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Fractional Calculus in Visco-Elasticity

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50+ Years of AIMETA

Abstract

The fractional calculus is now popular in the engineering community because of its capability, especially to predict the visco-elastic response of the various materials in both time and frequency domain. In the present paper results of the research group of Palermo, in this setting, are briefly summarized.

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References

  1. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus Theoretical Developments and Applications in Physics and Engineering. Springer (2007)

    Google Scholar 

  2. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  3. Carpinteri, A., Chiaia, B., Ferro, G.: Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Mater. Struct. 28, 311–317 (1995). https://doi.org/10.1007/BF02473145

  4. Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of visco-elastically damped structure. AIAA J. 21, 741–748 (1983)

    Article  MATH  Google Scholar 

  5. Bagley, R.L., Torvik, P.J.: Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23, 918–925 (1985)

    Article  MATH  Google Scholar 

  6. Flügge, W.: Viscoelasticity. Blaisdell Publishing Company, Massachusetts (1967)

    MATH  Google Scholar 

  7. Pipkin, A.: Lectures on Viscoelasticity Theory. Applied Mathematical Sciences. Springer (1972)

    Google Scholar 

  8. Christensen, R.M.: Theory of Viscoelasticity: An Introduction. Academic Press (1982)

    Google Scholar 

  9. Nutting, P.G.: A new general law deformation. J. Franklin Inst. 191, 678–685 (1921)

    Article  Google Scholar 

  10. Gemant, A.: A method of analyzing experimental results obtained by elasto-viscous bodies. Physics 7, 311–317 (1936)

    Article  Google Scholar 

  11. Di Paola, M., Pirrotta, A., Valenza, A.: Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mech. Mater. 43, 799–806 (2011)

    Article  Google Scholar 

  12. Samko, G.S., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Amsterdam (1993)

    MATH  Google Scholar 

  13. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering 198, Academic Press (1999)

    Google Scholar 

  14. Gonsovski, V.L., Rossikhin, Yu.A.: Stress waves in a viscoelastic medium with a singular hereditary kernel. J. Appl. Mech. Tech. Phys. 14(4), 595–597 (1973)

    Article  Google Scholar 

  15. Stiassnie, M.: On the application of fractional calculus on the formulation of viscoelastic models. Appl. Math. Model. 3, 300–302 (1979)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30(1), 133–155 (1986)

    Article  MATH  Google Scholar 

  18. Schmidt, A., Gaul, L.: Finite element formulation of viscoelastic constitutive equations using fractional time derivatives. Nonlinear Dyn. 29, 37–55 (2002)

    Article  MATH  Google Scholar 

  19. Mainardi, F., Gorenflo, R.: Time-fractional derivatives in relaxation processes: a tutorial survey. Fractional Calc. Appl. Anal. 10(3), 269–308 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Evangelatos, G.I., Spanos, P.D.: An accelerated Newmark scheme for integrating the equation of motion of nonlinear systems comprising restoring elements governed by fractional derivatives. In: Kounadis, A.N., Gdoutos, E.E. (eds.) Recent Advances in Mechanics 1, pp. 159–177 (2011)

    Google Scholar 

  21. Koeller, R.C.: Application of fractional calculus to the theory of visco-elasticity. ASME J. Appl. Mech. 51, 299–307 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mainardi, F.: Fractional relaxation in anelastic solids. J. Alloy. Compd. 211, 534–538 (1994)

    Article  Google Scholar 

  23. Shen, K.L., Soong, T.T.: Modeling of visco-elastic dampers for structural applications. J. Eng. Mech. 121, 694–701 (1995)

    Article  Google Scholar 

  24. Pritz, T.: Analysis of four-parameter fractional derivative model of real solid materials. J. Sound Vib. 195, 103–115 (1996)

    Article  MATH  Google Scholar 

  25. Papoulia, K.D., Kelly, J.M.: Visco-hyperelastic model for filled rubbers used in vibration isolation. J. Eng. Mater. Technol. 119, 292–297 (1997)

    Article  Google Scholar 

  26. Pirrotta, A., Kougioumtzoglou, I.A., Di Matteo, A., Fragkoulis, V.C., Pantelous, A.A., Adam, C.: Deterministic and random vibration of linear systems with singular parameter matrices and fractional derivative terms. J. Eng. Mech. (2021). https://doi.org/10.1061/(ASCE)EM.1943-7889.0001937

