Dynamic Fracturing Behavior of Layered Rock with Different Inclination Angles in SHPB Tests

The fracturing behavior of layered rocks is usually influenced by bedding planes. In this paper, five groups of bedded sandstones with different bedding inclination angles θ are used to carry out impact compression tests by split Hopkinson pressure bar. A highspeed camera is used to capture the fracturing process of specimens. Based on testing results, three failure patterns are identified and classified, including (A) splitting along bedding planes; (B) sliding failure along bedding planes; (C) fracturing across bedding planes. The failure pattern (C) can be further classified into three subcategories: (C1) fracturing oblique to loading direction; (C2) fracturing parallel to loading direction; (C3) mixed fracturing across bedding planes. Meanwhile, a numerical model of layered rock and SHPB system are established by particle flow code (PFC). The numerical results show that the shear stress is the main reason for inducing the damage along bedding plane at θ = 0∼75. Both tensile stress and shear stress on bedding planes contribute to the splitting failure along bedding planes when the inclination angle is 90. Besides, tensile stress is the main reason that leads to the damage in rock matrixes at θ = 0∼90.


Introduction
Transversely isotropic rock is widely encountered in rock mechanics and rock engineering.Many sedimentary rocks and some metamorphic rocks, such as slate, shale, gneiss, and sandstone, belong to transversely isotropic rocks.Generally, the mechanical behaviors of these rocks are highly related to the orientation of bedding planes, exhibiting strong strength anisotropy and deformation anisotropy [1][2][3][4][5].Several types of rocks with bedding planes (e.g., slate, shale, sandstone, and coal) tend to split or break along the bedding planes rather than in other orientations.
The research on the mechanical properties of transversely isotropic rocks has attracted the attention of many scholars.The single weakness plane theory proposed by Jaeger [6,7] is the first attempt to describe the strength anisotropy of transversely isotropic rock.In this theory, the failure behaviors are classified into two independent modes, including failure along the discontinuity and failure through intact rock.Xian and Tan [8] have carried out a theoretical investigation on the layered rock masses under static loading, indicating that the bedding plane and filling materials have straightforward effects on the failure mechanism.Ramamurthy et al. [9] have investigated the anisotropy of phyllites and proposed a strength criterion to predict nonlinear strength behavior of transversely isotropic rocks.Tien and Kuo [4] have modified Jaeger's criterion and the Hoek-Brown criterion to accommodate two different failure modes (a sliding mode along the discontinuity and a nonsliding failure mode) for transversely isotropic rocks.Saeidi et al. [10] have also modified the Hoek-Brown criterion and Ramamurthy criterion to predict the strength of transversely isotropic rock under triaxial loading condition.Both of them have found that the strength of transverse isotropic rocks exhibit a U-shaped trend with the inclination angle increasing under static loading conditions [10,11].For numerical simulation, Dan and Konietzky [12] have found that the failure patterns of transverse isotropic rocks can be classified into three types (pure tensile failure, matrix-bedding tensile-shear failure, and bedding tensileshear failure) with foliation-loading angle varying.By using the discrete element method, Tan et al. [13] have drew a diagram which reflects the relationship between tensile strength, failure patterns, and foliation-loading angle of transverse isotropic rocks.

