Influences of Thermal Relaxation Times on Generalized Thermoelastic Longitudinal Waves in Circular Cylinder
This paper is concerned with propagation of thermoelastic longitudinal vibrations of an infinite circular cylinder, in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). Three displacement potential functions are introduced to uncouple the equations of motion. The frequency equation, by using the traction free boundary conditions, is given in the form of a determinant involving Bessel functions. The roots of the frequency equation give the value of the characteristic circular frequency as function of the wave number. These roots, which correspond to various modes, are numerically computed and presented graphically for different values of the thermal relaxation times. It is found that the influences of the thermal relaxation times on the amplitudes of the elastic and thermal waves are remarkable. Also, it is shown in this study that the propagation of thermoelastic longitudinal vibrations based on the generalized thermoelasticity can differ significantly compared with the results under the classical formulation. A comparison of the results for the case with no thermal effects shows well agreement with some of the corresponding earlier results.
[1] I. A. Abbas and A. N. Abd-alla, "Effects of thermal relaxations on
thermoelastic interactions in an infinite orthotropic elastic medium with
a cylindrical cavity," Archive of Applied Mechanics, vol. 78 no. 4, pp.
283-293, 2008.
[2] I. A. Abbas and A. N. abd-alla, "A study of generalized thermoelastic
interaction in an infinite fibre-reinforced anisotropic plate containing a
circular hole," Acta Physica Polonica A, vol. 119, no. 6, pp. 814-818,
2011.
[3] J. D. Achenbach, Wave Propagation in Elastic Solids. North-Holland
Publishing Company- Amsterdam, 1973.
[4] P. Chadwick, Thermoelasticity: the Dynamic Theory, in: R. Hill, I.N.
Sneddon (Eds.), Progress in Solid Mechanics, North-Holland,
Amsterdam, 1960, vol. I, pp. 263-328.
[5] D. S. Chandrasekharaiah, "Thermoelasticity with second sound: A
review," Appl. Mech. Rev., vol. 39, no. 3, pp. 353-376, 1986.
[6] L. S. Chen and H. S. Chu, "Transient thermal stresses due to
periodically moving line heat source of composite hollow cylinder," J.
Thermal Stresses, vol. 13, pp.131-145, 1990.
[7] R. Chitikireddy, S. K. Datta, A. H. Shah, H. Bai, "Generalized
thermoelastic waves in cylinders due to localized heating," International
Journal of Solids & Structures, vol. 48, no. 21, pp. 3063-3074, 2011.
[8] R. S. Dhaliwal and K. L. Chowdhury, "Dynamical problems of
thermoelasticity for cylindrical regions," Arch. Mech. Stos. vol.1, no.20,
pp.47-66, 1968.
[9] M. Elhagary, "Generalized thermoelastic diffusion problem for an
infinitely long hollow cylinder for short times," Acta Mechanica;, vol.
218, no. 3/4, pp. 205-215, 2011.
[10] S. Erbay and E. S. Suhubi, "Longitudinal wave propagation in a
generalized thermoelastic cylinder," J. Thermal Stresses, vol. 9 pp. 279-
295, 1986.
[11] T. Furukawa, N. Noda and F. Ashida, "Generalized thermoelasticity for
an infinite solid cylinder," JSME International Journal, Series I, vol. 34,
no. 3, pp.281-286, 1991.
[12] A. E. Green and K. A. Lindsay, "Thermoelasticity," J. Elasticity, vol. 2,
pp. 1-7, 1972.
[13] S. M. Hosseini and M. H. Abolbashari, "Analytical solution for
thermoelastic waves propagation analysis in thick hollow cylinder based
on Green-Naghdi model of coupled thermoelasticity," Journal of
Thermal Stresses, vol. 35, no. 4, pp.363-376, 2012.
[14] D. Iesan and A.Scalia, Thermoelastic deformations, Series: Solid
Mechanics and its Applications, 48, Kluwer Acad. Pub. Group,
Dordrecht, 1996.
[15] H. W. Lord and Y. Shulman, "A generalized dynamical theory of
thermoelasticity," J. Mech. Phys. Solids, vol. 15, pp.299-309, 1967.
[16] N. Noda, "Transient thermoelastic contact stress in a short length
circular cylinder," J. Thermal Stresses, vol. 8, pp.413-424, 1985.
[17] W. Nowinski, Theory of Thermoelasticity with Applications, Sijthoff &
Noordhoof Int. Publishing Comp., Netherlands, 1978.
[18] P. Ponnusamy and M. Rajagopal, "Wave propagation in a transversely
isotropic thermoelastic solid cylinder of arbitrary cross-section," Acta
Mechanica Solida Sinica, vol. 24, no. 6, pp.527-538, 2011.
[19] H. H. Sherief and M. Anwar, "A problem in generalized thermoelasticity
for an infinitely long annular cylinder", Int. J. Engng. Sci. vol. 20, no. 3,
pp.301-306, 1988.
[20] K. P. Soldatos, "Review of three dimensional dynamic analyses of
circular cylinders and cylindrical shells," App. Mech. Rev., vol. 47, no.
10, pp. 501-516, 1994.
[21] Y. Yang, T. Wang and C. Chen, "Thermoelastic transient response of an
infinitely long annular cylinder," J. Thermal Stresses, vol. 9 pp.19-30,
1986.
[1] I. A. Abbas and A. N. Abd-alla, "Effects of thermal relaxations on
thermoelastic interactions in an infinite orthotropic elastic medium with
a cylindrical cavity," Archive of Applied Mechanics, vol. 78 no. 4, pp.
