Shear Deformable Bending of Carbon Nanotubes Based on a New Analytical Nonlocal Timoshenko Beam Model

YIN Chun-song; YANG Yang

doi:10.3879/j.issn.1000-0887.2015.06.004

Volume 36 Issue 6

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2015 > 36(6): 600-606

YIN Chun-song, YANG Yang. Shear Deformable Bending of Carbon Nanotubes Based on a New Analytical Nonlocal Timoshenko Beam Model[J]. Applied Mathematics and Mechanics, 2015, 36(6): 600-606. doi: 10.3879/j.issn.1000-0887.2015.06.004

Citation:

PDF( 603 KB)

Shear Deformable Bending of Carbon Nanotubes Based on a New Analytical Nonlocal Timoshenko Beam Model

doi: 10.3879/j.issn.1000-0887.2015.06.004

Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming 650051, P.R.China

Funds: The National Natural Science Foundation of China(11261026;11462010)

Received Date: 2014-11-24
Rev Recd Date: 2015-03-31
Publish Date: 2015-06-15

Abstract

Abstract

According to Hamilton’s principle, a new mathematical model was established and the analytical solutions to the nonlocal Timoshenko beam model (ANT) were obtained based on the nonlocal elastic continuum theory in view of shear deformation and nonlocal effects. The new ANT equilibrium equations and boundary conditions were derived for bending analysis on carbon nanotubes (CNTs) of simply supported, clamped and cantilever types. The ANT deflection solutions demonstrate that the CNT stiffness is enhanced by the presence of nonlocal stress effects, as is predicted by the widely accepted but complicated molecular dynamics model and proved by tests. Furthermore, the new ANT model indicates verifiable bending behaviors of a cantilever CNT with point load at the free end, which depends on the magnitude of nonlocal stress. Therefore, this new model conveniently gives better prediction about the mechanical performances of nanostructures.
- Hamilton’s principle,
- nonlocal stress,
- carbon nanotube,
- Timoshenko nanobeam,
- bending behavior

FullText(HTML)

References(15)

References

[1]	Iijima S. Helical microtubules of graphitic carbon[J].Nature,1991,354(7): 56-58.
[2]	Tomanek D, Enbody R J.Science and Application of Nanotubes [M]. US: Springer, 2000: 1-274.
[3]	Ajayan P M. Applications of carbon nanotubes[J].Topics in Applied Physics,2001,80: 391-425.
[4]	Lam D C C, Yang F, Chong A C M, Wang J, Tong P. Experiments and theory in strain gradient elasticity[J].Journal of the Mechanics and Phsics of Solids,2003,51(8): 1477-1508.
[5]	Wang Q, Liew K M, He X Q, Xiang Y. Local buckling of carbon nanotubes under bending[J].Applied Physics Letter,2007,91(9): 093128.
[6]	Ru C Q. Effective bending stiffness of carbon nanotubes[J].Physical Review B,2000,62: 9973-9976.
[7]	Aifantis E C. Gradient deformation models at nano micro and macro scales[J].Journal of Engineering Material Technology,1999,121(2): 189-202.
[8]	Ma H M, Reddy J N. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory[J]. Journal of the Mechanics and Physics of Solids,2008,56(1): 3379-3391.
[9]	Peddieson J, Buchanan G R, McNitt R P. Application of nonlocal continuum models to nanotechnology[J].International Journal of Engeerning Science,2003,41(3): 305-312.
[10]	Reddy J N. Nonlocal theories for bending, buckling and vibration of beams[J].International Journal of Engeerning Science,2007,45(2/8): 288-307.
[11]	Wang Q, Liew K M. Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures[J].Physical Letter A,2007,363(3): 236-242.
[12]	Eringen A C.Nonlocal Continuum Field Theories [M]. US: Springer, 2002: 71-87.
[13]	C W 林. 基于非局部弹性应力场理论的纳米尺度效应研究: 纳米梁的平衡条件、控制方程以及静态挠度[J]. 应用数学和力学, 2010,31(1): 35-50.（Lim C W. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection[J].Applied Mathematics and Mechanics,2010,31(1): 35-50.（in Chinese））
[14]	YANG Yang, Lim C W. A variation principle approach for buckling of carbon nanotubes based on nonlocal Timoshenko beam models[J].Nanotechnology,2011,6(4): 363-377.
[15]	Lim C W. Equilibrium and static deflection for bending of a nonlocal nanobeam[J].Advances in Vibration Engineering,2009,8(4): 277-300.