Introduction

Fracturing and cracking processes of rock mass are important in hot dry rock (HDR) geothermal energy extraction. In the process of geothermal energy extraction, temperature of artificial reservoir will drop, due to the heat bilges cold shrink effect, thermal cracking phenomenon occurs. As a result, new cracks will be generated, and the original cracks will be extended or penetrated, which will eventually affect the heat exchange efficiency of the artificial reservoir1. Therefore, it is necessary to carry out the experimental study on the influence of the thermal environment or heating on the fracture properties of the rock.

Fracture toughness (FT) of rocks, which is defined as the resistance to crack propagation, is one of the basic material parameters in fracture mechanics. It must depend on the nature of pre-existing microstructure in the rock2. At high temperature, there were many thermal cracks in the rocks and the FT change with temperature. The mode FT of a rock in high temperature environment increased slightly at approximately 100–200 ℃, and then decreased with increase in the temperature3,4,5. After heat treated, the mode FT increased at approximately 100–150 ℃ when compared with the room temperature, and decreased with the increase of temperature6,7,8,9,10. The fracture mechanism is explained in terms of the distribution and density of micro-cracks by microscopic monitoring system. The micro-cracks initiated at multiple locations at the notch tip, and they continuously propagated, and coalesced with each other to eventually form a main crack in a zigzag and irregular way11. Scanning electron microscopy (SEM) results of rock showed that the thermal crack density and structural disintegration of mineral grains increased with temperature. The increasing thermal crack density is the main mechanism by which FT and stiffness of the rocks reduced6,7,8,9.

Crack opening displacement (COD) is defined as the separation between the top and bottom surfaces of crack, which is important material property in fracture mechanics12. Each material has a critical value of COD, when the COD reaches this critical value does the crack expand unstably. The critical COD was considered as the theoretical boundary of the traction-free zone (real crack) and the fracture process zone (FPZ, cohesive crack)13,14,15. The COD is measured by a linear variable displacement transformer, an extensometer, a clip gauge, a Fiber Bragg Grating and digital image correlation (DIC) technology11,12,13,14,16,17,18,19,20,21,22. The crack tip opening displacement (CTOD) is defined as the COD of a notch tip, and the crack mouth opening displacement (CMOD) is defined as the COD of a notch end12,13,14. Increased attention has been paid to the study of the CTOD and CMOD for the following reasons. First, the value of CTOD at peak load (critical CTOD) and critical COD were theoretically very close, and critical CTOD (CTODc) was easier to measure. Usually, CTODc is used to replace the critical COD to determine the position of the FPZ end13,14. Second, the traction-displacement relationship for a cohesive zone is normally back calculated by the load-CMOD of the numerical and experimental specimens23.

The rock crack propagation process of mode I fracture is another important research focus. When new cracks are initiated, acoustic emission (AE) events are generated. As the AE signals and the crack propagation activities carry a one-to-one relation during fracture process24. AE technology is used widely to observe the formation process of cracks, the crack propagation path, and the crack failure mechanism during the loading process2,25,26. The DIC technology was used to identify the fracture process of rocks. The research found a good correlation between the DIC-computed fracture parameters and their experimental values18,19. According to the displacement fields of specimen surface obtained from experiment, propagation length, and the crack width during the fracture process can be derived, providing information on the evolution of the FPZ at the interface13,14,20,21,22.

Most of the above scholars focused on the influence of the thermal environment or heating on the FT and fracture mechanism of the rock, and the COD and crack propagation process evolution of rock during loading at room temperature. The influence of the thermal environment or heating on the COD evolution and quantitative study of crack morphology have rarely been reported.

In this paper, the fracture properties of heat-treated (from room temperature 20 to 900 ℃) granite were investigated in three point bending tests (TPB). The influence of heat-treated on the COD properties measured by DIC and crack morphology measured by profilometer were studied. Additionally, the brittle-ductile transition behaviors of heat-treated granite fracture were analyzed.

