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Material-dependent thermoelastic damping limited quality factor and critical length analysis with size effects of micro/nanobeams

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Abstract

Micro/nanobeam-based resonators have found extensive applications in the micro/nanoelectromechanical system industry. Thermoelastic damping (TED) is a major energy loss issue in micro/nanobeam resonators that limits their important performance parameter, namely, the TED limited quality factor (QTED). The critical length (Lc) of a micro/nanobeam is another significant parameter that accounts for the maximum peak in the energy dissipation curve at which QTED assumes a minimum value. To evaluate QTED and Lc explicitly when the size of devices is scaled down, size effects play a decisive role and classical theories are inadequate. In this work, a higher-order theory, namely, modified couple stress theory (MCST), is used to overcome the size effects by including one internal material length scale parameter (l). The material-dependent thermoelastic coupled equations for a deflected Euler-Bernoulli microbeam are presented using variational and Hamilton principles. Moreover, the solutions for QTED are developed on the basis of a complex frequency approach with the appropriate material indices. The effects of material length scale parameters, material performance indices, mechanical boundary conditions (clamped-clamped, simply supported, and cantilever types), mode switching, and plane stress/strain conditions on QTED and Lc are analyzed. Numerical results are extracted from the analytical expressions by using MATLAB R2015a to quantify thermoelastic energy dissipation. The numerically computed QTED and Lc values are fully investigated to design high-performance resonators. The analyses verify that QTED is enhanced by optimizing the structural material and augmenting the material length scale parameter. The material order in which QTED is enhanced is the same for classical theories and MCST, i.e., it is inversely related to the TED index parameter. The influences of boundary types and mode switching on QTED are relatively less in accordance with the analysis. The effect of plane stress condition compared with that of plane strain condition on QTED is also remarkable. The Lc of the beam is determined to be dependent on the thermal diffusion length of the material used. From an adequate material point of view, poly-silicon has been proven to provide the maximum quality factor while silicon carbide yields the maximum Lc. These observations are significant and extremely helpful when designing low-loss micro/nanobeam resonators with superior performance by suitably selecting their geometry and structural materials.

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References

  1. K. Meinel, M. Melzer, C. Stoeckel, A. Shaporin, R. Forke, S. Zimmermann, K. Hiller, T. Otto and H. Kuhn, 2D scanning micromirror with large scan angle and monolithically integrated angle sensors based on piezoelectric thin film aluminum nitride, Sensors (MDPI), 20(22) (2020) 6599.

    Article  Google Scholar 

  2. S. A. Razali, N. A. C. Sidik and H. Koten, Cellulose nanocrystals: a brief review on properties and general applications, Journal of Advanced Research Design, 60(1) (2019) 1–15.

    Google Scholar 

  3. Y. Li, H. Li, Y. Xiao, L. Cao and Z.-S. Guo, A compensation method for nonlinear vibration of silicon-micro resonant sensor, Sensors (MDPI), 21(7) (2021) 2545.

    Article  Google Scholar 

  4. A. Asri, M. Izzudin, M. Hasan, M. Ahmad, Y. Yunos, M. Ali and M. Sultan, MEMS gas sensors: a review, IEEE Sensors Journal (2021) 18381–18397.

  5. M. Mehdi, M. T. Ajani, H. Tahir, S. Tahir, Z. Alizai, F. Khan, Q. Riaz and M. Hussain, PUF-based key generation scheme for secure group communication using MEMS, Electronics, 10(14) (2021) 1691.

    Article  Google Scholar 

  6. Z. Chen, Q. Jia, W. Liu, Q. Yuan, Y. Zhu, J. Yang and F. Yang, Dominant loss mechanisms of whispering gallery mode RFMEMS resonators with wide frequency coverage, Sensors, 20(24) (2020) 7017.

    Article  Google Scholar 

  7. A. Persano, F. Quaranta, A. Taurino, P. A. Siciliano and J. Iannacci, Thin film encapsulation for RF MEMS in 5G and modern telecommunication systems, Sensors, 20(7) (2020) 2133.

    Article  Google Scholar 

  8. L. Wei, X. Kuai, Y. Bao, J. Wei, L. Yang, P. Song, M. Zhang, F. Yang and X. Wang, The recent progress of MEMS/NEMS resonators, Micromachines, 12(6) (2021) 724.

    Article  Google Scholar 

  9. R. Syms and A. Bouchaala, Mechanical synchronization of MEMS electrostatically driven coupled beam filters, Micromachines, 12(10) (2021) 1191.

    Article  Google Scholar 

  10. R. P. Aswathy and R. Resmi, Analysis of the effects of substrate parameters on the performance of RF MEMS tunable microstrip bandpass filters, 2015 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT) (2015) 51–56.

