Advertisement

Articular Contact Mechanics

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

Abstract

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 Hunt–Crossley model for articular contact is developed in the framework of the short-time contact model for thin bonded biphasic layers.

Keywords

Articular Cartilage Articular Surface Contact Model Shear Elastic Modulus Elliptic Paraboloid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abdel–Rahman, E.M., Hefzy, M.S.: A two-dimensional dynamic anatomical model of the human knee joint. J. Biomech. Eng. 115, 357–365 (1993)CrossRefGoogle Scholar
  2. 2.
    Abdel–Rahman, E.M., Hefzy, M.S.: Three-dimensional dynamic behaviour of the human knee joint under impact loading. Med. Eng. Phys. 20, 276–290 (1998)CrossRefGoogle Scholar
  3. 3.
    Aleksandrov, V.M., Vorovich, I.I.: Contact problems for the elastic layer of small thickness. J. Appl. Math. Mech. 28, 425–427 (1964)MathSciNetCrossRefGoogle Scholar
  4. 4.
    ap Gwynn, I., Wade, S., Ito, K., Richards, R.G.: Novel aspects to the structure of rabbit articular cartilage. Eur. Cell. Mater. 4, 18–29 (2002)Google Scholar
  5. 5.
    ap Gwynn, I., Wade, S., Kaab, M.J., Owen, G.R., Richards, R.G.: Freeze-substitution of rabbit tibial articular cartilage reveals that radial zone collagen fibres are tubules. J. Microsc. 197, 159–172 (2000)Google Scholar
  6. 6.
    Argatov, I.I.: The pressure of a punch in the form of an elliptic paraboloid on a thin elastic layer. Acta Mech. 180, 221–232 (2005)zbMATHCrossRefGoogle Scholar
  7. 7.
    Argatov, I.I.: Asymptotic modeling of the impact of a spherical indenter on an elastic half-space. Int. J. Solids Struct. 45, 5035–5048 (2008)zbMATHCrossRefGoogle Scholar
  8. 8.
    Argatov, I.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
  9. 9.
    Argatov, I.I.: Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage. Trib. Int. 63, 213–225 (2013)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Argatov, I., Daniels, A.U., Mishuris, G., Ronken, S., Wirz, D.: Accounting for the thickness effect in dynamic spherical indentation of a viscoelastic layer: Application to non-destructive testing of articular cartilage. Eur. J. Mech. A/Solids 37, 304–317 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Argatov, I., Mishuris, G.: Contact problem for thin biphasic cartilage layers: Perturbation solution. Quart. J. Mech. Appl. Math. 64, 297–318 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: exact solution for monotonic loading. J. Biomech. 44, 759–761 (2011)CrossRefGoogle Scholar
  14. 14.
    Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Armstrong, C.G., Mow, V.C.: Variations in the intrinsic mechanical properties of human articular cartilage with age, degeneration, and water content. J Bone Joint Surg. Am. 64, 88–94 (1982)Google Scholar
  16. 16.
    Ateshian, G.A.: A B-spline least-squares surface fitting method for articular surfaces of diarthrodial joints. J. Biomech. Eng. 115, 366–373 (1993)CrossRefGoogle Scholar
  17. 17.
    Ateshian, G.A.: The role of interstitial fluid pressurization in articular cartilage lubrication. J. Biomech. 42, 1163–1176 (2009)CrossRefGoogle Scholar
  18. 18.
    Ateshian, G.A., Ellis, B.J., Weiss, J.A.: Equivalence between short-time biphasic and incompressible elastic material responses. J. Biomech. Eng. 129, 405–412 (2007)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Ateshian, G.A., Soltz, M.A., Mauck, R.L., Basalo, I.M., Hung, C.T., Lai, W.M.: The role of osmotic pressure and tension-compression nonlinearity in the frictional response of articular cartilage. Transp. Porous Med. 50, 5–33 (2003)CrossRefGoogle Scholar
  21. 21.
