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Tensile strength measurements of frangible bullets using the diametral compression test
ARTICLE IN PRESS
International Journal of Impact Engineering 35 (2008) 511–520 www.elsevier.com/locate/ijimpeng
Tensile strength measurements of frangible bullets using the diametral compression test Steven P. Matesa,, Richard Rhorerb, Stephen Banovica, Eric Whitentonb, Richard Fieldsa a
Metallurgy Division, MSEL, National Institute of Standards and Technology, 100 Bureau Drive Stop 8553, Gaithersburg, MA 20899, USA Manufacturing Metrology Division, MEL, National Institute of Standards and Technology, 100 Bureau Drive Stop 8553, Gaithersburg, MA 20899, USA
b
Received 16 October 2006; received in revised form 20 April 2007; accepted 25 April 2007 Available online 8 May 2007
Abstract Frangible bullets are designed to disintegrate on impact against rigid surfaces to avoid ricochet hazards in recreational shooting ranges or law enforcement training facilities. Their impact behavior against protective soft body armor is therefore quite different than conventional lead bullets, which are designed to expand on impact rather than shatter into fragments. Models to predict the impact of frangible bullets on soft body armor are currently sought to aid in the development of new performance standards for the law enforcement community against this unusual ballistic threat. Modeling success rests on the availability of constitutive data for frangible materials used in these bullets, such as tensile strength. To supply these critical data, the tensile strength of a Cu–Sn frangible bullet material is measured using the diametral compression test at quasi-static ð1 mm=sÞ and high ð12:5 m=sÞ displacement rates. The latter tests are conducted using a Kolsky bar. Finite element modeling is used to calculate the stress in the specimen at failure. Using a maximum tensile strain criterion, the effective tensile strength was found to be 104 MPa 14 MPa. Tensile strength was not strongly sensitive to the displacement rate. Published by Elsevier Ltd. Keywords: Diametral compression test; Brazilian test; Frangible bullets; Kolsky bar; Tensile strength
1. Introduction Frangible bullets are designed to disintegrate on impact against rigid surfaces to mitigate hazards associated with random ricochets in public shooting ranges or weapons training facilities. Additionally, because they contain no lead, they are more environmentally acceptable for use in recreational shooting or training than traditional bullets. However, while intended for benign uses only, these bullets nevertheless pose a possible threat against law enforcement officers in the field. Consequently, their performance against protective soft body armor is of keen interest to law enforcement standards organizations that are charged with classifying the performance of the various protective armors against any and all possible ballistic threats [1]. The behavior of these frangible bullets when striking compliant soft body armor is understandably quite different than Corresponding author.
E-mail address: [email protected] (S.P. Mates). 0734-743X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijimpeng.2007.04.005
conventional lead bullets, which are designed to expand on impact rather than shatter into fragments. Finite element models are currently sought to better understand the impact behavior of these bullets against soft body armor targets. The results of these studies will be used to develop new performance standards for protective armor employed by law enforcement officers against this unusual ballistic threat. Frangibility is achieved by introducing brittle phases into the bullet that cause it to fracture in tension on impact. Tensile strength data are therefore needed to modeling the behavior of frangible bullets on impact. In this paper, tensile strength measurements are reported for a common, commercially available frangible bullet material made from a mixture of elemental copper and tin powders carefully sintered to produce a specific amount of brittle intermetallic phases and porosity [2]. The diametral compression test, also known as the Brazilian disk test [3], was chosen as the test method. In this test, a thin disk of the test material is loaded along its diameter, creating a tensile
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512
1
from the applied load P and the stresses s2 and s3 . An effective tensile stress, se;2 , can then be determined from this failure strain by simply multiplying 2 by E:
diametral axis
P
3 2
se;2 ¼ E2 ¼ s2 ns3 . σ2 σ3
P
|σ3|
(- stress)
|σ2| (+ stress)
Fig. 1. Ideal elastic analysis of the diametral compression test without flats.
stress in the middle that acts perpendicular to the applied load, as shown in Fig. 1. As the load is increased, the tensile stress rises until a crack forms in the middle of the sample, breaking it in two. Tensile strength is determined by estimating the stress in the disk when the crack is initiated. Because of its simplicity, this test is readily adaptable to the Kolsky bar technique [4] which provides the high displacement rates needed to assess the sensitivity of tensile strength to loading rate, a key concern for modeling ballistic impact. Assuming the disk deforms elastically prior to failure and that plane-stress conditions are realized, the tensile stress is related to the applied load P by [5] s2 ¼
2P , pDt
(1)
where D is the diameter of the disk sample and t is its thickness. s2 is the maximum principal stress in the disk. It has been argued that simply reporting this stress as the tensile strength of the material is incorrect because it fails to account for the biaxial nature of the stress state under which failure occurs [6]. As shown in Fig. 1, a strong compressive stress, s3 , acts along the loaded axis perpendicular to s2 in this test. Compressive stresses acting perpendicular to the tensile axis are known to significantly reduce the fracture strength of a wide variety of brittle materials [7]. To account for this weakening effect, Brandes [8] suggests a maximum tensile strain failure criterion in place of the tensile stress criterion that, for the present test, is represented by Eq. (1). Shaw et al. [6] used this criterion to reconcile diametral compression tests results on tungsten carbide with expanding ring test data in which the ring test strength was significantly larger. They also pointed out several other examples where materials exhibited lower strengths when tested diametrally compared to pure uniaxial tests. In the maximum tensile strain criterion, s3 increases the tensile strain acting to open the crack, 2 , from Hooke’s law: 2 ¼
1 ðs2 ns3 Þ, E
(2)
where E is the elastic modulus of the disk and n is Poisson’s ratio. Thus, a tensile strain at fracture can be calculated
(3)
se;2 is interpreted as the stress that would be needed to cause fracture in a uniaxial tension test, and it is significantly larger than the stress given by Eq. (1). The magnitude of the compressive stress at the center of the disk is related to the applied load by [5] s3 ¼
6P . pDt
(4)
At the center of the disk, therefore, s3 ¼ 3s2 and, for n ¼ 0:3, se;2 ¼ 1:9s2 . Shaw et al. [6] point out that diametral tests often yield tensile strengths that are about half as large as values determined from other tension tests where the stress state is uniaxial. They therefore suggest that Brandes’ maximum tensile strain criterion is the proper way to interpret diametral compression test results. Unfortunately, because some materials have a much smaller discrepancy between disk tests and bend tests [9], this tensile strain failure criterion may not be universal. The appropriate failure criterion to use for the present material could be determined from additional uniaxial tension tests performed on dogbone specimens made from individual bullets. However, tensile loading is not achievable without significant modifications to our Kolsky bar. In addition, since the stress state in an actual bullet impact event is much more complex than either biaxial or uniaxial, preferring the uniaxial tensile strength value for modeling a whole bullet fracturing on impact is not necessarily justified. Further, machining tensile specimens from individual bullets may introduce additional defects or damage that could produce misleading results. As such, we elected not to attempt uniaxial tension tests on this material. Instead, we describe our measurement results using both failure criteria, and leave it to users of these data to determine which criterion gives a more realistic simulation of whole bullet fracture on impact. Finally, rather than relying on the foregoing twodimensional stress analysis to interpret the experimental results, finite element analysis is used instead to calculate the stress in the diametral sample during compression testing. The two-dimensional stress analysis is insufficient since the specimen geometry used in our tests is decidedly non-ideal: the ends of the sample are flattened to avoid yielding and brittle damage at the load bearing contacts. Yielding is more of a concern with this material compared to the stronger ceramics usually tested with this method due to the large amount of pure copper in its microstructure. Extensive plastic flow or crushing damage at the contacts would make it difficult to accurately model the stress evolution, even using finite element analysis, since the material flow and damage models are not sufficiently accurate. While pre-flattening is known to alter the stress evolution in the disk compared to the ideal case, these
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effects are readily determined by finite element analysis provided the deformation is kept mostly to the elastic regime. Elastic, three-dimensional finite element calculations of diametral compression testing of alumina under dynamic loading conditions have already proven useful in analyzing test results [10].
