Vertical vibration of a rigid circular disc at the interface of a transversely isotropic bi-material

A theoretical formulation is presented for the determination of the dynamic interaction of a vertically loaded rigid disc embedded at the interface of a transversely isotropic bi-material full-space. With the aid of Hankel integral transforms, a relaxed treatment of the mixed-boundary value problem is formulated as dual integral equations, which can be reduced to a Fredholm integral equation of the second kind. The dynamic contact pressure under the disc and the impedance function are analytically evaluated in the general dynamic case. It is shown that the impedance functions are in exact agreement with the existing solutions for a an elastic half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy and compared with previously published solutions.