Frequency-dependent Complex Modulus Research Articles

A hybrid constitutive model of high-damping viscoelastic materials used for vibration control of civil structures

The dynamic mechanical properties of high-damping viscoelastic (VE) materials exhibit significant frequency and temperature dependences in the glass transition zone. However, the current state of VE constitutive models face a challenge in balancing the accuracy, simplicity, and ease of structural dynamic analysis. In this study, a power-law and exponent-law hybrid model is proposed to model the frequency-dependent and temperature-dependent complex modulus of VE materials. The equivalence of hybrid model and existing VE constitutive models is presented. A total of 12 groups of experimental data are collected to validate the accuracy of hybrid model. An efficient numerical method for the hybrid model is proposed and then validated through a viscoelastically damped frame. Results show that the hybrid model is equivalent to two rheological models or fractional derivative models owning to the unshared model coefficients. The hybrid model is superior to the eight-parameter fractional derivative model in predicting the complex modulus over a wide equivalent frequency range. The proposed numerical method performs quite well in calculating seismic responses of proportional and non-proportional damped frames.

The balanced proper orthogonal decomposition applied to a class of frequency-dependent damped structures

A model reduction technique aimed at computing efficiently Frequency Response Functions of damped structures is presented. The frequency-dependent complex moduli are approximated by a mini-oscillators model, known as the Golla–Hughes–MacTavish (GHM) model, which permits to recast the original problem as a more familiar second-order, constant-coefficient system of equations. The matrix system, although much larger, is then treated by application of the Balanced Proper Orthogonal Decomposition (BPOD) which aims at approximating the transfer function matrix, or equivalently the admittance matrix, connecting forces and displacements at a specified set of points of the vibrating structure. All the necessary ingredients of the reduction strategy as well as its efficiency measured in terms of data reduction, accuracy and computational cost are shown. Two illustrative examples of increasing complexity involving a clamped cantilever beam and a realistic windshield are presented. It is shown that the admittance matrix can be approximated by matrices of very small size which computation can be speeded up via diagonalization. It is concluded that the application of BPOD combined with the GHM decomposition of the frequency-dependent algebraic system proves extremely efficient for the modeling of vibrating structures made of different materials, either viscoelastic or purely elastic.

Determination of Frequency-Dependent Shear Modulus of Viscoelastic Layer via a Constrained Sandwich Beam.

Viscoelastic material can significantly reduce the vibration energy and radiated noise of a structure, so it is widely used in lightweight sandwich structures. The accurate and efficient determination of the frequency-dependent complex modulus of viscoelastic material is the basis for the correct analysis of the vibro-acoustic behavior of sandwich structures. Based on the behavior of a sandwich beam whose core is a viscoelastic layer, a combined theoretical and experimental study is proposed to characterize the properties of the viscoelastic layer constituting the core. In this method, the viscoelastic layer is bonded between two constraining layers. Then, a genetic algorithm is used to fit the analytical solution of the frequency¬ response function of the free–free constrained beam to the measured result, and then the frequency-dependent complex modulus is estimated for the viscoelastic layer. Moreover, by varying the length of the beams, it is possible to characterize the frequency-dependent complex modulus of the viscoelastic material over a wide frequency range. Finally, the characterized frequency-dependent complex modulus is imported into a finite element model to compute the complex natural frequencies of a sandwich beam, and a comparison of the simulated and measured results displays that the errors in the real parts are within 2.33% and the errors in the imaginary parts are within 3.31%. It is confirmed that the proposed method is feasible, accurate, and reliable. This provides essential technical support for improving the acoustic vibration characteristics of sandwich panels by introducing viscoelastic materials.

