STRUCTURAL GEOLOGY
Lecture 22
The Mechanics of Faulting
(The strength of the Earth’s crust)

Faults and fault zones are mechanical systems which lend themselves to rigorous analysis using both experimental and analytical approaches.  Deformation textures and fault patterns observed within natural fault zones serve as precise guides to the selection of appropriate constitutive equations for the prediction of the behavior of faults.  These constitutive relations are known through lab experiments and if properly scaled define the reaction of fault zones to various stress conditions.
The purpose of this lecture is to review for the structural geologist the analytical approaches to understanding the mechanics of faulting.  The observational and experimental approaches were covered in the lectures on the Coulomb Failure Criterion, The Griffith Failure Criterion, and Friction.  The ingredients for an analysis of fault zones using mechanics will include:  (1) a statement concerning state of stress along fault zones including evidence for magnitude and orientation along natural fault zones: (2) the yield criterion for slip on faults; (3) the mode of slip and nature of deformation mechanisms accompanying and following yield; and (4) the energy expended during the faulting process as inferred from both rock mechanics and seismology.
Anderson (1905) recognized that principal stress orientations could vary among geological provinces within the upper crust of the earth.  Armed with this insight Anderson was able to deduce the connection between three common fault types:  normal, strike-slip, and thrust and the three principal stress systems arising as a consequence of the assumption that one principal stress must be normal to the eart h's surface (Fig. 22-1).

(FIG. 22-1)

Anderson's classification of faulting shown in Figure 22-1shows principal stresses written with subscripts 1,2, and 3.  These subscripts immediately identify the relative magnitude of the three principal stresses with the subscript 1 denoting the maximum principal stress.  The principal stress may also be written in terms of a coordinate system where the subscripts are x, y, and z with these subscripts representing the orientation of the principal stresses rather than their magniatude.  The Anderson classification can be viewed quantitatively assuming that the stresses in the x, y, and z directions are the principal stresses.   Note that in this coordinate system the vertical direction is indicated by y. 
The state of stress within the crust of the earth may change according to local conditions.  For example in a magma chamber where the rock is molten the the three principal stresses are equal so that 
	
	syy  =  sxx  = szz  = Pm	
where Pm is the pressure of the liquid magma. This state of stress is found in a liquid where all three principal stresses increase with depth in the liquid.  Immediately upon cooling below the solidus the three principal stresses in the solid rock will also be equal.  Any time this situation occurs in solid rock the state of stress is said to be lithostatic.  Such a stress state is unusual in the crust of the earth.
Another unusual state of stress is one in which the horizontal components of stress contain no contribution from a tectonic stress. In this case the horizontal components proportional to but less than the vertical component of stress.  Using the theory of elasticity this situation is represented by a uniaxial strain model (Fig. 22-2).  This model is equivalent to placing a rock in a box with four sides but open top and bottom.  In such a model the lateral boundaries are assumed to be fixed so that they do not move regardless of the magnitude of the push against them.  The horizontal components of stress are the magntiude of the push against the fixed boundaries.

(Figure 22-2)

For the uniaxial strain model the overburden pressure is the vertical component of the stress

	syy  =  rgz                                                                            
The vertical component of stress arises from the weight of the rocks above the point in question.  Note that the natural tendency  for a cube of rock without fixed boundaries would be to expand in the horizontal directions when loaded top and bottom.  This expansion is known as the Poisson effect where the horizontal expansion is some fraction of the vertical shortening. The following equation represents the Poisson effect:

	u  =   (horizontal expansion)/(vertical shortening) < 0.5

where u is Poisson's ratio.  In uniaxial strain the vertical load causes a horizontal push which is represented by the equation

	sxx  =  syy  =  u /(1- u ) szz  =  u /(1- u ) rgz 
For anaylsis of faulting a plane strain model is used where two sides of a cubic piece of earth are fixed (i.e. not allowed to move in or out) (Fig. 23-3).  Plane strain differs from uniaxial strain by the small number (2) of fixed sides.  Plane strain is a reasonable representation of faulting because two parallel sides of a cube will converge whereas the second pair of sides remain fixed and the third pair of sides expand.

(Fig. 22-3)
  
For the plane strain model the first state of stress to consider is the stress developed by overburden weight without any tectonic stress.  The overburden pressure is the vertical component of the stress

	szz  =  rgz                                                                            
Note that the tectonic stress in the vertical direction is always zero.  Any deformation in the vertical direction is always taken up by a change in elevation to readjust so that only the weight of overburden is pressing down.  Horizontal stress in the y direction in the absence of tectonic compression st is

	syy  =  urgz 
                                                          
where u is  the Poisson's ratio of the rock at the point in question.  Note that the z-direction is normal to the fixed boundaries.  This is the stress caused purely by the load due to gravity.   The stress in the x-direction is, of course, zero because there is no boundary against which rock will push.  Note here that the tectonic stress (st ) is that stress component which adds to the state of stress caused by a gravitational load.  The following analysis is a variation on an analysis presented in Turcotte and Schubert (1982).
For  thrust faulting the horizontal compression in the x-direction must exceed the vertical overburden pressure by an excess stress Dsxx where

