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A viscoelastic model for describing the response of biological fibers

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Abstract

In this paper, we extend the modeling effort of our earlier work concerning the development of an implicit elastic model for describing the response of biological fibers. We put into place a fractional-order viscoelastic (FOV) solid model that can quantify the properties of biological fibers comprised of collagen fibrils and elastic filaments. The compliance version of the FOV memory function is regularized, thereby removing the singularity from the viscoelastic kernel. The ensuing model is used to describe stress relaxation in mitral-valve chordæ tendineæ. Numerical solutions for the convolution integral are acquired via a midpoint quadrature rule with a Laplace endpoint correction.

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References

  1. Viidik, A.: Functional properties of collagenous tissues. Int. Rev. Connect. Tissue Res. 6, 127–215 (1973)

    Article  Google Scholar 

  2. Carton, R.W., Dainauskas, J., Clark, J.W.: Elastic properties of single elastic fibers. J. Appl. Physiol. 17, 547–551 (1962)

    Google Scholar 

  3. Hunter, P.J.: Myocardial constitutive laws for continuum mechanics models of the heart. In: Sideman, S., Beyar R. (Eds.), Molecular and subcellular cardiology: effects of structure and function, Ch. 30. Plenum Press, New York, pp. 303–318 (1995)

  4. Köhler, T., Vollrath, F.: Thread biomechanics in the two orb-weaving spiders Araneus diadematus (Araneae, Araneidae) and Uloborus walckenaerius (Araneae, Uloboridae). J. Exp. Zool. Part A 271, 1–17 (1995)

    Article  Google Scholar 

  5. Bao, G., Suresh, S.: Cell and molecular mechanics of biological materials. Nat. Mater. 2, 715–725 (2003)

    Article  Google Scholar 

  6. Bustamante, C., Marko, J.F., Siggia, E.D., Smith, S.: Entropic elasticity of lambda-phage DNA. Science 265, 1599–1600 (1994)

    Article  Google Scholar 

  7. Maksym, G.N., Bates, J.H.T.: A distributed nonlinear model of lung tissue elasticity. J. Appl. Physiol. 82, 32–41 (1997)

    Google Scholar 

  8. Rajagopal, K.R.: On implicit constitutive theories. Appl. Math. 48(4), 279–319 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Freed, A.D., Rajagopal, K.R. (2016) A promising approach for modeling biological fibers. Acta Mech. doi:10.1007/s00707-016-1583-8

  10. Hildebrandt, J.: Pressure-volume data of cat lung interpreted by a plastoelastic, linear viscoelastic model. J. Appl. Physiol. 28, 365–372 (1970)

    Google Scholar 

  11. Fung, Y.-C.: Stress-strain-history relations of soft tissues in simple elongation. In: Fung, Y.-C., Perrone, N., Anliker, M. (eds.) Biomechanics: its foundations and objectives, Ch. 7, pp. 181–208. Prentice-Hall, Englewood Cliffs (1971)

    Google Scholar 

  12. Fung, Y.-C.: Biorheology of soft tissues. Biorheology 10, 139–155 (1973)

    Google Scholar 

  13. Boltzmann, L.: Zur Theorie der elastischen Nachwirkung, Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften. Wien 70, 275–300 (1874)

    Google Scholar 

  14. Kaye, A.: A non-Newtonian flow in incompressible fluids. Technical Report 134, The College of Aeronautics, Cranfield (1962)

  15. Bernstein, B., Kearsley, E.A., Zapas, L.J.: A study of stress relaxation with finite strain. Trans. Soc. Rheol. 7, 391–410 (1963)

    Article  MATH  Google Scholar 

  16. Johnson, G.A., Livesay, G.A., Woo, S.L.-Y., Rajagopal, K.R.: A single integral finite strain viscoelastic model of ligaments and tendons. J. Biomech. Eng. 118, 221–226 (1996)

    Article  Google Scholar 

  17. Pioletti, D.P., Rakotomanana, L.R.: Non-linear viscoelastic laws for soft biological tissues. Eur. J. Mech. A/Solids 19, 749–759 (2000)

    Article  MATH  Google Scholar 

  18. Wineman, A.: Nonlinear viscoelastic solids—a review. Math. Mech. Solids 14, 300–366 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Neubert, H.K.P.: A simple model representing internal damping in solid matrials. Aeronaut. Q. 14, 187–210 (1963)

    Article  Google Scholar 

  20. Freed, A.D.: Soft Solids: A Primer to the Theoretical Mechanics of Materials, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Basel (2014)

    Book  Google Scholar 

  21. Provenzano, P.P., Lakes, R.S., Corr, D.T., Vanderby Jr., R.: Application of nonlinear viscoelastic models to describe ligament behavior. Biomech. Model. Mechanobiol. 1, 45–57 (2002)

    Article  Google Scholar 

  22. Puso, M.A., Weiss, J.A.: Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation. J. Biomech. Eng. 120, 62–70 (1998)

    Article  Google Scholar 

  23. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  24. Suki, B., Barabási, A.-L., Lutchen, K.R.: Lung tissue viscoelasticity: a mathematical framework and its molecular basis. J. Appl. Physiol. 76, 2749–2759 (1994)

    Google Scholar 

  25. Doehring, T.C., Freed, A.D., Carew, E.O., Vesely, I.: Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity. J. Biomech. Eng. 127, 700–708 (2005)

    Article  Google Scholar 

  26. Drozdov, A.D.: Fractional differential models in finite viscoelasticity. Acta Mech. 124, 155–180 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Freed, A.D., Diethelm, K.: Caputo derivatives in viscoelasticity: a non-linear finite-deformation theory for tissue. Fract. Calc. Appl. Anal. 10, 219–248 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Liouville, J.: Mémoire sur quelques questions del géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions. J. l’École Polytech. 13(cahier 21), 1–66 (1832)

