John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138.
Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712.
Proc Natl Acad Sci U S A. 2017 Oct 31;114(44):11597-11602. doi: 10.1073/pnas.1709025114. Epub 2017 Oct 16.
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.
受叶子、花朵和其他组织的差异生长驱动形态发生的启发，人们对由对热、光、湿度等环境刺激做出响应的活性材料制成的这些可变形薄片的人工模拟越来越感兴趣。但是，我们如何确定生长模式，以从另一种形状获得给定的形状呢？我们通过提出并解决弹性能量最小化问题来解决确定各向同性弹性双层以生长成目标形状的生长因子和方向（度量张量）的这个几何反问题。通过提供关于最终形状的生长度量张量的代数表达式，双层和弯曲单层之间的数学等价性大大简化了反问题。这种方法还使我们能够证明，我们可以使用各向异性生长的双层从任何参考表面生长任何目标表面。我们通过数值模拟将平面片生长成一个面、圆柱片生长成一朵花以及将平面片生长成复杂的峡谷状结构来证明这一点。
