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Shock initiation of explosives investigated with small partition experiment and numerical simulation
Acta Mechanica Solida Sinica, Vol. 26, No. 4, August, 2013 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
SHOCK INITIATION OF EXPLOSIVES INVESTIGATED WITH SMALL PARTITION EXPERIMENT AND NUMERICAL SIMULATION Weijun Tao1
Shi Huan1
Fenglei Huang1,2
Guoping Jiang1
1
( Earthquake Engineering Research Test Center of Guangzhou University, Guangzhou 510405, China) (2 State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China)
Received 18 February 2011, revision received 14 June 2012
ABSTRACT In order to investigate the shock ignition of high energy solid explosives by shock waves, we carry out Lagrangian experiments with 2-D Lagrangian technique which uses composite manganin-constantan (CMC). The effects of the shock sensitivity of pressed solid high explosives, TNT, and the effect of the lateral rarefaction wave were studied. Based on the measured pressure histories and the radial displacements, we formulate the Ignition and Growth reactive flow models for the pressed TNT. The shock initiation process simulated by Ignition and Growth model agreed well with experimental data. This pressed TNT model can be applied to shock initiation scenarios which are highly unpredictable and have not been or cannot be tested experimentally.
KEY WORDS shock waves, explosive, Lagrangian analysis, state equation, rarefaction wave
I. INTRODUCTION Since safety issues play a dominant role in present-day energetic material technology, the concern on the relative safety of the shock initiation of solid high explosives is increasing. Hazardous scenarios can involve multiple stimuli, such as flyer plates accelerated impact explosives, producing strong shock waves in explosive or decaying to a deflagration wave or to a non-reacting wave. So the study of the property of energetic materials is very important to predicting whether a material will detonate or not. Previous research[1–6] using unconfined or weakly confined charges of LX-04-01 and LX-17 showed that the increase in shock sensitivity was primarily due to two effects: the increase in the number and size of reacting hot spots formed by shock compression and the increase in hot spot growth rate into the surrounding explosive particles. The lateral effects always decrease the shock waves especially for the shock initiation of small size. Since the traditional 1-D measuring method is adopted to measure the pressure of the small size of explosives during shock initiation, the measured pressure is not real, but the overall performance of the pressure and radial displacements. So, it is extremely important to consider the impact of lateral rarefaction wave. In this paper, the research by using unconfined charges of pressed TNT of small size was carried out, and Ignition and Growth reactive flow model for pressed TNT was formulated based on the measured pressure histories and radial displacement histories.
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 10972060) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20104410110003).
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II. EXPERIMENTS The 1-D planar conditions for shock initiation are difficult to satisfy since it is too expensive. The 2-D Lagrangian gauge was used, which is made of manganin and constantan, and the related analytical techniques[7, 8] were also adopted in earlier research. But the gauges were only embedded in the axis of symmetry for different Lagrangian positions. As the stretch effect was not studied before, the lateral effects were not analyzed. In this paper, the pressure and radial displacements at four Lagrangian positions were measured by using two 2-D Lagrangian gauges which were embedded in the same Lagrangian position with different radial positions. The distribution of two 2-D CMC Lagrangian gauges at the same Lagrangian position is shown in Fig.1. Gauge 1 was embedded in the axisymmetric position, and gauge 2 was embedded in the position whose distance to the axisymmetric position is r0 .
Fig. 1. The distribution of two 2-D CMC Lagrangian gauges at the same Lagrangian position. Fig. 2. Experimental set-up for small partition experiment.
The schematic diagram of experiment is shown in Fig.2. The donor explosive charge is TNT, and the diameter is about 20.0 mm. The acceptor explosive charge is pressed TNT consisting of several discs of 2-4 mm thickness and about 20.0 mm diameter, the density is ρ0 = 1.58 g/cm3 . The 2-D Lagrangian gauges were embedded in four Lagrangian positions, which are 0.0 mm, 3.01 mm, 5.14 mm, and 8.11 mm, respectively. And two radial positions, which are located on-axis at r0 = 0 mm and off-axis at r0 = 11.25 mm, respectively. The initial pressure amplitude can be controlled by the thickness of copper partition. For the relationship of thickness and pressure see our earlier studies[7] . So the copper partition is made into different thicknesses which are 5.0 mm, 5.5 mm, 6.0 mm, 6.5 mm, and 7.0 mm, respectively. The small partition experiment is the most common type of shock initiation test. The copper partition thickness is varied until the critical thickness at which 50% of the acceptor charges detonate is determined. Half of the critical detonation thickness measured for pressed TNT in the experiment is 5.5 mm[9] . Figure 3 shows that the 2-D CMC Lagrangian gauges invented by Shi Huan are more complex than 1-D gauges. The details of the 2-D CMC gauges were described in our previous publications[7, 8] . The signals can be obtained by oscilloscopes and can be changed into pressure by using the following formula[10] :
2 ΔR ΔR + 1.07 (1) P = 0.27 + 34.4 R0 R0 where P is the pressure, ΔR is the change of the resistance, R0 is the initial resistance. During the experiment, oscilloscopes measure the change of voltage caused by the resistance change in the 2-D CMC Lagrangian gauges, and were then converted to pressure using formula (1). Figures 4 and 5 show the pressure records and relative radial displacements of a shock loaded with high explosive pressed TNT. The solid lines denote data located on-axis at r0 = 0 mm, and the dashed lines denote data located off-axis at r0 = 11.25 mm.
