## 3 Uniaxial Compressive Constitutive Equation Establishment
### Applicability of the Modified Constitutive Equation  
Since there is no standard test method for the uniaxial compression test of earth materials, the existing constitutive equations are obtained by fitting the test data obtained by the researcher himself. There are few studies on the wide applicability of the equations. Therefore, the purpose of this study is to establish a constitutive equation of earth materials under uniaxial compression that is suitable in different regions of China. The stress-strain curve of the earth materials compression test in this paper and related typical literature in China is fitted with the proposed modified constitutive equation to verify its applicability. Considering that the material modification will have a large effect on the physicochemical properties of the earth materials, which in turn will lead to a large change in the compressive stress-strain curve trend compared to the unmodified earth material [37]. Additionally, the test loading rate has a lower effect on the strength coefficient of variation. Therefore, in screening data samples from relevant literature, the effect of loading rate is not considered. The uniaxial compressive test curves of 100-mm-cubic standard specimens with a curing age of 28 d composed of unmodified earth materials (later referred to as earth materials) were selected for fitting to evaluate the applicability of the modified constitutive equations. Based on this, 14 sets of 94 uniaxial compressive test curves of earth materials from different regions were extracted from the four articles that satisfied the above conditions [34; 36; 38; 39]. The original curves have been plotted in Figure 12. Among them, soil samples were taken from Xinjiang, Shanxi, Fujian, and Heilongjiang, where the earth's structure is widely prevalent [36]. The basic physical properties of earth materials from different regions used in the related literature are given in Table 6. Since the particle size range of the earth materials is missing in the relevant literature, only the Arterberg limit as well as the maximum dry density and optimum water content of the earth materials are listed in the table. It can be seen that all earth materials are low-liquid-limit clays. This is also consistent with the earth materials selected in this paper and related studies. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Source**} & \multirow{2}{*}{**Region**} & \multicolumn{3}{c}{**Alterberg Limits**} & \multirow{2}{*}{**Type of Soil**} & \multirow{2}{*}{**Forming Method**} & \multirow{2}{*}{**Maximum Dry**} & \multirow{2}{*}{**Optimal**} \\ \cline{3-3} \cline{5-8}  & & & **LL (\%)** & & & & **LL (\%)** & **Pl (\%)** \\ \hline This test & Xinjiang & 36.3 & 23.5 & 12.8 & \multirow{2}{*}{Light compaction method} & 2.04 & 23.0 \\ [38] & Shanxi & 33.6 & 17.2 & 16.4 & \multirow{2}{*}{Jack compaction} & 1.90 & 18.2 \\ [34] & Shanxi & 26.0 & 15.0 & 11.0 & & Jack compaction & 2.04 & 18.2 \\ [34] & Xinjiang & 25.5 & 13.6 & 11.9 & Low liquid & Jack compaction & 2.14 & 14.5 \\ [34] & Fujian & 45.0 & 26.0 & 19.0 & limit clay & Jack compaction & 1.74 & 29.0 \\ [34] & Heilongjiang & 28.5 & 14.0 & 14.5 & \multirow{2}{*}{Jack compaction} & 2.00 & 16.3 \\ [36] & Shanxi & 26.0 & 15.0 & 11.0 & & Jack compaction & 2.04 & 18.2 \\ [39] & Shanxi & 26.0 & 15.0 & 11.0 & & Jack compaction & 2.04 & 18.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Related literature and basic physical properties of earth materials. First, the test constitutive curve was normalized. Then, the equations were fitted according to the least squares principle using Origin 8.5 software to determine the values of the curve shape parameters for the rising and descending sections of each literature-fitted curve. Finally, the fitting effect of the present constitutive equation is judged according to the coefficient of determination R\({}^{2}\).