Damage Evolution and Fractal Characteristics of Granite under the Influence of Temperature and Loading

Abstract

To study the damage characteristics and evolution law of granite under the influence of temperature and loading rate, uniaxial compression tests at different loading rates were performed for granite samples at 30 °C, 40 °C, 60 °C, and 80 °C. The damage characteristics and evolution law of granite were studied in conjunction with acoustic emission ringing, acoustic emission b values, acoustic emission signaling constellations, and fractal dimensions. The results showed the following: (1) At a low loading rate (0.005 mm/s), the peak stress of the granite increases gradually as the temperature increases. At high loading rates (0.01 mm/s and 0.02 mm/s), the peak stress of the granite first decreases, then increases as the temperature increases. The elastic modulus at different loading rates decreases first, then increases as the temperature increases. The elastic modulus of the granite is the lowest at 40 °C; (2) At the same temperature, specimens subjected to higher loading rates exhibit accelerated internal damage progression, with a precipitous decrease in the b-value occurring earlier. At a loading rate of 0.005 mm/s, the total number of acoustic emission ring-down events is at least 1.7 times greater than that at other loading rates, indicating a more complex development of internal cracks in the granite specimens. Moreover, as the temperature increases under a constant loading rate, the concentration of acoustic emission ring-downs intensifies during the unstable crack development and failure phases, accompanied by progressively larger fluctuations in the b-value; (3) Under the same temperature conditions, the acoustic emission signals in the specimens loaded at 0.005 mm/s and 0.01 mm/s propagate from one end to the other. At 0.02 mm/s, emissions appear simultaneously at both ends of the specimen and converge toward the center. The proportion of damage and energy associated with compression differs with temperature; at 30 °C and 40 °C, damage and energy are concentrated in the early phase of compression, whereas at 60 °C and 80 °C, they are more significant during the later stages; and (4) The fractal dimension of the granite specimens generally decreases, then undergoes a phase of fluctuating growth, followed by a decline. With increasing loading rates at a constant temperature, the compaction of the granite specimens decreases; the orderliness of the crack patterns gradually increases. As the temperature increases, the distribution of cracks during the compaction and failure phases becomes more orderly. During the crack initiation and development phases, the dispersion of cracks and the formation of crack networks are more pronounced. These findings provide a theoretical reference for understanding the microscale damage and failure mechanisms of granite under varying loading rates and temperatures.

1. Introduction

In the process of deeply buried tunnel construction, deeply buried tunnels often experience high ground temperatures, and at the same time, due to the influence of blasting, boring or other construction factors, the surrounding rock strain rate will have a greater impact on the damage characteristics of the rock body [1]. Additionally, under high temperatures and different strain rates, granite is more prone to engineering hazards such as slab cracking and rock bursting. However, the current research on the effects of high temperature and strain rate on the mechanical properties and damage evolution of granite is insufficient. Therefore, an in-depth study of the damage evolution process of high-temperature granite under different loading rates is highly important for the design and construction of deeply buried tunnels.

The mechanical characteristics of rocks exhibit notable variations when subjected to different loading rates and temperatures. In their study, Xu Xichang et al. [2] explored the mechanical behavior of granite in the Three Gorges area. They found that the mechanical properties of granite initially increase, followed by a subsequent decrease as the temperature increases. Xu Xiaoli et al. [3] conducted experiments on granite under various loading rates and temperature conditions. Their findings delineated the changes in rock mechanical properties at high temperatures and loading rates, considering the influence of temperature stages and fitting curves. Yang Haotian et al. [4] conducted high-temperature triaxial compression tests on granite. They reported that the elastic modulus of granite decreased, then increased, then decreased with increasing temperature. In a high-temperature granite test, He Ailin et al. [5] reported that the peak strength of granite increases suddenly at 300 °C. Consequently, the mechanical attributes of rocks experience significant alterations under the combined influence of temperature and loading rate, exhibiting an irregular developmental trajectory. In their investigation of the damage behavior of high-temperature rocks subjected to various loading rates, Su Guoshao et al. [6] performed high-temperature triaxial compression tests on granite. They concluded that granite exhibits maximum brittleness and fragmentation at 300 °C. Additionally, Haijian Su et al. [7] noted a transition in the damage pattern of sandstone from mixed tensile–shear damage to oblique–shear damage with increasing temperature and loading rate. Furthermore, Li Tianbin et al. [8] observed a pronounced trend of hard and brittle damage in rocks at higher temperatures within the range of 100 °C. These findings underscore the substantial changes in the mechanical properties and damage patterns of rocks under varying temperatures and loading rates. Moreover, existing research predominantly focuses on the damage and deterioration of rocks at elevated temperatures (above 100 °C), whereas investigations into the development of internal cracks and eventual failure of granite within the range of 100 °C remain relatively limited.

