Page 292 - Physical Principles of Sedimentary Basin Analysis

274 Rheology: fracture and flow

deviatoric stress, and for this special situation (σ 2 = σ 3 ) the deviatoric stress is proportional
to the differential stress (see Section 3.10). The Arrhenius factor makes the rocks ability
to flow strongly temperature dependent. Experimental data shows that the Arrhenius factor
◦
may increase by several orders of magnitude for a temperature increase of 100 C (see
Exercise 8.6). The formulation (8.41) of power law creep applies only in the principal
stress system. Note 8.5 shows how power law creep can be written with respect to any
reference frame.
The strain rate (8.41) can be rewritten as an expression for the stress difference as a
function of strain rate and temperature:

1/n
˙ E

σ 1 − σ 3 = exp . (8.42)
A nRT
This stress difference can be combined with the stress difference necessary for brittle fail-
ure as expressed by Byerlee’s law (8.2) to decide the type of deformation – brittle failure
or flow by creep. If the stress difference for brittle failure is less than for ductile flow the
mode of deformation is brittle failure, otherwise it is by ductile flow. Figure 8.15 shows
examples of such plots, which show the yield strength as an “envelope.” This type of plot
is therefore called a yield strength envelope or simply YSE. Byerlee’s law applies for a
large range of rocks and it is also independent of strain rate and temperature. On the other
hand, the stress difference for ductile flow is dependent on lithology in addition to strain
rate and temperature. For example, silica rich crustal rocks are less strong than mantle
rocks.
Figure 8.15 shows how YSE depends on the temperature, strain rate and crustal thick-
ness. An increasing temperature makes both the lower part of the crust and the lithospheric
mantle more ductile. Figure 8.15a shows the YSE for the four different geotherms in
Figure 8.15b, which differ by the amount of the radioactive heat generation in the crust.
Table 8.1 gives the remaining parameters. The Moho is at the depth 35 km in Figures 8.15a
and 8.15b, and these figures show that both the upper part of the crust and the mantle
have differential stress by frictional sliding according to Byerlee’s law. This is the linear
part of the YSE. The point where Byerlee’s law ends is where the rock becomes ductile,
and this point is called the brittle–ductile transition. The figures show one such point for
the crust and one for the mantle. The nature of the brittle–ductile transition is not point-
like as in Figure 8.15, but it is a depth interval where the deformation gradually goes
from brittle to ductile. If the crust is subdivided into several layers of different litholo-
gies it is possible to have a brittle–ductile transition zone in each layer. Figure 8.15c
shows how an increasing strain rate increases the differential stress, and Figure 8.15d
shows that decreasing the thickness of the crust increases the differential stress. Man-
tle rock is stronger than crustal rocks, and thinning of the crust therefore leads to a
stronger lithosphere, assuming that the geotherm remains the same. It should be noted
that the YSE shown in Figure 8.15 applies for extension, where σ 1 is in the direction of
extension.