A Rigorous Derivation of the Equations for the Clamped Biot-Kirchhoff-Love Poroelastic plate

TL;DR

This paper rigorously derives simplified plate equations from three-dimensional Biot's poroelastic equations as the plate thickness approaches zero, capturing the coupled elastic and fluid pressure behaviors in thin structures.

Contribution

It introduces a new class of plate equations for poroelastic materials that incorporate coupling between bending, in-plane elasticity, and pore pressure effects.

Findings

Strong convergence of displacement, pressure, and stress to the plate equations

Derivation of coupled bending and pressure equations with pore pressure effects

Identification of parabolic pressure behavior in the vertical direction

Abstract

In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi's time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the 2D Navier's linear elasticity equations, with elastic moduli depending on Gassmann's and Biot's coefficients. The bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the plate thickness. The pressure equation is parabolic only in the vertical direction. As additional terms it contains the time derivative of the in-plane Laplacean of the vertical deflection of the plate and of the the elastic in-plane compression term.