1. Failure I

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 >T room ) The cylinder, jacketed by rubber or copper, is subjected to a uniform, fluid-exerted confining pressure Start with an isotropic state of stress (  1 =  2 =  3 )

4. Measuring the Strength of Rocks Triaxial Compression Apparatus

6. Measuring the Strength of Rocks The confining pressure (  c =  3 ) is increased to reach a value which is then kept constant while the axial stress (  1 =  a ) is increased The rate of increase of the axial load (  a ), T, P f, and the  c can all be controlled The strength of rocks is controlled by P, T, e., H 2 O, composition, etc. The results are then recorded on (  - 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 (  1 -  3 ) 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 (  1 -  3 ) At a certain differential stress, the rock fails by fracture. The  1 and  3 are recorded at failure The above steps are repeated for a new  c =  3

9. Coulomb Criterion

10. Measuring the Strength

11. Coulumb Failure Envelope The loading of the rock cylinder is repeated under progressively higher confining pressures (  3 ) i.e., we conduct a series of experiments For each set of  3 and  1, 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

15. Coulomb Criterion

19. Coulumb Failure Envelope Experiment shows that the fracture strength (  1 -  3 ), 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  s = C o +  i  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  s = C o +  i  n Note: Fracture does not occur on the plane with maximum shear stress (i.e., not at =   +45): The angle 2  for fractures  is not 90 o ; it is > 90 o  o  30 o 90 o < 2  o 45 >  > 30 The angle 2  (where  is the angle from  1 to the normal to fracture)  determines the orientation of the fracture plane

22. Coulomb Criterion The slope of the line is the Coulomb coefficient,  i The angle of slope is the angle of internal friction  i  i = tan  i i.e.  i = tan -1  i The intersection of the radius of each circle with the failure envelope gives the state of stress (  n,  s ) on the fracture plane The  s and  n 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  1 (whose normal is at  to the  1 )

23. Cohesion The cohesion, C o, is the intercept of the envelope with the  s axis For loose sand which lacks cohesion, the fracture line passes through the origin of the graph, i.e., C o = 0 Cohesive materials such as rocks have a finite shear strength C o 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 C o (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)  +  = 90 o 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