## 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.

## 1. Introduction

Plateau [1] observed that a rigid wire frame, regardless of its shape, is spanned by at least one soap film. In the famous Plateau problem, the frame is modelled as a given closed, rectifiable, Jordan curve and the soap film is modelled as a surface of zero mean-curvature. A century after Plateau's observation, Alt [2] posed a variation of the Plateau problem, known as the ‘thread problem’, in which one or more segments of the supporting frame are replaced by twist-free, inextensible threads of prescribed lengths, with shapes to be determined. The thread problem has interested mathematicians for several decades, resulting in considerable progress towards the understanding of the existence and regularity issues, as detailed in two publications by Dierkes *et al.* [3,4]. A specialization of the thread problem in which the frame contains no rigid segments was first considered by Bernatzki & Ye [5]. In this problem, the flexural resistance of the bounding loop competes with surface tension, which may deform the boundary to reduce the area of the spanning surface. Recognizing that it couples aspects of Euler's elastica with Plateau's problem, Giomi & Mahadevan [6] referred to this specialization of the thread problem as the ‘Euler–Plateau problem’. Issues related to the stability and bifurcation of flat, circular solutions of the Euler–Plateau problem were considered by Giomi & Mahadevan [6] and, subsequently, by Chen & Fried [7]. Experimental realizations of the problem, in which loops of fishing line were dipped into and extracted from soapy water, were considered by Giomi & Mahadevan [6] and Mora *et al.* [8].

The experiments and numerical simulations of Giomi & Mahadevan [6] predict a rich spectrum of equilibrium configurations, including non-planar saddle-like surfaces and figure-eights. The observation of such configurations immediately raises the question of what happens if the bounding loop resists twisting in addition to bending. Also of interest is the related question of what happens if the fishing line is twisted before its ends are joined to form a loop. These considerations suggest a generalization of the Euler–Plateau problem wherein the role of twisting energy is taken into account. This leads to challenges similar to those encountered in studies of twisted elastic rings by Michell [9], Zajac [10] and Benham [11].

Applications involving biofilaments have recently inspired extensive interest in twisted elastic rings, as exemplified by the works of Schlick [12], Goriely & Tabor [13], Coleman & Swigon [14], Hoffman *et al*. [15] and Thompson *et al*. [16]. Additionally, soap films have long served as prototypes for material interfaces and fluid membranes such as lipid bilayers. A linkage between filaments and soap films therefore provides a canonical model for various physical and biological systems in which the energies of surfaces and boundaries compete. Prominent examples of such systems include discoidal high-density lipoprotein particles, which consist of lipid bilayers bound by chiral protein belts and which Catte *et al*. [17] found may occupy both flat and saddle shaped configurations,^{1} and bacterial biofilms, which, as Whitchurch *et al*. [19] explain, combine hydrated matrices of microorganisms and DNA. Yet another example is the convoluted intestinal canal anchored by the dorsal mesentery membrane. Savin *et al.* [20] discovered that the force exerted by the attached mesentery is responsible for the chirality and looping of vertebrate intestine.

In this paper, the works of Giomi & Mahadevan [6] and Chen & Fried [7] are extended to explore how endowing the bounding loop with twisting energy influences the buckling behaviour of flat, circular configurations. The problem of performing a complete nonlinear bifurcation analysis, taking advantage as necessary of path following, is left to a later work, as are issues of existence and regularity analogous to those mentioned above in connection with the thread problem. The remainder of the paper is organized as follows. The basic assumptions regarding the models for the bounding loop and spanning surface appear in §2, along with some essential preliminaries from the differential geometry of curves and surfaces. For an unshearable and isotropic rod with uniform circular cross section, it suffices to consider the angle needed to rotate the normal and binormal of the Frenet frame of the centreline of the rod into the cross-sectional directors of the rod. A model net potential-energy is provided in §3. After Giomi & Mahadevan [6], we neglect intrinsic curvature. Furthermore, we disregard the analogous effect of intrinsic twist. Two dimensionless parameters in addition to that arising in the Euler–Plateau problem are identified. Whereas the first of these accounts for the ratio between the torsional and bending rigidities of the bounding loop, the second is a combination of the first and the dimensionless twist density. Emulating an approach common to the works of Alt [2], Hildebrandt [21,22], Nitsche [23] and Chen & Fried [7], a parametric formulation of the problem is presented in §4. In this formulation, the spanning surface is described by a smooth, bijective mapping of the unit disc onto a surface embedded in three-dimensional space, the boundary of which coincides with the centreline of the loop. In §5, the linearized versions of the general equilibrium conditions are derived and used to determine the values of the salient dimensionless parameters for which a flat, circular ground state exhibits buckling. Physical interpretations of the results are provided in §7 and concluding remarks appear in §8. A derivation of the equilibrium conditions for the bounding loop based on balance laws and a constitutive relation for the specific internal moment appears in appendix A.