  11. Z.-Y. Tsai, P.-J. Shih, Y.-C. Tsai and C.-L. Dai, Manufacturing and testing of radio frequency MEMS switches using the complementary metal oxide semiconductor process, Sensors, 21(4) (2021) 1396.

    Article  Google Scholar 

  12. U. S. Arathy and R. Resmi, Analysis of pull-in voltage of MEMS switches based on material properties and structural parameters, 2015 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT) (2015) 57–61.

  13. Q. Bao, J. Zhang, M. Tang, Z. Huang, L. Lai, J. Huang and C. Wu, A novel PZT pump with built-in compliant structures, Sensors, 19(6) (2019) 1301.

    Article  Google Scholar 

  14. F. Forouzandeh, A. Arevalo, A. Alfadhel and D. A. Borkholder, A review of peristaltic micropumps, Sensors and Actuators A: Physical, 326 (2021) 112602.

    Article  Google Scholar 

  15. C. Jenke, R. J. Pallejà, S. Kibler, J. Häfner, M. Richter and C. Kutter, The combination of micro diaphragm pumps and flow sensors for single stroke based liquid flow control, Sensors, 17(4) (2017) 755.

    Article  Google Scholar 

  16. J.-S. Yoon, J. Park, H.-R. Ahn, S.-J. Yoo and Y.-J. Kim, Microfluidic airborne metal particle sensor using oil microcirculation for real-time and continuous monitoring of metal particle emission, Micromachines, 12(7) (2021) 825.

    Article  Google Scholar 

  17. F. Xu, Y. Wei, S. Bian, H. Wang, D.-R. Chen and D. Kong, Simulation-based design and optimization of rectangular micro-cantilever-based aerosols mass sensor, Sensors, 20(3) (2020) 626.

    Article  Google Scholar 

  18. C. Li, B. Yang, X. Guo and X. Chen, Design, analysis and simulation of a MEMS-based gyroscope with differential tunneling magnetoresistance sensing structure, Sensors, 20(17) (2020) 4919.

    Article  Google Scholar 

  19. C. Ai, X. Zhao and D. Wen, Characteristics research of a high sensitivity piezoelectric MOSFET acceleration sensor, Sensors, 20(17) (2020) 4988.

    Article  Google Scholar 

  20. D. Zhang, A. Cai, Y. Zhao and T. Hu, Macro modeling of V-shaped electro-thermal MEMS actuator with human error factor, Micromachines, 12(6) (2021) 622.

    Article  Google Scholar 

  21. H. H. Hillmer, M. S. Q. Iskhandar, M. K. Hasan, S. Akhundzada, B. Al-Qargholi and A. Tatzel, MOEMS micromirror arrays in smart windows for daylight steering, J. Optical Microsystems, 1(1) (2021) 014502.

    Article  Google Scholar 

  22. S. Finny and R. Resmi, Analysis of squeeze film damping in piston mode micromirrors, 2016 International Conference on Inventive Computation Technologies (ICICT) (2016) 1–5.

  23. G. S. Wood, A. Torin, A. K. Al-Mashaal, L. Smith, E. Mastropaolo, M. J. Newton and R. Cheung, Design and characterization of a micro-fabricated graphene-based MEMS microphone, IEEE Sens. J., 19 (2019) 7234–7242.

    Article  Google Scholar 

  24. S. A. Zawawi, A. A. Hamzah, B. Y. Majlis and F. Mohd-Yasin, A review of MEMS capacitive microphones, Micromachines, 11(5) (2020) 484.

    Article  Google Scholar 

  25. L. Herzog and K. Augsburg, Study on friction in automotive shock absorbers, part 2: validation of friction simulations via novel single friction point test rigs, Vehicles, 3(2) (2021) 197–211.

    Article  Google Scholar 

  26. B.-H. Kang, J.-H. Hwang and S.-B. Choi, A new design model of an MR shock absorber for aircraft landing gear systems considering major and minor pressure losses: experimental validation, Applied Sciences, 11(17) (2021) 7895.

    Article  Google Scholar 

  27. E. Amer, M. Wozniak, G. Jönsson and F. Arrhén, Evaluation of shock tube retrofitted with fast-opening valve for dynamic pressure calibration, Sensors, 21(13) (2021) 4470.

    Article  Google Scholar 

  28. Z. Dou, J. Tang, Z. Liu, Q. Sun, Y. Wang, Y. Li, M. Yuan, H. Wu, Y. Wang, W. Pei and H. Chen, Wearable contact lens sensor for non-invasive continuous monitoring of intraocular pressure, Micromachines, 12(2) (2021) 108.

    Article  Google Scholar 

  29. A. Henriksson, L. Kasper, M. Jäger, P. Neubauer and M. Birkholz, An approach to ring resonator biosensing assisted by dielectrophoresis: design, simulation and fabrication, Micromachines, 11(11) (2020) 954.

