A comparative study on the crack development in rock-like specimens containing unfilled and filled flaws

Highlights

• 3DP specimens are analyzed by DIC and BPM to investigate the coalescence behavior.

• Two scenarios are considered, unfilled flaws and filled flaws.

• Strain transformation significantly influenced by the flaws filling material.

• A new coalescence type was found in 30° partially overlapped flaws.

• Fractured fillings cause higher peak loads in filled specimens than unfilled ones.

Abstract

Rock masses consist of various discontinuities that significantly affect the crack development patterns which dominates the ultimate failure of geostructures. The interaction of the pre-existing flaws with each other and with the newly formed cracks is complicated and demands a comprehensive investigation. This paper couples the 3D printing technology with the digital image correlation (DIC) and the bonded particle model (BPM) to study failure of rock-like specimens with pre-existing flaws. Systematic flaws configurations are considered including single, coplanar, partially overlapped and fully overlapped arrangements. The flaws are considered to be unfilled and filled with a weak material. The DIC strain and BPM displacement vectors analysis indicate the strong effects of the filling material on the deformation behaviour of the 3D printed specimens. The failure pattern of the single filled flaws is transformation from compressive failure (0°) to shear failure (15°–60°) and to tensile failure (75°–90°). However, in the coplanar flaws failure is transformation from compressive (0°) to mixed-mode compressive shear (15°-30°), then to shear (45°), to mixed-mode tensile-shear (60°–75°) and then to pure tensile (90°). However, the partially coplanar filled flaws all (except 0° which is compressive failure) exhibit mixed-mode failure in the order of compressive-shear (15°–30°) and transformation to the tensile-shear (45°–90°). BPM displacement vectors revealed five crack types in the specimens by the analysis of the relative movement of the vectors in some critical locations such as flaws tips, rock bridge and coalescence zone. Moreover, a new coalescence type was identified in the 15° flaw (either unfilled or filed) which is named Type X and is a mixed-mode shear tensile crack with a coplanar secondary shear crack. Furthermore, it is observed that the peak load of the filled specimens are much higher than that of the unfilled ones because the filling material requires extra energy to fracture and thus the filled specimens can carry larger load before fail.

Introduction

Intact rocks are rare and a perfect homogeneity and isotropic characteristic in rocks is hard to encounter. Natural rock masses contain a particular source of anisotropy which causes complex deformation and failure behaviour. Amongst the anisotropic sources, discontinuities such as flaws, cracks, joint sets, bedding and fault are very common and are always confronted in civil and mining projects. These geological features not only affect the overall stability of the rock masses, but also degrade their deformation behaviour and impose strong influences on the crack propagation mechanism [19], [63], [23]. Therefore, a thorough instability assessment of any project involved with construction in rock masses demands in-detail analysis of the possible effects of the pre-existing discontinuities on the failure behaviour.

The term flaw, which is devoted to explain a general small non-persistent discontinuity in rock, has been a subject of many studies. The flaws appear in different scales and geometries, with some conditions including unfilled, filled, single, multiple having various inclination angles; each having a different influence on the deformation behaviour of the host rock masses. To better understand the impacts of the presence of the flaws on the overall behaviour of rocks and rock-like materials, a number of research works have been carried out [29], [48], [26], [10], [27], [75], [73]. The previous studies have revealed that three types of cracks develop from the flaws under uniaxial loading. These three flaws types are tensile, shear and mixed tensile-shear mode which have been found by Wong and Einstein [50]. Each of these crack types have been categorised into different classes as follows: tensile cracks consist wing crack, anti-wing crack, out-of-plan crack and far tensile crack; shear cracks consist coplanar secondary cracks, oblique secondary cracks, out-of-plane shear cracks, and far shear cracks [50], [74]. Once the number of flaw exceeds one, in addition to the propagation of the mentioned cracks, the coalescence of the flaws emerges which makes the phenomenon more complex. The coalescence mechanism of the multiple pre-existing flaws has attracted much attention over the years [64], [13], [55], [57], [56], [51]. The main factor affecting the coalescence behaviour of rock masses or rock-like materials is the geometry and spatial distribution of the flaws with regard to their adjacent flaw(s). The flaws inclination angle, the angle of the rock bridge, and the arrangement of the flaws strongly attribute to the different coalescence types [59], [37]. These coalescence types were found to be a combination of the crack types mentioned earlier in this section. Moreover, many experimental attempts have been performed to closely look into the crack propagation and coalescence in rock and rock-like materials. For instant, Zhou et al. [77] conducted uniaxial compression tests PMMA specimens having two three dimensional pre-existing cross-embedded flaws. They mainly focused on the cracking behaviors of PMMA specimens with various inclination angles and pre-existing flaws’ non-overlapping lengths. In Zhou et al. [76] uniaxial compressive tests were performed and coupled with digital imaging and AE techniques on prismatic specimens containing pre-existing flaws to investigate strain evolution and crack development in granite. Zhou et al. [78] in a series of experimental works on ductile and brittle specimens containing multiple pre-existing flaws by using digital correlation (DIC) technique, compared the crack initiation, propagation and coalescence between ductile and brittle material. Moreover, the specimens containing multiple flaws were also studied experimentally to evaluate coalescence behavior of rock-like materials under uniaxial loading [75], [73]. They identified five types of cracks at or near the tips of pre-existing flaws and also ten types of coalescence were observed. In an interesting attempt, AE event rate were characterized in the flawed rocks at the unstable cracking phase by using uniaxial compressive test combined with acousto-optical monitoring techniques [68]. The identified coalescence types will be further explained in detail as the analysis progresses in this paper.

