The Green’s tensor in anisotropic elastic media

Abstract

I t is well known tha t dislocat ion problems are of fundamenta l impor tance in connexion with the s tudy of the seismic source mechanisms. Such kind of studies h a v e been general ly per formed for isotropic medi~. In the last years some interes t has arisen among seismologists about the effect of anisot ropy (1). In this paper we want to de te rmine the Green's tensor for an infinite anisot ropic elast ic con t inuum. The Green 's tensor so &wived can be used in the Bur r idge-Knopof f representat ion theorem (2) to calcula te the displacements from a dislocation, which is the basis of the ea r thquake mech~mism model. Wc wil l consider the c~se of weak anisot ropy and we will use per tu rba t ion methods and the theory of dis tr ibut ions, The first terms of the asympto t i c expansion of a t ime-harmonic Green 's funct ion for an infinite anisotropic elastic medium have been obta ined by BVCHWALD (a). In such a case the Green 's funct ion represents the vector d isplacement field genera ted by a t i lne-harmonic source of finite extent . The d isplucement field due to the source is represented in te rms of Four ie r integrals t ha t are eva lua ted asympto t i ca l ly and yields expl ic i t expressions for the d isp lacement at points far from the source. Here we will make no res t r ic t ive assumpt ions on the source, but we will consider only weak anisotropy, which leads to great s implif icat ions and is of interes t in seismological applicat ions. Throughou t this paper we employ a rec tangular Cartesian system wi th the abbreviat ion x = (x~, x~, x3) and the usual indicial notat ions. Fo r brevi ty , all the usual regular i ty hypotheses on the considered funct ions wil l be omi t ted . Consider an infinite anisotropic elastic nmdium governed by the equat ions of l inear e las todynamics