Extremal Light-weight Microstructures

We introduce a new class of high-porosity microstructures, called “single-scale laminates,” made from arrays of parallel walls. They are extremal, in the sense that no stiffer structure exists with the same total mass. They are simple, in the sense of being easy to describe, and perhaps to manufacture. And they are universal: for any high-porosity microstructure there is a single-scale laminate using at most as much material which is at least as stiff. Moreover, any nondegenerate high-porosity Hooke’s law can be bounded both above and below by a single-scale laminate of the same weight. We give a simple formula for the effective Hooke’s law of such a structure. It reduces the task of minimum-weight design in the high-porosity regime to a problem of linear programming.