The effect of flaw geometry and infilling on crack initiation, propagation, and coalescence of a parallel flaw pair in marble

Abstract

A series of uniaxial compression tests were conducted on unfilled or filled marble specimens with a parallel flaw pair to study the effect of the flaw geometry and infilling on the crack coalescence pattern. Special attention was paid to the crack initiation, propagation, and coalescence processes, which were traced by the CCD camera and digital image correlation (DIC) technique. The experimental results have shown that the appearance of the white patch and strain localization band precedes the initiation of macroscopic crack. The tensile crack evolves from a narrow strain band or white patches with clear boundaries, while the shear crack evolves from a wider strain band or white patch with blurry boundaries. The high-strength fillers suppress the initiation and propagation of the tensile cracks (i.e., T1, T2, and T3) and promote the appearance of the coplanar shear cracks and mixed tensile-shear cracks (i.e., M1, M2, and S1). The variation in the infilling condition leads to the difference in the fundamental crack types, hence resulting in different coalescence patterns for a given flaw geometry. For small rock bridge angles (i.e., β = 30°, 45°), no coalescence and indirect coalescence tends to occur between the open and gypsum-filled flaws, while shear coalescence is prone to occur between the cement- and resin-filled flaws. The coalescence patterns are dominated by the flaw geometry for medium rock bridge angles (i.e., β = 60°, 90°), but the crack coalescence stress and peak strength of specimens are greatly affected by the infilling conditions. For large rock bridge angle (i.e., β = 120°), the increase in the strength of the filler tends to generate straighter wing cracks for the tensile coalescence at the rock bridge and promotes the initiation of mixed tensile-shear cracks for additional coalescence outside the rock bridge.

Highlights

• Cracking behaviors, coalescence patterns, and peak strength of flawed specimens greatly depends on the flaw geometry and infilling condition.

• The high-strength fillers suppress the initiation and propagation of the tensile cracks and promote the appearance of the coplanar shear cracks and mixed tensile-shear cracks.

• The infilling conditions significantly alter the coalescence pattern in flawed specimens with small and large rock bridge angles, while the coalescence pattern in specimens with medium rock bridge angles is dominated by the flaw geometry.

1 Introduction

Engineering rock masses generally contain different scales of discontinuities from tens of kilometers down to several microns, such as faults, flaws, joints, voids, etc. The natural discontinuities are not always open or closed, but often contain infilling materials, such as sheared-off or broken rock fragments, and clay particles caused by regional geotectonic movement, water softening, and weathering (Zhao and Zhou 2016). In addition, grouting is widely used to improve rock stability and reduce the permeability of fractured rocks by stopping or retarding the crack from propagation (Zhuang and Zhou 2020). Understanding the crack initiation, propagation, and coalescence process can provide valuable guidance for the stability analysis of rock slopes, concrete dams, and foundations with embedded filled joints or cracks.

Most occurrences of discontinuity are non-persistent in rock engineering sites, such as rock slopes, tunnels, and mines (Fig. 1). Rock bridge is commonly encountered in rock engineering practices and plays a significant role in the stabilization of rock mass. The crack initiation, propagation, and coalescence in the rock bridge areas greatly decrease the stability of rock mass and even lead to catastrophic disasters (Gehle and Kutter 2003; Ghazvinian et al. 2012). Considerable efforts have been devoted to studying the crack coalescence behavior of rock or rock-like specimens containing pre-existing flaws, and different crack coalescence patterns are summarized by laboratory tests. The previous studies indicated that the orientation, spacing, continuity, rock bridge angle, the friction coefficient of the flaws, and material properties are important factors significantly affecting the crack coalescence pattern (Shen et al. 1995; Sagong and Bobet 2002; Wong and Einstein 2008; Park and Bobet 2010). For example, Shen et al. (1995) conducted a series of uniaxial compression tests on gypsum specimens containing two non-frictional and frictional fractures and found that specimens with frictional fractures has straighter wing cracks and higher coalescence load than specimens with non-frictional fractures. Wong and Einstein (2008) experimentally studied the cracking and coalescence behavior of gypsum and Carrara marble specimens containing two parallel pre-existing open flaws using a high-speed video system. Their observation reveals that tensile cracking occurred more frequently than shear cracking in marble, while shear cracking occurred more frequently than tensile cracking in gypsum. Different crack types leads to the difference in the coalescence patterns in two types of flawed specimens, especially for those with stepped parallel flaws. Park and Bobet (2009) compared the crack coalescence pattern in gypsum specimens with open and closed flaws under compression, and eight types of coalescence have been identified in their study. Those studies on the crack coalescence behaviors of unfilled flaws have provided a good research foundation for the investigation on the cracking and coalescence behaviors of filled flaws.

Fig. 1

Discontinuities in rock mass a faults in rock slop (Brideau et al. 2009), b fractures in underground engineering site (Feng et al. 2018)

The cracking and coalescence behaviors of rocks containing filled flaws have been rarely studied. Miao et al. (2018) conducted a fracture analysis of sandstone with a filled flaw under uniaxial compression and found that the infilling can reduce the stress concentration at the flaw tips and thus improve the crack initiation stress and peak strength of rock specimens. Chang et al. (2018) numerically studied the mechanical properties and cracking behaviors of rock specimens containing a filled flaw and found that the strength of filler plays an important role in the rock strength and crack types. Similar conclusions were drawn by Zhao and Zhou (2016), who numerically studied the effects of fillings on mechanical properties and failure modes of rock specimens with infilled flaws using a particle mechanics method. A comparison of the mechanical properties and crack coalescence behaviors between unfilled and filled specimens were experimentally and numerically studied by Zhuang and Zhou (2020), in which two infilling conditions, i.e., gypsum filling and sand filling, were considered. They found that the specimens with gypsum filling have greater crack initiation stress and peak strength than specimens with sand filling because the gypsum filling provides stronger friction of cracks than the sand filling. Recently, Sharafisafa et al. (2021) conducted a comparative study on crack development in rock-like specimens containing unfilled and filled flaws and found that the infilling affects not only the peak strength but also the failure modes of flawed rock specimens. It can be seen that the studies mentioned above mainly stressed the effect of filling on strength enhancement, while the effect of infilling on crack coalescence process has not been fully revealved. In addition, the cracking behavior and coalescence patterns are closely related to the properties of infilling materials and flaw geometry. However, the crack coalescence behaviors of flawed rock specimens considering different infilling cases and rock bridge angles have not been studied. Therefore, it is necessary to systematically investigate the effect of infilling on the cracking process and coalescence pattern of flawed rock specimens under different flaw geometries.

