Model

To date, the majority of models have been developed for macrocrack data. In the graph of rates of growth, observe that the rate of growth for the microcracks tends to slow down, and then speed up again. This is inconsistent with what happens for macrocracks. It is widely recognised that this slowing down is caused by the microstructural properties of the material. The crack hits a grain boundary, and either stops altogether, or is greatly slowed, until such time as it overcomes the boundary.

This phenomenon is modelled, starting from the following assumptions:

1. When the crack gets out of the microcrack phase, the growth rate can be modelled by a macrocrack technique;
2. It is possible to model the deviation of microcrack growth from macrocrack models, by using a collection of random variables;
3. The underlying physical cause for the variation is the grain boundary, and the presence of this boundary should be modelled directly.

Previous work has been done on modelling the effect of the grain boundary. In general, the method is to model the variation in rate from that expected by the macrocrack model according to some function of crack specific parameters, . e.g. Miller [31], Plumtree [39].

One of the main features of the data requiring modelling is the fact that there are a large number of cracks in the specimen. It is assumed that it is possible to encapsulate the information about each crack i in terms of parameters , and that the cracks are exchangeable, and a hierarchical model for the family of cracks in the specimen may be used.

• Hierarchical Population Model
• Directed Graph