Stress Uniformity Analyses on Nonparallel End-Surface Rock Specimen during Loading Process in SHPB Tests

School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan 232001, China Engineering Research Center of Underground Mine Construction, Ministry of Education, Anhui University of Science and Technology, Huainan 232001, China State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China


Introduction
Split Hopkinson pressure bar (SHPB) apparatus has been a very popular and promising experimental technique for evaluating the dynamic mechanical characteristics of rock [1,2], concrete [3], soil [4], and cemented sand [5] at high strain rates (10 2 ∼10 4 s −1 ).Conventional SHPB apparatus consists of three essential parts: a striker bar, an incident bar, and a transmission bar.A pending specimen is sandwiched between the incident bar and transmission bar.erefore, there are three contact surfaces in SHPB tests: the contact surface between the striker and incident bars, the contact surface between the incident bar and the pending specimen, and the contact surface between the transmission bar and the pending specimen [6].Parallel condition and friction situation of three contact surfaces play a great impact on dynamic mechanical characteristics of pending specimen in SHPB tests [7][8][9].
With the development of SHPB technique, how to improve the accuracy of SHPB test results and the precision of dynamic mechanical characteristics measurement is becoming a key issue for SHPB technique [6].For SHPB apparatus, there are two kinds of bar misalignment effect: processing accuracy and assembly precision of elastic pressure bars.Six major types of bar misalignment, offset of neutral axis, uneven support height, nonparallel impact endsurface, bar straightness, dome, and cone impact end-surface shapes, have been investigated by numerical simulations, and the distorted signal generated by bar misalignment produces unreliable data analysis for the presence of a flexural mode of vibration [8].To estimate the effects of imperfect conditions on incident waves in SHPB tests, four types of imperfect conditions, curved bar, axis o set between the striker and incident bars, inclination of impact endsurface, and indentation of impact end-surface, have been investigated by numerical simulation, and inclination and indentation of impact end-surface present a great impact on incident wave [9].For SHPB apparatus with special shape striker, axis o set and inclination of special shape striker lead to wave distortion and amplitude decrease, and system calibration of SHPB apparatus is classi ed into four steps: system adjustment, wave distortion identi cation, measurement calibration, and transmission calibration [10].
For SHPB apparatus, striker bar, incident bar, and transmission bar are factory calibrated and assembled coaxially before SHPB tests.erefore, the in uence of processing accuracy and assembly precision of elastic pressure bars can be ignored.For rock materials, each pending rock specimen is prepared into a short cylinder with length to diameter ratio of 0.5 through drilling, cutting, and grinding process [11,12].e processing accuracy is varied for rock specimen.For short cylinder rock specimen, the atness of both end-surfaces can be within 0.02 mm, while the perpendicularity of two end-surfaces to the axis is di cult to be controlled [13].
erefore, the nonparallel e ect of rock specimen end-surface in SHPB tests should be investigated.
For rock-like materials, the validity of SHPB tests is based on no brittle failure before achieving stress equilibrium.To evaluate the in uence of nonparallel end-surface on stress uniformity during loading process in rock SHPB tests, numerical simulation on SHPB tests for rock materials have been carried out by LS-DYNA when end-face nonparallelism is within 0.40% and Young's modulus ranges from 14 GPa to 42 GPa.
en both stress nonuniformity coe cient and stress equilibrium time are analyzed under various end-surface nonparallelism conditions.

Numerical Model of Rock SHPB Tests with
Nonparallel End-Surface 2.1.Nonparallel End-Surface.In rock SHPB tests, the common size of rock specimen is Φ50 × 25 mm.To simplify the analysis, it can be assumed that only one end-surface is not perpendicular to the rock specimen axis and the nonparallel end-surface is in contact with the transmission bar, which can be seen from Figure 1.End-surface nonparallelism, marked as c, is de ned as the ratio of maximum height deviation δ to average height h of rock specimen [6,13].And it can be derived as To explore the in uence of end-surface nonparallelism on stress uniformity during loading process in rock SHPB tests, 9 kinds of end-surface nonparallelisms are involved for rock specimens, which are 0%, 0.05%, 0.10%, 0.15%, 0.20%, 0.25%, 0.30%, 0.35%, and 0.40%, respectively.erefore, maximum height deviation δ varies from 0 to 0.100 mm.

