Notch Sensitivity

Notch Sensitivity and Fatigue Notch Factor (structures/fatigue/notch-sensitivity)

Use when a notched detail (hole, fillet, cutout, lug) must be accounted for in a fatigue assessment: compute the elastic stress concentration factor Kt, reduce it to the fatigue notch factor Kf with the Peterson or Neuber method, quantify the notch sensitivity q, and amplify the nominal stress amplitude before the endurance check.

Domain quick reference

• Stress concentration factor: the elastic peak stress at the notch root divided by the nominal section stress, sigma_peak = Kt * sigma_nominal. Kt is a geometry-only quantity, independent of the material.
• Elliptical hole in an infinite plate: Kt = 1 + 2a/b, with a the semi axis perpendicular to the load and b the semi axis parallel to it. A circular hole (a = b) gives the classical Kt = 3.
• Circular hole in a finite width plate: Kt = 3 - 3.14(d/w) + 3.667(d/w)^2 - 1.527(d/w)^3, a curve fit of the Howland solution valid for d/w <= 0.5, with d the hole diameter and w the plate width. As the hole shrinks (d/w toward 0) the factor returns to 3.
• Fatigue notch factor: the reduction of the fatigue strength caused by the notch, Kf = S_smooth / S_notched. It lies between 1 (notch ignored) and Kt (full sensitivity), and it is what the endurance check multiplies the nominal amplitude by.
• Peterson method: Kf = 1 + (Kt - 1) / (1 + a / rho), with rho the notch root radius and a the Peterson material constant in the same length unit.
• Peterson material constant from strength: a = 0.0254 * (2070 / Sut)^1.8 in mm with Sut in MPa (steel correlation). At Sut = 2070 MPa the constant is 0.0254 mm, and it grows as the strength falls.
• Neuber method: Kf = 1 + (Kt - 1) / (1 + sqrt(a' / rho)), with a' the Neuber material constant in the same length unit as rho. The square root form decays more slowly with root radius than the Peterson linear form, so for a/rho below 1 the Neuber Kf sits below the Peterson Kf at equal constants.
• Notch sensitivity: q = (Kf - 1) / (Kt - 1), the fraction of the elastic concentration that actually affects fatigue. q = 0 means Kf = 1 (no fatigue reduction), q = 1 means Kf = Kt (full sensitivity). Blunt notches and ductile materials give q near 1; sharp notches and high strength give lower q.
• Effective stress amplitude: sigma_eff = Kf * sigma_nominal. The endurance check compares sigma_eff with the endurance limit, not the nominal amplitude.
• FAR-25 and CS-25 frame the certification context for fatigue substantiation of transport airplane structure; the notch mechanics themselves are standard mechanical engineering methodology.

Workflow

1. Resolve the geometry into Kt. For an elliptical or circular hole use kt_elliptical_hole (a perpendicular, b parallel to the load) or kt_circular_hole_finite_width (d and w in the same unit) when the plate width is known.
2. Estimate the Peterson material constant with peterson_material_constant from the ultimate tensile strength in MPa, or supply the Neuber constant a' directly from the material database (near 0.25 mm for steel).
3. Reduce Kt to Kf with peterson_fatigue_notch_factor or neuber_fatigue_notch_factor using the notch root radius rho.
4. Quantify the loss with notch_sensitivity: q = (Kf - 1) / (Kt - 1).
5. Amplify the nominal stress amplitude with effective_stress_amplitude and compare sigma_eff with the endurance limit for the verdict; max_stress_at_notch gives the elastic peak stress when the local elastic stress is what matters.
6. Record Kt, Kf, q, and sigma_eff in the fatigue substantiation report together with the endurance check result.

Pitfalls

• Using Kt in the fatigue check instead of Kf: the endurance limit comparison must use sigma_eff = Kf * sigma_nominal, not Kt * sigma_nominal, or the margin is systematically too small.
• Reading Kt from the wrong hole formula: 1 + 2a/b assumes the load runs along b; swap a and b and Kt drops from 5 to 2 for a 2:1 ellipse. Keep the perpendicular semi axis on top.
• Applying the finite width fit beyond its range: the d/w polynomial is a curve fit of the Howland solution valid for d/w <= 0.5, and the code rejects d >= w outright.
• Mixing length units in the material constants: Peterson a and Neuber a' must share the unit of rho (mm is the common choice); a factor of 25.4 error in a changes Kf materially at small root radii.
• Forgetting the strength dependence: the Peterson constant is not a fixed 0.0254 mm; it scales with (2070/Sut)^1.8, and using the high-strength value on a soft alloy overstates the fatigue notch factor.
• Treating q as a material constant: q depends on the root radius and the material constant together through Kf; a blunt notch in a notched part can show q near 1 while a sharp notch in the same material shows q near 0.5.
• Confusing peak elastic stress with fatigue-relevant stress: Kt * sigma_nominal is the local elastic peak; Kf * sigma_nominal is what the endurance limit check consumes. Report both, apply Kf.
• Assuming Neuber and Peterson agree: at equal constants and a/rho below 1, Neuber gives the lower Kf; pick one method and state it in the substantiation instead of cherry-picking the convenient one.

Behavior contract (gate 3)

The notch math is exercised by the gate 3 contract test: scripts/test_notch_sensitivity.py against scripts/notch_sensitivity_logic.py (stdlib unittest, offline). Run: python3 scripts/test_notch_sensitivity.py

Compliance

• Standards referenced, not reproduced: FAR-25 (US government work, public domain) and CS-25 (free EASA download) frame the fatigue substantiation context; the stress concentration, Peterson, and Neuber formulas above are common mechanical engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.