Shear Wave Dispersion Research Articles

Error in estimates of tissue material properties from shear wave dispersion ultrasound vibrometry

Shear wave velocity measurements are used in elasticity imaging to find the shear elasticity and viscosity of tissue. A technique called shear wave dispersion ultrasound vibrometry (SDUV) has been introduced to use the dispersive nature of shear wave velocity to locally estimate the material properties of tissue. Shear waves are created using a multifrequency ultrasound radiation force, and the propagating shear waves are measured a few millimeters away from the excitation point. The shear wave velocity is measured using a repetitive pulse-echo method and Kalman filtering to find the phase of the harmonic shear wave at 2 different locations. A viscoelastic Voigt model and the shear wave velocity measurements at different frequencies are used to find the shear elasticity (mu(1)) and viscosity (mu(2)) of the tissue. The purpose of this paper is to report the accuracy of the SDUV method over a range of different values of mu(1) and mu(2). A motion detection model of a vibrating scattering medium was used to analyze measurement errors of vibration phase in a scattering medium. To assess the accuracy of the SDUV method, we modeled the effects of phase errors on estimates of shear wave velocity and material properties while varying parameters such as shear stiffness and viscosity, shear wave amplitude, the distance between shear wave measurements (delta r), signal-to-noise ratio (SNR) of the ultrasound pulse-echo method, and the frequency range of the measurements. We performed an experiment in a section of porcine muscle to evaluate variation of the aforementioned parameters on the estimated shear wave velocity and material property measurements and to validate the error prediction model. The model showed that errors in the shear wave velocity and material property estimates were minimized by maximizing shear wave amplitude, pulse-echo SNR, delta r, and the bandwidth used for shear wave measurements. The experimental model showed optimum performance could be obtained for delta r = 3 - 6 mm, SNR =35 dB, with a frequency range of 100 to 600 Hz, and with a shear wave amplitude on the order of a few microns down to 0.5 microm. The model provides a basis to explore different parameters related to implementation of the SDUV method. The experiment confirmed conclusions made by the model, and the results can be used for optimization of SDUV.

Sonic to ultrasonic Q of sandstones and limestones: Laboratory measurements at in situ pressures

Laboratory measurements of the attenuation and velocity dispersion of compressional and shear waves at appropriate frequencies, pressures, and temperatures can aid interpretation of seismic and well-log surveys as well as indicate absorption mechanisms in rocks. Construction and calibration of resonant-bar equipment was used to measure velocities and attenuations of standing shear and extensional waves in copper-jacketed right cylinders of rocks ([Formula: see text] in length, [Formula: see text] in diameter) in the sonic frequency range and at differential pressures up to [Formula: see text]. We also measured ultrasonic velocities and attenuations of compressional and shear waves in [Formula: see text]-diameter samples of the rocks at identical pressures. Extensional-mode velocities determined from the resonant bar are systematically too low, yielding unreliable Poisson’s ratios. Poisson’s ratios determined from the ultrasonic data are frequency corrected and used to calculate thesonic-frequency compressional-wave velocities and attenuations from the shear- and extensional-mode data. We calculate the bulk-modulus loss. The accuracies of attenuation data (expressed as [Formula: see text], where [Formula: see text] is the quality factor) are [Formula: see text] for compressional and shear waves at ultrasonic frequency, [Formula: see text] for shear waves, and [Formula: see text] for compressional waves at sonic frequency. Example sonic-frequency data show that the energy absorption in a limestone is small ([Formula: see text] greater than 200 and stress independent) and is primarily due to poroelasticity, whereas that in the two sandstones is variable in magnitude ([Formula: see text] ranges from less than 50 to greater than 300, at reservoir pressures) and arises from a combination of poroelasticity and viscoelasticity. A graph of compressional-wave attenuation versus compressional-wave velocity at reservoir pressures differentiates high-permeability ([Formula: see text], [Formula: see text]) brine-saturated sandstones from low-permeability ([Formula: see text], [Formula: see text]) sandstones and shales.

