**A novel method for studying the buckling of nanotubes considering
geometrical imperfections**

NMA Krishnan and D Ghosh, APPLIED PHYSICS A-MATERIALS SCIENCE & PROCESSING, 117, 945-953 (2014).

DOI: 10.1007/s00339-014-8489-x

Buckling of nanotubes has been studied using many methods such as molecular dynamics (MD), molecular mechanics, and continuum-based shell theories. In MD, motion of the individual atoms is tracked under applied temperature and pressure, ensuring a reliable estimate of the material response. The response thus simulated varies for individual nanotubes and is only as accurate as the force field used to model the atomic interactions. On the other hand, there exists a rich literature on the understanding of continuum mechanics-based shell theories. Based on the observations on the behavior of nanotubes, there have been a number of shell theory-based approaches to study the buckling of nanotubes. Although some of these methods yield a reasonable estimate of the buckling stress, investigation and comparison of buckled mode shapes obtained from continuum analysis and MD are sparse. Previous studies show that the direct application of shell theories to study nanotube buckling often leads to erroneous results. The present study reveals that a major source of this error can be attributed to the departure of the shape of the nanotube from a perfect cylindrical shell. Analogous to the shell buckling in the macro-scale, in this work, the nanotube is modeled as a thin-shell with initial imperfection. Then, a nonlinear buckling analysis is carried out using the Riks method. It is observed that this proposed approach yields significantly improved estimate of the buckling stress and mode shapes. It is also shown that the present method can account for the variation of buckling stress as a function of the temperature considered. Hence, this can prove to be a robust method for a continuum analysis of nanosystems taking in the effect of variation of temperature as well.

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