Experimental equipment and methods

Sample preparation

The granite samples were collected from Pingyi, Shandong Province in China. The samples were in natural water state with a density of 2.71 g/cm3. The uniaxial compressive strength, tensile strength, elastic modulus, and Poisson’s ratio of granite are 160 MPa, 17.9 MPa, 35 GPa, and 0.25, respectively, at room temperature. All samples were cut from the same block of rock. The sample was notched at the mid-span with a thin cutting blade of thickness 1.0 mm. The sample was prepared with length L = 180 mm (span S = 160 mm), height H = 40 mm, thickness B = 20 mm, and notch length a = 12 mm, as shown in Fig. 1. The specimen size ratio is S:H:B = 8:2:1, which meets ASTM standards requirements.

Fig.1
figure 1

Dimensions of sample, loading geometry, and observation region on the sample surface.

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The samples were heat-treated in a muffle furnace at a rate of 1 °C per minute to avoid thermal shock. After reaching the target temperature, each sample was held at that temperature for 3 h to ensure equilibrium, then allowed to cool naturally to room temperature. The selected heat treatment temperatures included room temperature (20 °C), 100 °C, 200 °C, 300 °C, 400 °C, 500 °C, 600 °C, 700 °C, 800 °C, and 900 °C. Figure 2 shows the samples after thermal treatment, where a clear effect of temperature on the color of granite can be observed. With increasing temperature, the granite color changes from its original white–gray to an oxidized red.

Fig.2
figure 2

Sample photographs after thermal treatment.

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For Digital Image Correlation (DIC) analysis, a speckled pattern was applied to the sample surface using black and white spray paint13. To ensure a random distribution of speckles, the sample surface was first coated with a layer of white paint, followed by a light mist of black paint to create a unique speckle pattern.

Three-Point bending test apparatus and procedures

The experimental setup included a universal testing machine (UTM) and a DIC system (Fig. 3a). The DIC system consisted of image capture equipment (two digital cameras) and a post-processing device. During the three-point bending (TPB) test, a pre-load of 50 N was applied to secure the sample before loading, with axial displacement control applied at a rate of 0.05 mm/min27. The DIC system captured the initiation and propagation of cracks during loading. The image acquisition frame rate is 3 Hz。

Fig.3
figure 3

Experimental setup.

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The observation region, which covered the entire ligament area, is shown in Fig. 1 as the potential crack propagation area. Using DIC technology, displacement and strain fields in the observation region were obtained along the x-direction (perpendicular to the crack surface) and y-direction (along the crack surface). Four points were selected on each side of the notch, as shown in Fig. 1. Points 1 and 2 were positioned near the notch tip (y = 0), while Points 3 and 4 were near the notch end. The incremental displacement in the x-direction at Points 1 and 2 represented the crack tip opening displacement (CTOD) of the granite sample during loading, and the incremental displacement in the x-direction at Points 3 and 4 represented the crack mouth opening displacement (CMOD) during loading.

Crack morphology measurement

The Nanovea ST400 profilometer was used to perform 3D reconstruction of the crack on the sample surface (Fig. 3b). The Nanovea ST400 testing system comprises a nano module, a demodulator, and related software.

After the three-point bending (TPB) tests, 3D reconstruction of the crack on the sample surface was conducted as follows: (1) placing the sample on the platform; (2) selecting the central area for scanning; (3) setting the scanning parameters; and (4) automatically scanning the sample surface.

A rectangular reconstruction area was selected on the sample surface, as shown in Fig. 1. For samples heat-treated at temperatures between 20 °C and 400 °C, measurement steps along the x-direction (perpendicular to the crack surface) and y-direction (along the crack surface) were set to 1 μm and 2 μm, respectively. For samples heat-treated between 500 °C and 900 °C, the measurement steps in the x-direction and y-direction were set to 3 μm and 5 μm, respectively.

