Flexural Wave Dispersion Research Articles

Flexural wave dispersion in finitely pre-strained solid andhollow circular cylinders made of compressible materials

Flexural wave dispersion in finitely pre-strained solid andhollow circular cylinders made of compressible materials

Nonlocal continuum mechanics based ultrasonic flexural wave dispersion characteristics of a monolayer graphene embedded in polymer matrix

Wave propagation in graphene sheet embedded in elastic medium (polymer matrix) has been a topic of great interest in nanomechanics of graphene sheets, where the equivalent continuum models are widely used. In this manuscript, we examined this issue by incorporating the nonlocal theory into the classical plate model. The influence of the nonlocal scale effects has been investigated in detail. The results are qualitatively different from those obtained based on the local/classical plate theory and thus, are important for the development of monolayer graphene-based nanodevices. In the present work, the graphene sheet is modeled as an isotropic plate of one-atom thick. The chemical bonds are assumed to be formed between the graphene sheet and the elastic medium. The polymer matrix is described by a Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation of the surrounding elastic medium. When the shear effects are neglected, the model reduces to Winkler foundation model. The normal pressure or Winkler elastic foundation parameter is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs where the foundation modulus is assumed equivalent to stiffness of the springs. For this model, the nonlocal governing differential equations of motion are derived from the minimization of the total potential energy of the entire system. An ultrasonic type of flexural wave propagation model is also derived and the results of the wave dispersion analysis are shown for both local and nonlocal elasticity calculations. From this analysis we show that the elastic matrix highly affects the flexural wave mode and it rapidly increases the frequency band gap of flexural mode. The flexural wavenumbers obtained from nonlocal elasticity calculations are higher than the local elasticity calculations. The corresponding wave group speeds are smaller in nonlocal calculation as compared to local elasticity calculation. The effect of y-directional wavenumber (ηq) on the spectrum and dispersion relations of the graphene embedded in polymer matrix is also observed. We also show that the cut-off frequencies of flexural wave mode depends not only on the y-direction wavenumber but also on nonlocal scaling parameter (e0a). The effect of ηq and e0a on the cut-off frequency variation is also captured for the cases of with and without elastic matrix effect. For a given nanostructure, nonlocal small scale coefficient can be obtained by matching the results from molecular dynamics (MD) simulations and the nonlocal elasticity calculations. At that value of the nonlocal scale coefficient, the waves will propagate in the nanostructure at that cut-off frequency. In the present paper, different values of e0a are used. One can get the exact e0a for a given graphene sheet by matching the MD simulation results of graphene with the results presented in this article.

Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler–Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell’s relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range; 0–20THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field.

Mapping formation radial shear-wave velocity variation by a constrained inversion of borehole flexural-wave dispersion data

We have developed a novel constrained inversion method for estimating a radial shear-wave velocity profile away from the wellbore using dipole acoustic logging data and have analyzed the effect of the radial velocity changes on dipole-flexural-wave dispersion characteristics. The inversion of the dispersion data to estimate the radial changes is inherently a nonunique problem because changing the degree of variation or the radial size of the variation zone can produce similar wave-dispersion characteristics. Nonuniqueness can be solved by developing a constrained inversion method. This is done by constraining the high-frequency portion of the model dispersion curve with another curve calculated using the near-borehole velocity. The constraint condition is based on the physical principle that a high-frequency dipole wave has a shallow penetration depth and is therefore sensitive to the near-borehole shear-wave velocity. We have validated the result of the constrained inversion with synthetic data testing. Combining the new inversion method with four-component crossed-dipole anisotropy processing obtains shear radial profiles in fast and slow shear polarization directions. In a sandstone formation, the fast and slow shear-wave profiles show substantial differences caused by the near-borehole stress field, demonstrating the ability of the technique to obtain radial and azimuthal geomechanical property changes near the wellbore.

