Linear viscoelastic analysis of a semi-infinite porous medium

This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, nonoverlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of Kolosov– Muskhelishvili’s potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.