1. IMPACT OF MULTILAYERED EXPLOSIVE CHARGE ON A RIGID WALL A.A. Shtertser Design & Technology Branch of Lavrentyev Institute of Hydrodynamics SB RAS Tereshkovoi Str., 29, Novosibirsk, 630090, Russia asterzer@mail.ru EPNM -2010

2. HE are used in experimental investigation of high pressure effect on substance properties, in explosive working of materials (explosive welding, hardening, forming and so on) and in explosive cutting and demolition. As a rule HE charge is positioned in contact with any material or structure. At explosion high impulsive pressure is generated in treated material as high as tens and hundreds of kbar. It is often needed to estimate (even approximately) a pressure profile and total pressure impulse effecting the structure. This problem becomes more complicated if multilayered charge is used consisting of several layers of different HE. Generally speaking, this problem can be solved by numerical calculations with the use of computer. Nevertheless, use of physical models with simple and convenient formulas can give us fundamental understanding of processes running at explosive loading, In this presentation we offer the approach and formulas for estimation of pressure and impulse. This approach is based on known solutions taken from physics of explosion, on results of numerical calculations, and on certain physical considerations. Problem Statement EPNM -2010

3. Problem Statement The engineering formulas should be derived to estimate pressure and pulse (impact momentum) effecting the rigid wall at detonation of one- and multi- layered HE charge in one-dimension and two-dimension (plane) geometry. Rigid wall simulates any structural design which is subjected to HE impact. Compressibility of wall material can be considered additionally, and corrections to results of calculations can be made if it’s needed. Explosive is characterized by density  е, detonation velocity D H, and adiabatic exponent  of detonation products. Pressure at the Chapman- Jouguet plane is considered to be as detonation pressure P H =  e D H 2 /(  +1), where  = 2.8, 2.5, and 2.2 for RDX, amatol, and amatol/ammonium nitrate- 1/1 mixture correspondingly. EPNM -2010

4. One-Dimensional Expansion of Detonation Products Single-Layer Explosive Charge Explosive charge on the wall: a- DW falls normally on the wall; b- DW moves off the wall. 1- wall, 2- explosive, 3- front of DW, 4- detonation products. Arrays show the direction of DW propagation. 1 4 3 2 1 2 3 4 a- ignition at the open side of HE charge b- ignition at the wall There are two possible variants depending on ignition place Lateral dimension of HE charge and wall are considered to be sufficiently large therefore side flow of detonation products is not taken into account EPNM -2010

5. DW falls normally on the wall Pressure profile for  = 3 is described by expression: Physics of Explosion / edited by K.P. Stanyukovich, 2 edition, Moscow: Nauka, 1975 Pressure impulse derived by integration ( γ = 3):  =  е / D H t P0P0 P(t) τ2τ 8 times pressure drop 333 64 27 PHPH τ t P =≈PHPH 2,37 τ t =P0P0 τ t (1) 8 27 m e D H J =≈0,296m e D H (2) Pressure profile. Reference time is the moment of ignition. EPNM -2010

6. DW falls normally on the wall For arbitrary γ initial (peak) pressure on the wall P 0 can be calculated using the expression * : * Landau L.D., Lifshitz E.M. Hydrodynamics. 4 edition, Moscow: Nauka, 1988 P 0 /P H ratio depends weakly on adiabatic exponent . It changes from 2.6 to 2.3 as  changes from 1 to . For example P 0 / P H = 2.40, 2.41 and 2.43 for 2.8 (RDX), 2.5 (amatol) and 2.2 (amatol/ammonium nitrate-1/1 mixture) correspondingly. Rounding to one decimal place, we get that peak pressure can estimated for any  by formula: P0P0 PHPH = 5γ + 1 + (17γ 2 + 2γ + 1) 1/2 4γ (3) P 0 =2.4PHPH (4) EPNM -2010

