Contact problem for a transversely isotropic half-space with an unknown contact region

Abstract

A three-dimensional contact problem with an unknown contact region for a transversely isotropic half-space, when the isotropy planes are perpendicular to the half-space boundary, is investigated. The method described makes it possible to evaluate effectively the Brinell and Vickers hardness and the contact strength of materials, the near-surface properties of which can substantially depend on the direction (titanium, zinc, beryllium, cobalt, aluminum and zinc oxides, graphite, wood, temoral bone, sapphire, etc; five elastic constants). We use the fact that the kernel of the integral equation of a contact problem can be represented in a form free of quadratures. Such a form of the kernel, in view of the simplicity of its regularization in singular points, makes it possible to apply the method of nonlinear boundary integral equations developed by Galanov to solve the contact problem with an unknown contact region. To debug the computer program, an exact solution of the contact problem for a stamp shaped like an elliptic paraboloid is used. Calculations are made for various materials with the incorporation of the stamp shaped as a rectangular pyramid. The contact regions, pressures, and the values of the impressing force with the specified stamp immersion are determined.