Nonlocal Elasticity Theory

From the previous works, one can observe that the effects of size dependent and surface energy studied separately and there are a few papers that both of these effects are coincidentally studied. For this purpose, we can referee the readers to Ref [39]. In that paper, the authors applied the surface energy and the nonlocal elasticity theory to predict the vibration characteristic of a non-uniform beam. The vibration frequencies of the non-uniform nonobeam are computed by the Rayleigh–Ritz technique. In according to that paper, the size dependent and surface effects play important roles on the vibration frequencies of the non-uniform nanobeam. The effects of the surface energy and the size dependent were investigated on the vibration frequency

Also, in according to the previous paragraphs, it is important that both these phenomena are considered in the vibration behaviors of nanobeams. Consequently, the formulation should include both these phenomena to make a precise vibration analysis. Motivated by this idea, we aim to study the combine effects of the nonlocal and the surface effects on the vibration and buckling behavior of the rotating nanobeam in the magnetic field and under compressive axial load. Also, we have studied the effect of the surface energy to derive the vibration governing equation of the nanobeam. According to this theory, three factors are considered that these factors are the surface density, the surface stress and the surface elasticity. Then, the vibration frequencies and buckling loads of the nanobeam are obtained by DQM. Four different boundary conditions, Simply-Simply (SS), Clamp- Simply (CS), Clamp-Clamp (CC) and Clamp-Free (CF)) are investigated. The effects of (a) angular velocity, (b) surface elastic constants, (c) small scale, (d) different cross section geometry, (e) different boundary conditions, (f) compressive axial load (g) magnetic field and (h) hub radius on the vibration frequencies of the nanobeam are examined. This present study would be helpful to improve the optimum design of NEMS/MEMS devices using