Propagation of Harmonic Plane Waves in a Rotating Elastic Medium under Two-Temperature Thermoelasticity with Relaxation Parameter
Mukhopadhyay Santwana, Prasad Rajesh *
Department of Applied Mathematics
Institute of Technology
Banaras Hindu University, Varanasi-221005, India
*Corresponding author: e-mail: mukhosant.apm@itbhu.ac.in
Received:
(Received: 17 July 2011; revised: 03 May 2012; accepted: 04 May 2012; published online: 11 June 2012)
DOI: 10.12921/cmst.2012.18.01.25-37
OAI: oai:lib.psnc.pl:422
Abstract:
The present work investigates the propagation of harmonic plane waves in an isotropic and homogeneous elastic medium that is rotating with uniform angular velocity by employing the two-temperature generalized thermoelasticity, recently introduced by Youssef (IMA Journal of Applied Mathematics, 71, 383-390, 2006). Dispersion relation solutions for longitudinal as well as transverse plane waves are obtained analytically. Asymptotic expressions of several important characterizations of the wave fields, such as phase velocity, specific loss, penetration depth, amplitude coefficient factor and phase shift of thermodynamic temperature are obtained for high frequency as well as low frequency values. A critical value of the two-temperature parameter for the low frequency case is obtained. Using Mathematica, numerical values of the wave fields at intermediate values of frequency and for various values of the twotemperature parameter are computed. A detailed analysis of the effects of rotation on the plane wave is presented on the basis of analytical and numerical results. An in-depth comparative analysis of our results with the corresponding results of the special cases of absence of rotation of the body and with the case of generalized thermoelasticity is also presented. The most significant points are highlighted.
Key words:
generalized thermoelasticity, penetration depth, phase velocity, rotating elastic medium, specific loss, two-temperature thermoelasticity
References:
[1] P.J. Chen, M.E. Gurtin, On a theory of heat conduction involving two-temperatures. J. Appl. Math. Phys. (ZAMP) 19, 614-627, (1968).
[2] P.J. Chen, M.E. Gurtin, W.O. Williams, On the thermodynamics of non-simple elastic materials with two temperatures. J. Appl. Math. Phys. (ZAMP) 20, 107-112 (1969).
[3] M.E. Gurtin, W.O. Williams, On the Clausius-Duhem inequality. Z. angew. Math Phys. 17, 626-633 (1966).
[4] D. Iesan, On the thermodynamics of non-simple elastic materials with two temperatures. J. Appl. Math. Phys. (ZAMP) 21, 583-591 (1970).
[5] W.E. Warren, P.J. Chen, Wave propagation in the two temperature theory of thermoelasticity. Acta Mech. 16, 21-33 (1973).
[6] W.E. Warren, Thermoelastic wave propagation from cylindrical and spherical cavities in the two-temperature theory. J. Appl. Phys. 43, 3595-3597 (1972).
[7] D.E. Amos, On a half-space solution of a modified heat equation. Quart. Appl. Math. 27, 359-369 (1969).
[8] S. Chakrabarti, Thermoelastic waves in non-simple media. Pure Appl. Geophys. 109, 1682-1692 (1973).
[9] T.W. Ting, A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45, 23-31 (1974).
[10] D. Colton, J. Wimp, Asymptotic behavior of the fundamental solution to the equation of heat conduction in two-temperature. J. Math. Anal. Appl. 69, 411-418 (1979).
[11] R. Quintanilla, On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mech. 168, 61-73 (2004).
[12] P. Puri, P.M. Jordan, On the propagation of harmonic plane waves under the two- temperature theory. Int. J. Engg. Sci. 44, 1113-1126 (2006).
[13] H.M. Youssef, Theory of two-temperature-generalized thermoelasticity. IMA J. Appl. Math. 71, 383-390 (2006).
[14] H.W. Lord, Y. Shulman, A Generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299-309 (1967).
[15] A.E. Green, K.A. Lindsay, Thermoelasticity. J. Elasticity 2, 1-7 (1972).
[16] A. Magana, R. Quintanilla, Uniqueness and growth of solutions in two temperature generalized thermoelastic theories. Math Mech. Solids 14, 622-634 (2009).
[17] H.M. Youssef, E.A. Al-Lehaibi, State-space approach of two-temperature generalized thermoelasticity of one dimensional problem. Int. J. Solids Struct. 44, 1550-1562 (2007).
