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Penetration of a Shaped Charge

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Penetration of a Shaped Charge Penetration of a Shaped Charge Chris Poole Corpus Christi College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2005 Acknowledgements This research was funded by the EPSRC and QinetiQ, both of whom I would like to thank for funding the project. I would also like to thank my supervisor, Jon Chapman, for his inspiration and guidance throughout this work. I am also indebted to John Curtis (QinetiQ) for his invaluable support on the project. I must also thank John Ockendon, who has been a fountain of enthusiasm and stimulation throughout the project. Sasha Korobkin must be thanked for his input and insight into the chapter on filling-flows. The metallurgical analysis would not have been possible without the help of Paula Topping and others in the Materials Department, whom I would like to acknowledge. Finally, on a Mathematical note, I would like to thank all those in OCIAM who have helped me throughout the project. On a more personal level, I would like to thank all of those who have supported me and given me encouragement (and distractions!) since I have been in Oxford. In particular, I must mention the support of my family, the joviality of my friends in OCIAM (especially the DH9/DH10 folk, past and present), the conviviality of the OUSCR (and other ringers), and any other friends I haven’t yet mentioned. It is the mixture of all these people that has kept me (relatively) sane and made my time in Oxford so enjoyable. Abstract A shaped charge is an explosive device used to penetrate thick targets using a high velocity jet. A typical shaped charge contains explosive material behind a conical hollow. The hollow is lined with a compliant material, such as copper. Extremely high stresses caused by the detonation of the explosive have a focusing effect on the liner, turning it into a long, slender, stretching jet with a tip speed of up to 12km s−1. A mathematical model for the penetration of this jet into a solid target is developed with the goal of accurately predicting the resulting crater depth and diameter. The model initially couples fluid dynamics in the jet with elastic- plastic solid mechanics in the target. Far away from the tip, the high aspect ratio is exploited to reduce the dimensionality of the problem by using slender body theory. In doing so, a novel system of partial differential equations for the free-boundaries between fluid, plastic and elastic regions and for the velocity potential of the jet is obtained. In order to gain intuition, the paradigm expansion-contraction of a circular cavity under applied pressure is considered. This yields the interesting possi- bility of residual stresses and displacements. Using these ideas, a more realistic penetration model is developed. Plastic flow of the target near the tip of the jet is considered, using a squeeze-film analogy. Models for the flow of the jet in the tip are then proposed, based on simple geometric arguments in the slender region. One particular scaling in the tip leads to the consideration of a two-dimensional paradigm model of a “filling-flow” impacting on an obstacle, such as a membrane or beam. Finally, metallurgical analysis and hydrocode runs are presented. Unresolved issues are discussed and suggestions for further work are presented. Contents 1 Introduction 1 1.1 What is a shaped charge? . 1 1.1.1 Mechanics of a shaped-charge jet . 1 1.1.2 Applications of shaped charges . 4 1.2 Background mathematics . 5 1.2.1 Elasticity . 5 1.2.2 Metal plasticity . 11 1.2.3 Asymptotics . 16 1.2.4 Slender body theory . 17 1.3 Thesis outline . 19 1.4 Statement of originality . 20 2 Shaped-charge literature 22 2.1 Hydrodynamic models . 22 2.1.1 Birkhoff jet impact . 22 2.1.2 Water jets . 