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From:
“Sanders shell model for buckling of single-walled carbon nanotubes with small length-to-diameter ratios”, by N. Silvestre(1), C.M. Wang(2), Y.Y. Zhang(3) and Y. Ziang(3)
(1) Department of Civil Engineering & Architecture, ICIST-IST, Technical University of Lisbon, Portugal
(2) Engineering Science Programme, National University of Singapore, Kent Ridge, Singapore
(3) School of Engineering, University of Western Sydney, New South Wales, Australia, in
Composite Structures, Vol. 93, No. 7, June 2011, pp. 1683-1691
ABSTRACT: In this paper, the buckling behaviour of single-walled carbon nanotubes (CNTs) is revisited by resorting to Donnell and Sanders shell models, which are put in parallel and shown to lead to very distinct results for CNTs with large aspect ratio (length-to-diameter). This paper demonstrates inability of the widely used Donnell shell theory while it shows the validity and accuracy of the Sanders shell theory in reproducing buckling strains and mode shapes of axially compressed CNTs with large aspect ratios. The results obtained by the later shell theory are close to molecular dynamics simulation results.The Sanders shell theory could capture correctly the length-dependent buckling strains of CNTs which the Donnell shell theory fails to achieve. In view of this study, researchers should adopt the Sanders thin shell theory from hereon instead of the Donnell theory when analyzing CNTs with large aspect ratios.
The next slide shows the different predictions from the approximate Donnell's theory and from the more exact Sanders' theory. Donnell's theory is accurate only if a typical buckle spans a fairly shallow portion of the circumference of the cylindrical shell.
(Comment from D. Bushnell: It appears to me that the Donnell theory would not be appropriate for any of the examples displayed on this slide, since the number of circumferential waves, n, in the buckle pattern appears to be at most n = 2.)
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