RD-E: 4602 Euler Formulation

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

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

Options and Keywords Used

Due to the symmetries of the model, a quarter of the cylinder is modeled. Boundary conditions are set on the yOz plan at x = 0 (Tx = 0) and on the xOz plan at y = 0 (Ty = 0) to simulate the symmetry.

A planar detonation wave is defined at the bottom of the cylinder. A scale factor of 0.5 (on time step for all elements) is used for this type of application.

In solid properties, qa =1.1 and qb =0.05. These values must be set to 0 for the Lagrange formulation.
  • Isolid is set to 0 for TNT and copper solid properties.
Radioss Card (TNT)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/PROP/SOLID/2
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
                 1.1                 .05                   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
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
                 1.1                 .05                   0                   0                   0
#             dt_min   istrain      IHKT
                   0         0         0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Input Files

The input files used in this example include:
Cylinder Test
<install_directory>/hwsolvers/demos/radioss/example/46_TNT_Cylinder_Expansion_Test/Euler/*

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. With an Euler formulation, the air has to be modeled to measure radial expansion.

Since this problem is axisymmetric, only a quarter of the cylinder is modeled.

ex46_problem_description
Figure 1. Problem Description for Cylinder Test

Units: cm, μ s, g, Mbar

Using the Multi-Material Solid, Liquid, and Gas material law (/MAT/LAW51), the copper cylinder material has the following characteristics:
Material Properties
Initial density
8.96
Initial pressure C0mat1
1e-6
Hydrodynamic coefficients
C1mat1= 1.38
C2mat1= 1.372
C4mat1= 0.87
C5mat1= 0.87
Elastic shear modulus G1mat1
0.519
Yield stress amat1
0.9e-3
Plastic yield factor bmat1
0.292e-2
Plastic yield exponent nmat1
0.31
Plastic strain rate factor cmat1
0.025
Plastic reference strain rate symbol_e_mat1
1e-6
Thermal exponent mmat1
1.09
Specific heat Rhocvmat1
3.461e-5
Tmelt
1656
Radioss Card (Copper)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/1
Copper

#    Iform
        10
#              P_ext                  NU              Nu_Vol
                   0                   0                   0
#        ALPHA0_mat1           RHO0_mat1             E0_mat1           Pmin_mat1             C0_mat1
                   1                8.96                   0                   0                1E-6
#            C1_mat1             C2_mat1             C3_mat1             C4_mat1             C5_mat1
                1.38               1.372                   0                 .87                 .87
#            G1_mat1              a_mat1              b_mat1              n_mat1
                .519                9E-4              .00292                 .31
#             c_mat1   EPSILON_DOT0_mat1
                .025                1E-6
#             m_mat1             T0_mat1          Tmelt_mat1           Tlim_mat1          Rhocv_mat1
                1.09                   0                1656                   0            3.461E-5
#       Epspmax_mat1      sigma_max_mat1             KA_mat1             KB_mat1
                   0                   0                   0                   0
#        ALPHA0_mat2           RHO0_mat2             E0_mat2           Pmin_mat2             C0_mat2
                   0               .0012              2.5E-6              -1E-20                   0
#            C1_mat2             C2_mat2             C3_mat2             C4_mat2             C5_mat2
                   0                   0                   0                  .4                  .4
#            G1_mat2              a_mat2              b_mat2              n_mat2
                   0                   0                   0                   0
#             c_mat2   EPSILON_DOT0_mat2
                   0                   0
#             m_mat2             T0_mat2          Tmelt_mat2           Tlim_mat2          Rhocv_mat2
                   0                   0                   0                   0                   0
#       Epspmax_mat2      sigma_max_mat2             KA_mat2             KB_mat2
                   0                   0                   0                   0
#        ALPHA0_mat3           RHO0_mat3             E0_mat3           Pmin_mat3             C0_mat3
                   0                   0                   0                   0                   0
#            C1_mat3             C2_mat3             C3_mat3             C4_mat3             C5_mat3
                   0                   0                   0                   0                   0
#            G1_mat3              a_mat3              b_mat3              n_mat3
                   0                   0                   0                   0
#             c_mat3   EPSILON_DOT0_mat3
                   0                   0
#             m_mat3             T0_mat3          Tmelt_mat3           Tlim_mat3          Rhocv_mat3
                   0                   0                   0                   0                   0
#       Epspmax_mat3      sigma_max_mat3             KA_mat3             KB_mat3
                   0                   0                   0                   0
#        ALPHA0_mat4           RHO0_mat4             E0_mat4           Pmin_mat4             C0_mat4
                   0                1.63                 .07              -1E-20                1E-6
#                  A                   B                  R1                  R2                   W
               3.712               .0323                4.15                 .95                  .3
#                  D                 PCJ             C1_mat4
                .693                 .21                .036
/EULER/MAT/1
#     Modif. factor.
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
Using Multi-Material Solid, Liquid, and Gas material law (/MAT/LAW51), the TNT material has the following characteristics:
Material Properties
Initial density
1.63
Explosive cavitation pressure Pminmat4
-1e-20
Initial explosive pressure C0mat4
1e-6
Explosive coefficient B1
3.712
Explosive coefficient B2
0.0323
Explosive coefficient R1
4.15
Explosive coefficient R2
0.95
Explosive coefficient ω
0.3
Explosive coefficient C1mat4
0.036
Chapman Jouget parameters compute detonation time and burn fraction evolution:
Detonation velocity D
0.693
Chapman Jouguet pressure PCJ
0.21
Initial explosive energy per unit initial volume E0mat4
0.07
Radioss Card (TNT)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/2
TNT