7. DW moves off the wall Landau L.D., Lifshitz E.M. Hydrodynamics. 4 edition, Moscow: Nauka, 1988 2γ/(γ-1) PHPH γ +1 2γ P 0 = (5)(5) Gaseous detonation products at the wall have zero velocity. For strong DW sound velocity at the wall is c = D H / 2 for any . From this it follows that Calculations with the use of this formula give P 0 /P H = 0.296, 0.299, 0.305, 0.311 for  =3; 2.8 (RDX); 2.5 (amatol); 2,2 (А/AN-1/1) correspondingly. Rounding to one decimal place, we get that initial pressure can be estimated for any HE using the formula: 0,3P H P 0 = (6)(6) EPNM -2010

8. DW moves off the wall P(t) t P0P0 τ = 3δ e /D H Pressure profile. Reference time is the moment of ignition. The time point t = 3δ e /D H is the sum of DW run time from the wall to the HE charge open end and of rarefaction wave run time from the charge open end to the wall. During this time pressure is constant and equal to P 0. At t = 3δ e /D H pressure drop begins. EPNM -2010

9. Gurney Energy To derive formula for pressure impulse effecting the wall we make use of R. W. Gurney approach which was developed in 1943-47 for estimation of grenade and shell debris velocities. This approach was later successfully used for calculation of velocities of plates accelerated by explosive layers 1, 2. Consider that all chemical energy (heat of explosion) is transformed into kinetic energy of detonation products which move off the wall. This energy per mass unit is calculated using the expression 1,2 : 1) Deribas A.A. Physics of explosive hardening and welding. Novosibirsk: Nauka, 1980. 2) De Carli P.S., Meyers M.A. Design of Uniaxial Strain Shock Recovery Experiments // Proceed. Int. Conf. "Shock Waves and High-Strain-Rate Phenomena in Metals", Albuquerque, NM, 1980. New York: Plenum Press, 1980. P.341-373, 1033-1039 (Appendix A). E = Q = DH2DH2 2( γ 2 – 1) (7)(7) E is referred to as Gurney energy. Here it is equal to heat of explosion Q EPNM -2010

10. Pressure Impulse 0 δeδe W = ρ e u 2 2 (8)(8) ∫ dx = ρ e δ e u s 2 6 Consider that mass velocity of gaseous detonation products (DP) changes linearly along the charge thickness  е and DP density is everywhere equal to initial HE density  e. Then DP kinetic energy is u s =(9)(9)(6E) 1/2 = 3 γ 2 - 1 DHDH 1/2 Here u s is DP velocity at HE charge free surface. From (7) and (8) we have EPNM -2010

11. Pressure Impulse 2 J = (10) k( γ ) m e D H m e u s = k( γ ) = (11) 3 4( γ 2 – 1) 1/2 As long as DP velocity distribution is linear, then average u value equals to u s /2. Therefore total impulse effecting the wall is where coefficient k( γ) is defined as For γ = 3 difference of J values calculated by formulas (2) and (10) is 3.4%. This confirms that Gurney approach is quite relevant. EPNM -2010

12. Comparison with numerical calculations To verify additionally usefulness of formulas (10, 11) for any γ Yu. P. Mesheryakov (DTB of LIH) has performed numerical calculations of pressure and total impulse for two situations: a) DW falls normally on the wall, b) DW moves off the wall. As example the results for amatol (ρ e = 1 g/cm 3, δ e = 10 mm, D H = 4 km/s, γ = 3) are given in the table below. DW falls normally on the wallDW moves off the wall Numerical calculation Calculation with use of (4) and (10) Numerical calculation Calculation with use of (6) and (10) P 0, Gpa10.2610.971.321.37 J, GPa  μs 15.5915.1216.4115.12 EPNM -2010

13. Comparison with numerical calculations It is evident that numerically calculated impulses are actually the same for falling down and moving off DW (the difference is  5%). Results of numerical calculations and estimations using the derived formulas (4, 6, 10) differ in less than 10%. Thus these formulas can be used in practice. Impulse affecting the wall does not depend on a point of ignition, irrespective of where it was done – at the wall or at the open end of HE charge. On the contrary, pressure profile and maximal pressure on the wall depends on a point of ignition substantially. EPNM -2010

