Notes on Projectile Impact Analyses [1st ed.] 978-981-13-3252-4;978-981-13-3253-1

Table of contents :
Front Matter ....Pages i-xiv
Rigid Projectile Impact on Concrete Target (Hao Wu, Yong Peng, Xiangzhen Kong)....Pages 1-56
Geometrical Scaling Effect of Hard Projectiles Impacting Concrete Targets (Hao Wu, Yong Peng, Xiangzhen Kong)....Pages 57-109
High-Fidelity Physics-Based Numerical Simulation of Concrete Structures Subjected to Intense Dynamic Loadings (Hao Wu, Yong Peng, Xiangzhen Kong)....Pages 111-165
Impact Performance of Shaped Charge Formed Jet into Concrete Targets (Hao Wu, Yong Peng, Xiangzhen Kong)....Pages 167-240
Rigid and Erosive Projectiles Impact on Metallic Targets (Hao Wu, Yong Peng, Xiangzhen Kong)....Pages 241-306
Interface Defeat of Long-Rod Projectile Impacting on Ceramic Targets (Hao Wu, Yong Peng, Xiangzhen Kong)....Pages 307-370

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Hao Wu · Yong Peng · Xiangzhen Kong

Notes on Projectile Impact Analyses

Notes on Projectile Impact Analyses

Hao Wu Yong Peng Xiangzhen Kong •

•

Notes on Projectile Impact Analyses

123

Hao Wu College of Civil Engineering Tongji University Shanghai, China

Yong Peng College of Liberal Arts and Sciences National University of Defense Technology Changsha, China

Xiangzhen Kong College of Defense Engineering Army Engineering University of PLA Nanjing, China

ISBN 978-981-13-3252-4 ISBN 978-981-13-3253-1 https://doi.org/10.1007/978-981-13-3253-1

(eBook)

© Springer Nature Singapore Pte Ltd. 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Concrete, metal, and ceramic materials have been widely used in protective structures of both civil engineering and armored vehicles, such as military fortifications, underground shelter, infantry fighting vehicle, tank, etc., which are designed to withstand the intentional or accidental impact loadings caused by projectiles, fragments, and so on. In this monograph, the authors present their theoretical, experimental, and numerical works on some important issues in the projectile impact analyses. The main contents are introduced as follows: Chapter 1: In this chapter, based on the mean resistance approach, the penetration model for projectile impact on various typical targets is firstly established and validated. Then, a unified projectile perforation model for both thick and thin concrete slabs is further developed and applied into the impact resistance evaluation of the segmented targets. Finally, considering that the resistance acting on the projectile is critical for the penetration analyses, a modified spherical cavity-expansion model is established and discussed in detail. Chapter 2: In this chapter, the size effect in rigid projectile penetrations into concrete target is discussed. Based on the available experimental data and discussions on the empirical formulae, it is verified that the replica scaling law is satisfied for depth of penetration (DOP) in rigid projectile penetrations, as long as the scaling is done strictly for both projectiles and concrete targets including the coarse aggregates. The coarse aggregates with invariant size (not replica-scaled) could account for the size effect in DOP found in tests and empirical formulae. To explore the size effect in DOP caused by aggregates, a 3D mesoscopic finite element model for concrete target is developed, and the related parameters are discussed in detail. Finally, based on the numerical results, a semi-analytical model for predicting DOP is proposed, which improved the model in Chap. 1 by further considering the size effect. Chapter 3: In this chapter, our recent progress in the high-fidelity physics-based numerical simulation of concrete structures under projectile penetrations is presented, which includes the following three aspects. Firstly, a new material model for concrete subjected to intense dynamic loadings such as blast and impact is proposed. It can well predict both the global and local responses, fracture and failure of v

