Inhomogeneous Folding Modes in Infinite Lattices of Rigid Triangulated Miura-ori

Infinite two-dimensional tessellations of triangulated Miura-ori with rigid panels are known to exhibit only homogeneous modes of folding, thereby limiting their usefulness in engineering applications. In this work, we show that the corresponding one-dimensional lattices are less restricted and can exhibit inhomogeneous folding modes of deformation. We demonstrate this by looking at the modes in the null space of Bloch-reduced compatibility matrix in a nodal-displacement-based formulation, that is typically employed in the context of origami structural analysis. We compute the deformation modes that vary non-uniformly across the lattice depending on their wavelength, and identify the minimal number of modes that can represent such deformations. We then present a more efficient formulation based on folding-angles to study the deformation modes of infinite one-dimensional rigid triangulated origami lattices. We derive the degrees of freedom of the tessellations in terms of the minimal number of folding-angles that are required to capture the periodic inhomogeneous deformations of the infinite lattices. Within this formulation, we provide the framework to analytically derive the stiffness matrix of the lattice. Finally, we verify the new formulation by comparing the results with the bar-and-hinge model that is based on nodal-displacements. The observations from our work could have implications for the use of rigid panel origami lattices as acoustic metamaterials.