Nanostructure‐dependent dispersion of carbon nanostructures: New insights into the modified couple stress theory

Abstract

This paper presents a size‐dependent analysis of carbon nanostructures to put forward some new insights into the conventional modified couple stress theory. In fact, a micro‐inertia length scale parameter is added to the equation of motion to compensate the deficiency of the theory in describing the size effect of the longitudinal dispersion. In this study, the longitudinal dispersions of carbon nanotubes, graphene, and graphite layers are investigated to obtain the micro‐inertia length scale parameter, and the flexural dispersions of carbon nanotubes and graphene layer are assessed to expose the size dependency of the couple‐stress length scale parameter. The results indicate that the micro‐inertia length scale parameter can directly be related to the lattice size, while the couple‐stress length scale parameter depends on some more factors like geometry, size, and material properties. Finally, some general relations are proposed for the first time to make a connection between these two distinct parameters and other features.

Torsional Characteristics of Carbon Nanotubes: Micropolar Elasticity Models and Molecular Dynamics Simulation

“…Khajueenejad and Ghanbari [ 48 ] proposed different values for internal length parameter in the range of 4 to 10 Å for armchair and zigzag CNTs. Akbarzadeh and Soltani [ 49 ] added another length scale parameter to the modified couple stress theory named micro-inertia length scale parameter. The original length parameter varied while they fixed the so-called micro inertia parameter to be in the order of the lattice size.…”

“…Although the majority of literature, that deals with continuum modelling of CNTs [ 36 , 37 , 38 , 39 , 40 , 41 ], employs explicitly non-local Eringen’s theory of elasticity, as it perfectly captures the long-range interaction between atoms by relating stress at a point to the strain of entire domain through an attenuation-type kernel function [ 42 ], few works are devoted to the application of implicitly non-local microcontinuum field theories. For instance, Xie and Long [ 43 ] adopted micropolar theory to calculate the fundamental frequencies of single-walled CNTs (SWCNTs) and double-walled CNTs (DWCNTs), while others applied couple stress continuum theory (i.e., a subcase of micropolar theory with an internal constraint; micro-rotations equal the macro-rotation [ 44 , 45 , 46 ]) [ 47 , 48 , 49 ]. Akgoz and Civalek [ 47 ] calculated the critical axial compressive buckling loads of CNTs using the Euler-Bernoulli beam model based on the modified couple stress theory, and presented the influence of length scale parameter on the buckling characteristics.…”

“…By correlating the derived buckling loads with the MD simulations data of Liew et al [ 50 ], they estimated the variable couple stress internal length parameter for zigzag and armchair nanotubes. Akbarzadeh and Soltani [ 49 ] added another length scale parameter to the modified couple stress theory to match the theoretical relation for longitudinal and flexural dispersion of CNTs to the data provided by Wang et al [ 51 ].…”