_{}

The function *f(K)*
describes the crack growth rate for the material and loading condition; *a*_{o} and *a*_{f} are the initial and critical crack sizes, respectively. Three elements are necessary for the life
calculation: (1) the function *f(K)*,
(2) the stress-intensity factor relationship for the geometry, and (3) the
critical crack size (*a*_{f}). Each element will be separated determined in
the paragraphs below; subsequently, LIFE will be determined.

__Function __*f(K)* Established

The function *f(K)*
describes crack growth rate as a function of a stress-intensity factor
parameter (such as *DK*). As a result of the constant amplitude
loading condition, the engineer would consult the Damage Tolerant Design (Data)
Handbook [Skinn, et al., 1994] to find data consistent with the material and
stress ratio conditions. The data in
Figure 8.11.3.1 of the DTDH are considered representative of the 7079-T6
aluminum alloy. While it is possible to
utilize the mean trend data given in tabular form, as presented in the figure,
in conjunction with computer codes that employ table look-up schemes, it is
instructive to plot the mean trend data and determine if a simple (power law)
crack growth rate equation, i.e.

_{}

describes the behavior.
Both the mean trend data for the *R*
= 0.05 data set and a power law equation that describes these data are
presented. The power law equation was
determined (graphically evaluated) to be

_{}

Because the stress ratio (*R*)
for the assumed loading (*R* = 0) and
the data set (*R* = 0.05) are
relatively close, no stress ratio correction factor is applied to Equation
9.6.5. If a stress ratio correction
must be applied to a handbook data set, it is suggested that a Walker type
correction be considered. The suggested
Walker correction factor for aluminum alloys is given by

_{}

where *R*_{data set}
and *R*_{desired} are the
stress ratios associated with the Handbook data set and the assumed loading,
respectively, and where *n* is the
power law exponent for the data set (*n*
= 4.09 for the 7079-T6 aluminum data set).
A quick evaluation of the Walker equation with the appropriate constants
shows that the crack growth rate expression given by the power law equation is
approximately 10 percent higher than a corresponding stress ratio corrected
expression, and thus not overly conservative for a first order approximation.

Data Page From the Damage Tolerant Design (Data) Handbook [Skinn, et al., 1994]

Used in Example 9.6.1

Data From Damage Tolerant Design (Data) Handbook
Plotted and

Compared to Graphically Established Power Law Equation

__Stress-Intensity
Factor Establishment__

The stress-intensity factor for the blend-out cracking problem
can be solved without access to exact finite element stress analyses through
the use of some work of Dowling [1979].
For the purpose of providing a methodology for estimating total fatigue
life (crack initiation plus crack propagation lives) of notched structures,
Dowling needed a transition crack length that separated the initiation life
analysis from the crack propagation life analysis. His studies of the conditions controlling small crack growth
behavior led him to the stress-intensity factor evaluation shown below. The point *M* identifies the condition where the short crack stress-intensity
factor (*K*_{s}) is equal to
the long crack solution (*K*_{}). Dowling noted
that these two crack solutions provided reasonably accurate estimates of the
finite element solution in their respective crack length regions. For crack initiation life analysis, Dowling
restricted crack length size measured in smooth fatigue samples to sizes less
than the crack length associated with the point *M* in the figure. This is
because the stress concentration effect dominates in this region.

Short and Long Limiting Cases and Numerical Solution,
for Crack Growing

from a Circular Hole in an Infinite Plate (Newman)

For the purpose of analysis, the engineer could estimate the
stress-intensity factor for the blend-out repair using a Dowling type approach
where for small cracks, a short crack stress-intensity factor would apply, and
for longer cracks, a long crack stress-intensity factor would apply. Thus, for the blend-out repair, the engineer
could describe the stress-intensity factor as:

_{}_{ }

with the short and long crack stress-intensity factors given
by:

_{}

and

_{}

where *k*_{t} is the stress concentration factor associated with
the blend-out shape.

The equations are written in a form slightly different than
those presented in the figure because (1) the geometry of the blend-out is more
in line with an edge crack rather than a central crack (Dowling’s solution) and
(2) the crack length *a* is measured
from the surface of the blend-out (see
the figure for a definition of *a* and *d*).

