Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.