On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies

We prove a relation between the scaling \(h^{\beta}\) of the elastic energies of shrinking non-Euclidean bodies \(\mathcal{S}_{h}\) of thickness \(h\to0\), and the curvature along their mid-surface \(\mathcal{S}\). This extends and generalizes similar results for plates (Bhattacharya et al., Arch. Ration. Mech. Anal. 221(1):143–181, 2016; Lewicka et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34:1883–1912, 2017) 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 Aharoni et al. (Phys. Rev. Lett. 108:235106, 2012) using a formal asymptotic expansion. The proof involves calculating the \(\varGamma \)-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 \(\varGamma\)-limit proves asymptotics calculated in Aharoni et al. (Phys. Rev. Lett. 117:124101, 2016).