Effective crack observation method helps trace the crack initiation, propagation, and coalescence and analyze the fracture mechanism of rock specimens. A high-speed camera is an effective tool for crack observation, with which the crack propagation path and crack mechanism (tensile/shear) can be simultaneously acquired. Wong and Einstein (2009) suggested that the crack mechanism can be determined precisely with proper frame rate and image solution by observing the displacement jumps across newly formed cracks, i.e. tensile opening or shear sliding. The high-speed camera has been widely applied to trace the crack behaviors of flawed rock or rock-like materials (Yang et al. 2019; Yan et al. 2020). It should be noted that the high-speed camera can capture macro cracks with significant opening or shearing, while the deformation localization zones and microcracks with few-pixel openings or shearing can not be identified. Thus, it may cause some subjective deviation in the identification of crack initiation sequence, initiation location, and initiation time. To overcome this obstacle, an innovative technique, named digital image correlation (DIC), is employed in this study. DIC has been considered to be an effective, non-contact optical method to experimentally study the deformation characteristics and cracking behaviors of rocks and rock-like materials (Nguyen et al. 2011; Zhang et al. 2012; Zhang and Zhao 2013). Its measurement accuracy can reach the sub-pixel level, so the crack initiation time, location, and crack sequence can be accurately determined.

This study focuses on the effect of infilling on the cracking and coalescence behavior of a parallel flaw pair with varied flaw geometries. A series of uniaxial compression tests are conducted on marble specimens containing two parallel flaws with varied flaw inclination angles, rock bridge angles, and infilling conditions. The cracking behaviors of rock specimens are traced using the DIC technique, and a sequence of high-resolution pictures captured by a CCD camera is then used to form a video for crack mechanism identification. Besides, the effect of different infilling conditions and rock bridge angle on crack coalescence stress and peak strength of flawed specimens is also analyzed.

2 Specimen preparation and testing

2.1 Specimen preparation

Marble from China Jinping Underground Laboratory (CJPL) is used in this experiment. The marble is a fine-grained heterogeneous material and mainly consists of dolomite (85.0%), calcite (10%), and clay minerals (5%), as shown in Fig. 2. The mechanical properties of the marble are given in Table 1. Cuboid specimens with dimensions of $100\mathrm{ mm}\times 57\mathrm{ mm}\times 20\mathrm{ mm}$ (length × width × thickness) were cut from a large block in the same orientation to eliminate the effects of anisotropy. The six surfaces of rock specimens were carefully polished. The tolerance of the flatness and perpendicularity of the granite specimens met the requirements of the ISRM-suggested method. Two parallel flaws with a length of 12.5 mm and a width of 1.5 mm, located in the center of the specimen, were prefabricated using a high-pressure water-jet cutting machine (Fig. 3). The flaw geometries of the two parallel flaws are defined by flaw inclination angles α and rock bridge angles β, both of which are measured from the horizontal plane (Fig. 3). The length of rock bridge L, i.e., the straight-line distance between two inner tips of two parallel flaws, was fixed and equal to the flaw length 2a. The flaw inclination angle α was set as 45° and 60°, and the rock bridge angle β was set as 30° (for α = 45°), 45°, 60°, 90°, and 120°. The test scheme is summarized in Table 2, and three specimens were prepared for each test scheme to avoid the scatter of experimental data. In this study, the term “flaw” will be used to describe pre-existing fractures, and the term “crack” will be used to describe newly formed fractures.

Fig. 2

Microscopic observations: a scanning electron microscopy image, and b polarized microscope image

Table 1 Mechanical properties of Jinping marble used in this study

Fig. 3

The geometries of the flawed specimens and four infilling cases

Table 2 Specimen preparation for mechanical test

To study the effects of different infilling conditions on the cracking and coalescence behavior of the flawed marble specimen, three infilling materials, i.e., gypsum filler, cement filler, and epoxy resin filler, were chosen for grouting. Among the three infilling materials, gypsum has low strength and stiffness and is generally used to imitate soft infilling materials, such as clay minerals and detrital grains (Zhuang et al. 2014). Cement grouting is a common approach for strength enhancement for fractured rock mass and has been widely applied in rock engineering sites (Lu et al. 2016). Due to its high strength, high stiffness, and strong adhesion, the resin can fit seamlessly with the rock matrix and form a high-stiffness artificial structural plane (Miao et al. 2018). Therefore, the difference in the properties of filler may result in different crack initiation and coalescence behavior, which will enhance the understanding of the crack coalescence between filled flaws. Cement slurry and gypsum slurry were respectively prepared with a water-to-gypsum mass ratio and water-to-cement mass ratio of 1:2. In addition, transparent epoxy resin was also used to fill the flaws for the resin-filled flaws. The mechanical parameters of the three types of filler are presented in Table 3. The previous studies by Pan et al. (2019) have also revealed that resin-filled flaws have the greatest stiffness, followed by cement-filled flaws and gypsum-filled flaws. The detailed infilling process is the same as Pan et al. (2019), and the filled flaw pair is shown in Fig. 3. After the maintenance period of these specimens, random speckles with a random grey intensity distribution were made by spraying black paint and white paint onto the specimen surface.

