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Experimental Study of Size Effects on the Deformation Strength and Failure Characteristics of Hard Rocks under True Triaxial Compression

Experimental Study of Size Effects on the Deformation Strength and Failure Characteristics of... Hindawi Advances in Civil Engineering Volume 2021, Article ID 6832775, 15 pages https://doi.org/10.1155/2021/6832775 Research Article Experimental Study of Size Effects on the Deformation Strength and Failure Characteristics of Hard Rocks under True Triaxial Compression Qiang Han , Yaohui Gao , and Yan Zhang Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, Liaoning 110819, China Correspondence should be addressed to Qiang Han; hanqnu@163.com Received 25 June 2021; Revised 4 September 2021; Accepted 17 September 2021; Published 4 October 2021 Academic Editor: Jia Lin Copyright © 2021 Qiang Han et al. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Size effect has always been the focus of rock mechanics as a bridge between laboratory test and engineering site. Previously, the research conditions and objects of the rock size effect have mostly focused on cylindrical rock samples with different height-to- diameter ratios (H/Ds) under uniaxial or conventional triaxial compression, while there has been little research on the rock size effect under true triaxial compression (TTC), especially rectangular rock samples with different sizes and the same length-to- width-to-height ratio. Based on this, the deformation, strength, and failure characteristics of Beishan (BS) granite and Baihetan (BHT) basalt with different sample sizes under TTC were studied by a comparative analysis method. +e size effect of deformation and failure characteristics under TTC are not obvious, including stress-strain curves, Young’s modulus, peak strains, failure angles, and macrofailure mode. However, the damage stress (σ ) and peak strength (σ ) have obvious size effect; that is, the cd p smaller the sample size is, the higher the strength is. Additionally, the relationship among the peak strength, sample size, and intermediate principal stress (σ ) is power function. In addition, by comparing the peak strength increment caused by the sample size of the two types of rocks, the σ of the fine-grained BHT basalt is more sensitive to sample size than that of the coarse-grained BS granite. Finally, by analyzing the relationship between the size of the mineral grains or clusters in the two types of hard rocks and the complexity of crack propagation in the fracture surface under TTC, it is suggested that the minimum side length of rock samples should not be less than 10 times the maximum mineral clusters (such as feldspar phenocrysts in BHT basalt). In addition, the method of estimating elastic strain is established by analyzing the relationship between the size of the rock sample σ and the elastic strain under TTC. studies [4–8] have focused more on the size effect of cy- 1. Introduction lindrical rock samples with different aspect ratios under For a long time, how to combine the laboratory test results uniaxial compression. Meanwhile, the reason why there with the monitoring results of the project has been a major were few studies on the size effect of rocks under conven- tional triaxial conditions was that most rocks have showed problem faced by rock mechanics. +e size difference be- tween indoor rock samples and engineering rock mass is the brittle ductile transition characteristics with the increase of most direct obstacle to this problem, and research on the size confining pressure [9]. effect is considered to be an important part of solving this It is well known that the in situ stress in field engineering problem. To date, three main theories on the size effect have typically satisfies σ > σ > σ (σ : the maximum principal 1 2 3 1 been proposed: (a) the Weibull statistical theory of the size stress, σ : the intermediate principal stress, and σ : the 2 3 effect represented by [1], (b) the energy release-based theory minimum principal stress). After [10, 11] designed the first of the size effect represented by [2], and (c) the fractal true triaxial testing machine for rock mechanics, the re- approach of the size effect represented in [3]. Previous search on and application of TTC testing machines have 2 Advances in Civil Engineering become popular in the field of rock mechanics because this 3%, and calcite 3%. Figure 2 shows the microstructures type of machine can reflect σ . However, due to different under cross-polarized illumination of the two types of rocks. research needs and the differences in research technologies, Figure 2(a) is the microstructure of the BHT basalt; feldspar the sample sizes are largely different. In the studies of minerals with idiomorphic structures are filled by pyroxene [10, 11], the sample size was 15 × 15 × 30 mm ; in [12], the minerals with allotriomorphic structures, with no clear sample size was 19 × 19 × 38 mm ; in [13], the sample sizes boundaries between the two. According to the image scale, 3 3 were 57 × 57 × 25 mm and 76 × 76 × 178 mm ; in [14–18], the grain size was 50∼150 μm. Figure 2(b) is the micro- the sample size was 50 × 50 × 100 mm ; in [19], the sample structure of the BS granite. +ere is an alternating ar- size was 150 × 60 × 30 mm ; and in [20], the sample size was rangement of feldspar minerals with idiomorphic or 100 × 100 × 100 mm . +e sample sizes used in the previously hypidiomorphic structures and irregular quartz. +e grain mentioned studies were quite different, and most studies size was 500∼1500 μm. only tested samples of one size. However, there have been few studies on rock samples with different sizes under TTC, 2.2. Scheme and Process. True triaxial tests at the same stress and only the studies [21–23] have investigated the me- level were carried out on each type of rock sample according chanical and failure characteristics of the same rock with to the sample size, in which σ was constant (σ � 5 MPa) and different aspect ratios under true triaxial unloading condi- 3 3 the ratios of σ : σ were 1 : 1, 1 : 6, 1 : 12 and 1 : 18. +e tions. Moreover, these studies did not provide the strength, 3 2 specific stress levels are shown in Table 2. +e experiment deformation, and failure characteristics of rock samples with was completed on the high-pressure hard rock true triaxial a fixed aspect ratio but different sizes under TTC. Mean- test system [17] developed by Northeastern University. while, the existing strength criterion does not considers the +e test process was carried out according to the stress path size effect of rock, and there have been few studies on certain shown in Figure 3(a), and the stress path was divided into the important issues, for example, which sample size is more following three stages: (a) under hydrostatic pressure, suitable for the study of crack propagation on fracture σ � σ � σ was loaded simultaneously at a rate of 0.5 MPa/s surfaces. 1 2 3 until σ reached the predetermined value; (b) σ was kept In this study, the deformation, strength, and failure of BS 3 3 constant, and σ and σ were loaded synchronously at a loading granite and BHT basalt with different sample sizes under 1 2 rate of 0.5 MPa/s until σ reached the target value; (c) σ and σ TTC were analyzed. Moreover, by analyzing the relationship 2 2 3 were kept constant, the stress-controlled loading method was between the complexity of crack propagation and mineral used to increase σ1 to approximately 60∼70% of the peak particle size in the fracture surface with different sample strength, and then the strain-controlled loading method was sizes, the recommended sample sizes for analyzing crack used until the rock sample was completely damaged. Figure 3(b) propagation in the fracture surface are determined. +e shows the strain measurement method, and Figure 3(c) shows results of this study can help to understand the size effect the measurement method of failure angle of the rock sample. under TTC. Note that the focus of this study was size effect, so when the stress-controlled loading method is changed to the 2. Test Scheme and Process strain-controlled loading method, the strain rate should be −6 the same (2.67 × 10 /s) for all the rock samples. According 2.1. Specimen Preparation. BS granite and BHT basalt are to the sample width (from largest to smallest), the controlled selected as the research objects. To prevent the dispersion of deformation rates were as follows: 0.008 mm/min, test results caused by the different rock samples, the samples 0.0056 mm/min, and 0.004 mm/min. +e detailed control of the same rock with different sizes are selected from the variables of each stage of the stress path are shown in Table 3. same parent rock, and the samples with large differences are eliminated, the rock samples with the same or similar P- wave velocity are selected for the test. +e specimens were 3. Test Results used the same processing technologies, and the length: width: height ratio of the specimens was strictly controlled to 3.1. Influence of Specimen Size on Deformation Behavior. 1 : 1 : 2. +e sizes of the samples were 25 × 25 × 50 mm (SS), Figure 4 shows the stress-strain curves of the BS granite 3 3 35 × 35 × 70 mm (SM), and 50 × 50 × 100 mm (SL), and (Figure 4(a)) and BHT basalt (Figure 4(b)) with different sample dimensional tolerance and perpendicularity tolerance were sizes under σ � 5 MPa and σ � 30 MPa. +e stress-strain 3 2 given as ±0.01 and 0.02 mm for each side, respectively. +e curves of the BS granite show the elastic-plastic-brittle defor- basic physical and mechanical parameters of these two types mation and failure process, while that of the BHT basalt shows of rocks are shown in Table 1. the elastic-brittle deformation and failure process. Meanwhile, Figure 1 shows the size and photos of rock selected in changing the size of rocks did not significantly affect the overall this study. Figure 1(a) is the grayish-green BHT basalt with deformation and failure processes (the stress-strain type) be- scattered white plagioclase on the surface, and Figure 1(b) is cause the microfractures were dominant in rocks before their the BS granite. X-ray diffraction (XRD) analysis showed that peak strength was reached. However, due to the heterogeneity the mineral composition of the BHT basalt was feldspar and the randomness of the location of macro cracks in the 41.96%, pyroxene 45.57%, clinochlore 6.25%, mica 4%, and sample, the postpeak stress-strain curve will show some dif- quartz 2.22%, while the mineral composition of the BS ferences, especially the BHT basalt with high brittleness [24], as granite was feldspar 51%, quartz 35%, biotite 8%, pyroxene shown in Figure 4(b). Advances in Civil Engineering 3 Table 1: Basic physical and mechanical parameters of the BHT basalt and BS granite samples. P-wave velocity Rock type Density (g/cm ) Young’s modulus (GPa) Poisson’s ratio (μ) Tensile strength (MPa) Grain size (μm) (m/s) BHT basalt 2.95 5650± 150 55∼60 0.22 18.4 50∼150 BS granite 2.69 5100± 120 50∼54 0.27 5.06 500∼1500 BHT Basalt BS Granite (a) (b) Figure 1: Photos of different sizes of two kinds of rocks. (a) BHT basalt and (b) BS granite. Figure 5 shows Young’s modulus under the influence of different sizes was significantly different. +e difference between rock size under TTC for the two types of rocks (the cal- the ε for the size of SL and SS was 0.206% and was significantly 3p culation method of Young’s modulus is based on [25]). higher than the changes in the peak strain under other stress When the rock size was constant, Young’s modulus in- states (σ � 5 MPa, 60 MPa, and 90 MPa), as shown in creased with increasing σ , but there is not a strict positive Figure 6(a), while the ε of the BHT basalt was hardly affected 2 3p correlation between Young’s modulus and rock size. Under by the sample size, and the changes in ε were always between 3p the same stress condition, when the sample size changed, the 0.03% and 0.07%. variation in Young’s modulus of BS granite was within Under the stress condition in this paper, the peak strain 5 GPa, while that of BHT basalt were basically within 3 GPa. ranges of the BS granite were −0.46< ε < 0.11 and 2p When the sample size changed, Young’s modulus always −0.72< ε < −0.42, and