Thermoelastic Damping and Frequency Shift in Kirchhoff Plate Resonators Based on Modified Couple Stress Theory With Dual-Phase-Lag Model

Document Type : Research Paper


1 Department of Mathematics, Abhilashi University, Mandi, Himachal Pradesh, India

2 Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India



The present investigation deals with study of thermoelastic damping and frequency shift of Kirchhoff plate resonators by using generalized thermoelasticity theory of dual-phase-lag model. The basic equations of motion and heat conduction equation are written with the help of Kirchhoff-Love plate theory and dual phase lag model. The analytical expressions for thermoelastic damping and frequency shift of modified couple stress dual-phase-lag thermoelastic plate have been obtained. A computer algorithm has been constructed to obtain the numerical results. Influences of modified couple stress dual-phase-lag thermoelastic plate, dual- phase-lag thermoelastic plate and Lord-Shulman (L-S, 1967) thermoelastic plate with few vibration modes on the thermoelastic damping and frequency shift are examined. The thermoelastic damping and frequency shift with varying values of length and thickness are shown graphically for clamped-clamped and simply-supported boundary conditions. It is observed from the results that the  damping factor and frequency shift have noticed larger value in the presence of couple stress for varying values of length but opposite effect are shown for varying values of thickness in case of both vibration modes and boundary conditions.


[1] Abd-Elaziz E.M., Marin M., Othman M.I.A., 2019, On the effect of Thomson and initial stress in a thermo porous elastic solid under GN electromagnetic theory, Symmetry 11(3): 413.
[2] Alashti R.A., Abolghasemi A.H., 2014, A size-dependent Bernoulli-Euler beam formulation based on a new model of couple stress theory, Transactions C: Aspects 27(6): 951-960.
[3] Chen W., Li X., 2014, A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model, Archive of Applied Mechanics 84: 323-341.
[4] Cosserat E., Cosserat F., 1909, Theory of Deformable Bodies, Hermann et Fils, Paris.
[5] Daliwal R.S., Singh A., 1980, Dynamical Coupled Thermoelasticity, Hindustan Publishers, Delhi, India.
[6] Guo F.L., Wang G.Q., Rogerson G.A., 2012, Analysis of thermoelastic damping in micro-and nanomechanical resonators based on dual-phase-lagging generalized thermoelasticity theory, International Journal of Engineering Science 60: 59-65.
[7] Guo F.L., Song J., Wang G.Q., Zhou Y.F., 2014, Analysis of thermoelastic dissipation in circular micro-plate resonators using the generalized thermoelasticity theory of dual-phase-lagging model, Journal of Sound and Vibration 333: 2465-2474.
[8] Kakhki E.K., Hosseini S.M., Tahani M., 2016, An analytical solution for thermoelastic damping in a micro-beam based on generalized theory of thermoelasticity and modified couple stress theory, Applied Mathematical Modeling 40(4): 3164-3174.
[9] Kumar R., Devi S., Sharma V., 2017, A problem of thick circular plate in modified couple stress thermoelastic diffusion with phase-lags, Multidiscipline Modeling in Materials and Structures 12(3): 478-494.
[10] Kumar R., Devi S., 2017, Thermoelastic beam in modified couple stress thermoelasticity induced by laser pulse, Computers and Concrete 19(6): 707-716.
[11] Koiter W.T., 1964, Couple-stresses in the theory of elasticity, Proceedings of the Royal Netherlands Academy of Sciences 67: 17-44.
[12] Kumar R., Devi S., 2017, Response of thermoelastic functionally graded beam due to ramp type heating in modified couple stress with dual phase lag model, Multidiscipline Modelling in Materials and Structures 13(3): 471-488.
[13] Kumar R., Devi S., Sharma V., 2017, Damping in microscale modified couple stress thermoleastic circular plate resonators, AAM 12(2): 924-945.
[14] Kumar R., Devi S., 2018, Damping and frequency shift in microscale modified couple stress thermoelastic plate resonators, Journal of Solid Mechanics 10(3): 621-636.
[15] Marin M., 1997, An uniqueness result for body with voids in linear thermoelasticity, Rendiconti di Matematica 17(7): 103-113.
[16] Marin M., Florea O., 2014, On temporal behavior of solutions in thermoelasticity of porous micropolar bodies, Analele Universitatii "Ovidius" Constanta 22(1): 169-188.
[17] Ma H.M., Gao X.L., Reddy J.N., 2008, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of Mechanics and Physics of Solids 56(12): 3379-3391.
[18] Mindlin R.D., Tiersten H.F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415-448.
[19] Mindlin R.D., 1963, Influence of couple stresses on stress-concentrations, Experimental Mechanics 3: 1-7.
[20] Mindlin R.D., 1964, Micro-structure in linear elasticity, Archives for Rational Mechanics and Analysis 15: 51-78.
[21] Park S.K., Gao X.L., 2006, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16: 23-55.
[22] Rao S.S., 2007, Vibration of Continuous Systems, John Wiley & Sons, Inc. Hoboken, New Jersey.
[23] Rezazadeh G., Vahdat A.S., Tayefeh-Rezaei S., Cetinkaya C., 2012, Thermoelastic damping in a micro beam resonator using modified couple stress theory, Acta Mechanica 223: 1137-1152.
[24] Roychoudhuri S.K., 2007, On a thermoelastic three-phase-lag model, Journal of Thermal Stresses 30: 231-238.
[25] Sharma J.N., 2011, Thermoelastic damping and frequency shift in micro/nanoscale anisotropic beams, Journal of Thermal Stresses 34(7): 650-666.
[26] Sourki R., Hoseini S.A.H., 2016, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory, Applied Physics A 122: 413.
[27] Sun Y., Tohmyoh H., 2009, Thermoelasic damping of the axisymmetric vibration of circular plate resonators, Journal of Sound and Vibration 319: 392-405.
[28] Toupin R.A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11: 385-414.
[29] Tsiatas G.C., 2009, A new Kirchhoff plate model based on a modified couple stress theory, Journal of Solids and Structures 46: 2757-2764.
[30] Tzou D.Y., 1995, A unified approach for heat conduction from macro-to-micro scales, Journal of Heat Transfer 117: 8-16.
[31] Tzou D.Y., 1997, Macro-to-Microscale Heat Transfer: The Lagging Behaviour, Series in Chemical and Mechanical Engineering, Taylor and Francis, Washington.
[32] Voigt W., 1887, Theoretische Studienuber die Elasticitatsverhaltnisse der Krystalle,
Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, German.
[33] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 2731-2743.