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Explicit form of relaxation tensor for isotropic extended Burgers model and its spectral inversion

Youjun Deng, Ching-Lung Lin, Gen Nakamura

TL;DR

This work provides an explicit relaxation tensor for a three-dimensional, inhomogeneous isotropic extended Burgers model and applies it to the Earth's FOE problem by modeling the planet as a unit ball. By decomposing into volumetric and deviatoric parts, the authors derive a closed-form convolutional stress–strain relation and construct spectral data in the form of cluster eigenvalues (C-ev's). They develop a C-ev framework that reduces FOE to polynomial eigenvalue problems, enabling analysis of eigenvalue structure and offering inversion formulas to recover q-EBM parameters from C-ev data. The results offer analytic tools for geophysical spectral analysis and potential parameter estimation from eigenvalue clusters, with planned numerical validation and generalization to more complex EBMs.

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.

Explicit form of relaxation tensor for isotropic extended Burgers model and its spectral inversion

TL;DR

This work provides an explicit relaxation tensor for a three-dimensional, inhomogeneous isotropic extended Burgers model and applies it to the Earth's FOE problem by modeling the planet as a unit ball. By decomposing into volumetric and deviatoric parts, the authors derive a closed-form convolutional stress–strain relation and construct spectral data in the form of cluster eigenvalues (C-ev's). They develop a C-ev framework that reduces FOE to polynomial eigenvalue problems, enabling analysis of eigenvalue structure and offering inversion formulas to recover q-EBM parameters from C-ev data. The results offer analytic tools for geophysical spectral analysis and potential parameter estimation from eigenvalue clusters, with planned numerical validation and generalization to more complex EBMs.

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.
Paper Structure (6 sections, 12 theorems, 96 equations, 1 figure)

This paper contains 6 sections, 12 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Derivation of the relaxation tensor for inhomogeneous isotropic q-EBM
  3. Special eigenfunctions for the instantaneous term
  4. C-ev's for the FOE
  5. Structure of C-ev's and recovering q-EBM from C-ev's
  6. Conclusions and discussions

Key Result

Lemma 2.1

$L_1^S$ is negative definite.

Figures (1)

  • Figure : 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.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 12 more