Nonlocal discrete systems with mass preserving springs: dynamics and design criteria for metamaterials

The dynamics of periodic structures and lattices are gaining increasing attention in engineering and physics in wave manipulation devices across various scales and applications. In mechanics, nonlocal configurations incorporating beyond-nearest-neighbour (BNN) couplings are especially promising, as they can show “roton-like” dispersion relations at a macroscale, thus phase and group velocities with opposite signs and backwards wave propagation. While it is true that BNN connections offer novel possibilities in designing new architected and meta-materials, existing models and prototypes seem to lack criteria for comparing local and nonlocal lattices or a sound framework for transitioning between these configurations. A key issue arises from the usage of monoatomic mass-spring models as theo-retical paradigms for analysing their dynamics, where usually the mass of atoms is considered for the inertia of the system, while the mass of the springs is neglected. However, as the number of nonlocal pairings increases, the mass of the springs grows, potentially reaching a magnitude competing with the oscillating masses, thus leading to possible model inconsistencies. To fill this gap, in the first part of this PhD thesis a physically reasonable principle that monoatomic one-dimensional chains must obey to pass from single- to multi-connected systems is proposed through a mass conservation law for elastic springs, thereby introducing a suitable real dimensionless parameter α to tune stiffness spatial distribution over the system domain. Therefore, the dispersion relations as a function of α and of the degree of nonlocality P are derived analytically, demonstrating that the proposed principle can be rather interpreted as a general mechanical consistency condition to preserve proper dynamics, fixing the spring to atom mass ratio. The analyses are carried out through analytical and numerical investigations on discrete nonlocal lattices, both finite and infinite, imposing Bloch-Floquet periodicity conditions. Numerical dispersion relations are obtained through two-dimensional Fast Fourier Transforms, which highlight the applicability of the proposed principle to finite nonlocal chains, for which boundary effects are also discussed. The second part of the thesis focuses on applications of the proposed principle to models which presents a slightly nonlinear behaviour, due to a nonlinear elastic restoring force. The relationship between nonlocality and nonlinearity in a one-dimensional monoatomic chain is investigated. The conservation law is then generalized and enforced for non-linear couplings, so the level of nonlocality P and the stiffness distribution parameter α are still used to govern the dynamics of the nonlinear nonlocal chains. The results obtained are employed to yield perturbed and nonlinear amplitude-dependent dispersion relations for nonlocal, mass-preserving monoatomic chains. In the third part of the thesis, the presence of defects is taken into account. In fact, another major leap forward in metamaterial design in the last decade was the discovery of the impact of crystal symmetries on the topological properties of materials. These properties are often based on defects or irregularities in the lattice symmetries, which can be usefully employed to obtain waveguides and localization phenomena. Finally, the last part concerns the application of the proposed modelling strategy to two-dimensional lattices, taking into consideration in-plane and out-of-plane displacements in the dynamic behaviour of flat structures. The conservation law with its tuning parameters α and P is expanded and applied to two-dimensional Bravais lattices. At the end, dispersion curves are retrieved for nonlocal two-dimensional lattices and the results are discussed, envisioning opportunities for conceiving new classes of nonlocal metamaterials.