Analysis of the Effects of Phase-lags on Propagation of Harmonic Plane Waves in Thermoelastic Media
Kumar Roushan, Mukhopadhyay Santwana *
Department of Applied Mathematics, Institute of Technology,
Banaras Hindu University, Varanasi-221005, India
*Corresponding author: e-mail: mukho_santwana@rediffmail.com
Received:
Received: 3 July 2009; revised: 16 December 2009; accepted: 30 December 2009; published online: 25 March 2010
DOI: 10.12921/cmst.2010.16.01.19-28
OAI: oai:lib.psnc.pl:711
Abstract:
The present paper attempts to investigate the propagation of harmonic plane waves of assigned frequency by employing the thermoelasticity theory with three phase-lags, recently proposed by Roychoudhuri (2007). The solutions of dispersion relation for the longitudinal plane waves are determined analytically and asymptotic expansions of several characterizations of the wave fields- phase velocity, specific loss and penetration depth of the dilatational waves are obtained for both the high frequency and low frequency values. Computational work for numerical values of the above quantities is also carried out with the help of Mathematica. A detailed analysis of the effects of phase-lags on the plane wave is presented by contrasting the theoretical as well as numerical results of the present work
with the corresponding results of the theory of thermoelasticity type III (Green and Naghdi, 1993) as reported earlier.
Key words:
generalized thermoelasticity, harmonic plane wave, thermoelasticity type III, thermoelasticity with three phase-lags
References:
[1] M. Biot, Thermoelasticity and Irreversible thermodynamics. J. Appl. Phys. 27, 240-253 (1956).
[2] H.W. Lord and Y. Shulman, Generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299-309 (1967).
[3] A.E Green, K.A Lindsay, Thermoelasticity. Journal of Elasticity 2, 1-7 (1972).
[4] A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermoemechanics. Proc. Roy. Soc. London, A432, 171-194 (1991).
[5] A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid. J. Thermal stresses 15, 253-264 (1992).
[6] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation. J. Elasticity 31, 189-209 (1993).
[7] D.Y. Tzou, A unified filed approach for heat conduction from Macro to Micro scales. ASME Journal of Heat transfer 117, 8-16 (1995).
[8] R. Quintanilla, R. Racke, A note on stability in dual-phaselag heat conduction, Int. J. Heat and Mass Transfer 49, 7-8, 1209-1213 (2006).
[9] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: A Review of recent literature. Appl. Mech. Rev. 39 (1998).
[10] R.B. Hetnarski, J. Ignaczak, Soliton-like waves in a low temperature and non linear thermoelastic solid. Int. J. Eng. Sci. 34, 1767-1787 (1996).
[11] R. B. Hetnarski, J. Ignaczak, Generalized thermoelasticity. J. Thermal stresses 22, 451-476 (1999).
[12] S.K. Roychoudhuri, On a thermoelastic three-phase-lag model. J. Thermal Stresses 30, 231-238 (2007).
[13] R. Quintanilla, R. Racke, A note on stability in three-phaselag heat conduction. Int. J. Heat and Mass Transfer 51, 24-29 (2008).
[14] A. Kar, M. Kanoria, Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect. European Journal of Mechanics
A/Solids 28, 757-767 (2009).
[15] A. Kar, M. Kanoria, Generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect, Appl. Math. Modelling 33, 3287-3298 (2009).
[16] M. Lessen, Motion of thermoelastic solid, Q. Appl. Maths. 15, 105-108 (1957).
[17] H. Deresiewicz, Plane waves in a thermoelastic solid, J. Acoust. Soc. Am. 29, 204-209 (1957).
[18] P. Chadwick, I.N. Sneddon, Plane waves in an elastic solid conducting heat. J. Mech. Phys. Solids 6, 223-230 (1958).
[19] P. Chadwick, Thermoelasticity: the dynamic theory, in: R. Hill, I.N. Sneddon (Eds.), Progress in Solid Mechanics, vol. I, North-Holland, Amsterdam 263-328 (1960).
[20] A. Nayfeh, S. Nemat-Nasser, Thermoelastic waves in solids with thermal Relaxation. Acta Mech.12, 53-69 (1971).
[21] P. Puri, Plane waves in generalized thermoelasticity, Int. J. Eng. Sci. 11, 735-744 (1973).
[22] V.K. Agarwal, On plane waves in generalized thermoelasticity. Acta Mech. 31, 185-198 (1979).
[23] J.B. Haddow, J.L. Wegner, Plane harmonic waves for three thermoelastic theories. Math. Mech. Solids 1, 111-127, (1996).
[24] C.S. Suh, C.P. Burger, Effects of thermomechanical coupling and relaxation times on wave spectrum in dynamic theory of generalized thermoelasticity. J. Appl. Mech. (Trans. ASME)
65, 605-613 (1998).
[25] D.S. Chandrasekharaiah, Thermoelastic plane waves without energy dissipation. Mech. Res. Commun. 23, 549-555 (1996).
