A physics-based minimal model for perforation of a plate by a spherical projectile

Abstract

In aerospace applications, ballistic limit equations (BLEs) are empirical correlations that give the largest projectile a protective target (shield) can stop. The NASA single-wall BLE relates the critical projectile diameter to the target thickness, hardness, density ratio, and impact velocity through power-law. As part of a high-school research project, we are interested in finding out if and to what extent that structure can be understood or replicated from basic physics. The problem is approached by considering two limiting idealizations of the target response, Hertzian indentation of a half-space and bending of a thin plate. Using energy balance, a geometric perforation criterion, and point-source description of impact cratering, a power law form for the critical diameter with density and velocity exponents is obtained for both idealizations, which is structurally comparable to the NASA BLE. Illustrative calculations are performed to compare and contrast the results from the models with those from the empirical BLE and experiments. We hope that this work can be a useful example of how familiar high-school physics and intermediate mechanics could help interpret an important empirical engineering equation.