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SIAM J. APPL. MATH. c© 2011 Society for Industrial and Applied MathematicsVol. 71, No. 1, pp. 20–40

MODELING THE EQUILIBRIUM CONFIGURATION OF APIECEWISE-ORTHOTROPIC PNEUMATIC ENVELOPE WITH

APPLICATIONS TO PUMPKIN-SHAPED BALLOONS∗

MICHAEL C. BARG† , JIEUN LEE‡ , AND FRANK BAGINSKI‡

Abstract. Large superlight structural systems that, for functional reasons, require large surfacesare composed at least in part of structural membranes. For efficiency of design, components thatexperience low stress can be made of lighter material, while those expected to experience high stresscan be reinforced with tendons or made from a stronger, albeit heavier, material. The design engineerseeks an efficient design without compromising structural performance and safety. The undercon-strained nature of such structural membranes poses analytical difficulties and leads to challengingmathematical problems in modeling, analysis, and numerical simulation. Motivated by the problemof modeling the shape of a high-altitude large scientific balloon, we present a model for a tendon-reinforced piecewise-orthotropic thin pressurized membrane. Using direct methods in the calculusof variations, a variational principle for a quasi-convex Caratheodory Lagrangian is developed, andrigorous existence theorems are established. Our model is implemented into a numerical code whichwe use to explore equilibrium configurations of a strained pumpkin-shaped balloon at low pressurewhere the symmetric shape becomes unstable.

Key words. membranes, calculus of variations

AMS subject classifications. 74K15, 58E30

DOI. 10.1137/100795875

1. Introduction. Due to their size, it may be difficult to build appropriatelyscaled physical models of large light-weight structures that will respond in a waythat is representative of the actual structure. For example, stratospheric large sci-entific balloons can be over 130 meters in diameter and are constructed from thinpolyethylene (PE) film 20–40 microns thick. It would be impractical to build a scalemodel 2 meters in diameter, since a sufficiently thin film does not exist. Similarly,it would be impossible to simulate a zero-gravity environment in a ground test of alarge-aperture antenna (say, 10 meters in diameter) that is to operate in outer space.In situations such as these, where physical tests cannot provide a complete picture ofthe full-scale structure, an accurate mathematical model is critical.

Motivated by the problem of modeling the shape of a high-altitude large sci-entific balloon, we present a mathematical model for a tendon-reinforced piecewise-orthotropic pressurized membrane. This model is readily adaptable to other struc-tures. For example, this approach was adapted to carry out a stress analysis of a thinnylon spherical pressure vessel that contained the scintillator for the solar neutrino ex-periment Borexino (see [8]). Although the scintillator was shielded by a thick barrierof buffer fluid of similar density and the temperature was regulated, experimenterswere concerned that, in the event of a power failure, a substantial temperature dif-

∗Received by the editors May 19, 2010; accepted for publication (in revised form) October 13,2010; published electronically January 4, 2011. The research presented here was supported in partby NASA awards NAG5-5353, NNX07AQ49G, and NNX07AR67G.

http://www.siam.org/journals/siap/71-1/79587.html†Department of Mathematics, Niagara University, Niagara, NY 14109 ([email protected]). This

author’s work was supported in part by a grant from the Niagara University Research Council inSummer 2009.

‡Department of Mathematics, The GeorgeWashington University, Washington, DC 20052 ([email protected], [email protected]).

20


ORTHOTROPIC PNEUMATIC ENVELOPES 21

ference between the inside and outside fluids would lead to a buoyant scintillator,possibly damaging the nylon shell or the supporting structure. Another applicationarises in spacecraft missions that require a large radio frequency (RF) antenna andhigh-power transmitters to accommodate elevated data rates. An inflatable antennasystem has unique features that make it a viable means to support such requirementswhile meeting the mass and packing constraints of a deep space mission (see [18]).Surface features directly affect RF performance, and to thoroughly probe the designspace, it is important to have an accurate mathematical model (see [13]).

To assess design validity and performance of inflatable membrane structures, amathematical model that captures the essential features of the structure is needed.Moreover, the model should allow for rigorous analysis with existence results thatensure the validity of numerical solutions. In [4], we established existence resultsfor pressurized isotropic membranes inflated with a lifting gas and reinforced withinextensible tendons. The results in [4] apply to closed and open balloon systems. Inthe present paper, we extend the results in [4, 7] to piecewise-orthotropic membraneswith elastic tendons. The tendons can be modeled as narrow ribbons located alongthe seam between adjacent gores. We derive an expression for the strain energy in theribbon that is equivalent to the strain energy in a linearly elastic tendon, enabling usto derive estimates that are needed to establish our existence results.

In section 2, we provide background for the balloon problem. In section 3, weformulate the problem of determining the shape of a strained elastic balloon with re-inforcing tendons. In section 4, we establish theoretical existence results. In section 5,we present numerical solutions for pumpkin-shaped balloons. For a stable design anda sufficiently high pressure, the cyclically symmetric fully developed pumpkin shape isachieved. When the pressure is reduced and positive lift is maintained, the symmetricshape becomes unstable, and the balloon seeks a nonsymmetric equilibrium configu-ration. We simulate these scenarios with a computer model and find good qualitativeagreement with test observations.

2. Background. The balloon is modeled as a nonlinearly elastic membraneshell (see [12, section 9.4]). Our strain energy density, W , is equivalent to the two-dimensional strain energy density of Koiter’s nonlinearly elastic membrane shell (see[12, p. 548]). In our formulation, we can ignore the contribution of the “flexuralenergy” since the balloon has negligible bending stiffness. In [4], the balloon wasassumed to be constructed of a single layer of an isotropic material. In the modelpresented here, the membrane is constructed of piecewise-orthotropic materials. Thetotal energy of the balloon system is

(2.1) I(x) =

∫Ω

(fP (x,∇x) +W ∗(u,∇x) + fw(u,x) + σ) dA,

where x is an admissible deformation and u = (u, v) ∈ Ω, a bounded, open, connectedset in R

2. In (2.1), dA denotes area-measure on Ω, fP is the hydrostatic pressurepotential density, fw is the film and tendon gravitational potential energy density,W ∗ is the relaxed film strain energy density, and σ is a constant to be specified later.Ω is referred to as the flat reference configuration. A tendon is treated as a thin ribbonwith one of its edges attached to the right edge of a gore (see Figure 2.1(a)–(b)). Inreality, a tendon is contained in a flattened sleeve of PE. The right edge of the ithgore, the right edge of the flattened sleeve, and the left edge of the (i+ 1)th gore aresealed together. We assume that the tendon does not slide within the sleeve. We canmodel a tendon as a narrow ribbon, and it provides a realistic representation of how a


22 MICHAEL C. BARG, JIEUN LEE, AND FRANK BAGINSKI

(a) (b) (c)

Fig. 2.1. (a) Gi is the reference configuration of the ith gore; subregions Gi ∩ Ωl (film) and

Gi ∩ Ωl′ (tendon ribbon). (b) Film facet T ∈ Gi ∩ Ωl and tendon facet T ∈ Gi ∩ Ωl′ . In a real

balloon, T ∈ Gi ∩Ωl′ is essentially a right triangle with base L and height h. (c) Material directions{ex, ey} and principal directions {n1,n2}.

real tendon behaves. When Ω is triangulated, each facet T ∈ Ω corresponds to eithera membrane component or a tendon ribbon component (see Figure 2.1(b)).

