Crack growth equation explained

A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of a fatigue crack can result in catastrophic failure, particularly in the case of aircraft. When many growing fatigue cracks interact with one another it is known as widespread fatigue damage. A crack growth equation can be used to ensure safety, both in the design phase and during operation, by predicting the size of cracks. In critical structure, loads can be recorded and used to predict the size of cracks to ensure maintenance or retirement occurs prior to any of the cracks failing. Safety factors are used to reduce the predicted fatigue life to a service fatigue life because of the sensitivity of the fatigue life to the size and shape of crack initiating defects and the variability between assumed loading and actual loading experienced by a component.

Fatigue life can be divided into an initiation period and a crack growth period.^[1] Crack growth equations are used to predict the crack size starting from a given initial flaw and are typically based on experimental data obtained from constant amplitude fatigue tests.

One of the earliest crack growth equations based on the stress intensity factor range of a load cycle (

\Delta K ) is the Paris–Erdogan equation^[2] {da \over dN} = C(\Delta K) ^{m} where a is the crack length and {\rm d}a/{\rm d}N is the fatigue crack growth for a single load cycle N . A variety of crack growth equations similar to the Paris–Erdogan equation have been developed to include factors that affect the crack growth rate such as stress ratio, overloads and load history effects.

The stress intensity range can be calculated from the maximum and minimum stress intensity for a cycle

\Delta K = K _{max} - K _{min}

A geometry factor

\beta is used to relate the far field stress \sigma to the crack tip stress intensity usingK = \beta \sigma \sqrt{\pi a} .

There are standard references containing the geometry factors for many different configurations.^{[3] [4] [5]}

History of crack propagation equations

Many crack propagation equations have been proposed over the years to improve prediction accuracy and incorporate a variety of effects. The works of Head,^[6] Frost and Dugdale,^[7] McEvily and Illg,^[8] and Liu^[9] on fatigue crack-growth behaviour laid the foundation in this topic. The general form of these crack propagation equations may be expressed as

{da \over dN} = f(\Delta \sigma, a, C _{i} ),

where, the crack length is denoted by

a , the number of cycles of load applied is given by N , the stress range by \Delta \sigma , and the material parameters by C _{i} . For symmetrical configurations, the length of the crack from the line of symmetry is defined as a and is half of the total crack length 2 a .

Crack growth equations of the form

da/dN are not a true differential equation as they do not model the process of crack growth in a continuous manner throughout the loading cycle. As such, separate cycle counting or identification algorithms such as the commonly used rainflow-counting algorithm, are required to identify the maximum and minimum values in a cycle. Although developed for the stress/strain-life methods rainflow counting has also been shown to work for crack growth.^[10] There have been a small number of true derivative fatigue crack growth equations that have also been developed.^{[11] [12]}

Factors affecting crack growth rate

Regimes

Figure 1 shows a typical plot of the rate of crack growth as a function of the alternating stress intensity or crack tip driving force

\Delta K plotted on log scales. The crack growth rate behaviour with respect to the alternating stress intensity can be explained in different regimes (see, figure 1) as follows

Regime A: At low growth rates, variations in microstructure, mean stress (or load ratio), and environment have significant effects on the crack propagation rates. It is observed at low load ratios that the growth rate is most sensitive to microstructure and in low strength materials it is most sensitive to load ratio.^[13]

Regime B: At mid-range of growth rates, variations in microstructure, mean stress (or load ratio), thickness, and environment have no significant effects on the crack propagation rates.

Regime C: At high growth rates, crack propagation is highly sensitive to the variations in microstructure, mean stress (or load ratio), and thickness. Environmental effects have relatively very less influence.

Stress ratio effect

Cycles with higher stress ratio

R = K _{min} }/

Notes and References

1. Schijve . J. . January 1979 . Four lectures on fatigue crack growth. Engineering Fracture Mechanics . 11 . 1 . 169–181 . 10.1016/0013-7944(79)90039-0 . 0013-7944 .
2. P. C. . Paris . F. . Erdogan . A critical analysis of crack propagation laws. . Journal of Basic Engineering . 18 . 4 . 528–534 . 1963 . 10.1115/1.3656900 . .
3. Book: Y. . Murakami . S. . Aoki . Stress Intensity Factors Handbook . Pergamon, Oxford . 1987.
4. Book: Compendium of Stress Intensity Factors . D. P. . Rooke . D. J. . Cartwright . Her Majesty’s Stationery Office, London. . 1976.
5. Book: The Stress Analysis of Cracks Handbook. Third . Tada . Hiroshi . Paris . Paul C. . Irwin . George R.. 1 January 2000. ASME . 0791801535 . Three Park Avenue New York, NY 10016-5990 . 10.1115/1.801535.
6. Head . A. K. . September 1953 . The growth of fatigue cracks . The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . 44 . 356 . 925–938 . 10.1080/14786440908521062 . 1941-5982.
7. Frost . N. E.. Dugdale . D. S.. January 1958. The propagation of fatigue cracks in sheet specimens . Journal of the Mechanics and Physics of Solids . 6 . 2 . 92–110 . 10.1016/0022-5096(58)90018-8 . 0022-5096 . 1958JMPSo...6...92F.
8. Book: McEvily . Arthur J. . Illg . Walter . A Method for Predicting the Rate of Fatigue-Crack Propagation . 112–112–8 . ASTM International . 9780803165793 . 10.1520/stp45927s . Symposium on Fatigue of Aircraft Structures . 1960.
9. Liu . H. W. . 1961. Crack Propagation in Thin Metal Sheet Under Repeated Loading . Journal of Basic Engineering. 83 . 1 . 23–31 . 10.1115/1.3658886 . 2142/111864 . 0021-9223. free .
10. Sunder . R. . Seetharam . S. A. . Bhaskaran . T. A. . 1984 . Cycle counting for fatigue crack growth analysis . International Journal of Fatigue . 6 . 3 . 147–156. 10.1016/0142-1123(84)90032-X .
11. S. . Pommier . M. . Risbet . Time derivative equations for mode I fatigue crack growth in metals . International Journal of Fatigue . 2005 . 27 . 10–12 . 1297–1306. 10.1016/j.ijfatigue.2005.06.034 .
12. Small time scale fatigue crack growth analysis . Zizi . Lu . Yongming . Liu . International Journal of Fatigue . 32 . 8 . 2010 . 1306–1321. 10.1016/j.ijfatigue.2010.01.010 .
13. Ritchie . R. O. . 1977 . Near-Threshold Fatigue Crack Propagation in Ultra-High Strength Steel: Influence of Load Ratio and Cyclic Strength . Journal of Engineering Materials and Technology. 99 . 3 . 195–204 . 10.1115/1.3443519 . 136642892 . 0094-4289 .