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concrete structures subjected to intense dynamic loadings covering a wide range of loading intensity. Then, to address the mesh-size dependency of concrete structures under intense dynamic loadings, a non-local approach for hydrodynamic elastoplastic material model is presented, using the modified K&C model as an example. Finally, dynamic failures in concrete structures are numerically simulated based on the SPH method, providing a promising way to predict the failures in concrete structures. Chapter 4: In this chapter, the penetration performance of EFP formed by shaped charge into concrete targets is firstly studied. Then, based on EFP penetration test, a series of rigid projectile penetration test on these pre-damaged concrete targets is further conducted, and the influences of the related parameters on the target damage subjected to the two-stage munitions are experimentally and numerically discussed. Additionally, the penetration performance of shaped charge JET into ultra-highperformance concrete (UHPC) target is examined. The influences of target material, standoff distance, and target configuration (stacked and spaced) on the impact performance of JET are experimentally assessed and numerically discussed. Chapter 5: In this chapter, the performance of the metallic targets against rigid, deformable, and erosive projectile impact is concerned. Firstly, a perforation model for rigid projectile is introduced and validated. Then, nineteen shots of hemispherical nosed D6A steel projectiles penetrating into 5A06-H112 aluminum targets are conducted with the striking velocities ranged from 696 to 1870 m/s. Furthermore, the applicability of the existing six models for predicting the rigid and eroding DOPs is evaluated, and a new penetration model for predicting DOP in the rigid, deformable, and erosive penetration regime is proposed. Finally, the numerical simulations are performed through Lagrange method, coupled SPH method, and finite element method (FEM). Chapter 6: In this chapter, the existing tests of interface defeat for long-rod projectile impacting ceramic targets carried out from 2000 to 2011 are firstly reviewed. Then, by modeling the projectiles and ceramic targets with the SPH particles and Lagrange finite elements, the above tests are numerically reproduced. Furthermore, the influences of projectile characteristics, constitutive models of ceramic, buffer, and cover plate characteristics, as well as the prestress of target on the transition velocity and dwell time are systematically discussed. Finally, a modified model to predict the upper bound of the transition velocity (dwell to direct penetration) is proposed. This monograph is written for the researchers and engineers working in the fields of protective structures and high-speed penetration mechanics. It can also be referred by the senior undergraduate and postgraduate students majored in the defense engineering, terminal ballistic, etc. We acknowledge our team members Feng Hu, Yukai Xiao, and Yangxiu Zhai from Army Engineering University of PLA for their contributions to the preparation of this book. Shanghai, China Changsha, China Nanjing, China

Hao Wu Yong Peng Xiangzhen Kong

Contents

1 Rigid Projectile Impact on Concrete Target . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Unified Penetration Model for Different Targets . . . . 1.2.1 Mean Resistance . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 DOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A Unified Perforation Model for Thick and Thin Concrete Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Perforation Process . . . . . . . . . . . . . . . . . . . . . 1.3.2 Height of Rear Crater . . . . . . . . . . . . . . . . . . . 1.3.3 Perforation Limit and Ballistic Limit . . . . . . . . . 1.3.4 Residual Velocity . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Perforation Test on the Thin Concrete Slabs . . . 1.3.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 A Modified Cavity-Expansion Theory . . . . . . . . . . . . . 1.4.1 Modified Spherical Cavity-Expansion Model . . . 1.4.2 Numerical Results Based on Modified Spherical Cavity-Expansion Model . . . . . . . . . . . . . . . . . 1.4.3 Rigid Projectile Normal Penetration Model . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Geometrical Scaling Effect of Hard Projectiles Impacting Concrete Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Irregularity of Scaling in the Empirical Formulae . . . . . 2.2.1 Representative Empirical Formulae . . . . . . . . . . 2.2.2 Discussions on the Empirical Formulae . . . . . . .