Results and discussion

Load–Displacement curves

Figure 4 shows the load–displacement curves of granite after different thermal treatments. The loading curve comprises three primary stages: the pre-peak linear phase, the pre-peak nonlinear phase, and the post-peak monotonic decline. The pre-peak linear phase represents the elastic deformation, while the pre-peak nonlinear phase corresponds to the development of the fracture process zone (FPZ), or plastic deformation zone, in front of the notch13,28. The post-peak phase represents crack initiation and propagation at the notch tip. Crack initiation is considered the first instance of a new traction-free surface, while crack propagation refers to the continued formation of this traction-free surface29. Despite crack initiation and propagation occurring in the third stage, the FPZ continues to develop ahead of the advancing crack13. Zuo et al.[30]proposed a new method for calculating elastic modulus in TPB tests, demonstrating that the elastic modulus of rock is positively correlated with the slope of the linear phase in the load–displacement curve.

Fig.4
figure 4

Load–displacement curves of granite after different thermal treatments.

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As observed in Fig. 4, the peak load initially increases as the heat treatment temperature rises from room temperature to 100 °C, and then gradually decreases as the temperature further increases from 100 °C to 900 °C. The range and slope (elastic modulus) of the initial linear phase decrease with increasing heat treatment temperature. Additionally, the range of the nonlinear phase and the displacement at peak load increase as temperature rises. After the peak load, the load–displacement curve declines more gradually with higher temperatures. In the temperature range of 600 °C to 900 °C, a plateau (yield phenomenon) appears at the peak load on the load–displacement curve. All these load–displacement behaviors indicate that the ductility of the rock increases with higher heat treatment temperatures; conversely, the brittleness of the rock decreases as the heat treatment temperature rises.

Fracture toughness

The FT KIC  of single-edge-cracked three-point bending sample was calculated using the following formula6,29:

$$K_{IC} = \frac{{F_{\max } S}}{{BH^{3/2} }}\left[ {2.9\left( \frac{a}{H} \right)^{1/2} \, - 4.6\left( \frac{a}{H} \right)^{3/2} \, + 21.8\left( \frac{a}{H} \right)^{5/2} \, - 37.6\left( \frac{a}{H} \right)^{7/2} \, + 38.7\left( \frac{a}{H} \right)^{9/2} } \right]$$
(1)

where Fmax is the peak load measured in the TPB experiment when the granite fractures; S is the span between the two supporting points; b is the width of the sample; h is the height of the sample; and a is the notch length of the sample, as shown in Fig 1.

According to Eq. (1), for samples of the same dimensions, fluctuations in fracture toughness (FT) should align with fluctuations in peak load. The experimentally obtained fracture toughness of granite after different heat treatments yielded average values as follows: 1.21 MPa·m1/2 (25 ℃), 1.44 MPa·m1/2 (100 ℃), 1.23 MPa·m1/2 (200 ℃), 1.15 MPa·m1/2 (300 ℃), 0.81 MPa·m1/2 (400 ℃), 0.75 MPa·m1/2 (500 ℃), 0.32 MPa·m1/2 (600 ℃), 0.26 MPa·m1/2 (700 ℃), 0.2 MPa·m1/2 (800 ℃), 0.13 MPa·m1/2 (900 ℃).

Overall, FT shows an initial increase (20–100 °C) followed by a decrease as temperature rises (Fig. 5), consistent with the experimental results of Zuo et al.6, Mahanta et al.7, Talukdar et al.8, and Shao et al.9. Fracture toughness increased by 19% from 20 °C to 100 °C, which may be attributed to an enhanced blunting effect3,9,31. When granite is heated to 100 °C, thermal stress induces the formation of a small number of microcracks, which create a “blunting” effect at the notch tip. This effect increases the toughness of the sample, thereby enhancing its fracture toughness. Additionally, the effective stress increase caused by the evaporation of water in mineral grains also contributes to the increase in fracture toughness at 100°C6.

Fig.5
figure 5

The relationship between temperature and fracture toughness of granite.

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Above 100 °C, the extent of the fracture process zone (FPZ), a region of strong nonlinear deformation, expands9,13, and the size and density of thermal cracks continue to increase6. This enhancement in ductility and plasticity of the rock leads to a gradual decline in fracture toughness from 100 °C to 900 °C.

Crack formation and evolution process

The Digital Image Correlation (DIC) method is extensively used to observe the patterns of crack propagation in various contexts such as cleavages32,33, voids34,35, multiple cavities36, rock mineral grains37, and the impact of temperature on rock fracture13. Further, this technique has been applied to develop quantitative standards for identifying types of crack failures, as detailed in several studies38,39,40.

