1. The Traction Vector and Stress Tensor

2. Sign Convention
Positive Shear Stress
Determined by direction of traction acting on negative coordinate face
Negative Shear Stress
Tractions are positive in negative coordinate direction

3. Principle Stresses
for any stress system, we can always find a principle axis coordinate system in which all shear stresses are zero
the three normal stresses are principle stresses denoted by: ≥ ≥
compression is positive
tension is negative

4. Stresses in a Principal Stress Frame

5. Mohr Circles

6. Stress in 3D stress tensor describes stress in 3D
diagonal terms = normal stresses
nondiagonal terms = shear stresses

7. Differential Stress
differential stress is the difference between the maximum stress (σ1) and the minimum stress (σ3)
the most important quantity in the failure of rocks
we only analyze the plane containing σ1 and σ3 in 2D

8. Coulomb Failure and Mohr Circles
stable
failure occurs

9. Real Space

10. Mohr Space

11. Normal and Shear Stresses

12. Maximum Shear Stress

13. Brittle Failure brittle failure could result in:
1) development of a new fracture surface in an intact rock
2) slip on a preexisting fracture in a previously broken rock
failure criterion specifies the stress state when failure occurs

14. Mohr-Coulomb Failure Criterion
linear approximation:
angle of internal friction
failure envelope C
shear & normal stresses at failure
cohesion
coefficient of internal friction