[18] H.M. Youssef, Problem of generalized thermoelastic infinite cylindrical cavity subjected to a ramp-type heating and loading. Arch. Appl. Mech. 75, 553-565 (2006).
[19] H.M. Youssef, Theory of two-temperature-generalized thermoelasticity. IMA J. Appl. Math. 71, 383-390 (2006).
[20] H.M. Youssef, Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Arch. Appl. Mech. 80, 1213-1224 (2010).
[21] R. Kumar, R. Prasad, S. Mukhopadhyay, Variational and reciprocal principles in two- temperature generalized thermoelasticity. J. Therm. Stresses 33, 161-171 (2010).
[22] S. Mukhopadhyay, R. Kumar, Thermoelastic interaction on two-temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity. J. Therm. Stresses 32, 341-360 (2009).
[23] R. Quintanilla, P.M. Jordan, A note on the two temperature theory with dual phase lag delay: some exact solutions. Mech. Res. Comm. 36, 796-803 (2009).
[24] R. Kumar, S. Mukhopadhyay, Effects of thermal relaxation time on plane wave propagation under two-temperature thermoelasticity. Int. J. Engg. Sci. 48, 128-139 (2010).
[25] R. Kumar, R. Prasad, S. Mukhopadhyay, Some theorems on two-temperature generalized thermoelasticity. Arch. Appl. Mech. 81, 1031-1040 (2011).
[26] M. Ezzat, F. Hamza, E. Awad, Electro-magneto-thermoelastic plane waves in micropolar solid involving two temperatures. Acta Mech. Solida Sinica. 23, 200-212 (2010).
[27] S. Mukhopadhyay, R. Prasad, R. Kumar, On the theory of two-temperature generalized thermoelasticity with dual phase lags. J. Therm. Stresses 34, 352-365 (2011).
[28] P. Chadwick, I.N. Sneddon, Plane waves in an elastic solid conducting heat. J. Mech. Phys Solids 6, 223-230 (1958).
[29] P. Chadwick, Thermoelasticity: the dynamic theory. in: R. Hill, I.N. Sneddon (Eds.), Progress in Solid Mechanics, North-Holland, Amsterdam, I, 263-328 (1960).
[30] A. Nayfeh, S. Nemat-Nasser, Thermoelastic waves in solids with thermal relaxation. Acta Mech. 12, 53-69 (1971).
[31] P. Puri, Plane waves in generalized thermoelasticity. Int. J. Eng. Sci. 11, 735-744 (1973).
[32] V.K. Agarwal, On plane waves in generalized thermoelasticity. Acta Mech. 31, 185-198 (1979).
[33] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation. J. Elast. 31, 189-208 (1993).
[34] D.S. Chandrasekharaiah, Thermoelastic plane waves without energy dissipation. Mech. Res. Commun. 23, 549-555 (1996).
[35] P. Puri, P.M. Jordan, On the propagation of plane waves in type-III thermoelastic media. Proc. Royal Soc. A 460, 3203- -3221 (2004).
[36] A.E. Green, M. Naghdi, On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253-264 (1992).
[37] M. Schoenberg, D. Censor, Elastic waves in rotating media. Quart. Appl. Math. 31, 115-125 (1973).
[38] P. Puri, Plane thermoelastic waves in rotating media. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 24, 137-144 (1976).
[39] D.S. Chandrasekharaiah, K.R. Srikantiah, Thermoelastic plane waves in a rotating solid. Acta Mech. 50, 211-219 (1984).
[40] S.K. Roychoudhari, Effect of rotation and relaxation times on plane waves in generalized thermoelasticity. J. Elasticity 21, 59-68 (1985).
[41] S.K. Roychoudhari, N. Bandyopadhyay, Thermoelastic wave propagation in rotating elastic medium without energy dissipation. Int. J. Math. and Math. Sci. 1, 99-107 (2005).
[42] D.S. Chandrasekharaiah, Thermoelastic plane waves without energy dissipation in a rotating body. Mech. Res. Comm. 24, 551-560 (1997).
[43] D.S. Chandrasekharaiah, Plane waves in a rotating elastic solid with voids. Int. J. Engg. Sci. 25, 591-596 (1987).
[44] Mohamed I.A. Othman, Effect of rotation on plane waves in generalized thermo-elasticity with two relaxation times. Int. J. Solids and Structures, 41, 2939-2956 (2004).
[45] J.L. Auriault, Body wave propagation in rotating elastic media. Mech. Res. Comm. 31, 21-27 (2004).
