Load–displacement curve analysis

Figure 4 shows the load–displacement curves obtained for the granite NSCB samples in the three-point bending tests. The curve usually consists of compaction stage, linear elastic stage and failure stage. During the compaction stage, the slope of the curve increases nonlinearly. After reaching the linear elastic stage, the load increases linearly with the displacement increases. Below 300 °C, the load drops abruptly after the peak and the sample fails drastically, exhibiting a typical brittle fracture. At 400 °C, the load–displacement curves at larger loading rates (0.1 mm/min and 0.01 mm/min) fluctuate before reaching the peak load (Fig. 4a, b), and the decreasing trend becomes slower in the failure stage (Fig. 4a). These phenomena may be caused by the decreasing brittleness of the samples under high temperature. It is worth noting that at a loading rate of 0.001 mm/min, the curve at 300 °C has fluctuated before the peak, and the failure stage at 400 °C begins to show progressive rupture characteristics (Fig. 4c). This indicates that at lower loading rates (0.001 mm/min), the brittleness of the samples begins to weaken at lower temperatures. When the temperature rises to 500 °C (Fig. 4a, b, c), the curves are generally jagged at the failure stage, and the sample progressively ruptured until complete failure. The samples still have the ability to resist the external load when they are damaged, which is related to the decreasing brittleness of the rock with increasing temperature.

Fig. 4
figure 4

Load–displacement curves at different temperatures and loading rates. a 0.1 mm/min loading rate. b 0.01 mm/min loading rate. c 0.001 mm/min loading rate

At the loading rate of 0.1 and 0.01 mm/min, there is no obvious difference in the compaction stage of the curve, and the failure displacement basically increases with the increase of temperature. When the loading rate is reduced to 0.001 mm/min, the compaction stage is almost invisible above 200 °C, and the failure displacement decreases first and then increases with increasing temperature. In addition, at the three loading rates, the failure displacement generally increases with increasing loading rate. The above comparison shows that not only the high temperature, but also the loading rate affects the mechanical behaviour of granite.

As shown in Fig. 5, the peak load decreases generally with increasing temperature at all three loading rates, which is related to the damage of granite caused by high temperature. However, the loading rate poses different influence on the peak load at low and high temperatures. Below 200 ℃, the peak load increases with the increase of loading rate. From 200 °C to 400 °C, the peak loads at loading rates of 0.1 mm/min and 0.01 mm/min are still positively related to the loading rate. However, at a loading rate of 0.001 mm/min, the peak load is the largest. Especially at 500 ℃, the peak load decreases with the increase of loading rate, which indicates that the high temperature reverses the loading rate effect on the peak loading.

Fig. 5
figure 5

Peak load versus loading rate at different temperatures

Fracture toughness analysis

The mode I fracture toughness KIC of NSCB samples is calculated according to the following formula given by ISRM (Kuruppu et al. 2014):

$$K_{{{text{IC}}}} = Y^{^{prime}}frac{{P_{max } sqrt {pi a} }}{2RB}$$

(1)

$$Y^{^{prime}} = – 1.297 + 9.516(frac{s}{2R}) – (0.47 + 16.457(frac{s}{2R}))beta + (1.071 + 34.401(frac{s}{2R}))beta^{2}$$

(2)

where (Y^{^{prime}}) is the non-dimensional stress intensity factor;(P_{max }) is the maximum load at sample failure;(R),(a) and (B) are the production dimensions of the samples;(s) is the support span,({{text{s}} mathord{left/ {vphantom {{text{s}} {2R}}} right. kern-nulldelimiterspace} {2R}} = 0.8);(beta) is the normalized length, ({{beta = {text{a}}} mathord{left/ {vphantom {{beta = {text{a}}} R}} right. kern-nulldelimiterspace} R} = 0.5).

According to the above Eqs. (1, 2), the fracture toughness was calculated for different real-time temperatures and loading rates. The average fracture toughness and standard deviation are illustrated in Table 1. Compared with the fracture toughness at 25 °C, when the temperature increases from 100 to 500 °C, the fracture toughness at the loading rate of 0.1 mm/min decreases by 13.22%, 22.21%, 32.25%, 34.50% and 50.10%, respectively; the fracture toughness at the loading rate of 0.01 mm/min decreases by 15.80%, 24.43%, 37.34%, 37.47% and 45.34%, respectively; and the fracture toughness at the loading rate of 0.001 mm/min decreases by 16.13%, 21.62%, 22.96%, 29.44% and 24.37, respectively. The high temperature obviously reduces the fracture toughness of granite.

Table 1 Results of the fracture toughness

Figure 6 shows the trend of granite fracture toughness with real-time high temperature. Exponential function is used to fit the variation of the average fracture toughness with temperature, and the following fitting equations are obtained:

$$begin{gathered} K_{{{text{IC(0}}{.1)}}} { = – 0}{text{.10119 + 1}}{.63492} times e^{{left( {T/( – 837.08347)} right)}} hfill \ R^{2} { = 0}{text{.97036}} hfill \ end{gathered}$$

(3)

$$begin{gathered} K_{{{text{IC(0}}{.01)}}} { = 0}{text{.70896 + 0}}{.81018} times e^{{left( {T/( – 242.50761)} right)}} hfill \ R^{2} { = 0}{text{.98282}} hfill \ end{gathered}$$

(4)

$$begin{gathered} K_{{{text{IC(0}}{.001)}}} { = 1}{text{.05077 + 0}}{.4888} times e^{{left( {T/ – 86.40703} right)}} hfill \ R^{2} { = 0}{text{.95775}} hfill \ end{gathered}$$

(5)

where (K_{{{text{IC(0}}{.1)}}}),(K_{{{text{IC(0}}{.01)}}}) and (K_{{{text{IC(0}}{.001)}}}) represent the fracture toughness at different loading rates,(T) represents temperature,(R^{2}) represents the correlation coefficient.

