## 1 Introduction
Some models for predicting the \(\rm M_{R}\) considering the influence of matric suction have been proposed successively. Such a prediction model was proposed by [PERSON] et al. [17], considering matric suction, minimum bulk stress, and octahedral shear stress. \[M_{\rm R}=k_{0}p_{a}\left(\frac{\psi_{m}}{p_{a}}+1\right)^{k_{1}}\left(\frac{ \theta_{m}}{p_{a}}\right)^{k_{2}}\left(\frac{\tau_{\rm oct}}{p_{a}}+1\right) ^{k_{3}} \tag{2}\]  
where \(\psi_{m}\) is the total suction, and \(\theta_{m}\) is the minimum bulk stress. This model avoids the problem of equal body stress values under different combinations of deviatoric stress and confining pressure. However, the regression coefficients of the model are obtained based on the physical parameters of the fully weathered granite. Therefore, its applicability to other types of soil remains to be further verified. [PERSON] et al. [18] proposed a model for the modulus of resilience of recycled unbonded materials that uses the density ratio to characterize the effect of soil density on soil resilience. \[M_{\rm R}=k\left(\frac{\sigma_{m}}{p_{r}}\right)^{k_{1}}\left(\frac{\tau_{ oct}}{\tau_{ref}}\right)^{k_{2}}\left(\frac{\psi_{m}}{p_{a}}\right)^{k_{3}} \left[\frac{\rm DDR(1-k_{4}RCM/100)}{100}\right]^{k_{3}} \tag{3}\]  
where \(\sigma_{m}\) is the mean normal stress (i.e., \((\sigma_{1}+\sigma_{2}+\sigma_{3}\ )/3\)), \(\tau_{ref}\) is the reference shear stress (i.e.,\(\sqrt{2/3}q_{f}\)), \(q_{f}\) is the peak shear strength, DDR is the dry density ratio (%), RCM is the percentage of recycled clay masonry (%), and k4 and k5 are regression parameters (factors and exponents). [PERSON] [19] proposed an expression for the effective stress of unsaturated soils, which extended the Terzaghi effective stress principle in traditional soil mechanics to the study of unsaturated soils. This principle has been widely used as a theoretical basis in the study of the strength and deformation of unsaturated soils, and as one of its applications, it is often introduced to improve existing elastic modulus models. This expression describes the effective stress in the soil as  
\[\sigma^{\prime}=(\sigma-u_{a})+\chi(u_{a}-u_{w}) \tag{4}\]where \((u_{a}-u_{w})\) is matric suction, \(u_{a}\) is pore air pressure, and \(u_{w}\) is pore water pressure. Subsequent researchers have proved that the parameter \(\chi\) is related to saturation. Many researchers have successively put forward some expressions similar to the formula, but the measured soil parameters have different values for different problems (volume deformation or shear strength), different stress paths, and different soil types. Although the effective stress parameters of unsaturated soils are not easy to measure accurately, the approximate expression of \(\chi\) can be obtained through experiments, and then the theoretical estimation of the stress-strain problem of unsaturated soils has practical value [20; 21]. In addition to the influence of suction, the subgrade in the seasonal frozen soil area also undergoes long-term freezing and thawing. The weathering effect caused by freezing and thawing will not only change the soil structure, which will affect the strength of the subgrade, but also lead to water migration, resulting in the softening and deformation of the subgrade. [PERSON]'s research [22] shows that the strength of unsaturated soil decreases first and then tends to be stable with the increase in the freeze-thaw cycles. [PERSON] et al. [23] found, from a microscopic point of view, that a decrease in the freeze-thaw temperature and an increase in the freeze-thaw cycles led to the gradual decomposition of large loses particles into small particles, and the arrangement of the particles changed. Therefore, when testing the service performance of the improved subgrade filler in seasonally frozen areas, it is necessary to simulate complex environmental loads.