The circle of Mohr¶
The circle of Mohr evaluates the normal and shear stress acting on a surface of a given orientation in a graphical manner. The method is restricted to two dimensions. In this way the stresses acting for example on a discontinuity can be calculated.
In two dimensions the Cauchy stress tensor can be written as:
$\boldsymbol{\sigma} = \left[ \begin{array}{cc} \sigma_{xx} & \sigma_{xy} \\ \sigma_{xy} & \sigma_{yy} \end{array}\right]$
The normal and shear stress acting on a surface with an inclination of $\alpha$ with respect to the horizontal are:
$\sigma = \frac{1}{2}(\sigma_{xx}+\sigma_{yy})-\frac{1}{2}(\sigma_{xx}-\sigma_{yy})\cos(2\alpha) +\sigma_{xy}\sin(2\alpha)$
$\tau = \frac{1}{2}(\sigma_{xx}-\sigma_{yy})\sin(2\alpha) -\sigma_{xy}\cos(2\alpha)$
Combining the equations gives the equation of the circle of Mohr:
$\left(\sigma - p\right)^2 + \tau^2 = q^2$
where $p = \frac{1}{2}(\sigma_{xx}+\sigma_{yy})$ and $q = \sqrt{\left(\frac{\sigma_{xx}-\sigma_{yy}}{2}\right)^2 + \sigma_{xy}^2}$
In principal directions, as shown in the figure above, the circle of Mohr can be sketched as shown below.
To find the normal and shear stress acting on a surface inclined at an angle $\theta$ in the counterclockwise direction from the surface on which the maximum normal stress in acting, a rotation of $2\theta$ is performed on the circle of Mohr. The vertical coordinate of the point gives the shear stress and the horizontal gives the normal stress.
In the widget below $\sigma_1$ is the maximum, vertical stress and $\sigma_3$ is the minimum, horizontal stress.
