2. Military Technical College, Kobry Elkobbah, Cairo, Egypt;
3. The University of Manchester, School of Mechanical Aerospace and Civil Engineering, Manchester, UK;
4. Mechanical Engineering Department, Modern Academy for Engineering and Technology, Cairo, Egypt
The model of an explosivelydriven metal is used to predict the fragmentation velocities, the flyer plate motions and the collapse velocities of the shaped charge liners. Many authors have attempted to deduce simple relations governing the driven metal velocity under the effect of detonation gaseous products. Gurney^{[1]} tried to identify the chemical energy liberated from the detonation of a high explosive that could be imparted to the metal in contact with the explosive, causing its acceleration and attaining terminal velocity^{[2]}. This energy is called Gurney energy (E) and is considered as an intrinsic property for each explosive. It was defined as the part of the total chemical energy of explosive that is released during detonation and converted to the kinetic energy of the metal. The final kinetic energy from the detonation of explosive was partitioned between the kinetic energy of the driven metal and the gaseous product by an estimated linear velocity profile. To simplify the calculations of the terminal velocity of the metal, following assumptions were normally taken:
The detonation products are assumed to expand uniformly with constant density; Rarefaction and shock wave effect within the solid metals due to shock waves are neglected; The total Gurney energy is divided into the kinetic energy of the gas expansion and the kinetic energy of the driven elements in contact with the explosive.
The Gurney model was essentially based on the principles of momentum and energy conservations. It could be applied to any onedimensional explosivemetal interaction system. The Gurney approximation exhibits high accuracy for the prediction of the final metal velocity over the range of mass ratio between metal (M) and explosive (C) from 0.1 to 10 ^{[2]}. Many investigators introduced their analytical formulae used to determine the Gurney velocity and discuss the effect of metals and explosives formulations^{[34]}. But for the common shaped charges, the liner collapse velocity was determined using the formulae derived by different researchers ^{[5]}. GuiXi ^{[6]} extended the Gurney model to small shaped charges, whereas he also deduced another formula considering the effect of the confinement of the charge on the collapse velocity. The detonation characteristics of several highly pressed plastic bonded explosives (PBXs) and different plastic explosives have been studied ^{[710]} where these parameters could be used to determine the explosive Gurney energy. Also the effect of the geometry of different configurations was also studied by several researchers ^{[1113]}.
These studies show that the terminal velocity of the metal depends on the configurations of the metalexplosive interactions, the explosive Gurney energy and the mass ratio M/C.
1 Theories and fundamentalsKennedy ^{[14]} provided an easy approximation to determine the Gurney energy for some explosives. This approximation is:
$ E = 0.7{Q_\rm{v}} $  (1) 
Where Q_{v} is the heat of explosion of the explosive. However, for most commonly used explosives, the Gurney energy; E is the energy from the detonation that is responsible for metal acceleration. E varies between 0.61 Q_{v} and 0.7 Q_{v}.
