# A Thorough Investigation of the Dynamic Properties of Granite under Cyclic Loading
## 5 Dynamic Response Model
### Influence of Loading Frequency  
Figure 21 illustrates the outcomes of fitting Equation (5) to describe the relationship between dynamic strain and dynamic stress. The results indicate that with an increase in the level of dynamic stress, the dynamic strain of the specimen also rises. Moreover, at the same dynamic stress level, the dynamic strain of the specimen increases with a decrease in the loading frequency.  
Figure 22 presents the outcomes of fitting Equation (6) to illustrate the relationship between the ratio of dynamic strain amplitude to dynamic stress and dynamic strain amplitude. The results demonstrate a gradual decrease in the ratio of dynamic strain amplitude to dynamic stress as the dynamic stress amplitude increases.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Parameter and Confining Pressure Relationship Formula** \\ \hline \multirow{2}{*}{Granite specimens} & \(a=0.0633-0.00056\sigma_{\text{c}}\) \\  & \(b=-0.128-7.76\times 10^{-4}\sigma_{\text{c}}+7.4\times 10^{-5}\sigma_{\text{c}}^{2}\) \\  & \(c=0.144-3.89\times 10^{-4}\sigma_{\text{c}}-5.7\times 10^{-5}\sigma_{\text{c}}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The change curves of parameters \(a\), \(b\), and \(c\) with confining pressure.  
Figure 21: Specimens’ fitting curves with loading frequencies.  
Table 5 displays the outcomes of fitting Equation (6) to investigate the relationship between the ratio of dynamic strain amplitude to dynamic stress and dynamic strain amplitude at different frequencies. The correlation coefficients of the formulas for frequencies ranging from 1 to 10 Hz all reach 0.98, supporting our hypothesis that the relationship follows a quadratic polynomial pattern.  
However, at a frequency of 20 Hz, the correlation coefficient of the fitted formula decreases to 0.75. We attribute this observation to the higher frequency exerting a more significant impact on the specimen compared with the confining pressure, which is consistent with the findings and conclusions presented in Sections 3 and 4. As the next step in our research, we aim to deduce the primary stress field by integrating information from the wave velocity measurements within the tunnel envelope and the characteristics of the internal stress field. This primary stress field will then be employed to modify the formula for material performance parameters, enabling us to derive the initial stress conditions of the specimen.  
Table 6 presents the experimental parameters \(a\), \(b\), and \(c\) concerning their variation with loading frequency. The analysis reveals a clear power-function relationship between these parameters (\(a\), \(b\), and \(c\)) and the loading frequency.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Parameter and Loading Frequency Relationship Formula** \\ \hline \multirow{2}{*}{Granite specimens} & \(a=0.0154+0.073\times 0.88^{f}\) \\  & \(b=0.03-0.29\times 0.90^{f}\) \\  & \(c=-0.102+0.39\times 0.93^{f}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: The change curves of parameters \(a\), \(b\), and \(c\) with loading frequencies.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Loading Frequency \(f\)** & **Fitting Formula** & **Fitting (\(R^{2}\))** \\ \hline \multirow{4}{*}{Granite specimens} & 1 & \(y=0.0784-0.222x+0.247x^{2}\) & 0.98 \\  & 5 & \(y=0.0584-0.163x+0.211x^{2}\) & 0.98 \\ \cline{1-1}  & 10 & \(y=0.0319-0.053x+0.062x^{2}\) & 0.99 \\ \cline{1-1}  & 20 & \(y=0.023-0.013x+0.004x^{2}\) & 0.75 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Equation (6) fitting curve with loading frequencies.