NASA Contractor Report 177980 

lCASE REPORT NO. 85-41 

ICASE 

QUALITATIVE PROPERTIES OF LARGE 

BUCKLED STATES OF SPHERICAL SHELLS 

Ke-Gang Shih 

Stuart S. Antman 

NASA Contract No. NASl-17070 

September 1985 

NASA-CR-177980 

19860004961 

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II I,:'LU r;ES~ARCrl CUllER 

UCR.'\r,y. t'ASC. 

, ,. 'A~TOt:l, VIRGIJIllt, 

INSTITUTE FOR CO~~UTER APPLICATIONS IN SCIENCE AND ENGINEERING 

NASA Langley Research Center, Hampton, Virginia 23665 

Operated by the Universities Space Research Association 

NI\SI\ 

National Aeronautics and 

Space Administration 

Langley Research Cent. 

Hampton Virginia 23665 

111111111111111111111111111111111111111111111 

___ NFOI009 

------------ ---
1 Report No NASA CR-177980 I 2 Government Accll$$lon No 3 Recipient's Catalog No 

ICASE Report No. 85-41 

4 Title and Subtitle 5 Report Oate 

QUALITATIVE PROPERTIES OF LARGE BUCKLED STATES OF C"' .t. lQQt; 

SPHERICAL SHELLS 6 Performing Orgamzatlon Code 

7 Author(sl 8 Performing Orgamzatlon Report No 

Ke-Gang Shih and Stuart S. Antman 85-41 

10 Work Unit No 

9 Performing Organization Name and Address 

Institute for Computer App11cations in Science 

and Engineering 11 Contract or Grant No 

Mail Stop 132C, NASA Langley Research Center NAS1-17070 

Hampton VA 23665 13 Type of Report and Period Covered 

12 Sponsoring Agency Name and Address 

National Aeronautics and Space Administration " n 

Washington, D.C. 20546 14 SponSOring Agency't5Ode 

505-31-83-01 

15 Su pplementary Notes 

Langley Technical Monitor: Submitted to Archive for Rational 

J. C. South Jr. Mechanics and Analysis 

Final Report 

16 Abstract 

A system of 6th-order quasi-linear Ordinary Differential Equations is 

analyzed to show the global existence of axisymmetrically buckled states. A 

surpr1sing nodal property is obta1ned wh1ch shows that everywhere along a branch 

of solutions that bifurcates from a simple eigenvalue of the linearized 

equat10n, the number of s1multaneously vanish1ng points of both shear resultant 

and circumferent1al bending moment resultant remains invariant, prov1ded that a 

certain auxiliary condit1on is satisfied. 

17 Key Words (Suggested by Author(s" 18 DistributIon Statement 

shell buckl1ng 59 - Mathematical & Computer Sciences 

bifurcation 31 - Engineering 

Unclassified - Unlimited 

19 Securoty aasslf (of thiS report I 20 Security Clawf (of thiS pagel 21 No of Pages 22 Price 

Unclassified Unclassified 75 A04 

For sale by the National TechOicallnformatlon SerVice, Springfield Virginia 22161 

NASA-Langley, 1985 
QUALITATIVE PROPERTIES OF LARGE BUCKLED STATES 

OF SPHERICAL SHELLS 

Ke-Gang Shih 

Institute for Computer Appl1cations 1n Science and Engineering 

Stuart S. Antman 

Univers1ty of Maryland 

Abstract 

A system of 6th-order quasi-l1near Ordinary D1fferential 

Equations is analyzed to show the global existence ofaxisymmetr1cally buckled states. A surprising nodal property is obtained 

which shows that everywhere along a branch of solutions that 

bifurcates from a simple e1genvalue of the linearized equat1on, 

the number of simultaneously van1shing points of both shear 

resultant and circumferential bending moment resultant remains 

1nvar1ant, prov1ded that a certa1n auxiliary condition 1S 

sat1sfied. 

Research was supported 1n part by the Nat10nal Aeronautics 

and Space Adm1nistrat1on under NASA Contract No. NASl-17070 wh1le 

the author was in residence at rCASE, NASA Langley Research 

Center, Hampton, VA 23665. 

1 
This Page Intentionally Left Blank 
CONTENTS 

1. Introduction 1-2 

2. Formulation of the Governing Equations 3-9 

3. ~rivia1 Solutions 10 

4. The Linearized Equations 11-18 

5. Global Bifurcation 19-31 

6. Nudal PropeL ties of solutions 32-47 

7. Properties of G 48-55 

8. Conclusion 56-58 

Acknowledgment 59 

9. Appendix A. Formulation of the Governlng 60 

Equations 

10. References 70 

iii 
1. INTROOUCT~nN 

In this paper we study the global qualitat1ve 

behavior of axisymmetric buckled states of homogeneous 

isotrop1c nonlinearly elastic shells that can suffer 

flexure, compresS1on, and shear. Our model is geometr1cally 

exact 1n the sense that a geometr1c quantity, such as sln8, 

is not replaced by an approximat1on, such as e or 8 - e l/6. (The 

usual justification for such a replacement is that e 1S known 

to be small for the physical situation under study. But 1t 

1S quite possible that a mathemat1cal model w1th exact 

geometry permits only small solutions, wh11e those with approx1mate geometry have large solutions.) We allow the mater1al 

propert1es to be descr1bed by a very general class of non11near const1tutive relations. Conse~uently our governing 

equations form a quasil1near sixth-order system of ord1nary 

differential equat1ons. 

Our main result is that nontr1v1al branches of solutions 

pa1rs can be globally character1zed by a novel nodal pattern 

relat1ng shear and bending effects. The detect10n of th1S 

pattern is greatly a1ded by the general1ty of our theory, 

Wh1Ch does not obscure the slmple mathemat1cal structure of 

the governing equat10ns with approx1mate geometr1cal relat1ons. 

indeed, the beautiful numer1cal results of Bauer, Re1ss, 

& Keller ~97~ for the1r techn1cal theory of shells 1nd1cate 

that their solution branches do not enJoy the nodal propert1es 

we d1scover for our exact theory. We surm1se that no such 

techn1cal theory would have these propert1es. Consequently, 
2 

the nature of the b1furcatlon diaqram away from the triv1al 

Solut1on for these technical theories could dlffer markedly 

from ours. 

There is a large literature on the buckl1nq of spherlcal 

shells. The articles by Bauer, Re1ss, & Keller (1970), 

Antman (1971), Lanq~ & Krieqsmann (1981), and Sh1h (1985) 

give extenS1ve lists of references. Our results d1ffer 

markedly from those developed 1n these references (except 

for the work of Antman) by v1rtue of the general1ty of our 

theory and the global character of our theorems. 

The heart of our paper 1S 1n Sect10ns 5 and 6. In 

Section 5 we show that our equations meet the hypotheses of 

Rablnow1tz's Global Bifurcation Theorem. In Sectlon 6 we 

prove a baS1C uniqueness theorem that enables us to show 

that solution branches preserve the nodal pattern they lnherlt 

from the tr1vial branch. The use of the lsotropy of the 

shell materlal is crucial ln enabllnq us to overcome the 

dliricultles posed by the polar slngularltles. 

Notat1on. Part1al derlvatives are denoted by subscrlpts. 

Ordlnary der1vatives wlth respect to the varlable s are 

denoted by pr1mes. If f and g are funct10ns of x and y, then 

denotes the matrlx of partlal derlvatlves of f and g 

wlth respect to x and y. 
3 

§2. FORMULATION OF THE GOVERNING EQUATIONS. 

