Equivalence between short-time biphasic and incompressible elastic material responses

Abstract

Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response deltat<<Delta(2) / //parallelC(4)//K//, where Delta is a characteristic dimension, C(4) is the elasticity tensor, and K is the hydraulic permeability tensor of the solid matrix. Certain notes of caution are provided with regard to implementation issues, particularly when finite element formulations of incompressible elasticity employ an uncoupled strain energy function consisting of additive deviatoric and volumetric components.

Figure 1

Results of unconfined compression analysis of a cylindrical disk. For this axisymmetric analysis, the mesh extends from r = 0 to r = 3 mm. Symbols represent the biphasic response at δt = 0.001 s and solid lines represent the analytical solution for the incompressible elastic response of Eq. (54), evaluated at λ_z = 0.8.

Figure 2

Schematic of the axisymmetric finite element contact analysis.

Figure 3

Normal traction at the contact interface for the first and second analyses (the latter with tension-compression nonlinearity), for biphasic and incompressible elastic cases.

Figure 4

Fluid pressure at the contact interface for the first and second analyses, for biphasic and incompressible elastic cases.

Figure 5

Contour plot of the fluid pressure for (a) the biphasic case and (b) the incompressible-elastic case, for the second analysis.

Figure 6

Radial normal Lagrangian strain E_rr for (a) the biphasic case and (b) the incompressible-elastic case, for the second analysis.

Figure 7

Axial normal Lagrangian strain E_zz for (a) the biphasic case and (b) the incompressible-elastic case, for the second analysis.