Advertisement

Biomechanics and Modeling in Mechanobiology

, Volume 17, Issue 6, pp 1875–1883 | Cite as

An elasto-viscoplastic model to describe the ratcheting behavior of articular cartilage

  • Yilin Zhu
Original Paper
  • 95 Downloads

Abstract

In the present work, a constitutive model for articular cartilage is proposed in finite elasto-viscoplasticity. For simplification, articular cartilage is supposed to be a typical composite composed of a soft basis and a fiber assembly. The stress tensor and free energy function are hence accordingly divided into two components. The high nonlinear stress-strain response is assumed to be mainly related to the fiber assembly and described by an exponential-type hypoelastic relation. Ratcheting is considered according to the viscoplasticity, the evolution rule of which is deduced from the dissipative inequality by the co-directionality hypotheses. Then, the capability of the proposed model is validated by comparing its predictions with related experimental observations. Results show that the ratcheting behavior and stress-strain hysteresis loops are reasonably captured by the proposed model.

Keywords

Articular cartilage Constitutive model Logarithmic stress rate Ratcheting 

Notes

Funding

This study was funded by the National Natural Science Foundation of China (11702036) and Chengdu University New Faculty Start-up Funding (2081915038).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Ahmadzadeh G, Varvanifarahani A (2015) Ratcheting prediction of Al 6061/SiCP composite samples under asymmetric stress cycles by means of the Ahmadzadeh-Varvani hardening rule. J Compos Mater 50(17):2389–2397CrossRefGoogle Scholar
  2. Armstrong CG, Lai WM, Mow VC (1984) An analysis of the unconfined compression of articular cartilage. J Biomech Eng 106:165–173CrossRefGoogle Scholar
  3. Ateshian GA (2017) Mixture theory for modeling biological tissues: illustrations from articular cartilage. Springer, LondonGoogle Scholar
  4. Athanasiou K, Darling E, Hu J (2009) Articular cartilage tissue engineering. Synth Lect Tissue Eng 1(1):1–182Google Scholar
  5. Barker MK, Seedhom BB (2001) The relationship of the compressive modulus of articular cartilage with its deformation response to cyclic loading: does cartilage optimize its modulus so as to minimize the strains arising in it due to the prevalent loading regime? Rheumatology 40:274–284CrossRefGoogle Scholar
  6. Bellucci G, Seedhom BB (2001) Mechanical behaviour of articular cartilage under tensile cyclic load. Rheumatology 40:1337–1345CrossRefGoogle Scholar
  7. Bruhns OT, Xiao H, Meyers A (1999) Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate. Int J Plast 15:479–520CrossRefGoogle Scholar
  8. Bursać PM, Obitz TW, Eisenberg SR, Stamenović D (1999) Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis. J Biomech 32:1125–1130CrossRefGoogle Scholar
  9. Chen X, Hui S (2005) Ratcheting behavior of PTFE under cyclic compression. Polym Test 24:829–833CrossRefGoogle Scholar
  10. Gao LL, Zhang CQ, Yang YB, Shi JP, Jia YW (2013) Depth-dependent strain fields of articular cartilage under rolling load by the optimized digital image correlation technique. Mater Sci Eng, C 33:2317–2322CrossRefGoogle Scholar
  11. Gao LL, Qin XY, Zhang CQ, Gao H, Ge HY, Zhang XZ (2015) Ratcheting behavior of articular cartilage under cyclic unconfined compression. Mater Sci Eng C Mater Biol Appl 57:371–377CrossRefGoogle Scholar
  12. García JJ, Cortés DH (2006) A nonlinear biphasic viscohyperelastic model for articular cartilage. J Biomech 39:2991CrossRefGoogle Scholar
  13. Guo S, Kang G, Zhang J (2013) A cyclic visco-plastic constitutive model for time-dependent ratchetting of particle-reinforced metal matrix composites. Int J Plast 40:101–125CrossRefGoogle Scholar
  14. Huang CY, Mow VC, Ateshian GA (2001) The role of flow-independent viscoelasticity in the biphasic tensile and compressive responses of articular cartilage. J Biomech Eng 123:410–417CrossRefGoogle Scholar
  15. Kang G (2008) Ratchetting: recent progresses in phenomenon observation, constitutive modeling and application. Int J Fatigue 30:1448–1472CrossRefGoogle Scholar
