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Continuum Mechanics and Thermodynamics

, Volume 31, Issue 3, pp 607–626 | Cite as

A variational homogenization approach applied to the multiscale analysis of the viscoelastic behavior of tendon fascicles

  • Thiago André Carniel
  • Eduardo Alberto FancelloEmail author
Original Article

Abstract

This work presents a variational homogenization approach based on representative volume elements (RVE) in order to investigate the macro- and microviscoelastic behavior of tendon fascicles. A three-dimensional hexagonal–helicoidal finite element RVE is proposed to properly account for the morphology of tendon fascicles observed in serial block-face scanning electron microscopy. Two material phases (collagen fibers and cells) comprising three finite strain variational viscoelastic models (fibrils, matrix of fibers and cells) compose the proposed multiscale model. The material parameters of the micromechanical models were identified with the aid of atomic force microscopy experiments extracted from the literature. A set of multiscale simulations of tensile tests under physiological strain amplitudes were performed, providing the following results. Firstly, numerical predictions corroborate experimental findings: collagen fibrils are the main load-bearing structures of tendons; the cellular matrix contributes neither to the stiffness nor to the energy dissipation of tendons. Secondly, the model brings insights about microscale mechanics of tendon fascicles not completely understood: prediction of uncoiling of fibers during axial loads which may explain the large apparent Poisson ratios and fluid loss, and significant strain localization in cells, which may lead to important mechanotransduction mechanisms. Moreover, the distribution of dissipated power became available, pointing out the fibrils as the main source of dissipation of fascicles under high macroscopic strain rates and during the unloading phase in cyclic regimes.

Keywords

Tendon Soft tissues Multiscale Homogenization Viscoelastic model Finite element method 

