Experimental Study on the True-Triaxial Mechanical Properties and Fracture Mechanisms of Granite Subjected to Cyclic Thermal Shock

Abstract

During reservoir stimulation and long-term operation of Enhanced Geothermal Systems (EGSs), repeated injection of cold fluids induces cyclic thermal shock in the surrounding rock mass, leading to progressive modification of mechanical properties and fracture behavior. However, the combined effects of cyclic thermal shock and true-triaxial stress conditions on granite strength and failure characteristics remain inadequately quantified. In this study, a series of true-triaxial compression tests were conducted on granite specimens subjected to cyclic thermal shock at 400 °C. Thermal shock cycles of 0, 1, 5, 10, and 15 were considered in conjunction with intermediate principal stress levels of 5, 20, 30, and 50 MPa to systematically evaluate their coupled influence on characteristic stresses and macroscopic failure behavior. The results show that the peak intensity increases with the rise of the intermediate principal stress, but with the increase in the number of thermal shocks, it first increases and then decreases. Macroscopic failure is dominated by asymmetric V-shaped fracture surfaces, roughly oriented along the ${\sigma }_2$ direction. As the intermediate principal stress increases, the failure mode transitions from tensile–shear mixed failure to shear-dominated failure, whereas thermal cycling promotes the persistence of tensile–shear cracking even under relatively high ${\sigma }_2$ conditions. Based on these observations, a modified Mogi–Coulomb strength criterion that accounts for thermal shock-induced damage is proposed to describe granite strength under true-triaxial stress conditions. The research results can provide a theoretical basis for optimizing the design of hydraulic fracturing in hot dry rock and evaluating reservoir stability.

1. Introduction

Dry hot rock geothermal resources, as a type of abundant and renewable energy source that is clean and pollution-free, hold significant strategic importance in the global energy structure transformation [1,2,3]. At present, the development of HDR resources relies primarily on Enhanced Geothermal Systems (EGSs), in which hydraulic stimulation is employed to create interconnected fracture networks within low-permeability rock formations. These artificially generated flow paths enable effective fluid circulation and heat exchange, thereby allowing thermal energy to be continuously extracted from deep geological formations [4].

During underground thermal energy storage (EGS) operation, reservoir rocks are subject to repeated thermal disturbances associated with cold fluid injection and subsequent heat recovery [5]. Rapid cooling induced by injected fluids generates strong thermal gradients within the surrounding rock mass (on the order of 50–200 °C/m) [4], while prolonged circulation and heat exchange lead to reheating. Such alternating thermal loading processes are cumulative rather than isolated events, progressively modifying the mechanical properties of reservoir rocks and potentially affecting fracture stability, permeability evolution, and long-term energy extraction efficiency. For instance, after 10 thermal cycles, instantaneous permeability was observed to be 2–3 times higher than in experiments with 0–3 cycles [6]. When these thermal effects act simultaneously with in situ triaxial stress conditions, the mechanical response of the rock mass becomes highly complex due to the interaction between the three principal stresses. Therefore, a thorough understanding of the true-triaxial mechanical behavior of granite and the evolution of its fractures under cyclic thermal shock is of great significance for assessing the structural integrity and operational performance of enhanced geothermal system (EGS) reservoirs.

A large number of laboratory experiments have shown that high temperatures and heat treatment can significantly weaken the mechanical properties of crystalline rocks [7]. For granite, exposure to high temperatures reduces uniaxial compressive strength, elastic modulus, and fracture toughness, with damage severity increasing with treatment temperature [8]. Beyond monotonic heating, thermal shock is an important damage mechanism. Experiments involving rapid cooling after high-temperature treatment demonstrate that crack initiation and propagation are strongly influenced by temperature gradients and mineralogical heterogeneity [9]. Repeated thermal shock can further exacerbate this effect, leading to accumulation of damage, manifested by a gradual decrease in strength, an increase in porosity, and the development of heterogeneous microcracks [10]. Under cyclic thermal shock, granite failure exhibits progressively evolving behavior, with tensile strength decreasing significantly during the first five cycles (especially in the initial cycle) and the number of cracks increasing markedly, followed by minor changes in subsequent cycles. In some cycles, crack growth ranged from 13.2% to 24.5%, consistent with the trend of the thermal shock damage factor [11]. Supplementary observations such as tensile tests and wave velocity measurements further confirm that cyclic thermal shocks significantly alter the internal structure of granite [12]. Collectively, these studies indicate that thermal damage in granite is both temperature-dependent and strongly influenced by the number of thermal shock cycles.

