The relationship between the average penetration velocity, U¯\documentclass[12pt]{minimal}
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\begin{document}$$ \bar{U} $$\end{document}, and the initial impact velocity, V0\documentclass[12pt]{minimal}
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\begin{document}$$ V_{ 0} $$\end{document}, in long-rod penetration has been studied recently. Experimental and simulation results all show a linear relationship between U¯\documentclass[12pt]{minimal}
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\begin{document}$$ \bar{U} $$\end{document} and V0\documentclass[12pt]{minimal}
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\begin{document}$$ V_{ 0} $$\end{document} over a wide range of V0\documentclass[12pt]{minimal}
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\begin{document}$$ V_{ 0} $$\end{document} for different combinations of rod and target materials. However, the physical essence has not been fully revealed. In this paper, the U¯-V0\documentclass[12pt]{minimal}
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\begin{document}$$ \bar{U} - V_{ 0} $$\end{document} relationship is comprehensively analyzed using the hydrodynamic model and the Alekseevskii–Tate model. In particular, the explicit U¯-V0\documentclass[12pt]{minimal}
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\begin{document}$$ \bar{U} - V_{ 0} $$\end{document} relationships are derived from approximate solutions of the Alekseevskii–Tate model. In addition, the deceleration in long-rod penetration is discussed. The deceleration degree is quantified by a deceleration index, α=2μ¯/(KΦJp)≈Ypρp-1/2ρp-1/2+ρt-1/2V0-2,\documentclass[12pt]{minimal}
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\begin{document}$$ \alpha = {{2\bar{\mu }} \mathord{\left/ {\vphantom {{2\bar{\mu }} {(K\varPhi_{Jp} )}}} \right. \kern-0pt} {(K\varPhi_{Jp} )}} \approx Y_{p} \rho_{p}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} \left( {\rho_{p}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} + \rho_{t}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} } \right)V_{0}^{ - 2} , $$\end{document} which is mainly related to the impact velocity, rod strength, and rod/target densities. Thus, the state of the penetration process can be identified and designed in experiments.
