Abstract
Recent experiments by Chopin and Kudrolli (Phys Rev Lett 111:174302, 2013) showed that a thin elastic ribbon, when twisted into a helicoid, may wrinkle in the center. We study this from the perspective of elastic energy minimization, building on recent work by Chopin et al. (J Elast 119(1–2):137–189, 2015) in which they derive a modified von Kármán functional and solve the relaxed problem. Our main contribution is to show matching upper and lower bounds for the minimum energy in the small-thickness limit. Along the way, we show that the displacements must be small where we expect that the ribbon is helicoidal, and we estimate the wavelength of the wrinkles.
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First appeared in Green (1937); now in the public domain
\(\varOmega ^{+}\): regions filled with positively sloped diagonal lines (blue). \(\varOmega ^{-}\): regions filled with negatively sloped diagonal lines (red). \(\varOmega ^{0}\): Two hexagons \(\varOmega ^{+}\cap \varOmega ^{-}\). Outlined with thick lines (purple) (Color figure online).
\(\varOmega ^{\pm },\,\varOmega ^{0}\): carried over from Fig. 3. \(x_1= \xi '_\mathrm {left}\) or \(\xi '_\mathrm {right}\): thick vertical lines (dark green). \(\blacktriangle ^{+}\): triangular shaded regions (light green) (Color figure online).
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Notes
There are at least two candidates. Firstly, this buckling may be related to the Poisson ratio-driven wrinkling of a stretched but untwisted sheet, as analyzed by Cerda and Mahadevan (2003). Alternatively, these could have to do with the fact that, using a nonlinear energy functional such as Eq. (5), the helicoid is not the relaxed solution. Our model sees neither effect: we exclude the first by assuming Poisson ratio 0, and the second by using a small-displacement, small-slope energy functional.
This is called tension field theory in the physics literature.
Note the similarity to Landau theories: a small but convex term regularizes a non-convex minimization problem.
This argument is due to Strauss (1973), who used it to show that control on \({{\mathrm{e}}}(\varvec{u})\) in \(L^1\) yields control on \(\varvec{u}\).
In the sequel, we will make many statements that hold for both \(\varvec{a}^+\) and \(\varvec{a}^-\). Any statement containing ± should be interpreted as two statements, one with ± everywhere replaced with \(+\), and a similar one with −.
This argument is closely connected to the trace inequality in BD. We have \(L^1\) control on \({{\mathrm{e}}}(\varvec{u})\) and would like control on the trace of \(\varvec{u}\). This does not follow directly from Korn’s inequality (which requires \(L^p\) control, for \(p>1\)), but it follows from the trace inequality in BD (Temam and Strang 1980; Babadjian 2015). Because we would like to know how the constant depends on \(\xi \) and \(l\), we prove the result directly.
\(\varvec{u}^{(h)}(\varvec{x})=\varvec{0}\) for \(|x_1| > \xi \).
There is no ‘off by one’ error at the upper limit of the summation: recall from Eq. (30) that \(f_{N_{h}+1} = 0\).
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Communicated by Felix Otto.
This work was partially supported by the National Science Foundation through Grants OISE-0967140 and DMS-1311833.
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Kohn, R.V., O’Brien, E. The Wrinkling of a Twisted Ribbon. J Nonlinear Sci 28, 1221–1249 (2018). https://doi.org/10.1007/s00332-018-9447-0
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DOI: https://doi.org/10.1007/s00332-018-9447-0