Surface effect on size-dependent wave propagation in nanoplates via nonlocal elasticity

Within the framework of nonlocal elasticity, the surface layer model is proposed to investigate the wave propagation characteristics in a single-layered nanoplate. The general solutions of nonlocal governing equations are expressed using partial wave technique and the nonclassical boundary conditions are derived. The dispersion relation with the effects of surface and nonlocal small-scale is obtained, and the size-dependent dispersion behaviour is demonstrated. The impacts of surface elasticity, residual surface stress and nonlocal parameter on the dispersion curves of the lowest-order two modes are illustrated. Numerical examples reveal that both the surface effect and nonlocal small-scale effect can obviously decrease the magnitude of phase velocity, and the thinner nanoplate corresponds to the smaller wave velocity and the narrower frequency bandwidth.     

Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This paper investigates a model of wave propagation in single-wall carbon nanotubes (SWCNTs) with small scale effects are studied. The equation of motion of the dilatation wave is obtained using the nonlocal elastic theory. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. The SWCNTs structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The SWCNT was the (40,0) zigzag tube with an effective diameter of 3.13 nm. Nonlinear frequency equations of wave propagation in SWCNTs are described through the effect of small scale. The phase velocity and the group velocity are derived, respectively. The nonlinear dispersion relation is analyzed with different wave numbers versus scale coefficient. It can be observed from the results that the dispersion properties of the dilatation wave are induced by the small scale effects, which will disappear in local continuous models. The dispersion degree can be strengthened by increasing the scale coefficient and the wave number. Furthermore, the characteristics for the group velocity of the dilatation wave in carbon nanotubes can also be tuned by these factors.     

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.     

Wave dispersion in viscoelastic single walled carbon nanotubes based on the nonlocal strain gradient Timoshenko beam model

Based on the nonlocal strain gradient theory and Timoshenko beam model, the properties of wave propagation in a viscoelastic single-walled carbon nanotube (SWCNT) are investigated. The characteristic equations for flexural and shear waves in visco-SWCNTs are established. The influence of the tube size on the wave dispersion is clarified. For a low damping coefficient, threshold diameter for shear wave (SW) is observed, below which the phase velocity of SW is equal to zero, whilst flexural wave (FW) always exists. For a high damping coefficient, SW is absolutely constrained, and blocking diameter for FW is observed, above which the wave propagation is blocked. The effects of the wave number, nonlocal and strain gradient length scale parameters on the threshold and blocking diameters are discussed in detail.     

Investigating Bulk Waves in Orthotropic Rectangular Nanoplates Based on Three Dimensional Elasticity Theory and Nonlocal Elasticity Theory

The propagation of bulk waves in rectangular nanoplates is studied on the basis of nonlocal three-dimensional elasticity theory. The nonlocal theory applies to both thin and thick rectangular orthotropic nanoplates. The dispersion relation for the waves is derived analytically. Our results are checked against data for macroplates. The influence of nonlocality and other parameters on the wave frequency and phase velocity is discussed.     

Propagation of waves in nonlocal porous multi-phase nanocrystalline nanobeams under longitudinal magnetic field

ABSTRACT

This article investigates wave propagation behavior of a multi-phase nanocrystalline nanobeam subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations which are solved by applying an analytical method. The frequencies are compared with those of nonlocal and couple stress-based beams. It is showed that wave frequencies and phase velocities of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, magnetic field, surface effect and nonlocality.     

Longitudinal wave propagation in a piezoelectric nanoplate considering surface effects and nonlocal elasticity theory

The propagation characteristics of the longitudinal wave in a piezoelectric nanoplate were investigated in this study. The nonlocal elasticity theory was used and the surface effects were taken into account. In addition, the group velocity and phase velocity were derived and investigated, respectively. The dispersion relation was analyzed with different scale coefficients, wavenumbers, and voltages. The results showed that the dispersion degree can be strengthened by increasing the wavenumber and scale coefficient.     