## 2. Preliminaries

Consider a closed flexible loop spanned by a soap film. Let the loop be modelled as an inextensible, unshearable, isotropic and linearly elastic rod with uniform circular cross section and centreline *a*>0 and torsional rigidity *c*>0 of the rod to be uniform. Let the soap film be modelled as a surface *σ*>0. Assume that

In the context of special Cosserat theory of rods, as presented by Antman [24], the configuration of a rod is characterized by its centreline *d*_{1},*d*_{2},*d*_{3}} of orthonormal directors. Whereas *d*_{1} and *d*_{2} reside in the normal cross section of the rod, *d*_{3} is perpendicular to that cross section. For an inextensible, unshearable, isotropic rod, the third director *d*_{3} and the unit tangent vector ** t** of the Frenet frame of

*d*_{1}and

*d*_{2}must lie in the plane spanned by the normal

**and binormal**

*n***of the Frenet frame of**

*b***and**

*n***by rotation through some angle**

*b**ψ*, so that

The curvature *κ*, torsion *τ* and twist density *Ω* of *s* along *d*_{1} can be expressed as _{3}, yields
*Ω* may be non-trivial even if the torsion *τ* vanishes; moreover, *Ω* generally differs from *τ* unless the angle *ψ* needed to rotate the normal ** n** and binormal

**of the Frenet frame into the cross-sectional directors**

*b*

*d*_{1}and

*d*_{2}does not vary along the centreline

*Ω*simply as the ‘twist’. This definition is used by Langer & Singer [27], Coleman & Swigon [14], Gorielly & Tabor [28] and many others. In place of the term ‘twist density’, Landau & Lifshitz [29], Alexander & Antman [25] and Antman [24], respectively, use the terms ‘torsion angle’, ‘local twist’ and ‘torsional strain’. Despite delicate issues that accompany Love's [26] definition and the formulation of end conditions for open rods, Alexander & Antman [25] recognized its utility for closed rods. Moreover, the generalized ‘local twist’ defined by Alexander & Antman [25], eqn (3.4) specializes to (2.6) if, as is assumed here, the directors of the rod are orthonormal.

## 3. Net potential-energy

In view of the foregoing discussion and granted that gravitational effects are negligible, the net potential energy *E* of the system comprised by the flexible loop and the soap film takes the form
*c*=0 of vanishing torsional rigidity, (3.1) reduces to the net potential-energy of the Euler–Plateau problem [6,7].

Suppose that the system occupies a flat, circular configuration of the system with radius *R*>0 and uniform twist density *Ω*. For such a configuration, the right-hand side of (3.1) specializes to
*R* scales with the lineal bending-energy *πa*/*R* of the boundary, (3.2) reveals the prospective significance of three dimensionless parameters:
*χ* simply measures the importance of the torsional rigidity *c* relative to the flexural rigidity *a*, *μ* amalgamates the competing influences encoded in *χ* and the dimensionless twist density *RΩ*. Finally, *ν*, which is familiar from the Euler–Plateau problem, accounts for the importance of the areal free-energy *πR*^{2}*σ* relative to the lineal bending-energy *πa*/*R*, as observed by Chen & Fried [7].

## 4. Parametrization

Suppose that *R* is the radius of a circle with perimeter equal to the length of ** ξ** is a four-times continuously differentiable, injective mapping defined on the closed unit disc and the dimensionless radius

*r*and the polar angle

*θ*provide polar coordinates on the closed unit disc as illustrated in figure 1. Let differentiation with respect to

*r*and

*θ*be indicated by subscripts, so that

*r*≤1 and 0≤

*θ*≤2

*π*. The smoothness of

**implies that**

*ξ***(**

*ξ**r*,⋅) must be periodic on [0,2

*π*] for each

*r*in (0,1], as must its derivatives with respect to

*θ*up to the fourth order; moreover,

**must satisfy closure conditions**

*ξ**r*are mapped to points that belong to distinct, non-intersecting curves on

With reference to (4.1), assumption (2.1) that ** ξ** must be consistent with

*E*defined in (3.1) has the form

*χ*>0 as defined in (3.3)

_{1}. Further, since the area of a surface element of

*dr*and

*dθ*is simply |

*ξ*_{r}

*dr*×

*ξ*_{θ}

*dθ*|=|

*ξ*_{r}×

*ξ*_{θ}|

*dr*

*dθ*, the parametrized representation of the areal contribution to

*F*is simply

*ν*>0 as defined in (3.3)

_{3}.

