Sensitivity Analysis of Articular Contact Mechanics

  • Ivan ArgatovEmail author
  • Gennady Mishuris
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 50)


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.


  1. 1.
    Akiyama, K., Sakai, T., Sugimoto, N., Yoshikawa, H., Sugamoto, K.: Three-dimensional distribution of articular cartilage thickness in the elderly talus and calcaneus analyzing the subchondral bone plate density. Osteoarthritis Cartilage 20, 296–304 (2012)CrossRefGoogle Scholar
  2. 2.
    Anderson, A.E., Ellis, B.J., Maas, S.A., Weiss, J.A.: Effects of idealized joint geometry on finite element predictions of cartilage contact stresses in the hip. J. Biomech. 43, 1351–1357 (2010)CrossRefGoogle Scholar
  3. 3.
    Argatov, I.I.: Pressure of a punch in the form of an elliptic paraboloid on a thin elastic layer. Acta Mech. 180, 221–232 (2005)zbMATHCrossRefGoogle Scholar
  4. 4.
    Argatov, I.: Development of an asymptotic modeling methodology for tibio-femoral contact in multibody dynamic simulations of the human knee joint. Multibody Syst. Dyn. 28, 3–20 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Argatov, I.: Contact problem for a thin elastic layer with variable thickness: Application to sensitivity analysis of articular contact mechanics. Appl. Math. Model. 37, 8383–8393 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: Exact solution for monotonic loading. J. Biomech. 44, 759–761 (2011)CrossRefGoogle Scholar
  7. 7.
    Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Argatov, I.I., Mishuris, G.S.: Contact problem for thin biphasic cartilage layers: perturbation solution. Quart. J. Mech. Appl. Math. 64, 297–318 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Barber, J.R.: Contact problems for the thin elastic layer. Int. J. Mech. Sci. 32, 129–132 (1990)zbMATHCrossRefGoogle Scholar
  11. 11.
    Barry, S.I., Holmes, M.: Asymptotic behaviour of thin poroelastic layers. IMA J. Appl. Math. 66, 175–194 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bei, Y., Fregly, B.J.: Multibody dynamic simulation of knee contact mechanics. Med. Eng. Phys. 26, 777–789 (2004)CrossRefGoogle Scholar
  13. 13.
    Chadwick, R.S.: Axisymmetric indentation of a thin incompressible elastic layer. SIAM J. Appl. Math. 62, 1520–1530 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, New York (1980)zbMATHGoogle Scholar
  15. 15.
    Hsiao, G.C., Steinbach, O., Wendland, W.L.: Domain decomposition methods via boundary integral equations. J. Comput. Appl. Math. 125, 521–537 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Itskov, M., Aksel, N.: Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. Acta Mech. 157, 81–96 (2002)zbMATHCrossRefGoogle Scholar
  17. 17.
    Jaffar, M.J.: Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation. Int. J. Mech. Sci. 31, 229–235 (1989)CrossRefGoogle Scholar
  18. 18.
    Nazarov, S.A.: Perturbations of solutions of the Signorini problem for a second-order scalar equation. Math. Notes 47, 115–126 (1990)zbMATHCrossRefGoogle Scholar
  19. 19.
    Siu, D., Rudan, J., Wevers, H.W., Griffiths, P.: Femoral articular shape and geometry: A three-dimensional computerized analysis of the knee. J. Arthroplasty 11, 166–173 (1996)CrossRefGoogle Scholar
  20. 20.
    Van Dyke, M.D.: Perturbation Methods in Fluid Mechanics. Academic Press, New York (1964)zbMATHGoogle Scholar
  21. 21.
    Vorovich, I.I., Penin, O.M.: Contact problem for an infinite strip of variable height [in Russian]. Solid Mech. 5, 112–121 (1971)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsAberystwyth UniversityAberystwythUK

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