    Barber, J.R.: Contact problems for the thin elastic layer. Int. J. Mech. Sci. 32, 129–132 (1990)zbMATHCrossRefGoogle Scholar
  22. 22.
    Bei, Y., Fregly, B.J.: Multibody dynamic simulation of knee contact mechanics. Med. Eng. Phys. 26, 777–789 (2004)CrossRefGoogle Scholar
  23. 23.
    Bell, J.S., Winlove, C.P., Smith, C.W., Dehghani, H.: Modeling the steady-state deformation of the solid phase of articular cartilage. Biomater 30, 6394–6401 (2009)CrossRefGoogle Scholar
  24. 24.
    Blankevoort, L., Kuiper, J.H., Huiskes, R., Grootenboer, H.J.: Articular contact in a three-dimensional model of the knee. J. Biomech. 24, 1019–1031 (1991)CrossRefGoogle Scholar
  25. 25.
    Bursać, P., McGrath, C.V., Eisenberg, S.R., Stamenović, D.: A microstructural model of elastostatic properties of articular cartilage in confined compression. J. Biomech. Eng. 122, 347–353 (2000)CrossRefGoogle Scholar
  26. 26.
    Carter, D.R., Wong, M.: Modelling cartilage mechanobiology. Phil. Trans. R. Soc. Lond. B 358, 1461–1471 (2003)CrossRefGoogle Scholar
  27. 27.
    Caruntu, D.I., Hefzy, M.S.: 3-D anatomically based dynamic modeling of the human knee to include tibio-femoral and patello-femoral joints. J. Biomech. Eng. 126, 44–53 (2004)CrossRefGoogle Scholar
  28. 28.
    Chadwick, R.S.: Axisymmetric indentation of a thin incompressible elastic layer. SIAM J. Appl. Math. 62, 1520–1530 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Dhaher, Y.Y., Delp, S.L., Rymer, W.Z.: The use of basis functions in modelling joint articular surfaces:application to the knee joint. J. Biomech. 33, 901–907 (2000)CrossRefGoogle Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Dyagel, R.V., Lapshin, V.V.: On a nonlinear viscoelastic model of Hunt–Crossley impact. Mech. Solids 46, 798–806 (2011)CrossRefGoogle Scholar
  33. 33.
    Eberhardt, A.W., Keer, L.M., Lewis, J.L., Vithoontien, V.: An analytical model of joint contact. J. Biomech. Eng. 112, 407–413 (1990)CrossRefGoogle Scholar
  34. 34.
    Eberhard, P., Spägele, T., Gollhofer, A.: Investigations for the dynamical analysis of human motion. Multibody Syst. Dyn. 3, 1–20 (1999)zbMATHCrossRefGoogle Scholar
  35. 35.
    Fox, A.J.S., Bedi, A., Rodeo, S.A.: The basic science of articular cartilage: structure, composition, and function. Sports Health 1, 461–469 (2009)CrossRefGoogle Scholar
  36. 36.
    Garcia, J.J., Altiero, N.J., Haut, R.C.: An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load. J. Biomech. Eng. 120, 608–613 (1998)CrossRefGoogle Scholar
  37. 37.
    Gilardi, G., Sharf, I.: Literature survey of contact dynamics modelling. Mech. Mach. Theory 37, 1213–1239 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Glocker, Ch.: Formulation of spatial contact situations in rigid multibody systems. Comput. Method. Appl. M. 177, 199–214 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Goldsmith, W.: Impact: The Theory and Physical Behaviour of Colliding Solids. Edward Arnold Ltd, London (1960)Google Scholar
  40. 40.
    Gonthier, Y., McPhee, J., Lange, C., Piedbœuf, J.-C.: A regularized contact model with asymmetric damping and Dwell-time dependent friction multibody. Syst. Dyn. 11, 209–233 (2004)zbMATHGoogle Scholar
  41. 41.
    Guess, T.M.: Forward dynamics simulation using a natural knee with menisci in the multibody framework. Multibody Syst. Dyn. 28, 37–53 (2012)CrossRefGoogle Scholar
  42. 42.
