# Material Point Method Simulation of the Equation of State of Polymer-Bonded Explosive under Impact Loading at Mesoscale

^{1}

^{2}

^{*}

## Abstract

**:**

_{0}) and its pressure derivative (${{K}^{\prime}}_{0}$) were calculated. Additionally, the pseudo particle velocity and pseudo shock velocity variations were used to obtain the bulk wave speed c and dimensionless coefficient s for the Mie–Grüneisen EOS. The simulations provide an alternative approach for determining an EOS that is consistent with experimental observations.

## 1. Introduction

_{0}) and its pressure derivative (${{K}^{\prime}}_{0}$) for PBXs. The isotherms are transformed to a pseudo particle velocity (u

_{s}) and pseudo shock velocity (u

_{p}) plane using the Rankine–Hugoniot jump conditions [4].

## 2. Methodology and Simulation Model

#### 2.1. Material Point Method

#### 2.2. Interface Contact

#### 2.3. Simulations Performed

## 3. Construction of Simulation Systems

^{5}s

^{−1}.

^{3}, respectively, as shown in Figure 2, were studied. The theoretical maximum density (TMD) of HMX/Estane PBX is 1.864 g/cm

^{3}. The thickness of the Estane binder ranges from 0.026 to 3.9 μm and was selected to cover the outer layer of HMX grains. The mass ratio between the HMX and the binder is 95:5, and the impact velocity for different porosities is 250 m/s.

## 4. Results and Discussion

#### 4.1. Justification of Model

_{s}–u

_{p}variations of HMX converted from the static PV data using Equations (14) and (15) are plotted in Figure 4b and also compared with experimental results [40,42,43] and numerical simulation data [44,45,46]. The calculated u

_{s}–u

_{p}data also show a good agreement with experimental observations. Pseudo particle velocity ${u}_{p}$ and pseudo shock velocity ${u}_{s}$ are related by a linear relation [47], which is a Hugoniot EOS:

_{s}–u

_{p}linear fits taken from [49,52]. The Hugoniot data can be fitted as u

_{s}= 2.356u

_{p}+ 2.71 in this study, which is coincident with u

_{s}= 2.3u

_{p}+ 2.65 given by Gustavsen et al. [49]. Moreover, simulation data in this study are close to the results from [50] at low u

_{p}(0 < u

_{p}< 0.6 km/s). The aforementioned comparisons indicate that results of the MPM mesoscale simulation are reliable and that this model is applicable to develop the equation of state for investigating the pressure–volume variation behaviors of HMX/Estane PBX, as well as microstructure effects on EOS of PBX in mesoscale.

#### 4.2. PV Isotherms and Hugoniot Analysis of HMX/Estane PBX

^{3}. Each PV isotherm shows two trends: a linear variation at an early time (high V/V

_{0}), followed by rapid curved growth (low V/V

_{0}). The linear variation is associated with the rearrangement and deformation of particles squeezing out void space. At this stage, a small compression pressure results in a large volumetric change. After void space is removed, the PBX becomes highly compact, and the volume change slows down with the increase in compression pressure. It is seen from Figure 6a that at the same compression ratio the compression pressure decreases with the increase of porosity. A sample with smaller porosity has a higher packing density and therefore lower compressibility. It requires a large compression pressure for the same compression ratio, as shown in the experimental observation [3].

^{3}, respectively. The linear fitted parameters of the Hugoniot data are listed in Table 2. It is seen from Table 2 that $c$ decreases gradually with increasing porosity. However, no significant changes are observed for $s$ when porosity increases. Experimental work by Gustavsen et al. [49] showed a substantial difference of 0.18 km/s in the parameter c of the calculated Hugoniots for PBX 9501 at typical densities (range from 1.80 to 1.837 g/cm

^{3}) and at the theoretical maximum density (TMD = 1.86 g/cm

^{3}).

_{s}–u

_{p}curve, namely the dimensionless parameter s of the Mie–Grüneisen EOS, also decreases. On the other hand, the parameter c does not have a great decrease. Table 3 suggests that mixing a small content of the binder with HMX particles reduces the parameters c and s of the EOS obviously, and further addition of the binder has little effect on the parameters’ values. Experimental shock loading analyses of the PBX [51] reveal that for the PBX with the amount of the binder beyond a threshold value, the binder’s properties rather than the binder’s volume fraction exert a measurable influence on the shock sensitivity of the PBX. The authors in [53] also reported that the EOS and constitutive relation of the PBX mainly depend on the properties of the binder (such as density, elastic modulus, etc.), but this has limited dependence on the binder volume fraction. This is consistent with the simulation results.

#### 4.3. Equation of State (EOS) Study

_{s}–u

_{p}plane of the PBX can be expressed as:

_{s}–u

_{p}curve at the reference porosity; ${m}_{\phi}$ and ${n}_{\phi}$ refer to porosity effective factors. Here, ${s}_{0,b}$ and ${c}_{0,b}$ are the parameters fitting to the u

_{s}–u

_{p}curve at the reference binder volume fraction, and ${m}_{b}$ and ${n}_{b}$ refer to the porosity effective factors. The parameters obtained by fitting the data in Table 2 and Table 3 to Equations (26) and (27) are given in Table 7.

#### 4.4. Bulk Modulus and Its First-Order Pressure Derivatives

- In the limit of infinite pressure, V/V
_{0}→ 0. - With the increase in pressure, the isothermal bulk modulus increases continuously, and in the limit of infinite pressure, $K$ → ∞
- ${K}^{\prime}$ must decrease progressively with the increase in pressure, and ${K}^{\prime}$ remains greater than 5/3 in the limit of infinite pressure.

