Derivation of Born/von Kármán difference equations through consistent lattice angular interactions

Abstract

This paper investigates statics and dynamics of two-dimensional (2D) linear elastic lattices and their continuum approximations. Focus is on the mixed differential-difference equations proposed by Born and von Kármán in 1912 for cubic lattices with both central and non-central interactions, applied here to 2D lattices. The non-central interaction introduced by Born and von Kármán, classified as a shear interaction, corresponds to a frame-dependent (non-objective) angular interaction. A consistent connection between discrete and continuum elasticity can be achieved by incorporating the proper non-central (angular) interactions introduced by Gazis et al. in 1960. Inspired by Hrennikoff's truss scheme, we constructed an alternative lattice model, based on Gazis et al. formulation, that is augmented with additional objective internal angular interactions. The difference operators associated with both consistent lattices, i.e. Gazis et al. lattice and the new Hrennikoff inspired lattice, are shown to differ. The conditions of positive definiteness of the associated potential energy of each lattice, are discussed. The new lattice successfully bridges the gap between discrete and continuum elasticity, while being governed by the same mixed differential-difference equations proposed by Born and von Kármán. It can reproduce macroscopic auxetic behaviour, while preserving the positive definiteness of its discrete potential energy. This finding resolves the long-standing inconsistency of Born and von Kármán's equations, which, originally derived from flawed physical assumptions, can now be justified through correct mathematical reasoning.