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2.3 3D Mesoscopic Model for Concrete Target . . . . . . . . . . . . . . . 2.3.1 3D Mesoscopic Concrete Model with Convex Polyhedron Coarse Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 3D Mesoscopic Concrete Model with Sphere Coarse Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Numerical Simulations of Projectile Penetrations into Concrete Targets with Regular Aggregates . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Influence of the Cement Strength . . . . . . . . . . . . . . . . . 2.4.2 Influence of the Aggregate Strength . . . . . . . . . . . . . . . 2.4.3 Influence of the Volume Fraction of Aggregates . . . . . . 2.4.4 A Semi-analytical Formula for DOP with Considering the Non-scaling Effect . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Numerical Simulations of Projectile Penetrations into Concrete Targets with Random Aggregates . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Effects of Two Coarse Aggregates Shapes on Concrete Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Validation of 3D Mesoscopic Concrete Model . . . . . . . 2.5.3 Scaling Effect Analyses . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 High-Fidelity Physics-Based Numerical Simulation of Concrete Structures Subjected to Intense Dynamic Loadings . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A New Concrete Material Model . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Single Element Numerical Tests . . . . . . . . . . . . . . . . . . 3.2.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Non-local Formulation to Resolve the Mesh-Size Dependency . 3.3.1 Limitations of the Smeared Crack Model . . . . . . . . . . . 3.3.2 Non-local Formulation of the MKC Model . . . . . . . . . . 3.3.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Predictions of Local Failures in Concrete Structures . 3.4.1 Limitations of the Element Erosion Algorithm . . . . . . . . 3.4.2 Numerically Predicted Failures Using SPH Method . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Impact Performance of Shaped Charge Formed Jet into Concrete Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Review of the Existing Studies . . . . . . . . . . . . . . . . . . . . 4.1.1 Impact Performance of SC Formed Jet . . . . . . . . . . . 4.1.2 Rigid Projectile Penetration into Target Pre-damaged by SC Formed Jet . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Impact Performance of EFP into Concrete Targets . . 4.2.1 EFP Impact Test . . . . . . . . . . . . . . . . . . . . . 4.2.2 Numerical Simulations and Comparisons . . . . 4.2.3 Parametric Influences . . . . . . . . . . . . . . . . . . 4.3 Rigid Projectile Penetration into Concrete Targets Pre-damaged by EFP . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Results and Discussions . . . . . . . . . . . . . . . . 4.3.3 Numerical Simulations . . . . . . . . . . . . . . . . . 4.3.4 Further Discussions of Two-Stage Munitions . 4.4 Impact Performance of JET on UHPC Targets . . . . . 4.4.1 JET Penetration Test . . . . . . . . . . . . . . . . . . 4.4.2 Results and Discussions . . . . . . . . . . . . . . . . 4.4.3 Numerical Simulations . . . . . . . . . . . . . . . . . 4.4.4 Parametric Analyses . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Rigid and Erosive Projectiles Impact on Metallic Targets . . . . . 5.1 Rigid Projectile Perforation Model . . . . . . . . . . . . . . . . . . . . . 5.1.1 Resistance for the Compressible Power-Law Strain-Hardening Target . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 A New Model for Projectile Perforating Metallic Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Influence of Impact Velocity on the Rear Free-Surface Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Eroding Penetrations: Test and Analysis . . . . . . . . . . . . . . . . 5.2.1 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Evaluations of the Existing Models . . . . . . . . . . . . . . . 5.3 Penetration Model with Rigid, Deformable, and Erosive Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Analysis of Existing Models . . . . . . . . . . . . . . . . . . . . 5.3.2 New Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Comparisons with the Test Data . . . . . . . . . . . . . . . . . 5.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Calibration of Johnson–Cook Model . . . . . . . . . . . . . . 5.4.2 Numerical Simulation with Lagrange Algorithm . . . . . 5.4.3 Numerical Simulation with SPH–FEM Algorithm . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Interface Defeat of Long-Rod Projectile Impacting on Ceramic Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Review of the Existing Experimental Studies of Interface Defeat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Behner’s Test (Behner et al. 2008, 2011, 2016) . . . . . . 6.2.2 Lundberg’s Test (Lundberg et al. 2000, 2006; Westerling et al. 2001; Lundberg and Lundberg 2005) . 6.2.3 Anderson’s Test (Anderson et al. 2011) . . . . . . . . . . . 6.3 Validation of Numerical Models of Pure Gold Projectile and SiC Ceramic Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Experimental Details of Penetration Test (Behner et al. 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Validation of Numerical Models . . . . . . . . . . . . . . . . . 6.4 Numerical Simulation of the Existing Tests on Interface Defeat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Behner’s Test (Behner et al. 2008, 2011, 2016) . . . . . . 6.4.2 Lundberg’s Test (Lundberg et al. 2000, 2006; Westerling et al. 2001; Lundberg and Lundberg 2005) . 6.4.3 Anderson’s Test (Anderson et al. 2011) . . . . . . . . . . . 6.5 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Projectile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Constitutive Model of Ceramic . . . . . . . . . . . . . . . . . . 6.5.3 Buffer and Cover Plate . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 A Modified Model to Predict the Upper Bound of the Transition Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Predictions for the Lower and Upper Bounds of the Transition Velocity . . . . . . . . . . . . . . . . . . . . . 6.6.2 A Modified Model to Predict the Upper Bound of the Transition Velocity . . . . . . . . . . . . . . . . . . . . . 6.6.3 Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

