In a two-dimensional stress analysis, the state of stress at a point $P$ is
$$\begin{bmatrix} \sigma \end{bmatrix} = \begin{bmatrix}\sigma_{xx} &\tau_{xy} \\ \tau_{xy} &\sigma_{yy} \end{bmatrix}$$
The necessary and sufficient condition for existence of the state of pure shear at the point $P$, is
- $\sigma_{xx}\sigma_{yy}-\tau^2_{xy}=0$
- $\tau_{xy}=0$
- $\sigma_{xx}+\sigma_{yy}=0$
- $(\sigma_{xx}-\sigma_{yy})^2+4\tau^2_{xy}=0$