## 3 Results and Discussion
### Composite Exponential Model of Shear Stress and Shear Displacement  
From Figures 4 and 6, it can be seen that the variation in the shear stress-shear displacement curves of the two soils are slightly different. The muddy clay with permeation grouting has an obvious peak shear stress and stress decline stage, while grouted silty clay only has peak shear stress at a high content of cement (15%), and the attenuation trend of shear stress is relatively slow. In order to find a corresponding relationship between the shear stress and shear displacement of the soil around piles with permeation grouting, for the curves in Figures 4 and 6, four curve models were used for fitting, including compound exponential, hyperbolic, polynomial, and exponential models. It was found that the compound exponential model had the best fit. The formula is as follows:  
\[\tau=a(1-e^{-b\delta}) \tag{6}\]  
According to Equation (6), when the shear displacement \(\delta\) tends to infinity, the shear stress \(\tau=a\). Since the shear failure of the soil around permeation grouting piles also follows the Mohr-Coulomb criterion of failure, the parameter \(a\) in Equation (6) can be expressed as \(a=\sigma\tan\varphi_{1}+c_{1}\). The model parameter \(b\) represents the development rate  
Figure 8: Variation in shear strength index with cement content. (**a**) Cohesion. (**b**) Friction angle.  
of shear stress with shear displacement, and its value is related to grouting parameters. The values of \(b\) under different grouting parameters were obtained by fitting, as shown in Table 6.  
From Table 6, it can be seen that the model parameter \(b\) is positively correlated with the cement content and negatively correlated with the normal stress. The fluctuation range of parameter \(b\) under the same cement content and different normal stresses is large and has no obvious regularity. Using its average value for analysis, it is found that the parameter \(b\) and grouting parameter \(w\) have a positive linear correlation (\(R^{2}=0.907\)), as shown in Equation (7).  
\[b=0.064w+0.566 \tag{7}\]  
Parameters \(a\) and \(b\) are substituted into Equation (6), and the composite exponential model of shear stress and shear displacement can be obtained, as shown in Equation (8).  
\[\tau=(\sigma\tan\varphi_{1}+c_{1})\Big{(}1-e^{-(0.064w+0.566)\delta}\Big{)} \tag{8}\]  
The compound exponential model in Equation (8) was used to simulate two soil samples with different cement content. The fitted curve of muddy clay deviates slightly from the test curve, while the fitting curve of silty clay is basically the same as the test curve. Among them, the R-squared of the silty clay when the cement content is 10% is the best, as shown in Figure 9. The fitting curves under other normal stresses are in good agreement with the test curves except for the s-shaped fluctuations in the curve at 200 kPa. This result indicates that the model can be used to describe the relationship between the shear stress and shear displacement of silty clay with permeation grouting.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Cement Content**} & \multicolumn{4}{c}{**Normal Stress (kPa)**} & \multirow{2}{*}{**Average**} \\ \cline{2-2} \cline{4-6}  & **50** & **100** & **200** & **400** & \\ \hline
0\% & 0.923 & 0.927 & 0.518 & 0.395 & 0.691 \\
5\% & 0.829 & 0.841 & 0.736 & 0.428 & 0.709 \\
10\% & 1.192 & 1.277 & 1.406 & 0.868 & 1.186 \\
15\% & 1.903 & 1.679 & 1.588 & 1.214 & 1.596 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Model parameter \(b\) under different grouting conditions.  
Figure 9: Comparison of test curve and fitting curve (\(w=10\%\)).