What is the difference between Kt and Kf?

Kt (theoretical stress concentration factor) is the ratio of peak elastic stress to nominal stress from analysis. Kf (fatigue notch factor) is the actual fatigue strength reduction observed in testing, and is always ≤ Kt. The notch sensitivity index q captures how sensitive a material is to notches: Kf = 1 + q(Kt − 1).

What is the difference between the Neuber and Peterson methods?

Both compute notch sensitivity q from notch radius ρ and a material constant a, but with different functional forms. Neuber: q = 1/(1 + √(a/ρ)); Peterson: q = 1/(1 + a/ρ). Neuber yields slightly higher q (more conservative). Engineers select the method based on available material data or company standards.

How is the Neuber material constant 'a' determined?

Neuber's material constant is estimated as a = 0.0254×(1379/Su)⁴ mm, where Su is the ultimate tensile strength in MPa. Higher-strength materials have smaller a values and are more notch-sensitive. Lower-strength materials have larger a and are more notch-insensitive.

How do I use the modified Goodman diagram with a notch?

Replace the unnotched endurance limit Se with the modified value Se_notch = Se/Kf on the Goodman diagram. Any operating point (σ_mean, σ_amplitude) that falls below the line connecting (Su, 0) and (0, Se_notch) is considered safe against infinite fatigue life.

Fatigue Notch Factor Kf Simulator — Neuber vs Peterson Method

Material Preset

Su — Ultimate strength (MPa) 800

Se — Endurance limit (MPa) 400

Notch Parameters

Kt (stress concentration factor) 2.5

Notch radius ρ (mm) 0.50

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q (Neuber)

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q (Peterson)

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Kf (Neuber)

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Se_notch (MPa)

Key Formulas

Neuber: $q = \dfrac{1}{1 + \sqrt{a/\rho}}$

Peterson: $q = \dfrac{1}{1 + a/\rho}$

$K_f = 1 + q(K_t - 1)$

$a = 0.0254 \left(\dfrac{1379}{S_u}\right)^4$ mm

$S_{e,\text{notch}} = S_e / K_f$

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a (mm)

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Kf (Neuber)

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Kf (Peterson)

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Se_notch (MPa)

Chart 1: Notch Sensitivity q vs Notch Radius ρ

Chart 2: Fatigue Notch Factor Kf vs Kt (current ρ & material)

What is the Fatigue Notch Factor (Kf)?

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What exactly is the fatigue notch factor, Kf? I know Kt is for stress concentration, so is Kf just the same thing for fatigue?

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Basically, they're related but different. Kt is a purely geometric factor telling you how much a notch amplifies stress. Kf tells you how much that same notch actually reduces the fatigue strength of a *real material*. In practice, materials aren't perfectly sensitive to tiny notches, so Kf is almost always less severe than Kt. Try moving the "Ultimate Strength (Su)" slider in the simulator to see how the material changes Kf even when Kt stays the same.

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Wait, really? So a sharper notch (smaller radius) gives a higher Kt, but the material itself decides how much that hurts its fatigue life? How do we calculate that?

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Exactly! We use a "notch sensitivity factor," q, to bridge Kt and Kf. The formula is $K_f = 1 + q(K_t - 1)$. If q=1, the material is fully sensitive (Kf=Kt). If q=0, it's insensitive (Kf=1). The big debate is how to calculate 'q'. Our simulator lets you compare the two main theories: Neuber and Peterson. Change the "Notch Radius" parameter to see how their predictions differ, especially for very small radii.

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So Neuber and Peterson give different 'q' values... which one should I trust in a real design? And what's that 'a' constant in the formulas?

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Great question. 'a' is a material constant related to its microstructure. A common empirical formula is $a = 0.0254 (1379 / S_u)^4$ mm, where $S_u$ is ultimate strength. As for which theory to trust: Neuber is often more conservative (predicts higher Kf) for high-strength steels. Peterson can be better for ductile materials like aluminum. The best way to see this is in the simulator—crank up the Ultimate Strength and watch the gap between the two methods grow!

Physical Model & Key Equations

The core model connects the geometric stress concentration factor (Kt) to the fatigue notch factor (Kf) via a material-dependent notch sensitivity factor, q.

$$K_f = 1 + q(K_t - 1)$$

Here, $K_t$ is the theoretical stress concentration factor (from geometry), and $q$ is the notch sensitivity factor ranging from 0 (insensitive) to 1 (fully sensitive). The key difference between the Neuber and Peterson methods lies in how they calculate $q$ based on notch root radius $\rho$ and material constant $a$.

The Neuber and Peterson methods provide different formulas for the notch sensitivity factor, q, leading to different predictions for Kf, especially for small notch radii.

$$\text{Neuber: }q = \frac{1}{1 + \sqrt{a/\rho}}\quad \quad \text{Peterson: }q = \frac{1}{1 + a/\rho}$$

$\rho$ is the notch root radius in mm. $a$ is a material constant (in mm) that increases with ductility. A common empirical relation for steels is $a = 0.0254 \left(\dfrac{1379}{S_u}\right)^4$, where $S_u$ is the ultimate tensile strength in MPa. A smaller 'a' means a more notch-sensitive material (q closer to 1).

Real-World Applications

Automotive Crankshaft Design: Crankshafts have oil holes and fillets which act as stress concentrators. Calculating an accurate Kf is critical for predicting fatigue life under millions of engine cycles. Engineers use these methods to choose the right material strength and fillet radius to avoid catastrophic failure.

Aircraft Structural Joints: Rivet and bolt holes in aircraft frames are unavoidable notches. Using the Peterson or Neuber method helps determine the effective reduction in the aluminum alloy's endurance limit, ensuring the airframe can withstand cyclic pressurization and gust loads over its service life.

Medical Implant Fatigue Analysis: Bone screws and prosthetic stems have machined threads and grooves. Their fatigue strength in the human body is paramount. These Kf calculations guide the selection between more ductile (e.g., titanium) versus higher-strength materials, balancing notch sensitivity against overall strength.

Heavy Machinery Axles: Axles for construction equipment often have keyways or sudden diameter changes. Estimating Kf correctly prevents unexpected fatigue failures from repeated heavy loading. The choice between methods can influence whether to use a costlier, high-strength steel or a more forgiving, ductile grade.