TERMINAL BALLISTICS. [3 ed.]
 9783030466114, 3030466116

Table of contents :
Preface
Introduction
Contents
Part I Experimental and Numerical Techniques
1 Experimental Techniques
1.1 The Terminal Ballistics Lab
1.1.1 Laboratory Guns
1.1.2 Projectiles and Targets
1.1.3 Diagnostics for Terminal Ballistics
1.2 Determination of the Dynamic Properties of Materials
1.2.1 Equation of State Measurements
1.2.2 Dynamic Strength Measurements
1.2.3 Diagnostics
1.3 The Common Threats in Terminal Ballistics
2 Material Models for Numerical Simulations
2.1 General Description
2.2 Material Properties
2.2.1 The Equation of State
2.2.2 The Constitutive Relations
2.2.3 Failure of Ductile Materials
2.2.4 Failure of Brittle Materials
2.2.5 The Spall Failure
Part II Penetration Mechanics
3 Rigid Penetrators
3.1 The Mechanics of Deep Penetration
3.2 The Penetration Model for Rigid Long Rods
3.2.1 Impact at the Ordnance Velocity Range
3.2.2 High Velocity Impact: The Cavitation Phenomenon
3.3 The Cavity Expansion Analysis
3.4 Deep Indentation Tests
3.5 The Penetration of Short Projectiles
3.5.1 The Influence of the Entrance Phase
3.5.2 A Numerically Based Model for the Entrance Phase Effect
3.6 The Impact of Spheres
3.6.1 Rigid Sphere Impact
3.6.2 The Impact of Non-rigid Spheres
3.7 The Effect of Friction
3.8 The Optimal Nose Shape
3.9 Concrete Targets
3.9.1 The Penetration Model
3.9.2 The Scaling Issue and the Size Effect
3.9.3 Other Important Issues
3.9.4 The Rate Dependence of Concrete-like Materials
3.9.5 The Equivalent Aluminum Target
3.10 The Deep Penetration of Deforming Rods
3.11 The Effect of Yaw on Rigid Rod Penetration
3.12 The Transition to Finite-Thickness Targets
4 Plate Perforation
4.1 General Description
4.2 The Perforation of Ductile Plates by Sharp Nosed Rigid Projectiles
4.3 Plate Perforation by Spherical Nosed Projectiles
4.4 Plate Perforation by Blunt Projectiles
4.5 Forced Shear Localization and Adiabatic Shear Failure
4.6 The Effective Strength of Metallic Targets
4.7 The Perforation of Concrete Slabs by Rigid Projectiles
4.8 The Catastrophic Failure of Thin Panels by Hydrodynamic Ram
5 Eroding Penetrators
5.1 General Considerations
5.2 The Penetration of Shaped Charge Jets
5.3 The Penetration of Eroding Long Rods
5.3.1 The Allen–Rogers Penetration Model
5.3.2 The Alekseeνskii-Tate Penetration Model
5.3.3 The Validity of the AT Model
5.3.4 The L/D Effect
5.3.5 Other Penetration Models
5.4 Scaling Issues in the Terminal Ballistics of Eroding Rods
5.5 Deep Penetration at the Hypervelocity Regime
5.6 Plate Perforation by Eroding Rods
Part III Defeat Mechanisms
6 Defeat by High Strength Targets
6.1 Definitions
6.2 Metallic Targets
6.3 Ceramics for Armor
6.3.1 Ceramics Against AP Projectiles
6.3.2 The Interaction of Ceramics with Long Rods
6.3.3 Numerical Simulations
6.3.4 Ceramics Against Shaped Charge Jets
6.4 Woven Fabrics as Armor Materials
6.5 Elastomer-Steel Laminates for Armor Applications
6.6 Fragment Simulating Projectiles (FSPs) Perforating Aluminum Plates
6.7 The Defeat of EFP Threats
7 Asymmetric Interactions
7.1 General Description
7.2 Defeating AP Projectiles
7.3 Defeating Long Rods
7.3.1 Perforation of Inclined Plates
7.3.2 Ricochet of Long Rods
7.3.3 The Interaction of Long Rods with Moving Plates
7.3.4 The Impact of Yawed Rods
7.4 Defeating Shaped Charge Jets
7.4.1 The Explosive Reactive Armor (ERA)
7.4.2 Passive Cassettes
7.4.3 The Hole Diameter in a Plate Impacted by a Jet
7.5 Defeat of Micrometeoroids by Thin Metallic Plates
References

Citation preview

Zvi Rosenberg Erez Dekel

Terminal Ballistics Third Edition

Terminal Ballistics

Zvi Rosenberg Erez Dekel •

Terminal Ballistics Third Edition

123

Zvi Rosenberg RAFAEL Ballistics Center Haifa, Israel

Erez Dekel RAFAEL Ballistics Center Haifa, Israel

ISBN 978-3-030-46611-4 ISBN 978-3-030-46612-1 https://doi.org/10.1007/978-3-030-46612-1

(eBook)

1st edition: © Springer-Verlag Berlin Heidelberg 2012 2nd edition: © Springer Science+Business Media Singapore 2016 3rd edition: © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of our parents, Shmuel and Henya Rosenberg Nisim and Marsel Dekel

Preface

The high velocity impact of solid bodies has attracted a large amount of research over the past century by both military and civil engineers. These impacts, at hundreds to thousands of meters per second, involve large deformations of the impacting bodies which can result in their total destruction around the impacted area. The impact of projectiles on armored vehicles (at 1–2 km/s) and the impact of meteorites at space stations at (10–20 km/s) are areas of much interest in this field. At impact velocities of a few meters per second, the structural response of the bodies is the relevant issue for safety engineers in the automotive industry. In order to study the effects of high velocity impacts, a special scientific discipline has been developed over the past 50 years, termed the dynamic response of solids to impulsive loading. This field involves several different disciplines such as: elasticity and plasticity theories, hydrodynamics, high pressure physics, material response to high strain rates, fracture mechanics, and failure analysis. Several symposia dedicated to these issues were established during the last decades, such as the Hypervelocity Impact Symposia series, the International Symposia on Ballistics, the APS conferences on Shock Compression of Solids (in the US) and the DYMAT conferences in Europe. In addition, several journals specifically dedicated to this field were established, such as the International Journal of Impact Engineering, since 1983, and the International Journal of Protective Structures (launched in 2010). All of these activities are focused on the dynamic response of solids to impulsive loading, by developing new experimental facilities and diagnostics, as well as advancing numerical simulations and analytical modeling. This book is focused on the subject of terminal ballistics which deals with the interaction between a moving object (the threat) and a protective structure (the target), at impact velocities in the range of a few hundreds to a few thousands of meters per second. At these velocities the damage induced in the target is local, extending laterally to several projectile diameters, but it is concentrated along the direction of projectile’s motion. Thus, the target can be either perforated as is the case for thin targets, or deeply penetrated (for thick targets). These penetration/ perforation issues are important for the armor engineer who looks for ways to minimize the extent of damage to the protected structure. Similarly, the anti-armor vii

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designer is concerned with the improvements in the lethality of the threats by increasing their velocities, masses, etc. The field of terminal ballistics covers a large range of scientific challenges and engineering applications, and we had to limit the number of the subjects which are discussed in this book. Naturally, most of the subjects we chose belong to armor issues, on which we worked for many years at the terminal ballistics laboratory in RAFAEL, a defense-related research institute in Israel. We wish to thank our colleagues for fruitful and exciting research during many years. In particular, we acknowledge the scientific collaborations with: Y. Yeshurun, D. Yaziv, M. Mayseless, Y. Ashuach, R. Kositski, A. Malka-Markovitz and Y. Partom from RAFAEL, S. J. Bless, M. J. Forrestal, and N. S. Brar from the US, and N. K. Bourne and J. F. Millett from England. We acknowledge the excellent work of: M. Siman, R. Kreif, M. Rozenfeld, Y. Reifen, D. Kanfer, N. Yadan, D. Mazar, I. Shaharabani, and Y. Zidon in performing many experiments in our lab for over 30 years. We also thank C. E. Anderson, A. J. Piekutowski, K. Poormon, T. J. Holmquist, T. Borvik, S. Chocron, K. Thoma, S. Cimpoeru, and A. Dancygier for helpful discussions during the preparation of this book, and for sharing some of their best shadowgraphs which add so much to this book. Haifa, Israel

Zvi Rosenberg Erez Dekel

Introduction

The science and engineering of impacting bodies have a large range of applications depending on their type and their impact velocities. At very low velocities, these impacts can be limited to the elastic range of response, with practically no damage to the impacted bodies. In contrast, at very high impact velocities these bodies experience gross deformation, local melting, and even total disintegration upon impact. Various scientific and engineering disciplines are devoted to specific areas in this field such as: vehicle impacts, rain erosion, armor and anti-armor design, spacecraft protection against meteorites and the impact of planets by large meteors at extremely high velocities. In order to follow these different events the researcher has to be acquainted with diverse scientific fields which include: elasticity and plasticity of solids, fracture mechanics and the physics of materials at high pressures and temperatures. Terminal ballistics is the generic name for the science and engineering of impacts which are of interest to armor and anti-armor engineers. The relevant impact velocities usually range between 0.5 and 2.0 km/s, the so-called ordnance velocity range. These are the velocities at which projectiles are launched against personnel armored vehicles and buildings, by rifles and guns. The impact velocities of shaped charge jets are within the hypervelocity range of 2.0–8.0 km/s, and their interaction with armor is also of major interest to both armor and anti-armor engineers. This book is devoted to the science of terminal ballistics, as it is defined here, through the various threats which operate in the battlefield. The science of terminal ballistics started with the works of the great mathematician Leonard Euler (1745) and the British engineer Benjamin Robins (1742), who analyzed data for the penetration of steel cannonballs in soil as a function of their impact velocity. In the following two centuries, until the Second World War, the field of terminal ballistics was based on empirically derived relations between the penetration depth and the impact velocity of various projectiles on different targets. The reviews of Hermann and Jones (1961) and Backman and Goldsmith (1978), summarize many of these empirical formulas which were suggested over this period. During the years of WW-II, scientists in the US and UK have analyzed the penetration process of shaped charge jets and rigid steel projectiles into armor plates, through analytical models which were based on physical ix

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considerations. These models identify the main force exerted on the projectile during penetration, which is then inserted in its equation of motion. The aim of these models is to reduce the mathematical description of a complicated three dimensional problem, to a simple form which retains the essential physics of the penetration process. This simplification results in either a low-dimensional system of ordinary differential equations, or a few one-dimensional partial differential equations which can be easily solved. The models can be tested by controlled experiments, in which the parameters are varied in a systematic way, in order to establish the non-dimensional parameters of the process. With these analytical models, data correlation is made easy and extrapolations to areas beyond the ability of experimental facilities are possible. On the other hand, analytical models require some compromise to be made, limiting their use to ideal cases where only a single mechanism is at work. Still, analytical models have been used successfully in order to account for the data and to reduce the number of the necessary experiments in terminal ballistics. Since the advancements in numerical simulations, the role of analytical models seems to decline as the codes are getting better and more efficient. Our strong belief is that analytical modeling is crucial for the field of terminal ballistics in order to understand the physics involved, and to highlight the important parameters which influence these processes. Throughout this book, we bring examples where physically-based models play a major role in simplifying complex interactions. These models are derived either through experimental observations or through numerical simulations. Our approach is that numerical simulations can be viewed as the “perfect experiments,” with which one can change a single parameter at a time and find its influence on the investigated process. We shall demonstrate the usefulness of these numerical studies for the construction of numerically-based models, and for the critical examination of existing models. Note that the term “model,” as it is often used in the literature, is somewhat confusing since it is applied for both the description of material properties (material modeling) and for the analytical account of the process itself (the engineering model). For both applications these models should be physically based, with parameters which can be calibrated by well-defined experiments. A valid material model should be applicable for a variety of experimental configurations, and an engineering model has to account for the behavior of different materials in a given experiment. The book is divided into three parts. Part I includes a description of the terminal ballistics laboratory and the main diagnostic tools which are used in this field. It also summarizes several material models which are frequently used in numerical simulations. Part II is devoted to the field of penetration mechanics, dealing with the basic processes which take place upon the impact and penetration of various projectile/target combinations. Part III describes the working principles behind several armor concepts which are designed to defeat some of the more common threats.

Contents

Part I Experimental and Numerical Techniques . . . . . . . . . . . . . . . . . . .

1

1

Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Terminal Ballistics Lab . . . . . . . . . . . . . . . . . . . 1.1.1 Laboratory Guns . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Projectiles and Targets . . . . . . . . . . . . . . . . . . 1.1.3 Diagnostics for Terminal Ballistics . . . . . . . . . 1.2 Determination of the Dynamic Properties of Materials 1.2.1 Equation of State Measurements . . . . . . . . . . . 1.2.2 Dynamic Strength Measurements . . . . . . . . . . 1.2.3 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Common Threats in Terminal Ballistics . . . . . . . .

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2

Material Models for Numerical Simulations 2.1 General Description . . . . . . . . . . . . . . 2.2 Material Properties . . . . . . . . . . . . . . . 2.2.1 The Equation of State . . . . . . . 2.2.2 The Constitutive Relations . . . . 2.2.3 Failure of Ductile Materials . . . 2.2.4 Failure of Brittle Materials . . . . 2.2.5 The Spall Failure . . . . . . . . . . .

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Part II Penetration Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Rigid Penetrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Mechanics of Deep Penetration . . . . . . . . . . . . . . . . . . 3.2 The Penetration Model for Rigid Long Rods . . . . . . . . . . . 3.2.1 Impact at the Ordnance Velocity Range . . . . . . . . . 3.2.2 High Velocity Impact: The Cavitation Phenomenon . 3.3 The Cavity Expansion Analysis . . . . . . . . . . . . . . . . . . . . . 3.4 Deep Indentation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.5

The Penetration of Short Projectiles . . . . . . . . . . . . . . . . 3.5.1 The Influence of the Entrance Phase . . . . . . . . . . 3.5.2 A Numerically Based Model for the Entrance Phase Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The Impact of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Rigid Sphere Impact . . . . . . . . . . . . . . . . . . . . . 3.6.2 The Impact of Non-rigid Spheres . . . . . . . . . . . . 3.7 The Effect of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 The Optimal Nose Shape . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Concrete Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 The Penetration Model . . . . . . . . . . . . . . . . . . . . 3.9.2 The Scaling Issue and the Size Effect . . . . . . . . . 3.9.3 Other Important Issues . . . . . . . . . . . . . . . . . . . . 3.9.4 The Rate Dependence of Concrete-like Materials . 3.9.5 The Equivalent Aluminum Target . . . . . . . . . . . . 3.10 The Deep Penetration of Deforming Rods . . . . . . . . . . . 3.11 The Effect of Yaw on Rigid Rod Penetration . . . . . . . . . 3.12 The Transition to Finite-Thickness Targets . . . . . . . . . . .

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Plate Perforation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Perforation of Ductile Plates by Sharp Nosed Rigid Projectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Plate Perforation by Spherical Nosed Projectiles . . . . . . . . . 4.4 Plate Perforation by Blunt Projectiles . . . . . . . . . . . . . . . . . 4.5 Forced Shear Localization and Adiabatic Shear Failure . . . . 4.6 The Effective Strength of Metallic Targets . . . . . . . . . . . . . 4.7 The Perforation of Concrete Slabs by Rigid Projectiles . . . . 4.8 The Catastrophic Failure of Thin Panels by Hydrodynamic Ram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Eroding Penetrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Penetration of Shaped Charge Jets . . . . . . . . . . . . . . 5.3 The Penetration of Eroding Long Rods . . . . . . . . . . . . . . 5.3.1 The Allen–Rogers Penetration Model . . . . . . . . . . 5.3.2 The Alekseemskii-Tate Penetration Model . . . . . . . 5.3.3 The Validity of the AT Model . . . . . . . . . . . . . . . 5.3.4 The L/D Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Other Penetration Models . . . . . . . . . . . . . . . . . . . 5.4 Scaling Issues in the Terminal Ballistics of Eroding Rods . 5.5 Deep Penetration at the Hypervelocity Regime . . . . . . . . . 5.6 Plate Perforation by Eroding Rods . . . . . . . . . . . . . . . . . .

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Part III Defeat Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 6

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Defeat by High Strength Targets . . . . . . . . . . . . . . . . . . 6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Metallic Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ceramics for Armor . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Ceramics Against AP Projectiles . . . . . . . . . . 6.3.2 The Interaction of Ceramics with Long Rods . 6.3.3 Numerical Simulations . . . . . . . . . . . . . . . . . 6.3.4 Ceramics Against Shaped Charge Jets . . . . . . 6.4 Woven Fabrics as Armor Materials . . . . . . . . . . . . . 6.5 Elastomer-Steel Laminates for Armor Applications . . 6.6 Fragment Simulating Projectiles (FSPs) Perforating Aluminum Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 The Defeat of EFP Threats . . . . . . . . . . . . . . . . . . .

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Asymmetric Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Defeating AP Projectiles . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Defeating Long Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Perforation of Inclined Plates . . . . . . . . . . . . . . . . 7.3.2 Ricochet of Long Rods . . . . . . . . . . . . . . . . . . . . 7.3.3 The Interaction of Long Rods with Moving Plates . 7.3.4 The Impact of Yawed Rods . . . . . . . . . . . . . . . . . 7.4 Defeating Shaped Charge Jets . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Explosive Reactive Armor (ERA) . . . . . . . . . 7.4.2 Passive Cassettes . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 The Hole Diameter in a Plate Impacted by a Jet . . 7.5 Defeat of Micrometeoroids by Thin Metallic Plates . . . . . .

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Part I

Experimental and Numerical Techniques

Chapter 1

Experimental Techniques

Abstract Terminal ballistics is the general name for a large number of processes which take place during the high velocity impact of various projectiles/target combinations. There are two related disciplines which deal with launching these projectiles. Interior Ballistics concerns their acceleration to the desired velocity, and Exterior Ballistics deals with their flight dynamics from the launcher to the target. The present chapter describes, very briefly, some of the equipment and techniques which are used to launch a projectile to the desired velocity, as well as the relevant diagnostics which are used to follow its flight and its impact. In addition, the chapter covers the main experimental techniques, which are used to determine the relevant properties of the impacting materials.

1.1 The Terminal Ballistics Lab 1.1.1 Laboratory Guns Most of the experimental work in terminal ballistics is performed through scaled experiments within indoor laboratories, using the so-called laboratory guns. The obvious advantages of scaled experiments are their significantly higher ease of operation and lower cost, as compared with full-scale experiments. The most common scaling factors are the 1:3 and 1:4 scales, and the experiments are usually performed with guns of 20–40 mm in diameter. These guns can launch projectiles to velocities of up to 2.0 km/s with gun powder, and up to 10 km/s by the two-stage light gas gun, which will be described here. Several reviews are devoted to the working principles and the design of these guns, such as Siegel (1955) and Stilp and Hohler (1990). All the different launchers consist of a high-pressure gas reservoir (the breech) attached to a long tube (the barrel) where the projectile is accelerated. The desired accelerating pressures are achieved either by burning a given amount of gun powder (the powder guns) or by the fast release of compressed gas (in gas guns). The acceleration process lasts as long as the projectile is in the barrel, but its velocity does not increase appreciably after a distance of about 120 diameters, which is the typical length of most laboratory guns. The basic equation for this acceleration process © Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_1

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relates the final kinetic energy of the projectile to the work done by the high pressure pulse p(x), through: 1 π M V 2 = Dg2 2 4

 p(x)d x

(1.1)

where M is the mass of the projectile and Dg is the inner diameter of the gun. Taking an average value of the pressure profile which acts on the projectile (pav ), one can write for its exit velocity:  V = Dg

π · pav L g 2M

(1.2)

where L g is the length of the gun barrel. This equation gives the main ingredients which influence the performance of guns, as far as their ability to reach the desired velocity for a given mass. The science of Interior Ballistics deals with the exact determination of these pressure profiles, and their average values for different gases and gun powders. Every type of accelerating gas has a maximum velocity which it can impart to the projectile. This is the escape velocity (V esc ) with which the high pressure gas can move if it is allowed to expand freely (no mass to accelerate except its own), which is given by: Vesc =

2c ζ −1

(1.3)

where c is the sound speed in the gas at the high pressure condition, and ζ is the ratio of its specific heats. The sound speed in a gas is inversely proportional to the square root of its molecular weight, which is the reason for using either helium or hydrogen in light gas guns. The escape velocity is also proportional to the temperature of the gas, which increases during its fast compression in light gas guns, enhancing their ability to reach higher velocities. These guns were developed in the 1950s in order to study the impact of meteorite-like projectiles on spacecraft bodies, at the hypervelocity regime of impact velocities (≈10 km/s). The design principles of the light gas gun are described by Stilp and Hohler (1990), and its schematic description is shown in Fig. 1.1. The first stage of this system includes a large chamber in which a gun powder is ignited and its products accelerate a large plastic piston along the pump tube, which is a long barrel with an inner diameter D1 . The piston has a conical section at its rear part in order to start its motion after some pressure buildup in the breech chamber. A predetermined amount of a light-weight gas, usually helium, is inserted into the pump tube at pressures in the range of 20–30 atmospheres. As the plastic piston accelerates in the pump tube it compresses and heats the light gas to high pressures and temperatures. As seen in the figure, a conical steel section is leading the plastic piston from the pump tube (the first stage) to the launch tube (the second stage). This conical section is a very important feature of this gun design, as it has to stop the

1.1 The Terminal Ballistics Lab

5

Fig. 1.1 A schematic description of the light gas gun

plastic piston and prevent it from extruding into the launch tube. The conical section is sealed at its far end with a steel diaphragm, which is designed to rupture at a certain predetermined pressure. The diaphragm divides the two parts of the system and the projectile itself is located in the launch tube, which has a smaller diameter (D2 ). In most light gas gun systems the ratio between the diameters of the pump and launch tubes (D1 /D2 ) is about 3–4, the diameters of the launch tubes are in the range of D2 = 20–40 mm and the total lengths of these guns are 10–15 m. As stated by Hohler and Stilp (1990) the interior ballistics of the light gas gun is quite complex and its performance is determined by several factors such as the geometrical details of the two tubes, the conical section, and the weights of the plastic piston and the package to be launched. These details determine the exact values of the weight of the gun powder and the amount of light gas which have to be used in order to achieve the desired velocity.

1.1.2 Projectiles and Targets The projectile is packaged within a plastic sabot, usually made of polycarbonate, in order to prevent metallic contact with the barrel. These sabots are made of several parts which are designed to separate as the package exits the gun barrel. Some gun systems have specially designed sabot strippers near the gun muzzle, which prevent the plastic sabots from reaching the target. Other systems are based on the sabots to open up by air drag, while the projectile is out of the gun. More details concerning sabot design, and their separation before impact, are given by Stilp and Hohler (1990). A metallic disc (the pusher plate) made of a high strength material, is placed behind the projectile in order to prevent the punching of the plastic sabot by the accelerating projectile. Since most laboratory guns are housed indoors, the targets are situated at a distance of about 1–5 m from the muzzle, in a specially designed target chamber which includes ports for high speed photography, pulsed X-ray shadowgraphy, velocity interferometry and other diagnostic devices. After

6

1 Experimental Techniques

impact, all the debris from the target and projectile is collected in a large catcher tank, where the residuals are softly decelerated and recovered for further analysis. The targets in terminal ballistics studies are classified according to their thickness as semi-infinite, intermediate, and thin. A semi-infinite target is large enough so that its lateral and back surfaces do not influence the penetration process. The measurements performed with semi-infinite targets are usually post-mortem determinations of the final penetration depth, the volume and the shape of the crater, and the characteristics of the residual projectile which is left at the crater bottom. Experiments with semi-infinite targets focus on the physics of the penetration process rather than simulating actual armor designs. The thickness of an intermediate target is enough to stop the projectile but it is not large enough to prevent damage at its back surface. The interactions of projectiles with targets of intermediate thickness are relevant for the armor designer, since they represent actual situations of projectile/armor encounters. This is a more complicated penetration process since it involves various failure mechanisms which can take place during the interaction. Some of these failure mechanisms are shown schematically in Fig. 1.2, from Backman (1976). Thin targets are perforated by the projectile and their relevance for terminal ballistics is through the damage they can induce to the projectile. Targets which induce a considerable damage to the impacting threat, by a well-defined defeat mechanism, are considered as candidates for add-on armor designs. These mechanisms, which are of main interest to the armor designer, will be discussed in the third part of this book. One of the most important properties of a given projectile/target combination is its ballistic limit velocity (V bl ), which marks the velocity threshold for target perforation. In principle, the value of V bl for a given projectile/target configuration can be determined by two experiments at impact velocities which are close enough. The high-velocity impact should result in a complete perforation of the target, while at the low-velocity impact the projectile should be stopped in the target. However, due to experimental errors in velocity measurements and variations in material properties, Fig. 1.2 Possible failure modes in different impact situations

1.1 The Terminal Ballistics Lab

7

it is practically impossible to determine V bl by just a couple of experiments. Instead, a probabilistic approach was adopted by most researchers according to which at least six experiments are performed at the estimated limit velocity. In half of these experiments the target should be perforated while for the other half it should stop the projectile. With the better statistics of this technique the limit velocity, which is termed the V 50 velocity, is taken as the average of the highest velocity with no perforation and the lowest velocity with projectile perforation. The definition of perforation itself depends on the different standards which were adopted by various institutes. The US army criterion states that a complete penetration is achieved even if the projectile is still stuck in the target but light can pass through it, while according to the U.S. Navy’s definition a complete perforation is achieved when the projectile fully emerges from the target.

1.1.3 Diagnostics for Terminal Ballistics There are several measurements which have to be performed at each experiment in terminal ballistics. The most important one is the projectile’s impact velocity, which has to be determined to within ±0.5% in order to have a meaningful result. This velocity is best determined by non-interfering optical techniques like laser beams (or screens) which are cut off by the flying projectile. Another useful technique is the pulsed X-ray shadowgraphy, which uses very short (about 30 ns) bursts of radiation from specially designed tubes. The short exposure time “freezes” the movement of the projectile, resulting in a sharp picture of its contour on specially designed films or screens. These flash X-rays are also used to follow the alignment of the projectile as it impacts the target, and for the determination of its shape and velocity after it exits the target. Thus, a terminal ballistic laboratory includes several pairs of flash X-ray tubes which follow the projectile during its free flight, its impact with the target and its exit from the back surface, as shown schematically in Fig. 1.3. The use of two

Fig. 1.3 Schematic description of the terminal ballistics experimental setup

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1 Experimental Techniques

Fig. 1.4 Schematic description of the rotating-mirror camera

simultaneous flashes at perpendicular orientations, is needed to follow the spatial orientation of the projectile at the impact moment (the pitch and yaw angles). High speed photography techniques are also very useful to follow the interaction between the projectile and the target, as described by Stilp and Hohler (1990) and by Swift (1982). The basic objective of these cameras is to capture the projectile/target interaction at different times by projecting them at different places on a single film. This is done by using a rotating mirror which spins at an extremely high velocity by the action of an air turbine, as shown in Fig. 1.4. The projection of the impact event on the film is achieved by a series of projecting lenses as shown in the figure. Another high-speed photography technique is based on video cameras which, with the proper background lighting, capture the projectile as it penetrates and emerges from the target. These pictures are very useful for the determination of the residual velocity and residual mass of the projectile, as well as following the material ejected from the back of the target. Figure 1.5, from Borvik et al. (1999), shows several frames from such an experiment where a steel projectile perforates an aluminum plate. Fig. 1.5 Pictures from high-speed video camera of a steel projectile perforating an aluminum plate

1.2 Determination of the Dynamic Properties of Materials

9

1.2 Determination of the Dynamic Properties of Materials A high velocity impact induces high pressures and temperatures in both projectile and target, for very short times (high strain rates) which characterize the impulsive nature of this event. The term “dynamic properties” is referred to the physical and mechanical properties of the solid which are sensitive to high loading rates, as well as to high pressures and temperatures. Strain rates which are higher than about 103 s−1 fall in the category of high rates and pressures are considered to be high when their magnitude is larger than about 10 GPa. Temperature changes have to be considered when they are high enough to cause significant changes of material strength, by the so-called thermal softening process. The relevant temperatures are different for various materials depending on their melting temperatures. It is commonly accepted that the threshold temperature for thermal softening of a solid is about half its melting temperature. The present chapter describes some techniques which are used to determine the influence of these parameters (strain rates, pressures and temperatures), on material properties which are relevant for terminal ballistics. These properties can be grouped in two classes, those which affect the material’s equation of state and those which determine its constitutive relations. The equation of state of a solid relates its internal energy to the pressure and density which characterize its state. The constitutive relation accounts for the strength and failure characteristics of the solid under the high loading rates. As will be discussed in this book, the strength and failure of the solid are its most important properties as far as terminal ballistics is concerned. Thus, a comprehensive account of their variation with loading rates, pressures, and temperatures is essential for the evaluation of a given material as a possible candidate for either armor or projectile design. Failure refers to situations where a given element in the structure reaches a critical state and its load capacity is reduced, either totally or partially. The failure of solids under dynamic loading conditions is one of the most important areas for terminal ballistics, and it has been the focus of much research.

1.2.1 Equation of State Measurements A high-velocity impact can induce very high pressures and temperatures for several microseconds in the projectile and in the target, through fast moving shock waves. The irreversible increase in the temperature of shock loaded materials can reach their melting and even vaporization states. The study of materials response to shock wave loading is focused on their equations of state, which relate their densities to the high pressures and temperatures. Equations of state were determined for many solids in a systematic study, as described by Zeldovich and Raizer (1965), Al’tshuler (1965), McQueen et al. (1970), Chhabildas (1987) and Asay and Kerley (1987). These reviews also cover much of the experimental techniques which are used in this field. The plate-impact experiment, shown schematically in Fig. 1.6, is the most

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1 Experimental Techniques

Fig. 1.6 The plate impact experiment

useful technique to obtain the important data for the shock equation of state of a given solid. This is a relatively simple experimental configuration which is easy to analyze mathematically, as shown. Basically, a planar disc (the impactor) is accelerated by a large diameter gun towards a stationary disc (the target) made of the investigated material, which is carefully aligned at the end of the gun in order to ensure maximum planarity of the impact. The impact velocity is measured by electrically shorting pins prior to impact, and the target is instrumented with various gauges and sensors, which follow the amplitude and the structure of the shock wave in the specimen, as described below. Upon impact, planar shock fronts, or shock waves as they are usually called, are initiated at the impact face and move into both the impactor and target plates. One can conceive of these shock waves as extremely thin fronts which sweep the material at a constant velocity. The material behind these fronts is compressed to a state of high pressure and density, which depend on the impact velocity and on the properties of the impacted materials. Since these waves are planar they are compressing the two plates in a uniaxial strain mode, by which the only non-zero strain in the specimen is along the impact direction. This special mode of compression is analyzed through the Rankine–Hugoniot (R–H) jump relations, which express the changes in density, pressure and internal energy of the shocked material. These relations are given in terms of two mechanical parameters which are measured in the experiment, the shock velocity (U s ) and the particle velocity (up ) of the material behind the shock front. In fact, the R–H relations are the conservation equations for mass, momentum and energy under the special loading mode of uniaxial strain, and they are given by:   Mass conservation: ρ0 U S = ρ U S − u p

(1.4a)

Momentum conservation: p = ρ0 U S u p

(1.4b)

1.2 Determination of the Dynamic Properties of Materials

Energy conservation: e =

p · 2

11



1 1 − ρ0 ρ

 (1.4c)

where ρ, p and e are the density, pressure and change in internal energy of the shocked material, respectively. The subscript 0 stands for these parameters at ambient conditions. The shock velocity (U s ) and the particle velocity (up ) are the measured parameters in each experiment. With several experiments at different impact velocities, one can determine the shock equation of state for the solid. One of the most useful ways to represent this equation of state, in a graphical display, is by plotting the relations between the pressure and the particle velocity behind the shock front. This p = p(up ) curve, called the “Hugoniot curve” of the solid, is the loci of all the points representing the states of the material which can be reached by a single shock wave. The Hugoniot curve is different from the isothermal compression curves of the solid, since each point on the Hugoniot is characterized by a different temperature (internal energy). Hugoniot curves, for several materials which are relevant to terminal ballistics, are given in Fig. 1.7. For symmetric impacts, where the two impacting plates are made of the same material, the particle velocity behind the shock front is equal to half the impact velocity, since these are ideally plastic impacts. Thus, by accurately measuring the impact velocity one obtains the particle velocity behind the shock front in symmetric impacts. The shock velocity in the target plate is measured either by optical techniques or by electrical transducers which short-circuit upon the arrival of the shock front. For many materials, both solids and liquids, the relation between shock and particle velocities has been found to be linear, according to: U S = C0 + Su p

(1.5)

where C 0 is very close to the bulk sound speed of the solid at ambient conditions. The experimentally determined slope in this linear relation (S) has values in the range of S = 1.0–1.5 for most solids. The values for C 0 and S for many materials are listed in Fig. 1.7 Hugoniot curves for several materials

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1 Experimental Techniques

the data base of all the commercial codes which are used for numerical simulations of terminal ballistics experiments.

1.2.2 Dynamic Strength Measurements The previous section gave a brief account for the behavior of materials under planar shock wave loading which is a one-dimensional strain configuration. The materials were assumed to behave as fluids in the sense that their strength was ignored in the analysis. Thus, the values of the principal stresses in the specimen are equal to each other, σ x = σ y = σ z = p, where p is the pressure in the solid. This approximation is valid for intense shock waves where the pressure is much higher than the strength of the material (Y ). When these pressures are less than about 10Y the analysis has to take into account the elasto-plastic properties of the material. The shock Hugoniot of a material with strength is shown schematically in Fig. 1.8. The addition of material strength means that the principal stresses in the shocked solid are not equal to each other. The stress along the shock propagation direction (σ x ) is higher by Y than the two lateral stresses σ y and σ z , which are equal to each other. Note the distinction between the elastic and the so-called plastic regimes. The elastic response of the solid is represented by the straight line, which terminates at the Hugoniot elastic limit (HEL). The “plastic” response is given by the Hugoniot curve which is superposed on the elastic straight line. This stress–strain curve is due to the fact that the material behind the shock front is compressed in a uniaxial straining mode, with εx = 0 and εy = εz = 0, where x is the shock propagation direction. The strain along the shock propagation direction is, in fact, the volumetric strain of the material (εv ), which is related to the density ratio (ρ/ρ 0 ) attained by the shock pressure. The value of the shock pressure is given as the average of the three principal stresses in the material, through p = (σ x + σ y + σ z )/3, where σ x is larger than σ y and σ z . In contrast, the hydrostat is the compression curve of the solid when the three principal stresses are equal to each other. Fig. 1.8 A schematic description of the Hugoniot and the Hydrostat curves for a solid with strength

1.2 Determination of the Dynamic Properties of Materials

13

At very low impact velocities the solid behaves elastically and upon stress unloading it returns to its original state. The state of the solid within its elastic range is given by the longitudinal stress (σ x ) and the two lateral stresses (σ y = σ z ) which, under the uniaxial straining mode, are given by: σ y = σz =

v σx 1−v

(1.6)

where ν is the Poisson constant of the solid. The elastic regime terminates at the Hugoniot elastic limit (HEL), which marks the onset of material yielding under the uniaxial strain loading. At this point the difference between the principal stresses (σ x – σ y ) reaches its maximal value which is equal to the dynamic strength of the solid (Y d ). This constraint is the result of the yield criteria of Tresca and von-Mises, which limit the stress difference in the plastic range. Using this yield condition, σ x – σ y = Y d together with (1.6), results in the following relation between the HEL of the solid and its dynamic strength: HEL =

1−ν · Yd 1 − 2ν

(1.7)

where the dynamic strength of the solid (Y d ), as derived from the measured HEL value, can be very different from its static strength especially for strain-rate sensitive materials. By recording the amplitude of the elastic precursor one can determine the dynamic strength of the shocked material at the highest strain rates, which are of the order of (106 –108 ) s−1 . The existence of two regimes in the response of a solid to shock waves, results in a discontinuity in the slope of the shock Hugoniot at the HEL, as shown in Fig. 1.8. This discontinuity leads to a splitting of the shock front into two waves, which propagate at different velocities in the solid. The precursor elastic wave with amplitude which is equal to the HEL, is moving at the longitudinal wave velocity given by:  CL =

K + 4G/3 ρ

(1.8)

where K and G are the bulk and shear moduli of the solid, respectively, and ρ is its density. The slower “plastic wave” is propagating at the shock velocity (U s ), as given by (1.5). With increasing shock amplitude the velocity of the plastic wave increases until it attains the value of C L , and for higher shock amplitudes a single wave is propagating in the solid. The threshold amplitude for the emergence of a single shock depends on the properties of the solid and it varies between 10 and 100 GPa for most solids. For example, these threshold stresses are about 10 and 60 GPa for aluminum and steel, respectively.

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Fig. 1.9 Loading–unloading curves for a solid with strength

In a similar manner, one can follow the release process of the shocked solid by the release waves which emanate from free surfaces. The process takes place along the unloading path of the specimen in the stress–strain plane, as shown schematically in Fig. 1.9. The solid first unloads elastically along a straight line whose slope is related to the longitudinal wave velocity (C L ) at the high pressure. At a certain point during the unloading process, this line meets the yield surface of the solid, below its hydrostatic compression curve. Further unloading is achieved along an isentrope, which is roughly parallel to the Hugoniot curve as shown in the figure. This part of the release, termed the “plastic release”, lies at a distance of 2Y d /3 below the hydrostatic compression curve. Since the shock Hugoniot is offset by the same distance above the hydrostatic compression curve, the vertical “distance” between the shock and release curves is equal to 4Y d /3, where the dynamic strength Y d (p, T ) is the solid’s strength at the high pressure and temperature in the shocked state. This value can be much higher than the quasi static strength of the solid, due to pressure-hardening effects. On the other hand, it can be lower than the quasi static value, if the temperature of the shocked material is high enough to induce its thermal softening. The discontinuity in the slopes of the elastic and “plastic” parts in the release curve, results in a structured release wave which can be monitored continuously with stress or particle velocity gauges. From the amplitude of the elastic release wave one can determine the strength of the solid at its shocked state, Y d (p, T ), as will be discussed in the next section.

1.2.3 Diagnostics The plate impact experiment is very useful for the determination of the important parameters in the constitutive relations of the solid. These relations account for the dependence of the material strength and its failure characteristics on the high rates, pressures, and temperatures in the shocked state. Quantitative information about these variations are obtained by various in situ measurements which record the shock wave profiles in plate impact experiments, as reviewed by Chhabildas (1987) and by Asay

1.2 Determination of the Dynamic Properties of Materials

15

(1997). The strain rates in these experiments can reach values of 108 s−1 and the pressures can range between few to several hundreds of GPa. The necessary information is obtained by embedding stress gauges in the target plate or monitoring its back surface velocity by interferometric techniques. Embedded stress gauges are made of either piezoresistive materials, whose electrical resistivity changes appreciably with pressure, or of piezoelectric materials which produce electric charge (current) when a pressure pulse is sweeping them. The most useful piezoresistive materials for both shock and static loading are manganin, Yitterbium and carbon. The most successful piezoelectric gauge is made of specifically poled foils of Polyvinyldifluoride (PVDF). The poling techniques of these gauges, as well as their calibration under shock loading, have been worked out by F. Bauer, as described in Bauer (1983, 1984). The most common piezoresistance stress gauge for shock wave studies is made of manganin which is an alloy containing 84% copper, 12% manganese and 4% nickel. This material has an almost constant piezoresistive response up to pressures of about 100 GPa. In fact, the review article by Chhabildas (1987) cites measurements by P. de-Carli to shock pressures as high as 120 GPa. The electrical resistivity of manganin is practically insensitive to temperature changes, which makes it an ideal stress gauge for dynamic experiments. The Nobel laureate physicist P. W. Bridgman installed manganin wires to record the pressures in the static high pressure cells, for his classical studies at the early years of the twentieth century. The first use of manganin as a pressure transducer in shock experiments was published by Fuller and Price (1964). They embedded thin foils of manganin between two planar discs, and monitored their resistance changes during the passage of shock waves through the assembly. With a pre-calibrated response curve, these profiles were translated to the pressure–time profiles in the shocked specimens. Calibration curves for foil and wire gauges have been obtained by Barsis et al. (1970), Kanel et al. (1978), Vantine et al. (1980), and Rosenberg et al. (1980). These calibrations agree with each other to within 2%, with an average piezoresistance coefficient of R/R0 = 2.4% per GPa, where R0 is the initial resistance of the gauge and R is its resistance change. A typical gauge record from a plate impact experiment on a copper specimen is shown in Fig. 1.10. The figure also details the location of the gauge in this experimental set-up, the back surface configuration, which is specially suited for the determination of the dynamic compressive and tensile strengths of the specimen. With this configuration the stress history is actually determined at the interface of the specimen and a Plexiglas backing plate. Several features in this record should be noted. The first concerns the rising part of the signal, which clearly shows that the shock wave in the specimen has been split into the elastic precursor wave and the following “plastic wave”. The elastic precursor, of amplitude which is equal to the HEL, serves to determine the compressive strength of the solid at these high strain rates. The elastic and plastic waves are followed by a constant shock level which is subsequently unloaded, by a structured release wave. The unloading is followed by several stress reverberations which are due to the spall failure inside the target plate. Spall is the onset of tensile failure which is due to the interaction between the two release waves inside the specimen, and it will be described in more detail later on. These release waves are initiated at

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1 Experimental Techniques

Fig. 1.10 The back-surface gauge setup (top) and a manganin gauge record (bottom)

the impactor’s free surface and at the specimen/Plexiglas interface. The amplitude of the first reverberation in the spall signal is related to the failure strength of the specimen (the spall strength) under dynamic tensile conditions. Thus, with this simple experiment one derives both the dynamic compressive strength of the specimen and its dynamic tensile failure threshold. A direct technique to determine the compressive strength of the solid, at high shock pressures, is the lateral gauge technique, which was first suggested by Bernstein et al. (1967). Two stress gauges are embedded in the specimen at perpendicular planes, as shown in Fig. 1.11. The longitudinal gauge is embedded in a plane parallel to the shock front and measures the longitudinal stress in the specimen (σ x ). The lateral gauge is embedded in a plane along the shock propagation direction, and it measures the lateral stress in the specimen (σ y ). The difference between the two measured stresses is equal to the strength of the specimen at its shocked state, hence: σ x – σ y = Y d (p, T ). The main difficulty with this technique stems from the fact that the two gauge orientations result in different calibration curves for the gauge, as was shown by Rosenberg et al. (1981) and by Rosenberg and Partom (1985). This difference can be observed by the gauge records in Fig. 1.11. These are the resistance-time records of the two gauges, longitudinal and lateral, embedded in a Plexiglas target which was shocked to about 2.5 GPa in our lab. Note the short rise time of the longitudinal gauge record as compared with the step-wise rise time of the lateral gauge’s record. These steps are the result of the shock wave sweeping the “legs” of the grid-like gauge, in its lateral orientation. The records clearly show that the resistance change of the lateral gauge is higher than that of the longitudinal gauge. Since the longitudinal stress is never lower than the lateral stress under shock loadings, it is clear that the calibration of these gauges must be different for the two orientations. With a proper calibration for the manganin gauge in the lateral orientation, as determined for by Rosenberg and Moshel (2014), this technique can be used to measure the strength of materials under high shock pressures. As we shall see in the third part of this

1.2 Determination of the Dynamic Properties of Materials

17

Fig. 1.11 The longitudinal (right) and lateral (left) gauge orientations and their records in a Plexiglas target shocked to about 2.5 GPa

book, the lateral gauge technique has been very useful in determining the strength of ceramic materials at high shock pressures. The most accurate experimental techniques in shock wave research are the various interferometric devices which were reviewed by Chhabildas (1987). The basic idea behind these techniques is to follow the velocity of the specimen’s back surface continuously, using the reflected light from a laser beam. The most popular interferometers are the VISAR (Velocity Interferometer System for Any Reflector) system which was developed by Barker and Hollenbach (1964), and the Fabry–Perot interferometer which was first applied Johnson and Burgess (1968). Both systems use Doppler-shifted light from the moving back surface, which is either specular or diffuse. The reflected light is split into two beams to form the different legs of a wide angle Michelson interferometer. The fringe sensitivity of the interferometer is dependent on the delay time between the two legs. With the VISAR system the delay is achieved by a glass etalon along the path of one of the legs, with a predetermined time delay. With the Fabry–Perot system the delay is achieved through a set of plates separated by a well defined distance. A superposition of the two beams results in a fringe system with a count which is proportional to the velocity of the target’s back surface. Both systems have fast time response which enables one to follow small changes in the specimen’s free surface velocity very accurately. However, these techniques suffer from several disadvantages which are related to the changes in the intensity of the reflected light (for the VISAR) and to the resolution of the recording film, for the Fabry–Perot system. The heterodyne velocimeter which was

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introduced by Strand et al. (2006), overcomes the difficulties of both the VISAR and the Fabry–Perot systems while keeping their main advantages. This technique, also referred to as the Photonic Doppler Velocimetry (PDV), is a powerful diagnostic for tracking velocities (from a few m/s to km/s) on the short time scales (nanoseconds to microseconds) of a dynamic compression experiment. Essentially a fiber-based Michelson interferometer, PDV combines Doppler shifted light from a moving reflector with a reference source (λ0 = 1550 nm), yielding a complete fringe when the reflector moves a distance λ0 /2. The derived velocity–time history is directly related to the frequency of the beat wave form. Thus, there is no need for extra components to resolve the difficulties arising from the fringe jump ambiguities. In addition, the data is recorded on digitizers having long recording time, avoiding camera distortion and other problems. With the high resolution of these techniques one can gain accurate values for the strength of shock compressed specimens by using the self consistent technique which was suggested by Asay and Lipkin (1977). A schematic diagram of this experimental setup together with the resulting VISAR records, are shown in Fig. 1.12. The technique is based on performing two similar experiments which differ by the backing disc behind the impactor, having either higher or lower acoustic impedance than the impactor. The acoustic impedance of a material is defined by the product ρ 0 · C L , where ρ 0 and C L are the density and longitudinal sound velocity of the material, respectively. With the same impact velocity, the amplitudes of the initial shock waves in the two experiments are the same. The initial shock wave is followed either by a second shock, when the impactor is backed by a higher impedance material, or by a release wave for a backing material with lower impedance. From the measured elastic parts of these reshock and release records, one can determine the dynamic strength of the specimen in its shocked state, namely, Y d = Y (p, T ). A concise summary of the results for various materials from Asay and Kerley (1987), is shown in Fig. 1.13. The data is given for their normalized dynamic strengths

Fig. 1.12 Experimental setup and the resulting record from unloading and reloading experiments

1.2 Determination of the Dynamic Properties of Materials

19

Fig. 1.13 The change in strength at high shock pressures for several materials

(relative to the static values) as a function of shock amplitude. The decreasing trend of the strength with shock pressures for some of these materials is due to the competing effects of pressure hardening and temperature softening. Obviously, at high shock pressures the temperature of the specimen can be high enough to induce its softening, as seen here for the aluminum alloys. The Kolsky bar system which was introduced by Kolsky (1949), and is often referred to as the split Hopkinson bar, is the most useful apparatus to determine the dynamic stress–strain curves of solids under uniaxial stress conditions in the 102 –104 s−1 range of strain rates. This is the relevant range of strain rates for the projectile and target materials around their interface during the penetration process. The Kolsky bar system consists of two long instrumented bars made of high-strength steel, which hold the specimen as shown in Fig. 1.14. The diameter of these bars is usually in the range of 12–25 mm and their corresponding lengths are 3–5 m. A short bar (the striker) is launched by an air gun towards the first bar (the receiver) at a velocity of 5–20 m/s, inducing a square-shaped stress pulse in the receiver bar. The duration of this pulse is equal to twice the transit time of an elastic wave in the striker bar (typically 100–200 µs). This stress pulse is recorded by the first strain gauge, which is glued on the receiver bar at some distance from the bar/specimen interface. The incident strain-time history, εi (t), is one of three strain signals which are necessary for the analysis. Upon reaching the specimen, the incident stress pulse

Fig. 1.14 A schematic description of the Kolsky bar system

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1 Experimental Techniques

is partly reflected back to the receiver bar and partly transmitted to the second bar (the transmitter). The reflected pulse, εr (t), is recorded by the first strain gauge which recorded the incident wave. The transmitted pulse εt (t) is recorded by a second strain gauge, which is glued to the transmitter bar at the same distance from the specimen as the first gauge. The diameter of the specimen disc should be smaller than the bar’s diameter in order to achieve significant stress amplification in the specimen. With proper dimensions even strong materials can be compressed to large strains, well beyond their elastic range of response. Due to the early stress reverberations inside the specimen, the analysis of the Kolsky bar system cannot account for the elastic part of the specimen’s response. The technique is considered to be accurate only for strains beyond several percent. The variables which control the deformation of the specimen, are the velocities V 1 (t) and V 2 (t) of the two bars at their interfaces with the specimen. The strain rate of the specimen is simply given by: ε˙ =

V1 (t) − V2 (t) dε = dt h

(1.9)

where h is the thickness of the specimen, which is also changing with time. The velocities V 1 (t) and V 2 (t) are derived from the measured strain-time histories by using the well-known relation for elastic waves in long bars, V = C b ε, where C b is the velocity of elastic waves in long bars. Thus, one can write for the two interface velocities: V2 (t) = Cb εt (t)

V1 (t) = Cb [εi (t) − εr (t)],

(1.10)

The strain of the specimen is obtained by inserting these expressions for V 1 and V 2 in (1.9), and integrating it according to: ε(t) =

Cb h

 [εi (t) − εr (t) − εt (t)]dt

(1.11)

Assuming stress equilibrium along the thickness of the specimen, the stress is derived by averaging the two stresses which act on the opposite sides of the specimen: σ (t) =

F1 (t) + F2 (t) 2 As

(1.12)

where As is the cross section area of the specimen, which is also changing with time. The forces F 1 and F 2 are related to the strains in the bars, as measured by the two strain gauges, through: F1 (t) = Ab E[εi (t) + εr (t)]

and

F2 (t) = Ab Eεt (t)

(1.13)

1.2 Determination of the Dynamic Properties of Materials

21

where Ab is the cross section of the bar and E is its Young’s modulus. With these stress and strain histories in the specimen, σ (t) and ε(t), one derives its dynamic stress–strain relation, σ = σ (ε), at the high strain rate under consideration. Changing the dimensions of the specimen or the impact velocity of the striker bar, results in different strain rates as well as different maximal strains for the specimen. When the analysis follows the transient values of the specimen’s thickness, the resulting σ (ε) curves are the so-called “true stress–strain curves”. If such a correction is not made they are referred to as the “engineering stress–strain curves”. In order to plastically deform a high strength specimen in the Kolsky bar system their area ratio (Ab /As ) should be large. This will result in considerable stress amplification so that the stress in the specimen will be larger than its yield strength. It is very important to accurately measure the dimensions of the specimens before the experiment and to determine the elastic wave velocity (C b ) in the long bars. A typical set of strain signals in a Kolsky bar experiment as measured in our laboratory, for a specimen of the 6061-T651 aluminum alloy, is shown in Fig. 1.15. The figure also shows the resulting true stress–strain curve which was derived from these strain histories. Fig. 1.15 a The measured strain signals, b the resulting true stress–strain response for a specimen of the 6061-T651 aluminum alloy

22

1 Experimental Techniques

The actual state of stress in the disc shaped specimens has been the subject of many works which analyzed the effects of specimen inertia and the friction at the specimen/bar interfaces, on the stress uniformity in the specimen. One of the important conclusions, by Gorham et al. (1984), is that in order to minimize inertia effects the aspect ratio of the specimen, its thickness to diameter ratio, should be about 0.5. In order to minimize friction at the interfaces it is customary to apply some grease to the specimen faces before the experiment. The issue of inertia is significant for materials like concrete, as discussed by Li and Meng (2003). Their analysis shows that the apparent rate sensitivity of concrete and concrete-like materials, at strain rates of 10–102 s−1 , is due to the specimen’s inertia rather than to a real increase in concrete strength at high strain rates. We shall discuss this issue in Chap. 3, when we deal with the rate-sensitivity of concrete-like materials. As far as metals are concerned these inertia effects are negligible, and the Kolsky bar system is the best experimental technique to derive dynamic stress–strain curves at strain rates which are relevant for terminal ballistics. In particular, phenomena like thermal softening and adiabatic shear banding can be easily identified by the shape of the stress–strain curve, as will be discussed in Chap. 2. Avinadav et al. (2011) developed a new technique for the direct measurement of the bar velocities in the Kolsky system, instead of the traditional strain measurements. They used the velocity interferometer technique of Strand et al. (2006), the photonic Doppler velocimeter (PDV), which was specifically adapted for low velocity measurements. These interferometers measure the velocities at the sides of the bars by using fiber focusers which illuminate them at an angle of about 30°, and collect reflected Doppler-shifted light which is interfered with a reference beam. The new technique results in smooth velocity histories of the two bars, from which the stress and strain histories of the specimen are derived through the equations outlined above. This is a more direct and easy way to determine the dynamic stress–strain curves and, as shown by Avinadav et al. (2011), the results are identical to those obtained by the strain gauges. The simple arrangement of the Kolsky bar system for dynamic compression tests has been modified in order to load the specimen under dynamic tension or torsion. The basic principles of these devices are the same but the fixtures of the specimen are different. For example, tensile loading is achieved with the collar-technique where the specimen, having the shape of a standard tensile specimen, is screwed into the ends of the two bars and a collar made of high strength steel is placed over it. Upon loading, the compressive wave in the receiver bar is transmitted through the collar, while the specimen does not take a part in the compressive loading stage. At a later time, the stress waves reach the end of the bars and reflect back towards the specimen. The interaction of the two release waves loads the specimen in tension. At this stage the collar is detached from the bars and the tension acts only on the specimen. The same analysis of the strain gauge signals is used to derive the stress–strain response of the specimen under dynamic tension. This test configuration was used by Rosenberg et al. (1986) to heat the specimen in order to obtain high temperature dynamic stress– strain curves for several metals. The heating of the specimens has to be both localized and fast in order to avoid excessive heating of the bars. These requirements were met by using induction heating coils, as shown schematically in Fig. 1.16. The collars

1.2 Determination of the Dynamic Properties of Materials

23

Fig. 1.16 The induction heating setup in a tensile Kolsky bar test

in these tests were made of a high strength ceramic material in order to avoid their heating by the eddy currents which should flow only in the specimen. The heating times of the specimens with this system are typically less than a minute and their temperatures, as measured by a thermocouple inserted through a hole in the collar, can reach 1,000 °C.

1.3 The Common Threats in Terminal Ballistics This book is focused on the terminal ballistics of the more common threats which are encountered in the battlefield. These threats belong to the following four families: (1) small arms projectiles, (2) fragment simulating projectiles (FSPs), (3) long rods, (4) shaped charge jets and explosively formed projectiles (EFPs). The small arm projectiles, which are shot from rifles and machine guns, have either soft (usually lead) or hard cores made of steel or tungsten-carbide. Their diameters range between 5 and 15 mm and their muzzle velocities are in the range of 600–1,000 m/s. The armor piercing (AP) projectile has a very hard core (around 60Rc), which enables it to penetrate metallic target as a rigid body. A typical AP projectile, the 0.3 APM2, with its different constituents, is shown in Fig. 1.17. Fig. 1.17 The structure of 0.3 APM2 projectile (dimensions are given in mm)

24

1 Experimental Techniques

Fig. 1.18 The ogive nose shape

The ogive shape of these cores is the most effective nose shape as far as the penetration depth of a rigid projectile is concerned. This special shape is formed by sections of two spheres whose centers are outside the projectile, as shown schematically in Fig. 1.18. The radius of these spheres (S) is usually a multiple of 2–4 of the projectile’s diameter, and it defines the ogival shape in terms of caliber-radius-heads (CRH). Thus, an ogive nose with S = 3D is termed CRH3 ogive. Fragment simulators are steel cylinders with an aspect ratio of about 1.0, having diameters in the range of 5–20 mm and hardness of about 30Rc. These projectiles represent the shape of steel fragments from artillery shells, and their front face has a chiseled shape as shown in Fig. 1.19. They are usually used for testing new armor concepts which have to stop these fragments at impact velocities of about 1.0 km/s. The shaped charge jet is formed by the collapse of a conical shell made of a ductile metal (usually copper), which is inserted in a conical cavity at the end of an explosive cylinder, as shown in Fig. 1.20a. Upon impact with the target the explosive cylinder is ignited and a detonation front is moving along its length. When this front reaches the copper cone it induces its symmetric collapse, as shown in Fig. 1.20b, forming a long copper jet which moves at velocities of several kilometers per second. In fact, these jets have a velocity gradient along their length which results in their continual stretching with time and distance. The “head” of these jets move at velocities around 8 km/s, and their “tail” at 2–3 km/s. These long and thin jets (a few mm in diameter) can penetrate armor steel targets which are as thick as their length, making them the most lethal warhead against armored vehicles. Two consecutive shadowgraphs Fig. 1.19 The fragment simulator

1.3 The Common Threats in Terminal Ballistics

25

Fig. 1.20 The formation and stretching of a shaped charge jets during its flight

of such a jet, from Liden et al. (2008), are shown in Fig. 1.20c. Note the shape of the leading portion of the jet, which is common for many shaped charge jets. When a spherically shaped liner is inserted in the explosive cylinder, it collapses in a different way to form the so-called EFP, and Fig. 1.21 shows such a copper EFP in flight. These are spherically-nosed projectiles with an aspect ratio of 3–5, which move at a velocity of about 2.0 km/s. The diameter of these EFPs, which are made of copper or tantalum, is in the range of 20–40 mm. Their high lethality is due to the large craters which they form in the target and the large amount of debris which they propel into the inner compartment of the attacked vehicle. The debris includes the remnants of the EFP, as well as target material which are ejected from the crater at high velocities. Long rod penetrators are made of high density materials, such as tungsten alloys or depleted uranium, with densities of about 17.5 g/cm3 . Their aspect ratios range between L/D = 20–30, and they are shot at velocities of about 1.7 km/s. Due to their pointed fronts they do not lose much of their velocity even after a flight distance of a couple of kilometers. These long rods replaced the older steel projectiles which had smaller aspect ratios, when it was realized that the penetration depths of these projectiles are linearly dependent on their length. Since the compressive strength of these rods is only about 1.5 GPa, they are eroding during their impact and penetration Fig. 1.21 A typical EFP in flight

26

1 Experimental Techniques

Fig. 1.22 A long rod penetrator in flight

in armor steel targets. A picture of such a rod in its early stages of flight is shown in Fig. 1.22. Note the three parts of the aluminum sabot, which was used to launch the rod in the gun, as they are separated by the air drag.

Chapter 2

Material Models for Numerical Simulations

2.1 General Description Numerical simulations of high-velocity impact events are performed with large computer codes called hydrocodes, because their early use was for high pressure problems in which materials were treated as fluids. These codes can handle impulsive loadings which include shock waves with extremely high pressures and short rise times, as well as high temperatures and large deformations. Several review articles about hydrocodes and their uses have been published over the past 30 years. As far as terminal ballistics is concerned, the reviews by Anderson (1987) and by Zukas (1990) are the most comprehensive and informative. The present chapter highlights some of the important issues concerning these codes for terminal ballistics studies. We follow Anderson’s statement that “hydrocodes are the best instrumented experiments”, since they can (and should) be used for sensitivity studies in terminal ballistics. Such studies can highlight the role of each parameter in the investigated process and the physics behind the process may be unveiled. We shall demonstrate the construction and validation/refutation of analytical models with such sensitivity studies, through the different issues presented in this book. As an example, consider the difficulty to determine experimentally the dependence of a projectile’s penetration depth on the strength of the target. Any experimental attempt to vary only the strength of a target, within a large range of values, will result in some changes in its other properties as well. On the other hand, a series of simulations where only the strength of the target is varied systematically is easy to perform and it can offer the required insight for this issue. Another important advantage of hydrocodes is that they can give information about events which are beyond any laboratory’s performance. For example, the impact of meteors at velocities of 20–40 km/s cannot be studied experimentally, but it can be easily investigated with these codes. Basically, the hydrocode is an efficient and accurate scheme for solving the set of conservation equations for mass, momentum and energy, under the initial and boundary conditions which characterize the physical event. In addition to these equations and problem constraints, there are two sets of data which have to be specified for all © Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_2

27

28

2 Material Models for Numerical Simulations

the materials involved. These are the equation of state of the material and its constitutive relations, which were discussed briefly in Chap. 1. The equations to be solved describe the behavior of a continuum but the code is solving them by discretization techniques for both space and time, through finite difference or finite element forms. With the finite difference technique, a grid is generated to represent material points for the given geometry of the participating bodies. The continuous spatial derivatives in the equations, such as df /dx, are replaced by difference equations like f /x, where the differences x are related to the cell dimensions in the grid. The same ideas are applied to the differentiation with respect with time. Values of the parameters at a certain time (t) are calculated by their values at an earlier time (t − t) and by their corresponding time derivatives. With the finite element technique the actual differential equations are solved in small interconnected sub-regions (elements). Nodes are assigned to the elements and an interpolation function is used to represent the variation of the variable over the element. Anderson (1987) discusses the four properties which have to be carefully considered in order to optimize these calculations, namely: consistency, accuracy, stability and efficiency. The grids themselves are either Eulerian or Lagrangian and each type has its own merits and disadvantages. The Eulerian description has a spatial nature whereby the grid points and, consequently, the cell boundaries remain spatially fixed with time, while material can flow through the mesh. The net flow of material into this fixed cell volume determines its mass content, pressure, velocity etc. On the other hand, with the Lagrangian scheme, the grid is attached to the material and moves with it. With this scheme the grid points follow material paths, so it is more appropriate for cases where the material elements are less distorted. The Eulerian scheme is more useful for heavily distorted materials. An optimal combination for a situation where a rigid projectile is penetrating a soft target is obtained by using the Lagrange scheme for the projectile and the Euler scheme for the target. One of the more important issues which have to be carefully considered is the issue of mesh size. Different results are obtained if the number of cells per unit length is not adequate. For example, it was found that for penetration studies with eroding long rods, the number of cells on the rod’s radius should be at least eleven. The same density of cells should be kept in the target, at least for several projectile radii around its symmetry axis. In order to save computing time, the cell size at farther zones can be gradually increased according to their distance from the symmetry axis. The mesh cell size depends on the specific problem. As an example, a small cell size should be considered in cases where there is a fracture in the projectile or in the target. It is recommended that while preparing the code for its final runs, the numerical convergence with respect to mesh cell size should be checked. Another important issue, especially when material elements are expected to deform considerably, is the issue of erosion with Lagrangian codes. At large deformations the code may run into trouble when treating heavily deformed elements. The use of the erosion threshold condition is then necessary in order to eliminate highly distorted elements, at a predetermined value of the plastic or geometric deformation. The amount of the eroded elements should be monitored continuously, and when it is too high one should replace the Lagrangian with an Eulerian version of the code.

2.2 Material Properties

29

2.2 Material Properties The high velocity impact of two solid bodies can induce extremely high pressures and temperatures in both bodies, around their impact interface. These extreme conditions are propagated away from the impact area through strong shock waves, which eventually decay to elastic waves at large distances. During the short times in which a solid experiences these high pressures and temperature, it may undergo various physical changes such as melting or evaporation, as well as mechanical changes which lead to strength loss, local fracture and even total shattering. Even though the state of the compressed solid under strong shock loading is an irreversible one, it is assumed to be of thermodynamic equilibrium nature. Thus, the thermodynamic states of the shocked solid, which are defined by its internal energy (e), specific volume (V ) and the pressure (p), are assumed to lie on the equation of state surface, e = e(V, s) or e = e(V, T ), where e and s are the internal energy and the entropy of the solid, respectively. To summarize, the relevant material properties for terminal ballistics belong either to their equations of state (EOS), which account for material compression at high pressures and temperatures, or to their constitutive relations. These relations follow the strength and failure characteristics of the solid under high pressures, temperatures and strain rates, as well as during the large deformations which it experiences. This chapter describes some of the equations of state and the constitutive relations which are implemented in various hydrocodes. These codes include a large data bank which is continually updated, for the relevant parameters of the EOS and the constitutive properties for many materials.

2.2.1 The Equation of State The equation of state (EOS) is a relation between the internal energy of the material, its density (ρ) or its specific volume V, and the pressure which characterize its state, through e = e(V, p), or in terms of the corresponding compression curve p = p(V, e). We have already seen that the Hugoniot relations result in compression curves which account for the state of the shocked material, pH = pH (V, eH ), where the subscript H denotes a state on the Hugoniot. Other compression curves can be considered as special cases of the general EOS, such as the various isotherms p = p(V ), at a given temperature. The seminal work of the Nobel laureate P. W. Bridgman includes isotherms of many materials (both solids and liquids) compressed to about 10 GPa, in the form: V = −Ap + Bp 2 V0

(2.1)

where A and B are positive coefficients, characteristic to the material. The compressibility of the material is given by: β = A − BP, from which we find that B is the

30

2 Material Models for Numerical Simulations

pressure derivative of the compressibility. Another useful relation for the isotherm is the Murnaghan equation which accounts for the data to much higher pressures. This relation is based on the assumption that the bulk modulus (K) increases linearly with pressure, thus: K = K 0 + np, where n is the pressure derivative of K. The resulting isotherms are expressed as: K0 p(ρ) = n



ρ ρ0

n

 −1

(2.2)

where ρ is the density (1/V ). As was shown by Ruoff (1967), the value of n can be obtained by the relation n = 4S − 1, where S is the slope of the linear relation between the shock and particle velocities in (1.5). Using a typical value of S = 1.5, the value for n should be about 5.0, which is in accord with measured pressure derivatives of the bulk moduli for many solids. In order to appreciate the relation between the Hugoniot and the room temperature isotherm of a given material, consider the schematic description of these curves in Fig. 2.1. The points on the Hugoniot curve lie above the corresponding points on the isotherm, because of the higher temperatures which characterize the shocked state. For a given value of the specific volume (V ) the following relation can be written for the points on the Hugoniot and the isotherm for a given temperature (T 0 ): p H = pisoth + pT

(2.3)

where pT is the so-called thermal pressure. This is the extra pressure which is needed to compress the material to the volume V in order to overcome the temperature difference (T H − T 0 ) between the Hugoniot and the isotherm. Written in another way, pT can be expressed as:  pT =

∂p ∂T

 (TH − T0 )

(2.4)

V

The partial temperature derivative of the pressure in (2.4) is a most important quantity in high-pressure physics, as is clearly evident from the above description. It was Fig. 2.1 A schematic description of the Hugoniot and isotherm curves

2.2 Material Properties

31

treated theoretically through solid-state physics considerations in order to account for its experimentally-observed values. As discussed by Zeldovich and Raizer (1965), this derivative is closely related to the volumetric derivative of the atomic vibration frequency around their equilibrium positions. With some simplifying assumptions, Gruneisen showed that:   ∂p th C V (2.5) = ∂T V V where C v is the heat capacity of the material and  th is the thermal Gruneisen coefficient, which is related to the above-mentioned atomic frequencies, and is defined by: 

∂p th = V ∂e

 (2.6) V

These theoretical derivations led to one of the more useful equations of state, the Mie-Gruneisen equation, which is given by: p = pref +

th (e − eref ) V

(2.7)

where the current state of the material is related to some known (the reference) state through this simple relation. This equation of state is often used in the hydrocodes for deriving off-Hugoniot states, such as the release curves, while the Hugoniot relations are used in order to define the reference states. It turns out that  has a value of about 2.0 for many solids at ambient conditions. Moreover,  is assumed to be independent on temperature and it is also assumed that the product  · ρ is constant for a given solid, within a large range of pressures. Rosenberg and Partom (1982) demonstrated experimentally that this assumption holds for Plexiglas, by considering a somewhat different definition of , given by:   ∂ ln T =− ∂ ln V s

(2.8)

in which the partial derivative is defined along the isentropic compression curve of the solid. Rosenberg and Partom (1982) performed direct measurements of the temperature rise and the volumetric strains of isentropically-compressed Plexiglas plates, by using in-situ temperature and strain gauges. The isentropic loading of these plates was achieved by a series of low-amplitude shock reverberations, and the gauges monitored the temperature and specific volume (strain) at each reverberation. The main conclusion of this work was that the product of  · ρ for Plexiglas is, indeed, constant within the range of pressures achieved in these experiments.

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2.2.2 The Constitutive Relations The inclusion of the elastio-plastic behavior of solids into hydrocodes dates back to the early work of M. Wilkins (1964). His basic idea was to treat the elements of the stress tensor (σ ij ) as composed of two parts, the hydrostatic part (p) which is derived by the equation of state, and the deviatoric part (sij ) which represents the geometric distortion of the material. According to this convention, the stress components are written as: σi j = p · δi j + si j

(2.9a)

where δ ij is the Kronecker delta. The rate of change in the deviatoric stresses is related to the strain rates through: s˙i j = 2G ε˙ i j

(2.9b)

where G is the shear modulus of the solid. These strain rates are then decomposed to their elastic and plastic parts as follows: p

ε˙ i j = ε˙ iej + ε˙ i j

(2.9c)

Instead of working with tensors it is easier to use the equivalent (or effective) stress and strain measures, σ eq and εeq , as defined by:  σeq =

3si j si j and εeq = 2



2εi j εi j 3

(2.10a)

and the constitutive relations are written in the form: σeq = σ (εeq , ε˙ , T)

(2.10b)

Wilkins (1964) showed that it is easier to work within the stress space where the axes are along the principal stresses of the problem, denoted by σ 1 , σ 2 , and σ 3 . The diagonal line in this three-dimensional stress space, for which σ 1 = σ 2 = σ 3 , denotes states on the hydrostatic pressure loading of the material. The equivalent stress is a measure for the distance of a given point from the diagonal line in this space. The various yield criteria define the maximum permissible value of this distance. In terms of the principal stresses, the border between elastic and plastic states is a cylindrically shaped envelope around the diagonal line, which corresponds to the hydrostatic loading conditions. All the states inside this yield envelope correspond to the solid’s elastic range, while those on the envelope correspond to its plastic (yielded) states. By definition, the states which correspond to points outside this yield envelope are not permissible. Wilkins (1964) suggested the radial-return procedure, by which

2.2 Material Properties

33

these states should be brought to the yield envelope in numerical simulations. The equation for the yield surface in the principal stress space is given by: (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = 2Y 2

(2.11a)

or, in its deviatoric form: s12 + s22 + s32 = 23 Y 2

(2.11b)

The various yield criteria are, in fact, imposing a maximum value for this equivalent stress which has to be specified in the code. For example, the well-known von-Mises criterion states that the equivalent stress of the material cannot be higher than its yield strength under simple tension, σ eq ≤ Y 0 . In terms of the yield surface in the stress space, this criterion is represented by the surface of a cylinder with a radius of Y 0 , which surrounds the main diagonal (σ 1 = σ 2 = σ 3 ). The von-Mises criterion accounts for the yielding behavior of many metals and alloys which exhibit elastic-perfectly-plastic characteristics. This is true for high strength materials, such as the strong aluminum alloys which are used in military and industrial applications. Softer metals often show an appreciable strain-hardening behavior where the strength is a strong function of the strain. The stainless steel 304L is an example for such a significantly hardening material, since its flow stress can increase by a factor of five, at strains of about 50%. On the other hand, steels which are referred to as rolled homogenous armor, show an almost ideal elastic-plastic behavior with no strainhardening. One should note that the interaction between a penetrating projectile and a target is always accompanied by large strains in both materials. Thus, it is the stress at large strains, the so-called “flow stress”, which is the important strength property of the material for terminal ballistics. The relevant strain rates for impacts at ordnance velocities (up to about 2.0 km/s) are 103 –105 s−1 . These are the rates covered by the Kolsky bar system which is the main reason for its extensive use. We shall further discuss these issues in Chap. 4, which deals with the effective (average) strains and strain-rates in a metallic plate subjected to ballistic impact and perforation. One has to consider also the fact that the strength of most solids increases at high confining pressures, while high temperatures reduce their strength (thermal-softening). Thus, a complete constitutive relation for the strength of the solid should include all the data concerning all the effects discussed here. Some of these constitutive models which are implemented in commercial hydrocodes are described next. The constitutive equation proposed by Johnson and Cook (1983) is one of the more popular in the field of terminal ballistics. This equation, which is referred to as the JC model, includes the effects of plastic strain, strain rate, and temperature on the material strength, as follows:       Y = A1 + A2 εnp · 1 + A3 ln ε˙ ∗p · 1 − THm

(2.12)

where A1 , A2 , A3 , n and m, are material constants. εp is the equivalent plastic strain, and ε˙ p∗ = ε˙ p ε˙ 0 , is the normalized plastic strain rate with ε˙ 0 = 1.0 s−1 as the

34

2 Material Models for Numerical Simulations

reference rate. The homologous temperature (T H ) is defined by the room and the melting temperatures (T room and T m ), according to: TH =

T − Troom Tmelt − Troom

(2.13)

Another material model which is frequently used for high pressure impacts is the rate-independent model of Steinberg (1987). According to this model the changes in the flow stress (Y ), with high pressures and temperatures, are equal to the corresponding changes in the shear modulus (G) of the material. The relations for Y (p, T ) and G(p, T ) are given by the following expressions: Y ( p, T ) = Y0 ·

n G( p, T ) 1 + βpε G0

(2.14a)

  G p p G T · + (T − 300) G( p, T ) = G 0 · 1 + G 0 η1/3 G0

(2.14b)

where β p and n are material parameters, ε is the effective plastic strain and η = ρ/ρ 0 is the compression ratio of the solid. The primed parameters, with the subscripts p and T, are the derivatives of G with respect to pressure and temperature, respectively, at the reference state (T = 300 °K, p = 0). Steinberg’s model is used for materials experiencing the high pressures and temperatures under shock conditions, rather than under ballistic impacts. A third family of constitutive equations is due to Zerilli and Armstrong (1987), shortly termed the ZA model. Unlike the JC model, which is basically an empirical one, the constitutive equations of the ZA model are based on dislocation dynamics theory. They treat face centered (FCC) and cubic centered (BCC) metals differently, since the rate and temperature sensitivities are different for these two families of materials. Their constitutive equations are given by: σeq = σ0 + kd −0.5 + C2 ε0.5 exp[−C3 T + C4 T ln ε˙ ]

for FCC materials (2.15a)

σeq = σ0 + kd −0.5 + C1 exp[−C3 T + C4 T ln ε˙ ] + C5 εn

for BCC materials (2.15b)

where C 1 , …, C 5 and n are material parameters. The Hall-Petch relation, between the strength of a metal and its average grain diameter (d), has been taken into account in this formulation through the second term in (2.15a, 2.15b). Both the JC and ZA models account for the rate dependence of the metal’s strength through a moderate logarithmic term. However, many workers in the past decades found a significant increase in the dynamic strengths of various metals, at strain rates around 103 s−1 . This phenomenon has been termed the “upturn effect” and it has been a source of controversy since the 1970s, when it was first discussed by Lindholm (1979) who used the Kolsky bar system to compress high-purity copper specimens.

2.2 Material Properties

35

In contrast with existing data, his tests resulted in a relatively small increase in the flow stress of these copper specimens, up to the limiting strain rate of his bar system of about 6.104 s−1 . Thus, he specifically stated that there is no evidence of a viscous damping mode in his tests on copper, in contrast with the earlier claim by Kumar and Kumble (1969). These authors explained the upturn in their data for copper, at a rate of about 103 s−1 , by the different regimes for the motion of dislocations. Namely, at low strain rates (103 s−1 ) the dislocations experience viscous damping. This two-regime explanation was later adopted by many workers, in order to account for the upturn phenomena which they observed in various metals and alloys. Rosenberg et al. (2019) highlighted this controversy by reviewing conflicting evidence for several materials, which were tested by different experimental techniques. They argued that strain rates of about 103 s−1 are much lower than the range of 106 –109 s−1 , where one may anticipate a significant change in the corresponding flow stresses due to dislocation dynamics considerations. Several plausible causes for this controversy were proposed by Rosenberg et al. (2019), highlighting the problems with various experimental techniques which were used by workers who observed these upturns. In order to avoid such controversies they recommend employing the regular compressive Kolsky bar system, with which very high strain rates can be achieved by smaller bar and specimen diameters. The specimen’s shape should be a disc with an aspect ratio of about 0.5, in order to avoid possible complexities in the data reduction process. In addition, the flow stress of the specimen should be taken at strains of at least 10% in order to avoid stress-equilibrium issues. Most importantly, one should be aware of the large scatter in the data gained by Kolsky bar systems, at their limiting performance range. One should also refrain from using shock wave data from plate impact tests, because they include significant high-pressure and high-temperature contributions, which may complicate the interpretation process for the rate dependence of the flow stresses.

2.2.3 Failure of Ductile Materials The models described above treat the changes in the material’s effective strength through its hardening and softening mechanisms. The characterization of these changes is important for ductile materials which can withstand large plastic strains before they fail, either under tension or by shear, when a certain threshold strain is reached. The failure of a specimen is defined by the loss of cohesion in its interior, leading to either its complete disintegration or to some inner damage, which is manifested by the appearance of new free surfaces inside the specimen. Fracture under tension is the end result of the coalescence of small voids, forming macroscopic free surfaces inside the specimen. The processes of void nucleation and growth in ductile materials under dynamic loading conditions have been investigated by many workers, and the review article of Curran (1982) gives a comprehensive account of this field. Fracture under shear is usually the end result of the appearance of shear

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bands in the specimen at the highest shear locations. These bands are very narrow zones in which voids are eventually developed, leading to failure by their coalescence when the shear strains in the bands exceed their limiting values. In contrast, brittle materials fail within their elastic response regime, with negligible straining, and their main mode of failure involves microcrack nucleation and coalescence, as discussed in the next section. Johnson and Cook (1985) proposed a fracture criterion for ductile materials, which is especially suited for numerical codes. This criterion is based on the maximum strain to fracture (εf ) of an element, which depends on its strain path, strainrate, and temperature. It also depends on the stress triaxiality, which is defined by σ * = σ m /σ eq , where σ m is the local pressure as calculated by the average of the three principal stresses in each element. This is a very important parameter since the propensity of ductile materials to fail is strongly dependent on the pressure exerted on them. Compressive stresses tend to prevent failure by closing voids and microcracks, while tensile stresses enhance failure by further opening them. The influence of stress triaxiality according to various models is based on the void growth model of Rice and Tracey (1969). Different stress triaxialities are not easily achieved in well defined experiments, since it is difficult to accurately define the real state of a specimen, especially under complex loading situations. One of the more popular techniques for generating controlled stress triaxialities, which was suggested by Bridgman (1952), is based on varying the notch radius in the midsection of a tensile specimen. This is a relatively simple and straightforward technique which is easy to analyze through numerical simulations. The Cockroft-Latham fracture model (1968) is a rather simple failure criterion which is especially suitable for numerical simulations with semi-ductile metals such as the strong aluminum alloys. The model assumes that fracture takes place in a given element when a certain value of the following integral reaches a threshold value W CL : ε f WC L =

σ1 dε

(2.16)

0

where σ 1 is the major principal tensile stress in the element. The value of the threshold value (W CL ) is determined by the tensile stress-strain curve of the material and the integral is taken over the equivalent plastic strain path up to failure (εf ). This fracture criterion is easy to implement in the code and together with the stress triaxiality, which determines the variations in εf , one can have very good agreement with experimental data, as is clearly seen in the work of Holmen et al. (2016) for example. The dependence of fracture strain on the stress triaxiality for the structural steel Weldox 460E, as obtained by Hopperstad et al. (2003), is shown Fig. 2.2. Note that according to this representation, tensile and compressive stresses correspond to positive and negative triaxialities, respectively, while pure shear corresponds to zero triaxiality. Obviously, under compressive stresses the fracture strain increases markedly with stress triaxiality since high pressures oppose the opening of voids and

2.2 Material Properties

37

Fig. 2.2 The dependence of fracture strain on the stress triaxiality for Weldox 460E steel

cracks. In fact, some models suggest that these graphs should rise asymptotically to infinite fracture strains at a compressive triaxiality of σ * = −1/3. The JC failure model defines a continuous degree of damage (Df ), in the element under consideration, which is given by: Df =

εeq εf

(2.17)

where εeq is the increment of accumulated equivalent plastic strain during an integration cycle and εf is the equivalent plastic strain at fracture. This value is reached when the damage function (Df ) reaches the value of 1.0. The failure strain is given by the following functional form:   ε f = [D1 + D2 exp(D3 σ ∗ )] · 1 + D4 ln ε˙ ∗p · [1 + D5 TH ]

(2.18)

where D1 , …, D5 are constants which have to be calibrated for each material. Many of the constants which appear in these constitutive equations, for strength and failure, are determined by the Kolsky bar system described in Chap. 1. Typical examples of dynamic stress-strain curves for three aluminum alloys, as obtained by the Kolsky bar system in our laboratory, are shown in Fig. 2.3. Note that the 6061-T6 alloy exhibits a truly ductile behavior, where the specimen does not fail even at compressive strains as high as 0.8. On the other hand, the stronger alloys, 2024-T3 and 7075-T6, show a clear softening behavior at strains of about 0.6 and 0.4, respectively. These strong alloys fail at much higher strains under static loading, which is an indication for a thermal softening effect operating at the high strain rates, as discussed by Rosenberg and Ashuach (2014). They showed that these strength decreases can be related to the fact that the specimens reach a temperature of about half their melting temperatures. At these high temperatures they lose a significant amount of their strength, which leads to subsequent deformation followed by even higher temperatures. The flow stresses of these alloys at a strain of about

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Fig. 2.3 Dynamic stress–strain curves for three aluminum alloys loaded in the Kolski bar system

0.3, and strain rates of (103 –104 ) s−1 , are: Y = 0.42, 0.65 and 0.7 GPa, for the 6061T651, 2024-T351 and 7075-T6 alloys, respectively. As we shall see in Chap. 4, these are the appropriate values which one should use for their compressive strength in numerical simulations which follow terminal ballistics events, especially if the simple von-Mises yield criterion is used. A dynamic stress–strain curve for a specimen of the titanium alloy Ti–6Al–4V, compressed by the Kolsky bar in our lab, is shown in Fig. 2.4. One can see that at a compressive strain of about 20% the disc shaped specimen lost its load capacity. The failure resulted in two pieces of the disc which were separated by a fracture along its diagonal. The early failure of this alloy is attributed to its propensity to fail by adiabatic shearing. This unique process has been investigated by numerous researchers over the years, as reviewed by Bai and Dodd (1992). We summarize now, very briefly, some of the main results from these studies. The dynamic failure of metals and alloys under impulsive loading conditions is usually manifested by gross shearing at material interfaces, as a result of large velocity gradients at those locations. In many cases these large shear strains appear in very narrow bands which have been termed adiabatic shear bands (ASB). This Fig. 2.4 The dynamic stress–strain curve for the Ti–6Al–4V alloy

2.2 Material Properties

39

failure mechanism, which is believed to be due to thermo-mechanical instabilities, is one of the more common failure modes under dynamic loading at strain rates above 103 s−1 . The narrow shear bands often contain material which experienced a structural phase transformation, in which case they are referred to as transformation bands. The phenomenon of adiabatic shear banding received a lot of attention because of their importance for both military and industry applications. Bai and Dodd (1992) summarize much of the data and the analysis concerning this phenomenon. Important articles concerning the nature of ASBs are those of Staker (1981), Rogers (1983), Timothy and Hutchings (1985), and Giovanola (1988a, 1988b). Although many aspects of this issue have been highlighted in these studies, its physical picture is not complete because, in addition to the instabilities involved, the shear bands often include microcracks or microvoids which initiate and grow during band evolution. The work of Giovanola (1988a, 1988b), on 4340 steel under pure shear, highlights the interconnection between the thermo-mechanical shear instability, which acts at the first stage of the process, and the subsequent coalescence of microvoids which leads to fracture within the band at the second stage. These stages are clearly seen in the work of Xue et al. (2002) who performed radial collapse experiments with specimens of pure titanium and Ti–6Al–4V alloy, and followed the evolution of shear bands. In these experiments hollow cylinders are subjected to a pressure impulse by a surrounding explosive layer. The recovered cylinders are examined for the number, length and spacing of the shear bands. Figure 2.5 shows shear bands in specimens of the titanium alloy which were subjected to different strains. One can clearly follow the sequence of events in which, at low strains the voids nucleate within the band as a result of tensile stresses, and for larger compressions (strains) they grow and eventually coalesce. The interesting observation from these pictures is that the voids Fig. 2.5 Void nucleation and growth within shear bands in thick-wall collapsing Ti–6Al–4V cylinders

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2 Material Models for Numerical Simulations

form within the band boundaries, due to tensile stresses which develop during the strong shearing of these bands. Other materials, especially strong steels, exhibit a similar evolution of the process with microcracks, rather than microvoids, initiating in the shear bands. In all of these works the initiation of shear bands has been treated through the classical work of Zener and Hollomon (1944), who proposed a mechanism for the thermo-mechanical instability which initiates them. The analysis starts by writing the differential for the shear stress, τ = τ (T, ε, ε˙ ), which depends on the temperature, strain, and strain-rate in the solid, in the following way:  dτ =

∂τ ∂T





ε,˙ε

dT +

∂τ ∂ε





ε˙ ,T

dε +

∂τ ∂ ε˙

 ε,T

d ε˙

from which one obtains:       dT d ε˙ ∂τ dτ ∂τ ∂τ + = + dε ∂ T ε,˙ε dε ∂ε ε˙ ,T ∂ ε˙ ε,T dε

(2.19a)

(2.19b)

The instability condition is defined by dτ /dε = 0 which, for a constant strain rate experiment, occurs when: 

∂τ ∂ε

 T

   dε −1 ∂τ =− ∂ T ε dT

(2.19c)

Since these bands form under adiabatic conditions the generated heat cannot diffuse out of the bands themselves. Thus, one can write the following equation for the temperature rise in the bands, due to their plastic deformation: ρCv dT = τ dε

(2.20a)

where ρ is the density of the specimen and C v is its specific heat. From this relation one obtains: dε/dT = ρCv /τ

(2.20b)

In order to predict the critical shear strain for adiabatic shearing from these equations, one needs to have the constitutive equation for the material. Assuming a power law dependence of the shear stress on shear strain, such as τ = N · εn , the threshold value of the strain (εi ), marking the onset of thermo-mechanical instability, is given by:  εi = nρCv

∂τ ∂T

 (2.21) ε,˙ε

2.2 Material Properties

41

which has been shown to account for the experimental results in many cases, but not for all of them. These issues were discussed by Bai and Dodd (1992) and by Woodward (1990) who summarizes other approaches to the formation of adiabatic shear bands. Some confusion about the true nature of adiabatic shear bands arises from the fact that in many cases they appear in specimens having sharp corners, where a geometrically forced stress-concentration is responsible for the shear failure. This issue will be further discussed in Chap. 4, where we highlight the importance of adiabatic shearing for plate perforations by flat-nosed projectiles. A somewhat different loading history of specimens in the Kolsky bar system is obtained by the so called upsetting (or multi-step) test. In these tests the disc shaped specimen is loaded to a relatively small strain, so that it does not experience a significant temperature rise. After some time it is then reloaded, again to a small strain, and the procedure is repeated several times. In fact, this is a dynamic loading with practically negligible heating, and the specimen is supposed to withstand much larger strains without failure. Ashuach and Rosenberg (2014) performed such multistep loadings on several metals and alloys and observed the difference between their stress-strain curves under single and multi-step loadings. Figure 2.6 shows the stress-strain curves for specimens of Al 2024-T351, subjected to single and multistep loadings. It is clearly seen that the specimen loaded by a single step, at a strain rate of 5 · 103 s−1 , experienced thermal softening at a strain of about 0.6. In contrast, the multi-step loading did not result in a loss of strength even at a strain of 0.8. In fact, the stress-strain curve under multi-step loading is very similar to the quasi-static curve, as expected, since the temperature rise should be very small under multi-step loading. When the temperature of the specimen reaches a certain threshold the specimen’s strength is expected to decrease, and thermal softening sets in. These threshold temperatures can be determined by static compression tests under various temperatures. Rosenberg and Ashuach (2014) showed that, indeed, the onset of dynamic failure for various metals and alloys coincides with the large decrease in their strength at the Fig. 2.6 The response of Al2024-T351 specimens to single and multi-step loadings

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2 Material Models for Numerical Simulations

Fig. 2.7 The response of pure titanium (left) and the alloy (right) to single and multi-step loadings

threshold temperatures. As a rough estimate, the softening threshold temperatures are equal to about half the respective melting temperature of the specimen. Most of the materials tested by Rosenberg and Ashuach (2014) showed no failure even at high strains, when loaded by the multi-step technique. In contrast, multi-step loadings of the titanium (Ti–6Al–4V) and magnesium (AM50) alloys, resulted in nearly the same failure strains as in their corresponding single-step loading. Figure 2.7 shows the results from Rosenberg and Ashuach (2014), of these tests for pure titanium and for its alloy, using single and multi-step loadings. The difference between the responses of these materials is very clear. Pure titanium shows the typical response for most materials, with a clear indication for thermal softening in the single step loading, while its multi-step loading shows no indication for failure even at much higher strains. In contrast, the multi-step loaded Ti–6Al–4V specimen failed during the fourth loading step, at a compressive strain which is very close its failure strain under single loading. The same result was obtained for the magnesium alloy. As the multi-step loading is, practically, an isothermal process, these results for both the titanium and magnesium alloys indicate that the temperature has very little to do with the failure of these alloys. Hence, one should look for other causes for the propensity of these alloys to fail at low strains by adiabatic shearing. The adiabatic shear failure is a very important issue in terminal ballistics because it affects the capability of a target plate to withstand the impact of flat-faced projectiles. As we shall see in Chap. 4, this is a real concern for high-strength targets whose thickness is of the order of the projectile’s diameter. Figure 2.8, from Olson et al. (1980), demonstrates this issue very well. Here we see the end results of impacts by high strength steel projectiles at two plates of 4340 steel, which have very different hardness (strength). The low-strength steel plate (Rockwell hardness RC 15) deformed heavily and was able to absorb the projectile’s kinetic energy through this large deformation. In contrast, the high-strength plate (Rockwell hardness RC 52), was hardly penetrated but it developed a shear band extending almost to its back surface. Clearly, this plate would be penetrated at a somewhat higher impact velocity, with a shear plug ejected from it. Thus, the high-strength plate is less efficient in this case,

2.2 Material Properties

43

Fig. 2.8 Different failure mechanisms in steel plates of different strength

due to its propensity to fail by adiabatic shear. The white color of shear bands in steels is usually observed by the etching process of sectioned plates.

