- Mohr-Coulomb theory
Mohr-Coulomb theory is a
mathematical model (seeyield surface ) describing the response of a material such as rubble piles orconcrete 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 differenteffective stress es.In
structural engineering it is used to determine failure load as well as the angle offracture of a displacement fracture in concrete and similar materials. Coulomb'sfriction 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 theprinciple 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 externalmechanical 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. Byconservation 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 [Coulomb, C. A. (1776). "Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture." Mem. Acad. Roy. Div. Sav., vol. 7, pp. 343–387. ] 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:au = sigma~ an(phi) + c where au is the shear strength, sigma is the normal stress, c is the intercept of the failure envelope with the au 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 - au_m sinphi ~;~~ au = au_m cosphi where:au_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:au_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{egin{align} pmcfrac{sigma_1 - sigma_2}{2} & = left [cfrac{sigma_1 + sigma_2}{2} ight] sin(phi) + ccos(phi) \ pmcfrac{sigma_2 - sigma_3}{2} & = left [cfrac{sigma_2 + sigma_3}{2} ight] sin(phi) + ccos(phi)\ pmcfrac{sigma_3 - sigma_1}{2} & = left [cfrac{sigma_3 + sigma_1}{2} ight] sin(phi) + ccos(phi)end{align} ight.The Mohr-Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space.
The expressions for au 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, au are:egin{align} sigma & = n_1^2 sigma_{1} + n_2^2 sigma_{2} + n_3^2 sigma_{3} \ au & = 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:au = sigma~ an(phi) + c for the six planes of maximum shear stress.
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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 :cfrac{sigma_1-sigma_3}{2} = cfrac{sigma_1+sigma_3}{2}~sinphi + ccosphi can be expressed as:left [sqrt{3}~sinleft( heta+cfrac{pi}{3} ight) - sinphicosleft( heta+cfrac{pi}{3} ight) ight] ho - sqrt{2}sin(phi)xi = sqrt{6} c cosphi Alternatively, in terms of the invariants p, q, r we can write:left [cfrac{1}{sqrt{3}~cosphi}~sinleft( heta+cfrac{pi}{3} ight) - cfrac{1}{3} anphi~cosleft( heta+cfrac{pi}{3} ight) ight] q - p~ anphi = c where:heta = cfrac{1}{3}arccosleft [left(cfrac{r}{q} ight)^3 ight] ~.
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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 [Citation needed] :g:= sqrt{(alpha~c_y~ anpsi)^2 + G^2(phi, heta)~ q^2}~ - p~ anphi where alpha is a parameter, c_y is the value of c when the plastic strain is zero (also called the initial cohesion yield stress), psi is the angle made by the yield surface in the Rendulic plane at high values of p (this angle is also called the dilation angle), and G(phi, heta) is an appropriate function that is also smooth in the deviatoric stress plane.ee 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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