Curved geometries from planar director fields - Solving the two-dimensional inverse problem

Abstract

Thin nematic elastomers, composite hydrogels and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date, there is no general solution to the inverse design problem: how to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.

arXiv comment10 pages, 6 figures

References in corpus (8)

• Making Faces: Universal Inverse Design of Surfaces with Thin Nematic Elastomer Sheets
• Blueprinting Nematic Glass: Systematically Constructing and Combining Active Points of Curvature for Emergent Morphology
• Encoding Gaussian curvature in glassy and elastomeric liquid crystal polymer networks
• Curvature by design and on demand in liquid crystal elastomers
• Curvature generation in nematic surfaces
• Programming complex shapes in thin nematic elastomer and glass sheets
• Nematic director fields and topographies of shells of revolution
• Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets

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• Multi-valued inverse design: multiple surface geometries from one flat sheet
• Interfacial metric mechanics: stitching patterns of shape change in active sheets
• Metric mechanics with non-trivial topology: actuating irises, cylinders and evertors
• Discontinuous metric programming in liquid crystalline elastomers
• Shape Morphing of Planar Liquid Crystal Elastomers
• Compatible director fields in ${\mathbb{R}}^3$
• Design of pre-stressed plate-strips to cover non-developable shells
• Active elastocapillarity in soft solids with negative surface tension