The process of rock damage inherently involves the generation and dissipation of energy. Acoustic emission technology serves as a real-time monitoring tool capable of sensitively tracking the generation and outcomes of microfractures within rock samples [9]. Throughout the rock damage process, parameters, such as the acoustic emission ringing energy, b-value, and RA/AF can be employed to investigate the propagation and evolution of cracks within the rock, while acoustic emission localization points offer specific insights into crack distribution. In the realm of acoustic emission parameter analysis, Kang Yumei et al. [10] discovered that the acoustic emission b-value and RA/AF can predict the rock damage point and its associated form. Moreover, Liu Xiling et al. [11] investigated the internal damage and damage patterns of graywackes at various stages under multiple loading conditions using acoustic emission ringing and acoustic emission energy. In an investigation of acoustic emission localization, Dong Zhikai et al. [12] utilized PFC simulation to validate the intrinsic correlation between acoustic emission localization and actual internal damage. Similarly, Li et al. [13] identified fractal characteristics in the distribution of acoustic emission localization points in sandstone through acoustic emission localization techniques. Building upon this, Gong Chimney et al. [14] suggested projecting acoustic emission localization points onto the X–Y and Y–Z planes and analyzing them using fractal dimensions for a more comprehensive depiction of the spatial evolution of these points. Overall, these studies illustrate how acoustic emission parameters can delineate changes in internal cracks during the rock damage process, while acoustic emission localization can portray the internal crack distribution. However, the current research still exhibits some limitations regarding the quantitative analysis of the positional distribution of localization points and the energy development process associated with these points.

At present, under the action of temperature and different loading rates, the change in the mechanical properties of granite and damage characterization of granite is generally focused on the damage of granite above 100 °C, but the highest engineering temperature encountered in current traffic tunnel construction is below 100 °C. Hence, uniaxial loading tests were conducted on granite below 100 °C, employing various loading rates. Using acoustic emission technology, this study analyzed the internal damage and development process of granite under the influence of temperature and loading rate. By integrating the acoustic emission b-value and ringing energy, along with the evolution of the acoustic emission localization during the granite loading process, this research aims to offer valuable insights into the strain-rate dependency of the internal microscopic damage and rupture mechanisms of granite under different temperatures and loading rates.

2. Experimental Methods

2.1. Test Equipment

The test apparatus is composed of a hydraulic universal testing machine with an acoustic emission system and a strain collector, as shown in Figure 1. Figure 1d shows that the WAW-1000B type testing machine has a maximum pressure of 1000 kN, which can be used for a variety of loading methods. Figure 1g shows that the expressed acoustic emission system has 6 channels and can accurately realize acoustic emission positioning and acoustic emission for the acquisition of various parameters. Figure 1h shows the vertical strain and lateral strain measurements using resistance strain gauges. Because the strain gauge resistance and sensitivity coefficient are small, to ensure accurate measurements, the DH3820 strain gauge with the smallest parameters in the device, which has a range of ±50,000 με and a minimum resolution of 0.5 με, was used. The strain indication error was not greater than 0.2% ± 2 με.

2.2. Test Scheme and Test Procedure

In the project, temperatures exceeding 30 °C in traffic tunnels significantly affect construction personnel and equipment maintenance [15,16]. Currently, high-temperature tunnels, both domestically and internationally, mostly maintain temperatures below 80 °C [17]. For testing purposes, temperatures of 30 °C, 40 °C, 60 °C, and 80 °C were selected, which are aligned with loading rates of 0.005 mm/s, 0.01 mm/s, and 0.02 mm/s, respectively, as suggested in prior studies [18] and recommendations [19].

