**Abstract:**The present problem deals with the thermo-elastic interaction of a gold nano-beam resonator induced by ramp-type heating under the two temperature theory of generalized thermoelasticity. The governing equations are constructed in the context of two-temperature three-phase-lag model (2T3P) and two-temperature Lord-Shulman (2TLS) model of generalized thermoelasticity. Using the Laplace transform, the fundamental equations have been expressed in the form of a vector-matrix differential equation which is then solved by Eigen value approach and Mathematica software package has been used as a tool. The inversion of Laplace transforms are computed numerically using the method of Fourier series expansion technique. Numerical results for lateral vibration, temperature, displacement, stress, and the strain energy are presented graphically for Lord-Shulman model and also for three-phase lag model. A numerical instance of gold nano-beam in femtoseconds scale has been calculated to present the effect of the ramping time parameter on the entire studied field. The effect of two-temperature parameter is also discussed on the physical fields.

**How to cite this paper**

Mondal, S., Sur, A & Kanoria, M. (2017). Modeling and analysis of vibration of a gold nano-beam under two-temperature theory.

*Engineering Solid Mechanics*, 5(1), 15-30.**Refrences**

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Ackerman, C. C., & Guyer, R. A. (1968). Temperature pulses in dielectric solids. Annals of Physics, 50(1), 128-185.

Bagri, A., & Eslami, M. R. (2004). Generalized coupled thermoelasticity of disks based on the Lord–Shulman model. Journal of Thermal Stresses, 27(8), 691-704.

Al-Huniti, N. S., Al-Nimr, M. A., & Naji, M. (2001). Dynamic response of a rod due to a moving heat source under the hyperbolic heat conduction model. Journal of Sound and Vibration, 242(4), 629-640.

Bagri, A., & Eslami, M. R. (2007). A unified generalized thermoelasticity formulation; application to thick functionally graded cylinders. Journal of Thermal Stresses, 30(9-10), 911-930.

Bagri, A., & Eslami, M. R. (2007). Analysis of thermoelastic waves in functionally graded hollow spheres based on the Green-Lindsay theory. Journal of Thermal Stresses, 30(12), 1175-1193.

Banik, S., & Kanoria, M. (2011). Two-temperature generalized thermoelastic interactions in an infinite body with a spherical cavity. International Journal of Thermophysics, 32(6), 1247-1270.

Banik, S., & Kanoria, M. (2012). Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity. Applied Mathematics and Mechanics, 33(4), 483-498.

Boley, B. A. (1972). Approximate analyses of thermally induced vibrations of beams and plates. Journal of Applied Mechanics, 39(1), 212-216.

Chandrasekharaiah, D. S. (1996). Thermoelastic plane waves without energy dissipation. Mechanics research communications, 23(5), 549-555.

Chandrasekharaiah, D. S. (1996). A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation. Journal of Elasticity, 43(3), 279-283.

Chen, P. J., & Gurtin, M. E. (1968). On a theory of heat conduction involving two temperatures. Zeitschrift für angewandte Mathematik und Physik (ZAMP), 19(4), 614-627.

Chen, P. J., & Williams, W. O. (1968). A note on non-simple heat conduction. Zeitschrift für angewandte Mathematik und Physik ZAMP, 19(6), 969-970.

Chen, P. J., Gurtin, M. E., & Williams, W. O. (1969). On the thermodynamics of non-simple elastic materials with two temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 20(1), 107-112.

Das, N. C., & Lahiri, A. (2000). Thermo-elastic interactions due to prescribed pressure inside a spherical cavity in an unbounded medium. International Journal of Pure and Applied Mathematics, 31, 19-32.

Dhaliwal, R. S., & SHERIEF, H. H. (1980). Generalized thermoelasticity for anisotropic media. Quarterly of Applied Mathematics, 38(1), 1-8.

Diao, J., Gall, K., & Dunn, M. L. (2004). Atomistic simulation of the structure and elastic properties of gold nanowires. Journal of the Mechanics and Physics of Solids, 52(9), 1935-1962.

El-Karamany, A. S., & Ezzat, M. A. (2011). On the two-temperature Green–Naghdi thermoelasticity theories. Journal of Thermal Stresses, 34(12), 1207-1226.

Elsibai*, K. A., & Youssef†, H. M. (2011). State-space approach to vibration of gold nano-beam induced by ramp type heating without energy dissipation in femtoseconds scale. Journal of Thermal Stresses, 34(3), 244-263.

