Homogenization of non-rigid origami metamaterials as Kirchhoff-Love plates

Origami metamaterials have gained considerable attention for their ability to control mechanical properties through folding. Consequently, there is a need to develop systematic methods for determining their effective elastic properties. This study presents an energy-based homogenization framework for non-rigid origami metamaterials, effectively linking their mechanical treatment with that of traditional materials. To account for the unique mechanics of origami systems, our framework incorporates out-of-plane curvature fields alongside the usual in-plane strain fields used for homogenizing planar lattice structures. This approach leads to a couple-stress continuum, resembling a Kirchhoff-Love plate model, to represent the homogeneous response of these lattices. We use the bar-and-hinge method to assess lattice stiffness, and validate our framework through analytical results and numerical simulations of finite lattices. Initially, we apply the framework to homogenize the well-known Miura-ori pattern. The results demonstrate the framework's ability to capture the unconventional relationship between stretching and bending Poisson's ratios in origami metamaterials. Subsequently, we extend the framework to origami lattices lacking centrosymmetry, revealing two distinct neutral surfaces corresponding to bending along two lattice directions, unlike in the Miura-ori pattern. Our framework enables the inverse design of metamaterials that can mimic the unique mechanics of origami tessellations using techniques like topology optimization.