In order to make statements about the probability that
no crack exceeds a particular threshold length, 
, it is
necessary to examine how cracks grow
within the material.
The area of primary research, was in the area of modelling crack growth. To this end various models were examined, leading to closer scrutiny of the Paris-Erdogan model. An adaptation of Paris-Erdogan is developed. Consideration is given to a random variable parameterisation of the model, and a hierarchical population model is developed for the family of microcracks within specimens [57].
As was seen above, models for long crack growth are plentiful in the literature. It was decided to examine other aspects of the problem rather than choosing to compare variations on other standard models in the literature.