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Vibrational Responses of Micro/Nanoscale Beams: Size-Dependent Nonlocal Model Analysis and Comparisons

Published online by Cambridge University Press:  12 August 2014

C. Li ,
L. Chen  and
J.P. Shen
C. Li
Affiliation:
School of Urban Rail Transportation, Soochow University, Suzhou, China
L. Chen
Affiliation:
School of Urban Rail Transportation, Soochow University, Suzhou, China
J.P. Shen*
Affiliation:
School of Urban Rail Transportation, Soochow University, Suzhou, China
*
*Corresponding author (shenjp@suda.edu.cn)
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Abstract

A size-dependent dynamical model is suggested to investigate the vibrational characteristics of an Euler-Bernoulli micro/nanoscale beam. The strain gradient type of nonlocal elasticity is employed and a small intrinsic length scale parameter is considered in the theoretical model. A partial differential equation that governs transverse motion is derived and the corresponding ordinary differential equation and its dispersion relation are determined from the governing equation by the method of separation of variables. The problem is solved for several sets of strain gradient nonlocal boundary conditions by use of the eigenvalue method. These examples show that strain gradient nonlocality affects the natural frequencies of micro/nanoscale beams significantly. The initial axial force is also proved to play an important role in the vibrational behaviors of a micro/nanoscale beam. The critical compression and critical strain are derived and they are compared with some other approaches. The material constant in nonlocal elasticity theory can be determined by comparing the nonlocal theoretical results with molecular dynamic simulation and it is consistent with the estimation in previous work. Some further comments on the mechanisms of size dependence and physical meaning of micro/nanoscale parameter are presented. Comparisons of natural frequencies by nonlocal theory, classical theory, molecular dynamic simulation and surface effects are also constructed and they indicate the validity of the model developed in the present study.

Keywords

MEMS/NEMSStrain gradient type of nonlocal elasticityMicro/nanoscale beamLateral vibrationPrinciple modeNatural frequency

Information

Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 1 , February 2015 , pp. 7 - 19
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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References

