مشکل تماس برای یک لایه الاستیک نازک با ضخامت متغیر: کاربرد برای حساسیت تجزیه و تحلیل از مکانیک تماس مفصلی

In the framework of the recently developed asymptotic models for tibio-femoral contact incorporating frictionless elliptical contact interaction between thin elastic, viscoelastic, or biphasic cartilage layers, we apply an asymptotic modeling approach for analytical evaluating the sensitivity of crucial parameters in joint contact mechanics due to small variations in the thicknesses of the contacting cartilage layers. The four term asymptotic expansion for the normal displacement at the contact surface is explicitly derived, which recovers the corresponding solution obtained previously for the 2D case in the compressible case. It was found that to minimize the influence of the cartilage thickness non-uniformity on the force–displacement relationship, the effective thicknesses of articular layers should be determined from a special optimization criterion.

Contact problems involving transmission of forces across biological joints are of considerable practical importance and a number of numerical models for articular contact are available [1] and [2]. At the same time, the necessity of analytical models becomes an important issue in developing improved understanding of load distribution in the normal and pathological joints, which affects the mechanical aspects of osteoarthritis [3] and [4]. Also, analytical modeling of the distributed internal forces generated by articular contact in tibio-femoral joints is required in multibody dynamic simulations of physical exercise of a human skeleton [5] and [6]. As a rule, analytical models of articular contact assume rigid bones and represent cartilage as a thin elastic layer of constant thickness resisting to deformation like a Winkler foundation consisting of a series of discrete springs with constant length and stiffness [7]. However, a subject-specific approach to articular contact mechanics requires developing patient-specific models for accurate predictions. Recently, a sensitivity analysis of finite element models of hip cartilage mechanics with respect to varying degrees of simplified geometry was performed in [8]. Based on the asymptotic analysis of the frictionless contact problem for a thin elastic layer bonded to a rigid substrate in the thin-layer limit [9] and [10], the following asymptotic model for contact interaction of two thin incompressible layers was established [11]: equation(1) View the MathML source-(E1-1h13+E2-1h23)Δyp(y)=δ0-φ(y),y∈ω, Turn MathJax on equation(2) View the MathML sourcep(y)=0,∂p∂n(y)=0,y∈Γ. Turn MathJax on Here, p(y)p(y) is the contact pressure density, hαhα and EαEα are the thickness and elastic modulus of the layer material, respectively, View the MathML sourceα=1,2,Δy=∂2/∂y12+∂2/∂y22 is the Laplace differential operator, δ0δ0 is the vertical approach of the rigid substrates, φ(y)φ(y) is the gap function defined as the distance between the layer surfaces in the vertical direction, ω is the contact area, Γ is the contour of ω,∂/∂nω,∂/∂n is the normal derivative. It was shown [12], [3] and [13] that the problem (1) and (2) describes the instantaneous response of thin biphasic layers to dynamic and impact loading. In [14], the elastic model (1) and (2) was generalized for the general viscoelastic case.

The present study results in asymptotic solutions to the three-dimensional unilateral contact problem for a bonded thin elastic compressible or incompressible layer of variable thickness. As the main result of the present paper, the four term asymptotic expansion (76) is obtained for the normal displacement at the contact surface. In the compressible case, the derived asymptotic expansion (79) for the contact pressure state recovers the corresponding solution obtained in [18] for the 2D case. The objective of this study was to apply an asymptotic modeling approach for evaluating the sensitivity of the asymptotic model of tibio-femoral contact due to small variations in the thicknesses of the contacting articular cartilage layers. It was found that to minimize the influence of the cartilage thickness non-uniformity on the force–displacement relationship, the effective geometrical characteristics h1h1 and h2h2 of articular layers, which enter the asymptotic model (1) and (2), should be determined from the introduced optimization criterion (97).