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Fig. 3. The 2-D composite manganin-constantan Lagrangian gauges.
III. LAGRANGIAN METHOD The record of the 2-D Lagrangian method is used to calculate the reactive flow field in tested materials. The experimental temporal curves are used to calculate the value-time relations of particle velocity, relative specific volume and internal energy per unit volume at each Lagrangian position. The relationship between experiment, theoretical model and the numerical simulation is established based on the Lagrangian method. The two-dimensional conservation equations are as follows[10] :
t2 2 l ∂P 1 u = u1 − dt (2) ρ0 t1 l0 ∂h t 2 l ∂u dt (3) v = v1 + l ∂h t 0 t1 t2 ∂v E = E1 − P (t) dt (4) ∂t h t1 where ρ0 is the density, u the particle velocity, u1 the particle velocity of shock front, v the relatively specific volume, v1 the relatively specific volume of shock front, E the internal energy per unit volume, E1 the internal energy per unit volume of shock front, and t1 , t2 the start time and end time, respectively, h the Lagrangian position, l the radial displacement, l0 the length of sensitive part, and l/l0 the relative radial displacement. The path lines and particle line method can be used to avoid the loss of useful information when integration along isochrones lines is made. The analogical points[11] (e.g. the characteristic points on the waves such as the end point of elastic wave, the peak point of plastic wave etc.) of the pressure-time curves are synthesized to establish the path lines. The particle lines are the curves of the parameters varying with the time recorded by the Lagrangian gauges. The integral along isochrones lines can be changed along the path lines and particle lines: dt dp ∂p ∂p = − (5) ∂h t dh j ∂t h dh j ∂u dt du ∂u = − (6) ∂h t dh j ∂t h dh j So, Eqs.(2), (3) and (4) can be written as follows:
t2 2 ∂P l ∂t 1 ∂P dt (7) − u = u1 − ρ0 t1 l0 ∂h j ∂t h ∂h j
t2
v = v1 + t1
E = E1 −
2 ∂u l ∂t ∂u dt − l0 ∂h j ∂t h ∂h j
t2
P (t) t1
t2
∂v ∂t
(8)
dt h
(9)
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The experiment of cast TNT has been carried out by Shi Huan[7] . The particle velocity, relative specific volume and specific internal energy histories are all obtained from Eqs.(7), (8) and (9) with integral along the times. The integral along isochrones lines changed to along the path lines and particle lines with no other assumptions by Eqs.(5) and (6). The error mainly stems from the fitting between experiment and the curves. The least square curve fitting method is adopted with the B-spline function as test function to prevent the error diffusing to the whole flow field. The details of the error analysis of the 2-D Lagrangian method were described in our previous publications[7–9] .
IV. SINGLE-TEMPERATURE MODEL The single-temperature model with mixture laws is used to calculate the mixture of unreacted explosive and reaction products defined by the fraction reacted F (F = 0 implies no reaction, F = 1 implies complete reaction). The temperature and pressure are assumed to be equal and the relative specific volumes are additive, i.e., Pm = Ps = Pg
(10)
Tm = Ts = Tg
(11)
vm = (1 − F )vs + F vg
(12)
em = (1 − F )es + F eg
(13)
where subscript ‘m’ represents the mixed state, subscript ‘s’ represents the un-reacted state and subscript ‘g’ stands for the reacted state.
V. EQUATION OF STATE The model uses two Jones-Wilkins-Lee (JWL) equations of state as a function of temperature. One is for the un-reacted explosive and the other for its reaction products. P = Ae−R1 V + Be−R2 V +
ωCV T V
(14)
A JWL equation of state defines the pressure in the un-reacted explosive as Pe = Ae e−R1 Ve + Be e−R2 Ve +
ωe CVe Te Ve
(15)
where Pe is the pressure in megabars, Ve and Te are the relative volume and temperature, respectively, of the un-reacted explosive, Ae , Be , R1 , R2 are constants, CVe is the average heat capacity of un-reacted high explosive and ωe is Gruneisen coefficient. Another JWL equation of state defines the pressure in the reaction products as Pp = Ap e−R3 Vp + Bp e−R4 Vp +
ωp CVp Tp Vp
(16)
where Pp is pressure in megabars, Vp and Tp are the relative volume and temperature, respectively, of the un-reacted explosive, Ap , Bp , R3 , R4 are constants, CVp is the average heat capacity of reaction products and ωp is Gruneisen coefficient. From Eqs.(10)-(13), (15) and (16), the reaction rate histories can be calculated, and used to determine the parameters of the Ignition and Growth models.
VI. EXPERIMENTAL RESULTS AND ANALYSIS By making two-dimensional experiments and using the above-mentioned theory, the pressure and radial displacement on the axis of symmetry (r0 = 0 cm, solid lines) and off-axis (r0 = 1.125 cm, dashed lines) are processed with four gauges recording, respectively.
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Fig. 4. A typical pressure gauges record for pressed TNT material.
