La caracterización de la respuesta de metales a la carga de una onda de choque de altos explosivo (HE) es un problema de investigación desde los primeros días de la investigación de las propiedades dinámicas de los materiales. Este documento trata de estudiar los procesos de interacción y daño de la onda de choque sobre muestras de cobre en contacto con explosivos, para buscar pruebas de procesos de recompresión. Se encontró, tanto experimental como computacionalmente que se podía adaptar la magnitud de los choques iniciales y la recompresión mediante la variación de la dirección de la explosión y el espesor de la muestra de cobre.
We have studied the dynamic spall process for copper samples in contact with detonating low-performance explosives. When a triangular shaped shock wave from detonation moves through a sample and reflects from the free surface, tension develops immediately, one or more damaged layers can form, and a spall scab can separate from the sample and move ahead of the remaining target material. For dynamic experiments, we used time-resolved velocimetry and x-ray radiography. Soft-recovered samples were analyzed using optical imaging and microscopy. Computer simulations were used to guide experiment design. We observe that for some target thicknesses the spall scab continues to run ahead of the rest of the sample, but for thinner samples, the detonation product gases accelerate the sample enough for it to impact the spall scab several microseconds or more after the initial damage formation. Our data also show signatures in the form of a late-time reshock in the time-resolved data, which support this computational prediction. A primary goal of this research was to study the wave interactions and damage processes for explosives-loaded copper and to look for evidence of this postulated recompression event. We found both experimentally and computationally that we could tailor the magnitude of the initial and recompression shocks by varying the explosive drive and the copper sample thickness; thin samples had a large recompression after spall, whereas thick samples did not recompress at all. Samples that did not recompress had spall scabs that completely separated from the sample, whereas samples with recompression remained intact. This suggests that the hypothesized recompression process closes voids in the damage layer or otherwise halts the spall formation process. This is a somewhat surprising and, in some ways controversial, result, and the one that warrants further research in the shock compression community.
Key Topics
Characterizing the response of metals to direct high explosive (HE) shock loading is a research problem that has been pursued since the earliest days of dynamic material properties research. Early interest was focused on, for the most part, the understanding how energy output from the detonating HE could be best coupled into moving metals in a well characterized way for a broad range of applications. However, as has been clearly shown in the past, spall in the metal is also of concern when loading metals with a detonation wave (which is generally triangular in shape), a shock followed immediately by a more gradual release. Damage or spall occurs because reflection of a triangular wave at a free surface causes the immediate development of tension.1 Because spall is a complex phenomenon, dependent upon several variables, it is not surprising that spall resulting from triangular wave forms can yield different results than research done using flat top shocks. Flat top shock waves are commonly produced using flyer-plate impacts, for example. The time under stress may be important because work hardening in a ductile metal depends upon the time available for plastic processes, such as dislocation multiplication and glide.2 Dislocation densities are correlated to increased shock hardening3–5 via increased work hardening, which has been linked to lower spall strengths in some materials. For triangular wave shapes, relatively less time is spent at peak stress, reducing the time for nucleation and growth of damage and possibly leading to a higher spall strength.6,7 The degree of spall and damage formation is also thought to depend on the peak stress, tensile strain rate, material microstructure, and locations of impurities.8,9 Of these effects, tensile strain rate is known to have a relatively large effect, and reported variations of spall strength with stress amplitude may actually be a manifestation of changing the tensile strain rate.
In previous studies,6,7 it was reported that copper targets subjected to compressive and tensile loading from flyer-plate impacts producing flat top and triangular shocks can exhibit free surface velocity profiles indicative of spall, depending on the details of the exact stress-time history applied. When there is a complete spall or a very extensive and continuous plane of damage in a sample, an acoustic wave is trapped in the spall scab and reflects back and forth, leading to a sample free surface velocity profile with oscillations (ringing). The oscillation period is twice the thickness of the spall scab divided by the sound speed. Having such a single-frequency trapped wave is strong evidence of either complete spall or a significant number of voids. For samples that do not spall or damage extensively, there can be a similar ringing, but in this case, the period is often consistent with the full sample thickness. In experiments where the velocimetry indicates spall scab ringing, the metallurgical analysis of the recovered copper samples for various experimental stresses and release rates reveals a variety of conditions, ranging from plastic strain without damage to complete spall. However, the location of the damage plane, whatever be the extent of the damage, is consistent with the ringing period.6,7 These observations suggest that a free surface velocity measurement is a good indicator of the location of the damage plane, but is not always a reliable indicator of complete spall separation.
