Shock Initiation Experiments with Modeling on a DNAN Based Melt-Cast Insensitive Explosive

Feiho Mio , Dndn Li ,*, Yngfn Cheng , Junjiong Meng , Lin Zhou

a School of Chemical Engineering, Anhui University of Science and Technology, Huainan 232001, China

b Composite Explosive Research Department, Xi'an Modern Chemistry Research Institute, Xi'an 710065, China

c State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

Keywords:2,4-Dinitroanisole (DNAN)Shock initiation Insensitive explosives Ignition and growth model Equation of state (EOS)

ABSTRACT 2,4-dinitroanisole (DNAN) is a good replacement for 2,4,6-trinitrotoluene (TNT) in melt-cast explosives due to its superior insensitivity.With the increasing use of DNAN-based melt-cast explosives, the prediction of reaction violence and hazard assessment of the explosives subjected to shock is of great significance.This study investigated the shock initiation characteristics for a DNAN-based melt-cast explosive, DHFA, using the one-dimensional Lagrangian apparatus.The embedded manganin gauges in the apparatus record the pressure histories at four Lagrangian positions and show that shock-todetonation transition in DHFA needs a high input shock pressure.The experimental data are analyzed to calibrate the Ignition and Growth model.The calibration is performed using an objective function based on both pressure history and the arrival time of shock.Good agreement between experimental and calculated pressure histories indicates the high accuracy of the calibrated parameters with the optimization method.

1.Introduction

The insensitive munitions require explosives to be sufficiently insensitive to unpredictable stimuli during manufacture, storage and transportation [1].The requirement accelerates the development of modern explosives research,notably insensitive explosives.The DNAN (2,4-dinitroanisole) based melt-cast explosive is a promising insensitive explosive[2]and has been widely used over the past decade,such as the PAX[3]and IMX[4]explosive family in the USA,and the ARX[5]explosive family in Australia.Therefore,it becomes necessary to assess the hazards and performance of DNAN based melt-cast explosives.Shock is a main inadvertent stimulus to explosives in service and may initiate violent reactions, or even detonation.The shock-to-detonation transition in explosives results from hot spots[6].Various models have been developed based on one or more hot-spot formation mechanisms, such as Ignition and Growth [7], the Wescott-Stewart-Davis model [8], Computational Reaction Evolution dependent on Entropy (S) and Time(CREST) model [9] and the Duan-Zhang-Kim model [10].These models divide the shock-to-detonation transition into three phases,unreacted explosive, detonation products and partially reacted explosive[11],and characterize the three phases by two equation of states (EOSs) and a reaction rate equation.The models adopt the similar procedures while these use different forms of reaction rates or EOSs.

The Ignition and Growth model may be the most widely used one and can be used to reproduce numerous experimental data sets on shock initiation and detonation [12].First utilization of the model needs to calibrate its parameters.The parameters of the Ignition and Growth model can be obtained from shock initiation experiments, where the Wedge test [13] and the Lagrangian test[14] are commonly used.Wedge test can give the run distances to detonation for several input shock pressures,i.e.,the POP plot[15].This plot provides enough shock initiation data for the calibration of the model.Lagrangian test embeds manganin or particle velocity gauges [16,17] in explosives at several positions.After the shock wave passes through it,the gauge moves with the explosive,so the tests are termed “Lagrangian”.The manganin and particle velocity gauges record the pressure and particle velocity histories at corresponding Lagrangian positions, respectively.The histories can characterize the growth of the shock front, and also provides abundant information on the flow field behind the shock front.Therefore,the Lagrangian test just needs a few,or perhaps even just one shot to calibrate the model.The shock initiation characteristics of several DNAN-based melt-cast explosives [18-20] have been reported based on the Lagrangian test.

The present study aims to investigate the shock initiation characteristics of a DNAN-based melt-cast explosive, DHFA,and to calibrate its Ignition and Growth model.Section 2 carries out Lagrangian tests for DHFA and gives a detailed description of the experimental apparatus.Section 3 presents the calibration methods and the parameters of the Ignition and Growth model.Finally, the Ignition and Growth model calculations are compared with the experimental data, and some important conclusions are drawn.

