Journal of Elasticity, Vol. 6, No. 2, April 1976 Noordhoff International Publishing - Leyden Printed in The Netherlands

Closed form solutions for small deformations superimposed upon the symmetrical expansion of a spherical shell

JAMES M. HILL*

Department of Theoretical Mechanics, University ~?f Nottingham, England

(Received October 10, 1974)

ABSTRACT

For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are super- imposed upon the symmetrical expansion of a spherical shell, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the n = 1 buckling criterion for thick-walled spherical shells which are subjected to uniform external pressure. They are also used to deduce an upper bound to the force deflection relation for small superimposed translational deflec- tions of bonded pre-compressed spherical rubber bush mountings.

ZUSAMMENFASSUNG

Fiir den Fall kleiner, axialsymmetrischer Deformationen isotropischer inkompressibler hyperelastischer Materialien, die der symmetrischen Dehnung einer sphgrischen Schale tiberlagert sind, werden L/Ssungen in geschlossener Form abgeleitet, ohne einschr~inkende Bedingungen ftir die Deformationsenergiefunktion. Mit Hilfe dieser L6sungen wird das Knick-Kriterium (n = 1) fiir dickwandige sph/irische Schalen gewonnen, die gleichf/Srmigem, ~iusserem Druck, ausgesetzt sind. Weiterhin wird mit Hilfe der L/Ssungen eine obere Grenze gewonnen fiir die Kraft-Ablenkungs-Relation im Falle iiberlagerter kleiner Translationen yon gebundenen, vorgespannten sph~irischen Gummibuchsen.

1. Introduction

For isotropic incompressible hyperelastic materials the author [1] has given closed form solutions for a plane deformation which is superimposed upon the simultaneous inflation and extension of a cylindrical tube. In this paper for the same materials we give corresponding closed form solutions for a small axially symmetric deformation which is superimposed upon the symmetrical expansion of a spherical shell. These solutions, as in [1], are derived without any restrictions on the strain-energy function of the material and are used to formulate the buckling criterion for the first (n= 1) unsymmetrical mode of a thick-walled spherical shell under uniform external pressure. We also use these solutions to note the force-deflection relation for small super- imposed translational deflections of a bonded pre-compressed complete spherical shell.

In the following section we give the general equations in spherical polar coordinates

* Present address: Department of Mathematics, The University of Wollongong, Australia.

Journal ~?f Elasticity 6 (I 976) 125-136

126 James M. Hill

for axially symmetric deformations of isotropic incompressible hyperelastic materials. The initial deformation and the governing fourth order system of ordinary differential equations for the small superimposed deformation are given in section 3. In section 4, by evaluating the resultant force on an arbitrary originally spherical surface we are able to integrate this fourth order system completely for the special case of n = 1. As with the solutions given in [1] we find that two of the four independent solutions do not involve the strain-energy function of the material and are thus controllable.

The problem of the stability of thick-walled spherical shells loaded by a uniform external pressure has been considered by both Wesolowski [2] and Wang and Ertepinar [3] and the reader is referred to these papers for further references to the problem. In both [2] and [3], numerical solutions of the governing ordinary differential equations are given. In section 5 we use the solutions given in section 4 to derive the buckling criterion (5.4) for the first (n= 1) unsymmetrical mode. This criterion involves the strain-energy function of the material and for particular examples we consider the neo-Hookean and Varga materials. For the neo-Hookean material it is a simple matter to show that buckling in the n= 1 mode cannot occur. For the Varga material (for which references are given in [1]), buckling in the n= 1 mode is possible under an applied internal pressure and we tabulate roots of the buckling criterion (5.12) and corresponding critical pressures for various radii of the shell. We remark that whether or not the n= 1 deformation mode is jactually achieved for the Varga material would depend upon the critical pressures for the higher deformation modes which we do not consider here.