513
8.7 mm
2. Experiment
9.0 mm
2.25 mm 4.5 mm
2.1. Material characterization A representative microstructure of the frangible copper–tin material is shown in Fig. 2. It consists of a pure copper matrix surrounding two intermetallic phases, Cu6 Sn5 and Cu3 Sn, which in turn bound some residual unreacted tin. The phases were identified by X-ray diffraction and qualitative chemical analysis by energy dispersive spectroscopy (EDS) [2]. The structure also contains approximately 5% porosity by volume. The mixture consists of 90% copper and 10% tin by mass. Accounting for the porosity, the theoretical density is 8040 kg=m3 . No density measurements were attempted. 2.2. Diametral compression tests The disk specimen dimensions are shown in Fig. 3. Individual disks are cut from 9 mm bullets using a wire EDM (electrical discharge machining) process, one disk per bullet. Thus, the test samples are similar in scale to an entire bullet, lending confidence that the measured fracture strengths will be appropriate for predicting bullet failure. However, strength variations with specimen size, while a concern for purely brittle materials, is not as likely here since this material exhibits some ductility. Flats equal to 25% of the disk diameter in width ðw=D ¼ 0:25Þ are added on opposite sides of the disk by hand sanding the samples while they are mounted in a precision jig. The flats are lubricated with common axle grease to reduce friction during the test.
Fig. 3. Dimensions of pre-flattened test specimen.
Quasi-static tests are carried out with a servohydraulic test machine fitted with a 50 kN load cell and maraging steel loading rams (HRC 54). Samples are loaded using position control, and compression is conducted with a constant ram velocity of 0.001 mm/s. Load versus time curves are sampled at 10 Hz using data acquisition equipment. High displacement rate tests are performed using a Kolsky bar with the arrangement shown in Fig. 4. The test sample is first placed between two compression bars. Then, the free end of the first bar is struck with a projectile (called a striker bar) to produce an elastic compression pulse. The pulse travels rapidly down the first bar and through the sample into the second bar, resulting in extremely fast compressive loading of the sample. The Kolsky bar used for these tests employs 1500 mm 15 mm diameter maraging steel compression bars. The 375 mmlong striker bar, also made of maraging steel, impacts the first bar at approximately 25 m/s. This impact velocity produces a peak displacement rate of approximately 12.5 m/s on the sample, or about 107 times the displacement rate in the quasi-static test. Strain gages (1000 O metal foil type) placed in the middle of each bar measure the incident and transmitted strain pulses, as shown in Fig. 3. The incident pulse is shaped so that the load applied to the sample increases steadily during the test so that the failure load can be clearly identified. The rising saw-tooth strain pulse is created by placing an annealed copper disk ð3 mm diameter 4 mm thickÞ, or pulse shaper, on the exposed end of the first bar, which softens the impact of the striker resulting in the desired wave shape. Annealed copper’s ample capacity for strain hardening combined with the thickness of the pulse shaper together produce the slow rising stress pulse necessary for performing the dynamic tests. The tensile strength is determined by computing the stress in the disk at the load carried just before failure. The load on the disk is measured by converting the transmitted strain signal recorded on the second bar, t , to a force using the following relation: P ¼ AEt ,
Fig. 2. Optical micrograph of the frangible bullet microstructure.
(5)
whereA is the cross-sectional area of the compression bar ð176:7 mm2 Þ, and E is the elastic modulus of the bar (200 GPa). The response of each strain gage is determined using a single-arm Wheatstone bridge arrangement
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Striker Bar
Test Specimen strain gage 1 incident wave
strain gage 2 transmitted wave strain
strain
Pulse Shaper
time
failure time
Fig. 4. Diametral compression test arrangement on a Kolsky bar.
incident bar
sample 2
3 1 ax surmedia is fac l e
symmetry planes
diametral plane
transmitted bar
Fig. 5. Domain and mesh for finite element analysis. Only one quarter of the sample and bars is modeled. Axis, medial and surface lines correspond to stress plots in Fig. 9.
powered by a 24 V DC supply. The strain signals are sampled at 2 MHz using a digital storage oscilloscope. 2.3. Finite element model ABAQUS/Explicit1 [11] finite element software was used to calculate the stress field in the diametral compression sample as it deforms in response to the compression pulse applied using the Kolsky bar. Quasi-static tests were not modeled since, although the loading rate is high in the Kolsky bar, the sample is in force equilibrium through most of the test and therefore modeling the same static load acting on the sample would yield identical results. Modeling the Kolsky bar test dynamically has the advantage that the modeled response of the sample can be compared to the experimental results over a large range of loads in a single analysis. Fig. 5 shows the mesh used for the simulations. Only one quarter of the test geometry is modeled, with symmetry imposed on surfaces facing the h1i and h2i directions. Both compression bars and the sample are modeled with 1 Commercial products are identified in this work to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by NIST, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
C3D8R hexahedral elements. Approximately 27,000 elements are used to model the quarter sample while 225 elements are used for each compression bar. This mesh density provided relatively smooth contact behavior between the sample and the compression bars and good resolution of the stress field in the sample. To reduce computation time, neither the striker bar nor the pulse shaper is modeled. Instead, the incident stress wave is created artificially by specifying a pressure boundary condition on the incident compression bar to closely match the experimental incident stress pulse. Time stepping is controlled automatically by the ABAQUS software using the element-by-element option. Contact between the compression bars and the sample is modeled using the kinematic hard contact formulation. Friction is neglected. The maraging steel compression bars are modeled as elastic isotropic solids with the following properties: E ¼ 200 GPa, n ¼ 0:29, and r ¼ 8100 kg=m3 . The elastic modulus of the frangible material was determined to be E ¼ 79 GPa by quasi-static compression tests performed on cylindrical specimens (9 mm in diameter and 4.5 mm thick) by a commercial testing laboratory. Poisson’s ratio is assumed to be n ¼ 0:3. An average flow stress of 294 MPa was determined from three quasi-static and three high strain rate (1500 1/s) compression tests, so strain rate effects were considered insignificant. Flow stress measurements were performed using the same cylindrical specimens used for the diametral compression tests but without flattened ends. No attempt was made to account for porosity or strain hardening effects. As such, the flow stress is equivalent to the yield stress in the simulations. The flow stress, elastic modulus and density are generally similar for common casting bronze alloys of similar composition [12]. 2.4. Experimental results Failure loads recorded for six quasi-static tests and 12 dynamic tests are listed in Table 1 in order of increasing failure load. The scatter in the measurements is small compared to purely brittle materials where failure loads can vary by a factor of two or more. Ductility likely plays an important role, since the ability to yield at the crack tip makes the material less susceptible to failure from large but statistically rare defects that can dominate the strength of
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purely brittle materials. This result also suggests that the microstructure is consistent from sample to sample, which is characteristic of materials produced by powder metallurgy techniques [13] and in agreement with microstructural analysis carried out on this material [2]. Because of the tightness of the results, there is little to be gained from describing the data using Weibull statistics. The average quasi-static failure load was found to be 3876 N 100 N (95% confidence), while the dynamic failure load was found to be 4113 N 87 N (95% confidence). Thus, the tensile strength of this material is not very sensitive to the Table 1 Quasi-static and dynamic failure load measurements Quasi-static
