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2.9 Equilibrium Swelling
When the interstitial fluid of a porous medium contains one or more solutes, an osmotic pressure may be produced in the fluid if the osmolarity of the interstitial fluid is non-uniform, or if it is different from that of the external bathing solution surrounding the porous medium. In general, since the osmolarity of the interstitial fluid may vary over time in transient problems, the analysis of such swelling effects may be addressed using, for example, the biphasic-solute material model described in Section 2.6↑. However, if we are only interested in the steady-state response for such types of materials, when solvent and solute fluxes have subsided, the analysis may be simplified considerably.
The Cauchy stress tensor for a mixture of a porous solid and interstitial fluid is given by where is he fluid pressure and is the stress in the solid matrix resulting from solid strain. When steady-state conditions are achieved, the fluid pressure results exclusively from osmotic effects and ambient conditions (i.e., it does not depend on the loading history). Thus, in analogy to eq.(2.6.2-1), where is the mechanical pressure resulting from ambient conditions and is the osmotic pressure resulting from the osmolarity of the solution.
The osmotic pressure may produce swelling of the solid matrix, which is opposed by the solid matrix stress. This becomes more apparent when considering, for example, the case of a traction-free body. The traction is given by , where is the unit outward normal to the boundary. When , the relation of eq.(2.9-1) produces , clearly showing that the osmotic pressure is balanced by the swelling solid matrix.
The interstitial osmolarity (number of moles of solute per volume of interstitial fluid) may be related to the solute and solid content according to where is the number of moles of solute per volume of the mixture in the reference configuration, is the volume fraction of the solid in the reference configuration, and is the volume ratio of the porous solid matrix. Neither nor depend on the solid matrix deformation, thus eq.(2.9-2) provides the explicit dependence of on . This relation shows that the osmolarity of the interstitial fluid is dependent on the relative change in volume of the solid matrix with deformation. Effectively, under equilibrium swelling conditions, the term in eq.(2.9-1) represents an elastic stress and may be treated in this manner when analyzing equilibrium swelling conditions.
Since also depends on the osmotic coefficient, if we assume that depends on the solid strain at most via a dependence on , we may thus state generically that under equilibrium swelling. It follows that the elasticity tensor for is where is the elasticity tensor of .