# Experimental Investigations on Crack Propagation Characteristics of Granite Rectangle Plate with a Crack (GRPC) under Different Blast Loading Rates

Authors: Xin Liu; Jun Yang; Zhenyang Xu; Lianjun Guo; Hongliang Tang
Journal: Shock and Vibration
 (2020)

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## Abstract

The experimental system of 3D digital image correlation (3D-DIC) is set up to eliminate a certain extent of out-of-plane motion for accurate measuring the full-field strain field during crack propagation, and the effect of blast loading rates of fracture behavior of granite rectangle plate with a crack (GRPC) is investigated. The experimental results indicated that the maximum values of the strain concentration zone do not fully represent the crack tip during the whole process of crack propagation. The axial strain threshold value tip (ASTVT) plotting with lines and coordinate contours corresponding with the actual crack at the shooting area can be used to describe the position of the crack. The axial strain 1.3% is more practical to obtain crack velocity and average crack velocity, and the average crack velocity decreases as the blast loading rates increase. Through observing the relationship between crack width and time, it can be found that there are three stages, and the crack width increases as the blast loading rates increase.


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## Body

## 1. Introduction

Blasting is one of the main rock breaking methods that is widely used in rock excavation, and rock blasting fragmentation time is an important component of the delay time of differential blasting, which is related to the fracture velocity and fracture process of rock [1]. On the one hand, granites are a common rock type in the open bench blasting, and thermal shocks, frost action, and salt crystallization leading to cracking of granites are widely studied [2–5]; on the other hand, the fracture characteristics of granite under explosive dynamic loading are one of the core problems of rock dynamic mechanics, and then the research in this area has practical application significance [6]. Contemporarily, the research on the rock failure mechanism under explosive loading mainly focuses on the singularity of the crack tip, the variation law of stress and strain fields, and the crack propagation behavior [7]. For example, Zhu et al. [8] proposed a new method to test the dynamic fracture toughness under explosive loading by utilizing a rectangular slab concrete specimen with mode-I crack; the crack tip initiation time was measured by using a strain gauge, and the pressure obtained from the two strain gauges near the hole was applied as the dynamic boundary condition by using the finite element method to calculate crack tip open displacements and crack tip dynamic intensity factors. Liu et al. [9] proposed a single internal crack circular disc (SICCD) specimen with poly methyl methacrylatemethacrylic acid (PMMA) to investigate the fracture toughness parameters and propagation behavior of mode-I crack under blasting load by using crack propagation gauges (CPGs); the crack propagation velocity, dynamic initiation toughness, propagation toughness, and fracture parameters of mode-I crack were obtained through the experimental-numerical method. Sun et al. [7] introduced symmetry experimental model with PMMA to implement the visualized positioning of the crack tip and obtained the more accurate record of full-field strain, and it is concluded that the maximum strain value point in the main strain field cannot be used as the judgment basis of the crack tip. Wan et al. [10] used PMMA to customize a rectangle plate with a crack and edge notches (RPCENs) to study the fracture toughness of mode-I crack under blasting load, and they concluded that the crack propagation velocity is variable and related to the fracture surface roughness, the nucleation rate, and propagation toughness. Xu et al. [11] studied the effect of blast waves on the precrack with a cracked medium, and they found that the precrack underwent different developments under three kinds of stress waves. The obvious characteristic of these models is that the borehole is the loading boundary of the blast stress wave, and then a precrack extending to the borehole wall is used to investigate the crack propagation behavior.In terms of experimental technique to measure the crack initiation and propagation process, there are strain gauge, CPGs, geometric grating method, moire interference method, photoelastic method, electronic speckle interference method, and two-dimension digital image correlation (2D-DIC) method [12–16]. Ju et al. [17] summarized the measurement techniques used to determine the crack initiation and crack tip location. Zhang Zhao [18] demonstrated that the measurements of strain gauge and 2D-DIC are in good agreement and 2D-DIC is reliable, especially, 2D-DIC can continue to measure the strain on the sample surface after the failure of the strain gauge. By comparing with various test methods, 2D-DIC has the advantages of simple sample preparation, easy measurement system construction, and surface deformation field measurement. And 2D-DIC is a common successful experimental technique to determine the crack processes that occur to the surface of a rock specimen under dynamic loading. For instance, by combining 2D-DIC with the split-Hopkinson pressure bar (SHPB), the notched semicircular bend (NSCB) was selected to obtain the vertical strain field and the vertical displacement field around the tip of the prenotched crack; besides, Zhang and Zhao [19] deemed that a high-speed camera is reliable than traditional SHPB methods of obtaining the crack propagation results from the range of higher loading rates. Sutton et al. [20] published a review on dynamic NSCB-SHPB tests with the 2D-DIC method, and they summarized the calculation principle of 2D-DIC and introduced what reliable information about the fracture propagation process can be provided by 2D-DIC. However, 2D-DIC theoretically is restricted to planar surfaces, predominantly in-plane deformations, and cases where the recording camera can be set perpendicular to the object surface [20], i.e., out-of-plane motions affect the measurement of the object deformation. 3D-DIC can eliminate this factor; for example, Xing et al. [21] proved that the distribution of strain developed by 2D-DIC and 3D-DIC at the same position on the sandstone surface under dynamic compression shows a different pattern with the maximum strain error of 0.18%.The three-dimensional digital image correlation (3D-DIC) method was put forward by Luo et al. [22], and the basic principle is to combine the principle of binocular stereovision with digital image correlation matching technology, restore the three-dimensional space coordinates of the points before and after deformation, and then obtain the surface morphology and three-dimensional deformation information of the object [23]. Completing a test with 3D-DIC, a few pointers helping us to get the best results in the shortest period time is worth pointing out [24]: (1) the preferred surface texture should be nonperiodic; (2) a simple task lamp of 50–100 watts will give excellent light levels for many moderately sized specimens; (3) it should be aware of the orientation and potential camera placement, and then the next step will be to set focus; (4) it will also be necessary to adjust the brightness of the image as you make the image sharp through the focus adjustment; (5) selecting a grid that approximately fills the field of view, and calibration images will be useful as long as all three of the hollow marker dots are visible; and (6) setting appropriate subset and step is important for DIC to analysis, and then output the data. At present, 3D-DIC has also been used in measuring dynamic crack propagation. For example, Fan et al. [25] proposed that 3D-DIC is a practical and reliable method to determine crack initiation time and dynamic fracture toughness. Xu et al. [26] applied a linear-shaped jet to a cylindrical concrete mold, and full-field three-dimensional deformation characteristics of the splitting development of the specimen were analyzed by 3D-DIC.From the existing research, the study of rock dynamic fracture response is focused on the fracture toughness parameters and propagation behavior of mode-I crack under the same blasting load. In this paper, a granite rectangular plate with a crack (GRPC) is introduced to investigate crack propagation behavior based on 3D-DIC under different blast loading rates.


