Evaluating computational performances of hyperelastic models on supraspinatus tendon uniaxial tensile test data

Document Type : Research Paper


Biomechanics Research Group, Department of Mechanical and Industrial Engineering, School of Engineering, College of Science, Engineering and Technology, University of South Africa, Private Bag X6, Florida, 1710, Johannesburg, South Africa.


Accurate modelling of the mechanical behaviour of tendon tissues is vital due to their essential role in the facilitation of joint mobility in humans and animals. This study focuses on the modelling of the supraspinatus tendon which helps to maintain dynamic stability at the glenohumeral joint in humans. It is observed that in sporting activities or careers that involve frequent arm abduction, injuries to this tendon are a common cause of discomfort. Therefore, this paper evaluates the relative modelling capabilities of three hyperelastic models, namely the Yeoh, Ogden and Martins material models on the tensile behaviour of three tendon specimens. We compare their fitting accuracies, convergence rates during optimisation, and the different forms of sensitivities to data-related features and initial parameter estimates. We find that the Martins model outperforms the other models in fitting accuracies; the Yeoh model has the most stable performance across all initial parameter estimates (with correlations above 99 %) and has the fastest convergence rates (above 20 and 8 times as fast as the Ogden and Martins models’ rates, respectively); and that the Ogden model does not depend on differences in the topological features of the test data. The material parameters of relevant constitutive model may be used for further development of computational models.