Table 3 Mechanical properties of the infilling materials

2.2 Testing system

Uniaxial compression tests were conducted on flawed rock specimens using a rock mechanics testing system (RMT-150C) with a maximum loading capacity of 1000 kN and maximum vertical travel of 50 mm. The axial load is applied in displacement control mode with a rate of 0.002 mm·s⁻¹, and axial load and displacement are measured using a load cell and LVDT, respectively. The DIC technique is used for tracking the damage and cracking process of flawed marble specimens. The digital image acquisition system contains a CCD camera with a resolution of 3376 $\times$ 2704 pixels, two white light sources, and a computer equipped with image acquisition software. A sequence of high-resolution images is recorded using the image acquisition software at a rate of 9 frames per second. The rock mechanics testing system and digital image acquisition system are presented in Fig. 4. The nature of a crack is identified by observing the relative displacement of two newly formed fracture surfaces and checking the morphological characteristics of the crack surface after rock failure.

Fig. 4

Rock mechanics testing system and digital image acquisition system

2.3 DIC technique

The DIC technique is a practical and effective tool for quantitative deformation measurement of a test surface. Full-field displacements and full-field strains are obtained by comparing the digital images acquired before and after deformation. It has been widely used in the field of experimental mechanics due to many attractive advantages compared to other strain measurement methods, such as simple experimental setup and specimen preparation, low requirements in the measurement environment, and high measurement precision. The basic principle of DIC is tracking the same subset between two images that were recorded before and after deformation, which is shown in Fig. 5. A square reference subset centered at point P from the reference image is chosen and used to track its corresponding location in the deformed image. To evaluate the similarity between the reference subset and the deformed subset (target subset), a correlation coefficient C should be predefined. The correlation function used in this study is given by,

$$c\left( {\Delta x,\Delta y} \right) = \frac{{\left\langle {f\left( {x,y} \right),g\left( {x + \Delta x,y + \Delta y} \right)} \right\rangle }}{{\left| {f\left( {x,y} \right)} \right| \cdot \left| {g\left( {x,y} \right)} \right|}}$$

(1)

where $f$ and $g$ are the reference and target image intensity function at location (x, y), respectively. “$\cdot$” is the standard scalar product. $\Delta x$ and $\Delta y$ are the distances in respective directions from the subset center P to Q.

Fig. 5

Schematic illustration of the principal of DIC technique

In the practice application of the DIC technique, the images at the 5% peak load are set as the reference image to avoid the relative movement and rotation between the rollers and specimens at low loads. Displacement fields are measured with a grid spacing of 7 pixels and a subset radius of 16 pixels. During the implementation of the DIC technique, the parameters of the iteration options, i.e. the maximum iteration and the threshold for ‖∆p‖, are set as 50 and 1e⁻⁶, respectively. The radius of the circular window for displacement gradient estimation is set as 7 pixels.

3 Results

3.1 Crack classification and coalescence pattern

Crack coalescence refers to the linkage of two pre-existing flaws by newly formed cracks. Since coalescence is mainly caused by the initiation, propagation, and linkage of some fundamental crack types, it is reasonable to relate the coalescence patterns to fundamental crack types from the single flaw. Figure 6 summarized a crack classification scheme that can simultaneously describe crack trajectories and fracture mechanisms. It includes tensile wing crack (T1), tensile sub-vertical crack (T2), tensile anti-wing crack (T3), coplanar shear crack (S1), mixed tensile-shear horsetail crack (M1), and mixed tensile-shear anti-wing crack (M2). The stress distribution around the inclined flaw is used to justify the crack classification scheme. According to Lajtai (1971), the tangential stress along the periphery of the inclined open flaw is depicted in Fig. 7, where the positive and negative denote compressive stress and tensile stress, respectively. For an inclined flaw, the tensile stress concentration occurs at $\upeta ={5}^{^\circ }\sim {20}^{^\circ }$, where the tensile wing crack initiates. The wing crack propagates in a curvilinear path and grows stably. The tensile sub-vertical crack initiates from the flaw tips where $\upeta <{5}^{^\circ }$ and its trajectory is approximately vertical to the direction of maximum compressive stress. The mixed tensile-shear horsetail cracks initiate from the pre-existing flaw tips and consist of a significant shear segment near the flaw tips and a tensile segment away from the flaw tips. High compressive stress promotes the generation of shear cracks near the flaw tips. The coplanar shear crack (S1) is caused by high compressive-shear stress along the flaw plane, and its formation is closely related to the tensile-compression strength ratio of rocks and flaw properties (Pan et al. 2019). Figure 7b shows that extremely high compressive stress occurs at $\upeta ={160}^{^\circ }\sim {180}^{^\circ }$. The great shear displacement across newly formed anti-wing cracks reveals that the generation of anti-wing cracks is also caused by high shear stress (Miao et al. 2021). Therefore, the anti-wing cracks undoubtedly have a shear segment near the flaw tips. As shown in Fig. 6, the two anti-wing cracks have different propagation directions. The tensile anti-wing cracks propagate toward the rock matrix above the pre-existing flaw, while the mixed tensile-shear anti-wing cracks propagate toward the rock matrix far away from the pre-existing flaw.

