## Fatigue & Fracture

#### An Overview:

For over a century, before much of the physics of fatigue was understood, fatigue has been described using an s-N diagram, relating demand (stress or strain, s) and capability (cycles-to-failure, N). All engineering s-N curves use a logarithmic axis for cycles, and the dependent variable, cycles, is plotted on the x-axis. s-N curves are sigmoidal over the entire life range from monotonic tensile strength (plotted at 1/4 cycle) and stress at fatigue runout, if such a fatigue limit exists. A linear relationship is at best appropriate in the middle life region, but this is often the range of interest.

Most engineering alloys are polycrystalline. (Some are not: modern turbine blades are grown from a single crystal that can be 6 inches or more in length.) The grainsizes can be large, a quarter inch in cast parts, or small, 0.001 inch in fine grain wrought superalloys. (The diameter of a human hair is 0.003 inch.) The micromechanics of fatigue proceeds as an involved sequence of events that begins with microslip in shear between adjacent planes of atoms in the crystal structure of a single grain, and usually oriented 45 degrees to the normal (perpendicular) load, and culminates with the propagation of a macrocrack many times larger than the grainsize, and oriented normal to the load. The behavior of the material in aggregate can be rather different from the behavior of an individual crystal. Still it is useful to describe the aggregate behavior in gross terms by relating some measure of durability like cycles-to-failure, with one or more of the durability-controlling parameters, such as applied stress or strain range. (Stress is the applied force normalized to a unit area; strain is elongation normalized to undeformed length.)

The material response is a function of more than just stress (or strain) range. Stress ratio, R, (R=min stress/max stress.), hold-times-at-load, and temperature (if isothermal) or thermal cycle also influence fatigue capability. The material’s chemistry, and forming history also play a role, as does the local three-dimensional geometry of the fatigued part. (It can be argued that the three-dimensional loading and resulting geometry-influenced state of stress is outside the purview of materials modeling, however.) Finally, the relative influence of all of these factors changes throughout the fatigue process.

##### s-N models can’t deal with cracks.

Unfortunately, s-N models can’t deal with cracks. After a fatigue crack is formed, the s-N curve is no longer useful. (This is somewhat ironic since some fraction of the life of a fatigue specimen is comprised of a propagating macrocrack, and the mechanics of crack propagation in low cycle fatigue (LCF) is well understood.) Fracture mechanics (F/M) considers the stress field (not just an average stress) AND its synergism with a material discontinuity (crack). For two decades fracture mechanics has been used with great success to describe the behavior of a potential crack, and thus mitigate its threat. Because of the very large variability in time-to-crack, many component lifetimes are determined using the anticipated behavior of a propagating crack which is assumed to be present from cycle one. (cf: “Cumulative Damage Fracture Mechanics Under Engine Spectra,” J. M. Larsen, B.J. Schwartz, and C. G. Annis, Jr., AFML-TR-79-4159, January 1980.)

##### High Cycle Fatigue is not Low Cycle Fatigue at a Higher Frequency.

If fracture mechanics is so wonderful, who cares about s-N curves? Some of the emphasis on F/M may change with the recognition that high cycle fatigue (HCF, i.e.: $$N >> 10^7$$ cycles) is not LCF at a higher frequency. The ability to describe the behavior of a crack may not be useful, if the propagation time is measured in minutes, even if the cycle count is measured in millions. (At 20KHz you can accumulate more than a million cycles in less than a minute.) Further, the variance of fatigue lifetime increases (or appears to increase) for longer and longer lives as a consequence of a random fatigue-limit. In any event, the precision for predicting fatigue lives under HCF is quite poor and inadequate for component design. But we still may be able to predict the probability of there being the HCF excitation. Thus a potential shift in emphasis from estimating a runout stress under low cycle fatigue to understanding the conditions under which HCF loading will produce failure. (Some HCF excitation cannot be avoided, but it can be mitigated, by damping, for example.)