Mechanical properties and damage characterization of cracked granite after cyclic temperature action

doi:10.21203/rs.3.rs-5007610/v1

To study investigates the mechanical properties and damage characteristics of granite under the influence of cyclic temperature and crack inclination, uniaxial compression tests were conducted on granite with prefabricated cracks at cyclic temperatures of 30 °C, 50 °C, 70 °C, 100 °C, and 130 °C. The mechanical properties and damage characteristics of the granite were analyzed using characteristic stress, acoustic emission parameters, damage variables, fractal dimensions, and SEM. The results indicate that: (1) At the same cyclic temperature, the granite with a 45 ° prefabricated crack exhibits the lowest peak stress, elastic modulus, and σci/σm, resulting in a significant reduction in the specimen's load-bearing capacity. At the same crack inclination, the granite at 70 °C shows the highest peak stress, elastic modulus, σ_cc/σ_m, and σ_ci/σ_m, with a noticeable reduction in internal microcracks. (2) At the same cyclic temperature, the b-value drop point of acoustic emissions in the 45 ° prefabricated crack specimen occurs earlier, indicating the earlier formation of the main crack inside the specimen. At the same crack inclination, the cumulative ring count and total energy of acoustic emissions are the lowest at 70 °C, indicating the least thermal damage to the specimen. (3) At the same cyclic temperature, the damage in the 45 ° prefabricated crack granite specimen develops more rapidly, with the damage variable reaching above 0.8 at the damage stress point, and the fluctuation in fractal dimension is minimal, indicating strong orderliness in the development of internal microcracks. At the same crack inclination, the fractal dimension of the granite specimen significantly increases at 70 °C, and the dispersion of damage proportion increases significantly after 70 °C. (4) At a cyclic temperature of 70 °C, the changes in cracks and pores in the granite uniaxial compression specimen are evident, with relatively mild damage; the final failure modes of specimens under various cyclic temperatures and crack inclinations differ significantly, transitioning from tensile-shear failure to shear failure and tensile failure as the crack inclination increases. The research results can provide theoretical references for understanding the mesoscopic damage and failure mechanisms of granite under different cyclic temperatures and crack inclinations.

Physical sciences/Engineering

Physical sciences/Engineering/Civil engineering

Single-crack granite

cyclic temperature

characteristic stress

acoustic emission parameters

damage

fractal dimension

In recent years, high temperatures have become an increasingly important issue in underground geotechnical engineering[1,2], with problems such as repeated temperature fluctuations occurring during high-temperature tunnel excavation, ultra-deep drilling, and geothermal exploitation[3]. Natural cracks in rock can guide and accelerate the damage and failure of surrounding rock, leading to engineering disasters like cracking and instability. Therefore, it is crucial to study the changes in mechanical properties and the damage and instability behavior of cracked rocks under cyclic temperature effects for the safe construction of geotechnical engineering.

Numerous studies have been conducted both domestically and internationally on the impact of cyclic temperature on rock. Li et al conducted Brazilian splitting tests on granite subjected to cyclic temperatures ranging from 100 ℃ to 700 ℃, finding that the mechanical properties of granite decline continuously as the temperature increases[4]. Yang et al performed uniaxial compression tests on granite subjected to cyclic heating and water cooling, discovering that the mechanical properties of the specimens improve to some extent at 50 ℃ and 150 ℃[5]. Other researchers have explained the changes in the mechanical properties of rocks under cyclic temperature from the perspective of internal structural and chemical changes. For example, Yang studied the impact of thermal-cold shocks at 600 ℃ on rock porosity, finding that pores in the rock develop severely and form clusters under these conditions[6]. Kumari conducted thermal shock tests on granite and found that the minerals inside the granite undergo phase transitions at 573 ℃[7]. These studies indicate that the effect of cyclic temperature on the mechanical properties of rock is not linear and significantly impacts the internal structure and chemical properties of the rock. In studies on the impact of cyclic temperature on rock damage, Xia et al confirmed through temperature cycling tests that the damage caused by cyclic temperature on rocks is irreversible and cumulative[8]. Zhu et al observed macrocracks, expansion, and penetration in granite through SEM analysis at temperatures exceeding 300 ℃[9]. Jia et al found that the failure mode of granite shifts from tensile-shear failure to tensile failure as the temperature increases from 200 ℃ to 400 ℃[10]. These findings suggest that rock damage after cyclic temperature is irreversible and that the failure mode changes. In research on the effects of repeated temperature cycling on cracked rocks, Wang et al used acoustic emission to study the impact of different crack angles on rock damage, finding that damage is greater at a 45 ° crack angle, with a significant decrease in mechanical properties[11]. Song et al studied freeze-thaw cycling in double-crack rocks and discovered that initial damage decreases as the crack angle increases, but damage grows rapidly in the later stages of compression[12]. Wang et al used numerical simulations to study granite with triangular cracks under high temperatures, finding that thermal stress is concentrated around the prefabricated cracks, and thermal stress fields become uneven[13]. Li et al studied the impact of thermal cycling on cracked rocks, finding that the damage area of cracked rocks is smaller than that of intact rocks after thermal cycling[14].These studies have primarily focused on mineral decomposition, changes in physical and mechanical properties, the impact of crack angle, and crack development characteristics. However, most of the temperatures studied have been above 100 ℃, while in current high-temperature tunnel excavations, the issue is the large-scale damage to surrounding rock caused by repeated cooling and heating cycles within 100 ℃. There are few reports on the changes in mechanical properties and damage characteristics of cracked rocks under cyclic temperatures between 30 ℃ and 100 ℃.

