Engineering Computations Computer simulation of two-dimensional linear-shaped charge jet using smoothed particle hydrodynamics Yang Gang Han Xu Hu De'an
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To cite this document: Yang Gang Han Xu Hu De'an, (2011),"Computer simulation of two-dimensional linear-shaped charge jet using smoothed particle hydrodynamics", Engineering Computations, Vol. 28 Iss 1 pp. 58 - 75 Permanent link to this document: http://dx.doi.org/10.1108/02644401111097028 Downloaded on: 29 April 2015, At: 18:12 (PT) References: this document contains references to 17 other documents. To copy this document:
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Received 29 November 2009 Revised 22 April 2010 Accepted 29 April 2010
Computer simulation of two-dimensional linear-shaped charge jet using smoothed particle hydrodynamics Yang Gang, Han Xu and Hu De’an State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, China Abstract Purpose – The purpose of this paper is to investigate the formation process of linear-shaped charge jet using the smoothed particle hydrodynamics (SPH). Different material yield models are embed to test the performance of SPH method in the simulation of explosive driven metal liner. The effects of different ignition model to the formation of metal jet have also been studied. Design/methodology/approach – The SPH method is used with the correction of artificial viscosity and penalty force to simulate the formation process of linear-shaped charge jet, which includes the process of explosion and interaction between explosive gas and metal liner. The numerical results which got by SPH method are compared with these obtained by mesh-based method. Different material yield models are implemented in the numerical examples to show the effect of material model to the formation process of metal jet. The single point and two point ignition models are used to study the effect of ignition models to the process of explosion and formation of metal jet. Findings – Compared with the original mesh-based method, the SPH method can simulate the physical process of linear-shaped charge jet naturally, as well as the capturing of explosive wave propagation. The implementation of different material yields models to obtain the same formation tendency of metal jet, but some numerical difference exists. In two-point ignition model the explosive pressure is superimposed at the location that two detonation waves intersect. Compared with two ignition models, the two point ignition model can form the metal jet faster and get the higher velocity metal jet. Originality/value – There are a few references that address the application of SPH to simulate shaped charge explosion process. The feasibility of the SPH method to simulate the formation process of linear shaped charged jet is tested and verified in this paper. From the results which compared with mesh-based method, it is shown that the SPH method has the advantage in tracking the large deformation of material and capturing the explosive wave propagation. The SPH method can be selected as a good alternative to traditional mesh-based numerical methods in simulating similar explosively driven metal material problems which can be referenced from this paper. Keywords Jets, Deformation, Hydrodynamics Paper type Research paper
Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 28 No. 1, 2011 pp. 58-75 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644401111097028
1. Introduction The linear-shaped charge jet is often used in transient cutting device for its high-speed performance. The detonation of high explosive and the interaction between detonation The financial supports from National Pre-research Project (9140A04040208JW3201) and National Defense Foundation of China (A1420080116) are gratefully acknowledged.
