Mathematical modeling of a rigid circular membrane in a trimaterial transversely isotropic full-space undergoing a prescribed displacement

A horizontal layer between two transversely isotropic half-spaces forms a trimaterial full-space, involved in this paper. A mathematical formulation is presented to determine the response of a rigid circular membrane, which is laid down at an interface of the tri-material transversely isotropic full-space and is considered to be under a prescribed horizontal displacement. The governing equations are expressed in the cylindrical coordinate system. With the aid of a system of two scalar potential functions, the governing equations of motion can be uncoupled into two separated partial differential equations, which may be transformed to some ordinary differential equations by applying the Hankel integral transforms in the radial direction and Fourier series along the angular coordinate. After determining the unknown functions by imposing the relaxed boundary conditions, they are transformed to a set of four coupled integral equations, which are reduced to two coupled Fredholm-Volterra integral equations of the second kind, from which both displacement and the stress fields are computed. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for a transversely isotropic half-space. In order to investigate the degree of material anisotropy, some numerical evaluations are given for different combinations of transversely isotropic region.