In 2002, Koch et al. ^{[15]} used the low of energy conservation to get a relation between the explosive detonation velocity and its Gurney velocity. They applied the energy conservation law to different explosive metal configuration, symmetrical plates, cylinder filled with explosive core and hollow metallic sphere filled with a solid explosive. For common explosives, it was found that the Gurney velocity =U_{D}/3.08, where U_{D} is the detonation velocity of the explosive, which generally agrees with existing estimations ^{[15]}. Furthermore, Gurney velocity has been experimentally determined for certain explosives, which are listed in Table 1. The average value calculated for these explosives of U_{D}/(2E)^{1/2} is 3.19 with a standard deviation of 0.2, which means that this formula presents a reliable tool to estimate the Gurney velocity for certain explosives based on their detonation velocity. In 2006, Keshavarz and Abolfazl ^{[16]} extended this definition to include the detonation gaseous and solid products and their heat of formation. The proposed new relationship between E and Q_{v} is similar to those calculated by the existing approximations for a range of different explosives.
Hurley ^{[18]} proposed a new function to predict the Gurney Velocity of ammonium nitrate based explosives based on their detonation velocity. This proposed relationship is applied to a weldability window. In 2012, Ludwig and Wibke discussed various expressions for the analytical and empirical Gurney formula ^{[19]}. In 2015, Frem suggested an approximation to the Gurney constant depending on the empirical thermochemical BKW equation of state of various high explosives ^{[20]}.
The introduced approximations have quite accurate estimation of the characteristic of Gurney velocity, but it does not include the specific impulse of an energetic material that represents the change in momentum of this energetic material, it has unit of velocity. The next section thus discusses the relation between the characteristic Gurney velocity and its dependence on the specific impulse of an explosive regardless the value of detonation velocity ^{[15, 18]} or the heat of formation of this energetic material ^{[16]}.
2 Results and discussionThe Gurney velocities of various explosive types are depicted in Figure 1 against the characteristic explosive Chapman Jouget pressure to its total impulse ratio. The following relation was obtained from the fitting of the previously calculated Gurney velocity against the P_{CJ}/I_{SP}.ρ_{0} ratio.
$ \sqrt {2E} = 0.2415\left( {\frac{{{P_{{\rm{CJ}}}}}}{{{I_{{\rm{SP}}{\rho _0}}}}}} \right)970.76 $  (2) 
where P_{CJ} is the Chapman Jouguet pressure (Pa), I_{SP} is the specific impulse of the explosive used as a monopropellant (N·s/kg) and ρ_{0} is the explosive density (kg/m^{3}). If the impulse of the explosive is not known, it can be calculated using the empiricalimpulse relation for C_{a}H_{b}N_{c}O_{d} explosives ^{[17]}:
$ \begin{array}{l} {I_{{\rm{SP}}}} = 2.42050.074a0.003\;6b + 0.002\;37c + \\ \;\;\;\;\;\;\;0.04a0.1001{n_{{\rm{N}}{{\rm{H}}_x}}}\_0.146\;6\left( {{n_{{\rm{Ar  1}}}}} \right) \end{array} $  (3) 
where I_{SP} in N·s/kg, is the number of NH_{2} and NH groups and is the number of aromatic rings in aromatic explosives. The presented equation exhibited quite accurate approximation for various pure and composite explosives as proved in Ref.^{[17]}. Various explosives (21 types) were studied and their pressure, impulse and densities are listed in Table 2. P_{CJ} values have been determined using Explo 5 thermodynamic code, where BKWN equation of state was applied ^{[7, 10]}. The calculated and the measured Gurney velocities as well as the deviation between them are also presented in Table 2. The greatest deviation between the measured and the calculated Gurney velocities based on Eq. (2) is 4.8%, which means that this approximation can be used accurately over a wide range of explosives.
In the present research, the verification of the suggested Gurney formulae has been performed using flash Xray radiograph and the multilayer electric aluminium panels that have been used to measure the jet tip velocity and to depict the jet profile at different times from the point of initiation. The used shaped charge has a conical copper liner of angle 45 degree, a wall thickness of 1.3 mm and a charge calibre of 36.6 mm. The used explosive charge is composition C4 ^{[24]} of loading density 1.61 g/cm^{3} and mass of 30 g, while the charge casing is made up of mild steel as shown in Figure 2.