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products and metallic liner are factored into the formation of linear-shaped charge jet, the process of which is too complex for the theoretical analysis. As for experimental study, which is usually not carried out in a cost-effective manner for researches on linear-shaped charge jet, sometimes certain physical phenomenon related to the transient process cannot be captured when conducting the experiments of explosive dynamics. Along with improvements in computer technology and numerical methods, more and more studies on detonation process and the linear shaped charge have been concentrated on numerical simulation, which confers the benefits of efficiency and cost (Mader, 1998; Walters et al., 1989; Gazonas et al., 1995). However, many traditional mesh-based methods have deficiency in solving the detonation process and the large deformation of material simultaneously. The Lagrange mesh-based methods have the difficulties to handle the extreme deformation of grid, which is occurred in the explosion and metal large deformation process. The Eulerian mesh-based mesh cannot trace the time history of metal material motion. The smoothed particle hydrodynamics (SPH) method, which is different from the traditional mesh-based methods, provides an alternative solution for numerical simulation of linear-shaped charge jet. SPH method was originally proposed to solve the astrophysical problems (Gingold and Monaghan, 1997; Lucy, 1977). Later, this method has been developed and extended for dynamic response with material strength (Liberskty et al., 1993). Owing to the advantages of SPH method in treating large deformation and capturing material interface and free surface, this method has been introduced to study the high explosive problem including the shaped charge detonation process (Swegle et al., 1995; Liu et al., 2002, 2003a). Many contact algorithms have also arisen in SPH method to deal with the unphysical penetration between particles from different materials in high-velocity impact and high-pressure problems (Johnson et al., 1996; Campbell et al., 2000). The penalty force (Liu et al., 2003b, c), which is similar to the molecular force of Lennard-Jones form, has been employed successfully to solve the material interface of underwater explosion problem. These advantages and developments of SPH method make it fairly attractive in simulating the process of detonation and the movement of metallic material driven by detonation products. There are a few references that address the application of SPH to simulate shaped charge explosion process. Early works (Liu et al., 2003a) have been focused on the flow behavior of the explosive gas rather than the formation of true metal jet during the detonation and explosion process of shaped charge. In this paper, the SPH method is applied to investigate the formation process of linear-shaped charge jet. Both the detonation process and deformation of metallic liner are reproduced. Different kinds of material yield models for metallic liner are employed in numerical examples. The simulation results show that the presented SPH method can predict the detonation process of shaped charge as well as the formation process of metallic jet. Major physics of the formation process of linear-shaped charge jet can be captured in the simulation. 2. Governing equations The basic configuration of linear-shaped charge is shown in Figure 1. This problem generally can be considered as plane symmetry for simplification in simulation. The governing equations can be written as:
Computer simulation using SPH 59
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dr ›v b ¼ 2r b dt ›x
ð1Þ
dv a 1 ›s ab ¼ dt r ›x b
ð2Þ
du s ab ›v a ¼ r ›x b dt
ð3Þ
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where d=dt ¼ ð›=›tÞ þ v b ð›=›x b Þ is the Lagrangian time derivative, r is the density, v a is the velocity, u is the specific internal energy per unit mass, x ab is the position vector, s ab is the total stress tensor and the Greek superscripts a and b are used to indicate the Cartesian coordinate direction. The total stress tensor s ab can be written as:
s ab ¼ 2pd ab þ S ab
ð4Þ
ab
where p is the isotropic pressure and S is traceless deviatoric stress tensor. Because the viscosity and material strength of charge have less effect on the process of detonation and the formation of the metallic jet, only the material strength of the metallic liner is considered. The Jaumann rate is adopted here for isotropic elastic-plastic metal material, the constitutive equation as following: S_ ab ¼ 2G1_ ab þ S ag Vgb þ S bg Vga 1 1_ ab ¼ 1_ ab 2 d ab 1_ gg 3 a 1 ›v ›v b þ 1_ ab ¼ 2 ›x b ›x a 1 ›v a ›v b Vab ¼ 2 2 ›x b ›x a
