Short note: Friction angle

The angle of friction ($\phi$) is a measure of the shear strength due to friction on a surface or in a soil or rock. It is the angle whose tangent is the ratio of shear stress to normal stress at failure on a surface with no cohesion:

$$\tau = \sigma_n \tan\phi \qquad \Rightarrow \qquad \phi = \tan^{-1}\left(\frac{\tau}{\sigma_n}\right)$$

With cohesion $c$ the Mohr-Coulomb equation is $\tau = c + \sigma_n \tan\phi$.

• Basic friction angle $\phi_b$: from smooth, clean saw-cut rock surfaces (25 to 35 degrees).
• Residual friction angle $\phi_r$: after large displacement; lower than the peak value. Clay-filled joints can have $\phi_r$ of 10-20 degrees.
• Peak friction angle: includes roughness: $\phi_p = \phi_b + i$, where $i$ is the roughness angle.
• Angle of repose: the steepest angle of a loose heap that stays stable, about $\phi$ for dry granular material.

Example: $\tau = 140$ kPa at $\sigma_n = 200$ kPa gives $\phi = \tan^{-1}(0.7) = 35.0^\circ$.

A slope with a discontinuity dipping steeper than $\phi$ and daylighting in the face can slide.