Understanding Stress and Failure: Principles and Criteria in Materials Engineering

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 aPrincipal 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