Nonlocal Elastic Shell Model Research Articles

Small scale effects on the stability of carbon nano-peapods under radial pressure

In this paper, a nonlocal elasticity formulation is presented for analyzing the instability of nano-peapods by modeling the nanotube as a shell. Using the nonlocal elasticity theory, the small scale characteristics of Carbon Nano-Peapods (CNPs) are taken into account. While the classical elastic shell model overestimates the critical pressure for the onset of structural instability of carbon nanotubes, the obtained results show that the nonlocal elastic shell model for nano-peapods can potentially provide better predictions. According to the results, it is concluded that the presence of C 60 inside (10,10) Carbon Nanotubes (CNTs) significantly increases the stability resistance of the single CNT.

A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm

The buckling behaviors of microtubules (MTs) in a living cell have been studied based on the nonlocal anisotropic shell theory and Stokes flow theory. The analytical expressions for the buckling load and the growth rate of the buckling are obtained and discussed. In addition, the pressure on MTs, resulting from cytosol motion, is derived on the basis of the Stokes flow theory. The influences of filament network elasticity and the shear modulus of MTs, especially the cytosol viscosity and MT small scale effects on MT buckling behaviors, are investigated. The analytical results show that the MT buckling growth rate increases with the MT small scale parameter, while decreases as the filament network elastic modulus, the MT shear modulus and cytoplasm viscosity increase. Although the cytosol viscosity has a significant influence on the value of the growth rate, it shows little effects on the range of the axial wave number of buckling as well as the critical axial wave number that corresponds to the maximal growth rate. Finally, the MT buckling growth rates have been calculated using the beam model, the isotropic shell model, and the anisotropic shell model. The results indicate that using the anisotropy shell theory to model the buckling behavior of MTs is necessary.

Mechanics behavior of microtubules based on nonlocal anisotropic shell theory

A nonlocal anisotropic elastic shell model of a microtubule (MT) has been established in this paper. The proposed model exactly expresses the persistence length, the growth rate for buckling and the buckling load which include the small scale effects in a closed form. The predicted results of our model indicate that the length-dependence of the persistence length is sensitive to the small scale effects when the length of the MT is short. The curves for the growth rate of a MT with the consideration of the small scale effects show that the larger the small scale parameter is, the larger the growth rate is. The buckling behavior of a MT with the consideration of the small scale effects and the filaments network surrounded the MT have been investigated, which presents the filaments network plays an important role in buckling behaviors of MTs.

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10 −9 and 0.95×10 −9 m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e 0 for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.

Vibration Characteristics of Multiwalled Carbon Nanotubes Embedded in Elastic Media by a Nonlocal Elastic Shell Model

In this paper, the vibrational behavior of the multiwalled carbon nanotubes (MWCNTs) embedded in elastic media is investigated by a nonlocal shell model. The nonlocal shell model is formulated by considering the small length scales effects, the interaction of van der Waals forces between two adjacent tubes and the reaction from the surrounding media, and a set of governing equations of motion for the MWCNTs are accordingly derived. In contrast to the beam models in the literature, which would only predict the resonant frequencies of bending vibrational modes by taking the MWCNT as a whole beam, the current shell model can find the resonant frequencies of three modes being classified as radial, axial, and circumferential for each nanotube of a MWCNT. Big influences from the small length scales and the van der Waals’ forces are observed. Among these, noteworthy is the reduction in the radial frequencies due to the van der Waals’ force interaction between two adjacent nanotubes. The numerical results also show that when the spring constant k0 of the surrounding elastic medium reaches a certain value, the lowest resonant frequency of the double walled carbon nanotube drops dramatically.

AXI-SYMMETRIC WAVE PROPAGATION OF CARBON NANOTUBES WITH NON-LOCAL ELASTIC SHELL MODEL

A non-local elastic shell model is proposed for the first time in this study for considering the small-scale effect in axi-symmetric wave propagation in carbon nanotubes (CNTs). Two coupled radial and longitudinal modes, and one decoupled torsional mode, are derived from the developed non-local shell model. The small-scale effect on wave propagation is numerically studied and discussed. In addition, the cut-off frequency based on the non-local shell model is obtained, which is found to be free of the small-scale effect. It is interesting to note that only one asymptotic phase velocity exists in the CNTs using the non-local shell model, whereas two asymptotic phase velocities can be predicted using the classical or local elastic shell model. It is hoped the research presented herein can be used as a benchmark for future studies on the wave propagation of CNTs with non-local continuum models.

Thermal Buckling of Multi-Walled Carbon Nanotubes by Nonlocal Elasticity

The small internal length scales of nanomaterials/nano-devices may call the direct application of classical continuum models into question. In this research, a nonlocal elastic shell model, which takes the small scale effects into account, is developed to study the thermal buckling behavior of multi-walled carbon nanotubes. The multi-walled carbon nanotubes are considered as concentric thin shells coupled with the van der Waals forces between adjacent nanotubes. Closed form solutions are formulated for two types of thermal buckling of a double-walled carbon nanotube: Radial thermal buckling (as in a shell under external pressure) and axial thermal buckling. The effects of small scale effects are demonstrated, and a significant influence of internal characteristic parameters such as the length of the C‐C bond has been found on the thermal buckling critical temperature. The study interestingly shows that the axial buckling is not likely to happen, while the “radial” buckling may often take place when the carbon nano-tubes are subjected to thermal loading. Furthermore, a convenient method to determine the material constant, “e0” and the internal characteristic parameter, “a,” is suggested.

Validation of the non-local elastic shell model for studying longitudinal waves insingle-walled carbon nanotubes

This study deals with the dispersion of longitudinal waves in a single-walled armchaircarbon nanotube, focusing on the effect of the microstructure of the carbon nanotube onthe wave dispersion. To reveal the effect analytically, the carbon nanotube is modelled as anon-local elastic cylindrical shell in the study. The dynamic equation of the non-localelastic cylindrical shell is established and the corresponding dispersion relation oflongitudinal waves is derived. These analytic results are verified via molecular dynamicssimulations, based on the Terroff–Brenner potential, for the propagation of variouslongitudinal waves in three single-walled armchair carbon nanotubes. The moleculardynamics simulations indicate that the longitudinal wave dispersion predicted by the modelof the non-local elastic cylindrical shell shows a good agreement with that ofmolecular dynamics simulations in a wide frequency range up to the terahertz region.They also show that both the microstructure of the carbon nanotube and thecoupling between the longitudinal wave and the radial motion play an importantrole in the dispersion of longitudinal waves in a single-walled armchair carbonnanotube.