Buckling of a soap film spanning a flexible loop resistant to bending and twisting

Abstract

A generalization of the Euler–Plateau problem to account for the energy contribution due to twisting of the bounding loop is proposed. Euler–Lagrange equations are derived in a parametrized setting and a buckling analysis is performed. A pair of dimensionless parameters govern buckling from a flat, circular ground state. While one of these is familiar from the Euler–Plateau problem, the other encompasses information about the ratio of the torsional rigidity to the bending rigidity, the twist density and the size of the loop. For sufficiently small values of the latter parameter, two separate groups of buckling modes are identified. However, for values of that parameter exceeding the critical twist density arising in Michell's study of the stability of a twisted elastic ring, only one group of buckling modes exists. Buckling diagrams indicate that a loop with greater torsional rigidity shows more resistance to transverse buckling. Additionally, a twisted flexible loop spanned by a soap film buckles at a value of the twist density less that the value at which buckling would occur if the soap film were absent.

Theoretical and experimental study of the stability of a soap film spanning a flexible loop

“…wherein β is a constant and ϑ is the contact angle defined in (11). The equilibrium conditions (28) coincide with equations derived previously by Biria and Fried (2014) as a consequence of a vectorial equilibrium condition arising in the context of a parameterized description akin to that utilized in Section 4. For σ = 0, the external force acting on the boundary vanishes and, hence, (28) 2 and (28) 3 specialize to the Euler-Lagrange equations for a Kirchhoff elastic rod, as developed by Langer and Singer (1996).…”

“…The aim of the present work is to build upon the previous work of Biria and Fried (2014) by performing a stability analysis and, further, to conduct an experimental test of the theory. Motivated by the application to DNA minicircles, a considerable literature has developed in relation to the stability of twisted rings.…”

Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses

“…We note that, alternatively, an energy formulation can be used to establish the governing equations; for example, the static buckling of twisted elastic rings within a soap film is studied in Refs. [12,14].…”

“…Many other elastic structures exhibit a similar phenomenology: For instance, the static buckling of an elastic ring under external pressure is a physical problem that has received much attention [6][7][8][9][10][11][12][13][14]. These seminal works provide an in-depth analysis of the stability of the ring, showing that the lowest mode is the "figure-of-eight" shape, determining higher modes and also offering different analytical and numerical methods through which to find the equilibrium shapes after bifurcation.…”

“…This leaves an unbalanced tension, γ ≈ 26.5 mN/m, acting on the inner edge of the ring. This corresponds to an effective pressure difference p = γ /h ≈ 26.5 Pa acting on the ring, or a dimensionless pressure, defined in (12), P ≈ 105. Here, the characteristic timescale t * ≈ 0.31 s so that the final image shown here corresponds to dimensionless time t/t * ≈ 0.1.…”