Low-frequency Shear Waves Research Articles

Study of Rheological-Mechanical Properties and Vibration Mechanics Bandgap of Row Pile Foundation

To analyze the rheological and mechanical properties as well as the vibration-mechanical forbidden zone effect of row pile foundations, this paper employs time-dependent modulus to examine the rheological mechanics of soil. Drawing from viscoelastic theory, we derive the expression of deformation modulus in the frequency domain to analyze the frequency dependence of the shear modulus of rheological soils. We construct a continuous medium dynamics model of the pile-soil periodic structure, taking into account soil rheology, and derive the dispersion equation of shear waves in the periodic structure using the multiple scattering method. The band gap characteristics and parameters that influence the law of shear waves in rheological soil-row pile foundations are studied through the analysis of arithmetic cases. The results show that under the loading condition, the zero-frequency shear modulus of soil is larger than the initial modulus value, and the real part of the shear modulus decreases monotonically with the increase of frequency and finally converges to the initial modulus value; under the unloading condition, the zero-frequency shear modulus of soil is smaller than the initial modulus value, and the real part of the shear modulus increases monotonically with the increase of frequency and finally converges to the initial modulus value; the larger the relaxation time of soil, the faster the convergence rate; the imaginary part of the shear modulus of soil The imaginary part of the soil shear modulus is positive under loading condition and negative under unloading condition, the value of the imaginary part increases and then decreases with increasing frequency and finally converges to 0. The imaginary part reaches the peak at the critical frequency, the larger the relaxation time the smaller the critical frequency, and the peak of the imaginary part is independent of the relaxation time. This study analyzed the dispersion curve of shear waves in a pile-soil periodic structure and found that increasing low-frequency shear wave velocity in rheological soil pile foundation shifts the band gap position to a higher frequency band, resulting in a smaller band gap width than in linear elastic soil. The relaxation time of soil affects the frequency position and width of the band gap, with larger relaxation times resulting in higher frequency positions and smaller widths. Additionally, soil rheology widens the forbidden vibration band gap of the pile periodic structure when the filling rate of the pile foundation is larger.

Low-frequency (105 Hz) shear modulus of nanosuspension

The low-frequency shear wave propagation in a suspension of silica nanoparticles in a polyethylsiloxane liquid was studied. The shear wave length was measured on ultrasonic interferometer, and it is equal to 55 μm. The value of the tangent of the mechanical losses angle is determined to be 0.18. These parameters were used to calculate the shear modulus of the investigated colloidal suspension; its value is 0.15‒105 Pa. The results obtained are in quite agreement with the data obtained by another way of the acoustic resonance method.

Actuators for MRE: New Perspectives With Flexible Electroactive Materials

Since 1995, Magnetic Resonance Elastography (MRE) has been constantly developed as a non-invasive diagnostic tool for quantitative mapping of mechanical properties of biological tissues. Indeed, mechanical properties of tissues vary over five orders of magnitude (the shear stiffness is ranging from 102 Pa for fat to 107 Pa for bones). Additionally, these properties depend on the physiological state which explains the granted benefit of MRE for staging liver fibrosis and its potential in numerous medical and biological domains. In comparison to the other modalities used to perform such measurement, Magnetic Resonance (MR) techniques offer the advantages of acquiring 3D high spatial resolution images at high penetration depth. However, performing MRE tissue characterization requires low frequency shear waves propagating in the tissue. Inducing them is the role of a mechanical actuator specifically designed to operate under Magnetic Resonance Imaging (MRI) specific restrictions in terms of electromagnetic compatibility. Facing these restrictions, many different solutions have been proposed while keeping a common structure: a vibration generator, a coupling device transmitting the vibration and a piston responsible for the mechanical coupling of the actuator with the tissue. The following review details the MRI constraints and how they are shaping the existing actuators. An emphasis is put on piezoelectric solutions as they solve the main issues encountered with other actuator technologies. Finally, flexible electroactive materials are reviewed as they could open great perspectives to build new type of mechanical actuators with better adaptability, greater ease-of-use and more compactness of dedicated actuators for MRE of small soft samples and superficial organs such as skin, muscles or breast.

