Continuum Mech. Thermodyn. (1995) 7: 249-258
Stability of Thick Spherical Shells
I-Shih Liu1
Instituto de Matemática, Universidade Federal do Rio de Janeiro
Caixa Postal 68530, Rio de Janeiro 21945-970, Brazil
The pressure-radius relation of spherical rubber balloons has been derived
and its stability behavior analyzed. Here we show that those features are
practically unchanged for thick spherical shells of Mooney-Rivlin materials.
In addition, we also show that eversion of a spherical shell is possible for any
incompressible isotropic materials if the shell is not too thick.
1
Introduction
Stability of rubber balloons is an interesting example in finite elasticity. It has been
investigated and observed in simple experiments [1, 2]. The main feature of the problem
lies in the non-monotone characteristic of the pressure-radius relation for rubber balloons.
On the other hand, rubber-like materials modeled by Mooney-Rivlin type constitutive
equation are among the most commonly treated theoretical models in finite elasticity. A
well-known example is the problem of inflation and eversion of spherical shells. From the
explicit solution for the stress field and the boundary conditions on the shell, it follows
immediately the pressure-radius relation, which reduces to the known one for thin shells
or rubber balloons in the approximation. It is more striking to notice that they are
practically the same curve irrespective of the thickness, and therefore they have similar
stability properties. As for eversion, we have notice that it is possible to turn a shell
inside out freely for any incompressible isotropic materials if the shell is not too thick.
1
e-mail: liu@im.ufrj.br
1
To my knowledge, previous results only confirm the free eversion for Mooney-Rivlin
materials.
2
Universal Solutions for Spherical Shells
We summarize here the results of a well-known universal solution for incompressible
isotropic elastic bodies. The details can be found in [3, 4]. Consider a deformation given
by
r = (A ± R3 )1/3 , θ = Θ, φ = ±Φ,
(2.1)
in terms of spherical coordinates (R, Θ, Φ) and (r, θ, φ) in the reference and the deformed
configurations, where A is a constant. Using orthonormal basis (er , eθ , eφ ) of the coordinate system, we obtain the deformation gradient,
F =±
R2
r
r
er ⊗ er + eθ ⊗ eθ ± eφ ⊗ eφ ,
2
r
R
R
and hence the ± signs must be associated, namely, taking either the upper signs or the
lower signs, to ensure that det F = 1.2 Then the left Cauchy-Green tensor is
R4
r2
r2
B = 4 er ⊗ er + 2 eθ ⊗ eθ + 2 eφ ⊗ eφ ,
r
R
R
and its invariants are
I1 =
R4
r2
+
2
,
r4
R2
I2 =
r4
R2
+
2
.
R4
r2
(2.2)
The equilibrium equation takes the form
∂Thrri
2
= − (Thrri − Thθθi ),
∂r
r
(2.3)
and the non-vanishing components of the stress tensor are given by
R4 r2 dr
s
−
s
,
1
−1 2
R2
r4
R r
r2
R2
R4 r4 = Thφφi = s1 2 − 4 + s−1 2 − 4 + Thrri ,
R
r
r
R
Thrri = 2
Thθθi
Z 2
r
−
2
(2.4)
We have employed the usual convention for spherical coordinate system defined by x = r sin θ cos φ,
y = r sin θ sin φ, z = r cos θ for r > 0, 0 ≤ θ ≤ π and −π ≤ φ ≤ π. Note that the deformation (2.1)
is different from the one given in [3, eq. (57.35)] and elsewhere in which the ± sign in φ is placed in θ
instead. It would be a mistake if the same convention is meant.
2
where the material parameter s1 and s−1 are functions of (I1 , I2 ) in general.
