Investigation of the effect of nonlocal scale on ultrasonic wave dispersion characteristics of a monolayer graphene

In this article, an ultrasonic wave propagation in graphene sheet is studied using nonlocal elasticity theory incorporating small scale effects. The graphene sheet is modeled as an isotropic plate of one-atom thick. 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 wave propagation model is also derived for the graphene sheet. The nonlocal scale parameter introduces certain band gap region in in-plane and flexural wave modes where no wave propagation occurs. This is manifested in the wavenumber plots as the region where the wavenumber tends to infinite or wave speed tends to zero. The frequency at which this phenomenon occurs is called the escape frequency. The explicit expressions for cutoff frequencies and escape frequencies are derived. The escape frequencies are mainly introduced because of the nonlocal elasticity. Obviously these frequencies are function of nonlocal scaling parameter. It has also been obtained that these frequencies are independent of y-directional wavenumber. It means that for any type of nanostructure, the escape frequencies are purely a function of nonlocal scaling parameter only. It is also independent of the geometry of the structure. It has been found that the cutoff frequencies are function of nonlocal scaling parameter (e(0)a) and the y-directional wavenumber (k(y)). 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 e(o)a are used. One can get the exact e(0)a for a given graphene sheet by matching the MD simulation results of graphene with the results presented in this paper. (C) 2010 Elsevier B.V. All rights reserved.

Analytical solution of the laminar mean flow wave equation in a lined or unlined two-dimensional rectangular duct

The prediction of the sound attenuation in lined ducts with sheared mean flow has been a topic of research for many years. This involves solving the sheared mean flow wave equation, satisfying the relevant boundary condition. As far as the authors' knowledge goes, this has always been done using numerical techniques. Here, an analytical solution is presented for the wave propagation in two-dimensional rectangular lined ducts with laminar mean flow. The effect of laminar mean flow is studied for both the downstream and the upstream wave propagation. The attenuation values predicted for the laminar mean flow case are compared with those for the case of uniform mean flow. Analytical expressions are derived for the transfer matrices.

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. (C) 2010 Elsevier B.V. All rights reserved.

Transverse plane wave analysis of short elliptical chamber mufflers: An analytical approach

Short elliptical chamber mufflers are used often in the modern day automotive exhaust systems. The acoustic analysis of such short chamber mufflers is facilitated by considering a transverse plane wave propagation model along the major axis up to the low frequency limit. The one dimensional differential equation governing the transverse plane wave propagation in such short chambers is solved using the segmentation approaches which are inherently numerical schemes, wherein the transfer matrix relating the upstream state variables to the downstream variables is obtained. Analytical solution of the transverse plane wave model used to analyze such short chambers has not been reported in the literature so far. This present work is thus an attempt to fill up this lacuna, whereby Frobenius solution of the differential equation governing the transverse plane wave propagation is obtained. By taking a sufficient number of terms of the infinite series, an approximate analytical solution so obtained shows good convergence up to about 1300 Hz and also covers most of the range of muffler dimensions used in practice. The transmission loss (TL) performance of the muffler configurations computed by this analytical approach agrees excellently with that computed by the Matrizant approach used earlier by the authors, thereby offering a faster and more elegant alternate method to analyze short elliptical muffler configurations. (C) 2010 Elsevier Ltd. All rights reserved.

A note on the effect of wall compliance on lowest?order mode propagation in fluid?filled/submerged impedance tubes

Wave propagation in fluid?filled/submerged tubes is of interest in large HVAC ducts, and also in understanding and interpreting the experimental results obtained from fluid?filled impedance tubes. Based on the closed form analytical solution of the coupled wave equations, an eigenequation, which is the determinant of an 8×8 matrix, is derived and solved to obtain the axial wave number of the lowest?order longitudinal modes for cylindrical ducts of various diameter and wall thickness. The dispersion behavior of the wave motion is analyzed. It is observed that the larger the diameter of the duct and/or the smaller its wall thickness, the more flexible the impedance tube leading to more coupling between the waves in the elastic media. Also, it is shown that the wave motion in water?filled ducts submerged in water exhibits anomalous dispersion behavior. The axial attenuation characteristics of plane waves along water?filled tubes submerged in water or air are also investigated. Finally, investigations on the sound intensity level difference characteristics of the wall of the air?filled tubes are reported.

Plane wave analysis of conical and exponential pipes with incompressible mean flow

The general equation for one-dimensional wave propagation at low flow Mach numbers (M less-than-or-equals, slant0·2) is derived and is solved analytically for conical and exponential shapes. The transfer matrices are derived and shown to be self-consistent. Comparison is also made with the relevant data available in the literature. The transmission loss behaviour of conical and exponential pipes, and mufflers involving these shapes, are studied. Analytical expressions of the same are given for the case of a stationary medium. The mufflers involving conical and exponential pipes are shown to be inferior to simple expansion chambers (of similar dimensions) at higher frequencies from the point of view of noise abatement, as was observed earlier experimentally.