## 2. Materials and Methods

### 2.1. Granite Materials

Experimental samples are Wulian Flower Granite (G3761; Wulian County, Rizhao City, Shandong province, China). Polarizing microscope tests (LEICA DM2500 p) are done in Yunenng geological service (Langfang, China) for obtaining the main minerals of granite, as shown in Figure1, and the test parameter is 10 × 2.5 (eyepiece multiples × objective multiples). The main minerals of the rock are plagioclase (Pl; the main grain size is 2–5 mm, some grain size is 0.5–2 mm, and the content is about 25%), potassium feldspar (Kfs; the main grain size is 2–5 mm, and some grain size is 5–10 mm, and the content is about 45%), and quartz (Qtz; the main grain size is 2–5 mm, some grain size is 0.5–2 mm, and the content is about 20%), while the secondary minerals are biotite (Bt; the main grain size is 0.2–2 mm, and the content is about 7%) and amphibole (Hbl; the main grain size is 0.5–2 mm, and the content is about 2%). The auxiliary minerals are magnetite, zircon (Zrn), and apatite (Spn). The altered minerals are sericite, tetrahedrite, and chlorite. Based on the International Society for Rock Mechanics (ISRM) standards, parallelism and perpendicularity of two end planes are smaller than 0.2 mm, cylindrical specimens cut into 50 × 100 mm and 50 × 25 mm for uniaxial static compressive strength tests and static tensile strength tests are cored from the same rock block without obvious joint surface, and the quasi-static properties of granite are tested by the hydraulic screen-display universal testing machine (WEP-600) and in the Laboratory of Civil and Resource Engineering School at University Science and Technology Beijing (Beijing, China). The BX 120–10AA strain gages (sensitive grid size is 10 × 2 mm) are used to test deformation. Five replicate tests are performed for each property; the static mechanical parameters of granite are shown in Table 1, and the P-wave velocity tested for sample 1 by using an acoustic emission detector (PCI-2) is 4386 m/s.Figure 1
Polarizing microscope images. (a) Plane-polarized light and (b) cross-polarized light.
(a)(b)Table 1
The static mechanical parameters of granite.
No.Density (kg/m3)Elastic modulus (GPa)Poisson’s ratioStatic compressive strength (MPa)Static tensile strength (MPa)1258138.250.15784.583.572256820.510.13550.952.553258647.540.16290.362.554258239.140.15252.483.575257331.720.14374.94.08Average257835.430.1570.653.26


### 2.2. Granite Rectangular Plate with a Crack (GRPC)

The GRPC is cut by professional rock cutting tools from a granite cuboid with 400 × 400 × 80 mm; the GRPC contains a mode-I crack and a circular hole, as shown in Figure2. The length of the crack tip with 10 mm is incised by using a diamond wire saw, and the width of the crack tip is approximately 1 mm. The reason for the choice of the crack tip is that the failure should take place only between the initial notch tip and the back-face of the specimen [27], as shown in Figure 3(a). A set of split-type smoke shielding devices is designed to avoid the impact of blasting gas on camera shooting effect (“Front” and “Back” in Figure 1), the cylindrical charge charging with PETN in plastic straw is used to ensure uniform force on the borehole wall, the charge length is selected the same as the thickness of GRPC (80 mm), and the charging mode is chosen as radial decoupling charge (the decoupling coefficients are 3.82, 3.5, and 3.0, respectively), as shown in Table 2. To ensure the cylindrical charge is located at the center of the borehole, the insulating tape is wound around the front and back (position in connection with the detonating cord) of the cylindrical charge. Finally, the detonator detonates the detonating cord, and then the detonating cord detonates the PETN.Figure 2
Schematic drawing of the GRPC and charging method (unit: mm).Figure 3
The failure pattern [27].
(a)(b)(c)Table 2
Charge parameters.
Test no.Charge length (mm)Charge diameter (mm)Charge weight (g)Density of PETN (g/cm3)Detonation velocity (m/s) [28]1805.51.510.79494762.132806.01.980.87585061.463806.52.420.91215195.77According to reference [28], detonation velocity (km/s) = 3.19 + 3.7 ∗Density of PETN (g/cm3)−0.37.