2.Shock initiation experiments

The shock initiation experiments were performed using the one-dimensional Lagrangian apparatus, which is similar to previous experiments [19].Fig.1 is the schematic of the experimental apparatus consisting of a plane wave lens, a TNT booster, an aluminum gap, manganin gauges, and disks of the DHFA samples.The plane wave lens, TNT boosters and samples are 50 mm in diameter.In experiments, a planar shock wave produced by the explosive lens successively propagates over booster, air gap and aluminum gap, and eventually into the samples.The pressure transmitted to the samples can be adjusted by changing the thickness of air gap and/or aluminum gap.In this work, the input shock pressure was adjusted only through the thickness of air gap.

Fig.2.Teflon encapsulated manganin gauge.

Manganin gauges were used to record the pressure histories at various distances into the sample.The manganin gauges are sandwiched between sample discs.The thickness of four sample discs are 3 mm,5 mm,5 mm and 20 mm and four manganin gauges are placed at Lagrangian positions of h1 (0 mm), h2 (3 mm), h3(8 mm), and h4 (13 mm).As shown in Fig.2, the manganin gauge has 2 input and 2 output leads and all leads emerge from one side to simplify the assembly process and the gauge is encapsulated in 0.2 mm of insulation to prevent gauge failure.The resistance of Teflon is several orders of magnitude higher than that of the active manganin elements in a fully detonating explosive and Teflon provides a good shock-impedance match to explosives.Therefore,Teflon was selected as insulation for the manganin gauges.

Two experiments were carried out with different thicknesses of air gap using the one-dimensional Lagrangian and the pressure histories measured at h1 (0 mm), h2 (3 mm), h3 (8 mm), and h4(13 mm) are shown in Fig.3.The DHFA is an insensitive explosive and needs high initiation pressure.The thickness of air gap was determined based on our previous shock initiation experiments to ensure incident shock growth.Fig.3 shows the growth of reaction in the two shots and the clean hole in the steel witness plate,shown in Fig.4,indicates the shock to detonation transition.The reduction of the thickness of air gap from 6 to 4 mm increases the input shock pressure about 1 GPa,which induces a faster reaction growth and a higher shock velocity.The arrival time of shock at h4 reduces from 3.00 to 2.77 μs as shown in Figs.3(a) and 3(b), respectively.

Fig.1.Schematic of the one-dimensional Lagrangian apparatus.

Fig.3.Pressure histories in DHFA: (a) Shot 8961 (The air gap 6 mm thick); (b) Shot 8962 (The air gap 4 mm thick).

Fig.4.The steel witness plates of shot 8961 (right) and shot 8962 (left).

3.Ignition and Growth modeling

3.1.Unreacted equation of state

In the Ignition and Growth model,the Jones-Wilkins-Lee(JWL)equation of state is used for both unreacted explosives and detonation products.The JWL can be written as

where p is pressure,V =v/v0is the relative volume,and E =e/v0is internal energy per unit initial volume.A, B, R1, R2and ω are all parameters to be calibrated.The parameters A and B have dimensions of pressure, while the other parameters are dimensionless.The unreacted JWL parameters can be calibrated by Hugoniot obtained from impedance matching experiments.Hugoniot for most materials can be expressed as a linear relationship between shock velocity D and particle velocity u

where c0and s are material-dependent parameters.If the D- u relationship of each component is known, c0and s of the mixture can be obtained by mass averaging [21],

where x is mass fraction, ρ0is initial density, Г0is the Grüneisenparameter, and the subscript i represents the i th component.The mass fraction and corresponding parameters of each component of the DHFA are shown in Table 1.

Table 1 The EOS parameters for DHFA and its components.

Eq.(3) was used to calculated the parameters of DFHA and the results are listed in Table 1.In the process of melting and casting,the explosive inevitably contains holes, microcracks and other defects.Therefore,the DHFA sample density,1.872 g/cm3,is less than the theoretical density calculated in Table 1.The D-u relationship of the low density explosive can be obtained according to the characteristics of porous materials.The porosity of a porous material can be expressed as m = v00/v0, where v00and v0are initial specific volume of porous and solid materials, respectively.The Hugoniot of porous and solid materials has the following relationships [21]：

where superscripts“p”and“s”refer to porous and solid materials,respectively.Applying Rankine-Hugoniot relations to both porous and solid materials, the shock velocity and particle velocity in the porous materials can be written as follows:

Given a current specific volume, the corresponding shock velocity and particle velocity in the porous material can be calculated by Eqs.(5)-(7).With a series of u and D,the c0and s were fitted for the DHFA samples with initial density 1.872 g/cm3, as shown in Fig.5.The Hugoniot in Fig.5 can be represented in p-v plane and provides high-pressure data for the calibration of JWL parameters.The expression of JWL in p-v plane was obtained by combining Hugoniot equation and Eq.(1) to eliminating E.

