Fracture Evaluation and Dynamic Stress Concentration of Granite Specimens Containing Elliptic Cavity under Dynamic Loading

Abstract

The Split-Hopkinson pressure bar (SHPB) was used to determine the fracture characteristics of a long bar rock specimen with an elliptical cavity under different axial ratios and dip angles. A high speed camera was applied to record the fracturing process of the granite specimen around the cavity. The experimental results showed that the fracture characteristics around the elliptical cavity were closely related to the axial ratio and dip angle. A three-dimensional numerical model was established using LS-DYNA to quantitatively analyze the dynamic stress state around the cavity. The numerical results indicate that the dip angle and axial ratio of the elliptical cavity significantly affected the dynamic stress concentration factor (DSCF), then affected the rock failure. The location of higher DSCF led to a higher possibility of spalling failure. The maximum DSCF remarkably decreased with a decreasing dip angle and increased the axial ratio. In the dynamic loading propagate process, the stress concentration distribution around the cavity formed by a compression stress wave had a certain damaging effect on the destruction of rock around the cavity, and the stress concentration generated by the tensile stress wave was the main factor of the rock fracture, which was most notable in the peak area of the stress concentration.

1. Introduction

The construction of cavities widely exists in civil and mining engineering. Under external disturbances such as blasting and earthquakes, the stress waves that spread to cavities will produce a series of reflection and transmission phenomena. The stress wave arrives at different diffraction degrees around the cavity and strengthens the local stress, which influences the stability of the engineering structure.

The Hopkinson bar is commonly used for testing rock dynamic properties [1]. Based on the Hopkinson bar, many scholars have studied the failure characteristics of rock and rock-like materials [2,3,4]. Regarding rock specimens with a cavity, Li et al. [5] conducted an experimental study on the dynamic failure characteristics of a rock specimen containing an elliptic hole based on the Split-Hopkinson pressure bar (SHPB). In their experiment, the dynamic compressive strength, failure mode, and crack propagation characteristics of a rock specimen with a hole under different dynamic loads were analyzed. Tao et al. [6,7] carried out a static and dynamic combination test of the prefabricated circular hole specimen by using an SHPB device and compared the results with theoretical calculations. They concluded that the dynamic stress concentration area was the first region damaged in the specimen.

Theoretically, the stress distribution around holes under dynamic incident loading is commonly studied as a diffraction or scattering of elastic waves. Pao et al. [8] systematically studied the basic principle of wave diffraction and the main directions of the related theory to determine the diffraction in various shapes of holes under steady-state and transient wave incidence conditions. Liu et al. [9] analyzed the scattering of SH waves by a rectangular cavity. Yi et al. [10,11] obtained the dynamic response of a circular lining tunnel as function of a cylindrical P-wave. Tao et al. investigated the dynamic stress concentration factor (DSCF) around a circular hole subjected to transient waves by using the complex variable method [12].

Additionally, some scholars have conducted thorough research on the cavity diffraction question from aspects of numerical simulation. Lin et al. [13] proposed a simple and easy numerical solution technique for the wave diffraction problem. She and Guo [14] studied the stress concentration factors under tensile stress in the elliptical hole wall of isotropic materials by the finite element method. Zhou et al. [15] verified that the dynamic stress state around the elliptical hole had a relationship with the wave number. Deng et al. [16] found that the joint position of rock mass had a great influence on the tunnel damage by using UDEC. Du et al. [17] analyzed specimen shape and cross-section effects on rocks properties. More recently, Fang et al. [18] analyzed the influence of a viscoelastic interface on the dynamic stress around the tunnel by means of numerical examples. Kung et al. [19] concluded that the influence of tunnel radius on the earthquake stress increment was stronger than that of the tunnel depth by using ABAQUS. Li et al. [20], combined with PFC-2D, analyzed the dynamic stress concentration and the energy evolution process surrounding the hole affected by the explosion stress wave. Additionally, Wang et al. [21,22] reported that the hole defects of brittle materials had a great influence on dynamic fracturing, and mechanized excavation face.

