Усредненная модель распространения малых возмущений в конфигурации упругое тело – пороупругая среда для двухскоростного континуума

Abstract

The article continues the series of the authors’ papers devoted to averaging of mathematical models describing the isothermal acoustic processes in a heterogeneous medium with two components separated by a common boundary. One of these components is an elastic body, and the other is a poroelastic continuum. Poroelastic medium is understood to mean an elastic body permeated with a system of pores filled with liquid. The exact mathematical model constructed proceeding from the classical laws of continuum mechanics is used as the initial model. The model’s differential equations contain rapidly oscillating coefficients that appear in making transition to dimensionless variables. It is assumed that there are finite or infinite limits of these coefficients as the small parameter “epsilon” tends to zero. The small parameter “epsilon” is taken equal to the ratio of the average pore size to the characteristic size of the region under consideration. It should be noted that the coefficients of differential equations and the geometry of the considered region both depend on these parameters. Various averaged (limit) models that do not contain rapidly oscillating coefficients have been derived. For the possibility of using the averaging theory and the known averaging results, simplifying geometric assumptions about the periodicity and connectivity of the pore space and elastic skeleton are added. Averaged models are understood to mean such boundary value problems for equations or systems with relatively slowly changing characteristics that the solutions of boundary value problems for the initial models converge (in a sense) to the solution of the corresponding equations for the averaged model when the period e of the considered periodic structure tends to zero. Depending on the characteristics of the continuum (whether the fluid is viscous, low-viscous, compressible, or incompressible; whether the skeleton is highly deformable, elastic, perfectly rigid, etc.), different limit modes are obtained. One of cases involving weakly compressible and low viscous liquid and a weakly deformable elastic skeleton in one region and an elastic body in the other region is investigated. The original mathematical model reflects the real physical process in a fairly accurate manner, but is so complex that the standard averaging scheme does not work for it. Therefore, the two-scale convergence method is used as the main tool. On the one hand, it is often impossible to calculate the model’s limit modes even in terms of weak convergence, but it is possible to do so in terms of two-scale convergence. On the other hand, the sequence of solutions is usually bounded but not compact, and in this case the sequence weak limit is not a satisfactory approximation to the solution of the initial mathematical model, and it is more preferable to use a two-scale limit. The results for an individually taken poroelastic region or for a region occupied by an elastic body were presented in the authors’ previous papers. In the case considered, the joint motion of an elastic body and porous elastic medium is studied, and the main problem lies in deriving the conditions at the common boundary of the elastic and poroelastic regions.