Therefore, considering the actual conditions in high-temperature tunnel construction, where the rock temperature rises again after temporary cooling measures, this paper conducts uniaxial compression tests on cracked granite under cyclic temperature effects. The mechanical properties and damage characteristics during the loading process are studied and analyzed. The results provide theoretical support and reference for stability control and support structure design of surrounding rock under cyclic temperature effects.

The experimental equipment mainly consists of a loading system, an acoustic emission system, and a strain analyzer, as shown in Figure 1. The loading system uses a TWA-2000 kN rock triaxial mechanical testing system, with a maximum pressure of 2000 kN, capable of directly measuring the axial stress-strain during the compression process, as shown in Figure 1(j). The SENSOR acoustic emission system accurately measures various acoustic emission parameters, as shown in Figure 1(g). The strain analyzer has a measurement range of ±50000 με, with a strain indication error of no more than 0.2% ± 2με, as shown in Figure 1(f).

Granite specimens from the Sangzhuling Tunnel were drilled and polished to obtain cylindrical specimens with a height of 100mm and a diameter of 50 mm. The specimen surface was then cut with a crack cutting machine to create cracks at angles of 0 °, 45 °, and 90 ° with respect to the horizontal direction. The cracks had a depth of 4mm, a length of 50 mm, and a width of 1mm, as shown in Figure 1(a). The cyclic temperatures selected were 30 ℃ 50 ℃, 70 ℃, 100 ℃, and 130 ℃ as the maximum heating temperatures, then cooled down to 15 ℃. Each temperature cycle was repeated five times. The specific experimental steps are as follows:

(1) Cyclic Heating-Cooling Test: Before heating, the specimen was wrapped in a

high-temperature waterproof bag to ensure that water does not affect the specimen during the temperature cycle, as shown in Figure 1(b). The specimen was then heated in an electric blast drying oven for 24 hours to ensure uniform heating, as shown in Figure 1(c). After heating, the specimen was cooled in a constant temperature refrigerator (15 ℃) for 4 hours, and its temperature was monitored with a temperature gun to ensure complete cooling. To ensure waterproofing, the high-temperature waterproof bag was replaced before each cycle, and the above steps were repeated until the required number of cycles was reached.

(2) Uniaxial Compression Test: The uniaxial compression was performed using a displacement loading method with an axial stress loading rate of 0.01 mm/s. During the compression process, the acoustic emission system and strain analyzer were used to monitor rock damage and deformation. Six acoustic emission probes were selected, distributed in two groups at the upper and lower ends of the specimen, each group placed approximately 20 mm from the end. The upper probes were arranged at 0 °, 90 °, and 180 °, while the lower probes were arranged at 180 °, 270 °, and 360 ° to enhance signal strength and improve accuracy, as shown in Figure 1(h). In the strain analysis system, strain gauges were attached laterally to the middle of the specimen, approximately 50 mm from both ends, to monitor lateral strain during compression, as shown in Figure 1(e).

The stability of the surrounding rock is closely related to the physical and mechanical properties of the rock. After five temperature cycles, the mechanical properties of granite specimens with different crack angles and cycle temperatures are shown in Figs. 2 and 3.

From Fig. 2, it is evident that after five temperature cycles, both the peak strength and elastic modulus of the rock samples initially decrease, then increase, and finally decrease again with the rise in temperature. The average peak strength values are 183.51 MPa, 161.10 MPa, 169.78 MPa, 157.43 MPa, and 111.17 MPa, respectively, while the average elastic modulus values are 33.40 GPa, 27.22 GPa, 28.56 GPa, 23.23 GPa, and 16.76 GPa, respectively. Between 30 ℃ and 70 ℃, the peak strength and elastic modulus of granite fluctuate, and beyond 70 ℃, both parameters rapidly decline. Although the peak strength and elastic modulus of cracked granite exhibit slight fluctuations with temperature increase, the overall trend shows a decrease.

Figure 3 indicates that as the crack angle increases, the average peak strength and elastic modulus of granite under different cycle temperatures show a trend of initial decrease followed by an increase. The average peak strength values are 163.7 MPa, 153.01 MPa, and 153.09 MPa, while the average elastic modulus values are 28.35 GPa, 23.68 GPa, and 25.47 GPa, respectively. The peak stress and elastic modulus reach their minimum values at a 45 ° crack angle. This is because the rock specimens with a prefabricated 45 ° crack predominantly exhibit compressive-shear failure, whereas those with a 90 ° prefabricated crack mainly undergo tensile failure (refer to Section 5.2 for failure modes). The difference in failure modes results in the regular variation of peak strength and elastic modulus. Overall, the ability of granite to resist deformation weakens with the increase in cycle temperature, its ductility increases, and as the prefabricated crack angle increases, the ability to resist deformation and ductility first weakens (weakest at 45 °) and then increases.

3.1 Analysis of Stress Characteristic Point Variations

While peak stress and elastic modulus can reflect the stability of the surrounding rock during engineering excavation, they cannot characterize the internal damage and microstructural changes within the rock. Therefore, by studying the stress damage thresholds (stress characteristic points) at each stage of rock compression, the internal damage and microstructural characteristics of granite under the influence of cyclic temperature and cracks can be determined.

