Buckling behavior of nonuniform carbon nanotubes using nonlocal elasticity theory and the differential transformation method


The buckling behavior of a nonuniform single-walled carbon nanotube (SWCNT), subjected to axially compressive load, is studied using the nonlocal elasticity theory. The differential transformation method (DTM) has been used to obtain the nonlocal buckling loads of the nonuniform SWCNT under various boundary conditions, namely simply supported, fixed–fixed, and fixed-simply supported. The nanotube’s nonlocal buckling load increases significantly with an increase in the tip’s diameter; however, it decreases substantially with increasing the small-scale parameter for both uniform and nonuniform SWCNTs. The results obtained from the DTM agree well with those reported in the literature for uniform SWCNTs. The accuracy of the results revealed that DTM is useful and convenient for investigating the buckling behavior of nonuniform CNTs with small-scale effects for various boundary conditions compared to other analytical methods. This work would provide helpful insights into the design of nonuniform CNT-based devices.

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\(x\), \(y\) :

Cartesian coordinates

\(L\) :

Length of carbon nanotube

\(d\) :

Diameter of carbon nanotube tip

\(d_{0}\) :

Diameter of carbon nanotube at \(x = 0\)

\(E\) :

Young's modulus

\(u\left( x \right)\) :

Flexural deflection at \(x\)

\(w\left( \xi \right)\) :

Dimensionless flexural deflection at \(\xi\)

\(W\left(k\right)\) :

Differential transformed deflection

\(\left( {\begin{array}{*{20}c} m \\ r \\ \end{array} } \right)\) :

Combination \(\left( {m!/\left( {m - r} \right)!r!} \right)\)

\(r_{a}\), \(r_{b}\) :

Outer and inner radii of carbon nanotube

\(I_{0}\) :

Moment of inertia at \(x = 0\)

\(I\left( x \right)\) :

Moment of inertia at \(x\)

\(M\left( x \right)\) :

Bending moment at \(x\)

\(V\left( x \right)\) :

Shear force at \(x\)

\(e_{0}\) :

Adjustable parameter

\(a\) :

Internal characteristic length

\(A\left( x \right)\) :

Cross-sectional area at \(x\)

\(F\) :

Buckling load

\(p\) :

Dimensionless load parameter

\(\xi\) :

Dimensionless variable (\(x/L)\)

\(\varepsilon \left( x \right)\) :

Strain at \(x\)

\(\beta\) :

Taper ratio

\(\sigma \left( x \right)\) :

Stress at \(x\)

\(\mu\) :

Small-scale parameter


Carbon nanotube


Single-walled carbon nanotube


Multi-walled carbon nanotube


Differential transformation method


Simply supported




Fixed-simply supported


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D. P. Rai acknowledges Core Research Grant from Department of Science and Technology SERB (CRG DST-SERB, New Delhi India) via Sanction no. CRG/2018/000009(Ver-1).

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Correspondence to P. K. Patra.

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Mawphlang, B.R.K.L.L., Ghimire, M.P., Rai, D.P. et al. Buckling behavior of nonuniform carbon nanotubes using nonlocal elasticity theory and the differential transformation method. Int Nano Lett (2020). https://doi.org/10.1007/s40089-020-00319-5

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  • Buckling behavior
  • Nonlocal elasticity theory
  • Nonuniform carbon nanotubes
  • Differential transformation method