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.

翻译：从平面薄片精确、可编程地调控复杂三维形状的能力，是软体机器人、可重构器件和功能材料等新兴技术的基础。本文提出一种针对双层系统的降阶数值与实验框架，该系统由刺激响应型热塑性薄片（Shrinky Dink）与基里伽米图案化的惰性塑料层粘合而成。在均匀加热条件下，活性层收缩而图案化层限制面内拉伸但允许面外弯曲，从而从简单的平面前驱体产生可编程的三维形态。我们的方法通过捕捉拉伸与弯曲耦合力学的单层简化模型，实现了三维形态的高效计算设计与可扩展制造。与传统双层建模不同，本框架将多层复合材料简化为单层节点与单元，减少了自由度并支持在二维几何上进行仿真。这是通过引入一种新颖的能量公式实现的，该公式捕捉了面内拉伸失配与面外弯曲之间的耦合——超越了简单的各向同性线弹性模型。在实验方面，我们利用刺激响应型热塑性材料和激光切割的惰性塑料层，建立了一套完全平面化、可重复的制造方案。编程的应变失配驱动了一系列三维形态（如碗状、舟状和花瓣状结构），均通过仿真和物理原型验证。