Waves in elastic metamaterials

Wave propagation in long-range elastic metamaterials produces important propagation phenomena, such as wave stopping, negative and supersonic group velocities. Metamaterials are particular materials that gain peculiar properties not because of the material itself, but thanks to the realization of periodic sub-structures, capable of modifying the behavior of the macro-structure. The advances in technological machining, especially in the field of additive manufacturing techniques, give the possibility to realize complex elastic connections between different cellular elements; the presence of micro- and nano-scale connected structures changes the dynamic response of materials composed by the aggregation of these lattices. Such solids are clustered as elastic metamaterials. In homogeneous elastic materials, each elementary cell is connected to its first neighbors, constituting short-range interactions; on the other hand, when the range of connections spreads, interesting phenomena in terms of mechanical wave propagation appear. The chance of controlling wave propagation by simply changing the topology of the elastic connections is an important achievement; for instance, such designed materials can manipulate the mechanical energy flow, by isolating a vibration sensitive area at the cost of routing the energy into less sensitive regions, or better the energy can be focused to be harvested and converted. A physical mathematical model has been constructed starting from some of the long-range interactions that can be found in nature, such as the presence in classical discrete system (Mass-Spring) of magnetostatic or electrostatic elements. The main feature of these interactions is the dependency on the absolute distance and not on the relative displacement; each element, indeed, exchange forces with all the elements of the system, not only with its close neighbors. Several approaches to the problem have been adopted to model long-range interaction metamaterials. The expertise in the homogenization of discrete problems involving complex structure has been the theoretical basis for building up a continuous model capable of describing long-range interactive metamaterials. Two opposite approaches have been adopted to describe the phenomenon: (i) a differential approach and (ii) an integral approach. The differential approach needs some assumptions and approximations, especially in the interaction range that needs to be limited in space, while the integral approach use some of the same assumptions and approximations but works with a full interaction range. For this reason, most of the mathematical efforts in modeling such materials has been dedicated to the development of the integral approach-based model. To better describe the system, the interaction model is based on two prototype forces: the Gauss-like and the Laplace-like. Technically speaking, these interactions, as the ones from which they come from, decay with the distance and respect the action-reaction principle holds. In addition, they have a known Fourier transform; it is not the case with magnetostatic and electrostatic forces, generally power-law based. The principal result of the investigation consists in finding some properties of phase and group velocities for one-dimensional infinite waveguides, where several unusual phenomena emerge. Namely we show and demonstrate in a close analytical form the following phenomena: wave-stopping, eigenstate migration, negative group velocity, leading to supersonic propagation. An analogous approach is used to evaluate wave propagation in long-range beams (Euler-Bernoulli), and in two dimensional membranes with circle-step long-range interaction, with the borne of analogous phenomena. The investigation includes also a huge experimental campaign, set in cooperation with Technion, Israel Institute of technology. An experimental setup for the testing of single and coupled long-range magnetic waveguide has been built, demonstrating interesting phenomena in terms of magnetically coupled waveguides that reproduce some of the results the theoretical investigation predicts