Longitudinal Vibration and Wave Propagation of Viscoelastic Nanorods Based on the Nonlocal Theory

TANG Guangze; YAO Linquan; LI Cheng; JI Changjian

doi:10.21656/1000-0887.390166

Volume 40 Issue 1

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TANG Guangze, YAO Linquan, LI Cheng, JI Changjian. Longitudinal Vibration and Wave Propagation of Viscoelastic Nanorods Based on the Nonlocal Theory[J]. Applied Mathematics and Mechanics, 2019, 40(1): 36-46. doi: 10.21656/1000-0887.390166

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Longitudinal Vibration and Wave Propagation of Viscoelastic Nanorods Based on the Nonlocal Theory

doi: 10.21656/1000-0887.390166

School of Rail Transportation, Soochow University, Suzhou, Jiangsu 215131, P.R.China

Funds: The National Natural Science Foundation of China（11572210）

Received Date: 2018-06-14
Rev Recd Date: 2018-07-21
Publish Date: 2019-01-01

Abstract

Abstract

The longitudinal dynamics of viscoelastic nanorods was investigated based on the nonlocal theory and the Kelvin viscoelastic theory, including axial free vibration and wave propagation. Firstly, the partial differential governing equations were derived and then the 1st 3 vibration properties were discussed under 3 kinds of typical boundary conditions with the dimensionless method. Finally, the relationships between the circular frequency, the wave speed and the wave number were obtained in the problem of wave propagation. The numerical results show that, the small-scale effect makes the 1st and 2nd frequencies decrease persistently and the 3rd frequency increase first and decrease later, which indicates that the nanostructural stiffness is weakened or strengthened. In particular, for a concentrated mass at the free end of the nanorod, the 2nd frequency has multiple values when the viscoelastic coefficient increases, which may cause instability. The numerical examples also prove that stronger nonlocal effect brings lower damping effect of viscoelastic materials. The longitudinal wave can propagate at high wave numbers due to occurrence of the escape frequency. The effects of viscoelastic coefficients on the damping ratio may be ignored at low wave numbers, however, be significant at high wave numbers.
- axial vibration,
- wave propagation,
- small-scale effect,
- nanorod,
- viscoelasticity

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References(27)

