J. Cent. South Univ. Technol. (2008) 15: 218-223

DOI: 10.1007/s11771-008-0042-7

Relationship between diameter of split Hopkinson pressure bar and minimum loading rate under rock failure

LI Xi-bing(李夕兵)¹, HONG Liang(洪 亮)^{1, 2}, YIN Tu-bing(尹土兵)¹,

ZHOU Zi-long(周子龙)¹, YE Zhou-yuan(叶洲元)¹

(1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Department of Civil Engineering, Hunan City University, Yiyang 413000, China)

Abstract: In order to investigate the relationship between bar diameter and loading rate of the split Hopkinson pressure bar(SHPB) setup under the failure of rock specimen and realize the medium strain rate loading of specimen, new SHPB setups with different elastic bar’s diameters of 22, 36, 50 and 75 mm were constructed. The tests were carried out on these setups at different loading rates, and the specimens had the same diameter of elastic bars and same ratio of length to diameter. The test results show that the larger the elastic bar’s diameter is, the less the loading rate is needed to cause specimen failure, they show good power relationship,        and that under the same strain rate loading, specimens are broken more seriously with larger diameter SHPB setup than with smaller one.

Key words: rock failure; Hopkinson pressure bar; diameter; minimum loading rate; medium strain rate

1 Introduction

Since 1970s, the split Hopkinson pressure bar (SHPB) has been widely used to measure the dynamic properties of materials at high strain rates^[1-2]. Conventional SHPB is mainly used to measure rock dynamic properties in the strain rate range of 10³-10⁴ s^-1. At greater strain rates, gas guns have been successfully deployed to measure the response of rocks. It is relatively easy to obtain the quasi-static strength at low     loading rates (10^-5-10^-2 s^-1) by normal hydraulic testing machines or servo-controlled testing machines, and the quasi-dynamic strength at moderate to relatively high strain rates (10^-2-10⁰ s^-1) can be realized by pneumatic power rapid loading machines^[1]. Nowadays, limited by the experimental and technical conditions, little is known about the mechanical characteristics of brittle rock at medium strain rate^[3-6]. However, these characteristics are essential to understand in the fields of mechanical rock drills, deep mining engineering, anti-explosion design of large span underground chamber and structure. Some domestic and foreign institutions, such as Los Alamos National Laboratory of America and the University of Science and Technology of China, have studied on the experimental techniques of materials at the medium strain rate. Using the hydraulic impact testing apparatus developed by the University of Science and Technology of China, tension dynamic characters of metal materials at the strain rate of 22 s^-1 have been obtained^[7]. The specimen size of rock cannot be processed as small as that of metal materials because of its non-uniform; the specimen processed with larger size will inevitably lead to strain rate descending in the dynamic test on rock.

Based on the fact that strain rate is in reverse proportion to the length of samples on the SHPB impact tests, the strain rate can be reduced by increasing the length of samples. Usually the ratio of length to diameter (L/D) is about 0.5, which is favorable to reducing the inertial effect and end effect (the sample’s L/D ratio is  when the inertial effect is zero, where μ is the Poisson ratio^[8]), so increasing the length of sample will result in larger diameter of sample. The diameter of the sample is slightly smaller than that of the Hopkinson bars. Increasing diameter of the elastic bars, the diameter and length of the sample will also increase accordingly. It is found that the optimum diameter of the bars should be selected to ensure rock fracture on the precondition of not dropping the impact velocity in SHPB, and the constitutive relationship of the rock at medium strain rate can be obtained in the preliminary experimental research^{[3, 9]}. Many SHPB systems of large diameter were used for testing rock or concrete in recent years^[10-14]. Yet the quantified relationships between the minimum loading rate and the diameter of SHPB, and the suitable diameter of bars used to realize medium strain rate loading of rock have not been investigated. The large

diameter of conventional SHPB certainly corresponds to long input and output bar due to the 2D effect, which is inconvenient and difficult to be commissioned in the test. A conventional loading method with large diameter does not overcome strong oscillation and dispersion of the incident wave, and the data must be rectified through dispersion correction method^[15-18]. In order to establish the relationship between the minimum loading rate and the diameter of SHPB, and provide the basis for determining the suitable diameter of bars to realize medium strain rate loading of the rock, the impact experiments were carried out at different loading rates on the self-developed SHPB equipments with elastic bars diameters of 22, 36, 50 and 75 mm, together with a striker that can initiate a half-sine waveform in the input bar and avoid P-C oscillation^[19-20]. The granite samples had the same diameter of elastic bars and the same ratio of length to radius in tests.

