Vibrational Responses of Micro/Nanoscale Beams: Size-Dependent Nonlocal Model Analysis and Comparisons

Abstract

A size-dependent dynamical model is suggested to investigate the vibrational characteristics of an Euler-Bernoulli micro/nanoscale beam. The strain gradient type of nonlocal elasticity is employed and a small intrinsic length scale parameter is considered in the theoretical model. A partial differential equation that governs transverse motion is derived and the corresponding ordinary differential equation and its dispersion relation are determined from the governing equation by the method of separation of variables. The problem is solved for several sets of strain gradient nonlocal boundary conditions by use of the eigenvalue method. These examples show that strain gradient nonlocality affects the natural frequencies of micro/nanoscale beams significantly. The initial axial force is also proved to play an important role in the vibrational behaviors of a micro/nanoscale beam. The critical compression and critical strain are derived and they are compared with some other approaches. The material constant in nonlocal elasticity theory can be determined by comparing the nonlocal theoretical results with molecular dynamic simulation and it is consistent with the estimation in previous work. Some further comments on the mechanisms of size dependence and physical meaning of micro/nanoscale parameter are presented. Comparisons of natural frequencies by nonlocal theory, classical theory, molecular dynamic simulation and surface effects are also constructed and they indicate the validity of the model developed in the present study.