Free Vibration Analysis of Micropolar Thermoelastic Cylindrical Curved Plate in Circumferential Direction

Subject Areas : Engineering

G  Partap ¹ (Department of Mathematics, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, Punjab)
R  Kumar ² (Department of Mathematics, Kurukshetra University)

Received: 2010-08-07 Accepted : 2010-11-02 Published : 2010-09-30

Keywords: Phase velocity, Micropolar, Circumferential wave number, Thermoelasticity type III, Thermoelasticity without energy dissipation,

Abstract :

The free vibration analysis ofhomogeneous isotropic micropolar thermoelastic cylindrical curved plate in circumferential direction has been investigated in the context of generalized themoelasticity III, recently developed by Green and Naghdi. The model has been simplified using Helmholtz decomposition technique and the resulting equations have been solved using separation of variable method. Mathematical modeling of the problem to obtain dispersion curves for curved isotropic plate leads to coupled differential equations and solutions are obtained by using Bessel functions. The frequency equations connecting the frequency with circumferential wave number and other physical parameters are derived for stress free cylindrical plate. In order to illustrate theoretical development, numerical solutions are obtained and presented graphically for a magnesium crystal.

References:

[1] Biot M.A., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27: 240- 253.

[2] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the mechanics and physics of solids 15: 299-309.

[3] Green A.E., Lindsay K. A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.

[4] Green A.E., Naghdi P.M., 1991, A re-examination of the basic postulates of thermomechanics, in: Proceedings of the Royal society of London A357: 253-270.

[5] Green A.E., Naghdi, P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15: 253-264.

[6] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of elasticity 31: 189-209.

[7] Eringen A.C., Suhubi E.S., 1964, Nonlinear theory of simple microelastic solid-I, International Journal of Engineering Science 2: 189-203.

[8] Eringen A.C., 1968, Theory of Micropolar Elasticity, Volume II, Chapter 7, Fracture, edited by H.Liebowitz, Academic Press, New York.

[9] Nowacki W., 1966, Couple stresses in the theory of thermoelasticity III, Bulletin of the Polish Academy of Sciences Technical Sciences 14: 801-809.

[10] Eringen A.C., 1970, Foundations of micropolar thermoelasticity, Intern, Centerfor Mechanical Studies, Course and Lectures, No. 23, Springer-Verlag, Wien.

[11] Viktorov I.A., 1958, Rayleigh type waves on a cylindrical surface, Soviet Physics-Acoustics4: 131-136.

[12] Qu J., Berthelot V., Li Z., 1996, Dispersion of guided circumferential waves in a circular annulus, Review of progress in Quantitative Nondestructive Evolution, edited by D.O. Thompson and D.E. Chimenti, Plenum, New York, 169-176 .

[13] Liu G., Qu J., 1998, Guided circumferential waves in a circular annulus, ASME Journal of Applied Mechanics65: 424-430.

[14] Liu G., Qu J., 1998, Transient wave propagation in a circular annulus subjected to impulse excitation on its outer surface, Journal of the Acoustical Society of America 103: 1210-1220.

[15] Valle C., Qu J., Jacobs L. J., 1999, Guided circumferential waves in layered cylinders, International Journal of Engineering Science 37: 1369-1387.

[16] Towfighi S., Kundu T., Ehsani M., 2002, Elastic wave propagation in circumferential direction in anisotropic cylindrical curved plates, ASME Journal of Applied Mechanics69: 283-291.

[17] Tajuddin M., Shah S.A., 2006, Circumferential waves of infinite hollow poroelastic cylinders, Journal of Applied Mechanics 73: 705-708.

[18] Sharma J.N., Pathania V., 2005, Generalized thermoelastic wave propagation in circumferential direction of transversely isotropic cylindrical curved plates, Journal of Sound and Vibration 281: 1117-1131.

[19] Tyutekin V.V., 2004, Circumferential normal modes in an empty elastic cylinder, Acoustical Physics 50: 742-750.

[20] Jiangong Y., Bin W., Cunfu H., 2010, Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation, Ultrasonic 50: 416-423.

[21] Eringen A.C., 1999, Microcontinuum Field Theories I. Foundations and Solids, Springer-Verlag, New York.

[22] Eringen A.C., 1984, Plane waves in non-local micropolar elasticity, International Journal of Engineering Science 22: 1113-1121.

[23] Dhaliwal R.S., Singh A., 1980, Dynamic Coupled Thermoelasticity, Hindustan Publication Corporation, New Delhi, India.