Crack growth equation

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_{\text{max}}-K_{\text{min}}

A geometry factor $\beta$ is used to relate the far field stress $\sigma$ to the crack tip stress intensity using

K=\beta \sigma {\sqrt {\pi a}}

.

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

^ Schijve, J. (January 1979). "Four lectures on fatigue crack growth". Engineering Fracture Mechanics. 11 (1): 169–181. doi:10.1016/0013-7944(79)90039-0. ISSN 0013-7944.
^ Paris, P. C.; Erdogan, F. (1963). "A critical analysis of crack propagation laws". Journal of Basic Engineering. 18 (4): 528–534. doi:10.1115/1.3656900..
^ Murakami, Y.; Aoki, S. (1987). Stress Intensity Factors Handbook. Pergamon, Oxford.
^ Rooke, D. P.; Cartwright, D. J. (1976). Compendium of Stress Intensity Factors. Her Majesty’s Stationery Office, London.
^ Tada, Hiroshi; Paris, Paul C.; Irwin, George R. (1 January 2000). The Stress Analysis of Cracks Handbook (Third ed.). Three Park Avenue New York, NY 10016-5990: ASME. doi:10.1115/1.801535. ISBN 0791801535.{{cite book}}: CS1 maint: location (link)

[1] Schijve, J. (January 1979). "Four lectures on fatigue crack growth". Engineering Fracture Mechanics. 11 (1): 169–181. doi:10.1016/0013-7944(79)90039-0. ISSN 0013-7944.

[paris63-2] Paris, P. C.; Erdogan, F. (1963). "A critical analysis of crack propagation laws". Journal of Basic Engineering. 18 (4): 528–534. doi:10.1115/1.3656900..

[3] Murakami, Y.; Aoki, S. (1987). Stress Intensity Factors Handbook. Pergamon, Oxford.

[4] Rooke, D. P.; Cartwright, D. J. (1976). Compendium of Stress Intensity Factors. Her Majesty’s Stationery Office, London.

[tada-5] Tada, Hiroshi; Paris, Paul C.; Irwin, George R. (1 January 2000). The Stress Analysis of Cracks Handbook (Third ed.). Three Park Avenue New York, NY 10016-5990: ASME. doi:10.1115/1.801535. ISBN 0791801535.{{cite book}}: CS1 maint: location (link)

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