The solutions given in section 4 can be shown to apply to other non-axially symmetric deformations which are superimposed upon the initial deformation and which moreover can be shown to give rise to the same buckling criterion (5.4). In particular for a hollow sphere which is pre-compressed by a deformation of the form (3.1) and is then "cold" bonded to effectively rigid inner and outer spherical metal shells, we can deduce from the solutions given in section 4 the force-deflection relation for small superimposed translational deflections which are effected when the outer metal shell is held fixed and the inner suffers a small uniform translation. This force-deflection relation provides an upper bound for spherical rubber bush mountings which are used extensively as engineering components and consist of thick spherical pre-compressed rubber shells with their polar caps removed. Since thecalculations associated with these non-axially symmetric superimposed deformations are extremely long and tedious, we simply note the main results for them in section 6.

2. Axially symmetric deformations of isotropic ,incompressible hyperelastic materials

In this section we outline the main results for axially symmetric deformations in spherical polar coordinates of is0tropic incompressible hyperelastic materials. These results can be derived in a manner similar to that described in [4] for axially symmetric deformations in cylindrical polar coordinates and the reader is referred to this paper for further details, l~0r: material and spatial spherical polar coordinates (R, O, ~) and (r, 0, 95) respectively, we consider the axially symmetric deformation

r=r(R,O), O=O(R,O) , 9 5 = ~ , (2.1)

Symmetrical expansion of a spherical shell 127

which for an incompressible material satisfies the condition

R 2 sin O rROo--roOR-- r2 sin O , (2.2)

where rR, 0o, etc. denote partial derivatives. It can be shown that the contravariant components of the Cauchy stress tensor for an isotropic incompressible hyperelastic material are given by

122= _ p, ( 0~'~ 7 + ¢1 0~+ RU'

= q51 ( rg0 R + ro0o'~ (2.3) t12 R 2 / '

133 P* (~2 - r 2 sin2~ + R 2 sin z O '

t 13 = t 23 =- 0 ,

where p* is the pressure function and q51 and ~b2 are response functions which are given by

q~l= 2 + )2 ,

= + 1 - 2 4 .

(2.4)

Here Z(I1, I2) is the strain-energy function of the material and 11 and I 2 are the usual first two invariants of the inverse Cauchy deformation tensor and are given by

1 I1 = 1+22, 1 2 = 2 2 I + 2~- , (2.5)

where I and 2 are defined b y

( 0~'~ r sin 0 (2.6) I = r ~ + rR--5° 2 + r z 02 + R2 }, 2 - R s i n ~ "

From (2.3) and the appropriate equilibrium equations we can deduce the following equations

{ ( ,o,o ,sin2O P* = ~)1 V Z r - r 02 + R z j j + ~91grR + R ~ -- q~2 R 2 s i n 2 0 ,

{ 2( ,oo< ,ooo s oocoso P~ -- q~l V20 -[- rROR + R2 / j + ~)IROR ~- R2 ~2 R2 sin 2 0 r 2

(2.7)

128 James M. Hill

where the Laplacian V 2 is given by

8 2 2 a 1 8 2 cot 0 8 V 2 = - + + - - - - + Re . (2.8) 8R 2 Rs-R~ R 2 802 80

If we introduce the first Piola-Kirchhoff stress tensor in the usual way, then from (2.3) and (2.2) we can show that the resultant force F* in the conventional z-direction, acting on a surface which is given originally by the sphere R = constant, is given by

i~ [ -p*r sin O(r sin 0)o+ RZsin 04)1(r cos 0)R]dO. (2.9) F* 2r~

Moreover we can show that the boundary conditions of a normal pressure P acting inwardly over a surface again given originally by the sphere R = constant become

( - p * + P)r 2 sin O0o+ R 2 sin 04)1r R = O, (2.10)

( - p * + P ) sin O r e - R 2 sin O~blO R= O.