Unbiased retrieval of frequency-dependent mechanical properties from noisy time-dependent signals

The mechanical response of materials to dynamic loading is often quantified by the frequency-dependent complex modulus. Probing materials directly in the frequency domain faces technical challenges such as a limited range of frequencies, long measurement times, or small sample sizes. Furthermore, many biological samples, such as cells or tissues, can change their properties upon repetitive probing at different frequencies. Therefore, it is common practice to extract the material properties by fitting predefined mechanical models to measurements performed in the time domain. This practice, however, precludes the probing of unique and yet unexplored material properties. In this report, we demonstrate that the frequency-dependent complex modulus can be robustly retrieved in a model-independent manner directly from time-dependent stress-strain measurements. While applying a rolling average eliminates random noise and leads to a reliable complex modulus in the lower frequency range, a Fourier transform with a complex frequency helps to recover the material properties at high frequencies. Finally, by properly designing the probing procedure, the recovery of reliable mechanical properties can be extended to an even wider frequency range. Our approach can be used with many state-of-the-art experimental methods to interrogate the mechanical properties of biological and other complex materials.

Modeling the Effect of Microscopic and Mesoscopic Heterogeneities on Frequency-Dependent Attenuation and Seismic Signatures

At seismic and sonic frequencies, the major cause of wave attenuation and dispersion in fluid-saturated rocks might be the wave-induced fluid flow on microscopic and mesoscopic scales. However, it is challenging to assess these effects as the attenuation mechanisms related to both heterogeneities that cannot be expected to be independent. This is due to the fact that, fluid flow taking placing at the mesoscopic scale may be impacted by squirt flow mechanism in the presence of microscopic heterogeneities via modifying the dry rock to be frequency-dependent complex moduli. Understanding the integrated effects, related to microscopic squirt flow and wave-induced fluid flow of mesoscopic heterogeneities, would be important for quantifying the relative contribution of the interdependent energy loss mechanisms. We introduce a procedure in this letter to estimate the frequency-dependent seismic attenuation and dispersion by considering the combined presence of microscopic and mesoscopic heterogeneities. The corresponding seismic reflections of a finely stratified model with a dispersive reservoir are calculated using a propagator matrix method in the frequency domain to study the sensitivity of seismic signatures to pore-fluid mobility and rock heterogeneities.

Assessment of the apparent bending stiffness and damping of multilayer plates; modelling and experiment

In the context of aeronautics, automotive and construction applications, the design of light multilayer plates with optimized vibroacoustical damping and isolation performances remains a major industrial challenge and a hot topic of research. This paper focuses on the vibrational behavior of three-layered sandwich composite plates in a broad-band frequency range. Several aspects are studied through measurement techniques and analytical modelling of a steel/polymer/steel plate sandwich system. A contactless measurement of the velocity field of plates using a scanning laser vibrometer is performed, from which the equivalent single layer complex rigidity (apparent bending stiffness and apparent damping) in the mid/high frequency ranges is estimated. The results are combined with low/mid frequency estimations obtained with a high-resolution modal analysis method so that the frequency dependent equivalent Young's modulus and equivalent loss factor of the composite plate are identified for the whole [40 Hz-20 kHz] frequency band. The results are in very good agreement with an equivalent single layer analytical modelling based on wave propagation analysis (model of Guyader). The comparison with this model allows identifying the frequency dependent complex modulus of the polymer core layer through inverse resolution. Dynamical mechanical analysis measurements are also performed on the polymer layer alone and compared with the values obtained through the inverse method. Again, a good agreement between these two estimations over the broad-band frequency range demonstrates the validity of the approach.

Squirt-Flow in Fluid-Saturated Porous Media: Propagation of Rayleigh Waves

Propagation of attenuated waves is studied in a squirt-flow model of porous solid permeated by two different pore regimes saturated with same viscous fluid. Presence of soft compliant microcracks embedded in the grains of stiff porous rock defines the double-porosity formation. Microcracks and pores respond differently to the compressional effect of a propagating wave, which induces the squirt-flow from microcracks to pores. Elastodynamics of constituent particles in porous aggregate is represented through a single-porosity formulation, which involves the frequency-dependent complex moduli. This formulation is deduced as a special case of double-porosity formation allowing the wave-induced flow of pore-fluid. This squirt-flow model of porous solid supports the attenuated propagation of two compressional waves and one shear wave. Superposition of these body waves, subject to stress-free surface, defines the propagation of Rayleigh wave. This wave is governed by a complex irrational dispersion equation, which is solved numerically after rationalising into an algebraic equation. For existence of Rayleigh wave, a complex solution of the dispersion equation should represent a leaky wave, which decays for propagation along any direction in the semi-infinite medium. A numerical example is solved to analyse the effects of squirt-flow on phase velocity, attenuation and polarisation of the Rayleigh waves, for different combinations of parameters. Numerical results suggest the existence of an additional (second) Rayleigh wave in the squirt-flow model of dissipative porous solids.

