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Rock brittleness evaluation method based on the complete stress-strain curve

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Rock brittleness evaluation method based on the complete stress-strain curve
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    C.Y. Liu et alii, Frattura ed Integrità Strutturale, 49 (2019) 557-567; DOI: 10.3221/IGF-ESIS.49.52    557 Rock brittleness evaluation method based on the complete stress-strain curve ChenYang Liu, Yong Wang, XiaoPei Zhang, LiZhi Du University of Jilin, Changchun 130012, China wyzrp613@jlu.edu.cn  A  BSTRACT .  Brittleness plays an important role in the brittle failure process of rocks, and is also one of the important mechanical properties of rocks and a key indicator in rock engineering such as hydraulic fracturing, tunnelling machine borehole drilling and rockburst prediction. Therefore, aiming at the applicability of the brittleness index, this paper summarizes and analyzes the existing brittleness indices based on different experimental methods. Through analysis, it is found that many of the existing methods have their limitations. On the other hand, the brittleness evaluation method based on the stress s-strain curve makes it easier to obtain key parameters and quantify them.  Therefore, this paper also adopts this practically widely used method. It proposes a brittleness index based on the post-peak stress drop rate of the rock stress-strain curve and the difficulty of brittle failure, verifies by the traditional triaxial surrounding rock pressure test the accuracy and superiority of B L  and further explores the differences between the brittleness indices B 8 , B 11 , B 12  and B L . Finally, the brittleness index B L  and B 13  are further contrasted by the existing experimental data. K  EYWORDS .  Rock mechanics; Failure; Brittleness; Complete stress-strain curve. Citation:  Liu, C.Y., Wang, Y., Zhang, X.P., Du, L.Z., Rock brittleness evaluation method based on the complete stress-strain curve, Frattura ed Integrità Strutturale, 49 (2019) 557-567. Received: 07.11.2018  Accepted: 19.02.2019 Published:  01.04.2019 Copyright:  © 2019 This is an open access article under the terms of the CC-BY 4.0,  which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal author and source are credited. I NTRODUCTION   rittleness is a fundamental parameter of rock mechanics and plays an important role in rock failure engineering. For example, the brittleness of rocks is an important indicator to evaluate the risks of rockburst. For underground engineering, rockburst is the most important issue [1]; in tunnel engineering, the rock brittleness determines the excavation efficiency of the TBM shield tunnelling machine; it also determines the efficiency of shale gas and oil production and has a great impact on the hydraulic fracturing of horizontal wells [2-5]. In order to reduce or even prevent the adverse B   C.Y. Liu et alii, Frattura ed Integrità Strutturale, 49 (2019) 557-567; DOI: 10.3221/IGF-ESIS.49.52 558 effects of disasters on underground engineering and improve mining efficiency, it is extremely important for scientific research to figure out how to accurately evaluate the brittleness of rocks.    =      ,     =        +   ,  3  =     2 ,    = √ 3      and    are the UCS and the tensile strength, respectively.      =  ′  " ,  ′  =     /      ,  "  =  + + , CS =        /  /        ε  , ε  , ε   are the peak strain, the peak strain maximum and the minimum  value of sample rock specimen, respectively.  ,  ,   are the standardized coefficients.    and     are the peak strength and the residual strength, respectively.     and    are the peak strain and the residual strain, respectively.    6  =  / ,  7  =/   M and E are the post-peak modulus and the pro-peak elastic modulus, respectively.    =      /  ,  9  =      /  ,    =   /       and    are the peak strength and the residual strength, respectively.     and    are the peak strain and the residual strain, respectively,    is the reversible strain of the stress-strain curve.    =  ′   " ,  ′  =     /  ,  "  =lg|  |/10      and    are the peak strength and the residual strength, respectively.    is the post-peak stress drop rate.    =  ′  + " ,  ′  =     /      ,  "  =           /           and    are the peak strength and the residual strength, respectively,    and    are the peak strain and the residual strain, respectively.  