Let { ~'2'~ } be a fixed right-handed orthonormal 

basls for the Euclldean 3-space E3. For each ~ ln R 

we deflne 

(2.1) el(¢) = cos ¢i + sin ¢2' e2(¢) = -sin ¢i + cos ¢j, 

e3 = k 

To each (s,¢) € [O,n] x [O,2n] there corresponds exactly 

one point on the sphere of radius 1 centered at the origin 

with position vector of the form 

(2.2) 

Note that s measures the arc length to r*(s,¢) from the 

south pole of the sphere. This convention will simplify 

some of the formulas. We interpret the sphere defined by 

(2.2) to be the natural reference state of the midsurface 

of a spherical shell. The coordinates (s,¢) identify 

material points on this surface. 

An axisymmetric-configuration of a spherical shell 

that can suffer flexure, extension, and shear is determined 

by a pair of vector functions 

(2.3) (s,¢) ~ r(s,¢), b(s,¢) 

with r(s,-) and b(s,-) having period 2n and with 
(2.4) 

4 

r(s,cp) • ez (cp) = 0 , 

b(s,cp) • e2 (cp) = o. 

I~(s,cp) I = 1 

The reference configuration of the shell is given by 

r = ~*' b = -~*. The vector b(s,cp) is interpreted as 

characteriz~ng the deforme~ configuration of the material 

points that in the natural state of the shell lie along the 

ray determined by ~*(s,cp) 

We set 

(2.5) r(s,cp) - r (S):l (cp) + z(s)k -

(2.6) a(s,cp) - cos 1/1 (S):l (cp) + s~n 1/I(s)~ 

b(s,cp) - - sin 1/1 (s) : 1 (cp) + cos 1/I(s)k 

(2.7) :s(s,cp> - v(s)a(s,cp) + n(s)b(s,cp) 

(2.8) T (s) res) - sin s 

~ (s) - 1/1 I (s) - 1 

cr (s) sin ~ (s) - 1 sin s -

The strain variables for our problem are 

(2.9) w - ( V, T, n, ~, cr) 

Let n1(s,cp) and m1(s,cp) denote the resultant 

contact force and contact couple per unit reference length 
5 

of the circle ¢ ~ :*(s,¢) of radius sin s that are 

exerted across this circular section at :*(s,¢) Let 

~2(S,¢) and m2(s,¢) denote the resultant contact force 

and contact couple per unit reference length of the circle 

s * :*(s,¢) of radius 1 that are exerted across this 

section at :*(s,¢) . Since we seek axisymmetric solutions, 

we require these resultants to have the form 

(2.10) nl(s,¢) = N(s)a(s,¢) + H(s)b(s,¢) 

(2.11) n2 (s,¢) = T(s)e2 (<1» 

(2.12) ml (s, <1» = - M(s) e2 (<1» 

(2.13) 

Then the equilibrium equations for the shell under an external 

hydrostatic pressure of intensity A per unit actual area 

are 

(2.14) [sin sN(s)] '-T(s)cos Ij!(s)-sin sH(s)Ij!' (s)-Ar(s)n(s)=O, 

(2.15) [sin sH(s)] '+T(s)sin Ij!(s)+sin sN(s)Ij!' (s)+Ar(s)v(s)=O, 

(2.16) [sin sM(s)] '-L(S)COS Ij!(s)+sin s[v(s)H(s)-n(s)N(s)]=O. 

We can combine (2.14) and (2.15) to obtain 

A 

(2.17) {sin stsin lj!(s)N(s)+cos lj!(s)H(s)]+Ar(s)2/2}'=0, 
6 

A A 

(2.18) {sin s[-cos 1jJ(s)N(s)+sin 1jJ(s)H(s)]}' + 

+ T(s)+Ar(s)z' (s)=O 

The material of the shell is homogeneous and nonlinearly elastic if there are constitutive functions N, 

T, H, M, r such that 

(2.19) N (s) = N (w (s) ), etc. , 

when (2.3)-(2.8) hold. 

We assume that these constitutive functions are thr~ce 

continuously differentiable on their common domain of 

definition 

(2.20 ) v = { w e: RS v > h 1~+11 , 1" > hlo+11 } 

where h is a given number in (0,1) , wh~ch may be interpreted as half the thickness of the shell. The inequal~t~es 

in (2.20) ensure that in a standard three-dimensional 

interpretation of our deformation variables the local ratio 

of deformed to reference volume never be zero. 

We require the constitutive functions to satisfy 

the monotonicity conditions: 

(2.21) a(N,H,M) 

a ( v, n, ~) 

a (T, L) 

a ( 1", 0) are positive-definite. 
7 

( This is the strict form of the strong-ellipticity condition 

for axisymmetric deformations of axisymmetric shells. Cf. 

Antman (1978).) These conditions ensure,e.g., that an 

increase in the bending strain ~ is accompanied by a 

corresponding increase in the bending couple M. We also 

use special forms of the following very natural inequalities 

(2.22) N + N > 0 

V T 

T + T > 0 T V 

We impose compatible growth conditions ensuring that extreme 

strains are accompanies by extreme resultants: 

(2.23) N(w) ~ + = as v ~ = if T-h\1+cr\has a positive 

lower bound and if n,~,cr are bounded, 

N(w) ~ - = as v ~ h\1+~\if T is bounded above and 

if n,~,cr, are bounded, 

T(w) ~ + = as T ~ = if v-h\1+~\ has a positive 

lower bound and if n,~,cr are bounded, 

T(w) ~ - = as v ~ h\1+cr\ if v is bounded above 

and if n,~,cr are bounded, 

H(w) ~ ± = as n ~ ±= if (V,T,~,cr) lies in a 

compact subset of {(V,T,~,cr) :v>h\1+cr\, 

T>h\1+cr\} 

M(w) ~ ± = as ~ ~ ±= if v >h\1+cr\ and if (T,n,~) 

lies in a compact subset of {(T,n,~): 

T>h\1+cr\} 
8 

r(w) + ± ~ as cr + ± ~ if T > hll+crland if (v,n,~) 

lies in a compact subset of {(v,n,~): 

v>hll+crl}. 

We finally require that the shell be isotropic and 

have a stress-free natural state : 

(2.24) n ~ H(w) is odd, n ~ N(w),T(w) ,M(w) ,r(w) are even, 

M(w) = 0 = r(w) if ~ = 0 = cr , 

N (w)=N (w)=T (w)=T (w)=O if n = ~ = cr = 0, cr - ~ - cr - ~_ 

N(V,T,O,~,cr) = T(T,V,O,cr,~), 

M(V,T,o,~,cr) = r(T,V,O,cr,~) 

N(1,l,O,O,O) = T(1,l,O,O,O) = 0 

Our boundary conditions essentially requ~re that the deformation be regular at the poles and that there be no net rigid 

displacement in the z-direction: 

(2.25a,b) reO) = 0 = r(rr) 

(2.26a,b) nCO) = 0 = n(rr) 

(2.27a,b) \/J(O) = 0, \/J(rr) = rr 

(2.28) f: z(s)ds = 0 
9 

substituting (2.25a) into the integral of (2.17) we obtain 

'" (2.29) sin s[N(s)sin ~(s)+H(S)cos ~(s)]+Ar(s)2/2 = o. 