ð5Þ ð6Þ ð7Þ ð8Þ
where G is the shear modulus and 1_ is the strain rate tensor, 1_ ab is the traceless part of 1_ and Vab is the rotation rate tensor. The provisional von Mieses flow stress (effect stress) J is computed using the deviatoric stress:
Linear shaped charge
Figure 1. The basic configuration of linear-shaped charge
Metallic liner
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ab ab J¼ S S 2
ð9Þ
When the effect stress J exceeds the yield strength Y which depended on problem, the deviatoric stresses have to be scaled back to the yield surface:
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S ab ¼
Y ab S J
ð10Þ
Different equations of state are used for shaped charge and metallic liner, respectively. The Jones-Wilkins-Lee (JWL) equation is applied to the detonation product. The pressure of the detonation product is: vh 2R1 =h vh 2R2 =h p ¼ D1 1 2 þ D2 1 2 þ vhr0 u ð11Þ e e R1 R2 where D1, D2, R1, R2 and v are fitting coefficients, h is the ratio of density h ¼ r/r0, u is the specific internal energy unit mass. The Mie-Gruneisen equation is adopted for the metallic liner: 1 pðr; eÞ ¼ 1 2 Gðh 2 1Þ pH ðrÞ þ Grðu 2 u0 Þ ð12Þ 2 8 < a0 ðh 2 1Þ þ b0 ðh 2 1Þ2 þ c0 ðh 2 1Þ3 h . 1 ð13Þ pH ðrÞ ¼ : a0 ðh 2 1Þ h#1 where subscript “H” refers to Hugoniot Curve and G is the Gruneisen parameter. The constants a0, b0 and c0 can be computed from the linear shock velocity relation: U S ¼ CS þ SS U P
ð14Þ
where US is the shock velocity, UP is the material particle velocity, CS is the sound speed and SS is the slope, then: a0 ¼ r0 C 2S
ð15Þ
b0 ¼ a0 ½1 þ 2ðS S 2 1Þ
ð16Þ
c0 ¼ a0 ½2ðS S 2 1Þ þ 3ðS S 2 1Þ2
ð17Þ
The sound speed of solid can be written as (Hallquist, 1998): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4G ›p c¼ þ 3r0 ›rentropy
ð18Þ
Two types of material yield model for metallic liner are employed in numerical simulation. The first one is the Johnson-Cook model (Johnson et al., 1983), which is stated as following:
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62
* *m 0 Y ¼ A0 þ B0 1pn ln 1 1 þ C 1 2 T _ eff p max 1 ; 1 _ _ min eff * 1_ ¼ 1_0
ð19Þ
ð20Þ
T 2 T room ð21Þ T melt 2 T room * where A0 , B0 , C0 , n and m are material constants, 1peff is the effective plastic strain, 1_ is 21 the dimensionless effective plastic strain rate for 1_0 ¼ 1:0 s , 1_min is the specified * minimum effective plastic strain rate, T is the homologous temperature, T room is the room temperature and T melt is the melting temperature of the material. When the * temperature T reaches or excesses the melting temperature T melt ðT $ 1:0Þ, the stress goes to zero for all strains and strain rates, the material no longer has strength and becomes hydrodynamic. The estimation of temperature T is defined as: u 2 u0 T ¼ T0 þ ð22Þ Cv T0 is the initial temperature, Cv is the specific heat and u0 is the initial energy per unit mass. Steinberg-Guinan model (Steinberg et al., 1980) is adopted as the second type of material yield model for metallic liner. In Steinberg-Guinan model, the shear modulus G before the material melting is defined as: 0 0 Gp GT p G ¼ G0 1 þ ðT 2 300Þ ð23Þ þ G0 h 1=3 G0 *
T ¼
E 2 Ec Cv Z h dh 1 Ec ¼ pðhÞ 2 2 300C exp a 1 2 h g0 2a h h 1 T¼
ð24Þ ð25Þ
3Rr0 ð26Þ A 0 0 where G0, Gp =G0 , GT =G0 , g0 and a are material parameters, p is the pressure, E is the specific internal energy, Ec is the cold compression energy, Cv is the specific heat, r0 is the initial density, R is the gas constant and A is the atomic weight. The melting energy Em is: Cv ¼
E m ðhÞ ¼ E c ðhÞ þ C v T M ðhÞ which is a function of the melting temperature T M ðhÞ : 1 h 2ðg0 2a2ð1=3ÞÞ T M ðhÞ ¼ T m0 exp 2a 1 2 h and T m0 is the melting temperature at r ¼ r0.
ð27Þ
ð28Þ
The yield strength is given by: 0 0 YP GT p Y ¼ Y 00 1 þ ðT 2 300Þ þ Y 0 h 1=3 G0
n Y 00 ¼ Y 0 1 þ gð1 þ 1i Þ 1
ð29Þ ð30Þ
where Y 0P =Y 0 , G0T =G0 , Y0, g and n1 are material parameters, 1i is the initial equivalent plastic strain, normally set to zero. When Y 00 exceeds Ymax, the maximum permitted yield strength, Y is set to Ymax.
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3. SPH formulation and some numerical aspects The SPH approximate equations for linear-shaped charge jet are formulated as follows: N X d ri ›W ij ¼ ri mj v bij dt ›x bi j¼1
ab N X Y sab dv ai i þ sj ¼ mj þ ij dt ri rj j¼1
ð31Þ !
