# How does the S-N curve (or Wöhler curve) of a material change with frequency?

How a Material fatigues can be characterized by a Wöhler curve where each datapoint corresponds to a failure with the stress on the y-Axis and the number of cycles on the x-axis. An example is the following Wöhler curve:

How does this curve change with the frequency with which a cycle count is reached?

• The Wohler curve shifts downward/leftward (fewer cycles for a given stress) with increasing temperature. I would expect the curve to shift downward/leftward in a similar way as frequency increases because the heat built up during the bending process would have less time to dissipate and would raise the temperature of the material. – Chris Mueller Jan 21 '16 at 12:54
• @ChrisMueller That sounds very plausible, although I would like to see some data which supports (which is probably not that easy to get ...) – John H. K. Jan 22 '16 at 10:49

If, for any reason, the fatigue process becomes time-dependent, then it also becomes frequency-dependent. Under normal conditions, fatigue failure is independent of frequency. But when corrosion or high temperatures, or both, are encountered, the cyclic rate becomes important. The slower the frequency and the higher the temperature, the higher the crack propagation rate and the shorter the life at a given stress level (1)

This quote is taken from Mechanical Engineering Design, 7th Ed. in reference to metallic parts. Please note - metallic fatigue stresses are taken at high cycle times. The standard R.R. Moore rotational testing machine has capacities up to 10,000 rpm, and thus the heat generated from fast cycling is usually already present in the sample data. Instead the quote states time dependency for fatigue, and a failure of the Wöhler curve to recognize low frequency crack propagation.

Time dependency means external, time dependent factors that increase the stresses slowly over time. For example:

• Corrosion removes thickness at a fixed rate with time. A part that is designed for endurance limit stress levels placed in a corrosive environment will only handle a finite number of cycles before the corrosion reduces the load bearing cross-section to a level below the endurance limit.
• At elevated temperature, the ultimate strength is reduced. A part that is near its endurance limit at room temperature could cycle into and out of high temperatures (for example, automobile parts that are stressed for a few hours per day). In this case, the frequency of the loading will become a factor, as the number of cycles taken during the times of high temperature will count towards fatigue and thus reduce life at low temperature simultaneously.

A different model is used for low-frequency, high stress loading. (So severe that plastic deformation / crack propogation begins almost immediately) This method, the Manson-Coffin Relationship, attempts to find the remaining cycles left after a crack has begun. If $\Delta \epsilon$ is the strain from loading, $\sigma^{'}_F$ is the true stress for first fracture, $\epsilon^{'}_F$ is the true strain for first fracture, then the estimated reversals (half-cycles) $(N)$ after first fracture can be predicted for all frequencies as:

$$\frac{\Delta \epsilon}{2} > \frac{\sigma^{'}_F}{E}(2N)^b + \epsilon^{'}_F(2N)^c$$

b and c are empirically determined coefficients. b is in the -0.055 to -0.09 range, while c is in the -1.0 to -0.66 range. More data can be gathered from (2) and (3).

In summary, while internal heat generation may very well be a key at high cycles speeds for plastics, metallic testing is taken at high frequencies already and the Wöhler curve actually presents problems at low frequencies, which is better covered by the Manson-Coffin model.

1. Mishke, Charles R., and Richard G. Budynas. "Endurance Limit Modifying Factors, Cyclic Frequency." Mechanical Engineering Design. By Joseph E. Shigley. 7th ed. New York: McGraw-Hill, 2004. 335. Print.

2. J.F. Tavernelli and L.F. Coffin, Jr., "Experimental Support for Generalized Equation Predicting Low Cycle Fatigue," and S. S. Manson, discussion, Trans. ASME, J. Basic Eng. vol. 84, no. 4, pp. 533-537.

3. N.E. Dowling's Mechanical Behavior of Materials, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J., 1999, Chap. 14.

Endurance limit decreases with frequency.

• Welcome to Engineering SE, I would encourage you explain your answer in addition to the image you have posted. This will greatly enhance your answer and benefit other posters – Mahendra Gunawardena Jun 5 '16 at 10:22