2.2.4 Failure of Brittle Materials The dynamic behavior of brittle materials such as glass and ceramics, is quite different and more complicated than that of ductile metals and alloys. First, there is a large difference between the compressive and tensile strengths of a brittle solid, which can reach a factor of 10 and more. By definition, brittle solids fail within their elastic range of response, with practically no plastic straining. Their failure behavior is pressure dependent, with relatively very small (if any) strain-rate sensitivity. Also, the damaged zone in a brittle material can spread from the impact point, through cracks which run in all directions, affecting the properties of a large volume in the brittle solid. The various constitutive equations which describe the failure thresholds of brittle materials, are based on the classical work of Griffith (1920, 1924), who analyzed the stresses around an open elliptic flaw in a plate under biaxial stress loading. Griffith followed the analysis of Inglis (1913), who showed that even when the far-end stresses are compressive, the stresses at the tips of the flaw are tensile. With increasing stress at the plate boundaries, these tensile stresses around the crack reach a critical value (σ 0 ) which is a material property, leading to uncontrolled crack growth. The value of σ 0 according to this analysis is equal to the tensile strength of the material under uniaxial loading. Denoting the two far-end stresses which act on the plate as σ 1 and σ 2 , Griffith derived the following relations expressing the critical conditions for the brittle failure:

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2 Material Models for Numerical Simulations

σ2 = σ0

if: 3σ2 + σ1 > 0

(σ1 − σ2 )2 = −8σ0 (σ1 + σ2 ) if: 3σ1 + σ2 < 0

(2.22a) (2.22b)

where tensile stresses are positive and compressive stresses are negative in this formulation. These equations describe the failure envelope of a brittle solid in the (σ 1 , σ 2 ) plane, as shown in Fig. 2.9. All the states which are described by points inside the two curves correspond to intact material, while those outside the curves correspond to failure situations. Hydrostatic compressions, which correspond to the third quadrant in this figure, enhance the integrity of the brittle solid, as expected. According to Griffith’s model the critical tensile strength (σ 0 ) of the solid is smaller by a factor of 8 than its compressive strength under uniaxial stress loading. Later workers, such as McClintock and Walsh (1962), found a factor of 10 between these two strength thresholds. Other refinements of these models, as well as data for concrete and rocks, can be found in Paterson (1978). The Griffith failure criterion which, in effect, defines the strength of a brittle solid under compression and tension, is very different from the various yield criteria for ductile materials which were discussed above. Thus, the relation between the Hugoniot elastic limit (HEL) of a brittle solid and its dynamic strength (Y d ) should be different than the corresponding relation for ductile materials, (1.7), which follows the von-Mises yield criterion. The relation for brittle solids should be based on the Griffith criterion, (2.22a, 2.22b), through which Rosenberg (1993) derived the following relation: Yd =

Fig. 2.9 The failure envelope for brittle materials, according to Griffith’s theory

(1 − 2v)2 · HEL 1−v

(2.23)

2.2 Material Properties

45

Fig. 2.10 The JH-1 model for the equivalent strength of brittle materials

where v is Poisson’s ratio of the brittle material. Thus, the dynamic strength of a brittle specimen is smaller by a factor of (1 − 2v) than the strength deduced from (1.7), which is based on the von-Mises yield criterion. The first numerical simulations for terminal ballistics applications involving brittle solids (ceramic tiles) were performed in the late 1960s by M. L. Wilkins and his colleagues, as described by Wilkins (1978). This work will be further discussed in the third part of this book, where we review the issue of ceramic armor. The first implementation of the Griffith model itself, for ceramic tiles in terminal ballistics simulations, was made by Mescall and Tracy (1986). Johnson and Holmquist (1990, 1993) further developed the Griffith model and included a damage function at every time step in the simulation. The various parameters in the first version of their model (JH-1) are shown in Fig. 2.10, for the equivalent strength of a brittle material. The model assumes that the tensile strength (spall) is much lower than the compressive strength, and that the fractured material has lost its tensile strength completely. The compressive strength of both intact and damaged materials is linearly dependent on pressure at the low pressure range, reaching asymptotic values which depend on the strain rate. The damage function is calculated continuously at each time step in the simulation, and it lowers the strength of the failed material. Note the various material properties, the slopes of the various lines and the threshold pressures (P1 and P2 ), which have to be calibrated for each ceramic. This issue will be further discussed in Chap. 6, when we deal with ceramics for armor applications.

2.2.5 The Spall Failure It was noted earlier that spalling is a unique failure phenomenon, which takes place under dynamic conditions. When shock waves are reflecting from free surfaces as release waves, they induce tensile stresses in the solid. If the amplitude of these tensile waves is high enough the material fails and a spall is created near its free surface. The relevance of this mode of failure for terminal ballistics can be seen in sectioned targets, which have been impacted by relatively blunt projectiles at high impact

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Fig. 2.11 Spall failure in an aluminum plate impacted by a glass sphere at 6.0 km/s

velocities. For example, the spall in an annealed 1100 aluminum plate impacted by a glass sphere, from the work of Horz et al. (1994), is clearly seen Fig. 2.11. The diameter of the sphere (3.18 mm) was about a quarter of the plate thickness, and its impact velocity was 6.0 km/s. The study of spall failure in solids has been the focus of a large amount of research for over 50 years. The early models assigned a threshold tensile strength to the specimen, which was calibrated by one-dimensional shock wave experiments, as discussed in Chap. 1. Subsequent developments showed that the spall process is more complicated, and that its time dependence should be taken into account, as was first noted by Tuler and Butcher (1968). Many models were developed in order to account for this type of failure and the article of Curran (1982) describes those which are based on the nucleation of defects and their growth (the so called NAG models). The time dependence of the spall process leads to scaling effects which were the focus of an intense research by Russian researchers, as summarized by Ogorodnikov et al. (1999). A particularly thorough work on this issue is that of Al’tshuler (1966), where the reciprocal relation between the critical tensile stress and its duration is very clearly demonstrated. The dependence of the spall failure on the tensile impulse, rather than on a threshold stress, has been highlighted by Gray et al. (2007). They showed that a square-shaped tensile pulse and a triangular-shaped pulse can cause the same amount of incipient spall, if they induce a similar tensile impulse at the spall plane in the specimen. Thus, according to Gray et al. (2007), the tensile impulse is the important factor for the spall phenomenon, rather than the amplitude of the tensile wave. Grady (1988) proposed energy balance considerations to establish the loading conditions at which spall occurs. This approach accounts for the transition between brittle and ductile spall failures, which has been observed for several materials when the loading conditions are changed. As an example, consider the sectioned 6061-T6 aluminum targets shown in Fig. 2.12, from the plate-impact experiments of Christman et al. (1971). A marked difference in the void shape is clearly seen in these plates, which were subject to tensile stresses of different durations and rates. The round shaped voids in the specimen at the top of the figure, suggest a ductile spall mechanism for the shorter pulse duration (higher rate). On the other hand, a brittle spall is evident in the cross section of the specimen at the bottom of the figure, which was subjected to a tensile pulse of a longer duration (and lower rate). The different loading rates

2.2 Material Properties

47

Fig. 2.12 Different void shapes in aluminum plates subjected to tensile pulses of different durations

in the two experiments resulted in from the different thicknesses of the impactor and target plates in these experiments. Note that in both cases the specimens are only partially spalled, a phenomenon which is termed “incipient spall”. A full spall process is characterized by the complete coalescence of the voids, leading to a clear opening inside the specimen, as shown in Fig. 2.11. The analysis of Grady (1988) is based on macroscopic material properties, such as the fracture toughness and the flow stress of the material, resulting in a predictive tool for the occurrence of spall under given loading conditions. This analysis leads to the following relations between the spall strengths, for both ductile and brittle materials, and their relevant properties and loading conditions:

1/3 for brittle spall Pspall = 3ρC0 K c2 ε˙

(2.24a)

1/2 Pspall = 2ρC0 Y ε f for ductile spall

(2.24b)

where ρ and C 0 are the specimen’s density and sound speed, respectively, Y and K c are its flow stress and fracture toughness, respectively. The strain rate (˙ε) in the experiment is important for the brittle fracture, while the strain to failure (εf ) plays an important role for the ductile failure mode. Rosenberg (1993) analyzed the spall failure in brittle materials through the Griffith failure criterion and derived the following relation between the spall strength and the Hugoniot elastic limit (HEL) of the brittle solid:

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2 Material Models for Numerical Simulations

σspall =

(1 − 2v)2 · HEL 8(1 − v)

(2.25)

where v is the Poisson ratio of the solid. This expression results in factors of 15–25 between the HEL and the spall strength of brittle materials having Poisson’s ratios in the range of v = 0.15–0.25. Such factors are often measured for strong ceramics, having spall strengths of 0.3–0.6 GPa and HEL values in the range of 6.0–15.0 GPa. A rather simple model for the spall strength of ductile materials was proposed by Rosenberg (1987), which is based on the cavity expansion analysis of Bishop et al. (1945). They derived an analytic expression for the minimal pressure which has to be applied on the wall of a small cavity in an elasto-plastic solid, in order to expand its volume indefinitely. The cavity expansion analysis will be described in detail in Chap. 3 when we deal with the deep penetration of rigid projectiles. In the spherical cavity expansion analysis the pressure is gradually introduced in the cavity until it reaches a threshold value which drives the cavity indefinitely. This threshold pressure depends on the elastic properties of the solid (E and ν) and on its strength (Y ). Assuming that spall is initiated by a process which is similar to the cavity expansion process, Rosenberg (1987) derived the following expression for the spall strength of ductile solids: σspall

   E 2Y 2 + ln = 3 3(1 − v)Y

(2.26)

where E, ν and Y are the Young’s modulus, Poisson’s ratio and the strength of the solid, respectively. The predicted spall strengths of various materials as calculated by this expression were found to be very close to their experimental values. For example, this expression results in: σ spall = 1.7, 3.6 and 4.6 GPa, for an aluminum alloy with strength of Y = 0.4 GPa, steel with Y = 0.8 GPa, and the titanium alloy Ti/6Al/4V with strength of Y = 1.2 GPa, respectively. These predicted values are very close to the experimentally derived spall strengths of these materials.

Part II

Penetration Mechanics

Chapter 3

Rigid Penetrators

Abstract This chapter discusses the penetration process of rigid penetrators into semi-infinite targets. This is the basic science behind the field of penetration mechanics. We start with the basic question regarding the nature of deceleration of a rigid projectile, and use both empirical data and numerical simulations to show that this deceleration is constant throughout the penetration process, independent on the projectile’s impact velocity at the ordnance range. At higher velocities, beyond a certain threshold, the inertia of the target sets in and the deceleration becomes velocity dependent. This issue is treated numerically, as is the effect of the entrance phase on the penetration process. Numerically-based analytical models are developed for these phenomena and are shown to account for the corresponding empirical results. We discuss the penetration of concrete targets by rigid projectiles. This process is also governed by a constant-deceleration of the projectile. The non-scaling phenomenon of penetrations into concrete targets is accounted for by a simple model. The transition from rigid to deforming-rod penetration is presented and accounted for by analytical considerations. Finally, the effect of yaw on the penetration of rigid projectiles is treated numerically.

3.1 The Mechanics of Deep Penetration The penetration of a very thick (semi-infinite) target by a rigid projectile has been the subject of intense research for over two centuries. A large number of empirical relations and engineering models have been proposed over these years, as reviewed Backman and Goldsmith (1978). The engineering models account for the penetration depth of a given projectile in terms of its impact velocity, and they differ from each other by the basic assumptions concerning the retarding forces (stresses) on the projectile. With the recent advancements in the quality and reliability of numerical simulations, these analytical models can be further validated, as will be demonstrated in this chapter. We start the discussion with a short review of the more popular models which have been proposed over the years. This is followed by a careful examination of a certain set of experimental data, which will lead to the proper choice of an analytical model for the deep penetration by rigid penetrators. Numerical simulations are then © Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_3

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used in order to enhance the validity of this model and to explore the role of its physical parameters. The penetration process of a rigid penetrator is determined by the retarding force which the target exerts on it during penetration. Since, by definition, the mass (M) of the rigid penetrator does not change during penetration, the process is governed by its deceleration (a). Thus, the following discussion concentrates on the deceleration of a rigid penetrator. Replacing the time variable (dt) with the penetration increment (dx), the deceleration is given by: a=

dV dV =V dt dx

(3.1a)

from which we get the penetration increment: dx =

V dV a(V )

(3.1b)

The penetration depth (P) of the rigid penetrator is obtained by integrating this equation between the boundaries: V = V 0 at x = 0 and V = 0 at x = P. Thus, one can write for the final penetration depth: 0 P= V0

V dV a(V )

(3.1c)

The only unknown quantity in this equation is the actual dependence of the deceleration on penetration velocity, a = a(V ). Various functional forms for a(V ) have been suggested over the years for different projectile/target combinations, and all of them are special cases of the general expression: a(V ) = C + AV + BV 2

(3.2)

where the constants A, B and C have been determined empirically for each set of experiments. The most popular forms for a(V ) are: Euler−Robins a = C = const Poncelet Resal

(3.3a)

a = C + BV 2

(3.3b)

a = AV + BV 2

(3.3c)

The first expression, with the constant deceleration, was suggested (independently) by the great mathematician Leonard Euler (1745) and the British gun engineer

3.1 The Mechanics of Deep Penetration

53

Benjamin Robins (1742). Obviously, Euler does not need an introduction. Robins is known as the inventor of the ballistic pendulum and for his use of Newtonian mechanics for exterior ballistics. The second and third equations are attributed to Poncelet (1835) and Resal (1895), respectively, who were well-known engineers in France. In fact, J. V. Poncelet, together with G. Coriolis, introduced the definition of kinetic energy as we know it (mV2 /2), during the early years of the 19th century. Inserting the above expressions for a(V ) in (3.1c) and integrating between the boundary conditions, results in the following expressions for the final penetration depth (P) in terms of impact velocity (V 0 ): P=

V02 2C

  1 BV02 P= ln 1 + 2B C   BV0 1 P = ln 1 + B C

(3.4a) (3.4b) (3.4c)

Clearly, the various expressions for a(V ) result in different relations for the final penetration depth of a rigid penetrator. There are several ways by which the actual dependence of a(V ) for a given projectile/target pair can be determined. The most obvious one is to measure the deceleration history during penetration. Such measurements have been performed for projectiles penetrating soil and concrete, as in Forrestal et al. (1984). However, these measurements are not accurate enough and they are difficult to perform with other target materials. Another way to determine the functional form of a(V ) is through theoretical models which are based on some physical principles, such as the conservation laws. Due to the complexity of the process, this route did not result in a clear answer either. The third possibility, which is the one we adopt here, is based on analyzing experimental data of P = P(V 0 ) for a given penetrator/target pair, in order to infer the actual dependence of the deceleration on velocity. The conclusions from this analysis will be further enhanced through numerical simulations. In order to determine the functional dependence of a(V ) one needs a reliable set of experiments where the values of P = P(V 0 ) are given for a large range of impact velocities. The extensive studies of M. Forrestal and his colleagues offer a wealth of such data, as in Forrestal et al. (1988), Forrestal et al. (1991), and Piekutowski et al. (1999), for example. These works include accurate measurements of impact velocities and penetration depths of hardened steel rods impacting thick aluminum targets. The aspect ratios of these rods were in the range of L/D = 10–15, where L and D are the rod’s length and diameter, respectively. The rods had different nose shapes (conical, ogive and spherical) and the aluminum targets were of the 6061T651 alloy. The largest range of impact velocities for which these rods remained rigid, was obtained by Piekutowski et al. (1999) for L/D = 10 ogive-nosed rods. Impact velocities in these experiments reached 1.82 km/s with no sign of rod deformation.

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3 Rigid Penetrators

The data from this set of experiments will be used now in order to construct the analytical model for the deep penetration of rigid long rods. We start the analysis with the simplest assumption possible, namely, that the average deceleration of these ogive-nosed rods does not depend on their velocity. This is the assumption behind the Euler–Robins model which leads to the simplest penetration relation, (3.4a). The validity of this assumption can be easily checked by using the experimental data of Piekutowski et al. (1999), in order to calculate the average rod deceleration (aav ) in each experiment, through the expression: aav =

V02 2P

(3.5)

Table 3.1 presents the data from Piekutowski et al. (1999) in terms of the impact velocity and penetration depth for each experiment, together with the corresponding values for aav as determined by (3.5). It is clear that, except for the low velocity experiments, this set of results strongly supports the Euler–Robins model since the average decelerations are practically independent on impact velocity. This independence over a wide range of impact velocities means that the deceleration of the rod was constant during each experiment, with an average value of: a = 3.55 · 10−3 mm/µs2 . Figure 3.1 demonstrates the agreement between the experimental results of Piekutowski et al. (1999) and the model’s predictions through (3.4a) with this value for the constant deceleration. The rods in this set of experiments were made of two different steels, the VAR 4340 steel and the stronger AerMet 100, which remained rigid at higher impact velocities. As was demonstrated by Rosenberg and Dekel (2009a), the same conclusion about the constant deceleration of long rods at deep penetrations, is obtained by examining the data for rods with other nose shapes (conical and spherical). Thus, we conclude that the deep penetration of rigid rods in semi-infinite metallic targets is characterized by Table 3.1 The data from Piekutowski et al. (1999), and the average decelerations of the ogive-nosed rods

V 0 (m/s)

P (mm)

aav (10−3 mm/µs2 )

570

55

2.95

679

72

3.20

821

102

3.30

966

140

3.33

1076

160

3.62

1147

190

3.46

1255

229

3.44

1348

254

3.58

1538

332

3.56

1654

389

3.52

1786

452

3.53

1817

462

3.57

3.1 The Mechanics of Deep Penetration

55

Fig. 3.1 Comparing the constant deceleration model with the data for ogive-nosed steel rods

a constant deceleration process. This conclusion holds for impact velocities beyond the low velocity range where the decelerations are somewhat lower due to the target’s free impact face, as discussed below. The impact face of the target exerts a lower retarding force on the penetrator since the target material around the rod can flow backwards. This is the so called “entrance phase effect”, which accounts for the relatively higher penetrations (lower decelerations) of rigid penetrators at the low impact velocity range. In contrast, at deep penetration the target material is “pushed” forward or sideways by the advancing penetrator. The low velocity experiments of Piekutowski et al. (1999) are within the influence range of the entrance phase, and they are underestimated by the constant deceleration model, as is clearly seen in Fig. 3.1. In the next sections we shall quantify the influence range of the impact face through a numerically-based model which accounts for the entrance phase effect. The next step is to use numerical simulations in order to validate the conclusion concerning the constant deceleration of the rigid rod, and to determine the physical parameters which govern this deceleration. These numerical simulations, with the AUTODYN-2D code, were described in detail in Rosenberg and Dekel (2009a) and we summarize their findings here. The rigid rods were simulated with the Lagrange processor of the code, using a value of 50 GPa for their yield strength, in order to prevent their plastic deformation at all impact velocities. The Eulerian processor which handles large deformations in a better way was used for the targets in these simulations. The constitutive relations for the targets were based on the simple elastoplastic model, using the von-Mises criterion with no strain hardening or strain rate effects in order to simplify the analysis. Consider first the penetration and velocity histories of L/D = 20 steel rods with different nose shapes, impacting a semi-infinite aluminum target with strength of 0.4 GPa, at V 0 = 1.0 km/s, as shown in Fig. 3.2. Clearly, the ogive-nosed rod is the most efficient one since its final penetration depth is the highest, while the flat-nosed rod is the least efficient, as expected. The slight decrease in penetration depths, seen at the later times of these penetration histories, is due to the elastic rebound of the target

56

3 Rigid Penetrators

Fig. 3.2 Simulation results of penetration and velocity histories for different nose shaped rods

which pushes the projectile backwards at the end of the process. Final penetration depths should be determined by their maximal values in the simulations. Figure 3.2b shows the velocity histories of these rods and it is clear that their corresponding decelerations, as determined by the slopes of these histories, are practically constant. Moreover, the corresponding slopes of these velocity histories suggest that the least deceleration is obtained for the ogive-nosed rod and the highest for the flat-ended rod. This relatively straightforward result needs further investigation because of its importance. We now follow the actual mechanism which governs the velocity changes of a rigid rod during its deep penetration. Any change in the velocity of the rod’s nose takes a certain time to reach its tail, since it must be transmitted by an elastic wave which travels along the rod. Upon reaching the rod’s tail this elastic wave travels back towards the nose of the rod as a release wave. Thus, the actual process of rigid rod deceleration is achieved by a series of elastic waves reverberating along the rod length at the bar velocity, C b = (E/ρ)0.5 , where E and ρ are the Young’s modulus and density of the rod material, respectively. These reverberations are clearly seen

3.1 The Mechanics of Deep Penetration

57

in Fig. 3.3, from Rosenberg and Dekel (2009a), for the nose and tail velocities of a spherical-nosed steel rod impacting an aluminum target. The rod in this simulation was 120 mm long and 6 mm in diameter, the aluminum target had strength of Y t = 0.4 GPa, and the impact velocity was 1.5 km/s. Note the constant amplitude of the steps in these velocity histories, and their corresponding time shift in Fig. 3.3b, which is equal to the transit time of an elastic wave through the length of the rod. Similar step-like velocity histories, for L/D = 20 rods with different nose shapes, impacting an 0.4 GPa aluminum target at 1.0 km/s, are shown in Fig. 3.4. The reverberating elastic waves have significantly different amplitudes, which depend on the nose shape of the rod. The low amplitude waves in the ogive-nosed rod result in the longest penetration time and, consequently, their lowest deceleration. In contrast, the high amplitude waves which reverberate in the flat ended rod result in the shortest penetration time and highest deceleration. Thus, the ogive nosed rod is expected to Fig. 3.3 a Simulation results for the nose velocity of a spherical-nosed steel rod impacting an aluminum target at 1.5 km/s and b both nose and tail velocities for the same impact

58

3 Rigid Penetrators

Fig. 3.4 Nose velocity histories for rods with different nose shapes

penetrate deeper than the others and the flat ended rod should result in the lowest penetration, in accord with the results shown in Fig. 3.2. The stresses corresponding to these waves can be calculated by the well-known relation between stress (σ ) and particle velocity (v) in a long elastic bar: σ = ρCb v

(3.6a)

where ρ and C b are the density and elastic wave velocity of the bar, respectively. In fact, the velocity steps of the rod’s nose, as seen in Fig. 3.4, are equal to twice the particle velocity (v) behind the reverberating wave fronts, since the particle velocity is doubled with each reflection from the rod’s back end. Thus, the velocity steps in Fig. 3.4 have amplitudes of v = 2v and the corresponding stresses (σ ) should be calculated through:  σ = ρCb v 2

(3.6b)

Using the corresponding values for the rod’s density and its elastic wave velocity, one can have an estimate for the amplitudes of these reverberating stress waves: σ = 1.9, 2.2 and 2.5 GPa, for the ogive, spherical and flat-nosed rods, respectively. As we shall see later on, these stresses are equal to the resisting stresses which the target exerts on these rods. Note also that the first step in the velocity histories of the rod’s nose has half the amplitude of the other steps. This velocity drop results in from the impact itself, while all the later velocity drops are subjected the doubling process discussed here. The next step in this numerical study is to follow the deceleration of the rigid rod impacting a given target at different velocities. Such deceleration histories, for an ogive nosed rod impacting the 0.4 GPa aluminum target, are shown in Fig. 3.5. The constant deceleration of this rod for an impact velocity of 1.0 km/s is clearly

3.1 The Mechanics of Deep Penetration

59

Fig. 3.5 a The deceleration history of an ogive-nosed rod penetrating 0.4 GPa aluminum target, at an impact velocity of 1.0 km/s, and b deceleration histories for several impact velocities

seen in Fig. 3.5a, and the independence of the deceleration on impact velocity is demonstrated in Fig. 3.5b. The deceleration of the rod at the early stages of penetration is lower than its value at deep penetration, as shown by the shaded area in Fig. 3.5a. This is due to the effect of the entrance phase, as discussed above. Rosenberg and Dekel (2009a) found through these simulations that for CRH3 ogive-nosed rods the entrance phase dominates the process for a penetration depth of six rod diameters. Figure 3.5b clearly shows that higher impact velocities result in a shorter duration of the entrance phase, since it takes less time for the rod to reach this penetration depth. For spherical nosed rods the deceleration achieves a constant value at a depth of about three rod diameters. These values of P = 6D and P = 3D are the dominance ranges of the entrance phase for the ogive and spherical nosed rods, respectively. The influence ranges of these entrance phases extend to much larger penetration depths, as will be demonstrated later by the numerically-based model for the entrance phase effect. One should note

60

3 Rigid Penetrators

that short rigid projectiles such as armor piercing (AP) projectiles, do not reach the constant deceleration phase since their penetration depths are of the order of a few projectile diameters. Thus, the whole penetration process of short projectiles is within the influence range of the entrance phase, and it should be analyzed differently, as will be discussed in Sect. 3.5. The advantage of using numerical simulations in order to follow the physics of deep penetrations is demonstrated by the following sets of simulations, from Rosenberg and Dekel (2009a). Simulation results for the deceleration of the ogive nosed rods, as they penetrate aluminum targets with different strength (0.4 and 0.8 GPa), are shown in Fig. 3.6. As expected, the stronger target exerts a higher force (deceleration) on the rod. From these simulations it is clear that these decelerations do not follow a linear dependence on target strength, as discussed in the next section. The role of rod density is highlighted in Fig. 3.7 which shows the nose velocities of L/D = 20 spherical-nosed aluminum and steel rods, impacting the 0.4 GPa aluminum target at 1.4 km/s. The velocity steps in the aluminum rod are about three times higher than those of the steel rod. This is the expected result considering (3.6b) for the amplitude of the reverberating elastic stress waves. We have already mentioned that the amplitudes of these stress waves are equal to the resisting stress which the target exerts on the rod. For the same resisting stresses, the reverberating waves in these rods have to be of equal stress amplitude. This means that their velocity steps should be inversely proportional to the corresponding products ρC b of the two rod materials. Since aluminum and steel have similar values for their wave speeds, the velocity steps should be inversely related to their density ratio, which is about three. This is exactly the result obtained in the simulations for the two rods, as seen in Fig. 3.7. Stated differently, this result indicates that the amplitudes of the reverberating stress waves in the two rods are equal. The larger velocity steps in the aluminum rod shorten its total penetration time and increase its deceleration, as compared with the steel rod. Fig. 3.6 The effect of target strength on the deceleration of an ogive-nosed rod

3.1 The Mechanics of Deep Penetration

61

Fig. 3.7 Nose velocity histories of the spherical-nosed aluminum and steel rods penetrating aluminum target

The simulation results for the deceleration of ogive-nosed steel rods of different lengths are shown in Fig. 3.8. These rods, with L = 58 mm and L = 120 mm, and the same diameter (D = 6 mm), impacted the 0.4 GPa aluminum target at 1.0 km/s. The deceleration of the shorter rod is higher simply because it takes less time for the elastic waves to reverberate along its length. In summary, the experimental results for rigid rod penetrations suggest that their deceleration is constant, beyond the dominance range of the entrance phase. The numerical simulations show that the constant deceleration of a rigid rod depends on its nose shape and on the strength of the target, and it is inversely proportional to the rod’s density and its length. Fig. 3.8 Deceleration histories for rods of different length

62

3 Rigid Penetrators

3.2 The Penetration Model for Rigid Long Rods We are now ready to present the analytical model for the penetration of rigid rods into semi-infinite targets. The model accounts for target materials which are characterized by well-defined flow stress and elastic moduli. Many metals and alloys show some hardening effects at increased strains and strain rates, as well as thermal softening in regions which experience large deformations. These complicating effects are ignored in the following discussion in order to simplify the analysis. However, the model presented here is still applicable for materials which moderately harden (or soften) if an effective (average) flow stress is used for the strength of these materials. These flow stresses should be taken from the corresponding stress–strain curves at high strain values. As discussed in Chap. 1, the Kolsky bar test is most appropriate for the determination of these flow stresses at the relevant strain rates, which the targets experience during penetration. The specific way to choose these effective flow stresses will be discussed later on.

3.2.1 Impact at the Ordnance Velocity Range With the observations discussed in Sect. 3.1, we can construct the analytical model for the deep penetration of rigid long rods in terms of their properties (density, length and nose shape), and those of the target (strength and elastic moduli). The model ignores the entrance phase effect and assumes that the deceleration of the rigid rod is constant throughout the whole penetration process. This means that the retarding force (F), which the target exerts on the rod, is also constant during its penetration. The retarding force can be expressed by the resisting stress (Rt ) which the target exerts on the rod, multiplied by its cross section area (π r 2 ). Thus, for a rod with mass M, an effective length of L eff and radius r, one can write: F = Ma = ρ p L eff · π r 2 a = const = πr 2 Rt

(3.7a)

where the effective length of the rod is defined by: L eff = M/(π r 2 ρ p ). Thus, the constant deceleration of the rod is related to the resisting stress (Rt ) through: a=

Rt ρ p L eff

(3.7b)

The inverse dependence of the rod’s deceleration on its density and length has been already demonstrated by the simulations presented above. We have also seen that the rod’s deceleration depends on its nose shape and on the strength of the target. Thus, we may assume that the resisting stress (Rt ) depends on these parameters as well. Inserting the above relation into the expression for the penetration depth, (3.4a), results in the following penetration equation for the rigid long rod:

3.2 The Penetration Model for Rigid Long Rods

P=

63

ρ p L eff V02 2Rt

(3.8a)

and the normalized penetration depth is given by: ρ p V02 P = L eff 2Rt

(3.8b)

Several points should be noted here. According to this equation the penetration depth of a rigid rod does not depend on the density of the target. It is only the target strength which determines the penetration depth, through its resisting stress (Rt ). In fact, its Young’s modulus also plays an important role in determining the value of Rt , as will be shown below. We have shown that the nose shape of the rod plays an important role in determining its deceleration, and one may expect that Rt depends on the rod’s nose shape as well. The influence of the entrance phase was ignored throughout the analysis. Thus, the predicted penetrations through (3.8a, 3.8b) are expected to underestimate the experimental data for low impact velocities. The simple penetration equation derived above is based on the target’s resistance to penetration (Rt ) which has to be determined for each rod/target combination. A well established theoretical model for Rt is still lacking, although several attempts were made over the years, as will be discussed below. The procedure which will be described now for the determination of Rt can be referred to as a “numerically based” model. With this approach the “data” is obtained through numerical simulations by varying each parameter at a time. For the present case consider the simulation results of Rosenberg and Dekel (2009a) for rods with different nose shapes. These simulations resulted in the rod’s deceleration for different target materials and strengths. These decelerations were inserted into (3.7b) and the corresponding values of Rt were determined for the various rod/target combinations. Simulation results for Rt values are given in Table 3.2, for rigid rods with different nose shapes impacting aluminum and steel targets of different strengths. The simulations of Rosenberg and Dekel (2010b) were aimed at finding a general relation between Rt and the relevant parameters of the target (Y t , E), for rods with different nose shapes. The strength of the aluminum and steel targets in these simulations was varied within a large range, Y t = 0.05–0.8 GPa. The resulting Rt values were presented through the dependence of Rt /Y t on ln(E/Y t ), for the different rod/target combinations, as shown in Fig. 3.9. Table 3.2 Values of Rt (in GPa) for different targets and rod nose shapes Nose shape

Aluminum targets

Steel targets

Y t = 0.4 GPa

Y t = 0.8 GPa

Y t = 0.4 GPa

Y t = 0.8 GPa

Ogive

1.87

3.12

2.28

3.96

Spherical

2.25

3.87

2.63

4.71

Conical

1.95

3.30

2.36

4.14

64

3 Rigid Penetrators

Fig. 3.9 Simulation results for the relation between Rt /Y t and E/Y t . Triangles and circles denote aluminum and steel targets, respectively

As is clearly seen, these Rt values follow a linear relation between the normalized values Rt /Y t and the corresponding values for ln(E/Y t ), according to:   E Rt −φ = 1.1 ln Yt Yt

(3.9)

where φ depends on the nose shape of the rod, having the values of: φ = 1.15, 0.93 and 0.2, for the CRH3 ogive, the conical and the spherical-nosed rods, respectively. The values of Young’s modulus for steel and aluminum in these simulations were E = 200 and E = 73 GPa, respectively. The ogive nosed rods in these simulations were of CRH3 and the conical nosed rod had a nose length of three rod radii. Different cone lengths and ogive radii are expected to result in different Rt values, as will be shown in Sect. 3.4. In order to assess the validity of this numerically-based model we compare its predictions with experimental data for rigid L/D = 10 steel rods, having different nose shapes, impacting 6061-T651 aluminum targets. These experiments are described in Forrestal et al. (1988) and in Piekutowski et al. (1999). The dynamic stress–strain curve of the 6061-T651 aluminum alloy, as obtained from the Kolsky bar system in our lab, is given in Fig. 1.15. It is clear that a flow stress of Y t = 0.42 GPa represents an average value for the dynamic strength of this material at large strains. Inserting this value of Y t in (3.9), together with E = 69 GPa for this alloy, results in the values of Rt = 1.87, 2.27 and 1.97 GPa for its resisting stresses against rods with ogive, spherical, and conical nose shapes, respectively. Inserting these Rt values in (3.8a, 3.8b) results in the corresponding relations for the penetration depths of these rods into 6061-T651 aluminum targets. These relations, together with the data from Piekutowski et al. (1999) and Forrestal et al. (1988), are shown in Fig. 3.10. The experiments of Piekutowski et al. (1999), with the ogive nosed rods, were described earlier and their results are given in Table 3.1 and Fig. 3.1. With the corresponding value of Rt = 1.87 GPa, we get from (3.8b) the predicted penetration

3.2 The Penetration Model for Rigid Long Rods Fig. 3.10 The agreement between our model’s predictions and the experimental data for: a ogive-nosed rods, b conical-nosed rods and c spherical-nosed rods, impacting 6061-T651 aluminum targets

65

66

3 Rigid Penetrators

relation for this rod/target pair: P/L eff = 2.12 V 20 (where V 0 is given in km/s). The excellent agreement between this relation and the experimental results of Piekutowski et al. (1999) is shown in Fig. 3.10a. The experiments of Forrestal et al. (1988) included both spherical and conical nosed rods. For the spherical nosed rods we find from (3.8b) that the penetration equation is: P/L eff = 1.76 V 20 , and the agreement with the experimental results of Forrestal et al. (1988) is shown in Fig. 3.10b. In a similar way, the penetration equation for the conical nosed rods, according to the model, is: P/L eff = 2.04 V 20 and its agreement with the data from Forrestal et al. (1988) is shown in Fig. 3.10c. The agreement between the predicted penetration formulas and the experimental data for these rods, with the different nose shapes, strongly enhances the validity of the penetration model presented here. Moreover, we have just demonstrated the usefulness of numerical simulations in constructing an analytical model, which was then validated by empirical data. This procedure is going to be repeated in the book for several other cases. The penetration capabilities of rigid rods with the different nose shapes are related to their corresponding Rt values as given by (3.9), which was derived through the numerical simulations of Rosenberg and Dekel (2010b). The specific choice for this relation was motivated by the fact that similar relations were obtained for the resisting stresses on rigid rods, by the analytical models of Tate (1986) and Yarin et al. (1995). These models follow the flow field of the target material around a rigid projectile which has the shape of a Rankine ovoid. This particular shape for a body of revolution enables a closed analytical solution for the potential function which determines the velocity field around this projectile. The shape of the Rankine ovoid is shown in Fig. 3.11, together with the equation which describes its surface. The projectile’s radius, at a large distance from its nose, is R. The analysis by Tate (1986) resulted in the following expression for the resisting stress on the ovoid-shaped rod:  Rt = Yt

  2 2E + ln 3 3Yt

(3.10a)

which can be rewritten as: Rt E = ln + 0.26 Yt Yt

(3.10b)

The analysis of Yarin et al. (1995), for an incompressible elastic–plastic solid, resulted in the following expression for the resisting stress on the ovoid-shaped rod: Fig. 3.11 The shape of the Rankine ovoid

3.2 The Penetration Model for Rigid Long Rods

   G Rt 3G = ln = ln 4/9 + 0.482 Yt 4 Yt Yt

67

(3.10c)

where G is the shear modulus of the target material. It is interesting to note that Tate’s model, (3.10b), results in Rt values which are very close to those derived from (3.9) for the spherical-nosed rod. For example, steel targets (E = 200 GPa) with compressive strengths of Y t = 0.5, 1.0 and 1.5 GPa, result in the following values from Tate’s model: Rt = 3.13, 5.56 and 7.73 GPa, respectively. The corresponding values, from (3.9) for the spherical nosed rod are: Rt = 3.2, 5.63, and 7.77 GPa, which are practically the same as those from Tate’s model. This is the expected result since the ovoid shape is similar to the spherical nose. Using a value of G = 80 GPa for the shear modulus of steel, the analysis of Yarin et al. (1995) results in somewhat lower values for Rt , by about 12.5%, for these steel targets.

3.2.2 High Velocity Impact: The Cavitation Phenomenon Up to this point we have analyzed experimental data and simulation results for rigid rods impacting semi-infinite targets at the ordnance range of impact velocities. We have seen that, except for the entrance phase, these rods experience a constant resisting stress (Rt ) which depends on the target’s strength and its Young’s modulus, as well as on the nose shape of the rod. The simple expressions, (3.8a, 3.8b) and (3.9), have been shown to account for penetration depths of rigid steel rods into aluminum and steel targets. The question which we have to ask at this point is whether these equations hold for higher impact velocity as well. Before answering this question we have to introduce the concept of cavitation, which has an important role for high velocity impacts. In a short summary of his early work during WWII, Hill (1980) used the term cavitation, from fluid mechanics, to describe situations for which the diameter of the penetration channel is larger than the diameter of the projectile. Hill’s analysis for the penetration of ogive-nosed projectiles resulted in a threshold velocity (V cav ) for the onset of cavitation, which means that at higher velocities the projectile “invests” some of its energy in expanding the channel’s diameter. According to Hill’s analysis the force on the ogive nosed projectile, at low impact velocities (V 0 < V cav ) should be constant, with no cavitation in the penetration channel. On the other hand, at higher impact velocities (V 0 > V cav ) the resisting force on the projectile depends on its velocity, through the addition of a target-inertia term (ρ t V 2 ). At these velocities the diameter of the penetration channel is larger than that of the projectile, as cavitation takes place for V 0 > V cav . This is an important observation since it relates the constant diameter of the penetration channel with a constant resisting stress which the target exerts on the projectile. The cavitation phenomenon is shown schematically in Fig. 3.12, where the diameter of the penetration channel is evidently larger than the projectile’s diameter.

68

3 Rigid Penetrators

Fig. 3.12 A schematic description of the cavitation phenomenon

In summary, Hill concluded that for ogive-nosed projectiles the inertia of the target should be added to its resisting stress only for high impact velocities (V 0 > V cav ). This assertion has been validated by the numerical simulations of Rosenberg and Dekel (2009a), as discussed below. On the other hand, Hill stated that for both spherical and conical nosed projectiles the inertia should be added at all impact velocities, resulting in the cavitation phenomenon at each test. This conclusion is in sharp contradiction with both experimental observations and numerical results, for penetration channels formed by rigid rods with these nose shapes. In fact, the onset of cavitation at a critical impact velocity occurs for all nose shapes, as will be demonstrated here. Post-mortem pictures of the 6061-T6 aluminum targets in the experiments of Forrestal and his colleagues, showed no cavitation for rods with either spherical or conical nose shapes, at impact velocities up to 1.2–1.4 km/s. At higher impact velocities the rods failed by plastically deforming or by breaking and eroding. For ogive–nosed rods, the penetration channels in the experiments of Piekutowski et al. (1999) showed no cavitation for velocities up to about 1.82 km/s. At impact velocities above 2.0 km/s these rods broke up as a result of the high impact pressures. Thus, there is no evidence for cavitation in the aluminum targets which were impacted by these rods with different nose shapes. Some of the post-mortem radiographs, from Forrestal et al. (1988) and Piekutowski et al. (1999), are shown in Fig. 3.13. Clearly, the channel diameters in these experiments were the same as the diameters of the rods. Moreover, the simulation results of Rosenberg and Dekel (2009a) for conical and spherical nosed rods did not show a cavitation phenomenon in aluminum targets for this range of impact velocities. Thus, the conclusion of Hill (1980) that cavitation is induced at all impact velocities for conical and spherical nosed projectiles, has been refuted by both the experimental results of Forrestal et al. (1988) and by the numerical simulations of Rosenberg and Dekel (2009a). In order to further investigate the issue of cavitation, Rosenberg and Dekel (2009a) performed 2D simulations for rigid steel rods, impacting steel and aluminum targets at high velocities. These simulations showed that a certain cavitation threshold velocity (V cav ) exists for every rod/target pair. For impact velocities below V cav the diameter of penetration channel was equal to the rod’s diameter. In contrast, for impact velocities above V cav the diameter of the penetration channel was larger than that of the rod. These cavitations take place at the initial part of the channel, which becomes narrower as the rod slows down at deeper penetrations. A close examination of the penetration

3.2 The Penetration Model for Rigid Long Rods

69

Fig. 3.13 Post mortem radiographs of conical, spherical and ogive nosed rods in aluminum targets

and velocity histories in these simulations showed that when the velocity of the rod reaches the threshold value (V cav ), the diameter of the channel is exactly equal to that of the rod. Figure 3.14 shows the shape of the channel at low (V < V cav ) and high rod velocities (V > V cav ) in these simulations. The simulations of Rosenberg and Dekel (2009a) showed that the threshold impact velocity (V cav ) depends on the nose shape of the rod and on the strength and density of the target, and that it does not depend on the density of the rod. The highest values of V cav were obtained for ogive-nosed rods and the lowest for flat-nosed rods. These simulations led to a simple relation for the value of V cav in terms of the density of the target (ρ t ), its resistance to penetration (Rt ), and the nose shape of the rod (b), as follows: 2 = Rt bρt Vcav

Fig. 3.14 The cavitation phenomenon in the simulations of Rosenberg and Dekel (2009a)

(3.11)

70 Table 3.3 Values for V cav , (in km/s) for rods of different nose shapes and different targets

3 Rigid Penetrators Nose shape

Aluminum (0.4 GPa)

Steel (0.4 GPa)

Ogive

2.17

1.385

Spherical

1.3

0.82

Conical

1.73

1.115

Flat

0.85

0.525

where the nose shape factor (b) was found to have the following values: b = 0.15, 0.24, 0.5 and 1.25, for the CRH3 ogive, conical, spherical and flat nose shapes, respectively. The conical head of the rods in these simulations had a length which was three times the rod’s radius. Table 3.3 lists several values for V cav which were obtained in these simulations for the various nose shapes and different targets. Note the high values of V cav for the 0.4 GPa aluminum targets, which account for the absence of cavitation in all of the experiments by Forrestal and his colleagues with the 6061-T651 aluminum targets, as discussed above. Rapoport and Rubin (2009) followed the analysis of Yarin et al. (1995) and showed that their model also predicts a cavitation threshold behavior, with V cav = 1.97 km/s for the ovoid-nosed projectile penetrating an aluminum target with strength of 0.4 GPa. Rubin et al. (2016) compared results for an ovoid-shaped projectile penetrating an aluminum target, from the analytical approach of Rapoport and Rubin (2009), with simulation results for the penetration of an ovoid-shaped projectile, using the 2D version of the LS-DYNA code. The elastic-perfectly-plastic aluminum target in these simulations, with strength of 0.4 GPa, was nearly incompressible (v = 0.49) in order to match the incompressibility condition of the analytical solution for the ovoidshaped nose. The simulations were performed for constant penetration velocities in the range of 0.5–3.0 km/s and the force on the projectile nose was recorded in each simulation. It turned out that this force was constant for all velocities below about 2.0 km/s, and it increased at higher penetration velocities. Considering the fact that the analytical solution resulted in a value of V cav = 1.97 km/s, one can state that an excellent agreement has been obtained between an analytical model and numerical simulations in this case. To emphasize the issue of cavitation threshold Rubin et al. (2016) performed more simulations with an aluminum target having realistic constitutive relations including compressibility, strain hardening, strain rate and temperature effects. The results of these simulations show that even when these features are added, the drag force on the projectile remains relatively constant for V < V cav , and that it is strongly increasing with velocity for V > V cav . Rubin et al. (2016) point out the shortcomings of analytical approaches which are based on cavity expansion analyses, for the target’s resistance to penetration, since these models do not result in the cavitation phenomenon at high velocities. Fan and Li (2019) also studied the penetration resistance of a rigid projectile with ogive nose shape in an incompressible elastic-perfectly-plastic target. They used a velocity field normal to the projectile’s nose surface to approximate the actual flow of the target material, and obtained an analytical solution for the target’s resistance on the projectile nose. It shows that the penetration resistance consists of three terms,

3.2 The Penetration Model for Rigid Long Rods

71

i.e. a constant term, a velocity-squared related term and an acceleration-related term. They also determined the critical cavitation velocity using Hill’s cavitation criterion. Their model shows that when the impact velocity is less than the critical cavitation velocity, the constant term is dominant, while when the impact velocity is higher than the critical velocity, cavitation associated with target-projectile separation happens, and both constant term and velocity-squared term dominate. Good agreement was obtained between their model’s results for V c and those obtained by the simulations of Rosenberg and Dekel (2009a), especially for the spherical nosed projectiles. The qualitative explanation for these observations is relatively simple. As long as the velocity of the projectile is lower than V cav , the lateral velocity imparted by the projectile to the target is not high enough to overcome its strength, and the channel diameter remains the same as that of the projectile. At higher impact velocities, the lateral momentum is high enough to initiate the cavitation phenomenon in the target. As a result, the projectile invests an extra amount of energy in order to open the cavity, in addition to the energy needed for its advancement. In other words, the inertia of the target (ρ t V 2 ) starts to play an important role at high impact velocities. The immediate consequence of these observations is that for V 0 > V cav , the deceleration of the rod cannot be constant. This is clearly seen in Fig. 3.15, which shows the simulation result for the deceleration history of an L/D = 10 spherical-nosed steel rod, impacting a 0.4 GPa aluminum target at 2.5 km/s (V 0 > V cav ). Note that at about 180 µs after impact, the deceleration reaches a constant level, since the velocity of the rod has reached the value of V cav at this time. With these observations we can construct now the analytical model for the penetration depths of rigid rods at high impact velocities. In order to derive an expression for the penetration depth of a rigid rod at impact velocities above V cav , one has to divide the process into two stages. The first stage starts with the impact at V = V 0 and ends when the rod slows down to V = V cav . During this stage the inertia of the target has to be added to its strength-related term (Rt ), in order to account for its increased resistance to penetration due to cavitation. Fig. 3.15 The deceleration history for an impact of a spherical nosed projectile at V 0 > V cav

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Fig. 3.16 The two penetration regimes for a high velocity impact

The second stage of the process starts when V = V cav and ends at V = 0. As discussed above, the deceleration of the rod is constant during this phase, and the rod penetrates to a certain depth (Pc ) which is derived below. The two penetration regimes are shown schematically in Fig. 3.16. We start the analysis by considering the second penetration regime, with the constant deceleration of the rod. Inserting V 0 = V cav in (3.8a), the penetration depth (Pc ) in this regime is given by: Pc = P(Vcav ) =

2 ρ p L eff Vcav 2Rt

(3.12a)

and with the expression for V cav from (3.11), we get: L eff ρ p 2bρt

(3.12b)

ρp Pc = L eff 2bρt

(3.12c)

Pc = or, in its normalized form:

This simple equation gives the maximal penetration depth (Pc ) for a given rod/target pair which can be achieved with no cavitation. The equation highlights the relevant parameters for Pc , namely, the density of the target and the rod, the effective length of the rod and its nose shape factor (b). It is interesting to note that Pc does not depend on target strength. This result can be understood by realizing that the strength of the target is responsible for two opposing effects. On the one hand, the penetration depth of a given rod decreases with target strength. On the other hand, the cavitation threshold increases with target strength, and the no-cavitation range of impact velocities is larger for the stronger target. The fact that Pc in (3.12c) does not depend on target strength means that the two effects cancel each other.

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73

In order to get an estimate for the values of Pc for some practical cases, consider a steel rod impacting an aluminum target, with a density ratio of ρ p /ρ t = 2.85. Since b = 0.5 for a spherical-nosed rod and b = 0.15 for an ogive-nosed rod, we get Pc = 2.85L eff and Pc = 9.5L eff for these rods, respectively. The large value of Pc for the ogive-nosed rod accounts for the fact that the penetration channels in the experiments of Piekutowski et al. (1999), showed no sign of cavitation even for an impact velocity of 1.82 km/s, with a penetration depth of nearly 7.0 L eff . The value of V cav = 2.17 km/s in Table 3.3, means that the ogive-nosed rods in these experiments could penetrate without cavitation even at higher impact velocities. However, at an impact velocity of about 2.0 km/s these rods started to break and erode, and the cavitation phenomenon could not be demonstrated in these experiments. We now consider the first penetration stage, with impact velocities above the cavitation threshold. As discussed above, at these velocities the target’s resistance to penetration has to include its inertia, and the resisting stress () should be given by the sum of two terms. The first term is the strength related resistance to penetration (Rt ), and the second is a velocity-dependent term which is due to the inertia of the target. Rosenberg and Dekel (2009a) assumed that a natural choice for the resistance to penetration would be: 2 )  = Rt + Bρt (V 2 − Vcav

(3.13a)

where B is a geometric factor which is related to the nose shape of the rod. A reasonable assumption is that the two shape factors, B and b, are closely related and a possible choice would be that B = b/2. The validity of this choice will be checked later on when we compare the predicted penetrations from the model, with simulation results for high impact velocities. Inserting this relation in (3.13a) together with the definition of V cav through (3.11), results in the following relation for : =

1 1 Rt + bρt V 2 , for: V ≥ Vcav 2 2

(3.13b)

which reduces to the expected value of  = Rt for V 0 = V cav . Relation (3.13b) can be rewritten in the form: =

 1  Rt 1 + (V Vcav )2 2

(3.13c)

The resisting force on the rod is obtained by multiplying this expression for  by the cross section of the rod (π r 2 ). Inserting this force in the equation of motion, (3.7a), and integrating between V = V 0 and V = V cav , results in the following relation for the penetration depth at the first stage of penetration, from V = V 0 to V = V cav :   2 ρ p Vcav 1+x P = · ln L eff Rt 2

(3.14a)

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where x is defined by:  x=

V0 Vcav

2 (3.14b)

Adding the penetration depth for the low velocity stage, as given by (3.12a), results in the following expression for the total penetration of a rigid projectile at an impact velocity above the cavitation threshold:    2 ρ p Vcav 1+x P for: V0 ≥ Vcav = · 1 + 2 ln L eff 2Rt 2

(3.15a)

and with the definition of Pc in (3.12a), the total penetration depth can be written in a more compact form:   1+x P for: x ≥ 1.0 = 1 + 2 ln Pc 2

(3.15b)

Using the above expressions for Pc and Rt it is easy to show that the penetration depth in the low velocity range is given by: ρ p L eff P= · 2bρt



V0 Vcav

2 for: V ≤ Vcav

(3.16a)

which, in its normalized form is given by: P = Pc



V0 Vcav

2 = x for: x ≤ 1.0

(3.16b)

As was noted above, the values of V cav are quite high for both ogive and conical nosed rods impacting aluminum and steel targets. At these impact velocities even high strength rods lose their rigidity by deforming or shattering. This is the reason for the absence of experimental data which show clear cavitation thresholds, and it is impossible to validate the analysis discussed above through experimental data. Instead, Rosenberg and Dekel (2009a) compared the predictions from the model with numerical simulations for various rods, impacting both aluminum and steel targets at velocities above their corresponding values of V cav . This comparison is shown in Fig. 3.17 demonstrating an excellent agreement between the model’s predictions and the simulation results. This agreement strongly enhances the validity of the assumptions made above, particularly the assumption concerning the relation B = b/2 between the two nose-shape factors. The penetration efficiency of a rigid projectile at impact velocities above V cav , is lower than its efficiency at low velocities, since the P = P(V 0 ) curve is getting shallower with increased velocities. This is the expected result since an increasing

3.2 The Penetration Model for Rigid Long Rods

75

Fig. 3.17 Comparing the analytical predictions with the simulation results for V 0 > V cav

amount of the projectile’s energy is spent on enlarging the penetration channel laterally. In contrast, for impact velocities below V cav , the kinetic energy of the projectile is spent only on forming the necessary channel for its passage. As discussed above, Hill (1980) noted that a penetration process with no cavitation must be achieved by a constant resisting stress, which depends on target strength. He defined this stress as “the mean work to form a unit volume of the resultant cavity by progressive penetration”. A simple way to understand this claim is as follows. Consider the projectile advancing by a small increment x into the target, pushing away a small volume of target material, π r 2 x, where r is the radius of the crater, which is the same as that of the projectile. By doing so the projectile is losing a certain amount of its kinetic energy, E k . Energy conservation dictates that this loss of energy has to be equal to the incremental work done on the target, which is linearly related to the volume displaced by the projectile. Thus, one can write E k = k 1 π r 2 x, where k 1 is some constant. This equation should hold for any increment along the penetration channel if it has a constant radius. Integrating this relation, between E k0 at x = 0 and E k = 0 at x = P (the penetration depth), results in: V 20 = k 2 P, where k 2 is another constant. Thus, by using a simple energy-conservation argument we find that the penetration depth is related to V 20 , which means that the resistance to penetration is constant along the whole penetration process, as discussed above. This is the logic behind the statement of Hill (1980), who identified the constant resisting stress with the no-cavitation type of penetration. This is exactly what was found in the simulations of Rosenberg and Dekel (2009a) for impact velocities below the corresponding cavitation threshold velocities (V cav ), of rigid rods with different nose shapes. Both Hill (1980) and Rosenberg and Dekel (2009a) conclude that a penetration process without cavitation is related to a constant resisting stress. The only difference between the two works is due to Hill’s assumption that for both conical and spherical-nosed projectiles, cavitation occurs at all impact velocities. In contrast, the numerical simulations of Rosenberg and Dekel (2009a) showed that a cavitation threshold exists for every

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nose shape. These simulation results are substantiated by the experimental results of Forrestal and his colleagues for rigid steel rods with spherical and conical nose shapes. In a sharp contrast with Hill’s prediction, no cavitation was observed in these experiments which were performed at impact velocities up to 1.2–1.4 km/s. Note that Hill’s prediction about the resisting stresses on rods with these nose shapes is based on a certain functional dependence of these stresses on the curvature of the projectile’s nose. Apparently, this distribution of stresses is only a rough approximation for the actual stresses.

3.3 The Cavity Expansion Analysis Several penetration models which are based on the work of Goodier (1965) have been suggested during the last decades. These analytical models assume that the resistance to penetration of a rigid projectile () should include the inertia of the target for all impact velocities, through a Poncelet-type expression:  = C + Bρ t V 2 , where B is some nose shape factor. For the strength-related constant C, Goodier used the cavity expansion analysis of Bishop, Hill, and Mott (1945). This classical work derived the threshold pressure which is needed to expand a small cavity in a large incompressible elasto-plastic body. Note that the third author of this paper, the Nobel laureate Sir Neville Mott, worked for the war ministry in the UK during the years of World War II, and contributed important insights to the dynamic response of solids under impulsive loadings. The cavity expansion analysis of Bishop et al. (1945) was developed in order to account for the relatively high force which is needed to push a conical-nosed rigid punch into large metallic blocks. The resisting stresses on these punches, as they were pushed into copper blocks, reached a constant value which was about five times the strength of the copper. The experimental setup for these deepindentation tests, and the resulting stress-distance curves are shown schematically in Fig. 3.18. Note the clear evidence for the entrance phase in Fig. 3.18b, which lasts to 4–5 rod diameters. In order to account for the constant resisting stress at the deep penetration, and for the factor of about five between this stress and the strength of the metallic block, Bishop et al. (1945) developed the static cavity expansion analysis. This analysis Fig. 3.18 A schematic description of a the experimental setup and b the resulting force-distance curves in Bishop et al. (1945)

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77

accounts for the pressure needed to expand a small cavity in an elasto-plastic solid, as will be described below. Hill (1950) followed the cavity expansion process for compressible elasto-plastic solids, and Hopkins (1960) derived a solution for the dynamic cavity expansion, which accounts for the pressure needed to propel the cavity wall at a certain velocity. This dynamic pressure is equal to the sum of the quasi-static threshold pressure, as derived by Hill (1950), and a term which is related to the material’s inertia (ρV 2 ), as derived by Hopkins (1960). The application of these analyses for terminal ballistics models was made, for the first time, by Goodier (1965). His starting point was that the resisting stress which the target exerts on the projectile is the sum of the two terms, which were derived by the dynamic cavity expansion analysis. As described above, the first term is related to the target’s strength, and it was taken from the static cavity expansion analysis of Bishop et al. (1945). The second term, with the target’s inertia, was taken from Hopkins (1960). With these assumptions by Goodier obtained a closed-form formula for the penetration depth of rigid projectiles which was, in fact, the first attempt to derive a penetration model from first principles. This was the main reason that many workers followed Goodier’s approach in order to account for their data, which included deep penetrations of rigid projectiles into metallic targets, concretes, and even ceramics. However, there are several problems with Goodier’s model which have to be carefully re-examined, as follows. The model is based on penetration data from Hermann and Jones (1961) for various metallic spheres impacting thick metallic targets at velocities up to about 1.5 km/s, and it treated these spheres as rigid bodies. Goodier found that the penetration depths for all of these spheres were much lower than those calculated by assuming a constant resistance to penetration, as derived from the static cavity expansion analysis of Bishop et al. (1945) and Hill (1950). Thus, he added the target’s inertia to the resisting stress, following the dynamic cavity expansion analysis of Hopkins (1960). The enhanced resistance to penetration resulted in a much better agreement with the experimental data, which was considered by Goodier as evidence for the validity of his model. However, the spheres in the experiments which Goodier analyzed were made of copper, aluminum, soft steel and lead. Obviously, these materials are not strong enough to warrant their rigidity at impact velocities of 0.5–1.5 km/s. In fact, all of these spheres deformed heavily during their impact and penetration, which is the main reason for their low penetration depths. Thus, the addition of target inertia to its penetration resistance was based on a flawed interpretation by Goodier, of the data he used for modeling. The second problematic issue in Goodier’s analysis concerns the assumption that the strength-related term, for the target’s resistance to penetration, can be derived from the quasi-static cavity expansion analyses of Bishop et al. (1945) or Hill (1950). As will be demonstrated below, the cavity expansion analysis results in much lower threshold pressures than the corresponding Rt values for the resisting stresses, as obtained by both experimental results and numerical simulations. They are also much lower than the values derived by analytical models, such as those of Tate (1986) and Yarin et al. (1995), which follow the target’s motion around the projectile’s nose. In order to shed more light on this issue we now discuss the cavity expansion analyses in some detail.

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Fig. 3.19 The spherical cavity expansion (left) and the asymptotic increase of the cavity pressure (right)

The cavity expansion analysis of Bishop et al. (1945) starts with a small cavity of radius a0 , either spherical or cylindrical, in a large elasto-plastic solid having welldefined values of strength (Y ), Young’s modulus (E) and Poisson’s ratio (ν). The cavity is loaded by an internal pressure, acting on its wall, and the analysis follows the state of the solid around the wall as the pressure is increased. At low pressures the solid is compressed elastically until the pressure reaches a value of 2Y /3, and the material at the cavity wall reaches its elastic limit and yields. With increasing pressures more material around the cavity yields and an elasto-plastic interface is moving away from the cavity wall. At these pressures the material between the cavity wall (which expands to radius—a) and the elasto-plastic interface (at radius—c), is in its plastic state, while the material beyond the plastic front is in its elastic state. The different states of the solid are shown schematically in Fig. 3.19a. The analysis follows the evolution of the different regimes and accounts for the ratio of c/a, which reaches an asymptotic value as the pressure increases. The most important result of this analysis is that when the pressure in the cavity reaches a limiting value, it drives the cavity’s wall indefinitely as shown schematically in Fig. 3.19b. This asymptotic behavior is a case of plastic instability, which characterizes the threshold pressure of the static cavity expansion process. The asymptotic nature of the pressure is manifested through its dependence on the ratio a0 /a, and for the spherical cavity analysis it is given by: Psph =

   E a3 2Y 1 + ln 1 − 03 3 (1 + ν)Y a

(3.17)

where a0 /a is ratio between the initial cavity radius and its transient value. Clearly, the term (a0 /a)3 becomes negligibly small as compared with 1.0, even for modest expansions. The asymptotic pressures for the spherical and cylindrical cavities are given by the following expressions: Psph and

  E 2Y 1 + ln for the spherical cavity = 3 (1 + ν)Y

(3.18a)

3.3 The Cavity Expansion Analysis

Pcyl

79

√ ⎤ ⎡ 3E Y ⎣ ⎦ for the cylindrical cavity = √ 1 + ln 2(1 + v)Y 3

(3.18b)

These expressions were obtained by Bishop et al. (1945) for an incompressible solid. For a work-hardening material these pressures have an additional term related to the hardening modulus of the material, which is the slope of the stress–strain curve of the solid in its plastic range of response. Hill (1950) derived the corresponding expression for the spherical cavity in a compressible solid. His analysis replaces the term (1 + ν) by the term 3(1 – ν), in the logarithmic expression of (3.18a). Thus, according to Hill (1950), the threshold (critical) pressure to expand a spherical cavity indefinitely, in a large compressible solid, is given by: Psph =

  E 2Y 1 + ln 3 3(1 − v)Y

(3.18c)

Rosenberg and Dekel (2008) performed numerical simulations of the cylindrical cavity expansion process, from which they determined the pressures which are needed to expand this cavity in compressible solids. These simulations showed that the term (1 + ν) in (3.18b) for the cylindrical case, should be replaced by 3(1 – ν) for the compressible solid, which is the same replacement as Hill’s for the spherical cavity. Thus, according to these simulations the pressure to expand a cylindrical cavity in a compressible elasto-plastic solid is given by: √ ⎤ 3E Y ⎦ = √ ⎣1 + ln 6(1 − v)Y 3 ⎡

Pcyl

(3.18d)

These analytical expressions for compressible solids result in critical pressures which are much lower than the Rt values, as derived from the numerically-based model of Rosenberg and Dekel (2009a) through (3.9), for spherical and ogive-nosed rigid rods. They are also lower than the threshold stresses obtained by the analytical models of Tate (1986) and Yarin et al. (1996) for the ovoid-shaped rigid rod. Consider, for example, these pressures for a steel target (E = 200 GPa, ν = 0.29) with strength of Y = 1.0 GPa. From (3.18c) we get Psph = 3.7 GPa, and from (3.18d) Pcyl = 3.12 GPa. These values are, indeed, much lower than the corresponding value of Rt = 5.6 GPa for a spherical-nosed rod penetrating this steel target. The asymptotic pressures, as derived from the cavity expansion analyses, are bound to underestimate the resisting stresses on rigid projectiles because of the different situations for which they are applied. The cavity expansion analyses account for cases where the material around the cavity is moving perpendicularly to the surface of the cavity. This is a one-dimensional case, where the material moves in a very different manner than its complex motion along flow lines during the penetration of rigid rods.

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In all the models discussed here the functional dependence of the normalized threshold pressures (Pthresh /Y ) on the relevant properties of the solid can be written as: E Pthresh = K 1 ln + K 2 Y Y

(3.19)

where K 1 and K 2 have different values in these models. Thus, K 1 = 2/3 and 0.577, for the spherical and cylindrical cavity expansion models, respectively. On the other hand, K 1 = 1.1 according to the numerically-based model of Rosenberg and Dekel (2009a), as given by (3.9), and K 1 = 1.0 according to the analysis of Tate (1986). The values for K 2 depend on the nose shape of the rigid rod, according to the numericallybased model presented above, and it depends on Poisson’s ratio of the solid according to both cavity expansion models and Tate’s model. The functional similarity of the expressions for Pthresh /Y is not surprising, since all the analyses presented here deal with the pressures which have to be applied inside the solid, in order to “push” it against a resisting stress which depends on its strength. The differences between the corresponding values of resisting stresses are related to the details concerning the actual flow of the material, as discussed above. The numerical simulations of Rosenberg and Dekel (2008) were aimed at a better understanding of the cavity expansion process, in order to highlight the differences between this process and the deep penetration of rigid rods. Their numerical simulations followed the expansion of small spherical cavities in large spheres of steel, aluminum and lead, having von-Mises strengths in the range of 0.5–1.5 GPa. Pressure was applied inside the spherical cavity, in each simulation, and a threshold (critical) value was defined as the pressure above which a steady-state velocity was imparted to the cavity’s wall. Figure 3.20 shows the simulation results for a 1.0 GPa steel sphere in which these pressures were increased, until a steady velocity of the cavity wall was achieved. As is seen in the figure, this threshold pressure is about 3.6 GPa Fig. 3.20 Simulation results for the cavity expansion process in a 1.0 GPa steel sphere

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81

for the 1.0 GPa steel. The figure depicts the features of the spherical cavity expansion analysis, which we discussed above. Namely, for pressures below the threshold, the cavity starts to move but it slows down to a halt as the plastic front moves away from the cavity into the solid. For cavity pressured above the threshold pressure, the cavity acquires a steady velocity which is dependent on the pressure. The values of these critical pressures for spherical cavities in the aluminum, steel, and lead spheres were in excellent agreement with those predicted by the analysis of Hill (1950), according to (3.18a). This agreement is shown in Fig. 3.21 and it demonstrates the usefulness of numerical simulations in verifying analytical models. On the other hand, the cavity expansion model has been well established for many years, and the agreement shown in Fig. 3.21 can be considered as a demonstration for the validity of these numerical simulations. The numerical simulations of Rosenberg and Dekel (2008) were also used in order to find a relation between the velocity of the cavity’s wall and the applied pressure (Pdyn ) on the wall. These results for steel, aluminum and lead, with strengths of 0.5, 1.0 and 1.5 GPa, are shown in Fig. 3.22a. The different curves shown in this figure can be normalized to a single curve as shown in Fig. 3.22b, and the normalized presentation of these simulation results is given by: Pdyn = Psph + 1.1875ρV 2

(3.20)

where Psph is the static pressure which is needed to open the spherical cavity, as given by (3.18a). It is worth noting that (3.20) applies for very different metals which were used in these simulations, and it should be accounted for by a proper analytical model for the dynamic cavity expansion process. As discussed above, such a model for the dynamic expansion of a spherical cavity, in an incompressible solid, was derived by Hopkins (1960). His result includes terms which are related to the expansion velocity of the cavity and to its acceleration. For a constant expansion velocity (V ), Fig. 3.21 The agreement between simulation results and the cavity expansion model for the critical pressures in compressible solids

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Fig. 3.22 a The relations between pressures and cavity wall velocities for the three materials, and b the normalized representation of these results

his dynamic term is equal to 3ρV 2 /2. The numerical factor (3/2) of the inertia term (ρV 2 ) in this expression is higher than the corresponding factor of 1.1875 in (3.20), probably because Hopkins considered an incompressible solid, while materials in numerical simulations are inherently compressible. The analytical results of Masri and Durban (2005) show that this inertia coefficient depends on Poisson’s ratio of the solid, and that its value is around 1.2 for the materials they considered. In summary, the cavity expansion analyses are important mathematical solutions for a complex elasto-plastic problem. Their main use is for the design of highpressure vessels, such as gun breeches, which have to remain elastic under the high inner pressures they experience. However, as discussed above, both the quasi-static and the dynamic cavity expansion analyses are irrelevant for terminal ballistics, due to several issues. Firstly, the material motion during these one-dimensional expansions is very different from the flow of a target material around a rigid projectile. This difference results in higher Rt values for the resisting stress, than both Pspher

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83

and Pcyl from the static cavity expansion of Bishop et al. (1945) and Hill (1950). These higher resisting stresses were inferred from the experimental results of Forrestal and his colleagues and from their numerical simulations, as shown above. The second issue concerns the addition of an inertia-related term (ρ t V 2 ) to the target’s resisting stress for all impact velocities. Both experimental data and numerical simulations show that the resisting stresses are independent on the projectile’s velocity, up to the cavitation threshold velocity (V cav ). The addition of target inertia in various penetration models was based on Goodier’s flawed interpretation of the data he analyzed for the deep penetrations of metallic spheres. In fact, these spheres were far from rigid and their low penetration depths were due to their significant deformation upon impact. Finally, the dependence (or independence) of the resisting stress on the projectile’s velocity is manifested by the shape of the penetration channel through simple energy-conservation arguments. A penetration channel with a diameter which is equal to that of the projectile is a clear indication for the independence of the resisting stress on the projectile’s velocity. This situation changes when the impact velocity is higher than the corresponding cavitation threshold (V cav ). All the experiments of Forrestal and his colleagues, with rigid steel rods and aluminum targets, showed no cavitations in their penetration channels. Thus, the data from these experiments cannot be accounted for by a velocity-dependent resisting stress, because of the energy-conservation argument presented here. We shall see in Sect. 3.8 that the same phenomena are observed in the deep penetrations of rigid projectiles in concrete targets. As a final note on this issue we wish to acknowledge the importance of the static deep penetration experiments of Bishop et al. (1945), as shown in Fig. 3.18. Our understanding is that such experiments with a hard cylindrical punch can result in the resisting stress (Rt ) which a given target exerts on a rigid projectile during penetration. This is especially true if the target’s strength is not sensitive to strain rates. In this case there should be no difference between the force-distance curve as measured by the static indentation test, and the resisting stress history on a projectile which impacts the target at a velocity below the cavitation threshold. Particularly, the value of Rt for various nose-shaped rods can be obtained by the constant level reached in these static tests. Obviously, the strength of the cylindrical punch has to be higher than the corresponding value of Rt for the metallic block. For example, with Al 6061-T6 targets one needs punches with strengths higher than about 2.0 GPa, which are difficult to obtain. On the other hand, with much weaker targets one can perform these tests with conventional high-strength steel punches. We turn now to some examples where we applied deep indentation tests to relatively soft materials, such as polycarbonate and a lead alloy. We shall return to this issue later on when we discuss the deep penetration in concrete targets.

3.4 Deep Indentation Tests Rosenberg and Kositski (2017a) used the deep indentation test to follow the resisting force which a large polycarbonate cylinder exerts on a conical-nosed rigid indenter,

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Fig. 3.23 The static indentation test (a) and the indenter’s dimensions, in mm (b)

as it is being pushed by an Instron machine. The force and the penetration depth were simultaneously recorded by the machine as the steel indenter, with a diameter of 7.1 mm and length of 60 mm was pushed at a rate of 1 mm per minute. Figure 3.23 shows the setup of this static indentation test. The force-distance curve as obtained by the machine is shown in Fig. 3.24 and one can clearly see that it has three distinct stages. The first stage lasts up to a distance of about 13 mm which is equal to the length of the conical nose of the indenter. The Fig. 3.24 The force-distance curve obtained by the static indentation test on PC

3.4 Deep Indentation Tests

85

second stage, which lasts to a distance of about 21.5 mm, is due to the entrance phase where the penetration resistance is still increasing, as found by Bishop et al. (1945) for their copper plate. The final stage with a linear increase in the force is due to the friction between the PC and the shank of the indenter. In order to arrive at the true resisting stress on the indenter’s nose, the friction force has to be subtracted from the total force, as was pointed out by Bishop et al. (1945). As we shall see later on, the effect of friction between a high velocity projectile and the target is negligible. In order to subtract the friction contribution Rosenberg and Kositski (2017a) used the slope of the force-distance curve at its third stage, which is due to the friction along the shank. Thus, one can write dF fric /dx = 2π rτ , where r is the indenter’s radius and τ is the shear stress responsible for the friction between the steel indenter and the PC. The value for τ in this test was 32.2 MPa and it was used to subtract the friction force along the whole penetration process, including the conical nose. Figure 3.25 shows the resulting stress-distance history on this indenter after friction subtraction, and one can see that the constant resisting stress at deep penetration amounts to 0.29–0.3 GPa. Rosenberg and Kositski (2017a) analyzed the data for penetration depths of 7.62 mm AP projectiles impacting large polycarbonate targets, as given by Weiss et al. (2013). These 9.91 g projectiles impacted the PC targets at velocities in the range of 600–900 m/s and their penetration depths were accurately determined. Rosenberg and Kositski (2017a) argued that the resisting stress exerted by the PC target on these projectiles was the same as the asymptotic value of the stress on the static indenter as given in Fig. 3.25. Thus, they used the value of Rt = 0.29 GPa in the penetration relation (3.8a), with the corresponding value for ρ p L eff for the AP projectile. The resulting agreement between the model’s predictions and the penetration data of Weiss et al. (2013) is shown in Fig. 3.26. The excellent agreement between model and data is somewhat surprising considering the fact that the strength of polycarbonate is known to be rate sensitive. However, the work of Rittel and Dorogoy (2008) showed that, when confined by high lateral stresses, the difference between static and dynamic strengths of PC specimens diminishes significantly. The elements around the penetrating bodies experience high lateral stresses, under both static indentation Fig. 3.25 The stress-distance curve obtained after subtraction of the friction force

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3 Rigid Penetrators

Fig. 3.26 The agreement between model predictions and the data for PC penetration

and projectile penetration, which may be the reason for the equality of the static and dynamic resisting stresses. A similar deep indentation test was performed by Rosenberg et al. (2018a) on a thick cylinder of a lead alloy (with 10% antimony). This alloy was chosen for its very low strength which enabled a deep indentation process with a CRH3 ogivenosed hard steel indenter. The resulting force-distance history and the corresponding resisting stress, after elimination of the friction effect, are given in Fig. 3.27. Note the fast increase in the resisting force for the first 12 mm, until the ogive nose of the indenter is fully embedded in the lead cylinder. The subsequent increase is more moderate, up to a distance of about 30 mm, which is the extent of the entrance phase of about 4.5D, as in the experiments of Bishop et al. (1945) shown in Fig. 3.18. Finally, the friction effect is clearly depicted by the linear increase in the resisting force at depths beyond 30 mm. The deep penetration resistance which this lead alloy exerted on the ogive-nosed indenter, after friction subtracted, is about 240 MPa.

Fig. 3.27 The force-distance history (left) and the resulting stress-distance (right) in the lead-alloy test

3.4 Deep Indentation Tests Table 3.4 Predicted and measured penetration depths in the lead alloy targets

87 V 0 (m/s)

Measured penetration (mm)

Predicted penetration (mm)

462

45

45.9

531

60

59.3

575

68

67.7

742

97.5

98.1

In order to account for the strain rate sensitivity of this lead alloy Rosenberg et al. (2018a) performed a static compression test on a cylindrical specimen, and a dynamic test with their Kolsky bar system, at a rate of about 6000 s−1 . The resulting stress strain curves showed an increase of about 36% in the dynamic strength, as expected from soft metallic specimens. In order to estimate the dynamic resistance to penetration of this lead alloy, they multiplied the observed static resistance from the indentation test by a factor of 1.36. This factor resulted in an estimated value of Rt = 0.326 GPa for the penetration resistance of this lead alloy against impacting projectiles with a CRH3 ogive nose shape. The next step of Rosenberg et al. (2018a) was to use this value of Rt in order to predict the penetration depths for the hard steel cores of 0.3 AP projectiles impacting thick targets of this lead alloy. This projectile/target pair is a unique case since the lead alloy has a relatively high density (ρ t = 10.6 g/cc) and a very low strength. Thus, according to (3.11) its cavitation threshold velocity (V cav ) is expected to be quite low. Substituting the values of ρ t and Rt for the lead alloy in (3.11) together with b = 0.15 for the projectile’s CRH3 nose shape, results in a value of V cav = 453 m/s. The impact velocities in the tests of Rosenberg et al. (2018a) were in the range of 460–740 m/s (above V cav ). Thus, the predicted penetration depths were calculated through (3.15a) rather than through (3.8a, 3.8b). Table 3.4 shows the agreement between predicted and measured penetration depths in these tests. The excellent agreement strongly enhances the validity of the analysis leading to the penetration depths at impact velocities beyond the cavitation threshold velocity.

3.5 The Penetration of Short Projectiles We mentioned above that the penetration process of short projectiles is different than that of long rods, because of the enhanced effect of the impact face on the penetration depths of short projectiles. In fact, it was shown that the impact face dominates the penetration characteristics of CRH3 ogive-nosed rods for about six rod diameters. The effect of the impact face is even more important for short projectiles impacting at velocities in the ordnance range, since they penetrate to depths of only a few projectile diameters. Thus, these projectiles are not expected to experience a constant deceleration during penetration. The present section deals with this issue in order

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3 Rigid Penetrators

to highlight the influence of the entrance phase and we shall present a numericallybased model which accounts for the influence of this phase. It will be shown that the predictions from this model are in good agreement with experimental data for short projectiles, demonstrating the usefulness of such numerically-based models.

3.5.1 The Influence of the Entrance Phase Rosenberg and Dekel (2009b) followed the influence of the entrance phase in a quantitative manner, through simulations for the penetration process of short rigid projectiles with different nose shapes. Simulation results for the penetration histories of L/D = 3, D = 6 mm ogive-nosed steel projectile, impacting a 0.4 GPa aluminum target, are shown in Fig. 3.28a. Impact velocities of V 0 = 0.5, 1.0 and 1.5 km/s resulted in penetration depths of: 17, 41.5, and 78.5 mm, respectively. A penetration depth of 17 mm is less than three projectile diameters, and is well within the dominance range of the entrance phase. Even for the 1.0 km/s impact, the penetration depth of Fig. 3.28 a Penetration histories for the short projectiles, b their deceleration histories

3.5 The Penetration of Short Projectiles

89

41.5 mm is very close to the six projectile diameters, which is the dominance range of the entrance phase. Thus, we do not expect a constant deceleration for these impacts, as is clearly manifested by the deceleration histories in Fig. 3.28b. Figure 3.28b shows that a constant deceleration was obtained only for the highest impact velocity (1.5 km/s), starting at about 25 µs after impact. This time corresponds to a penetration depth of 36 mm in this simulation, which is exactly equal to six projectile diameters, the dominance range of the entrance phase. A close examination of the early part in the deceleration history for the 1.5 km/s impact shows that the entrance phase consists of two stages. During the first stage, up to about t = 7 µs, the projectile experiences a fast change in its deceleration, due to the embedment process of its nose in the target. The second stage which lasts until t = 25 µs is due to the entrance phase effect, as discussed above. The simulations of Rosenberg and Dekel (2009b) showed that the entrance phase takes less time for blunt projectile (either spherical or flat nosed), because their noses are shorter and their corresponding embedment times are smaller. In fact, these simulations showed that the dominance range of the entrance phase for the spherical nosed projectile extends to about three projectile diameters. However, its influence range extends to much larger depths, as is also the case with the ogive nosed projectiles. One may consider the influence range of the entrance phase as the range beyond which its dominance range plays a negligible part in the process. These issues will be further highlighted in the next sections. In order to have a quantitative assessment for the influence of the entrance phase, we define an effective (average) resisting stress (Reff ) and an effective deceleration (aeff ), according to (3.8a, 3.8b), through the following relations: Reff =

ρ p L eff V02 2P

(3.21a)

V02 2P

(3.21b)

aeff =

for a given projectile impacting at V 0 . Using the corresponding penetration depths in the simulations presented above and the effective length of the projectile L eff = 14.4 mm, results in the following values for the effective resisting stresses: Reff = 0.84, 1.37 and 1.63 GPa for the impacts at V 0 = 0.5, 1.0 and 1.5 km/s, respectively. These values are substantially smaller than the value of Rt = 1.87 GPa which is the resisting stress exerted by a 0.4 GPa aluminum target on an ogive-nosed projectile at deep penetration. The increasing trend in Reff demonstrates the diminishing weight of the entrance phase with increasing impact velocities. Note that even for the highest impact velocity in this case, the average resistance is lower than the asymptotic value by 13%, because the penetration depth at this velocity is only twice the dominance range of the entrance phase. In order to further demonstrate the strong effect of the entrance phase, Rosenberg and Dekel (2009b) performed several simulations with aluminum targets which have a cylindrical hole around their symmetry axis, as shown in Fig. 3.29. These holes

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3 Rigid Penetrators

Fig. 3.29 The target with a central hole

were 50 mm deep (about eight times the projectile diameter) and their diameter was equal to the projectile’s diameter (6 mm). The basic idea behind these simulations was to reduce the effect of the free impact surface by moving the impact point deep into the target. Thus, upon impact the projectile should experience a resisting stress (deceleration) which is much closer to that at deep penetrations. Deceleration histories of the ogive-nosed projectile in both targets, with and without the hole, are shown in Fig. 3.30 for two impact velocities, V 0 = 1.0 and 1.5 km/s. One can clearly see that the effect of the entrance phase diminished by the addition of the hole, and the deceleration histories reach their asymptotic values shortly after impact. As expected, the penetration depths into the targets with the holes were lower than the corresponding depths in the flat-faced targets. The differences diminished with increasing impact velocities, but even for the highest velocity (1.5 km/s) the depth of penetration into a regular target is higher by 5% than the corresponding depth in the target with the hole. The same trend was found in the simulations for the spherical-nosed projectile. The corresponding differences between flat targets

Fig. 3.30 Deceleration histories for the ogive-nosed projectile in a regular target and in a target with a hole

3.5 The Penetration of Short Projectiles

91

and those with a hole were much lower, due to the smaller dominance range of the entrance phase for these nose shapes. A quantitative estimate for the strong effect of the entrance phase can be gained by analyzing the experimental results of Brooks and Erickson (1971). They shot ogivenosed tungsten-carbide projectiles at 4340 steel targets with a Brinell hardness of 294 BHN. These L/D = 4.92 projectiles with a diameter of 8.86 mm, stayed rigid even at an impact velocity of 1.34 km/s. This set of experiments is particularly interesting since the high density of the projectiles (14.72 g/cm3 ) resulted in relatively deep penetrations, well beyond the dominance range of the entrance phase. In order to asses the effect of the entrance phase we calculate the effective deceleration of the projectile for each experiment through (3.21b). With increasing impact velocities these decelerations are expected to increase towards their asymptotic value at deep penetration. Table 3.5 lists the relevant data from the experiments of Brooks and Erickson (1971) together with the effective decelerations (aeff ), in units of 10−3 mm/µs2 , as calculated by (3.21b). Clearly, the effective decelerations increase with impact velocity but they do not seem to reach an asymptotic value as the entrance phase is affecting the process in all of these experiments. This is the expected result since even the highest impact velocity resulted in a penetration depth of P = 12.6D, which is only twice the dominance range of the entrance phase. In order to have an estimate for the asymptotic value for the deceleration of this projectile at deep penetrations, consider the target’s flow stress of about Y t = 1.0 GPa in these experiments. Inserting this value in (3.9) one obtains a value of Rt = 4.68 GPa for the resisting stress, which this target should exert on an ogive nosed projectile at deep penetration. From the dimensions of the projectile and its mass (30.9 g) one gets L eff = 34 mm, for its effective length. Inserting the values for Rt , ρ p and L eff in (3.7b), results in a value of a = 9.35 · 10−3 mm/µs2 for the asymptotic value of the deceleration for this projectile/target combination. This value is higher by 16% than the maximal value for the effective deceleration in these experiments, as given in Table 3.5. These examples demonstrate the significant effect of the entrance phase on the penetration process of short projectiles. It is clear that the entrance phase influences their penetration depths much beyond its dominance range. On the other hand, there are situations where the penetration depths are much higher than 6D even for short projectiles, and the penetration formula (3.8a) should predict the correct penetration depths for these impacts. Obviously, such cases involve targets with low values of Rt , which can be penetrated to large depths, well beyond the influence range of the entrance phase. Table 3.5 The data from Brooks and Erickson (1971) and the effective decelerations (in 10−3 mm/µs2 ) V 0 (m/s)

434

568

691

805

920

1,027

1,138

P (mm)

24.1

34.3

44

52

64

75.5

87

1,343 112

P/D

2.72

3.87

5.0

5.87

7.22

8.52

9.82

12.64

aeff

3.91

4.70

5.43

6.23

6.61

7.00

7.44

8.05

92

3 Rigid Penetrators

In order to demonstrate this issue consider the impact of a 0.5 AP projectile at a large Plexiglas block at an impact velocity of V 0 = 0.9 km/s. Its penetration depth, as measured in our lab was P = 190 mm which amounts to P = 17D for the D = 11 mm hard steel core of this projectile. This penetration depth is well beyond the dominance range of the entrance phase (P = 6D), and the model presented above should account for it. The values of Young’s modulus for Plexiglas is about 5.5 GPa and its dynamic compressive strength is near Y t = 0.35 GPa. Inserting these values in (3.9) and noting that the hard steel core of the 0.5 projectile has an ogive shape with φ = 1.15 in (3.9), we obtain a value of Rt = 0.66 GPa for the resisting stress of Plexiglas. Inserting this value together with L eff = 38.2 mm for the hard steel core in (3.8a), results in a predicted penetration of P = 185 mm, which is very close to the experimental result of P = 190 mm. The excellent agreement between model and experiment for this short projectile is the result of its deep penetration in the target, as discussed here. Note that the value of V cav for this projectile/target pair as calculated by (3.11) is V cav = 1.55 km/s, and no cavitation was expected for this shot (at V 0 = 0.9 km/s) as was, indeed, the case in the experiment. Another set of data which was affected by the entrance phase is given by Forrestal et al. (1991). They impacted spherically nosed steel rods with L/D = 5, 10 and 15, at 6061-T651 targets with velocities in the range of V 0 = 0.25–1.0 km/s. The normalized penetration depths for the three types of rods are shown in Fig. 3.31, and it is clear that the results for the L/D = 10 and 15 rods fall around a single curve. In contrast, the L/D = 5 rods resulted in considerably higher values of P/L. As the maximal penetration depth of the shorter rods was only P = 9.5D, the entrance phase is expected to have influenced all the penetrations of the shorter rods. On the other hand, the maximal penetrations for the L/D = 10 and 15 rods were P = 18D and P = 23D, respectively, which are well beyond the dominance range of the entrance phase. Fig. 3.31 Normalized penetrations for the spherical-nosed steel rods with different aspect ratios

3.5 The Penetration of Short Projectiles

93

Figure 3.31 also includes the predicted penetration curve for the spherical nosed rod impacting the 6061-T651 aluminum targets as given by: P/L eff = 1.76 V 20 , which was calculated for Fig. 3.10b. As is clearly seen in Fig. 3.31 this curve accounts for the data of the L/D = 10 and 15 rods, while the data for the L/D = 5 rods lies significantly above this curve. This is a clear demonstration for the effect of the entrance phase and in the next section we compare these results with the predictions of the numerically-based model for the entrance phase effect.