Fig. 6

Fundamental crack types from a pre-existing flaw

Fig. 7

a Schematic of an inclined flaw subjected to far-field compressive stress, and b tangential stress distribution along the periphery of the inclined open flaw under axial compressive stress

Table 4 summarizes the coalescence patterns observed in specimens with flaws inclined at $\mathrm{\alpha }={45}^{^\circ }$ and ${60}^{^\circ }$ under different infilling conditions. In Table 4, six different crack coalescence types are classified mainly according to the flaw geometries. However, the same geometry with different flaw properties, i.e. infilling cases, does not always produce the same coalescence pattern, so each coalescence type can be further divided into some sub-patterns. The description of these coalescence patterns is given below: (1) Coalescence type I is no coalescence, which means that no linkage occurs between pre-existing flaws. This type of coalescence generally occurs when the rock bridge angle is pretty small. (2) Coalescence type II occurs outside the rock bridge region and was named indirect coalescence by Wong and Einstein (2009) and Zhang and Wong (2013). (3) Coalescence type III is direct shear coalescence with its coalescence crack coplanar with the pre-existing flaws, so the two inner flaw tips are linked up by either two S1 cracks or cracks S1 and M1. (4) Coalescence type IV is tensile coalescence or mixed tensile-shear coalescence. It occurs when two pre-existing flaws start to overlap or slightly overlap, and three different sub-patterns are observed. The first sub-pattern involves the linkage of two T2 cracks initiated from two inner flaw tips. The second sub-pattern occurs through the linkage of a T2 crack and an S1 crack, where a T2 crack first initiates from a flaw tip, then extends toward another flaw tip, and finally links up with the other flaw by an S1 crack. In the third sub-pattern, a sub-vertical tensile segment first appears in the bridge area, then simultaneously extends toward two flaw tips, and finally links up with two coplanar shear cracks initiated from two inner flaw tips. (5) Coalescence type V involves a direct tensile coalescence between two inner flaw tips by one or two T1 cracks and additional coalescence between an inner and an outer flaw tip by T3 or M2 cracks. The type of coalescence generally occurs in highly overlapped flaw pairs. (6) Coalescence type VI is produced by two M1 cracks, which initiate from the inner flaw tips and vertically extend toward the outer flaw tips. Note that no coalescence occurs between two inner flaw tips in this type. Similarly, this coalescence also occurs between highly overlapped flaws.

Table 4 Summarization of coalescence type observed in this study

Further, the influences of flaw geometries and infilling conditions on coalescence patterns are summarized in Table 5. As the rock bridge angle $\upbeta$ increases from ${30}^{^\circ }$ or ${45}^{^\circ }$ to ${120}^{^\circ }$, the coalescence type generally changes from type I to type VI. Additionally, the coalescence pattern also varies with the infilling conditions. For $\mathrm{\alpha }={45}^{^\circ }$ and $\upbeta ={30}^{^\circ }$, the coalescence pattern transforms from no coalescence (type I) to indirect coalescence (type II), and then to direct shear coalescence (type III) as the strength and stiffness of filler increase. For $\upbeta ={45}^{^\circ }$, coalescence pattern changes from indirect coalescence (type II) to direct shear coalescence (type III). For $\upbeta ={60}^{^\circ }$ or ${90}^{^\circ }$, the infilling conditions have little effect on the coalescence type, but minor differences still exist in the fundamental crack types involved in crack coalescence. For specimens with a rock bridge angle $\upbeta ={120}^{^\circ }$, the coalescence type is affected by both the flaw inclination angle and infilling conditions. Note that coalescence type V occurs in specimens containing flaws inclined at $\mathrm{\alpha }={45}^{^\circ }$, while for specimens containing flaws inclined at $\mathrm{\alpha }={60}^{^\circ }$, coalescence type V occurs between unfilled flaws and coalescence type VI occurs between filled flaws.

Table 5 Summary of coalescence types for marble specimens with different flaw geometries and infilling conditions

3.2 Crack initiation, propagation, and coalescence process

3.2.1 Rock bridge angle $\upbeta ={30}^{^\circ }$

Figure 8 shows the axial stress–strain curves for specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={30}^{^\circ }$ and different infilling conditions. Note that the axial stress–strain curves for flawed specimens under infilling conditions share several similarities, including the initial concave nonlinear segment, subsequent linear deformation segment, nonlinear deformation segment near the peak stress, and post-peak segment. Compared to unfilled specimens, the infilled specimens have greater peak strength, peak strain, and capacity for energy storage that is estimated by the areas formed by the stress–strain curves and the horizontal axis.

Fig. 8

Typical axial stress–strain curves for specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={30}^{^\circ }$ under different infilling conditions

Figure 9 presents the failure patterns and the fundamental crack types in specimens with flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={30}^{^\circ }$ under different infilling conditions. It can be seen that no coalescence occurs between two open flaws, and apparent axial splitting is observed at rock failure. For specimens with a gypsum-filled flaw pair, two cracks from two inner flaw tips are linked up by an inclined mixed tensile-shear crack, leading to the typical indirect coalescence. For cement-filled flaw pair and resin-filled flaw pair, the shear rupture at the moment of rock failure causes the direct shear coalescence between filled flaws.

Fig. 9

Failure patterns of specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={30}^{^\circ }$ under different infilling conditions