C-Elastography: In Vitro Feasibility Phantom Study

C-Elastography (CE) is a new ultrasound technique that locally maps the non-linear elasticity of soft tissue using low-frequency (150–250 Hz) shear waves generated by the acoustic radiation force (ARF). CE is based on a recent finding that the magnitude of the ARF in an isotropic tissue-like solid is related linearly to a third-order modulus of elasticity, C, which is responsible for the coupling between deviatoric and volumetric constitutive behaviors. The main objective of the work described here was to examine the feasibility of using and performance of C-elastography in differentiating and characterizing soft tissue via a pilot study on ex vivo tissue and tissue-mimicking inclusions cast in a gelatin block. In this vein, the CE technique deploys a combination of ultrasound motion sensing and 3-D visco-elastodynamic simulation to estimate the non-linear modulus C. As ultrasound focusing inherently confines the ARF to a small region, CE provides the means for measuring C within O(mm3) volumes. Equipped with such data analysis, we performed in vitro CE experiments on agar-based, xenograft and normal breast tissue samples embedded in a gelatin matrix. The compound C-elastograms indicate marked (and sharp) C-contrast, with average values of 1.9 and 5.6 at push points inside the featured soft and hard inclusions, respectively.

Ultrasonic Interferometer on Shear Waves in Liquids

The developed ultrasonic interferometer on low-frequency shear waves in liquids is described. The problem of the piezo-quartz crystal – liquid interlayer – cladding interaction has been solved, and formulas for calculating the viscoelastic parameters of the liquid have been obtained. The developed theory is experimentally confirmed on the example of polymer liquid PMS-100.

Wave Resonance and the Supersonic Generation of Shear Waves in Dissipative Media

A new means of the supersonic generation and resonant amplification of shear elastic waves in dissipative media is developed. The procedure is based on the remote formation of a source of low-frequency shear waves inside a biological tissue under the radiation pressure of a focused ultrasonic beam. Finite-difference modeling of the propagation of low-frequency shear waves in biological tissues is performed. Measurements and visualization of focused acoustic fields along with a Mach spiral structures in acoustic phantoms are performed that confirm the efficiency and feasibility of the developed means of the supersonic generation and resonant amplification of shear waves in biological tissues for ultrasonic diagnostics and therapy.

Using surface wave amplitude spectra to estimate the source depth of Sichuan Jiuzhaigou <italic>M</italic>7.0 earthquake

The accurate source depth of earthquake is important for understanding plate-tectonic processes and structure of seismic faults, identifying natural earthquake and non-natural earthquake. The amplitude formula of seismic surface wave shows that the amplitude of surface wave is closely related to the focal mechanism, the azimuth from epicenter to station and the source depth. When the focal mechanism and the observation position are determined, the excitation energy of surface wave is sensitive to source depth. The surface wave amplitude spectrum can be applied to determine the focal depth. The amplitude energy of shallow earthquake shows the “notch” phenomenon in band of frequency between 0.01 and 0.08 Hz in amplitude spectra, i.e. the attenuation of amplitude energy. The position of the frequency-dependent “notch” is related to the source depth. In this study, the relationship between the position of the “notch” and the source depth is analyzed by theoretical seismogram. We obtain the amplitude spectrums of the far-field at different source depths and then locate the source depth by means of comparing with observed seismic waveforms. By the amplitude spectrum of theoretical surface wave, small changes in the source depth can cause the variation of the amplitude spectrum so that the source depth of shallow earthquake can be accurately determined. The focal mechanism is coupled with the source depth. In order to analyze the influence of the disturbance of source mechanism to source depth determination, the focal mechanism of Jiuzhaigou earthquake was inversed, with disturbance of 10 degrees to strike, dip and rake. The results show only rather small changes in locating depth. In order to eliminate the influence of low frequency noise, shear wave and surface wave of short-period on amplitude spectra, this study adopts the AK135 velocity model to remove the dispersion. In this study, the seismic source depth of the Jiuzhaigou M 7.0 earthquake of 8 August 2017 in Sichuan province, China, was determined at 8.2 km by the energy attenuation characteristics of Rayleigh amplitude spectrum.

