Investigation of Material Nonlinearities in Surface Effects and Bulk on the Vibration Characteristics of Nanobeams
DOI:
https://doi.org/10.31181/Keywords:
Materially Nonlinearities, Surface Effects, Vibration of NanobeamsAbstract
The surface effects play an important role in the vibrational properties of nanostructures. Investigating the nonlinear vibrations of nanobeams, the nonlinear material effects on surface effects have not been considered. This study aims to do research on the effect of material nonlinear coming from the stress-strain nonlinear equation on surface effects in order to increase the calculation accuracy. The effect of materially nonlinear behaviors of the bulk and surface effects in the presence of nonlinear Von Kármán strains are considered simultaneously. The governing equation based on the Hamilton principle and then the governing nonlinear differential equation based on applying the Galerkin’s method have been extracted. The obtained nonlinear differential equation possesses cubic and quantic nonlinearities, which are due to the geometric and the materially nonlinear behaviors, respectively. The quantic nonlinearity is only due to the materially nonlinear behavior of bulk and surface effects. The temporal response and nonlinear frequency of nanobeams are obtained by solving the nonlinear differential equation based on the modified Lindstedt–Poincaré method. The results represent the simultaneous effects of the materially nonlinear behaviors of the bulk and surface layers on the temporal response and nonlinear frequency of the nanobeam. To examine the validity of the results, the obtained natural frequencies are compared with other studies in the absence of the materially nonlinear term.
References
Alam, M., & Mishra, S. K. (2021). Nonlinear vibration of nonlocal strain gradient functionally graded beam on nonlinear compliant substrate. Composite Structures, 263, 113447. https://doi.org/10.1016/j.compstruct.2020.113447
Alizadeh Hamidi, B., Khosravi, F., Hosseini, S. A., & Hassannejad, R. (2020). Closed form solution for dynamic analysis of rectangular nanorod based on nonlocal strain gradient. Waves in Random and Complex Media, 1-17. https://doi.org/10.1080/17455030.2020.1843737
Arash, B., & Wang, Q. (2013). Detection of gas atoms with carbon nanotubes. Scientific reports, 3(1), 1-6. https://doi.org/10.1038/srep01782
Bunch, J. S., Van Der Zande, A. M., Verbridge, S. S., Frank, I. W., Tanenbaum, D. M., Parpia, J. M., Craighead, H. G., & McEuen, P. L. (2007). Electromechanical resonators from graphene sheets. science, 315(5811), 490-493. https://doi.org/10.1126/science.1136836
Burton, T. (1984). A perturbation method for certain non-linear oscillators. International Journal of Non-Linear Mechanics, 19(5), 397-407. https://doi.org/10.1016/0020-7462(84)90026-X
Cammarata, R. C. (1994). Surface and interface stress effects in thin films. Progress in surface science, 46(1), 1-38. https://doi.org/10.1016/0079-6816(94)90005-1
Chen, C., Shi, Y., Zhang, Y. S., Zhu, J., & Yan, Y. (2006). Size dependence of Young’s modulus in ZnO nanowires. Physical review letters, 96(7), 075505. https://doi.org/10.1103/PhysRevLett.96.075505
Chen, T., Chiu, M.-S., & Weng, C.-N. (2006). Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids. Journal of Applied Physics, 100(7), 074308. https://doi.org/10.1063/1.2356094
Cheung, Y., Chen, S., & Lau, S. (1991). A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators. International Journal of Non-Linear Mechanics, 26(3-4), 367-378. https://doi.org/10.1016/0020-7462(91)90066-3
Dang, V.-H. (2020). Buckling and nonlinear vibration of size-dependent nanobeam based on the non-local strain gradient theory. Journal of Applied Nonlinear Dynamics, 9(3), 427-446. https://doi.org/10.5890/JAND.2020.09.007