Average (N) Uncertainty (N)
Dynamic
Rank
Load
Rank
Load
1 2 3 4 5 6
3716 3738 3908 3912 3953 4027
1 2 3 4 5 6 7 8 9 10 11 12 13
3945 3962 3992 4002 4013 4063 4096 4101 4130 4130 4261 4264 4508
3876 100
4113 87
515
displacement rate. Because of this, the remainder of the paper will mainly deal with dynamic test results, with the understanding that the quasi-static test results are similar. Doubt exists as to whether the diametral compression test yields a valid measure of a material’s strength, aside from the general question of whether brittle fracture strength can even be considered an inherent material property. The main issue is whether failure begins in the center of the disk under tensile stresses, as the test intends, or near the edge of the disk under a more complicated and less well characterized stress state [14]. To verify that failure in the Kolsky bar tests occurs by the intended mechanism of a Mode I (tensile) crack originating in the center of the disk, high speed video recordings ð105 frames=sÞ were made of each test. This recording speed was sufficient, in most cases, to capture the nucleation and growth of the crack over the span of several frames, as shown in Fig. 6. This figure shows a typical test in which fracture is observed to begin near the center of the disk and propagate outwards to each end. Secondary fractures, such as the ones visible in the third frame of Fig. 6 emanating from the loaded surfaces, were also often observed. However, these secondary fractures generally did not appear before the primary tensile crack emerged, as shown in Fig. 6. This figure also shows a typical broken specimen recovered after a dynamic test. It shows that large amounts of material are lost from the contact regions due to brittle crushing damage and cracking, the beginnings of which is evident in the last frame capture of the high speed video recording, shown in Fig. 6.
Fig. 6. High speed digital visible light video of primary tensile crack failure in the dynamic diametral compression test (left) and photograph of broken sample (right). Arrows indicate the direction of the load.
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Fig. 7. (a) Optical micrograph showing tensile crack arrested in quasi-static diametral test. White arrows indicate ends of visible crack. Black arrow indicates detail (b) showing crack path following prior particle boundaries.
In the quasi-static tests, proof that failure begins in the center of the disk during these tests was obtained by examining all the tested samples using an optical microscope. Unlike in the Kolsky bar tests, in the quasi-static tests the load on the sample can be removed as soon as the primary failure occurs, which prevents further damage to the sample and allows the initial failure to be examined. In quasi-static test samples, center fractures were observed to be completely arrested well inside the contact surfaces, as shown in Fig. 7. None of the tested samples broke into two pieces. To verify that fractures did not initiate near the middle of the contact surfaces, which would be undetectable from sideways observations, the ends of the samples were polished to a 1 mm finish and examined under an optical microscope. None of the tested samples showed evidence of fractures that either initiated at the contact surface or propagated there from within the sample. Fracture surfaces of the broken specimens revealed three separate failure mechanisms, as indicated in Fig. 8. Cleavage of the intermetallic phases, separation at prior particle boundaries and ductile fracture by microvoid coalescence are all observed. The dominant fracture mode was particle separation, followed by intermetallic cleavage. Fig. 7 shows the fracture path is circuitous rather than planar, skirting prior particle boundaries. Regions of microvoid coalescence were only observed in the few instances where cleavage zones were closely bounded by the copper matrix. Thus, ductile fracture plays a very minor role. These fracture surfaces are qualitatively similar to fracture surfaces of bullet fragments recovered from ballistic tests on soft body armor [2], adding confidence that the strength data obtained from the diametral tests will be useful for predicting bullet breakup.
2.5. Stress analysis results Fig. 9 compares experimental strain pulses from Kolsky bar tests with simulated strain pulses. The good agreement between model and experiment indicates the overall deformation of the sample is correctly captured in the simulation. In this particular experiment there is some yielding prior to fracture that is not predicted by the model, as indicated by the departure of the experimental transmitted pulse from the linear simulated pulse. However, this yielding is minor, and it is localized to the contact regions where the compressive stress is very high. At fracture, the transmitted strain signal drops suddenly. Because the finite element model does not include a failure criterion, the simulated transmitted strain continues to rise beyond the experimental fracture point. The stress distribution at failure is obtained from the model results at the time step where the simulated load is closest to the average measured failure load of 4113 N. Using this simulated failure load of 4070 N, which is well within the uncertainty of the measured failure load, avoids having to perform extensive interpolations of the model results. Fig. 10 compares the stress distribution at failure computed by finite element analysis to the two-dimensional elastic analysis (e.g. Eqs. (1), (2) and (4)). It reveals how the load distribution over the flattened contacts alters the stress distribution compared to the idealized case with point loads. In the central portion of the disk, the model results are in good overall agreement with the ideal solution. Here s2 is tensile and uniform, and the magnitude is comparable though a bit lower than the ideal value at the medial and axial locations. At the free surface, s2 is slightly higher, and it is uniform over a larger portion of the disk compared to
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Fig. 8. Scanning electron micrographs of fracture modes on surface of broken diametral test specimen: (a) cleavage and microvoid coalescence; (b) particle separation.
the medial and axial locations. At the ends of the disk, the model results deviate markedly from the ideal solution. The axial compressive stress s3 in the ideal solution becomes unbounded at the contacts, whereas in the finite element solution s3 remains bounded. s2 also deviates rapidly from the ideal solution toward the ends, becoming compressive rather than remaining constant and tensile. Similar effects have been demonstrated analytically and using finite element analysis in two dimensions with distributed loading [6,15]. Three-dimensional effects are also evident
in the stress distributions near the contact surfaces. For example, Fig. 10 indicates that the peak contact pressure, equivalent to s3 at the ends of the sample, is greater at the free surface than on the axis, contrary to expectation. This is due to the fact that the faces of the Kolsky compression bars bend slightly around the specimen as they are pressed together, resulting in a larger load on the perimeter of the contact surface. When the bar ends are modeled as rigid solids, this does not occur and the load is highest in the center of the contact surface. Three-dimensional effects
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518
0.0001
0.0003 0.0002
incident bar strain
0
0
-0.0001 -0.00005
-0.0002 -0.0003
-0.0001
-0.0004
Experiment: Incident Sim: Incident Experiment: Transmitted Sim: Transmitted
-0.0005
transmitted bar strain
0.00005
0.0001
-0.00015
-0.0006
sample failure
-0.0007 0.0063
0.0064
0.0065
0.0066 time (s)
0.0067
0.0068
-0.0002 0.0069
Fig. 9. Comparison of experimental incident and transmitted strain pulses with finite element simulation for a typical diametral compression test performed in the Kolsky bar.
