Mohr–Coulomb theory
Mohr–Coulomb theory is a mathematical model (see yield surface) describing the response of brittle materials such as concrete, or rubble piles, 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. Generally the theory applies to materials for which the compressive strength far exceeds the tensile strength. In geotechnical engineering 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 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. An alternative view derives the Mohr-Coulomb criterion as extension failure. History of the development. The Mohr–Coulomb theory is named in honour of Charles-Augustin de Coulomb and Christian Otto Mohr. Coulomb's contribution was a 1773 essay entitled ""Essai sur une application des règles des maximis et minimis à quelques problèmes de statique relatifs à l'architecture"" Mohr developed a generalised form of the theory around the end of the 19th century. As the generalised form affected the interpretation of the criterion, but not the substance of it, some texts  continue to refer to the criterion as simply the 'Coulomb criterion'. Mohr–Coulomb failure criterion. 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 where formula_2 is the shear strength, formula_3 is the normal stress, formula_4 is the intercept of the failure envelope with the formula_2 axis, and formula_6 is the slope of the failure envelope. The quantity formula_4 is often called the cohesion and the angle formula_8 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 formula_3 should be replaced with formula_10. If formula_11, the Mohr–Coulomb criterion reduces to the Tresca criterion. On the other hand, if formula_12 the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of formula_8 are not allowed. From Mohr's circle we have formula_14 where formula_15 and formula_16 is the maximum principal stress and formula_17 is the minimum principal stress. Therefore, the Mohr–Coulomb criterion may also be expressed as formula_18 This form of the Mohr–Coulomb criterion is applicable to  failure on a plane that is parallel to the formula_19 direction. Mohr–Coulomb failure criterion in three dimensions. The Mohr–Coulomb criterion in three dimensions is often expressed as The Mohr–Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space. The expressions for formula_2 and formula_3 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 where formula_24 are three orthonormal unit  basis vectors, and if the principal stresses formula_25 are aligned with the basis vectors formula_26, then the expressions for formula_27 are The Mohr–Coulomb failure criterion can then be evaluated using the usual expression formula_29 for the six planes of maximum shear stress.