## 4 Mathematical Model
### Generalized Model  
Integrating both the polynomial and rational functions into a unified model that incorporates the volume fraction and contrast as inputs significantly enhances our ability to model the intricate behavior of composite materials. The polynomial function is adept at tracing the general trend of material behavior as the volume fraction varies, whereas the rational function is more effective at capturing the changes to behavior as the contrast between the constituents of the composite material increases.  
Moreover, this combined approach allows for a more thorough consideration of the interplay between volume fraction and contrast. For instance, an increase in the volume fraction could intensify the interaction between the composite's constituents, potentially leading to a nonlinear relationship between behavior and contrast.  
By employing both polynomial and rational functions, we can adeptly represent both the overarching trends and the subtle, complex interactions between these two key variables. This dual-function approach culminates in a more precise and comprehensive model of the composite material's behavior. The final integrated function, which embodies this dual approach, is formulated as follows:  
\[f(x,y)=(ax^{2}+bx+c)(\frac{dy^{2}+ey+f}{gy^{2}+hy+k}) \tag{39}\]  
Figure 8: Comparison of Voigt–Reuss and Hashin–Shtrikman bounds with material property curves obtained by finite element data fitting as function of (**a**) volume fraction with a fixed contrast and (**b**) contrast with a fixed volume fraction.  
where:  
\(x\), \(y\) respectively represent the volume fraction and the contrast;  
\(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\), and \(k\) are adjustable coefficients.  
Figure 9 shows a 3D curve constructed using the function that represents the product of the polynomial function and the rational function in two variables: volume fraction and contrast. This function was developed to describe the behavior of composite materials as a function of these two parameters.  
The 3D curve generated in our study vividly illustrates the evolution of material properties in response to variations to the volume fraction and contrast. On this curve, the axes represent the two independent variables--volume fraction and contrast--while the surface of the curve maps the dependent variable, which in this case is the material properties.  
Significantly, the two red lines on the curve demarcate the boundary conditions where the volume fraction \(V_{f}\) equals zero and the contrast \(C\) is one. These lines mark the extreme limits of our study and highlight scenarios where the dispersed phase's volume fraction is nil and the contrast is at its minimum, which perfectly aligns with the matrix phase.  
Close examination of the 3D curve reveals regions where the material properties reach their highest and lowest points as functions of volume fraction and contrast. It also illustrates the global evolution of material properties when one parameter remains constant while the other varies. Consequently, this curve enhances our understanding of  
Figure 9: Three-dimensional plot of mechanical proprieties as a function of contrast and volume fraction, demonstrating the relationship between material composition and mechanical properties.  
the interrelationships between parameters and properties in composite materials and offers valuable insights for the design and manufacturing of optimized composites.  
Table 4 presents the optimal values of the coefficients for the resulting function that most effectively fit the curve; these values encapsulate the nuanced relationships.