'" '" If Nand H are bounded and if A > 0, 

then (2.27) implies that (2.25b) holds. If we further 

assume that T is bounded, then (2.29) and (2.27) imply 

, 

that H(O) = 0 = H(TI). The properties of H then ensure 

that (2.26) holds. By comparing the integrals of (2.18) 

over [O,s] and [s,TI] we readily obtain 

(2.30) f

TIo '" T(s) ds = -A f:r(S)Z' (s) ds 

Note that the integral on the right side is the signed area 

between the curve (r,z) and the k-axis. 

Our boundary value problem consists of the strain 

configuration equations (2.5)-(2.8), the equilibrium 

equations (2.14)-(2.16) the constitutive equations (2.19) 

and the boundary conditions (2.25)-(2.28). A reqular 

solution of the remaining equations would automatically 

satisfy (2.25b) and (2.26). 

The shell theory we employ was originated by the Cosserats 

(1909) . (Cf. Naghdi (1972) and Libal & SImmonds (1983).) Our 

formulation was foreshadowed by several papers of Reissner, 

typical of WhICh is his paper of 1963. The variables we 

employ, and the very simple form of constitutive equations they 

Induce, were Introduced by Antman (1971). 
10 

§3. TRIVIAL SOLUTIONS. 

We now seek solutions of our boundary value problem 

in which the shell remains spherical, unsheared, and uniformly stretched, so that v = T = k(const), n = 0, ~(s) = s. 

Thus ~ = ° = a. Under these conditions, (2.22) reduces 

(2.14)-(2.17) to 

(3.1 ) 

Then 

+\k 2 /2 

N(k,k,O,O,O) = T(k,k,O,O,O) = -\k2 /2 

(2.22) and (2.23) imply that k ..... N(k,k,O,O,O) + 

strictly increases from - ~ to ~ as k 1ncreases 

to ~ provided that \ > 0. Thus (3.1) has a from ° 

unique solution for k, denoted k(\) with 

(3.2) [O,~) 3 \ t+ k(A) £ (0,1] 

thrice continuously differentiable and strictly decreasing . 
11 

§4. THE LINEARIZED EQUATIONS. 

We adopt the convention that if R is any constitutive 

function (such as N, Nv ,····· ), then 

(4.1 ) RO (1..) - R(k(A) ,k(A) ,0,0,0). 

The linearization of our boundary value problem about the 

trivial solution is equivalent to the following boundary 

value problem for the linearized variables Vl, nl , ~l 

(4.2) 

(4.3) 

(4.4) 

(4.5) 

(4.6) 

rl(s)= J:{Vl(t)COS t-[nl(t)+k~l(t)]sin t} dt = 

= Tl(s)sin s 

NO[(Vlsin S)'-T1COS s]= [(No +Ho +Ak)n 1 + 

v T n 

+(Nok+Ak2/2)~1]sin s , T 

HO (nlsin s)' = - (No +No+Ak) lVl + Tl) sin s + 

n v T 

+ ( Ak 2 /2) (~l sin s)' , 

~l (0) = a = ~l (7T) 
12 

In deriving this system, we relied crucially on (2.24). 

From this system, we wish to extract a system for 

nl and ~l alone. To accomplish this efficiently we write 

(4.3), (4.4), and the derivative of (4.4) in the form 

(4.8) (vlsin s) '-TlCOS s = f, (Vl+T!lsin s = g, 

(vlsin s) '+VlCOS s= h , 

from which we immediately deduce that 

(4.9) g = (h-f)tan s 

We write out this equation, replacing (~l'sin s) '-~l/sin s 

by it expression from (4.5) and supplement the resulting 

equation with (4.5), (4.6) to get the system 

(4.10a) (Lnd (s) + Anl (s) sin s = -a~l (s) sin s 

(4.10b) (L~d (s) + -B~l (s) sin s = -bnl (s) sin s 

(4.11) 

where 

(4.12) (Lu) (s) = [u' (s) sin s]' - u(s) /sin s 

(4.13) A = (Ho)-1[~Ak2b+(No+No+Ak) (NO)-l(No-No + 

n v T v T v 

+Ak+Ho)] , n 
13 

a = 

, 

We now convert (4.10), (4.11) to a problem for a 

single complex-valued function , 

(4.14) 

where a and 8 are complex constants to be adJusted so 

that (4.10a,b) can be combined into a single complex 

equation. Thus we require that 

(4.15) 

where C is a complex constant to be determined. It 

follows from (4.15) that 

(4.16) aA + 8b = aC aa + 8B = 8C 

so that 

(4.17) aa 2 + (B-A)a8 - b8 2 = 0 
14 

Using (4.15) we reduce (4.10), (4.11) to 

(4.18) (Lcj» (s) + C<psin s = 0, <P(O) = 0 = <p(rr) 

which has a nontrivial solution if and only if 

(4.19) C = n(n+1) , n = 1,2,.· ••••• 

in which case 

(4.20) <P(s) = <Pn(s) = const.p~(cos s) 

where pI 

n 

is the associated Legendre function of the first 

kind of degree n and order 1. When (4.19) holds, 

system (4.16) has a nontrivial solution if and only if 

(4.21) g(~;n) - [A-n(n+1)] [B-n(n+1)] - ab = 0 

Note that A, B, a, beach depneds on the eigenvalue parameter 

~ For a given integer n there can be none, one, or many 

real solutions ~ of (4.21) . Noting that NO(A)= -Ak(~)2/2, 

we could readily fashion conditions on the constitutive 

assumptions to ensure any of these possib1lities. (Cf. 

Antman & Rosenfeld (1978) for a discussion of this issue.) 

We limit our attention to positive solutions A of (4.21). 

Negative solutions correspond to shear instabilities. (Cf. 
15 

Antman & Carbone (1977).) 

For A > 0, (2.21) implies that b > O. Thus (4.16) 

has no nontrivial solutions (a, S) with a = o. We can accordingly 

adopt the special normalization that a = 1. In this case, 

the roots S of (4.17) are 

(4.22) = B-A ± .; (A-B) 2+4ab 

S± 2b = 

B-A± [A+B-2n(n+1)] 

2b , 

the second equality coming from (4.21). Thus' S± are real 

when (4.21) holds. In this case, S± are distinct if and 

only if 

(4.23) A+B f 2n(n+1) 

When (4.21) and (4.23) hold, (4.14) and (4.20) reduce 

to a system of the form 

(4.24) 

If we take D± real and not both 0, then the distinctness 

of S± ensures that (4.24) can be solved uniquely for nl 

These solutions are proport1onal to pl (cos s) n 

To find D± ' i.e., to find the constants of proportionality, 

we normalize ~l by setting 

(4.25) = Epl (cos s) n 
16 

Then a comparison of (4.10) and (4.11) with (4.18) 

implies that 

(4.26 ) A + alE = n(n+1) = B + bE 

whence 

(4.27) E n(n+l)-B 

= b 

a = -n~(-n-+-:-l-:-) --A=--

The compatibility of these two expressions for E is 

ensured by (4.21) . 

Now we study the special case that 

(4.28) A + B = 2n(n+1), n = 1,2,····· 

Then (4.22) implies that B is unique with 

(4.29a,b) 2bB = B - A 

Consequently (4.20) yields 

(4.30) 

Note that (4.29b) implies that there are no such solutions 

if a < 0 
17 

First we treat the even more special subcase that 

(4.31) A = B = n (n+l) 

Then 6 = 0 , nl = DP~(COS s) 

that a = o. Since b > 0 

Moreover (4.16) implies 

the alternative theorem 

implies that (4.10b) has no solution unless D = 0 and 

hence nl = O. In this case, ~l = const P~(cos s) . Note 

that if we make the eminently reasonable assumption that 

B < 2 ,then (4.31) cannot hold. Likewise, (4.31) cannot 

hold if a:F 0 

Now we assume that (4.28) holds, but that 

(4.32) A :F B :F n(n+1) 

Hence 6:F o. Using (4.30) we cor.vert (4.10b) to 

(4.33) = -bDP1(COS s)sin s 

n 

since (4.28) and (4.29) imply that B - b6 = n(n+1) . 