	Dsxx  >  0 
                                                                      
(Fig. 23-4).  For high horizontal stress

	sxx   =   urgz  +  st  =  rgz  + Dsxx                                    

This analysis assumes a plane strain model where there is no strain in the y direction.  So, the excess  stress in the y direction Dsyy is

	Dsyy  = uDsxx                                                                  

and the horizontal stess in the y direction is 

	syy  =  rgz  +  uDsxx                                                          

(Fig. 22-4)

Thus the horizontal stress in the y direction exceeds the overburden pressure but is less than the horizontal stress in the z direction by the value of the Poisson's ratio of the rock at the point of consideration.  Then thrust faults satisfy the sterss condition

	sxx >  syy  >  szz 

For the case of normal faulting the overburden pressure is still

	szz  =  rgy                                                                          

and the excess stress in the horizontal direction is now negative (tensional) where

	Dsxx  <  0                                                                             

and so the horizontal stress sxx is now smaller than the vertical stress syy.  Here

	sxx   =   urgy  -  st  =  rgy  - Dsxx                                    

and 

	syy  =  rgy  -  uDsxx                                                       

Again using the plane strain assumption to calculate the syy the relationship between the three principal stresses is

	szz  >  syy  >  sxx.

For strike-slip faulting the vertical overburden pressure remains the same as above but the excess stresses are positive (compressional) in one direction and negative (tensional) in the other direction.  Here two cases can occur

	Dsyy  >  0  and  Dsxx  <  0                                                  

 or	

	Dsyy  <  0  and  Dsxx  >  0                                                  

so that in the case of strike-slip faulting the overburden pressure is always the intermediate in value between the horizontal principal stresses.
The Anderson theory of faulting was developed by combining the classification of faulting as outlined above with the failure criterion for slip on a fault known as Amonton's law for sliding friction.  Amonton's law, a simple constitutive relationship for slip on faults, states that the force parallel to a fault ft necessary to initiate slip on that fault is proportional to the normal force across the fault fn

	ft  =  µfn                                                                         

where the proportionality constant µ is the coefficient of static friction.  Dividing by the area of contact we have

	t  =  µsn                                                                          

where t  and sn are defined in the Mohr construction.  
Using Amonton's law it is possible to calculate the dependence of the angle of dip b on the coefficient of friction µ for normal and thrust faults and to calculate the dependence of deviatoric stress on the coefficient of friction for both normal and thrust faulting (Turcottte and Schubert, 1982).  For this analysis it is assumed that the vertical stress szz is the overburden pressure and the horizontal tectonic compression is the sum of the overburden pressure and a deviatoric stress Dsxx.  sxx and szz are related to t and sn by the equations for the Mohr construction

	sn  =  1/2( sxx  +  szz)  +  1/2(sxx  -  szz)cos2g

	t  =  -1/2(sxx  -  szz) sin2g
                                              
where g is the angle of the fault with respect to the vertical.  Upon substituting the equations for the vertical and horizontal stresses into the Mohr construction equations, the normal and shear stresses on the faults are

	sn  =  rgy  +  1/2(Dsxx)(1  +  cos 2g)

	t  =  -1/2(Dsxxsin 2g)                                                       

In general faults contain ground water at hydrostatic or higher pressure.  The pore water pressure Pw affects the shear stress necessary to initiate slip by Amonton's law such that
 
	t  =  µ(sn - Pw)                                                     

Substituting into Amonton's law for the sn and t for a stress system associated with either normal or thrust faults
	
	±  1/2Dsxxsin2g  =  µ[rgy   +  1/2Dsxx(1  +  cos 2g)] -  Pw  

where the equation is positive for thrust faults and negative for normal faults.  From above comes the expression for the deviatoric stress int erms of the angle of fault with respect to the vertical

	Dsxx  =  [2µ(rgy - Pw)]/[ ± sin2g  -  µ(1  +  cos2g)]  
                                                                                                                 
It is assumed by many that the upper crust contains faults of many orientations so slip will occur on that fault for which the Dsxx is minimum.  To calculate the angle of faulting g which minimizes the previous equation, we set 

	dDsxx/dg = 0                                                                     

and find that

	tan2g  =  ±  1/µ 
                                                                    
Figure 23-4 shows how the coefficient of friction and dip angle of both thrust and normal faults aare related to minimize Dsxx.  The tectonic stress corresponding to these angles of dip are found by substituting the previous equation into the equation for Dsxx.

	Dsxx  =  [ ±2µ(rgy  -  Pw)]/(1  +  µ2)1/2  ±  µ                  

 This relationship is also shown in Figure 23-5.

(Fig. 23-5)

If the behavior of the fault was dependent on just the constitutive relationship of the fault zone, then the fault would slip only when the yield criterion was exceeded.  But, because the fault zone is part of a system that includes a loading frame with an elasticity, slip on the fault is controlled in part by the loading system.  Within the earth the loading system is taken to be the wall rocks next to the fault zone.  As fault slip takes place the loading frame relaxes; the rate of this relaxation will determine whether the fault will slip continuously or by jerky steps.  If slip is by jerky steps then the fault is known to have earthquakes.