DOI: 10.1002/prep.201400106 Mach Wave Control in Explosively Formed Projectile Warhead Chuan-Sheng Zhu,[a] Zheng-Xiang Huang,*[a] Xu-Dong Zu,[a] and Qiang-Qiang Xiao[a] 1 Introduction Explosively formed projectiles (EFPs) are used in numerous modern ammunition systems because of their many advan- tages, such as their effective stand-off and strong secon- dary effects after penetration. However, designers often en- counter the problem where the wave shaper should be embedded passively in charge. For example, sensing ele- ments have to be embedded in charge to decrease the length of the warhead in many smart ammunition systems. Thus, the wave shaper has to be embedded in charge to cover these sensing elements [1] . In addition, the wave shaper can be actively embedded in charge such that the velocity of the penetrator can increase because the wave shaper can adjust detonation wave shape [2–5]. However, a Mach wave emerges at the top of the liner when a wave shaper is embedded in charge, and thus significantly affects the formation of an EFP. When the relationship between the Mach wave and liner fails to match, the nose of the EFP may be seriously broken [6–8]. Thus, the pressure behind the Mach wave should be controlled to achieve a matched relationship. David Bender et al. [6] found that optimizing the configuration of the liner can avoid breakage at the nose of the EFP because the location of the ring initiator is fixed. Miao Qin-shu et al. [9] carried out numerical studies of the effect of annular initiation position on EFP formation and found that increasing the diameter of annular initiation can significantly increase the velocity and length-diameter ratio of the EFP and that reducing the distance between the annular initiation and liner can reduce the velocity and length-diameter ratio of the EFP. However, these studies did not investigate how to avoid breakage at the nose of the EFP by matching the relationship between the Mach wave and liner. When a wave shaper is embedded in charge, detonation waves collide at the axis of the charge after climbing the wave shaper. When the incident angle of the detonation wave is below the critical angle for Mach re- flection, regular reflection occurs, and the pressure at the colliding point is 2.4 times the Chapman-Jouguet (CJ) pres- sure. Meanwhile, when the incident angle is above the criti- cal angle, Mach reflection occurs, and the pressure behind the Mach wave decreases from four times the CJ pressure to normal CJ pressure with increasing incident angle [10, 11]. Therefore, overdriven detonation is produced when a wave shaper is embedded in charge; the pressure behind the overdriven detonation wave is related to the incident angle of the detonation wave. The pressure behind the overdriven detonation wave can be controlled by varying the incident angle at the top of the liner. An analytical model for Mach wave parameter calculation is presented in this study based on three-shock theory. The parameters of Mach waves, such as the growth angle, their radius, their velocity along the plane of symmetry, and the pressure behind them, can be determined. Calculation re- sults show that the pressure behind the Mach wave and Abstract : A Mach wave emerges at the top of the liner when a wave shaper is embedded in charge, and thus seri- ously breaks the explosively formed projectile (EFP) nose. Thus, to avoid breakage at the EFP nose, the pressure behind the Mach wave should be controlled. An analytical model for calculating Mach wave parameters is presented based on three-shock theory. The parameters of Mach waves, such as their growth angles and radii, their velocity along the plane of symmetry, and the pressure behind them, can be determined. Calculation results show that when the diameter of the wave shaper is reduced or the distance between the wave shaper and liner increases, the incident angle of the detonation wave at the top of the liner increases and thereby lowers the pressure behind the Mach wave. Avoiding the occurrence of Mach waves by re- ducing the incident angle fails to avoid breakage at the nose of the EFP, but lowering the pressure behind the Mach wave by increasing the incident angle avoids break- age at the nose of the EFP. Calculation and simulation re- sults are validated through X-ray imaging experimentation. Keywords: Explosion mechanism · Explosively formed projectile (EFP) · Wave shaper · Mach wave [a] C.-S. Zhu, Z.-X. Huang, X.-D. Zu, Q.-Q. Xiao School of Mechanical Engineering Nanjing University of Science and Technology Xiaolingwei 200 Nanjing 210094, P. R. China *e-mail :