Half-Sine Loading Waveform.
For rock-like materials, half-sine loading waveform with a slow rising is an ideal loading waveform.e half-sine loading waveform not only attenuates oscillation and dispersion e ect, but also achieves an approximate constant strain rate deformation [10].During SHPB numerical simulation, a half-sine loading waveform with amplitude of 260 MPa and duration of 240 μs is directly applied on the impact end-surface of the incident bar.

2.3.
ree-Dimensional Numerical Model of SHPB Tests.Finite element model of SHPB test is based on Φ50 mm steel SHPB test apparatus.A three-dimensional numerical model including incident bar, transmission bar, and rock specimen is set up in ANSYS.en, a keyword le is output from ANSYS and revised by applying HJC model for the rock specimen.Finally, LS-DYNA is adopted to run the revised keyword and output SHPB numerical simulation results.In three-dimensional numerical model, both incident bar and transmission bar are straight bars with size of Φ50 × 2000 mm, and their axial directions are along the Z-axis.
e average height of rock specimen with nonparallel end-surface is 25 mm.According to the actual situation of SHPB apparatus, the incident bar, transmission bar, and rock specimen are constrained from X direction and Y direction.A hexahedral solid element SOLID164, de ned by 8 nodes, is used for three-dimensional modeling.And the number of solid elements for the incident bar, transmission bar, and nonparallel end-surface rock specimen is all 60000.
ree-dimensional numerical model including the incident bar, transmission bar, and rock specimen with nonparallel end-surface is shown in Figure 2.

Material Parameters for Rock Specimens and Elastic Steel
Pressure Bar.Isotropic linear elastic model is applied for  Advances in Civil Engineering elastic steel pressure bar.e density, Young's modulus, and Poisson's ratio for both the incident bar and transmission bar are 7.85 g/cm 3 , 210 GPa, and 0.30 respectively.Considering various Young's moduli of rock material, 5 kinds of Young's moduli are involved, which are 14 GPa, 21 GPa, 28 GPa, 35 GPa, and 42 GPa, respectively.Holmquist-Johnson-Cook (HJC) dynamic damage constitutive model is chosen for rock specimens [14].In HJC model, only shear modulus G and crushing volumetric strain μ c vary with Young's modulus and can be calculated by ( 2) and (3).e other parameters for HJC model of rock specimens keep constant for various Young's moduli.
e density ρ and Poisson's ratio ] of rock material are 2.47 g/cm 3 and 0.20, respectively.f c and T stand for quasistatic uniaxial compressive strength and maximum tensile hydrostatic pressure.A, B, C, N, and S max are normalized cohesive strength, normalized pressure hardening coe cient, strain rate coe cient, pressure hardening exponent, and normalized maximum strength.έ 0 and ε fmin are reference strain rate and plastic strain before fracture.p c , p lock , μ c , and μ lock are crushing pressure, locking pressure, crushing volumetric strain, and locking volumetric strain.D 1 and D 2 are damage constants.K 1 , K 2 , and K 3 are material constants.f s is failure type.When Young's modulus is 14 GPa, the material parameters for HJC model are listed in Table 1:

SHPB Numerical Simulation Results of
Nonparallel End-Surface Rock Specimens

Numerical Simulation Validation. SHPB technique is
based on two fundamental assumptions.One is onedimensional stress wave propagation, and the other is stress uniformity [1].And the stress uniformity assumption is the key to validate a SHPB test and can be checked by comparing the stress histories on both ends of rock specimens or checking the unbalance stress during SHPB tests.
In SHPB tests, the incident stress σ I (t), re ected stress σ R (t), and transmitted stress σ T (t) are acquired by strain gauges mounted on the incident bar and transmission bar.Two strain gauges are mounted symmetrically on the surface of incident bar or transmission bar to eliminate the exural vibration e ect.As shown in Figure 2, four elements, A, B, C,  and D, at the same cross section where two strain gauges mounted in actual SHPB tests are selected to export numerical simulation results for stress uniformity analyses.Two adjacent elements, A and B, correspond to maximum height point, and two adjacent elements, C and D, correspond to minimum height point.Four elements on the incident bar are 1000 mm away from the contact surface between the incident bar and rock specimen.While four elements on transmission bar are 400 mm away from the contact surface between the transmission bar and rock specimen.When Young's modulus is 28 GPa, the incident, re ected, transmitted, and unbalance stresses for the rock specimen with end-surface nonparallelism of 0% and 0.40% are shown in Figure 3. e unbalance stress can be calculated  p lock (GPa) Advances in Civil Engineering by σ I (t) + σ R (t) − σ T (t).When rock specimen element failure occurs, the original contact between the incident bar and rock specimen does not exist and become a free end-surface.Hence, there are two peaks in re ected stress wave, one in loading process, and the other after the failure of rock specimen.

