# Damage and Failure in a Statistical Crack Model
## 4 The Fracture Process in Words  
In all the results of this paper, I will simulate the fracture of a two-dimensional solid with horizontally bedded cracks undergoing normal tension at high strain rates. The fracture process proceeds in three phases, which can be termed elastic, hotspots, and avalanche. In the initial elastic phase, the tension ramps up but is not sufficient to cause the largest cracks to grow. Recall that the size of the largest cracks is determined by the cutoff I have applied in initializing the crack distribution. When the critical crack size of Equation (1) reduces to that of the largest cracks, the second phase begins. Because those largest cracks represent the extreme of the crack distribution, they are relatively few and can be termed hotspots. With the length scale parameters I have chosen for the simulations, most rows contain no hotspots. A few rows contain one or possibly more. In the hotspot phase, the hotspots grow in isolation under [PERSON]'s law until the first hotspots intersect their nearest horizontal neighbors--more specifically, when the process zones of the hotspot and nearest neighbor first touch. This initiates the third or avalanche phase in which hotspot(s) continue to grow at the asymptotic speed by the process of coalescence. Ultimately, a crack in one row will span the entire cell, which is the array (mesoscale) definition of failure. From the point of view of the computational cell, the important quantities are threshold stress for the fracture process to begin, total time to failure, and damage as a function of time. The threshold stress for failure depends directly on our choice of the cutoff crack size. The total time to failure consists of the time spent in the hotspot phase plus the time in the avalanche phase. Since the crack growth in the hotspot phase is an order of the subcell size \(dx\) while the crack growth in the avalanche phase is an order of the total cell \(DX\), the fracture process in the cell will spend most of its time in the avalanche phase for high strain rate processes. I will qualitatively define high- strain rate processes as those in which most time is spent in the avalanche phase. For lower strain rates, the hotspot cracks will take a longer time to accelerate to the asymptotic growth velocity and to initiate the avalanche process. In the next section, I will consider strain rates of \(10^{4}\,\mathrm{s}^{-1}\), \(10^{5}\,\mathrm{s}^{-1}\), and \(10^{6}\,\mathrm{s}^{-1}\), all of which qualify as high strain rates. In the avalanche phase, the avalanche moves with the maximum speed, i.e., the Rayleigh wave speed, cf. Equation (4), which depends on the material properties, but is independent of the strain rate. On the other hand, the total damage in the computational cell is dominated by the growth of cracks in all the rows, of which the damage in the particular row that fails is only a small part. One should expect that the total damage at the time of failure will depend on strain rate. A higher strain rate implies faster growth of the stress field and faster decrease of the critical crack size from Equation (1). In other words, a higher strain rate implies more cracks will grow in the avalanche phase and cause a larger value of damage at time of failure. Note that the time spent in the avalanche phase depends in a discontinuous fashion on the number of hotspots in the row that causes failure, which should be understood in terms of extreme statistics. One might expect that the time to failure for a row with two hotspots would be, roughly speaking, only half that of a row with one hotspot. Then the growth of the totality of subcells and the resulting damage in the case of two hotspots would be much less than in the case of one hotspot. Will averaging over an ensemble of failure processes with these bimodal, or perhaps multimodal statistics lead to smooth results for the damage as a function of time, or for final damage at the time of failure? If so, how many processes must one consider in the ensemble?