    Article  Google Scholar 

  27. Schiessel, H., Metzler, R., Blumen, A., Nonnemacher, T.F.: Generalized visco-elastic models: their fractional equations with solutions. J. Phys. A: Math. Gen. 28, 6567–6584 (1995)

    Article  MATH  Google Scholar 

  28. Alotta, G., Barrera, O., Cocks, A.C.F., Di Paola, M.: On the behavior of a three-dimensional fractional viscoelastic constitutive model. Meccanica 52, 2127–2142 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yao, Q.Z., Liu, L.C., Yan, Q.F.: Quasi-static analysis of beam described by fractional derivative kelvin visco-elastic model under lateral load. Adv. Mater. Res. 189–193, 3391–3394 (2011)

    Article  Google Scholar 

  30. Di Paola, M., Heuer, R., Pirrotta, A.: Fractional visco-elastic Euler-Bernoulli beam. Int. J. Solids Struct. 50(22–23), 3505–3510 (2013)

    Article  Google Scholar 

  31. Di Paola, M., Zingales, M.: Exact mechanical models of fractional hereditary materials (FHM). J. Rheol. 56(5), 983–1004 (2012)

    Article  Google Scholar 

  32. Di Paola, M., Pinnola, F., Zingales, M.: A discrete mechanical model of fractional hereditary materials. Meccanica 48(7), 1573–1586 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Di Paola, M., Pinnola, F., Zingales, M.: Fractional differential equations and related exact mechanical models. Comput. Math. Appl. 66, 608–620 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Di Paola, M., Galuppi, L., Royer Carfagni, G.: Fractional viscoelastic characterization of laminated glass beams under time-varying loading. Int. J. Mech. Sci. 196(4) (2021)

    Google Scholar 

  35. Di Matteo, A., Lo Iacono, F., Navarra, G., Pirrotta, A.: Innovative modeling of tuned liquid column damper motion. Commun. Nonlinear Sci. Numer. Simul. 23, 229–244 (2015)

    Article  Google Scholar 

  36. Di Matteo, A., Di Paola, M., Pirrotta, A.: Innovative modeling of tuned liquid column damper controlled structures. Smart Struct. Syst. 18(1), 117–138 (2016)

    Article  Google Scholar 

  37. Pirrotta, A., Cutrona, S., Di Lorenzo, S., Di Matteo, A.: Fractional visco-elastic Timoshenko beam deflection via single equation. Int. J. Numer. Meth. Eng. 104(9), 869–886 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pirrotta, A., Cutrona, S., Di Lorenzo, S.: Fractional visco-elastic Timoshenko beam from elastic Euler-Bernoulli beam. Acta Mech. 226, 179–189 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Di Lorenzo, S., Di Paola, M., Pinnola, F.P., Pirrotta, A.: Stochastic response of fractionally damped beams. Probab. Eng. Mech. 35, 37–43 (2014)

    Article  Google Scholar 

  40. Di Matteo, A., Pirrotta, A.: Generalized differential transform method for nonlinear boundary value problem of fractional order. Commun. Nonlinear Sci. Numer. Simul. 29, 88–101 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Di Paola, M., Reddy, J.N., Ruocco, E.: On the application of fractional calculus for the formulation of viscoelastic Reddy beam. Meccanica 55, 1365–1378 (2020)

    Article  MathSciNet  Google Scholar 

  42. Burlon, A., Alotta, G., Di Paola, M., Failla, G.: An original perspective on variable-order fractional operators for viscoelastic materials. Meccanica 56, 769–784 (2021)

    Article  MathSciNet  Google Scholar 

  43. Ramirez, L.E.S., Coimbra, C.F.M.: A variable order constitutive relation for viscoelasticity. Ann. Phys. (Leipzig) 16(7–8), 543–552 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Sun, H.G., Chen, W., Wei, H., Chen, Y.Q.: A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top. 193, 185–192 (2011)

    Article  Google Scholar 

  45. Ingman, D., Suzdaltnitsky, J.: Application of differential operator with servo-order function in model of viscoelastic deformation process. J. Eng. Mech. 131(7) (2005) https://doi.org/10.1061/(ASCE)0733-9399(2005)131:7(763)