2 Experimental

2.1 Equipments

There are four kinds of elastic bars in SHPB measurement systems that are 22, 36, 50 and 75 mm in diameter, respectively. Material of bars is 40Cr alloy steel whose density and P-wave velocity are separately  7 810 kg/m³ and 5 410 m/s. The lengths of input and output bars are both 1.20 m in d22 mm series, and 1.50 m in d36 mm series, 2.00 m in d75 mm series, while the lengths of input and output bar in d50 mm series are 2.00 m and 1.50 m respectively. Shapes of the strikers are shown in Fig.1. The material and maximum diameter of strikers are the same as those of the corresponding input/output bars. The generating device for stress wave consists of high-pressure gas tank, air pressure control valve, airflow control switch and striker launching set. The structure of striker is based on the inversion design where a half-sine wave loading can be generated on impact of the input bar^{[1, 3, 13]}. The typical shapes of four loading waves obtained from tests are shown in Fig.2.

Fig.1 Striker series of SHPB measurement equipment

Fig.2 Four kinds of typical loading waves obtained from tests

2.2 Samples

The samples used in dynamic testing were cored from the same granite block and processed by the ZS200 standing coring device, DQ-4 rock cutter and SHM-200 double end-faces abrader. The ratio of length to diameter is controlled in the scope of 0.5-0.6. All the test samples were labeled according to the cores from which the samples obtained in the processing. For example, test sample “2-1” indicates the first sample obtained from the second core. The dimensions, mass, P-wave velocity of samples were measured after air-dry. The samples were deserted whose density or P-wave velocity was abnormal.

3 Results and discussion

3.1 Experimental results

The SHPB impacting experiments were conducted to investigate damage of the granite specimen at different strain rate loadings. The strain rate of specimen was changed through changing the incident wave amplitude that was controlled by adjusting the position of the striker in the launch tube and the gas pressure. Based on the actual damages of test specimen, the strain rates of each bar series were divided into 3-5 levels. The strain rate that could only result in crack coalescence of sample was taken as the lowest. The number of each level was determined by the number of samples obtained from a core, 3 at least. The samples would be complemented if the test results had large dispersion. Test results by using d22 mm and d75 mm SHPB are shown in Fig.3 and Table 1, Fig.4 and Table 2, respectively.

Four loading sets were conducted in the impact test by using d36 mm SHPB, the range of average strain rate of samples measured from high to low was 184.51- 41.12, 135.49-169.24, 92.48-111.58 and 75.51-106.65 s^-1, respectively, and the fracture status of the    samples correspondingly was mainly chips that include

Fig.3 Typical fractures of samples at degressive loading rate by using d22 mm SHPB: (a) Sample 46-4; (b) Sample 45-2; (c) Sample 40-5; (d) Sample 48-3

Table 1 Test results of samples subjected to impact stress wave from d22 mm SHPB

Fig.4 Typical fractures of samples at degressive loading rate by using d75 mm SHPB: (a) Sample 13-3; (b) Sample 11-3; (c) Sample 4-1; (d) Sample 8-2; (e) Sample 9-3

Table 2 Test results of samples subjected to impact stress wave from d75 mm SHPB

4-7 fragments, 10-13 fragments including a little chips, 4-6 fragments and 2-3 fragments or crack coalescence. Three loading sets were conducted in the impact test by using d50 mm SHPB, the range of average strain rate of samples measured from high to low was 100.58-129.05, 79.94-113.36 and 44.75-62.94 s^-1, respectively, and the fracture status of samples correspondingly was mainly chips including 5-8 fragments, 3-8 fragments including a little chips, and 2 fragments or crack coalescence.

