Multimaterial design of physically nonlinear structures

Abstract

A theoretical consideration of optimization problems for physically nonlinear hyperelastic structures is carried out. The structures are subjected to a single static "dead" loading and a multimaterial design approach is analysed. Structural materials are assumed to be isotropic, with stress-strain relations being weakly concave. The problems considered are mass minimization with prescribed structural stiffness, stiffness maximization with prescribed structural mass, and mass minimization with constrained stresses. Optimality conditions for the problems are analysed. Generalizations of Maxwell's and Michell's theorems for the considered structures are proved. Some regularities inherent in the third problem are analysed using the analytical example of a three-rod physically nonlinear truss made of two materials. An algorithm for compliance decreasing in the case of prescribed structural mass is proposed. The monotonicity property of the algorithm is proved. Numerical examples (bi-material beam and airframe) are presented and corresponding results are analysed on a basis of the theoretical approaches developed.