### 2.3. Experimental System

As shown in Figure4, the experimental system of 3D-DIC is set up in the State Key Laboratory of Explosion Science and Technology at the Beijing Institute of Technology (Beijing), and the main components of this experimental system are two high-speed cameras (Photron Fastcam SA5), standard Nikon lens, VIC-3D analysis software, the lighting system (Godox SL-200W) synchronous trigger system, and a computer. The texture of the granite itself is chosen as a speckle pattern [29], the distance between the cameras and the GRPC is 2.0 m, and the shooting angle of the two cameras is 10 degrees. When the detonator detonates, a square signal transmits to the synchronous system, and thus the signal triggers the cameras.Figure 4
Schematic diagram of the experimental system.Calibrating a VIC-3D system is straightforward and has been streamlined into a nearly automatic process. The procedure for the 3D system involves moving, imaging, and analyzing a rigid calibration target in front of a stereo camera pair. This precisely calculates the cameras’ intrinsic and extrinsic parameters while triangulating the positions of the cameras and removing lens distortions. This ultimately removes any measurement bias and defines a three-dimensional coordinate system on the surface of the specimen. The standard calibration target set is provided with each VIC-3D™ system and covers fields of view from 30 mm and above. Each target features an ultramatte finish, a rigid and durable design, and they are specially coded for automatic spacing detection in VIC-3D v7 and later. These targets can be used for calibrating high-speed and low-speed systems and for both high and low-resolution cameras [30]. In this paper, calibration of the stereovision system is performed using 32 images of a translated and rotated planar dot pattern with a spacing of 10 mm (12 × 9–10.0 mm).As shown in Figure5, the camera shooting parameters and postprocess parameters are set as follows: the photograph frames are set as 60,000 μfps (the time interval between the two images is 16.67 μs), the image resolution is set as 320 pixels × 320 pixels, the origin of coordinates is located in the center of the shooting surface (black hand-painted frame), the shooting area is approximately 120 mm × 120 mm, the calculated zone of interest (ZOI) is approximately 100 mm × 110 mm, the line of interest (LOI) locates on the ZOI with the spacing of 10 mm, the reference subset of pixels in the 3D-DIC computation is chosen to be 33 pixels × 33 pixels, and a subset step of 3 pixels is used. 3D-DIC mainly analyzed the strain field in crack propagation but not crack initiation, and 3D-DIC analysis is not affected by blasting gas because the speed of crack propagation is higher than that of blasting gas expansion.Figure 5
Zone of interest (ZOI) and line of interest (LOI).


## 2.1. Granite Materials

Experimental samples are Wulian Flower Granite (G3761; Wulian County, Rizhao City, Shandong province, China). Polarizing microscope tests (LEICA DM2500 p) are done in Yunenng geological service (Langfang, China) for obtaining the main minerals of granite, as shown in Figure1, and the test parameter is 10 × 2.5 (eyepiece multiples × objective multiples). The main minerals of the rock are plagioclase (Pl; the main grain size is 2–5 mm, some grain size is 0.5–2 mm, and the content is about 25%), potassium feldspar (Kfs; the main grain size is 2–5 mm, and some grain size is 5–10 mm, and the content is about 45%), and quartz (Qtz; the main grain size is 2–5 mm, some grain size is 0.5–2 mm, and the content is about 20%), while the secondary minerals are biotite (Bt; the main grain size is 0.2–2 mm, and the content is about 7%) and amphibole (Hbl; the main grain size is 0.5–2 mm, and the content is about 2%). The auxiliary minerals are magnetite, zircon (Zrn), and apatite (Spn). The altered minerals are sericite, tetrahedrite, and chlorite. Based on the International Society for Rock Mechanics (ISRM) standards, parallelism and perpendicularity of two end planes are smaller than 0.2 mm, cylindrical specimens cut into 50 × 100 mm and 50 × 25 mm for uniaxial static compressive strength tests and static tensile strength tests are cored from the same rock block without obvious joint surface, and the quasi-static properties of granite are tested by the hydraulic screen-display universal testing machine (WEP-600) and in the Laboratory of Civil and Resource Engineering School at University Science and Technology Beijing (Beijing, China). The BX 120–10AA strain gages (sensitive grid size is 10 × 2 mm) are used to test deformation. Five replicate tests are performed for each property; the static mechanical parameters of granite are shown in Table 1, and the P-wave velocity tested for sample 1 by using an acoustic emission detector (PCI-2) is 4386 m/s.Figure 1
Polarizing microscope images. (a) Plane-polarized light and (b) cross-polarized light.
(a)(b)Table 1
The static mechanical parameters of granite.
No.Density (kg/m3)Elastic modulus (GPa)Poisson’s ratioStatic compressive strength (MPa)Static tensile strength (MPa)1258138.250.15784.583.572256820.510.13550.952.553258647.540.16290.362.554258239.140.15252.483.575257331.720.14374.94.08Average257835.430.1570.653.26


## 2.2. Granite Rectangular Plate with a Crack (GRPC)

The GRPC is cut by professional rock cutting tools from a granite cuboid with 400 × 400 × 80 mm; the GRPC contains a mode-I crack and a circular hole, as shown in Figure2. The length of the crack tip with 10 mm is incised by using a diamond wire saw, and the width of the crack tip is approximately 1 mm. The reason for the choice of the crack tip is that the failure should take place only between the initial notch tip and the back-face of the specimen [27], as shown in Figure 3(a). A set of split-type smoke shielding devices is designed to avoid the impact of blasting gas on camera shooting effect (“Front” and “Back” in Figure 1), the cylindrical charge charging with PETN in plastic straw is used to ensure uniform force on the borehole wall, the charge length is selected the same as the thickness of GRPC (80 mm), and the charging mode is chosen as radial decoupling charge (the decoupling coefficients are 3.82, 3.5, and 3.0, respectively), as shown in Table 2. To ensure the cylindrical charge is located at the center of the borehole, the insulating tape is wound around the front and back (position in connection with the detonating cord) of the cylindrical charge. Finally, the detonator detonates the detonating cord, and then the detonating cord detonates the PETN.Figure 2
Schematic drawing of the GRPC and charging method (unit: mm).Figure 3
The failure pattern [27].
(a)(b)(c)Table 2
Charge parameters.
Test no.Charge length (mm)Charge diameter (mm)Charge weight (g)Density of PETN (g/cm3)Detonation velocity (m/s) [28]1805.51.510.79494762.132806.01.980.87585061.463806.52.420.91215195.77According to reference [28], detonation velocity (km/s) = 3.19 + 3.7 ∗Density of PETN (g/cm3)−0.37.


