Dynamic Instability Analysis of Embedded Multi-walled Carbon Nanotubes under Combined Static and Periodic Axial Loads using Floquet–Lyapunov Theory

Subject Areas : Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering

Habib  Ramezannejad ¹ (Department of Mechanical Engineering, Ramsar branch, Islamic Azad University, Ramsar, Iran)
Hemad  Keshavarzpour ² (Department of Mechanical Engineering, Rasht branch, Islamic Azad University, Rasht, Iran)
Reza  Ansari ³ (Associate Professor, Department of Mechanical Engineering, University of Guilan)

Received: 2017-06-25 Accepted : 2017-11-16 Published : 2017-08-01

Keywords: Multi-walled Carbon Nanotubes, Dynamic instability, Mathieu-Hill model, Floquet&ndash, Lyapunov theory,

Abstract :

The dynamic instability of single-walled carbon nanotubes (SWCNT), double-walled carbon nanotubes (DWCNT) and triple-walled carbon nanotubes (TWCNT) embedded in an elastic medium under combined static and periodic axial loads are investigated using Floquet–Lyapunov theory. An elastic multiple-beam model is utilized where the nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. Moreover, a radius-dependent vdW interaction coefficient accounting for the contribution of the vdW interactions between adjacent and non-adjacent layers is considered. The Galerkin’s approximate method on the basis of trigonometric mode shape functions is used to reduce the coupled governing partial differential equations to a system of extended Mathieu-Hill equations. Applying Floquet–Lyapunov theory, the effects of elastic medium, length, number of layers and exciting frequencies on the instability conditions of CNTs are investigated. Results show that elastic medium, length of CNTs, number of layer and exciting frequency have significant effect on instability conditions of multi-walled CNTs.

References:

[1]        Q. Han, G. Lu, and L. Dai, “Bending instability of an embedded double-walled carbon nanotube based on Winkler and van der Waals models,” Compos. Sci. Technol., vol. 65, pp. 1337–1346, 2005.

[2]        J. Yoon, C. Q. Ru, and A. Mioduchowski, “Vibration and instability of carbon nanotubes conveying fluid,” Compos. Sci. Technol., vol. 65, , pp. 1326–1336, 2005.

[3]        J. Yoon, C. Q. Ru, and A. Mioduchowski, “Flow-induced flutter instability of cantilever carbon nanotubes,” Int. J. Solids Struct., vol. 43, pp. 3337–3349, 2006.

[4]        V. G. Hadjiev et al., “Buckling instabilities of octadecylamine functionalized carbon nanotubes embedded in epoxy,” Compos. Sci. Technol., vol. 66, pp. 128–136, 2006.

[5]        K. Y. Volokh and K. T. Ramesh, “An approach to multi-body interactions in a continuum-atomistic context: Application to analysis of tension instability in carbon nanotubes,” Int. J. Solids Struct., vol. 43, pp. 7609–7627, 2006.

[6]        A. Tylikowski, “Instability of thermally induced vibrations of carbon nanotubes,” Arch. Appl. Mech., vol. 78, pp. 49–60, Nov. 2007.

[7]        Q. Wang, K. M. Liew, and W. H. Duan, “Modeling of the mechanical instability of carbon nanotubes,” Carbon N. Y., vol. 46, pp. 285–290, 2008.

[8]        L. Wang and Q. Ni, “On vibration and instability of carbon nanotubes conveying fluid,” Comput. Mater. Sci., vol. 43, pp. 399–402, Aug. 2008.

[9]        L. Wang, Q. Ni, and M. Li, “Buckling instability of double-wall carbon nanotubes conveying fluid,” Comput. Mater. Sci., vol. 44, pp. 821–825, 2008.

[10]      Q. Wang, “Torsional instability of carbon nanotubes encapsulating C60 fullerenes,” Carbon N. Y., vol. 47, pp. 507–512, 2009.

[11]      Y. Fu, R. Bi, and P. Zhang, “Nonlinear dynamic instability of double-walled carbon nanotubes under periodic excitation,” Acta Mech. Solida Sin., vol. 22, pp. 206–212, 2009.

[12]      E. Ghavanloo, F. Daneshmand, and M. Rafiei, “Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation,” Phys. E Low-Dimensional Syst. Nanostructures, vol. 42, pp. 2218–2224, 2010.

