Multi-stability of the hexagonal origami hypar based on group theory and symmetry breaking

The origami hyperbolic paraboloid, or the hypar, is widely known for its characteristic non-zero Gaussian cur-vature and multi-stable states. Previous investigations have mainly considered two particular cases of the origami hypar patterns, namely the square and the circular hypars. As a representative example of general polygonal hypar patterns, the hexagonal origami hypar displays more desirable energy properties and subtler multi-stable configurations, which is however, rarely studied. In this paper, we investigate the multi-stability of the hexagonal origami hypar by combining a group-theoretic approach, a symmetry-breaking method, and a bar-and-hinge structural model, to simplify the kinematic analysis of this highly symmetric structure . Notably, the kine-matic path of the hexagonal origami hypar is divided into three bifurcation branches by symmetry breaking. Each branch corresponds to two symmetric stable states according to the equilibrium loading and potential energy simulated using the bar-and-hinge model. The non-rigid deformation of the hexagonal origami hypar is mainly controlled by the folding of creases and the bending of facets. Moreover, the energy barrier among the stable states becomes increasingly stronger with higher symmetry orders, thicker sheets of material, and longer creases. This work provides a new strategy for analyzing multi-stable origami structures with high symmetry orders, which can be useed in the design and development of novel adaptive or deployable engineering structures.