Optimal control of plate shape with incompatible strain fields

TL;DR

This paper develops a numerical method to determine optimal inhomogeneous growth fields that shape flat plates into desired three-dimensional forms, with applications demonstrated on various target geometries including paraboloids and saddles.

Contribution

It introduces a novel numerical approach for designing growth fields to achieve arbitrary target shapes in plates, including semi-analytical solutions for specific geometries.

Findings

Effective numerical solutions for non-symmetric and symmetric cases

Successful shaping of plates into paraboloidal, cylindrical, and saddle forms

Identification of a soft mode in non-axisymmetric deformations

Abstract

A flat plate can bend into a curved surface if it experiences an inhomogeneous growth field. In this article a method is described that numerically determines the optimal growth field giving rise to an arbitrary target shape, optimizing for closeness to the target shape and for growth field smoothness. Numerical solutions are presented, for the full non-symmetric case as well as for simplified one-dimensional and axisymmetric geometries. This system can also be solved semi-analytically by positing an ansatz for the deformation and growth fields in a circular disk with given thickness profile. Paraboloidal, cylindrical and saddle-shaped target shapes are presented as examples, of which the last two exemplify a soft mode arising from a non-axisymmetric deformation of a structure with axisymmetric material properties.