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Notch Sensitivity: Method

Notch Sensitivity and Fatigue Notch Factor

Problem Statement

A member with a notch, hole or fillet of root radius r has an elastic stress concentration factor K_t. Fatigue tests show that the fatigue limit of the notched member is reduced by a smaller factor, the fatigue notch factor K_f = (fatigue limit of the smooth specimen)/(fatigue limit of the notched specimen on its net section), with 1 ≤ K_f ≤ K_t. The notch sensitivity q = (K_f − 1)/(K_t − 1) measures how much of the elastic concentration the material feels: q = 1 is full sensitivity, q = 0 none. Two things set it: the root radius, because the material responds to the stress averaged over a small volume at the root and a sharp notch has a steep gradient, and the strength of the material, because strong materials have a smaller averaging volume. Small notches are the practical point: a scratch, a small hole or a sharp fillet in a ductile material has a K_f far below its K_t, and below some depth a notch does not reduce the fatigue limit at all.

The tool takes K_t, the root radius r, the flank angle ω of a V-notch (0 for round notches, holes and fillets), the material and its ultimate tensile strength F_tu (written S_ult in the NACA reports and S_ut in Shigley), and optionally the notch depth d, the fatigue limit S_e and the threshold stress intensity range ΔK_th. It returns q and K_f by the two forms below, the constants they used with their derivation, and the recommended K_f.

V-notch with root radius r, depth d and flank angle omega in a member under alternating stress S

Figure A: A notch of root radius r, depth d and flank angle ω under an alternating nominal stress S. K_t is the elastic peak at the root; K_f is the factor on the fatigue limit. ρ′ and α are the material constants of the two forms.

Key Assumptions

Long-life, fully reversed loading; wrought steels, aluminum and titanium alloys; constants tied to the ultimate strength. The relations are empirical fits to rotating-beam and axial fatigue tests with a scatter of roughly 20 % in q. They give the factor on the fatigue limit (10⁶ to 10⁷ cycles); at shorter lives the effective factor falls toward 1. They do not transfer to castings, polymers or composites without test data, and they assume the notch root stays elastic in the sense of the fatigue-limit test.

Neuber's Form

Neuber proposed that the material responds to the stress averaged over an elementary block of size ρ′ at the root. Kuhn and Hardrath (NACA TN 2805, 1952) wrote the resulting factor for a V-notch of flank angle ω as

q = 1 / [1 + (π/(π − ω)) √(ρ′/r)], K_f = 1 + q(K_t − 1),

which for a round root (ω = 0) is the familiar q = 1/(1 + √(ρ′/r)). The Neuber constant ρ′ is a length (Neuber's and Pilkey's symbol; Kuhn and Hardrath write A, Dowling β, Shigley tabulates its square root as √a). It falls steeply with strength: a weak alloy is nearly insensitive to a sharp notch, a strong one is not.

Peterson's Form

Peterson (1959) proposed the simpler hyperbola

q = 1 / (1 + α/r), K_f = 1 + q(K_t − 1),

with the Peterson constant α, also a length (Peterson wrote it a; Dowling and Pilkey write α, used here to keep it apart from Shigley's √a). Peterson's constants are given for bending and axial loads. The two forms agree within their scatter over the usual range of radii; the Neuber form rises more steeply from q = 0 at very small radii because of the square root.

Material Constants

The tool evaluates the constants from the material choice and F_tu and prints the working on the results page; a constant typed on the form is used instead, as entered.

Steels. ρ′ for bending and axial loads is read from Kuhn and Hardrath's own curve, Figure 3 of NACA TN 2805, digitized for the tool and interpolated linearly between the points of Table 1 (50 to 250 ksi; outside that range the nearer end is used, with a note). For torsion, for which the NACA report has no curve, the tool uses Shigley's torsional fit (eq. 6-35b): √ρ′ = 0.190 − 2.51(10⁻³)F_tu + 1.35(10⁻⁵)F_tu² − 2.67(10⁻⁸)F_tu³ √in with F_tu in ksi, 50 to 250 ksi; the torsional constant is smaller, so a notch is felt more in torsion. α comes from Dowling's fit to Peterson's steel data, log₁₀ α = 2.654(10⁻⁷)F_tu² − 1.309(10⁻³)F_tu + 0.01103 with α in mm and F_tu in MPa, 345 to 2070 MPa (0.27 mm at 500 MPa, 0.093 mm at 1000 MPa, 0.027 mm at 2070 MPa), which spans Peterson's own values of 0.01 in for annealed or normalized steels and 0.0025 in for quenched and tempered steels.

Table 1: Neuber constant √ρ′ (√in) of steels against tensile strength, digitized from NACA TN 2805 Figure 3 (bending and axial loads)
F_tu (ksi)	50	60	70	80	90	100	120	140	160	180	200	220	250
√ρ′	0.126	0.106	0.093	0.082	0.073	0.064	0.051	0.040	0.031	0.023	0.017	0.011	0.0055

Aluminum alloys. ρ′ comes from the curves Kuhn and Figge fitted to notch fatigue and static notch tests of wrought sheet (NASA TN D-1259), as reproduced in Kuhn's 1965 Figure 2(b) and digitized for the tool (Table 2). Heat-treated T tempers and annealed or strain-hardened O and H conditions have separate curves, the second lower, so they are separate choices in the Material list; the curves cover 13 to 90 ksi (T) and 12 to 46 ksi (O, H). α is Peterson's single value for aluminum alloys, 0.02 in (0.51 mm).

