Fatigue limit

Surface factor

The surface modifying factor, ${\displaystyle C_{s}}$, is related to both the tensile strength, ${\displaystyle S_{ut}}$, of the material and the surface finish of the machine component.

${\displaystyle C_{s}=aS_{ut}^{b}}$

Where factor a and exponent b present in the equation are related to the surface finish, and are determined experimentally.

Fatigue cracks usually initiate on the surface of the material. Stress concentrations are lower for smoother surfaces, resulting in a higher fatigue strength and hence, a higher fatigue limit. On the other hand, rougher surfaces results in increased stress concentrations, thus lowering the fatigue limit.^[17][18]

Gradient factor

Besides taking into account the surface finish, it is also important to consider the size gradient factor ${\displaystyle k_{G}}$. Stress gradient factor accounts for the effect of stress gradients on fatigue life, particularly at locations with stress concentrations like notches, by modifying the fatigue strength reduction factor. When it comes to bending and torsional loading, the gradient factor is also taken into consideration.

${\displaystyle C_{g}={\frac {S'_{e}}{S_{e}}}}$

where

• ${\displaystyle S'_{e}}$ = Modified endurance limit
• ${\displaystyle S_{e}}$ = Theoretical endurance limit from unnotched specimens
• ${\displaystyle C_{g}}$ = Stress gradient factor^[19]

Load factor

The strength values derived from the S-N (Wohler plot) is as a result from a reversing bending load as the test specimen is rotated. In rotating-bending tests, every point on the diameter surface of the specimen experiences a bending stress in one direction and then in the opposite direction, with only a limited area on the outer surface experiencing the peak stress level. The reversed axial loading scenario exhibits a significantly tougher condition because the entire cross-section is subjected to the full stress rather than just the surface elements. (The strength values noted for reversed axial loading have been documented at varying ratios ranging from 0.7 to 0.85 times those reported for reversed bending). In the reversed torsion scenario, the loading involves shear rather than bending.^[20]

Load modifying factor can be identified as.

${\displaystyle k_{L}=0.85}$ for axial

${\displaystyle k_{L}=1}$ for bending

${\displaystyle k_{L}=0.59}$ for pure torsion

Temperature factor

The temperature factor is calculated as

${\displaystyle k_{T}={\frac {S_{o}}{S_{r}}}}$

${\displaystyle S_{o}}$ is tensile strength at operating temperature

${\displaystyle S_{r}}$ is tensile strength at room temperature

Reliability factor

We can calculate the reliability factor using the equation,

${\displaystyle k_{R}=1-0.08Z_{a}}$

${\displaystyle z_{a}=0}$ for 50% reliability

${\displaystyle z_{a}=1.288}$ for 90% reliability

${\displaystyle z_{a}=1.645}$ for 95% reliability

${\displaystyle z_{a}=2.326}$ for 99% reliability

Miscellaneous Effects Modification Factor

This factor includes other factors that influences the endurance limit of a material, such as environmental conditions (e.g., corrosion, humidity), Residual stresses from manufacturing processes, and the presence of notches or other geometric discontinuities. These factors can either enhance or degrade fatigue performance, depending on their nature and severity.^[21]

Early Observations and the Birth of Fatigue Studies (1837-1870)

The concept of fatigue was first introduced in the early 19th century when Wilhelm Albert conducted the first recorded fatigue tests on mining chains in 1837. He observed that metal components could fail under repeated loading, even if the stresses were well below the material’s ultimate tensile strength (Schütz, 1996).^[22]

In 1854, the term "fatigue" was first used by Braithwaite to describe material degradation under cyclic loading. However, it was August Wöhler (1858-1870) who laid the foundation of modern fatigue analysis. As a railway engineer, Wöhler systematically studied the failure of railway axles, developing the S-N curve (Wöhler curve), which remains a fundamental tool in fatigue analysis today (Schütz, 1996). His work demonstrated that materials could fail due to cyclic stress even when stresses were below the yield strength.

Advancements in Fatigue Research (1870-1925)

The period following Wöhler’s work saw significant progress:

• Bauschinger Effect (1880)s: Johann Bauschinger discovered that cyclic loading could alter the yield strength of metals, influencing the development of fatigue life models.
• Stress Concentration Factor (1898): Kirsch calculated the stress concentration factor for holes in materials, explaining why cracks often initiate at discontinuities.
• Metallurgical Insights (1903): Ewing and Humfrey identified microscopic slip bands as the first evidence of fatigue damage in metals.

The Golden Era of Fatigue Studies (1925-1945)

Between 1925 and 1945, fatigue research expanded with contributions from multiple countries, particularly Germany:

• Thum and Bautz (1937): Introduced the concept of "Gestaltfestigkeit", emphasizing component shape over material properties in fatigue resistance.
• Gassner’s Variable Amplitude Fatigue Tests (1939): Revolutionized fatigue testing by replicating real-world service loads, leading to modern operational fatigue testing.
• Shot Peening (1940s): Methods to improve fatigue strength, such as cold rolling and shot peening, were explored.^[23]

Post-War Developments and Modern Fatigue Strength Research (1945-Present)

After World War II, fatigue failures in aircraft, bridges, and automobiles accelerated research:

• Miner’s Rule (1945): Introduced a cumulative damage hypothesis to predict fatigue life under variable loading.
• Paris Law (1960s): Paul Paris developed a model for crack propagation based on fracture mechanics.
• Low-Cycle Fatigue (1954): Manson and Coffin introduced strain-based fatigue life prediction methods, critical for high-temperature applications.^[23]

However, recent research suggests that endurance limits do not exist for metallic materials, that if enough stress cycles are performed, even the smallest stress will eventually produce fatigue failure.^[9][24]