Crack growth equation

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. 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 (formula_1) is the Paris–Erdogan equation
where formula_3 is the crack length and formula_4 is the fatigue crack growth for a single load cycle formula_5. 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
A "geometry factor" formula_7 is used to relate the far field stress formula_8 to the crack tip stress intensity using
There are standard references containing the geometry factors for many different configurations.
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, Frost and Dugdale, McEvily and Illg, and Liu 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 formula_3, the number of cycles of load applied is given by formula_5, the stress range by formula_13, and the material parameters by formula_14. For symmetrical configurations, the length of the crack from the line of symmetry is defined as formula_3 and is half of the total crack length formula_16.
Crack growth equations of the form formula_17 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. There have been a small number of true derivative fatigue crack growth equations that have also been developed.
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 formula_1 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.
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 formula_19 have an increased rate of crack growth. This effect is often explained using the crack closure concept which describes the observation that the crack faces can remain in contact with each other at loads above zero. This reduces the effective stress intensity factor range and the fatigue crack growth rate.
Sequence effects.
A formula_17 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:
where formula_23 is the plastic zone corresponding to the ith cycle that occurs post the overload and formula_24 is the distance between the crack and the extent of the plastic zone at the overload.
Crack growth equations.
Threshold equation.
To predict the crack growth rate at the near threshold region, the following relation has been used 
Paris–Erdoğan equation.
To predict the crack growth rate in the intermediate regime, the Paris–Erdoğan equation is used
Forman equation.
In 1967, Forman proposed the following relation to account for the increased growth rates due to stress ratio and when approaching the fracture toughness formula_27
McEvily–Groeger equation.
McEvily and Groeger proposed the following power-law relationship which considers the effects of both high and low values of formula_1
NASGRO equation.
The NASGRO equation is used in the crack growth programs AFGROW, FASTRAN and NASGRO software. It is a general equation that covers the lower growth rate near the threshold formula_31 and the increased growth rate approaching the fracture toughness formula_32, as well as allowing for the mean stress effect by including the stress ratio formula_33. The NASGRO equation is
where formula_35, formula_36, formula_37, formula_38, formula_39, formula_31 and formula_32 are the equation coefficients.
McClintock equation.
In 1967, McClintock developed an equation for the upper limit of crack growth based on the cyclic crack tip opening displacement formula_42
where formula_44 is the flow stress, formula_45 is the Young's modulus and formula_7 is a constant typically in the range 0.1–0.5.
Walker equation.
To account for the stress ratio effect, Walker suggested a modified form of the Paris–Erdogan equation
where, formula_48 is a material parameter which represents the influence of stress ratio on the fatigue crack growth rate. Typically, formula_48 takes a value around formula_50, but can vary between formula_51. In general, it is assumed that compressive portion of the loading cycle formula_52 has no effect on the crack growth by considering formula_53 which gives formula_54 This can be physically explained by considering that the crack closes at zero load and does not behave like a crack under compressive loads. In very ductile materials like Man-Ten steel compressive loading does contribute to the crack growth according to formula_55.
Elber equation.
Elber modified the Paris–Erdogan equation to allow for crack closure with the introduction of the "opening" stress intensity level formula_56 at which contact occurs. Below this level there is no movement at the crack tip and hence no growth. This effect has been used to explain the stress ratio effect and the increased rate of growth observed with short cracks. Elber's equation is
Ductile and brittle materials equation.
The general form of the fatigue-crack growth rate in ductile and brittle materials is given by
where, formula_37 and formula_38 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 formula_1 term, in ceramics on formula_63, and intermetallics have almost similar dependence on formula_1 and formula_63 terms.
Prediction of fatigue life.
Computer programs.
There are many computer programs that implement crack growth equations such as "Nasgro", 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 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.
Crack growth programs typically provide a choice of:
Analytical solution.
The stress intensity factor is given by
where formula_67 is the applied uniform tensile stress acting on the specimen in the direction perpendicular to the crack plane, formula_3 is the crack length and formula_7 is a dimensionless parameter that depends on the geometry of the specimen. The alternating stress intensity becomes
where formula_71 is the range of the cyclic stress amplitude.
By assuming the initial crack size to be formula_72, the critical crack size formula_73 before the specimen fails can be computed using formula_74 as
The above equation in formula_76 is implicit in nature and can be solved numerically if necessary.
Case I.
For formula_77 crack closure has negligible effect on the crack growth rate and the Paris–Erdogan equation can be used to compute the fatigue life of a specimen before it reaches the critical crack size formula_73 as
Crack growth model with constant value of formula_7 and R = 0.
For the Griffith-Irwin crack growth model or center crack of length formula_81 in an infinite sheet as shown in the figure 2, we have formula_82 and is independent of the crack length. Also, formula_83 can be considered to be independent of the crack length. By assuming formula_84 the above integral simplifies to
by integrating the above expression for formula_86 and formula_87 cases, the total number of load cycles formula_88 are given by
Now, for formula_90 and critical crack size to be very large in comparison to the initial crack size formula_91 will give
The above analytical expressions for the total number of load cycles to fracture formula_93 are obtained by assuming formula_94. For the cases, where formula_7 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 formula_96.
Case II.
For formula_97 crack closure phenomenon has an effect on the crack growth rate and we can invoke Walker equation to compute the fatigue life of a specimen before it reaches the critical crack size formula_73 as
Numerical calculation.
This scheme is useful when formula_7 is dependent on the crack size formula_3. The initial crack size is considered to be formula_102. The stress intensity factor at the current crack size formula_3 is computed using the maximum applied stress as
Now, by substituting the stress intensity factor in Paris–Erdogan equation, the increment in the crack size formula_109 is computed as
where formula_111 is cycle step size. The new crack size becomes
where index formula_113 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
Once this failure criterion is met, the simulation is stopped.
The schematic representation of the fatigue life prediction process is shown in figure 3.
Example.
The stress intensity factor in a SENT specimen (see, figure 4) under fatigue crack growth is given by
The following parameters are considered for the calculation
formula_121MPa,formula_122, formula_123.
The critical crack length, formula_124, can be computed when formula_125 as
By solving the above equation, the critical crack length is obtained as formula_127.
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 formula_129.