Experimental and numerical investigations of beryllium strength models using the Rayleigh-Taylor instability

Beryllium (Be) is a metal with excellent structural properties and unique radiation characteristics.¹ It has a high elastic modulus, a low Poisson ratio, a low density, and a high melting point. Be has an elastic stiffness comparable to steel, at a quarter the density of steel.² Its high strength-to-weight ratio and high melting point make it ideal for many defense and aerospace applications.^3–7 However, Be's low ductility at room temperature presents challenges for both manufacturing and conditions where it might experience extreme deformations, thus limiting its use to low strain applications. It would, therefore, be helpful to understand the dynamic behavior of Be under more extreme conditions of high pressure, strain, and strain-rate.

Early investigations of Be focused primarily on dynamic material properties of polycrystalline Be under tensile stress conditions.⁸ Initial Be equation of state descriptions and shock wave profiles up to 5 GPa provided material constants for early analytic models.⁹ Christman and Feistmann¹⁰ investigated the dynamic properties of Be, such as elastic constants and elastic precursor decay, which produced a yield plateau, strain hardening, and strain-rate behavior. Chhabildas et al.¹¹ studied the hcp-bcc phase transition in Be over the stress region 6–35 GPa using shock-release experiments. Using biaxial tensile tests, Lindholm et al.¹² and Lindholm and Yeakley⁸ observed yield, plastic flow, and failure of Be under plane stress conditions. Pope and Johnson¹³ performed the first attempt at studying yielding on primary slip planes of Be using shock loading planar impact of single crystal Be. They also studied the effects of material anisotropy on plane wave propagation. Jönsson and Beuers¹⁴ studied the dislocation microstructure of single crystal Be at 2% strain. Christian and Mahajan¹⁵ provided an extensive review of twinning in various crystal structures, including Be. Using Split-Hopkinson pressure bar (SHPB) experiments with strain-rates from 10⁻³ to 10⁴s⁻¹, Blumenthal et al.^1,16 studied the evolution of dynamic mechanical behavior and crystallographic texture to understand deformation mechanisms and the role of texture in polycrystalline Be. Brown et al.¹⁷ showed how active deformation mechanisms can be controlled via manipulation of straining direction,¹⁸ deformation temperature,¹⁹ deformation rate,² and crystallographic texture.²⁰ Brown et al.²⁰ extensively investigated the importance and relative contribution of twinning and slip in Be over a range of strain-rates (10⁻⁴–10⁴s⁻¹). Other experimental data² showed the dominance of twinning at strain-rates around 10⁴s⁻¹. Brown et al.²¹ recently performed Be ramp-release experiments to investigate Be shear stress in high strain rate (10⁶s⁻¹) and pressures (110 GPa).

Be failure modes and spall have been the focus of recent experimental work. Experiments of explosively loaded Be samples up to strain-rates of 10⁴–10⁵s⁻¹ were used to investigate Be spall fracture and showed a weak dependence between the spall strength and strain-rate.²² Adams et al.^23,24 observed elastic precursor decay as a function of target thickness in plate impactor experiments. In earlier experiments, their data indicated brittle spall behavior and a long rise-time in the elastic and plastic waves, which they attribute to twinning being the predominant initial deformation mechanism.²³ In another set of plate-impactor experiments, Mescheryakov et al.^25,26 instigated spallation in the Be sample and studied material tensile strength in the microsecond region of dynamic loading. Recent experiments^27,28 used post-mortem analysis of explosively driven Be to evaluate failure behaviors. Peak shock pressures of 15 GPa were observed but no definite source of failure in Be was identified.²⁷ 