Although a large number of studies have focused on thermal effects, most existing experiments are conducted under uniaxial or conventional triaxial stress conditions. In contrast, rocks in deep geothermal reservoirs are usually subjected to true-triaxial stress states, with significant differences in the three principal stresses [13]. Under such conditions, both deformation characteristics and failure mechanisms may deviate markedly from those observed in simpler stress configurations. To address this discrepancy, true-triaxial testing has increasingly been employed. Thermo-mechanically coupled true-triaxial experiments show that rock strength decreases noticeably with increasing temperature [14]. Real-time high-temperature true-triaxial tests further reveal that temperature not only reduces strength parameters but also alters failure patterns, promoting a transition from shear-dominated failure to mixed tensile-shear modes [15]. True-triaxial tests under full stress paths further reveal the crucial role of intermediate principal stress in controlling crack propagation and the brittle-ductile transition behavior [16]. Accordingly, strength criteria that explicitly consider the influence of intermediate principal stress, such as the Mogi-Coulomb strength criterion, have been shown to have better predictive capability [17]. More recent studies have further confirmed that thermal treatment significantly modifies the true-triaxial mechanical response and deformation characteristics of both sandstone and granite [18,19]. Although the effects of thermal treatment, cyclic thermal shock, and true-triaxial stress have been investigated individually, their combined influence remains insufficiently explored. In particular, experimental evidence regarding how cyclic thermal shock interacts with true-triaxial stress to govern strength degradation, characteristic stress evolution, and fracture mechanisms in granite is still limited. Moreover, constitutive or strength prediction models capable of accurately describing the mechanical behavior of thermally cycled rocks under true-triaxial stress conditions are scarce.

In order to accurately assess the mechanical response and structural stability evolution characteristics of dry hot rock reservoirs under cyclic thermal shock during geothermal development, it is necessary to systematically study the mechanical properties and failure modes of granite under high-temperature heating–cooling cycles under true-triaxial stress conditions. This study conducted true-triaxial compression tests on granite samples subjected to 400 °C cyclic thermal shock (0, 1, 5, 10, and 15 cycles) under different intermediate principal stress levels (${\sigma }_2=$ 5, 20, 30, and 50 MPa). The effects of the number of thermal shock cycles and the intermediate principal stress on the characteristic stress thresholds and macroscopic failure patterns of granite were systematically analyzed. Meanwhile, a three-dimensional laser scanner and a scanning electron microscope were employed to quantitatively characterize the surface roughness and microstructural features of the fractured surfaces of granite. Based on this, a modified Mogi–Coulomb strength criterion incorporating the parameter of cyclic thermal shock was proposed to predict the strength evolution law of granite subjected to cyclic thermal shock under the true-triaxial stress state.

2. Materials and Methods

2.1. Specimen Preparation

All granite samples used in this study were taken from the same quarry in Qingdao, Shandong Province, China, to ensure material consistency. The main physical properties of the intact rock samples were characterized before the mechanical tests. The granite exhibited a natural density of 2.613 g/cm³, a porosity of 0.56%, and an average P-wave velocity of 4902 m/s. Mineralogical composition was determined using X-ray diffraction (XRD) analysis on powdered samples. The results indicate that the granite was predominantly composed of feldspar (64%) and quartz (31%), with a minor content of mica (5%) (Figure 1). The high feldspar content suggests a potential susceptibility to thermally induced microcracking due to mineralogical heterogeneity.

The dimensions and surface conditions of the samples were controlled according to the general guidelines of the International Society for Rock Mechanics (ISRM) [20], with particular attention to the requirements of true-triaxial loading. Granite blocks were processed into prismatic specimens and subsequently finished to achieve a target geometry of $50\mathrm{mm}\times 50\mathrm{mm}\times 100\mathrm{mm}$. During the preparation of the specimen, geometric tolerances were strictly controlled to minimize boundary effects in the test. Dimensional deviations were maintained within 0.02 mm, perpendicularity between adjacent faces was controlled below 0.25°, and parallelism of the loading end faces was kept within 0.02 mm.

2.2. Experimental Apparatus

After different numbers of thermal shock cycles, a servo-controlled true-triaxial testing system was used to conduct true-triaxial compression tests on granite samples (Figure 2). The experimental platform integrates stress loading, deformation monitoring, and real-time data acquisition within a closed-loop computer-controlled framework. The system provides sufficient loading capacity for high-stress rock testing, with a maximum axial stress of 500 MPa and upper limits of 60 MPa for both lateral stress and confining pressure.

To reproduce realistic true-triaxial stress states, the three principal stresses were applied and controlled independently. The major principal stress (${\sigma }_1$) and the intermediate principal stress (${\sigma }_2$) were imposed through rigid loading platens, enabling stable stress-controlled loading along the corresponding directions. The minor principal stress (${\sigma }_3$) was applied by hydrostatic pressurization of the sealed chamber using hydraulic oil. Independent servo control of each stress component ensured accurate implementation of prescribed loading paths while minimizing mechanical interference among loading directions.

The axial and transverse deformation responses of the specimen were continuously monitored using six linear variable differential transformers (LVDTs) symmetrically arranged around the specimen. Each LVDT had a displacement resolution of 0.01 mm, allowing reliable capture of deformation evolution throughout the loading process. The combination of independent stress control and multi-point deformation measurement provides a reliable experimental basis for analyzing the mechanical response and failure behavior of granite under true-triaxial stress conditions.