›W ij ›xbi
N dui 1 X pi þ pj Y ›W ij Gi ab ab ¼ mj þ ij vbij þ 1 1 dt ri rj 2ri i i 2 j¼1 ›xbi
ð32Þ
ð33Þ
where N is the number of particles in the support domain of particle i, the index j denotes the neighbor particle of particle i, mj is the mass associated with particle j, Wij is the smoothing function of particle i evaluated at particle j, Pij is the artificial viscosity and v bij ¼ ðv bi 2 v bj Þ. The Gaussian function is employed as the smoothing function: W ðq; hÞ ¼
1 2q 2 e ph 2
ð34Þ
where q ¼ jx 2 x 0 j=h, x and x0 are the position vectors at different particles, h is the smoothing length. The Monaghan (1992) type artificial viscosity is used in the SPH methodology to stabilize the numerical scheme, which is: 8 2 > < 2aP cij fij þbP fij v ij · x ij , 0 rij ð35Þ Pij ¼ > v ij · x ij $ 0 :0 h ij v ij · x ij fij ¼ 2 ; r ij þw 2 v ij ¼ v i 2 v j ;
1 cij ¼ ðci þ cj Þ; 2 x ij ¼ x i 2 x j ;
1 rij ¼ ðri þ rj Þ 2 1 hij ¼ ðhi þ hj Þ 2
Computer simulation using SPH
ð36Þ
ð37Þ
63
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where aP and bP are constants that are both typically set around 1.0, the factor w ¼ 0:1hij is introduced to prevent singularity when two particles are approaching each other (xij ¼ 0), c represents the speed of sound. On account of the similar explosion process of high explosive, the Lennard-Jones form of penalty force is selected to represent the material interface: 8 x < D pNe 1 þ pNe 2 jx ijj2 pe $ 1 ij ð38Þ Fð pe Þ ¼ :0 p,1
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where the parameter D, N1 and N2 are taken as 106, 6 and 4, respectively. The penetration pe is detected when pe ¼
3ðhi þ hj Þ=2 $1 jx ij j
ð39Þ
The computer simulation steps of SPH method for linear-shaped charge jet are briefed as the following procedure: Step 1. Defining the position of material particles and initial information for shaped charge and metallic liner, respectively. Step 2. Searching the nearest neighboring particles for the preparation of numerical interpolation. The tree search algorithm is employed here. Step 3. Calculating the density change rate by SPH approximate equation (31). Step 4. Calculating the forces generated by the particles interactions. The forces calculated in this step include the internal force, artificial viscous force and penalty force. Note that the particle pressure is obtained by the equation of state, and the penalty force is only applied to the particles between different phases. Step 5. Calculating the change rate of momentum and energy. Update the particles position, velocity and internal energy. Step 6. Update the total time steps, then go to the Step 2 for the next time step until the time steps arrive the initial specified physical time. 4. Numerical examples Based on the SPH method mentioned above, a code is developed to simulate two-dimensional linear-shaped charge jet problems. The Comp-B and copper are chosen as the components of linear shaped charge in the numerical model. In order to verify the results obtained by SPH method, the LS-DYNA software also has been used to simulate the problem based on Lagrangian solution and Euler solution for comparison, respectively. 4.1 Single point ignition model The first numerical example discussed here is the single point ignition model. Figure 2 is the numerical sketch of shaped charge in the two-dimensional space. The ignition point locates on dot A1. The geometric dimensions of shaped charge are, L ¼ 80 mm, H ¼ 97 mm,
Computer simulation using SPH
L A2
A3 A1
65
H
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l2
Figure 2. The numerical sketch of shaped charge
l1
