Applying mass conservation constraints in the wave dispersion analysis of hexagonal non-local lattices with rotational springs

Due to the simplicity of their building blocks, mass–spring lattices are commonly used to explore the unconventional dynamic properties of ideal photonic/phononic crystals and metamaterials. Recent studies shown that adding long-range interactions in these chains leads to unusual mechanics, such as roton-like dispersion relations. Non-neighbouring interactions occur through springs connecting beyond-the-nearest masses, so that an increase in the springs total mass is provided. This could lead to inconsistent results when performing dynamic analysis of non-local systems and a sort of correction paradigm is required. In this context, this paper focuses on the theoretical analysis of elastic wave propagation in hexagonal diatomic non-local lattices satisfying the mechanical consistency condition recently introduced by some of the present authors for the one-dimensional case. The principle is based on a springs mass conservation rule, together with the imposition of an order of magnitude ratio between the masses of the springs and the oscillating atoms. Using Bloch’s theory, closed-form dispersion equations up to an arbitrary order of beyond-the-nearest connections are derived as a function of the order of non-locality and a properly defined scaling parameter to tune the springs stiffness at each non-local order. Different local and non-local scenarios are compared, revealing a more complex configuration of the dispersion curves with an increasing number of slope inversions inside the Brillouin’s boundaries. A local-to-non-local transition rule is also obtained from the analysis of emblematic cases highlighting the transition between local and non-local behaviour.