Few dynamic studies of Be have been performed in recent years and little is known about its strength properties at high strain (≳0.2), strain-rate (≳10⁴s⁻¹), and pressure (≳10 GPa), yet there are many competing material strength models that try to characterize its behavior by extrapolating to these more extreme conditions with often diverging results. Most strength models are informed by physics (e.g., strain hardening laws, rate dependencies on thermal activation and phonon drag, pressure and temperature dependence of shear moduli, etc.), but are ultimately predicated on certain ansatz to facilitate practical applications to phenomena in regimes beyond those easily accessible with current experimental techniques. As a result, the governing equations contain multiple parameters that are calibrated using data from experiments in low pressure, strain, and strain-rate regimes, e.g., data from Split-Hopkinson pressure bar experiments (at zero pressure, strains of 10–20%, and rates below 10⁴s⁻¹) or Taylor anvil experiments (∼10 GPa, strains of 0.1 (Ref. 29), and rates near 10⁵s⁻¹). Because the models are typically tied to phenomenology, they tend to diverge when confronted with data far from their calibration points, whether that is in pressure, temperature, strain or strain-rate space, see Figure 1. In this paper, we discuss a set of dynamic experiments to characterize Be strength behavior under extreme conditions and use the results to discriminate among different strength models. A Rayleigh-Taylor (RT) instability occurs at the interface between two materials accelerated such that the pressure and density gradients are anti-parallel.^30,31 Under conditions where the accelerated material has no strength (and low viscosity), a surface perturbation grows non-linearly as t², where t is time. However, in the instances where the material remains solid, the perturbation growth is mitigated and even halted depending on the mode of the interface perturbation.³² In 1974, Barnes et al.³³ took advantage of this observation and developed a technique using high explosives (HE) to accelerate aluminum and stainless steel plates with perturbations machined on the HE facing surface. By setting off an HE charge at a stand-off distance of ∼1.3–2.5 cm, they accelerated the plates without shocking the material and reached peak pressures of 10 GPa. By modeling the growth of imposed sinusoidal perturbations on the side of the material facing the expanding HE products, they inferred the influence of strength during the dynamic loading process. Others have since expanded the technique to other materials using modern diagnostic techniques in addition to traditional flash radiography,^34,35 while others still have adopted the technique using lasers to drive targets with plasma to achieve yet higher strain rates and pressures.^36,37 We designed HE-driven RT experiments for ramp loading of Be to reach pressures of 50 GPa and strain-rates of 10⁶s⁻¹. These are pressure and strain-rate regimes where data are sparse and strength models diverge in their predicted behavior.

This article is organized as follows. We describe the setup of the HE-driven RT experiments and present the experimental results in Sec. II. After discussing the existing strength models, we compare the experimental results to simulations of the experiments in Sec. III. We end with a discussion of Be recovery experiments. Though there are other effects besides plasticity, our analysis is predicated on the assumption that plasticity is the dominant effect governing the flow. We address and qualify this assumption in Sec. III.

The experimental setup is shown in Figure 2. A two-stage planar HE drive launches an iron impactor at a second charge of HE, overdriving it beyond the Chapman–Jouguet pressure. The detonated HE products expand across the vacuum gap and shocklessly load against the Be target, which has a machined sinusoidal perturbation on the loaded surface, Figure 3, with a quasi-isentropic compression wave, thereby initiating an RT instability at the HE-Be interface. A Plexiglas bracket initially holds the Be target in place before the target is accelerated by the HE. The Be targets are made of S200F Be composed of 98.5 wt. % pure Be with a maximum of 1.5% BeO. The targets were manufactured by Materion Electrofusion using a hot isostatic press. The rippled patterns were formed via wire electrical discharge machining. The Be microstructure was imaged using electron backscatter diffraction (EBSD), see Figure 4. The grain size distribution was calculated from the EBSD image, see Figure 5. The average grain size in the tested Be was 9.5 μm. EBSD scans indicate a weak (0001) basal plane texture aligned with the drive direction, see Figure 6. Nine Vickers hardness tests determined the average Be hardness to be 1830 ± 200 MPa (from a hardness of 186.9 kgf mm⁻² from a load of 1 kg over a ∼0.031 mm long diagonal spot). The perturbation is sinusoidal and the wavelength for all the RT experiments was λ = 4 mm. The perturbation peak-to-valley amplitude for four experiments with 2 mm substrates (as measured to the middle of the perturbations) was 0.48 mm. For two other experiments, the perturbation amplitude was 0.38 mm on 1.78 mm thick substrates. An example of a machined Be target used in these experiments is shown in Figure 3. The experiments were designed by conducting a series of simulations over an ensemble of strength models and Be equations of state. To optimize the dispersion of the Be strength models (thereby maximizing model differentiation), the design simulations varied the perturbation wavelength, amplitude, target thickness, and HE stand-off distance. Bevels were machined into the outer edge of the targets to enable side-on imaging of the ripple growth as the target accelerates and deforms. Without these bevels, the outer edge of the target would hide the perturbation growth from the imaging diagnostic.

