Modeling the non-linear elastic response of periodic lattice materials

Periodic elastic lattice materials are a class of cellular materials with unique properties that cannot be achieved with fully uniform solids. In this work we employ moderate-rotation theory, which is an expansion of linear elasticity that takes into account moderate displacements and rotations, to derive a fundamental multi-scale model that captures the non-linear response of infinitely periodic elastic lattice materials and allows the imposition of local constraints on lattice junctions. We begin by modeling the response of a single strut. Next, an infinitely periodic lattice structure is considered. The periodicity of the structure is accounted for through a unit cell (UC) and periodic lattice directions. A representative volume element (RVE) comprising one or more UCs is then defined. It is assumed that the RVE experiences the macroscopic displacement gradient. To determine the stress, we employ the principle of virtual work and homogenization methods. To illustrate the predictions of the model, we analyze two infinitely periodic lattice materials - a diamond lattice and a triangular lattice with rigid junctions. To validate our predictions, we compute the response for RVEs of several sizes until convergence is reached.