CUED Publications database

Hidden asymptotic symmetry in a long elastic structure

Pandurangi, SS and Healey, TJ and Triantafyllidis, N (2020) Hidden asymptotic symmetry in a long elastic structure. SIAM Journal on Applied Mathematics, 80. pp. 1083-1100. ISSN 0036-1399

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Abstract

Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling solutions; cf. Healey, Li, and Cheng [J. Nonlinear Sci., 23 (2013), pp. 777-805]. These correspond to arbitrary phase shifts of the wrinkled pattern in the transverse direction. While such behavior is normally associated with problems in the presence of a continuous symmetry group, an unloaded rectangular sheet possesses only a finite symmetry group. In order to understand this phenomenon, we consider a simpler problem more amenable to analysis-a finite-length beam on a nonlinear softening foundation under axial compression. We first obtain asymptotic results via amplitude equations that are valid as a certain nondimensional beam length becomes sufficiently large. We deduce that any two phase shifts of a solution differ from one another by exponentially small terms in that length. We validate this observation with numerical computations, indicating the presence of solution orbits for sufficiently long beams. We refer to this as "hidden asymptotic symmetry."

Item Type: Article
Subjects: UNSPECIFIED
Divisions: Div C > Materials Engineering
Depositing User: Cron Job
Date Deposited: 28 Oct 2020 20:18
Last Modified: 06 Apr 2021 01:47
DOI: 10.1137/19M125162X