Analysis of stress partitioning in biphasic mixtures based on a variational purely-macroscopic theory of compressible porous media: Recovery of terzaghi's law

The mechanics of stress partitioning in two-phase porous media is predicted on the basis of a variational purely-macroscopic theory of porous media (VMTPM) with compressible constituents. Attention is focused on applications in which undrained flow (UF) conditions are relevant, e.g., consolidation of clay soils and fast deformations in cartilagineous tissues. In a study of the linearized version of VMTPM we have recently shown that, as UF conditions are approached (low permeability or fast loading), Terzaghi's effective stress law holds as a general property of rational continuum mechanics and is recovered as the characteristic stress partitioning law that a biphasic medium naturally complies with. The proof of this property is obtained under minimal constitutive hypotheses and no assumptions on internal microstructural features of a particular class of material. VMTPM predicts that such property is unrelated to compressibility moduli of phases and admits no deviations from Terzaghi's expression of effective stress, in contrast with most of the currently available poroelastic theoretical frameworks. This result is presently illustrated and discussed. Simulations of compressive consolidation tests are also presented; they are obtained via a combined analytical-numerical integration technique, based on the employment of Laplace transforms inverted numerically via de Hoog et al.'s algorithm. The computed solutions consistently describe a transition from drained to undrained flow which confirms that Terzaghi's law is recovered as the limit UF condition is approached and indicate a complex mechanical behavior.