# Multiscale modeling of armor fragmentation due to the impact of an explosively formed projectile

## Contents |

# Introduction

An accurate understanding and model of the mechanisms at work in an impact and fragmentation system is key in the modern military environment. This understanding is necessary in the design and improvement of both armors and penetrating ballistics ^{[1]}.

The difficulties of modeling the impact of an Explosively Formed Projectile (EFP) begin with the projectile. An EFP begins as a concave spherical liner made of a dense, ductile material - often copper, tantalum, or a similar alloy ^{[2]} ^{[3]} ^{[4]}. The liner is backed by a highly explosive material as shown in Figure 1.

On detonation, the center of the liner is accelerated more quickly than the edge, thus shaping and propelling the projectile. The shape will be dependent on the location and number of detonators, and on the design parameters of the liner.

An EFP should be differentiated from a shaped charge, which acts by a similar mechanism, with a couple important differences. A shaped charge liner is commonly conical and is detonated only at close range to the target based on a standoff distance. The delayed detonation is necessary because the shaped charge undergoes more drastic deformation, and the resulting projectile is a jet of molten metal. This research will focus primarily on EFP's, though many of the concepts should overlap with a shaped charge impact situation.

Modeling the impact of EFP's with any target is challenging, as they can be traveling at velocities up to 2 km/s. The impact model must be able to accurately represent the material properties of both projectile and armor at high strain rates, pressures, and temperatures.

Modern armor construction also introduces complexities into the system. The armor is often layered with both brittle and ductile materials. The brittle outer layer deters penetration by blunting or defeating the projectile, and the ductile backing distributes the impact loading. Each layer has the potential for a different fragmentation mechanism, including catastrophic brittle fracture of the outer layer or back-spall of the inner layer.

## Analytical Fragmentation Models

A variety of analytical models have been used to describe and predict the results of brittle fragmentation. These can be summarized in two general categories: statistical or thermodynamic.

Statistical fragmentation models essentially describe the distribution of fragment sizes and the total number, given the impact conditions and material. Statistical models began as far back as Mott ^{[5]} during World War II. His work provided a statistical description of empirical fragmentation phenomena that served as the basis for fragmentation research for decades. Sternberg ^{[6]} used the Mott distribution as the part of his empirical piecewise distribution fit. Grady modified and expanded Mott's description of fragmentation to improve its performance in multiple dimensions ^{[7]} ^{[8]} ^{[9]}.

Energy balance models predict the size and velocity distributions, and the number of fragments through an energy balance that considers the energy from impact and the energy dissipated by creating a surface in the material. Grady ^{[10]} and Grady and Kipp ^{[11]} developed a model that was based on both the energy and the initial distribution of flaws. Grady ^{[12]} also developed a model for spall characteristics under different strain rates, including predicting a brittle-to-ductile spall point. Additionally, De Chant ^{[13]} provided a comparison of a computational implementation of a Grady-Kipp fragmentation model combined with an analytical strain-rate model developed by leveraging hydrodynamic concepts. Specifically, he draws an analog with a turbulent flow length scale to determine a length scale parameter for strain-rate effects.

## Numerical Impact and Fragmentation Models

In addition to the analytical fragmentation models, there are a number of different numerical methods. Often these models combine finite element models with different analytical fragmentation models to capture the variation while keeping the model tractable.

Cohesive law models essentially model each fracture explicitly by including special elements, or cohesive zones, through which a crack can grow. These elements are much smaller from the bulk elastic regions. These are primarily used in numerical simulations, though for a very simple case they are semi-analytically solvable ^{[14]}. Xu and Needleman ^{[15]} and Ortiz and Suresh ^{[16]} used this model by including both elastic elements and cohesive elements in the original mesh. Camacho and Ortiz ^{[17]} built on this using an adaptive mesh to add cohesive elements based on an effective stress intensity factor derived from the traction across a potential crack path. Miller ^{[18]} also used a cohesive model, including cohesive elements in the original mesh to examine the time-dependent fragmentation effects that energy balance models typically struggle to capture. Molinari Et Al. ^{[19]} used a dynamic insertion cohesive law model, similar to Camacho and Ortiz ^{[17]}, to study the energy convergence dependence on the mesh. Molinari found that a mesh with random element spacing improved energy convergence by roughly two orders of magnitude.

