Nonlinear oscillations of viscoelastic microcantilever beam based on modified strain gradient theory

Document Type : Article

Authors

School of Mechanical Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9567, Iran

Abstract

A viscoelastic microcantilever beam is analytically analyzed based on the modified strain gradient theory. The Kelvin Voigt scheme is used to model the beam viscoelasticity. Applying Bernoulli-Euler inextensibility of the centerline condition via Hamilton’s principle, the nonlinear equation of motion and related boundary conditions are derived based on shortening effect theory and discretized by Galerkin method. Inner damping, nonlinear curvature effect, and nonlinear inertia terms are applied. The generalized derived formulation in this article, allows modeling of any nonlinearity combinations such as nonlinear terms arises due to inertia, damping, and stiffness, as well as modeling the size effect via considering modified coupled stress or modified strain gradient theories. First mode nonlinear frequency and time response of the viscoelastic microcantilever beam are analytically evaluated utilizing multiple time scale method and validated by numerical findings. Results indicate that the nonlinear terms have an appreciable effect on natural frequency and time response of a viscoelastic microcantilever. Furthermore, the investigation reveals that due to the size effects, natural frequency enhances drastically, especially when the thickness of the beam and the length scale parameter are comparable. Outcomes clarify the importance of size effects in analyzing of the mechanical behavior of small scale structures.

