# Propagation of pop ups in kirigami shells

## TL;DR

This paper investigates how kirigami shells can undergo discontinuous phase transitions through snapping-induced curvature inversion, enabling controlled pop-up processes and advancing the design of responsive metamaterials.

## Contribution

It demonstrates the existence of discontinuous phase transitions in kirigami shells and shows how to control the transition zone and stress through geometric design.

## Key findings

- Discontinuous phase transitions occur in cylindrical kirigami shells.
- Transition width and triggering stress are controllable via geometry.
- A smart skin design enhances actuator crawling efficiency.

## Abstract

Kirigami-inspired metamaterials are attracting increasing interest because of their ability to achieve extremely large strains and shape changes via out-of-plane buckling. While in flat kirigami sheets the ligaments buckle simultaneously as Euler columns leading to a continuous phase transition, here we demonstrate that kirigami shells can also support discontinuous phase transitions. Specifically, we show via a combination of experiments, numerical simulations and theoretical analysis that in cylindrical kirigami shells the snapping-induced curvature inversion of the initially bent ligaments results in a pop-up process that first localizes near an imperfection and then, as the deformation is increased, progressively spreads through the structure. Notably, we find that the width of the transition zone as well as the stress at which propagation of the instability is triggered can be controlled by carefully selecting the geometry of the cuts and the curvature of the shell. Our study significantly expands the ability of existing kirigami metamaterials and opens avenues for the design of the next generation of responsive surfaces, as demonstrated by the design of a smart skin that significantly enhance the crawling efficiency of a simple linear actuator.