## 2 Methodology  
Based on the theory of multinomial continuum mechanics, the overall stress state of unsaturated soil can be represented by two independent tensors under the condition that soil pores and pore water are incompressible. \[ \begin{bmatrix}\sigma_{x}-u_{a}&\tau_{yx}&\tau_{zx}\\ \tau_{xy}&\sigma_{y}-u_{a}&\tau_{zy}\\ \tau_{xz}&\tau_{yz}&\sigma_{z}-u_{a}\end{bmatrix}\begin{bmatrix}u_{a}-u_{w}&0 &0\\ 0&u_{a}-u_{w}&0\\ 0&0&u_{a}-u_{w}\end{bmatrix} \]  
where \(u_{a}-u_{w}\) corresponds to suction, which is a common key variable in unsaturated soil analysis. Since the suction does not directly act on the soil skeleton element, the normal action area of the soil element is not equal to the external load and cannot be directly added to \(\sigma-u_{a}\). Due to the complexity of the elasto-plastic problem of unsaturated soils, this study was guided by the Bishop effective stress principle [19]. When constructing the elastic modulus model, the suction force was simplified as a three-dimensional isobaric external force. Under this premise, discuss the parameter \(\chi\) so that it can modify \(u_{a}-u_{w}\) and make it addable to \(\sigma-u_{a}\). In order to find a suitable \(\chi\) construction method for the modified soil, the soil-water characteristic curve of the compacted modified soil was measured in advance. The soil-water characteristic curve (SWCC) is often used to express the relationship between the water content and suction in unsaturated soil research, which can provide a reference for the research of other characteristics. In this study, the SWCC of oil shale waste residue-modified soil was measured using the filter paper method [24]. The test sample was made out of oil shale waste residue powder, fly ash, and silty clay with the same particle size, and the mass ratio of the three materials was 2:1:2. This ratio was obtained according to the research group's previous experimental research on the improvement of subgrade soil by oil shale waste residue [25]. Under this ratio, the road performance indexes (such as unconfined compressive strength and shear strength, etc.) of the improved soil reach the best. The basic physical properties of modified soil are shown in Table 1.  
The compacted modified soil samples with the initial volumetric moisture contents of 20%, 25%, 30%, 35%, and 40% were prepared, and the sample was a cylinder of a standard ring cutter size (diameter 61.6 mm, height 20 mm, volume 60 mm3). The experimental process was according to the American Society for Testing and Materials (ASTM) standard [24]. Since data measured by the filter paper method are discontinuous, in order to better observe the law between matric suction and water content, this study used the classic Fredlund model [10] to fit them. \[\theta=\frac{\theta_{s}}{\left\{\ln\left[e+\left(\frac{\pi}{a}\right)^{n} \right]\right\}^{m}} \tag{5}\]  
where \(\theta_{s}\) is the saturated moisture content coefficient (unless otherwise specified below, the moisture content mentioned is the volumetric moisture content), which represents the starting position of the Fredlund fitting curve, and m, n, and \(a\) are soil parameters that affect the soil-water characteristic curve. The fitted SWCCs were then plotted on a semi-logarithmic coordinate system (Figure 1). Auxiliary lines and feature points are also shown according to the morphological features of the SWCC. In Figure 1, the transitions from the boundary effect section to the transition section and from the transition section to the residual section occur at points \((\psi_{b},\theta_{b})\) and \((\psi_{s},\theta_{s})\), respectively. In the semi-logarithmic coordinates, the slope of the transition segment between these two locations is significantly different from the other two sections.