Like the net potential-energy (3.2) of a flat, circular configuration with given uniform twist density, the lineal and areal free-energies (4.7) and (4.8) scale with *a*/*R*. It is therefore convenient to work with the dimensionless net potential-energy *Φ*, as defined by

## 5. Equilibrium conditions

At equilibrium, the first-variation condition
** w**=

*δ*

**and**

*ξ**ι*=

*δψ*be smooth variations of

**and**

*ξ**ψ*. Prior to computing

*δΦ*, consider the implications of the constraint (4.5). In particular, on differentiating (4.5), it follows that

*ξ*_{θ}and

*w*_{θ}must jointly satisfy

*δψ*)

_{θ}=

*δ*(

*ψ*

_{θ}), from which it ensues that

**defined on the boundary of the unit disc and consistent with the condition**

*p***(0)=**

*p***(2**

*p**π*),

*q*is a scalar field defined on the boundary of the unit disc and consistent with

*q*(0)=

*q*(2

*π*).

Following Chen & Fried [7], varying the sum of the flexural and areal contributions to *Φ* yields
** m** is defined according to

*λ*

_{1}is a Lagrange multiplier needed to ensure satisfaction of the constraint (4.5) of inextensibility.

Next, consider the twisting contribution to (4.9). By (4.6)_{2} and (5.4),
*λ*_{2} is another Lagrange multiplier required by (4.5). In view of (5.4) and (5.7), varying the twisting contribution to (4.9) yields

Further, combining (5.5) and (5.8) yields
*λ* is a composite Lagrange multiplier formed by a difference
*μ* defined in (3.3)_{2} and the Lagrange multipliers *λ*_{1} and *λ*_{2} entering (5.5) and (5.8).

For ** w** compactly supported on the interior of the unit disc, applying the first-variation condition (5.1)–(5.9) yields the areal equilibrium condition

**normal to**

*m*Further, since ** w** and

*ι*may be chosen independently on the boundary of the unit disc, (5.1) and (5.9) yield two lineal equilibrium conditions. Whereas one of these conditions

*Ω*of

Since the areal contribution to the net free-energy considered here is identical to that of the Euler–Plateau problem (without twist), it is not surprising that (5.11) is equivalent to the dimensionless areal equilibrium condition of Chen & Fried [7], eqn (100), who denote the unit normal to ** n** instead of

**. Additionally, in the degenerate case where the torsional rigidity**

*m**c*obeys

*c*=0 (so that, by (3.3)

_{2},

*μ*=0), (5.13) reduces to the lineal equilibrium condition of the Euler–Plateau problem obtained by Chen & Fried [7].

In addition to establishing the equivalence between the equilibrium conditions obtained by Giomi & Mahadevan [6] and those arising in the parametrized setting, Chen & Fried [7] show that the areal equilibrium condition is equivalent to the requirement

To acquire some insight regarding the geometric content of the lineal equilibrium condition (5.13), note that arclength on the boundary of the unit disc is measured by
** ζ** such that

**to**

*m***and**

*n***of the Frenet frame and contact angle**

*b**ϑ*of

*κ*,

*τ*and

*Ω*, and the definition (5.18) of

**the lineal equilibrium condition (5.13) can then be expressed as**

*ν**β*a constant. For

*σ*=0, in which case the spanning surface is absent, (5.20)

_{2}and (5.20)

_{3}coincide with the Euler–Lagrange equations arising from (10) of Langer & Singer [27] if the quantities

*λ*

_{1},

*λ*

_{2}and

*λ*

_{3}entering that equation are related to

*a*,

*c*,

*Ω*and

*γ*by

*Ω*=0, in which case twist is absent, (5.20)

_{2}and (5.20)

_{3}agree with the equilibrium conditions (2.8a,b) of Giomi & Mahadevan [6] if the quantity

*α*entering those equations and

*a*are related by

_{2}and (5.20)

_{3}become (2.8a) and (2.8b) of Giomi & Mahadevan [6], respectively, with

*τ*replaced by

*β*replaced by

_{1}, this observation suggests identifying the quantity

*cΩ*/2

*a*as a uniform spontaneous torsion induced by the uniform twist density

*Ω*.

