Influence of a spanning liquid film on the stability and buckling of a circular loop with intrinsic curvature or intrinsic twist density

Abstract

A variational model is used to study the behavior of a flexible but inextensible loop spanned by a liquid film, with the objective of explaining the stability and buckling of flat circular configurations. Loops made from filaments with intrinsic curvature and/or intrinsic twist density are considered, but attention is restricted to filaments with circular cross sections and uniform mechanical properties. Loops made with intrinsic curvature but no intrinsic twist density exhibit in-plane and out-of-plane buckling modes corresponding to stable solution branches that bifurcate from the branch of flat circular solutions and out-of-plane buckling occurs at a lower value of the dimensionless surface tension of the liquid film than does in-plane buckling. Additionally, however, the destabilizing influence of the intrinsic curvature can be countered by increasing the torsional rigidity relative to the flexural rigidity. For a loop with both intrinsic curvature and intrinsic twist density, only one branch of stable solutions bifurcates from the flat circular solution branch, the in-plane and out-of-plane buckling modes are intertwined, and bifurcation occurs at a value of the dimensionless surface tension less than that governing the behavior of loops made from filaments that are intrinsically rectilinear. Moreover, increasing the torsional rigidity relative to the flexural rigidity has no or little stabilizing effect if the loop is either too short or too long and, in contrast to what occurs for loops with only intrinsic curvature, if the intrinsic twist density is sufficiently large then the destabilizing influence of the intrinsic curvature cannot be countered by increasing the torsional rigidity relative to the flexural rigidity, regardless of the length of the loop.

Buckling of chiral elastic rings spanned by fluid films

“…) where µ is an unknown constant. Evidently, the equilibrium criterion (A.5) is identical with (13) describing moment balance in the tangential direction on C. Therefore, to prove that the force balance condition (14) coincides with (A.1) 1 , it is sufficient to prove that…”

“…Giomi and Mahadevan [8] then experimentally and numerically identified the forms of the boundary using fishing strands and soapy fluid. Fried and co-workers [9][10][11][12][13] subsequently investigated theoretically the stability and bifurcation of the planar configuration.…”

Solution of the Kirchhoff–Plateau Problem

“…In recent years, some attention has been drawn to the Kirchhoff-Plateau problem following a paper by Giomi & Mahadevan [19], and the stability properties of flat circular solutions have been investigated, under various conditions regarding the material properties of the rod, by Chen & Fried [8], Biria & Fried [6,7], Giusteri, Franceschini & Fried [20], and Hoang & Fried [25]. An existence result for a similar problem was given by Bernatzky & Ye [5], but the elastic energy used therein fails to satisfy the basic physical requirement of invariance under superposed rigid transformations and thereby implicitly entails the appearance of unphysical forces.…”