Geometrically incompatible confinement of solids

TL;DR

This paper introduces a new theoretical principle called the Gauss-Euler elastica, explaining how thin solids confined to incompatible geometries can store minimal strain energy despite geometric constraints, simplifying complex mechanical analyses.

Contribution

It generalizes Euler's elastica to include incompatible confinement, providing a new framework for understanding the mechanics of thin solids under complex geometric constraints.

Findings

The ratio of strain to bending energy can be arbitrarily small in incompatible confinement.

Numerical and analytical results demonstrate the validity of the Gauss-Euler elastica.

The framework simplifies solving nonlinear equilibrium equations for thin solids.

Abstract

The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current day efforts to engineer flexible sensors that conform to the human body. We introduce a theoretical principle, broadly generalizing Euler's {\emph{elastica}} -- a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and has been widely applied to classical and modern studies of beams and rods. We define a class of {\emph{geometrically incompatible confinement}} problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' {\emph{Theorema Egregium}}, induces strain. Focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler {\emph{elastica}}". This emergent rule states that - despite the unavoidable strain in such an incompatible confinement - the ratio between the energies stored in straining and bending the solid may be arbitrarily small. The Gauss-Euler {\emph{elastica}} underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. This development thus opens new possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics.