S-N curve and high-cycle fatigue
Theory and Physics
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."
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? That "feeling of being pulled" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It cannot be omitted in impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—different concepts.
- External force term (load term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but applying "compression"—sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat through air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy for a smoother ride. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2–3 times. Recommended when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but too time-consuming for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
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).
Practical Checklist
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