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As shown in Figure 3, the unbalance stress for rock specimen with end-surface nonparallelism of 0% and 0.40% is very small and oscillates above and below 0. Relatively big uctuation only appears at the initial part and end part of unbalance stress.
erefore, nite element model for rock SHPB tests is correct and can be used for the following analyses.

Re ected Stress Wave and Transmitted Stress Wave.
According to one-dimensional stress wave theory, re ected wave and transmitted wave emerge when stress wave propagates to the contact surface between two materials with di erent wave impedances.And the area and geometry of contact surface show an impact on the re ected wave and transmitted wave [15].Typical re ected stress and transmitted stress for various nonparallel end-surface rock specimens are shown in Figures 4 and 5. Considering the rising time of half-sine incident stress wave, the time of loading process in SHPB numerical simulation is 120 μs.
As shown in Figure 4, an obvious uctuation presents in re ected stresses, especially in the middle part of the whole loading process.e uctuation e ect in re ected stresses is enhanced with end-surface nonparallelism and Young's modulus.
e amplitude of re ected stresses for rock specimens gradually rises with the increase of end-surface nonparallelism and decreases with the increase of Young's modulus.
As shown in Figure 5, a slight uctuation presents in transmitted stresses, especially in the middle part of the whole loading process.Both the amplitude and duration of uctuation in transmitted stresses are enhanced with endsurface nonparallelism, whereas both the amplitude and duration of uctuation in transmitted stresses are weakened with the increase of Young's modulus.e amplitude of transmitted stresses for various nonparallel end-surface rock specimens gradually decrease with the increase of endsurface nonparallelism and rise with the increase of Young's modulus.
Above all, uctuation e ect exists in both re ected stresses and transmitted stresses.For parallel end-surface rock specimens, re ected stress and transmitted stress propagate along the axis for an entire circular contact area.

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While for nonparallel end-surface rock specimens, the contact area is a part of circular section for a tiny gap between the rock specimen and transmission bar.When incident stress propagates to this place, most of the incident stress is re ected for the less contact area of the nonparallel end-surface, and the contact area at the nonparallel contact surface becomes larger for compressive deformation of the rock specimen under transmitted stress.Re ected stress and transmitted stress propagate along a certain angle to the axis for the nonparallel end-surface.After repeated re ection and transmission between two ends of rock specimen, a slight uctuation presents for propagation direction deviation.Due to the parallel contact surface between the incident bar and rock specimen, distorted re ected stress caused by the nonparallel contact surface can be well transmitted into the incident bar.erefore, re ected stress presents a relatively obvious uctuation e ect.Moreover, the uctuation e ect is enhanced with end-surface nonparallelism for both re ected stresses and transmitted stresses.
During stress wave propagation, the compressive deformation can compensate for the tiny gap between the rock specimen and transmission bar.On one hand, the compressive deformation can weaken the distorted re ected stress from the nonparallel contact surface, which means a weak uctuation e ect in re ected stresses.On the other hand, the compressive deformation can enhance the uctuation e ect in transmitted stress for the unstable contact area between the rock specimen and transmission bar during repeated re ection and transmission in rock specimen.It is well known that compressive deformation is related to its Young's modulus.Hence, the uctuation e ect in re ected stresses is enhanced with Young's modulus, while the uctuation e ect in transmitted stresses is weakened with Young's modulus.