    Article  Google Scholar 

  30. Ö. E. Aşırım, A. Yolalmaz and M. Kuzuoğlu, High-fidelity harmonic generation in optical micro-resonators using BFGS algorithm, Micromachines, 11(7) (2020) 686.

    Article  Google Scholar 

  31. J.-L. Zhang, S. Liao, C. Chen, X.-T. Yang, S.-A. Lin, F. Tan, B. Li, W.-W. Wang, Z.-X. Zhong and G.-G. Zeng, Research on trimming frequency-increasing technology for quartz crystal resonator using laser etching, Micromachines, 12(8) (2021) 894.

    Article  Google Scholar 

  32. A. Tsitlakidis, A. S. Tsingotjidou, A. Kritis, A. Cheva, P. Selviaridis, E. C. Aifantis and N. Foroglou, Atomic force microscope nanoindentation analysis of diffuse astrocytic tumor elasticity: relation with tumor histopathology, Cancers, 13(18) (2021) 4539.

    Article  Google Scholar 

  33. E. H. C. P. Sinnecker, J. M. García-Martín, D. Altbir, E. C. J. D’Albuquerque and J. P. Sinnecker, A magnetic force microscopy study of patterned T-shaped structures, Materials, 14(6) (2021) 1567.

    Article  Google Scholar 

  34. J. Fei, Y. Fang and Z. Yuan, Adaptive fuzzy sliding mode control for a micro gyroscope with backstepping controller, Micromachines, 11(11) (2020) 968.

    Article  Google Scholar 

  35. W. Zhao, Y. Cheng, S. Zhao, X. Hu, Y. Rong, J. Duan and J. Chen, Navigation grade MEMS IMU for a satellite, Micromachines, 12(2) (2021) 151.

    Article  Google Scholar 

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

    Article  Google Scholar 

  37. J. Yao, W. Qiang, X. Guo, H. Fan, Y. Zheng, Y. Xu and X. Yang, Defect filling method of sensor encapsulation based on micro-nano composite structure with parylene coating, Sensors, 21(4) (2021) 1107.

    Article  Google Scholar 

  38. R. Filters-Syms and A. Bouchaala, Mechanical synchronization of MEMS electrostatically driven coupled beam filters, Micromachines, 12(10) (2021) 1191.

    Article  Google Scholar 

  39. C. Tu, J.-Y. Lee and X.-S. Zhang, Dissipation analysis methods and Q-enhancement strategies in piezoelectric MEMS laterally vibrating resonators: a review, Sensors, 20(17) (2020) 4978.

    Article  Google Scholar 

  40. J. E. Lee, Engineering high Q-factor MEMS resonators and probing losses, 2017 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), (2017) 439–443.

  41. Z. Chen, T. Wang, Q. Jia, J. Yang, Q. Yuan, Y. Zhu and F. Yang, A novel Lamé mode RF-MEMS resonator with high quality factor, International Journal of Mechanical Sciences, 204 (2021) 106484.

    Article  Google Scholar 

  42. M. I. Younis and A. H. Nayfeh, Simulation of squeeze-film damping of microplates actuated by large electrostatic load, J. Comput. Nonlinear Dyn., 2(3) (2007) 232–242.

    Article  Google Scholar 

  43. S. Finny and R. Resmi, Material and geometry optimization for squeeze film damping in a micromirror, 2016 International Conference on Emerging Technological Trends (ICETT) (2016) 1–5.

  44. Y. Zhang, L. He, J. Yang, G. Zhu, X. Jia and W. Yan, Multi-objective optimization design of a novel integral squeeze film bearing damper, Machines, 9(10) (2021) 206.

    Article  Google Scholar 

  45. A. Albukhari and U. Mescheder, Investigation of the dynamics of a 2-DoF actuation unit cell for a cooperative electrostatic actuation system, Actuators, 10(10) (2021) 276.

    Article  Google Scholar 

  46. P. Xu, C. Si, Y. He, Z. Wei, L. Jia, G. Han, J. Ning and F. Yang, A novel high-Q dual-mass MEMS tuning fork gyroscope based on 3D wafer-level packaging, Sensors, 21(19) (2021) 6428.

    Article  Google Scholar 

  47. R. Resmi, M. R. Baiju and V. S. Babu, Material dependent thermoelastic damping limited quality factor analysis of disc resonators, 2017 International Conference of Electronics, Communication and Aerospace Technology (ICECA) (2017) 675–680.

  48. G. D. Vukasin et al., Effect of substrate thickness on anchor damping in MEMS devices, 2019 20th International Conference on Solid-State Sensors, Actuators and Microsystems & Eurosensors XXXIII (2019) 1843–1845.

  49. S. Mol and R. Resmi, Anchor loss limited Q factor analysis of disk resonator for varying disk geometry, 2017 International Conference on Intelligent Computing, Instrumentation and Control Technologies (ICICICT) (2017) 1033–1037.

