Poroelastic plate model obtained by simultaneous homogenization and dimension reduction

Abstract

In this paper, the starting point of our analysis is a coupled system of linear elasticity and Stokes equation. We consider two small parameters: the thickness $h$ of the thin plate and the pore scale $\varepsilon(h)$ which depends on $h$. We will focus specifically on the case when the pore size is comparatively small relative to the thickness of the plate. The main goal here is derive a model of a poroelastic plate, starting from the $3D$ problem as $h$ goes to zero, using simultaneous homogenization and dimension reduction techniques. The obtained model generalizes the poroelastic plate model derived by A. Mikeli\'c et. al. in 2015 using dimension reduction techniques from $3D$ Biot's equations in the sense that it also covers the case of contacts of poroelastic and (poro)elastic plate as well as the evolution equation with inertial term.