Figure 6 shows the incremental horizontal displacement field and main strain field for different loading. Notice the Fmax means peak load in TPB test, and 85% Fmax is the load at 85% of peak load. Figures 6a-d indicated the evolution process of the incremental horizontal displacement field during 85% Fmax to peak load. A zone of displacement discontinuity formed ahead of the notch, and a tip was identified in regions where the contours merged. As the load increases, the discontinuity of displacement increases and the displacement gradient ahead notch tip increases. Figures 6e-h indicates the evolution process of the main strain field during 85% Fmax to peak load. At 85% Fmax, the strain concentration was observed at notch tip. And the zone of strain concentration increases during load increases. There is a good consistency between the strain concentration zone and the discontinuous displacement zone. It is generally acknowledged that the displacement discontinuity zone (strain concentration zone) is the FPZ, and the location of the point of convergence is the tip of FPZ13,14,20. Therefore, the sample was followed an elastic response in initial linear phase. In nonlinear period of increasing load, the FPZ appeared and propagated during the loading, and the FPZ was fully developed at peak load. From micro-mechanism of fracture propagation, FPZ development was considered as the nucleation and coalescence of micro-cracks in the vicinity of the crack tip13,16. After the peak load, real crack extension was caused by unstable crack propagation. The true crack should be expanded along the path of FPZ.

Fig.6
figure 6

Displacement field and main strain during the fracture process.

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Crack opening displacement (COD)

Figure 7 presents the load-crack tip opening displacement (F-δ) and load-crack mouth opening displacement (CMOD) curves of granite after different heat treatments. These curves characterize various fracture behaviors associated with the overall mechanical response of the beam. The mechanical behavior can be divided into a pre-peak linear phase, a pre-peak nonlinear phase, and a post-peak phase, which are closely related to the crack propagation process within the material19. In the first phase, the linear relationship between load and CTOD and between load and CMOD indicates elastic deformation. In the second phase, the load-CTOD and load-CMOD curves exhibit nonlinear growth up to the peak load, at which point microcracks begin to randomly form within the maximum tensile strain zone above the notch16,19,20, as shown in Fig. 6. After the peak load, both CTOD and CMOD continue to increase while the load decreases, indicating the onset of unstable crack propagation.

Fig. 7
figure 7

Load-CTOD curve of granite after different thermal treatments.

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In Fig. 7, the range of the pre-peak linear phase decreases with increasing heat treatment temperature, while the range of the pre-peak nonlinear phase expands as temperature rises. After the peak load, the load–displacement curve declines more gradually with higher temperatures. In the temperature range of 600–900 °C, a plateau (yield phenomenon) appears at the peak load on the load-CTOD and load-CMOD curves. Notably, the significant nonlinear response suggests a larger fracture process zone (FPZ)41.

The critical crack tip opening displacement (CTODc, δc) values at different temperatures are shown in Fig. 8. For heat treatment temperatures of 20 °C, 100 °C, 500 °C, 600 °C, and 900 °C, the average δc values are 9.8 μm, 10.5 μm, 23.6 μm, 45.9 μm, and 124.4 μm, respectively. Compared to δc at 20 °C, the average δc increased by 7%, 141%, 368%, and 1169% at heat treatment temperatures of 100 °C, 500 °C, 600 °C, and 900 °C, respectively.

Fig. 8
figure 8

The relationship between heat-treated temperature and CTODc.

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CTODc increases with rising heat treatment temperature, consistent with the findings of Wu et al.13. As heat treatment temperature rises, the density of thermal cracks gradually increases, leading to a reduction in the stiffness of granite6,7,8,9,10. Consequently, the deformation of heat-treated granite during unstable fracture increases, indicating enhanced ductility in the material as temperature increases.

