Linear viscoelastic analysis of a semi-infinite porous medium

Abstract

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 KolosovMuskhelishvili'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.

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

“…(see [Rabotnov 1988]), where dξ /dξ = exp(−2iβ), and β is the angle between the axis O x and the tangent at the point ξ ∈ L k . The potentialsφ andψ for the far-field stress and the load applied at the boundary of the half-plane are given in [Pyatigorets et al 2008].…”

“…The potentialsφ andψ for the far-field stress and the load applied at the boundary of the half-plane are given in [Pyatigorets et al 2008].…”

“…The potentialsφ F (z; s) andψ F (z; s) are given in [Pyatigorets et al 2008] and the rest are obtained by applying the Laplace transform to their elastic analogs [Mogilevskaya 2000]:…”

“…In general, the coefficients‫ނ‬ mk are different from the coefficientsB mk as they may include the multipliers given by (23). The operatorsΥ kk,q ,Υ k j,q , k,q , and k j,q can be found in [Pyatigorets et al 2008] 1 if one takes into account the specific method of truncating Fourier series in the present approach. Separating the real and imaginary parts in complex equations (25) and taking into account that (25) is real for q = 0, one arrives at a linear system of N k=1 (4M k − 1) equations.…”

“…The method does not assume a time-independent viscoelastic Poisson's ratio and allows one to adopt a variety of viscoelastic models. The method described in the present paper is an extension of [Pyatigorets et al 2008], where the problem of an isotropic, linearly viscoelastic half-plane containing multiple holes was considered. The inversion of the Laplace transform in the latter problem could be performed analytically, which is not the case for the more general problem considered in the present paper.…”