Theorem for the design of deployable kirigami tessellations with different topologies

TL;DR

This paper presents a systematic design method for creating deployable kirigami tessellations with various topologies, enabling precise shape control and topological preservation during deployment.

Contribution

It introduces a unified parametrization and theorem-based approach for designing complex kirigami patterns with different topologies, filling a gap in systematic design principles.

Findings

Developed a linear equation-based parametrization of kirigami patterns

Formulated a unified theorem for deployability constraints across topologies

Demonstrated shape control along deployment paths while maintaining topology

Abstract

The concept of kirigami has been extensively utilized to design deployable structures and reconfigurable metamaterials. Despite heuristic utilization of classical kirigami patterns, the gap between complex kirigami tessellations and systematic design principles still needs to be filled. In this paper, we develop a unified design method for deployable quadrilateral kirigami tessellations perforated on flat sheets with different topologies. This method is based on the parametrization of kirigami patterns formulated as the solution of a linear equation system. The geometric constraints for the deployability of parametrized cutting patterns are given by a unified theorem covering different topologies of the flat sheets. As an application, we employ the design method to achieve desired shapes along the deployment path of kirigami tessellations, while preserving the topological characteristics of the flat sheets. Our approach introduces interesting perspectives for the topological design of kirigami-inspired structures and metamaterials.