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Handbook for Damage Tolerant Design

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Section 9.2.1. Variable Amplitude Crack Growth Behavior

Many airframe loading conditions are sufficiently repetitive over a number of flight (~100 to 500 flights) that the crack growth damage accumulates in a relatively continuous manner.  Figure 9.2.1 describes two examples of experimental crack growth data generated under typical flight-by-flight loadings involving multiple missions.  Figure 9.2.1a represents the behavior experienced at a hole subjected to a fighter wing stress history and Figure 9.2.1b represents the behavior observed at a hole subjected to a bomber aircraft wing stress history.  Both behaviors illustrate the regular and relatively continuous crack growth pattern exhibited by many flight-by-flight histories.

Figure 9.2.1a.  Experimental Propagation Behavior of Corner Crack with Full F-4E/S Wing Spectrum (68000 cycles/1000 flight hours) Scaled to Two Stress Levels (36 and 30.5 ksi).


Figure 9.2.1b.  Experimental Flight-By-Flight Fatigue Crack Growth Behavior for a B-1A Wing Spectrum Scaled to Three Levels (24.17, 31.12, and 36.31 ksi).

As a result of the regularity of such flight-by-flight induced crack growth behavior, there was a recognition as early as 1963 that aircraft stress histories can induce crack growth behavior similar to constant amplitude behavior.  This early recognition has led to a number of schemes for utilizing limited information to characterize the behavior of cracks in aircraft structure.  These schemes all focus on the translation of variable amplitude crack growth life data to variable amplitude crack growth "rate" data so that the simple analysis schemes for constant amplitude loadings can be used to establish life estimates.  These replace the more complicated numerical, computer-based algorithms used for a load interaction, cycle-by-cycle analysis of the complete stress history.

The translation of the variable amplitude crack growth life data to that of variable amplitude crack growth rate data follows most of the procedures used to convert constant amplitude crack growth life data to constant amplitude crack growth rate data (see Subsection 7.2.2 and Figure 7.2.9, which is repeated as Figure 9.2.2).  The major differences between describing variable amplitude rate behavior and constant amplitude rate behavior is in the choice of the rate variable and the characterizing stress history parameter.  In variable amplitude descriptions, the crack growth rate may be described as rate per flight, rate per flight hour, or rate per cycle.  Also, since the magnitude of the individual stress events in the stress history is a random variable the characteristic stress parameter that described the history is a statistical measure of the individual events in the history.

Figure 9.2.2.  Method for Reducing Fatigue Crack Growth Life Data to Fatigue Crack Growth Rate Data

Figure 9.2.3 describes a variable amplitude crack growth rate behavior (da/d(Flight)) as a function of a spectra dependent characteristic stress-intensity factor () for two transport wing histories.  The K is calculated based on the formula


where K/s is the stress-intensity factor coefficient for the geometry and  is the characteristic stress parameter, here chosen as the root mean square (RMS) of the maximum stresses in the history, i.e.


In equation 9.2.2, N is the number of stress events, and smax(i) denotes the maximum stress for the ith stress event.

Figure 9.2.3.  Fatigue Crack Growth Rate Data for Two Transport Spectra (A = Upper Wing, B = Lower Wing)

It is seen from Figure 9.2.3 that the crack growth rate (on a per-flight basis) behavior for the two spectra might be described by a power law equation of the form


The corresponding image integration equation can be expressed as


or as


where F is the number of flights required to grow the crack from a0 to a, and where Daj is evaluated for the current crack length using Equation 9.2.3.  The coefficients C and p are evaluated using least squares procedures applied to data of the type shown in Figure 9.2.3.

The value of the data shown in Figure 9.2.3 and its description with a simple equation, e.g. Equation 9.2.3, is that parametric studies can be conducted in a relatively simple manner.  Such parametric studies could cover other ranges of crack length for the same geometry, other structural geometries, and other stress magnification factors applied to the same spectra.