# 4 Mohr-Coulomb

The Mohr-Coulomb is the most basic model capable of reproducing the key features of soils. While more complex models in principle offer the possibility of matching real soil behaviour better, the simplicity of the MC model makes it an attractive alternative, provided that the material parameters are chosen appropriately.

# 4.1 Summary of material parameters

The key material parameters are summarized below. More specialized parameters are elaborated on in later sections.

# Stiffness

E: Young's modulus
\nu: Poisson's ratio

# Strength

c': cohesion
\phi': friction angle

# 4.2 Governing equations

# 4.2.1 Elasticity

Isotropic elasticity defined by E and \nu is used (see Elasticity).

# 4.2.2 Failure surface

The Mohr-Coulomb failure surface is given by

F = |\sigma_1-\sigma_3|+(\sigma_1'+\sigma_3')\sin\phi' - 2c'\cos\phi' \tag{4.1}

where c' and \phi' are the cohesion and friction angle respectively.
Some possible depictions of the MC failure surface are shown in Figure 5.1.

# 4.2.3 Flow rule

The flow potential is given by

G = |\sigma_1-\sigma_3|+(\sigma_1'+\sigma_3')\sin\psi \tag{4.2}

where \psi is the dilation angle.

Figure 4.1: Possible depictions of Mohr-Coulomb yield surface in principal stress space. In (a) and (b), the principal stress ordering is \sigma_1\leq\sigma_2\leq\sigma_3 while no particular ordering is assumed in (c).

In GX, the basic choice of flow rule is between Associated and Nonassociated. In the former case, \psi=\phi. In the latter case, \psi is specified along with a possible cap on the dilation. That is, it is possible to define a critical strain above which \psi is set to zero. For Dilation Cap = Yes, the critical strain may be specified either in terms of a volumetric strain, \varepsilon_{v,cr}, or a deviatoric strain \varepsilon_{q,cr} (see Notation for exact definition of these strains).

Figure 4.2: Softening Mohr-Coulomb material in triaxial compression.

# 4.2.4 Softening

Softening is activated by first choosing Softening = Yes. Three parameters then define the properties of then softening stress-strain curve: the residual friction angle, \phi_\mathsf{residual}, and two characteristic deviatoric strains, \varepsilon_{q,cr,1} and \varepsilon_{q,cr,2}. These are sketched in Figure 4.2 with respect to a triaxial compression test. The ultimate residual strength is:

q_{u,\mathsf{residual}} =|\sigma_1-\sigma_3|_{u,\mathsf{residual}} = \frac{2c'\sin\phi_{\mathsf{residual}}'}{1-\sin\phi_{\mathsf{residual}}'}(-\sigma_3'+c'/\tan\phi_{\mathsf{residual}}') \tag{4.3}

where it is noted that only the friction angle and not the cohesion is affected by the softening.

Figure 4.3: Response of Mohr-Coulomb in triaxial compression without and with a Volumetric Dilation Cap.

# 4.3 Model response

The response of the MC material in triaxial compression is shown in Figure 4.3. The ultimate deviatoric stress is given by

q_u =|\sigma_1-\sigma_3|_u = \frac{2c'\sin\phi'}{1-\sin\phi}(-\sigma_3'+c/\tan\phi') \tag{4.4}