A fractional nonlocal elastic model for lattice wave analysis. (November 2022)

• Record Type: Journal Article
• Authors: Challamel, Noël; Atanacković, Teodor; Zhang, Y.P.; Wang, C.M.
• Is Part Of: Mechanics research communications. Volume 126(2022)
• Publication Date: 2022-11
• Subjects: Wave propagation -- Scale effects -- Nanostructures -- Nonlocal elasticity -- Eringen model -- Fractional derivative -- Heterogeneous material -- Dispersive properties -- Born-Kármán model -- Lattice mechanics -- Discrete Elasticity; Mechanics, Applied -- Periodicals; Mécanique appliquée -- Périodiques; Mechanics, Applied; Periodicals; 530
• DOI: 10.1016/j.mechrescom.2022.103999
• Languages: English
• ISSNs: 0093-6413

Abstract

Highlights: The link between lattice elasticity and nonlocal continuum mechanics is studied for 1-d and 2-d wave propagation problems. A 3d generalization of Lagrange lattice is the cubic lattice of Gazis et al. composed of central and angular interactions. Lattice-based nonlocal continuum model differs from the isotropic Eringen's nonlocal model. The fractional generalization of Eringen's model accurately fits the wave dispersive properties of the cubic crystal. The fractional nonlocal model highlights an isotropic response for low frequencies and a cubic symmetry for high frequencies. Abstract: This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of the one-dimensional lattice are accurately fitted by Eringen's nonlocal differential elastic law using asymptotic or phenomenological arguments. A fractional version of Eringen's nonlocal model based on the introduction of a fractional Laplacian for the nonlocal differential operators can improve the accuracy in matching the wave dispersive curve of lattice dynamics. A multi-dimensional generalization of Lagrange lattice for cubic crystals is the lattice of Gazis et al. composed of central and angular interactions. We show in this paper that the differential Eringen's nonlocal elasticity differs from the lattice-based nonlocal elasticity continuum. In particular, Eringen's nonlocal elasticity preserves the isotropic property, whereas the cubic lattice is only isotropic in the low frequency regimes. The fractional generalization of Eringen's model accurately fitted the wave dispersive properties of the cubic crystal in the first Brillouin zone. The fractional generalization of Eringen's model highlights an isotropic response in the low frequency regimes and a cubic symmetry in the high frequency regime.