    Guess, T.M., Thiagarajan, G., Kia, M., Mishra, M.: A subject specific multibody model of the knee with menisci. Med. Eng. Phys. 32, 505–515 (2010)CrossRefGoogle Scholar
  43. 43.
    Hanley, K., Collins, F., Cronin, K., Byrne, E., Moran, K., Brabazon, D.: Simulation of the impact response of a sliotar core with linear and non-linear contact models. Int. J. Impact Eng. 50, 113–122 (2012)CrossRefGoogle Scholar
  44. 44.
    Hirokawa, S., Ueki, T., Ohtsuki, A.: A new approach for surface fitting method of articular joint surfaces. J. Biomech. 37, 1551–1559 (2004)CrossRefGoogle Scholar
  45. 45.
    Herbert, R.G., McWhannell, D.C.: Shape and frequency composition of pulses from an impact pair. J. Eng. Ind. 99, 513–518 (1977)CrossRefGoogle Scholar
  46. 46.
    Herzog, W., Federico, S.: Considerations on joint and articular cartilage mechanics. Biomech. Model. Mechanobiol. 5, 64–81 (2006)CrossRefGoogle Scholar
  47. 47.
    Hohe, J., Ateshian, G., Reiser, M., Englmeier, K.-H., Eckstein, F.: Surface size, curvature analysis, and assessment of knee joint incongruity with MRI in vivo. Magn. Reson. Med. 47, 554–561 (2002)CrossRefGoogle Scholar
  48. 48.
    Horvay, G., Veluswami, M.A.: Hertzian impact of two elastic spheres in the presence of surface damping. Acta Mech. 35, 285–290 (1980)zbMATHCrossRefGoogle Scholar
  49. 49.
    Hu, K., Radhakrishnan, P., Patel, R.V., Mao, J.J.: Regional structural and viscoelastic properties of fibrocartilage upon dynamic nanoindentation of the articular condyle. J. Struct. Biol. 136, 46–52 (2001)CrossRefGoogle Scholar
  50. 50.
    Huiskes, R., Van Dijk, R., de Lange, A., Woltring, H.J., Van Rens, Th.J.G.: Kinematics of the human knee joint. In: Berme, N., Engin, A.E., Correia da Silva, K.M. (eds.) Biomechanics of Normal and Pathological Human Articulating Joints, pp. 165–187. Martinus Nijhoff Publ., Dordrecht (1985)Google Scholar
  51. 51.
    Hughes, L.C., Archer, C.W., ap Gwynn, I.: The ultrastructure of mouse articular cartilage: collagen orientation and implications for tissue functionality. A polarised light and scanning electron microscope study and review. Eur. Cell. Mater. 9, 68–84 (2005)Google Scholar
  52. 52.
    Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. ASME J. Appl. Mech. 42, 440–445 (1975)CrossRefGoogle Scholar
  53. 53.
    Iatridis, J.C., ap Gwynn, I.: Mechanisms for mechanical damage in the intervertebral disc annulus brosus. J. Biomech. 37, 1165–1175 (2004)Google Scholar
  54. 54.
    Johnson, K.L.: Contact Mechanics. Cambridge Univ. Press, Cambridge, UK (1985)Google Scholar
  55. 55.
    Jurvelin, J.S.: Biomechanical properties of the knee articular cartilage under various loading conditions. Ph.D. thesis, University of Kuopio, Kuopio, Finland (1991)Google Scholar
  56. 56.
    Karvonen, R.L., Negendank, W.G., Teitge, R.A., Reed, A.H., Miller, P.R., Fernandez-Madrid, F.: Factors affecting articular cartilage thickness in osteoarthritis and aging. J. Rheumatol. 21, 1310–1318 (1994)Google Scholar
  57. 57.
    Khan, I.M., Gilbert, S.J., Singhrao, S.K., Duance, V.C., Archer, C.W.: Cartilage integration: evaluation of the reasons for failure of integration during cartilage repair. Rev. Eur. Cell. Mater. 16, 26–39 (2008)Google Scholar
  58. 58.
