### Velocity and Shape of Explosive Computation using Multi-Material and ALE Formulations

#### Abstract

In this paper, a mathematical and numerical description of the bulk viscosity for an equation of state that is linear in density is presented. The bulk viscosity is used in many academic and industrial dynamic codes, and there is no description concerning the smearing of the shock for engineers and researchers in the manuals or in published papers. To clearly show the usefulness of the bulk viscosity, a simple one dimensional problem is used, where a shock is developed through a pressure wave travelling inside a compressible fluid. By adding a viscous pressure to equilibrium equations, high oscillations in the front shock have been considerably attenuated, by thickening the shock over few element mesh sizes.

The method is developed mathematically for one dimensional hydrodynamic problem, but has been used successfully for more complex applications including high-impact problems, explosive detonation in air and underwater explosions. Application of the method to a complex problem is illustrated in calculation of the peak velocity and shape of an explosively-formed projectile (EFP).The symmetry common to most EFPs permits their characterization using 2D axisymmetric analysis. Formation of an EFP entails volumetric expansion of the explosive and extensive plastic flow of the metal plate, both of which can be calculated using an Arbitrary Lagrangian Eulerian (ALE) method. Accordingly, a 2D axisymmetric ALE was used to calculate the velocity and shape of an EFP. The methodology was validated against EFP velocity and shape measurements published in SAND-92-1879 [Hertel 1992].

The Jones-Wilkins-Lee (JWL) equation of state (EOS) were used for the LX-14 high explosive backing the copper plate. The explosive burn was initiated using a high explosive material which converts the explosive charge into a gas at high pressure and temperature. The copper plate and steel casing were included using the constitutive model developed by Johnson and Cook. An equation of state developed by Grüneisen for high-pressure simulation was used for the metals. The calculated peak velocity of the EFP was in excellent agreement with the peak velocity published by Hertel. However, the calculated shape did not agree with the experimental shadowgraph of the plate. Specifically, the calculated shape was elongated compared to the measurement and continued to elongate as long as the calculation was continued. In other words, the shape of the copper plate did not reach a dynamic equilibrium.

The methodology for calculating the EFP peak velocity and shape is described. The calculated results are compared to measurements from Hertel. Finally, possible sources for the inaccuracy of the calculated shape are investigated. These include the element size and formulation, initial geometry of EFP, explosive equation of state and the constitutive model for the copper plate*.** *

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DOI: http://dx.doi.org/10.21152/1750-9548.13.1.31

Copyright (c) 2019 M Souli, J Puryear, B Harrison

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