Bulk Wave Propagation Research Articles

Using the Partial Wave Method for Wave Structure Calculation and the Conceptual Interpretation of Elastodynamic Guided Waves

The partial-wave method takes advantage of the Christoffel equation’s generality to represent waves within a waveguide. More specifically, the partial-wave method is well known for its usefulness when calculating dispersion curves for multilayered and/or anisotropic plates. That is, it is a vital component of the transfer-matrix method and the global-matrix method, which are used for dispersion curve calculation. The literature suggests that the method is also exceptionally useful for conceptual interpretation, but gives very few examples or instruction on how this can be done. In this paper, we expand on this topic of conceptual interpretation by addressing Rayleigh waves, Stoneley waves, shear horizontal waves, and Lamb waves. We demonstrate that all of these guided waves can be described using the partial-wave method, which establishes a common foundation on which many elastodynamic guided waves can be compared, translated, and interpreted. For Lamb waves specifically, we identify the characteristics of guided wave modes that have not been formally discussed in the literature. Additionally, we use what is demonstrated in the body of the paper to investigate the leaky characteristics of Lamb waves, which eventually leads to finding a correlation between oblique bulk wave propagation in the waveguide and the transmission amplitude ratios found in the literature.

Analysis of the transient surface wave propagation in soft-solid elastic plates.

In dynamic elastography, the goal is to estimate the Young's modulus from audio-frequency wave propagation in soft-tissues. Within this frequency range, the shear wavelength is centimeter-sized while the compressional wavelength is meter-sized. Thus, the experimental data are usually collected in the near-field of the source. Near-field effects have been widely studied for bulk wave propagation. However, the near- and transient-fields of surface and guided waves have received less attention. In this work, the transient surface displacement field in soft-solid elastic plates in vacuum is analyzed. Due to the high Poisson's ratio, mode conversion has special characteristics in soft-solids. They are analyzed through this work where it is shown that the transient-field over the surface can be interpreted by tracing a few reflections. The authors show the existence of a critical distance needed for the formation of Rayleigh-Lamb modes. Below this distance, only direct surface waves propagate without contribution from reflected waves. Thus, the dispersion curve differs from that predicted by Rayleigh-Lamb modes. Instead, the authors propose a model based on the interference of surface waves, which agree with the experimental data. In addition, the conditions needed in order to retrieve the shear wave phase velocity from the surface field are given.

A Comparative Study on the Elastic Characteristics of an Aluminum Thin-Film Using Laser Optical Measurement Techniques

The increase of a surface area-to-volume ratio with the reduction of material dimensions significantly alters the characteristics of materials from their macroscopic status. Therefore, efforts have been made to establish evaluation techniques for nanoscale films. While contact mechanics-based techniques are conventionally available, non-contact and nondestructive methods would be preferable in case damages left on a sample after testing are not desirable, or an in situ assessment is required. In the present study, the Young’s modulus of an aluminum thin-film was evaluated using two different laser optical measurement techniques. First, microscale beam testing has been performed so that the resonant frequency change of a microfabricated cantilever beam induced by coating of a 153 nm thick aluminum layer on its top surface can be detected using a laser interferometer in order to evaluate the mechanical property through modal analysis using the finite element method. Second, picosecond ultrasonics were employed for cross-verification so that the mechanical characteristics can be evaluated through the investigation of the longitudinal bulk wave propagation behavior. Results show that the Young’s moduli from both measurements agree well with each other within 3.3% error, proving that the proposed techniques are highly effective for the study of nanoscale films.

On the surface motion characteristics in a single layered half-space excited by bulk waves

In this paper, a study of bulk wave propagation in a layered half-space and the corresponding surface motion characteristics is reported. Two different types of layer material, namely elastic and viscoelastic, are considered. In both cases, the half-space is modeled as an elastic medium. The displacement functions are obtained for the free surface material point motion. In general, the free surface material point motion in a layered half-space is elliptical with certain aspect ratio and orientation. The aspect ratio and orientation depend upon the incident wave and medium properties. The effects of various parameters of the viscoelastic media on the material point motion are studied. The dependence of phase velocity and quality factor on frequency is derived using the creep function of the viscoelastic medium. The effects of various controllable parameters on the characteristics of surface displacement and material point motion are presented and discussed.

Iterative solution to bulk wave propagation in polycrystalline materials.