Currently, methods for calculating the stress cracking point include the volumetric strain method[15], acoustic emission method[16], lateral strain difference method[17], and dissipated energy rate method[18]. Most scholars believe that the energy generated during the compression of rock is directly related to its deformation and failure[19]. Therefore, when determining stress characteristic points, the point at which the elastic energy rate reaches its maximum under axial pressure is defined as the closure stress (σ_cc). Afterward, as axial pressure continues to increase, the rock enters a period of calm crack development, where the dissipated energy decreases linearly. The end of this linear development of dissipated energy is defined as the crack initiation stress (σ_ci)[20]. After this, the cracks within the rock enter an unstable stage, and the proportion of dissipated energy increases. When damage fully develops, the dissipated energy rate reaches a minimum, thereby determining the damage stress (σ_cd)[21]. As can be seen from the above analysis, from the perspective of energy dissipation, the evolution of cracks and determination of stress characteristics have a solid theoretical background. Therefore, the dissipated energy rate method is selected to calculate the characteristic stress points, which allows for accurate determination of these points during the compression of rock specimens[22]. The specific calculation method and determination of characteristic points are as follows:

The total energy of the rock can be calculated using the following expression:

$${U_0}={U_{\text{e}}}{\text{+}}{U_{\text{d}}}{\text{=}}\int {{\sigma _{{\text{1i}}}}{\text{d}}{\varepsilon _{\text{1}}}{\text{=}}\sum\limits_{{{\text{i=1}}}}^{{\text{n}}} {\frac{{\text{1}}}{{\text{2}}}} } \left( {{\sigma _{{\text{1i}}}}{\text{+}}{\sigma _{{\text{1i-1}}}}} \right)\left( {{\varepsilon _{1{\text{i}}}} - {\varepsilon _{1{\text{i}} - 1}}} \right)$$

1

Where U₀ is the total energy generated by the failure of rock per unit volume (kJ·m^− 3); U_e is the elastic energy (kJ·m^− 3); U_d is the dissipated energy (kJ·m^− 3); σ_1i and σ_1i−1 are the stresses at the time of collection (MPa); ε_1i and ε_1i−1 are the axial strains at the time of collection; and i is the collection sequence.

The elastic energy U_e in Eq. (1) can be calculated using the following formula:

$${U_e}=\frac{1}{2}{\sigma _1}\varepsilon _{1}^{e} \approx \frac{{\sigma _{1}^{2}}}{{2{E_0}}}$$

2

Where σ₁ is the principal stress (MPa); and E₀ is the initial elastic modulus (GPa).

$${U_d}={U_0} - {U_e}=\sum\limits_{{{\text{i=1}}}}^{{\text{n}}} {\frac{{\text{1}}}{{\text{2}}}} \left( {{\sigma _{{\text{1i}}}}{\text{+}}{\sigma _{{\text{1i-1}}}}} \right)\left( {{\varepsilon _{1i}} - {\varepsilon _{1i - 1}}} \right) - \frac{{\sigma _{1}^{2}}}{{2{E_0}}}$$

3

The dissipated energy U_d generated by rock compression in Eq. (1) can be calculated using the following formula:

$${U_d}={U_0} - {U_e}=\sum\limits_{{{\text{i=1}}}}^{{\text{n}}} {\frac{{\text{1}}}{{\text{2}}}} \left( {{\sigma _{{\text{1i}}}}{\text{+}}{\sigma _{{\text{1i-1}}}}} \right)\left( {{\varepsilon _{1i}} - {\varepsilon _{1i - 1}}} \right) - \frac{{\sigma _{1}^{2}}}{{2{E_0}}}$$

3

An example of the determination of characteristic stress points using dissipated energy is shown in Fig. 4.

Characteristic stress points are thresholds in the development stage of rock damage. The ratio of characteristic stress to peak stress can indicate the extent of internal damage in the rock. The larger the ratio, the greater the proportion of total damage at that stress threshold. The ratio of closure stress to peak stress σ_cc/σ_m represents the development extent of pore closure during rock compression[23]. The ratio of crack initiation stress to peak stress σ_ci/σ_m is significantly affected by internal defects and can serve as a criterion for rock integrity[24]. The ratio of damage stress to peak stress σ_cd/σ_m directly reflects the long-term strength of the rock and indicates the stage where cracks within the rock enter instability[25]. The experimental results of characteristic stress and the ratio of characteristic stress to peak stress under temperature cycles are shown in Table 1 and Fig. 5.

Table 1

Characteristic stress values and ratios of crack-containing granite under the action of temperature cycling
Angle	Temperature	Characteristic stress value(MPa)				Characteristic stress/peak stress
Angle	Temperature	σ_cc	σ_ci	σ_cd	σ_m	σ_cc/σ_m	σ_ci/σ_m	σ_cd/σ_m
0°	30	3.81	51.18	165.2	182.29	2.09℅	28.07%	90.62%
	50	1.03	48.95	165.03	176.88	0.58%	27.67%	93.59%
	70	1.67	74.04	173.8	185.16	0.905℅	39.99%	93.86%
	100	1.97	54.49	157.37	166.98	1.18%	32.63%	94.24%
	130	4.52	32.49	102.07	111.88	4.043%	29.03%	91.22%
45°	30	2.21	47.90	171.35	183.25	1.20%	26.14%	93.50%
	50	0.83	44.07	157.73	173.63	0.47%	25.43%	90.84%
	70	4.81	50.82	157.18	168.19	1.36%	30.21%	93.55%
	100	2.55	40.93	135.03	145.32	1.759%	28.16%	92.92%
	130	1.00	24.68	97.56	112.64	1.88%	21.91%	90.61%
90°	30	3.03	48.84	170.33	185.32	1.63%	26.35%	91.91%
	50	2.97	38.37	144.98	156.49	1.54%	24.52%	92.64%
	70	1.44	54.68	142.53	157.45	0.91%	34.73%	90.52%
	100	5.90	50.63	140.66	161.31	3.68%	31.58%	90.74%
	130	0.55	30.83	106.54	117.21	3.47%	26.30%	90.90%

