Application of Fractional Order Theory of Thermoelasticity in an Elliptical Disk and Associated Thermal Stresses
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Department of Mathematics, K.I.T.S., Ramtek, India
Department of Mathematics, R.G.C. Engineering & Research, Nagpur, India
Online publication date: 2020-08-17
Publication date: 2020-09-01
International Journal of Applied Mechanics and Engineering 2020;25(3):169-180
In this article, a time fractional-order theory of thermoelasticity is applied to an isotropic homogeneous elliptical disk. The lower and upper surfaces of the disk are maintained at zero temperature, whereas the sectional heat supply is applied on the outer curved surface. Thermal deflection and associated thermal stresses are obtained in terms of Mathieu function of the first kind of order 2n. Numerical evaluation is carried out for the temperature distribution, Thermal deflection and thermal stresses and results of the resulting quantities are depicted graphically.
Abel N.H. (1823): Solution de Quelques Problèmes à I’aide D’intègrales Dèfinites. − Werkee, vol.1, pp.10.
Caputo M. and Mainardi F. (1971): A new dissipation model based on memory mechanism. − Pure Appl. Geophys., vol.91, pp.134-147.
Caputo M. and Mainardi F. (1971): Linear models of dissipation in an elastic solids. − Rivis ta del Nuovo ciment, vol.1, pp.161-198.
Caputo M. (1974): Vibration on an infinite viscoelastic layer with a dissipative memory. − J. Acous. Soc. Am., vol.56, pp.897-904.
Ning T.H. and Jiang X.Y. (2011): Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation. − Acta Mechanica Sinica, vol.27, No.6, pp.994-1000.
Lenzi E.K., Mendes R.S., Gonçalves G., Lenzi M.K. and da Silva L.R. (2006): Fractional diffusion equation and Green function approach: Exact solutions. − Physica A 360, pp.215-226.
Povstenko Y. (2013): Fractional heat conduction in an infinite medium with a spherical inclusion. − Entropy, vol.15, pp.4122-4133.
Povstenko Y. (2015): Fractional Thermoelasticity. − Switzerland: Springer.
Povstenko Y. (2016): Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses. − J. Therm. Stresses, vol.39, No.11, pp.1442-1450.
Abbas I.A. (2015): Eigen value approach on fractional order theory of thermoelastic diffusion problem for an infinite elastic medium with a spherical cavity. − Applied Mathematical Modelling, vol.39, pp.6196–6206 (2015).
Warbhe S.D., Tripathi J.J., Deshmukh K.C. and Verma J. (2017): Fractional heat conduction in a thin circular plate with constant temperature distribution and associated thermal stresses. − Journal of Heat Transfer, vol.139 / 044502-1-4.
Youssef H.M. (2010): Theory of fractional order generalized thermoelasticity. − J. Heat Transfer, vol.132, pp.1-7.
Youssef H.M. (2012): Two-dimensional thermal shock problem of fractional order generalized thermoelasticity. − Acta Mech., vol.223, pp.219-1231.
Khobragade N.L. and Lamba N.K. (2019): Modeling of thermoelastic hollow cylinder by the application of fractional order theory. − Research and Reviews: Journal of Physics; vol.8, Nio.1, pp.46-57.
Tripathi J.J., Kedar G.D. and Deshmukh K.C. (2016): Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds. − J. Therm. Stresses, vol.39, No.2, pp.220-230.
Kukla S. and Siedlecka U. (2017): An analytical solution to the problem of time-fractional heat conduction in a composite sphere. − Bulletin of the Polish Academy of Sciences Technical Sciences, vol.65, No.2, pp.179-186.
Khobragade N.L. and Lamba N.K. (2019): Thermal deflection and stresses of a circular disk due to partially distributed heat supply by application of fractional order theory. − Journal of Computer and Mathematical Sciences, vol.10, No.3, pp.429-437.
Khobragade N.L. and Lamba N.K. (2019): Study of thermoelastic deformation of a solid circular cylinder by application of fractional order theory. − Journal of Computer and Mathematical Sciences, vol.10, No.3, pp.438-444.
McLachlan N.W. (1945): Heat conduction in elliptical cylinder and an analogous electromagnetic problem. − Philosophical Magazine, vol.36, pp.600-609.
McLachlan N.W. (1947): Theory and Application of Mathieu function. − Oxford: Clarendon Press.
Dhakate T., Vinod Varghese and Lalsingh Khalsa (2017): Integral transform approach for solving dynamic thermal vibrations in the elliptical disk. − Journal of Thermal Stresses, DOI: 10.1080/01495739.2017.1285215.
Gupta R.K. (1964): A finite transform involving Mathieu functions and its application. − Proc. Net. Inst. Sc., India, Part A, vol.30, pp.779-795.
Bhad P., Vinod Varghese and Lalsingh Khalsa (2017): Heat source problem of thermoelasticity in an elliptic plate with thermal bending moments. − Journal of Thermal Stresses, vol.40, No.1, pp.96-107.
Dhakate T., Vinod Varghese and Lalsingh Khalsa (2018): A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate. − Vol. 41, No.2, pp.271-285.
Ishteva M., Scherer R. and Boyadjiev L. (2005): On the Caputo operator of fractional calculus and C-Laguerre functions. − Mathematical Sciences Research Journal, vol.9, No.6, pp.161-170.
Haubold H.J., Mathai A.M. and Saxena R.K. (2011): Mittag-Leffler functions and their applications. − Journal of Applied Mathematics, paper ID 298628.
Bhad P.P. (2017): Study of thermoelastic problems on elliptical objects with an internal heat source. − PhD Dissertation - Gondwana University, Gadchiroli, India.
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