This article reevaluates two foundational models for bulk ultrasonic wave propagation in polycrystals. A decoupling of real and imaginary parts of the effective wave number permits a simple iterative method to obtain longitudinal and shear wave attenuation constants and phase velocity relations. The zeroth-order solution is that of Weaver [J. Mech. Phys. Solids 38, 55-86 (1990)]. Continued iteration converges to the unified theory solution of Stanke and Kino [J. Acoust. Soc. Am. 75, 665-681 (1984)]. The converged solution is valid for all frequencies. The iterative method mitigates the need to solve a nonlinear, complex-valued system of equations, which makes the models more robust and accessible to researchers. An analysis of the variation between the solutions is conducted and is shown to be proportional to the degree of inhomogeneity in the polycrystal.

3-D phononic crystals with ultra-wide band gaps

In this paper gradient based topology optimization (TO) is used to discover 3-D phononic structures that exhibit ultra-wide normalized all-angle all-mode band gaps. The challenging computational task of repeated 3-D phononic band-structure evaluations is accomplished by a combination of a fast mixed variational eigenvalue solver and distributed Graphic Processing Unit (GPU) parallel computations. The TO algorithm utilizes the material distribution-based approach and a gradient-based optimizer. The design sensitivity for the mixed variational eigenvalue problem is derived using the adjoint method and is implemented through highly efficient vectorization techniques. We present optimized results for two-material simple cubic (SC), body centered cubic (BCC), and face centered cubic (FCC) crystal structures and show that in each of these cases different initial designs converge to single inclusion network topologies within their corresponding primitive cells. The optimized results show that large phononic stop bands for bulk wave propagation can be achieved at lower than close packed spherical configurations leading to lighter unit cells. For tungsten carbide - epoxy crystals we identify all angle all mode normalized stop bands exceeding 100%, which is larger than what is possible with only spherical inclusions.

Folding and faulting of an elastic continuum

Folding is a process in which bending is localized at sharp edges separated by almost undeformed elements. This process is rarely encountered in Nature, although some exceptions can be found in unusual layered rock formations (called ‘chevrons’) and seashell patterns (for instance Lopha cristagalli). In mechanics, the bending of a three-dimensional elastic solid is common (for example, in bulk wave propagation), but folding is usually not achieved. In this article, the route leading to folding is shown for an elastic solid obeying the couple-stress theory with an extreme anisotropy. This result is obtained with a perturbation technique, which involves the derivation of new two-dimensional Green's functions for applied concentrated force and moment. While the former perturbation reveals folding, the latter shows that a material in an extreme anisotropic state is also prone to a faulting instability, in which a displacement step of finite size emerges. Another failure mechanism, namely the formation of dilation/compaction bands, is also highlighted. Finally, a geophysical application to the mechanics of chevron formation shows how the proposed approach may explain the formation of natural structures.

Modeling for establishment of evaluation conditions of functional safety of the railway track

The paper examines the modeling of the lifecycle of the railway track elements for investigating the deformation processes as the basis for the regulatory framework of the track operation to ensure the reliability of railways. To this end, the foundations of the wave propagation theory are used in describing the track and rolling stock interaction. The basic theoretical provisions and principles to describe the features of deformation operation and the calculation algorithm to determine the stress-strain state of the track are presented. Propagation of bulk and surface waves is used as exciting pulses. The requirements to the minimum length of the site under study and distance between the forces that must be considered in modeling are defined. The requirements to evaluation conditions of functional safety of the track are formed. According to the proposed model, a dynamic process, which involves the movement of the track under the effects of horizontal and vertical movements of wheels on rails for a period of time is considered.

Electroelastic waves in dielectrics modeled as polarizable continua

On the basis of a dielectric microcontinuum model, we investigate the problem of bulk wave propagation in a dielectric crystal with hexagonal material symmetry. The present linear micropolar model allows to express electric polarization via mechanical macro and micro-strain measures so that the coupling between acoustic and polarization modes can be described in terms of intrinsic dipole and quadrupole densities. The governing differential systems for different coupled modes are equivalent to some previous results of the classical phenomenological approach to ferroelectrics but also hold for piezoelectric solids with null intrinsic polarization. Resonance couplings between polaritons and acoustic waves arise from the dispersion equations depending on suitable relations among the micropolar constitutive parameters. Exploiting the dynamical representation of polarization for the admitted modes, we obtain piezoelectric coefficients and electromechanical coupling factors as functions of the wavelength (or frequency). As an application, a numerical example is given for the hexagonal phase of zinc sulfide.