## 6. Solution to the linearized equilibrium conditions

Let ** e** and

*e*^{⊥}denote, respectively, the radial and transverse basis vectors associated with the dimensionless radius

*r*and polar angle

*θ*(figure 1). As observed by Chen & Fried [7], a planar disc bounded by a circle of unit radius provides a trivial solution to the dimensionless version of the Euler–Plateau problem. For this solution,

**and**

*ξ**λ*are given by

*e*_{θ}=

*e*^{⊥}and

*e*_{θθ}=−

**, it follows that**

*e*_{2−4},

Consider perturbations
** η** and

*ϵ*sufficiently small. Straightforward but lengthy calculations then lead to linearized versions

*μ*, a calculation of limited interest shows doing so does not alter the results up to the order of significance considered here.

The perturbation ** η** entering (6.4)

_{1}admits a decomposition of the form

*η*_{θ}to the boundary of the unit disc must obey

*θ*on both sides of (6.9) yields several useful identities:

Substituting (6.7) in the linear equilibrium equation (6.5) leads to the scalar Laplace equation
*w*. Similarly, substituting the decomposition (6.7), the inextensibility constraint (6.8), and the results (6.9) and (6.10) in the linearized boundary condition (6.6), and subsequently decomposing it into its azimuthal, radial and transverse components results in scalar equilibrium conditions of the form

Assuming that (6.11) admits separable solutions of the form
*ϱ* bounded and *Θ* periodic, produces solutions of the form
_{3} determines the perturbation *ϵ* entering (6.4)_{2} through
*C* a constant. Using (6.15) in (6.12)_{2} gives
*v*|_{r=1} of the radial perturbation *v* to the boundary of the unit disc admits a representation of the form
*c*_{0}=*C*/(1+*ν*). As a consequence of (6.17), the inextensibility constraint (6.8) requires that the azimutional perturbation *u* obeys

Additionally, for each *n*≥1, substituting (6.14) and (6.17) in the linearized boundary conditions (6.12)_{1} and (6.16) yields a pair
*a*_{n}, *b*_{n}, *c*_{n} and *d*_{n}. Interestingly, an argument due to Domokos & Healey [30] can be adapted to derive (6.19) and (6.20) as consequences of the rotational symmetry of the ground state. For (6.19) and (6.20) to possess non-trivial solutions, the determinants of the relevant coefficient matrices must vanish. Since those matrices differ only by the signs of their diagonal elements, their determinants are equal. A quick calculation shows that the associated solvability condition takes the form
*n*=1 describes rigid body rotations and, hence, is of no physical interest. For *n*≥2, (6.21) holds if and only if *ν* and *μ* satisfy
*n* and *μ* are viewed as given, (6.22) is a quadratic equation for *ν*. By (3.3)_{3}, *ν* must be positive and is thus determined by

On setting *μ*=0 and, thus, neglecting the influence of the twisting energy, (6.23) simplifies to
*ν*=*n*^{2}−1 and *ν*=*n*(*n*+1), consistent with the results of Chen & Fried [7] for in-plane and transverse buckling. In the absence of twist, the first buckling modes are therefore triggered at *ν*=3 and *ν*=6, corresponding to the choice *n*=2, as depicted in figure 3.

Another interesting specialization arises on neglecting the surface tension *σ* of the soap film, so that, by (3.3)_{3}, *ν*=0 and (6.21) reduces to
_{2} of *μ*, (6.25) yields the well-known critical value *Ω*=(*n*^{2}−1)*a*/*Rc* of the twist density for a twisted elastic ring, which was obtained independently by Michell [9], Zajac [10] and Benham [11]. In particular, (6.25) demonstrates that a twisted ring first buckles at *n*=2. This phenomenon has been termed the Michell instability by Goriely [31], who also provides an interesting description of the salient historical developments. With (6.23), the kernels of the coefficient matrices appearing in (6.19) and (6.20) can be found. On restricting attention to modes *n*≥2, these kernels can be used to obtain expressions for the coefficients in (6.17) and (6.20). Inserting these coefficients in (6.7) and introducing
** η** to the boundary of the unit disc.