From Fig. 5(a), it is observed that the σ_cc/σ_m of cracked granite specimens decreases first and then increases with temperature rise for the same crack angle. The fluctuation range of σ_cc/σ_m with crack angle variation is within 2%, indicating that temperature has a significant effect on σ_cc/σ_m, while the crack angle has a minimal impact. Figure 5(b) shows that σ_ci/σ_m of cracked granite increases first and then decreases with temperature rise for the same crack angle, with a maximum at 70 ℃. At the same temperature, σ_ci/σ_m decreases first and then increases with the increase of crack angle, with a larger fluctuation at a 0 ° crack angle, with variation values of 15.46% and 12.33%. Figure 5(c) shows that while temperature and crack angle affect the σ_cd/σ_m of cracked granite, the pattern is not evident, with a variation range of about 3% σ_m. This is because σ_cc/σ_m, σ_ci/σ_m, and σ_cd/σ_m are significantly influenced by internal pores, prefabricated cracks, natural cracks, the overall structure, and chemical properties of the rock. Compared to crack angle, temperature changes have a more significant impact on the overall structure of the rock.

3.2 Analysis of Acoustic Emission Parameters

The variation trends of rock mechanical properties can only qualitatively describe the impact of internal defects on rock damage and failure. However, it is difficult to quantitatively analyze the number of internal cracks and the severity of crack development during compression. Acoustic emission (AE) ringing represents the change in the number of internal cracks during rock loading, and AE energy describes the extent of crack propagation. Additionally, the fluctuation of the AE b-value can directly reflect the degree of internal damage development in the specimen[26], and the sudden drop point of the b-value can serve as a characteristic parameter indicating rock failure[27]. A smaller b-value suggests more severe development of microcracks within the rock. Therefore, by introducing AE ringing, AE energy, and AE b-value analyses, the damage and failure evolution within the rock can be studied. The b-value is calculated using the following formula:

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

4

In the equation, M represents the magnitude, which is calculated as the amplitude of acoustic emission divided by 20. N denotes the number of occurrences within the range of M, typically representing the number of impacts exceeding the acoustic emission amplitude. a is a constant, usually defined based on the type of rock, and b is the ratio of small amplitude events to large amplitude events[28].

After cyclic temperature effects, the changes in acoustic emission ringing, energy, b-value, and stress-strain curves during the compression process of granite specimens with different crack angles are shown in Fig. 6. The characteristic stress points and b-value drop points in the rock compression process are marked. In Fig. 6(a), 66% σ_m represents the stress percentage at the point where the acoustic emission b-value drops, and "30 − 0" means the specimen temperature is 30 ℃ with a crack angle of 0 °. The other numbers in the figures have similar meanings.

From Fig. 6, it can be seen that under different temperatures, the magnitude of the acoustic emission b-value first increases and then decreases with the increase of the crack angle. There is no significant change in the b-value during the σ_cc/σ_m stage, while during the σ_cc-σ_ci stage, the b-value rises continuously and reaches its maximum, indicating that a large number of microcracks appear inside the rock in this stage, but the degree of crack development and penetration is low. When the specimen enters the σ_ci-σ_m stage, the b-value fluctuates and decreases, indicating that microcracks develop rapidly until macroscopic cracks form, leading to complete rock failure and the b-value reaching its minimum. At the same temperature, as the crack angle increases, the average b-value drop point decreases from 80%σ_m (0 ° prefabricated crack) to 60% σ_m (45° prefabricated crack) and then increases to 75%σ_m (90 ° prefabricated crack), with fluctuations in the b-value drop point. This indicates that the prefabricated cracks influence and guide the development of internal cracks in the rock, significantly enhancing the intensity of internal damage development and leading to the early formation of large-scale cracks. At the same crack angle, the change in the b-value drop point with increasing temperature is not obvious, indicating that the temperature has little impact on the formation of large-scale cracks inside the specimen.

Furthermore, Fig. 6 shows that under temperatures of 30 ℃, 50 ℃, and 70 ℃, the acoustic emission ringing and energy distribution of each crack angle advance significantly with increasing temperature. Under 100 ℃ and 130 ℃, the acoustic emission ringing and energy distribution of the 0° crack first delay and then advance with increasing temperature, while for the 45 ° and 90 ° cracks, they continuously delay with increasing temperature, with the 90 ° crack showing a significant delay. This indicates that temperature has a considerable impact on the pores and natural cracks inside the rock, affecting the distribution of acoustic emission ringing and energy in the early stages of compression, and that crack angle can guide the development of internal cracks. Therefore, both temperature and crack angle have significant effects on the distribution of acoustic emission ringing and energy in the specimen.

