Dynamic Compressive Characteristics of Sandstone under Confining Pressure and Radial Gradient Stress with the SHPB Test

School of Civil Engineering, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China Hunan Provincial Key Laboratory of Geotechnical Engineering for Stability Control and Health Monitoring, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, China Work Safety Key Lab on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China School of Mechanical Engineering, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China


Introduction
As a strain-rate sensitive material, the mechanical properties of a rock are affected by dynamic loading and confining pressure and have received a great deal with attention in many engineering fields, particularly excavation-induced seismicity, rockbursts during tunnel excavation and mining activities [1][2][3][4][5].e dynamic mechanical properties of rock under confining pressure have been widely investigated using many approaches, such as numerical simulations and experimental methods.One commonly used method is a split Hopkinson pressure bar (SHPB), which has been proven as an ideal and a reliable loading technique to measure the dynamic properties of rock at a high strain rate (up to 10 1 -10 3 s −1 ) [6,7].Many researchers used Hopkinson bars to study the compressive strength and the dynamic tensile strength of rock-like materials and their rate dependency under uniaxial impact [8][9][10].Given the particular requirements, SHPB systems have been significantly improved upon by many researchers [11][12][13] to obtain a complete triaxial loading test.us, the dynamic properties affected by a confining pressure could be better understood using these modified SHPB systems.Both a high loading rate and high confining pressure could increase the compressive strength of rock material [14][15][16][17].Yin et al. [18] used SHPB to investigate the failure characteristics of rock under static and dynamic loading conditions as well as axial pre-pressure and con ning pressure.
e results from Yin et al. [18] showed that high-stress rocks under unloading con ning pressure were more likely to produce an unstable failure when exposed to the dynamic disturbance.Li et al. [19] proposed a new method for testing the spalling strength at di erent static precon ning pressures and found that the spalling strength decreased with an increase in the conning pressure.Recently, Guo [20] investigated the dynamic compressive mechanical properties of Beishan granite after high-temperature (25 °C to 800 °C) treatment under di erent pressures.In another work, Ma et al. [21] studied the dynamic stress-strain curves for frozen sandy clay which were measured using SHPB and varying the con ning pressures (e.g., 0.5, 1.0, and 1.5 MPa) at the freezing temperatures of −5 °C and −15 °C with strain rates from 160 s −1 to 265 s −1 .e rock was found to be a ratedependent, temperature-sensitive, and con ning pressuresensitive material.e dynamic peak stress increased with an increase in strain rate and the con ning pressure [20,21].Tao proposed an experimental method to explore the dynamic failure process of a prestressed rock specimen with a circular hole.Tao obtained the dynamic failure processes and characteristics of a granite specimen with a cavity under di erent coupled static prestress and dynamic loading experiments conducted on a modi ed Hopkinson pressure bar system [22,23].A signi cant number of studies have been done to date, but a paucity of research has been devoted to the dynamics of rock materials under con ning pressure.For example, in the past, the dynamic mechanical tests of rock under a con ning pressure were mainly focused on those measured under equal con ning pressure conditions.During excavation for underground structures, such as like tunnels and mine tunnels, the rock surrounding the excavation surface mainly exists in a state of zero normal stress.e hoop stress and the radial stress increase with an increase in radial distance until they achieve the stress of the primary rock [24].In general, the surrounding rocks of underground mine tunnels exist under radial gradient stress condition, in which they are a ected by dynamic loadings such as blasts or machinery during excavation.e stress state of the rock exists under the dynamic loading disturbance, and the radial gradient stress is where the stress increases along the radial gradient stress (see Figure 1).e stress state is di erent from the static and dynamic mechanisms of the rocks measured under the con ning pressure [10].In this study, a traditional SHPB con ning pressure test was done by drilling through the center of the specimen and con ning it.According to elastic theory, the stress of the specimen from the center hole to the periphery is a gradient transformation rather than an evenly distributed one, forming a radial gradient stress.e dynamic mechanical properties of the rock under the radial gradient stress were investigated in this study.For comparison, a conventional con ning pressure impact test under the same conditions was also performed systematically.

