Biomechanics and Modeling in Mechanobiology

, Volume 6, Issue 3, pp 189–197 | Cite as

A constitutive law for the failure behavior of medial collateral ligaments

  • Raffaella De VitaEmail author
  • William S. Slaughter
Original Paper


A constitutive model is proposed for the description of the tensile properties of medial collateral ligaments (MCLs). The model can reproduce the three regions – the toe region, the linear region, and the failure region – of the stress–stretch curve of ligamentous tissues. The collagen fibers are assumed to be the only load-bearing component of the tissues. They are all oriented along the physiological loading direction of the ligament. They are crimped in the slack configuration and are unable to sustain load. After becoming taut and before failing, each collagen fiber exhibits a linear elastic behavior. The fiber straightening and failure processes are defined stochastically by means of Weibull distributions. Published experimental data for the MCLs are employed to validate the constitutive relationship. Finally, the constitutive model is generalized in order to describe the three-dimensional mechanical behavior of the ligaments by following he structural approach.


Constitutive Model Medial Collateral Ligament Failure Behavior Failure Region Green Deformation Tensor 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitch SD, Papageorgiou CD, Debski RE, Clineff TD, Woo SLY (2003) A biomechanical and histological evaluation of the structure and function of the healing medial collateral ligament in a goat model. Knee Surg Sport Tr A 11:155–162Google Scholar
  2. Amiel D, Billings E Jr, Akenson WH (1990) Ligament structure, chemistry, and physiology. In: Daniel DM, Akeson WH, O’Connor JJ (eds) Knee ligaments: structure, function, injury and repair. Raven Press, New YorkGoogle Scholar
  3. Bollen S (2000) Epidemiology of knee injuries: diagnosis and triage. Br J Sport Med 34:227–228CrossRefGoogle Scholar
  4. Comninou M, Yannas IV (1976) Dependence of stress–strain nonlinearity of connective tissues on the geometry of collagen fibers. J Biomech 9:427–433CrossRefGoogle Scholar
  5. Decraemer WF, Maes MA, Vanhuyse VJ, Vanpeperstraete P (1980) A non-linear viscoelastic constitutive equation for soft biological tissues based upon a structural model. J Biomech 13:559–564CrossRefGoogle Scholar
  6. De Vita R, Slaughter WS (2006) A structural constitutive model for the strain rate-dependent behavior of anterior cruciate ligaments. Int J Solids Struct 43:1561–1570CrossRefzbMATHGoogle Scholar
  7. Diamant J, Keller A, Baer E, Litt M, Arridge RGC (1972) Collagen; ultrastructure and its relation to mechanical properties as a function of ageing. Proc Roy Soc Lond B Biol 180B:293–315CrossRefGoogle Scholar
  8. Findley WN, Lai JS, Onaran K (1976) Creep and relaxation of nonlinear viscoelastic material. Dover, New YorkGoogle Scholar
  9. Fung YC (1993) Biomechanics: mechanical properties of living tissues. Springer, Berlin Heidelberg New YorkGoogle Scholar
  10. Humphrey JD, Yin FC (1997) A new constitutive formulation for characterizing the mechanical behavior of soft tissues. Biophys J 52:563–570CrossRefGoogle Scholar
  11. Hurschler C, Loitz-Ramage B, Vanderby R (1997) A structurally based stress–stretch relationship for tendon and ligament. J Biomech Eng—T ASME 119:392–399Google Scholar
  12. Kastelic J, Galeski A, Baer E (1978) The multicomposite structure of tendon. Connect Tissue Res 6:11–23CrossRefGoogle Scholar
  13. Kastelic J, Palley I, Baer E (1980) A structural mechanical model for tendon crimping. J Biomech 13:887–893CrossRefGoogle Scholar
  14. Kukreti U, Belkoff SM (2000) Collagen fibril D-period may change as a function of strain and location in ligament. J Biomech 33:1569–1574CrossRefGoogle Scholar
  15. Kwan MK, Woo SLY (1989) A structural model to describe the nonlinear stress–strain behavior for parallel-fibered collagenous tissues. J Biomech Eng—T ASME 111:361–363Google Scholar
  16. Lanir Y (1979) A structural theory for the homogeneous biaxial stress–strain relationship in flat collageneous tissues. J Biomech 12:423–436CrossRefGoogle Scholar
  17. Lanir Y (1980) A microstructural model for the rheology of mammalian tendon. J Biomech Eng—T ASME 102:332–339Google Scholar
  18. Lanir Y (1983) Constitutive equations for fibrous connective . J Biomech 16:1–12CrossRefGoogle Scholar
  19. Liao H, Belkoff SM (1999) A failure model for ligaments. J Biomech 32:183–188CrossRefGoogle Scholar
  20. Nelder J, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313zbMATHGoogle Scholar
  21. Park SK, Miller KW (1988) Random number generators: good ones are hard to find. Commun ACM 31:1192–1201CrossRefMathSciNetGoogle Scholar
  22. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Numerical recipes in C: the art of scientific computing. Cambridge University Press, CambridgeGoogle Scholar
  23. Provenzano PP, Lakes R, Keenan T, Vanderby R (2001) Nonlinear ligament viscoelasticity. Ann Biomed Eng 29:908–914CrossRefGoogle Scholar
  24. Provenzano PP, Heisey D, Hayashi K, Lakes R, Vanderby R Jr (2002) Subfailure damage in ligament: a structural and cellular evaluation. J Appl Physiol 92:362–371Google Scholar
  25. Provenzano PP, Lakes RS, Corr DT, Vanderby R Jr (2002) Application of nonlinear viscoelastic models to describe ligament behavior. Biomech Model Mechanobiol 1:45–57CrossRefGoogle Scholar
  26. Sacks SM (2000) A structural constitutive model for chemically treated planar tissues under biaxial loading. Comput Mech 26:243–249zbMATHCrossRefGoogle Scholar
  27. Sasaki N, Odajima S (1996) Stress–strain curve and Young’s modulus of a collagen molecule as determined by x-ray diffraction technique. J Biomech 29:655–658CrossRefGoogle Scholar
  28. Schapery RA (1969) On the characterization of the nonlinear viscoelastic materials. Polym Eng Sci 9:295–310CrossRefGoogle Scholar
  29. Stouffer DC, Butler DL, Hosny D (1985) The relationship between crimp pattern and mechanical response of human patellar tendon–bone units. J Biomech Eng—T ASME 107:158–165CrossRefGoogle Scholar
  30. Thornton GM, Frank CB, Shrive NG (2000) Ligament creep behavior can be predicted from stress relaxation by incorporating fiber recruitment. J Rheol 45:493–507CrossRefGoogle Scholar
  31. Truesdell C, Noll W (1965) The non-linear field theories of mechanics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  32. Viidik A (1969) A rheological model for uncalcified parallel-fibered collagenous tissue. J Biomech 1:3–11CrossRefGoogle Scholar
  33. Wren TAL, Carter DR (1998) A microstructural model for the tensile constitutive and failure behavior of soft skeletal connective tissues. J Biomech Eng—T ASME 120:55–61Google Scholar
  34. Zioupos P, Barbenel JC (1994) Mechanics of native bovine pericardium II. A structural based model for anisotropic mechanical behaviour of the tissue. Biomaterials 15:374–382Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Mechanical EngineeringUniversity of PittsburghPittsburghUSA

Personalised recommendations