Damping and Frequency Shift in Microscale Modified Couple Stress Thermoelastic Plate Resonators

Document Type: Research Paper

Authors

1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, India

2 Department of Mathematics, Kurukshetra University, India

Abstract

In this paper, the vibrations of thin plate in modified couple stress thermoelastic medium by using Kirchhoff- Love plate theory has been investigated. The governing equations of motion and heat conduction equation for Lord Shulman (L-S) [1] theory are written with the help of Kirchhoff- Love plate theory. The thermoelastic damping of micro-beam resonators is analyzed by using the normal mode analysis. The solutions for the free vibrations of plates under clamped-simply supported (CS) and clamped-free (CF) conditions are obtained. The analytical expressions for thermoelastic damping of vibration and frequency shift are obtained for couple stress generalized thermoelastic and coupled thermoelastic plates. A computer algorithm has been constructed to obtain the numerical results. The thermoelastic damping and frequency shift with varying values of length and thickness are shown graphically in the absence and presence of couple stress for (i) clamped-simply supported, (ii) clamped-free boundary conditions. Some particular cases are also presented.

Keywords

Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[2] Mindlin R. D., 1963, Influence of couple-stresses on stress-concentrations, Experimental Mechanics 3: 1-7.
[3] Mindlin R. D., Tiersten H. F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415-448.
[4] Toupin R. A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11(1): 385-414.
[5] 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.
[6] Eringen A. C., 1966, Linear theory of micropolar elasticity, Journal of Mathematics and Mechanics 15: 909-923.
[7] Tsiatas G. C., 2009, A new Kirchhoff plate model based on a modified couple stress theory, International Journal of Solids and Structures 46: 2757-2764.
[8] Sun Y., Tohmyoh H., 2009, Thermoelasic damping of the axisymmetric vibration of circular plate resonators, Journal of Sound and Vibration 319: 392-405.
[9] Sun Y., Saka M., 2010, Thermoelasic damping in micro-scale circular plate resonators, Journal of Sound and Vibration 329: 338-337.
[10] Sharma J. N., Sharma R., 2011, Damping in micro-scale generalized thermoelastic circular plate resonators, Ultrasonics 51: 352-358.
[11] Ezzat M.A., El-Karamany A.S., Samaan A.A., 2001, State-space formulation to generalized thermoviscoelasticity with thermal relaxation, Journal of Thermal Stresses 24(9): 823-846.
[12] El-Karamany A.S., Ezzat M.A., 2002, On the boundary integral formulation of thermo-viscoelasticity theory, International Journal Engineering Sciences 40(17): 1943-1956
[13] Ezzat M.A., El-Karamany A. S., 2003, On uniqueness and reciprocity theorems for generalized thermoviscoelasticity with thermal relaxation, Canadian Journal of Physics 81(6): 823-833.
[14] Ezzat M.A., El-Karamany A.S., El-Bary A.A., 2017, Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer, Microsystem Technologies 24(2): 951-961.
[15] Fang Y., Li P., Wang Z., 2013, Thermoelasic damping in the axisymmetric vibration of circular microplate resonators with two dimensional heat conduction, Journal of Thermal Stresses 36: 830-850.
[16] Shaat M., Mahmoud F. F., Gao X. L., 2014, Faheem A. F., 2014, Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects, International Journal of Mechanical Sciences 79: 31-37.
[17] Simsek M., Aydm M., Yurtcu H. H., Reddy J. N., 2015, Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory, Acta Mechanica 226: 3807-3822.
[18] Darijani H., Shahdadi A. H., 2015, A new shear deformation model with modified couple stress theory for microplates, Acta Mechanica 226(8): 2773-2788.
[19] Gao X. L., Zhan G. Y., 2016, A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects, Continuum Mechanics and Thermodynamics 28: 195-213.
[20] Reddy J. N., Romanoff J., Loya J. A., 2016, Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory, European Journal of Mechanics- A/Solids 56: 92-104.
[21] Chen W., Wang Y., 2016, A model of composite laminated Reddy plate of the global-local theory based on new modified couple-stress theory, Mechanics of Advanced Materials and Structures 23(6): 636-651.
[22] Marin M., 1998, A temporally evolutionary equation in elasticity of micropolar bodies with voids, Scientific Bulletin Series A Applied Mathematics and Physics 60: 3-12.
[23] Marin M., 2010, Harmonic vibrations in thermoelasticity of microstretch materials, Journal of Vibration and Acoustics 132(4): 044501-044506.
[24] Sharma K., Marin M., 2013, Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space, Scientific Bulletin Series A Applied Mathematics and Physics 75(2): 121-132.
[25] Marin M., Agarwal R. P., Codarcea L., 2017, A mathematical model for three-phase-lag dipolar thermoelastic bodies, Journal of Inequalities and Applications 109: 1-16.
[26] Rao S. S., 2007, Vibration of Continuous Systems, John Wiley & Sons, Inc. Hoboken, New Jersey.
[27] Sharma J. N., Kaur R., 2014, Transverse vibrations in thermoelastic-diffusive thin micro-beam resonators, Journal of Thermal Stresses 37: 1265-1285.
[28] Sharma J. N., 2011, Thermoelastic damping and frequency shift in Micro/Nanoscale anisotropic beams, Journal of Thermal Stresses 34(7): 650-666.
[29] Daliwal R.S., Singh A., 1980, Dynamical Coupled Thermoelasticity, Hindustan Publishers, Delhi.