Characterization of soft marine sediments using vector sensors

Horizontally and vertically polarized interface waves with phase speeds of 45–375 m/s were observed in the North Sea using a horizontal linear array of three-component seismometers [H. Dong et al., J. Acoust. Soc. Am. 134, 176–184 (2013)]. The waves were generated by a low-frequency shear wave source on the seafloor, which operated in the 2–60 Hz band. Dispersion curves of the interface waves exhibit linear dependences between the logarithm of phase speed and logarithm of frequency, with distinct slopes at large and small phase speeds. This suggests a geoacoustic model consisting of two sediment layers with power-law dependences of the shear wave speed on depth below seafloor [O. A. Godin and D. M. F. Chapman, J. Acoust. Soc. Am. 110, 1890–1907 (2001)]. In this paper, dispersion curves of interface waves are inverted for the shear wave speed profile in the sediments. Observations of lateral wave arrivals are used to corroborate the geoacoustic model suggested by the surface wave dispersion and constrain c...

An analysis of intrinsic variations of low-frequency shear wave speed in a stochastic tissue model: the first application for staging liver fibrosis

Shear wave elastography is increasingly being used to non-invasively stage liver fibrosis by measuring shear wave speed (SWS). This study quantitatively investigates intrinsic variations among SWS measurements obtained from heterogeneous media such as fibrotic livers. More specifically, it aims to demonstrate that intrinsic variations in SWS measurements, in general, follow a non-Gaussian distribution and are related to the heterogeneous nature of the medium being measured. Using the principle of maximum entropy (ME), our primary objective is to derive a probability density function (PDF) of the SWS distribution in conjunction with a lossless stochastic tissue model. Our secondary objective is to evaluate the performance of the proposed PDF using Monte Carlo (MC)-simulated shear wave (SW) data against three other commonly used PDFs.Based on statistical evaluation criteria, initial results showed that the derived PDF fits better to MC-simulated SWS data than the other three PDFs. It was also found that SW fronts stabilized after a short (compared with the SW wavelength) travel distance in lossless media. Furthermore, in lossless media, the distance required to stabilize the SW propagation was not correlated to the SW wavelength at the low frequencies investigated (i.e. 50, 100 and 150 Hz). Examination of the MC simulation data suggests that elastic (shear) wave scattering became more pronounced when the volume fraction of hard inclusions increased from 10 to 30%.In conclusion, using the principle of ME, we theoretically demonstrated for the first time that SWS measurements in this model follow a non-Gaussian distribution. Preliminary data indicated that the proposed PDF can quantitatively represent intrinsic variations in SWS measurements simulated using a two-phase random medium model. The advantages of the proposed PDF are its physically meaningful parameters and solid theoretical basis.

Shear wave propagation in worm-like micellar fluids

In viscous Newtonian fluids, support of shear waves are limited to the viscous boundary layer. Non-Newtonian fluids which have shear modulus, however, support shear waves over much longer distances. The restoring force responsible for the shear wave propagation arises from the entanglement of high aspect ratio macromolecules. We report low frequency (30–60 Hz) shear wave studies of aqueous worm-like micellar fluids composed of cetyltrimethylammonium bromide (CTAB) for the surfactant and sodium salicylate (NaSAL) as the salt over a wide concentration range (20–500 mM CTAB). Shear speeds range from 75 to 700 mm/s over this concentration range at room temperature with evidence of two phase transitions at 200 mM and 375 mM CTAB. Shear stress attenuation and temperature resolved measurements between 20 and 40 C will also be presented.

