Mathematical Modeling of Thermoelastic State of a Thick Hollow Cylinder with Nonhomogeneous Material Properties

Document Type: Research Paper

Authors

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

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

Abstract

The object of the present paper is to study heat conduction and thermal stresses in a hollow cylinder with nonhomogeneous material properties. The cylinder is subjected to sectional heating at the curved surface. All the material properties except for Poisson’s ratio and density are assumed to be given by a simple power law in the axial direction. A solution of the two-dimensional heat conduction equation is obtained in the transient state. The solutions are obtained in the form of Bessel’s and trigonometric functions. For theoretical treatment, all the physical and mechanical quantities are taken as dimensional, whereas we have considered non-dimensional parameters, for numerical analysis. The influence of inhomogeneity on the thermal and mechanical behaviour is examined. The transient state temperature field and its associated thermal stresses are discussed for a mixture of copper and tin metals in the ratio 70:30 respectively. Numerical calculations are carried out for both homogeneous and nonhomogeneous cylinders and are represented graphically.

Keywords


[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] Arefi M., Rahimi G. H., 2011, General formulation for the thermoelastic analysis of an arbitrary structure made of functionally graded piezoelectric materials, based on the energy method, Mechanical Engineering 62: 221-235.

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

[4] Cho H., Kardomateas G.A., Valle C.S., 1998, Elastodynamic solution for the thermal shock stresses in an orthotropic thick cylindrical shell, Transactions of the ASME 65: 184-193.

[5] Deshmukh K.C., Khandait M.V., Kumar R., 2005, Thermal Stresses in a circular plate by a moving heat source, Material Physics and Mechanics 22: 86-93.

[6] Deshmukh K.C., Quazi Y.I., Warbhe S.D., Kulkarni V.S., 2011, Thermal stresses induced by a point heat source in a circular plate by quasi-static approach, Theoretical and Applied Mechanics Letters 1: 031007.

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

[8] Ghasemi A.R, Kazemian A., Moradi M., 2014, Analytical and numerical investigation of FGM pressure vessel reinforced by laminated composite materials, Journal of Solid Mechanics 6: 43-53.

[9] Ghorbanpour A. A., Arefi M., Khoshgoftar M. J., 2009, Thermoelastic analysis of a thick walled cylinder made of functionally graded piezoelectric material, Smart Materials and Structures 18: 115007.

[10] Ghorbanpour A. A., Haghparast E., Zahra K. M., Amir S., 2014, Static stress analysis of carbon nano-tube reinforced composite (CNTRC) cylinder under non-axisymmetric thermo-mechanical loads and uniform electro-magnetic fields, Composites Part B: Engineering 68: 136-145.

[11] Hata T., 1982, Thermal stresses in a non-homogeneous thick plate under steady distribution of temperature, Journal of Thermal Stresses 5: 1-11.

[12] Hosseini S.M., Akhlaghi M., 2009, Analytical solution in transient thermoelasticity of functionally graded thick hollow cylinders, Mathematical Methods in the Applied Sciences 32: 2019-2034.

[13] Jabbari M., Aghdam M.B., 2015, Asymmetric thermal stresses of hollow FGM cylinders with piezoelectric internal and external layers, Journal of Solid Mechanics 7: 327-343.

[14] Kassir K., 1972, Boussinesq Problems for Non-homogeneous Solid, Proceedings of the American Society of Civil Engineers,Journal of the Engineering Mechanics Division 98: 457-470.

[15] Kaur J., Thakur P., Singh S.B., 2016, Steady thermal stresses in a thin rotating disc of finitesimal deformation with mechanical load, Journal of Solid Mechanics 8: 204-211.

[16] Kim K.S., Noda N., 2002, Green's function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material, Acta Mechanica 156: 145-161.

[17] 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.

[18] Noda N., Hetnarski R. B., Tanigawa Y., 2003, Thermal Stresses, Taylor & Francis, New York.

[19] Noda N., Ootao Y., Tanigawa Y., 2012, Transient thermoelastic analysis for a functionally graded circular disk with piecewise power law, Journal of Theoretical and Applied Mechanics 50: 831-839.

[20] Ootao Y., Tanigawa Y., 1994, Three-dimensional transient thermal stress analysis of a nonhomogeneous hollow sphere with respect to rotating heat source, Transactions of the Japan Society of Mechanical Engineering 60: 2273-2279.

[21] Ootao Y., Akai T., Tanigawa Y., 1995, Three-dimensional transient thermal stress analysis of a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction, Journal of Thermal Stresses 18: 497-512.

[22] Ootao Y., Tanigawa Y., 2005, Transient thermoelastic analysis for a functionally graded hollow cylinder, Journal of Thermal Stresses 29: 1031-1046.

[23] Ootao Y., 2010, Transient thermoelastic analysis for a multilayered hollow cylinder with piecewise power law nonhomogenity, Journal of Solid Mechanics and Materials Engineering 4: 1167-1177.

[24] Ootao Y., Tanigawa Y., 2012, Transient thermoelastic analysis for a functionally graded hollow circular disk with piecewise power law nonhomogenity, Journal of Thermal Stresses 35: 75-90.

[25] Rezaei R., Shaterzadeh A.R., Abolghasemi S., 2015, Buckling analysis of rectangular functionally graded plates with an elliptic hole under thermal loads, Journal of Solid Mechanics 7: 41-57.

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

[27] Sugano Y., 1987, An expression for transient thermal stress in a nonhomogeneous plate with temperature variation through thickness, Ingenieur-Archiv 57: 147-156.

[28] Sugano Y., 1988, Transient thermal stresses in a nonhomogeneous doubly connected region, JSME International Journal Series 31: 520-526.

[29] Sugano Y., Akashi K., 1989, An analytical solution of unaxisymmetric transient thermal stresses in a nonhomogeneous hollow circular plate, Transactions of the Japan Society of Mechanical Engineers Part A 55: 89-95.

[30] Tanigawa Y., Jeon S.P., Hata T., 1997, Analytical development of axisymmetrical elastic problem for semi-infinite body with Kasser’s nonhomogeneous material property, The Japan Society of Mechanical Engineers 96:86-93.

[31] 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.

[32] Vimal J., Srivastava R.K., Bhatt A.D., Sharma A.K., 2015, Free vibration analysis of moderately thick functionally graded plates with multiple circular and square cutouts using finite element method, Journal of Solid Mechanics 7: 83-95.

[33] Wang Xi., 1993, The elastodynamic solution for a solid sphere and dynamic stress focusing phenomenon, Applied Mathematics and Mechanics 14: 777-785.

[34] Zamani N. 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: 366-377.