#    Iform
        10
#              P_ext                  NU              Nu_Vol
                   0                   0                   0
#        ALPHA0_mat1           RHO0_mat1             E0_mat1           Pmin_mat1             C0_mat1
                   0                8.96                   0                   0                1E-6
#            C1_mat1             C2_mat1             C3_mat1             C4_mat1             C5_mat1
                1.38               1.372                   0                 .87                 .87
#            G1_mat1              a_mat1              b_mat1              n_mat1
                .519                9E-4              .00292                 .31
#             c_mat1   EPSILON_DOT0_mat1
                .025                1E-6
#             m_mat1             T0_mat1          Tmelt_mat1           Tlim_mat1          Rhocv_mat1
                1.09                   0                1656                   0            3.461E-5
#       Epspmax_mat1      sigma_max_mat1             KA_mat1             KB_mat1
                   0                   0                   0                   0
#        ALPHA0_mat2           RHO0_mat2             E0_mat2           Pmin_mat2             C0_mat2
                   0               .0012              2.5E-6              -1E-20                   0
#            C1_mat2             C2_mat2             C3_mat2             C4_mat2             C5_mat2
                   0                   0                   0                  .4                  .4
#            G1_mat2              a_mat2              b_mat2              n_mat2
                   0                   0                   0                   0
#             c_mat2   EPSILON_DOT0_mat2
                   0                   0
#             m_mat2             T0_mat2          Tmelt_mat2           Tlim_mat2          Rhocv_mat2
                   0                   0                   0                   0                   0
#       Epspmax_mat2      sigma_max_mat2             KA_mat2             KB_mat2
                   0                   0                   0                   0
#        ALPHA0_mat3           RHO0_mat3             E0_mat3           Pmin_mat3             C0_mat3
                   0                   0                   0                   0                   0
#            C1_mat3             C2_mat3             C3_mat3             C4_mat3             C5_mat3
                   0                   0                   0                   0                   0
#            G1_mat3              a_mat3              b_mat3              n_mat3
                   0                   0                   0                   0
#             c_mat3   EPSILON_DOT0_mat3
                   0                   0
#             m_mat3             T0_mat3          Tmelt_mat3           Tlim_mat3          Rhocv_mat3
                   0                   0                   0                   0                   0
#       Epspmax_mat3      sigma_max_mat3             KA_mat3             KB_mat3
                   0                   0                   0                   0
#        ALPHA0_mat4           RHO0_mat4             E0_mat4           Pmin_mat4             C0_mat4
                   1                1.63                 .07              -1E-20                1E-6
#                  A                   B                  R1                  R2                   W
               3.712               .0323                4.15                 .95                  .3
#                  D                 PCJ             C1_mat4
                .693                 .21                .036
/EULER/MAT/2
#     Modif. factor.
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
Using the Multi-Material Solid, Liquid, and Gas material law (/MAT/LAW51), the Air material has the following characteristics:
Material Properties
Initial density
0.0012
Initial energy per unit initial volume E0mat2
2.5e-6
Hydrodynamic cavitation pressure Pminmat2
-1e-20
Hydrodynamic coefficient C4mat2
0.4
Hydrodynamic coefficient C5mat2
0.4
Radioss Card (Air)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/3
Air

