Notch Sensitivity and Fatigue Notch Factor

Fatigue notch factor Kf = 1 + q (Kt − 1) of a notch, hole or fillet from its elastic stress concentration factor Kt, its root radius r and the notch sensitivity q of the material. Two forms of q are evaluated: Neuber's, q = 1 / [1 + (π/(π − ω)) √(ρ′/r)] with the flank angle ω and the Neuber constant ρ′ (Kuhn and Hardrath 1952), and Peterson's, q = 1 / (1 + α/r) with the Peterson constant α (Peterson 1959). The constants come from the material and its ultimate tensile strength Ftu: for steels ρ′ is read from the NACA TN 2805 curve and α from Dowling's fit; for aluminum alloys ρ′ from the Kuhn and Figge curves (NASA TN D-1259) and α = 0.02 in; for titanium alloys α = 0.0015 in from the KAPL-3158 tests; any constant typed on the form overrides. Small notches are the point: as r shrinks q falls toward zero and Kf stays well below Kt, and an optional threshold bound Kf ≤ √(1 + d/a0) from the notch depth d, the fatigue limit Se and the threshold ΔKth caps the factor for notches too small to start a crack.

  • Notch sensitivity q by Neuber's and Peterson's forms from the material and Ftu (steel, aluminum and titanium alloys) or from entered constants ρ′ and α

  • Fatigue notch factor Kf = 1 + q (Kt − 1) for axial and bending loads or torsion, with the flank angle ω of V-notches

  • Optional threshold bound √(1 + d/a0) for small notches from Se and ΔKth

  • Every constant derived on the page with its source; charts of q and Kf against the root radius

Notch and Material

Unit System: /

Convention

Notched member with root radius r and depth d; the fatigue notch factor K_f lies between 1 and the elastic K_t

Notch





Material and Load






Threshold Bound (optional)



Notch Sensitivity
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Notch Sensitivity and Fatigue Notch Factor

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 Kt. Fatigue tests show that the fatigue limit of the notched member is reduced by a smaller factor, the fatigue notch factor Kf = (fatigue limit of the smooth specimen)/(fatigue limit of the notched specimen on its net section), with 1 ≤ Kf ≤ Kt. The notch sensitivity q = (Kf − 1)/(Kt − 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 Kf far below its Kt, and below some depth a notch does not reduce the fatigue limit at all.

The tool takes Kt, 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 Ftu (written Sult in the NACA reports and Sut in Shigley), and optionally the notch depth d, the fatigue limit Se and the threshold stress intensity range ΔKth. It returns q and Kf by the two forms below, the constants they used with their derivation, and the recommended Kf.

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. Kt is the elastic peak at the root; Kf 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)], Kf = 1 + q(Kt − 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), Kf = 1 + q(Kt − 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 Ftu 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)Ftu + 1.35(10−5)Ftu² − 2.67(10−8)Ftu³ √in with Ftu 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)Ftu² − 1.309(10−3)Ftu + 0.01103 with α in mm and Ftu 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)
Ftu (ksi)5060708090100120140160180200220250
√ρ′0.1260.1060.0930.0820.0730.0640.0510.0400.0310.0230.0170.0110.0055

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)
Ftu (ksi)131520253040465060708090
T condition0.5080.4670.3700.3110.2690.2140.190.1790.1590.1430.1300.119
O and H condition0.4040.3440.2560.2000.1640.1230.100

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 (Ftu 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 Kt = 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)KtKfqα = r(1/q − 1)
0.0016.93.30.390.0016 in
0.001510.06.10.570.0011 in
0.0055.74.50.740.0018 in
0.0153.42.60.670.0074 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 Kf, and for the specimen's Kt and root radius α = r(Kt − Kf)/(Kf − 1) and ρ′ = r[(Kt − Kf)/(Kf − 1)]². MIL-HDBK-5J carries notched fatigue curves (Kt = 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 = ΔKth/(Y√(πd)), whatever Kt is. El Haddad, Topper and Smith (1979) joined this to the smooth fatigue limit Δσe with the intrinsic crack length a0 = (1/π)(ΔKth/(YΔσe))², so that Δσ = ΔKth/(Y√(π(d + a0))) holds for every depth (the Kitagawa–Takahashi diagram). The fatigue notch factor can therefore never exceed

Kf,th = Δσe/Δσ = √(1 + d/a0).

The tool evaluates it with Y = 1.12 (edge crack), Δσe = 2Se 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 Se = 50 ksi and ΔKth = 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 Kf by each form, the threshold bound, and a recommended Kf: the higher of the two forms (conservative), capped by the bound when it applies. Two charts show q and Kf against the root radius with the user's notch marked.

Checks

For r → ∞ both forms give q = 1 and Kf = Kt; for r → 0 they give q = 0 and Kf = 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 ΔKth 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.



References

Kuhn, P., and Hardrath, H. F. (1952). An Engineering Method for Estimating Notch-Size Effect in Fatigue Tests on Steel. NACA TN 2805. Neuber's equation in the form KN = 1 + (Kt − 1)/[1 + (π/(π − ω)) √(ρ′/r)] with the flank angle ω, and Figure 3, the Neuber constant √A (= √ρ′) of steels against the ultimate strength Sult (the tool's Ftu), 50 to 250 ksi, fitted from rotating-beam tests; steels, bending and axial loads only. The tool uses the curve itself, digitized every 5 ksi (digitization and overlay filed with the copy in the site's References folder).

Neuber, H. (1946). Theory of Notch Stresses. J. W. Edwards, Ann Arbor (translation of Kerbspannungslehre, 1937). The technical stress concentration factor and the material "elementary block" behind the Neuber constant.