Theoretical research, experiments, and numerical simulations of the holes above showed that there are more studies on circular holes and that the various theoretical methods are mature. However, studies on the effect of the dynamic stress state formed by the diffraction of the stress wave through elliptical holes on the defective rock are rare. In practical engineering, defects exist in small cracks and large chambers. The problem of defective rocks can be described more comprehensively in terms of the shape of elliptical holes under different axial ratios and dip angles. In contrast to circular holes, the change of the axial ratio and dip angle of an elliptical hole will greatly affect the stress concentration distribution around the hole when the stress wave passes through the hole. Therefore, it is of great practical importance to study the stress concentration and failure characteristics of an elliptical-hole rock based on different axial ratios and dip angles under the influence of a stress wave.

In the present study, based on the SHPB experimental platform, spallation failure tests for long bar granite specimens with different elliptical holes were conducted. The fracture location and process were recorded by a high speed camera. Furthermore, numerical simulations were employed to discuss the dynamic stress distribution of the compressed and tensile stress waves propagating to the elliptical hole under the different dip angles and axial ratios.

2. Experimental Process and Result Analysis

2.1. Material Identification and Experimental Program

Granite samples were chosen of a relatively homogeneous granite from a quarry mine in Dingzi Town, Changsha, Hunan Province, China. Granite, which is plutonic acidic igneous rock, is generated from the condensation of magma. The dominant minerals identified were feldspar, quartz, mica, and a small amount of dark minerals. The complexity and diversity of the mineral component make the rock fracture uniquely inhomogeneous and the study of the rock fracture more difficult. To provide guidance for the microscopic analysis of the rock fracture, the mineralogical characteristics of the granite samples were studied before the spall tests.

After identification, the rock sample was named as gray-white biotite granite, as shown in Figure 1a and the mineral content is shown in Figure 1b. Among them, quartz accounted for about 50% of the total composition, which occurred as an irregular granular form with flaws.

The modified SHPB experiment apparatus was adopted in this study, and the details have been described in our previous studies [23,24,25]. The wavelength of the incident wave was about 72 mm, and wave period was about 240 μs. The rock specimen was cuboid with a cross-section of 35 mm × 35 mm and a length of 1200 mm; the density was 2694 kg/m³, the Poisson’s ratio was 0.19, the elasticity modulus was 45.3 GPa, P-wave velocity was 4104 m/s, uni-axial tensile strength was 9.4 MPa, and the uni-axial compression strength was 128 MPa. Six surfaces of the cuboid were carefully polished and non-parallelism of its surface were within 0.02 mm. An elliptical hole with a different axial ratio of the long axis to the short axis and dip angles was then drilled in the center of the lateral surface. Keeping the long axis of the elliptical hole as 10 mm, and the short axis as 2 mm, 4 mm, 7 mm, and 10 mm, respectively, the corresponding axial ratios were m = 5, m = 2.5, m = 1.43, and m = 1. The dip angle of holes is the angle between the long axis direction and the loading direction, and there are five kinds of dip angles of a specimen with an elliptical hole under each axial ratio, which are β = 0°, β = 30°, β = 45°, β = 60°, and β = 90°. The specimen was in contact with the input bar, and the schematic views of the modified SHPB device are shown in Figure 2.

Due to the existence of holes, the incident compressive stress wave will produce a diffraction phenomenon, leading to a dynamic stress concentration, thus causing the non-uniform stress distribution surrounding the hole. When the incident compressive stress wave passes through the hole and continues to spread toward the right free surface of the specimen, it is reflected as the tensile stress wave. When the reflected tensile stress wave spreads from right to left, it will also generate diffraction after encountering the hole, which will lead to dynamic stress concentration generated by a tensile wave in the periphery of the hole. The wavelength of the incident wave is considered in the design process, which avoids encountering incident and reflective stress waves near the holes. Given the length of the specimen is two times that of the wavelength, the reflective stress wave has no effect on the incident wave when a whole half-sine incident stress wave passes through a hole.

