*Article* **Experiment and Numerical Simulation on Friction Ignition Response of HMX-Based Cast PBX Explosive**

**Junming Yuan 1,\*, Yue Qin 1, Hongzheng Peng 2, Tao Xia 1, Jiayao Liu 1,2, Wei Zhao 2, Hu Sun <sup>1</sup> and Yan Liu <sup>1</sup>**


**Abstract:** In order to study the ignition process and response characteristics of cast polymer-bonded explosives (PBX) under the action of friction, HMX-based cast PBX explosives were used to carry out friction ignition experiments at a 90◦ swing angle and obtain the critical ignition loading pressure was 3.7 MPa. Combined with the morphology characterization results of HMX-based cast PBX, the friction temperature rise process was numerically simulated at the macro and micro scale, and the ignition characteristics were judged. The accuracy of the numerical simulation results was ensured based on the experiment. Based on the thermal–mechanical coupling algorithm, the mechanical–thermal response of HMX-based cast PBX tablet under friction was analyzed from the macro scale. The results show that the maximum temperature rise is 55 ◦C, and the temperature rise of the whole tablet is not enough to ignite the explosive. Based on the random circle and morphology characterization results of tablet, the mesoscopic model of HMX-based cast PBX was constructed, and the microcrack friction formed after interface debonding was introduced into the model. The temperature rise process at the micro scale shows that HMX crystal particles can be ignited at a temperature of 619 K under 4 MPa hydraulic pressure loaded by friction sensitivity instrument. The main reason for friction ignition of HMX-based cast PBX is the friction hot spot generated by microcracks formed after interface damage of the tablet mesoscopic model, and the external friction heat between cast PBX tablet and sliding column has little effect on ignition. External friction affects the ignition of HMX-based cast PBX by influencing the formation of internal cracks and the stress at microcracks.

**Keywords:** numerical simulation; friction sensitivity; ignition; cast PBX; mesoscopic model

#### **1. Introduction**

Cast PBX is a kind of high polymer-bonded explosive, which is widely used in highspeed penetration and damage weapons because of its good mechanical properties and low sensitivity. During the transportation, storage and use of casting PBX charge warheads, accidental ignition may occur due to external stimulation, which greatly affects the reliability and safety of weapons. Friction is one of the important stimulation sources. A lot of experiments and numerical simulations have been done on the ignition of explosives under friction. Min-cheol Gwak et al. [1] analyzed the friction ignition process of HTPB based solid propellant from the perspectives of reaction kinetics and friction heating, and built relevant models to predict the ignition time of propellant under friction action through numerical simulation. Sun Baoping et al. [2] carried out numerical simulation of PBX tablet friction-ignition experiment based on the finite element method, and analyzed the influence of pressure, velocity, and friction coefficient on ignition. Deng Chuan et al. [3] established a test method for friction sensitivity of pendulum impact-driven sand target friction explosive tablets, and tested the friction sensitivity of three PBX tablets. R. Charley et al. [4] studied the friction ignition process of solid propellant based on friction device, which

**Citation:** Yuan, J.; Qin, Y.; Peng, H.; Xia, T.; Liu, J.; Zhao, W.; Sun, H.; Liu, Y. Experiment and Numerical Simulation on Friction Ignition Response of HMX-Based Cast PBX Explosive. *Crystals* **2023**, *13*, 671. https://doi.org/10.3390/ cryst13040671

Academic Editors: Rui Liu, Yushi Wen and Weiqiang Pang

Received: 15 March 2023 Revised: 3 April 2023 Accepted: 7 April 2023 Published: 13 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

can obtain the overall deformation process of propellant by high-speed photography, and then simulate the response of solid propellant under friction by discrete element method. And compare the cloud image of numerical simulation with the topography of high-speed photography. The deformation process analyzed by the discrete element method is in good agreement with the actual deformation process and can reflect the meso-ignition process of the explosive. Dai Xiaogan et al. [5] carried out the friction ignition experiment on PBX and designed a device to calculate the friction work and friction power threshold of explosive ignition under friction and analyzed the ignition mechanism of explosive in the friction sensitivity experiment. The results show that it is difficult to heat PBX as a whole and make it ignite under friction. Zeman et al. [6] provided an overview of the main developments over the past nine years in the study of the sensitivity of energetic materials (EM) to impact, shock, friction, electric spark, laser beams and heat. Luo Yi et al. [7] and others carried out slide experiment and Numerical Simulation Research on PBX-8701, and analyzed the influence of external-friction coefficient on ignition delay time based on the theory of heat transfer and diffusion and reaction kinetics.

The ignition phenomenon of explosives under external stimulation can be explained by the hot spot theory. Dienes et al. [8] analyzed and compared four hot spot mechanisms (pore collapse; hole collapse; hole collapse) in the projectile target experiment, shock heating, shear band under plastic flow. The results show that the interface friction of closed crack is the main reason of hot spot. Randolph et al. [9] developed a friction ignition model to predict the thermal decomposition of condensed phase explosives when impacted at an oblique angle on a rigid target surface. Andersen et al. [10] established a mathematical relationship between the friction coefficient of materials and the parameters that affect friction during shearing. An et al. [11] used the ReaxFF force field to study the hot spot formation mechanism of PBXN-106 explosive under the action of shock waves. The shear relaxation of the micro-convex body at the interface of the density discontinuity caused by the impact resulted in energy deposition, and the local shear in this region formed a hot spot, which was eventually accompanied by chemical reactions leading to the explosion of the system. It is concluded that reducing the density of the binder (about 1/3 of the explosive density) can inhibit the hot spot generation. Cai et al. [12] used molecular dynamics to simulate the impact response characteristics of coarse-grained explosives and pointed out that both interparticle friction and shear deformation between particles can generate hot spots, and the direction of shear slip between particles significantly affects the frictional heat generation efficiency. Keshavarz et al. [13] presented a novel general simple model for prediction of the relationship between friction sensitivity and activation energy of thermolysis of cyclic and acyclic nitramines on the basis of their molecular structures. Richard et al. [14] studied the response to mechanical non-shock stimulation using explosive-driven deformation test and ballistic impact chamber. According to the experimental results, the shear rate threshold as a single parameter to describe the mechanical sensitivity is challenged, and preference is given to the development of an ignition criterion based on intergranular sliding friction under the action of a normal pressure. Hu et al. [15] presented a combined computational– experimental study of the mesoscale thermo-mechanical behavior of HTPB bonded AP composite energetic material subjected to dynamic loading conditions. The computational model considers the AP-HTPB interface debonding, post debonding interface friction and temperature rise due to viscoelastic dissipation as well as dissipative interfacial processes. Jafari et al. [16] introduced a reliable method to predict friction sensitivity of quaternary ammonium-based EILs, which are based on elemental composition of cation and anion of a desired ionic liquid as well as the contribution of specific cations and anions. Gruau et al. [17] simulated the behavior of the PBX by means of the elastic-plastic damage law and the ignition criterion due to localization of plastic strain in the microstructure, and the simulation results were consistent with the experimental results. Wu et al. [18] developed a micromechanical ignition model of the hot spot formation of HMX and PETN mixed powder explosives under the impact of a falling hammer in the fine structure, estimated the temperature rise caused by plasticity and frictional dissipation, and added the self-heating

reaction model of explosives to predict the hot spot ignition by thermal explosion. Joel G. Bennett et al. [19] proposed visco elastic statistical crack ignition model (visco-scram) for numerical simulation of non-shock ignition of PBX explosives. In this model, the friction heat of microcracks is taken as the main source of hot spot formation, and the mechanical behavior of PBX explosives is considered. As a classical non-shock ignition model, this model is widely used. However, there are many parameters in the model, and it is difficult to calibrate. For different explosive formulations, it is necessary to re determine the material parameters, so the pretreatment process of numerical simulation is relatively complex. Xue, H.J. et al. [20] established an improved combined microcrack and microvoid model (CMM) to study the damage and ignition behaviors of polymer-bonded explosive (PBX) under coupled impact and high-temperature loading conditions. Bai, Z.X., Li, H.T., Yin, Y. and Duarte, C.A. [21–24] have studied the friction behavior and hot spot formation of HMX explosive crystals, as well as the ignition and combustion behavior under hot spot conditions.

The research on friction ignition has gradually changed from macro to micro hot spot formation. Barua et al. [25–28] studied the temperature rise process of PBX9501 from micro scale by CFEM finite element method. Based on the digital image processing technology, a mesoscopic model was established, which was consistent with the actual situation. The friction heating of microcracks caused by the failure of the interface between particles and matrix was studied. The relationship between the particle-failure mechanism and the overall temperature rise of PBX has been researched. Amirreza Keyhani et al. [29] analyzed the ignition process of PBX9501 at meso level based on CFEM method and considered the contribution of friction and viscoelastic plastic dissipation energy of microcracks formed after damage to temperature rise. The research results show that the viscoelastic plastic dissipation energy has little effect, and the energy contributed by crack friction is the main reason for hotspot formation.

At present, there are many research studies on the ignition of pressed PBX, but relatively few studies on cast PBX, and the research on the micro ignition focuses on the impact overload, while studies on the friction overload are fewer. Therefore, HMXbased cast PBX samples were prepared and the friction-ignition experiment of cast PBX at a 90◦ swing angle were carried out in this paper. Based on the experimental results combined with the macro numerical simulation, the friction-ignition process and response characteristics of HMX-based cast PBX tablets are studied at the micro level, and the accuracy of the numerical simulation results is judged by friction ignition experiments.

#### **2. Materials and Methods**

#### *2.1. Experiment*

2.1.1. Preparation and Characterization of HMX-Based Cast PBX Sample

According to the typical HTPB casting PBX formulation, a small amount of casting PBX explosive samples were prepared. Cast PBX is composed of Al powder, HTPB bonding system and HMX crystal particles. All raw materials should be fully dried before preparation, and their water content must be strictly controlled. Aluminum powder and binder are thoroughly mixed and added to the main explosive in batches under heated conditions. After mixing evenly, the material is poured into the mold so that the explosive is cured at a constant temperature of 60 ◦C for 72 h. The formulation of cast PBX is shown in Table 1 referred from [30].

