In classical Cauchy elasticity, 3D materials exhibit six eigenmodes of deformation. Following the 1995 work of Milton and Cherkaev, extremal elastic materials can be classified by the number of eigenmodes, N, out of these six that are “easy”. Using Greek number words, this leads to hexamode (N = 6), pentamode (N = 5), tetramode (N = 4), trimode (N = 3), dimode (N = 2), and monomode (N = 1) materials. While hexamode materials are unstable in all regards, the possibility of pentamode metamaterials (“meta‐fluids”) has attracted considerable attention throughout the last decade. Here, inspired by the 2021 theoretical work of Wei, Liu, and Hu, microstructured 3D polymer‐based tetramode metamaterials are designed and characterized by numerical band‐structure calculations, fabricated by laser printing, characterized by ultrasound experiments, and compared to the theoretical ideal. An application in terms of a compact and broadband polarizer for acoustical phonons at ultrasound frequencies is demonstrated.

Three-dimensional (3D) chiral mechanical metamaterials enable behaviors not accessible in ordinary materials. In particular, a coupling between displacements and rotations can occur, which is symmetry-forbidden without chirality. In this work, we solve three open challenges of chiral metamaterials. First, we provide a simple analytical model, which we use to rationalize the design of the chiral characteristic length. Second, using rapid multi-photon multi-focus 3D laser microprinting, we manufacture samples with more than 105 micrometer-sized 3D chiral unit cells. This number surpasses previous work by more than two orders of magnitude. Third, using analytical and numerical modeling, we realize chiral characteristic lengths of the order of ten unit cells, changing the sample-size dependence qualitatively and quantitatively. In the small-sample limit, the twist per axial strain is initially proportional to the sample side length, reaching a maximum at the characteristic length. In the thermodynamic limit, the twist per axial strain is proportional to the square of the characteristic length. We show that chiral micropolar continuum elasticity can reproduce this behavior.

A theoretical paper based on chiral micropolar effective‐medium theory suggested the possibility of unusual roton‐like acoustical‐phonon dispersion relations in 3D elastic materials. Here, as a first novelty, the corresponding inverse problem is solved, that is, a specific 3D chiral elastic metamaterial structure is designed, the behavior of which follows this effective‐medium description. The metamaterial structure is based on a simple‐cubic lattice of cubes, each of which not only has three translational but also three rotational degrees of freedom. The additional rotational degrees of freedom are crucial within micropolar elasticity. The cubes and their degrees of freedom are coupled by a chiral network of slender rods. As a second novelty, this complex metamaterial is manufactured in polymer form by 3D laser printing and its behavior is characterized experimentally by phonon‐band‐structure measurements. The results of these measurements, microstructure finite‐element calculations, and solutions of micropolar effective‐medium theory are in good agreement. The roton‐like dispersion behavior of the lowest phonon branch results from two aspects. First, chirality splits the transverse acoustical branches as well as the transverse optical branches. Second, chirality leads to an ultrastrong coupling and hybridization of chiral acoustical and optical phonons at finite wavevectors.

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