#    Iform
        10
#              P_ext                  NU              Nu_Vol
                   0                   0                   0
#        ALPHA0_mat1           RHO0_mat1             E0_mat1           Pmin_mat1             C0_mat1
                   0                8.96                   0                   0                1E-6
#            C1_mat1             C2_mat1             C3_mat1             C4_mat1             C5_mat1
                1.38               1.372                   0                 .87                 .87
#            G1_mat1              a_mat1              b_mat1              n_mat1
                .519                9E-4              .00292                 .31
#             c_mat1   EPSILON_DOT0_mat1
                .025                1E-6
#             m_mat1             T0_mat1          Tmelt_mat1           Tlim_mat1          Rhocv_mat1
                1.09                   0                1656                   0            3.461E-5
#       Epspmax_mat1      sigma_max_mat1             KA_mat1             KB_mat1
                   0                   0                   0                   0
#        ALPHA0_mat2           RHO0_mat2             E0_mat2           Pmin_mat2             C0_mat2
                   1               .0012              2.5E-6              -1E-20                   0
#            C1_mat2             C2_mat2             C3_mat2             C4_mat2             C5_mat2
                   0                   0                   0                  .4                  .4
#            G1_mat2              a_mat2              b_mat2              n_mat2
                   0                   0                   0                   0
#             c_mat2   EPSILON_DOT0_mat2
                   0                   0
#             m_mat2             T0_mat2          Tmelt_mat2           Tlim_mat2          Rhocv_mat2
                   0                   0                   0                   0                   0
#       Epspmax_mat2      sigma_max_mat2             KA_mat2             KB_mat2
                   0                   0                   0                   0
#        ALPHA0_mat3           RHO0_mat3             E0_mat3           Pmin_mat3             C0_mat3
                   0                   0                   0                   0                   0
#            C1_mat3             C2_mat3             C3_mat3             C4_mat3             C5_mat3
                   0                   0                   0                   0                   0
#            G1_mat3              a_mat3              b_mat3              n_mat3
                   0                   0                   0                   0
#             c_mat3   EPSILON_DOT0_mat3
                   0                   0
#             m_mat3             T0_mat3          Tmelt_mat3           Tlim_mat3          Rhocv_mat3
                   0                   0                   0                   0                   0
#       Epspmax_mat3      sigma_max_mat3             KA_mat3             KB_mat3
                   0                   0                   0                   0
#        ALPHA0_mat4           RHO0_mat4             E0_mat4           Pmin_mat4             C0_mat4
                   0                1.63                 .07              -1E-20                1E-6
#                  A                   B                  R1                  R2                   W
               3.712               .0323                4.15                 .95                  .3
#                  D                 PCJ             C1_mat4
                .693                 .21                .036
/EULER/MAT/3
#     Modif. factor.
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Using the Multi-Material Solid, Liquid, and Gas material law (/MAT/LAW51), the Boundary material has the following characteristics.

Radioss Card (Boundary)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/4
Boundary

#    Iform
         3

#            ALPHA_1              RHO_01                E_01              P_min1                C_01
                   0                8.96                   0                   0                1E-6


#            ALPHA_2              RHO_02                E_02              P_min2                C_02
                   1               .0012              2.5E-6              -1E-20                1E-6


#            ALPHA_3              RHO_03                E_03              P_min3                C_03
                   0                   0                   0                   0                   0


/EULER/MAT/4
#     Modif. factor.
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Model Method

A 3D mesh is made of brick elements. The element size for the copper cylinder is approximately of 0.035 cm x 0.035 cm x 0.035 cm.

The mesh is dragged along the z direction (z = 30.5 cm). It is important to have no discontinuity in element volume in order to ensure a good propagation of detonation wave and shock wave.

Units: cm, μ s, g, Mbar

ex46-2_model_mesh
Figure 2. Model Mesh

Results

Curves and Animations

Figure 3 displays the variation of density in the cylinder, explosive and air.

ex46_density-2
Figure 3. Density Distributed in Copper and TNT at time = 33 μ s
The following diagram shows the comparison between the experimental and simulation measurement of radial expansion. The displacement values are estimated on the animations using the density contour.

ex46-2_comparison
Figure 4. Comparison Between Experimental Results and Simulation Results

Conclusion

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

References

1 Adiabatic Expansion of high explosive detonation products, LANL, Wilkins (1969)
2 A Constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, Gordon R. Johnson, William H. Cook