those of the BHT basalt were 3p changed small within the rock size range of this study, as −0.17< ε < 0.03 and −0.31< ε < −0.11. +e analysis 2p 3p shown in the light blue area in Figures 5(a) and 5(b), in- showed that the peak strain range of the BHT basalt was dicating that Young’s modulus of the two types of rocks was significantly smaller than that of the BS granite, which in- less affected by the sample size and the regularity was not dicates that the BS granite is prone to a large yield defor- obvious. mation under the same stress. To sum up, the stress-strain Figure 6 shows the influence of sample size on the peak curves, Young’s modulus, and peak strains for the BS granite strain (ε and ε ) in the direction of σ and σ under TTC (for and BHT basalt were related to the stress state and rock 3p 2p 3 2 example, peak strain ε refers to the strain when the stress in the properties, but these were not significantly affected by the 3p direction of σ reaches peak strength). For the BS granite and rock size. BHT basalt, when the rock size was constant, ε decreased with 3p increasing σ , which showed the rock was always under tensile deformation in the direction of σ during the loading process, 3.2. Influence of Rock Size on Characteristic Stress. Figure 7 shows the characteristics of damage stress (σ ) while ε increased under the same stress condition, which 2p cd showed the deformation in the direction of σ changed from under the influence of rock size. σ is the stress point cd corresponding to the turning point of the volume strain tensile to compression. Figure 6(a) shows the peak strain of the curve, which is the maximum point of the volume strain BS granite of different sizes in the direction of σ and σ under 3 2 TTC. It can be seen that, under the same stress condition, the curve before the peak and the calculation method refers to [25, 26]. Figure 7 shows that the σ of the two types of rocks peak strain ε in the direction of σ is very close and inde- cd 2p 2 pendent of the sample size. +e relationship between ε and showed an increasing with decreasing sample size under the 2p same stress level. However, for the BS granite, as shown in sample size of BHT basalt under TTC was the same as that of the BS granite, as shown in Figure 6(b). +erefore, the rock size Figure 7(a), when σ � 90 MPa, the σ of SM was slightly 2 cd lower than that σ � 60 MPa, which may be caused by two had no significant effect on ε within the scope of this study. 2p 2 However, when σ � 30 MPa, the ε of BS granite under reasons. On the one hand, when σ � 5 MPa, σ � 90 MPa 3 2 2 3p 100×50×50 mm 70×35×35 mm 50×25×25 mm 100×50×50 mm 70×35×35 mm 50×25×25 mm 4 Advances in Civil Engineering Clinochlore Plagioclase Plagioclase K-feldspar Mica Quartz Matrix : Diopside Plagioclase 500 µm 500 µm (a) (b) Figure 2: Microstructures under cross-polarized illumination of the two types of rocks. (a) BHT basalt and (b) BS granite. Table 2: Test scheme and results of the size effect for two rocks under TTC. Rock type Size (mm ) σ (MPa) σ (MPa) σ (MPa) σ (MPa) σ /σ ε (%) ε (%) θ ( ) A Failure mode 3 2 cd p cd p 3p 2p 5 144 202 0.71 −0.452 −0.452 74 201.82 Shear 30 186 268 0.69 −0.512 −0.080 79 270.33 Shear SL 5 60 193 295 0.65 −0.493 −0.006 80 294.79 Tension-shear 90 197 308 0.64 −0.577 0.061 80 305.97 Tension-shear 5 150 203 0.74 −0.438 −0.438 73 201.82 Shear 30 190 285 0.67 −0.602 −0.143 77 270.33 Shear BS granite SM 5 60 205 300 0.68 −0.501 −0.006 81 294.79 Tension-shear 90 204 313 0.65 −0.589 0.106 81 305.97 Tension-shear 5 151 205 0.74 −0.453 −0.453 72 201.82 Shear 30 200 292 0.69 −0.718 −0.163 78 270.33 Shear SS 5 60 212 311 0.68 −0.612 0.021 82 294.79 Tension-shear 90 211 325 0.65 −0.608 0.052 81 305.97 Tension-shear 5 250 250 1.00 −0.168 −0.168 74 247.94 Tension-shear 30 278 282 0.99 −0.185 −0.023 78 277.89 Tension-shear SL 5 60 324 324 1.00 −0.186 0.010 78 318.65 Tension-shear 90 341 350 0.97 −0.270 0.030 80 346.57 Tension-shear 5 261 261 1.00 −0.119 −0.119 75 265.25 Tension-shear 30 286 296 0.97 −0.202 −0.100 79 277.89 Tension-shear BHT basalt SM 5 60 339 339 1.00 −0.224 −0.019 79 318.65 Tension-shear 90 353 358 0.99 −0.230 0.028 80 346.57 Tension-shear 5 274 284 0.96 −0.165 −0.165 75 265.25 Tension-shear 30 329 334 0.99 −0.218 −0.073 78 277.89 Tension-shear SS 5 60 362 366 0.99 −0.233 −0.032 80 318.65 Tension-shear 90 367 386 0.95 −0.302 −0.029 88 346.57 Tension-shear was just near the turning point where σ first increased and preset σ did not reach the decreasing stage of σ in the two p 2 p then decreased [27]. On the other hand, reference [26] types of rocks under this condition, σ did not decrease). pointed out that the σ range of the BS granite is (0.64∼0.74) Moreover, the smaller the sample size was, the higher the σ cd p σ under TTC, which is within a reasonable range. In of the two types of rocks under the same stress, such as when comparison, the σ /σ of the BHT basalt under TTC was σ � 30 MPa. cd p 2 relatively large, approximately 0.95∼1.0 (Table 2), and the Figure 8(a) shows the σ of the BS granite with different turning point for it, where σ first increases and then de- sizes under TTC. When σ � σ � 5 MPa, as the sample size p 2 3 creases, is higher than BS granite. +erefore, this result is decreased from SL to SM and SS, the σ increment was very rarely found in BHT basalt: σ at σ � 90 MPa is slightly small, approximately 1 MPa or 2 MPa, which indicated that cd 2 lower than that of σ � 60 MPa (Figure 7(b)). the σ of the BS granite was almost unaffected by sample size. 2 P References [28–30] showed that when σ is constant, the In contrast, when the conventional triaxial stress condition σ increases first and then decreases with increasing σ (σ � σ ) changed to the true triaxial stress condition P 2 2 3 under TTC. Figure 8 shows the σ under the influence of (σ ≠ σ ), the size effect on the σ of the BS granite was P 2 3 P sample size for the BS granite and BHT basalt. As seen from significant. Figure 8, when σ � 5 MPa, the σ of the two types of rocks Compared with that of BHT basalt, changing the sample 3 P with different sizes increased with increasing σ (since the size (SL⟶ SM⟶ SS) of the BS granite will lead to the Biotite Advances in Civil Engineering 5 LVDTs Beam type strain gauge (b) (a) (c) Figure 3: Stress path and strain measurement of rock under TTC. (a) Stress path, (b) strain measurement method [17], and (c) measurement method of failure angle. Table 3: Controlling rates of each loading stage during the TTC test. Rock size (mm ) Loading rate of σ (MPa/s) Loading rate of σ (kN/s) Loading rate of σ (kN/s) Deformation rate (mm/min) 3 2 1 50 × 50 × 100 (SL) 0.5 2500 1250 0.008 35 × 35 × 70 (SM) 0.5 1225 612.5 0.0056 25 × 25 × 50 (SS) 0.5 625 312.5 0.004 unstable change of the peak strength increment the σ increment of the BS granite was approximately less ((σ − σ )/σ or (σ − σ )/σ ). For example, the than 10%, while that of the BHT basalt was approximately pSM pSL pSL pSS pSM pSM σ increment was 6.34% when the sample size of the BS 20%. For BHT basalt, the σ increment caused by the re- p p granite decreased from SL to SM under the stress condition duction of sample size from SM to SS was almost twice that that σ � 5 MPa and σ � 30 MPa, which was significantly caused by the reduction of sample size from SL to SM. 3 2 higher than the average 2.52% under other stress states. In Meanwhile, the σ increment of the BS granite also increased contrast, when the sample size of BHT basalt decreased from as the sample size changed, but the changes were very small. SL to SM and from SM to SS, the percentages of the σ +e smaller the sample size of the BHT basalt, the higher the increment were always maintained at approximately 4.1% sensitivity of the peak strength to the size effect, indicating and 9.3%, respectively, which was obtained by comparing that the sensitivity of the peak strength of BHT basalt to the the width of the blue or yellow areas enclosed by the changes size effect was higher than that of BS granite under the same in the peak strength caused by the sample size under each stress. stress in Figures 8(a) and 8(b). However, Figure 8(b) showed To clarify the relationship between the peak strength, the that the size effect on σ of BHT basalt was obvious in both sample size, and the stress state under TTC, the statistical the conventional triaxial and TTC. When the sample size analysis of the test results was carried out, as shown in decreased from SL to SM, change of the width of the blue Figure 9. +e volume of the SL sample was V, and the strip area was consistent with the change of the width of the volumes of the SM and SS samples were normalized ′ ′ yellow strip area when the sample size decreased from SM to according to V, such as V � 0.343 V and V � 0.125 V. In SM SS SS, and there were no abrupt changes in the σ increment Figure 9, the normalized results are plotted as the horizontal under a certain stress, which was different from the results axis, and the peak strength is the vertical axis. Figure 9(a) is for the BS granite. the results of BS granite, and Figure 9(b) shows those of BHT +e previously mentioned analysis showed that the basalt. sensitivity of σ for BS granite and BHT basalt to sample size As can be seen from Figure 9, the peak strengths of both was different. When the sample size decreased from SL to SS, types of rocks decreased with increasing sample size under Loading σ Loading σ Loading σ θ 6 Advances in Civil Engineering 350 400 ε ε ε 3 2 1 ε ε ε 3 2 v 280 320 210 240 140 160 70 80 0 0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 ε (%) ε (%) BS Granite BHT Basalt Size : SL σ = 5 MPa Size : SL σ = 5 MPa 3 3 Size : SM σ = 30 MPa Size : SM σ = 30 MPa 2 2 Size : SS Size : SS (a) (b) Figure 4: Full stress-strain curves of the BS granite and BHT basalt with different sample sizes under σ � 5 MPa and σ � 30 MPa. (a) BS 3 2 granite and (b) BHT basalt. aσ +bσ +c the same stress state. All test data were fitted by power (5) function, and the fitting variance of data was greater than σ � σ × 􏼠 􏼡 . p pV 0.9, indicating that the power function could well express the relationship between peak strength, sample size, and stress According to a, b, and c of the BS granite and BHT basalt state. Note that multiple curves were used to fit the ex- obtained in Figure 10, the binomial expression of B and σ perimental data under TTC because of the variable of σ , and can be expressed as follows: the general equation of the fitting curve was determined to be − 6 2 − 4 B � −2.778 × 10 σ + 7.167 × 10 σ − 0.06, Granite 2 (6) (1) R � 1, σ � A × 􏼠 􏼡 , − 6 2 − 3 B � −7.222 × 10 σ + 1.45 × 10 σ − 0.12, where B (B< 0) is a parameter related to the rock type and σ Basalt 2 2 2 (7) ′ ′ 2 and the units of V and V are mmQ. WhenV � V, R � 1. σ � A. (2) PV Equations (6) and (7) can be substituted into equation Substituting equation (2) into (1), we derive the (5) to obtain an expression relating σ , V, and σ for the BS p 2 following: granite and BHT basalt: −6 2 −4 ′ − 2.778×10 σ +7.167×10 σ − 0.06 V 2 (3) σ � σ × 􏼠 􏼡 . p pV σ � σ × 􏼠 􏼡 , V pGranite pV (8) −6 2 −3 − 7.222×10 σ +1.45×10 σ − 0.12 Because B is related to σ , the fitting relationship between 2 2 σ � σ × . them is obtained (Figure 10), which shows that the fitting 􏼠 􏼡 pBasalt pV result is well (R � 1). +us, the relationship between B and σ of the BS granite and BHT basalt can be expressed as +e relationship of the peak strength and damage follows: stress to the sample size of the BS granite and BHT basalt under TTC showed that the variation amplitude of the B � aσ + bσ + c, (4) 2 2 characteristic stress increment caused by the size effect in the fine-grained BHT basalt was obviously smaller than where a, b, and c are the fitting parameters related to li- thology, as shown in Figure 10. +e general expression of σ that of the medium- to coarse-grained BS granite, and the characteristic stress of the two types of rocks was was obtained by substituting equation (4) into (3): σ -σ (MPa) 1 3 σ -σ (MPa) 1 3 Advances in Civil Engineering 7 80 80 BS Granite σ = 5 MPa BHT Basalt σ = 5 MPa 3 3 70 70 60 60 50 50 40 40 30 30 0 15 30 45 60 75 90 0 15 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 5: Young’s modulus characteristics under the influence of rock sample size under TTC. (a) BS granite and (b) BHT basalt. 