There are many ways to incorporate a probabilistic aspect into a so-called deterministic model. One way to do this is to add random noise to the model. Another way is to associate a random distribution with the model parameters. Yang et. al. (1983) [58] demonstrate this for a hyperbolic crack growth rate function. A quick overview of the method demonstrates the idea.
Given the crack growth equation;
one can add in a random element in two ways. The first is to add a Gaussian random process, Z(X). The crack growth equation then becomes
The task then remains to evaluate the precise form that Z(X) might take.
The next is to consider 
, 
, and 
as random
variables. In this case the equation would take the same form as
Equation 
, where the coefficients are considered to
be random quantities.