2.2. Experimental Results and Analyses

In the experimental process, the constant gas pressure of 0.5 MPa was applied for the impact, and the whole rock damage process around the hole caused by the dynamic loading was observed using a high-speed camera employed to record the failure process. The resolution ratio of the high-speed camera adopted in this test was 128 × 128, and the videography frequency was 180,000 fps. Herein, the arrival time of the incident stress wave at the left end of the specimen was defined as the zero point of the time record, based on the geometry and wave velocity of the specimen. The complete failure processes are presented in Figure 3, where the left end is the impact incidence.

In addition, Figure 4 shows the rock fracture around the hole that changed with time at different axial ratios when β = 90°. When t = 139 μs, the incident compressive stress wave almost arrived at the left side of the hole. The incident wave completely passed through the hole at about t = 378 μs. When t = 533 μs and t = 589 μs, the reflective tensile wave spread through the hole. Up until the t = 828 μs moment, the reflective stress wave completely passed through the hole.

The high-speed camera images showed that the dip angle β = 90°. When the incident compressive stress wave passed through the hole, there was some rock debris ejection at the upper and lower positions of the holes, whereas such phenomena did not appear at other positions. This shows that powder-like particles pop up when the axial ratio is small. As the axial ratio increases, the number of ejected debris and debris size increase, but at this time only as small debris without cracks. When the reflective tensile stress wave gradually returns and re-incidents through the hole, a crack gradually forms at the upper and lower positions. Moreover, the crack slowly stretches along the upper and lower positions. Figure 4 shows that there were some differences in the time of the crack initiation and elongation around the hole under different axial ratios. The crack initiation at an axial ratio of 2.5 occurred earlier than that related to the other two axial ratios. At t = 533 μs, signs of a crack appeared in the upper part of the hole. At t = 589 μs, the crack notably formed and the width of the crack was greater than that of the other two axial ratios. Similarly, in comparison with the cracks of m = 1.43 and m = 1, signs of cracking appeared in the upper and lower parts of the hole at t = 589 μs and m = 1.43, which were especially notable in the upper part. At the same time, only a tiny crack appeared in the upper part of the hole related to m = 1; the signs of cracks in the lower part were not stronger. At t = 828 μs, the crack around the hole had completely formed for each axial ratio. In contrast, the width of the crack was significantly different, that is, the width of the crack with a high axial ratio was greater than that of the crack with a low axial ratio.

The results show that the specimen with the defective elliptic hole had different degrees of fracture around the hole under compressive and tensile stress waves, but that both failure positions were in the defective hole, and that the shape of the fracture changed with the variation in the axial ratio and dip angle of the hole. Figure 5 shows a comparison of the failure modes of partially prefabricated holes under different axial ratios and dip angles. When the dip angle β = 90°, the failure point around the holes under different axial ratios was at the top and bottom of the hole, and the damage position was symmetrical with the short axis, that is, basically on a straight line with the regular destruction form. As the dip angle decreased to β = 45° and β = 30°, the failure point appeared in the upper right and lower left corners of the hole. At this time, the damage position was no longer symmetrical with the short axis, but was close to the central hole, whereas the failure surface was still regular, that is, perpendicular to the spread direction of the stress wave.

When the dip angle β = 0°, the fracture position and destruction form were similar to that of β = 90°. Based on the damage form of all holes in the picture, we concluded that apart from the fracture around the hole, the other locations ere intact and only some of the holes had uneven pits. Compared with the failure form of the hole, the dip angle of the hole had a certain influence on the failure of the rock. Taking the center of the hole as the reference point, the angle of the failure point will fluctuate with the change of the dip angle.