References

[1]	IIJIMA S. Helical microtubules of graphitic carbon[J]. Nature,1991,354: 56-58.
[2]	ERINGEN A C, KIM B S. Stress concentration at the tip of the crack[J]. Mechanics Research Communications,1974,1(4): 233-237.
[3]	TREACY M M J, EBBESEN T W, GIBSON J M. Exceptionally high Young’s modulus observed for individual carbon nanotubes[J]. Nature,1996,381: 678-680.
[4]	ERINGEN A C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[J]. Journal of Applied Physics,1983,54(9): 4703-4710.
[5]	ERINGEN A C, EDELEN D G B. On nonlocal elasticity[J]. International Journal of Engineering Science,1972,10(3): 233-248.
[6]	杨武, 彭旭龙, 李显方. 锥形纳米管纵向振动固有频率[J]. 振动与冲击, 2014,33(2): 158-162.(YANG Wu, PENG Xulong, LI Xianfang. Natural frequencies of longitudinal vibration of cone-shaped nanotubes[J]. Journal of Vibration and Shock,2014,33(2): 158-162.(in Chinese))
[7]	黄伟国, 李成, 朱忠奎. 基于非局部理论的压杆稳定性及轴向振动研究[J]. 振动与冲击, 2013,32(5): 154-156.(HUANG Weiguo, LI Cheng, ZHU Zhongkui. On the stability and axial vibration of compressive bars based on nonlocal elasticity theory[J]. Journal of Vibration and Shock, 2013,32(5): 154-156.(in Chinese))
[8]	LI C, LI S, YAO L Q. Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models[J]. Applied Mathematical Modelling,2015,39(15): 4570-4585.
[9]	NARENDAR S, GOPALAKRISHNAN S. Nonlocal scale effects on ultrasonic wave characteristics of nanorods[J]. Physica E: Low-Dimensional Systems and Nanostructures,2010,42(5): 1601-1604.
[10]	PANG M, ZHANG Y Q, CHEN W Q. Transverse wave propagation in viscoelastic single-walled carbon nanotubes with small scale and surface effects[J]. Journal of Applied Physics, 2015,117(2): 024305.
[11]	WANG L F, HU H Y. Flexural wave propagation in single-walled carbon nanotubes[J]. Journal of Computational & Theoretical Nanoscience,2008,5(4): 581-586.
[12]	王碧蓉, 邓子辰, 徐晓建. 基于梯度理论的碳纳米管弯曲波传播规律的研究[J]. 西北工业大学学报, 2013,31(5): 774-778.(WANG Birong, DENG Zichen, XU Xiaojian. Modified Timoshenko beam models for flexural wave dispersion in carbon nanotubes with shear deformation considered[J]. Journal of Northwestern Polytechnical University,2013,31(5): 774-778.(in Chinese))
[13]	张宇, 邓子辰, 赵鹏. 辛体系下碳纳米管阵列中太赫兹波传播特性研究[J]. 应用数学和力学, 2016,37(9): 889-900.(ZHANG Yu, DENG Zichen, ZHAO Peng. Study of terahertz wave propagation in carbon nanotube arrays based on the symplectic formulation[J]. Applied Mathematics and Mechanics,2016,37(9): 889-900.(in Chinese))
[14]	尹春松, 杨洋. 考虑非局部剪切效应的碳纳米管弯曲特性研究[J]. 应用数学和力学, 2015,36(6): 600-606.(YIN Chunsong, 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.(in Chinese))
[15]	徐晓建, 邓子辰. 非局部因子和表面效应对微纳米材料振动特性的影响[J]. 应用数学和力学, 2013,34(1): 10-17.(XU Xiaojian, DENG Zichen. Surface effects of adsorption-induced resonance analysis of micro/nanobeams via nonlocal elasticity[J]. Applied Mathematics and Mechanics,2013,34(1): 10-17.(in Chinese))
[16]	LI C, LIU J J, CHENG M, et al. Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces[J]. Composites Part B: Engineering,2017,116: 153-169.
[17]	LI C. Nonlocal thermo-electro-mechanical coupling vibrations of axially moving piezoelectric nanobeams[J]. Mechanics Based Design of Structures & Machines,2017,45(4): 463-478.
[18]	SHEN J P, LI C, FAN X L, et al. Dynamics of silicon nanobeams with axial motion subjected to transverse and longitudinal loads considering nonlocal and surface effects[J]. Smart Structures and Systems,2017,19(1): 105-113.
[19]	LIU J J, LI C, FAN X, et al. Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory[J]. Applied Mathematical Modelling,2017,45: 65-84.
[20]	LIU J J, LI C, YANG C J, et al. Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory[J]. Journal of Vibration and Control,2017,23(20): 3327-3344.
[21]	KARAMI B, SHAHSAVARI D, LI L. Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory[J]. Physica E: Low-Dimensional Systems and Nanostructures,2018,97: 317-327.
[22]	EL-BORGI S, RAJENDRAN P, FRISWELL M I, et al. Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory[J]. Composite Structures,2018,186: 274-292.
[23]	XU M, FUTABA D N, YAMADA T, et al. Carbon nanotubes with temperature-invariant viscoelasticity from -196 ℃ to 1 000 ℃[J].Science, 2010,330(6009):1364-1368.
[24]	XU M, FUTABA D N, YUMURA M, et al. Tailoring temperature invariant viscoelasticity of carbon nanotube material[J]. Nano Letters,2011,11(8): 3279-3284.
[25]	杨挺青. 黏弹性理论与应用[M]. 北京: 科学出版社, 2004.(YANG Tingqing. Viscoelastic Theory and Application [M]. Beijing: Science Press, 2004.(in Chinese))
[26]	谢官模. 振动力学[M]. 北京: 国防工业出版社, 2007.(XIE Guanmo. Vibration Mechanics [M]. Beijing: National Defense Industry Press, 2007.(in Chinese))
[27]	