Shock and Vibration
As mentioned above, the failure patterns and the strength of transversely isotropic rocks under static loading condition will be affected by the inclination angle.However, under dynamic loading condition, the effects of inclination angle on transversely isotropic rock have been paid less attention.Compared with static loading, the mechanical behavior of layered rocks is more complex under dynamic loading.The dynamic strength of rocks shows obvious strain rate effect [14].Engineering practices also indicate that shock wave caused by earthquake or blasting excavation is an important factor for rock damage and rock failure [15].Therefore, the researches on the dynamic behavior of transversely isotropic rock are important and necessary.In the theory of stress wave propagation, Li et al. [16] have proposed an equivalent viscoelastic medium method to investigate stress wave propagating across layered rock masses, and they found that the incident angle and the thickness of layered rock mass can affect wave propagation properties.Zhu and Zhao [17] have used virtual wave source method to analyze the obliquely incident stress wave propagating across rock joints and have observed that the amplitude of superposed transmitted wave increases with increasing the incident angel of P-wave.
In dynamic experimental study, Liu et al. [18] have found that the energy consumption for vertical bedded coal-rock is greater than that of the parallel coal at the same fractal dimension.Li et al. [19] have found that Young's modulus of nonpenetrating fractured rock increases with inclination angle increasing.Jia and Zhu [20] have also observed the effects of joint dip angle on the occurrence locations of rock burst.Therefore, the effect of bedding planes on dynamic properties of transversely isotropic rocks is worth studying.
In order to further explore the influence of bedding planes on dynamic compressive fracturing behavior, different inclination angles of sandstone bedding planes have been considered.In tests, five groups of sandstones with different bedding inclination angles 0 ∘ , 15 ∘ , 45 ∘ , 75 ∘ , and 90 ∘ have been prepared to be tested for bedded sandstones under impact load by split Hopkinson pressure bar (SHPB).Besides, the discrete element method is a good mean to study the fracturing behavior of rock.Through the discrete element method, many scholars have reproduced the fracturing behavior of rock material observed in laboratory and have also investigated the failure evolution [21,22].Therefore, the numerical mode with different inclination angle would be established by PFC2D to analyze the micro fracturing mechanism.

Experiment
2.1.Testing Equipment.A modified SHPB system is used to carry out the dynamic experiments, as described by Li et al. [23].The diameter of the incident and transmitted bars is 50 mm.The bars are made of 40Cr alloy steel, with a density of 7.810 g/cm 3 , a P-wave velocity of 5400 m/s, and Young's modulus of 240 GPa.The striker is a specific spindle punch, which can generate a half sine wave.This waveform has enough load rising edge so that it can effectively avoid the stress uneven prior to failure [14,24].Strain gauges are pasted on the incident and transmitted bars to measure the strains when stress waves spread along elastic bars.Simultaneously, the oscilloscope and data acquisition system capture the signals of strain.In order to observe the fracture development during the impact test, a high-speed camera is used to synchronously capture the failure process of specimens.The frame rate is set to 100000 fps, which means that it can take a picture every 10 s.The dynamic fracturing process of the bedded sandstone can be recorded by the high-speed camera.

Specimen Preparation.
A yellow sandstone block containing even-spaced bedding planes is selected to prepare the specimens for SHPB test.The yellow sandstone block is obtained from Jiujiang, Jiangxi province, China.Cylindrical specimens are prepared with a nominal diameter of 50 mm and a height of 50 mm.Five different bedding inclination angles  (the angle between the normal direction of bedding surface and the impact loading direction, as shown in Figure 1) are prepared, including 0 ∘ , 15 ∘ , 45 ∘ , 75 ∘ , and 90 ∘ .Each group contains at least seven specimens.The specimens are numbered as - where  means the inclination angle of the bedding planes and x means the number of specimens.Typical bedded sandstone specimens with different inclination angles are shown in Figure 1.It can be seen that the bedding planes are parallel to each other, and the bedding colors alternate between light and dark.As shown in Figure 1, the two mutually perpendicular solid lines marked on the specimen represent the inclination line and the normal line of bedding planes, respectively, where the longer one is the inclination line and the shorter one is the normal line.The dotted line in these figures represents the impact loading direction.(A) Splitting along Bedding Planes.This pattern generates the macro main fractures which separate rock specimens along bedding planes.Commonly, these fractures are clean and smooth.Most fractures are parallel to the loading direction, showing the properties of splitting failure.In a few cases, some short branching fractures which are oblique to the loading direction may occur at the tips of main fractures.