3.5.2 A Numerically Based Model for the Entrance Phase Effect The influence of the entrance phase complicates the analysis to such a degree that it is very difficult to construct an analytical model for the penetration process of short projectiles. Thus, a different approach is suggested here which is based on numerical simulations. The recent improvements with hydrocodes enable one to construct numerically-based models for terminal ballistics. Obviously, the validity of these models should be checked by comparing their predictions with experimental results, as will be demonstrated in this case. The effect of the entrance phase will be quantified through the effective resisting stress (Reff ) as defined by (3.21a). We first rewrite this equation for the normalized penetration depth of a rigid projectile: ρ p V02 P = with: Reff ≤ Rt L eff 2Reff

(3.22a)

noting that Reff should increase with impact velocity towards its asymptotic value (Rt ) at deep penetrations. Rewriting (3.22a) we obtain: ρ p V02 Rt P = L eff 2Rt Reff

(3.22b)

which leads to the following relation: ρ p V02 L eff Reff P ·q = ≡ I where: q ≡ · D 2Rt D Rt

(3.23)

Up to this point we have only defined two new parameters, q and I, through the other “old” parameters. The parameter q as defined by the ratio Reff /Rt , expresses the effect of the entrance phase. Its value is always smaller than 1.0 and it is expected to approach this value at deep penetrations. The second parameter (I) is a measure for the penetration power of the particular projectile into the given target. This parameter is equal to the normalized penetration (P/D) of the projectile without the entrance phase effect, according to the constant deceleration model presented above. All the

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3 Rigid Penetrators

terms which determine the value of I are well defined, for a given projectile/target pair, and none of them depend on the details of the entrance phase. In fact, several workers have used similar parameters in order to account for their penetration data. Haldar and Hamieh (1984) presented penetration data for concrete targets using empirical relations between P/D and a parameter which they called the Impact Factor. This factor was defined through the mass (M), the diameter (D) of the projectile, and its impact velocity, according to: Impact Factor = N

M V02 D3 fc

(3.24)

where f c is the unconfined compressive strength of the concrete and N is a nose shape factor. It is easy to see that the impact factor as defined by (3.24) is similar to the parameter I as defined by (3.23), since the term M/D3 is equal to the term ρ p L eff /D. The two definitions differ by the expression for the target strength. Haldar and Hamieh (1984) use the concrete’s compressive strength (f c ) while the resistance to penetration (Rt ) is used in our definition. In addition, the target strength and the nose shape parameter appear separately in the definition of the impact factor through (3.24), while by our definition both of them are lumped together in Rt . Chen and Li (2002) also used an impact factor to account for the penetration depths of rigid rods in semi-infinite targets. Their impact factor includes a target strength term, which is derived from the cavity expansion analysis, and a nose-shape parameter in a similar way to that proposed by Haldar and Hamieh (1984). Thus, all the relevant parameters which influence the penetration depth of a given projectile are lumped into these closely related impact factors. As far as our approach is concerned, the impact factor is not a free parameter since it is the expected penetration efficiency (P/D) of the rigid projectile, according to the constant deceleration model. The basic idea behind our approach is to account for the lower resisting stresses at the entrance phase through a universal relation between q and I. This will be achieved by a numerically-based model as described below. The model is based on analyzing simulation results for the penetration depths of L/D = 3 projectiles impacting a given target at several velocities. With these penetration depths, the corresponding values for Reff and q can be calculated in order to determine the influence range of the entrance phase. As a next step, general relations between P/D and the impact factor (I) will be constructed for both ogive and spherical-nosed projectiles. The main assumption behind the analysis is that these relations can be treated as universal for all the projectiles with similar nose shapes. This assumption will be validated by the close agreement between the model’s predictions and experimental data. The simulations were performed for D = 6 mm rigid steel projectiles with CRH3 ogive and spherical noses, having effective lengths of L eff = 14.5 and 17.1 mm, respectively. These short projectiles impacted an aluminum target with strength of 0.4 GPa at velocities in the range of V 0 = 0.1–2.0 km/s for the ogive-nosed projectile, and V 0 = 0.1–1.25 km/s for the spherical-nosed projectile. These velocity ranges were chosen in order to avoid the cavitation phenomenon which may complicate

3.5 The Penetration of Short Projectiles

95

Fig. 3.32 Simulation results for the parameter q as a function of V 0 for both nose shapes

the analysis. Each simulation resulted in the corresponding value of P from which Reff values were determined through the relations given above. With the known values of Rt for both nose shapes the curves for q = Reff /Rt versus impact velocity were constructed, as shown in Fig. 3.32. As expected, the values for q increase asymptotically from q = 0 at V 0 = 0 towards a value of q = 1.0 for high impact velocities, as Reff approaches its asymptotic value of Rt . These simulation results are presented in Fig. 3.33 in terms of P/D as a function of qP/D = I and, as is clearly seen, the two curves are very close to each other. This important result means that all the sharp-nosed projectiles, conical and other ogive shapes should follow the curve derived here for the CRH3 ogive-nosed projectile. The two curves shown in Fig. 3.33 will be treated as universal relations, which account for the entrance phase effect for sharp and spherically-nosed projectiles. In order to verify that these curves are not dependent on the density of the target, an additional set of simulations was performed, for the ogive-nosed projectile impacting a 0.4 GPa steel target. The results of these simulations are shown in Fig. 3.34 together with those for the 0.4 GPa aluminum target, and it is clear that both sets of simulations results fall on a single curve. We also added the straight line P/D = I which represents Fig. 3.33 Simulation results for the relation between P/D and I for the two nose shapes

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Fig. 3.34 Simulation results for ogive-nosed projectiles impacting 0.4 GPa steel and aluminum targets

the penetration curve without the entrance phase effect. The distance between this line and the simulation points is a measure for the influence of the entrance phase. The two curves in Fig. 3.33 rise very sharply for low I values (up to about I = 1.0) and then increase more gradually for the higher values of I. These trends will be represented by different relations between P/D and I for the two regimes. For large values of I (I > 1.0) the following fits were chosen to represent the simulation results: P = 1.32 + 1.2I − 0.008I 2 D P = 0.65 + 1.3I − 0.025I 2 D

for sharp-nosed projectiles

(3.25a)

for spherical-nosed projectiles

(3.25b)

It is interesting to note that the curves corresponding to these expressions meet the diagonal line (P/D = I) at I = 30 and I = 14 for the sharp and spherical-nosed projectiles, respectively. These are the corresponding influence ranges of the entrance phase for these projectiles. The influence range for the sharp-nosed projectile (P/D = 30) is about twice as large as that of the spherical nosed projectile (P/D = 14). The same factor was obtained for the dominance ranges of the entrance phase, P = 6D and P = 3D, for the ogive and spherical-nosed projectiles, respectively. The values, of I = 30 and I = 14, define the applicability range of the numerically based model which accounts for the entrance phase effect. For higher values of the impact factor (I), one should use the simple penetration relation of (3.8a, 3.8b). In order to have a better estimate for the influence of the entrance phase, consider the two intermediate cases where I = 20 for an ogive-nosed projectile, and I = 10 for the spherical-nosed projectile. Using (3.25) we find that P/D = 22.1 and P/D = 11.15, for the ogive and spherical-nosed projectiles, respectively. These values are about 10% higher than their corresponding impact factors, demonstrating the significance of the entrance phase even for relatively high impact factors. Thus, experimental penetration depths

3.5 The Penetration of Short Projectiles

97

Fig. 3.35 Simulation results for the low values of the impact factor (I)

are expected to be higher than those calculated through (3.8a, 3.8b) by 10%, for ogive and spherical-nosed projectiles having impact factors of I = 20 and I = 10, respectively. The P/D = f (I) curves for the low values of I (I < 1.0), are shown in Fig. 3.35. The fact that the curve for the ogive-nosed projectiles is much higher than that of the spherical- nosed ones is due to the larger dominance range of the entrance phase for the ogive-nosed projectiles, as discussed above. The following expressions were chosen to represent the relations between P/D and I, for the low values of I (I < 1.0): P = 2.5I 0.5 D P = 1.9I 2/3 D

for sharp-nosed projectiles

(3.26a)

for spherical-nosed projectiles

(3.26b)

Inserting the definition of I from (3.23) into these expressions results in the following relations between P/D and V 0 , for the low range of impact factors (I ≤ 1.0):   ρ p L eff 0.5 P = 1.77 V0 for sharp-nosed projectiles D Rt D   ρ p L eff 2/3 4/3 P = 1.2 V0 for spherical-nosed projectiles D Rt D

(3.27a) (3.27b)

which predict a linear dependence of P/D on impact velocity for the sharp-nosed projectile, and a V 4/3 0 dependence for the spherical-nosed projectile. In order to demonstrate the validity of the equations derived above, we compare their predictions with experimental data for several projectile/target combinations. The data of Brooks and Erickson (1971) as given by Table 3.5 is a good case for such a validity check because both the projectile and the target are different from those used in the simulations presented above. These were ogive-nosed tungsten-carbide

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3 Rigid Penetrators

Table 3.6 Comparing the model’s predictions with the data of Brooks and Erickson (1971) V 0 (km/s)

0.434

0.568

0.691

0.805

0.920

1.027

1.138

I

1.14

1.95

2.88

3.91

5.11

6.37

7.82

1.343 10.89

P(calc) in mm

23.7

32.2

41.7

52.2

64.2

76.5

90.5

119

P(exp) in mm

24.1

34.3

44

52

64

75.5

87

112

projectiles with ρ p = 14.72 g/cm3 , D = 8.86 mm and L eff /D = 3.84. The targets were 4340 steel with hardness of 294 Brinell which corresponds to Rt = 4.68 GPa. Inserting these values of Rt , ρ p and L eff /D in (3.23) we get for this projectile/target pair: I = 6.04 · V 20 , where the velocity is given in km/s. The values of I for each experiment in this set are listed in Table 3.6. Since all the values of I are higher than 1.0 they were inserted in (3.25a) in order to obtain the corresponding values of P/D. The predicted penetration depths are compared with the experimental data in Table 3.6, and the good agreement between the two sets enhances the validity of the numerically-based model. Consider now the experimental results from Forrestal et al. (1991) for the L/D = 5 spherical-nosed steel rods impacting aluminum targets, which were presented in Fig. 3.31. Due to the entrance phase effect these short rods resulted in higher values for their normalized penetrations (P/L), as compared with the L/D = 10 and 15 rods. We compare now the predictions from the numerically-based model with the experimental results for the shorter rods. Using (3.23) with the following values: ρ p = 8.0 g/cm3 , L eff /D = 5.34, and Rt = 2.27 GPa for the 6061-T651 aluminum target (E = 69 GPa, Y t = 0.42 GPa), we find that for this rod/target combination: I = 9.39 · V 20 . Using (3.25b), for spherical-nosed projectiles, results in the predicted penetration curve shown in Fig. 3.36, together with the experimental results. The excellent agreeFig. 3.36 Comparing the model’s predictions (curve) with the data (squares) for the L/D = 5 spherical nosed steel rods impacting 6061-T651 aluminum targets

3.5 The Penetration of Short Projectiles

99

ment between the model’s predictions and the experimental data strongly enhances the validity of the numerically-based model. The penetration process of a long rod can be influenced by the entrance phase even at moderate impact velocities when the target has a high strength resulting in relatively low impact factors. This is the case with the experiments of Forrestal et al. (1992) for ogive nosed steel rods impacting 7075-T651 aluminum targets at velocities up to about 1.0 km/s. The dynamic flow stress of this aluminum alloy is about Y t = 0.7 GPa, and with E = 73 GPa for its Young’s modulus a value of Rt = 2.77 GPa is obtained from (3.9) for this rod/target pair. Together with ρ p = 8.02 g/cm3 , L eff = 77.8 mm, and D = 7.11 mm, relation (3.23) results in I = 15.84 V 20 for the impact factor in this case. We have already seen that an impact factor of I = 20 for ogive-nosed rods results in a correction of 10% in their penetration depth due to the entrance phase effect. Thus, these relatively low velocity experiments are expected to be significantly influenced by the entrance phase. The predicted penetration depths for these experiments are given in Table 3.7, together with the experimental results from Forrestal et al. (1992). We also added the calculated penetration depths from (3.8a, 3.8b), ignoring the entrance phase effect. This equation results in very low penetration depths as compared with the experimental data, demonstrating the strong effect of the entrance phase. The effect is especially strong in this case where these rods impacted a high strength target at relatively low velocities. On the other hand, the excellent agreement between the measured penetration depths and those predicted by (3.25a) demonstrates the validity of the numerically-based model for the entrance phase effect. Finally, we compare the predictions from the numerically-based model with the penetration data for 0.3” APM2 projectiles impacting 6061-T6 aluminum targets, from Chocron et al. (1999). They used a modified projectile which consisted mainly of the hard steel core, with D = 6.17 mm and L eff = 22.4 mm. For these aluminum targets (E = 69 GPa and Y t = 0.42 GPa) and for the ogive-nosed projectiles we have: Rt = 1.87 GPa from (3.9). Inserting these values in (3.23) results in: I = 7.6 V 20 for this projectile/target combination. With (3.25a) we calculate the penetration depths of these steel cores as shown in Fig. 3.37, and find that the agreement between model and data is very good. Table 3.7 Comparing calculated and measured penetrations (in mm) for the 7075-T651 Al targets V 0 (km/s)

0.372

0.695

0.978

1.067

P (exp)

26

70

127

147

P (from 3.25)

27.8

71.3

125.5

144.5

P (from 3.8)

15.6

54.4

107.7

128

100

3 Rigid Penetrators

Fig. 3.37 The model’s prediction (curve) and the data (circles), for the hard cores of 0.3 APM2 projectiles impacting 6061-T6 targets

3.6 The Impact of Spheres The entrance phase regime introduces an inherent complexity to the analysis of the penetration data for compact rigid projectiles such as spheres and L/D = 1 cylinders, since their penetration depths are well within the dominance range of the entrance phase. Another complicating issue concerns the role of target density. In the previous sections we have seen that, as far as rigid penetrators are concerned, the relevant target properties which determine its resistance to penetration are its strength and Young’s modulus. The density of the target does not play a role because the initial shock waves, which are generated upon impact, have negligible effect on the penetration process of rigid projectiles. Obviously, this assertion is much less accurate for very short projectiles having a blunt nose shape, and the transient shock phase can have a non-negligible effect on their penetration process. In fact, the initial penetration velocity of a thin plate impacting a given target depends on the ratio of their acoustic impedances (the products of their density and sound velocity). Since sound velocities in most metals have similar values, we expect the density ratio to play an important role on the early deceleration of very short projectiles. Indeed, experimental data for spherical projectiles impacting different metallic targets show a significant dependence of their penetration depths on their density ratio (ρ p /ρ t ). These complicating factors are the main reason for the lack of analytical models for these cases. Instead, the experimental data has been presented through empirical relations for the dependence of the normalized penetration depth (P/D) on impact velocity and on the density ratio. Obviously, such relations can be used only as an interpolation tool. A comprehensive review of such studies for various materials, at a large range of impact velocities, is given in Hermann and Jones (1961). Most of the data in this review is concerned with the hypervelocity range where the spheres are heavily deforming. However, several sets of data include the impact of rigid spheres,

3.6 The Impact of Spheres

101

which are presented empirically as P/D = f (ρ p /ρ t , Y t ) · V n0 , where the values for the exponent are around n = 1.5.

3.6.1 Rigid Sphere Impact The complexity of the interaction between an impacting sphere and a semi-infinite target can be overcome by using numerical simulations, in order to estimate the dependence of the normalized penetration depths (P/D) on the relevant parameters. We performed several sets of simulations, for various rigid spheres impacting aluminum and steel targets of different strengths. The results of these simulations showed that, indeed, the density of the target plays an important role in determining the penetration depths of these spheres. However, it turned out that the dependence of the penetration depths on the density ratio is very complex. Even the exponent n, in the expression for P/D as a function of V 0 , turned out to depend on the density ratio. Thus, it was practically impossible to derive a general relation to account for the penetration depths of rigid spheres, from these simulations. Still, for a limited set of experiments one can have some useful information, as we show next. Consider the simulation results for a rigid steel sphere impacting an aluminum target with strength of 0.4 GPa, as shown in Fig. 3.38. The maximal impact velocity in this set of simulations was V 0 = 1.25 km/s, in order to limit the discussion to the no-cavitation range (V 0 < V cav ). The normalized penetration depths (P/D) seem to follow a relation of the sort: P/D = kV n0 . A value of n = 4/3 for the exponent was chosen for the sake of consistency with (3.27b), which was derived from the numerically-based model for sphericalnosed projectiles, at low values of the impact factor (I). As shown in Fig. 3.38, the simulation results for steel spheres impacting the 0.4 GPa aluminum target follow an empirical relation of the form: Fig. 3.38 Simulation results for rigid steel sphere impacting 0.4 GPa aluminum target

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3 Rigid Penetrators

P 4/3 = k1 V0 D

(3.28)

where k 1 = 1.85 for this sphere/target combination, and V 0 is given in km/s. The value of the multiplying constant (k 1 ) depends on the strength of the target and on the density ratio (ρ p /ρ t ), as discussed above. At this stage we assume that the functional dependence of P/D on target strength should be similar to that given by (3.27b), as discussed above. However, we replace the term Rt in (3.27b) by Y t , since the penetration resistance at very low penetration depths is expected to be governed by Y t rather than by the resistance at deep penetration (Rt ). Thus, we write:  2 2/3 V P = k2 0 D Yt

(3.29)

with a new constant (k 2 ) which is assumed to depend on the densities of the sphere and the target. Inserting the value of Y t = 0.4 GPa for the aluminum target in these simulations, we find that k 2 = 1.0 for the impact of rigid steel spheres at the 0.4 GPa aluminum target. In order to check the validity of this result consider the experiments of Weimann (1974) who shot hard steel spheres at semi-infinite 2024-T351 aluminum targets. The diameter of these spheres ranged between 3 and 20 mm and their impact velocities reached 1.5 km/s. The flow stress of this aluminum alloy, at high strain rates, varies between 0.55 and 0.65 GPa and we take an average value of Y t = 0.6 GPa. Inserting this value in (3.29) results in the following expression for the normalized penetration in this case: P 4/3 = 1.4V0 D

(3.30)

where V 0 is given in units of km/s. The good agreement between this expression and the normalized penetrations of the rigid steel spheres of Weimann (1974) is shown in Fig. 3.39. Another important result from Weimann’s experiments concerns the crater’s diameter as measured at the impact face, for rigid steel spheres with D0 = 12.7 mm impacting the 2024-T3 aluminum targets. The crater diameter (Dc ) should be equal to that of the sphere for impact velocities below the cavitation threshold (V cav ). For higher velocities the cavitation phenomenon should result in an increase of the crater diameter with impact velocity. Using (3.9) with Y t = 0.6 GPa for the flow stress of the 2024 aluminum, results in a value of Rt = 3.05 GPa for the resisting stress which this target exerts on rigid spheres. From (3.11) one gets an estimated value of V cav = 1.48 km/s for a spherically nosed projectile (b = 0.5) impacting these aluminum targets. Thus, a plot of the experimental results for the crater diameters as a function of impact velocity is expected to show a marked increase near this impact velocity. The experimental results of Weimann (1974) for the normalized crater diameters

3.6 The Impact of Spheres

103

Fig. 3.39 Comparing the model’s predictions with the data for hard steel spheres impacting 2024-T3 aluminum targets

(Dc /D0 ) at their impact face are shown in Fig. 3.40. As expected, the crater diameters are relatively constant for impact velocities in the range of V 0 = 0.5–1.5 km/s and they increase markedly for higher impact velocities. For impact velocities below 0.5 km/s these spheres did not penetrate more than half of their diameter, and crater diameters were less than the sphere’s. The expected plateau at Dc /D0 = 1.0 for impact velocities below V cav is, in fact, increasing slightly, and the increase in Dc /D0 for higher velocities is not an abrupt one as predicted by the model. These features are probably due to effects which are not accounted for by the model. Such an effect is the temperature increase of the target as a result of high velocity impacts. An impact at 1.5 km/s of a steel sphere on an aluminum target creates a shock wave which induces a temperature rise of about 200 °C in the target. This temperature change can lower the flow stress of the target material around the crater, resulting in a decrease in the value of its Rt value and, Fig. 3.40 The normalized diameters of the craters at the impact face for the 12.7 mm rigid spheres

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3 Rigid Penetrators

consequently, a lower value for V cav . Still, the overall shape of the Dc /D0 curve in this figure is in general agreement with the expected trends, according to the model presented in this chapter.

3.6.2 The Impact of Non-rigid Spheres A large portion of the published data for the penetrations of spheres is actually given for deforming spheres. In order to have an estimate for the difference between the penetration capabilities of rigid and deforming spheres, consider the data of Weimann (1974), who used 12.7 mm steel spheres which were hardened to different levels. These spheres impacted 2024-T351 aluminum targets at velocities up to 1.5 km/s. The results for the fully hardened and for the half-hardened spheres are shown in Fig. 3.41. Note the lower penetrations of the half hardened (soft) spheres, which flattened to various degrees with increasing impact velocity. As expected, the deformation of these spheres has a strong effect on their penetration capability. We also added the experimental results from the survey of Hermann and Jones (1961) for steel spheres impacting 2024-T3 aluminum targets. These experimental results were used by Goodier (1965) for his penetration model of rigid spheres, as described above. Obviously, the penetration depths of these spheres are significantly lower than those of the rigid spheres in the experiments of Weimann (1974). In fact, they are even lower than the data for the half hard spheres in Weimann’s experiments. The methodical study of Weimann (1974) resulted in another important observation which concerns the volumes of the craters formed by rigid and deforming spheres. The experiments with rigid spheres of diameters in the range of D = 3– 20 mm, resulted in the following relation between crater volume (Vol) and impact velocity: Fig. 3.41 The effect of sphere hardness on its penetration efficiency

3.6 The Impact of Spheres

105

Vol (mm3 ) = 1.14D 3 V02

(3.31)

where D is in mm and V 0 in km/s. This relation shows that the crater volume is proportional to the initial kinetic energy of the projectile, which is the expected result since no energy was dissipated in deforming or breaking the spheres. Weimann (1974) also measured the crater volumes in experiments with deforming steel spheres having different hardness. The crater diameters in these experiments were larger than those with the rigid spheres and their penetration depths were smaller, as expected. However, the data for all the different hardness spheres, including the rigid ones, followed the same linear relation between the volume of the crater (Vol.) and the initial kinetic energy of the sphere (E k ), according to: Vol. (mm3 ) = 0.554E k (joules)

(3.32)

which means that the energy spent on plastically deforming the softer spheres is too small to have an appreciable influence on their penetration process. We can argue that this result should be generalized for other cases concerning the penetration of deforming projectiles. As a good approximation one can assume that such deformation energies do not contribute much to the energy balance in these events. The high velocity impact of very hard spheres results in their shattering, rather than their deformation, due to the low ductility of very strong materials. One can expect that a hard sphere impacting a given target will shatter at a well defined velocity, with some loss of its penetration capability. Such reductions in penetration depth with impact velocity have been reviewed by Hermann and Jones (1961), for tungstencarbide spheres impacting various targets. The data for these spheres impacting two types of aluminum targets is shown in Fig. 3.42. Note that the strength of the 2014-T6 alloy (0.5 GPa) is higher by an order of magnitude than the strength of the 1100 F aluminum, resulting in the significant difference between their corresponding Fig. 3.42 The penetration depths of tungsten-carbide spheres in two aluminum targets

106

3 Rigid Penetrators

penetration depths. On the other hand, the threshold impact velocities for sphere shattering are practically the same for the two types of aluminum targets, and one may conclude that these thresholds depend on target density rather than on its strength. In fact, the data for tungsten-carbide spheres impacting several targets, in Hermann and Jones (1961), show a decreasing tendency of the shattering threshold velocity (V shatt ) with increasing target density. The general shape of the penetration curves for projectiles which shatter at a certain impact velocity (V shatt ) is similar to that shown in Fig. 3.42. These curves exhibit a sharp decrease in penetration for impact velocities just above V shatt , which increases again with impact velocity after reaching a certain minimum. The range of velocities, from the onset of penetration decline to that where the penetration is as deep as that at the threshold, is known by the name of “shatter range” in the military literature. The relation between the threshold velocity for shattering (V shatt ) and the density of the target was further investigated by Senf (1974) for hardened steel spheres. He found that these thresholds are related to the target’s density by a simple equation: 2 = const ρt Vshatt

(3.33)

and his experimental results are shown in Fig. 3.43 for a large number of different targets, together with his fitting curve –ρ t V 2sh = 7.8 GPa. One can use this value in order to define a threshold shattering pressure through a Bernoulli type of relation: Pshatt = ρ t V 2sh /2 = 3.9 GPa. This value is higher by a factor of 2 than the compressive strength of the hard steel spheres which were used in this study (Y t = 1.9 GPa). One may argue that the shattering of these spheres should be related to their dynamic tensile strengths (spall), rather than their compressive strength. A spall strength of 3.9 GPa is a reasonable value for steel with a compressive strength of 1.9 GPa. The Fig. 3.43 The relation between shattering velocity and target density for hardened steel spheres

3.6 The Impact of Spheres

107

shattering of high strength projectiles deserves further analysis due to its importance for both armor and anti-armor designs.

3.7 The Effect of Friction One of the less understood issues in the field of terminal ballistics concerns the friction between the projectile and target surfaces, at the high sliding velocities which characterize the penetration process. It is usually assumed that friction effects are small, even negligible, which is the main reason for the scarcity of basic studies concerning this issue. Friction between the projectile and the target is manifested by the nose coloring of the recovered projectile which has often a snowcap characteristic. The area of this snow-cap can be reduced by lubricating the projectile but its appearance is an indication for surface melting during penetration. Metallurgical examinations of the target material near the crater surface often show that a thin layer has reached the melting point, which is also an indication for frictional forces. One of the more interesting studies in this field is that of Kraft (1955) who measured the friction forces on a spinning conical-nosed projectile. He used a torsion-type Kolsky bar system in order to measure the torsional adhesion of the projectile during penetration. The system consisted of a long metallic bar with several strain gauges glued to its surface, at different locations and different orientations. Longitudinal strain gauges were glued near the impact point, while gauges oriented at 45° to the bar’s axis were glued farther along the bar. With this arrangement the longitudinal and shear waves in the bar could be separated easily and the effect of the torsional forces could be determined. The conical nosed hard steel projectiles were fired from a spirally rifled 0.3 gun. The strain gauge records which followed the torsion effects showed a clear difference between spinning and non-spinning projectiles. The measured torque was compared with predictions from a theoretical model, which was based on the assumption that friction arises from molten surfaces. It turned out that the energy consumed by the torsional friction forces amounts to about 3% of the total energy consumption of the rigid projectile. These results agree with other experiments where the projectiles had different surface finish and lubrications, resulting in friction-induced energy losses of three to four percent. A closely related issue concerns the influence of friction on the projectile’s cylindrical body (the shank) during its penetration process. In order to have an estimate for the effect of this frictional force, Rosenberg and Forrestal (1988) performed a series of experiments with long conical-nosed steel rods having different shank designs (but the same mass) as shown in Fig. 3.44. These rods were shot at 25.4 mm thick 6061-T651 plates, and their residual velocities were determined. Clearly, the reduced-shank design avoids frictional forces on the shank, so that any difference between their residual velocities should be attributed to friction acting on the rod’s shank. Residual velocities were determined for impact velocities in the range of 0.3–0.9 km/s and the results are shown in Fig. 3.45. The residual velocities of the reduced-shank rods were somewhat higher,

108

3 Rigid Penetrators

Fig. 3.44 The two rod designs for the friction study

indicating that friction on the regular rods was small but not negligible. For the same reason, the V bl value for the reduced-shank (arrowhead) rod was somewhat lower, as seen in this figure. A set of deep penetration experiments, with spherical nosed steel rods having cut-back shanks was performed by Forrestal et al. (1988). The resulting penetration depths of these rods were very close to those for the regular rods, indicating that the friction on the rod’s shank is, indeed, very small in such experiments.

Fig. 3.45 Residual velocities for the rods with different shank design

3.8 The Optimal Nose Shape

109

3.8 The Optimal Nose Shape Several workers attempted to find an optimal nose shape for rigid projectiles which should result in maximal penetration depths. They start their analysis by writing an expression for the force on the projectile’s nose, and minimize it with respect to the relevant parameter which describes the nose shape. This was done, for example, by Jones et al. (1998) who included the inertia of the target (ρ t V 2 ) as well as its strength for the resisting force. Their conclusion was that the optimal nose shape has a small blunt section at the tip of the nose, which turns into a conical shape all the way to the cylindrical shank of the projectile. These nose shapes were not tested experimentally and, to our knowledge, no simulations were performed to enhance the claim that these nose shapes are optimal. It was shown above that the target’s resisting stress is constant for impact velocities below the corresponding threshold (V cav ), and that the inertia term plays a role only for V 0 > V cav . For a constant resisting stress there is no reason to expect an optimal nose shape, and one may assume that a sharper nose enhances the penetration capability of the rigid penetrator. The only question here is which of the sharp nose shapes, the cone or the ogive, is more efficient as far as rigid penetrations are concerned. In order to answer this question we conducted a series of simulations for ogive and conical-nosed rigid steel rods impacting aluminum targets with strength of 0.4 GPa. The sharpness of the rod’s nose can be defined as the ratio of its length ( ) to its diameter (D). These ratios were varied in the range of /D = 0.25–3, where /D = 0.5 for an ogive nose is the spherical nose shape. This ratio is equal to 2.95 for a CRH9 ogive nose and it compares with the ratio of /D = 3.0 for the conical nose, which was the sharpest nose we simulated. The main output from these simulations was the constant deceleration of the rod (a), from which we derived the corresponding value of Rt , through the relation: Rt = ρ p · L eff · a. The simulations were performed with impact velocities in the range of 0.75–1.0 km/s, and Fig. 3.46 shows the resulting values for Rt as a function of the nose sharpness ( /D). We also added the point corresponding to a flat nosed rod ( /D = 0) at Rt = 2.46 GPa, from our earlier simulations. Fig. 3.46 The effect of nose sharpness on the target’s resistance to penetration

110

3 Rigid Penetrators

One can clearly see that the results for the two nose shapes practically fall on a single curve. The main conclusion from these simulations is that sharper noses experience less resistance and penetrate more, as expected, while the details of the nose shape (ogive or cone) are much less important. It is also clear that the benefit of sharper noses is decreasing with increased sharpness, and it seems that values of

/D > 3.0 will not add much to the penetration capability of rigid penetrators.

3.9 Concrete Targets Concrete has been widely used for over 200 years by civil and military engineers, in order to protect structures against impact and blast loads. Potential hazards include projectiles and bomb fragments, aircraft crashes, and flying objects due to natural disasters like tornados and meteoroids. The recent requirements for the safety of nuclear reactors also motivated the study of concrete impact by various missiles. The vast amount of data concerning the impact of different missiles on concrete structures has been summarized by empirical relations since the early years of the 20th century, as reviewed by Li et al. (2005). They categorized these empirical relations according to several interactions between the projectile and the concrete target which include penetration, spalling (material ejected from the target’s front surface), scabbing (material ejected from the back surface) and perforation. Very few analytical models have been suggested for these interactions mainly because of the complex structure of concrete materials, which includes the cement, hard aggregates of different sizes, and steel rebars for strengthening purposes. This is also the reason for the scarcity of material models which can be implemented in hydrocodes in order to follow the interaction between an impacting object and a concrete structure. The compressive strength of concrete, and concrete-like materials, is about an order of magnitude larger than their tensile strength as is the case for other brittle materials. Their compressive strength is determined by uniaxial loading of large cubes or cylinders, with typical dimensions of about 100 mm. This is the so-called unconfined compressive strength (f c ) which is the main strength parameter in the study of concrete penetration. Most of the studies on concrete penetration have been done with concretes having strengths in the range of 30–100 MPa. However, stronger concretes have recently been examined following the developments of new manufacturing techniques. In the following discussion we outline the semi-analytical approach to concrete penetration of Rosenberg and his colleagues, through their publications in the last decade. As we shall see, their understanding evolved over the years from the first work of Rosenberg and Dekel (2010a) through the work by Rosenberg and Kositski (2016) to the latest by Rosenberg et al. (2020a).

3.9 Concrete Targets

111

3.9.1 The Penetration Model Rosenberg and Dekel (2010a) analyzed the penetration depths of rigid ogive-nosed rods impacting semi-infinite concrete targets, as given in several sets of experiments. They arrived at the same conclusion as for metallic targets, namely, that the deceleration of rigid projectiles in semi-infinite concrete targets is constant for penetration depths beyond the entrance phase. This conclusion is enhanced by noting that the penetration channels in these experiments, beyond a short entrance phase, had a diameter which was equal to that of the rigid projectile. As discussed above, this observation can be considered as an evidence for the constant resisting stress assumption. In order to demonstrate this issue consider Table 3.8, which lists the relevant data from the experiments of Frew et al. (1998) for L/D = 10 ogive-nosed steel rods impacting concrete targets with unconfined compressive strength of f c = 58.4 MPa. The table lists the average decelerations (aav ) of these rods as calculated through: aav = V 20 /2P resulting in velocity-independent values, except for the lowvelocity shots with somewhat lower decelerations due to the entrance phase effect. In order to calculate the resisting stress which these concrete targets exerted on the rods, we use their effective length L eff = 282.6 mm and the average value of their inferred decelerations: a = 2.53 · 10−4 mm/µs2 . Inserting these values in the relation Rt = ρ p · L eff · a results in Rt = 0.566 GPa which is about ten times the compressive strength. According to (3.9) this value of Rt is the penetration resistance of a weak aluminum target with strength of about 100 MPa. The correlation between concrete and aluminum targets will be further discussed later on. Rosenberg and Dekel (2010a) showed that the constant deceleration result holds for all the data from Forrestal et al. (1994, 1996) and Frew et al. (1998) for rigid ogivenosed rods penetrating concrete and grout targets, with compressive strengths in the range of 13–97 MPa. Namely, the inferred decelerations in each set of these experiments do not depend on the rod’s impact velocity, rendering a constant resisting-stress which depends on the compressive strength of the concrete. Using the simple relations discussed above, Rosenberg and Dekel (2010a) derived Rt values from each set of penetration depths data, for each of these concretes. As an example, consider the experiments of Forrestal et al. (1996) for grout targets with a compression strength of Table 3.8 The data from Frew et al. (1998), and the resulting inferred decelerations

V 0 (km/s)

P (mm)

aav (10−4 mm/µs2 )

0.445

460

2.15

0.584

790

2.16

0.796

1230

2.57

0.972

1960

2.41

0.980

1950

2.46

0.992

1960

2.51

1.176

2670

2.59

1.225

2830

2.65

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3 Rigid Penetrators

f c = 21.6 GPa. The penetration depths resulted in a value of Rt = 0.37 GPa, through the inferred decelerations of these rods. The agreement between the constant resisting stress model and the data of Forrestal et al. (1996) for these grout targets is shown in Fig. 3.47. Similar agreements were demonstrated by Rosenberg and Dekel (2010a) for all the other concrete targets in the experiments of Forrestal et al. (1994, 1996) and Frew et al. (1998). The values of Rt were found to be related to the concrete’s unconfined compressive strength (f c ) through the empirical relation: Rt = 0.22 ln( f c ) − 0.285

(3.34)

where Rt is given in GPa and f c in MPa. The strength of the concretes in these works ranged between 13 and 100 MPa, and the projectiles were ogive-nosed steel rods with aspect ratios in the range of L/D = 10–15. One should note that most of the rods in these experiments had diameters in the range of 20–30 mm. In a later work, Rosenberg and Kositski (2016) showed that the resistance to penetration of a given concrete target also depends on the projectile’s diameter. This is a clear scaling issue, or a “diameter effect”, as discussed later on, but we first discuss the issue of the entrance phase in concrete targets, which was also addressed in the analysis of Rosenberg and Kositski (2016). The effect of the entrance phase in concrete penetration is manifested by the fact that the inferred decelerations for the low-velocity tests are lower than those calculated for the high-velocity tests, as seen in Table 3.8. Thus, the stresses exerted by the target at the early stages of penetration are lower than those at deep penetration. Forrestal et al. (2003) accounted for this effect by assuming that the force on the projectile increases linearly with distance, from zero at the impact face to its final value at a depth of about two projectile diameters. The choice of 2D as the extent of Fig. 3.47 Comparing model and data for penetration depths in f c = 21.6 MPa grout targets

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113

the entrance phase was motivated by empirical observations concerning the crater depths in impacted concrete targets. These craters, which are due to large concrete layers spalled-off the impact area, have a thickness of about 2D when the projectile has a CRH3 ogive nose shape, as shown schematically in Fig. 3.48a. Thus, it is reasonable to assume that the entrance phase extends to a similar depth. This assertion was substantiated through direct measurements of the projectile’s declaration by Forrestal et al. (2003), using in situ accelerometers embedded in the projectiles. The measured decelerations in two types of concrete targets with f c = 23 and 39 MPa, exhibited a linear increase with distance up to a penetration depth of about 2D. Even more importantly, the measured decelerations reached a constant level at deeper penetrations as claimed by Rosenberg and Dekel (2010a). These features are clearly seen in Fig. 3.48b, which shows the deceleration record of a D = 76 mm projectile in an f c = 39 MPa concrete target. One can see the linear increase in deceleration (marked by the dotted line in Fig. 3.48b) up to a depth of Z = 150 mm (about 2D). For different impact velocities the decelerations reached roughly the same constant level which, as expected, was higher for the stronger concrete. These deceleration records led Warren et al. (2014) to conclude that, indeed, the deceleration beyond the entrance phase is constant, independent on the projectile’s velocity. Their model for the entrance phase effect resulted in the following relation for the penetration depth under the assumption of a constant resisting stress (Rt ) at deep penetration: Fig. 3.48 a The entrance phase and the penetration channel, b deceleration record in an f c = 39 MPa concrete, from Forrestal et al. (2003)

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P=

ρ p L eff V02 +D 2Rt

(3.35)

which takes into account the entrance phase effect. Note that for typical penetration depths in concrete targets we have P  D. Thus, values for the penetration depths as calculated by (3.35), or by the procedure outlined in Rosenberg and Dekel (2010a) ignoring the entrance phase, are very close to each other. The constant penetration resistance of a concrete target beyond the entrance phase, in the analysis of Rosenberg and Kositski (2016) is given by rewriting (3.35) as: Rt =

ρ p L eff V02 2(P − D)

(3.36)

We have seen that for metallic targets the actual values for Rt can be derived by either empirical relations based on penetration depths versus impact velocities, or by numerical simulations using proper material constitutive relation. In contrast, concrete is a much more complex material as compared with metals, and its constitutive relations are not well established. Thus, one is led to the empirical route in order to derive the relation between Rt and the compressive strength (f c ) of the concrete, as was done by Rosenberg and Dekel (2010a) who derived the empirical relation (3.34). This was also the approach of Rosenberg and Kositski (2016) who reviewed a much larger set of data, as compared with Rosenberg and Dekel (2010a). Specifically, the data included projectiles of diameters in the range of 6–76 mm and concrete targets with compressive strengths of up to 160 MPa. They found that Rt depends on the diameter of the projectile as well as on the strength of the concrete. This is clearly evident by their summary of the data, as presented in Fig. 3.49 where the Rt values were determined through (3.36). These values seem to fall on three different curves; Fig. 3.49 The data for Rt as a function of f c for different projectile diameters

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115

for the small projectiles with D = 6–13 mm, the medium-sized projectiles with D = 20–30 mm, and the large projectiles with D = 76 mm. The dependence of Rt on the projectile’s diameter is a clear indication of a scaling issue, or a “diameter effect”, which results in a lower penetration resistance for the larger projectiles. The question whether this is a “diameter effect” or a “mass effect” was discussed by Rosenberg and Kositski (2016) using penetration data of projectiles having the same diameter but different lengths (mass). The penetration depths of these projectiles, shot at concrete targets of the same strength, resulted in the same Rt values which is a clear indication that this is a “diameter effect” rather than a “mass effect”. The experimental work of Wu et al. (2015a) with corundum-aggregated concretes shows that an increase of the aggregate size results in lower penetration depths (higher resistance) for projectiles with the same diameter. Thus, Rosenberg and Kositski (2016) related the observed variations in Rt to the normalized projectile diameter (D/d), where d is the maximal size of the aggregates in the concrete. The data presented in Fig. 3.49 covers a large range of concrete strengths, 13–160 MPa, which was accounted for by Rosenberg and Kositski (2016) through the following empirical relation: Rt = 0.25 ln( f c /2.72) − 0.046(D/d)

(3.37)

where Rt is given in GPa, and f c in MPa. This relation accounts for the scaling issue in concrete penetration as far as the resistance to penetration is dependent on the projectile’s diameter. A plausible explanation for this result can be related to the fact that the aggregates are usually stronger than the cement in which they are embedded. Thus, one may assume that large aggregates can exert an enhanced resistance to penetration, especially when their size is of the order of the projectile’s diameter. On the other hand, if the projectile’s diameter is much larger than the aggregate size, one can assume that the resistance to penetration will be affected mainly by the average strength of the cement/aggregate pair. Accordingly, the resistance to penetration is expected to decrease with increasing ratios of D/d. Rosenberg and Kositski (2016) suggested that these trends can be highlighted by using mesoscopic numerical simulations for concretes with various aggregates, impacted by projectiles with different sizes. Such numerical simulations were performed by several groups as in Peng et al. (2018), Feng et al. (2018) and Wu et al. (2019), but they resulted in weak scaling effects which did not account for the experimental data. Thus, the source of the scaling problem was not unveiled by these simulations. Another problem with concrete penetration is related to fact that the unconfined compressive strength of a given concrete (f c ) is determined by test specimens, either cylindrical or cubical, having different size. Consequently, values for the unconfined compressive strength as cited by various workers may be very different even for similar concretes. This issue led several workers such as Forrestal et al. (2003), to conclude that the unconfined compressive strength (f c ) is not a real material property for concrete-like materials, since it depends on the shape and dimensions of the test specimen. This issue and the scaling problem were addressed by Rosenberg et al. (2020a) who reanalyzed the experimental data for concrete penetrations of Forrestal

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and his colleagues, as well as data from other sources. The main purpose of this work was to look for a new explanation for the scaling problem in the penetration of concrete targets by rigid projectiles.

3.9.2 The Scaling Issue and the Size Effect The basic assumption of Rosenberg et al. (2020a) was that the scaling issue is closely related to the size effect of concrete specimens according to which, their unconfined compressive strength is higher for smaller specimens. The size effect has been explained by fracture-mechanics theories for brittle solids, by attaching a characteristic length to the solid. This length is not a scalable entity and the size effect is a direct result of its influence on the compressive strength. The subject has been thoroughly researched for the past 50 years and the interested reader is referred to the review article by Bazant (2000). In order to have a quantitative assessment for the size effect, Rosenberg et al. (2020a) used the data of Sim et al. (2013) for normalweight concretes (with densities of 2.24–2.48 g/cc), having maximal aggregate sizes of: d = 8, 13 and 19 mm. The cylindrical test specimens in this work had an aspect ratio of H/D = 2.0 which is the more common shape for these test specimens. The data of Sim et al. (2013) was found by Rosenberg et al. (2020a) to follow the simple relation: f c (D)/ f c (D0 ) = (D0 /D)0.25

(3.38)

where D0 is the diameter of a reference cylinder with a well-defined compressive strength f c (D0 ). This relation closely follows the uniaxial compressive strength data in Sim et al. (2013) for specimen diameters in the range of D = 50–400 mm. In order to account for the scaling effect Rosenberg et al. (2020a) conjectured that a penetrating projectile, with a diameter of Dp , has to overcome a concrete cylinder with about the same diameter by pushing it aside throughout the penetration process. This heuristic assumption can be justified by realizing that the concrete material which “participates” in the penetration process is a cylinder with a diameter of about 2D as observed by Heuze (1990). He reviewed rock penetration data and their numerical simulations, and stated that “An order of magnitude decay in the stresses induced around the projectile’s body, takes place over a radial distance of about 2.5 times the projectile diameter; this gives the scale of the volume of target material involved in controlling the penetration”. One may assume that the same assertion is true for concrete targets as well. Rosenberg et al. (2020a) followed this observation and assumed that the concrete’s resistance to the penetration of a projectile with a diameter of Dp , should be related to the strength of a concrete cylinder with this diameter. They introduced a new strength parameter, f c∗ = f c (Dp ), which depends on both the target’s strength and projectile’s diameter. This new entity is the uniaxial strength of a concrete cylinder with a diameter of Dp , and it should be calculated

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by using the values of D0 and f c (D0 ) for the corresponding test specimen through (3.38), resulting in the following definition of f c∗ : f c∗ ≡ f c (D p ) = f c (D0 )(D0 /D p )0.25

(3.39)

The next step in the analysis was to use a large number of published sets of data in which penetration depths as a function of impact velocities are given for various projectile/concrete pairs. The diameters (D0 ) and compressive strengths f c (D0 ) of the cylindrical test specimens in all of these sets of data are reported in the corresponding papers. Moreover, they used cylindrical test specimens with aspect ratios of H/D = 2.0, which enabled Rosenberg et al. (2020a) to use (3.39), in order to derive the corresponding values of f c∗ for each projectile/concrete pair. The diameters of the cylindrical test specimens in these sets of data were in the range of 50–150 mm. In addition, they derived the corresponding values of Rt for each pair by using (3.36) through the empirical data for P(V 0 ). This procedure resulted in sixteen pairs of (Rt , f c∗ ) values for concretes with strengths in the range of f c = 13–210 MPa, and projectiles with diameters in the range of Dp = 12.9–76 mm. Most of the projectiles were ogive-nosed with CRH3 and a few had a CRH2 nose shape. Figure 3.50 shows these data points in terms of Rt = Rt ( f c∗ ) together with their best-fit curve according to: Rt = 0.3 ln( f c∗ /10)

(3.40)

where Rt is given in GPa and f c∗ in MPa. As demonstrated by Rosenberg et al. (2020a), this simple relation, together with (3.35), account very well for other sets penetration data. The approach presented by Rosenberg et al. (2020a) resolves a major problem with the terminal ballistics of concrete targets. This problem is related to the fact that the unconfined compressive strength (f c ) is not a physically-based property of Fig. 3.50 The data for Rt as a function of f c∗

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concrete and concrete-like materials, since it depends on the size and shape of the test specimen which is used to determine it. Thus, f c cannot be considered as a true material property, as was also pointed out by Forrestal et al. (2003). The new approach of Rosenberg et al. (2020a) circumvents this difficulty by introducing the concept of an effective compressive strength ( f c∗ ), which is more physically-based since it incorporates the size effect, as discussed here. Note that the analysis of Rosenberg et al. (2020a) does not include the size of the aggregates explicitly, since their effect is implicitly imbedded in the values of f c (D0 ) and f c∗ . Thus, one can use the analysis even when the aggregate size is not specified, as is the case for some of the data presented in Fig. 3.50. A similar scaling phenomenon has been observed by Frew et al. (1999) for the penetration depths of limestone targets with no aggregates, which means that the issue has little to do with the aggregate size. These rock targets were impacted by three similarly shaped CRH3 ogive-nosed projectiles having diameters of: 7.1, 12.7 and 25.4 mm, at velocities in the range of 0.5–1.5 km/s. It turned out that the normalized penetration depths (P/D) were higher for the larger projectiles. This scaling issue for the limestone rock was addressed by Rosenberg et al. (2020b) by relating it to the size effect in the compressive strengths of rock materials, similarly to the way the scaling issue in concrete was treated by Rosenberg et al. (2020a). Another important point in the work of Frew et al. (1999) is related to the fact that in all of their tests the diameters of penetration channels, beyond an entrance phase of 2D, were equal to the corresponding projectiles’ diameters. As discussed above, this feature is a strong indication for the constant resisting stresses which the limestone targets exerted on these projectiles. As shown in Fig. 3.51, the data of Frew et al. (1999) in terms of P = P(V 0 ) for the three different projectiles, can be accounted for by constant resisting stresses using (3.35). Note the decreasing values for Rt with increasing projectile diameter, which is the manifestation of the scaling issue for the limestone target. The scaling issue in concrete penetration was addressed by Canfield and Clator (1966) in a very careful and scientifically-based manner. They chose two sets of Fig. 3.51 Fitting the model with the data for limestone targets impacted by projectiles with D = 7.1 mm (squares), D = 12.7 mm (circles), and D = 25.4 mm (triangles)

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Fig. 3.52 Normalized penetrations for full scale and 1:10 scaled-down projectiles impacting the corresponding concrete targets, from Canfield and Clator (1966)

projectile/concrete pairs, which differed by a factor of 10 in their dimensions, and performed a series of tests on both targets at impact velocities in the range of 300– 800 m/s. In order to have a perfect scaling between the two sets, they reduced the aggregate size and the diameter of the steel reinforcements, by a factor of 10 in the scaled-down targets. They also applied a factor of 10 for the dimensions of the test specimens which were used to determine the unconfined compressive strengths of these targets. The cylinders had an aspect ratio of 2.0, with diameters of 203 mm and 20.3 mm for the large and the scaled-down targets, respectively. The corresponding compressive strengths were very close to each other with an average value of f c = 35.6 MPa. Their CRH1.5 ogive-nosed projectiles had diameters of 76.2 mm and 7.62 mm and weights of 5.9 kg and 5.9 g, respectively. Thus, a perfect scaling has been achieved for the two sets of projectile/target pairs in the work of Canfield and Clator (1966). The most important result of this work was that a single normalized penetration curve (P/Dp ) as a function of impact velocity accounted for both sets of data, as is clearly seen in Fig. 3.52. Thus, a linear scaling behavior has been demonstrated for the terminal ballistics of concrete targets, when a perfect scaling was applied to both targets and projectiles. This means that the “scaling issue” in concrete penetration is, in fact, the result of incorrect scaling when only the projectile’s dimensions are varied. Note the important role of the projectile/aggregate ratio (Dp /d) which was also used by Rosenberg and Kositski (2016), as well as the role of the ratio Dp /D0 which was used in the work of Rosenberg et al. (2020a).

3.9.3 Other Important Issues Several points concerning the approach described above have to be addressed here. The effect of the projectile’s nose shape on its penetration depth in concrete targets,

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has not received enough attention since most of the studies have been performed with the CRH3 ogive nose shape. Only a small part of the data analyzed above had CRH2 projectiles and these showed similar penetration depths to the CRH3 projectiles. This fact may indicate that the penetration depths of rigid ogive-nosed projectiles are not very sensitive to their exact nose shape. Some support to this conclusion can be gained through the experimental results of Hansson (2011) who impacted CRH3, CRH8 and CRH12 projectiles at different concretes. Penetration depths of the CRH12 projectiles were higher by only 12–22% than those of the CRH3 ogives. These results enhance the claim that the exact nose shape of theses ogives is not very significant, as far as deep penetrations are concerned. In order to validate these estimates we performed several deep-indentation tests on concrete cylinders 180 mm in diameter, encapsulated in thin-walled steel tubes. Such static tests were described in Sect. 3.4 above and will be further discussed in the next section. The constant stress levels (beyond the entrance phase) in these tests resulted in values of: 980, 840 and 810 MPa for ogive-nosed indenters with CRH3, CRH5 and CRH8, respectively. The ratios between these static stresses are close to those cited by Hansson (2011) for his penetration data. Another issue concerns the compressive strength of the aggregate material which may affect the concrete’s resistance to penetration. One may assume that a very strong aggregate material will result in a high Rt value, especially for a relatively small diameter projectile. The experimental results of Wu et al. (2015a) with concrete targets containing aggregates of either basalt or corundum seem to support this assertion. The penetration depths in the corundum-aggregated targets were lower than the corresponding depths in the concrete with the basalt aggregates having similar f c values. As an example, consider their tests on a basalt-aggregated concrete with f c = 125.2 MPa, and a corundum-aggregated concrete with f c = 110.7 MPa. The corundum aggregates have a much higher compressive strength (2000 MPa), as compared with a compressive strength of 120 MPa for the basalt aggregates. Both aggregates had the same maximal size of 20 mm, and the D = 25.3 mm projectiles impacted these targets at 510 m/s. In spite of the higher compressive strength of the basalt-aggregated concrete, the penetration depth into this target was higher by 25% than that in the corundum-aggregated concrete. This result seems to justify the claim of Wu et al. (2015a) that the resistance to penetration of a given concrete may depend on the aggregate material, and not just on the concrete’s compressive strength. On the other hand, the projectiles penetrating the corundum-aggregated concrete in this work suffered more from the mass abrasion phenomenon, so they were not truly rigid along the whole penetration process. Other phenomena which can take place at very high impact velocities should also be considered carefully. One may assume that at a certain impact velocity the onset of cavitation will take place, as in metallic targets which were discussed earlier in this chapter. Thus, at higher impact velocities the target’s inertia may have to be added to the resisting stress on the rigid projectile. This addition will influence the penetration capability of the rigid rod and we may expect a diversion from the simple quadratic relation between penetration depths and the impact velocities. In addition, projectile abrasion and mass loss at high impact velocities should be analyzed and

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accounted for by the models. Such mass losses have been observed in many works as in Silling and Forrestal (2007), for example, with mass losses of up to 10% which lead to similar reductions in the corresponding penetration depths. Moreover, the nose shape of recovered projectiles often showed a change from ogival to spherical due to the strong abrasion process, further reducing their penetration efficiency. This issue should be analyzed quantitatively through the changes in the nose shape and mass of the projectile, and their effect on its penetration efficiency. Reinforcement of concrete is an essential structural component for the carrying capacity of static loads and nearby blasts, which may cause a global response of the structure. Reinforcements can also be used to confine the concrete and, subsequently, increase its effective tensile strength. However, as far as the concrete’s resistance to penetration it was found that reinforcement ratios of up to about 1.5% do not enhance the penetration resistance of the concrete, as observed by Slitter (1980) and Dancygier et al. (2007). On the other hand, heavily reinforced concretes do show an improved resistance to penetration which was included in several empirical relations, such as in Young (2009) and Li et al. (2005). We should note that many of the concrete targets which were used in the experiments described above included relatively light reinforcements which, apparently, did not affect their penetration resistance. An important feature of steel reinforcements is to limit the damage caused by the projectile at the front (spalling) and back (scabbing) surfaces. Concrete mixtures which include steel fibers, the so-called Steel Fiber Reinforced Concretes (SFRC), also show a reduced amount of damage by bridging the cracks which develop in the concrete, as described by Bentur and Mindess (2007).

3.9.4 The Rate Dependence of Concrete-like Materials Concrete and concrete-like materials (mortars and rocks) have been considered for many years to be rate-sensitive at strain rates in the range of 1.0–102 s−1 , through dynamic strength measurements with the Kolsky bar system. This sensitivity is quantified by the dynamic increase factor (DIF), which is the ratio of the dynamic to the quasi-static compressive strength, as obtained by the unconfined compression test. However, the analysis of Li and Meng (2003) showed that the apparent increase in concrete strength at these strain rates is due to inertia effects, rather than to a real strengthening mechanism. They concluded that there is no physically based mechanism for the rate sensitivity of concrete-like materials, and that these strength enhancements are due to radial confinement, especially for large specimens subjected to the high strain rates in Kolsky bar tests. This important observation was manifested by performing numerical simulations for cylindrical specimens in the Kolsky system. Li and Meng (2003) followed the stress-strain characteristics in these simulations for various specimen geometries, while the constitutive relation of these concretelike specimens did not include rate dependence. Figure 3.53, from Li et al. (2009), shows the results of such simulations for two different situations, a direct impact of the striker bar on the transmitter bar and an impact on a wave-shaper in front of

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Fig. 3.53 Simulation results for the DIF of a concrete-like specimen from Li et al. (2009)

the transmitter bar. As is clearly seen in the figure, both cases resulted in a sharp increase in the DIF of the specimen at a strain-rate of about 10 s−1 , resembling the experimental results from which the rate sensitivity of concrete was derived. This sharp increase for a material which is, in fact, rate-insensitive strongly enhances the claim of Li and Meng (2003) that the measured DIF’s are related to the test characteristics rather than to material properties. Other workers, like Mu et al. (2012), reached similar conclusions by performing numerical simulations for various dynamic tests on concrete-like materials. These authors conclude that the DIF is not a physical property of these materials and that it is caused by the lateral confinement of the dynamically loaded specimens. The penetration experiments of Canfield and Clator (1966) which were described above, support this rate insensitivity claim although in a non-direct way. As shown in Fig. 3.52, the normalized penetrations (P/D) from the two sets of experiments with a factor of 10 in their scale are evenly scattered around a single curve. This finding enhances the claim of Li and Meng (2003) that concrete is not a strain-rate sensitive material. Rate sensitivity is expected to result in a measurable scaling effect in experiments with these different scales, since the strain rates in the smaller scale experiments are much higher than the rates at the large scale. Thus, the smaller targets should have resulted in lower values for the normalized penetrations, if concrete is really a rate-sensitive material. Another way to demonstrate that the strength of concrete materials is rate independent is through the static deep-indentation test with a high-strength punch, which was described in Sect. 3.4. Such a test on a concrete specimen is reported in Rosenberg et al. (2018a) with a concrete cylinder, 180 mm in diameter, encased in a 10 mm thick steel tube. The hard steel indenter, with a diameter of 7 mm had a conical nose which was similar to a CRH3 ogive nose-shape. The Instron machine was operated at a rate of 1 mm per minute, simultaneously recording the force on the indenter and its penetration depth. The friction on the indenter’s shank was subtracted from the actual force in the way outlined in Sect. 3.4. The friction stress, as computed

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Fig. 3.54 Resisting stress-distance histories in a static deep-indentation test on a concrete block

from the linear portion of the force–distance profile, was τ = 45 MPa in this test. The measured stress-distance profile and the stress-distance after friction subtraction are shown in Fig. 3.54. In the early part of the friction-subtracted profile, the stress increases almost linearly up to a depth of 15 mm which is about twice the indenter’s diameter, as expected. The asymptotic value of the static penetration resistance, beyond the entrance phase, was Rt = 0.75 GPa. Rosenberg et al. (2018a) also performed a ballistic test on a similar concrete cylinder using the hard core of a 0.3 AP projectile. The impact velocity in this test was 759 m/s and the penetration depth was P = 60 mm. Using relation (3.35) with the value of Rt = 0.75 GPa as obtained in the static indentation test, results in a predicted penetration depth of 60 mm, which is exactly the same as the experimental penetration depth. This excellent agreement strongly enhances the claim that the strength of concrete is rate independent, since the penetration depth of the projectile was accurately predicted by the static value of Rt . Moreover, this agreement demonstrates the usefulness of the static indentation test for terminal ballistics, as discussed in Sect. 3.4.

3.9.5 The Equivalent Aluminum Target The penetration process of rigid projectiles into both metallic and concrete-like targets, beyond their corresponding entrance phase, is governed by a constant resisting stress for impact velocities below the cavitation threshold velocity. This conjecture is based on energy conservation considerations, as discussed above, and it has been demonstrated through empirical data and their numerical simulations. As a consequence, the quadratic relations between penetration depths and impact velocities, P = kV 20 , for metallic and concrete-like targets are very similar as given by (3.8a, 3.8b) and (3.35), respectively. The major difference between these two families of materials lies in their constitutive relations, which are relatively simple for metals and very

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complex for brittle materials. One is led to assume that the various material parameters in the constitutive relations of concrete-like materials have to be accurately calibrated in order to be used in numerical simulations of deep penetration tests. However, this assumption has been refuted by the numerical simulations of Fossum (2004) for a limestone target impacted by a test projectile. The parameters in the limestone’s constitutive relation included the pressure and strain-rate dependencies of the rock, tensile cracking, nonlinear elasticity, kinematic hardening, compaction, and the Lode angle dependence. Fossum (2004) checked the role of these parameters by systematically turning off each one of them in the simulations, following the corresponding change in the penetration depth of his test projectile. He found that turning off the pressure dependence of the rock’s strength resulted in the most significant effect (68%) on the penetration depth. For all the other features of the rock he found much smaller changes in the penetration depth (from 0.1 to 6%). Thus, one may conclude that the most significant material property of concrete-like solids, for terminal ballistic studies, is their actual compressive strength during the penetration process, as is the case for metallic targets. In other words, concrete and rock targets are very similar to metallic targets as far as deep penetrations into semi-infinite targets are concerned. This observation led Rosenberg and his colleagues (2020a, 2020b) to introduce the concept of the equivalent aluminum target. The basic idea behind this concept is to replace the concrete-like target in numerical simulations with an aluminum target which has the same resistance to penetration (Rt ). This replacement can simplify the computation process especially for complex interactions, such as inclined and yawed impacts. They used the 3D Lagrangian version of the LS-DYNA code in these simulations, and Al-6061 was chosen for the equivalent targets since its density is close to the density of concretes and rocks. The constitutive relation for the aluminum target was the elasto-plastic von-Mises criterion, without rate or strain hardening dependencies. Once the value of Rt is known for a given projectile/concrete pair, the strength of the equivalent aluminum target (Y ) is determined through (3.9). This value is then used in simulations with the equivalent aluminum target impacted by the given projectile, in order to predict relevant information about the concrete penetration, without the complexity of the concrete’s constitutive relations. As an example consider the penetration tests of O’Neil et al. (1999) who impacted L/D = 9 rods at strong concrete targets (f c = 157 MPa). These impacts suffered from relatively large inclination angles (yaw and pitch) which necessitated 3D numerical simulations. The analysis of Rosenberg et al. (2020a) starts with the values of f c = 157 MPa and D0 = 102 mm, as reported by O’Neil et al. (1999), who used 906 g projectiles with a diameter of Dp = 26.9 mm. From (3.39) one gets a value of f c∗ = 219 MPa for this projectile/concrete pair, and from (3.40) one finds that the penetration resistance of this projectile is: Rt = 0.926 GPa. Using (3.9) one finds that Y = 170 MPa should be used as the compressive strength of the equivalent aluminum target in the numerical simulations. These 242 mm long, 4340 steel projectiles (hardened to 44RC ) had an inner grout cylinder, with a diameter of 10.2 mm, and density of 1.58 g/cc. This material was simulated as an elastic solid of the same density and a Young’s modulus of 20 GPa. The well-known Johnson-Cook constitutive relation was used for the

3.9 Concrete Targets Table 3.9 Comparing simulation results with the data of O’Neil et al. (1999)

125 V 0 (m/s)

γ (°)

P in concrete (mm)

P in aluminum (mm)

229

1.08

80.7

83.6

406

1.0

180.2

182.7

587

3.62

296

300

747

2.6

444

445.7

metallic part of these projectiles, in order to simulate its true properties at high strains and strain rates. The projectiles impacted the aluminum targets at inclination angles (γ ) which are equivalent to the measured yaw (γ 1 ) and pitch (γ 2 ), through the wellknown relation between these angles: sin2 γ = sin2 γ 1 + sin2 γ 2 . Table 3.9 shows the excellent agreement between the data for the penetration depths in the concrete targets, from O’Neil et al. (1999), and those from the corresponding simulations with the equivalent aluminum targets. This agreement enhances the validity and the usefulness of the equivalent aluminum concept. Moreover, it supports the analysis presented in Rosenberg et al. (2020a) concerning the effective strength concept ( f c∗ ) and its relation to the penetration resistance (Rt ) according to (3.40). Figure 3.55 shows a series of “snapshots” from the simulation of the test with the highest inclination angle (3.62°). One can see the strong bending of this rod which resulted in a 20% reduction in its penetration depth, as compared with a value of 358 mm for a normal impact according to a corresponding simulation. Fig. 3.55 Several “snapshots” from the simulation for the γ = 3.62° impact

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3.10 The Deep Penetration of Deforming Rods The deep penetration of rigid penetrators has a rather limited application since most of the practical cases involve eroding rods and shattering projectiles. Before we discuss the penetration of eroding rods we wish to explore the range where these rods deform rather than erode. Thus, we consider a rod which is rigid at low impact velocities, but beyond a certain threshold velocity (V d ) it deforms during penetration. At these impact velocities the length of the rod is shortening and the diameter of its front portion becomes larger than its original diameter. As a result, the rod loses its penetration capability and its penetration depths are decreasing with increasing impact velocities. As we shall see here, this deformation phase takes place in a narrow range of impact velocities between two threshold impact velocities which depend on the properties of the rod (strength and nose shape) and on the strength of the target. At the lower threshold velocity the projectile nose deforms and its penetration depth decreases simultaneously. This process continues until a higher threshold velocity above which rod erosion sets in and the penetration process becomes that of an eroding rod. This complex behavior was noted by Brooks (1974) in his work on short projectiles impacting steel targets. He termed the second threshold velocity, from deforming to eroding projectiles, as the hydrodynamic transition velocity (V HT ). The model of Tate (1977) for the hydrodynamic transition velocity in long rods assumes that this transition occurs when the rod’s erosion rate is equal to the velocity of the “plastic wave front” in the rod, which results in its gross plastic deformation. The transition from rigid to deforming rods was analyzed by Rosenberg and Dekel (2010b) through experimental results, and numerical simulations were used to construct an analytical model for the transition velocity. Before we get into the details of the analysis we review the experimental findings concerning the various aspects of the rigid-to-deforming transition. Consider first the results of Forrestal and Piekutowski (2000) for the penetration depths of spherically-nosed steel rods impacting 6061-T651 aluminum targets, as shown in Fig. 3.56. The general trend Fig. 3.56 The data for spherical-nosed steel rods impacting aluminum targets

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of the data shows that, unlike the accepted understanding (by many workers) the transition from rigid to eroding rod behavior does not take place at a well-defined threshold velocity. Instead, there is a certain range of impact velocities between the two regimes, which corresponds to heavily deforming rods with no mass loss. The penetration depths in this regime, which is marked by the shaded area in the figure, decrease with increasing impact velocities. Thus, the actual sequence of impact events with increasing impact velocities is as follows: (1) at low impact velocities the rod penetrates as a rigid body up to a certain threshold velocity (V d ) which corresponds to the onset of its deformation, (2) for higher impact velocities the rod is deforming with no mass loss resulting in reduced penetrations with increasing impact velocities, (3) at a second threshold velocity rod erosion sets in and for higher impact velocities the penetration depths are increasing again. The post-mortem radiographs in Fig. 3.57, from Forrestal and Piekutowski (2000), show three residual steel rods which impacted the aluminum targets at velocities just below and above the onset of rod deformation and its erosion. At the left side we see a rod on the verge of deformation after an impact at 936 m/s. The residual rod in the intermediate velocity shot at V 0 = 1,037 m/s, is heavily deformed at its front part with a diameter of about 1.5 times that of its undeformed back part. An impact velocity of 1,193 m/s resulted in an eroded rod with a much lower penetration depth as is clearly seen in the radiograph at the right hand side of the figure. The same behavior was found by Wickert (2007) for WHA rods impacting Al7020 targets. These were truncated conical-nosed rods that remained rigid up to velocities of about 600 m/s, and deformed without losing mass up to about 750 m/s, as shown in Fig. 3.58. The strength of these WHA rods was similar to the strength of the steel rods in the work of Forrestal and Piekutowski (2000), as were the strengths of the two aluminum alloys in these works. Thus, the two figures imply that the transition velocity from rigid to deforming rod behavior is inversely dependent on the rod’s density. We can already have a rough estimate for this dependence considering the threshold velocities of these rods, as seen in Figs. 3.56 and 3.58. These velocities are about 900 m/s for the steel rod and about 600 m/s for the WHA rod, resulting in a ratio of 1.5 for these threshold velocities. This value is very close to the inverse of the square root of their respective densities, a feature which will be borne out by the model presented below. These features were investigated by Rosenberg and Dekel (2010b) through numerical simulations for steel rods with strengths between 1.2 and 2.0 GPa, impacting semi-infinite aluminum targets with strengths in the range of 0.4–0.6 GPa. The simulations were carried out for impact velocities which correspond to the rigid and the

Fig. 3.57 Post-mortem radiographs of the residual rods in the aluminum targets

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Fig. 3.58 Penetration depths for WHA rods impacting aluminum targets

deforming-rod phases with the aim to better understand the deformation mechanism, and their results were used to construct an analytical model for the rigid-to-deforming transition velocity. The constitutive model for both rod and target materials was the simple von-Mises criterion with well-defined strength values, and without strainhardening or strain-rate effects. This choice of constitutive relation is in accord with the stress–strain relations of the high strength rods and the 6061-T651 aluminum targets which were used by Forrestal and Piekutowski (2000). Figure 3.59 shows the deformation process of a 2.0 GPa steel rod as it penetrates the 0.4 GPa aluminum target, for an impact velocity of 1.2 km/s. One can clearly see that the deformation process takes about 160 µs after which the rod reaches its final shape, with a clear Fig. 3.59 Simulation results for the deformation of a 2.0 GPa steel rod impacting the 0.4 GPa aluminum target at 1.2 km/s, times after impact are in microseconds

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129

distinction between the deformed and undeformed parts. Note the similar shapes of the deformed rod in this simulation and of the residual rod in the experiment with V 0 = 1.037 km/s, as shown in Fig. 3.57. A close examination of the nose deformation process in this simulation shows that after about 30 µs the diameter of the rod’s nose reached its final value. This is, roughly, the time which is needed for the rod to penetrate through the dominance range of the entrance phase. The growth of the nose diameter is, apparently, countered by the increasing resisting stress which the target exerts during the entrance phase stage. On the other hand, as the rod penetrates deeper the length of its plastically deformed part continues to grow, until about 160 µs after impact. At this time the velocities of its nose and tail attain similar values as shown in Fig. 3.60, which is the reason that the rod is not deforming further after this time. The velocity of the rod’s tail is decreasing through a series of steps having a well defined amplitude, in a similar way to that of a rigid rod. However, the elastic waves which reverberate in the rigid and the deforming rods are different in two respects. The first difference is in their amplitude which for a rigid rod is equal to the target’s resistance to penetration (Rt ). For the impact discussed here the strength of the rod is Y p = 2.0 GPa and the expected target’s resistance to penetration is Rt = 2.24 GPa. Obviously, the rod cannot carry an elastic wave with amplitude which is larger than its strength. The amplitude of the velocity steps in Fig. 3.60 is v = 0.1 mm/µs which, according to (3.6b), corresponds to stress waves of σ = 2.0 GPa, exactly the strength of the rod (Y p ) in this simulation. Thus, the deceleration of the rigid part of a deforming rod is controlled by its strength rather than by the target’s resistance to penetration. Another difference concerns the time durations of these velocity steps, which are equal to the transit time of an elastic wave moving between the rod’s tail and the propagating boundary of its deforming front. Obviously, these times are getting shorter at each reverberation due to the backward propagating plastic front. The deceleration of the deforming rod is expected to be much higher than that of a rigid rod because of these shorter stress reverberations, and this is clearly seen Fig. 3.60 The velocity histories of the deforming rod’s nose and tail

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Fig. 3.61 The deceleration history of the deforming rod

in Fig. 3.61. The deceleration achieved a constant value at about t = 80 µs, when its front and tail velocities are still varying in a very different manner, as seen in Fig. 3.60. The value of this constant deceleration is higher by a factor of two than the deceleration of a rigid rod with the same initial dimensions, since its effective length, after deformation is smaller by the same factor. The next sets of simulations were performed by Rosenberg and Dekel (2010b) in order to determine the threshold velocities for rod deformation. Figure 3.62 shows the simulation results for the penetration depths of spherical and ogive-nosed steel rods, Fig. 3.62 Simulation results for steel rods impacting 0.4 GPa aluminum targets for: a spherical-nosed rods and b ogive-nosed rods

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131

having an aspect ratio of L/D = 20. These rods, with strengths of 1.6 and 2.0 GPa, impacted aluminum targets with strength of 0.4 GPa. Note the corresponding values of Rt = 2.24 and 1.87 GPa, for the spherical and ogive-nosed rods, respectively. Thus, these simulations deal with cases where the rods’ strengths are close to the resisting penetration stresses of the target. The figures also include the penetration curves for the corresponding rigid rods using the model presented above. These curves help in determining the onset for the gross deformation of the rods, as their penetration depths start to decrease with impact velocity. These simulations show that the transition velocity from rigid to deforming rod increases with rod strength as expected. Also, the transition velocities for the ogive-nosed rod are higher by a factor of about two, than those for the spherical-nosed rod of the same strength, in agreement with the experimental results of Forrestal and Piekutowski (2000). In addition, the velocity range where rod deformation takes place is getting narrower with increasing rod strength, which was also found experimentally by Forrestal and Piekutowski (2000). In order to construct an analytical model for the deformation threshold velocity (V d ) Rosenberg and Dekel (2010b) considered the effect of the confining stresses which the target exerts laterally on the rod’s nose. Such stresses can prevent the deformation of the rod or postpone it to higher impact velocities. This idea was suggested by Segletes (2007) in order to account for the deformation pattern of spherical-nosed tungsten-alloy rods perforating aluminum plates. His basic assumption was that once the rod’s nose is embedded in the target, its bulging is constrained by the surrounding material which exerts these lateral stresses on the rod’s nose. In order to validate the effect of these laterally confining stresses, Rosenberg and Dekel (2010b) performed several simulations for spherical-nosed steel rods impacting aluminum targets, which had a cylindrical hole around their symmetry axis. The diameter of these holes was 6 mm (the same as the rod’s diameter) and their depths were varied between 6 and 30 mm. The rods in these simulations had strength of 2.0 GPa and their impact velocity was 1.2 km/s. Table 3.10 lists the resulting penetration depths, as well as the final nose diameters of the rods in these simulations. The results from the reference simulation with a flat-faced target, are added for the sake of comparison. It is evident that the lateral confinement of the rod’s nose limits its mushrooming and enhances its penetration capability. This confining effect is significant even for the Table 3.10 Simulation results (in mm) for steel rods impacting aluminum targets with a hole

Hole depth

Final nose diameter

Penetration depth

0

8.65

213

6

6.91

213.2

12

6.82

222.4

18

6.78

225.3

24

6.76

226

30

6.74

228

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3 Rigid Penetrators

6 mm hole depth (one rod diameter), since the final nose diameter in this simulation is considerably smaller than that for the impact on a flat target. The penetration depths of the rod in these simulations should be compared with those for a rigid rod impacting targets with holes of different depths. As was shown above, such impacts resulted in smaller penetration depths with increasing hole depths, due to the diminishing effect of the free impact face. The fact that the deforming rod penetrates deeper with increasing hole depth, strongly supports the assumption of Segletes (2007) concerning the role of the lateral stresses. Apparently, the effect of these stresses is strong enough to overcome the higher resisting stresses at deeper depths, and the confined rods penetrate deeper than the unconfined rod which impacts the flat target. Figure 3.63 shows schematically a deforming rod and the lateral stress (H lat ) which supports it radially. This lateral support arises from the fact that the rod is mushrooming, exerting an additional stress on the crater wall, as compared with a rigid rod for which the interaction with target is only at its nose. The model of Rosenberg and Dekel (2010b) is based on the assumption that the onset of rod deformation takes place when the pressures on both sides of the rod/target interface are equal. On the rod side the pressure is equal to the effective strength of the rod Y eff , which includes the lateral support from the target (H lat ). On the target side, the pressure at the interface is given by the target’s effective resistance to penetration (Reff ) and the dynamic (inertia) contribution, bρ t V 2 , where b is the nose shape factor which was defined above. Thus, the basic equation of this model is: Yeff = Y p + Hlat = Reff + bρt Vd2

(3.41)

Rosenberg and Dekel (2010b) noted that since the deformation process starts right after impact, the target resistance term (Reff ) cannot assume its value at deep penetration (Rt ). As will be shown in Chap. 5, the resistance of the target at the very early stage of penetration is equal to 3Y t . Thus, the effective resistance term (Reff ) was chosen as the average of these two resisting stresses, namely, 3Y t and Rt . For the confining lateral stress they assumed a value of H lat = 3Y t through some simplifying assumptions. These considerations led to the following expression for the threshold Fig. 3.63 Schematic description of the lateral support to the deforming rod

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velocity (V d ) between rigid and deforming rod behavior:  Vd =

Yp −  bρt

(3.42)

where the nose shape factors are b = 0.5 and b = 0.15 for the spherical and ogivenosed rods, respectively, as derived above for rigid rod penetration. The inverse dependence of V d on the square root of the rod’s density was pointed out above when we compared the results of Piekutowski et al. (1999) for steel rods, with those of Wickert (2007) for WHA rods. The large difference between the values of b accounts for the much higher values of V d for the ogive-nosed rod as obtained in these simulations. This was also the case for several sets of experiments with rods having different nose shapes, as shown by Rosenberg and Dekel (2010b). The term  in (3.42) is due to the averaging scheme for the resisting stresses within the entrance phase. Since the dominance range of this phase is different for the ogive and spherical-nosed rods, their corresponding expressions for  are somewhat different, as given by:  = (Rt − 3Y p )/2 for spherical-nosed rods

(3.43a)

 = (Rt − 4Y p )/2 for ogive-nosed rods

(3.43b)

These simple relations were found to account for all the values of V d obtained by the simulations of Rosenberg and Dekel (2010b), and for the experimental data of V d from several studies such as Piekutowski et al. (1999) and Scheffler (1997). For example, the ogive-nosed rods used by Piekutowski et al. (1999) were of two types of steel, the VAR 4340 with Y p = 1.14 GPa, and the AerMet100 with Y p = 1.72 GPa. These rods were shot at 6061-T6511 aluminum targets, which have a dynamic flow stress of about Y t = 0.42 GPa. Inserting these values of Y p in (3.42) together with the corresponding value of Rt = 1.87 GPa, results in V d = 1.57 and 1.98 km/s for the two rods. These predicted values are in excellent agreement with the experimental results of V d = 1.58 km/s and 1.92 km/s for the Y p = 1.14 GPa and the 1.72 GPa rods, respectively.

3.11 The Effect of Yaw on Rigid Rod Penetration The impact of yawed rigid rods at semi-infinite targets received very little attention as compared to the corresponding research with eroding rods, which will be discussed in Chap. 7. This lack of information may be due to the relatively few practical cases where long rods remain rigid at their interaction with armored vehicles such as main battle tanks. Still, missiles impacting concrete structures may keep their rigidity for the whole penetration process, and the issue has to be addressed properly. This was

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Fig. 3.64 The penetration of an ogive-nosed projectile impacting at a velocity of 1.0 km/s and yaw = 10°

done by Vayig and Rosenberg (2020) in a series of 3D numerical simulations with rigid steel rods of L/D = 5 and 10, impacting aluminum targets at velocities of 500, 1000 and 1500 m/s with yaw angles up to 20°. The 3D Lagrangian version of the LSDYNA code was used for this study with a proper meshing scheme which was checked for convergence. The rods were given high strength (100 GPa) in order to ensure their rigidity throughout the impact and penetration processes. The constitutive relation of the aluminum targets was the elasto-plastic von-Mises relation with Y = 0.43 GPa, which represents the effective dynamic strength of the 6061-T651 aluminum alloy. Figure 3.64 shows several snapshots from the simulation for the L/D = 10 ogive-nosed rod (CRH3), impacting the target at 1000 m/s with a yaw angle of 10°. One can see the large crater at the impact face of the target followed by the much narrower penetration tunnel, which has the same diameter as the rod. The relatively large crater around the impact point is, obviously, the result of the yaw angle and one may expect that during its formation the velocity and deceleration histories of the yawed rods will take very different shapes, as compared with the corresponding histories for normal impacts. The deceleration of the rod from the simulation described above is presented in Fig. 3.65 together with a normal-impact deceleration history of the same rod. As expected, the deceleration history of the

Fig. 3.65 Deceleration histories for 1.0 km/s impacts at normal incidence (red) and inclined impact (10°) (black)

3.11 The Effect of Yaw on Rigid Rod Penetration

135

Fig. 3.66 Normalized penetrations for ogive-nosed rods at 500 m/s (black), 1000 m/s (blue), 1500 m/s (red)

yawed rod shows a large spike at the early stages of penetration, corresponding to the creation of the large crater around the impact point. Such spikes were observed in all the other simulations, with different yaw angles, with heights which depend on the yaw angle and on the impact velocity, but they seem to last for about the same time for the different yaw angles. Simulations results for the reduction in penetration depth (P/P0 ) as a function of yaw angle, are shown in Fig. 3.66 for the L/D = 10 ogive-nosed rod at the three impact velocities. As expected, the effect of yaw angles is stronger at the higher velocities, which was also the result in the simulations for the spherical nosed rods. The numerical simulations for spherical nosed rigid rods were similar to those for the ogive-nosed rods at an impact velocity of 1000 m/s, as seen in Fig. 3.67 for the L/D = 5 and 10 rods. Numerical simulations with rigid rods impacting aluminum

Fig. 3.67 Normalized penetrations at 1.0 km/s, ogive-nosed rod (left) and spherical-nosed rod (right)

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targets at yaw angles of a few degrees are useful for cases where impact velocities are not too high and the rods are made of high-strength materials. This is the case for most of the tests performed by Forrestal and his colleagues with hard steel rods of various nose shapes, impacting aluminum and concrete targets. The simulations of Vayig and Rosenberg (2020) were able to account for their results for impact velocities up to impact velocities where these rods started to deform or erode. Pietukowski et al. (1999) report on a very unusual phenomenon which occurred in several tests with ogive-nosed rods made of the relatively-weak 4340 steel (hardness of 38Rc). These rods impacted aluminum 6061-T6511 targets with velocities of 1.58–1.77 km/s, at yaw angles of a few degrees. These velocities are slightly higher than the deformation threshold velocity of V d = 1.57 km/s, which was derived above for this projectile/target pair. Thus, the rods in these tests are expected to deform, at least partially, at the early stages of penetration. The post-mortem radiographs of the aluminum targets showed that the rods “turned around” during penetration, reaching a state where their back end is in front of their nose. Figure 3.68 shows a radiograph of such a rod after an impact of 1.77 km/s at a yaw of only 1.2°. Note the large diameter of the crater at its lower part, where the “turning around” must have happened. Clearly, the high velocity and the low strength of the rod resulted in its nose deformation which, together with the yaw angle, led to these unusual deviations in the rod’s penetration path. Vayig and Rosenberg (2020) performed several simulations of these tests in order to follow the “turn around” phenomenon. They applied the Johnson-Cook constitutive relation to these rods with: A = 1.14 GPa, B = 510 MPa, n = 0.19 and C = 0.01 in order to account for the properties of the 4340 steel. The rods impacted the targets at the corresponding velocities and yaw angles as in the tests of Piekutowski et al. (1999) and they, indeed, turned around during penetration. Figure 3.69 shows snapshots from one of these simulations and one can follow the stages in this unusual process. The final penetration depth in this test was 105 mm, while the simulation resulted in a depth of 103 mm. Note the similar shapes of the rods in the final simulation snapshot Fig. 3.68 The “turn around” rod after an impact at 1.77 km/s and yaw of 1.2°

3.11 The Effect of Yaw on Rigid Rod Penetration

137

Fig. 3.69 The “turn around” process for an impact velocity of 1.585 km/s, and a yaw of 5°

and in the radiograph in Fig. 3.68. A similar shape of the enlarged craters at the final penetration stages is seen in both test and simulation, attesting for the validity of these simulations.