3.2 Discussion

The above experimental results imply that with changing strain rate from high to lower level, the fracture status of samples is mainly from chips to block, and the fracture degree is on descending till occurring crack coalescence only in the same diameter SHPB test. But there always exists a lowest strain rate that can cause rock samples failure, such as, that of the d22 mm sample is 120.93-178.52 s^-1, d36 mm sample is 75.51-1 06.65 s^-1, d50 mm sample is 44.75-62.94 s^-1; d75 mm sample is 41.09-48.33 s^-1. The rock samples will not break under once impact if the strain rate loading is under the lowest one^[3]. The average strain rate of every level was chosen as the representative value. Fig.5 shows the distribution characteristics of strain rate that can cause rock fracture under different bar diameters.

Fig.5 Distribution of loading rates causing samples breakage subjected to SHPB with different diameter

Fig.5 indicates that the minimum average strain rate that causes rock breakage can be significantly reduced by increasing the diameter of Hopkinson bars. Curve fitting of the diameter of Hopkinson bar and their corresponding minimum average strain rates shows that both of them represent a favorable power relationship:

                                   (1)

where  D is the diameter of Hopkinson bar, a and b are the constants interrelated with rock characteristics. The values of a and b for the granite samples in this test are 3843.7 and 1.05, respectively, and the correlation coefficient is equal to 0.96. As a result, increasing the diameter of Hopkinson bar is one of the effective approaches to realize the medium strain rate loading of rocks.

Rock fracture status of the same loading rate under different Hopkinson bars was compared, for example, d75 mm samples are in chips, d50 mm samples are in blocks and some chips, d36 mm samples are in blocks, while d22 mm samples is not broken when the average strain rate is about 100 s^-1. Obviously, the specimens loaded by larger diameter bar are more seriously broken than the smaller one under the same loading rate. In addition, the variation range of average strain rate of d75 mm samples is 45.81-74.48 s^-1, that of d50 mm samples is 53.67-96.42 s^-1, that of d36 mm samples is 89.26- 152.63 s^-1, that of d22 mm samples is 157.29-263.42 s^-1 with the same fracture status transferring from the crack coalescence to chips and blocks, and the average strain rate is increased by 28.67, 42.75, 63.37 and 106.13 s^-1, respectively. It can be seen that under the same increment of strain rate, the breakage effect of specimens in larger SHPB system is more significant than that in smaller one. This has considerably referenced value in designing the impact rock machines and improving rock breakage efficiency.

4 Conclusions

1) Under the SHPB tests with the same diameter bars, the breakage degree of specimens turns weaker with decreasing the strain rate. There always exists a lower limit of strain rate causing specimen breakage in different diameter bars. The lowest loading rate for specimens breakage descends remarkably with increasing diameter, and they show a favorable power relationship. It is an effective approach to realize the rock medium strain rate by increasing the diameter of SHPB.

2) Under the same strain rate loading, specimens with larger diameter bar are more intensive broken than that with smaller one. This test result can be used as a reference in designing the impact rock machines and impacting rock breakage efficiently.

References

[1] LI Xi-bing, GU De-sheng. Rock impact dynamics[M]. Changsha: Central South University of Technology Press, 1994. (in Chinese)

[2] FIELD J E, WALLEY S M, PROUD W G, GOLDREIN H T, SIVIOUR C R. Review of experimental techniques for high rate deformation and shock studies[J]. International Journal of Impact Engineering, 2004, 30(7): 725-775.

[3] LI X B, LOK T S, ZHAO J. Dynamic characteristics of granite subjected to intermediate loading rate[J]. Rock Mechanics and Rock Engineering, 2005, 38(1): 21-39.

