Some fundamental aspects of non-local beam mechanics for nanostructures applications

Local elasticity is inherently size-independent. In contrast, non-local continuum mechanics allows us to account for the small-length scale effect that can become significant when dealing with small-scale structures (typically nanostructures). This is made possible because non-local elasticity models abandon the classical assumption of locality, and admit that stress depends not only on the strain at that point, but also on the strains of every point in the body. In fact, the small-length scale phenomenon is linked to the atomistic structure of the lattice material. Therefore, in recent years, nonlocal continuum mechanics has attracted the attention of many researchers who worked on the modeling and analysis of micro/nanostructures. More specifically, there has been a considerable interest in the extension of local beam theory to non-local beams. This chapter aims at describing the main engineering beam theories such as the Euler-Bernoulli and the Timoshenko (or Bresse-Timoshenko) theories with allowance for the non-local effects. The non-local constitutive law is classified as an integral-based Eringen's non-local model and gradient elasticity model. The equations are then specialized for buckling and free vibration problems of beams under archetypal boundary conditions such as the pinned-pinned boundary conditions. Some theoretical solutions are presented to provide a lucid understanding on how the small length scale effect influences the local beam solutions.