Finite deformation theory of hierarchically arranged porous solids. - II. Constitutive behaviour.

A constitutive formulation for finite deformation of porous solids, including an hierarchical arrangement of the pores is presented. An extended Darcy equation is derived by means of a formal averaging procedure. The procedure transforms the discrete network of pores into a continuum, without sacrificing essential information about orderly intercommunication of the pores. The distinction between different hierarchical levels of pores is achieved by means of a hierarchical parameter. The macroscopic equations are derived assuming that the pores are a network of cylindrical vessels in which Poiseuille-type pressure-flow relations are valid. The relationships between stress, strain, strain rate, fluid volume fraction, fluid volume fraction rate and time are derived from Lagrange equations of irreversible thermodynamics. The theory has applications, particularly in the field of the mechanics of blood perfused soft tissues, where the distinction between arterioles, capillaries and venules is essential for a correct quantification of regional blood perfusion of the tissue. Conductance of the medium depends on the local state of tissue deformation which is assumed to cause stretching and buckling of the vessels. Deformations are assumed quasi-static and isothermal. Both solid and fluid are assumed incompressible. It is shown that the theory is consistent with Biot's finite deformation theory of porous solids for the limiting case where the pore structure has no hierarchy.