# Experimental Study of the Dynamic Characteristics of PVA-Amended Expansive Soils
## 3 Analysis of Test Results
### Analysis of the Dynamic Triaxial Test Results
#### 3.2.2 Changing Law for the Dynamic Elastic Modulus
The relationship between the dynamic elastic modulus \(\mathrm{E_{d}}\) and the dynamic strain \(\varepsilon_{\mathrm{d}}\) of the soil can be expressed by fitting relationship (3) [27; 28]:  
\[\sigma_{\mathrm{d}}=E_{\mathrm{d}}\varepsilon_{\mathrm{d}}=\frac{E_{0}\varepsilon _{\mathrm{d}}}{1+\eta_{\mathrm{d}}\varepsilon_{\mathrm{d}}} \tag{2}\]  
\[E_{\mathrm{d}}=\frac{E_{0}}{1+\eta_{\mathrm{d}}\varepsilon_{\mathrm{d}}} \tag{3}\]  
where \(E_{0}\) is the initial dynamic elastic modulus and \(\eta_{d}\) is the fitting parameter  
The fitted curve between the dynamic modulus of elasticity and dynamic strain for PVA-amended expansive soil at a confining pressure of \(\sigma_{3}\) = 100 kPa and frequency \(f\) = 1 Hz is shown in Figure 9, where the specific parameters for the fitted equation are given by  
\[E_{\mathrm{d}}=\frac{0.46}{1+0.70\varepsilon_{\mathrm{d}}} \tag{4}\]  
At this time, the curve fitting coefficient is 0.98. According to the fitting results shown in Figure 9, Equation (3) can provide a more complete expression for the change rule between the dynamic elastic modulus \(\mathrm{E_{d}}\) and the dynamic strain \(\varepsilon_{\mathrm{d}}\) of PVA-amended expansive soil. The fitted parameters at other frequencies and envelope pressures are shown in Table 6.  
The initial dynamic elastic modulus \(E_{0}\) is the dynamic elastic modulus of the soil when subjected to initial dynamic loading and is usually known as the initial shear modulus or initial compression modulus. This dynamic elastic modulus is an important mechanical parameter reflecting the stiffness of the soil and can be used to assess the stability and load-bearing capacity of the improved soil. Figure 10 shows the initial dynamic elastic modulus versus the frequency of the enclosing pressure. Figure 10a shows that the initial dynamic elastic modulus increases with increasing enclosing pressure. Figure 10b demonstrates that the initial dynamic elastic \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Hz**} & \multicolumn{2}{c}{\(\sigma_{3}\)**=100 kPa**} & \multicolumn{2}{c}{\(\sigma_{3}\)**=200 kPa**} & \multicolumn{2}{c}{\(\sigma_{3}\)**=300 kPa**} \\ \cline{2-7}  & \(\mathbf{E_{0}}\) **(MPa)** & \(\mathbf{\eta_{d}}\) & \(\mathbf{E_{0}}\) **(MPa)** & \(\mathbf{\eta_{d}}\) & \(\mathbf{E_{0}}\) **(MPa)** & \(\mathbf{\eta_{d}}\) \\ \hline
1 & 42.74 & 0.64 & 59.45 & 0.98 & 90.41 & 1.33 \\
2 & 50.72 & 0.73 & 79.80 & 1.53 & 106.76 & 1.71 \\
3 & 59.26 & 0.86 & 88.01 & 1.56 & 126.33 & 2.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Modified soil dynamic elastic modulus and dynamic strain fitting parameters. Figure 9: Fitted curve for the dynamic elastic modulus \(\mathrm{E_{d}}\) versus dynamic strain \(\varepsilon_{\mathrm{d}}\). modulus increases with increasing frequency relative to the enclosing pressure, but the increase trend is slow. Therefore, the circumferential pressure has a large effect on the initial dynamic elastic modulus, while the frequency also has an effect on the initial dynamic elastic modulus, but the effect is small. This is because the increase in the circumferential pressure will lead to the specimen being subjected to a greater lateral constraint force, and the deformation of the specimen under dynamic loading will be smaller, so the initial dynamic elastic modulus will increase. The effect of the loading frequency on the initial dynamic elastic modulus of the specimen manifests during the loading process, and the initial dynamic elastic modulus is generated during the initial loading of the specimen. Thus, the effect of the frequency on the initial dynamic elastic modulus can be analyzed only within a very small strain interval. When the specimen is subjected to high-frequency loading, at the beginning of the initial dynamic loading, the specimen will experience stress and strain. At this time, the stress-strain curve is similar to a straight line, so the specimen viscosity is relatively small.