Ghosh, M. K., & Kanoria, M. (2008). Generalized thermoelastic problem of a spherically isotropic infinite elastic medium containing a spherical cavity. Journal of Thermal Stresses, 31(8), 665-679.

Ghosh, M. K., & Kanoria, M. (2009). Analysis of thermoelastic response in a functionally graded spherically isotropic hollow sphere based on Green–Lindsay theory. Acta mechanica, 207(1-2), 51-67.

Kanoria, M., & Ghosh, M. K. (2010). Study of dynamic response in a functionally graded spherically isotropic hollow sphere with temperature dependent elastic parameters. Journal of Thermal Stresses, 33(5), 459-484.

Green A. E., & Lindsay, K. A. (1972). Thermoelasticity, Journal of Elasticity, 2, 1-7.

Green, A. E., & Naghdi, P. M. (1991, February). A re-examination of the basic postulates of thermomechanics. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 432, No. 1885, pp. 171-194). The Royal Society.

Green, A. E., & Naghdi, P. M., (1992). An unbounded heat wave in an elastic solid, Journal of Thermal Stresses, 15, 253-264.

Green, A. E., & Naghdi, P. M. (1993). Thermoelasticity without energy dissipation. Journal of elasticity, 31(3), 189-208.

Gurtin, M. E., & Williams, W. O. (1966). On the clausius-duhem inequality, Z. angew. Math. Phys., 7, 626-633.

Gurtin, M. E., & Williams, W. O. (1967). An axiomatic foundation for continuum thermodynamics, Archive for Rational Mechanics and Analysis, 26, 83-117.

Honig, G., & Hirdes, U. (1984). A method for the numerical inversion of Laplace transforms. Journal of Computational and Applied Mathematics, 10(1), 113-132.

Ignaczak, J. (1979). Uniqueness in generalized thermoelasticity. Journal of Thermal Stresses, 2(2), 171-175.

Ignaczak, J. (1982). A note on uniqueness in thermoelasticity with one relaxation time. Journal of Thermal Stresses, 5(3-4), 257-263.

Ignaczak, J., & Ostoja-Starzewski, M. (2010). Thermoelasticity with finite wave speeds. Oxford University Press.

Islam, M., Kar, A., & Kanoria, M. (2013). Two-temperature generalized thermoelasticity in a fiber-reinforced hollow cylinder under thermal shock. International Journal for Computational Methods in Engineering Science and Mechanics, 14(5), 367-390.

Islam, M., Mallik, S. H., & Kanoria, M. (2011). Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces. Applied Mathematics and Mechanics, 32(10), 1315-1332.

Kar, A., & Kanoria, M. (2007a). Thermoelastic interaction with energy dissipation in a transversely isotropic thin circular disc. European Journal of Mechanics A/Solids, 26, 969-981.

Kar, A., & Kanoria, M. (2007b). Thermoelastic interaction with energy dissipation in an unbounded body with a spherical hole. International Journal of Solids and Structures, 44, 2961-2971.

Kar, A., & Kanoria, M. (2009). Generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect. Applied Mathematical Modeling, 33, 3287-3298.

Kar, A., & Kanoria, M. (2009). Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect. European Journal of Mechanics A/Solids, 28, 757-767.

Kidawa-Kukla, J. (2003). Application of the Green functions to the problem of the thermally induced vibration of a beam. Journal of Sound and Vibration, 262, 865-876.

Kumar, R., Prasad, R., & Mukhopadhyay, S. (2010). Variational and reciprocal principles in two-temperature generalized thermoelasticity. Journal of Thermal Stresses, 33(3), 161-171.

Kumar, R., Prasad, R., & Mukhopadhyay, S. (2011). Some theorems on two-temperature generalized thermoelasticity. Archive of Applied Mechanics, 81(8), 1031-1040.

Chen, P. J., Gurtin, M. E., & Williams, W. O. (1969). On the thermodynamics of non-simple elastic materials with two temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 20(1), 107-112.

Lord, H. W., & Shulman, Y. (1967). A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids, 15(5), 299-309.

Manolis, G. D., & Beskos, D. E. (1980). Thermally induced vibrations of beam structures. Computer Methods in Applied Mechanics and Engineering, 21(3), 337-355.

Puri, P., & Jordan, P. M. (2006). On the propagation of harmonic plane waves under the two-temperature theory. International Journal of Engineering Science, 44(17), 1113-1126.

