In this thesis, we unveil a third design path to manipulate elastic waves within architected media, distinct from the traditional phononic crystal and locally-resonant metamaterial concepts. The core innovation lies in the concept of nonlocal resonances, defined as zero-frequency modes possessing non-zero wave-vectors, resulting in anomalous dispersion cones within the wave spectrum. This distinct working principle allows nonlocally-resonant metamaterials to circumvent the bandwidth-subwavelengthness trade-off that characterises traditional architected wave media.

Introducing kinematic graphs as a visual design tool for zero-frequency modes in planar elastic metamaterials, we identify various classes of zero-mode scaling, with a specific focus on the oligomodal class, characterised by a fixed number of global deformation modes that remain independent of system size. Building upon this foundation, we obtain elastic nonlocal resonances by imposing a Bloch-wave requirement on zero-modes hosted by oligomodal geometries.

This opens up the door to a comprehensive exploration of the wave physics of nonlocally-resonant metamaterials, culminating in an inverse design approach to freely position anomalous cones within k-space. We then validate our theory through a combination of full-wave simulations, compression experiments and vibration experiments, establishing the practical viability of oligomodal geometries and nonlocally-resonant metamaterials.

Having firmly anchored the core concept of nonlocal resonance, we chart out its boundaries by studying various edge cases. We start by considering mass gaps, higher-frequency gaps and momentum gaps, and discover a connection between nonlocal resonances and exceptional-point dynamics in the process. We then depart from strict adherence to the Bloch wave requirement, thus revealing edge states and power-law spectral signatures. This allows us to extend our wave perspective to also include various non-Bloch zero-modes that naturally arise in the study of oligomodal geometries.

Finally, we connect back to the architected media that inspired the introduction of the nonlocal resonance concept, namely electromagnetic interlaced wire media, by studying their direct elastic equivalent. This enriches our discussion by allowing us to investigate the interplay between symmetry and nonlocal resonances within a three-dimensional framework. In particular, we show that representation theory naturally applies to the permutation space of the macroscopic interlaced components that constitute the interlaced wire medium.

In conclusion, this thesis presents a novel paradigm for manipulating elastic waves, which we term nonlocally-resonant metamaterials, offering new vistas for multifunctional materials and advanced wave control.