14. Pressure Impulse === γ + 1 3 J 1 = (12) 3P 0 δ e γ + 1 2 γ /( γ -1) m e D H D H 2 γ = Remark should be made concerning pressure profile for DW moving off the wall. Total impulse is the sum of two parts: rectangular part of pressure profile and tail part, where pressure drops, J = J 1 + J 2. Rectangular part gives contribution to the total impulse P(t) t P0P0 τ = 3δ e /D H J1J1 J2J2 1/2 3(γ – 1) λ = (13) J 1 J = 2 γ + 1 2γ /( γ -1) γ + 1 2 γ The J1/J ratio depends on adiabatic exponent γ only: For estimations one can take λ ≈ 0.7 actually for all explosives EPNM -2010

15. Pressure Impulse (data for calculations) Explosive  e, g/cm 3 D H, km/sP H, GPa  k(  ) (  ) RDX (cyclonite)1.0 1.25 1.8 6.2 6.84 8.75 10.0 17.0 35.0 2.8 2.44 2.94 0.33 0.39 0.31 0.714 0.685 0.722 Amatol1.04.25.02.50.380.691 Amatol/AN – 1/11.04.05.02.20.440.660 TNT1.62 1.56 1.3 1.2 1.0 0.7 6.95 6.7 6.5 5.7 5.0 4.3 18.5 20.4 14.0 10.0 7.0 3.0 3.23 2.44 2.92 2.89 2.57 3.31 0.28 0.39 0.32 0.37 0.27 0.738 0.685 0.721 0.719 0.697 0.741 Ammonium Nitrate (AN)1.722.73.03.180.290.735 Plastic explosive G-751.88.030.02.840.330.716 Plastic explosives: GP-74 GP-87 1.6 7.0 7.2 20.0 21.0 2.92 2.94 0.32 0.31 0.721 0.722 TNT/RDX - 36/64 TNT/RDX 50/50 1.7 1.6 8.0 7.6 29.0 23.0 2.75 3.01 0.34 0.31 0.710 0.726 PETN1.78.332.32.630.360.701 Tetril1.717.8528.02.770.340.711 Nitroguanidine1.557.6521.23.270.280.740 Nitroglicerin1.67.728.52.330.410.674 Low Velocity Dynamite0.94.45.52.170.440.656 Minol-2 (AN20+ TNT40+Al20) 1.685.8219.81.870.550.611 EPNM -2010

16. Multilayered Charge Multilayered HE charges give possibility to vary pressure profile shape in a wide diapason, at that mass of a charge is constant. Multilayered HE charges are used in experiments and can be used effectively in explosive metalworking. A.I.Gulidov (ITAM SB RAS) has performed numerical calculations for 2- and 3-layered charges with ignition point at the wall and at the charge open end. The next combinations were considered (count of layers from the wall): 1) Three-layered charge. Layer 1: RDX, δ e = 132 mm (  e =1 g/cm 3, D H = 6.2 km/s,  =2.8); layer 2: A/AN-1/1, δ e = 135 mm (  e =1 g/cm 3, D H =4.0 km/s,  =2.2); layer 3: uglenit E6, δ e = 80 mm (  e =1 g/cm 3, D H =2.0 km/s,  =2.2); 2) Three-layered charge. Layer 1 : Amatol/RDX-3/2, δe = 45 мм (  e = 1 г/см3, D H = 5.6 км/с,  = 2.8); Layer 2: A/AN - 1/1, δe = 45 mm (  e = 1 g/cm 3, D H = 4.0 km/s,  = 2.2); Layer 3: ANFO, δe = 90 mm (  e = 1 g/cm 3, D H = 2.9 км/с,  =2.2); 3) Two-layered charge. Layer 1: A/AN -1/1, δe = 60 мм (  e =1 g/cm 3, D H =4.0 km/s,  =2.2); Layer 2: ANFO, δe = 60 mm (  e =1 g/cm 3, D H =2.9 km/s,  =2.2); 4) Two-layered charge. Layer 1: Plastic EVV-11, δe = 5 mm (  e =1.6 g/cm 3 D H =7.6 km/s,  = 2.8); Layer 2: ANFO, δe = 120 mm (  e =1 g/cm 3, D H =2.9 km/s,  =2.2); EPNM -2010

