Propagation and Coalescence of Blast-Induced Cracks in PMMA Material Containing an Empty Circular Hole Under Delayed Ignition Blasting Load by Using the Dynamic Caustic Method

Zhongwen Yue, Yao Song, Zihang Hu and Yanlong Lu

(School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing 100083, China)

Abstract: In this paper, dynamic caustic method is applied to analyze the blast-induced crack propagation and distribution of the dynamic stress field around an empty circular hole in polymethyl methacrylate (PMMA) material under delayed ignition blasting loads. The following experimental results are obtained. ① In directional-fracture-controlled blasting, the dynamic stress intensity factors (DSIFs) and the propagation paths of the blast-induced cracks are obviously influenced by the delayed ignition. ② The circular hole situated between the two boreholes poses a strong guiding effect on the coelesence of the cracks, causing them to propagate towards each other when cracks are reaching the circular hole area. ③ Blast-induced cracks are not initiated preferentially because of the superimposed effect from the explosive stress waves on the cracking area. ④ By using the scanning electron microscopy (SEM) method, it is verified that the roughness of crack surfaces changes along the crack propagation paths.

Key words: crack propagation and coalescence; dynamic caustic method; delayed ignition; blast-induced cracks; dynamic stress intensity factor (DSIF)

Directional-fracture-controlled blasting technology has been widely used for smooth blasting and shock absorption inroadway excavation, road construction, hydropower engineering, and tunnel engineering. There are many factors affecting the propagation and coalescence of blast-induced cracks. In terms of the ignition time, for example, Simha[1]studied the stress wave interference between boreholes from delayed ignition using a dynamic photo elastic method. Zhu[2]studied the initiation, growth and coalescence of cracks between boreholes employing the dynamic photo elastic method. Zhang[3]studied the use of a method to calculate the rock blasting fragmentation time employing fractal theory. Tose[4]reviewed different criteria and methodologies used in mining operations. Tests conducted by Rorke[5]showed clear trends of improved fragmentation, with harder and more brittle materials being more responsive to very short ignition delays. Xie[6]investigated the crack coalescence mechanism in a coal seam under a delayed blasting load in a similar simulation. Most work in the literature on delayed blasting has focused on the damping effect, whereas there have been relatively few studies on crack propagation and coalescence under a delayed blasting load.In the study of the empty-hole effect, Mohanty[7-8]first introduced that an empty hole was situated between two boreholes for the purpose of controlling the directions of blast-induced crack paths. Nakamura[9]evaluated the effects of an empty hole and that with double notches on blast-induced crack propagation. Cho[10]analyzed the directional fracture in rock material with an empty hole by using numerical and experimental methods.

Among available experimental techniques for the analysis of stress, the optical method of caustics, owing to the simple pattern that it generates and its high sensitivity to strain gradients, has been one of the most effective approaches to determine the relationship between the characteristic parameters of singular stress fields and the fracture properties of materials[11-12]. In recent years, Yao et al.[13- 14]revealed the variations of fracture behaviors in polymethyl methacrylate (PMMA) material with offset-parallel cracks and graded material by using the dynamic caustic method, respectively. Yue[15-17]studied blast-induced crack propagation under simultaneous ignition by using the dynamic caustic method and compared the caustic method and strain gage method in the determination of fracture parameters. Liu et al.[18]employed the caustic method to investigate the stress intensity factor of cylindrical shell in PMMA material.

Fig.1 Schematic diagram of the dynamic digital laser caustic system

However, there is no published paper until now dealing with crack propagation and coalescence between boreholes with simultaneous existence of an empty hole and delayed ignition time. This paper employed a dynamic digital laser caustic system to investigate the blast-induced crack propagation and coalescence as well as the distribution of the dynamic stress field around an empty circular hole situated between two notched boreholes in PMMA specimens under delayed ignition blasting load. The caustic spots at the tips of blast-induced cracks were clearly recorded by a high-speed camera used in the testing system. Therefore, the crack propagation velocities at each moment were then calculated and the variations of dynamic stress intensity factors (DSIFs) around the crack tips with respect to the time were also obtained by using the dynamic caustic method. In addition, the stress differences at the vicinity of the empty circular hole were given to illustrate the stress field distribution there as well.

1 Experiment of Dynamic Caustics

1.1 Experimental system

A schematic diagram of the dynamic digital laser caustic system is shown in Fig.1. A photograph of the system is shown in Fig.2. The system consists of a laser generator, a beam expander, two field lenses, a loading device, a high-speed camera (Fastcam-SA5-16G: Photron, Japan), a multipoint pulse ignitor and a computer. The green laser light with the wave length of 532 nm was adopted in the experiment and the maximum power of the laser generator could reach 300 mW. The diameter and the focal length of the field lens are 300 mm and 900 mm, respectively. The high-speed camera with the highest frame rate of 106frame/s was used to record the dynamic caustic spots and the propagation process of the crack. The multipoint pulse ignitor was connected to the specimen through copper wires. With the polished tips of the copper wires plugged into the notched boreholes, the lead azide explosive loaded in the boreholes was ignited as the multipoint pulse ignitor reached the trigger voltage.