To quantitatively analyze the variation pattern of acoustic emission parameters under the influence of cyclic temperature and crack angle, the trends in cumulative ringing count and total energy of granite with cracks under the influence of temperature and crack angle are shown in Fig. 7.

From Fig. 7, it can be seen that with increasing cyclic temperature, the cumulative ringing count and total energy of granite with cracks first increase, then decrease, and increase again, reaching a minimum at 70 ℃. At the same temperature, the cumulative ringing count and total energy of acoustic emission first decrease and then increase with increasing crack angle, with a minimum at 45 °. The above analysis shows that under the same temperature, the cumulative ringing count and total energy of the specimen with a 45 ° crack angle are the smallest, indicating that the 45° crack angle facilitates the guidance of rock damage and failure. The internal microcracks in the rock quickly merge to form large cracks, while the guidance of internal crack development is significantly reduced in specimens with 0 ° and 90 ° cracks. As a result, large-scale cracks cannot form quickly, leading to better crack development and a significant increase in cumulative ringing count and total energy.

3.3 Analysis of Damage Development Characteristics

The acoustic emission b-value, cumulative ringing, total energy, and energy distribution can indicate the intensity, total amount, and distribution of damage influenced by cyclic temperature and prefabricated cracks. However, a quantitative analysis of the damage development process during the compression of rocks is still lacking. Therefore, a damage variable is introduced to further quantitatively analyze the damage development characteristics of granite with cracks under cyclic temperature. The sudden increase in the damage variable represents the large-scale initiation of microcracks inside the specimen[29].

The damage variable can be defined as:

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

4

Where N_d represents the cumulative ringing count of acoustic emission at time d from the start of the axial compression test, and N_m represents the cumulative ringing count of acoustic emission at the end of the axial compression test. Based on the above calculation, the damage variables of granite under cyclic temperature and crack angle were calculated, and the results are shown in Fig. 8. In Fig. 8(a), "30 − 0" indicates that the specimen temperature is 30 ℃ and the crack angle is 0 °, with the same meaning for the other values.

From Fig. 8, it can be seen that the damage variable of each specimen under different conditions shows a trend of slow development, followed by rapid growth, and finally stable development. There is no significant change in the damage variable during the 0-σ_ci stage; however, during the σ_cc-σ_ci stage at 70 ℃, the damage variable increases significantly, indicating a rapid increase in the number of microcracks inside the specimen during this stage, leading to an early development of damage. The σ_ci-σ_m stage is the phase where the damage variable rises sharply, indicating the initiation, expansion, and continuous accumulation of microcracks inside the specimen until the specimen reaches σ_m, where the damage variable approaches its maximum value of 1, and macroscopic cracks penetrate the specimen, leading to instability and failure. Under the same temperature, the damage variable of the granite specimen with a 45 ° crack increases earlier, and when the stress reaches σ_cd, the damage variable of the 45 ° crack specimen reaches above 0.8. The damage variable develops rapidly during the σ_ci-σ_cd stage, indicating that after reaching σ_ci, the internal cracks of the 45 ° crack specimen develop rapidly, with the 45 ° crack significantly enhancing the guidance of microcracks inside the specimen.

To further analyze the damage proportion in each characteristic stress stage under the influence of cyclic temperature and crack angle, the trend of damage proportion in each characteristic stress stage of granite under the influence of cyclic temperature and crack angle is shown in Fig. 9.

From Fig. 9, it can be seen that under cyclic temperatures of 30 ℃ and 50 ℃ and different prefabricated crack angles, the maximum damage proportion is almost entirely concentrated in the σ_ci-σ_cd stage, indicating that at these temperatures, the internal microcracks of the granite specimen initiate and expand rapidly from the crack initiation stress to the damage stress stage, leading to accelerated specimen damage. However, after the cyclic temperature exceeds 70 ℃, the damage proportion begins to show dispersed distribution, and the higher the temperature, the greater the dispersion of the damage proportion, indicating that the thermal damage effect inside the specimen becomes more evident after 70 ℃. This conclusion is consistent with that of section 4.1.

3.4 Analysis of Damage Fractal Characteristics

Fractal dimension, as a mathematical tool for analyzing irregular objects, can quantitatively describe irregular objects to reveal their intrinsic relationships. The crack development process during rock compression follows a nonlinear pattern with regularity. To describe the intrinsic relationships of this nonlinear process, the fractal dimension can be used for analysis.

To further analyze the damage distribution during the granite specimen's damage development process, the correlation dimension of the acoustic emission ringing count is introduced to characterize the chaos degree of the damage distribution within the rock[30, 31]. This approach is used to explore the influence of cyclic temperature and crack inclination on the damage distribution and development patterns within the rock at various stages. The correlation dimension calculation considers the acoustic emission ringing counts of granite specimens under compression during four stages: 0-σ_cc, σ_cc-σ_ci, σ_ci-σ_cd, and σ_cd-σ_m, with each stress stage being treated as a separate time window for analysis, leading to the calculation of the fractal dimension D.The calculation method is as follows:

The G-P correlation dimension is determined by first constructing a phase space using a sample of acoustic emission ringing counts n and forming an m-dimensional space (m < n ) as follows:

$$X=\left[ {{x_1}{x_2}x \cdots {x_n}} \right]$$

5

Where the first vector X₁ is created by taking the first m elements of the time series of the ringing counts. A phase space of dimension m is then constructed by sequentially selecting elements at intervals, leading to N = n - m + 1 vectors:

