RD-E: 4601 Lagrange Formulation

The purpose of this example is to show how to simulate the cylinder expansion test and compare the simulation result to experimental data.

Figure 1.


Detonation is initiated at the bottom of the explosive. Radial expansion of the cylinder is measured and compared to experimental data.

The following features are used in the model:
  • Axisymmetrical analysis (/ANALY)
  • Lagrange formulation
  • Quad elements

Options and Keywords Used

Input Files

Before you begin, copy the file(s) used in this example to your working directory.

Modeling Video

Model Description

A OFHC copper cylinder (1.53cm diameter, 0.26cm thickness, 30.5cm height) is filled with an explosive (TNT). Detonation is initiated at the bottom of the explosive. Radial expansion is measured at a length of 8*D cm.

Since this problem is axisymmetric, only a quarter of the cylinder is modeled.
Figure 2. Problem description for cylinder test

ex46_problem_description

Units: cm, μ s, g, Mbar

The TNT material uses Jones-Wilkins-Lee Material (/MAT/JWL) and Lagrange formulation with the following characteristics:
Material Properties
Value
Initial density
1.63
A
3.7121
B
0.0323
R1
4.15
R2
0.95
ω
0.3
Chapman Jouget parameters enable detonation time to compute and burn fraction evolution:
Detonation velocity D
0.693
Chapman Jouguet pressure PCJ
0.21
Detonation energy E0
0.07
Radioss Card (TNT)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/JWL/2
TNT
#              RHO_I
                1.63
#                  A                   B                  R1                  R2               OMEGA
              3.7121               .0323                4.15                 .95                  .3
#                  D                P_CJ                  E0                Eadd   I_BFRAC     Q_OPT
                .693                 .21                 .07                   0         0         0		
#                 P0                 Psh                B_un
                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
Using Hydrodynamic Johnson-Cook material law (/MAT/LAW4), the copper cylinder material has the following characteristics:
Material Properties
Value
Initial density
8.96
E-Module
1.24
Poisson's ratio
0.35
A
0.9e-3
B
0.292e-2
N
0.31
σ max
0.0066
C
0.025
ε ˙ 0
1e-5
M
1.09
ρ 0 C p
3.461e-3
Tmelt
1656
The Gruneisen equation of state (/EOS/GRUNEISEN) is used for copper with the following characteristics:
Material Properties
Value
C
0.394
S1
1.489
γ 0
1.97
a
0.47
E0
8.96
Radioss Card (Copper)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYD_JCOOK/1
Copper
#              RHO_I
                8.96
#                E0                  nu
                1.24                 .35
#                 A                   B                   n              epsmax              sigmax
              .9E-03            .292E-02                 .31                   0              0.0066
#              Pmin
              -1.E30
#                 C           EPS_DOT_0                   M               Tmelt                Tmax
             .25E-01              .1E-05                1.09              1656.0                1e30
#              RHOCP                                                          Tr
            3.461E-5                                                           0
/EOS/GRUNEISEN/1
Copper
#                  C                  S1                  S2                  S3
                .394               1.489                   0                   0
#             GAMMA0               ALPHA                  E0               RHO_0
                1.97                 .47                   0                8.96 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Model Method

A 2D axisymmetric mesh is made of Quad elements. The element size is approximately of 0.035 cm x 0.035 cm.

Using N2D3D=1 (axisymmetrical), the actual structure is the 2D mesh rotated around the z-axis. It is important to have 2D mesh in YZ plane and element normals have to be in the positive x-direction.
Figure 3. Model mesh

ex46_model_mesh

Boundary condition is set on the XZ plane at y = 0 (Tx = 0, Ty = 0) to ground the model.

A planar detonation wave is defined at the bottom of the cylinder.

In order to plot the curve of radial expansion, displacements of node ID 31240 z = 24.48 cm on the outer wall of the copper cylinder are saved in time history. It corresponds to L/D=8 in agreement with experimental protocol.

P j w l = A ( 1 ω R 1 V ) e R 1 V + B ( 1 ω R 2 V ) e R 2 V + ω ( E + Q ) V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadcfadaWgaaWcbaGaamOAaiaadEhacaWGSbaabeaakiabg2da 9iaadgeadaqadaqaaiaaigdacqGHsisldaWcaaqaaiabeM8a3bqaai aadkfadaWgaaWcbaGaaGymaaqabaGccaWGwbaaaaGaayjkaiaawMca aiaadwgadaahaaWcbeqaaiabgkHiTiaadkfadaWgaaadbaGaaGymaa qabaWccaWGwbaaaOGaey4kaSIaamOqamaabmaabaGaaGymaiabgkHi TmaalaaabaGaeqyYdChabaGaamOuamaaBaaaleaacaaIYaaabeaaki aadAfaaaaacaGLOaGaayzkaaGaamyzamaaCaaaleqabaGaeyOeI0Ia amOuamaaBaaameaacaaIYaaabeaaliaadAfaaaGccqGHRaWkdaWcaa qaaiabeM8a3naabmaabaGaamyraiabgUcaRiaadgfaaiaawIcacaGL PaaaaeaacaWGwbaaaaaa@610A@

A scale factor of 0.5 (on time step for all elements) is used for this type of application.

In solid properties, qa and qb default values are used. These values have to be changed depending of the formulation (ALE, Euler).

Radioss Card (TNT)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/PROP/SOLID/2
TNT
#   Isolid    Ismstr               Icpre  Itetra10     Inpts   Itetra4    Iframe                  dn
         0         0                   0         0         0         0         0                   0
#                q_a                 q_b                   h            LAMBDA_V                MU_V
                   0                   0                   0                   0                   0
#             dt_min   istrain      IHKT
                   0         0         0 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
Radioss Card (Copper)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/PROP/SOLID/1
Copper
#   Isolid    Ismstr               Icpre  Itetra10     Inpts   Itetra4    Iframe                  dn
         0         0                   0         0         0         0         0                   0
#                q_a                 q_b                   h            LAMBDA_V                MU_V
                   0                   0                   0                   0                   0
#             dt_min   istrain      IHKT
                   0         0         0 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Results

Curves and Animations

The following diagrams display the pressure and density in the cylinder and the explosive.
Figure 4. Pressure distributed in Copper and TNT at time = 11 μ s

ex46_pressure_dist
Figure 5. Density distributed in Copper and TNT at time = 11 μ s

ex46_density
Figure 6 shows the comparison between the experimental and simulation measurement of radial expansion.
Figure 6. Comparison between experimental results and simulation results

ex46_comparison

Conclusion

Good correlation between experimental and simulation results. A thinner meshing could improve the correlation between simulation and experimental curves.

Elapsed time for simulation: t = 11 441 s, 8514 cycles, (4 cpu intel core i7 Q 840 @ 1.87 GHz).

As the model is Lagrangian, the mesh becomes very distorted at the end of the simulation to obtain a proper mesh, it is possible to use the Euler method.