## 2.3. Experimental System

As shown in Figure4, the experimental system of 3D-DIC is set up in the State Key Laboratory of Explosion Science and Technology at the Beijing Institute of Technology (Beijing), and the main components of this experimental system are two high-speed cameras (Photron Fastcam SA5), standard Nikon lens, VIC-3D analysis software, the lighting system (Godox SL-200W) synchronous trigger system, and a computer. The texture of the granite itself is chosen as a speckle pattern [29], the distance between the cameras and the GRPC is 2.0 m, and the shooting angle of the two cameras is 10 degrees. When the detonator detonates, a square signal transmits to the synchronous system, and thus the signal triggers the cameras.Figure 4
Schematic diagram of the experimental system.Calibrating a VIC-3D system is straightforward and has been streamlined into a nearly automatic process. The procedure for the 3D system involves moving, imaging, and analyzing a rigid calibration target in front of a stereo camera pair. This precisely calculates the cameras’ intrinsic and extrinsic parameters while triangulating the positions of the cameras and removing lens distortions. This ultimately removes any measurement bias and defines a three-dimensional coordinate system on the surface of the specimen. The standard calibration target set is provided with each VIC-3D™ system and covers fields of view from 30 mm and above. Each target features an ultramatte finish, a rigid and durable design, and they are specially coded for automatic spacing detection in VIC-3D v7 and later. These targets can be used for calibrating high-speed and low-speed systems and for both high and low-resolution cameras [30]. In this paper, calibration of the stereovision system is performed using 32 images of a translated and rotated planar dot pattern with a spacing of 10 mm (12 × 9–10.0 mm).As shown in Figure5, the camera shooting parameters and postprocess parameters are set as follows: the photograph frames are set as 60,000 μfps (the time interval between the two images is 16.67 μs), the image resolution is set as 320 pixels × 320 pixels, the origin of coordinates is located in the center of the shooting surface (black hand-painted frame), the shooting area is approximately 120 mm × 120 mm, the calculated zone of interest (ZOI) is approximately 100 mm × 110 mm, the line of interest (LOI) locates on the ZOI with the spacing of 10 mm, the reference subset of pixels in the 3D-DIC computation is chosen to be 33 pixels × 33 pixels, and a subset step of 3 pixels is used. 3D-DIC mainly analyzed the strain field in crack propagation but not crack initiation, and 3D-DIC analysis is not affected by blasting gas because the speed of crack propagation is higher than that of blasting gas expansion.Figure 5
Zone of interest (ZOI) and line of interest (LOI).


## 3. Experimental Results

### 3.1. Full-Field Strain and Crack Evolution Process

Figure6 shows that the strain concentration zone has been beginning to behave obvious tendency at 2333 μs, the location of the crack tip is closest to the front end of the maximum strain concentration zone at 2383 μs, soon afterward, the location of the crack tip surpasses gradually the front end to the maximum strain concentration zone, and the crack tip locates approximately −10 mm on the Y-axis from the high-speed photographic image at 2433 μs, and at this time of the front end of the maximum strain, the concentration zone locates approximately 13 mm on the Y-axis. Except for the numerical differences, the full-field strain of the axial strain and the principal strain is almost the same in terms of transformation pattern. Through observing high-speed photographic images, the crack tip gradually extends and propagates before 2433 μs, and the location of the crack tip can be identified, especially at 2433 μs (point C in Figure 7(a)); after that, the location of the crack tip is relatively difficult to identify because subsequent cracks appear uniformly as a whole. Furthermore, the maximum displacement value corresponding to the maximum strain concentration zone at 2433 μs is 0.152 mm on the Z-axis.Figure 6
Crack evolution and full-field strain at different stages for test 1. (a) Crack evolution, (b) axial strain, and (c) principal strain.
(a)(b)(c)Figure 7
Actual crack and fracture appearance after the explosion. (a) Actual crack in the shooting area for test 1, (b) actual crack in the shooting area for test 2, (c) actual crack in the shooting area for test 3, (d) actual crack, and (e) fracture appearance.
(a)(b)(c)(d)(e)Figure8 shows that the strain concentration zone behaves obvious tendency as the crack propagates, the location of the crack tip is closest to the front end of the maximum strain concentration zone at 1800 μs, soon afterward, the location of the crack tip surpasses gradually the front end to the maximum strain concentration zone, and the crack tip locates approximately −12 mm on the Y-axis from high-speed photographic image at 1867 μs, and from this time on the front end of the maximum strain, the concentration zone locates approximately 7 mm on the Y-axis. Through observing the high-speed photographic images, the crack tip gradually extends and propagates before 1867 μs, and this time of the location of the crack tip is a turning point; it can be identified at 1867 μs (point C in Figure 7(c)), and after that, the subsequent cracks appear uniformly as a whole. Furthermore, the maximum displacement value corresponding to the maximum strain concentration zone at 1950 μs is 0.2 mm on the Z-axis.Figure 8
(a) Crack evolution and (b) full-field axial strain at different stages for test 3.
(a)(b)Figures6 and 8 present the crack evolution and the full-field strain (axial strain and principal strain) at different stages for test 1 and test 3, and the two groups of tests showed the following common characteristics: (1) the strain concentration zone is formed at the crack propagation location; (2) the maximum value of strain concentration zone (the red zone) does not fully represent crack tip for the whole process of crack propagation; (3) there are inflection points in the crack propagation; and (4) the full-field strain of the axial strain and the principal strain is almost the same in terms of transformation pattern except for the numerical differences.As shown in Figures7(a) and 7(c), the length of the crack from O to A is all 20 mm, and the length of the crack from O to E is 120 mm and 124 mm, respectively. It is assumed that the right deflection of the crack is positive and the left deflection is negative, and the crack deflection angle during crack propagation is described in the figure. It can be seen in Figure 7(e) that fracture appearance is flat for test 2 and test 3.


