Variational Estimates of the Poroelastic Coefficients

Saturated, isotropic, poroelastic materials are classically described by their elastic stiffiness, one Biot coefficient and one Biot modulus. The situation becomes more complex for unsaturated, isotropic, poroelastic materials that require multiple Biot coefficients and moduli. These poroelastic coefficients are dependent and linked by several linear relationships. Micromechanical estimates of these coefficients have been proposed by several authors. 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. Adopting stress-polarization fields that are similar to the eigenstress-free case 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.