Assessment of first and third order shear deformation beam theories for the buckling and vibration analysis of nanobeams incorporating surface stress effects

Abstract

A novel approach for the vibration and buckling analysis of nanobeams with surface stress effects is presented in this paper. The material model is derived in which the bulk core of a beam is assumed to be a single crystalline material and have certain crystallographic directions. The beams are assumed to be covered by atomic layers with the same crystallographic directions on the top and bottom surfaces. A generic third order shear deformable beam theory is employed to derive the total energy functional of the beams. Minimizing the total potential energy functional by applying the p-Ritz method, the eigenvalue equations for the buckling and vibration of nanobeams with surface stress effects are obtained. The buckling and vibration behaviours of the beams considering the surface stress effects are discussed in detail and are compared with the results obtained from the first order beam theory and the Reddy's third order beam theory. It is found that in certain cases, both the first order and the Reddy's third order beam theories may not be able to capture the surface effect accurately on the vibration and buckling behaviours of the nanobeams. The effects of surface stresses on the buckling and vibration behaviours for nanobeams made of Ni single crystals obtained by the generic third order beam theory showed opposite trends to the ones obtained by the first order or the Reddy's third beam theory.