A complex elastographic hyperbolic solver (CEHS) to recover frequency dependent complex shear moduli in viscoelastic models utilizing one or more displacement data-sets

In this paper, we present a linear marching scheme to recover frequency-dependent complex shear moduli in viscoelastic models utilizing two sets of single component displacement data. The proposed method is designed to provide stable and accurate estimation of the tissue viscoelastic stiffness parameters by solving a first-order complex partial differential equation. To control the exponential growth of the numerical error resulting from one of the complex coefficients in the inverse equation, a modified upwind discretization is utilized on the first-order derivative terms of the target parameter. The algorithm is fully stablized when: (1) carefully chosen multiple data-sets are combined to eliminate the remaining complex coefficient that contributes to exponential error growth; and (2) a modified Tikhonov regularization is applied to the inversion method. We obtain the stability result in the norm so that the numerical scheme is convergent at fractional 1 / 2 order. Its performance is compared with the performance of the Algebraic Inversion Model previously investigated. We present shear modulus reconstructions from synthetic data, from laboratory phantom data and match frequency-dependent complex moduli from phantom data to several viscoelastic models. Since we have previously presented phase wave speed images from interference patterns, we exhibit those images here for comparison.

Frequency-dependent mechanical modelling of poro-elastic materials for sound transmission loss simulations

Multilayer panels made of sheet steel, a poro-elastic layer and an impervious mass are widely used in noise control for increasing sound insulation. A complete description of the vibro-acoustical behavior of the poro-elastic material used in these panels requires knowledge of mechanical parameters (i.e. complex modulus and Poisson's ratio) in order to model the wave propagation through the elastic structure constituting its skeleton. Although many of the materials used for noise and vibration control have viscoelastic behavior, prediction models currently make use of the linear elasticity theory. As a consequence, acoustical quantities, such as sound transmission loss, are predicted by using static or quasi-static values of mechanical properties. In this research, an experimental time domain approach in combination with an analytical model for linear viscoelasticity is utilized for determining the frequency-dependent complex modulus and increasing the accuracy of sound transmission loss simulations in practical applications.

Aeroelastic dynamics stability of rotating sandwich annular plate with viscoelastic core layer

A dynamic model for a rotating sandwich annular plate with a viscoelastic core layer is developed. All fundamental equations and boundary conditions are established based on Hamilton’s principle, and the rotation effect and viscoelastic properties of the sandwich structure are taken into account. The aerodynamics force acting on the plate is described by a rotating damping model, and the constitutive behavior of the viscoelastic core layer is formulated by the frequency-dependent complex modulus. The effects of geometrical and material parameters on frequencies and damping of forward and backward traveling waves and the dynamic stability for the rotating sandwich plate are numerically analyzed by means of Galerkin’s method. The results show that the critical and flutter speeds of the rotating plate can be increased at some certain parameters of the viscoelastic core layer.

Dynamic Analysis of Three-Layer Sandwich Beams with Thick Viscoelastic Damping Core for Finite Element Applications

This paper presents an analysis of the dynamic behaviour of constrained layer damping (CLD) beams with thick viscoelastic layer. A homogenised model for the flexural stiffness is formulated using Reddy-Bickford’s quadratic shear in each layer, and it is compared with Ross-Kerwin-Ungar (RKU) classical model, which considers a uniform shear deformation for the viscoelastic core. In order to analyse the efficiency of both models, a numerical application is accomplished and the provided results are compared with those of a 2D model using finite elements, which considers extensional and shear stress and longitudinal, transverse, and rotational inertias. The intermediate viscoelastic material is characterised by a fractional derivative model, with a frequency dependent complex modulus. Eigenvalues and eigenvectors are obtained from an iterative method avoiding the computational problems derived from the frequency dependence of the stiffness matrices. Also, frequency response functions are calculated. The results show that the new model provides better accuracy than the RKU one as the thickness of the core layer increases. In conclusion, a new model has been developed, being able to reproduce the mechanical behaviour of thick CLD beams, reducing storage needs and computational time compared with a 2D model, and improving the results from the RKU model.

