Design of kirigami metamaterials with square-symmetric auxeticity under large stretching

Abstract

Auxetic kirigami metamaterials (KMs) exhibit vertical expansion under lateral stretching, making them ideal for applications requiring large-strain flexibility. However, existing thin-sheet KM designs face two major challenges, including loss of auxeticity due to out-of-plane bucking at high strains and anisotropic auxetic behavior limited to a single loading direction. This study proposes an optimization method to design innovative KMs with square-symmetric auxeticity. That is, the designed KMs possess consistent negative Poisson's ratios under large uniaxial stretching as independently loaded from two orthogonal directions. This is achieved by enforcing both mirror and rotational symmetries to the KM unit cells, which ensures identical geometries as viewed from these two directions. A structural optimization problem is formulated to determine cut shapes that can achieve desired negative Poisson's ratios under large stretching, considering potential out-of-plane buckling and yielding failures that affect in-plane properties. To avoid cut intersections, three distinct design domains are defined to constrain the movement of the control points. Numerical results demonstrate that the optimized KMs can achieve negative Poisson's ratios ranging from $-0.25$ to $-1.00$ under large stretching strains up to 10 %. Experimental results further validate the auxeticity of the proposed KM designs. In addition, the KMs with $-1.00$ are applied as flexible substrates for distortion-free image display under either lateral or vertical large stretching. The proposed method enables the design of innovative thin-walled structures with broad applications in flexible electronics, soft robotics and adaptive structures.