Failure I

1
Failure I
2
(No Transcript)
3
Measuring the Strength of Rocks

• A cored, fresh cylinder of rock (with no surface
  irregularities) is axially compressed in a
  triaxial rig (usually at T gtTroom)
• The cylinder, jacketed by rubber or copper, is
  subjected to a uniform, fluid-exerted confining
  pressure
• Start with an isotropic state of stress (s1 s2
  s3)

4
Measuring the Strength of Rocks
Triaxial Compression Apparatus
5
(No Transcript)
6
Measuring the Strength of Rocks

• The confining pressure (sc s3) is increased to
  reach a value which is then kept constant while
  the axial stress (s1 sa) is increased
• The rate of increase of the axial load (sa), T,
  Pf, and the sc can all be controlled
• The strength of rocks is controlled by P, T, e.,
  H2O, composition, etc.
• The results are then recorded on (s - e) (e
  t) diagrams and on a Mohr circle

7
Stress-Strain Diagram Fracture Experiment
8
Measuring the Strength of Rocks

• Mohr circles can be used to "map" the values of
  normal and shear stresses at failure
• Failure is the loss of cohesion of a material
  when the differential stress (s1-s3) exceeds some
  critical value that varies with different types
  of rocks
• As the axial stress is increased, the Mohr circle
  becomes larger, with a diameter (differential
  stress) of (s1 - s3)
• At a certain differential stress, the rock fails
  by fracture.
• The s1 and s3 are recorded at failure
• The above steps are repeated for a new sc s3

9
Coulomb Criterion
10
Measuring the Strength
11
Coulumb Failure Envelope

• The loading of the rock cylinder is repeated
  under progressively higher confining pressures
  (s3)
• i.e., we conduct a series of experiments
• For each set of s3 and s1, we get a limiting
  fracture-inducing Mohr circles
• A best-fit line connecting the failure values of
  normal and shear stress for several Mohr circles
  is termed the Mohr failure envelope

12
Coulumb Failure Envelope

• The envelope is drawn tangent to all of these
  Mohr circles, linking the stress conditions on
  each plane at failure
• The Mohr failure envelope is the locus of all
  shear and normal stresses at failure for a given
  rock material
• The Mohr failure envelope delineates stable and
  unstable states of stress for a given rock
  material

13
Each Experiment has a series of circles only
those of 1 are shown
14
(No Transcript)
15
Coulomb Criterion
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
Coulumb Failure Envelope

• Experiment shows that the fracture strength
  (s1-s3), that the rock can withstand before
  breaking, increases with confining pressure
  (i.e., circles become larger)
• Under moderate confining pressures (e.g., for
  granite, sandstone) and within the field of shear
  fracturing, the envelope defines a straight line

20
Coulumb Failure Envelope

• At higher pressure, rocks become more ductile
  (e.g., shale) and the line becomes more gently
  sloping and convex upward
• The equation of the straight line is given by the
  Coulomb criterion
• ss Co mi s n
• States of stress with Mohr circles below the
  envelope do not result in fracture (it should
  touch or exceed the envelope for fracturing)

21
Coulomb Criterion

• ss Co mi sn
• Note Fracture does not occur on the plane with
  maximum shear stress (i.e., not at q 45)
• The angle 2q for fractures is not 90o it is gt
  90o
• 0o lt f lt 30o 90o lt 2q lt 120o
• 45 gt ? gt 30
• The angle 2q (where q is the angle from s1 to
  the normal to fracture) determines the
  orientation of the fracture plane

22
Coulomb Criterion

• The slope of the line is the Coulomb coefficient,
  mi
• The angle of slope is the angle of internal
  friction fi
• mi tan fi
• i.e. fi tan-1 mi
• The intersection of the radius of each circle
  with the failure envelope gives the state of
  stress (sn, ss) on the fracture plane
• The ss and sn at the moment the material fails by
  shear are the components of a traction acting on
  a plane inclined at an angle of ? to the s1
  (whose normal is at ? to the s1)

23
Cohesion

• The cohesion, Co, is the intercept of the
  envelope with the ss axis
• For loose sand which lacks cohesion, the fracture
  line passes through the origin of the graph,
  i.e., Co 0
• Cohesive materials such as rocks have a finite
  shear strength Co which must be overcome before
  the material will yield, even at zero normal
  stress
• Thus for such cohesive materials the fracture
  line intersects the ordinate at Co (not at the
  origin!)

24
Tensile vs. Compressive Strength

• Most materials have a greater strength in
  compression than in tension
• The dihedral angle 2?, between the shear
  fractures (bisected by the ?1), decreases with
  decreasing confining pressure (i.e., 2?
  increases). Note
• ? is the angle between ?1 and each fracture plane
• ? is the angle from ?1 to the pole of each
  fracture (or between ?3 and the fracture)
• ? ? 90o
• For brittle rocks the ratio of the compressive
  strength to the tensile strength is as high as
  20-25, and the dihedral angle between the shear
  fractures is correspondingly acute
• Materials that have greater tensile strength than
  compressive strength are highly ductile, and the
  dihedral angle is obtuse about the principal axis
  of compression