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References

  1. 1.
    Svensson, R.B., Hassenkam, T., Grant, C.A., Peter Magnusson, S.: Tensile properties of human collagen fibrils and fascicles are insensitive to environmental salts. Biophys. J. 99(12), 4020–4027 (2010). ISSN 00063495ADSCrossRefGoogle Scholar
  2. 2.
    Vergari, C., Pourcelot, P., Holden, L., Ravary-Plumioën, B., Gerard, G., Laugier, P., Mitton, D., Crevier-Denoix, N.: True stress and Poisson’s ratio of tendons during loading. J. Biomech. 44(4), 719–724 (2011). ISSN 00219290CrossRefGoogle Scholar
  3. 3.
    Chernak, L.A., Thelen, D.G.: Tendon motion and strain patterns evaluated with two-dimensional ultrasound elastography. J. Biomech. 45(15), 2618–2623 (2012)CrossRefGoogle Scholar
  4. 4.
    Reese, S.P., Weiss, J.: Tendon fascicles exhibit a linear correlation between poisson’s ratio and force during uniaxial stress relaxation. J. Biomech. Eng. 135(3), 34501 (2013). ISSN 1528-8951CrossRefGoogle Scholar
  5. 5.
    Böl, M., Ehret, A.E., Leichsenring, K., Ernst, M.: Tissue-scale anisotropy and compressibility of tendon in semi-confined compression tests. J. Biomech. 48(6), 1092–1098 (2015)CrossRefGoogle Scholar
  6. 6.
    Lynch, H.A., Johannessen, W., Wu, J.P., Jawa, A., Elliott, D.M.: Effect of fiber orientation and strain rate on nonlinear tendon tensile properties. J. Biomech. Eng. 125, 726–731 (2003)CrossRefGoogle Scholar
  7. 7.
    Ahmadzadeh, H., Freedman, B.R., Connizzo, B.K., Soslowsky, L.J., Shenoy, V.B.: Micromechanical poroelastic finite element and shear-lag models of tendon predict large strain dependent Poisson’s ratios and fluid expulsion under tensile loading. Acta Biomaterialia 22, 83–91 (2014)CrossRefGoogle Scholar
  8. 8.
    Kannus, P.: Structure of the tendon connective tissue. Scand. J. Med. Sci. Sports 10(6), 312–320 (2000)CrossRefGoogle Scholar
  9. 9.
    Franchi, M., Trirè, A., Quaranta, M., Orsini, E., Ottani, V.: Collagen structure of tendon relates to function. TheScientificWorldJournal 7, 404–420 (2007)CrossRefGoogle Scholar
  10. 10.
    Thorpe, C.T., Birch, H.L., Clegg, P.D., Screen, H.R.C.: Tendon physiology and mechanical behavior: structure-function relationships. In: Gomes, M.E., Reis, R.L., Rodrigues, M.T. (eds.) Tendon Regeneration: Understanding Tissue Physiology and Development to Engineer Functional Substitutes, Chap. 1, pp. 3–39. Elsevier Academic Press, Cambridge (2015)CrossRefGoogle Scholar
  11. 11.
    Svensson, R.B., Hassenkam, T., Hansen, P., Peter Magnusson, S.: Viscoelastic behavior of discrete human collagen fibrils. J Mech Behav Biomed Mater 3(1), 112–115 (2010b)CrossRefGoogle Scholar
  12. 12.
    Shen, Z.L., Kahn, H., Ballarini, R., Eppell, S.J.: Viscoelastic properties of isolated collagen fibrils. Biophys J 100(12), 3008–3015 (2011)CrossRefGoogle Scholar
  13. 13.
    Yang, L., van der Werf, K.O., Dijkstra, P.J., Feijen, J., Bennink, M.L.: Micromechanical analysis of native and cross-linked collagen type I fibrils supports the existence of microfibrils. J Mech Behav Biomed Mater 6, 148–158 (2012)CrossRefGoogle Scholar
  14. 14.
    Blanco, P.J., Sanchez, P.J., de Souza Neto, E.A., Feijo, R.A.: Variational Foundations and generalized unified theory of RVE-based multiscale models. Arch Comput Methods Eng 23(2), 191–253 (2014). ISSN 18861784MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    de Souza Neto, E., Blanco, P.J., Sánchez, P.J., Feijóo, R.: An RVE-based multiscale theory of solids with micro-scale inertia and body force effects. Mech Mater 80, 136–144 (2015)CrossRefGoogle Scholar
  16. 16.