  50. Y. Tong and T. Han, Anchor loss reduction of lamb wave resonator by pillar-based phononic crystal, Micromachines, 12(1) (2021) 62.

    Article  Google Scholar 

  51. H. A. Alharthi, Characterization of the vibration and strain energy density of a nanobeam under two-temperature generalized thermoelasticity with fractional-order strain theory, Mathematical and Computational Applications, 26(4) (2021) 78.

    Article  Google Scholar 

  52. F. Serdean, M. Pustan, C. Dudescu, C. Birleanu and M. Serdean, Analysis of the thermoelastic damping effect in electrostatically actuated MEMS resonators, Mathematics, 8(7) (2020) 1124.

    Article  Google Scholar 

  53. Y. Tai, P. Li, Y. Zheng and J. Tian, Entropy generation and thermoelastic damping in the in-plane vibration of microring resonators, Entropy, 21(7) (2019) 631.

    Article  MathSciNet  Google Scholar 

  54. C. K. Thein and F. M. Foong, Material damping analysis of triangular cantilever beam for electromagnetic vibration energy harvesting applications, Engineering Proceedings, 10(1) (2021) 1.

    Google Scholar 

  55. H. M. Youssef, A. A. El-Bary and E. A. N. Al-Lehaibi, Characterization of the quality factor due to the static prestress in classical Caputo and Caputo-Fabrizio fractional thermoelastic silicon microbeam, Polymers, 13(1) (2021) 27.

    Article  Google Scholar 

  56. S. D. Senturia, Thermoelastic damping in fine grained polysilicon flexural beam resonators, J. Microelectromech. Syst., 11(5) (2002) 499–504.

    Article  Google Scholar 

  57. D. V. Parayil, S. S. Kulkarni and D. N. Pawaskar, Analytical and numerical solutions for thick beams with thermoelastic damping, Int. J. Mech. Sci., 94 (2015) 10–19.

    Article  Google Scholar 

  58. S. Vengallatore, Analysis of thermoelastic damping in laminated composite micromechanical beam resonators, J. Micromech. Microengg. (2005) 2398–2404.

  59. X. Li, L. Li, Y. Hu, Z. Ding and W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, 165 (2017) 250–265.

    Article  Google Scholar 

  60. I. Pradiptya and H. M. Ouakad, Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory, Int. J. Mech. Mater. Des., 14(3) (2018) 393–415.

    Article  Google Scholar 

  61. R. Resmi, V. S. Babu and M. R. Baiju, Impact of dimensionless length scale parameter on material dependent thermoelastic attenuation and study of frequency shifts of rectangular microplate resonators, 2021 IOP Conf. Ser.: Mater. Sci. Eng. (2021) 1091012067.

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

    Article  Google Scholar 

  63. R. Resmi, V. S. Babu and M. R. Baiju, Analysis of thermoelastic damping limited quality factor and critical dimensions of circular plate resonators based on axisymmetric and non-axisymmetric vibrations, AIP Advances, 11 (2021) 035108.

    Article  Google Scholar 

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

    Article  Google Scholar 

  65. W. Zuo, P. Li, J. Zhang and Y. Fang, Analytical modeling of thermoelastic damping in bilayered microplate resonators, Int. J. Mech. Sci., 106 (2016) 128–137.

    Article  Google Scholar 

  66. S. Liu, Y. Sun, J. Ma and J. Yang, Theoretical analysis of thermoelastic damping in bilayered circular plate resonators with two-dimensional heat conduction, Int. J. Mech. Sci., 135 (2018) 114–123.

    Article  Google Scholar 

  67. J. Kim and J. Reddy, A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis, Acta. Mech., 226(9) (2015) 2973.

    Article  MathSciNet  MATH  Google Scholar 

  68. Y. C. Pei, Thermoelastic damping in rotating flexible microdisk, Int. J. Mech. Sci., 61(1) (2012) 52–64.

    Article  Google Scholar 

  69. S. T. Hossain, S. McWilliam and A. A. Popov, An investigation on thermoelastic damping of high-Q ring resonators, Int. J. Mech. Sci., 106 (2016) 209–219.

    Article  Google Scholar 

  70. C. Zener, Internal friction in solids, I. theory of internal friction in reeds, Phys. Rev., 52(3) (1937) 230–235.

    Article  MATH  Google Scholar 

  71. C. Zener, Internal friction in solids, II. general theory of thermoelastic internal friction, Phys. Rev., 53(1) (1938) 90–99.

    Article  MATH  Google Scholar 

  72. T. V. Roszhart, The effect of thermoelastic internal friction on the Q of micromachined silicon resonators, IEEE Solid-State Sensor and Actuator Workshop (1990) 13–16.

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

    Article  Google Scholar 

  74. S. Evoy, A. Olkhovets, D. W. Carr, J. M. Parpia and H. G. Craighead, Temperature-dependent internal friction in silicon nanoelectromechanical systems, Appl. Phys. Lett., 77 (2000) 2397.

