Viscoelastic Wave Propagation Research Articles

Studies of the propagation of viscoelastic spherical divergent stress waves based on the generalized Maxwell model

The governing equations for viscoelastic spherical divergent stress waves are formulated, and the solutions for spherical stress waves are analytically given in the Laplace domain. Based on the generalized Maxwell model, the propagation coefficient for viscoelastic spherical stress waves is obtained analytically and the characteristics of the attenuation coefficient and the phase velocity are discussed. Meanwhile, the solutions for viscoelastic spherical stress waves are calculated by using the numerical method of the inverse Laplace transform in the case of cavity explosion. The propagating characteristics for strong discontinuous visco-elastic spherical waves and steady-state values caused by the cavity pressure are discussed using theoretical and numerical methods.

An efficient seismic modeling in viscoelastic isotropic media

Energy is absorbed and attenuated when seismic waves propagate in real earth media. Hence, the viscoelastic medium needs to be considered. There are many ways to construct the viscoelastic body, in which the generalized standard linear viscoelastic body is the most representative one. For viscoelastic wave propagation and imaging, it is very important to obtain a compact and efficient viscoelastic equation. Because of this, we derived a set of simplified viscoelastic equations in isotropic media on the basis of the standard linear solid body and the constitutive relation for a linear viscoelastic isotropic solid. The simplified equations were composed of the linear equations of momentum conservation, the stress-strain relations, and the memory variable equations. During the derivation of the equations, the Lamé differentiation matrix, which has a similar form to the stiffness matrix and indicates the relations between viscoelastic and elastic stiffness matrices, was introduced to simplify the memory variable equations. Analogous to the elastic equations, the simplified equations have symmetrically compact forms and are very useful for efficient viscoelastic modeling, migration, and inversion. Applied to a 2D simple model and the 2D SEG/EAGE salt model, the results show that our simplified equations are more efficient in computation than Carcione’s equations.

Optimized viscoelastic wave propagation for weakly dissipative media

The representation of viscoelastic media in the time domain becomes more challenging with greater bandwidth of the propagating waves and number of travelled wavelengths. With the continuously increasing computational power, more extreme parameter regimes become accessible, which requires the reassessment and improvement of the standard ‘memory variable’ methods to implement attenuation in time-domain seismic wave-propagation methods. In this paper, we propose a method to minimize the error in the wavefield for a fixed complexity of the anelastic medium. This method consists of defining an appropriate misfit criterion based on a first-order analysis of how errors in the discretized medium propagate into errors in the wavefield and a simulated annealing optimization scheme to find the globally optimal parametrization. Furthermore, we derive an analytical time-stepping scheme for the memory variables that encode the strain history of the medium. Then we develop the coarse grained memory variable approach for the spectral element method (SEM) and benchmark it using the 2.5-D code AxiSEM for global body waves up to 1 Hz. Showing very good agreement with a reference solution, it also leads to a speedup of a factor of 5 in the anelastic part of the code (factor 2 in total) in this 2.5-D approach. A factor of ≈15 (3 in total) can be expected for the 3-D case compared to conventional implementations.

Static versus viscoelastic wave propagation approach for simulating loading effects on flexible pavement structure

Conventional pavement design methods use static and elastic approaches for stress and strain analysis. The hot mix asphalt (HMA) layer is a typical viscoelastic/viscoplastic material as temperature and time dependent, and the more recently developed mechanistic-empirical pavement design guide (MEPDG) has adopted HMA’s dynamic moduli as parameter inputs. However, its stress and strain modeling is based on a static analysis for emulating the dynamic vehicle loading. This study aims to evaluate the effects of static versus viscoelastic wave propagation simulation approaches on pavement responses using finite element method. Four simulation approaches are proposed: (1) static elastic, (2) elastic wave propagation, (3) static viscoelastic, and (4) viscoelastic wave propagation approaches. Two pavement structures under the falling weight deflectometer and vehicle loadings are modeled. Frequency spectrum, modal shape, deflection, stress, and strain are analyzed. Results indicate that the dynamic effect significantly contributes to the time delay of deflections to loading and response oscillations at the rest time due to wave propagation. Material viscoelastic property significantly contributes to the response magnitude and pattern, resulting in higher deflection on the pavement top, higher tensile strain while lower tensile stress at the HMA bottom as compared to quasi-elastic analysis. The viscoelastic wave propagation simulation approach yields closer results with measurements, and thus is recommended for the future pavement analysis and design.

