Abstract
A three-dimensional unilateral contact problem for articular cartilage layers attached to subchondral bones shaped as elliptic paraboloids is considered in the framework of the biphasic cartilage model. The main novelty of the study is in accounting not only for the normal (vertical), but also for tangential vertical (horizontal) displacements of the contacting surfaces. Exact general relationships have been established between the contact approach and some integral characteristics of the contact pressure, including the contact force. Asymptotic representations for the contact pressure integral characteristics are obtained in terms of the contact approach and some integral characteristics of the contact zone. The main result is represented by the first-order approximation problem. We supply the theoretical description of the asymptotic method by numerical analysis of the model. Our calculations demonstrate good convergence of the numerical scheme in determination of the parameters. In particular, it is shown that accounting for the tangential displacement is important in cases where the contact zone is non-circular.
Highlights
Biomechanical contact problems involving transmission of forces across biological joints are of considerable practical interest
In [4], the unilateral contact problem for articular cartilages bonded to subchondral bones with a contact zone in the shape of an arbitrary ellipse has been considered, and a closed form analytic solution was found
We will address the main question of this analysis, the importance of accounting for the tangential displacement of the contact problem, without an assumption of axisymmetry
Summary
Biomechanical contact problems involving transmission of forces across biological joints are of considerable practical interest (see, e.g. [2, 3, 11, 14]). In [4], the unilateral contact problem for articular cartilages bonded to subchondral bones with a contact zone in the shape of an arbitrary ellipse has been considered, and a closed form analytic solution was found. Exploiting this exact result, Argatov and Mishuris [7] have performed perturbation analysis of the contact problem with approximate geometry of the contact surfaces. The principal originality of this work, with contrast to papers [6] and [4], is in the accounting for tangential displacements in the contact problem for cartilage layers while using a contact zone of elliptical shape, based on the biphasic model. On the basis of this discussion of the obtained numerical results we make some conclusions concerning the model
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