Substituting (6.27) in (6.4)_{1} determines the bounding curve of the surface governed by (6.11), and thereby delivers both a solution to the linearized version of the Euler–Lagrange equations and the fundamental modes that bifurcate from the disc-shaped ground state. Note that the coefficients *a*_{n} and *b*_{n} in (6.26) are modal amplitudes, only one of which is independent when a normalization condition on the perturbation field *η* is imposed. Also, for 1+*μ*^{2}<*n*^{2}, the choice of sign in the first two terms on the right-hand side of (6.27) leads to two distinguished families of solutions. Representative plots of elements of these families are provided in figure 4 and the corresponding projections onto the (*x*,*y*)-plane are provided in figure 5. The projections of the first group onto the (*x*,*y*)-plane approximate circles that reveal the *n*-fold rotational symmetry of mode *n* around the vertical axis. While the primary buckling mode (*n*=2) is a simple saddle, the next buckling mode is a monkey saddle involving three depressions, and, similarly, the mode corresponding to *n*=4 is a saddle-like surface with four depressions. Although members of the second family of solutions do not possess rotational symmetry, they are centrosymmetric. In addition, their projections onto the (*x*,*y*)-plane exhibit *n*-fold symmetry. The projection of the primary buckling mode *n*=2 onto the (*x*,*y*)-plane is ellipse-like and the subsequent modes yield *n*-fold star-shaped figures.

## 7. Discussion

Figure 3 compares the dimensionless parameters for the primary buckling mode *n*=2 in the present setting with those of the Euler–Plateau and Michell problems. In the absence of twisting energy, the curve on the right specializes to the result obtained for transverse buckling in the context of the Euler–Plateau problem by Chen & Fried [7]. Also, the associated buckling mode, as depicted in the first row and column of figure 3, agrees with their suggested saddle-shaped configuration. Although including twisting energy does not alter the buckling mode, it increases the critical value of *ν* for transverse buckling—consonant with might have been anticipated on intuitive grounds. Transverse buckling induces torsion on the boundary. Endowing the bounding loop with twisting energy effectively penalizes warping of the spanning surface. Buckling of this kind therefore occurs only if the value of *ν* exceeds that predicted without twist.

On the other hand, if twisting energy is neglected the curve on the left in figure 3 specializes to yield the planar elliptic bifurcation discussed by Chen & Fried [7]. Since a planar bifurcation is not accompanied by a change of the twisting energy, it is reasonable to ask why accounting for twisting energy makes any difference and, moreover, why doing so leads to buckling at values of *ν* smaller than those arising in the Euler–Plateau problem. An answer to this question emerges on tracing the curve on the left in figure 3 down to *ν*=0, where the value of *μ* corresponding to Michell's [9] instability is recovered. This indicates that the curve under consideration describes the buckling of a circular flexible loop which is twisted before joining its ends and is spanned by a soap film. To reduce its twisting energy, such a twisted loop buckles out of plane at *ν*. Some buckling modes for this situation are depicted in the second row of figure 4. These share characteristics of both the planar modes (ellipse-like projections onto the plane for the primary bifurcating mode) of the Euler–Plateau problem and the non-planar modes of Michell as described by Goriely & Tabor [28], p. 37. In the absence of twisting energy (*μ*=0), these buckling modes become planar with shapes similar to those depicted in the second row of figure 5.

## 8. Summary

The equilibrium of a flexible loop spanned by a soap film was investigated, accounting for the previously neglected effect of twist. The approach taken was inspired by various developments in Cosserat rod theory and by the parametrized formulation employed by pioneers in the study of the thread problem, including Alt [2], Hildebrandt [21,22] and Nitsche [23]. The net potential-energy of the system, comprised by the bending and twisting energies of the flexible loop and the surface energy of the soap film, was varied under the constraint of lineal inextensibility to obtain the Euler–Lagrange equations. A flat, circular configuration was found to provide a trivial ground state solution to these equations. Small perturbations of the ground state were considered, leading to a linearized version of the Euler–Lagrange equations. In this linearized setting, the equilibrium shape of the surface is determined by the Laplace equation for the transverse component of the perturbation field. Moreover, the transverse and in-plane components of the perturbation field are coupled by a system of boundary conditions—boundary conditions that involve derivatives up to the fourth order with respect to the polar angle.