    Article  Google Scholar 

  75. A. Duwel, R. N. Candler, T. W. Kenny and M. Varghese, Engineering MEMS resonators with low thermoelastic damping, J. Microelectromech. Syst., 15(6) (2006) 1437–1445.

    Article  Google Scholar 

  76. J. Yang, T. Ono and M. Esashi, Energy dissipation in submicrometer thick single-crystal silicon cantilevers, J. Microelectromech. Syst., 11(6) (2002) 775–783.

    Article  Google Scholar 

  77. G. Rezazadeh, A. Tahmasebi and M. Zubstov, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage, Microsyst.Technol., 12 (2006) 1163–1170.

    Article  Google Scholar 

  78. P. Sairam and V. Srikar, Thermoelastic damping in bilayered micromechanical beam resonator, J. Micromechanics and Microengineering, 17(3) (2007) 532–538.

    Article  Google Scholar 

  79. S. Prabhakar and S. Vengallatore, Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction, J. Microelectromech. Syst., 17(2) (2008) 494–502.

    Article  Google Scholar 

  80. K. Tunvir, C. Q. Ru and A. Mioduchowski, Thermoelastic dissipation of hollow micromechanical resonators, Physica E-low-dimensional Systems & Nanostructures — PHYSICA E, 42 (2010) 2341–2352.

    Google Scholar 

  81. K. Tunvir, Thermoelastic dissipation in stepped-beam resonators, Microsystem Technologies, 19 (2012) 721–731.

    Article  Google Scholar 

  82. K. Tunvir, C. Q. Ru and A. Mioduchowski, Effect of cross-sectional shape on thermoelastic dissipation of micro/nano elastic beams, International Journal of Mechanical Sciences, 62 (2012) 77–88.

    Article  Google Scholar 

  83. K. Tunvir, C. Q. Ru and A. Mioduchowski, Large-deflection effect on thermoelastic dissipation of microbeam resonators, Journal of Thermal Stresses, 35 (2012) 1076–1094.

    Article  Google Scholar 

  84. Z. Nourmohammadi, S. Prabhakar and S. Vengallatore, Thermoelastic damping in layered microresonators: critical frequencies, peak values, and rule of mixture, Microelectromechanical Systems, 22 (2013) 747–754.

    Article  Google Scholar 

  85. J. N. Sharma and R. Kaur, Transverse vibrations in thermoelastic-diffusive thin microbeam resonators, J. Thermal Stresses, 37 (2014) 1265–1285.

    Article  Google Scholar 

  86. Z. Y. Zhong, J. P. Zhou and H. L. Zhang, Thermoelastic damping in functionally graded microbeam resonators, IEEE Sensors J., 17(11) (2017) 3381–3390.

    Article  Google Scholar 

  87. E. Taati, On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment, International Journal of Engineering Science, 128 (2018) 63–78.

    Article  MathSciNet  MATH  Google Scholar 

  88. Y. Fu, L. Li and Y. Hu, Enlarging quality factor in microbeam resonators by topology optimization, J. Thermal Stress., 42(3) (2019) 341–360.

    Article  Google Scholar 

  89. A. Seyfi, M. Nouraei and P. Haghi, Influence of Magnetic Field on The Wave Propagation Response of Functionally Graded (FG) Beam Lying on Elastic Foundation in Thermal Environment, Waves in Random and Complex Media (2021).

  90. I. Kaur, P. Lata and K. Singh, Study of frequency shift and thermoelastic damping in transversely isotropic nano-beam with GN III theory and two temperature, Archive of Applied Mechanics, 91 (2021) 1–15.

    Google Scholar 

  91. S. Guha and A. K. Singh, Frequency shifts and thermoelastic damping in different types of nano-/micro-scale beams with sandiness and voids under three thermoelasticity theories, Journal of Sound and Vibration, 510 (2021) 116301.

    Article  Google Scholar 

  92. N. Fleck, G. Muller, M. Ashby and J. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metall. Mater., 42(2) (1994) 475–487.

    Article  Google Scholar 

  93. K. W. McElhaney, J. J. Vlassak and W. D. Nix, Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments, J. Mater. Res., 13 (1998) 1300–1306.

    Article  Google Scholar 

  94. I. Chasiotis and W. G. Knauss, A new microtensile tester for the study of MEMS materials with the aid of atomic force microscopy, Exp. Mech., 42 (2002) 51–57.

    Article  Google Scholar 

  95. T. P. Weihs, S. Hong, J. C. Bravman and W. D. Nix, Mechanical deflection of cantilever microbeams: a new technique for testing the mechanical properties of thin films, J. Mater. Res., 3 (1988) 931–942.

    Article  Google Scholar 

  96. A. Cauchy, Mémoiresur les systèmesisotropes de points matériels, Oeuvres Complètes, 1re Série — Tome II (1850a) 351–386.