**Table 1.** Formulation of HMX-based cast PBX explosive.


Figure 1 shows the cast PBX tablets and the electron microscopic morphology. After the preparation for the cast PBX is completed, slice work is carried out, and the slice size is about Φ10 mm × 1 mm. Because the formulation contains Al powder, the color of the tablet is grey, as shown in Figure 1a,b. The tablet was characterized by scanning electron microscope, and HMX crystal particles with a particle size range of 10~80 μm are shown in Figure 1c. The electron microscope morphology provides experimental support for the establishment of the mesoscopic model in the next friction ignition simulation.

**Figure 1.** (**a**,**b**) HMX-based Cast PBX tablets; and (**c**) 1000 times electron microscope morphology of HMX-based PBX explosive sample.

The photographs in Figure 1c indicate the presence of micropores in cast PBX near the HMX particles and the binder matrix. This is also reflected in Figure 1b, where pores are visible on the surface of the poured PBX tablet. The binder curing process itself has pores, as shown in Figure 2a. Cast PBX tablets without vacuum pumping and uneven mixing can cause pores. The presence of pores weakens the cohesive strength of the interface boundaries and contributes to the development of deformation localization and local heating at the boundaries of HMX particles under the action of stresses arising from friction of the samples. Figure 2c shows the microstructure of HMX-based cast PBX tablets. The picture clearly shows that the HTPB/Al matrix composed of Al powder particles and adhesive is tightly wrapped with many HMX crystal particles of different sizes, providing theoretical support for the construction of a simplified model.

**Figure 2.** (**a**) HTPB binder curing pores; (**b**) Pore of HMX-based cast PBX tablets; (**c**) 200 times electron microscope morphology of HMX-based PBX explosive sample.

#### 2.1.2. Test Process of Friction Sensitivity Experiment

The cast PBX tablet is loaded onto the sliding column end face of the friction device, and the upper sliding column is gently placed. The upper slide column is rotated for 1–2 cycles to evenly distribute the explosive sample between the end faces of the two slide columns. The loaded friction device is placed in the combustion equipment of the friction sensitivity instrument, at a 90◦ swing angle, and the gauge pressure raised to 4.0 MPa (or other different loading pressures). The pendulum is released, and the striking rod is struck. The striking rod causes the upper and lower sliding columns to move at high speed, causing sliding friction on the end faces of the upper and lower columns, resulting in intense friction

on the test sample. After completing this operation process, the experimenter repeats the process to perform the test on the next sample.

#### 2.1.3. Critical Ignition Loading Pressure

MGY-I friction sensitivity instrument is used to test the friction ignition of cast PBX [31]. In order to obtain the exact critical initiation conditions of friction, the Bruceton method [32] is used for reference to the high impact sensitivity characteristics. In the test of friction sensitivity, there are two variables controllable, one is the pendulum angle and the other is the loading pressure of hydraulic press. Considering the feasibility of numerical simulation, the paper uses the loading pressure as a variable. The Bruceton method is used to determine the critical ignition loading pressure of friction sensitivity. The method is calculated according to the following formula.

$$
\sigma\_{50}^f = \left[ \mathbf{A} + \mathbf{B} \left( \frac{\mathbf{C}}{\mathbf{D}} \pm \frac{1}{2} \right) \right] \tag{1}
$$

$$\mathbf{C} = \sum \mathbf{i} \times \mathbf{n}\_{\mathbf{i}} \tag{2}$$

$$\mathbf{D} = \sum \mathbf{n}\_{\mathbf{i}} \tag{3}$$

In the formula, σ<sup>f</sup> <sup>50</sup> is the critical ignition loading pressure of explosives; A is the minimum hydraulic pressure loaded by friction sensitivity instrument for initiation; B is the set loading pressure interval, 0.5 MPa in this paper; i is the height serial number; ni is the sum of the number of explosion times under the height of serial number i. The friction ignition experiment was carried out under a swing angle of 90◦ on cast PBX tablets with the same specification. In this experiment, the loading pressure range is 2 MPa~4 MPa, and the pressure gradient is 0.5 MPa; So, A = 2 MPa, B = 0.5 MPa. Finally, the critical ignition loading pressure calculated is σ<sup>f</sup> <sup>50</sup> = 3.7 MPa. The calculation parameters in Formula (1) are shown in Table 2.

**Table 2.** Sorting of test results.


2.1.4. Solution of Relative Slip Rate

In order to obtain the slip rate of friction ignition experiment, the traditional friction sensitivity instrument was improved, as shown in Figure 3. A force sensor is added at the end of the striking bar. The impact force and action time of the sliding column can be measured by the force sensor, and the displacement curve of the sliding column can be obtained by the motion equation.

When measuring the friction sensitivity, the upper sliding column starts to slide to the right under the action of pendulum. The sliding speed of the upper column is not uniform. According to the F-T curve measured by the sensor, the displacement x-t curve of the sliding column can be obtained by calculating the F-T integral. The impact force F obtained does not provide acceleration for the upper sliding column, but also overcomes friction resistance F.

**Figure 3.** Pendulum impact force test device.

The velocity curve of the sliding column can be used as the boundary condition of the numerical simulation to ensure that the numerical simulation is consistent with the experiment. The slip rate curves under different loading pressures are shown in Figure 4a. Over time, under different loading pressures, the slip rate of the upper sliding column first rises and then gradually stabilizes. Under a loading pressure of 2 MPa, the slip rate curve and displacement curve are shown in Figure 4b. As can be seen from the Figure 4b, the maximum sliding speed of the sliding column is 8 m/s, and the maximum sliding displacement is about 12 mm. This is a linear system of friction, and the sliding distance increases with time. In actual friction experiments, due to the limitation of the length of the striking rod and the constraint of the guide hole, the sliding displacement of the upper sliding column generally does not exceed 2.5 mm.

**Figure 4.** Slip rate and friction displacement curve of the upper sliding column.

#### *2.2. Numerical Simulation*

#### 2.2.1. Modeling

The macro simulation physical model of friction ignition experiment is shown in Figure 5. The model is established according to the equal proportion of the size of the experimental device, in which the size of the upper and lower striking columns is the same, which is a cylinder with a diameter of 10 mm and a height of 10 mm. The diameter of HMX-based cast PBX tablets ϕ the thickness is 1 mm. The vertical downward pressure load is applied on the top of the sliding column on the model, and the pressure load is 2–4 MPa, and the gradient is 0.5 MPa. The friction displacement is applied horizontally, and the displacement load is applied according to the curve. The sliding column is completely fixed.

**Figure 5.** Macroscopic friction model of cast PBX.

The main components of the cast PBX prepared in this paper are HMX/Al powder and HTPB binder. The particle size and gradation of each component need to be considered. The mesoscopic model of cast PBX is built based on the random circle model. The proportion of each particle in the model is closely approximate with that of the cast PBX based on the above Figure 2c. Due to the large number of particles, the mesh and contact action need to be divided, so appropriate simplification is adopted. Some literatures point out a treatment method: small particles and binder matrix are regarded as homogeneous binder system, only the spatial distribution of large particles is considered, and the random spatial coordinates of large particles are given by Matlab function programming. Combined with the electron micrograph of HMX-based PBX explosive sample in Figure 2c, the simplified mesoscopic model is shown in the Figure 6.

**Figure 6.** HMX-based cast PBX mesoscopic model: (**a**) random circular mesoscopic model; and (**b**) simplified mesoscopic model by Al and HTPB as a matrix.

A small part of the whole cast PBX tablet was taken and magnified to obtain the mesoscopic model. The friction coefficient f is 0.15. As the friction displacement of the sliding column is relatively small, it can be ignored. Pressure 1 is applied as the loading pressure as shown in Figure 7. The left boundary is fixed horizontally, and there are degrees of freedom in the vertical direction. In the model, the upper sliding column is fixed in the vertical direction, and a fixed displacement is applied to the right. The displacement and amplitude are applied according to the curve. The boundary conditions of the mesoscopic model are shown in Figure 7.

**Figure 7.** Boundary conditions of mesoscopic model for friction ignition.

#### 2.2.2. Material Model

(1) Viscoelastic material model.

A viscoelastic body is set at any time τ<sup>i</sup> have an instantaneous strain Δ*ε*i(τi), The instantaneous strain Δ*ε*i(τi), For the stress at any subsequent time t σ(t) Impact, in order to Δσi(t, τi) The subscript of Δ indicates that the effect is caused by the i-th strain increment Δ*ε*i(τi), because the research is linear system, the stress change Δσi(t, τi) and instantaneous strain increment Δ*ε*i(τi) Satisfy relationship [33]:

$$
\Delta \sigma\_{\mathbf{i}}(\mathbf{t}, \tau\_{\mathbf{i}}) = \mathbf{E}(\mathbf{t}, \tau\_{\mathbf{i}}) \, \Delta \varepsilon\_{\mathbf{i}}(\tau\_{\mathbf{i}}) \tag{4}
$$

Before time t, if the instantaneous strain increment Δ*ε*i(τi) is more than one (i = 1, 2, 3, ···, n), and each Δ*ε*<sup>i</sup> has no influence on each other. According to the principle of Boltzmann superposition, there are:

$$\sigma(\mathbf{t}) = \sum\_{\mathbf{i}=1}^{n} \Delta \sigma\_{\mathbf{i}}(\mathbf{t}, \tau\_{\mathbf{i}}) = \sum\_{\mathbf{i}=1}^{n} \mathbf{E}(\mathbf{t}, \tau\_{\mathbf{i}}) \, \Delta \varepsilon\_{\mathbf{i}}(\tau\_{\mathbf{i}}) \tag{5}$$

When the instantaneous strain is continuous, it becomes *ε*(τ), (n → ∞), The continuous strain can be obtained *ε*(τ), The expression of stress response is as follows:

$$\mathbf{u}(\mathbf{t}) = \int\_{-\infty}^{\mathbf{t}} \mathbf{E}(\mathbf{t}, \mathbf{r}) \, \mathrm{d}\varepsilon(\mathbf{r}) = \int\_{-\infty}^{\mathbf{t}} \mathbf{E}(\mathbf{t}, \mathbf{r}) \frac{\partial \varepsilon}{\partial \mathbf{r}} \mathrm{d}\mathbf{r} \tag{6}$$

$$\mathbf{E}(\mathbf{t}, \mathbf{\tau}) = \mathbf{2}(1+\mathbf{v}) \, \mathbf{G}(\mathbf{t}, \mathbf{\tau}) \tag{7}$$