Figure 4 and Figure 5 show that the ejection of rock debris caused by the incident compressive stress wave and the crack initiation and propagation caused by the reflective tensile stress wave all occurred in the upper and lower positions of the hole. Similar phenomena were not observed in other places. The rock debris ejected by the incident compressive stress wave through the hole led to the initial damage of the defective hole in a certain area, but the initial damage was not enough to break the defective specimen. When the tensile stress wave passed through the hole, it formed different types of cracks; at the same time, the crack formation around the hole under the high axial ratio was faster than that under the low axial ratio. In other words, when the same reflective tensile wave spread to the defect, the net tensile stress of the same magnitude was enough to cause an initial crack around the hole with a high axial ratio.

Based on this, the local stress concentration based on the same compressive and tensile stress waves was greater under a high axial ratio than that under a low axial ratio around the hole. The dynamic damage around the hole under a high axial ratio was stronger than that under a low axial ratio. Therefore, there was a larger amount and size of rock debris cuttings around the hole under the high axial ratio than that under the low axial ratio and that the crack initiation occurred earlier than that under low axial ratio.

3. Numerical Material Model and Validation

The experiment qualitatively explained that the failure point was related to the stress concentration, however, the form of destruction was not enough to absolutely ensure that the stress concentration distribution as stress wave passing through the hole could promote the fracture of the rock sample, or that the damage position had a correlation with the enhancement of local stress. In order to explain more meticulously that rock damage is closely related to axial ratio and dip angle, the finite element program LS-DYNA was employed to simulate the spalling process of rocks with a different elliptical hole, which has been proven to be suitable for rock simulation [26,27].

The model and test system were performed at a ratio of 1:1 and the experimental model is shown in Figure 6a, and the mesh around of the elliptical hole is shown in Figure 6b. The contact among the entities adopted single-sided automatic contact and the boundary was set as a free boundary.

Additionally, the elastic material model was used for the SHPB elastic bars. The continuous surface cap model (CSCM) was used to model the rock bar, which has been widely used for rocks [7,27]. Therefore, we used CSCM and the rock mechanic parameters to calculate the related simulation [27,28,29]. By comparing the results with the experiment, we aimed to verify the correctness of the material model. The damage patterns around the holes using numerical simulations and the experimental fracture are shown in Figure 7. The stress-time curve of the incident wave under the numerical simulation and experimental fracture are shown in Figure 8.

The above results indicate that the plastic deformation of the numerical simulation and experimental fracture around the hole were generally consistent. The period, amplitude, and variation trend were close to each other. In summary, the material model is suitable for simulation analysis of hard rock.

4. Numerical Simulation Results and Analyses

Based on the validated material model and its parameters, the damage around the elliptical holes was simulated under different axial ratios (m = 5, m = 2.5, m = 1.43, and m = 1) and different dip angles (β = 0°, β = 30°, β = 45°, β = 60°, and β = 90°). For each axial ratio, there were five groups of simulations under different dip angles except for a circular hole with m = 1. The failure mechanism of elliptical holes under different axial ratios and different dip angles were further analyzed.

4.1. Stress Evaluation and Failure Characteristics around the Hole

We defined the moment as the record zero point when the incident wave entered the specimen, based on the material parameters and geometry of the model, which was about 146 μs, and the incident wave spread from the left side of the hole to arrive at the hole. At approximately 300 μs, the incident wave spread through the hole. At about 386 μs, the incident wave completely passed through the hole. At about 438 μs, the reflective tensile wave spread via the reflection of the free surface to arrive at the right side of the hole. At about 678 μs, the reflective tensile wave completely passed through the hole.

With an increase in the axial ratio, the time of the failure of the hole moved ahead of schedule. Figure 9 shows that the larger the axial ratio, the greater the high effective strain produced by the compressive stress wave. When m = 5, the strain region that appeared at the upper and lower positions of the elliptical holes was larger than that of other axial ratios. These phenomena indicate that the axial ratio affected the dynamic stress concentration.

