21. Fractional order magneto‑thermoelasticity in a rotating media with one relaxation time

M. Bachher1, N. Sarkar2

1Media Girl’s High School, Gobardanga, 24-Pgs (N), West Bengal, India

2Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700 009, India

2Corresponding author

E-mail: 1mitali.ju08@gmail.com, 2nantu.math@gmail.com

(Received 14 April 2016; accepted 20 May 2016)

Abstract. The theory of generalized thermoelasticity based on the heat conduction equation with the Caputo time-fractional derivative is used to study magneto-thermoelastic response of a homogeneous isotropic two-dimensional rotating elastic half-space solid. The solution for the physical variables is obtained using the normal mode analysis together with an eigenvalue approach technique. Numerical computations are carried out for a hypothetical copper like material the numerical results are illustrated graphically. Some comparisons have been made in the graphical results to show the effect of fractional parameter, magnetic field and the rotation on the field variables.

Keywords: non-Fourier heat conduction, Caputo time-fractional derivative, rotating media, normal mode analysis, eigenvalue approach.

References

[1]        Biot M. Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics. Vol. 27, 1956, p. 240‑253.

[2]        Dreyer W., Struchtrup H. Heat pulse experiments revisited. Continuum Mechanics and Thermodynamics, Vol. 5, 1993, p. 3‑50.

[3]        Ignaczak J., Ostoja‑Starzewski M. Thermoelasticity with Finite Wave Speeds. Oxford University Press, New York, 2010.

[4]        Caputo M., Mainardi F. A. A new dissipation model based on memory mechanism. Pure and Applied Geophysics, Vol. 91, 1971, p. 134‑147.

[5]        Caputo M., Mainardi F. A. Linear models of dissipation in anelastic solids. Rivis ta del Nuovo Cimento, Vol. 1, 1971, p. 161‑198.

[6]        Caputo M. Vibrations of an infinite viscoelastic layer with a dissipative memory. The Journal of the Acoustical Society of America, Vol. 56, 1974, p. 897‑904.

[7]        Rabotnov Yu N. Creep of Structural Elements. Nauka, Moscow, 1966, (in Russian).

[8]        Mainardi F. Applications of fractional calculus in mechanics. Transforms Method and Special Functions, Bulgarian Academy of Sciences, Sofia, 1998, p. 309‑334.

[9]        Kimmich R. Strange kinetics, porous media and NMR. The Journal of Chemical Physics, Vol. 284, 2002, p. 253‑285.

[10]     Povstenko Y. Z. Fractional heat conduction equation and associated thermal stress. Journal of Thermal Stresses, Vol. 28, 2005, p. 83‑102.

[11]     Caputo M. Linear model of dissipation whose Q is almost frequency independent – II Geophysical. Royal Astronomical Society, Vol. 13, 1967, p. 529‑539.

[12]     Youssef H. Theory of fractional order generalized thermoelasticity. Journal of Heat Transfer, Vol. 132, 2010, p. 1‑7.

[13]     Youssef H., Al‑Lehaibi E. Fractional order generalized thermoelastic half space subjected to ramp type heating. Mechanics Research Communications, Vol. 37, 2010, p. 448‑452.

[14]     Sherief H. H., El‑Sayed A., El‑Latief A. Fractional order theory of thermoelasticity. International Journal of Solids and Structures, Vol. 47, 2010, p. 269‑275.

[15]     Ezzat M. A., Fayik M. A. Fractional order theory of thermoelastic diffusion. Journal of Thermal Stresses, Vol. 34, 2011, p. 851‑872.

[16]     Ezzat M. A., El-Karamany A. S. Fractional order theory of a perfect conducting thermoelastic medium. Canadian Journal of Physics, Vol. 89, 2011, p. 311‑318.

[17]     Ezzat M. A., El‑Karamany A. S. Theory of fractional order in electro-thermo-elasticity. European Journal of Mechanics – A/Solids, Vol. 30, 2011, p. 491‑500.

[18]     Bachher M., Sarkar N., Lahiri A. Generalized thermoelastic infinite medium with voids subjected to an instantaneous heat sources with fractional derivative heat transfer. International Journal of Mechanical Sciences, Vol. 89, 2014, p. 84‑91.

[19]     Bachher M., Sarkar N., Lahiri A. Fractional order thermoelastic interactions in an infinite voids material due to distributed time-dependent heat sources. Meccanica, Vol. 50, 2015, p. 2167‑2178.