Mechanical properties of importance are film thickness, Young’s modulus, andPoisson’s ratio. In [4], the balloon film was assumed to have a constant thickness andconstant mechanical properties so that W ∗(u,x,∇x) ≡ W ∗(∇x). It follows that theintegrand in (2.1) is continuous, enabling us to apply [15, Theorem 2.9, p. 180] toestablish existence results. However, this theorem does not apply in the present case,where the mechanical properties of the film may vary from region to region and theintegrand in (2.1) (denoted f∗

Tot) is no longer continuous. In addition, we assume thatthe film is orthotropic and not isotropic. We will show that f∗

Tot is a Caratheodoryfunction and obtain weak lower semicontinuity as a consequence of [1, Theorem II.4,p. 137]. In the piecewise-orthotropic case, W ∗ depends explicitly on u ∈ Ω. We willuse a modified version of [15, Theorem 2.9, p. 180], where weak lower semicontinuityis guaranteed by [1, Theorem II.4, p. 137]. The next step is to show that f∗

Tot satisfiesα|A|p ≤ f∗

Tot(u,x,A) ≤ q(u) + C(|x|p + |A|p), where p > 1, α > 0, and C ≥ 0are constants and q is a nonnegative locally summable function. In our analysis, wewill show that W ∗ has upper and lower bounds proportional to |∇x|4, leading to aproblem formulation in W 1,4(Ω,R3). Next, we record a few basic definitions.

Definition 2.1. A continuous function f : R2×3 → R is quasi-convex if for everyA ∈ R

2×3, for every open subset Ω ⊂ R2, and every function φ ∈ W 1,∞

0 (Ω,R3),

(2.2) f(A)meas(Ω) ≤∫Ω

f(A+∇φ(u)) dA.

Definition 2.2. If f : Ω×R3×R

2×3 → R is continuous in the third argument foralmost every u ∈ Ω and every x ∈ R

3, then f is quasi-convex in the third argumentif (2.2) holds for almost every u ∈ Ω and every x ∈ R

3.For our model, we consider a complete shape. Constraints are built directly

into our solution space of admissible deformations. Here, there is only one boundary



condition: the bottom of the balloon and tendons are attached to a rigid nadir end-fitting whose center is located at the origin of a Cartesian coordinate system, i.e.,

(2.3) x(u, v) = (x0(u, v), y0(u, v), 0) for (u, v) ∈ Γ,

where Γ parametrizes the boundary of the end-fitting. Equation (2.3) can be writtenin the form g = 0 ∈ R

3 with g(u) = x(u) − (x0(u), y0(u), 0); it is an example of alocal constraint. A volume constraint is an example of a global constraint. In [4], wederived estimates to show that a volume constraint can be included in the solutionspace. We set g4(x) = V (x) − ω0, where V (x) is the volume of the lifting gas andω0 is the target volume. In general, a global constraint has the form gi(x) = 0, wheregi ∈ C(X,R) for some i and X is the solution space (see section 3). X is a closedsubset of the Sobolev space W 1,4(Ω,R3) and incorporates the constraints directly. Weuse the convention g(x) = 0 to indicate all local and global constraints.

3. Mathematical formulation. The balloon design problem is to determinethe flat cutting pattern for a gore lobe so that when the complete balloon is assembledand pressurized, it will attain the desired shape (see Figure 5.1(a)). For our purposes,the cutting pattern is known. See [3, 17] for more on the design process. In thissection, we present results that are needed to establish Theorem 4.3.

3.1. Preliminaries. Normally, a large scientific balloon is made by sealing anumber of long tapered panels of PE edge-to-edge to form a complete shape. Suchpanels are called gores and are constructed as follows. Long circular cylinders of PEfilm are formed using a slot-die extrusion process (imagine blowing a long Delaunaysurface of PE that is flattened, then rolled up onto a spool). The axis of the cylinderis called the machine direction. The cylinder is cut along its axis, then laid out into aplane. The final gore pattern GF is then cut from these panels. See Figure 2.1. Thenumber of gores is ng. Tendons are located along the seams between adjacent gores.Suppose Ω =

⋃ng

i=1 Gi, where Gi is the flat reference configuration of the ith gore.The reference configuration of a gore is positioned so that ey and ex are parallel tothe machine and transverse directions, respectively.

In the present work, we will allow the thickness of the film, as well as its mechanicalproperties, to vary over subregions of Ω. Let u = (u, v) ∈ Ω. The properties ofinterest are film thickness t(u), film weight density wfilm(u), Young’s moduli E0

i (u),and Poisson’s ratios ν0i (u) for i = 1, 2, which are assumed to be piecewise-constantfunctions. The superscript 0 indicates that the material constants are specified withrespect to the unstressed principal material frame (see section 3.4). We assume that

there is a finite collection of distinct values tl, E0,l1 , E0,l

2 , ν0,l1 , ν0,l2 , wl for l = 1, . . . ,msuch that

(3.1) Ωl = {u ∈ Ω | t(u) = tl, wfilm(u) = wl, Ek(u) = E0,lk , νk(u) = ν0,lk , k = 1, 2}

forms a partition of Ω, i.e., Ω =⋃m

l=1 Ωl. It is convenient to set νl1 = νl21 and νl2 = νl12.

Using Ωl from (3.1), a partition of the ith gore consists of the sets Ωli = Gi ∩ Ωl for

l = 1, . . . ,m, i = 1, . . . , ng. Each Ωli is triangulated. T denotes a triangle in Ωl

i.Strictly speaking, along the boundary where two Ωl

i’s are joined together, thereis some overlap. However, for most interfaces this overlap is insignificant, as it doesnot influence the overall response of the structure. The exception to this occurs alonga seam where a load tendon is attached. The contribution of the load tendon will behandled separately. Without loss of generality, we can assume that each Ωl

i is open



and Ωl ∩Ωk = ∅ when l �= k. By construction, the material properties of the film areconstant in Ωl

i.

In our present model, the edges of the gores are identified as they were in [4]. Anarrow ribbon of constant width h is attached along the seam between every pair ofadjacent gores. The complete membrane is Ω =

⋃ng

i=1 Gi, where each Gi is a copyof the known cutting pattern: GF = {(u, v) | −H(v) < u < H(v), 0 < v < Lc}.See Table 5.1(b). We assume that H ∈ C0,1((0, Lc),R) so that GF is a C0,1 domain.Since a balloon normally has a bottom end-fitting, we assume that GF has an edgeat the bottom. Let ΓF = {(u, v) | −H(0) < u < H(0), v = 0} be the bottom edge ofGF . For this exposition, we assume that a typical gore comes to a point at the apex,and so u = H(Lc) = 0. In Figure 2.1(a), Gi is displayed. Suppose G1 = GF , Δ > 0,and Gi = {(u+ (i − 1)Δ, v) | (u, v) ∈ G1}, i = 2, . . . , ng. The balloon is modeled byconsidering the union of the Gi’s with the following identifications. The right seam ofGi is identified with the left seam of Gi+1 for i = 1, 2, . . . , ng − 1, and the right seamof Gng is identified with the left seam of G1. Thus, Ω =

⋃ng

i=1

⋃ml=1 Ω

li.

Due to the identifications of the Gi’s along their respective seams, we can treatΩ as a two-dimensional manifold. However, to avoid the complicated notation of themanifold setting, we will treat the flat reference configuration as an open subset in R

2.In an effort to clarify our presentation, we also use the following conventions. We write

x ∈ C1(◦Ω,R3) to mean that x is differentiable in the interior of the open sets Gi for

1 ≤ i ≤ ng. With an appropriate metric defined on Ω, we can consider the closureof Ω, denoted by Ω. The boundary of Ω is denoted by Γ and represents the edge ofthe balloon that is attached to the bottom end-fitting. With these conventions, we

consider mappings x ∈ C1(◦Ω,R3) ∩ C(Ω,R3).

A typical mapping is x : Ω → S ⊂ R3, where x(Ω) = S and x is an admissible

deformation (see Definition 3.1). Let S0 = x0(Ω) be an initial, possibly strained, con-figuration that is not necessarily in equilibrium. x0(u, v) = (x0(u, v), y0(u, v), z0(u, v))is a deformation that satisfies (2.3). As in [4] and [7], W 1,4(Ω,R3) = {x | ‖x‖1,4 < ∞}is the proper Banach space setting for the balloon problem. The norm ‖·‖1,4 is equiv-alent to the standard Sobolev norm (see, e.g., [4]). In the following, we will studythe balloon problem in a subset of W 1,4(Ω,R3). We can now define an admissibledeformation mapping.