Quantum effects on propagation of bulk and surface waves in a thin quantum plasma film

Abstract The propagation of bulk and surface plasma waves in a thin quantum plasma film is investigated, taking into account the quantum effects. The generalized bulk and surface plasma dispersion relation due to quantum effects is derived, using the quantum hydrodynamic dielectric function and applying appropriate additional boundary conditions. The quantum mechanical and film geometric effects on the bulk and surface modes are discussed. It is found that quantum effects become important for a thin film of small thickness.

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.

Rayleigh wave scattering from a vertical edge of isotropic substrates

The paper numerically studies the harmonic Rayleigh wave scattering at the 90-degree corner of isotropic substrate. The finite element method is used. The main attention is paid to two cases. The first one is the apex of the substrate corner is rounded off. The second one consists in that a layer of foreign material is deposited on the face which scatters the Rayleigh wave. The dependence of the reflection and the transmission coefficients on the Poisson ratio, the angle of incidence, the fillet radius, and the layer thickness are obtained. It is found that if the Rayleigh wave is incident perpendicularly to the substrate border, then the fillet of small radius as compared to the wavelength increases the reflection coefficient and decreases the transmission coefficient by factors 1.3–1.8. At normal incidence, the Poisson ratio does not change qualitatively the dependence of the reflection and transmission coefficients on the fillet radius. But the Poisson ratio can substantially affect the angle dependence of these coefficients if the wave is incident obliquely on the corner rounded off. It is also found out that a layer can modify the conditions of scattering such that the incident wave is totally reflected without transmission and conversion into bulk waves in a wide interval of angle of incidence, although, in principle, the bulk wave generation is allowed within a part of this angle interval.

Mutual conversion of bulk and surface acoustic waves in gratings of finite length on half-infinite substrates. II. FE analysis of bulk wave generation

The paper studies numerically the bulk acoustic wave generation by the surface acoustic wave propagating across a grating created on the surface of an elastically anisotropic half-infinite substrate. The computations are fully based on the finite element method. Applying the discrete Fourier transformation to the displacement field found inside the substrate and using an orthogonality relation valid for plane modes we determine separately the spacial spectrum of the quasi longitudinal and the quasi transverse bulk waves, that is, the dependence of the amplitudes of these waves on the tangential component of the wave vector. The dependence is investigated of the central spectral peak height and shape on the frequency of the incident surface wave as well as on the thickness, the width, and the number of strips forming the grating. In particular, it is found that under certain conditions the central peak can be approximated fairly precisely by the central peak of a sinc-function describing the spectrum of the bounded acoustic beam of rectangular shape and of width equal to the length of the grating.

Polarization of elastic waves in tellurium single crystal

We study special cases of elastic bulk wave propagation in a crystal of tellurium, in which the phase velocity of shear waves can exceed those of longitudinal waves. The impact of the piezoelectric effect on the phase velocity and polarization of waves is discussed.

Three-dimensional single particle tracking in dense dust clouds by stereoscopy of fluorescent particles

In dense dust clouds of a dusty plasma single particle trajectories are impossible to follow due to occlusion of particles and ambiguities in particle correspondences. By stereoscopic imaging of fluorescent tracer particles, we were able to reconstruct 3D single particle trajectories within dense dust clouds. Several measurements are shown that justify to regard the tracer particles as suitable representatives for the whole dust system. A first analysis of dust density waves in dense clouds already shows that these waves exhibit three-dimensional dynamics at larger wave amplitudes that cannot be resolved by 2D imaging techniques: a broad velocity distribution perpendicular to the oscillation plane due to dust-dust collisions is seen, while the velocity distribution in the oscillation direction is bimodal and shifted due to the bulk wave propagation.

