New trends in free boundary problems and new mathematical tools together with broadening areas of applications have led to attempts at presenting the state of art of the field in a unified way. In this monograph we focus on formal models representing contact problems for elastic and elastoplastic plates and shells. New approaches open up new fields for research. For example, in crack theory a systematic treatment of mathematical modelling and optimization of problems with cracks is required. Similarly, sensitivity analysis of solutions to problems subjected to perturbations, which forms an important part of the problem solving process, is the source of many open questions. Two aspects of sensitivity analysis, namely the behaviour of solutions under deformations of the domain of integration and perturbations of surfaces seem to be particularly demanding in this context. On writing this book we aimed at providing the reader with a self-contained study of the mathematical modelling in mechanics. Much attention is given to modelling of typical constructions applied in many different areas. Plates and shallow shells which are widely used in the aerospace industry provide good exam ples. Allied optimization problems consist in finding the constructions which are of maximal strength (endurance) and satisfy some other requirements, ego weight limitations. Mathematical modelling of plates and shells always requires a reasonable compromise between two principal needs. One of them is the accuracy of the de scription of a physical phenomenon (as required by the principles of mechanics).

Inhalt
1 Introduction.- 1 Elements of mathematical analysis and calculus of variations.- 2 Mathematical models of elastic bodies. Contact problems.- 2 Variational Inequalities in Contact Problems of Elasticity.- 1 Contact between an elastic body and a rigid body.- 2 Contact between two elastic bodies.- 3 Contact between a shallow shell and a rigid punch.- 4 Contact between two elastic plates.- 5 Regularity of solutions to variational inequalities of order four.- 6 Boundary value problems for nonlinear shells.- 7 Boundary value problems for linear shells.- 8 Dynamic problems.- 3 Variational Inequalities in Plasticity.- 1 Preliminaries.- 2 The Hencky model.- 3 Dynamic problem for generalized equations of the flow model.- 4 The Kirchhoff-Love shell. Existence of solutions to the dynamic problem.- 5 Existence of solutions to one-dimensional problems.- 6 Existence of solutions for a quasistatic shell.- 7 Contact problem for the Kirchhoff plate.- 8 Contact problem for the Timoshenko beam.- 9 The case of tangential displacements.- 10 Beam under plasticity and creep conditions.- 11 The contact viscoelastoplastic problem for a beam.- 4 Optimal Control Problems.- 1 Optimal distribution of external forces for plates with obstacles.- 2 Optimal shape of obstacles.- 3 Other cost functionals.- 4 Plastic hinge on the boundary.- 5 Optimal control problem for a beam.- 6 Optimal control problem for a fourth-order variational inequality.- 7 The case of two punches.- 8 Optimal control of stretching forces.- 9 Extreme shapes of cracks in a plate.- 10 Extreme shapes of unilateral cracks.- 11 Optimal control in elastoplastic problems.- 12 The case of vertical and horizontal displacements.- 5 Sensitivity Analysis.- 5.1 Properties of metric projection in Hilbert spaces.- 5.2 Shape sensitivity analysis.- 5.3 Unilateral problems in H20(?).- 5.4 Unilateral problems in H2(?) ? H10(?).- 5.5 Systems with unilateral conditions.- 5.6 Shape estimation problems.- 5.7 Domain optimization problem for parabolic equations.- References.