Fig.2 Photograph of the dynamic digital laser caustic system

1.2 Experimental details

PMMA was employed as the specimen material in this experiment. The dynamic mechanical and optical properties of PMMA are listed in Tab.1. The specimen dimension is 400 mm×300 mm×5 mm in length×width×thickness, as shown in Fig.3. There are two notched boreholes with the same diameter of 6 mm, and the distance between the center of these two boreholes is 120 mm. In the middle position of the notched boreholes, an empty circular hole with a diameter of 6 mm is situated. By such an arrangement in the specimen, the directional-fracture-controlled blasting can be realized and the crack propagation is easily ensured to be consistent with the notched locations after the explosion of the charge. The time differ-Ed-Young’s modulus,νd-Poisson’s ratio,Ct-Stress optical constant,C1-Longitudinal wave speed,C2-Shear wave speed.

Tab.1 Dynamic mechanical and optical properties

ence of detonation in the delayed blasting was set to 13.33 μs and the frame rate of the high-speed camera was 3×105frame/s. The resolution of the photograph recorded was 256×64 pixels and the spatial resolution was 1.8 mm/pixel. The laser power was set to 60 mW. The charge mass in each notched boreholes was 140 mg.

Fig.3 Schematic diagram of an experimental specimen

2 Evaluation of Dynamic Fracture Parameters

2.1 Crack propagation velocity

The caustic spot recorded by the digital high-speed camera can be used to locate the position of the propagating crack tip precisely.Thus, the crack lengths at each moment can be determined. The data-fitting methods proposed by Takahashi and Arakawa[19]are used to calculate the real crack lengthl(t) and the crack propagation velocity and relative fracture mechanical parameters can then be obtained. The nine-order polynomial ofl(t) with respect to timetcan be expressed as

(1)

Here, the coefficientliis obtained using the least-squares method, and the crack propagating speedνcan then be deduced by calculating the first derivative of the fitting curvel(t) with respect to timet

(2)

2.2 Dynamic mode-I stress intensity factor

Fig.4 Formation of the caustic spot

Fig.4 shows the formation of a caustic spot. It can be seen from Fig.4 that when the parallel light beams project onto the front side of the specimen, of which the thickness can be thinned at the crack surface as a consequence of the tensile stress applied on the boundaries of the specimen, the light paths can be changed, causing a non-uniform distribution on the reference plane. Therefore, a shadow area (caustic spot) can be formed and a bright curve (caustic curve) can also be found because of the convergence in light beams.

According to the dynamic caustic patterns of the propagating crack tip under the explosive load, the mode-I and mode-II load exists spontaneously during the dynamic fracture process. The dynamic mode-I and mode-II stress intensity factor at the crack tip can be expressed as[20]

(3)

(4)

wherez0(z0=1 200 mm) is the distance between the reference plane and the specimen surface,ctis the stress optical constant of the specimen material,vis the crack propagation speed, anddis the effective thickness of the specimen.Dmaxis the diameter of the caustic spot.λmis the magnification ratio of the optical devices andλm=1 in this paper.μrepresents the ratio of mixed mode stress intensity factor around the crack tip, which is determined by the maximum and minimum characteristic sizes (DmaxandDmin) in the caustic spots.F(v) is the correction factor of the dynamic crack propagating velocity, which accounts for velocity effects on the distribution of the dynamic stress field for a propagating crack, and may be given by

(5)

where the coefficientsβiare given by

(6)

Here,c1andc2are the velocities of the longitudinal and shear waves, respectively, andvis the velocity of crack propagation. It needs to be noted thatF(v) can be approximately equalled to 1 during the data processing[20].

2.3 Stress difference around a circular hole

An empty circular hole in a plate subjected to a biaxial stress field generated by an explosive load from two boreholes is simplified as shown in Fig.5a, whereas Fig.5b presents caustic curves for the considered stress concentration. The relation between the length parameter for characteristic points on the caustic curve around a circular hole and the load is expressed as[21]

(7)

Fig.5 Caustic curves for the considered stress concentration

wherep-qis the stress difference around an empty circular hole,Ris the radius of the empty circular hole,Dis the length parameter for characteristic points on the caustic curves, and the other parameters are the same as the ones in Eq.(3).

3 Experimental Results and Analysis

3.1 Blast-induced crack paths

Fig.6 shows the patterns of experimental results under the delayed blasting load. In Fig.6 the blast-induced cracks A and B are generated between notched boreholes and both slightly deviate from the empty circular hole. The path of blast-induced crack A isobviously curved. These results show that the influence from later ignition detonation on blast-induced crack path is stronger than that from earlier ignition.

