Mohr's circle

Definition

Consider a plane state of stress, given by the symmetric matrix $\begin{pmatrix} \sigma_x & \tau_{xy} \\ \tau_{xy} & \sigma_y \end{pmatrix}$ of a Cauchy stress tensor in the plane. For the unit normal $n = (\cos\theta, \sin\theta)$ and the unit tangent $m = (-\sin\theta, \cos\theta)$, let the normal and shear stresses on the line with normal $n$ be $$\sigma_n = n \cdot \sigma n = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos 2\theta + \tau_{xy} \sin 2\theta,$$ $$\tau_n = m \cdot \sigma n = -\frac{\sigma_x - \sigma_y}{2} \sin 2\theta + \tau_{xy} \cos 2\theta.$$ Mohr's circle is the circle in the $(\sigma_n, \tau_n)$-plane with centre $(c, 0)$ and radius $R$, where $$c = \frac{\sigma_x + \sigma_y}{2}, \qquad R = \sqrt{\left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2}.$$ As $\theta$ varies, the point $(\sigma_n, \tau_n)$ runs over the whole circle.