0.20 0.15 BS Granite BHT Basalt 0.00 0.00 -0.20 -0.15 -0.40 -0.30 -0.30 0.00 -0.45 -0.15 -0.60 -0.30 -0.75 -0.40 0 15 30 45 60 75 90 105 0 15 30 45 60 75 90 105 σ (MPa) σ (MPa) 2 2 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 6: Influence of sample size on the peak strain in the direction of σ and σ under TTC. (a) BS granite and (b) BHT basalt. 3 2 obviously affected by the sample size. +e relationships σ � 30 MPa and σ � 5 MPa, and Table 2 shows the failure 2 3 among the peak strength, rock sample size, and inter- modes and fracture angles of the rocks under TTC. It can be mediate principal stress could be represented by a power seen from Figure 11 that both BS granite and BHT basalt function. show macroshear failure under the same stress condition σ � 5 MPa and σ � 30 MPa, which indicates that the sample 3 2 size did not change the macroscopic failure mode for the two 3.3. Influence of Sample Size on Failure Characteristics. To types of rocks under the same stress. However, reducing the better compare the rock failure modes with different sizes, sample size may lead to secondary cracks near the main the failure pictures of the samples with different sizes were crack near the center of the sample, which are nearly parallel enlarged to the same size. Figure 11 shows the failure photos to the direction of σ , making the fracture surface more of BS granite and BHT basalt under different sizes at complex as shown in areas enclosed by red lines in ε (%) ε (%) 3p 2p E (GPa) ε (%) ε (%) E (GPa) 3p 2p 8 Advances in Civil Engineering 220 380 BS Granite BHT Basalt 140 240 0 15 30 45 60 75 90 0 15 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 σ = 5 MPa σ = 5 MPa 3 3 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 7: Damage stress characteristics under the influence of rock sample size. (a) BS granite and (b) BHT basalt. 330 400 BS Granite BHT Basalt 195 240 0 1530456075 90 0 1530456075 90 σ (MPa) σ (MPa) σ = 5 MPa σ = 5 MPa 3 3 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 8: Peak strengths of the two types of rocks with different sizes under TTC. (a) BS granite and (b) BHT basalt. Figures 11(c), 11(f ), and 12(a). Additionally, for the BHT clusters increased the complexity of the fracture surfaces of basalt with the size of SS, the cracks are easy to develop along the small-sized rock samples. the mineral cluster (feldspar phenocryst) during its prop- Figure 13 shows the failure angles of the two types of agation except for secondary cracks in the fracture surface, as rocks with different sizes under TTC, and the measurement shown in the areas enclosed by blue lines in Figures 11(e), of the failure angle refers to [17]. For the tortuous fracture 11(f ), and 12(b), which is possibly because the size of the surface, the near-linear measuring method was used, as mineral clusters was of the same order of magnitude as the shown in Figure 3(c). Figure 13 shows that when the sample length of the shortest edge of rock samples. +erefore, the size was constant and σ increased, the failure angle θ in- generation of secondary cracks and cracks along the mineral creased. However, changing the sample size did not σ (MPa) σ (MPa) cd σ (MPa) cd σ (MPa) p Advances in Civil Engineering 9 360 450 -0.018 2 σ =305.97(V′/V) , R =0.9170 -0.048 2 σ =346.57(V′/V) , R =0.9052 -0.027 2 350 σ =294.79(V′/V) , R =425 -0.041 2 σ =270.33(V′/V) , R =0.9009 -0.059 2 300 σ =321.94(V′/V) , R =0.9738 -0.083 2 σ =277.89(V′/V) , R =0.9376 -0.062 2 -0.007 2 σ =247.94(V′/V) , R =0.9608 180 σ =201.82(V′/V) , R =0.9586 BS Granite BHT Basalt 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.3 0.6 0.9 1.2 1.5 V′/V V′/V σ = 5 MPa σ = 60 MPa σ = 5 MPa σ = 60 MPa 2 2 2 2 σ = 30 MPa σ = 90 MPa σ = 30 MPa σ = 90 MPa 2 2 2 2 (a) (b) Figure 9: Relationship between the peak strength, sample size, and stress state of the two rocks under TTC (symbols of the same color represent the σ of different sizes under the same stress level). (a) BS granite and (b) BHT basalt. -0.01 -0.02 BS Granite BHT Basalt -0.02 -0.04 B -0.03 B -0.06 -0.04 -0.08 –6 2 –4 2 –6 2 –3 2 B = –2.778×10 σ + 7.167×10 σ –0.06, R = 1 B = –7.222×10 σ + 1.45×10 σ –0.12, R = 1 2 2 2 2 -0.05 -0.10 30 45 60 75 90 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 B B Fitting cure Fitting cure (a) (b) Figure 10: Fitting results between σ and B. (a) BS granite and (b) BHT basalt. significantly impact the angle of the fracture surface under 4. Discussion the same stress. For example, in Figure 13(a), when the 4.1. Microscopic Interpretation of Complex Fracture Surfaces sample size of the BS granite decreased from SL to SS under the same stress, the fracture angle only varied by approxi- Caused by the Size Effect. Section 3.3 showed that changing the sample size does not significantly affect the macroscopic mately 1∼2 ; in Figure 13(b), the variation in the fracture failure mode under the same stress. However, the crack angle of the BHT basalt was basically the same as that of the propagation in the fracture surface became more complex BS granite under the same condition, approximately 0∼3 . when the sample size decreased to SM or SS, as shown in +us, the small variation of the fracture angle in this paper Figures 11(c), 11(e), 11(f ), and 12. Under TTC, a macro- further shows that the sample size do not significantly scopic shear fracture plane with a “V” shape was easily change the macroscopic failure mode of these two types of rocks, while the reduction of sample size will lead to more generated [29, 31, 32]. On this type of fracture surface, especially near the center of the rock sample, almost no complex crack propagation on the fracture surface. σ (MPa) σ (MPa) P 10 Advances in Civil Engineering Size : SL Size : SM Size : SS (a) (b) (c) Size : SL Size : SM Size : SS (d) (e) (f ) Figure 11: Failure modes of rock samples with different sizes under the same stress state of σ2 � 30 MPa and σ3 � 5 MPa. (a–c) BS granite; (d–f ) BHT basalt. obvious secondary cracks nearly parallel to the direction of +e complex crack propagation on the failure surface σ were generated during the propagation of the main of the small-sized samples may be related to the mineral cracks, as shown in Figures 11(a), 11(b), and 11(d). How- grain size or the mineral grain aggregate size of the rocks. ever, this situation is likely to occur when the sample size It is well known that grain size is one of the most im- decreased from SL to SS under the same stress, as shown in portant microstructure parameters of rock mechanical Figures 11(c), 11(f ), and 12(a). properties. Taking the BHT basalt as an example, Advances in Civil Engineering 11 Size : SS Size : SS 5 mm σ = 60 MPa σ = 60 MPa 2 2 σ = 5 MPa σ = 5 MPa 3 3 (a) (b) Figure 12: Failure mode of two rock samples of SS size at the same true triaxial stress state of σ � 60 MPa and σ � 5 MPa. (a) BS granite and 2 3 (b) BHT basalt. Figure 2(a) shows that pyroxene, feldspar, and other of feldspar. +erefore, feldspar and other weaker minerals minerals were uniformly arranged in the matrix of the were more prone to brittle failure during the process of stress cracking, as demonstrated by the closed fractures BHT basalt and the size of feldspar grain was 50∼150 μm. During the diagenetic process, a large number of feldspar on the surface of feldspar phenocrysts (Figures 2(a) and 14(c)), which may explain why the crack propagation on grains aggregated to form lath-shaped white feldspar phenocrysts, with a length of 5 mm or larger, as shown in the fracture surface is more complex for small samples Figures 12(b), 14(a), and 14(b). Pyroxene is a silicate than for large samples (the complexity of the crack rock-forming mineral with a shear modulus of 64.9 GPa, propagation of large rock samples was much lower). and feldspar is a brittle rock-forming mineral with a shear In the samples with size SS in this study, the ratio of the modulus of 28.6 GPa. Section 2.1 showed that the total size of the large feldspar phenocrysts to the length or width composition of pyroxene and feldspar minerals in the of the sample reached 1/5 (Figure 12(b)), and the ratio was BHT basalt accounted for more than 87%, and the two even larger when multiple phenocrysts were aggregated. constituted the basic framework. Pyroxene is a mineral Reference [34] pointed out that grain size plays an important role in crack propagation and used numerical modeling to with an allotriomorphic structure, while feldspar and others are minerals with idiomorphic structures, and the show that the interactions of adjacent cracks can be used to structural relationship between them is similar to the inhibit crack propagation. Additionally, they also pointed relationship between water and stone in a river. Basalt is out that this inhibitory effect can gradually disappear with igneous rock and pyroxene (like water) can fill the holes the increase of grain size. For the BHT basalt, the size of the and gaps between minerals with an allotriomorphic feldspar phenocrysts remained unchanged, but the de- structure (like stone) in the process of diagenesis with no creased sample size was equivalent to indirectly increasing clear boundaries between the two. the size of the relatively weak feldspar phenocrysts Reference [33] showed that, in the 6 × 6 stiffness (Figures 14(a) and 14(b)), and the role of feldspar pheno- matrix represented by the Voigt notation, the stiffness of crysts in the structure could not be ignored. +erefore, the single-crystal pyroxene in all directions is larger than that inhibition effect of the surrounding cracks could be 12 Advances in Civil Engineering 84 84 BS Granite BHT Basalt 70 72 0 15 30 45 60 75 90 0 15 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 σ = 5 MPa σ = 5 MPa 3 3 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 13: Failure angles of two types of rocks with different sizes under TTC. (a) BS granite and (b) BHT basalt. Closed cracks 100 µm 50 mm 25 mm (c) (a) (b) Figure 14: Pictures of BHT basalt and its microstructures under cross-polarized illumination. (a) BHT basalt specimen of SL size; (b) local magnification of the specimen in (a); (c) microstructures of (b) under cross-polarized illumination part. weakened and the crack density could increase when the to the direction of σ . In the left main fracture plane cracks propagated to the vicinity of feldspar phenocrysts (Figure 12(b)), two groups of feldspar phenocrysts in the two with a relatively large size. References [35, 36] showed that, areas enclosed by blue dashed lines led to the propagation for fine-grained materials, an increase in crack density can direction of some cracks (all in the same direction), resulting be equivalent to an increase in the spatial heterogeneity of in poor symmetry of the left and right fracture planes and a the local stress field. +erefore, when cracks occur, cracks are “Y” shaped fracture plane. +e cracks that grew along the more likely to propagate along the weak feldspar pheno- feldspar phenocrysts were also observed in the area enclosed crysts, which results in a complex crack morphology on the by red dashed lines in the right main fracture surface. When fracture surface. For example, in the area enclosed by the the direction of the feldspar phenocrysts was close to the blue dashed line in Figure 11(e), it is obvious that a crack growth direction of cracks in the fracture surface, the cracks developed along the axis of feldspar phenocrysts. In the were more likely to grow along the feldspar phenocrysts. For small area enclosed by the blue dashed line in Figure 11(f ), the BS granite, the distribution of constituent minerals was the main crack passed through the axis of the feldspar relatively uniform, and the grain size reached 500∼1500 μm phenocrysts and produced secondary cracks nearly parallel or even larger. When the sample size was reduced to SS, the 7 mm High : 100 mm θ (°) High : 50 mm θ (°) Advances in Civil Engineering 13 0.57 0.62 BS Granite BHT Basalt 0.60 0.54 σ = 5 MPa σ = 5 MPa 3 3 0.58 0.51 0.56 0.48 0.54 0.45 0.52 0.42 eb eb 2 0.50 ε = 1.0035 + 0.7628ε ,R = 0.90 P O eb eb 2 ε = 1.5391 + 0.7423ε ,R = 0.85 P O 0.39 0.48 0.40 0.44 0.48 0.52 0.56 0.60 0.44 0.48 0.52 0.56 0.60 0.64 eb eb Observed ε (%) Observed ε (%) (a) (b) Figure 15: Prediction