A Novel Viscoelastic Material Modulus Function for Modifying the Golla-Hughes-McTavish Method

The growing popularity of finite element analysis in the 1980s and 1990s spawned new techniques for modelling damping in complex structures. The Golla-Hughes-McTavish (GHM) method is one technique developed during this era. This method adds non-spatial degrees of freedom to a finite element model in order to account for a viscoelastic material’s ability to dissipate energy. In the GHM method, a material modulus function is used to characterize the frequency-dependent complex modulus. This paper presents a novel material modulus function, thus modifying the GHM method. The advantages of this Modified Golla-Hughes-McTavish (MGHM) approach are a reduction in the curve fitting error and a standardized approach for computing material modulus coefficients. An additional parameter is introduced for each dissipation degree of freedom. This paper will compare the original GHM curve fit approach with this new standardized MGHM approach. Advantages of the MGHM approach and physical insight into the model are explained.

Estimation of material functions using system identification techniques

Material properties of an elastic material are characterized by the elastic modulus, which is real-valued and constant. For viscoelastic materials, such as plastics and polymers, the relationship between stress and strain is instead dynamic, and characterized by the complex-valued and frequency-dependent complex modulus. It is in this paper described how system identification techniques can be used to determine the complex modulus using strain data from wave propagation experiments on a test specimen. Modeling, derivation of estimators, and analysis of their numerical and statistical properties are included. Identifiability and experimental design are examined in some detail. Several practical examples are presented using real-world data, and a number of extensions are outlined.

Dynamic characteristics of traveling waves for a rotating laminated circular plate with viscoelastic core layer

The dynamic governing equations and the corresponding boundary conditions for a rotating thin laminated circular plate with a viscoelastic core layer are derived in this paper based on the Hamilton principle. The analysis on dynamic features of the forward and Backward Traveling Waves for the rotating laminated plate is performed by means of Galerkin's method. The frequency-dependent complex modulus model for describing the constitutive behavior of the viscoelastic core layer is employed. The dynamic characteristics of frequencies and dampings of traveling waves for the rotating plate are obtained numerically. The effects of geometrical and material parameters on the critical speed of the rotating laminated plate with viscoelastic core are discussed in detail.

Application of resonant acoustic spectroscopy to beam shaped asphalt concrete samples.

The dynamic modulus of asphalt concrete is a key parameter needed in modern pavement design and management. Traditional laboratory tests based on cyclic loading (0.1–25 Hz) at different testing temperatures are time consuming and require expensive equipment. There is therefore a need for more efficient non‐destructive methods to determine the dynamic modulus of asphalt concrete. This study applies resonant acoustic spectroscopy (RAS) to beam shaped asphalt concrete samples. Multiple modes of vibration are measured at each testing temperature using a miniature accelerometer and a small steel sphere as impact source. The complex modulus from each resonant frequency is calculated using the Rayleigh–Ritz method. The heterogeneous and viscoelastic nature of asphalt concrete presents challenges to the application of conventional RAS. The number of measurable modes decreases with increasing test temperature. In an attempt to extend the usable frequency and temperature range measured, transfer functions are inverted using the finite element method along with a frequency dependent complex modulus. Initial results indicate that RAS can be an efficient method for the prediction of the high‐frequency part of the asphalt concrete dynamic modulus mastercurve.

Investigation of temperature-dependent viscoelastic properties of thermal lesions in ex vivo animal liver tissue

The viscoelastic characteristics of thermal lesions in ex vivo animal liver are investigated in this paper. Characterization of the moduli of thermal lesions prepared at several temperatures will provide additional information for the elastographic monitoring of radio frequency ablation of hepatic tumors. In this study, the frequency-dependent complex modulus of thermal lesions prepared at temperatures ranging from 60 to 90 °C over a frequency range from 0.1 to 50 Hz are presented. Lesions were prepared using either radio frequency ablation or double immersion boiling. It was found that both the magnitude and phase of the modulus increase with frequency, a behavior that has been noted in the literature. A new result reported shows that the modulus dependence on temperature reveals a local maximum around 70–75 °C corresponding to the temperature at which tissue has released most of its water content. The modulus values at temperatures higher than 70 °C continued to increase, but the extent of increase depend on animal species and other factors.