Solutions to the linearized equilibrium equations provide the basis for a buckling analysis. A pair of dimensionless parameters were found to control buckling from the ground state, parameters denoted here by *ν* and *μ*. Whereas *ν* measures the strength of the areal free-energy of the soap film relative to the lineal bending-energy of the flexible loop and is familiar from the Euler–Plateau problem, *μ* combines the dimensionless twist density with the ratio of the torsional rigidity to the bending rigidity. Two sets of buckling modes were identified. Of these, one set of modes describes the buckling of a twisted flexible loop spanned by a soap film. Consideration of these modes shows that the soap film reduces the value of the critical dimensionless parameter at which buckling occurs in comparison to the situation involving only a flexible loop endowed with bending energy and twisting energy, namely the situation considered by Michell [9], Zajac [10] and Benham [11]. In the absence of twisting energy, these modes are identical to the planar buckling modes of the Euler–Plateau problem obtained by Chen & Fried [7]. The remaining set of buckling modes correspond to transverse buckling and demonstrate the stabilizing influence of the torsional rigidity.

Aside from its inherent physical and mathematical beauty, the problem considered here provides a simple model for various systems in which biological membranes are bonded by elastic filaments or tubes, such as high-density lipoproteins, biofilms and the dorsal mesentery. A natural generalization of the problem considered would therefore involve endowing the spanning surface with elastic resistance to bending. Under such conditions, the surface exerts not only a force but also a bending moment on the bounding loop, as observed and investigated by Biria *et al.* [32]. Other future directions would include studying stability and bifurcations from non-trivial equilibrium configurations, and solving the full nonlinear version of the problem—where the treatment of self-contact becomes inescapably important. Savin *et al.* [20] note that the interplay between the elastic energy of the intestinal tube and surface energy of the mesentery attached to it results in the coiled structure of the gut. While coiling is essential for fitting the gut into the limited space of the abdominal cavity, excessive bending and twisting of the intestines, followed by knotting or kinking, result in volvulus and medical emergencies associated with bowel obstruction and ischaemia. Solving the nonlinear version of the Euler–Plateau problem augmented by twisting energy might help alleviate health problems stemming from highly twisted intestines.

Applications of the current problem are perhaps not limited to biological systems. Since the boundary conditions of the Euler–Plateau problem have mathematical analogies to conditions arising in cosmology reminiscent of those considered by Eardley [33], it is reasonable to ask whether this is also true for the generalization of that problem considered here? If twisting energy is taken into consideration, are there any similarities between the shape of the bounding loop and the mysterious twisted ring of molecular cloud surrounding the region of hot gas and dust in the galactic centre observed by Molinari *et al.* [34]? These and other interesting questions remain open and seem worthy of consideration.

## Appendix A. Alternative derivation of the lineal equilibrium conditions

The lineal equilibrium conditions (5.12) and (5.20) can also be derived directly from the force and moment balances for the centreline ** ϕ** and

**, those balances read**

*μ***=−**

*k**σ*

**is the external force, per unit length, that the spanning surface**

*ν*In terms of the directors *d*_{1}, *d*_{2} and *d*_{3} (figure 1), the specific internal moment ** μ** corresponding to the lineal part of the net free-energy

*E*defined in (3.1) admits a representation of the form

*d*_{1},

*d*_{2}and

*d*_{3}and the elements

**,**

*t***and**

*n***of the Frenet frame, simplifies to**

*b*_{2}into components parallel to the tangent

**and normal**

*t***to**

*n**m*is a constitutively indeterminate scalar. Using (A 4) in (A 5)

_{1}and recalling the assumption that the torsional rigidity

*c*is positive shows that

*Ω*must satisfy

**.**

*μ*Further, using (A 4) in (A 5)_{2} leads to
_{1} results in a vectorial condition
*β* is the constant appearing in (5.20). Finally, on stipulating that *m* and the Lagrange multiplier *λ* obey the relation
_{1}, with distributed external force ** κ**=−

*σ*

**, and the constitutive relation (A 2) for the internal moment**

*ν***agree with those, (5.20), derived on variational grounds.**

*μ*## Footnotes

↵1 Prior to the introduction of the Euler–Plateau problem by Giomi & Mahadevan [6], Catte

*et al.*[17] suggested that observed buckling transitions between flat, circular shapes and saddle-shaped configurations of discoidal high-density lipoprotein particles might be aptly modelled by treating the lipid bilayer and protein double belt components of such a particle as a soap film and a twist-free, inextensible rod, respectively (see Maleki & Fried [18], who study an augmentation of the Euler–Plateau problem that accounts for the resistance of the bilayer to bending and the resistance of the protein belt to bending and kinking).

- Received May 7, 2014.
- Accepted September 8, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.