Effect of a non-linear boundary layer on the radiation from earthquakes and underground nuclear explosions

Abstract

A constitutive equation in the immediate source region is postulated. It is proposed that between the permanently deformed source volume and the linear zone there exists a microfractured boundary layer having the constitution of a five-parameter Murnaghan solid that exhibits both geometrical finite strain and physical cubic terms in the energy density function. Waves excited by the source carry the signature of the boundary layer into the linear zone, all the way to the far field. The mathematical realization of this programme consists of the application of the longwave Born–Rayleigh approximation to the Landau–Goldberg equation. The perturbed field, governed by an additional force system, is represented as a multipole expansion of eigenvector spherical harmonics of the unperturbed background field. Our solution aims at the spatial characteristics of the 3-D perturbed field (e.g. radiation patterns, amplitude amplification, etc.) together with its frequency-content features. The results show, in general, that for every spatial mode with indices (ℓ1, m1) fed into the non-linear boundary layer system, there emerges in the elastic zone the output modes (ℓ1±2s, 2m1), s=0,2,4,…. Compressional and shear modes generate each other (‘mode crossing’), and spatial field harmonics of lower and higher orders than originally existed in the primary field appear. In addition, one encounters the expected frequency doubling in the spectral displacements of the first-order perturbation. Detailed calculations are made, especially for dipolar sources, which include explosion (ℓ1=0,m1=0) and shear dislocations (ℓ1=2, m1=0,1,2). It is revealed that the boundary layer tends to amplify the source field by a factor that may reach 4 for explosions in pre-existing cavities and 2 for earthquakes. Hence, the amplification will effectively limit the true seismic decoupling of underground nuclear explosions to a limiting value of under 70 and tend to boost the true seismic moments of earthquakes. In the latter case the non-linear theory suggests the introduction of two new source parameters: a virtual radius, r0, associated with the yield limit of rocks, and a larger radius associated with the elastic limit of rocks. It is shown that for shear dislocations, the fault's area is interpreted geometrically as a virtual spherical surface of radius r0. As long as small finite strains are assumed inside a thin boundary layer, scrambling of the wavefield (that is displacement amplification, generation of higher-degree multipoles and mode crossing) is the most conspicious physical perturbation, independent of any non-classical modification of the constitutive law. In this sense, the salient gross features of our final results are model-independent. Our results serve as a bridge between the microscopic and macroscopic theories of fracture and hopefully pave the road towards a dynamical theory of earthquake-source mechanisms.