    Kim, Y.J., Bonassar, L.J., Grodzinsky, A.J.: The role of cartilage streaming potential, fluid flow and pressure in the stimulation of chondrocyte biosynthesis during dynamic compression. J. Biomech. 28, 1055–1066 (1995)CrossRefGoogle Scholar
  59. 59.
    Kłodowski, A., Rantalainen, T., Mikkola, A., Heinonen, A., Sievänen, H.: Flexible multibody approach in forward dynamic simulation of locomotive strains in human skeleton with flexible lower body bones. Multibody Syst. Dyn. 25, 395–409 (2011)zbMATHCrossRefGoogle Scholar
  60. 60.
    Koo, S., Andriacchi, T.P.: A comparison of the influence of global functional loads vs. local contact anatomy on articular cartilage thickness at the knee. J. Biomech. 40, 2961–2966 (2007)CrossRefGoogle Scholar
  61. 61.
    Kren, A.P., Naumov, A.O.: Determination of the relaxation function for viscoelastic materials at low velocity impact. Int. J. Impact Eng. 37, 170–176 (2010)CrossRefGoogle Scholar
  62. 62.
    Kücük, H.: The effect of modeling cartilage on predicted ligament and contact forces at the knee. Comput. Biol. Med. 36, 363–375 (2006)CrossRefGoogle Scholar
  63. 63.
    Landinez-Parra, N.S., Garzón-Alvarado, D.A., Vanegas-Acosta, J.C.: A phenomenological mathematical model of the articular cartilage damage. Comput. Meth. Prog. Bio. 104, e58–e74 (2011)CrossRefGoogle Scholar
  64. 64.
    Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112, 369–376 (1990)CrossRefGoogle Scholar
  65. 65.
    Lankarani, H.M., Nikravesh, P.E.: Continuous contact force models for impact analysis in multibody systems. Nonlinear Dyn. 5, 193–207 (1994)Google Scholar
  66. 66.
    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)CrossRefGoogle Scholar
  67. 67.
    Li, S., Patwardhan, A.G., Amirouche, F.M.L., Havey, R., Meade, K.P.: Limitations of the standard linear solid model of intervertebral discs subject to prolonged loading and low-frequency vibration in axial compression. J. Biomech. 28, 779–790 (1995)CrossRefGoogle Scholar
  68. 68.
    Lin, Y-Ch., Haftka, R.T., Queipo, N.V., Fregly, B.J.: Surrogate articular contact models for computationally efficient multibody dynamic simulations. Med. Eng. Phys. 32, 584–594 (2010)CrossRefGoogle Scholar
  69. 69.
    Ling, Z.-K., Guo, H.-Q., Boersma, S.: Analytical study on the kinematic and dynamic behaviors of a knee joint. Med. Eng. Phys. 19, 29–36 (1997)CrossRefGoogle Scholar
  70. 70.
    Machado, M., Flores, P., Claro, J.C.P., Ambrósio, J., Silva, M., Completo, A., Lankarani, H.M.: Development of a planar multibody model of the human knee joint. Nonlinear Dyn. 60, 459–478 (2010)zbMATHCrossRefGoogle Scholar
  71. 71.
    Machado, M., Moreira, P., Flores, P., Lankarani, H.M.: Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012)CrossRefGoogle Scholar
  72. 72.
    Mansour, J.M.: Biomechanics of cartilage. In: Oatis, C.A. (ed.) Kinesiology: the mechanics and pathomechanics of human movement, 2nd edn, pp. 66–79. Lippincott Williams and Wilkins, Philadelphia, PA (2008)Google Scholar
  73. 73.
    Marhefka, D.W., Orin, D.E.: Simulation of contact using a nonlinear damping model. In: Proceedings of the 1996 IEEE International Conference on Robotics and Automation. pp. 16621–6628. Minneapolis, Minnesota (1996)Google Scholar
  74. 74.
