4.6 Inflation of a Balloon

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This is a classical example discussed in different places in the literature (see e.g. (Holzapfel

2000), (Beatty 1987) and (Verron et al. 2001)). The geometry is depicted in Figure 4.8.

The undeformed radius and thickness are 10 m and 0.1 m respectively.

Assume that we select a St Venant–Kirchhoff material, that is, a linear elastic isotropic

material, satisfying

E = 1

E

[−ν(trS)G + (1 + ν)S] . (4.273)

Consider a material particle of the balloon on the X-axis. Because of symmetry conditions,

we have S12 = S23 = S13 = 0 and S22 = S33 = S. Furthermore, the balloon is assumed to

be in plane stress, S11 = 0. Accordingly, Equation (4.273) leads to the following strains:

E22 = E33 = _1 − ν

E

_

S (4.274)

E11 = −2ν

E

S (4.275)

E12 = E13 = E23 = 0. (4.276)

HYPERELASTIC MATERIALS 213

p

r = λR

tλt

σ σ

R = 10.0 m

t = 0.1 m

X, x

Y, y

Figure 4.8 Geometry of the balloon

The stretch can be obtained from Equation (1.45):

λ(N) = _(2EKL + 1)NKNL (4.277)

yielding for the circumferential stretch (take N in Y-direction)

λ =

_

2_1 − ν

E

_

S + 1 (4.278)

and for the thickness stretch

λt =

__1 − 4ν

E

S

_

. (4.279)

Taking λ = r/R as the independent variable during the inflation of the balloon, the Piola–

Kirchhoff stress of the second kind can be obtained from Equation (4.278):

S = E(λ2 − 1)

2(1 − ν)

. (4.280)

Expressing the equilibrium of a hemisphere, one obtains

pπr2 = S(2πR)t (4.281)

from which the pressure p results:

p = S

(2Rt)

r2

= Et

R(1 − ν)

_1 − 1

λ2

_

. (4.282)

214 HYPERELASTIC MATERIALS

This is a monotonic increasing function of λ. Substituting Equation (4.278) into Equation

(4.279), one obtains an expression for the thickness stretch of the balloon:

λt =

_

1 − 2ν(λ2 − 1)

(1 − ν)

. (4.283)

Surprisingly enough, λt is zero for

λ = (1 − ν

2ν

+ 1. (4.284)

If ν = 0.5, the thickness of the balloon is reduced to zero for λ =

√

3/2. This corresponds

to an infinite circumferential Cauchy stress. It is well known that the circumferential stretch

during inflation can reach values up to 10 and higher. Accordingly, the St Venant–Kirchhoff

material is not suited to model balloon behavior.

In Chapter 1, it was emphasized that ν = 0.5 represents isochoric deformation for

infinitesimal strains only. The real isochoric condition is J = 1. This can be illustrated

by noticing that for the balloon

J = λtλ2 = λ2

_

1 − 2ν(λ2 − 1)

(1 − ν)

. (4.285)

Substituting ν = 0.5, one obtains

Jν=0.5 = λ2_3 − 2λ2. (4.286)

The plot of this function in Figure 4.9 shows that ν = 0.5 is indeed a bad approximation

for isochoric behavior as soon as the stretch deviates markedly from λ = 1.

To model true balloon behavior, recourse must be taken to hyperelastic laws such as

the Neo-Hooke or Mooney–Rivlin law. The following constants are taken

neo-Hooke: B10 = 211 250. Pa, D1 = 0.2367 × 10−6 Pa−1 (4.287)

Mooney–Rivlin: B10 = 184 843.75 Pa, B01 = 26 406.25 Pa,

D1 = 0.2367 × 10−6 Pa−1. (4.288)

The isochoric constants are taken from (Holzapfel 2000), the volumetric constants are such

that the equivalent Poisson coefficient amounts to ν = 0.475. Only one-eighth of the balloon

was modeled using seventy-five 20-node brick elements with reduced integration (one layer

across the thickness). The pressure as a function of the circumferential stretch is plotted

in Figure 4.10 and agrees well overall with the analytical predictions for incompressible

material in (Holzapfel 2000). The neo-Hooke curve is about 6% lower than the analytical

prediction. This also applies to the Mooney–Rivlin model for small stretches up to the

local maximum at a stretch of approximately 1.5. For higher stretch, the present curve does

not show the local minimum with the renewed pressure increase obtained in (Holzapfel

2000). This is attributed to the volumetric term. Notice that both models predict a local

pressure maximum. This phenomenon, which was not predicted by the St Venant–Kirchhoff

HYPERELASTIC MATERIALS 215

0

0

1

1

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.2

1.2

Jν=0.5

λ

Figure 4.9 Jacobian determinant during deformation for a St Venant–Kirchhoff material

0

1

1

2

2

3

3

4

4

5

5

6

6

7 8 9

λ

p/1000 (Pa)

neo-Hooke

Mooney–Rivlin

Figure 4.10 Pressure in the balloon

216 HYPERELASTIC MATERIALS

material description, is well known: inflating a party balloon, the initial pressure is quite

high, but decreases significantly after some stretching takes place. Because its behavior is

not monotonic, the pressure cannot be taken as an independent variable during the finite

element calculation. The radial forces in the nodes of the mesh are taken instead since they

continuously increase.

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