Definition 3.1. Let x0(u, v) = (x0(u, v), y0(u, v), z0(u, v)) ∈ W 1,4(Ω,R3). Anadmissible deformation mapping x(u, v) = (x(u, v), y(u, v), z(u, v)) is such that

(a) x ∈ C1(◦Ω,R3) ∩ C(Ω,R3),

(b) x(u, v) = x0(u, v) for (u, v) ∈ Γ, and(c) x(0, Lc) = y(0, Lc) = 0.

The set of admissible deformations is denoted by D. Implicit in Definition 3.1(c)is the fact that the top of the balloon (ztop = z(0, Lc)) is free to slide up and downthe z-axis. We define our solution space,

(3.2) X ={x = x0 + x | x ∈ W 1,4

0 (Ω,R3)},

to be the completion of D with respect to the norm ‖ · ‖1,4. This is consistent withthe definition of Sobolev spaces in the manifold setting [2, p. 32]. For the balloonproblem, a solution is in the form x = x0+ x ∈ X, where x ∈ W 1,4

0 (Ω,R3). The space

W 1,40 (Ω,R3) is the completion of C1(

◦Ω,R3) ∩ C0(Ω,R

3) with respect to ‖ · ‖1,4.



3.2. Hydrostatic pressure potential. We consider a position dependent hy-drostatic pressure in the form −P (z) = bz + p0. −P (z) > 0 means that the insideof the gas bubble is pushing outward at a height z units above the fixed base of theballoon, z = 0. At its base, an open balloon system has pressure p0. The volume willadjust to meet this condition. In a closed balloon system, p0 is unknown a priori, andit must be determined from the volume constraint. The zero-pressure natural-shaped(ZPNS) balloon system is open, and the pumpkin-shaped balloon system is closed. Ifthe density of the atmosphere is denoted by ρa, the density of the lifting gas is ρg,and the gravitational acceleration is g, then the specific buoyancy of the lifting gas,b = g(ρa − ρg), is positive. From [4], with P (x) = −(bx ·k+ p0), the potential for thehydrostatic pressure is

EP (x,∇x) =

∫D

P (x) dV =

∫Ω

fP (u,x,∇x) dA, where

fP (u,x,∇x) = − ( 12b(x · k)2 + p0(x · k))k · adj2∇x,(3.3)

D ⊂ R3 is the gas bubble, and dV is volume-measure in R

3. The adjugate adj2 isdefined in [15, section 4.1]. Suppose |x| ≤ R. Since |k · adj2∇x| ≤ |∇x|2, we have

(3.4) − (12bR2 + |p0|R) |∇x|2 ≤ fP (u,x,∇x) ≤ ( 12bR2 + |p0|R

) |∇x|2.

fP is shown to be quasi-convex in [4]. The condition |x| < R is not a serious restriction.R is chosen sufficiently large so that the constitutive relations that we employ remainvalid for a solution x.

3.3. Gravitational potential energy due to weight. The gravitational po-tential energy due to the film weight is

Efilm(x,∇x) =

∫Ω

fw(u,x,∇x) dA,

where fw(u,x,∇x) = wfilm(u)(x · k). Note that tendon ribbons also contribute toEfilm. Assuming |x| ≤ R and setting w = maxl{wl}, we see

(3.5) −wR ≤ fw(u,x,∇x) ≤ wR.

For almost every u ∈ Ω and for all x ∈ R3, the mapping A �→ fw(u,x,A) is constant

and hence convex. Thus, fw is quasi-convex in the third argument.

3.4. Constitutive relations for orthotropic materials. We present a briefsurvey of key definitions related to orthotropic materials here. We will follow theconventions of [16, Chapter 1]. The reference state of an elastic material is a naturalstate without stress or strain; i.e., the components of the stress tensor σ are σij = 0,and the components of the strain tensor ε are εij = 0. In three-dimensional linearelasticity, σ is a linear function of ε; i.e., σ = C : ε, and

(3.6) σij = Cijklεkl for i, j, k, l = 1, 2, 3.

In a homogeneous medium, the Cijkl elements are the 81 components of the elasticmoduli tensor or stiffness tensor C. Inverting (3.6), we find ε = S : σ, where

(3.7) εij = Sijklσkl for i, j, k, l = 1, 2, 3.



The Sijkl elements are the 81 components of the elastic compliance in the compliancetensor S. Using the symmetries of the tensors as previously defined, and following[16, Chapter 1], we have the contracted notation I = i for i = j, I = 9 − (i + j) fori �= j, and J = k for k = l, J = 9 − (k + l) for k �= l. Thus, (3.6) can be rewrittenσI = CIJ εJ for I, J = 1, . . . , 6, where

σI = σij with I = i for i = j,σI = σij with I = 9− (i + j) for i �= j,

εJ = εkl with J = k for k = l,εJ = 2εkl with J = 9− (k + l) for k �= l.

With these conventions, (3.7) can be expressed in the form

(3.8)

⎡⎢⎢⎢⎢⎢⎢⎣

ε1ε2ε3ε4ε5ε6

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

S11 S12 S13 0 0 0S12 S22 S23 0 0 0S13 S23 S33 0 0 00 0 0 S44 0 00 0 0 0 S55 00 0 0 0 0 S66

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

σ1

σ2

σ3

σ4

σ5

σ6

⎤⎥⎥⎥⎥⎥⎥⎦

(see [21, (2.20)]). The 6×6 matrix in (3.8), denoted by [Sij ], involves nine independentconstants and is called the compliance matrix. We find

(3.9) [Sij ] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1E1

− ν21E2

− ν31E3

0 0 0

− ν12E1

1E2

− ν32E3

0 0 0

− ν13E1

− ν23E2

1E3

0 0 0

0 0 0 1G23

0 0

0 0 0 0 1G31

0

0 0 0 0 0 1G12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where E1, E2, E3 are Young’s moduli in the 1, 2, 3 directions, respectively; νij arePoisson’s ratios for transverse strain in the j direction when stressed in the i direction;and G23, G31, G12 are shear moduli in the 2-3, 3-1, and 1-2 planes, respectively. Sincethe compliance matrix is symmetric,

(3.10) νij/Ei = νji/Ej .

Using the notation in [16], where ε4 = γ23, ε5 = γ31, ε6 = γ12, σ4 = τ23, σ5 = τ31, andσ6 = τ12, we see that for a lamina in the 1-2 plane, a state of plane stress is definedby setting σ3 = 0, τ23 = 0, and τ31 = 0 in (3.8). This implies that auxiliary equationsε3 = S13σ1 + S23σ2, γ23 = 0, and γ31 = 0 can be used to reduce (3.8) to

(3.11)

⎡⎢⎣

ε1

ε2

γ12

⎤⎥⎦ =

⎡⎢⎢⎣

1E1

− ν21E2

0

− ν12E1

1E2

0

0 0 1G12

⎤⎥⎥⎦⎡⎢⎣

σ1

σ2

τ12

⎤⎥⎦ .

If the matrix in (3.11) is nonsingular, we have

(3.12)

⎡⎢⎣

σ1

σ2

τ12

⎤⎥⎦ =

⎡⎢⎢⎣

E1

1−ν12ν21ν12E2

1−ν12ν210

ν21E1

1−ν12ν21E2

1−ν12ν210

0 0 G12

⎤⎥⎥⎦⎡⎢⎣

ε1

ε2

γ12

⎤⎥⎦ .



(a) (b)

Fig. 3.1. (a) E1/Ex, E2/Ey, G12/Gxy, and νxy as functions of θ, where E1 = 137 MPa,E2 = 164 MPa, ν12 = 0.58, ν21 = 0.69, and G12 = 30 MPa; (b) (δ1, δ2)-domain for W l and (W l)∗.

