Variational Estimates of the Poroelastic Coefficients
Résumé
Saturated, isotropic, poroelastic materials are classically described by their elastic stiffness, one Biot coefficient and one Biot modulus (Coussy 2010; Dormieux, Molinari, and Kondo 2002). The situation becomes more complex for unsaturated, isotropic, poroelastic materials that require multiple Biot coefficients and moduli (Coussy and Brisard 2009). These poroelastic coefficients are dependent and linked by several linear relationships.
Micromechanical estimates of these coefficients have been proposed by several authors (Ulm, Constantinides, and Heukamp 2004; Pichler and Hellmich 2010). However, these estimates may fail to fulfill the linear relationships that relate the exact poroelastic coefficients. This might be regarded as an undesirable inconsistency of the model.
In this work, we propose new, consistent (in the sense that the above mentioned linear relationships are preserved) estimates of the poroelastic coefficients. Our point of departure is the principle of Hashin and Shtrikman, suitably extended to eigenstressed materials (Bornert et al. 2001). Adopting stress-polarization fields that are similar to the eigenstress-free case (Willis 1977; Ponte Castañeda andWillis 1995) allows us to derive variational estimates of the poroelastic coefficients, which can be shown to fulfill all known linear relationships required from the exact values.
After outlining the derivation within the general framework of eigenstressed, heterogeneous materials, the results will be specialized to poroelasticity. This will lead to a variational justification of the ad-hoc pore isodeformation assumption (Coussy and Brisard 2009).