Previously, an apparently anomalous result was reported for direct triangle wave loading of copper with the explosive Baratol.10 No ductile voids or evidence of void or crack coalescence were observed in the cross section of the recovered copper samples, in spite of the fact that the measured wave profiles showed a ringing signature indicative of a spall plane (either complete or with a large population of voids). Instead, the recovered sample showed a metallurgical feature that was at the time interpreted to be localized plastic strain features and high dislocation densities with no evidence for voids. This feature was at the location in the sample where spall damage was expected based upon ringing in the time-resolved free surface data. This data set has raised many questions concerning HE-driven spall damage; the overarching issue is reconciling the time-resolved data, which show strong evidence for considerable spall damage, with the microstructural evidence from the recovered sample.
Subsequent experiments10 with PBX-9501, a more energetic HE, showed multiple spall and damage layers consistent with the velocity profile. It was postulated that different release rates and/or local plastic deformation can alter the local impedance of the material enough for acoustic wave reflection without void formation. However, this has not been experimentally verified, and a change in local impedance (a damaged layer) almost certainly could not have caused the kind of ringing observed in the wave profiles. Such a layer would allow for both reflected and transmitted waves, which would alter the nature of the ringing signature. A spall plane would allow for only a reflected wave with a single ringing frequency observed. It is worth noting here that the recovered samples exhibit features from the entire process that the sample was subjected to, from the moment the shock enters the sample until it is recovered and sectioned for metallurgy. Recovery techniques are not capable of providing time resolution of the sample loading and unloading history.
A possible explanation of this behavior was postulated by Becker and LeBlanc.11 They suggested that the damaged zone could be recompressed after void formation using a shock wave of sufficiently high stress. Specifically, they proposed that if a sufficiently strong recompression wave follows tension, the recompression can drive the damaged layer back together, causing the voids to collapse and the spall-induced surfaces to “stick” back together. They further postulated that collapsed voids might not be readily apparent in subsequent metallurgical analysis of the recovered sample. They conducted gas gun experiments with a layered flyer plate to drive a recompression shock into the spalled target and determined that their experimental results support their hypothesis. They found highly strained material where the spall plane was expected, but there were no remaining voids in the optical images of the recovered samples. More detailed analysis using electron backscatter diffraction revealed highly localized plastic deformation and the remnants of what were interpreted to be collapsed voids. Others have also used layered flyer plates to produce spall and recompression.12
For HE drive, spall may occur while the sample is still being accelerated by the detonation product gases. Tension from release at the free surface can pull the spall scab away from the sample, and it can coast at a constant velocity for a while. If the HE product gases continue to accelerate the remaining sample sufficiently, it may impact the scab and cause recompression and acceleration of the scab. Details of this recompression will depend upon the thickness (or mass) of the remaining target and the explosives used. Fig. 1 is a notional time versus position diagram for an HE-driven experiment with spall and recompression. The initial shock front (a) is reflected at the free surface (b) as a release wave and interacts with the still oncoming Taylor wave release, creating tension and spall at (c) and perhaps also later at (d). A trapped wave (e) in the spall scab (f) causes the characteristic ringing in free surface velocity profiles, but on average, the scab travels with a constant speed. A trapped wave (g) rings in the remainder of the sample, which continues to accelerate because the HE product gases are still under pressure. Eventually, the sample can catch up to and impact (h) the spall scab and recompress the spall plane. After recompression, both waves (e) and (g) may be able to pass through the spall plane.
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The fundamental question in this argument arose: Can the spall scab and the remaining sample be recompressed together in such a way as to “weld” them back together and leave the metallurgical “scar” observed in the recovered sample? Answering this question was the primary motivation for this research. Detailed simulations using the CTH hydrodynamics code13 were done to see if the fundamental governing equations, as solved numerically, support this possibility. Results support the hypothesis that, depending upon sample thickness, a sample could be spalled, and then, the pieces pushed back together by continued drive from the HE product gases. The details were somewhat different, but the overall features were captured. We plan to document these results in a future publication.
In the Baratol experiment,10 the velocimetry data also showed an increase in particle velocity (usually indicative of a wave arrival) at late times, but the published data were truncated because it was believed that the increase was caused by edge releases. These data might also be interpreted to mean that the copper sample spalled, but later push by HE products caused a recompression event that essentially welded the sample back together. However, we note that in Ref. 10 the authors state that the post-recovery metallurgical analysis yields no evidence supporting such a recompression event. This discrepancy indicates a strong need to do further experimentation.
