Abstract
In Sect. 6.1, a three-dimensional deformation problem for an articular cartilage layer is studied in the framework of the linear biphasic model. The articular cartilage bonded to subchondral bone is modeled as a transversely isotropic biphasic material consisting of a solid phase and a fluid phase. In Sect. 6.2, the same problem is reconsidered with the effect of inherent viscoelasticity of the solid matrix taken into account. The frictionless unilateral contact problem for the articular cartilage layers is considered in Sect. 6.3. It is assumed that the subchondral bones are rigid and shaped like elliptic paraboloids. The obtained short-time leading-order asymptotic solution is valid for monotonically increasing loading conditions.
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Argatov, I., Mishuris, G.: Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface. Eur. J. Mech. A/Solids 29, 1051–1064 (2010)
Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: exact solution for monotonic loading. J. Biomech. 44, 759–761 (2011)
Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)
Argatov, I.I., Mishuris, G.S.: An asymptotic model for a thin biphasic poroviscoelastic layer. Quart. J. Mech. Appl. Math. (2015). doi:10.1093/qjmam/hbv008
Ateshian, G.A., Wang, H.: A theoretical solution for the frictionless rolling contact of cylindrical biphasic articular cartilage layers. J. Biomech. 28, 1341–1355 (1995)
Ateshian, G.A., Maas, S., Weiss, J.A.: Finite element algorithm for frictionless contact of porous permeable media under finite deformation and sliding. J. Biomech. Eng. 132, 061006 (13 pages) (2010)
Ateshian, G.A., Lai, W.M., Zhu, W.B., Mow, V.C.: An asymptotic solution for the contact of two biphasic cartilage layers. J. Biomech. 27, 1347–1360 (1994)
Barry, S.I., Holmes, M.: Asymptotic behaviour of thin poroelastic layers. IMA J. Appl. Math. 66, 175–194 (2001)
Donzelli, P.S., Spilker, R.L., Ateshian, G.A., Mow, V.C.: Contact analysis of biphasic transversely isotropic cartilage layers and correlations with tissue failure. J. Biomech. 32, 1037–1047 (1999)
Dortmans, L.J.M.G., van de Ven, A.A.F., Sauren, A.A.H.J.: A note on the reduced creep function corresponding to the quasi-linear visco-elastic model proposed by Fung. J. Biomech. Eng. 116, 373–375 (1994)
Eberhardt, A.W., Keer, L.M., Lewis, J.L., Vithoontien, V.: An analytical model of joint contact. J. Biomech. Eng. 112, 407–413 (1990)
Eberhardt, A.W., Keer, L.M., Lewis, J.L.: Normal contact of elastic spheres with two elastic layers as a model of joint articulation. J. Biomech. Eng. 113, 410–417 (1991)
Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1981)
Han, S.K., Federico, S., Epstein, M., Herzog, W.: An articular cartilage contact model based on real surface geometry. J. Biomech. 38, 179–184 (2005)
Hlaváček, M.: A note on an asymptotic solution for the contact of two biphasic cartilage layers in a loaded synovial joint at rest. J. Biomech. 32, 987–991 (1999)
Hlaváček, M.: Frictionless contact of two parallel congruent rigid cylindrical surfaces coated with thin elastic incompressible transversely isotropic layers: an analytic solution. Eur. J. Mech. A/Solids 25, 497–508 (2006)
Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)
Kelkar, R., Ateshian, G.A.: Contact creep of biphasic cartilage layers. J. Appl. Mech. 66, 137–145 (1999)
Lavrentyev, M.A., Shabat, B.V.: Methods of Complex Variable Functions. Nauka, Moscow (1987) (in Russian)
LePage, W.R.: Complex Variables and the Laplace Transform for Engineers. McGraw-Hill, New York (1961)
Li, G., Sakamoto, M., Chao, E.Y.S.: A comparison of different methods in predicting static pressure distribution in articulating joints. J. Biomech. 30, 635–638 (1997)
Mishuris, G., Argatov, I.: Exact solution to a refined contact problem for biphasic cartilage layers. In: Nithiarasu, P., Löhner, R. (eds.) Proceedings of the 1st International Conference on Mathematical and Computational Biomedical Engineering—CMBE2009, pp. 151–154. Swansea, June 29–July 1 2009
Wu, J.Z., Herzog, W.: On the pressure gradient boundary condition for the contact of two biphasic cartilage layers. J. Biomech. 33, 1331–1332 (2000)
Wu, J.Z., Herzog, W., Ronsky, J.: Modeling axi-symmetrical joint contact with biphasic cartilage layers—an asymptotic solution. J. Biomech. 29, 1263–1281 (1996)
Wu, J.Z., Herzog, W., Epstein, M.: An improved solution for the contact of two biphasic cartilage layers. J. Biomech. 30, 371–375 (1997)
Wu, J.Z., Herzog, W., Epstein, M.: Joint contact mechanics in the early stages of osteoarthritis. Med. Eng. Phys. 22, 1–12 (2000)
Yang, T., Spilker, R.L.: A Lagrange multiplier mixed finite element formulation for three-dimensional contact of biphasic tissues. J. Biomech. Eng. 129, 457–471 (2007)
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Argatov, I., Mishuris, G. (2015). Contact of Thin Biphasic Layers. In: Contact Mechanics of Articular Cartilage Layers. Advanced Structured Materials, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-20083-5_6
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