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Abbreviations

ACE ALE A-T BHN BRL CD CET CRH DEM DOP EFP EOS FE FEM FPZ FSI HC HEL HJC HSC ITZ JPC KE MKC NDRC NSC RHA RHT SC SHPB

Army corps of engineers Arbitrary Lagrange–Euler Alekseevskii–Tate Brinell’s hardness Ballistic Research Laboratory Charge diameter Cavity-expansion theory Caliber-radius-head Discrete element method Depth of penetration Explosively formed projectile Equation of state Finite element Finite element method Fracture process zone Fluid–structure interaction Hydrostatic compression Hugoniot elastic limit Holmquist–Johnson–Cook High-strength concrete Interfacial transition zone Jetting projectile charge Kinetic energy Modified K&C National Defense Research Committee Normal-strength concrete Rolled homogeneous armor Riedel–Hiermaier–Thoma Shaped charge Split Hopkinson pressure bar

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SHTB SOD SPH TCK TXC UHPC UHPCC UHP-SFRC UUC UUT UX WFRCAs

Abbreviations

Split Hopkinson tensile bar Standoff distance Smooth particle hydrodynamics Taylor–Chen–Kuszmaul Triaxial compression Ultra-high-performance concrete Ultra-high-performance cement-based composite Ultra-high-performance steel-fiber-reinforced concrete Unconfined uniaxial compression Unconfined uniaxial tension Uniaxial strain Woven fabric rubber composite armors

Nomenclature

Fx N1, N2 L d lm rs l M V0 I 0, N fc Ft E P Vbl Vr w rr, rө m K q0 S or R I H hper G DIFt DIFc J2

Instantaneous axial resistance acting on the projectile during penetration, N Projectile nose shape factor Projectile length, m Projectile’s shank diameter, m Sliding friction coefficient between the projectile and target Quasi-static target material strength, MPa Projectile nose length, m Mass of projectile, kg Initial striking velocity of projectile, m/s Dimensionless parameters Uniaxial compressive strength of concrete, Pa Uniaxial tensile strength of concrete, Pa Young’s modulus, MPa Depth of penetration, m Ballistic limit, m/s Residual velocity of projectile after perforating target, m/s Caliber-radius-head (CRH) of projectile Radial and circumferential Cauchy stress, MPa Poisson’s ratio Bulk modulus of elasticity Density of target, kg/m3 Dimensionless concrete strength parameter Dimensionless impact factor Thickness of target, m Perforation limit of target, m Shear modulus, MPa Dynamic increase factor for tension Dynamic increase factor for compression Second deviatoric stress invariant

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Gf efrac k r0 lm Rc VR VE

Nomenclature

Fracture energy, J Fracture strain Modified equivalent plastic strain Ratio of current meridian to compressive meridian Sliding friction coefficient Rockwell hardness Upper limit velocity of rigid penetration, m/s Lower limit velocity of erosive penetration, m/s

Chapter 1

Rigid Projectile Impact on Concrete Target

Abstract In this chapter, based on the mean resistance approach, the penetration model for projectile impact on various typical targets is firstly established and validated. Then, a unified projectile perforation model for both thick and thin concrete slabs is further developed and applied into the impact resistance evaluation of the segmented targets. Finally, considering that the resistance acting on the projectile is critical for the penetration analyses, a modified spherical cavity-expansion model is established and discussed in detail.

1.1 Introduction Terminal ballistic parameters of hard projectiles penetrating/perforating into concrete targets are the main focuses for both civil engineers and weapon designers. The related experimental, analytical, and numerical studies can be referred to the reviews by Li et al. (2005), Corbett et al. (1996) and Anderson and Bodner (1988). The empirical formulae received widely applications, and some of them are the fundamentals of the design codes (e.g., TM5-855-1, TM-1300) of protective structures. However, the empirical formulae are only applicable within the parametric ranges that the tests covered, and some of them even are unit-dependent. Numerical simulations can provide visible information on damage, stress, and deformation field, while the related parameters in material model of targets are difficult to be determined. Experiment-based analytical (semi-analytical) approaches could be the most efficient and economical way to study the local effects of the projectile impacting on concrete targets. For rigid projectile penetration, based on the cavity-expansion theory (CET) and assuming that the normal stresses acting on the projectile’s nose equal to the expansion stresses of the spherical or cylindrical cavity, Forrestal and Luk (1992), Forrestal and Tzou (1997), Forrestal et al. (1995), Forrestal and Warren (2008), Luk et al. (1991), and Warren et al. (2004) proposed a series of formulae to predict DOPs of ogive-nosed projectiles penetrating into soil, concrete, ductile metal, and rock targets. For the complexity of solving the cavity-expansion theory, based on the penetration test data, Forrestal et al. (1994, 1996) and Frew et al. (1998) curve fitted © Springer Nature Singapore Pte Ltd. 2020 H. Wu et al., Notes on Projectile Impact Analyses, https://doi.org/10.1007/978-981-13-3253-1_1