Since b:F 0 , the alternative theorem implies that (4.33) 

has no solution unless D = 0 , in which case (4.30) yields 

(4.34) 
18 

In summary, the eigenvalues of (4.10) , (4.11) 

are solutions A of (4.21). We limit our attention to 

positive· eigenvalues. If B < 2 , then the corresponding 

eigenfunctions are 

(4.35) (nl'~l) = ([n(n+1)-B]/b, 1)P~(cos s) 

Now the function P~(cos s) has exactly n+1 zeros on 

[O,~] , including those at 0 and ~ , each of which is 

simple. Thus every nontrivial solution (nl'~l) of (4.10) , 

(4.11) is characterized by the fact that nl and ~l have 

exactly the same zeros. 
19 

§5.GLOBAL BIFURCATION. 

To show that our equations meet the hypotheses of 

the bifurcation theorem and uniqueness theorem, which form 

the basis of our analysis, we must transform them into 

alternative forms that enable us to control the polar singularity manifested by the presence of sin s as a coefficient of the highest derivatives of our differential 

equations. We substitute (2.19) into (2.14)-(2.17), 

carry out the differentiation of the leading terms, use 

Cramer's rule (justified by (2.21» to solve the resulting 

equations for v'sin s, nIsin s, ~'sin s, and f1nally 

force this latest version of the system into a form suggested 

by the linear equations (4.3)-(4.5). Using (4.12) we 

obtain 

(5.1) (v sin s) , = T cos s +(No)-l(No+Ho+Ak)n v T T sin s + 

+ (No) -1 (N ok+Ak 2 /2) (1jJ-s) sin s + n , v T 

(5.2) (n sin s) , = -(HO)-l(No+No+Ak) (v+T)sin s + 

n v T 

+ (H ° ) -1 (A k 2 / 2) [ (1jJ - s) sin s1 ' + h , n 

(5.3) L( 1jJ-s ) = -B(1jJ-s)sin s - b sin s + m 

where 

. (5.4) n _ 6-1D1 +(V-T)COS S-(No)-l(No+Ho+Ak)n sin s -

v T n 

- (No) -1 (N ok+Ak 2 /2) (1jJ-s) sin s 

v T 
(5.5) 

(5.6) 

(5.7a) 

(5.7b) 

(5.Sa) 

(5. Sb) 

(5.Sc) 

20 

h -1 D2+nco9 - t:. s+ (HO)-l(No +No +:\k ) (V+T) sin 

n v T 

- (H 0) -1 ( :\k 2./ 2) [ (\jJ-s) sin s] , n 

-1 D3 (\jJ-s) 'cos ~-s m - t:. + s + B(1jJ- s)sin sin s 

+ bnsin s , 

t:. det a(N,H,M) - a (v,n,l.l) 

D is the determinant obtained from t:. by J 

replacing its jth column with (~) , y 

a _ -(N T'+N a')sin s+Tcos 1jJ-Ncos s+H1jJ'sin s+:\rn T a 

B - -(HTT'+Haa')sin s-Tsin 1jJ-Hcos S-N1jJ'sin s-:\rv 

y - -(M T'+M a')sin S+LCOS 1jJ-Mcos s-(vH-nN)sin s T a 

From (2.5)-(2.S) we find that 

(5.9) Lr = (vsin s) 'cos \jJ+ (vsin s) (cos 1jJ)' -

- (nsin s)' sin 1jJ- (nsin s) (sin 1jJ)' - T 

(5.10) (z'sin s)' = (vsin s)' sin 1jJ+ (vsin s) (sin 1jJ)' + 

+(nsin s) 'cos 1jJ+ (nsin s) (cos 1jJ)' • 

s 

s 
21 

We now introduce new variables (Ul' U2, us) - u by 

(S.lla) Ul (s) (sin 5) ~ = (Lr) (5) 

(S.llb) u2(s)(sins)~= (z'sins)' 

(S.llc) Us (s) (sin s) ~ = L (tjJ-s) (s) 

Let us assume that u is continuous. Then we can solve 

(5.lla,c), (2.25), (2.27) in terms of Ul and Us Now 

from (S.llb) we see that U2 must satisfy 

(5.12) JTI U2(S) (sin s)~ ds = 0 

o 

We thus find that 

(5.13) z' (s) = (sin s) -1 f: U2 (t) (sin t) ~ dt 

= - (sin s) _1 JTI U2 (t) (sin t) ~ dt 

s 

This representation shqws that z satisfies the boundary 

conditions z' (0) = 0 = z' (TI), which are consequences of 

(2.5)-(2.8), (2.26), (2.27). We integrate one of the 

equations of (5.13) and then use (2.28) to evaluate the 

constant of integration. We obtain 

(S.14a) res) = ITI K(S,t)U1(t) (sin t)~ dt - (Gud(s) 

o 
22 

(5.14b) z(s) 

-1 

[TI y-1+X[O,s] (y)] fY ~ sin y - u

2 (t) (sin t) dt dy 

o 

, 

(5.14c) 1jJ(s)-s = fTI K(S,t)U3 (t) (sin t)~dt - (GU3) (s) 

o 

where XeD,s] is the characteristic function for [0 , s] 

and K is Green's function for L subject to Dirichlet data: 

-~ 1+cos s sin t for t < 1+cos t sin s 

s 

(5.15) K(s,t) -

-~ 1+cos t sin s for t > 1+cos sin t s 

s 

(We could ~xpress (S.14b) in terms of a Green's function, 

but doing so would interfere with our analysis.) We also 

record that 

(5.16a) 

(5.16b) 

where. 

r' (s) = JTIKs(S,t)U l (t) (sin t)~dt = (GUl) , (s) 

o 

1jJ'(s)-1 = fTIKs(S,t)U3(t) (sin t)~dt _ (Gu 3)'(s) 

o 
23 

(5.17) 

~ l+cos s sin t . l+cos t sin2 for t < s 

s 

Ks(s,t) = 

-~ l+cos t 1 for t > l+sos s sin t s 

z' is given by (5.13) • From (2.7), (2.8) we then obtain 

(5.18a) , 

(5.18b) 

(5.18c,d) T = (Gul)/sin s , 0= [sin(s+Gu3)/sin s] - 1 

We now convert our differential equations (5.1)-

(5.3) into integral equations for u We replace the 

left sides of (5.9), (5.10), and (5.3) with (5.14) • We 

next substitute (5.1), (5.2) into the right sides of (5.9), 

(5.10). Then we replace the variables appearing on the right 

sides of (5.9), (5.10), (5.3) with their representations 

from (5.14)-(5.18). Finally, as a precautionary measure, 

we subtract from the right side of the resulting form of 

(5.11) its mean value, a move suggested by (5.12). We 

denote the resulting system of integral equations by 

(5.19) u = f(A,u) 
24 

Let 

(5.20a) (5.12) holds.} 

We wish to show that f(A,-) is a compact mapping from 

(5.20b) u (\{ u 

into itself ( V is defined in (2.20». 

