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.
![](../../../images/solvers/euler_formulation_model.png)
Figure 1.
Detonation is initiated at the bottom of the explosive. Radial expansion of the cylinder is measured and compared to experimental data.
- Multi-material
- Euler formulation
- Brick elements
Options and Keywords Used
- Multi-Material Solid, Liquid, and Gas material law (/MAT/LAW51 (MULTIMAT))
- Solid property (/PROP/TYPE14 (SOLID))
- Boundary condition (/BCS)
- Detonation plan (/DFS/DETPLAN)
Input Files
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. With an Euler formulation, the air has to be modeled to measure radial expansion.
![ex46_problem_description](../../../images/solvers/ex46_problem_description.png)
Figure 2. Problem description for cylinder test
Units: cm, s, g, Mbar
- Material Properties
- Value
- Initial pressure C0mat1
- 1e-6
- Hydrodynamic coefficients
- C1mat1= 1.38
- 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
- 1e-6
- Thermal exponent mmat1
- 1.09
- Specific heat Rhocvmat1
- 3.461e-5
- Tmelt
- 1656
#---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----|
- Material Properties
- Value
- 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
- 0.3
- Explosive coefficient C1mat4
- 0.036
- Detonation velocity D
- 0.693
- Chapman Jouguet pressure PCJ
- 0.21
- Initial explosive energy per unit initial volume E0mat4
- 0.07
#---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----|
- Material Properties
- Value
- 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
#---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.
#---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.
![ex46-2_model_mesh](../../../images/solvers/ex46-2_model_mesh.png)
Figure 3. Model mesh
![](../../../images/solvers/euler_formulation_bc.png)
Figure 4. Boundary condition
![](../../../images/solvers/euler_formulation_planar.png)
Figure 5. Planar detonation
- Isolid is set to 0 for TNT and copper solid properties.
#---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----|
#---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----|
Results
Curves and Animations
![ex46_density-2](../../../images/solvers/ex46_density-2.png)
Figure 6. Density distributed in Copper and TNT at time = 33s
![ex46-2_comparison](../../../images/solvers/ex46-2_comparison.png)
Figure 7. 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.