The granite was initially heated in an electric blast drying oven at a constant temperature for 24 h. Subsequently, it was removed from the oven, wrapped in a vinyl shell, and equipped with a temperature compensation device to maintain a steady temperature. Following the temperature stabilization process, a uniaxial loading test was conducted. Simultaneously, the damage progression of the rock under loading was monitored using a strain analyzer and an acoustic emission meter. The acoustic emission meter utilized six probes arranged diagonally above and below the specimen, positioned 20 mm from the top and bottom ends of the rock. The top three probes were uniformly placed at angles of 0°, 90°, and 180°, mirroring the arrangement at the bottom end (Figure 1i). Petroleum jelly was selected as the coupling agent to minimize reflection loss and friction between the probe surface and the rock. During testing, both the acoustic emission threshold and gain were set to 40 dB. The localization settings included an overdetermination value of 20, an event definition value of 200, and a time-locked value of 400 µs. The specific test steps are depicted in Figure 1.

3. Experimental Results and Analysis

3.1. Effects of the Loading Rate and Temperature on the Elastic Modulus and Peak Stress of Granite

The variations in the peak stress and elastic modulus of the rock can provide insights into its bearing capacity and deformation capacity to some extent. The test results illustrating these findings are presented in Figure 2.

Figure 2 shows that both the temperature and loading rate have significant effects on the mechanical properties of granite. Figure 2a shows that the peak stress of granite under a lower loading rate (0.005 mm/s) increases gradually with increasing temperature, and the peak stress of granite under a higher loading rate (0.01 mm/s, 0.02 mm/s) decreases first and then increases with increasing temperature. Figure 2b shows that with increasing temperature, the elastic modulus under different loading rates first decreases and then increases to 40 °C when the elastic modulus of the granite is the lowest because the temperature of the granite first decreases and then increases [4,20]. This is because the temperature of the granite first decreases and then increases the mechanical law of change [4,20]. A microscopic study revealed that with an increase in the temperature of the granite pores to 40 °C, a sudden decrease [21] occurs with an increase in the temperature. At the same time, in the temperature interval of 50–200 °C, the formation of cracks is not enough to offset the change in porosity caused by the closure of the cracks [22]; therefore, the decrease in the porosity of the pores at 40 °C leads to the rapid entry of the rock into the pressure-tight zone of rock, which in turn leads to a decrease in the modulus of elasticity of the rock and the peak stress; however, as the increase in the temperature porosity increases, the expansion of thermal damage cracks cannot offset the effect of increased porosity, which in turn leads to the development of damage within the granite requiring a greater axial pressure.

3.2. Analysis of the Acoustic Emission Parameters

The phenomenon of elastic waves generated in all directions by material damage is called material acoustic emission [14]. Its trend corresponds to the change in stress and strain in the damage process of rock under loading. Therefore, acoustic emission signals can be monitored to reflect the accumulation of damage, crack initiation, and crack penetration in the rock. Further, acoustic emission ringing can describe the number of cracks inside the rock according to the number of signals generated by rock damage, while the acoustic emission b-value can reflect the intensity of rock damage development through fluctuations [23,24]. It can be used as a characteristic parameter of the rock damage signs, thus introducing the acoustic emission b-value and acoustic emission ringing to analyze the development of the internal damage rule of change in the rock, as follows:

$$\lg N=\mathrm{a}-\mathrm{b}M$$

(1)

where M is the magnitude, the value of which can be obtained by dividing the acoustic emission amplitude by 20; N is the number of shocks within the range of values of M, which is usually taken to be the number of impacts exceeding the acoustic emission amplitude; a is a constant, usually depending on the type of rock; and b is the proportion of small-value events relative to large-value events.

To calculate the b value accurately, the minimum number of window samples should be greater than 2000 when using the least squares method [25], and the amplitude should be calculated as 20 dB. The b value derived from Equation (1) is depicted alongside the acoustic emission ringing, acoustic emission energy, and stress and strain curves in Figure 3. Additionally, curves illustrating the changes in the acoustic emission parameters at various temperatures under different loading rates are presented, with the peak value corresponding to the σ_m-stress.

