The science and engineering communities have significant interest in experimental platforms to evaluate and improve models for dynamic material deformation. While well-developed platforms exist, there are still gaps to fill for strain and strain rate conditions accessed during impact and other high-rate loading scenarios. To fill one such gap for strength measurements, a platform was recently developed that accesses high strain rate (105/s) and large strain (50%) conditions by measuring the transient closure of a cylindrical hole using in situ x-ray imaging. In the work reported here, further refinement of the platform is performed to reduce the potential effects of porosity and anelasticity on the measurement. This helps us to isolate the strength effects that are the focus of the experiment. The updated experimental configuration employs a two-layer flyer design and elongated target to reduce the magnitude of the tensile excursions associated with rarefaction wave interactions. This allows for a more direct assessment of strength models commonly used for dynamic simulations of metals. We apply the new technique to well-characterized tantalum material, allowing for a robust connection to other experimental techniques. Deformation localization can be a concern in large strain experiments, and to help inform future use of the experimental platform, we use simulations with a sub-zone treatment of shear banding to explore potential localization behavior. Overall, we develop and utilize an experimental configuration with improved isolation of strength effects that can be applied to an expanded range of materials.

While in defense-related application spaces there is significant interest in the large-strain and high-rate deformation response of materials, relatively few experimental techniques provide focused data under the extreme conditions of interest. In this work, we describe advances in an experimental platform to probe a material’s resistance to permanent deformation or strength. The experimental design is motivated by the desire to isolate strength-related aspects of the overall constitutive response, and computational methods are used both in the experimental design and in the interpretation of the results. Our experimental approach produces impact-driven deformation in a fully three-dimensional geometry, with imaging of the time-dependent closure of a cylindrical hole normal to the impact direction. The method here builds from recent work by Lind et al.1 and from the earlier work by Glazkov et al.2 The technique complements existing platforms, such as Kolsky bar,3 Taylor cylinder,4 traditional plate impact,5,6 ramp compression,7,8 Rayleigh–Taylor instability (RTI),9 and Richtmyer–Meshkov instability (RMI)10 techniques.

Simulations of the hole closure experimental platform employed here indicate that the material undergoes strains greater than 50% near the surface of the hole with strain rates approaching 106/s. Thus, the technique fills a gap left by other experimental methods, as shown in Fig. 1. For comparison, Fig. 1 shows results for simulations of RMI experiments,10 RTI experiments,9 and Taylor cylinder experiments.4 As in the other experiments shown in the figure,11 strength inferences from the hole closure experiments are based on measurement of the change in the shape of the material, with a stronger material resulting in a smaller or a slower change in the shape.

FIG. 1.

Histograms of pressure and equivalent plastic strain vs strain rate from various experimental platforms that assess the dynamic response of materials: Taylor cylinder in red, hole closure in green, Richtmyer–Meshkov instability (RMI) in blue, and laser-driven Rayleigh–Taylor instability (RTI) at the Omega facility in black. Histogram intensity contributions are weighted by the rate of plastic dissipation in a simulation of the given experimental type.

FIG. 1.

Histograms of pressure and equivalent plastic strain vs strain rate from various experimental platforms that assess the dynamic response of materials: Taylor cylinder in red, hole closure in green, Richtmyer–Meshkov instability (RMI) in blue, and laser-driven Rayleigh–Taylor instability (RTI) at the Omega facility in black. Histogram intensity contributions are weighted by the rate of plastic dissipation in a simulation of the given experimental type.

Close modal

While the three-dimensional nature of the loading condition allows access to large deformations, it also introduces a complex set of wave dynamics. Release waves from the hole and other surfaces can overlap to produce tensile states of stress and associated porosity growth or spall.12 In previous work,1 a relatively simple single-layer flyer was utilized, but that work was performed on copper, which is relatively resistant to porosity growth and spall. In this work, we are looking to enable the application of the technique to a wider class of materials, and, thus, we redesign the experiment to mitigate the potential effects of porosity growth. The wave dynamics can also produce significant reversals in the loading condition and, thus, induce sensitivity to anelastic effects that favor deformation in the reverse direction.13 Such effects can influence the velocity traces in the release portion of both shock and ramp compression experiments and can complicate the determination of strength.14 As part of more general studies of constitutive response, it is possible to calibrate a model that includes anelastic effects,15,16 but the need to include anelastic effects can increase uncertainty for material strength inferences. Therefore, an important aspect of this work is the use of finite element simulations in the design of experimental conditions that mitigate the influence of these other physical effects. The specific material studied here is high purity and well-pedigreed tantalum.11,17 As shown below, simulations of tantalum experiments using a single-layer flyer suggest that experimental results may be sensitive to porosity and anelasticity. The computational redesign to mitigate these effects led to a two-layer flyer and an elongated target. The flyer consists of a lower-impedance aluminum layer backed by a higher-impedance copper layer, with the new design changing the timing and structure of the release waves passing from the flyer into the target. There is a mild increase in the complexity of fielding the experiments, but a significant benefit in the additional control over the nature of the dynamic loading and in the isolation of the strength effects that are of interest. The utility of the new design is assessed in a series of experiments. First, we describe these experimental results and then we discuss the simulation results. The experimental results are compared to results for four different tantalum strength models. In these comparisons, we utilize both as-published parameter sets from the literature and model calibrations that are specific to the material employed here.17 Given that the experimental approach produces large deformations in the vicinity of the closing hole, strain localization can also be a concern. The work concludes with an examination of strain localization as a possible source of discrepancy between simulations and experimental measurements. Results do not indicate that localization is a significant concern for tantalum but should be considered for larger amounts of closure2 and materials prone to unstable deformation.

Tantalum samples are taken from a clock-rolled plate stock produced by HC Starck Inc. (now Taniobis) with the rolling direction (RD) taken as coinciding with the final roll.17 To complete the orthogonal system, the reference directions are defined as the normal direction (ND) being normal to the plate surface and the transverse direction (TD) is orthogonal to RD and ND. For the hole closure experiments, the cylindrical hole’s axis is aligned parallel with the plate TD and impact occurs along the plate RD. Given that there is significantly more variation in grain size and texture near the plate perimeter due to rolling, the samples are taken from the center region to reduce the effects of texture-induced anisotropy, although the grains do show the preferential alignment of [111] crystallographic directions with the TD direction.18 The texture of this material is comparatively mild and the material shows texture uniformity that is significantly improved over the long-range variations (banding) often seen in tantalum.17 In our simulations, we, thus, neglect texture and anisotropic effects and focus instead on variations within the form of the strength model. The grain size of this material is of the order of 35 μm.

Samples are removed from the plate via electrical discharge machining into right cubic prisms measuring 8×8×8mm3. A through-hole of either 0.6 or 0.8 mm diameter is drilled with its axis parallel to the normal of one face and with the axis 2 and 4 mm from the corner of the impact surface. Figure 2 shows an idealized rendering of the cube sample with the location of the hole relative to the impact face. Table I provides details of the hole sizes in the samples. For physical samples and flyer geometries, tolerances of better than 0.0005 in. (12.7 μm) are prescribed. To ensure that sample shape, hole size, and hole parallelism and position were within acceptable tolerances, various forms of metrology are performed on each sample.

FIG. 2.

(a) Computational rendering of the two-layer flyer and target configuration used in this study. Note the two-layer flyer with a aluminum impact layer backed by copper. For clarity, the sabot and target plate are not shown. (b) is the mesh in the region surrounding the hole. Mesh configuration is optimized to reduce mesh distortion at the hole surface to eliminate mesh-related artifacts.