Fig. 6. Particle velocity histories of pressed TNT.
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Fig. 5. A typical relative radial displacements record for pressed TNT material.
Fig. 7. Relative specific volume histories of pressed TNT.
The pressure and radial displacement curves shown in Figs.4 and 5 are measured with the twodimensional Lagrangian experimental set-up (Fig.2). Figure 4 shows that the shock wave reaches the axis of symmetry more quickly than the same Lagrangian position off-axis (r0 = 1.125 cm). The particle velocity histories, relative specific volume histories and internal energy per unit volume histories (Figs.6, 7 and 8) are obtained by the Lagrangian method. Each curve in Figs.3-7 typically represents that the axis of symmetry (r0 = 0 cm) has been slightly affected by lateral rarefaction wave while the off-axis (r0 = 1.125 cm) has been seriously affected. The more reaction happens, the greater time difference of arriving time of the shock front for two gauges at the same Lagrangian positions. Owing to the lateral effects, the pressure on the axis of symmetry is also higher than off-axis (r0 = 1.125 cm). The radial displacement off-axis (r0 = 1.125 cm) cannot be ignored and is actually very large. Owing to the lateral
Fig. 8. Specific internal energy histories of pressed TNT.
Fig. 9. Fraction reacted histories of pressed TNT.
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rarefaction waves, the reaction zone expanded and the particle velocity as well as the arriving time to the same Lagrangian positions increased. But the sound velocity and detonation velocity decreased. The slope of reaction rates is larger on the axis of symmetry than off-axis (r0 = 1.125 cm). The fraction reacted histories are shown in Fig.9.
VII. IGNITION AND GROWTH MODELING The impact of lateral rarefaction wave was discussed in the previous section. The existing parameters of the Ignition and Growth are mainly calibrated by 1-D tests without considering these effects. But for small partition experiment, the effects of lateral rarefaction wave should be taken into account. The Ignition and Growth reactive flow of shock initiation and detonation of solid explosives has been incorporated into several hydro-dynamic computer codes and used to solve many explosive, propellant safety and performance problems. For example, Green et al.[12] and Kury et al.[13] using the Ignition and Growth model for cast TNT would complement the model. Paul[14] using the model for Composition B and C-4 explosives, and the Composition B model was then tested on short pulse, initiation experiment, and projectile impact shock experiments with good results. The reaction rate, F˙ , is given by (17) F˙ = I(1 − F )b (ρ/ρ0 − 1 − a)x + G1 (1 − F )c F d P y + G2 (1 − F )e F g P z (12) where F is the fraction reacted, t is the time, ρ is the current density, ρ0 is the initial density, and I, G1 , G2 , a, b, c, d, e, g, x, y, and z are constants. As explained in previous papers, this rate law with three terms models the three stages of reaction generally observed in shock initiation of heterogeneous solid explosives. The first term represents the ignition of the explosive as it is compressed by a shock wave creating heated areas (hot spots) as the voids in the material collapse. The second term in Eq.(17) represents the growth of reaction from the hot spots into the remaining solid. During shock initiation, this term is responsible for the relatively slow spreading of reaction in a deflagration-type process of inward and outward grain burning. The third term in Eq.(17) describes the rapid transition to detonation observed when the growing hot spots begin to coalesce and transfer large amounts of heat to the remaining un-reacted explosive particles causing them to react very rapidly. In this paper, the Ignition and Growth model parameters of pressed TNT unconfined by small sizes were then calibrated on existing experimental data with good results. The goodness of fit R2 is 0.98. The model parameters for pressed TNT are listed in Table 1. Table 1. Ignition & growth parameters for pressed TNT
UNREACTED JWL PRODUCT JWL A = 17.98 MBar A = 3.712 MBar B = −0.9310 MBar B = 0.03231 MBar R1 = 4.15 R1 = 6.2 R2 = 3.1 R2 = 0.95 ω = 0.8926 ω = 0.3 Cv = 1.0 × 10−5 MBar/K Cv = 2.1 × 10−5 MBar/K T0 = 298 K E0 = 0.07 MBar Shear Modulus= 0.04 MBar Yield Strength= 0.002 MBar REACTION RATES a = 0.065 x=6 b = 0.667 y=1 c = 0.667 z=3 d = 0.667 Figmax = 0.015 e = 0.333 Fgrmax = 1.0 g = 0.333 Fgrmin = 0.0 I = 8 × 108 μs−1 G1 = 11.2 MBar−1 μs−1 G2 = 820 MBar−1 μs−1
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VIII. PRESSED TNT FLOW MODELING In order to verify the validity of the model, the whole process of shock initiation was modeled by hydro dynamic computer codes. Because of the symmetry, only half of the model was established which mainly consists of five parts. They were donor explosive, acceptor explosive and copper partition. The element type SOLID164 was used for all parts. Constraints were imposed on the plane of symmetry. The element type SOLID164 was used for 3-D modeling of solid structures. The eight-node element with the following degrees of freedom at each node is defined: translations, velocities, and accelerations in the nodal x, y, and z directions. The mesh resolution used was 50 cells per millimeter[15–17] . The donor explosive is defined with null material which allows the equations of state to be considered without calculating deviatoric stresses. The JWL equation of state is used for the un-reaction and reaction explosives. The parameters for un-reacted JWL and product JWL are shown in Table 1. The Ignition and Growth model with calibrated parameters is used to simulate the acceptor explosives. The pressure histories and radial displacement histories of the explosives have been calculated, from which we can see that the shock initiation characteristics revealed by experiment and numerical simulation are consistent.
