Scale Effect Of Fluctuations In Performance Studies Of Carbon Nanotubes

The research on carbon nanotubes(CNTs) is introduced, and then wave characteristics of CNTs are studied via the generalized nonlocal elasticity theory which is based on the nonlocal elasticity theory and nonlocal strain gradients theory.The difference of nonlocal elasticity theory from classical elasticity theory is that long range force is incorporated in the former, namely, stress at a point depends on strain in a region near that point. With the help of this idea, in space any field can be expressed by a weighted average with respect to its volume. So far, the nonlocal elasticity theory is practical and useful in analyzing CNTs. Howover, in treating dislocation problems in an elastic medium, application of the nonlocal elasticity theory only removes the singularity of the stress field near the dislocation core, and the strain field is still singular near the dislocation core. And it has been shown that both the stress field and the strain field are finite, not singular while generalized nonlocal elasticity theory is adopted. This indicates the superiority of the generalized nonlocal elasticity theory over the nonlocal elasticity theory and the nonlocal strain gradients theory. In fact, the two latter theories are special cases of the former.Based on the beam model of generalized nonlocal elasticity this paper studies the size effects of transverse wave propagation of carbon nanotubes (CNTs) in a free space,embedded in a surrounding elastic medium and with initial axial stress. For fexural waves propagating in single-walled CNTs, the phase and group velocities are obtained analytically and scale-dependent. In particular, waves in single-walled CNTs surrounded by an elastic medium can not propagate for frequencies lower than the cut-off frequency dependent on the surrounding medium. For double-walled CNTs, there are two wave speeds for the acoustic and optical modes, and the cut-off frequency is dependent on the van der Waals (vdW) force between two tubes and the surrounding medium when it is present, but independent of the scale parameters. For very low wavenumbers or frequencies, the surrounding elastic medium strongly affects the wave velocity, while the scale effects for higher wavenumbers or frequencies are notable. Dispersion relations are presented graphically to show the scale effects of wave speed with and without the surrounding elastic medium.For fexural waves propagating in CNTs with initial axial stress, the phase and group velocities are also obtained analytically, found to be scale-dependent, and dependent on the tension or compression form of initial axial force. For double-walled CNTs, it is found that the vander Waals (vdW) force between two tubes has significant effects on the cut-off frequency, while scale parameters do not. Moreover, there is no existing wave when wave frequency is lower than the cut-off frequency. Finally, dispersion curves are presented graphically to show the effects of scale parameters and initial axial loading on fexural waves for CNTs.