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1 Rigid Projectile Impact on Concrete Target

a dimensionless empirical parameter S, which is dependent only on the concrete compressive strength, to simplify the expression of the cavity-expansion stress. Furthermore, by considering the target boundary influences on the penetration process, Forrestal et al. (2003) and Frew et al. (2006) proposed another parameter R to describe the penetration resistance of concrete target. However, the equivalent target strength parameter R must be confirmed by the actual projectile penetration test data and the prediction function is lost. By introducing the dimensionless projectile-nosed geometry function and impact factor, as well as considering the projectile–target interfacial frictions, Chen and Li (2002) as well as Li and Chen (2003) extended the Forrestal formula to the dimensionless form to analyze projectiles with arbitrary nose profiles penetration into diversified targets. Since the target strength parameter S used in Forrestal et al. (1994, 1996), Frew et al. (1998), Chen and Li (2002) and Li and Chen (2003) was obtained by fitting the test data of ogive-nosed projectiles penetrating into normal strength concrete targets, Wu et al. (2012a) further regressed the expression of S for high-strength concrete (HSC) targets subjected to projectiles with various nose shapes, which were validated by the penetration tests on HSC targets with the maximum compressive strength of ~ 250 MPa. The resistances acting on the projectile during the penetration in the above studies are dependent on the instantaneous velocity of the projectile. For the derivation of DOP, this makes it more complicated in integrating the motion equation, especially for the complex projectile nose profile and consideration of the interfacial friction between the projectile and target. For rigid projectile perforations, based on the three-stage (cratering + tunneling + shear plugging) perforation model, Chen et al. (2008) and Li and Tong (2003) proposed the formulae for predicting the ballistic limit and perforation limit, as well as the residual velocity of the projectile after perforations. By further considering the kinetic energy carried by the ejected fragments on the rear face of the concrete slab, Wu et al. (2012a) proposed an expression to predict the residual velocity of the projectile after perforations. Recently, by considering the energy dissipated through the fracture of ejecting fragments into pieces, Grisaro and Dancygier (2014) proposed a modified energy balance model of the projectile perforating on the concrete slab. However, the perforation limit in this model is obtained by curve fitting of the test data and can only applied for residual velocity predictions. Besides, the height of rear crater and the ejecting velocity of rear shear fragment are the two key factors in the projectile perforation model for the concrete slab. However, the rear crater height equation proposed in Chen et al. (2008) and Li and Tong (2003) is very complex, and its accuracy is not validated for the lack of test data. More recently, Wu et al. (2015a) conducted a series of projectile perforation tests on concrete slabs with thicknesses ranged from 100 to 300 mm, and the striking and residual velocities of the projectiles as well as the dimensions of the front impact crater and rear shear crater were recorded in detail. In this chapter, based on the mean resistance approach, the penetration model for projectile impact on various typical targets is firstly established and validated in Sect. 1.2. Then, in Sect. 1.3, a unified projectile perforation model for both thick and thin concrete slabs is further developed and applied into the impact resistance evaluation of the segmented targets. Finally, considering that the resistance acting

1.1 Introduction

3

on the projectile is critical for the penetration model, more detailed discussions on it should be included, and thus, a modified spherical cavity-expansion model is established and discussed in Sect. 1.4.

1.2 A Unified Penetration Model for Different Targets In this section, aiming to establish a simple and effective projectile penetration model into various targets (e.g., concrete, metal, rock), by assuming that the resistance acting on the projectile keeps unchanged during the penetration process, a mean resistance approach based on dynamic cavity-expansion approximation is proposed. A simple and effective projectile penetration model is further given to predict the DOP of different nose-shaped hard projectiles penetrating into diversified targets. Besides, the related mean resistance coefficient is confirmed as 0.4 based on the parametric analyses.

