Contact problem for an arbitrarily oriented transversely isotropic half-space

Abstract

The typical contact problems considered in the past for a transversely isotropic half-space $z\ge 0$ used the presumption that z $=$ const. are the planes of isotropy. In 2011, the author has published the first solution to the non-traditional contact problem for a half-space $x\ge 0$. A more general problem is considered here. We use two systems of Cartesian coordinates: the usual major axes (x, y, z) and the arbitrarily oriented one $(x_{1},x_{2},x_{3})$, which is obtained by rotation of the major axes about axis Ox (axis $Ox_{1}$ coinciding with major axis Ox) by an arbitrary angle $\varphi$. The basis for any contact problem is the solution of the Boussinesq problem for the half-space $x_{3}\ge 0$, related to the axes $(x_{1},x_{2},x_{3})$. This problem is solved by two different methods, to show that the solution is correct. The point force solution leads to the governing integral equation for the contact problem in the case of an arbitrary domain of contact. Several basic contact problems are solved. The complete field of displacements and stresses is presented as contour integrals over a unit circle.