The Traction Vector and Stress Tensor Sign Convention

The Traction Vector and Stress Tensor

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

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 ≥

Stresses in a Principal Stress Frame

Mohr Circles

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

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 2 D

Coulomb Failure and Mohr Circles stable failure occurs

Real Space

Mohr Space

Normal and Shear Stresses

Maximum Shear Stress

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

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