This study investigates the correlation between the projectile nose velocity and tail velocity during the penetration of semi-infinite metallic targets by long rods in the velocity range of 1500–2500 m/s. Focusing on the high-velocity quasi-steady erosive penetration stage, a hybrid modeling framework integrating dimensional analysis and physical constraints is established. The results show that, within this regime, the nose–tail velocity relation can locally exhibit a relatively stable approximately linear structure, in which the slope is mainly governed by the material density ratio, whereas the intercept and its correction terms reflect the combined effects of initial velocity, strength difference, and geometric scale. To achieve a unified representation across different material groups, velocity levels, and length-to-diameter ratios, an effective stress scale is introduced, together with a material-group scale factor to coordinate the systematic magnitude differences among materials of different densities while preserving dimensional consistency. The proposed method adopts a physics-constrained learning strategy, in which the data-driven component is restricted to the identification of closure coefficients and a small number of scale parameters rather than the direct fitting of velocity variables, thereby balancing predictive accuracy and physical interpretability. Validation based on 2987 point-wise samples shows that the model achieves a coefficient of determination of 0.9963 and a root-mean-square error of 24.01 m/s on the test set.
Wu et al. (Wed,) studied this question.