Mohr's circle is a graphical representation of how normal and shear stress components change as the orientation of a plane changes.

For plane stress, plot normal stress $\sigma$ horizontally and shear stress $\tau$ vertically. The two points representing the stresses on perpendicular faces form opposite ends of a diameter.

Principal stresses

The orientations where shear stress is zero are the principal planes. The corresponding normal stresses are the principal stresses $\sigma_1$ and $\sigma_2$.

On Mohr's circle they occur where the circle crosses the normal-stress axis.

Maximum shear stress

The maximum in-plane shear-stress magnitude equals the radius of the circle.

The doubled angle

A physical rotation of the stress element by $\theta$ corresponds to a movement of $2\theta$ around Mohr's circle. This follows from the double-angle terms in the stress-transformation equations.

Mohr's circle turns an abstract coordinate transformation into a geometric picture and provides the stress quantities needed by many failure criteria.