A. Dubinsky

Ballistic Impact: Recent Advances in Analytical Modeling of Plate Penetration Dynamics–A Review

This review covers studies dealing with simplified analytical models for ballistic penetration of an impactor into different solid media, namely, metals, soil, concrete, and composites at high speeds, but not at hypervelocities. The overview covers mainly papers that were published in the last decade, but not analyzed in previous reviews on impact dynamics. Both mathematical models and their engineering applications are considered. The review covers 280 citations. DOI: 10.1115/1.2048626

On the ballistic resistance of multi-layered targets with air gaps

Abstract High velocity penetration of a 3-D rigid sharp impactor into a ductile layered target with air gaps between the plates is studied using the assumption about the localized projectile-target interaction. The special property of the penetration phenomenon for conical-nosed impactors is established, namely, that the ballistic performance of the target is independent on the air gap widths and on the sequence of the plates in the target. Similar results are also obtained for 3-D non-conical impactors on the basis of some class of models. These findings are in good agreement with available experimental results.

Optimal 3D impactors penetrating into layered targets

Abstract Optimization of 3D sharp high speed impactors with given form of a longitudinal contour, length, and volume, penetrating into layered ductile targets, both for conical and thin non-conical strikers using approximate models is studied. It is found that the impactor with the minimum drag moving in a homogenous target with a constant velocity penetrates to the maximal depth into a semi-infinite target and has the minimal ballistic limit when it penetrates into a finite thickness target, regardless of the distribution of the material properties of the target along its depth, the number of the layers, etc. Using the analogy with the hypersonic flow over the flying projectiles it is predicted that the optimal impactor should have a star-shaped form of the cross section. If an impactor has a polygonal cross sections allowing the inscribed circles, the ballistic limit and maximum depth of penetration are independent not only of the properties of the target but also of the form of the polygon in the cross section and equal to the corresponding values for the inscribed body of revolution.

The optimum arrangement of the plates in a multi-layered shield

Abstract A normal impact of a three-dimensional rigid conical impactor penetrating into a layered shield is studied using a simplified model for an impactor–shield interaction. The shield consists of adjacent plates manufactured from one of two possible materials, and the total thickness of the plates manufactured from every material is given. It is found that advancing any plate inside a shield in the direction of penetration causes a monotone change in the ballistic limit velocity. A criterion for increasing or decreasing of the ballistic limit velocity, which depends on the properties of the materials of the layers in the shield, is determined. A maximum ballistic limit velocity is attained for a two-layered shield without alternating the plates manufactured from different materials.

Optimization of two component ceramic armor for a given impact velocity

Abstract Using Florence’s model a problem of two-component ceramic-faced lightweight armors design against ballistic impact is solved. Approximate analytical formulas are derived for areal density and thicknesses of the plates of the optimal armor as functions of parameters determining the properties of the materials of the armor components, cross-section and mass of an impactor, and of the expected impact velocity.

On the order of plates providing the maximum ballistic limit velocity of a layered armor

Abstract The high-velocity impact of a three-dimensional rigid sharp impactor penetrating into a ductile layered armor is studied using a simplified model for an impactor–armor interaction. The goal of the study is to determine the order of the plates in the armor that provides the maximum ballistic limit velocity. It is found that the ratio of the distortion pressure to the density of the plate is the parameter that determines the order of the plates in the multi-layered armor that provides the maximum ballistic limit velocity, namely, the plates must be placed in the order of increasing this parameter, regardless of their thickness. The right order of the plates can be of prime importance if the values of this parameter vary strongly for different plates. Numerical simulation is performed in order to verify the theoretical findings and to obtain a quantitative estimate for the increase in the ballistic limit that results from changing the order of the plates in the armor.

Optimal nose geometry of the impactor against FRP laminates

A model is proposed for describing the penetration and perforation of monolithic FRP laminates struck transversely by a rigid projectile with an arbitrary shape. The shapes of the impactors with the minimum and the maximum ballistic limit velocities (BLVs) are found analytically. It is shown that the optimal impactor with the minimum BLV has a plane bluntness and its BLV is very close to that for the optimal blunt conical impactor. It is demonstrated that the impactor with the maximum BLV is a flat-nosed cylinder. Comparison of different shapes showed that the advantage of the optimal impactor over the sharp cone and ogive impactors is significant.

Shape optimization of penetrator nose

Abstract A simplified procedure is suggested to optimize a penetrators nose shape using localized impactor–target interaction model in the cases of penetration into a semi-infinite target and into a target with a finite thickness. Such a procedure establishes the similarity between the projectiles shape optimization in penetration dynamics and in aerodynamics of high speeds. The optimization of the impactors nose shape can thus be reduced to the variational problem considered previously in connection with the projectiles shape optimization in high speed gas dynamics. Two examples from the literature are analyzed when impactors shape optimization involved difficulties whereas the mathematically similar problem was solved before in aeromechanics. Some aspects of applying gas dynamics similarity for optimization of the impactors shape for bodies of revolution are discussed.

Numerical solution for shape optimization of an impactor penetrating into a semi-infinite target

The shape of the impactor with the maximum depth of penetration (DOP) for a given impact velocity is found using a numerical procedure for solving a corresponding non-classical variational problem. It is shown that the optimum shape in a general case is close to a blunt cone. The variation of the optimal shape of the impactor and the dependence of the DOP vs. the initial (impact) velocity and friction coefficient is studied. The analysis is performed also for optimal conical impactors.

A model for predicting penetration and perforation of FRP laminates by 3-D impactors

A model is proposed for describing the penetration of a monolithic semi-infinite and finite FRP laminates struck transversely by a rigid projectile with an arbitrary 3-D shape. Relationships for calculating the characteristics of penetration and perforation (ballistic limit velocity, depth of penetration, etc.) are derived. It is showed that some typical problems of optimization of the shape of the impactor for finite-thickness and semi-infinite shield can be reduced to the same variational problem. An advantage of a 3-D conical impactors over conical impactors having the shape of body of revolution is predicted using the suggested model.