Table 2: Neuber constant √ρ′ (√in) of wrought aluminum alloys against tensile strength, digitized from Kuhn 1965 Figure 2(b)
F_tu (ksi)	13	15	20	25	30	40	46	50	60	70	80	90
T condition	0.508	0.467	0.370	0.311	0.269	0.214	0.19	0.179	0.159	0.143	0.130	0.119
O and H condition	0.404	0.344	0.256	0.200	0.164	0.123	0.100

Titanium alloys. No published constant exists. The tool offers α = 0.0015 in (0.038 mm), back-calculated with α = r(1/q − 1) from the only systematic notch-radius series found in the open literature, Mowbray's KAPL-3158 tests of annealed Ti-6Al-6V-2Sn (F_tu 159 ksi) at 10⁶ cycles and room temperature (Table 3); the report itself rates the alloy's sensitivity as that of a 200 ksi steel. Other titanium data disagree strongly: Hosseini's rotating-bending tests of Ti-6Al-4V at K_t = 3.6 and r = 0.144 mm gave q of only 0.015 to 0.05, which would mean α of several millimetres. The titanium value is therefore the conservative end of a wide range. No Neuber constant is offered for titanium (Kuhn's 1965 Figure 2(c) gives tentative ones, marked for static notch factors only); enter ρ′ if a value is known.

Table 3: Peterson constant of titanium back-calculated from KAPL-3158 Table 4 (Ti-6Al-6V-2Sn, 75 °F, 10⁶ cycles)
r (in)	K_t	K_f	q	α = r(1/q − 1)
0.001	6.9	3.3	0.39	0.0016 in
0.0015	10.0	6.1	0.57	0.0011 in
0.005	5.7	4.5	0.74	0.0018 in
0.015	3.4	2.6	0.67	0.0074 in (not used)

Other materials, or a measured value. Enter the constants. Either one is back-calculated from a single notched fatigue test of the material: with the smooth and notched fatigue limits you have K_f, and for the specimen's K_t and root radius α = r(K_t − K_f)/(K_f − 1) and ρ′ = r[(K_t − K_f)/(K_f − 1)]². MIL-HDBK-5J carries notched fatigue curves (K_t = 2 and 3 specimens) for many alloys from which this can be done.

Threshold Bound for Small Notches

Neither form says when a notch stops mattering altogether. Smith and Miller (1978) showed that beyond a critical acuity a notch behaves like a crack of its own depth d: the fatigue limit of the member is the stress range at which such a crack reaches the threshold, Δσ = ΔK_th/(Y√(πd)), whatever K_t is. El Haddad, Topper and Smith (1979) joined this to the smooth fatigue limit Δσ_e with the intrinsic crack length a₀ = (1/π)(ΔK_th/(YΔσ_e))², so that Δσ = ΔK_th/(Y√(π(d + a₀))) holds for every depth (the Kitagawa–Takahashi diagram). The fatigue notch factor can therefore never exceed

K_f,th = Δσ_e/Δσ = √(1 + d/a₀).

The tool evaluates it with Y = 1.12 (edge crack), Δσ_e = 2S_e and the long-crack threshold at R = −1, when the fatigue limit, the threshold and the notch depth are entered, and caps the fatigue notch factor by it. For a steel with S_e = 50 ksi and ΔK_th = 6 ksi√in, a₀ = 0.0009 in: a notch 0.002 in deep cannot cost more than a factor of 1.8 however sharp it is, and one 0.0005 in deep less than 1.25.

What the Tool Reports

The constants with their derivation (the curve points or fit with the numbers substituted, or "entered"), q and K_f by each form, the threshold bound, and a recommended K_f: the higher of the two forms (conservative), capped by the bound when it applies. Two charts show q and K_f against the root radius with the user's notch marked.

Checks

For r → ∞ both forms give q = 1 and K_f = K_t; for r → 0 they give q = 0 and K_f = 1. The digitized curves sit on the figures (overlays filed with the sources). For a 0.1 in radius in a 100 ksi steel in bending the tool gives q = 0.84 by Neuber (√ρ′ = 0.064 √in) and 0.94 by Peterson (α = 0.0068 in), matching the published notch-sensitivity charts.

Limitations

The constants are statistical fits to test data on wrought alloys and carry about 20 % scatter; the titanium value rests on one alloy and is conservative. The relations describe the fatigue limit under fully reversed loading; mean stresses, finite life, surface finish, size and residual stresses are separate factors. Peterson's form is not evaluated in torsion unless a constant is entered. The threshold bound uses a long-crack ΔK_th at R = −1 and the edge-crack geometry factor; thresholds at other stress ratios or for short cracks differ, and the bound is a limit, not a prediction.