Optimization of a punch shape with a doubly connected contact domain

Abstract

The objective is to optimize the distribution of the normal pressure under a rigid punch (indenter) having a doubly connected contact domain close to a circular ring and interacting with a homogeneous isotropic elastic half-space. In the problem considered here, the required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the present pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a set of similar problems was obtained for a circular ring to determine the functions characterizing the normal pressure distribution over a non-circular ring contact domain under the punch, as well as the normal displacements, from where the optimal punch shape is determined.