For Mooney-Rivlin materials with constant s1 and s−1 , the integral for Thrri in (2.4)1
can easily be integrated and we obtain
R2
R
r
−
s
∓
2
+ K,
−1
2 r4
r
r2
R
(2.5)
r2
r4
1 R4
R
r
Thθθi = Thφφi = s1 2 −
±2
− s−1 4 ∓ 2
+ K,
R
2 r4
r
R
R
where K is an integration constant and either the upper signs or the lower signs are
taken associated with (2.1).
Thrri = s1
2.1
1 R4
±2
A Remark on Eversion of a Spherical Shell
If we take the lower signs in (2.1), we have
r = (A − R3 )1/3 ,
θ = Θ,
φ = −Φ.
This deformation is an eversion which turns the shell inside out. Since for R0 < R1 we
have A − R03 > A − R13 , and hence r0 = r(R0 ) > r1 = r(R1 ), in other words, the inner
surface becomes the outer surface after deformation and vice versa. Moreover, for it be
physically possible, it is necessary that r1 be positive, or the constant A must satisfy
A > R13 .
(2.6)
From (2.4) if we require that the shell be free of traction at the inner and the outer
surfaces in the everted state, Thrri (r1 ) = Thrri (r0 ) = 0, then
Z
r1 r0
R4 r2 dr
r2
−
s
−
s
= 0,
1
−1
R2
r4
R2 r
(2.7)
where R = (A − r3 )1/3 and the material parameters s1 and s−1 are functions of r for
incompressible isotropic materials in general.
The constant A is to be determined as a solution of the above equation for a tractionfree everted state. We can show that such a solution exists for any incompressible
isotropic materials if the shell is not too thick and the following inequalities hold,
s−1 ≤ 0,
s1 > 0,
(2.8)
which are known as E-inequalities in elasticity and seem to be supported by experimental
evidence. Indeed, by the mean value theorem of definite integral, we have from (2.7)
r̄2
R(r̄)4 r̄2 r1 − r0
−
s
(r̄)
−
s
(r̄)
= 0,
1
−1
R(r̄)2
r̄4
R(r̄)2
r̄
3
for some value r̄ such that r0 ≥ r̄ ≥ r1 . By (2.8) it implies that
r̄2
R(r̄)4
−
= 0.
R(r̄)2
r̄4
The only real positive root is R(r̄) = r̄, or (A − r̄3 )1/3 = r̄, which gives
A = 2r̄3 ,
and hence A must satisfy
2r03 ≥ A ≥ 2r13 ,
or equivalently 2R03 ≤ A ≤ 2R13 .
Since, from (2.6), the constant A must also be greater than R13 , we conclude that the
shell can always
be everted if R13 < 2R03 . In other words, if the thickness (R1 − R0 ) is
√
less than ( 3 2 − 1) times (or about 26% of) the inner radius R0 , it is possible to turn the
shell inside out freely.
It has been proved by Ericksen [5] that the eversion is possible for Mooney-Rivlin
materials.
3
Pressure-Radius Relation
For the inflation of a shell, the upper signs in (2.1) are taken,
r = (A + R3 )1/3 ,
θ = Θ,
φ = Φ.
(3.1)
For a shell of radius R0 ≤ R ≤ R1 , the inner and the outer surfaces must be maintained
by suitable pressures,
Thrri (r0 ) = −p0 ,
Thrri (r1 ) = −p1 ,
where r0 = r(R0 ) and r1 = r(R1 ) are the inner and the outer radii in the deformed
configuration. Let [[p]] = p0 − p1 denote the pressure difference between the inner and
the outer surfaces, then
[[p]] = Thrri (r1 ) − Thrri (r0 ).
(3.2)
We define the thickness parameter a and the expansion ratio λ by
a=
R1
,
R0
λ=
4
r1
.
R1
(3.3)
Of course, a > 1, and for expansion λ > 1. We have
r0 = (1 + a3 (λ3 − 1))1/3 R0 ,
r1 = aλR0 ,
(3.4)
where the second relation is obtained by the elimination of the constant A from (3.1)1 .
