Two-Dimensional Generalized Magneto-Thermo-Viscoelasticity Problem for a Spherical Cavity with One Relaxation Time Using Fractional Derivative

Year 2022, Volume: 25 Issue: 2, 89 - 97, 01.06.2022

Satish Khavale Kishor Gaikwad

https://doi.org/10.5541/ijot.1035396

Cited By: 1

Abstract

The present paper is aimed to studying the two-dimensional generalised magneto-thermo-viscoelasticity problem for a spherical cavity with one relaxation time using fractional derivative. The formulation is applied to generalised thermoelasticity based on the theory of generalised thermoelastic diffusion with one relaxation time. The spherical cavity of the solid surface is assumed to be traction free and subjected to both heating and an external magnetic field. The Laplace transform technique is used to obtain the general solution. The inverse Laplace transform is carried out using a numerical inversion method based. The temperature, displacement, and stresses are obtained and represented graphically with the help of Mathcad software.

Keywords

Fractional order, spherical cavity, electromagnetic field, magneto-thermo-viscoelasticity

Supporting Institution

Chhatrapati Shahu Maharaj Research, Training and Human Development Institute (SARTHI).

Project Number

CMSRF - 2019

References

• M. A. Biot, “Thermoelasticity and Irreversible Thermodynamics,” J. Appl. Phys., 27, 240-253, 1956.
• H. W. Lord, Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” J. Mech. Phys. Solids., 15, 299-307, 1967.
• M. Caputo, “Vibrations on an Infinite Viscoelastic Layer With a Dissipative Memory,” J. Acoust. Soc. Amer., 56, 897-904, 1974.
• M. A. Ezzat, “ Generation of generalized magnetother-moelastic waves by thermal shock in a perfectly conducting half-space,” J. Therm. Stress., 20, 617-633, 1997.
• M. A. Ezzat, “Magneto-thermoelasticity with thermo-electric properties and fractional derivative heat transfer,” Phys. B., 406, 30-35, 2011.
• S. K. Roychoudhuri, S. Banerjee, “ Magnetothermo-elastic interactions in an infinite viscoelastic cylinder of temperature rate dependent material subjected to a periodic loading,” Int. J. Eng. Sci., 36, 635-643, 1998.
• S. K. Roychoudhuri, S. Mukhopadhyay, “ Effect of rotation and relaxation times on plane waves in generalized thermo-viscoelasticity,” Int. J. Math. Math. Sci., 23, 497-505, 2000.
• H. H. Sherief, A. El-Sayed, A. A. El-Latief, “ Fractional order theory of thermoelasticity,” Int. J. Solids Struct., 47, 269-275, 2010.
• Y. Z. Povstenko, “ Thermoelasticity that uses fractional heat conduction equation,” J. Math. Sci., 162, 296-305, 2009.
• S. Deswal, K. K. Kalkal, “ A two-dimensional genera-lized electro-magneto-thermo-viscoelastic problem for a half-space with diffusion,” Int. J. Therm. Sci., 50, 749-759, 2011.
• A. M. Zenkour, D. S. Mashat, A. E. Abouelregal, “Generalized thermodiffusion for an unbounded body with a spherical cavity subjected to periodic loading,” J. Mech. Sci. Tech., 26, 749-757, 2012.
• K. R. Gaikwad, K. P. Ghadle, “Quasi-static thermo-elastic problem of an infinitely long circular cylinder,” Journal of the Korean Society for Industrial and Applied Mathematics., 14, 141-149, 2010.
• K. R. Gaikwad, K. P. Ghadle, “On a certain thermo-elastic problem of temperature and thermal stresses in a thick circular plate,” Australian Journal of Basic and Applied Sciences., 6,34-48, 2012.
• K. R. Gaikwad, K. P. Ghadle, “ Nonhomogeneous heat conduction problem and its thermal deflection due to internal heat generation in a thin hollow circular disk,” Journal of Thermal stresses., 35, 485-498, 2012.
• K. R. Gaikwad, “Analysis of thermoelastic deforma-tion of a thin hollow circular disk due to partially distributed heat supply,” Journal of Thermal stresses., 36, 207-224, 2013.
• H. Sherief, A. M. Abd El-Latief, “Application of fractional order theory of thermoelasticity to a 1d problem for a half-space,” ZAMM., 2, 1-7, 2013.
• W. Raslan, “Application of fractional order theory of thermoelasticity to a 1D problem for a cylindrical cavity,” Arch. Mech., 66, 257-267, 2014.
• K. K. Kalkal, S. Deswal, “Analysis of vibrations in fractional order magneto-thermoviscoelasticity with diffusion,” J. Mech., 30, 383-394, 2014.
• E. M. Hussain, “Fractional order thermoelastic problem for an infinitely long solid circular cylinder,” Journal of Thermal Stresses., 38, 133-145, 2015.
• W. Raslan, “Application of fractional order theory of thermoelasticity in a thick plate under axisymmetric temperature distribution,” Journal of Thermal Stresses., 38, 733-743, 2015.
• K. R. Gaikwad, “Two-dimensional steady-state temp-erature distribution of a thin circular plate due to uniform internal energy generation,” Cogent Mathematics., .3, 1-10, 2016.
• J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds,” Journal of Thermal Stresses, 39, 220-230, 2016.
• K. R. Gaikwad, “Axi-symmetric thermoelastic stress analysis of a thin circular plate due to heat generation,” International Journal of Dynamical Systems and Differential Equations., 9, 187-202, 2019.
• K. R. Gaikwad, S. G. Khavale, “Time fractional heat conduction problem of a thin hollow circular disk and its thermal deflection,” Easy Chair Preprint., 1672, 1-11, 2019
• S. G. Khavale, K. R. Gaikwad, “Generalized theory of magneto-thermo-viscoelastic spherical cavity problem under fractional order derivative: state space approach,” Advances in Mathematics:Scientific Journal., 9, 9769-9780, 2020.
• K. R. Gaikwad, Y. U. Naner, “Analysis of transient thermoelastic temperture distribution of a thin circular plate and its thermal deflection under uniform heat generation,” journal of thermal stress., 44(1),75-85, 2021.
• K. R. Gaikwad, V. G. Bhandwalkar, “ Fractional order thermoelastic problem for finite piezoelectric rod subjected to different types of thermal loading - direct approach,” Journal of the Korean Society for Industrial and Apllied Mathematics., 25, 117–131, 2021.
• K. R. Gaikwad, S. G. Khavale, “Fractional order transient thermoelastic stress analysis of a thin circular sector disk,” International Journal of Thermodynamics., 25(1), 1–8, 2022.
• I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
• H. Stehfest, Communication of the ACM, 13, 47, 1970.
• PTCMathcad Prime-7.0.0.0, [Online]. Available: https://support.ptc.com/help/mathcad/r7.0/en/ (accessed Dec. 1, 2021).