Shear Wave Spectroscopy for In Vivo Quantification of Human Soft Tissues Visco-Elasticity

In vivo assessment of dispersion affecting the propagation of visco-elastic waves in soft tissues is key to understand the rheology of human tissues. In this paper, the ability of the supersonic shear imaging (SSI) technique to generate planar shear waves propagating in tissues is fully exploited. First, by strongly limiting shear wave diffraction in the imaging plane, this imaging technique enables to discriminate between the usually concomitant influences of both medium rheological properties and diffraction affecting the shear wave dispersion. Second, transient propagation of these plane shear waves in soft tissues can be measured using echographic images acquired at very high frame. In vitro and in vivo experiments demonstrate that dispersion curves, which characterize the rheological behavior of tissues by measuring the frequency dependence of shear wave speed and attenuation, can be recovered in the 75-600 Hz frequency range. Based on a phase difference algorithm, the dispersion curves are computed in 1 cm2 regions of interest from the acquired propagation movie. In vivo measurements in Biceps Brachii muscle and liver of three healthy volunteers show important differences in the rheological behavior of these different tissues. Liver tissue appears to be much more dispersive with a phase velocity ranging from approximately 1.5 m/s at 75 Hz to approximately 3 m/s at 500 Hz whereas muscle tissue shows an important anisotropy, shear waves propagating longitudinally to the muscular fibers are almost nondispersive while those propagating transversally are very dispersive with a shear wave speed ranging from 0.5 to 2 m/s between 75 and 500 Hz. The estimation of dispersion curves is local and can be performed separately in different regions of the organ. This signal processing approach based on the SSI modality introduces the new concept of in vivo shear wave spectroscopy (SWS) that could become an additional tool for tissue characterization. This paper demonstrates the in vivo ability of this SWS to quantify both local shear elasticity and dispersion in real time.

Scattering of plane transverse waves by spherical inclusions in a poroelastic medium

SUMMARY The scattering of plane transverse waves by a spherical inclusion embedded in an infinite poroelastic medium is treated for the first time in this paper. The vector displacement wave equations of Biot's theory are solved as an infinite series of vector spherical harmonics for the case of a plane S-wave impinging from a porous medium onto a spherical inclusion which itself is assumed to be another porous medium. Based on the single spherical scattering theory and dynamic composite elastic medium theory, the non-self-consistent shear wavenumber is derived for a porous rock having numerous spherical inclusions of another medium. The frequency dependences of the shear wave velocity and the shear wave attenuation have been calculated for both the patchy saturation model (inclusions having the same solid frame as the host but with a different pore fluid from the host medium) and the double porosity model (inclusions having a different solid frame than the host but the same pore fluid as the host medium) with dilute concentrations of identical inclusions. Unlike the case of incident P-wave scattering, we show that although the fluid and the heterogeneity of the rock determine the shear wave velocity of the composite, the attenuation of the shear wave caused by scattering is actually contributed by the heterogeneity of the rock for spherical inclusions. The scattering of incident shear waves in the patchy saturation model is quite different from that of the double porosity model. For the patchy saturation model, the gas inclusions do not significantly affect the shear wave dispersion characteristic of the water-filled host medium. However, the softer inclusion with higher porosity in the double porosity model can cause significant shear wave scattering attenuation which occurs at a frequency at which the wavelength of the shear wave is approximately equal to the characteristic size of the inclusion and depends on the volume fraction. Compared with analytic formulae for the low frequency limit of the shear velocity, our scattering model yields discrepancies within 4.0 per cent. All calculated shear velocities of the composite medium with dilute inclusion concentrations approach the high frequency limit of the host material.

Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity

Characterization of tissue elasticity (stiffness) and viscosity has important medical applications because these properties are closely related to pathological changes. Quantitative measurement is more suitable than qualitative measurement (i.e., mapping with a relative scale) of tissue viscoelasticity for diagnosis of diffuse diseases where abnormality is not confined to a local region and there is no normal background tissue to provide contrast. Shearwave dispersion ultrasound vibrometry (SDUV) uses shear wave propagation speed measured in tissue at multiple frequencies (typically in the range of hundreds of Hertz) to solve quantitatively for both tissue elasticity and viscosity. A shear wave is stimulated within the tissue by an ultrasound push beam and monitored by a separate ultrasound detect beam. The phase difference of the shear wave between 2 locations along its propagation path is used to calculate shear wave speed within the tissue. In vitro SDUV measurements along and across bovine striated muscle fibers show results of tissue elasticity and viscosity close to literature values. An intermittent pulse sequence is developed to allow one array transducer for both push and detect function. Feasibility of this pulse sequence is demonstrated by in vivo SDUV measurements in swine liver using a dual transducer prototype simulating the operation of a single array transducer.