Dang, V. H., & Nguyen, T. H. (2021). Buckling and nonlinear vibration of functionally graded porous micro-beam resting on elastic foundation. Mechanics of Advanced Composite Structures. https://doi.org/10.22075/macs.2021.24098.1350
Ebrahimi, F., & Hosseini, S. H. S. (2021). Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study. Engineering with Computers, 37(1), 395-408. https://doi.org/10.1007/s00366-019-00830-0
Esfahani, S., Khadem, S. E., & Mamaghani, A. E. (2019). Nonlinear vibration analysis of an electrostatic functionally graded nano-resonator with surface effects based on nonlocal strain gradient theory. International Journal of Mechanical Sciences, 151, 508-522. https://doi.org/10.1016/j.ijmecsci.2018.11.030
Gheshlaghi, B., & Hasheminejad, S. M. (2011). Surface effects on nonlinear free vibration of nanobeams. Composites Part B: Engineering, 42(4), 934-937. https://doi.org/10.1016/j.compositesb.2010.12.026
Gurtin, M., Weissmüller, J., & Larche, F. (1998). A general theory of curved deformable interfaces in solids at equilibrium. Philosophical Magazine A, 78(5), 1093-1109. https://doi.org/10.1080/01418619808239977
Hamidi, B. A., Hosseini, S. A., Hassannejad, R., & Khosravi, F. (2020). Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green–Naghdi via nonlocal elasticity with surface energy effects. The European Physical Journal Plus, 135(1), 1-20. https://doi.org/10.1140/epjp/s13360-019-00037-8
Hamidi, B. A., Hosseini, S. A., Hayati, H., & Hassannejad, R. (2020). Forced axial vibration of micro and nanobeam under axial harmonic moving and constant distributed forces via nonlocal strain gradient theory. Mechanics Based Design of Structures and Machines, 1-15. https://doi.org/10.1080/15397734.2020.1744003
Hieu, D.-V., & Bui, G.-P. (2020). Nonlinear vibration of a functionally graded nanobeam based on the nonlocal strain gradient theory considering thickness effect. Advances in Civil Engineering, 2020. https://doi.org/10.1155/2020/9407673
Jazi, S. H. (2020). Nonlinear vibration of an elastically connected double Timoshenko nanobeam system carrying a moving particle based on modified couple stress theory. Archive of Applied Mechanics, 90(12), 2739-2754. https://doi.org/10.1007/s00419-020-01746-8
Kakei, A., Epaarachchi, J., Islam, M., & Leng, J. (2018). Evaluation of delamination crack tip in woven fibre glass reinforced polymer composite using FBG sensor spectra and thermo-elastic response. Measurement, 122, 178-185. https://doi.org/10.1016/j.measurement.2018.03.023
Kakei, A., Manuela, J., Srinivasan, V., Sharda, A., Islam, M., & Epaarachchi, J. (2019). Investigation of FBG sensor performance in detection of delamination damage in a half-conical shape composite component. In Proceedings of the 12th International Workshop on Structural Health Monitoring (Vol. 2, pp. 2973-2979). https://doi.org/10.12783/shm2019/32451
Khosravi, F., Hosseini, S. A., & Hamidi, B. A. (2020). Torsional Vibration of nanowire with equilateral triangle cross section based on nonlocal strain gradient for various boundary conditions: comparison with hollow elliptical cross section. The European Physical Journal Plus, 135(3), 1-20. https://doi.org/10.1140/epjp/s13360-020-00312-z
Khosravi, F., Hosseini, S. A., Hamidi, B. A., Dimitri, R., & Tornabene, F. (2020). Nonlocal torsional vibration of elliptical nanorods with different boundary conditions. Vibration, 3(3), 189-203. https://doi.org/10.3390/vibration3030015
Kuilla, T., Bhadra, S., Yao, D., Kim, N. H., Bose, S., & Lee, J. H. (2010). Recent advances in graphene based polymer composites. Progress in polymer science, 35(11), 1350-1375. https://doi.org/10.1016/j.progpolymsci.2010.07.005
Lee, C., Wei, X., Kysar, J. W., & Hone, J. (2008). Measurement of the elastic properties and intrinsic strength of monolayer graphene. science, 321(5887), 385-388. https://doi.org/10.1126/science.1157996
