In general, the crack-growth-damage-integration procedure consists of the following steps, schematically outlined in Figure 5.2.12.

 

Figure 5.2.12.  Steps Required for Crack Growth Integration

Step 1.    The initial crack size follows from the damage tolerance assumptions as a₁.  The stress range in the first cycle is Ds₁.  Then determine DK₁ = bDs₁Öpa₁by using the appropriate b for the given structural geometry and crack geometry.  Computer programs generally have a library of stress-intensity factors or schemes for tabular data input for determining the appropriate b.

Step 2.    Determine (da/dN)₁, at DK₁ from the da/dN -DK baseline information, taking into account the appropriate R value.  The da/dN - DK baseline information may use one of the crack growth equations discussed in Section 5.1.2.  The computer program may contain options for any of these equations, or it may use data in tabular form and interpolate between data points.  The crack extension Da₁ in cycle 1 is

                       

               The new crack size will be a₂ = a₁ + Da₁

Step 3.    The extent of the yield zone in Cycle 1 is determined as

                         

                        where  

                         for plane stress

                        or  for plane strain.

Step 4.    The crack size is now a₂.  The stress range in the next cycle is Ds₂.  Calculate DK with

                        .

Step 5.    Calculate the extent of the yield zone
         Y₂₂ = a₂ + r_p2 .

Step 6.    If Y₂₂ < Y₂:

                  When using the Wheeler model, calculate C_p according to Equation 5.2.2.

                  When using the Generalized Willenborg model, calculate or and R_eff according to Equations 5.2.3, and 5.2.5.

               Go to Step 9, skipping steps 7 and 8.

Step 7.    If Y₂₂ > Y₂, determine (da/dN)₂ from DK₂.  Determine the new crack size a₃

                       

Step 8.    Replace Y₂ by Y₂₂ , which is now called Y₂.  Replace a_oL =a₁ with a_oL =a₂.  Go to Step 10, skipping Step 9.

Step 9.    When using the Wheeler model, determine the amount of crack growth on the basis of DK₂ from the da/dN - DK data.  Find the new crack size from

                       

When using the Generalized Willenborg model, determine the amount of crack growth using the DK and R_eff value determine in Step 6 from the da/dN - DK data.  Determine the new crack size as

                       

Step 10.  Repeat Steps 4 through 9 for every following cycle, while for the i^th cycle replacing a₂ by a_i and a₃ by a_i+1.

 

This routine of cycle-by-cycle integration is not always necessary.  The integration is faster if the crack size is increased stepwise in the following way.

·        At a certain crack size, the available information is a_i, a_oL, Y₂.

·        Calculate Da_i for the i_th cycle in the same way as in Steps 4 through 9.

·        Calculate Da_j+1, . . . , Da_j, . . . , Da_n for the following cycles but let the current crack size remain a_i constant.  This eliminates recalculation of b every cycle.

·        Calculate Y_2k for every cycle.  If Y_2k > Y₂, then replace Y₂ by Y_2k and call it Y₂.  Then replace a_oL by a_i and call it a_oL.

·        Sum the crack-growth increments to give:
           

·        Continue increasing j until Da exceeds a previously determined size or until j = n and the cycles are exhausted.  Then increment the crack size by 

                  a= a_i + Da,

and repeat the procedure.

 

A reasonable size for the crack-growth increment is Da = 1/20 a_i; this choice of increment typically keeps the change in K small.  It can also be based on the extent of the yield zone, e.g., Da = 1/10 (Y₂ - a_i).  The advantage of the incremental crack-growth procedure is especially obvious if series of constant-amplitude cycles occur.  Since the crack size (a_i) is fixed, the stress intensity does not change.  Hence, each cycle produces the same amount of growth.  This means that all n constant-amplitude cycles can be treated as one cycle to give

           

The integration scheme is a matter of individual judgment, but may be dictated by available computer facilities.