The major share of the fatigue life of the component may be taken up in the propagation of crack. By applying fracture mechanics principles it is possible to predict the number of cycles spent in growing a crack to some specified length or to final failure.

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.

**Figure 1.**Extended service life of a cracked component

### Fatigue Crack Growth Curves

Typical constant amplitude crack propagation data are shown in Fig. 2. The crack length, a, is plotted versus the corresponding number of cycles,**N**, at which the crack was measured.

**Figure 2.**Constant amplitude crack growth data

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.

**Figure 3.**Remote stress range

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

**DK**values, crack growth rates are extremely high and little fatigue life is involved.

**Figure 4.**Three regions of crack growth rate curve

### Region II

Most of the current applications of LEFM concepts to describe crack growth behavior are associated with Region II. In this region the slope of the log**da/dN**versus log

**DK**curve is approximately linear and lies roughly between 10

^{-6}and 10

^{-3}in/cycle. Many curve fits to this region have been suggested. The Paris equation, which was proposed in the early 1960s, is the most widely accepted. In this equation

(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

**a**«

_{i}**a**). Large changes in

_{f}**a**result in small changes of

_{f}**N**as shown schematically in Fig. 5.

_{f}

**Figure 5.**Effect of final crack size on life