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Wave equation in fractional Zener-type viscoelastic media involving Caputo–Fabrizio fractional derivatives

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Abstract

We investigate propagation of waves in the Zener-type viscoelastic media through a model which involves fractional derivatives with a regular kernel. The restrictions on the coefficients in the constitutive equation that follow from the weak form of the dissipation principle are obtained. We formulate a problem of motion of a spatially one dimensional continuum in a dimensionless form. Then, it is considered in the frame of distribution theory. The existence and the uniqueness of a distributional solution as well as the analysis of its regularity are presented. Numerical results provide the illustration of our approach.

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Notes

  1. Recall that \(AC([0,\infty ))\) consists of continuous functions y having locally integrable derivatives (\(dy/dt=y^{(1)}\in L^1_{loc}([0,\infty )).\)

  2. Since \(\sigma =\varSigma +\varepsilon\), with (19), the stress strain relation (1) becomes

    $$\begin{aligned} \sigma (x,t)=\int _{-\infty }^{\infty }{\mathcal {F}}^{-1}\left[ \varPhi \left( \omega \right) \right] \left( t-\tau \right) \frac{\partial \varepsilon \left( x,\tau \right) }{\partial \tau }d\tau +\varepsilon (x,t), \end{aligned}$$

    which shows nonlocality of constitutive equation (1) with fractional derivative (3).

  3. We note that our definition of the Laplace transform differs from the one in [29] up to the rotation with the angle \(\pi /2\).

References

  1. Atanackovic TM, Pilipovic S, Zorica D (2018) Properties of the Caputo–Fabrizio fractional derivative and its distributional settings. Fract Calc Appl Anal 21:29–44

    Article  MathSciNet  MATH  Google Scholar 

  2. Bagley RL, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30(1):133–155

    Article  ADS  MATH  Google Scholar 

  3. Caputo M, Mainardi F (1971) A new dissipation model based on memory mechanism. Pure Appl Geophys 91:134–147. https://doi.org/10.1007/BF00879562

    Article  ADS  MATH  Google Scholar 

  4. Cuahutenango-Barro B, Taneco-Hernández MA, Gómez-Aguilar JF (2017) Application of fractional derivative with exponential law to bi-fractional-order wave equation with frictional memory Kernel. Eur Phys J Plus 132:515. https://doi.org/10.1140/epjp/i2017-11796-9

    Article  Google Scholar 

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

    Google Scholar 

  6. Caputo M, Fabrizio M (2015) Properties of a new fractional derivative without singular Kernel. Progr Fract Differ Appl 1(2):87–92

    Google Scholar 

  7. Caputo M, Fabrizio M (2017) On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica. https://doi.org/10.1007/s11012-017-0652-y

  8. El-Karamany AS, Ezzat MA (2011) On fractional thermoelasticity. Math Mech Solids 16:334–346

    Article  MathSciNet  MATH  Google Scholar 

  9. El-Karamany AS, Ezzat MA (2015) Two-temperature Green–Naghdi theory of type III in linear thermoviscoelastic anisotropic solid. Appl Math Modell 39:2155–2171

    Article  MathSciNet  Google Scholar 

  10. El-Karamany AS, Ezzat MA (2011) Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity. J Therm Stresses 34:264–284

    Article  Google Scholar 

  11. Eringen AC (2002) Nonlocal continuum field theories. Springer, New York

    MATH  Google Scholar 

  12. Enelund M, Mähler L, Runesson K, Lennart Josefson B (1999) Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws. Int J Solids Struct 36:2417–2442

    Article  MATH  Google Scholar 

  13. Ezzat MA, El-Bary AA (2016) Unified fractional derivative models of magneto-thermo-viscoelasticity theory. Arch Mech 68:285–308

    MathSciNet  MATH  Google Scholar 

  14. Fabrizio M, Lazzari B, Nibbi R (2017) Existence and stability for a visco-plastic material with a fractional constitutive equation. Math Meth Appl Sci 40:6306–6315

    Article  MathSciNet  MATH  Google Scholar 

  15. Fernndez-Saez J, Zaeraa R, Loya JA, Reddy JN (2016) Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116

    Article  MathSciNet  MATH  Google Scholar 

  16. Hristov J (2017) Derivatives with non-singular Kernels from the Caputo–Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. Front Fract Calc 1:270–342

    Google Scholar 

  17. Kochubei AN (2011) General fractional calculus, evolution equations, and renewal processes. Integr Equ Oper Theory 71:583–600

    Article  MathSciNet  MATH  Google Scholar 

  18. Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity. Imperial College Press-World Scientific, London

    Book  MATH  Google Scholar 

  19. Mainardi F, Spada G (2011) Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur Phys J Spec Top 193:133–160

    Article  Google Scholar 

  20. Ortigueira MD, Tenreiro Machado JA (2015) What is fractional derivative? J Comput Phys 293:4–13

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Ortigueira MD, Tenreiro Machado JA (2018) A critical analysis of the Caputo–Fabrizio operator. Commun Nonlinear Sci Numer Simul 59:608–611

    Article  ADS  MathSciNet  Google Scholar 

  22. Pallu De LA Barriere R (1980) Optimal control theory: a course in automatic control theory. Dower Publications Inc, New York

    MATH  Google Scholar 

  23. Reed M, Simon B (1980) Methods of modern mathematical physics, I: functional anaalysis. Academic Press, New York

    Google Scholar 

  24. Ross B (1975) A brief history and exposition of the fundamental theory of fractional calculus. In: Ross B (ed) Fractional calculus and its applications. Lecture notes in mathematics. Springer, Berlin, pp 1–36

    Chapter  Google Scholar 

  25. Sherief HH, El-Sayed AMA, Abd El-Latief AM (2010) Fractional order theory of thermoelasticity. Int J Solids Struct 47:269–275

    Article  MATH  Google Scholar 

  26. Stankovic B, Atanackovic TM (2004) On an inequality arising in fractional oscillator theory. Fract Calc Appl Anal 7:11–20

    MathSciNet  MATH  Google Scholar 

  27. Tarasov VE (2018) No nonlocality. No fractional derivative. Commun Nonlinear Sci Numer Simul 62:157–163

    Article  ADS  MathSciNet  Google Scholar 

  28. Vladimirov VS (1979) Generalized functions in mathematical physics. Mir Publishers, Moscow

    MATH  Google Scholar 

  29. von Ende S, Lion A, Lammering L (2011) On the thermodynamically consistent fractional wave equation for viscoelastic solids. Acta Mech 221:1–10

    Article  MATH  Google Scholar 

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Acknowledgements

This research was supported by the Serbian Academy of Sciences and Arts (TMA) and Serbian Ministry of Science Grants TR32035, III44003 (MJ) and 174024 (SP).

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Authors and Affiliations

  1. Serbian Academy of Arts and Sciences, Kneza Mihaila 35, Belgrade, 11000, Serbia

    Teodor M. Atanacković

  2. Institute of Mathematics, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Belgrade, 11000, Serbia

    Marko Janev

  3. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg D. Obradovića 4, Novi Sad, 21000, Serbia

    Stevan Pilipović

Authors
  1. Teodor M. Atanacković
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  3. Stevan Pilipović
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Correspondence to Teodor M. Atanacković.

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Atanacković, T.M., Janev, M. & Pilipović, S. Wave equation in fractional Zener-type viscoelastic media involving Caputo–Fabrizio fractional derivatives. Meccanica 54, 155–167 (2019). https://doi.org/10.1007/s11012-018-0920-5

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