This exercise is taken largely from Means, 1976, Stress and Strain, Basic Concepts of Continuum Mechanics For Geologists, Springer Verlag, p. 115-117.
As you descend in the crust the vertical traction on a horizontal plane will increase due to the increasing load of rock overhead (as a simple function of its density). This is known as the lithostatic gradient. Given average continental crustal densities this would be a gradient of .027 MPa/m, .267 bar/m, or 1.2 psi/ft. If you want a more specific number for another crust/density, then the density times the gravitational acceleration times the depth will yield the vertical load. In this way one can compute the vertical traction with depth. Without signficant topography, the earth's surface acts as a principal plane (one that has no shear stresses on it) and therefore the vertical traction would also be one of the three principal stresses.In the graph below it is denoted as sigma-vertical, and the units used are Mpal.
Elasticity theory indicates that if you push down vertically on an interior block with a certain load (traction), the rock will try to respond elastically by horizontal extension. Hoever, adjacent rock is in the way, and so a restraining horizontal stress traction develops in response to the vertical load and the horizontal confinement. With water, which is incompressible, the horizonal stress is the same as the vertical (i.e. a hydrostatic stress state). With rock the horizontal stress traction (on a vertical plane) is some fraction of the vertical load. A material property called Poisson's ratio (Pr) helps compute what fraction. Poisson's ratio captures the elastic sponginess of the rock, and the spongier it is the lower the value. The lower the value the lower the horizontal tractions that are produced. The horizontal traction will be equal to Pr/(1-Pr) times the vertical traction. A typical value for Poisson's ratio of rocks is .25, but it can vary considerably. However, in some situations it increases with depth (while we won't include that in our model it is simple to include such a change).
To build a more complete and realistic model, you can then add a horizontal tectonic stress in some direction (in this example in the E-W direction). Earthquakes often indicate tectonic stress magnitudes of 10-15 Mpa, and so we will use that value as the magnitude of sigma-tectonic. Given a common coordinate axis system, the resulting stress state from the two combined tensors (lithostatic and tectonic) is achieved with the simple addition of the components of the tensors. In Mean's model he does not consider how the tectonic load might induce vertical and an orthogonal horizontal components of its own, as was done for the lithostatic load (and we do not consider it either.Ostensibly the crust would be able to respond vertically and so an additional traction would not be generated in that direction. Material 'in the way' would exist in the horizontal direction. The additional horizontal traction would elevate the value of the other horizontal stress traction (by a third of the sigma-tectonic if Poisson's ratio is .25), but this would not change the switchover depth.
Note that at any given shallow level the E-W horizontal tectonic stress is the largest and therefore is sigma one, but at some depth the lithostatic gradient overcomes it and sigma one becomes the vertical traction. This implies that different levels of the crust can be in different dynamic states with different expected kinematics. We can compute that switchover depth of sigma one. A copy of the Excel sheet that produced these graphs is available in Blackboard for you to experiment with.
Below are results from an excel model with a 15 Mpa E-W traction added to a lithostatic gradient and a constant Poisson's ratio of .3. Note the switchover depth is about 800 meters. Increasing Poisson's ratio or the tectonic traction increases the switchover depth. Note also, how the deviatoric stress changes with depth.
It is also interesting in this case to look at how the differential and nondeviatoric components of the stress state change with descent. The results for the model depicted above are shown in the graph below, which has depth as the x axis, and stress traction magnitude in Mpal as the y axis. Remember that the differential component promotes deformation overall (generates the shear stresses), while the nondeviatoric component tends to inhibit it. Remember this is only a simple model, and we have not accounted for the effect of water pressures, changes in Poisson's ratio or other elastic parameters with depth, changes in temperature, and rock anisotropy.
Copyright Harmon D. Maher Jr., This may be used for non-profit educational purposes as long as proper attribution is given. Otherwise, please contact me. Thank you.