In calculating the curve of relaxation modulus, the stress and strain at different relaxation times need to be measured. According to the integral formula, a method is given in document [34,35]. The exponential series fitting formula is used. The equation is as follows:

$$\mathbf{G(t)} = \mathbf{G}\_{\infty} + \sum\_{i=1}^{n} \mathbf{G\_i} \mathbf{e^{\frac{-t}{t\_i^\*}}} = \mathbf{G\_0} (1 - \sum\_{i=1}^{n} \mathbf{g\_i^P} \mathbf{e^{\frac{-t}{t\_i^\*}}}) \tag{8}$$

$$\mathbf{g\_R(t)} = 1 - \sum\_{i=1}^{n} \mathbf{g\_i^P} \mathbf{e^{\frac{-t}{\tau\_i^P}}} \tag{9}$$

where G0 <sup>=</sup> <sup>G</sup><sup>∞</sup> <sup>+</sup> <sup>n</sup> ∑ i=1 Gi is the instantaneous relaxation shear modulus, G<sup>∞</sup> is the steadystate shear modulus, gR(t) = GR(t)/G0 is the dimensionless relaxation shear modulus, g<sup>P</sup> <sup>i</sup> = Gi/G0 is relative modulus of the i-th term. The softening effect of temperature on materials is usually described by Williams-Landel-Ferry (WFL) time-temperature equivalent equation:

$$
\pi\_{\mathbf{i}}^{\mathbf{G}} = \int\_0^{\mathbf{t}} \frac{\mathbf{d} \, \mathbf{t}'}{\mathbf{a}(\theta(\mathbf{t}'))} \, \tag{10}
$$

$$-\lg\left(\mathbf{a}\left(\theta\left(\mathbf{t'}\right)\right)\right) = \frac{\mathbf{a}\left(\mathbf{T} - \mathbf{T}\_{\text{ref}}\right)}{\beta + \mathbf{T} - \mathbf{T}\_{\text{ref}}}\tag{11}$$

where a is the time temperature transfer function, α and β is constant, Tref is the reference temperature, τ<sup>G</sup> <sup>i</sup> is the relaxation time. Mechanical and thermal parameters of cast PBX and HTPB/Al matrix are shown in Tables 3 and 4 respectively. Among the Tables 3 and 4, λ is the thermal conductivity coefficient. In addition, qm is the heat released by the complete decomposition of the unit mass explosive, and Ea is the activation energy in Table 3.

(2) Elastoplastic material model.

**Table 3.** Mechanical and thermal parameters of cast PBX [36,37].


**Table 4.** Mechanical and thermal parameters of HTPB/Al matrix [15,37,38].


Compared with HTPB bonding system, HMX crystal particles has higher hardness and modulus, so it is elastic-plastic. Mechanical and thermal parameters of HMX are shown in Table 5. The constitutive model is as follows:

$$\sigma = \begin{cases} \begin{array}{c} \text{E}\mathfrak{e} \\ \sigma\_0 + \mathrm{E}\_t(\mathfrak{e} - \mathfrak{e}\_0) \end{array} & \mathfrak{e} \le \mathfrak{e}\_0 \\ \end{cases} \tag{12}$$

where E is Young modulus, σ<sup>0</sup> is the yield stress, Et is plastic hardening rate. The material of upper sliding column and lower sliding column is special steel, the material model is same as HMX crystal particles. Material parameters of sliding column are shown in Table 6. Among Tables 5 and 6, Z is pre-exponential factor, and Cv is the specific heat capacity, respectively.

**Table 5.** Mechanical and thermal parameters of HMX crystal particles [39,40].


**Table 6.** Material parameters of sliding column [40].



In the micromechanical analysis, the interface between the binder and explosive plays an important role in the micromechanical properties and thermal response of PBX. On the one hand, the debonding of particles and binder at the interface will cause the mechanical damage of PBX. On the other hand, friction between the debonding explosive particles and the binder will form a high temperature concentration area, which may produce local hot spots. In the existing literature, a cohesive model is used to simulate the debonding phenomenon at the interface, which can better describe the debonding phenomenon between explosive particles and binder, as shown in Figure 8.

**Figure 8.** Bilinear cohesive zone model.

In the model, Tmax is the maximum traction force allowed; δ<sup>0</sup> is the separation amount of the interface during the initial damage, K is the stiffness matrix, K = Tmax <sup>δ</sup><sup>0</sup> . When the traction force T > Tmax, the interface begins to be damaged, and the strength of the model will decrease in the damage evolution stage. When the separation amount δ > δmax, the model is completely destroyed, and the bonding of the interface fails. Interface parameters of HMX and binder are shown in Table 7. Among the Table 7, μ is the coefficient of friction, and Kc is the heat distribution coefficient between two objects.

**Table 7.** Interface parameters of HMX crystal particle and binder [15].


The bilinear cohesive unit simulates the mechanical relationship between explosive particles and binder and can analyze the failure between explosive particles and binder. There are three ways to realize cohesive unit in ABAQUS: (a) By inserting a thick cohesive element in the interface layer; (b) Building a thick cohesive element by establishing a thin layer model; (c) Establishing a cohesive contact (Surface-based cohesive behavior) through a contact algorithm.

The first two are constructed by assigning cohesive properties to the units. Each unit is a cohesive mechanical unit. Unfortunately, the cohesive unit in ABAQUS only has mechanical properties and cannot adapt to the thermo-mechanical coupling algorithm. Therefore, the cohesive unit cannot perform thermal and temperature calculations. Calculations can only analyze the mechanical response process, so this paper adopted the last method cohesive contact method to realize the debonding of particles and binder and the friction effect after debonding.

#### (2) The friction of interface.

After the failure of the cohesive interface, microcracks will be formed, and friction will occur in microcracks. The heat generated by friction is the main reason for the temperature concentration. The heat generated by friction and the distribution law of heat [41] are as follows:

$$\mathbf{Q}\_{\mathbf{f}} = \mu \mathbf{F}\_{\mathbf{N}} \mathbf{v} \tag{13}$$

$$\mathbf{Q}\_{\mathbf{f}} = \mathbf{Q}\_{\mathbf{f}1} + \mathbf{Q}\_{\mathbf{f}2} \tag{14}$$

$$\mathbf{K}\_{\mathbf{c}} = \frac{\mathbf{Q}\_{\mathbf{f}1}}{\mathbf{Q}\_{\mathbf{f}2}} = \sqrt{\frac{\lambda\_1 \mathbf{C}\_{\mathbf{v}1} \rho\_1}{\lambda\_2 \mathbf{C}\_{\mathbf{v}2} \rho\_2}}\tag{15}$$

where, Qf is the heat produced by friction, FN is the normal pressure, v is the slip rate, μ is the coefficient of friction. Qf can be further divided into two parts, Qf1 and Qf2 represents the friction heat of two objects: Kc is the heat distribution coefficient between two objects, λ is the heat conductivity, Cv is the specific heat capacity, ρ is density.

#### 2.2.4. Self Heating Reaction Theory and Heat Transfer Equation of Explosives

The ignition process of explosive in low-speed friction can be explained by the hot spot theory, and the hot spot initiation can be described by the thermal initiation equation. Therefore, it is necessary to analyze the internal thermal balance process of explosive. The internal thermal balance equation of explosive is as follows [42].

$$
\rho \mathbf{C} \frac{\partial \mathbf{T}}{\partial \mathbf{T}} = \lambda \nabla^2 \mathbf{T} + \Phi\_\mathbf{V} \tag{16}
$$

where, ∇2T can be regarded as the inflow heat from the outside; <sup>λ</sup> is thermal conductivity coefficient and Φ<sup>V</sup> is the intensity of internal heat source. The intensity of internal heat source Φ<sup>V</sup> is determined by the exothermic intensity of explosive reaction. The thermal decomposition process of explosives is usually expressed as the rate by the first order Arrhenius equation. The total reaction heat per unit volume per unit time of explosives is as follows:

$$\Phi\_{\rm V} = \mathbf{Q}\_{\rm V} = \rho \mathbf{q}\_{\rm m} \mathbf{c}^{\rm n} \mathbf{k} = \rho \mathbf{q}\_{\rm m} \mathbf{Z} \exp(-\frac{\mathbf{E}\_{\rm a}}{\mathbf{RT}}) \tag{17}$$

In this paper, the heat-generation subroutine HETVAL is written to realize the heat generation in the thermal decomposition process of explosives. In the actual decomposition process, it is necessary to consider the initial decomposition temperature of explosives. In this paper, the initial decomposition temperature is 554 K. When the temperature of the elemental particles of the explosive is greater than the initial decomposition temperature TS (554 K), the thermal decomposition reaction of the explosive will be triggered. At lower temperatures, the thermal decomposition rate is lower, and the chemical heat generated is lower. During this process, the frictional heat of microcracks is still the main source of internal heat in explosives. If the temperature continues to rise, the reaction rate of the explosive accelerates, and the heat of chemical reaction becomes the main source of internal heat. Due to the rapid thermal decomposition process of cast PBX, the instantaneous heat flow generated will cause a sharp increase in temperature, which can be considered as the ignition response of the explosive under external stimulus.

#### **3. Results and Discussion**

#### *3.1. Response Analysis of PBX Tablet Simulation*

#### 3.1.1. Analysis of Tablet Deformation Process

The deformation process and Mises stress nephogram of cast PBX tablet under 3 MPa loading pressure are shown in Figure 9. It can be seen that the casting PBX tablet is first pressed by the hydraulic press. With the sliding of the upper sliding column, the stress begins to concentrate. The Mises stress of the concentrated unit is much greater than that of the hydraulic press. The maximum Misses stress values at 1 ms, 1.5 ms, 2.5 ms, 4 ms, 6 ms, and 10 ms are 0.32 MPa, 8.93 MPa, 15.64 MPa, 32 MPa, 50.34 MPa, and 55.33 MPa, respectively. As time goes on, the Misses stress at the interface gradually increases. At 2.5 ms, the tablet begins to deform, and the stress-concentration area is the most obvious. Combined with the displacement curve of the upper sliding column, the sliding speed is the fastest. At 4 ms, the deformation of the tablet is intensified, but no distortion occurs. Therefore, the stress concentration area is still in the center of the tablet. With the movement of the upper sliding column, the tablet began to deform. At 6 ms, the tablet was slightly distorted, and the stress concentration area diffused due to the distortion. At 10 ms, the tablet was seriously deformed.

