On body shapes providing maximum depth of penetration

The problem of maximization of the depth of penetration of rigid impactor into semi-infinite solid media (concrete shield) is investigated analytically and numerically using two-stage model and experimen- tal data of Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997). The shape of the axisym- metric rigid impactor has been taken as an unknown design variable. To solve the formulated optimization problem for nonadditive functional, we expressed the depth of penetration (DOP) under some isoperimetric constraints. We apply approaches based on analyti- cal and qualitative variational methods and numerical optimization algorithm of global search. Basic atten- tion for considered optimization problem was given to constraints on the mass of penetrated bodies, ex- pressed by the volume in the case of penetrated solid body and by the surface area in the case of pene- trated thin-walled rigid shell. As a result of performed investigation, based on two-term and three-term two stage models proposed by Forrestal et al. (Int J Impact Eng 15(4):396–405, 1994), Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997) and effectively developed by Ben-Dor et al. (Comp Struct 56:243–248, 2002, Comput Struct 81(1):9–14, 2003a, Int