Thermoelastic Interaction in A Spherically Symmetric Hollow Sphere via Three-Phase Lag and Memory Effect

Subject Areas : Mechanics of Solids

Jitendra Patil ¹ , Chandrakant Jadhav ² , Nitin Chandel ³ , Vinod Varghese ⁴

1 - K.B.C.North Maharashtra University
2 - K.B.C.North Maharashtra University
3 - Gondwana University Gadchiroli
4 -

Received: 2024-06-30 Accepted : 2024-11-21 Published : 2024-11-30

Keywords: Thermoelasticity , fractional-order, three-phase-lags, nonsimple media, hollow sphere, spherically symmetry system,

Abstract :

This model considers the impact of microscopic structure on a nonsimple thermoelastic hollow sphere with a spherically symmetry exposed to thermal shock on the inner and outer curved surface. The influence of temperature discrepancy factor and time-phase lag in the context of memory-dependent derivative is examined. The study employs a numerical Laplace inversion method to provide numerical solutions for various physical domains, illustrating transient reactions like temperature, displacement, and stress graphically. The study indicates that the memory-dependent derivative accurately represents the memory effect, which is the rate of change influenced by the previous state, potentially aiding in better heat management of nonsimple media. The physical perspectives presented in this article may prove useful in the creation of new materials to simulate nanoscale heat transfer issues in a variety of devices. This study aims to provide future researchers with a comprehensive understanding of nonsimple three-phase lags thermoelasticity, including a detailed analysis of the memory effect.

References:

