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  • Open Access

Stretching Theory of Hookean Metashells

Luca Giomi*

  • *Contact author: giomi@lorentz.leidenuniv.nl
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Phys. Rev. X 16, 021035 – Published 14 May, 2026

DOI: https://doi.org/10.1103/myqd-2677

Abstract

Despite being governed by the familiar laws of Hookean mechanics, elastic shells patterned with an internal structure (i.e., metashells) exhibit a wealth of unusual mechanical properties with no counterparts in unstructured materials. Here I show that much of this behavior can be captured by a real-valued analog of the inhomogeneous Schrödinger equation, with the lateral pressure experienced by the internal structure in the role of the wave function. In the fine structure limit—i.e., when the length scale associated with the internal structure is much smaller than the local radius of curvature—this approach reveals the existence of localized states, in which elastic deformations are prevented to diffuse away from their origin, thereby allowing the internal structure to smoothly adapt to the intrinsic geometry of the metashell. Leveraging on an analogy with scattering states in nearly free electrons, I further show that periodic metashells, obtained from the repetition of the same structural unit periodically in space, support elastic Bloch waves, corresponding to stationary periodic configurations of the internal structure and characterized by a geometry-dependent band structure. When applied to crystalline monolayers, this approach provides a generalization of the elastic theory of interacting topological defect to compressible systems.

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Mechanical metamaterials can display striking properties because their response is governed not only by the material they are made of, but also by the geometry of their internal structure. When such architectures are embedded on curved surfaces—forming so-called metashells—their microstructure must deform to accommodate the underlying curvature. In this work, we develop a continuum theory describing how these deformations arise and propagate. Surprisingly, the governing equation for the lateral pressure within the microstructure takes the form of a real-valued analog of the Schrödinger equation. This correspondence reveals unexpected behaviors. In the limit of very fine internal structure, elastic stresses become localized: curvature-induced deformations remain confined near their origin instead of spreading across the shell. In periodically structured metashells, the theory further predicts stationary deformation patterns analogous to Bloch waves in crystals, with geometry-dependent band structures. Beyond engineered metamaterials, this framework also applies to crystalline particle layers on curved interfaces, offering a unified perspective on how geometry shapes mechanical response.

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Phys. Rev. X 16, 021035– Published 14 May, 2026
  • Received 24 March 2025
  • Accepted 9 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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