To evaluate the effect of nonparallel end face of rocklike specimens in SHPB tests, the characteristics of energy dissipation are analyzed based on numerical simulations for endface nonparallelism from 0% to 0.40% and Young’s modulus from 14 GPa to 42 GPa. With the increment of endface nonparallelism, both energy consumption density and dissipated energy density show a slight increase trend, while releasable elastic strain energy density presents a slight decrease trend. Existence of elastic unloading in the damaged rocklike specimen leads to a reduction of energy consumption density and a constant dissipated energy density during total strain shrinkage. At peak dynamic stress, dissipated energy density presents a linear upward trend with the increment of endface nonparallelism and Young’s modulus, while releasable elastic strain energy density shows an inverse trend. A binary linear regression equation is deduced to estimate the energy dissipation ratio. Mechanical damage evolution of the rocklike specimen is divided into two regions in line with the two regions in dynamic stressstrain curves, and the transition between the slowgrowth region and rapidgrowth region is shifted to the right with the increment of endface nonparallelism. Due to the presence of nonparallel end face, fluctuation presents in energy density evolution and mechanical damage evolution. The fluctuation is enhanced with the increment of endface nonparallelism and weakened with the increase of Young’s modulus. Based on energy density evolution and mechanical damage evolution analyses, the maximum endface nonparallelism should be controlled within 0.20%, twice the value in ISRM suggested methods, which reduces the cost and time for processing rocklike specimens.
The split Hopkinson pressure bar (SHPB), also known as Kolsky bar, is an extensive, convenient, and reliable technique to characterize the behaviors of solid materials at a high strain rate, mainly in the range of 10^{2}–10^{4}s^{−1} [
With the development of SHPB technique, the accuracy and precision of dynamic mechanical characteristics are the key issues for SHPB tests. Based on numerical simulations, six types of incident bar misalignment in SHPB apparatus are investigated, and the distorted signal generated by bar misalignment is mainly induced by the presence of flexural modes of vibration and affects the SHPB test results adversely [
The deformation and failure of rock can be considered as an irreversible process of energy dissipation [
Considering the processing deviation of the rocklike specimen, numerical simulations of SHPB tests are conducted for nonparallel endface rocklike specimens with various Young’s moduli by LSDYNA. During numerical simulation, endface nonparallelism ranges from 0.0% to 0.40%, and Young’s modulus ranges from 14 GPa to 42 GPa. Then, energy density dissipation is analyzed to reveal the effect of endface nonparallellism. As mechanical damage evolution is closely related with energy dissipation, the influence of nonparallel end face on mechanical damage evolution is also studied.
Based on the physical Φ50 mm SHPB apparatus, a series of 3D finite element models without a striker are set up to conduct SHPB tests for rocklike materials. Basically, a typical SHPB apparatus consists of a striker, an incident bar, and a transmitted bar. As shown in Figure
A schematic diagram of SHPB setup in the 3D finite element model.
ANSYS is used to prepare 3D finite element models, and the SOLID164 element with one integration point is employed to save the computer time [
In physical SHPB tests, a compressive loading stress wave is generated by launching a striker impacting on the incident bar. For traditional rectangular compressive stress wave, premature failure of the rocklike specimen before stress equilibrium makes test results unreliable. Moreover, high signal oscillation presents in rectangular compressive stress wave due to the wave dispersion [
The rocklike specimen with a length to diameter ratio of 0.5 is modeled and sandwiched between the incident bar and transmitted bar [
As for the dynamic characteristics test of rocklike materials, the incident and transmitted bars in physical SHPB apparatus are all made of a homogenous and isotropic alloy steel, and they keep in a linear elastic deformation state during SHPB tests. Therefore, the elastic constitutive model for an isotropic elastic material in LSDYNA is selected for both incident and transmitted bars. According to the alloy steel properties in physical SHPB apparatus, the density, Young’s modulus, and Poisson’s ratio are set as 7.85 g/cm^{3}, 210 GPa, and 0.30, respectively.
Considering the high strain rate in SHPB tests, the HJC constitutive model, for materials subjected to large strain, high strain rate, and high pressure in LSDYNA, is employed for the rocklike specimen [
Material parameters of HJC constitutive model for rocklike material [