1.2.1 Mean Resistance The instantaneous axial resistance F x acting on the projectile during penetration process mainly consists of two parts, the quasi-static resistance term and the dynamic term (or the inertial term) induced by the projectile velocity Fx =

 π d2  N1 σs + B N2 ρ0 V 2 4

8μm N1 = 1 + 2 d

(1.1a)

l0 ydx

(1.1b)

0

8 N2 = 2 d

l0 0

yy 3 8μm dx + 2 1 + y 2 d

l0 0

yy 2 dx 1 + y 2

(1.1c)

where d and V are the shank diameter and the instantaneous velocity of projectile, respectively. ρ 0 is the density of the target, σ s is the quasi-static target material strength, B is the coefficient of dynamical resistance derived from dynamic cavityexpansion analysis, N 1 , N 2 are projectile nose shape factor, μm is the sliding friction coefficient between the projectile and target during impact, and y = y(x) is the nose shape function as shown in Fig. 1.1. Actually, the deceleration test data (Forrestal and Luk 1992; Forrestal et al. 2003) of ogive-nosed projectile increase almost linearly with the displacement x during the initial penetration stage, namely F x = kx and k is a constant. Thus, the instantaneous axial resistance expressed in Eqs. (1.1a, 1.1b, 1.1c) is only applicable for

4

1 Rigid Projectile Impact on Concrete Target

y

d/2

y(x)

0

b

x

Fig. 1.1 Half profile of longitudinal section of projectile with a general nose shape, reprinted from Peng et al. (2015), copyright 2019, with permission from Elsevier

the stable penetration phase, but not for the initial penetration stage. To describe this phenomenon, Forrestal et al. (1994) assumed that the initial penetration stage corresponds to the stage when the projectile entering into the frontal crater region of the concrete target and the depth of crater of which is adopted as 2d. In other words, the resistance F x = kx in the crater region, and if the projectile penetrating beyond the crater region, the resistance in Eqs. (1.1a, 1.1b, 1.1c) is adopted. Chen and Li (2002) held the similar opinion and regarded the frontal impact crater depth as 0.707d + l (where l is the length of the projectile nose). However, the aforementioned method is not suitable for the flat and truncated-nose projectile because the instantaneous resistance acting on these projectiles is far more than zero when the projectile impacts on the target frontal face. Of course, the aforementioned method is also not suitable for the targets made of ductile material since there are not obvious impact crater for these targets. Unlike the above method, Teland and Sjøl (2004) considered that the value of resistance in the initial penetration stage is proportional to the pressed area, and the initial penetration stage ends when the projectile nose has completely entered into the target. Based on this assumption, the reason why F x = kx for ogive-nosed projectile in the initial penetration stage is that the pressed area increases linearly with the displacement approximately. This methodology for the initial penetration stage is suitable for projectiles with arbitrary nose shapes and does not depend on the initial impact responses of the targets. Figure 1.2 shows the instantaneous images of a projectile penetrating into the concrete target (Li et al. 2013), and it can be seen that the target still has no spalling when the projectile nose completely invades the target. Thus, the assumption that the pressed area increases with the displacement is also correct for the concrete-like material which the impact crater will occur. For deep penetrations, the initial penetration stage has little influence on DOP and it could be neglected for simplification. Thus, the penetration depth hpen can be derived by integrating Eqs. (1.1a, 1.1b, 1.1c) with considering the Newton’s second law   2M B N2 ρ0 V02 P= ln 1 + π d 2 B N2 ρ0 N1 σs

(1.2)

1.2 A Unified Penetration Model for Different Targets

5 Projectile

Projectile

sabot fragments

sabot fragments

Target

Target

Fig. 1.2 Instantaneous images of a projectile penetrating into the concrete target, reprinted from Li et al. (2013), copyright 2019, with permission from Elsevier

where M and V 0 are the mass and the striking velocity of a projectile, respectively. The deviation of considering the initial penetration stage or not is about l/2. The resistance in Eq. (1.2) is complicated since it is dependent on the instantaneous projectile velocity. As described in the following, we will present a mean resistance function which keeps unchanged during the tunneling stage. Figure 1.3 illustrates the typical deceleration–instantaneous penetration depth curve (red dashed line 0–1–2–3) of the projectile penetrating into concrete targets, which could be obtained from Eqs. (1.1a, 1.1b, 1.1c, 1.2). For rigid projectile penetration, the ascending part 0–1 and descending part 1–2 correspond to the projectile cratering and tunneling stages, respectively, and the projectile stops penetration at the point 2. For simplicity, the mean deceleration during the whole penetration process which is only dependent on the initial striking velocity (black solid line 0–1 –2 –3) is proposed and illustrated in Fig. 1.3. The precondition of which is that the works done by the actual and mean resistances are equal, that is to say the areas A1 = A2 is satisfied in Fig. 1.3.