**Figure 9.** Nephogram of Misses stress and friction deformation of cast PBX tablet: (**a**) t = 1 ms; (**b**) t = 1.5 ms; (**c**) t = 2.5 ms; (**d**) t = 4 ms; (**e**) t = 6 ms; and (**f**) t = 10 ms.

#### 3.1.2. Effect of Loading Pressure on Tablets

Under different hydraulic conditions, the stress and temperature rise of the tablet under friction are also different. The hydraulic pressure of 2 MPa, 2.5 MPa, 3 MPa, 3.5 MPa and 4 MPa is applied, respectively, and the displacement amplitude under the corresponding hydraulic pressure is applied. The stress component (S22) and temperature rise of the central layer unit (Unit No. 1924) of the tablet under different hydraulic pressures are analyzed. The Figure 10 shows the stress component of S22 under hydraulic pressure. With the increase of hydraulic pressure, the stress also increases. The maximum stress under different pressures is 61 MPa, 78 MPa, 87 MPa, 88 MPa and 92 MPa. The figure shows the maximum temperature rise inside the tablet under different hydraulic pressures. The maximum temperature rises under 2–4 MPa hydraulic pressure are 34 ◦C, 47 ◦C, 54 ◦C, 50 ◦C and 55 ◦C, respectively. It can be seen that the temperature rise gradually increases with the increase of pressure, but the overall temperature rise is not high.

Through the simulation of friction sensitivity experiment, the influence of pressure on the temperature rise of tablets was analyzed. The simulation results show that the pressure has a great influence on the temperature rise of tablets. When the pressure is low, there is no large area of temperature rise concentration area. With the increase of pressure, the temperature concentration area begins to appear. The concentration area further moves and gradually shrinks, and the maximum temperature rise increases with the increase of pressure. The external friction can make the local temperature of the tablet rise, but it can't

continue. The overall temperature rises are not able to make the explosive ignite, so we need to further consider the ignition mechanism from the micro level.

**Figure 10.** (**a**) Stress components of tablet unit under different hydraulic pressures; and (**b**) temperature rise of tablet unit under different hydraulic pressures.

#### *3.2. Response Analysis of Tablet Mesoscopic Model*

#### 3.2.1. Analysis of Micro Ignition Response and Critical Loading Pressure

In order to understand the meso view of the fire process under the action of friction, take the temperature rise cloud diagram of the PBX mesoscopic model under 3.5 MPa and 4 MPa hydraulic pressure to analyze the temperature rise and ignition process are shown in Figure 11. Under 3.5 MPa hydraulic pressure, no explosive particles ignite. Under a loading pressure of 3.5 MPa, the maximum temperature values at 0.75 ms, 2.5 ms, 5 ms, 7 ms, 9 ms, and 10 ms are 298 K, 306 K, 453 K, 579 K, 425 K, and 396 K, respectively. At 0.25 ms, the friction heat at the bottom of the tablet is the main factor affecting the temperature rise. At 7 ms, it can be seen that the heat generated by the bottom friction has almost begun to diffuse into the HMX crystal particles. Due to the interface, the heat conductivity is small, and the heat diffusion is slow. The entire friction process ends at 10 ms, and the heat spreads to the entire explosive area. However, because the temperature accumulation is not obvious enough, and the friction generates little heat, the whole does not ignite.

Figure 11b shows the temperature rise under 4 MPa hydraulic pressure. Under a loading pressure of 4 MPa, the maximum temperature values at 0.75 ms, 2.5 ms, 5 ms, 5.5 ms are 299 K, 355 K, 1253 K, 1630 K. At 5.75 ms and 6 ms, the temperature rose sharply, and ignition occurred. During the sliding process, the distance increases, and the temperature hot spots gradually increase, especially the temperature at the interface between the sliding column and the tablet gradually rises. The temperature rise inside the tablet is mainly concentrated at the interface. The temperature of HTPB at the interface is higher than that of HMX crystal particles. However, as the temperature of HMX rises, the heat-generation subroutine begins. At this time, the self-heating reaction of HMX crystal particles generates a lot of heat, which breaks the original thermal equilibrium state, and the temperature rises sharply. This process can be considered as the process of ignition of HXM particles. It can be seen from the figure that the first smaller temperature rise area starts at 0.25 ms; the local temperature rise points increase with time, and the maximum temperature gradually becomes larger. At 5.5 ms, the temperature distribution cloud map before the critical ignition, the highest the temperature zone is distributed at the microcrack interface. At this time, the corresponding instantaneous temperatures of the matrix and particles are 1654 K and 619 K, respectively. At 5.75 ms, it can be seen that the color of the HMX crystal particles has deepened, indicating that the temperature has risen sharply, and ignition has occurred.

**Figure 11.** (**a**) Meso-level temperature rise cloud diagram at a loading pressure of 3.5 MPa; (**b**) meso-level temperature rise cloud diagram under a loading pressure of 4 MPa.

The stress distribution inside the PBX tablet is obtained through macro simulation. The tablet under different hydraulic pressures will generate different internal stresses. The stress component at the stress concentration area of the tablet is taken as the boundary load condition of the mesoscopic model, and the stress distribution and the ignition state of the mesoscopic model are analyzed. In order to control irrelevant variables, select the same unit under different pressures for analysis. Figure 12 shows the temperature rise curves of HMX crystal particles under different hydraulic pressures. It can be seen from the figure that with the increase of hydraulic pressure, the maximum temperature rise of HMX crystal particles gradually increases. When the hydraulic pressure is 4 MPa, the temperature of HMX crystal particles rises faster and the heat subroutine is generated. Therefore, it can be judged that the HMX crystal particles are ignited under the hydraulic pressure of 4 MPa. Point A (5.5 ms, 619 K) is the inflection point of temperature rise of explosive particles under 4 MPa hydraulic pressure. The abscissa reflects the time of ignition, and the ordinate is the temperature at the time of ignition. Therefore, it can be judged that the critical ignition hydraulic pressure is between 3.5 MPa and 4 MPa.

**Figure 12.** Temperature rises curve of mesoscopic model under different loading pressures.

3.2.2. Influence of Cohesive Interface Friction Coefficient on Ignition

The friction between the HMX crystal particles and the HTPB/Al matrix interface in the cast PBX is the main cause of ignition. At this time, the tablet is subjected to normal stress on the one hand, and frictional shear force on the other hand, under the combined action of pressure and shear force. The stress generated at the interface is relatively concentrated, by adjusting the friction coefficient μ at the cohesive interface to 0.22, 0.27, 0.37 and 0.42, respectively, and the pressure load is set to the minimum ignition pressure 4 MPa to analyze the effect of friction coefficient on ignition.

Figure 13 shows the temperature-rise curves of μ explosive particles with different interfacial friction coefficients. It can be seen from the figure that when the friction coefficient is less than 0.22, no ignition occurs. Where A, B, C, and D are the temperature inflection points when the friction coefficient is 0.27, 0.32, 0.37, 0.42, and the coordinates are A (6.5 ms, 671 K), B (5.5 ms, 620 K), C (5 ms, 704 K), D (4.5 ms, 675.5 K). The abscissa reflects the initial ignition time, and the ordinate reflects the ignition temperature. From the change of the abscissa, it can be judged that the ignition start time shortens with the increase of the friction coefficient. From the change of the ordinate, it can be seen that the friction coefficient has little effect on the initial ignition temperature.

**Figure 13.** The influence of external friction coefficient on the temperature rise of cast PBX.

The coefficient of friction between the particle and the matrix only affects the ignition time, but has little effect on the ignition temperature. It may be due to the increase in the friction coefficient and the increase in the resistance of the interface slippage, which reduces the sliding displacement, resulting in friction at the interface. The total heat is relatively constant, and the temperature change is relatively small when the total heat is relatively constant.

3.2.3. The Influence of Friction Coefficient between Slide Column and Tablet on Ignition

In the friction-ignition experiment of the cast PBX tablets, the friction between the tablet and the spool generated higher heat. As mentioned above, the maximum temperature rises of HMX crystal particles under the pressure of 2–4 MPa are 49 ◦C, 62.2 ◦C, 72.6 ◦C and 75.6 ◦C respectively. The main reason why the cast PBX tablet is ignited by impact is the frictional heat of microcracks at the interface between the particles and the substrate. However, in addition to the internal friction of the tablet under the action of friction, the frictional heat between the tablet and the spool cannot be ignored. Based on this analysis of the influence of external friction on the ignition of the PBX, the friction coefficient f was set to 0.05, 0.25, 0.35, 0.45. The friction coefficient f was set to 0.15 in the previous paragraph. The temperature rises under different friction coefficients under 3.5 MPa and 4 MPa hydraulic pressure were analyzed, as shown in Figure 14.

**Figure 14.** (**a**) The influence of external friction coefficient f on the temperature rise of cast PBX under 3.5 MPa loading pressure; and (**b**) the influence of external friction coefficient f on the temperature rise of cast PBX under 4 MPa loading pressure.

Figure 14a shows the temperature rise curve under 3.5 MPa hydraulic pressure. It can be seen that when the friction coefficient f = 0.05, 0.45, the explosive ignites. But when f = 0.15, 0.25, 0.35, no ignition occurs. The temperature rise curves in the ignition area basically coincide, and the friction coefficient has little effect on the temperature rise of explosive particles. From the analysis of the ignition area, A (6.2 ms, 545 K) and B (6.15 ms, 668 K) are f = 0.05 and 0.45, respectively. The turning point of temperature rise, point A, is suddenly ignited from the original temperature rise, and the ignition does not conform to the general temperature-rise law. It can be judged that the explosive particles of this unit are ignited due to the explosion of other explosive particles, and point B is caused by internal interface friction. The concentrated heat flow triggers the self-heating reaction of the explosive and ignites. Through the above analysis, it can be considered that the friction coefficient f has an effect on the ignition of the cast PBX. However, it does not directly affect the friction heat to affect the ignition of the explosive but affects the ignition by affecting the internal microcrack mechanical behavior.