The effect of surface stress on the propagation of Lamb waves

This work investigates the possibility of the propagation of Lamb waves in thin solid layers with external traction free surfaces, in the presence of surface elasticity, inertia and residual stress. It is demonstrated that such waves do exist and that their characteristics can be quite different from their classical counterparts. The governing equations with non-classical boundary conditions involving the bulk and surface stress are solved exactly in the frequency-wavenumber domain. This solution is utilized to compute the Lamb wave modes for different layer thicknesses. An efficient strategy to capture all the modes of Lamb waves within a given frequency window is outlined. It is shown that the effect of surface elasticity and inertia becomes significant with increasing frequency and decreasing layer thickness, where the number of modes participating within a given frequency window is more than that permitted by the classical theory. Further, it is observed that the nature of the Lamb wave modes (in terms of negative dispersion) in the presence of surface stress is similar to what predicted by the nonlocal theory and microstructure based continuum theory.     

Scale-dependent effects on wave propagation in magnetically affected single/double-layered compositionally graded nanosize beams

This article deals with the wave propagation analysis of single/double layered functionally graded (FG) size-dependent nanobeams in elastic medium and subjected to a longitudinal magnetic field employing nonlocal elasticity theory. Material properties of nanobeam change gradually according to the sigmoid function. Applying an analytical solution, the acoustical and optical dispersion relations are explored for various wave number, nonlocality parameter, material composition, elastic foundation constants, and magnetic field intensity. It is found that frequency and phase velocity of waves propagating in S-FGM nanobeam are significantly affected by these parameters. Also, presence of cut-off and escape frequencies in wave propagation analysis of embedded S-FGM nanobeams is investigated.     

Effects of viscous fluid on wave propagation in carbon nanotubes

In this Letter, the effects of the viscous fluid on the propagation characteristics of elastic waves in carbon nanotubes are studied. Based on the nonlocal continuum theory, the small scales effects are also considered. The equations of wave motion are derived and the dispersion relation is presented. Numerical simulations are performed with the consideration of different scale coefficients to discuss the influence of the viscous fluid. From the results, it can be observed that the dispersion relation can be changed by the fluid viscosity obviously. Moreover, due to the fluid viscosity, the wave frequency will be reduced to a low region and the elastic wave behaviors can be significantly influenced by the viscous fluid velocity.     

Longitudinal vibration and instabilities of carbon nanotubes conveying fluid considering size effects of nanoflow and nanostructure

In this study, the effects of small-scale of the both nanoflow and nanostructure on the vibrational response of fluid flowing single-walled carbon nanotubes are investigated. To this purpose, two various flowing fluids, the air-nano-flow and the water nano-flow using Knudsen number, and two different continuum theories, the nonlocal theory and the strain-inertia gradient theory are studied. Nano-rod model is used to model the fluid-structure interaction, and Galerkin method of weighted residual is utilizing to solve and discretize the governing obtained equations. It is found that the critical flow velocity decreases as the wave number increases, excluding the first mode divergence that it has the least value among of the other instabilities if the strain-inertia gradient theory is employed. Moreover, it is observed that Kn effect has considerable impact on the reduction of critical velocities especially for the air-flow flowing through the CNT. In addition, by increasing a nonlocal parameter and Knudsen number the critical flow velocity decreases but it increases as the characteristic length related to the strain-inertia gradient theory increases.     

Wave characteristics of single-walled fluid-conveying carbon nanotubes subjected to multi-physical fields

Wave propagation in single-walled carbon nanotubes (SWCNTs) conveying fluids and placed in multi-physical fields (including magnetic and temperature fields) is studied in this paper. The nanotubes are modelled as Timoshenko beams. Based on the nonlocal beam theory, the governing equations of motion are derived using Hamilton's principle, and then solved by Galerkin approach, leading to two second-order ordinary differential equations (ODEs). Numerical simulations are carried out to verify the analytical model proposed in the present study, and determine the influences of the nonlocal parameter, the fluid velocity and flow density, the temperature and magnetic field flux change, and the surrounding elastic medium on the wave behaviour of SWCNTs. The results show that the nonlocal parameter has a considerable influence on dynamic behaviour of the nanotube and the fluid flow inside it. The results also show that the magnetic and temperature fields play an important role on the wave propagation characteristics of SWCNTs.     

Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory

A nonlocal Euler–Bernoulli elastic beam model is developed for the vibration and instability of tubular micro- and nano-beams conveying fluid using the theory of nonlocal elasticity. Based on the Newtonian method, the equation of motion is derived, in which the effect of small length scale is incorporated. With this nonlocal beam model, the natural frequencies and critical flow velocities for the case of simply supported system and for the case of cantilevered system are obtained. The effect of small length scale (i.e., the nonlocal parameter) on the properties of vibrations is discussed. It is demonstrated that the natural frequencies are generally decreased with increasing values of nonlocal parameter, both for the supported and cantilevered systems. More significantly, the effect of small length scale on the critical flow velocities is visible for fluid-conveying beams with nano-scale length; however, this effect may be neglected for micro-beams conveying fluid.     

Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory

The governing equation of wave motion of viscoelastic SWCNTs (single-walled carbon nanotubes) with surface effect under magnetic field is formulated on the basis of the nonlocal strain gradient theory. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that the size-dependent effects on the phase velocity may be ignored at low wave numbers, however, is significant at high wave numbers. Phase velocity can increase by decreasing damping or increasing the intensity of magnetic field. The damping ratio considering surface effect is larger than that without considering surface effect. Damping ratio can increase by increasing damping, increasing wave number, or decreasing the intensity of magnetic field.     

Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory

A nonlocal Euler–Bernoulli elastic beam model is developed for the vibration and instability of tubular micro- and nano-beams conveying fluid using the theory of nonlocal elasticity. Based on the Newtonian method, the equation of motion is derived, in which the effect of small length scale is incorporated. With this nonlocal beam model, the natural frequencies and critical flow velocities for the case of simply supported system and for the case of cantilevered system are obtained. The effect of small length scale (i.e., the nonlocal parameter) on the properties of vibrations is discussed. It is demonstrated that the natural frequencies are generally decreased with increasing values of nonlocal parameter, both for the supported and cantilevered systems. More significantly, the effect of small length scale on the critical flow velocities is visible for fluid-conveying beams with nano-scale length; however, this effect may be neglected for micro-beams conveying fluid.     

Asymptotic Models of Diffusion Effects due to Nonlinear Resonant Interaction of Waves with Flows

We develop an asymptotic theory describing nonlocal effects caused by weak-diffusion processes in the case of resonant interaction of quasi-harmonic waves of small but finite amplitudes with flows of various physical nature in the case of an arbitrary relation between the nonlinearity and diffusion.We analyze the interaction of internal gravity waves with plane-parallel stratified shear flows in the nonlinearly-dissipative critical layer (CL) formed in the vicinity of the resonance level where the flow velocity is equal to the phase velocity of the wave. It is shown that the combined effect of the radiation force in the inner region of the CL and vorticity diffusion to the outer region results in the formation of a flow in which the asymptotic values of average vorticity at different sides of the CL are constant but different. If the criterion of the linear dynamic stability is satisfied (the Richardson number Ri>1/4), the resulting vorticity steps are comparable to the unperturbed vorticity. As a result, a wave reflected from the vorticity inhomogeneity in the CL is formed. As the amplitude of the incident wave increases, the average vorticity at the incidence side approaches the linear-stability threshold (Richardson number Ri > 1/4), and the reflection coefficient tends to -1.In the regime of nonlinear dissipative CL, we study the quasi-stationary asymptotic behavior of the flow formed by an internal gravity wave incident on a dynamically stable flow with velocity and density stratification, whose velocity at some level is equal to the phase velocity of the wave. It is shown that the vorticity diffusion results in the formation of a nonlocal transition region between the CL and the unperturbed flow, which we call the diffusive boundary layer (DBL). In this case, the CL is shifted toward the incident wave. We obtain a self-similar solution for the average fields, which is valid in the case of a constant vorticity step in the CL, and determine its parameters depending on the inner Reynolds number in the CL which describes the relation between the nonlinear and diffusive effects for the wave field in the resonance region. We determine the structure and temporal dynamics of the DBL formed by a rough surface streamlined by a stratified fluid whose velocity changes direction at some level.It is shown that in the case of the nonlinear resonance interaction of plasma electrons with a Langmuir wave, the electron diffusion in the velocity space leads to a significant nonlocal distortion of the electron distribution function outside the trapping region. We determine the distorted distribution function and calculate the rate of the nonlinear Landau damping of a finite-amplitude wave for an arbitrary ratio of the electron collision rate and the oscillation period of trapped electrons.     