FIG. 2.

(a) Computational rendering of the two-layer flyer and target configuration used in this study. Note the two-layer flyer with a aluminum impact layer backed by copper. For clarity, the sabot and target plate are not shown. (b) is the mesh in the region surrounding the hole. Mesh configuration is optimized to reduce mesh distortion at the hole surface to eliminate mesh-related artifacts.

Close modal
TABLE I.

List of hole closure experiments performed on tantalum.

Shot #Impact speedAl thicknessCu thicknessHole diameterFinal relative area
(m/s)(mm)(mm)(mm)(:)
317 2.00 2.00 0.60 0.34 
385 2.00 2.00 0.80 0.29 
386 2.00 2.00 0.60 0.08 
390 3.00 2.00 0.60 0.20 
400 2.00 2.00 0.60 0.05 
Shot #Impact speedAl thicknessCu thicknessHole diameterFinal relative area
(m/s)(mm)(mm)(mm)(:)
317 2.00 2.00 0.60 0.34 
385 2.00 2.00 0.80 0.29 
386 2.00 2.00 0.60 0.08 
390 3.00 2.00 0.60 0.20 
400 2.00 2.00 0.60 0.05 

Hole closure experiments are performed at the Dynamic Compression Sector (DCS) at the Advanced Photon Source (APS) at Argonne National Laboratory on tantalum samples from Sec. II A to probe their high strain rate and large plastic strain strength response. The hole closure experiments are essentially gas gun driver flyer plate-impact experiments with a multi-frame in situ imaging diagnostic. Upon impact, a mechanical stress wave of a few GPa amplitudes is driven into the target. When the stress wave reaches the small hole in the target, the hole is driven to close. The experiments are performed on a 12.7 mm diameter single stage light gas gun, and the bore diameter places some constraints on the experimental design. While there is some design latitude in target geometries, hole sizes, and flyer thicknesses; there are restrictions on the effective impact stresses, strain rates, and total plastic strain achievable in a test.

The flyers consist of a two-layer configuration with a 10.16 mm diameter 6061 aluminum alloy disk of various thicknesses are backed by a 10.16 mm diameter oxygen-free high thermal conductivity (OFHC) copper disk to provide different loading pulses. Idealized target geometry and two-layer flyer configuration are shown in Fig. 2(a) as well as the mesh in the region surrounding the hole in Fig. 2(b). The main objective is to measure the transient change in the area of a hole that is perpendicular to the impact direction using x-ray imaging techniques. The final hole size and timing to reach the final hole size relate to the total plastic strain and strain rate of the test. Hole closure tests are integrated tests in that they probe a range of strain rates and total plastic strain as demonstrated in Fig. 1. One of the main goals of these tests is to drive partial hole collapse, which allows investigation of the viscoplastic response evaluation of the predictive capabilities of various strength models. The primary hole size was 0.6 mm with an additional shot being performed with an 0.8 mm hole. For additional experimental details related to imaging and target alignment, see Lind et al.1 

Flyer velocities ranged from 300 to 400 m/s. All flyers had a 2 mm copper backer and the aluminum thickness was varied to control the amount of closure exhibited in the target. The aluminum layer thickness was varied from 2 to 3 mm with the thicker aluminum layer causing less closure due to lateral surface release waves interacting and limiting the pressure pulse duration. This is in contrast to traditional plate-impact experiments in which the lateral surfaces are far enough away from the midline to neglect lateral release during the measurement time interval. In the hole closure platform, the flyer width-to-thickness ratio plays a role in controlling the amount of closure, providing access to higher flyer velocities. The flyer layers are bonded using commercial AngstromBond® AB9110LV epoxy. They are further backed by a third layer of polycarbonate that is 10.16 mm in diameter and 6 mm in length. This low-density polycarbonate cylinder allows the flyer to sit far ahead of the brass sabot, removing any effects from the sabot on the loading of the sample during the time interval of interest. Design calculations including the full flyer and sabot geometry were performed and compared to the simplified case without the polycarbonate backing or sabot (as shown in Fig. 2) with no difference in the predicted hole closure response. This comparison confirms that the primary driver of hole collapse is due to the dual metal flyer geometry and not the full geometry of the projectile. Table I defines the specific shot details related to flyer velocity, layer thicknesses, initial hole diameter, and the relative area in the final frame. For definition, the relative area is the primary experimental observable that simulations are compared with, and it is a measure of the fractional area of the open hole at a given instant in time relative to its initial starting area. Given the hole area is expected to remain constant after the initial closure which happens over the order of a few μs, the final relative area is the late-time final measurement from the 8-frame imaging setup. Shots 1, 3, and 5 provide comparisons for the effects of flyer velocity, shots 3 and 4 compare aluminum layer thickness, and shots 2 and 3 compare hole size effects. Impact speed is measured by optical beam blockage by the projectile in the gun barrel at three different locations just prior to impact.

Figure 3 shows the x-ray images for shots 3 and 5 from Table I. The images, as discussed in Ref. 1, correspond to the narrowest aperture along the hole, which occurs at the midplane. Any minor misalignment of the imaging system is mitigated by the formation of the narrower aperture at the midplane. The first image correlates to the static images prior to impact and the slight ellipticity is due to the slight misalignment with respect to the x-ray beam. As the stress wave propagates around the hole, the closure is relatively symmetric about the horizontal midline. Note the surface roughening as the deformation progresses at later times. Initially, the hole surface is relatively smooth with minor local perturbations to the circular nature of the hole. As the plastic collapse becomes significant the local grain and anisotropic effects become increasingly pronounced, leading to vertically asymmetric features such as the flattening along the top left edge of the hole in Fig. 3(c) and the local penetration in Fig. 3(d) along the bottom of the hole. These features suggest the possibility that local instabilities are driving localization. They motivate a computation investigation into the potential source of discrepancy between the models, which are fit to uniaxial stress data, and the hole closure results. Electron backscatter diffraction (EBSD) examinations provide further evidence of the possibility of material localization. These features motivate post-shot microstructure characterization, described immediately below, and the computational investigation into potential effects of strain localization on the amount of hole closure, described in Sec. IV C.

FIG. 3.

Experimental images for transient hole closure in tantalum targets with (a) all frames for shot 3, (b) all frames for shot 5, (c) the final frame for shot 3, and (d) the final frame from shot 5. Each individual frame represents a field of view of 1×1mm2. These representative images highlight the potential localization present at large amounts of closure and associated with asymmetries in the hole closure.

FIG. 3.

Experimental images for transient hole closure in tantalum targets with (a) all frames for shot 3, (b) all frames for shot 5, (c) the final frame for shot 3, and (d) the final frame from shot 5. Each individual frame represents a field of view of 1×1mm2. These representative images highlight the potential localization present at large amounts of closure and associated with asymmetries in the hole closure.

