Small perturbations of two-phase fluid in pores: effective macroscopic monophasic viscoelastic behavior

The linearized model of joint motion of an elastic porous body and a two-phase viscous compressible liquid in pores is considered. The reciprocal deformation of liquid phases is governed by Rakhmatullin’s scheme. It is assumed that the porous body has a periodic geometry and that the ratio of the pattern periodic cell and the diameter of the entire mechanical system is a small parameter in the model. The homogenization procedure, i.e. a limiting passage as the small parameter tends to zero, is fulfilled. As the result, we find that the limiting distributions of displacements of the media serve as a solution of a well-posed initial-boundary value problem for the model of linear monophasic viscoelastic material with memory of shape. Moreover, coefficients of this newly constructed model arise from microstructure, more precisely, they are uniquely defined by data in the original model. Homogenization procedure is based on the method of two-scale convergence and is mathematically rigorously justified.