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Research Papers

# Nonlocal Shear Deformable Shell Model for Post-Buckling of Axially Compressed Double-Walled Carbon Nanotubes Embedded in an Elastic Matrix

[+] Author and Article Information
Hui-Shen Shen1

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China; State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of Chinahsshen@mail.sjtu.edu.cn

Chen-Li Zhang

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

1

Corresponding author.

J. Appl. Mech 77(4), 041006 (Apr 09, 2010) (12 pages) doi:10.1115/1.4000910 History: Received December 13, 2008; Revised October 08, 2009; Published April 09, 2010; Online April 09, 2010

## Abstract

Buckling and post-buckling analysis is presented for axially compressed double-walled carbon nanotubes (CNTs) embedded in an elastic matrix in thermal environments. The double-walled carbon nanotube is modeled as a nonlocal shear deformable cylindrical shell, which contains small scale effects and van der Waals interaction forces. The surrounding elastic medium is modeled as a tensionless Pasternak foundation. The post-buckling analysis is based on a higher order shear deformation shell theory with the von Kármán–Donnell-type of kinematic nonlinearity. The thermal effects are also included and the material properties are assumed to be temperature-dependent and are obtained from molecular dynamics (MD) simulations. The nonlinear prebuckling deformations of the shell and the initial local point defect, which is simulated as a dimple on the tube wall, are both taken into account. A singular perturbation technique is employed to determine the post-buckling response of the tubes and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the tube and the elastic medium. The small scale parameter $e0a$ is estimated by matching the buckling loads of CNTs observed from the MD simulation results with the numerical results obtained from the nonlocal shear deformable shell model. Numerical solutions are presented to show the post-buckling behavior of CNTs surrounded by an elastic medium of conventional and tensionless Pasternak foundations. The results show that buckling and post-buckling behavior of CNTs is very sensitive to the small scale parameter $e0a$. The results reveal that the unilateral constraint has a significant effect on the post-buckling response of CNTs when the foundation stiffness is sufficiently large.

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## Figures

Figure 1

An elastic shell model for a double-walled carbon nanotube embedded in an elastic matrix: (a) geometry and loading case and (b) coordinate system and point defect

Figure 2

Comparisons of post-buckling behavior for a (9,9)-tube under axial compression in thermal environments: (a) T=300 K and (b) T=700 K

Figure 4

Comparisons of post-buckling behavior for SWCNT and DWCNTs resting on tensionless elastic foundations: (a) load-shortening and (b) load-deflection

Figure 5

Effect of the foundation stiffness on the post-buckling behavior of a [(9,9), (14,14)]-tube resting on tensionless elastic foundations: (a) load-shortening and (b) load-deflection

Figure 6

Comparisons of post-buckling behavior for a [(9,9), (14,14)]-tube resting on different kinds of elastic foundations: (a) load-shortening and (b) load-deflection

Figure 7

The effect of temperature rise on the post-buckling behavior of a [(9,9), (14,14)]-tube resting on tensionless elastic foundations: (a) load-shortening and (b) load-deflection

Figure 3

Comparisons of post-buckling behavior of (9,9)-tube with different values of small scale parameter e0a subjected to axial compression: (a) load-shortening and (b) load-deflection

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