Close modal

To further examine grain-scale deformation, electron backscatter diffraction (EBSD) was performed at the midplane on recovered samples from shots 2, 4, and 5 from Table I to identify features related to localization at various levels of hole closure. EBSD results for shots 3, 4, and 5 are shown in Figs. 4(a)4(c), respectively. The SEM contrast image in Fig. 4(d) gives an indication of roughness on the hole surface and its potential association with grain-scale deformation heterogeneity. For reference, the impact was performed along the horizontal direction in the images with the impact plane situated to the left of the images. Recovery of these samples occurred without provisions for soft capture, and the post-shot state includes deformation history during both plate-impact loading and any subsequent collisions inside the gun tank. Comparison between the final frame of x-ray imaging and the post-shot microscopy showed on average an additional 7% change in the relative area of the hole. Thus, late-time collisions with the gun tank may influence hole size to a modest degree. However, with the hole closure occurring predominantly as the result of the flyer impact, we feel that it remains informative to evaluate the grain-structure morphology near the hole in assessing the closure process. Consistent with the information in Table I, the recovered samples in Figs. 4(a)4(c) show progressively more hole closure. Qualitatively, they exhibit correspondingly more effects of localization in the grain-scale deformation response. Grains are increasingly elongated in the vicinity of the hole, in contrast with more equiaxed grains in the bulk material. Furthermore, the results exhibit increasing overall asymmetry in the deformation response of the material at larger amounts of hole closure. As shown in Sec. IV C below, simulations of hole closure that utilize a sub-scale treatment of shear localization also develop asymmetries in the deformation response.

FIG. 4.

Electron backscatter diffraction images for transient hole closure in tantalum targets with (a) shot 2, (b) shot 4, and (c) shot 5 highlight the grain-scale localization near the hole. (d) is the SEM contrast image for shot 4, which highlights the potential effects grain-scale deformation heterogeneity on hole surface roughening.

FIG. 4.

Electron backscatter diffraction images for transient hole closure in tantalum targets with (a) shot 2, (b) shot 4, and (c) shot 5 highlight the grain-scale localization near the hole. (d) is the SEM contrast image for shot 4, which highlights the potential effects grain-scale deformation heterogeneity on hole surface roughening.

Close modal

To assist in the redesign and assessment of the hole closure experiments platform, finite element simulations were performed using an arbitrary-Lagrangian–Eulerian finite element code, ALE3D, developed at Lawrence Livermore National Laboratory.19 Pre-shot simulations were performed to help design the updated experimental configuration, with simulations examining flyer material selection, flyer layer thicknesses, and shot velocities. These simulations utilized previously published parameter sets for the strength of tantalum.20 In Secs. IV AIV C, we discuss how this updated design helps mitigate porosity and reduce the effects on anelasticity, making it easier to use the experiments to evaluate strength models. Post-shot simulations of the experiments are then used to evaluate strength models with as-published parameter values. Simulations with these as-published parameter sets under-predicted the amount of hole closure and, thus, over-predict the strength in the hole closure experiments. Strength data for a specific lot of tantalum used in this study are available from quasi-static and Kolsky bar experiments, and strength models calibrated to our specific material produce simulation results that compare more favorably with the hole closure experiments. Given the need to account for the lateral release effects, full 3D simulations were performed using quarter symmetry as shown in Fig. 5, with symmetry boundary conditions applied on two planes with normals perpendicular to the impact direction. These symmetry planes are the faces visible in the figure. Simulations neglect the sabot given that scoping calculations indicated incorporation of it did not affect the predictions and added unnecessary computational cost. As noted above, flyer material, flyer layer thickness, and target thickness were varied to mitigate potential effects of porosity on the hole closure, and Fig. 5(a) shows the updated experimental configuration. Figure 5(b) shows a single-layer flyer configuration similar to that used in earlier work.1 To illustrate the mitigation of potential porosity development, we include below results of finite element simulations utilizing a porosity model for both configurations shown in Fig. 5. Additionally, simulations indicated that the hole closure was insensitive to slight variations in flyer strength, such as the use of annealed vs half-hard copper.

FIG. 5.

Computational model configurations for (a) the newly designed experimental configuration with a two-layer flyer employing an aluminum layer backed by copper and (b) the previous experimental configuration with a single-layer aluminum flyer and shorter target. Simulations we performed using a quarter symmetry representation to reduce computation times. Symmetry boundary conditions are applied on the planar surfaces visible in the figure.

FIG. 5.

Computational model configurations for (a) the newly designed experimental configuration with a two-layer flyer employing an aluminum layer backed by copper and (b) the previous experimental configuration with a single-layer aluminum flyer and shorter target. Simulations we performed using a quarter symmetry representation to reduce computation times. Symmetry boundary conditions are applied on the planar surfaces visible in the figure.

Close modal

The strength models chosen for assessment have previously been used in studies of tantalum strength at elevated strain rates. The four models are Preston–Tonks–Wallace (PTW),21 Mechanical Threshold Stress (MTS),22 Zerrili–Armstrong (ZA),23 and the Livermore Multi-Scale (LMS)24 models. All of the strength models are suited to use in a classical J2 plasticity context. We use parameters from the previous publications as well as updated parameterizations determined through optimization to experiments performed on the same lot of Starck tantalum.17 The PTW, MTS, and ZA models were calibrated using a least squares approach to a collection of quasi-static and Kolsky experiments that cover a range of strain rates, from 0.001/s to 3500/s, and temperatures from 293 to 848 K.11,25 The Tri-Lab PTW models were calibrated to the same quasi-static and Kolsky bar data but also includes RMI experiments detailed in Prime et al.,11 which reach 107/s. All models, except for the Tri-Lab PTW parameterization, were calibrated using the same protocol and hole closure simulations are compared to the experimental data for model assessment and validation. The following computational analysis will investigate three salient features of the experiment: (1) flyer and target design on mitigating porosity and anelasticity, (2) flyer thickness, and (3) hole diameter to better understand the updated experimental configuration. The relevant equations for the four constitutive models are provided in the  Appendix, and more complete descriptions can be found in the references provided. Sections III B and III C describe the salient aspects of scaling strength for plane-strain deformation as well as pertinent equations for the porosity and anelasticity model forms.

In the hole closure platform, the experimental measurements capture the smallest aperture of the hole, which occurs at the midpoint along the hole.1 Given that the large-deformation flow of the material in this region of the target is close to a plane-strain condition, the apparent strength of the material may be reduced as compared to that in uniaxial stress experiments, such as quasi-static and Kolsky experiments.1,26 Due to this reduction in apparent strength, each model’s flow strength, which was calibrated for uniaxial deformation, is scaled for use in the hole closure simulations. This scaling is determined by the ratio of the Taylor factors for uniaxial stress and plane-strain loading. For cubic symmetry crystals, this ratio is approximately 0.9. All appropriate parameters are scaled by this value so that the effective flow strength follows the form τ¯=M¯τ where τ¯ is the effective strength, M¯=0.9 is the Taylor factor ratio, and τ is the flow strength for the given models calibrated to uniaxial data.

Constitutive models that capture additional deformation mechanisms can have formulations that are significantly more complex. To facilitate discussion, we introduce salient details of the porosity and anelasticity models considered here but we omit details of the formulations and their implementations.

For simulations in which we model porosity, the formulation includes both the volume fraction ϕ of porosity and the Number density of Pores (NP). Within this NP model, porosity evolution is based on the Cocks–Ashby form27,28 for rate dependent porosity evolution. As described in Refs. 29–31, the rate sensitivity n enters directly into the porosity kinetics and the dominant term in the kinetics has the form

ϕ˙=α1sinh[α2(n12n+12)(pτϕ)][1(1ϕ)n(1ϕ)]ϵ˙p,
(1)

where τϕ is the flow strength evaluated at a strain rate that includes contributions from plasticity associated with porosity evolution, p is the pressure, and α1 and α2 are order-unity material parameters.