Fig.6 Blast-induced crack paths

3.2 Dynamic caustic patterns under a blasting load

Fig.7 shows serial dynamic caustic patterns under a delayed ignition blasting load. The ignition time of borehole H2 is 13.33 μs earlier than that of borehole H1. Att=16.67 μs, caustic spots are formed around borehole H2. Whent=23.33 μs, explosive stress waves from borehole H2 interact with the empty circular hole. Meanwhile, the stress is concentrated around the empty circular hole and a crescent caustic spot appears on the right of the empty circular hole. Att=30.00 μs, the explosive stress waves from borehole H2 meet those from borehole H1 and, at this time, the variation of the caustic spot on the left of the empty circular hole is obviously seen. Att=40.00 μs, the explosive stress waves from borehole H2 meet the caustic spot at the tip of blast-induced crack A from borehole H1. Att=43.33 μs, the caustic spots at the tips of blast-induced cracks present oval shapes under the explosive stress wave load. Att=50.00 μs, the explosive stress waves from borehole H1 meet the caustic spot at the tip of blast-induced crack B from borehole H2. Att=96.67 μs, the blast-induced crack B coalesces with the empty circular hole. In the whole process, with the reflection, diffraction and other effects of explosive stress waves on the empty circular hole, the caustic spots move around the empty circular hole and change in size. The variation of the size and direction for the caustic spots around the empty circular hole intuitively reflects the stress concentration and stress field around the empty circular hole under the explosive stress wave load.

Fig.7 Serial dynamic caustics patterns under a delayed ignition blasting load

3.3 Velocity of main blast-induced cracks

Fig.8 and Fig.9 show the changing trend of blast-induced crack length versus time and the variations in the propagation velocity of blast-induced cracks under delayed ignition blasting loads, respectively. It can be seen from Fig.8 that the total lengths that crack A and crack B reach are both about 43 mm. By using Eq.(2), the velocities of the two blast-induced cracks can then be deduced. It can be observed from Fig.9 that, in the initial stage of blast-induced crack growth, the peak value of the crack propagating speed is about 600 m/s and the blast-induced crack propagating velocity soon decreases rapidly. Whent=43.34 μs, the velocity of blast-induced crack A under a delayed ignition blasting load is reduced to a minimal value of 321.48 m/s, while att=50.00 μs, the velocity of blast-induced crack B reached a minimal value of 237.50 m/s. Meanwhile, the conclusion can be made that the interaction between compressive P-waves and blast-induced running cracks can reduce the propagating velocity of blast-induced running cracks. Later, the propagating velocity of blast-induced cracks rapidly reached the peak values. Fromt=56.67 μs tot=100.00 μs, the propagating velocities of blast-induced cracks oscillates and has basically the same trends. Att=106.67 μs, blast-induced crack B coalesces with the empty circular

Fig.8 Blast-induced crack length vs time

hole. After approximately 10 μs, blast-induced crack B passes through the empty circular hole and its propagating velocity rapidly reaches another peak value of 529.81 m/s.

Fig.9 Velocity of the blast-induced crack propagation vs time

3.4 Evolution of DSIFs at the tips of blast-induced cracks

Fig.10 Variations of DSIFs at the crack tips with respect to time

3.5 Stress difference around a circular empty hole under a blasting load

Fig.11 Curve of stress difference vs. time around an empty circular hole

Fig.11 shows the stress difference around an empty circular hole under two-borehole delayed ignition blasting loads. It is seen that the variation in the stress difference around the empty circular hole indicates the strength of the stress field in the vicinity of the empty circular hole. When two boreholes are ignited with a delay, att=23.33 μs, the stress difference around the empty circular hole is 0.87 MN/m2. Fromt=53.33 μs tot=106.67 μs, the stress difference around the empty circular hole increases from 0.51 MN/m3to 4.07 MN/m2. Aftert=123.33 μs, the stress difference around the empty circular hole decreases in an oscillating manner.

3.6 Analysis of microscopic characterizations in specimens

For a further investigation, the micrographs of the crack surfaces are captured by using the scanning electron microscopy (SEM) method. SEM images of crack surfaces in the vicinity of both circular hole and boreholes are given in Fig.12. It can be concluded from Fig.12 that the crack surfaces become coarse when the crack propagates towards to the circular hole. In contrast, the crack surfaces are relatively smooth in the vicinity of boreholes. Obviously, the roughness of the crack surfaces varies along the crack propagation paths under the influence of the superposition of blasting stress waves and the guiding effect from the circular hole.

Fig.12 Typical SEM images of the fracture surfaces in blast-induced cracks

4 Conclusions

This paper employs the dynamic caustic method to investigate the blast-induced crack propagation and coalescence and the distribution of the dynamic stress field around an empty circular hole in PMMA material under the delayed ignition blasting load. The following conclusions are drawn from the results of the study.

①In directional-fracture-controlled blasting, the dynamic stress intensity factors at the tips of the blast-induced cracks from earlier ignition are smaller than that from later ignition. The explosive stress waves interact with the blast-induced crack and affect the propagation of the blast-induced crack.

②The blast-induced cracks gradually deflect toward the empty circular hole and coalesce with each other in the propagating process.It shows that the empty hole plays a good role in guiding the blast-induced cracks.

③Blast-induced cracks are not generated preferentially where the explosive stress waves are superimposed. Blast-induced cracks are generated from borehole walls and meet between the two notched boreholes. Cracks generated preferentially at the superposition of explosive stress waves need to be studied further.

④Due to the superposition of blasting stress waves and a guiding effect from the circular hole, the crack surfaces are coarse in the vicinity of circular hole while they are smooth when cracks start to propagate near the boreholes.