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# Section 9.2.3. Power Law Descriptions

A number of experimental and analytical investigations have revealed that the flight-by-flight crack growth rate behavior of military aircraft can be described with a power law relationship (Equation 9.2.3).  Specifically, the stress histories considered were developed to facilitate the design of a new structure or an analysis of an in-service aircraft for force management purposes.  As such, these stress histories represented an expected average usage based on a force wide composite mission mix; most of the stresses in these histories repeated after an application of a large block of flights or flight hours.  None of the histories involved any major mission change during the expected life of the aircraft.  For these histories, one might say that the operations today will be like the operation next year or five years from now.

Nevertheless, the generalized observations of power law flight-by-flight crack growth rate behavior here are immediately applicable to the study of parameters affecting structural repair.  Thus, the results of these studies are summarized in Tables 9.2.1 and 9.2.2 for bomber/transport behavior and for fighter/attack/trainer behavior, respectively.  Table 9.2.1 presents the coefficients for a crack growth rate per flight type equation, while Table 9.2.2 presents the coefficients for a crack growth rate per flight hour type equation.

The reader can note from Table 9.2.1 that the exponent p for bomber/transport aircraft wing stress histories only varies from about 3.0 to 3.5; Table 9.2.2 indicates a wider variation in the exponent for the aircraft and conditions indicated (2.2 £ p £ 3.7).  Based on a close analysis of the results, it can be said that the largest variations in the exponent p are generated due to the wide variations in spectrum content (load magnitude and frequency).

Table 9.2.1.  Bomber/Transport Behavior

 Aircraft History Flights/Block (ksi) C+ p Aluminum Alloy B1-B Wing pivot 100 27.3 4.91x10-8 3.025 2219-T851 C-5A Upper wing 100 11.7 1.70x10-8 3.111 7075-T651 C-5A Lower wing 300 12.3 1.05x10-7 3.183 7075-T651 B-52D Lower wing 200 16.4 2.61x10-8 3.529 7075-T651 KC-135 Proof test, Lower wing 200 17.8 5.97x10-9 3.454 7178-T6 KC-135 Lower wing 200 18.4 1.01x10-8 3.338 7178-T6

+ inch/flight, ksiÖin

Table 9.2.2.  Fighter/Attack/Trainer Behavior (Based on 1000 Flight

Hour Block Spectra)

 Aircraft History C+ p Aluminum Alloy T-38 Lower wing (baseline) 2.66x10-8 2.678 7075-T651 T-38 Lower wing (severe) 1.07x10-8 3.152 7075-T651 T-38 Lower wing (mild) 5.32x10-9 2.460 7075-T651 F-4 Lower wing (baseline) 1.68x10-8 2.242 7075-T651 F-4 Lower wing (high stress baseline) 1.77x10-8 2.242 7075-T651 F-4 Lower wing (severe) 1.76x10-8 2.395 7075-T651 F-4 Lower wing (mild) 5.77x10-9 2.395 7075-T651 F-16 Lower wing (mix) 6.92x10-10 3.62 7475-T7351 F-16 Tail (mix) 1.33x10-10 3.67 7475-T7351 F-16 Lower wing (air to air) 1.07x10-9 2.905 7475-T7351 F-16 Lower wing (air to ground) 8.94x10-11 3.464 7475-T7351

+ inch/flight, ksiÖin

Before employing a flight-by-flight crack growth rate type analysis to estimate the life of a repair, the analyst should be concerned with the adequacy of such an analysis.  The most important part of the analysis is the definition of the stress history that the repaired member will experience in the future.  If the history is anticipated to be statistically repetitive as a function of time-in-service then the results from a flight-by-flight rate analysis will be comparable to a cycle-by-cycle analysis.

If the mission type or mix is expected to change significantly as a function of time, then projecting a predefined rate of crack growth without detailed consideration of how the damage will be changing could lead to non-conservative errors.  One method for addressing mission type or mix changes is to utilize one rate curve before the time of change and another rate curve subsequently.  A more exact method for addressing mission changes is by using a cycle-by-cycle crack growth analysis applied to the stress history that accounts for the changes.

Rate methods have one inherent problem: they tend to minimize the effects of the infrequently applied large loads.  These large loads will cause retardation effects and tend to slow the growth process (if, in application, failure is not induced).  Thus rate methods will normally predict somewhat shorter (more conservative) lives than the cycle-by-cycle analyses.

Based on Tables 9.2.1 and 9.2.2 the analyst should note that the crack growth rate equation is a function of material, location, and usage.  An equation generated for the horizontal tail should not be used for the vertical tail (nor wing); an equation generated for air-to-ground operations should not be utilized for air-to-air operations.