Figure 14b shows the temperature rise curve at 4 MPa hydraulic pressure, regardless of the friction coefficient, the particles all ignite. Among them, D (5 ms, 539 K), E (5.5 ms, 620 K), F (5.8 ms, 681.5 K), G (5.8 ms, 665 K), H (5.2 ms, 696 K) are the temperature rise

turning points corresponding to the friction coefficient. The temperature at the transition point of the temperature rise at point D is 539 K, which is less than the thermal decomposition starting temperature of 554 K. It can be judged that the explosive at this point was not ignited due to crack friction but was smashed. From the coordinates of points E, F, G and H, it is difficult to determine the ignition law. When f is 0.05–0.35, the temperature-rise curves basically overlap, and it can be considered that the friction coefficient has no effect on the temperature rise but has an effect on the ignition.

#### **4. Conclusions**

The critical loading pressure of cast PBX tablet was obtained by friction ignition experiment, and the device was designed to analyze the actual friction rate of the tablet. Then, based on the thermal–mechanical coupling algorithm, the friction ignition process of cast PBX was numerically simulated at the macro and micro scale. The critical conditions of the friction ignition of cast PBX were analyzed by numerical models at different scales. The results of the numerical simulations were consistent with the experimental results, which shows that the numerical simulation is feasible. The main conclusions are as follows:


**Author Contributions:** Conceptualization, J.Y.; methodology, J.Y.; validation, H.P., T.X. and J.L.; formal analysis, W.Z.; data curation, J.Y.; writing—original draft preparation, J.Y. and T.X.; writing review and editing, J.Y. and Y.Q.; visualization, H.S. and Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Robust Munitions Center, CAEP (No. RMC2014B03), the Bottleneck Technology and JCJQ Foundation (No. 20210579) and Special Projects of Energetic Materials (No. 20221206).

**Data Availability Statement:** The data presented in this study are openly available.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**



#### **References**


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**Jiahao Liang 1, Jianxin Nie 1,\*, Rui Liu 1, Ming Han 2, Gangling Jiao 3, Xiaole Sun 4, Xiaoju Wang <sup>5</sup> and Bo Huang <sup>5</sup>**

<sup>1</sup> State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China


**Abstract:** To study the design method and pressure relief effect of the mitigation structure of a shell under the action of thermal stimulation, a systematic research method of theoretical calculationsimulation-experimental verification of the mitigation structure was established. Taking the shelled PBX charge as the test material, the pressure relief area that can effectively reduce the reaction intensity of the charge is obtained by theoretical calculation. The influence of the pressure relief hole area, distribution mode, and other factors on the pressure relief effect is calculated by simulation. The pressure relief effect of the mitigation structure was verified by the low-melting alloy plug with refined crystal structure for sealing the pressure relief hole and the cook-off test. The research results show that the critical pressure relief area is when the ratio of the area of the pressure relief hole to the surface area of the charge is AV/SB = 0.0189. When the number of openings increases to 6, the required pressure relief coefficient decreases to AV/SB = 0.0110; When the length/diameter ratio is greater than 5, the opening at one end cannot satisfy the reliable pressure relief of the shell. The designed low-melting-point alloy mitigation structure can form an effective pressure relief channel. With the increase in AV/SB from 0.0045 to 0.0180, the reaction intensity of the cook-off bomb is significantly reduced in both fast and slow cook-off, which improves the safety of the charge when subjected to unexpected thermal stimulation.

**Keywords:** mitigation structure; pressure relief area; pressure relief effect; cook-off; low-melting crystal

#### **1. Introduction**

During the process of production, transportation, storage, and usage, ammunition could be stimulated by unexpected sources, such as fire, which will lead to violent reactions and bring about major safety accidents. Accidental thermal stimulation is one of the most common accidental excitable sources encountered in the whole life cycle of ammunition. The charge ignites and burns under thermal stimulation. Huge personnel and economic losses would occur when the shell is sealed because the temperature and pressure will rise rapidly in the confined space of the charge, resulting in a chain reaction from combustion to deflation to detonation. Aiming at the thermal safety of ammunition, the improvement methods mainly include insensitive explosive and charging technology, shell pressure relief technology, thermal shock buffer technology, and other aspects [1]. The principle of shell pressure relief technology is that when ammunition is under a certain thermal stimulation condition, a pressure relief channel is formed through the mitigated structure on the shell to relieve the internal pressure of the ammunition, the self-heating reaction rate of the charge is suddenly reduced, so the severity of charge response is reduced. The decrease in the internal temperature, the decrease in the natural reaction rate, and the convection driven by the product bubbles collectively lead to a delay in the ignition time [2]; these factors

**Citation:** Liang, J.; Nie, J.; Liu, R.; Han, M.; Jiao, G.; Sun, X.; Wang, X.; Huang, B. Study and Design of the Mitigation Structure of a Shell PBX Charge under Thermal Stimulation. *Crystals* **2023**, *13*, 914. https:// doi.org/10.3390/cryst13060914

Academic Editor: Pavel Lukácˇ

Received: 5 May 2023 Revised: 30 May 2023 Accepted: 2 June 2023 Published: 5 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

reduce the reaction intensity of the charge, achieving the purpose of improving the safety performance of the assembly.

Research on the design of mitigation structures had already begun when the United States and European countries developed insensitive ammunition in the last century. After considerable technical accumulation, some mitigation structure design technologies have been successfully applied to model ammunition [3–7]. Therefore, the pressure relief technology of designing vent holes on the shell is an effective control method to improve the thermal safety of ammunition. William et al. [8] studied the response characteristics of explosives through a fast cook-off test and measured the internal pressure of ammunition through a pressure sensor. The results show that a vent hole can effectively reduce the response intensity of the ammunition under the condition of fast cooking. Glascoe et al. [9] conducted a slow cook-off test equipped with a pressure relief hole, compared a molten Composition-B explosive with an HMX-based agglomerated explosive, and found that the size of the molten cast explosive pressure relief hole was too small, which may improve the response intensity of the ammunition. The gas pressure inside the condensed explosive cannot be discharged in the form of bubbles, and the pressure relief hole has almost no effect. Kinney [10] calculated the critical area of the pressure relief channel according to the dynamic equilibrium relationship between the pressure increase rate when the charge facilitated the combustion reaction and the pressure decrease rate when the pressure relief channel was relieved. Hakan et al. [11] calculated the critical pressure relief area of the pressure relief channel according to the dynamic transport equilibrium relationship between the mass of the gas generated during the combustion of the charge and the mass of the gas discharged from the channel. Wardel et al. [12] set up two different venting methods to leak the gas inside the explosive from the top center and the top edge and found that the cook-off bomb with the vent set at the top center had a more severe response. Niu Gongjie [13] studied the influence of different distribution modes of pressure relief holes on the dose-response intensity and pointed out that the pressure relief hole should be set near the ignition position of the charge. Madsen et al. [14] studied the cook-off characteristics of four explosives, including molten Composition-B explosive, PAX-28, PBXN-109, and PBXN-9, under different vent hole sizes by scale testing and analyzed the selection of a low-melting-point material for plugging the pressure relief hole.

However, the abovementioned studies were only verified through theoretical calculations or experiments. The area of the pressure relief channel, the size of the pressure relief hole, or the feasibility of the low-melting-point material as a mitigation structure were not considered comprehensively, and the actual shell installation was not considered. It is difficult to provide a complete basis for the design of the mitigation structure of a shell without forming a systematic research method of the theoretical calculation-simulationexperimental verification of the mitigation structure.

Therefore, in this study, the pressure relief area and the distribution of pressure relief holes through the systematic design method of the mitigation structure was investigated, the appropriate filling material was selected, and the pressure relief effect under different pressure relief conditions and shell failure strength thresholds were analyzed by means of simulation and experimentation. Taking a polymer bonded explosive (PBX) charge as the research object, slow cook-off, and fast cook-off tests were used for verification, which can provide a reference for the design of insensitive munitions mitigation structures.

#### **2. Shell Mitigation Structure Design**

The key to the design of the mitigation structure is determining the area of the pressure relief channel, the arrangement of the pressure relief holes, and the selection of the filling material for the pressure relief holes. The material filling the pressure hole should not affect the use under normal working conditions so that it cannot only perform a sufficient role in exhaust and pressure relief but also meet the requirements of the structural strength of the shell.

Taking a standard cook-off bomb as an example, the shell mitigation structure was designed. The materials of the shell and end cover are #45 steel. The shell is 240 mm long, 60 mm in inner diameter, 66 mm in outer diameter, and 3 mm in thickness, and both ends of the body were machined with 45 mm external threads. The thickness of the end cover is 5 mm, the diameter is 73 mm, the internal thread is processed, the pitch is 1.5 mm, the charge is Φ60 mm × 240 mm, the length-diameter ratio is 4:1, the PBX pressed charge is filled, its content component is RDX/Al/Viton F2602 is 65.5/30/4.5, and the charge density is 1927 kg m−3. Among them, the RDX used is Class II, 0.075~0.300 mm. Al powder is 1–2 μm.

#### *2.1. Pressure Relief Area Design*

The pressure relief area of the mitigation structure needs to be designed according to different conditions to ensure the effect of the mitigation structure. The determination of the pressure relief area mainly considers the relationship between the pressure increase rate in the shell and the pressure release rate of the pressure relief passage. According to the research of Kinney [10], the pressure growth rate of the charge in the shell when burning is:

$$\frac{dp}{dt} = \frac{RT\_B}{V} \frac{dn}{dt} = \frac{RT\_B}{V} \frac{\rho}{M} \frac{\kappa}{(A - BT\_0)} S\_B P \tag{1}$$

where *TB* ( ◦C) is the flame temperature when the charge burns; *R* is the universal gas constant, which is 8.314 J·mol−1· ◦C<sup>−</sup>1; *V* is the volume (m3); *ρ* is the density of the charge (kg m<sup>−</sup>3); *M* is the average molar mass of the gas molecules during combustion (kg mol<sup>−</sup>1); *T0* is the temperature of the charge at ignition (◦C); *SB* is the surface area of the charge (m2); *P* is the absolute charge pressure (bar); and *α*, *A,* and *B* are constants for the charge burning rate and temperature.