[email protected] Propellants Explos. Pyrotech. 2010, 35, 1 – 8 � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim &1& These are not the final page numbers! �� Full Paper the Mach wave radius can be controlled by varying the wave shaper diameter and the distance between the wave shaper and liner, which determine the incident angle at the top of liner. The relationship between the Mach wave and the configuration of the liner can then be matched well. In this way, breakage at the nose of the EFP can be avoided. 2 Analytical Model for Mach Wave Parameter Calculation Figure 1 presents the geometry of classical Mach reflection. As Figure 1 shows, OI is the wave front of the incident detonation, OR is the wave front of the reflected shock, O is the triple point, and T is the collision point. The image can be separated into four regions by the detonation and shock waves: zone (0) is the unreacted explosive, zone (1) is the region behind the detonation wave, zone (2) is the region behind the reflection wave, and zone (3) is the region behind the Mach wave. The parameters in every zone are shown with the corresponding subscript. Based on descriptions of detonation reflection in the explosive [12], the parameters of each region are described as fol- lows: The Mach number M1 and deflection angle q in region (1) are defined by where g is the exponent in the polytropic equation of state for the explosive and yI is the incident angle of the detona- tion wave. Based on the equations of mass, momentum, and energy conservation in region (2), the following equation is ob- tained: where PCJ and 1CJ are the CJ detonation pressure and densi- ty, respectively, of the explosive and e is the deflection angle. Based on the equations of mass, momentum, and energy conservation in region (3), the following equation is de- rived: where DM and DCJ are the velocity of the Mach wave and detonation wave, respectively; 10 is the initial density of the explosive, b is the angle of the tangent line at the Mach wave to the symmetry axis, h is the ratio of the spe- cific chemical energy release of the explosive material that passes through the Mach wave to that of the material that goes through the CJ detonation front, and a is the deflec- tion angle. The medium in regions (2) and (3) can meet the condi- tions that the flow velocity is parallel and the pressures are equal; that is: We can obtain the parameters of P3, b, and a near the triple point by solving Eqs. (1) to (4). The growth angle of the Mach wave c can be obtained as follows [13]: c ¼ p=2�b ð5Þ The steps for calculating c are as follows: For every value when b decreases from p/2 to yI, a is calculated using Equation (3), and e is calculated according to P2=P3. The Figure 1. Flow setup used to describe Mach reflection. &2& www.pep.wiley-vch.de � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim Propellants Explos. Pyrotech. 2010, 35, 1 – 8 �� These are not the final page numbers! Full Paper C.-S. Zhu, Z.-X. Huang, X.-D. Zu, Q.-Q. Xiao www.pep.wiley-vch.de calculation stops when e+a is approximately equal to q, and b at this time is the required value. The growth angle of the Mach wave c can be obtained by substituting b into Equation (5). According to Lambourn [14] and Hull [15], the values of c obtained from Equation (5) are significantly greater than the experimental values because the straight Mach wave assumption is used. Based on the experimental values re- ported by Lambourn [14], the theoretical value of c can be modified. The experimental values and the values obtained from Equation (5) are shown in Table 1. Based on the data shown in Table 1, the theoretical value of c can be modified as where cM, c, and YI are the degree values. When a wave shaper is embedded in charge, detonation waves collide at the axis of the charge after climbing over the wave shaper. When the incident angle of the detona- tion wave is above the critical angle, Mach reflection occurs. An analytical model for Mach wave radius calcula- tion can be proposed when the growth angle of the Mach wave cM is obtained. As schematically shown in Figure 2, O is the triple point, T is the collision point, RWS is the radius of the wave shaper, and S is the distance between the liner and wave shaper. yI, RM, cM, and L represent the incident angle of the detona- tion wave, the radius of the Mach wave, the growth angle of the Mach wave, and the distance from the starting point of the Mach wave to the liner, respectively. The Mach wave usually starts at the point where the incident angle of the detonation wave is 418 [14], such that L can