Stress History on Two Ends of Rock Specimens. When
Young's modulus is 28 GPa, stress history on two ends of nonparallel end-surface rock specimens are presented in Figure 6.Stress history on the end contacted with the incident bar is marked as σ 1 (t), and stress history on the end contacted with the transmission bar is marked as σ 2 (t).σ 1 (t) and σ 2 (t) can be derived from incident, re ected, and transmitted stresses by the following equations: As shown in Figure 6(a), the noncoincidence between the stress history on two ends of parallel end-surface rock specimen is mainly in the initial loading stage, and the di erence is very small.As shown in Figures 6(b)-6(i), the noncoincidence between the stress history on two ends of nonparallel end-surface rock specimens is extended with the increase of end-surface nonparallelism in loading process, and the di erence is also enlarged.Comparing Figure 6(a) with Figures 6(b) and 6(c), it can be found that the di erence of stress history on two ends of nonparallel end-surface specimen is smaller than that for parallel end-surface rock specimen when end-surface nonparallelism not exceeds 0.10%.
To analyze the di erence between stress history on two ends quantitatively, stress deviation history Δσ(t) between stress history on two ends is involved and can be calculated by σ 1 (t) substracting σ 2 (t).
e stress deviation history Δσ(t) for various end-surface nonparallelisms and various Young's moduli are shown in Figure 7.And the maximum stress deviation Δσ max for each condition is scattered in Figure 8.
As shown in Figure 7, stress deviation history Δσ(t) presents a wave trend with the loading time, and stress deviation achieves its maximum value at about 40 μs.Under the same Young's modulus, the amplitude of stress deviation history Δσ(t) increases with the increase of end-surface nonparallelism.Under the same end-surface nonparallelism, the amplitude of stress deviation history decreases with the increase of Young's modulus.
As shown in Figure 8, under the same Young's modulus, the maximum stress deviation Δσ max rst decreases then increases with the increase of end-surface nonparallelism and reaches its lowest value at end-surface nonparallelism of 0.05%.
is may be explained by the fact that the dispersion e ect in SHPB test is further weakened by the uctuation e ect and shifting in re ected Advances in Civil Engineering stresses and transmitted stresses.Meanwhile, the maximum stress deviation Δσ max shows a little di erence when end-surface nonparallelism is within 0.10%.Under the same end-surface nonparallelism, the maximum stress deviation Δσ max decreases with the increase of Young's modulus.

Effect of Nonparallel End-Surface on Stress Uniformity during Loading Process
4.1.Stress Nonuniformity Coe cient.Stress nonuniformity coe cient α(t) is adopted to describe the stress uniformity during loading process in SHPB tests, and it can be derived by dividing absolute value of stress deviation to average value of stress history on two ends [16][17][18].Stress nonuniformity coe cient α(t) can be calculated by the following equation: Stress nonuniformity coe cient α(t) re ects the state of stress uniformity of rock specimens at various loading times.
e closer the stress uniformity nonuniformity coe cient α(t) to 0 is, the more uniform the internal stress in the rock specimen is.Stress nonuniformity coe cient α(t) for various end-surface nonparallelisms and various Young's moduli are shown in Figure 9.
As shown in Figure 9, under the same Young's modulus, stress nonuniformity coe cient α(t) for various end-surface nonparallelism attenuates in a serrated uctuation with the increase of loading time.And stress nonuniformity coe cient α(t) increases with the increase of end-surface nonparallelism.
e in uence of end-surface nonparallelism on stress nonuniformity coe cient α(t) is weakened with loading time going on and mainly concentrates on the rst 70 μs of loading process.

Stress Equilibrium Time.
When stress nonuniformity coe cient α(t) is equal to or less than 0.05, it is considered that the stress equilibrium state is achieved in loading process and the stress distribution in rock specimen meets the assumption of stress uniformity.Stress equilibrium time, marked as t u , is de ned as the time spent from the start of loading to the stress nonuniformity coe cient α(t) equal to or less than 0.05 [17].Stress equilibrium time t u for each condition is scattered in Figure 10.
As shown in Figure 10, under the same Young's modulus, the stress equilibrium time t u rst decreases then increases in a step type with the increase of end-surface nonparallelism.
erefore, nonparallel end-surface leads to two reverse results for stress uniformity during SHPB loading process, to extend stress equilibrium time and to shorten stress equilibrium time.When end-surface nonparallelism is 0.10%, stress equilibrium time achieves its lowest value whatever the Young's modulus is.And the lowest stress equilibrium time for Young's modulus of 14 GPa, 21 GPa, 28 GPa, 35 GPa, and 42 GPa are 46.2 μs, 44.5 μs, 43.7 μs, 42.6 μs, and 42.2 μs, respectively.