Reconstructing 3-D maps of the local viscoelastic properties using a finite-amplitude modulated radiation force

A modulated acoustic radiation force, produced by two confocal tone-burst ultrasound beams of slightly different frequencies (i.e. 2.0MHz±Δf/2, where Δf is the difference frequency), can be used to remotely generate modulated low-frequency (Δf⩽500Hz) shear waves in attenuating media. By appropriately selecting the duration of the two beams, the energy of the generated shear waves can be concentrated around the difference frequency (i.e., Δf±Δf/2). In this manner, neither their amplitude nor their phase information is distorted by frequency-dependent effects, thereby, enabling a more accurate reconstruction of the viscoelastic properties. Assuming a Voigt viscoelastic model, this paper describes the use of a finite-element-method model to simulate three-dimensional (3-D) shear-wave propagation in viscoelastic media containing a spherical inclusion. Nonlinear propagation is assumed for the two ultrasound beams, so that higher harmonics are developed in the force and shear spectrum. Finally, an inverse reconstruction algorithm is used to extract 3-D maps of the local shear modulus and viscosity from the simulated shear-displacement fields based on the fundamental and second-harmonic component. The quality of the reconstructed maps is evaluated using the contrast between the inclusion and the background and the contrast-to-noise ratio (CNR). It is shown that the shear modulus can be accurately reconstructed based on the fundamental component, such that the observed contrast deviates from the true contrast by a root-mean-square-error (RMSE) of only 0.38 and the CNR is greater than 30dB. If the second-harmonic component is used, the RMSE becomes 1.54 and the corresponding CNR decreases by approximately 10–15dB. The reconstructed shear viscosity maps based on the second harmonic are shown to be of higher quality than those based on the fundamental. The effects of noise are also investigated and a fusion operation between the two spectral components is applied to enhance the reconstruction quality. Finally, a modified shear-wave spectroscopy technique, shown to be more robust to noise, is described for the estimation of the viscoelastic properties inside and outside the spherical inclusion under conditions of increased noise.

Patient-specific modelling of the calf muscle under elastic compression using magnetic resonance imaging and ultrasound elastography

Compression therapy is traditionally employed to achieve a variety of therapeutic goals related to venous insufficiency but its mechanisms are neither clearly understood nor conclusively demonstrated. (Avril et al., 2010) have pointed out that the capacity of the treatment to achieve the desired medical goal is closely related to the question of transmission of pressure through the soft tissues, as a result of which biomechanical models have been developed (Dubuis et al., 2012). In these models, the prediction of pressure was made based on the rather strong assumption that the behaviour of soft tissues (muscles and adipose tissues) can be approximated as homogeneous materials. Recent developments in ultrasound imaging can be helpful to improve both our understanding of the effect of medical compression stockings (MCS) on the blood return and on the quality of our predictions. (Bercoff et al., 2004) introduced a new ultrasound based technology called Supersonic Shear Imaging (SSI), for real-time soft tissue elasticity mapping. This transient elastography technique remotely produces a low frequency shear wave inside soft tissue through a radiation force by focusing ultrasonic beams at a supersonic speed. The shear wave motion is captured at a frame rate of 5 MHz. The velocity field of the medium can be mapped an dassuming the medium to be elastic, Young's modulus can be estimated quantitatively. This non-invasive technique allows identifying the material property heterogeneities inside the different compartments of the human leg. Using this experimental technique combined with FE modeling, the present study aims at shedding a light on the influence of those heterogeneities of material properties on the pressure inside the leg under elastic compression.

Estimating the local viscoelastic properties from dispersive shear waves using time–frequency ridge analysis

Modulated low-frequency shear waves can be non-invasively generated locally within a medium, by the oscillatory acoustic radiation force resulting from the interference of two focused quasi-CW ultrasound beams of slightly different frequencies. The propagation of such shear waves within a viscoelastic medium is known to be affected by the dispersive effects of viscosity. Specifically, a low-frequency (LF) spectral component was shown to arise with increased viscosities and higher modulation frequencies and appear as a ‘slow’ wave at the end of the shear waveform. In this paper, the shear dispersion characteristics are studied based on the Pseudo–Wigner–Ville distribution (PWVD) in the time–frequency domain. The ridges of the PWVD are then extracted and used to calculate the frequency-dependent shear speed, by identifying the LF dispersive component both in time and frequency. Using numerical simulations, it is shown that this way of estimating the shear dispersion is more efficient and robust than the conventional phase-delay Fourier method. Thus, more accurate estimates of the local shear modulus and viscosity of the propagating medium could be achieved. The effects of noise on the proposed method are also discussed.