Fig. 10. Calculated pressure histories for pressed TNT on axis.
Fig. 11. Calculated pressure histories for pressed TNT offaxis.
Figures 10 and 11 show the calculated pressure histories (solid lines) in the center zone and offcenter zone of each embedded 2-D CMC Lagrangian gauge using the Ignition and Growth parameters for pressed TNT listed in Table 1 and the equations of state for the copper material listed in Table 2. And these figures also include experimental traces of pressure (dashed lines) for a direct comparison between experiment and calculation. Both measured and calculated growths of reaction agree closely with the experiment. Table 2. Gruneisen equation of state parameters for copper material using the following equation: P = ρ0 c2 μ[1 + (1 − γ0 /2)μ− (a/2)μ2 ]/[1 − (S1 − 1)μ − S2 μ2 /(μ + 1) − S3 μ3 /(μ + 1)2 ]2 + (γ0 + aμ)E, where μ = ρ/ρ0 − 1 and E is thermal energy
INERT Copper
ρ0 (g/cm3 ) 8.45
C (mm/μs) 3.834
S1 1.43
S2 0.0
S3 0.0
γ0 2.0
a 0.0
Figure 12 shows the calculated radial displacement (solid lines) in the off-center zone of each embedded 2-D CMC Lagrangian gauge using the Ignition and Growth parameters for pressed TNT listed in Table 1. And these figures also include experimental traces of radial displacement (dashed lines) for a direct comparison between experiment and calculation. Both measured and calculated growths of reaction agree closely with the experiment. As a result of being not or slightly affected by lateral rarefaction wave, the relative radial displacement histories on the axis of symmetry are basically equal to 1.
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Fig. 12. Calculated radial displacement histories for pressed TNT off-axis.
IX. CONCLUSIONS (1) The shock initiation characteristics of pressed TNT are obtained by experiment and numerical simulation. The equations of state adopted to describe explosive and products were all based on JWL state equations. The particle velocity, relative specific volume and specific internal energy at different Lagrangian positions can be obtained. The reaction rate equation of the Ignition and Growth model based on JWL state equations was determined. (2) The lateral effects on the shock waves were investigated, which were all obtained from the pressure histories, relative radial displacement, particle velocity, relative specify volume and reaction rate. The results show that the lateral effects must be considered for the small partition experiment. (3) The numerical simulation models were established which can simulate the process of the shock initiation of explosives. And Ignition and Growth modeling calculations resulted in good agreement with the experimental data. The pressed TNT model can be used to predict other shock initiation scenarios that are highly unpredictable otherwise and have not been or cannot be experimentally tested directly.
References [1] Tarver,C.M., Kury,J.W. and Breithaupt,R.D., Detonation waves in triaminotrinitrobenzene. Journal of Applied Physics, 1997, 82: 3771-3782. [2] Tarver,C.M. Hallquist,J.O. and Erickson,L.M., Modeling short pulse duration shock initiation of solid explosives. In: Eighth International Symposium on Detonation, 1985: 884-892. [3] Urtiew,P.A., Erickson,L.M., Aldis,D.F. and Tarver,C.M., Shock initiation of LX-17 as a function of its initial temperature. In: Ninth International Symposium on Detonation, 1989: 112-122. [4] Bahl,K., Bloom,G., Erickson,L., Lee,R., Tarver,C., Von Holle,W. and Weingart,R., Initiation studies on LX-17 explosive. In: Eightth International Symposium on Detonation, 1985: 729-739. [5] Urtiew,P.A., Tarver,C.M., Maienschein,J.L. and Tao,W.C., Effect of confinement and thermal cycling on the shock initiation of LX-17. Combustion and Flame, 1996, 105(1-2): 43-53. [6] Urtiew,P.A., Tarver,C.M., Forbes,J.W. and Garcia,F., Shock sensitivity of LX-04 at elevated temperatures. AIP Conference Proceedings, 1997, 429: 727-730. [7] Huan,S. and Ding,J., A two-dimensional Lagrangian technique for shock initiation diagnosis. In: Ninth International Symposium on Detonation, 1989: 77-82. [8] Huan,S. and Ding,J., A two-dimensional Lagrangian technique for flow field measurement under high dynamic pressure. Acta Mechanica Sinaica, 1990, 6(2): 188-192. [9] Huan,S., Tao,W.J., Huang,F.L. and Jiang,G.P., Critical initiating state and 2D Lagrangian analysis of Pressed TNT. Applied Mechanics and Materials, 2011, 94-96: 1715-1718. [10] Huan,S., Shock initiation and two-dimensional steady-state detonation in heterogeneous explosives. PhD thesis, Beijing: State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, 1988 (in Chinese). [11] Gaupta,Y.M., High strain-rate shear deformation of a polyurethane elastomer subjected to impact loading. Polymer Engineering & Science, 1984, 24(11): 851-861.
