## 3 Mechanisms of Macro- and Micro-Parameter Influence and Construction of Multiple Regression Models
### Analysis of the Influence Mechanism of Significant Micromechanical Parameters Interaction
#### 3.2.1 The Effects of \(R_{max}/R_{min}\), \(E\), and \(k\) on the \(E\)  
As shown in Figure 2(a), when the effective modulus is fixed, the elastic modulus decreases as the radius ratio (\(R_{max}/R_{min}\)) increases. When the radius ratio exceeds 1.5, the elastic modulus levels off and exhibits no significant changes. On the other hand, as depicted in Figure 2(b), with the radius ratio fixed, the elastic modulus increases linearly with increasing effective modulus. Notably, when the radius ratio is greater than 2.0, the curves corresponding to different radius ratios almost coincide, suggesting that, under these conditions, the elastic modulus is primarily governed by the effective modulus.  
As shown in Figure 3(a), when the stiffness ratio is held constant, the elastic modulus exhibits a decreasing trend with increasing \(R_{max}/R_{min}\), with the most significant change observed when the stiffness ratio equals one. As the stiffness ratio increases, the rate at which the elastic modulus decreases becomes less pronounced. Regardless of the stiffness ratio, when \(R_{max}/R_{min}\) exceeds 1.5, the downward trend of the elastic modulus levels off. At this stage, the elastic modulus is predominantly influenced by the stiffness ratio. As shown in Figure 3(b), under different \(R_{max}/R_{min}\) conditions, the elastic modulus decreases with increasing stiffness ratio. The \(R_{max}/R_{min}\) has a significant impact on the elastic modulus at a relatively low level. However, when \(R_{max}/R_{min}\) exceeds 2.0, the elastic modulus remains nearly unchanged as the stiffness ratio increases, indicating that the influence of \(R_{max}/R_{min}\) becomes negligible in this range.  
A significant interaction effect is observed between the effective modulus and stiffness ratio parameters. As illustrated in Figure 4(a), when the effective modulus is held constant, the elastic modulus decreases with increasing stiffness ratio. Notably, when the effective modulus is at a high level, the inhibitory effect of the stiffness ratio on the elastic modulus is significantly enhanced; conversely, when the effective modulus is at a low level, the influence of the stiffness ratio on the elastic modulus weakens. As shown in Figure 4(b), with the stiffness ratio held constant, the elastic modulus increases monotonically with  
Figure 3: The effects of \(R_{max}/R_{min}\) and \(E\) on the \(E\).  
increasing effective modulus. With increasing stiffness ratio, its inhibitory influence on the dominant effect of the effective modulus becomes increasingly evident.  
As can be seen, the elastic modulus increases with increasing effective modulus and decreases with increasing stiffness ratio. It is worth noting that a threshold exists for the effect of the radius ratio (\(R_{max}\)/\(R_{min}\)) on the elastic modulus. Below this threshold, the radius ratio exerts an inhibitory influence. However, beyond the threshold, its effect becomes negligible, and the elastic modulus is predominantly governed by the effective modulus and stiffness ratio. Based on this characteristic, selecting an appropriate radius ratio can help ensure computational accuracy while enabling dimensionality reduction of the computational model.