The linear isotropic variational theory and the recovery of biot’s equations

In this chapter, the general framework presented in Chap. 2 is specialized to address linear isotropic two-phase poroelasticity. The elastic moduli of the resulting isotropic theory are derived with the special forms achieved by the governing PDEs for hyperbolic and parabolic problems. Next, the hyperbolic system is deployed to analyze the propagation of purely elastic waves. The chapter is concluded with a section dedicated to a comparison between the hyperbolic isotropic equations resulting from the present theory and their counterparts in Biot’s theory. This comparison shows the recovery by the medium-independent VMTPM framework of the essential structure of Biot’s PDEs. This recovery is herein deductively achieved in absence of heuristic statements, proceeding from the consideration of individual strain energies of the solid and fluid phases and from the minimal kinematic hypotheses of Chap. 2. This study is complemented by an analysis of the bounds of the elastic moduli of the isotropic theory, which is undertaken deploying a generalization to the present two-phase context of the Composite Sphere Assemblage homogenization technique by Hashin.