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[12] Green,L.G., Tarver,C.M. and Erskin,D.J., Reaction zone structure in supracompressed detonating explosives. In: Ninth International Symposium on Detonation, 1989: 670-682. [13] Kury, John,W., Breithaupt,R.D. and Tarver,C.M., Detonation waves in Trinitrotoluene. Shock Waves, 1999, 9(4): 227-237. [14] Ultiew,P.A., Vandersall,K.S., Tarver,C.M., Garcia,F. and Forbes,J.W., Shock initiation experiments and modeling of composition B and C-4 explosives: experiments and modeling. Combustion and Explosion. 2008, 2(2): 162-171. [15] Tao,W.J., Research on shock initiation process and constitutive relation of solid explosive. Master Degree thesis, Guangzhou: Earthquake Engineering Research Test Center, 2008 (in Chinese). [16] Clutter,J.K. and Belk,D., Simulation of detonation wave interaction using an Ignition and Growth model. Shock Waves, 2002, 12: 251-263. [17] Kiyanda,C.B., Detonation Modeling of Non-ideal High Explosives. Urbana, Illinois: University of Illinois at Urbana-Champaign, 2010.
ISSN 0894-9166
SHOCK INITIATION OF EXPLOSIVES INVESTIGATED WITH SMALL PARTITION EXPERIMENT AND NUMERICAL SIMULATION Weijun Tao1
Shi Huan1
Fenglei Huang1,2
Guoping Jiang1
1
( Earthquake Engineering Research Test Center of Guangzhou University, Guangzhou 510405, China) (2 State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China)
Received 18 February 2011, revision received 14 June 2012
ABSTRACT In order to investigate the shock ignition of high energy solid explosives by shock waves, we carry out Lagrangian experiments with 2-D Lagrangian technique which uses composite manganin-constantan (CMC). The effects of the shock sensitivity of pressed solid high explosives, TNT, and the effect of the lateral rarefaction wave were studied. Based on the measured pressure histories and the radial displacements, we formulate the Ignition and Growth reactive flow models for the pressed TNT. The shock initiation process simulated by Ignition and Growth model agreed well with experimental data. This pressed TNT model can be applied to shock initiation scenarios which are highly unpredictable and have not been or cannot be tested experimentally.
KEY WORDS shock waves, explosive, Lagrangian analysis, state equation, rarefaction wave
I. INTRODUCTION Since safety issues play a dominant role in present-day energetic material technology, the concern on the relative safety of the shock initiation of solid high explosives is increasing. Hazardous scenarios can involve multiple stimuli, such as flyer plates accelerated impact explosives, producing strong shock waves in explosive or decaying to a deflagration wave or to a non-reacting wave. So the study of the property of energetic materials is very important to predicting whether a material will detonate or not. Previous research[1–6] using unconfined or weakly confined charges of LX-04-01 and LX-17 showed that the increase in shock sensitivity was primarily due to two effects: the increase in the number and size of reacting hot spots formed by shock compression and the increase in hot spot growth rate into the surrounding explosive particles. The lateral effects always decrease the shock waves especially for the shock initiation of small size. Since the traditional 1-D measuring method is adopted to measure the pressure of the small size of explosives during shock initiation, the measured pressure is not real, but the overall performance of the pressure and radial displacements. So, it is extremely important to consider the impact of lateral rarefaction wave. In this paper, the research by using unconfined charges of pressed TNT of small size was carried out, and Ignition and Growth reactive flow model for pressed TNT was formulated based on the measured pressure histories and radial displacement histories.
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 10972060) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20104410110003).
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II. EXPERIMENTS The 1-D planar conditions for shock initiation are difficult to satisfy since it is too expensive. The 2-D Lagrangian gauge was used, which is made of manganin and constantan, and the related analytical techniques[7, 8] were also adopted in earlier research. But the gauges were only embedded in the axis of symmetry for different Lagrangian positions. As the stretch effect was not studied before, the lateral effects were not analyzed. In this paper, the pressure and radial displacements at four Lagrangian positions were measured by using two 2-D Lagrangian gauges which were embedded in the same Lagrangian position with different radial positions. The distribution of two 2-D CMC Lagrangian gauges at the same Lagrangian position is shown in Fig.1. Gauge 1 was embedded in the axisymmetric position, and gauge 2 was embedded in the position whose distance to the axisymmetric position is r0 .
Fig. 1. The distribution of two 2-D CMC Lagrangian gauges at the same Lagrangian position. Fig. 2. Experimental set-up for small partition experiment.
