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Thickness dependence of viscoelastic stress relaxation of laminated microbeams due to mismatch strain

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Abstract

Time-dependent behaviors due to various mismatch strains are very important to the reliability of micro-/nano-devices. This paper aims at presenting an analytical model to study the viscoelastic stress relaxation of the laminated microbeam caused by mismatch strain. Firstly, Zhang’s two-variable method is used to establish a mechanical model for predicting the quasi-static stress relaxation of the laminated microbeam. Secondly, the related analytical solutions are obtained by combining the differential method and the eigenvalue method in the temporal domain. Finally, the influence of the substrate-to-film thickness/modulus ratio on the relaxation responses of the laminated microbeam subject to a step load of the mismatch strain is studied. The results show that the present predictions are consistent with the previous theoretical studies. Furthermore, the thickness dependence of stress relaxation time of the laminated microbeam is jointly determined by the intrinsic structural evolution factors and tension-bending coupling state; the stress relaxation time can be controlled by adjusting the substrate-to-film thickness/modulus ratio.

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References

  1. NIKPOURIAN, A., GHAZAVI, M. R., and AZIZI, S. Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance. Applied Mathematics and Mechanics (English Edition), 41(10), 1517–1538 (2020) https://doi.org/10.1007/s10483-020-2658-6

    Article  MathSciNet  Google Scholar 

  2. YAN, X., BROWN, W. L., LI, Y., PAPAPOLYMEROU, J., PALEGO, C., HWANG, J. C. M., and VINCI, R. P. Anelastic stress relaxation in gold films and its impact on restoring forces in MEMS devices. Journal of Microelectromechanical Systems, 13, 570–576 (2009)

    Article  Google Scholar 

  3. OH, S. R., YAO, K., CHOW, C. L., and TAY, F. E. H. Residual stress in piezoelectric poly (vinylidene-fluoride-co-trifluoroethylene) thin films deposited on silicon substrates. Thin Solid Films, 519, 1441–1444 (2010)

    Article  Google Scholar 

  4. LI, J., ZHENG, B., YANG, Q., and HU, X. Analysis on time-dependent behavior of laminated functionally graded beams with viscoelastic interlayer. Composite Structures, 107, 30–35 (2014)

    Article  Google Scholar 

  5. LI, Y. Challenges and issues of using polymers as structural materials in MEMS: a review. Journal of Microelectromechanical Systems, 27, 581–597 (2018)

    Article  Google Scholar 

  6. WANG, S., SHAN, Z., and HUANG, H. The mechanical properties of nanowires. Advanced Science, 4, 1600332 (2017)

    Article  Google Scholar 

  7. CAO, Z. and ZHANG, X. Experiments and theory of thermally-induced stress relaxation in amorphous dielectric films for MEMS and IC applications. Sensors and Actuators A, 127, 221–227 (2006)

    Article  Google Scholar 

  8. MA, Y., PENG, G. J., JIANG, W. F., CHEN, H., and ZHANG, T. H. Nanoindentation study on shear transformation zone in a Cu-Zr-Al metallic glassy film with different thickness. Journal of Non-Crystalline Solids, 442, 67–72 (2016)

    Article  Google Scholar 

  9. SMITH, D. A., HOLMBERG, V. C., LEE, D C., and KORGEL, B. A. Young’s modulus and size-dependent mechanical quality factor of nanoelectromechanical germanium nanowire resonators. The Journal of Physical Chemistry C, 112, 10725–10729 (2008)

    Article  Google Scholar 

  10. CHEN, B., GAO, Q., WANG, Y., LIAO, X., MAI, Y. H., TAN, H. H., ZOU, J., RINGER, S. P., and JAGADISH, C. Anelastic behavior in GaAs semiconductor nanowires. Nano Letters, 13, 3169–3172 (2013)

    Article  Google Scholar 

  11. FREUND, L. B. Some elementary connections between curvature and mismatch strain in compositionally graded thin films. Journal of the Mechanics and Physics of Solids, 44, 723–736 (1996)

    Article  Google Scholar 

  12. STONEY G. G. The tension of metallic films deposited by electrolysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 82, 172–175 (1909)

    Google Scholar 

  13. KO, S. C., LEE, S., and HSUEH, C. H. Viscoelastic stress relaxation in film/substrate systems—Kelvin model. Journal of Applied Physics, 93, 2453–2457 (2003)

    Article  Google Scholar 

  14. WU, J. Z. and ZHANG, N. H. Clamped-end effect on static detection signals of DNA-microcantilever. Applied Mathematics and Mechanics (English Edition), 42(10), 1423–1438 (2021) https://doi.org/10.1007/s10483-021-2780-6

    Article  MathSciNet  Google Scholar 

  15. RU, C. Q. Size effect of dissipative surface stress on quality factor of microbeams. Applied Physics Letters, 94, 051905–051907 (2009)

    Article  Google Scholar 

  16. DEHROUYEH-SEMNANI, A. M., DEHROUYEH, M., TOABI-KAFSHGARI, M., and NIKKHAH-BAHRAMI, M. A damped sandwich beam model based on symmetric-deviatoric couple stress theory. International Journal of Engineering Science, 92, 83–94 (2015)

    Article  Google Scholar 

  17. HOSSEINI-HASHEMI, S., BEHDAD, S., and FAKHER, M. Vibration analysis of two-phase local/nonlocal viscoelastic nanobeams with surface effects. The European Physical Journal Plus, 135, 190–208 (2019)

