Roton dispersion and displacement oscillatory for a nonlocal mechanical system

Recently, we revealed anomalous static response in metamaterials with strong beyond-nearest-neighbor interactions or nonlocal interactions. Therein, the displacement field of a metamaterial beam when stretched is not simply linear in space like for ordinary materials, but rather exhibits pronounced spatial oscillations. The unusual behavior originates from evanescent Bloch modes at zero frequency, or frozen evanescent modes, with large decaying length. Here, we start from a discrete nonlocal mass-and-spring model and adopt an effective-medium approach based on higher-order differential equation to describe the anomalous behaviors. We demonstrate that the theory well captures the frozen evanescent modes and predicates the exact spatial oscillations of the displacement field. The strong dependence of the displacement field on the beam length is also revealed. The feasibility of the effective-medium approach is validated by comparison with the uniaxial tensile test results of metamaterials designed to support the anomalous frozen evanescent phonons. This theory can potentially be used for exploring other intriguing phenomena in nonlocal materials.