3. The initial deformation and governing ordinary differential equations for the small superimposed deformation

We suppose that a spherical shell of inner and outer radii A and B respectively is subjected to the following symmetrical inflation (Green and Shield [5])

r = ( R 3 + K ) ~, 0 - - O , ~ b = ~ , (3.1)

where K is a constant. For this deformation we find from (2.6) that

g = + X o - £ (3.2) ' - - R '

where we define p to be given by

p = (R3+K) ~ . (3.3)

From (2.4) and (3.2) we note that the response functions q51 and q52 are the same for the deformation (3.1) and from (2.7)1 we can deduce that the pressure function corresponding to (3.1) is given by

( R ) 4 IR 2q51 K( 2~3+K ) P°(R)=q51 --JA ( - ~ ~ d e + a , (3.4)

where a is a constant. The stress tensor t~ for (3.1) is now easily obtained from (2.3). We now look for solutions of (2.2) and (2.7) of the form

r = (R 3 + K)++ eu.(R)P.(cos 0 ) ,

0 = 0 + evn(R ) dP,(cos O) dO ' (3.5)

p* = po(R) +~G(R)P, (cos O),

where e is some small parameter for which we can neglect terms of order e e and

Symmetrical expansion of a spherical shell 129

higher, n is a positive integer and P, is a Legendre function of degree n. Using the incompressibility condition (3.8)1, we find from (2.5), (2.6), (3.2) and (3.5) that the invariants 11 and I2 are given by

5 P- ' ' e o ) , I1 = Ilo+e 2KR Ilou, ,(cos

5 Y ' 'R o ) , I2 = I2o+e 2KR I2°un n(c°s

(3.6)

where 11o and 120 are the invariants of the initial deformation and primes denote differentiation with respect to R. Thus for the response functions 01 and 02 we have

/35 01(I1, 12)= 01(110, 120) +~ 2 ~ 0i(Ilo , !20)'u'nPn(cos 0 ) ,

/)5 02(11, ]2)= 02(110, /20) -~'~ ~ 02(110, I20)'u'.P.(cos 0 ) ,

(3.7)

and since 01(11o,/2o) =02(ho, I2o), the response functions therefore admit the same expansion up to order e. Hereafter 01 denotes the response function evaluated at the initial deformation, that is 41(ho, X2o). Using (3.5) and (3.7) we obtain from (2.2) and (2.7), on equating terms of order e, the following system of ordinary differential equations

(~)2 u '+2 -u-p n(n+l)v=O,

P '=01 u " + 2 ~ - n(n+l)R 2 R2 + 2 n ( n + l ) ~ 5 RZp5 j

+0'1 (I pR3u'' (4R6+3KR3-ZK2)2KRp 2 u'} + O'l'PR3u , (3.8)

P -~'01 t 02 v t t~2R - n(n+ l ) ~ + 2 ~ - + 2 7 v ' + 2 RZpS j

, ~ R 2 pTu'[ +01 P 2v' + - - + p2 u 2 g3 ,

where we have omitted the suffix n from the functions u,(R), v,(R) and p,(R). The fourth order system of ordinary differential equations (3.8) is evidently highly

non-homogeneous and as with the corresponding system given in [1] there are as yet no known closed form solutions of these equations for general n. In the following section we deduce the complete integral of (3.8) for the special case of n = 1.

4. Closed form solutions for the special case of n = 1

If we substitute (3.5) and (3.7)1 into (2.9), then we find that the resultant force F* has a term of order e only for the case of n = 1. For n = 1 (3.5) becomes

130 James M. Hill

r = (R3+K)}+gu(R) cos O ,

0 = O-ev(R) sin O , (4.1)

p* = po(R)+ cp(R) cos O ,

where u, v and p in (4.1) and hereafter refer to the functions Ul, vl and Pl. The resultant force F* is found to be given by

R3p 3 ] F* 4rcg _pp2 + , , 2 , = 3 ~ - ~ 1 u +R ~l(u+2pv) l , (4.2)

and since this expression must be constant we set

8roe F* -- - - - (4.3) 3 K271'

where 71 is a constant. Thus from (4.2) and (4.3) we have

p = 2 ~K2yl R3P2K-q)I u " ' R2 + + 7241(u+2pv)' , (4.4)

which is a first integral of (3.8) for n = 1. If we now substitute (4.4) into either (3.8)2 or (3.8)3 using (3.8)1 to eliminate v,

then we obtain after some simplifications the following equation

~bl O ' - - q- ~10=4K2~;1 7 , (4.5)

where ~(R) is defined by

O(R) = pu" + ( 4 R 3 - K ) u' (4.6) Rp2

Integrating (4.5) we obtain

R 2 q510 -- 2K71 + 2K72, (4.7)

