Hierarchical Laminates Cannot Reproduce All Effective Properties of Planar Two-Phase Composites: A Counterexample with a 16-Component Potential
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Lianghao Cao · arXiv: 2610.37899v1 · Summary page
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Layering does not exhaust composite design
Repeatedly layering constituent materials is a powerful way to design a composite, but can increasingly elaborate layering reproduce every attainable effective response? This preprint answers no for certain planar composites governed by linear systems with many potential components. The authors construct constituent properties for which every hierarchical laminate obeys an equality between effective energies, then build periodic microstructures that violate it.
The construction begins with locally layered material whose proportions vary smoothly, before replacing it with fine stripes of the original constituents. It also separates the quasiconvex and rank-one convex envelopes of a particular energy—mathematical objects used to describe relaxation and the onset of fine structure. The counterexample and the strict separation are formally verified in Lean 4. For materials design, the message is not that layering is ineffective, but that restricting a search to layered architectures can miss genuinely different responses.
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