Acta Mechanica

, Volume 230, Issue 3, pp 771–785 | Cite as

Modeling size-dependent thermoelastic energy dissipation of graphene nanoresonators using nonlocal elasticity theory

  • S. Rashahmadi
  • S. A. MeguidEmail author
Original Paper


Recent developments in nanostructured materials have led to the use of graphene sheets as resonators in advanced micro- and nanoelectromechanical systems. An important feature of micro- and nanoresonators is their ability to function with low power dissipation. The main intrinsic mechanism of energy loss in these advanced devices is thermoelastic damping (TED). In this article, we study TED effects in orthotropic graphene sheets of varied lengths operating at different temperatures using nonlocal elasticity theory. For this purpose, the fundamental thermoelastic relations are used to develop a system of coupled partial differential equations to describe the behavior of graphene nanoresonators. The orthotropic mechanical and thermal properties of graphene were taken into account in our model for zigzag and armchair chiralities operating at different temperatures. The free in-plane vibration of the graphene nanoresonator is analyzed using Galerkin method. Decidedly, we show that the developed system of equations is capable of describing the TED behavior of graphene nanoresonators along the two considered chiralities during thermoelastic vibration. Specifically, we examined the influence of size, chirality, and temperature upon thermoelastic damping, as measured by the so-called quality factor, of the graphene nanoresonator. Our results reveal that the nanoresonator experiences higher energy dissipation with increased temperature. They also reveal the dependence of the energy dissipation upon the size and chirality of the graphene sheet.


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The authors wish to thank the Natural Sciences and Engineering Research Council of Canada for the partial financial support of the current investigations. S. Rashahmadi wishes to thank the Department of Mechanical Engineering and Urmia University, Urmia, Iran, for approving his sabbatical leave at the University of Toronto, Canada.


