Nonlinear stress propagation in the Earth's upper mantle

This paper consists of two parts. The first is theoretical and extends Elsasser's theory of stress propagation in the upper mantle to an asthenosphere with nonlinear rheology. Exact solutions of the nonlinear equations are found for two geologically important problems. The second part uses these theoretical results as the basis for a measurement of the rheology of the asthenosphere. The seaward migration pattern of aftershocks from the February 4, 1965, Rat Island earthquake is analyzed, and strong evidence for a nonoNewtonian stress-strain relation in the asthenosphere is presented. It is found that an individual large earthquake can influence the regional stress pattern only to a distance of about 300 km perpendicular to the line of rupture. Excellent agreement is found between the stress propagation coefficient calculated from the aftershock migration pattern and that calculated from laboratory measurements of high-temperature creep in olivine. We thus arrive at a picture of stress propagation in the upper mantle which is consistent both with theoretical expectation and with observational evidence. INTRODUCTION In 1967, Elsasser 1967 propounded a theory of stress propagation in an elastic lithosphere overlying a viscous asthenosphere. Elsasser showed that provided both the lithosphere and the asthenosphere are thin in comparison to the breadth of the stress distribution, the displacement of the lithosphere from an unstressed state is described by a diffusion-type equation. Recent work on the migration of earthquake epicenters Anderson, 1975 has used Elsasser's theory, and it seems likely that this)ype of theory is well adapted to the description of stresses in lithospheric plates. Recent progress in understanding high-temperature creep of rock Weertman and Weertman, 1975 has, however, led to doubt about the validity of assuming that the asthenosphere exhibits Newtonian viscosity. It seems far more likely that the stress-strain relation in the asthenosphere is nonlinear, of the form where o is the strain rate tensor, ,4 is a dimensional parameter (which is an exponential function of the absolute temperature), o is the deviatoric stress tensor, and = Tr(o) is the second invariant of . The value of n for possible mantle materials is uncertain but probably lies between 2 and 6 Weenman and Weertman, 1975. If the rheology of the asthenosphere is nonlinear (n 1), then the Elsasser theory must be modified. Such modification is the subject of the first part of this paper. The nonlinear equations of motion are derived in the first section. Subsequent sections show that exact solutions of these equations may be obtained in two cases of geologic interest. The first case is that of a sudden change in position of the edge oI a lithospheric plate, approximating conditions in the oceanic plate following a decoupling earthquake. The second case describes the displacements in a plate whose edge suddenly begins to move with constant velocity, approximating a change in forces acting on the edge of a lithospheric plate, or the average effect of a large number of decoupling earthquakes.