l1 ¼ 40 mm, l2 ¼ 35.5 mm, u ¼ 1028. The number of particles used in this simulation is 89,501, respectively, 81,185 for the charge and 8,316 for the liner. The initial smoothing length of charge particles is h10 ¼ 0.462 £ 102 3 except those near the liner particles which have the same initial smoothing length h20 ¼ 0.231 £ 102 3 as the liner particles. The parameters listed in Tables I and II are the material constants of copper for Johnson-Cook yield model and Steinberg-Guinan yield model, respectively. The parameters of Comp-B in JWL equation are listed in Table III. Table IV is the parameters for Mie-Gruneisen equation of state. Initial particles distribution is shown in Figure 3. The pressure distributions are shown in Figure 4. The results obtained by SPH method are compared with those got by LS-DYNA (Lagrangain solution and Euler solution). From this figure, it can be seen that the detonation process can be captured by SPH method as well as the mesh-based method. After the detonation shock reaching the copper liner A0 (MPa)
B0 (MPa)
C0
n
m
Tmelt (K)
Troom (K)
Cv ( J/kg K)
292
0.025
0.31
1.09
1,356
298
875
90
G0 (GPa) 47.7
Y0 (GPa)
Ymax (GPa)
g
n1
Gp0 /G0 (GPa2 1)
GT0 /G0 (kK2 1)
Yp0 /Y0 (GPa2 1)
Tm0 (K)
g0
a
0.12
0.025
36
0.45
0.028
20.38
0.028
1,790
2.02
1.5
r0 (kg/m3)
VD (m/s)
pCJ (GPa)
u0 (kJ/kg)
D1 (GPa)
D2 (GPa)
R1
R2
v
1,710
7,790
28.3
4,860
524.3
7.67
4.2
1.1
0.34
Table I. Material constants in Johnson-Cook yield model for copper
Table II. Material constants in Steinberg-Guinan yield model for copper
Table III. Material parameters and coefficients in JWL equation for Comp-B
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Table IV. Material parameters and coefficients in Mie-Gruneisen equation for copper
r0 (Kg/m3)
CS (m/s)
SS
G
8,960
3,940
1.489
2.0
0.09 0.08 0.07 0.06 y/m
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66
(the instants after 15 ms in Figure 4), the liner starts being broken and the metal jet begins forming. Since the surrounding outside space of the shaped charge is assumed to be a vacuum, with the propagation of detonation wave through the charge, the rarefaction wave also propagates into the gaseous products and result in the decrease of pressure. Figure 5 shows the formation process of metallic jet. With the increasing quantity of material converged to forming the jet, the necking phenomenon occurs due to the existence of velocity gradient along the jet body. The jet is stretched gradually until being ruptured. The Lagrangian solution of LS-DYNA is interrupted after t ¼ 55 ms, due to the large deformation of metallic liner. The SPH method can simulate this process naturally. From the results got by Euler solutions and SPH method, it can be seen that the jet has been broken into fragments at the time after t ¼ 70 ms. Figure 6 shows the velocity-time history for the front tip of jet. Figure 7 is the local enlarged image of Figure 6. The curve obtained by SPH method for Steinberg-Guinan yield model accords well with the result obtained by Lagrangian solution of LS-DYNA before the termination of computation. A sharp jump happens after t ¼ 10 ms and the velocity tends to stabilization as time increases. In the stable interval of velocity, the maximal value is which got by Johnson-Cook model of SPH method, the Euler solution is the minimum, and the result given by SPH method for Steinberg-Guinan yield model is at the middle. The distributions of vertical velocity that along the vertical direction of center line of jet at different instants are shown in Figures 8 and 9, respectively. The distribution of velocity for Lagrangian solution of LS-DYNA tends to linear earlier than those for other solutions shown in Figure 8.The vertical velocity distributions all tends to linearity with passage of time. Through the locations of velocity distribution points, the length of metal jet can be approximately estimated, e.g. the length of jet is 0.04-0.05 m at 30 ms.
