Quasi-Static Strain and Stress Fields due to a Moment Tensor in Elastic–Viscoelastic Layered Half-Space

Abstract

We derived explicit expressions in the time domain for 3-D quasi-static strain and stress fields, due to a point moment tensor source in an elastic surface layer overlying viscoelastic half-space under gravity. The expressions of strain in the elastic surface layer were directly obtained from the expressions of displacement in our previous paper. The conversion of strain into stress is easy, because the stress–strain relation of elastic material is linear. In the viscoelastic substratum, the expressions of strain were obtained by applying the correspondence principle of linear viscoelasticity to the associated elastic solution. The strain–stress conversion is not straightforward, as the stress–strain relation of viscoelastic material is usually given in a differential form. To convert strain into stress, we used an integral form of the stress–strain relation instead of the usual differential form. The expressions give the responses of elastic half-space at \( t = 0 \), and the responses of an elastic plate floating on non-viscous liquid at \( t = \infty \). The moment tensor is rationally decomposed into the three independent force systems, corresponding to isotropic expansion, shear faulting and crack opening, and so the expressions include the strain and stress fields for these force systems as special cases. As the first numerical example, we computed the temporal changes in strain and stress fields after the sudden opening of an infinitely long vertical crack cutting the elastic surface layer. Here, we observe that the stress changes caused by the sudden crack opening gradually decay with time and vanish at \( t = \infty \) everywhere. After the completion of stress relaxation, a characteristic pattern of shear strain remains in the viscoelastic substratum. Since the strain and stress fields at \( t = \infty \) can be read as the strain- and stress-rate fields caused by steady crack opening, respectively, this numerical example demonstrates the realization of a steady stress state supported by steady viscous flow in the asthenosphere, associated with steady seafloor spreading at mid-ocean ridges. For the second numerical example, we computed the temporal changes in strain and stress fields after the 2011 Tohoku-oki mega-thrust earthquake, which occurred at the North American-Pacific plate interface. In this numerical example, the stress changes caused by coseismic fault slip vanish at \( t = \infty \) in the viscoelastic substratum, but remain in the elastic surface layer. The coseismic stress changes (and also strain changes) in the elastic surface layer diffuse away from the source region with time, due to gradual stress relaxation in the viscoelastic substratum.