Therefore for a given thickness parameter a, the pressure difference [[p]] is a function of
the expansion ratio λ. For Mooney-Rivlin materials, this function can be written out
explicitly from (2.5)1 . Such pressure-radius relations, for s−1 /s1 = −0.1 and several
values of a, are shown in Fig. 1, where we have plotted the dimensionless pressure
difference [[p∗ ]] against dimensionless mean expansion ratio r∗ defined by
[[p∗ ]] =
[[p]] R0 + R1
,
s1
2D
r∗ =
r 0 + r1
.
R0 + R1
(3.5)
Both [[p∗ ]] and r∗ are explicit functions of λ.
2
1.5
[[p∗ ]] 1
0.5
a = 1.500
a = 1.200
a = 1.001
0
1
2
3
4
r∗
5
6
7
Figure 1: Pressure-radius relation
We notice that the pressure-radius curves for a = 1.200 and a = 1.001 are almost
identical, while even for a very thick shell a = 1.500, a thickness half of the inner radius,
the curve is only slightly different. Therefore the pressure-radius curves for thick shells
are characteristically similar to the one for a thin shell, here say a = 1.001.
The steep rise of the pressure curve at small expansion ratio is accompanied by
a drastic reduction of thickness of the shell as is shown in Fig. 2, where the ratio of
5
thickness, d = r1 −r0 and D = R1 −R0 at the reference and the inflated states respectively,
is plotted against the mean expansion ratio r∗ . It is even more appealing to notice that
the curves are almost identical irrespective of thickness, for a = 1.500 through a = 1.001.
For thin shells, the expression (3.2) for the pressure difference can be approximated
by the Taylor expansion,
[[p]] = Thrri (r1 ) − Thrri (r0 ) =
∂Thrri d + o(d2 ).
∂r r0
By the use of the equations (2.3) and (2.4) and by neglecting the higher order terms, it
becomes
2
2 r2
R4 r2 [[p]] = d (Thθθi − Thrri ) = d 2 − 4 s1 − s−1 2 ,
r
r R
r
R
which can also be written as
[[p]] = 2
D R R7 r2 − 7 s1 − s−1 2 .
R r
r
R
(3.6)
In the above expression since the shell is thin, we have dropped the subindex 0 in both
r and R for simplicity, and employed the relation r2 d ' R2 D which follows from (3.1).
Note that the derivation of the pressure-radius relation (3.6) for thin spherical shell is
valid for any incompressible isotropic materials. The present derivation is essentially that
of Green and Shield [6] (see also [4, eq. (3.10.27)]). Other derivations can be found in
[1, 2]. For Mooney-Rivlin materials, the pressure-radius relation is practically the same
curve shown in Fig. 1 for a = 1.001, where
[[p∗ ]] =
[[p]] R
,
s1 D
r∗ =
r
,
R
in the thin shell approximation.
4
Stability under Constant Pressures
The pressure-radius curve for the inflation of a spherical shell, as shown in Fig. 1, is
non-monotone, which usually may leads to certain unstable behavior. We shall consider
one such implication here.
Let the spherical shell lie in a region V between ∂V0 and ∂V1 , where V0 and V1 are
spherical balls with radii r0 and r1 respectively. Assume that the shell is subject to
uniform temperature and constant internal and external pressures, p0 and p1 . In order
to be able to maintain the prescribed constant pressure in the interior, we have tacitly
assumed that a suitable device is provided, such as a tube connected to a constant
6
1
a = 1.500
a = 1.200
a = 1.001
0.8
0.6
d
D
0.4
0.2
0
1
2
3
4
5
r∗
6
7
Figure 2: Thickness-radius relation
pressure chamber. However, as long as the interior is maintained at constant pressure
such a device is irrelevant to the problem.