Figure 10 shows the crack initiation, propagation, and coalescence processes for these specimens with flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={30}^{^\circ }$ under different infilling conditions. For a specimen with an open flaw pair (Fig. 10a), fracture process zones appear at each flaw tip due to the tensile stress concentration at $0.72{\sigma }_{p}$. When the axial load increases to $0.93{\sigma }_{p}$, another two high-strain bands appear at the outer tips of two flaws, indicating the trajectories of impending cracks. After entering the post-peak stage, three anti-wing cracks, including an M2 crack from the upper flaw and two T3 cracks from the lower flaw, propagate vertically toward the loading directions, leading to the splitting failure of the specimen. For the specimen with a gypsum-filled flaw pair (Fig. 10b), similar cracking behaviors are observed, but the indirect coalescence takes place at the moment of ultimate failure (Fig. 9). For a specimen with a cement-filled flaw pair (Fig. 10c), strain localization bands appear at the tips of cement-filled flaws at $0.87{\sigma }_{p}$, and then develop into tensile cracks at $0.93{\sigma }_{p}$. Note that cracks initiated from inner flaw tips is shorter than that from outer flaw tips. At the peak, except for the high strain bands, feather-shaped strain bands of great width appear around the crack tips due to the formation of microcracks induced by high compressive stress. Later, mixed tensile-shear anti-wing crack M2 appears in each feather-shaped strain band at the post-peak stage. At the moment of rock failure, the shear sliding of two pre-existing flaws drives the shear coalescence between two inner flaw tips without any omen (Figs. 9, 10c). For the specimen with a resin-filled flaw pair (Fig. 10d), X-shaped strain bands appear at each resin-filled flaw tip before the macro cracks initiate. These significant strain bands prove the high compressive stress concentration at the resin-filled flaws. Note that no discernible cracks are identified until the peak, and some short cracks appear within these strain localization bands at the post-peak stage. Note that the initiation of tensile cracks, such as T1, T2, and T3, are suppressed at the resin-filled flaws. Similar to the cement-filled flaw pair, shear coalescence occurs between the resin-filled pair at rock failure, and a wide range of shear spalling around the rock bridge area is observed from the back of the specimen due to the huge energy release. It can be seen that no significant deformation localization bands appear in the rock bridge area before the coalescence crack links up with two inner flaw tips or initiated cracks. That is, the crack coalescence occurs in an unstable manner and not in a progressive process.

Fig. 10

Cracking behaviors and coalescence process in specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={30}^{^\circ }$ under different infilling conditions: a open flaws, b gypsum-filled flaws, c cement-filled flaws, and d resin-filled flaws (* indicates the post-peak stage)

3.2.2 Rock bridge angle $\upbeta ={45}^{^\circ }$

For $\upbeta ={45}^{^\circ }$, the coalescence between pre-existing flaws transfers from indirect coalescence (type II) to shear coalescence (type III) in response to the variable infilling conditions (Table 5). Indirect coalescence occurs in specimens with open and gypsum-filled flaws, and direct shear coalescence develops in specimens with cement-filled and resin-filled flaws.

Taking flaw geometry $\mathrm{\alpha }={60}^{^\circ },\upbeta ={45}^{^\circ }$ as an example, Fig. 11 shows the failure patterns of specimens containing a flaw pair under different infilling conditions, in which Fig. 11a shows the maximum principal strain contour before rock failure and Fig. 11b illustrates the crack path and fracture mechanism at rock failure. It is worth mentioning that some white spots in strain contours are caused by the spalling of speckled surfaces or the appearance of the macrocracks with great displacement jumps. As shown in Fig. 11, the indirect coalescence between two open flaws results from a mixed tensile-shear crack, which links up two mixed tensile-shear anti-wing cracks from two inner flaw tips. For the specimen with a gypsum-filled flaw pair, the indirect coalescence takes place outside of the rock bridge region by the direct link of two mixed tensile-shear cracks from two inner crack tips. For the cement-filled flaw pair, two S1 cracks emanating from two outer flaw tips extend a great distance toward the corners of the specimen, and a shear plane tends to form along the pre-existing flaws and the two shear cracks. The shear sliding of the cement-filled flaw also causes the stress concentration at the inner crack tip, leading to the direct shear coalescence through the linkage of the S1 crack and M1 crack. Almost all initiated cracks from the resin-filled flaws are mixed tensile-shear or shear cracks, and the tensile cracks are almost inhibited. Besides, a wide strain localization zone appears in the rock bridge area. Then, two coplanar shear cracks initiate from the inner tips, propagate toward the central part of the bridge region, and finally coalesce with each other, leading to the direct shear coalescence between resin-filled flaws.

Fig. 11

Failure patterns of specimens with a flaw geometry $\mathrm{\alpha }={60}^{^\circ },\upbeta ={45}^{^\circ }$ under different infilling conditions: a maximum principal strain contour before rock failure, and b illustration of the crack path and fracture mechanism at rock failure

3.2.3 Rock bridge angle $\upbeta ={60}^{^\circ }$

For the rock bridge angle $\upbeta ={60}^{^\circ }$, direct shear coalescence (type III) occurs in all specimens containing flaws inclined at $\mathrm{\alpha }={45}^{^\circ }$ and ${60}^{^\circ }$, as listed in Table 5. Taking specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={60}^{^\circ }$ as an example, Fig. 12 presents the failure patterns of flawed specimens under different infilling cases. Note that the direct shear coalescence occurs between all flaw pairs, and the failure of these specimens is caused by the shear rupture. Besides, the shear coalescence between cement-filled and resin-filled flaws is accompanied by violent energy release and loud sounds.

Fig. 12

Failure patterns of specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={60}^{^\circ }$ under different infilling conditions

Figure 13 shows the cracking behaviors and coalescence process for specimens with flaw geometry $\mathrm{\alpha }={45}^{^\circ }$, $\upbeta ={60}^{^\circ }$ under different infilling conditions. For the open flaw pair (Fig. 13a), the fracture process zone appears at each flaw tip at $0.74{\sigma }_{p}$. As the axial stress further increases, two tensile cracks initiated from the outer flaw tips propagate toward the loading direction, while the propagation of tensile cracks from the inner flaw tips is inhibited. Moreover, shear strain bands, quasi-coplanar with the pre-existing flaw, emanate from the two inner flaw tips and merge in the bridge zone at $0.86{\sigma }_{p}$. Afterward, coplanar shear cracks begin to initiate from the flaw tips and propagate inside the wide strain band in the bridge zone. The linkage of the two coplanar shear cracks at the post-peak stage leads to the shear coalescence in the rock bridge area. For the gypsum-filled flaw pair (Fig. 13b), the tensile stress concentration at the flaw tips results in the appearance of strain localization bands at $0.75{\sigma }_{p}$, which further develops to T1 cracks at $0.88{\sigma }_{p}$. Meanwhile, two wide strain bands that emanate from the two inner flaw tips meet in the bridge zone, and shear coalescence caused by the linkage of an S1 crack and an M1 crack leads to the final rupture of the rock specimen at the peak. For the cement-filled flaw pair (Fig. 13c) and the resin-filled flaw pair (Fig. 13d), the strain band appears in the bridge area at about $0.85{\sigma }_{p}$, and it gradually widens and extends as the axial stress increases. Then, the strain localization band further develops into a high-strain band that links two inner flaw tips at the peak. After the peak, the high-strain band in the bridge region evolves into macro shear crack, resulting in shear coalescence in the bridge region. As shown in Fig. 13, differences in cracking behaviors caused by different infilling conditions are also observed. The extension of the tensile cracks at the cement-filled and resin-filled crack tips is inhibited, resulting in shorter crack length than that from open and gypsum-filled flaw tips. Moreover, coplanar shear cracks and mixed tensile-shear cracks are more prone to initiate from the outer tips of cement-filled and resin-filled flaws.

