Elasticity and permeability of porous fibre-reinforced materials under large deformations

Soft biological tissues with collagen reinforcement can be represented by a porous matrix saturated by a fluid and reinforced by a network of statistically oriented, impermeable fibres. This paper aims at determining the effect of the fibres on both the elastic properties and the permeability of the system, under large deformations, and represents the unification and generalisation of previous works in which the elasticity was studied for a pure solid, in the absence of the pore fluid, and the permeability was studied in the neighbourhood of the undeformed configuration. Throughout this work, the saturation constraint is assumed to hold, and the solid and fluid phases are assumed to be intrinsically incompressible. These hypotheses imply that the volumetric fraction of one of the two phases (e.g., the solid) is sufficient to determine the distribution of solid and fluid mass at every point of the homogenised porous medium. Overall incompressibility is achieved at compaction, i.e., when pores are closed and all the fluid has escaped. A new form of the elastic strain energy potential is proposed, based on the sum of a given base potential and a correction potential, function of the volumetric deformation, which serves solely to impose the incompressibility constraint at compaction. The large-strain overall permeability is obtained by employing a pull-back of the structure tensor to the reference configuration. This gives rise to an integral form of the permeability that needs to be calculated at each increment of deformation. The presentation is entirely covariant, so that general curvilinear coordinates can immediately be employed, if needed. (C) 2011 Elsevier Ltd. All rights reserved.