fatigue-life

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.