Fundamental solutions of circular inclined rigid punch on a half plane with an oblique edge crack

Abstract

A circular rigid punch is located on a half plane with an oblique edge crack. The punch is acted by a vertical load at its center, and frictional force is assumed to exist on the contact region and applies in the horizontal direction. A pair of concentrated forces is assumed to act at arbitrary point in the half plane. Owing to the existence of the crack, the frictional force and the concentrated forces or point dislocations, the punch is usually inclined and the inclined angle is decided by the condition that the resultant moment about the center on the contact region must be vanished. The punch is supposed to contact with the half plane with two sharp corners at both ends, therefore the length of the contact region has been given. To find the analytical solution of the problem, a rational mapping function which maps the half plane with an oblique edge crack into a unit circle is used According to the loading and displacement conditions, the problem can be transformed into the Riemann-Hilbert problem. To solve the problem, the complex stress functions are divided into two parts; one is the principle part, which corresponds to the fundamental solution of the half plane with an oblique edge crack; the other is the holomorphic part of the problem, which can be derived explicitly by solving Riemann-Hilbert equation.