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

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.

💡 Research Summary

The paper tackles the problem of modeling Earth’s free oscillations (FOE) using a quasi‑static extended Burgers model (q‑EBM), an integro‑differential system that captures the anelastic behavior of mantle materials. The authors first develop a systematic method to derive an explicit expression for the relaxation tensor (G(t)) associated with a spatially inhomogeneous but isotropic q‑EBM. By decomposing the infinitesimal strain tensor into volumetric and deviatoric parts, they introduce orthogonal projection operators (I_m) (volumetric) and (J_m) (deviatoric). This decomposition allows the original coupled system of equations to be rewritten as two independent block‑matrix ordinary differential systems, one for the volumetric mode and one for the deviatoric mode. The resulting matrices (L_0) (volumetric) and (L_1) (deviatoric) are ((n+1)\times(n+1)) real symmetric matrices whose entries are linear combinations of the elastic Lamé parameters (\lambda_i,\mu_i) and the viscosities (\eta_i) of the springs and dashpots in the extended Burgers chain.

The authors then diagonalize (L_0) and (L_1). Both matrices are shown to be negative‑definite, guaranteeing real positive eigenvalues (-\kappa_j) and (-\tau_j) (with (j=0,\dots,n)). Corresponding orthonormal eigenvectors (q_j) and (v_j) are used to construct the matrix exponentials (e^{tL_0}) and (e^{tL_1}). The (1,1) components of these exponentials, denoted (g_{00}(t)) and (g^{0}_{00}(t)), are expressed as weighted sums of simple exponentials: \