Recently, non-local configurations have been proposed by adding beyond nearest neighbour couplings among elements in lattices to obtain roton-like dispersion relations and phase and group velocities with opposite signs. Even though the introduction of non-local elastic links in metamaterials has unlocked unprecedented possibilities, literature models and prototypes seem neither to provide criteria to compare local and non-local lattices nor to discuss any related rules governing the transition between the two configurations. A physically reasonable principle that monoatomic one-dimensional chains must obey to pass from single- to multi-connected systems is here proposed through a mass conservation law for elastic springs thereby introducing a suitable real dimensionless parameter alpha to tune stiffness distribution. Therefore, the dispersion relations as a function of alpha and of the degree of non-locality P are derived analytically, demonstrating that the proposed principle can be rather interpreted as a general mechanical consistency condition to preserve proper dynamics, involving the spring-to-bead mass ratio. Finally, after discussing qualitative results and deriving some useful inequalities, numerical simulations and two-dimensional FFTs are performed for some paradigmatic examples to highlight key dynamics features exhibited by chains with finite length as the parameters alpha and P vary.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.
Local-to-Non-Local Transition Laws for Stiffness-Tuneable Monoatomic Chains Preserving Springs Mass / Guarracino, Flavia; Fraldi, Massimiliano; Pugno, Nicola M.. - In: PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY OF LONDON SERIES A: MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES. - ISSN 1364-503X. - ELETTRONICO. - 2024, 382:2279(2024), pp. 1-21. [10.1098/rsta.2024.0037]
Local-to-Non-Local Transition Laws for Stiffness-Tuneable Monoatomic Chains Preserving Springs Mass
Flavia Guarracino;Massimiliano Fraldi
;Nicola M. Pugno
2024-01-01
Abstract
Recently, non-local configurations have been proposed by adding beyond nearest neighbour couplings among elements in lattices to obtain roton-like dispersion relations and phase and group velocities with opposite signs. Even though the introduction of non-local elastic links in metamaterials has unlocked unprecedented possibilities, literature models and prototypes seem neither to provide criteria to compare local and non-local lattices nor to discuss any related rules governing the transition between the two configurations. A physically reasonable principle that monoatomic one-dimensional chains must obey to pass from single- to multi-connected systems is here proposed through a mass conservation law for elastic springs thereby introducing a suitable real dimensionless parameter alpha to tune stiffness distribution. Therefore, the dispersion relations as a function of alpha and of the degree of non-locality P are derived analytically, demonstrating that the proposed principle can be rather interpreted as a general mechanical consistency condition to preserve proper dynamics, involving the spring-to-bead mass ratio. Finally, after discussing qualitative results and deriving some useful inequalities, numerical simulations and two-dimensional FFTs are performed for some paradigmatic examples to highlight key dynamics features exhibited by chains with finite length as the parameters alpha and P vary.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.File | Dimensione | Formato | |
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