  1. 1.
    Oshidari, Y., Hatakeyama, T., Kometani, R., Warisawa, S., Ishihara, S.: High quality factor graphene resonator fabrication using resist shrinkage-induced strain. Appl. Phys. Express 5, 117201 (2012)CrossRefGoogle Scholar
  2. 2.
    Jiang, R.W., Shen, Z.B., Tang, G.J.: A semi-analytical method for nonlocal buckling and vibration of a single-layered graphene sheet nanomechanical resonator subjected to initial in-plane loads. Acta Mech. 228, 1725–1734 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Yu, J.Y., Tian, X.G., Liu, J.: Size-dependent damping of a nanobeam using nonlocal thermoelasticity: extension of Zener, Lifshitz, and Roukes’ damping model. Acta Mech. 228, 1287–1302 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Nourmohammadi, Z.: Thermoelastic damping in micromechanical and nanomechanical resonators. Department of Mechanical Engineering, PhD Thesis, McGill University, Montreal, Quebec (2014)Google Scholar
  5. 5.
    Nayfeh, A.H., Younis, M.I.: Modeling and simulations of thermoelastic damping in microplates. J. Micromech. Microeng. 4, 1711–1717 (2004)CrossRefGoogle Scholar
  6. 6.
    Roszhart, T.V.: Proceedings of the Solid-State Sensor and Actuator Workshop. Hilton Head Island, SC, IEEE, New York (1990)Google Scholar
  7. 7.
    Vahdat, A.S., Rezazadeh, G., Ahmadi, G.: Thermoelastic damping in a micro-beam resonator tunable with piezoelectric layers. Acta Mech. Solida Sin. 25, 73–81 (2012)CrossRefGoogle Scholar
  8. 8.
    Severine, L.: Stochastic finite element method for the modeling of thermoelastic damping in microresonators. Ph.D., Department of Aerospace and Mechanics, University of Liege (2006)Google Scholar
  9. 9.
    Zener, C.: Internal friction in solids I, theory of internal friction in reeds. Phys. Rev. 52, 230–235 (1937)CrossRefzbMATHGoogle Scholar
  10. 10.
    Zener, C.: Internal friction in solids II, general theory of thermoelastic internal friction. Phys. Rev. 53, 90–99 (1938)CrossRefzbMATHGoogle Scholar
  11. 11.
    Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., Ward, P.: Experimental study of thermoelastic damping in MEMS gyros. Sensors Actuators A Phys. 103, 70–75 (2003)CrossRefGoogle Scholar
  12. 12.
    Lifshitz, R., Roukes, M.L.: Thermoelastic damping in micro-and nanomechanical systems. Phys. Rev. B 61, 5600–5609 (2000)CrossRefGoogle Scholar
  13. 13.
    Prabhakar, S., Vengallatore, S.: Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction. J. Microelectromech. Syst. 17, 494–502 (2008)CrossRefGoogle Scholar
  14. 14.
    Bostani, M., Karami Mohammadi, A.: Thermoelastic damping in microbeam resonators based on modified strain gradient elasticity and generalized thermoelasticity theories. Acta Mech. 229, 173–192 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Murmu, T., Pradhan, S.: Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model. Physica E Low Dimens. Syst. Nanostruct. 41, 1628–1633 (2009)CrossRefGoogle Scholar
  16. 16.
    Jiang, R.W., Shen, Z.B., Tang, G.J.: Vibration analysis of a single-layered graphene sheet-based mass sensor using the Galerkin strip distributed transfer function method. Acta Mech. 227, 2899–2910 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wallbank, J.: Electronic Properties of Graphene Heterostructures with Hexagonal Crystals. Springer, New York (2014)CrossRefGoogle Scholar
  18. 18.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  19. 19.
    Despotovic, N.: Stability and vibration of a nanoplate under body force using nonlocal elasticity theory. Acta Mech. 229, 273–284 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tiwari, A., Balandin, A.A. (eds.): Innovative Graphene Technologies: Evaluation and Applications, vol. 2. Smithers Rapra Technology Ltd., Akron (2013)Google Scholar
  21. 21.
    Ansari, R., Shahabodini, A., Rouhi, H.: A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions. Curr. Appl. Phys. 15, 1062–1069 (2015)CrossRefGoogle Scholar
  22. 22.
    Abd-Alla, A., Abo-Dahab, S., Hammad, H.: Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field. Appl. Math. Model. 35, 2981–3000 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Casiano M.J.: Extracting damping ratio from dynamic data and numerical solutions. Marshall Space Flight Center, Huntsville, Alabama, NASA/TM-218227 (2016)Google Scholar
  24. 24.
    Yi, Y.B.: Geometric effects on themoelastic damping in MEMS resonators. J. Sound Vib. 309, 588–599 (2008)CrossRefGoogle Scholar
  25. 25.
    Lee, C., Wei, X., Kysar, J.W., Hone, J.: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008)CrossRefGoogle Scholar
  26. 26.
    Huang, Y., Wu, J., Hwang, K.C.: Thickness of graphene and single-wall carbon nanotubes. Phys. Rev. B 74, 245413 (2006)CrossRefGoogle Scholar
  27. 27.
    Xu, Y., Shen, H., Li Zhang, C.: Nonlocal plate model for nonlinear bending of bilayer graphene sheets subjected to transverse loads in thermal environments. Compos. Struct. 98, 294–302 (2013)CrossRefGoogle Scholar
  28. 28.
    Pop, E., Varshney, V., Roy, A.K.: Thermal properties of graphene: fundamentals and applications. MRS Bull. 37, 1273–1281 (2012)CrossRefGoogle Scholar
  29. 29.
    Seoànez, C., Guinea, F., Castro Neto, A.H.: Dissipation in graphene and nanotube resonators. Phys. Rev. B 76, 125427 (2007)CrossRefGoogle Scholar
  30. 30.
    Kim, S.Y., Park, H.S.: Multilayer friction and attachment effects on energy dissipation in graphene nanoresonators. Appl. Phys. Lett. 94, 101918 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mechanics and Aerospace Design LaboratoryUniversity of TorontoTorontoCanada
  2. 2.Department of Mechanical EngineeringUrmia UniversityUrmiaIran

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