0.05 0.04 0.03 0.02 0.01
Figure 3. The initial particle distribution
–0.06
–0.04
–0.02
0 x/m
0.02
0.04
0.06
t = 2 ms Fringe levels
Fringe levels
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06
2.073e–01 1.866e–01 1.658e–01
1.948e–01
1.451e–01 1.244e–01
1.558e–01
1.036e–01 8.291e–02 8.218e–02
1.169e–01
4.145e–02 2.072e–02
7.790e–02
Computer simulation using SPH
1.753e–01
1.363e–01
67
9.738e–02
5.843e–02
–8.125e–06
3.895e–02
0.04
1.948e–02
0.02
–3.316e–07
t = 8 ms Fringe levels
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Fringe levels 0.25 0.2 0.15
2.882e–01
2.533e–01
2.594e–01
2.279e–01
2.306e–01
2.026e–01
2.017e–01
1.773e–01
1.729e–01
1.520e–01
1.441e–01
1.266e–01
1.153e–01
1.013e–01
8.645e–02
0.1
7.598e–02
5.763e–02
5.065e–02
2.881e–02
0.05
2.532e–02
–1.708e–05
–3.316e–07
t = 12 ms 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Fringe levels
Fringe levels 3.920e–01
4.084e–01
3.476e–01
3.619e–01
3.032e–01
3.153e–01
2.588e–01
2.688e–01
2.145e–01
2.223e–01
1.701e–01
1.757e–01
1.257e–01
1.292e–01
8.132e–02
8.264e–02
3.694e–02 –7.444e–03
3.610e–02
–5.183e–02
–1.043e–02 –5.697e–02
t = 15 ms Fringe levels 0.07
Fringe levels
0.06
8.237e–02
7.798e–02 6.458e–02
6.859e–02
0.05
5.482e–02
5.119e–02
0.04
4.105e–02
3.780e–02
0.03
2.727e–02
2.441e–02
1.350e–02
0.02
1.102e–02
–2.726e–04
0.01
–1.405e–02
0
–2.782e–02
–2.375e–03 –1.577e–02 –2.916e–02
–4.159e–03
–0.01
–4.255e–02
–5.537e–02
–5.594e–02
(a)
(b)
(c)
Notes: (a) The solution of SPH method; (b) the Lagrangian solution of LS-DYNA; (c) the Euler solution of LS-DYNA
Figure 4. The pressure magnitude contour £ 1.0 £ 1011 Pa at different instants for the detonation process of shaped charge which ignited at the dot A1
Figure 5. The configuration of metal jet at different time instants
(I)
(II)
t = 55 ms
t = 30 ms
t = 15 ms
(III)
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68 (IV) (continued)
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(I)
(IV)
Notes: (I) The SPH result for Steinberg-Guinan yield model; (II) the SPH result for Johnson-Cook yield model; (III) the Lagrangian solution of LS-DYNA; and (IV) the Euler solution of LS-DYNA
(I)
t = 100 ms
t = 80 ms
t = 70 ms
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Figure 5.
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4,500 4,000 3,500 3,000 |vy|/ (m/s)
70
2,500 2,000
Lagrangian solution of LS-DYNA Euler solution of LS-DYNA SPH solution for Johnson-Cook yield model SPH solution for Steinberg-Guinan yield model
1,000 500 0
Figure 6. Velocity-time histories for the front tip of jet
0
10
20
30
40
50 t/ms
70
60
80
90
100
Note: The velocity showed in this picture is the absolute value of vertical velocity for the front tip of jet
4,000 3,900 3,800 3,700 |vy|/ (m/s)
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1,500
3,600 3,500 Lagrangian solution of LS-DYNA Euler solution of LS-DYNA SPH solution for Johnson-Cook yield model SPH solution for Steinberg-Guinan yield model
3,400 3,300 3,200 3,100
Figure 7. The partial enlarged drawing of Figure 6
3,000 10
15
20
25
30
35 t/ms
40
45
50
55
4.2 Two points ignition model The geometric model, particle distribution and material parameters in this example are the same as example 1, except that the ignition points are located on dot A2 and A3 as shown in Figure 2. Only Steinberg-Guinan model is considered in this example.