  97. A. Cauchy, Mémoiresur les vibrations d’un double système de molécules et de l’éthercontinudans un corps cristallisé, 1re Série — Tome II (1850) 338–350.

  98. W. Voigt, Theoretischestudienüber die elasticitätsverhältnisse der krystalle. i. ableitung der grundgleichungenaus der annahmemitpolaritätbegabtermoleküle, Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaftenzu Göttingen, 24 (1887) 3–52.

    Google Scholar 

  99. R. D. Mindlin and H. F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11(1) (1962) 415–448.

    Article  MathSciNet  MATH  Google Scholar 

  100. R. D. Mindlin, Influence of couple-stresses on stress concentrations, Experimental Mechanics, 3(1) (1963) 1–7.

    Article  Google Scholar 

  101. R. D. Mindlin, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16(1) (1964) 51–78.

    Article  MathSciNet  MATH  Google Scholar 

  102. R. A. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, 11(1) (1962) 385–414.

    Article  MathSciNet  MATH  Google Scholar 

  103. R. A. Toupin, Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17(2) (1964) 85–112, doi: https://doi.org/10.1007/BF00253050.

    Article  MathSciNet  MATH  Google Scholar 

  104. W. T. Koiter, Couple-stresses in the theory of elasticity, I and II, Proceedings Series B, Koninklijke Nederlandse Akademie van Wetenschappen, 67 (1964) 17–47.

    MathSciNet  MATH  Google Scholar 

  105. E. Cosserat and F. Cosserat, Theory of Deformable Bodies, Hermann et Fils, Paris (1909).

    MATH  Google Scholar 

  106. A. C. Eringen, Theory of micropolar elasticity, Microcontinuum Field Theories, Springer, New York (1999) 101–248.

    Chapter  MATH  Google Scholar 

  107. A. C. Eringen and D. G. B. Edelen, On nonlocal elasticity, International Journal of Engineering Science, 10(3) (1972) 233–248.

    Article  MathSciNet  MATH  Google Scholar 

  108. A. C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 10(5) (1972) 425–435.

    Article  MATH  Google Scholar 

  109. F. Yang, A. C. M. Chong, D. C. C. Lam and P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10) (2002) 2731–2743.

    Article  MATH  Google Scholar 

  110. M. Fathalilou and G. Rezazadeh, Effects of the length scale parameter on the thermoelastic damping of a micro-beam considering the couple stress theory, International Journal of Applied Mechanics, 8(6) (2016) 1650083.

    Article  Google Scholar 

  111. B. A. Hamidi, S. A. Hosseini, R. Hassannejad and F. Khosravi, An exact solution on gold microbeam with thermoelastic damping via generalized Green-Naghdi and modified couple stress theories, Journal of Thermal Stress, 43(2) (2020) 157–174.

    Article  Google Scholar 

  112. A. Rahmani, S. Faroughi, M. I. Friswell and A. Babaei, Eringen’s nonlocal and modified couple stress theories applied to vibrating rotating nanobeams with temperature effects, Mechanics of Advanced Materials and Structures (2021).

  113. D. C. C. Lam, F. Yang, A. C. M. Chong, J. Wang and P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51(8) (2003) 1477–1508.

    Article  MATH  Google Scholar 

  114. S. Kong, S. Zhou, Z. Nie and K. Wang, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, Int. J. Eng. Sci., 47(4) (2009) 487–498.

    Article  MathSciNet  MATH  Google Scholar 

  115. F. Allahkarami, M. Nikkhah-bahrami and M. G. Saryazdi, Magneto-thermo-mechanical dynamic buckling analysis of a FGCNTs-reinforced curved microbeam with different boundary conditions using strain gradient theory, Int. J. Mech. Mater. Des., 14 (2018) 243–261.

    Article  Google Scholar 

  116. I. Pradiptya and H. M. Ouakad, Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory, Int. J. Mech. Mater. Des., 14(3) (2018) 393–415.

    Article  Google Scholar 

  117. J. Y. Yu, X. G. Tian and J. Liu, Size-dependent damping of a nanobeam using nonlocal thermoelasticity: extension of Zener, Lifshitz, and Roukes’ damping model, Acta. Mech., 228 (2017) 1287–1302.

    Article  MathSciNet  MATH  Google Scholar 

  118. S. Rashahmadi and S. A. Meguid, Modeling size-dependent thermoelastic energy dissipation of graphene nanoresonators using nonlocal elasticity theory, Acta. Mech., 230(3) (2019) 771–785.

    Article  MathSciNet  MATH  Google Scholar 

  119. V. Borjalilou, M. Asghari and E. Taati, Thermoelastic damping in nonlocal nanobeams considering dual-phase-lagging effect, Journal of Vibration and Control, 26(11–12) (2020) 1042–1053.