3.12 The Transition to Finite-Thickness Targets As was stated above, the semi-infinite target has a tutorial role in the field of penetration mechanics. In most of the actual situations the targets are either perforated, or they are thick enough to stop the projectile near their back surface. In these cases the back surface of the target plays an important role through its interaction with the projectile, as we shall see in the next chapters. This interaction lowers the resisting stresses on the penetrator as it approaches the back surface of the target. Moreover, the interaction can induce various failure modes in the rear part of the target, which further lower the resisting forces on the penetrator. The immediate result of these back surface interactions is that the thickness of the plate which is needed to stop a given projectile is always larger than the corresponding penetration depth of the projectile in a semi-infinite target. This issue was studied by Senf and Weimann (1973) with 12.7 mm rigid steel spheres impacting finite plates of 2024-T3 aluminum. They found that the ratios of the penetration depths into the semi-infinite targets (Pn ), and the thicknesses of the finite plates which are needed to stop these spheres (H BL ), are about 0.73. One should note that the values of H BL depend on the specific criterion chosen for plate perforation. Thus, different values may appear in the literature for the

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Pn /H BL ratio even for the same projectile/target combinations. This issue was pointed out by Dehn (1986) who reviewed a large number of projectile/target combinations and quotes values of Pn /H BL = 0.33–0.9 for this ratio. Kinard et al. (1958) investigated this issue at much higher impact velocities (1.2– 4 km/s) by impacting aluminum spheres at aluminum targets and steel spheres at copper targets. These spheres perforated plates which were 1.5 times thicker than their penetration depth in semi-infinite targets (Pn ). Thus, their experiments resulted in a ratio of Pn /H BL = 0.67. They also found that the penetration depth (P) at a given impact velocity is strongly dependent on the plate thickness (H). These experimental results were summarized, empirically, by the relation:  P = 1.3 · H

P − Pn Pn

(3.44)

and with this relation it is possible to predict the penetration depth of a spherical projectile in a given plate, if its penetration into a semi-infinite target is known. The minimal plate thickness which is needed to stop a given projectile is obtained by setting P/H = 1 in (3.44), resulting in a ratio of Pn /H BL = 0.63. Another set of experiments, with relatively soft L/D = 1 steel cylinders impacting armor steel targets, both finite and semi-infinite, is given by Hohler and Stilp (1990). Impact velocities reached 4.0 km/s and the semi-infinite penetrations were substantially smaller than the corresponding values for H BL . The corresponding ratios (Pn /H BL ) increased gradually from about 0.33 to 0.68, for impact velocities in the range of 1.0–4.0 km/s.

Chapter 4

Plate Perforation

Abstract The perforation of finite-thickness plates by rigid projectiles, are discussed here through analytical models and numerical simulations. The models consider the projectile’s non-constant deceleration during the perforation process, due to the effects of the front and back surfaces of the plate. These considerations are shown to account for many sets of empirical data for both metallic plates and concrete slabs. In particular, the models result in accurate predictions for the ballistic limit velocities of many projectile/plate combinations, and for the residual velocities of the projectile after plate perforation.

4.1 General Description The process of plate perforation is the most important issue in terminal ballistics for armor engineers who seek to optimize the weight and cost of their protective designs. This subject has been the focus of a large number of studies, concentrating on two issues which are of major interest to the armor engineer, (1) the ballistic limit velocity for the given projectile/plate combination and (2) the projectile’s residual velocity and mass, as it exists from the target. The perforation process is influenced by the back surface of the target which, together with the impact face, results in a time-varying force on the projectile during perforation. This addition complicates the interaction and introduces more difficulties, as far as analytical modeling is concerned. Moreover, the back surface of the plate can lead to different modes of perforation, and their energy absorption capabilities have to be carefully analyzed, especially when the process involves more than a single mode. Such failure modes during perforation of finite-thickness plates are spalling, petaling, discing, and plugging and they depend on the impact velocity, the properties of the plate material, and the loading geometry (plate thickness, projectile diameter and its nose shape). In addition, when thin plates are perforated they tend to stretch and bend around the impact area, absorbing a significant part of the projectile’s kinetic energy through these deformations. These issues have been discussed by Wilkins (1978), Woodward (1990), Corbett et al. (1996), Liu and Stronge (2000) who highlighted the inherent complexities in the perforation process of thin plates. Obviously, different analytical approaches have to © Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_4

139

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4 Plate Perforation

be applied here, as compared with the simpler process of deep penetration by rigid penetrators, which was discussed in Chap. 3. Some of the failure modes in perforated plates are shown in Fig. 4.1, from Wingrove (1973), demonstrating the wealth of phenomena that take place by changing the nose shape of the projectile. In this work, 10 mm thick 2014-T6 aluminum plates were impacted by hard steel projectiles, 7 mm in diameter, with different nose shapes. This high-strength aluminum alloy is known to behave in a semi-brittle manner. The picture on the left side of the figure shows the plug formed through a shearing process, by the impact of a flat-ended projectile at 180 m/s. The diameter of the plug is equal to that of the projectile and its thickness is somewhat smaller than the plate’s thickness. The spherical nosed projectile induced a cone-shaped plug with a thickness of about half the plate thickness, as seen in the middle picture. The ogive-nosed projectile induced a failure through the discing phenomenon, which is evidenced by the detached plates at the back of the target. Discing is a common failure mode for high strength materials, and it is due to the bending of the back part of the plate. This bending induces tensile failure along weak planes parallel to the face of the plate, as discussed by Woodward (1990). Wingrove (1973) measured the resisting stress which the plate exerts on the flat ended projectile, during a quasi-static perforation process, by continuously monitoring the load on a projectile which was pushed into the plate by an Instron machine. He found that this force is increasing up to a point where the penetration is about 3.5 mm, after which it decreases gradually until the breakout of the projectile. This means that the process of plug formation is over by the time the projectile is at a distance of about one diameter from the back surface of the target. We shall discuss this issue later on, when we analyze the perforation of plates by flat-ended projectiles. In summary, these pictures demonstrate the wealth of different failure modes which take place during perforation, complicating the analysis as far as engineering models are concerned. The present chapter deals with perforation processes for which a reasonable understanding exists through numerical simulations and analytical models. A comprehensive review of various analytical models for plate perforations has been given by Corbett et al. (1996). We shall focus on perforation processes of metallic plates

Fig. 4.1 Failure modes during plate perforation by rigid projectiles of different nose shapes

4.1 General Description

141

by rigid projectiles of different nose shapes (sharp, spherical and blunt). We shall also discuss the perforation of concrete slabs by ogive-nosed rigid projectiles, and present an analytical model which accounts for their ballistic limits and residual velocities. The perforation of ductile metallic plates by sharp nosed projectiles is the simplest process for analytical modeling, since it is achieved by the so-called ductile-hole-enlargement mechanism. This interaction will be discussed in detail in the next section. The perforation of thin plates is characterized by the dishing process, through which the plate is heavily stretched and bent (globally deformed) in a large area around the impact point. At certain impact conditions the large bending of thin plates is followed by a petaling phenomenon, which results in a fracture of the deformed plate around the perforation hole. The petaling process was analyzed by Landkof and Goldsmith (1985), Atkins et al. (1998) and Wierzbicki (1999) by considering the energy involved in hoop stretching. Since our interest is focused on terminal ballistics for armor applications we shall not discuss the petaling process, and the interested reader is referred to these articles. The perforation of plates by blunt projectiles is much more complex since several failure modes take place during this process, and the borderlines between the corresponding regimes are unclear. The specific type of failure mode depends on the normalized thickness of the plate (H/D), where D is the diameter of the projectile. This ratio is often used in order to classify a target as either thin or thick, as we shall see here.

4.2 The Perforation of Ductile Plates by Sharp Nosed Rigid Projectiles The perforation of ductile plates by sharp nosed projectiles is the simplest perforation process, since it does not involve a failure mechanism in the plate. The passage of the projectile through the plate is achieved by pushing the target material to the side, through the ductile-hole-enlargement mechanism. Such a process, for a conical nosed steel projectile perforating a mild steel plate, from Woodward (1978), is shown in Fig. 4.2. One can see that a significant amount of target material has been pushed along the projectile’s direction of motion, creating the front and back “lips” in the Fig. 4.2 A conical nosed projectile perforating a mild steel plate

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4 Plate Perforation

target. The aluminum plate shows no bending and there is no evidence for material failure inside the plate. Rosenberg and Dekel (2009b, 2010c) followed the forces which act on the projectile during such perforations by using numerical simulations. In order to simplify the analysis the constitutive relations of the plate materials were based on the simple von-Mises yield criterion without the complicating effects of strain rate and strain hardening. Since most materials show a certain amount of strain hardening, we recommend that the strength of such materials in the numerical simulations will be taken as their dynamic flow stresses. These are the high strain values of their stress–strain curves as determined by the Kolsky bar system. As discussed in Chap. 1, these tests can reach strain rates in the range of 103 –104 s−1 , which are the relevant rates for impact velocities at the ordnance range. Later on, we shall deal with the desired dynamic test for material properties. We start the analysis by considering the deceleration characteristics of the rigid projectile, as obtained by the numerical simulations of Rosenberg and Dekel (2009b). These simulations for a conical nosed tungsten projectile with D = 8.3 mm and Leff = 25.6 mm, impacting aluminum plates with strength of 0.4 GPa, followed the experiments of Forrestal et al. (1990). The simulation results for the residual velocities of the projectiles were very close to the experimental data for all the plate thicknesses. Thus, they were used by Rosenberg and Dekel (2009b) in order to highlight the physics of the perforation process. The first issue which can be demonstrated with such simulations, concerns the nature of the deceleration histories for these projectiles during perforation. The deceleration histories for the conicalnosed projectiles, impacting plates of different thicknesses at V0 = 800 m/s, are shown in Fig. 4.3a. It is clearly seen that, except for the H = 100 mm plate, these decelerations do not reach a constant level, due to the combined influence of the front and back free surfaces. Even for the 100 mm plate, with H/D = 12, a constant deceleration is achieved only for a very short time, which terminates at about 120 microseconds after impact, as shown in Fig. 4.3a. This time marks the onset of the influence of the plate’s back surface, and it is clear that even for the thickest plate the perforation is achieved mostly through a time-varying deceleration. In contrast, the deceleration of this projectile in a semi-infinite target reaches a constant value after about 100 microseconds, as shown in Fig. 4.3b. This is the time duration for a penetration depth of about 6D, which is the dominance range of the entrance phase for sharp nosed penetrators, as discussed in Chap. 3. Rosenberg and Dekel (2010c) found that the back surface starts to exert its influence on the projectile when it is about three diameters away from this surface. Since the dominance range of the impact phase is 6D, we may conclude that the thickness of a finite plate should be at least 10D, in order to exert its full resisting stress. Even for these large thicknesses most of the penetration process is achieved through a non-constant deceleration, as shown in Fig. 4.3. Due to this time variation, these decelerations cannot be used to construct a simple analytical model for finite plate perforation. This is the main reason for the fact that researchers use integral approaches, which are based on energy and momentum conservations, such as the model of Recht and Ipson (1963). This is a relatively simple model which accounts for a vast range

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143

Fig. 4.3 The deceleration histories of a short projectile impacting several aluminum plates at 800 m/s

of experimental data for both rigid projectiles and long rods. The model, which will be termed the RI model, is described below and its validity will be demonstrated through both experimental results and numerical simulations. Moreover, the basic approach of this model, namely, the use of energy conservation, will be adopted for many cases of plate perforation, as descried in this chapter. The RI model considers the reduction in the projectile’s kinetic energy, as a result of plate perforation, through a simple energy conservation argument. Since the projectile is assumed to be rigid this energy reduction has to be equal to the work invested in the target in order to open a hole for the passage of the projectile. The kinetic energy of the plate and the energy invested in its bending or stretching, are ignored by the RI model. Thus, the energy balance between the projectile’s kinetic energy and the work invested in the target (W p ), is given by:

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4 Plate Perforation

1 1 M V02 = M Vr2 + W p 2 2

(4.1a)

where V 0 and V r are the impact and the residual velocities, respectively. By definition, for an impact at the ballistic limit velocity the residual velocity is equal to zero, and one can write. 1 M Vbl2 = W p 2

(4.1b)

Combining the two equations results in the following relation: Vr2 = V02 − Vbl2

(4.2a)

and its normalized version: Vr = Vbl



V0 Vbl

2 −1

(4.2b)

which has been found to be extremely successful in accounting for the data concerning sharp-nosed rigid penetrators perforating ductile plates, as was shown by Rosenberg and Dekel (2009b). Figure 4.4 summarizes experimental results from Forrestal et al. (1987, 1990) and Piekutowski et al. (1996), for ogive and conical nosed steel rods perforating aluminum plates. One can clearly see that the model’s predictions are in excellent agreement with the data for a large range of impact velocities. The nature of (4.2b) leads to an asymptotic approach of V r to V 0 with increasing impact velocities, since the work needed to perforate the target is getting much smaller than the kinetic energy of the projectile. The inherent assumption of the RI model is that the work needed to perforate the plate does not depend on the impact velocity. Fig. 4.4 The agreement between the RI model, (4.2), and data from several sources

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145

The numerical simulations of Rosenberg and Dekel (2009b) for different projectiles, targets, and impact velocities support the basic assumption of the RI model, that the energy spent on perforating a given plate is practically independent on the impact velocity. In order to demonstrate this issue, consider the simulation results for the conical nosed tungsten projectile perforating a 12.7 mm aluminum plate, with strength of 0.4 GPa. For impact velocities of V 0 = 300, 500 and 800 m/s, these simulations resulted in residual velocities of: V r = 207, 449 and 769 m/s, respectively. Using (4.2a) we get the inferred value for the ballistic limit velocities: V bl = 217.1, 220, and 220.5 m/s for these simulations, respectively. Thus, the inferred ballistic limit of this projectile/target combination is V bl = 218.8 ± 1.7 m/s. The small variation in the value of Vbl enhances the claim that the work needed to perforate this plate, as inferred by (4.1b), is practically independent on impact velocity. Obviously, a significant variation in the inferred values of V bl would have indicated that the work needed to perforate the target is velocity dependent, and complicate the analysis. Similar results were obtained with simulations for the other plate thicknesses, which also resulted in small variations (less than 2%) in the inferred values of V bl for different impact velocities. In conclusion, these simulation results support the main assumption behind the RI model, as manifested by (4.1a) and (4.1b). These results also demonstrate the usefulness of numerical simulations for the validation of basic assumptions behind analytical models. One may consider relations (4.2) as “universal” in the sense that they apply to all perforation events which are based on the ductile-hole-enlargement process. Thus, the curve shown in Fig. 4.4 should account for all projectile/target configurations which satisfy this requirement. The main significance of the RI model, as manifested by (4.2a) and (4.2b), is that there is no need to perform many experiments in order to determine the V r (V 0 ) curve for a given projectile/plate combination, if the perforation is achieved by the ductile hole enlargement process. In fact, one has to determine the value of V bl , accurately enough, and then use (4.2b) to derive the residual velocity for any impact velocity. Thus, the whole physics of this perforation process is manifested by the value of V bl for the given projectile/plate pair. In fact, the value of V bl can be determined by a single experiment in which the impact and residual velocities are measured to a great accuracy, by inserting these velocities in (4.2a). With this inferred value of V bl the normalized curve of V r /V bl as a function of V 0 /V bl is obtained through (4.2b). It is important to note that the normalized equation has a very steep slope for impact velocities slightly above the ballistic limit velocity, since the curve is practically perpendicular to the horizontal axis at V 0 = V bl . Thus, small variations in material properties may cause a large scatter in the measured residual velocities at this range, as is often the case with the experimental results. By analyzing the various sources for experimental errors, Rosenberg and Dekel (2009b) recommended that the single experiment, for the determination of V bl , should be performed at an impact velocity of about 1.5 V bl . Such an impact velocity is beyond the steep rise of the V r (V 0 ) curve, and it is not high enough to be in the range where small errors in the measured residual velocity cause large variations in the inferred ballistic limit velocities.

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In order to demonstrate the importance of these issues consider the data of Dey et al. (2004) concerning the perforation of 12 mm thick plates, of three different steels, by ogive-nosed steel projectiles with L eff = 80 mm and D = 20 mm. The plates were made of the Weldox 460E, 700E, and 900E steels. Four experiments were performed for each type of steel at impact velocities above the corresponding ballistic limits, and several experiments were performed below these limits. Dey et al. (2004) determined the values of V bl by averaging the lowest impact velocity which resulted in plate perforation, and the highest velocity for which there was no perforation. This is the accepted technique to determine the value of the ballistic limit velocity, as explained in Chap. 1. With these values of V bl they fitted their data for V r /V bl through an empirical relation which is different than the physically-based relation of the RI model, (4.2a). We wish to emphasize that such an empirical fit does not serve the science of terminal ballistics, even when it results in a better agreement with the data, as compared with a physically-based model. The impact velocities in most of the experiments of Dey et al. (2004) were only slightly higher than the corresponding V bl values, and repeated experiments with the same impact velocities did not result in the same residual velocities. These facts led to the result that the data did not adhere to the RI model, which led Dey et al. (2004) to use empirical fitting procedures. A more physically-based approach to these experiments is suggested next. The first step in the analysis is to determine the inferred value of V bl for each experiment through (4.2a), and average these values in order to determine the value of V bl for each plate. This procedure results in the following values for the experiments of Dey et al. (2004): V bl = 282.0, 317.6, and 312.5 m/s for the 460E, 700E, and 900E steel plates, respectively. These inferred ballistic limits are slightly different than those determined by Dey et al. (2004): V bl = 290.6, 318.1, and 322.2 m/s, but they are consistent with the physically-based RI model. Moreover, these values for the inferred ballistic limits can highlight important issues concerning the perforation mechanisms of the different plates. As an example, consider the inferred values of V bl for the 700E and the 900E steels which result in a non-monotonic change of V bl with target strength. This trend indicates that the perforation process of the hardest steel plate cannot be described by the ductile-hole-enlargement process, probably because of some failure mechanism which took place during its perforation. The same non-monotonic trend was also observed for the conical nosed projectiles in this study, and it was even more pronounced in the experimental results for the flat ended projectiles. As we shall see below the perforation process by the ductile-holeenlargement mechanism should result in a monotonic increase of V bl with both plate thickness and its strength. Thus, by analyzing the data through the physically-based RI model one can follow such deviations from the ductile hole-enlargement-process, as described here. The following discussion concentrates on this penetration process in metallic plates, with the aim of establishing a relation between the compressive strength of the plate material and its resistance to peroration. Such a relation will lead to a numerically-based analytical model for the ballistic limit velocity of a given projectile/plate pair, in terms of their relevant physical properties.

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There have been several attempts at constructing analytic models for the ballistic limit velocity which are based on the work invested in the target during perforation. This issue has been analyzed by the great physicists, the Nobel laureate Bethe (1941) and Taylor (1948), who worked at the war ministries in the U.S. and in England during the Second World War. Short summaries of their work can be found in Woodward (1978), Corbett et al. (1996) and in Billington and Tate (1981). Specifically, these two approaches were aimed at determining the resisting stress which has to be overcome in order to open a hole in a metallic plate. In order to simplify their analysis, Bethe and Taylor assumed plane stress conditions in the plate and that the diameter of the hole is of the order of the plate thickness (H). The expressions which they obtained for this work (W p ) were similar but differed by their numerical factors, according to. W p = πr 2 H · 2Yt

Bethe [1941]

(4.3a)

W p = πr 2 H · 1.33Yt

Taylor [1948]

(4.3b)

where Y t is the strength of the plate material and r is the radius of the hole which is, in fact, the radius of the perforating projectile. One can interpret these equations as defining the effective stresses which resist the opening of the hole, which are equal to 2Y t and 1.33Y t according to the analyses of Bethe and Taylor, respectively. These values are lower than the threshold pressures which are needed to open a spherical or cylindrical cavity in a large elasto-plastic solid, as calculated (also during WW II) by Bishop et al. (1945). As was described in Chap. 3, the cavity expansion analysis results in values of the order of (3–4)Y t for these threshold pressures. The values obtained from (4.3a) and (4.3b) are much lower than the resisting stresses on a rigid projectile during its penetration in a semi-infinite target, which amount to Rt = (5–6) Y t , as derived in Chap. 3. The lower resisting stresses from (4.3) are due to the fact that the stress state in a finite thickness plate is close to a plane stress, where the stress normal to the plate is zero. On the other hand, the three dimensional state of stress in a semi-infinite target is closer to a plane strain situation, where all stresses are non-zero, resulting in much higher resisting stresses. Rosenberg and Dekel (2010c) presented a numerically-based approach in order to derive the effective resisting stress on the perforating projectile, by using 2D numerical simulations. Their basic idea was to replace the actual time-varying stresses by a constant (effective) stress, which results in the same energy loss for the projectile. This effective resisting stress (σ r ) was defined through the equation of motion for the rigid projectile as it perforates a plate of thickness H. F=M

dV dV = MV = πr 2 σr = const. dt dx

(4.4)

where M is the projectile’s mass and r is its radius. Integrating this equation between the boundaries: V 0 = V bl at x = 0, and V = 0 at x = H, results in:

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4 Plate Perforation

M Vbl2 = π · r 2 H σr 2

(4.5a)

Using the definition of L eff , through M = ρ p πr 2 L eff , and rearranging (4.5a), lead to the following relation for V bl :  Vbl =

2H σr ρp Lef f

(4.5b)

This simple equation can be used in order to derive the ballistic limit velocity for any projectile/plate pair, if the corresponding value of σ r is known. In principle, one can determine the value for V bl even without performing a single experiment if a reliable model is constructed for σ r , in terms of the relevant parameters for the projectile and the target plate. One of these parameters is, obviously, the thickness of the plate (H). The dependence of σ r on H should be quite strong, considering the values which σ r attains for the extreme cases of thin and thick plates, as discussed above. A plausible assumption is that the relevant measure for the plate thickness is the ratio H/D, where D is the diameter of the projectile. Thus, thin plates are those with H/D  1.0, plates of intermediate have H/D = 1, and thick plates are those with H/D  1.0. The analytical model of Thomson (1955) for thin plates, gives the lower bound on σ r by considering the perforation process through the dishing mechanism, which is shown schematically in Fig. 4.5. According to Thomson’s model the resisting stress on the projectile approaches the value of σ r = Y t /2 for very thin plates. The dishing mode of perforation involves also plate stretching and bending, which have to be analyzed through further modeling, as in Woodward and Cimpoeru (1998). We can summarize now the analytical expressions for the resisting stresses from the models presented above. According to Thomson’s model the limiting value of the resisting stress is equal to σ r = Y t /2 for very thin plates. For intermediate plate thicknesses, of the order of the projectile’s diameter, the models of Bethe (1941) and Taylor (1948) result in values in the range of σ r = (1.33–2) Y t . As the thickness of the plate increases its free surfaces become less influential, and σ r should approach values of Rt = (5–6) Y t , as derived in Chap. 3. Thus, by moving from very thin plates to very thick ones, the effective resisting stresses (σ r ) are expected to increase by

Fig. 4.5 Schematic description of the dishing process in thin plates

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an order of magnitude. Another reasonable assumption is that the proper measure of a plate’s resistance to perforation is the normalized stress σ r /Y t , where Y t is the compressive strength of the plate material. The following discussion of the perforation process by the hole enlargement process, is aimed at determining the relations between the normalized effective stress (σ r /Y t ) and the normalized thickness of the plate (H/D). The best way to perform a rigorous parametric study for these relations is through numerical simulations with well-defined variations for each parameter. This was done by Rosenberg and Dekel (2010c) who constructed a numerically-based model for σ r , through a large number of simulations with different projectile/plate combinations. With these simulations they were able to determine the dependence of σ r on the plate’s thickness and on the strength of the plate material. The results were analyzed in terms of the relation between the normalized parameters σ r /Y t and H/D, for steel and aluminum plates of different strengths, and thicknesses in the range of H/D = 0.026–15. The nose shapes of the projectiles were either conical or ogival with different sharpness, in order to cover a large range of sharp-nosed projectiles. The value of σ r for each projectile/target combination was determined through the following steps: (1) several simulations with impact velocities above V bl resulted in the corresponding values for the residual velocities (V r ), (2) these values of V r were inserted in (4.2a) and the inferred ballistic limit velocity was obtained, (3) this value of V bl was inserted in (4.5b), and σ r was determined for the particular projectile/plate pair. The normalized values of σ r /Y t as a function of H/D for all the different projectile/plate combinations, in Rosenberg and Dekel (2010c), are shown in Fig. 4.6. These simulation results can be grouped into three families according to their corresponding values for H/D. For thin plates (H/D ≤ 1/3), the results follow a straight line with a relatively steep slope, which is given by:   H H 1 2 σr for ≤ = +4 Yt 3 D D 3

(4.6)

For intermediate plate thicknesses (1/3 ≤ H/D ≤ 1.0), the results fall on a horizontal line, according to: H σr 1 ≤ 1.0 = 2.0 for ≤ Yt 3 D

(4.7)

and since σ r = 2Y t for this range of plate thicknesses, a simple relation is obtained for their V bl values, from (4.5b), as follows: V bl = 2(HY t /ρp L eff )0.5 . For thick plates with H/D ≥ 1.0, the values of σ r increase asymptotically towards their corresponding Rt values for semi-infinite targets. Rosenberg and Dekel (2010c) chose the following relation between (σ r /Y t ) and (H/D), to represent their numerical results for these plates:   H H σr for ≥ 1.0 = 2.0 + 0.8 ln Yt D D

(4.8)

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4 Plate Perforation

Fig. 4.6 The simulation results for σ r /Y t versus H/D. a Thick plates with H/D ≥ 1.0 and b intermediate and thin plates with H/D < 1.0

Inserting the values for σ r , as obtained from these expressions, in (4.5b), results in the V bl for each of the three cases described here. A simple curve fitting procedure, through the three branches of H/D, results in the following relation between V bl and H/D, which can be used for convenience purposes for H/D values up to about 5.0:  1.5  0.5   Vbl H H H − 0.066 = 0.66 + 0.426 V∗ D D D

(4.9a)

where the normalizing parameter (V *) is the ballistic limit velocity for a plate thickness which is equal to the projectile’s diameter, as given by: V∗ = 2



DYt ρp Lef f

0.5 (4.9b)

Several points are worth noting here. The numerical results of σ r for all the projectile/plate combinations are within ±5% of these expressions. Thus, the maximum

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151

error in determining V bl , through (4.5b), should be ±2.5% which can be considered as a reasonable agreement between model and data. As will be demonstrated below, Rosenberg and Dekel (2010c) showed that such an agreement is, indeed, obtained between the predicted V bl values and several sets of experimental data. Note that the constant value of σ r /Y t = 2.0, for the intermediate plate thicknesses with 1/3 < H/D < 1.0, is exactly the value calculated by Bethe (1941) as given by (4.3a). This agreement is very satisfying since Bethe’s analysis deals with plates of intermediate thicknesses, where H ≈ D. The constant plateau in the values of σ r /Y t , for the intermediate plate thicknesses, is probably due to the plane stress conditions which characterize the stress state of these plates. The transition to the plane strain condition takes place, apparently, at a value of H/D = 1.0. Obviously, these observations need further analytical modeling. The asymptotic increase in σ r /Y t with H/D for thick plates is quite expected since the values of σ r should approach the corresponding values of Rt = (4–6) Y t , which were derived in Chap. 3. As was mentioned above, according to the model of Thomson (1955) the limiting value of σ r , for diminishing plate thicknesses, is Y t /2. This value is close to the limiting value of 2Y t /3 from these simulations, as given by (4.6). One may wonder about the physics behind a mechanism which assigns a finite resisting stress to a plate whose thickness approaches zero. A qualitative explanation can be given by considering the fact that as the plate is getting thinner, a larger plate area is participating in the dishing process, resulting in a non- vanishing value for the energy absorbed by the dishing process. The simulations of Rosenberg and Dekel (2010c) for thin plates (H/D ≤ 1/3) showed that the plates were heavily bent and that, as expected, a dishing process characterized the perforated area. In fact, for a normalized plate thickness of about H/D = 1/3, the perforation process changed from dishing to the ductile hole enlargement, as shown in Fig. 4.7. These simulations were performed for aluminum plates perforated by an ogive- nosed projectile, with D = 6 mm. The plate thicknesses were H = 1.5, 1.8 and 3 mm, corresponding to normalized values of H/D = 0.25, 0.3 and 0.5, respectively. The ductile hole enlargement process, in the thicker plate (H/D = 0.5), is characterized by a very small plate bending and by the appearance of front and Fig. 4.7 The change from dishing to the hole enlargement process with increasing plate thickness

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4 Plate Perforation

back “lips” around the hole. The dishing process is clearly evident for the thinnest plate (H/D = 0.25), and to a lesser extent for the H/D = 0.3 plate. Considering these simulation results, one may conclude that the value of H/D = 1/3 marks the transition, in the perforation mechanism, from dishing to ductile hole enlargement. Due to the scatter in the simulation results, as shown in Fig. 4.6b, this value should be viewed as an average value for different projectile/plate combinations. In conclusion, the results from this numerically-based model are in agreement with the analytical models of Thomson (1955) and Bethe (1941), and with the expected increase of σ r for very thick targets. These numerical simulations covered a large variety of projectiles and plates and the resulting relations for σ r , as given by (4.6), (4.7) and (4.8), can be used for the derivation of V bl values for many projectile/plate combinations, through (4.5b). In order to demonstrate the predictive capability of this numerically-based model for V bl , in conjunction with the RI model for V r /V bl , we compare their predictions with several sets of experimental data from different sources, which cover a large range of H/D and Y t values. Consider first the experiments of Borvik et al. (2004) for conical nosed steel projectiles, with L eff = 80 mm and D = 20 mm, perforating 5083-H116 aluminum plates with thicknesses in the range of H = 15–30 mm. The corresponding H/D ratios ranged between H/D = 0.75–1.5. The experimental data for the normalized velocities, V r /V bl versus V 0 /V bl are given in Fig. 4.8. The ballistic limit velocities, as obtained by these experiments, were: V bl = 216.8, 249, 256.6, and 314.4 m/s for the H = 15, 20, 25, and 30 mm plates, respectively. The predicted curve for V r /V bl in terms of V 0 /V bl from (4.2b) of the RI model is also shown in this figure, and it is clearly seen that this model accounts for the experimental results. In order to check the ability of the numerically-based model to account for the experimental values of V bl , the flow stress of the aluminum plates has to be known. The different plates which were used by Borvik et al. (2004) had different flow stresses, as depicted by their corresponding stress–strain curves in Borvik et al. Fig. 4.8 Comparing model with data for the normalized residual velocities of conical-nosed projectiles perforating aluminum plates

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(2009). Thus, the following values for Y t were considered in the model: Y t = 0.48 GPa for the 15 mm plate, Y t = 0.45 GPa for the 20 mm and 30 mm plates, and Y t = 0.4 GPa for the 25 mm plate. Inserting these values in (4.7) for the 15 mm and 20 mm plates, and in (4.8) for the 25 mm and the 30 mm plates, result in the following values for the effective resisting stresses: σ r = 0.96, 0.9, 0.87 and 1.06 GPa, for the H = 15, 20, 25 and 30 mm plates, respectively. Note that the values of σ r for the thin plates (15 and 20 mm thick) are higher than the corresponding value for the 25 mm plate. This is due the lower flow stress of the 25 mm plate as given by Borvik et al. (2009). Inserting these values of σ r in (4.5b) results in the predicted values of: V bl = 214, 239.5, 263.2 and 318.2 m/s, for the 15, 20, 25 and 30 mm plates, respectively. These values are in excellent agreement with the corresponding experimental values of Borvik et al. (2004): V bl = 216.8, 249, 256.6, and 314.4 m/s. As far as these experiments are concerned, the RI model for the residual velocities, and the numerically-based model for V bl from Rosenberg and Dekel (2010c), successfully account for the data of sharp-nosed projectiles perforating ductile metallic plates. Cheeseman et al. (2008) determined V bl values for relatively thick 2139-T8 aluminum plates, with compressive strength of Y t = 0.5 GPa, which were impacted by 0.3 APM2 and 0.5 APM2 projectiles. The normalized thicknesses of the aluminum plates were in the range of: H/D = 3.6–6.6, for which (4.8) has to be used in order to determine the effective resisting stress (σ r ) for each plate. This value is then inserted in (4.5b) in order to derive V bl for the specific projectile/plate pair. The model’s predictions were calculated for the hard steel cores of these projectiles, with D = 6.2 mm, L eff = 22.4 mm for the 0.3 core, and D = 10.8 mm, L eff = 34.8 mm for the 0.5 core. Table 4.1 shows the excellent agreement between the predicted V bl values and the experimental V 50 values from Cheeseman et al. (2008). Gupta et al. (2007) used thin (0.5–3.0 mm) plates of 1100-H12 aluminum, with strength of Y t = 0.28 GPa, which were perforated by ogive nosed hollowed steel projectiles, with D = 19 mm and L eff = 23.6 mm. This is a case study for very low H/D values, ranging from 0.0263 to 0.158. The predicted values of σ r were calculated by (4.6), which applies for normalized thicknesses in the range of H/D ≤ 1/3. Table 4.2 shows the agreement between these predicted V bl values and the experimental data which were bounded by the highest impact velocity with no perforation (V 1 ), and the Table 4.1 Comparing the model’s predictions for V bl (in m/s) with the data from Cheeseman et al. (2008) 0.3 projectiles

0.5 projectiles

H (mm)

V 50 (exp)

V bl (pred)

H (mm)

V 50 (exp)

V bl (pred)

25.2

682.6

672

39.0

657.3

653.0

32.3

783.2

780

40.0

668.0

663.4

39.0

860.9

876

40.9

677.4

673.0

40.9

893.0

902

52.1

785.4

783.3

57.2

819.5

830.0

64.1

873.5

884.5

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4 Plate Perforation

Table 4.2 Comparing the model’s predictions for V bl with the data from Gupta et al. (2007) H (mm)

H/D

V 1 (m/s)

V 2 (m/s)

V bl (pred)

3.0

0.158

90.4

96.9

108.4

2.5

0.1315

79.4

96.7

94.6

2.0

0.105

67.2

83.7

81.0

1.5

0.079

54.3

62.9

66.4

1.0

0.0526

45.3

51.3

51.5

0.71

0.03

38.4

44.3

41.4

0.5

0.0263

33.7

40.7

34.1

lowest impact velocity with perforation (V 2 ). The good agreement between model’s predictions and experimental results enhances the validity of the numerically-based model for very low H/D values, where dishing is the main perforation mechanism. Up to this point both the simulations and the experiments dealt with materials which behave in an ideal elastic-perfectly plastic manner, obeying the von-Mises yield criterion. Many metals and alloys experience a significant strain-hardening effect, without having a well-defined value for their flow stress. In order to investigate the effect of strain-hardening on the perforation process, Rosenberg and Dekel (2010c) performed a series of numerical simulations for conical nosed rigid steel projectiles, perforating plates of the 304 stainless steel. The constitutive parameters for the 304 steel in these simulations were taken from Steinberg’s model, as given in the code library: Y = Y 0 (1 + βp εp )n , with Y 0 = 0.34 GPa, βp = 43, and n = 0.35. This material exhibits a significant hardening effect with a strength increase from Y 0 = 0.34 GPa at yield, to Y max = 2.5 GPa at failure. These simulations, for various plate thicknesses, were analyzed by the method described above. Namely, the impact and residual velocities were used to determine V bl for each plate thickness, and this value was inserted in (4.5b) in order to derive the corresponding value of σ r . The results of these simulations, in terms of σ r as a function of H/D, are shown in Fig. 4.9. Obviously, the values for σ r were not normalized in this presentation, since Y t does not have a well-defined value for this strain-hardening steel. Figure 4.9a presents the simulation results for the whole range of plate thicknesses, and Fig. 4.9b is an expanded view of the results for plate thicknesses in the range of H/D ≤ 1.5. All the features observed for the elastic-perfectly-plastic solid are also evident for the strain-hardening 304 steel. These features include the linear rise of σ r with H/D for low values of H/D, a constant plateau for H/D = 0.4–1.5, and the subsequent increase of σ r for H/D > 1.5. The fact that the transition from dishing to the hole enlargement process takes place at about H/D = 0.4, rather than at H/D = 1/3, is probably due to the strain hardening property of this steel. The same argument holds for the span of the plateau region which extends to H/D = 1.5, rather than to H/D = 1.0 for the elastic-perfectly-plastic plates. The simulations resulted in a value of σ r = 1.66 GPa at the plateau in Fig. 4.9, between H/D = 0.4 and H/D = 1.5. We now assume that the same ratio of σ r /Y t = 2.0, which was found for the elasto-plastic metals, also holds for strain-hardening

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Fig. 4.9 Simulation results for 304 stainless steel plates perforated by conical-nosed rigid projectiles

materials, which results in an effective strength of Y eff = 0.83 GPa for the 304 stainless steel. Steinberg’s constitutive relation which was used in these simulations, shows that this value of Y eff corresponds to an effective value of εp = 0.27 for the plastic strain. This issue will be dealt with in Sect. 4.6, where we analyze the stresses and strains in the plate’s elements around the perforation hole. Extrapolating the straight line in Fig. 4.9b at the lower range of H/D to the limit of H/D = 0, results in a value of σ r = 0.55 GPa. This value corresponds to a ratio of σ r /Y eff = 2/3 for the limiting stress of very thin stainless steel plates, which is the same ratio we obtained for the elastic-perfectly-plastic plates. A linear fit through the simulation results for H/D ≤ 0.4, as shown in Fig. 4.9b, gives a slope of 3.33 which is somewhat lower than the corresponding slope of 4.0 for the von-Mises materials. One can conclude that these results account for the response of other strain-hardening materials, and that their σ r /Y eff values for thin plates should be bounded between (4.6) and the corresponding relation for the 304 steel: H 2 σr = + 3.33 Ye f f 3 D

for H/D ≤ 0.4

(4.10)

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4 Plate Perforation

and the exact values for Y eff depend on the parameters which describe the stress– strain curve of the strain-hardening material (Y 0 , Y max , βp and n). More work is needed in order to properly account for the numerically-derived relations of strain hardening solids. The analysis presented here applies for ductile plates which experience no failure during the perforation process. When quasi-brittle plates, such as the strong aluminum alloys 2024-T351 or 7075-T651, are impacted by sharp nosed rigid projectiles they may experience significant failure, with large pieces of material sheared off their back sides through scabbing and discing. An example for such a discing process, in a 2014-T6 aluminum plate impacted by an ogive nosed projectile, was shown in Fig. 4.1. Such a failure results in a reduced effective thickness of the target, which lowers its ballistic limit velocity. This was the case with the experiments of Borvik et al. (2010), who impacted ogive nosed projectiles with L eff = 80 mm and D = 20 mm, at 20 mm thick 7075-T651 aluminum plates. Cross sections of these plates showed that a large volume from their back side was detached through this quasi-brittle failure. The resulting value for the ballistic limit velocity for this plate was V bl = 208.7 mm. This value is much smaller than the corresponding value of V bl = 249 mm for a 20 mm thick 5083-H116 aluminum plate, which was perforated by a conical nosed projectile of similar dimensions, as determined by Borvik et al. (2004). Thus, in spite of an increase by a factor of nearly 2.0 in their strength, the value of V bl for the stronger 7075 alloy is lower (by about 15%) than that of the ductile 5083 plate, due to the back surface failure of the quasi-brittle aluminum. The damaged volume at the back of a brittle plate is directly related to the diameter of the impacting projectile. Thus, the failure in a given 7075-T651 aluminum plate is expected to be less important for perforation tests with smaller projectiles, and the results of these experiments are expected to follow the trends discussed above. In particular, the ballistic limit velocities of plates with large H/D values should be proportional to their corresponding strengths through (4.5b). This was, in fact, the result obtained by Forrestal et al. (2010) for 20 mm and 40 mm aluminum plates of 7075-T651 and 5083-H116 alloys, which were impacted by 0.3 AP projectiles. For example, the experimental values of V bl were: 513 and 633 m/s for the 20 mm plates of the 5083 and 7075 alloys, respectively. Thus, the stronger plate resulted in a higher value for V bl against the hard steel cores of the 0.3 AP projectile (D = 6.17 mm), as expected by the model. We can also compare the predictions from the numerically-based model, with these experimental results. Using (4.8) one obtains for these 20 mm plates (with H/D = 3.24) a value of σ r /Y t = 2.94 for the normalized resisting stress. The corresponding stress–strain curves for the two alloys show that their flow stresses are 0.4 and 0.68 GPa, for the 5083 and 7075 alloys, respectively. Thus, the resisting stresses of these projectile/plate pairs are σ r = 1.176 and 1.716 GPa, respectively. Using (4.5b) we get the predicted values of V bl = 517.5 and 639 m/s, for the 5083 and 7075 plates, respectively. These values are in excellent agreement with the experimental results of 513 and 633 m/s, strongly enhancing the validity of the numerically-based model. Moreover, this agreement supports the claim that the quasi-brittle nature of relatively thick plates does not affect their ballistic performance.

4.2 The Perforation of Ductile Plates by Sharp Nosed Rigid Projectiles

157

One of the practical questions in armor design is whether a laminated target is more efficient than a monolithic target of the same total thickness. One can use the above relations in order to highlight this issue for sharp nosed projectiles perforating ductile targets. Consider a set of N plates each of thickness h, separated so that the projectile exits each plate before impacting the subsequent one. The residual velocity of the projectile as it exits from each plate is equal to its impact velocity at the next one. Thus, one can write the energy balance, (4.2a), for each plate and obtain a set of N equations: 2 Vri2 = V0i2 − Vbli

(4.11a)

Vri = V0(i+1)

(4.11b)

where

Summing up these equations and noting that identical plates (thickness and material) have the same V bl values, we find that the final residual velocity is given by: Vr2 = V02 − N Vbl2 (h)

(4.12a)

where V bl (h) is the value for a plate of thickness h. Thus, the ballistic limit velocity of this widely-spaced laminated target is given by: Vbl (N × h) =



N · Vbl (h)

(4.12b)

One can estimate the efficiency of a laminated target by considering its V bl value, as given by (4.12b), in comparison with that of a monolithic target of the same total thickness (H), namely V bl (H). This estimate can be made by comparing the terms (σ r H)0.5 and (σ r h)0.5 which appear in (4.5b) for V bl . If σ r has the same value for the monolithic plate (of thickness H) and for the single plate in the laminate (of thickness h), the two targets should have the same V bl values. This situation occurs only when both H/D and h/D are in the plateau range for σ r , with normalized thicknesses between 1/3 and 1.0. For all the other cases σ r (H) is higher than σ r (h), and the monolithic target should have a higher value for V bl than the laminated one. An intermediate case is obtained when the plates are closely packed and they support each other during perforation. Thus, one can write: Vbl (H ) ≥ Vbl (N × h) closely packed ≥ Vbl (N × h) widely spaced =



N Vbl (h) (4.13)

A quantitative estimate for these inequalities can be obtained by considering an example for a plate of thickness H = 4D, which is split in several ways. For the monolithic plate, with H/D = 4, we find from (4.8) that σ r = 3.11Y which, by (4.5b), results in V bl = 5(YD/ρp L eff )0.5 for this plate. For the sake of simplicity

158

4 Plate Perforation

let us define the term V * = (YD/ ρp L eff )0.5 , so that for the monolithic plate (with H = 4D) we obtain V bl = 5 V*. The value of V bl for a target composed of two plates, each of thickness 2D, should be V bl = 4.52 V*, according to (4.12b). We have used (4.8) to calculate σ r for the H = 2D plate, and inserted this value in (4.5b) to find that V bl = 3.2 V* for this plate. Repeating the same procedure for a target composed of two plates, of thickness D and 3D, results in V bl = 4.61 V* for this pair. Obviously, the same value is obtained for the two ordering possibilities. Finally, for a target consisting of four plates, each of thickness H = D, we find that V bl = 4 V*. This example gives an estimate of the expected reductions in V bl for different plate laminations, as compared with the value of V bl for the monolithic plate. These estimates also show that the ordering of the plates plays no role, as far as their V bl values are concerned. As mentioned above, these estimates are relevant for sharp nosed projectiles perforating ductile targets through the hole-enlargement process. On the other hand, perforations by blunt projectiles may result in different ordering of the ballistic efficiency, as will be discussed in Sect. 4.4. Dey et al. (2007) reviewed much of the published data on laminated targets and concluded that they show contradicting results. These discrepancies can be explained, at least partly, by the fact that the works they analyzed included projectiles of different nose shapes and targets with different H/D values. In order to demonstrate the validity of the analysis presented here, consider the experiments of Dey et al. (2007). They shot ogive-nosed steel projectiles, with D = 20 mm and L eff = 80 mm, at plates of Weldox 700E steel, and determined the ballistic limit velocities for several plate configurations. The three different setups and their corresponding V bl values were: (1) monolithic plates, 6 mm and 12 mm thick with V bl = 198 m/s and 318 m/s, respectively, (2) two widely separated 6 mm plates with V bl = 280 m/s, and (3) two closely packed 6 mm plates with V bl = 288 m/s. As predicted by (4.12b), the value √ of V bl = 280 m/s for the two widely spaced 6 mm plates, is equal to exactly 2 = 1.414 times the value of V bl = 198 m/s for the single 6 mm plate. For the closely packed 6 mm plates, the experimental value of V bl = 288 m/s is somewhat higher than V bl = 280 m/s for the widely spaced plates, as expected by the inequality (4.13). Thus, these experimental results enhance the validity of the approach presented above. Ben-Dor et al. (2006) summarized the results of several studies concerning the ballistic performance of monolithic and layered targets of the same total thickness. In general, these works support the observations discussed above. For example, Almohandes et al. (1996) impacted monolithic and layered mild steel targets having total thicknesses in the range of 8–14 mm, with a standard 0.3 projectile. They found that monolithic plates were more effective than laminated targets, for all configurations. The difference in performance diminished with impact velocity and the effectiveness of the laminated targets, whether in contact or spaced, decreased with the number of plates comprising each target. Another set of experiments, with aluminum and steel targets, was performed by Gupta and Madhu (1997). They determined their ballistic efficiencies against 0.3 AP projectiles through residual velocities measurements. The experimental results showed that for plates with thicknesses of H/D > 1.0, there is no significant change in the ballistic performance due to layering. For

4.2 The Perforation of Ductile Plates by Sharp Nosed Rigid Projectiles

159

thinner plates they found that the layered targets resulted in higher residual velocities, thus, lower ballistic limits, as compared with the monolithic targets. This is in accord with the analysis presented above as depicted by the inequality (4.13). In addition, spaced plates resulted in somewhat lower ballistic limits than the closely stacked targets. Similar results were obtained by Marom and Bodner (1979) for aluminum plates. The monolithic target performed better than the closely stacked one but its performance was somewhat lower than that of a widely spaced laminate. These results for widely spaced laminates can be explained by considering the fact that the projectile is deflected somewhat by the individual plates, and these deflections are further enhanced by successive impacts. In conclusion, the laminated targets should be less effective against sharp nosed projectile, as compared with monolithic targets of the same total thickness. In Sect. 4.4 we arrive at a different conclusion for the perforation of these targets by blunt nosed projectiles.

4.3 Plate Perforation by Spherical Nosed Projectiles This is an intermediate case between the impacts of sharp and flat-nosed projectiles. It has been observed, empirically, that spherically nosed projectiles perforate a ductile plate in a similar way to that of a sharp nosed projectile, as long as the plate is not too thin. This was shown, for example, by Borvik et al. (2002) who shot hard steel projectiles, with D = 20 mm, at 12 mm plates of Weldox 460E steel, and found that the ballistic limit velocities for the conical and spherically nosed projectiles were practically the same. A close examination of the sectioned plates, in that work, indicates that their perforations were achieved through the ductile-hole-enlargement process for both nose shapes. The only difference between the two sets of experiments was due to the ejection of a small plug from the target ahead of the spherical nosed projectile, which is a common feature for perforations by blunt projectiles. Thus, the perforation process by a spherical nosed projectile is, indeed, an intermediate case between blunt and sharp nosed projectiles. An important property of the experimental V r (V 0 ) curves for blunt projectiles is that they do not tend asymptotically towards the V r = V 0 line, as is the case with sharp nosed projectiles. Instead, the data asymptotically approach a straight line below the V r = V 0 line, with an offset which increases with plate thickness. This offset is due to the creation of the plug, and its magnitude is related to the kinetic energy of the plug, which has to be accounted for in the energy balance equation of these perforations. In order to construct a simple model for the residual velocities of spherically nosed projectiles we follow the energy balance relation in a similar manner to that with sharp nosed projectiles, as given by (4.1a). The only difference is due to the kinetic energy of the plug which has to be accounted for, and one can write: 1 1 1 M V02 = M Vr2 + mVr2 + W p 2 2 2

(4.14)

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4 Plate Perforation

where m is the mass of the plug, which is assumed to move at the same velocity as the exiting projectile (V r ). This is only a rough approximation since these plugs often move ahead of the projectile at a somewhat higher velocity. At the ballistic limit velocity we have V r = 0 which results in MV 2bl /2 = Wp as in (4.1b), and with (4.14) we obtain:  1 · 1 + m/M

Vr = Vbl



V0 Vbl

2 −1

(4.15)

This equation accounts for the asymptotic approach of the experimental V r (V 0 ) curves to straight lines which are offset below the V r = V 0 line. The magnitude of this offset depends on the mass ratio (m/M) as is evident by (4.15). For thin plates, with H ≤ D, one may assume that the plug thickness is equal to that of the plate and that its diameter is close to that of the projectile. For this case the mass ratio is given by: m ρt H =λ = M ρp Lef f

(4.16)

where L eff is the effective length of the projectile. Inserting this value in (4.15) results in the following expression for the residual velocity: Vr = Vbl



1 · 1+λ



V0 Vbl

2 −1

(4.17a)

and if the projectiles are rigid spheres, with an effective length of L eff = (2/3) D, this equation takes the form: Vr = Vbl



1 1+

3ρt H 2ρ p D

 ·

V0 Vbl

2 −1

(4.17b)

In order to demonstrate the validity of this analysis we show in Fig. 4.10 a comparison between measured and calculated values of V r (V 0 ) for the experiments of Senf and Weimann (1973). In these experiments hard steel spheres, with D = 12.7 mm, perforated 2024-T3 aluminum plates with thicknesses in the range of H = 3–25 mm. The following experimental values of V bl were used in (4.17b): V bl = 0.2, 0.34, 0.5, 0.66 and 0.95 km/s, for the H = 3, 6, 10, 15 and 25 mm plates, respectively. Figure 4.10 shows that the predicted curves account for the experimental results and that at high velocities they approach parallel straight lines, which are offset from the V r = V 0 line by the predicted distance. Another set of measurements which can be used to demonstrate the applicability of this model, is given by Borvik et al. (2002) for spherical-nosed steel projectiles,

4.3 Plate Perforation by Spherical Nosed Projectiles

161

Fig. 4.10 Comparing predicted and measured residual velocities for hard steel spheres perforating aluminum plates

with L eff = 80 mm and D = 20 mm, perforating 12 mm thick Weldox 460E steel plates. Choosing a value of V bl = 285 m/s for this projectile/target combination, the comparison between the measured and calculated values of V r , from (4.17a), is given in Table 4.3. The mass of the plug and its velocity which were measured in these experiments are different than those assumed by the model. In fact, the plugs were cup shaped rather than cylindrical, and their thicknesses ranged between (0.5–0.7) H, where H is the plate thickness. The smaller thickness of these plug indicate that the perforation process started with a ductile hole enlargement, and the plugs were formed when the projectiles reached depths of (0.3–0.5) H. The velocities of the plugs in these experiments were higher than the residual velocities of the projectiles by factors which ranged between 1.07 and 1.64. Thus, their kinetic energies were in the range of 0.6–1.8 times the values which are calculated by the model presented above, assuming that their thickness is H and that their velocities are equal to V r . The error introduced by these simplifying assumptions is not large since the mass of the plug was only about 15% of the projectile’s mass in these experiments. In conclusion, the perforation model for spherical-nosed projectiles which was introduced here is similar to the RI model for sharp nosed projectiles, except for the presence of the plug. Once the value of V bl is determined, the model accounts for the residual velocities of the projectile and the ejected plug, assuming that they move at the same velocity. An analytical model for the value of V bl is still needed, and a numerically based model may be the easiest approach. Another issue which has to be resolved concerns the actual thickness of the plug ejected from plates which are Table 4.3 Predicted and measured residual velocities (in m/s) for spherically nosed projectiles V0

300.0

326.7

362.9

420.6

452.0

V r (exp.)

97.2

154.8

220.2

284.3

325.1

V r (calc.)

88.2

148.9

209.5

287.7

327.1

162

4 Plate Perforation

perforated by spherically-nosed projectiles. For very thin plates (H/D 1, plugging may occur at the later stages of perforation when the projectile is at a distance of about D from the back surface of

4.4 Plate Perforation by Blunt Projectiles

163

the plate. As discussed by Woodward (1990), a blunt projectile starts its perforation process through the ductile hole enlargement process and, at a certain depth, the penetration mode changes to the plugging mechanism which is less “costly” for the projectile. He analyzed a situation where the projectile is inside the target and the remaining thickness to be penetrated is h. In order to penetrate an extra depth of δh by the ductile-hole-enlargement process, the projectile has to invest an increment of work (δW ) given by: δW =

π 2 D Yt δh 2

(4.18)

which is derived through the analysis of Bethe (1941) for the work needed to open a hole in the target, (4.3a), where the effective resisting stress is equal to 2Y t . According to Woodward (1990), the incremental work which is needed to shear a plug √ of thickness h along a distance of δh, by overcoming a shear stress of τ = Y t / 3, is given by: π δW = √ Yt Dhδh 3

(4.19)

√ Equating the two expressions results in a threshold thickness of h = ( 3/2) D, where the penetration mode by plugging is more favorable for the projectile. Thus, it is expected that a blunt projectile impacting a thick target (H/D > 1) will first penetrate by the ductile- hole-enlargement process, pushing target material to the sides and compressing it in front of the projectile. At a distance of about 0.87 D from the free surface of the target the process is expected to change to plug formation and its ejection. Plug formation is due to the large shear strains which develop in the target at the periphery of the projectile and extend to the target’s back surface. In a later work Woodward and Cimpoeru (1998) assigned a value of 2.7 Y t to the resistive stress on the projectile during its early penetration. This value was adopted from the theory of Tabor (1951) for the indentation of plates by a flat ended punch, and it results in a transition from hole-enlargement to plugging at distance of h = 1.17 D from the back surface of the target. Thus, one may expect these transitions to occur at a distance of the order of h = D from the back surface, in accord with many experimental results, and a few examples are given below. Figure 4.11, from Woodward et al. (1984), shows sectioned aluminum plates which were impacted by flat-ended rigid steel projectiles at different velocities. The diameter of the projectile was 4.76 mm and the thicknesses of these 7039-T6 aluminum plates were 9 mm. It is clearly seen that up to an impact velocity of about 260 m/s, the penetration is achieved through the ductile-hole-enlargement process. The target material is compressed ahead of the projectile and pushed to the side, forming “lips” at the plate’s front and a visible bulge at its back. At an impact velocity of 315 m/s, when the projectile is about one diameter from the target’s back surface, the concentrated shearing develop to visible fractures around the plug, which is going to be ejected at higher impact velocities. In fact, two plugs are formed in the

164

4 Plate Perforation

Fig. 4.11 Cross sections of aluminum plates impacted by blunt projectiles at different velocities

plate, at different locations and at different times. The upper plug with the conical shape is the so-called “dead zone”, formed early in the process due to the large shear strains which develop directly below the projectile. The more familiar cylindrical plug is formed when the projectile is at a distance of about one diameter from the back surface of the target, and its thickness is close to the diameter of the projectile. This is the common perforation process for materials which have high strength and relatively low failure strains. In contrast, low strength plates deform considerably around their impacted area, avoiding the shear failure. We performed numerical simulations in order to follow the perforation process of steel plates having different thicknesses, by an L/D = 3 rigid cylindrical projectile with a diameter of 10 mm. The Johnson-Cook model was used for the constitutive relation of the plate with Y 0 = 0.8 GPa. These simulations were performed in order to follow the shearing process of these plates by flat-ended projectiles. In addition, they demonstrate the importance of the failure mechanisms of the plate, by varying its maximal strain to failure (εf ) or its spall strength (pmin ). When the equivalent strain of a given element in the simulation reaches the threshold value (εf ), its strength is reduced to zero at the subsequent time steps. In a similar way, the assignment of a threshold spall pressure (pmin ) in the code leads to an immediate loss of strength in the element, if it experiences a tensile pressure which is larger than pmin . Figure 4.12 shows simulation results for the perforation process of a 10 mm thick steel plate by the rigid steel cylinder, when the failure strain of the plate is εf = 1.5. The first stage of perforation is achieved through the ductile-hole-enlargement

4.4 Plate Perforation by Blunt Projectiles

165

Fig. 4.12 Simulation results for the perforation of a plate with H = D, with εf = 1.5

process, as described above. A bulge is evident in the plate already after 10 μs, and the first sign of plate failure takes place, as observed for the aluminum plate in Fig. 4.11. At 20 μs after impact the shear failure in the plate is already evident through the cracks which move towards its back surface. These cracks form the plug whose thickness is about 8.6 mm in this case. This thickness amounts to 0.86D which is in excellent agreement with the predicted plug thickness by the model of Woodward (1990), as discussed above. The velocity history of the projectile in this simulation is shown in Fig. 4.13 and it is clear that it contains new features which were not observed in the corresponding velocity histories of sharp nosed projectiles. Particularly, the flat levels in this velocity history are due to temporary detachments between the projectile and the plate and they are unique for this mode of perforation. The first plateau appears at about t = 10 μs, which corresponds to the initiation of the shear cracks in the plate around the projectile’s periphery. These features show that the deceleration history of the projectile is more complex than we have seen before, with sharp nosed penetrators. Thus, a simple model for plate perforation, which is based on its deceleration, would be much more difficult to construct. Fig. 4.13 The velocity history of a rigid steel projectile perforating a 10 mm thick plate

166

4 Plate Perforation

Fig. 4.14 Simulation results for perforation of a thin plate (H/D = 0.5) and a thick plate (H/D = 1.5)

Simulations results for other plate thicknesses are shown in Fig. 4.14. As is clearly seen the plug in the H = 5 mm plate is sheared off at an early time, and its thickness is about the same as the plate’s thickness. The plug in the H = 15 mm plate is formed at about 30 μs after impact, and it seems to have sheared into a central plug and a surrounding ring. Similar features were obtained by Borvik et al. (2003) for the perforation of Weldox-460E steel plates, both experimentally and numerically. Their projectile’s diameter was 20 mm and for plates of thickness up to about 16 mm the plug’s thickness was equal to the plate’s thickness. For thicker plates the plugs’ thicknesses tended asymptotically to a value which was close to the diameter of the projectile, in general agreement with the model of Woodward (1990). The perforation process is quite different for materials which have a high propensity for adiabatic shearing, such as the Ti/6Al/4V alloy. This was clearly demonstrated in Woodward et al. (1984) by impacting rigid steel cylinders, with D = 4.76 mm, at 6 mm thick plates of this alloy. The plates developed adiabatic shear failures even at low impact velocities, and the plugs were formed at the very early stages of perforation, as shown in Fig. 4.15. Thus, plugging by adiabatic shear is the expected mode of failure for materials with inherent tendency to localize their shear bands, such as carbon steels and titanium alloys. The localization itself weakens the material in and around the shear bands, due to the large temperature increase which develop inside the bands. The propensity of this titanium alloy to fail by adiabatic shearing is clearly manifested by its dynamic stress–strain curve, as shown in Fig. 2.4. The failure takes place when the compressive strain of the specimen, in the Kolsky bar system, reaches a

4.4 Plate Perforation by Blunt Projectiles

167

Fig. 4.15 Plugging in a Ti/6Al/4V plate impacted by a blunt projectile

value of about 0.2. These early failures can be reproduced by numerical simulations which incorporate low failure strains (εf ) for the plate material. In fact, these simulations can reproduce the basic features of adiabatic shear bands if the cell size is small enough. In the simulations presented next, the cell size was 50 μm which is close to the typical width of 20 μm for an adiabatic shear band. Simulation results for the perforation of the 10 mm steel plate with strain to failure values of εf = 0.2, 0.5 and 1.0 are shown in Fig. 4.16. The L/D = 3 steel cylinder with D = 10 mm impacted these plates at V0 = 300 m/s and the JC model with Y 0

Fig. 4.16 Simulation results for the perforation of a steel plate having different strains to failure

168

4 Plate Perforation

= 0.8 GPa, was used for their constitutive relation, as above. One can clearly see the strong effect which the failure strain parameter has on the perforation process. In particular the low value of εf = 0.2 results in an early plug formation which is very similar to the plugging in the titanium alloy plate, as shown in Fig. 4.15. In fact, this simulation shows that the plug acquires a high enough velocity and it moves ahead of the projectile even before it is ejected from the plate. The residual velocities of the projectiles in these simulations decreased with increasing values of εf , as expected. Their resulting values were V r = 196, 123, and 0 m/s for the cases of εf = 0.2, 0.5 and 1.0, respectively. The projectile did not perforate the plate in the simulation with εf = 1.0, but we can see that the plug was formed by the time of 30 μs. The large differences between the values of V r are due to the corresponding differences in the work which the projectile has to invest in the target in order to perforate it. This issue will be discussed below when we describe the analytical model of Recht and Ipson (1963) for V r , and the model of Bai and Johnson (1982) for the perforation work. Finally, we wish to demonstrate the ability of numerical simulations to reproduce other failure mechanisms which take place during the perforation of thin plates. The simulation results shown in Fig. 4.17 were obtained by using a different failure threshold for the target material. This is the pmin threshold according to which, elements in the target fail when they experience a tensile strength which is equal to the value of pmin . For the particular simulation discussed here we used a value of pmin = 0.8 GPa, which is equal to the yield strength of this steel. One can clearly see the scabbing failure at the back of the plate as well as the cracks which develop around the penetration channel. These cracks may develop to the discing rings, as discussed by Woodward (1990), and they are very similar to those in the 2014 aluminum plate which is shown at the right hand side of Fig. 4.1. Considering the different failure modes which were described here, it is clear that a simple perforation model for blunt projectiles is not easy to construct. Moreover, in many cases the strength of these blunt projectiles is not high enough to guarantee their rigidity and they deform, erode or even shatter at relatively low velocities, adding more complications to the analysis. As mentioned above, several models were suggested over the years in order to account for the ballistic limit velocities

Fig. 4.17 The failure characteristics of a 10 mm steel plate with pmin = 0.8 GPa

4.4 Plate Perforation by Blunt Projectiles

169

and the residual velocities of blunt projectiles perforating metallic plates. However these multi-stage models are very complex since they rely on several parameters, which have to be calibrated for each stage in the perforation process. In contrast, the model of Recht and Ipson (1963) for the residual velocity of blunt projectiles treats the process in an integral manner, using only energy and momentum considerations. With this approach, the exact details of the failure modes are not important since they are lumped in the value for the plastic work (W p ) at the ballistic limit velocity. The basic assumptions of this model and its resulting expression for the residual velocities are outlined next. The model starts with the notion that the impact between the projectile and the target is basically a plastic impact. As a first stage, the model considers the impact of the projectile with a free plug with mass (m), surrounded by the rest of the target with no shearing processes. Conservation of momentum results in the loss of projectile’s energy (E fn ) due to this plastic impact, according to: E fn =

1 m · M V02 M +m 2

(4.20)

This “lost energy” is taken into account in the energy balance of the perforation process, which takes the form: 1 1 1 M V02 = M Vr2 + mVr2 + E f n + W p 2 2 2

(4.21)

where Wp is the work done on the target in order to form and push the plug out. Note that the model assumes that the velocity of the plug is equal to the residual velocity of the projectile (V r ). For V 0 = V bl the residual velocity is zero and one can write: 1 M Vbl2 = E f n + W p 2

(4.22)

These equations lead to the following relation for the residual velocity: Vr =

M · M +m



V02 − Vbl2 =

1 · 1+λ



V02 − Vbl2

(4.23)

where λ = m/M as defined by (4.15), and the ballistic limit velocity is given by:  Vbl =

2W p (M + m) M2

(4.24)

Equation 4.23 is similar to the corresponding relation for the spherical nosed projectile, (4.16), except for the fact that the square root of the mass ratio appears in (4.16). For M  m, the two expressions result in very close values, which is probably the main reason that (4.23) was used successfully for spherical nosed projectiles as well. This equation does not consider the physical properties of the target material

170

4 Plate Perforation

Fig. 4.18 Comparing model and data for steel sphere perforating thin titanium plates

such as its strength, hardening and rate sensitivities, and its failure strain. These properties determine the work needed to perforate the plate (W p ), and they are lumped into the value of V bl , as is clearly evident by (4.24). In order to demonstrate the applicability of (4.23) for the case of spherical projectiles impacting thin targets (M  m) consider the perforation of 1.32 mm Ti/6Al/6 V plates by rigid steel spheres with a diameter of 3.16 mm, as given by Dehn (1986). These experimental results can be accounted for by (4.23), as is clearly shown in Fig. 4.18. By writing (4.21) one assumes that the velocity of the plug is equal to the residual velocity of the projectile. This is seldom the case and the plugs often have much higher velocities than the projectiles. However, for relatively thin plates the predictions from (4.23) should be in good agreement with the data for V r , since the difference between the actual and calculated kinetic energies of the plugs plays a minor role in the total energy balance. In order to account for the actual velocity of the plug (V pl ), we can define an effective plug mass (m*) in the following way: m∗ = m



V pl Vr

2 (4.25)

This effective mass of the plug should be inserted in (4.23) instead of its actual mass (m), in order to have a better agreement with the data for V r when the plug’s velocity (V pl ) is significantly higher than the projectile’s residual velocity (V r ). In order to demonstrate the validity of this approach, one needs a set of experiments for which the velocities and the masses of the plugs were accurately determined. This was done by Borvik et al. (1999) who shot rigid steel cylinders, with L = 80 mm and D = 20 mm, at 12 mm thick Weldox 460E steel plate. The data shows that the ratio Vpl /Vr decreases from a value of about 1.5 for impact velocities near the ballistic limit, to a value of about 1.2 for the highest velocity in this work. Thus, an average value of V pl /V r = 1.35 can be taken for this set of experiments. The mass of the plug is given in Borvik et al. (1999) for each experiment and their average value is

4.4 Plate Perforation by Blunt Projectiles

171

Table 4.4 Comparing the predicted values for V r (in m/s) with the data from Borvik et al. (1999) V0

303.5

285.4

244.2

224.7

200.4

V r (exp.)

199.7

181.1

132.6

113.7

71.4

V r (pred.)

202.4

182.3

134.4

107.7

65.7

m = 27.8 g. This value is somewhat lower than the value of m = 29.7 g for a plug whose diameter is exactly equal to that of the projectile, and its thickness is the same as that of the plate. Using a value of V bl = 184.5 m/s for this set of experiments, as determined by Borvik et al. (1999), we find a good agreement between the data and the model’s predictions for V r , as shown in Table 4.4. This agreement enhances the validity of the model, as given by (4.23), with the correction for the effective plug mass as given by (4.25). It is clear that there is a real need for an analytical model to account for the higher velocities of these plugs which are ejected by flat ended projectiles. Borvik et al. (2003) determined V bl values and residual velocities for the same projectiles perforating Weldox 460E plates with thicknesses in the range of 6–20 mm. In all of these experiments the plug velocity was higher than that of the residual projectile by factors which were similar to those for the 12 mm plate, as described above. Using the same analysis for the plates of thicknesses in the range of 8–20 mm, shows that 4.23 and 4.25 account for all the data in Borvik et al. (2003) as far as V r (V 0 ) is concerned. On the other hand, the model does not account for the data concerning the H = 6 mm plates, probably because these thin plates experienced a large permanent deflection which is not accounted for by the model. Specifically, the energy which the projectile invests in deforming the thin plate is missing in the model and, as stated above, it plays a significant role in the energy balance of the perforation process in thin plates. The various models which were presented in this chapter resulted in similar functional dependences of V r on V 0 . In fact, Lambert and Jonas (1976) reviewed a large number of perforation theories for non-deforming projectiles, and concluded that they adhere to a basic form which is given by: Vr =k Vbl



V0 Vbl

2 −1

(4.26)

where k is an empirical constant which depends on the plug/projectile mass ratio. We have shown that this equation can be derived by analytical considerations and that it applies for the different cases discussed above. Thus, with k = 1 Eq. (4.26) reduces to (4.2) of Recht and Ipson (1963), for sharp-nosed projectiles. With k = [M/(M + m)]0.5 this equation reduces to (4.17) for the spherical nosed projectile, and with k = M/(M + m) we get (4.23) for the flat-faced projectile. In conclusion, the residual velocities of rigid projectiles can be accounted for by the energy and momentum conservation equations without specifying the actual

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4 Plate Perforation

failure modes of the plates. We emphasized the importance of using these physicallybased equations in order to account for the experimental data, rather than using empirical best-fitting procedures. The situation becomes more complex when the projectile loses mass through the process, in which case it is impossible to use such energy and momentum considerations. We shall discuss this issue in Chap. 5, where we deal with plate perforations by eroding rods. The next issue which has to be addressed now is that of determining the value of V bl for blunt-nosed projectiles, through analytical models such as the RI model. The model resulted in a general expression for V bl which is given by (4.24). The difficulty with this equation stems from the fact that the work needed to perforate the plate (W p ) is not easy to derive analytically, as will be discussed below. We have shown in Sect. 4.2 that numerical simulations can be used to derive a numericallybased model for V bl of sharp-nosed projectiles, perforating ductile metallic plates through the hole enlargement process. It is more difficult to construct such a model for the perforation process by blunt projectiles, because of the various failure modes involved. Several attempts at deriving analytical expressions for V bl have been presented in the literature, for either thin or thick plates, as discussed above. Borvik et al. (2003) tried to account for their experimental results with some of these models. However, none of these models could account for the experimental values of V bl of all the plate thicknesses which were used in their study. For thin plates, the values of V bl can be accounted for by the model of Wen and Jones (1996), which takes global bending into account. The physically-based model of Bai and Johnson (1982) was more successful in accounting for the V bl values of moderately thick plates, as described next. The model of Bai and Johnson (1982) addresses the work (W p ) needed to perforate a plate of thickness H by a rigid punch having a diameter which is close to the plate’s thickness. The analysis starts with a general expression for the constitutive relation between the shear stress (τ ) and shear strain (γ ) of the plate material, according to: τ = τ0 (1 − αT T )γ n

(4.27)

where τ 0 , α T and the hardening coefficient (n) are material constants, and T is the temperature. The effect of strain rate is not taken into account in this analysis. The temperature is calculated according to the usual adiabatic assumption which converts the plastic work increment, dW p = τ dγ , to a temperature increase through: dW p = ρ C v dT, where C v is the specific heat of the plate material. Using these relations and the definition of the instability strain (γ i ) as the strain for which dτ = 0, Bai and Johnson (1982) obtained the following relation for the critical shear strain:  γi =

 1 nρCv ( 1+n ) αT τ0

and for the maximum shear stress (τ m ) they obtained:

(4.28)

4.4 Plate Perforation by Blunt Projectiles

173

τm = τ0 (1 + αT T0 ) γin exp(

−n ) 1+n

(4.29)

where T 0 is the room temperature. The analysis is based on the material parameters (n, γi , and τ m ) as well as on the thickness of the plate (H) and on the diameter of the punch (D), and it follows the plugging process via two consecutive stages. At first, the projectile is penetrating to a certain depth, as in the hole-enlargement process. During this stage the shear stress in the target, around the periphery of the projectile, provides the main resistance to penetration. This stress increases according to the strain hardening rule, (4.27), until it reaches the value of τ m , and the shear strain reaches its maximum value (γi ). At this point the projectile has penetrated a certain distance (Pi ), which has to be determined. The model assumes that beyond this point the target offers no further resistance as a result of its shear failure. Thus, the equation of motion for the first stage is: (M + m)

dV = −π D H τ R dt

(4.30)

where M and m are the masses of the projectile and the plug, respectively, and τ R is the shear stress in the target around the projectile’s periphery. The value of τ R is determined through the equations above and (4.30) is then integrated between x = 0 and x = Pi , which is the depth of penetration when instability sets in. In order to have an estimate of the strain distribution in the target around the punch’s periphery, Bai and Johnson (1982) used the following expression for the shear strain (γ) in the plate at a distance r outside the punch radius (R): γ = γR

 1/n R r

for r ≥ R

(4.31)

where γR is the target strain at r = R, which is assumed to be the maximum value of the shear strain. The resulting expression for the penetration depth (Pi ), when the strain reaches its instability point, is: n Pi = γR R 1+n

(4.32)

This equation results in values for Pi of the order of 0.3R, since the value of γR is limited by the maximum strain (γi ), which is around 1.0 for most ductile materials, and the values of the exponent (n) are usually bound in the range of 0.3– 0.5. Note that this value of Pi corresponds to a value of 0.15 D, which means that the plug’s thickness should be equal to about 0.85 D, which is practically the same value derived by the simple approach of Woodward (1990), as discussed above. The second stage in the perforation process starts when the depth of penetration reaches the value of Pi and the shear strain has reached its maximal value (γi ). High values of γi result in a delay in the appearance of the instability, which is of major

174

4 Plate Perforation

importance for the energy absorbed by the plate. Bai and Johnson (1982) derive equations for the energy absorbed in the plugging process, from which they can have an estimate for the ballistic limit velocities. The analysis results in two limiting cases for which V bl is proportional either to H (D/M)0.5 or to D (H/M)0.5 . As is often found in practical cases, the ballistic limit velocities do follow, empirically, one of these simple relations. The most important outcome of this dimensionless analysis is that the energy absorbed by the plate is dependent on: n, γi and H/D. The end result is demonstrated in Fig. 4.19, where the normalized absorbed energy (ωp ) is plotted as a function of the normalized plate thickness (H/D) for several pairs of (n, γi ) values. The normalized absorbed energy is defined by ωp = 2W p /πD2 Hτm . As is clearly evident in this figure, in order to achieve a better resistance to plugging, an increase of plate thickness is helpful only for materials with a high failure strain. The energy absorbed by materials with low γi values increases only linearly with the thickness of the plates. The expected ballistic limits for such plates will vary as (H/ρp L eff )0.5 . On the other hand, for materials with high values of γi , the perforation energy is related to H2 , and the ballistic limit velocities are expected to be related to H (1/ρp L eff D)0.5 . These plates will absorb much more energy under this type of loading, which is the reason that materials with a high propensity for shear banding, like the Ti/6Al/4V alloy, do not perform as well as metals which exhibit high values of γi . These conclusions are substantiated by the numerical simulations presented above, for the perforation process of steel plates having different values of the failure strain (εf ). We have seen that this parameter plays a major role in determining the time for plug formation which, in turn, determines the amount of the projectile’s energy absorbed by the plate and its residual velocity. A simplified version of the Bai-Johnson model was suggested by Woodward (1990), for perforations through the plugging process. According to his model the work done by the projectile in shearing a plug of thickness H and diameter D is

Fig. 4.19 The energy absorbed by the plate according to the model of Bai and Johnson (1982)

4.4 Plate Perforation by Blunt Projectiles

175

W p = πDH2 τ, where τ is the shear strength of the plate material. This expression is obtained by assuming that a constant shear stress (τ) acts on the plug’s cylindrical area (πDH), resisting the penetration for a distance of H, until the plug is ejected. In fact, this expression is a simplified version of the Bai and Johnson (1982) model for the √ case of high values of γi . Woodward (1990) recommends using the relation τ = Y / 3 for the shear stress, where Y is the compressive strength of the target material. The plastic work to perforate the plate through this process is equated with the kinetic energy of the projectile at its ballistic limit velocity: 1 M Vbl2 = π D H 2 τ 2

(4.33a)

and with M = πD2 ρp L eff /4, this equation results in:  Vbl = 2H ·

2τ ρp Lef f D

(4.33b)

We should note that a similar equation was used already in the 1880’s by British engineers, who analyzed the perforation of wrought iron targets with large steel projectiles. Browne (1884) reports on several experiments which were analyzed by Fairbairn’s equation, which looks like (4.33a), except that it has an empirical constant (k) instead of the shear strength (τ). The article compares several iron and steel plates as candidates for armor in ships, and it emphasizes the differences between soft and hard targets as far as cracking and bending are concerned. It is interesting to find that issues like elongation, fracture toughness and hardness were known to these engineers, although in different names, and they realized their importance for terminal ballistics. A linear relation between V bl and plate thickness is expected through the simple plugging model depicted by (4.33b). In order to check how well it predicts the experimental results of Borvik et al. (2003), we chose a value of τ = 0.35 GPa for the Weldox 460E steel plates, using their flow stress of Y = 0.6 GPa. Inserting this value of τ together with L eff = 80 mm D = 20 mm for the projectiles in (4.33b), results in the predicted values for V bl which are listed in Table 4.5, together with the experimental results from Borvik et al. (2003). The close agreement between the model’s predictions and the data, especially for the thicker plates, is due to the fact that the failure of these plates was mainly by a simple shearing mechanism. The Table 4.5 Comparing predicted and measured V bl (in m/s) for the experiments of Borvik et al. (2003) H (mm)

6

8

10

12

16

20

V bl (exp.)

145.5

154

165.3

184.5

236.9

293.9

V bl (pred.)

89.6

119.4

149.3

179.2

238.9

298.6

176

4 Plate Perforation

Fig. 4.20 Cross sections of perforated steel plates of different thickness

agreement is getting worse for the thinner plates (H ≤ 10 mm), due to the extra energy which the projectiles invested in the global deflection, as discussed above. The different deformations of these plates are clearly evident in the pictures of the sectioned plates in Fig. 4.20, from Borvik et al. (2003). It is clear that the 6 mm thick plate, with H/D = 0.3, experienced a significant global deformation. The deformation is much smaller for the 8 mm plate, with H/D = 0.4, and it is negligible for the 12 mm plate with H/D = 0.6. The large deformations resulted in higher V bl for the thin plates, as compared with the model of Woodward (1990), since the model does not include global deflections. These results suggest that the value of H/D = 0.5 marks the onset of global deflections for plates which are perforated by blunt projectiles. In order to have a better estimate for the onset of global deflections consider the simulation results from Borvik et al. (2003) for plates with thicknesses between 4 and 12 mm, as shown in Fig. 4.21. The simulations were performed for impact velocities which were higher by about 20% than the corresponding ballistic limit values of these plates. The global deflections are decreasing with plate thickness, as expected, and they are practically negligible for plates with H ≥ 10 mm. Thus, a rough estimate

Fig. 4.21 Simulation results for different plate thicknesses perforated by flat nosed projectiles

4.4 Plate Perforation by Blunt Projectiles

177

for the transition can, indeed, be set at H/D = 0.5. This transition is similar to the results presented in Sect. 4.2 for plate perforation by sharp-nosed projectiles, where the transition from dishing to ductile hole enlargement was found to be at about H/D = 1/3. The two processes are different in the sense that the perforation by blunt projectile does not involve a transition between penetration modes. The global deformation of the thin plates simply adds to the energy loss of the projectile. This addition plays a major role in the performance of laminated targets as compared with monolithic ones, as we show next. The perforation process of laminated targets by blunt projectiles can be very different than their response to the impact of sharp-nosed projectiles. The difference is due to the fact that blunt projectiles have to invest more energy by perforating thin plates, through their global deflections. In addition, the plugs ejected from the first plates can influence the perforation of the subsequent plates, as noted by Gogolewski et al. (1996), Woodward and Cimpoeru (1998) and Dey et al. (2007). These works compared ballistic limit velocities for cylindrical steel projectiles perforating monolithic and laminated targets, which were made of two or three closely stacked plates. The values of V bl for the monolithic plates were considerably lower than those of the laminated plates, and we focus now on this enhanced performance of the laminates in these works. Woodward and Cimpoeru (1998) performed V bl tests for several double layered 2024-T351aluminum targets and compared them with the V bl value for a monolithic plate of the same total thickness. All the double layered targets resulted in higher V bl values as shown in Fig. 4.22, from Woodward and Cimpoeru (1998). The value of V bl for the target with equal thickness plates was the highest and the improvement in V bl , as compared with the monolithic plate, was by 13.5%. Moreover, the laminate with the thin back plate showed a better ballistic performance than the similar laminate

Fig. 4.22 Sectioned aluminum targets perforated by flat-ended projectiles

178

4 Plate Perforation

with the thick back plate. This difference can be attributed to the pronounced global deflection in the back plate, as is clearly seen in the figure. We consider now the improvement in V bl values for the layered targets in the three works cited above, in terms of the normalized thickness (h/D) of the corresponding back plates. Woodward and Cimpoeru (1998) used a cylindrical projectile with D = 6.35 mm, and obtained an improvement of only 13.5% in V bl for the double-layered target of 2024-T351 aluminum plates, with h/D = 0.75. Gogolewski et al. (1996) used a D = 6.7 mm projectile and found an increase of 23% in V bl for a laminated target consisting of three 2.235 mm thick 6061-T6 aluminum plates, for which h/D = 0.3. The experiments of Dey et al. (2007) were with monolithic and double layered targets of Weldox 700E steel plates with a total thickness of 12 mm. The resulting values for V bl showed a significant increase (about 47%) for the laminated target, with h/D = 0.3, as compared with the monolithic target. Thus, it is already clear that both the strength of the plates and their normalized thickness (h/D) contribute to the improved ballistic performance of the laminated targets. The deflection and stretching of the back plate is clearly evident in the simulations of Dey et al. (2007), for the spaced laminate shown in Fig. 4.23. These simulation results highlight the important features of the perforation process for laminated targets. These simulations were performed with impact velocities which were slightly higher than the corresponding ballistic limits. The black colors in the targets correspond to high plastic strains in the plates around the projectile’s periphery. The simulations for the spaced target capture all the features which were seen in the experiments. In particular, they show that the ejected plug from the first pate prevents the shear localization in the second plate, which experiences a large deformation and membrane stretching. The plug itself is stopped at the second plate and adds an extra mass which has to be pushed by the projectile, leading to a further increase in

Fig. 4.23 Simulations results for monolithic (12 mm) and double-layered (6 mm + 6 mm) targets

4.4 Plate Perforation by Blunt Projectiles

179

the value of V bl . The fact that the first plate does not experience a large deflection is due to the impact velocity which is much higher than the ballistic limit velocity corresponding to this plate. This is a well-established result for high velocity impacts of thin plates by blunt projectiles, where a plug is sheared with no plate deflection. In fact, Goldsmith and Finnegan (1971) found that the amount of plate deflection is maximal at the ballistic limit velocity, and it decreases monotonically with increasing impact velocities. In summary of the three works discussed here, the ballistic limit velocities of the laminated targets, impacted by blunt projectiles, were significantly higher than the ballistic limits for monolithic plates of the same total thickness. The improvement depends on the corresponding h/D values of the laminated targets, where h is the thickness of the back plate. This is the plate which absorbs an extra amount of kinetic energy from the projectile through its large deflection. The two configurations with h/D = 0.3 resulted in much higher improvements than that with h/D = 0.75. This is the expected result since thinner plates are more effective as energy absorbers, as is clearly seen in Fig. 4.22. Comparing the results of Dey et al. (2007) and Gogolewski et al. (1996), with h/D = 0.3 in both cases, we find that the strong steel plates of Weldox E700 resulted in a much larger improvement (by 47%), as compared with the improvement achieved with the 6061-T6 aluminum plates (23%). This difference is probably due to the higher strength of the steel plates. An estimate for the effect of h/D alone can be obtained through the simulation results of Teng et al. (2007) for two sizes of flat-nosed projectiles perforating the same targets. The ballistic limit of a 12 mm thick steel plate was compared with that of a double-layered target with 6 mm thick plates. The projectiles in these simulations were L/D = 4 steel cylinders with diameters of 20 and 7.6 mm. The properties of the plates in these simulations represented those of Weldox 460E steel, which has a lower strength than the Weldox 700E steel in the experiments of Dey et al. (2007). The ballistic limit velocities for the double-layered targets in these simulations were higher than those of the monolithic targets for both projectiles, as expected. The improvement in V bl for the larger projectile, with h/D = 0.3, was about 25%, while for the smaller projectile, with h/D = 0.79, it amounted to only 7%. These simulations demonstrated the much larger deflection of the back plate in the double layered target which was impacted by the larger projectile. These results confirm the conjecture that the improved ballistic limits of laminated targets are due to the large deflections of their back plates, and it is more significant for plates with small h/D values, as obtained experimentally. Also, the improvement is strongly dependent on target strength, since it amounted to 25% in the simulations of Teng et al. (2007) for the Weldox 460E steel, while the experiments of Dey et al. (2007) resulted in an improvement of 47% for the stronger Weldox 700E plates. Based on this rather limited information, we may conclude that the lamination and the strength of the plates play roughly equal roles, as far as the improvements in ballistic limits of laminated targets are concerned. The experimental data and the simulation results confirm the notion that the significant improvements in the values of V bl , for laminated targets impacted by blunt projectiles, are due to the deflections of their

180

4 Plate Perforation

back plates. These deflections are significant for plates with normalized thicknesses of about h/D = 0.3. The simulations of Teng et al. (2007) for spaced laminates, impacted by the projectiles with the different sizes, D = 20 mm and D = 7.6 mm, resulted in a decrease of about 0.5–1.0% in their corresponding values of V bl , as compared with the closely stacked laminates. These minor differences enhance the claim that the large improvements in V bl which are often obtained experimentally for spaced laminates, are probably due to the deflection of the projectiles by the first plates, rather than to a physical effect. Teng et al. (2007) also performed numerical simulations for conical nosed projectiles with diameters of 20 and 7.6 mm, perforating the same monolithic and laminated steel targets. They found that for both projectiles the ballistic limit velocities were somewhat higher for the monolithic targets, as compared with the double layered target, which is the expected result by the analysis in Sect. 4.2. Also, for both cases the spaced plates resulted in a lower ballistic limit (by less than 1%), as compared with the closely stacked plates, which is also in agreement with the analysis presented there. These simulations confirm the main conclusion derived in this chapter, namely, that the effect of lamination is very different for the two types of projectile nose shapes.

