Nano-scale effects on nonlocal boundary conditions for exact buckling analysis of nano-beams with different end conditions

Abstract

This paper presents a mathematical approach for applying the nano-scale effects on higher-order nonlocal boundary conditions for exact buckling strains of nano-beams based on a modified nonlocal Timoshenko beam theory (MNTBT) for various end conditions. Researchers usually neglect these higher-order boundary conditions in their analysis. Hence, the strain gradient approach and variational method are implemented in MNTBT for deriving these higher-order boundary conditions and exact closed-form critical buckling strains for different end conditions. On this basis, exact numerical results are presented for buckling analysis of single-walled carbon nanotubes (SWCNTs). It is investigated that nonlocal boundary conditions have the effect of reducing the critical buckling strains. This effect is the most significant for doubly clamped nano-beams and the least significant for cantilever nano-beams. Furthermore, it is shown that presented model based on higher-order nonlocal boundary conditions can capture correctly the length-dependent buckling strains of SWCNTs as compared with the other nonlocal beam theories. Finally, the results are compared with molecular dynamics simulations and the Eringen’s nonlocal coefficient (e0) is calibrated for buckling problems of SWCNTs with good accuracy as compared with literatures.