Ultrasound stimulated vibrometry for measuring tissue properties

Harmonic or pulsed radiation pressure and subsequent measurement of the tissue response can be used to accurately quantitatively measure fundamental material properties of tissue. The measurement of shear wave dispersion can be used to estimate elastic shear moduli of tissue using an appropriate model. Fundamental model free properties such as complex modulus can also be measured. Ultrasound radiation pressure is used to induce free propagating shear waves. The measurable properties of the shear waves such as speed are sensitive to only the local material properties of the tissue under certain circumstances. A model relating the shear wave speed as a function of frequency is related to the elastic and viscous moduli within small regions of tissue according to, for instance, the Voigt model. Shear and elastic moduli in tissue are measured with high accuracy and precision given appropriate models of wave propagation within the geometry of the tissue. Complex modulus in liver of live pig and other tissues have been made. Careful use of shear wave propagation and subsequent measurements can provide fundamental quantitative measurements of tissue mechanical properties if models are accurate. An application of this method is the noninvasive measurement of liver stiffness as a surrogate for fibrosis.

Dispersion and attenuation in thermoelastic multisize particulate composites

This paper deals with the problem of multiple scattering by a random distribution of spherical solid particles in a solid. The material properties of both media are taken as thermoelastic. The radii of the inclusions may be different. The self-consistent method in its variant of the effective medium is used to find the dispersion and attenuation of quasi-elastic, quasi-thermal and shear waves. The single scattering problem required by this technique is solved approximately by means of the Galerkin method applied to an integral equation using the Green function. Numerical results display a characteristic resonance phenomena which appears in the interval where the results are approximately valid, that is, for very long waves down to wavelengths about twice the largest diameter of the spheres. Examples are shown, for composites with two sets of inclusions, which have either a very similar or dissimilar size. Comparisons are made with the elastic counterpart. Among the material properties, the mass density ratio, inclusion to matrix, seems to play an important and simple role. Frequency intervals are distinguished and shown to depend on that ratio, where the attenuation and dispersion of quasi-elastic and P-waves are either very close to each other or not at all. The same applies to shear waves in either composite. The mass density ratio also displays a simple monotonic decreasing behaviour as a function of the frequency at the first attenuation maximum and velocity minimum. These results may be of interest for the nondestructive testing characterization of particulate composites.

Propriedades Sísmicas Anisotrópicas Derivadas da Orientação Cristalográfica Preferencial de Muscovita-Quartzo Milonitos

Seismic wave propagation in organized matter usually results in azimuthal variations of longitudinal waves (Pwaves), as well as the effect of birefringence in transversal waves (S-waves), which results in two orthogonal shear waves with contrasting velocities. In this paper we present the results of the anisotropic seismic properties of five samples of muscovitequartz mylonites collected in different parts of a fold in the Saas Fee region, Western Internal Alps. The P-wave velocities in these rocks varies from 5.73 to 6.32 km/s, whereas the high-velocity shear wave (S1) varies from 3.82 to 4.22 km/s and the low velocity (S2) from 3.73 to 4.09 km/s. The anisotropy in these rocks is relatively high and reaches values from 9.5% for P-waves, and almost 11% for shear wave splitting. Both anisotropy and propagation directions seem to be related to from the strong preferred orientation of quartz and muscovite but also depend of muscovite modal content within the different specimens. Development of preferred orientation of minerals destroys and disperses the single crystal seismic properties, which causes a decrease of wave velocities and a dispersion of propagation directions, of both compressional and shear waves. Since the preferred orientation of quartz and muscovite can be directly related to the main macroscopic structures in these rocks (foliation, lineation, and pole of foliation) and the anisotropic seismic properties are related to the preferred orientation, it is possible to determine the propagation directions in terms of these structures. Due to the relatively high muscovite content, many of the maximum propagation velocities are parallel/subparallel to the foliation and some parallel to the lineation of the reference frame. On the other hand, directions of minimum propagation cannot be directly related to the foliation pole. The presence of folds in the mid-to lower crust can exert changes in the propagation directions due to the foliation variation around such structures, mainly in the P-waves.

