Size-Dependent Properties of Miura-ori Tessellations

Abstract

We investigate the size-dependent behavior of Miura-ori-based origami tessellations by changing the number of origami unit cells. For large tessellations, the Miura-ori sheet generally exhibits a negative in-plane Poisson's ratio, whereas if the size of the Miura-ori tessellations becomes small, the transition between positive and negative Poisson's ratio emerges in the middle of the folding process. Here, we show that such a transitioning point, i.e., zero Poisson's ratio, yields a kinematic locking state. We also experimentally demonstrate the tunable locking behavior altered by tessellation sizes. Extending the analysis to three-dimensional origami tessellations, we find that the direction of kinematic locking changes depending on the tessellation size. Varying tessellation size thus enables control over both the onset and the direction of locking in origami metamaterials.

Figures (4)

• Figure 1: Geometry of the Miura-ori. (a) Miura-ori unit cell. (b) Folding motion of $4 \times 4$ Miura-ori whose overall width and breadth are denoted by $L_x$ and $L_y$, respectively.
• Figure 2: Size-dependent behavior of the Miura-ori. We compare two Miura-ori tessellations with different sizes in the $y$ direction: $N_y=1$ (left) and $N_y=3$ (right). For both cases, $N_x=1$ and $(b/a, \alpha)=(2, 70^\circ)$. (a) Normalized length $L_y/L_{y, \mathrm{max}}$ versus $\lambda/\lambda_{\mathrm{max}}$; insets show corresponding deployment sequences. (b) Poisson's ratio $\nu_{xy}$ versus $\lambda/\lambda_{\mathrm{max}}$. Red: positive, Blue: negative. (c) Nondimensionalized stiffness $\widetilde{E}_y/k_\theta$ versus $\lambda/\lambda_{\mathrm{max}}$. (d) Colormaps of $\nu_{xy}$ in the $\alpha-\lambda/\lambda_{\mathrm{max}}$ plane; the black solid curve marks the locking trajectory.
• Figure 3: Experimental validation of size-dependent locking in Miura-ori under uniaxial compression. (a) Mechanical boundary conditions. Inset: Miura-ori tube fabrication from two identical Miura-ori sheets. (b) Photo of the experiment. (c),(d) Comparison of theoretical and experimental mean normalized length $L_y/L_{y, max}$ versus $\lambda/\lambda_{\mathrm{max}}$ for $N_y=1, 3$; black arrows indicate the compression direction. (e),(f) Corresponding theoretical and experimental mean $F_y$-$\lambda/\lambda_{\mathrm{max}}$ curves. Experimental curves show the mean of five trials; standard deviations are typically no larger than the line width and omitted for clarity. All samples share identical geometry with $\alpha=70^\circ$, $b/a=2$, and $N_x=2$.
• Figure 4: Poisson's ratios and nondimensionalized stiffness of Miura-ori structures with different numbers of unit cells $N_y$ and identical unit-cell geometry ($\alpha=70^\circ$, $b/a=2$). Poisson's ratios $\nu_{xy}$, $\nu_{xz}$, $\nu_{zy}$ are plotted as functions of $\lambda/\lambda_{\textrm{max}}$. Insets show the nondimensionalized stiffness distributions $\widetilde{E}_n$ (diverging stiffness clipped at $\widetilde{E}_{cl}$ for visualization) and the rendered structures at $\lambda/\lambda_{\textrm{max}}=0.86$; color intensity indicates stiffness magnitude, and gray arrows mark representative directions on the divergence surfaces of $\widetilde{E}_n$. All insets share a common coordinate system. (a) $N_y=1$; arrow (i) indicates locking along the $y$ axis. (b) $N_y=3$; (c) $N_y\to\infty$.