be calculated geometrically. According to the mathematical description of the sche- matic in Figure 2, the radius of the Mach wave, RM, can be calculated as follows: 3 Theoretical Results and Analysis As shown in Figure 2, the incident angle of the detonation wave yI is determined by the wave shaper radius RWS and the distance S between the wave shaper and liner. To ex- plore the effect of RWS and S on the Mach wave parameters, such as the pressure behind the Mach wave and the Mach wave radius, many incident angles of the detonation wave on the top of the liner can be obtained by keeping RWS constant and changing S (way I) or by keeping S constant and changing RWS (way II). Five angles can be obtained in both ways: 558, 608, 658, 708, and 758. In way I, the radius of the wave shaper, RWS, is 32 mm, and in way II, the dis- tance between the wave shaper and liner S is 45.7 mm. The pressure behind the Mach wave and the Mach wave radius are calculated for the five incident angles (Figure 3). Table 1. Experimental and theoretical values of c. YI/[8] 60 65 70 75 80 Experimental values 2.4 2.75 3.2 4.15 4.8 Theoretical values 9.85 11.62 12.15 11.2 8.71 Figure 2. Schematic used to calculate the Mach wave radius. Figure 3. Pressure behind Mach wave and Mach wave radius for five incident angles. Propellants Explos. Pyrotech. 2010, 35, 1 – 8 � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.pep.wiley-vch.de &3& These are not the final page numbers! �� Mach Wave Control in Explosively Formed Projectile Warhead www.pep.wiley-vch.de As shown in Figure 3, every angle has two points. The left point is the pressure at the colliding point T, whereas the right point is the pressure at the triple point O. The horizontal distance between the two points represent the radius of the Mach wave. When the incident angle of the detonation wave at the top of the liner is small, the hori- zontal distance is short, but the pressures are high, indicat- ing that the impact range is small but the impact strength is high (Figure 3). The horizontal distance increases whereas the pressures decrease with the increasing incident angle of the detonation wave, indicating that increasing the inci- dent angle increases the impact range and decreases impact strength. When the incident angle of the detona- tion wave is large, the pressures approximate the CJ pres- sure, indicating that the impact strength of the Mach wave approximates that of the detonation wave. In terms of the formation of EFP, when the impact range is small but the impact strength is high, the axial collapse velocity of the element at the roof of the liner is significant- ly higher than that of other elements, thus breaking the nose of the EFP. When the impact strength of the Mach wave approximates the impact strength of the detonation wave, the axial collapse velocity of the element at the roof of the liner approximates that of the other elements, which is favorable for keeping the nose of the EFP integrated but unfavorable for increasing EFP length. Only when the impact strength by the Mach wave is suitable is it desirable to both avoid breakage at the nose of the EFP and increase EFP length. Therefore, reducing the wave shaper radius or increasing the distance between the wave shaper and liner increases the incident angle of the detonation wave at the top of the liner and lowers the impact strength of the Mach wave, thereby avoiding breakage at the nose of the EFP. 4 Simulation Research and Experiment Validation EFP penetration capability can be enhanced when a wave shaper is placed in charge. However, a Mach wave emerges at the top of the liner when a wave shaper is embedded in charge, significantly affecting the formation of EFP. When the relationship between the Mach wave and liner fails to match, the nose of the EFP may be seriously broken. Thus, the pressure behind the Mach wave should be controlled to match the relationship well. 4.1 Avoid Mach Wave Given that the Mach wave adversely affects the formation of EFP, one way to avoid this effect is to avoid the occur- rence of a Mach wave. According to the theory of the re- flection of detonation waves in a condensed explosive, the critical angle for Mach reflection is approximately equal to 458 [10]. When the incident angle of the detonation wave is below the critical angle, Mach reflection is avoided. To verify whether breakage at the nose of the EFP can be avoided when no Mach wave is found in the charge, a simu- lation was carried out for a warhead by using LS-DYNA software. The configuration of this warhead is shown in Figure 4, where the diameter