Small Parameter Analysis of the Modified Tate Equations

Report No. ARL-RP-199
Authors: Jesse A. Huguet; Sarah Reichwein; Stanley E. Jones; William P. Walters
Date/Pages: February 2008; 14 pages
Abstract: The Tate Theory of penetration of armor targets by long rod penetrators [1,2] has been the benchmark one-dimensional model of this event for decades. The model is applied to metal-on-metal normal impact of cylindrical rod penetrators. The key physical parameters in the model are the penetrator and target strengths and densities (assumed constant), as well as the penetrator length. With these parameters and the impact speed, penetration depth for all combinations of the parameters can be evaluated. In a recent paper, Walters et al [3] showed that all of the important effects of the classic Tate Theory could be captured by a regular perturbation solution of the fundamental equations. The small parameter that they used was ε =Yppv02, where Yp is the penetrator yield strength, ρp is the penetrator density and v0 is the impact speed. In 1987, Jones et al [4] modified the equation of motion of the undeformed section to include mass loss and mushrooming at the interface with the target. The changes to the theory that result from these modifications bring the strengths of the target and penetrator into line with laboratory levels while achieving reasonable penetration depths. In this report, we will show that the regular perturbation analysis used by Walters et al [3] can be extended to the modified system of equations from Jones et al [4] using the same small parameter mentioned in the previous paragraph. The perturbation process is carried out to terms of first order and an approximate analytical solution is found. This solution is then used to repeat the reduction of data given by Wilson et al [5] for Aluminum and Steel alloy penetrators normally impacting Aluminum and Steel targets. Another case involving heavy metal penetrators impacting Rolled Hard Armor (RHA) targets is also presented.
Distribution: Approved for public release
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Last Update / Reviewed: February 1, 2008