We performed several SHPB experiments to characterize the dynamic behavior of the Be samples in strain-rate regimes between ∼2000 and 5000 s⁻¹ and compare with models calibrated to previous data.³⁸ These experiments were performed at ambient conditions. The targets were made from the same batch of pressed S200F as the RT targets, and the SHPB tests were conducted at LLNL. The SHPB data usually end at low strains in the samples due to brittle failure. Figure 7 shows the stress-strain curve for an experiment with a strain-rate of 2000 s⁻¹ (the data from a single experiment are shown for illustration purposes and the SHPB experimental data for all the experiments are available in MIDAS (Ref. 38)). The strength models based on previous SHPB results seem accurately to describe the stress-strain response at these relatively low strain-rates. It is not possible to discriminate among the different models in these regimes for these particular samples. The multiple parameter sets for the Preston-Tonks-Wallace (PTW) model all fit the SHPB data but they predict different behavior for the higher strain-rates observed in the RT experiments, see Figure 7(b). The RING model also fits the data well in this regime, though it predicts lower stress at higher strains than the data. The lack of differences in the models at these low strain-rates, and strains, and their significant divergence at high strain, strain rates, and pressures is the main motivation for performing these RT strength experiments. The RT experiments are designed to provide data in these regimes to help discriminate among the different available strength models.

Two diagnostic techniques were used to capture the experimental data. X-ray radiographs at the Eridan-3 facility at RFNC-VNIIEF imaged the target perturbation growth, see Figure 8. A 1-MeV, 0.15 μs pulse flash x-ray was used to record one image for each experiment, which was captured on a ADC-CR photochromatic screen.³⁹ Measurements of the free surface velocity of the targets were performed with a Velocity Interferometer System for Any Reflector (VISAR) during each experiment.⁴⁰ The VISAR spot size is 200 μm and the spot is centered on the back of the target to minimize the effects of potential bowing of the target as it is being driven (radiographs shown there is very little bowing until late in time for some shots, see Sec. II C). The pressure of the explosives on the loaded surface of the target determines the free surface velocity of the target, and hence the expected RT growth. Comparing the VISAR data with simulations indicates that the drive conditions in the simulations match that of the experiments so a proper interpretation of the growth data can be made (see Sec. III).

We performed a total of six HE-driven Be RT experiments with x-ray diagnostics to measure the perturbation growth as a function of distance travelled. We use distance travelled since it can be measured directly in the experiment without having to account for fiducial timing in the HE drive. Time and distance are, of course, easily related through the velocimetry. The radiographs are shown in Figure 9. The clear white region in the center of the radiograph is the Be liner. The bright area on the bottom of the radiograph is the HE, and the bevels observed on the side are the Plexiglas brackets holding the liner. The apparent absence of any visible voids in the Be suggests that the targets have not spalled at image time, though there is the possibility that cracks formed at length scales below the camera detection limit. It is possible that the last radiograph shows signs of failure in the bubble, though this is not very clear. The perturbations exhibit non-linear growth at larger distances, see Figures 9(e) and 9(f). The evolution of the perturbation growth with increasing distance travelled was measured using the six radiographs, and the growth factors are shown in Figure 10.