If a membrane is taut, then the 3 × 3 matrices in (3.11) and (3.12) are invertible.However, in the presence of compressive stresses, a thin membrane will wrinkle andthe matrices in (3.11) and (3.12) are singular. To model wrinkling, we will follow theenergy relaxation work of Pipkin [23] as described in section 3.5. It is important tonote that applying standard membrane theory and allowing “small” stresses may notaccurately describe the response of a thin membrane under all relevant loading condi-tions. The beauty of Pipkin’s approach is that it avoids the complications of dealingwith a singular stiffness matrix. For other applications using Pipkin’s approach, see[20, 24].

Properties such as Young’s modulus and Poisson’s ratio for a thin membranemust be determined via mechanical testing on the film of interest. Otherwise, theconstitutive relation will not describe the appropriate membrane response to a givenstrain. For example, invoking the assumption of isotropy and applying standard three-dimensional elasticity theory, one is led to the conditions 0 < ν < 1

2 (see [11, p. 129]).However, a Poisson’s ratio greater than 1

2 is to be expected for thin PE films (see, e.g.,[9, 19]). Two-dimensional membrane and shell theories can be developed by asymp-totic methods [12, Chapter 1] by letting the thickness go to zero. However, meaningfulresults can be obtained using appropriate mechanical properties and assuming thatthe thickness is piecewise-constant.

If the natural coordinates are not aligned with the principal material coordinates,then it is possible to relate quantities in the different frames. In the case of planestress, where the 1-2 principal frame is rotated an amount θ from the xy-frame, wehave

(3.13)

⎡⎣ σx

σy

τxy

⎤⎦ =

⎡⎣ cos2 θ sin2 θ −2 sin θ cos θ

sin2 θ cos2 θ 2 sin θ cos θsin θ cos θ − sin θ cos θ cos2 θ − sin2 θ

⎤⎦⎡⎣ σ1

σ2

τ12

⎤⎦

(see [16, section 1.5]). In the case of plane stress, the authors in [21, (2.97)] derivedrelations for an orthotropic membrane that is stressed in nonprincipal xy-coordinates,e.g., 1/Ex = cos4 θ/E1+(1/G12−2ν12/E1) sin

2 θ cos2 θ+sin4 θ/E2. Using [21, (2.97)],we present graphs of E1/Ex, E2/Ey, G12/Gxy, and νxy with 0 ≤ θ ≤ 1

2π for a typicalPE film at room temperature in Figure 3.1(a). In principle, one should be able toestimate the mechanical properties of the membrane for the principal material direc-tions and then use the matrix in (3.13) to transform the stresses of (3.12) when the



membrane is stressed in nonprincipal xy-coordinates. However, the mechanical prop-erties are only estimates, and they may lead to nonrealistic values in the transformedframe. Moreover, G12 is difficult to measure in the lab for thin PE, and the esti-mates for G12 are arguably unreliable. In our analysis, we will assume that E1(u, θ),E2(u, θ), and ν12(u, θ) for u ∈ Ωl are known functions of θ determined by mechanicaltesting on Ωl-specimens (in practice, they will be determined for several directions:θi =

12π(i− 1)/k for i = 1, . . . , k + 1). For the situation that is of primary interest to

us, thin membranes with no resistance to bending, there are four independent materialproperties E1, E2, ν12, and G12 when we account for the relation ν12/E1 = ν21/E2.In a state of plane stress, normal stresses dominate, and there is negligible resistanceto shearing. For these reasons, we do not include the shear stiffness in our model. Weare led to

(3.14)

[σ1

σ2

]=

1

1− ν12ν21

[E1 ν12E2

ν21E1 E2

] [ε1ε2

].

Assuming that the membrane thickness t(u) is piecewise-constant, it will be conve-nient to relate the strains to stress resultants (see (3.15)). The principal directionsare based on the geometry as defined by (x,∇x). The response of the membrane tostrain will depend on (u,∇x). Strictly speaking, we should write E1(u,G), E2(u,G),and ν12(u,G), where G is the Green strain. However, to simplify notation, we willsometimes suppress G (or θ) and write E1(u), E2(u), and ν12(u). If the differencebetween the Ei’s and the difference between the νi’s are small, it may be more com-putationally efficient to average the Young’s moduli and Poisson’s ratios and treatthe film as if it were piecewise-isotropic. We follow this approach when we presentnumerical results in section 5.

3.5. Film strain energy. The Green strain is G = 12 (C− I), where C = FTF

is the right Cauchy–Green strain and F = ∇x. The eigenvalues of C are called theCauchy strains, denoted by λ2

i for i = 1, 2. The corresponding unit eigenvectors aredenoted by ni and are referred to as the principal directions. The eigenvalues of Gare called the principal strains and are denoted by δ1 and δ2, where δi =

12 (λ

2i − 1)

for i = 1, 2. Multiplying (3.14) by t(u) and using the Green strain tensor, we findthat the tensor of the second Piola–Kirchoff stress resultants (denoted by S) can bewritten in the form

(3.15)

[μ1(u,G)μ2(u,G)

]= τ(u,G)

[E1(u,G) ν12(u)E2(u,G)

ν21(u,G)E1(u,G) E2(u,G)

] [δ1(G)δ2(G)

]

for u ∈ Ω and τ(u,G) = t(u)/(1− ν12(u,G)ν21(u,G)). It will be convenient to haveS in matrix form:

(3.16) S(u,G) = a(u,G)(A(u,G)G+ b(u,G)Cof(G)T

),

where

A(u,G) =

[1

E2(u,G) 0

0 1E1(u,G)

], a(u,G) =

E1(u,G)E2(u,G)

1− ν12(u,G)ν21(u,G)t(u),

and

(3.17) b(u,G) = ν12(u,G)/E1(u,G) = ν21(u,G)/E2(u,G).



The eigenvalues of S(u,G) are the principal stress resultants:

μ1(u,G) = a(u,G) (δ1/E2(u,G) + b(u,G)δ2),μ2(u,G) = a(u,G) (δ2/E1(u,G) + b(u,G)δ1).

Since G and S are symmetric, they have the spectral representations

G = δ1n1 ⊗ n1 + δ2n2 ⊗ n2,(3.18)

S(u,G) = μ1(u,G)n1 ⊗ n1 + μ2(u,G)n2 ⊗ n2,(3.19)

respectively, where a⊗ b = (aibα)1≤i≤m,1≤α≤n for a ∈ Rm and b ∈ R

n.The balloon film strain energy density can be written W (u,G) = 1

2G : S(u,G),which, in terms of the principal strains, gives

(3.20) W (u, δ1, δ2) =1

2a(u,G)

(1

E2(u,G)δ21 +

1

E1(u,G)δ22 + 2b(u,G)δ1δ2

).

Our aim is to obtain estimates on W in terms of |F|2 = λ21 + λ2

2. To accomplish this,

we first estimate W on some Ωl. Let u ∈ Ωl. Let al =El

1El2

1−νl12ν

l21tl, Al =

[ 1/El2 0

0 1/El1

],

and bl =νl12

El1=

νl21

El2. Define W l := W (u, δ1, δ2) for u ∈ Ωl, i.e.,

(3.21) W l =1

2al(

1

El2

δ21 +1

El1

δ22 + 2blδ1δ2

)for u ∈ Ωl.

In terms of the Cauchy strains, W l is

W l(u, λ1, λ2) =al

8

[(1

El2

λ41 +

1

El1

λ42 + 2bl

(λ21λ

22 − (λ2

1 + λ22)))

(3.22)

− 2

(1

El2

λ21 +

1

El1

λ22

)+

1

El1

+1

El2

+ 2bl].

Suppose El1 < El

2. It is straightforward to show

(3.23)νl21|F|4/El

2 ≤ λ41/E

l2 + λ4

2/El1 + 2blλ2

1λ22 ≤ |F|4/El

1,

(1 + νl21)|F|2/El2 ≤ bl(λ2

1 + λ22) + λ2

1/El1 + λ2

2/El2 ≤ (1 + νl12)|F|2/El

1.

