3.1 Introduction to High-Cycle Fatigue
Fatigue is the progressive, localized, permanent structural damage that occurs when a material is subjected to fluctuating stresses and strains. It is the dominant failure mode in rotating machinery, aircraft structures, automotive suspensions, and welded joints — accounting for an estimated 80–90% of mechanical failures.
The stress-life (S–N) approach, also called the high-cycle fatigue (HCF) method, is appropriate when:
- Stresses remain predominantly elastic (no significant plastic strain per cycle).
- Failure lives exceed roughly $10^4$–$10^5$ cycles.
- The controlling parameter is stress amplitude rather than strain amplitude.
The method was pioneered by Wöhler (1850s) and systematized by Basquin, Marin, and Goodman. It remains the workhorse of machine design (Shigley, Budynas, Bannantine) because of its simplicity and extensive empirical databases.
3.2 The S–N Curve and Wöhler's Experiments
Wöhler tested railway axles under rotating bending and plotted the alternating stress amplitude $S_a$ (or stress range) against the number of cycles to failure $N_f$ on log–log axes.
3.2.1 Key features of the S–N curve
- Finite life region ($10^3 < N < 10^6$): approximately linear on log–log plot.
- Endurance limit $S_e$ (ferrous alloys): below a threshold stress, the specimen does not fail even after $10^7$+ cycles. This "knee" appears around $10^6$ cycles for steels.
- No endurance limit (most non-ferrous alloys, especially aluminum): the curve continues downward; a fatigue strength $S_f$ at a specified life (e.g., $5 \times 10^8$ cycles) replaces $S_e$.
3.2.2 Rotating-beam specimen
The laboratory endurance limit $S_e'$ is measured from a polished, unnotched rotating-beam specimen ($d \approx 7.6$ mm) in fully reversed bending ($R = \sigma_{\min}/\sigma_{\max} = -1$, so $\sigma_m = 0$). Real machine parts differ in surface finish, size, loading, temperature, and stress concentration — requiring correction factors.
3.3 Basquin's Power Law (Finite Life)
For the finite-life branch ($N > 10^3$ cycles), Basquin proposed:
$$\boxed{S_a = \sigma_f' (2N_f)^b}$$
where:
- $\sigma_f'$ is the fatigue strength coefficient (MPa), approximately equal to the true fracture strength.
- $b$ is the fatigue strength exponent (Basquin exponent), typically $-0.05$ to $-0.12$ for steels.
- $2N_f$ is the number of stress reversals (two reversals per cycle for fully reversed loading).
On log–log axes, this is a straight line with slope $b$. The line is anchored at:
- $(N = 10^3, S_a \approx 0.9 S_{ut})$ (approximate intercept for steels).
- $(N = 10^6, S_a = S_e)$ (endurance limit for ferrous alloys).
3.3.1 Determining $b$ and $\sigma_f'$
From two points on the S–N curve:
$$b = \frac{\log(S_2/S_1)}{\log(N_2/N_1)}, \qquad \sigma_f' = \frac{S_1}{(2N_1)^b}.$$
For a typical carbon steel with $S_{ut} = 600$ MPa and $S_e = 300$ MPa:
$$b = \frac{\log(300/540)}{\log(10^6/10^3)} = \frac{-0.255}{3} = -0.085.$$
3.4 Marin Endurance Limit Modification Factors
The rotating-beam endurance limit $S_e'$ must be corrected to the actual part conditions:
$$\boxed{S_e = k_a \, k_b \, k_c \, k_d \, k_e \, k_f \, S_e'}$$
| Factor | Symbol | Physical Effect |
|---|---|---|
| Surface finish | $k_a$ | Machined, hot-rolled, or as-forged surfaces have micro-notches that initiate cracks earlier |
| Size | $k_b$ | Larger sections have more volume for defect statistics (Weibull effect) |
| Load type | $k_c$ | Axial loading vs bending vs torsion |
| Temperature | $k_d$ | High temperature reduces strength |
| Reliability | $k_e$ | Statistical scatter in fatigue data |
| Miscellaneous | $k_f$ | Residual stress, corrosion, plating, shot peening |
3.4.1 Surface factor $k_a$
From Shigley Table 6-3 (approximate):
- Ground/polished: $k_a \approx 1.0$
- Machined: $k_a \approx 0.75$–$0.90$ (depends on $S_{ut}$)
- Hot-rolled: $k_a \approx 0.50$–$0.70$
- As-forged: $k_a \approx 0.40$–$0.60$
Empirical fit: $k_a = a S_{ut}^b$ with $(a,b)$ from tables.
