Fractional viscoelastic models with novel variable and constant order fractional derivative operators

Authors

  • Krunal Kachia Charotar University of Science and Technology (CHARUSAT), Changa, Anand 388421, Gujarat, India
  • Francisco Gomez CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México. http://orcid.org/0000-0001-9403-3767

DOI:

https://doi.org/10.31349/RevMexFis.68.020703

Keywords:

Fractional Viscoelastic Models, Variable-order derivatives

Abstract

This paper deals with the application of a novel variable-order and constant-order fractional derivative without singular kernel of Atangana-Koca type to describe the fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model. For each fractional viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under the variable-order and constant-order fractional derivative without singular kernel. Our results show that the relaxation modulus and creep compliance exhibit viscoelastic behaviors producing temporal fractality at different scales. For each viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under novel variable-order and constant-order fractional derivative with no singular kernel.

References

B. T. Alkahtani, I. Koca and A. Atangana, A novel approach of variable order derivative: Theory and Methods, J. Nonlinear Sci. Appl., 9 (2016), 4867–4876.

B.S.T. Alkahtani, Chua’s circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons and Fractals, 89 (2016), 547–551.

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20(2) (2016), 763-769.

A. Atangana and I. Koca, New direction in fractional differentation, Math. Nat. Sci., 1 (2017), 18–25.

A. Atangana and J.F. Gomez Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 133–166.

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

G. S. Blair, The role of Psychophysics in Rheology, Journal of Colloid Science, 2(1) (1947), 21–32.

M Caputo and M Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2) (2015), 73–85.

M Caputo and M Fabrizio, On the notion of fractional derivative and applications to the hysteresis phenomena, Meccanica, 52(13) (2017), 3043–3052.

B. Carmichael, H. Babahosseini, H. Mahmoodi and S. N. Agah, The fractional viscoelastic response of human breast tissue

cells, Phys. Biol., 12(4) (2015), 046001.

Y. Carrera, G. Avila-de la Rosa, E. J. Vernon-Carter and J. Alvarez-Ramirez, A fractional order Maxwell model for NonNewtonian fluids, Physica A: Stat. Mech. Appl., 482 (2017), 276–285.

J. D. Ferry, Viscoelastic properties of Polymers, 3rd edition, John Wiley and Sons, New York, 1980.

Y. C. Fung, Biomechanics: mechanical properties of living tissues., 2nd edition, Springer-Verlag, New York, 1993.

J. F. Gomez-Aguilar, Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and

without singular kernels, Eur. Phys. J. Plus., 133 (2018) 133.

W. G. Glockle and T. F. Nonnenmacher, Fractional integral operators and Fox functions in the theory of viscoelasticity,

Macromolecules, 24 (1991), 6424–6434.

W. G. Glockle and T. F. Nonnenmacher, Fractional relaxation and the time-temperature superposition principle, Rheol. Acta,

(1994), 337–343.

W. G. Glockle, and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68 (1995), 46–53.

M. E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal., 11 (1962), 291–356.

J Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models, Frontiers in fractional calculus, Bentham Science Publishers, Sharjah (2017), pp. 270-342.

L. Jianmin, X. Rui and C. Wen, Fractional viscoelastic models with non-singular kernels, Mechanics of Materials, 27 (2018),

–64.

D.C. Labora, J.J. Nieto, R. Rodriguez-Lopez. Is it possible to ´construct a fractional derivative such that the index law holds.,

Progr Fract Differ Appl, 4(1) (2018), 1–3.

R. Lakes, Viscoelastic Materials, Cambridge University Press, Cambridge, 2009.

K. Ervin Lenzi, A. Tateishi Angel and V. Haroldo Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Front. Phys., 5 (2017), 1–9.

D. Lei, Y. J. Liang and R. Xiao, A fractional model with parellel fractional Maxwell elements for amorphous thermoplastic, Physica A: Stat. Mech. Appl., 490 (2018), 465–475.

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

V. F. Morales-Delgado, J. F. Go´mez-Aguilar, M. A. TanecoHerna´dez and R. F. Escobar-Jime´nez, A novel fractional derivative with variable and constant order applied to a massspring-damper system, Eur. Phys. J. Plus., 133 (2018), 78.

T. R. Prabhakar, A singular integral equation with a genearlized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.

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Published

2022-03-01

How to Cite

[1]
K. Kachia and F. Gomez, “Fractional viscoelastic models with novel variable and constant order fractional derivative operators”, Rev. Mex. Fís., vol. 68, no. 2 Mar-Apr, pp. 020703 1–, Mar. 2022.

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Section

07 Gravitation, Mathematical Physics and Field Theory