The NP model also includes provisions for the softening of the material due to the increase in porosity.30 The porosity model is used together with the MTS strength model. Parameters α1 and α2 and the nucleation behavior are calibrated to the free surface velocity data, specifically the pullback velocity, for plate-impact-driven spall experiments reported in Jones et al.6 that were conducted on the same lot of Starck tantalum as used in the hole closure experiments. The calibrated parameters are unitless and were determined to be α1=1.0 and α2=0.55. A more complete discussion of the porosity model along with the nucleation parameters can be found in Qamar et al.31 Consequent to loading transients and reversals, pinned dislocation loops can bow out, with the swept area contributing additional inelastic deformation that is not captured by traditional strength model implementations. These effects have been examined in both quasi-static and dynamic loading scenarios.13–15,32,33 This anelastic deformation mechanism can be modeled by the incorporation of a back stress, β, that follows the evolution equation,

β˙=(fβ)(σβ).
(2)

Anelastic strain rate takes a similar form,

ϵa˙=(fa)(σβ).
(3)

In these equations, the coefficients fa=12c2 and fbeta=8Gc1 are calibration parameters, with G being the shear modulus, that are related to dislocation content in the material.13,15 For the results shown below, specific parameter values are drawn from Ref. 14, where c1 is 100 GPaμs and c2 is 5.6 GPaμs.

In this section, we discuss both the computational redesign of the experimental configuration and the use of the experimental results to evaluate the aforementioned collection of strength model parameterizations. For the experimental redesign, we outline the salient simulation results that lead to the layered flyer design, material selection, and elongated target. The discussion focuses on how the design update mitigates porosity and anelastic effects and, thus, improves the isolation of the strength response of the material. In the design studies, simulation results are evaluated principally in terms of the time evolution of the predicted relative hole area, as the relative hole area relates directly to the experimental observations. For evaluation of the strength model parameterizations, we compare all models to a small subset of the experimental shots and then compare the best-performing model and parameter set combination to the complete set of experimental data.

A key aspect of the experimental design is the mitigation of potential porosity formation within the target. Examination of results from preliminary experiments (not discussed in detail here) using a single-layer flyer indicated that porosity may have had a significant influence on the experimental results. A computationally driven redesign leads to the two-layer flyer, composed of aluminum backed by copper, as well as the elongated target. Elongation of the target delays wave reflection from the rear of the target until after stresses driving the hole closure have been significantly relaxed by the action of other waves.

Efficacy of the experimental redesign as predicted by the simulations is shown using results from a pair of two-layer and single-layer simulations with the velocity in the single-layer simulation adjusted to produce roughly the same amount of hole closure as in the two-layer simulation. The two-layer condition corresponds to experimental shot 3, and the corresponding single-layer flyer velocity was 550 m/s with the aluminum layer being the same thickness for both configurations. While the increased velocity will produce a higher peak compressive pressure, the primary goal of hole closure was to provide an experimental platform that high strain rates and large strain conditions. This comparison highlights that the updated configuration accomplishes this goal while reducing the complicating effects of porosity occurring in the target away from the hole. The higher-impedance copper layer in the flyer sends an additional compression wave into the target [Fig. 6(a)], rather than allowing significant tensile waves to develop in the flyer and target early in time [Fig. 6(c)]. The aluminum layer thickness is adjusted so that the compression wave mitigates release wave effects [Fig. 6(b)] that would otherwise produce significant tensile stress excursions [Fig. 6(d)]. In the following example, the updated configuration with the two-layer flyer and elongated target produces a reduced shock pressure of 4 GPa as compared to the single-layer configuration at 6 GPa. The duration of the pulse is longer at 1μs for the two-layer configuration vs 0.75μs for the single-layer configuration. This allows for similar amounts of closure leading to similar strain accumulation in the vicinity of the hole at similar strain rates. Of particular importance in regard to mitigating porosity is the peak tensile pressure in the target. For the two-layer configuration, it is 1 GPa, whereas in the single-layer configuration, it reaches 3.6 GPa. The reduction in tensile excursion corresponds directly to a significant reduction in predicted porosity highlighting the utility of the two-layer flyer and elongated target vs the previous design.

FIG. 6.

Pressure fields for both the two-layer [(a) and (b)] and single-layer [(b) and (d)] configurations. Figures (a) and (c) compare the pressure field at 0.8 μs and show the difference in pressure states particularly in the flyer. Figures (b) and (d) show the pressures as the hole reaches the maximum amount of closure, which for (b) the two-layer is at 1.8 μs and for (d) the single-layer is at 1.4 μs. The plots show pressure in gigapascals, with the maximum pressures off-scale given that the plot bounds are chosen to emphasize variations around zero pressure. Note all figures use the same plot bounds provided in (a).

FIG. 6.

Pressure fields for both the two-layer [(a) and (b)] and single-layer [(b) and (d)] configurations. Figures (a) and (c) compare the pressure field at 0.8 μs and show the difference in pressure states particularly in the flyer. Figures (b) and (d) show the pressures as the hole reaches the maximum amount of closure, which for (b) the two-layer is at 1.8 μs and for (d) the single-layer is at 1.4 μs. The plots show pressure in gigapascals, with the maximum pressures off-scale given that the plot bounds are chosen to emphasize variations around zero pressure. Note all figures use the same plot bounds provided in (a).

Close modal

While metallographic and EBSD characterization of the recovered targets focused around the hole, it is worth noting that we did not observe significant porosity in the targets fired in the redesigned experimental configuration. We note that the DCS facility that provides imaging capabilities does not allow for soft recovery of the target of the sort that would traditionally be used in examining spall in ductile metals.34Figure 7(a) shows the effects of porosity evolution and anelasticity over the duration of the experimental measurement and Fig. 7(b) shows the maximum porosity in the target. Notably, for the single-layer configuration, there is a significant difference in the relative area predictions when including porosity or anelasticity in the constitutive description. Conversely, for the two-layer flyer configuration, the simulations predict that porosity evolution and anelasticity effects are reduced to negligible levels. In the single-layer configuration, porosity evolution changes the release wave structure and, compared to the simulation without porosity, there is less reopening of the holder at around 2.0 μs after impact. In contrast, anelastic effects that favor deformation upon loading reversal increase the amount of hole reopening at around 2.0 μs. Simulations at lower impact velocities still exhibit a significant influence of porosity on hole closure but showed a significantly diminished effect from anelasticity.

FIG. 7.

(a) Comparisons of the simulation results for the single-layer flyer (sl) and two-layer flyer (tl) configurations, using a reference model and models with the addition of either porosity (pore) or anelastic (anel) effects. (b) The porosity evolution showing that for the two-layer configuration the maximum porosity in the target remains negligible at well under 1% for the duration of the simulation, while the single-layer flyer configures produces a much higher maximum porosity.

FIG. 7.

(a) Comparisons of the simulation results for the single-layer flyer (sl) and two-layer flyer (tl) configurations, using a reference model and models with the addition of either porosity (pore) or anelastic (anel) effects. (b) The porosity evolution showing that for the two-layer configuration the maximum porosity in the target remains negligible at well under 1% for the duration of the simulation, while the single-layer flyer configures produces a much higher maximum porosity.