TABLE I.
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The experimental configuration is shown schematically in Fig. 2. We used a 5- or 6-layer stack of 25 mm diameter by 1.7 mm thick sheets of Detasheet to produce a peak shock stress very close to that of Baratol, which is no longer readily available. This was done to allow comparison with the previous experiments10 that used Baratol drive. The HE is axially detonated with an RP-1 detonator. This yields a slightly divergent, nearly 1-D shock wave in the sample.
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To minimize the effects of wave releases from the edge of the 25 mm diameter HE drive, we used only the center 10 mm of the target for our analysis. The copper target was a 10 mm diameter disk press fit into a guard ring14 (40 mm outer diameter and 10 mm inner diameter) of similar copper with an interference fit and no measureable gap. After assembly, the target was polished flat to the final thickness of 0.6 to 4.3 mm. The guard ring formed a momentum trap for edge releases, allowing planar compression but no significant radial tension in the central sample, thereby minimizing 2-D perturbations. Often, momentum-trapping rings used on gas gun experiments require several components.15 Since our HE drive has a slightly curved shock front, we are able to use 2-D hydrocode simulations to design a single guard ring that quickly pulls away from the sample, leaving a gap between the sample and ring while the sample remains relatively flat. All targets were prepared from 99.99% pure oxygen-free, high-conductivity (OFHC) copper (c10100 specification). The center 10 mm portion was from a sample annealed under vacuum at 600 °C for 1 h, resulting in an average grain size of 40 μm.
A steel stripper (a steel plate with a hole that allows only the center 10 mm sample to pass) kept the guard ring fragments from impacting the target during soft recovery in a ballistic gel. After the sample passed the steel stripper and pellicle turning mirror, a single-pulse flash x-ray system provided a radiographic image of the target before it entered the ballistic gel and was captured. These images were taken about 100 μs after detonation to verify the shape and trajectory of the 10 mm center of the target. After an HE experiment, the sample was recovered from the ballistic gel. The shock stress generated in the samples when striking the gel ranged from 2 to 4 GPa because of the relatively high velocity imparted to the sample by the HE drive. These are significant reshocks.
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The shock breakout velocity decreases with increasing sample thickness, as expected for decaying shock waves, because the releasing wave overtakes the leading shock as it propagates. Peak shock stresses near the free surface just prior to the shock breakout were about 27 GPa for the 0.6 mm sample and decreased to about 17 GPa for the 4.3 mm sample. The release rate immediately after the shock breakout also decreases with sample thickness, from 2100 m s−1 μs−1 at 0.6 mm to 630 m s−1 μs−1 for 4 mm thickness. Consequently, the damage layer is expected to form deeper into the sample for thicker samples. The ringing period shows that the putative damage layer forms at 0.17 mm for a sample thickness of 0.6 mm and at 0.43 mm for a sample thickness of 4.3 mm. The depth of the spall signature from each experiment is used to estimate the spall strength, which shows some dependence on the sample thickness and has values around 3.5 GPa (Table I). This approximate value was calculated using the momentum shock jump condition
where ρ0
The 3.0 mm and 4.3 mm samples, shots 4 and 5, did not show any late-time reshock in the velocimetry records, suggesting that these heavier bulk samples never caught up to the spall scabs. X-ray images, described below in Section II E, show a spall scab that was nearly detached from the sample for shot 4 and a scab that was completely detached for shot 5.
The measured spall scab coast velocities and asymptotic bulk sample velocities are plotted as a function of sample thickness in Fig. 4. In the case of the 4.3 mm sample, the spall scab completely detached from the bulk sample, so we estimated the bulk sample velocity from the timing information obtained from the x-ray image of this experiment. For samples thinner than 2.2 mm, the asymptotic velocity was higher than the spall scab coast velocity, and the bulk sample impacted the spall scab and recompressed the sample. For samples thicker than 2.2 mm, recompression cannot occur and complete spallation is expected, as shown in Fig. 4. It is important to note that the details of the coast and asymptotic velocities are dependent on the geometry of the HE package and are specific to our experiments. Experiments done with different kinds of HE would differ in detail. However, similar qualitative trends are expected for a wide variety of HE experiments.
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It is interesting to compare shots 5 and 6, which were a 4.3 mm thick copper sample driven by a Detasheet shock and a 2.2 mm thick sample driven by NM, respectively. The velocimetry from these shots is plotted in Fig. 5. As can be readily observed, the peak shock stresses and the release rates were similar. Consequently, we expect the initial damage should be similar as well. The recovered samples are shown in Fig. 6. The recovered sample from shot 5 shows a separate spall scab, while the sample from shot 6 does not. (This will also be evident in the x-rays, Section II D.) The principal difference is that only shot 6 had a recompression wave. As described above, the thicker sample does not accelerate enough to overtake its spall scab.
