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 e₀ 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.     

Small scale effect on flow-induced instability of double-walled carbon nanotubes

Small scale effect on flow-induced instability of double-walled carbon nanotubes (DWCNTs) is investigated using an elastic shell model based on Donnell’s shell theory. The dynamic governing equations of DWCNTs are formulated on the basis of nonlocal elasticity theory, in addition, the van der Waals (vdW) interaction between the inner and outer walls is taken into account in the nonlocal shell modeling. The instability of DWCNTs that is induced by a pressure-driven steady flow is investigated. The numerical computations indicate that as the flow velocity increases, DWCNTs have a destabilizing way to get through multi-bifurcations of the first and second bifurcations in turn. It is concluded that the natural frequency of DWCNTs and the critical flow velocity of the flow-induced instability are strictly related to the ratio of the length to the outer radius of DWCNTs, the pressure of the fluid and the small scale effects. Furthermore, it is interesting to observe that as the small scale effects are considered, the natural frequencies and the critical flow velocities of DWCNTs decrease as compared to the results with the classical (local) continuum mechanics, therefore, the small scale effects play an important role on performing the instability analysis in the fluid-conveying DWCNTs.     

Postbuckling prediction of double-walled carbon nanotubes under axial compression

An elastic double-shell model is presented for the buckling and postbuckling of a double-walled carbon nanotube subjected to axial compression. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. The van der Waals interaction between the inner and outer nanotubes and the nonlinear prebuckling deformations of the shell are both taken into account. A boundary layer theory of shell buckling is extended to the case of double-walled carbon nanotubes under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the single-walled carbon nanotube and the double-walled carbon nanotube both have an unstable postbuckling behavior.     

Spectral element based model for wave propagation analysis in multi-wall carbon nanotubes

A spectrally formulated finite element is developed to study elastic waves in carbon nanotubes (CNT), where the frequency content of the exciting signal is at terahertz level. A multi-walled nanotube (MWNT) is modelled as an assemblage of Euler–Bernoulli beams connected throughout their length by distributed springs, whose stiffness is governed by the van der Waals force acting between the nanotubes. The spectral element is developed using the recently developed formulation strategy based on the solution of polynomial eigenvalue problem (PEP). A single element can model a MWNT with any number of walls. Studies are carried out to investigate the effect of the number of walls on the spectrum and dispersion relation. Effect of the number of walls on the frequency response function is investigated. Response of MWNT for terahertz level loading is analyzed for broad-band shear pulse.     

Shell element based model for wave propagation analysis in multi-wall carbon nanotubes

A spectrally formulated finite element is developed to study very high frequency elastic waves in carbon nanotubes (CNTs). A multi-walled nanotube (MWNT) is modelled as an assemblage of shell elements connected throughout their length by distributed springs, whose stiffness is governed by the van der Waals force acting between the nanotubes. The spectral element is formulated using the recently developed strategy based on singular value decomposition (SVD) and polynomial eigenvalue problem (PEP). The element can model a MWNT with any number of walls. Studies are carried out to investigate the effect of the number of walls on the spectrum and dispersion relation. The importance of shell element based model over the beam model is established. The zone of validity of the previously developed beam model is also investigated. It is shown that the shell model is required to capture the symmetric Lamb wave modes. It is also shown through numerical examples that the developed element efficiently captures the response of MWNT for Tera-hertz level frequency loading.     

Electromechanical coupling effect on electronic properties of double-walled boron nitride nanotubes

We report on a first-principles study of a novel band modulation in zigzag double-walled boron nitride nanotubes (DBNNTs) by applying radial strain and coupled external electric field. We show that the band alignment between the inner and outer walls of the DBNNTs can be tuned from type I to type II with increasing radial strain, accompanied with a direct to indirect band gap transition and a substantial gap reduction. The band gap can be further significantly reduced by applying a transverse electric field. The coupling of electric field with the radial strain makes the field-induced gap reduction being anisotropic and more remarkable than that in undeformed DBNNTs. In particular, the gap variation induced by electric field perpendicular to the radial strain is the most remarkable among all the modulations. These tunable properties by electromechanical coupling in DBNNTs will greatly enrich their versatile applications in future nanoelectronics.     

The thermal effect on axially compressed buckling of a double-walled carbon nanotube

The thermal effect on axially compressed buckling of a double-walled carbon nanotube is studied in this paper. The effects of temperature change, surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model with thermal effect is presented for axially compressed buckling of a double-walled carbon nanotube embedded in an elastic matrix under thermal environment. Based on the model, an explicit formula for the critical axial stress is derived in terms of the buckling modes of the shell and the parameters that indicate the effects of temperature change, surrounding elastic medium and the van der Waals forces. Based on that, some simplified analysis is carried out to estimate the critical axial stress for axially compressed buckling of the double-walled carbon nanotube. Numerical results for the general case are obtained for the thermal effect on axially compressed buckling of a double-walled carbon nanotube. It is shown that the axial buckling load of double-walled carbon nanotube under thermal loads is dependent on the wave number of axially buckling modes. And a conclusion is drawn that at low and room temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube increase as the value of temperature change increases, while at high temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube decrease as the value of temperature change increases.     

