# Damage and Failure in a Statistical Crack Model
## The Array Code
### Crack Interaction Model  
There is a practical reason why BFRACT and BCM do not represent local interactions in their crack growth models. The central concept of a statistical distribution of the cracks is not sufficient to keep information about a crack's nearest neighbors. However, in the array code, this information is available. So a basic idea of this paper is to introduce a simple crack interaction model into the array code, accumulate statistics on damage and failure, and to formulate modifications to the basic growth laws from the numerical results. I have based a simple crack interaction model on two assumptions:  
1. I will assume that the crack distribution is initially random in space and only evolves correlations through the interaction law.
2. I will assume there is a maximum crack size that occurs infrequently in the computational cell. These few cracks are hotspots where crack growth initiates and which drive rapid mesoscopic crack growth through coalescence.  
The first assumption seems very reasonable. However, the second assumption leads to a picture of high strain rate fracture that is very different than that in SCMs like BFRACT and BCM. My purpose in this paper is to understand the behavior induced by these assumptions. The validity and utility of the consequent model will rest on comparison with experimental data and is not addressed in this paper.  
In the model, I allow for a process zone, a region associated with the crack tip where there are concentrations of high stresses and weakening of the material on the length scale of the microcracks. Initially, the rare hotspot cracks being the largest are most susceptible to growth under [PERSON]'s law. When the process zone of a hotspot crack first touches the process zone of a neighbor, it starts the neighbor's growth. The neighbor crack then starts to grow with the maximum speed until it overlaps with its other-side neighbor so touching off a chain reaction that I term an avalanche.  
The hotspot cracks must only grow to lengths of the order \(dx\) while the avalanche must grow to the length \(DX\). So one expects that the hotspot phase of fracture will be a relatively small fraction of the total time to failure and that most the time will be spent in the avalanche phase. Then, since the large cracks grow with asymptotic speed independent of strain rate, the time to failure may be expected to remain relatively constant as a function of strain rate; in fact, Figure 3 shows the time to failure decreases with increasing strain rate.  
This interaction model has only two new parameters. The process zone is defined to be a fraction \(PZ\) of the crack radius. I will explore the sensitivity of \(PZ\) in Figure 6 with simulations assuming \(PZ=0\), \(1/10\) and \(2/10\). There is relatively small sensitivity since those choices mainly affect time spent in the hotspot phase. The process zone defines the length scales over which two cracks interact, but because it is proportional to the crack size, it does not represent a new length scale.  
The second parameter is the maximum initial crack size, which defines a length scale that is the average distance between hotspots in a row. Tying the fracture process to a physical length scale rather than the computational cell size will reduce possible mesh dependence.