Buckling-Induced Kirigami

We investigate the mechanical response of thin sheets perforated with a square array of mutually orthogonal cuts, which leaves a network of squares connected by small ligaments. Our combined analytical, experimental and numerical results indicate that under uniaxial tension the ligaments buckle out of plane, inducing the formation of 3D patterns whose morphology is controlled by the load direction. We also find that by largely stretching the buckled perforated sheets, plastic strains develop in the ligaments. This gives rise to the formation of kirigami sheets comprising periodic distribution of cuts and permanent folds. As such, the proposed buckling-induced pop-up strategy points to a simple route for manufacturing complex morphable structures out of flat perforated sheets.

Figure 1

(a) Schematic of the system: an elastic sheet of thickness $t$ perforated with a square array of mutually orthogonal cuts. (b) In the thick limit (i.e., for large values of $t/\delta$) the perforated sheet deforms in plane and identically to a network of rotating squares [28]. (c)–(d) For sufficiently small values of $t/\delta$ mechanical instabilities triggered under uniaxial tension result in the formation of complex 3D patterns, which are affected by the loading direction. The 3D patterns obtained for $\gamma =45°$ and $\gamma =0°$ are shown in (c) and (d), respectively. Scale bars: 6 mm.

Figure 2

(a) Experimental stress-strain curves for perforated sheets characterized by different normalized hinge width $\delta /l$ and normalized sheet thickness $t/\delta$ for $\gamma =45°$. Note that the stress is normalized by the effective in-plane Young’s modulus $\overline{E}=2/3E(\delta /l{)}^2$. (b) Snapshots of the sample with $\delta /l=0.06$ and $t/\delta \simeq 0.085$ at ${ϵ}_x=0$, 0.12, and 0.24. (c) Critical strain ${ϵ}_c$ as a function of $(t/\delta {)}^2$ as obtained from experiments (markers) and predicted analytically (dashed line).

Figure 3

The buckling-induced Miura kirigami sheet (a) is flat foldable, (b) forms a saddle shape with a negative Gaussian curvature upon nonplanar bending, (c) twists under antisymmetric out-of-plane deformation, and (d) has much higher bending rigidity than the corresponding flat perforated sheet (inset). Note that the $127\mu \mathrm{m}$ thick Miura kirigami sheet shown here supports a 20 g weight.

Figure 4

Effect of loading direction $\gamma$ on the mechanical response of the perforated sheets. Evolution of the (a) normalized stress ${\sigma }_x/\overline{E}$, (b) the in-plane macroscopic Poisson’s ratio ${\nu }_{yx}$ and (c) the opening angle of cuts $2{\theta }_{o_1}$ and $2{\theta }_{o_2}$ as a function of the applied strain ${ϵ}_x$ for different values of $\gamma$. Note that ${\nu }_{yx}$ is negative only for ${ϵ}_x<{ϵ}_c$ (as at this stage the deformation of the structure is purely planar and identical to that of a network of rotating squares) and that it increases sharply and reaches positive values once the instability is triggered. (d) Numerical snapshots of 3D patterns obtained at ${ϵ}_x=0.125$ for different values of $\gamma$. The contours shows the normalized out-of-plane displacements.