考虑非线性本构和小初始变形的单壁碳纳米管的Bernoulli-Euler梁模型

在有限变形条件下石墨烯的应力应变关系是非线性的,而且石墨烯卷曲形成的碳纳米管(carbon nanotubes,CNTs)在应用时通常会产生小的初始变形.但是,在已有的碳纳米管力学性质的研究中还没有同时考虑这两种因数影响.基于石墨烯的非线性本构关系,建立了新的包含小初始变形的Bernoulli-Euler梁模型,然后应用该模型研究了单壁碳纳米管(single-walled carbon nanotubes,SWCNTs)的静态弯曲和一阶受迫振动.结果表明:在静力弯曲时,当初始变形与外荷载的方向一致时,初始变形增大了系统的刚度.当初始变形与外荷载的方向相反时,初始变形使碳纳米管的静力变形变得复杂.在受迫振动时,初始变形和本构中的非线性项都能改变振幅的分岔点的位置.初始变形可以使碳纳米管的力学性质由硬非线性变为软非线性.因此,非线性本构和初始变形均对碳纳米管力学行为有显著影响.

Bernoulli-Euler beam model of single-walled carbon nanotubes with small initial deformation and nonlinear constitutive model

The stress-strain relationship of graphene is nonlinear for finite deformations, and the carbon nanotubes (CNTs), which are formed by curling graphene, are usually in small initial deformations in practice. However, these two factors have been not considered in existing researches up to now. In present paper, the Bernoulli-Euler beam model with small initial deformation was established based on the nonlinear constitutive relation of graphene, and then the static bending and first-order forced vibration of single-walled carbon nanotubes (SWCNTs) were studied using the new model. The results show that the initial deformation is consistent with the direction of the external load in static bending, the initial deformation increases the rigidity of the system. The initial deformation is opposite to the direction of the external load, the initial deformation complicates the static deformation of CNTs. For the forced vibration, both the initial deformation and the nonlinear term in the constitutive can change the position of the bifurcation point of the amplitude. The initial deformation can change the basic properties of CNTs from a hard spring to a soft spring. Therefore, the nonlinear terms induced by the nonlinear constitutive and the initial deformation have significant effects on the mechanical behaviors of CNTs.