Detecting stress fields in an optimal structure Part I: Two-dimensional case and analyzer

In this paper, we investigate the stress fields in optimal elastic structures by the necessary conditions of optimality, studying two-phase elastic composites in two dimensions. The necessary conditions show that an optimal design is characterized by three zones: Zone 1 of pure weak Material 1; Zone 2 of pure strong Material 2; and Zone 3, where Material 1 and Material 2 mix to form an optimal microstructure. To characterize these zones, we introduce two rotationally invariant norms N ₁ and N ₂ of a stress tensor. The derived optimality conditions state that the inequality N ₁()< constant ₁ holds in Zone 1; the inequality N ₂()> constant ₂ holds in Zone 2; and two equalities hold simultaneously in Zone 3: N ₁()= constant ₁, in Material 1; N ₂()= constant ₂ in Material 2. Using this result, we analyze suboptimal projects and find how close the fields are to the regions of optimality.