Continuum Field Theory for the Deformations of Planar Kirigami

Mechanical metamaterials exhibit exotic properties that emerge from the interactions of many nearly rigid building blocks. Determining these properties theoretically has remained an open challenge outside a few select examples. Here, for a large class of periodic and planar kirigami, we provide a coarse-graining rule linking the design of the panels and slits to the kirigami’s macroscale deformations. The procedure gives a system of nonlinear partial differential equations expressing geometric compatibility of angle functions related to the motion of individual slits. Leveraging known solutions of the partial differential equations, we present an illuminating agreement between theory and experiment across kirigami designs. The results reveal a dichotomy of designs that deform with persistent versus decaying slit actuation, which we explain using the Poisson’s ratio of the unit cell.

Figure 1

Response of planar kirigami to the heterogeneous loading conditions shown by the arrows. (a) Rotating squares pattern. (b) Another pattern with rhombi slits. Insets: a typical unit cell before and after deformation. The central slit opens through an angle $2\xi$, and the cell rotates by $\gamma$.

Figure 2

Coarse graining a mechanism. (a) Vectors ${\mathbf{s}}_i$, ${\mathbf{t}}_i$ define the unit cell, which tessellates along $\mathbf{s}$ and $\mathbf{t}$ to produce the pattern. (Note ${\mathbf{s}}_1=-{\mathbf{t}}_4$ and ${\mathbf{s}}_4={\mathbf{t}}_3$.) In a mechanism, panels rotate by the rotation matrices $\mathbf{R}(\gamma \pm \xi )$. (b) Coarse graining through the lattice defines the effective deformation gradient ${\mathbf{F}}_{\mathrm{eff}}$. Soft modes agree locally with this picture.

Figure 3

Effective Poisson’s ratio ${\nu }_{21}$ as a function of slit actuation $\xi$ for different rhombi-slit kirigami. The plot fixes $\alpha =-0.9$ and varies $\beta$ from 0 to 0.9. The RS pattern on the lower left sits at the lower extreme $\beta =0.9$. It is purely dilational (${\nu }_{21}=-1$) and is auxetic for all $\xi$. The upper extreme $\beta =0$ arises for the design on the upper left. It is nonauxetic (${\nu }_{21}>0$) for all relevant $\xi >0$. Some designs transition between auxetic and nonauxetic behavior as a function of $\xi$.

Figure 4

Comparison between theory and experiments of rhombi-slit kirigami. (a),(d) Two $16\times 16$ cell patterns before deformation, with opposite Poisson’s ratios and types. Top row is nonauxetic and hyperbolic. Bottom row is auxetic and elliptic. (b),(e) Left entries are experimental samples pulled along their centerlines. Right entries show theoretical panel motions, obtained from exact solutions of the effective PDEs by the procedure in the Supplemental Material, Sec. SM.3 [39]. (c),(f) Annular deformations produced experimentally (left) and using the theory (right). Color maps show the slit actuation angle $\xi (\mathbf{x})$, extracted from the experiment per Supplemental Material, Sec. SM.7 [39].