Skip to main content
SpringerLink
Log in

Investigation on thermoelastic damping of micro-plate resonators based on the modified couple stress theory incorporating the memory-dependent derivative heat transfer model

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

As a main source of energy dissipation, thermoelastic damping (TED) cannot be ignored in designing resonators with high-quality factor. However, due to the size-dependent effect and the thermal relaxation, the TED models formulated in the classical theory are no longer able to accurately estimate the energy dissipation in the micro-/nano-resonators. To fill theses gaps, the present work aims at developing a new TED model for micro-plate resonators based on the modified couple stress theory incorporating memory-dependent derivative heat conduction model. The corresponding governing equations are derived, and the analytical solution for the TED is obtained. In order to discover the variation law of TED of micro-plate resonators, the numerical results corresponding to different materials and different theories are illustrated and compared. In calculation, the effects of the material parameters, the boundary conditions, the length scale parameters and the length on the peak value as well as the critical thickness of TED are further studied. It is expected that this novel model may provide a theoretical basis for designing high-performance micro-plate resonators under complex working conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Li, X., Bhushan, B., Takashima, K., Baek, C.W., Kim, Y.K.: Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 97(1), 481–494 (2003)

    Article  Google Scholar 

  2. Eom, K., Kwon, T.Y., Yoon, D.S., Lee, H.L., Kim, T.S.: Dynamical response of nanomechanical resonators to biomolecular interactions. Phys. Rev. B Condens. Matter 76(11), 113408 (2007)

    Article  Google Scholar 

  3. Lee, I., Lee, J.: Measurement uncertainties in resonant characteristics of MEMS resonators. J. Mech. Sci. Technol. 27(2), 491 (2013)

    Article  Google Scholar 

  4. Pelesko, J.A., Bernstein, D.H.: Modeling Mems and Nems. CRC Press, Boca Raton (2002)

    Book  MATH  Google Scholar 

  5. Ciminelli, C., Dell’Olio, F., Armenise, M.N.: High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation. IEEE Photonics J. 4(5), 1844–1854 (2012)

    Article  Google Scholar 

  6. Lin, L.W., Howe, R.T., Pisano, A.P.: Microelectromechanical filters for signal processing. Micro Electron. Mech. Syst. 7(3), 286–294 (1992)

    Article  Google Scholar 

  7. Ekinci, K.L., Roukes, M.L.: Nanoelectromechanical systems. Science 76(6), 25–30 (2005)

    Google Scholar 

  8. Beek, J.V., Puers, R.: A review of MEMS oscillators for frequency reference and timing applications. J. Micromech. Microeng. 22(1), 013001 (2012)

    Article  Google Scholar 

  9. Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., Ward, P.: Experimental study of thermoelastic damping in MEMS gyros. Sensors Actuators A Phys. 15(1), 70–75 (2003)

    Article  Google Scholar 

  10. Zener, C.: Internal friction in solids II: general theory of thermoelastic internal friction. Phys. Today 47(2), 117–118 (1938)

    MATH  Google Scholar 

  11. Lifshitz, R., Roukes, M.L.: Thermoelastic damping in micro- and nano mechanical systems. Phys. Rev. B 61(8), 5600–5609 (2000)

    Article  Google Scholar 

  12. Nayfeh, A.H., Younis, M.I.: Modeling and simulations of thermoelastic damping in microplates. J. Micromech. Microeng. 14(12), 1711–1717 (2004)

    Article  Google Scholar 

  13. Sun, Y.X., Saka, M.: Thermoelastic damping in micro-scale circular plate resonators. J. Sound Vib. 329(3), 328–337 (2010)

    Article  Google Scholar 

  14. Fang, Y.M., Li, P., Wang, Z.: Thermoelastic damping in the axisymmetric vibration of circular microplate resonators with two-dimensional heat conduction. J. Therm. Stress. 36(8), 830–850 (2013)

    Article  Google Scholar 

  15. Fang, Y.M., Li, P., Zhou, H.Y.: Thermoelastic damping in rectangular microplate resonators with three-dimensional heat conduction. Int. J. Mech. Sci. 133, 578–589 (2017)

    Article  Google Scholar 

  16. Zuo, W.L., Li, P., Zhang, J.R.: Analytical modeling of thermoelastic damping in bilayered microplate resonators. Int. J. Mech. Sci. 106, 128–137 (2016)

    Article  Google Scholar 

  17. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  18. Aifantis, E.C.: Gradient deformation models at nano, micro, and macro scales. J. Eng. Mater. Technol. 121(2), 189–202 (1999)

    Article  Google Scholar 

  19. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)

    Article  MATH  Google Scholar 

  20. Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46(13), 2757–2764 (2009)

    Article  MATH  Google Scholar 

  21. Zhong, Z.Y., Zhang, W.M., Meng, G., Wang, M.Y.: Thermoelastic damping in the size-dependent microplate resonators based on modified couple stress theory. J. Microelectromech. Syst. 24(2), 431–445 (2015)

    Article  Google Scholar 

  22. Segovia, F.J., Piazza, G.: Analytical and numerical methods to model anchor losses in 65-MHz AlN contour mode resonators. J. Microelectromech. Syst. 25, 459–468 (2016)

    Article  Google Scholar 

  23. Maxwell, J.C.: On the dynamical theory of gases. Phil. Mag. 157, 49–88 (1972)

    Google Scholar 

  24. Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. C. R. Phys. 247, 431–433 (1958)

    MATH  Google Scholar 

  25. Vernotte, P.M., Hebd, C.R.: Paradoxes in the continuous theory of the heat conduction. C. R. Phys. 246, 3154–3155 (1958)

