Higher-order elasticity theories have recently been used to predict the dispersion characteristics of flexural waves in carbon nanotubes (CNTs). In particular, nonlocal elasticity and gradient elasticity (with unstable strain gradients) have been employed within the framework of classical Euler-Bernoulli or improved Timoshenko beam theory to capture the dynamical behavior of CNTs. Qualitative agreement with the predictions of related molecular-dynamics (MD) simulations was observed, whereas the MD results departed significantly from those obtained with classical elasticity calculations. The present contribution aims to alert that the aforementioned higher-order models may yield questionable results for the higher wave numbers. As an alternative, gradient elasticity (with stable strain gradients), by also incorporating inertia gradients for dynamical applications, is used in combination with both Euler-Bernoulli and Timoshenko beam theories and shown to describe flexural wave dispersion in CNTs realistically for the small-to-medium range of wave numbers, i.e., the range for which MD results are available.

The term “gradient nanomechanics” is used here to designate a generalized continuum mechanics framework accounting for “bulk-surface” interactions in the form of extra gradient terms that enter in the balance laws or the evolution equations of the relevant constitutive variables that govern behavior at the nanoscale. In the case of nanopolycrystals, the grain boundaries may be viewed either as sources/sinks of “effective” mass and internal force or as a separate phase, interacting with the bulk phase that it surrounds, and supporting its own fields, balance laws, and constitutive equations reflecting this interaction. In either view, a further common assumption introduced is that the constitutive interaction between bulk and “interface” phases enters in the form of higher order gradient terms, independently of the details of the underlying physical mechanisms that bring these terms about. The effectiveness of the approach is shown by considering certain benchmark problems for nanoelasticity, nanoplasticity, and nanodiffusion for which standard continuum mechanics theory fails to model the observed behavior. Its implications to interpreting size-dependent stress-strain curves for nanopolycrystals with varying grain size are also discussed.