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

Markus Gahn

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.

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 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 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.
Paper Structure (19 sections, 26 theorems, 181 equations)

This paper contains 19 sections, 26 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. The microscopic model
  4. Microscopic geometry
  5. Microscopic equations and weak formulation
  6. Assumptions on the data
  7. Main results and the Biot-law
  8. Existence and uniqueness of weak solutions
  9. A priori estimates for the microscopic solutions
  10. Compactness microscopic solutions
  11. Derivation of the Biot-Plate-Model
  12. The generalized Darcy-law
  13. The plate equation
  14. Uniqueness for the macroscopic model
  15. Properties of the space $\mathcal{V}$
  16. ...and 4 more sections

Key Result

Theorem 1

There exists a unique weak solution $(v_{\varepsilon},p_{\varepsilon},u_{\varepsilon})$ of the microscopic problem MicroModel. This solution fulfills Additionally, it holds the initial condition $v_{\varepsilon}(0) =0$ and $p_{\varepsilon}(0) = 0$.

Theorems & Definitions (53)

  • Definition 1: Weak solution
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 43 more