Glossary term

Von Mises Criterion

A yield criterion that predicts the onset of plastic deformation in ductile materials under multiaxial stress states.

Branch: Mechanical Engineering
Glossary type: method
Content: Glossary term
Updated: May 22, 2026
Revision: v0.2.0 · reviewed

Definition

method

The von Mises criterion is a yield criterion that predicts that a ductile material begins to yield when the von Mises equivalent stress — a scalar measure of the distortional energy in the stress state — reaches the material's uniaxial yield strength.

In uniaxial loading, yielding is straightforward to predict: it occurs when the applied stress reaches the yield strength. Under multiaxial loading — where normal stresses and shear stresses act simultaneously in different directions — a single scalar criterion is needed to map the combined stress state onto the same yield threshold. The von Mises criterion achieves this by computing the distortional strain energy density and comparing it to the value at yield in a uniaxial test. It is the most widely used yield criterion for ductile metals in engineering design and finite element analysis.

When a material element is subjected to a multiaxial stress state, yielding depends not on any single stress component but on the combined effect of all normal and shear stresses acting at the point. The von Mises criterion provides a scalar measure — the von Mises equivalent stress $\sigma_e$ , also called the effective stress — that condenses the full stress tensor into a single number comparable to the uniaxial yield strength $\sigma_y$ .

The von Mises equivalent stress

In terms of the six independent stress components of the stress tensor, the von Mises equivalent stress is:

\displaystyle \sigma_e = \sqrt{\frac{1}{2}\left[(\sigma_{xx}-\sigma_{yy})^2 + (\sigma_{yy}-\sigma_{zz})^2 + (\sigma_{zz}-\sigma_{xx})^2 + 6(\tau_{xy}^2+\tau_{yz}^2+\tau_{zx}^2)\right]}

In terms of the three principal stresses $\sigma_1$ , $\sigma_2$ , $\sigma_3$ — for which all shear components are zero — this simplifies to:

\displaystyle \sigma_e = \sqrt{\frac{1}{2}\left[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\right]}

The yield condition is:

\sigma_e \geq \sigma_y

where $\sigma_y$ is the uniaxial yield strength determined from a tensile test. When $\sigma_e < \sigma_y$ , the material remains in the elastic regime. When $\sigma_e$ reaches $\sigma_y$ , plastic deformation begins.

Physical interpretation: distortional energy

The von Mises criterion has a clear physical basis. The total strain energy density stored in an elastically deformed body can be decomposed into two parts: the volumetric (dilatational) part, associated with uniform hydrostatic pressure and change in volume, and the distortional (deviatoric) part, associated with shape change at constant volume. Experiments on ductile metals show that hydrostatic pressure alone — even at very high magnitudes — does not cause yielding. Yielding is driven by the distortional component. The von Mises equivalent stress is precisely proportional to the square root of the distortional strain energy density:

\displaystyle U_\text{distortion} = \frac{\sigma_e^2}{6G}

where $G$ is the shear modulus. Yielding occurs when this distortional energy reaches the value it has at yield in a uniaxial test.

Comparison with the Tresca criterion

The Tresca criterion (maximum shear stress criterion) predicts yielding when the maximum shear stress reaches half the uniaxial yield strength:

\displaystyle \tau_\text{max} = \frac{\sigma_1 - \sigma_3}{2} \geq \frac{\sigma_y}{2}

For most ductile metals under combined loading, the von Mises criterion is more accurate and less conservative than Tresca. In the principal stress plane, the von Mises criterion forms an ellipse while the Tresca criterion forms a hexagon inscribed within it. The maximum difference between the two predictions is about 15%, occurring under pure shear. For conservative design — particularly in pressure vessel codes — the Tresca criterion is sometimes preferred precisely because it is more conservative.

Application in engineering design and FEA

In structural design, the von Mises equivalent stress is used to check whether a component remains elastic under service loads. The design condition is:

\displaystyle \sigma_e \leq \frac{\sigma_y}{S}

where $S$ is the safety factor. Finite element analysis software automatically computes and displays $\sigma_e$ at every point in the model, making it the standard output for design checks in mechanical, aerospace, civil, and pressure equipment design. The criterion applies to isotropic ductile metals — steels, aluminium alloys, titanium alloys — under static loading. It is not appropriate for brittle materials, highly anisotropic materials, or fatigue assessment, each of which requires different criteria.

Common mistakes

A common mistake is accepting a design because peak von Mises stress is below yield while ignoring fatigue, buckling, fracture, contact pressure, weld details, or local singularities. Another is applying the criterion to brittle materials, composites, soils, foams, or strongly anisotropic materials without checking validity. A strong von Mises review states material model, yield strength basis, stress extraction location, mesh sensitivity, load combination, safety factor, temperature, plasticity assumptions, and whether the governing failure mode is actually yielding.

REF

Disciplines

Von Mises Criterion

Definition

See also