Computer simulation using SPH
4,000 Lagrangian solution of LS-DYNA Euler solution of LS-DYNA SPH solution for Johnson-Cook yield model SPH solution for Steinberg-Guinan yield model
3,500
71
|vy|/ (m/s)
3,000
2,500
1,500
1,000
0
–0.01
–0.02
–0.03 y/m
–0.04
–0.05
Figure 8. The distribution of vertical velocity along the vertical direction of center line which from the end part to the front tip of jet at 30 ms
4,000 Lagrangian solution of LS-DYNA Euler solution of LS-DYNA SPH solution for Johnson-Cook yield model SPH solution for Steinberg-Guinan yield model
3,500
3,000 |vy|/ (m/s)
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2,000
2,500
2,000
1,500
1,000
0
–0.05 –0.06 –0.07 –0.08 –0.09 –0.1 y/m
–0.11 –0.12 –0.13 –0.14 –0.15
In Figure 10, the pressure magnitude contours are shown at different time instants. With the propagating of shock wave in two directions, the shock wave will intersect and superpose, and the contour at 8 ms is one of the representations. Owing to the superimposed effect, the detonation pressure produced in this example is higher than
Figure 9. The distribution of vertical velocity along the vertical direction of center line which from the end part to the front tip of jet at 50 ms
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72
0.2
0.5
0.18
0.45
0.16
0.4
0.14
0.35
0.12
0.3
0.1
0.25
0.08
0.2
0.06
0.15
0.04
0.1
0.02
0.05 t = 8 ms
t = 2 ms
0.7 0.12 0.6 0.1 0.5 0.08 0.4 0.06
Figure 10. The pressure magnitude contour ( £ 1.0 £ 1011 Pa) at different instants for the detonation process of shaped charge which ignited at dot A2 and dot A3
0.3 0.04 0.2 0.02 0.1 0 0 –0.02 t = 12 ms
t = 15 ms
that in single point ignition model. Figure 11 shows the configurations of jet. The necking and breakdown are happened earlier than single point ignition model, because the higher pressure is generated in two points ignition model. The velocity-time histories for the front tip of jet are shown in Figure 12. Comparing with the result given by the single point ignition model, the absolute velocity in two-point ignition model is greater than that in single point ignition model. Figure 13 is the distributions of vertical velocity along the vertical direction of center line of jet at the time of 50 ms. The tendencies of vertical velocity distribution along the jet are similar in different ignition models, but the length of jet in two ignition models is longer than the single point ignition model at the same instant. From the results showed in this example, it is found that the mode of ignition employed in the shaped charge jet has a great effect on the performance of metal jet. Among the two numerical examples presented above, the model ignited at two points works better than the first one.
Computer simulation using SPH
t = 15 ms
73
t = 30 ms
t = 80 ms
Figure 11. The configuration of metal jet at different time instants
5,000 4,500 4,000 3,500 3,000 |vy|/ (m/s)
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t = 55 ms
2,500
The result of one point ignition The result of two point ignition
2,000 1,500 1,000 500 0
0
10
20
30
40
50 60 70 80 90 100 t/us Note: The results of single point center ignition and two points ignition are shown, respectively
5. Conclusions The implementation of SPH method to simulate the two-dimensional linear-shaped charge jet has been presented in this paper. The propagation of detonation wave in the linear shaped charge and the response characteristics of metallic jet are revealed for practical model of linear-shaped charge jet. The effect of different ignition mode for the performance of shaped charge jet has also been studied. From the numerical results which have been compared with LS-DYNA, it can be seen that the SPH method can be applied to simulate
Figure 12. Velocity-time histories for the front tip of jet
EC 28,1
5,000 4,500
The result of one point ignition The result of two point ignition
4,000
74 |vy|/ (m/s)
3,500 3,000
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2,500 2,000
Figure 13. The distribution of vertical velocity along the vertical direction of center line which from the end part to the front tip of jet at 50 ms
1,500 1,000 –0.04
–0.06
–0.08
–0.1 y/m
–0.12
–0.14
–0.16
Note: The results of single point center ignition and two points ignition are shown, respectively
the linear-shaped charge jet stably and accurately. The deficiencies of Lagrangian and Euler solution of LS-DYNA have also been exposed in the numerical results. The Lagrangian solution of LS-DYNA cannot simulate the deformation of liner completely and the computation process is interrupted by the distortion of mesh. Although the Euler solution of LS-DYNA can represent the physical process of linear-shaped charge jet, the time-history of material point is hard to track. Numerical results also indicate that the mode of ignition has a great effect on the performance of metal jet, and demonstrate that the SPH method can be a good alternative method for linear-shaped charge jet problem. But the severe mismatch of the impedance between metal and charge and the inhomogeneous distribution of particles will bring some difficulties to the calculation. The future works should pay more attention to the effect of boundary problems. References Campbell, J., Vignjevic, R. and Libersky, L.D. (2000), “A contact algorithm for smoothed particle hydrodynamics”, Computer Methods in Applied Mechanics and Engineering, Vol. 184 No. 1, pp. 49-65. Gazonas, G.A., Segletes, S.B., Stcgall, S.R. and Paxton, C.V. (1995), “Hydrocode simulation of the formation and penetration of a linear shaped demolition charge into an RHA plate”, Report No. AD-A777992, July, available at: www.stormingmedia.us/77/7779/A77792.html Hallquist, J.O. (1998), LS-DYNA Theoretical Manual, Livermore Software Technology Corporation, Livermore. Johnson, G.R., Stryk, R.A. and Beissel, S.R. (1996), “SPH for high velocity impact computations”, Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 347-73.
