Von Mises Stress

Theory

The von Mises criterion predicts yielding when the distortion energy density reaches the value at uniaxial yield. For 3D stress state: σ_vM = √[(σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²] / √2. Yielding occurs when σ_vM ≥ σ_y. For uniaxial tension σ_vM = σ. For pure shear, yielding occurs at τ = σ_y / √3 ≈ 0.577 σ_y. This is the most widely used yield criterion for ductile metals and thermoplastics.

Application

FEA software displays von Mises stress contour plots as the default output: coloured maps showing where σ_vM approaches σ_y indicate risk of yielding. For topology optimisation of FDM brackets, targets are typically σ_vM < σ_y / SF (safety factor 1.5-3). Von Mises stress is indispensable for multi-axial loading scenarios (combined bending, torsion, and axial loads).

Common mistakes

Von Mises criterion is designed for ductile, isotropic materials. For brittle materials (ceramics, PLA in Z-direction), the maximum principal stress criterion is more appropriate. For anisotropic FDM parts, von Mises comparisons to bulk σ_y are inaccurate in Z-direction. High von Mises stress at a sharp corner in FEA indicates a stress singularity: refine the mesh and add fillets before judging.

Related terms: Stress, Yield Strength, Shear Stress, Stress-Strain Curve