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Mohr-Coulomb theory is a mathematical model (see yield surface) describing the response of a material such as rubble piles or concrete to shear stress as well as normal stress. Most of the classical engineering materials somehow follow this rule in at least a portion of their shear failure envelope.## Mohr-Coulomb failure criterion

### Mohr-Coulomb failure criterion in three dimensions

The Mohr-Coulomb criterion in three dimensions is often expressed as

In geology it is used to define shear strength of soils at different effective stresses.

In structural engineering it is used to determine failure load as well as the angle of fracture of a displacement fracture in concrete and similar materials. Coulomb's friction hypothesis is used to determine the combination of shear and normal stress that will cause a fracture of the material. Mohr's circle is used to determine which principal stresses that will produce this combination of shear and normal stress, and the angle of the plane in which this will occur. According to the principle of normality the stress introduced at failure will be perpendicular to the line describing the fracture condition.

It can be shown that a material failing according to Coulomb's friction hypothesis will show the displacement introduced at failure forming an angle to the line of fracture equal to the angle of friction. This makes the strength of the material determinable by comparing the external mechanical work introduced by the displacement and the external load with the internal mechanical work introduced by the strain and stress at the line of failure. By conservation of energy the sum of these must be zero and this will make it possible to calculate the failure load of the construction.

A common improvement of this model is to combine Coulomb's friction hypothesis with Rankine's principal stress hypothesis to describe a separation fracture.

The Mohr-Coulomb failure criterion represents the linear envelope that is obtained from a plot of the shear strength of a material versus the applied normal stress. This relation is expressed as

- $$

tau = sigma~tan(phi) + cwhere $tau$ is the shear strength, $sigma$ is the normal stress, $c$ is the intercept of the failure envelope with the $tau$ axis, and $phi$ is the slope of the failure envelope. The quantity $c$ is often called the cohesion and the angle $phi$ is called the angle of internal friction . Compression is assumed to be positive in the following discussion. If compression is assumed to be negative then $sigma$ should be replaced with $-sigma$.

If $phi\; =\; 0$, the Mohr-Coulomb criterion reduces to the Tresca criterion. On the other hand, if $phi\; =\; 90^circ$ the Mohr-Coulomb model is equivalent to the Rankine model. Higher values of $phi$ are not allowed.

From Mohr's circle we have

- $$

sigma = sigma_m - tau_m sinphi ~;~~ tau = tau_m cosphiwhere

- $$

Therefore the Mohr-Coulomb criterion may also be expressed as

- $$

tau_m = sigma_m sinphi + c cosphi ~.

This form of the Mohr-Coulomb criterion is applicable to failure on a plane that is parallel to the $sigma\_2$ direction.

- $$

The expressions for $tau$ and $sigma$ can be generalized to three dimensions by developing expressions for the normal stress and the resolved shear stress on a plane of arbitrary orientation with respect to the coordinate axes (basis vectors). If the unit normal to the plane of interest is

- $$

- $$

& = sqrt{n_1^2 n_2^2 (sigma_1-sigma_2)^2 + n_2^2 n_3^2 (sigma_2-sigma_3)^2 +n_3^2 n_1^2 (sigma_3 - sigma_1)^2} end{align} The Mohr-Coulomb failure criterion can then be evaluated using the usual expression

- $$

tau = sigma~tan(phi) + cfor the six planes of maximum shear stress.

Derivation of normal and shear stress on a plane Let the unit normal to the plane of interest be - $$

- $$

- $$

- $$

- $$

- $$

- $$

& = sqrt{n_1^2 n_2^2 (sigma_1-sigma_2)^2 + n_2^2 n_3^2 (sigma_2-sigma_3)^2 +

n_3^2 n_1^2 (sigma_3 - sigma_1)^2} end{align}## Mohr-Coulomb failure surface in Haigh-Westergaard space

The Mohr-Coulomb failure (yield) surface is often expressed in Haigh-Westergaad coordinates. For example, the function- $$

- $$

- $$

- $$

Derivation of alternative forms of Mohr-Coulomb yield function We can express the yield function - $$

- $$

- $$

- $$

We can express the yield function in terms of $p,q$ by using the relations

- $$

## Mohr-Coulomb yield and plasticity

The Mohr-Coulomb yield surface is often used to model the plastic flow of geomaterials (and other cohesive-frictional materials). Many such materials show dilatational behavior under triaxial states of stress which the Mohr-Coulomb model does not include. Also, since the yield surface has corners, it may be inconvenient to use the original Mohr-Coulomb model to determine the direction of plastic flow (in the flow theory of plasticity).A common approach that is used is to use a non-associated plastic flow potential that is smooth. An example of such a potential is the function

- $$

## See also

- 3-D elasticity
- Christian Otto Mohr
- Henri Tresca
- Lateral earth pressure
- von Mises stress
- Shear strength
- Shear strength (soil)
- Strain (materials science)
- Stress (physics)
- Yield (engineering)
- Yield surface

## References

- http://fbe.uwe.ac.uk/public/geocal/SoilMech/basic/soilbasi.htm
- http://www.civil.usyd.edu.au/courses/civl2410/earth_pressures_rankine.doc

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Last updated on Monday September 01, 2008 at 07:04:06 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday September 01, 2008 at 07:04:06 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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