TH-B-L100E-01: Ultrasound Stimulated Vibrometry for Measuring Tissue Properties

Background: Ultrasound stimulated Vibro-acoustography has been used to image tissues using the acoustic response of tissues to localized harmonic radiation pressure. The method provides high resolution and high dynamic range images of tissues. The parameter being imaged is a complex combination of scattering, attenuation, and nonlinearity. However specific use of harmonic or pulsed radiation pressure and subsequent measurement of the tissue response can be used to measure fundamental material properties of tissue. The measurement of shear wave dispersion can be used to estimate elastic shear moduli of tissue. Fundamental model free properties such as elastic storage and loss constants can also be measured. Methods: Ultrasound radiation pressure is used to induce free propagating shear waves. The measurable properties of the shear waves are sensitive to only the material properties of the tissue under certain circumstances. A model relating the shear wave speed as a function of frequency is related to the elastic and viscous moduli within small regions of tissue. Results: Shear and elastic moduli in tissue are measured with high accuracy and precision given appropriate models of wave propagation within the geometry of the tissue. Conclusion: Careful use of shear wave propagation and subsequent measurements can provide fundamental measurements of tissue mechanical properties if models are accurate. An application of this method is the noninvasive measurement of liver stiffness as a surrogate for fibrosis. Educational Objectives 1. Understand Shear Wave moduli. 2. Understand Vibrometry. 3. Understand measurement of elasticity and viscosity with ultrasound.

Ultrasound stimulated vibroacoustography and vibrometry for measuring tissue properties

Ultrasound stimulated vibroacoustography has been used to image tissues using the acoustic response of tissues to localized harmonic radiation pressure. The method provides high resolution and high dynamic range images of tissues. The parameter being imaged is a complex combination of scattering, attenuation, and nonlinearity. However, specific use of harmonic or pulsed radiation pressure and subsequent measurement of the tissue response can be used to measure fundamental material properties of tissue. The measurement of shear wave dispersion can be used to estimate elastic shear moduli of tissue. Fundamental model free properties such as elastic storage and loss constants can also be measured. Ultrasound radiation pressure is used to induce free propagating shear waves. The measurable properties of the shear waves are sensitive to only the material properties of the tissue under certain circumstances. Shear moduli in tissue are measured with high accuracy and precision given appropriate models of wave propagation within the geometry of the tissue. Careful use of shear wave propagation and subsequent measurements can provide fundamental measurements of tissue mechanical properties if models are accurate.

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.

Assessment of errors induced by the anomalous dispersion of high frequency shear waves in rheometrical measurements on gelling systems

The principal objectives of the work reported in this paper were two-fold. Firstly, to assess, by means of the numerical simulation of transverse wave propagation within a linear viscoelastic medium, the degree to which wavegroup propagation involving a finite frequency interval (2Δ ω) corresponds to an ‘ideal’ beat (i.e. one for which 2Δ ω → 0). Secondly, to compare the results of shear wave propagation measurements with predictions based on a rheometrical theory, which describes the gelation of materials in terms of their wave dispersion characteristics. The results of the numerical simulations indicate that for a sufficiently small value of the frequency interval Δ ω/ ω, the nodal and anti-nodal regions of the amplitude modulation envelope of wavegroups propagating within a critical-gel may be defined with sufficient accuracy to permit calculation of the corresponding value of the group velocity U. The results of high frequency shear wave dispersion measurements made on a system known to display critical-gel behaviour provide encouraging agreement with theory and the results of the simulations.