2.3. Experimental Procedure

2.3.1. Thermal Shock Test

Before undergoing mechanical loading, the granite samples were first subjected to rigorous high-temperature treatment to simulate potential thermal disturbances that could occur in a geothermal environment. Thermal shock conditioning was conducted using a muffle furnace with a target temperature of 400 °C. To limit additional damage associated with rapid heating, the furnace temperature was increased at a constant rate of 5 °C/min, following commonly adopted heating protocols in previous studies [21,22]. Once the target temperature was reached, the specimens were held at 400 °C for 4 h to ensure uniform temperature distribution within the specimen volume [23]. Subsequently, the heated sample was rapidly transferred to a water bath at 20 °C to simulate the thermal shock process. After the sample had completely cooled, it was removed from the water. Finally, the surface moisture was wiped off, and the sample was placed in a drying oven for further drying. Surface moisture was then removed, and the specimens were subsequently placed in a constant-temperature drying oven and dried at 105 °C for 24 h to eliminate residual water prior to further testing [24]. For clarity and consistency, the sequence comprising high-temperature heating, water quenching, and post-quench drying was defined as one complete thermal shock cycle, as schematically illustrated in Figure 3.

According to the above process, there are five working conditions for the granite samples, with the number of thermal shocks being 0, 1, 5, 10, and 15, where 0 thermal shocks represent the samples in their natural state. It is worth noting that each experimental condition was examined using a single test, and some variability among individual results was therefore expected. Nevertheless, clear overall trends can still be identified, which is consistent with observations reported in previous true-triaxial experimental studies.

2.3.2. True-Triaxial Compression Test

After completing the thermal shock treatment, the granite specimens were installed between specially designed loading plates for true-triaxial compression testing. The lower plate had a groove connected to the lifting platform, which allowed the specimens to be securely fixed during installation and effectively reduced eccentric loading during subsequent testing [25]. To capture deformation along the three principal directions (${\epsilon }_1$, ${\epsilon }_2$, and ${\epsilon }_3$), mounting ports were integrated into the platen edges for displacement sensor installation. Two linear variable differential transformers (LVDTs) were arranged in each major direction, and average readings were used to improve measurement reliability and reduce local deviation. The surface of the splint was coated with a mixture of stearic acid and petroleum jelly in a 1:1 ratio to reduce the effect of end friction [26]. To prevent the hydraulic oil from penetrating into the interior of the rock and affecting its strength, the gap between the sample and the clamping plate was sealed with rubber, and the surface of the sample exposed in the confining pressure chamber was evenly coated with rubber. Before the test began, the LVDT was installed according to the corresponding connection holes, and then the lifting platform was raised to send the sample into the pressure chamber.

The true-triaxial loading scheme adopted in this study is shown in Figure 4. Stress application was designed to progressively establish the target true-triaxial stress state prior to failure. At the initial loading stage, the three principal stresses (${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$) were synchronously applied at a uniform rate of 0.5 MPa/s until the minimum principal stress reached the preset level of ${\sigma }_3=5$ MPa. Thereafter, ${\sigma }_3$ was kept constant to provide a stable confining condition. Subsequently, the intermediate and major principal stresses were increased at the same rate until the target intermediate principal stress levels (${\sigma }_2=$ 5, 20, 30, and 50 MPa) were reached, after which ${\sigma }_2$ was kept constant. After the prescribed true-triaxial stress state was established, loading was continued by increasing the major principal stress ${\sigma }_1$ under displacement control at a rate of 0.05 mm/min, while ${\sigma }_2$ and ${\sigma }_3$ remained constant. When the specimen reached macroscopic failure, the loading process was terminated. Macroscopic failure was determined by observing sudden drops in the real-time stress–strain curve, combined with the occurrence of cracking sounds from the specimen.

3. Results and Discussion

3.1. P-Wave Velocity

Figure 5 presents the variation of P-wave velocity in granite as a function of thermal shock cycles. A pronounced reduction in wave velocity is observed as thermal cycling progresses, reflecting the continuous degradation of the internal structure. The intact specimens exhibit an average P-wave velocity of 4902 m/s. After 1, 5, 10, and 15 thermal shock cycles, the average velocities decrease to 2427 m/s, 1773 m/s, 1712 m/s, and 1656 m/s, respectively, corresponding to relative reductions of 49.51%, 63.83%, 65.08%, and 66.22% compared with the intact condition.

This is mainly because, as the number of thermal shocks increases, the thermal cracks inside the granite samples gradually increase, hindering the propagation of sound waves within the samples and causing a decrease in the longitudinal wave velocity of the granite samples [5]. A distinct phase-related characteristic can be clearly observed. The first thermal shock cycle causes a sudden decrease in speed, with the drop accounting for nearly half of the total decrease observed after 15 cycles. In contrast, the reduction in subsequent cycles gradually diminishes, and the speed stabilizes after about five cycles. This behavior indicates that thermal damage develops rapidly in the initial stage and gradually approaches a saturation state with continued exposure to thermal shock.

