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

Biria, Aisa; Fried, Eliot

doi:10.1016/j.ijengsci.2015.05.002

Cited by 17 publications

(29 citation statements)

References 26 publications

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“…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].…”

Section: A Model Problemmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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Slender elastic objects such as a column tend to buckle under loads. While static buckling is well understood as a bifurcation problem, the evolution of shapes during dynamic buckling is much harder to study. Elastic rings under normal pressure have emerged as a theoretical and experimental paradigm for the study of dynamic buckling with controlled loads. Experimentally, an elastic ring is placed within a soap film. When the film outside the ring is removed, surface tension pulls the ring inward, mimicking an external pressurization. Here we present a theoretical analysis of this process by performing a postbifurcation analysis of an elastic ring under pressure. This analysis allows us to understand how inertia, material properties, and loading affect the observed shape. In particular, we combine direct numerical solutions with a postbifurcation asymptotic analysis to show that inertia drives the system towards higher modes that cannot be selected in static buckling. Our theoretical results explain experimental observations that cannot be captured by a standard linear stability analysis.

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Section: A Model Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2020

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“…For κ 0 = 0 and ω 0 = 0 , (12) reduces to the expression considered by Biria and Fried [16,17]. Otherwise, (12) involves a vectorial measure κ − κ 0 d of bending strain and a scalar measure ω − ω 0 of twisting strain.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let r be an arclength parameterization of C , so that r ′ = t and let u = δ r denote the variation of r . Then, using (A.27) and (A.33) of Biria and Fried [17] to replace δ κ and δ τ in (21) by relations involving u , u ′ , u ″ , and ( δ ψ ) ′ , we integrate by parts twice while invoking the periodicity of ψ , p , …”

Section: First and Second Variation Conditionsmentioning

confidence: 99%

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

Tuan

Fried

2016

Mathematics and Mechanics of Solids

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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.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Solution of the Kirchhoff–Plateau Problem

2017

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The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of noninterpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments.

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

Cited by 17 publications

References 26 publications

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

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

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

Solution of the Kirchhoff–Plateau Problem

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