### 3.2. Axial Strain on the LOI

Figure9 and 10 present the axial strain on the line of interest at different times for test 1 and test 3. From the peak position in the graphs, the maximum value position of the strain concentration zone corresponding to the X-axis value can be obtained and the X-axis value of crack position is also obtained. If the LOI line is used as the monitoring line and the crack is considered to be a linear scale, then the crack path can be described. For example, for test 1, the X-axis values are 3.66 mm (LOI 1), 5.44 mm (LOI 2), 6.92 mm (LOI 3), 6.75 mm (LOI 4), 5.84 mm (LOI 5), 3.73 mm (LOI 6), 2.82 mm (LOI 7), and 7.95 mm (LOI 8), respectively; for test 3, the X-axis values are 1.83 mm (LOI 2), −1.04 mm (LOI 3), −0.98 mm (LOI 4), −1.14 mm (LOI 5), −5.78 mm (LOI 6), −10.87 mm (LOI 7), and −14.49 mm (LOI 8), respectively. Then, the location of the crack can be known by incorporating the X-axis values measured by LOI and the Y-axis values of LOI, and the crack path can be obtained by connecting these coordinate points. Therefore, the shorter the LOI spacing is, the more accurate the data likely to be are.Figure 9
Axial strain on the line of interest at a different time for test 1. (a) LOI 1, (b) LOI 2, (c) LOI 3, (d) LOI 4, (e) LOI 5, (f) LOI 6, (g) LOI 7, and (h) LOI 8.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Axial strain on the line of interest at a different time for test 3. (a) LOI 1 (abnormal data points are eliminated), (b) LOI 2, (c) LOI 3, (d) LOI 4, (e) LOI 5, (f) LOI 6, (g) LOI 7, and (h) LOI 8.
(a)(b)(c)(d)(e)(f)(g)(h)Furthermore, the LOI 6 locates at the inflection point (point D in Figure7(a) and point C in Figure 7(c)) in test 1 and test 3. For test 1, the axial strain value at LOI 6 is 0.0154 at 2450 μs, and the axial strain values on the LOI 5 is 0.0139 at 2450 μs. According to Figure 6(b), the front end of the maximum strain concentration zones matches approximately the crack tip at 2383 μs, and the maximum value of the axial strain is 0.0143. On the contrary, point C in Figure 7(a) locates approximately within LOI 5, and the crack tip propagates approximately to point C at 2433 μs. The axial strain value at LOI 5 is 0.0109 at 2433 μs. According to Figures 9(e) and 9(f), the axial strain values on LOI 6 is 0.0102 at 2417 μs, and the axial strain values on LOI 5 is 0.0084 at 2417 μs. For test 3, the axial strain value on LOI 7 is 0.0155 at 1933 μs, and the axial strain values on LOI 6 is 0.0125 at 1933 μs. Similar to the analysis above, the axial strain value in Figure 10(b) is 0.0146 at 1800 μs. Something like this, it can be identified that the crack occurs preferentially on LOI 6 in test 1 and on LOI 7 in test 3.


### 3.3. Crack Velocity

It has been known that the maximum value of the strain concentration zone does not fully represent the crack tip during the whole process of crack propagation, and the critical values of axial strain are 0.0143, 0.0146, and 0.0109 when the maximum axial strain concentration zone matches approximately the crack tip. Therefore, 1.0%, 1.1%, 1.2%, 1.3%, and 1.4% of axial strain is chosen as a threshold value to detect the axial strain threshold value tip (ASTVT) to measure the position of the crack tip and crack velocity. The evolutionary processes of axial strain for different strain threshold values for test 1 and test 3 are shown in Figures11 and 12. It can be observed that the length of ASTVT and the width of the axial strain concentration zone increase with time when the axial strain threshold value is fixed, and the length of ASTVT and the width of axial strain concentration zone decrease as the axial strain threshold value increases.Figure 11
Evolutionary process of axial strain for different strain threshold values for test 1. (a) 1.0%, (b) 1.1%, (c) 1.2%, (d) 1.3%, and (e) 1.4%.
(a)(b)(c)(d)(e)Figure 12
Evolutionary process of axial strain for different strain threshold values for test 3. (a) 1.0%, (b) 1.1%, (c) 1.2%, (d) 1.3%, and (e) 1.4%.
(a)(b)(c)(d)(e)ASTVT plotting with lines and coordinate contours corresponding with the actual crack in the shooting area can be used to describe the position of the crack, as shown in Figure13. It can be found that ASTVT plotting with lines in test 3 matches well with the actual crack position, and the trajectory of ASTVT is consistent with the crack morphology for test 1 and test 3. In addition, through observing the coordinate contours corresponding with the actual crack (the line of the red arrow representing the Y-axis), it can also be found that ASTVT plotting with lines in test 1 has an error of 2 mm with the actual crack position, which is maybe due to a mismatch between the coordinates established by the camera and those set by the human (black block diagram in Figure 5). In a word, it can be affirmed that ASTVT plotting with lines is helpful to determine the actual crack position.Figure 13
The position of axial strain threshold value tip. (a) Test 1 and (b) test 3.
(a)(b)The coordinates of the crack tip (visible method) or ASTVT at timeti is (xi, yi, zi), and within the schedule Δti = ti−ti−1, the crack velocity vi can be approximately regarded as the instantaneous velocity when the crack grows to the measuring point ci. It satisfies the following equation:(1)si=xi−xi−12+yi−yi−12+zi−zi−12,(2)vi=siΔti,(3)vavg=∑i>2vii−1,where si is the crack tip or ASTVT displacement and vavg is the average crack velocity.The crack velocity is 400 m/s, 500 m/s, and 410 m/s from point A to point E in Figure7(a), and the crack velocity is 484.8 m/s, 882.4 m/s, and 518.1 m/s from point A to point D in Figure 7(c). The average crack velocity is 436.7 m/s for test 1 and 628.4 m/s for test 3. As shown in Figure 14, the crack velocity experiences a peak process in both visible method and ASTVT. It can also be seen that the peak value of ASTVT mismatches the visible method for test 1, the peak value of ASTVT matches the visible method when the axial strain threshold value is 1.3% for test 3, and the horizontal stage of crack velocity after the peak is all in the vicinity of 200 m/s by using ASTVT to measure. Through calculating the average crack velocity, the following can be obtained: the average crack velocity is 400.8 m/s (1.0%), 433.8 m/s (1.1%), 469 m/s (1.2%), 438.6 m/s (1.3%), and 481.6 m/s (1.4%) for test 1, and the average crack velocity is 293.8 m/s (1.0%), 310 m/s (1.1%), 359.6 m/s (1.2%), 332 m/s (1.3%), and 307.2 m/s (1.4%) for test 3. It is found that the average crack velocity matches both visible method and ASTVT when the axial strain threshold values are 1.1% and 1.3% for test 1. Combining the position of crack and ASTVT at 2433 μs, 1.3% is more appropriate.Figure 14
Crack velocity. (a) Test 1 and (b) test 3.
(a)(b)


