Strain-rate dependence of cartilage stiffness in unconfined compression: the role of fibril reinforcement versus tissue volume change in fluid pressurization

Abstract

The strain and strain-rate-dependent response of articular cartilage in unconfined compression was studied theoretically. The transient stress and stiffness of cartilage were determined for strain rates ranging from zero to infinity. It is shown, for a given compressive strain, that the axial stress initially increases quickly as a function of strain rate, and then increases progressively more slowly towards the stress corresponding to the instantaneous response. The volume change of the tissue does not give its transient stiffness uniquely, because of the strong strain-rate dependence. The variation of tissue stiffness is primarily determined by the transient stiffness of the radial fibrils. Load sharing between the solid matrix and fluid pressurization also depends on the strain rate. At 15% axial compression, the matrix bears more than 80% of the applied load at a strain rate of 0.005%/s, while the fluid pressurization contributes more than 80% of the load at a strain rate of 0.15%/s. These results show the interplay between fibril reinforcement and fluid pressurization in articular cartilage: the fluid drives fibril stiffening which in turn produces high pore pressure at high strain rates. As a secondary objective of the present work, a fibrillar continuum element was formulated to replace the fibrillar spring element used previously in fibril-reinforced modeling, in order to eliminate the deformation incompatibility between the spring system and the nonfibrillar matrix. The results obtained using the two fibrillar elements were compared with the closed-form solutions for the static and instantaneous responses for the case of large deformation. It was found for unconfined compression that using the spring elements did not generally result in greater numerical errors than using the fibrillar continuum elements.