Mohr’s circle
Every 2D stress state at a point, on every possible rotated plane, maps to a single circle. Drag the stress element to rotate it and watch the corresponding point travel around Mohr’s circle. Principal stresses, maximum shear — the circle reveals them all at a glance. A beautiful geometric encoding of tensor transformation.
σ′ = (σx+σy)/2 + (σx−σy)/2 cos2θ + τxy sin2θ
What is stress transformation?
At any point in a loaded material, the stress state is described by a symmetric 2×2 tensor — three independent numbers: two normal stresses (σx, σy) and one shear stress (τxy). But these numbers depend on which coordinate axes you choose. Rotate your axes by an angle θ, and you get different values of normal and shear stress on the rotated planes. The question is: what are all the possible stress states you can see by rotating? The answer, discovered by Christian Otto Mohr in 1882, is that they all lie on a circle.
The geometry of Mohr’s circle
Plot normal stress σ on the horizontal axis and shear stress τ on the vertical axis. The stress state on any rotated plane is a point (σ′, τ′) in this space. As the rotation angle θ varies from 0 to 180°, this point traces out a complete circle. The center of the circle is at ((σx + σy)/2, 0) — the average normal stress, which is a tensor invariant (it doesn’t change with rotation). The radius is √[((σx − σy)/2)² + τxy²]. Physical rotation by θ corresponds to moving around the circle by 2θ — the famous “double angle” relationship.
Principal stresses
The principal stresses σ1 and σ2 are the maximum and minimum normal stresses — the rightmost and leftmost points on the circle. At these orientations, the shear stress is zero. Every stress state has a principal orientation where the material experiences only tension or compression, no shear. The principal stresses are eigenvalues of the stress tensor, and the principal directions are eigenvectors. Mohr’s circle gives you these directly: σ1,2 = center ± radius.
Maximum shear stress
The maximum shear stress is the topmost (and bottommost) point on the circle — equal to the radius itself. It occurs at 45° to the principal directions (90° on the circle, due to the double-angle relationship). This is why materials often fail along planes at 45° to the principal stress directions — a fact visible in the diagonal fracture patterns of brittle materials under tension and the slip lines in ductile metals.
Why it works: the algebra is a circle
The transformation equations are σ′ = C + R cos(2θ − 2θp) and τ′ = R sin(2θ − 2θp), where C is the center and R the radius. Eliminating the parameter 2θ, you get (σ′ − C)² + τ′² = R² — the equation of a circle. The stress transformation is mathematically identical to circular motion. This is not a coincidence; it is a consequence of the fact that 2D stress is a rank-2 symmetric tensor, and rotation of such a tensor in the (σ, τ) plane traces a circle. The same geometry appears in any 2D symmetric tensor: strain, moment of inertia, dielectric permittivity.
Mohr’s legacy
Christian Otto Mohr published this construction in 1882, and it became one of the most widely taught graphical methods in engineering. Before computers, engineers used Mohr’s circle with compass and straightedge to solve stress transformation problems by hand. Today, the circle remains valuable not for computation but for insight: it makes the abstract algebra of tensor transformation visually intuitive. You can see that rotating 45° from the principal directions gives maximum shear. You can see that the average normal stress never changes. You can see the entire family of stress states at once, encoded in a single geometric figure.