Combining (3.23) and (3.22), we obtain

al

8

[νl21El

2

|F|4 − 2(1 + νl12)

El1

|F|2 +(

1

El1

+1

El2

+ 2bl)]

≤ W (u, λ1, λ2)

≤ al

8

[1

El1

|F|4 − 2(1 + νl21)

El2

|F|2 + 1

El1

+1

El2

+ 2bl].

(3.24)

It is helpful to observe νl12/El1 = νl21/E

l2 < 1/El

2 and νl21/El2 = νl12/E

l1 < 1/El

1. IfEl

2 < El1, we can interchange the 1 and 2 indices in (3.24) to obtain similar inequalities.

Upon defining El = mini{Eli} and E

l= maxi{El

i}, we find

albl

8|F|4 − al

4

(1

El+ bl)|F|2 + al

8

(1

El2

+1

El1

+ 2bl)

≤ W l

≤ al

8

[1

El|F|4 − 2

(1

El+ bl)|F|2 +

(1

El1

+1

El2

+ 2bl)]

.

(3.25)



In an effort to further clarify our exposition, we set

(3.26)E = minl{El},E = maxl{El},

a = minl{al},a = maxl{al},

b = minl{bl},b = maxl{bl}.

Since a thin membrane is unable to support compressive stresses, we follow Pip-kin’s approach [22, 23] and replace (3.20) with its quasi-convexification. The balloonshape is approximated by a faceted surface based on a triangulation of Ω. T ∈ Ω;we consider T → x(T ) and calculate the corresponding principal stresses and strains.The membrane can be decomposed into three disjoint sets:

Slack: S = {u | δ1, δ2 < 0},Taut: T = {u | μ1(u) ≥ 0, μ2(u) ≥ 0},

Wrinkled: U = {u | u /∈ T and u /∈ S}.

U can be decomposed further as U = U1 ∪U2, where U1 = {u | δ1 ≥ 0, μ2(u) ≤ 0} andU2 = {u | δ2 ≥ 0, μ1(u) ≤ 0}.

The relaxation of W is the largest convex function not exceeding W . In [22],Pipkin shows that the relaxation of W is the quasi-convexification of W if W is aconvex function of G. The convexity of (δ1, δ2) �→ W (u, δ1, δ2) is equivalent to theconvexity of the function Ψ(u, x, y) = x2/E2(u)+y2/E1(u)+2b(u)xy. In this context,we are speaking of convexity for fixed u ∈ Ω and fixed θ. Let α, β ≥ 0, α + β = 1,u ∈ Ωl, and 0 < νl12, ν

l21 < 1. Since 0 < νl12ν

l21 < 1 or, equivalently,

(3.27) 0 < El1E

l2(b

l)2 < 1,

we have det(D2Ψ) = 4(1 − νl12νl21)/(E

l1E

l2) > 0. We find that Ψ is convex; i.e.,

Ψ(u, αx1 + βx2, αy1 + βy2) ≤ αΨ(u, x1, y1) + βΨ(u, x2, y2). We conclude that W isa convex function of G.

Let W ∗ be the the relaxed strain energy density. On T, W ∗ = W ; on S, W ∗ ≡ 0;on U, Pipkin shows that there exists a matrix G∗ such that W ∗(u,G) = W (u,G∗),

(3.28) S∗(u,G) = S(u,G∗).

We can write the matrix G∗ as

(3.29) G∗ = G+ β2n⊗ n, where β ∈ R, n ∈ R2, |n| = 1.

−β2n ⊗ n is the wrinkling strain, and G∗ is the elastic strain (see [23]). This leadsto a uniaxial tension on U in the form

(3.30) S∗ = μt⊗ t,

where μ > 0 and t ∈ R2 is a unit vector orthogonal to n. One can demonstrate (3.30)

by determining β and n. To this end, suppose

n · S∗n = 0,(3.31)

n · S∗t = 0.(3.32)

Here (3.31) ensures that the elastic strain produces a uniaxial tension on U; i.e., theeigenvalues of S∗(u,G) are nonnegative principal stress resultants. Equation (3.32)



is true since t and n are orthogonal directions. Adapting the analysis in [14] to anorthotropic membrane, we let K(u) = a(u)

(A(u)n ⊗ n+ b(u)Cof(n ⊗ n)T

). Using

(3.16), (3.28), and (3.29), we find

(3.33) S∗(u) = S(u) + β2(u)K(u).

From the orthogonality of n and t, one has the identity Cof(n ⊗ n)T = t ⊗ t. Itfollows that K(u) = a(u)A(u)n ⊗ n+ b(u)a(u)t ⊗ t, where

K(u)n = a(u)A(u)n,(3.34)

K(u)t = b(u)a(u)t.(3.35)

From (3.31), (3.33), and (3.34), we have

(3.36) β2(u) = − n · S(u)na(u)n ·A(u)n

.

Now, if u ∈ U2 = {δ2 ≥ 0, μ1(u) ≤ 0}, then we should take t = n2 and n = n1.In this case, β2(u) = −δ1 − E2(u)b(u)δ2 = −δ1 − ν21(u)δ2. Similarly, if u ∈ U1 ={δ1 ≥ 0, μ2(u) ≤ 0}, then take t = n1 and n = n2 so that β2(u) = −δ2 − ν12(u)δ1.Since G∗(u) = G+ β2(u)n ⊗ n, we have

(3.37) G∗(u) =

⎧⎪⎪⎨⎪⎪⎩

0 if u ∈ S,δ2(n2 ⊗ n2 − ν21(u)n1 ⊗ n1) if u ∈ U2,δ1(n1 ⊗ n1 − ν12(u)n2 ⊗ n2) if u ∈ U1,δ1n1 ⊗ n1 + δ2n2 ⊗ n2 if u ∈ T.

On U, the principal strains of G∗ are in the form {δ2,−ν21(u)δ2} or {δ1,−ν12(u)δ1}.Following Pipkin’s approach, we can model wrinkling by replacing W with W ∗, where

(3.38) W ∗(u, δ1, δ2) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, δ1, δ2 < 0,12 t(u)E2(u)δ

22 , μ1(u) ≤ 0, δ2 ≥ 0,

12 t(u)E1(u)δ

21 , μ2(u) ≤ 0, δ1 ≥ 0,

a(u)2

(δ21

E2(u)+

δ22E1(u)

+ 2b(u)δ1δ2

), μ1(u), μ2(u) ≥ 0.

System (3.38) is obtained by substituting the principal strains of G∗ into (3.20). Wecan calculate S∗ = ∂

∂G∗W∗ or use (3.16) and (3.28) with (3.29), obtaining

(3.39) S∗(u) =

⎧⎪⎪⎨⎪⎪⎩

0, δ1, δ2 < 0,t(u)E2(u)δ2n2 ⊗ n2, μ1(u) ≤ 0, δ2 ≥ 0,t(u)E1(u)δ1n1 ⊗ n1, μ2(u) ≤ 0, δ1 ≥ 0,μ1(u)n1 ⊗ n1 + μ2(u)n2 ⊗ n2, μ1(u), μ2(u) ≥ 0.

Next, we establish upper and lower inequalities forW ∗. SinceW ∗ is the relaxationof W , we have the pointwise estimate W ∗ ≤ W and (see [23, section 5])

(3.40) (W l)∗ ≤ W l.

To obtain a lower bound onW ∗, we will argue as in [4]. Figure 3.1(b) shows admissiblesubsets of the principal strain space (δ1, δ2). For fixed l, consider the following subsets:

Λl = {(δ1, δ2) | δ1 ≥ − 12 , δ2 ≥ − 1

2},Λl1 = {(δ1, δ2) ∈ Λl | δ2 > −δ1/ν

l21, δ2 > −νl12δ1},

Λl2 = Λl \ Λl

1.