Numerical analysis of wave propagation in fluid‐filled deformable tubes

AbstractThe theory of Biot describing wave propagation in fluid saturated porous media is a good effective approximation of a wave induced in a fluid‐filled deformable tube. Nonetheless, it has been found that Biot's theory has shortcomings in predicting the fast P‐wave velocities and the amount of intrinsic attenuation. These problems arises when complex mechanical interactions of the solid phase and the fluid phase in the micro‐scale are not taken into account. In contrast, the approach proposed by Bernabe does take into account micro‐scopic interaction between phases and therefore poses an interesting alternative to Biot's theory. A Wave propagating in a deformable tube saturated with a viscous fluid is a simplified model of a porous material, and therefore the study of this geometry is of great interest. By using this geometry, the results of analytical and numerical results have an easier interpretation and therefore can be compared straightforward. Using a Finite Difference viscoelastic wave propagation code, the transient response was simulated. The wave source was modified with different characteristic frequencies in order to gain information of the dispersion relation. It was found that the P‐wave velocities of the simulations at sub‐critical frequencies closely match those of Bernabe's solution, but at over‐critical frequencies they come closer to Biot's solution. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Viscowave – a new solution for viscoelastic wave propagation of layered structures subjected to an impact load

The forward models frequently adopted for use in the dynamic backcalculation of the falling weight deflectometer (FWD) data are based on the solutions that utilise discrete transforms. In this paper, a new computational algorithm, namely ViscoWave, has been developed and implemented for modelling the pavement dynamics under the FWD impact load. The primary advantage of the proposed solution over some of the existing solutions is that it uses continuous integral transforms (Laplace and Hankel transforms) that are more appropriate for the FWD time histories whose signal characteristics are transient, non-periodic and truncated. Sample runs of the ViscoWave and the validation efforts presented in this paper showed that the proposed algorithm is capable of modelling the dynamics of a layered structure with elastic or viscoelastic material with or without a halfspace, indicating the potential of ViscoWave as a forward model for dynamic backcalculation of flexible pavement properties.

High-order thin layer method for viscoelastic wave propagation in stratified media

Wave propagation in stratified media has wide applications in petroleum exploration, geophysical inversion, nondestructive evaluation of highway and airport pavement structures, countermine technology, structural health monitoring, and vehicle weigh-in-motion systems. A high-order thin layer method is developed to improve the accuracy and robustness of thin layer method for analyzing viscoelastic wave propagation in stratified media. It approximates the stiffness matrix involving transcendental functions by truncating the Taylor series of the stiffness matrix to the fourth order term. A generalized eigenvalue problem is then formulated, which allows an efficient numerical algorithm to be designed in a computer program DynaThinLayer. The high-order thin layer method is compared to the thin layer method, the stiffness matrix method and the exact Zoeppritz method in terms of accuracy and efficiency. It is shown that the high-order method requires the same level of computation time as does the thin layer method, but the former produce more accurate result than the latter. The computation time of the high-order thin layer method increases linearly when the number of layers increases. The new method is most applicable to situations where a large number of layers is involved or to situations where some natural layers have large thickness.

Back-calculation of elastic modulus of soil and subgrade from portable falling weight deflectometer measurements

In the field of nondestructive testing of pavement, the portable deflectometer devices have gained, in the recent years, a wide use for in situ assessment of elastic properties of soils, subgrade and pavement foundations. However, if the use of the elasto-static model based on the Boussinesq’s theory does not constitute a shortcoming in the back-calculation procedure of homogenous elastic modulus, questions have been arose about the reliability and accuracy of the peak value method commonly used to extract the static stiffness of soils and subgrade from the dynamic transient data. This paper deals with the use of minimization technique, based on least square algorithm as an alternative method for data analysis and soil elastic stiffness identification. Details of the mathematical basis, the implementation and different steps for elastic modulus back calculation, are presented. In this method, an equivalent spring-mass-dashpot system, where the complex dynamic soil behavior is reduced to a viscoelastic one-dimensional wave propagation problem, is used for modeling the loading plate/soil system. The comparative study shows the suitability and the accuracy of the minimization method for soil elastic stiffness identification and elastic modulus back calculation.