[1]     J.H‑C. Wang, Mechanobiology of tendon, Journal of Biomechanics, Vol. 39, No. 9, pp.1563-1582, 2006.
[2]     A. Viidik, 1987, Biomechanics of tendons and other soft connective tissues. Testing methods and structure-function interdependence, in Biomechanics: Basic and Applied Research edited by G. Bergmann, R. Köbel, A. Rohlmann, Kluwer Academic Publishers, Dordrecht.
[3]     J.G. Snedeker, J. Foolen, Tendon injury and repair – A perspective on the basic mechanisms of tendon disease and future clinical therapy, Acta Biomaterialia, Vol. 63, pp. 18-36, 2017.
[4]     Y.C. Fung, 1993, Biomechanics: Mechanical properties of living tissues, Springer, New York.
[5]     F.J. Masithulela, 2016, Computational biomechanics in the remodelling rat heart post myocardial infarction, PhD Thesis, University of Cape Town.
[6]     F. Masithulela, Bi-ventricular finite element model of right ventricle overload in the healthy rat heart, Bio-medical Materials Engineering, Vol. 27, No. 5, pp. 507-525, 2006.
[7]     F. Nemavhola, Detailed structural assessment of healthy interventricular septum in the presence of remodelling infarct in the free wall – A finite element model, Heliyon, Vol. 5, No. 6, e01841, 2019.
[8]     F. Nemavhola, Fibrotic infarction on the LV free wall may alter the mechanics of healthy septal wall during passive filling, Bio-medical Materials Engineering, Vol. 28, No. 6, pp. 579-599, 2017.
[9]     Z. Ndlovu, F. Nemavhola, D. Desai, Biaxial mechanical characterization and constitutive modelling of sheep sclera soft tissue, Russian Journal of Biomechanics, Vol. 24, No. 1, pp. 84‑96 , 2020.
[10]   F. Nemavhola, Biaxial quantification of passive porcine myocardium elastic properties by region, Engineering Solid Mechanics, Vol. 5, No. 3, 155-166, 2017.
[11]   F. Masithulela, The effect of over-loaded right ventricle during passive filling in rat filling heart: A biventricular finite element model, ASME International Mechanical Engineering Congress and Exposition, Vol. 3, 57380, V003T03A005, 2015.
[12]   F. Masithulela, Analysis of passive filling with fibrotic myocardium infarction, ASME International Mechanical Engineering Congress and Exposition, Vol. 3, 57380, V003T03A004, 2015.
[13]   A.H. Lee, S.E. Szczesny, M.H. Santare, D.M. Elliott, Investigating mechanisms of tendon damage by measuring multi-scale recovery following tensile loading, Acta Biomaterialia, Vol. 57, pp. 363‑372, 2017.
[14]   A.R. Akintunde, K.S. Miller, Evaluation of microstructurally motivated constitutive models to describe age-dependent tendon healing, Biomechanics and Modeling in Mechanobiology, Vol. 17, pp. 793-814, 2018.
[15]   J.L. Cook, E. Rio, C.R. Purdam, S.I. Docking, Revisiting the continuum model of tendon pathology: what is its merit in clinical practice and research? British Journal Sports Medicine, Vol. 50, No. 19, pp. 1187-1191, 2016.
[16]   J.S. Lewis, Rotator cuff tendinopathy: a model for the continuum of pathology and related management, British Journal Sports Medicine, Vol44, No. 13, pp. 918-923, 2010.
[17]   B.R. Freedman, J.A. Gordon, L.J. Soslowsky, The Achilles tendon: fundamanetal properties and mechanisms governing healing, Muscles Ligaments Tendons Journal, Vol. 4, No. 2, pp. 245-255, 2014.
[18]   N.L. Leong, J.L. Kator, T.L. Clemens, A. James, M. Enamoto‑Iwamoto, J. Jiang, Tendon and ligament healing and current approaches to tendon and ligament regeneration, Journal of Orthopaedic Research, Vol. 38, No. 1, pp. 7-12, 2020.
[19]   S.E. Szczesny, D.M. Elliot, Incorporating plasticity of the interfibrillar matrix in shear lag models is necessary to replicate the multiscale mechanics of tendon fascicles, Journal of the Mechanical Behaviour of Biomedical Materials, Vol. 40, pp. 325-338, 2014.
[20]   B.N. Safa, A.H. Lee, M.H. Santare, D.M. Elliott, Evaluating plastic deformation and damage as potential mechanisms for tendon inelasticity using a reactive modeling framework, Journal of Biomechanical Engineering, Vol. 141, No. 10, 1010081-10100810, 2019.
[21]   P.A.L.S. Martins, R.M. Natal Jorge, A.J.M. Ferreira, A comparative study of several material models for prediction of hyperelastic properties: application to silicone‐rubber and soft tissues, Strain, Vol. 42, No. 3, 135-147, 2006.
[22]   R.S. Rivlin, Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Philosophical Transactions of the Royal Society of London A, Vol. 241, No. 835, pp. 379-397, 1948.
[23]   O.H. Yeoh, Some forms of the strain energy function for rubber, Rubber Chemistry and Technology, Vol. 66, No. 5, pp. 754-771, 1993.
[24]   R.W. Ogden, 1984, Non-linear elastic deformations, Dover Publications, New York.
[25]   Mathworks® Inc. MATLAB, Documentation: Least-squares (model fitting) algorithms. Accessed from https://www.mathworks.com/help/optim/ug/least-squares-model-fitting-algorithms.html on 30/12/2019.
[26]   T. Wren, S. Yerby, G.S. Beaupré, D.R. Carter, Mechanical properties of the human Achilles tendon, Clinical Biomechanics, Vol. 16, No. 3, pp. 245-251, 2001.
[27]   G.A. Holzapfel, T.C. Gasser, R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of Elasticity and the Physical Science of Solids, Vol. 61, pp. 1-48, 2000.
[28]   R.W. Ogden, G. Saccomandi, I. Sgura, Fitting hyperelastic models to experimental data, Computational Mechanics, Vol. 34, pp. 484-502, 2004.
[29]      M. Mooney, A theory of large elastic deformation, Journal of Applied Physics, Vol. 11, No. 9, pp. 582-592, 1940.
Volume 52, Issue 1
March 2021
Pages 27-43
  • Receive Date: 22 September 2020
  • Revise Date: 05 November 2020
  • Accept Date: 02 December 2020