Stiffness maximization of plates and shells made of elastic material with fixed Kelvin moduli

摘要

The aim of this paper is to put forward a new formulation of the Free Material Design (FMD) for minimal compliance of elastic plates and shells. We are interested in solving the optimal orientation problem for eigentensors (or so-called proper states) of the elastic constitutive tensor whose Kelvin moduli values are kept fixed. Both membrane and bending stiffnesses are involved in the representation of an optimal Hooke's tensor thus coupling in-plane and anti-plane deformations. Our method of analysis with respect to the proper states of a fourth rank doubly symmetric tensor reduces an optimum design study to the equilibrium problem of an effective plate or shell with hyperelastic physical properties. The important point to note here is that although the hyperelastic potential of an optimal structure is nonlinear, the relevant constitutive equations are analytically and explicitly derived in both primal and dual settings hence paving the way for developing the Newton-like iterative scheme for treating the equilibrium problem. Its solution determines all membrane and bending components of the optimized stiffness tensor.