3.2. True-Triaxial Stress–Strain Curves

The stress–strain curves obtained from true-triaxial compression tests provide key insights into the deformation evolution, strength development, and failure mechanisms of granite. Representative stress–strain curves under true-triaxial loading are shown in Figure 6. In agreement with earlier true-triaxial studies [27,28], the mechanical response of granite can be divided into five successive stages:

1.

Microcrack compaction and closure (OA): At low stress levels, pre-existing microcracks and micropores are progressively compressed and closed, resulting in a nonlinear compaction response in the stress–strain curve.

2.

Elastic deformation (AB): With increasing load, the stress–strain relationship becomes approximately linear, indicating that the internal structure of the specimen remains essentially intact. When the applied stress reaches the crack initiation stress ${\sigma }_{ci}$ at point B, new microcracks begin to nucleate and propagate.

3.

Stable microcrack growth (BC): Between ${\sigma }_{ci}$ and the damage stress ${\sigma }_{cd}$, microcracks propagate in a stable manner. As the applied stress approaches ${\sigma }_{cd}$, crack interaction intensifies, and the material gradually transitions toward unstable behavior. Point C corresponds to the damage stress ${\sigma }_{cd}$, marking the onset of unstable crack growth.

4.

Unstable microcrack growth (CD): Once the applied stress exceeds ${\sigma }_{cd}$, crack propagation becomes unstable and accelerates rapidly. Microcracks coalesce and form dominant fracture zones, leading to a significant reduction in load-bearing capacity as the stress approaches the peak strength ${\sigma }_p$.

5.

Post-peak behavior (DE): After reaching the peak stress ${\sigma }_p$, a rapid stress drop occurs due to the coalescence of microcracks into macroscopic fractures, accompanied by a pronounced loss of load-bearing capacity.

Figure 6. True triaxial stress–strain curves of granite.

Figure 6. True triaxial stress–strain curves of granite.

Figure 7 shows the true-triaxial stress–strain curves of granite subjected to cyclic thermal shock under different intermediate principal stresses. At a constant intermediate principal stress ${\sigma }_2$, increasing the number of thermal shock cycles prolongs the initial nonlinear compaction stage of the curves. This behavior is caused by the progressive formation of microcracks due to repeated thermal shocks, and greater axial strain is required to close these cracks during the early loading stage.

For a given number of thermal shock cycles, the deformation responses along the ${\epsilon }_2$ and ${\epsilon }_3$ directions are strongly influenced by the magnitude of ${\sigma }_2$. When ${\sigma }_2={\sigma }_3=5$ MPa, both directions exhibit similar dilatant behavior. In contrast, when ${\sigma }_2>{\sigma }_3$, the stress–strain responses along the two lateral directions become increasingly asymmetric. Specifically, the peak strain along the ${\epsilon }_2$ direction decreases with increasing ${\sigma }_2$, indicating a transition from dilation-dominated to compression-constrained deformation. For example, for specimens subjected to 10 thermal shock cycles, increasing ${\sigma }_2$ from 5 MPa to 50 MPa changes the peak strain along ${\epsilon }_2$ from $-0.60\%$ to $+0.19\%$. This trend can be attributed to the enhanced lateral confinement at higher ${\sigma }_2$, which suppresses random microcrack propagation while promoting preferential crack growth along the ${\epsilon }_3$ direction [29,30].

Correspondingly, Figure 8 shows that the elastic modulus E exhibits a non-monotonic evolution with increasing thermal shock cycles under different ${\sigma }_2$ conditions. Specifically, E initially increases and then decreases with thermal shock cycling for all investigated intermediate principal stresses. Relatively higher elastic moduli are observed at five thermal shock cycles, and higher intermediate principal stress consistently results in a larger modulus. When ${\sigma }_2$ equals 5, 20, 30, and 50 MPa, the corresponding elastic moduli at five thermal shock cycles are 34.69 GPa, 36.19 GPa, 36.87 GPa, and 38.85 GPa, respectively, representing increases of 5.90%, 3.91%, 2.86%, and 4.50% compared with those at ambient temperature.

As the number of thermal shock cycles further increases to 15, the elastic modulus decreases significantly, with reductions of 12.92%, 8.18%, 6.13%, and 4.71% relative to the corresponding ambient values for ${\sigma }_2$ = 5, 20, 30, and 50 MPa, respectively. These results indicate that the overall stiffness of granite subjected to cyclic thermal shock exhibits significant nonlinear changes under different intermediate principal stresses, highlighting the close relationship between thermal damage accumulation and stress constraint effects [31].

