Fractional calculus in one-dimensional isotropic thermo-viscoelasticity

Magdy A. Ezzat; Ahmed S. El-Karamany; Alaa A. El-Bary; Mohsen A. Fayik

doi:10.1016/j.crme.2013.04.001

Fractional calculus in one-dimensional isotropic thermo-viscoelasticity

Magdy A. Ezzat ¹ ; Ahmed S. El-Karamany ² ; Alaa A. El-Bary ³ ; Mohsen A. Fayik ¹

¹ Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
² Department of Mathematical and Physical Sciences, Nizwa University, Nizwa 611, P.O. Box 1357, Oman
³ Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt

Comptes Rendus. Mécanique, Volume 341 (2013) no. 7, pp. 553-566.

Résumé

A new fractional relaxation operator is derived using the methodology of fractional calculus. The governing coupled fractional differential equations in the frame of the thermo-viscoelasticity with fractional order heat transfer are applied to the one-dimensional problem with heat sources. Laplace transform and state space techniques are used to get the solution. According to the numerical results and its graphs, conclusion about the new theory of thermo-viscoelasticity has been constructed. The theories of coupled thermo-viscoelasticity and of generalized thermo-viscoelasticity with one relaxation time follow as limit cases.

Reçu le : 2011-07-12
Accepté le : 2013-04-03
Publié le : 2013-04-30

DOI : 10.1016/j.crme.2013.04.001

Mots clés : Thermo-viscoelasticity, Fractional relaxation function, Non-Fourier heat conduction, State space approach, Laplace transforms, Fractional calculus

Affiliations des auteurs :

Magdy A. Ezzat ¹ ; Ahmed S. El-Karamany ² ; Alaa A. El-Bary ³ ; Mohsen A. Fayik ¹

¹ Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
² Department of Mathematical and Physical Sciences, Nizwa University, Nizwa 611, P.O. Box 1357, Oman
³ Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt

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     title = {Fractional calculus in one-dimensional isotropic thermo-viscoelasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {553--566},
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     volume = {341},
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     year = {2013},
     doi = {10.1016/j.crme.2013.04.001},
     language = {en},
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Magdy A. Ezzat; Ahmed S. El-Karamany; Alaa A. El-Bary; Mohsen A. Fayik. Fractional calculus in one-dimensional isotropic thermo-viscoelasticity. Comptes Rendus. Mécanique, Volume 341 (2013) no. 7, pp. 553-566. doi : 10.1016/j.crme.2013.04.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.04.001/

Bibliographie
Cité par

[1] A.M. Freudenthal Effect of rheological behavior on thermal stresses, J. Appl. Phys., Volume 25 (1954), pp. 1110-1117

[2] Y.C. Fung Foundation of Solid Mechanics, Prentice–Hall, Englewood Cliffs, NJ, 1980

[3] B. Gross Mathematical Structure of the Theories of Viscoelasticity, Hermann, Paris, 1953

[4] A.J. Staverman; F. Schwarzl (H.A. Stuart, ed.), Die Physik der Hochpolymeren, vol. 4, Springer-Verlag, 1956 (Chap. 1)

[5] T. Alfrey; E.F. Gurnee (F.R. Erich, ed.), Rheology: Theory and Applications, vol. 1, Academic Press, New York, 1956

[6] J.D. Ferry Viscoelastic Properties of Polymers, J. Wiley and Sons, New York, 1970

[7] A.A. Ilioushin; B.E. Pobedria Fundamentals of the Mathematical Theory of Thermal Visco-Elasticity, Nauka, Moscow, 1970

[8] B.E. Pobedria Coupled problems in continuum mechanics, J. Durability Plasticity, Volume 2 (1984), pp. 224-253

[9] R.M. Christensen Theory of Viscoelasticity. An Introduction, Academic Press, London, 1982

[10] C. Cattaneo Sur une forme de lʼéquation de la chaleur éliminant le paradoxe dʼune propagation instantanée, C. R. Acad. Sci. Paris, Volume 247 (1958), pp. 431-433

[11] H. Lord; Y. Shulman A generalized dynamical theory of thermo-elasticity, J. Mech. Phys. Solids, Volume 15 (1967), pp. 299-309

[12] R. Dhaliwal; H. Sherief Generalized thermoelasticity for anisotropic media, Quart. Appl. Math., Volume 33 (1980), pp. 1-8

[13] J. Ignaczak Generalized thermoelasticity and its applications (R.B. Hetnarski, ed.), Thermal Stresses, vol. III, Elsevier, New York, 1989, pp. 279-354

[14] D.S. Chandrasekharaiah Hyperbolic thermoelasticity. A review of recent literature, Appl. Mech. Rev., Volume 51 (1998), pp. 705-729

[15] R.B. Hetnarski; J. Ignaczak Nonclassical dynamical thermoelasticity, Int. J. Solids Struct., Volume 37 (2000), pp. 215-224

[16] J. Ignaczak; M. Ostoja-Starzewski Thermoelasticity with Finite Wave Speeds, Oxford University Press, 2009

[17] A.S. El-Karamany; M.A. Ezzat Thermal shock problem in generalized thermo-viscoelasticity under four theories, Int. J. Eng. Sci., Volume 42 (2004), pp. 649-671

[18] M.A. Ezzat; M.I. Othman; A.S. El-Karamany State space approach to generalized thermo-viscoelasticity with two relaxation times, Int. J. Eng. Sci., Volume 40 (2002), pp. 283-302

[19] M.A. Ezzat; A.S. El-Karamany On uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with thermal relaxation, Can. J. Phys., Volume 81 (2003), pp. 823-833