0
100
-100 -200
0
σ3 [MPa]
σ2 [MPa]
50
-50 -100
Axis Medial Surface 2D Ideal Elastic Analysis
-150
-300 -400 -500 -600
-200
-700 0
0.2
0.4 0.6 Normalized Distance
0.8
1
0
0.2
0.8
1
0.4 0.6 Normalized Distance
0.8
1
0.004 0.003
ε2
0.002 0.001 0 -0.001 -0.002 0
0.2
0.4 0.6 Normalized Distance
Fig. 10. Comparison of the stress and strain distributions on the diametral plane at the failure load computed in three-dimensional by finite elements with the ideal elastic stress analysis. (See Fig. 5 to locate plots on sample mesh.)
therefore have an important influence on the development of the stresses near the contact surfaces. However, in the central portion of the disk where fracture occurs, the stress distribution in these thick, pre-flattened specimens is qualitatively rather similar to the ideal two-dimensional plane stress solution that is traditionally used to analyze diametral compression tests.
Because stress in the center of the sample is not strictly uniform, as is usually assumed when employing the ideal stress analysis method, some rationale is needed to convert this varying stress field into a single failure criterion that adequately captures the test results. Assuming the flaws that determine the tensile strength of the material are uniformly distributed, one expects that failure will initiate
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in the region where the effective tensile stress is highest and then propagate through the rest of the sample. The task is then to decide how to quantify the extent of this maximum tensile stress region in order to adequately capture the conditions of failure. Currently there is no standard way to define this region. Weakest-link methods have been used to assess the effective volume of the test and account for stress gradients and multi-axial loading in diametral compression testing of purely brittle materials [16,17]. The present material, however, does not behave in a purely brittle fashion, as evinced by the ability to exhibit crack arrest behavior (Fig. 7a) and its narrowly scattered failure loads. Consequently, these methods are not appropriate here. Shaw et al. [6] considered the test volume to be where the effective tensile stress was within 10% of the peak stress. Using this method, they were able to reconcile the tensile strengths obtained from diametral tests with those obtained using three-point-bend and expanding ring tests. However, using the 10% rule is unsatisfactory here since it captures just the two small regions near the free surface where 2 experiences shallow maxima (see Fig. 10c). Fracture does not originate in these areas. As an alternative, values of 2 within 25% of the peak value that are also found within the middle 63% of the specimen are averaged to estimate the effective tensile strain at failure. While somewhat arbitrary, this method keeps the peak strains from biasing the result, and avoids the regions near the ends of the sample where 2 rapidly decreases and turns compressive. Using this method to define the effective test volume, the averaged stresses in the disk calculated at a load of 4070 N are given in Table 2. The average of s3 in the test volume is roughly four times the average s2 value, leading to an effective tensile strength, se;2 , about twice s2 at failure. This ratio is close to the discrepancy between diametral test results and direct tension test results for many [6] but not all [9] brittle materials. The ratio of compressive yield strength to the effective tensile strength ðsy =se;2 Þ is 2.8, confirming the brittle nature of this material in tension. Using the alternative maximum tensile stress failure criterion, this ratio increases to 6.3. The uncertainty in the failure strength is taken to be twice the standard deviation of the relevant failure stress measure over the 30 mm3 test volume. Using this method, the uncertainty in Table 2 Comparison of average stress and strain determined from finite element model and ideal plane stress analysis Quantity
Finite element model
Ideal elastic analysis
P (N) 2 (Std.) (%) s2 (Std.) (MPa) s3 (Std.) (MPa) se;2 (Std.) (MPa) sy =se;2 Test volume ðmm3 Þ
4070 0.132 (0.009) 47 (7) 191 (23) 104 (7) 2.8 30
4070 0.154 64 192 121.6 2.4 –
519
se;2 is 14 MPa while the uncertainty in s2 is 7 MPa. This method of calculating uncertainty results in a larger value compared to taking the difference in the average stress at the lowest and highest experimental failure loads reported in Table 1. Thus, the uncertainty in the test is chiefly due to the variability in the stress distribution in the disk rather than the variability in the observed failure load. At failure, s2 is 27% lower than the two-dimensional analysis result using Eq. (1) while se;2 is 14% below the ideal analysis result using Eq. (3). Thus, using finite element analysis can be said to have improved the test results by these amounts depending on the failure criterion considered. 3. Conclusions Diametral compression tests were performed quasistatically and dynamically using a Kolsky bar to determine the tensile strength of a commercial frangible bullet material made from a sintered copper–tin powder mixture. Unlike purely brittle materials, the failure loads measured in repeated tests on this material were not widely scattered, owing to ductility in the copper-rich phase and a homogeneous defect population. In the Kolsky bar tests, high-speed video recordings confirmed that failure occurred by the intended mechanism of a tensile crack nucleating in the disk center and propagating to the ends. In quasi-static tests, where the load could be removed immediately after failure, the center fracture was found to be completely arrested well inside the contact surfaces on the sample, indicating a good test. Using the maximum tensile stress failure criterion, the tensile strength, as determined by finite element analysis, was 47 MPa 7 MPa. This value is 27% below the two-dimensional ideal elastic prediction owing to the distributed loading over the pre-flattened ends of the sample. Using the maximum tensile strain failure criterion, the effective tensile strength was 104 MPa 14 MPa, or only 14% below the twodimensional ideal analysis value. The failure strength was not strongly sensitive to loading rate. Fracture is dominated by particle separation, with cleavage of intermetallic phases playing a secondary role. Fracture surfaces were similar to those observed in bullet fragments recovered from ballistic testing on soft body armor, adding confidence that the fracture strengths determined here are relevant to modeling bullet impact. However, since the state of stress in an impacting bullet is generally different than that which exists in the diametral compression test, using either of the tensile strength values presented here to model the frangible bullet impact may be justified on the basis of the level of agreement achieved between such a model and actual bullet impact experiments. Acknowledgments This work was supported by Office of Law Enforcement Standards at NIST and the National Institute of Justice. The authors wish to thank Mr. Mike Kennedy of the MEL
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at NIST for general support and assistance with the experiments, Dr. William Luecke at NIST for performing quasi-static diametral compression tests, and Prof. Wayne Chen of Purdue University for valuable consultations. References [1] NIJ Standard-0101.04. Ballistic resistance of personal body armor. Washington, DC, USA: US Department of Justice; 2000. [2] Banovic SW, Mates SP. Microstructural characterization and mechanical behavior of Cu–Sn frangible bullets. Mater Sci Eng A, submitted May 2007. [3] Rudnick A, Hunter AR, Holden FC, An analysis of the diametralcompression test. Mater Res Stand 1963; 283–9. [4] Gray III GT. Classic split-Hopkinson pressure bar testing. In: ASM handbook, vol. 8. Materials Park, OH: The American Society for Metals; 1990. p. 462–76. [5] Frocht MM. Photoelasticity, vol. II. New York: Wiley; 1948. p. 125. [6] Shaw MC, Braiden PM, DeSalvo GJ. The disk test for brittle materials. ASME J Eng Ind 1975; 77–87.