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In the Baratol-based experiments,10 a metallurgical “scar” was observed at the approximate distance from the free surface as predicted for spall to have occurred (based upon the ringing period in the velocimetry data). Nevertheless, the authors rule out the possibility that their sample was recompressed because there was no evidence of ductile failure, such as void formation or coalescence. Although there was evidence for localized plastic strain, the grain structure surrounding the metallurgical feature remained largely undisturbed. It is worth noting that a recent reexamination of the velocimetry results for times beyond where the velocity waveform was truncated in Ref. 10 showed a recompression wave very similar to that observed in our Detasheet experiments. In Fig. 8, we show data from that experiment as reanalyzed over a longer time frame. Early times show a typical triangular-wave spall signature with associated ringing, and late times show a recompression pulse. The late-time (∼5 μs) increase in particle velocity was ignored at that time, believed to be caused by edge release waves.
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We also note that micrographs made using optical imaging and orientation imaging microscopy (Fig. 3 of Ref. 10) show a metallurgical feature about 1 mm from the free surface, which is similar to the distance calculated from the ringing structure in the time-resolved data (Table I of Ref. 10). The authors concluded that these features were not evidence for spall having occurred. This brings into focus the fundamental issue: the time-resolved data showed clear evidence for spall damage, but the microstructural analysis did not.
We have looked at some of our recovered samples using optical imaging. The analysis of the complete set of recovered samples is an ongoing process and will continue into the future as resources allow. We show here early results from our 1.9 mm sample driven by Detasheet explosive. Fig. 9 is a photograph, made with an optical microscope, of a cross section of the center part of this sample. The sample was cut through a radius and then polished and etched. We see evidence for a band of perturbed microstructure similar to that reported by both Becker11 and Koller10 in their results. This band is very close to the location predicted from the period of ringing in the velocimetry from this experiment.
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Before shock deformation, the copper metal used in our experiments contained grains of sizes that ranged from 10 μm to greater than 100 μm, as shown in the optical micrographs of Figs. 10(a) and 10(b). They also had some texture to them, as observed in the scanning electron micrograph (SEM) of Fig. 10(c). The texturing is more clearly observed on the grains of darker contrast, which happen to be optimally oriented for best texture imaging. The lighter grains exhibit a homogeneous surface that is lightly etched.
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After impact, the samples show significant deformation, depending upon the sample thickness. SEM pictures of the 1.9 mm specimen (Fig. 11) were found to be completely different in morphology when compared to the pristine copper specimens. This sample shows definite inhomogeneities reminiscent of highly deformed and recrystallized copper in the presumed spall region, not surprising if one assumes a significant increase in temperature17,18 during the tensile strain process. Note that the shock that initially compresses the target carries a stress that is estimated to cause a relatively minor (<200 K) temperature rise. But as the rarefaction waves interact in the sample, stretch the material, and presumably create voids or a complete spall plane, significant plastic deformation is occurring; this can cause a larger (but hard to estimate) temperature rise. If this spall damage is recompressed (as we hypothesize), an even larger temperature rise may be expected (again, hard to estimate) as is typically observed for compression of porous materials. The high strain-rate deformations and subsequent temperature rises experienced by this specimen can result in a number of metallurgical reactions, especially in the tensile region, including localized recrystallization such as we observe here. This microstructure is very similar to what has been found on polycrystalline specimens of copper during deformation at 473 K (Ref. 19) as well as aluminum powder that has been dynamically compacted.20 Compacted aluminum powders were shown to have regions where localized heating during the porous compaction process caused localized melting.
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In the SEM of the recovered 3 mm specimen (Fig. 12), grains with varying orientations and some porosity are evident. The connected pores in Fig. 12(a) correspond to the region that presumably separated and reconnected during the dynamic history of the sample (i.e., the presumed spall region). This specimen has similarities to the pristine copper in that the grains have a lightly etched and homogeneous surface. However, the texturing of the sample is no longer present, suggesting some level of recrystallization. It is worth noting here, as mentioned earlier, that this sample was observed in flight (between the initial HE loading and the recovery gel, Fig. 7) to have a spall region for which the x-ray image clearly shows a low-density band in the sample. When entering the ballistic gel, it will be subjected to a reshock that can be a few gigapascals (estimated using a CTH simulation). The fact that this sample was recovered as one piece rather than two is suggestive that the shock it sustained during recovery was high enough in stress to somehow reattach the two pieces. This process is different in detail from that for the 1.9 mm sample, possibly explaining the slightly different observed microstructures. We also note that because of the high velocities, the samples obtained as a result of the HE drive process, and the existence of a significant reshock when entering the recovery medium, the recovered sample microstructure contains information about the sum total of all the stress excursions experienced from the time the initial shock enters until the sample is sectioned for microstructural analysis. This complexity of history would seem to indicate that we must be careful in interpreting the microstructural results.
