Poroelastic plate model obtained by simultaneous homogenization and dimension reduction

TL;DR

This work develops a poroelastic plate model by simultaneously homogenizing the 3D Biot–Stokes system and reducing the dimension in the plate’s thickness, under the regime $\varepsilon(h)\ll h$. It introduces a Griso-based decomposition to obtain compactness and passes to a limit that yields a coupled Biot plate system with bending and in-plane components linked through effective tensors $\mathbb{A}^{\text{hom}}$, $\mathbb{B}^{H}$, $\mathbb{K}$, and $M_0$, while deriving a vertically-dominant Darcy law $\partial_3 p$ and interface conditions for pressure continuity or Neumann data depending on flow. The quasi-static and inertial cases are treated, with existence, uniqueness, energy identities, and strong convergence established; the model extends Mikelić (2015) by incorporating interfacial contacts and inertial effects, and clarifies the role of microscopic boundary conditions in the macroscopic limit. Overall, the paper provides a rigorous multiscale derivation of Biot-type plate equations, including limit Darcy behavior and pressure-interface conditions, and discusses generalizations such as different boundary conditions and surface loads, highlighting the absence of memory effects in the bending regime. These results offer a mathematically solid foundation for predicting the behavior of poroelastic plates with complex microstructures in engineering and geophysical applications.

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. Mikelić 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.

Figures (3)

• Figure 1: Illustration of the poroelastic plate, i.e. the fluid and the solid phases
• Figure 2: (a) is an example of interface where there is no flow, while (b) is an example of interface such that there is a flow. Note also that the cells in (b) don't have the opposite boundaries matching.
• Figure 3: In this figure we explain poroelastic region and the domain of vertical permeability. For simplicity of the picture we drew that each $I_i$, $i=1,\dots,8$ has only one (layer of) cell. Here $J_p=J_2 \cup J_3 \cup J_4\cup J_5$ and $J_K=J_2 \cup J_3 \cup J_4$ and $J_i$, for $i=1,\dots,5$ are assumed to be open intervals.

Theorems & Definitions (90)

• Remark 2.1
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Proposition 2.6
• Remark 2.7
• Lemma 2.8
• proof
• Lemma 2.9
• proof