Emergent morphogenesis via planar fabrication enabled by a reduced model of composites

Abstract

The ability to engineer complex three-dimensional shapes from planar sheets with precise, programmable control underpins emerging technologies in soft robotics, reconfigurable devices, and functional materials. Here, we present a reduced-order numerical and experimental framework for a bilayer system consisting of a stimuli-responsive thermoplastic sheet (Shrinky Dink) bonded to a kirigami-patterned, inert plastic layer. Upon uniform heating, the active layer contracts while the patterned layer constrains in-plane stretch but allows out-of-plane bending, yielding programmable 3D morphologies from simple planar precursors. Our approach enables efficient computational design and scalable manufacturing of 3D forms with a single-layer reduced model that captures the coupled mechanics of stretching and bending. Unlike traditional bilayer modeling, our framework collapses the multilayer composite into a single layer of nodes and elements, reducing the degrees of freedom and enabling simulation on a 2D geometry. This is achieved by introducing a novel energy formulation that captures the coupling between in-plane stretch mismatch and out-of-plane bending - extending beyond simple isotropic linear elastic models. Experimentally, we establish a fully planar, repeatable fabrication protocol using a stimuli-responsive thermoplastic and a laser-cut inert plastic layer. The programmed strain mismatch drives an array of 3D morphologies, such as bowls, canoes, and flower petals, all verified by both simulation and physical prototypes.