The circle of Mohr for fractures

As already mentioned in the widget handling the circle of Mohr, it evaluates the normal and shear stress acting on a surface of a given orientation in a graphical manner. The method is restricted to two dimensions. In this way the stresses acting for example on a discontinuity can be calculated.

In two dimensions the Cauchy stress tensor can be written as:

$\boldsymbol{\sigma} = \left[ \begin{array}{cc} \sigma_{11} & \sigma_{13} \\ \sigma_{13} & \sigma_{33} \end{array}\right]$ and visualized as:

The normal and shear stress acting on a surface with an inclination of $\theta$ with respect to the horizontal are:

$\sigma = \frac{1}{2}(\sigma_{11}+\sigma_{33})-\frac{1}{2}(\sigma_{11}-\sigma_{33})\cos(2\theta) +\sigma_{13}\sin(2\theta)$

$\tau = \frac{1}{2}(\sigma_{11}-\sigma_{33})\sin(2\theta) -\sigma_{13}\cos(2\theta)$

or can be read from the following diagram, as the coordinates of the red point.

In principal directions, as shown in the figure above, the circle of Mohr can be sketched as shown below.

A discontinuity will slide, when the failure criterion crosses the point describing the stress state on the discontinuinity, as shown in the following figure.

This is different from intact rock or soil, in that the failure criterion can cut the circle of Mohr without failure necessarily taking place. The point is further illustrated in the figure bellow.

• Sliding takes place on the red discontinuity.
• The stress state on the blue discontuity is not possible.
• No movement takes place on the green and yellow discontinuities.

The implication for fractured rock mass is that

• sliding can take place on discontinuities, if their orientation allows it. This is the case when the failure criterion for the discontinuity crosses the point describing the stress state on the discontinuity.
• or new fractures can be created, when the failure criterion for the intact rock touches the circle of Mohr.

This means that, depending on the disconuity orientation and the strength of the fracture and the intact rock, the discontinuity, the intact rock or none of the two may fail.

In the widget below $\sigma_1$ is the maximum, vertical stress and $\sigma_3$ is the minimum horizontal stress. The friction angle of the discontinuity has a subscript K, while the cohesion and friction angle of the intact rock have no subscript. The cohesion of the fracture is assumed to be zero. The inclination of the discontinuity is $\theta$.

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References

[1] Brady, B.H.G. and Brown, E.T. (2006) Rock Mechanics for Underground Mining. Springer, Berlin.