A variational deduction of second gradient poroelasticity I: general theory

Abstract

Second gradient theories have to be used to capture how local micro heterogeneities macroscopically affect the behavior of a continuum. In this paper a configurational space for a solid matrix filled by an unknown amount of fluid is introduced. The Euler-Lagrange equations valid for second gradient poromechanics, generalizing those due to Biot, are deduced by means of a Lagrangian variational formulation. Starting from a generalized Clausius-Duhem inequality, valid in the framework of second gradient theories, the existence of a macroscopic solid skeleton Lagrangian deformation energy, depending on the solid strain and the Lagrangian fluid mass density as well as on their Lagrangian gradients, is proven.

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

“…Strain gradient models (see e.g. [2,65] for classical references and [66,67] for more recent results) are good candidates for this purpose, as shown in [68][69][70]. A review of results on the theoretical foundation of a variational approach for higher gradient theories is [71].…”

A continuum model for deformable, second gradient porous media partially saturated with compressible fluids

“…Indeed, what is currently done in continuum poromechanics (see e.g. [13], [14], [20], [39]) is to assume that only the solid and fluid placements are independent kinematical fields, while porosity is usually indirectly calculated by means of so-called "saturation hypotesis" according to which porosity is obtained as the ratio between the "apparent" density of the fluid (which is indeed one of the unknowns of classical poromechanical models) and the "true" or "intrinsic" density (which is usually arbitrarily assigned and is not an unknown field of the problem). If this hypothesis is sensible in the case of saturated media, it constitutes a too strict constraint for partially saturated ones.…”

“…A more general theory accounting for the presence of second gradient of the fluid placements φ w and φ a (i.e. of first gradient of the fluid densities) is needed to treat some kind of problems and can be obtained in the spirit of what done in [39].…”

“…Other examples of macroscopic variational principles for multi-phase continua can be found e.g. in [20,18,19,39]. The study of saturated porous media considered as macroscopic systems was originated by Biot (see e.g.…”

From fractal media to continuum mechanics

“…(i) By taking as the starting point the "microscopic models" for interactions between the fluid flow and a deformable porous matrix [70][71][72][73] so as to obtain higher gradient fractal models.…”