Closed-form dispersion relations for non-local rectangular mass–spring lattices satisfying mass conservation constraints

Periodic mass–spring lattices represent an ideal playground to investigate wave propagation in a wide range of mechanical systems, including photonic/phononic crystals and metamaterials. A great deal of theoretical, numerical and experimental research has been conducted on this topic, mainly based on traditional mass–spring models characterised by masses connected to their nearest neighbours only. Recently, researchers added long-range interactions, which have been proved to be responsible for observing new mechanics, such as roton-like dispersion relations. This increase in couplings, realised by introducing additional springs connecting beyond-the-nearest masses, obviously increases the springs total mass and disregarding this aspect could lead to inconsistent results when performing dynamic analysis of non-local systems. A rational comparison between local and non-local lattices is also not assured. In this context, this paper applies a mechanical consistency condition, recently introduced by some of the present authors for the one-dimensional case, to preserve proper dynamics of non-local rectangular lattices and assuring, at the same time, a more accurate comparison between local and non-local models. 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, a closed-form expression for the dispersion equation up to an arbitrary order of beyond-the-nearest connections is initially derived as a function of the order of non-locality and properly defined scaling parameters to tune the springs mass and stiffness at each non-local order. The analysis and comparison of different local and non-local scenarios reveals a more complex configuration of the dispersion curves, which display an increasing number of local minima/maxima inside the Brillouin's boundaries as the order of non-locality is increased. Emblematic cases highlighting the transition between local and non-local behaviour are also observed, from which a general local-to-non-local transition rule is obtained.