Fig. 13

Cracking behaviors and coalescence process in specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={60}^{^\circ }$ under different infilling conditions: a open flaws b gypsum-filled flaws c cement-filled flaws and d resin-filled flaws (* indicates that the stress level is at the post-peak stage)

3.2.4 Rock bridge angle $\upbeta ={90}^{^\circ }$

As the rock bridge angle $\upbeta ={90}^{^\circ }$, coalescence type IV occurs in all specimens with flaw inclination angles $\mathrm{\alpha }={45}^{^\circ }$ and ${60}^{^\circ }$ (Table 5). Taking specimens with flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={90}^{^\circ }$ as an example, the failure patterns and the crack types for flawed specimens under different infilling conditions are presented in Fig. 14. For the open flaw pair, the tensile coalescence is caused by two T1 cracks from the inner flaw tips, and the extension of the tensile and mixed tensile-shear cracks from the outer tips of flaws leads to the ultimate tensile splitting of the specimen. For the gypsum-filled and cement-filled flaw pair, the crack coalescence in the rock bridge area is caused by a T2 crack and an S1 crack. A similar coalescence pattern has also been reported by Shen et al. (1995). The coalescence between two inner flaw tips subsequently promotes the initiation and propagation of cracks that initiate from the outer flaw tips. For the specimen with a resin-filled flaw pair, a T2 crack initiates from the inner tip of the lower flaw and propagates towards another inner flaw tip. When it comes close to another inner tip, the tensile crack links up with a coplanar shear crack. Similarly, the initiation and propagation of the shear cracks and mixed tensile-shear cracks from the outer flaw tips are promoted by the crack coalescence in the rock bridge area. Note that the crack coalescence is dominated by tensile cracks as the rock bridge angle $\beta =90^\circ$, and the shear cracks only occupy a small proportion.

Fig. 14

Failure patterns in specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={90}^{^\circ }$ under different infilling conditions: a maximum principal strain contour before rock failure, and b illustration of the crack path and fracture mechanism at rock failure

White patches resulting from the generation of micro-cracks appear in marble before the initiation of macroscopic cracks, which has been proven by laboratory tests (Wong and Einstein 2009). Figure 15 presents the development of white patches in a specimen containing an open flaw pair with a flaw geometry $\mathrm{\alpha }={60}^{^\circ },\upbeta ={90}^{^\circ }$, which exhibits cracking behaviors in another form. It can be seen that two wing-shaped and two sub-vertical white patches emanate from two inner flaw tips at 0.65 ${\sigma }_{p}$. The two sub-vertical white patches approach each other in the bridge area. The sub-vertical crack from the lower flaw tip consists of several branches and a wide white zone near the flaw tip. As the axial stress further increases, the upper sub-vertical white band evolves into the macro tensile crack, and the white patch from the lower flaw tips becomes wider and more distinct. Two sub-vertical cracks meet with each other at 0.95 ${\sigma }_{p}$, and the sliding of the pre-existing flaw promotes the crack coalescence, accompanied by the generation of two petal-shaped shear zones. It can be seen that crack coalescence involves the complex development of microcracks within the rock matrix.

Fig. 15

White patches and crack development in specimen containing an open flaw pair with the flaw geometry $\mathrm{\alpha }={60}^{^\circ },\upbeta ={90}^{^\circ }$

3.2.5 Rock bridge angle $\upbeta ={120}^{^\circ }$

The coalescence type V occurs in specimens with flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={120}^{^\circ }$, and the failure patterns for flawed specimens considering different infilling conditions are presented in Fig. 16. Even though crack coalescence between two inner flaw tips is mainly caused by one or two T1 cracks, the trajectories from flaws with different fillers are different. In addition, the crack types involved in additional coalescence are also affected by infilling conditions. For the open flaw pair, a T1 tensile crack from the inner tip of the lower flaw propagates and coalesces with the inner tip of the other flaw in the rock bridge area, and the additional coalescence outside the rock bridge area is achieved by two T3 cracks. For the gypsum-filled flaw pair, a T1 crack initiates from one inner flaw tip, extends towards another inner tip, and finally link up with another flaw at the peak. In addition, additional coalescence is achieved by two M2 cracks that emanate from the outer flaw tips and propagate toward the nearest inner flaw tips. For the cement-filled and resin-filled flaw pair, one or two T1 cracks with less curved trajectories initiate from the inner flaw tips and then coalesce with the center of another flaw surface. Meanwhile, additional coalescence between an outer flaw tip and an inner flaw tip is achieved by an M2 crack. Figure 17 shows the characteristics of additional coalescence patterns between unfilled and filled flaws. Note that the tensile anti-wing cracks from the outer flaw tips are observed, and its propagation path is smooth and absent of pulverized power (Fig. 17a). On the other hand, the macro cracks can not be easily identified with the naked eye in specimens with a filled flaw pair due to the appearance of wide white patches (Fig. 17b). Besides, crushed debris is observed near the tips of the filled flaw due to the intense shearing.

