Structural Geology
Lecture 14
Microcracks and Shear Fracturing
(Rock Failure)

The Coulomb criterion is empirical requiring several rock deformation experiments at different s1 - s3 conditions.   From these experiments the failure of a certain type of rock can then be predicted.  The criterion contains no information on the mechanism or the microscopic process involved in rock fracture.  Interestingly,  the Coulomb criterion was formulated about 1770 and more than 150 years passed before the actual process of rock failure was understood.  About 1920 an engineer named Griffith first developed a mechanistic failure criterion which could be applied to rocks.  Griffith's theory was originally formulated to explain fracture in glass.

Griffith's contribution  

Griffith's criterion was based on the observation that glass is not internally homogeneous but rather contain pores.  The same can be said for rocks.  Pores in rocks can be original spaces between grains that did not completely fill with cement or they can be microcracks that develop when the rock is stressed.  These voids play an important role in rock failure directly through weakening the rock and indirectly through hosting pore water which under pressure can also effect the rock strength.  The effect of voids on the elastic properties of a rock can include:  1.)  The Young's modulus (E) of a rock containing voids and cracks is less than the intrinsic E of a solid rock (an intrinsic property refers to that property in a body without flaws or cracks);  2.)  The Poisson's ratio (u) of a rock containing cracks is less than its intrinsic u;  3.)  cracks give rocks a different modulus during loading and unloading (this difference in behavior is known as hysteresis).

Griffith observed that prior to glass (i.e. rock) failure microcracks and pore cavities within rocks grow in length or propagate.  Many of these microcracks must propagate and then intersect before the rock can fail along a shear zone.  Griffith examined the conditions under which individual microcracks propagated.  First, he observed that cracks must be pull apart to cause propagation.   In other words, the microcracks must fail in tension.  Griffith's approach was to examine the amount of energy available to cause the crack to propagate.  Recall that work, a scalar, (W) is the dot product of a force vector f and displacement vector d.
 
W  =  f (dot) d.                                                                    

Griffith (1924) recognized that during crack propagation surface energy, dUS, is necessary to create new surface area.   As the crack propagates, a rock may undergo a change in strain energy, dUE which will contribute to the surface energy.  Boundary tractions may also contribute to the surface energy in the form of mechanical work, dWR Both dWR and dUE are forms of mechanical energy.  The total change in energy for crack propagation is 

.	[14-1]

Under certain circumstances crack propagation may take place without changing the total energy of the rock-crack system.  This is known as the Griffith energy-balance concept for crack propagation where the standard equilibrium requirement is that for an increment of crack extension dc, 

.	[14-2]

For equilibrium, the mechanical (-dWR + dUE) and surface energy (dUS) terms within the rock-crack system must balance over a unit crack extension dc.  The total energy of the crack-rock system is at a maximum at equilibrium.  Any perturbation of the system such as an increase in tensile load will cause the crack to propagate spontaneously without limit.  During crack propagation the crack walls move outward to some new lower energy configuration upon removal of the restraining tractions across an increment of crack.  In effect, the motion of the crack walls represent a decrease in mechanical energy while work is expended to remove the restraints across the crack increment.  The work to remove the restraints is the surface energy for incremental crack propagation.  
To evaluate the three energy terms relating to crack propagation, Griffith cited a theorem of elasticity which states that, once an elastic body is subject to a load, its boundaries will displace from the unloaded state to the equilibrium state so that

.	[14-3]

For a body of rock containing an elliptical crack with major axis perpendicular to a uniform tension (i.e., Sh < 0), Griffith calculated that 

,	[14-4]

provided that the rock is thick enough for plane strain conditions to hold.  According to Jaeger and Cook (1977) the effect of a crack on the elastic properties of the rock shows up on the p and crack length terms.  For the surface energy of the crack, Griffith defined crack length as 2c and recognized the crack propagation produces two crack faces.  Therefore,

	[14-5]

where g is the free surface energy per unit area.  Substituting Equations [14-3], [14-4], and [14-5] into [14-1] and then applying Equation [14-2],  Griffith solved for the critical stress for crack propagation

	[14-6]

where T0 is the uniaxial tensile strength of the rock.  Here  for a dry rock in the near surface.  Equation 3-22 is the famous Griffith crack propagation criterion.                                                          

This is the famous Griffith criterion for rock failure in tension (Figure 14-1). 