3.3. Strength Characteristics

Under true-triaxial compression, granite undergoes a progressive failure process, which is controlled by the initiation, propagation, and interaction of microcracks. To quantitatively describe this process, specific stress thresholds are defined to mark the key stages of deformation and damage development during loading. In this study, three stress parameters are used to represent these stages: the crack initiation stress (${\sigma }_{ci}$), the damage stress (${\sigma }_{cd}$), and the peak stress (${\sigma }_p$). The crack initiation stress ${\sigma }_{ci}$ was identified using the lateral strain response (LSR) method [32], which detects the onset of microcrack activity through changes in lateral strain evolution. The damage stress ${\sigma }_{cd}$ was determined as the axial stress corresponding to the maximum volumetric strain, marking the transition from stable crack growth to accelerated damage development. The peak stress ${\sigma }_p$ is defined as the maximum axial stress reached prior to macroscopic failure. A summary of the characteristic stress values obtained under different thermal shock cycles and true-triaxial stress conditions is provided in

Table 1. Characteristic stress values of granite under different conditions.

${\mathit{\sigma }}_2$ (MPa)	Thermal Shock Cycles	E (GPa)	${\mathit{\sigma }}_{\mathit{ci}}$ (MPa)	${\mathit{\sigma }}_{\mathit{cd}}$ (MPa)	${\mathit{\sigma }}_{\mathit{p}}$ (MPa)
5	0	32.76	77.59	161.55	226.50
	1	33.54	76.39	157.11	229.72
	5	34.69	47.84	147.87	232.39
	10	29.30	60.49	126.64	224.69
	15	28.52	30.07	103.69	211.62
20	0	34.83	95.00	175.00	255.00
	1	35.74	128.81	185.68	260.54
	5	36.19	80.54	160.13	261.16
	10	32.28	103.36	174.77	250.74
	15	31.98	79.90	147.47	248.17
30	0	35.84	146.49	186.96	261.62
	1	36.22	150.80	188.88	263.00
	5	36.87	134.00	213.00	266.94
	10	34.76	109.20	200.68	259.91
	15	33.65	75.07	161.14	255.35
50	0	30.69	161.51	193.12	278.87
	1	38.41	194.72	240.38	280.86
	5	38.85	153.06	222.67	292.40
	10	37.17	169.41	207.68	270.29
	15	35.42	145.46	188.39	269.24

.

Figure 9 presents the variation curves of characteristic stresses in cyclically thermal-shocked granite under different intermediate principal stresses. It can be observed that when the number of thermal shock cycles is constant, all characteristic stresses increase with an increase in the intermediate principal stress ${\sigma }_2$. However, under the same ${\sigma }_2$, the characteristic stresses exhibit a nonlinear variation with the number of thermal shock cycles. Among them, the characteristic stresses ${\sigma }_{ci}$ and ${\sigma }_{cd}$ follow essentially similar variation patterns. When ${\sigma }_2=5$ MPa, both ${\sigma }_{ci}$ and ${\sigma }_{cd}$ show an overall decreasing trend as the number of thermal shock cycles increases, declining from 77.59 MPa and 161.55 MPa to 30.07 MPa and 103.69 MPa, respectively, corresponding to reductions of 61.3% and 35.8%. This can be attributed to the relatively low intermediate principal stress, under which the initiation and propagation of internal microcracks are primarily governed by the number of thermal shock cycles. The cumulative thermal cracks generated by cyclic thermal shocks significantly degrade the integrity of the rock, leading to the reduction in ${\sigma }_{ci}$ and ${\sigma }_{cd}$ [31]. With the increase in intermediate principal stress (${\sigma }_2>5$ MPa), both ${\sigma }_{ci}$ and ${\sigma }_{cd}$ exhibit a trend of initial increase followed by decrease. Taking ${\sigma }_2=50$ MPa as an example, ${\sigma }_{ci}$ and ${\sigma }_{cd}$ rise from 169.43 MPa and 207.68 MPa to 194.72 MPa and 240.38 MPa, respectively, after the first thermal shock cycle, and eventually decrease to 145.46 MPa and 188.39 MPa with further cycling. This behavior is mainly attributed to the increasingly pronounced inhibitory effect of higher intermediate principal stress on crack propagation [29], which leads to a slight increase in ${\sigma }_{ci}$ and ${\sigma }_{cd}$ for specimens subjected to a low number of thermal shock cycles. However, as the number of thermal shock cycles increases, the accumulation of thermal damage surpasses the strengthening effect induced by ${\sigma }_2$, resulting in a reduction in both characteristic stresses.

