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

Abstract

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.

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

“…• For the problem of an infinite or semi-infinite perforated viscoelastic plane subjected to constant loading, the stresses in the matrix do not depend on time and are exactly the same as those for the corresponding elastic problem [Huang et al 2006b;Pyatigorets et al 2008]. Thus, one may consider the problem in which the inhomogeneities have very small shear moduli so they can simulate the holes in the numerical analysis.…”

Linear viscoelastic analysis of a semi-infinite porous medium

“…The review of different numerical approaches, used to solve time-dependent problems, can be found in Huang et al (2006a). The authors considered the problem of an infinite viscoelastic plane containing multiple holes.…”

“…In the present work we extend the method of Huang et al (2006a) to the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, non-overlapping circular holes. 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.…”

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

“…The present method retains the advantages of the previous complex variable boundary integral method for an infinite viscoelastic plane [17]. For example, it has the flexibility in adopting a variety of physical models.…”

“…While more efficient than the conventional collocation boundary element method, the approach was based on the restrictive assumption of a constant viscoelastic Poisson's ratio. To overcome this disadvantage we proposed [17] a new two-dimensional complex variable boundary integral method for viscoelasticity. It combined with the analytical Laplace transform and its inversion to obtain a semi-analytical solution for the problem of an infinite viscoelastic plane containing circular holes.…”