Figure 3 shows that the acoustic emission characteristic parameters of each granite specimen under different loading rates and temperatures closely align with the stress–strain curves. The b-value of granite under each condition exhibits an overall fluctuating and decreasing trend. During the elastic deformation and crack development stages, a higher loading rate correlates with greater volatility in the b-value. At each temperature, an increase in the loading rate leads to a sudden decrease in the b-value from 90% to 80% of the peak stress, indicating accelerated internal damage of the specimen. This decrease in the b-value occurs earlier with larger loading rates, highlighting its potential as a characteristic parameter for detecting rock damage signs. Furthermore, the b-value serves as a key indicator of rock damage. Under the same loading rate, the magnitude of the b-value change is approximately 0.3 at 30 °C, 0.5 at 40 °C, 0.6 at 60 °C and 80 °C, suggesting that the magnitude of the b-value change increases with temperature, exacerbating internal rock damage.

Figure 3 clearly shows that under the same loading rate, higher temperatures result in greater acoustic emission ringing counts during the unstable crack development and damage stages. Granite specimens at 60 °C and 80 °C exhibit a certain degree of centralized and delayed internal damage. Additionally, at the same temperature, the total number of ringing events at a loading rate of 0.005 mm/s is 1.7 times greater than that at 0.01 mm/s and 0.02 mm/s, indicating more complex internal crack development in the granite specimens at the 0.005 mm/s loading rate.

3.3. Granite Damage Development and Three-Dimensional Evolution of Cracks

To some degree, acoustic emission ringing and energy provide insights into the extent of crack development in rock at specific stages, while the b-value predicts rock damage. However, these metrics do not comprehensively represent the crack development trends, energy magnitudes, and resultant damage across different locations. To delve deeper into the crack development and evolution patterns of granite under varying temperatures and loading rates, we further analyzed the crack development process. This analysis combines the ratio of acoustic emission damage and energy with the distribution of acoustic emission signal points at each stage of the granite loading process.

The real-time signal point size represents the acoustic emission energy, with various colors indicating the acoustic emission ringing counts. Equations (2) and (3) were utilized to calculate the damage and energy shares at each stage of rock fracture development [26], as follows:

$$D=\frac{N_{\mathrm{d}}}{N_{\mathrm{m}}}$$

(2)

$$R=\frac{E_{\mathrm{d}}}{E_{\mathrm{m}}}$$

(3)

where N_d represents the number of acoustic emission ringing events at a specific moment in pcs; N_m represents the total number of acoustic emission ringing events after the conclusion of rock compression; E_d represents the total energy at a specific stage; E_m represents the total energy generated after the conclusion of rock compression; D signifies the rock damage variable; and R represents the energy share.

By applying the aforementioned analytical methods to scrutinize the acoustic emission spatial signal point variation pattern of granite at each stress stage, we obtained the acoustic emission spatial signal point evolution process of granite specimens at each stress stage under various temperatures and loading rates. These findings are illustrated in Figure 4, Figure 5, Figure 6 and Figure 7.

Figure 4 shows that when the temperature is 30 °C, the acoustic emission signals are first distributed discretely in the lower left corner of the specimen at 0.2 under a loading rate of 0.005 mm/s. With increasing axial loads, the number of acoustic emission signals increases and increases to the upper right corner of the specimen, while the acoustic emission signals become increasingly intense at the lower left corner of the specimen. The acoustic emission signals become increasingly powerful, which means that the cumulative damage of the rock specimen is accompanied by cumulative damage. With increasing specimen strength, the penetrating main crack inside the specimen gradually forms. Meanwhile, when the specimen is pressurized, the main crack in the specimen forms, and the bearing capacity begins to decrease. At a stress value of 0.9, the number of acoustic emission signals continues to increase, albeit with low energy, suggesting that the specimen retains its load-bearing capacity. Microcracks persist within the specimen until eventual failure, with analytical findings aligning closely with the final test results (Figure 4). At loading rates of 0.005 mm/s and 0.01 mm/s, the damage-to-energy ratio primarily emphasized the early and middle stages of fracturing, indicating greater damage during these phases than during the later stages. This trend is consistent with the distribution of the acoustic emission signals, which are particularly evident during the middle and early stages of uniaxial compression under both loading rates. These observations suggest that higher loading rates are more detrimental to the rock specimen during the early and late stages of the compression process.

