criterion The Mohr–Coulomb 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 is the shear strength, is the normal stress, is the intercept of the failure envelope with the axis, and is the slope of the failure envelope. The quantity is often called the cohesion and the angle 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 should be replaced with . If , 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 are not allowed. From Mohr's circle we have where and is the maximum principal stress and 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 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 and 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 are aligned with the basis vectors , then the expressions for are The Mohr–Coulomb failure criterion can then be evaluated using the usual expression for the six planes of maximum shear stress. {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of normal and shear stress on a plane |- |Let the unit normal to the plane of interest be where are three orthonormal unit basis vectors. Then the traction vector on the plane is given by The magnitude of the traction vector is given by Then the magnitude of the stress normal to the plane is given by The magnitude of the resolved shear stress on the plane is given by In terms of components, we have If the principal stresses are aligned with the basis vectors , then the expressions for are |} Mohr–Coulomb failure