Von Mises stress, explained

The idea

A tensile test gives one number, the yield strength S_y, for one stress state: pure tension. A real part is loaded in several directions at once. The von Mises criterion says a ductile metal yields when its distortion energy, the energy that changes its shape rather than its volume, reaches the value it has at yield in that tensile test.

Hydrostatic pressure, equal in all directions, changes volume and not shape. That is why it drops out of the formula, and why ductile metals do not yield under pure pressure.

Von Mises against Tresca

Tresca says yield starts when the maximum shear stress reaches half the tensile yield strength. It is simpler and slightly more conservative. Plotted in principal stress space, Tresca is a hexagon inside the von Mises ellipse, and the two never differ by more than about 15%. Most design codes and FEA packages use von Mises, because it matches test data on ductile metals more closely.

When not to use it

Brittle materials such as cast iron, ceramics and glass fail by fracture, not yield, and are much stronger in compression than in tension. Use maximum normal stress or Mohr–Coulomb instead. Fatigue needs the alternating and mean stresses treated separately. And at a sharp re-entrant corner in an FEA model, the stress is a singularity: the peak keeps rising as you refine the mesh, so that number is not a result.

A worked number

A shaft surface sees bending stress σ = 100 MPa and torsional shear τ = 50 MPa. With σ_y = 0, the plane-stress form gives σ_vm = √(100² + 3 × 50²) = √17,500 ≈ 132 MPa. For a steel with S_y = 250 MPa the factor of safety against yield is 250/132 ≈ 1.9.

Common mistakes

• Reading von Mises as tension or compression. It is always positive and carries no sign.
• Applying it to a brittle material.
• Reporting the peak at a mesh singularity as the part’s maximum stress.