7.2 S-N Diagrams and Endurance Limits

S-N Diagrams and Endurance Limits

Fatigue failure is a leading cause of mechanical component failure, and it often happens without warning. S-N diagrams give engineers a way to predict how long a part will survive under repeated loading by mapping the relationship between stress amplitude and the number of cycles to failure. Endurance limits take this a step further: they define a stress threshold below which certain materials can theoretically survive infinite cycles.

This section covers how to read and use S-N curves, what endurance limits actually mean in practice, and the modification factors that adjust idealized lab data for real-world conditions.

S-N Curve Components

Stress-Life Relationship

An S-N curve (also called a Wöhler curve) plots stress amplitude on the vertical axis against number of cycles to failure on the horizontal axis. Both axes typically use logarithmic scales because the data spans several orders of magnitude.

• Stress amplitude ($\sigma_a$) is the magnitude of the alternating stress applied during each loading cycle. It's half the range between the maximum and minimum stress in a fully reversed cycle.
• Number of cycles ($N$) is how many complete stress cycles the material endures before fracture.

On a log-log plot, the finite-life portion of the curve appears roughly linear, which makes it much easier to interpolate fatigue life at a given stress level.

Fatigue Life Regions

The S-N curve divides into two distinct regions:

• Finite life region: The downward-sloping portion of the curve. Higher stress amplitudes correspond to fewer cycles before failure. This region is where most design problems live when components experience high cyclic loads for a limited service life.
• Infinite life region: The flat (horizontal) portion of the curve, where the stress amplitude is low enough that the material can theoretically survive an unlimited number of cycles. The stress level at which the curve flattens out is the endurance limit.

The transition between these regions typically occurs around $10^6$ to $10^7$ cycles for ferrous metals (steels and cast irons). One important caveat: not all materials exhibit a clear endurance limit. Aluminum alloys and many non-ferrous metals don't have a true horizontal asymptote. For those materials, engineers instead report a fatigue strength at a specified cycle count (usually $10^7$ or $10^8$) and design accordingly.

Endurance Limit and Fatigue Strength

Fatigue Resistance Parameters

Endurance limit ($\sigma_e$) is the stress amplitude below which a material can withstand an infinite number of cycles without failure. For steels, a common first approximation is:

$\sigma_e' \approx 0.5 \, S_{ut}$

where $S_{ut}$ is the ultimate tensile strength, and $\sigma_e'$ is the uncorrected (ideal specimen) endurance limit. This approximation holds reasonably well for steels with $S_{ut}$ up to about 1400 MPa; above that, the endurance limit tends to plateau.

Fatigue strength ($S_f$) is the stress amplitude that causes failure at a specific number of cycles. You use this when designing for a finite life or when working with materials that lack a true endurance limit.

The distinction matters: endurance limit is a threshold for infinite life, while fatigue strength is tied to a particular cycle count.

Notch Effects on Fatigue

Real components have holes, fillets, keyways, and other geometric features that concentrate stress. The fatigue notch factor ($K_f$) captures how much these features reduce fatigue strength.

$K_f = \frac{\text{Fatigue strength of unnotched specimen}}{\text{Fatigue strength of notched specimen}}$

A few things to keep in mind:

• $K_f$ is always $\geq 1$. A value of 1 means the notch has no effect; higher values mean greater sensitivity.
• $K_f$ is related to, but not the same as, the theoretical stress concentration factor $K_t$. The relationship depends on the notch sensitivity ($q$) of the material: $K_f = 1 + q(K_t - 1)$, where $q$ ranges from 0 (no sensitivity) to 1 (full theoretical sensitivity).
• Harder, higher-strength materials tend to be more notch-sensitive (higher $q$), while softer, more ductile materials are more forgiving.

Stress Considerations

Mean Stress Effects

Most real loading isn't fully reversed. The stress ratio $R$ describes the nature of the loading cycle:

$R = \frac{\sigma_{min}}{\sigma_{max}}$

• $R = -1$: Fully reversed loading (zero mean stress). This is the baseline condition for standard S-N data.
• $R = 0$: Repeated loading from zero to a maximum (pulsating tension).
• $R = 1$: Static load (no alternating component).

The mean stress ($\sigma_m$) is the average of the max and min stresses: $\sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2}$. Its effect on fatigue life is significant:

• Tensile mean stress reduces fatigue life. It holds cracks open and promotes growth.
• Compressive mean stress can actually improve fatigue life by pressing crack faces together and retarding propagation. This is the principle behind shot peening and other surface treatments.

To account for mean stress in design, engineers use criteria like the Goodman line, Gerber parabola, or Soderberg line, each of which defines a safe combination of alternating and mean stress. The modified Goodman criterion is the most commonly used in practice for its balance of conservatism and simplicity.

Stress-Life Modification Factors

S-N data comes from carefully controlled lab tests on small, polished, round specimens loaded in rotating bending. Real parts differ from these ideal conditions, so you apply modification factors to adjust the endurance limit:

$\sigma_e = k_a \cdot k_b \cdot k_c \cdot k_d \cdot k_e \cdot \sigma_e'$

where $\sigma_e'$ is the uncorrected endurance limit and each $k$ factor accounts for a specific real-world condition:

Each factor is $\leq 1$, so the corrected endurance limit is always lower than (or equal to) the ideal specimen value. In practice, the combined effect of these factors can reduce the endurance limit substantially, sometimes by 50% or more. That's why you can't just use textbook endurance limit values directly in design without applying these corrections.