Split-Hopkinson-bar Tests on Rock under Confining Pressure Description of apparatus and test results is given. Attention is given to complete measurement of principal strain and resolution of initial stress-strain data by R. J. Christensen, S. R. Swanson and W. S. Brown

ABSTRACT The effect of dynamic loading on the stressstrain and failure characteristics of nugget sandstone was investigated at strain rotes from 102 to 103 per second and confining pressures to 30 k.si. The apparatus developed for these experiments consists of a conventional split Hopkinson bar enclosed in o pressure vessel. The apparatus permitted determination and recording of all principal stresses and strains from the onset of loading to failure. A description of the experimental techniques used in obtaining data including the method of determining stress and strain, the method of reducing data, and use of various shaped projectiles to tailor the input-stress wave ore reported together with experimental results for nugget sandstone. All rocks tested exhibited an increase in strength as the loading rate increased. The dynamic stress-strain curves were similar in shape to curves from quasi-static testing of the same material.

Introduction A t t e m p t s to m a k e realistic calculations of rock be-

R. 1. Christensen, S. R. Swanson and W . S. Brown are Research Assistant, Assistant Research Professor and Professor o~ Mechanical Engineering, respectively, College o~ Engineering, University of Utah, Salt Lake City, Utah 84112. Paper scheduled for presentation at 1972 Fall Meeting due to be held in Seattle, Wash. on October 17-20. Orlginal manuscript received: ]anuary 18, 1972. Final version received: March 15, 1972.

havior and r o c k - s t r u c t u r e interaction under loadings such as blast or e a r t h q u a k e stress waves require valid stress-strain relations for the rock. Previous l a b o r a t o r y m e a s u r e m e n t s of the complete stressstrain b e h a v i o r of rock have been p e r f o r m e d at strain rates of 10 -'~ to 10-1/sec under confining pressures up to 7 Kb*. TM Tests at higher strain rates have been p e r f o r m e d w i t h o u t confining pressure using the Hopk i n s o n - b a r t e c h n i q u e 5,6 although, in general, these tests have not b e e n i n s t r u m e n t e d to m e a s u r e all strain components. E x t r e m e l y high pressure and strain rates have been produced in flying-plate impact tests 7 which result in a condition of o n e - d i m e n s i o n a l strain. The sensitivity of rock strength and stress-strain b e h a v i o r to confining pressure is a w e l l - k n o w n effect. Rock has also been found to exhibit a n u m b e r of interesting inelastic effects including the bulking or v o l u m e - i n c r e a s e p h e n o m e n o n that occurs as the rock is stressed beyond a p p r o x i m a t e l y o n e - h a l f to t h r e e fourths of the failure stress. This effect, often t e r m e d dilatancy s,14, is associated w i t h microcrack generation and grain sliding. It has b e e n conjectured that this and other features of rock b e h a v i o r may be strongly influenced by strain rate and that dilatancy * One kilobar = 14,500 psi.

Fig. 1--Dynamic apparatus

508 I November 1972

might be suppressed at higher strain rates. The p r e s e n t study was u n d e r t a k e n to answer these questions b y d e t e r m i n i n g all principal stresses and strains in triaxial tests on rock conducted at higher strain rates.

