Theory of attenuation and finite propagation speed in viscoelastic media (original) (raw)

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

Journal of Mathematical Physics, 2010

It is shown that the dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum have a sublinear growth rate at very high frequencies. A local dispersion relation in parametric form is found. The exact limit between attenuation growth rates compatible and incompatible with finite propagation speed is found. Incompatibility of superlinear frequency dependence of attenuation with finite speed of propagation and with the assumption of positive relaxation spectrum is demonstrated.

Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli

Wave Motion, 2013

It is shown that viscoelastic wave dispersion and attenuation in a viscoelastic medium with a completely monotonic relaxation modulus is completely characterized by the phase speed and the dispersion-attenuation spectral measure. The dispersion and attenuation functions are expressed in terms of a single dispersion-attenuation spectral measure. An alternative expression of the mutual dependence of the dispersion and attenuation functions, known as the Kramers-Kronig dispersion relation, is also derived from the theory. The minimum phase aspect of the filters involved in the Green's function is another consequence of the theory. Explicit integral expressions for the attenuation and dispersion functions are obtained for a few analytical relaxation models.

Effects of Newtonian viscosity and relaxation on linear viscoelastic wave propagation

2019

In an important class of linear viscoelastic media the stress is the superposition of a Newtonian term and a stress relaxation term. It is assumed that the creep compliance is a Bernstein class function, which entails that the relaxation function is LICM. In this paper the effect of Newtonian viscosity term on wave propagation is examined. It is shown that Newtonian viscosity dominates over the features resulting from stress relaxation. For comparison the effect of unbounded relaxation function is also examined. In both cases the wave propagation speed is infinite, but the high-frequency asymptotic behavior of attenuation is different. Various combinations of Newtonian viscosity and relaxation functions and the corresponding creep compliances are summarized.

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

The effect of Newtonian viscosity and relaxation on linear viscoelastic wave propagation

2019

The effect of Newtonian viscosity superposed on relaxation on wave propagation in a linear viscoelastic medium is examined. It is shown that the Newtonian viscosity is a dominates over the features resulting from relaxation. Since a generic linear viscoelastic medium has a Newtonian component it follows that the properties of wave motion derived here are generic. For comparison we also study the effect of unbounded relaxation function. In both cases the wave propagation speed is infinite.

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).

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.

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).