ptimal Forms in Mechanics of Contact Interaction

摘要

The problem of optimization of the pressure distri_bution under a rigid stamp that interacts without fric_tion with an elastic medium filling the half-space is investigated. The stamp shape is accepted as a desirable design variable, and the root-mean-square deviation of the pressure appearing under the stamp from a certain specified distribution plays the role of a minimized functional. In this case, the summary forces and moments applied to the stamp are assumed as specified, which leads to limitations imposed on the pressure dis_tribution by the equilibrium conditions. It is shown that the formulated problem of optimization admits decom_position into two sequentially solved problems. The first problem (a) lies in seeking the pressure distribution giving a minimum to the optimizing quality functional, and its solution is found analytically. The second prob_lem (b) is reduced to the immediate finding of the opti_mal shape of the stamp, for which the found pressure distribution is realized. The solution of this problem can be found using the potential of a simple layer. The optimization problem is investigated analytically for stamps of different shapes in the plane. As an example, an exact solution for the stamp with a rectangular basis is presented.