Figure 5 reveals that at 40 °C, the acoustic emission signals under loading rates of 0.005 mm/s and 0.01 mm/s initially concentrate centrally in the lower right corner of the specimen. Conversely, under a loading rate of 0.02 mm/s, these signals first emerge at the upper end of the specimen before gradually propagating toward the inner part, with their number and energy intensifying while the density strengthens. Concurrently, at the same stress level of the specimen, both the quantity and energy of the acoustic emission signals increase under a loading rate of 0.02 mm/s, resulting in the most severe damage to the specimen. The ultimate damage pattern observed in the compression test of the specimen further corroborates the accuracy of the acoustic emission development process. Regarding the distribution of damage and energy percentages under each loading rate, the trends in the acoustic emission energy and damage percentages mirror those observed at 30 °C and do not necessitate repetition.

As shown in Figure 6, at 60 °C, under loading rates of 0.005 mm/s and 0.01 mm/s, discrete acoustic emission signals initially manifested in the lower left and lower right corners of the specimen. These signals then propagate inward, with a continuous increase in both number and energy, indicating the formation of through cracks and impending specimen damage. Conversely, under the 0.02 mm/s loading rate, acoustic emission signals first emerge at the ends of the specimen and progress toward the middle, ultimately leading to the formation of cracks in the upper and lower parts of the specimen, resulting in specimen damage. The acoustic emission monitoring process closely mirrors the test results. The distributions of damage and energy shares at loading rates of 0.005 mm/s and 0.01 mm/s, respectively, primarily occur in the middle and late stages of the fracturing process. However, at a loading rate of 0.02 mm/s, they predominantly occur in the early and late stages of rock fracturing. This pattern differs from that observed at 30 °C and 40 °C, where the damage and energy shares at a loading rate of 0.02 mm/s are mainly concentrated in the early and late stages of rock fracturing.

As observed in Figure 7, at 80 °C, under loading rates of 0.005 mm/s and 0.01 mm/s, the acoustic emission signal points in the early stage of specimen compression initially exhibit a discrete distribution at the lower end and then progress toward the middle and upper parts, respectively. At the 0.02 mm/s loading rate, acoustic emission signals simultaneously appear at the top and bottom ends of the specimen and then propagate toward the middle, ultimately resulting in through-crack formation. These findings closely align with the test results, demonstrating the effectiveness of acoustic emission monitoring. The distributions of damage and energy shares under loading rates of 0.005 mm/s and 0.01 mm/s, respectively, are primarily concentrated in the middle and late stages of rock fracturing. Conversely, under the 0.02 mm/s loading rate, they mainly occur in the early and late stages of uniaxial compression, mirroring the observations at 60 °C.

According to Figure 4, Figure 5, Figure 6 and Figure 7, according to the distributions of the damage and energy ratio under each loading rate, the damage is positively correlated with the energy ratio and the distribution of the acoustic emission signal. Under each temperature condition, the acoustic emission evolution laws at the different loading rates exhibit certain differences. At 0.005 mm/s and 0.01 mm/s, the acoustic emission signal first appeared at one end of the specimen and then developed to the other end; at 0.02 mm/s, the acoustic emission signal first appeared from the upper and lower ends of the specimen at the same time and then developed to the middle of the specimen. At 30 °C and 40 °C, the damage and energy percentages are mainly concentrated in the middle and early stages of uniaxial compression. Further, at 60 °C and 80 °C, the damage and energy percentages are mainly concentrated in the late stage of uniaxial compression. At 30 °C and 40 °C, the damage and energy percentages are mainly concentrated in the middle and early stages of uniaxial compression, and at 60 °C and 80 °C, the damage and energy percentages are mainly concentrated in the middle and late stages of uniaxial compression. At 60 °C and 80 °C, the damage and energy ratio are mainly concentrated in the early and late stages of uniaxial compression, respectively. This shows that the transient impact of a low loading rate on the rock specimen is small, the specimen is easily damaged from one end, cracks develop until destruction occurs, the transient impact of a high loading rate on the rock specimen is greater, the specimen is damaged at both ends first, and then the cracks are extended until destruction. At the same time, the sensitivity of temperature to damage to granite is not the same.

