Skip to main content
Top

2015 | OriginalPaper | Chapter

6. Contact of Thin Biphasic Layers

Authors : Ivan Argatov, Gennady Mishuris

Published in: Contact Mechanics of Articular Cartilage Layers

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
2.
go back to reference Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: exact solution for monotonic loading. J. Biomech. 44, 759–761 (2011)CrossRef Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: exact solution for monotonic loading. J. Biomech. 44, 759–761 (2011)CrossRef
3.
go back to reference Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)MATHMathSciNetCrossRef Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)MATHMathSciNetCrossRef
4.
5.
go back to reference 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)CrossRef 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)CrossRef
6.
go back to reference 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., 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)
7.
go back to reference 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)CrossRef 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)CrossRef
9.
go back to reference 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)CrossRef 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)CrossRef
10.
go back to reference 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)CrossRef 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)CrossRef
11.
go back to reference Eberhardt, A.W., Keer, L.M., Lewis, J.L., Vithoontien, V.: An analytical model of joint contact. J. Biomech. Eng. 112, 407–413 (1990)CrossRef Eberhardt, A.W., Keer, L.M., Lewis, J.L., Vithoontien, V.: An analytical model of joint contact. J. Biomech. Eng. 112, 407–413 (1990)CrossRef
12.
go back to reference 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)CrossRef 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)CrossRef
13.
go back to reference Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1981)CrossRef Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1981)CrossRef
14.
go back to reference 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)CrossRef 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)CrossRef
15.
go back to reference 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)CrossRef 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)CrossRef
16.
go back to reference 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)MATHMathSciNetCrossRef 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)MATHMathSciNetCrossRef
18.
go back to reference Kelkar, R., Ateshian, G.A.: Contact creep of biphasic cartilage layers. J. Appl. Mech. 66, 137–145 (1999)CrossRef Kelkar, R., Ateshian, G.A.: Contact creep of biphasic cartilage layers. J. Appl. Mech. 66, 137–145 (1999)CrossRef
19.
go back to reference Lavrentyev, M.A., Shabat, B.V.: Methods of Complex Variable Functions. Nauka, Moscow (1987) (in Russian) Lavrentyev, M.A., Shabat, B.V.: Methods of Complex Variable Functions. Nauka, Moscow (1987) (in Russian)
20.
go back to reference LePage, W.R.: Complex Variables and the Laplace Transform for Engineers. McGraw-Hill, New York (1961)MATH LePage, W.R.: Complex Variables and the Laplace Transform for Engineers. McGraw-Hill, New York (1961)MATH
21.
go back to reference 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)CrossRef 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)CrossRef
22.
go back to reference 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 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
23.
go back to reference 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)CrossRef 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)CrossRef
24.
go back to reference 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)CrossRef 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)CrossRef
25.
go back to reference Wu, J.Z., Herzog, W., Epstein, M.: An improved solution for the contact of two biphasic cartilage layers. J. Biomech. 30, 371–375 (1997)CrossRef Wu, J.Z., Herzog, W., Epstein, M.: An improved solution for the contact of two biphasic cartilage layers. J. Biomech. 30, 371–375 (1997)CrossRef
26.
go back to reference Wu, J.Z., Herzog, W., Epstein, M.: Joint contact mechanics in the early stages of osteoarthritis. Med. Eng. Phys. 22, 1–12 (2000)CrossRef Wu, J.Z., Herzog, W., Epstein, M.: Joint contact mechanics in the early stages of osteoarthritis. Med. Eng. Phys. 22, 1–12 (2000)CrossRef
27.
go back to reference 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)CrossRef 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)CrossRef
Metadata
Title
Contact of Thin Biphasic Layers
Authors
Ivan Argatov
Gennady Mishuris
Copyright Year
2015
DOI
https://doi.org/10.1007/978-3-319-20083-5_6

Premium Partners