Kravchuk A.S., Neittaanmäki P.J. Variational and Quasi-Variational Inequalities in Mechanics

Springer-Verlag Berlin, 2007. — 338 p. — ISBN 978-1-4020-6376-3.The aim of this hugely practical book is to consider a wide set of problems arising in the mathematical modeling of mechanical systems under unilateral constraints. In these investigations elastic and non-elastic deformations, friction and adhesion phenomena are taken into account. All the necessary mathematical tools are given: local boundary value problem formulations, construction of variational equations and inequalities and their transition to minimization problems, existence and uniqueness theorems, and variational transformations (Friedrichs and Young-Fenchel-Moreau) to dual and saddle-point search problems. Important new results concern contact problems with friction. The Coulomb friction law and some others are considered, in which relative sliding velocities appear. The corresponding quasi-variational inequality is constructed, as well as the appropriate iterative method for its solution. Outlines of the variational approach to non-stationary and dissipative systems and to the construction of the governing equations are also given. Examples of analytical and numerical solutions are presented. Numerical solutions were obtained wit.