Focused, radially polarized shear wave beams in tissue-like media

In the past decade there has been a surge in the optics literature regarding the unique characteristics of focused, radially-polarized light beams. Of particular interest is the existence of a longitudinal component to the electric field in the focal region of the beam, of comparable amplitude to the radial component and yet with a smaller beamwidth [cf. Q. Zhan, Adv. Opt. Photon. 1, 1-57 (2009)]. In the linear approximation there exists a direct analogy between these light beams and radially-polarized shear wave beams in incompressible elastic media, and hence we may interpret the results found in the optics literature as applying to low-frequency shear waves propagating through tissue-like media. When considering nonlinear effects, however, the fact that the gradient of the field experiences a larger gain due to focusing than the field itself implies that the shear wave case is more susceptible to nonlinear behavior than its optical analog. Second-harmonic generation in the focal region of a focused, radially-polarized shear wave beam in a soft solid is investigated. [Work supported by the ARL:UT McKinney Fellowship in Acoustics and by NIH DK070618.]

Shear waves in viscoelastic wormlike micellar fluids over a broad concentration range

Low frequency (61 Hz) shear wave speeds have been measured in viscoelastic wormlike micellar (WM) fluids for a concentration range of 20/12-500/300 mM CTAB/NaSAL where CTAB is the surfactant and NaSAL is the salt and the concentration ratio was fixed at 0.6 for all experiments. The birefringent property of the WM fluids was exploited to visually track the the shear pulse using crossed optical polarizing filters and high speed video. Several scalings of shear wave speed as a function of concentration were discovered: c(s) ~ √C for 20-200 mM and c(s) ~ C for higher concentrations, but with a break in the slope at 400 mM CTAB. Over this full concentration range, the shear wave speed varied from 0.08-0.7 m/s. The shear wave speed was also found to be sensitive to the time between fluid synthesis and measurement indicating a long equilibrium time. Further, comparison with elastic and loss moduli obtained from rheology data show that shear wave propagation is dominated by the elastic modulus for this frequency range. Also briefly discussed are potential applications of this fluid in elastography.

Estimating material viscoelastic properties based on surface wave measurements: A comparison of techniques and modeling assumptions

Previous studies of the first author and others have focused on low audible frequency (<1 kHz) shear and surface wave motion in and on a viscoelastic material comprised of or representative of soft biological tissue. A specific case considered has been surface (Rayleigh) wave motion caused by a circular disk located on the surface and oscillating normal to it. Different approaches to identifying the type and coefficients of a viscoelastic model of the material based on these measurements have been proposed. One approach has been to optimize coefficients in an assumed viscoelastic model type to match measurements of the frequency-dependent Rayleigh wave speed. Another approach has been to optimize coefficients in an assumed viscoelastic model type to match the complex-valued frequency response function (FRF) between the excitation location and points at known radial distances from it. In the present article, the relative merits of these approaches are explored theoretically, computationally, and experimentally. It is concluded that matching the complex-valued FRF may provide a better estimate of the viscoelastic model type and parameter values; though, as the studies herein show, there are inherent limitations to identifying viscoelastic properties based on surface wave measurements.