After eliminating the heat flux from the energy equation, the entropy inequality
becomes
Z
d Z
1
ρ(ψ + v · v)dv −
T v · n da ≤ 0,
(4.1)
dt V
2
∂V
where no body force and external heat supply are considered. In addition, from the
boundary conditions, we have
Z
∂V
T v · n da = p0
Z
∂V0
v · n da − p1
Z
v · n da
∂V1
d Z
d 4 3 d Z
dv − p1
dv =
πr [[p]] .
= [[p]]
dt V0
dt V1 −V0
dt 3 0
(4.2)
In this derivation, we have used the incompressibility condition that the volume of V =
V1 − V0 does not change. Combing (4.1) and (4.2), we obtain
Z r1
d
1
4
4π
ρ(ψ + v · v)r2 dr − πr03 [[p]] ≤ 0.
dt
2
3
r0
Therefore, we can define the availability A, a monotone decreasing function of time, in
a quasi-static problem (thus neglecting the kinetic energy), for a spherical shell subject
7
to constant pressures as
r1
4
ρψr2 dr − πr03 [[p]].
(4.3)
3
r0
An equilibrium state is said to be stable if a small disturbance away from it will
eventually disappear, in other words, the body tends to return to the original equilibrium
state at the end. Since the availability is non-increasing in the process, it must then tend
to a local minimum at a stable equilibrium state. We shall explore this stability criterion
more specifically for Mooney-Rivlin materials. The free energy function ψ for a MooneyRivlin material can be written as
s1
s−1
ρψ = (I1 − 3) −
(I2 − 3),
(4.4)
2
2
A = 4π
Z
where I1 and I2 are now given by (2.2).
4.1
Thin Shell Approximation
For thin shells, from (4.3) the availability A can be approximated by
4
A = 4πr2 d ρψ − πr3 [[p]].
3
Since A is a function of the radius r, if r = r̄ corresponds to a stable equilibrium, then
the necessary and sufficient conditions for A to be a minimum at r = r̄ are
d2 A ≥ 0.
dr2 r=r̄
dA = 0,
dr r=r̄
These conditions can easily be evaluated. From the expressions (4.4) and (2.2) for the
free energy and the relation r2 d ' R2 D, we obtain after simple differentiations,
dA
= 4πr2 F (r) − [[p]] ,
dr
d2 A
dF (r)
=
8πr
F
(r)
−
[[p]]
+ 4πr2
,
2
dr
dr
where
R
r
D
R7 R5 F (r) = 2
s1
− 7 − s−1
− 5
.
R
r
r
R
r
(
)
Therefore the first condition implies that in equilibrium [[p]] = F (r̄), which is merely the
pressure-radius relation (3.6), while the other condition implies that the pressure-radius
curve F (r) must have a positive slope at the equilibrium.
8
4.2
Thick Spherical Shells
The stability conditions for thick shells, subject to prescribed constant pressures, are
practically the same as that for thin shells as we shall see now. By employing the
thickness parameter a = R1 /R0 and the expansion ratio λ = r1 /R1 introduced in (3.3)
and denoting ξ = r/R, it follows that
ξ = r(r3 − A)−1/3 ,
A = r13 − R13 = (λ3 − 1)a3 R03 .
(4.5)
Hence for a given thickness parameter a, ξ = ξ(r, λ) and the availability A in (4.3)
becomes a function of λ only,
A(λ) = 4π
Z
r1 (λ)
r0 (λ)
4
ρψ(ξ(r, λ))r2 dr − πr0 (λ)3 [[p]],
3
(4.6)
where r1 (λ) and r0 (λ) are given by (3.4). Therefore, A(λ) must atain its minimum at a
stable equilibrium state characterized by the expansion ratio λ.
From (4.6), we have
dA
= 4π J1 (λ) + J2 (λ) ,
(4.7)
dλ
where
dr1
dr0
dr0
J1 (λ) = ρψ(ξ(r1 , λ))r12
− ρψ(ξ(r0 , λ))r02
− [[p]]r02
,
dλ
dλ
dλ
Z r1 ∂ψ ∂I1
∂ψ ∂I2 ∂ξ 2
J2 (λ) =
ρ
+
r dr.