Shear wave attenuation and dispersion in melt‐bearing olivine polycrystals: 1. Specimen fabrication and mechanical testing

Five melt‐bearing polycrystalline olivine aggregates have been newly prepared by hot isostatic pressing and tested at high temperature and pressure with torsional forced‐oscillation and microcreep methods. Cylindrical specimens, varying in average grain size from 7 to 52 μm, were annealed and then tested during slow staged cooling under 200 MPa pressure from maximum temperatures of 1240–1300°C where they contained basaltic melt fractions ranging from ∼0.0001 to 0.037. For temperatures ≥1000°C, pronounced departures from elastic behavior are evident in strain energy dissipation Q−1 and associated dispersion of the shear modulus G. In marked contrast with the high‐temperature viscoelastic behavior of melt‐free materials, a broad dissipation peak is observed for each of the melt‐bearing specimens ‐ superimposed upon a melt‐enhanced level of monotonically frequency‐ and temperature‐dependent “background” dissipation. The oscillation period at which the peak is centered decreases systematically with increasing temperature. A “global” model comprising an Andrade‐pseudoperiod background plus Gaussian peak accounts adequately for the variation of Q−1 with frequency, temperature, average grain size and melt fraction. In the following paper (Part II) a microstructural explanation for the observed viscoelastic behavior is sought and the global model is used to extrapolate the experimental data to the conditions of teleseismic wave propagation in the Earth's upper mantle.

Shear wave attenuation and dispersion in melt‐bearing olivine polycrystals: 2. Microstructural interpretation and seismological implications

The torsional forced oscillation tests of melt‐bearing olivine aggregates reported by Jackson et al. [2004] consistently show a peak in attenuation that is absent from melt‐free aggregates tested under similar conditions and grain sizes. Characterization by SEM shows that the melt resides in triple junction tubules and larger pockets as previously described. TEM imaging and EDS analysis reveals that olivine‐olivine grain boundaries are characterized by a region ≤1 nm wide which is structurally and chemically distinct from olivine grain interiors. From the possible mechanisms that can produce an anelastic attenuation peak, melt squirt can be eliminated for our samples and experimental conditions. We attribute the observed attenuation peak to elastically accommodated grain boundary sliding, requiring that the grain boundaries are weak relative to olivine grain interiors but have a significantly higher viscosity than bulk melt. While the nanometer scale grain boundary structure in the melt‐bearing aggregates is essentially the same as for melt‐free aggregates studied previously, elastically accommodated sliding in the latter is apparently inhibited by tight three‐grain edge intersections. The exponentially increasing high temperature background attenuation in both types of aggregate is attributed to diffusionally accommodated grain boundary sliding. Extrapolation to mantle grain sizes shows that the broad peak may be responsible for nearly frequency independent attenuation in partially molten regions of the upper mantle.

Lattice Boltzmann method for viscoelastic fluids.

A lattice Boltzmann model for viscoelastic flow simulation is proposed. Elastic effects are taken into account within the framework of a Maxwell model. To test the approach, we estimate the transverse velocity autocorrelation function for a freely evolving system, and find clear manifestations of shear at large frequencies. We then characterize boundary-driven shear waves, and the resonant enhancement of shear oscillations in a periodically driven fluid confined within a capillary. The measured shear-wave dispersion relation is compared to that obtained from the Navier-Stokes equation with a Maxwell viscoelastic term, and good agreement is obtained.

An approximate Biot model for marine sediments with nonuniform pore sizes

The issue of attenuation and dispersion in marine sediments has been investigated by Hamilton [Geophysics (1972)] in detail. In his paper, Hamilton stated that an adequate model for marine sediments must be anelastic in which viscous damping and velocity dispersion should be considered if the pore water moves significantly relative to mineral structure. He introduced an attenuation parameter k which is empirically related to mean grain size and porosity. Based on the experimental measurements in frequencies within 14–100 kHz, he suggested a nearly elastic model with linear frequency dependency of attenuation and negligible velocity dispersion. Then, he explained the peak value of k for fine sands under the context of frictional energy losses between grains. In this study, an approximate Biot model is proposed. The new model includes both frame anelasticity and viscous damping in marine sediments with nonuniform pore sizes. For most marine sediments, the approximate model predicts similar frequency dependency of attenuation and velocity dispersion of compressional and shear waves to those of predicted by the Biot model. It has been shown that mean grain size and porosity dependency of parameter k can be also predicted by the new model without introducing an empirical relation. [Work supported by ONR.]