  46. Li, M., Pu, H., Cao, L.: Variable-order fractional creep model of mudstone under high temperature. Therm. Sci. 21, 343–349 (2017)

    Article  Google Scholar 

  47. Bouras, Y., Zorica, D., Atanackovic, T.M., Vrcelj, Z.: A non-linear thermo viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete. Appl. Math. Model. 55, 551–568 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Meng, R., Yin, D., Drapaca, C.S.: A variable order fractional constitutive model of the viscoelastic behavior of polymers. Int. J. Nonlinear. Mech. 113, 171–177 (2019)

    Article  MATH  Google Scholar 

  49. Meng, R., Yin, D., Zhou, C., Wu, H.: Fractional description of time-dependent mechanical property evolution in materials with strain softening behavior. Appl. Math. Model. 40, 398–406 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Alotta, G., Di Paola, M.: Fractional viscoelasticity under combined stress and temperature variations. In: Carcaterra, A., Paolone, A., Graziani, G. (eds.) Proceedings of XXIV AIMETA Conference 2019, Lect. Notes Mech. Eng., pp. 1703–1717, Springer International Publishing (2020)

    Google Scholar 

  51. Di Paola, M., Alotta, G., Burlon, A., Failla, G.: A novel approach to nonlinear variable-order fractional viscoelasticity. Philos. Trans. Roy. Soc. A 378(2172), 20190296 (2020)

    Google Scholar 

  52. Bologna, E., Di Paola, M., Deseri, L., Dayal, K., Zingales, M.: Fractional-order nonlinear hereditariness of tendons and ligaments of the human knee. Philos. Trans. Roy. Soc. A 378(2172), 20190294 (2020)

    Google Scholar 

  53. Marchiori, G., Lopomo, N.F., Bologna, E., Spadaro, D., Camarda, L., Berni, M., Visani, A., Zito M., Zaffagnini, S., Zingales, M.: How preconditioning and pretensioning of grafts used in ACLigaments surgical reconstruction are influenced by their mechanical time-dependent characteristics: Can we optimize their initial loading state? Clin. Biomech. 83, 105294 (2021)

    Google Scholar 

  54. Bologna, E., Deseri, L., Graziano, F., Zingales, M.: Power-Laws hereditariness of biomimetic ceramics for cranioplasty neurosurgery. Int. J. Nonlinear Mech. 115, 61–67 (2019)

    Article  Google Scholar 

  55. Bologna, E., Marchiori, G., Lopomo, N., Zingales, M.: A non-linear stochastic approach of ligaments and tendons fractional-order hereditariness. Probab. Eng. Mech. 60, 103034 (2020)

    Google Scholar 

  56. Alotta, G., Di Paola, M., Pinnola, F.P., Zingales, M.: A fractional nonlocal approach to nonlinear blood flow in small-lumen arterial vessels. Meccanica 55, 891–906 (2020)

    Article  MathSciNet  Google Scholar 

  57. Deseri, L., Pollaci, P., Zingales, M., Dayal, K.: Fractional hereditariness of lipid membranes: instabilities and linearized evolution. J. Mech. Behav. Biomed. Mater. 58, 11–27 (2016)

    Article  Google Scholar 

  58. Deseri, L., Di Paola, M., Zingales, M., Pollaci, P.: Power-law hereditariness of hierarchical fractal bones. Int. J. Numer. Meth. Biomed. Eng. 29, 1338–1360 (2013)

    Article  MathSciNet  Google Scholar 

  59. Alaimo, G., Zingales, M.: Laminar flow through fractal porous materials: the fractional-order transport equation. Commun. Nonlinear Sci. Numer. Simul. 22, 889–902 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors fully acknowledge all the research group of Palermo, for the precious and fruitful cooperation during the studies on fractional visco-elasticity: G. Alotta, E. Bologna, A. Burlon, A. Di Matteo, G. Failla, D. Fiandaca, F. Lo Iacono, C. Masnata, G. Navarra, F. P. Pinnola, S. Russotto, M. Zingales.

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Correspondence to Antonina Pirrotta .

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Di Paola, M., Pirrotta, A. (2022). Fractional Calculus in Visco-Elasticity. In: Rega, G. (eds) 50+ Years of AIMETA. Springer, Cham. https://doi.org/10.1007/978-3-030-94195-6_16

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  • DOI: https://doi.org/10.1007/978-3-030-94195-6_16

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