Mohr

Mohr-Coulomb theory

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.

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.

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

 tau = sigma~tan(phi) + c

where 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 cosphi

where
tau_m = cfrac{sigma_1-sigma_3}{2} ~;~~ sigma_m = cfrac{sigma_1+sigma_3}{2}

and sigma_1 is the maximum principal stress and sigma_3 is the minimum principal stress.

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.

Mohr-Coulomb failure criterion in three dimensions

The Mohr-Coulomb criterion in three dimensions is often expressed as
left{begin{align} pmcfrac{sigma_1 - sigma_2}{2} & = left[cfrac{sigma_1 + sigma_2}{2}right]sin(phi) + ccos(phi) pmcfrac{sigma_2 - sigma_3}{2} & = left[cfrac{sigma_2 + sigma_3}{2}right]sin(phi) + ccos(phi) pmcfrac{sigma_3 - sigma_1}{2} & = left[cfrac{sigma_3 + sigma_1}{2}right]sin(phi) + ccos(phi) end{align}right. The Mohr-Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space.

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

mathbf{n} = n_1~mathbf{e}_1 + n_2~mathbf{e}_2 + n_3~mathbf{e}_3

where mathbf{e}_i,~~ i=1,2,3 are three orthonormal unit basis vectors, and if the principal stresses sigma_1, sigma_2, sigma_3 are aligned with the basis vectors mathbf{e}_1, mathbf{e}_2, mathbf{e}_3, then the expressions for sigma,tau are
begin{align} sigma & = n_1^2 sigma_{1} + n_2^2 sigma_{2} + n_3^2 sigma_{3} tau & = sqrt{(n_1sigma_{1})^2 + (n_2sigma_{2})^2 + (n_3sigma_{3})^2 - sigma^2}
        & = 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) + c

for the six planes of maximum shear stress.

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