Local-to-Nonlocal Transition Laws: Mass Conservation for Stiffness-Tuneable Weakly Nonlinear Monoatomic Chains

Dynamics of periodic structures and quasi-crystalline lattices is attracting a growing interest in physics and engineering at different scale levels and for a wide range of applications. Very recently, nonlocal metamaterials have been proposed by adding, in the lattices, multiple beyond nearest neighbour couplings among elements to obtain dispersion relations allowing for roton-like behaviours and phase and group velocities with opposite signs. Moreover, metamaterials with nonlinear pairings have been studied in the attempt to increase the tailorable features, e.g., through amplitude dependent frequency responses and therefore enhance the design spectrum for wave-manipulation devices. However, mathematical models and real prototypes in literature seem neither to provide criteria to compare local and nonlocal lattices nor to discuss any related rules governing the transition between the two configurations. In this framework, the present study extends a previously proposed conservation law for the elastic connections in mass-spring systems by the present authors, which are the most employed building blocks for a vast number of metamaterials. The relationship between nonlocality and nonlinearity in a one-dimensional monoatomic chain is here investigated. The conservation law is enforced for nonlinear couplings, so the level of nonlocality P and the stiffness distribution parameter α are used to govern the design of the nonlinear nonlocal chain. Effects of weak cubic nonlinearities are observed through perturbation techniques. The results attained are employed to yield perturbed and nonlinear amplitude-dependent dispersion relations for nonlocal, mass-preserving monoatomic chains.