# Damage and Failure in a Statistical Crack Model
## 7 Scaling the Standard Model  
The primary task of the SCM is to predict damage and failure as a function of the loading. The size effect described in the previous section would appear to complicate the prediction of failure as it  
Figure 8: Time of failure is plotted as a function of the cell width \(DX\). The strain rate for all points is \(\dot{\epsilon}=1\times 10^{5}\) s\({}^{-1}\), the ensemble size is 100 and the process zones is 10% of the crack radius. In the upper curve, \(DY=100\)\(\,\mathrm{\SIUnitSymbolMicro m}\) for all points. In the lower curve, \(DY=DX\). On both curves, wider cells take longer times to fail. Taller cells take less time.  
might seem that a new \"standard\" model is required for every size and shape (aspect ratio) of a computational cell. In this section, I will consider variations of the width of \(\widehat{SCM}\) and ignore the smaller effects of not having square cells, cf. Table 3.  
Figure 9 shows the accumulation of damage as a function of time for 4 square cells of dimensions \(100\times 100\), \(200\times 200\), \(300\times 300\) and \(400\times 400\) (all units in microns). The strain rate is \(10^{5}\,\mathrm{s}^{-1}\). The circles represent the individual times of failure of the four cells, with larger cells showing more damage at the time of failure. The notable feature is that the damage growth in all cells is described by the same \"backbone\" curve. That is, for fracture at \(10^{5}\,\mathrm{s}^{-1}\), the increase of damage in time follows the quadratic curve in Figure 9 until time of failure for all square cells, \(DX=DY\). A plot (not shown) for all cells with \(DY=100\)\(\upmu\)m is qualitatively similar, i.e., damage increases for all cells on a slightly different backbone quadratic curve until time of failure.  
Perhaps the most surprising result can be seen in Figure 10, where data of damage vs. time to failure for both shapes of cells are combined in a single plot. The resulting curve is a single straight line. Further, additional data for a wide variety of cell sizes and shapes also lie on the same straight line. To summarize, there is a linear relationship between damage and time to failure for all cells fracturing at a strain rate of \(10^{5}\,\mathrm{s}^{-1}\). Similar linear relationships will exist for all high strain rate processes and the slope of the line will, for dimensional reasons, depend linearly on the strain rate.  
To implement the \(\widehat{SCM}\) for cells of arbitrary width, first, recall that the relevant curves in Figures 7-10 are all fit by simple polynomials whose coefficients are readily scaled for any strain rate. Now suppose one is given the strain rate and the width of the cell. From the curve in Figure 4, one finds the damage at time of failure for a cell with width \(100\,\upmu\)m. Then, from the (lower) curve in Figure 7, scaled to the given strain rate, one finds the damage at time of failure for a cell of the given width. The time to failure of the cell is then found from the (scaled) curve in Figure 10. Finally, the damage as a function of time is given in the (scaled) curve of Figure 9. In practice, all these steps can be composed.  
Figure 9: The ensemble-averaged damage is plotted as a function of time for an imposed strain rate \(\dot{e}=1\times 10^{5}\)\(\mathrm{s}^{-1}\), an ensemble size of 100 and the process zone is 10% of crack radius. The four dots represent damage at time of failure for \(DX=100,200,300,400\,\upmu\)m and \(DY=DX\). The damage backbone is the same for all cell widths, but the time to failure increases with the cell width.