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 straingradient 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.

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.

Second gradient theories have been developed in mechanics for treating different phenomena as capillarity in fluids, plasticity and friction in granular materials or shear band deformations. Here, there is an attempt of formulating a second gradient Biot like model for porous materials. In particular the interest is focused in describing the local dilatant behaviour of a porous material induced by pore opening elastic and capillary interaction phenomena among neighbouring pores and related micro-filtration phenomena by means of a continuum microstructured model. The main idea is to extend the classical macroscopic Biot model by including in the description second gradient effects. This is done by assuming that the surface contribution to the external work rate functional depends on the normal derivative of the velocity or equivalently assuming that the strain work rate functional depends on the porosity and strain gradients. According to classical thermodynamics suitable restrictions for stresses and second gradient internal actions (hyperstresses) are recovered, so as to determine a suitable extended form of the constitutive relation and Darcy's law. Finally a numerical application of the envisaged model to one-dimensional consolidation is developed; the obtained results generalize those by Terzaghi; in particular interesting phenomena occurring close to the consolidation external surface and the impermeable wall can be described, which were not accounted for previously. (C) 2007 Elsevier Ltd. All rights reserved

Fluid saturated porous media are modelled by the theory of mixtures and the placement maps of the solid and of the fluid are considered. The momentum balance equations are derived in the framework of a variational approach: We take an action functional and two families of variations and assume that the sum of the virtual work of the external forces and the variation of such an action along each variation are zero. Constitutive equations for the two Cauchy stress tensors and for the interaction force are derived taking into account a general state of pre-stress for the solid and for the fluid species. Governing equations are therefore formulated, however, for the sake of simplicity, only the case of pure initial pressure is further investigated. The propagation of bulk (transversal and longitudinal) waves and the influence of pre-stress are studied: In particular, stability analyses are carried out starting from dispersion relations and the role of pre-stress is investigated. Finally, a numerical example is established for a given state of pre-stress, deriving the phase velocities and the attenuation coefficients of transversal and longitudinal waves.

International audienc

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.

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).

Size-effects characterize the fracture process of many engineering materials. Their modelling calls for material constitutive relations which are not indifferent, as standard elasticity, to variations of scale: strain-gradient elasticity or plasticity have often served the purpose. The three classical crack opening problems of fracture mechanics are here solved within the framework of linear strain-gradient elasticity for the most general isotropic material. Apart from the Lame constants, this is completely identified by five additional moduli, modelling its micro-structural characteristics. This general setting allows for recovering, as particular cases, previous analyses in (Gourgiotis and Geogiadis in J. Mech. Phys. Solids 57(11): 1898-1920, 2009), relative to modes I and II for the so-called Simplified Mindlin materials (17), and in (Radi in Int. J. Solids Struct. 45(10): 3033-3058, 2008), relative to mode III for couple-stress materials. More importantly, having a rather refined material description allows for understanding how the energy release rate is affected by the actual values of the characteristic lengths. Hence we demonstrate the strengthening effect of suitable microstructures and their optimality in order to provide cohesive like actions on the crack lips. Despite being limited to the linear elastic hypothesis, the solutions found constitute an useful insight to develop more complex, possibly nonlinear, constitutive relationships for the strain-gradient terms as for plastic or viscoelastic materials