Homogenization of motion equations for a medium consisting of an elastic material and an incompessible Kelvin-Voigt fluid

Abstract

We consider an initial-boundary problem describing the motion of a two-phase medium with a periodic structure. The first phase of the medium is an isotropic elastic material and the second phase is an incompressible viscoelastic Kelvin-Voigt fluid. This problem consists of the partial differential equations of the second and fourth order, the conditions of continuity of displacements and stresses at the phase boundaries, and the homogeneous initial and boundary conditions. Using the Laplace transform method, we derive a homogenized problem as an initial-boundary value problem for the system of fourth order partial integro-differential equations with constant coefficients. The coefficients and convolution kernels of the homogenized equations are found by using solutions of auxiliary periodic problems on the unit cube. In the case of a layered medium, the solutions of the periodic problems are written explicitly, and this allows us to find analytic expressions for the homogenized coefficients and convolution kernels. In particular, we establish that the type and properties of the homogenized convolution kernels depend on the volume fraction of the fluid layers inside the periodicity cell.