Fig. 6
figure 6

Fracture toughness versus real-time high temperature

From 25 to 400 °C, the fracture toughness at the loading rate of 0.1 mm/min is greater than that at the loading rate of 0.01 mm/min. The increase of loading rate has a certain toughening effect on granite, but this strengthening effect is not obvious, and the difference of fracture toughness between the two is within 0.117 MPa·m1/2. It is worth noting that at the loading rate of 0.001 mm/min, the fracture toughness is not significantly affected by temperature above 200 °C, decreasing by only 0.039 MPa·m1/2 from 200 to 500 °C. The fracture toughness does not strictly follow a monotonic decrease with temperature, which may be caused by the inhomogeneity of the granite material (Yin et al. 2018). It can also be seen that the variation law of fracture toughness and peak load with real-time temperature or loading rate is consistent, indicating that the fracture toughness is proportional to peak load.

Macro-fracture traces analysis

The NSCB samples after tests are shown in Fig. 7, which clearly shows that the colour of the samples changed from blue–grey below 300 ℃ to slightly yellow at 400 ℃. At 500 ℃, the samples turn to be beige in colour and some debris can be observed at the notch of prefabricated crack. The phenomenon is attributed to that the biotite-rich granite turns to be yellow in colour at high temperature (Vazquez et al. 2016). Figure 8 shows the traces of the fracture plane, which initiates from the middle of the straight notch and then propagates along the axial direction.

Fig. 7
figure 7

NSCB samples after testing

Fig. 8
figure 8

The maximum deviation distance of the crack (Fig. 8) is defined as the perpendicular distance from the crack to the center line of the prefabricated straight notch plane (Wong et al. 2019). As shown in Fig. 9, temperature has a significant effect on the maximum deviation distance, and the temperature reduces the deviation distance of the cracks. The reason is that the high temperature reduces the strength of the rock, and cracks are more likely to expand axially along the prefabricated straight notch under load (Feng et al. 2017). Based on the research of Kuruppu et al. (2014), the maximum deviation distance of the crack should be less than 0.05D, otherwise the sample is subjected to the torsion and shear (i.e., I–II mixed mode fracture occurs). Our test results in Fig. 9 shows that the maximum deviation distance of the crack is 1.71 mm, indicating that pure tensile failure occurs. The maximum deviation distance of the cracks is close to each other for NSCB samples under different loading rates, indicating the loading rate has insignificant effect on the deviation degree of the crack. Exponential function is used to fit the variation of the maximum crack deviation distance with temperature, and the following results are obtained:

$$begin{gathered} M_{{{(0}{text{.1)}}}} { = 0}{text{.68512 + 0}}{.86351} times e^{{left( { – 0.00333T} right)}} hfill \ R^{2} { = 0}{text{.90489}} hfill \ end{gathered}$$

(6)

$$begin{gathered} M_{{{(0}{text{.01)}}}} { = 0}{text{.74216 + 1}}{.5796} times e^{{left( { – 0.00618T} right)}} hfill \ R^{2} { = 0}{text{.9044}} hfill \ end{gathered}$$

(7)

$$begin{gathered} M_{{{(0}{text{.001)}}}} { = 0}{text{.47492 + 1}}{.23584} times e^{{left( { – 0.00247T} right)}} hfill \ R^{2} { = 0}{text{.96131}} hfill \ end{gathered}$$

(8)

where (M_{(0.1)}),(M_{(0.01)}) and (M_{(0.001)}) is the maximum deviation distance of the crack at different loading rates.

Fig. 9
figure 9

Maximum deviation distance of the crack at different temperatures

Micro-damage analysis

Moreover, micro-damages on the fracture surface were also observed by SEM from the microscopic scale. The images of microcracks are shown in Fig. 10, where the number of micro cracks in granite obviously increases with temperature. The main mineral components of the granite in this study are albite, quartz and biotite. At 25 ℃, the micro-structure of the granite is comparatively intact. The pre-existing cracks and weak boundaries between mineral grains in granite are locally damaged when the material is stressed, resulting in a small number of cracks. From 25 ℃ to 300 ℃, the number of microcracks begins to increase, and the aperture of the cracks also tends to increase (Fig. 10a–d). They are mainly intergranular cracks, which occur in quartz particles, feldspar particles, or between quartz and feldspar particles (Yang. 2022). The main reason for the formation of microcracks at high temperatures is that the granite is composed of mineral particles with different thermal expansion coefficients and thermoelastic coefficients, and the mineral particles produce uneven expansion at high temperatures (Sun et al. 2015). At 400 °C, the number of intergranular cracks increases sharply and transgranular cracks begin to appear (Fig. 10e). When the temperature is increased to 500 °C, a large number of transgranular cracks appear in the sample, and the intergranular cracks and transgranular cracks are interconnected to form a large broken area (Fig. 10f). Transgranular cracks mainly appear in feldspar particles, which is due to the lower strength of feldspar than quartz (Yang. 2022). Below 500 °C, intergranular cracks appear rarely between biotite particles. The reason is that the distinctive layered microstructure of biotite, which causes cracks to be generally hindered by biotite grains and to develop around their boundaries (Li et al. 2002).

Fig. 10
figure 10

SEM images of granite at 25–500 ℃ and 0.01 mm/min loading rate (The red lines indicate the intergranular cracks, the yellow lines indicate the transgranular cracks, and the red circles indicate the broken area.). a 25 ℃. b 100 ℃. c 200 ℃. d 300 ℃. e 400 ℃. f 500 ℃

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