[7] Bridgman PW. Studies in large plastic flow and fracture. New York: McGraw-Hill; 1952. [8] Brandes M. The effect of high hydrostatic pressures on the cohesive strength of brittle materials. Int J Fract Mech 1965;1(1):56–63. [9] Berenbaum R, Brodie I. Measurement of the tensile strength of brittle materials. Br J Appl Phy 1959;10:281–7. [10] Galvez F, Galvez VS. Numerical modelling of SHPB splitting tests. J Phys IV 2003;110:347–52. [11] hwww.abaqus.comi. [12] ASM handbook, vol. 2. Materials Park, OH: The American Society for Metals; 1990. [13] German RM. Powder metallurgy science. Princeton, NJ: Metal Powder Industries Federation; 1994. [14] Darvell BW. Uniaxial compression tests and the validity of indirect tensile strength. J Mater Sci 1990;25(2A):757–80. [15] Fahad MK. Stresses and failure in the diametral compression test. J Mater Sci 1996;31:3723–9. [16] Bruckner-Foit A, Fett T, Munz D, Schirmer K. Discrimination of multiaxiality criteria with the Brazilian disc test. J Eur Ceram Soc 1997;17:689–96. [17] Neergaard LJ, Neergaard DA, Neergaard MS. Effective volume of specimens in diametral compression. J Mater Sci 1997;32:2529–33.
International Journal of Impact Engineering 35 (2008) 511–520 www.elsevier.com/locate/ijimpeng
Tensile strength measurements of frangible bullets using the diametral compression test Steven P. Matesa,, Richard Rhorerb, Stephen Banovica, Eric Whitentonb, Richard Fieldsa a
Metallurgy Division, MSEL, National Institute of Standards and Technology, 100 Bureau Drive Stop 8553, Gaithersburg, MA 20899, USA Manufacturing Metrology Division, MEL, National Institute of Standards and Technology, 100 Bureau Drive Stop 8553, Gaithersburg, MA 20899, USA
b
Received 16 October 2006; received in revised form 20 April 2007; accepted 25 April 2007 Available online 8 May 2007
Abstract Frangible bullets are designed to disintegrate on impact against rigid surfaces to avoid ricochet hazards in recreational shooting ranges or law enforcement training facilities. Their impact behavior against protective soft body armor is therefore quite different than conventional lead bullets, which are designed to expand on impact rather than shatter into fragments. Models to predict the impact of frangible bullets on soft body armor are currently sought to aid in the development of new performance standards for the law enforcement community against this unusual ballistic threat. Modeling success rests on the availability of constitutive data for frangible materials used in these bullets, such as tensile strength. To supply these critical data, the tensile strength of a Cu–Sn frangible bullet material is measured using the diametral compression test at quasi-static ð1 mm=sÞ and high ð12:5 m=sÞ displacement rates. The latter tests are conducted using a Kolsky bar. Finite element modeling is used to calculate the stress in the specimen at failure. Using a maximum tensile strain criterion, the effective tensile strength was found to be 104 MPa 14 MPa. Tensile strength was not strongly sensitive to the displacement rate. Published by Elsevier Ltd. Keywords: Diametral compression test; Brazilian test; Frangible bullets; Kolsky bar; Tensile strength
1. Introduction Frangible bullets are designed to disintegrate on impact against rigid surfaces to mitigate hazards associated with random ricochets in public shooting ranges or weapons training facilities. Additionally, because they contain no lead, they are more environmentally acceptable for use in recreational shooting or training than traditional bullets. However, while intended for benign uses only, these bullets nevertheless pose a possible threat against law enforcement officers in the field. Consequently, their performance against protective soft body armor is of keen interest to law enforcement standards organizations that are charged with classifying the performance of the various protective armors against any and all possible ballistic threats [1]. The behavior of these frangible bullets when striking compliant soft body armor is understandably quite different than Corresponding author.
E-mail address: [email protected] (S.P. Mates). 0734-743X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijimpeng.2007.04.005
conventional lead bullets, which are designed to expand on impact rather than shatter into fragments. Finite element models are currently sought to better understand the impact behavior of these bullets against soft body armor targets. The results of these studies will be used to develop new performance standards for protective armor employed by law enforcement officers against this unusual ballistic threat. Frangibility is achieved by introducing brittle phases into the bullet that cause it to fracture in tension on impact. Tensile strength data are therefore needed to modeling the behavior of frangible bullets on impact. In this paper, tensile strength measurements are reported for a common, commercially available frangible bullet material made from a mixture of elemental copper and tin powders carefully sintered to produce a specific amount of brittle intermetallic phases and porosity [2]. The diametral compression test, also known as the Brazilian disk test [3], was chosen as the test method. In this test, a thin disk of the test material is loaded along its diameter, creating a tensile
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1
from the applied load P and the stresses s2 and s3 . An effective tensile stress, se;2 , can then be determined from this failure strain by simply multiplying 2 by E:
diametral axis
P
3 2
se;2 ¼ E2 ¼ s2 ns3 . σ2 σ3
P
|σ3|
(- stress)
|σ2| (+ stress)
Fig. 1. Ideal elastic analysis of the diametral compression test without flats.
stress in the middle that acts perpendicular to the applied load, as shown in Fig. 1. As the load is increased, the tensile stress rises until a crack forms in the middle of the sample, breaking it in two. Tensile strength is determined by estimating the stress in the disk when the crack is initiated. Because of its simplicity, this test is readily adaptable to the Kolsky bar technique [4] which provides the high displacement rates needed to assess the sensitivity of tensile strength to loading rate, a key concern for modeling ballistic impact. Assuming the disk deforms elastically prior to failure and that plane-stress conditions are realized, the tensile stress is related to the applied load P by [5] s2 ¼
2P , pDt
(1)
where D is the diameter of the disk sample and t is its thickness. s2 is the maximum principal stress in the disk. It has been argued that simply reporting this stress as the tensile strength of the material is incorrect because it fails to account for the biaxial nature of the stress state under which failure occurs [6]. As shown in Fig. 1, a strong compressive stress, s3 , acts along the loaded axis perpendicular to s2 in this test. Compressive stresses acting perpendicular to the tensile axis are known to significantly reduce the fracture strength of a wide variety of brittle materials [7]. To account for this weakening effect, Brandes [8] suggests a maximum tensile strain failure criterion in place of the tensile stress criterion that, for the present test, is represented by Eq. (1). Shaw et al. [6] used this criterion to reconcile diametral compression tests results on tungsten carbide with expanding ring test data in which the ring test strength was significantly larger. They also pointed out several other examples where materials exhibited lower strengths when tested diametrally compared to pure uniaxial tests. In the maximum tensile strain criterion, s3 increases the tensile strain acting to open the crack, 2 , from Hooke’s law: 2 ¼
1 ðs2 ns3 Þ, E
(2)
where E is the elastic modulus of the disk and n is Poisson’s ratio. Thus, a tensile strain at fracture can be calculated
(3)
se;2 is interpreted as the stress that would be needed to cause fracture in a uniaxial tension test, and it is significantly larger than the stress given by Eq. (1). The magnitude of the compressive stress at the center of the disk is related to the applied load by [5] s3 ¼
6P . pDt
(4)
At the center of the disk, therefore, s3 ¼ 3s2 and, for n ¼ 0:3, se;2 ¼ 1:9s2 . Shaw et al. [6] point out that diametral tests often yield tensile strengths that are about half as large as values determined from other tension tests where the stress state is uniaxial. They therefore suggest that Brandes’ maximum tensile strain criterion is the proper way to interpret diametral compression test results. Unfortunately, because some materials have a much smaller discrepancy between disk tests and bend tests [9], this tensile strain failure criterion may not be universal. The appropriate failure criterion to use for the present material could be determined from additional uniaxial tension tests performed on dogbone specimens made from individual bullets. However, tensile loading is not achievable without significant modifications to our Kolsky bar. In addition, since the stress state in an actual bullet impact event is much more complex than either biaxial or uniaxial, preferring the uniaxial tensile strength value for modeling a whole bullet fracturing on impact is not necessarily justified. Further, machining tensile specimens from individual bullets may introduce additional defects or damage that could produce misleading results. As such, we elected not to attempt uniaxial tension tests on this material. Instead, we describe our measurement results using both failure criteria, and leave it to users of these data to determine which criterion gives a more realistic simulation of whole bullet fracture on impact. Finally, rather than relying on the foregoing twodimensional stress analysis to interpret the experimental results, finite element analysis is used instead to calculate the stress in the diametral sample during compression testing. The two-dimensional stress analysis is insufficient since the specimen geometry used in our tests is decidedly non-ideal: the ends of the sample are flattened to avoid yielding and brittle damage at the load bearing contacts. Yielding is more of a concern with this material compared to the stronger ceramics usually tested with this method due to the large amount of pure copper in its microstructure. Extensive plastic flow or crushing damage at the contacts would make it difficult to accurately model the stress evolution, even using finite element analysis, since the material flow and damage models are not sufficiently accurate. While pre-flattening is known to alter the stress evolution in the disk compared to the ideal case, these
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effects are readily determined by finite element analysis provided the deformation is kept mostly to the elastic regime. Elastic, three-dimensional finite element calculations of diametral compression testing of alumina under dynamic loading conditions have already proven useful in analyzing test results [10].
