Notch Sensitivity⚠ unverified
Mechanical / Fatigue · Compute the notch-adjusted stress-concentration factor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| q | q | — | 1.0 | Notch sensitivity (dimensionless), between 0 and 1 |
| Kt | Kt | — | 1.0 | Theoretical (geometric) stress-concentration factor (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Kf | — | Fatigue stress-concentration factor Kf (dimensionless) |
The science & history
Understanding the Parameters
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Notch sensitivity $q$ — a material property (with the notch radius): $q = 0$ means fully insensitive ($K_f = 1$, the notch has no fatigue effect — as for very ductile or very small-radius notches in some materials), while $q = 1$ means fully sensitive ($K_f = K_t$, the full geometric peak acts). Hardened, high-strength steels approach $q = 1$; soft steels and small radii give lower $q$.
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Geometric factor $K_t$ — the purely elastic stress raiser from geometry alone (fillet, hole, groove); read from Peterson/Roark charts. It is what $q$ scales.
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Fatigue factor $K_f$ — always between $1$ and $K_t$. It is applied to the alternating stress in every fatigue calculation (Goodman, shaft/bolt/weld fatigue), which is why getting it right is central.
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Why $q < 1$ — the peak elastic stress at a notch acts over a tiny, steeply-graded volume; fatigue initiation needs a critically-stressed region, not just a point, so sharp notches are less damaging than $K_t$ alone predicts.
Derivation (Approaching a Proof)
By definition the notch sensitivity interpolates $K_f$ between its two physical bounds — no effect ($K_f = 1$) and full elastic effect ($K_f = K_t$):
$$q \equiv \frac{K_f - 1}{K_t - 1} \;\Longrightarrow\; K_f = 1 + q\,(K_t - 1).$$
The bounds are exact: at $q = 0$, $K_f = 1$; at $q = 1$, $K_f = K_t$. The physical content is in $q$ itself, which the Neuber and Peterson models predict from the notch root radius $r$ and a material length constant $a$:
$$q_{\text{Neuber}} = \frac{1}{1 + \sqrt{a}/\sqrt{r}}, \qquad q_{\text{Peterson}} = \frac{1}{1 + a/r}.$$
Both express the same idea — sharper notches (small $r$) and tougher/softer materials (larger $a$) give smaller $q$, so the fatigue concentration falls below the geometric one.
Dimensional check. $K_f = 1 + q\,(K_t - 1)$ is dimensionless — $q$, $K_t$, and $K_f$ are all pure ratios.
History and Development
The notch-sensitivity concept reconciles the elastic stress-concentration theory (Kirsch's hole solution, 1898; Neuber's notch analysis) with the observation that fatigue notches are "less bad" than elasticity predicts. Heinrich Neuber and later R.E. Peterson provided the material-length models for $q$ that, combined with $K_t$ charts, give the $K_f$ used throughout Shigley-style fatigue design. It is the bridge between a geometric feature and its real fatigue penalty.
Related Concepts: Fatigue Stress Concentration, Fatigue Notch Factor, Stress Concentration, Marin Endurance Limit, Shaft Fatigue Factor, Modified Goodman Factor
Notes: Outputs $K_f$ (not $q$); $q$ is an input. Duplicate of Fatigue Stress Concentration. $1 \le K_f \le K_t$. Get $q$ from Neuber/Peterson (notch radius + material constant); $K_t$ from Peterson charts. $K_f$ multiplies the alternating stress.