The many faces of shear Alfvén waves

One of the fundamental waves in magnetized plasmas is the shear Alfvén wave. This wave is responsible for rearranging current systems and, in fact all low frequency currents in magnetized plasmas are shear waves. It has become apparent that Alfvén waves are important in a wide variety of physical environments. Shear waves of various forms have been a topic of experimental research for more than fifteen years in the large plasma device (LAPD) at UCLA. The waves were first studied in both the kinetic and inertial regimes when excited by fluctuating currents with transverse dimension on the order of the collisionless skin depth. Theory and experiment on wave propagation in these regimes is presented, and the morphology of the wave is illustrated to be dependent on the generation mechanism. Three-dimensional currents associated with the waves have been mapped. The ion motion, which closes the current across the magnetic field, has been studied using laser induced fluorescence. The wave propagation in inhomogeneous magnetic fields and density gradients is presented as well as effects of collisions and reflections from boundaries. Reflections may result in Alfvénic field line resonances and in the right conditions maser action. The waves occur spontaneously on temperature and density gradients as hybrids with drift waves. These have been seen to affect cross-field heat and plasma transport. Although the waves are easily launched with antennas, they may also be generated by secondary processes, such as Cherenkov radiation. This is the case when intense shear Alfvén waves in a background magnetoplasma are produced by an exploding laser-produced plasma. Time varying magnetic flux ropes can be considered to be low frequency shear waves. Studies of the interaction of multiple ropes and the link between magnetic field line reconnection and rope dynamics are revealed. This manuscript gives us an overview of the major results from these experiments and provides a modern prospective for the earlier studies of shear Alfvén waves.

Mapping the local shear modulus and viscosity using a transient finite-amplitude modulated radiation force

Localized narrowband low-frequency shear waves can be non-invasively generated within tissue, by a modulated finite-amplitude radiation force, resulting from the interference of two focused quasi-CW ultrasound beams of slightly different frequencies. Assuming a Voigt viscoelastic model, this paper describes the use of a finite-element-method model, to simulate two-dimensional shear-wave propagation in viscoelastic media, containing circular inclusions (lesions). Using this model, an inverse approach is used to extract maps of the local shear modulus and viscosity. The performance is evaluated based on three metrics: the lesion contrast, the contrast-transfer-efficiency (CTE), and the contrast-to-noise ratio (CNR). Modified definitions of these metrics are proposed and used in order to account for the time-varying nature of the shear waves and the inverse reconstruction algorithm. In the absence of any noise, it is shown that accurate reconstruction can be achieved not only with the fundamental, but also with the higher harmonics, as well as, with a low-frequency component that occurs for high viscosity and high modulation frequencies. For low-viscosity conditions, the lesion contrast, CTE, and CNR are shown to exhibit very good performance not only for the fundamental, but also, for the higher harmonics. In the case of increased viscosities and modulation frequencies, the generated low-frequency component is shown to provide superior contrast performance even when compared to that of the fundamental. The effects of noise on the reconstruction quality are examined. Depending on the lesion and background properties, it is shown that noise can seriously degrade reconstruction from the higher harmonics.

Shear waves in viscoelastic wormlike micellar fluids

Low frequency (61 Hz) shear wave speeds have been measured in viscoelastic wormlike micellar (WM) fluids for a concentration range of 20/12-160/96 mM CTAB/NaSAL. The strain induced birefringence of the WM fluids was exploited to optically track the shear pulse using crossed polarizing filters and high speed video. It was found that shear speed increases roughly linearly with concentration at a rate of 3.5 mm s(-1) mM(-1) CTAB. Further, comparison with elastic and loss moduli obtained from rheology data show that shear wave propagation is dominated by the elastic modulus for this frequency range.

Surface response of a fractional order viscoelastic halfspace to surface and subsurface sources

Previous studies by the second author published in this journal focused on low audible frequency (40-400 Hz) shear and surface wave motion in and on a viscoelastic material representative of biological tissue. Specific cases considered were that of surface wave motion on a halfspace caused by a finite rigid circular disk located on the surface and oscillating normal to it [Royston et al., J. Acoust. Soc. Am. 106, 3678-3686 (1999)] and compression, shear, and surface wave motion in a halfspace generated by a subsurface finite dipole [Royston et al., J. Acoust. Soc. Am. 113, 1109-1121 (2003)]. In both studies, a Voigt model of viscoelasticity was assumed in the theoretical treatment, which resulted in agreement between theoretical predictions and experimental measurements over a limited frequency range. In the present article, the linear viscoelastic assumption in these two prior works is revisited to consider a (still linear) fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time. It is shown that in both excitation source configurations, the fractional order Voigt model assumption improves the match of theory to experiment over a wider frequency range (in some cases up to the measured range of 700 Hz).