3.4.2 Size factor $k_b$
For bending/torsion, $d \le 7.6$ mm: $k_b = 1.0$. For larger diameters:
$$k_b = \begin{cases} 0.85 & 7.6 < d \le 51 \text{ mm} \\ 0.75 & 51 < d \le 254 \text{ mm} \\ 0.6 & d > 254 \text{ mm} \end{cases}$$
For axial loading: $k_b = 1.0$ (size effect less pronounced).
3.4.3 Load factor $k_c$
- Bending (rotating beam reference): $k_c = 1.0$
- Axial: $k_c = 0.85$
- Torsion: $k_c = 0.59$
3.4.4 Reliability factor $k_e$
For 99% reliability (1% failure): $k_e \approx 0.81$. For 99.9%: $k_e \approx 0.75$. Based on log-normal scatter of fatigue data with standard deviation $s = 0.08$–$0.10$ in log-$N$ space.
3.5 Mean Stress Effects: Goodman, Soderberg, and Gerber
Real service loads rarely produce fully reversed stress ($R = -1$). Mean stress $\sigma_m$ (or $S_m$) significantly affects fatigue life: tensile mean stress is harmful (crack faces stay open, promoting crack growth); compressive mean stress is beneficial (crack faces clamp shut).
The safe operating region in the $\sigma_m$–$\sigma_a$ plane is bounded by a failure locus. Three classical models:
3.5.1 Modified Goodman (linear, conservative)
$$\boxed{\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n}}$$
Intercept on mean-stress axis: $S_{ut}$. Intercept on alternating-stress axis: $S_e$.
3.5.2 Soderberg (most conservative)
$$\boxed{\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n}}$$
Uses yield strength $S_y < S_{ut}$ as the mean-stress intercept, shrinking the safe region.
3.5.3 Gerber (parabolic, less conservative)
$$\boxed{\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_{ut}}\right)^2 = \frac{1}{n}}$$
Better fit to experimental data for ductile steels at moderate mean stresses, but unconservative at high mean stress.
3.5.4 Haigh diagram
The Haigh diagram (or Smith diagram) plots the safe/unsafe boundary in $\sigma_m$–$\sigma_a$ coordinates. Goodman is a straight line; Gerber is a parabola; Soderberg is a steeper straight line. The diagram also shows the yield locus (von Mises or Tresca) to check simultaneous yield and fatigue.
3.6 Stress Concentration and Fatigue Notch Factor $K_f$
Geometric notches (fillets, holes, keyways, threads) amplify local stress. The theoretical stress concentration factor $K_t$ is computed from elasticity (Peterson's charts, FEA):
$$K_t = \frac{\sigma_{\max}}{\sigma_{\mathrm{nom}}}.$$
In fatigue, the effective notch sensitivity reduces $K_t$ to the fatigue notch factor:
$$\boxed{K_f = 1 + q(K_t - 1)}$$
where $q$ is the notch sensitivity ($0 \le q \le 1$), depending on material, notch radius, and grain size.
3.6.1 Notch sensitivity
For steels with $S_{ut} > 500$ MPa: $q \approx 0.8$–$1.0$ (notch is nearly fully effective). For ductile aluminum: $q \approx 0.3$–$0.6$. Empirical fit (Neuber, Peterson):
$$q = \frac{1}{1 + \sqrt{a}/\sqrt{r}}$$
where $a$ is a material constant ($\approx 0.025$ mm for steels) and $r$ is the notch root radius.
3.6.2 Application
Always apply $K_f$ to the nominal stress before entering the fatigue criterion:
$$\sigma_a = K_f \cdot \sigma_{a,\mathrm{nom}}, \qquad \sigma_m = K_f \cdot \sigma_{m,\mathrm{nom}}.$$
Critical rule: Never apply $K_t$ to the endurance limit $S_e$ and also to the applied stress — double-counting the notch effect.