### 3.4. Evolution of Horizontal Displacement Field and Crack Width

The crack propagation process is characterized by the continuous evolution of the measured surface displacement field from disorder to order and the continuous displacement field concentration in the crack area, as shown in Figure15.Figure 15
Evolution of horizontal displacement field. (a) Test 1 and (b) test 3.
(a)(b)Wu et al. [31] proposed the method to determine the crack opening displacement; in this paper, the crack opening displacement along LOI 4 is calculated to avoid the effects of blasting gas. It satisfies the following equation:(4)W=UR−UQ,where W is the crack width; U is the horizontal displacement, and the points R and Q in Figures 16(a) and 16(b) are chosen according to the method of Wu et al. [31].Figure 16
Displacements along LOI 4. (a) Test 1, (b) test 3, and (c) crack width.
(a)(b)(c)As shown in Figure16(c), through observing the relationship between crack width and time, it can be found that there are three stages, i.e., initial stage (under line A), abrupt stage (rectangle B) or delayed stage (rectangle C), and linear stage, and the crack width of test 3 is larger than test 1. The linear stage coincides with the crack velocity (1.3%) rising phase in time. The average crack velocity of test 3 is smaller than test 1 according to Figure 14, and the width of the axial strain field of test 3 is larger than test 1 according to Figures 11 and 12. Using the equation of state of ideal gases to evaluate of borehole wall pressure [32] and the ratio of charge length to detonation velocity to estimate peak pressure rise time, the blast loading rates are obtained that 40.48 MPa/16.8 μs = 2.41 MPa/μs for test 1 and 91.27 MPa/15.4 μs = 5.93 MPa/μs for test 3. Then it can be demonstrated that, the crack width increases as the blast loading rates increase, and the average crack velocity decreases as the blast loading rates increase.


### 3.5. Fracture Process Zone

The fracture process zone (FPZ) in rocks is defined as the region affected by microcracking and frictional slip surrounding the visible crack tip propagating under stress [33]. Strain or displacement can be used as the defining standard of FPZ region [31]. If the axial strain threshold value (1.0%, 1.1%, 1.2%, 1.3%, and 1.4%) is used as the defining standard of the FPZ region, it can be found that the width of FPZ (axial strain concentration zone) increases with time when the axial strain threshold value is fixed. Then the width of FPZ is the same on the same X coordinate after 2450 μs for test 1 and 1867 μs for test 3, and the width of FPZ is also basically the same on the different Y coordinate except ASTVT. For example, the width of FPZ is approximately 18 mm (1.0%) and 15 mm (1.4%) at 2500 μs for test 1. If displacement is used as the defining standard of FPZ region, the X coordinate of point Q and point R is different according to Figures 15(a) and 15(b) (XR–XQ < 20 mm for test 1, and XR−XQ > 20 mm for test 3, where XR is the X coordinate of point R and XQ is the X coordinate of point Q). Wu [31] has demonstrated that the final macrocrack trajectory is in good agreement with the propagation path of the FPZ determined by the DIC technique; therefore, it can be demonstrated that the width of FPZ increases as the blast loading rates increase and FPZ generates along the trajectory of crack.


## 3.1. Full-Field Strain and Crack Evolution Process

Figure6 shows that the strain concentration zone has been beginning to behave obvious tendency at 2333 μs, the location of the crack tip is closest to the front end of the maximum strain concentration zone at 2383 μs, soon afterward, the location of the crack tip surpasses gradually the front end to the maximum strain concentration zone, and the crack tip locates approximately −10 mm on the Y-axis from the high-speed photographic image at 2433 μs, and at this time of the front end of the maximum strain, the concentration zone locates approximately 13 mm on the Y-axis. Except for the numerical differences, the full-field strain of the axial strain and the principal strain is almost the same in terms of transformation pattern. Through observing high-speed photographic images, the crack tip gradually extends and propagates before 2433 μs, and the location of the crack tip can be identified, especially at 2433 μs (point C in Figure 7(a)); after that, the location of the crack tip is relatively difficult to identify because subsequent cracks appear uniformly as a whole. Furthermore, the maximum displacement value corresponding to the maximum strain concentration zone at 2433 μs is 0.152 mm on the Z-axis.Figure 6
Crack evolution and full-field strain at different stages for test 1. (a) Crack evolution, (b) axial strain, and (c) principal strain.
(a)(b)(c)Figure 7
Actual crack and fracture appearance after the explosion. (a) Actual crack in the shooting area for test 1, (b) actual crack in the shooting area for test 2, (c) actual crack in the shooting area for test 3, (d) actual crack, and (e) fracture appearance.
(a)(b)(c)(d)(e)Figure8 shows that the strain concentration zone behaves obvious tendency as the crack propagates, the location of the crack tip is closest to the front end of the maximum strain concentration zone at 1800 μs, soon afterward, the location of the crack tip surpasses gradually the front end to the maximum strain concentration zone, and the crack tip locates approximately −12 mm on the Y-axis from high-speed photographic image at 1867 μs, and from this time on the front end of the maximum strain, the concentration zone locates approximately 7 mm on the Y-axis. Through observing the high-speed photographic images, the crack tip gradually extends and propagates before 1867 μs, and this time of the location of the crack tip is a turning point; it can be identified at 1867 μs (point C in Figure 7(c)), and after that, the subsequent cracks appear uniformly as a whole. Furthermore, the maximum displacement value corresponding to the maximum strain concentration zone at 1950 μs is 0.2 mm on the Z-axis.Figure 8
(a) Crack evolution and (b) full-field axial strain at different stages for test 3.
(a)(b)Figures6 and 8 present the crack evolution and the full-field strain (axial strain and principal strain) at different stages for test 1 and test 3, and the two groups of tests showed the following common characteristics: (1) the strain concentration zone is formed at the crack propagation location; (2) the maximum value of strain concentration zone (the red zone) does not fully represent crack tip for the whole process of crack propagation; (3) there are inflection points in the crack propagation; and (4) the full-field strain of the axial strain and the principal strain is almost the same in terms of transformation pattern except for the numerical differences.As shown in Figures7(a) and 7(c), the length of the crack from O to A is all 20 mm, and the length of the crack from O to E is 120 mm and 124 mm, respectively. It is assumed that the right deflection of the crack is positive and the left deflection is negative, and the crack deflection angle during crack propagation is described in the figure. It can be seen in Figure 7(e) that fracture appearance is flat for test 2 and test 3.