R 7

where Y2 is a constant. Using (4.6) we see that (4.7) can be written as

u' = 2KTa ~ + 2K72 ~bl(R)" (4.8)

Performing two further integrations and interchanging the order of integration in the double integrals that result, we can deduce the following

R 44 i " - u(R)= I <(¢)

+72/Tg-jA~R2 (R({3q-K)-~ d { - f ~ 2 ( ~ 3 + K ) } d ~ } ~ / 5 1 ~ - ~bl(~) (4.9)

R 2 -~Y3 ~2- q- 74,

Symmetrical expansion'of a spherical shell 131

where 73 and 7, are arbitrary constants. From (4.9) and (3.8)1 we find that the function v(R) is given by

{ l f ~ ~2(~3+K) ~ d~ - I (R ~4 v ( R ) = 71 qbl (~) p 3A ~ d ~

{ I I R ( { 3 + K ) ' d ~ - 1-fR~2(~3+K)}d{} (4.10)

+ ~ + ~ R p

while the function p(R), if needed, can be obtained from (4.4), (4.9) and (4.10). As with the corresponding results given in [1], we note that the solutions associated

with the constants 73 and 74 are independent of the strain-energy function of the material and are therefore controllable. In the following two sections we give two applications of the solutions given here.

5. The first buckled mode of thick-walled spherical shells subjected to uniform external pressure

If a spherical shell is subjected to a uniform external pressure P, then the deformation (3.1) describes the symmetrical deformed configuration where the constants K and a are determined by the boundary conditions of zero pressure at the inner radius and normal pressure G at the outer radius. From (2.10), (3.1) and (3.4) we find that a is zero and that K is determined by the condition

IB d~ (5.1) 2Kq~1(~)(2~3 + K)

p : _ -- A (4 3 + K ) 7/3 "

In order to find the critical pressures for which unsymmetrical solutions of the boundary value problem may exist, we use the method of adjacent equilibria for which further references can be found in [2] and [3]. We look for unsymmetrical solutions of the form (3.5) and for n= 1 we find from (2.10), (3.7)1 and (4.1) that the boundary conditions of normal pressure at the inner and outer radii become

R 3 p . . . . R 2

p = qqu +2 ~ q ~ l u ' 2K p"

at R = A, B, (5.2) e 2

~ ) i u p4

where we remind the reader that p is defined by (3.3). Since there are no resultant forces we have that 71 is zero and from (4.4) on using (3.8)1 with n = 1 we see that (5.2)1 reduces to (5.2)2.

From (4.9), (4.10) and (5.2)2 we obtain the following equations for the constants 72 and Y3

" (#3+K) d~ - (5.3) 72[)A q S * ( ~ ) 01(R)(2R3+K)J + 73=0 at R=A,B,

132 James M. Hill

which have non-trivial solutions if and only if

f B(¢3+K)~ d4 = (5.4) B(B3 + K) 7/3 A(A3 + K) 7/3

A ~ ~,(S)(2B3+K) ~x(A)(2A3+K) '

and this is the buckling criterion for the first (n = 1) unsymmetrical mode. By integration by parts we can show that (5.4) becomes

f B 4(¢3 q_K)s,- ~6ys-(~3q-K)(243q-K) ~b~(~)} d4 =0 (5.5) JA ~b,(~)(243+K)2{ " q61(4) "

For the neo-Hookean material for which q~l is a constant we obtain

f B ~6(~3 .~_ K) ff A (2¢3+K) 2 d 4 = 0, (5.6)

for which the integrand can evidently not vanish for any 40 such that A<<.4o<...B if both A and A 3 + K are to be strictly positive. Thus for the neo-Hookean material buckling in the n= 1 mode can never occur. For the Varga material for which references are given in [1] the strain-energy function is given by

Z = 2#(21 + 22 + 2 3 - 3), (5.7)

where # is the usual infinitesimal shear modulus and 2~ ( i= 1, 2, 3) are the principal stretches which'for the deformation (3.1) are given by

2 1 = , 2 2 - - 2 3 = ~ - .

