Nonlocal Finite Element Analysis of CNTs with Timoshenko Beam Theory and Thermal Environment

Pradhan, S.C.; Mandal, Utpal

doi:10.1007/s40032-012-0041-1

Cited by 5 publications

(7 citation statements)

References 46 publications

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“…35 In the present discussion, buckling and vibration results are investigated considering the material and geometric properties of carbon nanotubes as mentioned by Reddy and Pang. 36 The Young's modulus E , mass density and Poisson's ratio are assumed to be 1.0 TPa, 2300 Kg/m 3 .…”

Section: Resultsmentioning

confidence: 99%

Radial Scale Effect of Nonlocal Parameter in Carbon Nanotubes

Mandal¹,

Pradhan²

2014

Jnl of Comp & Theo Nano

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In this article, small scale effect of carbon nanotubes in radial direction is studied. Finite element analysis including nonlocal effect is carried out for the buckling and vibration of carbon nanotubes. It is found that there is significant influence of nonlocal parameter on the radial structural response of carbon nanotubes. The influence of nonlocal parameter in the radial direction is defined as radial scale effect. Radial scale effect is also observed for the carbon nanotubes with four different sets of boundary conditions. Thus, in the analysis of carbon nanotubes, radial scale effect needs to be incorporated to accurately predict the structural response of carbon nanotubes. Further this radial scale effect significantly increases as the carbon nanotube diameter decreases below 50 nm.

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Section: Resultsmentioning

confidence: 99%

Radial Scale Effect of Nonlocal Parameter in Carbon Nanotubes

Mandal¹,

Pradhan²

2014

Jnl of Comp & Theo Nano

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“…The following relations are given for the transverse motion of bending finite elements without transverse shear displacement

w_{b} goodbreak= ϕ boldd bold, \frac{\partial w_{b}}{\partial x} goodbreak= D^{k} w_{b} goodbreak= boldBd

where d is the transverse displacement vector,

D^{k}

is the kinematic operator, and

ϕ

is the shape function of finite element. For the finite element analyses of the Timoshenko beams, shape functions including the shear effect are used in the scientific literature 84,85,102 . However, since the equation of motion is derived only in terms of transverse bending displacement, the shape functions of the EBBT 95 can be employed for the Timoshenko beam model examined:

({}, ϕ) goodbreak= {\{\begin{matrix} ϕ_{1} \\ ϕ_{2} \\ ϕ_{3} \\ ϕ_{4} \end{matrix}\}}^{T} goodbreak= {\{\begin{matrix} 1 - 3 ξ^{2} + 2 ξ^{3} \\ l_{e} (- ξ - 2 ξ^{2} + ξ^{3}) \\ 3 ξ^{2} - 2 ξ^{3} \\ l_{e} (- ξ^{2} + ξ^{3}) \end{matrix}\}}^{T}, boldd goodbreak= {\{w_{1} w_{2} w_{3} w_{4}\}}^{T}

where

ξ = x / l_{e}

is the nondimensional longitudinal coordinate.…”

Section: Size–dependent Finite Element Solutionmentioning

confidence: 99%

“…For the finite element analyses of the Timoshenko beams, shape functions including the shear effect are used in the scientific literature. 84,85,102 However, since the equation of motion is derived only in terms of transverse bending displacement, the shape functions of the EBBT 95 can be employed for the Timoshenko beam model examined:…”

Section: Size-dependent Finite Element Solutionmentioning

confidence: 99%

“…In addition, natural frequency behavior of thick nanoplates was examined by a nonlocal finite element method (NL‐FEM) 83 . The nonlocal bending, buckling, and vibration analyses based on TBT of carbon nanotubes are given 84,85 . Nonlocal statics and dynamics problems of Euler–Bernoulli beams are formulated with nonlocal finite element 86,87 .…”

Section: Introductionmentioning

confidence: 99%

“…83 The nonlocal bending, buckling, and vibration analyses based on TBT of carbon nanotubes are given. 84,85 Nonlocal statics and dynamics problems of Euler-Bernoulli beams are formulated with nonlocal finite element. 86,87 Free dynamics of functionally graded Timoshenko nanobeams is examined taking into consideration neutral axis position by Eltaher et al 88 Demir and Civalek 89 studied transversely thermal vibration of silica carbide nanotubes resting on elastic foundation.…”

Section: Introductionmentioning

confidence: 99%

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A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method

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Akgöz

et al. 2021

Math Methods in App Sciences

125

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In this study, size‐dependent thermo‐mechanical vibration analysis of nanobeams is examined. Size‐dependent dynamic equations are obtained by implementing Hamilton's principle based on Timoshenko beam theory and then combined with stress equation of nonlocal elasticity theory. The separation of variables total method and finite element formulation is utilized to solve the eigenvalue problem. Local and nonlocal stiffness and mass matrices are firstly derived by using a weighted residual method for the finite element analysis. The accuracy of the finite element solution is demonstrated by comparisons with the earlier studies. Then, nondimensional frequencies of nanobeams with different boundary conditions based on a nonlocal finite element method are presented for vibration analysis that cannot be analytically solved under different parameters. It is aimed to emphasize the importance of the nonlocal finite element method in the size‐dependent vibration behavior of nanobeams which form different components of nano‐electro‐mechanical systems.

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State-of-the-Art of Vibration Analysis of Small-Sized Structures by using Nonclassical Continuum Theories of Elasticity

Nuhu

Safaei

2022

Arch Computat Methods Eng

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Nonlocal Finite Element Analysis of CNTs with Timoshenko Beam Theory and Thermal Environment

Cited by 5 publications

References 46 publications

Radial Scale Effect of Nonlocal Parameter in Carbon Nanotubes

Radial Scale Effect of Nonlocal Parameter in Carbon Nanotubes

A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method

State-of-the-Art of Vibration Analysis of Small-Sized Structures by using Nonclassical Continuum Theories of Elasticity

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