Validation of a detonation product equation of state for an insensitive high explosive via slab geometry expansion tests Open Access

A detonation in a high explosive (HE) generates large pressures (up to a few 10 s of GPa) due to the rapid shock-driven combustion process^1–3 of the condensed phase HE. The energy accumulated by the products of detonation is then available to push on the surrounding HE confinement as the products expand volumetrically. Thus, accurately calibrated models of the detonation products equation of state (EOS) are essential for typical HE applications. Such EOSs are normally calibrated from data obtained in a cylindrical expansion (CYLEX) test.^4,5 In the CYLEX experiment, a cylinder of HE with a large length-to-diameter aspect ratio is encased in a thin-walled, annealed, copper (Cu) tube. Modern implementations of the CYLEX test use photon-Doppler-velocimetry (PDV) diagnostics to measure the velocity component of the outer wall of the expanding Cu tube in the direction of the PDV probe as the HE is detonating in a steady-state motion. These expansion data are then used to calibrate the parameters of a given form of detonation product EOS. By far, the most commonly calibrated detonation product EOS model is the Jones–Wilkins–Lee (JWL) EOS form, which has been calibrated for a large number of HEs, for instance, cyclotrimethylenetrinitramine (RDX), cyclotetramethylene-tetranitramine (HMX), triamino-2,4,6-trinitrobenzene (TATB), and ammonium nitrate (AN) based materials.^4,6–9

A variety of JWL EOS calibration methods have been developed previously based on data obtained in the CYLEX test configuration. The majority of these involve hydrodynamic simulations based on programmed burn (PB) calculations, where, for timing, the normal component of the detonation velocity is assumed constant, and where comparisons of PB simulations with experimental Cu wall expansion data are made at a discrete number of expansion volumes.^4,6,10 In recent work,^9,11–13 the present authors have developed an advanced methodology for calibrating the JWL equation of state of detonation products via CYLEX tests. It involves the use of a PB approach employing accurate detonation timing calculations via velocity-curvature detonation shock dynamics (DSD) modeling, combined with velocity-adjusted JWL EOS hydrodynamic simulations of the expanding HE detonation products driving the CYLEX Cu wall motion. The EOS parameters are iterated on over several hundreds of hydrodynamic simulations, until the complete experimental Cu wall expansion trajectory is reproduced to within some predefined metric.⁹ However, to date, there has been little attention paid to the validation of the JWL product EOSs derived from CYLEX-geometry tests. In particular, the ability to demonstrate the predictive capability of HE models calibrated in one geometry and then applied to other geometries is an essential component of HE detonation modeling.¹⁴ 

In order to examine this capability, we use the slab expansion (SLABEX) test geometry, consisting of a rectangular slab of HE confined by two rectangular metal plates. The SLABEX test is the two-dimensional planar analog of the CYLEX geometry. Previous studies on the SLABEX configuration include Gimenez et al.,¹⁵ who examined the viability of the SLABEX geometry for calibrating JWL product EOSs as an alternative to the CYLEX geometry. Tarver et al.¹⁶ used a SLABEX configuration with a small length-to-thickness ratio to evaluate the performance of a previous derived JWL product EOS for the HMX-based HE LX-14 and found poor agreement with the experimentally measured plate expansion velocity. Hill¹⁷ studied a modification of the SLABEX test design, which involved heavy clamping to eliminate the need for a bonding layer between the HE and metal plates, with the notion of accessing higher pressure regions of the product EOS.

In the following, we describe a series of three PBX 9502 SLABEX tests to provide such a validation of the CYLEX test-based calibration process for the JWL detonation product EOS developed in Refs. 9 and 11–13. Hydrodynamic simulations of the Cu plate expansion in the SLABEX geometry are subsequently conducted using the JWL product EOS for PBX 9502 recently calibrated in the CYLEX test geometry⁹ and then compared to the measured lateral component of the velocity of the expanding Cu plates in the SLABEX experiments.