In uence of End-Surface Nonparallelism on
Stress Uniformity 4.3.1.c ≤ 0.10%.As shown in Figure 10, when end-surface nonparallelism is equal to or less than 0.10%, the stress equilibrium time slightly reduces with the increase of endsurface nonparallelism.erefore, the in uence of endsurface nonparallelism on stress equilibrium time is very small.Meanwhile, Young's modulus also presents little inuence on stress equilibrium time.e stress equilibrium time also slightly reduces with the increase of Young's modulus.
Due to the uctuation e ect and shifting in re ected stresses and transmitted stresses, the dispersion e ect in SHPB test is further weakened.Meanwhile, both stress deviation and average stress of two ends are changed.It is considered that the tiny gap between the rock specimen and transmission bar can improve the stress distribution in rock specimens during loading process in SHPB tests.When endsurface nonparallelism is equal to or less than 0.10%, nonparallel end-surface can shorten the stress equilibrium time, avoid premature failure of rock-like materials, and improve stress uniformity of rock specimen.Compared with the parallel end-surface rock specimen, the stress equilibrium time for the rock specimen with end-surface nonparallelism of 0.10% is reduced by 4.3%, 5.7%, 5.8%, 6.8%, and 7.0% corresponding to Young's modulus of 14 GPa, 21 GPa, 28 GPa, 35 GPa, and 42 GPa.e e ect of nonparallel endsurface is very small, which can be negligible.Hence, endsurface nonparallelism of rock specimen is suggested to be controlled within 0.10% when conducting SHPB tests.4.3.2.0.10% ≤ c ≤ 0.40%.As shown in Figure 10, when endsurface nonparallelism ranges from 0.10% to 0.40%, the stress equilibrium time increases in a step type.And step change range varies for various Young's moduli.For Young's modulus of 14 GPa, one step change occurs when end-surface nonparallelism ranges from 0.10% to 0.20%.For Young's modulus of 21 GPa, two step changes occur when Advances in Civil Engineering Under the same end-surface nonparallelism, stress equilibrium time decreases with the increase of Young's modulus.And the descent of stress equilibrium time is mainly occurred when Young's modulus ranges from 14 GPa to 28 GPa.When Young's modulus is equal to or greater than 28 GPa, the stress equilibrium times for various Young's moduli are roughly the same.When end-surface nonparallelism is 0.30%, the stress equilibrium time are 76.1 μs, 71.6 μs, 60.7 μs, 60.1 μs, and 59.9 μs corresponding to Young's modulus of 14 GPa, 21 GPa, 28 GPa, 35 GPa, and 42 GPa.It can be found that when Young's modulus is ranging from 28 GPa to 42 GPa, the in uence of Young's modulus on stress equilibrium time is very small and can be negligible.

Conclusions
(1) Fluctuation e ect exists in both re ected stresses and transmitted stresses, and it is enhanced with the increase of end-surface nonparallelism.With the increase of Young's modulus, the uctuation e ect in re ected stresses is enhanced, while the uctuation e ect in transmitted stresses is weakened.Under the same Young's modulus, the amplitude of re ected stresses gradually rises, while the amplitude of transmitted stresses gradually decreases with the increase of end-surface nonparallelism.(2) Stress deviation history presents a wave trend with the loading time, and stress deviation achieves its maximum value at about 40 μs.Under the same Young's modulus, the amplitude of stress deviation history increases with the increase of end-surface nonparallelism.Under the same end-surface nonparallelism, the amplitude of stress deviation history decreases with the increase of Young's modulus.(3) Under the same Young's modulus, stress nonuniformity coe cient for various end-surface nonparallelism attenuates in a serrated uctuation with the increase of loading time.And stress nonuniformity coe cient increases with the increase of end-surface nonparallelism.e in uence of endsurface nonparallelism on stress nonuniformity coe cient is weakened with loading time going on, and it mainly concentrates on the rst 70 μs of loading process.(4) Under the same Young's modulus, the stress equilibrium time rst decreases slightly then increases in a step type with the increase of end-surface nonparallelism.When end-surface nonparallelism is 0.10%, stress equilibrium time achieves its lowest value whatever the Young's modulus is.When Young's modulus exceeds 28 GPa, the in uence of Young's modulus on stress equilibrium time can be negligible.(5) Nonparallel end-surface leads to two reverse results for stress uniformity during SHPB loading process.One is extending stress equilibrium time, and the other is shortening stress equilibrium time.In uence on shortening stress equilibrium time is weak and can be negligible, while in uence on extending stress equilibrium time is great.Hence, end-surface nonparallelism of rock specimen is suggested to be controlled within 0.10% when conducting SHPB tests.

Figure 2 :
Figure 2: ree-dimensional numerical model including incident bar, transmission bar, and rock specimen.

Figure 8 :
Figure 8: Relation between maximum stress deviation Δσ max and end-surface nonparallelism c.

Figure 10 :
Figure 10: Relation between stress equilibrium time t u and endsurface nonparallelism c.

Table 1 :
Material parameters for HJC model of rock material with Young's modulus of 14 GPa.ρ (g/cm3)G (GPa) f c (MPa) T (MPa)