  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.
