On angular and surface interactions in two-dimensional elastic lattices

This paper deals with two-dimensional elastic lattices which asymptotically converge towards isotropic linear elastic continua. The square lattice composed of central and angular interactions, pioneered by Gazis et al. (1960) for 3D cubic lattices, allows the modelling of isotropic linear elastic continua (in the continuous asymptotic limit) with equivalent Poisson's ratios smaller than 1/3 in a plane stress assumption. This paper presents an alternative or complementary elastic model based on surface elastic interactions, which can be viewed as a complementary lattice model for values of Poisson's ratios greater than 1/3 in plane stress. The physical background of this non-central potential is discussed, basing on the existence of a dual surface pressure. The model can be also introduced from variational arguments. This surface model of Fuchs type is shown to be equivalent to a perimetric elastic model in the linear range. Additionally, this paper explores the possibility of coupling both surface and angular lattice elasticity for square lattices, with the potential to cover a large variety of elastic continua in the asymptotic limit. Gradient elasticity and lattice-based nonlocal elastic continua are constructed, starting from this simple square lattice composed of angular, surface and central interactions. The static behaviour of these square lattices (with central, angular and surface interaction) is investigated in pure compression and pure shear loading, with exact analytical solutions for the discrete displacement field.