Derivation of a Biot-Plate-System for a thin poroelastic layer

Abstract: We study incompressible fluid flow through a thin poroelastic layer and rigorously derive a macroscopic model when the thickness of the layer tends to zero. Within the layer we assume a periodic structure and both, the periodicity and the thickness of the layer, are of order $\varepsilon$ which is small compared to the length of the layer. The fluid flow is described by quasistatic Stokes-equations and for the elastic solid we consider linear elasticity equations, and both are coupled via continuity of the velocities and the normal stresses. The aim is to pass to the limit $\varepsilon \to 0$ in the weak microscopic formulation by using multi-scale techniques adapted to the simultaneous homogenization and dimension reduction in continuum mechanics. The macroscopic limit model is given by a coupled Biot-Plate-system consisting of a generalized Darcy-law coupled to a Kirchhoff-Love-type plate equation including the Darcy pressure.