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The samples did not suffer any changes in composition (see x-ray diffraction pattern in Fig. 13), but there are differences in the relative intensity of the peaks, suggesting changes in grain orientation in line with what is seen in the SEM. Thus, the changes in sample morphology are necessarily connected to the different stresses and temperatures experienced by the samples during their complex dynamic histories. The 3 mm specimen reached a peak stress of 19.5 GPa, whereas the 1.9 mm specimen reached a peak stress of 22.4 GPa (Table I). This small difference in stress will not result in a large difference in initial shock temperature. What will be different are the details of what happens in the putative recompaction process, since it is earlier and stronger for the 1.9 mm sample than for the 3 mm sample. These differences may be responsible for the elimination of texturing and partial recrystallization in the band of perturbed microstructure of the 3 mm specimen (Fig. 12), while the 1.9 mm specimen experienced full recrystallization in the band of perturbed microstructure (Fig. 11). In any case, the microstructural details in the perturbed region for both samples show clear evidence for some amount of recrystallization, which may have been caused by local temperature increases or perhaps other unknown dynamic processes.
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We note that in Ref. 10 the authors state: “These micrographs show that the areas of plastic strain do not preferentially follow the grain boundaries, but also slice through whole grains leaving the surrounding material undisturbed. This indicates the material did not crack or form voids and recompress during recovery as this type of process would lead to much more disruption of the surrounding grain structure.” This is true for the samples recovered here as well. We observe a localized metallurgical “scar” that is not confined to grain boundaries. So, there is still an important question to be answered: Can the postulated spall and recompression process leave this kind of metallurgical feature? This remains a topic of active research.
This research has focused on a previously identified issue in directly driving copper plates with HE in 1-D and nearly 1-D geometries.10 HE drive of metal coupons results in the metal sample being subjected to decaying triangular wave loading. When such a wave shape arrives at a free surface and reflects through itself, tension develops, and it can become very large in amplitude (depending upon the initial shock compression stress). When high enough, it can cause damage in a localized region and potentially the formation of a spall scab. For copper, this process is known to happen through the nucleation of ductile voids, which can coalesce if there is sufficient tension. Without damage to relieve it, the estimated tension (from CTH simulations) for Baratol- or Detasheet-driven copper samples would be approximately 3 times the spall strength as determined from the time-resolved data. This leads to a natural question: Why do the samples recovered in the work of Koller10 show no clear metallurgical evidence of a spall plane, or even any ductile voids?
To study this issue, we performed several HE-driven spall experiments on copper samples using Detasheet or NM as the shock drive and velocimetry as the principal time-resolved diagnostic tool. Input shock stress values ranged from 15.7 to 27.1 GPa. In one case, we started with a 1.9 mm thick sample and observed wave profiles that are very similar to those in Ref. 10. Both our wave profiles (for the relatively thin samples) and those from Ref. 10 show a late time rise in particle velocity. In Ref. 10, this reshock was thought to be due to edge effects, but from our data and CTH simulations, this wave is now thought to be consistent with a recompression. It is possible to explain the time-resolved data by the following. For thin samples, a spall scab is created in the initial spall process and moves away from the rest of the sample with a constant velocity. HE product gases then continue to push on the remainder of the sample until it overtakes and impacts the spall scab, causing a shock wave to be created and move forward into the sample. The recompression shock is observed as a rise in particle velocity. This possibility is further supported by the ringing in the spall scab, which at an early time has a period consistent with a ∼0.5 mm thickness, but after the rise in particle velocity is consistent with the full thickness of the sample.
For our Detasheet-driven experiments, we then varied the target thickness, which caused details of the velocimetry data to change and led to the observation that for 3 and 4 mm target thicknesses, no late time rise in the particle velocity was observed. This correlates with the late time x-ray results, which show clear evidence of spall separation for 3 and 4 mm thick samples. This change in recompression response was also observed in hydrodynamic calculations. For experiments with sample thickness of 2 mm or less, we still observe the late time rise in the particle velocity.