Let w + R(w) be any constitutive function. If 

R Eel (V ), then 

(5.21a) R(y) = R(x) + R. (x,y) (y.-x.) 1. _ _ 1. 1. 

with 

(S.21b) R. (x,y) 1. _ -

1 

- f Rw. (t¥+(l-t)~) dt 

o 1. 

Here and below, twice repeated indices are summed from 1 to 

5. If R E C2 ( V ), then 

(5.21c) 

with 

(S.21d) 

R(y_)=R(X_) + R (x) (y.-x.)+~R .. (x,y) (y.-x.) (y -x.) wi - 1. 1. 1.J - - 1. 1. J J 

R .. (x,y) 

1.J - - =( o 

R w.w. (ty+(l-t)x) dt _ _ 1. J 
25 

We let R (w) w. - and R W.w. (w) _ ~ ~ J 

be any values of R. ~ 

and R .. 

~J 

in which the arguments consist solely of entries from w. 

E. g. , R (w) 

w. - could be taken to be Ri«T,\),O,O',~), 

~ 

(\),T,n,~,O'» . 

Since we whall use only the continuity of {R ,R } w. W.w. ~. ~ J 

and not the specific functional forms, we allow their 

meaning to vary even in the same expression. 

In the spirit of (5.21) we define the continuously 

differentiable functions C and S by 

(5.22a) 

(5.22b) 

Then 

(5.23a) cos w- cos s = - (w-s)sin s + U 

(5.23b) u - - (W-S)2C(W-S)cos s + (W-s) 3S(w-s)sin s 

(5,23c) sin W - sin s = (W-s)cos s + V 

(5.23d) 

Thus 
26 

(5.23e) T I sin s = (V-T) cos s - (1Ji-s) (vsin s+ncos s) -

- nsin s + vU - nV 

(5.23f) a'sin s = (~-a)cos s + 1Ji1 [U-(1Ji-s) sin s1 

Let 

, 

(5.24) w* = (T, V , a , a , ~ ) 

By using the formulas (5.21)-(5.23) we can convert a 

given by (5.8a), to 

(5.25) 

-[N (w*) (v-T)+N (w*) (~-a)+~N .. (w*,w) (w.-w.) * 

v - ~ - ~J - - ~ ~ 

* - * - * (w.-w)+N (V-T) (w.-w.)+N (~-a) (w.-w )]cos s J J TW • ~ ~ aw . ~ ~ ~ ~ 

+ N (w) [(1Ji-s) (vsin s+ncos s)+nsin s-vU+nV] - T -

-N (w)1Ji 1 [U-(1Ji-s) sin s] + H n1Ji ' sin s + Arn 

a - n 

We obtain similar representations for Band y Note 

that a + [N (w*) (V-T) +N (w*) (~-a) 1 -n (H 1Ji I sin s+Ar) consists 

v - ~ - n 

of sums of product of continuous functions of (w,1Ji-s) with - quadratic functions of v -T, n, ~-a , 1Ji-s By a lengthy 

computation we find that the terms in f consist of products 

of continuous functions of (w,1Ji-s)with quadratic functions 

-k -k -k of (sin s) ~ (V-T), (sin s) ~n (sin s) 4 (~-a) , and 
27 

with a number of other such functions of (w, ~-s) Let - us examine -k (sin s) "(ll-a), which (5.23c), (5.14c), (5.16b) 

reduce to 

(5.26a) -J.. (sin s) "[~. (s) - 1 -(~(s)-s)cot s - V(s)csc s ] 

= J7r J(S,t)U3(t) dt - (sin s)-5/1+ V(s) 

° 

where 

(5.26b) 

. t < s, 

J(s,t) = 

_L (I-cos S)2(sin t]9 / 1+ (. t)-3/1+ t > s ~ I-cos t sin s s~n . 

The Arzela-Ascoli Theorem supports a straightforward proof 

that the kernel J generates a compact and continuous 

operator from CO([O,7r]) to itself. Note that the kernel 

-J.. (sin s) "Ks See (5.17» does not. It is the cancellation 

of terms in J that is responsible for this compactness. 

Now (5.23d) implies that 

(S.26c) 

51 _11 -(sin s)- I+V = [(sin s) 9(~-S)]2C(~-S) + 

+ [(sin S)-5/12(~_S)]3S(~_S)COS s 

Since the Arzela-Ascoli Theorem implies that the integral 

representation for (sin S)-5/12(~_S) obtained from (5.14c) 
28 

in a compact mapping of u ,it follows that (S.26c) 

defines a compact mapping. Since compositions of continuous 

and =ompact mappings are compact, we find that (S.26c), 

_1/ (S.26a), and (sin s) 2(~_cr)2 correspond to compact 

operators taking CO{[O,TIl) into itself. We emphasize that 

the appearance of terms such as (~-cr) in f is a consequence 

of the isotropy conditions. 

By extensive use of arguments such as these, We 

find that f is a compact and continuous mapping of E 4 

itself. Moveover, the form of t shows that it is --------

into 

Frechet differentiable at the trivial branch of solutions . 

Let UO{A) be the trivial solution of (S.19) for fixed - A; Uo corresponds to the trivial solutions discussed in 

Section 3. The linear1zation of (S.19) about the trivial 

state Uo , namely -

(5.27) v = f -v (A,UO _ _ CA»v 

is equivalent to the equations of Section 4. We accordingly 

can invoke the global bifurcation theory of Rabinowitz(1971) 

and the local theory of Crandall & Rabinow1tz (1971) to 

obtain 

5.28. Theorem. For a given positive integer n let In 

be a solution of (4.21) of odd algebraic multiplicity,i.e., 

let there be an odd positive integer k such that , 
29 

(5.29) j=O, •• k-l, 

ak _ 

---k g(A ,U) # 0 . aA n -

Then bifurcating from the point_ JX'n'::o (X'n» on the trivial 

+ - branch of solutions are two connected sets C - (A ) of nonn 

trivial solution pairs (A,U) of (5.19) that enjoy at - -- least one of the following properties : 

+ - (i) C-(An ) cannot be confined to any compact subset 

of E 

(ii) The closure of + - C-(A ) contains another point n 

(Z;,UO(z;» where z; satisfies (4.21) 

Suppose, moreover, that In is simple,i.e., that k=l 

in (5.29) . Let be a normalized eigenfunction of (5.27) • 

Let P (X ) be the projection of U onto span(v). Then there 

- n _n 

number - a £ > 0 and there are continuous functions K 

n 

[-£", £"] t+ R, ¥n : [-E, £] t+ [I-P("X' )] u with K (0) = 0 

n -- n 

is a solution pair of (5.19) for lEI < £" Moreover, 

there is a neighborhood 

(A,U) is a solution pair of (5.19) lying in N

n 

' then 

either (A,U) = (A ,UO(A» or else there ~s an £ £ [-£",E] 

such that (A, u) = (X + K (d, u a ct

n

+ K n ( £» + £ [v +y ( £ ) ]) • n n _ _n _n 

Note also that (5.14)-(5.18) imply that w corresponding to any solution of (5.1~) is continuous and in 

1.S 
30 

particular 1JJ e: C1 ([O,TI]) Moreover 

(5.30) 1 im [ v ( s) - T (s)] = ° , 1 im [)J ( s) - a ( s)] = ° 

s-+O, TI . s-+O , TI 

It follows from (5.1)-(5.3) that u generates a classical 

solution to our boundary vlaue problem. In fact, the requirement that the constitutive functions N, T, ~, M, L 

be thrice continuously differentiable on V implies that 

the strains w generated by u are twice continuously 

differentiable on the open interval (O,TI) . To determine 

the boundary behavior of n' ,which is needed in the 

sequel, we divide (2.29) by sin2 s and use (2.19) and 

(5.21) to obtain 

(5.31) N(W(s))si~ 1JJ(s) +H (w) ~(s)cos 1JJ(s) +AT(S)2/2 = ° - S1n s n - S1n s 

on (O,TI). Since n satisfies (2.26), we can let s -+ O,TI 

in (5.31) to conclude that n is differentiable on [O,TI]. 