Strong nonlocalization induced by small scale parameter on terahertz flexural wave dispersion characteristics of a monolayer graphene

This paper presents the strong nonlocal scale effect on the flexural wave propagation in a monolayer graphene sheet. The graphene is modeled as an isotropic plate of one atom thick. Nonlocal governing equation of motion is derived and wave propagation analysis is performed using spectral analysis. The present analysis shows that the flexural wave dispersion in graphene obtained by local and nonlocal elasticity theories is quite different. The nonlocal elasticity calculation shows that the wavenumber escapes to infinite at certain frequency and the corresponding wave velocity tends to zero at that frequency indicating localization and stationary behavior. This behavior is captured in the spectrum and dispersion curves. The cut-off frequency of flexural wave not only depend on the axial wavenumber but also on the nonlocal scaling parameter. The effect of axial wavenumber on the wave behavior in graphene is also discussed in the present manuscript.

Flexural wave dispersion in multi-walled carbon nanotubes conveying fluids

Motivated by the great potential of carbon nanotubes for developing nanouidic devices, this paper presents a nonlocal elastic, Timoshenko multi-beam model with the second order of strain gradient taken into consideration and derives the corresponding dispersion relation of exural wave in multi-walled carbon nanotubes conveying uids. The study shows that the moving flow reduces the phase velocity of exural wave of the lowest branch in carbon nanotubes. The phase velocity of exural wave of the lowest branch decreases with an increase of flow velocity. However, the effects of flow velocity on the other branches of the wave dispersion are not obvious. The effect of microstructure characterized by nonlocal elasticity on the dispersion of exural wave becomes more and more remarkable with an increase in wave number.

Gradient elasticity and flexural wave dispersion in carbon nanotubes

Higher-order elasticity theories have recently been used to predict the dispersion characteristics of flexural waves in carbon nanotubes (CNTs). In particular, nonlocal elasticity and gradient elasticity (with unstable strain gradients) have been employed within the framework of classical Euler-Bernoulli or improved Timoshenko beam theory to capture the dynamical behavior of CNTs. Qualitative agreement with the predictions of related molecular-dynamics (MD) simulations was observed, whereas the MD results departed significantly from those obtained with classical elasticity calculations. The present contribution aims to alert that the aforementioned higher-order models may yield questionable results for the higher wave numbers. As an alternative, gradient elasticity (with stable strain gradients), by also incorporating inertia gradients for dynamical applications, is used in combination with both Euler-Bernoulli and Timoshenko beam theories and shown to describe flexural wave dispersion in CNTs realistically for the small-to-medium range of wave numbers, i.e., the range for which MD results are available.

Nonlocal elastic beam models for flexural wave propagation in double-walled carbon nanotubes

Flexural wave in a double-walled carbon nanotube is studied. The scale effect of the carbon nanotube on the wave dispersion is investigated through the nonlocal elastic beam theories. The flexural wave dispersion predicted by the nonlocal elastic Timoshenko beam theory has a good agreement with that by molecular dynamics simulations in a wide frequency range up to the terahertz region. The results show that only the nonlocal elastic Timoshenko beam model can predict the small-scale effect on the dispersion of flexural wave in double-walled carbon nanotube when the wave number is large. Moreover, an estimation of the scale coefficient e0 for the nonlocal elastic double Timoshenko beam model is suggested by validations from the molecular dynamics simulations. The noncoaxial flexural vibration of the double-walled carbon nanotube can be observed from molecular dynamics simulations at high frequency range. The van der Waals interaction is found to have little effect on the noncoaxial flexural vibration of the carbon nanotube, and the nonlocal elastic Timoshenko beam theory is found to be inapplicable in modeling the noncoaxial wave propagation in carbon nanotubes.

Flexural wave dispersion in orthotropic plates with heavy fluid loading

Orthotropic plates support flexural waves with wavenumbers that depend on their angle of propagation. The present work investigates the effect of fluid loading on this angular dependence, and finds that the effect is relatively small for typical composite plate materials in contact with water. This finding results from an analytical model of the fluid-loaded plate, in which the plate is modeled by classical laminated plate theory and the fluid is modeled as an ideal acoustic fluid. The resulting dispersion relation is a tenth-order polynomial in the flexural wavenumber. Direct numerical solution, as well as analysis at frequencies below coincidence, reveals that the angular dependence of wavenumber is magnified but not significantly distorted by the addition of fluid loading.