Now from (2.4)1, (2.5) and (3.2) we have that for any strain-energy function

pV R3S ' (5.9) ~1 - 2K2(2R3 + K) ,

and thus from this equation and (5.7) and (5.8) we find that the response function q~l for a Varga material is given by

2#Rp2 (5.10) q~l- (2R3+K) •

Using (5.10), (5.1) can be shown to become

p B 2 A 2

- 2~ = (B3+K) } (A3+K) } ' (5.11)

while the buckling criterion (5.5) yields

f~ ~-(343- K)(43 + K) ~ d4 = 0. (5.12)

From (5.12) we see that K=3~g for some ~o such that A<~o~B and thus K is positive and hence for the Varga material buckling in the n = 1 mode can occur only under an applied internal pressure (that is for P negative).

The integral (5.12) can be evaluated in terms of elementary functions. If we define the function f(x) by

Symmetrical expansion of a spherical shell

( 7 + i ) 1 )~ 1 ~ (2x+ 1) x/3 2- X 2 ~ x S f(x) = ½ log (x + ~ - t a n - + ,

and if in addition we define

B K a=~, k=y,, and

~ = l + k / , f i = 1 +

then we find that k is determined as a root of the equation

f ( a ) = f ( f i ) •

TABLE 1 Values of K/A 3 and - P / 2 # for various values of B/A.

6 = B/A k = K/A s - P/21~

1.1 3.48 0.06

1.2 4.01 0.11

1.3 4.60 0.15

1.4 5.25 0.20

1.5 5.97 0.23

2.0 10.64 0137

3.0 26.41 0.53

4.0 52.90 0.60

5.0 92.57 0.64

10.0 568.73 0.73

133

(5.13)

(5.14)

(5.15)

(5.16)

Table 1 gives the roots of (5.16) and the corresponding values of -P/2# obtained from (5.11) for typical values of 6. We remind the reader that although we have shown that the n = 1 mode is a possible deformed configuration for the Varga material, this deformation mode may never be achieved in practice if the critical pressures for the higher deformation modes are less than that for the n = 1 mode.

6. Non-axially symmetric superimposed deformations

More generally the solutions (4.9) and (4.10) can be shown to be applicable to the following deformation

r = (R ~ + I~)*~ +~u(R) S,

0 = O+~v(R)So, (6.1)

~(R) q~ = ~ + ~ s ® ,

p* = po(R) +~p(R) S,

134 James M. Hill

where S(O, q)) is any spherical harmonic of degree one, that is any solution of the equation

Soo + cot O So + cosec 20 See + 2S = 0. (6.2)

We note that (4.1) is retrieved from (6.1) by taking S= cos O and that it can be shown that the buckling criterion (5.4) is applicable to any S which is a solution of (6.2). The calculations for these non-axially symmetric modes are extremely long and will be ommitted here.

The case of S=sin O cos q~ is of particular interest in connection with the pre-compressed bonded spherical rubber bush mountings described in the introduction. In this case (6.1) becomes

r = ( R 3 + K ) ~ + e u ( R ) sin O cos ~ ,

0 = O+~v(R) cos O cos q~, (6.3)

~) = ~ - e v ( R ) --sin q~ sin O '

p* = po(R)+ep(R) sin O cos 45,

and in this context we can identify the constant ~4 appearing in (4.9) and (4.10) as representing a uniform rigid translation of the sphere R = constant in the conventional x-direction. Moreover the constant 71 is now associated with the resultant force in the conventional x-direction acting on the sphere R = constant.