17. Pressure Impulse of Multilayered HE Charge J = ∑ J i = ∑ k( γ i ) m ei D Hi (14) Analysis of numerical calculations show that total impulse can be estimated by additivity method using the formula J i – impulse of i-layer of explosive charge. Summation is made by number of explosive layers. For DW falling down on the wall P(t) is well described by formula (1) derived for one-layer charge: P(t) = P 0 ( τ/t) 3. At that, time parameter  = 2J/P 0, where J is calculated by (14). Initial pressure P 0 is find using formula (3) or estimated as P 0 = 2.4 P H with accuracy reasonable for practice. For DW moving off the wall pressure profile has a step-wise shape. The number of steps equals to number of explosive layers. Pressure at any step can be estimated using formula (5) or more approximately as P0i = 0.3 PH. Time duration of any can be calculated using the expression τ i = λ i J i / P 0i (15) EPNM -2010

18. Pressure Impulse of Multilayered HE Charge (results of numerical and analytical calculations) Impulse, GPa  µs Numerical CalculationCalculation by additive method (14) Combination NoDW falls down on the wallDW moves off the wall 12341234 525 245 178 157 531 249 179 158 578 277 182 173 Average difference between results of analytical and numerical calculations is about 8%. EPNM -2010

19. Sliding Detonation. One-Layer HE charge Sliding detonation is mostly used in explosive working of materials 4 1 2 3 Peak pressure on the wall is equal to detonation pressure P 0 = P H, and pressure profile is well described by exponential function DW is perpendicular to the wall. 1- rigid wall; 2- explosive layer; 3- DW front; 4- detonation products flying away. Array shows the direction of DW movement. 3( γ + 1) 4( γ – 1) 1/2 ee DHDH τ = (23) P = P 0 exp (- t / τ )(16) EPNM -2010

20. Sliding Detonation. One-Layer HE charge E = Q – u 2 /2 = Q - DH2DH2 2( γ + 1) 2 (17) As the DP flow is two-dimensional gas velocity has two components – normal and tangential to the load wall surface. Therefore total impulse affecting the wall is less than in one-dimensional case. And Gurney energy is also less than heat of explosion. De Carly P.S. and Meyers M.A. have suggested the formula for calculation of Gurney energy. De Carli P.S., Meyers M.A. Design of Uniaxial Strain Shock Recovery Experiments // Proceed. Int. Conf. "Shock Waves and High-Strain-Rate Phenomena in Metals", Albuquerque, NM, 1980. New York: Plenum Press, 1980. P.341-373, 1033-1039 (Appendix A). Here u- mass velocity of DP in Chapman-Jouguet point. The meaning of (17) is that kinetic energy of DP connected with mass velocity parallel to the wall surface is subtracted from heat of explosion. From (17) and (7) we get E = DH2DH2 ( γ – 1)( γ + 1) 2 (18) And for coefficient k( γ) 1 k ( γ ) = 3 2( γ - 1) (19) ( γ + 1) 1/2 EPNM -2010

21. Sliding Detonation. Arbitrary angle of DW collision with the wall DW falls down on the wall at the angle α (a), DW moves off the wall at the angle α (b), irregular mode with Mach stem (c). 1- rigid wall, 2- explosive layer, 3- DW front, 4- DP, 5- Mach stem, α- angle between DW front and the wall. Array shows the direction of DW movement, D= D H /Sinα. Stationary sliding detonation with inclined DW front can be realized only with use of additional explosive layer with greater detonation velocity. On picture (a) this imagined layer is on the top of layer 2, whereas on picture (b) this imagined additional layer is on the wall. Thickness of additional layer is supposed to be much less than thickness of a charge 2, therefore its input in total impulse is negligible. α 4 1 2 3 a- regular mode α 4 1 2 3 b α 4 1 2 3 c- irregular mode 5 EPNM -2010

