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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 4.4.5–1). Thus, the Mohr-Coulomb criterion can be written as
is the shear stress, 
is the normal stress (negative in compression), c is the cohesion of the material, and 
is the material angle of friction.From Mohr's circle, 
and , the Mohr-Coulomb criterion can be rewritten as 
The constitutive model described here is an extension of the classical Mohr-Coulomb failure criterion. It is an elastoplastic model that uses a yield function of the Mohr-Coulomb form; this yield function includes isotropic cohesion hardening/softening. However, the model uses a flow potential that has a hyperbolic shape in the meridional stress plane and has no corners in the deviatoric stress space. This flow potential is then completely smooth and, therefore, provides a unique definition of the direction of plastic flow.
An additive strain rate decomposition is assumed:
where
is the total strain rate, 
is the elastic strain rate, and 
is the inelastic (plastic) strain rate.The Mohr-Coulomb criterion written above in terms of the maximum and minimum principal stresses can be written for general states of stress in terms of three stress invariants. These invariants are the equivalent pressure stress,
is the stress deviator, defined as 
is the friction angle of the material in the meridional stress plane, where 
is the temperature and 
are other predefined field variables; 
represents the evolution of the cohesion of the material in the form of isotropic hardening (or softening); 
is the equivalent plastic strain, its rate defined by the plastic work expression 
is the Mohr-Coulomb deviatoric stress measure defined as 
is the deviatoric polar angle (Chen and Han, 1988) defined as 
, also controls the shape of the yield surface in the deviatoric plane as shown in Figure 4.4.5–3. The range of values the friction angle can have is 0° 
90°. In the case of 
0° the Mohr-Coulomb model reduces to the pressure-independent Tresca model with a perfectly hexagonal deviatoric section. In the case of 90° the Mohr-Coulomb model would reduce to the “tension cut-off” Rankine model with a triangular deviatoric section and 
(this limiting case is not permitted within the Mohr-Coulomb model described here).Potential flow is assumed, so
where g can be written as
is the dilation angle measured in the p–
plane at high confining pressure; 
is the initial cohesion yield stress; and 
is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). This flow potential, which is continuous and smooth in the meridional stress plane, ensures that the flow direction is defined uniquely in this plane. The function asymptotically approaches a linear flow potential at high confining pressure stress and intersects the hydrostatic pressure axis at 90°. A family of hyperbolic potentials in the meridional stress plane is shown in Figure 4.4.5–4.The flow potential is also continuous and smooth in the deviatoric stress plane (the 
-plane); we adopt the deviatoric elliptic function used by Menétrey and Willam (1995):
is the deviatoric polar angle defined previously, 
, and e is a parameter that describes the “out-of-roundedness” of the deviatoric section in terms of the ratio between the shear stress along the extension meridian (
) and the shear stress along the compression meridian (
). The elliptic function has the value 
along the extension meridian and has the value 
along the compression meridian; this ensures that the flow potential matches the yield surface at the triaxial compression and extension in the deviatoric plane provided that e is defined appropriately (see further discussion later). Although the elliptic function is defined only in the sector 
, the polar radius 
extends to all polar directions 
using the three-fold symmetry shown in Figure 4.4.5–5.By default, the out-of-roundedness parameter, e, is dependent on the friction angle ; it is calculated by matching the flow potential to the yield surface in both triaxial tension and compression in the deviatoric plane:
. The upper limit, 
(or 0°), leads to 
, which describes the Mises circle in the deviatoric plane. The lower limit, 
(or 90°), leads to 
and would describe the Rankine triangle in the deviatoric plane (this limiting case is not permitted within the Mohr-Coulomb model described here).Flow in the meridional stress plane can be close to associated when the angle of friction, , and the angle of dilation, 
, are equal and the eccentricity parameter, , is very small; however, flow in this plane is, in general, nonassociated. Flow in the deviatoric stress plane is always nonassociated. Therefore, the use of this Mohr-Coulomb model generally requires the solution of nonsymmetric equations.