The shaped charge is attached to the top of the laminated electric foil layers separated by 25 cm provided with electric circuits that are used to determine the jet tip velocity as it passes through them as shown in Figure 3.
The flash Xray trial was performed in COTEC (Cranfield Ordnance Test and Evaluation Centre) field. Various heads were used to capture photos of the jet profile at different times. Figure 4 shows the jet shapes obtained from Xray radiograph and the numerical jet formations using Autodyn hydrocode at 34 μs and 122 μs, respectively. A curved shape is observed for the real Xray jet, which may be caused either by some asymmetries in the liner positioning during the manual filling of the charge or due to the nonuniform explosive mass distribution inside the charge cavity. Nevertheless, the general aspects of the jet shapes are similar.
The proposed approach has been used within the unsteady state PER theory ^{[2]} for the complete jetting calculations using the value of the Gurney velocity obtained from Eq. (2) for the Comp. C4 explosive that was found to be 2 553 m/s. Thus, the jet tip velocities were found to be 6 100 m/s and 6000 m/s from the real measurements and the analytical model based on unsteady state PER theory, respectively, as shown in Fig. 5. The jet velocity was calculated via the direct explicit relation among the collapse velocity V_{0}, the stagnation velocity V_{1}, the flow velocity V_{2} and the collapse and deflection angles as well. For shaped charges, an oftenused formula is that of a single flat plate backed by a slab of explosive, the collapse velocity V_{0} is calculated by references ^{[25]}:
$ {V_0} = \sqrt {2E} {\left[{\frac{1}{{\left[{4_\mu ^2 + 5\mu + 1} \right]}}} \right]^{0.5}} $  (4) 
where μ is the metal/explosive mass ratio and V_{0} is the collapse velocity of the liner element.
The collapse velocity is then used to estimate both the flow velocity and the stagnation velocity as follow:
$ {V_1} = \frac{{{V_0}{\rm{cos}}\left( {\frac{{\beta\alpha }}{2}} \right)}}{{{\rm{sin}}\beta }} $  (5) 
$ {V_2} = \frac{{{V_0}{\rm{cos}}\left( {\frac{{\beta+\alpha }}{2}} \right)}}{{{\rm{sin}}\beta }} $  (6) 
where V_{1} and V_{2} are stagnation and flow velocities during jet formation respectively whereas α is the half cone apex angle and β is the collapse angle.
Thus the jet velocity is calculated from:
$ {V_{\rm{j}}} = {V_1} + {V_2} $  (7) 
while the jet mass m_{j} is calculated from Eq.
$ {m_\rm{j}} = \frac{m}{2}\left( {1{\rm{cos}}\beta } \right) $  (8) 
where m is the total liner mass.
Further details of the analytical shaped charge modelling can be obtained from Ref. ^{[2, 6, 22]}. Besides, the obtained jet velocities of few elements on its axis show the relevant accuracy of the analytical model depending on the approached Gurney velocity of the chosen C4 explosive. This result means that the error percent in the jet tip velocity is only 1.6%, which implies that the suggested Gurney formulae can be used effectively to calculate the shaped charge jet characteristics with reasonable accuracy.
4 ConclusionThe relation between the characteristics Gurney velocity of a range of explosive materials and the ratio between the Chapman Jouget pressures to the total impulse is studied. It was found that the presented empirical relation exhibits a relative accurate tool that can used to estimate the characteristic Gurney velocity, where the maximum deviation between the measured and the calculated Gurney velocities was nearly 6%. Thus, the calculated Gurney constant was implemented in the analytical method to calculate the jet tip velocity using small caliber shaped charge, where it showed quite accurate model based on the small difference between the measured and the calculated tip velocity using the suggested Gurney approach.
[1] 
Gurney R.
The Initial Velocities of Fragments from Bombs, Shell and Grenades[M]. Maryland: Army Ballistic Research Labrary, Aberdeen Proving Ground MD, 1943.

