A solid-fluid mixture model allowing for solid dilatation under external pressure

Abstract

To cite this version:Giulio Sciarra, Francesco Dell'Isola, Kolumban Hutter. A solid-fluid mixture model allowing for solid dilatation under external pressure. Continuum Mechanics and Thermodynamics, Springer Verlag, 2001, pp.20 A sponge subjected to an increase of the outside fluid pressure expands its volume but nearly mantains its true density and thus gives way to an increase of the interstitial volume. This behaviour, not yet properly described by solid-fluid mixture theories, is studied here by using the Principle of Virtual Power with the most simple dependence of the free energy as a function of the partial apparent densities of the solid and the fluid. The model is capable of accounting for the above mentioned dilatational behaviour, but in order to isolate its essential features more clearly we compromise on the other aspects of deformation. Specifically, the following questions are addressed: (i) The boundary pressure is divided between the solid and fluid pressures with a dividing coefficient which depends on the constituent apparent densities regarded as state parameters. The work performed by these tractions should vanish in any cyclic process over this parameter space. This condition severely restricts the permissible constitutive relations for the dividing coefficient, which results to be characterized by a single material parameter.(ii) A stability analysis is performed for homogeneous, pressurized reference states of the mixture by postulating a quadratic form for the free energy and using the afore mentioned permissible constitutive relations. It is shown that such reference states become always unstable if only the external pressure is sufficiently large, but the exact value depends on the interaction terms in the free energy. The larger this interaction is, the smaller will be the critical (smallest unstable) external pressure. (iii) It will be shown that within the stable regime of behaviour an increase of the external pressure will lead to a decrease of the solid density and correspondingly an increase of the specific volume, thus proving the wanted dilatation effects. (iv) We close by presenting a formulation of mixture theory involving second gradients of the displacement as a further deformation measure (Germain 1973); this allows for the regularization of the otherwise singular boundary effects (dell'Isola and Hutter 1998, dell'Isola, Hutter and Guarascio 1999).

Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

“…Only in Wilmanski (2000) the aforementioned initial pressure is explicitly taken into account, however neglecting its effect on the dependence of the deformation tensor on the stress tensor. Having in mind some special applications in geomechanics and in particular the design of pressurised salt caverns (Cosenza et al, 1999), the situation in which the initial pressure of saturating fluid is large enough to be able to originate some instability phenomena (see also Smeulders et al, 1992;Sciarra, 2001;Sciarra et al, 2001), is explicitly considered. As a matter of facts, it is well known that high pressures of confined fluids in sealed salt caverns may cause highly destructive phenomena.…”

“…Moreover, the prestress is prescribed on the mixture and its partition between the solid and the fluid part can be chosen in a certain range, see e.g. Sciarra et al (2001). We In the range of pressures that is represented in the graphic it is evident that the hyperbola can be approximated by a straight line.…”

A damage constitutive model for sliding friction coupled to wear

“…This would imply, in the description of abrasive wear and damage, the introduction of model of the interface as a thick region (with non-vanishing volume) endowed with micro-structures properties, as done, for example, in [39,40,[58][59][60]. Of course, in cited models, the structure of the continua constituting thick interfaces must be richer than that of Cauchy continua; in particular, when considering second gradient materials, the state of stress needs to be described by a second-order Cauchy stress and by a third-order hyperstress (see, e.g., [61,62]). …”

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

“…Note that the partial Cauchy stress tensor T (s) is symmetric and ∂ /∂F (s) equals the partial first Piola-Kirchhoff stress tensor of the solid constituent. Equations (15)- (18) are derived in [34] where Germain's [23] arguments are used to obtain constitutive relations for a second gradient porous matrix filled with an ideal fluid. Equation (13) …”