Measurements of the free surface velocity during each of the experiments indicate consistent drive conditions, see Figure 11. Time t = 0 is the HE arrival time at the Be/HE interface. For a small part of the trace, there are some spurious fringes around 1 μs due to the large VISAR spot tracking different parts of the target with slightly different velocities. The velocimetry profile recovers at 1.2 μs and those fluctuations disappear.

We model the experiments with Ares, an Arbitrary Lagrangian Eulerian hydrodynamics code.⁴¹ The mesh resolution for all the simulations is 8 μm, at which point the simulation results are converged. We assumed planar symmetry and performed two-dimensional simulations of a half wavelength slice of the system, thereby neglecting the release at the edges of the system. The second layer of HE, which unloads against the target, is 90 mm in diameter while the target itself is 50 mm across. The gap between the two is only 2 mm (see Figure 2) so that even if the release in the unloading HE products was fast enough to decrease the planarity of the drive at a 45° angle, the target would still see a 1D planar drive when the HE products arrived at the front surface. Assuming a sound speed in Be of 13 000 m s⁻¹, it takes the release wave from the edge of the Be about 2 μs to travel the radius of the target, affecting only the latest few data points, which were obtained after seeing limited growth at the earlier times taken first in the sequence. 2D simulations were performed in the target design to determine the shape of the bevel at the edge of the target, which was specifically designed to prevent bowing in the target that could interfere with the side-on view of the diagnostic. The results shown in Figure 9 demonstrate the planar behavior of the target even at long travel distances.

The simulated system consists of the iron impactor, the HE, the vacuum gap, and a half wavelength ripple on the Be target. The iron impactor initiates the HE detonation, with its impact velocity determining the peak pressure in the HE explosion. The HE was modeled with a JWL++ reactive flow equation of state⁴² using the parameters in Table I. All other materials used a tabulated equation of state from the LEOS data library based on a QEOS-like model.⁴³ We also compare the results using an analytic Gruneisen EOS. The drive, Figure 12(a), and the growth, Figure 12(b), are very similar regardless of the form of the equation of state. Though contributions to the results beyond plasticity might exist, we did not use a damage model for the Be as we assume that strength is the dominant effect in these experiments. Our analysis assumes that the observables are a direct result of plastic flow. We address this assumption at the end of this section.

We examined the behavior of several strength models, which typically have very different dependencies on strain, strain-rate, and shear modulus, as they relate to the Be flow strength. The Steinberg-Cochran-Guinan⁴⁴ (SCG) model is rate-independent but assumes “high” rates of order 10⁵s⁻¹. The flow strength goes as the strain to the nth power, where n is a work hardening parameter, and the shear modulus includes linear pressure and thermal terms. The Steinberg-Lund⁴⁵ (SL) model is based on the SCG model and adds an additional strain-rate dependence in the thermal activation regime. The strain-rate depends on the inverse of the sum of an exponential of the thermal component of the stress with the inverse of the athermal component of the stress. The PTW (Ref. 46) model describes material behavior in both the thermal activation and phonon drag regimes over many orders of magnitude of strain-rate. At low strain-rates (<10⁴s⁻¹), two different expressions for the work hardened saturation stress and flow strength are used to describe the thermal regime and vary as the error function of the logarithm of the inverse of the strain rate. At high strain-rates, the phonon drag regime for dislocation motion is described using the theory of overdriven shocks where the saturation stress and yield stress are set equal and are related to a power of the strain-rate divided by the atomic vibration frequency. The stress in the transition region between the low and high strain-rate regimes is the maximum of the low and high strain-rate regime stresses. A single model can have different parameter sets to describe a given material. In this paper, we use four versions of the PTW model, each differing in their model parameters: the original values,⁴⁶ and those proposed by Blumenthal et al.,⁴⁷ are shown for Be in Table II. The relaxation model of beryllium strength^48,49 (RING) model includes relaxation terms, a term to account for twinning and terms to account for recovery at elevated temperatures. Finally, the mechanical threshold stress⁵⁰ model, valid at strain-rates up to the phonon drag limit, includes thermal activation effects. The stress is a linear combination of different stresses caused by dislocation barriers. These are scaled via factors representing the structure functions for the various dislocation barriers. The scaling factors are highly non-linear functions of temperature and strain-rate. These models have been calibrated to data from low pressures and low strain and strain-rate experiments. Predicting the RT growth in higher pressure and strain-rate regimes is therefore particularly challenging.

