Derivation of a poroelastic flexural shell model

TL;DR

This paper derives a new poroelastic flexural shell model by analyzing the limit behavior of quasi-static Biot's equations as shell thickness approaches zero, extending classical elastic shell results to poroelastic materials.

Contribution

It extends classical shell models to poroelastic shells, deriving coupled bending and pressure equations with new technical methods.

Findings

Strong convergence of displacement, pressure, and stress to shell equations

Coupled bending and pressure equations including pore pressure effects

Pressure equation is parabolic in the normal direction

Abstract

In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic flexural shells as the thickness of the shell tends to zero and extend the results obtained for the poroelastic plate by Marciniak-Czochra and Mikeli\'c. We choose Terzaghi's time corresponding to the shell 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 shell equations. The derived bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the shell thickness. The effective pressure equation is parabolic only in the normal direction. As additional terms it contains the time derivative of the middle-surface flexural strain. Derivation of the model presents an extension of the results on the derivation of classical linear elastic shells by Ciarlet and collaborators to the poroelastic shells case. The new technical points include determination of the 2 X 2 strain matrix, independent of the vertical direction, in the limit of the rescaled strains and identification of the pressure equation. This term is not necessary to be determined in order to derive the classical flexural shell model.