In order to analyse the deformation field and crack development patterns in rock and rock-like materials, optic methods have been widely employed to provide information regarding crack types and coalescence mechanism. Among the optical techniques, the digital image correlation (DIC) method has demonstrated its capabilities in capturing cracks and strain evolution trends. The DIC is a non-contact, full field, non-destructive, optical measurement method which is able to track strain and deformation evolutions in real-time. The DIC has been used in many fields including rock engineering, particularly in studying fracture process zone in granite [52], rock damage evolution under cyclic loading [9], rock failure under coupled static-dynamic loading [32], hole-flaw interaction [53], transversely isotropic rocks [2], [3] rock blasting [58], failure of 3D printed specimens [41], [42], [43], fracture toughness [22], hydraulic fracturing [62], failure of flawed specimens [61], [84], [85], [86] and many more.

Although some finding can be revealed by experimental analysis, it is difficult to obtain precise information in micro-scale by the experiments or the DIC. These methods are capable of tracking and illustrating overall failure pattern, but they face difficulties when it comes to displacement and load analysis in a very small area of interest. Therefore, numerical simulation can be employed to aid to investigate cracking processes in micro scale. Among the many numerical methods, discrete element method (DEM) has been demonstrated to be rigorously capable of the analysis of deformation field and crack development trajectories in any scale and under any circumstance. In this regard the BPM embedded in particle flow code (PFC) which is based on the DEM approach has been utilized by many researchers in modelling grain based materials including rock masses. This method has proven its strong capacities in modelling various aspects of rock engineering problems including complex geometry simulation, different loading types (quasi-static, dynamic, creep, thermal, etc.), crack type detection, propagation and coalescence identification and more. For instance some successful applications of this method are crack development from pre-existing cracks [34], [33], triaxial loading [25], joint shear failure [68], rock mass cavability [35], blasting [60], mining sequences [44], layered rocks [49] and many more. Cao et al. [6] carried out DEM simulations on jointed rock-like specimens having multiple joints under compression-shear loading. They observed four types of failure modes namely shear cracking via a plane (Mode I), shear cracking through intact material (Mode II), shear crack obliquely linking failure (Mode III) and failure with stepped trajectory (Mode IV). In an attempt made by Zhang et al. [71] by DEM on rock-like materials with non-parallel flaws under compression, five different coalescence patterns were identified: tensile crack coalescence, tensile crack coalescence with shear linkage at a flaw tip, shear crack coalescence, a mixed mode coalescence (shear-tensile mode) and crack coalescence indirectly. All the previous studies have demonstrated that the DEM is capable of modeling complex crack development patterns.

There are some limitations associated with the DEM which can affect the results which are categorized and discussed in detail in Donze et al. [12]. For example, there exist particlesize effect which arises from fracture representation by elements. This element sizes control the behavior of a fracture which may be different from a natural fracture. Second the real dimensions and shapes of natural grains are different with the DEM element which is so-called cross effect. This results in difference between numerical porosity and real porosity. Another challenge in DEM simulation is material property calibration to represent real macroscopic behavior of rocks. In order to establish the relationship between the local and macroscopic constitutive laws, data obtained from classical geomechanical tests which may be impractical are used. According to Donze et al. [12], constructing the relationships between local/macroscopic constitutive laws by utilizing only data acquired from classical experimental methods such as triaxial test seems insufficient to set a single solution. They recommended using new investigation methods such as tomography [31] capable of providing extra information to improve accuracy of the simulations. Despite these limitations, the wide application and verification of the DEM in rock engineering has demonstrated its extraordinary capabilities in dealing with small and large scale problems. This method not only allows modeling the pre-existing defect such as crack, flaws, bedding and faults, but is also capable of simulating newly formed cracks in micro- and macro-scale with high precision.

Although both the DIC and DEM have extraordinary capabilities in studying rock failure, coupling both methods has attracted some attention and has been successfully utilized in rock engineering. For instance in Fakhimi et al. [14] DIC was coupled with the bonded particle model to capture the global and local mechanical behavior of Berea sandstone specimen. In Zhao et al. [72] digital imaging and DEM were employed to study mechanical behaviors of rock aggregates. Cao et al. [7] coupled the DIC with the 2D DEM in order to investigate failure behavior of ubiquitous-joint rock-like specimens under uniaxial loading. In Huang et al. [24] the FDM-DEM method was coupled with the DIC method to deal with the failure mechanism of tunnel excavation in rock mass.