## 3.2. Axial Strain on the LOI

Figure9 and 10 present the axial strain on the line of interest at different times for test 1 and test 3. From the peak position in the graphs, the maximum value position of the strain concentration zone corresponding to the X-axis value can be obtained and the X-axis value of crack position is also obtained. If the LOI line is used as the monitoring line and the crack is considered to be a linear scale, then the crack path can be described. For example, for test 1, the X-axis values are 3.66 mm (LOI 1), 5.44 mm (LOI 2), 6.92 mm (LOI 3), 6.75 mm (LOI 4), 5.84 mm (LOI 5), 3.73 mm (LOI 6), 2.82 mm (LOI 7), and 7.95 mm (LOI 8), respectively; for test 3, the X-axis values are 1.83 mm (LOI 2), −1.04 mm (LOI 3), −0.98 mm (LOI 4), −1.14 mm (LOI 5), −5.78 mm (LOI 6), −10.87 mm (LOI 7), and −14.49 mm (LOI 8), respectively. Then, the location of the crack can be known by incorporating the X-axis values measured by LOI and the Y-axis values of LOI, and the crack path can be obtained by connecting these coordinate points. Therefore, the shorter the LOI spacing is, the more accurate the data likely to be are.Figure 9
Axial strain on the line of interest at a different time for test 1. (a) LOI 1, (b) LOI 2, (c) LOI 3, (d) LOI 4, (e) LOI 5, (f) LOI 6, (g) LOI 7, and (h) LOI 8.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Axial strain on the line of interest at a different time for test 3. (a) LOI 1 (abnormal data points are eliminated), (b) LOI 2, (c) LOI 3, (d) LOI 4, (e) LOI 5, (f) LOI 6, (g) LOI 7, and (h) LOI 8.
(a)(b)(c)(d)(e)(f)(g)(h)Furthermore, the LOI 6 locates at the inflection point (point D in Figure7(a) and point C in Figure 7(c)) in test 1 and test 3. For test 1, the axial strain value at LOI 6 is 0.0154 at 2450 μs, and the axial strain values on the LOI 5 is 0.0139 at 2450 μs. According to Figure 6(b), the front end of the maximum strain concentration zones matches approximately the crack tip at 2383 μs, and the maximum value of the axial strain is 0.0143. On the contrary, point C in Figure 7(a) locates approximately within LOI 5, and the crack tip propagates approximately to point C at 2433 μs. The axial strain value at LOI 5 is 0.0109 at 2433 μs. According to Figures 9(e) and 9(f), the axial strain values on LOI 6 is 0.0102 at 2417 μs, and the axial strain values on LOI 5 is 0.0084 at 2417 μs. For test 3, the axial strain value on LOI 7 is 0.0155 at 1933 μs, and the axial strain values on LOI 6 is 0.0125 at 1933 μs. Similar to the analysis above, the axial strain value in Figure 10(b) is 0.0146 at 1800 μs. Something like this, it can be identified that the crack occurs preferentially on LOI 6 in test 1 and on LOI 7 in test 3.


## 3.3. Crack Velocity

It has been known that the maximum value of the strain concentration zone does not fully represent the crack tip during the whole process of crack propagation, and the critical values of axial strain are 0.0143, 0.0146, and 0.0109 when the maximum axial strain concentration zone matches approximately the crack tip. Therefore, 1.0%, 1.1%, 1.2%, 1.3%, and 1.4% of axial strain is chosen as a threshold value to detect the axial strain threshold value tip (ASTVT) to measure the position of the crack tip and crack velocity. The evolutionary processes of axial strain for different strain threshold values for test 1 and test 3 are shown in Figures11 and 12. It can be observed that the length of ASTVT and the width of the axial strain concentration zone increase with time when the axial strain threshold value is fixed, and the length of ASTVT and the width of axial strain concentration zone decrease as the axial strain threshold value increases.Figure 11
Evolutionary process of axial strain for different strain threshold values for test 1. (a) 1.0%, (b) 1.1%, (c) 1.2%, (d) 1.3%, and (e) 1.4%.
(a)(b)(c)(d)(e)Figure 12
Evolutionary process of axial strain for different strain threshold values for test 3. (a) 1.0%, (b) 1.1%, (c) 1.2%, (d) 1.3%, and (e) 1.4%.
(a)(b)(c)(d)(e)ASTVT plotting with lines and coordinate contours corresponding with the actual crack in the shooting area can be used to describe the position of the crack, as shown in Figure13. It can be found that ASTVT plotting with lines in test 3 matches well with the actual crack position, and the trajectory of ASTVT is consistent with the crack morphology for test 1 and test 3. In addition, through observing the coordinate contours corresponding with the actual crack (the line of the red arrow representing the Y-axis), it can also be found that ASTVT plotting with lines in test 1 has an error of 2 mm with the actual crack position, which is maybe due to a mismatch between the coordinates established by the camera and those set by the human (black block diagram in Figure 5). In a word, it can be affirmed that ASTVT plotting with lines is helpful to determine the actual crack position.Figure 13
The position of axial strain threshold value tip. (a) Test 1 and (b) test 3.
(a)(b)The coordinates of the crack tip (visible method) or ASTVT at timeti is (xi, yi, zi), and within the schedule Δti = ti−ti−1, the crack velocity vi can be approximately regarded as the instantaneous velocity when the crack grows to the measuring point ci. It satisfies the following equation:(1)si=xi−xi−12+yi−yi−12+zi−zi−12,(2)vi=siΔti,(3)vavg=∑i>2vii−1,where si is the crack tip or ASTVT displacement and vavg is the average crack velocity.The crack velocity is 400 m/s, 500 m/s, and 410 m/s from point A to point E in Figure7(a), and the crack velocity is 484.8 m/s, 882.4 m/s, and 518.1 m/s from point A to point D in Figure 7(c). The average crack velocity is 436.7 m/s for test 1 and 628.4 m/s for test 3. As shown in Figure 14, the crack velocity experiences a peak process in both visible method and ASTVT. It can also be seen that the peak value of ASTVT mismatches the visible method for test 1, the peak value of ASTVT matches the visible method when the axial strain threshold value is 1.3% for test 3, and the horizontal stage of crack velocity after the peak is all in the vicinity of 200 m/s by using ASTVT to measure. Through calculating the average crack velocity, the following can be obtained: the average crack velocity is 400.8 m/s (1.0%), 433.8 m/s (1.1%), 469 m/s (1.2%), 438.6 m/s (1.3%), and 481.6 m/s (1.4%) for test 1, and the average crack velocity is 293.8 m/s (1.0%), 310 m/s (1.1%), 359.6 m/s (1.2%), 332 m/s (1.3%), and 307.2 m/s (1.4%) for test 3. It is found that the average crack velocity matches both visible method and ASTVT when the axial strain threshold values are 1.1% and 1.3% for test 1. Combining the position of crack and ASTVT at 2433 μs, 1.3% is more appropriate.Figure 14
Crack velocity. (a) Test 1 and (b) test 3.
(a)(b)