The problem of fragmentation has also been studied by use of various 2 or 3 dimensional hydrocodes. Johnson et al. ^{[20]} developed a fragment distribution algorithm for the EPIC-2 hydrocode. Fahrenthold and Yew ^{[21]} implemented a Grady-Kipp fragmentation model into an Eulerian CTH hydrocode. Wu et al. ^{[22]} modeled an EFP from formation to penetration using LS Dyna3D with the Johnson-Cook material model ^{[23]} and an arbitrary Langrangian-Eularian (ALE) method to couple solid-liquid interactions.

Clayton ^{[24]} also presented a small elastic strain continuum model for impact and fragmentation that incorporates a thermodynamic and statistical prediction of fragmentation.

## Potential Modeling Methodology

The proposal is to apply the concepts of integrated computational materials engineering (ICME) ^{[25]} in order to implement a multiscale model capable of capturing the major results of impact and fragmentation, while also quantifying the related uncertainty. The relationship of the different scales is described in the following sections.

**Downscaling Requirements**

The first step in the ICME process is to downscale the requirements of the highest scale models, in order to determine what information must be supplied by the lower scale models. For this problem, this must be done for both the impact and the fragmentation model.

The impact model requires material properties, plasticity behavior, and damage parameters. The impact model gets bulk material properties such as the elastic moduli from the electronic scale. Since plasticity is driven primarily by dislocations, the model gets plasticity behavior from the nanoscale and microscale models. The microscale and mesoscale models provides the required damage nucleation, growth, and coalescence parameters.

The fragmentation model requires material properties, damage, and observable variables from the impact model. The material properties and damage parameters are provided by the identical scales as for the impact model, though the specific embodiment of the upscaled information may differ. The observable state variables that the model requires are provided by the impact model, including triaxial stress state, temperature, and local strain in the fragmenting region.

**Upscaling Methodology**

The next step in the ICME process involves upscaling the required information from the lower length scales. This section briefly describes the models that will be used.

Ab Initio principles at the electronic scale, such as Density Functional Theory (DFT) implemented in VASP ^{[26]} ^{[27]} ^{[28]} ^{[29]}, are used to obtain elastic properties and lattice structure. The elastic moduli are incorporated as material properties into the impact and fragmentation model. The moduli and the lattice structure are used to calibrate a pair potential at the next length scale, Molecular Dynamics (MD).

At the nanoscale, MD is used with a modified embedded atom method (MEAM) potential ^{[30]} ^{[31]} to determine some plasticity properties, such as dislocation mobility and nucleation, and surface energies ^{[32]}. The dislocation nucleation can be upscaled as a plasticity parameter to the impact model and the dislocation mobility to the Dislocation Dynamics (DD) model at the next length scale. The surface energies will be upscaled to the fragmentation model.

At the microscale, the DD model will be used to determine dislocation forest hardening properties and dislocation motion. Dislocation motion will be upscaled to the impact model as another plasticity parameter. The forest hardening properties will be passed to crystal plasticity at the next length scale.

Crystal plasticity at various length scales ranging from the scale of an interstitial particle to the poly crystalline scale, will be used to determine crack nucleation, growth, and interactions, as well as interactions between cracks and particles. These crack properties will be upscaled as damage and damage growth parameters to the impact and fragmentation model ^{[33]} ^{[34]}. At the polycrystalline scale, a cohesive zone fragmentation model ^{[15]} ^{[17]} ^{[18]} could also be used to quantify crack interactions under catastrophic fracture, to be incorporated into the fragmentation model.

# References

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^{2.0}^{2.1}Alain Kerdraon and Michel Vives. Performance explosive-formed projectile, June 2001. Patent. - ↑ Kimball J. Norman and Dan W. Pratt. Shaped charge liner and method of manufacture, January 2000. Patent.
- ↑ G. Hussain, A. Hameed, J. G. Hetherington, A. Q. Malik, and K. Sanaullah. Analytical performance study of explosively formed projectiles. J Appl Mech Tech Phy, 54(1):10-20, February 2013.
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^{15.0}^{15.1}X. P. Xu and A. Needleman. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids, 42(9):1397-1434, September 1994. - ↑ M. Ortiz and S. Suresh. Statistical properties of residual stresses and intergranular fracture in ceramic materials. Journal of Applied Mechanics, 60(1):77-84, March 1993.
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