4.5 Forced Shear Localization and Adiabatic Shear Failure The issue of adiabatic shear failure in metals and alloys received a lot of attention for many years, as described in Chap. 2. The importance of this failure mode for terminal ballistics has been emphasized in the present chapter, particularly for the perforation of high strength plates by blunt projectiles. As discussed in Chap. 2, this mode of failure has been attributed to a shear instability which takes place under dynamic loading as a result of two opposing effects, the strain hardening of the material and its thermal softening. Various materials are often claimed to fail by adiabatic shear when, in fact, their failure under dynamic loading is very similar to that under static loading conditions. Thus, workers made the distinction between materials which fail by shear instabilities, and those which fail by a distinctive adiabatic shear mechanism, such as the Ti/6Al/4V alloy. This titanium alloy fails at a strain of about 30% under static compression, while its failure strain is only 20% in a Kolsky bar experiment, at dynamic strain rates of about 103 s−1 . In contrast, strong alloys such as the 2024T3 and 7075-T651 aluminum alloys, fail at much higher compressive strains under static loading, as compared with dynamic loadings. The difference between these failure strains is attributed to localized thermal softening process which takes place at the sites of maximal shear in the specimen. This thermo-mechanical failure is also referred to as adiabatic shear failure. However, the classification of dynamic failures as adiabatic should be free of geometrically-induced stress concentrations in the specimen. Materials are prone to fail at relatively low shear strains when such geometrical constraints are part of their testing configuration, and the measured failure strain should not be interpreted as a physical property of the material. Such

4.5 Forced Shear Localization and Adiabatic Shear Failure

181

Fig. 4.24 Specimen geometries with inherent stress concentration zones, a the hat shaped specimen, b a truncated cone, and c the notched cylinder

stress concentrations appear in the hat-shaped specimen of Chen et al. (1999), in the truncated cone specimen of Li et al. (2003), and in the notched specimen used by Rittel et al. (2008), which are shown in Fig. 4.24. All of these geometries introduce well-defined locations of stress concentration, which mask the true nature of the adiabatic shear failure. For example, the shear failure is forced to occur along the dotted lines shown in the hat-shaped specimen, and it will take place for all materials at relatively low strain levels. Li et al. (2003) tested cylindrical specimens of a tungsten alloy in the Kolsky bar system, and observed no failure even at very high strains. On the other hand, the trapezoidal shaped specimen of this alloy, seen in Fig. 4.24b, failed at very low strains along the shaded area shown in the figure. The main conclusion of Rosenberg et al. (2010) is that the quest for a truly physical account of the adiabatic shear phenomenon should be focused only on materials which have the inherent propensity for adiabatic shearing, such as the titanium alloy. These properties should be highlighted through experimental configurations which are free of stress concentrations, as discussed here. Flockhart et al. (1991) discussed the appearance of shear failure under various axisymmetric loading conditions, including indentation by a flat punch, perforation and deep penetration by blunt projectiles and the compression of cylindrical specimens in the Kolsky bar system. They emphasize the difference between adiabatic shear failure by thermo-mechanical instabilities of the Zener and Hollomon (1944) model, which was described in Chap. 2, and the failure by shear fracture through void nucleation, followed by their growth and coalescence. The distinction between the two mechanisms is not clear, and in many actual cases the failure is by shear fracture within narrow bands of maximal shear strain, rather than by the adiabatic shear mechanism. They claim that in several actual situations the failure is not by an adiabatic shear mechanism, as in the appearance of a dead zone in brittle materials, and the occurrence of plugging in ductile materials. The numerical simulations of Flockhart et al. (1991) followed the velocity discontinuities in the specimens which were loaded by different techniques. They found that shear failure sites corresponded with the shear strain-rate maxima in the simulations. These maxima identify closely with the observed failures in compression tests, plugging during plate perforations, etc. We stress this issue here because there are many cases in the published literature where the term adiabatic shear is used very freely instead of referring to a simple

182

4 Plate Perforation

shear failure. The true nature of adiabatic shear is still an unresolved issue, and one should look for the physics behind this specific phenomenon. The Ti/6Al/4V alloy is an excellent example for a material which is considered to fail by a true adiabatic shear mechanism, as mentioned above. Its compressive failure strain is about 20%, under dynamic loading in the Kolsky bar, as was shown in Fig. 2.5. Obviously, this value of strain is too low to induce a significant uniform temperature increase in the specimen, as discussed in Chap. 2. Thus, its failure through shear banding cannot be simply related to strain instability, resulting from the strain-hardening and the thermal softening of the bulk specimen. Instead, one may consider the adiabatic shear failure as due to a significant heating within the localized shear band, and the simultaneous development of microvoids, or microcracks, which further raise the local temperatures and enhance the softening mechanism. This approach was taken by Rice and Levy (1969) who developed a physicallybased model which accounts for adiabatic shear failure by following the temperature increase inside the shear bands, through the nucleation and growth of these imperfections. They derived the following expression for the local temperature rise ahead of a microcrack which moves at a velocity V: (1 − υ)2 K C Y · T = E

 2V ρck

(4.34)

where ρ, E, v and Y are the density, Young’s modulus, Poisson’s ratio and the strength of the material, respectively, K c is its fracture toughness, c is the sound speed, and k is its thermal conductivity. The model is based on the following assumptions: (1) the material has no strain-hardening features, (2) deformation takes place in a small zone ahead of the crack tip, and (3) there are no thermo-mechanical coupling effects. Equation 4.34 contains the relevant thermo-physical parameters which control the temperature rise ahead of a moving crack, and it can account for a large temperature increase inside a narrow band. Using published data for these parameters Rice and Levy (1969) found that for a given crack velocity, the temperature rise in a Ti/6Al/4V specimen is an order of magnitude higher than the corresponding values for mild steel and the 2024 aluminum alloy. This difference can explain the propensity for adiabatic shearing in the titanium alloy.

4.6 The Effective Strength of Metallic Targets All the simulations discussed above as well as the analytical models derived by them, treated metallic plates as elasto-perfectly-plastic solids through the von-Mises constitutive relation. This procedure simplified the process by which one can construct an analytical model from a set of simulations. However, the actual strengths of most metals exhibit a non-ideal behavior through their strain-hardening, strain-rate sensitivity and temperature softening. The main question to be asked at this stage is what

4.6 The Effective Strength of Metallic Targets

183

Fig. 4.25 Map of equivalent plastic strains around the perforation hole

value should one use for the strength parameter (Y ) in all the expressions derived above. In other words, what is the effective strength of a metallic plate which is subjected to a ballistic event? This issue has been raised in several works such as in Riegel and Davison (2016), which used a best-fit analysis for penetration data of WHA rods impacting RHA targets. This section summarizes the work of Rosenberg et al. (2018b) who addressed the issue through a series of numerical simulations, by following the state of the elements in a metallic plate around the perforation hole, as it is perforated by a rigid projectile. The first step in the numerical work of Rosenberg et al. (2018b) was to follow the plastic strains around the hole in a metallic plate after it has been perforated by a rigid projectile. Aluminum and steel plates, 6 mm thick, were impacted by a short CRH3 ogive-nosed steel projectile with a diameter of 6 mm, at velocities of 800 and 1100 m/s, The von-Mises strengths of these plates were in the range of 0.4–1.6 GPa. The distribution of the equivalent plastic strain (εp ) around the hole in a typical simulation is shown in Fig. 4.25. One can clearly see that the maximal value of this strain is concentrated in a very narrow range around the hole in the plate. The radial distribution of εp at the middle section of the plate is shown in Fig. 4.26 for various simulations. One can see that the same trend was obtained for all cases Fig. 4.26 Plastic strain distributions around the hole

184

4 Plate Perforation

studied in this work, regardless of the type of metal (density, strength) or the impact velocity. Moreover, these simulations show that the plastic strain is negligible beyond a distance of about 2.4 projectile radii from the edge of the hole. The next step was to calculate the mean plastic strain in a ring around the midsection of the plate, with an inner radius of r 0 = R (the projectile’s radius) and an outer radius of r f = 2.4 R, where the plastic strains are practically zero. The strain at each location was multiplied by the area of a ring at that radius, in order to have a weighted value of the plastic strain at the midsection of the plate. These calculations were then performed for all the rings across the thickness of the plate, and averaged according to the following relation: 1 εp = H

H 0

⎡ 1

r f





ε p (r, x)2πr dr ⎦dx π r 2f − r02 r

(4.35)

o

The final result of these calculations was that the effective (average) plastic strain around the perforation hole is εp = 0.27 for all the cases simulated in this study. Thus, Rosenberg et al. (2018b) concluded that the effective strength of a perforated metallic plate should be taken as the corresponding flow stress of this metal at a strain of: εp = 0.27. A similar calculation was performed for the strain rates in the elements around the hole, during the perforation process. The highest rates, at the edge of the hole, reached values of about 105 s−1 , and the weighted value for the average strain rate was found to be about 104 s−1 . The conclusion from these calculations is that the effective strength of an impacted metallic plate should be taken as its flow stress at a strain of εp = 0.27, as determined by a Kolsky bar test at a rate of about 104 s−1 . Note that the same effective strain of 0.27 was obtained by analyzing the simulation results for the perforation of a 304 stainless steel plate, as depicted in Fig. 4.9. In order to enhance these assertions, Rosenberg et al. (2018b) performed a series of perforation experiments with the hard steel cores of 0.3 AP projectiles impacting several metallic plates. The impact velocities (V 0 ) and the residual velocities (V r ) were determined by a high-speed video camera. These were plates of RHA, 304L stainless steel, 6061-T651 aluminum, and C-101 copper in the as-received condition. In addition, Kolsky-bar tests at high rates were performed on L/D = 0.5 cylindrical specimens taken out from these plates and the results for the two steels are shown in Fig. 4.27. One can clearly see that both materials are practically insensitive to the rates in this range. The RHA steel behaves in an elastic-perfectly-plastic mode while the stainless steel exhibits a significant strain hardening behavior. Using these stress-strain curves Rosenberg et al. (2018b) derived the effective flow stresses at a strain of 0.27, as Y eff = 1.42 GPa and 1.05 GPa for the RHA and the stainless steel, respectively. These values were used in (4.7) and (4.5) to derive the predicted values of V bl for the perforation tests of the 6 mm steel plates, resulting in: V bl = 510 m/s and 420 m/s for the RHA and the stainless steel plates, respectively. With these values, the predicted curves for V r (V 0 ) were calculated through the Recht

4.6 The Effective Strength of Metallic Targets

185

Fig. 4.27 Stress-strain curves for a RHA and b 304L stainless steel, rates are given in units of s−1

and Ipson model, as given by (4.2). Figure 4.28 shows their agreement with the test results for the residual velocities, supporting the validity of the procedure outlined by Rosenberg et al. (2018b) for the effective strength determination. Another set of perforation tests was performed with 8 mm aluminum and copper plates. These materials differ significantly by their density but have very similar strengths as can be seen in Fig. 4.29, which presents their Kolsky-bar stress-strain curves at a rate of 4000 s−1 . Clearly, the two curves overlap at strains between 0.25 and 0.4 and at a strain of εp = 0.27 the two metals have the same effective strength of Y eff = 0.43 GPa. Using the procedure described above, Rosenberg et al. (2018b) calculated a value of V bl = 332 m/s for both plates and Fig. 4.30 shows the agreement between the predicted and measured residual velocities for the two plates. Again, the good agreement enhances the validity of their approach as far as the choice of the effective strength is concerned. The procedure outlined above for the determination of the effective strength of metallic plates, has been shown to be very useful for thick plates as well. For example, consider the data of Roisman et al. (1999), who shot tungsten-alloy ogive-nosed projectiles, with M = 88 g and D = 11.3 mm, at stacks of three 6061-T651 aluminum Fig. 4.28 Comparing the predicted (dotted curves) and measured residual velocities for the two steel plates

186

500 4000 s -1

400

stress (MPa)

Fig. 4.29 Dynamic stress-strain curves for the C-101 copper and the 6061-T651 aluminum specimens

4 Plate Perforation

300 200 Al

100

Cu

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

strain

plates with a total thickness of 120 mm. From (4.8) one finds that that σr = 3.89 Y for this case and with Y = 0.43 GPa for the 6061-T651 alloy, one gets from (4.5) a ballistic limit velocity of 676 m/s for this projectile/target pair. With this value of V bl the calculated V r (V 0 ) curve in Fig. 4.31 demonstrates the excellent agreement between the model’s predictions and the data. Rosenberg et al. (2016) applied this procedure for a large set of peroration tests on various aluminum alloys impacted by 7.62 mm AP projectiles. Through numerical simulations and empirical data, they found that the V bl values for the hard steel cores of these projectiles are higher by about 7% than the corresponding values for the full projectiles (with the soft jacket). They also showed that layering a target by two or three closely-packed plates can reduce the corresponding values of V bl by 2.5% and by 5%, respectively. Their model’s predictions were compared with published data for eight aluminum alloys of various heat treatments, and for plate thicknesses ranging between 20 mm and 60 mm.

Fig. 4.30 Comparing the model with tests results for aluminum and copper plates

4.6 The Effective Strength of Metallic Targets

187

Fig. 4.31 Comparing the model’s predictions with data from Roisman et al. (1999)

Overall, the model was found to agree with more than 50 data points as shown in Fig. 4.32. The values for V bl are normalized through V bl * which is the corresponding value for a plate thickness of H = D, as defined by (4.9b). Finally, Rosenberg et al. (2018b) checked whether the same effective strengths should be also used in numerical simulations for rigid rod penetrations in large targets. They simulated several impacts from the work of Piekutowski et al. (1999), with CRH3 ogive-nosed rods impacting aluminum 6061-T6511 targets. The projectiles in these simulations were given a very high strength in order to assure their rigidity, while the von-Mises yield criterion, with Y = 0.43 GPa, was used for the aluminum targets. Table 4.6 compares the simulation results with the data for some of these shots, and this agreement further enhances the validity of the specific choice of the effective strain, strain-rate and flow stress for numerical simulations and analytical modeling. In order to check for the sensitivity of these simulations to small variations in the von-Mises strength of the target, they repeated the simulation for the 966 m/s impact with Y = 0.39 GPa and Y = 0.47 GPa, which are about 9% different from the value of Y = 0.43 GPa. The penetration depths in these simulations were 132 mm and 151 mm Fig. 4.32 Comparing model predictions (dotted curve) and data for 7.62 mm AP projectiles perforating aluminum plates of various alloys

188 Table 4.6 Comparing data with simulation results for the deep penetrations in aluminum targets

4 Plate Perforation V 0 (m/s)

Pexp (mm)

Pcalc (mm)

569

58

59

966

140

142

1237

224

220

1817

462

453

for the higher and lower strengths, respectively, demonstrating the sensitivity of these simulations to the specific value of the effective strength for the target material.

4.7 The Perforation of Concrete Slabs by Rigid Projectiles The deep penetration of rigid projectiles in concrete targets was discussed in Chap. 3, by following the three stages in the work of Rosenberg and his colleagues during the last decade. In the first stage, Rosenberg and Dekel (2010a) demonstrated that the resisting stress which a concrete target exerts on a rigid projectile (Rt ), is constant throughout the penetration process and is independent on the impact velocity. In the next stage Rosenberg and Kositski (2016) included the effect of the entrance phase on Rt , and accounted for the influence of the aggregates in the concrete. In the third stage, Rosenberg et al. (2020a) dealt with the scaling issue, which is manifested by the fact that larger projectiles experience lower resisting stresses, resulting in larger normalized penetration (P/D) in a given concrete target. They related this scaling issue to the well-known size effect in the compressive strength of concrete specimens. This analysis led to a new strength parameter (f c *) for the concrete, which takes into account the projectile’s diameter, and the geometry of the test specimen which is used to determine the compressive strength of the target. This analysis resulted in the empirical relation: Rt = 0.3 ln(f c */10), which was shown to account for a large set of penetration data with projectiles of various sizes. We turn now to modeling the perforation process of finite-thickness concrete plates (slabs). As stated earlier, the most important parameter in the perforation process of thin plates is the value of their ballistic limit velocity (V bl ), for a given projectile/plate configuration. In addition, the residual velocity of the projectile imacting the plate at a higher velocity, has to be accounted for by the analytical model. We should note that not all the works reviewed here provide the size and geometry of their test specimens for fc determinations. Thus, the values for f c * cannot be determined for the corresponding projectile/concrete pairs. For these cases we follow the analysis in Rosenberg and Kositski (2016), which is based on the projectile/aggregate diameter ratio (D/d). The perforation of finite-thickness concrete slabs is complicated by the effect of the slab’s free back-face which has to be accounted for in the model. This is also the case with metallic targets but for concrete it is even more complex, due to the large damaged zone which is created at the back of the slab, from which a lot of concrete material is ejected at the end of the perforation process. Even before this stage, when the projectile reaches a certain location in the target it “feels” the existence of the back free surface and, consequently, its deceleration decreases. This distance can

4.7 The Perforation of Concrete Slabs by Rigid Projectiles

189

Fig. 4.33 Decelerationdistance history from a numerical simulation

be referred to as the exit phase and it is plausible to assume that it is related to the projectile’s diameter. Direct measurements of projectile decelerations in perforation experiments are very scarce and difficult to analyze. Instead, Rosenberg and Kositski (2016) used numerical simulations in order to estimate the effect of the concrete’s back face. The AUTODYN code’s library includes the RHT constitutive equations for concrete, and they were used in these simulations. This constitutive model, by Riedel et al. (1999), was calibrated for concretes with compressive strengths of about 35 MPa. The projectile in the simulations of Rosenberg and Kositski (2016) was an ogive-nosed rigid rod with a diameter of 76 mm, and the targets were 760 mm thick with a strength of f c = 35 MPa. A typical deceleration-distance history, from one of these simulations, is shown in Fig. 4.33. The dotted lines follow the nearly linear rise and fall of the projectile’s deceleration during the entrance and exit phases, respectively, and the relatively constant deceleration during the tunneling phase. From these simulations Rosenberg and Kositski (2016) concluded that the entrance phase extends to 1.5D–2D, while the exit phase extension is between 3D and 4D. The main result from these simulations is that the resisting stress on the projectile can be described schematically by a trapezoid, as shown in Fig. 4.34. The extents of Fig. 4.34 A schematic description of the resisting stress-distance profile in a concrete slab

190

4 Plate Perforation

the entrance and exit phases are Z 1 and Z 2 , respectively, and the penetration depth with the constant deceleration is: H−(Z 1 + Z 2 ), where H is the thickness of the target. The area below this curve is, in fact, the work done by the target on the projectile during the perforation process. This work can be represented by an equivalent rectangle whose length is H, and its height (Reff ) is equal to an average (effective) value of the varying resisting stress. Energy conservation dictates that the area of this equivalent rectangle has to be equal to the area below the corresponding trapezoid, for the given projectile/slab configuration. In fact, this is the same construction which was performed for the perforation of metallic plates in Sect. 4.2, with the definition of the effective resisting stress (σ r ). By equating the areas of the trapezoid and the equivalent rectangle, one obtains the following relation for the effective resisting stress (Reff ) which the concrete slab exerts on the projectile: Re f f = Rt (1 −

Z1 + Z2 ) 2H

(4.36a)

Using their simulation results, Rosenberg and Kositski (2016) chose an average value of (Z 1 + Z 2 ) = 5.5D, for the combined extents of the entrance and exit phases. With this value one gets for the effective resistance.   2.75D Re f f = Rt 1 − H

(4.36b)

The following analysis is identical with that for metallic plate peroration, as depicted by (4.5a) and (4.5b). These equations are based on the understanding that the projectile’s kinetic energy at its ballistic limit velocity (V bl ), has to be equal to the work needed to perforate the target. Thus, Reff replaces σ r in (4.5b) and the resulting relation for V bl of a given projectile/slab configuration is:  Vbl =

2Re f f H ρp Lef f

(4.37)

which has been shown by Rosenberg and Kositski (2016) to account for several sets of data for concrete slabs perforated by rigid steel projectiles. Note the similarity between this equation and the corresponding relation (4.5b) for V bl of rigid projectiles perforating metallic plates. Both relations were derived through the same procedure, in which an effective (constant) resisting stress represents the time-varying stress which a thin target actually exerts on the projectile during perforation. One can use (4.36) and (4.37) in order to derive a relation for the ballistic limit thickness (H bl ) of a given concrete/projectile pair. This is the minimal thickness of the concrete slab which is needed to stop a given projectile, impacting the slab at a velocity V 0 . Inserting the expression (4.36b) for Reff in (4.37), and replacing V bl with V 0 , results in the following relation for H bl :

4.7 The Perforation of Concrete Slabs by Rigid Projectiles

Hbl =

ρ p L e f f V02 ρ p L e f f V02 = 2Re f f 2Rt (1 − 2.75D/Hbl )

191

(4.38a)

which simplifies to: Hbl = P0 + 1.75D

(4.38b)

where P0 is the penetration depth of the given projectile in the semi-infinite concrete at an impact velocity of V 0. , as calculated by (3.35). Thus, according to this analysis the thickness of a concrete slab which is needed to stop a given projectile is larger by 1.75 D than the corresponding penetration depth of this projectile in the thick target. Similar results are obtained for metallic plates, as will be shown in Chap. 5. The validity of relation (4.38b) can be demonstrated by using the data of Cargile et al. (1993), who derived experimental values of H bl for their f c = 38.2 MPa slabs. Projectiles with M = 2.34 kg and D = 50.8 mm were shot at these concrete slabs at velocities of 313, 370 and 471 m/s. The data resulted in H bl values of 284, 342 and 452 mm for these impact velocities, respectively. As shown by Rosenberg and Kositski (2016), the analysis through (4.38b) results in the values of H bl = 276.5, 331 and 449 mm, which are in excellent agreement with the data. Before we give more examples for the agreement between experimental data and the model’s predictions, we consider the issue of residual velocities for projectiles perforating concrete slabs, which is similar to the analysis for metallic plates, as discussed in Sect. 4.2. The analysis starts with the energy conservation relation for a rigid projectile of mass M impacting a concrete slab at V 0 and exiting at V r , assuming that the work needed to perforate the slab (W ) is independent on impact velocity. As is always the case with concrete perforation, a lot of concrete debris is ejected both from the rear and front faces of the slab. These are concrete chunks of various sizes which move at different velocities, and it is practically impossible to follow their size and their velocity by direct measurements or by numerical simulation. Thus, one has to make some plausible assumptions concerning the total kinetic energy of the concrete debris. Rosenberg and Kositski (2016) assumed that this energy can be represented by some equivalent mass (m) moving at the residual velocity of the projectile (V r ). The equivalent mass was taken as the mass of a concrete cylinder of diameter D and height of 2D. This estimate was based on the size of the back scabbing layers in concrete slabs, as reported for many experiments. Thus, the energy-based equation for concrete perforation is: 1 1 πρt D 3 1 M V02 = M Vr2 + mVr2 + W with m = 2 2 2 2

(4.39)

where ρ t is the density of the concrete and W is the work needed to perforate the slab. For V 0 = V bl we have V r = 0. Thus, the work needed to perforate the target is given by:

192

4 Plate Perforation

W =

M Vbl2 2

(4.40)

which, together with (4.39), results in the following relation:   Vr = b V02 − Vbl2 where b =

M M +m

0.5 (4.41a)

and this relation can be rewritten in a normalized form as:   V0 2 Vr = −1 bVbl Vbl

(4.41b)

The assumptions made in this model concerning the total energy of the concrete debris, were justified by the good agreements between the data and the model’s predictions for residual velocities, as shown by Rosenberg and Kositski (2016). These comparisons also test the validity of the model regarding the values for V bl , as given by (4.37). Some of these comparisons between measured residual velocities and the model’s predictions are given next. The experimental work of Hanchak et al. (1992) is considered as one of the best sets of data for concrete perforation because they used very different concretes in their study, with f c = 48 MPa and f c = 140 MPa. The CRH3 ogive-nosed steel projectiles, with D = 25.4 mm and M = 500 g, impacted slabs of thickness H = 178 mm at velocities in the range of 300–1000 m/s. In order to analyze the results of these tests we start with the determination of the corresponding f c * values, by using the diameters Dp = 25.4 mm and D0 = 63.5 mm of the projectiles and the test specimens, respectively. Relation (3.39) results in f c * = 60.4 MPa and 176 MPa for the 48 MPa and 140 MPa concretes, respectively, and by (3.40) one finds that Rt = 0.539 GPa and 0.86 GPa for these concretes. Then one uses (4.36b) in order to determine the corresponding Reff values for the slabs, and these are inserted in (4.37), with H/D = 7, in order to obtain V bl values for these concrete slabs. Table 4.7 summarizes the relevant properties of these concretes and the resulting values for V bl , as obtained by the procedure outlined above. One should note that the large difference in the fc values of the two concretes in this study (a factor of 3) resulted in an increase of only 26% in their corresponding Table 4.7 Relevant properties for the concretes in Hanchak et al. (1992) and their V bl values

f c (MPa)

48

140

ρ t (g/cc)

2.44

2.52

f c * (MPa)

60.4

176

Rt (GPa)

0.539

0.86

Reff (GPa)

0.327

0.522

V bl (m/s)

343.5

434

4.7 The Perforation of Concrete Slabs by Rigid Projectiles

193

V bl values. This fact was observed by many workers who analyzed these experiments and tried to account for this unintuitive result. A partial explanation may be obtained by realizing that V bl is dependent on the square root of Rt . Thus, factor of 3 between values of resisting stresses should result in differences of about 70% between their corresponding V bl limits. In fact, this observation holds for metallic targets as well. The question is why a factor of 3 in the f c values results in a difference of about 60% between the Rt values, as is evident in Table 4.7. This small deference is, apparently, due to the logarithmic dependence of Rt on f c *, as given by the empirical relation (3.40) in Chap. 3. Thus, the real issue concerns the physics behind this logarithmic relation, which results in a diminishing increase in Rt as f c * is increased. Hanchak et al. (1992) tried to account for this issue by suggesting that the inherent resistance to penetration of a concrete target is probably not its unconfined compressive strength (f c ). Rather, the strength under confined pressure may be more adequate as a material property, since the concrete ahead of the projectile experiences a certain amount of pressure during penetration. They quote static measurements which showed that the shear strengths of the two concretes in their study increase significantly, under confining pressures up to 600 MPa. However, the difference between these shear strengths diminish with increasing pressures, and at a pressure of 600 MPa the difference between the strengths of the two concretes amounts to only 20%. This asymptotic-like behavior of the shear strength may be the reason for the logarithmic dependence of Rt on f c * which we see in Fig. (3.49), resulting in diminishing changes in Rt with increasing f c * values. The agreement between the model’s predictions for residual velocities and the data of Hanchak et al. (1992) is shown in Fig. (4.35). The values of V r for the concrete slabs of both types were calculated through (4.41) using the corresponding values for ρ t , as given in Table 4.7. The good agreement between model and data enhances the validity of the model, as far as the expressions for V bl and V r are concerned. Fig. 4.35 Comparing the model with data for residual velocities from Hanchak et al. (1992)

194

4 Plate Perforation

Particularly, the assumptions concerning the kinetic energy of the ejected debris are enhanced by this agreement between model and data. Another set of experiments is from Li et al. (2013), with M = 4.8 kg and D = 64 mm projectiles perforating concrete slabs having f c = 34.3 MPa, with thicknesses between 300–600 mm, and impact velocities were around 410 m/s. The maximal aggregate size of these slabs was 10 mm, which means that D/d = 6.4 for these experiments. The calculated value of Rt for these slabs according to (3.37) is 0.34 GPa, and the relevant parameters for the different slabs are given in Table 4.8. The table includes the predicted and measured values of V r for the slabs in Li et al. (2013), with a good agreement between them. The relatively large discrepancy between predicted and measured V r values for the 300 mm thick slab is probably due to the fact that this is a relatively thin slab. As will be shown later on, the analysis presented above has to be modified for thin slabs. Finally, consider the experiments of Wu et al. (2015b) with M = 428 g, D = 25.3 mm ogive-nosed projectiles perforating f c = 41 MPa slabs of thicknesses between 100 and 300 mm. This is a very elaborate work which includes data for slabs which were supported at their back by 1 mm thick steel plates (liners). Rosenberg and Kositski (2016) accounted for the data of the steel-backed slabs by evaluating the stopping power of these 1 mm thick liners, as will be discussed below. As for the unsupported slabs, the analysis starts with calculating the Rt value through (3.37), which results in: Rt = 0.562 GPa for the corresponding projectile/concrete configuration. For the 150 and 200 mm slabs we get: Reff = 0.3 GPa, 0.366 GPa and V bl = 325 m/s, 414.5 m/s for the two slab thicknesses, respectively. With these values of V bl Rosenberg and Kositski (2016) found good agreement between the calculated V r (V 0 ) curves and the data for both slab thicknesses, as shown in Fig. 4.36. With decreasing slab thicknesses the extent of the tunneling region gets smaller, while the extents of the entrance and exit phases do not change. The trapezoidshaped curve for the resisting stress, in Fig. 4.34, will turn into a triangle when the total extent of these phases is equal to the thickness of the slab, namely, when: (Z 1 + Z 2 ) = H. According to the model of Rosenberg and Kositski (2016) this occurs when H = 5.5 D. For this slab thickness, the variation of the resisting stress with distance has a triangle shape with a base of Z 1 + Z 2 and a height of Rt , as depicted Table 4.8 Comparing model predictions with data from of Li et al. (2013), velocities are given in m/s H (m/m)

300

400

500

H/D

4.74

6.25

7.8

600 9.75

Reff (GPa)

0.141

0.19

0.22

0.244

V bl

238

319

384

443

V0

405.7

400.8

415.4

405.1

V r (meas.)

250

220.6

149.5

30

V r (pred.)

298

220

144

0

4.7 The Perforation of Concrete Slabs by Rigid Projectiles

195

Fig. 4.36 Comparing model (curves) and data from Wu et al. (2015b)

by the large triangle in Fig. 4.37. For thinner slabs, with a thickness of less than 5.5 D, Rosenberg and Kositski (2016) assumed that the height of the equivalent triangle is getting smaller accordingly, while its slopes remain the same as those for the large triangle. Thus, the two triangles are assumed to be geometrically similar, as shown schematically in Fig. 4.37. Using simple geometrical considerations for the areas of these triangles, one obtains the following expression for the effective resisting stress (Reff ) of a thin slab, with thickness H = Z D which is less than 5.5 D: Re f f =

Z Rt 11

(4.42)

as derived by equating the triangle’s area with the area of a rectangle which has a width of H = Z D and a height of Reff . This simple relation was found to account for the perforation of the 100 mm slabs (H/D = 3.95) in the work of Wu et al. (2015b). The calculated value of Reff for these slabs, Reff = 0.202 GPa, results in a value of V bl = 217.6 m/s for this projectile/slab configuration. This value for the ballistic Fig. 4.37 The triangular shape of the resistance-distance profile for thin slabs

196

4 Plate Perforation

Fig. 4.38 Comparing the model with the data for the thin slabs in Wu eta al. (2015b)

limit velocity was used to calculate the predicted V r (V 0 ) curve, and its comparison with the data of Wu et al. (2015b) is shown in Fig. 4.38. A very good agreement is obtained between predicted and measured values for the residual velocities. The results of the experiments discussed here can be presented on a single curve, which may be considered a “universal curve”, as shown in Fig. 4.39. The figure includes data from Hanchak et al. (1992) with the two concretes (48 and 140 MPa), and from Wu et al. (2015b) for slab thicknesses of 100, 150 and 200 mm. The figure displays the good agreement between the predictions from (4.41) and the data for the residual velocities in these experiments. The universality of the dependence of V r /(bV bl ) on V 0 /V bl is similar to that depicted in Fig. 4.11, for metallic plates. According to the models presented in this chapter, the two figures should account Fig. 4.39 The universal curve for the normalized residual velocities

4.7 The Perforation of Concrete Slabs by Rigid Projectiles

197

for all finite-thickness targets, as far as the residual velocities of ogive-nosed rigid projectiles are concerned. As mentioned above, several slabs in the work of Wu et al. (2015b) were backed by 1 mm thick steel liners. Such liners were found to be very effective in reducing the amount of debris which is ejected from the back face of concrete slabs. Grisaro and Dancygier (2014) used a rather simple approach to evaluate the effect of such liners on the ballistic limit velocities of steel-backed slabs. They assume that the ballistic limit velocity of the supported concrete slab is increased by the ballistic limit velocity of the steel liner itself. Thus, according to this simple approach one can write: Vbl (lined) = Vbl (free) + Vbl (steel)

(4.43)

where V bl (steel) is the ballistic limit velocity of the steel liner at the back of the concrete slab. Grisaro and Dancygier (2014) compared V bl (steel) values from the analysis of Rosenberg and Dekel (2010c) with an empirical formula which was developed in BRL. They found good agreement between the two formulations, as far as the values of V bl for thin steel plates are concerned. For the unsupported (free) concrete slab they used V bl values from several empirical relations. Rosenberg and Kositski (2016) calculated both V bl values, for the free slab and for the steel liner, through their corresponding analytical models. Thus, the valued of V bl (free) for the unsupported slabs were calculated through (4.37), while for the thin steel liners they used the relations (4.5b) and (4.6) in Sect. 4.2. Designating the thickness of the steel liner by h, these equations take the form:  Vbl =

2σr h ρp Lef f

(4.44)

and σ r for thin plates (h/D < 1/3) is given by:  σr = Yt

2 h +4 3 D

 (4.45)

where Y t is the flow stress of the liner material. Using a value of Y t = 0.4 GPa for the steel liners in the work of Wu et al. (2015b), results in a value of V bl (steel) = 28 m/s for the projectiles used in this study and the h = 1 mm steel liners. Thus, according to this analysis, the supported concrete slabs should have ballistic limit velocities which are higher by 28 m/s than the unsupported slabs. With the addition of 28 m/s to the calculated V bl values for the 150 and 300 mm thick slabs, Rosenberg and Kositski (2016) calculated the respective V r (V 0 ) curves, for the experiments of Wu et al. (2015b). Figure 4.40 shows the excellent agreement between the predicted curves for V r and the data for these supported slabs. The figure also includes the V r (V 0 ) curves as obtained from the model without considering the liner’s contribution. As is clearly seen, the increase by 28 m/s in the values of V bl accounts for the data quite well (Fig. 4.40).

198

4 Plate Perforation b

a

600

600

H=150 mm

H=300 mm 500

400

model without liner

Vr (m/s)

Vr (m/s)

500

300

300

model without liner

300 200

200 100

400

100

model wi th li ner 400

500

600

500

V 0 (m/s)

model with liner

600

700

800

V 0 (m/s)

Fig. 4.40 Comparing the model with data for lined concrete slabs, a H = 150 mm b H = 300 mm

4.8 The Catastrophic Failure of Thin Panels by Hydrodynamic Ram The process by which projectiles penetrating liquid-filled cells transfer their kinetic energy to the cell walls is termed hydrodynamic ram (HR). Upon impact the projectile penetrates the cell wall and a high pressure is transmitted to the fluid by strong shock waves which decay shortly afterwards. Subsequently, the projectile is moving in the fluid, experiencing its drag force, inducing lower pressures in the fluid which last for much longer times. These pressures load the cell walls for long times, often causing structural damage to the whole cell. The main concern with this failure process has been for aircraft fuel tanks which are hit by fragments from exploding shells. The walls of these aluminum tanks, which are about 2 mm thick, often fail catastrophically and the tank explodes as the fuel is gushing out. These concerns led the U.S. defense establishments to support a thorough study during the 1970s, in several ballistics labs. Some of these studies are summarized in Bless and Holten (1977), Bless et al. (1977) and Cross (1980). This section describes, briefly, the experimental findings from such studies and the semi-analytical model of Rosenberg et al. (1987) which accounts for their results in a phenomenological way. The experimental setup in these experiments consisted of a large water-filled tank with a square opening in its front face, where the test panel was attached. Steel cubes or spheres, 5–10 mm in diameter, were shot at these panels as shown schematically in Fig. 4.41. The experiments included pressure measurements at several locations in the water, using Lithium-Niobate pressure transducers, and high speed pictures of the front panel to follow its movement and failure. There are several points to note concerning the experimental observations. The most important one is that for every projectile/panel configuration there was a certain impact velocity (V c ) above which the panel failed catastrophically, with cracks running from the entrance hole to the sided of the panel. The water spilled out from

4.8 The Catastrophic Failure of Thin Panels by Hydrodynamic Ram

199

Fig. 4.41 Schematic description of the HR experiment

the large opening in the panel, resembling the catastrophic failure of aircraft fuel tanks. On the other hand, for impact velocities below this critical velocity (V < V c ), the panel had a square (or round) hole with water spilling out from that hole. Thus, a certain physical parameter has reached its critical value at this threshold velocity. The high speed pictures showed that the cracks formed within 15 μs after impact and that they moved at about 0.3 km/s. The catastrophic failure was due to the large cracks emanating from the corners of the square holes in the panels, reaching the panel edges as seen in Fig. 4.42, which shows 1.6 mm thick 7075-T6 panels after impacts of 5.67 g steel cubes. Impact velocities were 719 m/s with no failure, and 764 m/s with two large cracks emanating from the corners of the hole. The two velocities are very close to each other indicating that the catastrophic failure occurs at a well-defined impact velocity, possibly at 740 m/s for this projectile/panel combination. The experiments showed that the corresponding V c ’s were higher for panels of 2024-T3 aluminum, as compared with panels of the stronger alloys 7075-T6 or 2024T81. For example, a cube of 7.8 g impacting 1.6 thick panels, resulted in a critical velocity of V c = 1000 m/s in the 2024-T3 plate, which is much higher than V c = 760 and 700 m/s for the 2024-T81 and 7075-T6 panels, respectively. Moreover, the values of V c were much higher for spherical projectiles, as compared with the their values

Fig. 4.42 The square holes in 7075-T6 panels impacted below the critical velocity (right) and above (left)

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4 Plate Perforation

for cubes. Rosenberg et al. (1987c) rounded the edges of several cubes and found that the resulting V c values were much higher than the corresponding velocities with the regular cubes. All of these observations point to the role of stress concentrations in the catastrophic failure of these panels. High speed pictures showed that the failure occurred at about 50 ms after impact, when the panels were moving backwards by the action of the pressurized water. These observations led Rosenberg et al. (1987c) to conclude that the catastrophic failure is due to the stretching of the plates, as a result of the water action, as depicted in Fig. 4.43. According to the evidence gathered in these experiments, upon impact the panel is pushed towards the water as a result of the momentum imparted by the projectile. Subsequently, the panel compresses the adjacent water until the pressure is high enough to resist further compression, and it relaxes by pushing the panel backwards. This occurs at a time scale of tens of milliseconds and the panel can be considered as being stretched bi-axially. As stated above, the experimental observations point to a panel failure by the high stress concentrations around the hole at its center, due to the impacting projectile. These stress concentrations are built up during the stretching of the panel, as a result of the water action, as depicted in the figure. The next step of Rosenberg et al. (1987c) was to analyze the bi-axial-stretching of a thin plate with a hole at its center. This biaxial stretching of a thin plate is shown schematically in Fig. 4.43c. Fracture Mechanics considerations were applied in order to quantitatively account for stress concentrations around these holes, either square or round. The applied stresses (σ a ) are equivalent to the stress buildup in the panel as it is being pushed by the water pressure in its middle section, while it is constrained at its boundaries to the frame in the water tank. Thus, one can use the elastic stressstress relation σ a = Eε to find the relation between the stress in the plate and its deflection (δ) in its midsection:

Fig. 4.43 Stages in the panel’s loading during the HR process

4.8 The Catastrophic Failure of Thin Panels by Hydrodynamic Ram

 W δ 2 (t) σa (t) = E −1 ∼ =E W0 2W02

201



(4.46)

where 2W 0 and 2 W are the original and the strained widths of the plate, respectively, and E is its Young’s modulus. Note that both σ a and δ are time varying during the action of the water pressure. The next step in the analysis was to account for δ by some simplifying assumptions. Rosenberg et al. (1987c) assumed that δ is proportional to the impulse imparted by the projectile to the plate, p t, and is inversely proportional to the plate’s areal density (ρ h). The corresponding pressure is dependent on V 20 while t is determined by the time duration which the projectile “spends” around the entrance hole, which is related to b/V 0 , where b is the size of the cube (or sphere). With these assumptions they obtained the following phenomenological relation for the plate’s extension (δ) prior to its catastrophic failure: δ=A

V0 b ρh

(4.47)

where A is some constant. The maximal deflection of the plate (δ c ), at its catastrophic failure, occurs when the impact velocity (V 0 ) reaches its critical value (V c ). In order to determine the maximal stress in the bi-axially plate, with a hole in its center, the following fracture-mechanics considerations were applied. The treatment of a large plate with a small hole in its center is similar to that of bi-axially stressed plate with a penny-shaped crack, with stresses concentrated at its edges. The stress concentration factor is related to σ a (πb)0.5 where 2b is the size of the crack, and its limiting value is the fracture toughness of the plate material (K c ). The main assumption of Rosenberg et al. (1987c) was that the critical deflection of the plate is reached when the stress intensity factor reaches its critical value (K c ). These considerations led to the following relation between δ c and the relevant parameter of the plate: W0 δc = B 1/4 b



Kc E

0.5 (4.48)

where B is another numerical constant. Equating the two expressions for δ c , from (4.47) and (4.48), one finally obtains: Vc = C

ρhW0 b5/4



Kc E

0.5 (4.49)

where C is a constant which lumps all the others. This phenomenological relation was shown by Rosenberg et al. (1987c) to account for many aspects which were observed in the experiments on this issue. For example, the linear relation between V c and the panel’s thickness (h) is demonstrated by 5.8 g cubes impacting 7075-T6 panels, 1.6 mm and 3.2 mm thick, resulting in V c = 740 m/s

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and 1490 m/s, respectively. Also, larger plates widths (W ) resulted in higher V c values, and large projectiles (b) resulted in lower values of V c . The most significant result was that for the more ductile 2024-T351 aluminum alloy V c values were higher than those for the 2024-T81 and the 7075-T6 panels, by a factor which is very close to the square root of their corresponding K c values, as predicted by this relation. The most interesting result from these experiments refers to the way these panels failed. The relatively brittle aluminum alloys, 2024-T81 and 7075-T6, failed catastrophically at well-defined values of V c , while the 2024-T351 panels showed different failure characteristics. At a certain impact velocity, small cracks appeared along the corners of the square holes in these panels, but the panels did not fail catastrophically. An increase in impact velocity resulted in longer cracks which were arrested near the hole, until their length reached about one inch. A further increase of the impact velocity resulted in a catastrophic failure in these plates. Thus, the 2024-T351 alloy exhibited the crack-arrest phenomenon which is typical for ductile material. In fracture-mechanics terminology this behavior is due to the tearing zone in ductile fractures, which results in an increase of the stress intensity factor and is responsible to a more stable cracking process. The increase with crack length in an aluminum 2024-T351 plate, is shown in Fig. 4.44 which was taken from a metal handbook. Note the increase by a factor of about 4 in the value of K as the crack length increases up to about one inch. According to (4.49), a factor of 2 is then expected between the impact velocities to induce a catastrophic failure and to just initiate a crack in these panels. As Rosenberg et al. (1987c) note, this was exactly the velocity ratio for 1.6 mm panels impacted by 11.7 g steel cubes; 650 m/s to initiate the cracks and 1330 m/s to induce the catastrophic failure in the 2024-T351 panels. We elaborated on this issue in order to demonstrate the usefulness of phenomenological approaches when analytical treatments are too difficult to apply. The catastrophic failure of metallic plates due to hydrodynamic ram is different from all the other cases we deal with in this book, since it involves the structural response of the Fig. 4.44 The stress intensity increase with crack length in a 2024-T351 aluminum plate

4.8 The Catastrophic Failure of Thin Panels by Hydrodynamic Ram

203

plates at much longer times. Still, we think that such case studies are worth reviewing because they add much insight to the physics of these impact processes. In particular, we showed that fracture mechanics considerations are very helpful in such cases, highlighting the roles of the various parameters involved.

Chapter 5

Eroding Penetrators

Abstract The chapter describes the penetration processes of shaped-charge jets and long rod penetrators impacting metallic targets. These two types of eroding penetrators are considered as the most lethal threats for armored vehicles. Thus, in order to design efficient armors against them one has to understand the various aspects of their penetration characteristics, which are based on simple physical principles that were developed during and since WWII. We start with a short account of the socalled hydrodynamic theory of penetration, which was first applied for the penetration process of shaped charge jets and was further developed for the penetration of eroding long rods. Numerical simulations are presented in order to examine the validity of these models and to offer ways to improve them.

5.1 General Considerations This chapter deals with the penetration process of eroding penetrators, namely, long rods and shaped charge jets. The subject is of much interest for both warhead designers who try to improve the penetration capabilities of these threats, and for armor engineers who seek to defeat them with lighter and cheaper designs. As described in Chap. 1, a shaped charge jet is formed by the collapse of a thin walled conical liner, usually made of copper, which is inserted in a conical cavity at the end of a cylindrical explosive charge. Following the detonation of the explosive the copper liner is collapsing on itself forming a long jet, a few millimeters in diameter, which moves at an extremely high velocity. In fact, the jet elements acquire different velocities during the collapse of the metallic cone, and they move at a continuous velocity gradient from tip to tail. The head of the jet can reach velocities in excess of 8 km/s while its tail is moving at about 3 km/s. A slower and much thicker part of the jet (called slug) is trailing behind at a velocity of about 2.0 km/s. Due to this velocity gradient the jet can reach lengths of the order of 1 m, resulting in similar penetrations depths in armor steels. Long rod penetrators have diameters of 15–25 mm and aspect ratios in the range of 20–30. These rods are made of high density materials such as tungsten alloys (WHA) or depleted uranium (DU), having densities of 17.0–18.0 g/cm3 .

© Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_5

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Long rods can penetrate armor steel targets of thicknesses which are more than their lengths at impact velocities in the range of 1.5–1.7 km/s.

5.2 The Penetration of Shaped Charge Jets During the years of the Second World War, groups in the U.S. and in England analyzed the structure of high velocity shaped charge jets and (independently) developed the so-called hydrodynamic penetration theory, in order to account for their penetration depths in large metallic blocks. These works, which were published after the war by Birkhoff et al. (1948) and by Pack and Evans (1951), treated the metallic jet as a high velocity fluid which generates extremely high pressures upon impact, so that the strength of both jet and target could be ignored in the analysis. The penetration of the jet was treated as a steady-state process and it was analyzed in terms of fluid mechanics. Figure 5.1a describes the essence of this process for a jet of length (L) and density (ρ j ), impacting a large target of density (ρ t ) at a velocity (V ). Figure 5.1b shows a flash radiograph of a jet penetrating a polymeric target, demonstrating the main features of the hydrodynamic analysis. According to this analysis, the head of the jet is deformed to a mushroom-like shape upon impact, and it penetrates the target at a constant velocity (U) while getting continuously eroded during penetration. It is simpler to view this process from a system of coordinates which moves with a velocity U, as shown in the lower part of Fig. 5.1a. The hydrodynamic penetration model assumes that Bernoulli’s equation, for the equality of pressures at both sides of the jet/target interface along their centerline, applies for this steady-state process. Thus, one can write: 1 1 ρ j (V − U )2 = ρt U 2 2 2

(5.1a)

and the solution of this equation gives U in terms of V according to:

Fig. 5.1 a The hydrodynamic description of jet penetration. b Radiograph of a jet penetrating a polymeric target

5.2 The Penetration of Shaped Charge Jets

U=

207

V where: μ = 1+μ



ρt ρj

(5.1b)

The erosion rate of the jet (dL/dt) is given by (V – U) which has a constant value for constant V and U. The final penetration depth (P) is calculated via the total penetration time T = L/(V –U) when the jet is totally eroded, resulting in the following relation: P=L·

U =L· V −U



ρj ρt

(5.1c)

This elegant result means that the penetration depths of shaped charge jets are linearly dependent on their lengths and on the corresponding densities, but they do not depend on their velocities. However, even in their early publications both Birkhoff et al. (1948) and Pack and Evans (1951) noted that some modifications to the simple hydrodynamic theory are needed. For example, the penetration depths into very soft targets (such as lead) were much higher than those calculated by (5.1c). This extra penetration which was termed the secondary penetration in those works was attributed to the large momentum which the target acquires. This momentum overcomes the strength of a soft target and leads to the extra penetration after complete jet erosion, which was later referred to as the “third phase of penetration” by Orphal (1997). Pack and Evans (1951) realized the importance of target strength (Y t ), by observing the difference between jet penetrations into armor steel and soft steel. They suggested that the total penetration depth is the sum of two contributions, the primary penetration (Pprime ) which depends on target strength (Y t ), and the secondary penetration Psec , as follows: Ptotal = Pprime + Psec

  ρj ΩYt = L 1− + Psec ρjV 2 ρt

(5.2)

where Ω is an empirical constant which depends on the jet and target densities, and the term with the target’s strength can reduce the primary penetration by as much as 30%. The secondary penetration (Psec ) was taken by most workers as equal to half of the average crater diameter, which was also found to be dependent on the strength of the target material. Another modification to the simple hydrodynamic theory was due to the fact that jets have large velocity gradients along their length, as noted by Birkhoff et al. (1948) who followed the particulation of the jet due to this velocity gradient. Eichelberger (1956) considered the combined effects of target strength and jet particulation. He added a strength term (σ ) to the right-hand side of (5.1a), which expressed the difference between the target and jet strengths. He also noted that the strength term (σ ) should be more important for the slower parts of the jet where the impact pressures are not too high. In order to account for jet particulation he multiplied the jet density in (5.1a) by a statistical factor (g), and suggested a modified Bernoulli equation as follows:

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5 Eroding Penetrators

1 1 gρ j (V − U )2 = ρt U 2 + σ 2 2

(5.3)

where the values of g ranged between 1 and 2. In summary, several issues which are related to the non-ideal properties of the jet were already identified during the early stages of the research. Much of the historical developments concerning jet formation, its breakup, and its penetration process were summarized by Dehn (1986) and by Chou and Flis (1986). An updated review of the research on this subject is given by Walters and Zukas (1989). The velocity gradient of the jet elements is responsible for the observed shape of its penetration versus standoff-distance curve, as shown schematically in Fig. 5.2. The first part of this curve depicts the steep rise in penetration depths with increasing standoff, which is due to the stretching of the jet. With further standoff increase, jet particulation results in the more gradual change of the slope in this curve. At a certain standoff the jet elements deviate from their original flight direction and hit the crater wall, resulting in reduced penetration depths with increasing standoffs. This process is responsible for the maxima in these curves, as seen in Fig. 5.2. Obviously, in order to achieve large penetrations these shaped charge devices should be manufactured with high precision and symmetry to ensure their collinear motion even after jet breakup. In order to account for the shape of these curves the concept of “virtual origin” was introduced by several authors during the early years of research. This concept states that all the jet elements can be considered as having a mutual origin in space and time, from which they start moving at different velocities. Allison and Vitali (1963) followed these ideas for the continuous jet and were able to account, analytically, for the increase of penetration with standoff distance up to its breakup. For larger standoffs, they introduced the concept of the impact-velocity-cutoff (V min ), below which a jet element does not penetrate the target. This concept was replaced with the penetration-velocity-cutoff (U min ) by DiPersio et al. (1965), who were able Fig. 5.2 A schematic description of the penetration depth of a shaped charge jet versus its standoff distance

5.2 The Penetration of Shaped Charge Jets

209

to account for the decreasing trend in penetration depths at large standoffs. These concepts provided useful penetration versus standoff formulas for different jet/target combinations. For example, when the jet completes the penetration process before its breakup, the penetration depth (P) as a function of standoff distance (X) is given by:  P=X·

V0 Vmin



1/μ

−1

(5.4a)

where V 0 is the velocity of the jet tip. A different approach, with U min instead of V min , resulted in the following relation between penetration depth and standoff distance:  P=X·

V0 (1 + μ) Umin



1/μ

−1

(5.4b)

These equations are very useful in accounting for the experimental data but they do not address the physical causes for jet breakup. The influence of jet breakup on its penetration capability has been also considered through its breakup time. The process of jet break up was analyzed by several groups through the inclusion of viscosity and inertia, as reviewed by Walters and Zukas (1989).

5.3 The Penetration of Eroding Long Rods The deep penetration of a long metallic rod into a semi-infinite target has been studied intensively since the 1960s when such rods, made of dense materials, replaced the much shorter steel projectiles as battle-tank ammunition. By the term long-rods we mean penetrators with aspect ratios of at least ten, while short projectiles have aspect ratios of five or less. The deep penetration of long rods made of tungsten heavy alloys (WHA) or depleted uranium (DU), into armor steels, is accompanied by their continued erosion which terminates when the rod is almost totally consumed. Thus, the penetration process of long eroding rods is similar to that of shaped charge jets, and is very different from the penetration process of rigid long rods which was discussed in Chap. 3. The process itself has been envisioned by Christman and Gehring (1966) to be divided into four stages, as shown schematically in Fig. 5.3 for the pressure at the rod/target interface. The first stage in the process is the transient entrance phase which includes the high pressure shock waves generated in the rod and in the target, and the large deformation of the rod’s nose. This phase lasts for a very short time by which the rod penetrates to a depth of a few rod diameters. The analytical description of this phase is very complex and a proper account for its contribution to the total penetration depth is still lacking. The second stage is the quasi-steady process of penetration which has been the focus of much research, both analytically and numerically, as will be described

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Fig. 5.3 The four penetration stages of long rods

in this chapter. This is the dominant mode of penetration for high velocity rods of large aspect ratios. The third stage is called “secondary” by some researchers or “after-flow” by others, and it takes place after the rod is totally consumed. Orphal (1997) distinguishes between two different mechanisms which can operate at the end of the primary phase. The “secondary” penetration is related to the extra penetration which the inverted rod can achieve in certain cases. The “after-flow” penetration is similar to that observed by Pack and Evans (1951) for shaped charge jets penetrating low strength targets. Namely, the momentum imparted to the target material is high enough to overcome its strength, and the crater continues to grow even after the rod has been completely eroded. We shall discuss these mechanisms later on and present some numerical results which follow these processes. The fourth stage is due to the target’s recovery at the end of the penetration process by its rebound due to its strength. This process has a very small effect on the total penetration depth and it is usually ignored. A “semi-infinite” target is large enough so that both its lateral dimensions and its thickness do not affect the penetration process. These free surfaces can influence both experimental and numerical results, and it is important to determine the minimal values of the target’s dimensions in order to avoid their effects. This issue was investigated by Rosenberg and Dekel (1994a) through numerical simulations for L/D = 10 and 20 WHA rods, impacting steel targets at velocities in the range of 1.4–2.2 km/s. The Johnson-Cook constitutive model was used for the strength of the targets, with initial yield strength of either 0.8 GPa or 1.2 GPa in order to represent actual armor steels. The first set of simulations was performed for very large targets in order to determine the reference values of the penetration depths (P0 ) in semiinfinite targets. The changes in the penetration depths with the target’s diameter and thickness, were then determined numerically and the results are shown in Fig. 5.4. As is clearly seen in Fig. 5.4a the diameter of the target (Dt ) should be at least 25 times larger than the rod diameter (Dp ) in order to reduce the effect of lateral surfaces to within 1%. As far as the target thickness (H) is concerned, Fig. 5.4b shows that it should be about twice as thick as the penetration depth in order to be considered as a semi-infinite target. As stated above, these guidelines should be considered for both experiments and numerical simulations. Similar results were obtained by

5.3 The Penetration of Eroding Long Rods

211

Fig. 5.4 Simulation results for the effects of target dimensions on the penetration depths of long rods: a the effect of target diameter and b the effect of target thickness

Littlefield et al. (1997) who performed both numerical simulations and experiments with WHA rods impacting steel targets having different diameters, at a velocity of 1.5 km/s. They found that the ratio Dt /Dp has to be larger than about 15 in order to avoid lateral release effects, in general agreement with the findings of Rosenberg and Dekel (1994a).

5.3.1 The Allen–Rogers Penetration Model The first published account for the penetration of eroding long rods was by Allen and Rogers (1961). They used a two-stage light gas gun in order to launch 7075-T6 aluminum cylinders at very slender stationary rods, about 1 mm in diameter, which were made of different materials. This, so called “reverse ballistics” technique, is most suitable for high velocity impacts where the accelerations in the gun are too high

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for relatively soft and slender projectiles. The rods in these experiments were made of magnesium, aluminum, tin, copper, lead and gold, and their aspect ratios were in the range of L/D = 6.3–12.2. The analysis of Allen and Rogers (1961) follows that of Eichelberger (1956) for shaped charge jets, setting g = 1.0 in (5.3) for the lack of velocity gradients in these rods. Since the rods were made of relatively soft materials the strength term (σ) in (5.3) was related to the target’s strength. Solving (5.3) for the penetration velocity U results in the following expression:  V − μ V2 + Q U= 1 − μ2

(5.5a)

where: Q = 2σ ·

1 − μ2 ; μ= ρt



ρt ρp

(5.5b)

and ρ t /ρ p is the target/rod density ratio. Integrating (5.5a) over time, Allen and Rogers (1961) obtained the following expression for the normalized penetration depth (P/L) of these rods.  U V − μ V2 + Q P = =  (5.6a) L V −U μ V 2 + Q − μ2 V For very high impact velocities the term Q in this equation can be neglected, and one obtains the so called “hydrodynamic limit” for the penetration of these rods, as given by: P 1 = = L μ



ρp ρt

0.5 (5.6b)

which is also the penetration depth of an ideal shaped charge jet, as given by (5.1c). For the special case of ρ p = ρ t = ρ, (5.3) results in: U=

σ V − 2 ρpV

(5.6c)

Leading to the following relation for the normalized penetration depth in this special case: V 2 − 2σ/ρ p P = 2 L V + 2σ/ρ p

(5.6d)

The experimental results of Allen and Rogers (1961) are shown in Fig. 5.5, together with the P/L curves as calculated by (5.6a). These curves were calculated

5.3 The Penetration of Eroding Long Rods

213

Fig. 5.5 The data of Allen and Rogers (1961) and the curves predicted by their model

with a value of σ = 1.95 GPa for the strength term of the 7075-T6 aluminum targets, as suggested by Allen and Rogers (1961). The choice of this value for the strength related term (σ ) in the modified Bernoulli equation, will be discussed later on. One can clearly see that, except for the high-velocity experiments with the gold rods, this simple model is very successful in accounting for the experimental results of various long rods. The model predicts that the normalized penetration depths (P/L) should approach their corresponding hydrodynamic limits (ρ p /ρ t )0.5 at high impact velocities, where the strength of the target is less important. This is clearly seen in Fig. 5.5 as the experimental results approach these limits for each rod material (the horizontal lines in the figure). This asymptotic approach means that as far as the penetration depths of eroding rods are concerned, there is no benefit in increasing their impact velocity beyond a certain value. This feature is in sharp contrast with the inherent propensity of rigid rods for higher penetrations with increasing velocities. In fact, rigid rods are much more efficient than eroding rods as far as their corresponding values for P/L are concerned. In order to demonstrate this issue, consider the penetration depths of the ogive nosed steel rods into 6061-T651 aluminum targets in the experiments of Piekutowski et al. (1999). These rods penetrated to depths of about seven times their length at an impact velocity of 1.8 km/s. In contrast, the maximal penetration depth of an eroding steel rod impacting an aluminum target is expected be only 1.7 times its length. A very important observation from both the analysis and the experiments of Allen and Rogers (1961) is the existence of a threshold impact velocity (V c ), which marks the onset of penetration for a rod of density ρ p to a given target. This threshold (critical) velocity is obtained by substituting U = 0 in (5.3) which results in:  Vc =

2σt ρp

(5.7)

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The term σ t is related to the target strength expressing its resistance to penetration. In essence, the critical velocity is achieved when the dynamic stagnation pressure (ρ p V 2 /2) is equal to the target’s resistance to penetration (σ t ). The experimental values of Allen and Rogers (1961) for the threshold velocities of the different rods, resulted in a value of σ t = 1.95 ± 0.5 GPa for their target’s resistance to penetration. This value is about three times the dynamic compressive strength of the 7075-T6 alloy, which was used for the targets in this study. As noted by Allen and Rogers (1961), this is the expected result considering the indentation model of Tabor (1951). According to Tabor’s analysis, the indentation pressure of a solid by a cylindrical punch is equal to about three times its compressive strength. Thus, the minimum value for the dynamic pressure which the rod has to apply in order to achieve some penetration, should be equal to three times the dynamic compressive strength of the target. With this definition of the critical velocity, the normalized penetration of a rod into a target of the same density through (5.6d), takes the form: z2 − 1 V0 P = 2 where : z = L z +1 Vc

(5.8)

The various relations discussed above have been derived through the assumption that the rod strength is negligible and the only strength which enters the analysis is that of the target, through its resistance to penetration (σ t ). Thus, according to the hydrodynamic model all the data for zero strength rods impacting targets of the same density should fall on a single curve which is given by (5.8), where each rod has its own V c value according to (5.7). In order to further highlight the role of these threshold velocities we performed numerical simulations with the AUTODYN-2D code, for L/D = 10 zero-strength steel rods impacting steel targets of different (vonMises) strengths: Y t = 0.4, 0.8 and 1.2 GPa. Figure 5.6a shows the results for the three sets of simulations in terms of the normalized penetration depths versus impact velocities. The results of these simulations enhance the validity of the hydrodynamic theory through several aspects, as follows. Below a certain critical velocity the rod does not penetrate the target and the critical velocities increase with target strength, as expected. In addition, the penetration depths approach the hydrodynamic limit (in this case P/L = 1) with increasing impact velocities, for all target strengths. A close examination of the simulation results shows that the critical velocities are: V c = 0.55, 0.78 and 0.95 km/s for the Y t = 0.4, 0.8 and 1.2 GPa targets, respectively. Using (5.7) we find that the corresponding values for σt for each target are exactly equal to 3Y t , in agreement with the results of Allen and Rogers (1961) and with the indentation analysis of Tabor (1951). Once the threshold velocities are determined, one can construct the normalized penetration curve (P/L) in terms of the normalized velocities (z = V 0 /V c ) according to (5.8), as shown in Fig. 5.6b. It is clear that these simulation results can be represented by a single normalized curve. However, this curve does not coincide with the curve predicted by the hydrodynamic model, (5.8), as is clearly evident in Fig. 5.6b. This comparison between a model’s predictions and simulation results is

5.3 The Penetration of Eroding Long Rods

215

Fig. 5.6 a Simulation results for L/D = 10 zero-strength steel rods impacting steel targets of different strengths, b the normalized presentation of these results, P/L versus V 0 /V c

an excellent example for the important role of numerical simulations in the validation of analytical models. Thus, although the general trends of the model’s prediction and the simulation results are similar to each other, their slopes are quite different and their corresponding values for P/L can be very different for certain velocities. In particular, at high impact velocities the numerical simulations result in higher penetration depths than the hydrodynamic limits which are predicted by the model. Such higher values are often observed experimentally, as we shall see later on. Still, as far as a simplified 1D model is concerned, the hydrodynamic model can be considered as very successful in capturing the main features of the penetration process. An important result of this model is that the threshold velocity (V c ) does not depend on the density of the target. In order to check this issue we performed an additional set of simulations for the impact of zero-strength steel rods, at a semi-infinite aluminum target with strength of 0.4 GPa. These simulations resulted in a value of V c = 0.55 km/s which is exactly the same value obtained for the 0.4 GPa steel target. Thus, one may conclude that (5.7) can be used in order to predict threshold velocities for low strength rods and shaped charge jets. We have noted in the Introduction that one of the main roles of numerical simulations is to validate the assumptions behind analytical models. We have already seen examples for such validity tests in the previous chapters, and from the analysis

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described here we learn about the validity limits of the hydrodynamic theory for penetration. A similar study was performed by Anderson and Orphal (2008), through numerical simulations for zero strength steel and WHA rods, impacting zero strength steel targets. They compared the penetration velocity histories from these simulations to those calculated by the hydrodynamic theory. Their main conclusion is that the differences between these velocity histories are related to the material’s compressibility which is ignored in the hydrodynamic theory. With increasing impact velocities the effect of compressibility increases and for an impact velocity of 6.0 km/s of the steel rod, they found that the interface pressure is higher by about 10% than the corresponding value as calculated by the hydrodynamic theory. Another check of an analytical model is presented next, for the secondary penetration conjecture concerning the high-velocity gold rods in the analysis of Allen and Rogers (1961). The relatively deep penetrations of the gold rods at their highest velocities in the work of Allen and Rogers (1961), were attributed by them to a “secondary penetration” process, according to which the inverted gold rods kept on penetrating the target after the primary phase ended. Their analysis showed that an inverted rod can penetrate the target if the rod/target density ratio (ρ p /ρ t ) is large enough, and if the impact velocity is above a certain threshold. This issue has been reanalyzed by Rosenberg and Dekel (2000) who suggested a different explanation for the extra penetrations of these high-velocity gold rods. According to their interpretation the cylindrical aluminum targets which were launched at the stationary gold rods, were not thick enough to be considered as semi-infinite targets. Due to their high density, the gold rods penetrated to depths which were near the total length of these aluminum cylinders. Consequently, the back surface of these cylinders lowered their resistance to penetration at the later stages of penetration, resulting in the extra penetration depths for the high-velocity impacts. In order to enhance this explanation we performed numerical simulations for the impact of low strength (0.2 GPa) gold rods at semi-infinite 7075-T6 aluminum targets. The rods in these simulations had the same aspect ratio as those in the work of Allen and Rogers (1961), and the strength of the target in the simulation was set at Y t = 0.7 GPa, using the von-Mises yield criterion. This value represents the flow stress of the 7075-T6 alloy at high strain rate, as discussed in Chap. 4. Impact velocities in these simulations were varied between 0.4 and 2.9 km/s as in Allen and Rogers (1961), and Fig. 5.7 compares our simulation results together with their experimental data. The simulations follow the experimental results very closely for impact velocities up to about 2.2 km/s. For higher impact velocities the experimental data fall above the simulation results, which showed no indication for a secondary penetration process with the inverted rods. These simulations enhance the claim that the extra penetrations of the high velocity gold rods were due to some experimental artifact, rather than to a new penetration mechanism. Christman and Gehring (1966) impacted aluminum and steel projectiles with aspect ratios of L/D = 0.17–25 at various targets, with velocities in the range of 0.3–6.7 km/s. Some of their experiments included flash radiographs which followed the rods during the penetration process. They analyzed the four penetration phases which are shown schematically in Fig. 5.3, and concluded that: “the secondary phase

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Fig. 5.7 Simulation results for gold rods impacting aluminum targets and the experimental data from Allen and Rogers (1961), the dotted line is the hydrodynamic limit

begins after the rod has been completely deformed and removed from the system as a source of energy. This phase lasts until the energy density behind the expanding shock wave becomes too small to overcome the intrinsic resistance of the target.” They noted that both the primary and secondary phases may exist simultaneously during penetration. They also realized that at low impact velocities (V 0 < 2 km/s) these rods were not totally consumed, which implied that the back of the rod is decelerating during penetration. Thus, a steady state process was not achieved at the lower impact velocities. Christman and Gehring (1966) accounted for their results through a semi-empirical approach, which was similar to the “after-flow” concept of Pack and Evans (1951) for shaped charge jets. According to this model, the primary penetration is achieved through a steady-state hydrodynamic process which lasts until the rod has a length of about L/D = 1. This last portion of the rod adds an extra penetration, which depends on its velocity and on the strength of the target, as expressed by its Brinell hardness number (Bh ). Their semi-empirical relation for the total penetration depth includes the contributions from both phases, as follows: 

ρp P = (L − D) ρt

 21



ρp + 2.42D ρt

 23 

ρt V 2 Bh

 13 (5.9a)

where the secondary penetration term can be rewritten in a somewhat different form as: 1   13  ρpV 2 3 ρp Psec = 2.42 D ρt Bh

(5.9b)

and this relation will be discussed in more detail when we deal with the hypervelocity impact of metallic spheres. It is interesting to note that relation (5.9b) is very similar to the well-known empirical relation of Charters and Summers (1958) for the penetration depths of metallic spheres impacting semi-infinite metallic targets at the hypervelocity range. The main difference between the two relations is that

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instead of the hardness (Bh ) in (5.9b) the C-S relation has an empirical term (S t ) which should be determined by a single calibration test. This strength parameter (S t ) has been shown to be linearly related to the hardness of the target (Bh ), making the two relations practically the same. The Charters-Summers relation has been very successful in accounting for many sphere/target pairs, as shown, for example, by Payne (1966). Rosenberg and Dekel (2001b) examined secondary penetrations at high impact velocities using numerical simulations for zero-strength steel rods with aspect ratios of L/D = 3–30, impacting semi-infinite aluminum targets with strength of 0.4 GPa. Figure 5.8 show simulation results for an L/D = 10 rod impact at 7.0 km/s. The pressure history at the rod/target interface, in Fig. 5.8a, shows the features described schematically in Fig. 5.3. The simulations show a practically constant variation with time of rod length and its penetration depth with a clear break towards the end of the penetration process, as shown in Fig. 5.8b, while Fig. 5.8c shows the velocity-time histories of the rod’s head and tail. Note the time which marks the end of primary penetration in Fig. 5.8a, and the corresponding start of the secondary penetration. This

Fig. 5.8 Simulation results for an L/D = 10 steel rod impacting an aluminum target at 7.0 km/s. a pressure at the interface, b penetration depth and rod length, and c head and tail velocities

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time was then used to calculate the penetration depths in the primary and secondary phases, as marked in Fig. 5.8b. The pressure and velocity histories show the expected behavior with an initial transient phase which lasts for several microseconds, followed by the primary steady-state penetration with a constant interface pressure, and constant head and tail velocities. The primary penetration phase is followed by a sharp decrease within a short time. This is the time duration of the secondary phase which adds the extra penetration to the steady-state primary phase. Rosenberg and Dekel (2001b) chose the break in the pressure–time history (marked by the arrow in Fig. 5.8a) as the onset of the secondary phase, in order to distinguish between penetration depths during the primary and the secondary phases. Their simulation results agreed with the general concepts of Christman and Gehring (1966), according to which the penetration at the end of the primary phase is very close to the hydrodynamic limit for a rod of length (L−D), and that the secondary penetrations of these L/D = 10–30 rods are due to their last portion with L/D = 1. The secondary penetrations were found to be linearly dependent on impact velocity in the range of V 0 = 3.0–7.0 km/s. Changing the strength of the aluminum targets in their simulations they found that the secondary penetrations are proportional to the ratio (V 2 /Y t )0.5 . This result is not very different from the ratio of (V 2 /Y t )1/3 in the empirical relation of Christman and Gehring (1966) as given by (5.9a). Similar results were obtained by Anderson and Orphal (2003) in their simulations for zero strength WHA rods impacting steel targets who found that the length of the residual rod which is responsible for the extra penetration beyond the primary phase is about 1.25 D, and that this last portion of the rod continues to penetrate until its length diminishes to zero. In summary, these simulations showed that the last portion of the rod with L/D = 1.0 is responsible for the extra penetrations beyond the hydrodynamic limits of long rods. The extra penetrations depend on the rod’s impact velocity and on the target’s strength. No evidence was found in these simulations for the “after-flow” phenomenon, where the crater continues to grow after the rod is totally consumed. We should stress here that the situation is quite different for compact projectile such as spheres, or L/D = 1 cylinders, impacting semi-infinite targets at high velocities. As will be discussed later, these projectiles are consumed very soon after impact, and the momentum which they impart to the target is responsible for the growth of a hemi-spherically shaped crater. As we shall see later on, the final size of this crater depends on the kinetic energy of the projectile and on the strength of the target, even at impact velocities as high as 10 km/s.

5.3.2 The Alekseeνskii-Tate Penetration Model The next stage in the theoretical approach to the penetration of long eroding rods was made independently, and almost simultaneously, by Alekseevskii (1966) and Tate (1967, 1969). Their analysis has been considered as one of the more successful models in terminal ballistics. The so-called AT model (after Alekseevskii and Tate),

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follows the penetration of an eroding long rod having a finite strength (Y p ), which is continuously decelerating during penetration. This model is also based on the modified Bernoulli equation of Allen and Rogers (1961), but it distinguishes between the dynamic strength of the rod (Y p ) and the target’s resistance to penetration (Rt ). The deceleration of the rigid part of the rod is achieved through elastic waves of amplitude Y p , reverberating between its eroding front and its back surface. Thus, an equation for the deceleration of the back portion of the rod was added to the modified Bernoulli equation and to the relation for the erosion rate, which were used by Allen and Rogers (1961). The modified Bernoulli equation, according to the AT model, takes the following form: 1 1 ρ p (V − U )2 + Y p = ρt U 2 + Rt 2 2

(5.10)

The other two equations of the AT model express the deceleration of the rigid part of the rod, of length l, using Newton’s second law: ρ pl

dV = −Y p dt

(5.11)

and the kinematic equation for the rod’s erosion rate: dl = −(V − U ) dt

(5.12)

These equations have to be solved numerically in order to obtain the penetration depths of a given rod into a semi-infinite target for different impact velocities. The model distinguishes between two different situations concerning the rod and target strength parameters, either Y p > Rt or Y p < Rt . Each case has its own solution and characteristic shape for the penetration vs. velocity curves, as follows: 1. Y p > Rt . The strength of the rod is higher than the target’s resistance to penetration and at low impact velocities the rod penetrates as a rigid body. The rigid penetration phase takes place until the impact velocity is high enough and the rod enters its eroding phase. Tate (1969) assumed that the retarding stress on the rigid rod is due to both the strength of the target (Rt ) and its inertia (ρ t V 2 /2), so that the rod’s equation of motion during its rigid penetration phase is:

ρp L

  1 dV = − ρt V 2 + Rt dt 2

(5.13)

It was shown in Chap. 3 that the deceleration of a rigid rod should be constant for impact velocities below the cavitation threshold velocity (V cav ) and that target inertia should be added only when V 0 > V cav . There is no account for the cavitation phenomenon in the AT model. Moreover, the resisting stress on a rigid rod depends

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on its nose shape, as shown in Chap. 3, an issue which is also absent in the AT model. We have also seen that the transition from rigid to eroding rod penetration does not occur at a well defined threshold velocity as suggested by Tate (1969). Rather, a certain range of impact velocities where the rod is deforming without losing mass, precedes the eroding-rod phase. Due to these deficiencies we shall not deal with this part of the AT model. We consider now the second case with Y p < Rt , which is also the more relevant case for armor applications. 2. Y p < Rt . The strength of the rod is lower than the target’s resistance to penetration. This means that the rod does not penetrate the target unless its impact velocity is higher than a threshold value (V c ) which, from (5.10) with U = 0, is given by:  Vc =

2(Rt − Y p ) ρp

(5.14)

For V 0 > V c the rod can penetrate the target and it starts to deform at the impact moment, achieving a mushroom-like shape within a short transient phase. Subsequently, the rod is continuously eroding while its back portion is decelerating. This mode was assumed to be quasi-steady penetration mode with a penetration velocity (U), which decreases with time at the same rate as the velocity of the rod’s back (V ). The modified Bernoulli Eq. (5.10) results in the following relation between the two velocities:  V − μ2 V 2 + (1 − μ2 )Vc2 (5.15) U= 1 − μ2 where μ = (ρ t /ρ p )0.5 as before and V c is the critical velocity defined by (5.14). The rigid part of the rod is continuously decelerating by the reverberating elastic waves, as explained in Chapter 3, and it is continuously shortened by the erosion process. Penetration ceases either when the velocity of the rigid part reaches the value of Vc , or when the rod is completely consumed. Solving the three Eqs. (5.10– 5.12) numerically, results in an S-shaped penetration curve which starts at Vc and approaches, asymptotically, the hydrodynamic limit (ρ p /ρ t )0.5 at high impact velocities. This behavior is shown in Fig. 5.9 for WHA rods (ρ p = 17.3 g/cc) impacting two steel targets (ρ t = 7.85 g/cc) with different strengths. These curves were calculated for Y p = 1.0 GPa for the WHA rod, and Rt = 3.0 and 5.0 GPa for the two steel targets. The main features of these curves have been observed in many experimental studies with different rod/target combinations. These features include (1) the existence of a threshold velocity (V c ) below which the rod does not penetrate the target, (2) the fact that Vc increases with target strength according to (5.14), (3) the S-shape of the normalized penetration curve and its steep rise at impact velocities in the ordnance range (1.0–2.0 km/s), and (4) the asymptotic approach to the hydrodynamic limit (ρ p /ρ t )0.5 at higher velocities. Figure 5.10 shows the experimental results of

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Fig. 5.9 The normalized penetration curves from the AT model for an eroding WHA rod impacting two steel targets of different strength

Hohler and Stilp, from the database of Anderson et al. (1992), for L/D = 10 WHA rods impacting four steel targets with different hardness (strength) at velocities up to 4.0 km/s. One can clearly see that all the features of the AT model, as discussed above, appear in these experimental curves including their S-shape and their asymptotic approach to the hydrodynamic limit which is shown by the dotted line in the figure. Penetrations take place at velocities above a certain threshold (V c ) which depends on the strength of the target, as expected. The most important issue concerning both the AT model and the experimental data, is the strong dependence of the penetration depth on target strength in the ordnance velocity range, as shown in Fig. 5.10. For example, at an impact velocity of 1.5 km/s, an increase by a factor of two in target strength Fig. 5.10 Experimental result for L/D = 10 WHA rods impacting steel targets of different strength

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results in a reduction in penetration depth by a similar factor. At higher impact velocities the effect of target strength diminishes, as predicted by the AT model. Experimental results from other sources with different rod/target combinations as in Perez (1980) show the same features, strongly enhancing the validity of the AT model. However, at high impact velocities the data tend to cross the hydrodynamic limit especially for low strength targets. As discussed above, these extra penetrations beyond the hydrodynamic limit are due to the later stages in the penetration process. It has been shown both experimentally and numerically that the strength of the rod has a much smaller effect on the penetration depth, as compared with the effect of the target’s strength. This is partly due to the fact that higher rod strengths induce an increased deceleration of its rear part which reduces its penetration capability. Thus, the possible gain in penetration depth by higher rod strengths is diminished by their enhanced decelerations. As we shall see later, the failure properties of the rod material are much more important than its compressive strength, as far as the penetration efficiency of these rods is concerned. In order to compare the predictions from the AT model with actual penetration data, we chose two sets of experiments of Hohler and Stilp which are given in the data base of Anderson et al. (1992). The steel targets in these experiments had hardness of 180 Brinell and the L/D = 10 rods were either steel or WHA rods. The results for the normalized penetration (P/L) versus impact velocity for the two sets of data are shown in Fig. 5.11. Clearly, the higher density WHA rods penetrate more than the steel rods and the critical velocity (V c ) is higher for the steel rods, as predicted by (5.13). Both penetration curves tend asymptotically to their corresponding hydrodynamic limits of (ρ p /ρ t )0.5 = 1.5 and 1.0 for the WHA and the steel rods, respectively, as shown by the dotted horizontal lines in Fig. 5.11. The figure also includes the predicted P/L curves from the AT model using Rt = 3.5 GPa for the target and Y p = 1.2 GPa for Fig. 5.11 Comparing AT model’s predictions (curves) with the data (symbols) for WHA and steel rods impacting a steel target

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both the steel and WHA rods. Good agreement is evident for the steel on steel impact at all impact velocities. On the other hand, for the WHA rods the agreement is good only for the low velocity range. We have already mentioned a similar discrepancy between numerical simulations and the hydrodynamic model’s predictions. Thus, the AT model results in lower penetration depths than those obtained by both experiments and simulations, which are in reasonable agreement with each other. As we noted above, these deviations are typical for low strength targets and the agreement is getting better for targets of higher strength. We turn now to the issue of similarity between different sets of rod/target combinations, as obtained by experimental data and their numerical simulations. Well established similarity relations can save a lot of experimental work, in addition to their importance in understanding the physics of a given process. In such a similarity study Rosenberg and Dekel (2001a) showed that different sets of data can be conveniently normalized by using the threshold velocity V c as a normalization factor for the impact velocity. They suggested that experimental data should be presented as P/L versus V 0 /V c , and they used the data of Hohler and Stilp in Fig. 5.10 to demonstrate this issue. The experiments with the WHA rods impacting two different steel targets which are shown in Fig. 5.12a, resulted in values of V c = 0.55 and 0.75 km/s for the 180 and the 388 Brinell targets, respectively. Using the normalized values of V 0 /V c one finds that the two sets of data fall on a single normalized curve as shown in Fig. 5.12b. The numerical simulations of Rosenberg and Dekel (2001a) showed that all the penetration curves for a given rod impacting targets which differ only by their strength, can be represented by a single normalized curve for P/L as a function of V 0 /V c . Also, multiplying the strengths of rod and target by the same factor, leads to the same normalized curve for both pairs. Moreover, the simulations showed that the same normalization scheme applies for different rod/target combinations if their corresponding density ratios are the same.

Fig. 5.12 a The data for WHA rods impacting steel targets with different hardness and b the normalized presentation of these experimental results

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Fig. 5.13 The normalized penetrations of two rod/target combinations having the same density ratio

In order to demonstrate that such normalization schemes are applicable for experimental data, consider Fig. 5.13. The data is taken from Perez (1980), who impacted L/D = 11 titanium-alloy rods at aluminum targets and aluminum rods at magnesium targets. The density ratios of the two pairs are practically the same, ρ p /ρ t = 1.66 and 1.64, respectively, which is the main reason that the data points, for the two sets of experiments, fall on a single curve for P/L versus impact velocity. The perfect similarity between the two sets of data is also due to the fact that their strength ratios Y t /Y p have similar values. Note that the data points in both sets of experiments cross the hydrodynamic limit of (ρ p /ρ t )0.5 = 1.28, as was found for other targets with relatively low strength.

5.3.3 The Validity of the AT Model The validity of the one-dimensional AT model has been the subject of a large number of works over the years. The main incentive of these works was to account for several discrepancies between experimental results and the model’s predictions, some of which were discussed above. Anderson and Walker (1991) performed numerical simulations which were specifically focused on the question whether the penetration process according to the AT model follows their numerical simulations. Figure 5.14 shows the comparison between their simulation results and the model’s predictions for the nose and tail velocities, of an L/D = 10 WHA rod impacting an armor steel target at 1.5 km/s. There are several differences between model and simulation results. The strong deceleration of the rod’s last portion at the final stages of the process in the simulation, is not captured by the AT model. Also, the model predicts that the rod should be fully eroded at impact velocities in the range of 1.5–2.0 km/s, while both simulations and experiments show that a residual rod with L/D = 1 remains at the bottom of the crater. Additional simulations by Anderson et al. (1996c) showed that the difference

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Fig. 5.14 Comparing simulation results with the AT model’s predictions

(Rt –Y p ) is velocity dependent. Thus, one may conclude that due to its simplicity the AT model does not fully account for the complex process of eroding rod penetration. The validity of the strength parameters of the AT model has been also examined through numerical simulations in Rosenberg and Dekel (1994b) for the target’s resistance to penetration (Rt ), and in Rosenberg and Dekel (2000) for the rod’s strength (Y p ). In order to check the validity of Rt , they followed the penetration process of zero strength L/D = 20 rods made of different materials, such as steel, WHA, copper, and aluminum. The targets were of either steel or tungsten-alloy, with yield strengths (von-Mises criterion) which varied between 0.4 and 2.0 GPa. Such simulations, with zero strength rods have two advantages which are, in fact, related to each other. First, the term Y p is eliminated from the modified Bernoulli relation, (5.10), and one can focus on the nature of the target strength parameter (Rt ). In addition, with Y p = 0 the rod is not decelerating during penetration, and both V and U are constant throughout the whole process. These issues simplify the analysis for the validity of Rt , as demonstrated by the following example. The simulation results for the rod’s tail and head velocity histories of a zerostrength L/D = 20 WHA rod, impacting a 0.8 GPa steel target at a velocity of 2.2 km/s, are shown in Fig. 5.15. It is clear that after a short transient phase, of about 20 μs, the penetration velocity reaches a constant value of U = 1.16 km/s. Inserting these values for V and U in (5.10) together with the densities of the rod and target, results in a value of Rt = 3.9 GPa for this target. Thus, the derivation of Rt values from simulations for zero strength rods is a relatively straightforward procedure, which results in a well-defined value for Rt . One should note that the transient phase in this simulation corresponds to a penetration depth of 3–4 rod diameters, which is equal to the dominance range of the entrance phase for spherical nosed rigid rods, as discussed in Chap. 3. Similar simulations were performed for several rod/target pairs where the strength of the target was varied within a large range, in order to derive relations between

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Fig. 5.15 Simulation results for a zero strength WHA rod penetrating a steel target

the target’s resistance to penetration (Rt ) and its von-Mises strength (Y t ). These simulations were performed for L/D = 20 zero strength WHA rod, impacting steel targets of different strength at an impact velocity of V 0 = 2.2 km/s. The resulting relation between Rt /Y t and ln(E/Y t ) for steel targets is shown in Fig. 5.16. It is clear that the resulting values of Rt can be presented through a normalized form with Rt /Y t as a function of ln(E/Y t ), where E = 200 GPa is Young’s modulus of the steel targets. This form of presentation was chosen in order to follow the numerical and analytical results for rigid rods, which were discussed in Chap. 3. With this choice of presentation, the normalized values of Rt /Y t for the different steel targets follow a similar dependence on ln(E/Y t ), according to:   E 2 Rt + 1.21 = ln Yt 3 Yt Fig. 5.16 Simulation results for the dependence of Rt on target strength

(5.16)

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The simulations of Rosenberg and Dekel (1994b) for different rod and target materials, showed that the parameter Rt depends on target strength, as expected, and that it is independent on either rod or target density. Moreover, for impact velocity up to about 3.0 km/s, the values of Rt were practically independent on impact velocity. The main conclusion from this study was that the parameter Rt can be considered as a valid physical property of the target, especially for the ordnance range of impact velocities (up to about 2.0 km/s). In contrast, the simulations presented by Rosenberg and Dekel (2000) for rods with finite strengths, showed that the rod strength parameter (Y p ) of the AT model, depends on both impact velocity and on the rod’s aspect ratio (L/D). They concluded that the parameter Y p in the AT model is its “weak” point, and it cannot be considered as a well defined physical entity. In conclusion, the AT model was found to have several deficiencies due mainly to its simplicity. These deficiencies could have been overcome by additional corrections, if not for the so called “the L/D effect” which is the main problem of the AT model, as discussed below. Before we turn to this issue we wish to highlight the difference between the target’s resistance to the penetration of rigid and eroding rods. It is clear that the values for Rt as obtained from the simulations of zero-strength rods through (5.16), are significantly lower than the corresponding Rt values for spherical-nosed rigid rods as derived from (3.9). We can have an appreciation for these differences by comparing the corresponding Rt values for steel targets of different strength which are penetrated by the two types of rods. For the eroding rod relation (5.16) results in: Rt = 2.6, 4.74, and 6.7 GPa for steel targets with: Y t = 0.5, 1.0 and 1.5 GPa, respectively, while from (3.9) one gets: Rt = 3.2, 5.63, and 7.77 GPa for the spherical-nosed rigid rod penetrating the same steel targets. These differences, which amount to 16–23%, are due to the different flow fields of the target material in the two cases. Thus, the intuitive assumption that spherical nosed rigid rods experience the same resisting stresses as eroding rods is refuted by these simulation results. In order to demonstrate the dependence of the target’s resistance to penetration on the nature of the flow field around the penetrator, consider the simulation results of Rosenberg and Dekel (2008). These simulations followed the motion of spherical cavities inside a large elasto-plastic solid which were loaded by an internal pressure only partially, rather than on their whole surface. The partial loading was achieved by applying the pressure to a high strength thin shell (0.1 mm thick) at the cavity’s wall, as shown schematically in Fig. 5.17. Fig. 5.17 A schematic description of the partially-loaded cavity

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Fig. 5.18 The critical expansion pressures as a function of shell angle

The shells in these simulations extended to various angles (θ) around the symmetry axis of the 2D numerical scheme: θ = 13°, 27°, 54°, 90° and 180°. Due to the 2-D symmetry of these simulations the 180° shell covers the whole area of the spherical cavity, while the 90° shell corresponds to the case where exactly half of the cavity area is loaded. The simulation results showed that the critical (threshold) pressures which are needed to expand these partially loaded cavities are strongly dependent on the area on which the pressure is applied. This dependence for a 1.0 GPa steel target is shown in Fig. 5.18. The critical pressures range from Pcrit = 3.63 GPa for the fully loaded spherical cavity, to Pcrit = 5.64 GPa when the cavity is loaded by the smallest shell (with an angle of 13°). These significantly different values are due to the different ways by which the material is flowing around the loaded area, much like the stream lines of the target around rods of different nose shapes. For the 180° case where the whole cavity surface is loaded, the critical pressure is exactly equal to the value of Psph = 3.63 GPa which is derived from the spherical cavity expansion analysis for a compressible steel of strength 1.0 GPa. This is the expected result because the material in both cases is moving along radial directions, perpendicularly to the surface of the cavity. For all the other cases where the cavity is loaded only partially, we get higher values for the critical pressures. This result clearly shows that the spherical cavity expansion analysis underestimates the actual resisting stress exerted by a target on a penetrating projectile as discussed in Chap. 3. The most interesting result of these simulations concerns the 90° shell, which corresponds to a pressure applied on half of the cavity’s area. The simulations for the partially loaded cavity resulted in a value of Pcrit = 4.7 GPa, which is very close to the value of Rt = 4.74 GPa for the resisting stress of a steel target with Y t = 1.0 GPa, as obtained by (5.16) from the numerical simulations for eroding rods. The agreement is due to the similarity in the flow fields of the target material in the two processes, namely, the deep penetration by an eroding rod and the expansion of a cavity by a constant pressure which is exerted on half of its area. This similarity is shown in Fig. 5.19 presenting the flow field of the 1.0 GPa steel target which was impacted by a copper rod at 2.2 km/s, and the flow field around an expanding cavity which was loaded

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Fig. 5.19 Velocity fields around the crater during rod penetration and around the expanding cavity for the 90° shell

at half of its surface (the 90° shell).The various colors in this figure correspond to different material velocities, and one can clearly see the similarity of the two flow fields.