Wave propagation analysis in nonlinear curved single-walled carbon nanotubes based on nonlocal elasticity theory

Theoretical predictions are presented for wave propagation in nonlinear curved single-walled carbon nanotubes (SWCNTs). Based on the nonlocal theory of elasticity, the computational model is established, combined with the effects of geometrical nonlinearity and imperfection. In order to use the wave analysis method on this topic, a linearization method is employed. Thus, the analytical expresses of the shear frequency and flexural frequency are obtained. The effects of the geometrical nonlinearity, the initial geometrical imperfection, temperature change and magnetic field on the flexural and shear wave frequencies are investigated. Numerical results indicate that the contribution of the higher-order small scale effect on the shear deformation and the rotary inertia can lead to a reduction in the frequencies compared with results reported in the published literature. The theoretical model derived in this study should be useful for characterizing the mechanical properties of carbon nanotubes and applications of nano-devices.     

Rayleigh wave propagation along the boundary of a non-Newtonian fluid-saturated porous medium

The Frenkel-Biot theory is used to study the reflection of elastic waves from the boundary of a non-Newtonian (Maxwell) fluid-saturated porous medium. The velocity and attenuation of a Rayleigh surface wave propagating along the boundary of the medium are determined. Two models of a fluid-saturated porous medium are used for calculation: with pore channels of a fixed diameter and with a lognormal distribution of pore channels in size. The results of calculations show that, when the fluid in the porous medium is characterized by a small Deborah number (i.e., exhibits non-Newtonian properties), the velocity of Rayleigh waves exhibits a considerable frequency dispersion. The results also suggest that, in principle, it is possible to estimate the Deborah number from the measured frequency dispersion of the Rayleigh wave velocity.     

Longitudinal waves in carbon nanotubes in the presence of transverse magnetic field and elastic medium

In the present paper, the coupling effect of transverse magnetic field and elastic medium on the longitudinal wave propagation along a carbon nanotube (CNT) is studied. Based on the nonlocal elasticity theory and Hamilton's principle, a unified nonlocal rod theory which takes into account the effects of small size scale, lateral inertia and radial deformation is proposed. The existing rod theories including the classic rod theory, the Rayleigh-Love theory and Rayleigh-Bishop theory for macro solids can be treated as the special cases of the present model. A two-parameter foundation model (Pasternak-type model) is used to represent the elastic medium. The influence of transverse magnetic field, Pasternak-type elastic medium and small size scale on the longitudinal wave propagation behavior of the CNT is investigated in detail. It is shown that the influences of lateral inertia and radial deformation cannot be neglected in analyzing the longitudinal wave propagation characteristics of the CNT. The results also show that the elastic medium and the transverse magnetic field will also affect the longitudinal wave dispersion behavior of the CNT significantly. The results obtained in this paper are helpful for understanding the mechanical behaviors of nanostructures embedded in an elastic medium.     

Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory

In this paper, the viscoelastic wave propagation in an embedded viscoelastic single-walled carbon nanotube (SWCNT) is studied based on the nonlocal strain gradient theory. The characteristic equation for the viscoelastic wave in SWCNTs is derived. The emphasis is placed on the influence of the tube diameter on the viscoelastic wave dispersion. A blocking diameter is observed, above which the wave could not propagate in SWCNTs. The results show that the blocking diameter is greatly dependent on the damping coefficient, the nonlocal and the strain gradient length scale parameters, as well as the Winkler modulus of the surrounding elastic medium. These findings may provide a prospective application of SWCNTs in nanodevices and nanocomposites.