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

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].

References in corpus (4)

• Elastic theory of unconstrained non-Euclidean plates
• Elasticity and Fluctuations of Frustrated Nano-Ribbons
• Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics
• Symmetry breaking in indented elastic cones

Cited by in corpus (4)

• Cumulative geometric frustration in physical assemblies
• Asymptotic rigidity for shells in non-Euclidean elasticity
• Stability of isometric immersions of hypersurfaces
• Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case