Quasi-static internal deformation due to a dislocation source in a multilayered elastic/viscoelastic half-space and an equivalence theorem

Abstract

We have obtained general expressions for quasi-static internal deformation fields due to a dislocation source in a multilayered elastic/viscoelastic half-space under gravity by applying the correspondence principle of linear viscoelasticity to the associated elastic solution (Fukahata & Matsu'ura 2005). The use of the upgoing propagator matrix for the region below the source and the downgoing propagator matrix for the region above the source in the derivation of mathematical expressions guarantees the numerical stability of the obtained viscoelastic solution over the whole region. The viscoelastic deformation fields due to a dislocation source tend to a certain steady state with the progress of viscoelastic stress relaxation. The completely relaxed viscoelastic solution can be directly obtained from the associated elastic solution by taking the rigidity of the elastic layer corresponding to a Maxwell viscoelastic layer to be zero. We gave an explicit mathematical proof of this theoretical relationship, which we named the equivalence theorem, on the basis of the correspondence principle of linear viscoelasticity and the limiting value theorem of the Laplace transform. The equivalence theorem is applicable not only to the elastic-viscoelastic stratified medium but also to general elastic and linear-viscoelastic composite media. As numerical examples we show the quasi-static internal displacement fields due to strike-slip motion on a vertical fault and dip-slip motion on a subduction plate boundary in an elastic surface layer overlying a viscoelastic half-space. The temporal variation of the computed deformation fields shows that the effective relaxation time of the elastic-viscoelastic system is much longer than the Maxwell relaxation time defined by the ratio of viscosity to rigidity in the viscoelastic layer.