In the construction of a model for this data, it is assumed that the cracks are exchangeable, that is, that the length of one crack is conditionally independent on the lengths of the other cracks, given some set of parameters.
It is possible to focus on the growth of a single crack, and to explicitly model that. The model, is that the expected crack length is determined from a rate of growth given by;
The observed length A(N) is normally distributed with mean a(N). Two types of
variance were considered, namely constant variance 
and multiplicative
variance 
. The constant variance model was computationally easier, but
after consultation with the engineers it was decided to use multiplicative variance.
The model proposed, explicitly takes into account the grain boundary. In the limit, as the length of the crack becomes much larger than D, the model tends toward Paris-Erdogan.
The effect of 
on rate of growth is shown for different
values of in Figure 
. Large values of
yield a slower rate for a given length. Thus in the plot, 
is toward the top of the graph and 
is toward the
bottom. The plotted values are 
. The
actual values on the x and y axes of the plots give an
indication of scale only, and do not refer to any real data.
Figure: Effect of
on Rate of Growth.
Figure: Effect of m on
Rate of Growth.
In a similar fashion to , the effect of m on rate of
growth is shown in Figure . The other parameters are
held fixed. Thus in the plot, m = 10 is toward the top of the
graph and m=0.01 is toward the bottom. The plotted values are m
= 10,1,0.1,0.01.
Thus, dictates the depth of the trough in growth rate,
whereas m how wide it is. It should be noted that since depth is
fixed, widening the trough means lowering the rates on either
side.
To summarise, the rate of growth is modelled by an adaptation of
Paris-Erdogan, such that the observed length after N cycles for
crack i within the specimen, 
, is distributed with mean
value 
, where 
is a collection of
crack specific parameters, 
.