О модели изотермической акустики для двухкомпонентной среды

Abstract

A mathematical model describing the processes of isothermal acoustics in a heterogeneous medium with two components separated by a common boundary is studied. One of the components is an elastic body, and the other one is a poroelastic medium (for example, it may be a liquid-saturated soil). The poroelastic medium is permeated with a system of pores filled with viscous weakly compressible liquid. The differential equations of the model describing the motion of an elastic body and the joint motion of a solid skeleton and liquid in the pores are based on the classical laws of continuous medium mechanics and adequately reflect the physical processes. However, these equations contain rapidly oscillating coefficients that depend on a small parameter equal to the ratio of the mean pore size to the size of the region under study. The existence of such coefficients prevents the use of the model for carrying out numerical calculations. The generalized solution of the initial boundary-value problem is given; and the theorem for existence and uniqueness of the generalized solution is presented together with its a priori estimates. For performing the homogenization procedure, the standard assumption about the periodicity of the pore space and solid skeleton is adopted. The obtained a priori estimates and the N. Nguetseng's two-scale convergence method were used as a basis for deriving the averaged equations and the initial boundary conditions (that is, the limit equations with the small parameter tending to zero). Different limiting modes depending on the continuous medium parameters are obtained. An averaged model for a special case that does not contain rapidly oscillating coefficients and can be used for numerical calculations is presented.