Mathematics & Physics 2015, 8(3), 356–370 УДК 517.958 Homogenization of Acoustic Equations for a Partially Perforated Elastic Material with Slightly Viscous Fluid Alexey S. Shamaev∗ Vladlena V.Shumilova† Institute for Problems in Mechanics of RAS Vernadskogo, 101–1, Moscow, 119526 Russia Received 15.04.2015, received in revised form 10.05.2015, accepted 25.06.2015 In this paper a mathematical model describing small oscillations of a heterogeneous medium is considered. <...> The medium consists of a partially perforated elastic material and a slightly viscous compressible fluid filling the pores. <...> For the given model the corresponding homogenized problem is constructed by using the two-scale convergence method. <...> The boundary conditions connecting equations of the homogenized model on the boundary between the continuous elastic material and the porous elastic material with fluid are found. <...> In applications, such problems describe some physical processes in heterogeneous media, for example, a diffusion process in highly heterogeneous media [2] or a joint motion of an elastic skeleton and a slightly viscous fluid [7]. <...> Recently, themethod of two-scale convergence is widely applied in the homogenization of various mathematical problems that arise in mechanics of heterogeneous media (see, e.g., [8–14]). <...> In this paper, we consider a mathematical problem that describes small oscillations of a heterogeneous medium consisting of a partially perforated elastic material and a slightly viscous compressible fluid filling the pores. <...> We assume that the elastic material is inhomogeneous with ε-periodic microstructure, and the structure of the perforation in the porous part of the elastic material is also ε-periodic. <...> The mathematical problem under consideration involves the linear elasticity system describing the motion of the elastic material, and the Stokes system describing the motion of the fluid. <...> Using the method of two-scale convergence and the Laplace transforms, we construct the corresponding homogenized problem and find the boundary conditions which connect equations of the homogenized problem on the boundary between the continuous elastic material and the porous elastic material with fluid. <...> All rights reserved c – 356 – Alexey S.Shamaev, Vladlena V.Shumilova Homogenization of Acoustic Equations for a Partially . elastic part of the heterogeneous medium is completely perforated, the corresponding homogenization problem was analyzed in [7, 9, 12] and [15]. <...> In addition, we denote by Y <...>