Planar calculations with simplified geometry

In one-dimensional planar geometry, a calculation representative of the microstructure of EDC37 would contain alternating HMX and binder layers of various thicknesses. The results of such a calculation are rather complicated, so it was decided to start by investigat- ing simplified geometries. These are relatively easy to understand and allow confidence to be gained, before trust is placed in Peruse simulations of more realistic geometries. Since the explosives PBX9501 and EDC37 are composed mainly of HMX, with a small percentage by weight of binder, only geometries dominated by HMX are studied here. HMX and binder material properties were taken from chapter3.

Firstly, simulations were performed in which a single HMX crystal was fired into a rigid wall. This drives a shock into the HMX crystal in a computationally efficient manner, as there is no need to represent an impactor in the simulation. The geometry is illustrated in figure5.2. The impact velocity of 1.7 km/s was used for all the simulations in this section and corresponds to an input pressure of∼20 GPa. This is a high pressure for the shock to detonation regime, so detonation would be expected to occur promptly in the plastic-bonded explosive (although not in a single crystal as discussed in section4.2). The Pop-plots of PBX9501 and EDC37 extrapolate to run distances of∼1 mm at this pressure.

HMX

3 mm Rigid wall

1.7 km/s

Figure 5.2: Geometry of single HMX crystal calculation in Peruse with plane geome- try. The impact velocity of 1.7 km/s corresponds to ∼20 GPa and was used for all the simulations described in this section.

This pressure is used as an upper limit for the shock to detonation regime: if shock heating of crystals and binder is not a feasible hotspot mechanism at this pressure, then the non- linear behaviour of the Arrhenius reaction rate equations means that it certainly will not be feasible at lower pressures.

The computational geometry is 3 mm in length to provide an adequate computational domain extent to allow reactions to build up, with some leeway over the 1 mm run dis- tance expected from the Pop-plot. In order to prevent excessive computational run-times, 10µm meshing was used. An investigation into the sensitivity of the results to mesh den- sity is reported later. Virtual gauges are positioned every 0.15 mm through the geometry to record histories of temperature, pressure, species mass fractions, etc. Temperature his- tories from the single-crystal simulation are shown in figure5.3. The axes are chosen to be comparable with later plots. The temperature histories show that a shock is propagated through the HMX crystal but that very little reaction occurs behind it. This is confirmed by examination of the burn fraction histories (not shown).

The first gauge, coloured black in figure5.3, is positioned in the cell nearest the rigid wall. It is shocked to a lower temperature than the deeper cells because of wall heating errors in the simulations (see section2.2.1). Here, wall cooling occurs because the initial velocity applied in Peruse smears the shock over too many (rather than too few) compu- tational cells. Wall heating errors affect the first gauge in all the figures in this section, so the first temperature trace which is usually coloured black should be disregarded. This calculation shows that a single HMX crystal with these material properties (and with no defects or hotspots) will not react when shocked to 20 GPa. This is consistent with single crystal Pop-plot data [11] in which “reasonable” run-to-detonation distances (<10 mm) are only produced for shock pressures higher than∼35 GPa.

To add a level of complication, a thin layer of binder was incorporated next to the rigid wall. This geometry is illustrated in figure5.4. The binder thickness of 0.20 mm was chosen to be comparable with the largest HMX crystal sizes in these explosives and so

Figure 5.3: Temperature histories for a single HMX crystal calculation in Peruse. The different colours correspond to gauges located at 0.15 mm depths through the crystal and are labelled here to help with the description of later figures. The temperature at each gauge does not increase after the shock wave has passed, showing that no reaction occurs when HMX (without hotspots) is shocked to∼20 GPa.

2.8 mm 0.2 mm

HMX Binder

Figure 5.4: Geometry of binder/crystal calculation in Peruse.

that it could be adequately resolved with the 0.01 mm meshing used in these simulations. Temperature histories are shown in figure5.5. Note that the black gauge is in the binder next to the rigid wall, the red gauge (at 0.15 mm) is also in the binder, but the remaining gauges are in the HMX. The temperature histories once again show that very little reaction occurs during the simulation. Ignoring the black gauge, the traces demonstrate the effect of multiple shocks on the temperature achieved. To help explain this, a time-distance plot is given in figure 5.6. The frame is moving at 1.7 km/s towards the rigid wall, so that initially the HMX and binder regions appear stationary and the rigid wall (now acting as a piston) appears to have a forward velocity. This is purely for ease of understanding. The pressure histories from the PBX9501 simulation are shown in figure5.7. Both figures should be used together to understand the explanation below.

The green, blue and yellow gauges in the HMX are positioned close to the binder. They achieve significantly lower ultimate temperatures than the deeper gauges because they are double shocked to the final pressure of 20 GPa by a first shock at 15 GPa. This second shock is caused by a shock reverberation through the binder layer, as illustrated in figure5.6. In contrast, the pink and orange gauges, for example, are positioned deeper into the HMX, where the second shock has amalgamated with the first shock. These gauges are single shocked straight to 20 GPa. Since multiple shocks to the same pressure result in lower temperatures than single shocks, there is a large variation in the temperature reached at different gauge positions through the HMX. For instance, HMX temperatures between 770 and 1020 K are observed in PBX9501. Figure5.5 shows that the binder in EDC37 reaches a higher temperature than the binder in PBX9501 owing to the different equations of state of the two materials.

