## 4 Mechanical Property Evolution Analysis
### Shear Stiffness, \(K\)  
During the reciprocating shear tests, the soil sample's shear stiffness, \(K\), can be calculated with Equation (2), where \(K_{1}\) and \(K_{2}\) are the forward failure shear stiffness and reverse failure shear stiffness, respectively. The variation of the shear stiffness ratio, \(K/K_{f}\), with the number of reciprocating shears, \(N\), can be obtained after eliminating the dimension of the soil's shear stiffness using the ratio method. At the same time, the relationship between the shear stiffness ratio, \(K/K_{f}\), and the reciprocating shear times, \(N\), was fitted and analyzed, and the results are shown in Table 6.  
It can be seen from Table 6 that the logarithmic fitting of the relationship between the shear stiffness ratio, \(K/K_{f}\), and the reciprocating shear times, \(N\), is the best, as shown in Figure 12.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Model** & **Fitting Model Expression** & \begin{tabular}{c} **Degree of** \\ **Freedom** \\ \end{tabular} & \
\begin{tabular}{c} **Residual Sum of** \\ **Squares** \\ \end{tabular} &
\begin{tabular}{c} **Mean Square** \\ **Deviation** \\ \end{tabular} & **Fitting Coefficient** \\ \hline Exponential model & \(K/K_{f}=1.1201e^{-0.1449N}\) & 2 & \(9.06\times 10^{-4}\) & 1.28 & 0.9779 \\ Linear model & \(K/K_{f}=-0.1133N+1.0733\) & 2 & \(6.46\times 10^{-4}\) & 0.06 & 0.9851 \\ Logarithmic model & \(K/K_{f}=-0.243\ln N+0.9834\) & 1 & \(4.19\times 10^{-6}\) & 0.03 & 0.9901 \\ Power model & \(K/K_{f}=0.9859N^{-0.20}4\) & 2 & \(1.58\times 10^{-3}\) & 1.28 & 0.9635 \\ \hline \end{tabular}
\end{table}
Table 6: The fitting relationship between failure shear stiffness ratio, \(K/K_{f}\), and reciprocating shear times, \(N\).  
Figure 11: \(K_{f}\)-\(P_{5}\) Relationship fitting curve.  
Figure 12: \(K/K_{f}\)-\(N\) relationship fitting curve.  
As shown in Figure 12, the \(K/K_{f}\) has obvious nonlinear characteristics with the increase of the \(N\). Fitting with the logarithmic function yields:  
\[K/K_{f}=-0.243\ln N+0.9834 \tag{6}\]  
Combining Equations (5) and (6), an empirical formula for the variation of the shear stiffness, K, with the coarse grain content and the number of reciprocating shears in the argillaceous siltstone rockfill materials can be established, as expressed in Equation (7).  
\[K=(-0.0053P_{5}-0.3479)\ln N+0.0213P_{5}+1.4078 \tag{7}\]  
where \(N\) is the number of reciprocating shears, \(1\leq N\leq 4\); \(P_{5}\) is the coarse grain content, \(0\%\leq P_{5}\leq 75\%\).  
The soil sample's shear stiffness, \(K\), was calculated using Equation (7) and compared with the test values, as shown in Figure 13. The experimental error analysis is shown in Figure 14.  
According to Figures 13 and 14, the calculated shear stiffness of the soil sample has a small error with the test value, indicating that Equation (7) can be used to characterize the variation of the shear stiffness of argillaceous siltstone rockfill material with the number of reciprocating shears and coarse grain contents.  
Figure 14: Error analysis chart.  
Figure 13: Comparison diagram between calculated value and test value.