On angular and volumetric interactions in elastic cubic lattices

This paper investigates statics and natural vibration of linear elastic cubic lattices, together with their continuum approximations. The lattice endowed with central and angular interactions, referred to as Gazis et al.’s, is considered first: since the stiffness of each lattice phase must be positive, the equivalent macroscopic Poisson’s ratio must be lower than its central limit 1/4. A volumetric interaction based on a volume-dependent internal pressure is introduced as an additional non-central interaction for a complete calibration of the equivalent Poisson’s ratio up to its incompressibility limit 1/2. This volumetric interaction can also be classified as Fuchs-type, providing a potential energy that depends on the volume variation of each cell. The mixed differential-difference equations of the associated lattice derive from Hamilton’s principle applied to the discrete energies. The algebraic properties of the stiffness matrix of the discrete cell provide information on the positive definiteness of the potential energy, for each lattice with central and non-central interactions. The convergence of this finite lattice towards a linear elastic continuous right parallelepiped is shown in several static loading schemes. The discrete Lame´ problem for the free vibration of this parallelepiped is solved for all the considered lattices. It is concluded that discrete and continuum elasticity can be connected by this cubic lattice within a complete range of elasticity parameters.