An effective general framework for the homogenisation of lattice metamaterials

We present a homogenisation framework, applicable to both stretching- and bending-dominated architectures, to derive equivalent continuum models of three-dimensional lattice metamaterials in a fully automatic way and without making any a-priori assumption on the nature of the homogenised medium. Enforcing a standard energetic equivalence between a discrete lattice model and a continuum medium, the constitutive tensors of the homogenised continuum are obtained in a systematic way by invoking only the topological data of the unit cell, i.e. strut directions and distances, cross-sectional and material properties, thus avoiding ad-hoc formulations. In particular we prove that stretching-dominated lattice architectures can be effectively described by a standard Cauchy model and that the resulting elasticity tensor is positive definite provided that the unit cell is internally stable, i.e. if no internal mechanisms exist when all nodes are modelled as spherical hinges. In this way a rigorous link is established between the kinematic properties of the discrete structure and the well-posedness of the equivalent continuum model. The proposed computational framework is validated by recovering, as a special case, the known expression of the constitutive tensor for a lattice based on octet–truss and SC/BCC unit cells. For bending-dominated architectures, the analytical expression of the derived constitutive tensor, which appears to be new, is numerically validated by FE analyses. In particular, we show that the strain-gradient contribution becomes significant with respect to the first-gradient terms only when internal mechanisms, associated with lack of rigidity of the nodal connection, are present in the lattice.