# Effect of real-time high temperature and loading rate on mode I fracture toughness of granite – Geothermal Energy

#### ByKe Yang, Fan Zhang, Fan-zhen Meng, Da-wei Hu and Xian-feng Tan

Aug 23, 2022 ### 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.

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.

### 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.

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.

### 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).

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