5.3.4 The L/D Effect It has been a well-established fact that the penetration efficiency as defined by the penetration depth per unit rod length (P/L) is lower for a long eroding rod than that of a short rod. In particular, an L/D = 1 cylinder has a higher efficiency than an L/D = 5 rod, which has a higher efficiency than an L/D = 10 rod. This trend is clearly seen in Fig. 5.20, from Tate et al. (1978), for WHA rods impacting armor steel plates at the ordnance range of impact velocities. These rods had aspect ratios of L/D = 3, 6 and 12, and one can clearly see that the corresponding efficiencies, in terms of P/L, are decreasing with increasing aspect ratios. Fig. 5.20 The decreasing trend in P/L with increasing aspect ratio for WHA rods impacting armor steel targets

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Fig. 5.21 Experimental results for WHA rods impacting RHA targets, demonstrating the L/D effect

The accepted assumption among researchers was that this decreasing trend saturates at about L/D = 10, and that the data for all L/D ≥ 10 rods should fall on a single curve of P/L versus impact velocity. The independence of P/L on L/D is an inherent feature of the AT model for eroding long rods, and much of its popularity was attributed to this feature. However, the experimental data of Hohler and Stilp (1987) showed that the penetration efficiency of long rods continues to decrease even for L/D = 20 and 30 rods. Their results for steel and WHA rods impacting steel targets revealed a strong dependence of P/L on L/D, especially for impact velocities within the ordnance range. This dependence of P/L on L/D was termed “the L/D effect” and it has been the subject of intense research during the 1990’s. Figure 5.21 shows the data from Hohler and Stilp (1987) for WHA rods impacting armor steel targets (RHA), and one can clearly see that the difference between the values of P/L for L/D = 10 and 30 rods reaches 50% at impact velocities around 1.5 km/s. The “L/D effect” became a major subject for further analysis by several workers who used numerical simulations in order to find the mechanisms responsible for this effect. The simulations of Rosenberg and Dekel (1994c, 2001b) reproduced the L/D effect and showed that the differences in P/L are diminishing as the L/D ratios become larger. These simulations showed that even zero strength rods resulted in the same behavior, as is clearly seen in Fig. 5.22 for zero strength WHA rods impacting RHA targets at 1.4 km/s. Since zero strength rods do not decelerate during their penetration, one may assume that the “L/D effect” is due to the entrance phase which exerts a lower resisting stress on the rod. Since short rods penetrate less than the longer ones they probably experience, on the average, a reduced resisting stress, and their corresponding P/L values should be higher. The numerical simulations of Anderson et al. (1995, 1996a) showed that, in fact, several contributions are responsible for the “L/D effect” at different stages of the penetration process. For low impact velocities the non-steady nature of the process at the early stages is responsible for the effect. At the high velocity range, above 2.0 km/s, the terminal phase of penetration adds another contribution to the effect.

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Fig. 5.22 Simulation results for zero strength WHA rods impacting RHA targets at 1.4 km/s

Figure 5.23 summarizes experimental results from several sources for WHA rods impacting steel targets. These data were assembled by Anderson et al. (1995) in order to demonstrate that the L/D effect is practically independent on impact velocity within the ordnance range. In conclusion, the one-dimensional penetration model of Alekseevskii and Tate has been found to be too simplistic to give a fair account for the penetration process of eroding rods, mainly because it does not deal with the transient phases at the start and at the end of the process. The steady state (primary) penetration phase, which is the main subject of the model, is established only after the rod penetrates to a depth of several rod diameters, as shown by the simulations of Anderson and Orphal (2008). The entrance phase for eroding rods includes the transition to the mushroomed-shaped nose, which also takes some time. With this transitory nature Fig. 5.23 The L/D effect for different impact velocities in the ordnance range

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233

of the entrance phase one can have a qualitative estimate for the extent of the “L/D effect”. As was shown above, eroding rods penetrate to depths of about 1.0–1.5 times their original length. Thus, for an L/D = 10 rod the total penetration depths are in the range of (10–15) D. The entrance phase may last for several rod diameters, which means that the steady state process for an L/D = 10 rod, accounts only for a part of the penetration process. Thus, it is not surprising that this “L/D effect” is influencing the penetration depths of L/D = 10–30 rods. The numerical simulations of Anderson et al. (1996a) highlighted another important point in this issue by considering the penetration efficiencies of rods with different aspect ratios. The penetration efficiency can be defined by dP/dL, which expresses the amount of penetration achieved per unit length of eroded rod. Using the equations for the modified Bernoulli model, this efficiency is given by U/(V −U). Anderson et al. (1996a) and Anderson (2003) followed this efficiency in their simulations for WHA rods with L/D = 3 – 30, impacting armor steel targets at several velocities. Figure 5.24 shows the results of these simulations for impact velocities of 1.5 and 2.0 km/s. The relatively large decrease in the penetration efficiency for V 0 = 1.5 km/s can explain the strong “L/D effect” at this velocity, since the average penetration efficiency of the L/D = 10 rod is higher than those of the longer rods. In contrast, at an impact velocity of 2.0 km/s the simulations show that the penetration efficiencies Fig. 5.24 The penetration efficiency as a function of penetration depths for WHA rods impacting a steel target at a 1.5 km/s, b 2.0 km/s

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change very little with penetration depths. Thus, the normalized penetration depths (P/L) of the different rods should be the same for this impact velocity (and for higher velocities) as was observed experimentally. The large difference between the specific efficiency (P/L) for L/D = 1 and L/D = 10 or 20 rods motivated several researchers to consider the segmented-rod concept as a more efficient projectile. This rod consists of ten or twenty L/D = 1 cylinders which are widely spaced. The basic idea here is that this separation enables each segment to complete its penetration before the subsequent segment impacts the crater. This way, the deceleration of the back part of the rod is avoided and the contribution of the back segments enhances the total penetration depth. Numerical simulations, by Orphal and Franzen (1989) for example, showed a promising gain with this arrangement especially at high impact velocities. Tate (1990) reviewed the theory and the experimental data concerning this subject and the practical issues involved with launching a segmented rod. He concluded that the relatively few trials which were published over the years did not reveal an appreciable improvement with segmented rods over continuous ones for impact velocities within the ordnance range.

5.3.5 Other Penetration Models The deficiencies of the one-dimensional AT model which were discussed above prompted several attempts to improve this model. These approaches consider the forces acting on the rod/target interface rather than the stresses along their centerline, as was suggested by Dehn (1987), Rosenberg et al. (1990), and Grace (1993). These models consider the area of the mushroomed nose as another free parameter and derive equations which are based on the resisting forces. The model of Dehn (1987) accounts for a large variety of penetrators (long and short, rigid and eroding), but it involves the knowledge of the projectile/target interface area, which has to be determined empirically for each penetrator/target combination. The non-steady penetration model of Grace (1993) is based on a new concept by defining a finite target volume (thickness), which is pushed by the rod and participates in the penetration process. An equation of motion for this volume has to be added to the deceleration and erosion equations for the rod. A more rigorous treatment of the penetration process for an eroding long rod has been given by Wright and Frank (1988). They reduced the full balance laws for mass, momentum, and energy into a set of approximate penetration equations. They divided the rod material into two parts, the rigid part (R) and the deforming nose (S) as shown in Fig. 5.25, and treated the different parts accordingly. The target is specified by its density (ρ t ) and the cross section area of the crater (At ). In region R, the rigid part of the rod with density (ρ p ), length , and cross section (Ap ), moves at a velocity (V ). Region S is assumed to be constant in size and shape (steady- state situation) and the material is assumed to flow plastically at the R-S boundary, reversing its velocity direction (W ) and attaining different values for its density of (ρ d ) and cross sectional area (Ad ). The target material lies in region T

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Fig. 5.25 The different regions in the Wright and Frank (1988) model

and the cross section of the crater in the target is At . Both R-S and S-T boundaries simply translate horizontally with velocity U. Wright and Frank (1988) consider the conservation equations for the different parts of the rod and make several simplifying assumptions in order to reduce the number of free parameters. For part R they use the deceleration and erosion equations of the AT model, (5.11) and (5.12). For region S they consider the forces which act on the two interfaces R-S and S-T, using the different areas Ap , Ad and At . Conservation of mass and momentum result in the following equations: mass: ρ p A p (V − U ) = ρd Ad (U + W ) momentum: ρ p A p (V − U )2 + ρ p Ad (U + W )2 = Ft − Fr − Fd = F

(5.17a) (5.17b)

The three forces in the momentum equation are: F r —the axial force on the R-S boundary, F d —the axial force on the rearward flow of the inverted rod, and F t —the force on the S-T boundary. These forces have to be specified by some simplifying assumptions. For example, the compressive force on the S-T boundary for a steady state process has the form: Ft = At · (a1 Yt + b1 ρt U 2 )

(5.18)

where the constant b1 plays the role of a shape factor. The most interesting result which emerges from this analysis is that the energy equation takes the form:

1 At 2 At a1 Yt 1 2 ρ p (V − U ) + Y p = ρt b1 · U + + f (Y p , ρ p ) 2 4 Ap 4Ap

(5.19a)

which can be re-written in a more concise way, resulting in a relation similar to the modified Bernoulli equation: 1 1 ρ p (V − U )2 + Y p = ρt∗ U 2 + Rt∗ 2 2

(5.19b)

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where Rt * is the sum of the terms in the square brackets of (5.19a), and ρ t * is an effective target density defined by: ρt∗ =

ρt b1 At 2Ap

(5.19c)

With b1 = 0.5 for the nose shape factor and assuming that At /Ap = 4, the true and effective target densities are equal. For At /Ap = 4, the first term in the square brackets of (5.19a) reduces to the well-known resistance to penetration: Rt = a1 Y t . The last term in the square brackets of (5.19a) is related to the rod, which means that the target resistance to penetration (Rt *) includes the properties of the rod. However, one can move this term to the left side of the equation and add it to the rod strength (Y p ). This result may explain the difficulties discussed above, concerning the validity of the parameter Y p in the AT model. The time-dependent model of Walker and Anderson (1995) is based on a more elaborate description of the deformations in the rod and the target, around their interface. This model starts with the momentum balance equation and uses numerical simulations in order to obtain the velocity gradients in the rod and in the target, away from their interface. The shape of the mushroomed end of the rod is similar to that of the Wright and Frank (1988) model, and the plastic zone in the rod extends to a certain distance (s) from the rod/target interface. The elastic waves decelerating the rod reverberate between its free end and the plastic zone. A velocity profile for the material within the plastic zone is obtained from numerical simulations, and the velocity of the rod is gradually increasing from U to V within this zone. The velocity profile in the target is also taken from numerical simulations through another parameter (α p ) which determines the extent of the plastic zone in the target. The hemi-spherical plastic zone in the target extends to a distance of α p Dc where Dc is the crater diameter. The parameter α p is velocity dependent and, consequently, it changes with time during penetration. The extent of the plastic zone in the projectile (s) is related to the parameter α p through the relation:   Dc V 1 −1 1− 2 s= 2 U αp

(5.20)

and the extent of the plastic zone in the target (α p ) is obtained as the solution of the following transcendental equation:   ρt U 2 α 2p  ρt U 2 2 K t − ρt U α p = 1 + K t − ρt U 2 1+ Yt 2G t

(5.21)

where K t is the pressure dependent bulk modulus of the target material and Gt is its shear modulus. The equations which have to be solved numerically are the equations for rod erosion, the deceleration of its rigid part and the stress balance across the rod/target

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interface. The rod erosion is given by the same equation as in the AT model, (5.11), while the equation for the deceleration of the rigid part is given by:

Yp V −U 1 ds dV =− · 1+ + · dt ρ p (L − s) cb cb dt

(5.22)

where cb is the elastic wave speed in the rod. The third equation includes V and U, their time derivatives and other terms which are related to s and α p . Since α p is both velocity and time dependent the main result of this model is that Rt is not constant during a given penetration process and that it depends on impact velocity. On the other hand, the time variation of α p is not very large, as shown by Walker and Anderson (1995), and one can choose an average value to represent this parameter for a given penetration. The governing equations of this model reduce to those of the AT model if the three-dimensional terms are set to zero and the speed of the elastic waves in the rod becomes very large. In fact, the expression for the target’s resistance to penetration (Rt ) reduces to: Rt =

7Yt lnα p 3

(5.23)

The model predicts, at the very minimum, that Rt varies with the extent of the plastic zone (α p ) and with the flow stress of the target (Y t ). Walker and Anderson (1995) suggested using the cavity expansion analysis from Bishop et al. (1945), in order to provide an a priori estimate for the value of α p . They found that the expression for α p from the cylindrical cavity expansion analysis provides a better agreement with the data, as compared with the spherical expansion analysis. Particularly, they found that α p is close to the ratio of (r y /r) between the radii of plastic zone and the cavity in the cylindrical analysis. This issue was further discussed by Chocron et al. (1998) who confirmed the better agreement with the cylindrical analysis. The analysis of Bishop et al. (1945), which was discussed in Sect. 3.3, gives the following expressions for the extent of the yielded material (r y ) around an expanding cavity of radius r, in an elasto-plastic material with Young’s modulus (E), Poisson’s ratio (ν), and strength (Y ):

1/3 E for the spherical cavity r sph (1 + υ)Y  √ 0.5 r  3E y = for the cylindrical cavity r cyl (2 + υ)Y r  y



=

(5.24a)

(5.24b)

Using (5.24b) together with α p = (r y /r)cyl , one gets from (5.23) the following relation for Rt according to Walker and Anderson (1995):

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√   7 E 7 Rt 3 = ln + ln Y 6 Y 6 2(1 + υ)

(5.25a)

which, for steel targets with ν = 0.29 results in:   E Rt = 1.167 ln − 0.465 Y Y

(5.25b)

The spherical expansion term for r y /r, (5.24a), results in the following expressions for Rt /Y according to this model:     7 E 7 1 Rt = ln + ln Y 9 Y 9 1+υ

(5.26a)

which, for steel targets gives:   Rt E − 0.2 = 0.778 ln Y Y

(5.26b)

These expressions result in values of Rt = 5.72 GPa (cylindrical) and Rt = 3.9 GPa (spherical) for a steel target (E = 200 GPa) with strength of Y = 1.0 GPa. Using relation (5.16) from the numerical simulations described above, results in an intermediate value of Rt = 4.74 GPa for this steel target. It is worth noting that the linear relations presented here, between Rt /Y and ln(E/Y ), are similar to (5.16) which was derived by numerical simulations for the eroding rod. They are also similar to the equations derived for the rigid rods in Chap. 3, through our numerical simulations and the analytical models of Tate (1986) and Yarin et al. (1996).

5.4 Scaling Issues in the Terminal Ballistics of Eroding Rods Up to this point we did not encounter any scaling problems in terminal ballistics since all the penetration data, for both rigid and eroding penetrators, could be normalized through either P/D or P/L as a function of V 0 . Namely, the data is linearly scaled with the dimensions of the penetrator. Scaling issues have an important practical aspect since most of the experimental work in terminal ballistics is done with scaled experiments using in-doors laboratory guns. Typical working scales are 1:3 and 1:4 and one should be aware of any non-scaling issue which may affect the conclusions from down-scaled experiments. This section deals with two closely related scaling issues which were highlighted by Magness and Farrand (1990) and by Magness and Leonard (1993). In their work on depleted uranium (DU) and WHA rods, Magness and Farrand (1990) found that the penetration efficiency of DU rods is higher by an

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239

Fig. 5.26 The normalized penetration of DU and WHA rods in RHA targets

appreciable amount, although the densities and strengths of these rods are nearly the same. Their results for the corresponding penetrations in RHA targets are shown in Fig. 5.26. One can clearly see that the data points fall on two distinct lines, which are practically parallel within the ordnance range of impact velocities (around 1.5 km/s). This difference is a material scaling effect which must be due to some well-defined difference between the properties of depleted uranium and the tungsten alloy. At higher impact velocities the data seem to merge into a single curve, at about V 0 = 2.0 km/s. The different penetration efficiencies were attributed by Magness and Farrand (1990) to the “self sharpening” property of the DU rod, due the propensity of this material to fail by adiabatic shear. According to this interpretation, the nose shape of an eroding DU rod is more pointed than that of a WHA rod, since it fails at relatively lower strains by the adiabatic shear failure mode. The difference in nose shapes was seen in the flash shadowgraphs of residual rods in perforation experiments with finite thickness plates. Moreover, long rods made of depleted-uranium alloy (U-6Nb) which contains 6% niobium, did not perform as well as the DU rods containing 0.75% titanium. The residual rods of the U-6Nb alloy showed no evidence for adiabatic shear failure, strongly enhancing the connection between the ballistic efficiency of long rods and their actual nose shape during penetration. Thus, one can clearly claim that this scaling issue is related to the different properties of these rod materials. The second scaling issue was unveiled when Magness and Leonard (1993) compared data from full-scale experiments with data from 1:4 scaled experiments, for both DU and WHA rods impacting RHA targets. For both materials the normalized penetrations of the full-scale rods were substantially higher than those for the scaleddown rods, as seen in Fig. 5.27. This is a geometrical scaling issue which appears to be similar for both of these rod materials. At a first sight, the geometrical non-scaling seems to be related to the different strain rates which the targets experience in these experiments. Both target and projectile, in the two different scales, experience the same strain histories during the

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Fig. 5.27 Data for full-scale and quarter-scale experiments with DU and WHA rods

penetration process, but times are scaled according to their size. Thus, the strain rates in a small-scale experiment are higher than those in the full-scale experiment. This difference may lead to a higher strength of the smaller targets especially if the target material is rate-sensitive. Consequently, one can expect lower values for the normalized penetrations in the small-scale experiments as was found in these tests. In order to investigate this issue Anderson et al. (1993) performed a series of numerical simulations for WHA rods impacting armor steel targets at 1.5 km/s. The constitutive equation of their target included a strain-rate term which allowed for a quantitative assessment of rate effects. A comparison of the penetration depths for the full scale and the 1:10 model in these simulations resulted in a difference of only 5% for the normalized penetration depths. This was also the magnitude of the strainrate effect for the residual velocities and rod lengths which perforated finite targets in these simulations. Thus, Anderson et al. (1993) concluded that the large geometrical scaling effects which were obtained for both tungsten alloy and depleted uranium cannot be related to rate effects. Rather, they assumed that this scaling effect is due to the accumulated damage in the targets, which is a strong function of the absolute times rather than the scaled times. According to their interpretation, the larger target accumulates more damage since it is loaded for a longer time. These issues were further discussed by Rosenberg et al. (1997a) and Rosenberg and Dekel (1999a) and a possible mechanism was suggested for the two scaling issues described above. The basic idea in these works was that small changes in the nose shape of these rods can result in a significant difference in their penetration capability. This issue was demonstrated by Rosenberg and Dekel (1999a) through numerical simulations for L/D = 6 rigid WHA rods with different nose shapes, impacting steel targets at 1.2 and 1.7 km/s. The results of these simulations are shown in Fig. 5.28, and it is clear that the nose shape of the rod dominates its penetration efficiency. Considering the large differences between the penetration depths of these rods, one may expect that even a small difference in the nose shape of the penetrating rod can lead to a significant change in its penetration depth.

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Fig. 5.28 The influence of nose shape on the penetration efficiency of rigid rods

Rosenberg et al. (1997a) performed several perforation experiments with rods of different materials, in order to examine their nose shapes during penetration. These L/D = 10 rods had the same dimensions and they were made of copper, WHA and the titanium alloy (Ti/6Al/4V), which is well-known for its tendency to fail by adiabatic shearing. This titanium alloy rod was used in order to check the thesis that the failure by adiabatic shearing results in a much sharper nose shape. The recovered residuals of these rods, after perforating finite thickness steel plates, are shown in Fig. 5.29. One can clearly see the pointed shape of the titanium-alloy rod which failed by adiabatic shear, and the round nose of the copper rod which deformed in an ideal hydrodynamic manner. The nose shape of the tungsten-alloy rod is intermediate between these two extremes. The figure also includes the schematic description with which Magness and Farrand (1990) described the different failure mechanism in rods made of various materials. They refer to the three types of nose shapes as: early shear

Fig. 5.29 Recovered residual projectiles of different materials

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failure for DU-like rods which fail by adiabatic shearing, the late shear failure of the semi-brittle WHA rod, and the hydrodynamic mushrooming for a ductile material. Figure 5.29 also shows, schematically, that the different rods are expected to open craters of different diameters in a given target. Thus, a ductile rod should open the widest crater while the self-sharpening rod is expected to result in the narrowest crater, as shown by the arrows in the figure. The work in Rosenberg et al. (1997a) confirmed the assertions of Magness and Farrand (1990), as far as material properties are responsible for the different nose shapes of these rods. In their later work, Rosenberg and Dekel (1999a) recovered the residuals of two L/D = 10 WHA rods with diameters of 8 and 16 mm which perforated steel plates. As seen in Fig. 5.30, the nose shape of the larger rod was less “mushroomed” as compared with the nose of the small rod. In fact, the diameter of the smaller rod’s head was 38% larger than the rod’s diameter, while the larger rod’s head was only 12% larger than its original diameter. Thus, one concludes that the larger rod is “sharper” than the smaller one. Based on their simulation results in Fig. 5.28, they claimed that such a difference can account for the higher ballistic efficiency of the larger rod, and that this is the source for the geometric scaling effect as observed experimentally. These findings strongly support the thesis that both phenomena, the geometrical non-scaling and the superiority of DU over WHA rods, are related to the same effect, namely, the sharpening of the rod’s nose. These rods experience considerable plastic strains around their nose area, much beyond their corresponding failure strains. Thus, the periphery of the nose is continuously failing by shear as they penetrate the target, leading to nose sharpening. The DU and the titanium-alloy rods have lower values for their strain to failure as compared with the WHA rod. Thus, they tend to form much sharper nose shapes resulting in the superiority of DU over WHA, as suggested by Magness and Farrand (1990). On the other hand, the degree of sharpening depends on the rod’s dimensions, as shown in Fig. 5.30. Larger rods tend to sharpen more than the smaller ones, leading to the geometrical non-scaling for both WHA and DU rods, as shown in Fig. (5.27). In contrast, copper is a very ductile material with a large strain to failure, which was responsible for the rounding of the copper rod’ nose. Consequently, one may expect that the normalized penetrations for copper rods of different size will be independent on their dimensions. This was exactly the result obtained by Rosenberg et al. (1997a) for L/D = 10 copper rods which differed by a factor of two in their dimensions, having diameters of 8 mm and 16 mm. Fig. 5.30 The shape of the residual WHA rods of different size

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243

The normalized penetrations (P/L) for these rods, at impact velocities in the range of 0.9–1.6 km/s, were practically the same as shown in Fig. 5.31. This result supports the conjecture of Rosenberg et al. (1997a) concerning the geometric scaling issue, which is related to the properties of the projectile. In order to account for the different mushrooming shapes of the WHA rods with the different size, Rosenberg et al. (1997a) considered the size of the plastic zone ahead of a crack (r p ) as defined by fracture mechanics. According to this theory, materials can behave as either ductile or brittle depending on their size, as compared with their r p values. If the dimensions of a specimen are much smaller than its corresponding r p value, it is expected to behave in a ductile manner under large deformations. In contrast, specimens much larger than their corresponding r p values will behave in a brittle manner. According to fracture mechanics, the size of the plastic zone ahead of a crack is given by: 1 rp = 2π



Kc σy

2 (5.27)

where K c is the fracture toughness of the √ material and σ y is its yield strength. Using typical values of K c = 30 and 150 MPa m for the fracture toughness, together with σ y = 1.2 and 0.2 GPa for WHA and copper, respectively, one obtained: r p = 0.4 mm for the tungsten-alloy and r p = 90 mm for copper. The large value of r p for copper explains the fact that the copper rods behaved in a ductile manner at the two scales as described above. In contrast, WHA rods with a diameter of the order of a few millimeters are expected to behave in a quasi-brittle mode, with different amounts of ductility for the two rod sizes, as obtained in these experiments. The issue of geometrical scaling in situations which involve both plastic flow and fracture received a lot of attention both experimentally and analytically. Such analyses have been summarized by Atkins (1988) and Ivanov (1994) for experimental configurations which are similar to the cases discussed here. In conclusion, the various experimental features of the non-scaling effects, which were highlighted by Magness and his colleagues, can be attributed to the strong dependence of the penetration depth on the exact shape of the rod’s nose. This shape can be very different in rods of different Fig. 5.31 Normalized penetrations for copper rods of two scales

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size, for quasi-brittle materials which are characterized by plastic zones (r p ) of the order of the rod’s diameter. An important observation concerns the diminishing magnitude of the non-scaling effects, for both the superiority of DU rods and the geometric scaling issue, at impact velocities of about 2.0 km/s as seen in Figs. 5.26 and 5.27. In order to explain this issue consider the numerical simulations of Rosenberg and Dekel (1998a) where a failure threshold was added to the constitutive equation of the rod. This is the maximum strain to failure (εf ) which the material can sustain, and it is a common feature of material models which are used in hydrocodes. We showed its influence on the perforation process of plates which are impacted by blunt-nosed projectiles in Chap. 4. Once the strain in a computational element reaches this limiting value, its strength is reduced to zero in the following computational cycles. Rosenberg and Dekel (1998a) varied the value of εf for the WHA rods in their simulations from εf = 0.2 to εf = 5.0, and obtained significant changes of their penetration depths into steel targets, as shown in Fig. 5.32. These simulations were performed for an L/D = 20 WHA rod impacting an RHA target at several velocities. As is clearly seen here, with increasing impact velocity the decrease in penetration depths (due to εf ) is diminishing. This is also the trend seen in the data of Magness and Farrand (1990), as shown in Fig. 5.26. In fact, the data indicates that at a velocity of about 2.0 km/s, the penetration capabilities of DU and WHA penetrators should be practically the same. This experimental observation is accounted for by the simulation results shown in Fig. 5.32, since at an impact velocity of 2.1 km/s the penetration depths for different values of εf are very close to each other. Anderson et al. (1999) showed that the addition of a failure strain parameter (εf ) to the rod’s constitutive relation improves the agreement between simulation results and the experimental data for high-strength steel rods, as far as their residual velocities and lengths are concerned. They assumed that the plastic work per unit volume, which can be absorbed by the projectile material before failure, is constant. Thus, Fig. 5.32 Simulation results for the penetration depth dependence on the strain to failure (εf )

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the product Y p εf , for the different steel rods which they used, was given a constant value of 1.5 GJ/mm3 in their simulations. This value corresponds to a failure strain of εf = 1.0 for a rod strength of Y p = 1.5 GPa. Such a relation between material strength and its maximum failure strain is in accord with experimental results, which show that stronger steels have lower elongations under tensile tests. In conclusion, the failure strain parameter (εf ) in numerical simulations accounts for several important features which have been observed experimentally. As suggested by Anderson et al. (1999), the values for εf in these simulations should be closely related to the real material property, namely, the maximum strain which it can tolerate under dynamic loadings. The importance of attaching real material properties to the target and the projectile in numerical simulations was further demonstrated by Rosenberg and Dekel (1998a). In these simulations for WHA rods impacting RHA targets, the strength of the rods (von Mises) was varied between 0.2 and 2.5 GPa. When the constitutive equations for the rods did not include a failure threshold, the simulations resulted in clear maxima in the penetration depth versus rod strength curves, as shown in Fig. 5.33. The simulations for an impact velocity of 1.4 km/s showed that the deepest penetration was obtained for a rod strength of about 0.8 GPa. This unexpected result is due to the two competing effects which are related to the rod’s strength. Clearly, a higher strength rod is expected to penetrate more into a given target. On the other hand, with increasing rod strength its deceleration during penetration is higher, which can lead to less penetration. Thus, for some rod strength the two contradicting effects balance each other, resulting in these maximal penetrations. When a thermal softening mechanism was added to the constitutive relation of the rod, these maxima disappeared, as is clearly seen in this figure. In contrast, the addition of a failure mechanism to the target did not remove the maxima in the simulation results, and it only shifted the P/L curve to higher values since the target became weaker with this addition. Fig. 5.33 Simulation results for the penetration efficiency of rods with and without thermal softening

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It is interesting to note that the simulations which included thermal softening of the rod show that its penetration efficiency (P/L) is very sensitive to its strength at low strength values. On the other hand, for rod strengths beyond about 1.2 GPa this sensitivity practically disappears. According to these simulations there is little gain by increasing the strength of the rod from 1.2 to 2.0 GPa. This saturation effect is probably the result of the compromise between the two opposing mechanisms with increasing rod strength, namely, the increased penetration capability of the rod on the one hand and its higher deceleration on the other.

5.5 Deep Penetration at the Hypervelocity Regime The survey of Hermann and Jones (1961) includes many experimental results for the penetration depths of compact projectiles, impacting semi-infinite targets at the hypervelocity regime (2.0–10 km/s). Such experiments were also reviewed by Hermann and Wilbeck (1987). The projectiles in these experiments were either spheres or short cylinders with an aspect ratio of about 1.0. The target craters at these high velocities had a hemispherical shape, which is the expected result when all the energy of the projectile is spent almost instantly upon impact. In fact, the crater in such an impact is formed by the radially expanding intense shock wave which is generated upon impact. In contrast, at low impact velocities the crater formed by a compact projectile is cylindrical, and its depth is either larger or smaller than its diameter, depending on the densities and strengths of the materials involved. With increasing impact velocities the craters become more hemispherical in shape. Several workers even used this observation in order to define the threshold impact velocity which marks the transition to the hypervelocity regime. The ratio between the volume of these craters and the kinetic energy of the projectiles was found to be constant for impacts in the hypervelocity range. The value of this ratio depends on the ultimate strength of the target material or, as summarized by Hermann and Wilbeck (1987), on its Brinell hardness (Bh ) according to the relation: Ek = 27Bh Vol

(5.28)

where the kinetic energy of the projectile (E k ) is given in joules, and the volume of the crater (Vol) in cm3 . It may be difficult to accept that such a relation can hold for velocities up to 10 km/s, where target strength is expected to play a minor role in the process. In fact, several works which were presented at the early hypervelocity symposia (from 1955 to 1965), tried to find the threshold velocity above which the strength of the target becomes negligible. In contrast with early conceptions, the resulting data showed that the effect of target strength is not diminishing even at impact velocities above 10 km/s. For example, Kineke and Richards (1963) extended the earlier work of Halperson and Atkins (1962), by impacting aluminum spheres at two types of aluminum targets. These were the 2014-T6 alloy with strength of

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about 0.5 GPa, and the much softer 1100-F alloy with strength of about 0.05 GPa. Their results showed that the crater volumes in the two targets had a constant ratio of about 0.37, even for impact velocities as high as 9.7 km/s. The 4.76 mm aluminum spheres resulted in a constant difference of about one projectile diameter, between the crater depths into the two aluminum targets for impact velocities in the whole range of 3–9 km/s, as shown in Fig. 5.34. Thus, even at these extremely high velocities the strength of the target plays a dominant role, as far as the crater dimensions are concerned. The data for metallic spheres impacting semi-infinite targets, as summarized in Hermann and Jones (1961) and in Belyakov et al. (1964), show that the normalizing parameter which controls the penetration depths of these spheres, is the term ρ p V 20 /Bh , where Bh is the hardness of the target. This parameter is often termed the Best number, after the 19th century ballistician, who investigated the relation between the kinetic energy of projectiles and the volume of the craters which they produced in clay targets. Hermann and Jones (1961) presented semi-empirical models for the penetration depths of high velocity spheres, which use the Best number as one of the non-dimensional parameters, and the projectile/target density ratio as a second parameter. Note that the empirical formula of Christman and Gehring (1966), (5.9b), for the secondary penetration of long rods at high impact velocities, also includes the Best number and the rod/target density ratio. According to Christman and Gehring (1966), the secondary penetration is due to the last segment of the rod, with L/D = 1. Thus, it is not surprising that its penetration depth is dependent on the Best number, as is the case with the high velocity impacts of compact projectiles. In order to demonstrate the validity of this normalization scheme consider the data for the two aluminum alloys, 1100-F and 2014-T6, as shown in Fig. 5.34. The normalized presentation of the data with Bh = 26 and 137 BHN for these alloys is shown in Fig. 5.35, and one can clearly see that this is a viable normalization scheme. Belyakov et al. (1964) showed that the P/D data for symmetrical impacts, with Fig. 5.34 Experimental results for aluminum spheres impacting two different aluminum targets

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Fig. 5.35 Penetration data from Fig. 5.34 as a function of Best’s number for two aluminum targets

spheres of iron, copper, aluminum and lead impacting targets of the same materials, fall on a single curve which extends to values of more than 400 for the Best number (ρ p V02 /Bh ). Assuming a constant ratio between kinetic energy and crater volume as given by (5.28), it is quite straightforward to derive the following relation for the normalized penetrations (P/D) for spherical craters: P 1 = k(ρ p V02 ) 3 D

(5.29)

where the constant k depends on the specific properties of the projectile/target pair. 2/3 One should note that the dependence of P on V0 results in a curve which is very close to the curve of P as a function of ln(V 0 ). Thus, it is not surprising that many workers used this logarithmic relation for P = P(V 0 ), in their semi-empirical models for hypervelocity impacts, as summarized in Hermann and Jones (1961). As was shown by Holsapple (1987) the particular dependence of P/D on ρ p and V 0 as given by (5.29) has a physically-based justification through dimensional analyses, which are based on the point source approximation. The basic assumption here is that the energy delivered by the projectile to the target, is the same as that of point-source acting on its surface at the moment of impact. The point-source approach has been used successfully since the 1940s for gases and fluids. However, its application for solids was possible only after proving that a single scalar measure of the form D(V 0 )δ1 (ρ)δ2 , is the general measure of all point source solutions. Here D and ρ are the projectile’s diameter and density, respectively, and V 0 is its impact velocity. The exponents δ 1 and δ 2 determine the resulting scaling rules according to the initial assumptions of the theory. Holsapple (1987) showed that energy scaling leads to the result given by this relation with δ1 = 2/3 and δ 2 = 1/3, which is what we have in (5.29). The interested reader is referred to the articles in Kinslow (1970) for more

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information concerning the theory of scaling rules and point source analyses, and for more numerical and experimental data concerning hypervelocity impacts. As mentioned above, there are several differences between the shapes of craters produced by compact projectiles and those by long rods, for impacts at the hypervelocity range. With compact projectiles these craters are roughly spherical while long rods produce narrow and deep craters. Another difference is related to the limiting penetration depths at very high velocities. The penetration depths of long rods asymptotically approach the hydrodynamic limit of (ρ p /ρ t )0.5 , or to a somewhat higher value when strength effects are important. On the other hand, we have seen that for spherical projectiles the crater depth is practically unbounded, as it grows according to V 2/3 0 . This is also the case for short cylindrical projectiles with aspect ratios of about 1.0. It is interesting to find the threshold value of L/D which marks the transition from compact projectiles to long rods. In order to explore this issue, at least qualitatively, consider the simulation results of Rosenberg and Dekel (2001b) as shown in Fig. 5.36. These simulations were performed for zero strength steel projectiles with L/D = 3–30, impacting 0.4 GPa aluminum targets at velocities up to 7.0 km/s. Note that the normalized penetration depths (P/L) are multiplied by (ρ t /ρ p )0.5 in order to set all the values of the asymptotic limits at 1.0. One can clearly see that the shortest projectiles, with L/D = 3 and 5, behave like the spherical projectiles discussed above, resulting in a continued increase of penetration depths with impact velocity. On the other hand, the L/D = 20 and 30 rods can be considered as long rods, since their penetration depths tend asymptotically to the hydrodynamic limit. The L/D = 10 rod shows an intermediate behavior and one may conclude that the threshold value of L/D, which marks the transition from a compact projectile to a long rod, is somewhere between L/D = 5 and L/D = 10. In order to support this conjecture, consider the experimental results for the penetration depths of L/D = 5 tantalum and L/D = 10 WHA rods impacting 6061-T6 aluminum targets, as shown in Fig. 5.37. The data for the tantalum rods is taken from Bless et al. (1987), and the data of Hohler and Stilp for the WHA rods is from Anderson et al. (1992). Both sets of experiments cover a large range of impact velocities, and the normalized penetration (P/L) are multiplied by their corresponding density Fig. 5.36 Simulation results for zero strength steel projectiles impacting 0.4 GPa aluminum target

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Fig. 5.37 Normalized penetrations in aluminum targets of L/D = 5 tantalum rods and L/D = 10 WHA rods

factors (ρ t /ρ p )0.5 as in Fig. 5.37. One should note that the densities of tantalum and the tungsten-alloy rods are very close to each other. Clearly, at high velocities the penetration depths of the L/D = 5 tantalum rods keep increasing with impact velocity. On the other hand, the penetration depths of the L/D = 10 WHA rods reach an asymptotic value in accord with the penetration theory for long rods. These different behaviors enhance the above statement that the division between short and long rods is somewhere between L/D = 5 and L/D = 10. The normalized penetrations of the L/D = 10 WHA rods reach an asymptotic value which is considerably higher than the predicted value from the hydrodynamic penetration model. This behavior was observed with other rod/target combinations and it is due to the high strength ratio for this rod/target pair. Within the ordnance velocity range the normalized data for both sets of experiments fall on the same line, although there is a large difference between their corresponding L/D values. This is probably due to several competing effects which are influencing the penetration process in different ways. On the one hand, the L/D effect should have resulted in higher normalized penetrations for the shorter tantalum rods. On the other hand, these tantalum rods have a lower strength as compared with the WHA rods, resulting in their lower penetrations. In addition, tantalum is a ductile metal while the tungsten alloy has quasi-brittle properties. This difference can add to the deeper penetrations of the WHA rods through its nose sharpening, as discussed above. These contradicting effects influence the penetration process of long rods especially at impact velocities in the ordnance range, and it seems that they compensate each other in this case with the two types of rods.

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5.6 Plate Perforation by Eroding Rods The perforation of plates by eroding rods is by far the most important issue for the armor designer since this is the case encountered in most practical situations. At the same time, this is also the most complicated interaction between projectiles and targets, as it involves many issues which are difficult to describe analytically. These issues include the non-steady nature of the penetration and erosion processes, the reduction of the resisting stress by the target’s back surface, and the various failure mechanisms which take place in the plate even before the projectile reaches its back surface. These complications are the main reason for the lack of simple analytical models which deal with ballistic limit velocity, residual velocity, and residual mass of the eroding penetrator. Still, several analytical models, such as those of Ravid and Bodner (1983) and Walker (1999), try to follow the penetration of eroding rods into finite plates, and to account for the bulging of their back surface. The Ravid–Bodner model is based on plasticity theory while Walker’s model considers the velocity field ahead of the penetrator and the influence of the target’s back surface on this field. Chocron et al. (2003) combined these two perforation/bulging models and applied them for a case where an eroding rod is perforating a number of separated steel plate. However, all of these models are far from simple and they rely heavily on numerical simulations. The complexity of this interaction is demonstrated in Fig. 5.38, which shows the penetration crater in a steel plate impacted by an L/D = 10 steel rod at 2.03 km/s, from Stilp and Hohler (1990). The thin-walled cylinder of the inverted rod is clearly seen in the figure as it lines the crater wall. Note the bulging of the plate and the evidence for its tensile failure at the back surface. Also, the increased diameter of the crater near the back surface is an indication for the lower resistance of the target at the later stage of penetration. Much of the work on finite plates has been aimed at determining the minimal impact velocity (V bl ) which is needed to perforate a given target of thickness (H). The data can be also interpreted as giving the minimal plate thickness (HBL ) which is needed to stop the rod at a given impact velocity. As expected, the limiting thicknesses of these plates are higher than the corresponding penetration depths into semi-infinite targets. The data compiled by Anderson et al. (1992) contains many results of such HBL (V 0 ) relations for various rods and plates. Figure 5.39 shows the data for HBL (V 0 ), by Hohler and Stilp, for L/D = 10 steel rods impacting steel plates with hardness of 180 Brinell. The normalized penetration depths (P/L) into semi-infinite steel targets Fig. 5.38 The crater and the bulge in a steel target impacted by a steel rod at 2.03 km/s

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Fig. 5.39 The data for L/D = 10 steel rods impacting semi-infinite and finite steel targets

of the same hardness, are also shown in the figure. One can clearly see that the difference between the two sets of data is quite appreciable even at high impact velocities. In fact, the values of the ratio P/HBL increase from 0.63, at an impact velocity of 1.0 km/s, to about 0.85 for a velocity of 3.0 km/s. Translating these differences to units of rod diameters, we find that HBL is larger than P by one to two rod diameters in this velocity range. Similar results were obtained by Tate et al. (1978) for WHA rods impacting RHA targets. Thus, one can use these observations as a “rule of thumb” in order to asses HBL values from the corresponding penetration depths into semi-infinite targets. It is also interesting to note that similar relations between HBL and the penetration into semi-infinite target, were found for concrete targets, as discussed in Chap. 4. Hohler and Stilp (1975) performed a set of experiments with L/D = 10 steel rods impacting targets which consisted of two equal-thickness steel plates separated by some distance. The shape of the craters in one of these experiments, from Stilp and Hohler (1995), are shown in Fig. 5.40. The shape of the residual rod and the debris cloud from the plate as they exit the first plate is also seen in this shadowgraph. The impact velocity of the rod in this experiment was 2.63 km/s, and each plate was 25.2 mm thick.

Fig. 5.40 The resulting craters and the exiting rod in an experiment with a double-plate target

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Fig. 5.41 The data for a double plate configuration (circles) compared with the curves for finite-thickness and semi-infinite targets

The data from these experiments of Hohler and Stilp (1975) is plotted in Fig. 5.41. in terms of the combined thickness of the two plates (T 12 ) which is needed to stop these projectiles, as a function of their impact velocity. The curves for HBL /L and P/L from Fig. 5.39 are also shown in Fig. 5.41. At low impact velocities (V 0 < 1.2 km/s) the data for the double plate configuration (T 12 /L) follows the curve for the finitethickness plate (HBL /L), and it shifts towards the curve for the semi-infinite target (P/L) at higher impact velocities. The residual velocities of long eroding rods have been summarized by Grabarek (1971) for a large number of experiments with armor steel plates. He suggested the following empirical relation for the residual velocities: 1.1y 2 + 0.8y + 2y 0.5 Vr V0 = −1 where: y = Vbl 1+y Vbl

(5.30)

This relation was found to hold for both normal and oblique impacts and it represents the data for impact velocities in the range 1.0 < V 0 /V bl < 2.5. For impact velocities above 2.5 V bl the residual velocities are practically the same as the impact velocities. This result can be understood by noting that the deceleration of long eroding rods is relatively small for most of their penetration process, both in finite thickness and in semi-infinite targets. As was discussed above only when the rod length is of the order of its diameter, the deceleration becomes appreciable. Thus, when the impact velocity is much higher than the corresponding ballistic limit, the rod “spends” a relatively short time in the target with a negligible reduction of its velocity. This observation is clearly evident by the experimental results of Hohler et al. (1978) for L/D = 10 WHA rods impacting steel plates of different thickness at 1.52 km/s. Figure 5.42 shows their results for both normal impacts and for an obliquity of β = 60o , where β is the angle between the rod’s axis and the normal to the plate. For a plate of thickness H which is inclined at an angle β, the relevant plate dimension is the line-of-sight thickness given by H/cosβ. Thus, the normalized thicknesses of these plates are H/(Lcosβ) as shown in the figure. The two curves

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Fig. 5.42 The normalized residual velocity as a function of plate thickness for normal and oblique impacts

enhance the features discussed above concerning the small reduction of the residual velocities, for targets which are much thinner than HBL , and the large drop in V r when the target thickness approaches HBL . The values of HBL are shown by the small arrows on the ordinate axis of Fig. 5.42. The value of HBL for the inclined target is somewhat higher than that of the normal target, indicating that plate inclination improves its ballistic performance beyond the simple increase in the line-of-site distance. Anderson et al. (1996b) used Grabarek’s functional dependence, (5.30), for their data on steel rods with different strengths perforating armor plates of two hardness values. Their best fit curve resulted in the following values for the numerical coefficients in (5.30): 0.9, 1.3 and 1.6 for the y2 , y and y0.5 terms, respectively. These values are different than those given by Grabarek’s empirical equation: 1.1, 0.8 and 2.0, respectively. It is clear that such empirical fits with no physical basis are of little use if they do not account for a variety of experiments with the same set of empirical constants. A different relation for the residual velocities of eroding long rods has been suggested by Lambert (1978). His semi-empirical approach is based on some analytical considerations which are described by Zukas (1982). The relation suggested by Lambert (1978) for the residual velocity of an eroding rod is: Vr = k0 · Vbl



V0 Vbl

m

1/m −1

(5.31)

where k 0 and m are empirical constants which have to be determined for each projectile-target combination. Using this equation with k 0 = 1 and m = 2.5, we find that it accounts very well for the data of Anderson et al. (1996b), as shown in Fig. 5.43. Burkins et al. (1996) measured the residual velocities of L/D = 10 WHA and DU rods perforating Ti/6Al/4 V plates and found that (5.31) with k 0 = 1 and m = 2.6 accounts for their data quite well. Considering the two empirical approaches which were discussed here, the Lambert relation (5.31) is preferable to Grabarek’s

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Fig. 5.43 Residual velocities for steel rods perforating RHA targets and their fit to Lambert’s equation with k 0 = 1 and m = 2.5

relation (5.30), since it includes less parameters and it is based on some physical considerations for the process. In order to further investigate this issue we performed a set of numerical simulations for L/D = 10, D = 6 mm steel rod impacting a 35 mm thick steel target. The von-Mises strengths of the rod and the target in these simulations were 1.0 GPa and 0.8 GPa, respectively. Impact velocities ranged between 1.5 and 2.2 km/s and the resulting normalized values for V r /V bl as a function of V 0 /V bl , are shown in Fig. 5.44. The value for V bl from these simulations was 1.55 km/s. The curve shown in Fig. 5.44 follows Lambert’s relation (5.31) with k 0 = 1 and m = 2.5, and it seems to account for the simulation results quite well. This agreement enhances the claim that the Lambert relation captures the essential features of the perforation process, as far as residual velocities of eroding rods are concerned. The parameter m in this relation determines the steepness of the curve at low values of V 0 /V bl , while the high values of V r /V bl are less sensitive to this parameter. In order to demonstrate these points, consider the values of V r /V bl for V 0 /V bl = 1.1 from (5.31). Fig. 5.44 The normalized residual velocities from numerical simulations

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The resulting values are V r /V bl = 0.458 and 0.69 for m = 2 and m = 3, respectively. These values differ by 50%, demonstrating the strong dependence of V r /V bl on the parameter m for impact velocities which are close to V bl . On the other hand, for V 0 /V bl = 3 the Lambert relation results in V r /V bl = 2.83 and 2.96 for m = 2 and m = 3, respectively, which differ by only 4% from each other. It is also clear that the Lambert relation results in the fast approach of V r to V 0 at impact velocities of V 0 = 2.5 V bl , as was also observed by Grabarek (1971). The residual length of an eroding long rod decreases proportionally with the plate’s thickness, as expected. The experimental results of Hohler et al. (1978) for L/D = 10 WHA rods perforating armor steel plates at different impact velocities are shown in Fig. 5.45 for both normal and oblique impacts. As discussed above, the normalized thickness of an inclined plate is given by H/L0 cosβ. Impact velocities in these experiments were about 1.52 km/s and the data from both sets of experiments, for the relative change in rod length ( L/L 0 ), fall on a single curve. The asymptotic value for the change in rod’s length is L/L 0 = 0.9, which means that the residual length is one rod diameter. This is in agreement with numerical results discussed above for residual rod lengths at the ordnance range of impact velocities. These experimental results exhibit a linear relationship with a slope of 0.9 between L/L 0 and the plate’s thickness, as seen in Fig. 5.45. In order to account for these features consider the simulation results of Anderson and Walker (1991) in Fig. 5.14, which shows the time variation of the penetration velocity (U) and the velocity of the rod’s tail (V ), for a WHA rod impacting a semi-infinite RHA target at 1.5 km/s. These velocity histories are nearly parallel to each other, with an almost constant relation between U and V for most of the penetration process. This observation can be used for a simple analysis, as follows. We start with the simplifying assumption that the ratio (U/V ) is constant also for the case of plate perforation. Thus, we write: U = αV Fig. 5.45 The length reduction of as a function of plate thickness, from Hohler et al. (1978)

(5.32)

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where α < 1, and we use the definition of rod erosion:   U dL = −(V − U ) = − − U = −γ U where: γ = (1 − α)/α dt α

(5.33a)

which can be rewritten as: dL = −γ U dt

(5.33b)

Integrating this equation, with the notion that the integral of Udt is the plate thickness H, we finally get: L/L 0 = γ (H/L 0 )

(5.34)

which is the desired result, exhibiting the linear relation discussed above. In order to account for the value of γ consider the simulation results in Fig. 5.14. We find that the ratio α = U/V is about 0.53, for the relevant rod/plate configuration. Inserting this value of α in (5.33a) results in a value of γ = 0.89, which is very close to the experimental value of 0.9 for the slope in Fig. (5.45). One may assume that similar data for other rod/plate configurations will exhibit linear reduction in the rod’s length with plate thickness, and that the slope of these lines can be estimated by the simple analysis presented above.

Part III

Defeat Mechanisms

Chapter 6

Defeat by High Strength Targets

Abstract This chapter discusses the defeat mechanisms which are based on the high strength of the armor material. The armor designer is looking for low-density materials in order to reduce the armor’s weight, and such materials are described here. Particularly, ceramic plates and high-strength reinforced polymers are in common use in modern armor designs. The interaction between various threats and these designs is explored through experimental data, numerical simulations and analytical models.

6.1 Definitions The protective capability of a structure against a given threat can be evaluated by several measures. One of them is the ballistic limit velocity (V bl ) of the specific armor/threat combination. Obviously, the aim of the armor designer is to increase the value of V bl without increasing the weight of the structure. The relevant measure of the armor weight is its areal density (AD), given in units of kg/m2 , which is simply the density of the protective structure multiplied by its thickness. The ballistic efficiency of a given structure is defined by its areal density, as compared with that of a reference target which is needed to defeat the given threat. Frank (1981) suggested several measures for the ballistic efficiency of improved structures, through their mass (E m ) and space (E s ) efficiencies, according to: Em =

(AD)r (AD)s

(6.1a)

Pr Ps

(6.1b)

Es =

where the indices (r) and (s) denote the reference target and the improved structure, respectively, and P is the minimal thickness of the target which is needed to defeat the threat. It is clear that the ballistic efficiency should be higher than 1.0 and the task of the armor designer is to increase these efficiencies to higher values. One of the more practical ways to defeat a given threat is by adding a relatively lightweight © Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_6

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Fig. 6.1 The add-on armor and the relevant parameters for the differential ballistic efficiency

structure in front of the protected object, as shown in Fig. 6.1. Such an add-on armor structure can reduce the penetration capability of a given threat through an effective defeat mechanism, as will be demonstrated in this part of the book. The most popular measure of the ballistic efficiency for add-on armors is the differential efficiency factor (DEF), as defined by Yaziv et al. (1986): DEF =

ρr · (Pr − Pres ) ρs Hs

(6.2)

where ρ s H s is the areal density of the add-on armor. Pres and Pr are the penetration depths into the main structure (of density ρ r ) with and without the add-on armor, respectively.

6.2 Metallic Targets The strength of the target is the most important physical parameter in determining the penetration depth of both rigid and eroding penetrators, as was demonstrated in Part II. The penetration efficiency (P/L) of rigid penetrators decreases with increasing values of the target’s resistance to penetration (Rt ), which depends on its compressive strength. The density of the target was shown to play no role in its resistance to penetration, for impact velocities of rigid projectiles in the ordnance range (below the cavitation thresholds). This is demonstrated in Fig. 6.2, which summarizes the penetration depths of 0.5 APM2 projectiles impacting various metallic targets at muzzle velocity. It is clear that these penetration depths depend on the strength of the target and not on its density. As far as their areal densities are concerned, the aluminum targets are more efficient than the steel targets against these projectiles. This is obvious by comparing the corresponding products of density and penetration depth for the different targets. For example, the areal density of the 7075-T6 aluminum alloy is lower by a factor of 1.6

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Fig. 6.2 Penetration depths of 0.5 APM2 projectiles impacting various metallic targets

than that of the RHA steel. Thus, as far as these projectiles are concerned, it is much better to build the main body of the vehicle with high strength aluminum alloys. This was one of the considerations in using the 5083 aluminum alloy, instead of the traditional armor steel, for the main body in some lightweight armored personnel carriers (APC’s). The superiority of low-density targets holds also for the penetration depths of shaped charge jets into monolithic armors, as depicted by the hydrodynamic penetration theory discussed in Chap. 5. The theory predicts that the penetration depth of a jet is given by L(ρ j /ρ t )0.5 , where L is the jet length. Multiplying this expression by the target’s density results in an areal density of AD = L(ρ j · ρ t )0.5 which is needed to completely erode this jet. Thus, a low density target should defeat a given jet with a lower areal density, resulting in a higher ballistic efficiency. The effectiveness of a low-density target, as compared with a reference target, should be equal to the inverse square root of their corresponding densities. For example, an aluminum target should be more efficient than a steel target by a factor of (7.85/2.75)0.5 = 1.69. The addition of strength to the target improves its ballistic efficiency but it does not change the general trend, and for targets of equal strength the lower density target should be more efficient against shaped charge jets. The same conclusion holds for eroding long rods although it is less obvious in this case. In order to demonstrate this fact, consider our simulation results for eroding rods impacting two different targets which have the same strength. Figure 6.3 presents the normalized penetration depths of L/D = 10 zero-strength steel rods, impacting aluminum and steel targets having strengths of 0.4 GPa. As is clearly seen in this figure, the normalized penetrations asymptotically approach their corresponding hydrodynamic limits of (ρ p /ρ t )0.5 at high impact velocities (marked by the dashed lines in the figure). These are also the penetration depths for shaped charge jets, as discussed above. Thus, the lower density target is more efficient by the same factor against high-velocity eroding rods. These simulation

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Fig. 6.3 Simulation results for zero-strength steel rods impacting 0.4 GPa aluminum and steel targets

resulted in the same value of the critical impact velocity (V c = 0.55 km/s) for the two targets, in agreement with the AT model through (5.11). As a result, for impact velocities which are slightly higher than the critical velocity, the penetration depths in both targets are practically the same. Thus, the ballistic efficiency of the low-density target as determined by its areal density is higher by a factor which is equal to their inverse density ratio. Simulations for rods with finite strengths show the same trend. In conclusion, for equal strength targets the ratio of their corresponding areal densities which are needed to defeat an eroding rod should change gradually from ρ 1 /ρ 2 at low impact velocities to a value of (ρ 1 /ρ 2 )0.5 for high velocities. This is exactly the trend observed in the data of Gooch et al. (1995), who shot long rods of WHA at semi-infinite steel (RHA) and Ti/6Al/4V targets. Their results for L/D = 10 rods are given in Fig. 6.4 in terms of the corresponding areal densities, as calculated by the corresponding depths of penetration. Note that the areal density Fig. 6.4 Areal densities as calculated from the penetration depths of L/D = 10 WHA rods impacting RHA and Ti–6Al–4V targets

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of the titanium-alloy targets is lower than that of the RHA target throughout the whole range of impact velocities. Moreover, their areal density ratio changes from a value of 1.75 at an impact velocity of 1.0 km/s, to a value of 1.45 at an impact velocity of 2.0 km/s. The steel/titanium density ratio is 1.76, and its square root is 1.33, which are very close to the experimental ratios, as predicted by the analysis presented above. Consider now the perforation process of finite thickness plates by either rigid or eroding projectiles. We have seen in Chap. 4 that the resistance to penetration of these plates is determined by their effective strength (σ r ), which was shown to be related to their compressive strength. The plate’s density does not affect its perforation process, as it only determines the mass of the ejected plugs. Thus, in order to improve the ballistic efficiency of add-on structures which are based on thin plates, one should use high-strength materials having low densities. This conclusion prompted a large amount of research on low-density metallic alloys of high strength, such as the strong aluminum and titanium alloys mentioned above. These materials are said to have a high specific strength, which is defined by their strength/density ratio. Another material which has been recently considered for armor applications is the magnesium alloy AZ31B, which contains about 3% aluminum and 1% zinc as the main additives. This material has a very low density (1.75 g/cc) and its ultimate tensile strength is about 0.26 GPa. Its specific strength is close to that of the aluminum alloy 5083-H131, which is used for the construction of APC’s. The work of Jones et al. (2007) showed that the ballistic performance of the magnesium-alloy against 0.3 AP is comparable to that of the 5083 aluminum alloy, and even to that of armor steel plates. Van-Wegen and Carton (2008) tested several magnesium alloys against various projectiles. The measured V bl for 14.5 mm AP projectiles and 20 mm fragment simulating projectiles (FSP) impacting plates of Elektron 675, resulted in mass efficiencies of E m = 1.3– 1.5 and E m = 2.0, respectively. These efficiency measures were calculated with reference to the armor aluminum alloy 5083-H32, demonstrating that magnesium alloys have a real potential for armor applications. As reported by Knoeller (2019), recent advancements in this field resulted in two magnesium alloys which have high strengths, both alloyed with yttrium and zirconium, reaching 350 MPa and approximately 500 MPa. The increase in strength for all of these potentially improved armor materials is accompanied by an inevitable reduction of their ductility. Since high strength materials are more brittle, they may experience a large volume of shattered material around the impact point. Moreover, the onset of brittleness can change the perforation mode for a given target, which may lead to a significant reduction of the target’s ballistic efficiency with increasing strength. This transition in the penetration mode is especially important for thin plates which tend to fail by adiabatic shear. A schematic description for the contradicting effects of strength and brittleness is shown in Fig. 6.5, for high strength steel plates perforated by AP projectiles. The low-hardness steel plates are perforated by the ductile hole enlargement process with negligible damage to the hard steel cores. Their ballistic efficiency increases monotonically up to a plate hardness of about 370 Brinell. This is the range of armor steels (RHA) which are used for the main body of battle tanks. As the hardness of the steel plate is increased

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Fig. 6.5 The ballistic efficiency of steel plates against hard-cored projectiles, as a function of their hardness

further it fails by adiabatic shear plugging and discing, resulting in a clear reduction of its ballistic efficiency. At hardness values of about 450 Brinell this trend is reversed and the ballistic efficiency increases again, since these high-hardness steels (HHS) are strong enough to shatter the hard cored projectile upon impact. High-hardness plates are usually very brittle and they experience a large damaged zone around the impact point, which may reduce their multiple-hit capability. In order to overcome the inherent deficiency of high-hardness steels, armor designers came up with the concept of the dual-hardness target. This configuration is based on cladding a very hard front plate to a more ductile backing plate. The basic principle behind this concept is that the first plate shatters the projectile while the second plate can deflect and arrest the debris from the projectile and from the front plate. Moreover, the second plate minimizes the damage to the first plate by supporting it during perforation, resulting in a smaller comminuted volume in the high hardness plate. With an optimized ratio of the hardness and thickness of these plates, an increased ballistic efficiency can be achieved. The dual hardness concept was found to be quite effective against various AP projectiles, but manufacturing problems in the welding process of these plates limit their application as add-on armor. Currently, dual hardness configurations have a front plate with hardness of 600–700 Brinell and a back plate of about 500 Brinell. Recent improvements in manufacturing techniques resulted in steel alloys which have high hardness without a significant loss of ductility. Such alloys have Brinell hardness of over 650, with tensile strengths of 2.0–2.25 GPa and elongations of about 8%. As shown by Showalter et al. (2008) these steels have 10–20% higher values of V50 against 0.3 and 0.5 AP projectiles, as compared with the standard high hardness plates. Obviously, the potential to improve the ballistic performance of these steels is the focus of research in many countries. High strength steels are also very efficient against shaped charge jets, as was demonstrated by Held (1993). His experimental results for the penetration depths of a given jet, in terms of charge diameters (CD), are listed in Table 6.1. It is clear that significant improvements in the ballistic efficiency can be achieved by using very hard steels against shaped charge jets.

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Table 6.1 Penetration depths of a shaped charge jet into steel targets of different strengths Y t (GPa)

0.3

0.45

0.7

2.05

2.9

P (CD)

5.9

5.25

4.8

3.25

2.2

6.3 Ceramics for Armor A natural extension of the ideas presented above is to replace the hard steel front plate in the dual hardness target by a ceramic tile, and replacing the backing steel plate by a lighter one made of aluminum or a composite material. Ceramics have relatively low densities, in the range of 2.5–4.5 g/cc, and their compressive strengths are higher than those of high-strength steels. Thus, it is not surprising that ceramicbased armors were found to be very effective against AP projectiles and long rods, as will be discussed in the following sections. A vast amount of research on this issue has been published during the last decades and the interested reader is referred to the articles presented in McCauley et al. (2002), which discuss various aspects of ceramic armor. In particular, the review article of Gooch (2002) outlines many of the ceramic-based designs for body armor, helicopter seats, and armored personnel carriers. The following sections describe the interaction between different threats and ceramic tiles. Each section is devoted to a specific threat in order to highlight their specific defeat mechanisms.

6.3.1 Ceramics Against AP Projectiles The pioneering work of Mark Wilkins and his colleagues on ceramic armor is considered as one of the classical works in the field of terminal ballistics. This work was summarized in several internal reports as in Wilkins (1968) and in Wilkins et al. (1970), for example. The authors combined experiments and numerical simulations in order to follow the interaction of 0.3 hard steel projectiles with targets composed of a ceramic tile backed by a ductile metallic plate. The experiments included flash radiography which followed the interaction at different times, and measurements of residual velocities of the outgoing projectiles. The analyses of failure modes for both projectile and ceramic tile through 2D numerical simulations, with the finite difference code HEMP, were the first of their kind in the field of terminal ballistics. Moreover, the model which was implemented in the code for the dynamic response of ceramics has been subsequently used by many researchers as the basis for their models. Short accounts of this work have been published by Wilkins (1978) and by Anderson (2002, 2006). The experimental work focused on the ballistic evaluation of various ceramics including alumina, boron-carbide, titanium-diboride, and siliconcarbide. The simulations incorporated the constitutive properties of the aluminum backing plates and the hard steel projectile, as determined by tensile stress–strain

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tests. The bulk and shear moduli of the ceramics were determined by ultrasonic measurements, and their compressive strengths were derived from the corresponding Hugoniot elastic limits as measured by plate impact experiments. The failure model of these ceramics was based on the Griffith model for brittle materials, which was described in Chap. 2. The model incorporated the following features: (1) Fracture initiates at a computational cell when one of the principal tensile stresses in the plane of motion, exceeds a threshold value which was set at 0.3 GPa. (2) There is a time delay to complete the fracture in a cell. (3) Fracture can initiate only at surfaces (including material interfaces) or at a neighboring cell which has been completely fractured. (4) A cell can fracture only after a certain time which depends on the velocity assigned to the fracture propagation, about half the shear wave velocity. (5) The strength of a fractured cell is reduced to zero in the following time step of the simulation. Anderson (2006) discussed these constraints and pointed out that the different models which were developed in the 1990s, assigned some finite strength to the comminuted material, in order to have a better fit with more experimental data. In addition, he demonstrated that the crack velocity, or damage advancement velocity, should be much lower than half the shear wave velocity in order to get a better agreement with experiments. In spite of these modifications, the basic understanding for the sequence of events, from impact to perforation, has not changed since the early observations by Wilkins and his colleagues in the 1960s. The initiation and development of fracture in the ceramic tile and the shortening of the projectile, from one of those early simulations, are shown in Fig. 6.6. In this simulation a 0.3 conical-nosed hard steel projectile impacts a target at 760 m/s. The target consists of an alumina tile, 8.13 mm thick, backed by an aluminum plate, 6.35 mm thick. Fig. 6.6 Simulation results for the impact of a hard steel projectile on an Al2 O3 /Al target

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Upon impact, a fracture conoid initiates at the impact face, advancing into the ceramic tile at an angle and velocity which are very similar to those observed experimentally. At the same time the projectile’s nose starts to break while its back part is slowing down. Thus, its kinetic energy is reduced by both the mass loss of its tip, and the deceleration of its back part. During this stage the projectile does not penetrate the ceramic plate, a phenomenon which was called “dwell” some 30 years later when it was rediscovered through the research on long rods impacting thick ceramic tiles. The fracture conoid limits the amount of ceramic which takes place in transmitting the load to the back plate. The aluminum plate is compressed by the ceramic material inside the cone and starts to deflect at a certain time after the impact. This deflection relieves the high pressures at the ceramic/aluminum interface and the stress in the ceramic becomes tensile at this location. As a result, an axial crack is formed at the back of the tile and it propagates backwards, towards the projectile, as shown by the dark areas in Fig. 6.6. Up to this time the projectile is continuously losing mass and velocity while it is “dwelling” at the impact face. Subsequently, all the ceramic material within the conoid breaks up and the remaining projectile can penetrate this comminuted material as a rigid body. With these simulations Wilkins (1978) demonstrated that a stack of two alumina tiles, backed by an aluminum plate, is less efficient than a single tile of the same total thickness. The simulations showed that the first tile fails at its interface with the second tile at a relatively early time, resulting in a reduced ballistic efficiency for this double-tile arrangement. From this series of events it is clear that the most important issue in this interaction is the dwell time, by which all the material within the conoid is fractured. Thus, in order to maximize the ballistic performance of these targets it is essential to prolong their dwelling process as much as possible. One can have an appreciation for the methodical work of Wilkins and his colleagues, by considering the amount of work (and its precision) which was needed to collect the data presented in Fig. 6.7. Here, the ballistic limit velocities, to within Fig. 6.7 The ballistic limit velocities of AD85 alumina tiles backed by 6061 aluminum plates

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±15 m/s, are given for targets made of AD85 alumina tiles backed by 6061-T6 aluminum plates. One should note that each value of V bl in Fig. 6.7 was obtained by several shots at different velocities in order to determine the value of V bl . In addition, at least six shots had to be performed at V 0 = V bl and half of them should result a projectile penetration. The AD85 alumina includes about 85% alumina and 15% silica, and its density is about 3.4 g/cm3 . This density is significantly lower than that of pure alumina (near 4.0 g/cm3 ), but the low cost of the AD85 tiles make them ideal for parametric studies which involve a large amount of experiments. A linear change in V bl with the thickness of the backing plate (δ) was obtained for several tile thicknesses (), as shown in Fig. 6.7. The distinctive breaks in these curves at δ = 5–6 mm are due to a change in the failure mode of the aluminum plate, from dishing and petaling for the low thicknesses, to shear plugging for the thicker plates, as noted by Wilkins (1978). One of the main conclusions of Wilkins and his colleagues was that the ballistic performance of a given ceramic could be improved if its ductility is enhanced. This conclusion prompted an intense research which lasted for several decades, and was aimed at adding a few percent of soft metal to the ceramic in order to improve its ductility. These so-called cermets performed better than their pure ceramic counterparts, but the improvements were not dramatic and did not justify their manufacturing cost. Another innovation which was initiated by Wilkins and his colleagues was that of graded ceramics. The realization that the ceramic failure starts at its interface with the metallic backing, led researchers to manufacture these graded ceramics where the ceramic-metal proportions change gradually, from a pure ceramic at their impacted face to a pure metal at their back. Graded materials have also shown improved ballistic performance but their manufacturing problems have yet to be solved. Anderson (2002, 2006) further analyzed the defeat mechanisms which were described by Wilkins (1978), namely, the erosion of the projectile and its deceleration during the dwell phase. He showed that these processes contribute roughly equal parts to the loss of the projectile’s kinetic energy, which can be reduced by as much as 50% at the end of the dwell phase. An excellent set of flash radiographs from Anderson (2006) is shown in Fig. 6.8. One can clearly follow the dwell phase of the 0.3 APM2 projectile at the interface of a 7.62 mm boron-carbide tile, which was glued to a 6.6 mm thick 6061-T6 aluminum plate. Impact velocities in these shots were around 820 m/s and the radiographs were obtained by 1 MeV X-ray pulsers. Several features are clearly seen in this figure: the conoid fracture of the ceramic tile, the breakup of the projectile’s nose during the dwell phase which lasts for about twenty microseconds, and the subsequent penetration of the residual projectile (as a rigid body) into the comminuted ceramic. In order to simulate these experiments with the CTH code, Anderson and Walker (1999) modified the original failure model of Wilkins which was described above. Their model has five material parameters which describe the failure of the ceramic tile: the compressive strength of the intact material, the slope and cap of a Drucker– Prager yield surface for the damaged ceramic, and a constant which governs the propagation speed of the damage in the ceramic tile. They found that in order to account for the experimental results, this speed should be about 0.025 times the shear

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Fig. 6.8 Flash radiographs of 0.3 APM2 projectile impacting a B4 C tile backed by an aluminum plate

wave velocity of the ceramic, rather than half the shear velocity which was used by Wilkins in his simulations. In addition, Anderson and Walker (1999) assigned a finite strength to the comminuted material, in contrast with its zero strength in Wilkins’ model. These modifications resulted in an excellent agreement between the simulations and the experimental radiographs for the positions of the projectile’s nose and tail. Such an agreement is shown in Fig. 6.9, from Anderson (2006), for the simulation of the experiment shown in Fig. 6.8. The data collected by Wilkins and his coworkers for the different ceramic tiles backed by a 6.35 mm aluminum plate, resulted in a hierarchy according to which the best performance was obtained by the lowest density ceramic, beryllium-boride, while the poorest performance was obtained for the high density ceramics, titaniumdiboride and titanium-carbide. This hierarchy in the ballistic performance has been

Fig. 6.9 The agreement between simulation results (lines) and experiments (circles) for the 0.3 steel projectile impacting the B4 C/Al target

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Fig. 6.10 The relative ballistic efficiencies as a function of the ceramic density

explained by Rosenberg et al. (2009), who plotted the figures of merit of these ceramics as a function of their density, as shown in Fig. 6.10. The ballistic efficiencies of the different ceramics are compared with the boron-carbide tile, which was given a figure of merit of 1.0. The other tiles obtained their figure of merit (relative efficiency) according to their ballistic limit velocity, as compared with that of boron-carbide. A clear correlation is obtained between the relative ballistic efficiency of these tiles and their density, as shown in the figure. The alumina tiles are marked by squares, the beryllium compounds by crosses, and the titanium compounds are marked by triangles. The decreasing trend of tile efficiency with its density has to be due to a real physical phenomenon since the two different ceramics B4 C and BeO + B, which have practically the same density of 2.5 g/cm3 , resulted in exactly the same ballistic efficiency. In addition, the lighter titanium compound TiBe12 is much more effective than the denser titanium compounds, TiB2 and TiC. Thus, it is clear that the density of the tile, rather than its constituents or strength properties, is responsible for the trend we see in Fig. 6.10. In order to understand this trend one has to realize that these ceramics were graded by their corresponding values of V bl , while keeping a constant weight for all the different tile/plate combinations. Since the aluminum plates in these targets were of the same thickness, the only changing parameter was the thickness of the ceramic tile which, obviously, varied with its density in order to keep a constant weight for the target. Thus, the Be2 B tile with the lowest density (2.0 g/cm3 ), was the thickest tile in these tests, while the TiB2 and TiC tiles with densities of 4.5 and 4.9 g/cm3 , respectively, were the thinnest. All of these ceramics are graded as armor materials, having minimal porosities and compressive strengths which are higher than that of the steel projectile. Thus, one may assume that all of them failed by the same mechanism which was described above. It was emphasized earlier that the dwell time is the most important parameter for the defeat process of these projectiles. Thus, one should look for differences in the dwelling times of these

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tiles, in order to account for their different ballistic efficiencies. Clearly, a thicker tile will result in a longer dwelling time since both the conoid formation and the comminuting of the material inside the conoid, are bound to take more time. Thus, a better performance is expected for the thicker tiles with longer dwell times, which had the lower densities in the experiments. This simple explanation accounts, at least qualitatively, for the decreasing trend in the relative ballistic efficiencies of the denser ceramics, as shown in Fig. 6.10. In order to demonstrate that the compressive strength of these ceramics are high enough to defeat AP projectiles, Rosenberg et al. (2009) impacted a 0.5 AP projectile at a large block of alumina, 300 mm on each side. The projectile shattered upon impact and was completely stopped at the impact face, hardly scratching the block. This experiment also supports the thesis that the failure of thin tiles is due to tensile stresses which originate at their free surfaces, or at their interfaces with other plates. Such a failure was avoided in the test with the large ceramic block, since its free surfaces were too far from the impact point to have a significant influence on the process. In conclusion, the different ballistic efficiencies of the ceramic tiles in the experiments of Wilkins and his colleagues are due to their test configuration, namely, the V bl test for constant-weight targets. The V bl test is a design-specific test which gives a measure for the ballistic performance of a certain configuration against a given threat. It includes various details such as the properties of the backing plate, which are not related to the ceramic tile. More importantly, the compressive strength of the ceramic tile did not manifest itself upon its V bl value, as shown by the results of Wilkins and his colleagues. In addition, the V bl test is quite expensive since it demands several shots at different velocities around the value of V bl . In order to overcome the deficiencies of the V bl test, a new test configuration was suggested by Rosenberg et al. (1987a). This is the thick-backing technique, shown schematically in Fig. 6.11, which has been later termed the Depth of Penetration test (DOP). This simple configuration has been widely used since its inception as a standard ranking method for ceramic tiles. The new configuration also provides relevant information concerning the performance of ceramics under realistic armor designs against long rods, as we shall see in the next section. With the DOP test, the ceramic tile of thickness hc and density ρ c , is glued to a thick metallic block of a reference material, such as aluminum or steel. The residual Fig. 6.11 The DOP test

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penetration (Pres ) of the projectile into the thick backing determines the ballistic efficiency of the ceramic tile, as compared with the backing material. In their work with different AP projectiles, Rosenberg et al. (1987a) and Rosenberg and Yeshurun (1988) used 2024-T351 aluminum blocks as their reference materials. They applied the DEF measure as defined by (6.2), to obtain the ballistic efficiency (η) of the ceramic tile through: η=

ρ Al (PAl − Pres ) ρc h c

(6.3)

where PAl is the penetration depth of the projectile into the bare aluminum block, and ρ c hc is the areal density of the tested tile. One of the main advantages of this configuration is due to the support which the thick backing supplies to the ceramic tile, preventing its early failure by tensile stresses or, at least, postponing this failure to much later times. More importantly, this configuration results in an efficiency measure for the ceramic through a single test, as will be shown below, simplifying the evaluation process as compared with the V bl test. In order to check whether this technique can be based on a single experiment for a given ceramic, Rosenberg et al. (1987a) performed several experiments with various AP projectiles impacting AD85 alumina tiles of different thicknesses, which were backed by thick blocks of 2024-T351 aluminum. The residual penetrations in these shots are given in Fig. 6.12, and it is clearly seen that they fall on well-defined straight lines. The fact that the results for different tile thicknesses fall on a single line means that their ballistic efficiency is independent on tile thickness, and it can be determined by a single test using (6.3). The three AP projectiles in this study have similar ogivenosed steel cores which differ only by their size. Also, their muzzle velocities, in the range of 830–970 m/s, are close enough to obscure any velocity-dependent effect. Fig. 6.12 The DOP test results for 0.3 , 0.5 , and 14.5 mm AP projectiles

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These similarities are the main reasons for the fact that the lines in Fig. 6.12 are very nearly parallel to each other. Once the data points were found to fall on straight lines, a somewhat different expression for the ballistic efficiency can be defined, which is based on the extrapolation of these lines to a residual penetration of Pres = 0. The new efficiency measure is defined by the following ratio of the areal densities: η=

ρ Al PAl ρc h ∗c

(6.4a)

where hc * is the minimal tile thickness for which the residual penetration is zero. If we plot the results of these tests in terms of areal densities rather than penetration depths, we find that the ballistic efficiency (η) is equal to the slope of the straight line through the experimental points. Thus, one can rewrite (6.4a) in terms of the angle (θ ) which these straight lines form with the ordinate, as follows: η = tg(θ )

(6.4b)

This display of the ballistic efficiencies is shown in Fig. 6.13 for several ceramics which were impacted by 0.5 APM2 projectiles, as given by Rosenberg and Yeshurun (1988). These ceramics included silicon-carbide, boron-carbide, and two types of alumina ceramics (AD85 and the somewhat denser BC90G). The resulting efficiencies are clearly dependent on both the density and the strength of the tile, in contrast with the results obtained by the V bl tests which were summarized in Fig. 6.10. For example, the two alumina ceramics, AD85 and BC90G, have densities of 3.42 and 3.56 g/cc, respectively. Their Hugoniot elastic limits, as measured by plate impact experiments in our lab, are 6.0 GPa and 7.0 GPa for the AD85 and the BC90G, respectively. The higher strength BC90G alumina is, probably, the reason for its higher ballistic efficiency of η = 4.9, as compared with Fig. 6.13 The ballistic efficiencies for different ceramics, defined by the slope of the experimental lines

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the value of η = 4.0 for the AD85 alumina. In fact, Rosenberg and Yeshurun (1988) showed that the ballistic efficiencies of different ceramics, as determined by the DOP test, are related to their specific strength which is defined by their strength/density ratio. It is also worth noting that the ballistic efficiencies of these ceramics in the DOP test, can reach relatively high values such as η = 7.7 for the boron-carbide tile. These high efficiencies are due to the thick metallic backings which prevent the early failure of the tile by tensile stresses, as explained above. Such high efficiencies should motivate the armor designer to attach the ceramic tiles directly to the main body of their vehicle. In fact, this idea was implemented for several armored vehicles, as reviewed by Gooch (2002). One should be aware of the main causes for the possible scatter in the results of these DOP tests, which is responsible for inconsistencies between results from different sources. The reproducibility of the same material properties for different tile thicknesses is one of the issues which can lead to a thickness-dependent ballistic efficiency. Also, relatively thick tiles result in small residual penetrations, often with a large scatter between repeated tests. On the other hand, very thin tiles can cause some ambiguity in the results because of their small effect on the process. In order to avoid these difficulties one should use tile thicknesses which result in a residual penetration of about half the penetration of the test projectile into the bare metallic block. It has been found that the lateral dimensions of the tiles should be at least ten times larger than the diameter of the projectile, in order to avoid the early failure by release waves from their lateral surfaces. The tiles can have smaller lateral dimensions if they are embedded in the metallic backing itself, with a close fitting between tile and backing. Rosenberg et al. (1990) noted an unexpected result in some of their experiments with 0.5 AP projectiles impacting 10 mm thick alumina AD85 tiles, which were backed by aluminum blocks. Most of these tests resulted in residual penetrations of 15–16 mm in the thick backing. However, in several experiments the residual penetrations were 32–33 mm. This result did not seem to be a matter of experimental scatter, and they attributed it to the effect of the brass jacket which surrounds the hard steel cores of these projectiles. Their assumption was that as the jacket deforms upon impact it surrounds the nose tip of the steel core, preventing its shattering upon impact on the ceramic tile. In order to test this assumption Rosenberg et al. (1990) removed the front end of the brass jacket and repeated the experiment with projectiles having exposed tips. The residual penetrations in these tests were around 15 mm, as expected, since the tips had no lateral support. In a second series of experiments they glued a copper plate, 3 mm thick, to the front of the ceramic tile and used the projectiles with the exposed tips. All of these experiments resulted in residual penetrations of about 32 mm. Thus, the copper plate played the same role as the brass jacket, namely, laterally supporting the sharp tip of the hard steel cores. Apparently, this lateral support takes place in a statistical manner rather than in all tests. The importance of the lateral supports which is provided by the so-called soft caps has been known to ballisticians since the late decades of the 19th century. Tresidder (1908) gives a very detailed account of the workings of these caps, which is also a historical account for their developments.

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The ballistic efficiency values cited above are related to the 2024-T351 aluminum backing which was used by Rosenberg et al. (1987a) and Rosenberg and Yeshurun (1988). Other backing materials or different projectiles can result in different values for the ballistic efficiencies for the same ceramics, but their ranking order is not supposed to change. In fact, with 0.3 projectiles made of sintered tungsten (Kennametal W-2), Rosenberg et al. (1987a) obtained a value around η = 8.0 for the ballistic efficiency of AD85 tiles backed by aluminum blocks. This value is higher by a factor of two than the value obtained for these tiles impacted by the regular 0.3 AP projectiles with hard steel core. Vural et al. (2002) used the much softer 6061T0 aluminum as the backing material for their tests on alumina tiles against 0.3 AP projectiles. Their efficiency merits were very high (η = 8–10), which is a clear demonstration for the effect of the strength of the backing material. Finally, Senf et al. (1998) used armor steel as a backing material for their AD995 alumina tiles, of density of 3.95 g/cc. These L/D = 3.2 spherical-nosed tungsten-alloy projectiles, with a diameter of 10 mm, impacted the targets at velocities of 1.0–2.0 km/s. The efficiency measure for these tiles, against the tungsten-alloy projectile was about 2.0 as compared with the armor steel backing. The residual penetrations vs. tile thickness from Senf et al. (1998) are shown in Fig. 6.14. For a given impact velocity the data for different tile thicknesses fall on a straight line, as in the case of AP projectiles. This important result enhances the claim that the DOP test can give an efficiency measure for a given tile through a single experiment. Moreover, the data for different impact velocities fall on parallel lines, indicating that the ballistic efficiency of this alumina does not depend on impact velocity, at least for this range of velocities. The relatively large scatter in the data for the thicker tiles is due to the factors discussed above. In conclusion, the examples presented here show that the efficiency merits of a given ceramic are strongly dependent on all the above mentioned parameters. Thus, Fig. 6.14 DOP results for alumina/steel targets impacted by short WHA projectiles at different velocities

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the DOP test should be regarded as a convenient screening test for the relative ranking of ceramic tiles through a very simple and relatively cheap experiment.

