A high velocity plate penetration hole diameter relationship based on late time stagnation point flow concepts

摘要

Here we derive a plate penetration hole diameter relationship using late time (steady) stagnation point flow concepts. High-speed impact of thin plates and the ultimate development of penetration holes is an important problem for the practical design of impact/shield/armor systems. Though computational and empirical models provide information concerning plate penetration, a theoretical, late time model provides the convenience and efficiency (computation of the full time history of the impact process is not necessary) of a closed form model with the rational basis of a physics based mathematical relationship. To model the decidedly unsteady behavior of impact and penetration a time dependent lower order solution, i.e. 1-d damped kinematic wave, is used to estimate temporal behavior in the more complete 2-d unsteady stagnation point problem, giving a completely time independent stagnation point model. Shear behavior in the stagnation point model, as well as, damping in the 1-d kinematic wave model are modeled using a nonlinear resistance model which draws upon classical viscous fluid flow relationships. No empirical constants are introduced in the associated resistance relationship. Having obtained the late time stagnation point flow equation, a self-similar solution is sought by posing a separation type solution form. Self-similarity of the stagnation point flow is admitted by treating the penetration coincident flow in the same manner as the 1-d kinematic wave solution. The associated self-similar cross-steam flow equation is solved approximately and combined with the separation solution form to yield late time penetration flow velocity field. Examination of the eigenvalues of the velocity solution gives an explicit relationship for the size of the plate penetration hole. The plate penetration hole diameter solution shows good agreement with CTH simulations (a well validated hydrocode) and experimental data.