Semi-analytical solution for a viscoelastic plane containing multiple circular holes

The paper considers the problem of an infinite, homogeneous, isotropic viscoelastic plane containing multiple circular holes. Constant or time-dependent loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The solution of the problem is based on the use of the correspondence principle, and the governing equation in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements at the boundaries of the holes. The main feature of this equation is that the material parameters are only involved as multipliers for the terms other than the integrals of transformed displacements. The unknown transformed displacements are approximated by truncated complex Fourier series with coefficients dependent on the transform parameter. A system of linear algebraic equations is formulated using Taylor series expansion for determining these coefficients. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of Kolosov-Muskhelishvili potentials, and an inverse Laplace transform is used to provide the time domain solution. All the operations (space integration, Laplace transform and its inversion) are performed analytically. The method described in the paper enables the consideration of a variety of viscoelastic models. For the sake of illustration, examples are given for the cases where the viscoelastic solid responds as (i) a Boltzmann model in shear and elastically in dilatation, (ii) a Boltzmann model in both shear and dilatation, and (iii) a Burgers model in shear and elastically in dilatation. The accuracy and efficiency of the approach are demonstrated by comparing selected results with the solutions obtained by the finite element method (ANSYS) and the time stepping boundary element approach.