    Article  MathSciNet  Google Scholar 

  120. X. Li, L. Li, Y. Hu, Z. Ding and W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, 165 (2017) 250–265.

    Article  Google Scholar 

  121. L. Lu, X. Guo and J. Zhao, A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms, International Journal of Engineering Science, 119(Supplement C) (2017) 265–277.

    Article  Google Scholar 

  122. B. A. Hamidi, S. A. Hosseini and H. Hayati, Forced torsional vibration of nanobeam via nonlocal strain gradient theory and surface energy effects under moving harmonic torque, Waves in Random and Complex Media, 32(1) (2020) 318–333.

    Article  MathSciNet  Google Scholar 

  123. M. Fathalilou, M. Sadeghi, G. Rezazadeh, M. Jalilpour, A. Naghiloo and S. Ahouighazvin, Study on the pull-in instability of gold micro-switches using variable length scale parameter, Journal of Solid Mechanics, 3 (2011) 114–123.

    Google Scholar 

  124. M. Rahaeifard, M. Kahrobaiyan, M. Asghari and M. Ahmadian, Static pull-in analysis of microcantilevers based on the modified couple stress theory, Sens. Actuat. A, 171(2) (2011) 370–374.

    Article  MATH  Google Scholar 

  125. M. Asghari, M. H. Kahrobaiyan, M. Nikfar and M. T. Ahmadian, A sizedependent nonlinear Timoshenko micro-beam model based on the strain gradient theory, Acta Mechanica., 223(6) (2012) 1233–1249.

    Article  MathSciNet  MATH  Google Scholar 

  126. G. Rezazadeh, A. S. Vahdat, S. Tayefehrezaei and C. Cetinkaya, Thermoelastic damping in a micro-beam resonator using modified couple stress theory, Acta. Mechanica., 223(6) (2012) 1137–1152.

    Article  MathSciNet  MATH  Google Scholar 

  127. A. S. Vahdat, G. Rezazadeh and G. Ahmadi, Thermoelastic damping in a micro-beam resonator tunable with piezoelectric layers, Acta Mech. Solida. Sin., 25 (2012) 73–81.

    Article  Google Scholar 

  128. M. Shaat and S. Mohamed, Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories, Int. J. Mech. Sci., 84 (2014) 208–217.

    Article  Google Scholar 

  129. B. Akgöz and Ö. Civalek, A new trigonometric beam model for buckling of strain gradient microbeams, International Journal of Mechanical Sciences, 81 (2014) 88–94.

    Article  Google Scholar 

  130. E. Taati, M. M. Najafabadi and H. B. Tabrizi, Size-dependent generalized thermoelasticity model for Timoshenko microbeams, Acta Mech., 225(7) (2014) 1823–1842.

    Article  MathSciNet  MATH  Google Scholar 

  131. A. A. Emami and A. Alibeigloo, Exact solution for thermal damping of functionally graded Timoshenko microbeams, J. Thermal Stress, 39(2) (2016) 231–243.

    Article  Google Scholar 

  132. H. Zhang, T. Kim, G. Choi and H. H. Cho, Thermoelastic damping in micro-and nanomechanical beam resonators considering size effects, Int. J. Heat Mass Transf., 103 (2016) 783–790.

    Article  Google Scholar 

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

    Article  Google Scholar 

  134. J. Lei, Y. He, S. Guo, Z. Li and D. Liu, Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM, Journal of Thermal Stresses, 40(6) (2017) 665–689.

    Article  Google Scholar 

  135. X. Li and Y. Luo, Size-dependent postbuckling of piezoelectric microbeams based on a modified couple stress theory, International Journal of Applied Mechanics, 9(4) (2017) 1750053.

    Article  Google Scholar 

  136. M. Bostani and A. K. Mohammadi, Thermoelastic damping in microbeam resonators based on modified strain gradient elasticity and generalized thermoelasticity theories, Acta. Mech., 229 (2018) 173–192.

    Article  MathSciNet  MATH  Google Scholar 

  137. M. Kandaz and H. Dal, A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams, Archive of Applied Mechanics, 88 (2018) 2051–2070.

    Article  Google Scholar 

  138. M. Al-shujairi and Ç. Mollamahmutoğlu, Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect, Composites Part B: Engineering, 154 (2018) 292–312.

    Article  Google Scholar 

  139. V. Borjalilou and M. Asghari, Small-scale analysis of plates with thermoelastic damping based on the modified couple stress theory and the dualphase-lag heat conduction model, Acta Mechanica., 229(9) (2018) 3869–3884.

    Article  MathSciNet  MATH  Google Scholar 

  140. F. Allahkarami, M. Nikkhah-bahrami and M. G. Saryazdi, Magneto-thermomechanical dynamic buckling analysis of a FGCNTs-reinforlifshiced curved microbeam with different boundary conditions using strain gradient theory, International Journal of Mechanics and Materials in Design, 14(2) (2018) 243–261.