Quintanilla, R. (2004a). Exponential stability and uniqueness in thermoelasticity with two temperature. Dynamics of Continuous Discrete and Impulsive Systems A, 11, 57-68.

Quintanilla, R. (2004). On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mechanica, 168(1-2), 61-73.

Quintanilla, R. (2008). A well-posed problem for the dual-phase-lag heat conduction. Journal of Thermal Stresses, 31(3), 260-269.

Quintanilla, R. (2009). A well-posed problem for the three-dual-phase-lag heat conduction. Journal of Thermal Stresses, 32(12), 1270-1278.

Choudhuri, S. R. (2007). On a thermoelastic three-phase-lag model. Journal of Thermal Stresses, 30(3), 231-238.

Sur, A., & Kanoria, M. (2014). Thermoelastic interaction in a viscoelastic functionally graded half-space under three-phase-lag model. European Journal of Computational Mechanics, 23(5-6), 179-198.

Soh, A. K., Sun, Y., & Fang, D. (2008). Vibration of microscale beam induced by laser pulse. Journal of sound and vibration, 311(1), 243-253.

Sun, Y., Fang, D., Saka, M., & Soh, A. K. (2008). Laser-induced vibrations of micro-beams under different boundary conditions. International Journal of Solids and Structures, 45(7), 1993-2013.

Sur, A., & Kanoria, M. (2014b). Vibration of a gold nano beam induced by ramp-type laser pulse under three-phase-lag model. International Journal of Applied Mathematics and Mechanics, 10(5), 86-104.

Sur, A., & Kanoria, M. (2014). Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell. Latin American Journal of Solids and Structures, 11(7), 1132-1162.

Sur, A., & Kanoria, M. (2014). Fractional order generalized thermoelastic functionally graded solid with variable material properties. Journal of Solid Mechanics, 6(1), 54-69.

Sur, A., & Kanoria, M. (2012). Fractional order two-temperature thermoelasticity with finite wave speed. Acta Mechanica, 223(12), 2685-2701.

Sur, A., & Kanoria, M. (2014). Finite thermal wave propagation in a half-space due to variable thermal loading. Applications and Applied Mathematics, 9(1), 94-120.

Warren, W. E., & Chen, P. J. (1973). Wave propagation in the two temperature theory of thermoelasticity. Acta Mechanica, 16(1-2), 21-33.

Youssef, H. M., & Al-Lehaibi, E. A. (2007). State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem. International journal of solids and structures, 44(5), 1550-1562.

Youssef, H. M. (2006). Theory of two-temperature-generalized thermoelasticity. IMA Journal of Applied Mathematics, 71(3), 383-390.

Youssef, H. M. (2008). Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating. Computational Mathematics and Modeling, 19(2), 201-216.

Ackerman, C. C., & Guyer, R. A. (1968). Temperature pulses in dielectric solids. Annals of Physics, 50(1), 128-185.

Bagri, A., & Eslami, M. R. (2004). Generalized coupled thermoelasticity of disks based on the Lord–Shulman model. Journal of Thermal Stresses, 27(8), 691-704.

Al-Huniti, N. S., Al-Nimr, M. A., & Naji, M. (2001). Dynamic response of a rod due to a moving heat source under the hyperbolic heat conduction model. Journal of Sound and Vibration, 242(4), 629-640.

Bagri, A., & Eslami, M. R. (2007). A unified generalized thermoelasticity formulation; application to thick functionally graded cylinders. Journal of Thermal Stresses, 30(9-10), 911-930.

Bagri, A., & Eslami, M. R. (2007). Analysis of thermoelastic waves in functionally graded hollow spheres based on the Green-Lindsay theory. Journal of Thermal Stresses, 30(12), 1175-1193.

Banik, S., & Kanoria, M. (2011). Two-temperature generalized thermoelastic interactions in an infinite body with a spherical cavity. International Journal of Thermophysics, 32(6), 1247-1270.

Banik, S., & Kanoria, M. (2012). Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity. Applied Mathematics and Mechanics, 33(4), 483-498.

Boley, B. A. (1972). Approximate analyses of thermally induced vibrations of beams and plates. Journal of Applied Mechanics, 39(1), 212-216.

Chandrasekharaiah, D. S. (1996). Thermoelastic plane waves without energy dissipation. Mechanics research communications, 23(5), 549-555.