The next level of complication is a three-region geometry, illustrated in figure 5.8. Temperature histories from these simulations are shown in figure5.9. Note that the black gauge is in the HMX next to the rigid wall, the red gauge (at 0.15 mm) is also in the HMX, the green gauge (at 0.30 mm) is in the binder, and the remaining gauges are in the HMX. These simulations show that reaction proceeds in the binder region, although rather slowly, but not in the HMX regions. The behaviour of the temperature traces, with

Figure 5.5: Temperature histories for binder/HMX calculations for PBX9501 (above) and EDC37 (below), showing that very little reaction occurs in HMX or binder regions.

t _piston x binder HMX second shock first shock

Figure 5.6: Time-distance plot for the binder/HMX simulations to aid the interpretation of figure5.5.

HMX Binder

HMX

0.2 mm 2.6 mm

0.2 mm

Figure 5.8: Geometry of crystal/binder/crystal calculation in Peruse.

a minimum temperature achieved at the turquoise gauge, is rather complicated and is explained below. Figure5.10shows a time-distance plot for the HMX/binder/HMX sim- ulation; once again, the frame is moving at 1.7 km/s towards the rigid wall. Figure5.11 shows pressure histories from the simulation.

The impact of the piston drives a shock into the first HMX layer, giving rise to state 1 in the HMX at a pressure of 20 GPa. This shock is transmitted into the binder and a rarefaction propagates back into the HMX, giving rise to state 2 in the first HMX layer and in the binder at 15 GPa. This rarefaction is eventually reflected back from the piston, taking the first HMX layer down to 10 GPa and propagating on through the geometry as rarefaction A. Meanwhile, once it has traversed the binder layer, the original shock propagates onwards into the second HMX layer as shock B, taking the binder and HMX to state 3 at 20 GPa. A reflected shock reverberates through the binder layer and propagates forwards into the second HMX layer as shock C.

The first gauge in the second HMX layer is coloured blue in figure5.11. The pressure history shows that it is shocked up to state 3 at 20 GPa and then rarefaction A arrives, tak- ing the pressure down to 15 GPa. Since it has been shocked to a high pressure, and shocks are more effective at generating heat than rarefactions are at removing it, the temperature of the blue gauge remains close to 1000 K. Rarefaction A travels faster than shock B and catches up with it, eroding its strength as it propagates further into the HMX. By the time shock B has reached the location of the turquoise gauge, it has been entirely overtaken by rarefaction A and has decayed to a strength of 15 GPa. Meanwhile, second shock C has been propagating through the second HMX layer. The turquoise gauge experiences a double shock and achieves only a fairly low temperature of∼750 K. The second shock C travels faster than the first shock B and so catches up with it around the location of the orange gauge. A single shock to a given pressure achieves a higher temperature than a double shock to the same pressure, and so the orange gauge and those deeper into the HMX are shocked to a higher temperature∼1000 K.

This fairly simple simulation demonstrates the wide range of temperatures that can be achieved in a single HMX layer simply because of the shock wave history it has been

Figure 5.9: Temperature histories for HMX/binder/HMX calculations for PBX9501 (above) and EDC37 (below), showing that reaction occurs in the binder (green gauge) only.

t _piston 1 2 2 3 3 A C x HMX binder HMX B

Figure 5.10: Time-distance plot for the HMX/binder/HMX simulations, to aid the inter- pretation of figure5.9.

Binder HMX 0.2 mm HMX Binder etc. 0.2 mm 0.2 mm 3.0 mm 0.2 mm

Figure 5.12: Geometry of alternating crystal/binder calculation in plane geometry. exposed to. In a calculation with Arrhenius reaction rates, which have a strong temper- ature dependence, this phenomenon could lead to significant variations in reaction rate through a single crystal. With a more representative geometry, involving multiple HMX and binder layers, the computational results are difficult to understand in detail and trust must be placed in the overall conclusions of the simulation (e.g. whether detonation oc- curs). This demonstrates the power of the computational approach for such materials.

The final level of complication described in this section is an idealised geometry comprising alternating layers of HMX crystals and binder, as illustrated in figure5.12. Temperature histories from these simulations are shown in figure 5.13. The PBX9501 calculation shows that no significant reaction occurs, since the temperature histories do not rise above∼1200 K. Once again, there is considerable variation in the temperatures reached by different HMX and binder layers. For EDC37, some of the binder regions do react but only slowly.

Comparing the results of these simplified geometry calculations, none of the HMX regions react to any significant extent within the duration of the simulation, and build up to detonation has certainly not occurred. In contrast, at this pressure, the experimental Pop-plot predicts a run distance of 1 mm (equivalent to the gauge which is shocked at

∼0.2µs). Since the simulations do not produce a run-to-detonation distance consistent with the experimental data, these results indicate that shock heating of crystals and binder is not a feasible hotspot mechanism.