Bulk wave propagation in layered piezomagnetic/piezoelectric plates with initial stresses or interface imperfections

Bulk wave propagation in laminated piezomagnetic/piezoelectric plates with initial stresses and imperfect interface is investigated using the state space approach. Either electrically and magnetically open or shorted conditions on the top and bottom surfaces are considered. The phase velocity and frequency spectra of the wave are numerically calculated for various electric and magnetic conditions, and normalized modes are also analyzed. Although numerical results do indicate that the initial compressive stress can reduce the phase velocity of the wave propagating in a layered multiferroic structure, the effect is not as obvious as that reported in literature where the initial stresses were taken far beyond the strength of real materials. While the imperfect interface affect the frequency spectra much greater than initial stresses, because imperfection decreases the structural stiffness.

Finite element models in Lamb’s problem

We consider a complex of issues on finite element modeling of the propagation of bulk and surface waves caused by harmonic concentrated actions applied to the boundary of an elastic half-space or elastic half-plane (Lamb’s external problem).

Relationships between the anisotropy of longitudinal wave velocity and hydroxyapatite crystallite orientation in bovine cortical bone

Quantitative ultrasound (QUS) is now widely used for evaluating bone in vivo, because obtained ultrasonic wave properties directly reflect the visco-elasticity. Bone tissue is composed of minerals like hydroxyapatite (HAp) and a collagen matrix. HAp crystallites orientation is thus one parameter of bone elasticity. In this study, we experimentally investigated the anisotropy of ultrasonic wave velocity and the HAp crystallites orientation in the axial–radial and axial–tangential planes in detail, using cylindrical specimens obtained from the cortical bone of three bovine femurs. Longitudinal bulk wave propagation was investigated by using a conventional ultrasonic pulse system. We used the one cycle of sinusoidal pulse which was emitted from wide band transmitter. The nominal frequency of the pulse was 1 MHz. First, we investigated the anisotropy of longitudinal wave velocity, measuring the anisotropy of velocity in two planes using cylindrical specimens obtained from identical bone areas. The wave velocity changed due to the rotation angle, showing the maximum value in the direction a little off the bone axis. Moreover, X-ray pole figure measurements also indicated that there were small tilts in the HAp crystallites orientation from the bone axis. The tilt angles were similar to those of the highest velocity direction. There were good correlations between velocity and HAp crystallites orientation obtained in different directions. However, a comparatively low correlation was found in posterior bone areas, which shows the stronger effects of bone microstructure. In the radial–tangential plane, where the HAp crystallites hardly ever align, weak anisotropy of velocity was found which seemed to depend on the bone microstructure.

On the Modelling of Ultrasonic Bulk Wave Propagation in Epoxies

Bulk wave attenuation in epoxies has been found experimentally to be a linear function of frequency. Anecdotal evidence suggests that the formulations used for bulk wave attenuation to be input to computational propagation models, including models for guided waves, can take many forms: Either the simple linear relationship between attenuation and frequency can be input directly, or any of a number of analytic models could, in principle, be used – examples being the Maxwell, Kelvin-Voigt and Zener anelastic solid models, or the hysteretic damping model sometimes employed for studies of structural vibrations. This paper considers the physical bases of these various models, as well as multiple degree of freedom models that have their origins in dielectric theory. It is shown that a simple two degree of freedom adaptation of the Zener model provides a good simulation of experimentally observed data for both attenuation and phase speed over a wide frequency band.

Surface True and Leaky Waves in Subsonic and Supersonic Regions. Explicit Solution of a Simple Model

The presence of surfaces breaks the translational invariance of the material and, therefore, gives rise to new vibrational states showing amplitudes that decrease with the depth into the bulk. The kind of motion would be precluded if the material had no bounds because the amplitude then would increase infinitely on the side of vacuum. When the system remains spatially periodic in the surface plane the wave vectors parallel to the surface are “good quantum numbers” so that the localized vibrations propagate along the surface as surface waves. The phenomenon occurs in a large range of scales starting from surfaces of stars and of the Earth, to those of crystals and nanoparticles. Acoustic surface waves have been described for the first time by Lord Rayleigh in 1887. Their frequencies are always lower than those of all the bulk waves with the same component of the wave vector parallel to the surface. Modifications of surfaces by some adlayers or, on the atomic scale, adatoms, may however, bring the frequencies of surface vibrations to the regions where the propagation of bulk waves conveys the energy to the bulk. Then the surface wave acquires a finite lifetime and becomes a surface resonance. In the next section we give the rudiments of the mathematical treatment of the surface waves. In Sect. 3 we construct a simple model showing both, surface waves and surface resonances. The model allows one to obtain all the characteristics of the