Fig.5.Shock velocity D versus particle velocity u for the DHFA.

3.2.Product equation of state

The JWL for detonation products can be constructed by a thermochemical code.The thermochemical code only needs some basic information such as density, enthalpy of formation and elemental composition of explosives to calculate the thermodynamic equilibrium states.With the calculation capability to the state points,the thermochemical code can predict the isentrope and Hugoniot of detonation products and calibrate the JWL parameters.To predict the state curves we developed a thermochemical code, TCHEM,which uses the minimization of Gibbs free energy technique to solve the thermodynamic equilibrium.The TCHEM code have incorporated several EOSs,including BKW[29],KHT[30],VLW[31],LJD [32] and JCZ [33], and been validated with experimental data for well-known explosives.The detonation performance of a series of DNAN based melt-cast explosives were calculated by those EOSs to select a suitable EOS for the DHFA.The predicted results and my co-author’s experimental data[34] are compared in Fig.7.

Fig.6.The Hugoniot of unreacted DHFA in p-v plane.

Table 2 The JWL parameters for unreacted explosive.

The experimental values are listed with red dotted lines.The Xaxis label is explosive formulation where HMX, RDX, Al represent octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine, hexahydro-1,3,5-trinitro-1,3,5-triazine and aluminum powder, respectively.As shown in Fig.7, BKW overestimates the detonation velocity and pressure, while VLW, LJD and JCZ underestimate the detonation performance.KHT gives more accurate prediction and the predicted detonation velocity and pressure are shown to be within 1%and 2%of experimental values for the explosives without aluminum powder.The calculation for aluminized explosives in Fig.7 takes the influence of partial reaction of aluminum into account.In the calculation,the reaction fraction of aluminum is adjusted in steps of 5%until the difference between the calculated detonation heat and the experimental value reaches a minimum.The detonation pressure and velocity of the aluminized explosives are calculated with the reaction fraction using this method.Fig.7 shows that KHT also gives a more accurate prediction for the aluminized explosives.

Therefore, KHT was used to calculate the state curves of DHFA.The detonation velocity and heat of DHFA were measured as 7058 m/s and 7027 kJ/kg,respectively.KHT calculates the reaction fraction of aluminum as 63%, which was determined with the method used in Fig.7.The corresponding detonation velocity is 7009 m/s and it is in good agreement with the experimental value.Furtherly, the isentrope and Hugoniot of DHFA were calculated by KHT and the JWL parameters were calibrated based on the state curves as shown in Fig.8.The product JWL parameters are listed in Table 3.

Fig.7.Detonation velocity and pressure of DNAN based melt-cast explosives.

Fig.8.The isentrope and Hugoniot of DHFA.

Table 3 The product JWL parameters.

3.3.Reaction rate equation

In the Ignition and Growth model, the reaction rate equation is written as

where F is the fraction reacted,I,b,a,x,G1,c,d,y,G2,e,g,z,Figmax,FG1maxand FG2minare all parameters to be calibrated.The three terms on the right-hand side represent the three stages of reaction observed in shock initiation experiments and modeling and can be turned off/on by Figmax, FG1maxand FG2min.The reaction rate parameters can be calibrated by shock initiation experiments in Section 2.LS-OPT software [35] was used for the calibration, and the process is shown as Fig.9.The shock initiation modeling and the expression of objective function are the significant parts in the optimization process.The shock initiation was simulated by the one-dimensional hydrodynamic code, Dynamic1D [11], and the schematic of Dynamic1D model is shown in Fig.10.

The simulations take the insulation of encapsulated gauges into account and its state is described by Grüneisen EOS.

where μ=(ρ/ρ0-1)and other parameters are list in Table 4.DHFA uses the ignition growth model where the JWL parameters for unreacted explosive and detonation products have been calibrated and listed in Tables 2 and 3 The experimental pressure history in h1 is taken as the boundary condition.When a simulation gives a normal termination, Dynamic1D outputs the pressure history and the arrival time of shock.The output is used to construct the objective function.

