## 1 Introduction  
As transportation networks continue to improve, road and tunnel construction projects are gradually extending into cold regions with harsh climatic conditions [1]. These regions are widely covered by permafrost, which refers to various types of rocks and soils containing ice at or below 0 \({}^{\circ}\)C [2; 3]. The study of frozen rock has emerged as a more specialized direction from permafrost research. Rocks, as porous media composed of solid skeleton particles and intergranular pores, have structures made up of interconnected solid skeleton particles, with the pores typically filled by one or more fluids [4; 5; 6]. In cold regions, temperatures often drop below zero, and seasonal changes can cause drastic temperature fluctuations, subjecting rocks to freeze-thaw cycles [7; 8; 9]. Under cyclic influences of temperature stress and pore ice have forces, rock skeleton particles experience uneven expansion and contraction, which expands the internal pores, resulting in damage to the macroscopic structure and a decline in mechanical properties [10; 11; 12]. Freeze-thaw deformation disrupts the original equilibrium state of the rock mass, affecting the stability and durability of underground rock engineering in cold regions [13; 14; 15]. Many scholars have conducted extensive studies on rock deformation under freeze-thaw cycles. [PERSON] et al. [16] conducted freeze-thaw cycle experiments on cracked red sandstone and found that under freeze-thaw conditions, the cracked red sandstone exhibited frost heave deformation. [PERSON] et al. [17] studied the deformation patterns of rocks with different crack lengths, widths, and lithologies over time and temperature during freeze-thaw cycles, finding that deformation damage was mainly due to cumulative effects from temperature damage, frost heave, and thermal expansion. [PERSON] et al. [18] investigated the deformation characteristics of rocks in low-temperature environments and simulated frost heave deformation in cold-region tunnels. By testing strain characteristics in dry and water-saturated rock specimens under low temperatures, they found that water-saturated specimens exhibited residual strain during contraction, frost heave, thawing, and thermal expansion stages, while dry specimens only displayed linear elasticity. [PERSON] et al. [19] discovered that specimens with 50% and 90% saturation exhibited significant deformation during freeze-thaw cycles. [PERSON] et al. [20] conducted experiments on the strain process of cracked rocks with different lithologies under freeze-thaw cycles, finding differences between saturated and dry states, with freeze-thaw cycle frequency affecting strain values. [PERSON] et al. [21] conducted unidirectional freezing experiments on saturated cracked rock, demonstrating that the frost heave effect caused by cracks differed significantly from that of the rock matrix, with cracked rock masses displaying strong heterogeneous frost heave behavior. [PERSON] et al. [22] developed a new elasto-plastic model to estimate the long-term frost heave and deformation of porous rocks in cold regions under freeze-thaw conditions, considering factors such as pore size, matrix yield stress, and water saturation. [PERSON] et al. [23] proposed a thermo-mechanical coupled elasto-plastic model for porous rocks under freeze-thaw conditions, which can accurately predict the temperature changes and freeze-thaw deformation processes of porous rocks. Most scholars use strain gauge methods for monitoring deformation, as exemplified by [PERSON] et al. [24], who studied the principle of measuring linear expansion coefficients using strain gauges and compared this method with traditional quartz tube dilatometers. Their results showed that strain gauge monitoring is simple, reliable, and accurate. In summary, numerous scholars have conducted extensive experimental and theoretical studies on the freeze-thaw deformation of cracked rocks, primarily focusing on variables such as crack length, crack width, lithology, and rock saturation.