    Lanir, Y.: Multi-scale structural modeling of soft tissues mechanics and mechanobiology. J Elast 129(1–2), 7–48 (2017).  https://doi.org/10.1007/s10659-016-9607-0. ISSN 15732681MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fang, F., Lake, S.P.: Modelling approaches for evaluating multiscale tendon mechanics. Interface Focus, 6 (2016)Google Scholar
  18. 18.
    Marino, M., Vairo, G.: Stress and strain localization in stretched collagenous tissues via a multiscale modelling approach. Comput. Methods Biomech. Biomed. Eng. 1–20 (2012)Google Scholar
  19. 19.
    Herchenhan, A., Kalson, N.S., Holmes, D.F., Hill, P., Kadler, K.E., Margetts, L.: Tenocyte contraction induces crimp formation in tendon-like tissue. Biomech. Model. Mechanobiol. 11(3–4), 449–459 (2012)CrossRefGoogle Scholar
  20. 20.
    Fallah, A., Ahmadian, M.T., Firozbakhsh, K., Aghdam, M.M.: Micromechanics and constitutive modeling of connective soft tissues. J. Mech. Behav. Biomed. Mater. 60, 157–176 (2016).  https://doi.org/10.1016/j.jmbbm.2015.12.029 CrossRefGoogle Scholar
  21. 21.
    Ganghoffer, J.F., Laurent, C., Maurice, G., Rahouadj, R., Wang, X.: Nonlinear viscous behavior of the tendon’s fascicles from the homogenization of viscoelastic collagen fibers. Eur. J. Mech. A/Solids 59, 265–279 (2016).  https://doi.org/10.1016/j.euromechsol.2016.04.006 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ortiz, M., Stainier, L.: The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 7825(98), 419–444 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fancello, E., Ponthot, J.-P., Stainier, L.: A variational formulation of constitutive models and updates in non-linear finite viscoelasticity. Int. J. Numer. Methods Eng. 65(11), 1831–1864 (2006)CrossRefzbMATHGoogle Scholar
  24. 24.
    Mosler, J., Bruhns, O.T.: On the implementation of rate-independent standard dissipative solids at finite strain—Variational constitutive updates. Comput. Methods Appl. Mech. Eng. 199(9–12), 417–429 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Vassoler, J.M., Reips, L., Fancello, E.A.: A variational framework for fiber-reinforced viscoelastic soft tissues. Int. J. Numer. Methods Eng. 89(13), 1691–1706 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kalson, N.S., Lu, Y., Taylor, S.H., Starborg, T., Holmes, D.F., Kadler, K.E.: A structure-based extracellular matrix expansion mechanism of fibrous tissue growth. eLife, 4 (2015). ISSN 2050-084XGoogle Scholar
  27. 27.
    Aifantis, K.E., Shrivastava, S., Odegard, G.M.: Transverse mechanical properties of collagen fibers from nanoindentation. J. Mater. Sci.: Mater. Med. 22(6), 1375–1381 (2011)Google Scholar
  28. 28.
    Haga, H., Sasaki, S., Kawabata, K., Ito, E., Ushiki, T., Sambongi, T.: Elasticity mapping of living fibroblasts by AFM and immunofluorescence observation of the cytoskeleton. Ultramicroscopy 82, 253–258 (2000). ISSN 03043991CrossRefGoogle Scholar
  29. 29.
    Sirghi, L., Ponti, J., Broggi, F., Rossi, F.: Probing elasticity and adhesion of live cells by atomic force microscopy indentation. Eur. Biophys. J. 37(6), 935–945 (2008). ISSN 01757571CrossRefGoogle Scholar
  30. 30.
    Raman, A., Trigueros, S., Cartagena, A., Stevenson, P.Z., Susilo, M., Nauman, E., Antoranz Contera, S.: Mapping nanomechanical properties of live cells using multi-harmonic atomic force microscopy. Nat. Nanotechnol. 6(12), 809–814 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    Nawaz, S., Sánchez, P., Bodensiek, K., Li, S., Simons, M., Schaap, I.A.T.: Cell visco-elasticity measured with AFM and optical trapping at sub-micrometer deformations. PLoS ONE 7(9), (2012). ISSN 19326203Google Scholar
  32. 32.
    Hecht, F.M., Rheinlaender, J., Schierbaum, N., Goldmann, W.H., Fabry, B., Schäffer, T.E.: Imaging viscoelastic properties of live cells by AFM: power-law rheology on the nanoscale. Soft Matter 11(23), 4584–4591 (2015). ISSN 1744-6848ADSCrossRefGoogle Scholar