Close modal

As shown in Fig. 7(b), the single-layer flyer configuration results in a peak porosity of approximately 30% at 2 μs. While Fig. 8 contains plots of the porosity distribution for both the two-layer [Fig. 8(a)] and single-layer [Fig. 8(b)] configurations. Conversely, as shown in Fig. 7(b), the maximum porosity in the two-layer flyer case remains less than 0.5% for the duration of the experiment, without developing any regions of strongly localized porosity evolution [Fig. 8(a)]. The release wave overlap structure in the single-layer configuration concentrates the porosity in two locations [Fig. 8(b)]. One region of elevated porosity falls between the impact surface and the hole, and the second region falls between the hole and the target’s rear surface. While the two-layer design is unlikely to completely eliminate porosity and anelasticity in the target, these results indicate that the updated configuration with the two-layer flyer and elongated target provide the potential to significantly reduce the effects that could confound the interpretation of the hole size measurements.

FIG. 8.

Comparisons of simulated porosity distribution using the number density porosity model for the (a) two-layer and (b) single-layer flyer configuration at 4 μs. The two-layer flyer shows a significant reduction in maximum porosity as well as a diffuse distribution throughout the target.

FIG. 8.

Comparisons of simulated porosity distribution using the number density porosity model for the (a) two-layer and (b) single-layer flyer configuration at 4 μs. The two-layer flyer shows a significant reduction in maximum porosity as well as a diffuse distribution throughout the target.

Close modal

As noted above, for all models, we use both an as-published (nominal) parameter set and one or two parameter sets calibrated to the specific lot of tantalum employed here. Figure 9 contains a comparison of uniaxial stress–strain curves for the various models at strain rates relevant to the hole closure platform. Figure 9(a) is the comparison of all models and associated parameter sets at a strain rate of 105/s, and Fig. 9(b) shows the various Starck-optimized parameterizations at a strain rate of 106/s. While uniaxial stress conditions are an obvious simplification relative to the dynamically evolving stress state near the hole, the plots in Fig. 9 allow for a direct comparison of each model’s flow stress behavior. The strengths predicted by the various models are consistent with experimental data from Casem et al.,35 which reaches strain rates over 105/s and up to 30% strain. Notably, these figures show the important competition of strain and strain rate hardening vs thermal softening when a material is deformed under adiabatic conditions to large strains.

FIG. 9.

(a) Comparison of the simulated stress–strain curves for all models and associate parameter sets under uniaxial adiabatic compression at 105/s. (b) Comparison of the Starck-optimized models at 106/s. Dashed lines indicate the nominal parameterizations, solid lines indicate the models optimized to the Starck quasi-static and Kolksy bar data, and the dotted line is the PTW Tri-Lab optimization.

FIG. 9.

(a) Comparison of the simulated stress–strain curves for all models and associate parameter sets under uniaxial adiabatic compression at 105/s. (b) Comparison of the Starck-optimized models at 106/s. Dashed lines indicate the nominal parameterizations, solid lines indicate the models optimized to the Starck quasi-static and Kolksy bar data, and the dotted line is the PTW Tri-Lab optimization.

Close modal

We note that a misinterpretation of the stress–strain data published in Ref. 20 reduces the quality of published parameterizations, but that it is still worth using those parameterizations given that they are in common use. Corrected data were subsequently published in Ref. 36.

Figure 10 compares simulation results of the previously published and Starck tantalum parameterizations against experimental data for two representative shots. The figure has results for two hole closure shots that used similar aluminum layer thicknesses, in this case 2 mm, to provide a direct comparison between flyer velocities of 317 and 386 m/s. As can be seen, the nominal calibrations under-predicted the amount of hole closure for all models. Given that the material strength resists the closure of the hole, under-prediction of the amount of hole closure is consistent with an over-prediction of the strength. The updated parameterizations improved the predictions for all models other than ZA (Zerrili–Armstrong), with ZA showing little difference after calibration to the uniaxial stress data. For the models calibrated to only quasi-static and Kolsky bar data, the comparison to hole closure experiments is an extrapolation to strain rates that are roughly two orders of magnitude higher. With the types of experimental data used in the PTW (Preston–Tonks–Wallace) Tri-Lab model calibrations, and particularly the use of RMI data (see Fig. 1), the PTW Tri-Lab calibration utilizes data spanning a range of strain rates including those probed in the hole closure shots reported here. However, the material in the vicinity of the closing hole experiences strains larger than those in the RMI shots, and we see that the PTW Tri-Lab calibration does not have the closest match to the experimental data despite having been calibrated across a wide range of strain rates. This result highlights the utility of hole closure experiments to assess material strength at high strain rates and large strains where there is significant competition between strain hardening and thermal softening due to the temperature rise in high-rate experiments. Ultimately, the hole closure platform is complementary to existing experimental platforms given the unique range of conditions accessed.

FIG. 10.

Model comparison to experimental results for two shots with 2 mm thick aluminum layers. Flyer velocities compared are (a) 317 m/s (shot 1) and (b) 386 m/s (shot 3). Nominal parameter sets are the dashed lines and solid lines are the updated parameterization specific to the Starck tantalum used for the target material in this study.

FIG. 10.

Model comparison to experimental results for two shots with 2 mm thick aluminum layers. Flyer velocities compared are (a) 317 m/s (shot 1) and (b) 386 m/s (shot 3). Nominal parameter sets are the dashed lines and solid lines are the updated parameterization specific to the Starck tantalum used for the target material in this study.

Close modal

Comparing Figs. 10 and 9, one sees that the ordering of the hole closure curves is not the same as that for the stress–strain curves at constant strain rate. This may be related to the complexity of the hole closure loading condition. We note that the standard form of the PTW model utilizes a hardening equation [Eq. (7) in Ref. 21] that builds in an assumption of a constant strain rate path. Thus, the variable strain rate in the hole closure experiments may be one factor in the different ordering between Figs. 10 and 9.

As shown in Fig. 10, the updated MTS (Mechanical Threshold Stress) model came the closest to predicting the hole closure data. Given the deficiency of the nominal calibrations, we focus the remainder of this discussion on the updated parameterizations. Error bars for the time axis are static throughout the experiment with the laser-to-impact distance being repeatable to 0.025 mm and the velocity being known within 1% reported value. The relative area is known within ±5 μm of areal equivalent radius, so the uncertainty varies frame to frame.

Figure 11 shows results for the updated MTS parameterization, compared against experimental data for all of the shots listed in Table I. Note the deviation from unity for the initial frames is due to a minor misalignment prior to impact. Given that the smallest aperture develops at the midplane of the hole this does not affect the measurements postimpact as shown in Lind et al.1 As flyer thickness increases the effects of lateral release waves becomes increasingly pronounced. This effect is highlighted when comparing shots 3 and 4, which have similar flyer velocities, but have aluminum layer thicknesses of 2 and 3 mm, respectively. The ability to control the amount of closure by varying both the velocity and aluminum layer thickness allows for another control parameter to drive various amounts of closure or to access an expanded range of velocities. Given gas guns have a lower bound on attainable exit velocities, this can allow one to better control the desired outcome of causing differing amounts of hole collapse between shots.

FIG. 11.

Complete comparison of all hole closure shots in Table I to the MTS constitutive model using the parameter sets that were optimized to the Starck tantalum quasi-static and Kolsky bar experiments.

FIG. 11.

Complete comparison of all hole closure shots in Table I to the MTS constitutive model using the parameter sets that were optimized to the Starck tantalum quasi-static and Kolsky bar experiments.