The schematic diagram of experiment is shown in Fig.2. The donor explosive charge is TNT, and the diameter is about 20.0 mm. The acceptor explosive charge is pressed TNT consisting of several discs of 2-4 mm thickness and about 20.0 mm diameter, the density is ρ0 = 1.58 g/cm3 . The 2-D Lagrangian gauges were embedded in four Lagrangian positions, which are 0.0 mm, 3.01 mm, 5.14 mm, and 8.11 mm, respectively. And two radial positions, which are located on-axis at r0 = 0 mm and off-axis at r0 = 11.25 mm, respectively. The initial pressure amplitude can be controlled by the thickness of copper partition. For the relationship of thickness and pressure see our earlier studies[7] . So the copper partition is made into different thicknesses which are 5.0 mm, 5.5 mm, 6.0 mm, 6.5 mm, and 7.0 mm, respectively. The small partition experiment is the most common type of shock initiation test. The copper partition thickness is varied until the critical thickness at which 50% of the acceptor charges detonate is determined. Half of the critical detonation thickness measured for pressed TNT in the experiment is 5.5 mm[9] . Figure 3 shows that the 2-D CMC Lagrangian gauges invented by Shi Huan are more complex than 1-D gauges. The details of the 2-D CMC gauges were described in our previous publications[7, 8] . The signals can be obtained by oscilloscopes and can be changed into pressure by using the following formula[10] :
2 ΔR ΔR + 1.07 (1) P = 0.27 + 34.4 R0 R0 where P is the pressure, ΔR is the change of the resistance, R0 is the initial resistance. During the experiment, oscilloscopes measure the change of voltage caused by the resistance change in the 2-D CMC Lagrangian gauges, and were then converted to pressure using formula (1). Figures 4 and 5 show the pressure records and relative radial displacements of a shock loaded with high explosive pressed TNT. The solid lines denote data located on-axis at r0 = 0 mm, and the dashed lines denote data located off-axis at r0 = 11.25 mm.
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Fig. 3. The 2-D composite manganin-constantan Lagrangian gauges.
III. LAGRANGIAN METHOD The record of the 2-D Lagrangian method is used to calculate the reactive flow field in tested materials. The experimental temporal curves are used to calculate the value-time relations of particle velocity, relative specific volume and internal energy per unit volume at each Lagrangian position. The relationship between experiment, theoretical model and the numerical simulation is established based on the Lagrangian method. The two-dimensional conservation equations are as follows[10] :
t2 2 l ∂P 1 u = u1 − dt (2) ρ0 t1 l0 ∂h t 2 l ∂u dt (3) v = v1 + l ∂h t 0 t1 t2 ∂v E = E1 − P (t) dt (4) ∂t h t1 where ρ0 is the density, u the particle velocity, u1 the particle velocity of shock front, v the relatively specific volume, v1 the relatively specific volume of shock front, E the internal energy per unit volume, E1 the internal energy per unit volume of shock front, and t1 , t2 the start time and end time, respectively, h the Lagrangian position, l the radial displacement, l0 the length of sensitive part, and l/l0 the relative radial displacement. The path lines and particle line method can be used to avoid the loss of useful information when integration along isochrones lines is made. The analogical points[11] (e.g. the characteristic points on the waves such as the end point of elastic wave, the peak point of plastic wave etc.) of the pressure-time curves are synthesized to establish the path lines. The particle lines are the curves of the parameters varying with the time recorded by the Lagrangian gauges. The integral along isochrones lines can be changed along the path lines and particle lines: dt dp ∂p ∂p = − (5) ∂h t dh j ∂t h dh j ∂u dt du ∂u = − (6) ∂h t dh j ∂t h dh j So, Eqs.(2), (3) and (4) can be written as follows:
t2 2 ∂P l ∂t 1 ∂P dt (7) − u = u1 − ρ0 t1 l0 ∂h j ∂t h ∂h j
t2
v = v1 + t1
E = E1 −
2 ∂u l ∂t ∂u dt − l0 ∂h j ∂t h ∂h j
t2
P (t) t1
t2
∂v ∂t
(8)
dt h
(9)
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The experiment of cast TNT has been carried out by Shi Huan[7] . The particle velocity, relative specific volume and specific internal energy histories are all obtained from Eqs.(7), (8) and (9) with integral along the times. The integral along isochrones lines changed to along the path lines and particle lines with no other assumptions by Eqs.(5) and (6). The error mainly stems from the fitting between experiment and the curves. The least square curve fitting method is adopted with the B-spline function as test function to prevent the error diffusing to the whole flow field. The details of the error analysis of the 2-D Lagrangian method were described in our previous publications[7–9] .
IV. SINGLE-TEMPERATURE MODEL The single-temperature model with mixture laws is used to calculate the mixture of unreacted explosive and reaction products defined by the fraction reacted F (F = 0 implies no reaction, F = 1 implies complete reaction). The temperature and pressure are assumed to be equal and the relative specific volumes are additive, i.e., Pm = Ps = Pg
(10)
Tm = Ts = Tg
(11)
vm = (1 − F )vs + F vg
(12)
em = (1 − F )es + F eg
(13)
where subscript ‘m’ represents the mixed state, subscript ‘s’ represents the un-reacted state and subscript ‘g’ stands for the reacted state.
