S-N curve and high-cycle fatigue
S-N curve and high-cycle fatigue: Theoretical Foundations
What is an S-N Curve?
Professor, the S-N curve is fundamental to fatigue, right?
The S-N curve (Wöhler curve) shows the relationship between stress amplitude $S$ vs. number of cycles to failure $N$. It was established by August Wöhler (1860) through his research on fatigue failure of railway axles.
Or in logarithmic form: $\log N = \log C - m \log S$
Fatigue Limit
Steel has a fatigue limit (fatigue limit) $S_e$. If $S < S_e$, the material will not fail even under infinite cycles ($N > 10^7$). Aluminum alloys do not have a clear fatigue limit.
What percentage of tensile strength is $S_e$ approximately?
For steel, $S_e \approx 0.4 \sim 0.5 \sigma_u$. However, it is reduced by surface finish, size effect, and mean stress. In actual design, correction factors are applied.
Effect of Mean Stress
When mean stress $\sigma_m \neq 0$, the Goodman diagram is used:
$$ \frac{S_a}{S_e} + \frac{\sigma_m}{\sigma_u} = 1 $$
Steel has a fatigue limit (fatigue limit) $S_e$. If $S < S_e$, the material will not fail even under infinite cycles ($N > 10^7$). Aluminum alloys do not have a clear fatigue limit.
What percentage of tensile strength is $S_e$ approximately?
For steel, $S_e \approx 0.4 \sim 0.5 \sigma_u$. However, it is reduced by surface finish, size effect, and mean stress. In actual design, correction factors are applied.
When mean stress $\sigma_m \neq 0$, the Goodman diagram is used:
$S_a$ is the stress amplitude, $\sigma_m$ is the mean stress. Tensile mean stress reduces fatigue life.
Summary
Key Points:
- $S = AN^{-1/m}$ — Stress amplitude vs. life
- Fatigue limit $S_e$ — For steel $\approx 0.4\sigma_u$. Does not exist for aluminum.
- Goodman diagram — Effect of mean stress. Life decreases under tensile stress.
- High-cycle fatigue ($N > 10^4$) — The domain of S-N curves.
- FEM stress → Life prediction via S-N curve — Basic flow of fatigue analysis.
Wöhler's Ten-Year Experiment
August Wöhler, who established the S-N curve, conducted fatigue tests on railway axles from the 1860s to the 1870s, collecting the world's first systematic fatigue data. His testing machine rotated a 30mm diameter iron specimen under load at 60 rpm (steam-powered at the time). Wöhler was also the first to call the stress level at which no failure occurred even after 10^7 cycles the "fatigue limit."
Computational Methods for S-N curve and high-cycle fatigue
S-N Fatigue in FEM
Flow from FEM stress to S-N evaluation in fatigue software:
1. Calculate stress distribution with FEM (static analysis or frequency response).
2. Input stress results into fatigue software (nCode, fe-safe, FEMFAT).
3. Specify S-N curve (material database or test data).
4. Define load history (constant amplitude or variable amplitude).
5. Mean stress correction (Goodman, Gerber, Soderberg).
6. Output fatigue life $N$ (contour display).
Solver/Tools
Summary
S-N Curve Logarithmic Regression and Confidence Intervals
S-N curves are approximated as straight lines on semi-log or log-log plots, expressed by the power law σ^m × N = C (m is a material constant). Statistical confidence intervals can be set at 50%, 95%, and 99%; 97.7% (2σ) is commonly used for design. With fewer than 10 data points, confidence intervals are wide and uncertainty is high, so it is recommended to establish the curve with at least 15–20 data points.
S-N curve and high-cycle fatigue in Practice
S-N Fatigue in Practice
Used for fatigue evaluation of automotive suspensions, aircraft structures, pressure vessels, welded structures, and mechanical components.
Stress Concentration Factor $K_t$
FEM stress includes stress concentration, so S-N curves are evaluated based on notch stress. Alternatively, correct using $K_f$ (fatigue notch factor).