[论文解读] Explicit form of relaxation tensor for isotropic extended Burgers model and its spectral inversion

Youjun Deng, Ching-Lung Lin+1 more|arXiv (Cornell University)|Feb 11, 2026

Nonlinear Waves and Solitons被引用 0

一句话总结

本文推导出非均匀各向同性准静态扩展Burgers模型（q-EBM）的显式松弛张量，并分析地球自由振动的特征值簇（C-ev’s），包括用于从C-ev’s辨识q-EBM的反演方法。

ABSTRACT

Concerning the anelastic nature of Earth, the quasi-static extended Burgers model (abbreviated by q-EBM), an integro-differential system, is used to study the free oscillation of Earth (abbreviated by FOE). In this paper, we first provide a general method to obtain an explicit form of the relaxation tensor for inhomogeneous isotropic q-EBM. Then, we apply it to compute the eigenvalues of the free oscillation of Earth, assuming that Earth is a unit ball modeled as a homogeneous and isotropic q-EBM. So far, an analytical and systematic way to compute the eigenvalues of the FOE has been missing when modeling Earth as a q-EBM. In particular, we compute some clusters of eigenvalues (abbreviated by C-ev's). To be more precise, integrating by parts with respect to time of the q-EBM under the assumption that the initial strain is zero, the q-EBM becomes the sum of two terms. The first term, called the instantaneous term, doesn't have any integration with respect to time, but the second term, called the memory term, has such an integration. Then, consider the eigenvalues of the instantaneous part of the q-EBM. The eigenfunctions of C-ev's share the same eigenfunctions of the instantaneous part. However, the C-ev's may be shifted from the eigenvalues of the instantaneous part. Further, we analyze the structure of C-ev's and provide an inversion formula identifying the q-EBM from the C-ev's.

研究动机与目标

• 通过扩展Burgers模型（EBM）来激发并建模地球的非弹性行为。
• 提供非均匀各向同性EBM的显式松弛张量形式。
• 建立计算并分析地球自由振动的特征值簇（C-ev’s）的谱框架。
• 推导从观测C-ev’s识别q-EBM的反演公式。

提出的方法

• 通过体积和偏差分解推导非均匀各向同性EBM的显式松弛核G(t)。
• 用一个完全显式的松弛张量及其分解为瞬时项和记忆项来表达应力-应变关系。
• 将问题化简为准静态EBM并分析相应的Boltzmann型黏弹性系统（BVS）。
• 在牵引无边界条件下，求解对瞬时项和q-EBM都为特征函数的特殊径向特征函数。
• 通过对简化系统的特征分解，计算并分析控制记忆项的矩阵并给出C-ev’s的表达式。
• 提供一个反演框架，将C-ev’s与潜在的q-EBM参数联系起来。

The extended Burgers model consists of one Maxwell model $M_{0}$ and $n$ numeber of the Kelvin-Voigt models $K_{i}$ , $i=1,\cdots,n.$ Also, they are connected in series. Further, the superscripts $s$ and $d$ refer to springs and dashpots, respectively.

实验结果

研究问题

• RQ1 如何为三维非均匀各向同性EBM获得显式松弛张量？
• RQ2 由q-EBM建模的地球自由振动的C-ev簇的结构是什么？
• RQ3 能否从观测的C-evs识别或反演出q-EBM？
• RQ4 有哪些特殊的径向特征函数对瞬时项和q-EBM同时是特征函数？
• RQ5 体积分量和偏差分量如何对松弛及边界牵引条件贡献？

主要发现

• 得到非均质各向同性EBM的显式松弛张量形式。
• q-EBM的应力-应变关系结合了瞬时项和记忆项，从而实现解析谱分析。
• 构造出同时是瞬时项和q-EBM特征函数的特殊径向特征函数。
• 通过简化系统的特征分解得到C-evs的表达式，将其与松弛张量中的参数联系起来。
• 提供一个反演框架，从C-evs识别出q-EBM，阐明C-ev结构如何编码模型参数。