3  =  3′  + 3′′ ,  3′  =      i     ⁄  i   ,  3′′  =  (    r )             and    are the peak strength and the residual strength, respectively.     and    are the peak strain and the residual strain, respectively,  i  and  i  are the crack initiation stress and the crack initiation strain    =  ,    =45° +/2      is the inner friction angle determined from Mohr’s envelope.    6  = 0.5   +0.5  ,    =  1/81×100 ,    = 0.41/0.40.15  E and   are the post-peak modulus and the elastic modulus, respectively.    7  =      /   where H   μ   is the micro-indentation hardness, H  m   is the macro-indentation hardness, and c   is the constant      =   /    where H  a   is hardness and K  c   is fracture toughness.    9  =   /    where H  a  and K  c  are same as those in B 18 ,  E   is Young’s modulus    9  =   /   +   +   ), W  qtz  , W  carb   and W  clay   are the content of quartz, clay and carbonate minerals, respectively.    Table 1: Formula meaning and explanation.    C.Y. Liu et alii, Frattura ed Integrità Strutturale, 49 (2019) 557-567; DOI: 10.3221/IGF-ESIS.49.52    559 However, there is currently no standard or widely accepted concept for the evaluation of the rock brittleness index. Researchers hold different views for different research purposes. Morley [6] and Hetényi [7] argue that rock brittleness is characterized by low elongation or low strain value due to the lack of ductility or compressibility. Ramasy [8] defines brittleness as the loss of cohesion in a rock as it deforms within the elastic range. Similarly, Obert and Duaval [9] consider brittleness to be a phenomenon in which a material (such as cast iron or rock) breaks or only slightly exceeds the yield stress.  Tarasov and Potvin [10] proposed that brittleness is the ability to self-maintain the macroscopic damage in the post-peak area under the compressive load due to the accumulation of elastic energy. With the development of rock mechanics, a lot of research has been done on the evaluation of rock brittleness, and the commonly used evaluation methods are shown in  Tab. 1. Brittleness evaluation method based on the rock stress-strain curve In Tab. 1, the brittleness indices B 1 - B 4  are proposed based on the relationship between UCS (uniaxial compressive strength) and uniaxial tensile strength. According to the experimental results, Kahraman [11] believed that the penetration rate of the rotary drill bit has a strong exponential relationship with the brittleness indices  B 1  and B 2 . Altindag [12] proposed the brittleness indices B 3 - B 4  based on the tensile-compressive strength curve to quantitatively evaluate the brittleness of rocks.  These two indices are often used to predict the drillability of rocks. However, the brittleness indices B 1 - B 4 based on UCS and uniaxial tensile strength is not applicable to the analysis of rock brittleness under complex stress conditions. According to the post-peak stress drop rate and peak strength, Li Qinghui et al [13] proposed the brittleness index B 5 , which is based on the three standard coefficients for the post-peak stress drop, but only applies to a certain type of rocks, and what is more, a lot of experiments need to be done to obtain an accurate value. At the same time, Tarasov and Potvin [14] proposed the brittleness indices B 6 - B 7  based on the post-peak secant modulus and the pre-peak elastic modulus. Xia Yingjie believed that these two brittleness indices cannot effectively distinguish the brittle characteristics of different stress-strain curves [15]. R. Altindag [16] proposed the brittleness indices B 7  and B 8  based on the peak stress-strain and residual stress-strain relations. Hucka and Das [17] proposed the brittleness indices B 10  based on the ratio of recoverable strain to peak strain before the peak; however, these methods consider only a few mechanical parameters and cannot fully reflect the strain-strain process of the entire rock. Therefore, these methods require further improvement. Meng et al. [18] proposed the brittleness index B 11  based on the relative magnitude and absolute rate of the post-peak stress drop, and further verified the accuracy of the index by doing comparison test on different types of rocks under different surrounding rock pressures, but