Rewriting Equation 2.4.1 such that the integration is conducted between the initial crack length (a_o) and any intermediate crack length (a_K) between a_o and the critical crack length results in

where t(N) is the elapsed time (number of load cycles) corresponding to growing the crack a_o to the intermediate crack length a_K.  The next cycle of the applied stress (the N + 1 cycle) induces a crack length growth increment Da_N+1.  The damage integration model provides the analysis capability to determine this crack length growth increment.  The growth increment Da_N+1 is equated to the constant amplitude crack growth rate, which in turn is determined from a function of stress intensity factor range (DK) and stress ratio (R), i.e.,

The stress intensity factor range and stress ratio in Equation 2.4.4 are determined by using the maximum and minimum stresses in the N+1 cycle of the given stress history and evaluating the stress intensity factor coefficients associated with the given structural geometry at the crack length a_K.  Subsequent to the direct calculation of the two crack tip parameters DK and R, and prior to their insertion in Equation 2.4.4, DK and R are modified to account for the effect of prior load history using retardation models.  Retardation models account for high-to-low load interaction effects, i.e., the phenomena whereby the growth of a crack is slowed by application of a high load in the spectrum.  Failure to account for high-to-low load interaction via a retardation model leads to conservative (~2 to 5 times shorter) life. 

There are numerous functional forms of Equation 2.4.4 and numerous models describing retardation.  The following list describe the general scheme of the crack growth calculation.

Step 1 - Knowing crack length a_K, determines the stress intensity factor coefficient, K/s.

Step 2 - For the given stress cycle, Ds, and the coefficient K/s, determine the stress intensity factor cycle, DK, and stress ratio R.

Step 3 - Utilizing the retardation model, modify the stress-intensity cycle DK and R to account for previous load history.

Step 4 - Determine the growth rate for the stress-intensity factor cycle to establish the crack growth increment.

Section 5 provides a current state-of-the-art summary of the procedures and techniques that are used in damage integration models.