The pressure release rate of the pressure relief channel is calculated by the following formula:

$$-\frac{dp}{dt} = \frac{A\_V C\_D}{V} a' P\tag{2}$$

where *AV* is the area of the pressure relief hole (m2); *CD* is the exhaust coefficient, which is taken as 0.82 [15]; and *a'* is the speed of the air passing through the air hole (m/s).

Combining Equations (1) and (2), when the pressure increase rate in the shell and the pressure release rate of the pressure relief channel are equal, the calculation formula of the critical area of the pressure relief channel can be obtained as:

$$\frac{\text{AV}}{\text{S}\_{\text{B}}} = \frac{a\rho R T\_{\text{B}}}{M \text{C}\_{D} a'(A - B T\_{0})} \tag{3}$$

Equation (3) shows that the ratio of the area of the exhaust passage to the combustion area of the charge can be directly obtained from the relevant physical and chemical parameters of the charge, and the critical pressure relief area can be determined.

The molar mass of the gaseous product of the elemental explosive RDX (C3H6N6O6) in the PBX charge is [16]:

$$\mathbf{M} = \frac{56c + 88d - 8b}{2c + 2d + b} = \frac{27.2g}{mol} \tag{4}$$

According to ref. [17] of the burning rate parameter:

$$\frac{1}{r} = A - BT\_0 \tag{5}$$

In the formula, *r* is the burning speed of the charge. The other parameters were calibrated from previous studies [18] and experiments and the size of the pressure relief hole required for the ignition of PBX charges at different temperatures was calculated, as shown in Table 1.


**Table 1.** The size of the pressure relief holes required for the ignition of PBX explosives at different temperatures.

Note: \* is the ignition temperature obtained from the preliminary test.

Therefore, for a PBX charge with a charge size of Φ60 × 240 mm, the minimum AV/SB is 0.0189 when ignited at a temperature of 237 ◦C.

#### *2.2. Design of the Pressure Relief Hole Distribution*

At present, most of the pressure relief holes are designed to be distributed in the tail of the projectile, the wall of the projectile, etc. The distribution mode has an important influence on the charge reaction level, and the design of the pressure relief channel near the ignition position has a significant effect.

A numerical calculation model was established based on the cook-off experiment and the data in the literature [20], and the simulation model was a standard cook-off bomb of Φ60 × 240 mm. CFD fluent software was used to calculate the response position of the PBX charge at heating rates of 3.3 ◦C/h (0.055 ◦C/min), 0.1 ◦C/min, 0.5 ◦C/min, 1 ◦C/min, and 3.3 ◦C/min. The position distribution of the pressing holes provides the design basis. Figure 1 shows the position distribution of the ignition time. Figure 2 shows the changes in ignition position under different heating rates. When the charge material and the charge structure size are determined, the ignition position is mainly affected by the heating rate. With the acceleration of the heating rate, the reaction position first moved from the center of the charge to the two ends along the axis, then moved to the corner of the cylindrical section and the end cap.

**Figure 1.** Ignition positions at different heating rates.

**Figure 2.** Schematic diagram of the ignition position change.

Since the thermal decomposition temperature of the charge is constant when the charge reaches the decomposition temperature, the charge begins to decompose, and the released heat is transferred to the low-temperature charge and the outside of the shell. When the heating rate is low, there is more time for slow thermal decomposition to occur. At the same time, because the heat of charge decomposition is greater than the heat provided by the heating of the shell, the inward transfer of heat will lead to the decomposition of the

internal charge. In contrast, the outward transfer is relatively smooth, and the temperature increase process in the shell is mainly caused by the heat of the charge decomposition control, resulting in the continuous movement of the reaction center to the charge center. When the temperature of the reaction center reaches the ignition temperature, ignition occurs, and the reaction center at this moment becomes the ignition position. When the heating rate is fast, ignition occurs before the high-temperature point is transferred to the center of the charge. The pressure relief channel is designed near the ignition position, which is conducive to the timely dissipation of the heat generated by the decomposition and reduces the temperature of the charge.

Therefore, for the PBX charge with a charge size of Φ60 × 240 mm, the pressure relief hole design can be considered at one or both ends close to the ignition position.

#### *2.3. Design of Pressure Relief Hole Filling Material Design*

Low-melting alloys are heat-sensitive, and their mechanical strength decreases at high temperatures. Under a certain ambient temperature and internal pressure, the mitigation structure of low-melting alloys are disabled and destroyed. These alloys can be used in the starting device of mitigation structures. Design requirements for mitigation structures of low-melting alloy plugs are as follows [21]:


The slow cook-off response temperature of the PBX charge (RDX/Al/binder mass fraction of 65.5/30/4.5) used in this paper was obtained from the preliminary test of approximately 237 ◦C. Strickland et al. [22] pointed out that to effectively suppress the deflagration to detonation transition of energetic materials, the opening temperature of the pressure relief channel should be more than 60 ◦C lower than the slow cook-off response temperature; that is, the opening temperature of the pressure relief channel is approximately 177 ◦C. Therefore, the melting point of the low-melting-point alloy for the mitigation structures of the shell should be below the slow cook-off response temperature of the charge, and the mechanical strength of the low-melting alloy material is significantly reduced at a temperature of 177 ◦C. The structure is destroyed in this situation of internal pressure to open the pressure relief channel.

The melting point of the Sn-Zn binary alloy system is 198.5 ◦C, which meets the design requirements of mitigation structures. At the same time, the low-melting-point alloy must meet the requirements of temperature environment adaptability; that is, it must have good mechanical properties from −50~70 ◦C. Therefore, the Sn9Zn-3Al0.2La low-melting-point alloy with good mechanical properties was selected as the filling material of the pressure relief channel [23], and its mechanical properties are shown in Table 2. The addition of La element is to refine its crystal structure and improve its mechanical properties.

**Table 2.** Mechanical properties of Sn9Zn-3Al0.2La.


#### **3. Analysis of Factors Affecting the Pressure Relief Effect**

#### *3.1. Simulation Model*

The pressure relief process of the shell mitigation structure is a competitive process of unburned charge combustion and gas release, and different mitigation structures (such as the length/diameter ratio of the shell, the number of pressure relief holes, and the location) have a great influence on the pressure relief of confined spaces. The explosion process time of the charge is very short, the internal pressure changes greatly, and the explosion process is dangerous. The experimental research will be limited by the conditions of the site, testing methods, and safety. The simulation calculation can easily change the conditions, such as different mitigation structures, and a comprehensive analysis of the internal pressure of the shell can be conducted. The commonly used commercial software ANSYS Fluent was used to simulate the pressure relief process of the shelled charge under three-dimensional conditions. The purpose is to obtain the internal pressure changes in the shell during the pressure relief process in different sustained-release structural conditions and provide data information support for the design of mitigation structures.

The inlet mass source term can be defined as the product of the total reaction volume *Aburn*, the explosive burning velocity *rburn*, and the combustion product density *ρburn* in the shell [11]: .

$$
\dot{m}\_{\text{irlet}} = A\_{burn} r\_{burn} \rho\_{burn} \tag{6}
$$

The simple expression for the mass outflow after opening the pressure relief channel is:

$$
\dot{m}\_{\text{outlet}} = A\_{\text{vent}} \iota^\* \rho^\* \mathbb{C}\_D \tag{7}
$$

In the formula, *CD* is the exhaust coefficient, which is taken as 0.82 [15], and the other terms are solved by the following isentropic equations:

$$
\rho^\* = \frac{P\_{\text{vent}}}{RT\_{\text{vent}}} \tag{8}
$$

$$
\mu^\* = \sqrt{kRT\_{\text{vent}}}M\tag{9}
$$

$$P^\* = \frac{P\_{channeler}}{\left[1 + \frac{k-1}{2}M^2\right]^{\frac{k}{k-1}}}\tag{10}$$

$$T^\* = \frac{T\_{chamber}}{\left[1 + \frac{k-1}{2}M^2\right]^{\frac{k}{k-1}}}\tag{11}$$

A schematic diagram of the description of the exhaust gas pressure relief process of the projectile is shown in Figure 3. The pressure-rising stage of the charge ignition stage is used as the input condition, and the UDF is loaded into the software to simulate the pressure-rising process. The pressure rise process caused by the combustion of the charge is calibrated with reference to the test results [24] (the pressure curve is shown in Figure 4). Test method: During the cook-off test, a piezoelectric pressure sensor is used to measure the pressure time-history curve inside the cook-off bombshell.

**Figure 3.** Schematic diagram of the exhaust pressure relief process of the projectile. (**a**) suffer from thermal stimulation; (**b**) cook-off bomb; (**c**) thermal decomposition diagram.

**Figure 4.** Pressure rise curve during charge ignition. (A—thermal decomposition; B—explosive combustion; C—shell rupture).

#### *3.2. Simulation Results*

#### 3.2.1. Influence of the Number of Pressure Relief Holes

Figure 5 shows the variation in pressure with time when there is no mitigation structure and the critical dimension AV/SB = 0.0189. The different colored lines represent the pressure curves of different monitoring points, and the yellow stars represent the positions of the monitoring points. The curve with a sharp rise in pressure is the pressure curve designed without the mitigation structure, and monitoring points 1–7 are the critical pressure relief area pressure curve when the pressure relief channel is opened. The pressure rises when the charge starts to respond quickly is recorded as 0 times. It can be seen from the pressure curve without the mitigation structure that when the pressure relief channel is not opened, the pressure rises exponentially according to the set pressure rise rate. At 0.45 ms, the pressure reached 20.8 MPa, and finally exceeded the shell burst pressure and exploded. When the pressure relief channel is opened, when the breaking pressure of the low-meltingpoint alloy plug is 20.8 MPa, the pressures of monitoring points 1 and 2 near the pressure relief channel fluctuate rapidly, then fluctuate and rise and finally stabilize at 7.71 MPa and 17.08 MPa at 6 ms, respectively. Due to the pressure hysteresis effect, monitoring point 7 away from the pressure relief channel will first rise and reach a maximum value of 28.05 MPa at 1.07 ms, then oscillate and decrease. Then, the pressure in the shell is gradually decreased, then fluctuates. At approximately 4 ms, the pressure remains basically unchanged and stabilizes at 19.79 MPa. The pressure at other monitoring points has the same trend as monitoring point 7. The pressure first rises to a maximum value, then oscillates down due to the hysteresis effect over time, and the pressure at the last stable point is below 20 MPa, at this time, all areas in the shell exhibit equilibrium pressure relief. The timely opening of the pressure relief channel can effectively reduce the pressure in the shell, make the charge undergo a relatively stable combustion reaction, and reduce the probability of a more severe reaction. Since the pressure relief channel is a type of pressure opening, and the pressure at monitoring point 7, far from the pressure relief channel, is the innermost pressure point. If the equilibrium pressure of monitoring point 7 meets the design requirements, then other positions in the shell will also meet the design requirements. In the subsequent research and analysis of the pressure relief effect, only the pressure curve at monitoring point 7 away from the pressure relief channel was analyzed.

