# Damage and Failure in a Statistical Crack Model
## 6 Size Effect  
All of the numerical experiments of the previous section were performed on a finite volume computational cell with dimensions 100 microns by 100 microns. Do the various scaling relations derived there depend on the actual size or shape of the cell? It is straightforward to test this in the array code by varying \(DX\) and \(DY\) while keeping all other length scales, e.g., \(\bar{c}\), \(dx\), \(dy\), unchanged. In this section, all the simulations assume a constant applied strain rate of \(10^{5}\) s\({}^{-1}\). Figures 7 and 8 show array code results for the dependence of time to failure and of damage at time of failure for two variations of cell size. In one variation, the cell remains square so that \(DX=DY\). In the second variation, only \(DX\) varies while \(DY=100\) um. The point at \(DX=100\) um is common to both sets of data. The graphed results show a systematic increase in the apparent strength of the larger computational cells, as defined by the level of damage that can be supported before failure; that is the case for both variations of increasing the cell size. Further, each of the four curves in those figures can be accurately modeled by a scale-invariant power law, which is shown on the graph. Both figures also show that the thinner cells, i.e., where \(DY<DX\) are stronger than the square cells where \(DX=DY\). Figures 7 and 8 imply that in addition to the size effect, there is a shape effect. However, Table 3 shows that both the time to failure and the damage at time of failure are much more sensitive to varying the width than they are to varying the height. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Cell IJ** & Time to failure (s) & Damage \\ \hline \(100\times 100\) & \(4.60\times 10^{-9}\) & 0.1096 \\ \hline \(200\times 100\) & \(7.53\times 10^{-9}\) & 0.1484 \\ \hline \(100\times 200\) & \(4.25\times 10^{-9}\) & 0.1059 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Doubling the width of the statistical crack model (\(\widehat{SCM}\)) (100x100) leads to large changes in the time to failure and damage at failure. Doubling the height of the \(\widehat{SCM}\) leads to much smaller changes in both quantities. Figure 7: Damage at 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 support more damage at failure. Taller cells support less damage at failure. When considering discrete physical samples, it is well known that bigger samples are weaker than smaller samples. That result is understood from the argument that (1) bigger cracks begin to grow at lower stress levels; and (2) bigger samples are more likely to contain bigger cracks. However, that argument does not apply to the array model where I have limited the radius of the largest crack to \(0.5\,dx\) which remains constant in the simulations. The apparent strength of wider samples lies in the assumed hotspot nature of the fracture process and the details of the interaction model in the array code. It is illustrative to consider how those details lead to the result that wider cells are stronger. 1. In the array code, one calculates an ensemble of fracture processes, each with a different realization of the cracks seeded. 2. Cracks are assumed to grow with the Rayleigh wave speed in the avalanche phase. Thus, the time to failure for any individual fracture process is roughly determined by the largest actual distance (gap) between hotspots found in the critical row, i.e., in the row or rows that have the maximum number of hotspots among all the rows of the computational cell. 3. The largest distance between hotspots in the critical row is limited by the size of the computational cell. 4. A mathematical calculation of the conditional probabilities involved would be a monumental task.