By tuning the velocity of the iron impactor detonating the second stage HE, Figure 2, we ensure that the simulations have the same drive conditions as the experiments. Comparisons between the velocimetry data of the Be targets measured by VISAR and a simulation using the PTW model are presented in Figure 11. Simulations with different strength models do not present significantly different predictions of the free surface velocity because the drive conditions are essentially independent of the strength models. The simulations are sufficient to allow for an interpretation of the growth data with an adequate acceleration profile. Small discrepancies between the experimental drives and simulated drives have little impact on our results. We performed sensitivity analysis of the drives by increasing and decreasing the simulated drives by one root mean square deviation (as measured between the simulated and experimental drives, illustrated with representative error bars in Figure 11). The impact of these changes on the growth factors is within the experimental error bars.

Simulations of the experiments indicate that the Be targets reached ∼50 GPa and strain-rates of ∼10⁶s⁻¹, the phase-space where the model predictions of the perturbation growth differ. The simulations that best match the experimental data show a peak yield stress in Be of 2.2 GPa, which is comparable to the values of 1.8 GPa reported by Chhabildas et al.¹¹ and 16.5 reported by Brown et al.²¹ Though there is some agreement between these experiments, the comparison is not perfect because the strain rates and loading paths are different between an RT experiment and these shock-release and ramp-loading experiments. The rise time of the first stress wave is greater than 15 ns and, therefore, is not a shock. The simulations corroborate that this wave is a ramp and that the temperature is lower than that generated by a shock. According to the simulations, the temperature in the Be in these experiments is inferred to be approximately 700 K. In the SHPB experiments of Blumenthal et al., there is no evidence of Be failure even at strains as high as 1.0. Therefore, we have no evidence that the Be failure is occurring in the RT experiments, for which the peak strains do not exceed 1.0 (with the possible exception of very late in time). Simulations of the RT experiments indicate that the sample is under compressive strain during the entire process under which we take data. While a release wave propagates from the rear free surface of the target, the stress is continuously increased as the compressive ramp wave from the HE moves through the target. As a consequence, the Be target never experiences tension in the RT bubble. Additionally, the stress vs. strain curves from the SHPB experiments are smooth and do not show evidence of failure under compression (although microcracks develop at high strains (>0.2), these will not affect the results unless the sample is under tension).We present pseudo-color plots of pressure and strain-rate in Figure 13. Initially, a compression wave induced by the iron impactor travels through the HE, causing detonation, see Figure 13(a). The HE then expands through the vacuum gap and loads against the Be target, see Figure 13(b). The ripples at the HE-Be interface grow as the target is accelerated by expanding HE gas, see Figures 13(d) and 13(e).

Figure 10 also presents a comparison between the predicted growth factors of the perturbations using different strength models and the experimental data points as a function of target displacement. The data suggest that the Be ripples grew close to classically and are consistent with either low strength in the Be, or significant, but unobserved, damage. “Classically” usually refers to a constant acceleration at a non-viscous liquid/gas interface. In this context, we use it to describe growth in the presence of no strength or viscosity regardless of the dynamic loading profile. From the radiographs, we observe that the ripples have a significant mushroom shape, indicative of a classical RT growth in the non-linear regime. Consequently, most of the strength models under-predict the growth of the perturbations. The Mechanical Threshold Stress model predicts very little growth, which indicates that the work hardening is over-predicted for this region of the phase space. The MTS stress-strain curve, as shown in Figure 1, clearly illustrates this as it is steeper than the other stress-strain curves in this region of phase space. The RING model, as adopted in Ares, is the strength model with results closest to the experimental data. The experiments enable us to discriminate against certain models, such as MTS, which do not capture the data in this regime while the results indicate that the RING and SCG models are adequate in this regime. The small differences among the PTW models are not as significant as the differences between PTW and the other models in general. The data indicate that the models in general are inadequate for capturing the high strain, strain-rate, and pressure regime of the experiments. These models require better physics-based components to underwrite their validity. This is due in part to the fact that they are based on observations made in different parts of the phase space and they cannot be relied upon to predict material behavior away from their calibration points without accepting the inherent risk associated with extrapolation.

