# Mohr-Coulomb model

 The Abaqus Mohr-Coulomb plasticity model uses the classical Mohr-Coulomb yield function, which includes isotropic cohesion hardening and softening. It also uses a smooth flow potential that has a hyperbolic shape in the meridional stress plane and a piecewise elliptic shape in the deviatoric stress plane.The following topics are discussed:
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The Mohr-Coulomb failure or strength criterion has been widely used for geotechnical applications. Indeed, a large number of the routine design calculations in the geotechnical area are still performed using the Mohr-Coulomb criterion.

The Mohr-Coulomb criterion assumes that failure is controlled by the maximum shear stress and that this failure shear stress depends on the normal stress. This can be represented by plotting Mohr's circle for states of stress at failure in terms of the maximum and minimum principal stresses. The Mohr-Coulomb failure line is the best straight line that touch es these Mohr's circles (Figure 1). Thus, the Mohr-Coulomb criterion can be written as

$τ=c-σ⁢tan⁡ϕ,$

where $τ$ is the shear stress, $σ$ is the normal stress (negative in compression), c is the cohesion of the material, and $ϕ$ is the material angle of friction.

Figure 1. Mohr-Coulomb failure criterion.

From Mohr's circle,

$τ=s⁢cos⁡ϕ,σ=σm+s⁢sin⁡ϕ.$

Substituting for $τ$ and $σ$, the Mohr-Coulomb criterion can be rewritten as

$s+σm⁢sin⁡ϕ-c⁢cos⁡ϕ=0,$

where

$s=12⁢(σ1-σ3)$

is half of the difference between the maximum and minimum principal stresses (and is, therefore, the maximum shear stress) and

$σm=12⁢(σ1+σ3)$

is the average of the maximum and minimum principal stresses (the normal stress). Thus, unlike the Drucker-Prager criterion, the Mohr-Coulomb criterion assumes that failure is independent of the value of the intermediate principal stress. The failure of typical geotechnical materials generally includes some small dependence on the intermediate principal stress, but the Mohr-Coulomb model is generally considered to be sufficiently accurate for most applications. This failure model has vertices in the deviatoric stress plane (see Figure 2).

Figure 2. Mohr-Coulomb model in the deviatoric plane.