Mohr-Coulomb failure criterion ($|\tau |=c+{\mu }_f{\sigma }_n^{\prime }$)

A frictional shear-strength approximation $|\tau |=c+{\sigma }_n^{\prime }\tan \phi$, with cohesion $c$ and internal friction angle $\phi$. Effective compression increases shear resistance. A separate tensile cutoff is generally needed at low confinement. The names refer to Christian Otto Mohr and Charles-Augustin de Coulomb.

For a circular tunnel of radius $a$ in a plane-strain elastic medium, isotropic far compression $p_{\infty }$ and wall bracing pressure $P$ give ${\tau }_{rr}=-p_{\infty }-(P-p_{\infty })a^2/r^2$ and ${\tau }_{\theta \theta }=-p_{\infty }+(P-p_{\infty })a^2/r^2$. The Mohr circle center is $-p_{\infty }$ and its radius is $|P-p_{\infty }|a^2/r^2$. Its maximum is at the wall, proving that first yield occurs there. Cohesionless frictional tangency requires this radius to equal $p_{\infty }\sin \varphi$. Lowering $P$ from hydrostatic confinement yields the displayed threshold and stress ratio ${\tau }_{\theta \theta }/{\tau }_{rr}=(1+\sin \varphi )/(1-\sin \varphi )$. Increasing bracing instead reaches $P=p_{\infty }(1+\sin \varphi )$ with the reciprocal ratio. The two cases must not be confused when identifying the first loading threshold.

The angle whose tangent gives the slope of a linear frictional failure envelope. It controls both the confinement dependence of strength and the optimum new shear-fracture orientation.

The shear-strength intercept $c$ at zero effective normal stress in a specified frictional failure envelope. It is a model parameter and is not automatically the tensile strength.