Rayleigh-Lamb wave propagation on a fractional order viscoelastic plate

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a 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, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.     

FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

High frequency shear ultrasonic properties of water/sorbitol solutions

Dynamic viscoelastic properties (G′ and G′′), ultrasonic shear velocity and attenuation were measured for aqueous solutions of sorbitol at 5 MHz. For pure sorbitol, the shear ultrasonic velocity reached 1470 m s⁻¹ with a density of 1500 kg m⁻³, consequently leading to a high acoustical impedance compared with “classical” polymers (polystyrene, nylon, polyethylene, Teflon, etc.). We demonstrate that this surprisingly high shear ultrasonic velocity for a viscoelastic material was due to the fact that the glass transition begins at a concentration above 85% of sorbitol in water. Hence, pure sorbitol is an ideal coupling material for high frequency shear experiments.     

Harmonic wave propagation in an infinite viscoelastic medium with a periodic array of cylindrical elastic fibers

The work involves the propagation of plane harmonic waves in an infinite isotropic medium in which a doubly periodic array of cylindrical fibers is embedded. The direction of propagation is perpendicular to the fibers and the matrix material is taken to be viscoelastic in shear, modeled through hereditary integrals. A finite element method based on Galerkin's technique is employed, which leads to a non-linear eigenvalue problem. An iterative scheme is used to obtain two modes of dispersion, for both real and imaginary wave numbers, for a specific composite.     

Dynamics of two-dimensional composites of elastic and viscoelastic layers

Models of frequency response, acoustic transmission, and transient wave propagation are presented for a two-dimensional composite of elastic and viscoelastic layers, simply supported at the two boundaries. The three models adopt transfer matrices to relate state variables over the two faces of a layer. In the frequency domain, a viscoelastic constitutive law is derived by nonlinear fitting a Padé series to measured data of complex shear modulus. For an elastic material, the eigenproblem admits positive real eigenvalues and their negatives. For a viscoelastic material, it admits positive complex eigenvalues and their negative conjugates. The imaginary part of the eigenvalue acts as a velocity-dependent viscous damper. Modal analysis solving transient response utilizes the complex eigenquantities and the static-dynamic superposition method.     

Ultrasonic interferometry for the measurement of shear velocity and attenuation in viscoelastic solids

A method for the measurement of the shear properties of solid viscoelastic materials is presented. The viscoelastic material is cut into a cylindrical sample which is clamped between two rods. The transmission and reflection coefficient spectra of the fundamental torsional mode through the sample are measured by means of two pairs of piezoelectric transducers placed at the free ends of the rod-sample-rod system. Such spectra exhibit maxima and minima which occur approximately at the resonance frequencies of the free viscoelastic cylinder. Therefore, the shear velocity can be obtained by measuring the frequency interval between two consecutive maxima or minima. The shear attenuation is derived by best fitting the analytical expression of the reflection and transmission coefficients to the experimental spectra. The test is very quick to set up as the sample is simply clamped between the two rods.     

Simulation based estimation of dynamic mechanical properties for viscoelastic materials used for vocal fold models

In order to obtain a deeper understanding of the human phonation process and the mechanisms generating sound, realistic setups are built up containing artificial vocal folds. Usually, these vocal folds consist of viscoelastic materials (e.g., polyurethane mixtures). Reliable simulation based studies on the setups require the mechanical properties of the utilized viscoelastic materials. The aim of this work is the identification of mechanical material parameters (Young's modulus, Poisson's ratio, and loss factor) for those materials. Therefore, we suggest a low-cost measurement setup, the so-called vibration transmission analyzer (VTA) enabling to analyze the transfer behavior of viscoelastic materials for propagating mechanical waves. With the aid of a mathematical Inverse Method, the material parameters are adjusted in a convenient way so that the simulation results coincide with the measurement results for the transfer behavior. Contrary to other works, we determine frequency dependent functions for the mechanical properties characterizing the viscoelastic material in the frequency range of human speech (100–250 Hz). The results for three different materials clearly show that the Poisson's ratio is close to 0.5 and that the Young's modulus increases with higher frequencies. For a frequency of 400 Hz, the Young's modulus of the investigated viscoelastic materials is approximately 80% higher than for the static case (0 Hz). We verify the identified mechanical properties with experiments on fabricated vocal fold models. Thereby, only small deviations between measurements and simulations occur.     

Radiation impedance of a finite circular piston on a viscoelastic half-space with application to medical diagnosis

In a recent study a new analytical solution was developed and validated experimentally for the problem of surface wave generation on a linear viscoelastic half-space by a rigid circular disk located on the surface and oscillating normal to it. The results of that study suggested that, for the low audible frequency range, some previously reported values of shear viscosity for soft biological tissues may be inaccurate. Those values were determined by matching radiation impedance measurements with theoretical calculations reported previously. In the current study, the sensitivity to shear viscoelastic material constants of theoretical solutions for radiation impedance and surface wave motion are compared. Theoretical solutions are also compared to experimental measurements and numerical results from finite-element analysis. It is found that, while prior theoretical solutions for radiation impedance are accurate, use of such measurements to estimate shear viscoelastic constants is not as precise as the use of surface wave measurements.     