513
8.7 mm
2. Experiment
9.0 mm
2.25 mm 4.5 mm
2.1. Material characterization A representative microstructure of the frangible copper–tin material is shown in Fig. 2. It consists of a pure copper matrix surrounding two intermetallic phases, Cu6 Sn5 and Cu3 Sn, which in turn bound some residual unreacted tin. The phases were identified by X-ray diffraction and qualitative chemical analysis by energy dispersive spectroscopy (EDS) [2]. The structure also contains approximately 5% porosity by volume. The mixture consists of 90% copper and 10% tin by mass. Accounting for the porosity, the theoretical density is 8040 kg=m3 . No density measurements were attempted. 2.2. Diametral compression tests The disk specimen dimensions are shown in Fig. 3. Individual disks are cut from 9 mm bullets using a wire EDM (electrical discharge machining) process, one disk per bullet. Thus, the test samples are similar in scale to an entire bullet, lending confidence that the measured fracture strengths will be appropriate for predicting bullet failure. However, strength variations with specimen size, while a concern for purely brittle materials, is not as likely here since this material exhibits some ductility. Flats equal to 25% of the disk diameter in width ðw=D ¼ 0:25Þ are added on opposite sides of the disk by hand sanding the samples while they are mounted in a precision jig. The flats are lubricated with common axle grease to reduce friction during the test.
Fig. 3. Dimensions of pre-flattened test specimen.
Quasi-static tests are carried out with a servohydraulic test machine fitted with a 50 kN load cell and maraging steel loading rams (HRC 54). Samples are loaded using position control, and compression is conducted with a constant ram velocity of 0.001 mm/s. Load versus time curves are sampled at 10 Hz using data acquisition equipment. High displacement rate tests are performed using a Kolsky bar with the arrangement shown in Fig. 4. The test sample is first placed between two compression bars. Then, the free end of the first bar is struck with a projectile (called a striker bar) to produce an elastic compression pulse. The pulse travels rapidly down the first bar and through the sample into the second bar, resulting in extremely fast compressive loading of the sample. The Kolsky bar used for these tests employs 1500 mm 15 mm diameter maraging steel compression bars. The 375 mmlong striker bar, also made of maraging steel, impacts the first bar at approximately 25 m/s. This impact velocity produces a peak displacement rate of approximately 12.5 m/s on the sample, or about 107 times the displacement rate in the quasi-static test. Strain gages (1000 O metal foil type) placed in the middle of each bar measure the incident and transmitted strain pulses, as shown in Fig. 3. The incident pulse is shaped so that the load applied to the sample increases steadily during the test so that the failure load can be clearly identified. The rising saw-tooth strain pulse is created by placing an annealed copper disk ð3 mm diameter 4 mm thickÞ, or pulse shaper, on the exposed end of the first bar, which softens the impact of the striker resulting in the desired wave shape. Annealed copper’s ample capacity for strain hardening combined with the thickness of the pulse shaper together produce the slow rising stress pulse necessary for performing the dynamic tests. The tensile strength is determined by computing the stress in the disk at the load carried just before failure. The load on the disk is measured by converting the transmitted strain signal recorded on the second bar, t , to a force using the following relation: P ¼ AEt ,
Fig. 2. Optical micrograph of the frangible bullet microstructure.
(5)
whereA is the cross-sectional area of the compression bar ð176:7 mm2 Þ, and E is the elastic modulus of the bar (200 GPa). The response of each strain gage is determined using a single-arm Wheatstone bridge arrangement
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Striker Bar
Test Specimen strain gage 1 incident wave
strain gage 2 transmitted wave strain
strain
Pulse Shaper
time
failure time
Fig. 4. Diametral compression test arrangement on a Kolsky bar.
incident bar
sample 2
3 1 ax surmedia is fac l e
symmetry planes
diametral plane
transmitted bar
Fig. 5. Domain and mesh for finite element analysis. Only one quarter of the sample and bars is modeled. Axis, medial and surface lines correspond to stress plots in Fig. 9.
