Generalized viscoelastic wave equation (original) (raw)
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An acoustic wave equation based on viscoelasticity
An acoustic wave equation for pressure accounting for viscoelastic attenuation is derived from viscoelastic equations of motion. It is assumed that the relaxation moduli are completely monotonic. The acoustic equation differs significantly from the equations proposed by Szabo (1994) and in several other papers. Integral representations of dispersion and attenuation are derived. General properties and asymptotic behavior of attenuation and dispersion in the low and high frequency range are studied. The results are compatible with experiments. The relation between the asymptotic properties of attenuation and wavefront singularities is examined. The theory is applied to some classes of viscoelastic models and to the quasi-linear attenuation reported in seismology.
Geophysical Journal International
In previous studies, the auxiliary differential equation (ADE) method has been applied to 2-D seismic-wave propagation modelling in viscoelastic media. This method is based on the separation of the wave propagation equations derived from the constitutive law defining the stress-strain relation. We make here a 3-D extension of a finite-difference (FD) scheme to solve a system of separated equations consisting in the stress-strain rheological relation, the strain-velocity and the velocity-stress equations. The current 3-D FD scheme consists in the discretization of the second order formulation of a non-linear viscoelastic wave equation with a time actualization of the velocity and displacement fields. Compared to the usual memory variable formalism, the ADE method allows flexible implementation of complex expressions of the desired rheological model such as attenuation/viscoelastic models or even non-linear behaviours, with physical parameters that can be provided from dispersion analysis. The method can also be associated with optimized perfectly matched layers-based boundary conditions that can be seen as additional attenuation (viscoelastic) terms. We present the results obtained for a non-linear viscoelastic model made of a Zener viscoelastic body associated with a non-linear quadratic strain term. Such non-linearity is relevant to define unconsolidated granular model behaviour. Thanks to a simple model, but without loss of generality, we demonstrate the accuracy of the proposed numerical approach.
Dispersion and Attenuation for an Acoustic Wave Equation Consistent with Viscoelasticity
Journal of Computational Acoustics, 2014
An acoustic wave equation for pressure accounting for viscoelastic attenuation is derived from viscoelastic equations of motion. It is assumed that the relaxation moduli are completely monotonic (CM). The acoustic equation differs significantly from the equations proposed by Szabo (1994) and in several other papers. Integral representations of dispersion and attenuation are derived. General properties and asymptotic behavior of attenuation and dispersion in the low and high-frequency range are studied. The results are compatible with experiments. The relation between the asymptotic properties of attenuation and wavefront singularities is examined. The theory is applied to some classes of viscoelastic models and to the quasi-linear attenuation reported in seismology.
Nonlinear viscoelastic wave propagation: an extension of Nearly Constant Attenuation models
Journal of Engineering …, 2009
Hysteretic damping is often modeled by means of linear viscoelastic approaches such as “nearly constant Attenuation (NCQ)” models. These models do not take into account nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior. The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On the one hand, the shear modulus is a function of the excitation level; on the other, the description of viscosity is based on a generalized Maxwell body involving non-linearity. This formulation is implemented into a 1D finite element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude ground motion response. For larger excitation levels, the analysis of seismic wave propagation in a nonlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content).
Wave propagation simulation in a linear viscoelastic medium
Geophysical Journal International, 1988
A new formulation for wave propagation in an anelastic medium is developed. The phenomenological theory of linear viscoelasticity provides the basis for describing the attenuation and dispersion of seismic waves. The concept of a spectrum of relaxation mechanisms represents a convenient description of the constitutive relation of linear viscoelastic solids; however, Boltzmann's superposition principle does not have a straightforward implementation in time-domain wave propagation methods. This problem is avoided by the introduction of memory variables which circumvent the convolutional relation between stress and strain. The formulae governing wave propagation are recast as a first-order differential equation in time, in the vector represented by the displacements and memory variables. The problem is solved numerically and tested against. the solution of wave propagation in a homogeneous viscoelastic medium, obtained by using the correspondence principle.
Nonlinear viscoelastic wave propagation: an extension of Nearly Constant Attenuation (NCQ) models
Journal of Engineering Mechanics-asce, 2009
Hysteretic damping is often modeled by means of linear viscoelastic approaches such as "nearly constant Attenuation (NCQ)" models. These models do not take into account nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior. The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On the one hand, the shear modulus is a function of the excitation level; on the other, the description of viscosity is based on a generalized Maxwell body involving non-linearity. This formulation is implemented into a 1D finite element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude ground motion response. For larger excitation levels, the analysis of seismic wave propagation in a nonlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content).
Viscoelastic Waves in Layered Media
The Journal of the Acoustical Society of America, 2009
This book is a rigorous, self-contained exposition of the mathematical theory for wave propagation in layered media with arbitrary amounts of intrinsic absorption. The theory, previously not published in a book, provides solutions for fundamental wave-propagation problems in the general context of any media with a linear response (elastic or anelastic). It reveals physical characteristics for two-and three-dimensional anelastic body and surface waves, not predicted by commonly used models based on elasticity or one-dimensional anelasticity. It explains observed wave characteristics not explained by previous theories. This book may be used as a textbook for graduate-level courses and as a research reference in a variety of fields such as solid mechanics, seismology, civil and mechanical engineering, exploration geophysics, and acoustics. The theory and numerical results allow the classic subject of fundamental elastic wave propagation to be taught in the broader context of waves in any media with a linear response, without undue complications in the mathematics. They provide the basis to improve a variety of anelastic wave-propagation models, including those for the Earth's interior, metal impurities, petroleum reserves, polymers, soils, and ocean acoustics. The numerical examples and problems facilitate understanding by emphasizing important aspects of the theory for each chapter.
Seismic modeling in viscoelastic media
Geophysics, 1993
Anelasticity is usually caused by a large number of physical mechanisms which can be modeled by different microstructural theories. A general way to take all these mechanisms into account is to use a phenomenologic model. Such a model which is consistent with the properties of anelastic media can be represented mechanically by a combination of springs and dashpots. A suitable system can be constructed by the parallel connection of several standard linear elements and is referred to as the general standard linear solid rheology. Two relaxation functions that describe the dilatational and shear dissipation mechanisms of the medium are needed. This model properly describes the short and long term behaviors of materials with memory and is the basis for describing viscoelastic wave propagation.
A model for longitudinal and shear wave propagation in viscoelastic media
The Journal of the Acoustical Society of America, 2000
Relaxation models fail to predict and explain loss characteristics of many viscoelastic materials which follow a frequency power law. A model based on a time-domain statement of causality is presented that describes observed power-law behavior of many viscoelastic materials. A Hooke's law is derived from power-law loss characteristics; it reduces to the Hooke's law for the Voigt model for the specific case of quadratic frequency loss. Broadband loss and velocity data for both longitudinal and shear elastic types of waves agree well with predictions. These acoustic loss models are compared to theories for loss mechanisms in dielectrics based on isolated polar molecules and cooperative interactions.
The Transmission Problem of Viscoelastic Waves
Acta Applicandae Mathematicae, 2000
In this paper we consider the transmission problem of viscoelastic waves. That is, we study the wave propagations over materials consisting of elastic and viscoelastic components. We show that for this types of materials the dissipation produced by the viscoelastic part is strong enough to produce exponential decay of the solution, no matter how small is its size. We also