V. EQUATION OF STATE The model uses two Jones-Wilkins-Lee (JWL) equations of state as a function of temperature. One is for the un-reacted explosive and the other for its reaction products. P = Ae−R1 V + Be−R2 V +
ωCV T V
(14)
A JWL equation of state defines the pressure in the un-reacted explosive as Pe = Ae e−R1 Ve + Be e−R2 Ve +
ωe CVe Te Ve
(15)
where Pe is the pressure in megabars, Ve and Te are the relative volume and temperature, respectively, of the un-reacted explosive, Ae , Be , R1 , R2 are constants, CVe is the average heat capacity of un-reacted high explosive and ωe is Gruneisen coefficient. Another JWL equation of state defines the pressure in the reaction products as Pp = Ap e−R3 Vp + Bp e−R4 Vp +
ωp CVp Tp Vp
(16)
where Pp is pressure in megabars, Vp and Tp are the relative volume and temperature, respectively, of the un-reacted explosive, Ap , Bp , R3 , R4 are constants, CVp is the average heat capacity of reaction products and ωp is Gruneisen coefficient. From Eqs.(10)-(13), (15) and (16), the reaction rate histories can be calculated, and used to determine the parameters of the Ignition and Growth models.
VI. EXPERIMENTAL RESULTS AND ANALYSIS By making two-dimensional experiments and using the above-mentioned theory, the pressure and radial displacement on the axis of symmetry (r0 = 0 cm, solid lines) and off-axis (r0 = 1.125 cm, dashed lines) are processed with four gauges recording, respectively.
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Fig. 4. A typical pressure gauges record for pressed TNT material.
Fig. 6. Particle velocity histories of pressed TNT.
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Fig. 5. A typical relative radial displacements record for pressed TNT material.
Fig. 7. Relative specific volume histories of pressed TNT.
The pressure and radial displacement curves shown in Figs.4 and 5 are measured with the twodimensional Lagrangian experimental set-up (Fig.2). Figure 4 shows that the shock wave reaches the axis of symmetry more quickly than the same Lagrangian position off-axis (r0 = 1.125 cm). The particle velocity histories, relative specific volume histories and internal energy per unit volume histories (Figs.6, 7 and 8) are obtained by the Lagrangian method. Each curve in Figs.3-7 typically represents that the axis of symmetry (r0 = 0 cm) has been slightly affected by lateral rarefaction wave while the off-axis (r0 = 1.125 cm) has been seriously affected. The more reaction happens, the greater time difference of arriving time of the shock front for two gauges at the same Lagrangian positions. Owing to the lateral effects, the pressure on the axis of symmetry is also higher than off-axis (r0 = 1.125 cm). The radial displacement off-axis (r0 = 1.125 cm) cannot be ignored and is actually very large. Owing to the lateral
Fig. 8. Specific internal energy histories of pressed TNT.
Fig. 9. Fraction reacted histories of pressed TNT.
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rarefaction waves, the reaction zone expanded and the particle velocity as well as the arriving time to the same Lagrangian positions increased. But the sound velocity and detonation velocity decreased. The slope of reaction rates is larger on the axis of symmetry than off-axis (r0 = 1.125 cm). The fraction reacted histories are shown in Fig.9.
VII. IGNITION AND GROWTH MODELING The impact of lateral rarefaction wave was discussed in the previous section. The existing parameters of the Ignition and Growth are mainly calibrated by 1-D tests without considering these effects. But for small partition experiment, the effects of lateral rarefaction wave should be taken into account. The Ignition and Growth reactive flow of shock initiation and detonation of solid explosives has been incorporated into several hydro-dynamic computer codes and used to solve many explosive, propellant safety and performance problems. For example, Green et al.[12] and Kury et al.[13] using the Ignition and Growth model for cast TNT would complement the model. Paul[14] using the model for Composition B and C-4 explosives, and the Composition B model was then tested on short pulse, initiation experiment, and projectile impact shock experiments with good results. The reaction rate, F˙ , is given by (17) F˙ = I(1 − F )b (ρ/ρ0 − 1 − a)x + G1 (1 − F )c F d P y + G2 (1 − F )e F g P z (12) where F is the fraction reacted, t is the time, ρ is the current density, ρ0 is the initial density, and I, G1 , G2 , a, b, c, d, e, g, x, y, and z are constants. As explained in previous papers, this rate law with three terms models the three stages of reaction generally observed in shock initiation of heterogeneous solid explosives. The first term represents the ignition of the explosive as it is compressed by a shock wave creating heated areas (hot spots) as the voids in the material collapse. The second term in Eq.(17) represents the growth of reaction from the hot spots into the remaining solid. During shock initiation, this term is responsible for the relatively slow spreading of reaction in a deflagration-type process of inward and outward grain burning. The third term in Eq.(17) describes the rapid transition to detonation observed when the growing hot spots begin to coalesce and transfer large amounts of heat to the remaining un-reacted explosive particles causing them to react very rapidly. In this paper, the Ignition and Growth model parameters of pressed TNT unconfined by small sizes were then calibrated on existing experimental data with good results. The goodness of fit R2 is 0.98. The model parameters for pressed TNT are listed in Table 1. Table 1. Ignition & growth parameters for pressed TNT
UNREACTED JWL PRODUCT JWL A = 17.98 MBar A = 3.712 MBar B = −0.9310 MBar B = 0.03231 MBar R1 = 4.15 R1 = 6.2 R2 = 3.1 R2 = 0.95 ω = 0.8926 ω = 0.3 Cv = 1.0 × 10−5 MBar/K Cv = 2.1 × 10−5 MBar/K T0 = 298 K E0 = 0.07 MBar Shear Modulus= 0.04 MBar Yield Strength= 0.002 MBar REACTION RATES a = 0.065 x=6 b = 0.667 y=1 c = 0.667 z=3 d = 0.667 Figmax = 0.015 e = 0.333 Fgrmax = 1.0 g = 0.333 Fgrmin = 0.0 I = 8 × 108 μs−1 G1 = 11.2 MBar−1 μs−1 G2 = 820 MBar−1 μs−1
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VIII. PRESSED TNT FLOW MODELING In order to verify the validity of the model, the whole process of shock initiation was modeled by hydro dynamic computer codes. Because of the symmetry, only half of the model was established which mainly consists of five parts. They were donor explosive, acceptor explosive and copper partition. The element type SOLID164 was used for all parts. Constraints were imposed on the plane of symmetry. The element type SOLID164 was used for 3-D modeling of solid structures. The eight-node element with the following degrees of freedom at each node is defined: translations, velocities, and accelerations in the nodal x, y, and z directions. The mesh resolution used was 50 cells per millimeter[15–17] . The donor explosive is defined with null material which allows the equations of state to be considered without calculating deviatoric stresses. The JWL equation of state is used for the un-reaction and reaction explosives. The parameters for un-reacted JWL and product JWL are shown in Table 1. The Ignition and Growth model with calibrated parameters is used to simulate the acceptor explosives. The pressure histories and radial displacement histories of the explosives have been calculated, from which we can see that the shock initiation characteristics revealed by experiment and numerical simulation are consistent.
