## 4 Results and Discussion
### Discussion
For example, in the case of the plot shown in Figure 7d, using a program that fits the parameters of the exponential curve to the points found during simulation, \(\tau=21.63\) s is determined. The coefficient of determination, or \(R^{2}\), which provides information about a model's goodness of fit, equals 0.9999, indicating a very good curve fit. Due to the system's non-stationarity, the description of heterogeneous textiles is much more complicated. The results of many fitting attempts lead to the following conclusions. Two steps must be performed to determine the fitting curve. The first determines the fitting of the fast-change part of the characteristic, followed by the slow-change part. Such an approach yields a value of \(R^{2}>0.99\). The equation  
\[\frac{E}{E_{max}}=A_{0}+A_{1}\times\mathrm{e}^{-t/\tau_{1}}+A_{2}\times \mathrm{e}^{-t/\tau_{2}}+A_{3}\times\mathrm{e}^{-t/\tau_{3}} \tag{4}\]  
approximates the plot, where \(\tau_{1}\) and \(\tau_{2}\) are related to the fast-change part of the plot and \(\tau_{3}\) is related to the slow-change part. To illustrate the approach, 1.8 mm thick nonwoven fabric with \(\sigma_{1}=3\times 10^{-11}\) S/m and \(\sigma_{2}=10,000\) S/m is considered in detail (see Figure 9a at \(\sigma_{1}=3\times 10^{-11}\) S/m). Figure 22 shows the plot of the fast-change part of the plot. Figure 21: Charge decay plot for nonwoven fabric with a thickness of 1.80 mm at \(\sigma_{1}=1\times 10^{-9}\) S/m and \(\sigma_{2}=17,500\) S/m (result of simulation); (**a**) the full scale; (**b**) the zoomed initial part of the plot. As a result of fitting this part of the plot, \(A_{1}=0.351\), \(\tau_{1}=9.95\) ps, \(A_{2}=0.233\), and \(\tau_{2}=52.3\) ps (\(R^{2}=1\)) are determined. Moreover, a constant value (0.4156) is found. In the second stage, the constant is replaced by the description of the slow-change part of the charge decay plot. The parameters for fitting the second part of the plot are \(A_{3}=0.384\), \(\tau_{2}=2.34\) s, and a constant of 0.0042 (\(R^{2}=0.9980\)). After elementary manipulations, the final approximation is found as follows:  
\[\frac{E}{E_{max}}=0.032+0.351\times\mathrm{e}^{-t/9.95\times 10^{-12}}+0.233 \times\mathrm{e}^{-t/52.3\times 10^{-12}}+0.384\times\mathrm{e}^{-t/2.34}. \tag{5}\]  
Figure 23 shows the original plot obtained by simulation (black dots) and the plot corresponding to the obtained approximation (solid blue line). Table 2 presents the determined values of the \(\tau_{1}\), \(\tau_{2}\), and \(\tau_{3}\) parameters of approximation (4) for \(\sigma_{1}=3\times 10^{-11}\) S/m and different values of \(\sigma_{2}\). In Table 2, the results demonstrate an almost linear relationship of \(\tau_{1}\) and \(\tau_{2}\) with \(\sigma_{2}\). A tenfold increase in \(\sigma_{2}\) leads to a proportional decrease in \(\tau_{1}\) and \(\tau_{2}\). However, \(\tau_{3}\) shows a slight decrease with an increase in \(\sigma_{2}\). When comparing the value of \(\tau_{3}\) with the time constant (\(\tau\)) of a homogeneous material with the same \(\sigma_{1}\), it becomes apparent that using a 25 mm \(\times\) 25 mm grid of conductive fibers causes the charge to dissipate approximately 8.5 times faster than in a homogeneous textile. The proposed modeling approach makes it possible to observe the effects of different damage to the specimen structure on the charge decay plot and shielding characteristics. To illustrate these possibilities, damage in the shape of a 40 mm long and 10 mm wide tear is considered. The damage leads to a break in three conductive threads (Figure 24a). Figure 24b,c show slices in the \(y-z\) plane, demonstrating the effect of this damage on the values of the electric field norm. Figure 25 shows the charge decay plots for the textile before and after the damage. A significant shift between the plots can be observed.