Optimal design of nonlinear-elastic and elastic-plastic bars in torsion

We consider the problem of optimal distribution of several materials over the cross section of a prismatic bar (or rod) in torsion. The materials are assumed to be nonlinear-elastic or elastic-plastic, and the amount of each material is specified. The aim of design is to maximize or minimize the stored elastic energy for fixed angle of twist of a unit length of the bar. We show that this problem reduces to the solution of the torsion problem for some nonlinear material, whose law of state is determined by the data for the optimal problem. The design is characterized by zones of three-dimensional sliding control modes, which comprise regions occupied by infinitely frequently alternating layers of the original materials. We give examples of optimal designs of bars of circular and square cross section, fabricated from two elastic-plastic materials.