Structural Geology
Lecture 20
Friction
(The behavior of faults)

Rock friction is of interest because it controls many processes in the earth's crust including: flexural slip folding; earthquakes; landslides; subduction of continental crust; strike-slip motion of the earth's major plate as they slide past each other.  Ultimately it is the frictional strength of the crust that controls the magnitude of differential stress within the earth's crust.  Friction is an active mechanism of deformation above the brittle-ductile transition within the earth's crust.  It determines the level of shear stress (t) required to induce slip along any existing discontinuity.
The plate tectonics model suggests that large portions of the crust of the earth (lithosphere) move over the mantle (asthenosphere).  This process is believed to be driven by the gravitational forces developed as thermal cooling of oceanic lithosphere takes place next to mid-oceanic ridges.  Motion of the lithosphere is resisted by slip along strike-slip plate boundaries and sbuduction zones.  The collision of plates allows stress to build up within the plates.  The question of commonly asked, "What limits the buildup of differential stress in the lithosphere?".  
In previous lectures we made the point that the earth's crust is pervaded by discontinuities, most of which were mode I cracks (extension fractures, joints, tension cracks, etc.).  Because of these discontinuities a buildup in earth stress is controlled by friction rather than the intact strength of rocks.  In a sense, the generation of fractures, particularly by shear failure, is a local phenomena occurring on relatively small scales.  Measurement of in situ stress (i.e. earth stress) by techniques such as hydraulic fracturing shows that the differential stress throughout much of the crust is less than or equal to the frictional strength of local rocks.
The information that we have on the frictional strength of rocks comes from laboratory tests such as those in Fig. 20-1.  Samples A and B are cross sections of cylinders cut for triaxial testing.  A is a sample sliding on a saw cut in a triaxial test.  B is a sample  sliding on a previously induced shear fracure in a triaxial test.  C is a conventional shear  test without confining pressure but with a normal stress indicated by the vertical arrow.  D is a double shear test with one plate sliding between two other plates.  E is a torsion test.  In tests C and D the normal stress sn and the shear stress t is measured directly.  In the triaxial tests sn and t must be calculated from the principal stresses s1 and s3.  For the torsion test the t  must be integrated over the surface of the cylinder.  The advantage of the torsion test over the other four is that infinite (very large) displacement may be achieved during a test.  Large displacements permit the evolution of steady state conditions along the sliding surface.  In all cases friction, µ, for a single experiment is a the ratio t / sn.

	µ  =  t / sn. 	(20-1)                                                                         

 If sn and t for several experiments are plotted the line connecting the data points may converge at the origin.  In this case µ is the slope of the line and friction has the same meaning as for the single experiment.  But if the line does not converge to the origin, it will cut the t axis at the value S0 which represents the cohesive strength of the fault zone.  Again µ is the slope of the line but

	µ  =  (t - S0)/ sn.	(20-2)

Fig. 20-1

There is a distinction between sliding friction and internal friction that is best illustrated in Mohr Space.  Shear fractures do not form at 45° to s1 and in the plane of tmax but rather in a plane whose normal is closer to s3 than s1.  The reason for this behavior is found in a closer examination of the Coulomb criterion where
	
	t  =  S0  +  µ0sn

where sn and t the normal and shear stress on the plane of failure respectively and S0 is the cohesive strength of the rock (Fig. 20-2).  µ0 is the angle of internal friction

	µ0  =  tan f 

Tan f can not be meassured directly but, rather, is derived from the slope of the Coulomb failure envelop.  µ0 should be distinguished from  the coefficient of sliding friction (µ) which relates t and sn during slip of a fault

	µ  =  t/sn.
                                                                             
(µ is the subject of this lecture).

 µ0 predicts the angle of shear failure whereas the m predicts the angle of frictional slip.

µ0  =  (t - S0)/sn.

(Fig. 20-2a)

 
(Fig.20-2b)

Using one of the five sample configurations shown in figure 20-1, force-displacement curves are derived for various lithologies.  Figure 20-3 illustrates the various types of force-displacement curves.  A shows the ideal curve for a sliding friction test.  Here the force increases until the rock slips.  Once slip is initiated the rock will continue to slip without further increase in the force necessary to maintain slip.  B illustrates a more realistic force-displacement curve where there is a slight increase in force necessary to maintain slip.  Curves A and B are examples of a behavior called stable sliding.  C illustrates a situation where force builds and then suddenly drops.  This same cycle is repeated many times throughout the experiment.  D is the same behavior measured at the sliding surface.  This contrasts with C where displacment was measured at some distance from the sliding surface.  The displacement shown in C is a reflection of the elastic distortion of the load frame during force buildup.  The rock slips instantaneously with a major drop in force as shown in C.  If displacment is measured at the rock sample, it becomes obvious that the force drop accompanies the slip of the rock sample.  This behavior is known as stick-slip and is believed to be similar to earthquakes along a major fault zone within the crust.

Fig.  20-3a

Fig.  20-3b

As we learned during Lecture 21, stick-slip occurs because unloading of the elastic system which was responsible for the force buildup can not follow the force drop of the rock.  Figure 20-3b illustrated this behavior.  The straight line AB is the uncontrolled unloading of the elastic load frame holding the sample (the earth on either side of a fault zone).  Here the earthquake (stick-slip event) starts at point A where the shearing force along the fault zone drops faster than the unloading of the load frame.  This creates excess potential energy which must be absorbed by the loading system.  Slip on the fault will stop when the available excess potential energy represented by the shaded area below the line AB is absorded by the extra work represented by the shaded area above the line AB.  At point B the load frame must reloaded (shearing stress increased) to the point C before slip starts anew.  Stick-slip starts again at point D.