  33. 33.
    Thorpe, C.T., Udeze, C.P., Birch, H.L., Clegg, P.D., Screen, H.R.C.: Specialization of tendon mechanical properties results from interfascicular differences. J. R. Soc. Interface 9(July), 3108–3117 (2012)CrossRefGoogle Scholar
  34. 34.
    de Aro, A.A., de Campos Vidal, B., Rosa Pimentel, E.: Biochemical and anisotropical properties of tendons. Micron 43(2–3), 205–214 (2012)Google Scholar
  35. 35.
    Starborg, T., Kalson, N.S., Lu, Y., Mironov, A., Cootes, T.F., Holmes, D.F., Kadler, K.E.: Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization. Nat. Protoc. 8(7), 1433–48 (2013). ISSN 1750-2799CrossRefGoogle Scholar
  36. 36.
    Provenzano, P.P., Vanderby, R.: Collagen fibril morphology and organization: implications for force transmission in ligament and tendon. Matrix Biol. 25(2), 71–84 (2006). ISSN 0945053XCrossRefGoogle Scholar
  37. 37.
    Buehler, M.J.: Nature designs tough collagen: explaining the nanostructure of collagen fibrils. Proc. Nat. Acad. Sci. USA 103, 12285–12290 (2006)ADSCrossRefGoogle Scholar
  38. 38.
    Svensson, R.B., Mulder, H., Kovanen, V., Peter Magnusson, S.: Fracture mechanics of collagen fibrils: influence of natural cross-links. Biophys. J. 104(11), 2476–2484 (2013)ADSCrossRefGoogle Scholar
  39. 39.
    Gurtin, M.E., Anand, L.: The decomposition F = FeFp, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous. Int. J. Plast. 21(9), 1686–1719 (2005). ISSN 07496419CrossRefzbMATHGoogle Scholar
  40. 40.
    Nguyen, T.D., Jones, R.E., Boyce, B.L.: Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites. Int. J. Solids Struct. 44, 8366–8389 (2007)CrossRefzbMATHGoogle Scholar
  41. 41.
    Schröder, J., Neff, P.: Invariant formulaiton of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct. 40(2), 401–445 (2003). ISSN 00207683CrossRefzbMATHGoogle Scholar
  42. 42.
    Merodio, J., Ogden, R.W.: Mechanical response of fiber-reinforced incompressible non-linearly elastic solids. Int. J. Non-Linear Mech. 40(2–3), 213–227 (2005). ISSN 00207462ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Ehret, A.E., Itskov, M.: A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues. J. Mater. Sci. 42(21), 8853–8863 (2007). ISSN 00222461ADSCrossRefGoogle Scholar
  44. 44.
    Holzapfel, G., Ogden, R.W.: Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 367(1902), 3445–75 (2009). ISSN 1364-503XADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Balzani, D., Neff, P., Schröder, J., Holzapfel, G.: A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43(20), 6052–6070 (2006). ISSN 00207683MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    de Souza Neto, E.A., Peric, D., Owen, D.R.J.: Computational methods for plasticity: Theory Appl. (2009)Google Scholar
  47. 47.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  48. 48.
    Carniel, T.A., Fancello, E.A.: Modeling the local viscoelastic behavior of living cells under nanoindentation tests. Latin Am. J. Solids Struct. 1–19 (2017)Google Scholar
  49. 49.
    Vaz Jr., M., Cardoso, E.L., Stahlschmidt, J.: Particle swarm optimization and identification of inelastic material parameters. Eng. Comput. 30(7), 936–960 (2013). ISSN 0264-4401CrossRefGoogle Scholar
  50. 50.
    Screen, H.R.: Hierarchical approaches to understanding tendon mechanics. J. Biomech. Sci. Eng. 4(4), 481–499 (2009)CrossRefGoogle Scholar
  51. 51.
    Svensson, R.B., Herchenhan, A., Starborg, T., Larsen, M., Kadler, K.E., Qvortrup, K., Peter M.S.: Evidence of structurally continuous collagen fibrils in tendon. Acta Biomaterialia, 1–9 (2017)Google Scholar
  52. 52.
