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

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Abstract

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).

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Notes

  1. Note that this does not imply that \(\mathcal{S}\) is flat, which is \(\mathcal{R}^{\mathcal{S}}\equiv0\).

  2. Note that for different choices of \(Q_{h}\) we can have that \(v_{h}\) converge to different functions; however we can further require that \(\int_{B} f = 0\), \(\int_{B} \operatorname{Skew}(df) = 0\). In this case there is no ambiguity.

  3. The main theorem in [36] only states the uniqueness of \(F\), however its proof (specifically, the last paragraph on p. 34) shows the uniqueness of \(q^{\perp }\) as well.

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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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  1. Department of Mathematics, University of Toronto, Toronto, Canada

    Cy Maor

  2. Institute of Mathematics, The Hebrew University, Jerusalem, Israel

    Asaf Shachar

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  1. Cy Maor
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Correspondence to Asaf Shachar.

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Maor, C., Shachar, A. On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies. J Elast 134, 149–173 (2019). https://doi.org/10.1007/s10659-018-9686-1

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