Stability of Thick Spherical Shells
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1 Continuum Mech. Thermodyn. (1995) 7: Stability of Thick Spherical Shells I-Shih Liu 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro Caixa Postal 68530, Rio de Janeiro , 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 liu@im.ufrj.br 1
2 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 ± R 3 ) 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 (e r, e θ, e φ ) of the coordinate system, we obtain the deformation gradient, F = ± R2 r 2 e r e r + r R e θ e θ ± r R e φ e φ, 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 and its invariants are B = R4 r 4 e r e r + r2 R 2 e θ e θ + r2 R 2 e φ e φ, I 1 = R4 r r2 R 2, The equilibrium equation takes the form I 2 = r4 R 4 + 2R2 r 2. (2.2) T rr r = 2 r (T rr T θθ ), (2.3) and the non-vanishing components of the stress tensor are given by ( r 2 T rr = 2 R R4 )( r 2 s1 s ) dr 2 r 4 1 R 2 r, T θθ = T φφ = s 1 ( r 2 R R4 ) ( R 2 + s 1 2 r 4 r 2 r4 R 4 ) + T rr, (2.4) 2 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
3 where the material parameter s 1 and s 1 are functions of (I 1, I 2 ) in general. For Mooney-Rivlin materials with constant s 1 and s 1, the integral for T rr in (2.4) 1 can easily be integrated and we obtain T rr = s ( 1 R r ± 2R ) ( R 2 s 1 4 r r 2 r ) + K, 2 R T θθ = T φφ = s ( r 2 1 R 1 R r ± 2R ) ( r 4 s 1 4 r R 2 r (2.5) ) + K, 4 R where K is an integration constant and either the upper signs or the lower signs are taken associated with (2.1). 2.1 A Remark on Eversion of a Spherical Shell If we take the lower signs in (2.1), we have r = (A R 3 ) 1/3, θ = Θ, φ = Φ. This deformation is an eversion which turns the shell inside out. Since for R 0 < R 1 we have A R 3 0 > A R 3 1, and hence r 0 = r(r 0 ) > r 1 = r(r 1 ), 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 r 1 be positive, or the constant A must satisfy A > R 3 1. (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, T rr (r 1 ) = T rr (r 0 ) = 0, then r1 r 0 ( r 2 R R4 )( r 2 s1 s ) dr 2 r 4 1 R 2 r = 0, (2.7) where R = (A r 3 ) 1/3 and the material parameters s 1 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, s 1 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) ( r 2 R( r) R( r)4 2 r 4 )( s1 ( r) s 1 ( r) 3 r 2 ) r 1 r 0 R( r) 2 r = 0,
4 for some value r such that r 0 r r 1. By (2.8) it implies that r 2 R( r) 2 R( r)4 r 4 = 0. The only real positive root is R( r) = r, or (A r 3 ) 1/3 = r, which gives and hence A must satisfy A = 2 r 3, 2r 3 0 A 2r 3 1, or equivalently 2R 3 0 A 2R 3 1. Since, from (2.6), the constant A must also be greater than R 3 1, we conclude that the shell can always be everted if R 3 1 < 2R 3 0. In other words, if the thickness (R 1 R 0 ) is less than ( 3 2 1) times (or about 26% of) the inner radius R 0, 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 + R 3 ) 1/3, θ = Θ, φ = Φ. (3.1) For a shell of radius R 0 R R 1, the inner and the outer surfaces must be maintained by suitable pressures, T rr (r 0 ) = p 0, T rr (r 1 ) = p 1, where r 0 = r(r 0 ) and r 1 = r(r 1 ) are the inner and the outer radii in the deformed configuration. Let [p] = p 0 p 1 denote the pressure difference between the inner and the outer surfaces, then [p] = T rr (r 1 ) T rr (r 0 ). (3.2) We define the thickness parameter a and the expansion ratio λ by a = R 1 R 0, λ = r 1 R 1. (3.3) 4
5 Of course, a > 1, and for expansion λ > 1. We have r 1 = aλr 0, r 0 = (1 + a 3 (λ 3 1)) 1/3 R 0, (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 /s 1 = 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 Both [p ] and r are explicit functions of λ. 2 [p ] = [p] s 1 R 0 + R 1 2D, r = r 0 + r 1 R 0 + R 1. (3.5) 1.5 [p ] a = a = a = r Figure 1: Pressure-radius relation We notice that the pressure-radius curves for a = and a = 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 = 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
