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

TL;DR

The paper tackles scalable design of 3D morphologies from planar bilayer sheets by introducing a reduced-order, single-layer discrete-elastic-shell framework that couples in-plane stretch with out-of-plane bending under thermal actuation. The core innovation is an energy formulation $E(X,T)$ and a DES algorithm that captures stimulus-dependent rest configurations, enabling accurate prediction of permanent morphing in Shrinky Dink/kirigami-PLA bilayers with a fraction of the computational cost of full 3D models. Experimental validation across three patterns shows strong agreement (SSIM around $0.8$) with numerically predicted shapes and corroboration with finite-element simulations, demonstrating both predictive capability and computational efficiency. The approach supports repeatable planar fabrication and rapid design of complex 3D forms for applications in soft robotics and deployable structures, with potential generalization to other architected materials beyond bilayers.

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.

Figures (10)

• Figure 1: Examples demonstrating the validation between 3D shapes manufactured by physical experiments using the protocol developed here and the ones generated by numerical simulations using the proposed energy in Equation \ref{['eq:bilayer_energy']}. Scalebar: 1 cm.
• Figure 2: Examples demonstrating the morphing of 2D bilayer composites into 3D shapes via thermal actuation. The first row shows the 2D designs: each design consists of a specific kirigami pattern (PLA layer) on a thermoplastic (Shrinky Dink) substrate of defined shape. The second row displays the resulting 3D shapes. Subfigures A, B, and C correspond to three representative cases: bowl, canoe, and flower petal morphologies, respectively. For each design, the left column (i) presents results from physical experiments, and the right column (ii) shows the corresponding numerical simulations using the new energy model in Equation \ref{['eq:bilayer_energy']}.
• Figure 3: Experimental setup. (A) Design of the kirigami pattern (inert PLA layer) and selection of the substrate domain (thermoplastic Shrinky Dink sheet). (B) 3D printing of the kirigami pattern. (C) Assembly of the bilayer composite by bonding the 3D-printed PLA pattern to the substrate. (D) Thermal actuation in an oven for shape morphing. (E) Measured variation of the shrink ratio ($L/L_0$) as a function of normalized temperature ($T/T_g$), where $T_g = 200^\circ$F ($366.5$K) is the glass transition temperature of polystyrene. The sheet begins to shrink when $T/T_g > 1$, and the shrink ratio stabilizes above $T/T_g \approx 1.15$ (indicated by the vertical dashed line). Scale bars in (B), (C), and (D) denote 1 cm.
• Figure 4: Reference-configurations meshes for designs A-C in Figures \ref{['fig:1']} and \ref{['fig:2']}. A-C(i) Thermoplastic PLA bilayer boundaries (solid lines) on discrete mesh. A-C(ii) The distribution of thermal strains as defined in Equation \ref{['eq:thermal_strain']}. See Section \ref{['sec:energy']} for definitions of $\Omega$, $\mathcal{B}$, $\varepsilon_{\rm pre}^{\mathrm{th}}$, and $d_i$. D. Local node and edge IDs for a bending pair. The number of nodes $N_n$, number of edges $N_e$, and number of triangular elements $N_t$, $(N_n, N_e, N_t)$ are (970, 2800, 1831) for pattern A, (1215, 3505, 2291) for pattern B, and (388, 1087, 700) for pattern C, respectively. E. Variation of normalized thermal strain $\varepsilon^{\mathrm{\rm th}}/\varepsilon_{\rm pre}^{\mathrm{\rm th}}$ with normalized distance ${d}/{d_{\max}}$ for different values of $\gamma$, corresponding to the characteristic length $L_c=d_{\max}/\gamma$ in Equation \ref{['eq:Lc']}. F(i)-(iii) Deformed configurations of pattern A with $\varepsilon_{\rm pre}^{\mathrm{\rm th}}=0.7$ for $\gamma=0.1$, 1, and $5$, respectively. G. Top views of the corresponding configurations shown in F.
• Figure 5: Representative snapshots, starting with the pattern of Figure \ref{['fig:2']}A during the thermally actuated manufacturing process. From the top to the bottom rows, the duration of the thermal stimulus increases. In each column, (i) shows the experimentally scanned shape, (ii) shows the numerical simulation, (iii) shows the distribution of the absolute value of the dihedral angle $|\theta|$, and (iv) shows the distribution of the axial strain $\varepsilon$.