6.3.2 The Interaction of Ceramics with Long Rods The interaction of long rods with ceramic tiles and their defeat mechanisms are very different from those for hard-cored AP projectiles. Long rods are made of heavy alloys which have much lower strengths than the hard steel cores of the AP projectiles. Thus, their defeat mechanism has to be based on the continued erosion of the rod, rather than on its shattering. The thickness of a ceramic tile which is needed to completely erode a long rod has to be correspondingly large. This fact imposes several difficult-to-achieve constraints for the armor designer and, as a result, the concept of ceramics against long rods is still in its early stages of implementation. Still, much work has been done on this subject since the 1980s, when hydrocodes became available. Numerical simulations are heavily used to follow the specific details in the rod/ceramic interaction, as will be reviewed here. The use of the DOP test for the screening of ceramic tiles against eroding rods, was first reported by Bless et al. (1987) who impacted AD85 alumina tiles, glued to thick aluminum backings, with L/D = 5 tantalum rods. Mellgrad et al. (1989) and Woolsey et al. (1989) suggested the use of armor steel as backing materials, in order to reduce the residual penetrations of the rods, and to obtain information about the possible use of ceramics for real armor applications. The DOP configuration with the armor steel backing is used by most researchers, with tiles which are either bare or have a thin cover plate at their front. Some works were aimed at providing a better lateral confinement to the tile by embedding them in the steel backing, in order to achieve optimal confinement conditions, as described by Hauver et al. (1992). The experimental results of Hohler et al. (1995) with alumina tiles (density of 3.8 g/cm3 ) backed by hard steel blocks, are shown in Fig. 6.15. The thickness of the tiles ranged between 10 and 80 mm, and the impact velocities of the L/D = 12 WHA rods were: 1.25, 1.7 and 3.0 km/s. As shown in Fig. 6.15, the residual penetrations fall on nearly parallel straight lines. Thus, the efficiency of these tiles against long rods was found to be independent on their impact velocity within a large range of these velocities. The ballistic efficiency of these alumina tiles against the long rods is η = 1.7 which is somewhat smaller than the value of η = 2.0 obtained by Senf et al. (1998), for similar tiles impacted by the shorter projectiles, as shown in Fig. 6.14. The difference is probably due to the different strengths of the steel backings used in the two sets of experiments, which was higher for the long-rod experiments of Hohler et al. (1995). We have seen above that a backing material with a higher strength results in a lower ballistic efficiency for a given ceramic, since the tile’s efficiency as it is defined above is related to the backing material. On the other hand, backing plates of higher strengths result in a stronger effect on the eroding rod as they are less likely to deform, enhancing the integrity of the ceramic tile for longer times. This effect has been demonstrated by the numerical simulations of Gour et al. (2017) with

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Fig. 6.15 DOP results for L/D = 12 WHA rods impacting alumina/steel targets at various velocities

WHA rods impacting SiC tiles, which were embedded in steel backing plates. The strength of this backing was varied and it was found that the higher strength backings resulted in less damage to the ceramic tile which, in turn, resulted in shorter residual penetrators. Thus, stronger backing materials seem to be preferable, as far as armor applications are concerned. Another set of experiments with various ceramics backed by high-hardness steel blocks, is described by Rosenberg et al. (1998). The ceramics tiles were 20–80 mm thick and their lateral dimensions ranged between 75 mm and 150 mm. The tiles were impacted by L/D = 12.5 WHA rods, with D = 5.8 mm, at a velocity of 1.7 km/s. This set of experiments included tiles of silicon and boron carbides, titanium-diboride and aluminum-nitride. The resulting depths of penetrations as a function of the tile’s areal density are shown in Fig. 6.16. The residual penetrations for the tiles with small lateral dimensions resulted in lower values of η, which is a clear indication for the effect of lateral release waves in reducing the high pressures around the rod/tile interface. Various experiments indicated that in order to avoid the interfering effects of lateral boundaries, the width of the tiles should be at least five times larger than their thickness. This constraint was not kept in the experiments described in Fig. 6.16. Another way to overcome these lateral release effects is to embed the tile in the backing, or to apply a metallic frame around the tile. Improvements in the ballistic efficiency have been achieved by close fitting the tile into a cavity in the backing block. Several researchers even used special glue around the tile which included metallic powder, in order to have a better acoustic match between the tile and its backing. The ballistic efficiencies of the ceramics presented in Fig. 6.16 are bounded between η = 1.7 and η = 2.7, as shown by the two lines in the figure. These are typical values for the efficiency of strong ceramics as compared with high-hardness steel backings, as was also shown by Reaugh et al. (1999). They shot L/D = 4 WHA rods at various ceramic tiles backed by high-strength steel blocks. Their ballistic

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Fig. 6.16 DOP results for various ceramics backed by high hardness steel blocks

efficiencies values for different tiles, at an impact velocity of 1.7 km/s were: η = 2.0 for alumina, η = 2.5 for both AlN and SiC, and η = 3.1 for TiB2 tiles. The experiments of Hohler et al. (1995) with alumina tiles were performed with flash X-ray radiographs which followed the penetrating rod in the ceramic tiles, as shown in Fig. 6.17. It is clearly seen that the rod is eroding during its penetration in the ceramic in a similar way to its erosion in a thick metallic target. In particular, the quasi steady-state mode of penetration and the rod’s erosion rate were found to be similar to those in metals. Thus, one may conclude that the penetration model of Fig. 6.17 The penetration and erosion of a WHA rod impacting an alumina tile at V 0 = 2.5 km/s

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Alekseevskii and Tate (the AT model) can also account for the penetration of rods into thick ceramic tiles. The applicability of the AT penetration model for the interaction of long rods with thick ceramic tiles, has been demonstrated by Rosenberg and Tsaliah (1990). Their first step was to check whether a penetration threshold velocity (V c ) can be defined for thick ceramics, in the same way as for semi-infinite metallic targets. They shot L/D = 10 rods of different materials (copper, steel and WHA), at large alumina blocks of two different types, AD85 and BC90G having densities of 3.42 and 3.56 g/cc, respectively. By varying the impact velocity of these rods, they found that below a certain threshold velocity the alumina blocks were not penetrated, with only a small indentation shown around the impact point. These threshold velocities for the different rods impacting the AD85 alumina blocks were: V c = 1.15, 0.99 and 0.66 km/s for the copper, steel and WHA rods, respectively. Using the strengths of these rods (Y p ) and the relation V c = [2(Rt − Y p )/ρ p ]0.5 for the threshold velocity, from the AT model, a value of Rt = 5.5 ± 0.5 GPa was obtained for the AD85 alumina. A similar set of experiments on the stronger alumina (BC90G), resulted in a value of Rt = 7.0 GPa for this alumina. These values of Rt for the two types of alumina, are very close to their Hugoniot elastic limits (HEL), as measured by plate impact experiments. In addition, Rosenberg and Tsaliah (1990) showed that when these values for Rt are inserted in the AT model, they account for the residual penetration in the steel backing, for several experiments with alumina tiles backed by steel blocks. Thus, they concluded that the assignment of Rt ≈ HEL for a given ceramic accounts for its resistance to penetration by long eroding rods. The thresholdvelocity behavior of long rods, which was demonstrated by Rosenberg and Tsaliah (1990), was later termed the “interface defeat” phenomenon and became the focus of intense research, as will be described below. The experiments of Subramanian and Bless (1995) with L/D = 20 tungsten rods impacting large AD995 alumina tiles, resulted in a value of Rt = 8.0 ± 1.0 GPa, which is close to the value of HEL for this alumina. This value was obtained by analyzing the measured penetration velocities of the rods through flash X-ray radiographs. On the other hand, the experiments by Behner et al. (2008), with gold rods impacting thick SiC tiles, resulted in a threshold impact velocity of about 0.9 km/s. The corresponding stagnation pressure at this velocity is: ρ p V 2c /2 = 7.8 GPa, which is significantly lower than the reported value of the HEL for this ceramic (11 GPa). Obviously, a theoretical account of the threshold velocity for ceramics is very much in need. Since high-grade ceramics can have HEL values of 8–12 GPa, a rough estimate results in values of V c = 0.85–1.1 km/s, for the threshold velocities of WHA rods (with ρ p = 17.5 g/cc and Y p = 1.5 GPa) impacting large ceramic blocks. As mentioned above, Subramanian and Bless (1995) used flash X-ray radiography, in order to follow the eroding tungsten rod as it penetrated into thick AD995 alumina targets. Their experimental results showed that the rod’s penetration velocity (U) is linearly dependent on its impact velocity (V 0 ), as shown in Fig. 6.18. The open circles in this figure were obtained by linear fits to the data from each experiment, while the filled circles were obtained by fits to the combined data of six shots at the nominal velocity.

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Fig. 6.18 The results for penetration velocities of tungsten rods into AD995 alumina tiles

The solid line in Fig. 6.18 is a linear fit to the data, and the dashed line is the predicted relation according to the hydrodynamic model (no strength) through the relation: Uhyd =

V   = 0.69V 1 + ρt ρ p

(6.5)

These experiments resulted in the following relation between penetration velocities (U) and impact velocities (V 0 ), for V 0 ≥ 1.5 km/s: U = 0.836V0 − 0.742

for AD995 Alumina

(6.6a)

where U and V 0 are given in km/s. A linear relation between penetration velocity and impact velocity was also obtained by Subramanian et al. (1995), for tungsten rods impacting 6061-T651 aluminum targets at velocities in the range of 1.5–4.2 km/s. Similar results were obtained for other ceramics penetrated by WHA rods, for impact velocities between 1.5 and 5.0 km/s. These experiments were summarized by Orphal et al. (1996) for aluminum-nitride, Orphal et al. (1997) for boron-carbide, and Orphal and Franzen (1997) for silicon-carbide. The linear fits for the data in these works are as follows: U = 0.575V0 − 0.406

for B4 C

(6.6b)

U = 0.781V0 − 0.51

for SiC

(6.6c)

U = 0.792V0 − 0.524

for AlN

(6.6d)

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As discussed by Orphal and Anderson (2006), these linear relations may seem to contradict the predictions of the AT penetration model. They argued that this discrepancy highlights the shortcomings of the AT model, since similar relations were obtained for metallic targets as well. However, a close examination of the AT model shows that, in fact, it does predict an effectively linear relationship between U and V 0 for high density rods and low density targets, especially for high impact velocities. Note that at high impact velocities these rods do not decelerate appreciably during most of the penetration process. Thus, the velocity of their uneroded parts (V ) is expected to be close to their impact velocity (V 0 ). An effectively linear relation between U and V 0 (or V ) is obtained from the AT model by considering the expression for U(V ), as derived from the modified Bernoulli relation (5.10):  V − μ V 2 + A0 U= (1 − μ2 )

(6.7a)

where A0 and μ are given by: 2(Rt − Y p )(1 − μ2 ) A0 = ; μ= ρt



ρt ρp

(6.7b)

Using representative values for the densities of the materials involved in the experiments discussed above, ρ t = 3.0 g/cc and ρ p = 17.5 g/cc, together with typical values of Rt = 10.0 GPa and Y p = 1.0 GPa, result in: A0 = 5 and μ = 0.414. Inserting these values in (6.7a), it is easy to show that the following relation accounts for the calculated values of U, for impact velocities in the range of: V 0 = 1.5–5.0 km/s: U = 0.8V0 − 0.7

(6.8)

which is similar to the linear relations obtained experimentally for the different ceramics, as given by (6.6a–6.6d). In conclusion, the data for ceramic penetration by long rods can be analyzed by the AT model and the ballistic efficiencies of different ceramics can be related to their corresponding Rt values, as derived from the measured penetration velocities of eroding rods. The next step in this line of research was achieved by Bless et al. (1992) who showed that by properly encapsulating the ceramic tile, a much higher threshold velocity is obtained for the onset of rod penetration into a large ceramic block. This issue has been the focus of intensive research during the last decades, due to its potential benefit for ceramic armor against long rods. The works of Hauver et al. (1993, 1994), Subramanian and Bless (1995), Lundberg et al. (1998, 2000) and Westerling et al. (2001) demonstrated that the penetration of high-density rods can be prevented even at impact velocities as high as 1.6 km/s. Most of these experiments were performed with the reverse ballistics technique, where the rods are stationary and the ceramic target is launched towards them by a light gas gun. This configuration is especially suited for high velocity impacts, which are impossible to achieve by

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accelerating slender rods with either powder or gas guns, while it is quite simple to accelerate a thick ceramic cylinder confined in a metallic sabot to velocities of up to 5.0 km/s with light gas guns. However, since these guns have diameters of about 40 mm the experiments have to be highly scaled, and the diameter of the stationary rod is limited to D = 1–2 mm. Thus, scaling issues had to be considered and Hauver et al. (2005) noted that larger scale experiments, with D = 6 mm for example, reproduced the main findings of the small-scale studies. Scaled experiments have another advantage concerned with flash X-ray radiography. Due to the small size of the targets, X-ray flashes can easily penetrate the small ceramic cylinders, and follow the shape and advancement of the rod during penetration. This is clearly shown in Fig. 6.19, from Holmquist et al. (2008), who shot silicon-carbide cylinders at slender gold rods (D = 1 mm, L = 70 mm). The analysis of such X-ray pictures led Lundberg et al. (1998) to the realization that the interface-defeat mechanism is followed by a narrow range of impact velocities where the rod is actually “dwelling” at the interface for a certain time before it starts penetrating. Note that such a dwell phenomenon was first observed by Wilkins and his colleagues in their classical work with AP projectiles, as described above. When the dwelling phase is over, the rod penetrates the ceramic target at a relatively low penetration velocity. For higher impact velocities the dwell phase is diminished, and penetration proceeds at a high velocity, practically right after impact. The results of Lundberg et al. (1998) for boron-carbide tiles impacting stationary WHA rods are shown in Fig. 6.20. One can clearly see the range of impact velocities around V 0 = 1.5 km/s which corresponds to the dwell mechanism (the shaded range in the figure). This is the transition range between the interface-defeat of the rod and its regular penetration process.

Fig. 6.19 X-ray radiographs in a reverse-ballistics test of a SiC cylinder impacting a stationary gold rod at V 0 = 1.38 km/s

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Fig. 6.20 The interface-defeat and dwell regions for boron carbide tiles

The experiments of Bless et al. (1992), Hauver et al. (1993), Lundberg et al. (1998, 2000), and Hauver et al. (2005), showed that much higher values of the impact velocities are obtained for the interface-defeat velocity by the proper use of specific cover plates. Such plates can successfully contain the rod material around the impact point, applying a continuous pressure on the tile for a relatively long time. This dynamic confinement prevents the early failure of the tile by tensile stresses which initiate at the impact surface. The second role of the cover plate is to spread the initial load of the rod as it impacts the ceramic tile, by enlarging its nose area. Lundberg et al. (1998) used a steel cover plate with a central protrusion having diameter and height of several rod diameters. Numerical simulations by Holmquist and Johnson (2002) showed that a small copper disc glued to the tile, is enough to result in a significant increase of the interface-defeat velocities. The diameter and the thickness of these copper buffers in their simulations were equal to 2D and 4D, respectively, where D is the diameter of the rod. The role of the copper buffer will be discussed in the next section where we review the simulations concerning the impact of long rods on ceramic tiles. Lundberg et al. (2001) showed that the dwell process can last for long times (300 µs), by using very long rods (over 770 mm long), which were composed of several welded shorter rods. The aim of these experiments was to show that with the proper design, the interface-defeat mechanism could be viable also against full scale rods.

6.3.3 Numerical Simulations A large amount of research has been focused on numerically simulating the various phenomena described above, in order to highlight the physical parameters which control the response of ceramics to impact and penetration. Several material models

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for ceramics were implemented in commercial hydrocodes, and all of them have the same features of the model developed by M. Wilkins, as described in Sect. 6.3.1. The most elaborate one is the JH-1 model of Johnson and Holmquist (1990), which was described in Chap. 2. Other models for brittle materials include those of Curran et al. (1993), Rosenberg et al. (1995), Rajendran and Grove (1996), Walker and Anderson (1996) and Walker (2003). These models are based on the realization that upon impact, high amplitude stress waves are generated in the ceramic tile inducing its fracture and comminution, in a large volume ahead of the penetrating projectile. Thus, in order to simulate the behavior of the ceramic tile it is necessary to account for the properties of both the intact and the fractured material. This unique feature of brittle materials was convincingly demonstrated by Shockey et al. (1990), who impacted metallic spheres at steel covered silicon-nitride tiles, at relatively low velocities, so that the spheres did not penetrate the steel covers. A cross section of the tiles after impact revealed a wealth of different cracks and comminuted material below the impact point, as shown in Fig. 6.21. The extensive damage in this particular tile was due to the impact of a tungsten-carbide sphere at a velocity of 231 m/s. Since the sphere did not penetrate the steel cover-plate, the damage in the tile has to be attributed to the strong stress waves, both compressive and tensile, which were initiated by the impact itself. Thus, a penetrating projectile is due to encounter a severely damaged tile, and it is evident that the properties of the damaged ceramic control the penetration process. The comminuted material ahead of a penetrating projectile is often referred to as the Mescall zone, after John Mescall who contributed much to this research. Compton et al. (2013) analyzed the sequence of occurrence of the four major failure modes which they identified in such situations. These modes include: (1) lattice plasticity through glide dislocations or twinning, (2) intergranular or transgranular cracking leading to comminution, (3) radial cracking emanating either from the plastically deformed region, below the impact point, or from the tile’s back surface directly below the impact point, and (4) cone cracking starting from the impact point. They used numerical simulations to follow the time-history of the contact pressure and contact force for a wide range of projectile strengths and impact velocities. Also, the temporal and spatial evolutions of subsurface stresses were recorded in these simulations. The results were then used to assess and calibrate analytical models Fig. 6.21 Damage to a ceramic tile from a non-penetrating sphere impact

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for peak stresses and forces, which were combined with established criteria for the onset of each of the failure modes. They found that plasticity and comminution are controlled by the maximum contact pressure, radial cracking by the maximum contact force, and cone cracking by the maximal stress intensity factors. The analysis resulted in the minimal impact velocities of the spherical projectiles to induce the different failures, which were in good agreement with their observations for steel, brass and aluminum spheres impacting alumina tiles. Such studies are important for the proper description of the physical state of an impacted tile, shortly after impact, in order to follow the resisting pressures which the tile exerts on the projectile at much later times. The numerical modeling of a ceramic tile undergoing these different failures is very complex mainly because, for example, the cracked material has different properties than the comminuted one. These differences should be accounted for, especially for well confined tiles, since the confinement tends to prolong the high pressures in the tile for a certain time. A cracked tile under high confining pressures is not expected to lose its resistance to penetration. Thus, the potential damage induced by tensile stresses can be prevented or delayed to much later times by properly confining the ceramic. Much of the experimental work on the constitutive relations of ceramics has been devoted to the properties of fractured and comminuted ceramics. In fact, the main differences between the various models cited above, are due to the different values which they assign to the strength of the damaged material, as reviewed by Anderson (2006). For example, the different strength values for siliconcarbide which are used by the JH-1 model and the model of Walker (2003) are shown in Fig. 6.22. The two models use the same value for the high-pressure strength of the intact ceramic (13 GPa), which was taken from various experiments. On the other hand, the strength of the fractured ceramic is 1.3 GPa in the JH-1 model, and 3.7 GPa in Walker’s model. These very different values were obtained by adjusting the corresponding models to different sets of experimental results, enhancing the need for well-calibrated properties of failed ceramics. Fig. 6.22 The JH-1 model for SiC and the Walker–Anderson failure surface

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Fig. 6.23 Simulation results for the impact of the gold rod at a SiC cylinder, shown in Fig. 6.19

In order to demonstrate the predictive capability of these models, consider the simulation results from Holmquist et al. (2008) for a SiC tile impacted by a gold rod at a velocity of 1.38 km/s, as shown in Fig. 6.23. The radiographs of the corresponding experiment are shown in Fig. 6.19. One can clearly see the excellent agreement between the simulation and the experiment concerning the advancement of the penetration process and the damage around the crater. Such an agreement strongly enhances the validity of the material model which was used for this ceramic. The effect of the buffer discs has been investigated through numerical simulations by Holmquist and Johnson (2002), and Holmquist et al. (2005, 2008). A comparison between the simulation results and the experimental radiographs from Holmquist et al. (2008), for a buffered SiC tile impacting a gold rod at 1.484 km/s, is shown in Fig. 6.24. The copper buffer in both experiment and simulation was 5 mm in diameter and 4 mm thick. The diameter of the rod was 1.0 mm and its length 70 mm, as in the impact on the bare tile at 1.38 km/s, shown in Figs. 6.19 and 6.23. Clearly, the copper buffer prevented the penetration of the tile at the higher impact velocity, while the bare tile was easily penetrated by this rod at the lower velocity. Thus, both experiments and numerical simulations demonstrate that the addition of a small copper buffer leads to a significant increase in the range of impact velocities for the interface-defeat phenomenon. In order to further demonstrate the effect of the copper buffer consider the simulation results of Holmquist et al. (2005), as shown in Fig. 6.25. These simulations were performed for bare and buffered silicon-carbide tiles, impacted by a D = 0.75 mm, L = 30 mm gold rod, at various velocities. The results of these simulations agree with the experimental data, for the interface-defeat threshold velocities and the corresponding dwell times. Both experiments and simulations show that a bare SiC tile defeats these gold rods at impact velocities up to about V 0 = 0.9 km/s, which can be defined as the threshold velocity (V c ) for penetration. The stagnation (Bernoulli) pressure which corresponds to this impact velocity is: P = ρ p V 2c /2 = 7.8 GPa, as was mentioned earlier. On the other hand, the buffered SiC tile was shown, both

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Fig. 6.24 Experimental (top) and simulation results for a buffered SiC cylinder impacted by a gold rod at 1.484 km/s

experimentally and numerically, to defeat the gold rods at an impact velocity of V 0 = 1.65 km/s, corresponding to a stagnation pressure of about 26 GPa. Thus, the addition of a copper buffer increased this threshold pressure by a factor of 3.3. Naturally, such enhancements of the ballistic resistance draw much attention among armor designers for the practical application of ceramics tiles. According to Holmquist et al. (2005) the buffer should be large enough to attenuate the impact shock which is delivered to the tile, and to induce a mushroom shape to the rod’s nose in order to spread its load over a larger area. The buffer also applies some compressive pressure over the impacted area for a short time, which is enough to prevent the early failure of the tile by tensile stresses around the impact zone. The buffer material should have a high density to reduce the loading stress on the ceramic face. Simulations with a polycarbonate buffer resulted in a higher and steeper loading profile at the surface of the tile, as compared with those induced by either copper

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Fig. 6.25 Simulation results for bare and buffered SiC tiles impacted by gold rods at different velocities

or tungsten buffers. These simulations were performed for a gold rod impacting the buffered silicon-carbide tile at 1.6 km/s, and the stress histories at the symmetry axis for the three different buffers, are shown in Fig. 6.26, from Holmquist et al. (2005). The estimated impact shock which the gold rod imparts to the bare silicon carbide at V 0 = 1.6 km/s, is 38 GPa. The addition of copper or tungsten buffers reduces the amplitude of this shock to a level of 26.3 GPa, and spreads its rise time considerably, as is clearly seen in the figure. With the material parameters which were used in these simulations, the transition from interface defeat to the dwelling phase seems to take place at a stress level of about 26 GPa. We should note that this value is based on numerical simulations of the experimental results. A physically-based analysis is still needed in order to fully account for this value for the threshold pressure. Some insight into the physics of this issue was given by several works, as we describe below. Malaise et al. (2000) obtained a significant increase in the interface threshold velocity by using a different experimental setup. They compared the penetration performance of L/D = 10, D = 5 mm WHA rod encased in a larger polycarbonate cylinder, with the penetration characteristics of a free-flying rod. These rods impacted 30 mm thick SiC tiles backed by steel blocks (DOP testing), at impact velocities of about 1.45 km/s. The free-flying rod perforated the whole ceramic tile and left a

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Fig. 6.26 Simulation results of the loading stresses for different buffer materials

residual penetration of 4.5 mm in the steel backing. On the other hand, the impact of the encased rod resulted in a dent of only 3 mm at the impact face of the ceramic tile, demonstrating the strong effect of the plastic encasement. They attributed the effect to the mushrooming of the plastic cylinder around the rod, applying a compressive stress on the impacted face for a relatively long time, thus preventing its early failure. Sarva et al. (2007) offered another possible source for the strong effect of frontface covers on the ballistic performance of ceramic tiles. They demonstrated that the ballistic efficiency of ceramic tiles can be vastly improved by wrapping them with thin membranes of polymeric composites or metal sheets. Experiments were conducted on armor-grade Al2 O3 and SiC tiles, which were restraint by very thin shields of various thicknesses. These included different materials such as: E-glass/epoxy, carbon-fiber/epoxy and the Ti/3Al/2.5V alloy. Flat-faced WHA rods with D = 6 mm and L = 20 mm, impacted the various targets at 900 m/s. By restraining the impactface of the tile, with a membrane of suitable tensile strength, the ballistic efficiency improved by nearly 25% for a mere 2.5% increase in areal density. The high speed videos of the bare tiles showed that right after impact a pulverized zone is formed ahead of the projectile, due to the intense stress conditions around the impacted area. This material is ejected backwards, clearing the way to the penetrator in its penetration process. At the later times, these pictures showed that a lot of ceramic debris is ejected from the back face of the tile, even before the penetrator reaches there. In contrast, much less debris was ejected from the front face of the restrained tile, indicating that the penetrator has to overcome more material in this case. Also, the initiation of the back surface debris was delayed by 10–25 µs for the wrapped tiles, as compared with the bare ones. These observations indicate that the penetrators had a more “torturous” path within the wrapped ceramic tiles, accounting for their reduced residual velocities and residual lengths, which were shorter by half.

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The wrapping of ceramic tiles by a few polymeric layers was used by Reddy et al. (2008) to demonstrate their applicability for body armor applications. They wrapped alumina tiles, 7 mm thick, with several layers of either Dyneema or aramid (Twaron) fabrics. The wrapped tiles were bonded to a 10 mm thick fiberglass laminate, which served as the backing for the tile. These wrappings added only a few percent to the total areal density of the targets. The targets were shot at with 7.62 mm armor piercing projectiles at muzzle velocities (820 m/s), and their energy absorption efficiency was determined through the residual kinetic energy of the outgoing projectile. The targets with bare tiles resulted in a reduction of about 44% in the projectile’s energy, while the energy absorbed by the Dyneema-wrapped tiles were between 65 and 87%, depending on the number of fiber layers (2, 4 or 8). The Twaron wrappings resulted in energy reductions of 76–95% for the same number of layers. Thus, with an increase of only 10% in the areal densities, Reddy et al. (2008) obtained a factor of about 2 in the energy absorption of the wrapped alumina tiles, as compared with the corresponding targets with the bare tiles. The role of ceramic confinement was investigated by Andersson et al. (2007) who shot SiC cylinders at stationary WHA rods, with diameter of 2 mm and length of 120 mm. The confinement of the ceramic cylinders increased the threshold velocity between interface defeat and penetration, from about from about 1030 m/s to 1550 m/s. This is an appreciable increase which has yet to be accounted for. In another study, Lundberg et al. (2013) used WHA rods of different size and found that these threshold velocities varied significantly with the size of the rod. For rod diameters of 0.5, 1.0, 2.0 and 5 mm the threshold velocities were: 1480, 1270, 1030 and 840 m/s, respectively. This apparent size effect is very important for future applications of ceramics for armor, and it has to be modeled as well. In conclusion, the role of the various buffers/covers which were described here is a combination of several features which contribute to the improved ballistic behavior of ceramic tiles. They reduce the amplitude of the initial shock pressure and prolong the compressive loading of the impact face. They spread the load over a large area through the mushroomed shape of the rod, and they prevent the early ejection of the ceramic debris. We should emphasize that the material parameters for the damaged ceramics, in all the simulations described above, are actually calibrated by experimental results. Thus, the fact that the simulation results agree with experimental data does not mean that a full understanding of the damage process has been achieved. Clearly, a theoretically-based model for damage initiation and its evolution in impacted ceramics is very much needed in order to understand the empirical observations detailed here. Uth and Deshpande (2013) offered a different explanation for the dwell phenomenon. They challenged the widely accepted explanation which is based on cracking-induced loss of penetration strength, by noting three experimental difficulties with this explanation. First, whereas microcracks were observed in recovered silicon-carbide tiles, they were not seen in tiles made of tungsten and boron carbides, although they displayed the dwell phenomenon. Secondly, the process of microcracking takes times of the order of a microsecond, while the dwell can last for tens of microseconds. Finally, experiments show that the penetration rates during dwell

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are not zero, indicating that the target surface is being pushed in, or removed during dwell. They also noted that Renstorm et al. (2004) showed that the loading of the ceramic surface by a long metallic rod, is predominantly inertial (hydrodynamic) in nature. All of these facts led them to consider the interaction between the rod and the tile through a fluid-structure sort of interaction, in which the force exerted on the target face depends on the angle between the incoming fluid jet and the outgoing fluid spray. The force is minimal when the spray spreads laterally, on the target’s surface, and it increases as the fluid jet “pushes” the target material resulting in a backward flowing jet. In order to test this hypothesis they performed scaled experiments with low velocity water jets (6–10 m/s), impacting a soft and transparent target. They used high speed imaging and a piezoelectric disc at the bottom of the target, which recorded the force history exerted by the water jet. Their experimental results show all the features of the dwell phenomenon with a close linkage to the shape of the backward flow of the water jet. Thus, they conclude that the fluid-structure interaction may be operating during the dwell phenomenon, also in ceramics. The initial deformation of the ceramic tile upon impact may induce this backward motion of the eroding rod, increasing its pressure on the interface, which leads to the dwell. This model may account for the significant effect of the buffer in increasing the threshold velocities of the interface-defeat and dwell phenomena. A simplified version of the JH-1 model was suggested by Rosenberg et al. (1995). Their model accounts for much of the penetration data for long rods impacting ceramic tiles which are backed by thick metallic blocks. The model is based on experimental data for the constitutive properties of the intact material which include its compressive strength (Y c ) under uniaxial stress conditions, its tensile strength (spall), and its compressive strength under high shock pressures (Y max ). The values of Y max are measured by the lateral gauge technique which was described in Chap. 1. For materials which exhibit a constant strength at high pressures, the failure curve of the intact material is bilinear as shown in Fig. 6.27, where the properties of the intact and failed ceramic are shown schematically. When the stress state in an element of the numerical scheme reaches the upper curve, its strength is reduced to the lower curve which accounts for the properties of the failed material. The failed-material curve starts from the origin (no tensile strength) and it increases linearly with a certain Fig. 6.27 The simplified model for ceramics

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slope up to the maximal strength of the failed material. The slope and the maximal strength of the failed material are obtained by multiplying the corresponding slope and the maximal value (Y max ) of the intact material by a parameter f < 1.0, which is the only free parameter in the model. The value of this “failure parameter” for a given ceramic is obtained by the following procedure. Given a set of penetration experiments choose one of them as the calibration experiment. Change the value of the failure parameter (f ), in the corresponding simulation, until a match is obtained with the experimental penetration depth. All the other experimental results should be accounted for by simulations with the same value of the failure parameter. This procedure resulted in a value of f = 0.3 for the AD85 alumina tiles in the experiments of Rosenberg et al. (1995), and a value of f = 0.45 for the AD995 alumina in the experiments of Rosenberg et al. (1997b). The value for the high-pressure strength (Y max ) of the intact ceramic in the model of Rosenberg et al. (1995), can be obtained by using longitudinal and lateral stress gauges in plate impact experiments, as described in Chap. 1. These gauges determine the two principal stresses (σ x and σ y ) in the shocked specimen, and its strength under high pressures should be derived through a proper yield criterion, such as the vonMises criterion Y max = σ x − σ y . The experimental results for AD85 alumina from Rosenberg et al. (1987a), and for titanium-diboride from Rosenberg et al. (1991a), are shown in Fig. 6.28. It is clearly seen that the alumina specimens exhibit a constant strength at high pressures, which was also the case for aluminum-nitride as measured by Rosenberg et al. (1991b). On the other hand, the TiB2 ceramic showed a pressure-hardening effect, with a significant increase in strength with increasing shock pressures. Up to this point we have dealt with the normal impact of scaled rods impacting various ceramic tiles through the reverse-ballistics configuration. Experiments by Hauver et al. (2005) were designed to increase the diameter of the rods and to examine the issue of oblique impacts. Their two-decade work is described in detail in this report and it includes many tests with normal impacts and scaled rods as well. At first, they used a rather heavy target configuration which included various steel parts, fitted by heat-shrinking, as well as front attenuating-layers which consisted

Fig. 6.28 The principal stress difference of AD85 alumina (left) and TiB2 (right) as determined by longitudinal and lateral stress gauges in plate impact experiments

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of aluminum/acrylic layers. These experiments confirmed the main conclusions discussed above, concerning the improvements in the threshold interface-velocities with a proper front cover. In order to move from the theoretical research to a more applicable one they used larger WHA and DU rods, with D = 4.93 mm L/D = 20 mm, in their inclined impact tests, as well as lighter target configurations. The metallic parts of these targets were made of either aluminum 7075-T6 or the Ti/6Al/4V alloy. The structure of such a titanium-based target is shown in Fig. 6.29. Note the thin graphite layer which was added in front of the ceramic tile, in order to let the mushrooming rod expand laterally, as recommended by Bless et al. (1992). It is also important to note that in these experiments the armor-graded alumina and the TiB2 tiles performed well, but the B4 C tiles did not exhibit the interface-defeat mechanism. This result may be due to the fact that in spite of its initial high compressive strength, boron-carbide is known to lose its strength under high dynamic pressures. The diameter and thickness of the TiB2 tiles used in these experiments were 72 mm and about 25 mm, respectively, and the thickness of the graphite layer was 2.38 mm. The tile was shrink-fitted by a tube of 17-4 PH steel, with a wall thickness of 5 mm, which was heated in order to accommodate the ceramic tile. The design of the attenuator layer consisted of a front plate, 6.3 mm thick, of porous alumina and 31 alternating layers of low-density polyethylene (0.57 mm thick) and 0.52 mm thick aluminum plates. This complex target design performed very well against a WHA rod, with D = 4.93 mm, impacting the TiB2 tile at a velocity of 1.6 km/s and

Fig. 6.29 The light target configuration of Hauver et al. (2005)

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6 Defeat by High Strength Targets

Fig. 6.30 Interface-defeat for a WHA rod obliquely attacking a TiB2 tile confined in the titanium-alloy target

obliquity of 45°, as shown in Fig. 6.30. At the time corresponding to this radiograph the rod has lost about 78% of its length and there is no evidence of penetration, or loss of any material from the ceramic surface. The excellent interface flow of the rod was later recognized to correlate with the rather thick (57 mm) titanium-alloy backing. Similar results were obtained for DU rods impacting a lighter target based on the 7075 aluminum alloy, at a similar velocity and at the same obliquity.

6.3.4 Ceramics Against Shaped Charge Jets We have seen that the ballistic efficiency of various ceramics against high-density long rods is of the order of η = 2.0–3.0, as compared with armor steel targets. These efficiencies are due to their high compressive strength and their low densities. Experimental results for both ceramic and glass blocks showed that they are even more efficient against shaped charge jets, resulting in values in the range of η = 4.0–5.0, as compared with armor steels. This fact was demonstrated for glasses by Hornemann (1989), Solve and Cagnoux (1990), Finch (1990), Moran et al. (1991) and Kozhushko and Rykova (1994), as well as for several ceramics by Kozhushko et al. (1992). It may seem that the high strength of these materials is more effective against the high-speed jets, in contrast with the accepted knowledge that at high impact velocities the strength of the target becomes less important. High-speed pictures and X-ray radiographs in the works cited above showed that the penetrating jet elements are significantly deflected from their original line of flight during penetration, by the action of some lateral forces. Thus, it was concluded that the high efficiency of these materials against shaped charge jets is due to this lateral action, rather than to their high strength. These lateral disturbances were related to the dilatancy phenomenon in brittle materials, referring to the extra expansion of their volume after stress unloading, as a result of crack initiation and growth during the loading phase. Specifically, the enhanced efficiency of glass targets against shaped charge jets was attributed to the lateral impact of glass debris with the penetrating jet elements.

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This issue has been investigated by Rosenberg et al. (1995) using numerical simulations for high velocity zero-strength rods, impacting ceramic tiles which were backed by thick steel blocks. An impact velocity of 5.0 km/s was chosen for these simulations in order to represent the average velocity of the different jet elements, and a 10 mm steel plate was placed in front of the 40 mm alumina tile in these simulations. The failure model of the ceramic tile was the one described in the previous section with f = 1.0, in order to avoid the complications resulting from early failure. The simulations showed that the final crater depths in the steel backings are practically the same for ceramic strengths in the range of 3.0–9.0 GPa, confirming the notion that at these impact velocities the strength of the target plays a minor role on its resistance to penetration. However, the diameter of the crater in the tile decreased appreciably with increasing tile strength. More importantly, the crater walls in these simulations were collapsing laterally on the jet, and more material was launched towards the axis with increasing tile strength. The laterally moving material included both ceramic debris from the crater walls, and the inverted jet elements which lined the wall after penetration. This lateral motion for the 6.0 GPa alumina is shown in Fig. 6.31. One can clearly see that sometime between 40 and 60 µs after impact, a large portion of the inverted rod material reached the symmetry axis with a significant lateral velocity. This laterally moving material can deflect the slower parts of a longer jet, resulting in the enhanced ballistic efficiency of ceramics and glasses against shaped charge jets. The numerical simulations of Rosenberg et al. (1995) highlighted the correlation between the strength of the tile and the degree of these lateral disturbances, through the amount of material which is launched laterally towards the axis, as shown in Fig. 6.32. The figure shows the state of the tiles and the debris along the penetration axis at relatively late times, in order to emphasize the differences between the three cases with different ceramic strength. The simulation for the 3.0 GPa tile showed Fig. 6.31 Simulation results for the ceramic debris and inverted rod motion towards the symmetry axis

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Fig. 6.32 The effect of ceramic strength on the lateral motion of the debris

that no material was propelled towards the axis. The 9.0 GPa tile propelled more material and at much earlier times, as compared with the 6.0 GPa tile. As is clearly seen here, the residual penetrations of the steel rods into the steel backing, in these simulations, were practically the same for the different strength tiles, because their penetration process was over before the arrival of the lateral debris at the axis. With much longer jets, this debris is bound to impact their slower elements and reduce their penetration capability. Thus, the increased efficiency of high strength ceramics and glasses against shaped charge jets has to do with the lateral impacts of the inverted jet debris, as depicted by these simulations.

6.4 Woven Fabrics as Armor Materials We have seen that the optimal design for add-on armors is based on layered targets with low density materials, where the first layer has a high compressive strength (such as a ceramic tile), and the second has a high tensile strength in order to absorb the debris from the projectile and the front plate. Composite materials which are based on woven fabrics have been found to be very efficient as the backing materials for both ceramics and hard metallic plates, due to their improved specific strengths (their strength/density ratios). The first composites were introduced in the 1960s and they included glass fibers which reinforced various plastics (GFRP). More efficient materials are based on aramide fibers (trade name of Kevlar® ), the ultra-high molecular weight polyethylene fibers (trade names Spectra® and Dyneema® ), and the PBO fiber which was first marketed by the name Zylon® . The low density of these organic fibers, in the range of 1.0–1.4 g/cm3 , and their high tensile strength (around

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299

3.0 GPa) led to their extensive use for various ballistics applications, from the protection of space stations to body armor vests. These fibers have high initial elastic moduli (around 100 GPa) and tensile failure strains of 3–5%, resulting in excellent energy absorption capabilities which surpass the energy absorption by metals. The fibers are bundled into yarns and these are woven into layers which are then stacked together, with or without impregnation, in order to produce the armor structure. The fibers are woven into 2D or 3D structures through different weaving techniques, as described by Zaera (2011). Comprehensive accounts of the properties and manufacturing processes of fibers, yarns, and woven fabrics, and their use in different armor systems are given in Bhatanagar (2006) and in Abrate (2011). The most extensive use of fabrics is for vests which are manufactured either with fabric laminates alone, or with ceramic tiles attached to their front. Fabric laminates are also attached to the inner walls of armored vehicles as “spall liners”, in order to stop a large part of the behind-armor debris (BAD) which is generated by the penetrating threat. The fabrics themselves are used as either “dry” sheets sewn together, or impregnated with resins such as polyester or epoxy. An example for the effectiveness of these fabric liners is given in the work of Weber et al. (2001). They used 50 mm thick liners bonded to plates of RHA, 70 mm thick, and observed the debris behind the plate, at about 400 µs after the impact of a WHA rod, 20 mm in diameter and 120 mm long. Two shadowgraphs from these experiments, with the polyamide liner and without it, are shown in Fig. 6.33. One can clearly see the reduced amount of debris behind the lined plate, demonstrating the usefulness of these spall liners. In order to follow the ballistic response of a panel made of several sheets of woven fabrics, researchers usually start with the response of a single yarn to the lateral Fig. 6.33 The amount of debris behind a bare plate (top) is reduced by the addition of a spall liner (bottom)

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6 Defeat by High Strength Targets

impulse of a ballistic impact. The impact at a woven sheet is then considered, with some assumptions concerning the cross joints of the fibers. Finally, the response of a bundle of sheets, either with or without the resin, has to be considered. Cheeseman and Bogetti (2003) review the important factors which control the perforation process of fabrics and compliant composites. These factors include the material properties of the yarns, the fabric structure, the shape and impact velocity of the projectile, the interaction of multiple plies, the far-field boundary conditions, and the friction between the yarns and the projectile. This variety of different factors complicates the analysis of the penetration and perforation processes, and the conclusions from a given set of experiments usually cannot be generalized for other cases. For example, Jacobs and Van Dingenen (2001) observed that unidirectional layers of Dyneema® fibers, placed at 0°/90° orientation on top of each other, perform much better than woven fabrics under the impact of blunt projectiles. They claim that this result is due to the fact that the 0°/90° arrangement allows for a much wider area of the structure to take part in the interaction. This conclusion holds only for blunt-nosed projectiles and, probably, for these fibers alone. Obviously, these complex factors are the main reason for the scarcity of analytical approaches to these projectile/composite interactions, although much progress has been made through empirical studies and numerical simulations. The complexity of the interaction between projectiles and textiles can be appreciated by the post-mortem pictures from Irenmonger (1999), as shown in Fig. 6.34. These are sectioned targets of Dyneema® which were impacted by 5.56 mm projectiles having a lead core and a hard steel tip. Note the shearing process of the thin target (H = 11 mm) which was easily perforated by the projectile, resulting in a crater diameter which is close to that of the projectile. This is the typical failure mode of relatively thin targets which are perforated by sharp-nosed projectiles at high impact velocities. The right hand side of the figure shows the more complex interaction for the H = 22 mm target, which was thick enough to defeat this projectile. The

Fig. 6.34 Cross sections of Dyneema targets impacted by 5.56 mm projectiles

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301

sectioned target shows that a shearing action by the projectile took place during the first stages of its penetration. At the later stages, the target exerted its high tensile strength on the projectile, as is evident by the extensive deflection and delamination of the back plies. This is the more common mode of operation for fabrics when they are used as backing layers in composite targets. With these configurations the fabrics exhibit their superior tensile properties in stopping debris, leading to their high ballistic efficiency. The large crater diameter at the later stage of penetration is due to the gross deformation of the projectile during this stage. This deformation resulted in a much larger volume of fabric participating in the process, as seen in the picture. The geometry of the projectile and its impact velocity determine the failure mode of the fabric, and it is obvious that sharp-nosed projectiles actually wedge through the fabric with relatively low resistance to their motion. With higher impact velocities the yarns are sheared by the projectile, as shown above, and they can even rupture upon impact, as described by Cheeseman and Bogetti (2003). Thus, the best performance of these composites is achieved against low-velocity blunt projectiles, where the yarns absorb a large part of the projectile’s kinetic energy through large transverse deflections. It should be noted that for ballistic applications, the resin content in the composite should be around 20% in order to enable large deflections of the plies. For the same reason, the fiber-matrix adhesion should not be too strong. When a multiple-ply target is impacted by the projectile the first few layers are sheared, as shown on the right hand side of Fig. 6.34. The remaining layers behave as a membrane and they absorb the projectile’s energy through their deflection, exerting their optimal properties and high efficiencies. These features are clearly demonstrated by the experiments and numerical simulations of Gama and Gillespie (2011) with thick S-2 glass/SC15 laminates, which were impacted by L/D = 1 steel cylinders. In order to follow the relevant material properties for these projectile/fabric interactions we summarize some of the important issues concerning these interactions. The early work on single-yarn impacts by J. C. Smith and his colleagues in the 1950s is considered as the basic theory for transverse loading of a woven sheet, as reviewed by Rakhmatulin (1966). The salient features of this theory, as given in Smith et al. (1958), can be stated with reference to Fig. 6.35 which describes a transversely impacted yarn. The figure describes the shape of the loaded yarn as obtained by

Fig. 6.35 Schematic description of a textile yarn impacted by a rigid projectile

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6 Defeat by High Strength Targets

high speed photography, demonstrating that the yarn takes a triangular shape which increases with time until failure takes place. The mathematical treatment of this developing triangular shape assumes that upon impact, longitudinal and transverse strain waves are propagated outward from the impact point at different velocities. The longitudinal wave, moving at the elastic wave velocity c0 = (E/ρ)0.5 , puts the material under certain tensile stress (σ 0 ) and strain (ε0 ). The high elastic moduli of these fabrics, together with their low densities, result in very high values for c0 . This means that a large portion of the target is engaged in the interaction with the projectile, enhancing its energy absorption capability. The material behind this wave front moves at a certain velocity (w) towards the impact point. The transverse wave, which propagates behind the longitudinal wave at a velocity of c1 , changes the direction of material motion towards that of the projectile. Behind the transverse wave front all particle velocities are equal to the velocity of the projectile (V ), and the constant-angle triangle continues to grow with time. For a material with a constant modulus (E) the following relation was derived between the strain in the fiber (ε0 ) and the projectile’s velocity (V ), as described by Roylance (1977):

V2 =

   Eε0 2 ε0 + ε02 − ε0 ρ

= c02 f (ε0 )

(6.9)

and this relation can be solved numerically in order to compute ε0 in terms of V, for the purpose of design curves of a given fabric. Once ε0 is known, all the other parameters of the problem, such as σ 0 , c1 and w, can be determined for a given impact velocity. For example, the transverse wave velocity is given by: c1 = c0

  ε0 + ε02 − ε0

(6.10)

With this simplified analysis one can calculate the strain energy σ 0 ε0 /2 and correlate it with the energy absorption rate of the fiber. When the maximal values for the tensile stress and strain are inserted in (6.9), the maximal value for V which marks the onset of fiber failure is obtained. However, these critical velocities are seldom realized in the experiments and, in fact, the actual critical velocities are lower by a factor of two than the theoretical values. Several explanations have been suggested for this discrepancy, as in Bazhenov et al. (2001), but the issue is still unresolved. The work of Tan et al. (2003) examined the influence of the projectile’s nose shape

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303

on the ballistic limits, the energy absorption behavior, and the perforation mechanisms of a single ply of plain woven aramide target. The experiments showed that the same trends were obtained for all the different projectile shapes, namely, an increase in energy absorption up to a critical velocity and a subsequent decrease at higher velocities. In order to demonstrate the high tensile elongations which can be achieved by a single yarn, consider the high-speed pictures of a blunt-nosed projectile impacting a nylon yarn, from Jameson et al. (1962), as shown in Fig. 6.36. The images in this figure are slightly displaced in order to highlight the fiber’s extension in the way depicted by Fig. 6.35. One should note that the ballistic efficiency cannot be increased indefinitely by increasing fiber stiffness as this is accompanied by a decrease in failure (breaking) strain. Thus, a complete model should have a relation between the two parameters in order to determine the optimal stiffness of the fabric. Considering the ballistic performance of a textile structure which consists of several plies, one has to take into account some conflicting trends which will adversely affect the performance of the system, as described by Cunniff (1992). Adding plies may constrain the transverse deflection of the first plies, amplify the concentration of tensile stresses, and reduce the ballistic efficiency of the structure. Cunniff (1992) analyzed several aspects of the structure performance, such as the aperture size of the fixture frame which constrains the fabric. As expected, a larger aperture resulted in a higher ballistic limit velocity, since the plies could undergo larger deflections. A large amount of data concerning the ballistic behavior of different fabrics has been summarized by Cunniff (1996) and in several papers which he presented at the 18th International Symposium on Ballistics (1999). Most of the data relates to right circular cylinders with an aspect ratio of L/D = 1, made of either steel or tungsten alloy. The textiles he analyzed included different Kevlar and Spectra fabrics as well as nylon (6,6), E-glass and some others. As with metallic plates there are two issues

Fig. 6.36 High speed pictures of a nylon yarn impacted by a projectile

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Fig. 6.37 The residual velocities of steel fragments perforating an 8-ply Kevlar 29 target

which concern the armor designer, namely, the ballistic limit velocity (V bl ) of a given projectile/fabric combination, and the projectile’s residual velocity (V r ) as a function of its impact velocity for V 0 > V bl . The general behavior of V r as a function of V 0 is demonstrated in Fig. 6.37 from Cunniff (1996), for an 8-ply Kevlar 29 target impacted by steel cylinders of different mass: M = 0.13, 0.26, 1.04 and 4.16 g. One can see that the data follow similar trends as for metallic plates, with a steep rise of V r for impact velocities just above the corresponding ballistic limits, and an asymptotic approach to a straight line which is parallel to the V r = V 0 line (denoted by the dotted lines in the figure). With increased projectile mass the ballistic limit velocities decrease and the straight asymptotes approach the V r = V 0 line, as for metallic targets. The actual values for the ballistic limit velocities, as shown by Cunniff (1999a, 1999b) for a large number of projectile/target pairs, can be simply related to their areal density ratio. He showed that the values of V bl can be scaled through a normalizing velocity parameter (U*) which is typical for each fabric. This velocity is given in terms of the strain energy of the fabric (σ f εf /2) where σ f and εf are its tensile failure strength and strain, respectively, and by its sound velocity (c0 ) and density (ρ), according to: U∗ =



σ f ε f c0 2ρ

1/3 (6.11)

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Table 6.2 Important physical parameters for various composites Spectra 1000

E (GPA)

ρ (g/cc)

σ r (GPa)

εf

U* (m/s)

120

0.97

2.57

3.5

801

600 Denier Kevlar KM2

82.6

1.44

3.4

3.55

682

840 Denier Kevlar 129

99.1

1.44

3.24

3.25

672

1500 Denier Kevlar 29

74.4

1.44

2.9

3.38

625

1.8

3.8

1.76

593

Carbon fiber

227

Cunniff (1999b) showed that the results for V bl /U*, for a large number of projectile/fabric combinations, are correlated with their areal density ratios. Thus, his empirical observations reduce to a universal relation between V bl /U* and the areal density ratio (ADt /ADp ): 

Vbl ADt , Φ U ∗ AD p

 =0

(6.12)

where AD is areal density and the subscripts (t) and (p) relate to the target (the fabric) and the projectile, respectively. These areal densities are, in fact, equal to ρ t H and ρ p L eff for the target and the projectile, respectively. The tensile failure stresses (σ f ) for the various fabrics analyzed by Cunniff (1999b), are in the range of 3.0–5.0 GPa, and their failure strains are about 3–4%. Several physical parameters for various composites are summarized in Table 6.2, from Cunniff (1999b), including the values for Young’s modulus (E) and the normalizing velocity U*. The same trends were found by Van-Gorp et al. (1993) who analyzed V bl data for several fabrics impacted by rigid FSP’s of different size. They showed that the data can be represented through a linear relation between the energy absorbed by the fabric and the areal density ratio. Thus, their results for the ballistic limit velocity of a given projectile/fabric pair can be written as: Vbl2 = C

ADt AD p

(6.13)

where the slope (C) depends on the fabric type. This empirical equation means that the kinetic energy which is needed to perforate a given composite target is proportional to its thickness. In view of the discussions in Chap. 3, this result indicates that these targets exert effectively a constant resisting stress on the projectile. Gellert et al. (2000) showed that this simple picture has to be somewhat modified. They analyzed several sets of data for FSP projectiles with different size impacting different composites, and found a bi-linear relation between the energy at V bl and the composite thickness. The data showed that for composite targets which are thicker than the projectile’s diameter, the slopes of these lines depended on the type of the composite. For thin targets they found a different behavior according to which the density of the composite does not play a significant role.

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Walker (2001) offered a simple analysis for the perforation process of composites, and derived an expression for V bl of a given projectile/fabric combination. He used momentum conservation and obtained the following relation between the impact velocity (V 0 ) and the velocity of the projectile and the fabric right after impact (V ): V =

M V0 M + ψ A pρ f

(6.14)

where M and Ap are the mass and the projected areas of the projectile, respectively, and ρ f is the density of the fabric. The factor ψ denotes the actual area of the fabric under the projectile which takes part in the interaction, and Walker (2001) chose a value of ψ = 2.5 for his model. Realizing that the fabric can be stretched to a maximum value of εf , he arrived at the following relation for the ballistic limit velocity:  Vbl = 1.8 1 + ψ X f c0 ε f

(6.15)

where X f is the corresponding projectile/fabric areal density ratio. This simple relation accounts for much of the data for the different fabrics which were impacted by L/D = 1 steel cylinders. The addition of resin to the fabrics results in a complex behavior which was further analyzed by Walker (2001). For a few sheets of fabrics this addition degrades the ballistic performance of the target. On the other hand, as the areal density of the fabric increases the fabric/resin panel outperforms the equivalent-areal-density dry fabric. It was found experimentally that this crossover in the ballistic performance is in the region where the amount of fabric involved in the momentum balance is equal to the mass of the projectile. In order to account for this complex behavior Walker (2001) considered the two contradicting effects of the resin. On the one hand, its addition is at the expense of fabric material for equal-areal-density targets, and it is expected to lower the ballistic limit of the fabric/resin panel. On the other hand, the addition of resin improves the bending resistance of the composite, increasing its ballistic limit velocity. He derived the following relation for V bl (X f , r f ), in terms of the ballistic limit velocity of the dry fabric V bl (X f , 0):  Vbl X f , r f =

  3  1 − r f + r f ψ X f Vbl X f , 0

(6.16)

where r f is the mass fraction of the resin. Walker (2001) showed that this equation accounts for the ballistic limits of “dry” and “wet” Kevlar 29 panels, and for the crossover value of X f which was observed experimentally, concerning the change in the superiority of the dry over the wet panels. The excellent performance of ultra-high-molecular-weight polyethylene composites (Dyneema) against FSP threats has been demonstrated by Nguyen et al. (2016) in a series of tests with 12.7 and 20 mm FSP’s against various fabrics, as well

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307

Fig. 6.38 Ballistic limit velocities for the two FSP’s impacting Dyneema targets

as metallic plates. These tests were performed on Dyneema® HB26, a unidirectional composite which consists of approximately 80% volume fraction of fibers in a polyurethane matrix. The plies were approximately 60 µm thick, laid in a [0/90]n stacking sequence, and hot pressed under 125 °C and a pressure of 14 MPa. Figure 6.38 summarizes the V bl values for the two types of fragments impacting these targets. The thickness of these targets are presented by their corresponding ratios of (AD)t /(AD)p , between the areal densities of the target and the projectile. Note the straight line through the data points, which is a clear evidence for the linear scaling in this interaction, as was found empirically by Cunniff (1999b). This linear relation between V bl and the areal density ratio is different from the quadratic relation (6.13) of Van-Gorp et al. (1993). Nguyen et al. (2016) compared the performance of their Dyneema targets with the performance of other fabrics impacted by L/D = 1 steel and tungsten cylinders, from the work of Cunniff (1999b) which was described above. These fabrics included: Kevlar® KM2/polyvinylbutyral (AFRP), E-glass/polyester (GFRP) and carbon fiber/epoxy (CFRP). The superiority of the Dyneema fabric is clearly demonstrated in Fig. 6.39. Nguyen et al. (2016) also compared the performance of Dyneema targets with the ballistic performance of several metallic plates of RHA and HH steels, with Brinell hardness of 350 and 500, respectively. They added data for the 5059-H131 aluminum alloy and the results are shown in Fig. 6.40. Compared with these metals, the Dyneema composite has a significantly better ballistic performance against the two types of FSP’s. The published literature on numerical simulations for the impact on fabrics is very extensive, and they can be classified as either discreet or continuum simulations, as summarized by Chocron et al. (2011). The discreet simulations started with a 1D bar element which was pin-jointed to other elements in order to simulate the woven fabric, as in Shim et al. (1995). The 2D discreet approach uses shell elements which

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Fig. 6.39 Comparing the Dyneema composite with other fiber reinforced targets

provide a more realistic interaction between the yarns, as well as between the layers. The 3D discreet approaches, where the fabric is described by solid elements which represent the yarns, are very useful. Duan et al. (2006) and Rao et al. (2009) studied the effect of friction and performed sensitivity studies of various parameters. The continuum approach avoids the need to describe the yarns individually. Tensile and shear tests on the fabrics are used to evaluate the unit cell properties, and they are then tested against the data from ballistic tests. Chocron et al. (2011) used a mixed mode in the sense that the fabric was simulated in a discreet yarn level, and the yarns are composed of solid elements, as was first done by Shockey et al. (1999). The numerical models used by Chocron et al. (2011) for Kevlar KM2, Dyneema and PBO, were thoroughly checked against various experiments. These experiments included single Fig. 6.40 Comparing Dyneema composites with metallic targets against FSP’s of 20 mm (open symbols) and 12.7 mm (closed symbols)

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309

Fig. 6.41 Simulation result for the evolving deflection of a composite target, 35 µs after impact of an FSP

yarn and single layer impacts, as well as multilayer and ballistic limit determinations. Good agreement (of the order of 10%) were obtained between the simulation results and the experiments, which enable one to use such simulations as a design tool and save a lot of experimental work. An example for the capabilities of such simulations is given in Fig. 6.41, from Chocron et al. (2011). This is a perspective view of the pyramidal development in a Dyneema target composed of ten layers, which was impacted by a fragment simulating projectile (FSP). As was mentioned above, the main advantage of fabric materials stems from their high specific strength, as defined by their tensile-strength/density ratio. Their high ballistic efficiency against blunt-nosed fragments makes them ideal as backing materials for either ceramic tiles or metallic plates. Basically, the hard facing material shatters the projectile, while the fabrics stop the fragments by applying their improved tensile properties. Obviously, the fabrics do not perform as well when they are directly impacted by sharp-nosed rigid projectile. This is the main reason for the fact that only a few experimental studies were devoted to the direct impact of these projectiles on textile targets, such as in Zhu et al. (1992a, 1992b). They impacted conical nosed hard steel projectiles at targets of Kevlar 29 plies which were impregnated by a thermosetting polyester resin. They determined V bl values for two types of projectiles and for target thicknesses in the range of 3.1–12.7 mm. We wish to demonstrate now that their experimental results can be accounted for by the models which apply for metallic targets, as discussed in Chap. 4. The projectiles in the works of Zhu et al. (1992a, 1992b) had diameters of 12.7 and 9.525 mm and masses of 28.5 and 12.5 g, respectively. The measured residual velocities, as a function of impact velocities, for the larger projectile (D = 12.7 mm) impacting the 6.35 and 3.125 mm targets, are shown in Fig. 6.42. The figure also shows the corresponding curves from the model of Recht and Ipson (1963), through the expression V r = (V 20 − V 2bl )0.5 , with V bl values which were determined by the experiments. The good agreement

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6 Defeat by High Strength Targets

Fig. 6.42 Residual velocities (symbols) for the 12.7 mm projectiles perforating the two Kevlar 29 targets and the Recht-Ipson model (curves)

between the model and the experimental results enhances the assertion that sharpnosed projectiles perforate the fabrics in a similar way to their perforation of a metallic plate. The measured values of V bl are given in Table 6.3 for the two types of projectiles perforating the Kevlar targets with different thicknesses. These values for V bl show a similar dependence on target thickness as obtained for sharp nosed projectiles perforating metallic plates. In order to demonstrate this point, consider the effective stresses (σ r ) which a target exerts on the projectile, as defined in Chap. 4: σ r = ρ p L eff V 2bl /2H, where H is the thickness of the target. The effective lengths of the D = 12.7 and 9.5 mm projectiles in this study were: L eff = 28.7 and 22.4 mm, respectively. The calculated values of σ r , as a function of the ratio H/D, are shown in Fig. 6.43. One can clearly see that the same features which were found for metallic targets perforated by sharp nosed projectiles are also evident here. Namely, there is a range of H/D values, between about 0.3 and 1.0, where σ r is practically constant, while it increase with H/D for target thicknesses outside this range, namely at H/D < 0.3 and H/D > 1.0. Obviously, the few experimental results in this work are not enough to prove these similarities, and the subject deserves further experiments and analysis. The conclusion concerning the similarity between metallic and fiber-based targets is corroborated by the work of Heisserer et al. (2013), who impacted hard steel spheres, 5.58 mm in diameter, at targets made of Dyneema SK76 plies. These targets were 12.8 and 25.8 mm thick and impact velocities ranged between 0.25 and 1.5 km/s. After impact the targets were sectioned and the penetration channels were found to Table 6.3 The values for V bl (in m/s) from Zhu et al. (1992a) H (mm)

3.125

6.35

9.525

12.7

V bl (D = 12.7 mm)

78

116

143

170

V bl (D = 9.5 mm)

97

135

168

210

6.4 Woven Fabrics as Armor Materials

311

Fig. 6.43 The relation between σ r and H/D for Kevlar 29 targets perforated by sharp-nosed projectiles

have a constant diameter, which is equal to that of the impacting sphere as is the case for metallic targets. The kinetic energy absorbed in perforating the thin targets at the higher velocities, turned out to have a constant value as calculated through: m(V 20 − V 2r )/2. Generalizing this observation, one may conclude that the work to perforate a given composite target does not depend on the impact velocity, as is also the case for metallic targets. The thinner targets had larger deflections at their back layers due to the excessive bending and stretching of the last layers. This is probably the reason that the thinner targets resulted in slightly lower penetration depths, as seen in Fig. 6.44 which summarizes the results of this work. A close examination of the data shows that the penetration depths increase with impact velocities according to: 4/3 P = k V 1.5 0 . This relation is close to the V 0 dependence, which was found in Chap. 3 for rigid spheres impacting metallic targets. As described above, one of the major applications of these composites, for armor configurations, is as spall liners behind the walls of the protected vehicle. These liners Fig. 6.44 Penetration depths of 5.58 mm hard steel spheres in Dyneema targets of different thickness

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have been found to be very effective in reducing the number of fragments which penetrate the target and their spatial extent, as shown in Fig. 6.33. The damage to the interior of the vehicle can be reduced significantly by adding a relatively lowweight liner, as demonstrated by Lampert and Jeanquartier (1999) who shot L/D = 1 WHA cylinders at 13.7 mm steel plates, with and without Kevlar liners. The liners’ thicknesses ranged between 5 and 20 mm and the ballistic limit velocities were determined for each target. For the bare steel plates they obtained V bl = 970 m/s, while for the steel plate with liners of 5, 10 and 20 mm they obtained V bl = 1350, 1630 and 1900 m/s, respectively. These significant improvements in the stopping power of the steel/liner targets were accompanied by a substantial reduction in the number of fragments, and their spatial extent, at impact velocities above the corresponding V bl values. Finally, consider the ballistic efficiency of thick targets made of composite materials, against long rod penetrators. The published data on this subject is very scarce, and the study of Ernst et al. (2003) is one of the very few publications. They determined the ballistic efficiency of glass reinforced plastics (GFRP) against L/D = 20, D = 7.25 mm WHA rods. The DOP test configuration with a thick RHA backing was used in these tests. The thickness of the GFRP layers varied between 22 and 116 mm and impact velocities were around 1.8 km/s. The most interesting result from these tests concerns the increasing trend of the ballistic efficiency with the thickness of the fabric, reaching a value of about η = 1.9 as compared with RHA, for the large thicknesses. This is a relatively high value for the ballistic efficiency against long rods and, as recommended by Ernst et al. (2003), these materials should be considered as possible candidates for passive armor designs against these threats.

6.5 Elastomer-Steel Laminates for Armor Applications One of the more successful approaches in armor design is the application of elastomeric coatings to hard substrates. The initial work on these ideas was performed by the US Air Force, with polyurea coatings applied to the walls of buildings in order to retain their structural integrity during a bomb blast. The function of the coating in this application is to prevent building fragments produced by the blast from being propelled through the interior of the building. Subsequently the US Navy has pursued the use of polyurea coatings to mitigate the damage from ballistic fragmentation and projectiles. These coatings were applied to the front sides of armor plates, increasing their resistance to penetration from gunfire and fragmenting explosives. Roland et al. (2010) demonstrated that many types of elastomers are as effective as polyurea in improving the ballistic performance of high-hardness steel plates. They performed ballistic limit tests for steel plates covered with the following base polymers: polyisobutylene, polyurea, polynorbornene, nitrile rubber, 1,4-polybutadiene, and both synthetic and natural 1,4-polyisoprenes, all of which are high-molecularweight organic polymers. These compounds were applied to the front face of steel plates with hardness in the range 470–500 Brinell, and all of them demonstrated large improvements in the corresponding ballistic limit velocities against 0.5 fragment

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simulators (FSPs). In fact, these polymer coatings resulted in 40–50% improvements in values of V bl for plates with thicknesses in the range of 5–12.7 mm. For example, the value of V bl = 600 m/s of a bare 6.2 mm plate, increased to 830 m/s and 900 m/s with coating thicknesses of 5 mm and 19 mm, respectively. As Roland et al. (2010) point out large improvements were found by other workers for high-hardness steels, while improvements were much less drastic when softer steels were used as substrates. The performance of polyurea coating is quite remarkable as seen in Fig. 6.45, from Roland and Giller (2015). The figure summarizes the V bl values for 0.5 FSP’s impacting 5.13 mm HH steel plates, as a function of polyurea thickness. One can clearly see that the addition of a few millimeters of polyurea resulted in a substantial increase in the ballistic performance of these plates. On the other hand, the additional benefit of thicker layers is very small. Other workers found that against pointed projectiles such as hard-cored AP projectiles, these coatings were easily penetrated exhibiting no improvements in the ballistic performance of these plates. Roland et al. (2010) also showed that by using two elastomeric layers the improvements are even better, as compared with a single layer with the same total thickness. For example, the ballistic limit velocity of a bare HHS plate, 12.7 mm thick, was found to be 1184 m/s. The addition of a 12.7 mm thick coating increased this value to 1483 m/s, with a very small addition to the areal density of the target. Adding two layers each of 6.4 mm, resulted in a value of V bl = 1820 m/s, which is a substantial improvement. Roland et al. (2013) used both polyurea and vulcanized butyl-rubber coatings with various metallic substrates and showed that, indeed, the best improvements in the V bl values are for the harder (stronger) substrates. In fact they found a linear relation between the increased values of V bl and the hardness of the substrates, which included several steels and aluminum alloys, as well as a titanium substrate. They also found that the diameters of the penetration holes in coated plates are larger, by about a factor of two, than the holes in the corresponding bare plates. This Fig. 6.45 Improvements in V bl values of 5.13 mm thick HH steel plates with polyurea coatings

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is, apparently, the result of the longer interaction time during which the projectile is deforming substantially before it starts to penetrate the substrate. During these times the projectile is also slowed down considerably, losing a substantial portion of its kinetic energy. It may be that these are the important issues responsible to the ballistic efficiency of the elastomer coatings, rather than phenomena such as a viscoelastic phase transition, which were suggested by several workers. We should note that Roland et al. (2010) found that rubber coatings were as efficient as the polyurea coating. Also, an experimental study performed by Ashuach (2005) showed that a Neopren rubber coating, of either 1 mm or 10 mm thick, resulted in significant improvements in the V bl values for 12 mm plates of HH steel, against 20 mm FSP’s. The ballistic limit velocity for the bare plate (590 m/s) increased to 740 m/s and to 840 m/s for the two rubber thicknesses, respectively. These improvements are similar to those shown in Fig. 6.45 for the polyurea-coated HH plates. Moreover, Dyatkin and Gamache (2018) showed that a Teflon coating of a few mm resulted in similar (if not better) improvements in the stopping power of high-hardness steel plates, against 0.5 FSP’s. They coated steel plates, 4.7 and 6.3 mm thick, with Teflon layers having thicknesses in the range of 2.3–19.7 mm. Improvements through higher V bl values were similar to those shown in Fig. 6.45 for the polyurea coating. They note that the Teflon coating resulted in higher ballistic limits, as compared with the other polymer coatings. Even the thinnest layer (of about 2 mm) resulted in higher V bl values by 55% and 80% for the 4.7 mm and the 6.3 mm thick plates, respectively. These results led these authors to conclude that the energy absorption mechanism of these polymers is, probably, not the result of viscoelastic phase transitions.

6.6 Fragment Simulating Projectiles (FSPs) Perforating Aluminum Plates Over the last two decades a large research program at the ARL (in the US) was aimed at improving the ballistic performance of aluminum alloys. This effort was motivated by their excellent ballistic advantages when utilized as primary armors on tactical trucks and combat vehicles. The primary alloy used in US combat vehicles is the aluminum-magnesium alloy 5083-H131 that is defined by a set of mechanical properties, corrosion requirements and ballistic standards. However, this alloy has a rather low strength relative to other aluminum alloys, and the effort is to find other alloys which could have a better ballistic performance, without giving up the advantages of the 5083 alloy. Plates of various thicknesses are tested for their ballistic limit velocities against both AP projectiles and fragment simulators (FSPs). Two types of fragments are usually used, with nominal diameters of 0.5 and 20 mm, which have very similar shapes as shown by their cross-sections in Fig. 6.46. They are made of 30Rc steel, weigh 13.4 and 53.8 g, and they are shot at velocities in the range of 500–1200 m/s. We discuss now some of the important findings concerning

6.6 Fragment Simulating Projectiles (FSPs) Perforating Aluminum …

315

Fig. 6.46 The dimensions of the 0.5 caliber and the 20 mm simulators

the V bl values of several aluminum alloys that were studied as possible replacement for the 5083 alloy. The first point to note is the fact that the data for the different projectile sizes, in terms of V bl as a function of the normalized plate thickness (H/D), fall on the same curve. As an example, Fig. 6.47 shows the results for the V bl values of the two FSPs impacting plates of the 6055-T651 alloy, from Gallardy (2015). There are two important points which immerge from this figure. The first is the fact that there is a clear geometric scaling of the results for the two fragments, as far as their normalized thicknesses are concerned, which can save many tests for the armor designer. The second point is the fact that the resulting V bl values are linearly dependent on H/D, in the range of 1.0–2.5, a trend which has to be accounted for by analytical modeling. One should also be aware of the fact that plates of different thicknesses may have different strengths. This is also the case here, as is clear by the hardness values reported in Gallardy (2015). The average hardness values ranged between 126 Brinell for the 20 mm plates, and 107 Brinell for the 50 mm plates. Thus, the interpretation of the results in these tests, as well as in similar tests on other alloys, should consider variations of the order of 15% in the strength of the plates.

Fig. 6.47 Scaling in the V bl values of Al-6055 = T651 plates impacted by FSPs of different size

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The experimental results for different aluminum alloys are summarized in various ARL reports, as in Gallardy (2012, 2015, 2016), Gooch et al. (2007), Showalter et al. (2008), and Cheeseman et al. (2008). Additional data are given in Fras et al. (2016) and in Guo et al. (2019). A close examination of all the data from these sources, in terms of V bl (H/D), shows that they are very close to each other. Thus, the differences in the compressive strengths of these alloys do not affect the corresponding values of their V bl . In order to demonstrate this issue we chose the data for six alloys from three series which span the different strengths of these aluminum alloys. These are the high-strength 7000 alloys (7085-TE01 and 7039-T64) with dynamic compressive strengths near 0.7 GPa, the intermediate-strength 2000 alloys (2139-T8 and 2060-T8) with strengths of around 0.55 GPa, and the relatively soft 6000 alloys (6061-T651 and 6055-T651) with strengths of about 0.4 GPa. Figure 6.48 shows the corresponding results for plates of different thicknesses, in terms of V bl as a function of their normalized thicknesses (H/D). The data includes both the 12.7 and 20 mm FSPs, and the results fall within a narrow range of 10% around an averaged straight line, as shown by the dotted lines in the figure. A possible reason for this result is the fact that two different penetration modes are at play here. At the first stages of penetration, the fragment is pushing the target material aside in a ductile-hole enlargement process, which depends on the dynamic strength of the plate material. This process is followed by a shear failure in the plate, around the projectile’s periphery, leading to plug formation and its ejection. The plugging mechanism is dependent on the maximal shear strain that the plate can sustain, which is a decreasing function of its strength. Thus, while a stronger plate may exert a higher resistance to penetration during the first stage, it is reasonable to expect that it will be easier to fail by shearing at the later stage. One can assume that the energy to perforate a given plate depends on the plastic work which can be stored in the plate’s material, namely, the area below the corresponding stress-strain curve. Since a higher strength is accompanied by a lower failure strain, these energies for the different aluminum alloys may be close to each other, as demonstrated by Holmen et al. (2016). Fig. 6.48 Data for aluminum alloys from different series: 6000 (red), 2000 (black) and 7000 (blue)

6.6 Fragment Simulating Projectiles (FSPs) Perforating Aluminum …

317

Fig. 6.49 Tensile stress-strain curves for the four different treatments

The work of Holmen et al. (2016) is an excellent demonstration for the conflicting roles of strength and failure-strain in the perforation process of strong aluminum plates. They impacted four differently-treated aluminum 6070 plates 20 mm thick, with flat-nosed rigid projectiles having a diameter of 20 mm. Plugs were ejected from all the plates and the corresponding V bl were determined by using the measured residual velocities through the Recht–Ipson model. They also provide static tensile stress-strain curves for the four heat treatments which show significant differences in yield strength, strain hardening, and failure strain as shown in Fig. 6.49, where the x marks the point of failure in each test. Clearly, the weakest treatment (T0) has the highest strain to failure while the strongest treatments had the lowest failure strains. The T4 treatment resulted in an intermediate behavior as is seen in the figure. The different failure strains resulted in V bl values which were very close to each other, in spite of the large differences in the corresponding tensile strengths of these plates. These values were: V bl = 155 m/s, 174 m/s, 144.4 m/s and 166.5 m/s for the T0, T4, T6 and T7 specimens, respectively. The T4 alloy resulted in the highest value for V bl due to its intermediate properties and the softest alloy (T0) resulted in a higher V bl than the strongest alloy (T6), which is an excellent example for the contradicting effects of strength and ductility. Figure 6.50 highlights the different Fig. 6.50 Sectioned plates of the T0 and T6 treatments perforated by ogive-nosed projectiles

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failure modes which operated in the perforation processes of these plates by ogivenosed projectiles, 20 mm in diameter, from the study of Holmen et al. (2016). The ductile hole enlargement mode is clearly evidenced in the case of the T0 perforation, while spalling and scabbing failures due to the small failure strain, are seen in the brittle T6 plate. According to Holmen et al. (2016) the values of V bl correlate with the corresponding values of the Cockroft–Latham parameter which is a measure of the area below the tensile stress-strain curves. The Cockroft–Latham fracture model (1968) is a straightforward failure criterion which is mostly used for semi-ductile metals such as the aluminum alloys. The model assumes that fracture takes place when a certain value of the following integral reaches a threshold value W CL :

ε f WC L =

σ1 dε

(6.17)

0

where σ 1 is the major principal tensile stress and the integral is taken over the equivalent plastic strain path up to failure strain (εf ). The value of this integral is determined by a uniaxial tensile stress test. The experimental values of W CL for the four treatments were: 155, 211, 115, and 128 MPa for the T0, T4, T6, and T7 specimens, respectively. Clearly, this analysis accounts for the fact that the T4 alloy with the highest value of W CL parameter had the highest value of V bl .

6.7 The Defeat of EFP Threats One of the most lethal threats against armor vehicles is the explosively formed projectile (EFP) which was described in Chap. 1. This is a short projectile with a relatively large diameter and an aspect ratio of 1–3, moving at a velocity of about 2 km/s. It is very difficult to defeat these threats since they are not prone to being shattered, broken or deflected. Thus, the best one can do against these “chunky” projectiles is to slow them down and deform them, with a minimal weight of addon armor. An example for such an attempt is given by Liu et al. (2018) who shot copper EFPs at steel plates of medium strength. The copper cones (liners) had a diameter of 66 mm and they were attached to explosive cylinders having a diameter of 70 mm and length of 60 mm. The flight velocities of these EFP’s were in the range of 1.28–1.33 km/s. Their reference test employed a 40 mm thick steel plate which was penetrated by the EFP, and the velocity of the ejected steel plug according to their simulations, was about 100 m/s. Thus, this test can be considered as close to the ballistic limit of the given projectile/plate pair. The most interesting results of this work were obtained with two or four steel plates, having a total thickness of 40 mm. As shown in Fig. 6.51 these multi-plate targets were not penetrated by the EFPs. The strong bending of the thinner plates resulted in a penetration of only half of their

6.7 The Defeat of EFP Threats

319

Fig. 6.51 Cross sections of the three targets with the same total thickness

thickness, resulting in a very efficient way to absorb the projectile’s kinetic energy. One may conclude that a layered target with a total thickness of less than 40 mm could have stopped this projectile rendering a lighter armor design. Obviously, this kind of energy-absorbance requires relatively ductile plates which are usually of medium-strength. Such a compromise between strength and ductility accompanies the armor designer all the time, and the works described below address this issue in some detail. The work of Klavzar et al. (2017) was aimed at comparing the performance of steel plates with different strength and ductility, in reducing the lethality of EFP’s. Instead of using explosive devices they machined copper EFP-simulants and launched them in a powder gun using aluminum sabots. Figure 6.52a shows such a projectile installed in its sabot, and Fig. 6.52b is a double-exposure X-ray film of a flying EFP at different times. Residual velocity as determined by these flashes, was chosen as a measure for the effectiveness of different targets in reducing the lethality of these stimulants. Klavzar et al. (2017) launched these copper simulants at steel plates of various thicknesses as machined from thicker plates of eight different types of steel. This variety of steels ranged between structural steels with tensile strengths of 550 MPa, and very strong plates with hardness around 600 Brinell. As is clearly seen in Fig. 6.53 the residual velocities of the EFP simulants did not depend on the mechanical properties of these plates, since all the data seem to follow a straight line. The authors explained this result by the conflicting penetration mechanisms which these plate experience, the ductile hole-formation in the softer steels, and the adiabatic shearing

Fig. 6.52 a The EFP simulant in its sabot and b flash shadowgraphs during its flight

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Fig. 6.53 Residual velocities of the EFP stimulants perforating various steel plates

of the stronger plates. As with the FSP’s impact at aluminum plates, the two modes compete with each other resulting in the linear behavior we see in the figure. Glikin and Rosenberg (2019) further explored this issue through numerical simulations, by varying the velocity of their EFPs as well as the strength and density of the plates, in order to find an optimal design for this interaction. The basic idea behind these simulations was to assess the effectiveness of these interactions by comparing the penetration depths of the residual projectiles in the aluminum witness blocks, rather than looking for their residual velocities as was done by Klavzar et al. (2017). A short copper cylinder, with L = D = 20 mm, represented the EFP in all of these simulations. In the reference simulations this projectile impacted a copper plate 10 mm thick at 1.0–2.0 km/s, and the residual projectile penetrated a thick witness plate, of the aluminum 6061-T6 alloy, which was placed 80 mm from the copper plate. In order to simplify the analysis, all the parts in the simulations were described by the elastic-perfectly-plastic von-Mises constitutive model. A shock Mie-Gruneisen EOS with a linear relation between shock and particle velocities was used for all the materials involved. The projectile in all simulations had strength of 0.4 GPa, which was also the strength of the aluminum witness block. Figure 6.54 shows several “snapshots” from the reference simulation with an impact velocity of V 0 = 2.0 km/s, and

Fig. 6.54 Snapshots from the reference simulation

6.7 The Defeat of EFP Threats

321

a plate’s strength of 0.4 GPa. This simulation was repeated for impact velocities in the range of 1.0–3.0 km/s, resulting in penetration depths of 5–23 mm in the witness block. Glikin and Rosenberg (2019) varied the von-Mises strength of the copper plates in the range of 0.2–2.0 GPa, and found that the penetration depths of the residual projectiles increased by a very small amount from 12.5 mm to 13.8 mm, respectively. This counterintuitive result was explained by the shape of the residual projectiles. It turned out that stronger plates resulted in a “chunkier” projectile, whose mass is concentrated along the center of symmetry. On the other hand, the residual velocity of the projectile decreased with increased plate strength. Apparently, this decrease was not enough to warrant a lower penetration depth. Thus, one may conclude that the concentrated shape of the projectile emerging from a strong target is more influential as far as the residual penetration is concerned. This finding confirms the choice of Glikin and Rosenberg (2019) to follow the penetration depths, rather than residual velocities of these projectiles, as a measure of ballistic efficiency. Another set of simulations was performed for target plates of different densities, in the range of 2.23–17.86 g/cm3 (25–200% of the reference density). In all of these simulations the strength of the plates was kept at 0.4 GPa and the impact velocity was 2.0 km/s. The thicknesses of these plates were varied in order to keep their areal densities equal to that of the 10 mm thick reference copper plate. The resulting depths of penetration increased monotonically from 7 mm to about 16 mm, for plate densities in the range of 2.23–17.86 g/cc. Clearly, the low-density plates are more efficient, as far as residual penetrations are concerned. Glikin and Rosenberg (2019) were able to account for this result by following the shapes and velocities of the residual projectiles, as with their results for the variations in the strength of the plate. The most interesting results were obtained with simulations where the plate was moving towards the projectile at a certain velocity, in the range of 0.5–1.5 km/s. The projectile in all of these simulations moved at 2.0 km/s. The relative velocity is, obviously, the sum of the projectile’s and the plate’s velocities. Glikin and Rosenberg (2019) compared the resulting penetration depths in the witness blocks with those from the simulations in which the plate was stationary. This is shown in Fig. 6.55 in terms of penetration depths as a function of the relative velocity, and it is clear that Fig. 6.55 Penetration depths as a function of relative velocities

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the ballistic efficiency of the moving plate is significant. One can follow this result by comparing the impact of a 2.5 km/s projectile at a stationary plate with the impact of a 2.0 km/s projectile at a plate moving at a velocity of 0.5 km/s. The penetration depths in the witness blocks were 9 mm and 17 mm for the moving plate and the stationary plate, respectively. The simulations showed that the projectiles emerged with the same shape and both of them lost about 0.6 km/s. This means that the residual projectiles had different velocities after the interaction, namely, 1.9 km/s and 1.4 km/s for the stationary and the moving plates, respectively. This difference accounts for their different penetration depths, demonstrating the advantage of moving the plate towards the projectile in a possible armor design.

Chapter 7

Asymmetric Interactions

Abstract The asymmetric interactions of various threats with inclined plates, either static or moving, are discussed in this chapter. The resulting asymmetric forces can enhance the defeat mechanisms to these threats through their deflection, erosion or shattering. Most advanced armor designs, such as the explosive reactive armor (ERA) concept, are based on these asymmetric interactions. The basic defeat mechanisms behind these interactions are explored through 3D numerical simulations, which are shown to be very useful for the optimal design of armor systems.

7.1 General Description This chapter deals with defeat mechanisms which are due to asymmetrical interactions between the threat and the target, and their possible applications for armor systems. A major advancement in understanding and optimizing such defeat mechanisms is due to the continual improvements of 3D codes, which can follow these interactions accurately and efficiently. The basic asymmetry is obtained by inclining the target at a certain angle to the projectile’s line of flight. There are two ways to define the impact angle of the projectile, as shown in Fig. 7.1a, and we shall use both of them throughout this chapter. These are the impact angle (α) between the velocity vector of the projectile and the target’s face, and the obliquity angle (β) between the projectile’s velocity vector and the normal to the plate and, obviously,

Fig. 7.1 a The definition of impact angles and b the curved trajectory of a long steel rod impacting a thick aluminum target at an obliquity of 45° © Springer Nature Switzerland AG 2020 Z. Rosenberg and E. Dekel, Terminal Ballistics, https://doi.org/10.1007/978-3-030-46612-1_7

323

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α + β = 90°. The asymmetry of the impact at inclined plates induces a net force on the projectile perpendicularly to the target’s surface. This force induces a torque on the projectile which can either break or bend it, resulting its ricochet at obliquities beyond a certain threshold. Such an asymmetric interaction was demonstrated by Warren and Poormon (2001) for ogive-nosed steel rods, impacting thick aluminum targets at different obliquities. The trajectory of the rod for an impact velocity of 1.184 km/s and an obliquity of 45° is shown in Fig. 7.1b. The asymmetry of the force on the projectile at the early stages of its penetration, is due to the strong changes in the resisting stresses at the entrance phase, resulting in its curved trajectory in the target, and its pronounced bending which further enhanced its curved trajectory. As shown by the numerical simulations of Warren and Poormon (2001), rod bending and its subsequent ricochet was much less pronounced when the rod strength was increased by only 0.1 GPa. For finite thickness plates the asymmetric forces on the projectile’s nose act at both impact and back faces. These coupled forces result in a change in the flight direction of the projectile towards the normal to the plate, as shown schematically in Fig. 7.2a. An actual example for such a deflection of steel projectiles perforating aluminum plates is shown in Fig. 7.2b. The left side of this figure shows high speed pictures from Piekutowski et al. (1996), of an ogive-nosed steel projectile perforating an aluminum plate at an impact angle of α = 60°. The corresponding simulation results, by Coleau et al. (1998), are shown on the right hand side of Fig. 7.2b. The influence of the asymmetric interaction is clearly evident by the strong deviation Fig. 7.2 a The asymmetric forces exerted by an inclined plate, b experimental and numerical results for an ogive-nosed steel projectile perforating an aluminum plate

7.1 General Description

325

in the projectile’s line of flight. One can also follow the appreciable bending of this projectile in the experiment, which is reproduced by the numerical simulations. These bending moments can induce projectile breakup and even its total shattering, as will be demonstrated in this chapter. The asymmetric forces can induce several defeat mechanisms which have to be well understood in order to optimize their effects for armor designs. Thick inclined plates can induce ricochet of long rods while thin plates of high hardness steel can totally shatter the hard cores of AP projectiles. The reactive and active armor concepts enhance these defeat mechanisms by adding velocity to the inclined plates, which prolongs the asymmetric interaction and enhances the protective effectiveness of these devices. Much of the published work on asymmetric interactions was reviewed by Goldsmith (1999) and additional examples are given in Held (2005) and in Rosenberg et al. (2009). There are two armor configurations which induce asymmetric interactions even though the armor itself is not inclined to the projectile’s trajectory. The perforated armor plate which was suggested by Ben-Moshe et al. (1986), is based on evenly drilled holes in a high hardness steel plate. The diameter of the holes and their spacing are so designed that the projectile always impacts an area which includes some part of a hole. As a result, the asymmetric forces on the AP projectile either break its hard core or deflect its path considerably. These perforated plates are very effective in reducing the penetration capability of the AP projectile, as demonstrated by Heritier et al. (2010). They used steel plates with hardness of 600 Brinell which were perforated to about 50% of their area, in order to stop AP projectiles up to 14.5 mm in diameter. In addition to the high ballistic efficiency of these plates they demonstrated an excellent multiple-hit capability, with plates which were impacted by twenty shots with no loss of their performances. Another concept is based on ceramic spheres which induce a significant asymmetric interaction with AP projectiles (except for impacts at their exact center), as was first suggested by Yeshurun and Ziv (1992). Rosenberg et al. (1996) demonstrated the effect of a spherical surface by using L/D = 10 ceramic rods (D = 10 mm) which were glued together side by side. The front parts of the impacting 0.3 AP steel cores were shattered and their unbroken residuals acquired a significant angular velocity. These experiments demonstrated that spherical surfaces of hard ceramics are very effective in defeating AP projectiles. Obviously, the diameter of the ceramic spheres (or cylinders), should correspond to the size of the AP projectile in a practical armor design. Other workers replaced the ceramic spheres with L/D = 1 cylinders having a hexagonal cross section and a spherical cap. These closely packed cylinders have a better coverage of the area, with no holes around their contact points. These designs have an improved multiple hit capability as compared with flat ceramic tiles, since only a small area of the target around the impact point is destroyed at each shot. These innovations were designed for personnel protection vests, where the ceramics are usually backed by woven fabrics made of aramide (Kevlar® ) or high-strength polyethylene laminates (Spectra® and Dyneema® ). The following sections discuss the main defeat mechanisms which are due to inclined plates, both static and moving. Each section deals with specific threats

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according to their classification as AP projectiles, long rods, and shaped charge jets. Some of the issues will be presented in a more detailed manner because they have an added tutorial value. These examples emphasize the benefits from combining experiments, numerical simulations, and analytical models in order to highlight the physics behind such interactions.

