We show that anomalous static elastic behavior of periodic metamaterials is fundamentally connected to minima in their (real-valued) phonon band structure via the Cauchy-Riemann equations for analytical functions. This connection allows us to systematically engineer anomalously large characteristic decay lengths in static elasticity. We discuss different theoretical examples and an experimental validation based on 3D laser microprinted metamaterials with roton-like acoustic phonon band structure. Herein, statically pulling on one end of a metamaterial beam, while fixing the other end, leads to pronounced spatial oscillations of the displacement field along the beam axis. Such behavior also violates Saint Venant's principle.
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Wegener et al. (2024) studied this question.
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