A Conewise Linear Elasticity mixture model for the analysis of tension-compression nonlinearity in articular cartilage

Abstract

A biphasic mixture model is developed that can account for the observed tension-compression nonlinearity of cartilage by employing the continuum-based Conewise Linear Elasticity (CLE) model of Curnier et al. (J. Elasticity, 37, 1-38, 1995) to describe the solid phase of the mixture. In this first investigation, the orthotropic octantwise linear elasticity model was reduced to the more specialized case of cubic symmetry, to reduce the number of elastic constants from twelve to four. Confined and unconfined compression stress-relaxation, and torsional shear testing were performed on each of nine bovine humeral head articular cartilage cylindrical plugs from 6 month old calves. Using the CLE model with cubic symmetry, the aggregate modulus in compression and axial permeability were obtained from confined compression (H-A = 0.64 +/- 0.22 MPa, k2 = 3.62 +/- 0.97 x 10(-16) m4/N.s, r2 = 0.95 +/- 0.03), the tensile modulus, compressive Poisson ratio, and radial permeability were obtained from unconfined compression (E+Y = 12.75 +/- 1.56 MPa, v- = 0.03 +/- 0.01, kr = 6.06 +/- 2.10 x 10(-16) m4/N.s, r2 = 0.99 +/- 0.00), and the shear modulus was obtained from torsional shear (mu = 0.17 +/- 0.06 MPa). The model was also employed to predict the interstitial fluid pressure successfully at the center of the cartilage plug in unconfined compression (r2 = 0.98 +/- 0.01). The results of this study demonstrate that the integration of the CLE model with the biphasic mixture theory can provide a model of cartilage that can successfully curve-fit three distinct testing configurations while producing material parameters consistent with previous reports in the literature.

Figure 1

For the octantwise orthotropic CLE model of the solid phase of cartilage, the three preferred directions of material symmetry are taken to be: a₁ parallel to the cartilage split line direction, a₂ perpendicular to the split line direction, and a₃ normal to the articular cartilage surface.

Figure 2

(a) Apparatus for performing confined and unconfined compression tests. A different testing chamber was employed for (b) confined, and (c) unconfined compression.

Figure 3

Apparatus for performing torsional shear tests.

Figure 4

Experimental confined compression stress-relaxation response $[F_c(t)/\pi r_0^2]$ and corresponding theoretical curvefit for a typical specimen.

Figure 5

Experimental unconfined compression stress-relaxation response $[F_u(t)/\pi r_0^2]$ and corresponding theoretical curvefit for the same specimen as in Fig. 4. The experimental interstitial pressure at the specimen center [p(r = 0,t)] and corresponding theoretical prediction are also presented.

Figure 6

Interstitial fluid load, P_u(t), total load F_u(t), and ratio of fluid to total load support, P_u/ F_u, as a function of time for unconfined compression stress-relaxation, using the average material properties listed in Table 1.