FLAC3D Theory and Background • Constitutive Models

Mohr-Coulomb Tension Crack (MohrT) Model*

Note

*This model is used in FLAC3D only.

The Mohr-Coulomb model is not well suited for simulation of tensile failure (fracturing). One non-physical behavior of the Mohr-Coulomb constitutive model regarding the tensile failure is that the tensile plastic strain, \(\varepsilon^p_i\), is irreversible. Consequently, any reversal of the plastic strain that caused failure in tension results in immediate generation of the compressive stresses. Typically, the irreversible part of the tensile plastic strain in the geological materials is relatively small (i.e., can be assumed to be zero). In other words, tensile fractures need to close completely before compressive stresses are generated in the direction perpendicular to the fracture. One consequence of this approximation in the Mohr-Coulomb law is that whenever material fails in tension, it excessively dilates, or increases in volume. This can be a problem, for example, in case of dynamic response (e.g., earthquake or blasting) or deformation of the overburden above the undercut that involves opening of the cracks as the immediate roof collapses followed by closure of the cracks as the entire overburden subsides. The constitutive model presented here is an extension of the Mohr-Coulomb constitutive model that considers tensile plastic strains to be reversible and prevents generation of compressive normal stresses (perpendicular to cracks) before cracks close.

Model Description

The model assumes that a zone can have up to three mutually perpendicular cracks. Each crack completely cuts throughout a zone. If there is no tensile failure or all tensile cracks are closed, the model behaves the same as a perfectly plastic Mohr-Coulomb constitutive model. If the tensile strength (which is initially isotropic) is exceeded,

(1)\[f^t = \sigma^t - \sigma_3 < 0\]

the following happens:

1. A crack is formed perpendicular to the tensile principal stress, \(\sigma_3\). Model parameter number-cracks for the number of cracks in the zone is set to one.
2. The information on the crack orientation (i.e., orientation of the unit normal), is calculated and stored as model parameters, \(n_{1,x}\), \(n_{1,y}\) and \(n_{1,z}\).
3. The tensile strength (which becomes anisotropic) perpendicular to the crack is set to zero, \(\sigma_1 = 0\), as a result of instantaneous softening.
4. The stresses are corrected (i.e., normal stress perpendicular to the crack is set to zero) according to the tensile failure envelope and associated flow rule (both defined as in the Mohr-Coulomb model), yielding the value of parameter \(\lambda^t\).
5. Crack opening, \(\varepsilon^p_1\) (model parameter strain-tension-plastic-1), is incremented by \(\lambda^t\), \(\Delta \varepsilon^{p}_1 = \lambda^{t}\) in each step.

In subsequent calculation steps, as long as \(\varepsilon^p_1\) is greater than zero, the stresses are corrected as if material is yielding in tension in the direction perpendicular to the crack and \(\lambda^t\) is calculated and added to \(\varepsilon^p_1\). Also, the shear stresses on the plane of the crack are set to zero (or stress state within the zone is such that the traction on the crack plane is zero). After the crack closes (i.e., \(\varepsilon^p_1 = 0\)), the model behaves as if the crack does not exist, except that the tensile strength perpendicular to the crack is zero.

If there is one crack in the zone, the second crack will be perpendicular to it. (The orientation of the first crack is arbitrary, determined by the orientation of the principal stresses when the crack forms.) Also, the tensile strength for the second (and the third) crack is unaffected by the previous cracks in the zone. There can be the maximum of three mutually perpendicular cracks within the zone. Thus, when cracks one and two are formed, the orientation of the third crack is predefined. Otherwise, the second and third cracks behave the same as the first crack. The opening is tracked separately for each crack. Parameter number-cracks provides the total number of cracks in the zone. The opening of each crack is stored in \(\varepsilon^p_i \mbox{ , } i=1{\sim}3\). The orientations of the cracks are stored in \((n_{i,x}, n_{i,y}, n_{i,z}) \mbox{ , } i=1{\sim}3\).

mohr-coulomb-tension Model Properties

Use the following keywords with the zone property command to set these properties of the Mohr-Coulomb tension model.

mohr-coulomb-tension

bulk f

elastic bulk modulus, \(K\)

cohesion f

cohesion, \(c\)

dilation f

dilation angle, \(\psi\). The default is 0.0.

friction f

internal angle of friction, \(\phi\)

poisson f

Poisson’s ratio, \(\nu\)

shear f

elastic shear modulus, \(G\)

tension f

tension limit, \(\sigma^t\). The default is 0.0.

young f

Young’s modulus, \(E\)

normal-1 v (r)

normal direction of the crack plane 2, (\(n_{1,x}\), \(n_{1,y}\), \(n_{1,z}\))

normal-2 v (r)

normal direction of the crack plane 2, (\(n_{2,x}\), \(n_{2,y}\), \(n_{2,z}\))

normal-3 v (r)

normal direction of the crack plane 3, (\(n_{3,x}\), \(n_{3,y}\), \(n_{3,z}\))

number-cracks i (r)

number of crack sets developed

strain-tension-plastic-1 f (r)

accumulated plastic tensile strain in the direction \(n_1\)

strain-tension-plastic-2 f (r)

accumulated plastic tensile strain in the direction \(n_2\)

strain-tension-plastic-3 f (r)

accumulated plastic tensile strain in the direction \(n_3\)

Key

(r) Read-only property.

This property cannot be set by the user. Instead, it can be listed, plotted, or accessed through FISH.

Notes

• Only one of the two options is required to define the elasticity: bulk modulus \(K\) and shear modulus \(G\), or Young’s modulus \(E\) and Poisson’s ratio \(\nu\). When choosing the latter, Young’s modulus \(E\) must be assigned in advance of Poisson’s ratio \(\nu\).
• The tension cut-off is \({\sigma}^t = min({\sigma}^t, c/\tan \phi)\).