    Google Scholar 

  29. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  30. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, fractional differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 of Mathematics in Science and Engineering. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  31. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Vol. 2004 of Lecture Notes in Mathematics. Springer, Heidelberg (2004)

    Google Scholar 

  32. Kastelic, J., Galeski, A., Baer, E.: The multicomposite structure of tendon. Connect. Tissue Res. 6, 11–23 (1978)

    Article  Google Scholar 

  33. Sasaki, N., Odajima, S.: Elongation mechanism of collagen fibrils and force–strain relations of tendon at each level of structural hierarchy. J. Biomech. 29, 1131–1136 (1996)

    Article  Google Scholar 

  34. Fung, Y.C.: Elasticity of soft tissues in simple elongation. Am. J. Physiol. 28, 1532–1544 (1967)

    Google Scholar 

  35. Lanir, Y.: A structural theory for the homogeneous biaxial stress–strain relationships in flat collagenous tissues. J. Biomech. 12, 423–436 (1979)

    Article  Google Scholar 

  36. Babu, A.R., Byju, A.G., Gundiah, N.: Biomechanical properties of human ascending thoracic aortic dissections. J. Biomech. Eng. 137, 081013 (2015)

    Article  Google Scholar 

  37. Shen, Z.L., Kahn, H., Ballarini, R., Eppell, S.J.: Viscoelastic properties of isolated collagen fibrils. Biophys. J. 100, 3008–3015 (2011)

    Article  Google Scholar 

  38. Gautieri, A., Vesentini, S., Redaelli, A., Buehler, M.J.: Hierarchical structure and nanomechanics of collagen microfibrils from the atomistic scale up. Nano Lett. 11, 757–766 (2011)

    Article  Google Scholar 

  39. Lillie, M.A., Gosline, J.M.: Swelling and viscoelastic properties of osmotically stressed elastin. Biopolymers 39, 641–652 (1996)

    Article  Google Scholar 

  40. Aaron, B.B., Gosline, J.M.: Elastin as a random-network elastomer: a mechanical and optical analysis of single elastin fibers. Biopolymers 20, 1247–1260 (1981)

    Article  Google Scholar 

  41. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)

    Article  MATH  Google Scholar 

  42. Gorenflo, R., Rubin, B.: Locally controllable regularization of fractional derivatives. Inverse Prob. 10, 881–893 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. Freed, A.D., Diethelm, K.: Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 5, 203–215 (2006)

    Article  Google Scholar 

  44. Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues, 2nd edn. Springer, New York (1993)

    Book  Google Scholar 

  45. Humphrey, J.D.: Cardiovascular Solid Mechanics; Cells, Tissues, and Organs. Springer, New York (2002)

    Book  Google Scholar 

  46. Liao, J.: Mechanical and structural properties of mitral valve chordæ tendineæ, Ph.D. thesis, Cleveland State University, Cleveland, OH (2003)

  47. Liao, J., Priddy, L.B., Wang, B., Chen, J., Vesely, I.: Ultrastructure of porcine mitral valve chordæ tendineæ. J. Heart Valve Dis. 18, 292–299 (2009)

    Google Scholar 

  48. Freed, A.D., Doehring, T.C.: Elastic model for crimped collagen fibrils. J. Biomech. Eng. 127, 587–593 (2005)

    Article  Google Scholar 

  49. Braß, H.: Quadraturverfahren, Vol. 3 of Studia Mathematica. Vandenhoeck and Ruprecht, Göttingen (1977)

    Google Scholar 

  50. Diethelm, K., Freed, A.D.: An efficient algorithm for the evaluation of convolution integrals. Comput. Math. Appl. 51, 51–72 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zener, C.: Elasticity and Anelasticity of Metals. University of Chicago Press, Chicago (1948)

    MATH  Google Scholar 

  52. Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971)

    Article  Google Scholar 

  53. Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  54. Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, 133–155 (1986)

    Article  MATH  Google Scholar 

  55. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)

    Article  MathSciNet  Google Scholar 

  56. Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  58. Ford, N.J., Simpson, A.C.: The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26, 333–346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  59. Mittag-Leffler, G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène. Acta Math. 29, 101–168 (1904)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zaremba, S.: Sur une forme perfectionnée de la théorie de la relaxation. Bull. l’Acad. Crac. 594–614 (1903)

  61. Jaumann, G.: Geschlossenes System physikalischer und chemischer Differentialgesetze. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften: Mathematisch-naturwissenschaftliche Klasse 120, 385–530 (1911)

    MATH  Google Scholar 

  62. Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  63. Freed, A.D., Diethelm, K., Luchko, Y.: Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus: first annual report, TM 2002–211914, NASA Glenn Research Center at Lewis Field, Cleveland (2002)

  64. Freed, A.D., Diethelm, K.: Tensor fields for use in fractional-order viscoelasticity. In: Le Mehauté, A., Tenreiro Machado, J.A., Trigeassou, J.C., Sabatier, J. (eds.) Fractional Differentiation and Its Applications, pp. 169–182. UBooks Verlag, Neusäß (2005)

    Google Scholar 

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  1. Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77843-3123, USA

    Alan D. Freed & K. R. Rajagopal

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  1. Alan D. Freed
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Freed, A.D., Rajagopal, K.R. A viscoelastic model for describing the response of biological fibers. Acta Mech 227, 3367–3380 (2016). https://doi.org/10.1007/s00707-016-1673-7

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