Mohr–Coulomb failure criterion

Figure 1: View of Mohr–Coulomb failure surface in 3D space of principal stresses for

The Mohr–Coulomb ^[5] failure criterion represents the linear envelope that is obtained from a plot of the shear strength of a material versus the applied normal stress. This relation is expressed as

where τ is the shear strength, σ is the normal stress, c is the intercept of the failure envelope with the τ axis, and φ is the slope of the failure envelope. The quantity c is often called the cohesion and the angle φ is called the angle of internal friction . Compression is assumed to be positive in the following discussion. If compression is assumed to be negative then σ should be replaced with − σ.

If φ = 0, the Mohr–Coulomb criterion reduces to the Tresca criterion. On the other hand, if the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of φ are not allowed.

From Mohr's circle we have

where

and σ₁ is the maximum principal stress and σ₃ is the minimum principal stress.

Therefore the Mohr–Coulomb criterion may also be expressed as

This form of the Mohr–Coulomb criterion is applicable to failure on a plane that is parallel to the σ₂ direction.

Mohr–Coulomb failure criterion in three dimensions

The Mohr–Coulomb criterion in three dimensions is often expressed as

The Mohr–Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space.

The expressions for τ and σ can be generalized to three dimensions by developing expressions for the normal stress and the resolved shear stress on a plane of arbitrary orientation with respect to the coordinate axes (basis vectors). If the unit normal to the plane of interest is

where are three orthonormal unit basis vectors, and if the principal stresses σ₁,σ₂,σ₃ are aligned with the basis vectors , then the expressions for σ,τ are

The Mohr–Coulomb failure criterion can then be evaluated using the usual expression

for the six planes of maximum shear stress.

Figure 2: Mohr–Coulomb yield surface in the π-plane for

Figure 3: Trace of the Mohr–Coulomb yield surface in theσ1 − σ2-plane for

Mohr–Coulomb failure surface in Haigh–Westergaard space

The Mohr–Coulomb failure (yield) surface is often expressed in Haigh–Westergaad coordinates. For example, the function

can be expressed as

Alternatively, in terms of the invariants p,q,r we can write

where