Articular cartilage with intra and extrafibrillar waters – Simulations of mechanical and chemical loadings by the finite element method

Abstract

In line with previous constitutive developments, a three-phase multi-species electro-chemo-mechanical model of articular cartilage that accounts for the effect of two water compartments is pursued. Here, a finite element formulation is presented in order to solve initial and boundary value problems and to simulate the time and space heterogenous fields generated by mechanical and chemical loadings in laboratory samples. The formulation capitalizes upon (1) the chemo-mechanical model in Loret and Simões [B. Loret, F.M.F. Simões, Articular cartilage with intra- and extra-fibrillar waters. A chemo-mechanical model, Mech. Mater. 36 (5–6) (2004) 515–541; B. Loret, F.M.F. Simões, Mechanical effects of ionic replacements in articular cartilage. I. The constitutive model, Biomech. Model. Mechanobiol. 4 (2–3) (2005) 63–80; Mechanical effects of ionic replacements in articular cartilage. II. Simulations of successive substitutions of NaCl and CaCl 2 , Biomech. Model. Mechanobiol. 4 (2–3) (2005) 81–99] which was restricted to successive equilibria, and from which both time and space were excluded; and (2) the model in Loret and Simões [B. Loret, F.M.F. Simões, Articular cartilage with intra- and extra-fibrillar waters. Mass transfer and generalized diffusion, Eur. J. Mech.-A/Solids 26 (1995) 759–788], where the equations of mass transfer between the two fluid phases and the generalized diffusion equations in the extrafibrillar phase have been established. A thin cartilage layer, laterally confined, is loaded through time varying mechanical and chemical conditions applied on the lower and upper boundaries. Two types of chemical loadings are simulated. First the concentration of NaCl in the bath in contact with the cartilage is changed according to several time schemes, including in particular free swelling and cyclic changes. A more complex process consists in replacing NaCl by CaCl 2 . Chemical loading gives rise to heterogeneous fields during a transient period, whose extend depends on geometry and on a number of characteristic material properties, e.g. Darcy’s seepage time, Fick’s diffusion time, and typical times for intercompartmental mass transfer. Mechanical confined compression and swelling tests are simulated as well. Spatial profiles of the various chemical species and mechanical, chemical and electrical entities highlight the influences of the existence of two water compartments. At steady state, the strains, stresses, pressures and the distribution of water and ions in the two compartments obtained in Loret and Simões [B. Loret, F.M.F. Simões, Articular cartilage with intra- and extra-fibrillar waters. A chemo-mechanical model, Mech. Mater. 36 (5–6) (2004) 515–541; B. Loret, F.M.F. Simões, Mechanical effects of ionic replacements in articular cartilage. I. The constitutive model, Biomech. Model. Mechanobiol. 4 (2–3) (2005) 63–80; Mechanical effects of ionic replacements in articular cartilage. II. Simulations of successive substitutions of NaCl and CaCl 2 , Biomech. Model. Mechanobiol. 4 (2–3) (2005) 81–99] are recovered.