A Pop-plot gives the run-to-detonation distance as a function of input pressure for an explosive, neatly summarising its shock initiation behaviour [10]. The Pop-plot data for HMX [11] in figure4.4 illustrate the importance of microstructure on the shock-to- detonation transition. Wedge-shaped single crystals of HMX that are “free of significant voids” [182] give 3 mm run-to-detonation distances at ∼40 GPa pressures. In contrast, solvent-pressed HMX samples with complex microstructures give 3 mm run distances at 8 GPa. The solvent-pressed HMX is more sensitive (i.e. it requires a lower input pressure for the same run distance) because of the influence of hotspots on the reaction. The single crystals contain few hotspots, so they behave as a homogeneous explosive and need to be strongly shocked before reaction will build up into detonation. The solvent pressed material contains voids, crystal boundaries and other defects and so many hotspots are produced when it is shocked. It behaves as a heterogeneous explosive and can grow to detonation at modest input pressures.
Pop-plot data are often used to calibrate or validate reactive burn models, in particular their reaction rates. This is because hydrocode simulations of shock initiation are sensitive to the reaction rate and therefore to the unreacted equation of state on which the reaction rate depends, but are relatively insensitive to the equation of state of the reaction products. With validated unreacted equations of state, Pop-plot data can be used to test the reaction rates from section3.4 in the shock initiation regime. The reaction rates in a mesoscale model are intended to represent the defect-free behaviour of each of the components, since the effect of hotspots will be accounted for separately. Since they contain few hotspots, the Pop-plot data from HMX single crystals will be used to validate the HMX material model in this section. It is not possible to validate the binder reaction rates because Pop- plot data are not available.
The single crystal Pop-plot data in reference 11 were determined from magnesium flyer plate impact experiments [182]. No attempt has been made to reproduce the ex- perimental configuration, but Pop-plot points were extracted from simulations of a single HMX crystal hitting a rigid wall, to give a flat-topped shock initiation. The results from Peruse (which solves the reactive-flow equations 2.7 neglecting species diffusion) are shown in black in figure4.4. The HMX model does not appear to agree very well with the experimental data; the simulations require∼10 GPa lower input shock strength to give similar run to detonation distances, i.e. the model is more sensitive than the data.
Figure 4.4: Pop-plot data for single HMX crystals (blue) and solvent pressed HMX for comparison (red) from reference11. Peruse simulations for sustained shock initiation of a single HMX crystal (black) show that the HMX model is more sensitive than the data.
A first step towards improving the match of the HMX model to the Pop-plot data is to evaluate which of the material properties data in chapter 3 could influence the sen- sitivity. One candidate is the solid heat capacitycv,s, which determines the shock tem- perature at a given input pressure and therefore the reaction rate. To check whethercv,s can make a big enough difference to the sensitivity of HMX to account for the 10 GPa difference in the Pop-plot, additional Peruse simulations were run at the upper limit of
cv,s = 2.1 J/g K. While a 38.0 GPa shock causes prompt detonation with the nominal value ofcv,s =1.1 J/g K, little reaction occurs during the simulation forcv,s =2.1 J/g K. Even for sustained shock pressures as high as 43.2 GPa, reaction is still only building up by the time the shock wave reaches the end of the 3 mm crystal. For comparison, the experimental Pop-plot has a run distance of 0.80 mm at 43.5 GPa. Withcv,s =2.1 J/g K, the HMX model is less sensitive than the experimental data.
Therefore, the HMX model brackets the single crystal Pop-plot data with reasonable variations to the solid heat capacitycv,s. Although this does not prove thatcv,s is respon- sible for the discrepancy between the simulations and the experiments, it shows that it is a likely candidate. Other possibilities include a multi-step chemical reaction scheme with an endothermic first rate. The constant value ofcv,sused in this work could be adjusted to improve the fit to Pop plot data, but it is better to continue using the value derived from
experimental data to avoid masking other modelling deficiencies. Ways to improve the models will be suggested in section8.2. For this work, it is considered sufficient that the HMX model brackets the Pop-plot data whencv,sis varied within its error bounds. These bounds will be used in later chapters to establish the sensitivity to uncertainties in the material properties data, so the “true” response of HMX should lie within the range of the calculated results.
A better way to validate the reaction rates for HMX and the binder materials would be to use embedded gauge data. In recent years, embedded particle velocity gauge ex- periments [e.g.,183] have provided a wealth of data on shock initiation in plastic-bonded explosives. Owing to the spatial and temporal-resolution these data provide, chemical reaction rates can be more tightly constrained using embedded gauge data than Pop-plot data. Unfortunately, the reaction rates for HMX and the binder materials can not yet be validated in this way owing to a lack of available data.