On optimal microstructures for a plane shape optimization problem

The purpose of this article is to discuss some properties of optimal microstructure that are used in the homogenization approach for structural optimization problems involving compliance as the design criterion. The key ingredient for the homogenization method is to allow for microperforated composite materials as admissible designs. Some authors use periodic holes while others rely on optimal microstructures such as the so-called rank-2 layered materials which achieve optimality in the 2-D Hashin-Shtrikman bound. We prove that, in two space dimension, when the eigenvalues of the average stress have opposite signs, there is no optimal periodic microstructure. We also prove in this case that any optimal microstructure is degenerate, like the rank-a layered material, i.e. it cannot sustain a nonaligned shear stress. When the eigenvalues of the average stress have the same sign, we exhibit higher order layered material that is optimal and not degenerate.