Close modal

A larger initial hole size can also allow for a wider range flyer velocities ultimately making experiments with gas-gun-based platforms easier to perform. This is due to the lower limit of possible velocities along with decreasing the effect of grain-scale features such as grain size and local orientation. Shots 2 and 3 have approximately the same velocity but the initial hole size is larger in shot 2. This pair of shots, thus, illustrates that increasing the hole size tends to reduce the relative amount of closure. Simulation discrepancies are largest for the shots with the most hole closure and, thus, with the largest amount of strain accumulation in the vicinity of the hole. Discrepancies could be related to the onset of localized deformation for holes that are more fully closed. Figure 12 is the simulated effective plastic strain and the shape can be compared to the final frame for shot 5 [Fig. 3(d)]. The simulation result shows a roughly elliptical hole with a slight strain field asymmetry between the front and back of the hole. It is notably lacking the intrusions at the top left and bottom of the hole found in the experimental image. The strain predicted for a hole that has almost fully closed reaches over 150%. While the strain accumulation is higher in front of the hole where the wave first arrives, strain accumulation at the back of the hole reaches upward of 100% during the relatively prolonged hole closure process.

FIG. 12.

Comparison of (a) final frame for shot 5 and (b) simulated effective plastic strain. The simulation predicts a smooth hole surface that is roughly elliptical in shape with a mild strain field asymmetry between the front and back of the hole.

FIG. 12.

Comparison of (a) final frame for shot 5 and (b) simulated effective plastic strain. The simulation predicts a smooth hole surface that is roughly elliptical in shape with a mild strain field asymmetry between the front and back of the hole.

Close modal

At a given grain size, the closure of a hole with a smaller initial size will involve fewer grains and the closure may be more susceptible to heterogeneities introduced by the anisotropic response of the grains, which is a potential source for the intrusion noted previously in Fig. 3(d). Grain-scale heterogeneity in the deformation is also known to produce surface roughness during deformation, such as orange peel effects in aluminum, and deformation-induced roughness along the length of the hole would tend to make the projected hole area smaller than the actual hole area at a given cross section. In Sec. IV C below, we describe further simulations that were used to investigate the potential influence of strain localization on hole closure.

Previous experiments on tantalum with explosively driven hole closure showed potential strain localization in vicinity of the hole.37 Motivated by both this previous observation and by the results in Sec. II C, we employ an enhancement to the model to evaluate potential effects of strain localization. Because it is computationally impractical to mesh-resolve shear bands in three-dimensional simulations of centimeter-scale samples, the study here is accomplished using a sub-zone treatment of shear banding. This sub-zone model is based on a mixture theory that satisfies the local traction balance between the band material and the surrounding bulk material, and the use of the model has been shown to facilitate the localization of strain into the relatively thin shear bands.38 The approach has some commonalities with the earlier work by Needleman and co-workers.39 In our approach, material that is prone to shear banding is inserted at the beginning of the simulation with band thickness on a length scale well below the nominal finite element size. The band thickness in the following simulations was assumed to be 300 nm which is consistent with the band width observed in Ref. 37. The bands are oriented in a spiral pattern akin to that seen in Ref. 37, and this pattern aligns the bands with planes of maximum shear stress that are experienced in the idealization of the hole closing in an axisymmetric plane-strain deformation pattern. Overall, the sub-zonal treatment used here38 allows for shear localization into bands of a given characteristic thickness.

Adiabatic shear bands are unstable deformations that occur when the rate of thermal softening is greater than the rate of strain hardening. EBSD imaging in Sec. II C shows significant grain elongation in the radial direction near the hole, but signs of localization are less pronounced here than in Ref. 37. To determine whether this localization is driven by similar competition between thermal softening and strain hardening, we simulate the hole closure experiments using the traction-balance-based mixture approach. Figure 13 shows that the sub-zone shear localization model has a modest effect on hole closure and, thus, does not explain the discrepancy between the MTS model prediction and the experimental results.

FIG. 13.

Relative hole area for shots 4 in Table I assessing the potential for strain localization in tantalum to describe the discrepancy between the common model predictions (MTS-opt) and experimental results. Including shear localization capability (MTS-sb) produces a minimal difference for shot 4, indicating shear banding does not have a significant effect on the amount of hole closure for tantalum.

FIG. 13.

Relative hole area for shots 4 in Table I assessing the potential for strain localization in tantalum to describe the discrepancy between the common model predictions (MTS-opt) and experimental results. Including shear localization capability (MTS-sb) produces a minimal difference for shot 4, indicating shear banding does not have a significant effect on the amount of hole closure for tantalum.

Close modal

Certain materials, such as 4340 Steel and Ti–6Al–4V, are known to readily shear bands, whereas tantalum does not traditionally exhibit this same propensity. To round out our understanding of the tendency for shear band formation, simulations were performed using a constitutive model for 4340 steel;40 and these simulations indicate that the hole closure platform can indeed have the potential to drive significant shear banding. This should be considered when designing experiments for materials with a propensity for unstable deformation. Figure 14(b) shows the temperature in the shear band material for 4340 steel, with the majority of the temperature rise associated with plastic work heating. Compared to the tantalum results in Fig. 14(a), the steel shows significant additional deformation in the bands, leading to the formation of sawtooth-like features on the surface of the hole. While this behavior may be of significant interest in some settings, it does complicate the interpretation of the experiments and the behavior would not be captured using simpler constitutive models at traditional mesh resolutions.

FIG. 14.

Comparison of temperature in the vicinity of the hole at large deformation for the (a) tantalum and (b) 4340 steel simulations. The temperature field shown is for the shear band material. Notice the asymmetrical distribution of shear localization around the hole in (a) that bears similarities to the experimental asymmetries observed in Fig. 4.

FIG. 14.

Comparison of temperature in the vicinity of the hole at large deformation for the (a) tantalum and (b) 4340 steel simulations. The temperature field shown is for the shear band material. Notice the asymmetrical distribution of shear localization around the hole in (a) that bears similarities to the experimental asymmetries observed in Fig. 4.

Close modal

In this study, tantalum strength is investigated using the recently developed hole closure platform.1 As in other methods for examining strength at high rates,11 design and interpretation of the experiments involves significant use of simulations. The results show that adjustments to aspects of the experimental configuration, such as impactor layer thickness and target hole diameter, allow for improved control over the hole closure. Changes in the experimental configuration also allowed for the utilization of higher flyer impact velocities in line with the velocities more commonly available at gas gun facilities. While x-ray imaging was employed in the technique development work shown here, in the future, similar capabilities could be developed with the use of more generally accessible non-x-ray high-speed imaging collimated visible light. Additionally, changes in the hole size relative to characteristic microstructural length scales allow for either a more homogenized probe of material strength with larger initial hole sizes or for a probe of heterogeneities and the influence of grain-scale anisotropy with smaller initial hole sizes.