ENDPLATE "O'-RINGSEALS PISTON

Apparatus A n overall view of the test apparatus developed for these experiments is shown in Fig. 1. The apparatus is essentially a split Hopkinson bar enclosed by a pressure vessel with the i n p u t bar extending from the vessel. The specimen is loaded with a stress wave produced by impacting the loading b a r with a projectile fired b y a l o w - p r e s s u r e - a i r gun. This made it possible to determine stresses at each end of the s p e c i m e n at the strain rates utilized in this study. Both the pressure vessel and the g u n b a r r e l were m o u n t e d on rails on a heavy I - b e a m to facilitate alignment. The apparatus is somewhat similar in basic design to t h a t used previously b y Chalupnik and Ripperger. 9 F i g u r e 2 illustrates the method of m o u n t i n g the specimen, the i n p u t b a r and the output b a r in the pressure vessel. The vessel is 17-in. long and is constructed from 4340 steel heat treated to RC 50 to permit operation at confining pressures up to 2 Kb. The base plug containing the electrical leads and its seal was similar to those used in the pressure vessels in previous static work. 1-a A double O - r i n g was used for the i n p u t - b a r seal. This seal is c o n v e n i e n t and adequate for the pressure range of interest in these experiments. Pressure up to 30 ksi was obtained from a Haskel a i r - d r i v e n fluid p u m p with kerosene as the confining medium. The a l i g n m e n t of the elastic bars is critical in d y namic compression tests. A l i g n m e n t was established by centering the bars with brass bushings, which were located by the p r e c i s i o n - g r o u n d inside surface of the pressure vessel. It was established in prel i m i n a r y tests that the distortion of the compression pulse b y the bushings and O - r i n g seals was negligible. In fact, the entire pressure-vessel a r r a n g e m e n t t u r n e d out to be "clean" in the sense that little interaction occurred between the vessel and the loading bars. The c o m p r e s s e d ' a i r g u n shown in Fig. 3 consisted of an air t a n k to store compressed air, a pressure gage to m o n i t o r the air pressure, a fast opening A m e r i c a n S t a n d a r d solenoid valve, and the gun barrel. Operating at a charge pressure of 300 psi results in a velocity of approximately 185 ft/sec for a 6-in.-long steel projectile. At this velocity, the stress is approxim a t e l y 170 ksi, which is approaching the yield point of the i n p u t bar.

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Specimens The original s p l i t - H o p k i n s o n - b a r tests b y Kolsky I0 used v e r y t h i n (wafer) specimens since this i n creased the strain rate and simplified the data reduction as a x i a l - w a v e effects could be neglected. This practice has been modified b y several investigators since the end effects i n short specimens influence the test results. 11 This is p a r t i c u l a r l y true in rocks, since the failure strength increases with the m e a n compressive stress. End friction produces lateral restraint and compressive stresses and, therefore, the a p p a r e n t failure strength can increase in short speci-

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Experimental Mechanics I 509

mens and give erroneously high results. For this reason, the rock specimens used in this investigation were made i n the shape of right circular cylinders 1/2 in. in diameter b y 1 in. i n length. The specimens were cored and ground with e n d s b e i n g parallel to • in. using procedures previously developed. 2 The specimens were sealed from the confining fluid with laboratory plastic tubing.

of the stress at each end of the specimen gives the stress at the axial center of the specimen. Results are shown in Fig. 4 where the stress at nine specimen

_~ i00 50

Instrumentation Stress at each end of the specimen was obtained by m e a s u r i n g axial strain on the steel i n p u t and output bars a distance of 0.7 in. from the specimen, using 1/8-in. gage-length foil MM BT125 strain gages. Stress at the specimen-elastic b a r interface was t h e n calculated using o n e - d i m e n s i o n a l elastic-wave theory. The rock strains were m e a s u r e d with MM BT125 strain gages bonded directly to the rock in both the axial and t r a n s v e r s e directions at the specim e n axial center. The size of the gages employed is a compromise b e t w e e n h a v i n g a small gage for fidelity of d y n a m i c response, and having a large gage to average readings over the grain i n h o m o g e n i t y of the rock. All readings were recorded on HewlettP a c k a r d 181A and T e k t r o n i x Model 556 oscilloscopes.

Experimental Techniques and Data Interpretation Although of lesser interest in metals, the m e a s u r e m e n t of both axial and lateral strains is of prime importance i n rock. Because of the s y m m e t r y of the specimens, this gives all three principal strains and permits the calculation of the volume strain. As m e n tioned previously, strains were measured with strain gages in both principal directions at the specimen axial center. This procedure also minimizes errors due to a x i a l - s t r a i n variation n e a r the specimen ends. The necessity of using specimens with an L / D of 2 to avoid frictional end effects requires a consideration of a x i a l - w a v e propagation. Closely coupled with this is the n a t u r e of the rock stress-strain response. I n metals, the elastic portion of the d y n a m i c stressstrain curve is often of less interest and is generally not measured accurately. I n rocks, however, the transition to inelastic behavior is often gradual with no clear point of deviation from the initial (elastic) behavior. Thus, the entire stress-strain curve is of interest. This p r o b l e m was investigated b y both theoretical ( n u m e r i c a l ) and e x p e r i m e n t a l techniques. To investigate the a x i a l - w a v e propagation in the specimen theoretically ( n u m e r i c a l l y ) , a computer p r o g r a m was f o r m u l a t e d that followed the stresswave loading t h r o u g h o u t the specimen. Although strictly one dimensional, the program was w r i t t e n to accept a r b i t r a r y i n p u t - s t r e s s waves and material properties which are a f u n c t i o n of stress. The result was that the axial wave propagation could be computed. A similar procedure has been used b y Conn, 12 using a small n u m b e r of loading steps. P r e s e n t computers make it a trivial m a t t e r to use enough steps (say 500 or 1000) to ensure an adequate r e p r e s e n t a tion of the material a n d the loading. A n u m b e r of cases were investigated and typical results are shown in Fig. 4. F r o m these results it was concluded that, while significant variations i n stress exist axially in the specimen, the stress distribution in most cases is v e r y close to being linear. Thus, a simple average