## 5. Conclusions

_{0}) and its pressure derivative (${K}_{0}^{\prime}$) for the PBX were determined by analyzing the obtained isotherms using the Birch–Murnaghan equation of state. The results demonstrate that the established effective EOS meets the criteria based on the basic thermodynamic discussion. The bulk wave speed c and dimensionless parameter s in the Mie–Grüneisen EOS of the PBX obtained by fitting u

_{s}-u

_{p}variations with the different porosities and different binder volume fractions are also effective.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**)–(

**d**) are specimens with an initial porosities of 0.113, 0.073, 0.033, and 0.003, respectively.

**Figure 6.**PV isotherms of HMX/Estane PBX at (

**a**) different porosities and (

**b**) different binder volume fractions.

**Figure 7.**Calculated Hugoniots for HMX/Estane PBX at (

**a**) different porosities and (

**b**) different binder volume fractions.

**Figure 8.**PV curves of the PBX after compression pressure normalization at (

**a**) different porosities and (

**b**) different binder volume fractions.

**Figure 9.**(

**a**) Variation of isothermal bulk modulus with high pressure and (

**b**) variation of pressure derivative of isothermal bulk modulus with high pressure.

Material | Parameter’s | Values |
---|---|---|

HMX | Shear modulus G (GPa) | 10.0 |

Density ${\mathsf{\rho}}_{0}$ (g/cm^{3}) | 1.90 | |

Yield stress ${\sigma}_{0}$ (GPa) | 0.10 | |

${\mathrm{C}}_{\mathrm{c}}$ (-) | 0 | |

${\mathrm{P}}_{\mathrm{c}}$ (-) | 0 | |

$\mathrm{c}$ (m/s) | 2740 | |

Grüneisen ${\mathsf{\gamma}}_{0}$ (-) | 1.10 | |

$\mathrm{s}$ (-) | 2.60 | |

Estane | Bulk modulus K (GPa) | 4.50 |

Shear modulus G (GPa) | 2.70 | |

Density ${\rho}_{0}$ (g/cm^{3}) | 1.186 | |

$c$ (m/s) | 2320 | |

Grüneisen ${\gamma}_{0}$ (-) | 1.00 | |

$\mathrm{s}$ (-) | 1.70 |

Porosity (φ) | s | c |
---|---|---|

0.003 | 2.356 | 2.711 |

0.033 | 2.198 | 2.656 |

0.073 | 2.276 | 2.459 |

0.113 | 2.392 | 2.047 |

Binder (b) | s | c |
---|---|---|

0% | 2.545 | 2.759 |

5% | 2.356 | 2.711 |

10% | 2.319 | 2.714 |

20% | 2.282 | 2.678 |

Porosity (φ) | ${\mathit{P}}_{\mathit{\eta}}\left(\mathbf{GPa}\right)$ |
---|---|

0.003 | 9.698 |

0.033 | 8.022 |

0.073 | 6.792 |

0.113 | 5.086 |

Binder Volume Fraction (b) | ${\mathit{P}}_{\mathit{\eta}}\left(\mathbf{GPa}\right)$ |
---|---|

0% | 11.382 |

5% | 9.698 |

10% | 9.316 |

20% | 8.458 |

Related Parameters | ${\mathit{K}}_{0,\mathit{n}}$ | ${\mathit{K}}_{0,\mathit{n}}^{\prime}$ | ${\mathit{P}}_{0,\mathit{\phi}}$ | $\mathit{\alpha}$ | ${\mathit{P}}_{0,\mathit{b}}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|

Values | 0.8765 GPa | 23.27 | 9.668 GPa | 0.0168 | 10.02 GPa | 0.0724 |

Porosity (φ) | Binder Volume Fraction (b) | ||
---|---|---|---|

Parameters | Values | Parameters | Values |

${s}_{0,\phi}$ | 2.273 | ${s}_{0,b}$ | 2.476 |

${c}_{0,\phi}$ | 2.786 | ${c}_{0,b}$ | 2.746 |

${m}_{\phi}$ | −8.16 × 10^{−4} | ${m}_{b}$ | 1.5 × 10^{−3} |

${n}_{\phi}$ | 7.19 × 10^{−3} | ${n}_{b}$ | 3.99 × 10^{−4} |

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**MDPI and ACS Style**

Ge, S.; Zhang, W.; Sang, J.; Yuan, S.; Lo, G.V.; Dou, Y.
Material Point Method Simulation of the Equation of State of Polymer-Bonded Explosive under Impact Loading at Mesoscale. *Processes* **2020**, *8*, 983.
https://doi.org/10.3390/pr8080983

**AMA Style**

Ge S, Zhang W, Sang J, Yuan S, Lo GV, Dou Y.
Material Point Method Simulation of the Equation of State of Polymer-Bonded Explosive under Impact Loading at Mesoscale. *Processes*. 2020; 8(8):983.
https://doi.org/10.3390/pr8080983

**Chicago/Turabian Style**

Ge, Siyu, Wenying Zhang, Jian Sang, Shuai Yuan, Glenn V. Lo, and Yusheng Dou.
2020. "Material Point Method Simulation of the Equation of State of Polymer-Bonded Explosive under Impact Loading at Mesoscale" *Processes* 8, no. 8: 983.
https://doi.org/10.3390/pr8080983