Translating experimental crack growth life data to flight-by-flight crack growth rate only provides one method for generating the flight-by-flight power law growth rate relationship given by Equation 9.2.3.  The power law rate equation can also be generated using two different analytic methods.  One popular analytical method is to calculate the RMS range and maximum parameters and to substitute these parameters into a constant amplitude - stress ratio equation.  This method results in a single curve that describes the effects of this stress combination.  The following example illustrates the procedure.

 

 

EXAMPLE 9.2.1         RMS Power Law Analysis

The constant amplitude crack growth equation for a particular alloy is given by

           

where the effective stress-intensity factor has been determined to be of the form:

           

When the values of maximum stress (s_max) and stress range (Ds) for a given constant amplitude loading are known, these values are used with the stress-intensity factor coefficient (K/s) for the geometry of interest to generate K_max,

and with the stress ratio formula

  

to generate the parameters defined in the growth equation.  To obtain the power law relation that would result for a flight-by-flight spectrum, the RMS values are substituted into the equations.

If the RMS range (Ds) is 4.45 ksi and maximum (s_max) stresses for a given stress history are 12.30 ksi, the RMS stress ratio (R) is given by

and the RMS maximum stress-intensity factor is

The growth rate is

where

Notice that the rate da/dN is given on a per cycle basis so one must multiply this rate by the average number of cycles per flight or flight hour to obtain the corresponding average growth rate per flight or flight hour.

 

 

A method that substitutes RMS (or other statistically derived) parameters into constant amplitude equations has one major limitation.  This limitation is that load interaction effects (retardation or acceleration) are ignored.  Thus, the analyst must be wary of comparisons between spectra when using this method, since it will only provide first order approximations of spectra effects.

It is possible to account for load interaction effects with a cycle-by-cycle analysis, but as indicated above, the processing of the complete stress history requires extensive numerical analysis.  An approach was suggested in the early 1970’s for processing a limited portion of a stress history with a cycle-by-cycle analysis for the purpose of generating crack increments at several crack lengths.  Most of the details for generating crack increments for such an analysis were discussed in subsection 5.2.5 relative to Figures 5.2.10 and 5.2.11.  Figure 5.2.10 is repeated here as Figure 9.2.4, and the corresponding crack growth rate data is presented in Figure 9.2.5.  The choice of methods that one might employ for the cycle-by-cycle analysis is dictated by the success that a given analysis has had in predicting crack growth behavior of the type under consideration.  In Section 9.3, a detailed example of an analytical analysis of the crack growth behavior (life and rate) of three transport wing stress histories is conducted.  This example should provide additional insight into how the simplified rate method can be used to assess spectra and their differences.

 

Figure 9.2.4.  Crack Incrementation Scheme Based on Cycle-by-Cycle Crack Growth Analysis

Figure 9.2.5.  Crack Growth Rate Description of Crack Incrementation Data for Two Transport Wing Stress Histories (DF = 50 Flights)