Mohr–Coulomb theory

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''" . 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 : \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 \tan(\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 \sin\phi ~;~~ \tau = \tau_m \cos\phi 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 \sin\phi + c \cos\phi ~. 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} \pm\cfrac{\sigma_1 - \sigma_2}{2} & = \left[\cfrac{\sigma_1 + \sigma_2}{2}\right]\sin(\phi) + c\cos(\phi) \\ \pm\cfrac{\sigma_2 - \sigma_3}{2} & = \left[\cfrac{\sigma_2 + \sigma_3}{2}\right]\sin(\phi) + c\cos(\phi)\\ \pm\cfrac{\sigma_3 - \sigma_1}{2} & = \left[\cfrac{\sigma_3 + \sigma_1}{2}\right]\sin(\phi) + c\cos(\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_1\sigma_{1})^2 + (n_2\sigma_{2})^2 + (n_3\sigma_{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. : == 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}~\sin\phi + c\cos\phi can be expressed as : \left[\sqrt{3}~\sin\left(\theta+\cfrac{\pi}{3}\right) + \sin\phi\cos\left(\theta+\cfrac{\pi}{3}\right)\right]\rho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi. Alternatively, in terms of the invariants p, q, r we can write : \left[\cfrac{1}{\sqrt{3}~\cos\phi}~\sin\left(\theta+\cfrac{\pi}{3}\right) + \cfrac{1}{3}\tan\phi~\cos\left(\theta+\cfrac{\pi}{3}\right)\right]q - p~\tan\phi = c where \theta = \cfrac{1}{3}\arccos\left[\left(\cfrac{r}{q}\right)^3\right] ~. : \lambda(\mathbf{r^\prime}) gives the charge per unit length at position \mathbf{r^\prime}, and dl^\prime is an infinitesimal element of length, dq = \lambda(\mathbf{r^\prime})dl^\prime [10] For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where \sigma(\mathbf{r^\prime}) gives the charge per unit area at position \mathbf{r^\prime}, and dA^\prime is an infinitesimal element of area, dq = \sigma(\mathbf{r^\prime})\,dA^\prime\, For a volume charge distribution (such as charge within a bulk metal) where \rho(\mathbf{r^\prime}) gives the charge per unit volume at position \mathbf{r^\prime}, and dV^\prime is an infinitesimal element of volume, dq = \rho(\mathbf{r^\prime})\,dV^\prime[9] The force on a small test charge q^\prime at position \mathbf{r} is given by \mathbf{F} = q^\prime\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3} --> == 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 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} \tan\psi)^2 + G^2(\phi, \theta)~ q^2} - p \tan\phi 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(\phi,\theta) is an appropriate function that is also smooth in the deviatoric stress plane. == Typical values of cohesion and angle of internal friction ==

Cohesion (alternatively called the cohesive strength) and friction angle values for rocks and some common soils are listed in the tables below. ==See also==