Mechanics Research Communications 38 (2011) 29–33 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom A general solution of the axisymmetric contact problem for biphasic cartilage layers Ivan Argatov Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK article info Article history: Received 14 July 2010 Received in revised form 18 November 2010 Available online 24 December 2010 Keywords: Contact problem Cartilage layer Biphasic material model Analytical solution abstract A unilateral axisymmetric contact problem for articular cartilage layers is considered. The articular car- tilages bonded to subchondral bones are modeled as biphasic materials consisting of a solid phase and a fluid phase. It is assumed that the subchondral bones are rigid and shaped like bodies of revolution with arbitrary convex profiles. The obtained closed-form analytical solution is valid over time periods compared with the typical diffusion time and can be used for increasing loading. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Biomechanical contact problems involving transmission of forces across biological joints are of considerable practical impor- tance in orthopedic surgery and many numerical models for contact interaction of articular cartilage surfaces in joints are available (Wilson et al., 2005; Han et al., 2005; Anderson et al., 2008). At the same time, the necessity of analytical models becomes an important issue in developing improved understanding of load dis- tribution in the normal and pathological joints, which affects the mechanical aspects of osteoarthritis (Wu et al., 2000). It is known that the joint degradation in the early stages of osteoarthritis may be reflected in changes of material properties of articular cartilage layers, which were observed to become thicker, softer, and more permeable (see references given by Wu et al. (2000)). Thus, hav- ing at hand an analytical model of the joint, it is easy to predict the corresponding behavior of the important contact parameters such as the maximum contact pressure and the contact area dur- ing the evolution of osteoarthritis in its early stages. Modeling of articular cartilage replacement materials also requires an analyti- cal description of articular contact mechanics (Stoffel et al., 2009). In particular, analytical models would be useful in studying the structural optimization problem for synthetic implants for the local repair of full-thickness cartilage defects (Messner and Gillquist, 1993). On the other hand, analytical solutions are used to test the accuracy of numerical models (Wu et al., 1997a). E-mail address: ivan.argatov@gmail.com As it was observed by Genda et al. (2001), a practical and easy-to- use analysis technique for studying the patient’s hip joint contact pressure distribution would be useful to assess the effect of abnor- mal biomechanical conditions in the hip joint where the role of an ideal sphericity in normal function of hip joint is very important. It should be noted that the method of Genda et al. (2001) is based on the discrete element analysis and the contact interaction between the articular joint surfaces is modeled through the interaction of a series of normal and shear springs. There is a large body of literature associated with contact inter- action of thin layers (Eberhardt et al., 1991; Barry and Holmes, 2001; Hlavᡠcek, 2008). Even in the case of pure elastic behavior of the material, the contact problem presents significant difficul- ties for analytical solution (Argatov, 2005). But the contact problem for biphasic layers is time-dependent, and such problems have not been widely investigated before. Ateshian et al. (1994) obtained an asymptotic solution for the axisymmetric contact problem of two identical biphasic cartilage layers attached to two rigid imperme- able spherical bones of equal radii modeled as circular paraboloids. Wu et al. (1996) extended this solution by combining the joint contact model for the contact of two biphasic cartilages with the assumption of the kinetic relationship from classical contact mechanics (Johnson, 1985). The biphasic cartilage constitutive model proposed by Mow et al. (1980) has proved successful in describing the mechani- cal response of articular cartilage. That is why, the asymptotic model developed by Ateshian et al. (1994) and Wu et al. (1996) has received much attention in the recent years. In particular, Wu et al. (1997b) obtained an improved solution for the contact of two biphasic cartilage layers which can be used for dynamic loading. 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.11.006