## 4 The Values of the Parameters of the Constitutive Equation  
### Curve Shape Parameters  
In the previous paper, the modified uniaxial compressive constitutive equation was fitted and analyzed using the uniaxial compressive test curves of the earth from relevant typical Chinese literature to verify its applicability. However, it can be seen from Table 7 that the shape parameters of the constitutive equation obtained by different test curves are different. To facilitate the application, further construction of the method for taking values of each parameter in the present constitutive equation is needed. It has been shown [1; 36] that the trend and shape parameter values of the compressive stress-strain curve of the earth are affected to different degrees by factors, such as material composition, specimen shape, size, loading rate, and curing age. Additionally, the effects of different factors are reflected in changes in the compressive strength values of the earth materials. Therefore, based on the test sample data, a method for calculating the shape parameters of the constitutive equation curve based on compressive strength characterization is established in this study.  
Firstly, the dimensionless stress-strain curves were obtained by normalizing the data of the 94 curve samples extracted in the previous paper. Then, the curve shape parameters of each sample were obtained by fitting them using the modified constitutive equation in this paper. Finally, by conducting a regression analysis between the obtained sample curve shape parameters and the compressive strength values, the expressions for calculating the curve shape parameters based on the compressive strength characterization were established. Among them, the curve obtained by fitting the compressive strength to the curve shape parameter for these 94 samples is shown in Figure 14. The equations for calculating the curve shape parameters are shown in Table 8.  
As can be seen in Figure 14 and Table 8, the compressive strengths of the earth specimens are concentrated in the range of 1-4 MPa. There is a nonlinear relationship between each curve parameter and the compressive strength, which can be expressed as a quadratic polynomial function. Among them, the coefficients of determination R\({}^{2}\) of the rising section parameters a\({}_{1}\) and b\({}_{1}\) are 0.834 and 0.802, respectively, and the average value is 0.818, which is a relatively good fit. The coefficients of determination R\({}^{2}\) of the parameters a\({}_{2}\), b\({}_{2}\), and c\({}_{2}\) of the descending section are 0.779, 0.777, and 0.717, respectively,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Shape Parameter** & **Fitting Equation** \\ \hline \(a_{1}\) & \(a_{1}=-0.0768/c^{2}_{c}+0.6076/c-0.8198\) \\ \(b_{1}\) & \(b_{1}=0.2729/c^{2}_{c}-1.9406/c+5.572\) \\ \(a_{2}\) & \(a_{2}=1.0544/c^{2}_{c}-5.6251/c_{c}+4.4891\) \\ \(b_{2}\) & \(b_{2}=-2.1019/c^{2}_{c}+11.376/c-5.9548\) \\ \(c_{2}\) & \(c_{2}=1.615/c^{2}_{c}-8.7239/c+4.3801\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Calculation formula for the shape parameters of the constitutive equation curve.  
with a mean value of 0.758, which is a relatively lower fit. However, it is acceptable in view of the slightly discrete earth material.