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Meccanica
Published in: Meccanica 1-2/2019

17-11-2018

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

Authors: Teodor M. Atanacković, Marko Janev, Stevan Pilipović

Published in: Meccanica | Issue 1-2/2019

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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.
previous article Lattice Boltzmann simulation of nanofluid conjugate heat transfer in a wide microchannel: effect of temperature jump, axial conduction and viscous dissipation
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Footnotes
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 \(\pi /2\).
 
Literature
1.
Atanackovic TM, Pilipovic S, Zorica D (2018) Properties of the Caputo–Fabrizio fractional derivative and its distributional settings. Fract Calc Appl Anal 21:29–44MathSciNetCrossRefMATH
2.
Bagley RL, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30(1):133–155ADSCrossRefMATH
3.
4.
5.
Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular Kernel. Prog Fract Differ Appl 1(2):73–85
6.
Caputo M, Fabrizio M (2015) Properties of a new fractional derivative without singular Kernel. Progr Fract Differ Appl 1(2):87–92
7.
8.
9.
El-Karamany AS, Ezzat MA (2015) Two-temperature Green–Naghdi theory of type III in linear thermoviscoelastic anisotropic solid. Appl Math Modell 39:2155–2171MathSciNetCrossRef
10.
El-Karamany AS, Ezzat MA (2011) Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity. J Therm Stresses 34:264–284CrossRef
11.
Eringen AC (2002) Nonlocal continuum field theories. Springer, New YorkMATH
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 36:2417–2442CrossRefMATH
13.
Ezzat MA, El-Bary AA (2016) Unified fractional derivative models of magneto-thermo-viscoelasticity theory. Arch Mech 68:285–308MathSciNetMATH
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 40:6306–6315MathSciNetCrossRefMATH
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 99:107–116MathSciNetCrossRefMATH
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
17.
18.
Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity. Imperial College Press-World Scientific, LondonCrossRefMATH
19.
Mainardi F, Spada G (2011) Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur Phys J Spec Top 193:133–160CrossRef
20.
21.
Ortigueira MD, Tenreiro Machado JA (2018) A critical analysis of the Caputo–Fabrizio operator. Commun Nonlinear Sci Numer Simul 59:608–611ADSMathSciNetCrossRef
22.
Pallu De LA Barriere R (1980) Optimal control theory: a course in automatic control theory. Dower Publications Inc, New YorkMATH
23.
Reed M, Simon B (1980) Methods of modern mathematical physics, I: functional anaalysis. Academic Press, New York
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 1–36CrossRef
25.
Sherief HH, El-Sayed AMA, Abd El-Latief AM (2010) Fractional order theory of thermoelasticity. Int J Solids Struct 47:269–275CrossRefMATH
26.
Stankovic B, Atanackovic TM (2004) On an inequality arising in fractional oscillator theory. Fract Calc Appl Anal 7:11–20MathSciNetMATH
27.
28.
Vladimirov VS (1979) Generalized functions in mathematical physics. Mir Publishers, MoscowMATH
29.
von Ende S, Lion A, Lammering L (2011) On the thermodynamically consistent fractional wave equation for viscoelastic solids. Acta Mech 221:1–10CrossRefMATH
Metadata
Title
Wave equation in fractional Zener-type viscoelastic media involving Caputo–Fabrizio fractional derivatives
Authors
Teodor M. Atanacković
Marko Janev
Stevan Pilipović
Publication date
17-11-2018
Publisher
Springer Netherlands
Published in
Meccanica / Issue 1-2/2019
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI
https://doi.org/10.1007/s11012-018-0920-5

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