Learning topic
Material Fatigue and Endurance Limit
Learn fatigue analysis with stress amplitude and mean stress, Wöhler S–N curves, endurance limit, stress concentration, Goodman relation, and fatigue-life checks.
Material fatigue is the process of damage accumulation under repeatedly varying stresses, which may end in crack initiation, crack propagation, and final fracture. Fatigue failure can occur at maximum stresses below the static ultimate strength.
Stress-cycle parameters
A variable normal-stress cycle is defined by its maximum and minimum values $\sigma_{max}$ and $\sigma_{min}$. The main parameters are:
$$\sigma_m=\frac{\sigma_{max}+\sigma_{min}}{2},$$
$$\sigma_a=\frac{\sigma_{max}-\sigma_{min}}{2},$$
$$R=\frac{\sigma_{min}}{\sigma_{max}}.$$
- $\sigma_{max}$ — maximum cycle stress;
- $\sigma_{min}$ — minimum cycle stress;
- $\sigma_m$ — mean stress;
- $\sigma_a$ — stress amplitude;
- $R$ — stress ratio.
For a fully reversed cycle, $R=-1$ and $\sigma_m=0$. For a pulsating cycle from zero to a positive maximum, $R=0$.
The amplitude $\sigma_a$ characterizes the cyclic part of loading, while mean stress $\sigma_m$ shifts the entire cycle toward tension or compression. Therefore, two cycles with the same $\sigma_a$ but different $\sigma_m$ may have different fatigue severity.
S–N curve
An S–N fatigue curve shows the relationship between cyclic stress level or amplitude $S$ and the number of cycles $N$ to failure for specified test conditions and stress ratio $R$.
The endurance limit is a fatigue-resistance characteristic for a large number of cycles, defined for a particular material, stress cycle, test basis, and specimen condition. For materials without a distinct horizontal region of the S–N curve, a fatigue strength at a specified number of cycles $N$ is used instead.
An S–N curve should not be treated as a universal property independent of surface condition, size, stress concentration, mean stress, temperature, and environment.
To obtain an S–N curve, groups of specimens are tested at different cyclic stress levels and the number of cycles $N$ to failure is recorded. The number of cycles is commonly plotted on a logarithmic scale.
Endurance limit
For materials with a distinct endurance limit under a specified stress cycle, a stress level can be identified below which a laboratory specimen survives the large reference number of cycles adopted by the test method without fatigue failure. For many nonferrous alloys and other materials without a clear horizontal asymptote, fatigue strength is instead specified at a particular number of cycles $N$.
Stress concentration
Holes, threads, keyways, grooves, and abrupt section transitions create local stress maxima and promote fatigue-crack initiation. In fatigue design, the theoretical geometric stress-concentration factor is not always identical to the effective fatigue-strength reduction factor because material notch sensitivity must also be considered.
| Factor | Typical effect | What is considered in design |
|---|---|---|
| Stress concentration | Reduces fatigue strength | Fillets, holes, grooves, threads; theoretical and effective concentration factors |
| Surface condition | Rough or damaged surfaces generally reduce endurance | Roughness, machining, defects, surface strengthening |
| Absolute size | Effective endurance often decreases as size increases | Size factor for the adopted method |
| Mean stress | Tensile mean stress generally reduces allowable amplitude | Goodman, Gerber, Soderberg diagrams or code relations |
| Temperature and environment | May substantially change fatigue life | Corrosion, elevated temperature, service environment |
| Residual stress | Compressive surface residual stress may improve resistance to crack initiation | Shot peening, rolling, and other technologies |
Mean-stress effect
When $\sigma_m$ is nonzero, allowable amplitude is assessed using experimental or code-based diagrams. Common educational approximations include the Goodman and Soderberg lines and the Gerber parabola. The selected relation must be appropriate for the material and adopted method.
For example, a linear Goodman relation for tensile mean stress is often written as:
$$\frac{\sigma_a}{\sigma_{-1}}+\frac{\sigma_m}{\sigma_u}\le\frac{1}{n},$$
where $\sigma_{-1}$ is a reference endurance limit for fully reversed loading under the specified conditions, $\sigma_u$ is ultimate strength, and $n$ is the selected safety factor. For a real component, the reference endurance characteristic is corrected according to the adopted design method.
Typical fatigue-failure development
- Localization of cyclic plasticity or damage in a critical region.
- Initiation of a small crack, often at the surface or a stress concentrator.
- Progressive crack growth over many cycles.
- Final rapid fracture of the remaining section when it can no longer carry the load.
Verification procedure
- Determine $\sigma_{max}$ and $\sigma_{min}$ at the critical point.
- Calculate $\sigma_a$, $\sigma_m$, and $R$.
- Determine the reference fatigue characteristic for the required life $N$ and stress cycle.
- Account for stress concentration, surface condition, size, temperature, and environment.
- Account for mean stress using the selected criterion.
- Determine the fatigue safety factor or allowable life.
Limitations
Fatigue assessment depends strongly on experimental data and the adopted standard. Variable-amplitude loading, multiaxial fatigue, low-cycle fatigue, and crack-growth analysis require specialized models beyond the basic S–N approach.
About this topic
Repeated cyclic stress variations lead to structural fatigue failure at stress levels far below ultimate tensile strength. This page covers cyclic stress parameters, endurance limits, Wöhler S-N curves, stress concentration effects (notches, fillets), and size factors.