ability of equation (11) for total elastic strain in the direction of σ for two types of samples. (a) BS granite and (b) BHT basalt. fracture surface became complex, as shown in Figures 11(c) aσ +bσ +c σ × V /V − σ pV 2 eb eb and 12(a). +is may be one of the reasons why the Inter- (11) ε � ε + . f σ 􏼁 national Society for Rock Mechanics (ISRM) recommends 2 that the minimum side length of the sample should be more As shown in Figure 15, the linear regression coefficients than 10 times the maximum grain size of minerals. Based on of the total elastic strain of the BS granite and BHT basalt the previously mentioned analysis, it is better to choose a with sizes of SM and SS predicted by formula (11) in the large rock sample when studying crack propagation on the 2 2 direction of σ were R � 0.90 and R � 0.85, respectively, fracture surface under TTC. +e minimum side length of the indicating that the prediction ability of formula (11) was sample should be at least 10 times larger than the maximum reasonable. grain size of the mineral (ISRM) and the maximum grain size of the mineral aggregates with an idiomorphic structure (such as feldspar phenocrysts) to avoid a complex fracture 5. Conclusion surface. In this study, BS granite and BHT basalt with the same length : width : height ratio and different sizes were used to 4.2. Relationship between Elastic Strain and Sample Size. study the size effect under TTC conditions. +e following conclusions are drawn: +e test results in Section 3.2 showed that there was little relationship between Young’s modulus and sample size. +e (1) Regarding the deformation and failure characteris- maximum Young’s modulus of the same samples with tics within the range of rock size for this study, different sizes under the same stress state was fitted with σ , including the stress-strain curve, Young’s modulus, and the relationship between them was obtained as follows: peak strain in the directions of σ and σ , fracture 3 2 angle, and macrofailure mode, there was almost no E � dσ + e, (9) obvious size effect. However, the characteristics of deformation for the two types of rocks were related where d and e are related to rock type. to the rock properties and external stress conditions. According to the calculation method of elastic strain (2) +e peak strength and damage stress of the BS under TTC proposed in [24], granite and BHT basalt were significantly affected by the sample size and σ under TTC. As the sample size σ − σ p 2 eb eb (10) ε � ε + , decreased, the σ and σ increased. For these two p cd types of rocks, there was a power function rela- eb tionship among the peak strength, sample size, and where ε is the total elastic strain in the direction of σ and eb σ under TTC. Under the same conditions, the ε is the elastic strain in the biaxial loading stage in the sensitivity of the peak strength of the fine-grained direction of σ under TTC, that is, the elastic strain in the BHT basalt to the sample size was higher than that of process of Section 2.2 stress path b. the medium- to coarse-grained BS granite. Taking equation (5) into (10), the following can be obtained for calculating the total elastic strain in the di- (3) +e complex crack propagation on the fracture rection of σ related to the sample size under TTC: surface of smaller rock samples was due to indirectly eb Predicted ε (%) eb Predicted ε (%) 14 Advances in Civil Engineering [8] P. A. Cundall, M. E. Pierce, and D. M. Ivars, “Quantifying the increasing the mineral grain size or mineral cluster size effect of rock mass strength,” in Proceedings of the size in the rocks. Moreover, this study suggests that Southern Hemisphere International Rock Mechanics the minimum side length of rock samples should be Symposium, Perth, Australia, January 2008. at least 10 times the maximum size of the mineral [9] D. D. Hunt, “+e influence of confining pressure on size clusters when studying crack propagation on a effect,” Master of Sciences in Civil Engineering, Massachusetts fracture surface. Institute of Technology, Cambridge, MA, USA, 1973. (4) +e estimation method of elastic strain in a certain [10] K. Mogi, “Effect of the triaxial stress system on rock failure,” range of sample sizes was established by analyzing Rock Mechanics Japan, vol. 1, pp. 53–55, 1970. the relationship among sample size, peak strength, [11] K. Mogi, “Fracture and flow of rocks under high triaxial intermediate principal stress, and elastic strain in the compression,” Journal of Geophysical Research, vol. 76, no. 5, direction of σ , and the prediction result was well. pp. 1255–1269, 1971. [12] B. Haimson and C. Chang, “A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock Data Availability strength and deformability of Westerly granite,” International Journal of Rock Mechanics and Mining Sciences, vol. 37, no. 1- +e data used to support the findings of this study are 2, pp. 285–296, 2000. available from the corresponding author upon request. [13] W. R. Wawersik, L. W. Carlson, D. J. Holcomb, and R. J. Williams, “New method for true-triaxial rock testing,” Conflicts of Interest International Journal of Rock Mechanics and Mining Sciences, vol. 34, pp. 3-4, 1997. +e authors declare that they have no conflicts of interest. [14] P. Michelis, “A true triaxial cell for low and high pressure experiments,” International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, vol. 22, no. 3, Acknowledgments pp. 183–188, 1985. +e authors sincerely acknowledge the financial support [15] M. Takahashi and H. Koide, “Effect of the intermediate from the Liao Ning Revitalization Talents Program under principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than 2000 m,” Grant no. XLYC1801002, the National Natural Science International Journal of Rock Mechanics and Mining Science & Foundation of China under Grant no. 51839003, and the 111 Geomechanics Abstracts, vol. 27, pp. 19–26, 1989. Project under Grant no. B17009. +e authors are grateful to [16] J. T. Chen and X. T. Feng, “True triaxial experimental study on Prof. Xiating Feng, Prof. Xiwei Zhang, Prof. Chengxiang rock with high geostress,” Chinese Journal of Rock Mechanics Yang, Mr. Zhaofeng Wang, and Ms. Xinyue Wang for their and Engineering, vol. 25, pp. 1537–1543, 2006, (in Chinese). great assistance. [17] X.-T. Feng, X. Zhang, R. Kong, and G. Wang, “A novel mogi type true triaxial testing apparatus and its use to obtain References complete stress-strain curves of hard rocks,” Rock Mechanics and Rock Engineering, vol. 49, no. 5, pp. 1649–1662, 2016. [1] W. Weibull, “A statistical distribution function of wide ap- [18] L. Shi, X. B. Li, B. Bing, A. Wang, Z. Zeng, and H. He, “A plicability,” Journal of Applied Mechanics, vol. 18, no. 3, mogi-type true triaxial testing apparatus for rocks with two pp. 293–297, 1951. moveable frames in horizontal layout for providing orthog- [2] Z. P. Bazant and J. Planas, Fracture and Size Effect in concrete onal loads,” Geotechnical Testing Journal, vol. 40, pp. 542–558, and Other Quasibrittle Materials, CRC Press, Boca Raton, FL, USA, 1998. [19] M. C. He, J. L. Miao, and J. L. Feng, “Rock burst process of [3] A. Carpinteri, “Scaling laws and renormalization groups for limestone and its acoustic emission characteristics under true- strength and toughness of disordered materials,” Interna- triaxial unloading conditions,” International Journal of Rock tional Journal of Solids and Structures, vol. 31, no. 3, Mechanics and Mining Sciences, vol. 47, no. 2, pp. 286–298, pp. 291–302, 1994. [4] H. R. Pratt, A. D. Black, W. S. Brown, and W. F. Brace, “+e [20] X. Li, K. Du, and D. Li, “True triaxial strength and failure effect of speciment size on the mechanical properties of modes of cubic rock specimens with unloading the minor unjointed diorite,” International Journal of Rock Mechanics principal stress,” Rock Mechanics and Rock Engineering, and Mining Science & Geomechanics Abstracts, vol. 9, no. 4, vol. 48, no. 6, pp. 2185–2196, 2015. pp. 513–516, 1972. [21] X. G. Zhao and M. Cai, “Influence of specimen height-to- [5] B. C. Liu, J. S. Zhang, Q. Z. Du, and J. F. Tu, “Size effect of width ratio on the strainburst characteristics of Tianhu granite compressive strength of rock,” Chinese Journal of Rock Me- under true-triaxial unloading conditions,” Canadian Geo- chanics and Engineering, vol. 17, pp. 611–614, 1998, (in technical Journal, vol. 52, pp. 890–902, 2014. Chinese). ¨ [22] F. Zhao and M. C. He, “Size effects on granite behavior under [6] E. Tuncay, N. T. Ozcan, and A. Kalender, “An approach to unloading rockburst test,” Bulletin of Engineering Geology and predict the length-to-diameter ratio of a rock core specimen the Environment, vol. 76, no. 3, pp. 1183–1197, 2016. for uniaxial compression tests,” Bulletin of Engineering Ge- [23] X. B. Li, F. Feng, D. Y. Li, K. Du, P. G. Ranjith, and J. Rostami, ology and the Environment, vol. 78, no. 7, pp. 5467–5482, 2019. “Failure characteristics of granite influenced by sample [7] J. Fladr ´ and P. B´ ıly, ´ “Specimen size effect on compressive and flexural strength of high-strength fibre-reinforced concrete height-to-width ratios and intermediate principal stress under true-triaxial unloading conditions,” Rock Mechanics and Rock containing coarse aggregate,” Composites Part B: Engineering, vol. 138, pp. 77–86, 2018. Engineering, vol. 51, pp. 1–25, 2018. Advances in Civil Engineering 15 [24] X. T. Feng, J. Zhao, Z. F. Wang, C. X. Yang, Q. Han, and Z. Zheng, “Effect of high differential stress and mineral properties on deformation and failure mechanism of hard rocks,” Canadian Geotechnical Journal, vol. 58, 2020. [25] C. D. Martin, Ae strength of massive Lac du Bonnet granite around underground openings, Ph.d. thesis, Department of Civil Engineering, University of Manitoba, Winnipeg, Can- ada, 1993. [26] Y.-H. Gao, X.-T. Feng, X.-W. Zhang, G.-L. Feng, Q. Jiang, and S.-L. Qiu, “Characteristic stress levels and brittle fracturing of hard rocks subjected to true triaxial compression with low minimum principal stress,” Rock Mechanics and Rock Engi- neering, vol. 51, no. 12, pp. 3681–3697, 2018. [27] Y. Zhang, Energy evolution mechanism of failure process of hard rock in deep tunnel and discrimination of typical hazard types, Ph.d thesis, Northeastern University, Shenyang, China, [28] B. Haimson and J. W. Rudnicki, “+e effect of the interme- diate principal stress on fault formation and fault angle in siltstone,” Journal of Structural Geology, vol. 32, no. 11, pp. 1701–1711, 2010. [29] X. Ma, J. W. Rudnicki, and B. C. Haimson, “Failure char- acteristics of two porous sandstones subjected to true triaxial stresses: a,” Journal of Geophysical Research: Solid Earth, vol. 122, no. 4, pp. 2525–2540, 2017. [30] R. Kong, X.-T. Feng, X. Zhang, and C. Yang, “Study on crack initiation and damage stress in sandstone under true triaxial compression,” International Journal of Rock Mechanics and Mining Sciences, vol. 106, pp. 117–123, 2018. [31] J. Zhao, X.-T. Feng, X.-W. Zhang, Y. Zhang, Y.-Y. Zhou, and C.-X. Yang, “Brittle-ductile transition and failure mechanism of Jinping marble under true triaxial compression,” Engi- neering Geology, vol. 232, pp. 160–170, 2018. [32] Y. Zhang, X. T. Feng, X. W. Zhang, Z. F. Wang, M. Sharifzadeh, and C. X. Yang, “A novel application of strain energy for fracturing process analysis of hard rock under true triaxial compression,” Rock Mechanics and Rock Engineering, vol. 52, pp. 1–16, 2019. [33] J. D. Bass, “Elasticity of minerals, glasses, and melts,” American Geophysical Union, vol. 2, pp. 45–63, 1995. [34] E. Eberhardt, D. Stead, B. Stimpson, and E. Z. Lajtai, “+e effect of neighbouring cracks on elliptical crack initiation and propagation in uniaxial and triaxial stress fields,” Engineering Fracture Mechanics, vol. 59, no. 2, pp. 103–115, 1998. [35] J. T. Fredrich, B. Evans, and T.-F. Wong, “Effect of grain size on brittle and semibrittle strength: implications for micro- mechanical modelling of failure in compression,” Journal of Geophysical Research, vol. 95, no. B7, pp. 10907–10920, 1990. [36] J. Peng, L. N. Y. Wong, and C. I. Teh, “Influence of grain size heterogeneity on strength and microcracking behavior of crystalline rocks,” Journal of Geophysical Research: Solid Earth, vol. 122, no. 2, pp. 1054–1073, 2017. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Civil Engineering Hindawi Publishing Corporation