[4] PERKIN R D, GREEN S J, FRIEDMAN M. Uniaxial stress behavior of porphritic tonalite at strain rates to 10³/sec[J]. Int J Rock Mech Min Sci, 1970, 7(5): 527-535.

[5] GREEN J S, PERKINS R D. Uniaxial compression tests at varying strain rates on three geologic materials[J]. Basic and Applied Rock Mechanics, 1970, 3(5): 35-37.

[6] BLABTON T L. Effect of strain rates from 10^-2-10 s^{-1 in triaxial compression tests on three rocks[J]. Int J Rock Mech Min Sci, 1981, 18(1): 47-62.}

[7] ZHANG Xue-feng, XIA Yuan-ming. Development of material testing apparatus for intermediate strain rate test[J]. Journal of Experimental Mechanics, 2001, 16(1): 13-18. (in Chinese)

[8] DAVIES E D H, HUNTER S C. The dynamic compression testing of solids by the method of the split Hopkinson pressure bar[J]. J Mech Phys Solids, 1963, 11(3): 155-179.

[9] LOK T S, LI X B, ZHAO P J. Uniaxial compression tests on granite and its stress-strain relationship at high strain rate [C]// Proceedings of the 2001 ISRM International Symposium. WANG S J. Lisse: AA Balkema, 2001: 85-87.

[10] ROSS C A, TEDESCO J W, KUENNEN S T. Effect of strain rate on concrete strength[J]. ACI Materials Journal, 1995, 92(1): 37-47.

[11] ZHENG S, HAUSSLER-COMBE U, EIBL J. New approach to strain rate sensitivity of concrete in compression[J]. Journal of Engineering Mechanics, 1999, 125(12): 1403-1410.

[12] EIBL J, SCHMIDT-HURTIENNE B. Strain-rate-sensitive constitu- tive law for concrete[J]. Journal of Engineering Mechanics, 1999, 125(12): 1411-1420.

[13] LI Xi-bing, ZHOU Zi-long, WANG Wei-hua. Construction of ideal striker for SHPB device based on FEM and neural network[J]. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(23): 4215-4218. (in Chinese)

[14] LOK T S, LI X B, LIU D, ZHAO P J. Testing and response of large diameter brittle materials subjected to high strain rate[J]. Journal of Materials in Civil Engineering, 2002, 14(3): 262-269.

[15] LI X B, LOK T S, ZHAO J, ZHAO P J. Oscillation elimination in the Hopkinson bar apparatus and resultant complete dynamic stress-strain curves for rocks[J]. International Journal of Rock Mechanics and Mining Sciences, 2000, 37(7): 1055-1060.

[16] GONG J C, MALVERN L E, JENKINS D A. Dispersion investigation in the split Hopkinson pressure bar[J]. Journal of Engineering Materials and Technology, 1990, 112(3): 309-314.

[17] ZHAO Han, GERARD G. On the use of SHPB techniques to determine the dynamic behavior of materials in the range of small strains[J]. International Journal of Solids and Structures, 1996, 33(23): 3363-3375.

[18] LIFSHITZ J M, LEBER H. Data processing in the split Hopkinson pressure bar tests[J]. International Journal of Impact Engineering, 1994, 15(6): 723-733.

[19] LI Xi-bing, LIU De-shun, GU De-sheng. Effective method of eliminating the oscillation of rock dynamic stress-strain-strain rate curves[J]. J Cent South Univ Technol: Natural Science, 1995, 26(4): 457-460. (in Chinese)

[20] LI Xi-bing, GU De-sheng, LAI Hai-hui. On the reasonable loading stress waveforms determined by dynamic stress-strain curves of rocks by SHPB[J]. Explosion and Shock Waves, 1993, 13(2): 125-130. (in Chinese)

(Edited by CHEN Wei-ping)

Foundation item: Project(10472134) supported by the National Natural Science Foundation of China

Received date: 2007-07-22; Accepted date: 2007-09-16

Corresponding author: LI Xi-bing, PhD, Professor; Tel: +86-731-8879612; E-mail: xbli@mail.csu.edu.cn