Investigation of the spatial distribution pattern of 3D microcracks in single-cracked breakage

Abstract

The implementation of multi-crack recognition requires a theoretical basis from single-crack failure results. This study investigates the spatial distribution pattern of microcracks in single-crack-damaged coal specimens subjected to uniaxial compression. The distribution mechanism of the induced microcracks is analysed in terms of macrocracks and stress states. Gaussian and Weibull distributions are selected to explore the spatial distribution of microcracks, and quantile–quantile plots are used to study the probability density plots. The results show that the updated data better demonstrate the distribution characteristics of the ellipsoid shape visually, and the microcracks are observed to converge near the envelope. Linear distribution features are found in the scattered points of the quantile–quantile plots. The goodness-of-fit tests indicate that both Gaussian and Weibull distributions can be assigned to describe the microcrack distribution on the new axis. The probability density distribution results show that the shape is preferred to obey Gaussian distribution, whereas Weibull distribution exhibits a greater possibility of shape variation. The study results can enhance the existing knowledge on crack spatial distribution of acoustic emission positioning results associated with rock materials. In addition, the statistical results can interpret the cracks by recognising them with a specified distribution pattern.

Introduction

The spatial distribution of induced microcracks is important for revealing the ruptures formed inside rock samples subjected to loading.1, 2, 3 The cluster and coalescence phenomena of microcracks can lead to the formation of macrocracks and eventually failure planes, and thus contribute to increasing the initial connectivity of a fractured rock mass.4, 5, 6 Therefore, the pattern investigation of the spatial distribution characteristics of clustered microcracks is important for the understanding of rock failure.

Various methods have been applied to investigate crack distribution, such as computed tomography,⁷ infrared thermal imaging technology,⁸ fluorescent fluid tracking technology,⁹ optical analysis technology,¹⁰ digital image correlation technology¹¹ and acoustic emission (AE).^6,12, 13, 14, 15, 16 In particular, AE technology monitors the energy released by microdefect dislocations in the form of sound waves.¹⁷ The source location (i.e. the microcrack location) can be defined by the characteristics of the P(S) waves detected using different sensors.¹⁴ Owing to the convenience of 3D data acquisition and monitoring operability, AE technology is widely used in rock failure monitoring and damage area analysis.^15,18

The study of microcracks in rock fracturing processes has shown that microcracks are randomly generated at the beginning of the failure process, with the cluster being initiated in the form of tensile microcracks.¹⁹ Moreover, microcracks in the process zone exhibit similar mechanical mechanisms of initiation, growth and coalescing.^1,20, 21, 22 For instance, the development and coalescence of microcracks in granite rock types are related to the applied loading stress.²² As the load increases, the clustered microcracks become related to inelastic dissipation, which leads to macroscopic fracture and material splitting.²³ Therefore, the distribution characteristics and clustered pattern of microcracks are worthy of investigation to link the potential mechanical behaviour with the visualised microcrack area in 2D (3D in space) in the rock failure process.

A statistical investigation of crack size indicated that microcracks occur in sets that exhibit well-defined statistical properties.² In different rock types and movement modes, datasets of original faults and new fractures showed a power-law scaling,²⁴ and the data of microcracks were evaluated to characterise the macrocrack properties.²⁵ Furthermore, a close relation between the lengths of microcracks and macrocracks was observed as an exponential distribution²⁶ and power-law distribution.²⁷ The above studies on microcracks provide meaningful guidance for understanding macroscopic rock failure. However, the detailed shape pattern of clustered microcracks has scarcely been investigated. Therefore, it is important to study approaches that can facilitate a more reliable characterisation of these clustered microcracks.

To obtain potential shape characteristics from the clustered microcracks, the density of the spatial distribution of microcracks has been investigated under different loading conditions.^16,28 A hydraulic fracturing test was conducted in a laboratory set up 400 m underground, and the microcracks were projected onto three vertical planes to indicate the fracture distribution. The result obtained using the kernel density estimator on each plane showed that the high-density area presents an approximately elliptical distribution in the centre of the cracking zone, whereas the low-density area is almost scattered in the remaining space.¹⁶ In addition, to visualise the AE event in the spatial distribution over time, a 3D kernel density estimation was adopted to identify the magnitude and spatial scale of the 3D microcrack zone. The high-AE-density areas in the 3D envelope were observed to move in the rock.²⁹ The K-means algorithm was used to divide the 3D microcracks, and the centroid of the cluster far away from the end section was considered as the crack tip position.⁵

At a larger scale, such as microseismic monitoring, the spatial distributions of foreshocks are usually presented in both cross-sectional and subhorizontal views.³⁰ The main cracking zones were determined by the density contour of the aftershocks.¹³ In addition, regardless of the front or side view, the high-density area of microseismic events in an iron mine was observed to have an ellipsoidal shape, which was confirmed to correspond to the location where the rock mass was seriously damaged.³¹ Different cracking areas and shapes of microcracks have been successfully obtained, and the results confirm the significance of using clustered microcrack distribution to interpret the rock failure results. However, the distribution pattern obtained from these results, not limited to areas distinguished by high density, needs to be quantitatively investigated in 3D space.

These microcracks are often regarded as induced by the failure of one macrocrack at different scales. In most cases, however, they are attributed to the joint action of various macrocracks.^5,13,32 Thus, a detailed interpretation of the results of multiple macrocracks using the overall microcracks needs to be refined. Moreover, these microcracks need to be distinguished and interpreted using an appropriate distribution relation.^2,33,34 Therefore, a statistical analysis of the microcracks originating from a single macrocrack is required to propose a spatial distribution pattern of clustered microcracks. Moreover, exploration of the single-crack distribution pattern is expected to play an important role in the recognition of multi-crack distribution.⁶ For instance, in fields that are sensitive to crack distribution, such as the nuclear waste shielding, the triggering of local crack propagation and fracturing need to be strictly controlled.³⁵ However, it is preferred to have crack propagate for other applications such as geothermal mining or coalbed methane mining, where the crack penetration of fracturing is favoured.^36,37

AE technology is widely used for detecting material fracture under multiaxial compression, which leads to complex macrocracks and currently focuses on tensile fracture.²⁸ Therefore, the uniaxial compression test and AE monitoring of intact rock samples were performed simultaneously to obtain the single-cracked shear failure and data sample composed of microcrack coordinates. In addition, a statistical description and an inspection of potential distributions were carried out and compared. This will help to unify the microcrack distribution description of a single-cracked failure and support the understanding of multi-cracked failure.