3.4. Fractal Features of the AE Signal Points

The fractal dimension can be used to quantitatively evaluate the development process of complex phenomena. A larger fractal dimension indicates greater chaos at this stage; conversely, a reduced fractal dimension suggests improved order. Thus, the fractal dimension can, to some extent, describe the relationship between the local laws of phenomena and their overall development process. The development of acoustic emission signal points was analyzed in the previous section. However, a mere qualitative analysis of the development chart for acoustic emission signal points yields only a general trend of specimen damage and crack development. To further quantitatively analyze the development of acoustic emission signal points in rock, the coordinates of these points are plotted in both top and side views. Additionally, the correlation dimension is introduced to explore the fractal development of cracks in the X–Y and X–Z planes (Figure 8).

The projection of acoustic emission signal points onto the top (XOY plane) and side (XOZ plane) views involves measuring the distance, L, from the coordinate to the origin on the side view, indicating the signal point’s proximity to the lower end of the rock specimen. Similarly, the distance, R, from the coordinate to the origin on the top view indicates the distance from the center of the test specimen. Here, the X- and Y-axes are situated on the bottom surface of the cylinder, with the origin of the XOY plane located at the center of the cylinder’s bottom surface and the Z-axis running parallel to the cylinder’s axis and extending upward from the base.

The G–P correlation dimension method is utilized, employing a function called MATLAB 2020, to calculate the fractal dimensions of the R and L values of the acoustic emission signal points from the origin. This calculation aims to analyze the fractal characteristics of the signal point development within the rock under various loading rates and temperatures.

The core principle of the G–P correlation dimension method involves ordering the distances of each acoustic emission signal point from the origin according to their temporal sequence, as follows:

$$X=\left[x_1{x}_{2}x\cdots x_n\right]$$

(4)

The time series’ first vector of distance from the origin for m elements, denoted as X₁, can construct an m-dimensional phase space based on the sequential time delay interval, where (N = n − m +1), and m-dimensional vectors are constructed as follows:

$$X_{\mathrm{i}}=\left[X_i,X_{i+1}\cdots X_{i+m-1}\right]$$

(5)

The G-P function is defined by the space as follows:

$$C_m\left(r\right)=\frac{1}{N^2}{\displaystyle \sum _i^N{\displaystyle \sum _i^{N}H\left[r-‖X_i-X_j‖\right]}}$$

(6)

where C represents the correlation function, H denotes the Heaviside function, and r is the scale factor, expressed as follows:

$$H=\left\{\begin{array}{l}0,x<0\\ 1,x\ge 0\end{array}\right.$$

(7)

$$r\left(k\right)=k\frac{i}{N^2}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N\left|X_i-X_j\right|}}$$

(8)

where k is the observation coefficient, and based on the value of k, the value of r is determined in the interval to obtain the association dimension D, where the expression of D is as follows:

$$D=\underset{r\to 0}{\lim }\frac{\ln C\left(r\right)}{\ln r}$$

(9)

The linear regression of the calculation outcome {lnr, lnC} more closely resembles a straight line, which implies a tendency toward fractal characteristics in the data. Points are taken at 15 s intervals, and the minimum embedding dimension m = 1 is selected for the calculation. Each correlation dimension D in the calculation results is fitted using 20 {lnr, lnC} points, ensuring a fit interval across all the data results within the range of 95–99%. This suggests that during the process of damage destruction, the distance from the acoustic emission signal points to the origin of the granite, which exhibits fractal characteristics.

These computations yield the fractal dimension of the acoustic emission signal under various temperatures and loading rates, as depicted in Figure 9.