Crack morphology

Studies on crack morphology can be divided into two main approaches. The first approach involves mapping the macroscopic cracks after sample failure9,10,29,42, but this method does not support quantitative analysis. The second approach uses optical and scanning electron microscopy to observe rock fracture cracks4,6,7,8,11,30. Although these microscopic observations provide high precision, they are typically limited to small areas and only reflect localized crack characteristics. While it is possible to piece together microscopic images to view the entire crack2,43, this process is complex. In this study, a 3D reconstruction of the surface cracks was performed after the three-point bending tests. The accuracy of the 3D morphology reconstruction method lies between the two aforementioned methods, allowing for both quantitative analysis of macroscopic crack morphology and ease of operation. Crack length and width were measured using post-processing software. Crack length is defined as the extension of the main crack along the y-direction, while crack width is defined as the width of the main crack in the cross-section (x-direction) within 0–5 mm of the notch tip, as shown in Fig. 9.

Fig.9
figure 9

Crack morphology of granite samples after different thermal treatments.

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Figure 9 shows the crack morphology revealed by examining the crack shapes on the surfaces of all fractured samples. Based on the stress concentration effect, cracks initiate at locations with a larger radius of curvature following the principle of minimum energy dissipation, consistent with the experimental results of Zuo et al.43. The main crack generally extends perpendicular to the direction of maximum tensile stress (x-direction in Fig. 1) and propagates along the notch direction (y-direction in Fig. 1), eventually forming a jagged and irregular primary crack.

In the temperature range of 25 °C to 500 °C, the crack propagation path is characterized by a lack of branching. However, at temperatures between 600 °C and 1000 °C, thermal cracks lead to increased crack linkage and branching, resulting in greater irregularity (roughness) of the crack propagation path, which becomes more complex(As shown in Fig. 9). As the heat treatment temperature rises, both the length and width of the main crack increase. Therefore, it can be inferred that the propagation trajectory of the main crack is significantly influenced by thermal cracking at elevated temperatures.

Scanning electron microscope (SEM) tests by Zuo et al.[6]showed that the crack path in heat-treated granite is influenced by the distribution of thermal microcracks and mineral grains. When the temperature is below 200 °C, the crack propagation path is primarily governed by the shape of mineral grains due to the small size of thermal cracks. At temperatures above 300 °C, the main crack forms by linking thermal cracks and develops a band-shaped fracture zone along the main crack path6. Below 300 °C, crack lengths range between 5–10 μm, while above 300 °C, most cracks exceed 10 μm in length44. As temperature increases, both the density and length of thermal cracks increase45, and the width of the band-shaped fracture zone also expands.

The influence of different heat treatment temperatures on crack length is significant, as shown in Fig. 10. All experimental results fall within a linear strip region, with crack length increasing linearly with heat treatment temperature, consistent with the findings of Nasseri et al.46. Figure 11 illustrates the width of the main crack at various heat treatment temperatures, where crack width exhibits bilinear growth with temperature, with an inflection point at 400 °C. The increases in both crack length and width correspond to the rise in thermal cracking and increased branching of cracks.

Fig. 10
figure 10

Crack length of granite samples after different thermal treatments.

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Fig. 11
figure 11

Crack width of granite samples after different thermal treatments.

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Microcracks coalesce into a stepped propagation pattern, with hard mineral grains affecting the macroscopic crack propagation path. Hard mineral grains can cause crack deflection, and in some cases, due to high local stress, the mineral grains themselves may fracture. In this situation, aggregate interlock occurs directly in front of the notch, and during the stage of aggregate interlock, crack propagation is almost negligible (Figs. 10 and 11).

Discussion on the Brittle-Ductile Transition of Rock Fracture

In experimental studies of rock deformation, failure modes are typically categorized as brittle or ductile. Numerous studies on rock deformation have focused on identifying the experimental conditions—often pressure and temperature—that facilitate a transition from brittle to ductile behavior47. As temperature increases, the stress thresholds for crack initiation and damage decrease, and the stable crack propagation phase extends. Additionally, higher temperatures lead to earlier crack initiation, indicating that ductile or quasi-brittle characteristics become dominant at elevated temperatures48,49.