Instantaneous deceleration

50000

Deceleration (g)

Fig. 1.3 Diagram of projectile deceleration versus instantaneous penetration depth, reprinted from Peng et al. (2015), copyright 2019, with permission from Elsevier

1

40000

1'

Mean deceleration A1=A2

A2

A1

30000

2' 2

20000 10000 0 0.0

0.2

0.4

x (m)

0.6

0.8

3 hpen

1.0

6

1 Rigid Projectile Impact on Concrete Target

Introducing a mean resistance coefficient μ into Eqs. (1.1a, 1.1b, 1.1c), the mean resistance F m can be written as Fm = δ=

π d2 (1 + μδ)N1 σs 4

(1.3a)

I0 B N2 ρ0 V02 = N N1 σs

(1.3b)

The parameter δ denotes the ratio of dynamic to quasi-static resistance. I 0 and N are two dimensionless parameters proposed by Chen and Li (2002) I0 =

M V02 , N1 σs d 3

N=

M Bρ0 d 3 N2

(1.4)

It can be found that a larger N corresponds to a sharper and slender projectile, and I 0 is mainly dependent on the striking velocity according to Eq. (1.4). As for concrete and metal targets, the related parameters in Eqs. (1.3a, 1.3b) were suggested as follows: Concrete (Forrestal et al. 1994; Frew et al. 1998): σs = S f c , S = 82.6( f c /106 )−0.544 ,

B=1

(1.5a)

Metal (Forrestal et al. 1995; Forrestal and Warren 2008): σs =

  2E n 2Y 1+( ) I , 3 3Y

b I = 0

3Y (− ln x)n dx, b = 1 − , 1−x 2E

B = 1.5 (1.5b)

where f c is the uniaxial compressive strength of concrete Pa, E is Young’s modulus, Y is the yield stress, and n is the strain-hardening exponent to be determined by the curve fitting to the compression stress–strain test data of metal targets. We can also find the related parameters for soil, limestone targets, etc. in Forrestal and Luk (1992) and Warren et al. (2004). The nose shape factors N 1 and N 2 for common projectiles are listed in Table 1.1, in which the effect of friction is not included. For concrete-like targets, the friction between the projectiles and targets is negligible, and the nose shape factors in Table 1.1 could be employed directly. For metal targets, the friction at the projectile–target interface could not be neglected, and we multiply the nose shape factor N 1 and N 2 listed in Table 1.1 by a coefficient of 1.1 in the following predictions for simplicity. Other nose shape factor such as truncated-ogive nose and truncated-conical nose, as shown in Fig. 1.4, can be obtained by integrating Eqs. (1.1a, 1.1b, 1.1c). For truncated-ogive-nosed projectiles, the nose shape factors are

1.2 A Unified Penetration Model for Different Targets

N1 = 1; N2 =

ζ32

+ 8ψ

2

7

 

1 (ζ1 − ζ 2 )4 (2ψ − 1 + ζ3 )3 2ψ − 1 2ψ − 1 + ζ3 2 − + − 4 ψ4 2ψ 2ψ 24ψ 3 3 (1.6a)

where ψ = r/d, ζ 1 = l/d, ζ 2 = l 1 /d, ζ 3 = d 1 /d. When ζ 3 = 1, then ζ 1 = ζ 2 , Eq. (1.6a) becomes the nose shape factor for flat-nosed projectile; When ζ 2 = ζ 3 = 0, Eq. (1.6a) is the nose shape factor for ogive-nosed projectile. For truncatedconical-nosed projectiles:  N1 = 1;

N2 =

ζ32

+

1 − ζ32 4ζ12 + 1

 (1.6b)

When ζ 3 = 1, Eq. (1.6b) becomes the nose shape factor for flat-nosed projectile, and when ζ 2 = ζ 3 = 0, Eq. (1.6b) is the nose shape factor for conical-nosed projectile.

