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A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of fatigue cracks can result in catastrophic failure, particularly in the case of aircraft. 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.

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.

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

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

Crack growth equations of the form d a / d N {\displaystyle 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]

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 Î” K {\displaystyle \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 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.

A d a / d N {\displaystyle da/dN} equation gives the rate of growth for a single cycle, but when the loading is not constant amplitude, changes in the loading can lead to temporary increases or decreases in the rate of growth. Additional equations have been developed to deal with some of these cases. The rate of growth is retarded when an overload occurs in a loading sequence. These loads generate are plastic zone that may delay the rate of growth. Two notable equations for modelling the delays occurring while the crack grows through the overload region are:[16]

where r pi {\displaystyle r_{\text{pi}}} is the plastic zone corresponding to the ith cycle that occurs post the overload and r max {\displaystyle r_{\text{max}}} is the distance between the crack and the extent of the plastic zone at the overload.

The NASGRO equation is used in the crack growth programs AFGROW, FASTRAN and NASGRO software.[20] It is a general equation that covers the lower growth rate near the threshold Î” K th {\displaystyle \Delta K_{\text{th}}} and the increased growth rate approaching the fracture toughness K crit {\displaystyle K_{\text{crit}}} , as well as allowing for the mean stress effect by including the stress ratio R {\displaystyle R} . The NASGRO equation is

where, n {\displaystyle n} and p {\displaystyle p} are material parameters. Based on different crack-advance and crack-tip shielding mechanisms in metals, ceramics, and intermetallics, it is observed that the fatigue crack growth rate in metals is significantly dependent on Î” K {\displaystyle \Delta K} term, in ceramics on K max {\displaystyle K_{\text{max}}} , and intermetallics have almost similar dependence on Î” K {\displaystyle \Delta K} and K max {\displaystyle K_{\text{max}}} terms.

There are many computer programs that implement crack growth equations such as Nasgro,[24] AFGROW and Fastran. In addition, there are also programs that implement a probabilistic approach to crack growth that calculate the probability of failure throughout the life of a component.[25][26]

Crack growth programs grow a crack from an initial flaw size until it exceeds the fracture toughness of a material and fails. Because the fracture toughness depends on the boundary conditions, the fracture toughness may change from plane strain conditions for a semi-circular surface crack to plane stress conditions for a through crack. The fracture toughness for plane stress conditions is typically twice as large as that for plane strain. However, because of the rapid rate of growth of a crack near the end of its life, variations in fracture toughness do not significantly alter the life of a component.

where Ïƒ {\displaystyle \sigma } is the applied uniform tensile stress acting on the specimen in the direction perpendicular to the crack plane, a {\displaystyle a} is the crack length and Î² {\displaystyle \beta } is a dimensionless parameter that depends on the geometry of the specimen. The alternating stress intensity becomes

By assuming the initial crack size to be a 0 {\displaystyle a_{0}} , the critical crack size a c {\displaystyle a_{c}} before the specimen fails can be computed using ( K = K max = K Ic ) {\displaystyle {\big (}K=K_{\text{max}}=K_{\text{Ic}}{\big )}} as

For the Griffith-Irwin crack growth model or center crack of length 2 a {\displaystyle 2a} in an infinite sheet as shown in the figure 2, we have Î² = 1 {\displaystyle \beta =1} and is independent of the crack length. Also, C {\displaystyle C} can be considered to be independent of the crack length. By assuming Î² = constant , {\displaystyle \beta ={\text{constant}},} the above integral simplifies to

The above analytical expressions for the total number of load cycles to fracture ( N f ) {\displaystyle {\big (}N_{f}{\big )}} are obtained by assuming Y = constant {\displaystyle Y={\text{constant}}} . For the cases, where Î² {\displaystyle \beta } is dependent on the crack size such as the Single Edge Notch Tension (SENT), Center Cracked Tension (CCT) geometries, numerical integration can be used to compute N f {\displaystyle N_{f}} .

This scheme is useful when Î² {\displaystyle \beta } is dependent on the crack size a {\displaystyle a} . The initial crack size is considered to be a 0 {\displaystyle a_{0}} . The stress intensity factor at the current crack size a {\displaystyle a} is computed using the maximum applied stress as

where index i {\displaystyle i} refers to the current iteration step. The new crack size is used to calculate the stress intensity at maximum applied stress for the next iteration. This iterative process is continued until 2b1af7f3a8