Materials and Methods
2.1.Specimen Preparation.According to the requirements for the mechanical test performance of rock, the size of the static load specimen shall be Φ50 mm × 100 mm.Based on the dynamic test principles and existing research results [25], the Φ50 mm × 25 mm was selected as an optimum size for the dynamic load test specimen.According to the requirements for this test, a circular hole with an aperture of 8 mm was drilled along the axis of the specimen in the middle of the cross section (see Figure 2).A 2S-200 vertical coring machine, DQ-4 rock cutting machine, and SHM-200 double-end face grinding machine were used to drill, cut, and polish the rock specimens, to make sure that the surface roughness of the treated specimens was less than 0.02 mm, and the nonparallelism of the upper and lower end faces was less than 0.02 mm. e RMT-150C tester was used to measure the basic mechanical properties of the modi ed specimens.e static mechanical parameters used in this study are shown in Table 1.

Test Methods.
e impact test was performed on a largediameter SHPB device (diameter of 50 mm).A ow diagram showing the major steps for the SHPB test is shown in Figure 3 [26].Hydrostatic con nement was done by connecting two pressure cylinders with two tie rods [7,11].e test system was characterized by the medium, and high strain rates for loading were adapted to the heterogeneous brittleness of the rock materials.e strain rate of the obtained specimens ranged from 10 1 to 10 3 s −1 .A half-sine stress wave was generated on impact using a spindle-type structure punch [27].
To investigate the dynamic mechanical properties of rocks under a radial gradient stress, the major steps employed in the impact test were as follows: (1) Placing the specimen.e specimen was placed between two long elastic pressure rods, and the endfriction e ect was minimized by a smearing butter and installation of the con ning pressure device.Since axial pressure was not observed, the two ends were adjusted to clamp the specimen.It should be noted that when the con ning pressure was loaded, the specimen was subjected to a certain amount of stress in the axial direction due to Poisson's ratio e ect.However, in the ranges of con ning pressure and rock strength, its in uence would be considered insigni cant. 2

Advances in Civil Engineering
(2) Loading confining pressures were 5 MPa and 10 MPa in this test.(3) Adjusting the pressure and taking a single impact.e data acquisition system collected the signal and saved the data until the devices achieved the desired results.Under the same conditions, the impact of the sandstone under conventional confining pressure was also tested.e typical single-impulse waveforms recorded in the test are shown in Figure 4. Parameters such as the stress-strain curve, strain rate, incident energy, transmission energy, and absorption energy of the specimen can be calculated using the following expressions: where A e is the elastic rod area, A s is the area of the specimen, ρ e is the elastic rod density, C e is the velocity of the elastic rod, L s is the length of the specimen, t is the duration of the stress wave, E s is the absorption energy of the specimen, and E I , E R , and E T are the incident, reflected, and transmitted energies, respectively.It should be noted that, due to exist of the confining pressure, the specimen was not prone to crack or break.All of the specimens in the impacted experiments tested were not obviously cracked or broken and just presented with slight cracks.e effects of the gradient stress on the stressstrain curve, strength, deformation, and energy absorption characteristics of the specimen were investigated without obvious failure in this study.

Results
Figure 4 shows the impact waveforms for the two test specimens.From the figure, it can be seen that the waveforms of the specimen (denoted as specimen B) under radial gradient stress had a larger reflected wave and a smaller transmitted wave than those of the specimen (denoted as specimen A) under radial uniform stress.When combined with stress wave theory and Equations ( 1) and ( 3), it shows that at the same level of impact, specimen B had a lower strength and a higher strain rate.
Figure 5 shows the dynamic stress-strain curves for sandstone impacted under different radial stresses.It shows that their curve shapes are similar.In comparison with specimens under radial uniform stress, the stress-strain curve of the sandstone under radial gradient stress was slower, and lower elastic modulus, lower strength, and higher peak strain were found, especially in the initial stage and in the postpeak stage.e above phenomenon could be found to be more significant as the level of impact increases, whereas it would be reduced with an increase in the confining pressure.It can be seen from Figure 5 that the strength decreased but the peak strain was much larger when the level of impact increases or decreases with an increase in the confining pressure.e differences between Figures 4 and 5 suggest that the transflective capacity of the stress waves and stress-strain curve are more moderate for the specimen with a hole, and the dynamic mechanical properties of the sandstone under radial gradient stress were different from those occurring under conventional confining pressure conditions.