1.Eringen, A. C. and Edelen, D. G. B., “On Nonlocal Elasticity,” International Journal of Engineering Science, 10, pp. 233–248 (1972).Google Scholar
2.Eringen, A. C., “On Differential Equations of Non-local Elasticity and Solutions of Screw Dislocation and Surface Waves,” Journal of Applied Physics, 54, pp. 4703–4710 (1983).Google Scholar
3.Aifantis, E. C., “Gradient Deformation Modes at Nano, Micro, and Macro Scales,” Journal of Engineering Materials and Technology, 121, pp. 189–202 (1999).Google Scholar
4.Peddieson, J., Buchanan, G. R. and Mcnitt, R. P., “Application of Nonlocal Continuum Models to Nanotechnology,” International Journal of Engineering Science, 41, pp. 305–312 (2003).Google Scholar
5.Sudak, L. J., “Column Buckling of Multiwalled Carbon Nanotubes Using Nonlocal Continuum Mechanics,” Journal of Applied Physics, 94, pp. 7281–7287 (2003).CrossRefGoogle Scholar
6.Wang, C. M., Zhang, Y. Y., Ramesh, S. S. and Kitipornchai, S., “Buckling Analysis of Micro- and Nano-Rods/Tubes Based on Nonlocal Timoshenko Beam Theory,” Journal of Physics D: Applied Physics, 39, pp. 3904–3909 (2006).CrossRefGoogle Scholar
7.Lu, P., Lee, H. P. and Lu, C., “Dynamic Properties of Flexural Beams Using a Nonlocal Elasticity Model,” Journal of Applied Physics, 99, p. 073510 (9 pages) (2006).Google Scholar
8.Yang, J., Jia, X. L. and Kitipornchai, S., “Pull-In Instability of Nano-Switches Using Nonlocal Elasticity Theory,” Journal of Physics D: Applied Physics, 41, p. 035103 (8 pages) (2008).Google Scholar
9.Lee, H. L. and Chang, W. J., “Surface Effects on Frequency Analysis of Nanotubes Using Nonlocal Timoshenko Beam Theory,” Journal of Applied Physics, 108, p. 093503 (3 pages) (2010).Google Scholar
10.Wang, C. Y., Murmu, T. and Adhikari, S., “Mechanisms of Nonlocal Effect on the Vibration of Nanoplates,” Applied Physics Letters, 98, p. 153101 (3 pages) (2011).CrossRefGoogle Scholar
11.Wang, C. M., Zhang, Y. Y. and He, X. Q., “Vibration of Nonlocal Timoshenko Beams,” Nanotechnology, 18, p. 105401 (9 pages) (2007).CrossRefGoogle Scholar
12.Wang, Q., “Wave Propagation in Carbon Nanotubes Via Nonlocal Continuum Mechanics,” Journal of Applied Physics, 98, p. 124301 (6 pages) (2005).Google Scholar
13.Wang, C. M. and Duan, W. H., “Free Vibration of Nanorings/Arches Based on Nonlocal Elasticity,” Journal of Applied Physics, 104, p. 014303 (8 pages) (2008).CrossRefGoogle Scholar
14.Yan, Y., Zhang, L. X., Wang, W. Q., He, X. Q. and Wang, C. M., “Dynamical Properties of Multi-Walled Carbon Nanotubes Based on a Nonlocal Elasticity Model,” International Journal of Modern Physics B, 22, pp. 4975–4986 (2008).Google Scholar
15.Heireche, H., Tounsi, A. and Benzair, A., “Scale Effect on Wave Propagation of Double-Walled Carbon Nanotubes with Initial Axial Loading,” Nanotechnology, 19, p. 185703 (11 pages) (2008).Google Scholar
16.Aydogdu, M., “Axial Vibration of the Nanorods with the Nonlocal Continuum Rod Model,” Physica E, 41, pp. 861–864 (2009).CrossRefGoogle Scholar
17.Li, C., Lim, C. W., Yu, J. L. and Zeng, Q. C., “Transverse Vibration of Pre-Tensioned Nonlocal Nanobeams with Precise Internal Axial Loads,” SCIENCE CHINA Technological Sciences, 54, pp. 2007–2013 (2011).Google Scholar
18.Li, C., Lim, C. W. and Yu, J. L., “Dynamics and Stability of Transverse Vibrations of Nonlocal Nanobeams with a Variable Axial Load,” Smart Materials and Structures, 20, p. 015023 (7 pages) (2011).CrossRefGoogle Scholar
19.Chang, T.-P., “Thermal-Nonlocal Vibration and Instability of Single-Walled Carbon Nanotubes Conveying Fluid,” Journal of Mechanics, 27, pp. 567–573 (2011).CrossRefGoogle Scholar
20.Fotouhi, M. M., Firouz-Abadi, R. D. and Haddadpour, H., “Free Vibration Analysis of Nanocones Embedded in an Elastic Medium Using a Nonlocal Continuum Shell Model,” International Journal of Engineering Science, 64, pp. 14–22 (2013).CrossRefGoogle Scholar
21.Xu, M., “Free Transverse Vibrations of Nano-To-Micron Scale Beams,” Proceedings of the Royal Society A, 462, pp. 2977–2995 (2006).Google Scholar
22.Lu, P., Lee, H. P., Lu, C. and Zhang, P. Q., “Application of Nonlocal Beam Models for Carbon Nanotubes,” International Journal of Solids and Structures, 44, pp. 5289–5300 (2007).CrossRefGoogle Scholar
23.Zhang, Y. Q., Liu, G. R. and Xie, X. Y., “Free Transverse Vibrations of Double-Walled Carbon Nanotubes Using a Theory of Nonlocal Elasticity,” Physical Review B, 71, p. 195404 (7 pages) (2005).Google Scholar
24.Wang, Q. and Varadan, V. K., “Vibration of Carbon Nanotubes Studied Using Nonlocal Continuum Mechanics,” Smart Materials and Structures, 15, pp. 659–666 (2006).Google Scholar
25.Mobley, R. K., Vibration Fundamentals, Newnes, Boston (1999).Google Scholar
26.Öz, H. R. and Pakdemirli, M., “Vibrations of an Axially Moving Beam with Time-Dependent Velocity,” Journal of Sound and Vibration, 227, pp. 239–257 (1999).CrossRefGoogle Scholar
27.Pakdemirli, M. and Öz, H. R., “Infinite Mode Analysis and Truncation to Resonant Modes of Axially Accelerated Beam Vibrations,” Journal of Sound and Vibration, 311, pp. 1052–1074 (2008).CrossRefGoogle Scholar
28.Park, S. H., Kim, J. S., Park, J. H., Lee, J. S., Choi, Y. K. and Kwon, O. M., “Molecular Dynamics Study on Size-Dependent Elastic Properties of Silicon Nanocantilevers,” Thin Solid Films, 492, pp. 285–289 (2005).Google Scholar
29.Cao, G. and Chen, X., “Buckling of Single-Walled Carbon Nanotubes Upon Bending: Molecular Dynamics Simulations and Finite Element Method,” Physical Review B, 73, p. 155435 (10 pages) (2006).Google Scholar
30.Wang, L. F., Hu, H. Y. and Guo, W. L., “Validation of the Non-Local Elastic Shell Model for Studying Longitudinal Waves in Single-Walled Carbon Nanotubes,” Nanotechnology, 17, pp. 1408–1415 (2006).CrossRefGoogle Scholar
31.Miller, R. E. and Shenoy, V. B., “Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotechnology, 11, pp. 139–147 (2000).Google Scholar
32.Picu, R. C., “On the Functional Form of Non-Local Elasticity Kernels,” Journal of the Mechanics and Physics of Solids, 50, pp. 1923–1939 (2002).CrossRefGoogle Scholar
33.Ru, C. Q., “Size Effect of Dissipative Surface Stress on Quality Factor of Microbeams,” Applied Physics Letters, 94, p. 051905 (3 pages) (2009).CrossRefGoogle Scholar
34.Nam, C. Y., Jaroenapibal, P., Tham, D., Luzzi, D. E., Evoy, S. and Fischer, J. E., “Diameter-Dependent Electromechanical Properties of Gan Nanowires,” Nano Letters, 6, pp. 153–158 (2006).Google Scholar