Contact Problem For Thin Biphasic Cartilage Layers: Perturbation Solution

Advanced Structured Materials

2015

In this chapter, an asymptotic modeling methodology for tibio-femoral contact is developed, based on the asymptotic models of frictionless unilateral contact interaction between thin cartilage layers. In Sect. 7.1, the normal contact forces, which are needed for multibody dynamics simulations, are determined analytically based on the exact solution for elliptical contact between thin cartilage layers generally modeled as viscoelastic incompressible layers. In Sect. 7.2, the equivalent HuntCrossley model for articular contact is developed in the framework of the short-time contact model for thin bonded biphasic layers.

“…This question is explored in [12], where a perturbation solution is obtained under the assumption that the subchondral bones are rigid and shaped closely to elliptic paraboloids.…”

Section: Approximation Of the Articular Femur And Tibia Geometries Bymentioning

confidence: 99%

Articular Contact Mechanics

Argatov

Advanced Structured Materials

2015

“…(9.19) Following Argatov and Mishuris [8], we construct an asymptotic solution for the three-dimensional contact problem formulated by Eq. (9.13), under the monotonicity condition (9.19).…”

Section: (Y)h (T) − δ ε (T)mentioning

confidence: 99%

Sensitivity Analysis of Articular Contact Mechanics

Argatov

Advanced Structured Materials

2015

Asymptotic models of articular contact developed in the previous chapters assume, in particular, that the cartilage layers are of uniform thickness and are bonded to rigid substrates shaped like elliptic paraboloids. In this final chapter, treating the term "sensitivity" in a broad sense, we study the effects of deviation of the substrate's shape from the elliptic (Sect. 9.1) and of nonuniform thicknesses of the contacting incompressible layers (Sect. 9.2). It is shown that these effects in multibody dynamics simulations can be minimized if the geometric parameters in question (in particular, the layer thicknesses) are determined in a specific way to minimize the corresponding error in the force-displacement relationship.

“…An asymptotic modeling approach to study the contact problem for biphasic cartilage layers has been performed by Argatov and Mishuris in a series of articles (see [4,5,7]). In particular, it was shown [4] that accounting for the tangential displacements is important in the case of diseased cartilage where the measurement of indentation depth may differ even as much as 10% in comparison with the healthy case.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], 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. Other analytic solutions for the contact problem were found using the viscoelastic cartilage model for elliptic contact zone in [6].…”

Section: Introductionmentioning

confidence: 99%

“…Note that the perturbation method proposed in [7] could be one of the options for the analysis, however, the procedure is too complex to perform even a few asymptotic steps. Here, employing some technique and ideas from [4] and [5], we propose another way to construct the asymptotics which utilises the assumption that the shape of the contact zone is an ellipse at the initial stage of deformation and can be regarded as a small perturbation of the ellipse at any other stage of deformation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of the Unilateral Contact Problem for Biphasic Cartilage Layers With an Elliptic Contact Zone and Accounting for Tangential Displacements

Rogosin

Mathematical Modelling and Analysis

Koroleva

et al. 2016

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 (horisontal) 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.