To quantify
the sensitivity of fatigue crack growth rates and service life to various input parameters
such as stress level and initial crack length.

This
problem focuses on quantifying the sensitivity of certain fatigue performance factors such
as crack growth rate and operational service life to variations and/or uncertainties in
input parameters such as cyclic stress amplitude and initial crack length. Simplified problems are first studied
analytically. They are then followed by a
complex application in the windshield area of a military airplane.

Introduction

Damage tolerance analyses (DTA)
of structures are often challenging and time consuming endeavors because of the types of
data required and the sophistication of the techniques necessary to obtain it. Examples of required data include stress states in
parts so complex that finite element (FE) analyses are necessary to obtain them. The fatigue crack growth behavior of materials may
require lengthy experimental characterization.

The high demands of time and
expertise can easily exceed the resources available.
In such cases, it becomes necessary to prioritize the efforts devoted to obtaining
various pieces of information according to each one’s level of importance to the
final analysis. Critical factors should
receive large resources to enable their accurate determination. On the other hand, quick approximations may prove
satisfactory for factors of secondary importance. But
the question arises, “Which factors are critical, and which are secondary?” This example problem aims to address that
question.

Mathematical Background: Sensitivity
Analysis

Before one can determine what
factors are and are not important in a DTA, one must choose a method of quantifying the
qualitative term, *important*. Here, we
have chosen to use a sensitivity analysis approach. It
relates the percentage change in a system’s input to the resulting percentage change
in the system’s output, the ratio of the two being the sensitivity parameter. As an example, consider the following equation

where x is the input, y is the
output, and A and n are constants. The
sensitivity of y with respect to x is therefore the ratio of the percentage change in y
resulting from a given percentage change in x. The
percentage change in the output y would be expressed as

and likewise for the input
variable x. Defining the sensitivity
parameter, S_{y/x}, as the ratio of the percentage changes gives

and taking the limit as _{} gives the analytical definition of S_{y/x}.

Applying Eq. (4) to Eq. (1)
gives the result

which states that the percentage
change in y is simply n times the percentage change in x regardless of the values of A and
x. So if n=3 and x is increased by 10%, then
y would increase 30%. This is a very useful
result because of its simplicity. It will be
used extensively in the following applications of damage tolerance analysis. Of course many equations exist that are not in the
form of Eq. (1). In these cases, Eq. (4) must be applied on an individual basis.

Applications to Damage Tolerance Analysis

__Stress Intensity Factor__

One of the most fundamental
steps of any DTA is calculation of the stress intensity factor, K, using Eq. (6)

where b is the geometry factor, s is stress, and a is crack length. The sensitivity of K to the various parameters is
then

indicating that accurate values
of b and s are equally important to the calculation of K, and that the
sensitivity to crack length is less.

__Crack Growth Rates__

The situation becomes more
interesting when crack growth rates are analyzed. A
Paris Law dependence on DK will be assumed as follows

where N is the number of cycles,
and C and n are Paris Law constants. Note
that C and n are material properties with associated measurement uncertainties. Since the sensitivity of crack growth rate to the b-factor and stress is equal to n in both cases, it is worth
reviewing typical values. Figure MERC-3.1 shows crack growth data for Al 7075-T6. The Paris Law forms a straight line on the
logarithmic graph with n equal to the slope and C equal to da/dN at DK=1. It is seen
that in this case, n=3.6. (3 £ n £ for most materials) This
value has critical implications on the accuracy of crack growth predictions. It means that a 10% error in the estimate of the b-factor results in a 36% error in the prediction of da/dN. The same sensitivity applies to the stress as
well. It is this high sensitivity of da/dN
to DK, reflected in the value of n, which presents a major
challenge to the accurate prediction of crack growth rates.

