Document Type : Article

**Authors**

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

**Abstract**

In the current study, an analytical solution based on the modified couple stress theory for a nonlinear model describing the couple 3D motion of a functionally graded tapered micro-bridge is presented. The small scale effects and the nonlinearity arising from the mid-plane stretching are taken into consideration. Governing equations of motions are derived utilizing the modified couple stress theory and applying Hamilton principle. Dynamic and static analyses to determine the effects of lateral distributed forces and mid-plane stretching are investigated. To this aim, analytical Homotopy-pade technique is employed to capture the nonlinear natural frequencies in high amplitude vibrations of tapered micro-bridges with different types of geometries and material compositions. The obtained results of frequencies propose that there is a good agreement between the present analytical results and the numerical ones in opposed to well-known multiple-scale method. Furthermore, comparing the results in 2D and 3D analyses shows that in 2D analysis, the stiffness and natural frequency of the micro-beam is underestimated and it is found that increasing the tapered ratio has different impacts on natural frequencies for micro-beams with different slender ratios.

**Keywords**

**Main Subjects**

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26. Baghani, M., Mazaheri, H., and Salarieh, H. "Analysis of large amplitude free vibrations of clamped tapered beams on a nonlinear elastic foundation", Applied Mathematical Modelling, 38(3), pp. 1176-1186 (2014).

27. Raju, S.S., Umapathy, M., and Uma, G. "High-output piezoelectric energy harvester using tapered beam with cavity", Journal of Intelligent Material Systems and Structures, 29(5), pp. 1-16 (2017).

28. Mohammadsalehi, M., Zargar, M., and Baghani, M. "Study of non-uniform viscoelastic nanoplates vibration based on nonlocal first-order shear deformation theory", Meccanica, 52(4-5), pp. 1063-1077 (2017).

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31. Mojahedi, M., Ahmadian, M.T., and Firoozbakhsh, K. "The inبمuence of the intermolecular surface forces on the static deflection and pull-in instability of the micro/nano cantilever gyroscopes", Composites Part B: Engineering, 56, pp. 336-343 (2014).

32. Mojahedi, M. and Rahaeifard, M. "A size-dependent model for coupled 3D deformations of nonlinear microbridges", International Journal of Engineering Science, 100, pp. 171-182 (2016).

33. Park, S.K. and Gao, X. "Bernoulli-Euler beam model based on a modified couple stress theory", IOP science, 2355, pp. 1-5 (2006).

34. Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations, Springer (2011).

35. Carrera, E., Giunta, G., and Petrolo, M., Beam Structures, Beam Structures: Classical and Advanced Theories, John Wiley and S ons (2011).

36. Baghani, M., Mohammadsalehi, M., and Dabaghian, P.H. "Analytical couple-stress solution for sizedependent large-amplitude vibrations of FG taperednanobeams", Latin American Journal of Solids and Structures, 13(1), pp. 95-118 (2014).

37. Liao, S., Beyond Perturbation: An Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC (2004).

38. Zhang, B., He, Y., Liu, D., Gan, Z., and Shen, L. "Sizedependent functionally graded beam model based on an improved third-order shear deformation theory", European Journal of Mechanics, A/Solids, 47, pp. 211-230 (2014).

2. Chong, A.C.M., Yang, F., Lam, D.C.C., and Tong, P. "Torsion and bending of micron-scaled structures", Journal of Materials Research, 16(4), pp. 1052-1058 (2001).

3. Stolken, J.S. and Evans, A.G. "A microbend test method for measuring the plasticity length scale", Acta Material, 46(14), pp. 5109-5115 (1998).

4. Ma, Q. and Clarke, D.R. "Size dependent hardness of silver single crystals", J. Mater. Res., 10(4.4), pp. 853-863 (1995).

5. Chong Arthur, C.M. and Lam, D.C.C. "Strain gradient plasticity effect in indentation hardness of polymers", Materials Research, 14(10), pp. 4103-4110 (1999).

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

7. Mindlin, R.D. "Second gradient of strain and surfacetension in linear elasticity", International Journal of Solids and Structures, 1(4), pp. 417-438 (1965).

8. Fleck, N.A. and Hutchinson, J.W. "A phenomenological theory for strain gradient effects in plasticity", Journal of the Mechanics and Physics of Solids, 41(12), pp. 1825-1857 (1993).

9. Fleck, N.A. and Hutchinson, J.W. "Strain gradient plasticity", Advances in Applied Mechanics, 33, pp. 295-361 (1997).

10. Fleck, N.A. and Hutchinson, J.W. "A reformulation of strain gradient plasticity", Journal of the Mechanics and Physics of Solids, 49(10), pp. 2245-2271 (2001).

11. Ansari, R., Faraji Oskouie, M., and Rouhi, H. "Studying linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/ nano-beams using the strain gradient theory", Nonlinear Dynamics, 87(1), pp. 695-711 (2016).

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

13. Mindlin, R.D. and Tiersten, H.F. "Effects of couplestresses in linear elasticity", Archive for Rational Mechanics and Analysis, 11(1), pp. 415-448 (1962).

