Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions

2015 ◽  
Vol 122 ◽  
pp. 390-404 ◽  
Author(s):  
Chih-Ping Wu ◽  
Wei-Wen Lai
Keyword(s):  
Carbon Nanotube ◽  
Boundary Conditions ◽  
Elastic Medium ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Single Walled Carbon Nanotube ◽  
Various Boundary Conditions ◽  
Nonlocal Timoshenko Beam Theory ◽  
Reissner’S Mixed Variational Theorem

RMVT-Based Nonlocal Timoshenko Beam Theory for Stability Analysis of Embedded Single-Walled Carbon Nanotube with Various Boundary Conditions

2016 ◽  
Vol 16 (10) ◽  
pp. 1550068 ◽  
Author(s):  
Chih-Ping Wu ◽  
Jyun-Yu Liou
Keyword(s):  
Carbon Nanotube ◽  
Stability Analysis ◽  
Boundary Conditions ◽  
Critical Load ◽  
Elastic Medium ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Single Walled Carbon Nanotube ◽  
Various Boundary Conditions ◽  
Nonlocal Timoshenko Beam Theory

On the basis of Reissner’s mixed variational theorem (RMVT), a nonlocal Timoshenko beam theory (TBT) is developed for the stability analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium, with various boundary conditions and under axial loads. Eringen’s nonlocal elasticity theory is used to account for the small length scale effect. The strong formulations of the RMVT-based nonlocal TBT and its associated possible boundary conditions are presented. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Pasternak foundation models. The critical load parameters of the embedded SWCNT with different boundary conditions are obtained by using the differential quadrature (DQ) method, in which the locations of [Formula: see text] sampling nodes are selected as the roots of [Formula: see text]-order Chebyshev polynomials. The results of the RMVT-based nonlocal TBT are compared with those obtained using the principle of virtual displacement (PVD)-based nonlocal TBT available in the literature. The influences of some crucial effects on the critical load parameters of the embedded SWCNT are examined, such as different boundary conditions, Winkler stiffness and shear modulus of the foundation, aspect ratios, and the nonlocal parameter.

An efficient GDQ model for vibration analysis of a multiwall carbon nanotube on Pasternak foundation with general boundary conditions

2011 ◽  
Vol 225 (7) ◽  
pp. 1730-1741 ◽  
Author(s):  
P Soltani ◽  
P Bahar ◽  
A Farshidianfar
Keyword(s):  
Carbon Nanotube ◽  
Boundary Conditions ◽  
Elastic Medium ◽  
Differential Quadrature Method ◽  
Elastic Beam ◽  
Beam Theory ◽  
Multiwall Carbon Nanotube ◽  
Various Boundary Conditions ◽  
Aspect Ratios ◽  
Multiwall Carbon

In this article, the free transverse vibrational behaviour of a multiwall carbon nanotube (MWNT) surrounded by a Pasternak-type elastic medium has been determined using a very generalized model. The model has been made on the basis of Timoshenko elastic beam theory which allows the effects of shear deformation and rotary inertia and supports non-coaxial vibration of the adjacent layers of MWNT using interlayer van der Waals forces. The boundary conditions used in this simulation are such that not only standard and conventional kinds, but also all possible forms, of end conditions are applicable. A generalized differential quadrature method is utilized to solve the governing equations with assorted aspect ratios, various boundary conditions, and different foundation stiffnesses. This study shows that the resonant frequencies of MWNTs are strongly dependent on the stiffness of the elastic medium, aspect ratios, and number of walls in carbon nanotubes and, for short nanotubes, the boundary stiffness plays a significant role on the natural frequencies.

scholarly journals Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Ismail Kucuk ◽  
Ibrahim S. Sadek ◽  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Multiwalled Carbon Nanotubes ◽  
Timoshenko Beam ◽  
Variational Principles ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Small Scale ◽  
Multiwalled Carbon ◽  
Nonlocal Timoshenko Beam Theory

Variational principles are derived for multiwalled carbon nanotubes undergoing linear vibrations using the semi-inverse method with the governing equations based on nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. Physical models based on the nonlocal theory approximate the nanoscale phenomenon more accurately than the local theories by taking small scale phenomenon into account. Variational formulation is used to derive the natural and geometric boundary conditions which give a set of coupled boundary conditions in the case of free boundaries which become uncoupled in the case of the local theory. Hamilton's principle applicable to this case is also given.

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