## 5 Results and Analysis
### Statistical Analysis of Strain Rate-Strength Relationships  
A comprehensive statistical analysis was conducted based on the spall strength and corresponding local strain rate determined for each of the 18 experiments. The complete dataset, including calculated dynamic increase factors (DIF), is summarized in Table 5. The objectives of the analysis were to characterize the material's strain rate sensitivity and to quantitatively assess the influence of the reinforcement configuration. Figure 20 plots the spall strength as a function of strain rate (on a logarithmic scale) for all three specimen types. A clear positive correlation is evident, and the data for all  
Figure 19: Example of local axial strain history (\(\epsilon(t)\)) and calculated strain rate history (\(\epsilon(t)\)) at the primary spall crack location for specimen Z001-29. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test No.** & **Specimen Type** & **Test ID** & **Strain Rate (s\({}^{-1}\))** & **Spall Strength (Mpa)** & **DIF** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 5: Comprehensive experimental results from spalling tests, including local strain rates from DIC and calculated spall strengths and DIFs. configurations appear to follow a single common trend. A logarithmic regression on the combined dataset yields the following empirical relationship:  
\[\sigma_{spall}=4.1+4.7\log(\hat{\epsilon})\quad\text{(MPa)} \tag{7}\]  
It should be noted that this empirical relationship is validated for the range of local strain rates investigated in this study, approximately from 3 s\({}^{-1}\) to 55 s\({}^{-1}\). To rigorously test whether this relationship is statistically consistent across the three material groups, an analysis of covariance (ANCOVA) was performed. This analysis evaluates the influence of the specimen type while controlling for the effect of the strain rate. The results of the ANCOVA are summarized in Table 6.  
The ANCOVA results provide two key insights. First, the interaction term, Specimen Type:Log Strain Rate, yields a \(p\)-value of 0.2182. As this value is significantly greater than the significance level of \(\alpha=0.05\), it indicates that the slopes of the regression lines for the three groups are not statistically different. This means the rate-dependency of spall strength is consistent across all tested configurations. It is worth noting, however, that with an F-statistic of 1.73, the possibility of a very small interaction effect cannot be entirely ruled out by the current dataset; a larger dataset would be required to definitively confirm the complete absence of any influence of reinforcement type on strain rate sensitivity. Second, the main effect for Specimen Type has a \(p\)-value of 0.2701, which confirms that there \begin{table}
\begin{tabular}{l r r r r} \hline \hline
**Source** & **Sum Sq.** & **d.f.** & **Mean Sq.** & **F** & **Prob \textgreater{} F** \\ \hline Specimen Type & 4.0661 & 2 & 2.0331 & 1.46 & 0.2701 \\ Log Strain Rate & 20.3741 & 1 & 20.3741 & 14.66 & 0.0024 \\ Specimen Type:Log Strain Rate & 4.8170 & 2 & 2.4085 & 1.73 & 0.2182 \\ Error & 16.6802 & 12 & 1.3900 & & \\ Total & 59.4312 & 17 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the analysis of covariance (ANCOVA) for spall strength. Figure 20: Spall strength as a function of strain rate for all specimen types. The solid line represents the logarithmic fit to the combined dataset, with its equation and R-squared value shown. is no significant difference in the overall magnitude of spall strength between the groups after accounting for the effect of strain rate. Collectively, these statistical results provide robust evidence for a principal finding of this study: the relationship between dynamic tensile strength and strain rate is governed primarily by the properties of the concrete matrix and is not significantly affected by the presence or type of reinforcement under the tested conditions. It is also worth noting a subtle trend within the experimental data.