# Mechanical Properties and Damage Evolution of Heated Granite Subjected to Liquid Nitrogen Cooling
## 1 Introduction  
The exploitation of geothermal energy is currently facing great challenges, as the utilization of geothermal energy is limited by the current fracturing capacity [1, 2]. The efficiency of thermal extraction depends on the complexity of the fracture network [1]. The application of liquid nitrogen (LN\({}_{2}\)) in fracturing engineering is being focused on and studied as a supplement or alternative to conventional hydraulic fracturing [3, 4, 5, 6]. The low-temperature properties of LN\({}_{2}\) can lead to thermal stresses in contact with rock, especially high temperature rocks, which can induce thermal cracks within rock [7, 8]. This can be very helpful in improving the efficiency of reservoir fracturing. The degrading effect of LN\({}_{2}\) on the mechanical properties of rock is generally investigated experimentally. It has been demonstrated that LN\({}_{2}\) can degrade the mechanical properties of different types of rocks [9, 10, 11]. The sensitivity to LN\({}_{2}\) damage is different for rock types [11]. Especially, geothermal reservoir rocks have high temperature properties [2]. The mechanical properties of high temperature rock are different from those of room temperature rock [12, 13, 14]. The initial temperature of the rocks affects the degradation of LN\({}_{2}\)[15, 16, 17]. Benefiting from the current experimental means, the relationship betweenthe temperature and the degradation of LN\({}_{2}\) has been investigated objectively from various aspects. Many studies [15; 16; 17; 18] found that the mechanical properties degraded by LN\({}_{2}\) can be enhanced with an increasing temperature. Especially over 400 \({}^{\circ}\)C, the degradation of LN\({}_{2}\) is obvious [16; 17; 19]. This is explained in that way that an increase in the initial temperature can enhance the induced thermal stress, and thus a more significant degradation effect [20]. In addition to the experimental means, numerical simulation is adopted to study the thermal treatment process [7; 21; 22] and reveals the positive correlation between the thermal damage and temperature in the process of LN\({}_{2}\) cooling. Current studies revealing the deterioration effect of LN\({}_{2}\) are mainly divided into two aspects: one is the mechanism of the initial damage caused by LN\({}_{2}\)[17; 20; 23; 24], and the other is the mechanical behavior of rocks after the LN\({}_{2}\) damage [18; 19; 25]. The current studies mostly focus on the effect of LN\({}_{2}\) on mechanical parameters, such as porosity, P-wave velocity, strength, and peak strain. The effect of LN\({}_{2}\) on the whole failure process is not sufficient. The mechanical behavior can be described as a constitutive model [26]. There are many approaches to building a constitutive model, such as the Neural network-based constitutive model [26], elastic damage constitutive model [27], and statistical damage constitutive model [28; 29]. In the mesoscopic perspective, the mechanical parameters of rock matrixes can be regarded as satisfying a certain mathematical function distribution [30]. Assuming that the mechanical parameters of the rock matrix satisfy the Weibull distribution [31; 32; 33]. The failure process of rock is represented by the number of failure matrices; the number of failure matrices is calculated using the statistical method [30]. Combining the theory of continuum mechanics, the statistical damage constitutive model can be established. The statistical damage constitutive model has been widely adopted [32; 33; 34]. The damage evolution of rock can be analyzed by establishing a damage constitutive model. At present, investigating the degradation of LN\({}_{2}\) on rock by the damage constitutive model has not been focused on. To help fill those gaps, the deterioration effect of LN\({}_{2}\) on heated granite was investigated from experimental and theoretical perspectives. The effect of LN\({}_{2}\) on the whole failure process was investigated by a statistical damage constitutive model.