For each l, Λl1 is unbounded, but (W l)∗ = W l for (δ1, δ2) ∈ Λl

1, and we havealready established a lower bound onW l. Furthermore, the compactness of Λl

2 impliesthat there exists a real number dl such that

dl = max(δ1,δ2)∈Λl

2

{W l(δ1, δ2)− (W l)∗(δ1, δ2)}.

W l(δ1, δ2) has a critical point when δ1 + νl21δ2 = 0 and δ2 + νl12δ1 = 0, which im-plies that (δ1, δ2) = (0, 0) is the only such critical point. Thus, dl is achieved onthe boundary of Λl

2 at a corner point. By inspection, one finds that dl is achievedat (−0.5,−0.5), (W l)∗(−0.5,−0.5) = 0, and dl = 1

8al(

1El

1+ 1

El2+ 2bl

). Therefore,

W l ≤ (W l)∗ + 18a

l(

1El

1+ 1

El2+ 2bl

)for (δ1, δ2) ∈ Λl. From (3.40), we have

(3.41) W l − 1

8al(

1

El1

+1

El2

+ 2bl)

≤ (W l)∗ ≤ W l.

Applying (3.25)–(3.26) yields

(3.42)1

8a b|F|4 − a

4

(b+

1

E

)|F|2 ≤ W ∗ ≤ a

8E|F|4 + 1

4a

(1

E+ b

), u ∈ Ω.

3.6. Tendons. In a real balloon, the load tendon is fixed within a PE sleeve.Only one edge of the flattened sleeve is attached to the edge of the gore. For thisreason, we will model the tendon as a narrow ribbon of film attached along a seam.The strain energy in a ribbon segment (see Figure 2.1(b)) consisting of two identicalright triangles T ∈ Gi ∩ Ωl′ with Poisson’s ratio νl

′and Young’s modulus El′ can be

approximated by

(3.43) Sfilm = 2

∫T

W ∗ dA ≈ 2(

12 t

l′El′δ21

) (12hL

)= 1

2 tl′El′δ21hL.

The strain energy in a tendon segment with stiffness Kt and length L is

(3.44) St =

∫L

12Ktε

2(S) dS ≈ 12Ktδ

21L,

where ε(S) = 12 (|α′(S)|2−1), and α(S) is a parameterization of the deformed tendon.

The differential of arc length along α(S) is dS. Expression (3.44) follows from theapproximations (ds+dS)/(2dS) ≈ 1, ε ≈ δ1, and ε = (ds−dS)/dS ·(ds+dS)/(2dS) ≈(ds−dS)/dS. Choosing El′ = Kt/(t

l′h), we see that (3.43) and (3.44) are equivalent.Remark 3.2. The approximation in (3.43) is valid because the film, sleeve, and

tendon are very compliant. For stiffer membrane elements, these simplifications maynot be realistic.

4. Existence results. In this section, we will establish rigorous existence re-sults. The total energy of the balloon system is

(4.1) I(x) =

∫Ω

(W ∗(u,∇x) + fP (x,∇x) + fw(u,x)) dA.

Combining (3.4), (3.5), and (3.42) and simplifying, we obtain

1

8a b|F|4 −

(1

4a

(b +

1

E

)+

(1

2bR2 + |p0|R

))|F|2 − wR

≤ W ∗ + fP + fw ≤ 1

8a1

E|F|4 +

(1

2bR2 + |p0|R

)|F|2 + 1

4a

(1

E+ b

)+ wR.

(4.2)



Next, we restate a technical result that is proved in [4, Lemma 2].Lemma 4.1. Let κi > 0 for i = 1, . . . , 4. (i) There exist γ1, ρ1 such that

(4.3) κ1|u|4 + κ2|u|2 ≤ γ1|u|4 + ρ1 for 0 < κ1 < γ1 and ρ1 = κ22/4(γ1 − κ1).

(ii) There exist constants α, ρ2, γ2 such that

(4.4) α|u|4 − ρ2 ≤ κ3|u|4 − κ4|u|2 ≤ γ2|u|4

for 0 < α < κ3 < γ2 and ρ2 = κ24/4(κ3 − α).

Let κ3 = 18a b, κ4 = 1

4a(1E + b

)+ 1

2bR2 + |p0|R, and apply Lemma 4.1(ii) to the

first inequality in (4.2). Choose α such that 0 < α < 18a b and define b′ by α = 1

8ab′.

Then 0 < b′ < b, and by Lemma 4.1(ii)

18ab

′|F|4 − ρ2 ≤ κ3|F|4 − κ4|F|2,

where ρ2 =(a(b+ 1

E )+ 2bR2+4|p0|R)2

/ (8a(b− b′)). Consider the second inequality

in (4.2). Choose θ′ > 1 and apply Lemma 4.1(i) to obtain

1

8a1

E|F|4 +

(1

2bR2 + |p0|R

)|F|2 ≤ 1

8a1

Eθ′|F|4 + ρ1,

where ρ1 =(bR2 + 2|p0|R

)2/(2a 1

E (θ′ − 1)). Using the estimates in (4.2), we have

(4.5)ab′

8|F|4 − ρ2 − wR ≤ W ∗ + fP + fw ≤ aθ′

8E|F|4 + ρ1 +

a

4

(1

E+ b

)+ wR.

To obtain the proper form, we rewrite (4.5) as

1

8ab′|F|4 ≤ W ∗ + fP + fw + ρ2 + wR

≤ aθ′

8E|F|4 + ρ2 +

a

4

(1

E+ b

)+ ρ1 + 2wR

(4.6)

and, with σ = ρ2 + wR, define

(4.7) f∗Tot(u,x,∇x) = W ∗(u,∇x) + fP (x,∇x) + fw(u,x) + σ.

We can now state and prove a lemma needed for our existence results.Lemma 4.2. If f∗

Tot is as defined in (4.7) and |x| ≤ R, then(i) α|A|4 ≤ f∗

Tot(u,x,A) ≤ C (1 + |A|4), where α > 0 and C ≥ 0 are constants;(ii) f∗

Tot is a Caratheodory function;(iii) A �→ f∗

Tot(u,x,A) is quasi-convex for every x and almost every u.Proof. With C = max

{aθ′/(8E), ρ2 + a

(1/E + b

)/4 + ρ1 + 2wR

}, part (i)

follows from (4.6) with α = 18ab

′. To show part (ii), we begin as follows. For a fixed(x,A) ∈ R

3 × R2×3, we claim that u �→ f∗

Tot is measurable. Since any finite sumof measurable functions is measurable, it suffices to show that fP , σ, fw, and W ∗

are measurable. The mappings u �→ fP and u �→ σ are continuous and, therefore,measurable. u �→ fw is measurable since fw(u,x,A) = wfilm(u)(x · k) is piecewise-constant. To show that u �→ W ∗ is measurable, we need an expression for the principal



strains δi(A) for i = 1, 2. We then obtain an expression for W ∗(u,A) from (3.38).With

A = (A1,A2) =

⎛⎝ A1

1 A12

A21 A2

2

A31 A3

2

⎞⎠ ∈ R

2×3,

let A1 ·A2 =∑3

j=1 Aj1A

j2. It follows that

δ1(A) = 14 (|A|2 − 2) + 1

4

√(|A1|2 − |A2|2)2 + 4(A1 ·A2)2,

δ2(A) = 14 (|A|2 − 2)− 1

4

√(|A1|2 − |A2|2)2 + 4(A1 ·A2)2.

To determine W ∗(u,A), we substitute δ1(A), δ2(A) into (3.38) and find

(4.8) W ∗(u,A) = W ∗(u, δ1(A), δ2(A)).

If δ1 < 0 and δ2 < 0, then u �→ W ∗ ≡ 0 is measurable. For all other cases in (3.38),u �→ W ∗ is measurable since it is a linear combination of finite products of measurablefunctions t(u), a(u), b(u), and Ei(u). Thus, u �→ W ∗(u,A) is measurable, and weconclude that for fixed A ∈ ×R

2×3, u �→ f∗Tot is measurable.