Experimental Study of Size Effects on the Deformation Strength and Failure Characteristics of Hard Rocks under True Triaxial Compression

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Hindawi Advances in Civil Engineering Volume 2021, Article ID 6832775, 15 pages https://doi.org/10.1155/2021/6832775 Research Article Experimental Study of Size Effects on the Deformation Strength and Failure Characteristics of Hard Rocks under True Triaxial Compression Qiang Han , Yaohui Gao , and Yan Zhang Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, Liaoning 110819, China Correspondence should be addressed to Qiang Han; hanqnu@163.com Received 25 June 2021; Revised 4 September 2021; Accepted 17 September 2021; Published 4 October 2021 Academic Editor: Jia Lin Copyright © 2021 Qiang Han et al. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Size effect has always been the focus of rock mechanics as a bridge between laboratory test and engineering site. Previously, the research conditions and objects of the rock size effect have mostly focused on cylindrical rock samples with different height-to- diameter ratios (H/Ds) under uniaxial or conventional triaxial compression, while there has been little research on the rock size effect under true triaxial compression (TTC), especially rectangular rock samples with different sizes and the same length-to- width-to-height ratio. Based on this, the deformation, strength, and failure characteristics of Beishan (BS) granite and Baihetan (BHT) basalt with different sample sizes under TTC were studied by a comparative analysis method. +e size effect of deformation and failure characteristics under TTC are not obvious, including stress-strain curves, Young’s modulus, peak strains, failure angles, and macrofailure mode. However, the damage stress (σ ) and peak strength (σ ) have obvious size effect; that is, the cd p smaller the sample size is, the higher the strength is. Additionally, the relationship among the peak strength, sample size, and intermediate principal stress (σ ) is power function. In addition, by comparing the peak strength increment caused by the sample size of the two types of rocks, the σ of the fine-grained BHT basalt is more sensitive to sample size than that of the coarse-grained BS granite. Finally, by analyzing the relationship between the size of the mineral grains or clusters in the two types of hard rocks and the complexity of crack propagation in the fracture surface under TTC, it is suggested that the minimum side length of rock samples should not be less than 10 times the maximum mineral clusters (such as feldspar phenocrysts in BHT basalt). In addition, the method of estimating elastic strain is established by analyzing the relationship between the size of the rock sample σ and the elastic strain under TTC. studies [4–8] have focused more on the size effect of cy- 1. Introduction lindrical rock samples with different aspect ratios under For a long time, how to combine the laboratory test results uniaxial compression. Meanwhile, the reason why there with the monitoring results of the project has been a major were few studies on the size effect of rocks under conven- tional triaxial conditions was that most rocks have showed problem faced by rock mechanics. +e size difference be- tween indoor rock samples and engineering rock mass is the brittle ductile transition characteristics with the increase of most direct obstacle to this problem, and research on the size confining pressure [9]. effect is considered to be an important part of solving this It is well known that the in situ stress in field engineering problem. To date, three main theories on the size effect have typically satisfies σ > σ > σ (σ : the maximum principal 1 2 3 1 been proposed: (a) the Weibull statistical theory of the size stress, σ : the intermediate principal stress, and σ : the 2 3 effect represented by [1], (b) the energy release-based theory minimum principal stress). After [10, 11] designed the first of the size effect represented by [2], and (c) the fractal true triaxial testing machine for rock mechanics, the re- approach of the size effect represented in [3]. Previous search on and application of TTC testing machines have 2 Advances in Civil Engineering become popular in the field of rock mechanics because this 3%, and calcite 3%. Figure 2 shows the microstructures type of machine can reflect σ . However, due to different under cross-polarized illumination of the two types of rocks. research needs and the differences in research technologies, Figure 2(a) is the microstructure of the BHT basalt; feldspar the sample sizes are largely different. In the studies of minerals with idiomorphic structures are filled by pyroxene [10, 11], the sample size was 15 × 15 × 30 mm ; in [12], the minerals with allotriomorphic structures, with no clear sample size was 19 × 19 × 38 mm ; in [13], the sample sizes boundaries between the two. According to the image scale, 3 3 were 57 × 57 × 25 mm and 76 × 76 × 178 mm ; in [14–18], the grain size was 50∼150 μm. Figure 2(b) is the micro- the sample size was 50 × 50 × 100 mm ; in [19], the sample structure of the BS granite. +ere is an alternating ar- size was 150 × 60 × 30 mm ; and in [20], the sample size was rangement of feldspar minerals with idiomorphic or 100 × 100 × 100 mm . +e sample sizes used in the previously hypidiomorphic structures and irregular quartz. +e grain mentioned studies were quite different, and most studies size was 500∼1500 μm. only tested samples of one size. However, there have been few studies on rock samples with different sizes under TTC, 2.2. Scheme and Process. True triaxial tests at the same stress and only the studies [21–23] have investigated the me- level were carried out on each type of rock sample according chanical and failure characteristics of the same rock with to the sample size, in which σ was constant (σ � 5 MPa) and different aspect ratios under true triaxial unloading condi- 3 3 the ratios of σ : σ were 1 : 1, 1 : 6, 1 : 12 and 1 : 18. +e tions. Moreover, these studies did not provide the strength, 3 2 specific stress levels are shown in Table 2. +e experiment deformation, and failure characteristics of rock samples with was completed on the high-pressure hard rock true triaxial a fixed aspect ratio but different sizes under TTC. Mean- test system [17] developed by Northeastern University. while, the existing strength criterion does not considers the +e test process was carried out according to the stress path size effect of rock, and there have been few studies on certain shown in Figure 3(a), and the stress path was divided into the important issues, for example, which sample size is more following three stages: (a) under hydrostatic pressure, suitable for the study of crack propagation on fracture σ � σ � σ was loaded simultaneously at a rate of 0.5 MPa/s surfaces. 1 2 3 until σ reached the predetermined value; (b) σ was kept In this study, the deformation, strength, and failure of BS 3 3 constant, and σ and σ were loaded synchronously at a loading granite and BHT basalt with different sample sizes under 1 2 rate of 0.5 MPa/s until σ reached the target value; (c) σ and σ TTC were analyzed. Moreover, by analyzing the relationship 2 2 3 were kept constant, the stress-controlled loading method was between the complexity of crack propagation and mineral used to increase σ1 to approximately 60∼70% of the peak particle size in the fracture surface with different sample strength, and then the strain-controlled loading method was sizes, the recommended sample sizes for analyzing crack used until the rock sample was completely damaged. Figure 3(b) propagation in the fracture surface are determined. +e shows the strain measurement method, and Figure 3(c) shows results of this study can help to understand the size effect the measurement method of failure angle of the rock sample. under TTC. Note that the focus of this study was size effect, so when the stress-controlled loading method is changed to the 2. Test Scheme and Process strain-controlled loading method, the strain rate should be −6 the same (2.67 × 10 /s) for all the rock samples. According 2.1. Specimen Preparation. BS granite and BHT basalt are to the sample width (from largest to smallest), the controlled selected as the research objects. To prevent the dispersion of deformation rates were as follows: 0.008 mm/min, test results caused by the different rock samples, the samples 0.0056 mm/min, and 0.004 mm/min. +e detailed control of the same rock with different sizes are selected from the variables of each stage of the stress path are shown in Table 3. same parent rock, and the samples with large differences are eliminated, the rock samples with the same or similar P- wave velocity are selected for the test. +e specimens were 3. Test Results used the same processing technologies, and the length: width: height ratio of the specimens was strictly controlled to 3.1. Influence of Specimen Size on Deformation Behavior. 1 : 1 : 2. +e sizes of the samples were 25 × 25 × 50 mm (SS), Figure 4 shows the stress-strain curves of the BS granite 3 3 35 × 35 × 70 mm (SM), and 50 × 50 × 100 mm (SL), and (Figure 4(a)) and BHT basalt (Figure 4(b)) with different sample dimensional tolerance and perpendicularity tolerance were sizes under σ � 5 MPa and σ � 30 MPa. +e stress-strain 3 2 given as ±0.01 and 0.02 mm for each side, respectively. +e curves of the BS granite show the elastic-plastic-brittle defor- basic physical and mechanical parameters of these two types mation and failure process, while that of the BHT basalt shows of rocks are shown in Table 1. the elastic-brittle deformation and failure process. Meanwhile, Figure 1 shows the size and photos of rock selected in changing the size of rocks did not significantly affect the overall this study. Figure 1(a) is the grayish-green BHT basalt with deformation and failure processes (the stress-strain type) be- scattered white plagioclase on the surface, and Figure 1(b) is cause the microfractures were dominant in rocks before their the BS granite. X-ray diffraction (XRD) analysis showed that peak strength was reached. However, due to the heterogeneity the mineral composition of the BHT basalt was feldspar and the randomness of the location of macro cracks in the 41.96%, pyroxene 45.57%, clinochlore 6.25%, mica 4%, and sample, the postpeak stress-strain curve will show some dif- quartz 2.22%, while the mineral composition of the BS ferences, especially the BHT basalt with high brittleness [24], as granite was feldspar 51%, quartz 35%, biotite 8%, pyroxene shown in Figure 4(b). Advances in Civil Engineering 3 Table 1: Basic physical and mechanical parameters of the BHT basalt and BS granite samples. P-wave velocity Rock type Density (g/cm ) Young’s modulus (GPa) Poisson’s ratio (μ) Tensile strength (MPa) Grain size (μm) (m/s) BHT basalt 2.95 5650± 150 55∼60 0.22 18.4 50∼150 BS granite 2.69 5100± 120 50∼54 0.27 5.06 500∼1500 BHT Basalt BS Granite (a) (b) Figure 1: Photos of different sizes of two kinds of rocks. (a) BHT basalt and (b) BS granite. Figure 5 shows Young’s modulus under the influence of different sizes was significantly different. +e difference between rock size under TTC for the two types of rocks (the cal- the ε for the size of SL and SS was 0.206% and was significantly 3p culation method of Young’s modulus is based on [25]). higher than the changes in the peak strain under other stress When the rock size was constant, Young’s modulus in- states (σ � 5 MPa, 60 MPa, and 90 MPa), as shown in creased with increasing σ , but there is not a strict positive Figure 6(a), while the ε of the BHT basalt was hardly affected 2 3p correlation between Young’s modulus and rock size. Under by the sample size, and the changes in ε were always between 3p the same stress condition, when the sample size changed, the 0.03% and 0.07%. variation in Young’s modulus of BS granite was within Under the stress condition in this paper, the peak strain 5 GPa, while that of BHT basalt were basically within 3 GPa. ranges of the BS granite were −0.46< ε < 0.11 and 2p When the sample size changed, Young’s modulus always −0.72< ε < −0.42, and those of the BHT basalt