| 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).