Fig. 10. Calculated pressure histories for pressed TNT on axis.
Fig. 11. Calculated pressure histories for pressed TNT offaxis.
Figures 10 and 11 show the calculated pressure histories (solid lines) in the center zone and offcenter zone of each embedded 2-D CMC Lagrangian gauge using the Ignition and Growth parameters for pressed TNT listed in Table 1 and the equations of state for the copper material listed in Table 2. And these figures also include experimental traces of pressure (dashed lines) for a direct comparison between experiment and calculation. Both measured and calculated growths of reaction agree closely with the experiment. Table 2. Gruneisen equation of state parameters for copper material using the following equation: P = ρ0 c2 μ[1 + (1 − γ0 /2)μ− (a/2)μ2 ]/[1 − (S1 − 1)μ − S2 μ2 /(μ + 1) − S3 μ3 /(μ + 1)2 ]2 + (γ0 + aμ)E, where μ = ρ/ρ0 − 1 and E is thermal energy
INERT Copper
ρ0 (g/cm3 ) 8.45
C (mm/μs) 3.834
S1 1.43
S2 0.0
S3 0.0
γ0 2.0
a 0.0
Figure 12 shows the calculated radial displacement (solid lines) in the off-center zone of each embedded 2-D CMC Lagrangian gauge using the Ignition and Growth parameters for pressed TNT listed in Table 1. And these figures also include experimental traces of radial displacement (dashed lines) for a direct comparison between experiment and calculation. Both measured and calculated growths of reaction agree closely with the experiment. As a result of being not or slightly affected by lateral rarefaction wave, the relative radial displacement histories on the axis of symmetry are basically equal to 1.
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Fig. 12. Calculated radial displacement histories for pressed TNT off-axis.
IX. CONCLUSIONS (1) The shock initiation characteristics of pressed TNT are obtained by experiment and numerical simulation. The equations of state adopted to describe explosive and products were all based on JWL state equations. The particle velocity, relative specific volume and specific internal energy at different Lagrangian positions can be obtained. The reaction rate equation of the Ignition and Growth model based on JWL state equations was determined. (2) The lateral effects on the shock waves were investigated, which were all obtained from the pressure histories, relative radial displacement, particle velocity, relative specify volume and reaction rate. The results show that the lateral effects must be considered for the small partition experiment. (3) The numerical simulation models were established which can simulate the process of the shock initiation of explosives. And Ignition and Growth modeling calculations resulted in good agreement with the experimental data. The pressed TNT model can be used to predict other shock initiation scenarios that are highly unpredictable otherwise and have not been or cannot be experimentally tested directly.
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[12] Green,L.G., Tarver,C.M. and Erskin,D.J., Reaction zone structure in supracompressed detonating explosives. In: Ninth International Symposium on Detonation, 1989: 670-682. [13] Kury, John,W., Breithaupt,R.D. and Tarver,C.M., Detonation waves in Trinitrotoluene. Shock Waves, 1999, 9(4): 227-237. [14] Ultiew,P.A., Vandersall,K.S., Tarver,C.M., Garcia,F. and Forbes,J.W., Shock initiation experiments and modeling of composition B and C-4 explosives: experiments and modeling. Combustion and Explosion. 2008, 2(2): 162-171. [15] Tao,W.J., Research on shock initiation process and constitutive relation of solid explosive. Master Degree thesis, Guangzhou: Earthquake Engineering Research Test Center, 2008 (in Chinese). [16] Clutter,J.K. and Belk,D., Simulation of detonation wave interaction using an Ignition and Growth model. Shock Waves, 2002, 12: 251-263. [17] Kiyanda,C.B., Detonation Modeling of Non-ideal High Explosives. Urbana, Illinois: University of Illinois at Urbana-Champaign, 2010.