Chandrasekharaiah, D. S. (1996). A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation. Journal of Elasticity, 43(3), 279-283.

Chen, P. J., & Gurtin, M. E. (1968). On a theory of heat conduction involving two temperatures. Zeitschrift für angewandte Mathematik und Physik (ZAMP), 19(4), 614-627.

Chen, P. J., & Williams, W. O. (1968). A note on non-simple heat conduction. Zeitschrift für angewandte Mathematik und Physik ZAMP, 19(6), 969-970.

Chen, P. J., Gurtin, M. E., & Williams, W. O. (1969). On the thermodynamics of non-simple elastic materials with two temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 20(1), 107-112.

Das, N. C., & Lahiri, A. (2000). Thermo-elastic interactions due to prescribed pressure inside a spherical cavity in an unbounded medium. International Journal of Pure and Applied Mathematics, 31, 19-32.

Dhaliwal, R. S., & SHERIEF, H. H. (1980). Generalized thermoelasticity for anisotropic media. Quarterly of Applied Mathematics, 38(1), 1-8.

Diao, J., Gall, K., & Dunn, M. L. (2004). Atomistic simulation of the structure and elastic properties of gold nanowires. Journal of the Mechanics and Physics of Solids, 52(9), 1935-1962.

El-Karamany, A. S., & Ezzat, M. A. (2011). On the two-temperature Green–Naghdi thermoelasticity theories. Journal of Thermal Stresses, 34(12), 1207-1226.

Elsibai*, K. A., & Youssef†, H. M. (2011). State-space approach to vibration of gold nano-beam induced by ramp type heating without energy dissipation in femtoseconds scale. Journal of Thermal Stresses, 34(3), 244-263.

Ghosh, M. K., & Kanoria, M. (2008). Generalized thermoelastic problem of a spherically isotropic infinite elastic medium containing a spherical cavity. Journal of Thermal Stresses, 31(8), 665-679.

Ghosh, M. K., & Kanoria, M. (2009). Analysis of thermoelastic response in a functionally graded spherically isotropic hollow sphere based on Green–Lindsay theory. Acta mechanica, 207(1-2), 51-67.

Kanoria, M., & Ghosh, M. K. (2010). Study of dynamic response in a functionally graded spherically isotropic hollow sphere with temperature dependent elastic parameters. Journal of Thermal Stresses, 33(5), 459-484.

Green A. E., & Lindsay, K. A. (1972). Thermoelasticity, Journal of Elasticity, 2, 1-7.

Green, A. E., & Naghdi, P. M. (1991, February). A re-examination of the basic postulates of thermomechanics. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 432, No. 1885, pp. 171-194). The Royal Society.

Green, A. E., & Naghdi, P. M., (1992). An unbounded heat wave in an elastic solid, Journal of Thermal Stresses, 15, 253-264.

Green, A. E., & Naghdi, P. M. (1993). Thermoelasticity without energy dissipation. Journal of elasticity, 31(3), 189-208.

Gurtin, M. E., & Williams, W. O. (1966). On the clausius-duhem inequality, Z. angew. Math. Phys., 7, 626-633.

Gurtin, M. E., & Williams, W. O. (1967). An axiomatic foundation for continuum thermodynamics, Archive for Rational Mechanics and Analysis, 26, 83-117.

Honig, G., & Hirdes, U. (1984). A method for the numerical inversion of Laplace transforms. Journal of Computational and Applied Mathematics, 10(1), 113-132.

Ignaczak, J. (1979). Uniqueness in generalized thermoelasticity. Journal of Thermal Stresses, 2(2), 171-175.

Ignaczak, J. (1982). A note on uniqueness in thermoelasticity with one relaxation time. Journal of Thermal Stresses, 5(3-4), 257-263.

Ignaczak, J., & Ostoja-Starzewski, M. (2010). Thermoelasticity with finite wave speeds. Oxford University Press.

Islam, M., Kar, A., & Kanoria, M. (2013). Two-temperature generalized thermoelasticity in a fiber-reinforced hollow cylinder under thermal shock. International Journal for Computational Methods in Engineering Science and Mechanics, 14(5), 367-390.

Islam, M., Mallik, S. H., & Kanoria, M. (2011). Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces. Applied Mathematics and Mechanics, 32(10), 1315-1332.

Kar, A., & Kanoria, M. (2007a). Thermoelastic interaction with energy dissipation in a transversely isotropic thin circular disc. European Journal of Mechanics A/Solids, 26, 969-981.