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

6

The G-P function is then defined as:

$${C_m}\left( r \right)=\frac{1}{{{N^2}}}\sum\limits_{i}^{N} {\sum\limits_{i}^{N} {H\left[ {r - \left\| {{X_i} - {X_j}} \right\|} \right]} }$$

7

Where C is the correlation function, H is the Heaviside function, and r is a scale factor. The correlation dimension D is then given by:

$$H=\left\{ \begin{gathered} 0,x<0 \hfill \\ 1,x \geqslant 0 \hfill \\ \end{gathered} \right.$$

8

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

9

The equation includes the observational coefficient k, which is typically chosen between 10 and 20, to determine the value of r within a certain interval, thereby obtaining the correlation dimension D. The expression for D is as follows:

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

10

Based on the G-P correlation dimension model calculation described above, the computed data points {lnr, lnC} were linearly fitted. The closer the fit to a straight line, the more fractal characteristics the data set exhibits. In this study, each stress stage during the compression of the specimen was treated as an interval, and the minimum embedding dimension m = 1 was chosen for a single calculation. By fitting 15 {lnr, lnC} calculation points, the fitting interval of the data was determined to be between 93% and 99%, indicating that the ringing count of the granite specimen under the influence of temperature and crack angle exhibits fractal characteristics across the four stages of 0-σ_cc、σ_cc-σ_ci、σ_cd-σ_m、σ_m-post-peak.

Using this model, the ringing counts under different stress stages were calculated for the granite specimens subjected to varying cyclic temperatures and crack inclinations. The results are shown in Fig. 10.

Analysis of Fig. 10, it is observed that under the same crack inclination, the correlation dimension D exhibits a trend of first decreasing, then increasing, and finally decreasing again with increasing temperature. This indicates that the internal damage development in the rock initially becomes more ordered, then less ordered, and finally more ordered again, with the maximum correlation dimension occurring at 70 ℃, which acts as a turning point for temperature. For the same temperature, the impact of crack inclination on the correlation dimension D is not significant during the 0-σ_cc stage. However, during the σ_cc-σ_ci, σ_ci-σ_cd, and σ_cd-σ_m stages, the fluctuation in D with increasing crack inclination decreases from 0.19 (0 °) to 0.12 (45 °) and then increases to 0.39 (90 °).

This suggests that at 70 ℃, the distribution of microcracks within the specimen is chaotic and disordered, unable to form a clear development trend. The fluctuations in D during the σ_cc-σ_ci, σ_ci-σ_cd, and σ_cd-σ_m stages are more significant under 0 ° and 90 ° crack inclinations due to poor guidance of internal crack development, leading to dispersed damage development and less distinct crack growth paths.

4.1 Damage Mechanism Under Cyclic Temperature

Cyclic temperature directly affects the microscopic physical properties of the rock, such as pores and microcracks. Scanning Electron Microscopy (SEM) was used to further analyze the damage mechanism of granite under cyclic temperature. The granite specimens were subjected to cyclic temperatures of 30 ℃, 50 ℃, 70 ℃, 100 ℃, and 130 ℃, and then slices were taken from the middle part of the specimens for SEM analysis. Given space limitations, only partial results for each specimen group are presented in Fig. 11, with micro-pores and microcracks marked. The SEM images reveal that internal pore and microcrack changes are not significant at 30 ℃ and 70 ℃. However, at 50 ℃, 100 ℃, and 130 ℃, the number of internal pores and microcracks increases significantly, indicating a marked increase in rock damage. This observation is consistent with the conclusions drawn earlier in Sections 3.3 and 3.4.

To quantitatively analyze the SEM images under different temperatures, the images were binarized, and the box dimension was used to evaluate the surface roughness of the granite[32]. A higher box dimension F indicates lower surface roughness, and vice versa. The results are shown in Fig. 12. Analysis of Fig. 12 shows that the box dimension F decreases, then increases, and then decreases again with increasing temperature. This trend suggests that the changes in microcracks and micropores within the granite initially increase, then decrease, and then increase again with rising temperature, with a significant drop in the box dimension observed between 70 ℃ and 100 ℃, indicating that the internal microcracks and micropores are most pronounced between these temperatures. Thus, 70 ℃ serves as a temperature turning point.

According to the reference[33], at 70 ℃, the free water and weakly-bound water within the specimen tend to evaporate and escape, leading to increased friction between the crystal particles inside the granite. As a result, during loading, the granite specimen requires higher axial pressure to resist internal crystal friction, which in turn enhances the mechanical properties such as strength, elastic modulus, and crack initiation stress. However, at 50 ℃, 100 ℃, and 130 ℃, phenomena such as mineral crystal expansion, surface spalling, and significant increases in pores and microcracks occur within the granite, indicating a substantial development of thermal damage[34]. Consequently, the mechanical properties of the granite specimen significantly decline, consistent with the analysis conclusions in Sections 3, 3.1, and 3.2.

4.2 Damage Mechanism Under Crack Inclination

The crack inclination has a guiding effect on the damage and failure of granite. To further analyze the impact of crack inclination on damage and failure, the failure patterns of granite specimens with different cyclic temperatures and crack inclinations were examined, as shown in Fig. 13.