**Figure 5.** Pressure graph of a single pressure relief hole.

The influence of different numbers of relief holes (the number of relief holes is 1 to 6) on the pressure relief effect was studied, and the relief area at the same equilibrium pressure is shown in Figure 6. When the number of relief holes is 2, AV/SB = 0.0144 can achieve equilibrium pressure relief, and when the number of relief holes is 3~6, AV/SB is smaller, and only 0.0110 can achieve equilibrium pressure relief. This is because when the number of pressure relief holes increases, the air convection will be accelerated, the pressure will drop faster, and the degree of weakening of the shell will be reduced when the opening area is small. Therefore, the number of wells is 6, and AV/SB = 0.0110 is the control data.

**Figure 6.** Pressure relief area required to reach equilibrium pressure with different numbers of relief holes.

#### 3.2.2. Influence of Pressure Relief Area

Taking the number of openings as 6 and AV/SB = 0.0110 as the control, the influence of the area of the pressure relief holes (AV/SB are 0.0015, 0.0045, 0.0110, 0.0180, and 0.0268, respectively) on the pressure relief effect was investigated. The curve is shown in Figure 7. It can be seen that when the pressure relief channel is opened at 20.8 MPa, the pressure rise rate in the shell changes. However, because the monitoring point is located in the inner position, the pressure drop is delayed, so the pressure will continue to rise, and the pressure reaches a maximum value of 28 MPa at 1.06 ms. Then, according to the size of the pressure

relief area, the subsequent pressure will oscillate down or up. When the aperture is 11 mm, the equilibrium pressure is maintained at 19.74 MPa at approximately 4 ms. It can be seen that the area of the pressure relief channel at this time is the equilibrium pressure threshold in this condition. When the apertures are 14 mm and 17 mm, the area is in an equilibrium pressure relief state, and the equilibrium pressures at 6 ms are 18.24 MPa and 17.00 MPa, respectively. When the aperture is 4 mm and 7 mm, disequilibrium pressure relief occurs at this time, and the pressures at 6 ms are 36.93 MPa and 22.72 MPa, respectively. Then, the pressure inside the shell will continue to rise, eventually reaching the burst pressure of the shell. It can be seen that the timely opening of the pressure relief channel can effectively reduce the pressure in the shell, and with the reduction in the pressure relief area, the pressure in the shell will change from equilibrium pressure relief to disequilibrium pressure relief.

**Figure 7.** Pressure curves in different pressure relief areas.

#### 3.2.3. Influence of the Length/Diameter Ratio

The pressure curves in different length/diameter ratios (we kept the hole size unchanged at this time, and the hole size under different length/diameter ratios was 11 mm) are shown in Figure 8. It can be seen that with an increasing length/diameter ratio, the pressure hysteresis effect is more serious after the pressure relief channel is opened. When the length/diameter ratio is 4, the equilibrium pressure relief state will finally be formed, and the pressure is 19.74 MPa at 6 ms. When the length/diameter ratio is less than 4, the equilibrium pressure relief is satisfied, and the pressure is all less than 20.8 Mpa. As the length/diameter ratio decreases, the pressure in the shell at equilibrium is also smaller. When the length/diameter ratio is 5 and 8, the pressure first oscillates and drops to a certain value, then continues to rise while the pressure relief channel is opened. The rising rate is related to the length/diameter ratio. The larger the length/diameter ratio, the faster the rising rate. At 6 ms, the pressure in the shell is 23.97 MPa and 35.01 MPa, which cannot provide a good pressure relief effect, resulting in unequal pressure relief. When the length/diameter ratio is 4, it is the pressure relief threshold in this condition, and when the length/diameter ratio is greater than 4, equilibrium pressure relief cannot be formed.

**Figure 8.** Pressure curves under different length/diameter ratios.

The equilibrium pressure in different length/diameter ratios was studied, the AV/SB required to achieve equilibrium pressure relief in different length/diameter ratios was calculated, and the results are shown in Figure 9. It can be seen that with an increasing length/diameter ratio, a larger pressure relief area is needed, and the increase is approximately exponential. When the length/diameter ratio is 5, the pressure relief area has reached the maximum critical area at this time (one end can no longer open a larger aperture). When the aspect ratio is 8, the equilibrium pressure relief cannot be performed in the condition that one end is fully open, indicating that the opening of one end cannot meet the pressure relief conditions at this time, and it needs to be considered in combination with other mitigation structure designs.

**Figure 9.** Balance factor at different length/diameter ratios.

3.2.4. Influence of the Pressure Relief Hole Location

Taking the hole diameter as 11 mm and comparing the influence of the position of the pressure relief hole on the pressure relief effect in different length/diameter ratios, the pressure curve is shown in Figure 10. When the length/diameter ratio is 4, an equilibrium pressure relief will be formed, and at 6 ms, the equilibrium pressure of the openings at both ends is 11.71 MPa, which is lower than the equilibrium pressure of the openings at one end of 19.74 MPa. When the ratio is 8, one end of the hole cannot meet the pressure relief requirements, but when the two ends are opened, the pressure is 18.95 MPa at 6 ms, which meets the equilibrium pressure relief requirements. When the length/diameter ratio is too large, the pressure relief method can be adopted at both ends to relieve pressure. The reason for the analysis is that when the holes are opened at both ends, the pressure at the central position can be released from both ends at the same time, which actually reduces the length/diameter ratio of the shell. Therefore, the pressure relief effect of the holes at both ends is better than that of the holes at one end. Therefore, when the holes are opened at both ends, the disequilibrium pressure relief can be transformed into an equilibrium pressure relief in certain conditions, and the pressure originally in the equilibrium pressure relief state can be lower so that the pressure can be released faster and more effectively.

**Figure 10.** Pressure curves under different pressure relief hole locations.

#### **4. Test of Pressure Relief Hole Plugging Structure Strength**

#### *4.1. Hydrostatic Pressure in the High-Temperature Test*

Figure 11 shows the metallographic structure of six Sn-Zn-Al La alloys with different Al contents. It is found that as the Al content increases, the gray matrix is an Sn-Zn eutectic phase, while the dotted black is a rich Zn phase. With the addition of Al, the microstructure becomes coarser and coarser. Black dendritic tissue gradually increases and the microstructure distribution of each alloy phase in the eutectic alloy Sn9Zn is relatively uniform. As the Al content continues to increase, needle-like or dendritic Al phases gradually appear at grain boundaries or interdendritic boundaries. When the Al content increases to 10%, the black dendritic structure gradually decreases, and increasingly circular silver phases are formed in the alloy phase. The finer the crystal, the stronger the mechanical properties. Therefore, we chose Sn9Zn-3Al0.2La for the design of the sustained-release structure.

According to the self-developed hydrostatic pressure of the high-temperature test system [25], a comparative experiment was carried out to test the actual pressure-bearing capacity and pressure relief effect of low-melting-point alloys in a set high-temperature environment. The hydrostatic pressure in the high-temperature test system is shown in Figure 12. Three experiments were carried out, namely, the high-temperature blasting pressure of the shell without a mitigation structure and the normal-temperature/hightemperature blasting pressure of the Sn9Zn-3Al0.2La mitigation structure. The test methods are shown in Table 3.

**Figure 11.** Microstructure of Sn-Zn-Al-La with Different Al and La Content. (**a**) Sn9Zn; (**b**) Sn9Zn-0.8Al0.2La; (**c**) Sn9Zn-3Al0.2La; (**d**) Sn9Zn-10Al0.3La; (**e**) Sn9Zn-20Al0.3La; (**f**) Sn9Zn-20Al0.3La.

**Figure 12.** Schematic diagram of the hydrostatic pressure in the high-temperature test.


**Table 3.** Test method for hydrostatic pressure in a high-temperature test system.

#### *4.2. Results of the Low-Melting-Point Alloy Plug Strength Test*

The pressure time-history curve of the hydraulic experiment of the shell-simulated sample is shown in Figure 13. It can be seen from the comparison that the mitigation structure has a complete structure at normal temperature. In the temperature environment of 175 ◦C, through the design of the pressure relief diaphragm mitigation structure, the restraint strength of the simulation shell at 175 ◦C is reduced by nearly 50%, from 40 MPa to 21.21 MPa, which basically meets the start-up requirements of the pressure relief channel. It can be seen from the photos of the wreckage of the pressure relief diaphragm after the experiment that the mitigation structure loses most of its strength under the action of high temperature, fails under the action of internal pressure, and forms a pressure relief channel, which can be applied to the design of the mitigation structure of the shell. This experiment verifies that the low-melting-point alloy mitigation structure can open the pressure relief exhaust channel in advance under the action of a particular temperature environment and internal pressure, which meets the design requirements of the low-melting-point alloy plug mitigation structure.

**Figure 13.** Pressure curve of the hydrostatic pressure in the high-temperature test of the simulation shell. (**a**) simulation shell 1, without mitigation Structure; (**b**) simulation shell 2, Sn9Zn-3Al0.2La; (**c**) simulation shell 3, Sn9Zn-3Al0.2La.