Peterson, R. E. (1959). "Notch Sensitivity". Chapter 13 in Sines, G., and Waisman, J. L. (eds.), Metal Fatigue, McGraw-Hill, New York, pp. 293–306. The form q = 1/(1 + a/r) (the tool writes the constant α, after Dowling and Pilkey) and the constants a = 0.0025 in for quenched and tempered steels, 0.01 in for annealed or normalized steels and 0.02 in for aluminum alloys.

Budynas, R. G., and Nisbett, J. K. (2011). Shigley's Mechanical Engineering Design, 9th ed. McGraw-Hill, New York. Section 6-10, Eq. (6-35b): the torsional Neuber constant fitted to the ultimate strength, √a = 0.190 − 2.51(10−3)Sut + 1.35(10−5)Sut² − 2.67(10−8)Sut³ (Sut in kpsi, the tool's Ftu; √a in √in, 50 to 250 kpsi), a fit to the torsional notch-sensitivity chart (Figure 6-21, Peterson's data), used by the tool for steels in torsion, for which NACA TN 2805 has no curve. Eq. (6-35a), the bending and axial fit, agrees with the NACA curve to about 200 kpsi and is not used.

Dowling, N. E. (2013). Mechanical Behavior of Materials, 4th ed. Pearson, Boston. Chapter 10, "Stress-Based Approach to Fatigue: Notched Members": Peterson's constant of steels fitted to the ultimate strength, log10 a = 2.654(10−7)σu² − 1.309(10−3)σu + 0.01103 (a in mm, σu in MPa, 345 to 2070 MPa), the use of a = 0.51 mm for aluminum alloys, and the discussion of notch sensitivity at small radii.

El Haddad, M. H., Topper, T. H., and Smith, K. N. (1979). "Prediction of Non Propagating Cracks". Engineering Fracture Mechanics, Vol. 11, pp. 573–584. The intrinsic crack length a0 = (1/π)(ΔKth/(YΔσe))² that joins the fatigue limit to the threshold; the tool's bound √(1 + d/a0) follows from it.

Smith, R. A., and Miller, K. J. (1978). "Prediction of Fatigue Regimes in Notched Components". International Journal of Mechanical Sciences, Vol. 20, pp. 201–206. Blunt and sharp (crack-like) notch regimes: beyond a critical acuity the fatigue limit of a notched member is that of a crack of the notch depth, independent of Kt.

Kitagawa, H., and Takahashi, S. (1976). "Applicability of Fracture Mechanics to Very Small Cracks or the Cracks in the Early Stage". Proceedings of the 2nd International Conference on Mechanical Behavior of Materials, ASM, pp. 627–631. The diagram of threshold stress against crack size that defines the small-flaw regime the bound describes.

Kuhn, P. (1965). The Prediction of Notch and Crack Strength under Static or Fatigue Loading. NASA Langley Research Center, paper for the SAE–ASME Air Transport and Space Meeting, New York, April 1964 (SAE 640500); NTRS 19650025604. Figure 2: Neuber constants √ρ′ against tensile strength for (a) low-alloy steels (from NACA TN 2805), (b) wrought aluminum alloys, two curves, heat-treated T condition (13 to 90 ksi) and annealed O or strain-hardened H condition (12 to 46 ksi), from NASA TN D-1259, and (c) tentative curves for titanium alloys, marked for static notch factors only. The tool's aluminum Neuber constants are the digitized curves of Figure 2(b) (table in the explanation). Copy filed in the site's References folder.

Kuhn, P., and Figge, I. E. (1962). Unified Notch-Strength Analysis for Wrought Aluminum Alloys. NASA TN D-1259. The source of the aluminum Neuber-constant curves, fitted to notch fatigue and static notch-strength tests of sheet; the curves are used here as reproduced in Kuhn 1965 because the report itself was not obtainable by script.

Mowbray, D. F. (1966). Fatigue Design Data for the Titanium Alloy, Ti-6Al-6V-2Sn. KAPL-3158, Knolls Atomic Power Laboratory (General Electric) for the U.S. Atomic Energy Commission, OSTI 4476225. Table 4 and Figure 7: fatigue notch factors at 106 cycles for circumferentially notched round bars of annealed Ti-6Al-6V-2Sn (UTS 159 ksi) with root radii 0.001 to 0.015 in at 75 and 600 °F; the room-temperature notch sensitivities q = 0.39, 0.57, 0.74 at r = 0.001, 0.0015, 0.005 in give Peterson's α = r(1/q − 1) = 0.0016, 0.0011, 0.0018 in, the tool's titanium value 0.0015 in (0.038 mm); the report rates the alloy's sensitivity as that of heat-treated steel of 200 ksi (125 ksi at 600 °F). Copy filed in the site's References folder.

Hosseini, S. (2012). "Fatigue of Ti-6Al-4V". Chapter 3 in Hudak, R. (ed.), Biomedical Engineering: Technical Applications in Medicine, InTech, pp. 75–92. Table 4: notch sensitivity of Ti-6Al-4V round bars in rotating bending, q = 0.05 to 0.015 for UTS 900 to 1150 MPa at Kt = 3.6 and a root radius of 0.144 mm, far below the steel value 0.59 for the same radius; shows how widely titanium alloys and test conditions scatter. Copy filed in the site's References folder.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Section 1.9, "Notch Sensitivity": the definitions of q and Kf, Chart 1.1 (notch sensitivity of steels and aluminum alloys against the notch radius) and the remarks on the scatter of the constants.

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