Similarly, it can also be observed that under different intermediate principal stresses ${\sigma }_2$, the peak stress ${\sigma }_p$ exhibits a trend of initial increase followed by decrease with increasing number of thermal shock cycles. The peak stress reaches its maximum value at five thermal shock cycles, and a higher intermediate principal stress ${\sigma }_2$ corresponds to a greater peak stress. When ${\sigma }_2$ is 5 MPa, 20 MPa, 30 MPa, and 50 MPa, the corresponding peak stresses are 232.39 MPa, 261.16 MPa, 266.94 MPa, and 292.40 MPa, respectively. Compared with the peak stress under ambient temperature conditions, these represent increases of 2.6%, 2.42%, 2.03%, and 4.85%. This behavior is mainly attributed to the fact that when the number of thermal shock cycles is relatively small ($N\le 5$), despite the generation of numerous thermal cracks inside the specimen, the constraint effect provided by the intermediate principal stress ${\sigma }_2$ promotes particle rearrangement within the granite and enhances the overall strength through crack compaction and closure [33,34]. When the number of thermal shock cycles increases to 15, the peak stresses corresponding to ${\sigma }_2$ values of 5 MPa, 20 MPa, 30 MPa, and 50 MPa are 211.62 MPa, 248.17 MPa, 255.35 MPa, and 269.24 MPa, respectively. Compared with the peak stresses under ambient temperature conditions, these represent reductions of 5.58%, 2.68%, 2.40%, and 3.40%. This can be primarily attributed to the fact that under a higher number of thermal shock cycles, the cumulative effect of thermal cracking becomes the dominant factor. The initiation and propagation of microcracks within the specimen lead to the formation of localized fracture surfaces, and the strengthening effect provided by the intermediate principal stress ${\sigma }_2$ is insufficient to compensate for the strength degradation caused by repeated thermal shocks. Consequently, the peak stress of granite exhibits a slight declining trend [33].

3.4. Macroscopic Failure Modes

The macroscopic fracture characteristics of granite subjected to cyclic thermal shock and true-triaxial compression were investigated to clarify the combined effects of thermal damage and intermediate principal stress on failure mechanisms. The fracture angle, $\theta$, is defined as the angle between the dominant macroscopic fracture plane and the horizontal direction, following the definition commonly adopted in previous studies [14]. Figure 10 presents the post-failure morphologies of granite specimens after true-triaxial compression under various thermal shock cycles and intermediate principal stress levels. Regardless of testing conditions, all specimens develop multiple fracture surfaces after failure, among which a dominant macroscopic fracture plane can be clearly distinguished. This principal fracture surface is roughly parallel to the ${\sigma }_2$ direction and generally forms an asymmetric V-shaped structure. It intersects the ${\sigma }_3$ direction at a certain inclination and ultimately traverses the entire specimen.

At ambient temperature, failure behavior is strongly influenced by the intermediate principal stress ${\sigma }_2$. When ${\sigma }_2$ is relatively low (≤20 MPa), specimens predominantly undergo tensile–shear mixed failure. Under these conditions, lateral tensile strain promotes the initiation of numerous microcracks roughly parallel to the ${\sigma }_1$ direction. As the loading continues, these microcracks keep expanding and merging, eventually forming a macroscopic tensile–shear fracture surface. As ${\sigma }_2$ increases, its restraining effect on lateral deformation becomes increasingly apparent, thereby effectively inhibiting the initiation and propagation of tensile microcracks. As a result, deformation is increasingly accommodated by intergranular sliding and frictional resistance [35], and failure occurs primarily through shear-dominated mechanisms along the direction of maximum shear stress. Quantitatively, under ambient conditions, increasing ${\sigma }_2$ from 5 MPa to 50 MPa reduces the fracture angle $\theta$ from 84° to 76°, reflecting a gradual transition from tensile–shear mixed failure toward shear-dominated failure.

Following cyclic thermal shock treatment, the effect of ${\sigma }_2$ on macroscopic failure remains evident, with specimens subjected to different numbers of thermal cycles still showing a general transition from tensile–shear mixed failure to shear failure as ${\sigma }_2$ increases. Notably, for specimens tested at ${\sigma }_2=30$ MPa, cyclic thermal shock changes the failure behavior from shear-dominated at ambient temperature to tensile–shear mixed failure. This change is attributed to the extensive formation of thermally induced fractures within the rock matrix. Under such conditions, the stress difference (${\sigma }_2-{\sigma }_3$) becomes insufficient to effectively restrain the initiation of propagation. Moreover, the microcracks exhibit a preferential alignment, thereby forming a macroscopic tensile–shear fracture surface.

In addition, under relatively low intermediate principal stress conditions, specimens subjected to cyclic thermal shocks are more prone to secondary cracking, indicating increased damage accumulation and enhanced crack interaction. As ${\sigma }_2$ increases from 5 MPa to 50 MPa, the fracture angle $\theta$ decreases by approximately 6°, 9°, 2°, and 10° for specimens subjected to 1, 5, 10, and 15 thermal shock cycles, respectively, demonstrating a consistent decreasing trend with increasing ${\sigma }_2$ across different thermal shock histories. In addition, for a given ${\sigma }_2$, the change in fracture angle with the number of thermal shock cycles is not linear, highlighting the complex interaction between the evolution of thermal damage and the true-triaxial stress conditions, which in turn affects the macroscopic failure behavior.

3.5. Microstructural Analysis

Scanning electron microscopy (SEM) was employed to examine microstructural changes in granite induced by cyclic thermal shock. Fracture surface morphologies of specimens subjected to different numbers of thermal shock cycles are shown in Figure 11. As illustrated in Figure 11a, granite in the intact state exhibits a dense and coherent microstructure. Mineral grains are tightly bonded, microcracks are rarely observed, and fracture surfaces appear relatively smooth, with no obvious pore structures.

