Deployment kinematics of axisymmetric Miura origami: Unit cells, tessellations, and stacked metamaterials

Origami is emerging as a promising paradigm for deployable structures. The theoretical analysis on the relationship between geometry of the crease patterns and the induced deployment mechanisms is the key to the rational design of origami-inspired deployable structures. In this paper, we systematically investigate the deployment kinematics of axisymmetric Miura origami from unit cells to single-layer tessellations and multi-layer stacked metamaterials. We show that the axisymmetric Miura cells can be classified into two categories according to the monotonicity of the angular motion, and the axisymmetric Miura origami tessellations can be classified into seven categories based on the developability, flat-deployability, and the ability to form closed-ring shapes. We also derive the conditions for the deployability of the multi-layer metamaterials constructed by stacking axisymmetric Miura origami tessellations. Both the singly and doubly curved cases are studied for the tessellations and metamaterials. Additionally, based on the kinematic formulations, we develop an easy-to-implement optimization framework for the design of doubly curved axisymmetric Miura origami tessellations that approximate surfaces of revolution. The optimized shape-morphing tessellations can also be used to assemble multi-layer stacked metamaterials. Taken together, these results provide a comprehensive guidance to design axisymmetric deployable structures based on origami principles.