Restrictions on wave equations for passive media

Hidden jerk in universal creep and aftershocks

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

Wave equations for porous media described by the Biot model

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatial loss terms, whereas in the high frequency regime, it is represented with fractional order temporal and spatial loss terms. In the current work, a time domain representation in terms of a partial differential equation is proposed for all three wave solutions of the Biot model across all frequencies, and it is derived from the material response function of the Biot poroelastic medium with suitable approximations for the compressional modes and the dynamic permeability. The dynamic permeability in the time domain is represented by a fractional pseudo-differential operator. Optimal correction factors are introduced into the wave equation to compensate for the discrepancies in the compressional wave dispersion and attenuation. Additionally, the method for incorporating the squirt flow mechanism into the wave equation via the Extended Biot poroviscoelastic model is described. The proposed wave equation has a physical basis and satisfies the passivity criterion.

Time Domain Characterization of the Cole-Cole Dielectric Model

The Cole-Cole model for a dielectric is a generalization of the Debye relaxation model. The most familiar form is in the frequency domain and this manifests itself in a frequency dependent impedance. Dielectrics may also be characterized in the time domain by means of the current and charge responses to a voltage step, called response and relaxation functions respectively. For the Debye model they are both exponentials while in the Cole-Cole model they are expressed by a generalization of the exponential, the Mittag-Leffler function. Its asymptotes are just as interesting and correspond to the Curie-von Schweidler current response which is known from real-life capacitors and the Kohlrausch stretched exponential charge response.

Towards a consensus on rheological models for elastography in soft tissues

A rising wave of technologies and instruments are enabling more labs and clinics to make a variety of measurements related to tissue viscoelastic properties. These instruments include elastography imaging scanners, rheological shear viscometers, and a variety of calibrated stress-strain analyzers. From these many sources of disparate data, a common step in analyzing results is to fit the measurements of tissue response to some viscoelastic model. In the best scenario, this places the measurements within a theoretical framework and enables meaningful comparisons of the parameters against other types of tissues. However, there is a large set of established rheological models, even within the class of linear, causal, viscoelastic solid models, so which of these should be chosen? Is it simply a matter of best fit to a minimum mean squared error of the model to several data points? We argue that the long history of biomechanics, including the concept of the extended relaxation spectrum, along with data collected from viscoelastic soft tissues over an extended range of times and frequencies, and the theoretical framework of multiple relaxation models which model the multi-scale nature of physical tissues, all lead to the conclusion that fractional derivative models represent the most succinct and meaningful models of soft tissue viscoelastic behavior. These arguments are presented with the goal of clarifying some distinctions between, and consequences of, some of the most commonly used models, and with the longer term goal of reaching a consensus among different sub-fields in acoustics, biomechanics, and elastography that have common interests in comparing tissue measurements.

A multiple relaxation interpretation of the extended Biot model

The biphasic extended Biot poroviscoelastic model takes into account the squirt flow in grain-grain contacts and introduces the bulk and shear relaxation modes associated with it. This model has been criticized for its empirical approach, but here the constitutive equations and the time domain wave equations of the model are derived. This also makes it possible to find single phase viscoelastic equivalents for all three wave solutions of the extended Biot model. Particularly, the viscoelastic equivalents for shear wave propagation can be obtained with considerably fewer parameters than the original model. These equivalents are linear viscoelastic models with springs and dampers for the low frequencies and contain half-order spring-pots for high frequencies. For high frequencies, the non-physicality of the shear relaxation mode is highlighted. The relaxation modes of the extended Biot model are interpreted in the framework of multiple relaxation mechanisms showing that the P- and S-wave modes of the model are not much more complex than that for seawater. The model's near linear frequency dependent attenuation in the intermediate frequency range is the result of weighting each relaxation mechanism appropriately.

The dispersion formula and the Green's function associated with an attenuation obeying a frequency power law

An attenuation obeying a frequency power law scales as |ω|β , where ω is angular frequency and β is a real constant. A recently developed dispersion formula predicts that the exponent β can take only certain values in well defined, disjoint intervals. It is shown here that these admissible values of β are consistent with the physical requirement, stemming from the second law of thermodynamics, that the work done during the passage of a wave must always be positive. Since the dispersion formula, which is derived from the strain-hardening wave equation, is a causal transform, it is expected that the associated Green's function should also satisfy causality for all the permitted values of β. Such is not the case, however: the Green's function is maximally flat at the time of source activation, and hence is causal, but only for values of β in the interval (0.5, 1). This restriction supersedes the weaker constraints on β derived from the dispersion formula alone. For the previously admissible values of β outside the interval (0.5, 1), although the dispersion formula satisfies causality, the Green's function is non-causal. Evidently, causality may be satisfied by the dispersion formula but violated by the Green's function.