Testing Results and Discussion
(B) Sliding along Bedding Planes.The fractures along bedding planes which are oblique to the loading direction are the main fractures for this pattern.In most cases, the macro fractures directly go through the ends of specimens, and the rock specimens slide along the main fractures.In a few cases, a Vshaped failure zone occurs at the end of rock specimen due to the generation of a new oblique crack, which has different direction with bedding planes.(C) Fracturing across Bedding Planes.For this pattern, the macro main fractures penetrate the bedding planes and the failure mainly occurs in rock matrix.In general, the macro main fracture may be oblique to the loading direction or parallel to loading direction.According to the geometric relationship between main fractures and the loading direction, the fracturing across bedding planes can also be classified into three subcategories including (C1) fracturing oblique to loading direction; (C2) fracturing parallel to loading direction; (C3) mixed fracturing across bedding planes.
For the subfailure pattern (C1), the oblique fractures dominate the failure process.The surfaces of main fractures are relatively rough and many tiny dislocations have developed, as shown in Figure 2(C1).
For the specimens with the failure pattern (C2), they have obvious splitting fractures across bedding planes.The splitting fractures paralleled to loading direction are the main macro fractures.Figure 2(C2) shows the typical fracturing process under impact loading.From these pictures, it can be found that the specimen initially generates two splitting fractures across bedding plane at 100 s.Afterwards, macro failure occurs at about 160 s.
For the failure pattern (C3), the main fracture may be divided into two parts: the cracks parallel to loading direction and the cracks oblique to loading direction, as shown in Figure 2(C3).

Effect of Inclination
Angle on Failure Patterns.The occurrence ratio of each failure pattern seems to be related to the bedding inclination angles.Figure 3 shows the relationship between failure patterns ratio and the inclination angles.It can be seen that the failure pattern (C1) only is generated at  = 45 ∘ for all specimens, and similar phenomenon also occurs at  = 0 ∘ and 75 ∘ .
As shown in Figure 3, fracturing across bedding planes is more likely to occur under a lower inclination angle.For example, almost all the specimens of  = 0 ∘ ∼45 ∘ appear to be fracturing across bedding planes.However, when the inclination angle increases to a certain degree such as 75 ∘ , bedding planes will significantly affect the fracturing process and the specimens tend to break along bedding planes.
In conclusion, the variation of the failure pattern with inclination angle has three characteristics.(1) For a low inclination angle (less than 45 ∘ ), fracturing across bedding planes is the main failure pattern.With the increase of inclination angle, failure pattern (C2) will gradually convert into pattern (C1).(2) For a medium inclination angle (45 ∘ -75 ∘ ), the failure pattern will transform from pattern (C1) to pattern B with   the increase of inclination angle.(3) As the inclination angle continues to increase, the failure pattern will change from pattern B to pattern A. When it approaches to 90 ∘ , most specimens fail in splitting along bedding planes.

Numerical Investigations
Laboratory tests can well reflect the macroscopic deformation behaviors of rock, but they have a shortage in revealing microscopic deformation and fracturing mechanism.From Figure 2, the macroscopic failure patterns have been identified by high-speed camera photograph.In order to further investigate the micro fracturing mechanism, a particle flow code (PFC) numerical model is used to simulate the dynamic failure of layered rock in SHPB test.

Basic Methodology of PFC
4.1.1.Bonded-Particle Model.The bonded-particle model is represented by a dense packing of nonuniform sized circular particles bonded at their contact points.All particles are bonded together with parallel bond (PB) or contact bond (CB).The two basic bond models have specific forcedisplacement law.A CB model can provide the behavior of an infinitesimal, linear elastic, and either frictional or bonded interface, which carries a point force and does not resist relative rotation.The PB model provides the forcedisplacement behavior of a finite-sized piece of cementitious material deposited between two pieces in the vicinity of the contact location, acting in parallel with a linear model; particle can freely move in the shear and normal direction and even rotate between particles.With the PB model, the moment induced by these rotations can be resisted by the bond.With the CB model, however, the moment cannot be resisted.The mechanical behavior of model is governed by Newton's second law of motion and these force-displacement laws.Thus, the micro fracturing process can be simulated as the movement of particles and individual bond breakages [25,26].

