Generating multiple surfaces from a single inhomogeneous anisotropically deforming sheet*

Can we make a flat sheet transform first into Rodin’s thinker and then Michelangelo’s David?
Here, we derive a general solution to this inverse design problem for inhomogeneous, anisotropically deforming materials. Such actuating materials include 3D printed hydrogels that swell or "Baromorphing" pneumatic elastomers. In these materials local variations of the director field and deformation factors along and across the director field produce global shape changes. These multiple local degrees of freedom allow a single sheet to deform into multiple desired surface geometries in response to external actuation. Actuation by two external parameters, enables the sheet to cycle through a closed loop in conformation space to swim or do work. To solve the inverse problem, we use the curvatures of the target shapes to derive an integrable system of differential equations for the sheet's local degrees of freedom. We then provide an algorithm for integration of this system of equations that allows us to systematically find all solutions to the problem. This approach paves the way to find solutions optimized for different criteria including ease of manufacture, deformation pathway, and work efficiency.