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

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## 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

- 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
- Armstrong CG, Lai WM, Mow VC (1984) An analysis of the unconfined compression of articular cartilage. J Biomech Eng 106:165–173CrossRefGoogle Scholar
- Ateshian GA (2017) Mixture theory for modeling biological tissues: illustrations from articular cartilage. Springer, LondonGoogle Scholar
- Athanasiou K, Darling E, Hu J (2009) Articular cartilage tissue engineering. Synth Lect Tissue Eng 1(1):1–182Google Scholar
- 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
- Bellucci G, Seedhom BB (2001) Mechanical behaviour of articular cartilage under tensile cyclic load. Rheumatology 40:1337–1345CrossRefGoogle Scholar
- 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
- 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
- Chen X, Hui S (2005) Ratcheting behavior of PTFE under cyclic compression. Polym Test 24:829–833CrossRefGoogle Scholar
- 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
- 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
- García JJ, Cortés DH (2006) A nonlinear biphasic viscohyperelastic model for articular cartilage. J Biomech 39:2991CrossRefGoogle Scholar
- 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
- 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
- Kang G (2008) Ratchetting: recent progresses in phenomenon observation, constitutive modeling and application. Int J Fatigue 30:1448–1472CrossRefGoogle Scholar
- Kang G, Wu X (2011) Ratchetting of porcine skin under uniaxial cyclic loading. J Mech Behav Biomed Mater 4:498–506CrossRefGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- Matzat SJ, Van TJ, Gold GE, Oei EH (2013) Quantitative MRI techniques of cartilage composition. Quant Imaging Med Surg 3:162–174Google Scholar
- 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
- 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
- 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
- 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
- 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
- Soltz MA, Ateshian GA (2000b) Interstitial fluid pressurization during confined compression cyclical loading of articular cartilage. Ann Biomed Eng 28:150–159CrossRefGoogle Scholar
- Sophia Fox AJ, Bedi A, Rodeo SA (2009) The basic science of articular cartilage: structure, composition, and function Sports. Health 1:461–468Google Scholar
- Wilson W, van Donkelaar CC, Van RB, Huiskes R (2005) A fibril-reinforced poroviscoelastic swelling model for articular cartilage. R.G. Landes CoGoogle Scholar
- Xiao H, Bruhns DIOT, Meyers DIA (1997a) Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech 124:89–105MathSciNetCrossRefGoogle Scholar
- Xiao H, Bruhns OT, Meyers A (1997b) Hypo-elasticity model based upon the logarithmic stress rate. J Elast 47:51–68MathSciNetCrossRefGoogle Scholar
- 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
- 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
- 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