6 thickness, d = r 1 r 0 and D = R 1 R 0 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 = through a = For thin shells, the expression (3.2) for the pressure difference can be approximated by the Taylor expansion, [p] = T rr (r 1 ) T rr (r 0 ) = T rr r d + o(d 2 ). r0 By the use of the equations (2.3) and (2.4) and by neglecting the higher order terms, it becomes [p] = 2 r d (T θθ T rr ) = 2 r 2 r d( R R4 )( r 2 ) s1 s 2 r 4 1, R 2 which can also be written as [p] = 2 D R ( R r R7 r 7 )( s1 s 1 r 2 R 2 ). (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 r 2 d R 2 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. ( )]). 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 in the thin shell approximation. [p ] = [p] s 1 R D, r = r R, 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 V 0 and V 1, where V 0 and V 1 are spherical balls with radii r 0 and r 1 respectively. Assume that the shell is subject to uniform temperature and constant internal and external pressures, p 0 and p 1. 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
7 1 0.8 a = a = a = d D r 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 d ρ(ψ + 1 dt V 2 v v)dv T v n da 0, (4.1) V where no body force and external heat supply are considered. In addition, from the boundary conditions, we have V T v n da = p 0 v n da p 1 v n da V 0 V 1 = [p] d d dv p 1 dt V 0 dt V 1 V 0 dv = d dt ( 4 3 πr3 0 [p] ). (4.2) In this derivation, we have used the incompressibility condition that the volume of V = V 1 V 0 does not change. Combing (4.1) and (4.2), we obtain { d r1 4π ρ(ψ + 1 dt r 0 2 v v)r2 dr 4 } 3 πr3 0 [p] 0. 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
8 to constant pressures as r1 A = 4π ρψr 2 dr 4 r 0 3 πr3 0 [p]. (4.3) 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 Mooney- Rivlin material can be written as where I 1 and I 2 are now given by (2.2). 4.1 Thin Shell Approximation ρψ = s 1 2 (I 1 3) s 1 2 (I 2 3), (4.4) For thin shells, from (4.3) the availability A can be approximated by A = 4πr 2 d ρψ 4 3 πr3 [p]. Since A is a function of the radius r, if r = r corresponds to a stable equilibrium, then the necessary and sufficient conditions for A to be a minimum at r = r are da dr = 0, r= r d 2 A dr 2 0. r= r These conditions can easily be evaluated. From the expressions (4.4) and (2.2) for the free energy and the relation r 2 d R 2 D, we obtain after simple differentiations, where da dr = 4πr2( F (r) [p] ), d 2 A dr = 2 8πr( F (r) [p] ) 2 df (r) + 4πr, dr F (r) = 2 D { s ( R 1 R r R7 ) ( } r s 1 r 7 R R5 ). r 5 Therefore the first condition implies that in equilibrium [p] = F ( r), 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
9 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 = R 1 /R 0 and the expansion ratio λ = r 1 /R 1 introduced in (3.3) and denoting ξ = r/r, it follows that ξ = r(r 3 A) 1/3, A = r 3 1 R 3 1 = (λ 3 1)a 3 R 3 0. (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, r1 (λ) A(λ) = 4π ρψ(ξ(r, λ))r 2 dr 4 r 0 (λ) 3 πr 0(λ) 3 [p], (4.6) where r 1 (λ) and r 0 (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 dλ = 4π( J 1 (λ) + J 2 (λ) ), (4.7) where J 1 (λ) = ρψ(ξ(r 1, λ))r1 2 dr 1 J 2 (λ) = r1 Since r 3 1 r 3 0 = R 3 1 R 3 0, it gives and hence from (4.4) and (2.2), we get r 0 J 1 (λ) = a 3 R 3 0λ 2 ( 1 2 dλ ρψ(ξ(r 0, λ))r 2 0 ρ ( ψ I 1 I 1 ξ + ψ I ) 2 ξ I 2 ξ λ r2 dr. r0 2 dr 0 dλ = dr 1 r2 1 dλ = a3 R0λ 3 2, On the other hand, by the use of (4.5), we have dr 0 dλ dr 0 [p]r2 0 dλ, ( s1 (ξ 4 + 2ξ 2 ) s 1 (ξ 4 + 2ξ 2 ) ) r 1 /R 1 ξ dλ r2 dr = a 3 R 3 0λ 2 r 4 R 4 dr r = a3 R 3 0λ 2 ξ 3 1 ξ 3 dξ, r 0 /R 0 [p] ). 9
10 and hence J 2 (λ) = 1 2 a3 R0λ 3 2 r1 /R 1 ( s1 ( 4ξ 5 + 4ξ) s 1 (4ξ 3 4ξ 3 ) ) ξ 3 1 ξ dξ 3 r 0 /R 0 = 2a 3 R 3 0λ 2( s 1 (ξ ξ2 ) + s 1 (ξ ξ4 ) ) r 1 /R 1 By putting J 1 (λ) and J 2 (λ) together, (4.7) becomes where Comparison with (2.5) 1 leads to r 0 /R 0. da dλ = 4πa3 R 3 0λ 2( F (λ) [p] ), (4.8) F (λ) = ( s 1 ( 1 2 ξ 4 + 2ξ 1 ) s 1 (ξ 2 2ξ) ) r 1 /R 1 F (λ) = T rr (r 1 ) T rr (r 0 ). r 0 /R 0. 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 d 2 A dλ 2 = 8πa3 R0λ ( 3 F (λ) [p] ) + 4πa 3 R0λ 3 2 df (λ) dλ. Since it must be non-negative at equilibrium it implies the following stability condition: df (λ) dλ 0. (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
11 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, (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, (1955) [6] A. E. Green & R. T. Shield, Finite elastic deformation of incompressible isotropic bodies, Proc. Roy. Soc. London A 202, (1950) 11
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