Hierarchical Laminates Cannot Reproduce All Effective Properties of Planar Two-Phase Composites: A Counterexample with a 16-Component Potential

Abstract

Designing a composite requires understanding which effective properties can be achieved through varying its microstructure, i.e., the spatial arrangement of given constituent materials. Hierarchical laminates, created by successive layering, reach many known limits on these properties, making it natural to ask whether they can reproduce every effective response obtainable from the same constituents. We answer this question negatively for planar two-phase composites governed by linear systems whose potentials have 16 or more components. We construct constituent material properties and, at any prescribed nontrivial phase fraction, periodic microstructures whose effective properties cannot be approximated by hierarchical laminates, even when layer directions, proportions, and the number of layering steps are unrestricted. We design the constituents so that all hierarchical laminates have equal effective energies at two prescribed mean gradients. The key step is to construct a medium whose local properties are those of horizontal laminates with smoothly varying phase fractions, but whose effective energies violate this equality. Approximating these energies with sufficiently fine periodic stripes of the original constituents yields the two-phase counterexample. This construction also gives a negative answer to the two-well relaxation problem: the minimum of two positive-definite homogeneous quadratic forms on $\mathbb{R}^{2\times32}$ can have distinct quasiconvex and rank-one convex envelopes. The counterexample and the strict inequality between the envelopes are formally verified in Lean 4.