The objective function describes the deviation of the calculated values from the test values, which is defined in the "Composites"part(see Fig.9).Parameter calibration is the process of minimizing the objective function through optimization algorithm.Therefore,a reasonable objective function is necessary to obtain an accurate calibration.The deviation of experimental and calculated pressure history in shock initiation is generally used as the objective function.Gambino et al.[37]shows that an objective function based on both pressure history and the arrival time of shock gives a more accurate calibration.We also took the arrival time of shock into account and defined the objective function as

Fig.9.The flow chart of parameter calibration.

Fig.10.Dynamic1D model for the shock initiation experiments.

Table 4 The parameters of Grüneisen EOS [36].

where φpand φtrepresent the objectives corresponding to pressure history and the arrival time of shock, respectively, and w is the weight factor.Eq.(11) shows that the calibration is a multiobjective optimization in nature and its physical meaning is shown in Fig.11.As shown in Fig.11, φpuses the experimental pressure history as target curve and is defined as the area between the target curves and simulation curves.

where the superscript “^” denotes simulation values, the subscript“ij” indicates values recorded by j th gauge of i th experiment,and tenddenotes the time of gauge failure.

Our previous research[22,38]indicates that there is a L-shaped or V-shaped stage in the pressure history, shown in Fig.12, due to the impedance mismatch between Teflon and explosives.The time interval of the traditional objective function is[0,tend]including the stage, which may result in undesired calibration.The duration of the stage is about 0.1 μs based on our experiments and simulations.Although it varies with shock pressure and impedance of explosives,the duration does not exceed 0.2 μs for most shock initiation experiments.Therefore, φpsets the time interval as [0.2 μs,tend]corresponding to the limits of integration in Eq.(12).

Table 5 The weight factors in objective function.

Minimizing φpensures that simulation curves converge to target curve in the y-axis direction.A reasonable objective is necessary to avoid undesired shift of simulation curves in the x-axis direction as shown in Fig.11.φtis defined as

where τ is the arrival time of shock.The experimental pressure history in h1 is taken as the boundary condition, and so φpand φtexcludes it.Eqs.(11)-(13) construct the objective function.The weight factors in those equations are determined using a trial and error approach and listed in Table 5.The multi-objective optimization was performed using genetic algorithm.The final calibrated reaction rate equation parameters are listed in Table 6.Tables 2, 3 and 6 give the all parameters of Ignition and Growth model for DHFA.With the parameter sets, a good agreement between experiment and modeling is obtained for the two experiments, as shown in Fig.13.

Fig.11.Schematic of the objective function.

Fig.12.The L-shaped and V-shaped stages in the pressure histories: (a) L-shaped stage; (b) V-shaped stage.

Table 6 The parameters of reaction rate equation.

Fig.13.Pressure histories in DHFA: (a) Shot 8961 (The air gap 6 mm thick); (b) Shot 8962 (The air gap 4 mm thick).

4.Conclusions

Two experiments were carried out with different thickness of air gap using the one-dimensional Lagrangian for DHFA and the pressure histories at four Lagrangian positions were measured by manganin gauges.The Ignition and Growth model was used to reproduce the experimental results and its parameters were determined based on experimental data.The unreacted JWL parameters were calibrated to the Hugoniot porous mixture, which was obtained by applying Rankine-Hugoniot relations to both porous and solid materials.We developed TCHEM thermochemical code to construct product JWL.TCHEM’s calculations indicate that KHT gives more accurate prediction for a series of DNAN based melt-cast explosives.Therefore, TCHEM predicted the isentrope and Hugoniot of DHFA using KHT and the product JWL parameters were calibrated to the state curves.For the calibration of reaction rate parameters, an objective function based on both pressure history and the arrival time of shock was built to avoid the influence of L-shaped or V-shaped stage in the pressure history.With these parameters, a good agreement between experimental and calculated pressure histories is achieved for two experiments.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by Scientific Research Foundation for High-level Talents of Anhui University of Science and Technology(Grant No.2021yjrc38), Anhui Provincial Natural Science Foundation (Grant No.2208085QA27) and National Natural Science Foundation of China (Grant Nos.11972046, 12002266), and the authors would like to thank these foundations for financial support.