Fractional Order Two Temperature Thermo-Elastic Behavior of Piezoelectric Materials

Essam Bassiouny, Refaat Sabry

Department of Mathematics, Faculty of Science and Humanitarian Studies, Salman Bin AbdulAziz University, Al Kharj, KSA.

DOI: 10.4236/jamp.2013.15017                

Abstract

A new mathematical model of time fractional order heat equation and fractional order boundary condition have been constructed in the context of the generalized theory of thermo piezoelasticity. The governing equations have been applied to a semi infinite piezoelectric slab. The Laplace transform technique is used to remove the time-dependent terms in the governing differential equations and the boundary condition. The solution of the problem is first obtained in the Laplace transform domain. Furthermore, a complex inversion formula of the transform based on a Fourier expansion is used to get the numerical solutions of the field equations which are represented graphically.

Keywords

Fractional Order; Two Temperature; Generalized Thermoelectricity; Weak Diffusion; Thermal Loading; Piezoelectric Materials; Ceramics

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	P. J. Chen and M. E. Gurtin, “On a Theory of Heat Conduction Involving Two Temperatures,” Zeitschrift für Angewandte Mathematik und Physik ZAMP, Vol. 19, No. 4, 1968, pp. 614-627. http://dx.doi.org/10.1007/BF01594969
[2]	P. J. Chen, M. E. Gurtin and W. O. Williams, “A Note on Non-Simple Heat Conduction,” Zeitschrift für Angewandte Mathematik und Physik ZAMP, Vol. 19, No. 6, 1968, pp. 969-970. http://dx.doi.org/10.1007/BF01602278
[3]	P. J. Chen, M. E. Gurtin and W. O. Williams, “On the Thermodynamics of Non-Simple Elastic Materials with Two Temperatures,” Zeitschrift für Angewandte Mathematik und Physik ZAMP, Vol. 20, No. 1, 1969, pp. 107-112. http://dx.doi.org/10.1007/BF01591120
[4]	B. A. Boley and I. S. Tolins, “Transient Coupled Thermoelastic Boundary Value Problems in the Half-Space,” Journal of Applied Mechanics, Vol. 29, No. 4, 1962, pp. 637-646. http://dx.doi.org/10.1115/1.3640647
[5]	W. E. Warren and P. J. Chen, “Wave Propagation in the Two-Temperature Theory of Thermoelasticity,” Acta Mechanica, Vol. 16, No. 1-2, 1973, pp. 21-33. http://dx.doi.org/10.1007/BF01177123
[6]	H. M. Youssef, “Theory of Two-Temperature Generalized Thermoelasticity,” IMA Journal of Applied Mathematics, Vol. 71, No. 3, 2006, pp. 383-390. http://dx.doi.org/10.1093/imamat/hxh101
[7]	M. Caputo, “Linear Models of Dissipation Whose Q Is almost Frequently Independent II,” Geophysical Journal International, Vol. 13, No. 5, 1967, pp. 529-539. http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.x
[8]	F. Mainardi, “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” In: A. Carpinteri and F. Mainardi, Eds., Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, 1997, pp. 291-348.
[9]	I. Podlubny, “Fractional Differential Equations,” Academic Press, New York, 1999.
[10]	R. Hilfer, “Application of Fraction Calculus in Physics,” World Scientific, Singapore, 2000.
[11]	M. Caputo and F. Mainardi, “Linear Model of Dissipation in Inelastic Solids,” Rivis Ta El Nuovo Cimento, Vol. 1, No. 2, 1971, pp. 161-198. http://dx.doi.org/10.1007/BF02820620
[12]	M. Caputo, “Vibrations of an Infinite Viscoelastic Layer with a Dissipative Memory,” The Journal of the Acoustical Society of America, Vol. 56, 1974, pp. 897-904. http://dx.doi.org/10.1121/1.1903344
[13]	R. L. Bagley and P. J. Torvik, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” Journal of Rheology, Vol. 27, 1983, pp. 201-210. http://dx.doi.org/10.1122/1.549724
