## 4 Mathematical Model
Figure 5 illustrates the curve-fitting process employing both the polynomial and exponential functions and showcases the versatility and effectiveness of these fitting techniques. Figure 4: Variation in bulk modulus of heterogeneous material as a function of volume fraction with a contrast value equal to 200. Upon careful analysis of the results derived from the curve-fitting process, we observed that both polynomial and exponential fitting methods yield similar outcomes. However,  
Figure 5: Prediction of bulk, shear, and thermal conductivity as a function of volume fraction with varying contrast using polynomial and exponential curve fitting. a critical observation is that the polynomial fitting method aligns more closely with the precise values. This alignment with exact values led us to choose the polynomial fitting method for our analysis. The higher degree of accuracy and precision offered by the polynomial fitting method is instrumental for making well-informed decisions. Table 2 presents the coefficients \(a_{0}\), \(a_{1}\), and \(a_{2}\) for the polynomial fitting function corresponding to different contrast values. These coefficients' values change in relation to the contrast, mirroring the variations in material behavior as the contrast increases. An in-depth examination of these coefficients provides valuable insight into the material's characteristics and how it responds to different contrasts. Utilizing these coefficients in the fitting function enables us to predict the material's behavior more accurately by substituting the input with a specific volume fraction. Upon examining the curve and the parameters outlined in the table, a noteworthy observation is made. The formulated equation effectively satisfies the imposed boundary conditions. Notably, when the volume fraction \(V_{f}\) equals zero, the behavior of the material corresponds with that of the matrix phase. This alignment is clearly reflected in the value of \(a_{0}\), which closely approximates the known matrix bulk modulus \(K_{m}\) at 83.33. This congruence demonstrates the equation's accuracy and reliability at depicting the material's anticipated response. It also affirms the equation's capability to precisely represent scenarios for which the dispersed phase's contribution is minimal or null, reinforcing the model's validity for diverse material conditions. Modeling of Effective Elastic and Thermal Properties of Heterogeneous Material as a Function of the Contrast between Both Phases  
In the second phase of our study, we aim to model the various moduli as functions of contrast by maintaining a constant volumetric fraction. The analysis of the corresponding curve reveals a hyperbolic shape with a horizontal asymptote as the contrast value approaches infinity. This pattern aligns with the anticipated behavior of materials: beyond a certain contrast threshold, the material's behavior becomes increasingly dominated by the inclusion phase. Essentially, the greater the disparity in properties between the matrix and the inclusion, the more pronounced the influence of the inclusion on the overall properties of the material. This effect is distinctly illustrated by the curve's shape, which offers insights into the interaction between various parameters and their impact on the material's properties. Figure 6 visually demonstrates how the bulk modulus varies in response to changes in contrast. \begin{table}
\begin{tabular}{c c c c} \hline
**Contrast** & \(\mathbf{a_{0}}\) & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) \\ \hline
10 & 83.354 & 109.73 & 167.637 \\ \hline
20 & 83.369 & 117.306 & 216.81 \\ \hline
50 & 83.383 & 121.325 & 256. One commonly used mathematical function that can accurately model nonlinear behavior while remaining relatively simple to manipulate and interpret is rational functions. Rational functions are ratios of two polynomials, which gives them great flexibility to adapt to experimental data that may exhibit complex variations.