this index does not represent the pre-peak mechanical characteristics. Xia Yingjie et al. proposed the brittleness index B 12  based on the post-peak stress drop rate and the ratio of the elastic energy released by the instability failure to the total energy stored before the peak. Chen Guoqing et al. [19] proposed the rock brittleness index B 13  based on the post-peak stress drop rate and the stress growth rate between the pre-peak initiation stress and the peak stress. The method uses the stress growth rate between the initiation stress and the peak stress to characterize the pre-peak brittle state. At present, there are two effective ways to determine the initiation stress, one being acoustic emission and the other based on the strain inflection point of the crack volume [20]. Acoustic emission is often affected by noise, making it difficult to determine the moment of crack initiation. The second method, which determines the initiation stress through the strain inflection point of the crack volume, often depends on the mineral composition and particle size, so it is also difficult to determine an accurate value. In order to further verify the accuracy of the method proposed in this paper, the following section will make comparisons with the experimental data by Chen Guoqing et al. Brittleness evaluation method based on internal friction angle Hucka and Das [21] proposed evaluating the brittleness indices B 14  and B 15  of rocks considering the internal friction angle.  At the same time, Tarasov and Potvin [22] found through experiments that the brittleness index B 13  is positively correlated  with the rock fracture angle. However, the brittleness indices B 14  and B 15  only apply to the same type of rocks; and it is also difficult to obtain an accurate rock fracture angle. Brittleness evaluation method based on elastic modulus and Poisson's ratio Rockman et al. [23] proposed the brittleness index B 16  based on shale reservoir; however, the brittleness index B 16  only takes into account the elastic modulus and Poisson’s ratio, but ignores many important mechanical parameters. In order to obtain accurate parameters, mechanical experiments still need to be conducted on a lot of rock samples. All these factors limit the development of the brittleness index B 16 .   C.Y. Liu et alii, Frattura ed Integrità Strutturale, 49 (2019) 557-567; DOI: 10.3221/IGF-ESIS.49.52 560 Brittleness evaluation method based on hardness Hucka and Das proposed a rock brittleness evaluation method based on the difference between rock micro-hardness and macro-hardness. Lawn and Marshall [24] established the brittleness index B 18  for the ceramic engineering field. Quinn and Quninn [25] described an index B 19  to measure the rock brittleness based on the ratio of the deformation energy per unit  volume to the fracture surface energy per unit area. This brittleness evaluation index considers too few factors and is only applicable to the ceramic field, so its accuracy and applicability should be further considered in applications. R  OCK BRITTLENESS INDEX BASED ON STRESS - STRAIN DROP AND PEAK STRAIN   n the brittleness evaluation in hydraulic fracturing and rockburst prediction, the existing brittleness evaluation methods consider only a few mechanical parameters. What is more, many of them are only applicable to uniaxial load conditions, and not suitable for high surrounding rock pressure in deep tunnel construction. The stress-strain curve, on the other hand, reflects the whole process of the rock from deformation failure to the ultimate loss of bearing capacity under external load, and is applicable to the state analysis of rock failure under surrounding rock pressure. Based on the stress-strain curve of the rock failure, quantitative brittleness parameters can be obtained. Therefore, the post-peak stress-strain shape obtained in the laboratory is the main method for researchers to qualitatively understand the rock brittleness. Based on the above, this paper proposes a brittleness evaluation method based on post-peak stress drop rate and pre-peak brittle failure. Figure 1: Simplified stress-strain curve   In Fig. 1, the polyline OABC is a simplified Class I stress-strain curve. Point A (    , ε   ) corresponds to the peak point, and    and ε   are the peak intensity and the peak strain, respectively; point B (    , ε   ) corresponds to the residual point, and    and ε   are the residual stress and the residual strain, respectively. It is obvious in Fig. 1 that the polyline OABC is divided by point A (    , ε   ) and point B (    , ε   ), so that the corresponding mechanical parameters can be quantitatively obtained.  