[46] J.N. Sharma, Mohamed I.A. Othman, Effect of rotation on generalized thermo-viscoelastic Rayleigh–Lamb waves. Int. J. Solids and Structures 44, 4243-4255 (2007).
[47] S. Punnusamy, Foundation of complex analysis. Narosa Publishing House (2001).
The present work investigates the propagation of harmonic plane waves in an isotropic and homogeneous elastic medium that is rotating with uniform angular velocity by employing the two-temperature generalized thermoelasticity, recently introduced by Youssef (IMA Journal of Applied Mathematics, 71, 383-390, 2006). Dispersion relation solutions for longitudinal as well as transverse plane waves are obtained analytically. Asymptotic expressions of several important characterizations of the wave fields, such as phase velocity, specific loss, penetration depth, amplitude coefficient factor and phase shift of thermodynamic temperature are obtained for high frequency as well as low frequency values. A critical value of the two-temperature parameter for the low frequency case is obtained. Using Mathematica, numerical values of the wave fields at intermediate values of frequency and for various values of the twotemperature parameter are computed. A detailed analysis of the effects of rotation on the plane wave is presented on the basis of analytical and numerical results. An in-depth comparative analysis of our results with the corresponding results of the special cases of absence of rotation of the body and with the case of generalized thermoelasticity is also presented. The most significant points are highlighted.
Key words:
generalized thermoelasticity, penetration depth, phase velocity, rotating elastic medium, specific loss, two-temperature thermoelasticity
References:
[1] P.J. Chen, M.E. Gurtin, On a theory of heat conduction involving two-temperatures. J. Appl. Math. Phys. (ZAMP) 19, 614-627, (1968).
[2] P.J. Chen, M.E. Gurtin, W.O. Williams, On the thermodynamics of non-simple elastic materials with two temperatures. J. Appl. Math. Phys. (ZAMP) 20, 107-112 (1969).
[3] M.E. Gurtin, W.O. Williams, On the Clausius-Duhem inequality. Z. angew. Math Phys. 17, 626-633 (1966).
[4] D. Iesan, On the thermodynamics of non-simple elastic materials with two temperatures. J. Appl. Math. Phys. (ZAMP) 21, 583-591 (1970).
[5] W.E. Warren, P.J. Chen, Wave propagation in the two temperature theory of thermoelasticity. Acta Mech. 16, 21-33 (1973).
[6] W.E. Warren, Thermoelastic wave propagation from cylindrical and spherical cavities in the two-temperature theory. J. Appl. Phys. 43, 3595-3597 (1972).
[7] D.E. Amos, On a half-space solution of a modified heat equation. Quart. Appl. Math. 27, 359-369 (1969).
[8] S. Chakrabarti, Thermoelastic waves in non-simple media. Pure Appl. Geophys. 109, 1682-1692 (1973).
[9] T.W. Ting, A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45, 23-31 (1974).
[10] D. Colton, J. Wimp, Asymptotic behavior of the fundamental solution to the equation of heat conduction in two-temperature. J. Math. Anal. Appl. 69, 411-418 (1979).
[11] R. Quintanilla, On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mech. 168, 61-73 (2004).
[12] P. Puri, P.M. Jordan, On the propagation of harmonic plane waves under the two- temperature theory. Int. J. Engg. Sci. 44, 1113-1126 (2006).
[13] H.M. Youssef, Theory of two-temperature-generalized thermoelasticity. IMA J. Appl. Math. 71, 383-390 (2006).
[14] H.W. Lord, Y. Shulman, A Generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299-309 (1967).
[15] A.E. Green, K.A. Lindsay, Thermoelasticity. J. Elasticity 2, 1-7 (1972).
[16] A. Magana, R. Quintanilla, Uniqueness and growth of solutions in two temperature generalized thermoelastic theories. Math Mech. Solids 14, 622-634 (2009).
[17] H.M. Youssef, E.A. Al-Lehaibi, State-space approach of two-temperature generalized thermoelasticity of one dimensional problem. Int. J. Solids Struct. 44, 1550-1562 (2007).
[18] H.M. Youssef, Problem of generalized thermoelastic infinite cylindrical cavity subjected to a ramp-type heating and loading. Arch. Appl. Mech. 75, 553-565 (2006).
[19] H.M. Youssef, Theory of two-temperature-generalized thermoelasticity. IMA J. Appl. Math. 71, 383-390 (2006).