Floating ladder track response to a steadily moving load

AbstractFloating ladder rail tracks, which can significantly reduce traffic vibration and noise, have already been installed at several railway sites in and around Tokyo. The steel rails are fixed onto successive ladder‐like sections with two parallel longitudinal reinforced concrete sleepers, which are then mounted upon discrete resilient supports on a concrete bed. A simple mathematical model in which a continuous horizontal Bernoulli–Euler beam on periodic discrete elastic supports represents each floating ‘combined rail’ (i.e. rail and longitudinal sleeper), used earlier to discuss the low‐frequency free vibrations in the system, is again adopted to investigate the response due to a steadily moving load. We demonstrate that Fourier transforms can be invoked to obtain the forced deflexion, which depends upon the load speed. A contribution from the periodic supports determines the steady component of the deflexion moving with the load, and the other contributions from the supports produce oscillations. As is the case for a load moving over a beam or plate with continuous support, the response may be characterized using the free flexural wave dispersion relation—although there is now a countably infinite number of dispersion curves, corresponding to the existence of propagation bands in the periodic structure. The lowest wavenumber local minimum in the phase speed (coincident with the group speed) defines the primary critical load speed of most interest, at which the magnitude of the steady component accompanying the moving load becomes large. This primary critical load speed depends upon the relative elasticity of the discrete supports, which must not be too low if the floating ladder track is to be safe for fast rail systems. Copyright © 2007 John Wiley & Sons, Ltd.

Acoustoelastic effects on mode waves in a fluid-filled pressurized borehole in triaxially stressed formations

Based on the nonlinear theory of acoustoelasticity, considering the triaxial terrestrial stress, the fluid static pressure in the borehole and the fluid nonlinear effect jointly, the dispersion curves of the monopole Stoneley wave and dipole flexural wave propagating along the borehole axis in a homogeneous isotropic formation are investigated by using the perturbation method. The relation of the sensitivity coefficient and the velocity-stress coefficient to frequency are also analyzed. The results show that variations of the phase velocity dispersion curve are mainly affected by three sensitivity coefficients related to third-order elastic constant. The borehole stress concentration causes a split of the flexural waves and an intersection of the dispersion curves of the flexural waves polarized in directions parallel and normal to the uniaxial horizontal stress direction. The stress-induced formation anisotropy is only dependent on the horizontal deviatoric terrestrial stress and independent of the horizontal mean terrestrial stress, the superimposed stress and the fluid static pressure. The horizontal terrestrial stress ratio ranging from 0 to 1 reduces the stress-induced formation anisotropy. This makes the intersection of flexural wave dispersion curves not distinguishable. The effect of the fluid nonlinearity on the dispersion curve of the mode wave is small and can be ignored.

Flexural wave propagation in single-walled carbon nanotubes

The paper presents the study on the flexural wave propagation in a single-walled carbon nanotube through the use of the continuum mechanics and the molecular dynamics simulation based on the Terroff-Brenner potential. The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the model of a traditional Timoshenko beam, but also by the nonlocal elasticity characterizing the microstructure of carbon nanotube in a wide frequency range up to THz. For this purpose, the paper starts with the dynamic equation of a generalized Timoshenko beam made of the nonlocal elastic material, and then gives the dispersion relations of the flexural wave in the nonlocal elastic Timoshenko beam, the traditional Timoshenko beam and the Euler beam, respectively. Afterwards, it presents the molecular dynamics simulations for the flexural wave propagation in an armchair (5,5) and an armchair (10,10) single-walled carbon nanotubes for a wide range of wave numbers. The simulation results show that the Euler beam holds for describing the dispersion of flexural waves in the two single-walled carbon nanotubes only when the wave number is small. The Timoshenko beam provides a better prediction for the dispersion of flexural waves in the two single-walled carbon nanotubes when the wave number becomes a little bit large. Only the nonlocal elastic Timoshenko beam is able to predict the decrease of phase velocity when the wave number is so large that the microstructure of carbon nanotubes has a significant influence on the flexural wave dispersion.