Effect of small size on dispersion characteristics of wave in carbon nanotubes

Effect of small size on dispersion characteristics of waves in multi-walled carbon nanotubes is investigated using an elastic shell model. Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal elastic theory. The relationship between wavenumber and frequency of wave propagation is obtained from the solution of the eigenvalue equations. The numerical results show that the dispersion characteristics of wave in the multi-walled carbon nanotube are affected by the small size. Effect of small size is not obvious for the smaller wavenumber, and it will arise and increase gradually with the increase of the wavenumber. Effect of the small size will decrease as the inner radius of carbon nanotubes increases. In addition, the explicit expressions of the cut-off frequencies are derived. The results show that the cut-off frequencies cannot be influenced by the small size of carbon nanotubes.     

Wave propagation in carbon nanotubes embedded in an elastic matrix

This paper reports the results of an investigation into the characteristics of wave propagation in carbon nanotubes embedded in an elastic matrix, based on an exact shell model. Each of the concentric tubes of multi-walled carbon nanotubes is considered as an individual elastic shell and coupled together through the van der Waals forces between two adjacent tubes. The matrix surrounding carbon nanotubes is described as a spring element defined by the Winkler model. The effects of rotatory inertia and elastic matrix on the wave velocity, the critical frequency, and the amplitude ratio between two adjacent tubes are described and discussed through numerical examples. The results obtained show that wave propagation in carbon nanotubes may appear in a critical frequency at which the wave velocity changes suddenly; the elastic matrix surrounding carbon nanotubes debases the critical frequency and the wave velocity, and changes the vibration modes between two adjacent tubes; the rotatory inertia based on an exact shell model obviously influences the wave velocity at some wave modes. Finally, a comparison of dispersion solutions from different shell models is given. The present work may serve as a useful reference for the application and the design of nano-electronic and nano-drive devices, nano-oscillators, and nano-sensors, in which carbon nanotubes act as basic elements.     

Small-scale effect on torsional buckling of multi-walled carbon nanotubes

The small-scale effect on the torsional buckling of multi-walled carbon nanotubes coupled with temperature change is investigated in this paper. A nonlocal multiple-shell model for the multi-walled carbon nanotubes surrounded an elastic medium under torsional and thermal loads is established, and then general solutions are obtained from the governing equations. The influence of the nonlocal effect on critical shear force and change in temperature is investigated. It is demonstrated that the critical shear force could be overestimated by the classical continuum theory and the nonlocal effect on critical buckling force decreases as the change in temperature increases at room or low temperature but increases as the change in temperature increases at higher temperature. Meanwhile, the effect of small size-scale is dependent on the buckling mode under different thermal environments. It is also shown that the innermost radius and the number of layer can affect the small-scale effect on critical change in temperature and buckling shear force. When the ratio of tube length and outmost radius are given, the critical shear force in each layer decreases and the nonlocal effect on the critical shear force becomes weaker as the innermost radius and the layer number increase.     

Postbuckling of double-walled carbon nanotubes with temperature dependent properties and initial defects under combined axial and radial mechanical loads

Buckling and postbuckling analysis is presented for a double-walled carbon nanotube subjected to combined axial and radial loads in thermal environments. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual orthotropic shell with presence of van der Waals interaction forces and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include thermal effects. Temperature-dependent material properties, which come from molecular dynamics simulations, and initial point defect, which is simulated as a dimple on the tube wall, are both taken into account. A singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, double-walled carbon nanotubes subjected to combined axial and radial mechanical loads under different sets of thermal environments. The results reveal that temperature change only has a small effect on the postbuckling behavior of the double-walled carbon nanotube. The axially-loaded double-walled carbon nanotube subjected to radial pressure has an unstable postbuckling path, and the structure is imperfection–sensitive. In contrast, the pressure-loaded double-walled carbon nanotube subjected to axial compression has a very weak “snap-through” postbuckling path, and the structure is virtually imperfection–insensitive.     

Acoustoelastic effect and wave propagation in heterogeneous weakly anisotropic materials

A theoretical analysis of the acoustoelastic effect is presented. It is based upon the theory of sound wave propagation in a stressed heterogeneous weakly anisotropic elastic medium composed of grains. The effect of residual stress is included, and shown to be different from that of applied stress. The statistics of grain orientation and of grain correlation are taken into account. The acoustoelastic coefficients and the effects of dispersion, attenuation and symmetry of the medium are determined.     

