Initial rigid response and softening transition of highly stretchable kirigami sheet materials

Abstract

We study, experimentally and theoretically, the mechanical response of sheet materials on which line cracks or cuts are arranged in a simple pattern. Such sheet materials, often called kirigami (the Japanese words, kiri and gami, stand for cut and paper, respectively), demonstrate a unique mechanical response promising for various engineering applications such as stretchable batteries: kirigami sheets possess a mechanical regime in which sheets are highly stretchable and very soft compared with the original sheets without line cracks, by virtue of out-of-plane deformation. However, this regime starts after a transition from an initial stiff regime governed by in-plane deformation. In other words, the softness of the kirigami structure emerges as a result of a transition from the two-dimensional to three-dimensional deformation, i.e., from stretching to bending. We clarify the physical origins of the transition and mechanical regimes, which are revealed to be governed by simple scaling laws. The results could be useful for controlling and designing the mechanical response of sheet materials including cell sheets for medical regeneration and relevant to the development of materials with tunable stiffness and mechanical force sensors.

Figure 1

(a) A sheet of paper perforated with many cuts, kirigami, used for the protection of a bottle of wine. (b) “Airvase” (Torafu Architects, Japan) sold in museum shops worldwide, made from a sheet of kirigami. (c) Planer stretching of a sheet of kirigami with similar perforation geometry. The lack of circular symmetry leads to inhomogeneously stretched cuts.

Figure 2

(a) Kirigami pattern investigated in the present study. (b) Force F vs. extension Δ. The initial regime shown in the inset is linear, which is followed by the second soft regime and the final hardening regime. (c) In-plane deformation of the unit plate in the initial regime. (d) Out-of-plane deformation in the second regime: perspective view in the top and lateral view in the left bottom. (e) Illustration of bending of a plate to discuss the deformation energy.

Figure 3

(a) Stiffness constant K₁ vs thickness h for various cut length w and spacing d. The lines are guide for the eyes. (b) K₁/(hE) vs (d/w)³ demonstrating collapse of the data in (a) by rescaling of the both axes in (a) according to Eq. (3). The slight deviation of the open (green) symbols is consistent with the prediction: for these data the condition is satisfied whereas the theory requires the condition d ≪ w.

Figure 4

(a) Critical spacing Δ_c vs thickness h for various length w and spacing d. The curves are guide for the eyes. (b) δ_c/d vs. (h/d)² demonstrating collapse of the data in (a) by rescaling of the both axes in (a) according to Eq. (1). The slight deviation of the open (green) symbols is again consistent with the prediction.