7.2 Defeating AP Projectiles The asymmetric bending forces exerted by inclined high strength plates on AP projectile can cause their total shattering, as was shown by Yeshurun and Rosenberg (1993). They found that a relatively thin high-hardness steel plate, inclined at about α = 45°, can totally shatter the hard steel cores of 0.5 and 14.5 mm AP projectiles. These brittle cores are made of very hard steel or tungsten-carbide and they shatter at a certain threshold obliquity, which depends on the plate’s hardness and thickness. A series of X-ray shadowgraphs for impacts of 0.5 AP projectiles on a 4.4 mm hard steel plate, is shown in Fig. 7.3. Fig. 7.3 The defeat of 0.5 AP projectile by an inclined high-strength steel plate

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The hard steel core emerged intact for an impact inclination of α = 70° and, as expected, its axis was diverted towards the normal to the plate. For an impact at α = 60° the core broke up into two large pieces in addition to its deflection. The most interesting results are shown in the lower shadowgraphs, for α = 45° and α = 30°, where the core emerged totally shattered. Note that the symmetry axis of the shattered projectiles retained its original line of flight, indicating that the shattering process occurred during the early stages of perforation. Yeshurun and Rosenberg (1993) found that a high hardness steel plate, 4.4 mm thick, induces this shattering effects at a threshold angle of about α = 40° for 14.5 mm projectiles (both for steel and WC cores) and at α = 50° for the 0.5 steel core. The residual penetrations of these shattered projectiles into thick witness blocks placed some distance away, were reduced by 80% as compared with the penetration depths of intact projectiles. As expected, softer steel plates did not induce shattering to these cores at all inclination angles. Also, ball projectiles which have soft cores do not experience these shattering effects and the inclined plates are much less effective against them. In order to gain more insight into these effects Rosenberg et al. (2009) performed numerical simulations with the AUTODYN 3D code. Such simulations can save much experimental effort for practical design purposes, by changing the values of the relevant parameters over a large range instead of performing expensive tests. In order to reproduce the experimental values for the threshold obliquities, the tensile strength of the projectile material in these simulations had to be lower than its compressive strength. This brittle behavior is in accord with actual properties of the hard cores in these projectiles, having hardness of about 60RC . Figure 7.4 shows simulation results for a 14.5 mm core impacting a 4 mm high-hardness (HH) plate at two obliquities. One can clearly see that the basic features in the experimental shadowgraphs of Fig. 7.3 are reproduced in these simulations, namely, the total shattering at an inclination of α = 30° and the moderate core breaking at α = 60°. Fig. 7.4 Simulation results for a hard steel core impacting a high-hardness steel plate

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Paris et al. (2013) expanded on this issue by following the disintegration process of these projectiles at later times. Their main assumption was that due to the spinning of the projectiles, their fragments should disperse significantly as they move away from the inclined plate. This was, indeed, the case with 14.5 mm projectiles impacting hard steel plates of different thicknesses. Figure 7.5 shows shadowgraphs at 108 and 540 μs after impact at an inclination of α = 45°. One can see the amount of dispersion in the projectile’s debris cloud at the later time, due to the spinning of the projectiles. The effect of plate thickness on the amount of damage induced in the projectile is clearly seen in these shadowgraphs. Various defeat mechanisms of AP projectiles impacting inclined metallic plates have been identified by the systematic work of Recht and Ipson (1962). They varied the impact velocity and the obliquity for several projectile/plate combinations, and summarized the results in graphical representations such as the one shown in Fig. 7.6. The different areas in this “configuration map” and their corresponding borders, identify the conditions for a given defeat mechanism which can aide the armor designer. One should note that in order to derive this configuration map, Recht and Ipson (1962) had to perform a large number of experiments, measure impact velocities very accurately and recover the residual projectiles after each shot. This is a very costly and ineffective way to follow these interactions and 3D simulations should be used, after a proper calibration, in order to reduce the number of experiments which are needed to construct such maps.

Fig. 7.5 Shadowgraphs at 108 and 540 μs showing the dispersion of projectile’s debris cloud

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Fig. 7.6 The configuration map for 0.3 AP projectiles impacting a 6.35 mm thick 2024 aluminum plate

A ricocheting projectile exits the inclined plate at an angle which is very close to its incidence angle, creating a rather long and shallow crater on the surface of the plate. As reviewed by Goldsmith (1999), the strength of the target is its most important property for the ricochet process of AP projectiles. In a series of experiments with inclined polymeric plates Yeshurun et al. (2001) found that Plexiglas plates can induce both ricochet and an appreciable deflection to 0.3 AP projectiles at obliquities which can be used for practical armor designs. Other polymeric materials were tested in this study, such as polycarbonate, PVC, and an epoxy resin but only the Plexiglas plates induced the enhanced ricochet which is shown in Fig. 7.7. These shadowgraphs Fig. 7.7 The enhanced ricochet of a 0.3 AP projectile from an inclined Plexiglas target

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follow an 0.3 AP projectile impacting a 20 mm thick Plexiglas plate at an angle of α = 30°, and emerging perpendicularly to its original line of flight. This enhanced deflection was reproduced with 3D simulations by Rosenberg et al. (2005) who performed a parametric study of this interaction by varying the relevant properties of the polymeric plate. The simulations showed that this enhanced ricochet is due to the combination of high compressive strength and low tensile strength of the brittle Plexiglas plate. The compressive strength of Plexiglas under high strain rates can reach a value of about 0.4 GPa, while its dynamic tensile strength (spall) is an order of magnitude lower. By assigning such values to the Plexiglas plates in their simulations, Rosenberg et al. (2005) reproduced the main features observed in these experiments. The basic mechanism which operates during this interaction can be understood by following the state of the plate material surrounding the projectile, as shown in Fig. 7.8a. During the early stages of penetration the target material above the projectile’s trajectory is failing heavily, due to the tensile stresses induced by the expansion of the free impact surface. In contrast, the target material below the projectile’s trajectory fails to a much less degree, since it is under compression during these stages. Obviously, the failed material cannot exert a resistive force on

Fig. 7.8 a Simulation results for the ricochet process, b the asymmetric force on the projectile

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the projectile and it is only the material below the projectile which can affect its trajectory, as shown in Fig. 7.8b. Thus, the upturning of the projectile inside the target is the result of the different states attained by the brittle Plexiglas, above and below the projectile’s trajectory. In order to appreciate the importance of the material properties for this interaction, consider the experimental result and the numerical simulations, from Rosenberg et al. (2005), for an impact on a 20 mm polycarbonate (PC) plate at the same inclination of α = 30°, as shown in Fig. 7.9. The 0.3 AP projectile perforated the PC plate with the expected change in its flight direction. In the corresponding simulation both the compressive and tensile strengths were set at 0.1 GPa, which are typical values for polycarbonate. The simulation snapshot in Fig. 7.9 corresponds to the early time (t 1 ) in the shadowgraphs on the left. In order to validate their interpretation for the enhanced ricochet, Rosenberg et al. (2005) performed a special experiment and a corresponding numerical simulation. This is a “validation test” which is an important step whenever one wishes to enhance the validity of a phenomenological model. A rectangular slot, 20 mm deep, was cut along the front face of a 40 mm thick Plexiglas target for about half of its length. The slot was somewhat wider than the projectile’s diameter in order to avoid its impact with the walls of the slot, as shown in Fig. 7.10a. The projectile impacted at the end of this slot, so that its subsequent penetration path was totally encapsulated by the target. In essence, the free impact face was actually removed from the process and the forces on the projectile had to be more symmetric, according to the mechanism proposed by Rosenberg et al. (2005). This impact was expected to result in a complete perforation of the slotted Plexiglas plate which was, indeed, the case in both the experiment and the corresponding numerical simulation, as shown in Fig. 7.10. The simulation shows that the damaged volume in the Plexiglas around the penetrating projectile is much more symmetric, as compared to the shape of the damaged material in Fig. 7.8. The position and orientation of the projectile in the experiment and in the simulation, some 190 μs after impact, are very similar to each other. The resulting perforation of the plate strongly enhances the interpretation for the enhanced ricochet, as described above. We wish to emphasize the importance of such experiments and numerical simulations as validity tests. These tests should be performed in order to enhance the

Fig. 7.9 Experimental shadowgraphs (left) and the simulation result (right), for an inclined impact of 0.3 AP projectile at a polycarbonate plate

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Fig. 7.10 a The slotted target experiment, b the corresponding simulation, c the experimental shadowgraph

validity of the physical interpretation for a given phenomenon. We recommend that such validity tests be performed outside the range of the experiments which were performed during the course of the study, and they should be considered as “critical” in the sense that their results can either support or negate the proposed model.

7.3 Defeating Long Rods 7.3.1 Perforation of Inclined Plates The effects of the asymmetrical forces exerted by an inclined plate on an eroding long rod, are similar to those discussed above for rigid projectiles. This is clearly seen in Fig. 7.11, from Hohler et al. (1978), which shows three shadowgraph of an L/D = 10, D = 5.8 mm WHA rod impacting an 8 mm thick inclined steel plate. The nose of the emerging rod is significantly bent, as seen here, and often it is broken into several pieces. Rods which do not break at their front end may emerge with a larger deflection due to the prolonged action of the asymmetric forces. A deflected rod, as seen in Fig. 7.11, will subsequently impact a second target at a large yaw angle resulting in a significant reduction of its penetration capability. This is the main defeat mechanism achieved by inclined stationary plates against eroding long

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Fig. 7.11 Shadowgraphs of a WHA rod perforating an inclined steel plate

rods. As discussed in Chap. 5, the residual lengths and velocities of eroding rods perforating inclined plates, follow the similar variations with plate thickness as for normal impacts. Thus, as far as these parameters are concerned, obliquities do not enhance the ballistic efficiency of inclined plates by simply adding more material in the path of the rod. It is the asymmetry of the forces which results in rod rotation, as well as its nose bending and breaking, which enhance the ballistic efficiency of inclined plates.

7.3.2 Ricochet of Long Rods The thick and highly sloped fronts of main battle tanks are designed to offer maximal protection for the crew, by exerting asymmetrical forces on the penetrators in order to induce their ricochet. The ricochet of compact projectiles has been investigated by many workers, as reviewed by Goldsmith (1999), while the issue of long-rod ricochet received much less attention. Tate (1979) treated the ricochet process of a rigid rod which rotates around its center of mass, by considering the asymmetrical forces on its nose at the impact face. While the rod tip is locally deformed according to the model, ricochet occurs only if the rod is rigidly rotated so as to produce a net velocity of the rod tip parallel to the surface of the target. The geometry of the deformed rod tip and the asymmetric force acting on its nose, according to Tate (1979), are shown in Fig. 7.12. Note that the obliquity angle (β) is defined as the angle between the rod’s axis and the normal to the target. Fig. 7.12 The asymmetric force on the rod tip, according to Tate’s model

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Tate’s ricochet model is based on the modified Bernoulli equation from the AT model, which relates the penetration velocity (U) with the impact velocity (V ), and with the strength parameters of the rod (Y p ) and the target (Rt ). The force (F) which the target exerts on the rod is calculated through multiplying Y p by the tip area on which it is applied (the eroded area of the rod). The component of this force which is perpendicular to the axis of the rod is multiplied by half the length of the rod, in order to obtain the moment of this force about its centre of gravity. This moment induces a rotation rate which results in the following ricochet condition for the obliquity angle (β): 2ρ p V 2 · tan β > 3Y p 3



L D + D L

 ·

V V −U

(7.1)

The main difficulty with this model is that it treats the rod as a rigid body. Experimental observations showed that long rods are considerably deformed during inclined impacts, forming a plastic hinge around their contact area with the target. The existence of such a plastic hinge was observed by Senf et al. (1981), with L/D = 10 mild steel rods ricocheting from RHA plates. Figure 7.13 shows such a ricochet at an impact velocity of 968 m/s and an obliquity of β = 75°. The plastic hinge at the interaction area is clearly seen in the high speed pictures and in the numerical simulations performed by Senf et al. (1981). In the ricochet model of Rosenberg et al. (1989) the asymmetric force (F) from Tate’s model, is assumed to act only upon the mass of the rod which is engaged by the target. This alteration was made in order to account for the plastic hinge observed during rod ricochet. Their analysis resulted in the following ricochet criterion for the minimum obliquity angle for ricochet: tan2 β >

  ρpV 2 V + U Rt V −U

Fig. 7.13 Experimental and simulation results for the ricochet of a mild steel rod

(7.2)

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which is quite different than the ricochet condition of Tate’s model (7.1). The new expression includes the target resistance (Rt ) as the controlling strength parameter, rather than the rod strength (Y p ) in Tate’s model. Also, the length of the rod does not appear in this model since it concentrates on the plastic hinge rather than on the rod’s moment of inertia. One can define the critical angle for ricochet β c as the value of β which is obtained when the inequality (7.2) is replaced by the corresponding equality. Thus, the ricochet condition of this model can be described by the following curve in the (β, V ) plane:   ρpV 2 V + U tan βc = Rt V −U 2

(7.3)

The model of Rosenberg et al. (1989) was shown to account for their experimental results with L/D = 10 WHA rods impacting armor steel targets, at velocities in the range of 0.65–1.3 km/s and obliquities of β = 55°–75°, as shown in Fig. 7.14a. The plastic hinge in the ricocheting rod is clearly seen in the three consecutive shadowgraphs shown in Fig. 7.14b, from one of these experiments. Fig. 7.14 a Comparing the ricochet model with experiments for WHA rods and RHA targets, b shadowgraphs following the ricochet in one of these experiments

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Rosenberg et al. (2007) further developed these ideas, and found that an interesting situation arises when the impact velocity is lowered until it reaches the critical impact velocity (V c ) for the penetration of a given rod/target combination. At this impact velocity the rod does not penetrate the target (U = 0), and the following expression is obtained for the minimal value of the ricochet threshold obliquity (β min c ) from (7.3): tan

2

βcmin

  ρ p Vc2 Yp at V0 = Vc = =2 1− Rt Rt

(7.4)

where use was made of the relation V c = [2(Rt − Y p )/ρ p ]0.5 according to the AT model. This result means that for every rod/target combination there is a minimum obliquity angle (β min c ), which corresponds to the case of no penetration (V 0 = V c or U = 0). Thus, the ricochet curve in the (β, V ) plane should start at the point (β min c , V c ). This minimal obliquity depends only on the rod/target strength ratio which, for typical values of Y p and Rt is equal to about 1/3, resulting in a minimal obliquity = 49°. Another interesting result is obtained for the case of ρ p = ρ t , which of β min c results in the following ricochet curve: tan2 βc = 2z 2 ·

  Yp 3z 2 − 1 · 1 − z2 + 1 Rt

where z = V0 /Vc

(7.5)

This ricochet criterion accounts for the data of Johnson et al. (1982) with plasticine rods impacting plasticine targets, as shown in Fig. 7.15. The relevant material parameters were taken from Johnson et al. (1982), together with the value of V c = 45 m/s for the critical velocity of this rod/target combination. The aim of Johnson et al. (1982) was to demonstrate that soft materials with low densities, such as plasticine, behave in a similar way to metals when their corresponding damage number (J) is kept the same. The damage number was defined by Johnson (1972) as: Fig. 7.15 Comparing the ricochet model with experimental results for plasticine rods impacting plasticine targets

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J=

ρt V02 Yd

(7.6)

where ρ t and Y d are the density and the dynamic strength of the target, respectively, and V 0 is the impact velocity. This non-dimensional number is commonly used to classify the different regimes of projectile impacts and their severity, as far as the damage to the target and the projectile is concerned. The work of Johnson et al. (1982) on plasticine rods and targets demonstrated the validity of using this nondimensional parameter, as far as rod deformation and their penetration depths are concerned. The 3D numerical simulations of Rosenberg et al. (2007) resulted in ricochet thresholds for WHA rods impacting armor steel targets, at obliquities which agreed with the model’s predictions. Figure 7.16 shows two of these simulation results for an L/D = 20 rod with strength of 1.2 GPa, impacting armor steel targets at a velocity of 1.45 km/s and obliquities of β = 74° and 80°. The difference between the two cases is manifested as the rod penetrates the target for an impact obliquity of β = 74°, while it is only gouging its surface for the β = 80° impact. Up to this point, the analysis and the experiments dealt with very thick targets ignoring back surface effects. These effects, which are important for the interaction with thin plates, were investigated by Lee et al. (2002) both experimentally and numerically. They found good agreement between their simulations and experiments for L/D = 10.7 WHA rods, with D = 7 mm, impacting 6.25 mm thick plates of armor steel at velocities in the range of 1.0–2.0 km/s. Figure 7.17 shows simulation results for an impact velocity of 1.0 km/s and obliquities of β = 76° and 80°. The impact at

Fig. 7.16 Simulation results for long rod ricochet

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Fig. 7.17 Simulation results for a long WHA rod impacting a steel plate at V 0 = 1.0 km/s

β = 80° shows all the features of a ricochet, with a plastic hinge in the rod and little damage to the target. In contrast, for the β = 76° impact the rod seems to ricochet at the early stages of the interaction, but it penetrates the target at later times due to the effect of the back surface. Obviously, the back surfaces of thin plates affect their threshold ricochet obliquities (β c ), which are expected to be larger than those for a semi-infinite target. Lee et al. (2002) compared their simulation results for β c with those predicted from the model of Rosenberg et al. (1989), as shown in Fig. 7.18. The two curves have a similar shape throughout the whole velocity range in this study, and the values of β c for the thin plates are higher than those predicted by the model, as expected. However, the differences in the threshold obliquities amount to 3° only. Fig. 7.18 Comparing simulation results for the ricochet thresholds from thin plates, with the analytical model for thick targets

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7.3.3 The Interaction of Long Rods with Moving Plates The work described above demonstrated the strong dependence of the threshold obliquities for ricochet on the impact velocity of the rod. Obviously, it is easier to induce these ricochets if the plate is propelled in the direction of rod’s velocity, as a result of their reduced relative velocity. This is one of the basic mechanisms by which a reactive armor design can be used in order to defeat long rod penetrators. These add-on armor configurations which will be further discussed in Sect. 7.3, have two metallic plates with a layer of explosive between them. The explosive layer is initiated by the impact of the rod and the explosive products propel the two plates in opposite directions. The back plate, which moves in the same direction as the rod, can induce its ricochet for a larger range of inclinations, due to their low relative velocity. This effect was demonstrated by Rosenberg et al. (2009) who used an explosive layer to propel an armor steel plate to a velocity of 270 m/s. The impact of a WHA rod at a velocity of 1.0 km/s and obliquity of β = 63°, resulted in its ricochet as shown in Fig. 7.19. Such an impact would have resulted in a complete penetration of a stationary plate. Thus, one can conclude that add-on armor designs which are based on moving plates can reduce the lethality of long rods, as they do with shaped charge jets. The potential to defeat long rods with moving plates was demonstrated by Rosenberg and Dekel (2004). They presented results from a series of experiments and numerical simulations, with WHA rods impacting moving RHA plates, as shown in Fig. 7.20. The impact angles in these experiments were varied between α = 25° and α = 40°, and the nominal velocity of the rods was V p = 1.4 km/s. The 4.3 mm thick armor steel plates had a nominal velocity of V t = 430 m/s. We describe this work in detail because it is an excellent example for a study where experiments, numerical simulations, and analytical modeling are combined in order to understand the basic physics of a given phenomenon, such as the defeat mechanism in this case. In each experiment the rod/plate interaction was followed by three X-ray shadowgraphs at t = 30, 170 and 310 μs after impact. These times were chosen to follow the rod closely after impact, at about the middle of the interaction process and near its end, respectively. This set of experiments resulted in rod disruptions which were Fig. 7.19 The ricochet of a long WHA rod from a moving steel plate

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Fig. 7.20 The experimental set up for the interaction of a long rod with a moving plate

either mild or very severe. The most important observation was that all the severe disruptions took place in experiments with impact angles of α ≤ 35°, while for all the experiments with α ≥ 40° the rods experienced only a mild disruption. The different modes of rod disruption are clearly seen in Fig. 7.21, through the three shadowgraphs of experiments which resulted in the severe disruption at α = 35°, and the mild disruption at α = 40°. The plates in these shadowgraphs seem to be rather thick because of the parallax effect which takes place when the X-ray tube is not aligned optimally with the plate. The mild disruption resulted in a residual rod which was broken into several pieces, but its penetration capability was very close to that of an intact rod. In contrast, the penetration capability of the severely disrupted rod was very small. Clearly, a very efficient disruption mechanism is initiated at a certain impact angle, somewhere between α = 35° and α = 40°. In order to understand this result, Rosenberg and Dekel (2004) performed 3D numerical simulations of these impacts for plate inclinations of α = 30° and α = 40°. These simulations reproduced the experimental results quite successfully, as shown in Fig. 7.22, enhancing the claim that the onset of severe disruption has a real physical basis.

Fig. 7.21 Rod/plate interaction at α = 40° (mild disruption) and α = 35° (severe disruption)

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Fig. 7.22 Simulation results for the impacts at α = 30° (left), and at α = 40° (right)

In order to emphasize the excellent agreement between simulations and experimental results, consider the latest “snapshots” for the two cases discussed above, as shown in Fig. 7.23. One can see that the simulations reproduce all the fine details which have been observed experimentally. This is an important issue if one wishes to draw practical conclusions from such simulations, as will be described next. A close examination of the simulation results in Fig. 7.23, reveals a basic difference in the rod/plate interaction between the two impacts. This difference is manifested by the orientation of the steel strip which is sheared off the moving plate by its continuous interaction with the rod. Once we noticed these strips in the simulations we “found” them also in some of the shadowgraphs, as seen on the upper left side of

Fig. 7.23 The latest snapshots comparing experiments (top) and simulations (bottom), for the two impacts at different obliquities

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Fig. 7.24 The different orientations of the steel strips for the two inclinations and the shape of the recovered steel plates

Fig. 7.23. The simulation “snapshots” in Fig. 7.23 show that this strip is either pushed in the direction of plate movement or forced down towards the rod. This important observation is shown schematically in Fig. 7.24, in order to highlight the main cause for the difference between the two interactions. The figure also shows a schematic description of the recovered plates, which had a long slot along the interaction area with the projectiles. The width of these slots was exactly equal to the rod’s diameter, as expected. The severe disruption of the rod is caused by a strip which is being pushed backwards, as in the left side of the figure, while the minor disruption is caused by a strip which is pushed downwards as shown in the right side of the figure. Both the experimental shadowgraphs and the simulation results for the α = 30° impact, show that the rod material is dispersed around its interaction point as it continuously shears the strip. In fact, the rod does not penetrate the moving plate in the sense that its debris remain on the same side of the strip throughout the interaction. On the other hand, for an impact at α = 40° the rod penetrates the plate as it pushes the strip downwards. Thus, the substantial disruption of the rod is, apparently, related to the issue of rod/target penetration threshold. Note that the different directions of these strips could not be seen in the X-ray shadowgraphs because of the parallax mentioned above. Thus, here we have an excellent example where the most important issue concerning the source of this disruption mechanism, was actually unveiled by the numerical simulations. The analytical model presented by Rosenberg and Dekel (2004) considers the relative motion of the rod and the plate, as shown schematically in Fig. 7.25, depicting the relevant velocities in this interaction. Particularly, the component V t /sin α of the plate’s velocity plays the major role as it accounts for the relative velocity between the rod and the moving interaction point in the plate. The plate is assumed to move in a perpendicular direction to its face at a velocity V t . The most relevant velocity for this interaction is the relative velocity between the rod and the moving interaction point, as given by: Vrel = V p −

Vt sin α

(7.7)

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Fig. 7.25 A schematic description of the relative rod/plate velocity

In order to achieve a substantial disruption to the rod, this velocity has to be smaller than the threshold velocity for penetration (V c ) of the given rod/target pair. The condition V rel < V c for severe rod disruption leads to the following inequality:   Vc Vt · sin α > 1− Vp Vp

(7.8a)

A second constraint on the relative velocity is that it should be positive, V rel > 0, in order to obtain a continued interaction between the rod and the target. This is a purely geometrical constraint which leads to: Vt < sin α Vp

(7.8b)

Combining the two constraints results in the following conditions for the strong disruption process described above: sin α >

  Vc Vt · sin α > 1− Vp Vp

(7.9)

where the critical velocity is given by the AT model as V c = [2(Rt − Y p )/ρ p ]0.5 . It is convenient to present these constraints as two curves in the “disruption plane,” with axes which correspond to V t /V p and α, as defined by Rosenberg and Dekel (2004). The two inequalities (7.9) define the “disruption zone” which includes all of their experiments resulting in severe disruption for the rod. The experimental results given in Chap. 5 for WHA rods impacting RHA targets, show that V c = 640 m/s for this rod/target pair. For the experiments described here with a rod velocity of V p = 1.4 km/s, one obtains (1 − V c /V p ) = 0.54, which should be inserted in the right hand side of inequality (7.9). The graphical representation of these inequalities is shown in Fig. 7.26, which includes the experimental results of Rosenberg and Dekel (2004), in terms of severe rod disruption (filled circles) and minor disruptions (open circles). The borderline between the two disruption modes, as predicted by the model, passes between the two sets of experiments. Considering the simple assumptions behind the model, this is an excellent agreement between a phenomenological model and the experimental results. The shaded area in this figure, the “disruption zone,” is of high importance to the armor designer. Clearly, a larger disruption zone should

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Fig. 7.26 The disruption zone and the experimental results for rod interaction with a moving plate

result in severe rod disruptions at lower plate velocities, and for a wider range of plate inclinations. A larger disruption zone can be obtained by using higher strength plates with higher critical velocities (V c ), resulting in lower values for the term (1 − V c /V p ) on the right hand side of inequality (7.9). Thus, the lower borderline curve in Fig. 7.26 is expected to be even lower for a plate with a higher strength, resulting in a larger disruption zone. In order to demonstrate this issue Rosenberg and Dekel (2004) repeated the α = 40° experiment with a high hardness (HH) steel plate, instead of the RHA plates which they used in the other tests. Note that the impact at α = 40° with the RHA plate resulted in a minor disruption to the rod, as shown above. The critical penetration velocity for the WHA rod impacting an HH steel target was estimated as V c = 720 m/s through the experimental data presented in Chap. 5. The increase of V c from 640 to 720 m/s redefines the low-border of the disruption zone as: V t /V p = 0.486 sin α. The larger disruption zone was expected to encompass the two experiments, marked by the open circles in Fig. 7.26, which resulted in a minor disruption. Consequently, a significant disruption was expected for the α = 40° experiment with this HH steel plate which was, in fact, the result obtained in this experiment, as is clearly seen in Fig. 7.27. Thus, with stronger steel plates the

Fig. 7.27 Rod disruption by the HH steel plate at an angle of α = 40°

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Fig. 7.28 Simulation results for rod disruption as a function of plate velocity and thickness

disruption zone can be expanded to include a larger range of impact velocities and inclination angles. The critical velocity for penetration (V c ) was defined by the AT model for long rods impacting semi-infinite targets. It is unclear whether one can use the same values of V c for the process of rod disruption by finite thickness plates. This issue was explored by Rosenberg and Dekel (2005) through numerical simulations in which the thickness and the velocity of the plate were systematically varied. The simulations resulted in either a strong disruption or a mild one, as observed before, but the border line between the two cases shifted systematically with plate thickness. Simulation results for WHA rods impacting armor steel plates of different thickness, at an inclination of α = 35°, are shown in Fig. 7.28. The rod’s velocity in these simulations was 1.4 km/s and the velocity of the plates ranged between 390 and 520 m/s. The triangles in this figure correspond to minor rod disruptions, the squares to severe disruptions, and the crosses represent borderline cases. These simulations show that the plate velocities which are needed to induce a meaningful disruption to the rod are higher for thinner plates as expected, due to the enhanced influence of their back surface. For example, in order to induce a severe disruption the velocity of a 3.0 mm thick plate should be at least 500 m/s, according to these simulations. In order to validate these results Rosenberg and Dekel (2005) performed an additional experiment, with a WHA rod impacting a 3.0 mm RHA plate at V 0 = 1.4 km/s and at α = 35°. The velocity of the plate was 530 m/s which, according to these simulations, should have resulted in a strong disruption to the rod. This was, in fact, the result of this validation experiment as shown in Fig. 7.29. We elaborated on this issue of rod disruption by a moving plate, because it is an excellent example for the benefits gained by combining all the available tools for a given problem in terminal ballistics. At first, a rather surprising difference for rod disruption was found in two experiments which differed by only 5° in their plate obliquities. The 3D numerical simulations revealed that this difference is due to the

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Fig. 7.29 The three shadowgraphs in the experiment with the 3.0 mm plate moving at 530 m/s

different directions by which the steel strips which interact with the rod are sheared off the plates. This realization led to a phenomenological model for the interaction of rods with moving plates, which is based on the critical velocity for penetration (V c ) of the given rod/target combination. The model was able to account for all the experimental data and its validity was enhanced by additional experiments which were performed as validation tests. These rod/plate interactions are important elements for the design of reactive armors, which is based on plates propelled by an explosive layer. As will be discussed in Sect. 7.3, the explosive reactive armor configuration (ERA) is very effective against shaped charge jets and it has been applied as add-on armor to various armored vehicles. On the other hand, reactive armors are still in their early stage of research as far as long rods are concerned. Held (2005) analyzed the various defeat mechanisms which can take place by the combined effects of the front and rear plates in a heavy ERA configuration. According to his results, with a proper design of the ERA cassette, the penetration capability of the long rod can be reduced significantly. The difference between the disruption potential of the two plates, which move in opposite directions in a reactive armor design, has been highlighted by Shin and Yoo (2003). They studied the impact of L/D = 15, D = 5 mm WHA rods with inclined steel plate, 5 mm thick, moving at different velocities either towards the rod or away from it. Impact velocities of the rods in these simulations were 1.5 and 2.5 km/s and the impact angle was α = 30°. The simulations clearly showed that the disruption by the plate which moves towards the rod is very small and is, in fact, similar to that caused by a stationary plate. In contrast, plates which move away from the rod can induce a significant disruption even when their velocities are as low as 200 m/s. A similar numerical study was performed by Paik et al. (2007) for L/D = 15 and 30, D = 5 mm WHA rods interacting with 2.5 mm thick steel plates of a reactive cassette. The impact velocities were 1.5 and 2.5 km/s in these simulations, the inclination angle α = 30°, and the plate velocities were varied between 0.2 and 0.5 km/s. The differential efficiencies of these cassettes, as calculated by the residual penetrations into a thick steel witness block, increased with plate velocity and were significantly higher against the longer rods. At plate velocities of 0.3–0.5 km/s, these efficiencies attained values in the range of η = 3–6, and the higher efficiencies were obtained for the lower velocity rods. The shapes of the disrupted rods in these simulations are similar to

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those observed in the experiments of Held (2005), enhancing his observations for the various defeat mechanisms which take place during these interactions.

7.3.4 The Impact of Yawed Rods One of the more complex issues in the terminal ballistics of long rods concerns the penetration process of a yawed rod, especially at large obliquities. We shall first discuss the impact of yawed rods at normal impacts and then consider oblique yawed impacts. The yaw of a rod is defined by the angle between its longitudinal axis and the velocity vector of its center of mass, as shown schematically in Fig. 7.30. There are two orthogonal attitudes which define the flight characteristics of the rod, the pitch (θ 1 ) and the yaw (θ 2 ). The equivalent attitude angle (θ ) is defined by the relation: tan2 θ = tan2 θ 1 + tan2 θ 2 . The combined yaw and pitch angles is referred to as the attack angle. The data gathered on the impact of yawed rods shows that there is a narrow range for which their penetration depths are not affected significantly by the yaw angle, and that this range is smaller for rods with larger aspect ratios (L/D). Silsby et al. (1983) realized that the penetration capability of a long rod is not affected by yaw as long as its rear portion does not impact the wall of the crater. This geometrical constraint defines a threshold value, commonly referred to as the critical yaw (θ c ), according to: sin θc =

Dc − D p 2L

(7.10)

where Dc and Dp are the diameters of the crater and the rod, respectively. The values of θ c for L/D = 10–20 rods fall in the range of 1.5°–3°. Thus, an add-on armor system which induces yaw angles of several degrees to the impacting rod can be a very effective design. It is clear that the crater diameter plays an important role in this issue and, as described by Bierke et al. (1992), there is a wealth of different interactions between the rod and the crater walls for impacts at increasing yaw angles. Fig. 7.30 Schematics of a yawed rod impact

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Fig. 7.31 The crater shape and rod debris for a moderate yaw impact

For example, at striking yaws just above θ c there is evidence for the gouging of the rod, through the depressions on one side of the crater wall. With increasing yaw angles there is evidence for a rebound of the rod’s rear part through the depressions on the opposite side of the wall. Thus, the rod’s rear portion is bouncing from side to side during its penetration, resulting in an enhanced crater diameter and a marked reduction in penetration depth. A typical shape of the crater at moderate yaw angles is shown in Fig. 7.31, from Bierke et al. (1992). Gee and Littlefield (2001) performed a series of 3D numerical simulations in which they investigated the penetration process of L/D = 10 and 30 WHA rods, impacting armor steel targets at velocities of 1.4, 2.1 and 2.6 km/s. They observed the gouging phenomenon of the rod’s tail, and its subsequent realignment following the interaction with the crater walls. They checked the possibility to normalize the numerical results by presenting them in terms of P/Pn as a function of θ /θ c , where Pn is the penetration depth at normal impact. The need for a normalized presentation is evident when one compares their simulation results for L/D = 10 and 30 rods, impacting a steel targets at 2.1 km/s and at a yaw angle of θ = 30°, as seen in Fig. 7.32. Obviously, the crater shapes are very different for these impacts which, according to Gee and Littlefield (2001), is the result of the different values for the normalized yaw angles: θ /θ c = 8.8 and 30, for the L/D = 10 and 30 rods, respectively. The normalization scheme was successful for yaw angles in the range of θ /θ c < 10, while for higher yaw angles the normalized results for the L/D = 10 and 30 rods differed by a considerable amount. The experimental works of Bless et al. (1978), Yaziv et al. (1992), Bukharev and Zhurkov (1995) and Hohler and Behner (1999) showed that the penetration depths of yawed rods decrease drastically with yaw angles when these are greater than the corresponding critical values. Several phenomenological models were suggested in these works which account for the data quite well, as far as the strong dependence of penetration on yaw is concerned. However, these models are applicable only for the specific ranges of materials, impact velocities, and rod aspect ratios for which they were calibrated. These models realize that for yawed impacts near the critical angle, the penetration depth decreases according to the change in the rod’s effective length, which is proportional to L cos θ . On the other hand, for large yaw angles the penetration depth is influenced by the rod diameter and it is proportional to Dp /sin θ .

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Fig. 7.32 Simulations for L/D = 30 (up) and L/D = 10 (down) rods, impacting at V 0 = 2.1 km/s and θ = 30°

A different approach was taken by Lee and Bless (1998) who considered the yawed rod as comprised of a series of cylindrical disks. They assumed that the degradation in the penetration capability of such a disk is proportional to the portion of its diameter which overlapps the crater edge. The diameter of the crater in this model was determined by a momentum balance principle which was derived by Lee and Bless (1996). They used the AT model to calculate the penetration depth of the rod’s front part until sidewall grazing occurs. Good agreement was found between their model’s predictions and the experimental results of Yaziv et al. (1992), for L/D = 10 WHA rods impacting armor steel targets at 1.4 km/s, and a large range of yaw angles. Rosenberg et al. (2006) used a similar approach which was based on dividing the rod into n segments, each of length . They treated separately the segments which penetrate without gouging the crater wall, and those which impact the wall. The geometry of the model is described in Fig. 7.33. The crater diameter (Dc ) in terms of rod diameter (Dp ) was taken from the empirical fit by Walker et al. (2001), for a large amount of data of WHA rods impacting armor steel targets, as given by: Dc = D p · (1 + 0.7V0 )

(7.11)

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Fig. 7.33 The geometric model for yawed penetration

where the impact velocity (V 0 ) is given in km/s. Another empirical fit by Walker et al. (2001), for the normalized penetrations of these rods, was used in the model in order to account for the L/D effect: L P = 0.65 + 1.06 ln V0 − 0.05 ln L D

(7.12)

where V 0 is given in km/s. This equation accounts for a large range of penetration data, for WHA rods of different L/D values and impact velocities in the range of V 0 = 0.75–1.9 km/s. The next step was to calculate, geometrically, the effective diameters of the elements which do not impact the crater wall. The effective diameter of the nth element can be written as:  if nl tan θ + D p /(2 cos θ ) ≤ Dc /2 D p / cos θ Deff (n) = D p /2 + D p /(2 cos θ) − nl tan θ otherwise (7.13) The maximal number of segments is obtained when the lower corner of the nth segment aligns with the crater side. Thus, N eff is limited by the projectile’s length or diameter, according to: ⎧ ⎪ ⎨

Neff

Dc + D p / cos θ L L if ≥ l cos θ 2 tan θ cos θ = D D + D / cos θ / cos θ + D / cos θ D L ⎪ c p p c p ⎩ if < < . 2l tan θ sin θ 2 tan θ cos θ

(7.14)

The last element to contribute is the one for which its edge hits the crater sidewall. For simplicity, the above equations do not include cases where the rod hits the target nearly sideways (tgθ > L/Dp ), since these angles are larger than 80° even for short rods with L/D = 5. The effective projectile diameter is obtained by averaging the Deff (n) values, from (7.13), up to n = N eff : Deff =

1 Deff (n) Neff n

(7.15)

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Fig. 7.34 Model’s predictions for the effect of yaw on rods with different aspect ratios

and the effective length of the rod is defined by: L eff = Neff · l

(7.16)

These equations for Deff and L eff are then inserted in (7.12), in order to obtain the penetration depth of the yawed rod with its effective aspect ratio of L eff /Deff . Applying this model to WHA rods of different aspect ratios, impacting RHA targets at 2.1 km/s, resulted in the normalized penetration curves shown in Fig. 7.34. Note that the results in this figure are presented in terms of P/D versus yaw angle. Each curve has a flat top which means that there is a certain range of yaw angles with no reduction in the rod’s penetration capability, and this range decreases with increasing L/D, as observed experimentally. The common asymptote of P/D for large yaw values means that at these angles the rod diameter determines its penetration depth, rather than its length Rosenberg et al. (2006) also performed 3D numerical simulations for various sets of published data, in order to check whether a good agreement is obtained between these simulations, their analytical model, and various experimental results. Figure 7.35 shows a series of “snapshots” from a simulation for a WHA rod impacting RHA target at V 0 = 1.8 km/s, with a yaw angle of θ = 60°. It is clear that for this yaw angle the rod is penetrating sideways rather than along its axis. The agreement between the model and these 3D simulations for WHA rods of different L/D values, impacting RHA targets at 1.8 km/s, is shown in Fig. 7.36. The agreement is satisfactory considering the simple assumptions behind the analytical model of Rosenberg et al. (2006), as presented above. The agreement between the model’ predictions and experimental data for WHA rods impacting RHA targets, is shown in Fig. 7.37. The data from Bukharev and Zhurkov (1995) for impacts at 2.1 km/s is shown in the upper part of the figure, and the data from Yaziv et al. (1992) for L/D = 10 rods impacting at 1.4 km/s is shown in the lower part. Note the good agreement between model prediction, experimental data and numerical simulations for this case, as shown in the lower part of Fig. 7.37.

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Fig. 7.35 3D numerical simulation for a yawed WHA rod impact at V 0 = 1.8 km/s and θ = 60°

So far, we discussed the impact of yawed rods on semi-infinite targets where the rod velocity vector is normal to the target’s face. A much more complex interaction takes place when the target is inclined to this vector, involving both yaw and obliquity. Relatively few works have been performed with thick targets impacted obliquely by yawed rods, and the work of Behner et al. (2002) is the most elaborate of these studies. They impacted L/D = 20, D = 6.3 mm, blunt-nosed WHA rods on thick RHA targets which were inclined at α = ±30°, as shown in Fig. 7.38. The impact velocity was 1.65 km/s, yaw angles ranged between 0° and 90°, and the different techniques which were used to reach these yaw angles are also described by Behner et al. (2002). The results of these experiments in terms of P/P0 as a function of yaw angle are shown in Fig. 7.39, where P0 is the penetration depth for a zero-yaw impact. The difference between the two sets of experiments with the α = +30° and α = –30° obliquities, is clearly evident here. Fig. 7.36 Comparison between model predictions and simulation results for WHA rods impacting RHA targets at 1.8 km/s

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Fig. 7.37 Comparing the model’s predictions with experimental data for WHA rods impacting RHA targets

As mentioned above, an add-on armor which induces a yaw of several degrees to an L/D = 20 rod can be a very effective design. However, the high-density long rods which are used in military applications have high moments of inertia, and it is not simple to induce a significant yaw in these rods. One of the more efficient techniques to achieve this goal is to launch a flying plate towards the rod, so that the impact takes place at a distance of several meters away from the protected vehicle. With a proper impulse delivered to the rod a significant yaw rate can be induced in the rod, resulting in its impact at a high enough yaw angles. This is the basic principle behind the “active armor” concept, as discussed by Wollmann et al. (1996), Ernst et al. (1999) and Sterzelmeier et al. (2002), who presented various techniques to launch these plates towards the oncoming rod. Obviously, in order to obtain a significant deflection of the rod, the flying plate should be thick enough and its velocity vector

Fig. 7.38 Yawed rod impact on inclined surfaces

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Fig. 7.39 The effect of combined yaw and obliquity on the penetration depths of long rods

should be inclined to the axis of the rod. The continued action of this plate as it sweeps across the rod in a grazing mode, can induce a significant rotation of the rod and even its breakup to several pieces. Cagliostro et al. (1990) realized that a similar grazing action along the rod’s shank is obtained by its oblique impact on a stationary plate, at a certain yaw angle. Bless et al. (1999) and Liden et al. (2006) used this technique in order to induce a continuous “grazing interaction” between an inclined plate and the yawed rod. In order to follow these interactions, consider the three situations described in Fig. 7.40, from Cagliostro et al. (1990), which define the positive and negative yaw angles. The yaw angle (θ ) is given by θ = γ 0 − β, where γ 0 is the angle between the velocity

Fig. 7.40 The equivalence of inclined yawed impact and the impact with moving plates

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direction of the rod and the normal to the plate, and β is the obliquity angle between the rod’s axis and the normal to the plate as shown in the figure. It is easy to show that, as far as the rod/plate interaction is concerned, the cases of positive and negative yaw are equivalent to the cases of plates moving away and towards the rod, respectively, as shown in lower part of the figure. For a rod velocity of V 0 , an obliquity of β, and a plate velocity of v the equivalent yaw (θ ) according to Liden et al. (2006) is given by: tan θ =

ν sin β V0 ± ν cos β

(7.17)

where the (+) and (–) signs relate to plates moving towards and away from the rod, respectively. This analysis considers the relative velocity between the rod and the plate as V rel = V 0 ± v cos β, which is different from the expression (7.7) that was used by Rosenberg et al. (2004). This difference is due to the fact that (7.7) considers the rod/plate relative velocity in relation to their interaction point, while in the present case the interaction point is moving along the whole length of the rod (the grazing process). Cagliostro et al. (1990) performed 3D numerical simulations in order to analyze their experimental results for L/D = 10, spherical-nosed rods made of an uranium alloy with 0.75% titanium, which impacted moving RHA plates. These experiments are described schematically in Fig. 7.41. The reference experiment is the one where the rod impacts a stationary plate at an impact velocity of 1.29 km/s and obliquity of β = 65°. The two other experiments were performed with moving plates, at different velocities and plate obliquities, as shown in the figure. These experiments Fig. 7.41 The experiments with moving plates and the equivalent impacts on stationary targets with the corresponding yaw angles

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are equivalent to impacts on stationary targets with a velocity of 1.29 km/s and obliquity β = 65°, at yaw angles of θ = –9.3°, and θ = +10.3°. The difficulties encountered in performing similar experiments, with the desired velocity of the moving plate, prompted the use of the reverse-ballistics technique. With this technique, an inclined plate is launched at a stationary rod which is aligned at a certain yaw with respect to the plate’s velocity direction. The usefulness of this technique was demonstrated by Bless et al. (1999) who launched inclined steel plates towards stationary WHA rods. These L/D = 36 rods were inclined to the plate’s line of flight at the desired yaw angle. The combined yaw and inclination resulted in a significant rotation of the rods, which also broke up into several pieces during the grazing interaction. Liden et al. (2006) used the same technique to follow the interaction of a moving plate with a long rod. They used 2 mm thick steel plates which constituted an integral part of their sabots, and obliquities of: β = 20°, 30° and 60°. The plates were launched by a two-stage gas gun to velocities in the range of 1.83–2.18 km/s towards WHA rods with L/D = 15, and D = 2 mm. The rods were positioned at yaw angles of θ = 2.5°–8° in order to achieve equivalent plate velocities of 100, 200 and 300 m/s, for a relative rod/plate velocity of 2.0 km/s. Figure 7.42 shows two sets of X-ray shadowgraphs from experiments with β = 30° and yaw angles of θ = +2.6° and θ = –3.1°, which correspond to effective plate velocities of +200 m/s and –200 m/s, respectively. The shadowgraphs show the stationary rods before impact, and the rotating rods at 100, 150 and 200 μs after impact. It is clear that the continued grazing interaction with the plates induces a significant rotation to these rods. At a larger yaw angle of θ = 7° rod breakup was observed for the case where the plate was, effectively, moving away from the rod.

Fig. 7.42 Shadowgraphs from impacts at β = 30°, left: positive yaw θ = 2.6°, right: negative yaw θ = 3.1°

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Fig. 7.43 Simulation results for impacts at inclined plates, positive yaw (top) and negative yaw (bottom)

In order to follow these interactions Rosenberg et al. (2009) presented 3D numerical simulations for two cases in which the plates are moving at 2.0 km/s towards rods which are tilted at θ = +7° and –7°, representing the experiments performed by Liden et al. (2006). The resulting yaw rates from such simulations should be taken into consideration for an active armor design, in order to determine the necessary distance to the protected vehicle. The interaction described on the top of Fig. 7.43 corresponds to a plate moving away from the rod (positive yaw). This interaction resulted in a higher yaw rate, as compared with the rate induced by the plate moving towards the rod (negative yaw), which is shown on the bottom of this figure. Liden (2008) added a fracture model to the Johnson-Cook strength model of the WHA rods in their simulations. Good agreement was obtained between these simulations and the experimental results for the number of rod fragments and their positions along the rod, as shown on the right hand side of Fig. 7.44. The simulations also reproduced the experimental results where the rods were significantly bent but did not fracture, as shown on the left side of Fig. 7.44.

Fig. 7.44 Comparing simulations results (grey) with experiments (black) of yawed impacts, for plates moving towards the rod (left) and away from it (right)

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7.4 Defeating Shaped Charge Jets 7.4.1 The Explosive Reactive Armor (ERA) Shaped charge jets are the most lethal threats for battle tanks since their penetration capabilities exceed the wall thicknesses of these vehicles. For many years, armor designers were looking for defeat mechanisms which will reduce the lethality of these long jets. The invention of the explosive reactive armor (ERA) by Manfred Held (1970) is, undoubtedly, the most successful breakthrough in this field. Much of the published data on ERA as well as their interpretation, were summarized by Held et al. (1998) and by Mayseless (2010). The basic principle behind the ERA is very simple and its protective capability is so impressive that it is considered as the most efficient add-on armor technology for both battle tanks and armored personnel carriers. The ERA cassette includes two, relatively thin, steel plates with an explosive interlayer between them. Typically, the thickness of each one of these layers is 2– 3 mm. The cassette is placed at an angle of about 30° to the horizon, at a certain distance from the wall of the protected vehicle. The explosive layer is initiated upon jet impact and its products propel the two plates at opposite directions to velocities of about 1.0 km/s. The moving plates interact with the jet for a long time, eroding and deflecting its elements and reducing their penetration capability. The significant disruption to the jet is evident in the shadowgraph shown in Fig. 7.45, from Mayseless et al. (1984). The jet is moving from left to right in this shadowgraph, and it is clearly seen that its fast parts are severely disturbed as they emerge from the moving plates. One can also observe the damage induced to the slower parts of the jet by the front plate before they reach the rear plate. Right after impact, the jet opens a hole in both front and back plates. The leading part of the jet, the so-called “precursor”, escapes the interaction without being disrupted and it is responsible for the residual penetration in the protected body. The size of the holes in the plates depends on their strength and on the diameter of the jet tip, which is usually larger than the diameter of the rest of the jet, as seen in Fig. 1.20. As the plates are propelled by the explosive products they interact with the jet in a grazing mode, inducing small disturbances which grow with time as discussed by Held (2005), and as shown schematically in Fig. 7.46. These disturbances in the jet

Fig. 7.45 Jet disruptions by a reactive cassette

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Fig. 7.46 Schematic description of the interaction between the jet and the moving plates

are directed in opposite directions, depending on which plate interacts with the jet. The fact that some parts of the jet are missing in the shadowgraph is due, according to Held (2005), to the rear plate’s interaction with the disturbances which were produced by the front plate. As stated above, due to their high effectiveness against shaped charge jet, these ERA cassettes have been applied to various armored vehicles. However, there are relatively few publications which detail information concerning their performance as a function of the relevant parameters, such as plate thicknesses and their velocities. Held (2006) summarized some of his results and those of Ismail et al. (2004), while Mayseless (2010) analyzed these results by applying a phenomenological approach. In both sets of experiments, the residual penetrations (Pres ) of the jets into steel witness blocks, were determined as a function of explosive layer thickness (h) in a symmetric cassette b/h/b. The plate thicknesses (b) were 1 and 3 mm, in the works of Held (2006) and Ismail et al. (2004), respectively. The penetration depths (P0 ) of these jets into a thick steel target (without the cassette), were used to evaluate the effectiveness of these cassettes. Thus, the efficiency of a given cassette is inversely proportional to the ratio Pres /P0 . Although the two sets of experiments were performed with different explosives they seem to fall on a single curve for the ratio Pres /P0 as a function of explosive thickness (h), as shown in Fig. 7.47. The coincidence of these two sets of data is probably fortuitous, as argued by Held (2006) who reasoned that the velocity of the plates should be the controlling parameter, rather than the thickness of the explosive layer. Plate velocities can be calculated by the analysis of Gurney (1943), for the velocity of metallic plates which are propelled by explosive layers. Detailed descriptions of Gurney’s analysis for spherical, cylindrical, and planar configurations are given in Walters (1990) and in Meyers (1994). According to this analysis the velocity of the plates in a symmetric cassette, which consists of two equal mass (M) plates and an explosive interlayer of mass C exp , is given by:

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Fig. 7.47 The effect of explosive thickness on cassette efficiency for two types of cassettes

2E exp V = 2M/Cexp + 1/3

(7.18)

where the term (2E exp )0.5 is given in units of velocity, and E exp is related to the chemical energy of the specific explosive being used. With this equation, Held (2006) derived the velocities of the plates for the 1/h/1 and the 3/h/3 cassettes, and plotted the corresponding Pres /P0 ratios in terms of plate velocities, as shown in Fig. 7.48. This display of the experimental results shows that the efficiency of a given cassette depends on the velocity of the plates (V p ) as well as on their thickness (b). A close examination of the data points in this figure shows that for a given value of Pres /P0 , the two cassettes have equal values of the product bV 2p . This means that the data for the two cassettes should fall on a single curve of Pres /P0 as a function of the Fig. 7.48 The relation between cassette efficiency and plate velocity for the two types of cassettes

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kinetic energy of the plates. In fact, this is the expected result from Fig. 7.47 where the explosive thickness is the free parameter since, since according to the Gurney relation the plate’s energy is directly related to the explosive thickness. Thus, it seems that an energy-based interaction may account for the efficiency of reactive cassettes against shaped charges. Koch and Bianchi (2010) performed a parametric study by varying the thicknesses of the plates and the explosive layers. They tested cassettes with different steel plates in order to study the effect of plate strength on the cassette’s efficiency. Their 50 mm charge was placed at a distance of 250 mm from the reactive cassette, and a thick witness block was placed at a distance of 220 mm from the cassette, in order to determine residual penetrations. The thicknesses of the plates in these cassettes were: b = 1.0, 3.0 and 6.0 mm and the thicknesses of the PETN-based explosive layers, with density of 1.4 g/cm3 , were in the range of h = 1.0–7.0 mm. In all of these tests the inclination angle of the cassette was α = 45°. As expected, plates with higher strength resulted in a higher efficiency, due to the smaller holes which the jet opened in these plates. The relation between the diameter of these holes and the strength of the plate will be discussed at the end of this chapter. The resulting mass efficiencies (E m ) of these cassettes were E m = 2.85, 3.0, and 4.03 for steel plates with compressive strengths of: Y = 0.4, 0.6 and 2.0 GPa, respectively, enhancing the conjecture that stronger plates result in higher efficiencies. The results of Koch and Bianchi (2010) for three types of symmetrical cassettes, in terms of their mass efficiency as a function of explosive thickness, are shown in Fig. 7.49. The mass efficiency increases with explosive thickness due to the higher velocities imparted to the plates, in accord with the other cassette designs which were described above. The interaction of the jet with moving plates is a very complex process since it involves many issues such as the dynamics of hole growth in a thin plate, the grazing interaction of the jet elements with the edge of the hole, the growth of small Fig. 7.49 The mass efficiency of different cassettes as a function of explosive thickness

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disturbances along the jet with time and distance, and the effect of the explosive products on jet disruption. Obviously, this interaction is too complicated for an analytical treatment, which is the main reason for the scarcity of published models in the open literature. Mayseless (2010) applied momentum-based considerations in order to account for the efficiency of reactive cassettes. This analysis is based on an earlier phenomenological model by Mayseless et al. (1984) for the efficiency of a given cassette, in terms of the mass flux ratio between the jet and the plate. The mass flux ratio for a given jet/plate pair is given by: m˙ p k = m˙ j V j tan α/V p ± 1/ cos α

(7.19)

where the (+) and (–) signs stand for the front and rear plates, respectively. The parameter k includes the jet/plate density ratio and the ratio of jet diameter to plate thickness. The efficiency of the cassette according to this phenomenological model is due mainly to the rear plate, which moves in the direction of the jet. As shown by Mayseless (2010) the calculated efficiency, as defined by this mass-flux ratio, follows the general trend in the results of Held (2006) and Ismail et al. (2004). The high efficiency of ERA cassettes against shaped charge jets has been demonstrated in many publications by M. Held. For example, Held (1999a) found that the differential efficiency, as defined by (6.2), of cassettes with thick explosive layers can reach values of more than 20. The specific cassette in this work consisted of two steel plates, 2.5 mm thick, and a 12 mm thick datasheet explosive. The residual penetration of the jet into a semi-infinite steel target was 143 mm, as compared with a penetration depth of 460 mm without the cassette. As mentioned above, Held (1999b) also demonstrated the efficiency of heavy cassettes against long rod penetrators. He noted that a useful cassette should be effective against both shaped charge jets and long rods. Thus, he used a 40/10/25 cassette, with an explosive thickness of 10 mm, against both long rods and shaped charge jets in full-scale experiments. The thick plates were propelled to relatively low velocities in order to prolong their interaction with the long rod, which is much slower than the interaction with the jet. The differential efficiency of these heavy cassette against a large shaped charge jet was about 7, and about 4.5 against a long rod. One of the main problems concerning the use of reactive cassettes is due to the collateral damage which the flying plates induce to both the protected vehicle and to its surroundings. In order to reduce this damage Mayseless (2007) proposed the use of ceramic tiles instead of steel plates, realizing that the brittle nature of ceramics should result in their total comminuting a short time after jet impact. This idea has been followed by Bianchi et al. (2008) and by Koch and Bianchi (2010), who compared the efficiencies of alumina tiles with those of steel plates for equal weight reactive cassettes. The thicknesses of the alumina tiles were: h = 2.9, 5.9 and 11.8 mm, matching the weights of the h = 1, 3 and 6 mm steel plates, respectively. As far as the explosive thickness is concerned, the same general trends in the ballistic efficiency were found for these ceramic-based cassettes, as shown in Fig. 7.50.

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Fig. 7.50 Ballistic efficiencies of alumina-based symmetric cassettes as a function of explosive thickness

In fact, Koch and Bianchi (2010) obtained higher efficiencies with the ceramic tiles as compared with the equal-weight steel cassettes. For example, a symmetric steel cassette with 3 mm thick plates and an explosive layer of 5 mm, resulted in a mass efficiency of E m = 5, as compared with an efficiency of E m = 8.8 for an equal-weight ceramic cassette. This difference can be attributed to the smaller hole which the jet opens in the ceramic tile due to its high strength. A smaller hole results in a shorter jet which “escapes” the interaction, as was already demonstrated for the higher strength steel plates. It is also worth noting that these experiments resulted in an enhanced efficiency for asymmetric cassettes with thicker front plates, especially for cassettes with thick explosive layers. In particular, the highest efficiency in this study, of E m = 14.3, was obtained for a cassette with an explosive thickness of 7 mm, a front ceramic tile of 11.8 mm and a 2.9 mm thick back tile. The high efficiency is due, apparently, to several parameters which are optimized with this cassette. For example, its asymmetric structure ensures that the back plate (the thin one) will move at a high velocity which is also enhanced by the thick explosive layer. The front plate which does not contribute significantly to the disruption of the jet is playing as a buffer, by enhancing the velocity of the thin back plate. Also, the explosive products from a thick explosive layer can add to the disruption of the jet, as will be discussed later on. Finally, the holes generated by the jet in these ceramic tiles are probably much smaller than the holes in steel plates, due to their higher strength and their lower densities. Bianchi et al. (2008) tested reactive cassettes with composite plates, made of various fibers (glass, carbon, and aramid). These cassettes were also very efficient against the shaped charges, as compared with steel cassettes of the same weight. The best performance was for a hybrid cassette in which the front plate was a carbon fiber reinforced composite (CFRP) and the back plate was of AD99 alumina. The high efficiency of the composite plates can be attributed to their low density and relatively high strength. As will be discussed below, these factors minimize the diameter of the hole produced by the perforating jet, enhancing the jet’s disruption.

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Fig. 7.51 Shadowgraphs of the interaction between a jet and a reactive cassette with ceramic plates

In summary, the ceramic tiles showed a clear advantage over the steel plates, as far as mass efficiency is concerned. In addition, these tiles were pulverized during their early stages of flight as observed by the X-ray shadowgraphs in Fig. 7.51, from Bianchi et al. (2008). It is clear that at about 75 μs after impact the ceramic plates are totally pulverized, resulting in much less damage to the surroundings. Nevertheless, these ceramic fragments are able to disrupt the jet appreciably, as one can clearly see in these shadowgraphs. The effect of the explosive products on jet disruption has been investigated by Brown and Finch (1989) in a series of experiments where the jets impacted the cassettes at normal incidence. With this configuration, most of the jet elements are not disturbed by the plates. Thus, any disturbance to the main part of the jet should be attributed to its interaction with the explosive products. The cassettes in this study consisted of mild steel plates with an explosive sheet (type SX2) between them. X-ray shadowgraphs were taken after the whole jet emerged from its interaction with the cassette. Some shadowgraphs from this work are shown in Fig. 7.52 for 6/h/6 cassettes, which consisted of 6 mm plates of mild steel and explosive layers of different thicknesses: h = 0, 3, 6, and 12 mm. These shadowgraphs were taken at 152 μs after jet impact. It is clear that a minimal explosive thickness, of about 6 mm, is needed in order to induce a meaningful disruption to the jet. Note that the explosive products from a layer of 12 mm significantly disrupted the jet, which may explain the high efficiencies in the works cited above for cassettes with explosive layers thicker than 6–8 mm. At an inclination angle of 30° or 45° these thicknesses are equivalent to 12 mm or so.

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Fig. 7.52 Jet disruptions as a function of explosive thickness (h)

In order to investigate this disruption process by the explosive products, Rosenberg and Dekel (1999b) performed a series of 2D numerical simulations for a similar cassette configuration. The copper jets in these simulations, with a diameter of 3 mm, were given a uniform velocity of 7.0 km/s in order to simplify the simulations. The relevant properties of the steel plates and the explosive sheets, from Brown and Finch (1989), were reproduced in the simulations to a reasonable degree. The initiation of the explosive was performed through a threshold pressure which resulted in its almost immediate detonation. The basic disruption mechanism is captured by following the results of one of these simulations, as shown in Fig. 7.53 for a 6/6/6 cassette, which consists of equal thicknesses (6 mm) for all the layers. During the first few microseconds the jet penetrates the cassette, opens a large hole in the plates, and initiates the explosive. Between 10 and 15 μs after impact, the inverted jet material together with the plate debris and the explosive products, arrive at the symmetry axis and impact the jet, disrupting its later parts. The disturbance to the jet was found to be related to the timing of this lateral impact at the axis. Thus, the more effective disturbances, resulting in shorter precursors which “escaped” the interaction, were obtained when the debris arrived earlier at the axis. This part of the jet is denoted as L esc in the figure. These simulations qualitatively reproduced most of the experimental results from Brown and Finch (1989). For example, replacing the explosive with a stronger one resulted in a much shorter precursor in both experiments and simulations. In addition, the effectiveness of an explosive “pill” inserted in the steel plates, was compared with that of the explosive sheet. Both the experiments of Brown and Finch (1989) and the simulations by Rosenberg and Dekel (1999b), showed that these explosive discs induce the same jet disturbances obtained by continuous explosive sheets. Replacing the steel plates with aluminum plates of equal-weight resulted in very similar interactions, which means that the confinement of the explosive layer is important for this interaction. Rosenberg and Dekel (1999b) also performed a sensitivity study

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Fig. 7.53 Simulation results for the interaction of a jet with a reactive cassette at normal incidence

in which they looked for the optimal thicknesses of the steel plates, for a constantweight cassette with a total thickness of 12 mm. These cassettes were checked for the residual length of the “escaping” jet for different explosive layers and the results are shown in Fig. 7.54. One can clearly see that for the three types of cassettes, with explosive layers of h = 3, 6 and 12 mm, the shortest residual jets are obtained for the cassettes with the thinnest front plates. Extrapolation of these results gives a value of about 2 mm, for the front plate thickness which reduces the length of the escaping Fig. 7.54 The intact jet length as a function of front plate thickness in constant-weight cassettes

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jet to zero. These parametric studies are not intended to serve as design tools but they can save a lot of experimental effort for the armor engineer. One should regard these simulations as a demonstration of their usefulness in following the mechanism responsible for a given defeat process, as well as highlighting the role of material properties in the interaction. The work of Mickovic et al. (2016) is a very interesting attempt to combine several models with experimental observations into a code which can account for the disruption of a jet by a given cassette. They considered the effects of the front and the back plate on the jet’s disruption as well as the effect of the explosive products. Hole diameters in both front and back plates are estimated by the models of Naz (1989) and Szendrei (1983). The code accounts for the obliquity angle between the jet and the cassette, the exact location of the impact point on the cassette (edge effects) and even the effect of the cassette’s distance from the protected vehicle. In order to validate their approach they performed several tests with jets produced by charges, 120 mm in diameter, and found good agreement between their model and the test results, as shown in Fig. 7.55. The figure summarizes the penetrations of the residual jets into RHA witness plates, for several cassettes, with different explosive layers (H ex ). The thicknesses of the front and back plates in these calculations were 8 mm and 1 mm, respectively. One can see that the two experimental results agree very well with the model’s prediction. Another demonstration of the code’s capabilities is shown in Fig. 7.56, for the penetration depths of the same jet after its interaction with another family of cassettes. Here the thickness of the front plate and the explosive layer is 8 mm, while the back plate’s thickness (h) was the free parameter. Again, good agreement was obtained between the code’s predictions and the test results.

Fig. 7.55 Residual penetrations after jet interaction with 8/H/1 cassettes at different obliquities

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Fig. 7.56 Residual penetrations after jet interaction with 8/8/h cassettes at different obliquities

7.4.2 Passive Cassettes The reactive armor cassettes were found to be very effective in disrupting shaped charge jets but they have an inherent disadvantage due to the damage they inflict on their surroundings, as discussed above. A much less violent reaction is obtained if the explosive layer in the cassette is replaced by an inertial material, as was first suggested by Held (1973). These, so-called “passive cassettes”, consist of a low impedance interlayer between the two steel plates and, obviously, they are less effective than the reactive cassettes. However, with the proper choice of materials the passive cassette can disrupt a major part of the jet, as shown in Fig. 7.57. Their main advantage is the much less collateral damage which they induce, because the steel plates do not acquire high velocities. In fact, the plates do not fly apart during the interaction with the jet, which results in their bulging around the impact point, as is seen in the figure. The interaction of the jet with the passive cassette has been investigated by Gov et al. (1992), Thoma et al. (1993), Held (1993) and Yaziv et al. (1995). The jet-plate interaction is based on the bulging of the steel plates across a large area around the impact point, which is due to the high pressures and temperatures induced in the interlayer material. In addition, the rear plate is accelerated by the impact of the jet, and its bulging is enhanced as compared with the front plate. The local motion of the plates results in a continuous interaction with the jet, disrupting it in a similar way

Fig. 7.57 The disruption of a jet by a passive cassette

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to that of the reactive cassette. The main difference between the actions of the two cassettes is in the much shorter time duration which the passive cassette interacts with the jet. Gov et al. (1992) used 2D numerical simulations to follow the bulging process of the steel plates in a cassette which included a rubber interlayer. By comparing the bulging in their 2D simulations with X-ray shadowgraphs of inclined cassettes, they showed that the process of plate acceleration is insensitive to the obliquity of the impact. Since 2D simulations can deal only with the normal impacts, they do not account for the jet/plate continued interaction. Still, Rosenberg and Dekel (1998b) showed that 2D simulations can be very useful for a parametric study of the passive cassette. They followed the relative velocities of the metallic plates at several locations away from the symmetry axis, and related them to the efficiency of the cassette. Their basic assumption was that faster moving plates will induce enhanced disruption to the jet, so they looked for the maximal relative velocities of the two plates. They arbitrarily chose three control points at distances of z = 13.2, 16 and 20 mm from the impact point, where the relative velocities were determined. The bulging process of a 3/10/3 cassette, impacted by a copper jet with a diameter of 3 mm, at 7.0 km/s, is shown in Fig. 7.58. The 10 mm thick interlayer material in this simulation was Plexiglas with strength of 0.1 GPa. Note the much larger deflection of the back plate which moves in the direction of the jet’s motion, as discussed above. The following figures describe some of the results from Rosenberg and Dekel (1998b) in terms of the relative plate velocities, as a function of the relevant parameters of the passive cassettes. These velocities were determined at the three locations above the symmetry axis, z = 13.2, 16 and 20 mm, as described above. Figure 7.59 shows the relative velocities of the two plates for 3/H/3 cassettes as a function of

Fig. 7.58 Simulation results for the bulging process of a 3/10/3 passive cassette

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Fig. 7.59 The dependence of the relative velocity of the plates on the interlayer’s thickness

the interlayer thickness (H). The thickness of the Plexiglas interlayer was varied between 3 mm and 16 mm, and it is clear that for a thicknesses of about 8.0 mm these velocities reach their asymptotic values. Thus, according to these simulations there is no advantage in using a thicker interlayer for this particular setup. Varying the strength of the Plexiglas interlayer resulted in increased relative velocities with decreasing interlayer strength. On the other hand, these velocities were insensitive to the strength of the steel plates. Note that the efficiency of an inclined cassette with stronger steel plates is expected to be higher because the diameter of the holes in the plates will be smaller. With smaller holes, the stronger plates interact more frequently with the jet and the disruption is expected to be more severe. Another set of simulations was performed for constant-weight cassettes, where the total thickness of the front and back plates was 6 mm. As seen in Fig. 7.60, the relative velocities of the symmetric cassettes are the lowest, which means that the asymmetric cassettes are expected to perform better against the jet. As far as this configuration is concerned, cassettes with thin back plates are expected to perform better than those with thin front plates. Finally, the density and thickness of the interlayer in the cassette were varied in such a way that their product (the areal density) was held constant, at 19 kg/m2 . Figure 7.61 shows the results of this set of simulations, and it is evident that the highest plate velocities are obtained for interlayer densities in the range of 2.0–2.5 g/cm3 . We should emphasize, again, that such simulations are not aimed at an actual design of armor systems. Rather, they identify the important parameters of the process, in order to minimize the experimental work which is needed for such studies. We have noted above that the lower ballistic efficiency of the passive cassette as compared with the reactive cassette, is compensated by the much less damage which it induces to the surrounding. Several laboratories are engaged in developing hybrid cassettes which contain interlayers of energetic materials. These cassettes have a

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Fig. 7.60 Simulation results for constant-weight cassettes with total steel plate thickness of 6 mm

reduced efficiency as compared with reactive cassettes, but they are expected be more effective than the passive cassettes since their interlayer material is undergoing some reaction. Holzwarth and Weimann (2001) examined the performance of such an interlayer, which consisted of a mixture of RDX explosive with the energetic binder glycidyl-azide-polymer (GAP), and other materials. Their reference interlayer material for a 2/H/2 cassette, was rubber with a density of 1.4 g/cm3 . Tests with different energetic mixtures showed that a cassette with a mixture of the GAP binder with 50% RDX, resulted in a significant improvement as compared with the performance of the reference cassettes. Fig. 7.61 Simulation results for cassettes with a constant interlayer areal density

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7.4.3 The Hole Diameter in a Plate Impacted by a Jet One of the more important issues which determine the ballistic efficiency for both reactive and passive cassettes, is related to the growth dynamics and the maximal diameter of the hole which the jet induces in the plate. We have seen that large holes result in an appreciable jet which escapes the interaction with the plate without being disturbed. This so called “precursor” is responsible for the damage in the target behind the cassette. Thus, in order to minimize the length of these precursors one should aim at plates with minimal hole diameters. Moreover, the continued interaction of the jet and the cassette is more effective when these holes are of minimal size. This is a direct consequence of the grazing character of this interaction. In order to have an estimate for the parameters which influence the size of these holes, consider the model of Szendrei (1983) who addressed the dynamics of crater enlargement in a semi-infinite target impacted by a high-velocity jet. His analysis resulted in the following expression for the maximum diameter which the crater achieves (Dmax ) in terms of the jet’s velocity (V ), the jet and the target densities, and the target’s strength (Y t ): Dmax V =√ √ √ Dj 2Yt 1/ ρ j + 1/ ρt

(7.20)

where Dj is the diameter of the jet. Minor modifications to this expression were suggested by Shinar et al. (1995) and Held (1995), but the basic dependence of Dmax on the relevant parameters is the same as in (7.20). There is a certain ambiguity about the final expression for the crater diameter and the definition √ of Y t , in the works of Szendrei (1983) and Shinar et al. (1995) who used σ t = 2Y t / 3 for the strength of the target. Their expression for the maximal crater diameter is: Dmax = Dj

1+

V2 √ √ 2 2σt 1/ ρ j + 1/ ρt

(7.21)

which reduces to Szendrei’s expression, (7.20), for high impact velocities. In fact, for typical jet velocities of 3–8 km/s the two expressions result in similar values, as was shown by Shinar et al. (1995). One can simplify both equations for cases where copper jets (ρ j = 8.9 g/cm3 ) impact steel targets (ρ t = 7.85 g/cm3 ) by writing ρ t ≈ ρ t = ρ. Equation (7.20) reduces to the following form: Dmax V = Dj 2



ρ 2Yt

(7.22a)

and (7.21) simplifies to: 0.5  Dmax ρV 2 = 1+ Dj 8σt √ where σ t = 2Y t / 3, as mentioned above.

(7.22b)

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Naz (1989) performed a set of experiments with L/D = 10, D = 4 mm, copper rods which impacted various steel targets at velocities in the range of 2.5–5.0 km/s. The steel targets had compressive strengths in the range of 0.64–1.4 GPa. As shown by Shinar et al. (1995) the final crater diameters were in excellent agreement with the calculated values from (7.22a) and (7.22b). As is clearly seen in these relations, the strength of the target inhibits the lateral expansion of the crater, as expected. Another important consequence from these analyses is that for plates of equal strength, the one with the lower density is expected to result in a smaller cavity diameter. This observation can be understood by the fact that the inertia of the target, which is responsible to its lateral expansion, is linearly dependent on its density. The dynamics of crater growth as a result of jet impact has been also analyzed by Szendrei (1983). His result for the time dependence of the crater diameter, D = D(t), has been verified by several workers who followed this growth process with flash X-ray shadowgraphs, such as in Held and Kozhushko (1999). The analysis of Shinar et al. (1995) for the time dependence of the crater growth resulted in a close agreement with similar experiments by De-Rosset and Merendino (1984). An analytical model for the grazing interaction between the jet and the moving plates, of either reactive or passive cassettes, have to consider these time-dependent results from the analysis of Szendrei (1983). We have seen that, indeed, Mickovitc et al. (2016) included these relations in their code. On the other hand, both Szendrei (1983) and Shinar et al. (1885) considered semi-infinite targets, while the plates in reactive and passive cassettes are relatively thin. Thus, an analytical model for the hole’s diameter in thin plates is needed in order to study the grazing interaction between a jet and a thin plate. In order to construct such a model for the hole diameter in finite-thickness plates Kositski and Rosenberg (2013) and Rosenberg and Kositski (2017b) used 2D numerical simulations with the AUTODYN multi-material Eulerian solver. Their first step was to check whether these simulations reproduce the analytical models which were discussed above, for semi-infinite targets. The steel jet in these simulations, with diameter of 6 mm and length of 100 mm, was given zero strength in order to simplify the analysis. Figure 7.62 shows the process at two different times from one of these simulations in which the steel rod impacts a steel target at 7.0 km/s. The crater in the target has a well-defined diameter which can be compared with the predictions from the analytical models of Szendrei (1983) and Shinar et al. (1995). The next step was to perform a series of simulations where these rods impacted steel and aluminum targets, with different strengths, at a 7.0 km/s. The von-Mises

Fig. 7.62 The penetration of a fast moving steel rod in a semi-infinite steel target

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Fig. 7.63 Comparing simulation results with the analytical models for aluminum (right) and steel (left) targets

strength model was used in these simulations and the strengths of the steel and aluminum targets were varied between 0.2 and 2.0 GPa. Figure 7.63 shows the good agreement between simulation results and those predicted by the models of Szendrei (1983) and Shinar et al. (1995), for the hole diameter as a function of target strength. Another set of simulations was performed for these steel rods impacting a steel target with Y t = 1.0 GPa, at velocities of: V 0 = 3.0, 5.0 and 7.0 km/s. Figure 7.64 shows that the simulations follow the analytical models quite closely. Such comparisons between model and simulations are very important for the present issue, because they can determine which model should be chosen for further analysis. It seems that predictions from the two models presented above are in good agreement with the simulation results. However, as shown in these figures, the model of Shinar et al. (1995) is somewhat closer to the simulations results of Kositski and Rosenberg (2013). Fig. 7.64 Comparing simulation results (dots) with the analytical models for different rod velocities

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Fig. 7.65 The shape of the hole in a finite thickness plate

The numerical simulations of Kositski and Rosenberg (2013), for finite thickness plates, showed that the holes in these plates do not have a constant cross-section. Rather, they were narrower near the impact face, as shown in Fig. 7.65. In order to keep consistency throughout the work, they chose to follow the minimal diameters of these holes, as shown in the figure. Rosenberg and Kositski (2017b) performed a set of simulations which followed the temporal increase in the hole’s diameter for aluminum plates (Y = 0.5 GPa) of different thickness (h), impacted by zero-strength 100 mm long steel rods with a diameter of 6 mm, at a velocity of 7.0 km/s. We should note that for thin plates the mechanics of hole growth is the same for impacts by jets or by spheres of the same diameter (Dp ). Thus, we shall use both Dj and Dp in the following discussions. Fig. 7.66 shows the results of these simulations, in terms of the time evolution of the hole’s diameter, Dh (t), together with their best fit curves according to the following equation: Fig. 7.66 Hole growth histories for different plate thicknesses. Simulation results (symbols) compared with curves following (7.23)

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Dh (t) t = qh − (qh − 1) exp−K h Dp

(7.23)

where qh is the asymptotic value of the normalized hole diameter in a thin plate, for a given h/Dp value. The parameter K h in (7.23) has the units of (μs)–1 , and it controls the rate of the hole growth process. Similar exponential relations were used by Turpin and Carson (1970) to account for their experimental results, as determined by flash shadowgraphs. A close examination of the curves in Fig. 7.66 shows that the holes practically reach their asymptotic values at about t = 6, 10 and 20 μs for plates with h/Dp = 0.5, 1.0 and 2.0, respectively. As expected, these times are shorter for thinner plates due to the enhanced effect of their back surfaces. The relation in (7.23) was chosen to describe the hole’s growth rate because it has the essential features of a fast increase right after impact, followed by a slower increase at later times. It also has the correct values for the limiting cases at very short and very long times. Right after the plate’s perforation we expect to have Dh (0)/Dp = 1.0, while at much later times the normalized hole diameter should approach its asymptotic value of qh . The asymptotic values for the normalized hole diameter (qh ), from the set of simulations shown in Fig. 7.66 are: qh = 2.83, 4.08 and 5.0 for the h/Dp = 0.5, 1.0 and 2.0 plates, respectively, and they are smaller for thinner plates, as expected. It is reasonable to assume that as the plate’s thickness increases, the value of qh will reach its asymptotic value (Q) for a semi-infinite target, as derived by the simulations of Kositski and Rosenberg (2013). It is clear that when a fast-moving jet (or sphere) impacts a plate with thickness which is much smaller than the impactor’s diameter, the hole’s final diameter will be equal to that of the impactor. On the other hand for very thick targets the diameter of the hole should approach the diameter in a semi-infinite target, is given by Dmax in the analytical models described above. Rosenberg and Kositski (2017b) assumed that the increasing trend of the final hole diameter with plate thickness (h) can be characterized by the following relation for the hole diameter (Dh ) in a finite-thickness plate:   h Dh = Q − (Q − 1) exp −α Dj Dj

Q=

Dmax Dj

(7.24)

where the asymptotic value in a semi infinite target, Q = Dmax /Dj , as given by the models described above. This functional dependence of the hole diameter on the plate’s thickness has the proper values at the limiting cases of very thin and very thick plates. The parameter α should include the other properties of the plate and the jet (or the sphere). It turned out that for the simulation with a steel jet impacting a 1.0 GPa steel target at 7.0 km/s, the best-fit value for α is 1.0. Figure 7.67 shows the agreement between the model’s predictions with α = 1.0 and the results for this calibration set of simulations. According to the basic assumption of Rosenberg and Kositski (2017b), the lateral expansion of the hole in the plate is controlled by two opposing factors. Clearly, the strength of the target (Y t ) tends to inhibit the hole’s expansion, while the target’s

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Fig. 7.67 Correlation between model (curve) and calibration set of simulations (symbols) with α = 1.0

inertia (ρ t V j 2 ) is the main source for this expansion. Thus, they used the following expression for the parameter α: 

Yt α=A ρt V j2

0.5 (7.25)

where the “empirical” parameter (A) was determined by the calibration set of numerical simulations described above. Using the value of α = 1.0 from this set, with a steel jet impacting a steel target (Y t = 1.0 GPa), at V j = 7.0 km/s, the value of A = 19.61 was obtained. This value of A was then used in all the other simulations, as shown in Fig. 7.68 for various values of the physical parameters. Figure 7.68a compares the model with simulation results for the zero strength steel rods impacting a steel target (Y t = 1.0 GPa) at three velocities 3.0, 5.0 and 7.0 km/s. The comparison of the calibration set of simulations (Y t = 1.0 GPa) with those for a steel plate of lower strength (Y t = 0.5 GPa), with α = 0.7, is shown in Fig. 7.68b, and 7.68c shows the comparison between model and simulations for aluminum and steel targets having the same strength (0.5 GPa), which demonstrates the rather non-intuitive result for smaller hole diameters in a plate with lower density. This result explains, at least partly, the high ballistic efficiencies of reactive and passive cassettes which are composed of light materials, as discussed above. The high efficiency of ceramic plates in reactive armor is better understood now, since these materials have both high strengths and low densities. As the model and the simulations show, both of these properties contribute to minimize the hole induced by the jet, resulting in an enhanced grazing effect between the jet and the plate. Note that Bianchi et al. (2008) and Koch and Bianchi (2010) tested reactive cassettes with fabric composites plates. The resulting efficiencies were higher than those of steel plates, probably because of the low densities and relatively high strengths of these composites.

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Fig. 7.68 Comparing the model (curves) for hole diameter with numerical simulations (dots) for: a different impact velocities, b different target strength, c different targets of the same strength

7.5 Defeat of Micrometeoroids by Thin Metallic Plates The definition of the term “hypervelocity” was changed over the years, as noted in the review article by Swift (1982). Originally, the velocity of the projectile was used to determine this impact regime, but it is more customary now to consider the mechanisms which the impact invokes in determining its nature. Thus, hypervelocity impacts are those which cause a complete pulverization of the projectile and the target around the impact point. In this case both materials can be treated as fluids, neglecting their original strength, and the analysis becomes relatively easy. As summarized by Fair (1987), the early studies on hypervelocity impacts, during the 1950s, were motivated mainly by the need to protect the sensitive equipment in spacecraft from accidental impacts of micrometeoroids. Whipple (1947) described his idea for a low-weight shield which is based on a double wall system. This, so-called “Whipple Shield”, consists of a thin sacrificial outer plate, so-called the bumper shield, which is placed at some distance from the protected wall, as seen in Fig. 7.69. Upon impact, both the micrometeoroid and the bumper completely disintegrate around the impact

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Fig. 7.69 The Whipple bumper shield

point, either by vaporization or melting. The resulting debris is dispersed through a cloud which expands behind the shield. The impact of this finely dispersed debris upon the wall of the spacecraft induces much less damage to the wall, thus saving the equipment inside the spacecraft. Obviously, the bumper plate should be made of a low melting-point material and of high acoustic impedance, in order to induce high shock pressures (and temperatures) in the impacting micrometeoroid. It turned out, already in the early stages of the research, that cadmium has the optimal properties as a bumper shield. However, the research on hypervelocity impacts has been expanded for other materials, as reviewed by Hermann and Wilbeck (1987). A somewhat lighter design than the original bumper shield is based on two thin shields spaced apart, which further reduce the amount of debris and its spatial extent when it hits the main wall of the spacecraft. Other materials were investigated as possible candidates for bumper shields such as porous aluminum plates in Onose et al. (2012), and a two-layered structure of fiberreinforced plastic backed by an aluminum honeycomb core in Taylor et al. (1999). All the experiments within this velocity regime were performed with light gas guns (described in Chap. 1) which were developed particularly for this space-oriented research. The summary of Piekutowski (1996) is especially informative since it presents many flash X-ray shadowgraphs of the debris clouds formed by different projectiles (spheres and cylinders of various sizes) impacting thin plates at velocities up to about 7.0 km/s. These shadowgraphs reveal interesting details in the structure of the debris bubble, with which one can follow the trajectories of the debris particles from the projectile and the target. This is clearly seen in Fig. 7.70, which shows such a bubble at two different times after the impact of 12.7 mm aluminum sphere on a 2.03 mm thick aluminum plate, at a velocity of 6.38 km/s. One can follow the complex structure of the bubble and determine the size and velocity of the different fragments comprising the debris cloud.

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Fig. 7.70 The bubble formed by the impact of an aluminum sphere on an aluminum plate

The most interesting shadowgraphs are those which highlight the difference between materials which remain solid, and those which melt upon impact or during the subsequent release process. This difference is clearly evident in the two shadowgraphs in Fig. 7.71, from Anderson and Mullin (1988). These debris clouds resulted from impacts of a lead sphere on a lead plate, and a molybdenum sphere on a molybdenum plate. Both experiments were performed at an impact velocity of 6.58 km/s and the flashes were taken some 40 μs after impact. One should note that the densities of these materials are very close; ρ = 10.21 g/cc and 11.35 g/cc for molybdenum and lead, respectively. Thus, the shock pressures in the two impacts were practically the same, resulting in similar shock-induced temperatures. However, the melting temperature of lead is much lower which is the reason for the significant difference in the shape of these bubbles. The lead-on-lead impact shows a considerable amount of vapor and molten material in the debris cloud, while the molybdenum impact resulted in the regular solid fragmentation with no evidence for melting of either sphere or plate. The efficiency of the bumper plate, as compared with a single plate, is determined by comparing the critical (minimal) diameters of the spheres (Dc ) which can penetrate the two arrangements, at a given impact velocity. Typical results are shown in Fig. 7.72, for aluminum spheres impacting aluminum plates. One can clearly see the advantage of the bumper plate arrangement for impact velocities above 3 km/s.

Fig. 7.71 Shadowgraphs of Molybdenum impact (left) and lead impact (right) after about 40 μs

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381

Fig. 7.72 The advantage of the Whipple shield

One of the major interests of the research into hypervelocity impacts concerns the diameter of the holes created by various spheres in thin plates. As discussed above, the diameter of the hole in a thin plate is smaller than the diameter of a crater produced by the same projectile in a thick (semi-infinite) target. This is the result of the much faster decay of the shock pressures in thin plates due to the additional release waves from their back surface. This basic feature has been found experimentally in many studies such as in Eichelberger and Gehring (1961), who used flash X-ray radiography to follow the growth of the hole’s diameter in aluminum and steel plates. The work of Piekutowski and his colleagues spans many years of careful experimentation and data collection for aluminum spheres with diameters between 1.4 and 19 mm, impacting aluminum plates with normalized thicknesses of 0.008–0.618. Impact velocities were in the range of V 0 = 2.0–9.9 km/s, using two-stage and three-stage light gas guns. Rosenberg and Kositski (2017b) analyzed the data from Piekutowski and Poormon (2015) and Fig. 7.73 shows the comparison between their model’s predictions and the normalized hole diameters in aluminum 6061-T6 plates. Fig. 7.73 Comparing the model’s predictions with data for Al-6061-T6 plates. The dashed lines denote 5% deviations

382

7 Asymmetric Interactions

The dynamic strength of the aluminum alloy in these calculations was set at Y t = 0.42 GPa, as discussed previously. The vast majority of the data points are within five percent from the model’s predictions, as shown by the two dotted lines in the figure with slopes of 1.05 and 0.95. Similar agreements between model and data for the hole diameter, were obtained by Rosenberg and Kositski (2017b) for other metals such as copper and cadmium. In summary, the semi-analytical model in Rosenberg and Kositski (2017b) accounts for a large set of experimental data for the hole diameter in thin plates impacted by metallic spheres at the hypervelocity range. The model shows that the normalized hole diameter (Dh /Dp ) is determined by three non-dimensional parameters: the density ratio (ρ p /ρ t ), the normalized plate thickness (h/Dp ) and the inertia/strength ratio of the plate material (ρ t V 2 /Y t ), namely: Dh = f Dp



ρ p h ρt V 2 , , ρt D p Y

 (7.26)

where the target’s strength (Y t ) is its flow stress at strain-rates of the order of 104 s−1 . Obviously, for like-material impacts (ρ p /ρ t = 1) there are only two non-dimensional parameters which control the hole growth process and determine the hole diameters in thin plates.

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