Fig. 16

Comparison of failure patterns in specimens with a flaw geometry $\mathrm{\alpha }={45}^{^\circ },\upbeta ={120}^{^\circ }$ under different infilling conditions: a maximum principal strain contour before rock failure, and b illustration of the crack path and fracture mechanism at rock failure

Fig. 17

Characteristics of additional coalescence between unfilled and filled flaws

For flaw geometry $\mathrm{\alpha }={60}^{^\circ },\upbeta ={120}^{^\circ }$, the coalescence type transfers from type V for specimens with open flaws to type VI for specimens with filled flaws (Table 5). To reveal the coalescence characteristic of type VI, a specimen with a gypsum-filled flaw pair is taken as an example, and its failure pattern, crack types, and coalescence process is presented in Fig. 18a–c, respectively. At $0.80{\sigma }_{p}$, a blurry strain band appears between the inner and outer flaw tips. As axial stress further increases, M1 cracks initiate from the inner flaw tips and propagate toward the outer tips of another flaw. In addition, the propagation of T1 cracks from the outer flaw tips is promoted as the axial stress further increases.

Fig. 18

Flaw geometry $\mathrm{\alpha }={60}^{^\circ },\upbeta ={120}^{^\circ }$ and gypsum-filled flaws: a coalescence pattern, b coalescence types, and c cracking evolution process

3.3 Stress analysis

Figure 19 shows the effect of different flaw geometries and infilling conditions on the peak strength of specimens containing two parallel flaws. It can be seen that the peak strength of specimens with filled flaws is always greater than those with open flaws at a given flaw geometry. Among the specimens containing filled flaws, the rock specimens with resin-filled flaws have the greatest strength, followed by specimens with cement-filled flaws, and finally, gypsum-filled flaws. Besides, the rock bridge angle plays a great role in the peak strength of the flawed rock specimens. As the rock bridge angle increases, the strength first decreases and then increases, and the flawed specimens get a minimum value at ${\upbeta =90}^{^\circ }$ for a given infilling condition. In addition, the difference in peak strength between flawed specimens of varied rock bridge angle decreases as the strength and stiffness of the infilling materials increase. Moreover, compared to specimens containing flaws oriented at $\mathrm{\alpha }={45}^{^\circ }$, the difference in peak strength introduced by the variation in the flaw geometries and infilling materials is reduced for specimens containing flaws oriented at $\mathrm{\alpha }={60}^{^\circ }$. The variations in strength of flawed rock specimens with flaw geometries are consistent with results reported by Wong and Chau (1998) and Park and Bobet (2009).

Fig. 19

Peak strength for specimens with varied flaw geometries and infilling condition: a $\mathrm{\alpha }={45}^{^\circ }$, and b $\mathrm{\alpha }={60}^{^\circ }$

Table 6 shows crack coalescence stress (level) for specimens with varied flaw geometries and infilling cases, and the symbol * indicates that the stress level is at the post-peak stage. Crack coalescence stress refers to the axial stress at the crack coalescence, and the coalescence stress level is the ratio of the crack coalescence stress to the corresponding peak strength of the specimen. Consistency is observed between specimens with flaw inclination angles $\mathrm{\alpha }={45}^{^\circ }$ and $\mathrm{\alpha }={60}^{^\circ }$. For rock bridge angles $\upbeta ={{30}^{^\circ }, 45}^{^\circ }$ and ${60}^{^\circ }$, coalescence occurs either at or after peak stress, so the rock rupture is caused by the crack coalescence between two flaws. For rock bridge angles $\upbeta ={90}^{^\circ }$ and ${120}^{^\circ }$, coalescence occurs either before or at the peak stress, and the coalescence stress increases as the strength and stiffness of the pre-existing flaws increase.

Table 6 Crack coalescence stress for specimens with varied flaw geometries and infilling cases

4 Discussion

4.1 The effect of flaw property on cracking behavior

As revealed by the cracking process in Sect. 3.2, tensile cracks prevail in specimens with an open flaw pair and always extend a long distance before rock failure. However, the propagation of tensile cracks from filled flaw tips is suppressed, so shorter tensile cracks are observed in specimens with a filled flaw pair. In addition, shear cracks and mixed tensile-shear cracks are more prone to initiate from the filled flaw tips. For example, taking the anti-wing cracks as an example, crack T3 prevails in specimens with open flaws, whereas M2 cracks are more prone to initiate from filled flaws. The initiation of the anti-wing cracks sometimes is even inhibited from the filled flaw tips. As reported by Miao et al. (2018), the difference in the cracking behaviors can be attributed to the difference in the normal and shear stiffness of the flaws (Fig. 20). For an inclined flaw under uniaxial compression, the axial stress can be decomposed into normal stress $\sigma$, which is perpendicular to the flaw, and shear stress $\tau$, which is parallel to the flaw (Fig. 20a). The most dominant role of normal stress perpendicular to the pre-existing flaw is to induce the anti-wing cracks (Fig. 20b). It can be seen that most of the anti-wing cracks are mixed tensile-shear cracks, and the others are tensile cracks, generally from the open flaws. Significant shear sliding across the newly-formed cracks is accompanied by the appearance of anti-wing cracks. It implies that the initiation of anti-wing cracks is driven by the induced shear stress near the flaw tip. The rapid propagation of anti-wing cracks is observed after they initiate from the tips of open flaws, which only takes a few seconds. However, the progressive development of strain localization bands is observed during the initiation and propagation of the anti-wing cracks from the filled flaw with high-stiffness filler. This is because the filling material with high rigidity (i.e., cement and resin) can transfer partial normal stress, which alleviates the shear stress concentration and makes the stress distribution more uniform. Therefore, the length of the anti-wing cracks from the filled flaws is significantly reduced compared to that from the open flaws, and sometimes the anti-wing cracks are even suppressed (see Fig. 13). When considering a flaw under pure shear stress $\tau$ (Fig. 20c), the flaw is in mode II loading. The shear sliding induced by the shear stress leads to both the tensile stress and shear stress concentration at the flaw tips. A laboratory test performed by Mutlu and Pollard (2008) indicated that the tensile cracks first initiated in a direction approximately ${70}^{^\circ }$ from the pre-existing flaw, rather than the coplanar shear cracks, due to the great difference between the tensile strength and shear strength of rocks. This is also consistent with our experimental observation. Compared to the open flaw, the filler can provide frictional resistance along the flaw plane, so the effective shear stress used for driving the shear slip is reduced. Thus, the tensile stress concentration is released, leading to shorter wing crack length and greater crack initiation stress. Accordingly, the specimen can withstand higher axial stress, which creates the possibility for the initiation of a coplanar shear crack at the tips of the filled flaws. Wang et al. (2022) studied the mesoscopic failure mechanism of grout-infilled sandstone under uniaxial compression using an improved AE localization technique. Their results showed that compared to the specimens without grouting, more shear microcracks were detected in the grout-infilled specimens before the final failure occurs. Therefore, it can be concluded that shorter tensile cracks and coplanar shear cracks tend to initiate from filled flaws (Fig. 20d).

