Modeling and Control in Solid Mechanics

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).

1 Introduction.- 1 Elements of mathematical analysis and calculus of variations.- 1.1 Functional spaces. Simple properties.- 1.2 Variational inequalities.- 1.3 Minimization problems for convex functionals.- 1.4 Derivative of a convex functional.- 1.5 Minimization problems for nonsmooth functionals.- 1.6 Weak convergence. Compactness principles.- 1.7 Weak semicontinuity of functionals.- 1.8 Existence of solutions to the minimization problem.- 1.9 The case of Hilbert space.- 1.10 Elements of measure theory.- 2 Mathematical models of elastic bodies. Contact problems.- 2.1 Linear elastic bodies and shallow shells.- 2.2 Mathematical models of contact problems.- 2 Variational Inequalities in Contact Problems of Elasticity.- 1 Contact between an elastic body and a rigid body.- 1.1 Problem formulation.- 1.2 Regularity of solutions. Construction of measures.- 2 Contact between two elastic bodies.- 2.1 Formulation of the problem. Regularity of solutions.- 2.2 Construction of a measure.- 3 Contact between a shallow shell and a rigid punch.- 3.1 Existence of solutions.- 3.2 Regularity of solutions.- 3.3 Absence of concentrated forces.- 3.4 Parallel punch.- 4 Contact between two elastic plates.- 4.1 Problem formulation. Properties of the solution.- 4.2 Connectedness of the noncoincidence domain.- 5 Regularity of solutions to variational inequalities of order four.- 5.1 The contact problem of a plate with a membrane.- 5.2 The contact problem for a shell.- 6 Boundary value problems for nonlinear shells.- 6.1 General remarks.- 6.2 Inequalities on the boundary. Convergence of solutions.- 7 Boundary value problems for linear shells.- 8 Dynamic problems.- 8.1 Variational inequality for a beam.- 8.2 Variational inequality for a shell.- 3 Variational Inequalities in Plasticity.- 1 Preliminaries.- 2 The Hencky model.- 2.1 The three-dimensional elastoplastic body.- 2.2 The perfect plastic body.- 3 Dynamic problem for generalized equations of the flow model.- 4 The Kirchhoff-Love shell. Existence of solutions to the dynamic problem.- 4.1 Problem formulation.- 4.2 The main result.- 5 Existence of solutions to one-dimensional problems.- 5.1 Elastoplastic problems for a beam and cylindrical shell.- 5.2 The perfectly plastic problem for a beam.- 6 Existence of solutions for a quasistatic shell.- 6.1 Formulation of the problem.- 6.2 Theorem of existence.- 7 Contact problem for the Kirchhoff plate.- 7.1 Elastoplastic problem.- 7.2 The perfectly plastic problem.- 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.- 1.1 Cost functionals with measures.- 1.2 Cost functionals with norms.- 2 Optimal shape of obstacles.- 2.1 Cost functionals with norms.- 2.2 Cost functionals with measures.- 2.3 Finite set of pointwise restrictions.- 3 Other cost functionals.- 4 Plastic hinge on the boundary.- 4.1 Cost functionals with displacements.- 4.2 Cost functionals with measures.- 5 Optimal control problem for a beam.- 6 Optimal control problem for a fourth-order variational inequality.- 6.1 Fourth-order operator.- 6.2 Second-order operator.- 6.3 The passage to the limit.- 7 The case of two punches.- 7.1 Optimal control for a plate.- 7.2 Optimal control for a membrane.- 7.3 The passage to the limit.- 8 Optimal control of stretching forces.- 8.1 Optimal control for a plate.- 8.2 Optimal control for a membrane.- 8.3 Transition from a plate to a membrane.- 9 Extreme shapes of cracks in a plate.- 10 Extreme shapes of unilateral cracks.- 10.1 Interior cracks.- 10.2 Boundary cracks.- 10.3 A more precise nonpenetration condition.- 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.- 5.2.1 Material derivatives.- 5.2.2 Material derivatives on the boundary ?.- 5.2.3 Shape derivatives on the boundary ?.- 5.2.4 Displacement derivatives on S.- 5.2.5 Derivatives of shape functionals.- 5.3 Unilateral problems in H20(?).- 5.3.1 The tangent cone.- 5.3.2 Differentiability of metric projections.- 5.3.3 Applications to optimal design.- 5.4 Unilateral problems in H2(?) ? H10(?).- 5.4.1 Obstacle problem for simply supported Kirchhoff plate.- 5.5 Systems with unilateral conditions.- 5.6 Shape estimation problems.- 5.6.1 Admissible domains with norm constraints on graphs.- 5.6.2 Admissible domains with local constraints on graphs.- 5.6.3 Differentiability of metric projection.- 5.6.4 Shape estimation problem for the wave equation.- 5.7 Domain optimization problem for parabolic equations.- 5.7.1 Parabolic equation in a variable domain.- 5.7.2 Differentiability of the cost functional.- 5.7.3 Shape sensitivity analysis.- 5.7.4 Optimization problem.- 5.8 Shape sensitivity analysis of thin shells.- 5.8.1 Thin shells.- 5.8.2 Displacement derivatives.- 5.8.3 Shape sensitivity analysis of thin shells.- 5.8.4 Computation of derivatives of cost functional.- 5.8.5 Computation of the second derivative.- References.