Strength.
Rock is a strain rate sensitive material, and both the tensile strength and compressive strength increase with increasing strain rate [10,[14][15][16][17]. Figure 6 shows the variation in the dynamic strength of sandstone with strain rate under different confining pressures.It can be seen that under different radial stress states, the dynamic strength undergoes an exponential increase with an increase in the strain rate, and it was also found that under higher confining pressures, the dynamic strength was increased more significantly.is finding was consistent with most previous re-  search [11][12][13].e data in the red circle in Figure 6 are the dynamic strength at the same impact level under di erent con ning pressure conditions.Figure 6 shows that under the same impact level, the dynamic strength under radial gradient stress was smaller than under radial uniform stress, and the corresponding strain rate was relatively higher.is result was similar to those occurring under static loading conditions [28].As seen from the tting functions shown in Figure 6, although the strength of the specimen under radial gradient stress was lower, it was more sensitive to the strain rate than the specimen under radial uniform stress under two levels of con ning pressure.e phenomenon becomes more apparent with increases in the con ning pressure.e ndings implied that the dynamic strength of the sandstone under radial gradient stress was more sensitive to the strain rate and the con ning pressure.erefore, the dynamic strength of the rock under gradient stress would be di erent from those of the specimens under conventional con ning pressure conditions.

Deformation.
Figure 7 shows the variation in peak strain with strain rate under di erent radial stress conditions.e peak strain had a linearly increase in the strain rate under di erent con ning pressures [29].As shown in Figure 7, the growth rate of the peak strain decreased with the increase in strain rate and the con ning pressure under radial gradient stress conditions.It was sensitive to the increased con ning pressure under radial uniform stress.e sensitive of peak strain to the strain rate for the specimen with a hole is smaller compared with that of intact specimen under two levels of con ning pressure conditions.A possible explanation for the change in sensitivity would be that the exit of the hole provides a free surface for the specimen with a hole and allows the specimen to have better deformability.It is less sensitive to strain rate, and with the increase in the con ning pressure, the deformability caused by the free surface had become smaller.
e deformability for the specimen with a hole is improved.e red circle in the gure shows the peak strain of sandstone at the same level of impact.
e ndings showed that the peak strain of the sandstone under radial gradient stress was greater than under radial uniform stress, but the di erence between them would decrease with an increase in the con ning pressure.
ese results implied that the deformation capability of the rock under radial gradient stress was enhanced, which could be proven using the results in Figure 5. e main reason for the higher strain rate in the rock under radial gradient stress at the same level of impact was possibly attributed to a good capability for deformation.
Since the stress-strain curve under dynamic loading was not a straight line, we analyzed the secant modulus (E 50 ).Figure 8 shows the changes in E 50 with strain rate.It can be seen from the gure that the dynamic E 50 of the sandstone under di erent radial stress conditions showed two modes with the increase of the strain rate.Although the E 50 of the rock under the radial uniform stress increased with the increase in strain rate, the change was not obvious and was consistent with the ndings reported by Gong [30].e E 50 of rock under a radial gradient stress decreased rst and then increased with an increase in the strain rate.
e phenomenon was mainly due to that the exit of the hole provides greater deformability for the specimen with a hole compared with the intact specimen.And the E 50 of the rock under the radial uniform stress would decrease.Due to the small size of the hole, the deformability provided by the free faces of the hole is limited.With the increase of strain rate, the E 50 would increase, but the change was also not obvious.What is more, the di erence in sensitivity between the dynamic strength and peak strain at the strain rate may have an in uence on the E 50 increased with the strain rate.From Figure 8, it can be seen that the E 50 of the sandstone under a radial gradient stress showed a discrete in comparison with that of the sandstone under radial uniform stress.at may be due to the drilling process for the specimen which will inevitably a ect the mechanical properties of the specimen.e data in red circle are the E 50 of the sandstone that was impacted at the same level under di erent con ning pressure conditions.e E 50 of the sandstone under radial gradient stress was far less than that under radial uniform stress.When combined with the results in Figures 7 and 8, it could be concluded that the ductility of the sandstone under radial gradient stress was improved.

