# Acoustic Emission-Based Method for IFSS Characterization in Single-Fiber Fragmentation Tests
## 2 Materials and Methods
### IFSS Calculation
#### 2.7.2 AE Method: Fragment Average Length  
The calculation used to obtain the average fragment length using the AE method is based on the probability density function derived from the optical measurements described in the previous subsection. This probability function determines which statistical distribution best fits the measured data. To this end, Q-Q (quantile-quantile) plots were employed to visually compare the empirical data distribution against various theoretical models. Specifically, the fragment length data were compared to Weibull, Lognormal, Exponential, Gamma, and Gaussian distributions. The Q-Q plots help assess the goodness-of-fit by displaying how closely the data quantiles align with those of a given theoretical distribution. Deviations from the reference line indicate discrepancies, allowing for a qualitative assessment prior to any statistical testing. A Gamma distribution was selected in this case, as the observed fragment length data exhibit a clear positive skew. Additionally, the Kolmogorov-Smirnov (K-S) test was used to validate each distribution's fit quantitatively. The Gamma distribution yielded the lowest K-S statistic among the tested models, confirming its suitability for representing the probability density of the fragment lengths. The probability density function used is shown in Equation (5):  
\[f(x_{i};k,\theta)=\frac{x^{k-1}e^{-\frac{x_{i}}{\theta}}}{\theta^{k}\Gamma(k)}, \tag{5}\]where \(k\) is the shape parameter, \(\theta\) is the scale parameter, \(x_{i}\) is the value at which the gamma distribution will be evaluated, and \(\Gamma\) is the gamma function. To fit the gamma distribution to the values of the fragment lengths \(X\), the negative log-likelihood function of the gamma distribution is minimized, see Equation (6),  
\[\updownarrow(k,\theta)=(k-1)\sum_{i=1}^{m}\ln(X_{i})-\frac{1}{\theta}\sum_{i= 1}^{m}X_{i}-m\cdot k\cdot ln(\theta)-m\cdot\ln(\Gamma(k)), \tag{6}\]  
where \(\updownarrow(k,\theta)\) is the negative of the log-likelihood of the gamma distribution, \(m\) is the number of fragments in a sample, \(X_{i}\) is the i-th fragment length in a sample. The result of minimizing Equation (6) gives the parameters \(k\) and \(\theta\) of the gamma distribution that best fits the fragment length data measured with the microscope. The obtained gamma distribution will be used to correct the length of the fragments detected with the AE method. In the AE method, bursts are detected, which represents the release of a sufficiently large amount of energy to be detected by the sensor. Therefore, it is assumed that the occurrence of one of these events is due to the formation of a crack in the fiber, i.e., when two AE events occur, it implies that the fiber was broken into a single fragment within the calibrated zone. Consequently, the number of fragments as a function of AE events is shown in Equation (7):  
\[n_{AE,frag}=n_{AE,burst}-1, \tag{7}\]  
where \(n_{AE,frag}\) is the number of fragments counted using the AE method, and \(n_{AE,burst}\) is the number of AE-bursts detected in the AE signal. The next step is to correct the previously obtained gamma distribution. This is based on the number of fragments detected using the AE method \(n_{AE,frag}\), since the fragment length is directly dependent on their quantity. As the number of fragments increases, the average length decreases, causing the distribution to shift to the left. On the other hand, if the number of fragments decreases, the distribution shifts to the right. To correct the gamma distribution according to the number of fragments, a set of data points is generated from the previously obtained distribution, equal to the number of fragments \(n_{AE,frag}\).