Meta-Truss and Generalized Elastic Lattices with Long-Range Interactions

This paper aims to enhance comprehension of two-dimensional (2D) lattice elasticity by analyzing the intricate interplay between short-range and long­-range interactions. The 2D square elastic lattices considered herein asymptotically converge towards linear isotropic elasticity. Pioneered by (Gazis et al. in Phys Rev 119:533-544, 1960), the lattice comprising centrai and angular interactions is gener­alized to include short- and long-range interactions. This generalized 2D lattice can be also viewed as a particular case of a Molecular Dynamic System (MDS) with discrete short- and long-range interactions. It is shown herein that the associ­ated mixed differential-difference equations of this generalized lattice asymptotically converge towards Navier's elastodynamic partial differential equations. Owing to the long-range interactions, the mixed differential-difference equation takes on a higher ­order type for spatial dependence. The short- and long-range lattice parameters may be calibrated to enforce the asymptotic continuum to behave as a homogeneous linear elastic isotropic medium with free Poisson's ratio. A higher-order expansion of the mixed differential-difference equation provides gradient elasticity. The wave disper­sive behavior of this generalized lattice with long-range interactions is investigated in an exact form, with a particular emphasis on the short- to long-range interaction ratio. Additionally, static homogeneous compression and shear of square specimens exhibiting short- and long-range interactions are analytically and numerically investigated. It is shown that the homogeneous response of the higher-order lattice strongly ­depends on the calibration ofthe higher-order stiffness in the vicinity ofthe boundary, as previously commented in the literature for one-dimensional higher-order lattices.