## 3 Results
### Statistical Model  
Due to the availability of only three replications per width-to-diameter ratio, the specimens were subdivided in two groups based on their width, which was considered as the scale parameter's dominating size effect. This assumption was corroborated by the results of the model, as will be noted later. Two separate empirical and model-estimated statistical distributions were, therefore, constructed--one for the 24 mm wide specimens and one for the 36 mm wide ones. A probability-probability plot can be used as a tool to compare the two cumulative distribution functions and assess the goodness of fit of the model to the empirical data. Figure 3 shows the probability-probability plot of the estimated CDF (\(F_{mod}\)) versus the experimental empirical CDF (\(F_{exp}\)), after having optimized the four parameters of the model. \begin{table}
\begin{tabular}{c c c c c} \hline
**Specimen** & **Pexp,max [KN]** & **dexp [mm]** & **dFEM [mm]** & **d err\%** \\ \hline C21\_1 & 30.929 & \(-\)1.631 & \(-\)1.616 & 0.96\% \\ C21\_2 & 30.976 & \(-\)1.663 & \(-\)1.622 & 2.49\% \\ C21\_3 & 30.778 & \(-\)1.656 & \(-\)1.608 & 2.92\% \\ C22\_1 & 26.811 & \(-\)1.528 & \(-\)1.430 & 6.44\% \\ C22\_2 & 25.768 & \(-\)1.427 & \(-\)1.374 & 3.71\% \\ C22\_3 & 25.768 & \(-\)1.461 & \(-\)1.372 & 6.04\% \\ C23\_1 & 20.957 & \(-\)1.385 & \(-\)1.209 & 12.69\% \\ C23\_2 & 20.749 & \(-\)1.276 & \(-\)1.200 & 5.96\% \\ C23\_3 & 22.075 & \(-\)1.366 & \(-\)1.277 & 6.47\% \\ C31\_1 & 48.570 & \(-\)1.654 & \(-\)1.701 & \(-\)2.85\% \\ C31\_2 & 50.791 & \(-\)1.740 & \(-\)1.778 & \(-\)2.17\% \\ C31\_3 & 47.376 & \(-\)1.593 & \(-\)1.657 & \(-\)4.01\% \\ C32\_1 & 39.633 & \(-\)1.424 & \(-\)1.428 & \(-\)0.25\% \\ C32\_2 & 37.551 & \(-\)1.357 & \(-\)1.353 & 0.35\% \\ C32\_3 & 41.111 & \(-\)1.452 & \(-\)1.481 & \(-\)2.00\% \\ C33\_1 & 32.262 & \(-\)1.380 & \(-\)1.303 & 5.54\% \\ C33\_2 & 31.440 & \(-\)1.302 & \(-\)1.270 & 2.41\% \\ C33\_3 & 31.150 & \(-\)1.286 & \(-\)1.270 & 1.25\% \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental failure load, experimental failure displacement, FEM failure displacement, and the percentage error between them for each specimen. \begin{table}
\begin{tabular}{c c c c c} \hline
**Property** & **E1 [GPa]** & **E2 [GPa]** & **G12 [GPa]** & \(\mathbf{\upsilon_{12}}\) \\ \hline Value & 102.388 & 15.519 & 1.891 & 0.050 \\ \hline \end{tabular}
\end{table} 
Table 3: Material properties that were identified through optimization and then used in finite element models. The coefficient of determination of the statistical predictions with the experimental data is \(R^{2}=94\%\), as reported in Figure 3, demonstrating that the calibrated model accurately describes the size effect governing the failure of the notched specimens. All of the data points lie close to the bisector; however, the model tends to slightly overestimate the probability for the narrower specimens and underestimate it for the wider ones. At the cost of additional computational cost and complication, an ever better model fit could potentially be obtained, introducing additional specimen volume-dependent parameters, e.g., the shape parameter \(\beta\). The model parameters identified with the optimization procedure are listed in the following two tables: the four equivalent stress parameters are listed in Table 4 and the three statistical model parameters are listed in Table 5.  
The four \(\varphi\) parameters in Table 4 govern how the stresses are combined in the equivalent stress formulation. It can be observed how \(\varphi_{1}\), the coefficient to the fiber-direction stresses \(\sigma_{1}\), is an order of magnitude smaller than the other terms. This mirrors the ratio between the longitudinal and transverse stresses in the material and indicates that the relative importance of the two in the failure mechanism is similar. The calibrated model \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\varphi_{1}\) & \(\varphi_{2}\) & \(\varphi_{3}\) & \(\varphi_{4}\) \\ \hline
0.0613 & 0.5823 & 0.3033 & 1.1198 \\ \hline \hline \end{tabular}
\end{table}