If the perturbation growth is due solely to plasticity, the RING model is the closest to the data but still misses the late-time behavior. However, an alternate theory is possible if damage occurs early in the experiment and manifests itself through the appearance of high growth. The data are insufficient to discriminate solely between plastic-driven growth and a combination of plasticity and damage. The recovery experiments (see Sec. IV) seem to indicate support for the idea of damage induced growth. Either way, the data indicate that existing strength models are insufficient in their current forms to capture properly the behavior of Be under extreme loading conditions.

Current plasticity models generally assume deformation mechanisms driven by dislocations. However, Be is known to incur substantial twinning and can experience brittle failure under room temperature conditions. For example, previous work¹⁶ on hot isostatically pressed S200F indicates that at strains up to ∼20% under uniaxial compression, the dominant deformation mechanism is basal slip. The contributions from twins peak at 10% strain but never become dominant. To understand the extent of twinning and failure that might be present in higher strain-rate regimes, such as our RT experiments, we also performed Be recovery experiments, where Be samples were loaded and then recovered for analysis. The goal of these experiments was to explore the effect of loading and shock strength on the Be microstructure and to determine Be dislocation dynamics. Unfortunately, it is not possible to recover the RT targets themselves since their thin nature causes them essentially to disintegrate at late times before they can be recovered. As such, thicker targets were used with different loading profiles from the RT experiments making an exact comparison of the deformation not possible; nonetheless, the path we chose is close enough to describe the behavior of Be under both uniaxial loading and at least similar drive conditions.

We performed two types of recovery experiments. In the first, a Be target disk of ø60 × 15 mm (diameter × thickness) was sandwiched between two layers of aluminum (ø120 × 0.5 mm for the front disk and ø120 × 5 mm for the back disk) and placed near a charge of HE. The Be sample experienced quasi-isentropic loading resulting from the detonation of the HE. The compression wave steepened to a shock as it traveled in the Be. This experiment was designed to keep the pressure in the material constant, but to have a varying strain-rate (from 10⁵ to 10¹⁰s⁻¹). In the second experiment, a Be sample of the same dimensions was also placed between two layers of aluminum, but this time an HE charge launched an aluminum impactor (ø120 × 2 mm) resulting in a shock wave that decreased in strength as it traveled in the Be. In this case, the strain-rate was constant (approximately 10⁵s⁻¹), but the pressure varied from 15 to 10 GPa. Manganin-based pressure sensors (MPS) were used to measure the pressure in the sample. For both experiments, to minimize edge rarefactions, the Be target is surrounded by an aluminum (Al) sleeve. The impedance of the Be and Al is well matched due to the high Al sound speed (the intensity reflection coefficient is 0.03) and a simple one-dimensional hydrodynamic analysis shows little effect from interface rarefaction. Given the reputation of brittle failure in Be and uncertainty in the dominant deformation mechanism, a priori expectations from the experiments ranged from mild deformation to complete disintegration.

A microsection of the recovered Be sample for the first type of experiment is shown in Figure 14. The pressure sensors indicated a peak pressure of 25 GPa in the sample. The microsection reveals a fine-grained structure with an average grain size of 14 ± 6.7 μm. A crack is clearly visible. The observed twinning fraction was slightly less than 50%. A recovered sample from the second type of experiment illustrates the partial destruction of the Be sample under this type of loading, see Figure 15. In this case, the peak pressure in the sample was around 14 GPa. From these two types of recovery experiments, it is clear that the samples fractured but did not completely disintegrate.