To show that n e: C1([0,TI]) we use the expression for n' 

obtained from (5.2) and examine its limits as s -+ O,TI 

This process is simplified by the observation that the 

limiting forms for n' can equivalently be obtained by 

substituting (2.19) into (2.15), carrying out the 

differentiations, dividing the resulting equation by sin s , 

and exploiting (2.24) and (5.30) to show that the limits 

n' (0) and n' (TI) agree with those found from (5.31). 
31 

~n essentlally equivalent version of Theorem 5.28 can 

be obtalned by uSlnq the theory of Fitzpatrlck & PejsachowlcZ 

(1985), WhlCh qenerallzes that of Crandall and ~ablnowltz. The 

former theory does not requlre the use of Green functlons. 

The role of isotropystill remalns paramount. 
32 

§6. NODAL PROPERTIES OF SOLUTIONS 

The determination of nodal properties of 

solutions of our boundary value problem is a more subtle 

question than that for the boundary value problem for 

plates. The main reason for this difference is that in the 

plate theory (cf. Antman (1978» n is the product of a 

certain function with sin ~ and so the nodal properties of 

n are determined directly from those of ~ On the other 

hand, in our shell theory v T n ~ are all 

coupled by nonlinear differential equations, which lack an 

obvious nodal pattern. To approach the question for shells 

we first need a candidate for a nodal pattern that is 

preserved on solution branches. 

The solutions of the 11nearized problem, treated in 

Section 4 have the property that nl and ~l have the 

same zeros, which are simple. The perturbat10n solut10ns 

of Section 8 also have this feature. Thus we are motivated 

to study whether the number of simultaneous zeros of nand 

~ - s is fixed on each bifurcating branch. 

We immediately confront a serious difficulty. Let 

Sn' n = 1, 2, ••• be the collection of all real-valued 

functions p in C1([0,n)] having zeros at 0, nand 

having exactly n - I zeros on (O,n), each zero being 

simple. Then 

function in S 

n 

S is open in C1([0,n)]. Given any n 

it follows that all sufficiently nearby 

functions enjoy the same nodal properties, i.e., have the 
33 

same number n + 1 of zeros, each of which is simple. 

Moreover, the boundary as consists of functions 

n 

having at least one double zero on [O,n]. These observations 

support the nodal theory introduced by Crandall & Rabinowitz 

(1970) for bifurcation problems governed by nonlinear 

Sturm-Liouville equations. To study pairs of functions with 

simultaneous zeros, we let T be the collection of all 

n 

complex-valued functions in [C 1 ([0,n])]2 having zeros at 

0, n and having exactly n - 1 zeros on (O,n), each zero 

being simple. But T is not open in [C 1 ([0,n])]2 and 

n 

there is no useful characterization of aT 

n 

To 

circumvent this diff~culty, we must exploit the fact that 

(n, ~-s) is not merely an arbitrary element of 

is also a solution of a boundary value problem. 

T , but 

n 

Our strategy is as follows. We first describe an 

auxiliary condition ensuring that if there is a solution of 

our boundary value problem for which n n' , ~ -s, 

(~-s)' all vanish at a point So in [O,n], then V-T 

also vanishes at so. Next we prove a basic uniqueness 

theorem stating that if there is a solution of our boundary 

value problem to which v - T, n ~ - s , , (~-s) all 

van~sh at a point So in [O,n], then the solution is 

trivial. This result does not enable us to assert that on a 

nontrivial solution branch (n, ~-s) remains in T n if 

(n, ~-s) is in T at any solution pair on the branch. 

n 

To remedy this difficulty, we set 
34 

(6.1 ) n(s) = p(s) cos 8(s), 

1JJ (s) - s = p (s) sin 8 (s) 

If n(so) = 0 = 1JJ(so) - so, then p(so) = O. Thus the zeros 

of the real-valued function p are the simultaneous zeros 

of nand 1JJ - s. If nand 1JJ are part of a nontrivial 

solution of our boundary value problem for which the auxiliary 

condition holds, we use the uniqueness theorem to prove that 

p can be defined so that it belongs to Cl([O,~]). We 

can accordingly determine where p E In particular, 

if at a point on a nontrivial solution branch P E 

then p remains in S until it has a double zero ( by n 

virtue of our discussion of as ). n 

By the remarks at the beginning of the last paragraph, 

we conclude that the solution is trivial if p has a double 

zero ( and hence nand 1JJ - s have simultaneous double 

zeros ) provided the aux1liary condit10n in met. Let Xn be 

a simple solution of (4.21) and let C±( Xn) be the two 

nontrivial branches bifurcating from (X ,uo(X ». We wish n n 

to show that on C± ( T ) • A the funct10n p remains in 

n 

S as long as the auxiliary condition is met. To do this, n 

we must show that p admits a linearization and enjoys 

properties like those stated in the last part of Theorem 5.28. 

We thereby demonstrate that on c± ( A), the function p n 

inherits its nodal properties from those of its linearization 

and preserves them as long as the auxiliary condition is met, 

i.e. , on P E Sn as long as the auxiliary 
35 

condition is met. Thus on X) the function nand 

n 

~ - shave n - 1 simultaneous zeros on (O,n) as long as 

the auxiliary condition is met. 

We now proceed to carry out this program. We assume 

throughout the ensuing discussion that (A,~,W) generates a 

solution pair for our boundary value problem. 

Suppose that there is an So E (O,n) such that 

n (s 0) = n' (s 0) = ~ (s 0) - So = ~' (s 0) - 1 =ll (s 0) = O. Then 

(2.24) ensures that (2.15), (2.19), (2.29) reduce to 

(6.2) N + T + AT(Soh>(so) = 0, 

(6.3) 

where the arguments of Nand Tare (v(so), T(So), 0, 0, 0). 

We know from Section 3 that (6.2), (6.3) admit a solution 

of the form v(so) = T(So) = k(A) and that this is the only 

solution of these equations for which v(so) = T(So). By 

virtue of (2.20) the variables v T A are confined 

to (h,co) x (h,co) x (O,co). Let G be the subset of this 

region for which the only solution of (6.2), (~.3) is 

v (s 0) = T (s 0) = k (A) • We then have the following result, 

which is virtually a tautology. 

6.4. LEMMA. 

= 0 and if (v (s 0 ), T (s 0) , A) E G then 

v(so) = T (s 0) = k (A) • 
36 

In the next section we shall study the nature of G 

the "auxiliary condition", alluded to above is just the 

requirement that (v(so) ,T(So) ,A) E G Our basic 

uniqueness theorem is 

6.5. Theorem. If there is an So E [D,n] such that 

(6.6) 

= t/J' (so) - 1 = D 

then the solut10n is trivial . 