The drawbacks of the brittleness indices B 10 - B 11  are already discussed above. Based on these two methods, this paper proposes a new method  L , which considers both the stress drop B 8  and the strain drop B 9 . At the same time, the faster the stress drop rate, the higher the brittleness, so the difference between the peak stress and the residual stress is proportional to the brittleness index and the difference between the peak strain and the residual strain is inversely proportional to the brittleness index. In addition, in order to emphasize the final increase of the post-peak strain, the residual strain is used to replace the peak strain in the denominator of the brittleness index B 9 . First, the post-peak brittleness index  L  is:  L  =   (1) I    C.Y. Liu et alii, Frattura ed Integrità Strutturale, 49 (2019) 557-567; DOI: 10.3221/IGF-ESIS.49.52    561  where, ε r  and ε   are the residual strain and the peak strain, respectively; σ   and σ r   are the peak stress and the residual stress, respectively. Peak strain can reflect the difficulty of brittle failure [23]. In addition, according to previous studies, as the peak strain increases, the rock tends to transition from being brittle to being ductile, i.e., the brittleness index will become lower. Considering this factor, it is proposed that the pre-peak brittleness index B L2  should be the reciprocal of the peak strain. Second, the index of the difficulty of pre-peak brittle failure is as follows:  L  =     (2) Based on this, the new rock brittleness index B L considers both the post-peak stress drop rate and the difficulty of pre-peak brittle failure. Finally, the brittleness index B L  is expressed as follows:  L  =  L  L  (3) In summary, the new rock brittleness evaluation method B L  considers the post-peak stress drop rate and also introduces the residual strain to emphasize the final increase of post-peak strain. Moreover, it incorporates the difficulty of brittle failure to characterize the pre-peak brittle characteristics. C OMPARISON AND VERIFICATION OF BRITTLENESS INDICES   onsidering the great impact of surrounding rock pressure on rock brittleness in underground engineering, this section will explore the variations of rock brittleness under different surrounding rock pressure conditions. In order to verify the accuracy of the brittleness index B L , Tab. 2 determines the relevant mechanical parameters and calculates the brittleness index B L  according to the stress-strain curve in Fig. 4. At the same time, for comparison with other brittleness indices, the brittleness indices B 6 - B 12  and B 14 are selected from Tab. 1 (as B 13  needs initiation stress and initiation strain, it is not selected here for comparison. The following sections will use the data by Chen Guoqing et al. for further comparison). Confining pressure /MPa σ  / (100MPa) ε  /10 -3 σ r / (100MPa) ε r /10 -3   ε R /10 -3  Rupture angle(°) B 6   B 7   B 8   B 9   B 10   B 11   B 12   B 14   B L  0 0.419 2.59 0.268 3.47 1.656 85 0.037 0.963 0.361 0.340 0.639 0.081 0.294 0.996 0.549 3 0.467 2.71 0.363 4.23 2.106 79 -1.549 2.549 0.223 0.561 0.777 0.041 0.194 0.982 0.229 9 0.586 3.39 0.484 4.5 2.799 75 -0.908 1.908 0.174 0.327 0.826 0.034 0.149 0.966 0.208 12 0.605 3.63 0.357 4.9 2.142 70 0.080 0.920 0.410 0.350 0.590 0.094 0.339 0.939 0.436 15 0.658 3.84 0.563 5.26 3.286 70 -1.793 2.793 0.144 0.370 0.856 0.026 0.120 0.939 0.139 18 0.7956 4.51 0.677 6.12 3.837 65 -1.657 2.657 0.149 0.357 0.851 0.028 0.127 0.906 0.126 25 0.982 5 0.8085 7.79 4.117 60 -2.292 3.292 0.177 0.558 0.823 0.032 0.161 0.866 0.099  Table 2: Conventional triaxial compression experimental data of rock specimens from phyllite Preparation and experimental conditions of rock samples Rock brittleness has a profound effect on the stability of deep buried tunnels. In order to accurately evaluate the brittleness of rocks, the effects of surrounding rock pressure must be considered. This paper takes the phyllite samples obtained from C
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