Indentation Analysis of Biphasic Articular Cartilage: Nonlinear Phenomena Under Finite Deformation

Abstract

The nonlinear indentation response of hydrated articular cartilage at physiologically relevant rates of mechanical loading is studied using a two-phase continuum model of the tissue based on the theory of mixtures under finite deformation. The matrix equations corresponding to the governing mixture equations for this nonlinear problem are derived using a total Lagrangian penalty finite element method, and solved using a predictor-corrector iteration within a modified Newton-Raphson scheme. The stress relaxation indentation problem is examined using either a porous (free draining) indenter or solid (impermeable) indenter under fast and slow compression rates. The creep indentation problem is studied using a porous indenter. We examine the finite deformation response and compare with the response obtained using the linear infinitesimal response. Differences between the finite deformation response and the linear response are shown to be significant when the compression rate is fast or when the indenter is impermeable. The finite deformation model has a larger ratio of peak-to-equilibrium reaction force, and higher relaxation rate than the linear model during the early relaxation period, but a similar relaxation time. The finite deformation model predicts a slower creep rate than the linear model, as well as a smaller equilibrium creep displacement. The pressure distribution below the indenter, particularly near the loaded surface is also larger with the finite deformation model.