~ 1 0 ] November 1972

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PUT BAR ITIAL WAVEFORM IAL STRAIN

stress at the input end b y assuming a linear stress distribution. All test cases calculated in this m a n n e r d e m o n s t r a t e d t h a t this r e l a t i v e l y simple d a t a - r e d u c tion m e t h o d of a v e r a g i n g the stress on the two ends of the sample could be used w i t h good accuracy providing the stress rate was less t h a n 1010 psi/sec. This r u l e of t h u m b is f a i r l y c o n s e r v a t i v e but m a y h a v e to be changed slightly for m a t e r i a l s w i t h stress-strain curves which differ radically f r o m this material. It was also m a d e a p p a r e n t in both the n u m e r i c a l studies and e x p e r i m e n t a l data t h a t the accuracy and resolution of the initial portion of the stress-strain curves could be i m p r o v e d if the i n p u t - s t r e s s w a v e could be m a d e m o r e like a r a m p t h a n a step-input. This was p a r t i a l l y accomplished b y using conical projectiles r a t h e r t h a n t h e usual cylinder. A useful p r o jectile configuration was found to be a combination t r u n c a t e d cone and cylinder, fired so t h a t the small end of the cone is the i m p a c t end. A simplified analysis of t h e cone and cylinder 18 m a k e it possible to predict the s t r e s s - t i m e history resulting f r o m t h e t r u n c a t e d - c o n e impact. The comparison of analysis and e x p e r i m e n t is shown in Fig. 5. The loading is divided into t h r e e regions: t h e first corresponding to the initial impact of the t r u n c a t e d cone, the second c o r r e s p o n d i n g to the transition r e gion that is d e t e r m i n e d b y the cone angle, and the final region that depends on the area of the c y l i n d e r joined to the cone. F u r t h e r details on cone i m p a c t w i l l be f o r t h c o m i n g in a f u t u r e paper. A n illustration of the types of loadings t h a t can be e x p e c t e d f r o m different cone configurations is illustrated in Fig. 6. It can be seen that t h e input-stress w a v e can be v a r i e d over a considerable range w i t h this technique. The discussion so far has considered only o n e - d i mensional effects. Recent n u m e r i c a l w o r k b y K a r n e s and B e r t h o l f 14 has d e m o n s t r a t e d t h a t t w o - d i m e n s i o n a l effects can be of m a j o r i m p o r t a n c e n e a r the impact end of a cylinder. F u r t h e r , it is w e l l k n o w n f r o m the classic w o r k of Davies TM t h a t s h o r t - w a v e l e n g t h h a r m o n i c w a v e s lead to a v a r i a t i o n in stress across the rod d i a m e t e r as w e l l as other effects. The u p p e r limit of strain rates t h a t can be achieved e x p e r i m e n tally without h a v i n g to consider t w o - d i m e n s i o n a l effects in the data i n t e r p r e t a t i o n is perhaps still open to question. However, it appears f r o m the dynamic f i n i t e - e l e m e n t studies of I s e n b e r g 16 t h a t the stressw a v e loadings used in the present study w e r e sufficiently slow so as to essentially p r e c l u d e t w o - d i m e n s i o n a l - w a v e effects.

