A semi-analytical model incorporating scaling effects for deformable penetration of flat-nosed long rods into semi-infinite concrete targets

Abstract

Existing models have not consistently captured the scaling effects associated with deformable penetration of flat-nosed long rods into concrete. This paper proposes a novel semi-analytical model that explicitly incorporates the projectile diameter-to-aggregate size ratio. The projectile is treated as a control volume. Based on conservation laws and wave impedance conditions, an analytical model for the residual diameter is derived. A scaling-informed penetration resistance is used to define a yield velocity that accounts for the projectile diameter-to-aggregate size ratio. This velocity is then incorporated into a Forrestal-type resistance model, resulting in a closed-form solution for penetration depth. The model was validated against experimental data and numerical simulations. It captures the transition to the deformable regime and the subsequent reduction in penetration depth due to nose bulging. The model also captures two key scaling laws: (1) the normalized residual diameter decreases as the projectile diameter increases, and (2) the normalized penetration depth increases monotonically. Overall, the proposed model provides a unified framework that links scaling effects with deformable penetration behavior, and can be used as a useful tool for practical protective design.