and length of the charge are 80 mm, the diameter of the wave shaper is 64 mm, the dis- tance between the wave shaper and liner is 23.74 mm, and the incident angle of the detonation wave at the top of the liner is 36.578. The 8701 explosive is used in the calculation, and the liner material is copper. The parameters of all mate- rial models used in the calculations are found in the litera- ture [16, 17]. The simulation result is shown in Figure 5. Figure 5 shows that the nose of the EFP is broken, indi- cating that breakage emerges even when no Mach wave occurs in the charge. This phenomenon is attributed to the fact that when regular reflection occurs in the charge, the pressure at the colliding point is 2.4 times the CJ pressure, which is high enough to split the nose of the EFP. Thus, breakage at the nose of the EFP cannot be avoided by avoiding the occurrence of a Mach wave. 4.2 Increase of Incident Angle of Detonation Wave Avoiding the occurrence of Mach waves cannot prevent the breaking of the nose of the EFP. However, calculation Figure 4. Configuration of EFP warhead for avoiding a Mach wave. Figure 5. Simulation result of avoiding a Mach wave. &4& www.pep.wiley-vch.de � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim Propellants Explos. Pyrotech. 2010, 35, 1 – 8 �� These are not the final page numbers! Full Paper C.-S. Zhu, Z.-X. Huang, X.-D. Zu, Q.-Q. Xiao www.pep.wiley-vch.de results show that the pressure behind the Mach wave and the Mach wave radius can be controlled by varying the wave shaper diameter and the distance between the wave shaper and liner to avoid breakage at the nose of the EFP. To verify the reliability of the calculation results, simulation research is carried out for three warheads by using the LS- DYNA software. The configurations of these warheads are shown in Figure 6; the diameter and length of all the charges are 80 mm. The parameters of these warheads, such as the diameter of the wave shaper DWS, the distance between the wave shaper and liner S, and the incident angle of the detonation wave at the top of the liner YI, are shown in Table 2. The simulation results are shown in Figure 7. The calcula- tion time is 350 ms, at which moment EFPs form. As shown in Figure 7, the noses of the EFP in both types I and II are broken, but the nose in type III is integrat- ed. This result is attributed to the fact that the incident angles in both types I and II are so small that the pressures behind the Mach waves are high, thus breaking the nose of the EFPs. Owing to the large incident angle in type III, the pressure is weaker than that in types I and II ; the weaker pressure fails to split the nose of the EFP. Compared with the nose of the EFP in type II, that in type I is broken more seriously because the large diameter of the wave shaper in type I increases the radius of the Mach wave and thus ag- gravates breakage at the nose of the EFP although the pressures in the two warheads are approximately equal. Figure 6. Configuration of three warheads. Figure 7. Simulation results for three warheads. Table 2. Parameters of three warheads. Type DWS/[mm] S/[mm] YI/[8] I 64.00 48.13 56.38 II 40.00 28.13 54.58 III 40.00 48.13 67.43 Propellants Explos. Pyrotech. 2010, 35, 1 – 8 � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.pep.wiley-vch.de &5& These are not the final page numbers! �� Mach Wave Control in Explosively Formed Projectile Warhead www.pep.wiley-vch.de 4.3 Experiment Validation To verify the reliability of the simulation results, X-ray ex- periments on the two warheads in types I and II are carried out. The charge configurations used in the X-ray experi- ments are the same as those used in the simulation. The layout of the X-ray experiment is shown in Figure 8, and X- ray images of the EFPs are shown in Figure 9. These images are taken at 350 ms after the explosive is initiated, at which time EFPs form. Figure 9 shows that the nose of the EFP in type I frac- tures into two pieces, whereas that in type II slightly splits. The shapes of the EFPs in Figure 7 and Figure 9 are in good agreement, indicating that the simulation can recon- struct the experimentally generated projectile forms suc- cessfully and the simulation results are believable. 