    Article  Google Scholar 

  141. S. Rashahmadi and S. A. Meguid, Modeling size-dependent thermoelastic energy dissipation of graphene nanoresonators using nonlocal elasticity theory, Acta Mech., 230(3) (2019) 771–785.

    Article  MathSciNet  MATH  Google Scholar 

  142. R. Resmi, M. R. Baiju and V. S. Babu, Thermoelastic damping dependent quality factor analysis of rectangular plates applying modified coupled stress theory, AIP Conference Proceedings, 2166 (2019) 020029.

    Article  Google Scholar 

  143. V. Borjalilou, M. Asghari and E. Bagheri, Small-scale thermoelastic damping in micro-beams utilizing the modified couple stress theory and the dual-phase-lag heat conduction model, J. Thermal Stress., 42 (2019) 801–814.

    Article  Google Scholar 

  144. V. Borjalilou, M. Asghari and E. Taati, Thermoelastic damping in nonlocal nanobeams considering dual-phase-lagging effect, J. Vib. Control, 26(11–12) (2020) 1042–1053.

    Article  MathSciNet  Google Scholar 

  145. Z. Yang, D. Cheng, G. Cong, D. Jin and V. Borjalilou, Dual-phase-lag Thermoelastic Damping in Nonlocal Rectangular Nanoplates, Waves in Random and Complex Media (2021).

  146. S. Shuanhu, T. He and F. Jin, Thermoelastic damping analysis of size-dependent nano-resonators considering dual-phase-lag heat conduction model and surface effect, International Journal of Heat and Mass Transfer, 170 (2021) 120977.

    Article  Google Scholar 

  147. F. Ebrahimi, A. Seyfi, M. Nouraei and P. Haghi, Influence of Magnetic Field on the Wave Propagation Response of Functionally Graded (FG) Beam Lying on Elastic Foundation in Thermal Environment, Waves in Random and Complex Media (2021).

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Authors and Affiliations

  1. LBS Institute of Technology for Women, University of Kerala, Thiruvananthapuram, Kerala, 695012, India

    R. Resmi

  2. College of Engineering Trivandrum, APJ Abdul Kalam Technological University, Thiruvananthapuram, Kerala, 670644, India

    V. Suresh Babu

  3. University of Kerala, Kerala Public Service Commission, Thiruvananthapuram, Kerala, 695004, India

    M. R. Baiju

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R. Resmi obtained her B.Tech. in Electronics Engineering from Cochin University of Science and Technology, India in 2000 and her M.Tech. from the College of Engineering Trivandrum, University of Kerala. She is currently working as an Assistant Professor at the LBS Institute of Technology for Women, Kerala, India. Her research interests include RF MEMS resonators and sensors design, energy dissipation analysis in MEMS/NEMS structures, biomedical signal processing and instrumentation, image processing, and VLSI systems design. She is currently completing her Ph.D. in the Department of Electronics and Communication, College of Engineering Trivandrum, under the University of Kerala. Orcid Id: http://orcid.org/0000-0002-0849-5509.

V. Suresh Babu obtained his B.Tech. from the Electronics and Communication Engineering Department of Kerala University, Trivandrum, India in 1989, his M.Tech. in Integrated Electronic Devices and Circuits from IIT Madras in 1996, and his Ph.D. in Electronics and Communication in 2013 from Kerala University. Form 1990 to 1991, he was with the Kerala State Electronics Development Corporation Ltd., Trivandrum, India. Since 1991, he has been a Member of the Faculty of the Department of Electronics and Communication Engineering in Government Engineering Colleges in Kerala. He is currently a Professor at the Electronics and Communications Engineering Department, College of Engineering Trivandrum, Kerala, India. His areas of interest are solar photovoltaics and nano-electronic devices. Orcid Id: http://orcid.org/0000-0001-8024-2621.

M. R. Baiju obtained his B.Tech. in Electronics and Communication Engineering from the College of Engineering, Trivandrum, India, in 1988, M.Tech. in Electronics Design and Technology, and Ph.D. in Power Electronics from the Center for Electronics Design and Technology, Indian Institute of Science, Bangalore, India, in 1997 and 2004, respectively. From 1988 to 1991, he was with the National Thermal Power Corporation Ltd., New Delhi, India. Since 1991, he has been a Member of the Faculty of the Department of Electronics and Communication Engineering, College of Engineering Trivandrum, Kerala. His areas of interest include multilevel inverter control strategies, MEMS, and VLSI systems.

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Resmi, R., Babu, V.S. & Baiju, M.R. Material-dependent thermoelastic damping limited quality factor and critical length analysis with size effects of micro/nanobeams. J Mech Sci Technol 36, 3017–3038 (2022). https://doi.org/10.1007/s12206-022-0533-8

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