This document discusses concepts of failure in materials including tensile failure, shear failure, and failure criteria. It specifically examines the Mohr-Coulomb failure criterion, which states that failure depends on the material's cohesion and internal friction angle. The criterion can be represented on a Mohr's circle diagram, where failure occurs if the circle contacts the linear failure envelope line. Pore fluid pressure is also accounted for using effective stress. Triaxial tests are described that apply different confining pressures to measure failure properties over a range of stress conditions.

1. James A. Craig
Omega 2011

2.  Concepts of Failure
 Tensile Failure
 Shear Failure
 Failure Criteria
 Mohr-Coulomb Failure Criterion

3.  Failure occurs to any solid material when:
 Sufficiently large stress is applied.
 The material does not return to its original state after
stress relief.
 Mode of failure depends on:
 Stress state
 Type and geometry of material
 Fatigue makes failure to occur below the stress
level.

4. Uniaxial Test
 Stress is applied to the end
faces of the specimen.
 No radial (confining stress)
 Also called Unconfined
Compression Test.

5. Elastic region
Specimen
returns to its
original state
after stress
relief.
Yield Point
Permanent
changes beyond
this point.
Specimen does
not return to its
original state
after removal of
stress.
Uniaxial
compressive
strength
The peak
stress.
Ductile region
Permanent
deformation, but can
still support load.
Brittle region
Ability to withstand
stress decreases
rapidly as deformation
increases.

6. Triaxial Test
 In addition to axial stress,
confining pressure of
different magnitude is
applied to the
circumference of the
cylinder (by a confining oil
bath).

7.  Two of the principal stresses are equal.
 Process:
 Axial & confining loads are increased simultaneously
until a prescribed hydrostatic stress level is reached.
 Confining pressure is kept constant while axial load
increases until failure occurs.

8. Difference in principal stresses is plotted against axial deformation.
Specimen can still support load after failure due to high confining
pressure. It is called Work Hardening or Strain Hardening.

9. X → abrupt brittle
failure
Uniaxial test

10.  Tensile failure occurs when the
effective tensile stress across
some plane is the sample
exceeds a critical limit called
Tensile Strength.

11.  Tensile failure is caused by the stress concentrations
at the edges of thin cracks oriented normal to the
direction of the least compressive principal stress.
 For isotropic rocks, conditions for failure will always
be fulfilled first for the lowest principal stress.

 3   3  P  To
To = tensile strength (in Pa, atm, psi or bar).

12.  Most sedimentary rocks have a rather low tensile
strength, typically only a few MPa or less.
 Standard approximation for several applications is
that the tensile strength is zero

13.  It occurs when the shear stress
along some plane in the sample is
too large.

14.  Mohr–Coulomb
 Hoek–Brown
 Drucker–Prager
 Griffith (tensile)

15.   f  
  So  
 So = cohesion or inherent shear strength of material (in
Pa, atm, psi or bar).
 µ = coefficient of internal friction.
 Shear stress must overcome the cohesion plus the
internal friction in order to produce a macroscopic
shear failure.

16. Failure Line
Slope =
tan   
Mohr
Circle
So
A
 So cot 
tan 
If the Mohr’s circle lies below the failure line, the rock does not
fail and remains intact.

17.  φ = angle of internal friction. It varies from 0 to 90o
(approx. 30o)
 A = attraction (in Pa, atm, psi or bar).
 β = angle that fulfils the failure criterion. It gives
orientation of the failure plane. Varies between 45o and
90o.
 At point P:
 Angle 2β gives the position of coincidence of Mohr’s
circle and the failure line.
 Coordinates are given as:
1
   1   3  sin 2
2
1
1
  1   3   1   3  cos 2
2
2
2  90o  

OR


4


2

19.  Co = uniaxial compressive strength (in Pa, atm, psi or
bar).

20. 2 So cos 
a
 Co  2 So tan 
1  sin  
1  sin  
b  tan  
1  sin  
 tan   1
sin  
 tan   1
 1  a  b 3
 1  Co   3 tan 2 

21. © Haimson and Song (1995)

22.  Principle of effective stress is introduced, i.e.
subtract fluid pressure from the total stress.
 Previously:
 1  a  b 3
 1   1   Pf

 Then:

And 3   3   Pf

 1  a  b 3
1  sin  
2So cos 
1   Pf 
  3   Pf 
1  sin  
1  sin  

23.  Pore fluid can affect the failure of the rock in 2 ways:
 Mechanical effect of pore pressure.
 Chemical interactions between the rock and the fluid.

24.  Effect of pore pressure on failure:
 Shear stress is unaffected by the pore pressure
 Minimum & maximum principal stresses are
decreased by the same amount.
 Radius of the Mohr circle in unchanged.
 Center of the circle has shifted to the left.
 Circle moves towards the failure line when the fluid
pressure is increased for a material obeying the
criterion.