What of the sensitivities to the
Paris Law constants? The sensitivity to C is
unity since that is its exponent in Eq. (8). It
is necessary to apply Eq. (4) to Eq. (8) to determine the sensitivity to the exponent, n. Doing so gives

Since ln(DK) is usually greater than one in engineering analyses, it
is clear that the sensitivity of predicted crack growth rates to the accurate
determination of the slope of the da/dN–DK data in Figure MERC-3.1 is
even greater than to b-factors and stresses.
In summary, the results are as follows

*Service Life – Cycles to
Failure*

The quantity of primary interest
in a DTA is the service life of a component, measured in cycles to failure, N_{Life}. An analytical expression for N_{Life} can
be obtained if one neglects crack retardation and assumes that the b-factor and stress range are both constant throughout a
component’s life. Integrating Eq. (8)
and solving for N_{Life} gives

where
a_{o} is initial crack length, and a_{f} is final crack length at which
point failure takes place. From Eq. (11), it
is seen that the sensitivity of N_{Life} to certain parameters is simply negative
of the crack growth rate’s sensitivity to them.

So a 10% __increase__ in the b-factor
or stress would produce a 36% __decrease__ in service life assuming n=3.6. Eq. (4) must be applied to Eq. (11) to determine
the sensitivity of N_{Life} to initial and final crack lengths. Doing so gives

and

Eqs.(13) and (14) are plotted
versus a_{o}/a_{f} in Figure MERC-3.2 for
three values of n. The sensitivity to initial
crack length depends on both n and a_{f}, but is approximately –1 for common
values of these factors. So a 10% increase
in initial crack length results in a 10% reduction in predicted fatigue life. On the other hand, predicted life is relatively
insensitive to final crack length, showing only ~10% sensitivity. So a 10% increase in a_{f} produces only
~1% increase in predicted life. Since a_{f}
is usually chosen to equal the critical crack length, a_{crit}, this demonstrates
that variations in a_{crit} have a small impact on N_{Life} estimates.

*Variable b-Factors
– Numerical Example*

The final example demonstrates that fatigue life sensitivity to a b-factor can depend on its relative value, with lower
values being more critical than larger values. This
analysis will be performed numerically rather than analytically because of the
complexities of integrating non-constant b-factors. The horizontal leg of an aircraft longeron will be
chosen for this example. A finite element
model of it is shown in Figure MERC-3.3. The part is subjected to tension, bending, and
fastener forces. The crack begins at the
fastener hole and proceeds to the part edge as shown in the Figure. The b-factor
is plotted in Figure MERC-3.4.

Figure MERC-3.4 shows that the b-factor is approximately three at short crack lengths
because of the stress concentration at the fastener hole.
The b-factor then decreases to
approximately one with increasing crack length and then increases again as the crack
approaches the free surface. The predicted
life using the b-factor in Figure MERC-3.4 will be compared to two others having the
following modifications.

Case 1. Large values of the b-factor increased.
b values ³3 were
increased by 10%, b values £1 were not
changed, intermediate values were scaled proportionately, i.e., b values = 2 were increased by 5%.

Case 2. Small values of the b-factor increased.
b values £1 were
increased by 10%, b values ³3 were not
changed, intermediate values were scaled proportionately, i.e., b values = 2 were increased by 5%.

AFGROW was used to predict the
fatigue life of the part using the three different b-factor
cases. Other inputs include: (1) Ds=10ksi with R=0, (2) a_{0}=0.05 in. and a_{f}=1.25
in., (3) material da/dN–DK data taken from Figure MERC-3.1. Results
are shown in Figure MERC-3.5.

The 10% increase in small b-values
produced a 25% decrease in predicted fatigue life, yielding a sensitivity of -2.5. The sensitivity to the increase in large b-values is -0.4. This
demonstrates that fatigue life can be more sensitive to variations in small b-values than larger ones. It can therefore be more important to accurately
determine small b-factor values than larger ones. This is a potentially counter-intuitive result
since most analyses focus on large parameter values rather than small ones. This situation exists because cracks spend the
majority of their life growing slowly at lengths with corresponding small b-factors.

Summary

A sensitivity analysis of factors affecting fatigue life
predictions has been presented. It was
demonstrated that certain factors have a large impact on predicted life, while others do
not. Important factors include stress and b-factors. In
most cases, a 10% increase in either one leads to ~35% decrease in predicted life. This high sensitivity is directly related the high
sensitivity of da/dN to DK, which is a material
property. On the other hand, factors having a
relatively small impact on predicted life are critical crack length and large b-values that occur when a crack approaches a free
surface.