The Aboudi micromechanical model for shape design of structuresM

The micromechanical approach to topology design consists in deening a domain of porous material, including the loads and supports, and nding an optimal distribution of the densities. The usual method is to mesh the structure and assign a diierent density and material orientation to every element. Optimal topologies emerge when using the densities and orientations as design variables in a mathematical programming formulation in conjunction with a nite element analysis program. This paper presents the Aboudi method of cells, an analytical method to compute the elastic properties of composite material, and applies it to determine the mechanical properties of porous material as a function of the density of material. It is shown that the results are similar to those obtained by special nite element analysis set up to compute the elasticity matrix. The procedure is visualized on simple cantilever design problems and used within the context of a stress-ratio redesign scheme. 1. Introduction The ultimate structural design problem is probably to de-ne a set of loads and a reaction surface, and to look for the 'best' structure which will transfer the applied loads to the supports. Ground braking work by Michell (1906) yielded solutions for minimum volume structures under stress constraints years before Mathematical Programming techniques emerged as an engineering tool for structural design. The 'Michell' structures are often considered as theoretical lower bounds since they are diicult to implement in practice. Suboptimal solutions were proposed by assuming that the structure is a truss and by removing bars from an initial ground structure which considers all possible connectivities between a set of assumed nodes. A comprehensive review of these and related papers can be found in Rozvany, Bendsoe and Kirsch 6]. In an interesting paper, Bendsoe and Kikuchi 3] have used an alternative approach which consists in starting from a bloc of material (membranes for 2D problems) connecting the applied loads to the ground and gradually removing structurally ineecient material. The novel idea