An Exact Solution for Kelvin-Voigt Model Classic Coupled Thermo Viscoelasticity in Spherical Coordinates

Document Type : Research Paper


Mechanical Engineering Department, Islamic Azad University, South Tehran Branch, Tehran, Iran


In this paper, the classic Kelvin-Voigt model coupled thermo-viscoelasticity model of hollow and solid spheres under radial symmetric loading condition is considered. A full analytical method is used and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, and the heat source are considered in the most general forms and where no limiting assumption is used. This generality allows simulate varieties of applicable problems. At the end, numerical results are presented and compared with classic theory of thermoelasticity.       


[1] Lahiri A., Kar T. K., 2007, Eigenvalue approach to generalized thermoviscoelasticity with one relaxation time parameter, Tamsui Oxford Journal of Mathematical Sciences 23(2): 185-218.
[2] Hetnarski R. B., 1964, Solution of the coupled problem of thermoelasticity in the form of series of functions, Archiwum Mechaniki Stosowanej 16: 919-941.
[3] Hetnarski R. B., Ignaczak J., 1993, Generalized thermoelasticity closed-form solutions, Journal of Thermal Stresses 16: 473-498.
[4] Hetnarski R. B., Ignaczak J., 1994, Generalized thermoelasticity: re-sponse of semi-space to a short laser Pulse, Journal of Thermal Stresses 17: 377-396.
[5] Georgiadis H. G., Lykotrafitis G., 2005, Rayleigh waves generated by a thermal source: a three-dimensional transient thermoelasticity solution, Journal of Applied Mechanics 72: 129-138.
[6] Wagner P., 1994, Fundamental matrix of the system of dynamic linear thermoelasticity, Journal of Thermal Stresses 17: 549-565.
[7] Bahtui A., Eslami M. R., 2007, Coupled thermoelasticity of functionally graded cylindrical shells, Mechanics Research Communications 34: 1-18.
[8] Bagri A., Eslami M. R., 2004, Generalized coupled thermoelasticity of disks based on the lord-shulman model, Journal of Thermal Stresses 27: 691-704.
[9] Abd-Alla A.M., Hammad H. A. H., Abo-Dahab S.M., 2004, Magneto-thermo-viscoelastic interactions in an unbounded body with a spherical cavity subjected to a periodic loading, Applied Mathematics and Computation 155: 235-248.
[10] Knopoff L., 1955, The interaction between elastic wave motions and a magnetic field in electrical conductors, Journal of Geophysical Research 60: 441-456.
[11] Chadwick P., 1957, Elastic waves propagation in a magnetic field, Proceeding of the International Congress of Applied Mechanics, Brusseles, Belgium.
[12] Nowacki W., Francis P.H., Hetnarski R.B., 1975, Dynamic Problems of Thermoelasticity, Noordhoff, Leyden.
[13] Misra J. C., Samanta S. C., Chakrabarti A. K., 1991, Magneto-thermomechanical interaction in an aeolotropic viscoelastic cylinder permeated by a magnetic field subjected to a periodic loading, International Journal of Engineering Science 29 (10): 1209-1216.
[14] Misra J. C., Chatopadhyay N. C. , Samanta S. C., 1994, Thermo-viscoelastic waves in an infinite aeolotropic body with a cylindrical cavity-a study under the review of generalized theory of thermoelasticity, Composite Structures 52 (4): 705-717.
[15] Abd-alla A. N. , Yahia A.A., Abo-Dahab S. M., 2003, On the reflection of the generalized magneto-thermo-viscoelastic plane waves, Chaos, Solitons & Fractal 16: 211-231.
[16] Kaleski S., 1963, Aborpation of magneto-viscoelastic surface waves in a real conductor in a magnetic field, Proceedings of Vibration Problems 4 : 319-329.
[17] Abd-Alla A. M., Mahmoud S. R., 2011, Magneto-thermo-viscoelastic interactions in an unbounded non-homogeneous body with a spherical cavity subjected to a periodic loading, Applied Mathematical Sciences 5(29):1431- 1447.
[18] Song Y. C., Zhang Y. Q., Xu H. Y., Lu B. H., 2006, Magneto-thermo-viscoelastic wave propagation at the interface between two micropolar viscoelastic media, Applied Mathematics and Computation 176: 785-802.
[19] Abo-Dahab S.M., 2012, Effect of magneto-thermo-viscoelasticity in an unbounded body with a spherical cavity subjected to a harmonically varying temperature without energy dissipation, Meccanica 47:613-620.
[20] Sharma J. N., 2005, Some considerations on the rayleigh-lamb wave propagation in visco-thermoelastic plate, Journal of Vibration and Control 11: 1311- 1335.
[21] Sharma J. N., Singh D., Kumar R., 2004, Propagation of generalized visco-thermoelastic Rayleigh-Lamb waves in homogeneous isotropic plates, Journal of Thermal Stresses 27: 645- 669.
[22] Roy-Chudhuri S. K., Mukhopdhyay S., 2000, Effect of rotation and relaxation on plane waves in generalized thermo-viscoelasticity, International Journal of Mathematics and Mathematical Sciences 23: 497-505.
[23] Othman M. I. A., Abbas I. A., 2012, Fundamental solution of generalized thermo-viscoelasticity using the finite element method, Computational Mathematics and Modeling 23 (2):158-167.
[24] Kar A., Kanoria M., 2009, Generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect,. Applied Mathematical Modelling 33: 3287-3298.
[25] Ezzat M. A., Othman M. I., El Karamany A.S., 2002, State space approach to generalized thermo-viscoelasticity with two relaxation times, International Journal of Engineering Science 40: 283-302.
[26] Ezzat M. A., El Karamany A. S., Smaan A. A., 2001, State space formulation to generalized thermo-viscoelasticity with thermal relaxation, Journal of Thermal Stresses 24: 823- 846.
[27] Jabbari M., Dehbani H., Eslami M. R., 2010, An exact solution for classic coupled thermoelasticity in spherical coordinates, Journal of Pressure Vessel Technology 132 (3): 031201.