We have also recovered most of the samples and done a preliminary analysis on some of them. The results for our 1.9 mm sample experiment appear to be very similar to those in Ref. 10 for Baratol-driven copper.
Overall, the results are interesting and somewhat surprising. We note that:
- (1) Our data and the data of Ref. 10 support the hypothesis for spall and recompaction as presented in Ref. 11 by Becker et al., except for the details of the microstructure.
- (2) In the band of perturbed microstructure observed in both our experiments and those of Ref. 10, it is surprising (as pointed out in Ref. 10) just how narrow and localized the regions of disturbed grain structure are. Our results show evidence for recrystallization in these regions, where the orientation imaging microscopy results of Ref. 10 show black bands where grain structure was not resolved.
- (3) We believe that the microstructural results by themselves do not necessarily refute the spall and recompaction hypothesis and that more microstructural analysis is warranted, perhaps along with more focused experiments. As resources allow, we will attempt to do this in future research.
- (4) For Detasheet drive and a 1.9–2.0 mm copper sample geometry, it would be very useful to make a dynamic imaging measurement to see if a spall plane is observed before recompression. If such a measurement can be made, and if a spall plane is observed and a full thickness sample is recovered, this would directly support the spall/recompaction hypothesis.
- (5) It is possible that the 1.9 mm and thinner samples never did spall. That leads one to question how that is possible given the amount of tension generated in the copper sample. It also requires an explanation of what led to the band of perturbed microstructure that is observed at the precise location where a damage plane is indicated by early time ringing in the velocimetry. Finally, it requires another explanation for the features observed in the velocimetry data, where results from considerable previous spall research have been used to interpret the observed features.
In conclusion, this research has led to some very interesting results and has shown a clear need for yet more research in this area. If we cannot currently explain these results unambiguously, this indicates a need to improve the current understanding in the shock physics community of the physics of dynamic damage. We look forward to more research being done in this area.
This manuscript has been authored by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the U.S. Department of Energy and supported by the Site-Directed Research and Development Program. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published forms of the manuscript, or allow others to do so, for United States Government purposes. The U.S. Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan): DOE/NV/25946–2814. We wish to acknowledge Mike Grover of Special Technologies Laboratory for his significant part in this research and J. D. Montalvo and J. C. Foley, both of Los Alamos National Laboratory, who provided the optical imagery. We are very grateful to G. T. Gray, D. Koller, and V. Livescu for many helpful discussions and for supplying us with high-purity, annealed OFHC copper for these experiments.
APPENDIX: CALCULATION OF MAXIMUM DISTENSION BETWEEN BULK SAMPLE AND SPALL SCAB BEFORE RECOMPRESSIONIn this appendix, we introduce a model for estimating the maximum distension between the spall scab and the remainder of the sample in our spall and recompression experiments. This distension estimate can be thought of as an upper limit to the size of any voids formed prior to recompression and is useful for understanding the features found in the recovered samples.
In ideal 1-D triangular wave spallation, there are multiple planes of spall damage separated at regular intervals (as seen in the schematic drawing in Fig. 14, which shows the first two spall planes and the simplified velocity of these two planes). After spallation, scab 1 (the spall plane closest to the free surface) moves with the coast velocity Vcoast, which differs from the peak breakout velocity by the pullback velocity Vpullback. At time trecompress, the recompression wave reaches scab 1 and is detected in the velocity measurement. The spall stress for scab 2 is assumed to be the same as the spall stress for scab 1, so we assume that the pullback velocity for scab 2 is the same as the pullback velocity for scab 1. Therefore, the difference in velocities between scab 1 and scab 2 is also the pullback velocity Vpullback. Maximum distension (Xmax) occurs at time tmax, which is the moment that scab 2 is recompressed by the bulk sample.
where we set the spall time tspall = 0 and have neglected the pullback times and the time for the sound wave to cross scab 2. We further assume that when the bulk material recollects scab 2, it moves with the same recompression velocity as observed for the scab 1 recompression. Therefore, the maximum distension can also be written as
where Vrecompress − Vcoast is the closure velocity and trecompress − tmax is the closure time. Equating the above two equations allows us to solve for tmax
Rewriting Vrecompress − Vcoast as the bump velocity Vbump, the maximum distension in our simplified model based on measurable quantities is
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The calculated maximum distension values for the copper experiments with recompression are provided in Section II C.
Fuente: http://scitation.aip.org
