Structural Geology
Lecture 13
Coulomb Failure Criterion
(Mohr Envelope)

Structural geology is the study of deformed rocks.  Deforamtion is the response of the rock to a state of stress.  Because natural rock deformation is a slow process, laboratory experiments must be used to "speed" time.  One component of nature lost in the laboratory is size.  The deformation of small rock samples can be "mapped" in the laboratory by curves showing the relation between the magnitude of stress applied to the rock and the amount of strain or deformation the rock has suffered.  These curves are known as stress-strain curves.  Strain (e) is a measure of deformation when the rock deforms uniformly.  The simplest stress-strain curve (s-e curve) is one where 

	ds/de =  E

where E is a constant.  This differential equation is Hooke's Law for the linear-elastic behavior of materials.  In the case of Hooke's law the constant E is known as Young's modulus.  The reaction to a stress applied to an elastic body such as a rubber ball is a predictible and repeatable amount of strain.  Elastic strain is recoverable.  By this we mean that once the stress is remove the body will return to its original shape.  A meterial is described as linear-elastic if Hooke's Law holds accurately.
No material including rocks will sustain an indefinitely large amount of strain.  After a certain amount of elastic strain rocks will deform in some other manner.  The point of transition from elastic behavior to another type of behavior is called the yield point.  Yield connotes a non recoverable deformation.  One common mode of rock behavior after yield is brittle fracture and another mode of rock behavior is ductile flow.  A rock that has failed by brittle fracture has usually broken into more than one peice.
In the earth each small volume of rock is surrounded by other rock.  The effect of the surrounding rock is to confine the volume in question and apply a confining pressure (sometimes called a lithostatic stress).  In order for rock deformation to take place the principal stress in one direction (s1) must exceed the other two principal stresses (s2, and s3) which are at right angles to s1.  This difference between s1 and, say, s3 is called the differential stress to which the sample is subjected.  In the analysis of rock deformation s3 is equivalent to the confining pressure.  
Our knowledge of the behavior of rocks comes from experiments in the laboratory.  One common rock mechanics experiment uses a cylinder of rock placed in a rock deformation machine.  Such a cylinder is shown in cross section (Fig. 13-1).  Pistons contact the end of the cylinder and create the stresses necessary to deform the rock.  The cylinder is surrounded by a confining medium which is prevented from flowing into the pores of the rock sample by an impermeable jacket.  In Figure 13-1 the sample is stippled, the confining medium is represented by the inward pointing arrows, and the loading pistons are shown as the darker objects on either end of the rock sample.  The application of differential pressure is shown by the dark, vertical arrows on either end of the piston.

(Fig. 13-1)

A typical stress-strain curve is shown to the right of the rock deformaiton experiments.  As stress increases on the rock, the rock strains. Once the fracture strength of the rock is reached the rock fails along one or more fracture planes.  The failure is denoted by a sudden stress drop in the stress-strain curve.

(Fig. 13-2)

In the late 18th century a French naturalist, Coulomb, observed for rocks that shear stress |t| necessary to cause brittle failure across a plane is resisted by the cohesion of the material So and by a constant *µ times the normal stress sn across that plane:

	|t|  =  So  +  *µsn.
 
The constant *µ is called the coefficient of internal friction.  *µ is not to be mistaken for µ, the coefficient of sliding friction which will be discussed in detail in a future lecture.  Because a shear stress was parallel to the plane of failure,  the brittle fracture is commonly called a shear fracture.  This mode of fracture should be distinguished from a tensile crack that opens normal to the least principal stress (s3). Recall that 

	sn  =      1/2(s1  +  s3)  +  1/2(s1  -  s3) cos2q
and
	t     =    - 1/2(s1   -   s3) sin2q       (see Fig. 13-2).

Substituting into the Coulomb relationship

t - *µsn   =   1/2(s1 -  s3)[sin2q - *µ.cos2q]  -  1/2*µ(s1 + s3)

For failure the difference between t and sn must be maximum which occurs when

	tan2q  =  -1/*µ 

2q lies between 90° and 180°.  Thus

	sin2q  =  (*µ2  + 1)-1/2,
and
	cos2q  =  - *µ(*µ2  +  1)-1/2.

Again substituting

	t  -  *µsn  =  1/2(s1 -  s3)[*µ2 + 1]-1/2  -  1/2*µ(s1 -  s3)

which gives the maximum value of t  - *µßn.  Failure occurs when

	2S0  =  s1{(*µ2  +  1)1/2  -  *µ}  -  s3{(*µ2  +  1)1/2  +  *µ}.

This is the famous Coulomb criterion for brittle failure.   The Coulomb failure envelope can be plotted on a s1 vs.  s3 graph (Fig 13-3).  In fig. 13-3 the uniaxial compressive strength is C0 where

C0  =  2S0[( *µ2  +  1)-1/2  +  *µ]

(Fig. 13-3)

Another representation of the Coulomb criterion is obtained by introducing mean stress sm and maximum shear stress tm where

	sm  =  1/2(s1 +  s3)    and   tm  =  1/2(s1 -  s3).

Then the Coulomb criterion becomes

	(*µ2   +  1)1/2tm  -   *µ.sm  =  S0

Now if *µ  =  tan f where f is the angle of internal friction

	tm  =  smsin f  +  cos f

which is a line in the sm -tm  plane of inclination tan-1(sin f) and intercept S0 cos f on the tm-axis  (Fig 13-4).
This result is similar to the Mohr circle representation where the Coulomb criterion is represented by a straight line of slope *µ  =  tan f and intercept S‚ on the |t| axis.  If the s1 -  s3 circle touches the line of the Coulomb criterion then brittle failure.  
Mohr's hypothesis was that when shear failure takes place across a plane, the normal stress sn and shear stress t are related by a function

	|t|  =  f(sn).

Failure takes place if the circle of center C on the s1 -  s3 just touches AB.  The failure curve is obtained experimentally and is the envelop of many Mohr circles corresponding to failure under a variety of confining pressure (s3) conditions. 

(Fig. 13-4)