Programming complex shapes in thin nematic elastomer and glass sheets

Nematic elastomers and glasses are solids that display spontaneous distortion under external stimuli. Recent advances in the synthesis of sheets with controlled heterogeneities have enabled their actuation into nontrivial shapes with unprecedented energy density. Thus, these have emerged as powerful candidates for soft actuators. To further this potential, we introduce the key metric constraint which governs shape-changing actuation in these sheets. We then highlight the richness of shapes amenable to this constraint through two broad classes of examples which we term nonisometric origami and lifted surfaces. Finally, we comment on the derivation of the metric constraint, which arises from energy minimization in the interplay of stretching, bending, and heterogeneity in these sheets.

Figure 1

Interfaces and junctions on cooling. If each half of the sheet (a) is allowed to independently deform spontaneously, it has the shape in panel (b), where ${\left({\mathbf{\ell }}_{{\mathbf{n}}_i}^{1/2}\right)}_{3\times 2}:=r^{-1/6}[{\mathbf{I}}_{3\times 2}+(\sqrt{r}-1){\mathbf{n}}_i\otimes {\mathbf{n}}_i^{\prime }]$. The interface can be unbroken by rotating one side relative to the other if and only if (3) holds. (c) Symmetric junction. (d) Truncated junction.

Figure 2

Selected examples of nonisometric origami: The line diagrams show the design with the arrows representing the constant director prescribed in each region. The color images show the deformed shape upon cooling. We note that the designs in panels (a) and (b) are compositions of a number of symmetric junctions shown in Fig. 1 $[k=3$ in panel (a) and $k=5$ in panel (b)]. The design in panel (c) is a composition of symmetric junctions with $k=6$ and a generalization of the truncated junctions shown in Fig. 1. This design can be continued periodically.

Figure 3

The deformed shape and designs for lifted surfaces. The vector plots show the director orientation in the design. The amplitude of each vector denotes the planar component ${\mathbf{n}}_0^{\prime }$ of the director. The color images show the topographic map of the sheet after deformation with the colors representing height (hot colors are high). The designs are generated from Eq. (5) by taking $\phi$ to be a smoothed and rescaled grayscale of the desired image.