Shape optimization of penetrator nose G. Ben-Dor, A. Dubinsky, T. Elperin * The Pearls  tone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel Abstract A simpli®ed procedure is suggested to optimize a penetrator's nose shape using localized impactor±target interaction model in the cases of penetration into a semi-in®nite target and into a target with a ®nite thickness. Such a procedure establishes the similarity between the projectile's shape optimization in penetration dynamics and in aerodynamics of high speeds. The optimization of the impactor's nose shape can thus be reduced to the variational problem considered previously in connection with the projectile's shape optimization in high speed gas dynamics. Two examples from the literature are analyzed when impactor's shape optimization involved diculties whereas the mathematically similar problem was solved before in aeromechanics. Some aspects of applying gas dynamics similarity for optimization of the impactor's shape for bodies of revolution are discussed. Ó 2001 Elsevier Science Ltd. All rights reserved. 1. Introduction The localized interaction approach [1±4] allows the analytical study of some problems involving optimization of impactor's shape. However, even these relatively simple models for analyzing mo- tion of a rigid impactor penetrating normally into a target imply the dependencies of the coecients of equation of motion on the velocity and on the coordinate determining the location of the pene- trator inside the target. Since these dependencies are expressed in terms of integrals, the criterion for optimization maximum depth of penetration, minimum ballistic limit velocity BLV)) deter- mined as a solution of the equation of motion is quite involved [4]. Additional simpli®cations are required in order to simplify the formulas which determine this criterion. There are only a few publications on penetra- tor's shape optimization using analytical methods. In the ®rst studies on these problems it was as- sumed that an impactor moves inside a target with a constant velocity [5±8]. Although such model may be viewed as a legitimate approach to the penetration problem, it is in fact the problem of optimization of a drag of the projectile moving in a medium with a constant speed. The drag force dependence on the impactor's velocity was taken into account in [9±13] where the problems of shape optimization were investigated. Three-dimensional impactors [9±11] and bodies of revolution [12,13] were considered applying the criterion of the maximum depth of penetration into a semi-in®nite target while the resistance of the target at the ini- tial stage of penetration was neglected. The latter factor was taken into account in [2] where the optimization of three-dimensional conical impac- tors was studied. The suggested methodology was further developed in [14±16] and applied to the targets with a ®nite thickness minimum of the www.elsevier.com/locate/tafmec Theoretical and Applied Fracture Mechanics 35 2001) 261±270 * Corresponding author. Fax: +972-7-6472813990. E-mail address: elperin@menix.bgu.ac.il T. Elperin). 0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0167-844201)00050-7