[2] 
Walters P, Zukas J.
Fundamentals of Shaped Charge[M]. New York: Wiley Interscience Publication, 1989.

[3] 
Grady D.
Fragmentation of Rings and Shells the Legacy of N.F. Mott[J]. Berlin Heidelberg:SpringerVerlag, 2006.

[4] 
Locking P. Gurney velocity relationships[C]//In Proceedings of 29th International Symposium on Ballistics. Edinburgh: [s. n. ], 2016. 
[5] 
Elshenawy T, Elbeih A, Li Q L.
A modified penetration model for coppertungsten shaped charge jets with nonuniform density distribution[J]. Central European Journal of Energetic Materials, 2016, 13(4): 927–943.
DOI:10.22211/cejem/65141 
[6] 
GuiXi L.
The simplified model for predicting shaped charge jet parameters[J]. Propellants, Explosives, Pyrotechnics, 1995, 20(5): 279–282.
DOI:10.1002/(ISSN)15214087 
[7] 
Elbeih A, Wafy T, Elshenawy T.
Performance and detonation characteristics of polyurethane matrix bonded attractive nitramines[J]. Central European Journal of Energetic Materials, 2017, 14(1): 77–89.

[8] 
Elbeih A, Zeman S.
Characteristics of melt cast compositions based on cis1, 3, 4, 6Tetranitrooctahydroimidazo[4, 5d] imidazole (BCHMX)/TNT[J]. Central European Journal of Energetic Materials, 2014, 11(4): 501–514.

[9] 
Zeman S, Yan QL, Elbeih A.
Recent advances in the study of the initiation of energetic materials using the characteristics of their thermal decomposition Part Ⅱ. Using simple differential thermal analysis[J]. Central European Journal of Energetic Materials, 2014, 11(3): 395–404.

[10] 
Elbeih A, Mokhtar M M, Wafy T.
Sensitivity and detonation characteristics of selected nitramines bonded by Sylgard binder[J]. Propellants, Explosives, Pyrotechs, 2016, 41: 1044–1049.
DOI:10.1002/prep.v41.6 
[11] 
Backofen J E, Chris A W.
Obtaining the gurney energy constant for a twostep propulsion model[J]. AIP Conference Proceedings, 2002, 620(1): 958.

[12] 
Wlodarczyk E, Fikus B.
The initial velocity of a metal plate explosivelylaunched from an openfaced sandwich (OFS)[J]. Engineering Transactions, 2016, 64(3): 287–300.

[13] 
Elek P, Jaramaz S, Mickovic D.
Modeling of the metal cylinder acceleration under explosive loading[J]. Science Technology Review, 2013, 63(2): 39–46.

[14] 
Kennedy J E. Gurney energy of explosives: estimation of the velocity and impulse imparted to driven metal. SCRR——70790[R]. Albuquerque, NM: Sandia National Laboratories Report, 1970. 
[15] 
Koch A, Arnold N, Estermann M.
A simple relation between the detonation velocity of an explosive and its Gurney energy[J]. Propellants, Explosives, Pyrotechnics, 2002, 27(6): 365–368.
DOI:10.1002/prep.200290007 
[16] 
Keshavarz M H, Semnani A.
The simplest method for calculating energy output and Gurney velocity of explosives[J]. Journal of Hazardous Materials, 2006, 131(1): 1–5.

[17] 
Keshavarz M H.
Prediction method for specific impulse used as performance quantity for explosives[J]. Propellants, Explosives, Pyrotechnics, 2008, 33(5): 360–364.
DOI:10.1002/prep.v33:5 
[18] 
Hurley C. Development of ammonium nitrate based explosives to optimize explosive properties and explosive welding parameters used during explosion cladding[D]. Colorado: Colorado School of Mines, 2007. 
[19] 
Ludwig W. Investigation of the effect of convergent detonation on metal acceleration and Gurney. NPSPH12001[R]. Monterey, California: Naval Postgraduate School, June 2012. 
[20] 
Frem D.
A simple relationship for the calculation of the Gurney velocity of high explosives using the BKW thermochemical code[J]. Journal of Energytic Material, 2015, 33(2): 140–144.
DOI:10.1080/07370652.2014.940474 
[21] 
Hardesty D R, Kennedy J E.
Thermochemical estimation of explosive energy output[J]. Combustion and Flame, 1977, 28: 45–59.
DOI:10.1016/00102180(77)900074 
[22] 
Vigil M G. Optimized conical shaped charge design using the SCAP code. SAND881790[R]. Albuquerque, New Mexico: Sandia National Laboratories, 1988. 
[23] 
Zukas J A, Walters W, Walters W P.
Explosive Effects and Applications[M]. New York: Springer Science & Business Media, 2002.

[24] 
Pelikán V, Zeman S, Yan Q L, et al.
Concerning the shock sensitivity of cyclic nitramines incorporated into a polyisobutylene matrix[J]. Central European Journal of Energetic Materials, 2014, 11(2): 219–235.

[25] 
Chou P C, Flis W J.
Recent developments in shaped charge technology[J]. Propellants, Explosives, Pyrotechnics, 1993, 11: 99–114.
