Solving the punch problems by analogy with the interface crack problems

Abstract

By cutting through the contact/crack surface, the contact/crack problems can be treated as a half-plane problem with the displacements/tractions prescribed along a specified region. Through the conversion to half-plane problems, the analogy between the contact problems and crack problems has been noticed before. However, to the authors' knowledge, no detailed analogous relations have been provided. Moreover, no contact problems have been solved directly by just using the corresponding solutions of the crack problems, or vice versa. In this paper, special consideration will be focused upon the connections between the punch problems and the interface crack problems with one of the materials to be rigid. Similar to the analogy between line forces and line dislocations, cracks and rigid fine inclusions, or holes and rigid inclusions, we may now solve the punch problems by analogy with the interface crack problems. In addition, very simple explicit solutions for the contact pressure and the surface deformation are derived in this paper. Moreover, a general procedure to get the real form solution is also described. Finally, three representative punch problems are solved completely. They are the indentation by a flat-ended punch, a flat-ended punch tilted by a couple, and the indentation by a parabolic punch. The explicit full field solutions, and the real form solutions for the contact pressure and surface deformation of these three problems are all provided. Based upon these closed-form solutions, several numerical examples were done and their related stresses contours, surface deformations and contact pressures were also plotted to help us see more clearly the physical behaviors of the punch problems.