| name | fatigue-life |
| description | Complete fatigue life analysis — S-N curve, endurance limit, stress concentration, Goodman/Gerber/Morrow/SWT criteria, cycle life estimation, variable amplitude (Miner's rule), multiaxial fatigue. |
| metadata | {"priority":9,"promptSignals":{"phrases":["fatigue","cyclic loading","S-N curve","endurance limit","Goodman","Gerber","cycle life","fatigue failure","alternating stress"],"minScore":4}} |
Fatigue Life Analysis — Complete Skill
Step 1: Stress State Classification
- Fully reversed: σ_m = 0, R = -1, σ_a = σ_max
- Repeated: σ_min = 0, R = 0, σ_m = σ_a = σ_max/2
- Fluctuating: σ_m = (σ_max+σ_min)/2, σ_a = (σ_max-σ_min)/2
- Stress ratio: R = σ_min/σ_max
Step 2: Endurance Limit
Estimate (steel): S'_e ≈ 0.5Sut (if Sut ≤ 1400 MPa / 200 ksi)
S'_e ≈ 700 MPa (if Sut > 1400 MPa)
Correction factors (Marin equation):
Se = k_a · k_b · k_c · k_d · k_e · k_f · S'_e
- k_a = surface factor: a·Sut^b (machined: a=1.58,b=-0.085; hot-rolled: a=57.7,b=-0.718)
- k_b = size factor: 1.0 (d≤8mm), 1.24d^-0.107 (8<d≤250mm)
- k_c = load factor: 1.0 (bending), 0.85 (axial), 0.59 (torsion)
- k_d = temperature factor: 1.0 (T≤450°C), degrades above
- k_e = reliability factor: 0.868 (90%), 0.814 (95%), 0.753 (99%), 0.702 (99.9%)
- k_f = miscellaneous: residual stress, plating, etc.
Other materials:
- Aluminum: S_e ≈ 0.4Sut (at 5×10⁸ cycles), no true endurance limit
- Cast iron: S_e ≈ 0.4Sut
- Copper alloys: S_e ≈ 0.35Sut
Step 3: Stress Concentration
Fatigue stress concentration factor:
Kf = 1 + q(Kt - 1)
q = notch sensitivity = 1/(1 + √(a/r)) where a = √(a_Neuber)
Neuber constant √a (steel):
√a = 0.246 - 3.08×10⁻³Sut + 1.51×10⁻⁵Sut² - 2.67×10⁻⁸Sut³ (Sut in ksi)
Effective alternating stress:
σ'_a = Kf · σ_a (apply Kf to alternating component only for ductile materials)
σ'_m = Kfm · σ_m (Kfm = 1 for ductile yielded notch root, = Kf for brittle)
Step 4: Fatigue Criteria (choose appropriate)
Goodman (conservative, recommended for design):
σ_a/Se + σ_m/Sut = 1
n = 1 / (σ_a/Se + σ_m/Sut)
Gerber (less conservative, better fits data):
σ_a/Se + (σ_m/Sut)² = 1
Soderberg (very conservative, uses Sy):
σ_a/Se + σ_m/Sy = 1
ASME Elliptic:
(σ_a/Se)² + (σ_m/Sy)² = 1
Morrow (mean stress effect):
σ_a/Se + σ_m/σ_f' = 1 (σ_f' = fatigue strength coefficient)
Smith-Watson-Topper (SWT, multiaxial):
σ_max · ε_a · E = (σ_f')² / E · (2N_f)^2b + σ_f' · ε_f' · (2N_f)^(b+c)
Step 5: S-N Curve Construction
High cycle (10³ < N < 10⁶):
At N = 10³: S_1000 = 0.9Sut (bending), 0.75Sut (axial)
At N = 10⁶: S_e
Log-log interpolation: S_f = a · N^b
b = -1/3 · log(S_1000/Se), a = S_1000 / (10³)^b
Low cycle (N < 10³): Use strain-life:
ε_a = σ_f'/E · (2N_f)^b + ε_f' · (2N_f)^c
(Coffin-Manson: b ≈ -0.085, c ≈ -0.6, ε_f' ≈ 0.5-1.0)
Step 6: Cycle Life Estimate
N_f = (Se/σ_a)^(1/b) · 10⁶ (if S_e < σ_a < S_1000)
Or read from S-N curve
Step 7: Variable Amplitude — Miner's Rule
Σ(n_i/N_i) = 1 (failure when sum = 1)
Conservative: use Σ(n_i/N_i) = 0.7-0.9 for design
Rainflow counting: Required for irregular load histories. Count stress ranges and means, then apply damage per cycle type.
Step 8: Multiaxial Fatigue
Von Mises equivalent: σ_a,eq = √(σ_a² - σ_a·τ_a + τ_a²) (for proportional loading)
Non-proportional: use critical plane methods (Fatemi-Socie, Smith-Watson-Topper)
Step 9: Design Improvements
If n < target:
- Increase section (reduce σ_a)
- Improve surface finish (raise k_a → Se)
- Shot peen (introduce compressive residual stress, raise k_f)
- Reduce stress concentrations (increase fillet radius)
- Change material (higher Se)
- Apply cold working
Output
State: σ_a, σ_m, Se (with all Marin factors), governing criterion, n_f, N_f (cycles to failure), failure mode.
