Thermoelastic Analysis of a Rectangular Plate with Nonhomogeneous Material Properties and Internal Heat Source

Document Type: Research Paper


1 Department of Mathematics, RTM Nagpur University, Nagpur, India

2 Department of Mathematics, Shri Lemdeo Patil Mahavidyalaya, Nagpur, India


This article deals with the determination of temperature distribution, displacement and thermal stresses of a rectangular plate having nonhomogeneous material properties with internal heat generation. The plate is subjected to sectional heating. All the material properties except Poisson’s ratio and density are assumed to be given by a simple power law along x direction. Solution of the two-dimensional heat conduction equation is obtained in the transient state. Integral transform method is used to solve the system of fundamental equation of heat conduction. The effects of inhomogeneity on temperature and thermal stress distributions are examined. For theoretical treatment, all the physical and mechanical quantities are taken as dimensional, whereas for numerical computations we have considered non-dimensional parameters. The transient state temperature field and its associated thermal stresses are discussed for a mixture of copper and zinc metals in the ratio 70:30 respectively. Numerical calculations are carried out for both homogeneous and nonhomogeneous cases and are represented graphically.


[1] Al-Hajri M., Kalla S.L., 2004, On an integral transform involving Bessel functions, Proceedings of the International Conference on Mathematics and its Applications.

[2] Birkoff G., Rota G. C., 1989, Ordinary Differential Equations, Wiley, New York.

[3] Churchill R.V., 1972, Operational Mathematics, Mc-Graw Hill.

[4] Ding S.H., Li X., 2015, Thermoelastic analysis of nonhomogeneous structural materials with an interface crack under uniform heat flow, Applied Mathematics and Computation Archive 271: 22-33.

[5] Edited by the Japan Society of Mechanical Engineers, 1980, Elastic Coefficient of Metallic Materials, Japan Society of Mechanical Engineers.

[6] Gupta A.K., Singhal P., 2010, Thermal effect on free vibration of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness, Applied Mathematics 1: 456-463.

[7] Gupta A.K., Saini M., Singh S., Kumar R., 2014, Forced vibrations of non-homogeneous rectangular plate of linearly varying thickness, Journal of Vibration Control 20: 876-884.

[8] Hata T., 1983, Thermal stresses in a nonhomogeneous thick plate with surface radiation under steady distribution of temperature, The Japan Society of Mechanical Engineers 49: 1515-1521.

[9] Kassir M.K., 1972, Boussinesq problems for nonhomogeneous solid, Proceedings of the American Society of Civil Engineers,Journal of the Engineering Mechanics Division 98: 457-470.

[10] Kawamura R., Huang D., Tanigawa Y., 2001, Thermoelastic deformation and stress analyses of an orthotropic nonhomogeneous rectangular plate, Proceedings of the Fourth International Congress on Thermal Stresses.

[11] Kumar Y., 2012, Free vibrations of simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness and elastically restrained edges against rotation using Rayleigh-Ritz method, Earthquake Engineering and Engineering Vibration 11: 273-280.

[12] Lal R., Kumar Y., 2013, Transverse vibrations of nonhomogeneous rectangular plates with variable thickness, Mechanics of Advanced Materials and Structures 20: 264-275.

[13] Manthena V.R., Lamba N.K., Kedar G.D., 2016, Transient thermoelastic problem of a nonhomogeneous rectangular plate, Journal of Thermal Stresses 40: 627-640.

[14] Martynyak R.M., Dmytriv M.I., 2010, Finite-element investigation of the stress-strain state of an inhomogeneous rectangular plate, Journal of Mathematical Sciences 168: 633-642.

[15] Morishita H., Tanigawa Y., 1998, Formulation of three dimensional elastic problem for nonhomogeneous medium and its analytical development for semi-infinite body, The Japan Society of Mechanical Engineers 97: 97-104.

[16] Muravskii G.B., 2008, Response of an elastic half-space with power-law nonhomogeneity to static loads, Archive of Applied Mechanics 78: 965.

[17] Pandita B.B., Kulkarni V.S., 2015, Finite difference approach for nonhomogeneous problem of thermal stresses in cartesian domain, International Journal of Advances in Applied Mathematics and Mechanics 3: 100-112.

[18] Sharma S., Gupta U.S., Singhal P., 2012, Vibration analysis of nonhomogeneous orthotropic rectangular plates of variable thickness resting on winkler foundation, Journal of Applied Science and Engineering 15: 291-300.

[19] Sugano Y., 1987, Transient thermal stresses in a nonhomogeneous doubly connected region, The Japan Society of Mechanical Engineers 53: 941-946.

[20] Tanigawa Y., Ootao Y., Kawamura R., 1991, Thermal bending of laminated composite rectangular plates and nonhomogeneous plates due to partial heating, Journal of Thermal Stresses 14: 285-308.

[21] Tanigawa Y., 1995, Some basic thermoelastic problems for nonhomogeneous structural materials, Applied Mechanics Reviews 48: 287-300.

[22] Tanigawa Y., Kawamura R., Ishida S., 2002, Derivation of fundamental equation systems of plane isothermal and thermoelastic problems for in-homogeneous solids and its applications to semi-infinite body and slab, Theoretical and Applied Mechanics 51: 267-279.

[23] Tokovyy Y., Ma C-C., 2009, Analytical solutions to the 2D elasticity and thermoelasticity problems for inhomogeneous planes and half-planes, Archive of Applied Mechanics 79: 441-456.

[24] Wang C.Y., Wang C.M., 2011, Exact solutions for vibrating nonhomogeneous rectangular membranes with exponential density distribution, The IES Journal Part A: Civil & Structural Engineering 4: 37-40.

[25] Yang Q., Zheng B., Zhang K., Zhu J., 2013, Analytical solution of a bilayer functionally graded cantilever beam with concentrated loads, Archive of Applied Mechanics 83: 455-466.