Fig. 20

Schematic illustration of cracking behaviors for open and filled flaws

4.2 The effect of flaw property on crack coalescence

Coalescence patterns are further summarized to study the effect of infilling conditions on crack coalescence. For flaw geometry $\mathrm{\alpha }={45}^{^\circ }$ and ${\upbeta =30}^{^\circ }$, the coalescence type varies drastically with different infilling cases (Table 5). In this geometry, no coalescence occurs between the open flaw pair, whereas indirect coalescence or direct shear coalescence is more prone to occur between the filled flaw pair. However, no strain localization or crack initiation occurs as a precursor before crack coalescence takes place between the filled flaws (Fig. 10). Therefore, whether indirect or direct shear coalescence occurs in this geometry is a spinoff at rock failure. For flaw geometry $\mathrm{\alpha }={45}^{^\circ }, {60}^{^\circ }$ and ${\upbeta =45}^{^\circ }$, strain localization and crack initiation are observed in the bridge area where indirect or direct coalescence will occur (Figs. 11, 13). For flaw geometry $\mathrm{\alpha }={45}^{^\circ }, { 60}^{^\circ }$ and ${\upbeta =60}^{^\circ }, { 90}^{^\circ }$, though consistent coalescence type occurs in flawed specimens under different infilling conditions, differences in coalescence stress and involved crack types still exist due to the difference in the flaw properties (Tables 5, 6). For flaw geometry $\mathrm{\alpha }={45}^{^\circ }$ and ${\upbeta =120}^{^\circ }$, both the tensile coalescence inside the bridge region and additional coalescence outside the bridge region are observed (Fig. 16). The trajectories of tensile coalescence between two inner flaw tips depend on flaw properties. For specimens with an open flaw pair and a gypsum-filled flaw pair, the tensile crack coalescence at the rock bridge between flaws is induced by a curved tensile wing crack. However, less-curved tensile wing cracks initiate from the cement-filled or resin-filled flaw pair and propagated sub-vertically towards the center of another filled flaw, leading to the tensile crack coalescence at the rock bridge area. In addition, the nature of cracks involved in additional coalescence also varies from tensile cracks to mixed tensile-shear cracks as the stiffness of filler increases. For flaw geometry $\mathrm{\alpha }={60}^{^\circ }$ and ${\upbeta =120}^{^\circ }$, coalescence between an open flaw pair is achieved by type V, while type VI occurs between a filled flaw pair. It indicates that tensile concentration in the rock bridge is released by the infilling materials, so only additional coalescence occurs outside the rock bridge area in specimens with filled flaws.

5 Conclusions

A series of uniaxial compression tests were conducted on marble specimens containing a parallel flaw pair, and the effects of flaw geometry and infilling on crack initiation, propagation, and coalescence are studied with the help of the DIC technique. Some main conclusions are listed below.

• 1. The crack types and coalescence patterns vary with the flaw geometries and properties. Six different crack coalescence patterns are classified depending on flaw geometry, and each coalescence pattern is further divided into some sub-types due to the difference in the involved crack types resulting from different flaw properties.

• 2. The increase in the strength and stiffness of the fillers suppresses the initiation and propagation of tensile cracks (i.e., T1, T2, and T3) from the flaw tips, leading to shorter and straighter tensile cracks and greater crack initiation stress. In addition, the crack coalescence between flaws of greater stiffness is more likely to involve the coplanar shear crack and mixed tensile-shear cracks (i.e., M1, M2, and S1) for a given flaw geometry.

• 3. White patches and strain localization bands containing several microcracks appear before macroscopic crack initiation. The initiation of a tensile crack evolves from a narrow strain band or white patches with clear boundaries, while the appearance of a shear crack evolves from a wider strain band or white patch with blurry boundaries.

• 4. As the rock bridge angle increases from ${30}^{^\circ }$ or ${45}^{^\circ }$ to ${120}^{^\circ }$, the peak strength of flawed specimens first decreases and then increases for a given infilling condition, and a minimum value is acquired at ${\upbeta =90}^{^\circ }$. The peak strength of specimens with a filled flaw pair is always greater than those with an open flaw pair of the same flaw geometry. Among the specimens with the filled flaws, the specimens with resin-filled flaws have the greatest peak strength, followed by the specimens with cement-filled flaws, and finally by the specimens with cement-filled flaws. The differences in peak strength of flawed specimens are reduced as the flaw inclination angle and stiffness of filler increase.