Love's circular patch problem revisited: Closed form solutions for transverse isotropy and shear loading

This paper considers the title problem of uniform pressure or shear traction applied over a circular area on the surface of an elastic half space. The half space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is adopted where the elastic field is written in terms of three harmonic functions. The known point force potential functions are used to find the solution for uniform pressure or shear traction over a circular area by quadrature. Using methods developed by Love (1929) and Fabrikant (1988), the elastic displacement and stress fields for normal and shear loading are evaluated in terms of closed form expressions containing complete elliptic integrals of the first, second, and third kinds. The solution for uniform normal pressure on an isotropic half space was previously found by Love (1929). The present results for transverse isotropy including shear loading are new. During the course of this research, a new relation has been discovered between different forms of the complete elliptic integral of the third kind. This has allowed the present solution to be put in a more convenient form than that used by Love. Following a limiting procedure allows the isotropic solution to be obtained. It is shown that for normal loading the present results agree with Love's solution, while the results for shear loading of an isotropic half space are also apparently new. Special consideration is also given to derive the limiting form of the elastic field on the z-axis and the surface (z = 0).