If a rubber spherical shell which is pre-compressed by a deformation of the form (3.1) and is "cold" bonded to effectively rigid inner and outer metal shells, undergoes small superimposed translational deflections in the conventional x-direction which are effected by fixing the outer metal shell and moving the inner, then a deformation of the form (6.3) is applicable where the functions u(R) and v(R) satisfy the following boundary conditions

1 u(A) -- 1, v(A) - (A 3 + K) ~ ,

u(B) = O, v(B) = 0, (6.4)

and e is the small distance moved by the inner metal shell. The force F required to maintain the deformation is given by

F = 8rte ~ - K 2 2 1 , (6.5)

where the constant 71 is found from (4,9), (4.10) and (6.4) to be given by

(~3 + Kfi- d~ fA (#1 (¢) ~1 = {f~ (~3lt-S)' d* f~ ,4 (~B ,2(,3+K) , d,)2}" (6.6)

(bx(~) ~ - 7 ~ d: - - \ O a ~-7(~)

Equations (6.5) and (6.6) constitute the force-deflection relation for small super- imposed translational deflections of a complete spherical shell. This relation provides

Symmetrical expansion of a spherical shell 135

an upper bound for translational deflections of bounded pre-compressed spherical rubber bush mountings which consist of spherical rubber shells with their polar caps removed. In the context of (6.3) the solutions (4.9) and (4.10) have been derived by the author in [6] for the special case of the neo-Hookean material and by a different method to that used here. The reader is referred to this paper for further details of spherical rubber bush mountings and for numerical results for the force-deflection relation of a neo-Hookean material.

7. Conclusion

For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are superimposed upon the symmetrical expansion of a spherical shell we have derived new closed form solutions without any restrictions on the strain- energy function. We have used these solutions to formulate the n-- 1 buckling criterion for thick-walled spherical shells which are subjected to uniform external pressure. Using this criterion we have established that the n= 1 mode is not a possible deformed configuration for the neo-Hookean material. For the Varga material we have shown that the n = 1 mode is a possible deformed configuration only under an applied internal pressure and we have tabulated roots of the buckling criterion and corresponding critical pressures for various radii of the shell. In addition we have noted that the solutions derived here are also applicable to certain superimposed non-axially symmetric deformations one of which describes small translational deflections of a pre-compressed bonded spherical shell and we have noted the force-deflection relation for deformations of this type. This relation provides an upper bound for small superimposed translational deflections of bonded pre-compressed spherical rubber bush mountings which are commonly used as engineering components.

We might summarise the importance of the solutions given here and in [1] as follows. Firstly they are closed form solutions of highly non-homogeneous ordinary differential equations for which only numerical solutions have been given previously. Secondly they are applicable to any strain-energy function and illustrate clearly the manner in which the small superimposed deformation depends on the strain-energy function of the material. Finally these solutions should at least throw some light on the difficult problem of determining solutions for the higher deformation modes.

Acknowledgements

The author gratefully acknowledges the financial support of the Science Research Council.

136 J a m e s M . Hi l l

REFERENCES

[1] Hill, J. M., Closed form solutions for small deformations superimposed upon the simultaneous inflation and extension of a cylindrical tube, Journal of Elasticity 6 (1976) 113-123.

[2] Wesolowski, Z., Stability of an elastic, thick-walled spherical shell loaded by an external pressure, Arch. Mech. Stosow, 19 (1967) 3-44.

[3] Wang, A. S. D. and A. Ertepinar, Stability and vibrations of elastic thick-walled cylindrical and spherical shells subjected to pressure, Int. J. Nonlinear Mech., 7 (1972) 539-555.

[4] Hill, J. M., Partial solutions of finite elasticity axially symmetric deformations, Z. angew. Math. Phys., 24 (1973) 409-418.

[5] Green, A. E. and R. T. Shield, Finite elastic deformation of incompressible isotropic bodies. Proc. Roy. Soc. Lond., A202 (1950) 407-419.

[6] Hill, J. M., Load-deflection relations of bonded pre-compressed spherical rubber bush mountings, Q. Jl. Mech. appl. Math., 28 (1975) 261-270.