Using the MPS data, we calibrated the simulations to obtain the same experimental conditions, see Figure 16. The different loading paths and target strains are clearly visible when comparing Figures 17 and 18. In the first experiment, the simulation indicates that the pressure inside the target reached 26 GPa and remained constant throughout the material for about 1 μs before decreasing smoothly. The strain inside the target reached 0.1, remained constant, and then increased again. In the second experiment, the pressure at the leading edge of the Be target increased rapidly to approximately 15 GPa and decreased sharply thereafter to a near-constant value of 10 GPa. The amplitude of the shock wave decreased as it traveled in the Be. The strain varied from 0.15 to 0.11 depending on the depth in the Be target. The simulations also indicate different strain-rate behaviors in the two experiments, see Figure 19. In the first experiment, an initial spike in strain-rate is followed by a constant rate of ∼2 × 10⁴s⁻¹. In the second experiment, the strain-rate is around 10⁵s⁻¹ for about 0.25 μs and then decreases rapidly to approximately 2 × 10³s⁻¹.

We performed six HE-driven Be RT experiments to discriminate among different strength models. These experiments were designed to reach a phase space where the models' growth predictions differed. Relative to the predicted behavior, the data suggest that the Be ripples growth was only slightly mitigated by strength, indicating weaker than anticipated strength. The RING model does reasonably well predicting the growth for the larger initial amplitude experiments. The other models under-predict the perturbation growth. The experimental results challenge the underlying assumptions of the existing strength models. Once the material enters a strain, strain-rate, and pressure phase space far from the calibration regimes of the current models, its predicted behavior breaks down. In part, the models rely on a limited range of data, but also limited physical assumptions, mostly having to do with how strain and strain rate carry the plastic flow. For example, the results raise questions about the ansatz formulations, such as what are the proper rate hardening relationships in the thermal activation and phone drag regimes; where do the regimes even cross; are strain and strain rate the proper independent variables or should they be explicitly replaced with dislocation density and velocity? To complicate the challenge of developing a complete constitutive model for Be, the recovery experiments showing a twinning fraction of slightly less than 50% suggest that twining should not be overlooked as an important physical mechanism in the material flow. Furthermore, while the RT experiments show no observable spall or cracks at length scales that could be imaged, the recovery experiments do. Granted, the loading profiles between the two experiments differ, and the recovery experiments by their nature are done late in time, long after release waves have traversed the samples. However, the recovery experiments do suggest failure mechanisms should be included in any advanced Be plasticity model. As such, experiments might be done that are specifically designed to catch material failure under loading to determine if the behavior observed in these experiments is more a result of failure mechanisms, such as shear localization, or if indeed the plastic flow is truly a result of weaker constitutive properties than those predicted by most models.

If similar experiments are proposed for future work, we recommend adjusting the existing models to match the data set presented here and then driving the targets through different regions of stress-strain-rate phase space by adjusting drive or initial perturbations, or by tamping the target to maintain the Be at pressure for longer periods of time. Varying the initial perturbation wavelength would lead to a dispersion curve that could increase our understanding of Be strength in these extreme regimes while higher temperature experiments could also be a means to distinguish the models in future experiments.

The authors wish to thank Nathan Barton, Dana Goto, and Rob Rudd for their comments on early drafts of this paper. The authors would also like to express their gratitude to William Fritchie and Todd Stephens for help in navigating the export control maze so that these targets could be delivered to RFNC-VNIIEF. They also wish to thank Shou-Rong Chen for allowing us to share the updated Los Alamos PTW parameters for Be. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. Work at RFNC-VNIIEF was supported in part by Contract No. B590737 between LLNL and RFNC-VNIIEF. This research was supported in part by the DOE/NNSA under the predictive Science Academic Alliance Program by Grant No. DEFC52-08NA28616.