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

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 incre ase 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 funct ion of the partial apparent densities of the solid and the fluid. The model is capable of accounting for the above mentioned dilatational behaviour, b ut in order to isolate its essential features more clearly we compromise on the other aspects of deformation. Specifically, the following questions ar e addressed: (i) The boundary pressure is divided between the solid and flu id pressures with a dividing coefficient which depends on the constituent a pparent densities regarded as state parameters. The work performed by these tractions should vanish in any cyclic process over this parameter space. T his condition severely restricts the permissible constitutive relations for the dividing coefficient, which results to be characterized by a single ma terial parameter. (ii) A stability analysis is performed for homogeneous, p ressurized 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 depe nds on the interaction terms in the free energy. The larger this interactio n is, the smaller will be the critical (smallest unstable) external pressur e. (iii) It will be shown that within the stable regime of behaviour an inc rease of the external pressure will lead to a decrease of the solid density and correspondingly an increase of the specific volume, thus proving the w anted dilatation effects. (iv) We close by presenting a formulation of mixt ure theory involving second gradients of the displacement as a further defo rmation measure (Germain 1973); this allows for the regularization of the o therwise singular boundary effects (dell'Isola and Hutter 1998, dell'Isola, Hutter and Guarascio 1999).