Derivation of a poroelastic elliptic membrane shell model

ABSTRACT

A derivation of the model for a poroelastic elliptic membrane shell is undertaken. The flow and deformation in a three-dimensional shell domain is described by the quasi-static Biot equations of linear poroelasticity. We consider the limit when the shell thickness goes to zero and look for the limit equations. Using the technique developed in the seminal articles by Ciarlet, Lods, Miara et al.and the recent results on the rigorous derivation of the equations for poroelastic plates and flexural poroelastic shells by Marciniak-Czochra, Mikelić, and Tambača, we present a rigorous derivation of the linear poroelastic elliptic membrane shell model. After rescaling, the corresponding velocity and the pressure field are close in the $C([0,T];{(H_x^1)}^2\times {(L_x^2)}^2)$ norm and the stresses in $C([0,T];{(L_x^2)}^9)$ norm. We note the major difference with respect to the flexural case: (i) it is not anymore the rescaled total stress divided by the scaling parameter, but the rescaled total stress itself which converges ; (ii) the same comment applies to the pore fluid pressure; and (iii) there is a deterioration of the convergence for the vertical component of the rescaled displacement. Consequence of the above differences is that the effective model remains of the 2nd order in space. In the case of a spherical membrane shell, we confirm the results by Taber from the literature.

KEYWORDS:

• Membrane poroelastic shell
• Biot’s quasi-static equations
• elliptic–parabolic systems
• asymptotic methods

AMS SUBJECT CLASSIFICATIONS:

• 35B25
• 74F10
• 74K25
• 74Q15
• 76S

Notes

No potential conflict of interest was reported by the authors.

This contribution is dedicated to the memory of Professor Vasilii V. Zhikov.

Additional information

Funding

The research of Andro Mikelić was supported in part by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ‘Investissements d’Avenir’ (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).