Numerical Model of Transversely Isotropic
Rocks in SHPB Test.As shown in Figure 4, the numerical SHPB system was established.All components of SHPB system in experiment were simulated by PFC2D.In order to reduce the amount of calculation, the CB model was assigned on the transmitted bar, incident bar, and striker.Besides, the lengths of transmitted bar and incident bar were set to 0.75 m and 1.5 m.The density of bars particles and striker particles is 7810 kg/m 3 , and the contact bond strength was selected to be high enough to avoid damage during the impact test.The previous studies have been revealed that these settings are feasible [27,28].Because the impact velocity of striker in laboratory test was 9.4∼10.2m/s, and the impact velocity of analogue striker was set to be 10 m/s.Cylindrical specimens (7283 particle, 50 mm × 50 mm) were produced, and the particle radius of specimen is in the range of 0.3 mm∼0.6 mm.Special particles with radius of 1 mm were aligned to the end of striker, incident bar, specimen, and transmitted bar to improve the contact condition [27].The features of layered rock were modeled by two steps.Firstly, a simplified numerical model of layered rock was adopted.The particles were divided into two groups to simulate two kinds of rock layers which have different stiffness and strength.The thickness of layers is 10 mm.Most studies have shown that the PB model is more suitable to simulate rock materials than the CB model [26,29].Therefore, the particles in analogue specimens are bonded by the PB model.Secondly, the smooth joint (SJ) model was used to simulate the behavior of bedding planes.In previous study, the SJ model has been successfully used to simulate the behavior of bedding planes, joints, foliations, and schistosity [30,31].It is suitable for assigning smooth joint model to the interfaces between the two groups of particles.The behavior of a frictional or bonded joint can be modeled by assigning smooth joint models to all contacts between particles that lie on opposite sides of the joint.As shown in Figure 4, it provides the micro behavior of an interface regardless of the local particle contact orientations along the interface.The smooth joint can be envisioned as a set of elastic springs uniformly distributed over a circular cross-section, centered at the contact point and oriented parallel with the joint plane.Once the SJ model is created, preexisting parallel bonds would be deleted and replaced with SJ model.And some parameters such as normal and shear stiffness, tensile, and shear strength need to be reset.The micro parameters of bars, striker, and specimen are listed in Table 1.The elastic bars and striker use the undamped model, in which the damping coefficients were set to zero.And the damping coefficients of analogue layers are set to 0.15.Such Table 1: Micro parameters of SHPB system and layer rock model.arrangements allow the stress wave to propagate effectively, especially the undamped model which ensures that the stress wave propagates through an elastic bar without significant attenuation.Figure 5 shows the comparison of the incident waves obtained by the numerical model and experiment.The good fit between simulation wave and experimental wave ensures the accuracy of the numerical testing.