Balme et al.5 found a correlation between the brittle-ductile transition in rock and fracture toughness as temperature varies, as shown in Fig. 12. In Region 1, rising temperatures induce thermally generated microcracks, which “blunt” the crack tip and increase plastic energy dissipation in the crack propagation zone. In Region 2, further heating induces more microcracking, leading to crack linkage and a marked reduction in fracture toughness. In Region 3, as the brittle-ductile transition is approached, the crack propagation mechanism shifts to “tearing,” allowing fracture to occur at lower stress intensities.

Fig. 12
figure 12

Schematic of how fracture toughness is expected to be influenced by temperature5.

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In Fig. 5, Region 1 corresponds to the temperature range of 20 °C to 100 °C, Region 2 to 100 °C to 600 °C, and Region 3 to 600 °C to 900 °C. Therefore, in this study, the temperature threshold for the brittle-to-ductile transition is set at 600 °C. Moreover, a sudden drop in fracture toughness is often associated with the brittle-ductile transition50. From Fig. 5, fracture toughness of granite decreases by 89% as temperature rises from 20 °C to 900 °C, with the most significant reduction of 36% occurring in the 500 °C to 600 °C range.

Brittle and ductile are classifications used to describe the macroscopic behavior of rock. Most studies suggest that the yield plateau in the stress–strain curve during compressive deformation serves as the criterion for determining ductile failure in rocks. In theory, the stress–strain curve of rocks under tensile failure can more accurately reflect the brittle-ductile transition. However, due to experimental limitations, the tensile properties of rock materials are generally measured indirectly through methods such as Brazilian splitting, three-point bending, or four-point bending tests, making it challenging to obtain direct stress–strain curves.

In practice, there is a strong correlation between the stress–strain relationship and the load-deformation relationship. In three-point bending tests, the brittle-ductile transition can be interpreted through the load–displacement and load-CTOD curves. The load–displacement curve reflects the relationship between load and the displacement of the testing machine head in the direction of crack propagation (y-direction in Fig. 1). As shown in Fig. 4, the samples exhibit significant plastic response in the temperature range of 600 °C to 900 °C, with the brittle-ductile transition occurring between 500 °C and 600 °C.

The load-CTOD curve reflects the relationship between load and sample deformation in the direction perpendicular to crack propagation (x-direction in Fig. 1). As shown in Fig. 1, segment 12 is under unidirectional tensile stress, and CTOD represents the deformation of segment 12 in the x-direction. The average tensile strain of segment 12 can be expressed as CTOD divided by the length of segment 12. Additionally, the tensile stress of segment 12 is a function of FFF. Therefore, the load-CTOD curve provides a more accurate reflection of the constitutive relationship of granite in three-point bending tensile tests.

In Fig. 7a, a clear yield plateau appears in the load-CTOD curve within the temperature range of 600 °C to 900 °C, indicating a plastic response of granite, with the brittle-ductile transition occurring between 500 °C and 600 °C.

In the cohesive crack model (Fig. 13), there is no macroscopic failure of the sample before the peak load. The value of crack tip opening displacement (CTOD) δ includes recoverable elastic deformation δe and irreversible plastic deformation δp, which can be expressed by Eq. (1), where δc (CTODc) represents the theoretical minimum value for unstable crack propagation in rock. After reaching the peak load, the crack undergoes unstable propagation, leading to the formation of a macroscopic crack. The CTOD value then equals the sum of elastic deformation, plastic deformation, and rigid displacement. Rigid displacement can be divided into recoverable rigid displacement de and irreversible rigid displacement dp, which can be expressed by Eq. (2).

Fig. 13
figure 13

Cohesive crack model.

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Once the three-point bending test is complete and the load is removed, the recoverable rigid displacement de is restored. The measured crack width after the test includes both irreversible plastic deformation δp and irreversible rigid displacement dp, which can be expressed by Eq. (3). The crack width www represents the irrecoverable deformation retained in the sample along the direction of segment 12 (tensile stress direction) after the experiment, that is, the actual minimum deformation (CTOD) measured after sufficient unstable crack propagation. The difference Δ between crack width and CTOD reflects the irreversible minimum deformation of segment 12 in the x-direction (tensile stress direction) from the onset of unstable crack propagation to the formation of a macroscopic crack. This difference represents the rock’s irreversible deformation under residual stress, thus reflecting the ductility of the rock.

