Von Mises Stress (2D)⚠ unverified

Understanding the Parameters

• The two normal stresses $\sigma_x$, $\sigma_y$ — the direct (tension/compression) stresses on the element. Note the cross term $-\sigma_x\sigma_y$: when both are the same sign (biaxial tension), it reduces $\sigma_v$, so a material under equal biaxial tension is further from yield than either stress alone would suggest. When they are opposite in sign (tension–compression), the term adds, pushing toward yield — pure shear is exactly this case.

• The shear stress $\tau_{xy}$ — enters with a factor of 3 under the root, so shear is far more potent at causing yield than an equal normal stress. Pure shear ($\sigma_x=\sigma_y=0$) gives $\sigma_v=\sqrt 3\,\tau$, so yielding in shear occurs at $\tau_y = \sigma_y/\sqrt 3 \approx 0.577\,\sigma_y$ — the von Mises shear-yield prediction, confirmed by experiment for ductile metals.

• The output $\sigma_v$ — a positive scalar with units of stress. It is the tensile stress that would produce the same distortion energy as the actual multiaxial state (see the derivation). Below yield the part is safe; at yield it begins to flow plastically.

• Plane-stress assumption. This 2-D form assumes the third principal stress is zero (thin skins, panels, webs — the aerospace norm). For a full 3-D state the criterion generalises to differences of all three principal stresses.

Derivation (Approaching a Proof)

Von Mises is the distortion-energy criterion: yielding begins when the energy of shape change (not volume change) reaches a critical value. Any stress state splits into a hydrostatic part $\sigma_m = \tfrac13(\sigma_1+\sigma_2+\sigma_3)$ that changes volume but not shape, and a deviatoric part that changes shape. Ductile metals yield by shear-driven dislocation slip, which is insensitive to hydrostatic pressure — so only the deviatoric energy matters.

Writing the criterion in principal stresses, the distortion energy is proportional to

$$(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2.$$

Setting this equal to its value in a uniaxial tension test at yield ($\sigma_1=\sigma_y$, others zero) defines the von Mises equivalent stress

$$\sigma_v = \sqrt{\tfrac12\big[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\big]}.$$

For plane stress set $\sigma_3=0$ and express $\sigma_1,\sigma_2$ back in the general $x$–$y$ frame. The principal-stress invariants give $\sigma_1+\sigma_2=\sigma_x+\sigma_y$ and $\sigma_1\sigma_2=\sigma_x\sigma_y-\tau_{xy}^2$, and substituting collapses the expression to the working form

$$\sigma_v = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2}. \qquad\blacksquare$$

The factor-of-3 on $\tau_{xy}^2$ and the $-\sigma_x\sigma_y$ cross term are direct fingerprints of the distortion-energy origin. In principal axes ($\tau_{xy}=0$) it reads $\sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2}$.

Dimensional check. Every term under the root is a stress squared ($\text{Pa}^2$), so the root is $\text{Pa}$. $\checkmark$

History and Development

• Maxwell, Huber, von Mises, Hencky. The distortion-energy idea was suggested by James Clerk Maxwell (1865), formalised by Tytus Huber (1904), stated cleanly by Richard von Mises (1913), and given its energy interpretation by Heinrich Hencky (1924). The criterion carries von Mises's name but is genuinely a century-long collaboration.

• Why it beat Tresca. The older Tresca (maximum-shear) criterion is simpler and always conservative, but von Mises matches ductile-metal test data more closely (Tresca under-predicts shear yield by ~15%). Von Mises is now the default in essentially all metal-structure analysis and finite-element post-processing (the ubiquitous colour "stress" plot is almost always $\sigma_v$). Compare Tresca Stress and Octahedral Shear Stress.

• Aerospace ubiquity. Because airframes are dominated by thin, biaxially-loaded skins and shear webs, the plane-stress von Mises form is the single most-used hand check in airframe sizing, feeding directly into margins of safety.

Related Concepts: Von Mises Stress, Tresca Stress, Octahedral Shear Stress, Max Principal Stress, Min Principal Stress, Factor of Safety yield, Margin Of Safety, Wing Bending Stress

Notes: Registry calculator von-mises-aerospace (unverified). Plane-stress (2-D) distortion-energy criterion — correct as shipped; the third principal stress is assumed zero. Duplicates the Physics/Materials Von Mises Stress page (same criterion). Yield in pure shear at $\tau_y=\sigma_y/\sqrt3\approx0.577\sigma_y$.