__ __
Geometry of Blend-Out
with Edge Crack Present

Based on an analysis of these equations, one can see that the
blend-out geometry affects the stress-intensity factor solutions through the
stress concentration factor (*k*_{t})
and blend-out depth (*d*). An estimate of the stress concentration *k*_{t} for a blend-out repair is
made using the solution of an elliptical cut out in a plate. For an ellipse oriented with the major axis
in line with the direction of the stress axis, the stress concentration is
given by [Mushhelishvili, 1954; Peterson, 1974]:

_{}

where

_{}

with *L* and *d* defined as the major and minor radii
of the ellipse. As can be noted from
the figure, *L* and *d* define a segment of a circle that we
are approximating with a semi-ellipse.
Thus, if one has a measure of *L*
and *d* for a blend-out repair, one can
estimate *k*_{t} and the
corresponding short crack stress-intensity factor using the equations above.

__Critical Crack Size, __*a*_{f}

The critical crack size for both
as-manufactured (blueprint) and repaired structure will be based on the Irwin
hypothesis for abrupt failure, i.e., when

_{}

failure occurs. The critical stress-intensity factor is
obtained by estimating the stress-intensity factor range required to achieve a
growth rate of 1000 microinches/cycle. Solving
the power law equation in an inverse manner, i.e., solving

_{}

yields *DK*
= 33.34 ksiÖin As a lower bound to this estimate, one might
choose *K*_{cr} = 30 ksiÖin
for convenience. *K*_{cr} = 30 ksiÖin corresponds to a crack growth rate of 642
microinches/cycle.

The stress-intensity factor for the blueprint structure is
given by

_{}

whereas that for the repaired structure is given by

_{}

(Note that the long crack solution is being used for the
repair). In both equations, *a* is measured from the surface. Solving for *K*_{cr} = 30 ksiÖin and the above
stress-intensity factor solutions yields

*a*_{f}_{ }= 0.325 inch
for the blueprint critical size and

*a*_{f} = (0.325-*d*) inch for the repair critical size.

__Life Estimating__

While the LIFE equation could be used directly for life
estimates of the as-manufactured (blueprint) structure, the stress-intensity
factor analysis requires that the integral equation be broken into two
intervals. For this repair analysis, LIFE
is calculated using

_{}

where the crack size *a*_{M}
is associated with the transition between the short and long crack
stress-intensity factor solutions. This
crack size is obtained by equating the two solutions and solving for *a*_{M}, thus

_{}

in conjunction with the stress intensity equations results in

_{}

Since for a blend-out repair *k*_{t} would be greater than 1.0 and hopefully less than
1.4, *a*_{M} will be greater
than *d*.

__Numerical Details of
Blueprint Life__

For an edge crack problem with the material crack growth rate
response given by a power law expression, i.e. the LIFE equation can be written
as

_{}

When integrated, this equation becomes

_{}

Given the growth rate constants *C* and *n*, the critical
crack size *a*_{f}, the given
stress (*s* = 27 ksi) and the
given initial crack size (*a*_{o}
= 0.050 inch), the crack growth life for the blueprint conditions is determined
to be

*LIFE *=
2910 cycles

of zero-tension loading.

__Numerical Details of
Repair Life__

For the blend-out repair with the material crack growth rate
response given by a power law expression, the *LIFE* equation can be expressed as:

_{}

When integrated, this equation becomes

_{}

Given the growth rate constants *C* (=5.84 x 10^{-10}) and *n
*(=4.09), the critical crack size, the given stress (*s*
= 27 ksi), and the given initial crack size (*a*_{o} = 0.050 inch), one can estimate the LIFE for defined
values of *k*_{t} and *d*.
For example, when *d* and *L* are 0.08 inch and 1.0 inch, *k*_{t} is 1.16, *a*_{f} = 0.245 inch, *a*_{M} = 0.231 inch, and LIFE =
2033 cycles of zero-tension loading.
This is approximately 30 percent lower than that given for the blueprint
life.

__Comparative Analysis
of Shape Effect__

To summarize the analysis for different blend-out shapes, the
LIFE equation was repetitively solved for several different length (*2L*) and depth (*d*) conditions for *a*_{o}
= 0.050 inch. These results are
presented in the following table in the form of life ratios and utilize the
blueprint life obtained above. Focusing
on three crack depths (0.050, 0.100 and 0.150 inches) as representative, one
can immediately note from the table that even for the more gradual blend-out
case, the life is substantially reduced (to approximately 80, 65, and 50
percent, respectively) of the original life estimate.

The life ratios presented in the table show the close
correlation between life and the stress concentration factor. These results only reinforce common sense
since they show that the more gradual the blend-out, the closer to initial life
one achieves.