Fatigue and S-N Curve Fundamentals

Fatigue

Fatigue is a time-dependent fracture failure mechanism observed in ductile materials. For fatigue to occur, an initial crack must be present on either the surface or within the material. When a tensile load is applied, the stress at the crack tip exceeds the nominal stress due to stress concentrations. If the stress concentration factor is high enough, this stress can surpass the yield strength, leading to localized yielding.

Localized yielding acts to inhibit catastrophic crack growth; it slows down the rate at which the crack propagates. However, when another tensile load is exercised, it can prompt the crack to extend in a new direction. This cycle of localized yielding and crack growth can repeat numerous times. Eventually, this may lead to a significant decrease in the component's cross-sectional area, resulting in tensile rupture. A notable example is the progression of crack growth across a component's cross-sectional area prior to fast fracture, which can take years.

Fatigue occurs under periodic loading conditions where the stress magnitude is lower than a material-specific threshold, denoted as Sₛ. Since cracks are often introduced during manufacturing and handling processes, preventing crack formation is virtually impossible; hence, fatigue failure is a common risk for mechanical components, making it a critical failure mechanism.

S-N Curve

Bending stress is instrumental in fatigue testing due to its variation between tensile and compressive stress around a rod's circumference when a constant bending moment is applied. This characteristic can be harnessed in rotating bending fatigue tests using setup specified by ASTM Standards E466-E468.

An S-N curve, which is plotted on a log-log scale, typically comprises two segments: a straight line segment with a negative slope, which depicts the relationship between stress and number of cycles (from 10³ to 10⁶ cycles), and a horizontal line segment beyond 10⁶ cycles. This curve is crucial because it demarcates regions corresponding to the absence of fatigue (where the stress is below 0.5Sₛ) versus regions that indicate fatigue failure.

For most steels, Sₜ (endurance limit) is around 0.5Sₛ, while for many Al alloys that do not exhibit a saturation effect in their S-N characteristics, 10⁶-cycle fatigue strength is often substituted with 10⁸ cycles.
The general equation governing the S-N curve can be modeled as:

S = αN^β,
where alpha (α) and beta (β) are curve-fitting parameters determined from the fatigue strength at specified cycles.

Endurance Limit Modifying Factors

The endurance limit, representing the fatigue strength at 10⁶ cycles, can be influenced by various factors, including:

  • Surface Factor (Cₛ): Related to surface roughness and treatment, affecting crack initiation potential.

  • Gradient Factor (Cₜ): Affected by specimen size, specifically highlighting the hardenability and uniformity in tensile strength across a specimen.

  • Load Factor (Cₕ): Assigned based on the loading type (bending, axial).

  • Temperature Factor (Cₜ): Significant when operating conditions approach high temperatures where creep is a risk.

  • Reliability Factor (Cₗ): Important for multi-component systems where failure probability needs assessment.

These factors can be combined to derive a modified version of the endurance limit:

S' = Cₜ Cₕ Cₛ Cₚ Cₗ Sₛ,
where each characteristic contributes a penalty relative to the benchmark endurance limit (e.g., 0.5Sₛ for steels).

Mean Stress Effect on Fatigue Strength

Mean stress effects can be modeled using Goodman, Soderberg, and Gerber equations. Goodman’s linear relationship, particularly, allows for practical use in design analysis, capturing the degradation of fatigue life under alternating load conditions. The interaction of mean and alternating stresses can shift the fatigue limit, which necessitates adaptations in the S-N curves to reflect true service conditions.

In scenarios where the mean stress is non-zero, stress amplitude and mean stress can be effectively analyzed to ensure robust design parameters that accommodate fatigue behavior without excessive conservatism.

Stress Concentration Effects on Fatigue Strength

Notches increase local stress magnitudes, notably around geometrical discontinuities. The fatigue stress concentration factor (K₊) is defined as:

Notched Sₜ = Unnotched Sₜ / K₊
This quantifies how notches amplify nominal stress. The effects of residual stresses and the likelihood of crack formation must also be considered to complete analysis for components subjected to variable loads. Effective design mandates mitigatory strategies against stress concentrations and fracture possibilities due to microstructural weaknesses.

Understanding these elements is fundamental to ensuring the integrity of materials subjected to repeated load cycles, thus requiring evaluations under both static and dynamic conditions. This includes recognition of the nuances associated with connection systems, especially regarding fasteners. Strategies to counteract fatigue phenomena including optimized thread designs and coatings are critical in high-reliability applications, such as in aerospace and automotive industries.