Geometric Mechanics of Origami Patterns Exhibiting Poisson’s Ratio Switch by Breaking Mountain and Valley Assignment

Exploring the configurational space of specific origami patterns [e.g., Miura-ori (flat surface with parallelogram crease patterns), eggbox] has led to notable advances in science and technology. To augment the origami design space, we present a pattern, named “Morph,” which combines the features of its parent patterns. We introduce a four-vertex origami cell that morphs continuously between a Miura mode and an eggbox mode, forming an homotopy class of configurations. This is achieved by changing the mountain and valley assignment of one of the creases, leading to a smooth switch through a wide range of negative and positive Poisson’s ratios. We present elegant analytical expressions of Poisson’s ratios for both in-plane stretching and out-of-plane bending and find that they are equal in magnitude and opposite in sign. Further, we show that by combining compatible unit cells in each of the aforementioned modes through kinematic bifurcation, we can create hybrid origami patterns that display unique properties, such as topological mode locking and tunable switching of Poisson’s ratio.

Figure 1

(Top) Expanded design space of the Morph pattern (yellow shading) with standard eggbox (red line) and Miura-ori (blue line) as particular cases. (Middle) Fundamental modes of the Morph pattern: eggbox mode (left) and Miura mode (right). (Bottom) Configuration space showing transition of the Morph unit cell from one flat-folded state to another (see Supplemental Video 1). The crease line shown in red morphs from a mountain fold in the eggbox mode to a valley fold in the Miura mode.

Figure 2

Geometric configuration and in-plane mechanics of the Morph pattern. (a) Schematic of the unit cell with the description of geometric parameters and vertices. (b),(c) The configuration space and Poisson’s ratio in stretch, respectively, for different choices of $\alpha$ considering $\alpha +\beta =100°$. The solid and dashed lines represent the eggbox and Miura modes, respectively. (d) Stretching stiffness in $\mathbf{W}$ and $\mathbf{L}$ directions for $\alpha =60°$, $\beta =40°$. The markers represent numerical results from origami structural analyses using the bar-and-hinge reduced-order model [19]. We assume that $a=c=1$.

Figure 3

Out-of-plane bending of the Morph pattern. (a),(b) Bent shapes of the pattern in eggbox and Miura modes, respectively, obtained using the bar-and-hinge origami model. (c),(d) Triangular face tilts creating a net angle change across length $L$ of the Morph pattern in eggbox and Miura modes, respectively. The new coordinates of vertices $O_7$ and $O_9$ after bending (i.e., $O_7^{\prime }$ and $O_9^{\prime }$, respectively) can be calculated using the Rodrigues rotation formula [21].

Figure 4

Hybrid origami assemblages associated with the Morph pattern. (a) Alternating strips of Miura ($M$) and eggbox ($E$) modes. (b) Half pattern with strips in Miura ($M$) mode and the other half in eggbox ($E$) mode. (c) Creation of hybrid patterns from the Morph through kinematic bifurcation. (d) Change of Poisson’s ratio with respect to varying number of Miura mode strips ($n_m$) in a hybrid mode with $100\times 100$ unit cells. The notation ${\nu }_{WL,h}^s$ denotes Poisson’s ratio under stretch for the hybrid pattern. (c),(d) We assume $\alpha =60°$, $\beta =40°$. (e),(f) Mode locking due to extension in $\mathbf{L}$ direction when ${\nu }_{WL,h}^s>0$. The positive global Poisson’s ratio implies contraction in the $\mathbf{W}$ direction, resulting in decrease of ${\psi }_e$ and ${\psi }_m$. The oppositely signed unit cell Poisson’s ratios of the two modes indicates that while ${\varphi }_e$ increases, ${\varphi }_m$ decreases, meaning the Miura mode cells are axially contracting, opposite to the global axial deformation. The Miura mode cells with decreasing ${\varphi }_m$ are locked because such cells can no longer smoothly transition to their eggbox mode in a rigid origami motion. (f) Contrasting global and local deformations that occur in hybrid patterns leading to mode-locking behavior. (e),(f) The green lines represent the panel diagonals, whose projections provide a clean way of sketching the motions.