Optimization of parameters of layered plates during dynamic hard indenter penetration with friction and weakining effect of free surfaces

The problem of optimal braking hard indenter of the inhomogeneous plate during the impact of the normal was first formulated in 1978 (Aptukov V.N.). Results published later, based on the Pontryagin maximum principle was derived criteria for the optimal structure of inhomogeneous plate a minimum weight for different projectile shapes. At the present time this problem in a similar or different formulations studied by various researchers, some examples are presented in this paper. Variant of the viscous crater formation is implemented for medium velocity of the impact of the little-deformed sharp indenter into plastic target with a medium hardness. For this condition known empiric dependence of the penetration resistance is widely using. The dependence is applied under certain limitations on velocity, thicknesses of the target, shape of the indenter, mechanical characteristics, this fact is confirmed by numerous experiments that is carried out in the Stepanov V.A. laboratory in the Ioffe LPTI (at present Ioffe Physical-Technical Institute of the Russian Academy of Sciences in the St. Petersburg). We used numerical algorithm and we attempt to refine the problem of optimization taking into account the effect of free surfaces of the plate and friction on resistance to penetration in this paper. Gradual increase of complexity of the model by including new factors helps to approach a more realistic description of the penetration process. This improvement allows to further study the problem in a new improved model. Method of acicular variations is used to solve the problem. We received final solution of the problem and we formulated criteria for the optimal structure of the target in some cases. We have not received of the analytical solution in other cases, but we presented the results of the numerical calculation. We showed that the inclusion of additional effect theoretically lead to a qualitatively new type of solution compared to previously known solutions in some cases. We derived the algorithm for determining the optimal structure of the slab to the problem of the impact of the cone with n materials.