Fractional-order nonlinear hereditariness of tendons and ligaments of the human knee

E. Bologna, M. Di Paola, K. Dayal, L. Deseri, M. Zingales

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


In this paper the authors introduce a nonlinear model of fractional-order hereditariness used to capture experimental data obtained on human tendons of the knee. Creep and relaxation data on fibrous tissues have been obtained and fitted with logarithmic relations that correspond to power-laws with nonlinear dependence of the coefficients. The use of a proper nonlinear transform allows one to use Boltzmann superposition in the transformed variables yielding a fractional-order model for the nonlinear material hereditariness. The fundamental relations among the nonlinear creep and relaxation functions have been established, and the results from the equivalence relations have been contrasted with measures obtained from the experimental data. Numerical experiments introducing polynomial and harmonic stress and strain histories have been reported to assess the provided equivalence relations. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Original languageEnglish (US)
Article number20190294
Pages (from-to)20190294
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2172
StatePublished - May 29 2020


  • nonlinear hereditariness
  • quasi-linear viscoelasticity
  • single-integral
  • tendons and ligaments
  • Knee
  • Biomechanical Phenomena
  • Tendons/cytology
  • Humans
  • Mechanical Phenomena
  • Nonlinear Dynamics
  • Ligaments/cytology

ASJC Scopus subject areas

  • Engineering(all)
  • Physics and Astronomy(all)
  • Mathematics(all)


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