Propagation and Evaluation of Micro Cracks.
In PFC, once the contact stresses between particles exceed the shear strength or normal strength of the contact, the cracks would be counted and demonstrated.Therefore, the fracturing process of analogue specimen with different inclination angle could be recorded.Besides, the type of cracks is also counted by commands.If the normal tensile stress of a contact exceeds its normal strength, a tensile crack would be registered.Similarly, a shear crack was registered when the shear stress exceeds its shear strength.In addition, the tensile crack and shear crack were further subdivided according to the type of contact.If a PB contact registers a crack, it would be counted as a PB tensile crack or PB shear crack.If a SJ contact registers a crack, it would be counted as a SJ tensile crack or SJ shear crack.The micro cracks of layered rock mode with different inclination angle under impact loading were listed in Table 2.
Figure 6(a) shows the propagation of micro cracks of the analogue specimen with an inclination angle  = 90 ∘ .At the time of 34 s, some micro cracks begin to develop.Afterwards, the numbers of micro cracks rapidly increase at 52 s and they are gradually stabilized until 181 s.The macro failure patterns are similar to the experimental results.It can be seen that the main fractures basically extend along the analogue bedding planes.From the viewpoint of crack number, the SJ tensile cracks and SJ shear cracks contribute to the failure of analogue specimens together (listed in Table 2).And the effect of SJ tensile cracks is slightly larger than that of SJ shear cracks.
Figure 6(b) shows that the analogue bedding planes also play an important role in the fracturing process of the analogue specimen with an inclination angle  = 75 ∘ .The micro cracks initially generate and rapidly increase along the bedding plane at 31 s.The macro main fractures are composed of SJ shear cracks and SJ tensile cracks, which ultimately penetrated the ends of analogue specimen at 72 s.Afterwards, some PB cracks including some PB tensile cracks and a few PB shear cracks are generated in analogue rock matrixes due to the action of the subsequent stress wave.Overall, the SJ cracks, especially the SJ shear cracks, dominate the macro fractures of analogue specimen, and then the PB cracks facilitate the analogue specimen to damage further.
Figure 6(c) shows the fracturing process of the analogue specimen with an inclination angle  = 45 ∘ .There are two macro main fractures and some small branching fractures in analogue specimen.One of the macro main fracture is mainly composed of PB tensile cracks and PB shear cracks, which is oblique to the analogue bedding planes.The other macro main fracture is mainly composed of SJ shear cracks and SJ tensile cracks, which extends along the analogue bedding plane.It is different from the experimental results that the analogue bedding planes have actually affected the failure of analogue specimen.There were two features of micro cracks during the fracturing process: (1) both of the SJ cracks and the PB cracks facilitate the analogue specimens to fail.And the action has no obvious stage during the whole of fracturing process; (2) the PB tensile cracks predominate in   the analogue rock matrixes, and the number of PB tensile cracks is more than that of PB shear cracks.The SJ tensile cracks predominate along the analogue bedding planes, and the number of SJ shear cracks is more than that of SJ tensile cracks.Figure 6(d) shows the fracturing process of the analogue specimen with an inclination angle  = 15 ∘ .It can be seen that the micro cracks are initially generated at 61 s and ultimately stabilized until 200 s.The macro main fractures are oblique to the analogue bedding planes.The result agrees well with the experimental phenomenon as shown in Figure 2(C1).Macroscopically, it belongs to the fracturing across bedding planes.The PB tensile cracks predominate the fracturing process.In addition, the SJ cracks are scattered along the analogue bedding planes and these cracks cannot develop to the macro main fracture.
Figure 6(e) shows the propagation of micro cracks of the analogue specimen with an inclination angle  = 0 ∘ .The results show that there are two stages during the fracturing process of analogue specimen.In the first stage, the distributions of most micro cracks are parallel to the loading direction.The micro cracks initially extend toward the ends of analogue specimen before 46 s and ultimately penetrate the analogue specimen at 92 s.It is similar to the physical testing results as shown in Figure 2(C2).It belongs to the fracturing parallel to loading direction.In the second stage, many new micro cracks deviate from the location of the macro main fracture.The branching fractures oblique to the loading direction gradually develop and facilitate the analogue specimen to damage further.Figure 7 quantitatively shows the variation of crack number.As shown in Figure 7, the micro cracks emerge at about 25 s and rapidly increase after 68 s.The crack numbers are basically stable at about 180 s.The PB tensile cracks always predominate during the fracturing process.
Overall, the macro failure processes of analogue specimens are similar to the testing results of physical specimens.For the macro main fractures of analogue specimens and physical specimens, the development paths and the geometric relationship with bedding planes are basically the same.Obviously, whether it is experiment or numerical simulation, the bedding planes play an absolute dominant role in the fracturing process at  = 75 ∘ ∼90 ∘ and has less effect on the fracturing process at  = 0 ∘ ∼15 ∘ .There are also a few differences between experimental results and numerical results: (a) a few branch fractures oblique to bedding planes are occasionally generated in physical specimens at  = 90 ∘ ; (b) a macro main fracture along the bedding plane is generated in analogue specimens at  = 45 ∘ .An important reason leading to the difference is that the interlayer spaces of physical specimens are smaller than that of analogue specimens.For the physical specimens, when the cracks propagate along bedding planes, it may be easy to change the propagation direction due to the stress concentration at the tips of cracks.Thus, the influence of bedding planes on the fracturing process of physical specimens is also relatively weaker than that of analogue specimens.Correspondingly, the macro main fractures along the bedding planes has not been observed in physical specimens at  = 45 ∘ , as shown in Figure 2(C1).In general, the numerical simulation results are reliable and agree well with the experiments results.The experimental and the numerical results reveal the dynamic fracturing behavior of layered rock mass effectively from both macro and micro viewpoints.  of SJ tensile cracks always increases with the increase of inclination angle.In addition, the number of SJ shear cracks increases at first.When the inclination angle approaches to 90 ∘ , the number of SJ shear cracks decreases.The case of total crack is also similar to the results of PB tensile cracks.Generally, the more the number of crack type, the greater the damage caused by that cracks in specimen.Therefore, the above results can reflect the effect of the inclination angle to a certain degree.As shown in Figure 8(f), the ratio of SJ tensile crack and SJ shear crack increases with the increase of inclination angle, which indicates that the greater the inclination angle is, the more the possibility of fracturing along the bedding plane the sample is.On the contrary, the PB cracks decrease with the increase of inclination angle.This shows that the fracturing across the bedding planes is more likely to occur when the inclination angle decreases.Besides, the damage is also more likely to be caused in the rock matrixes rather than along bedding planes.The SJ shear crack is more than the SJ tensile crack when the inclination angle varies from 0 ∘ to 75 ∘ .This shows that the shear stress is the main reason of the damage along bedding planes at  = 0 ∘ ∼75 ∘ .Similarly, the tensile stress is the main reason of the damage in rock layers at  = 0 ∘ ∼90 ∘ since the PB tensile cracks are always more than PB shear cracks.

