Simple layer potential expansion for optimization of contact interaction taking into account friction and adhesion

Abstract

Friction plays a crucial role in the formation of contact problems, particularly through adhesion. This paper focuses on a quasi-static three-dimensional problem of a punch movement along the boundary of an elastic half-space. The investigation considers friction and adhesion forces, employing a two-term friction law. The objective is to optimize the pressure distribution beneath the punch. The shape of the punch serves as the design variable, while the deviation of the pressure distribution, originating from a given one, is minimized. The optimization problem can be divided into two sequentially solvable sub-problems. The first task involves finding a pressure distribution that minimizes the performance functional, which has a known solution. The second problem entails searching for the optimal shape of the punch to achieve the previously determined pressure distribution. A numeric-analytical solution is developed based on the expansion of the simple layer potential. The coefficients characterizing friction and adhesion act as small parameters. The proposed method gives the ability to obtain closed-form formulas in each approximation, enabling convenient qualitative analysis and practical engineering applications. The calculations and analytical dependence reveal an asymmetric distribution of pressure on the contact area, during the movement of an axisymmetric punch.