Lattice modes of periodic origami tessellations with voids

Elastic modes of origami lattices are of scientific value in the context of metamaterial applications. Miura-ori is a well-studied origami pattern, especially in engineering. A deeper understanding of spatially homogeneous deformations can be useful to homogenization-based material characterization. Miura-ori with rigid parallelogram panels deforms exclusively through crease-folding as a single degree of freedom (DOF) system. Substituting parallelograms with rigid triangular panels introduces two additional DOFs per vertex and could admit a rich space of lattice deformations. In this paper, we investigate the lattice modes of rigid triangulated Miura-ori (RTM) with enclosed voided regions within the tessellations. We use two widely adopted approaches — the bar and hinge framework (BHF) and a folding-angle framework (FAF), that are typically used for the analysis of origami lattices. Unlike the 2D RTM lattice without voids, we find that for the origami lattices with voids, the compatibility constraints based on the crease folding-angles alone are insufficient to capture the admissible deformation modes. Additional loop-closure constraints, based on Denavit–Hartenberg analysis of spatial linkages, must be imposed on creases around each enclosed void. We observe that the homogeneous modes with accumulation of deformations across the lattice are exclusive to the space of Bloch-wave modes within the FAF approach and are not straightforwardly obtained using BHF approach of modeling origami lattices. The 2D RTM lattices with voids, irrespective of the size and aspect ratio of enclosed voids, are found to exhibit exactly six such exclusive FAF modes which can be further characterized using intuitively defined relations between crease angle perturbations.