    MATH  Google Scholar 

  26. Tzou, D.Y.: A unified field approach for heat conduction from macro-to-micro-scales. J. Heat Transf. 117(1), 8–16 (1995)

    Article  Google Scholar 

  27. Choudhuri, S.K.: On a thermoelastic three-phase-lag model. J. Therm. Stress. 30(3), 231–238 (2007)

    Article  Google Scholar 

  28. Lord, H.W., Shulman, Y.A.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (2007)

    Article  MATH  Google Scholar 

  29. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2(1), 1–7 (1972)

    Article  MATH  Google Scholar 

  30. Yu, Y.J., Hu, W., Tian, X.G.: A novel generalized thermoelasticity model based on memory-dependent derivative. Int. J. Eng. Sci. 81, 123–134 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhou, H.Y., Li, P.: Thermoelastic damping in micro-and nanobeam resonators with non-Fourier heat conduction. IEEE Sens. J. 17(21), 6966–6977 (2017)

    Article  Google Scholar 

  32. Guo, X., Yi, Y.B., Pourkamali, S.: A finite element analysis of thermoelastic damping in vented MEMS beam resonators. Int. J. Mech. Sci. 74, 73–82 (2013)

    Article  Google Scholar 

  33. Zhou, H.Y., Shao, D.F., Song, X.R., Li, P.: Three-dimensional thermoelastic damping models for rectangular micro/nanoplate resonators with nonlocal-single-phase-lagging effect of heat conduction. Int. J. Heat Mass Transf. 196, 123271 (2022)

    Article  Google Scholar 

  34. Wang, Y.W., Chen, J., Zheng, R.Y., Li, X.F.: Thermoelastic damping in circular microplate resonators based on fractional dual-phase-lag model and couple stress theory. Int. J. Heat Mass Transf. 201, 123570 (2023)

    Article  Google Scholar 

  35. Borjalilou, V., Asghari, M., Bagheri, E.: Small-scale thermoelastic damping in micro-beams utilizing the modified couple stress theory and the dual-phase-lag model. J. Therm. Stress. 42(7), 1–14 (2019)

    Article  Google Scholar 

  36. Bhagwan, S., Harendra, K., Santwana, M.: Analysis of size effects on thermoelastic damping in the Kirchhoff’s plate resonator under Moore–Gibson–Thompson thermoelasticity. Thin-Walled Struct. 180, 109793 (2022)

    Article  Google Scholar 

  37. Kakhki, E.K., Hosseini, S.M., Tahani, M.: An analytical solution for thermoelastic damping in a micro-beam based on generalized theory of thermoelasticity and modified couple stress theory. Appl. Math. Model. 40(4), 3164–3174 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gu, B.D., He, T.H., Ma, Y.B.: Thermoelastic damping analysis in micro-beam resonators considering nonlocal strain gradient based on dual-phase-lag model. Int. J. Heat Mass Transf. 180, 121771 (2021)

    Article  Google Scholar 

  39. Stephen, T.: Theory of Plates and Shells. McGraw-Hill, New York (1959)

    Google Scholar 

  40. Dym, C.L., Shames, I.H.: Solid mechanics: a variational approach. Heidelberg Dordrecht, London (1980)

  41. Li, P., Fang, Y.M., Hu, R.F.: Thermoelastic damping in rectangular and circular microplate resonators. J. Sound Vib. 331(3), 721–733 (2012)

    Article  Google Scholar 

  42. Zhong, Z.Y., Zhang, W.M., Meng, G.: Thermoelastic damping in the size-dependent microplate resonators based on modified couple stress theory. J. Microelectromech. Syst. 24(2), 431–445 (2015)

    Article  Google Scholar 

  43. Chakraverty, S., Pradhan, K.K.: Free vibration of functionally graded thin rectangular plates resting on Winkler elastic foundation with general boundary conditions using Rayleigh–Ritz method. Int. J. Appl. Mech. 6(4), 1450043 (2014)

    Article  Google Scholar 

  44. Shi, S.H., He, T.H., Jin, F.: Thermoelastic damping analysis of size-dependent nano-resonators considering dual-phase-lag heat conduction model and surface effect. Int. J. Heat Mass Transf. 170(6), 120977 (2021)

    Article  Google Scholar 

  45. Borjalilou, V., Asghari, M.: Small-scale analysis of plates with thermoelastic damping based on the modified couple stress theory and the dual-phase-lag heat conduction model. Acta Mech. 229, 3869–3884 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Babaei, A., Noorani, M.S., Ghanbari, A.: Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory. Microsyst Technol 23, 4599–4610 (2017)

    Article  Google Scholar 

  47. Babaei, A., Rahmani, A.: On dynamic-vibration analysis of temperature-dependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mech. Adv. Mater. Struct. 27(16), 1451–1458 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

This study was supported by the National Natural Science Foundation of China (11972176, 12062011).

Author information

Authors and Affiliations

  1. School of Science, Lanzhou University of Technology, Lanzhou, 730050, China

    Guobin Zhao & Tianhu He

Authors
  1. Guobin Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tianhu He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianhu He.

Ethics declarations

Conflict of interest

The authors declare that the data have no conflicting interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, G., He, T. Investigation on thermoelastic damping of micro-plate resonators based on the modified couple stress theory incorporating the memory-dependent derivative heat transfer model. Arch Appl Mech 93, 3495–3509 (2023). https://doi.org/10.1007/s00419-023-02450-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-023-02450-z

Keywords

Search

Navigation