Geol 542: Advanced Structural Geology
Fall 2013
Problem Set #7: Coulomb and Griffith Failure Criteria
1. The Coulomb criterion for failure of intact rock may be written in terms of the Coulomb Stress S as follows:
S = |s| - in = So (at failure)
where s is shear stress, i is the coefficient of internal friction, and n is the normal stress. Using the Mohr
equations, this expression may be rewritten as:
S=
1
[(s1 - s 3 )(sin2b - mi cos2b ) - mi (s1 + s 3 )]
2
where  is the angle between the maximum compressive stress (1) and n.
(a) The maximum value of the Coulomb stress Smax will occur when the rate of change of S with respect to  is
zero:
Calculus alert!! Use this relationship to show that this critical value of  (called f), satisfies the condition:
cos2b f + mi sin2b f = 0.
(5)
(b) Although you don’t have to prove it, from the above result, it can be calculated that:
bf =
æ1ö
1
- tan-1ç ÷ where i ≥ 0.
2 2
è mi ø
p
Use the above equation to find f where i= 0.6.
(2)
(c) Show that the maximum Coulomb Stress Smax at a depth of 3 km in a sandstone section having i= 0.6 and
density  = 2300 kg/m3 is -6.4 MPa. Assume that g = 9.8 ms-2. In calculating the stresses at this depth,
assume zero pore pressure and a tectonic stress component of 120 MPa that causes the maximum
compressive stress, 1, to be horizontal and 3 to be vertical. Show all calculations!
(8)
(d) If the sandstone has an inherent shear strength of S o = 8 MPa, why is a fault not likely to form in the
sandstone for the stress field described in question 3? How much more Coulomb stress will be needed?
(3)
(e) Draw a Mohr circle to represent the conditions described in questions 3 and 4. Include the Coulomb
criterion frictional failure line for the sandstone. Use an entire sheet of graph paper in landscape format.
(8)
(f) Based on your Mohr circle diagram, write a sentence or two to corroborate your answer in question 4.
(2)
Geol 542: Advanced Structural Geology
Fall 2013
(g) On the same Mohr diagram, draw the frictional failure line for a pre-existing fault that has zero cohesion
and a coefficient of friction  = 0.5. Use your diagram to predict which range of fault orientations (in terms
of ) are prone to failure in the existing stress field (i.e., the range of  defined by the portion of the Mohr
circle crossing the lower failure line).
(6)
(h) Will a pre-existing fault with a dip of 50° be prone to slip? Explain your answer.
(4)
2. The table below shows theoretical (intrinsic) and measured strengths of several types of geologic materials.
NaCl
INTRINSIC STRENGTH
Ti (MPa)
3000
MEASURED STRENGTH
Tu (MPa)
5 - 70
Mica
10,000
200
Glass
4000
200
Sandstone
2000
10
Granite
3000
40
MATERIAL
The relationship between measured strengths (Tu) and intrinsic strengths (Ti) was determined by Griffith to be:
12
T é r(min) ù
Tu = s » i ê
2 ë a úû
r
1
a) Given the above relationship, calculate the lengths of the microcracks (“Griffith flaws”) that are characteristic
of each of the above materials. Assume that the minimum radius of curvature at the crack tip is constant in
each case, on the order of molecular dimensions (5 x 10-8 m).
(10)
b) Why are the lengths of the Griffith flaws different in each of these materials?
(2)
c) Speculate on what kinds of specific features may be acting as Griffith flaws in:
(i)
(ii)
(iii)
glass
sandstone
granite
(6)
3. Compile a list of 8 different features of a rock mass that could be treated as flaws that weaken the rock. Include
two examples in each of the following scales of observation:
(i)
(ii)
(iii)
(iv)
a single mineral grain
a hand sample
an outcrop
the entire thickness of the crust
(8)
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