## 4 Mathematical Model
### The Analytical Bounds  
When analyzing numerical data and utilizing fitting functions to model relationships between various measured variables, it is imperative to ensure that the results align with theoretical expectations and bear physical significance. The first- and second-order analytical bounds offer theoretical limitations for the properties of composite materials relative to the volume fraction of each component. These bounds are instrumental for validating fitting functions and enhancing the credibility of experimental data analyses. The first-order bounds, known as the Voigt-Reuss bounds, provide foundational limits. The Voigt bound is derived under the assumption that all phases are elastically isotropic and results in the application of the Voigt mixing rule for the overall elastic property calculation. Conversely, the Reuss bound is based on the same isotropy assumption but employs the Reuss mixing rule. Moving to the second-order bounds, known as the Hashin-Shtrikman bounds, we find the upper bound (HS+) and lower bound (HS-). The HS+ bound is established assuming rigidly connected inclusions using the Hashin-Shtrikman mixing rule and is particularly relevant for fiber-reinforced or ceramic matrix composites. The HS- bound, meanwhile, assumes completely isolated inclusions and also employs the Hashin-Shtrikman mixing rule. To validate the outcomes derived from fitting functions, it is critical to juxtapose the fitting curves with these analytical bounds. If the curves reside within the established first- and second-order bounds, it signifies consistency with theoretical principles and affirms that the fitting function yields physically meaningful parameters. Conversely, curves lying outside these bounds may suggest issues with the fitting function or discrepancies with theoretical assumptions. In Figure 8, we have plotted the bulk modulus of a composite material with a high contrast ratio of 200 as a function of volume fraction. Additionally, we have plotted the same modulus as a function of contrast ratio for a fixed volume fraction of 25%. In both plots, we have included the Voigt-Reuss bounds and Hashin-Shtrikman bounds of order 1 and 2 for comparison. To better visualize the behavior of these bounds, we have applied a logarithmic scale to the y-axis. The fitting curves derived from the graphs clearly demonstrate that the effective properties of the material are accurately captured by the selected functions. The alignment of these curves within the first- and second-order Voigt-Reuss and Hashin-Shtrikman bounds is a testament to the precision of our fitting results. This not only validates the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Volume** & \(a_{0}\) & \(b_{1}\) & \(b_{2}\) & \(c_{0}\) & \(c_{1}\) & \(c_{2}\) \\ \hline
10\% & 1046.42 & 1245.90 & 5.009 & 14.512 & 12.618 & 0.050 \\ \hline
15\% & 1175.60 & 1271.47 & \(-\)2.202 & 16.843 & 11.716 & \(-\)0.020 \\ \hline
20\% & \(-\)311.25 & \(-\)282.73 5 & 2.722 & \(-\)4.378 & \(-\)2.352 & 0.022 \\ \hline
25\% & \(-\)6775.65 & \(-\)4450.62 & 178.847 & \(-\)88.797 & \(-\)32.136 & 1.349 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal coefficients (\(b_{i}\) and \(c_{i}\)) of rational curve fitting for different volume fraction values to predict bulk modulus. fitting process but also reinforces the reliability of the homogenization methods employed for analyzing the properties of composite materials. The implications of these accurate fitting results extend into the realms of design and engineering of composite materials. Precise characterization of material properties enables more accurate predictions of composite material behavior under various load conditions, facilitating optimized design processes. In conclusion, the successful fitting of material property curves using second-degree polynomial and rational functions, coupled with their compliance with the Voigt-Reuss and Hashin-Shtrikman bounds of order 1 and 2, unequivocally validates the efficacy of this novel approach for the analysis of heterogeneous materials.