During the high-speed penetration of projectiles into concrete targets （the impact velocity ranges from 1.0 to 1.5 km/s）, important factors such as the incident oblique and attacking angles, as well as the asymmetric abrasions of the projectile nose induced by the target-projectile interactions, may lead to obvious deviation of the terminal ballistic tra- jectory and reduction of the penetration efficiency. Based on the engineering model for the mass loss and nose-blunting of ogive-nosed projectiles established, by using the Differ- ential Area Force Law （DAFL） method and semi-empirical resistance function, a finite differential approach was pro- grammed （PENTRA2D） for predicting the terminal ballistic trajectory of mass abrasive high-speed projectiles penetrating into concrete targets. It accounts for the free-surface effects on the drag force acting on the projectile, which are attributed to the oblique and attacking angles, as well as the asymmetric nose abrasion of the projectile. Its validation on the prediction of curvilinear trajectories of non-normal high-speed pene- trators into concrete targets is verified by comparison with available test data. Relevant parametric influential analyses show that the most influential factor for the stability of ter- minal ballistic trajectories is the attacking angle, followed by the oblique angle, the discrepancy of asymmetric nose abrasion, and the location of mass center of projectile. The terminal ballistic trajectory deviations are aggravated as the above four parameters increase.
The mass loss and nose blunting of a projectile during high-speed deep penetration into concrete target may cause structural destruction and ballistic trajectory instability of the penetrator,obviously reducing the penetration efficiency of penetrator.Provided that the work of friction between projectile and target is totally transformed into the heat to melt penetrator material at its nose surface,an engineering model is established for the mass loss and nose-blunting of the ogive-nosed projectile.A dimensionless formula for the relative mass loss of projectile is obtained by introducing the dimensionless impact function I and geometry function N of the projectile.The critical value V c0of the initial striking velocity is formulated,and the mass loss of projectile tends to increase weakly nonlinearly with I/N when V0〉V c0,whilst the mass loss is proportional to the initial kinetic energy of projectile when V0
The initial oblique and attacking angles as well as the asymmetrical nose abrasion may lead to bending or even fracture of a projectile,and the penetration efficiency decreases distinctly.The structural stability of a high-speed projectile non-normally penetrating into concrete and the parametric influences involved are analyzed with the mass abrasion taken into account.By considering the symmetrical or asymmetrical nose abrasion as well as the initial oblique and attacking angles,both the axial and the transverse drag forces acting on the projectile are derived.Based on the ideal elastic-plastic yield criterion,an approach is proposed for predicting the limit striking velocity（LSV）that is the highest velocity at which no yielding failure has occurred and the projectile can still maintain its integral structural stability.Furthermore,some particular penetration scenarios are separately discussed in detail.Based on the engineering model for the mass loss and nose-blunting of ogive-nose projectiles established in Part I of this study,the above approach is validated by several high-speed penetration tests.The analysis on parametric influences indicates that the LSV is reduced with an increase in the asymmetrical nose abrasion,thelength-diameter-ratio,and the concrete strength,as well as the oblique and attacking angles.Also,the LSV raises with an increase in the initial caliber-radius-head（CRH）and the dimensionless cartridge thickness of a projectile.
A decay function for the layering effect during the projectile penetrating into layered targets is constructed, which is obtained via the theoretical solution of a dynamically expanding layered spherical cavity with finite radius in the layered targets that are assumed to be incom- pressible Mohr-Coulomb materials. By multiplying the decay function with the semi-empirical forcing functions that account for all the constitutive behavior of the targets, the forcing functions for the layered targets are obtained. Then, the forcing functions are used to represent the targets and are applied on the projectile surface as the pressure boundary condition where the projectile is modeled by an explicit transient dynamic finite element code. This methodology is implemented into ABAQUS explicit solver via the user subroutine VDLOAD, which eliminates the need for discretizing the targets and the need for the complex contact algorithm. In order to verify the proposed layering effect model, depth-of-penetration experiments of the 37 mm hard core pro-jectile penetrating into three sets of fiber concrete and soil layered targets are conducted. The predicted depths of penetration show good agreement with the experimental data. Furthermore, the influence of layering effect on projectile trajectory during earth penetration is investigated. It is found that the layering effect should be taken into account if the final position and trajectory of the projectile are the main concern.
With a target treated as the incompressible Tresca and Mohr-Coulomb material, by assuming that cavity expansion produces plastic-elastic and plastic-cracked-elastic response region, the decay function for the free-surface effect is constructed for metal and geological tar- gets, respectively. The forcing function for oblique penetration and perforation is obtained by multiplying the forcing function derived on the basis of infinite target assumption with the de- cay function. Then the projectile is modeled with an explicit transient dynamic finite element code and the target is represented by the forcing function as the pressure boundary condition. This methodology eliminates discretizing the target as well as the need for a complex contact algorithm and is implemented in ABAQUS explicit solver via the user subroutine VDLOAD. It is found that the free-surface effect must be considered in terms of the projectile deformation, residual velocity, projectile trajectory, ricochet limits and critical reverse velocity. The numerical predictions are in good agreement with the available experimental data if the free-surface effect is taken into account.
