Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes

Flexural wave propagation in carbon nanotubes (CNTs) can be described through higher-order elasticity theories so as to capture the dispersive behavior induced by the inherent nanoscale heterogeneity. Motivated by experimental dispersion characteristics of metal nano-structured crystals, a new three-length-scale gradient formulation has been recently developed by the authors. In addition to the Laplacian of the strain, this model incorporates two higher-order inertia gradients for an improved dispersion behavior. A discrete medium with lumped masses at multiple scales of observation and combination of lumped mass and distributed mass at the macro-scale is introduced here to provide a micro-mechanical background to the proposed three-length-scale gradient model. The next aim of this paper is to assess the ability of this model to simulate flexural wave dispersion occurring in CNTs. We employ gradient-enriched Euler-Bernoulli and Timoshenko beam theories incorporating either stress gradients, or a combination of both strain gradients and inertia gradients - the latter leading to novel gradient-enriched beam theories. It is demonstrated that the proposed three-length-scale gradient elasticity formulation is able to capture the wave dispersion characteristics of armchair single-walled (5,5) and (10,10) CNTs arising from Molecular Dynamics simulations with high accuracy for a wide range of wave numbers. Advantages over alternative formulations of higher-order beam theories with stress gradients or combined strain-inertia gradient enrichments are discussed for comparative purposes.