The SLABEX test geometry is described in Fig. 1. Each of the three tests consists of a rectangular slab of PBX 9502, specifically virgin lot HOL88H891–008 (lot 008),⁹ with thickness T, length L, and width W. The PBX 9502 slab thicknesses tested are $T=8.04,$ $12.02$⁠, and $16.00$ mm (Table I), with $L=130$ mm and $W=150$ mm in each case (Fig. 1), giving $L/ T≈16,$ $11$⁠, and $8.$ The densities of each of the lot 008 PBX 9502 slabs are also given in Table I and closely match those used in the recent PBX 9502 CYLEX-geometry experiments.⁹ The slabs are confined by annealed Cu plates of thicknesses $0.82$ mm (for $T=8.04$ mm), $1.28$ mm (for $T=12.02$ mm), and $1.585$ mm (for $T=16.00$ mm). The silicone elastomer Sylgard is used to bond the Cu plates to the PBX 9502 slab assemblies, eliminating air gaps due to HE and Cu surface irregularities. The Cu plates extend past the PBX 9502 slabs by 50.8 mm in each direction (Fig. 1). Along the sides, Delrin plastic is used to fill the gaps between the plates, preventing detonation products from sweeping around the plates and interfering with the optical PDV diagnostics. A line wave generator is used to uniformly initiate a detonation in a Composition B booster of dimensions $T× T× W$ (Fig. 1), which subsequently initiates a detonation in the PBX 9502 slab. The $L/ T$ ratio in each case is sufficient to establish a steady propagating detonation wave in the direction of travel (Fig. 1), which, because of the magnitude of the width $W$⁠, remains two-dimensional (2D) planar near the central axis of the slab, as described in Chiquete et al.¹⁸ 

On one of the Cu plates, three PDV probes were placed at a distance of (2/3)L (86.7 mm) from the leading, booster-attached edge of the PBX 9502 slab (Fig. 1). One of the probes was offset from the central axis by 10 mm and the other by 20 mm, as shown, to confirm that the flow remains 2D near the central axis for the duration of the test. On the other Cu plate, a single probe was positioned at the equivalent location of the central axis probe in Fig. 1. The PDV probes record the component of motion (velocity vs time) in the lateral direction to the axis. Detonation timing wires were also placed along an axial line near the central axis of the outer surface of one of the Cu plates to record the arrival times of the detonation. The derived steady detonation speeds $D 0$ (using the fitting process detailed by Voelkel et al.⁹) for each test are given in Table I, along with the standard errors,⁹ and show an increase in $D 0$ with increasing T as expected. The timing wire data were also used to verify that the detonation had relaxed to steady-state propagation prior to reaching the axial PDV probe locations at (2/3)L. The use of the timing wires and PDV probes is very similar to that described by Voelkel et al.⁹ for the corresponding CYLEX tests on PBX 9502.

The lateral components of the outer surface velocities of the Cu plates are shown in Fig. 2 as a function of time relative to the time of wall motion start for each of the three SLABEX tests, along with an average of the data following the process described by Voelkel et al.⁹ For the $T=8.04$ and $12.02$ mm tests, one probe on each test returned a slightly shifted signal, likely due to issues with initial alignment,⁹ and the corresponding probe data were not used for the analysis. Figure 2 also shows the standard deviation of the velocity traces from the average for each test, showing that the deviation is within 10 m/s after the early stages of ringing in the Cu plates. Moreover, the overlap of the off-center traces (from the PDV probes labeled 2 and 4 in Fig. 1) with the central axis traces shows that three-dimensional flow effects from the Delrin confined sides of the SLABEX experiments had not penetrated to the central axis regions of 2D flow for the duration of the tests.

Figure 3(a) shows a comparison of the probe-averaged Cu plate motions for each test. Due to machining constraints at the time of the tests, the ratios of the slab or Cu plate thicknesses do not geometrically scale exactly between tests. In each case, the Cu plate thickness is approximately, but not exactly, a tenth of the HE slab thickness, while the slab thickness ratios $s= T/8.04$ mm are approximately 1.5 and 2 for the $T=12.02$ and $T=16.00$ mm tests. In order to account for the small variations in geometric scales in evaluating the relative motions of the Cu plates between the tests, in Fig. 3(b), we have plotted the kinetic energy of one of the Cu plates (⁠ $m C u u 2/2$⁠) for each test per unit mass of PBX 9502 (⁠ $m H E$⁠) against time scaled with the ratio of slab thicknesses $s.$ Here, $m C u= ρ C u T C u$ is the mass of the Cu plate per unit length and width, with $ρ C u$ being the density of the Cu plate (⁠ $=8.933$ g/cm $3$⁠) and $T C u$ being the Cu plate thickness. Also, $m H E= ρ 0 T/2$⁠. The close evolution of the trajectories shown in Fig. 3(b) indicates that the plate push is driven primarily by the detonation products;⁵ i.e., nearly complete combustion is being achieved in each of the SLABEX tests. We also note the marginally higher value of the scaled kinetic energy for the $T=12.02$ mm test over the $T=8.04$ and $16.00$ mm cases.