[20] H.M. Youssef, Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Arch. Appl. Mech. 80, 1213-1224 (2010).
[21] R. Kumar, R. Prasad, S. Mukhopadhyay, Variational and reciprocal principles in two- temperature generalized thermoelasticity. J. Therm. Stresses 33, 161-171 (2010).
[22] S. Mukhopadhyay, R. Kumar, Thermoelastic interaction on two-temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity. J. Therm. Stresses 32, 341-360 (2009).
[23] R. Quintanilla, P.M. Jordan, A note on the two temperature theory with dual phase lag delay: some exact solutions. Mech. Res. Comm. 36, 796-803 (2009).
[24] R. Kumar, S. Mukhopadhyay, Effects of thermal relaxation time on plane wave propagation under two-temperature thermoelasticity. Int. J. Engg. Sci. 48, 128-139 (2010).
[25] R. Kumar, R. Prasad, S. Mukhopadhyay, Some theorems on two-temperature generalized thermoelasticity. Arch. Appl. Mech. 81, 1031-1040 (2011).
[26] M. Ezzat, F. Hamza, E. Awad, Electro-magneto-thermoelastic plane waves in micropolar solid involving two temperatures. Acta Mech. Solida Sinica. 23, 200-212 (2010).
[27] S. Mukhopadhyay, R. Prasad, R. Kumar, On the theory of two-temperature generalized thermoelasticity with dual phase lags. J. Therm. Stresses 34, 352-365 (2011).
[28] P. Chadwick, I.N. Sneddon, Plane waves in an elastic solid conducting heat. J. Mech. Phys Solids 6, 223-230 (1958).
[29] P. Chadwick, Thermoelasticity: the dynamic theory. in: R. Hill, I.N. Sneddon (Eds.), Progress in Solid Mechanics, North-Holland, Amsterdam, I, 263-328 (1960).
[30] A. Nayfeh, S. Nemat-Nasser, Thermoelastic waves in solids with thermal relaxation. Acta Mech. 12, 53-69 (1971).
[31] P. Puri, Plane waves in generalized thermoelasticity. Int. J. Eng. Sci. 11, 735-744 (1973).
[32] V.K. Agarwal, On plane waves in generalized thermoelasticity. Acta Mech. 31, 185-198 (1979).
[33] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation. J. Elast. 31, 189-208 (1993).
[34] D.S. Chandrasekharaiah, Thermoelastic plane waves without energy dissipation. Mech. Res. Commun. 23, 549-555 (1996).
[35] P. Puri, P.M. Jordan, On the propagation of plane waves in type-III thermoelastic media. Proc. Royal Soc. A 460, 3203- -3221 (2004).
[36] A.E. Green, M. Naghdi, On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253-264 (1992).
[37] M. Schoenberg, D. Censor, Elastic waves in rotating media. Quart. Appl. Math. 31, 115-125 (1973).
[38] P. Puri, Plane thermoelastic waves in rotating media. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 24, 137-144 (1976).
[39] D.S. Chandrasekharaiah, K.R. Srikantiah, Thermoelastic plane waves in a rotating solid. Acta Mech. 50, 211-219 (1984).
[40] S.K. Roychoudhari, Effect of rotation and relaxation times on plane waves in generalized thermoelasticity. J. Elasticity 21, 59-68 (1985).
[41] S.K. Roychoudhari, N. Bandyopadhyay, Thermoelastic wave propagation in rotating elastic medium without energy dissipation. Int. J. Math. and Math. Sci. 1, 99-107 (2005).
[42] D.S. Chandrasekharaiah, Thermoelastic plane waves without energy dissipation in a rotating body. Mech. Res. Comm. 24, 551-560 (1997).
[43] D.S. Chandrasekharaiah, Plane waves in a rotating elastic solid with voids. Int. J. Engg. Sci. 25, 591-596 (1987).
[44] Mohamed I.A. Othman, Effect of rotation on plane waves in generalized thermo-elasticity with two relaxation times. Int. J. Solids and Structures, 41, 2939-2956 (2004).
[45] J.L. Auriault, Body wave propagation in rotating elastic media. Mech. Res. Comm. 31, 21-27 (2004).
[46] J.N. Sharma, Mohamed I.A. Othman, Effect of rotation on generalized thermo-viscoelastic Rayleigh–Lamb waves. Int. J. Solids and Structures 44, 4243-4255 (2007).
[47] S. Punnusamy, Foundation of complex analysis. Narosa Publishing House (2001).