Results

TPUT BAR ANSMERSE STRAIN

Fig. 7--Typical test records

A record of the e x p e r i m e n t a l oscillograph traces f r o m a typical test is shown in Fig. 7. The top trace is f r o m a gage ~ust before the specimen; the n e x t trace is the input-stress w a v e ; t h e t h i r d trace is t h e axial strain; the fourth trace is the w a v e just after the specimen; and the last trace is the t r a n s v e r s e strain m e a s u r e d at the center of the specimen. E x p e r i m e n tal traces of this kind w e r e r e c o r d e d for each test and reduced according to the procedures discussed previously. Thus, stress-strain curves w e r e established for each test and the f r a c t u r e stress v a l u e established for specimens loaded to failure. The values of d y n a m i c f r a c t u r e stress m e a s u r e d for the sandstone are shown in Fig. 8 as a function of

Experimental Mechanics ] 511

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Fig. 8--Failure envelope for nugget sandstone showing comparison between static and dynamic results 150

confining pressure. Because of the very high strength of this quartzitic sandstone, the dynamic stresses necessary to produce rock fracture above about 14ksi confining pressure had a t e n d e n c y to damage e q u i p m e n t ; therefore, applied stresses were limited to this level. A comparison with static (10-4/sec) fracture values also given in Fig. 8 indicates a consistent 15 to 20 percent increase over static values for all values of confining pressure. A comparison of the unconfined strength in the dynamic and quasi-static tests on the nugget sandstone is shown i n Fig. 9. Also shown i n this figure is a value at the i n t e r mediate strain rate of 0.3/sec obtained b y G r e e n J ~ It can be seen that the values at the 3 strain rates are consistent. The rate of increase is approximately 21/2 percent per decade increase in strain rate. This rate of fracture-stress increase is consistent with results summarized by H a n d i n is and Brace ,19 for other brittle rocks. S t r e s s - s t r a i n results from tests using the cylindrical and conical projectiles are compared in Fig. 10. The initial rate of loading is dependent on which projectile is used. However, because of the relative i n sensitivity of the nugget sandstone to small changes in strain rate, the chief effect of the conical projectile is to spread Out the recorded traces so as to improve the e x p e r i m e n t a l accuracy in the early portion of the stress-strain curves. Thus, the close agreement in results shown in Fig. 10 is a n indication of the accuracy of the stress-strain curves. The principal stress-strain response from the dynamic tests was resolved into shear and dilatation plots; typical results are shown in Figs. 11 and 12. I n the dilatation response shown in Fig. 12, the initial hydrostatic loading has been added as shown. The Comparison with the static results in these figures i n dicates little change i n the qualitative aspects of the

512 I November 1972

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d y n a m i c - l o a d curves. I n particular, the coupling of the shear and volume t h r o u g h the p h e n o m e n a of dilatancy is evident i n the d y n a m i c response to app r o x i m a t e l y the same e x t e n t that it is seen in the static response. This result is important as the effect of rate of loading on the microstructural mechanisms producing the dilatancy has been u n c e r t a i n and it has been conjectured that dilatancy might be suppressed at the higher loading rates. Clearly, this is not indicated by the present results.

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Conclusion E x p e r i m e n t a l apparatus and test techniques for s t r e s s - w a v e loading of rocks under confining pressure w e r e developed w h i c h permitted strain m e a s u r e m e n t w i t h strain gages placed directly on the rocks in both principal-strain directions. Stress w a s m e a s u r e d on both the input and output bars w i t h the stress at the center approximated b y assuming a linear stress distribution, thus averaging the stresses on the ends of the sample. This m e t h o d was verified n u m e r i c a l l y a n d g a v e a c c u r a t e r e s u l t s to i n p u t - s t r e s s r a t e s of u p to 1010 p s i / s e c .

A series of tests was run on nugget sandstone u n der various confining pressures f r o m 0 to 30 ksi. The f o l l o w i n g observations could be m a d e f r o m these data: 1. The failure locus of nugget sandstone at the high strain rate was 15-20 percent higher than that obtained in quasi-static tests for all values of confining pressure. 2. The m a i n features of the stress-strain response w e r e similar to those seen at l o w rates. In particular, the p h e n o m e n o n of dilatancy was not significantly affected b y the change f r o m 1 0 - 4 / sec to .5 • 108/sec axial-strain rate.