We now verify the second condition that a Caratheodory function must satisfy.Let ∂Ω =

⋃l,i ∂Ω

li, where {Ωl

i} is a partition of Ω as defined in section 3.1. Note thatmeas(∂Ω) = 0. For a fixed u ∈ Ω\∂Ω, we claim that (x,A) �→ f∗

Tot is continuous. Toshow this, we will observe that each of fP , σ, fw, and W ∗ is continuous. (x,A) �→ σis continuous since it is a constant mapping. Since the Ωl

i’s are disjoint, u ∈ Ω \ ∂Ωimplies that u ∈ Ωl

i for some i and l. For u ∈ Ωli, (x,A) �→ fw(u,x,A) = wl(x · k) is

continuous since it is linear. (x,A) �→ fP (u,x,A) = − (12b(x ·k)2+p0(x ·k))k ·adj2A

is continuous since A �→ adj2A is continuous (see [4, Lemma 3]). Finally, one canverify that W ∗ is a continuous function of (δ1, δ2). Since A �→ δi(A) for i = 1, 2 arecontinuous, (x,A) �→ W ∗ is continuous. Thus, for almost every u ∈ Ω, (x,A) �→f∗Tot(u,x,A) is continuous, and f∗

Tot is a Caratheodory function.For the proof of part (iii), fix u ∈ Ω\∂Ω and x ∈ R

3. As above, u ∈ Ωli for some i

and l. We will show that f∗Tot is quasi-convex in the third argument by showing that

each of fP , σ, fw, and W ∗ is quasi-convex in the third argument. From Definitions2.1–2.2, one verifies that A �→ σ and A �→ fw(u,x,A) = wfilm(u)(x · k) are quasi-convex since both are constant inA. By construction, A �→ W ∗ is quasi-convex. SinceA �→ fP is quasi-convex and the finite sum of quasi-convex functions is quasi-convex,(iii) is satisfied, and the proof of the lemma is complete.

Since all of the forces are conservative, equilibrium is achieved at a minimumof the energy functional over X . As the term σ only increases W ∗ + fP + fw by aconstant, the minimizer of I in (4.1) is the same as the minimizer of I with integrandW ∗+ fP + fw+σ. With f∗

Tot given by (4.7) and I(x) redefined, we state an existenceresult for a closed balloon system. The arguments are easily adjusted to accommodatean open balloon system and other boundary conditions.

Theorem 4.3. If |x| ≤ R, then

(B) inf

{I(x) =

∫Ω

f∗Tot(u,x,∇x) dA

∣∣∣ x ∈ X

}

admits at least one solution.



Proof. D is nonempty since we can always find an x ∈ x0 +W 1,40 (Ω,R3) param-

eterizing a spherical cap with volume ω0. To complete the proof, we need to showthat f∗

Tot is a quasi-convex Caratheodory function and verify Theorem A.2(i). This isaccomplished in Lemma 4.2. Since X without the volume constraint is weakly closed,there exists x ∈ X without the volume constraint such that I(x) = inf{I(x) | x ∈ X}.We must show that x satisfies the volume constraint. As in the proof of Theorem A.2(see [7, p. 23]), there is a minimizing sequence {xm} ⊂ X satisfying xm ⇀ x weaklyin W 1,4(Ω,R3). By Lemma A.1, V (x) = ω0. Thus, x ∈ X when g = 0 includes thevolume constraint. We conclude that (B) has at least one solution.

Remark 4.4. The boundary condition from (2.3) is a convex constraint. Itfollows that x ∈ X when g includes the boundary condition defined in (2.3).

5. Numerical solutions. To demonstrate the robustness of our model, we sim-ulate balloon ascent shapes and investigate low pressure regimes where the symmetricconfiguration becomes unstable and undergoes a localized collapse. We used the de-sign code Planetary Balloon [17] developed by Farley/NASA Goddard Space FlightCenter to determine the cutting pattern GF (see Table 5.1). We used Brakke’s Sur-face Evolver [10] to determine stability of a symmetric state. At the design pressureP0,d, the pumpkin shape is fully developed and the symmetric shape is stable. Tomimic ascent shapes, we decrease the nadir pressure P0. We find that the symmetricshapes are stable for P0 > 1 Pa and unstable for P0 < 1 Pa.

Table 5.1

Design parameters for a pumpkin-shaped balloon.

(a) Key properties. (b) Cutting pattern for GF . Units are in meters.

Variable Valueng 60P0,d 155 Pah 38 μmν 0.636

E 150.5 MPaEt 693 kN

v 2H(v)0.0000 0.03461.6401 0.12174.9295 0.29316.5743 0.37828.2192 0.46329.8641 0.548111.5134 0.633214.8033 0.801416.4483 0.883918.0933 0.964719.7383 1.043521.3878 1.1223.0329 1.193224.6781 1.2629

v 2H(v)27.9687 1.389129.6141 1.442232.9095 1.52834.5552 1.571636.2007 1.618937.846 1.665839.4912 1.707242.7835 1.741444.4271 1.727846.0706 1.704247.714 1.671249.3574 1.629351.0006 1.579252.6481 1.5216

v 2H(v)55.9339 1.388459.2192 1.236460.8617 1.155562.5086 1.072164.1509 0.987165.7931 0.90169.0774 0.72770.7196 0.639872.3661 0.55374.0082 0.466975.6502 0.381177.2921 0.295378.9341 0.209482.2221 0.0351

We assume that the balloon is constructed from PE film with ν1 = 0.58, ν2 =0.692, E1 = 137 MPa, and E2 = 164 MPa. These values correspond to a filmtemperature of 253 K and are based on empirical tests carried out at the Balloon Labat NASA’s Wallops Flight Facility. Note that (3.17) is only approximately satisfied.Since the values of νi and Ei differ by no more than 10%, we will assume that theballoon film is isotropic and use a Young’s modulus of E = (E1+E2)/2 and a Poisson’sratio of ν = (ν1 + ν2)/2. See Table 5.1(a) for other parameter values. The massdensity of PE is 920 kg/m3 and the cap length is 30 m. The equatorial bulge radiusof a fully inflated strained gore lobe is estimated to be 2.87 m, and the correspondingequatorial bulge angle is 70 degrees. The tendon has a mass density of 0.0127 kg/m.A tendon is assumed to be 0.3% slack, so it must strain this amount before comingunder tension. For our calculations, we use a standard linearly elastic string model



Table 5.2

Summary of strained balloon shape calculations.

Description/Pressure P0 (Pa) 155 50 10 2.5 0.5 -2.00Max. height (m) 39.4333 39.5344 40.6711 43.0638 46.4921 52.0328Apex height (m) 39.3381 39.4350 40.5465 42.89486 45.7803 51.0624Diameter (m) 63.4805 62.8063 62.1420 60.9320 59.5393 56.0676Max. principal stress 1 (MPa) 8.2900 3.6901 1.9318 0.2416 1.0558 0.4579Max. principal stress 2 (MPa) 9.1129 4.1770 2.2101 1.4754 1.3963 1.1406Max. principal strain 1 (m/m) 0.02716 0.00827 -0.00058 -0.00197 0.00039 0.00135Max. principal strain 2 (m/m) 0.03675 0.02177 0.01293 0.00317 0.00596 0.00307Max. tendon tension (N) 7539 2474 609 277 229 135Max. tendon strain (m/m) 0.0109 0.0036 0.0009 0.0004 0.003 0.0002Volume ratio (ω/ω0) 1.000 0.977 0.964 0.944 0.904 0.7724

for the tendons. The design shape has a height of 40.8 m and a diameter of 63 m. Weassume bd = 0.1034 N/m3. The apex fitting has a mass density of 42 kg. The nadirend-fitting has a diameter of 1.32 m.

