Circular and annular two-phase plates of minimal compliance

Abstract

The paper deals with optimal design of thin plates. The plate thickness assumes two possible values: h₂ and h_l and the plate volume is given. The problem of minimizing the plate compliance needs relaxation. The relaxed formulation was found by Gibiansky and Cherkaev in 1984 [13] . In the present paper a finite element approximation of this problem is presented in the framework of rotationally symmetric bending of circular and annular plates. The problem is composed of a nonlinear equilibrium problem coupled with a minimum compliance problem. The aim of the present paper is to analyze the forms of the optimal solutions, in particular, to look into the underlying microstructures. It is proved that in some solutions a ribbed microstructure occurs with ribs non-coinciding with both the radial and circumferential directions. Due to non-uniqueness of the sign of an angle of inclination of ribs the appearance of this microstructure does not contrasts with the radial symmetry of the problem. In the degenerated problem when the smallest thickness hl vanishes the above interpretation of the inclined ribbed microstructure becomes incorrect; in these regions one can assume that the plate is solid but with a varying thickness. The degenerated case of h₁ = 0 was considered in the papers by Rozvany et al. [26] and Ong et al. [25] but there such a microstructure was not taken into account. One of the aims of the paper is to re-examine these classical and frequently cited results.