Tunable Mechanical Metamaterials through Hybrid Kirigami Structures

Abstract

Inspired by the art of paper cutting, kirigami provides intriguing tools to create materials with unconventional mechanical and morphological responses. This behavior is appealing in multiple applications such as stretchable electronics and soft robotics and presents a tractable platform to study structure-property relationships in material systems. However, mechanical response is typically controlled through a single or fractal cut type patterned across an entire kirigami sheet, limiting deformation modes and tunability. Here we show how hybrid patterns of major and minor cuts creates new opportunities to introduce boundary conditions and non-prismatic beams to enable highly tunable mechanical responses. This hybrid approach reduces stiffness by a factor of ~30 while increasing ultimate strain by a factor of 2 (up to 750% strain) relative to single incision patterns. We present analytical models and generate general design criteria that is in excellent agreement with experimental data from nanoscopic to macroscopic systems. These hybrid kirigami materials create new opportunities for multifunctional materials and structures, which we demonstrate with stretchable kirigami conductors with nearly constant electrical resistance up to >400% strain and magnetoactive actuators with extremely rapid response (>10,000% strain s^-1) and high, repeatable elongation (>300% strain).

Figure 1

Tunable kirigami materials through hybrid structures. (a) Photographs of kirigami sheets with (left) only major cuts (Fig. 2a (design i)) and (right) hybrid structures of major and minor cuts (Fig. 2a (design v)). The same increasing weights (10 g, 20 g, and 40 g) are attached to each kirigami sheet to demonstrate mechanical response. Major cuts are highlighted in blue and the minor cuts are highlighted in red in the insets. (b) Load (P) versus strain (ε) for kirigami materials with design i of w = 3 mm (blue curve) and with design v of w = 3 mm and l_m/2w = 0.75 (red curve). Geometric parameters are described in Fig. 2a. The inset shows the initial regime of the P - ε plot and the slope indicates the effective in-plane stiffness.

Figure 2

Hybrid kirigami stiffness and ultimate strain. (a) A schematic of a kirigami structure with geometric parameters and different designs. Data point symbol represents each beam size, which is defined by major cuts, data point interior represents different minor cut designs, and data point color represents the ratio of minor cut length over width of beam. (b) Effective in-plane stiffness $\overline{K}$ versus l_m/(2w) for varying pattern designs, with l_M = 20 mm, w = d = 2 mm, and l_m = 0–3 mm. Dashed/dotted/solid lines represent Equation 2 with FFE (α = 16), PPE (α = 4), and non-prismatic beam/pinned end condition (α ≈ 0.8), respectively. (c) $\overline{K}/(Et)$ versus (w/l_M) with varying major and minor cut conditions, dashed/dotted/solid lines represent the same α values as in component (c). (d) Ultimate strain (ε_ULT) versus w/l_M, where lines represent Equation 3 with various γ values as specified in the legend.

Figure 3

Generalized design criteria across length scales. (a) Mechanical property ${\epsilon }_{ULT}/\overline{K}$ versus a combined geometric parameter for all the data in this work, where lines represent Equation 3. (b) Scaling plot showing agreement between Equation 3 and experimental/simulation data from kirigami materials in the literature and from the current work, across a wide range of length scales and material classes.

Figure 4

Multifunctional materials through hybrid kirigami materials. (a) Load versus strain plot for kirigami stretchable electrodes with various hybrid designs. The inset shows the tunable stiffness in the initial regime (l_M = 20 mm, l_m = 4.5 mm, w = 3 mm). (b) Normalized resistance versus strain plot showing that the electrical resistance for each design remains largely constant through failure. (c) Demonstration of hybrid kirigami structures as stretchable interconnects under axial deformation up to 380% strain. (d) High speed photographs of a magnetoactive hybrid kirigami structure made of Fe-PDMS with design v (l_M = 15 mm, l_m = 1.3 mm, w = 0.9 mm, and t = 0.82 mm) in response to a magnetic field. (e) Strain versus time plot of the patterned sample during the first cycle and (f) during the initial cycling and after ~1000 cycles (final cycles) maintaining a large deformation (~330%). (g) Strain rate versus time plot of the sample during the initial cycles and final cycles, showing rapid motion in response to a magnetic field.