Figure 10 shows the change in the plastic zone around the hole with time when the axial ratio m = 2.5 and the dip angle β = 0°, β = 30°, β = 45°, β = 60°, and β = 90.

Figure 10 shows that the larger the dip angle, the greater the strain produced by the compressive stress wave passing the hole. The position of the damage is consistent with the position of the effective strain, which shows that the change of the dip angle has great influence on the rock failure around the hole.

Figure 9 and Figure 10 show that the effective strain around the hole due to the spread of the stress wave will cause a different magnitude of strain of the rock around the hole, so the position will also be different. With the arrival of the reflective tensile wave, the effective tensile strain gradually increased and finally reached a maximum value. Subsequently, intense strain occurred around the hole, causing plastic deformation that led to rock failure. Based on the damage position and changing region of the effective strain, the strain area was subjected to dynamic damage due to the stress concentration. When the tensile stress arrived, the first damage occurred in the damaged area, showing that the first-damage position is the weakest part of the rock.

4.2. Dynamic Stress Concentration Factor (DSCF) around the Hole

A stress field was gradually produced around the holes due to the scattering or diffraction of the incident wave, and the dynamic stress was concentrated during the dynamic loading process. The magnitude of the dynamic stress concentration can be characterized by the dynamic stress concentration factor (DSCF), which is determined by normalizing the hoop stress by the magnitude of the radial stress of the incident wave at the same point in the medium with no opening. [10]

$$k_{\mathrm{DSCF}}=\frac{{\sigma }_{\theta }}{{\sigma }_r}$$

(1)

where k_DSCF is the dynamic stress concentration factor; k_MDSCF denotes the maximum dynamic stress concentration factor; ${\sigma }_{\theta }$ is the hoop stress value around the hole; and ${\sigma }_r$ is the axial stress propagated by stress waves when the medium has no holes.

When calculating the distribution of the DSCF around the hole, the calculation model is normally simplified as a plane problem to facilitate the analysis of the description of the phenomenon and results. The established plane coordinates and corresponding simplified model are shown in Figure 11.

The vector parameters $\overrightarrow{r_{\theta }}$, $\overrightarrow{r_{\theta \theta }}$, and $\overrightarrow{r}$ are introduced here. The parameter $\left|\overrightarrow{r_{\theta }}\right|$ is defined as the distance between the center of the hole and any point of the surrounding rock; $\left|\overrightarrow{r_{\theta \theta }}\right|$ is defined as the distance from the center of the hole to the hole boundary; and $\left|\overrightarrow{r_{}}\right|$ is defined as the vertical distance from any point of the surrounding rock to the boundary of the hole. The relationship between these three parameters is as follows:

$$\overrightarrow{r_{\theta }}=\overrightarrow{r_{\theta \theta }}+\overrightarrow{r_{}}$$

(2)

4.2.1. DSCF at the Same Dip Angle and Different Axial Ratios

Figure 12 and Figure 13 present the change of the hoop dynamic stress concentration factor with different axial ratios and the same dip angle around the holes under compressive and tensile stress waves when $\overrightarrow{r}$ = 0, i.e., $\overrightarrow{r_{\theta }}=\overrightarrow{r_{\theta \theta }}$, respectively. Figure 14 shows the change of the maximum dynamic stress concentration factor with the axial ratio at the same dip angle.

Figure 12 and Figure 13 show the DSCF around the hole. The results indicate that the axial ratio was the main factor affecting the stress concentration distribution under the same dip angle, but that minimum stress occurred near $\theta$= 0°.

Figure 14 shows the maximum DSCF value with varying axial ratios at the same dip angle. When the incident wave was reflected as tensile stress wave, the change in the maximum stress concentration was close to that of the compressive stress wave. The k_DSCF first increased, and then decreased with the increase of the axial ratio.