The analysis of Fig. 13 shows that the final failure modes of rocks with different crack inclinations at the same temperature are not the same. The failure mode under a 0° crack is dominated by tensile-shear failure, under a 45 ° crack by distinct shear failure, and under a 90 ° crack by tensile failure, with a significant reduction in shear failure development. Under a 45 ° crack, the final failure pattern retains better integrity, and the specimen is not completely pulverized after removing the protective film. This is because, during compression, the rock first develops along internal flaws[35]. Thus, the 45° crack guides the development of distinct diagonal cracks, leading to clear shear failure and rapid penetration of localized cracks. Therefore, the internal microcracks under a 45 ° crack are not fully developed, resulting in better integrity of the final failure pattern. This observation is consistent with the conclusions drawn earlier in Sections 3.1, 3.2, and 3.4.

1. At the same cyclic temperature, the peak stress, elastic modulus, and σ_ci/σ_m of granite initially decrease and then increase with the increase in crack inclination, reaching their minimum values at a crack inclination of 45 °. At the same crack inclination, the peak stress, elastic modulus, σ_cc/σ_m, and σ_ci/σ_m of granite initially decrease, then increase, and decrease again with increasing temperature, with 70 °C as the turning point.

2. At the same cyclic temperature, the damage drop point occurs earlier at 45°, with the 45 ° prefabricated crack guiding the development of internal cracks in the rock. At the same crack inclination, the cumulative ring count and total energy of acoustic emissions are the lowest at 70 °C. indicating the minimal damage to the rock specimen. The temperature has no significant impact on the b-value drop point.

3. At the same cyclic temperature, the rate of increase in the damage variable initially increases and then slows down with the increase in crack inclination. At the same crack inclination, the dispersion of damage proportion continuously increases with increasing temperature. Additionally, at a crack inclination of 45°, the fluctuation in fractal dimension is minimal, with strong orderliness and directionality in the development of internal microcracks. At 70°C, the fractal dimension is at its maximum, with internal microcracks in the specimen becoming more dispersed.

4. At a cyclic temperature of 70 °C, the number of internal microcracks and pores is relatively small. The crack inclination significantly influences the internal crack development trend in the specimen. As the crack inclination increases from 0 ° to 90 °, the specimen's failure mode transitions from tensile-shear failure (0 °) to shear failure (45°) and tensile failure (90 °). With increasing temperature, the internal microcracks and pores in the granite specimen initially increase, then decrease, and increase again, with the specimen's damage following a similar trend, with 70°C as the turning point.

Date availability

All data generated or analysed during this study are included in this published article [and its supplementary information files].

Funding

This research is supported by the Basic Scientific Research Business Fee Project of Universities Directly under the Inner Mongolia Autonomous Region(Grant No.2024XKJX009);Natural Science Foundation of Inner Mongolia of China(Grant No.2024LHMS05044);

Author Contribution

Xiankai Bao and Lingyu Wang designed the study and supervised the project. Guangqin Gui, Baolong Tian and Jianlong Qiao collected data. Shunjia Huang and Lizhi Wang performed the statistical analysis. Xiankai Bao and Lingyu Wang wrote the initial paper. All the authors read and approved the final manuscript.

Acknowledgements

The authors of the article sincerely appreciate and thank all the people who made it possible to implement this project.

Data Availability

All data generated or analysed during this study are included in this published article [and its supplementary information files].

