Lattice materials are a class of cellular materials characterized by a regular, periodic microstructure that can be idealized as a network of slender beams or rods. Likewise all cellular materials, they combine properties such as lightness, stiffness, strength and high energy absorbing capabilities that cannot be achieved by uniform fully solid material. In addition, due to their regular and controlled microstructure, lattice materials can be designed to fulfil specific material requirements, such as prescribed stiffness and strength along given directions and predetermined collapse modes. Recent advances in manufacturing techniques allow the production of lattice materials from a variety of solid materials, at a very fine scale, with high accuracy and within acceptable costs. Such technologies make lattice materials a viable option for use in the design of consumer products, and have driven the interest in modelling tools for analysis of complex components made of lattice materials.

The literature on the modelling of the mechanical properties of lattice materials is generally restricted to the geometrically linear regime. Applications that exploit the design of bending dominated lattices for morphing structures, and the development of stretching dominated lattices suitable for reconfigurable and smart actuated structures typically require modelling lattice materials into the nonlinear regime. In this study, we are concerned with the derivation of a constitutive model for the analysis of the geometrically non-linear behaviour of lattices. A number of studies have analysed simple topologies and obtained closed form expressions of the lattice stiffness and strength by analytically solving the equilibrium problem of the unit cell. This approach cannot in general be extended to include geometrical non-linearity, because closed form solutions are typically unavailable for a beam under large displacement.  In other homogenisation approaches, the equivalent stiffness of the lattice is determined by comparing selected physical quantities of the discrete lattice, such as the dispersion relation of harmonic waves, or the coefficient of the elastic equilibrium equation, to those of an equivalent continuous medium. These models are necessarily restricted to the linear regime and cannot be extended to consider the effects of geometric or material non-linearity.

The approach presented in this study belongs to the class of representative volume element (RVE) methods, which evaluate the constitutive relationships of a heterogeneous medium from the analysis of a small portion of it. The RVE consists in a limited region of the domain that contains the main microstructural features of the material and responds as the infinite medium, if uniform strain, or stress, boundary conditions are imposed. In general, RVE methods resort to a two-scale approach. On one hand, we have the macroscopic finite element model of the component, whose boundary conditions are defined by the general problem, where the material is treated as a continuum. On the other hand, we have the microscopic model of the RVE, which numerically evaluates the stress strain relationship, whose boundary conditions are generated by the macroscopic model.

The RVE model is interrogated at every integration point of the component model, a process that allows the assembly of the macroscopic internal force vector and of the tangent stiffness matrix, as it is done for a fully solid material. In this study, we use an approach that allows determining the macroscopic stress as the gradient of the strain energy density with respect to the components of the macroscopic displacement gradient. This formulation leads to a compact matrix expression for the macroscopic stress as a function of the macroscopic displacement gradient that can handle both geometrical and material non-linearity. The choice of the RVE plays an important role in the framework of a computational homogenization approach. Most of the studies in literature focus on materials with a stochastic microstructure, and are aimed at determining the conditions that ensure a statistical representativeness of the RVE, both for the purpose of numerical homogenization and for the definition of the size, shape, and number of samples required for experimentally measuring the material properties. Since our focus is on periodic lattices, a natural choice for the RVE is the unit cell, which we intend as the minimal entity capable of generating the lattice. Such choice, however, is not unique; any collection of contiguous unit cells can be used as RVE. Hence, the following question arises: how does the size of the RVE affect the response of the material? This study addresses this issue with particular reference to the effect of geometric non-linearity. We show that the size of the RVE has no influence on the model prediction, until a bifurcation point is encountered in the load path. After passing bifurcations, the predicted post bifurcation behaviour depends on the size of the RVE; hence, preliminary investigations should be carried out for a proper selection of the RVE.

The constitutive material model developed in this paper is validated by comparing the results of a discrete and a continuous model of a rectangular plate with a central hole under in-plane loads. In one case, a direct numerical simulation has been carried out on the discrete lattice, whose elements have been individually modelled as beams. In the other case, the domain was modelled with continuous plane stress elements whose constitutive law was numerically evaluated using the homogenisation approach developed in this study. The results from the discrete models have been compared to the prediction of the continuous models. We have found a good qualitative and quantitative agreement among the models. The continuous model of the hexagonal lattice could capture its typical compliant behaviour in compression, and the stiffening effect due to the reorientation of the struts along the load direction in tension.