[26] P. Puri, P.M. Jordan, On the propagation of plane waves in type-III thermoelastic media. Proc. R. Soc. A 460, 3203- 3221, 2004.
[27] S. Punnusamy, Foundation of complex analysis. Narosa Publishing House (2001).
The present paper attempts to investigate the propagation of harmonic plane waves of assigned frequency by employing the thermoelasticity theory with three phase-lags, recently proposed by Roychoudhuri (2007). The solutions of dispersion relation for the longitudinal plane waves are determined analytically and asymptotic expansions of several characterizations of the wave fields- phase velocity, specific loss and penetration depth of the dilatational waves are obtained for both the high frequency and low frequency values. Computational work for numerical values of the above quantities is also carried out with the help of Mathematica. A detailed analysis of the effects of phase-lags on the plane wave is presented by contrasting the theoretical as well as numerical results of the present work
with the corresponding results of the theory of thermoelasticity type III (Green and Naghdi, 1993) as reported earlier.
Key words:
generalized thermoelasticity, harmonic plane wave, thermoelasticity type III, thermoelasticity with three phase-lags
References:
[1] M. Biot, Thermoelasticity and Irreversible thermodynamics. J. Appl. Phys. 27, 240-253 (1956).
[2] H.W. Lord and Y. Shulman, Generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299-309 (1967).
[3] A.E Green, K.A Lindsay, Thermoelasticity. Journal of Elasticity 2, 1-7 (1972).
[4] A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermoemechanics. Proc. Roy. Soc. London, A432, 171-194 (1991).
[5] A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid. J. Thermal stresses 15, 253-264 (1992).
[6] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation. J. Elasticity 31, 189-209 (1993).
[7] D.Y. Tzou, A unified filed approach for heat conduction from Macro to Micro scales. ASME Journal of Heat transfer 117, 8-16 (1995).
[8] R. Quintanilla, R. Racke, A note on stability in dual-phaselag heat conduction, Int. J. Heat and Mass Transfer 49, 7-8, 1209-1213 (2006).
[9] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: A Review of recent literature. Appl. Mech. Rev. 39 (1998).
[10] R.B. Hetnarski, J. Ignaczak, Soliton-like waves in a low temperature and non linear thermoelastic solid. Int. J. Eng. Sci. 34, 1767-1787 (1996).
[11] R. B. Hetnarski, J. Ignaczak, Generalized thermoelasticity. J. Thermal stresses 22, 451-476 (1999).
[12] S.K. Roychoudhuri, On a thermoelastic three-phase-lag model. J. Thermal Stresses 30, 231-238 (2007).
[13] R. Quintanilla, R. Racke, A note on stability in three-phaselag heat conduction. Int. J. Heat and Mass Transfer 51, 24-29 (2008).
[14] A. Kar, M. Kanoria, Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect. European Journal of Mechanics
A/Solids 28, 757-767 (2009).
[15] A. Kar, M. Kanoria, Generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect, Appl. Math. Modelling 33, 3287-3298 (2009).
[16] M. Lessen, Motion of thermoelastic solid, Q. Appl. Maths. 15, 105-108 (1957).
[17] H. Deresiewicz, Plane waves in a thermoelastic solid, J. Acoust. Soc. Am. 29, 204-209 (1957).
[18] P. Chadwick, I.N. Sneddon, Plane waves in an elastic solid conducting heat. J. Mech. Phys. Solids 6, 223-230 (1958).
[19] P. Chadwick, Thermoelasticity: the dynamic theory, in: R. Hill, I.N. Sneddon (Eds.), Progress in Solid Mechanics, vol. I, North-Holland, Amsterdam 263-328 (1960).
[20] A. Nayfeh, S. Nemat-Nasser, Thermoelastic waves in solids with thermal Relaxation. Acta Mech.12, 53-69 (1971).
[21] P. Puri, Plane waves in generalized thermoelasticity, Int. J. Eng. Sci. 11, 735-744 (1973).
[22] V.K. Agarwal, On plane waves in generalized thermoelasticity. Acta Mech. 31, 185-198 (1979).
[23] J.B. Haddow, J.L. Wegner, Plane harmonic waves for three thermoelastic theories. Math. Mech. Solids 1, 111-127, (1996).
[24] C.S. Suh, C.P. Burger, Effects of thermomechanical coupling and relaxation times on wave spectrum in dynamic theory of generalized thermoelasticity. J. Appl. Mech. (Trans. ASME)
65, 605-613 (1998).
[25] D.S. Chandrasekharaiah, Thermoelastic plane waves without energy dissipation. Mech. Res. Commun. 23, 549-555 (1996).
[26] P. Puri, P.M. Jordan, On the propagation of plane waves in type-III thermoelastic media. Proc. R. Soc. A 460, 3203- 3221, 2004.
[27] S. Punnusamy, Foundation of complex analysis. Narosa Publishing House (2001).