$$\delta_{{{\text{pre}} - {\text{peek}}}} = \delta_{{\text{e}}} + \delta_{{\text{p}}}$$
(2)
$$\delta_{{{\text{pos}} - {\text{peek}}}} = \delta_{{\text{e}}} + \delta_{{\text{p}}} + d_{{\text{e}}} + d_{{\text{p}}}$$
(3)
$$w = \delta_{{\text{p}}} + d_{{\text{p}}}$$
(4)
$$\Delta = w_{{}} - \delta_{{\text{c}}}$$
(5)

In Fig. 8, CTODc increases with rising heat treatment temperature, with a more pronounced increase rate above 500 °C. In Fig. 12, crack width shows bilinear growth with heat treatment temperature, displaying an inflection point at 400 °C. In Fig. 14, as the heat treatment temperature increases, Δ initially increases slowly (20–400 °C), then rises rapidly (400–700 °C), and finally decreases between 700 °C and 900 °C. The experimental samples naturally cooled in air. The cohesive strength between mineral particles significantly weakens, resulting in a slower increase in crack width (as shown in Fig. 12), a reduction in rigid displacement (dp), and a faster increase in CTODc between 700 °C and 900 °C, which ultimately leads to a decrease in Δ.

Fig. 14
figure 14

Average irreversible deformation in residual stress of granite samples after different thermal treatments.

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Therefore, based on fracture toughness, CTODc, load–displacement curves, and load-CTOD curves of granite, the brittle-ductile transition temperature range is identified as 500 °C to 600 °C. Based on crack width and irreversible deformation under residual stress (Δ), the brittle-ductile transition temperature is around 400 °C. However, since crack width and Δ also include irreversible rigid displacement, setting the brittle-ductile transition temperature range at 500 °C to 600 °C is considered reasonable.

In the reconstruction of geothermal reservoirs in hot dry rocks (HDR), most fractures are mode I fractures, and their propagation characteristics and geometries significantly influence fluid flow within the rock mass fractures51. An increase in the number of fractures enhances the permeability of the HDR rock mass, while the growing complexity of the fractures improves the heat exchange efficiency, thereby increasing thermal energy extraction efficiency.

As the temperature of the HDR reservoir rises, fracture propagation gradually transitions toward ductile behavior. Under identical external conditions, the length, width, and roughness of the fractures all increase, favoring higher efficiency in geothermal energy extraction from HDR. Brittle failure occurs instantaneously, releasing substantial energy in a short period and posing risks of microseismic events that could destabilize the HDR reservoir. In contrast, ductile fracture in high-temperature rock occurs slowly, with ductile deformation absorbing the energy that would otherwise be released during failure.

Therefore, theoretically, geothermal energy extraction from higher-temperature HDR reservoirs is more efficient, safer, and more economical.

Conclusions

This study investigated the mode I fracture properties and crack initiation and propagation processes of heat-treated granite (20 °C to 900 °C) using three-point bending tests (TPB) based on Digital Image Correlation (DIC) technology and profilometry. The geometric characteristics of crack propagation were quantitatively analyzed. The main findings are as follows:

  1. 1.

    After high-temperature natural cooling, the fracture toughness (FT) of granite initially increases within the 20 °C to 100 °C range and then decreases with further temperature rise.

  2. 2.

    The critical crack tip opening displacement (CTODc) increases with rising heat treatment temperature.

  3. 3.

    The main crack propagation path is influenced by thermal cracks. In the 25 °C to 500 °C range, the crack propagation path shows no significant branching. However, in the 600 °C to 1000 °C range, thermal cracks lead to more crack linkages and branching, increasing the roughness of the crack path.

  4. 4.

    Crack length increases linearly with temperature, while crack width shows bilinear growth with heat treatment temperature, exhibiting an inflection point at 400 °C.

  5. 5.

    Based on fracture toughness (FT), load–displacement curves, load-CTOD curves, and irreversible deformation under residual stress, the brittle-ductile transition temperature for heat-treated granite fracture is determined to be between 500 °C and 600 °C.