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Abstract This study investigates the programming of elastic wave propagation bandgaps in periodic and multi‐stable metamaterials by intentionally and uniquely sequencing its constitutive mechanical bits. To this end, stretched kirigami is used as a simple and versatile testing platform. Each mechanical bit in the stretched kirigami can switch between two stable equilibria with different external shapes (aka. “(0)” and “(1)” states). Therefore, by designing the sequence of (0) and (1) bits, one can fundamentally change the underlying periodicity and thus program the phononic bandgap frequencies. This study develops an algorithm to identify the unique periodicities generated by assembling “n‐bit strings” consisting ofnmechanical bits. Based on a simplified geometry of thesen‐bit strings, this study also formulates a theory to uncover the rich mapping between input sequencing and output bandgaps. The theoretical prediction and experiment results confirm that the (0) and (1) bit sequencing is effective for programming the phonic bandgap frequencies. Moreover, one can additionally fine‐tune the bandgaps by adjusting the global stretch. Overall, the results of this study elucidate new strategies for programming the dynamic responses of architected material systems. 

Yoshimura origami is a classical folding pattern that has inspired many deployable structure designs. Its applications span from space exploration, kinetic architectures and soft robots to even everyday household items. However, despite its wide usage, Yoshimura has been fixated on a set of design constraints to ensure its flat foldability. Through extensive kinematic analysis and prototype tests, this study presents a new Yoshimura that intentionally defies these constraints. Remarkably, one can impart a unique meta-stability by using the Golden Ratio angle ( ${\cot }^{-1}1.618\approx {31.72}^{\circ }$) to define the triangular facets of a generalized Yoshimura (with $n=3$, where $n$is the number of rhombi shapes along its cylindrical circumference). As a result, when its facets are strategically popped out, a ‘Golden Ratio Yoshimura’ boom with $m$modules can be theoretically reconfigured into $8^m$geometrically unique and load-bearing shapes. This result not only challenges the existing design norms but also opens up a new avenue to create deployable and versatile structural systems. This article is part of the theme issue ‘Origami/Kirigami-inspired structures: from fundamentals to applications’. 

Origami — the ancient art of paper folding — has been widely adopted as a design and fabrication framework for many engineering applications, including multi-functional structures, deployable spacecraft, and architected materials. These applications typically involve complex and dynamic deformations in the origami facets, necessitating high-fidelity models to better simulate folding-induced mechanics and dynamics. This paper presents the formulation and validation of such a new model based on the Absolute Nodal Coordinate Formulation (ANCF), which exploits the tessellated nature of origami and describes it as an assembly of flexible panels rotating around springy creases. To estimate the crease folding, we mathematically formulate a “torsional spring connector” in the framework of ANCF and apply it to the crease nodes, where the facets meshed by ANCF plate elements are interconnected. We simulate the dynamic folding of a Miura-ori unit cell and compare the results with commercial finite element software (ABAQUS) to validate the modeling accuracy. The ANCF model can converge using significantly fewer elements than ABAQUS without sacrificing accuracy. Therefore, this high-fidelity model can help deepen our knowledge of folding-induced mechanics and dynamics, broadening the appeals of origami in science and engineering.