Proof. Let 

(6.7a) 

= ((V-T) sin s, (v+T-2k) sin s, nsin s, llsin s, t/J-s) , 

(6. 7b) 

Then we can combine (5.1)-(5.3), (2.8), and (5.23) to 

obtain the system 

(6.8a) 

(6. 8b) 

-1 -n(vscos s+V) + ~ Dl 
(6.8c) 

(6.8d) 

(6.8e) 

37 

-1 

vs = V,+ (sin s) 

If So E (O,TI) and if v satisfies (6.6) and (6.8) 

then v = 0 , because the right-hand side of (6.8) is a 

well-behaved function of s and ~ . Consequently the 

initial-value problem has a unique solution, which is the 

trivial solution v = 0 

Thus we need only concern ourselves with the cases 

So = 0, TI, in which the right-hand side of (6.8) is 

singular in s. We just treat the former case. Throughout 

our discussion we let v represent an arbitrary, given 

solution of our boundary value problem ( known to exist by 

the results of Section 5) which we subject to the 

additional requirement that it satisfy (6.6) . In the 

analysis to follow, we let c denote a typical positive 

constant depending on the fixed but arbitrary solution v. 

The meaning of c can change from place to place. We tacitly 

restrict s to lie in an interval of the form [O,E] where 

E is small and positive. 

Since sin s ~s cos s on [O,E], we find that 

(6.9) fS(Sin t)-~dt < 2(sin s)~/cos s 

a 

< c(sin s)~ = 
38 

Using this result in (5.13)-(5.18) and using the fact that 

lui ~ c , we obtain after careful analysis that 

(6.10) IV-TI + Inl + I~-ol ~ c(sin s)~ 

IIP-si < csin s I~I + 101 < c 

Since (5.23) implies that 

(6.11) 

we find from (5.25) and related expressions that 

(6.12) lal + lsi + IYI ~ c(sin s)~ 

Let us now examine (6.Ba) . From (6.10) we 

immediately obtain 

(6.13) < cv < cv(sin s)-~ = 

Note that (2.21) and (2.23) imply that 

Accounting for (2.24) by using the notation introduced after 

(5.21) we obtain 
39 

(6.15) D1 = aH M - aH M n2 - (8M -yH)N n + 

n ~ ~n nn ~ ~ nn 

+ 8N M n - yH (N n+N ~+N 0) ~ nn n ~n ~~ ~o , 

(6.16) 

• Inequalities (6.10)-(6.12), (6.14)-(6.16) imply that 

(6.17) IL.l A-1D1_rv"'" (N )-11 < cv (. 

s~n s )-~ 'V = 

If in (5.25) we replace N (w*) * 

'V -

with N (w ) +N (w. -w . ) 'V - 'Vwi ~ ~ 

and replace 

from (6.10), 

(6.18) 

N (w*) ~ -

(6.11) 

with N n+N ~+N 0 ~n ~~ ~o 

that 

< cv(sin s)-~ = 

, then we obtain 

Inequalities (6.13), (6.17), (6.18) imply that 

(6.19) < cv(sin s)-~ = 

A virtually identical analysis yields 

(6.20) 

The treatment of (6.8c) 'is just slightly different. 

Using the same ideas as those leading to (6.17) we obtain 
40 

(6.21) 

Since H - H n T - Tn' (2.24), equation (5.8) 

yields 

(6.22) lsi ~ cv(sin s)-~ +1 T+N+AvTlsin s 

Equation (3.1) Lmplies that 

(6.23) T + N + ATV = X = X (v-k) + X (T-k) v T 

Combining (6.8c) , (6.21)-(6.23) , we obtain 

(6.24 ) 

We remark that the only place that the variable V2 appears 

in (6.8a,c,d,e) is in the right side of (6.8c). Similar 

techniques produce 

(6.25) 

, IVit - vscot sl < cv(sin s)-~ = 

Combining (6.19), (6.20), (6.24), (6.25), (6.8e) we 

obtain 

(6.26) 
41 

the f~rst ~nequality being a standard consequence of the 

absolute continuity of v Since (6.6) implies that 

(6.27) v(s)/s ~ 0 as s ~ 0 

we can invoke the standard uniqueness theorems associated 

with differential inequalities (cf. Hartman(1964, Sec.III.6)) 

to conclude that v = 0 0 

It is important to note that if we were to insert a 

constant exceeding 1 in front of vcot s in (6.26), then 

we could not conclude that v = 0 . That the coefficient 

in front of vcot s is 1 is a consequence of the isotropy 

condition. 
42 

We are now ready to study the polar coordinates p 

and e introduced in (6.1). We state their properties in 

a series of lemmas. 

6.28. Lemma. Let the boundary value problem have a nontrivial solution pair with (V,T,A) taking values only in 

G and with p and e defined in any manner consistent 

with (6.1). Then the zeros of p on [D,n] are isolated. 

Proof. If not, they would accumulate at a point So £ [D,n], 

at which 'n and ~ -s would consequently have simultaneous double zeros. By Lemma 6.4 and Theorem 6.5, the 

solution would then be trivial. 0 

Now (6.1) implies that 

(6.29) tan e= (~-s)/n . 

Suppose that the solution ~s not trivial and that (V,T,A) take 

values only in G Let Sl and S2 be a pair of adJacent 

zeros of p If n vanishes nowhere on (Sl,S2), then 

(6.29 implies that tan e is continuously differentiable 

on (Sl, S2) and we can define e itself (mod n) to be 

continuously differentiable on (Sl, S2). If n vanishes at 

S3 £ (Sl, S2), then we define 
43 

(6.30) o ( s) _ 'IT /2 + f s {n (t) [1/1' (t) -1) - [1/1 (t) -1) n' (t) } d t (mod 'IT) 

S3 n(t)2+[1/I(t)-t]2 

for s E (Sl,S2). If s~ is a zero of p and if n' (s~) 

and 1/1' (s~) - 1 do not both vanish, then I'Hopital's rule 

enables us to deduce from (6.29) that 

(6.31) tan 0(s) ~ 1/1' (s4)-1 

n'(s~) 

the limit existing as an extended real number. (If n' (s~)= 

= 0 = 1/1' (s4)-1, then we could use Lemma 6.4 and Theorem 6.5 

to show that the solution is trivial when the condition on 

G holds.) Then 0(S4) is defined mod 'IT. By selectinq 

the value of 0 at such zeros of p appropriately, we can 

define 

(6.32) p = ncos 0 + (1/I-s) sin 0 

Hence Lemma 6.4 and Theorem 6.5 imply 

6 . 3 3 • Lemma. Let the boundary value problem have a nontrivial solution pair with (v,n,A) taking values only in 

G Then p and e satisfying (6.1) can be chosen to 

be continuous on [O,'IT]. 

Since n,1/I E C1 ([0,'IT), the constructions we have 

made above ensure that p and 0 are continuously differen-
44 

tiable where p does not vanish provided the hypotheses of 

Lemma 6.33 hold. Differentiating (6.1) we obtain 

(6.34a,b) p' = n'cos 0 + (~'-l)sin 0 

p0' = -nisin 0+ (~'-l)cos 0 

where p does not vanish. The continuity of the right 

sides of (6.34) then yields 

6.35. Lemma. If the hypotheses o£ Lemma 6.33 hold, then 

p' , pSI E CO ([D,n]). 

We require the following refinement of this lemma 

6.36. Lemma. If So is a zero of p and if n' (so) 

and ~'(so)-1 are not both zero, then p(so)0' (so) = D. 