4.2.2. DSCF at the Same Axial Ratio and Different Dip Angles

The previous section showed that the axial ratio had a significant effect on the DSCF around the hole. Under the same axial ratio, this section compares and analyzes the influence of the DSCF for each dip angle. Figure 15 and Figure 16 show the change of the DSCF for the same axial ratio and different dip angles around the holes under compressive and tensile stress waves, respectively, when $\overrightarrow{r}$ = 0 (i.e., $\overrightarrow{r_{\theta }}=\overrightarrow{r_{\theta \theta }}$).

Figure 17 shows the change in the maximum DSCF with the dip angle at the same axial ratio. Based on the change trend, the angles of the peak stress appeared to first decrease, and then increase with an increase in the dip angle of the hole.

Based on Figure 15, Figure 16 and Figure 17, the dynamic stress concentration factors for each axial ratio can be described in detail as follows:

First, when the axial ratio m = 1, the hole was circular, the stress distribution was not correlated with the dip angle β, and the maximum stress appeared at 90°. At this point, the change in the stress concentration under compressive and tensile stress waves was the same, but the maximum DSCF largely differed, which showed that k_CMDSCF was 2.22 times higher than that of k_TMDSCF. When the axial ratio m ≠ 1, the elliptical hole under the same axial ratio increased with the dip angle and the stress distribution changed from an initial y-axis symmetry to a central symmetry. When β = 90°, the distribution was symmetrical with the y-axis and the compressive stress wave had a larger range of stress concentration than the tensile stress wave.

Second, in the case of m = 1.43 and 2.5, k_CMDSCF under the compressive stress wave resembled linear growth with a gradually increasing dip angle and the linear rate under the high axial ratio was higher than that under a lower axial ratio. The k_TMDSCF under the tensile stress wave considerably fluctuated with the increasing dip angle and the whole trend of m = 1.43 steadily rose. However, in the case of m = 2.5, k_TMDSCF was different before and after β = 60°. It resembles linear growth before β = 60° and the growth rate was large, while it decreased after β = 60° and the growth rate was small.

Third, when m = 5, the growth rate of k_CMDSCF under the compressive stress wave notably varied before and after β = 60° with an increase in the dip angle of the hole. In the early stage, it showed rapid growth but turned flat in the later stage, with a small growth range. Under the function of the tensile stress wave, k_TMDSCF rapidly decreased, reaching the maximum value at ~45°. The maximum DSCF increased rapidly before 45°, with a growth rate larger than that of other axial ratios, while it decreased after 45°, finally reaching the minimum value when β = 90°. At this point, the stress concentration was smaller than that under a small axial ratio when compared with the curve of the maximum stress concentration for the same axial ratio that changed with the dip angle under the compressive stress wave. When the compressive wave passes through the hole, it will cause initial damage, and the results indicated that the factors causing the decrease in the maximum DSCF related to the initial damage. In comparison with the change in the maximum DSCF of the three types of axial ratios corresponding to the dip angle of β = 90°, under the tensile stress wave, the result coincided with the situation in the previous section, that is, the larger the axial ratio, the smaller the tensile stress concentration. Therefore, the effect of compressive stress wave is one of the reasons for this situation. In addition, in the same axial ratios, with the increase in dip angle, the void area perpendicular to the direction of stress wave gradually increased, and the weak surface of the defect rock was larger and larger, so the increase in the weak surface also promotes the occurrence of destruction.

5. Conclusions

This study indicates that rock failure is affected by the dip angle and axial ratio of an elliptical hole. Under the same axial ratio, the change of the compressive and tensile stress concentrations varied with the increase in the dip angle of the holes. The compressive stress concentration showed an increasing trend, while the tensile stress concentration showed a decrease in the high axial ratio. The numerical results indicate that the stress concentration produced by the compressive stress wave passing through the hole promotes damage to the rock surrounding the hole. However, the damage is insufficient to solely cause the fractured rock of the holes. When the tensile stress wave arrives, the position of the dynamic tensile stress concentration coincides with the previous incident compressive stress wave.