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Internal length parameter and buckling analysis of carbon nanotubes using modified couple stress theory and Timoshenko beam model

The internal length parameter of the modified couple stress theory for single walled carbon nanotubes (CNTs) is determined in this paper. Buckling of CNTs have been studied using Timoshenko beam model and modified couple stress theory. The governing equations for three different end conditions, simple–simple, clamped–clamped and clamped-free, are solved using variational methods and an exact solution is provided for the buckling load. The effects of the internal length parameter on the buckling load of various CNT length and diameters are studied. It is observed that the internal length parameter has larger influence on the higher modes of buckling and for shorter nanotubes. A method presented to obtain the internal length parameter of higher order theories. By correlating the obtained results with the more accurate molecular dynamics simulations, the internal length parameter has been calculated for zigzag and armchair nanotubes. It is observed that the internal length parameter has slight dependency on the size of the CNTs and an average value is provided.