14. Jafari-talookolaei, R., Ebrahimzade, N., and Rashidijuybari, S. "Bending and vibration analysis of delaminated Bernoulli-Euler microbeams using the modified couple stress", Scientia Iranica, 25, pp. 675-688 (2018).

15. Jalali, M.H., Zargar, O., and Baghani, M. "Sizedependent vibration analysis of FG microbeams in thermal environment based on modified couple stress theory", Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 43, pp. 761- 771 (2011).

16. Bhattacharya, S. and Das, D. "Free vibration analysis of bidirectional-functionally graded and doubletapered rotating micro-beam in thermal environment using modified couple stress theory", Composite Structures, 215, pp. 471-492 (2019).

17. Baghani, M. "Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory", International Journal of Engineering Science, 54, pp. 99-105 (2012).

18. Lu, C.F., Lim, C.W., and Chen, W.Q. "Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory", International Journal of Solids and Structures, 46(5), pp. 1176-1185 (2009).

19. Fu, Y., Du, H., and Zhang, S. "Functionally graded TiN/TiNi shape memory alloy films", Materials Letters, 57(20), pp. 2995-2999 (2003).

20. Rahaeifard, M., Kahrobaiyan, M.H., and Ahmadian, M.T. "Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials", ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 539-544 (2009).

21. Bashirpour, M., Forouzmehr, M., and Hosseininejad, S.E. "Improvement of terahertz photoconductive antenna using optical antenna array of ZnO nanorods", Scientific Reports, pp. 1-8 (2019).

22. Baghani, M. and Fereidoonnezhad, B. "Limit analysis of FGM circular plates subjected to arbitrary rotational symmetric loads using von-Mises yield criterion", Acta Mechanica, 224(8), pp. 1601-1608 (2013).

23. Baghani, M., Mazaheri, H., and Salarieh, H. "Analysis of large amplitude free vibrations of clamped tapered beams on a nonlinear elastic foundation", Applied Mathematical Modelling, 38(3), pp. 1176-1186 (2014).

24. Sadeghi, H., Baghani, M., and Naghdabadi, R. "Strain gradient elasticity solution for functionally graded micro", International Journal of Engineering Science, 50(1), pp. 20-23 (2011).

25. Bashirpour, M., Kefayati, A., Kolahdouz, M., and Aghababa, H. "Tuning the electronic properties of symetrical and asymetrical boron nitride passivated graphene nanoribbons: Density function theory", Journal of Nano Research, 54, pp. 35-41 (2018).

26. Baghani, M., Mazaheri, H., and Salarieh, H. "Analysis of large amplitude free vibrations of clamped tapered beams on a nonlinear elastic foundation", Applied Mathematical Modelling, 38(3), pp. 1176-1186 (2014).

27. Raju, S.S., Umapathy, M., and Uma, G. "High-output piezoelectric energy harvester using tapered beam with cavity", Journal of Intelligent Material Systems and Structures, 29(5), pp. 1-16 (2017).

28. Mohammadsalehi, M., Zargar, M., and Baghani, M. "Study of non-uniform viscoelastic nanoplates vibration based on nonlocal first-order shear deformation theory", Meccanica, 52(4-5), pp. 1063-1077 (2017).

29. Zhao, X.W., Hu, Z.D., and van der Heijden, G.H.M. "Dynamic analysis of a tapered cantilever beam under a travelling mass", Meccanica, 50(6), pp. 1419-1429 (2015).

30. Mojahedi, M., Ahmadian, M.T., and Firoozbaksh, K. "Effects of casimir and van der waals forces on the pull-in instability of the nonlinear micro and nanobridge gyroscopes", International Journal of Structural Stability and Dynamics, 14(2), pp. 135-159 (2014).

31. Mojahedi, M., Ahmadian, M.T., and Firoozbakhsh, K. "The inبمuence of the intermolecular surface forces on the static deflection and pull-in instability of the micro/nano cantilever gyroscopes", Composites Part B: Engineering, 56, pp. 336-343 (2014).

32. Mojahedi, M. and Rahaeifard, M. "A size-dependent model for coupled 3D deformations of nonlinear microbridges", International Journal of Engineering Science, 100, pp. 171-182 (2016).

33. Park, S.K. and Gao, X. "Bernoulli-Euler beam model based on a modified couple stress theory", IOP science, 2355, pp. 1-5 (2006).

34. Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations, Springer (2011).

35. Carrera, E., Giunta, G., and Petrolo, M., Beam Structures, Beam Structures: Classical and Advanced Theories, John Wiley and S ons (2011).

36. Baghani, M., Mohammadsalehi, M., and Dabaghian, P.H. "Analytical couple-stress solution for sizedependent large-amplitude vibrations of FG taperednanobeams", Latin American Journal of Solids and Structures, 13(1), pp. 95-118 (2014).

37. Liao, S., Beyond Perturbation: An Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC (2004).

38. Zhang, B., He, Y., Liu, D., Gan, Z., and Shen, L. "Sizedependent functionally graded beam model based on an improved third-order shear deformation theory", European Journal of Mechanics, A/Solids, 47, pp. 211-230 (2014).

Transactions on Mechanical Engineering (B)

November and December 2020Pages 2889-2901