283-293, 2008.
[2] I. A. Abbas and A. N. abd-alla, "A study of generalized thermoelastic
interaction in an infinite fibre-reinforced anisotropic plate containing a
circular hole," Acta Physica Polonica A, vol. 119, no. 6, pp. 814-818,
2011.
[3] J. D. Achenbach, Wave Propagation in Elastic Solids. North-Holland
Publishing Company- Amsterdam, 1973.
[4] P. Chadwick, Thermoelasticity: the Dynamic Theory, in: R. Hill, I.N.
Sneddon (Eds.), Progress in Solid Mechanics, North-Holland,
Amsterdam, 1960, vol. I, pp. 263-328.
[5] D. S. Chandrasekharaiah, "Thermoelasticity with second sound: A
review," Appl. Mech. Rev., vol. 39, no. 3, pp. 353-376, 1986.
[6] L. S. Chen and H. S. Chu, "Transient thermal stresses due to
periodically moving line heat source of composite hollow cylinder," J.
Thermal Stresses, vol. 13, pp.131-145, 1990.
[7] R. Chitikireddy, S. K. Datta, A. H. Shah, H. Bai, "Generalized
thermoelastic waves in cylinders due to localized heating," International
Journal of Solids & Structures, vol. 48, no. 21, pp. 3063-3074, 2011.
[8] R. S. Dhaliwal and K. L. Chowdhury, "Dynamical problems of
thermoelasticity for cylindrical regions," Arch. Mech. Stos. vol.1, no.20,
pp.47-66, 1968.
[9] M. Elhagary, "Generalized thermoelastic diffusion problem for an
infinitely long hollow cylinder for short times," Acta Mechanica;, vol.
218, no. 3/4, pp. 205-215, 2011.
[10] S. Erbay and E. S. Suhubi, "Longitudinal wave propagation in a
generalized thermoelastic cylinder," J. Thermal Stresses, vol. 9 pp. 279-
295, 1986.
[11] T. Furukawa, N. Noda and F. Ashida, "Generalized thermoelasticity for
an infinite solid cylinder," JSME International Journal, Series I, vol. 34,
no. 3, pp.281-286, 1991.
[12] A. E. Green and K. A. Lindsay, "Thermoelasticity," J. Elasticity, vol. 2,
pp. 1-7, 1972.
[13] S. M. Hosseini and M. H. Abolbashari, "Analytical solution for
thermoelastic waves propagation analysis in thick hollow cylinder based
on Green-Naghdi model of coupled thermoelasticity," Journal of
Thermal Stresses, vol. 35, no. 4, pp.363-376, 2012.
[14] D. Iesan and A.Scalia, Thermoelastic deformations, Series: Solid
Mechanics and its Applications, 48, Kluwer Acad. Pub. Group,
Dordrecht, 1996.
[15] H. W. Lord and Y. Shulman, "A generalized dynamical theory of
thermoelasticity," J. Mech. Phys. Solids, vol. 15, pp.299-309, 1967.
[16] N. Noda, "Transient thermoelastic contact stress in a short length
circular cylinder," J. Thermal Stresses, vol. 8, pp.413-424, 1985.
[17] W. Nowinski, Theory of Thermoelasticity with Applications, Sijthoff &
Noordhoof Int. Publishing Comp., Netherlands, 1978.
[18] P. Ponnusamy and M. Rajagopal, "Wave propagation in a transversely
isotropic thermoelastic solid cylinder of arbitrary cross-section," Acta
Mechanica Solida Sinica, vol. 24, no. 6, pp.527-538, 2011.
[19] H. H. Sherief and M. Anwar, "A problem in generalized thermoelasticity
for an infinitely long annular cylinder", Int. J. Engng. Sci. vol. 20, no. 3,
pp.301-306, 1988.
[20] K. P. Soldatos, "Review of three dimensional dynamic analyses of
circular cylinders and cylindrical shells," App. Mech. Rev., vol. 47, no.
10, pp. 501-516, 1994.
[21] Y. Yang, T. Wang and C. Chen, "Thermoelastic transient response of an
infinitely long annular cylinder," J. Thermal Stresses, vol. 9 pp.19-30,
1986.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60630", author = "Fatimah A. Alshaikh", title = "Influences of Thermal Relaxation Times on Generalized Thermoelastic Longitudinal Waves in Circular Cylinder", abstract = "This paper is concerned with propagation of thermoelastic longitudinal vibrations of an infinite circular cylinder, in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). Three displacement potential functions are introduced to uncouple the equations of motion. The frequency equation, by using the traction free boundary conditions, is given in the form of a determinant involving Bessel functions. The roots of the frequency equation give the value of the characteristic circular frequency as function of the wave number. These roots, which correspond to various modes, are numerically computed and presented graphically for different values of the thermal relaxation times. It is found that the influences of the thermal relaxation times on the amplitudes of the elastic and thermal waves are remarkable. Also, it is shown in this study that the propagation of thermoelastic longitudinal vibrations based on the generalized thermoelasticity can differ significantly compared with the results under the classical formulation. A comparison of the results for the case with no thermal effects shows well agreement with some of the corresponding earlier results.
", keywords = "Wave propagation, longitudinal vibrations, circular
cylinder, generalized thermoelasticity, Thermal relaxation times.", volume = "6", number = "12", pages = "1752-7", }