Geophysical parameter estimation

Seismic exploration for oil and gas is a parameter estimation problem. Geological properties and fluid content of subsurface reservoirs are sensed from the earth's surface. Seismic data are acquired by generating elastic waves at the surface and recording the reflections off subsurface targets using large receiver arrays. At present, it is not practical to determine earth parameters by iteratively refining an earth model until simulated seismic data match recorded field data. The computational demands of accurately simulating viscoelastic wave propagation in a 3-D heterogeneous earth are prohibitive. Instead, seismic data are processed to remove many complexities of wave propagation in the real environment, like multipath, attenuation, dispersion, shear waves, and interface waves. With these complexities removed, the processed data conform to simpler propagation models which can be solved in reasonable computer time. The cost of each simplification, however, is the loss of some information about the subsu...

1D harmonic response of layered inhomogeneous soil: Analytical investigation

The seismic response of inhomogeneous soil deposits is explored analytically by means of one-dimensional viscoelastic wave propagation theory. The problem under investigation comprises of a continuously inhomogeneous stratum over a homogeneous layer of higher stiffness, with the excitation defined in terms of vertically propagating harmonic S waves imposed at the base of the system. A generalized parabolic function is employed to describe the variable shear wave propagation velocity in the inhomogeneous layer. The problem is treated analytically leading to an exact solution of the Bessel type for the natural frequencies, mode shapes and base-to-surface response transfer function. The model is validated using available theoretical solutions and finite-element analyses. Results are presented in the form of normalized graphs demonstrating the effect of salient model parameters such as layer thickness, impedance contrast between surface and base layer, rate of inhomogeneity and hysteretic damping ratio. Equivalent homogeneous soil approximations are examined. The effect of vanishing shear wave propagation velocity near soil surface on shear strains and displacements is explored by asymptotic analyses.

Relaxation, dispersion, attenuation, and finite propagation speed in viscoelastic media

Dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum are given by integral representations in terms of a positive Radon measure satisfying a growth condition. Kramers–Kronig dispersion relations with one subtraction can be derived from the integral representations of the dispersion and attenuation functions. The dispersion and attenuation functions have sublinear growth in the high frequency range. The wave number vector can have a linear component in addition to the dispersion function. In this case the viscoelastic waves propagate with a bounded speed. In the other cases viscoelastic wave propagation has a diffusion-like character.

Rational approximation for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media

The article presents a numerical inversion method for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media using reconstruction of relaxation spectrum from measured or computed complex velocity or modulus of the medium. Mathematically, the problem is formulated as an inverse problem for reconstruction of relaxation spectrum in the analytic Stieltjes representation of the complex modulus using rational approximation. A rational (Pade) approximation to the relaxation spec trum is derived from a constrained least squares minimization problem with regularization. The recovered stress-strain relaxation spectrum is applied to numerical calculation of frequency-dependent Q factor and frequency-dependent phase velocity for known analytical models of a standard linear viscoelastic solid (Zener) model as well as a nearly constant-Q model which has a continuous spectrum. Numerical results for these analytic models show good agreement between theoretical and predicted values and demonstrate the validity of the algorithm. The proposed method can be used for evaluating relaxation mechanisms in seismic wavefield simulation of viscoelastic media. The constructed lower order Pade approximation can be used for determination of the internal memory variables in time-domain finite difference numerical simulation of viscoelastic wave propagation.

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.

Convolution Quadrature Time-domain Boundary Element Method and Acceleration by the Fast Multipole Method in 2-D Viscoelastic Wave Propagation

Convolution Quadrature Time-domain Boundary Element Method and Acceleration by the Fast Multipole Method in 2-D Viscoelastic Wave Propagation

Operational quadrature time-domain boundary element method and its acceleration by fast multipole method in 2-D viscoelastic wave propagation

Operational quadrature time-domain boundary element method and its acceleration by fast multipole method in 2-D viscoelastic wave propagation

550 粘弾性面内波動問題における高速多重極法を用いたCQ-BEMについて(OS12.境界要素法の高度化と最新応用(3),オーガナイズドセッション)

This paper presents a new time-domain boundary element method (BEM) in 2-D viscoelastic wave propagation problems. In the proposed method, the convolution quadrature method (CQM) developed by Lubich is applied to establish the stability behavior of the time-stepping scheme. In addition, the fast multipole method (FMM) is adapted to improve the computational efficiency for large size problems.

