory of Besov spaces is explained in chapter 2. In chapter 3, results for static contact problems obtained by time discretization as well as for quasi-static contact problems (if the motion of the body is very slow) are presented. The existence results for problems with one or two bodies are given provided the coefficients of friction are small enough. The size of the coefficients is then calculated. Chapter 4 is devoted to dynamic contact problems. The authors start with results for purely elastic bodies, which are limited to strings, polyharmonic problems and to special problems on half-space. In the rest of the chapter, an additional viscosity is added to properties of the body. Materials with short and long memory are also studied.
In the last chapter, the authors return to problems involving the Coulomb law for materials with a short memory. In order to overcome the lack of regularity, they replace Signorini conditions by their first order approximations with respect to time. Also existence is obtained if the friction coefficient is small and its size is explicitly computed. As there is dissipation of energy due to viscosity, it is natural to add an equation for this dissipated energy to the system. This is done in the last section of this chapter.
The book provides a self contained overview of the state of research in the given area. It will surely be appreciated by scientists working in this field and it will motivate them to further research. As there are explicitly computed bounds for the friction coefficient, the book will also be useful for engineers working on numerical approximation of contact problems.