SF_{MAX} 





2.47  0.13  0.79  1.60  0.007  4.0  11.67  0.045  1.0  0.61 


EF_{MIN} 











0.005  7.07  43.33  0.00278  1  0.1  85  −171  208  0.004 
As five kinds of Young’s moduli is considered, related material parameters of the HJC constitutive model should be modified with Young’s modulus. In the HJC constitutive model, shear modulus
In line with the physical SHPB test, four hexahedron elements at the same cross section referring to pair strain gages symmetrically mounted on the surface of the bars are chosen to export
Acquired incident, reflected, and transmitted stresses when Young’s modulus is 28 GPa.
As shown in Figure
The fundamental assumptions of SHPB technique are onedimensional stress wave propagation and stress uniformity [
Stresstime histories on two ends of parallel endface rocklike specimens.
As shown in Figure
When halfsine loading stress wave propagates in an elastic steel bar, both elastic deformation and motion are generated in elastic steel bars. The energy carried by stress wave is composed of elastic strain energy and kinetic energy, and the elastic strain energy is basically equal to the kinetic energy for the elastic stress wave [
An isothermal process is assumed in SHPB tests, and there is no heat exchange with the external environment. According to the first law of thermodynamics, also known as the energy conservation law, the absorption energy of rocklike specimen can be expressed as follows by neglecting the energy loss in SHPB tests [
According to the fundamental assumption of SHPB technique, stress uniformity, equation (
Based on equations (
Energytime histories during SHPB numerical simulation.
As illustrated in Figure
As only one type of halfsine loading stress wave is considered, the incident energy, namely, the whole input energy, is a constant value, 388.78 J. Based on SHPB numerical simulations, the variation of reflected energy and transmitted energy with the increment of endface nonparallelism
Variation of reflected energy and transmitted energy with endface nonparallelism.
As shown in Figure
Figure
Variation of total absorption energy with endface nonparallelism.
As illustrated in Figure
To illustrate the effect of energy dissipation per unit volume, energy consumption density, also known as specific energy absorption, is defined as the energy consumed for breaking the rocklike specimen per unit volume. Therefore, the energy consumption density presents a similar variation trend to absorption energy. In onedimensional loading condition, energy consumption density is defined as the area of dynamic stressstrain curve and can by calculated as follows:
According to the research of Wang et al. [
Based on equations (
Energy density evolution curves of the rocklike specimen.
Obviously from Figure
Due to the nonparallel end face, fluctuation presents in reflected stresses and transmitted stresses [
Under uniaxial compression, a typical complete stressstrain curve consists of five stages, crack closure, elastic, cracking, postfailure, and residual, and the idealized stressstrain curve can be basically divided into two regions, prefailure region and postfailure region [
Releasable elastic strain energy density and dissipated energy density at peak dynamic stress (unit: J·cm^{−3}).

14 GPa  21 GPa  28 GPa  35 GPa  42 GPa  










 
0  1.808  0.260  1.372  0.344  1.101  0.416  0.947  0.409  0.818  0.422 
0.05  1.802  0.254  1.368  0.378  1.100  0.431  0.941  0.453  0.809  0.442 
0.10  1.803  0.319  1.356  0.394  1.098  0.420  0.937  0.445  0.804  0.459 
0.15  1.791  0.280  1.350  0.393  1.092  0.447  0.937  0.450  0.806  0.462 
0.20  1.790  0.303  1.343  0.407  1.092  0.468  0.937  0.471  0.809  0.476 
0.25  1.786  0.348  1.342  0.415  1.093  0.473  0.936  0.492  0.808  0.511 
0.30  1.773  0.384  1.330  0.448  1.081  0.507  0.938  0.526  0.805  0.548 
0.35  1.742  0.439  1.316  0.483  1.070  0.539  0.935  0.565  0.808  0.581 
0.40  1.708  0.471  1.286  0.534  1.051  0.582  0.923  0.599  0.799  0.629 
As shown in Table
In order to study the influence of nonparallel end face on energy dissipation characteristics, the energy dissipation ratio
With the increment of endface nonparallelism, energy dissipation ratios for various Young’s moduli are drawn in Figure
Energy dissipation ratio versus endface nonparallelism.
As clearly illustrated in Figure
Intercept
Hence, with endface nonparallelism and dimensionless Young’s modulus as variables, a binary linear regression equation is deduced for energy dissipation ratio and is expressed as follows:
As deformation and failure progress of the rocklike material is also the progress of energy dissipation, mechanical damage can be defined as the ratio of dissipated energy density to total energy consumption density, which can be calculated as follows [
For a certain dynamic stressstrain curve, the total energy consumption density
Energy consumption density for mechanical damage calculation (unit: J·cm^{−3}).




