Thermoelastic Behaviour in a Multilayer Composite Hollow Sphere with Heat Source

Document Type : Research Paper


1 Department of Mathematics, S. N. Mor College, Tumsar (MS), India

2 Department of Mathematics, R. T. M. Nagpur University, Nagpur (MS), India



This paper deals with the mathematical approach to discuss the radially varying transient temperature distribution in a multilayer composite hollow sphere subjected to the time independent volumetric generation of heat in each layer. Initially the layers are at arbitrary temperature and the analysis assumes all the layers of the body are thermally isotropic and having a perfect thermal contact. It is novel to obtain the exact solution for temperature field by the separation of variables by splitting the problem into two parts homogeneous transient and non-homogeneous steady state. The set of equations obtained are solved by using the rigorous applications of analytic techniques with the help of eigen value expansion method. The thermoelastic response is studied in the context of uncoupled Thermoelasticity. The results obtained pointed out that the magnitude and distribution of the temperature and thermal stresses are greatly influenced by the layered heat generation parameter. The accuracy and feasibility of the proposed model is demonstrated by an example of three layered hollow sphere of Aluminium, Copper and Iron subjected to given conditions. The results presented in this article could be found hardly in an open literature despite of extensive search.


[1] Bulavin P.E., Kascheev V.M., 1965, Solution of the non-homogeneous heat conduction equation for multilayer bodies, Chemical Engineering 1(5): 112-115.
[2] Yener Y., Ozisik M.N., 1974, On the solution of unsteady heat conduction in multi region media with time dependent heat transfer coefficient, Proceedings of the 5th International Heat Transfer Conference, Tokyo.
[3] Lu X., Tarvola P., Viljanen M., 2006, Transient analytical solution to heat conduction in multidimensional composite cylindrical slab, Heat and Mass Transfer 49: 1107-1114.
[4] Jain P.K., Singh S., Rizwan-uddin, 2008, Analytical solution to transient Asymmetric heat conduction in a multilayer Annulus, Journal of heat Transfer 131(9): 011304.
[5] Kukla S., Siedlecka U., 2013, Heat conduction problem in a two layered hollow cylinder by using the Green’s function method, Journal of Applied Mathematics and Computational Mechanics 12(2): 45-50.
[6] Chen C.K., Yang Y.C., 1986, Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials, Journal of Engineering Science 24: 569-581.
[7] Jen K.C., Lee Z.Y., 1999, Thermoelastic transient response of an infinitely long multi-layered cylinder, Mechanics Research Communications 26(6): 709-718.
[8] Lee Z.Y., 2004, Coupled problem of thermoelasticity for multilayer spheres with time dependent boundary conditions, Journal of Marine Science and Technology 12(2): 93-101.
[9] Ootao Y., 2009, Transient thermoelatic analysis for a multilayered hollow cylinder with piecewise power law nonhomogeneity, Asian Pacific Conference Foe Materials and Mechanics, Yokohama, Japan.
[10] Koo J., Valgur J., 2008, Analysis of thermoelastic stresses in a layered plates, 6th International DAAAM Baltic Conference, Industrial Engineering, Tallinn.
[11] Zamani Nejad M., Rastgoo A., Hadi A., 2014, Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics 6(4): 366-377.
[12] Pawar S.P., Deshmukh K.C., Kedar G.D., 2015, Thermal stresses in functionally graded hollow sphere due to non-uniform Internal Heat Generation, Applications and Applied Mathematics: An International Journal 10(1): 552-569.
[13] Pawar S.P., Deshmukh K.C., Jyoti V., 2017, Thermal behavior of functionally graded solid sphere due to non-uniform internal heat generation, Journal of Thermal Stresses 40(1): 86-98.
[14] Guerrache F., Kebli B., 2019, An axisymmetric contact problem of a thermoelastic layer on a rigid circular base, Journal of Solid Mechanics 11(4): 862-885.
[15] Ozisik M.N., 1993, Heat Conduction, Wiley and Sons.
[16] Noda N., Hetnarski R.B., Tanigawa Y., 2003, Themal Stresses, Taylor and Francis, New York.
[17] Carslaw H.S., Jaeger J.C., 1986, Conduction of Heat in Solids, Oxford Clarendon.
[18] Vasiliev V.V., Morozov E.V., 2007, Advanced Mechanics of Composite Materials, Elsevier.
[19] Yener Y., Kakac S., 2008, Heat Conduction, Taylor and Francis.
[20] Hetnarski R. B., Reza Eslami M., 2009, Thermal Stresses-Advance Theory and Applications, Springer.