On the role of curvature in the elastic energy of non-Euclidean thin bodies

TL;DR

This paper establishes a fundamental relation between the elastic energy scaling of shrinking non-Euclidean bodies and their curvature, extending previous results to higher dimensions and confirming the natural $h^4$ scaling for rods.

Contribution

It generalizes the understanding of elastic energy scaling laws to arbitrary dimensions and co-dimensions, proving the $h^4$ scaling for non-Euclidean rods using $ ext{Gamma}$-convergence.

Findings

Elastic energy scales as $h^4$ for non-Euclidean rods.

The $ ext{Gamma}$-limit of scaled energies relates to curvature norms.

Results confirm previous asymptotic calculations.

Abstract

We prove a relation between the scaling $h^\beta$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the natural scaling for non-Euclidean rods with smooth metric is $h^4$, as claimed in [AAE+12] using a formal asymptotic expansion. The proof involves calculating the $\Gamma$-limit for the elastic energies of small balls $B_h(p)$, scaled by $h^4$, and showing that the limit infimum energy is given by a square of a norm of the curvature at a point $p$. This $\Gamma$-limit proves asymptotics calculated in [AKM+16].