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
for plane stress
for plane strain.
Step 4. The
crack size is now a2. The stress range in the next cycle is Ds2. Calculate DK
Step 5. Calculate
the extent of the yield zone
= a2 + rp2 .
Step 6. If
Y22 < Y2:
When using the Wheeler model,
calculate Cp according to
When using the Generalized
Willenborg model, calculate or and Reff
according to Equations 5.2.3, and 5.2.5.
Go to Step 9, skipping steps 7
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 aoL =a1 with aoL =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 Reff
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 ith cycle replacing a2 by ai and a3
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 ai, aoL, Y2.
for the ith cycle in the
same way as in Steps 4 through 9.
. . . , Daj, . . . ,
Dan for the
following cycles but let the current crack size remain ai constant.
This eliminates recalculation of b
for every cycle. If Y2k > Y2, then replace Y2
by Y2k and call it Y2. Then replace aoL
by ai and call it aoL.
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
ai + Da,
repeat the procedure.
A reasonable size for the crack-growth increment is Da = 1/20 ai;
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 - ai). The advantage of the incremental
crack-growth procedure is especially obvious if series of constant-amplitude
cycles occur. Since the crack size (ai) 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.