25 2.2 Models from solid mechanics . 27 2.2.1 Models of penetration . 27 2.2.2 Plastic instability and jet particulation . 29 2.2.3 Numerical models . 30 2.3 Aims of thesis . 31 3 An axisymmetric elastic-plastic model for penetration 32 3.1 Philosophy of the model . 32 3.1.1 Parameter estimates . 34 3.2 The jet . 34 3.2.1 Boundary conditions . 36 3.3 Plastic region . 36 3.3.1 Boundary conditions . 38 i 3.4 Elastic region . 39 3.4.1 Boundary conditions . 40 3.5 Different scalings . 40 3.5.1 Geometric and other scaling ideas . 41 4 Slender and outer analysis 44 4.1 Asymptotic analysis of nondimensional slender equations . 44 4.1.1 Jet region . 45 4.1.2 Plastic region . 48 4.1.3 Inner elastic region . 54 4.2 Outer region . 56 4.2.1 Quasistatic outer region . 56 4.2.2 Matching with a fully inertial outer region . 62 4.2.3 Comments . 65 4.3 Travelling-wave solution . 65 4.4 Inertial effects . 70 4.4.1 Modified inertial equations . 70 4.5 Comments on elastic-plastic modelling . 72 5 Gun-barrel mechanics 75 5.1 Linear elastic perfect-plastic cavity model . 76 5.1.1 Elastic expansion . 76 5.1.2 Elastic-plastic expansion . 76 5.1.3 Simple plastic contraction with no residual stress . 80 5.1.4 A possible model for cavity-contraction permitting a residual stress 80 5.1.5 “Elastic contraction” . 83 5.1.6 Elastic-plastic contraction . 85 5.1.7 Cyclic loading-unloading . 89 5.2 A nonlinear problem . 93 5.2.1 Elastic expansion . 95 5.2.2 Elastic-plastic expansion . 95 5.2.3 Contraction with no re-yielding . 97 5.2.4 Contraction with further plastic flow . 98 5.2.5 Comment . 98 5.3 Application to shaped-charge penetration . 99 5.3.1 Remarks . 100 5.4 An asymmetric perturbation to the gun-barrel problem . 101 5.4.1 Perturbation εY (θ, t) = ε(1 + cos θ)Q(t) to the cavity pressure . 103 5.4.2 Perturbation εY (θ, t) = ε(1 + cos Nθ)Q(t), N ≥ 2 to the cavity pressure . 107 5.4.3 Remarks . 110 5.4.4 Effects of applying a non-radially-symmetric perturbation with vary- ing sign as a function of θ to a plasticised annulus . 111 5.4.5 Remarks . 112 6 Ideas for a full elastic-plastic model 114 6.1 A full elastic-plastic model for the tip . 114 6.1.1 Tip jet region . 115 6.1.2 Plastic region . 116 6.1.3 Elastic region . 117 6.1.4 ‘Plasticised’ elastic region . 117 6.1.5 Remarks . 118 6.2 Squeeze film analogy . 120 6.2.1 Viscous squeeze film . 120 6.2.2 Elasto-plastic squeeze film under horizontal tension . 122 6.2.3 Elastic-plastic squeeze film under horizontal compression . 129 6.2.4 Elasto-plastic squeeze film under compression, with known, varying base . 130 6.2.5 Remarks . 133 7 Paradigm tip models for the jet 135 7.1 Simple two-dimensional filling-flow models for the jet . 136 7.1.1 A filling flow in a channel with constant height with various end- conditions . 137 7.1.2 Comments . 153 7.2 A model for a filling flow impacting a pre-stressed membrane . 154 7.2.1 Inner and outer analysis . 156 7.2.2 Global travelling wave solution . 163 7.3 A model for a filling flow impacting a pre-stressed beam . 168 7.3.1 Inner and outer analysis . 169 7.3.2 Global travelling wave solution . 171 7.4 A filling-flow impact model with general constitutive law p = p(H).... 173 7.4.1 Similarity solution . 175 7.5 Axisymmetric filling flows . 177 7.5.1 General constitutive law p = p(R).................. 180 7.5.2 Similarity solution . 181 7.6 Remarks . 182 8 Metallurgical and hydrocode analysis 184 8.1 Shaped charge metallurgy literature . 184 8.2 Metallurgical analysis . 185 8.2.1 Microscopic analysis . 189 8.3 Pictures of the microstructure . 190 8.3.1 Images from ‘Face 1’ . 190 8.3.2 Images from ‘Face 2’, with penetration occurring ‘into’ the paper . 197 8.3.3 Observations . 199 8.3.4 Remarks . 201 8.4 Hardness . 201 8.4.1 Hardness testing of the steel specimen . 203 8.4.2 Remarks on hardness test results . ..
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