SUMMARY

Linear and non-linear viscoelastic (VE) models such as the standard linear solid (SLS) and the generalized SLS (GSLS) are broadly used to represent the anelasticity of materials and Earth's media. However, although the VE approach is often satisfactory for any given observation, the inferred physical causes of anelasticity may be significantly misrepresented by this paradigm, and its predictions may be wrong or inaccurate in other cases. This problem is particularly important in heterogeneous media, including most cases of interest for seismology. For example, in homogenous media, VE and mechanics-based models predict identical quality-factor Q(f) and phase velocity c(f) spectra, but in heterogenous media, these models yield different time-stepping equations and interactions with material–property boundaries. The commonly used VE algorithms for modelling seismic waves rely on postulated convolutional integrals in time, whereas physically, models of rock rheologies should still be based on spatial interactions. To understand how VE models relate to mechanics, it is instructive to consider which physical properties of the medium are constrained reliably and which of them remain unconstrained by a pair of Q(f) and c(f) spectra, that is by VE properties. Despite its popular association with ‘attenuation,’ the peak value of Q⁻¹(f) is actually a purely elastic property representing the existence of two (for SLS) or multiple (for GSLS) elastic moduli. These moduli are analogous to the drained and undrained moduli in poroelasticity or isothermal and adiabatic moduli in thermodynamics. By virtue of the Kramers–Krönig relations, the peak Q⁻¹ is related to the total velocity dispersion, which is also caused by the difference between elastic moduli. By contrast, true anelasticity-related physical properties like viscosity are represented not by Q⁻¹ values but by the frequencies of Q⁻¹(f) peaks in the data. However, these frequencies also depend on multiple material properties that are not recognized or arbitrarily selected in the SLS and GSLS models. Inertial, body-force friction and the corresponding boundary effects are also ignored in VE models, which may again be improper for layered media. Thus, for physically accurate interpretation of laboratory experiments and numerical modelling of seismic waves, first-principle equations of mechanics should be used instead of VE models.

中文翻译：

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地球介质粘弹性模型的力学分析

概要

线性和非线性粘弹性（VE）模型，例如标准线性固体（SLS）和广义SLS（GSLS），广泛用于表示材料和地球介质的无弹性。但是，尽管VE方法对于任何给定的观察结果通常都是令人满意的，但这种范例可能会严重地误导所推断的无弹性物理原因，在其他情况下其预测可能是错误的或不准确的。这个问题在非均质介质中尤其重要，包括大多数地震学感兴趣的情况。例如，在均质介质中，基于VE和力学的模型可预测相同的品质因数Q（f）和相速度c（f）光谱，但是在异质介质中，这些模型会产生不同的时间步长方程以及与材料-特性的相互作用边界。用于建模地震波的常用VE算法在时间上依赖于假定的卷积积分，而在物理上，岩石流变学模型仍应基于空间相互作用。要了解VE模型与力学之间的关系，考虑介质的哪些物理特性受到可靠约束以及哪些Q（f）和c（f）频谱不受VE特性约束是有益的。尽管与“衰减”相关联，但Q的峰值 考虑介质的哪些物理特性受到可靠的约束以及哪些Q（f）和c（f）谱对（即VE特性）不受约束是有益的。尽管与“衰减”相关联，但Q的峰值 考虑介质的哪些物理特性受到可靠的约束以及哪些Q（f）和c（f）谱对（即VE特性）不受约束是有益的。尽管与“衰减”相关联，但Q的峰值^-1（f）实际上是纯弹性，表示存在两个（对于SLS）或多个（对于GSLS）弹性模量。这些模量类似于多孔弹性中的排水和不排水模量，或者类似于热力学中的等温和绝热模量。借助Kramers-Krönig关系，峰值Q ^-1与总速度色散有关，这也由弹性模量之间的差异引起。与此相反，如粘度真滞弹性相关的物理性质不会由Q代表^-1的值，但通过Q的频率^-1（f）数据中的峰值。但是，这些频率还取决于在SLS和GSLS模型中无法识别或任意选择的多种材料特性。惯性，体力摩擦和相应的边界效应在VE模型中也被忽略，对于分层介质而言，这又可能是不合适的。因此，为了物理上准确地解释实验室实验和地震波的数值模型，应使用力学的第一原理方程式代替VE模型。