Stoneley and flexural modes in pressurized boreholes

The propagation of Stoneley and flexural waves in a fluid‐filled borehole is adequately described by the linear equations of elasticity. However, when the borehole fluid is pressurized either due to the hydrostatic head at a given depth or with the aid of packers at the wellhead, both the fluid and the surrounding formation are subjected to biasing stresses. Under this situation, wave propagation along the borehole is described by the equations of motion for small dynamic fields superposed on a bias. The resulting formulation allows us to study the influence of a change in the fluid pressure on the Stoneley and flexural mode dispersion curves. Since the biasing stresses in the surrounding formation exhibit a radial decay away from the borehole, it is expedient to employ a perturbation technique to calculate changes in the borehole Stoneley and flexural wave dispersion curves as a function of hydrostatic pressure change in the fluid. A key advantage of this perturbation technique is that it separates contributions of the acoustoelastic effect due to the borehole fluid and that due to the formation. Insofar as the fluid nonlinear properties at a given pressure and temperature are known, the model provides a procedure for estimating the acoustoelastic coefficient of the formation for the borehole Stoneley and flexural wave velocities for a given change in the fluid pressure. The formation acoustoelastic coefficient can be expressed as a fractional change in the acoustic wave velocity caused by a unit change in the borehole pressure. Computational results show that acoustoelastic coefficients for the Stoneley and flexural waves are larger for formations with higher degree of nonlinearity which is typically associated with poorly consolidated rocks.

Effect of a thin low-velocity layer underneath a floating plate on flexural waves

Time-domain pulsed broadband laboratory experimental results are presented demonstrating that a thin low velocity fluid layer (thickness <H/10) at the bottom of a floating plate (thickness H) drastically decreases the group velocity of flexural waves on the right side of the dispersion curve maximum Umax. The low Scholte wave velocity of the layer affects the Scholte wave branch dispersion curve and causes it to deviate towards a lower value. Hunkins’ seismic studies of sea-ice [J. Geophys. Res. 65(10) (1960)] revealed two discrepancies between measured and calculated flexural wave group velocity dispersion: (a) sea-ice thickness determined from flexural wave dispersion is generally lower than the actual thickness, and (b) flexural wave components on the right of Umax have considerable lower velocity than predicted. Ultrasonic modeling studies [J. R. Chamuel and G. H. Brooke, J. Acoust. Soc. Am. Suppl. 1 79, S57 (1986)] provided a possible explanation for the first discrepancy namely that the presence of a large number of small cracks in a floating plate reduces U and shifts the corresponding dispersion curves towards an ‘‘effective’’ thinner plate. The current paper provides an explanation for the second discrepancy in Hunkins’ results which cannot be compensated for by changing ‘‘H.’’ Bogorodskii [Sov. Phys. Acoust. 22, 158–159 (1976)] showed the existence of a ‘‘clear-ice’’ layer underneath the ice, during the rapid growth of the ice cover, which has a compressional wave velocity of about 1050 m/s substantially less than the velocity of sound in water. [Work supported by ONR.]

Arctic low‐frequency propagation loss from shallow cracks in ice plates

At the 111th Meeting of the Acoustical Society of America, the authors reported on the effects of shallow cracks in floating ice plates on lowering the flexural wave dispersion curve creating discrepancies in plate thickness determination. In this paper, the effects of shallow cracks in sea‐ice plates on low‐frequency propagation loss in the water waveguide underneath the ice are demonstrated using laboratory ultrasonic models. Air‐filled shallow cracks extending near vertically in the ice plate present sharp impedance discontinuities to the low‐frequency water acoustic waves coupled to the ice plates. Upward refraction in arctic water increases the interaction of the acoustic waves with the cracks causing distributed backscatter along the water waveguide. The importance of the cracks increases as the effective water waveguide depth is decreased. Experimental laboratory results are presented for shallow water waveguide conditions. The new findings may explain the very high backscatter levels and high propagation losses of low frequencies observed in the Arctic. [Work supported by DREP.]

