## 4 Discussion  
### Development of Disintegration Model  
Quantitatively characterizing the disintegration rate of stabilized soils involves understanding the physical processes and the soil characteristics that contribute to soil disintegration. This section necessitates an in-depth discussion of methodologies for buffering a disintegration model tailored to stabilized soils. A disintegration model has been formulated using the Weibull distribution [44], encompassing considerations of soil compactness and stabilizer content, with the aim of quantifying the disintegration rate across three distinct stages. It could easily infer the distribution parameters with small samples [45; 46; 47] and can be applied to disintegration and fragmentation studies of soil particles [48]. To explore the evolution of disintegration breakage, this work employs a two-parameter Weibull distribution model for stabilized silty clay under static disintegration conditions. The current disintegration ratio \(P_{i}\) (0 < \(P_{i}\leq 1\)) of soil particles could be expressed by the Weibull cumulative distribution function, given as follows:  
\[P_{i}=1-e^{-\left(t_{i}/d\right)^{k}}, \tag{8}\]  
\[t_{i}=t/t_{\text{max}}, \tag{9}\]  
where \(t_{i}\) is the normalized disintegration time and \(d\) and \(k\) are the parameters of the Weibull distribution function, respectively. Although the maximum disintegration time was not consistent for different soil samples, a variation of the disintegration ratio with normalized disintegration time was obtained under different additions and compactness, as shown in Figures 16 and 17. This is consistent with the changing trend shown in Figures 12 and 14, where the disintegration ratio is positively correlated with the content of lime or lime and cement and negatively correlated with compactness. Hence, a unified model has been developed to elucidate the process of disintegration breakage for all stabilized soil samples. The parameters of the evolution model described in Equation (8) were determined by employing the least squares technique [49]. \[\sum_{i=1}^{n}\left[P_{i}-\left(1-e^{-\left(t_{i}/d\right)^{k}}\right)\right]^ {2}=\min_{j=1}^{m}\left\{\sum_{i=1}^{n}\left[P_{i}-\left(1-e^{-\left(t_{i}/d \right)^{k}}\right)\right]^{2}\right\} \tag{10}\]  
Figure 15: Maximum disintegration time of stabilized silty clay using the additions of lime and cement. The disintegration parameters for stabilized soil samples were derived using the experimental outcomes and a least squares approach outlined in Equation (6), as summarized in Table 3. To comprehensively assess Equation (4) in capturing the disintegration breakage of stabilized soil, the data from Figures 12 and 14 were recalibrated and presented in Figures 16 and 17. The relationship between the disintegration ratio (\(P_{i}\)) and the normalized time (\(t_{i}\)) is effectively described by the disintegration model using the Weibull distribution, as validated by the correlation coefficient \(R^{2}\) approaching 1. This compelling similarity serves as a strong confirmation of the efficacy of the Weibull distribution in characterizing the process of disintegration breakage. When the compaction level remains constant, an increase in stabilizer content causes both parameters \(d\) and \(k\) to rise progressively, resulting in a downward shift of the disintegration curve. Consequently, at the same normalization time, the disintegration ratio decreases, indicating an enhanced resistance to complete disintegration of the stabilized silty clay. To delve deeper into the importance of the Weibull parameters (\(d\) and \(k\)), Equation (8) can be applied to facilitate a detailed examination:  
\[\ln[\ln(1/(1-P_{i}))]=k\ln t_{i}-k\ln d \tag{11}\]  
A clear positive correlation is evident between \(P_{i}\) and \(\ln[\ln(1/(1-P_{i}))]\), with a linear association observed between \(\ln[\ln(1/(1-P_{i}))]\) and \(\ln t_{i}\).