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 stress-parameter-characterization.  The actual service stress histories are replaced with equivalent constant amplitude stress histories for the analytical prediction of component life [Smith, et al., 1968].

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, 1975].

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 properties

·        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:

(1.5.1R)

where Da_j is the growth increment associated with the j^th time increment, a_o is the initial crack length, a_cr is the critical crack length and t_f is the life of the structure.  The determination of t_f 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.

Cycle-by-cycle method

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:

(5.2.8)

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), i.e.,

(5.2.9)

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

(5.2.10)

where N_f is the number of flights corresponding to crack length a_k, and Da_j 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.

Crack-Incrementation Scheme

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 N_G flights:

(5.2.11)

where there are N_j stress cycles in the j^th flight.  The most direct method for applying the equation is called the simple crack-incrementation-mini-block approach.  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 K_max 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 mean Da/DF vs.  values and the variation of Da/DF at each K_max 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, 1976]

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 be integrated.

Summary

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.