基于非局部理论的黏弹性纳米杆轴向振动与波传播研究
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摘要
根据非局部理论和Kelvin黏弹性理论,针对黏弹性纳米杆自由振动和波传播的轴向动力学问题进行研究.首先,推导了黏弹性纳米杆的轴向动力学微分控制方程.然后,通过无量纲化讨论了3种典型边界纳米杆的前三阶振动特性.最后,研究黏弹性纳米杆波的传播问题,导出了圆频率、波速与波数之间的关系.数值结果表明,非局部效应使第一、二阶固有频率持续减小,第三阶频率先增大再减小,出现结构刚度削弱和增强两种趋势.特别地,对于自由端存在集中质量的情形,第二阶频率随着黏性系数增大出现了多值情况,易导致杆件失稳.数值算例还说明了非局部效应的增强可有效降低黏性材料的阻尼效应,产生逃逸频率,使得纵波能够在高波数段传播.另外,黏性系数在低波数段对阻尼比影响可忽略不计,而在高波数段下,黏性系数越大则阻尼比越大.
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 1 st 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 1 st and 2 nd frequencies decrease persistently and the 3 rd 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 2 nd 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. T he 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.
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 1 st 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 1 st and 2 nd frequencies decrease persistently and the 3 rd 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 2 nd 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. T he 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.
引文
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[20]LIU J J,LI C,YANG CJ,et al.Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory[J].Journal of V ibration 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,RAJENDRANP,FRISWELL MI,et al.Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory[J].Composite Structures,2018,186:274-292.
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[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 A pplication[M].Beijing:Science Press,2004.(in Chinese))
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[2]ERINGENA C,KIMB S.Stress concentration at the tip of the crack[J].Mechanics Research Communications,1974,1(4):233-237.
[3]TREACY MMJ,EBBESENT W,GIBSONJ M.Exceptionally high Young’s modulus observed for individual carbon nanotubes[J].Nature,1996,381:678-680.
[4]ERINGENA C.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[J].Journal of A pplied Physics,1983,54(9):4703-4710.
[5]ERINGENA C,EDELEND 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 coneshaped nanotubes[J].Journal of V ibration 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.(HU ANG Weiguo,LI Cheng,Z HU Z hongkui.O n the stability and axial vibration of compressive bars based on nonlocal elasticity theory[J].Journal of V ibration 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].A pplied Mathematical Modelling,2015,39(15):4570-4585.
[9]NARENDAR S,GOPALAKRISHNANS.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,CHENWQ.Transverse wave propagation in viscoelastic single-walled carbon nanotubes with small scale and surface effects[J].Journal of A pplied Physics,2015,117(2):024305.
[11]WANG L F,HU H Y.Flexural wave propagation in single-walled carbon nanotubes[J].Journal of Computational&T heoretical 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.(Z HANG Yu,DENG Z ichen,Z HAO Peng.Study of terahertz wave propagation in carbon nanotube arrays based on the symplectic formulation[J].A pplied 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.(YINChunsong,YANG Yang.Shear deformable bending of carbon nanotubes based on a newanalytical nonlocal T imoshenko beam model[J].A pplied 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 Z ichen.Surface effects of adsorption-induced resonance analysis of micro/nanobeams via nonlocal elasticity[J].A pplied 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]SHENJ P,LI C,FANX 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,FANX,et al.Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory[J].A pplied Mathematical Modelling,2017,45:65-84.
[20]LIU J J,LI C,YANG CJ,et al.Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory[J].Journal of V ibration 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,RAJENDRANP,FRISWELL MI,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 A pplication[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))