The aircraft industry has leaded the effort to understand and predict fatigue crack growth. They have developed the safe-life or fail-safe design approach. In this method, a component is designed in a way that if a crack forms, it will not grow to a critical size between specified inspection intervals. Thus, by knowing the material growth rate characteristics and with regular inspections, a cracked component may be kept in service for an extended useful life.

The aircraft industry has leaded the effort to understand and predict fatigue crack growth. They have developed the safe-life or fail-safe design approach. In this method, a component is designed in a way that if a crack forms, it will not grow to a critical size between specified inspection intervals. Thus, by knowing the material growth rate characteristics and with regular inspections, a cracked component may be kept in service for an extended useful life. This concept is shown schematically in Fig. 1.

As shown, most of the life of the component is spent while the crack length is relatively small. In addition, the crack growth rate increases with increased applied stress.

The crack growth rate, **da/dN**, is obtained by taking the derivative of the above crack length, **a**, versus cycles, **N**, curve. Two generally accepted numerical approaches for obtaining this derivative are the spline fitting method and the incremental polynomial method. These methods are explained in detail in many numerical methods textbooks. Values of **log da/dN** can then be plotted versus **log DK**, for a given crack length, using the equation

(1) |

where **Ds** is the remote stress applied to the component as shown in Fig. 3.

A plot of log **da/dN** versus log **DK**, a sigmoidal curve, is shown in Fig. 4. This curve may be divided into three regions. At low stress intensities, Region I, cracking behavior is associated with threshold, **DK _{th}**, effects. In the mid-region, Region II, the curve is essentially linear. Many structures operate in this region. Finally, in the Region III, at high

(2) |

where **C** and **m** are material constants and **DK** is the stress intensity range Kmax - Kmin.

Values of the exponent, **m**, are usually between 3 and 4. These range from 2,3 to 6,7 with a sample average of m = 3,5. In addition, tests may be performed. ASTM E647 sets guidelines for these tests.

The crack growth life, in terms of cycles to failure, may be calculated using Eq. (2). The relation may be generally described by

Thus, cycles to failure, **Nf**, may be calculated as

(3) |

where **a _{i}** is the initial crack length and af is the final (critical) crack length. Using the Paris formulation,

(4) | |

Because **DK** is a function of the crack length and a correction factor that is dependent on crack length [see Eq. (1)], the integration above must often be solved numerically. As a first approximation, the correction factor can be calculated at the initial crack length and Eq. (4) can be evaluated in closed form.

As an example of closed form integration, fatigue life calculations for a small edge-crack in a large plate are performed below. In this case the correction factor, f(g) does not vary with crack length. The stress intensity factor range is

(5) |

Substituting into the Paris equation yields

(6) |

Separating variables and integrating (for m<>2) gives

(7) |

Before this equation may be solved, the final crack size, af, must be evaluated. This may be done using as follows:

(8) |

For more complicated formulations of **DK**, where the correction factor varies with the crack length, **a**, iterative procedures may be required to solve for **a _{f}** in Eq. (8).

It is important to note that the fatigue-life estimation is strongly dependent on **a _{i}**, and generally not sensitive to af (when

May, 2001

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