Fractional Order Two Temperature Thermo-Elastic Behavior of Piezoelectric Materials

Show more

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