Yan, J. et al. Inoculation and characters of rockbursts in extra-long and deep-lying tunnels located on Yarlung Zangbo suture.Chinese Journal Of Rock Mechanics and Engineering.38(04),769-781 (2019).
Duan,ZF. et al. Potential estimation of hot dry rock geothermal resources in sedimentary basins: a case study from Jianhu Uplift，Subei Basin. Journal Of China University Of Petroleum( Edition of Natural Science).48(01), 46-54 (2024)
Yan, J. et al. Inoculation and characters of rockbursts in extra-long and deep-lying tunnels located on Yarlung Zangbo suture. Chinese Journal Of Rock Mechanics And Engineering.38(04),769-781 (2019).
Li, C. et al. Brazilian split characteristics and mechanical property evolution of granite after cyclic cooling at different temperatures. Chinese Journal Of Rock Mechanics And Engineering.39(09),1797-1807 (2020).
Yang, M. et al. Mechanical Properties and Failure Modes of Granite under the Condition of Cyclic Heating and Water-cooling. Science Technology And Engineering.21(32),13828-13836 (2021).
Yang, X.X. et al. Fracture characteristics and pore connectivity of sandstone under thermal shock. Chinese Journal Of Geotechnical Engineering.44(10),1925-1934 (2022).
Kumari P.G.W., Beaumont, M.D., Ranjith, G.P. et al. An experimental study on tensile characteristics of granite rocks exposed to different high-temperature treatments. Geomechanics And Geophysics For Geo-Energy And Geo-Resources.5(1),(2019).
Xia, C.C., Zhou, S.W., Hu, Y.S. et al. Preliminary study on mechanical property of basalt subjected to cyclic uniaxial stress and cyclic temperature. Chinese Journal Of Geotechnical Engineering.37(06),1016-1024 (2015).
Zhu, Z.N., Tian, H., Dong, N.N. et al. Experimental study of physico-mechanical properties of heat-treated granite by water cooling. Rock And Soil Mechanics.39(S2),169-176 (2018).
Jia, P., Wang, Y., Li, B. et al. Experimental Study on Mechanical Properties of Water-Cooled High Temperature Rock Under Cyclic Loading. Transactions Of Beijing Institute Of Technology.43(02),126-134 (2023).
Yuan, Y., W, H.C.D, Yong, D. et al. Study on crack dynamic evolution and damage-fracture mechanism of rock with pre-existing cracks based on acoustic emission location. Journal Of Petroleum Science And Engineering.201 (2021).
Song, Y.J., Cheng, K.Y., Meng, F.D.Research on acoustic emission characteristics of fractured rock damage under freeze-thaw action. Journal Of Mining ＆ Safety Engineering.40(02),408-419 (2023).
Wang, Y., Chen, W.H.Study on thermal stress field of triangle fracture of granite in high temperature environment. Chinese Journal Of Rock Mechanics And Engineering.40(S2),3074-3083 (2021).
Li, M,. Liu, X.S,. Pan, Y.H.Mechanical properties of fractured sandstone after cyclic thermal shock. Rock And Soil Mechanics.44(05),1260-1270 (2023).
Yuan, C,. Lei, X,. Meng, Y.Z. et al. Experimental investigation on the influence on mechanical properties and acoustic emission characteristics of granite after heating and water-cooling cycles. Geomechanics And Geophysics For Geo-Energy And Geo-Resources.9(1),(2023).
Gong, C,. Bao, H,. Wang, W.J. et al. Evolution law and dominant frequency characteristics of acoustic emission sources during red sandstone failure process. Journal Of Chian Coal Society.47(06),2326-2339 (2022).
Nicksiar, M,. Martin, C.D. Evaluation of methods for determining crack initiation in compression tests on low-porosity rocks. Rock Mechanics And Rock Engineering.45,607–617 (2012).
Xiao, Y.M,. Qiao, Y.F,. Li, H.R. Strain Rate Effect and Acoustic Emission Characteristics of Carbonaceous Slates in Uniaxial Compression.Journal Of Tongji University.50(09),1276-1285 (2022).
LI, X.W,. Yao, Z.S,. Huang, X.W. Investigation of deformation and failure characteristics and energy evolution of sandstone under cyclic loading and unloading.Rock And Soil Mechanics.42(06),1693-1704 (2021).
Ning, J.G. et al. Estimation of Crack Initiation and Propagation Thresholds of Confined Brittle Coal Specimens Based on Energy Dissipation Theory. Rock Mechanics And Rock Engineering.51(1),(2018).
Liu, X.H. et al. Study on determination of uniaxial characteristic stress of coal rock under quasi-static strain rate.Chinese Journal Of Rock Mechanics And Engineering.39(10),2038-2046 (2020).
Liu, X.X. et al. Energy evolution and failure characteristics of single fissure carbonaceous shale under drying-wetting cycles.Rock And Soil Mechanics.43(07),1761-1771 (2022).
Peng, J. et al. Experimental study of effect of water pressure on progressive failure process of rocks under compression.Rock And Soil Mechanics.34(04),941-946+954 (2013).
Liu, L,. Wang, S.J,. Yang, W.Ch.Strain rate effects on characteristic stresses and acoustic emission properties of granite under quasi-static compression.Frontiers In Earth Science.10.(2022).
Li, C.B. et al. Experimental and theoretical study on the shale crack initiation stress and crack damage stress.Jouranl Of China Coal Society.42(04),969-976 (2017).
Liu, X.L. et al. Acoustic emission b-values of limestone under uniaxial compression and Brazilian splitting loads.Rock And Soil Mechanics.40(S1),267-274 (2019).
Su, X.B. et al. Relationship between spatial variability of rock strain and b value under splitting condition.Jouranl Of China Coal Society.45(S1),239-246 (2020).
Dong, L.J,. Zhang, L.Y.Error Analysis of b-value of Acoustic Emission for Rock Fracture.Journal Of Yangtze River Scientific Research Institute.37(08),75-81 (2020).
Cai, M,. Kaiser, K.P,. ,Tasaka, Y.Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations.International Journal Of Rock Mechanics And Mining Sciences.41(5),(2004).
Yang, D.J,. Hu, J.H,. Zeng, P.P.Uniaxial acoustic emission from granite after triaxial cyclic predamage and Chaotic Fractal Analysis.The Chinese Journal Of Nonferrous Metals.33(02),619-629 (2023).
Wang, W. et al. Experiment of Fractal Characteristics of Acoustic Emission of Sandstone Under Triaxial Cyclic Loading and Unloading.Advancend Engineering Sciences.54(02),90-100 (2022).
Wang, S. et al. Inside a highly concentrated full-tailed sand slurry containing APAM Structural evolution characteristics.The Chinese Journal Of Nonferrous Metals.32(11),3553-3566 (2022).
Xu, X.L. et al. Experimental study of the effect of loading rates on mechanical properties of granite at real-time high temperature.The Chinese Journal Of Nonferrous Metals.36(08),2184-2192 (2015).
Xie, Q.T. et al. A study of crack propagation measurement on sandstone with a single inclined flaw under uniaxial compression.Rock And Soil Mechanics.32(10),2917-2921+2928 (2011).

No competing interests reported.

Mechanical properties and damage characterization of cracked granite after cyclic temperature action

Status:

Version 1

Abstract

Figures

Introduction

1. Experimental Equipment and Procedure

2. Mechanical Properties of Cracked Granite under Temperature Cycles

3. Damage Characteristics of Granite under Cyclic Temperature