Mahmoud, M. A. (2016). Validity and accuracy of resonance shift prediction formulas for microcantilevers: a review and comparative study. Critical Reviews in Solid State and Materials Sciences, 41(5), 386-429. https://doi.org/10.1080/10408436.2016.1142858
Nageswara Rao, B. (1992). Large-amplitude free vibrations of simply supported uniform beams with immovable ends. Journal of Sound Vibration, 155(3), 523-527. https://ui.adsabs.harvard.edu/abs/1992JSV...155..523N/abstract
Nazemnezhad, R., & Hosseini-Hashemi, S. (2014). Nonlocal nonlinear free vibration of functionally graded nanobeams. Composite Structures, 110, 192-199. https://doi.org/10.1016/j.compstruct.2013.12.006
Nejad, M. Z., Hadi, A., & Rastgoo, A. (2016). Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory. International Journal of Engineering Science, 103, 1-10. https://doi.org/10.1016/j.ijengsci.2016.03.001
Noroozi, M., & Ghadiri, M. (2020). Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 0954406220959104. https://doi.org/10.1177/0954406220959104
Qasim, B. M., Khidir, T. C., Hameed, A. F., & Abduljabbar, A. A. (2018). Influence of heat treatment on the absorbed energy of carbon steel alloys using oil quenching and water quenching. Journal of Mechanical Engineering Research and Developments, 41(3), 43-46. https://doi.org/10.26480/jmerd.03.2018.43.46
Rao, S. S. (2007). Vibration of continuous systems (Vol. 464). Wiley Online Library. https://wp.kntu.ac.ir/hrahmanei/Adv-Vibrations-Books/Continuous-Vibrations-Rao.pdf
Shafiei, N., Kazemi, M., Safi, M., & Ghadiri, M. (2016). Nonlinear vibration of axially functionally graded non-uniform nanobeams. International Journal of Engineering Science, 106, 77-94. https://doi.org/10.1016/j.ijengsci.2016.05.009
Şimşek, M. (2014). Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Composites Part B: Engineering, 56, 621-628. https://doi.org/10.1016/j.compositesb.2013.08.082
Singh, G., Rao, G. V., & Iyengar, N. (1990). Re-investigation of large-amplitude free vibrations of beams using finite elements. Journal of Sound and Vibration, 143(2), 351-355. https://doi.org/10.1016/0022-460X(90)90958-3
Sourani, P., Hashemian, M., Pirmoradian, M., & Toghraie, D. (2020). A comparison of the Bolotin and incremental harmonic balance methods in the dynamic stability analysis of an Euler–Bernoulli nanobeam based on the nonlocal strain gradient theory and surface effects. Mechanics of Materials, 145, 103403. https://doi.org/10.1016/j.mechmat.2020.103403
Tauchert, T. R. (1974). Energy principles in structural mechanics. McGraw-Hill Companies. https://cir.nii.ac.jp/crid/1130282271700195968
Wang, G.-F., & Feng, X.-Q. (2007). Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Applied physics letters, 90(23), 231904. https://doi.org/10.1063/1.2746950
Wang, G.-F., & Feng, X.-Q. (2009). Surface effects on buckling of nanowires under uniaxial compression. Applied physics letters, 94(14), 141913. https://doi.org/10.1063/1.3117505
Wang, K., & Wang, B. (2011). Vibration of nanoscale plates with surface energy via nonlocal elasticity. Physica E: Low-dimensional Systems and Nanostructures, 44(2), 448-453. https://doi.org/10.1016/j.physe.2011.09.019
Zhao, D., Wang, J., & Xu, Z. (2021). Surface Effect on Vibration of Timoshenko Nanobeam Based on Generalized Differential Quadrature Method and Molecular Dynamics Simulation. Nanomanufacturing and Metrology, 4(4), 298-313. https://doi.org/10.1007/s41871-021-00117-3
Zhao, K., Pharr, M., Hartle, L., Vlassak, J. J., & Suo, Z. (2012). Fracture and debonding in lithium-ion batteries with electrodes of hollow core–shell nanostructures. Journal of Power Sources, 218, 6-14. https://doi.org/10.1016/j.jpowsour.2012.06.074
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Reports in Mechanical Engineering
This work is licensed under a Creative Commons Attribution 4.0 International License.