∂I1 ∂ξ
∂I2 ∂ξ ∂λ
r0
Since r13 − r03 = R13 − R03 , it gives
r02
dr0
dr1
= r12
= a3 R03 λ2 ,
dλ
dλ
and hence from (4.4) and (2.2), we get
r
J1 (λ) = a
3
R03 λ2
/R
!
1 1
1
s1 (ξ −4 + 2ξ 2 ) − s−1 (ξ 4 + 2ξ −2 ) − [[p]] .
2
r0 /R0
On the other hand, by the use of (4.5), we have
∂ξ 2
r4 dr
ξ3
r dr = a3 R03 λ2 4
= a3 R03 λ2
dξ,
dλ
R r
1 − ξ3
9
and hence
Z r1 /R1 ξ3
1
J2 (λ) = a3 R03 λ2
s1 (−4ξ −5 + 4ξ) − s−1 (4ξ 3 − 4ξ −3 )
dξ
2
1 − ξ3
r0 /R0
r
/R
1 2
1 4 1 1
3 3 2
−1
= 2a R0 λ s1 (ξ − ξ ) + s−1 (ξ + ξ ) .
2
4
r0 /R0
By putting J1 (λ) and J2 (λ) together, (4.7) becomes
dA
= 4πa3 R03 λ2 F (λ) − [[p]] ,
dλ
where
(4.8)
r
/R
1 1
1
F (λ) = s1 ( ξ −4 + 2ξ −1 ) − s−1 (ξ −2 − 2ξ) .
2
r0 /R0
Comparison with (2.5)1 leads to
F (λ) = Thrri (r1 ) − Thrri (r0 ).
From (4.8), the condition that A(λ) be minimum at λ = λ requires vanishing derivative there. Therefore, we have [[p]] = F (λ), which agrees with the boundary condition
(3.2). In addition, from (4.8) we have
d2 A
dF (λ)
3 3
=
8πa
R
λ
F
(λ)
−
[[p]]
+ 4πa3 R03 λ2
.
0
2
dλ
dλ
Since it must be non-negative at equilibrium it implies the following stability condition:
dF (λ)
≥ 0.
dλ
(4.9)
In terms of the dimensionless quantities introduced in (3.5), the stability condition
(4.9) is equivalent to d[[p∗ ]]/dr∗ ≥ 0, since dr∗ /dλ > 0. Therefore, the range with negative
slope in the pressure-radius curves shown in Fig. 1 corresponds to unstable equilibrium
states under prescribed constant internal and external pressures.
5
Final Remarks
Unlike the stability analysis given in [1], we have not included the gas in the interior as
part of the thermodynamic system. The simplicity in the present analysis, especially in
the thin shell approximation, reflects the fact that the stability of the shell is a property
of its own and its boundary conditions. The nature of the gas in the surrounding is
irrelevant when the boundary conditions can be given explicitly.
10
References
[1] I. Müller, Thermodynamics, Pitman Publishing, London (1985)
[2] D. R. Merritt & F. Weinhaus, The pressure curve of a rubber balloon. Am. J. Phys.
46, 976-977 (1978)
[3] C. Truesdell & W. Noll, Non-Linear Field Theories of Mechanics, Handbuch de
Physik Vol. III/3, Ed. by S. Flügge, Springer Verlag, Berlin (1965)
[4] A. E. Green & W. Zerna, Theoretical Elasticity, Clarendon Press, Oxford (1954)
[5] J. L. Ericksen, Inversion of a perfectly elastic spherical shell, Z. Angew. Math. Mech.
35, 382-385 (1955)
[6] A. E. Green & R. T. Shield, Finite elastic deformation of incompressible isotropic
bodies, Proc. Roy. Soc. London A 202, 407-419 (1950)
11