Energy Absorption.
Figure 9 shows the energy absorption properties of sandstone under di erent con ning pressure conditions.As seen in Figure 9, the absorption energy per unit volume of sandstone increased with an increase in incident energy.e change in absorption energy of most rock-like materials also followed the above trend [31,32].e magnitude increase would be reduced with an increase in con ning pressure in this study, regardless of whether the specimen was under radial uniform stress or under radial gradient stress.e relationship between the absorption energy per unit volume and the incident energy was not signi cantly a ected by the radial gradient stress.at is mainly because the specimens were not obviously cracked or broken and just presented with a slight crack, and the absorbed energy is mainly consumed by the expansion and formation of microcracks.

Discussion
Due to the existence of a hole, specimen B has a free surface, which will allow the specimen to have greater deformability.
e initial stress of the specimen is radial gradient stress and implies that the holes had a greater in uence on the mechanical properties of the specimen according to previous research [29,[33][34][35].e holes allowed a greater deformation Figure 5: Stress and strain curves for the sandstone under dynamic loading with (a) impact at 0.6 MPa pressure under a con ning pressure of 5 MPa, (b) impact at 0.7 MPa pressure under a con ning pressure of 5 MPa, (c) impact at 0.6 MPa pressure under a con ning pressure of 10 MPa, and (d) impact at 0.7 MPa pressure under a con ning pressure of 10 MPa.
Advances in Civil Engineering of the specimen in the radial direction.e con ning pressure e ect was also smaller and resulted in a higher strain rate and a lower strength of the sandstone under radial gradient stress at the same impact level.More importantly, the dynamic strength of the sandstone under radial gradient stress was more sensitive to the strain rate.Many researchers have observed that at higher rates of loading, stronger concretes exhibit a smaller percentage gain in compressive strength than weaker concretes [36].Cowell suggested that the improvement noted for weak concretes may be in uenced more by their lower static strength when expressed as a proportion of this smaller value [37].e hole diameter has no e ect on the uniaxial compressive strength of hollow sandstone when the hole diameter is smaller than 10 mm [28,29]. is means that the di erence in strain rate sensitivity was not due to static strength di erences.Hence, the sensitivity of the strain rate for the sandstone with a hole may be improved due to greater deformability and radial gradient stress and needs to be investigated through further research.Although the exit of the hole provides greater deformability for the specimen with a hole compared with the intact specimen, due to the small size of the hole, the deformability provided by the free faces of the hole is limited.What is more, the di erence in sensitivity between dynamic strength and peak strain at a particular strain rate may have an in uence on the E 50 increased with the 6 Advances in Civil Engineering strain rate.Both of them would result in the E 50 of rock under radial gradient stress showing a trend that decrease first and then increase with the increase in the strain rate.e above findings provide guidance for improvement on practical engineering projects, such as tunnels and roadways, since the surrounding rock is in similar gradient stress conditions.In this study, only the dynamic compressive characteristics of sandstone were investigated at two levels of confining pressure.
e failure modes under different confining pressures and stress distributions for different specimens were not examined.e hole size effect was also not investigated.erefore, many other types of rocks of interest should be investigated using the aforementioned methodology in the future.

Conclusions
In this study, impact compression tests of sandstone under different confining pressures were conducted with SHPB.
e effect of the mechanical properties of the specimens under radial gradient stress was compared with those under radial uniform stress.e conclusions we obtained in this study are as follows: (1) e dynamic strength and the strain rate of sandstone had an exponential increase.e dynamic strength of the rock under radial gradient stress was smaller than that occurring under radial uniform stress, but the former was more sensitive to strain rate.
(2) e increase in peak strain was linearly correlated with the strain rate under different confining pressures, while the sensitivity of the peak strain to confining pressure was lower for the sandstone with a hole.e values of the E 50 were decreased, but further increasing the stain rate would lead to an increase in the level of the elastic modulus, which is different than under radial uniform stress.Also, the ductility of the sandstone under radial gradient stress could be improved.
(3) Although the absorption energy per unit volume of sandstone increased with an increase in incident energy and its growth rate was decreased with the increase in the confining pressure, the relationship between the absorption energy per unit volume and the incident energy was not significantly affected by the radial gradient stress.

Figure 1 :
Figure 1: Sketch of the loading state of the rock.

Table 1 :
Physical-mechanical parameters for sandstone under static load.