    Article  Google Scholar 

  18. SOBHY, M. and ZENKOUR, A. M. The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations. Mechanics of Advanced Materials and Structures, 27, 525–539 (2020)

    Article  Google Scholar 

  19. OSKOUIE, M. F. and ANSARI, R. Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects. Applied Mathematical Modelling, 43, 337–350 (2017)

    Article  MathSciNet  Google Scholar 

  20. EBRAHIMI, F. and BARATI, M. R. Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Composite Structures, 159, 433–444 (2017)

    Article  Google Scholar 

  21. CHEN, J., FANG, L., ZHANG, M., PENG, W., SUN, K., and HAN, J. Stress relaxation behaviors of monocrystalline silicon coated with amorphous SiO2 film: a molecular dynamics study. Acta Mechanica Solida Sinica, 34, 506–515 (2021)

    Article  Google Scholar 

  22. HEINRICH, S. M., WENZENL, M. J., JOSSE, F., and DUFOUR I. An analytical model for transient deformation of viscoelastically coated beams: applications to static-mode microcantilever chemical sensors. Journal of Applied Physics, 105, 124903 (2009)

    Article  Google Scholar 

  23. XI, Y. Y., LYU, Q., ZHANG, N. H., and WU, J. Z. Thermal-induced snap-through buckling of simply-supported functionally graded beams. Applied Mathematics and Mechanics (English Edition), 41(12), 1821–1832 (2020) https://doi.org/10.1007/s10483-020-2691-7

    Article  MathSciNet  Google Scholar 

  24. ZHANG, N. H. and CHEN, J. Z. An alternative two-variable model for bending problems of multilayered beams. Journal of Applied Mechanics, 75, 044503 (2008)

    Article  Google Scholar 

  25. LIU, H. L., ZHANG, C. Y., and ZHANG, N. H. Thermal stress and internal force analysis of PV module (in Chinese). Acta Energiae Solaris Sinica, 41, 215–220 (2020)

    Google Scholar 

  26. YUE, Y. M., XU, K. Y., ZHANG, X. D., and WANG, W. J. Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam. Applied Mathematics and Mechanics (English Edition), 39(7), 953–966 (2018) https://doi.org/10.1007/s10483-018-2346-8

    Article  MathSciNet  Google Scholar 

  27. YANG, T. Q. Theory of Viscoelasticity (in Chinese), Huazhong University of Science and Technology Press, Wuhan (1990)

    Google Scholar 

  28. ZHANG, N. H. and XING, J. J. Vibration analysis of linear coupled thermoviscoelastic thin plates by a variational approach. International Journal of Solids and Structures, 45, 2583–2597 (2008)

    Article  Google Scholar 

  29. LYU, Q., LI, J. J., and ZHANG, N. H. Quasi-static and dynamical analyses of a thermo-viscoelastic Timoshenko beam using the differential quadrature method. Applied Mathematics and Mechanics (English Edition), 40(4), 549–562 (2019) https://doi.org/10.1007/s10483-019-2470-8

    Article  MathSciNet  Google Scholar 

  30. ZHANG, N. H. and CHENG, C. J. A time domain method for quasi-static analysis of viscoelastic thin plates. Applied Mathematics and Mechanics (English Edition), 22(10), 1109–1117 (2001) https://doi.org/10.1007/BF02436446

    Article  Google Scholar 

  31. REN, J., WARD, M., KINNELL, P., CRADDOCKAND, R., and WEI, X. Plastic deformation of micromachined silicon diaphragms with a sealed cavity at high temperatures. Sensors, 16, 204–216 (2016)

    Article  Google Scholar 

  32. ZHANG, Y. and ZHAO, Y. P. Determining both adhesion energy and residual stress by measuring the stiction shape of a microbeam. Microsystem Technologies, 21, 919–929 (2015)

    Article  Google Scholar 

  33. PARK, J. S., YONG, S. H., CHOI, Y. J., and CHAE, H. Residual stress analysis and control of multilayer flexible moisture barrier films with SiNx and Al2O3 layers. AIP Advances, 8, 085101 (2018)

    Article  Google Scholar 

  34. BALASUBRAMANIANA, S., PRABAKARA, K., ADITIB, MUKHIYAB, R., and SUNDARI, S. T. Effect of biaxial curvature on the resonance frequency of uncoated microcantilevers. Sensors and Actuators A: Physical, 304, 111857 (2020)

    Article  Google Scholar 

Download references

Funding

Project supported by the National Natural Science Foundation of China (Nos. 12172204, 11772182, 11272193, and 10872121) and the Program of Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018)

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China

    Xiaosheng Cai, Nenghui Zhang & Hanlin Liu

Authors
  1. Xiaosheng Cai
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  2. Nenghui Zhang
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  3. Hanlin Liu
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Corresponding author

Correspondence to Nenghui Zhang.

Additional information

Citation: CAI, X. S., ZHANG, N. H., and LIU, H. L. Thickness dependence of viscoelastic stress relaxation of laminated microbeams due to mismatch strain. Applied Mathematics and Mechanics (English Edition), 43(4), 467–478 (2022) https://doi.org/10.1007/s10483-022-2841-5

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Cai, X., Zhang, N. & Liu, H. Thickness dependence of viscoelastic stress relaxation of laminated microbeams due to mismatch strain. Appl. Math. Mech.-Engl. Ed. 43, 467–478 (2022). https://doi.org/10.1007/s10483-022-2841-5

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  • DOI: https://doi.org/10.1007/s10483-022-2841-5

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