5 Conclusions Overdriven detonation is produced when a wave shaper is embedded in charge, and the pressure behind the over- driven detonation wave can be controlled by varying the wave shaper diameter and the distance between the wave shaper and liner. Avoiding the occurrence of a Mach wave by reducing the incident angle fails to prevent breakage at the nose of the EFP, but lowering the pressure behind the Mach wave by increasing the incident angle avoids break- age at the nose of the EFP. The incident angle of the deto- nation wave at the top of liner can be increased in two ways: by reducing the wave shaper diameter and by in- creasing the distance between the wave shaper and liner. Reducing the wave shaper diameter is an effective strategy because it not only lowers the pressure behind the Mach wave but also reduces the Mach wave radius. Given that system constraints generally limit the increase of warhead length, increasing the distance between the wave shaper and liner is also an effective strategy until the wave shaper diameter is small enough. References [1] J. Men, J. Jiang, L. Jian, Numerical Simulation Research on the Influence of Sensing Elements on EFP Forming (in Chinese), J. Ballistics 2005, 17, 67–71. [2] K. Weimann, Research and Development in the Area of Explo- sively Formed Projectiles Charge Technology, Propellants Explos. Pyrotech. 1993, 18, 294–298. [3] Z.-X. Huang, Mechanism Study on Jetting Projectile Charge For- mation, PhD Thesis, Nanjing University of Science & Technolo- gy, Nanjing, P.R. China, 2003 (in Chinese). [4] X. Zhang, H. Chen, Y. Zhao, Study on Shaped Charge Tech- nique of Small Diameter Which Have High Velocity EFP (in Chinese), J. Projectiles Rockets Missiles Guidance 2003, 23, 107– 109. [5] Y. Zhang, X. Zhang, Y. He, L. Qiao, Detonation Wave Propaga- tion in Shaped Charges With Large Wave-Shaper, 27th Interna- tional Symposium on Ballistics, Freiburg, Germany, April 22–26, 2013, p. 770–782. [6] D. Bender, R. Fong, W. Ng, B. Rice, Dual Mode Warhead Tech- nology for Future Smart Munitions, 19th International Symposi- um on Ballistics, Interlaken, Switzerland, May 7–11, 2001, p. 679–684. [7] R. Fong, W. Ng, K. Weimann, Nonaxisymmetric Waveshaped EFP Warheads, 20th International Symposium on Ballistics, Or- lando, FL, USA, September 23–27, 2002, p. 582–588. [8] M. Murphy, K. Weimann, K. Doeringsfeld, J. Speck, The Effect of Explosive Detonation Wave Shaping on EFP Shape and Per- formance, 13th International Symposium on Ballistics, Stock- holm, Sweden, June 1–3, 1992, p. 449–456. [9] Q. Miao, W. Li, X. Wang, Effect of Annular Initiation Position on Formation of EFP (in Chinese), J. Ballistics 2012, 24, 58–62. Figure 8. Layout of X-ray imaging experiment. Figure 9. X-ray images of EFP formation. &6& www.pep.wiley-vch.de � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim Propellants Explos. Pyrotech. 2010, 35, 1 – 8 �� These are not the final page numbers! Full Paper C.-S. Zhu, Z.-X. Huang, X.-D. Zu, Q.-Q. Xiao www.pep.wiley-vch.de [10] B. Dunne, Mach Reflection of Detonation Waves in Condensed High Explosives II, Phys. Fluids 1964, 7, 1707–1711. [11] X. Zhang, Z. Huang, L. Qiao, Detonation Wave Propagation in Double-layer Cylindrical High Explosive Charges, Propellants Explos. Pyrotech. 2011, 36, 210–218. [12] C. Bing, Study on the Phenomenon of Charge Detonation and EFP Formation Under the Condition of Multi-Point Ignition at the Upper End of the Charge, PhD Thesis, Nanjing University of Science and Technology, Nanjing, P.R. China 1998 (in Chinese). [13] J. Wang, Two-Dimensional Nonsteady Flow and Shock Waves, Science Press, Beijing 1994, p. 91 (in Chinese). [14] B. D. Lambourn, P. W. Wright, Mach Interaction of Two Plane Detonation Waves, 4th International Detonation Symposium, Arlington, VA, USA, 1965, p. 142–152. [15] L. M. Hull, Mach Reflection of Spherical Detonation Waves, 10th International Detonation Symposium, Boston, MA, USA, July 12–16, 1993, p. 11–18. [16] M. J. Murphy, E. L. Lee, Modeling Shock Initiation Composition B, 10th International Detonation Symposium, Boston, MA, USA, July 12–16, 1993, p. 963–970. [17] J. Wu, J. Liu, Y. Du, Experimental and Numerical Study on the Flight and Penetration Properties of Explosively-Formed Pro- jectile, Int. J. Impact. Eng. 2007, 34, 1147–1162. Received: May 4, 2014 Published online: && &&, 0000 Propellants Explos. Pyrotech. 2010, 35, 1 – 8 � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.pep.wiley-vch.de &7& These are not the final page numbers! �� Mach Wave Control in Explosively Formed Projectile Warhead www.pep.wiley-vch.de FULL PAPERS C.-S. Zhu, Z.-X. Huang,* X.-D. Zu, Q.-Q. Xiao && –&& Mach Wave Control in Explosively Formed Projectile Warhead &8& www.pep.wiley-vch.de � 2010 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim Propellants Explos. Pyrotech. 2010, 35, 1 – 8 �� These are not the final page numbers! Full Paper C.-S. Zhu, Z.-X. Huang, X.-D. Zu, Q.-Q. Xiao www.pep.wiley-vch.de