14 GPa  21 GPa  28 GPa  35 GPa  42 GPa  
0  2.658  2.439  2.142  1.630  1.546 
0.05  2.627  2.427  2.190  1.661  1.576 
0.10  2.728  2.436  2.246  1.691  1.607 
0.15  2.731  2.462  2.273  1.725  1.641 
0.20  2.838  2.482  2.296  1.769  1.673 
0.25  2.846  2.526  2.337  1.904  1.729 
0.30  2.886  2.576  2.407  2.012  1.754 
0.35  2.930  2.646  2.407  2.272  1.895 
0.40  2.969  2.689  2.456  2.345  2.229 
Figure
Mechanical damage evolution curves of the rocklike specimen.
As clearly seen from Figure
Transition between two regions of damage evolution is obvious and easy to determine in small Young’s modulus, while it becomes difficult to determine in large Young’s modulus, as illustrated in Figure
For parallel end face rocklike specimens, the dynamic stressstrain curve for various Young’s moduli is illustrated in Figure
Dynamic stressstrain curve for various Young’s moduli.
As obvious in Figure
Dynamic characteristics, energy density evolution, and mechanical damage evolution are desired by conducting SHPB tests. Both nonparallel end face and Young’s modulus have a great impact on SHPB test results of rocklike specimens. Young’s modulus of the rocklike material is an intrinsic characteristic of pending tested rocklike materials, and it is unknown before the test. In order to make the SHPB test results reliable, the errors induced by the rocklike specimen processing deviation should be controlled within an acceptable level. It is infeasible to give an allowable processing deviation for various Young’s moduli of rocklike materials. Therefore, a common practice is given an allowable processing deviation without regard to Young’s modulus.
When endface nonparallelism is 0.20%, the curve shape of both energy density evolution and mechanical damage evolution remain unchanged, and the error induced by nonparallel end face is small. According to above analyses, maximum endface nonparallelism can be controlled within 0.20%, namely, the allowable processing deviation is 0.05 mm for 25 mm height rocklike specimen, which is twice the value in ISRM suggested methods [
Regarding nonparallel end face of rocklike specimens in SHPB tests, numerical simulations have been performed with endface nonparallelism varying from 0% to 0.40% and Young’s modulus ranging from 14 GPa to 42 GPa. Then, the characteristics of energy dissipation and mechanical damage are analyzed to evaluate the effects of nonparallel end face. The main conclusions are summarized as follows:
With the increment of endface nonparallelism, both absorption energy and reflected energy show a slight increase trend, while transmitted energy presents a slight decrease trend.
Both energy consumption density and dissipated energy density increase with the increment of endface nonparallelism, while releasable elastic strain energy density reduces slightly. Due to the presence of nonparallel end face, fluctuation presents in the evolution of both releasable elastic strain energy density and dissipated energy density. The fluctuation is enhanced with the increment of endface nonparallelism and weakened with the increase of Young’s modulus.
At the peak dynamic stress, dissipated energy density presents a linear upward trend with the increment of endface nonparallelism and Young’s modulus, while releasable elastic strain energy density shows a linear downward trend. A binary linear regression equation is deduced to estimate energy dissipation ratio with endface nonparallelism and Young’s modulus.
In line with two regions in the dynamic stressstrain curve, mechanical damage evolution of the rocklike specimen is also divided into to two regions, slowgrowth region and rapidgrowth region. In the slowgrowth region, fluctuation presents due to the presence of nonparallel end face, and it weakens with the increase of Young’s modulus. Transition between two regions is shifted to the right with the increment of endface nonparallelism, which indicates an increase of both strain and damage threshold values.
Based on energy density evolution and mechanical damage evolution analyses, maximum endface nonparallelism can be controlled within 0.20%, namely, the allowable processing deviation is 0.05 mm for 25 mm height rocklike specimen. The suggested allowable processing deviation is twice the value in ISRM suggested methods, which reduces the cost and time for processing rocklike specimens.
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the National Natural Science Foundation of China (no. 51774011), Anhui Provincial Natural Science Foundation (no. 1808085QE148), project funded by China Postdoctoral Science Foundation (no. 2018M642504), Natural Science Research Project of Colleges and Universities in Anhui Province (no. KJ2017A097), Young Teacher Scientific Research Project of Anhui University of Science and Technology (no. QN201607), Doctoral Fund Project of Anhui University of Science and Technology (no. 11674), Science and Technology Project of Department of Housing and UrbanRural Development of Anhui Province (no. 2017YF08), National Innovation and Entrepreneurship Training Program for College Students (no. 201810361029), and Anhui Provincial Innovation and Entrepreneurship Training Program for College Students (no. 201810361174).