Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorodsanotubes and verification of different nonlocal models

摘要

Recently, there have been an increasing number of studies on nonlocal theoretical models, of which different kinds of nonlocal elasticity approaches including softening and hardening models have been investigated extensively. In the present work, the longitudinal dynamic behaviors of some common one-dimensional nanostructures (e.g. nanorods/ nanotubes) were examined using the hardening nonlocal approach. The hardening nonlocal longitudinal stress was employed and an eigenvalue analysis method was utilized in deriving some dynamic characteristics of the minute structures at nanoscale. The effects of a dimensionless nonlocal small scale parameter at molecular level unavailable in classical rods/tubes were investigated. Subsequently, nonlocal longitudinal vibration responses under various boundary conditions including soft and hard constraints were determined through a numerical method. The correlations between natural frequencies and the nonlocal nanoscale parameter were obtained and discussed in detail. Within the framework of hardening nonlocal stress theory, it concludes for the first time that the longitudinal free vibration frequencies of nanorodsanotubes are higher than those based on the classical continuum mechanics but they are quite different from the softening nonlocal model. The strengthening effects on nonlocal stiffness of nanorodsanotubes are observed and they are consistent with some other theoretical approaches or atomistic simulations. In particular, a case study on nonlocal natural frequencies of carbon nanotubes (CNTs) was presented and the results were calculated by atomistic simulations, classical continuum theory, softening and hardening nonlocal models, respectively. The validity of both existing softening and hardening nonlocal models was confirmed in comparison with the atomistic simulations. In addition, a comparative calculation for dimensional natural frequencies with respect to length of CNTs by different methodologies was provided to explain why the softening and hardening nonlocal models are both correct in nonlocal elasticity theory.