Kar, A., & Kanoria, M. (2007b). Thermoelastic interaction with energy dissipation in an unbounded body with a spherical hole. International Journal of Solids and Structures, 44, 2961-2971.

Kar, A., & Kanoria, M. (2009). Generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect. Applied Mathematical Modeling, 33, 3287-3298.

Kar, A., & Kanoria, M. (2009). Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect. European Journal of Mechanics A/Solids, 28, 757-767.

Kidawa-Kukla, J. (2003). Application of the Green functions to the problem of the thermally induced vibration of a beam. Journal of Sound and Vibration, 262, 865-876.

Kumar, R., Prasad, R., & Mukhopadhyay, S. (2010). Variational and reciprocal principles in two-temperature generalized thermoelasticity. Journal of Thermal Stresses, 33(3), 161-171.

Kumar, R., Prasad, R., & Mukhopadhyay, S. (2011). Some theorems on two-temperature generalized thermoelasticity. Archive of Applied Mechanics, 81(8), 1031-1040.

Chen, P. J., Gurtin, M. E., & Williams, W. O. (1969). On the thermodynamics of non-simple elastic materials with two temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 20(1), 107-112.

Lord, H. W., & Shulman, Y. (1967). A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids, 15(5), 299-309.

Manolis, G. D., & Beskos, D. E. (1980). Thermally induced vibrations of beam structures. Computer Methods in Applied Mechanics and Engineering, 21(3), 337-355.

Puri, P., & Jordan, P. M. (2006). On the propagation of harmonic plane waves under the two-temperature theory. International Journal of Engineering Science, 44(17), 1113-1126.

Quintanilla, R. (2004a). Exponential stability and uniqueness in thermoelasticity with two temperature. Dynamics of Continuous Discrete and Impulsive Systems A, 11, 57-68.

Quintanilla, R. (2004). On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mechanica, 168(1-2), 61-73.

Quintanilla, R. (2008). A well-posed problem for the dual-phase-lag heat conduction. Journal of Thermal Stresses, 31(3), 260-269.

Quintanilla, R. (2009). A well-posed problem for the three-dual-phase-lag heat conduction. Journal of Thermal Stresses, 32(12), 1270-1278.

Choudhuri, S. R. (2007). On a thermoelastic three-phase-lag model. Journal of Thermal Stresses, 30(3), 231-238.

Sur, A., & Kanoria, M. (2014). Thermoelastic interaction in a viscoelastic functionally graded half-space under three-phase-lag model. European Journal of Computational Mechanics, 23(5-6), 179-198.

Soh, A. K., Sun, Y., & Fang, D. (2008). Vibration of microscale beam induced by laser pulse. Journal of sound and vibration, 311(1), 243-253.

Sun, Y., Fang, D., Saka, M., & Soh, A. K. (2008). Laser-induced vibrations of micro-beams under different boundary conditions. International Journal of Solids and Structures, 45(7), 1993-2013.

Sur, A., & Kanoria, M. (2014b). Vibration of a gold nano beam induced by ramp-type laser pulse under three-phase-lag model. International Journal of Applied Mathematics and Mechanics, 10(5), 86-104.

Sur, A., & Kanoria, M. (2014). Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell. Latin American Journal of Solids and Structures, 11(7), 1132-1162.

Sur, A., & Kanoria, M. (2014). Fractional order generalized thermoelastic functionally graded solid with variable material properties. Journal of Solid Mechanics, 6(1), 54-69.

Sur, A., & Kanoria, M. (2012). Fractional order two-temperature thermoelasticity with finite wave speed. Acta Mechanica, 223(12), 2685-2701.

Sur, A., & Kanoria, M. (2014). Finite thermal wave propagation in a half-space due to variable thermal loading. Applications and Applied Mathematics, 9(1), 94-120.

Warren, W. E., & Chen, P. J. (1973). Wave propagation in the two temperature theory of thermoelasticity. Acta Mechanica, 16(1-2), 21-33.

Youssef, H. M., & Al-Lehaibi, E. A. (2007). State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem. International journal of solids and structures, 44(5), 1550-1562.

Youssef, H. M. (2006). Theory of two-temperature-generalized thermoelasticity. IMA Journal of Applied Mathematics, 71(3), 383-390.

Youssef, H. M. (2008). Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating. Computational Mathematics and Modeling, 19(2), 201-216.