Ultrasonic Energy Transfer and Stress Concentrations in a Single-Fiber Composite with Absorbing Interface Layer

Elastic wave scattering from coated and layered cylindrical scatterers encased in an elastic matrix is reviewed. Interaction of time harmonic monochromatic longitudinal and shear ultrasonic waves with a single fiber coated by homogeneous dissipative (polymeric) materials and embedded in an unbounded elastic matrix is studied using the classical method of eigenfunction expansion. The novel features of Havriliak—Negami model is employed to include the effect of hysteresis-type of compressional and shear waves' absorption in the interphase material by introduction of proper frequency-dependent complex moduli. The pertinent integrals for the time-averaged total fiber energy induced by the elastic waves refracted along the fiber are derived and numerically evaluated. The numerical results reveal the important effects of the absorptive fiber—matrix bond as well as interphase thickness on fiber/matrix energy transfer and dynamic stress concentrations. The calculated results may be interpreted in the context of phenomena observed in acousto-ultrasonic (AU) experiments on fiber reinforced composite materials by correlating certain parameters in the frequency spectrum of the detected waveform to mechanical characteristics of the dissipative interface layer. Limiting cases are considered and good agreements with well-known solutions are established.

Frequency-dependent complex modulus of the uterus: preliminary results

The frequency-dependent complex moduli of human uterine tissue have been characterized. Quantification of the modulus is required for developing uterine ultrasound elastography as a viable imaging modality for diagnosing and monitoring causes for abnormal uterine bleeding and enlargement, as well assessing the integrity of uterine and cervical tissue. The complex modulus was measured in samples from hysterectomies of 24 patients ranging in age from 31 to 79 years. Measurements were done under small compressions of either 1 or 2%, at low pre-compression values (either 1 or 2%), and over a frequency range of 0.1–100 Hz. Modulus values of cervical tissue monotonically increased from approximately 30–90 kPa over the frequency range. Normal uterine tissue possessed modulus values over the same range, while leiomyomas, or uterine fibroids, exhibited values ranging from approximately 60–220 kPa.

NONPARAMETRIC IDENTIFICATION OF COMPLEX MODULUS USING A NON-EQUILIBRIUM SHPB PROCEDURE

We investigate non-parametric identification of the frequency dependent complex modulus of a viscoelastic material, using a non-equilibrium split Hopkinson pressure bar technique. Three different procedures for estimating the complex modulus from experimental data are considered, and an analysis of the quality of the estimates is carried out. The validity of the theoretical results are confirmed by numerical studies and experimental tests.

Micromechanical Modeling of Particulate Composites for Damping of Acoustic Waves

The self-consistent (SC) micromechanical model of a composite containing coated micro-inclusions, originally proposed in the static regime by Cherkaoui et al. (1994, J. Eng. Mater. Technol., 116, 274–278), is implemented in the quasistatic regime by the introduction of frequency dependent complex moduli for the matrix material. The original model is improved by using dilute strain concentration tensor (DSCT) formulation. It is shown that these concentration tensors can be used to approximate effective composite behavior of composites containing ellipsoidal inclusions having a known orientation distribution or of composites containing multiple types of coated inclusions. The DSCT formulation is also shown to be capable of modeling the effects of multiple scales (submicron-meso-macro), as well as that of a distribution of inclusion coating thicknesses. Various potential material modeling applications are verified through comparison with experimental data in the literature. Notably, the DSCT SC model is applied in the quasistatic regime for calculation of acoustic transmission loss of a slab of viscoelastic composite submerged in water for the range of frequencies between 0-100kHz and compared with experimental data of Baird et al. (1999, J. Acoust. Soc. Am., 105, 1527–1538).