Programming evolution of geometry in shape-morphing sheets via spatiotemporal activation

Shape-programmed sheets morph into curved surfaces upon activation by stimuli such as illumination, and have attracted huge interest for their potential engineering applications, especially in soft robotics. Complex shape changes can be achieved by patterning a simple local active deformation, to generate differential growth. Usually the material itself is designed --- for example by patterning a molecular director --- such that a particular shape change occurs upon exposure to a spatially uniform stimulus. A limitation of this paradigm is that typically only one target geometry can be attained as the stimulus is adjusted. We show that this limitation can be overcome by patterning the stimulus itself, thereby exercising spatiotemporal control over local deformation magnitudes. Thus a single physical sample can be induced to traverse a continuous family of target geometries, opening the door to precise shape adjustments, new functionalities, and designable non-reciprocal loops in shape space. We illustrate these exciting possibilities with examples including active parabolic reflectors, chiral flow guides, and bending channels. In each case the actuation is mechanically strong because it involves changes in metric, rather than mere pure-bend deformation. Finding the necessary patterns of activation involves solving families of metric inverse problems; we solve these by reduction to ODEs in an axisymmetric setting, then present a novel numerical scheme to solve them in generality.