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**

1. Lot, M. and Moghimi, M. Transient behavior

and dynamic pull-in instability of electrostaticallyactuated

uid-conveying microbeams", International

Journal of Microsyst Technologies, 23(13), pp. 6015{

6023 (2017).

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 eect 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 eects 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).

2900 Sh. Haddad et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2889{2901

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. Eects 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 modi

ed 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 modied 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 modied 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

modied 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 lms based on

generalized rened 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 lms", 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",

Scientic 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 rst-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.

Eects 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 in

uence of the intermolecular surface forces

on the static de

ection 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 modied couple stress theory", IOP science,

2355, pp. 1{5 (2006).

34. Liao, S., Homotopy Analysis Method in Nonlinear

Dierential 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).

Sh. Haddad et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2889{2901 2901

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).

Volume 27, Issue 6 - Serial Number 6

Transactions on Mechanical Engineering (B)

November and December 2020Pages 2889-2901