The balloon is approximated by a faceted surface consisting of triangles whose(x, y, z) coordinates are free to move. Let q = (q1, q2, . . . , qN ) be the vector consistingof the (x, y, z) coordinates of the facet nodes, and let I(q) represent the resultingenergy of the balloon system based on the discretization of I(x). With 320 trianglesper gore, there are 28,981 degrees of freedom. The quantities I(q), DI = [∂I/∂qi],D2I =

[∂2I/∂qj∂qi

]are calculated analytically and are used by Matlab’s optimization

toolbox function fminunc to solve problem (B) when there are only linear constraints.Our model is also incorporated into Surface Evolver [10]. See [5] for other Evolverballoon applications. We used Evolver to determine that the cyclically symmetricshape is stable for the design conditions at float, i.e., P0,d = 155 Pa. A solution q∗ isstable if all the eigenvalues of D2I(q∗) are positive. At P0 ≈ 1 Pa, D2I(q∗) has onenegative eigenvalue.

We first calculated the strained shape for P0 = 155 Pa. The numerical val-ues of quantities such as the height, diameter, maximum principal stresses, etc., arepresented in Table 5.2. The corresponding equilibrium shape is presented in Figure5.1(a). We varied P0 and determined the corresponding strained shapes. Data ispresented in Table 5.2 for P0 = 155, 50, 10, 2.50, 0.50,−2.00 Pa. These shapes arerepresentative of ascent configurations. In each case, the buoyancy b was adjustedso that bV ≈ bdVd. To the eye, there is very little difference between the shapes for10 ≤ P0 ≤ 155. However, quantities do change significantly (see Table 5.2). Theshape at P0 = 2.5 Pa is cyclically symmetric (see Figure 5.1(b)), but it is not flat atthe nadir. At about P0 = 1 Pa, all 60 gores are not able to fully deploy, and in the caseof our simulation, two gores collapse and are swallowed into the interior of the bal-loon. This is a typical feature in partially inflated balloons. See Figure 5.1(c) wherean equilibrium shape with P0 = 0.50 Pa is presented. Figure 5.2 contains images ofa 27 m diameter pumpkin balloon with ng = 200 that were taken during inflationtests carried out by NASA’s Balloon Program Office at the TCOM Manufacturingand Flight Test Facility in Elizabeth City, NC in 2007. Figure 5.2(a) is an imageof a pumpkin balloon inflated to less than 5% of full capacity. It is characteristic ofthe shape of an ascending balloon shortly after launch. In Figure 5.2(b), the sameballoon at about 66% of full capacity is displayed. There are a number of locationsabout the circumference of the balloon where contiguous gores are swallowed inward.In general, these locations are not distributed symmetrically. In Figure 5.2(c), thenadir pressure has been increased, and all the gores are deployed, even though the



(a) (b)

(c) (d)

Fig. 5.1. Family of ascent shapes. (a) P0 = 155 Pa; (b) P0 = 2.5 Pa; (c) P0 = 0.5 Pa, twogores collapse; (d) P0 = −2.0 Pa, taut regions are shaded dark; wrinkled regions are bright.

pumpkin shape is not fully developed. In Figure 5.2(d), the nadir pressure has beenincreased to P0,d, and the pumpkin shape is fully developed. If we attempt to calculateequilibrium configurations for V/Vd < 0.66, the self-contact regions are more compli-cated, requiring more sophisticated contact models. In the present work, we handlethe contact problem by manually gathering excess material along certain curves andevolving equilibrium shapes with local constraints as described in [6]. Gathering doesnot impede the solution process and provides a realistic way to store excess materialin a uniaxial state.

When 50 ≤ P0 ≤ 155, most triangles are taut. However, when P0 is near 10 Pa,one notices an increase in wrinkling near the tendons in the southern hemisphere of theshape. As P0 decreases, the amount of wrinkling becomes more pronounced. WhenP0 = −2.0 Pa, we color the triangles according to their state (see Figure 5.1(d)). Ataut triangle is colored dark blue. The edge of the cap is indicated by the red curve.Wrinkled triangles are colored according to the wrinkle strain β2 from (3.36). If β2

is small, the triangle is colored light blue. Triangles with increasing values of β2 arecolored cyan, green, yellow, and red, respectively. A few triangles are slack, and theseare colored white. The wrinkle pattern in our numerical solution is very consistentwith what is observed in a real balloon (see Figure 5.2). It would be natural to assumesymmetry when modeling shapes such as those in Figure 5.2(c)–(d). However, suchan assumption leads to the exclusion of nonsymmetric shapes like those in Figures



(a) V/Vd < 0.05. (b) V/Vd ≈ 0.66.

(c) V/Vd ≈ 0.95. (d) V/Vd ≈ 1.00.

Fig. 5.2. Ascent shapes: (a) at launch, (b) some gores are internally ingested, (c) not fullydeveloped, (d) pumpkin shape is fully developed. Images courtesy of NASA’s Balloon Program Office.

5.2(a)–(b). As Figure 5.1 shows, we are able to capture a variety of features bymodeling a complete shape. Since there may be many wrinkle/fold patterns that giverise to the same stress distribution in the film, solutions like those of Figure 5.1(c)–(d)need not be unique. However, they provide the balloon designer with estimates offilm stresses in these off-design states.

6. Conclusions. In this paper, we present a mathematical model for a tendon-reinforced piecewise-orthotropic pressurized membrane. We apply our methods tothe problem of determining the equilibrium shape of a pumpkin-shaped balloon withan external cap, and calculate numerical solutions for a complete balloon withoutsymmetry assumptions. We establish estimates on the relaxed film strain energydensity and show that it is quasi-convex. We apply direct methods in the calculus ofvariations to rigorously show that solutions exist to this class of problems. Our results



hold when the constant differential pressure is unknown and a volume constraint isapplied. Linear constraints, like those used to handle contact problems, can also beincluded. We find that our numerical model is able to capture many nonstandardfeatures, such as wrinkling and self-contact, that are observed in real balloons.

Appendix A. Direct methods in the calculus of variations. For the con-venience of the reader, we record here results that appear with proof in [7].

Lemma A.1. If xk ⇀ x weakly in W 1,4(Ω,R3), then V (xk) → V (x), whereV (x) = 1

3

∫Ω x · xu × xv dA is the volume functional.

To prove Lemma A.1, one can show for a constant K > 0

3|V (xk)− V (x)| ≤ K‖xk − x‖∞ +

∣∣∣∣∫Ω

x · (adj2∇xk − adj2∇x) dA

∣∣∣∣ .Letting k → ∞, the proof follows as a consequence of the compact imbeddingC0,β(Ω,R3) ⊂⊂ W 1,4(Ω,R3) with 0 ≤ β < 1

2 , the uniform boundedness of {|∇xk|} inL4(Ω) and L2(Ω), and the weak convergence adj2∇xk ⇀ adj2∇x in L2(Ω,R3).

Theorem A.2. Let Ω ⊂ Rn be a bounded open set. Let f : Ω×R

m×Rn×m → R

be a quasi-convex Caratheodory function satisfying (i) α|A|p ≤ f(u,x,A) ≤ q(u) +C(|x|p + |A|p), where p > 1, α > 0 are constant, q is a nonnegative locally summablefunction, and C is a nonnegative constant. If

(P) inf

{I(x) =

∫Ω

f(u,x(u),∇x(u)) dA∣∣∣ x ∈ x0 +W 1,p

0 (Ω,Rm)

},

then (P) admits at least one solution.

Theorem A.2 is a modification of [15, Theorem 2.9, p. 180]. To establish theresult, one must demonstrate that I(x) is coercive and sequentially weakly lowersemicontinuous. The coercivity follows from the first estimate in (i). Sequential weaklower semicontinuity of I(x) is guaranteed by [1, Theorem II.4, p. 137]. The proof ofTheorem A.2 is standard and can be found in [7, p. 23] and [15, p. 181].

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