were 3p changed small within the rock size range of this study, as −0.17< ε < 0.03 and −0.31< ε < −0.11. +e analysis 2p 3p shown in the light blue area in Figures 5(a) and 5(b), in- showed that the peak strain range of the BHT basalt was dicating that Young’s modulus of the two types of rocks was significantly smaller than that of the BS granite, which in- less affected by the sample size and the regularity was not dicates that the BS granite is prone to a large yield defor- obvious. mation under the same stress. To sum up, the stress-strain Figure 6 shows the influence of sample size on the peak curves, Young’s modulus, and peak strains for the BS granite strain (ε and ε ) in the direction of σ and σ under TTC (for and BHT basalt were related to the stress state and rock 3p 2p 3 2 example, peak strain ε refers to the strain when the stress in the properties, but these were not significantly affected by the 3p direction of σ reaches peak strength). For the BS granite and rock size. BHT basalt, when the rock size was constant, ε decreased with 3p increasing σ , which showed the rock was always under tensile deformation in the direction of σ during the loading process, 3.2. Influence of Rock Size on Characteristic Stress. Figure 7 shows the characteristics of damage stress (σ ) while ε increased under the same stress condition, which 2p cd showed the deformation in the direction of σ changed from under the influence of rock size. σ is the stress point cd corresponding to the turning point of the volume strain tensile to compression. Figure 6(a) shows the peak strain of the curve, which is the maximum point of the volume strain BS granite of different sizes in the direction of σ and σ under 3 2 TTC. It can be seen that, under the same stress condition, the curve before the peak and the calculation method refers to [25, 26]. Figure 7 shows that the σ of the two types of rocks peak strain ε in the direction of σ is very close and inde- cd 2p 2 pendent of the sample size. +e relationship between ε and showed an increasing with decreasing sample size under the 2p same stress level. However, for the BS granite, as shown in sample size of BHT basalt under TTC was the same as that of the BS granite, as shown in Figure 6(b). +erefore, the rock size Figure 7(a), when σ � 90 MPa, the σ of SM was slightly 2 cd lower than that σ � 60 MPa, which may be caused by two had no significant effect on ε within the scope of this study. 2p 2 However, when σ � 30 MPa, the ε of BS granite under reasons. On the one hand, when σ � 5 MPa, σ � 90 MPa 3 2 2 3p 100×50×50 mm 70×35×35 mm 50×25×25 mm 100×50×50 mm 70×35×35 mm 50×25×25 mm 4 Advances in Civil Engineering Clinochlore Plagioclase Plagioclase K-feldspar Mica Quartz Matrix : Diopside Plagioclase 500 µm 500 µm (a) (b) Figure 2: Microstructures under cross-polarized illumination of the two types of rocks. (a) BHT basalt and (b) BS granite. Table 2: Test scheme and results of the size effect for two rocks under TTC. Rock type Size (mm ) σ (MPa) σ (MPa) σ (MPa) σ (MPa) σ /σ ε (%) ε (%) θ ( ) A Failure mode 3 2 cd p cd p 3p 2p 5 144 202 0.71 −0.452 −0.452 74 201.82 Shear 30 186 268 0.69 −0.512 −0.080 79 270.33 Shear SL 5 60 193 295 0.65 −0.493 −0.006 80 294.79 Tension-shear 90 197 308 0.64 −0.577 0.061 80 305.97 Tension-shear 5 150 203 0.74 −0.438 −0.438 73 201.82 Shear 30 190 285 0.67 −0.602 −0.143 77 270.33 Shear BS granite SM 5 60 205 300 0.68 −0.501 −0.006 81 294.79 Tension-shear 90 204 313 0.65 −0.589 0.106 81 305.97 Tension-shear 5 151 205 0.74 −0.453 −0.453 72 201.82 Shear 30 200 292 0.69 −0.718 −0.163 78 270.33 Shear SS 5 60 212 311 0.68 −0.612 0.021 82 294.79 Tension-shear 90 211 325 0.65 −0.608 0.052 81 305.97 Tension-shear 5 250 250 1.00 −0.168 −0.168 74 247.94 Tension-shear 30 278 282 0.99 −0.185 −0.023 78 277.89 Tension-shear SL 5 60 324 324 1.00 −0.186 0.010 78 318.65 Tension-shear 90 341 350 0.97 −0.270 0.030 80 346.57 Tension-shear 5 261 261 1.00 −0.119 −0.119 75 265.25 Tension-shear 30 286 296 0.97 −0.202 −0.100 79 277.89 Tension-shear BHT basalt SM 5 60 339 339 1.00 −0.224 −0.019 79 318.65 Tension-shear 90 353 358 0.99 −0.230 0.028 80 346.57 Tension-shear 5 274 284 0.96 −0.165 −0.165 75 265.25 Tension-shear 30 329 334 0.99 −0.218 −0.073 78 277.89 Tension-shear SS 5 60 362 366 0.99 −0.233 −0.032 80 318.65 Tension-shear 90 367 386 0.95 −0.302 −0.029 88 346.57 Tension-shear was just near the turning point where σ first increased and preset σ did not reach the decreasing stage of σ in the two p 2 p then decreased [27]. On the other hand, reference [26] types of rocks under this condition, σ did not decrease). pointed out that the σ range of the BS granite is (0.64∼0.74) Moreover, the smaller the sample size was, the higher the σ cd p σ under TTC, which is within a reasonable range. In of the two types of rocks under the same stress, such as when comparison, the σ /σ of the BHT basalt under TTC was σ � 30 MPa. cd p 2 relatively large, approximately 0.95∼1.0 (Table 2), and the Figure 8(a) shows the σ of the BS granite with different turning point for it, where σ first increases and then de- sizes under TTC. When σ � σ � 5 MPa, as the sample size p 2 3 creases, is higher than BS granite. +erefore, this result is decreased from SL to SM and SS, the σ increment was very rarely found in BHT basalt: σ at σ � 90 MPa is slightly small, approximately 1 MPa or 2 MPa, which indicated that cd 2 lower than that of σ � 60 MPa (Figure 7(b)). the σ of the BS granite was almost unaffected by sample size. 2 P References [28–30] showed that when σ is constant, the In contrast, when the conventional triaxial stress condition σ increases first and then decreases with increasing σ (σ � σ ) changed to the true triaxial stress condition P 2 2 3 under TTC. Figure 8 shows the σ under the influence of (σ ≠ σ ), the size effect on the σ of the BS granite was P 2 3 P sample size for the BS granite and BHT basalt. As seen from significant. Figure 8, when σ � 5 MPa, the σ of the two types of rocks Compared with that of BHT basalt, changing the sample 3 P with different sizes increased with increasing σ (since the size (SL⟶ SM⟶ SS) of the BS granite will lead to the Biotite Advances in Civil Engineering 5 LVDTs Beam type strain gauge (b) (a) (c) Figure 3: Stress path and strain measurement of rock under TTC. (a) Stress path, (b) strain measurement method [17], and (c) measurement method of failure angle. Table 3: Controlling rates of each loading stage during the TTC test. Rock size (mm ) Loading rate of σ (MPa/s) Loading rate of σ (kN/s) Loading rate of σ (kN/s) Deformation rate (mm/min) 3 2 1 50 × 50 × 100 (SL) 0.5 2500 1250 0.008 35 × 35 × 70 (SM) 0.5 1225 612.5 0.0056 25 × 25 × 50 (SS) 0.5 625 312.5 0.004 unstable change of the peak strength increment the σ increment of the BS granite was approximately less ((σ − σ )/σ or (σ − σ )/σ ). For example, the than 10%, while that of the BHT basalt was approximately pSM pSL pSL pSS pSM pSM σ increment was 6.34% when the sample size of the BS 20%. For BHT basalt, the σ increment caused by the re- p p granite decreased from SL to SM under the stress condition duction of sample size from SM to SS was almost twice that that σ � 5 MPa and σ � 30 MPa, which was significantly caused by the reduction of sample size from SL to SM. 3 2 higher than the average 2.52% under other stress states. In Meanwhile, the σ increment of the BS granite also increased contrast, when the sample size of BHT basalt decreased from as the sample size changed, but the changes were very small. SL to SM and from SM to SS, the percentages of the σ +e smaller the sample size of the BHT basalt, the higher the increment were always maintained at approximately 4.1% sensitivity of the peak strength to the size effect, indicating and 9.3%, respectively, which was obtained by comparing that the sensitivity of the peak strength of BHT basalt to the the width of the blue or yellow areas enclosed by the changes size effect was higher than that of BS granite under the same in the peak strength caused by the sample size under each stress. stress in Figures 8(a) and 8(b). However, Figure 8(b) showed To clarify the relationship between the peak strength, the that the size effect on σ of BHT basalt was obvious in both sample size, and the stress state under TTC, the statistical the conventional triaxial and TTC. When the sample size analysis of the test results was carried out, as shown in decreased from SL to SM, change of the width of the blue Figure 9. +e volume of the SL sample was V, and the strip area was consistent with the change of the width of the volumes of the SM and SS samples were normalized ′ ′ yellow strip area when the sample size decreased from SM to according to V, such as V � 0.343 V and V � 0.125 V. In SM SS SS, and there were no abrupt changes in the σ increment Figure 9, the normalized results are plotted as the horizontal under a certain stress, which was different from the results axis, and the peak strength is the vertical axis. Figure 9(a) is for the BS granite. the results of BS granite, and Figure 9(b) shows those of BHT +e previously mentioned analysis showed that the basalt. sensitivity of σ for BS granite and BHT basalt to sample size As can be seen from Figure 9, the peak strengths of both was different. When the sample size decreased from SL to SS, types of rocks decreased with increasing sample size under Loading σ Loading σ Loading σ θ 6 Advances in Civil Engineering 350 400 ε ε ε 3 2 1 ε ε ε 3 2 v 280 320 210 240 140 160 70 80 0 0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 ε (%) ε (%) BS Granite BHT Basalt Size : SL σ = 5 MPa Size : SL σ = 5 MPa 3 3 Size : SM σ = 30 MPa Size : SM σ = 30 MPa 2 2 Size : SS Size : SS (a) (b) Figure 4: Full stress-strain curves of the BS granite and BHT basalt with different sample sizes under σ � 5 MPa and σ � 30 MPa. (a) BS 3 2 granite and (b) BHT basalt. aσ +bσ +c the same stress state. All test data were fitted by power (5) function, and the fitting variance of data was greater than σ � σ × 􏼠 􏼡 . p pV 0.9, indicating that the power function could well express the relationship between peak strength, sample size, and stress According to a, b, and c of the BS granite and BHT basalt state. Note that multiple curves were used to fit the ex- obtained in Figure 10, the binomial expression of B and σ perimental data under TTC because of the variable of σ , and can be expressed as follows: the general equation of the fitting curve was determined to be − 6 2 − 4 B � −2.778 × 10 σ + 7.167 × 10 σ − 0.06, Granite 2 (6) (1) R � 1, σ � A × 􏼠 􏼡 , − 6 2 − 3 B � −7.222 × 10 σ + 1.45 × 10 σ − 0.12, where B (B< 0) is a parameter related to the rock type and σ Basalt 2 2 2 (7) ′ ′ 2 and the units of V and V are mmQ. WhenV � V, R � 1. σ � A. (2) PV Equations (6) and (7) can be substituted into equation Substituting equation (2) into (1), we derive the (5) to obtain an expression relating σ , V, and σ for the BS p 2 following: granite and BHT basalt: −6 2 −4 ′ − 2.778×10 σ +7.167×10 σ − 0.06 V 2 (3) σ � σ × 􏼠 􏼡 . p pV σ � σ × 􏼠 􏼡 , V pGranite pV (8) −6 2 −3 − 7.222×10 σ +1.45×10 σ − 0.12 Because B is related to σ , the fitting relationship between 2 2 σ � σ × . them is obtained (Figure 10), which shows that the fitting 􏼠 􏼡 pBasalt pV result is well (R � 1). +us, the relationship between B and σ of the BS granite and BHT basalt can be expressed as +e relationship of the peak strength and damage follows: stress to the sample size of the BS granite and BHT basalt under TTC showed that the variation amplitude of the B � aσ + bσ + c, (4) 2 2 characteristic stress increment caused by the size effect in the fine-grained BHT basalt was obviously smaller than where a, b, and c are the fitting parameters related to li- thology, as shown in Figure 10. +e general expression of σ that of the medium- to coarse-grained BS granite, and the characteristic stress of the two types of rocks was was obtained by substituting equation (4) into (3): σ -σ (MPa) 1 3 σ -σ (MPa) 1 3 Advances in Civil Engineering 7 80 80 BS Granite σ = 5 MPa BHT Basalt σ = 5 MPa 3 3 70 70 60 60 50 50 40 40 30 30 0 15 30 45 60 75 90 0 15 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 5: Young’s modulus characteristics under the influence of rock sample size under TTC. (a) BS granite and (b) BHT basalt. 