In geology it is used to define shear strength of soils and rocks 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

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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

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Therefore the Mohr-Coulomb criterion may also be expressed as

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

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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

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

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= sqrt{ (n_j~sigma_{1j})^2 + (n_k~sigma_{2k})^2 + (n_l~sigma_{3l})^2} ~~~text{(repeated indices indicate summation)} Then the magnitude of the stress normal to the plane is given by - $$

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in the $pi$-plane for $c\; =\; 2,\; phi\; =\; -20^circ$ | in the $sigma\_1-sigma\_2$-plane for $c\; =\; 2,\; phi\; =\; -20^circ$ |

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

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We can express the yield function in terms of $pq$ by using the relations- $$

A common approach that is used is to use a

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- 3-D elasticity
- Christian Otto Mohr
- Henri Tresca
- Lateral earth pressure
- von Mises stress
- Shear strength
- Shear strength
- Strain
- Stress
- Yield
- Yield surface
- Drucker Prager yield criterion — a smooth version of the M–C yield criterion

- http://fbeuweacuk/public/geocal/SoilMech/basic/soilbasihtm
- http://wwwcivilusydeduau/courses/civl2410/earth_pressures_rankinedoc