A Rational Procedure for the Experimental Evaluation of the Elastic Coefficients in a Linearized Formulation of Biphasic Media with Compressible Constituents

Abstract

This article illustrates the specialization to infinitesimal perturbations of the geometrically nonlinear formulation describing the macroscopic dynamic behavior of biphasic media presented in (Serpieri and Rosati J Mech Phys Solids 59(4): 2011) where, based on Lagrangian mechanics arguments, a consistent mathematical theory of fluidsaturated poroelastic material is developed accounting for the property that both constituent materials are microscopically compressible. The primary contribution of this work is the employment of the linearized model in the development of a general procedure for the determination of the relevant elastic coefficients on the basis of the data provided by a set of experimental measures analogous to the one originally considered by Biot and Willis consisting of a shear test, an unjacketed compressibility test and two jacketed tests in which drainage is either completely allowed or prevented. The linearization of the formulation is first presented for a generic configuration in which both phases are undergoing a generic motion. Subsequently, a specific study is presented on the case that the media is motionless, homogeneous, and isotropic in the reference configuration, which is typical of common experimental set-ups for static tests. A numerical example is shown and a general relation between the elastic coefficients and Biot's constant is derived which consistently accounts for the compressibility of all materials.

Variationally consistent derivation of the stress partitioning law in saturated porous media

“…The Variational Macroscopic Theory of Porous Media (VMTPM) (Serpieri and Rosati, 2011;Serpieri, 2011) is a purely mechanical continuum theory for fully-saturated biphasic media with intrinsically compressible constituents in which the dynamic momentum balance equations are derived on the basis of a Hamiltonian description of the dynamic balances. As a purely mechanical theory, all forces are represented by a potential so that Hamilton's principle, written for the solid porous phase, takes the form:…”

“…The framework of VMTPM has been presented and derived in its more general form at finite deformations (Serpieri and Rosati, 2011) while the linearized theory (infinitesimal displacements) has been derived in a parallel study (Serpieri, 2011). A brief account of the linearized VMTPM is reported in the present section while the reader is referred to the two references above to find a systematic presentation of the kinematics and mechanics of VMTPM both in a finite kinematics context and in a infinitesimal one, and to find a systematic derivation of the corresponding governing equations, strain measures, stress measures and stiffness moduli.…”

“…Such a minimal mechanical description is identified in a macroscopic variational theory of immiscible biphasic mixtures (Bedford and Drumheller, 1979;Serpieri and Rosati, 2011;Serpieri, 2011) in linearized kinematics. Compared to upscaling approaches (Gray and Miller, 2005;Gray et al, 2013;Hassanizadeh and Gray, 1979;Hassanizadeh and Gray, 1980), a macroscopic description of the poroelastic problem with a minimum of microstructural hypotheses determines greater generality, in that the absence of an explicit connection with the porescale physics demands no specific assumption on the microscopic realization of a particular medium.…”

“…Similarly to Bedford and Drumheller, linearized VMTPM fully accounts for the compressibility of both phases and employs a minimal kinematic and mechanical description: the solid deformation is entirely described by the macroscopic displacement field, u ðsÞ , and by the infinitesimal intrinsic volumetric strain field of the solid,ê ðsÞ . As reported, (Serpieri and Rosati, 2011;Serpieri, 2011), proceeding from the simplest possible strain energy potential functions that can be defined on the basis of such u ðsÞ -ê ðsÞ kinematic description, the momentum balance equations of the solid phase are derived in VMTPM exploiting no further physical rationale or assumption, apart from a rigorous deduction based on the least-action principle. The involved stress measures and stiffness moduli of the solid phase, in particular, are consistently introduced in terms of a straightforward work-association with the primary strain measures.…”

“…In Section 2 a summary is presented of the characteristic features of linearized VMTPM (Serpieri, 2011) which is retrieved in presence of QSF. The main objective of this summary is to provide a succinct overview with particular emphasis on the physical insights on the strain measures (Section 2.1), stress measures (Section 2.2), and balance equations (Section 2.3) of this novel theory, and to point out the difference with their counterparts in traditional single-continuum mechanics.…”