Mohr–Coulomb theory

Remarks

The stresses on all planes lie in the largest Mohr's circle, with centre $(\sigma_1 + \sigma_3)/2$ and radius $(\sigma_1 - \sigma_3)/2$. This circle touches the line $\abs{\tau} = c - \sigma_n \tan \phi$ exactly when its radius equals the distance from its centre to the line. This is why the two forms agree. Example: in uniaxial compression, $\sigma_1 = \sigma_2 = 0$ and $\sigma_3 = -q$, the criterion gives the compressive strength $q = 2c \cos\phi / (1 - \sin \phi)$; in uniaxial tension the strength is $2c\cos\phi/(1 + \sin\phi)$, which is smaller when $\phi > 0$. With $\phi = 0$ it reduces to the Tresca criterion with $c = \sigma_y / 2$. Non-example: a hydrostatic compression $\sigma_1 = \sigma_2 = \sigma_3 = -p$ with $p > 0$ never meets the criterion when $c > 0$. The left side is $0$, and the right side is $c\cos\phi + p \sin\phi > 0$. The yield surface in principal stress space is an irregular hexagonal pyramid with axis the line $\sigma_1 = \sigma_2 = \sigma_3$, and the intermediate stress $\sigma_2$ plays no role.