# Constitutive Characteristics of Rock Damage under Freeze-Thaw Cycles
## 1 Introduction  
In many engineering applications, rocks are subject to freeze-thaw cycles (FTC). Under the action of freeze-thaw cycle and low temperature frostbite, rock and soil will have irreversible damage. This is an important safety hazard and disaster inducement for geotechnical engineering in cold areas [1; 2]. For example, frost heave rupture of highway tunnels in cold regions leads to instability. Freeze-thaw collapse leads to landslide, multi-slope collapse of frozen rock in mining areas, dynamic disturbance of frozen rock caused by earthquakes, etc. The compressive strength of rock plays an important role in the construction of rock engineering and is a key parameter in the construction process [3; 4]. Therefore, the influence of FTC on the compressive strength of rock is discussed, and the intrinsic relationship between rock strength and FTC is understood. This is of great significance to the research and practice of rock engineering [5]. The compressive strength of rock before and after different FTCs has been obtained, and it was found that the compressive strength of rock will decrease more along with the increase in the number of FTCs [6]. In addition, it was found that, after the freezing temperature of the rock sample was reduced, the compressive strength decreased more significantly [7]. Berea rock was selected for FTC treatment, and the compression coefficient of the rock decreased by about 16% and 24%, respectively, when the temperature ranged from \(-4\)\({}^{\circ}\)C to \(6\)\({}^{\circ}\)C [8; 9; 10]. Different kinds of granites were selected for the FTC and ultrasonic test study [11], and it was found that the wave velocity of different kinds of granites decreased after each FTC, and the decreasing trend was related to lithology, freeze-thaw time and cycle times, with the maximum decrease of about 15%. According to the above analysis, these changes are mainly caused by the expansion of pores and cracks in rocks [12]. However, in many engineering practices, as mentioned above, rock causes a dynamic stress. There has been limited research conducted on the dynamic mechanical properties of rocks following FTC [13]. For example, through the comparison of a quasi-static uniaxial experiment and a SHPB impact experiment, it was found that the compressive strength of granite and limestone under high strain rate is greater than that under static load [14]. Generally speaking, the increase in the number of FTCs or a decrease in the loading rate will lead to a decrease in the fracture strength [15]. The failure process in rock under the action of the FTC is a change process of microstructural damage [4; 16; 17]: (1) the expansion force generated when the water contained in the rock voids is converted into ice, (2) thermal stress caused by the temperature difference between the surface and the interior of the rock, (3) the water pressure inside the rock caused by the freezing of the surface. In addition, there is also an expansive force phenomenon when ice crystals are formed [18; 19]. Looking at the current research status in China and abroad, there are many research projects on freeze-thaw rock, but few on damage caused by the mechanical properties of freeze-thaw rock. Most are confined to the unidirectional stress state, without considering the influence of confining pressure. The parameters of the model are mostly based on the fitting of test data, lack clear physical meaning, and can not reflect the general change law of the selected physical quantity. Drawing inspiration from the traditional statistical stress-strain model of rock, this paper postulates that the micro-strength of rock subjected to FTC adheres to the Weibull distribution. By incorporating a cumulative damage variable based on the number of FTCs, we have developed a constitutive model for cumulative damage in rock shear strength decay that aligns with the Mohr-Coulomb criterion.