Flow-induced deformation of kirigami sheets

We investigate the deformation in a uniform water flow of thin sheets perforated with parallel cuts. Through combined experiments and theory, we show that deployment is governed by the competition between fluid forces and sheet stretchability, which is prescribed by the cut pattern. Importantly, fluid loading is modulated by the local three-dimensional geometry of pores that open upon stretching. This mesostructure offers a lever to influence macroscopic morphing, notably leading here to asymmetric deformations of symmetric planar sheets. It brings to the fore the kirigami cutting technique as a promising framework for the design of flow-responsive components.

Figure 1

Flow-induced deformation of a kirigami sheet. (a) Deployment in a water crossflow. (b) Typical shape profiles extracted from top-view pictures of the structure of length $L$, when increasing the flow speed $U=[4,8,13,17,21]$ cm/s (denoted by the grayscale). (c) The tilting of buckled cut units (outlined in black) induces tangential fluid forces, resulting in asymmetric profiles prescribed by the buckling direction (reversed in the inset)

Figure 2

Fluid-elastic interaction. (a) Typical force-displacement curve for a kirigami sheet. (b) Effective stiffness $K$ measured in the intermediate linear regime [gray zone in (a)], and compared to the predicted scaling $K_{\text{sc}}$ [see Eq. (1)]. Cutting parameters (see schematics) are varied within $L_s=1.7–4.9$ cm, $d_x=2–9$ mm, and $d_y=2–18$ mm. (c) Inset: Dimensionless deformation amplitude $y_{\text{max}}/L$ as a function of flow speed $U$, for specimens with increasing slit length $L_s$. Results replotted as a function of the Cauchy number $C_y$ [see Eq. (2)] for all 19 cut patterns. Color gradients differentiate specimens within each parametric series, denoted by symbols.

Figure 3

Theoretical modeling. (a) Model of elastic membrane with distributed fluid loading ${\mathit{f}}_N$ and ${\mathit{f}}_T$. (b) Simplified system of parallel flat blades used to characterize fluid loading experimentally. (c) Fluid force coefficients $C_N$ and $C_T$ for the normal and tangential forces on the array (see main text), as a function of the blades' angle $\theta$ (see inset). The grayscale denotes flow speed $U=[7,10,14,17]$ cm/s, and symbols correspond to different center-to-center distances between the 8-mm-width blades: 1.1 ($⊳$), 1.3 ($+$), 1.6 ($\square$), 2.2 ($\circ$) cm. (d) Theoretical shape profiles for logarithmically increasing values of $C_y$ in ${10}^{-3}–10$ (denoted by the grayscale). (e) Corresponding evolution of the dimensionless deformation amplitude $y_{\text{max}}/L$ with $C_y$, compared to experimental data points [same legend as in Fig. 2]. (f) Top: Evolution with $C_y$ of the local elongation along the curvilinear coordinate $S$ (defined in the unstrained reference configuration). Bottom: The latter is compared to experimental elongations for the typical specimen of Fig. 1.

Figure 4

Programming shape through fluid force polarity. Profiles of sheets designed with dual tilting orientation of cut units, in a flow with speed $U=21$ cm/s: (a) The left half rotates clockwise while the right one rotates counterclockwise (see inset schematics), and (b) inversely.