After one thermal shock cycle (Figure 11b), noticeable microstructural alterations are observed. Microcracks begin to develop preferentially along grain boundaries, accompanied by localized intergranular separation and partial grain detachment. A limited number of micropores also appear, indicating that thermally induced stresses start to weaken grain-boundary cohesion.

More pronounced damage is evident after five thermal shock cycles (Figure 11c). Microcracks increase in both length and density and progressively interconnect to form a crack network. Both intergranular and transgranular cracking modes are present, with intergranular cracks remaining dominant. In addition, abrasion and fragmentation along mineral grain edges generate fine debris, and localized through-going fractures can be identified in some regions. These features collectively reflect an accelerated accumulation of thermal damage.

As the number of thermal cycles increases, the degradation of the microstructure becomes more apparent. After ten cycles (Figure 11d), cementation between mineral grains is noticeably weakened [36], and microcracks extend across multiple grains, forming a well-connected fracture network. Following fifteen cycles (Figure 11e), the microstructure shows dense crack intersections and localized grain disintegration. The crack paths become increasingly irregular and tortuous, indicating severe internal damage. This gradual deterioration is caused by the cumulative effects of repeated thermal shocks, where mismatched thermal expansion between mineral components promotes the formation, propagation, and coalescence of irreversible cracks, ultimately creating a highly connected fracture network and leading to a significant loss of structural integrity.

3.6. Modified Mogi–Coulomb Strength Criterion

Accurately predicting rock strength is of great significance for practical engineering such as EGS. Considering that rock strength under true-triaxial loading is significantly influenced by the intermediate principal stress, for rocks subjected to true-triaxial loading, strength criteria based on the octahedral shear stress have been shown to effectively capture the influence of the intermediate principal stress. Among these, the Mogi strength criterion and its linear extension, commonly referred to as the Mogi–Coulomb criterion, are adopted in this study to characterize the experimental results, as they can more reasonably reflect the constraining effect of the intermediate principal stress on rock failure behavior [37,38,39]. The two formulations can be expressed as:

$${\tau }_{\mathrm{oct}}=A{\sigma }_{m,2}^n$$

(1)

$${\tau }_{\mathrm{oct}}=a{\sigma }_{m,2}+b$$

(2)

where ${\tau }_{\mathrm{oct}}$ denotes the octahedral shear stress and ${\sigma }_{m,2}$ represents the mean effective principal stress. In the nonlinear Mogi criterion, A and n are empirical fitting parameters, whereas in the linear Mogi–Coulomb formulation, a and b correspond to material constants related to internal friction and cohesion, respectively.

Figure 12 presents the fitting results of the nonlinear Mogi criterion and the linear Mogi–Coulomb criterion in the ${\tau }_{\mathrm{oct}}$–${\sigma }_{m,2}$ space, with the corresponding fitting parameters listed in

Table 2. Parameters of the Mogi and Mogi–Coulomb strength criteria under different numbers of thermal shock cycles.

Thermal Shock Cycles	Mogi Criterion			Mogi–Coulomb Criterion
	A	n	R2	a	b (MPa)	R2
0	4.202	0.677	0.988	0.594	36.092	0.986
1	4.318	0.673	0.976	0.591	37.006	0.973
5	3.425	0.721	0.997	0.633	32.401	0.996
10	4.919	0.643	0.982	0.565	39.065	0.980
15	3.095	0.737	0.982	0.648	27.683	0.980

. For all investigated numbers of thermal shock cycles, both criteria provide a good approximation of the experimental strength data, with coefficients of determination exceeding $R^2=0.973$. This strong agreement indicates that both formulations are capable of capturing the primary true-triaxial strength characteristics of granite after cyclic thermal shock. However, in their original forms, neither criterion explicitly accounts for the coupled effects of intermediate principal stress and progressive damage induced by repeated thermal shock cycles.

To account for thermal shock-related degradation, a modified Mogi–Coulomb strength criterion is introduced in this study. Experimental observations indicate that key mechanical properties of rocks often evolve nonlinearly with the number of thermal shock cycles, and exponential relationships have been reported to provide a reasonable approximation of such degradation processes [5,40]. Accordingly, it is assumed that the parameters a and b in the linear Mogi–Coulomb criterion vary as exponential functions of the thermal shock cycle number N, which can be expressed as:

$$a=a_0{e}^{-k_{a}N}$$

(3)

$$b=b_0{e}^{-k_{b}N}$$

(4)

where $a_0$ and $b_0$ denote the strength parameters corresponding to intact granite under natural conditions ($N=0$). The coefficients $k_a$ and $k_b$ quantify the sensitivity of the friction-related and cohesion-related parameters, respectively, to damage accumulation induced by cyclic thermal shock.