0.20 0.15 BS Granite BHT Basalt 0.00 0.00 -0.20 -0.15 -0.40 -0.30 -0.30 0.00 -0.45 -0.15 -0.60 -0.30 -0.75 -0.40 0 15 30 45 60 75 90 105 0 15 30 45 60 75 90 105 σ (MPa) σ (MPa) 2 2 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 6: Influence of sample size on the peak strain in the direction of σ and σ under TTC. (a) BS granite and (b) BHT basalt. 3 2 obviously affected by the sample size. +e relationships σ � 30 MPa and σ � 5 MPa, and Table 2 shows the failure 2 3 among the peak strength, rock sample size, and inter- modes and fracture angles of the rocks under TTC. It can be mediate principal stress could be represented by a power seen from Figure 11 that both BS granite and BHT basalt function. show macroshear failure under the same stress condition σ � 5 MPa and σ � 30 MPa, which indicates that the sample 3 2 size did not change the macroscopic failure mode for the two 3.3. Influence of Sample Size on Failure Characteristics. To types of rocks under the same stress. However, reducing the better compare the rock failure modes with different sizes, sample size may lead to secondary cracks near the main the failure pictures of the samples with different sizes were crack near the center of the sample, which are nearly parallel enlarged to the same size. Figure 11 shows the failure photos to the direction of σ , making the fracture surface more of BS granite and BHT basalt under different sizes at complex as shown in areas enclosed by red lines in ε (%) ε (%) 3p 2p E (GPa) ε (%) ε (%) E (GPa) 3p 2p 8 Advances in Civil Engineering 220 380 BS Granite BHT Basalt 140 240 0 15 30 45 60 75 90 0 15 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 σ = 5 MPa σ = 5 MPa 3 3 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 7: Damage stress characteristics under the influence of rock sample size. (a) BS granite and (b) BHT basalt. 330 400 BS Granite BHT Basalt 195 240 0 1530456075 90 0 1530456075 90 σ (MPa) σ (MPa) σ = 5 MPa σ = 5 MPa 3 3 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 8: Peak strengths of the two types of rocks with different sizes under TTC. (a) BS granite and (b) BHT basalt. Figures 11(c), 11(f ), and 12(a). Additionally, for the BHT clusters increased the complexity of the fracture surfaces of basalt with the size of SS, the cracks are easy to develop along the small-sized rock samples. the mineral cluster (feldspar phenocryst) during its prop- Figure 13 shows the failure angles of the two types of agation except for secondary cracks in the fracture surface, as rocks with different sizes under TTC, and the measurement shown in the areas enclosed by blue lines in Figures 11(e), of the failure angle refers to [17]. For the tortuous fracture 11(f ), and 12(b), which is possibly because the size of the surface, the near-linear measuring method was used, as mineral clusters was of the same order of magnitude as the shown in Figure 3(c). Figure 13 shows that when the sample length of the shortest edge of rock samples. +erefore, the size was constant and σ increased, the failure angle θ in- generation of secondary cracks and cracks along the mineral creased. However, changing the sample size did not σ (MPa) σ (MPa) cd σ (MPa) cd σ (MPa) p Advances in Civil Engineering 9 360 450 -0.018 2 σ =305.97(V′/V) , R =0.9170 -0.048 2 σ =346.57(V′/V) , R =0.9052 -0.027 2 350 σ =294.79(V′/V) , R =425 -0.041 2 σ =270.33(V′/V) , R =0.9009 -0.059 2 300 σ =321.94(V′/V) , R =0.9738 -0.083 2 σ =277.89(V′/V) , R =0.9376 -0.062 2 -0.007 2 σ =247.94(V′/V) , R =0.9608 180 σ =201.82(V′/V) , R =0.9586 BS Granite BHT Basalt 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.3 0.6 0.9 1.2 1.5 V′/V V′/V σ = 5 MPa σ = 60 MPa σ = 5 MPa σ = 60 MPa 2 2 2 2 σ = 30 MPa σ = 90 MPa σ = 30 MPa σ = 90 MPa 2 2 2 2 (a) (b) Figure 9: Relationship between the peak strength, sample size, and stress state of the two rocks under TTC (symbols of the same color represent the σ of different sizes under the same stress level). (a) BS granite and (b) BHT basalt. -0.01 -0.02 BS Granite BHT Basalt -0.02 -0.04 B -0.03 B -0.06 -0.04 -0.08 –6 2 –4 2 –6 2 –3 2 B = –2.778×10 σ + 7.167×10 σ –0.06, R = 1 B = –7.222×10 σ + 1.45×10 σ –0.12, R = 1 2 2 2 2 -0.05 -0.10 30 45 60 75 90 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 B B Fitting cure Fitting cure (a) (b) Figure 10: Fitting results between σ and B. (a) BS granite and (b) BHT basalt. significantly impact the angle of the fracture surface under 4. Discussion the same stress. For example, in Figure 13(a), when the 4.1. Microscopic Interpretation of Complex Fracture Surfaces sample size of the BS granite decreased from SL to SS under the same stress, the fracture angle only varied by approxi- Caused by the Size Effect. Section 3.3 showed that changing the sample size does not significantly affect the macroscopic mately 1∼2 ; in Figure 13(b), the variation in the fracture failure mode under the same stress. However, the crack angle of the BHT basalt was basically the same as that of the propagation in the fracture surface became more complex BS granite under the same condition, approximately 0∼3 . when the sample size decreased to SM or SS, as shown in +us, the small variation of the fracture angle in this paper Figures 11(c), 11(e), 11(f ), and 12. Under TTC, a macro- further shows that the sample size do not significantly scopic shear fracture plane with a “V” shape was easily change the macroscopic failure mode of these two types of rocks, while the reduction of sample size will lead to more generated [29, 31, 32]. On this type of fracture surface, especially near the center of the rock sample, almost no complex crack propagation on the fracture surface. σ (MPa) σ (MPa) P 10 Advances in Civil Engineering Size : SL Size : SM Size : SS (a) (b) (c) Size : SL Size : SM Size : SS (d) (e) (f ) Figure 11: Failure modes of rock samples with different sizes under the same stress state of σ2 � 30 MPa and σ3 � 5 MPa. (a–c) BS granite; (d–f ) BHT basalt. obvious secondary cracks nearly parallel to the direction of +e complex crack propagation on the failure surface σ were generated during the propagation of the main of the small-sized samples may be related to the mineral cracks, as shown in Figures 11(a), 11(b), and 11(d). How- grain size or the mineral grain aggregate size of the rocks. ever, this situation is likely to occur when the sample size It is well known that grain size is one of the most im- decreased from SL to SS under the same stress, as shown in portant microstructure parameters of rock mechanical Figures 11(c), 11(f ), and 12(a). properties. Taking the BHT basalt as an example, Advances in Civil Engineering 11 Size : SS Size : SS 5 mm σ = 60 MPa σ = 60 MPa 2 2 σ = 5 MPa σ = 5 MPa 3 3 (a) (b) Figure 12: Failure mode of two rock samples of SS size at the same true triaxial stress state of σ � 60 MPa and σ � 5 MPa. (a) BS granite and 2 3 (b) BHT basalt. Figure 2(a) shows that pyroxene, feldspar, and other of feldspar. +erefore, feldspar and other weaker minerals minerals were uniformly arranged in the matrix of the were more prone to brittle failure during the process of stress cracking, as demonstrated by the closed fractures BHT basalt and the size of feldspar grain was 50∼150 μm. During the diagenetic process, a large number of feldspar on the surface of feldspar phenocrysts (Figures 2(a) and 14(c)), which may explain why the crack propagation on grains aggregated to form lath-shaped white feldspar phenocrysts, with a length of 5 mm or larger, as shown in the fracture surface is more complex for small samples Figures 12(b), 14(a), and 14(b). Pyroxene is a silicate than for large samples (the complexity of the crack rock-forming mineral with a shear modulus of 64.9 GPa, propagation of large rock samples was much lower). and feldspar is a brittle rock-forming mineral with a shear In the samples with size SS in this study, the ratio of the modulus of 28.6 GPa. Section 2.1 showed that the total size of the large feldspar phenocrysts to the length or width composition of pyroxene and feldspar minerals in the of the sample reached 1/5 (Figure 12(b)), and the ratio was BHT basalt accounted for more than 87%, and the two even larger when multiple phenocrysts were aggregated. constituted the basic framework. Pyroxene is a mineral Reference [34] pointed out that grain size plays an important role in crack propagation and used numerical modeling to with an allotriomorphic structure, while feldspar and others are minerals with idiomorphic structures, and the show that the interactions of adjacent cracks can be used to structural relationship between them is similar to the inhibit crack propagation. Additionally, they also pointed relationship between water and stone in a river. Basalt is out that this inhibitory effect can gradually disappear with igneous rock and pyroxene (like water) can fill the holes the increase of grain size. For the BHT basalt, the size of the and gaps between minerals with an allotriomorphic feldspar phenocrysts remained unchanged, but the de- structure (like stone) in the process of diagenesis with no creased sample size was equivalent to indirectly increasing clear boundaries between the two. the size of the relatively weak feldspar phenocrysts Reference [33] showed that, in the 6 × 6 stiffness (Figures 14(a) and 14(b)), and the role of feldspar pheno- matrix represented by the Voigt notation, the stiffness of crysts in the structure could not be ignored. +erefore, the single-crystal pyroxene in all directions is larger than that inhibition effect of the surrounding cracks could be 12 Advances in Civil Engineering 84 84 BS Granite BHT Basalt 70 72 0 15 30 45 60 75 90 0 15 30 45 60 75 90 σ (MPa) σ (MPa) 2 2 σ = 5 MPa σ = 5 MPa 3 3 Size : SL Size : SL Size : SM Size : SM Size : SS Size : SS (a) (b) Figure 13: Failure angles of two types of rocks with different sizes under TTC. (a) BS granite and (b) BHT basalt. Closed cracks 100 µm 50 mm 25 mm (c) (a) (b) Figure 14: Pictures of BHT basalt and its microstructures under cross-polarized illumination. (a) BHT basalt specimen of SL size; (b) local magnification of the specimen in (a); (c) microstructures of (b) under cross-polarized illumination part. weakened and the crack density could increase when the to the direction of σ . In the left main fracture plane cracks propagated to the vicinity of feldspar phenocrysts (Figure 12(b)), two groups of feldspar phenocrysts in the two with a relatively large size. References [35, 36] showed that, areas enclosed by blue dashed lines led to the propagation for fine-grained materials, an increase in crack density can direction of some cracks (all in the same direction), resulting be equivalent to an increase in the spatial heterogeneity of in poor symmetry of the left and right fracture planes and a the local stress field. +erefore, when cracks occur, cracks are “Y” shaped fracture plane. +e cracks that grew along the more likely to propagate along the weak feldspar pheno- feldspar phenocrysts were also observed in the area enclosed crysts, which results in a complex crack morphology on the by red dashed lines in the right main fracture surface. When fracture surface. For example, in the area enclosed by the the direction of the feldspar phenocrysts was close to the blue dashed line in Figure 11(e), it is obvious that a crack growth direction of cracks in the fracture surface, the cracks developed along the axis of feldspar phenocrysts. In the were more likely to grow along the feldspar phenocrysts. For small area enclosed by the blue dashed line in Figure 11(f ), the BS granite, the distribution of constituent minerals was the main crack passed through the axis of the feldspar relatively uniform, and the grain size reached 500∼1500 μm phenocrysts and produced secondary cracks nearly parallel or even larger. When the sample size was reduced to SS, the 7 mm High : 100 mm θ (°) High : 50 mm θ (°) Advances in Civil Engineering 13 0.57 0.62 BS Granite BHT Basalt 0.60 0.54 σ = 5 MPa σ = 5 MPa 3 3 0.58 0.51 0.56 0.48 0.54 0.45 0.52 0.42 eb eb 2 0.50 ε = 1.0035 + 0.7628ε ,R = 0.90 P O eb eb 2 ε = 1.5391 + 0.7423ε ,R = 0.85 P O 0.39 0.48 0.40 0.44 0.48 0.52 0.56 0.60 0.44 0.48 0.52 0.56 0.60 0.64 eb eb Observed ε (%) Observed ε (%) (a) (b) Figure 15: Prediction ability of equation (11) for