Crack growth equation
The growth of a fatigue crack can result in catastrophic failure, particularly in the case of aircraft.
[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 (
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.
to the crack tip stress intensity using There are standard references containing the geometry factors for many different configurations.
[3][4][5] 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.
are not a true differential equation as they do not model the process of crack growth in a continuous manner throughout the loading 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.
This reduces the effective stress intensity factor range and the fatigue crack growth rate.
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.
Two notable equations for modelling the delays occurring while the crack grows through the overload region are:[16] where
To predict the crack growth rate at the near threshold region, the following relation has been used[17] To predict the crack growth rate in the intermediate regime, the Paris–Erdoğan equation is used[2] In 1967, Forman proposed the following relation to account for the increased growth rates due to stress ratio and when approaching the fracture toughness
[18] McEvily and Groeger[19] proposed the following power-law relationship which considers the effects of both high and low values of
[20] It is a general equation that covers the lower growth rate near the threshold
To account for the stress ratio effect, Walker suggested a modified form of the Paris–Erdogan equation[22] where,
is a material parameter which represents the influence of stress ratio on the fatigue crack growth rate.
In very ductile materials like Man-Ten steel, compressive loading does contribute to the crack growth according to
[23] Elber modified the Paris–Erdogan equation to allow for crack closure with the introduction of the opening stress intensity level
Elber's equation is[16] The general form of the fatigue-crack growth rate in ductile and brittle materials is given by[21] where,
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.
Crack growth programs typically provide a choice of: The stress intensity factor is given by where
is the applied uniform tensile stress acting on the specimen in the direction perpendicular to the crack plane,
will give The above analytical expressions for the total number of load cycles to fracture
This iterative process is continued until Once this failure criterion is met, the simulation is stopped.
The schematic representation of the fatigue life prediction process is shown in figure 3.
The stress intensity factor in a SENT specimen (see, figure 4) under fatigue crack growth is given by[5] The following parameters are considered for the calculation
Now, invoking the Paris–Erdogan equation gives By numerical integration of the above expression, the total number of load cycles to failure is obtained as