 (Fig 14-1)

Another way of illustrating Griffith's criterion is to consider the simple crack system shown in figure 14-2.  The mechanical energy of this system is (-WL  +  UE) where UE is the potential energy stored in the system and WL is the work due to displacement of the outer boundary.  The lower end is fixed rigidly and the upper end is loaded with force F.  The crack system will behave as an elastic spring by Hook's Law

u  =  lF

where l is the elastic compliance of the system and u is the displacement of the top end of the system.  The strain energy of the system is then 

UE  =  1/2Fu  =  1/2F2l  =  1/2u2/l

provided we do not let the crack (2c) extend to dc.  If the crack does extend by dc when force F is applied, then the elastic compliance of the system increases as can be shown by differentiating

du  =  ldF  +  Fdl

For dc > 0, dF < 0 and dl > 0.  If the crack extends, the mechanical energy of the system should decrease.

dWL  =  Fdu  +  udF   =    Fdu

The total mechanical energy change is therefore 

d(-WL  +  UE)  =  -1/2F2dl

(Fig. 14-2)

This leads back to consideration the distinction between failure in tension and failure in compression.  Griffith's criterion gave a convenient physical explanation for crack propatation when the crack is subject to tensile stresses.  The crack propagates in the plane normal to the tensile stress or least princial stress (T =  -s3).  However, we have learned that whole-rock stresses including s3 increase with depth in the crust of the earth.  Yet,  rock fracture in shear depends on microcrack propagation where the propagation is caused by tensile stresses.  Where are the tensile stresses generated when rock is deep within the crust of the earth?  The answer is that on a microscopic scale grains press against each other at sharp contacts.  These contacts are called "stress risers".  Under these microscopic contacts very large tensile stresses can be generated even though the whole-rock stress in highly compressive.

Figure 14-3 shows an example of a stress riser.  The point contacts of each disk serve to concentrate the stress which was distributed over the end plattens.  The increase in stress is shown by the circular lines originating from the point contact.  These lines represent lines of equal stress difference.  The numbers indicate that the stress difference increases as the point contacts are approached.

  
(Fig. 14-3)

Another characteristic of point contacts is that a portion of the volume under the point contact is in a state of tension.  Such an volume is illustrated in figure 14-3.  When the rock is compressed under high s1 - s3 conditions, many of these microscopic contacts are activated and microscopic crack propagation is common.  As the population of microcracks increases a zone of weakness develops near but not along a plane of maximum shear stress (tmax).  This zone of microcrack intersection eventually  hosts the through-going shear fracture.

Shear fractures do not form at 45° to s1 and in the plane of †max 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. 14-4).  µ0 is the angle of internal friction

	µ0  =  tan f 

(Fig. 14-4)

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 will be discussed in a future lecture) (Fig. 14-5).

 µ0 predicts the angle of shear failure

µ0  =  (t - S0)/sn.

A graphical plot of t, sn, and  (t - S0)/sn shows that  shear failure occurs on the plane where (t - S0)/sn is maximized (Fig. 14-6).

 
(Fig.14-5)

(Fig. 14-6)

Shear fracturing requires a high differential stress.  Such stresses are not common within the crust.  Places where s1 - s3 may be high include:

1.) Fault zones --  Along fault zones asperities which interlock can act as local stress concentrators.  In fact, studies of earthquakes show that many faults lock and then slip violently.  Asperities cause the locking and more often than not the earthquake is a manisfestation of the asperity being sheared off.

2.)  Man-made structures --  Pillars in mines often have to bear loads many times that of neighboring rock left undisturbed by mining.   Mine shaft openings also act as stress concentrators in much the same manner as microscopic point contracts within rock.  The stresses around holes in rock can be many times as large as was found in undisturbed rock. 

3.  Folds -- Rocks can act like elastic plates which bend.   It is known from the theory of elasticity that part of the bent plate is compressed.  In this zone of compression within folded rocks compressive stresses can become so large that the shear strength of the rock is exceeded. 

There are several rock structures associated with shear fracturing.  Cataclastic rock is the product of concentrated microfracturing within a shear zone.  Cataclasis is the process of mechanical reduction of grain size by brittle fracture and rigid grain rotation.  Microfractures develop across the shear fracture in the direction of s1.  Continued slip on the shear fracture cause more microfracturing which leads to a wider zone of cataclasis.  Comminution is another term applied to the mechanical reduction of grain size.  As the pressure on the rock increases, tension gashes form the the shear zone becomes semiductile.  Tension gashes are filled macrocracks or extension fractures.  The origin of extension fractures is the subject of a future lecture.  Repeated shear fracturing causes deformation bands.  These are zones of cataclastic material most commonly observed in sandstones.   Deformation bands occur because each successive shear fracture or zone of cataclasis is permeable to cements which make the zone stronger than surrounding sandstone.  A successive shear fracture must then rupture intact rock or cemented fracture and is not focused by a previous plane of weakness.  

The surface of a shear fracture can either be a slickensided surface or a fiber-coated surface.  The latter has been erroneously called a slickensided surface.  Slickensided surfaces are the product of brittle polishing and commonly have carrot-shaped wear grooves.  Fiber-coated surfaces are the product of ductile slip where the deformation mechanism is diffusion mass transfer.  Pressure solution dissolves asperities where are redeposited as fibers pointing in the direction of slip.