Mixed boundary-value problems and dynamic impedance functions due to vibrations of a rigid disc on a thermoelastic transversely isotropic half-space

The mixed boundary-value problems and impedance functions due to forced vibrations of a rigid disc on a thermoelastic transversely isotropic half-space are presented analytically and numerically for the first time. Dynamic interaction problems of the disc with the thermoelastic half-space are solved by utilizing an indirect boundary-element method (BEM) which uses the dynamic Green's functions for uniform circular disc loads acting on the surface of a thermoelastic transversely isotropic half-space. The horizontal, vertical, rocking, and torsional vibrations of the disc are investigated and the results are presented in terms of several dual-and Fredholm-integral equations. For the four vibration modes, the related thermo-elastodynamic mixed boundary-value problems are solved and the contact normal and shear stresses, the resultant forces and moments acting on the disc as well as the impedance and compliance functions are derived analytically. By zeroing some param-eters, we proved that the obtained analytical thermo-elastodynamic solutions are reduced to the corresponding elastodynamic results reported in the literature. The numerical procedures for the involved semi-infinite integrals of the kernel functions as well as the numerical quadrature methods for evaluating the impedance functions are further presented. The discretization method is required to be applied only to the interface of the rigid disc with the thermoelastic half-space. A computer program is coded in Mathematica where the singularities due to some branch points and the Rayleigh pole were treated. The Gauss-Legendre quadrature method is used for numerical evaluation of impedance functions, and its advantages are presented and compared with other methods. Some graphical results are presented for the dimensionless impedance functions versus the dimensionless frequency. Effects of thermal and anisotropic properties of the thermoelastic materials; and frequency of excitation on the impedance functions are presented and compared with the elastic materials. Numerical results indicate the significant effects of material anisotropy on the impedance functions.