Keywords

References

References
1.      Zand, M.M. and Ahmadian, M.T., "Application of homotopy analysis method in studying dynamic pull-in instability of microsystems", Mechanics Research Communications, 36(7), pp. 851-858 (2009).
2.      Ghommem, M. and Abdelkefi, A., "Nonlinear reduced-order modeling and effectiveness of electrically-actuated microbeams for bio-mass sensing applications", International Journal of Mechanics and Materials in Design, 15(1), pp. 125-143 (2019).
3.      McFarland, A.W. and Colton, J.S., "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", Journal of Micromechanics and Microengineering, 15(5), pp. 1060-1067 (2005).
4.      Mindlin, R.D., "Second gradient of strain and surface-tension in linear elasticity", International Journal of Solids and Structures", 1(4), pp. 417-438 (1965).
5.      Koiter, W.T., "Couple stresses in the theory of elasticity, I and II", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series B, pp. 6717-6744 (1964).
6.      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).
7.      Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P., "Experiments and theory in strain gradient elasticity", Journal of the Mechanics and Physics of Solids, 51(8), pp. 1477-1508 (2003).
8.      Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T. and Firoozbakhsh, K., "Strain gradient formulation of functionally graded nonlinear beams", International Journal of Engineering Science, 65(0), pp. 49-63 (2013).
9.      Rahaeifard, M., Ahmadian, M. and Firoozbakhsh, K., "Vibration analysis of electrostatically actuated nonlinear microbridges based on the modified couple stress theory", Applied Mathematical Modelling, 39(21), pp. 6694-6704 (2015).
10.    Abbasi, M. and Mohammadi, A.K., "Study of the sensitivity and resonant frequency of the flexural modes of an atomic force microscopy microcantilever modeled by strain gradient elasticity theory", Journal of Mechanical Engineering Science, Proceedings of the Institution of Mechanical Engineers, Part C, 228(8), pp. 1299-1310 (2015).
11.    Lazopoulos, A.K., Lazopoulos, K.A. and Palassopoulos, G., "Nonlinear bending and buckling for strain gradient elastic beams", Applied Mathematical Modelling, 38(1), pp. 253-262 (2014).
12.    Karami, B., Shahsavari, D., Janghorban, M. and Li, L., "Influence of homogenization schemes on vibration of functionally graded curved microbeams", Composite Structures, 216, pp. 67-79 (2019).
13.    Zhang, B., He, Y., Liu, D., Gan, Z. and Shen, L., "Non-classical Timoshenko beam element based on the strain gradient elasticity theory", Finite elements in analysis and design, 79, pp. 22-39 (2014).
14.    Chen, X. and Li, Y., "Size-dependent post-buckling behaviors of geometrically imperfect microbeams", Mechanics Research Communications, 88, pp. 25-33 (2018).
15.    Akgöz, B. and Civalek, Ö., "Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory", Archive of Applied Mechanics, 82(3), pp. 423-443 (2012).
16.    Akgöz, B. and Civalek, Ö., "Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams", International Journal of Engineering Science, 49(11), pp. 1268-1280 (2012).
17.    Miandoab, E.M., Yousefi-Koma, A. and Pishkenari, H.N., "Poly silicon nanobeam model based on strain gradient theory", Mechanics Research Communications, 62, pp. 83-88 (2014).
18.    Mahmoodi, S., Khadem, S.E. and Kokabi, M., "Non-linear free vibrations of Kelvin–Voigt visco-elastic beams", International Journal of Mechanical Sciences, 49(6), pp. 722-732 (2007).
19.    Yazdi, F.C. and Jalali, A., "Vibration behavior of a viscoelastic composite microbeam under simultaneous electrostatic and piezoelectric actuation", Mechanics of Time-Dependent Materials, 19(3), pp. 277-304 (2015).
20.    Zhu, C., Fang, X. and Liu, J., "A new approach for smart control of size-dependent nonlinear free vibration of viscoelastic orthotropic piezoelectric doubly-curved nanoshells", Applied Mathematical Modelling, 77, pp. 137-168 (2020).
21.    Zhu, C., Fang, X. and Yang, S., "Nonlinear free vibration of functionally graded viscoelastic piezoelectric doubly curved nanoshells with surface effects", The European Physical Journal Plus, 134(10), pp. 486 (2019).
22.    Zhu, C.-S., Fang, X.-Q., Liu, J.-X. and Li, H.-Y., "Surface energy effect on nonlinear free vibration behavior of orthotropic piezoelectric cylindrical nano-shells", European Journal of Mechanics-A/Solids, 66, pp. 423-432 (2017).
23.    Eduok, U., Faye, O. and Szpunar, J., "Recent developments and applications of protective silicone coatings: A review of PDMS functional materials", Progress in Organic Coatings, 111, pp. 124-163 (2017).
24.    Le Digabel, J., Ghibaudo, M., Trichet, L., Richert, A. and Ladoux, B., "Microfabricated substrates as a tool to study cell mechanotransduction", Medical & biological engineering & computing, 48(10), pp. 965-976 (2010).
25.    Li, L. and Hu, Y., "Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory", Computational Materials Science, 112, pp. 282-288 (2016).
26.    Li, L., Hu, Y. and Ling, L., "Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory", Physica E: Low-dimensional Systems and Nanostructures, 75, pp. 118-124 (2016).
27.    Attia, M. and Mohamed, S., "Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory", Applied Mathematical Modelling, 41, pp. 195-222 (2017).
28.    Attia, M. and Mahmoud, F., "Analysis of viscoelastic Bernoulli–Euler nanobeams incorporating nonlocal and microstructure effects", International Journal of Mechanics and Materials in Design, 13(3), pp. 385-406 (2017).
29.    Attia, M.A. and Rahman, A.A.A., "On vibrations of functionally graded viscoelastic nanobeams with surface effects", International Journal of Engineering Science, 127, pp. 1-32 (2018).
30.    Ansari, R., Oskouie, M.F. 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 (2017).
31.    Fu, Y., Zhang, J. and Bi, R., "Analysis of the nonlinear dynamic stability for an electrically actuated viscoelastic microbeam", Microsystem technologies, 15(5), pp. 763 (2009).
32.    Fu, Y. and Zhang, J., "Electromechanical dynamic buckling phenomenon in symmetric electric fields actuated microbeams considering material damping", Acta Mech, 215(1-4), pp. 29-42 (2010).
33.    Zhang, J. and Fu, Y., "Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory", Meccanica, 47(7), pp. 1649-1658 (2012).
34.    Ghayesh, M.H., "Stability and bifurcation characteristics of viscoelastic microcantilevers", Microsystem technologies, 24(12), pp. 4739-4746 (2018).
35.    Ghayesh, M.H. and Farokhi, H., "Viscoelastically coupled size-dependent behaviour of imperfect extensible microbeams", International Journal of Mechanics and Materials in Design, 13(4), pp. 569-581 (2017).
36.    Ghayesh, M.H. and Farokhi, H., "Size-dependent internal resonances and modal interactions in nonlinear dynamics of microcantilevers", International Journal of Mechanics and Materials in Design, 14(1), pp. 127-140 (2018).
37.    Ghayesh, M.H. and Farokhi, H., "Size-dependent large-amplitude oscillations of microcantilevers", Microsystem technologies, 23(8), pp. 3477-3488 (2017).
38.    Farokhi, H., Ghayesh, M.H. and Hussain, S., "Large-amplitude dynamical behaviour of microcantilevers", International Journal of Engineering Science, 106, pp. 29-41 (2016).
39.    Farokhi, H. and Ghayesh, M.H., "Nonlinear resonant response of imperfect extensible Timoshenko microbeams", International Journal of Mechanics and Materials in Design, 13(1), pp. 43-55 (2017).
40.    Ghayesh, M.H., "Mechanics of viscoelastic functionally graded microcantilevers", European Journal of Mechanics-A/Solids, 73, pp. 492-499 (2019).
41.    Nayfeh, A.H. and Mook, D.T., "Nonlinear oscillations", John Wiley & Sons (2008).
42.    Nayfeh, A. and Nayfeh, S., "On nonlinear modes of continuous systems", Journal of Vibration and Acoustics, 116(1), pp. 129-136 (1994).
43.    Kong, S., Zhou, S., Nie, Z. and Wang, K., "Static and dynamic analysis of micro beams based on strain gradient elasticity theory", International Journal of Engineering Science, 47(4), pp. 487-498 (2009).
Scientia Iranica
Volume 28, Issue 2
Transactions on Mechanical Engineering (B)
March and April 2021
Pages 785-794

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