  16. Kang G, Wu X (2011) Ratchetting of porcine skin under uniaxial cyclic loading. J Mech Behav Biomed Mater 4:498–506CrossRefGoogle Scholar
  17. Kerin AJ, Coleman A, Wisnom MR, Adams MA (2003) Propagation of surface fissures in articular cartilage in response to cyclic loading in vitro. Clin Biomech 18:960CrossRefGoogle Scholar
  18. Kurz B, Lemke AK, Fay J, Pufe T, Grodzinsky AJ, Schünke M (2005) Pathomechanisms of cartilage destruction by mechanical injury. Ann Anat 187:473–485CrossRefGoogle Scholar
  19. Kwan MK, Lai WM, Mow VC (1990) A finite deformation theory for cartilage and other soft hydrated connective tissues–I. Equilibrium results. J Biomech 23:145–155CrossRefGoogle Scholar
  20. Li LP, Herzog W, Korhonen RK, Jurvelin JS (2005) The role of viscoelasticity of collagen fibers in articular cartilage: axial tension versus compression. Med Eng Phys 27:51–57CrossRefGoogle Scholar
  21. Lu F, Kang G, Zhu Y, Xi C, Jiang H (2016) Experimental observation on multiaxial ratchetting of polycarbonate polymer at room temperature. Polym Test 50:135–144CrossRefGoogle Scholar
  22. Matzat SJ, Van TJ, Gold GE, Oei EH (2013) Quantitative MRI techniques of cartilage composition. Quant Imaging Med Surg 3:162–174Google Scholar
  23. Mow VC, Kuei SC, Lai WM, Armstrong CG (1980) Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J Biomech Eng 102:73CrossRefGoogle Scholar
  24. Pierce DM, Ricken T, Holzapfel GA (2013) A hyperelastic biphasic fibre-reinforced model of articular cartilage considering distributed collagen fibre orientations: continuum basis, computational aspects and applications. Comput Methods Biomech Biomed Eng 16:1344–1361CrossRefGoogle Scholar
  25. Responte DJ, Natoli RM, Athanasiou KA (2007) Collagens of articular cartilage: structure, function, and importance in tissue engineering. Crit Rev Biomed Eng 35:363–411CrossRefGoogle Scholar
  26. Seifzadeh A, Oguamanam DC, Trutiak N, Hurtig M, Papini M (2012) Determination of nonlinear fibre-reinforced biphasic poroviscoelastic constitutive parameters of articular cartilage using stress relaxation indentation testing and an optimizing finite element analysis. Comput Methods Programs Biomed 107:315–326CrossRefGoogle Scholar
  27. Soltz MA, Ateshian GA (2000a) A conewise linear elasticity mixture model for the analysis of tension-compression nonlinearity in articular cartilage. J Biomech Eng 122:576CrossRefGoogle Scholar
  28. Soltz MA, Ateshian GA (2000b) Interstitial fluid pressurization during confined compression cyclical loading of articular cartilage. Ann Biomed Eng 28:150–159CrossRefGoogle Scholar
  29. Sophia Fox AJ, Bedi A, Rodeo SA (2009) The basic science of articular cartilage: structure, composition, and function Sports. Health 1:461–468Google Scholar
  30. Wilson W, van Donkelaar CC, Van RB, Huiskes R (2005) A fibril-reinforced poroviscoelastic swelling model for articular cartilage. R.G. Landes CoGoogle Scholar
  31. Xiao H, Bruhns DIOT, Meyers DIA (1997a) Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech 124:89–105MathSciNetCrossRefGoogle Scholar
  32. Xiao H, Bruhns OT, Meyers A (1997b) Hypo-elasticity model based upon the logarithmic stress rate. J Elast 47:51–68MathSciNetCrossRefGoogle Scholar
  33. Zhu Y, Kang G, Kan Q, Bruhns OT (2014a) Logarithmic stress rate based constitutive model for cyclic loading in finite plasticity. Int J Plast 54:34–55CrossRefGoogle Scholar
  34. Zhu Y, Kang G, Kan Q, Yu C (2014b) A finite viscoelastic-plastic model for describing the uniaxial ratchetting of soft biological tissues. J Biomech 47:996CrossRefGoogle Scholar
  35. Zhu Y, Kang G, Kan Q, Bruhns OT, Liu Y (2016) Thermo-mechanically coupled cyclic elasto-viscoplastic constitutive model of metals: theory and application. Int J Plast 79:111–152CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Architectural and Civil EngineeringChengdu UniversityChengduPeople’s Republic of China
  2. 2.School of Electromechanical Automobile EngineeringYantai UniversityYantaiPeople’s Republic of China

Personalised recommendations