We prove a relation between the scaling h β of the elastic energies of shrinking non-Euclidean bodies S h of thickness h→ 0 , and the curvature along their mid-surface 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 Γ-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 Γ-limit proves asymptotics calculated in Aharoni et al. (Phys. Rev. Lett. 117:124101, 2016).
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Acknowledgements We thank Robert Jerrard for some useful advice and suggestions during the preparation of this paper, and Raz Kupferman for his critical reading of the manuscript. The second author was partially funded by the Israel Science Foundation (Grant No. 661/13), and by a grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research, the Russian Federation.
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- Gauss-Codazzi equations
- Incompatible elasticity
- Non-Euclidean plates
- Non-Euclidean rods