Fatigue limit

The fatigue limit or endurance limit is the stress level below which an infinite number of loading cycles can be applied to a material without causing fatigue failure.^[1] Some metals such as ferrous alloys and titanium alloys have a distinct limit,^[2] whereas others such as aluminium and copper do not and will eventually fail even from small stress amplitudes. Where materials do not have a distinct limit the term fatigue strength or endurance strength is used and is defined as the maximum value of completely reversed bending stress that a material can withstand for a specified number of cycles without a fatigue failure.^[3][4] For polymeric materials, the fatigue limit is also commonly known as the intrinsic strength.^[5][6]

Representative curves of applied stress vs number of cycles for   steel (showing an endurance limit) and   aluminium (showing no such limit).

Definitions

The ASTM defines fatigue strength, ${\displaystyle S_{N_{f}}}$, as "the value of stress at which failure occurs after ${\displaystyle N_{f}}$ cycles", and fatigue limit, ${\displaystyle S_{f}}$, as "the limiting value of stress at which failure occurs as ${\displaystyle N_{f}}$ becomes very large". ASTM does not define endurance limit, the stress value below which the material will withstand many load cycles,^[1] but implies that it is similar to fatigue limit.^[7]

Some authors use endurance limit, ${\displaystyle S_{e}}$, for the stress below which failure never occurs, even for an indefinitely large number of loading cycles, as in the case of steel; and fatigue limit or fatigue strength, ${\displaystyle S_{f}}$, for the stress at which failure occurs after a specified number of loading cycles, such as 500 million, as in the case of aluminium.^[1][8][9] Other authors do not differentiate between the expressions even if they do differentiate between the two types of materials.^[10][11][12]

Typical values

Typical values of the limit (${\displaystyle S_{e}}$) for steels are one half the ultimate tensile strength, to a maximum of 290 MPa (42 ksi). For iron, aluminium, and copper alloys, ${\displaystyle S_{e}}$ is typically 0.4 times the ultimate tensile strength. Maximum typical values for irons are 170 MPa (24 ksi), aluminums 130 MPa (19 ksi), and coppers 97 MPa (14 ksi).^[2] Note that these values are for smooth "un-notched" test specimens. The endurance limit for notched specimens (and thus for many practical design situations) is significantly lower.

For polymeric materials, the fatigue limit has been shown to reflect the intrinsic strength of the covalent bonds in polymer chains that must be ruptured in order to extend a crack. So long as other thermo chemical processes do not break the polymer chain (i.e. ageing or ozone attack), a polymer may operate indefinitely without crack growth when loads are kept below the intrinsic strength.^[13][14]

The concept of fatigue limit, and thus standards based on a fatigue limit such as ISO 281:2007 rolling bearing lifetime prediction, remains controversial, at least in the US.^[15][16]

Modifying factors of fatigue limit

The fatigue limit of a machine component, Se, is influenced by a series of elements named modifying factors. Some of these factors are listed below.

Surface factor

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

${\displaystyle k_{S}=aS_{ut}^{b}}$

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

Gradient factor

Besides taking into account the surface finish, it is also important to consider the size gradient factor ${\displaystyle k_{G}}$. When it comes to bending and torsional loading, the gradient factor is also taken into consideration.

Load factor

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

History

The concept of endurance limit was introduced in 1870 by August Wöhler.^[17] 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][18]

See also

• Fatigue (material)
• Smith fatigue strength diagram [de], a diagram by British mechanical engineer James Henry Smith [de]

References

1. Beer, Ferdinand P.; E. Russell Johnston Jr. (1992). Mechanics of Materials (2 ed.). McGraw-Hill, Inc. p. 51. ISBN 978-0-07-837340-4.
2. "Metal Fatigue and Endurance". Archived from the original on 2012-04-15. Retrieved 2008-04-18.
3. Jastrzebski, D. (1959). Nature and Properties of Engineering Materials (Wiley International ed.). John Wiley & Sons, Inc.
4. Suresh, S. (2004). Fatigue of Materials. Cambridge University Press. ISBN 978-0-521-57046-6.
5. Robertson, C.G.; Stocek, R.; Mars, W.V. (27 November 2020). The Fatigue Threshold of Rubber and Its Characterization Using the Cutting Method. Springer. pp. 57–83. doi:10.1007/12_2020_71. ISBN 978-3-030-68920-9. Retrieved 24 July 2024.
6. Bhowmick, A. K. (1988). "Threshold fracture of elastomers". Polymer Reviews. 28 (3–4): 339–370. doi:10.1080/15583728808085379.
7. Stephens, Ralph I. (2001). Metal Fatigue in Engineering (2nd ed.). John Wiley & Sons, Inc. p. 69. ISBN 978-0-471-51059-8.
8. Budynas, Richard G. (1999). Advanced Strength and Applied Stress Analysis (2nd ed.). McGraw-Hill, Inc. pp. 532–533. ISBN 978-0-07-008985-3.
9. Askeland, Donald R.; Pradeep P. Phule (2003). The Science and Engineering of Materials (4th ed.). Brooks/Cole. p. 248. ISBN 978-0-534-95373-7.
10. Hibbeler, R. C. (2003). Mechanics of Materials (5th ed.). Pearson Education, Inc. p. 110. ISBN 978-0-13-008181-0.
11. Dowling, Norman E. (1998). Mechanical Behavior of Materials (2nd ed.). Printice-Hall, Inc. p. 365. ISBN 978-0-13-905720-5.
12. Barber, J. R. (2001). Intermediate Mechanics of Materials. McGraw-Hill. p. 65. ISBN 978-0-07-232519-5.
13. Lake, G. J.; P. B. Lindley (1965). "The mechanical fatigue limit for rubber". Journal of Applied Polymer Science. 9 (4): 1233–1251. doi:10.1002/app.1965.070090405.
14. Lake, G. J.; A. G. Thomas (1967). "The strength of highly elastic materials". Proceedings of the Royal Society of London A: Mathematical and Physical Sciences. 300 (1460): 108–119. Bibcode:1967RSPSA.300..108L. doi:10.1098/rspa.1967.0160. S2CID 138395281.
15. Erwin V. Zaretsky (August 2010). "In search of a fatigue limit: A critique of ISO standard 281:2007" (PDF). Tribology & Lubrication Technology: 30–40. Archived from the original (PDF) on 2015-05-18.
16. "ISO 281:2007 bearing life standard – and the answer is?" (PDF). Tribology & Lubrication Technology: 34–43. July 2010. Archived from the original (PDF) on 2013-10-24.
17. W. Schutz (1996). A history of fatigue. Engineering Fracture Mechanics 54: 263-300. DOI
18. Bathias, C. (1999). "There is no infinite fatigue life in metallic materials". Fatigue & Fracture of Engineering Materials & Structures. 22 (7): 559–565. doi:10.1046/j.1460-2695.1999.00183.x.