Curvature effects on axially compressed buckling of a small-diameter double-walled carbon nanotube

The curvature effects of interlayer van der Waals (vdW) forces on axially compressed buckling of a double-walled carbon nanotube (DWNT) of diameter down to 0.7 nm are studied. Unlike most existing models which assume that the interlayer vdW pressure at a point between the inner and outer tubes depends merely on the change of the interlayer spacing at that point, the present model considers the dependence of the interlayer vdW pressure on the change of the curvatures of the inner and outer tubes at that point. A simple expression is derived for the curvature-dependence of the interlayer vdW pressure in which the curvature coefficient is determined. Based on this model, an explicit formula is obtained for the axial buckling strain. It is shown that neglecting the curvature effect alone leads to an under-estimate of the critical buckling strain with a relative error up to −7%, while taking the average radius of two tubes as the representative radius and the curvature effect leads to an over-estimate of the critical buckling strain with a relative error up to 20% when the inner radius downs to 0.35 nm. Therefore, the curvature effects play a significant role in axially compressed buckling problems only for DWNTs of very small radii. In addition, our results show that the effect of the vdW interaction pressure prior to buckling of DWNTs under pure axial stress is small enough and can be negligible whether the vdW interaction curvature effects are neglected or not.     

Quasi-kinematic flood wave propagation

Summary In the flood waves of natural streams, the plot of discharge versus simultaneous water level values at a river station usually show a little shift from the socalled steady rating curve. In other terms, natural flood waves normally have small loops and may be considered to be quasi-kinematic. Very often, on the basis of the steady rating curve and of the observed hydrograph at an upstream gage station, we can quickly and easily forecast the values and time of flood peaks at one or more downstream channel sections. This article describes a set of very simple formulas first to check whether the flood wave may be considered to be quasi-kinematic, and then to estimate the speed and the attenuation with which it moves downstream. For a number of prismatic channels, the results were compared with those obtained with equally simple, currently used formulas (Jones, Henderson) and with the exact solutions. The comparisons showed that the formulas proposed in this article are more general and accurate than the other known simple formulas and that, for flood loops of less than 10%, give results that are very near to being exact.

---

Sommario Nei fenomeni di piena dei corsi d'acqua naturali la successions nel tempo dei valori di portata e livello idrometrico in una sezione d'alveo presenta, nella gran parte dei casi, modesti scostamenti dalla cosiddetta scala di deflusso di moto permanente: in altri termini normalmente le onde naturali di piena hanno cappi modesti e possono ritenersi quasi cinematiche. Molto spesso, data una stazione di monte in cui sia nota la scala di deflusso el'drogrammadei livelli, occorre prevedere semplicemente e rapidamente i valori e gli istanti dei colmi di portata e livello in una o più sezioni a valle. In questo articolo vengono presentate formule molto semplici per verificare dapprima che l'onda di piena possa ritenersi quasi cinematica, e stimare quindi la velocità e l'attenuazione con cui i colmi si propagano verso valle. I confronti fatti, per alcuni alvei cilindrici, con la soluzione esatta e con i risultati ottenibili con formule altrettanto semplici e di uso corrente (Jones, Henderson), mostrano che le formule proposte sono più generali e corrette di queste ultime, e danno risultati quasi esatti finchè i cappi di piena relativi sono inferiori al 10%.

---

List of symbols P(x, z) steady rating curve - Q volumetric rate of discharge - q=Q-P flood loop - g gravitational acceleration - I water surface slope - c=dP/dA kinematic wave velocity - y water depth - p, s time and space steps - x, t channel distance starting upstream; time - z water surface height above datum - A, B channel cross section area and surface width - F=B Q ²/gA ³ Froude number - L,L ₀,L ₁,L ₂,L ₃ characteristic lengths of the channel - T=L/c characteristic time of the channel - f _x =f/x, f _f =f/t etc for the partial derivatives - Df/Dt,Df/Dx for the derivatives of functionf following the discharge peak     

Area change effects on shock wave propagation

Experimental testing was conducted for a planar shock wave of incident Mach number \(M_\mathrm{s} = 1.33\) propagating through one of three compound parabolic profiles of 130, 195 or 260 mm in length, all of which exhibit an 80 % reduction in area. Both high-resolution single shot and low-resolution video were used in a schlieren arrangement. Results showed three main types of flow scenarios for propagation through a gradual area reduction, and an optimal net increase of 12.7 % in shock Mach number was determined for the longest profile, which is within 5 % of theoretical predictions using Milton’s modified Chester–Chisnell–Whitham relation.     

Effect of a shock wave on heat propagation

The nature of the propagation of a thermal wave produced by a powerful explosion was described in a number of papers, for example, [1–6]. It was shown by a numerical method [4] that a shock wave is present together with the thermal wave. In this paper, the effect of a homothermal shock wave on heat propagation is evaluated by an approximate method.