Analysis of Laser-generated Rayleigh wave on viscoelastic materials

In order to understand the viscoelasticity of material, this research has been conducted to study the propagation characteristics of viscoelastic Rayleigh wave theoretically. A model is presented for the pulsed laser generation of ultrasound on viscoelastic medium surface. Referred to the Kelvin model, the frequency equation and the normal displacement of viscoelastic Rayleigh wave were derived, the influence of the viscoelastic modulus on dispersion and attenuation was discussed. From the theoretical calculation, it is shown that the effect of viscoelasticity on the attenuation of Rayleigh wave is more than that on its dispersion. In the case of a weak viscosity, the attenuation of viscoelastic Rayleigh wave is directly proportional to viscosity modulus; the effect of shear viscosity on the attenuation is much more than that of bulk viscosity. The transient response of viscoelastic Rayleigh wave was also simulated using Laplace and Hankel inversion transform, which are showed in good agreement with the theoretic predictions. The model provides a useful tool for the determination of viscoelastic parameters of medium.     

Oscillations of polymeric microbubbles: effect of the encapsulating shell

A model for the oscillation of gas bubbles encapsulated in a thin shell has been developed. The model depends on viscous and elastic properties of the shell, described by thickness, shear modulus, and shear viscosity. This theory was used to describe an experimental ultrasound contrast agent from Nycomed, composed of air bubbles encapsulated in a polymer shell. Theoretical calculations were compared with measurements of acoustic attenuation at amplitudes where bubble oscillations are linear. A good fit between measured and calculated results was obtained. The results were used to estimate the viscoelastic properties of the shell material. The shell shear modulus was estimated to between 10.6 and 12.9 MPa, the shell viscosity was estimated to between 0.39 and 0.49 Pas. The shell thickness was 5% of the particle radius. These results imply that the particles are around 20 times more rigid than free air bubbles, and that the oscillations are heavily damped, corresponding to Q-values around 1. We conclude that the shell strongly alters the acoustic behavior of the bubbles: The stiffness and viscosity of the particles are mainly determined by the encapsulating shell, not by the air inside.     

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. 2.0 MHz ± Δf/2, where Δf is the difference frequency), can be used to remotely generate modulated low-frequency (Δf ? 500 Hz) 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 30 dB. If the second-harmonic component is used, the RMSE becomes 1.54 and the corresponding CNR decreases by approximately 10–15 dB. 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.     

Kinetic study of silica gels by a new rheological ultrasonic investigation

The last decades have seen the development of sol-gel (SG) process currently used to develop new materials in a wide range of scientific applications. The SG process leads to an oxide macromolecular network through a sol (liquid phase) to gel transition. To optimize this process, the control of the kinetic of the chemical reaction is required. This kinetic can be deduced from the temporal evolution of the viscoelastic parameters. Upto date no complete investigation during the SG formation can be achieved by a unique non-destructive technique. In this paper, we present an ultrasonic technique to measure the viscoelastic parameters (storage G' and loss G' shear moduli) of the gel material during its formation. By using a suitable model which takes into account the mass loading on the surface, the viscoelastic parameters of these materials are accurately deduced. In order to study the efficiency of this technique, silica gels transition is monitored at various formation temperatures and for different initial hydrolysis molar ratio (h). In addition, the monitoring is performed at different oscillatory shear measurements in the 6-54 MHz frequency range to determine a new characteristic time t(vs) corresponding to the moment when the material is no more a newtonian liquid. This characteristic time is then compared to the gelation time t(g) determined by rheological or acoustic audible range methods. Thus the new characteristic time is also a good criterion to characterize earlier the SG matrix transition. Our AT-cut quartz technique using our model can also be used as a high frequency rheometer for the sol-gel materials.     

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

Guided waves propagating in sandwich structures made of anisotropic,viscoelastic, composite materials

The propagation of Lamb-like waves in sandwich plates made of anisotropic and viscoelastic material layers is studied. A semi-analytical model is described and used for predicting the dispersion curves (phase velocity, energy velocity, and complex wave-number) and the through-thickness distribution fields (displacement, stress, and energy flow). Guided modes propagating along a test-sandwich plate are shown to be quite different than classical Lamb modes, because this structure does not have the mirror symmetry, contrary to most of composite material plates. Moreover, the viscoelastic material properties imply complex roots of the dispersion equation to be found that lead to connections between some of the dispersion curves, meaning that some of the modes get coupled together. Gradual variation from zero to nominal values of the imaginary parts of the viscoelastic moduli shows that the mode coupling depends on the level of material viscoelasticity, except for one particular case where this phenomenon exists whether the medium is viscoelastic or not. The model is used to quantify the sensitivity of both the dispersion curves and the through-thickness mode shapes to the level of material viscoelasticity, and to physically explain the mode-coupling phenomenon. Finite element software is also used to confirm results obtained for the purely elastic structure. Finally, experiments are made using ultrasonic, air-coupled transducers for generating and detecting guided modes in the test-sandwich structure. The mode-coupling phenomenon is then confirmed, and the potential of the air-coupled system for developing single-sided, contactless, NDT applications of such structures is discussed.     

