Finite element formulation for nano-scaled beam elements

Abstract

In the present study, size-dependent buckling and free vibration behaviors of single-walled boron nitride nanotube (SWBNNT) are performed in conjunction with various size-dependent elasticity theories. Modified couple stress theory (MCST) and Eringen's nonlocal elasticity theory are used for size-dependent models of SWBNNT. Also, the buckling loads and frequencies are obtained by using local theory to emphasize the effects and differences of these size-dependent theories. Consequently, three different elasticity theories (two non-classical and one classical) are utilized to achieve the detailed buckling and vibration analyses of SWBNNT. In this study, the buckling loads and frequencies of SWBNNTs are obtained via presented finite element formulation. In the finite element procedures based on two different size-dependent elasticity theories, matrices containing the small size parameter are derived. With these matrices containing the small size parameters, eigenvalue problems for buckling and free vibration analyses are formed. The buckling loads and frequency values of the SWBNNTs under the size effect are obtained. The influences of the dimensionless nonlocal parameter, dimensionless material length scale parameter, length-to-diameter ratio and boundary conditions on nanotube's buckling and vibration characteristics are investigated. In addition to these influences, the rotary inertia effect neglected in many other studies is also examined.