Ultrasonic wave characteristics of a monolayer graphene on silicon substrate

The present work deals with an ultrasonic type of wave propagation characteristics of monolayer graphene on silicon (Si) substrate. An atomistic model of a hybrid lattice involving a hexagonal lattice of graphene and surface atoms of diamond lattice of Si is developed to identify the carbon-silicon bond stiffness. Properties of this hybrid lattice model is then mapped into a nonlocal continuum framework. Equivalent force constant due to Si substrate is obtained by minimizing the total potential energy of the system. For this equilibrium configuration, the nonlocal governing equations are derived to analyze the ultrasonic wave dispersion based on spectral analysis. From the present analysis we show that the silicon substrate affects only the flexural wave mode. The frequency band gap of flexural mode is also significantly affected by this substrate. The results also show that, the silicon substrate adds cushioning effect to the graphene and it makes the graphene more stable. The analysis also show that the frequency bang gap relations of in-plane (longitudinal and lateral) and out-of-plane (flexural) wave modes depends not only on the y-direction wavenumber but also on nonlocal scaling parameter. In the nonlocal analysis, at higher values of the y-directional wavenumber, a decrease in the frequency band gap is observed for all the three fundamental wave modes in the graphene-silicon system. The atoms movement in the graphene due to the wave propagation are also captured for all the tree fundamental wave modes. The results presented in this work are qualitatively different from those obtained based on the local analysis and thus, are important for the development of graphene based nanodevices such as strain sensor, mass and pressure sensors, atomic dust detectors and enhancer of surface image resolution that make use of the ultrasonic wave dispersion properties of graphene. (C) 2011 Elsevier Ltd. All rights reserved.

Finite element simulation of BAW propagation in inhomogeneous plate due to piezoelectric actuation

A set of finite elements (FEs) is formulated to analyze wave propagation through inhomogeneous material when subjected to mechanical, thermal loading or piezo-electric actuation. Elastic, thermal and electrical properties of the materials axe allowed to vary in length and thickness direction. The elements can act both as sensors and actuators. These elements are used to model wave propagation in functionally graded materials (FGM) and the effect of inhomogeneity in the wave is demonstrated. Further, a surface acoustic wave (SAW) device is modeled and wave propagation due to piezo-electric actuation from interdigital transducers (IDTs) is studied.

Ultrasonic Lamb wave based monitoring of corrosion type of damage in plate using a circular array of piezoelectric transducers

In this paper we propose a concept and report experimental results based on a circular array of Piezoelectric Wafer Active Sensors (PWASs) for rapid localization and parametric identification of corrosion type damage in metallic plates. Implementation of this circular array of PWASs combines the use of ultrasonic Lamb wave propagation technique and an algorithm based on symmetry breaking in the signal pattern to locate and monitor the growth of a corrosion pit on a metallic plate. Wavelet time-frequency maps of the sensor signals are employed to obtain an insight regarding the effect of corrosion growth on the Lamb wave transmission in time-frequency scale. We present here a method to eliminate the time scale, which helps in identifying easily the signature of damage in the measured signals. The proposed method becomes useful in determining the approximate location of the damage with respect to the location of three neighboring sensors in the circular array. A cumulative damage index is computed from the wavelet coefficients for varying damage sizes and the results appear promising. Damage index is plotted against the damage parameters for frequency sweep of the excitation signal (a windowed sine signal). Results of corrosion damage are compared with circular holes of various sizes to demonstrate the applicability of present method to different types of damage. (C) 2011 Elsevier Ltd. All rights reserved.

A Revisit to Capture the Entire Behavior of Ultrasonic Wave Dispersion Characteristics of Single Walled Carbon Nanotubes Based on Nonlocal Elasticity Theory and Flugge Shell Model

In this paper, an ultrasonic wave propagation analysis in single-walled carbon nanotube (SWCNT) is re-studied using nonlocal elasticity theory, to capture the whole behaviour. The SWCNT is modeled using Flugge's shell theory, with the wall having axial, circumferential and radial degrees of freedom and also including small scale effects. Nonlocal governing equations for this system are derived and wave propagation analysis is also carried out. The revisited nonlocal elasticity calculation shows that the wavenumber tends to infinite at certain frequencies and the corresponding wave velocity tends to zero at those frequencies indicating localization and stationary behavior. This frequency is termed as escape frequency. This behavior is observed only for axial and radial waves in SWCNT. It has been shown that the circumferential waves will propagate dispersively at higher frequencies in nonlocality. The magnitudes of wave velocities of circumferential waves are smaller in nonlocal elasticity as compared to local elasticity. We also show that the explicit expressions of cut-off frequency depend on the nonlocal scaling parameter and the axial wavenumber. The effect of axial wavenumber on the ultrasonic wave behavior in SWCNTs is also discussed. The present results are compared with the corresponding results (for first mode) obtained from ab initio and 3-D elastodynamic continuum models. The acoustic phonon dispersion relation predicted by the present model is in good agreement with that obtained from literature. The results are new and can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Ultrasonic Wave Dispersion Characteristics of Fluid-Filled Single-Walled Carbon Nanotubes Based on Nonclassical Shell Model