Figure 9 shows that the fractal dimension fluctuates under different loading rates and temperatures, and the fractal dimension generally decreases, then fluctuates, and then decreases. The stages of decreasing fractal dimension in the granite specimens are the compaction stage and the destruction stage. This is attributed to the closure of internal pores during the rock compaction stage, leading to a reduction in fractal dimension. Subsequently, in the rock damage stage, as the main cracks penetrate through, the fractal dimension further decreases. Moreover, with increasing temperature, the magnitude of change in the fractal dimension during the destruction stage increases. The fractal dimension fluctuation stage includes the elastic deformation stage, stable crack development stage, and unstable crack development stage. During this stage, the crack initiation, expansion, or closure of the rock specimen, and orderly, disorderly changes of the cracks occur. This is an overall change in the direction of increasing order; thus, the fractal dimension fluctuation changes, and the amplitude of the overall crack development increases. Under identical temperature conditions, an increase in the loading rate results in a decrease in the magnitude of the fractal dimension within the range of 0.1–0.7. Specifically, a higher loading rate correlates with a lower degree of compactness in the granite specimens. The fluctuations in fractal dimension occur within the range of 0.02–0.52, transitioning between 0.15–0.52 with increasing loading rates, then to 0.1–0.3, further transforming to 0.002––0.320, and subsequently to 0.002–0.520. Notably, the fractal fluctuation between 0.002–0.212 significantly diminishes, indicating a reduction in cracks within the specimen during compression. At constant loading rates, as the temperature increases, the fractal dimension decreases within the range of 0.1–0.4, signifying stages of specimen compression density and crack distribution during the damage stage. Additionally, the amplitude of fractal dimension fluctuations increases from 0.05–0.52 at temperatures of 30 °C and 40 °C to 0.1–0.3 at temperatures of 60 °C and 80 °C. The amplitude of the change in amplitude decreases, indicating that the greater the temperature of the crack development process is, the greater the crack distribution, and the greater the temperature is, the greater the crack distribution. In the development process, with a more decentralized crack distribution, cracks cannot form in an obvious direction, and the cracks clearly form a network.

The analysis indicates that the fractal dimension of R and L can somewhat portray the rock’s traits across stages, such as compression density, stable crack development, unstable crack development, and rock damage. Furthermore, the crack evolution within the rock follows a pattern of initial enhancement, subsequent reduction, and then another enhancement. Thus, the fractal dimension provides a numerical depiction of crack evolution and destruction intensity in granite. Such an understanding can assist in predicting rock destruction and evolution processes in practical engineering scenarios.

4. Discussion

In an actual project, tunnel excavation will occur due to upper building construction, adjacent tunnel construction, and other reasons, leading to different degrees of disturbance in the surrounding rock of the tunnel, which can cause to rock damage. This paper revealed that a high-temperature tunnel at 40 °C under the influence of internal rock damage caused by a greater reduction in the mechanical properties of the rock increases the strength of the rock, but the development and destruction of rock damage are more concentrated in the late stage, resulting in tunnels being more prone to rock bursts and other engineering disasters. Using acoustic emission localization and acoustic emission localization fractals to further analyze the form of rock damage development, the loading rate of the larger internal damage of the rock multilocation development of the final crack into the network was determined. In summary, high-temperature tunnels at 40 °C should be prepared in advance to prevent tunnel peripheral rock from becoming prone to large cracks, while tunnels at 60 °C and 80 °C should experience controlled rock disturbance to reduce peripheral rock explosions and other engineering disasters.

5. Results

(1)

Under a loading rate of 0.005 mm/s, the peak stress of the granite increases gradually with increasing temperatures; under loading rates of 0.01 mm/s and 0.02 mm/s, the peak stress of the granite decreases first and then increases with increases in temperature; the modulus of elasticity under different loading rates shows a tendency to decrease first and then increase with increasing temperatures, and the modulus of elasticity is the smallest at 40 °C;

(2)

At the same temperature, the higher the loading rate is, the earlier the b value decreases, and the total number of ringing events decreases. Under the same loading rate, as the temperature increases, the change in the b-value gradually increases, and the acoustic emission ringing counts are increasingly concentrated in the unstable crack development stage and damage stage;

(3)

At loading rates of 0.005 mm/s and 0.01 mm/s, acoustic emission signals initially manifested at one end of the specimen and propagated toward the opposite end as the axial pressure changed. Conversely, with a loading rate of 0.02 mm/s, signals simultaneously emerged at both the upper and lower ends before advancing toward the middle. Under the same loading rates, damage and energy distributions at 30 °C and 40 °C were predominantly observed in the early and middle stages of uniaxial compression, respectively. In contrast, at 60 °C and 80 °C, damage and energy were more concentrated in the later stages of compression;

(4)

Under the same temperature conditions, as the loading rate increased, the compressive density of the granite specimen obviously decreased, and the cracks in the process of compression decreased; at the same loading rate, as the temperature increased, the compressive density of the specimen increased, and the cracks in the distribution of the order became more damaged to enhance the development process. As the temperature increased, the cracks in the more dispersed cracks in the network became more obvious.