Higher-order 2D lattices with long-range interactions and their nonlocal continua

Abstract

This paper presents the formulation of two-dimensional (2D) square elastic lattices consisting of point-like material particles and incorporating central and angular (non-central) short- and long-range interactions of arbitrary order p. Each particle is assumed to interact with all other particles of the discrete medium along arbitrary directions. At the first-order asymptotic limit, these lattices converge to simple linear elastic continua. The lattice parameters are calibrated such that the asymptotic continuum behaves as a homogeneous, linear elastic, isotropic medium with a free Poisson’s ratio under both plane stress and plane strain conditions. When higher-order terms are retained in the asymptotic expansion, the discrete medium is shown to correspond to higher-order gradient elasticity or, equivalently, at the desired order, to a nonlocal elastic continuum. The exact wave dispersion properties of the generalized lattice with long-range interactions are investigated, with particular emphasis on the role of the discrete kernel. It is demonstrated that, for kernels with monotonically decreasing influence functions, the wave dispersion curves in the first Brillouin zone are monotonic for any interaction order p. We give a first proof from the analytical determination of the roots of the group velocity function, up to p = 5 interactions. Another proof is presented, whatever the number p of interactions, by expressing this gradient function through the Dirichlet kernel. Finally, the paper provides a calibration of nonlocal elasticity models to reproduce the wave dispersion characteristics of the higher-order lattice with long-range interactions. For both isotropic and anisotropic nonlocal models, the characteristic length scales are shown to depend on the interaction order and the shape of the discrete kernel.