Generalization of the Singum Model for the Elasticity Prediction of Lattice Metamaterials and Composites

Abstract

The recently developed singum model was extended to lattice metamaterials and composites for prediction of the effective elasticity based on the stiffness of the lattice components and the structure of the lattice, in which the load is transferred through the lattice network represented by unit cells that can contain one or more singums. The equilibrium of the singums was considered under a displacement variation, and the relation between the variations of averaged stress and strain can be evaluated to predict the elasticity. It was proved that the stiffness of any unit cells is the same as that of the primitive cell. A generalized formulation was developed to calculate the effective elasticity of lattice metamaterials and composites, which reflects the symmetry and anisotropic feature of the lattice more accurately. A hydrostatic load does not change the shape of the singum but changes its elasticity, although the bonds are linear elastic. The formulation shows the prestress-dependent elasticity for lattice metamaterials. When a large uniform biaxial tension is applied, a honeycomb lattice can exhibit a negative Poisson’s ratio under a pre-tension. Case studies of auxetic and body-centered cubic lattices were conducted to demonstrate the negative Poisson’s ratio and anisotropic elasticity, respectively.