#### **5. Mitigation Design and Cook-Off Test of the Cook-Off Bomb**

To verify the pressure relief effect of the designed mitigation structure, a standard cook-off bomb was designed and sealed with low-melting-point alloy plugs. The test site was arranged to conduct fast cook-off and slow cook-off tests to study the mitigation effect of insensitive munitions for reference.

#### *5.1. Design of the Mitigation Structure of the Shelled PBX Charge*

A schematic diagram of the structure of the cook-off bomb is shown in Figure 14. Its structure is mainly composed of a shell, front and rear covers, a charge column, and a low-melting-point alloy plug. Six circular holes were opened along the periphery of one end face, and four pressure relief areas were chosen for the circular holes (without the mitigation structure, AV/SB were 0.0045, 0.0110, and 0.0180, respectively). The mitigation structure design of the shell was carried out under preset working conditions. The low melting point alloy material used to seal the pressure relief channel must meet the structural strength requirements. Therefore, the pressure relief mitigation structure adopted a threaded mechanical connection. The material of the low-melting-point alloy plug is Sn9Zn-3Al0.2La, which was processed according to the size of different pressure relief channels.

#### *5.2. Test Conditions*

The layout of the test site is shown in Figure 15, and the design test is shown in Table 4. In the slow cook-off test, the bomb was heated by remote control, and the temperature rose at the rate of 1 ◦C/min until the cook-off bomb experienced combustion and an explosion reaction or the temperature reached 400 ◦C and no reaction occurred. After the test was completed, the cook-off bomb was destroyed. The prepared cook-off bomb was hung horizontally on the support frame. The center was 300 mm directly above the combustion source. Aviation kerosene and an appropriate amount of gasoline were injected into the oil tank to the specified scale line, and the electric ignition device was placed into the base and energized. At the same time, a ground field overpressure sensor was placed 5 m before and after the cook-off bomb. Place the Revealer High-Speed Camera X113, the

maximum shooting rate is 25,000 FPS, the maximum memory is 64 GB, the full resolution is 1280 × 1024, and the minimum exposure time is 1 μs, as shown in Figure 16. The response characteristics of the bomb under the state of cook-off were evaluated by the state of the cook-off bomb after the test, the deformation of the shell, and other effective verification methods. Refer to the US military standard MIL-STD-2105D "Nonnuclear ammunitions of risk assessment" [26] to determine the response level.

**Figure 14.** Standard cook-off bomb structure diagram.

**Figure 15.** The layout of the cook-off site. (**a**) Layout of fast cook-off Test; (**b**) Oil pool layout plan; (**c**) Physical image of fast cook-off test; (**d**) Layout of slow cook-off Test; (**e**) cook-off bomb layout plan; (**f**) Physical image of slow cook-off test.


**Table 4.** Cook-off bomb design.

**Figure 16.** Revealer High-Speed Camera X113.

*5.3. Results and Analysis*

5.3.1. Slow Cook-Off Test

To study the pressure relief effects of the mitigation structure in the condition of slow cook-off, the cook-off bomb S-1 without mitigation structure and the cook-off bomb S-2 with AV/SB = 1.10% and the mitigation structure were tested. Relevant experiments were carried out, and the reaction characteristics and the effectiveness of the mitigation structure were studied.

During the test, a large amount of gas was first observed in the mitigation structure cook-off bomb, as shown in Figure 17. It can be seen that the exhaust effect of the mitigation structure is obviously. Due to the burning rate of the charge in the later stage is too fast, a thrust is formed, which makes the cook-off bomb break away from the shackles of the heating belt, which stops the heating environment, as shown in Figure 17d. The charge was kept away from the fire environment, and it failed to react completely. The residual charge is shown in Figure 17e. It was further confirmed that the mitigation structure could open well and form a pressure relief channel.

**Figure 17.** Effect diagram of the mitigation structure during the slow cook-off. (**a**–**d**) Slow cook-off test process; (**e**) Residual charge.

The slow cook-off results of the cook-off bomb are shown in Table 5, and the shell fragments are shown in Figure 18. The end cover at one end of the shell was punched open after the reaction of the S-1 cook-off bomb, and the end cover at the other end was still connected to the shell. The side wall of the shell was severely torn and broken into several large fragments, there was almost no residual explosive inside, no shock wave overpressure was detected in the test, and the reaction level was deflagration. When the temperature of the S-2 cook-off bomb is approximately 193 ◦C, the shell end cap screws are punched out. During the initial reaction of the charge, the gas generated by thermal decomposition forms a thrust, which keeps the bomb shell away from the fire environment but does not react completely. The end caps at both ends of the shell are slightly deflected, but the shell is intact and not damaged, and there is residual explosive inside that has not reacted completely. No shock wave overpressure is detected, and the reaction level is combustion and below. Therefore, it can be seen that the mitigation structure can effectively reduce the internal pressure of the shell during the reaction of the charge, keep the shell away from the fire environment, effectively reducing the intensity of the thermal reaction of the PBX charge, and improve the safety of the slow cook-off of the charge


**Figure 18.** Slow cook-off test wreckage. (**a**) S-1 (without mitigation structure); (**b**) S-2 (mitigation structure).

#### 5.3.2. Fast Cook-Off Test

**Table 5.** Slow cook-off test results.

To study the pressure relief reaction of the mitigation structure in the condition of fast cook-off, relevant experiments were carried out on three kinds of mitigation structure cookoff bombs with pressure relief areas (AV/SB are 0.45% (F-1), 1.10% (F-2) and 1.80% (F-3)), and the reaction characteristics and the pressure relief effect of the mitigation structure were studied. During the test, a large amount of gas can be observed from the mitigation structure cook-off bomb, as shown in Figure 19. For the cook-off bomb with the mitigation structure, there is a clear gas flow discharged from the pressure relief channel before the reaction, which confirms that the pressure relief channel can perform a good role in pressure relief.

Video screenshots of each cook-off bomb at different times are shown in Figure 20. It can be seen that it experienced two explosions at 58 s and 71 s, respectively, while F-3 experienced two explosions at 52 s and 66 s, respectively. This is because the response of the cook-off bomb is divided into two stages. In the first stage, after the internal pressure of the charge reaches the pressure threshold of the alloy plug, the alloy plug is destroyed, the gas product breaks through the mitigation structure, the pressure relief channel is opened, and a large-scale fireball is formed, which lasts for approximately 13~14 s. In the second stage, the area of the pressure relief channel formed by the mitigation structure is small,

210

and the pressure release rate of the pressure relief channel is less than the increased rate of the pressure in the shell, resulting in a secondary explosion. The upper-end cover is slightly deformed to create a deflection until the charge in the shell is completely burned. The F-1 cook-off bomb only exploded once at 65 s because the pressure relief area is small, so the pressure release rate is much smaller than the pressure growth rate in the shell, which causes direct damage to the shell, as shown in Figure 20a at 70 s. The shell fragments impacted and destroyed the oil sump. Therefore, when AV/SB is more than 1.10%, it can be clearly observed that the pressure relief channel is open, while when AV/SB is 0.45%, the pressure relief area is small, and the pressure relief cannot be fully discharged.

**Figure 19.** Effect diagram of the effect of the mitigation structure in the process of fast cook-off. (**a**–**f**) Fast cook-off test process.

**Figure 20.** Video screenshot of the response process of the cook-off bomb. (**a**) F-1 video screenshot of fast cook-off response process; (**b**) F-2 video screenshot of fast cook-off response process; (**c**) F-3 video screenshot of fast cook-off response process.

The fast cook-off results of the cook-off bomb are shown in Table 6, and the shell fragments are shown in Figure 21. The difference in the reaction time of different mitigation structures is very small, and the reaction levels are also combustion, but the intensity of combustion is different. The shell of the F-1 cook-off bomb is punched open, the two end caps are washed away, the shell is torn and deformed, and the reaction level is deflagration. The end cap screws of the F-2 cook-off bomb are punched out, the shell is completely without tearing occurs, the end caps at both ends are deformed, and the reaction level is combustion. The end cap screws of the F-3 cook-off bomb are punched open, the shell is intact, and no tearing occurs. The end cap at the end with the pressure relief channel is slightly deformed, and the reaction level is burning. Deflagration and the following reactions all occur for cook-off bombs with mitigation structures. When the pressure relief area increases, although the combustion reaction occurs, the severity of the reaction gradually decreases from the rupture of the product on site and the video. This shows that the increase in the pressure relief area has a certain effect on reducing the reaction level of the ammunition, which can improve the thermal safety of the ammunition.

**Table 6.** Fast cook-off test results.


**Figure 21.** Cook-off bomb wreckage. (**a**) F-1 Cook-off bomb wreckage; (**b**) F-2 Cook-off bomb wreckage; (**c**) F-3 Cook-off bomb wreckage.

Figure 22 shows the wreckages of the low-melting-point alloy plug with a diameter of 14 mm in the F-3 cook-off bomb after the test. The alloy plugs of the F-1 and F-2 cook-off bombs were not found. The overall structure of the alloy plug is basically intact. The reason why the screw is ejected is because the high temperature softens the alloy plug and reduces the mechanical strength. During the action of the internal pressure of the shell, the thread fails and is damaged, and the pressure relief channel is opened to achieve effective pressure relief inside the shell. The damage to the screw cap is caused by spraying out during the first explosion and hitting the witness board or other objects, and experiencing damage. In summary, the above phenomena show that the designed mitigation structure can reliably relieve pressure during the cook-off process and improve the thermal safety of the charge.

**Figure 22.** Wreckages of 14 mm alloy plug after fast cook-off.

#### **6. Conclusions**

In this paper, the design method of the mitigation structure is established, the influence of different mitigation structure designs on the pressure relief effect is simulated and calculated, and the mitigation structure is designed and verified by the cook-off experiment. The study found:


The work of this paper provides a reference research method for the design of the mitigation structure of a shelled charge under the action of thermal stimulation and the mitigation structure of the shell.

**Author Contributions:** Conceptualization, J.N. and R.L.; methodology and formal analysis, J.L.; investigation, J.L. and M.H.; resources, G.J.; data curation, X.S.; writing—original draft preparation, J.L.; writing—review and editing, J.N.; supervision, X.W. and B.H.; funding acquisition, J.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China [grant numbers 11772058] (Jianxin Nie).

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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