Viscoelastic state of a semi-infinite medium with multiple circular elastic inhomogeneities

This paper is concerned with the problem of an isotropic, linear viscoelastic half-plane containing multiple, isotropic, circular elastic inhomogeneities.Three types of loading conditions are allowed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, and a force uniformly distributed over the whole boundary of the half-plane.The half-plane is subjected to farfield stress that acts parallel to its boundary.The inhomogeneities are perfectly bonded to the material matrix.An inhomogeneity with zero elastic properties is treated as a hole; its boundary can be either traction free or subjected to uniform pressure.The analysis is based on the use of the elastic-viscoelastic correspondence principle.The problem in the Laplace space is reduced to the complementary problems for the bulk material of the perforated half-plane and the bulk material of each circular disc.Each problem is described by the transformed complex Somigliana's traction identity.The transformed complex boundary parameters at each circular boundary are approximated by a truncated complex Fourier series.Numerical inversion of the Laplace transform is used to obtain the time domain solutions everywhere in the half-plane and inside the inhomogeneities.The method allows one to adopt a variety of viscoelastic models.A number of numerical examples demonstrate the accuracy and efficiency of the method.