A study of crack initiation and source mechanism in the Brazilian test based on moment tensor
Introduction
Tensile strength is widely known to play an important role in analysing the failure mechanism of rock and in the design of rock engineering projects. Tensile strength can be determined mainly through the use of the direct tension test and the Brazilian test. Due to the simplicity of preparing the specimen and the convenience of conducting the Brazilian test, the Brazilian test has been widely used to obtain the tensile strength of brittle material [1].
There has been no uniform standard method for performing the Brazilian test over the past 70 years, and different countries or associations use different load contact conditions to conduct Brazilian tests. Specially designed curved loading jaws with sizes equal to 1.5 times the disc radius or flat loading platens are recommended for using in the Brazilian test (i.e., non-line loads) [2], [3], while flat loading platens with two small-diameter steel rods are suggested by China’s national standards or industry standards (i.e., line loads). Although different test devices are used, the recommended tensile strength calculation formula is the same, which is as follows:where P denotes the peak load; D and t represents the diameter and thickness of the specimen, respectively.
The Brazilian tensile strength (BTS) is typically calculated based on the assumption that failure occurred at the centre of the disc of maximum tensile stress. However, there are many factors that affect the BTS, such as the loading condition and the heterogeneity of the rock. Previous researchers reported that the Brazilian test may underestimate the true tensile strength[4], [5], give a reasonable value[6], [7], [8]or overestimate the tensile strength of geo-materials[9], [10], [11], [12], [13]. However, these studies about this difference between the direct and Brazilian tensile tests were based on a macroscopic phenomenon and are constrained to a microscopic and quantifiable interpretation of this difference. According to the Griffith theory, the crack initiation point should be located at the centre of the Brazilian disc. However, the crack initiation points have frequently occurred away from the centre of the specimen in laboratory tests. Some studies[14], [15], [16], [17], [18] observed that material the heterogeneity or inherent defects of rocks play dominant roles in influencing the BTS in both laboratory test and numerical simulations. Li and Wong [19] and Sarfarazi et al. [20] reported that the discrepancy between the theoretical assumptions and laboratory tests may be due to both the failure criteria, and the heterogeneity of the rocks. The crack patterns (including crack initiation and propagation) in the Brazilian test are still a popular area of research. However, it is still difficult to judge the crack patterns, i.e., whether the crack initiates and propagates from the loading points to the centre or from the centre to the loading points.
In recent years, some researchers showed that the contact between the specimen and the loading device had an effect on the BTS values obtained by different methods[21], [22], [23], [24]. Hudson et al. [25] found that the cracks always initiated directly near the loading points if flat platens were applied to load the disc. Markides et al. [23] found that the stress field was strongly dependent on the contact length and the distribution of the radial load, which changed according to the deformability of both the steel jaws and the specimen in practice. Erarslan and Williams [1] concluded that both the crack patterns and BTS values of rocks were critically influenced by diametric loading devices from the experimental studies. In light of the above conclusions, it becomes evident that the reliability of the results of the Brazilian test strongly depends on the actual contact conditions.
The acoustic emission (AE) technique is effective for monitoring and analysing the real-time initiation, propagation and coalescence of micro-cracks in brittle rock[26], [27], [28]. The three-dimensional distribution of AE events can be located by picking up the arrival time. Further, the AE source mechanism can be understood by studying the focal mechanism based on the moment tensor[29], [30]. Falls et al. [31] performed the Brazilian tests to analyse the microscopic fracture mode and stress distribution of the Brazilian disc using ultrasonic methods. However, the specimen was considered to have a two-dimensional geometry, leading to relatively inaccurate three-dimensional locations. Zhang et al. [32] conducted the linear loading Brazilian tests to study the three-dimensional evolution of failure in a sandstone specimen using active/passive ultrasonic techniques. The authors found that the crack initiation positions occurred away from the disc centre, indicating the BTS may underestimate the true tensile strength. Wang et al. [13] conducted direct tensile tests and non-linear loading Brazilian tests to evaluate the relationship between the AE source mechanism and the tensile strength. However, those studies may lead to low positioning accuracy based on two-dimensional model considerations or loss of AE events because the signal is not continuously monitored during the loading process in this test. With respect to the Brazilian test, few studies have applied an ultrasonic technique to study the failure process and the source mechanism under different loading conditions.
Most early studies focused on the analysis of the tensile stress distribution and failed to analyse the micro-failures occurring during the loading process, and little is known about that the influence of the contact conditions and material heterogeneity on the micro-mechanism in the Brazilian test. In this paper, an attempt to quantify the influence of these factors on crucial aspects of the Brazilian test using active and passive ultrasonic techniques and a 3D flat-joint model is described. Two sets of Brazilian tests with linear and non-linear loading were conducted to study the temporal and spatial evolution of the source mechanism. Four numerical tests were generated to investigate the effects of loading conditions and heterogeneity on the micro-mechanical behaviour of Brazilian discs. Based on moment tensor inversion, the crack initiation positions, the source mechanism, and the obtained tensile strength are analysed and discussed in detail.
Section snippets
Specimen preparation
The specimens were sampled from red sandstone cores obtained from the Zigong, Sichuan province of China. This sandstone had been used to carry out a series of experiments by the author’s team[33], [32]. The thin section analysis shows that it is composed of fine-grained rock with 0.2 mm average grain size, comprising 60% feldspar, 35% quartz, 5% magnesium oxide, and small amount of clay as cementation. The specimens prepared are standard Brazilian disks (suggested by ISRM (2007–2014) [34]) with
Rock tensile strength
The disc specimens were tested under linear and non-linear loading conditions with AE monitoring, producing peak forces of 11.10 and 15.02 kN, respectively. In addition, three more discs were further tested under both loading conditions without acoustic emission monitoring, and average peak forces of 9.96[32] and 14.67 kN were obtained. The Brazilian tensile strength values acquired under each loading condition are quite consistent, giving the average values of 5.21 and 7.51 MPa, respectively,
FJM3D modelling of red sandstone
The particle flow code (PFC) simulates the interaction behaviour between particles and can directly represent the fracture by bond breakage without complex crack-criteria [44], [46], [45], [47], [48]. To investigate the effects of loading conditions and heterogeneity on the micro-mechanical behaviour of Brazilian discs, a three-dimensional flat-joint model (FJM3D) was used in this section, which essentially represents the rock matrix as a 3D dense packing assembly of spherical particles. Four
Discussion
To further investigate the effects of the loading conditions and heterogeneity on the crack initiation of the Brazilian test, middle and surface columns with 17 measurement spheres are symmetrically placed along the loaded diameter to record the horizontal stress, σx (Fig. 18). It should be noted that each measurement sphere contains 2–4 grains, and the average stress is recorded.
Fig. 19a, b show the horizontal stresses of the middle and surface columns in the four types Brazilian tests when
Conclusions
While discrepancies between the theoretical analysis and laboratory tests have been frequently observed, few studies focus on the factors affecting the crack initiation and source mechanism in Brazilian tests. In the present study, the active and passive ultrasonic techniques and a 3D flat-joint model are used to investigate the effects of the contact conditions and the heterogeneity of rock on the Brazilian test with a microcosmic and quantitative perspective. Moment tensor inversion is used
Declaration of Competing Interest
The authors declared that there is no conflict of interest.
Acknowledgements
The Support, received from the National Natural Science Foundation of China (grant numbers 51774020; 51934003), the National Key Research and Development Program of China (2017YFC0805300), Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province, is gratefully acknowledged. The authors would also like to thank Itasca for providing technical support for this article.
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