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
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, DKth, 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
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
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
where ai is the initial crack length and af is the final
(critical) crack length. Using the Paris formulation,
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
Substituting into the Paris equation yields
Separating variables and integrating (for m<>2) gives
Before this equation may be solved, the final crack size, af, must be
evaluated. This may be done using as follows:
For more complicated formulations of DK,
where the correction factor varies with the crack length, a, iterative
procedures may be required to solve for af in Eq. (8).
It is important to note that the fatigue-life estimation is strongly
dependent on ai, and generally not sensitive to af
(when ai«af). Large changes in
result in small changes of Nf as shown schematically in Fig. 5.
Figure 5. Effect of final crack size on life
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