von-mises

name	von-mises
description	Von Mises yield criterion — principal stress transformation, 2D/3D von Mises stress, Mohr's circle, distortion energy theory, comparison with Tresca, failure envelope.
metadata	{"priority":8,"promptSignals":{"phrases":["von Mises","von mises stress","yield criterion","principal stress","Mohr's circle","distortion energy","Tresca","equivalent stress"],"minScore":3}}

Von Mises Stress & Yield Criteria — Complete Skill

Stress State — General 3D

Stress tensor [σ]: | σ_x τ_xy τ_xz | | τ_xy σ_y τ_yz | | τ_xz τ_yz σ_z |

Six independent components: σ_x, σ_y, σ_z, τ_xy, τ_yz, τ_xz

Principal Stresses

Principal stresses σ₁ ≥ σ₂ ≥ σ₃: eigenvalues of stress tensor Characteristic equation: σ³ - I₁σ² + I₂σ - I₃ = 0

Invariants: I₁ = σ_x + σ_y + σ_z (hydrostatic) I₂ = σ_xσ_y + σ_yσ_z + σ_zσ_x - τ_xy² - τ_yz² - τ_xz² I₃ = det[σ]

2D state (σ_z = τ_xz = τ_yz = 0): σ₁,₂ = (σ_x + σ_y)/2 ± √[(σ_x - σ_y)²/4 + τ_xy²] τ_max = (σ₁ - σ₂)/2

Von Mises (Distortion Energy) Criterion

Material yields when distortion energy reaches yield value:

3D general form: σ'_VM = √[½((σ₁-σ₂)² + (σ₂-σ₃)² + (σ₃-σ₁)²)]

2D (σ₃ = 0): σ'_VM = √(σ₁² - σ₁σ₂ + σ₂²)

Cartesian form (most useful in practice): σ'_VM = √(σ_x² + σ_y² + σ_z² - σ_xσ_y - σ_yσ_z - σ_zσ_x + 3τ_xy² + 3τ_yz² + 3τ_xz²)

Plane stress + shear (most common case): σ'_VM = √(σ_x² - σ_xσ_y + σ_y² + 3τ_xy²)

Bending + torsion (shaft design): σ'_VM = √(σ_b² + 3τ²) where σ_b = Mc/I, τ = Tc/J

Yield condition: Yielding begins when σ'_VM = Sy

Factor of safety: n = Sy / σ'_VM

Tresca Criterion (Maximum Shear Stress)

Yield when: τ_max = Sy/2 → σ₁ - σ₃ = Sy (yielding condition)

Comparison: Von Mises: τ_yield = Sy/√3 = 0.577 Sy Tresca: τ_yield = Sy/2 = 0.500 Sy

Von Mises is 15% more accurate (matches experiments) Tresca is conservative (safe, simpler, used in pressure vessel codes) Both agree at uniaxial tension and equal biaxial compression/tension

Mohr's Circle — 2D

Center: C = (σ_x + σ_y)/2 Radius: R = √[(σ_x - σ_y)²/4 + τ_xy²]

Principal stresses: σ₁ = C + R, σ₂ = C - R Max shear: τ_max = R (in-plane) Absolute max shear: τ_abs = max(|σ₁-σ₂|, |σ₂-σ₃|, |σ₃-σ₁|)/2

Sign convention: +τ plots downward on Mohr's circle (clockwise rotation) Angle 2θ on circle = 2× physical angle to principal plane

3D Mohr's Circle: three circles defined by (σ₁,σ₂), (σ₂,σ₃), (σ₁,σ₃) Absolute max shear = radius of largest circle = (σ₁-σ₃)/2

Failure Envelope Comparison

Biaxial stress space (σ₁, σ₂ plane): Von Mises: ellipse → σ₁² - σ₁σ₂ + σ₂² = Sy² Tresca: hexagon inscribed in Von Mises ellipse Modified-Mohr (brittle): asymmetric, uses Sut in tension, Suc in compression

Brittle materials: use Coulomb-Mohr or Modified-Mohr (not Von Mises) Criteria: σ₁/Sut - σ₂/Suc ≤ 1 (when σ₁>0, σ₂<0)

Hydrostatic vs. Deviatoric Decomposition

Total stress = hydrostatic + deviatoric: σ_hyd = I₁/3 = (σ₁+σ₂+σ₃)/3 (no distortion, no yield) σ_dev = σ - σ_hyd·I (causes distortion → yield)

Von Mises criterion: J₂ = Sy²/3 where J₂ = second deviatoric invariant J₂ = ½[(s₁²+s₂²+s₃²)] = (σ'_VM)²/3

Worked Example

Shaft: σ_x = 120 MPa (bending), τ_xy = 70 MPa (torsion), Sy = 350 MPa

σ'_VM = √(120² - 0 + 0 + 3×70²) = √(14400 + 14700) = √29100 = 170.6 MPa n = 350/170.6 = 2.05 ✓

Principal stresses: σ₁,₂ = 60 ± √(3600 + 4900) = 60 ± 92.2 σ₁ = 152.2 MPa, σ₂ = -32.2 MPa

τ_max = 92.2 MPa, τ_abs = (152.2+32.2)/2 = 92.2 MPa

Output

Report: σ'_VM [MPa], n = Sy/σ'_VM, principal stresses (σ₁,σ₂,σ₃), τ_max, safety criterion (Von Mises or Tresca).