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Failure I. 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 >T room ) The cylinder, jacketed by rubber or copper, is subjected to a uniform, fluid-exerted confining pressure

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measuring the strength of rocks
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 >Troom)
  • 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)
measuring the strength of rocks4
Measuring the Strength of Rocks

Triaxial Compression Apparatus

measuring the strength of rocks6
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
measuring the strength of rocks8
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 ands3 are recorded at failure
  • The above steps are repeated for a new sc = s3
coulumb failure envelope
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
coulumb failure envelope12
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
coulumb failure envelope19
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
coulumb failure envelope20
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 sn

  • States of stress with Mohr circles below the envelope do not result in fracture (it should touch or exceed the envelope for fracturing)
coulomb criterion21
Coulomb Criterion

ss = Co+ mi sn

  • Note: Fracture does notoccur on the plane with maximum shear stress (i.e., not at = q+45):
  • The angle 2q for fracturesis not 90o; it is > 90o

0o<f<30o 90o<2q< 120o

45 >> 30

  • The angle 2q (where qis the angle from s1to thenormal to fracture)determines the orientation of the fracture plane
coulomb criterion22
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 = tan-1mi

  • 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)
  • 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 strengthCo 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!)
tensile vs compressive strength
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 1and each fracture plane
    •  is the angle from 1to the pole of each fracture (or between 3and 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