New techniques in sol–gel characterisation – mechanical measurements and fractal characteristics

We report theoretical and experimental work which demonstrates how the viscoelastic dispersion of shear waves may be exploited in studies of the rheological properties of systems such as `critical-gels' – systems at the gel point. At the gel point a material undergoing gelation changes, from one in which only short range connectivity is present, to one in which structural self-similarity is sample-spanning. Experiments are described in which marked changes in high-frequency shear-wave dispersion are recorded during the viscoelastic-liquid to viscoelastic solid transition about the gel point. The results accord with theoretical treatments of the sol–gel transition, based on a modified form of the Gross–Marvin network model. We report how this modified model has been used to investigate the interdependence of the high- and low-frequency features of the evolving relaxation time spectra associated with the growth of discrete, mechanically self-similar nodal networks. An analysis of the growth of the networks reveals a scale-invariant characteristic of the underlying gel microstructure and an associated fractal dimension, d f, in the range 1< d f<1.8, the maximum value of which corresponds to that reported in computer simulations of `cluster–cluster' aggregation processes. The limitations of conventional rheometry in identifying the inner `cut-off' length scale of such a fractal characteristic are discussed. Shear wave dispersion measurements are also reported for aqueous dispersions of a synthetic clay colloid which, like the simulated viscoelastic networks, combines the characteristics of high-shear elasticity and low-wave attenuation with a maximum fractal dimension of 1.8.

Axial shear waves in fiber-reinforced composites with multiple interfacial layers between fiber core and matrix

Effects of interfacial material properties on the overall dispersion and attenuation of axial shear waves in a multiphase fiber-matrix composite are investigated. The fiber-matrix composite consists of identical fibers that are embedded in a matrix. Multiphase interfacial regions are assumed to exist between the fiber core and the matrix, and their material properties (shear modulus μ and density ρ) are assumed to have linear or exponential variation between fiber core and matrix. Only the case of axial shear waves is studied; thus, the effective axial shear modulus ( M), axial shear wave phase speed ( B), and axial shear wave specific damping capacity ( Ψ) are obtained. The analyses indicate, among others, that the effective mechanical properties of the multiphase composite depend on (in addition to the fiber volume, c, and dynamic parameters I n ) the variations of material properties in the interfacial region. Numerical computations show the following: 1. The effective axial shear wave phase speed ( B) of the composite with multi-interfacial region does not vary with driving frequencies when the frequency range is under 30 Khz. Specifically, for frequency range under 30 Khz, the multiphase composite is indispersive. 2. For the same composite, a higher frequency range will beget higher attenuation. 3. Under the same frequency, certain composites will have higher attenuation; composites with thicker interfacial region, composites with less smooth variation of the interfacial region, and composites with exponential variation (compared with those of linear variation) of the interfacial region.

Shear wave dispersion and mechanical relaxation spectra

It is shown that a mechanical stress relaxation spectrum may be obtained directly from measurements of the dispersion of shear waves in viscoelastic media. The effectiveness of this spectrum, in terms of its sensitivity and accuracy in parametrizing distributions of stress relaxation in the frequency domain, is demonstrated with reference to several published spectra. Of the three different forms of stress relaxation spectrum which may be obtained directly from oscillatory shear measurements, only that based on the group and phase velocities of shear waves is found adequate in terms of sensitivity to the presence of adjacent relaxation mechanisms in the systems considered herein.

Rheometrical aspects of the viscoelastic dispersion of shear waves in gel-like mechanical networks

A modified form of the Gross-Marvin ladder model is used to simulate gelation processes in which α, the viscoelastic stress relaxation exponent in the Gel Equation, can be varied in the range O< α<1. The model has been used to investigate the interdependence of the high- and low-frequency features of the evolving relaxation time spectra associated with the growth of discrete, mechanically self-similar nodal networks. Analysis of the growth of the networks in terms of their wavelengths reveals a fractal characteristic of the underlying gel microstructure. The results obtained may provide a viable rheometrical basis for examining the gel-like characteristics of pre-gel point viscoelastic liquid systems in terms of the Gel Equation but in the absence of a frequency independent loss tangent.