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In [[materials science]] and [[fatigue (material)|fatigue]], the '''Goodman relation'''<!--, named after [[????? Goodman]],--> is an equation used to quantify the interaction of mean and alternating [[stress (physics)|stresses]] on the [[Fatigue (material)|fatigue]] life of a material.
 
A '''Goodman diagram''',<ref>
Herbert J. Sutherland and John F. Mandell.
[http://www.coe.montana.edu/composites/documents/aiaa_2005_0196.pdf "Optimized Goodman diagram for the analysis of fiberglass composites used in wind turbine blades"].
</ref><ref>
David Roylance.
[http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/fatigue.pdf "Fatigue"].
2001.
</ref>
sometimes called a '''Haigh diagram'''<ref>
[http://www.mscsoftware.com/training_videos/patran/reverb3/index.html#page/Fatigue%20Users%20Guide/fat_theory.15.3.html "Fatigue Theory > Total Life (S-N) Analysis"]
</ref>
or a '''Haigh-Soderberg diagram''',<ref name="udomphol">
Tapany Udomphol.
[http://www.sut.ac.th/engineering/metal/pdf/MechMet/12_Fatigue%20of%20metals.pdf "Fatigue of metals"].
2007.
</ref>
is a graph of (linear) mean stress vs. (linear) alternating stress, showing when the material fails at some given number of cycles.
 
A scatterplot of experimental data shown on such a plot can often be approximated by a parabola known as the '''Gerber line''',
which can in turn be (conservatively) approximated by a straight line called the '''Goodman line'''.
<ref name="udomphol" /><ref>
[http://www.eng.auburn.edu/users/marghitu/chapter3.pdf "Fatigue"]
Figure 3.9
</ref>
 
==Mathematical description==
[[Image:GoodmanRelation.jpg|thumb|200px|The area below the curve indicates that the material should not fail given the stresses. The area above the curve represents likely failure of the material.]]
The Goodman relation can be represented mathematically as:
 
:<math>\sigma_\text{a} = \sigma_\text{fat}\times\left(1-\frac{\sigma_\text{m}}{\sigma_\text{ts}}\right).</math>
 
Where <math>\sigma_\text{a}</math> is the alternating stress, <math>\sigma_\text{m}</math> is the mean stress, <math>\sigma_\text{fat}</math> is the fatigue limit for completely reversed loading, and <math>\sigma_\text{ts}</math> is the [[ultimate tensile stress]] of the material. The general trend given by the Goodman relation is one of decreasing fatigue life with increasing mean stress for a given level of applied 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 applied 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.<ref>Hertzberg, p. 530-31.</ref>
 
==References==
{{Reflist}}
 
===Bibliography===
*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
==Further reading==
*{{cite book|last=Mott|first=Robert L.|title=Machine elements in mechanical design|year=2004|publisher=Pearson Prentice Hall|location=Upper Saddle River, N.J.|isbn=0130618853|pages=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}}
 
[[Category:Materials science]]
[[Category:Fracture mechanics]]
[[Category:Rubber properties]]
 
{{classicalmechanics-stub}}

Latest revision as of 06:53, 23 November 2012

In materials science and fatigue, the Goodman relation is an equation used to quantify the interaction of mean and alternating stresses on the fatigue life of a material.

A Goodman diagram,[1][2] sometimes called a Haigh diagram[3] or a Haigh-Soderberg diagram,[4] is a graph of (linear) mean stress vs. (linear) alternating stress, showing when the material fails at some given number of cycles.

A scatterplot of experimental data shown on such a plot can often be approximated by a parabola known as the Gerber line, which can in turn be (conservatively) approximated by a straight line called the Goodman line. [4][5]

Mathematical description

The area below the curve indicates that the material should not fail given the stresses. The area above the curve represents likely failure of the material.

The Goodman relation can be represented mathematically as:

Where is the alternating stress, is the mean stress, is the fatigue limit for completely reversed loading, and is the ultimate tensile stress of the material. The general trend given by the Goodman relation is one of decreasing fatigue life with increasing mean stress for a given level of applied 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 applied 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.[6]

References

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Bibliography

  • 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

Further reading

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    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Template:Classicalmechanics-stub

  1. ↑ Herbert J. Sutherland and John F. Mandell. "Optimized Goodman diagram for the analysis of fiberglass composites used in wind turbine blades".
  2. ↑ David Roylance. "Fatigue". 2001.
  3. ↑ "Fatigue Theory > Total Life (S-N) Analysis"
  4. ↑ 4.0 4.1 Tapany Udomphol. "Fatigue of metals". 2007.
  5. ↑ "Fatigue" Figure 3.9
  6. ↑ Hertzberg, p. 530-31.
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