[13]      E. Ghavanloo and S. A. Fazelzadeh, “Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid,” Phys. E Low-Dimensional Syst. Nanostructures, vol. 44, pp. 17–24, 2011.

[14]      T. Natsuki, T. Tsuchiya, Q. Q. Ni, and M. Endo, “Torsional elastic instability of double-walled carbon nanotubes,” Carbon N. Y., vol. 48, pp. 4362–4368, 2010.

[15]      W. H. Duan, Q. Wang, Q. Wang, and K. M. Liew, “Modeling the Instability of Carbon Nanotubes: From Continuum Mechanics to Molecular Dynamics,” J. Nanotechnol. Eng. Med., vol. 1, pp. 11001, 2010.

[16]      L.-L. Ke and Y.-S. Wang, “Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory,” Phys. E Low-dimensional Syst. Nanostructures, vol. 43, pp. 1031–1039, Mar. 2011.

[17]      T.-P. Chang and M.-F. Liu, “Flow-induced instability of double-walled carbon nanotubes based on nonlocal elasticity theory,” Phys. E Low-dimensional Syst. Nanostructures, vol. 43, pp. 1419–1426, Jun. 2011.

[18]      T.-P. Chang and M.-F. Liu, “Small scale effect on flow-induced instability of double-walled carbon nanotubes,” Eur. J. Mech. - A/Solids, vol. 30, pp. 992–998, Nov. 2011.

[19]      Y. X. Zhen, B. Fang, and Y. Tang, “Thermalmechanical vibration and instability analysis of fluid-conveying double walled carbon nanotubes embedded in visco-elastic medium,” Phys. E Low-Dimensional Syst. Nanostructures, vol. 44, pp. 379–385, 2011.

[20]      J.-X. Shi, T. Natsuki, X.-W. Lei, and Q.-Q. Ni, “Buckling Instability of Carbon Nanotube Atomic Force Microscope Probe Clamped in an Elastic Medium,” J. Nanotechnol. Eng. Med., vol. 3, p. 20903, 2012.

[21]      M. A. Kazemi-Lari, S. A. Fazelzadeh, and E. Ghavanloo, “Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation,” Phys. E Low-Dimensional Syst Nanostructures, vol. 44, pp. 1623–1630, 2012.

[22]      J. Choi, O. Song, and S.-K. Kim, “Nonlinear stability characteristics of carbon nanotubes conveying fluids,” Acta Mech., vol. 224, pp. 1383–1396, 2013.

[23]      A. Ghorbanpour Arani, M. R. Bagheri, R. Kolahchi, and Z. Khoddami Maraghi, “Nonlinear vibration and instability of fluid-conveying DWBNNT embedded in a visco-Pasternak medium using modified couple stress theory,” J. Mech. Sci. Technol., vol. 27, pp. 2645–2658, Sep. 2013.

[24]      M. M. Seyyed Fakhrabadi, A. Rastgoo, and M. Taghi Ahmadian, “Size-dependent instability of carbon nanotubes under electrostatic actuation using nonlocal elasticity,” Int. J. Mech. Sci., vol. 80, pp. 144–152, Mar. 2014.

[25]      Y.-Z. Wang and F.-M. Li, “Dynamical parametric instability of carbon nanotubes under axial harmonic excitation by nonlocal continuum theory,” J. Phys. Chem. Solids, vol. 95, pp. 19–23, Aug. 2016.

[26]      X. Wang, W. D. Yang, and S. Yang, “Dynamic stability of carbon nanotubes reinforced composites,” Appl. Math. Model., vol. 38, pp. 2934–2945, 2014.

[27] F. Agha-Davoudi , M. Hashemian, “Dynamic Stability of Single Walled Carbon Nanotube Based on Nonlocal Strain Gradient Theory” Journal of Solid Mechanics in Engineering, Volume 8, pp 1-11, 2015.

[28] S. Safari , M. Hashemian, “Dynamic Stability of Nano FGM Beam Using Timoshenko Theory” Journal of Solid Mechanics in Engineering, Volume 8, pp 239-250, 2015.

[29]      P. Friedman, C. E. Hammond, and T. H. Woo, “Efficient Numerical Treatment of Periodic Systems with Application to Stability Problems,” Int. J. Numer. Methods Eng., vol. 11,  pp. 1117–1136, 1977.