3.7 Variable Amplitude Loading and Miner's Rule
When stress amplitude varies over the service life, Miner's linear damage rule accumulates damage:
$$\boxed{D = \sum_i \frac{n_i}{N_i} = 1 \text{ at failure}}$$
where $n_i$ is the number of cycles at stress level $i$ and $N_i$ is the life at that level (from S–N curve or Goodman).
3.7.1 Limitations
- Linear superposition ignores load-order effects (overload retardation, underload acceleration).
- Does not account for mean stress changes between blocks.
- For conservative design, use $D \le 0.7$–$0.8$.
3.8 Design Procedure: S–N Fatigue Analysis
A systematic procedure for HCF design:
- Determine loading spectrum: $\sigma_a$, $\sigma_m$ at the critical section (include $K_f$).
- Compute corrected endurance limit: $S_e = k_a k_b k_c k_d k_e S_e'$ (typically $S_e' \approx 0.5 S_{ut}$ for steels).
- Apply mean-stress criterion: Goodman (standard), Soderberg (conservative), or Gerber.
- Compute safety factor: $n = 1/(\sigma_a/S_e + \sigma_m/S_{ut})$ (Goodman).
- Check finite life if $n < 1$: use Basquin to find $N_f$.
- Check yield: ensure $\sigma_a + \sigma_m < S_y/n$ (Soderberg locus doubles as a yield check).
3.9 Statistical Aspects and Reliability
Fatigue data exhibit significant scatter — identical specimens at the same stress level may fail between $10^5$ and $10^7$ cycles. The log-normal distribution of $N_f$ is standard:
$$f(N) = \frac{1}{N s \sqrt{2\pi}} \exp\Big[-\frac{1}{2}\Big(\frac{\log N - \log \hat{N}}{s}\Big)^2\Big]$$
Typical standard deviation $s = 0.08$–$0.10$ in log-$N$. The reliability factor $k_e$ shifts $S_e$ downward for higher required reliability.
3.10 Summary and When to Use S–N
| Condition | Use S–N? |
|---|---|
| Lives $> 10^4$–$10^5$ cycles | Yes |
| Predominantly elastic cycling | Yes |
| Notched or smooth ductile parts | Yes (with $K_f$) |
| Local plasticity per cycle | No → use ε–N (Ch.4) |
| Multiaxial non-proportional loading | Use equivalent stress + critical plane |
| Corrosion-fatigue | Apply additional knockdown or use corrosion S–N data |
3.11 Case Study: Shaft Fatigue Design Walkthrough
Consider a stepped shaft (filleted) in a gearbox:
- Loading: Bending moment varies $M = 200 \pm 150$ N·m; torque steady $T = 80$ N·m.
- Critical section: Fillet, $d = 35$ mm, $D = 50$ mm, $r = 2$ mm → $K_t = 1.7$ (Peterson).
- Nominal stress: $\sigma_a = 32 M_a / (\pi d^3) = 32 \times 150 / (\pi \times 0.035^3) = 35.6$ MPa; $\sigma_m = 32 \times 200 / (\pi \times 0.035^3) = 47.4$ MPa.
- Material: 4140 QT steel, $S_{ut} = 900$ MPa, machined, $d = 35$ mm.
- Endurance limit: $S_e' = 450$ MPa. $k_a = 0.735$, $k_b = 0.85$, $k_e = 0.81$ → $S_e = 236$ MPa.
- Notch: $q = 0.90$ → $K_f = 1 + 0.90 \times 0.7 = 1.63$.
- Corrected stresses: $\sigma_a = 1.63 \times 35.6 = 58.0$ MPa; $\sigma_m = 1.63 \times 47.4 = 77.3$ MPa.
- Goodman: $n = 1/(58.0/236 + 77.3/900) = 1/(0.246 + 0.086) = \mathbf{3.01}$.
- Conclusion: Adequate fatigue margin. If $n < 1.5$, increase fillet radius (reduce $K_t$) or use ground surface ($k_a \to 1.0$).
Design takeaway: The S–N method is deceptively simple. The engineering rigor lies in correctly computing $S_e$ (Marin factors), applying $K_f$ at the notch, and choosing an appropriate mean-stress criterion. Goodman is the industry default; Soderberg when yield interaction matters.