M-C envelopes for different soils

Mohr-Coulomb theory: the shear strength of a soil on a plane is a linear function of the normal stress on it, $\tau_f = c + \sigma\tan\phi$. Failure occurs when the Mohr circle of stresses touches this straight-line envelope.

Strength envelopes

1. Cohesive-frictional ($c$-$\phi$) soil (silty or sandy clay): the line has an intercept $c$ on the $\tau$-axis and slope $\phi$.
2. Cohesionless soil ($c=0$, e.g. clean sand): the line passes through the origin, $\tau_f = \sigma\tan\phi$.
3. Purely cohesive soil ($\phi=0$, e.g. saturated clay in undrained loading): the line is horizontal, $\tau_f = c$.

 tau            tau             tau
  |      /       |      /        |
  |    /         |    /          |---------- tau=c
  |  /  phi      |  /  phi       |
c |/             |/              |
  +---------s    +---------s     +---------s
 c-phi soil      sand (c=0)      pure clay
 tau=c+s*tan(phi) tau=s*tan(phi) (phi=0)