Growth patterns for shape-shifting elastic bilayers

Abstract

Inspired by the differential-growth-driven morphogenesis of leaves, flowers, and other tissues, there is increasing interest in artificial analogs of these shape-shifting thin sheets made of active materials that respond to environmental stimuli such as heat, light, and humidity. But how can we determine the growth patterns to achieve a given shape from another shape? We solve this geometric inverse problem of determining the growth factors and directions (the metric tensors) for a given isotropic elastic bilayer to grow into a target shape by posing and solving an elastic energy minimization problem. A mathematical equivalence between bilayers and curved monolayers simplifies the inverse problem considerably by providing algebraic expressions for the growth metric tensors in terms of those of the final shape. This approach also allows us to prove that we can grow any target surface from any reference surface using orthotropically growing bilayers. We demonstrate this by numerically simulating the growth of a flat sheet into a face, a cylindrical sheet into a flower, and a flat sheet into a complex canyon-like structure.

Keywords: 4D printing; form; growth; inverse physical geometry; morphogenesis.

Fig. 1.

(Left) A growing bilayer is considered as two independently, possibly inhomogeneously, growing layers, characterized by their own respective metrics ${𝐚}_{r1}$ and ${𝐚}_{r2}$, that are glued together at a shared midsurface. In this example, each layer grows in only one direction, orthogonal to that of the other layer, with the linear growth factor $s>\mathrm{\hspace{0.17em}0}$. After each layer is grown, the bilayer embedding that minimizes the total elastic energy, characterized by first and second fundamental forms ${𝐚}_c$ and ${𝐛}_c$, can be computed. (Right) The surface $M$ is defined as an embedding $\overrightarrow{m}$ of an arbitrary region of the plane $U$ into ${ℝ}^3$. The embedding provides a normal field $\overrightarrow{n}$, as well as the first and second fundamental forms ${𝐚}_c$ and ${𝐛}_c$, as described in the text.

Fig. 2.

Inverse design of vegetable, animal, and mineral surfaces. A snapdragon flower petal starting from a cylinder (Left), a face starting from a disk (Center), and the Colorado River horseshoe bend starting from a rectangle (Right). For each example, we show the initial state (top), the final state (bottom) and two intermediate grown states in between. In each state, the colors show the growth factors of the top (left) and bottom (right) layer, and the thin black lines indicate the direction of growth. The top layer is viewed from the front, and the bottom layer is viewed from the back, to highlight the complexity of the geometries. The target shape for each case is given in Inset at the bottom: a snapdragon flower (image courtesy of E. Coen); a computer-render of a bust of Max Planck (model is provided courtesy of Max Planck Institute for Informatics by the AIM@SHAPE Shape Repository); a satellite photo of the actual river bend (image courtesy of Google Earth). The height of the actual snapdragon flower is $\sim$30 mm (19), whereas the depth of the canyon is 393 m according to USGS elevation data. (See SI Appendix for animations and details.)