Mechanical effects of ionic replacements in articular cartilage. Part I: The constitutive model

Abstract

A three-phase multi-species electro-chemo-mechanical model of articular cartilage is developed that accounts for the effect of two water compartments, namely intra-fibrillar water stored in between collagen fibrils and extra-fibrillar water covering proteoglycans. The collagen fibers constitute the solid phase while intra-fibrillar water and dissolved NaCl and CaCl(2) on one hand and extra-fibrillar water, ions Na(+), Ca(2+) and Cl(-) and proteoglycans on the other hand, form the two fluid phases. The complete picture that includes time-dependent mass transfers between the two fluid phases, diffusion of water and ions and electrical flow emerges from the Clausius-Duhem inequality but it is deferred to further study. The analysis is restricted to equilibrium states. The present work complements the mechanical model developed in Loret and Simões (Mech Material 36(5-6): 515-541, 2004a) where the presence of the sole NaCl was considered. In its current version, the model can handle mechanical and chemical loadings and unloadings involving the two salts, NaCl and CaCl(2). In order to reproduce experimental data, the shielding effects are made cation-dependent. Strong orientation of collagen fibers parallel to the joint surface implies anisotropic mechanical properties. Electro-chemo-mechanical couplings result in a chemistry-dependent apparent tensile Poisson's ratio, that increases to large values as the solution gets fresher. The model captures these aspects as well. The features of the model are first exposed in an infinitesimal strain context. Subsequently, large strains that typically occur in uniaxial traction under deionized water are accounted for, and a nonlinear anisotropic hyper-elastic behavior is developed. Parametric identification and simulations of actual loading processes are described in a companion paper, Loret and Simões (Biomech Model Mechanobiol, in press, DOI 10.1007/s10237-004-0063-6).