Foldable cones as a framework for nonrigid origami

The study of origami-based mechanical metamaterials usually focuses on the kinematics of deployable structures made of an assembly of rigid flat plates connected by hinges. When the elastic response of each panel is taken into account, novel behaviors take place, as in the case of foldable cones ($f$-cones): circular sheets decorated by radial creases around which they can fold. These structures exhibit bistability, in the sense that they can snap through from one metastable configuration to another. In this work, we study the elastic behavior of isometric $f$-cones for any deflection and crease mechanics, which introduce nonlinear corrections to a linear model studied previously. Furthermore, we test the inextensibility hypothesis by means of a continuous numerical model that includes both the extended nature of the creases, stretching and bending deformations of the panels. The results show that this phase-field-like model could become an efficient numerical tool for the study of realistic origami structures.

Figure 1

(a) Imprinted crease pattern on a flat plate. (b) Deformed state of the $i\mathrm{th}$ panel of a $f$-cone. The curve $\mathrm{\Gamma }$, the material frame, and the Euler-like angles are defined.

Figure 2

Definition of (a) mountain and (b) valley creases. The vectors ${\mathbf{t}}_i^{\pm }, {\mathbf{n}}_i^{\pm }$ and the crease angle $\psi$ are defined.

Figure 3

The integration constant $c$ as function of the folding angle $\psi$ for (a) rest states and (b) snapped states for all-mountain $f$-cones with one, two, and four creases.

Figure 4

(a) Schematic definition of the polar angle ${\theta }_c$ for $n^{R,S} f$-cones ($n>1$). (b) Case $1^{R,S}$: The angle $\beta \left(\psi \right)=\pi -{cos}^{-1}[\mathbf{u}\left(0\right)·\mathbf{u}\left(\pi \right)]$ that describes the deviation from the flat configuration. (c) Case $2^{R,S}$: The polar angle ${\theta }_c\left(\psi \right)$ at the creases compared with the prediction of the linear model for the snapped state. (d) Case $4^{R,S}$: The polar angle ${\theta }_c\left(\psi \right)$ at the creases.

Figure 5

[(a)–(c)] Equilibrium shapes for $\overline{k}=1$ and ${\psi }_0=\pi /2$. (d) Final crease angles $\psi$ as function of $\overline{k}$ for ${\psi }_0=\pi /2$. Rest (blue lines) and snapped (red lines) states for all-mountain $f$-cones with one, two, and four creases (respectively dashed, dotted, and plain lines).

Figure 6

(a) Energy landscape of an all-mountain $f$-cone with four creases as function of ${\theta }_c$ for $\overline{k}=1$ and ${\psi }_0=\pi /2$. The blue and red curves correspond to the bending energy of the facets for the rest and snapped states, respectively. The plain gray curve is the crease energy given by Eq. (12). The black curve is the total elastic energy. (b) Normalized moment $M\left({\theta }_c\right)$ applied on the creases. The black dots correspond to the rest and snapped states of the $f$-cone.

Figure 7

Transverse view of a plate with a narrow slice whose thermal and mechanical properties differ from those of the rest of the plate. (a) Reference configuration. (b) Curvature-induced equilibrium configuration due to a linear thermal gradient across the thickness.

Figure 8

Schematics of the indentation protocol. Equal moments $M$ are applied on each crease such that the center rim moves vertically in the downward direction, while the end points of the symmetry planes (denoted by red dots) are free to move in the $xy$ plane. The average displacement of the center rim is imposed and the moment $M$ is computed as a function of the polar angle ${\theta }_c$.

Figure 9

[(a) and (b)] Indentation paths ${\theta }_c\left(\psi \right)$ as computed by the numerical model for $h_c/h=1,1/4$ and $E_c/E=2,1,1/2,1/10$. The theoretical curves ${\theta }_c\left(\psi \right)$ for a $4^{R,S} f$-cone are reproduced from Fig. 4. The dots correspond to the rest and snapped states of each indentation path.

Figure 10

Lines of smallest principal curvature of a symmetric foldable cone with four creases. The upper (respectively, lower) row corresponds to a stiff (respectively, soft) crease case. The equilibrium rest (left column) and snapped (right column) states are shown. The boxes show zoomed regions next to the vertex. Black dotted line indicates the location of the crease and the color code corresponds to the elastic energy density.

Figure 11

(a) Moment $M$ as function of ${\theta }_c$ during the indentation of a stiff crease with $h_c=h, E_c=2E$ ($—$), and a soft crease, with $h_c=h/4, E_c=E/2$ ($—$). (b) The corresponding total stretching energy. Inset: Normalized bending energy of the facets compared to the theoretical result of Fig. 6.