total elastic strain in the direction of σ for two types of samples. (a) BS granite and (b) BHT basalt. fracture surface became complex, as shown in Figures 11(c) aσ +bσ +c σ × V /V − σ pV 2 eb eb and 12(a). +is may be one of the reasons why the Inter- (11) ε � ε + . f σ 􏼁 national Society for Rock Mechanics (ISRM) recommends 2 that the minimum side length of the sample should be more As shown in Figure 15, the linear regression coefficients than 10 times the maximum grain size of minerals. Based on of the total elastic strain of the BS granite and BHT basalt the previously mentioned analysis, it is better to choose a with sizes of SM and SS predicted by formula (11) in the large rock sample when studying crack propagation on the 2 2 direction of σ were R � 0.90 and R � 0.85, respectively, fracture surface under TTC. +e minimum side length of the indicating that the prediction ability of formula (11) was sample should be at least 10 times larger than the maximum reasonable. grain size of the mineral (ISRM) and the maximum grain size of the mineral aggregates with an idiomorphic structure (such as feldspar phenocrysts) to avoid a complex fracture 5. Conclusion surface. In this study, BS granite and BHT basalt with the same length : width : height ratio and different sizes were used to 4.2. Relationship between Elastic Strain and Sample Size. study the size effect under TTC conditions. +e following conclusions are drawn: +e test results in Section 3.2 showed that there was little relationship between Young’s modulus and sample size. +e (1) Regarding the deformation and failure characteris- maximum Young’s modulus of the same samples with tics within the range of rock size for this study, different sizes under the same stress state was fitted with σ , including the stress-strain curve, Young’s modulus, and the relationship between them was obtained as follows: peak strain in the directions of σ and σ , fracture 3 2 angle, and macrofailure mode, there was almost no E � dσ + e, (9) obvious size effect. However, the characteristics of deformation for the two types of rocks were related where d and e are related to rock type. to the rock properties and external stress conditions. According to the calculation method of elastic strain (2) +e peak strength and damage stress of the BS under TTC proposed in [24], granite and BHT basalt were significantly affected by the sample size and σ under TTC. As the sample size σ − σ p 2 eb eb (10) ε � ε + , decreased, the σ and σ increased. For these two p cd types of rocks, there was a power function rela- eb tionship among the peak strength, sample size, and where ε is the total elastic strain in the direction of σ and eb σ under TTC. Under the same conditions, the ε is the elastic strain in the biaxial loading stage in the sensitivity of the peak strength of the fine-grained direction of σ under TTC, that is, the elastic strain in the BHT basalt to the sample size was higher than that of process of Section 2.2 stress path b. the medium- to coarse-grained BS granite. Taking equation (5) into (10), the following can be obtained for calculating the total elastic strain in the di- (3) +e complex crack propagation on the fracture rection of σ related to the sample size under TTC: surface of smaller rock samples was due to indirectly eb Predicted ε (%) eb Predicted ε (%) 14 Advances in Civil Engineering [8] P. A. Cundall, M. E. Pierce, and D. M. Ivars, “Quantifying the increasing the mineral grain size or mineral cluster size effect of rock mass strength,” in Proceedings of the size in the rocks. Moreover, this study suggests that Southern Hemisphere International Rock Mechanics the minimum side length of rock samples should be Symposium, Perth, Australia, January 2008. at least 10 times the maximum size of the mineral [9] D. D. Hunt, “+e influence of confining pressure on size clusters when studying crack propagation on a effect,” Master of Sciences in Civil Engineering, Massachusetts fracture surface. Institute of Technology, Cambridge, MA, USA, 1973. (4) +e estimation method of elastic strain in a certain [10] K. Mogi, “Effect of the triaxial stress system on rock failure,” range of sample sizes was established by analyzing Rock Mechanics Japan, vol. 1, pp. 53–55, 1970. the relationship among sample size, peak strength, [11] K. Mogi, “Fracture and flow of rocks under high triaxial intermediate principal stress, and elastic strain in the compression,” Journal of Geophysical Research, vol. 76, no. 5, direction of σ , and the prediction result was well. pp. 1255–1269, 1971. [12] B. Haimson and C. Chang, “A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock Data Availability strength and deformability of Westerly granite,” International Journal of Rock Mechanics and Mining Sciences, vol. 37, no. 1- +e data used to support the findings of this study are 2, pp. 285–296, 2000. available from the corresponding author upon request. [13] W. R. Wawersik, L. W. Carlson, D. J. Holcomb, and R. J. Williams, “New method for true-triaxial rock testing,” Conflicts of Interest International Journal of Rock Mechanics and Mining Sciences, vol. 34, pp. 3-4, 1997. +e authors declare that they have no conflicts of interest. [14] P. Michelis, “A true triaxial cell for low and high pressure experiments,” International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, vol. 22, no. 3, Acknowledgments pp. 183–188, 1985. +e authors sincerely acknowledge the financial support [15] M. Takahashi and H. Koide, “Effect of the intermediate from the Liao Ning Revitalization Talents Program under principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than 2000 m,” Grant no. XLYC1801002, the National Natural Science International Journal of Rock Mechanics and Mining Science & Foundation of China under Grant no. 51839003, and the 111 Geomechanics Abstracts, vol. 27, pp. 19–26, 1989. Project under Grant no. B17009. +e authors are grateful to [16] J. T. Chen and X. T. Feng, “True triaxial experimental study on Prof. Xiating Feng, Prof. Xiwei Zhang, Prof. Chengxiang rock with high geostress,” Chinese Journal of Rock Mechanics Yang, Mr. Zhaofeng Wang, and Ms. Xinyue Wang for their and Engineering, vol. 25, pp. 1537–1543, 2006, (in Chinese). great assistance. [17] X.-T. Feng, X. Zhang, R. Kong, and G. Wang, “A novel mogi type true triaxial testing apparatus and its use to obtain References complete stress-strain curves of hard rocks,” Rock Mechanics and Rock Engineering, vol. 49, no. 5, pp. 1649–1662, 2016. [1] W. Weibull, “A statistical distribution function of wide ap- [18] L. Shi, X. B. Li, B. Bing, A. Wang, Z. Zeng, and H. He, “A plicability,” Journal of Applied Mechanics, vol. 18, no. 3, mogi-type true triaxial testing apparatus for rocks with two pp. 293–297, 1951. moveable frames in horizontal layout for providing orthog- [2] Z. P. Bazant and J. Planas, Fracture and Size Effect in concrete onal loads,” Geotechnical Testing Journal, vol. 40, pp. 542–558, and Other Quasibrittle Materials, CRC Press, Boca Raton, FL, USA, 1998. [19] M. C. He, J. L. Miao, and J. L. Feng, “Rock burst process of [3] A. Carpinteri, “Scaling laws and renormalization groups for limestone and its acoustic emission characteristics under true- strength and toughness of disordered materials,” Interna- triaxial unloading conditions,” International Journal of Rock tional Journal of Solids and Structures, vol. 31, no. 3, Mechanics and Mining Sciences, vol. 47, no. 2, pp. 286–298, pp. 291–302, 1994. [4] H. R. Pratt, A. D. Black, W. S. Brown, and W. F. Brace, “+e [20] X. Li, K. Du, and D. Li, “True triaxial strength and failure effect of speciment size on the mechanical properties of modes of cubic rock specimens with unloading the minor unjointed diorite,” International Journal of Rock Mechanics principal stress,” Rock Mechanics and Rock Engineering, and Mining Science & Geomechanics Abstracts, vol. 9, no. 4, vol. 48, no. 6, pp. 2185–2196, 2015. pp. 513–516, 1972. [21] X. G. Zhao and M. Cai, “Influence of specimen height-to- [5] B. C. Liu, J. S. Zhang, Q. Z. Du, and J. F. Tu, “Size effect of width ratio on the strainburst characteristics of Tianhu granite compressive strength of rock,” Chinese Journal of Rock Me- under true-triaxial unloading conditions,” Canadian Geo- chanics and Engineering, vol. 17, pp. 611–614, 1998, (in technical Journal, vol. 52, pp. 890–902, 2014. Chinese). ¨ [22] F. Zhao and M. C. He, “Size effects on granite behavior under [6] E. Tuncay, N. T. Ozcan, and A. Kalender, “An approach to unloading rockburst test,” Bulletin of Engineering Geology and predict the length-to-diameter ratio of a rock core specimen the Environment, vol. 76, no. 3, pp. 1183–1197, 2016. for uniaxial compression tests,” Bulletin of Engineering Ge- [23] X. B. Li, F. Feng, D. Y. Li, K. Du, P. G. Ranjith, and J. Rostami, ology and the Environment, vol. 78, no. 7, pp. 5467–5482, 2019. “Failure characteristics of granite influenced by sample [7] J. Fladr ´ and P. B´ ıly, ´ “Specimen size effect on compressive and flexural strength of high-strength fibre-reinforced concrete height-to-width ratios and intermediate principal stress under true-triaxial unloading conditions,” Rock Mechanics and Rock containing coarse aggregate,” Composites Part B: Engineering, vol. 138, pp. 77–86, 2018. Engineering, vol. 51, pp. 1–25, 2018. Advances in Civil Engineering 15 [24] X. T. Feng, J. Zhao, Z. F. Wang, C. X. Yang, Q. Han, and Z. Zheng, “Effect of high differential stress and mineral properties on deformation and failure mechanism of hard rocks,” Canadian Geotechnical Journal, vol. 58, 2020. [25] C. D. Martin, Ae strength of massive Lac du Bonnet granite around underground openings, Ph.d. thesis, Department of Civil Engineering, University of Manitoba, Winnipeg, Can- ada, 1993. [26] Y.-H. Gao, X.-T. Feng, X.-W. Zhang, G.-L. Feng, Q. Jiang, and S.-L. Qiu, “Characteristic stress levels and brittle fracturing of hard rocks subjected to true triaxial compression with low minimum principal stress,” Rock Mechanics and Rock Engi- neering, vol. 51, no. 12, pp. 3681–3697, 2018. [27] Y. Zhang, Energy evolution mechanism of failure process of hard rock in deep tunnel and discrimination of typical hazard types, Ph.d thesis, Northeastern University, Shenyang, China, [28] B. Haimson and J. W. Rudnicki, “+e effect of the interme- diate principal stress on fault formation and fault angle in siltstone,” Journal of Structural Geology, vol. 32, no. 11, pp. 1701–1711, 2010. [29] X. Ma, J. W. Rudnicki, and B. C. Haimson, “Failure char- acteristics of two porous sandstones subjected to true triaxial stresses: a,” Journal of Geophysical Research: Solid Earth, vol. 122, no. 4, pp. 2525–2540, 2017. [30] R. Kong, X.-T. Feng, X. Zhang, and C. Yang, “Study on crack initiation and damage stress in sandstone under true triaxial compression,” International Journal of Rock Mechanics and Mining Sciences, vol. 106, pp. 117–123, 2018. [31] J. Zhao, X.-T. Feng, X.-W. Zhang, Y. Zhang, Y.-Y. Zhou, and C.-X. Yang, “Brittle-ductile transition and failure mechanism of Jinping marble under true triaxial compression,” Engi- neering Geology, vol. 232, pp. 160–170, 2018. [32] Y. Zhang, X. T. Feng, X. W. Zhang, Z. F. Wang, M. Sharifzadeh, and C. X. Yang, “A novel application of strain energy for fracturing process analysis of hard rock under true triaxial compression,” Rock Mechanics and Rock Engineering, vol. 52, pp. 1–16, 2019. [33] J. D. Bass, “Elasticity of minerals, glasses, and melts,” American Geophysical Union, vol. 2, pp. 45–63, 1995. [34] E. Eberhardt, D. Stead, B. Stimpson, and E. Z. Lajtai, “+e effect of neighbouring cracks on elliptical crack initiation and propagation in uniaxial and triaxial stress fields,” Engineering Fracture Mechanics, vol. 59, no. 2, pp. 103–115, 1998. [35] J. T. Fredrich, B. Evans, and T.-F. Wong, “Effect of grain size on brittle and semibrittle strength: implications for micro- mechanical modelling of failure in compression,” Journal of Geophysical Research, vol. 95, no. B7, pp. 10907–10920, 1990. [36] J. Peng, L. N. Y. Wong, and C. I. Teh, “Influence of grain size heterogeneity on strength and microcracking behavior of crystalline rocks,” Journal of Geophysical Research: Solid Earth, vol. 122, no. 2, pp. 1054–1073, 2017.

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