Low‐frequency transmission losses and effective flexural rigidity of cracked ice cover

Upward refraction in the artic causes substantial interactions of underwater acoustic waves with the ice cover. Although several research papers focused on the effects of ice cover roughness on transmission losses, apparently no studies have been reported on the effects of cracks in the ice cover on reducing the flexural rigidity of the ice plate in relation to leaky Lamb waves and energy exchange between the water and the ice. Coupling between the acoustic waves in the water and the elastic waves in the ice plate depends on the relative densities and phase velocities of the waves. Depending on the number, size, and spatial distribution of the cracks, the flexural wave velocity of a low‐frequency component may be higher or lower than the water compressional velocity. The steep sloped region of the flexural wave dispersion curve makes the low‐frequency transmission characteristics very sensitive to the presence of cracks which reduce the flexural rigidity. Laboratory ultrasonic models are used to provide physical insights into a potentially important low‐frequency transmission loss mechanism caused by the effective reduced flexural rigidity of the cracked ice cover. Thin plates with machined multiple shallow cracks are utilized. The crack spacing is about half a wavelength and the crack depth is less than 1/6 the plate thickness. [Work supported by DREP.]

Effect of roughness on subsonic flexural waves in a thin plate overlying a compressible fluid

The interaction between a thin plate (kH≲1, k being the wavenumber and H the plate thickness) and a compressible fluid half-space through a slightly rough boundary may be represented by the standard thin plate equations plus a smoothed boundary condition for the displacement of the fluid normal to the plate. This replaces the rough boundary by a fictitious smooth plane at which the relation ∂φs/∂z =ηφ1 must be obeyed, where φs is the perturbation of the smooth boundary solution due to roughness, φ1 is the total acoustic potential in the fluid at the smooth plane z=0, and η is a constant which, for isotropic roughness or for corrugations, is defined by two scatter parameters. This condition is isomorphic to those used for other types of rough interfaces [I. Tolstoy, J. Acoust. Soc. Am. 75, 1–22 (1984)]—the main difference residing in the numerical values of the parameters. The most important effects are connected with the flexural modes of the plate. Here, the roughness causes energy losses into incoherent acoustic scatter and introduces an important level of attenuation. Harmonic point source solutions are obtained for several types of interface roughness, showing that, for moderate and large ranges r (kr≳102), this effect introduces an effective cutoff frequency for κh≲1, κ being the horizontal wavenumber of the flexural waves and h the spacing between the bosses (or corrugations of the roughness model). The effect of the roughness on the flexural wave dispersion is of the order of several percent, and should be measurable for plates of relatively low flexural rigidity.

Flexural Wave Studies on the Basis of Single-Sensor Recordings

Abstract The method described here provides a means of measuring flexural-wave dispersion and attenuation appropriate to a selected area, on the basis of single-sensor recordings made remote from the area of interest. The measured dispersion may then be inverted in the conventional manner to obtain thickness and elastic properties of the ice within the selected area. The procedure eliminates the need for equipment within the study area and minimizes the need for personnel within the area. As only a single sensor is necessary, the requirements for recording equipment and sensor cabling are also minimized. The method is suited to any study of flexural-wave dispersion from artificial sources and is ideally suited to the study of areas where personnel and equipment safety are a major concern.

Flexural Wave Studies on the Basis of Single-Sensor Recordings

AbstractThe method described here provides a means of measuring flexural-wave dispersion and attenuation appropriate to a selected area, on the basis of single-sensor recordings made remote from the area of interest. The measured dispersion may then be inverted in the conventional manner to obtain thickness and elastic properties of the ice within the selected area. The procedure eliminates the need for equipment within the study area and minimizes the need for personnel within the area. As only a single sensor is necessary, the requirements for recording equipment and sensor cabling are also minimized. The method is suited to any study of flexural-wave dispersion from artificial sources and is ideally suited to the study of areas where personnel and equipment safety are a major concern.

Seismic studies of sea ice

During the International Geophysical Year, elastic wave propagation in sea ice was studied near Drifting Station Alpha in the central Arctic Ocean. Velocities of longitudinal and transverse waves in ice showed a marked seasonal change which is largely attributable to variations in ice temperature. With these velocity data, plus additional data on density, the calculation of various elastic constants of sea ice throughout the year can be made. Flexural wave dispersion was investigated for different ranges, charge sizes, and ice parameters. Experimental results are in general agreement with theory. Thickness, as determined from the dispersion of flexural waves and from air-coupled flexural waves, is characteristically lower than that found by direct measurement. Both longitudinal and flexural waves crossing leads suffer severe attenuation. Transmission across leads shortened the duration of the normal air-coupled flexural wave train.