A principal motivation for updating the design of the hole closure platform was to mitigate potential effects from porosity and anelasticity. Using a strength and porosity model calibrated to incipient spall from plate-impact testing, we were able to compare the updated and original experimental configurations. Thousands of simulations can be run at a nominal cost and provided guidance for the experimental configuration including material selection, layer thicknesses, hole size, flyer dimensions, and flyer velocity. These simulations probe the likelihood that each experimental configuration will produce porosity at a level significant enough to affect the observable of interest. A particular focus was put on limiting the tensile excursion experienced in the target throughout the duration of the measurement. This was achieved by designing a two-layer flyer with a lower-impedance aluminum impact layer backed by a higher-impedance copper layer. Additionally, we adjusted the impact layer thickness such that the lateral release waves overlap within the target and further reduce the tensile excursion. Through a computational investigation, the dual-layer flyer and elongated target show the potential to reduce the local maximum porosity by two orders of magnitude. With the new dual-layer flyer, simulation results predict that porosity will have a negligible effect on the experimentally observable transient hole closure. Additionally, modeling results found that the dual flyer also has the potential to reduce the sensitivity of the experimental results to anelastic effects associated with load reversals, especially in the late-time portion of the experiments. We assessed models in common use for the simulation of dynamic strength response by first fitting the models to a collection of quasi-static and Kolksy bar data that spanned a wide range of strain rate and temperature conditions. Results from the hole closure platform were then used to evaluate the predictive capabilities of the models for even higher-rate conditions. The MTS model proved to be the most consistent and effective model at predicting the shots performed on tantalum but still showed a small discrepancy when simulating the hole closure experiments. Future efforts could include model parameterization using Bayesian approaches,16,41,42 with modern computing resources making it possible to run the thousands of calculations needed to support such a workflow. Experimental results indicated there was potential localization occurring in the vicinity of the hole. This was observed both in distortion of the grain shapes and in breaking of the symmetry in the shape of the hole. The observed localization in the impact-based hole closure experiment does not appear as significant as what was observed by Nesterenko et al.37 Their research pointed to localization being driven by local thermal effects at large strain, and to evaluate this possibility, we performed additional modeling using a mixture approach wherein localization can occur at a subgrid scale. Sample calculations indicate that a small amount of localization could be leading to material softening and driving additional hole closure for these tantalum experiments. Other materials may be more susceptible to shear localization given the large strains achieved near the hole; and when testing other materials, it would be appropriate to consider their specifics, such as rate dependence, strain hardening, and thermal dependence. Additionally, experimental results in this study do indicate that grain-scale effects could influence the response near the hole. Given that spatial extent is limited by the bore size of available gas guns, fine-grained material may be attractive in promoting more uniform deformation in such hole closure experiments.

Overall, material strength under complex loading, high strain rate, and large strain conditions is not well understood, so new platforms such as RMI and hole closure continue to be developed to provide the ability to fill gaps in our understanding. For exploring high-rate hardening, this platform complements other observations, as shown in Fig. 1. In particular, the platform provides access to large-strain response due to the multi-dimensional nature of the loading condition, and the redesign to utilize a layered flyer improves the isolation of strength effects.

In general, there is interest in whether deformation at high strain rates produces an increase in apparent strain hardening of the material,22,24,43–46 with some work suggesting a marked increase in hardening with strain rate while other work does not. The tantalum models examined here consistently under-predicted the amount of hole closure, which means that they over-predicted the strength. This would indicate that while there might be a transition to higher hardening at high rates, we are not seeing significant evidence in the experiments here that access strain rates are in excess of 105/s. The previously published hole closure results on copper similarly did not show signs of a transition.1 

The authors would like to thank Alison Kubota for the assistance in setting up the anelastic simulations and valuable discussions about the results. This work was performed under the auspices of the U.S. Department of Energy (DOE) by Lawrence Livermore National Laboratory under Contract No. DEAC52-07NA27344 (LLNL-JRNL-836574-DRAFT) and supported in part by the Joint DoD/DOE Munitions Technology Development Program. This publication is based in part upon work performed at the Dynamic Compression Sector at the Advanced Photon Source supported by the U.S. Department of Energy (DOE), National Nuclear Security Administration, under Award No. DE-NA0002442 and operated by Washington State University (WSU). This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specic commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

The authors have no conflicts to disclose.

Matthew Nelms: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Jonathan Lind: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). Jonathan Margraf: Data curation (supporting); Methodology (equal); Software (equal); Writing – review & editing (supporting). Sayyad Basim Qamar: Data curation (supporting); Investigation (supporting); Writing – review & editing (supporting). Joshua Herrington: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). Andrew Robinson: Data curation (equal); Investigation (supporting); Writing – review & editing (supporting). Mukul Kumar: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Project administration (equal); Writing – review & editing (supporting). Nathan Barton: Conceptualization (equal); Formal analysis (supporting); Investigation (equal); Methodology (equal); Project administration (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The Zerrili–Armstrong (ZA) model is meant to account phenomenologically for the behavior of dislocations in a crystal lattice. It relies on the strong dependence of temperature and strain rate effects for body-centered cubic metals, which results in an isotropic model for metal plasticity [Eq. (A1)]. The specific form used in this study is a slightly modified version that couples the effects of strain hardening to the temperature and rate dependency. In Eq. (A1), c0c6 are model constants, T is the temperature, ϵ¯˙p is the effective plastic strain rate, ϵ¯˙0 is the reference strain rate, and ϵ¯p is the effective plastic strain,

τ=(c0+c1exp[(c2+c3ln(ϵ¯˙pϵ¯˙0))T]+c4exp[(c5+c6ln(ϵ¯˙pϵ¯˙0))T]ϵ¯pn).
(A1)

Table II contains a nominal parameter set taken from the work of Chen et al.20 as well as the parameters calibrated for the Stack tantalum.

TABLE II.

Zerrili–Armstrong model parameters—nominal calibration20 and calibration to the Starck quasi-static and Kolsky bar data.

ParameterNominalStarckUnits
c0 30.0 31.52 MPa 
c1 1.125 1.18 GPa 
c2 0.005 35 0.0052 K−1 
c3 0.000 327 0.000 33 K−1s−1 
c4 31.0 33.1 MPa 
c5 0.0 0.00012 K−1 
c6 0.0 −9.96 × 10−6 K−1s−1 
n 0.44 0.039 … 
ParameterNominalStarckUnits
c0 30.0 31.52 MPa 
c1 1.125 1.18 GPa 
c2 0.005 35 0.0052 K−1 
c3 0.000 327 0.000 33 K−1s−1 
c4 31.0 33.1 MPa 
c5 0.0 0.00012 K−1 
c6 0.0 −9.96 × 10−6 K−1s−1 
n 0.44 0.039 … 

The Preston–Tonks–Wallace (PTW) model is an extension of the Voce hardening law with physically motivated scalings meant to provide utility across a wide range of strain rate, pressure, and temperature conditions. The model includes terms meant to phenomenologically capture thermally activated dislocation glide. For simplicity, we have omitted the model features associated with more extreme strain rates where the model can transition to a form motivated by phonon drag. It is believed that phonon drag effects are not significant at the experimental conditions examined here.44,47 The flow stress takes the form,

τ=2τ^G(P,T),
(A2)

where G(P,T) is the pressure and temperature dependent shear modulus and τ^ is the non-dimensional shear strength,

τ^=τs^+1p(s0τy^)ln[1[1exp(pτs^τy^s0τy^)]exp(pθψ(s0τy^)exp(pτs^τy^s0τy^))],
(A3)

where the work hardening of the saturation and yield stress, τs^ and τy^ respectively, are rate and thermally dependent. Model constants s0 and s are the values τs^ takes at zero temperature and very high temperatures, respectively. Similarly, y0 and y are the yield strength, τy^, at zero and very high temperatures, respectively,

max{τs^=s0(s0s)erf[kT^ln(γξ˙/ϵ¯˙p)],min[y1(ϵ¯˙p/γξ˙)y2,s0d(ϵ¯˙p/γξ˙)β]}.
(A4)
max{τy^=y0(y0y)erf[kT^ln(γξ˙/ϵ¯˙p)],s0d(ϵ¯˙p/γξ˙)β}.
(A5)

In Eqs. (A4) and (A5), the strain rate normalization factor, ξ˙, takes the form,

ξ˙=12(4πρ3M)13(Gρ)12.
(A6)

Note that Eq. (A6) is a modification to the originally published model21 with ρ being the mass density and M is the molar mass. PTW model parameters are given in Table III, and the parameters that are not specified in the table are left at their nominal values from Ref. 21.

