This book covers the boundary value problems for a wide range of mathematical models of the mechanics of deformable bodies, in particular, the boundary value problems concerning plates and shells, crack theory, and elastoplastic bodies. An essential feature of the discussed boundary value problems is the availability of inequality type constraints imposed on solutions such as the nonpenetration condition for contact problems, and the yield plasticity condition. As a consequence, the presence of free boundaries is typical of the boundary value problems concerned. The objective of the book is to display some new methods of analyzing such problems, as well as to perform research on new models evolved from engineering practice. The book is intended for postgraduates, scientists and engineers. It will also be useful to students interested in problems of modelling and optimal control in the mechanics of deformable bodies.