Size dependent analysis of tapered FG micro-bridge based on a 3D beam theory

Document Type : Article


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


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.


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 e_ect 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 e_ects 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. E_ects 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 modi_ed 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 modi_ed 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  modi_ed 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 re_ned 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",  Scienti_c 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.  E_ects 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 inuence of the intermolecular surface forces  on the static deection 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 modi_ed couple stress theory", IOP science,  2355, pp. 1{5 (2006).  34. Liao, S., Homotopy Analysis Method in Nonlinear  Di_erential 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).