powered by a 24 V DC supply. The strain signals are sampled at 2 MHz using a digital storage oscilloscope. 2.3. Finite element model ABAQUS/Explicit1 [11] finite element software was used to calculate the stress field in the diametral compression sample as it deforms in response to the compression pulse applied using the Kolsky bar. Quasi-static tests were not modeled since, although the loading rate is high in the Kolsky bar, the sample is in force equilibrium through most of the test and therefore modeling the same static load acting on the sample would yield identical results. Modeling the Kolsky bar test dynamically has the advantage that the modeled response of the sample can be compared to the experimental results over a large range of loads in a single analysis. Fig. 5 shows the mesh used for the simulations. Only one quarter of the test geometry is modeled, with symmetry imposed on surfaces facing the h1i and h2i directions. Both compression bars and the sample are modeled with 1 Commercial products are identified in this work to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by NIST, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
C3D8R hexahedral elements. Approximately 27,000 elements are used to model the quarter sample while 225 elements are used for each compression bar. This mesh density provided relatively smooth contact behavior between the sample and the compression bars and good resolution of the stress field in the sample. To reduce computation time, neither the striker bar nor the pulse shaper is modeled. Instead, the incident stress wave is created artificially by specifying a pressure boundary condition on the incident compression bar to closely match the experimental incident stress pulse. Time stepping is controlled automatically by the ABAQUS software using the element-by-element option. Contact between the compression bars and the sample is modeled using the kinematic hard contact formulation. Friction is neglected. The maraging steel compression bars are modeled as elastic isotropic solids with the following properties: E ¼ 200 GPa, n ¼ 0:29, and r ¼ 8100 kg=m3 . The elastic modulus of the frangible material was determined to be E ¼ 79 GPa by quasi-static compression tests performed on cylindrical specimens (9 mm in diameter and 4.5 mm thick) by a commercial testing laboratory. Poisson’s ratio is assumed to be n ¼ 0:3. An average flow stress of 294 MPa was determined from three quasi-static and three high strain rate (1500 1/s) compression tests, so strain rate effects were considered insignificant. Flow stress measurements were performed using the same cylindrical specimens used for the diametral compression tests but without flattened ends. No attempt was made to account for porosity or strain hardening effects. As such, the flow stress is equivalent to the yield stress in the simulations. The flow stress, elastic modulus and density are generally similar for common casting bronze alloys of similar composition [12]. 2.4. Experimental results Failure loads recorded for six quasi-static tests and 12 dynamic tests are listed in Table 1 in order of increasing failure load. The scatter in the measurements is small compared to purely brittle materials where failure loads can vary by a factor of two or more. Ductility likely plays an important role, since the ability to yield at the crack tip makes the material less susceptible to failure from large but statistically rare defects that can dominate the strength of
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purely brittle materials. This result also suggests that the microstructure is consistent from sample to sample, which is characteristic of materials produced by powder metallurgy techniques [13] and in agreement with microstructural analysis carried out on this material [2]. Because of the tightness of the results, there is little to be gained from describing the data using Weibull statistics. The average quasi-static failure load was found to be 3876 N 100 N (95% confidence), while the dynamic failure load was found to be 4113 N 87 N (95% confidence). Thus, the tensile strength of this material is not very sensitive to the Table 1 Quasi-static and dynamic failure load measurements Quasi-static
Average (N) Uncertainty (N)
Dynamic
Rank
Load
Rank
Load
1 2 3 4 5 6
3716 3738 3908 3912 3953 4027
1 2 3 4 5 6 7 8 9 10 11 12 13
3945 3962 3992 4002 4013 4063 4096 4101 4130 4130 4261 4264 4508
3876 100
4113 87
515
displacement rate. Because of this, the remainder of the paper will mainly deal with dynamic test results, with the understanding that the quasi-static test results are similar. Doubt exists as to whether the diametral compression test yields a valid measure of a material’s strength, aside from the general question of whether brittle fracture strength can even be considered an inherent material property. The main issue is whether failure begins in the center of the disk under tensile stresses, as the test intends, or near the edge of the disk under a more complicated and less well characterized stress state [14]. To verify that failure in the Kolsky bar tests occurs by the intended mechanism of a Mode I (tensile) crack originating in the center of the disk, high speed video recordings ð105 frames=sÞ were made of each test. This recording speed was sufficient, in most cases, to capture the nucleation and growth of the crack over the span of several frames, as shown in Fig. 6. This figure shows a typical test in which fracture is observed to begin near the center of the disk and propagate outwards to each end. Secondary fractures, such as the ones visible in the third frame of Fig. 6 emanating from the loaded surfaces, were also often observed. However, these secondary fractures generally did not appear before the primary tensile crack emerged, as shown in Fig. 6. This figure also shows a typical broken specimen recovered after a dynamic test. It shows that large amounts of material are lost from the contact regions due to brittle crushing damage and cracking, the beginnings of which is evident in the last frame capture of the high speed video recording, shown in Fig. 6.
Fig. 6. High speed digital visible light video of primary tensile crack failure in the dynamic diametral compression test (left) and photograph of broken sample (right). Arrows indicate the direction of the load.
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Fig. 7. (a) Optical micrograph showing tensile crack arrested in quasi-static diametral test. White arrows indicate ends of visible crack. Black arrow indicates detail (b) showing crack path following prior particle boundaries.
In the quasi-static tests, proof that failure begins in the center of the disk during these tests was obtained by examining all the tested samples using an optical microscope. Unlike in the Kolsky bar tests, in the quasi-static tests the load on the sample can be removed as soon as the primary failure occurs, which prevents further damage to the sample and allows the initial failure to be examined. In quasi-static test samples, center fractures were observed to be completely arrested well inside the contact surfaces, as shown in Fig. 7. None of the tested samples broke into two pieces. To verify that fractures did not initiate near the middle of the contact surfaces, which would be undetectable from sideways observations, the ends of the samples were polished to a 1 mm finish and examined under an optical microscope. None of the tested samples showed evidence of fractures that either initiated at the contact surface or propagated there from within the sample. Fracture surfaces of the broken specimens revealed three separate failure mechanisms, as indicated in Fig. 8. Cleavage of the intermetallic phases, separation at prior particle boundaries and ductile fracture by microvoid coalescence are all observed. The dominant fracture mode was particle separation, followed by intermetallic cleavage. Fig. 7 shows the fracture path is circuitous rather than planar, skirting prior particle boundaries. Regions of microvoid coalescence were only observed in the few instances where cleavage zones were closely bounded by the copper matrix. Thus, ductile fracture plays a very minor role. These fracture surfaces are qualitatively similar to fracture surfaces of bullet fragments recovered from ballistic tests on soft body armor [2], adding confidence that the strength data obtained from the diametral tests will be useful for predicting bullet breakup.
2.5. Stress analysis results Fig. 9 compares experimental strain pulses from Kolsky bar tests with simulated strain pulses. The good agreement between model and experiment indicates the overall deformation of the sample is correctly captured in the simulation. In this particular experiment there is some yielding prior to fracture that is not predicted by the model, as indicated by the departure of the experimental transmitted pulse from the linear simulated pulse. However, this yielding is minor, and it is localized to the contact regions where the compressive stress is very high. At fracture, the transmitted strain signal drops suddenly. Because the finite element model does not include a failure criterion, the simulated transmitted strain continues to rise beyond the experimental fracture point. The stress distribution at failure is obtained from the model results at the time step where the simulated load is closest to the average measured failure load of 4113 N. Using this simulated failure load of 4070 N, which is well within the uncertainty of the measured failure load, avoids having to perform extensive interpolations of the model results. Fig. 10 compares the stress distribution at failure computed by finite element analysis to the two-dimensional elastic analysis (e.g. Eqs. (1), (2) and (4)). It reveals how the load distribution over the flattened contacts alters the stress distribution compared to the idealized case with point loads. In the central portion of the disk, the model results are in good overall agreement with the ideal solution. Here s2 is tensile and uniform, and the magnitude is comparable though a bit lower than the ideal value at the medial and axial locations. At the free surface, s2 is slightly higher, and it is uniform over a larger portion of the disk compared to
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Fig. 8. Scanning electron micrographs of fracture modes on surface of broken diametral test specimen: (a) cleavage and microvoid coalescence; (b) particle separation.