Table 1.1 Nose shape factor (without consideration of friction), reprinted from Peng et al. (2015), copyright 2019, with permission from Elsevier Nose shape

N1

N2

Definition

Ogival

1

1/3ψ − 1/24ψ 2 (0 < N 2 < 0.5)

ψ is CRH

Conical

1

1/(1 + 4ψ 2 ) (0 < N 2 < 1)

ψ = l/d

1

1 − 1/8ψ 2

ψ = r/d and r is the radius of sphere

Blunt

(0.5 < N 2 < 1)

d

(a)

d

(b)

x

x s

d1

σr

l1

σr

σr

l

θ1

l

σθ

s

θ

θ0

l1

σr

θ

στ

dθ

d1

Fig. 1.4 Geometry for projectile with a truncated-ogive nose, b truncated-conical nose

8

1 Rigid Projectile Impact on Concrete Target

1.2.2 DOP For deep penetration, considering the work done by the mean resistance equals to the initial kinetic energy of a projectile, it is obtained that Fm P =

1 M V02 2

(1.7)

Substituting Eqs. (1.3a, 1.3b) into Eq. (1.7) gives the dimensionless DOP as 2 I0 P = d π (1 + μδ)

(1.8)

When μ = 0, the inertial term is neglected, Eq. (1.8) is simplified to P 2 = I0 d π

(1.9)

Equation (1.9) was proposed by Forrestal et al. (2003) and Chen and Li (2002) to predict the low-to-mid speed impact of non-flat projectiles. It should be noted that Eq. (1.8) is only suitable for deep penetration, and adding l/2 to the DOP predicted by Eq. (1.8) is suggested for concrete targets when the final penetration depth is relatively shallow. The value of the mean resistance factor μ is derived by substituting Eq. (1.2) into the DOP in Eq. (1.8) μ=

1 1 − ln(1 + δ) δ

(1.10)

For a given target, parameter δ is closely related to the striking velocity V 0 and nose shape factor N 2 based on Eq. (1.3b). Figure 1.5 shows the variation of δ for various projectiles penetration into a concrete target with f c = 40 MPa. The plane (V 0 , N 2 ) in Fig. 1.5 is limited within a practical variation range of rigid projectiles (Li and Chen 2003), and it shows that the value of δ is less than 5. Actually, δ < 1 is always satisfied for ogive-nosed or conical-nosed projectiles. Figure 1.6 plots the relationship of the parameters μ and δ according to Eq. (1.10). It indicates that μ drops slightly with the increase of δ, and the value of μ is almost around 0.4 when δ < 5 as shown in Fig. 1.5. We attempt to set a constant μ1 instead of the coefficient μ expressed in Eq. (1.10), by substituting μ1 and Eq. (1.10) into Eq. (1.8), the deviations of the two predicted DOPs can be expressed as follows =

|μ1 − μ|δ × 100% 1 + μ1 δ

(1.11)

1.2 A Unified Penetration Model for Different Targets

5 4 3

δ

Fig. 1.5 Dependence of δ on striking velocity V 0 and nose shape factor N 2 , reprinted from Peng et al. (2015), copyright 2019, with permission from Elsevier

9

2 1 0 1

0.8

0.6

N2

Fig. 1.6 Relationship of μ with δ, reprinted from Peng et al. (2015), copyright 2019, with permission from Elsevier

0.4

0.2

0

0

200

400

600

800

1000

V0 (m/s)

1.0

Eq. (1.10) 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Figure 1.7 shows the above deviation versus the parameter δ, which is predicted by Eq. (1.11) with μ1 = 0.4. It indicates that the deviation is less than 7% when δ < 5. Especially, for the ogive-nosed or conical-nosed slender projectiles (δ < 1), the deviation is further less than 3%. Therefore, μ = 0.4 is assumed reasonably. Now it is interesting to examine the application range of the simple formula (Eq. 1.9), which neglects the inertial term. Substituting Eq. (1.10) and μ1 = 0 into Eq. (1.11) will give the deviation of the DOP predictions by Eq. (1.9). If the deviation of 10% is accepted, the application range of Eq. (1.9) proposed by Forrestal et al. (2003) and Chen and Li (2002) is limited to δ < 0.2. When the striking velocity is relatively low, the nose of the projectile could not penetrate into the target completely, there is only the initial penetration stage, and the instantaneous axial resistance F x acting on the projectile is x Fx = Fm , x ≤ l l

(1.12a)

10

1 Rigid Projectile Impact on Concrete Target

Fig. 1.7 Relationship of  with δ, reprinted from Peng et al. (2015), copyright 2019, with permission from Elsevier

8

Eq. (1.11) with

7

1 =0.4

(%)

6 5 4 3 2 1 0 0

1

2

3

4

5

Then the DOP could easily be obtained based on the conservation of energy P = d



4I0 l , π d

P