Anderson's theory of faulting

Remarks

Proof idea: a free surface has zero shear traction, so the vertical is an eigenvector of the stress tensor. Take a plane whose normal lies in the $\sigma_1 \sigma_3$ plane at angle $\beta$ to $\sigma_1$. Its normal and shear stresses lie on Mohr's circle of $\sigma_1, \sigma_3$. The Coulomb quantity $\tau - \mu \sigma_n$ is largest when $2\beta = 90^\circ + \phi$. The intermediate stress $\sigma_2$ does not enter, so the failure plane contains the $\sigma_2$ direction, and it makes the angle $90^\circ - \beta = 45^\circ - \phi/2$ with $\sigma_1$. Example: with $\mu = 0.6$, $\phi \approx 31^\circ$, so normal faults dip at about $60^\circ$ and thrust faults at about $30^\circ$, as often observed. The theory is named after E. M. Anderson, who set it out in his work on the dynamics of faulting. It describes new faults in homogeneous rock; slip on old weak faults of other orientations is not covered.

Going deeper

For a student

The theory combines a boundary condition, the traction-free surface, with the Mohr–Coulomb failure law. It turns the observed type and dip of faults into information about the orientation of stress in the crust. With Byerlee's law for friction on existing faults, it also bounds the ratio of the largest and smallest principal stresses that the crust can support.