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.where e i ,     i = 1 , 2 , 3 {displaystyle mathbf {e} _{i},~~i=1,2,3} are three orthonormal unit basis vectors. Then the traction vector on the plane is given byas 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. 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'. 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 τ {displaystyle au } is the shear strength, σ {displaystyle sigma } is the normal stress, c {displaystyle c} is the intercept of the failure envelope with the τ {displaystyle au } axis, and t a n ( ϕ ) {displaystyle tan(phi )} is the slope of the failure envelope. The quantity c {displaystyle c} is often called the cohesion and the angle ϕ {displaystyle 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 σ {displaystyle sigma } should be replaced with − σ {displaystyle -sigma } . If ϕ = 0 {displaystyle phi =0} , the Mohr–Coulomb criterion reduces to the Tresca criterion. On the other hand, if ϕ = 90 ∘ {displaystyle phi =90^{circ }} the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of ϕ {displaystyle phi } are not allowed.