Flow birefringence,stress optical rule and rheology of four micellar solutions with the same low shear viscosity

The flow birefringence and the rheological properties of four viscoelastic solutions having nearly the same zero shear viscosity and subjected to shear flows are investigated in the linear and non-linear domains. The surfactant used for the samples is the cetyltrimethylammonium chloride in water at the concentration of 100 mmol/l with an organic salt, the sodium salicylate. The low shear viscosity curve versus the salt concentration is non-monotonic and has two maxima separated by a minimum forming four domains in which the salt concentration is chosen. For the two solutions belonging to the inner branch, i.e. between the two maxima, a simple Maxwellian behaviour is observed and shear banding occurs as confirmed by the flow birefringence pictures. Contrary to the results of P. Fisher (1996) where the unstable flow regime is restricted to the first decreasing part of the low shear viscosity curve of a cetylpyridinium chloride solution, we show that shear banding exits in a wider domain of the salt concentration. Received 18 November 2002 / Published online: 1 April 2003 RID="a" ID="a"e-mail: Decruppe@lpli.sciences.univ-metz.fr     

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.     

Minimization of sound radiation from baffled beams through optimization of partial constrained layer damping treatment

An optimization study is presented with aim to minimize the sound power radiated by a simply supported, baffled beam with constrained layer damping (CLD) treatment. The governing equation of motion for the calculation of time-harmonic response of a partially CLD covered beam is derived first on the basis of energy approach. Assumed-modes method is used to solve the equation with obtained frequency response functions at different beam locations, which are further used for the calculation of its radiated sound power into half free-space by using Rayleigh’s integral. The optimization problem is then formulated to minimize the sound power radiated by the beam over a frequency range of interest covering multiple resonant modes. A genetic algorithm-based penalty function method is employed to search for the optimum of location/length of the CLD patch and the shear modulus of viscoelastic layer. Optimal results show that for a simply supported beam with a transverse force applied at its central location, it is not necessary to fully cover the structure using CLD patch in order to achieve the largest reduction in the sound power radiated by the beam over a frequency range. With inclusion of the amount of damping material to be minimized, the optimal CLD coverage length is only one-fourth of the base beam’s. Moreover, the optima of three design variables, the CLD coverage length, location on the beam and the shear modulus of viscoelastic layer, are highly relevant to each other.     

A simple-shear rheometer for linear viscoelastic characterization of vocal fold tissues at phonatory frequencies

Previous studies reporting the linear viscoelastic shear properties of the human vocal fold cover or mucosa have been based on torsional rheometry, with measurements limited to low audio frequencies, up to around 80 Hz. This paper describes the design and validation of a custom-built, controlled-strain, linear, simple-shear rheometer system capable of direct empirical measurements of viscoelastic shear properties at phonatory frequencies. A tissue specimen was subjected to simple shear between two parallel, rigid acrylic plates, with a linear motor creating a translational sinusoidal displacement of the specimen via the upper plate, and the lower plate transmitting the harmonic shear force resulting from the viscoelastic response of the specimen. The displacement of the specimen was measured by a linear variable differential transformer whereas the shear force was detected by a piezoelectric transducer. The frequency response characteristics of these system components were assessed by vibration experiments with accelerometers. Measurements of the viscoelastic shear moduli (G' and G") of a standard ANSI S2.21 polyurethane material and those of human vocal fold cover specimens were made, along with estimation of the system signal and noise levels. Preliminary results showed that the rheometer can provide valid and reliable rheometric data of vocal fold lamina propria specimens at frequencies of up to around 250 Hz, well into the phonatory range.     

Shear stress at the interface of coaxial composites determined by the resonance of low-amplitude vibrations

The aim of this work is to determine the viscoelastic behaviour of the interface in a coaxial composite material made of a tough shield and a ductile core. The elastic modulus and the amplitude-independent internal friction are measured using a longitudinal oscillating resonant system at 50 kHz. The contribution of the interface is modelled as a shear stress that modifies the elastic behaviour of the constituents. The value of this shear stress is determined for different interfaces (epoxy resin-brass, epoxy resin-Pyrex and paraffin-Pyrex). The model is autovalidated by the excellent agreement between the calculated and experimental values of the internal friction (damping) of the composites.     

Spontaneous thermal runaway as an ultimate failure mechanism of materials

The first theoretical estimate of the shear strength of a perfect crystal was given by Frenkel [Z. Phys. 37, 572 (1926)10.1007/BF01397292]. By assuming that two rigid atomic rows in the crystal would move over each other along a slip plane, he derived the ultimate shear strength to be about one-tenth of the shear modulus. Here we present a theoretical study showing that catastrophic failure of viscoelastic materials may occur below Frenkel's ultimate limit as a result of thermal runaway. The thermal runaway failure mechanism exhibits progressive localization of the strain and temperature profiles in space, thereby producing a narrow region of highly deformed material, i.e., a shear band. We calculate the maximum shear strength sigma_{c} of materials and then demonstrate the relevance of this new concept for material failure known to occur at scales ranging from nanometers to kilometers.