Giulio Sciarra

Generalized Hooke's law for isotropic second gradient materials

In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortuné & Vallée. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain-gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski. Using a suitable decomposition of the strain-gradient, it is found a necessary and sufficient condition, to be verified by the elastic moduli, assuring positive definiteness of the stored elastic energy. The problem of warping in linear torsion of a prismatic second gradient cylinder is formulated, thus obtaining a possible measurement procedure for one of the second gradient elastic moduli.

Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid

We study infinitesimal deformations of a porous linear elastic body saturated with an inviscid fluid and subjected to conservative surface tractions. The gradient of the mass density of the solid phase is also taken as an independent kinematic variable and the corresponding higher-order stresses are considered. Balance laws and constitutive relations for finite deformations are reduced to those for infinitesimal deformations, and expressions for partial surface tractions acting on the solid and the fluid phases are derived. A boundary-value problem for a long hollow porous solid cylinder filled with an ideal fluid is solved, and the stability of the stressed reference configuration with respect to variations in the values of the coefficient coupling deformations of the two phases is investigated. An example of the problem studied is a cylindrical cavity leached out in salt formations for storing hydrocarbons.

Phase coexistence in consolidating porous media

The appearance of the fluid-rich phase in saturated porous media under the effect of an external pressure is investigated. For this purpose we introduce a two field second gradient model allowing the complete description of the phenomenon. We study the coexistence profile between poor and rich fluid phases and we show that for a suitable choice of the parameters nonmonotonic interfaces show up at coexistence.

Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media

Abstract We consider a saturated porous medium in the solid–fluid segregation regime under the effect of an external pressure applied on the solid constituent. We prove that, depending on the dissipation mechanism, the dynamics is described either by a Cahn–Hilliard or by an Allen–Cahn-like equation. More precisely, when the dissipation is modeled via the Darcy law we find that, provided the solid deformation and the fluid density variations are small, the evolution equation is very similar to the Cahn–Hilliard one. On the other hand, when only the Stokes dissipation term is considered, we find that the evolution is governed by an Allen–Cahn-like equation. We use this theory to describe the formation of interfaces inside porous media. We consider a recently developed model proposed to study the solid–liquid segregation in consolidation and we give a complete description of the formation of an interface between the fluid-rich and the fluid-poor phase.

Kink localization under asymmetric double-well potentials.

We study diffuse phase interfaces under asymmetric double-well potential energies with degenerate minima and demonstrate that the limiting sharp profile, for small interface energy cost, on a finite space interval is in general not symmetric and its position depends exclusively on the second derivatives of the potential energy at the two minima (phases). We discuss an application of the general result to porous media in the regime of solid-fluid segregation under an applied pressure and describe the interface between a fluid-rich and a fluid-poor phase.

Phase transition in saturated porous media: Pore-fluid segregation in consolidation

Abstract Consider the consolidation process typical of soils; this phenomenon is expected not to exhibit a unique state of equilibrium, depending on external loading and constitutive parameters. Beyond the standard solution also, pore-fluid segregation can arise. Pore-fluid segregation has been recognized as a phenomenon typical of the short time behavior of a saturated porous slab or a saturated porous sphere, during consolidation. In both circumstances, the Biot three-dimensional model provides time increasing values of the water pressure (and fluid mass density) at the center of the slab (or of the sphere), at early times, if the Lame constant μ of the skeleton is different from zero. This localized pore-fluid segregation is known in the literature as the Mandel–Cryer effect. In this paper, a nonlinear poromechanical model is formulated. The model is able to describe the occurrence of two states of equilibrium and the switching from one to the other by considering a kind of phase transition. Extending the classical Biot theory, a more than quadratic strain energy potential is postulated, depending on the strain of the porous material and the variation of the fluid mass density (measured with respect to the skeleton reference volume). When the consolidating pressure is strong enough, the existence of two distinct minima is proven.

Phase field modeling of partially saturated deformable porous media

Abstract A poromechanical model of partially saturated deformable porous media is proposed based on a phase field approach at modeling the behavior of the mixture of liquid water and wet air, which saturates the pore space, the phase field being the saturation (ratio). While the standard retention curve is expected still^ to provide the intrinsic retention properties of the porous skeleton, depending on the porous texture, an enhanced description of surface tension between the wetting (liquid water) and the non-wetting (wet air) fluid, occupying the pore space, is stated considering a regularization of the phase field model based on an additional contribution to the overall free energy depending on the saturation gradient. The aim is to provide a more refined description of surface tension interactions. An enhanced constitutive relation for the capillary pressure is established together with a suitable generalization of Darcys law, in which the gradient of the capillary pressure is replaced by the gradient of the so-called generalized chemical potential, which also accounts for the “force”, associated to the local free energy of the phase field model. A micro-scale heuristic interpretation of the novel constitutive law of capillary pressure is proposed, in order to compare the envisaged model with that one endowed with the concept of average interfacial area. The considered poromechanical model is formulated within the framework of strain gradient theory in order to account for possible effects, at laboratory scale, of the micro-scale hydro-mechanical couplings between highly localized flows (fingering) and localized deformations of the skeleton (fracturing).

Compacton formation under Allen–Cahn dynamics

We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non–constant higher order stiffness. We analytically solve the stationary problem and deduce the existence of so-called compactons, namely, connections on a finite interval between the two phases. The dynamics problem is numerically solved and compacton formation is described.

Temperature-driven volume transition in hydrogels: Phase-coexistence and interface localization

Abstract We study volume transition phenomenon in hydrogels within the framework of Flory–Rehner thermodynamic modelling; we show that starting from different models for the Flory parameter different conclusions can be achieved, in terms of admissible coexisting equilibria of the system. In particular, with explicit reference to a one-dimensional problem we establish the ranges of both temperature and traction which allow for the coexistence of a swollen and a shrunk phase. Through consideration of an augmented Flory–Rehner free-energy, which also accounts for the gradient of volume changes, we determine the position of the interface between the coexisting phases, and capture the connection profile between them.