Figure 4 shows a comparison of the averaged experimental and simulated motion of the outer surface of the Cu plates for each SLABEX test geometry (⁠ $T=8.04,$ $T=12.02$⁠, and $T=16.00$ mm). Also shown are the relative velocity differences between the experiments and simulations. In each case, the simulations capture the experimental data well, with maximum velocity differences of $O(10−20)$ m/s after the early ringing stages in the Cu plate motion. The $T=12.02$ mm case is the most accurately captured one, with the simulated Cu plate expansion velocities marginally higher than the experimental data for the $T=8.04$ and $16.00$ mm tests at late times. Interestingly, the slightly more accurate prediction of the $T=12.02$ mm case also corresponds with the higher scaled kinetic energy found for the $T=12.02$ mm test over the $T=8.04$ and $16.00$ mm tests [Fig. 3(b)].

Of note is that the initial ringing behavior in the motion of the Cu plates in each SLABEX test (as observed in Fig. 4), resulting from compressible wave reflections between the inner and outer surfaces of the Cu plates, is moderately well captured by the SLABEX geometry simulations. This is especially interesting when considering that the ringing effects in the Cu tube in the corresponding CYLEX tests were not as well captured by the simulations despite the use of the CYLEX tests as the PBX 9502 JWL product EOS calibration geometry.⁹ Moreover, the SLABEX geometry PDV probe data (Fig. 2) do not show significant evidence of spall from the Cu plates (typically manifested as a plateau in the velocity traces at the end of a ringing stage⁹), as opposed to the equivalent traces in the CYLEX geometry.⁹ This could potentially indicate that spall plays a greater role in the CYLEX geometry than the SLABEX geometry. In practice, the degree of tension and strain rate induced in the Cu wall and the potential appearance of spall will depend on a number of factors, including HE type, detonation energy release characteristics, and the geometry and dimensions of the experiment.

Although small— $O(10−20)$ m/s—the SLABEX geometry Cu plate velocity differences between the simulation and experiment are marginally larger than the experimental standard deviations shown in Fig. 2, these being representative of the probe-to-probe velocity differences encountered experimentally. There are a number of factors that could contribute to the small velocity differences, particularly at late expansion times. For instance, in the CYLEX geometry, the strain rate in the Cu tube is the largest at late expansion times, and there are minimal calibration data for the copper flow stress in these regimes. Additionally, due to the small thicknesses (tens of $μ$m) of the Sylgard bonding layers, we are unable to include the presence of the bonding layer in our simulations and, instead, have replaced the thin bonding layers with HE, as noted above. Moreover, the same thickness of the bonding layer in the CYLEX and SLABEX geometries will generally correspond to different masses of Sylgard polymer per unit mass of HE and, thus, affect the Cu tube/plate push differently in the two geometries. A simple Gurney analysis²² gives an order estimate of 3–8 m/s difference in the limiting Cu wall velocity for the SLABEX tests between treating the Sylgard layer as HE or absorbing the Sylgard mass into the Cu layer. Last, the PBX 9502 densities for the SLABEX tests (Table I) are marginally lower than that in the CYLEX test for lot 008,⁹ and thus, it is possible that the push on the Cu plates by the HE will be marginally smaller in the SLABEX experiments than the simulations based on the calibration of the JWL EOS for the slightly higher density used in the CYLEX test. Again, a simple Gurney analysis indicates that the density change would affect the Cu wall velocity by 2–3 m/s. While each one of the above effects will contribute to the observed small velocity difference, they are hard to quantify precisely. Nonetheless, the results shown in Fig. 4 show that to within the uncertainties in our simulation methodologies, the detonation product EOS obtained from our recently developed advanced calibration process based on one geometry can reasonably predict the Cu wall expansion motion when applied to a separate geometry.

We have shown that the HE product JWL EOS for PBX 9502, populated by the hydrodynamic iteration technique described by Voelkel et al.⁹ in the CYLEX test geometry, can reasonably predict confiner motion in a 2D planar SLABEX geometry, given our modeling uncertainties. Future work on this topic includes experiments and simulations on Cu confined 2D arc geometry tests for PBX 9502,^23–25 as well as application and validation of the product EOS calibration process to other classes of HEs. Additionally, we are also examining the effect of the Cu EOS and strength modeling on the simulation results.

This work was supported by the U.S. Department of Energy through Los Alamos National Laboratory (Contract No. 89233218CNA000001).

The authors have no conflicts to disclose.

Eric K. Anderson: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Stephen J. Voelkel: Conceptualization (equal); Formal analysis (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Mark Short: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Carlos Chiquete: Conceptualization (equal); Formal analysis (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Scott I. Jackson: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

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