[1] M. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., Vol. 27, No. 3, 1956, pp. 240-253, 1956. DOI:10.1063/1.1722351
[2] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, Vol. 15, No. 5, pp. 299-309, 1967. doi:10.1016/0022-5096(67)90024-5
[3] R. Dhaliwal and H. Sherief, Generalized thermoelasticity for anisotropic media, Q. Appl. Math., Vol. 33, pp. 1-8, 1980. DOI: 43637007
[4] A. E. Green and N. Laws, On the entropy production inequality, Arch Ration Mech Anal, Vol. 45, No. 1, 1972, pp. 47-53. doi:10.1007/BF00253395
[5] A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elast., Vol. 2, No. 1, pp. 1-7, 1972. DOI: 10.1007/BF00045689
[6] A. E. Green and P. M. Naghdi, Thermoelasticity with out energy dissipation, Journal of Elasticity, Vol. 31, No.3, pp. 189-208, 1993. DOI: 10.1007/BF00044969
[7] D. Y. Tzou, Macro-to microscale heat transfer: the lagging behavior, Taylor and Francis, Washington, 1996
[8] D. Y. Tzou, A unified approach for heat conduction from macro- to micro- scales, J. Heat Transf., Vol. 117, No. 1, 1995, pp. 8-16. DOI: 10.1115/1.2822329
[9] D. Y. Tzou, Experimental support for the lagging behavior in heat propagation, J. Thermophys. Heat Transf., Vol. 9, pp. 686-693, 1995. DOI: 10.2514/3.725
[10] S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Therm. Stresses, Vol. 30, No. 3, pp. 31–238, 2007. DOI: 10.1080/01495730601130919
[11] A. Compte, and R. Metzler, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A Math. Gen., Vol. 30 No. 21, pp. 7277-7289, 1997. DOI: 10.1088/0305-4470/30/21/006
[12] H. M. Youssef, Theory of fractional order generalized thermoelasticity, J. Heat Transf., Vol. 132, No. 6, pp. 061301, 2010. DOI: 10.1115/1.4000705
[13] M. A. Ezzat and A. S. El-Karaman, Fractional-order heat conduction law in magnetothermoelasticity involving two temperatures, Z. fur Angew. Math. Phys., Vol. 62 No. 5, pp. 937-952, 2011. DOI: 10.1007/s00033-011-0126-3
[14] M. A. Ezzat and A. S. El-Karaman, Fractional thermoelectric viscoelastic materials, J. Appl. Polym. Sci., Vol. 124 No. 3, pp. 2187-2199, 2012. DOI: 10.1002/app.35243
[15] H. H. Sherief, A. M. A. El-Sayed, and A. M. A. El-Latief, Fractional-order theory of thermoelasticity, Int. J. Solids Struct., Vol. 47 No. 2, pp. 269-275, 2010. DOI: 10.1016/j.ijsolstr.2009.09.034
[16] H. H. Sherief, and A. M. A. El-Latief, Effect of variable thermal conductivity on a half-space under the fractional-order theory of thermoelasticity, Int. J. Mech. Sci., Vol. 74, pp. 185-189, 2013. DOI: 10.1016/j.ijmecsci.2013.05.016
[17] K. Diethelm, The Analysis of Fractional Differential Equations: An application-oriented exposition using differential operators of Caputo Type, Springer, Berlin, 2010. DOI: 10.1007/978-3-642-14574-2
[18] M. Caputo, Linear models of dissipation whose Q is almost frequency independent—II, Geophysical Journal International, Vol. 13 No. 5, pp. 529-539, 1967.
[19] Y. J. Yu, W. Hu, and X. G. Tian, A novel generalized thermoelasticity model based on memory dependent derivative, Int J Eng Sci., Vol. 81, pp. 123–134, 2014.
[20] S. Mondal, P. Pal, and M. Kanori, Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative, Acta Mech., Vol. 230, pp. 179–199, 2019. DOI: 10.1007/s00707-018-2307-z
[21] D. Ghosh, A. K. Das, and A. Lahiri, Modelling of a three dimensional thermoelastic half space with three phase lags using memory dependent derivative, Int J Appl Comp Math., Vol. 5, pp. 154, 2019. DOI: 10.1007/s40819-019-0731-y
[22] M. A. Fahmy, A new LRBFCM-GBEM modeling algorithm for general solution of time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues, Numer. Heat Transf. A, Vol. 75, pp. 616–626, 2019. DOI: 10.1080/10407782.2019.1608770
[23] M. A. Fahmy, Boundary element algorithm for nonlinear modeling and simulation of three-temperature anisotropic generalized micropolar piezothermoelasticity with memory-dependent derivative, Int J Appl Mech., Vol. 12, pp. 2050027, 2020.
[24] M. A. Ezzat and A. A. El-Bary, 2015, Memory dependent derivatives theory of thermo-viscoelasticity involving two temperatures, J Mech Sci Technol, Vol. 29, No. 10, pp. 4273-4279, 2015. DOI: 10.1007/s12206-015-0924-1
[25] M. A. Ezzat and A. S. El-Karaman, and A. A. El-Bary, Generalized thermo-viscoelasticity with memory-dependent derivatives, Int. J. Mech. Sci., Vol. 89, pp. 470-475, 2014. DOI: 10.1016/j.ijmecsci.2014.10.006
[26] H. H. Sherief and A. Abd El-Latief, Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity, Int. J. Mech. Sci., Vol. 74, pp. 185-189, 2013. DOI: 10.1016/j.ijmecsci.2013.05.016
[27] M. A. Ezzat and A. A. El-Bary, Two-temperature theory of magneto-thermo-viscoelasticity with fractional derivative and integral orders heat transfer, J. Electromagn. Waves Appl., Vol. 28, pp. 1985-2004, 2014. DOI: 10.1080/09205071.2014.953639
[28] A. E. Green and P.M. Naghdi, On undamped heat waves in an elastic solid, J. Therm. Stresses, Vol. 15, No. 2, pp. 253–264, 1992. DOI: 10.1080/01495739208946136
[29] A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elast., Vol. 31, pp. 189–208, 1993. DOI: 10.1007/BF00044969
[30] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., Vol. 19, pp. 559-577, 1968. DOI: 10.1007/BF01594969
[31] S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Therm. Stresses, Vol. 30, No. , pp. 31–238, 2007. DOI: 10.1080/01495730601130919
[32] M. A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., Vol. 27, No. 3, pp. 240–253, 1956. DOI: 10.1063/1.1722351
[33] K. Diethelm, The analysis of fractional differential equations, Springer Berlin, Heidelberg, 2010. DOI: 10.1007/978-3-642-14574-2
[34] J. L. Wang, H.F. Li, Surpassing the fractional derivative: concept of the Memory dependent derivative, Comput. Math. Appl., Vol. 62, pp. 1562–1567, 2011.
[35] L. Debnath and D. Bhatta, Integral transforms and their applications, Chapman and Hall/CRC, New York, 2006. DOI: 10.1201/9781420010916
[36] G. Honig and U. Hirdes, A method for the numerical inversion of the Laplace transform, J Comput ApplMath, Vol. 10, pp. 113–132, 1984.
[37] F. Durbin, Numerical inversion of Laplace transforms: An efficient improvement to Dubner and Abate's method, The Computer Journal, Vol. 17, No. 4, pp. 371–376, 1974. DOI: 10.1093/comjnl/17.4.371
[38] A. E. Abouelregal, On Green and Naghdi thermoelasticity model without energy dissipation with higher order time differential and phase-lags, J. Appl. Comput. Mech., Vol. 6, No. 3, pp. 445-456, 2020.
[39] A. E. Abouelregal, Generalized thermoelasticity for an isotropic solid sphere in dual-phase-lag of heat transfer with surface heat flux, Int. J. Comput. Methods Eng. Sci., Vol. 12, No. 2, pp. 1550-2287, 2011. DOI: 10.1080/15502287.2010.548172