46.2 - Euler Formulation

Title

Cylinder Expansion Test with Euler formulation

ex46_2

Number

46.2 Brief Description

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

Keywords

•	Euler formulation

•	Multi-Material, Fluid and Explosive material law (/MAT/LAW51)

•	Brick elements

RADIOSS Options

•	Axisymmetrical analysis (/ANALY)

•	Solid property (/PROP/SOLID)

•	Boundary condition (/BCS)

•	Detonation plan (/DFS/DETPLAN)

Input File

Cylinder Test: <install_directory>/demos/hwsolvers/radioss/46_TNT_Cylinder_Expansion_Test/Euler/*

Technical / Theoretical Level

Advanced

Overview

Physical Problem Description

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

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

Fig 1: Problem description for cylinder test

Units: cm, µs, g, Mbar

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

•	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 = 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

# 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.

#---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:

•	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

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

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/MAT/LAW51/2

TNT

# Iform

# 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.

#---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:

•	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

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/MAT/LAW51/3

Air

# Iform

# 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.

#---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

# 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.

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Analysis, Assumptions and Modeling Description

Modeling Methodology

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, µs, g, Mbar

ex46-2_model_mesh

Fig 2: Model mesh

RADIOSS Options 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.

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

/PROP/SOLID/2

copper

# Isolid Ismstr Icpre Inpts Itetra Iframe dn

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----|

/PROP/SOLID/1

TNT

# Isolid Ismstr Icpre Inpts Itetra Iframe dn

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----|

Simulation Results and Conclusions

Curves and Animations

The following figure displays the variation of density in the cylinder, explosive and air.

ex46_density-2

Fig. 3: Density distributed in copper and TNT at time = 33µs.

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

Fig. 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