    Monteiro, N.M.B., da Silva, M.P.T., Folgado, J.O.M.G., Melancia, J.P.L.: Structural analysis of the intervertebral discs adjacent to an interbody fusion using multibody dynamics and finite element cosimulation. Multibody Syst. Dyn. 25, 245–270 (2011)CrossRefGoogle Scholar
  75. 75.
    Mow, V.C., Gu, W.Y., Chen, F.H.: Structure and function of articular cartilage and meniscus. In: Mow, V.C., Huiskes, R. (eds.) Basic orthopaedic biomechanics and mechano-biology, 3rd edn, pp. 181–258. Lippincott Williams and Wilkins, Philadelphia, PA (2005)Google Scholar
  76. 76.
    O’Hara, B.P., Urban, J.P.G., Maroudas, A.: Influence of cyclic loading on the nutrition of articular cartilage. Ann. Rheum. Dis. 49, 536–539 (1990)CrossRefGoogle Scholar
  77. 77.
    Pandy, M.G., Sasaki, K., Kim, S.: A three-dimensional musculoskeletal model of the human knee joint. Part 1: Theoretical construction. Comput. Method. Biomec. 1, 87–108 (1997)Google Scholar
  78. 78.
    Peña, E., Calvo, B., Martínez, M.A., Doblaré, M.: A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy humen knee joint. J. Biomech. 39, 1686–1701 (2006)CrossRefGoogle Scholar
  79. 79.
    Pérez-González, A., Fenollosa-Esteve, C., Sancho-Bru, J.L., Sánchez-Marín, F.T., Vergara, M., Rodríguez-Cervantes, P.J.: A modified elastic foundation contact model for application in 3D models of the prosthetic knee. Med. Eng. Phys. 30, 387–398 (2008)CrossRefGoogle Scholar
  80. 80.
    Pipkin, A.C.: Lectures on Viscoelastic Theory. Springer Verlag, New York (1986)Google Scholar
  81. 81.
    Silva, M.P.T., Ambrósio, J.A.C., Pereira, M.S.: Biomechanical model with joint resistance for impact simulation. Multibody Syst. Dyn. 1, 65–84 (1997)zbMATHCrossRefGoogle Scholar
  82. 82.
    Sørensen, S.E., Hansen, M.R., Ebbesen, M.K., Mouritsen, O.Ø.: Implicit identification of contact parameters in a continuous chain model. Model. Ident. Control 32, 1–15 (2011)Google Scholar
  83. 83.
    van Ruijven, L.J., Beek, M., van Eijden, T.M.G.J.: Fitting parametrized polynomials with scattered surface data. J. Biomech. 32, 715–720 (1999)CrossRefGoogle Scholar
  84. 84.
    Wang, J.H-C., Ryu, J., Han, J.-S., Rowen, B.: A new method for the representation of articular surfaces using the influence surface theory of plates. J. Biomech. 33, 629–633 (2000)Google Scholar
  85. 85.
    Wilson, W., van Donkelaar, C.C., van Rietberger, R., Huiskes, R.: The role of computational models in the search for the mechanical behaviour and damage mechanisms of articular cartilage. Med. Eng. Phys. 27, 810–826 (2005)CrossRefGoogle Scholar
  86. 86.
    Wismans, J., Veldpaus, F., Janssen, J., Huson, A., Struben, P.: A three-dimensional mathematical model of the knee-joint. J. Biomech. 13(677–679), 681–685 (1980)Google Scholar
  87. 87.
    Wu, J.Z., Herzog, W., Epstein, M.: An improved solution for the contact of two biphasic cartilage layers. J. Biomech. 30, 371–375 (1997)CrossRefGoogle Scholar
  88. 88.
    Wu, J.Z., Herzog, W., Epstein, M.: Evaluation of the finite element software ABAQUS for biomechanical modelling of biphasic tissues. J. Biomech. 31, 165–169 (1997)CrossRefGoogle Scholar
  89. 89.
    Zhang, Y., Sharf, I.: Validation of nonlinear viscoelastic contact force models for low speed impact. J. Appl. Mech. 76, 051002 (12 pages) (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsAberystwyth UniversityAberystwythUK

Personalised recommendations