TABLE III.

PTW model parameters—nominal calibration21 and calibration to the Starck quasi-static and Kolsky bar data.

ParameterNominalStarckTri-LabUnits
θ 0.02 0.008 54 0.0132 … 
p 5.143 … 
s0 0.012 0.0119 0.007 702 … 
s 0.003 25 0.003 12 0.002 921 … 
κ 0.6 0.637 0.6212 … 
γ 4 × 10−5 4 × 10−5 5 × 10−7 … 
y0 0.01 0.009 95 0.006 520 … 
y 0.001 23 0.001 21 0.000 606 … 
y1 0.012 0.012 0.0127 … 
y2 0.4 0.4 0.4 … 
β 0.23 0.23 0.23 … 
ParameterNominalStarckTri-LabUnits
θ 0.02 0.008 54 0.0132 … 
p 5.143 … 
s0 0.012 0.0119 0.007 702 … 
s 0.003 25 0.003 12 0.002 921 … 
κ 0.6 0.637 0.6212 … 
γ 4 × 10−5 4 × 10−5 5 × 10−7 … 
y0 0.01 0.009 95 0.006 520 … 
y 0.001 23 0.001 21 0.000 606 … 
y1 0.012 0.012 0.0127 … 
y2 0.4 0.4 0.4 … 
β 0.23 0.23 0.23 … 

A multi-institution examination of tantalum strength across a wide range of conditions has produced a PTW calibration based on quasi-static, ramp-release, and RMI data. We make use of that model, designated “PTW Calibrated” in Ref. 11. The Starck material utilized in the hole closure shots here is from the same lot of tantalum as used in Ref. 11. The “PTW Calibrated” parameters are designated “Tri-Lab” Table III, and within the multi-institution working group, the associated model has been designated as “common model 3” or simply “CM3.”

The Tri-Lab model utilizes SESAME equation of state 93 524.48 The tabular equation of state includes a melt temperature Tm that is used with the Tri-Lab PTW model. For the “Starck” model, a constant melt temperature of 3287 as well as a Mie–Gruneisen equation of state. Below a pressure of roughly 100 GPa, the shear modulus in the Tri-Lab model is based on a zero-temperature shear modulus vs density Gρ(ρ) that is included with SESAME 93 524. That zero-temperature shear modulus is consistent with an analytic form and is combined with a thermal softening function so that the overall shear modulus is G(ρ,T)=(Gρ)(1αTTm). As in Ref. 49, the Tri-Lab model uses a constant α of 0.21 for thermal softening, rather than a density-dependent α as in Refs. 50 and 51. At higher pressures (that are well above the pressure range accessed in the hole closure experiments), the shear modulus in the Tri-Lab model increases more rapidly with pressure. See Sec. 3.2 of Ref. 11 and the discussion in Ref. 51 for motivation and caveats associated with this high-pressure modification. That modified Gρ can be captured with a ninth-order polynomial function of compression μ=ρρ01 with coefficients (0.716 461 4, 2.222 986 5, 3.005 476 9, 2.579 578 0, 0.658 319 0, 0.655 901 7, 2.498 559 9, 1.578 045 7, 2.267 470 1, 1.709 541 8) so that Gρ|ρ=ρ0=0.7164614. The reference density ρ0 is taken as 16.68 (the same as the reference density in SESAME 93 524). This polynomial fit is over a compression range from 0.75 to 1.2482.

The Mechanical Threshold Stress (MTS) model is based on Kocks–Mecking thermal activation theory where the work hardening and strain rate evolution are phenomenological representations of dislocation mediated plasticity.52 The model includes the additive decomposition of athermal, lattice, and work hardening effects with the notation, σa, σi, and σ^, respectively, and takes the form given in Eqs. (A7) and (A8), where kb is the Boltzmann constant, T is the temperature, G(P,T) is the pressure and temperature dependent shear modulus, and b is the magnitude of the Burgers vector,

τ=σa+f(ϵp¯˙,P,T,σ^),
(A7)
f=σ^(1(kbTg0dG(T)b3log(ψ0i˙ψ˙))1qd)1pd+σ^i(1(kbTg0iG(T)b3log(ψ0i˙ψ˙))1qi)1pi.
(A8)

The mechanical threshold stress evolves as a function of plastic strain, strain rate, and temperature with the generalized form,

dσ^dϵ¯p=[1F(σ^,σ^s)]ξΘ0(ϵ¯˙,T),
(A9)
F(σ^,σ^s)=tanh(ασ^σ^s)tanh(α).
(A10)

Given that many of the material constants are zero in the as-referenced model, the hardening factor Θ0(ϵ¯˙,T) collapses to a single constant c1 and ξ is taken to be unity. Table IV provides the parameters that were changed during the calibration to the Starck tantalum as well as the nominal values from Chen et al.20 

TABLE IV.

MTS model parameters—nominal calibration20 and calibration to the Starck quasi-static and Kolsky bar data.

ParameterNominalStarckUnits
σa 20 21.24 MPa 
σ^s0 650 441.99 MPa 
σ^i 1050.0 1021.61 MPa 
c1 3000.0 2183.19 MPa 
g0d 1.6 1.6 … 
g0i 0.133 0.133 … 
b 2.86 × 10−8 2.86 × 10−8 cm 
qd 1.0 1.0 … 
pd 0.667 0.667 … 
qi 1.5 1.5 … 
pi 0.5 0.5 … 
ParameterNominalStarckUnits
σa 20 21.24 MPa 
σ^s0 650 441.99 MPa 
σ^i 1050.0 1021.61 MPa 
c1 3000.0 2183.19 MPa 
g0d 1.6 1.6 … 
g0i 0.133 0.133 … 
b 2.86 × 10−8 2.86 × 10−8 cm 
qd 1.0 1.0 … 
pd 0.667 0.667 … 
qi 1.5 1.5 … 
pi 0.5 0.5 … 

The last model analyzed is Livermore Multiscale Strength (LMS), which uses a dislocation-density-based formulation and was originally motivated by and calibrated to sub-scale computational models.47 The form was subsequently modified to better capture strength response over the range of strain rates for which quasi-static and Kolsky bar data are available.24 The flow strength takes the general form,

τ=M[τ(v,p,θ)τp(p)+τ^(ρ,p)+τa].
(A11)

In Eq. (A11), M is the Taylor factor, τp is the pressure dependent Peierls stress, τ is a scaling factor related to thermal (τ~T) and drag (τ~D) effects, τ^ is the dislocation-based work hardening, and τa is the athermal resistance to dislocation motion. In Eq. (A12), τ^ describes the rate and temperature independent strengthening due to interacting dislocations with the classical Taylor form where G is a temperature and pressure dependent shear modulus, b is the Burgers vector magnitude, and β is a material parameter on the order of unity,

τ^(ρ,p)=βbGρ.
(A12)

The transition from thermally activated to drag-limited dislocation glide behavior is captured by the form τ=(τ~Tq+τ~Dq)1/q and the remaining aspects of the model are described in Ref. 24 and parameters can be found in Ref. 11.

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