Nonlinear and structurally nonlinear problems in the theory of plates

The contact problem of two parallel rectangular plates located at some distance one above the other is considered. The upper plate is under normal load. The deflection of the upper plate is prevented by the lower one, so that some contact zone appears. Problems of this kind refer to the problems of the the ory of elasticity with an unknown region of active interaction of structural elements. Such problems are constructively nonlinear, since their mathematical interpretation uses inequalities or undifferen tiable functions. The problem is reduced to a certain variational problem with deflection constraints in the form of inequalities. For finite dimensional approximation, the finitedifference method is used, as a result of which the convex quadratic programming problem is obtained. In the study of convex programming problems, duality theory can be effectively applied. Using the saddle point theorem of the Lagrange function, a dual problem of mathematical programming is formulated, the solution of which is the desired reaction force of the interaction of two plates. The method used in the work assumes a preliminary inversion of the equilibrium equations operators of the contacting elements. The solution of the equations of mechanics of rods, plates and shells is in itself a rather complicated problem. However, for this you can use all the classical methods. In the proposed paper, the results were obtained using an iterative procedure, which is a gradient projection method applied to the dual problem, the convergence of which was proved in the framework of the linear theory of rods, plates and shells. This method is also implemented in solving the nonlinear Karman problem. The results were compared. The comparison showed that the difference in the values of the contact interaction reaction forces in the linear and nonlinear cases is small, but the deflection values differ by almost one and a half times.