Goodman relation

File:Fatigue limit diagram.png

The relations can be represented mathematically as:

$(\backslash frac\{n\backslash sigma\_\backslash text\{m\}\}\{\backslash sigma\_\backslash text\{b\}\})^2\; +\; \backslash frac\{n\backslash sigma\_\backslash text\{a\}\}\{\backslash sigma\_\backslash text\{w\}\}\; =\; 1$

, Gerber Line (parabola)

:$\backslash frac\{\backslash sigma\_\backslash text\{m\}\}\{\backslash sigma\_\backslash text\{b\}\}\; +\; \backslash frac\{\backslash sigma\_\backslash text\{a\}\}\{\backslash sigma\_\backslash text\{w\}\}\; =\; \backslash frac\{1\}\{\{n\}\}$

, Goodman Line

:$\backslash frac\{\backslash sigma\_\backslash text\{m\}\}\{\backslash sigma\_\backslash text\{y\}\}\; +\; \backslash frac\{\backslash sigma\_\backslash text\{a\}\}\{\backslash sigma\_\backslash text\{w\}\}\; =\; \backslash frac\{1\}\{\{n\}\}$

, Soderberg Line

where $\backslash sigma\_\backslash text\{a\}$ is the stress amplitude, $\backslash sigma\_\backslash text\{m\}$ is the mean stress, $\backslash sigma\_\backslash text\{w\}$ is the fatigue limit for completely reversed loading, $\backslash sigma\_\backslash text\{b\}$ is the ultimate tensile strength of the material and $n$ is the factor of safety.

The Gerber parabola is indication of the region just beneath the failure points during experiment.

The Goodman line connects $\backslash sigma\_\backslash text\{b\}$ on the abscissa and $\backslash sigma\_\backslash text\{w\}$ on the ordinate. The Goodman line is much safer consideration than the Gerber parabola because it is completely inside the Gerber parabola and excludes some of area which is nearby to failure region.

The Soderberg Line connects $\backslash sigma\_\backslash text\{y\}$ on the abscissa and $\backslash sigma\_\backslash text\{w\}$ on the ordinate, which is more conservative consideration and much safer. $\backslash sigma\_\backslash text\{y\}$ is the yield strength of the material.{{Cite book|last=Bhandari|first=V. B.|url=https://books.google.com/books?id=d-eNe-VRc1oC|title=Design of Machine Elements|publisher=Tata McGraw-Hill Education|year=2007|isbn=9780070611412|pages=184, 185}}{{Cite book|title=Shigley's Mechanical Engineering Design|publisher=The Mc-Graw-Hill Companies|year=2011|isbn=9780073529288|pages=305}}

The general trend given by the Goodman relation is one of decreasing fatigue life with increasing mean stress for a given level of alternating stress. The relation can be plotted to determine the safe cyclic loading of a part; if the coordinate given by the mean stress and the alternating stress lies under the curve given by the relation, then the part will survive. If the coordinate is above the curve, then the part will fail for the given stress parameters.Hertzberg, pp. 530-31.

{{Reflist}}

• Goodman, J., Mechanics Applied to Engineering, Longman, Green & Company, London, 1899.
• Hertzberg, Richard W., Deformation and Fracture Mechanics and Engineering Materials. John Wiley and Sons, Hoboken, NJ: 1996.
• Mars, W.V., Computed dependence of rubber's fatigue behavior on strain crystallization. Rubber Chemistry and Technology, 82(1), 51–61. 2009.

• {{cite book |last=Mott |first=Robert L. |title=Machine elements in mechanical design |url=https://archive.org/details/machineelementsm00mott |url-access=limited |year=2004 |publisher=Pearson Prentice Hall |location=Upper Saddle River, N.J. |isbn=0130618853 |pages=[https://archive.org/details/machineelementsm00mott/page/n205 190]–192 |edition=4th}}
• {{cite book |last=Nisbett |first=Richard G. Budynas, J. Keith |title=Shigley's mechanical engineering design |year=2008 |publisher=McGraw-Hill Higher Education |location=Boston [Mass.] |isbn=9780073121932 |pages=295–300 |edition=8th}}

• [https://github.com/gunnstein/fatpack Fatpack. Fatigue analysis in python with Goodman mean stress correction implementation.]

Category:Materials science

Category:Fracture mechanics

Category:Rubber properties