the medial and axial locations. At the ends of the disk, the model results deviate markedly from the ideal solution. The axial compressive stress s3 in the ideal solution becomes unbounded at the contacts, whereas in the finite element solution s3 remains bounded. s2 also deviates rapidly from the ideal solution toward the ends, becoming compressive rather than remaining constant and tensile. Similar effects have been demonstrated analytically and using finite element analysis in two dimensions with distributed loading [6,15]. Three-dimensional effects are also evident
in the stress distributions near the contact surfaces. For example, Fig. 10 indicates that the peak contact pressure, equivalent to s3 at the ends of the sample, is greater at the free surface than on the axis, contrary to expectation. This is due to the fact that the faces of the Kolsky compression bars bend slightly around the specimen as they are pressed together, resulting in a larger load on the perimeter of the contact surface. When the bar ends are modeled as rigid solids, this does not occur and the load is highest in the center of the contact surface. Three-dimensional effects
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518
0.0001
0.0003 0.0002
incident bar strain
0
0
-0.0001 -0.00005
-0.0002 -0.0003
-0.0001
-0.0004
Experiment: Incident Sim: Incident Experiment: Transmitted Sim: Transmitted
-0.0005
transmitted bar strain
0.00005
0.0001
-0.00015
-0.0006
sample failure
-0.0007 0.0063
0.0064
0.0065
0.0066 time (s)
0.0067
0.0068
-0.0002 0.0069
Fig. 9. Comparison of experimental incident and transmitted strain pulses with finite element simulation for a typical diametral compression test performed in the Kolsky bar.
0
100
-100 -200
0
σ3 [MPa]
σ2 [MPa]
50
-50 -100
Axis Medial Surface 2D Ideal Elastic Analysis
-150
-300 -400 -500 -600
-200
-700 0
0.2
0.4 0.6 Normalized Distance
0.8
1
0
0.2
0.8
1
0.4 0.6 Normalized Distance
0.8
1
0.004 0.003
ε2
0.002 0.001 0 -0.001 -0.002 0
0.2
0.4 0.6 Normalized Distance
Fig. 10. Comparison of the stress and strain distributions on the diametral plane at the failure load computed in three-dimensional by finite elements with the ideal elastic stress analysis. (See Fig. 5 to locate plots on sample mesh.)
therefore have an important influence on the development of the stresses near the contact surfaces. However, in the central portion of the disk where fracture occurs, the stress distribution in these thick, pre-flattened specimens is qualitatively rather similar to the ideal two-dimensional plane stress solution that is traditionally used to analyze diametral compression tests.
Because stress in the center of the sample is not strictly uniform, as is usually assumed when employing the ideal stress analysis method, some rationale is needed to convert this varying stress field into a single failure criterion that adequately captures the test results. Assuming the flaws that determine the tensile strength of the material are uniformly distributed, one expects that failure will initiate
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in the region where the effective tensile stress is highest and then propagate through the rest of the sample. The task is then to decide how to quantify the extent of this maximum tensile stress region in order to adequately capture the conditions of failure. Currently there is no standard way to define this region. Weakest-link methods have been used to assess the effective volume of the test and account for stress gradients and multi-axial loading in diametral compression testing of purely brittle materials [16,17]. The present material, however, does not behave in a purely brittle fashion, as evinced by the ability to exhibit crack arrest behavior (Fig. 7a) and its narrowly scattered failure loads. Consequently, these methods are not appropriate here. Shaw et al. [6] considered the test volume to be where the effective tensile stress was within 10% of the peak stress. Using this method, they were able to reconcile the tensile strengths obtained from diametral tests with those obtained using three-point-bend and expanding ring tests. However, using the 10% rule is unsatisfactory here since it captures just the two small regions near the free surface where 2 experiences shallow maxima (see Fig. 10c). Fracture does not originate in these areas. As an alternative, values of 2 within 25% of the peak value that are also found within the middle 63% of the specimen are averaged to estimate the effective tensile strain at failure. While somewhat arbitrary, this method keeps the peak strains from biasing the result, and avoids the regions near the ends of the sample where 2 rapidly decreases and turns compressive. Using this method to define the effective test volume, the averaged stresses in the disk calculated at a load of 4070 N are given in Table 2. The average of s3 in the test volume is roughly four times the average s2 value, leading to an effective tensile strength, se;2 , about twice s2 at failure. This ratio is close to the discrepancy between diametral test results and direct tension test results for many [6] but not all [9] brittle materials. The ratio of compressive yield strength to the effective tensile strength ðsy =se;2 Þ is 2.8, confirming the brittle nature of this material in tension. Using the alternative maximum tensile stress failure criterion, this ratio increases to 6.3. The uncertainty in the failure strength is taken to be twice the standard deviation of the relevant failure stress measure over the 30 mm3 test volume. Using this method, the uncertainty in Table 2 Comparison of average stress and strain determined from finite element model and ideal plane stress analysis Quantity
Finite element model
Ideal elastic analysis
P (N) 2 (Std.) (%) s2 (Std.) (MPa) s3 (Std.) (MPa) se;2 (Std.) (MPa) sy =se;2 Test volume ðmm3 Þ
4070 0.132 (0.009) 47 (7) 191 (23) 104 (7) 2.8 30
4070 0.154 64 192 121.6 2.4 –
519
se;2 is 14 MPa while the uncertainty in s2 is 7 MPa. This method of calculating uncertainty results in a larger value compared to taking the difference in the average stress at the lowest and highest experimental failure loads reported in Table 1. Thus, the uncertainty in the test is chiefly due to the variability in the stress distribution in the disk rather than the variability in the observed failure load. At failure, s2 is 27% lower than the two-dimensional analysis result using Eq. (1) while se;2 is 14% below the ideal analysis result using Eq. (3). Thus, using finite element analysis can be said to have improved the test results by these amounts depending on the failure criterion considered. 3. Conclusions Diametral compression tests were performed quasistatically and dynamically using a Kolsky bar to determine the tensile strength of a commercial frangible bullet material made from a sintered copper–tin powder mixture. Unlike purely brittle materials, the failure loads measured in repeated tests on this material were not widely scattered, owing to ductility in the copper-rich phase and a homogeneous defect population. In the Kolsky bar tests, high-speed video recordings confirmed that failure occurred by the intended mechanism of a tensile crack nucleating in the disk center and propagating to the ends. In quasi-static tests, where the load could be removed immediately after failure, the center fracture was found to be completely arrested well inside the contact surfaces on the sample, indicating a good test. Using the maximum tensile stress failure criterion, the tensile strength, as determined by finite element analysis, was 47 MPa 7 MPa. This value is 27% below the two-dimensional ideal elastic prediction owing to the distributed loading over the pre-flattened ends of the sample. Using the maximum tensile strain failure criterion, the effective tensile strength was 104 MPa 14 MPa, or only 14% below the twodimensional ideal analysis value. The failure strength was not strongly sensitive to loading rate. Fracture is dominated by particle separation, with cleavage of intermetallic phases playing a secondary role. Fracture surfaces were similar to those observed in bullet fragments recovered from ballistic testing on soft body armor, adding confidence that the fracture strengths determined here are relevant to modeling bullet impact. However, since the state of stress in an impacting bullet is generally different than that which exists in the diametral compression test, using either of the tensile strength values presented here to model the frangible bullet impact may be justified on the basis of the level of agreement achieved between such a model and actual bullet impact experiments. Acknowledgments This work was supported by Office of Law Enforcement Standards at NIST and the National Institute of Justice. The authors wish to thank Mr. Mike Kennedy of the MEL
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