Up to date a number of numerical tools have been proposed and developed to study fracture propagation in rocks. Amongst the many numerical methods, some have been newly developed and have proven to be robust and capable of modeling complex fracturing processes in rocks. For instant, peridynamics is a relatively new method which has demonstrated its capabilities in modeling complex problems. This method is a mechanical non-local theory in which points of a material in a continuum media or in a group consists of multiple discrete particles interact with each other. The interaction is via forces which are material functions and kinematic variables [38], [39]. The peridynamics theory has three types; bond-based, ordinary-state-based, and non-ordinary-state-based. All of these types have been successfully in rock engineering and have attained good results in terms of crack development and failure. In Gao et al. [15] the peridynamics was utilized to simulate surrounding rock damage while tunnel excavation. Chen et al. [8] employed a modified peridynamics to simulate rock cracking mechanism during drilling operation. Crack propagation in ring-shaped rock specimens under dynamic loading was studied by this method by Zhang et al. [67]. Rabczuk and Ren [36] used the peridynamics to investigate rock quasi-static fracture and contact. Also this method was used to model crack development in rock under compression [18]. In Zhang et al. [68], Zhou and Shou [80] and Zhou and Wang [79] the non-ordinary state-based peridynamics was successfully utilized to simulate crack initiation and coalescence in Brazilian disc containing pre-cracks. Other successful applications of the peridynamics in rock engineering are in hydraulic fracture propagation [30], rock blasting [69], rock thermal failure behaviour [45], cracks in brittle solids [46], [47], another numerical tool with strong capabilities in dealing with rock fractures is general particle dynamics (GPD). The GPD is developed from smoothed particle hydrodynamics (SPH) and overcomes disadvantages of SPH. This method has also been employed to study crack propagation in rocks such as Zhou et al. [81] and Bi et al. [5]. Another robust method which uses advanced meshing method is the extended finite element method (XFEM) which was originally proposed by Belytschko and Black [4]. This method models the cracks without remising which in turn reduces computational cost and time and increase the computation efficiency. This method has also been employed to study fracture propagation in rock mechanics such as in Zhou and Yang [82], Sharafisafa and Nazem [40] and Haeri et al. [17].

Most of the studies carried out to investigate the failure of rocks containing pre-existing flaws have been conducted on rock-like specimens prepared with an artificial material such as cement, gypsum and a mixture with sand, water and other materials. In order to embed flaws, usually metal shims are used to generate flaws with flexible geometry generation. These methods are reliable and effective, but they usually impose some limitations such as imprecise flaw geometry and material inhomogeneity which may generate some unwanted micro-cracks that ultimately affect the desired results negatively. To overcome these limitations, 3D printing method has been proven to be able to precisely preparing rock-like specimens with complex geometries and a number of materials. Sharafisafa et al. [42] compared the brittleness index of the 3D printed powder based material with those of natural rocks and concluded that this material is an alternative option which can mimic real behavior of natural rocks. In Zhu et al. [83], internal defects found in natural rocks were replicated using 3D printing method. Xia et al. [54] utilized the 3D printing method to reconstruct rock masses with irregular columnar joints. In Ishibashi et al. [20] the 3D printing method was employed to replicate hydro-mechanical characteristics of fractures produced by the 3D printed material. There are numerous attempts to employ the 3D printing technology to replicate and mimic real behavior of natural rock masses which further confirm the extraordinary capabilities of this method in rock engineering field.

To date, the significance of the flaws filling have been neglected despite the strong influences imposed by the filling. The filling material not only influences the mechanical behaviour, but also alters the crack development patterns (initiation, propagation, coalescence and ultimate failure) which are of great importance in design and analysis of geostructures. Moreover, these geological structures are usually ignored in field measurements prior to geo-construction and thus their impacts are ignored. This insufficient attention to these important features leads to imprecise designs and may lead to improper support design. Past studies have mainly focused on the mechanic properties variations by a filling material and no in-depth and systematic study has attempted to fully look into the issue. Moreover, the existing studies usually discussed the ultimate failure patterns with no discussion on the differences in crack initiation, crack types and coalescence patterns. Therefore, the aim of the present study is to compare the crack evolution and coalescence patterns in the 3D printed rock-like specimens containing various flaws arrangements under unfilled and filled conditions. The flaws are filled with a weak material and all the unfilled and filled specimens are loaded under Brazilian loading. To analyse the deformation field, crack evolution and coalescence behavior of the flaws, the DIC and DEM are both utilized and the obtained results from each method is then compared to the results of the other method. The two methods are chosen to comprehensively look into the phenomenon of the failure of the pre-existing flaws when are filled with a different material.