## 3.4. Evolution of Horizontal Displacement Field and Crack Width

The crack propagation process is characterized by the continuous evolution of the measured surface displacement field from disorder to order and the continuous displacement field concentration in the crack area, as shown in Figure15.Figure 15
Evolution of horizontal displacement field. (a) Test 1 and (b) test 3.
(a)(b)Wu et al. [31] proposed the method to determine the crack opening displacement; in this paper, the crack opening displacement along LOI 4 is calculated to avoid the effects of blasting gas. It satisfies the following equation:(4)W=UR−UQ,where W is the crack width; U is the horizontal displacement, and the points R and Q in Figures 16(a) and 16(b) are chosen according to the method of Wu et al. [31].Figure 16
Displacements along LOI 4. (a) Test 1, (b) test 3, and (c) crack width.
(a)(b)(c)As shown in Figure16(c), through observing the relationship between crack width and time, it can be found that there are three stages, i.e., initial stage (under line A), abrupt stage (rectangle B) or delayed stage (rectangle C), and linear stage, and the crack width of test 3 is larger than test 1. The linear stage coincides with the crack velocity (1.3%) rising phase in time. The average crack velocity of test 3 is smaller than test 1 according to Figure 14, and the width of the axial strain field of test 3 is larger than test 1 according to Figures 11 and 12. Using the equation of state of ideal gases to evaluate of borehole wall pressure [32] and the ratio of charge length to detonation velocity to estimate peak pressure rise time, the blast loading rates are obtained that 40.48 MPa/16.8 μs = 2.41 MPa/μs for test 1 and 91.27 MPa/15.4 μs = 5.93 MPa/μs for test 3. Then it can be demonstrated that, the crack width increases as the blast loading rates increase, and the average crack velocity decreases as the blast loading rates increase.


## 3.5. Fracture Process Zone

The fracture process zone (FPZ) in rocks is defined as the region affected by microcracking and frictional slip surrounding the visible crack tip propagating under stress [33]. Strain or displacement can be used as the defining standard of FPZ region [31]. If the axial strain threshold value (1.0%, 1.1%, 1.2%, 1.3%, and 1.4%) is used as the defining standard of the FPZ region, it can be found that the width of FPZ (axial strain concentration zone) increases with time when the axial strain threshold value is fixed. Then the width of FPZ is the same on the same X coordinate after 2450 μs for test 1 and 1867 μs for test 3, and the width of FPZ is also basically the same on the different Y coordinate except ASTVT. For example, the width of FPZ is approximately 18 mm (1.0%) and 15 mm (1.4%) at 2500 μs for test 1. If displacement is used as the defining standard of FPZ region, the X coordinate of point Q and point R is different according to Figures 15(a) and 15(b) (XR–XQ < 20 mm for test 1, and XR−XQ > 20 mm for test 3, where XR is the X coordinate of point R and XQ is the X coordinate of point Q). Wu [31] has demonstrated that the final macrocrack trajectory is in good agreement with the propagation path of the FPZ determined by the DIC technique; therefore, it can be demonstrated that the width of FPZ increases as the blast loading rates increase and FPZ generates along the trajectory of crack.


## 4. Conclusions

In this study, the GRPC is introduced to investigate the crack propagation characteristics under different blast loading rates, and an experimental system of 3D-DIC is set up to eliminate a certain extent of out-of-plane motion for accurate measuring the full-field strain field during crack propagation. The main conclusions obtained from the study are as follows:(1)
There is a moment that the maximum value of strain concentration zone matches the visible crack tip; however, with the passage of time, the maximum value of the strain concentration zone does not fully represent the crack tip during the whole process of crack propagation. And then the location of the crack can be described by incorporating theX-axis values and the Y-axis values measured from the peak value of the axial strain in the LOI.(2)
The length of ASTVT and the width of the axial strain concentration zone increase with time when the axial strain threshold value is fixed, and the length of ASTVT and the width of the axial strain concentration zone decrease as the axial strain threshold value increases.(3)
ASTVT plotting with lines and coordinate contours corresponding with the actual crack in the shooting area can be used to describe the position of the crack. The trajectory of ASTVT is consistent with the crack morphology.(4)
The different strain threshold value and the visual method calculating average crack velocity are compared; in this study, the axial strain 1.3% is more practical to obtain crack velocity and average crack velocity, and the average crack velocity decreases as the blast loading rates increase.(5)
Through observing the relationship between crack width and time, it can be found that there are three stages which are initial stage, abrupt stage or delayed stage, and linear stage, and the crack width increases as the blast loading rates increase.



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*Source: 8885582-2020-10-24.xml*