A mesh-free vibration analysis of strain gradient nano-beams

This paper focuses on developing a mesh-free method to analyze vibrational behavior of strain gradient nano-beams. For this purpose, the paper starts with the dynamic equation of a strain gradient Euler beam, and then the moving least-square (MLS) approximation is used to construct the shape function and its second- and third-order derivatives. A mesh-free numerical simulation scheme is develop, in which the higher-order gradient of strain is directly approximated with the nodal components due to the higher-order continuity of the shape function. The reliability of the mesh-free method is illustrated by an example of the simply-supported beam. Numerical simulations are carried out to study the small scale effect on both natural frequencies and vibration mode shapes of a single-walled carbon nanotube (SWCNT) which can be modeled as a nano-beam. The results of the mesh-free analysis are in good agreement with the theoretical results in analyzing the simply-supported SWCNT. The difference of natural frequency between that predicted by the strain gradient elastic beam and the classical beam rises with the increasing of the mode order and decreasing of the length.