A study on mechanical characteristics of nanotubes based on new nonlocal elasticity models

Abstract

The nonlocal elasticity theory in continuum mechanics is a nonlinear elastic theory about the mechanical behaviors of micro and nanostructures. Because the nonlocal constitutive equation is an appropriate relation to describe the small scale effects of nanotubes in nanostructures, there are increasingly more elastic beam and shell models based on this nonlocal theory recently in the study on the mechanical properties of nanotubes. For most of these models, however, the nonlocal quantities such as nonlocal stresses, nonlocal bending moments and nonlocal shear forces are directly substituted into the classical governing equations and boundary conditions. In such cases, the classical equations and boundary conditions are taken for granted as tenable in the nano environments. Some contradictory predictions and surprising conclusions have been reported according to these nonlocal models. For example, the stiffness of carbon nanotubes predicted by such nonlocal models contradicts with the results by molecular dynamic simulation. Furthermore, the bending behaviors of nano cantilevers with a point load at the free end are not influenced at all by the nonlocal effect, which is not reasonable obviously. Therefore, the direct application of classical models with nonlocal quantities is not a valid approach. The nonlocal models established by this method are termed the partial nonlocal models (PN). To rectify these partial nonlocal models, a new analytical nonlocal model (AN) is established in this thesis based on the variational principle. The nonlocal governing equations and nonlocal boundary conditions are derived rigorously from constitutive equations. Some higher-order nonlocal terms are present in the new AN model, which truly describes the nonlocal effects of nanostructures. The reliability of AN model is confirmed by different means. In this thesis, the physical properties and engineering applications of carbon nanotubes are first investigated. The methodologies for studying the mechanical behaviors of carbon nanotubes are also elaborated, including details of comparison of the AN and PN models. The AN model is derived from the nonlinear constitutive equation based on the variational principle. The integration form of strain energy density for AN model is applied, instead of following the traditional linear strain energy density. The governing equations and boundary conditions thus derived contain nonlocal higher-order terms which are not present in the PN models. The contribution of the higher-order terms are so significant that the PN models could not accurately simulate the influence of nonlocal effect. By considering the shear effect of carbon nanotubes, the AN-Timoshenko beam model (ANT) are established in this thesis. Based on the ANT model, the mechanical behaviors for bending, buckling, wave propagation and free vibration of single-walled carbon nanotubes and the wave propagation of double-walled carbon nanotubes are analyzed in this thesis. All results confirm the stiffness enhancement of carbon nanotubes contributed by nonlocal effect. The buckling analysis shows that carbon nanotubes with fewer boundary constrains are more sensitive to the nonlocal effect. In wave propagation analysis of carbon nanotubes, the nonlocal effect is reduced at the high frequencies. The outer layers of double-walled carbon nanotubes are more sensitive to nonlocal effect than the inner layers. Beside the ANT model, the AN-Euler-Bernoulli beam model (ANE) without shear effect is also applied to study wave propagation of single-walled carbon nanotubes. Similar results with respect to ANT are also obtained where, for higher frequencies in wave propagation, the ANT model yields better results than the ANE model because shear deformation is neglected in ANE. Moreover, an AN shell model (ANS) is also established to study the axisymmetry wave propagation behaviors in carbon nanotubes and this is the first ever study of a two-dimensional AN model. Stiffness enhancement and high frequency wave decay are observed using this ANS model. Through verification by molecular dynamic simulation, the ANS model is yields more accurate solutions for carbon nanotubes with small diameter-to-length ratio. Finally, the thesis includes a study for the application of the ANE model in the analysis of wave propagation for fluid-filled carbon nanotubes and it is the first time that such study is presented. Similar to the previous studies, the nonlocal effect influences significantly the wave propagation behaviors, where the fluid velocity changes with the phase velocity for elastic wave in nanotubes when compared with the PNE model. The simulation results show that the wave frequency increases first and subsequently decreases with fluid velocity. The results provide useful references for engineering applications of carbon nanotubes. In conclusion, the ANT and ANS models are established according to the variational principle where the nonlocal and shear deformation effects in carbon nanotubes are considered. The mechanical characteristics of nanostructures, including carbon nanotubes, are first analyzed based on the ANT model. Subsequently, axisymmetry wave propagation is investigated by the ANS model. In a furthe example, wave propagation of fluid filled carbon nanotubes are studied by the ANE model. Through comparison and analysis for all results, the characteristics of the ANE, ANT, and ANS models are established. The influence of shear deformation and nonlocal effect on the mechanical behaviors of carbon nanotubes are also discussed in detail. Finally, the rationality of the AN model is confirmed by comparing with molecular dynamic simulations.