Some constraints on levels of shear stress in the crust from observations and theory

In situ stress determinations in North America, southern Africa, and Australia indicate that on the average the maximum shear stress increases linearly with depth to at least 5.1 km measured in soft rock, such as shale and sandstone, and to 3.7 km in hard rock, including granite and quartzite. Regression lines fitted to the data yield gradients of 3.8 MPa/km and 6.6 MPa/km for soft and hard rock, respectively. Generally, the maximum shear stress in compressional states of stress for which the least principal stress is oriented near vertically is substantially greater than in extensional stress regimes, with the greatest principal stress in a vertical direction. The equations of equilibrium and compatibility can be used to provide functional constraints on the state of stress. If the stress is assumed to vary only with depth z in a given region, then all nonzero components must have the form A+Bz, where A and B are constants which generally differ for the various components. This implies that the deviatoric stress changes linearly with depth, and the general solution also allows the directions of the horizontal principal stresses to change monotonically with depth. Solutions to the equations, a churning stress to vary with both z and x, were fit to the observations of Zoback and Roller, who measured stress along a horizontal profile near Palmdale, California. The results indicate that the average shear stress in the upper 8 km of the fault zone is about 3.5 MPa less than the shear stress in the far field, but this far field term, which is part of the solution and has the form A+Bz, cannot be evaluated using the existing constant depth data.