22. Regular and Irregular DW reflection M. Adamec, B.S. Zlobin, A.A. Shtertser. Reflection of Oblique Detonation Waves from Metal Backings // Combustion, Explosion, and Shock Waves. 1991. Vol. 27, No. 3. P. 385-387 Light emission of detonation front fixed using SNEF-4 camera (exposure time 50 ns) 1- DW front with greater velocity; 2- DW front with lesser velocity; 3- the Mach stem Calculations for regular reflection using gas-dynamic equations show that P 0 /P H ratio depends weakly on γ and α. At collision angle α > α cr regular reflection is impossible and shock-wave configuration with Mach stem appears. For widely used HE critical angle lies between 40 0 and 45 0. For example, for TNT/RDX -50/50 (  =3), powdered RDX (  =2.8), amatol (  =2.5), and A/AN- 50/50 (  =2.2) α cr = 40.7, 41.3, 42.4 and 44.1 0 correspondingly. In the region of DW irregular reflection (45 0    55 0 ) pressure behind the Mach stem is greater than pressure arising at normal collision of DW with the wall. When  changes from 55 0 to 90 0 P/P H drops monotone to 1. EPNM -2010

23. Regular and Irregular DW reflection (dependence of pressure on collision angle) , grad 5153040455055607585 P/PHP/PH 2,412,352,332,373,412,802,362,001,351,10 Calculations for γ = 3 EPNM -2010

24. Sliding Detonation. DW moves off the wall at the angle α Calculations using gas-dynamic equations show that P 0 /P H ratio depends weakly on γ. P/PHP/PH , grad  = 3  = 2.8  = 2.5  = 2.2 50.30 150.30 0.31 300.330.320.330.35 450.360.370.38 600.44 0.45 750.580.59 850.78 Diagram is graphed for γ = 3 EPNM -2010

25. Sliding Detonation. Arbitrary angle of DW collision with the wall. Multilayered charge. Sin 2 αDH2DH2 2( γ + 1) E = 1 γ - 1 (24) - γ + 1 1/2 γ - 13 4( γ 2 - 1) k ( γ, α) = 1(25)- γ + 1 Sin 2 α Using the aforesaid De Carli and Meyers approach we can get for Gurney energy and k( γ) D H1 D H2 D H3 α1α1 α2α2 α3α3 Sliding detonation in multilayered HE charge. D H1  D H2  D H3. J = Σ J i = Σ k( γ i, α i ) m ei D Hi Angles of DW fronts are connected with detonation velocities by expressions Sin  2 = D H2 / D H1, Sin  3 = D H3 / D H1. Whole of DW configuration moves with velocity equal to maximal D H. Total impulse can be find using the formula EPNM -2010

26. CONCLUSION The developed approach, obtained formulas, and data presented in tables and diagrams permit to make quickly estimation of pressure profile and total impulse affecting the rigid wall for one-layered and combined multilayered HE charge Thank you for your attention EPNM -2010

27. Pressure Impulse of One-Layer HE Charge 1- DW falls down on the wall4 2- DW moves off the wall EPNM -2010

28. Pressure Impulse of Multilayered HE Charge (DW moves off the wall) 1) Three-layered charge. Layer 1: RDX, δe =132 mm (  e =1 g/cm3, DH = 6.2 km/s,  =2.8); layer 2: A/AN-1/1, δe = 135 mm (  e =1 g/cm3, DH =4.0 km/s,  =2.2); layer 3: uglenit E6, δe = 80 mm (  e =1 g/cm3, DH =2.0 km/s,  =2.2); 2) Three-layered charge. Layer 1 : Amatol/RDX-3/2, δe = 45 мм (  e = 1 г/см3, DH = 5.6 км/с,  = 2.8); Layer 2: A/AN - 1/1, δe = 45 mm (  e = 1 g/cm3, DH = 4.0 km/s,  = 2.2); Layer 3: ANFO, δe = 90 mm (  e = 1 g/cm3, DH = 2.9 км/с,  =2.2); 3) Two-layered charge. Layer 1: A/AN -1/1, δe = 60 мм (  e =1 g/cm3, DH =4.0 km/s,  =2.2); Layer 2: ANFO, δe = 60 mm (  e =1 g/cm3, DH =2.9 km/s,  =2.2); 4) Two-layered charge. Layer 1: Plastic EVV-11, δe = 5 mm (  e =1.6 g/cm3 DH =7.6 km/s,  = 2.8); Layer 2: ANFO, δe = 120 mm (  e =1 g/cm3, DH =2.9 km/s,  =2.2); Calculations we made till 1000 µs EPNM -2010