Conclusions
Dynamic impact compression tests have been successfully conducted on bedded sandstone specimens.Five groups of specimens with different inclination angles have been tested, including 0 ∘ , 15 ∘ , 45 ∘ , 75 ∘ , and 90 ∘ .Under dynamic loading condition, failure pattern of bedded sandstone is related to the inclination angle of bedding.Three failure patterns are observed for the impact tests: (A) splitting along bedding planes; (B) sliding failure along bedding planes; (C) fracturing across bedding planes.For the failure pattern (C), three subcategories are also observed: (C1) fracturing oblique to loading direction; (C2) fracturing paralleled to loading direction; (C3) mixed fracturing across bedding planes.For a low inclination angles ( ⩽ 45 ∘ ), specimens mainly break across bedding planes, and the occurrence ratio of shear failure will increase with the increase of inclination angle.For a high inclination angles ( ⩾ 75 ∘ ), the specimens tend to break along bedding planes.
The numerical mode of SHPB system established by PFC2D reproduces the fracturing process of the layered rock under impact loading.The numerical results show that shear stress is the main reason of the damage along bedding plane at  = 0 ∘ ∼75 ∘ .Tensile stress and shear stress on bedding planes both contribute to the splitting along bedding planes together when inclination angle is 90 ∘ .Besides, tensile stress is the main reason of the damage in rock matrixes at  = 0 ∘ ∼90 ∘ .

3. 1 .
Identification of Macro Failure Patterns.According to the fracture development paths and the geometric relationship between main fractures and layer planes, the macro failure pattern can be classified into three types: (A) splitting along bedding planes; (B) sliding along bedding planes; (C) fracturing across bedding planes.The typical pattern of the specimens are shown in detail in Figure 2. The characteristics of the fracturing behavior of the five failure patterns are described as follows.

Figure 2 :
Figure 2: Typical failure patterns of bedded sandstone specimens under impact loading.(Note: P represents the fractures paralleled to loading direction; O represents the fractures oblique to loading direction; V represents V-shaped fractures zone; D represents tiny dislocation.)

Figure 6 :
Figure 6: Dynamic fracturing process of layered rock model with different inclination angles.

Figure 7 :
Figure 7: The variation of crack number with time of the analogue specimen ( = 0 ∘ ).
Failure.Figures8(a)-8(e) quantitatively show the variation of micro crack number with inclination angle.It can be seen that the number of PB tensile cracks increases when the inclination angle varies from 0 ∘ to 45 ∘ .When the inclination angle increases from 45 ∘ to 90 ∘ , the number begins to decrease.The maximum crack number occurs at the inclination angle 45 ∘ .The case of PB shear crack is similar to the results of PB tensile crack, but the maximum crack number occurs at the inclination angle 15 ∘ .As shown in Figure8(c)

Figure 8 :
Figure 8: The variation of micro cracks number of analogue layered rock with different inclination angles.

Table 2 :
The statistics of micro cracks.