[20] A.S. El-Karamany; M.A. Ezzat Discontinuities in generalized thermo-viscoelasticity under four theories, J. Therm. Stress., Volume 27 (2004), pp. 1187-1212

[21] A.S. El-Karamany; M.A. Ezzat On the boundary integral formulation of thermo-viscoelasticity theory, Int. J. Eng. Sci., Volume 40 (2002), pp. 1943-1956

[22] A.S. El-Karamany; M.A. Ezzat Uniqueness and reciprocal theorems in linear micropolar electro-magnetic thermoelasticity with two relaxation times, Mech. Time-Depend. Mater., Volume 13 (2009), pp. 93-115

[23] M.A. Ezzat The relaxation effects of the volume properties of electrically conducting viscoelastic material, Sci. Eng., B, Solid-State Mater. Adv. Technol., Volume 130 (2006), pp. 11-23

[24] R. Gorenflo; F. Mainardi Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997

[25] K.S. Miller; B. Ross An Introduction to the Fractional Integrals and Derivatives Theory and Applications, John Wiley & Sons Inc., New York, 1993

[26] S.G. Samko; A.A. Kilbas; O.I. Marichev Fractional Integrals and Derivatives Theory and Applications, Gordon & Breach, Longhorne, PA, 1993

[27] K.B. Oldham; J. Spanier The Fractional Calculus, Academic Press, New York, 1974

[28] M. Caputo; F. Mainardi Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), Volume 1 (1971), pp. 161-198

[29] R.C. Koeller Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech., Volume 51 (1984), pp. 299-307

[30] R.L. Bagley; P.J. Torvik On the fractional calculus model of viscoelastic behavior, J. Rheol., Volume 30 (1986), pp. 133-155

[31] Yu.A. Rossikhin; M.V. Shitikova Applications of fractional calculus to dynamic problems of linear and nonlinear heredity mechanics of solids, Appl. Mech. Rev., Volume 50 (1967), pp. 15-67

[32] S. Grimnes; O.G. Martinsen Bioimpedance and Bioelectricity Basics, Academic Press, San Diego, CA, 2000

[33] R.S. Lakes Viscoelastic Solids, CRC Press, Boca Raton, FL, 1999

[34] G. Honig; U. Hirdes A method for the numerical inversion of the Laplace transform, J. Comput. Appl. Math., Volume 10 (1984), pp. 113-132

[35] R. Gorenflo; F. Mainardi Time-fractional derivatives in relaxation processes: a tutorial survey, Fract. Calc. Appl. Anal., Volume 10 (2007), pp. 283-299

[36] I. Podlubny Fractional Differential Equations, Academic Press, New York, 1999

[37] Y.Z. Povstenko Fractional heat conductive and associated thermal stress, J. Therm. Stress., Volume 28 (2005), pp. 83-102

[38] Y.Z. Povstenko Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses, Mech. Res. Comm., Volume 37 (2010), pp. 436-440

[39] H.H. Sherief; A.M. El-Sayed; A. Abd El-Latief Fractional order theory of thermoelasticity, Int. J. Solids Struct., Volume 47 (2010), pp. 269-275

[40] H.M. Youssef Theory of fractional order generalized thermoelasticity, J. Heat Transfer, Volume 132 (2010), pp. 1-7

[41] M.A. Ezzat Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Physica B, Volume 405 (2010), pp. 4188-4194

[42] M.A. Ezzat Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B, Volume 406 (2011), pp. 30-35

[43] G. Jumarie Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time, application to Mertonʼs optimal portfolio, Comput. Math. Appl., Volume 59 (2010), pp. 1142-1164

[44] A.S. El-Karamany; M.A. Ezzat Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity, J. Therm. Stress., Volume 34 (2011), pp. 264-284

[45] M.A. Ezzat; A.S. El-Karamany Fractional order theory of a perfect conducting thermoelastic medium, Can. J. Phys., Volume 89 (2011), pp. 311-318

[46] M.A. Ezzat; A.S. El-Karamany Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures, Z. Angew. Math. Phys., Volume 62 (2011), pp. 937-952

[47] L.Y. Bahar; R.B. Hetnarski State space approach to thermoelasticity, J. Therm. Stress., Volume 1 (1978), pp. 135-145

[48] M.A. Ezzat State space approach to solids and fluids, Can. J. Phys. Rev., Volume 86 (2008), pp. 1241-1250

[49] F. Mainardi Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, Volume 7 (1996), pp. 1461-1477

[50] G. Doetsch Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, 1974

[51] F. Mainardi; R. Gorenflo On Mittag-Leffler-type function in fractional evolution processes, J. Comput. Appl. Math., Volume 118 (2000), pp. 283-299

[52] F. Mainardi Fractional calculus: some basic problems in continuum and statistical mechanics (Carpinteri; F. Mainardi, eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, New York, 1997, pp. 291-348

[53] A. Erdelyi; W. Magnus; W. Oberhetinger; F.G. Tricomi Higher Transcendental Functions, Bateman Project, vol. 3, McGraw-Hill, New York, 1955 (Chap. 18)

[54] M. Caputo Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Astron. Soc., Volume 13 (1967), pp. 529-539

[55] M. Biot Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., Volume 27 (1956), pp. 240-253

[56] M.A. Ezzat; A.S. El-Karamany; A.A. Smaan State space formulation to generalized thermoviscoelasticity with thermal relaxation, J. Therm. Stress., Volume 24 (2001), pp. 823-846

[57] M.I. Othman; M.A. Ezzat; S.A. Zaki; A.S. El-Karamany Generalized thermo-viscoelastic plane waves with two relaxation times, Int. J. Eng. Sci., Volume 40 (2002), pp. 1329-1347

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