The first approach is based on extensive spectrum crack growth
data. Tests that incorporate the
important stress levels, part geometry, crack shape details and loading
sequences are run to determine the effect of the particular variables of
interest on the component life.
A second approach, and one used extensively, is the
cycle-by-cycle crack growth analysis where crack rates are integrated over the
crack length of interest as a function of stress and crack length [Gallagher,
1976; Brussat, 1971].
A third approach is based on the statistical
actual service stress histories are replaced with equivalent constant amplitude
stress histories for the analytical prediction of component life [Smith, et
A fourth approach, recently developed, utilizes a
crack-incrementation scheme to analytically generate “mini-block” crack growth
rate behavior prior to predicting life.
It combines some features of the
first three methods [Gallagher, 1976; Brussat, 1971; Gallagher & Stalwaker,
The application of the second through fourth approaches
requires methods for integrating the crack growth rate relations requires the
knowledge of the following items:
An initial flaw distribution
The aircraft loading spectrum
Constant amplitude crack growth rate material
Crack tip stress-intensity factor analysis
A damage integrator model relating crack growth to
applied stress and which accounts for load-history interactions
The criteria which establishes the life-limiting end
point of the calculation
These items are described in detail in Section 1.5 of this
handbook. The basic damage integrating
equation is also presented as equation 1.5.1 but is repeated here:
is the growth increment associated with the jth time increment, ao is the initial crack
length, acr is
the critical crack length and tf
is the life of the structure. The
determination of tf is the
objective of this equation.
Of the integration methods described above, the second and
third are most frequently used. The
generation of the data required for the first method is very expensive and is
only recommended for extremely critical parts.
The second method, the cycle-by-cycle integration method, uses
a type of integrating relation whereby the effect of each cycle is considered
separately. This is generally the least
efficient method, but if the spectrum under consideration cannot be considered
as statistically repetitive, it may be the most accurate of the analytical
methods. This method is covered in
detail in subsection 5.2.3.
Statistical Stress-Parameter Characterization
The third method, using a statistical characterization of a
crack growth parameter is based on the similarity of certain variable amplitude
crack growth behavior to the constant amplitude function relationship:
where (da/dF) is the
flight-by-flight crack growth behavior and is a stress-intensity
factor parameter that is derived using the product of a statistically
characterizing stress parameter () and the stress-intensity factor coefficient (K/s),
The statistically characterizing parameters that have been employed
in the past to some success are derived using a root mean square (RMS) or
similar type analysis of the stress range or stress maximum. The crack growth behavior of both fighter
and transport aircraft stress histories have been described using various forms
of equation 5.2.8.
One might imply from equations 5.2.8 and 5.2.9 that the use of
a single stress characterizing parameter for stress histories would allow one
to utilize equivalent constant amplitude histories to derive the same crack
growth rate behavior. Unfortunately,
relating constant amplitude behavior to variable amplitude behavior has not
been that successful.
The damage integration Equation (1.5.1R) is now expressed for
the flight as
is the number of flights corresponding to crack length ak, and Daj
is computed from Equation 5.2.8 evaluated for the given conditions. The parameters C and p of Equation 5.2.8
are determined by a least squares curve fit to previously determined data. The value that comes from employing the
third method comes from the fact that a somewhat limited variable amplitude
data base might be extended to cover other crack lengths, structural geometry,
or stress level differences.
The fourth approach provides an analytical extension of the
cycle-by-cycle analysis to predict flight-by-flight crack growth rates. In essence, this approach combined some of
the best features of the other three methods.
The basic element in this analysis is what is referred to as a
mini-block which is taken to be a flight (includes takeoff, landing and all
intermediate stress events) or a group of flights. The approach hinges on the identification of the statistically
repeating stress group that approximates the loading and sequence effects for
the complete spectrum.
The basic damage integration equation can be written in the
mini-block form to compute the crack increment (Da) due to application
of NG flights:
where there are Nj
stress cycles in the jth flight.
The most direct method for applying the equation is called the simple
Successive crack increments are obtained at successively larger
initial-crack-lengths. Figure 5.2.10 illustrates this method. The resulting values of Da/DF and the corresponding Kmax values are fit with a curve of the desired type,
usually similar to Equation 5.2.8, which can now be used to compute life.
Figure 5.2.10. Simple Crack-Incrementation Scheme Used to
Determine Crack Growth Rate Behavior [Gallagher, 1976]
An alternate method, called the statistical
crack-incrementation-mini-block approach, is illustrated in Figure 5.2.11.
This method allows evaluation of the effect of mini-block group-to-group
variation in the crack growth rate behavior.
A number of different mini-block groups are used at each initial crack
length. A curve can be fit through the
vs. values and the
variation of Da/DF
at each Kmax can be
observed. Confidence limits can be
determined for each set of data.
Figure 5.2.11. Statistical Crack-Incrementation Scheme Used
to Determine Spectrum Induced Variations in Crack Growth-Rate Behavior [Gallagher,
The fourth approach provides a more efficient integration
scheme than the cycle-by-cycle analysis.
However, its use is determined by the type of stress history that has to
In summary, there are a number of integration schemes
available. These schemes all employ
modeling approaches based on either limited or extensive variable amplitude
databases so that the analyst might properly account for loading and sequence
effects in the most direct and most accurate manner.