In this paper, ultrasonic wave propagation analysis in fluid filled single-walled carbon nanotube (SWCNT) is studied using nonlocal elasticity theory. The SWCNT is modeled using Flugge's shell theory, with the wall having axial, circumferential and radial degrees of freedom and also including small scale effects. The fluid inside the SWCNT is assumed as water. Nonlocal governing equations for this system are derived and wave propagation analysis is also carried out. The presence of fluid in SWCNT alters the ultrasonic wave dispersion behavior. The wavenumber and wave velocity are smaller in presence of fluid as compared to the empty SWCNT. The nonlocal elasticity calculation shows that the wavenumber tends to reach the continuum limit at certain frequencies and the corresponding wave velocity tends to zero at those frequencies indicating localization and stationary behavior. It has been shown that the circumferential. waves will propagate non-dispersively at higher frequencies in nonlocality. The magnitudes of wave velocities of circumferential waves are smaller in nonlocal elasticity as compared to local elasticity. We also show that the cut-off frequency depend on the nonlocal scaling parameter and also on the density of the fluid inside the SWCNT, and the axial wavenumber, as the fluid becomes denser the cut-off frequency decreases. The effect of axial wavenumber on the ultrasonic wave behavior in SWCNTS filled with water is also discussed.

Guided-wave based structural health monitoring of built-up composite structures using spectral finite element method

The paper discusses basically a wave propagation based method for identifying the damage due to skin-stiffener debonding in a stiffened structure. First, a spectral finite element model (SFEM) is developed for modeling wave propagation in general built-up structures, using the concept of assembling 2D spectral plate elements and the model is then used in modeling wave propagation in a skin-stiffener type structure. The damage force indicator (DFI) technique, which is derived from the dynamic stiffness matrix of the healthy stiffened structure (obtained from the SFEM model) along with the nodal displacements of the debonded stiffened structure (obtained from 2D finite element model), is used to identify the damage due to the presence of debond in a stiffened structure.

Ultrasonic Guided Wave Characterization and Damage Detection in Foam-core Sandwich Panel using PWAS and LDV

Lamb wave type guided wave propagation in foam core sandwich structures and detectability of damages using spectral analysis method are reported in this paper. An experimental study supported by theoretical evaluation of the guided wave characteristics is presented here that shows the applicability of Lamb wave type guided ultrasonic wave for detection of damage in foam core sandwich structures. Sandwich beam specimens were fabricated with 10 mm thick foam core and 0.3 mm thick aluminum face sheets. Thin piezoelectric patch actuators and sensors are used to excite and sense guided wave. Group velocity dispersion curves and frequency response of sensed signal are obtained experimentally. The nature of damping present in the sandwich panel is monitored by measuring the sensor signal amplitude at various different distances measured from the center of the linear phased array. Delaminations of increasing width are created and detected experimentally by pitch-catch interrogation with guided waves and wavelet transform of the sensed signal. Signal amplitudes are analyzed for various different sizes of damages to differentiate the damage size/severity. A sandwich panel is also fabricated with a planer dimension of 600 mm x 400 mm. Release film delamination is introduced during fabrication. Non-contact Laser Doppler Vibrometer (LDV) is used to scan the panel while exciting with a surface bonded piezoelectric actuator. Presence of damage is confirmed by the reflected wave fringe pattern obtained from the LDV scan. With this approach it is possible to locate and monitor the damages by tracking the wave packets scattered from the damages.

Guided wave based detection of damage in honeycomb core sandwich structures

We report on the Lamb wave type guided wave propagation in honeycomb core sandwich structures. An experimental study supported by theoretical evaluation of the guided wave characteristics is presented that proves the potential of Lamb wave type guided wave for detection of damage in sandwich structures. A sandwich panel is fabricated with planar dimension of 600 mm x 600 mm, having a core thickness of 7 mm, cell size of 5 mm and 0.1 mm thick aluminum face sheets. Thin piezoelectric patch actuators and sensors are used to excite and sense a frequency band limited guided wave with a central frequency. A linear phased array of piezoelectric patch actuators is used to achieve higher signal strength and directivity. Group velocity dispersion curves and corresponding frequency response of sensed signal are obtained experimentally. Linearity between the excitation signal amplitude and the corresponding sensed signal amplitude is found for certain range of parameters. The nature of damping present in the sandwich panel is monitored by measuring the sensor signal amplitude at various different distances measured from the center of the linear phased array. Indentation and low velocity impact induced damages of increasing diameter covering several honeycomb cells are created. Crushing of honeycomb core with rupture of face sheet is observed while introducing the damage. The damages are then detected experimentally by pitch-catch interrogation with guided waves and wavelet transform of the sensed signal. Signal amplitudes are analyzed for various different sizes of damages to differentiate the damage size/severity. Monotonic changes in the sensor signal amplitude due to increase in the damage size has been established successfully. With this approach it is possible to locate and monitor the damages with the help of phased array and by tracking the wave packets scattered from the damages. (C) 2012 Elsevier Ltd. All rights reserved.

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 (eta(q)) 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 (e(0)a). The effect of eta(q) and e(0)a 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 e(0)a are used. One can get the exact e(0)a for a given graphene sheet by matching the MD simulation results of graphene with the results presented in this article. (c) 2012 Elsevier Ltd. All rights reserved.