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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 (physics)|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 Juvinal Robert C. & Marshek Kurt M.; Fundamentals of machine component design -2nd ed 1991 pp. 217 ISBN 0-471-62281-8
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

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 1776 essay entitled "Essai sur une application des règles des maximis et minimis à quelques problèmes de statique relatifs à l'architecture"AMIR R. KHOEI; Computational Plasticity in Powder Forming Processes; Elsevier Amsterdam; 2005; 449 ppMohr developed a generalised form of the theory around the end of the 19th centuryMAO-HONG YU; "Advances in strength theories for materials under complex stress state in the 20th Century"; Applied Mechanics Reviews; American Society of Mechanical Engineers New York USA; May 2002; 55 (3): pp. 169–218As 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 NIELS SAABYE OTTOSEN and MATTI RISTINMAA; The Mechanics of Constitutive Modeling; Elsevier Science Amsterdam The Netherlands; 2005; pp. 165ff

Mohr-Coulomb failure criterion

Figure 1: View of Mohr-Coulomb failure surface in 3D space of principal stresses for c=2 phi=-20^circ
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
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}rightThe 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=123 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 sigmatau 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
Derivation of normal and shear stress on a plane
Let the unit normal to the plane of interest be
mathbf{n} = n_1~mathbf{e}_1 + n_2~mathbf{e}_2 + n_3~mathbf{e}_3 where mathbf{e}_i~~ i=123 are three orthonormal unit basis vectors Then the traction vector on the plane is given by
mathbf{t} = n_i~sigma_{ij}~mathbf{e}_j ~~~text{(repeated indices indicate summation)} The magnitude of the traction vector is given by
= 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
sigma = mathbf{t}cdotmathbf{n} = n_i~sigma_{ij}~n_j ~~text{(repeated indices indicate summation)} The magnitude of the resolved shear stress on the plane is given by
tau = sqrt{|mathbf{t}|^2 - sigma^2} In terms of components we have
begin{align} sigma & = n_1^2 sigma_{11} + n_2^2 sigma_{22} + n_3^2 sigma_{33} + 2(n_1 n_2 sigma_{12} + n_2 n_3 sigma_{23} + n_3 n_1 sigma_{31}) tau & = sqrt{(n_1sigma_{11} + n_2sigma_{12} + n_3sigma_{31})^2 + (n_1sigma_{12} + n_2sigma_{22} + n_3sigma_{23})^2 + (n_1sigma_{31} + n_2sigma_{23} + n_3sigma_{33})^2 - sigma^2} end{align} 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 sigmatau 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}

Figure 2: Mohr-Coulomb yield surface
in the pi-plane for c = 2, phi = -20^circ
Figure 3: Trace of the Mohr-Coulomb yield surface
in the sigma_1-sigma_2-plane for c = 2, phi = -20^circ

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(theta+cfrac{pi}{3}right) - sinphicosleft(theta+cfrac{pi}{3}right)right]rho - 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(theta+cfrac{pi}{3}right) - cfrac{1}{3}tanphi~cosleft(theta+cfrac{pi}{3}right)right]q - p~tanphi = c where
theta = cfrac{1}{3}arccosleft[left(cfrac{r}{q}right)^3right] ~.
Derivation of alternative forms of Mohr-Coulomb yield function
We can express the yield function
cfrac{sigma_1-sigma_3}{2} = cfrac{sigma_1+sigma_3}{2}~sinphi + ccosphi as
sigma_1~cfrac{(1-sinphi)}{2~c~cosphi} - sigma_3~cfrac{(1+sinphi)}{2~c~cosphi} = 1 ~. The Haigh-Westergaard invariants are related to the principal stresses by
sigma_1 = cfrac{1}{sqrt{3}}~xi + sqrt{cfrac{2}{3}}~rho~costheta ~;~~ sigma_3 = cfrac{1}{sqrt{3}}~xi + sqrt{cfrac{2}{3}}~rho~cosleft(theta+cfrac{2pi}{3}right) ~. Plugging into the expression for the Mohr-Coulomb yield function gives us
-sqrt{2}~xi~sinphi + rho[costheta - cos(theta+2pi/3)] - rhosinphi[costheta+cos(theta+2pi/3)] = sqrt{6}~c~cosphi Using trigonometric identities for the sum and difference of cosines and rearrangement gives us the expression of the Mohr-Coulomb yield function in terms of xi rho theta
We can express the yield function in terms of pq by using the relations
xi = sqrt{3}~p ~;~~ rho = sqrt{cfrac{2}{3}}~q and straightforward substitution

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 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
g:= sqrt{(alpha~c_mathrm{y}~tanpsi)^2 + G^2(phi theta)~ q^2}~ - p~tanphi where alpha is a parameter c_mathrm{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(phitheta) is an appropriate function that is also smooth in the deviatoric stress plane

See also

  • 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


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