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Liberskty, L.D., Petschek, A.G., Carney, T.C., Hipp, J.R. and Allahdadi, F.A. (1993), “High strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response”, Journal of Computational Physics, Vol. 109 No. 1, pp. 67-75. Liu, M.B., Liu, G.R. and Lam, K.Y. (2002), “Investigation into water mitigations using a meshless particle method”, Shock Waves, Vol. 12 No. 3, pp. 181-95. Liu, M.B., Liu, G.R. and Lam, K.Y. (2003a), “Meshfree particle simulation of the explosion process for high explosive in shaped charge unlined cavity configurations”, Shock Waves, Vol. 12 No. 6, pp. 509-20. Liu, M.B., Liu, G.R., Zong, Z. and Lam, K.Y. (2003b), “Computer simulation of the high explosive explosion using smoothed particle hydrodynamics methodology”, Computers & Fluids, Vol. 32 No. 3, pp. 305-22. Liu, M.B., Liu, G.R., Zong, Z. and Lam, K.Y. (2003c), “Smoothed particle hydrodynamics for numerical simulation of underwater explosions”, Computational Mechanics, Vol. 30 No. 2, pp. 106-18. Lucy, L.B. (1977), “A numerical approach to testing of the fission hypothesis”, Astronomical Journal, Vol. 82, pp. 1013-24. Mader, C.L. (1998), Numerical Modeling of Explosives and Propellants, 2nd ed., CRC Press, Boca Raton, FL. Monaghan, J.J. (1992), “Smoothed particle hydrodynamics”, Annual Review of Astronomical and Astrophysics, Vol. 30, pp. 543-74. Steinberg, D.J., Cochran, S.G. and Guinan, M.W. (1980), “A constitutive model for metals applicable at high-strain rate”, Journal of Applied Physics, Vol. 51 No. 3, pp. 1498-504. Further reading Gingold, R.A. and Monaghan, J.J. (1977), “Smoothed particle hydrodynamics: theory and application to non-spherical stars”, Monthly Notices of the Royal Astronomical Society, Vol. 181, pp. 375-89. Johnson, G.R. and Cook, W.H. (1983), “A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures”, paper presented at the Seventh International Symposium on Ballistics, The Hague, April. Swegle, J.W. and Attaway, S.W. (1995), “On the feasibility of using smoothed particle hydrodynamics for underwater explosion calculations”, Computational Mechanics, Vol. 17 No. 3, pp. 151-68. Walters, W.P. and Zukas, J.A. (1989), Fundamentals of Shaped Charges, Wiley, New York, NY, pp. 203-9. About the authors Yang Gang is currently a Candidate for his PhD degree at the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, China. His research interests include meshfree method, SPH method, explosion shock, etc. Han Xu is currently a Professor at the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, China. His research interests include meshfree method, inverse problem, optimization technology, etc. Han Xu is the corresponding author and can be contacted at:
[email protected] Hu De’an is currently an Associate Professor at the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, China. His research interests include meshfree method, optimization technology, etc. To purchase reprints of this article please e-mail:
[email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints
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