An explicit solution of the Alekseevski–Tate penetration equations

The Alekseevski–Tate equations are typically used to predict the penetration, penetration velocity, rod velocity, and rod length of long rod penetrators and similar projectiles impacting targets. These nonlinear equations were originally solved numerically and more recently by the exact analytical solution of Walters and Segletes. However, due to the nonlinear nature of the equations, the penetration was obtained implicitly as a function of time, so that an explicit functional dependence of the penetration on material properties was not obtained. The current paper obtains the velocities, length, and penetration as an explicit function of time by employing a perturbation solution of the nondimensional Alekseevski–Tate equations. Simple (algebraic) analytical equations are given. Perturbation solutions of the Alekseevski–Tate equations were first undertaken by Forrestal et al., up to the first–order, and good agreement with the exact solutions was shown for relatively short times. In retrospect, this solution was only valid for penetrators impacting weak targets. The current study obtains a third-order perturbation solution and includes both penetrator and target strength terms, and is applicable for strong targets. The paper compares the exact solution to the perturbation solutions, and a typical comparison between the exact and approximate solutions for a tungsten rod impacting a semi-infinite steel armor target is shown. Also, alternate ways are investigated to normalize the governing equations in order to obtain an optimum perturbation parameter. In most cases, the third-order perturbation solution shows good agreement with the exact solution of the Alekseevski–Tate equations. In addition, simple equations based on the first–order perturbation solution are presented, which are accurate for the perforation of finite thickness (short penetration time) targets.