## 1 Introduction
Some researchers have attempted to indirectly estimate the impact of \(FT\) on rock mechanical properties based on laboratory \(FT\) test results and field investigations of rock weathering conditions. By employing empirical models such as rock mass classification indices (e.g., RMR, \(GSI\)) or the Hoek-Brown strength criterion, these studies offer a practical approach to parameter estimation in the absence of in situ testing and provide theoretical and empirical support for stability assessments and inverse analysis of cold-region rock engineering. [PERSON] et al. [15] used the Hoek-Brown empirical criterion to calculate slope safety factors before and after \(FT\) cycles, showing that \(FT\) significantly reduced the compressive strength of granite and thus decreased the overall slope safety factor. [PERSON] et al. [16] established a linear relationship between the Geological Strength Index (\(GSI\)) and the Tianshan Slope Mass Rating (TSMR), further analyzing the mechanical behavior and stability state of slope rock masses. [PERSON] et al. [17] investigated the degradation of Hoek-Brown parameters in multi-fractured rock masses under \(FT\) cycles, derived the shear expression of the Hoek-Brown criterion, and proposed a new estimation method based on shear tests. Results indicated that both \(GSI\) and the material constant \(m\) declined with increasing \(FT\) cycles, with a decreasing rate that followed a negative exponential trend. [PERSON] et al. [18], aiming to support \(FT\) damage prevention and mitigation in tunnels, proposed an elastoplastic solution for frost heave pressure based on the nonlinear Hoek-Brown criterion. Numerical analysis showed that as \(GSI\) decreased and disturbance parameter \(D\) increased, both the plastic zone and frost heave pressure expanded. [PERSON] and [PERSON] [19] applied the generalized Hoek-Brown criterion to evaluate the optimal slope stability of the Angouran open-pit mine. Using the estimated rock mass properties, a numerical model of the pit slope was developed, and its stability conditions were analyzed. The results showed good agreement between monitored displacements and numerical predictions, with a coefficient of determination \(R^{2}=0.80\). [PERSON] et al. [20] proposed a thermo-hydro-mechanical slope stability analysis model (THM-SSA) for evaluating slope stability. This model calculates the local factor of safety by incorporating the isotropic tensile strength of unfrozen soil and its evolution under multiple freeze-thaw cycles. [PERSON] et al. [21], based on the Hoek-Brown criterion, determined the mechanical parameters of rock masses and employed FLAC3D numerical simulations to analyze stress-deformation characteristics and the trend of safety factor variation under freeze-thaw effects. Their simulation results were validated against field monitoring data. To investigate the internal physical and mechanical changes in expansive soil slopes, [PERSON] et al. [22] conducted four freeze-thaw cycle experiments on slope models with slope ratios of 1:1.5, 1:1, and 1:0.5 in an environmental simulation chamber. The results indicated that steeper slopes lead to more intense temperature fluctuations, more pronounced crack development, and increased thermal gradients and moisture migration rates during freeze-thaw processes. [PERSON] et al. [23] used finite element simulations to analyze the impact of frost heave on slope deformation and stability, revealing the spatiotemporal characteristics of deformation mechanisms under freeze-thaw action. [PERSON] et al. [24] combined a THM model with field measurements to explore the coupled effects of freezing, leakage, and mechanical behavior on the stability of structural slopes and tunnels. In [PERSON] et al. 's research [25], the maximum safe slope angle of fractured rock masses was optimized by eliminating slope angles inducing unstable blocks (key blocks) and by using the stereographic projection method of the key block theory. [PERSON] et al.