[20]     Kaliski S., Nowacki W. Combined elastic and electro-magnetic waves produced by thermal shock in the case of a medium of finite electric conductivity. International Journal of Engineering Science, Vol. 1, 1963, p. 163‑175.

[21]     Massalas C., Dalamangas A. Coupled magneto-thermoelastic problem in elastic half-space having finite conductivity. International Journal of Engineering Science, Vol. 21, 1983, p. 991‑999.

[22]     Paria G. On magneto-thermoelastic plane waves. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 58, 1962, p. 527‑531.

[23]     Paria G. Magneto-elasticity and magneto-thermoelasticity. Advances in Applied Mechanics, Vol. 10, 1967, p. 73‑112.

[24]     Ezzat M. A., Youssef H. Generalized magneto-thermoelasticity in a perfectly conducting medium. International Journal of Solids and Structures, Vol. 42, 2005, p. 6319‑6334.

[25]     Ezzat M. A., Youssef H. Generation of generalized magneto-thermoelastic waves by thermal shock in a half-space of finite conductivity. Italian Journal of Pure and Applied Mathematics, Vol. 19, 2005, p. 9‑26.

[26]     Youssef H. Generalized magneto-thermoelasticity in a conducting medium with variable material properties. Applied Mathematics and Computation, Vol. 173, 2006, p. 822‑833.

[27]     Baksi A., Bera R. K., Debnath L. A study of magneto-thermoelastic problems with thermal relaxation and heat sources in a three-dimensional infinite rotating elastic medium. International Journal of Engineering Science, Vol. 43, 2005, p. 1419‑1434.

[28]     Choudhury Roy S. K., Debnath L. Magneto-thermoelastic plane waves in rotating media. International Journal of Engineering Science, Vol. 21, 1983, p. 155‑163.

[29]     Choudhury Roy S. K., Debnath L. Magneto-thermoelastic plane waves in infinite rotating media. Journal of Applied Mechanics, Vol. 50, 1983, p. 283‑287.

[30]     Choudhury Roy S. K., Chattopadhyay M. Magneto-viscoelastic plane waves in rotating media in the generalized thermoelasticity II. International Journal of Mathematics and Mathematical Sciences, Vol. 11, 2005, p. 1819‑1834.

[31]     Choudhury Roy S. K., Chattopadhyay M. Magneto-elastic plane waves in rotating media in thermoelasticity of type II (G-N model). International Journal of Mathematics and Mathematical Sciences, Vol. 71, 2004, p. 3917‑3929.

[32]     Akbarzadeh A. H., Babaei M. H., Chen Z. T. Thermopiezoelectric analysis of a functionally graded piezoelectric medium. International Journal of Applied Mechanics, Vol. 3, 2011, p. 47‑68.

[33]     Brischetto S., Carrera E. Thermomechanical effect in vibration analysis of one-layered and two‑layered plates. International Journal of Applied Mechanics, Vol. 3, 2011, p. 161‑185.

[34]     Xiong Q. L., Tian X. G. Transient magneto-thermoelastic response for a semi-infinite body with voids and variable material properties during thermal shock. International Journal of Applied Mechanics, Vol. 3, 2011, p. 891‑902.

[35]     Choudhury Roy S. K. Magneto-thermoelastic plane waves in infinite rotating media. International Journal of Engineering Science, Vol. 22, 1984, p. 519‑530.

[36]     Othman M. I. A., Zidan M. E. M., Hilal M. I. M. The influence of gravitational field and rotation on thermoelastic solid with voids under Green-Naghdi theory. Journal of Physics, Vol. 2, 2013, p. 22‑34.

[37]     Othman M. I. A., Zidan M. E. M., Hilal M. I. M. Effect of rotation on thermoelastic material with voids and temperature dependent properties of type III. Journal of Thermoelasticity, Vol. 1, 2013, p. 1‑11.

[38]     Sarkar N., Lahiri A. The effect of gravity field on the plane waves in a fiber-reinforced two‑temperature magneto-thermoelastic medium under Lord-Shulman theory. Journal of Thermal Stresess, Vol. 36, 2013, p. 895‑914.

[39]     Sarkar N. Analysis of magneto-thermoelastic response in a fiber-reinforced elastic solid due to hydrostatic initial stress and gravity field. Journal of Thermal Stresses, Vol. 37, 2014, p. 1‑18.

Cite this article

Bachher M., Sarkar N. Fractional order magneto‑thermoelasticity in a rotating media with one relaxation time. Mathematical Models in Engineering, Vol. 2, Issue 1, 2016, p. 56‑68.

 

Mathematical Models in Engineering. June 2016, Volume 2, Issue 1

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