Acknowledgments This study was sponsored by the D e f e n s e Nuclear Agency. The support and cooperation of Clifton McFarland of that agency are gratefully a c k n o w l edged. Technical discussions w i t h S. J. Green of Terra-Tek, Inc. w e r e e x t r e m e l y helpful.

References i . Swanson, S. R. and Brown, W . S., "'The Influence of State of Stress on the Stress-Strain Behavior of Rocks," Paper No. 71Met-AA to be published in ]nl. of Basic Engineering (1972). 2. Brown, W . S. and Swanson, S. R., "'Constitutive Equations for Westerly Granite and Cedar City Tonalite for a Variety of Loading Conditions," University of Utah Final Report DASA-2473

(March 1970). 3. Brown, W . S. and Swanson, S. R., "'Influence of Load Path and State of Stress on Failure Strength and Stress-Strain Properties of Rocks," Technical Report :ff:AFWL-TR-70-53, Kirtland Air Force Base, N. M. (]anuary 1971). 4. Green, S. J , Leasia, 1. D., Jones, A. H. and Perkins, R. D., "Multiaxial Stress Behavior of Solenhofen Limestone and Westerly Granite at High Strain Rates," submitted for publication in the lnl. of Geophysical Research (1971). 5. Green, S. 1. and Perkins, R. D., "'Uniaxial Compression Tests at Strain Bates from IO-~/sec. to lO~/sec, on Three Geologic Materials." Final Report DASA-2199. Also presented at the Tenth Symposium on Rock Mechanics, Austin, Tex. (1968). 6. Kumar, A., "'The Effects of Stress and Temperature on the Strength of Basalt and Granite," Geophysics, 33, 501-510 (1968). 7. Jones, A. H., lsbeU, W . M., Shipman, F. H., Perkins, R. D., Green, S. 1. and Maiden, C. 1, "'Material Properties Measurements for Selected Materials," Interim Report, Contract No. NAS2-3427, Ames Research Center, Moffett Field, Calif. (1968). 8. Brace, W . R., Paulding, B. W . and Seholz, C., "'Dilatancy in the Fracture of Crystalline Rocks," lnL of Geophysical Research, 71, 3939-3953 (1966). 9. Chalupnik, 1. D. and Ripperger, E. A., "'Dynamic Deformation of Metals under High Hydrostatic Pressure," EXPERIMENTAL IV[ECrIANICS, 6 (11), 547-554 (1966). 10. Kolsky, H., Stress W a v e s in Solids, Dover Publications, Inc., N e w York (1963). 11. Davis, D. H., and Hunter, S. C., "'The Dynamic Compression Testing of Solids by the Method of the Spllt Hopldnson B a r , " lnl. of Mech, and Phys. Solids, 11, 115-179 (1963). 12. Conn, A. F., "On the Use of Thin Wafers to Study Dynamic Properties of Metals," ]nl. of Mech. and Phys. Solids, 13, 311-327 (1965). 13. Christensen, R. I , "'Dynamic Properties of Rocks," MS Thesis, University of Utah (1971). 14. Karnes, C. H. and Bertholf, L. D., "'Numerical Investigation of Two-Dimensional, Axisymmetric Elastlc-Plastic W a v e Propagation Near the Impact End of Identical 1100-1 Aluminum Bars," Battelle Colloquium on Inelastic Behavior of Solids (1969). 15. Davies, R. M., "',4 Critical Study of the Hopkinson Pressure Bar," Philosophical Transactions of the Royal Society of London, 240. 375-452 (1948). 16. "'Progress Report on Stress and Strain Distribution in Strain Rate Testing," Prepared by Agbabian-]acobsen Associates. Los Angeles, Calif., R-71110-1787 (February 4, 1971). 17. Green, S. 1, Personal Communication, Terra-Tek, Inc., Salt Lake C~ty, Utah (January 1971). 1S. Handin, 1., "'Strength and Ductility," Handbook of Physical Constants, Sydney P. Clark, ed. Geological Society of America Merooir, 97, 223-289 (1966). 19. Brace, W . F. and .[ones, A. H., "Come, arisen of Unlaxlal Deformation in Shock and Static Loading of Three Bocks," Submitted for publication in the ]nl. of Geophysical Research (1971).

Experimental Mechanics [ 513