[14]	R. C. Koeller, “Applications of Fractional Calculus to the Theory of Viscoelasticity,” Journal of Applied Mechanics, Vol. 51, No. 2, 1984, pp. 299-307. http://dx.doi.org/10.1115/1.3167616
[15]	Yu. A. Rossikhin and M. V. Shitikova, “Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Heredity Mechanics of Solids,” Applied Mechanics Reviews, Vol. 50, No. 1, 1997, pp. 15-67. http://dx.doi.org/10.1115/1.3101682
[16]	H. H. Sherief, A. El-Said and A. Abd El-Latief, “Fractional Order Theory of Thermoelasticity,” International Journal of Solids and Structures, Vol. 47, No. 2, 2010, pp. 269-275. http://dx.doi.org/10.1016/j.ijsolstr.2009.09.034
[17]	M. A. Ezzat and A. S. El-Karamany, “On the Fractional Thermo-Elasticity,” Mathematics and Mechanics of Solid, 2011.
[18]	I. Podlubny, “Fractional Differential Equations,” Academic Press, New York, 1999.
[19]	R. Kimmich, “Strange Kinetics, Porous Media, and NMR,” Journal of Chemical Physics, Vol. 284, 2002, pp. 243-285.
[20]	Y. Fujita, “Integrodifferential Equation which Interpolates the Heat Equation and Wave Equation (II),” Osaka Journal of Mathematics, Vol. 27, 1990, pp. 797-804.
[21]	Y. Fujita, “Integrodifferential Equation which Interpolates the Heat Equation and Wave Equation (I),” Osaka Journal of Mathematics, Vol. 27, 1990, pp. 309-321.
[22]	Y. Z. Povstenko, “Fractional Heat Conductive and Associated Thermal Stress,” Journal of Thermal Stresses, Vol. 28, No. 1, 2004, pp. 83-102. http://dx.doi.org/10.1080/014957390523741
[23]	Y. Z. Povstenko, “Theories of Thermal Stresses Based on Space-Time-Fractional Telegraph Equations,” Computers and Mathematics with Applications, Vol. 64, No. 10, 2012, pp. 3321-3328. http://dx.doi.org/10.1016/j.camwa.2012.01.066
[24]	C. Cattaneo, “Sur une Forme de I’equation de la Chaleur Eliminant le Paradoxe d’une Propagation Instantanee’,” Comptes Rendus de l’Académie des Sciences, Vol. 247, 1958, pp. 431-433.
[25]	M. A. Ezzat and A. S. El-Karamany, “Discontinuities in Generalized Thermoviscoelasticity under Four Theories,” Journal of Thermal Stresses, Vol. 27, No. 12, 2004, pp. 1187-1212. http://dx.doi.org/10.1080/014957390523598
[26]	M. A. Ezzat and A. S. El-Karamany, “Fractional Order Theory of a Prefect Conducting Thermoelastic Medium,” Canadian Journal of Physics, Vol. 89, No. 3, 2011, pp. 311-318. http://dx.doi.org/10.1139/P11-022
[27]	M. A. Ezzat and A. S. El-Karamany, “Theory of Fractional Order in Electro-Thermoelasticity,” European Journal of Mechanics A/Solids, Vol. 30, No. 4, 2011, pp. 491-500. http://dx.doi.org/10.1016/j.euromechsol.2011.02.004
[28]	A. S. El-Karamany and M. A. Ezzat, “Convolutional Variational Principle, Reciprocal and Uniqueness Theorems in Linear Fractional Two-Temperature Thermoelasticity,” Journal of Thermal Stresses, Vol. 34, No. 3, 2011, pp. 264-284. http://dx.doi.org/10.1080/01495739.2010.545741
[29]	H. Youssef, “Theory of Fractional Order Generalized Thermoelasticity,” Journal of Heat Transfer, Vol. 132, No. 6, 2010, pp. 1-7. http://dx.doi.org/10.1115/1.4000705
[30]	H. M. Youssef and E. Bassiouny, “Two-Temperature Generalized Thermopiezoelasticity for One Dimensional Problems—State Space Approach,” Computational Methods in Science and Technology, Vol. 14, No. 1, 2008, pp. 55-64.
[31]	G. Honig and U. Hirdes, “A Method of the Numerical Inversion of Laplace Transform,” Journal of Computational and Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132. http://dx.doi.org/10.1016/0377-0427(84)90075-X