By incorporating Equations (3) and (4) into the linear Mogi–Coulomb framework, the strength criterion is extended to explicitly account for thermal shock effects, yielding the following modified expression:

$${\tau }_{\mathrm{oct}}=a_0{e}^{-k_{a}N}{\sigma }_{m,2}+b_0{e}^{-k_{b}N}$$

(5)

Fitting the proposed model to the experimental dataset using a least squares approach leads to an empirical relationship between the octahedral shear stress, the thermal shock cycle number, and the mean effective principal stress:

$${\tau }_{\mathrm{oct}}=0.622e^{-0.000519N}{\sigma }_{m,2}+33.215e^{-0.00506N}$$

(6)

As illustrated in Figure 13, the fitted response surface shows a close agreement between the modified Mogi–Coulomb criterion and the experimental strength data, with a coefficient of determination of $R^2=0.974$. This result indicates that the proposed formulation is able to simultaneously account for variations in intermediate principal stress and strength degradation associated with cyclic thermal shock. It can provide a new method for optimizing the design of actual dry geothermal rock fracturing and evaluating the stability of reservoirs, as well as predicting the strength of granite.

4. Engineering Implications

The experimental observations suggest that cyclic injection of low-temperature fluids prior to hydraulic fracturing can be used to deliberately introduce thermal shock effects into the reservoir rock. Repeated thermal perturbations promote the development of tensile and tensile–shear mixed microcracks, which act to reduce the overall mechanical integrity of the rock mass. As a result, the initiation pressure required for subsequent hydraulic fracturing is expected to decrease. In addition, thermally induced cracking increases the likelihood of interaction between pre-existing natural fractures and hydraulically generated fractures, favoring the development of a more complex and well-connected fracture network that is beneficial for sustained heat extraction. The results further indicate that fracture orientation is strongly controlled by the intermediate principal stress ${\sigma }_2$, as reflected by the systematic decrease in fracture angle with increasing ${\sigma }_2$. This emphasizes the necessity of accurately describing the in situ three-dimensional stress state when performing hydraulic fracturing operations under true-triaxial conditions. Incorporating reliable estimates of the magnitudes and orientations of the three principal stresses into numerical models can improve the accuracy of predictions for fracture geometry and inclination. This approach helps optimize the configuration of injection and production wells and increases the likelihood of establishing effective hydraulic connectivity within the stimulated reservoir volume. It should be noted that this study is based on laboratory-scale granite tests, and the experimental results are all within a stress range of ${\sigma }_2\le 50$ MPa. Therefore, further validation is still required in engineering applications in conjunction with specific geological conditions. However, the revealed mechanical mechanisms and modeling ideas provide valuable theoretical support for the design and stability assessment of hot dry rock reservoirs.

5. Conclusions

This study investigated the true-triaxial mechanical behavior of granite subjected to cyclic thermal shock at 400 °C under different intermediate principal stress conditions. By combining laboratory experiments with strength modeling, the coupled effects of thermal shock damage and intermediate principal stress on characteristic stresses, failure modes, and strength evolution were clarified. The main conclusions can be summarized as follows:

1.

The peak strength of granite exhibits a strong dependence on the intermediate principal stress, increasing systematically with higher ${\sigma }_2$. In contrast, its evolution with thermal shock cycling is non-monotonic, showing an initial increase followed by a gradual decrease. The maximum peak strength is attained after five thermal shock cycles. At relatively low thermal shock levels ($N\le 5$), strength enhancement is mainly associated with particle rearrangement and microcrack closure promoted by intermediate stress confinement. With continued thermal cycling, progressive accumulation of thermally induced microcracks becomes dominant, and the strengthening effect of intermediate stress is no longer sufficient to offset thermal damage.

2.

Macroscopic failure is characterized by asymmetric V-shaped fracture patterns with dominant fracture planes oriented approximately parallel to the ${\sigma }_2$ direction. Increasing ${\sigma }_2$ leads to a gradual transition from tensile–shear mixed failure toward shear-dominated behavior. Nevertheless, cyclic thermal shock facilitates the persistence of tensile–shear cracking even under relatively high intermediate principal stress. The fracture angle decreases with increasing ${\sigma }_2$ and varies nonlinearly with the number of thermal shock cycles.

3.

Cyclic thermal shock induces pronounced microstructural degradation in granite. With increasing thermal shock cycles, microcrack density rises markedly, and the dominant cracking mode evolves from predominantly intergranular to a mixed intergranular–transgranular pattern. Progressive linkage of dispersed microcracks results in the formation of an interconnected fracture network, which is consistent with the observed reduction in P-wave velocity and the associated decline in mechanical strength at the macroscopic scale.

4.

By incorporating thermal shock-related damage into the strength parameters, the modified Mogi–Coulomb criterion proposed in this study provides a quantitative description of granite strength evolution under coupled true-triaxial stress and cyclic thermal shock conditions. The formulation offers a practical basis for evaluating strength variations of granite in geothermal reservoirs and other high-temperature rock engineering applications.