Quasi-static and impact tests of honeycomb

In this paper the quasi-static and instrumented compression impact testing of two kinds of aluminum-alloy honeycomb are reported. Those two types of honeycomb called Hard (H) and Soft (S) were tested. The specimens in cubical form of dimensions 60 mm x 60 mm x 120 mm were made with and without the front aluminum alloy plates (thickness 1.0 mm) cemented to the specimen two faces. The tests have been performed along the largest dimension that is 120 mm, which is parallel to the aluminum sheet profiles forming the honeycomb. A wide range of compression velocities from the quasi-static rate (V0 = 10 mm/min) to the highest impact velocity V6 = 120 m/s were applied. The total number of velocities applied, including the quasi-static loading, was six. Several series of tests were performed. The first two were carried out with the flat-ended strikers of specific masses, which were adequate to each impact velocity. In order to obtain an adequate displacement of crushing the condition of constant kinetic energy of a striker was assumed. In addition, conical strikers were applied with the cone angle 120. Application of the direct impact arrangement along with properly instrumented 9m long Hopkinson bar of Nylon with diameter 80 mm enabled for a wave dispersion analysis to be applied. The crushing force versus time could be exactly determined at the specimen-bar interface by application of an inverse technique along with the theory of visco-elastic wave propagation.

3D Ground-Motion Estimation in Rome, Italy

Paleoseismic evidence and seismic-hazard analysis suggest that the city of Rome, Italy, has experienced considerable earthquake ground motion since its establishment more than 2000 years ago. Seismic hazards in Rome are mainly associated with two active seismogenic areas: the Alban Hills and the Central Apennines regions, located about 20 km southeast and 80–100 km east of central Rome. Within the twentieth century, M 6.8 and M 5.3 earthquakes in the Apennines and the Alban Hills, respectively, have generated intensities up to Mercalli-Cancani-Sieberg scale (mcs) VII in the city. With a lack of strong-motion records, we have generated a 3D velocity model for Rome, embedded in a 1D regional model, and estimated long-period (<1 Hz) ground motions for such scenarios from finite-difference simulations of viscoelastic wave propagation. We find 1-Hz peak ground velocities (PGVs) and peak ground accelerations (PGAs) of up to 14 cm/sec and 44 cm/sec2, respectively, for a M 5.3 Alban Hills scenario, largest near the northwestern edge of the Tiber River. Our six simulations of a M 7.0 Central Apennine scenario generate 0.5-Hz PGVs in Rome of up to 9 cm/sec, as well as extended duration up to 60 sec. The peak motions are similar to, but the durations much longer than those from previous studies that omitted important wave-guide effects between the source and the city. The results from the two scenarios show that the strongest ground-motion amplification in Rome occurs in the Holocene alluvial areas, with strong basin edge effects in the Tiber River valley. Our results are in agreement with earlier 2D SH - wave results showing amplification of peak velocities by up to a factor of 2 in the alluvial sediments, largest near the contact to the surrounding Plio-Pleistocene formations. Our results suggest that both earthquakes from the Alban Hills and the Central Apennines regions contribute to the seismic hazards in Rome. Although earthquakes from the former area may generate the larger peak motions, seismic waves from the latter region may generate ground motions with extended durations capable of causing significant damage on the built environment.

Causal analysis of transient viscoelastic wave propagation

Dispersion due to internal frictions as wave propagates is a consequence of the second principle of thermodynamics. When the wavelengths are several times higher than the mesoscopic inhomogeneities, internal diffractions can be ignored and the propagation medium can then be considered as a continuum at the scale of these wavelengths. Here, we consider the dissipation mechanism due to viscosity only. By mean of Laplace transforms both on time and space, a causal analysis leads us to a closed-form solution, which we think is the simplest analytical form. This is illustrated by searching the viscoelastic Green’s function associated with the horizontal shear wave generated by a uniform impulsive line source in an infinite homogeneous medium, whose example is almost mathematically equivalent to the study of the scalar wave generated in viscous fluid. The described method is thus restricted to one-wave propagation problems and is probably not generally applicable when the source generates several waves. In the course to obtain a transient analytical expression of pulsed wave through a dispersive medium, this study proposes a method for transient cylindrical waves, while most previous methods concern plane waves.

A mixed finite element approach for viscoelastic wave propagation

In this paper, we are interested in the modeling of wave propagation in viscoelastic media. We present a family of models which generalize the Zener’s model. We achieve its mathematical analysis: existence and uniqueness of solutions, energy decay and propagation with finite speed. For the numerical resolution, we extend a mixed finite element method proposed in [8]. This method combines mass lumping with a centered explicit scheme for time discretization. For the resulting scheme, we prove a discrete energy decay result and provide a sufficient stability condition. For the numerical simulation in open domains we adapt the perfectly matched layers techniques to viscoelastic waves [23]. Various numerical results are presented.