    Nguyen, V.D., Béchet, E., Geuzaine, C., Noels, L.: Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput. Mater. Sci. 55, 390–406 (2012). ISSN 09270256CrossRefGoogle Scholar
  53. 53.
    Hansen, K., Weiss, J., Barton, J.K.: Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. J. Biomech. Eng. 124(1), 72–77 (2002). ISSN 01480731CrossRefGoogle Scholar
  54. 54.
    Connizzo, B.K., Yannascoli, S.M., Soslowsky, L.J.: Structure–function relationships of postnatal tendon development: a parallel to healing. Matrix Biol. 32(2), 106–116 (2013)CrossRefGoogle Scholar
  55. 55.
    Cheng, V.W.T., Screen, H.R.C.: The micro-structural strain response of tendon. J. Mater. Sci. 42(21), 8957–8965 (2007). ISSN 00222461ADSCrossRefGoogle Scholar
  56. 56.
    Goh, K.L., Holmes, D.F., Lu, H.-Y., Richardson, S., Kadler, K.E., Purslow, P.P., Wess, T.J.: Ageing changes in the tensile properties of tendons: influence of collagen fibril volume fraction. J. Biomech. Eng. 130(2), 021011 (2008). ISSN 01480731CrossRefGoogle Scholar
  57. 57.
    Screen, H.R.C., Seto, J., Krauss, S., Boesecke, P., Gupta, H.S.: Extrafibrillar diffusion and intrafibrillar swelling at the nanoscale are associated with stress relaxation in the soft collagenous matrix tissue of tendons. Soft Matter 7(23), 11243–11251 (2011). ISSN 1744-683XADSCrossRefGoogle Scholar
  58. 58.
    Szczesny, S.E., Elliott, D.M.: Incorporating plasticity of the interfibrillar matrix in shear lag models is necessary to replicate the multiscale mechanics of tendon fascicles. J. Mech. Behav. Biomed. Mater. 40, 325–338 (2014)CrossRefGoogle Scholar
  59. 59.
    Legerlotz, K., Riley, G.P., Screen, H.R.C.: Specimen dimensions influence the measurement of material properties in tendon fascicles. J. Biomech. 43(12), 2274–2280 (2010)CrossRefGoogle Scholar
  60. 60.
    Haraldsson, B.T., Aagaard, P., Krogsgaard, M., Alkjaer, M., Magnusson, S.P.: Regoin-specific mechanical properties of the human patella tendon. J. Appl. Physiol. 98, 1006–1007 (2005)CrossRefGoogle Scholar
  61. 61.
    Akhtar, R., Schwarzer, N., Sherratt, M.J., Watson, R.E.B., Graham, H.K., Trafford, W., Mummery, P.M., Derby, B.: Nanoindentation of histological specimens: mapping the elastic properties of soft tissues. J. Mater. Res. 24(3), 638–646 (2009). ISSN 0884-2914ADSCrossRefGoogle Scholar
  62. 62.
    Hammer, N., Huster, D., Fritsch, S., Hädrich, C., Koch, H., Schmidt, P., Sichting, F., Franz Xaver Wagner, M., Boldt, A.: Do cells contribute to tendon and ligament biomechanics? PLoS ONE (2014). ISSN 19326203.  https://doi.org/10.1371/journal.pone.0105037
  63. 63.
    Herbert, A., Brown, C., Rooney, P., Kearney, J., Ingham, E., Fisher, J.: Bi-linear mechanical property determination of acellular human patellar tendon grafts for use in anterior cruciate ligament replacement. J. Biomech. 49(9), 1607–1612 (2016).  https://doi.org/10.1016/j.jbiomech.2016.03.041. ISSN 18732380CrossRefGoogle Scholar
  64. 64.
    Depalle, B., Qin, Z., Shefelbine, S.J., Buehler, M.J.: Influence of cross-link structure, density and mechanical properties in the mesoscale deformation mechanisms of collagen fibrils. J. Mech. Behav. Biomed. Mater. 1–13 (2014)Google Scholar
  65. 65.
    Liu, Y., Ballarini, R., Eppell, S.J.: Tension tests on mammalian collagen fibrils. Interface Focus 6(1), 20150080 (2016). ISSN 2042-8898CrossRefGoogle Scholar
  66. 66.
    Simo, J.C., Taylor, R.L.: Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Eng. 48(1), 101–118 (1985)ADSCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.GRANTE - Department of Mechanical EngineeringFederal University of Santa CatarinaFlorianópolisBrazil
  2. 2.LEBm - University HospitalFederal University of Santa CatarinaFlorianópolisBrazil

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