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Confined elasticae and the buckling of cylindrical shells

Confined elasticae and the buckling of cylindrical shells AbstractFor curves of prescribed length embedded into the unit disk in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds 2⁢π{2\pi}and in the large length limit. In the small excess length case, we prove convergence to a fourth-order obstacle-type problem with integral constraint on the real line which we then solve. From the solution, we obtain the energy expansion 2⁢π+Θ⁢δ13+o⁢(δ13){2\pi+\Theta\delta^{\frac{1}{3}}+o(\delta^{\frac{1}{3}})}when a curve has length 2⁢π+δ{2\pi+\delta}and determine first order coefficient Θ≈37{\Theta\approx 37}. We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Calculus of Variations de Gruyter

Confined elasticae and the buckling of cylindrical shells

Wojtowytsch, Stephan
Advances in Calculus of Variations , Volume 14 (4): 33 – Oct 1, 2021
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Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1864-8266
eISSN
1864-8266
DOI
10.1515/acv-2019-0033
Publisher site
See Article on Publisher Site

Abstract

AbstractFor curves of prescribed length embedded into the unit disk in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds 2⁢π{2\pi}and in the large length limit. In the small excess length case, we prove convergence to a fourth-order obstacle-type problem with integral constraint on the real line which we then solve. From the solution, we obtain the energy expansion 2⁢π+Θ⁢δ13+o⁢(δ13){2\pi+\Theta\delta^{\frac{1}{3}}+o(\delta^{\frac{1}{3}})}when a curve has length 2⁢π+δ{2\pi+\delta}and determine first order coefficient Θ≈37{\Theta\approx 37}. We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell.

Journal

Advances in Calculus of Variations – de Gruyter

Published: Oct 1, 2021

Keywords: Euler’s elastica; confined elastic curve; obstacle problem; integral constraint; energy scaling; cylindrical shell; buckling; 53A04; 49J40; 74K25; 74P20; 49K30

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