Proof. The Mean Value Theorem Y1elds 

(6 .37 ) rJ' 

n (s) = l 0 n' (as+(1-a)SO)da] (s-so) 

1/ds) - s = [( ~(as+(1-a)so)da ] (s-so) 

On any small one-sided neighborhood of So , (6.34b) thus 

yield 
45 

(6. 38 ) ±p(s)8' (s) 

1 1 

lJ (S) fa n ' da - n ' (S) fa lJ da 

= 1 1 

[[JOn'd~12+ [JO~d~12l~ 

where the arguments of n' and lJ in the integrands are 

as + (l-a)so . Since n' and lJ are un1formly continuous 

on [O,n] , we find that f: n' (as+(l-a)so)da + n' (so) as 

s + So ,etc. Thus in the limit as s + So ,the numerator 

on the right side of (6.38) approaches a ,while the 

denominator approaches a positive number. 0 

P E 

In view of Lemma 6.35 we can examine whether 

S • By the introductory remarks of this section, funcn 

tions p in as 

n 

have a double zero. But this fact alone 

does not guarantee that n and ~-s have a simultaneous 

double zeros, as an examination of (6.1) and (6.34a) shows. 

But Lemma 6.36 coupled with (6.34) implies that at a 

double zero of p the functions n and ~-s have a simultaneous double zero, a result that enables us to invoke 

Lemma 6.4 and Theorem 6.5 . Marshalling these facts in 

the manner proposed in the 1ntroductory remarks in this 

section, we obtain 

6~9.Theorem. Let K be a connected set of nontrivial 

solution pairs with the property that (V,T,A) takes values 
46 

only in G for each solution pair in K 

solution pair in K the function P E Sn 

If for one 

then P 

remains in Sn for every solution pair ±n K Thus n 

and W-s have exactly n+1 simultaneous zeros on K 

Let An be a simple root of (4.21) (so that (5.29 

holds with k = 1). Then the last part of Theorem 5.28 

together with the representations (s.14c) and (s.18b) 

imply that on the branch C( An) near the bifurcation point 

( An' a ), A n W have representations of the form 

(6.40) A=1' + K(E),n=Enl+o(E), n 

as E ~ 0, where K is continuous and K(O) = O. If we 

substitute (6.40) into (6.29) and use (4.25 we 

obtain 

(6.41 ) tan 8 = W 1 / n 1 + 0 ( 1) = E -1 + 0 ( 1) . 

, 

Thus we may take 

(6.42 ) -1 8 = arctan (E ) + 0(1). 

Then (6.32) implies that 

(6.43 ) P = E P 1 + 0 ( e: ) = £ [ P 1 +0 ( £) £ - 1 ] 
with 

(6.44 ) 

47 

Pl = (l+E )-~[Enl +~l] 

= (1+E )~ pl (cos s). n 

It follow that Pl £ Sn and that P £ for £ 

sufficiently small. Incidentally, we show in the next 

section that (V,T,A) generically belongs to G for e: 

sufficiently small essentially because v - T= 0 on the 

trivial branch. ) 

6.45. Theorem. 

We accordingly have 

Let A 

n 

be a simple root of 

On the maximal connected subset of the closure of 

(4.21). 

containing 

only in G 

An' ~o( An» for which (V,T,A) takes values 

nand ~ -s have exactly n + 1 simultaneous zeros. 
48 

§7. PROPERTIES OF G. 

We now study (6.2), (6.3), which we rewrite as 

(7.1) 

(7.2) N(V,T) + T(V,T) = -AvT 

, 

from which we obtain 

(7.3) T(V,T) = AT Pn-V) 

Here we have suppressed the last three arguments (0,0,0) 

of the functions Nand T The set G consists of 

all (v,T,A) in (h,~) x (h,~) x (O,~) for which the only 

solution of (7.1), (7.2) is v = T = k(A) . We make a 

few remarks about (7.1), (7.2) in general before studying 

a specific example. 

Since Nv > ° by (2.21) and s~nce N satisfies 

the growth conditions of (2.23), equation (7.1) can be 

solved uniquely for v in terms of T and A 

(7.4) v = f(TiA), with 1 = f(liO) 

Since f = -(N +AT)/N ,we find that f < ° if N > ° T T V T T 

and that f (1 i 0) > - 1 if N (1,1) < N (1,1) T T v 

The isotropy condition (2.24) ensures that the 

curve in the (v,T)-p1ane defined by (7.2) is symmetric 
49 

about the line v = T If T + N 

v > v 0, if T + N -+ - 00 

as v -+ 0 for fixed T , and if liminf T + N > 0 as 

= 

v -+ 00 for fixed T , then (7.2) can be uniquely solved 

for v in terms of T and A 

(7.5) v = g(T;A) with 1 = g(1;0) 

The results of Section 3 ensure that the graphs of f(·;A) 

and g(·;A) intersect on the line v = T. Since 

(7.6) 

(Ag+NT +TT) 

(A +N +T ) T V T 

it follows that gT < 0 under our assumptions and, not 

surprisingly, that f (T;A) = -1 when T = g(T;A) Thus 

T 

the graphs of f(·;A) and g(·;A) cross transversally 

on the line v = T when f # -1 here, i.e., when 

T 

(7.7a) 

Using (7.1) we can rewrite (7.7a) as a condition involv-

~ng only the constitutive function N 

(7.7b) N (T, T) - N (T, T) + 2N (T , T) / T # 0 

V T 

Wherever (7.7) holds, there is a neighborhood about 

(T,T,A) lying in G ( See the parenthetical remark 
50 

preceding Theorem 6.45.) 

There are several ways to analyze (7.1), (7.2) 

further by making general restrictions on Nand T 

leading to suitable estimates. It is also illuminating to 

view (7.1) and (7.2) as each describing the intersection of two graphs over the (v,T)-plane. Alternatively 

we could assume that Nand T are derivable from a 

stored energy function (as in (7.9) below), characterize 

the equations as Euler-Lagrange equations for a constrained 

problem, and study the extremization of a real valued 

function on a constraint curve. There are several ways to 

choose the constraint. It is more illuminating, however, 

to analyze a specific class of materials. 

(7.8) 

(7.9) 

Let A, B, C, a be positive numbers and let 

N(V,T) = Wv(V,T) - Wv (1,1) 

T(V,T) = WT(V,T) - WT (1,1) 

These equations give a reasonable model for material behavior 

in compression for h = 0 • When (7.8), (7.9) hold, 

equations (7.1), (7.2) reduce to 

(7.10) 
51 

(7.11) - Av - (a+1) T - (a+1) (V+T) + (2B+C) (V+T) +2A-

-2 (2B+C) = -AVT 

We now solve (7.11) for _A(VT)-(a+1) and substitute the 

resulting expression into (7.10), obtaining an equation with 

a factor v - T . Since we wish to study the locations of 

possible solutions of (7.1), (7.2) with v # T we 

cancel this factor obtaining 

(7.12) (B+C) (V+T) + A -2B - C = ~ AT2 • 

Let us denote the solution of (7.12) for v in terms of 

T by Vi(T,A) If we replace v in (7.10) with VieT) 

we obtain 

(7.13) AT- (a+1) = (AT-C) [ vi (T, A)] cx+1 

For A > 0 , trivial solut~ons of our boundary value 

problem are characterized by v = T = k(A) £ (0,1) , 

N = T =-~Ak(A) 2 .We accordingly limit our attention to a 

neighborhood (possibly quite large) of the trivial solutions 

by seeking solutions of (7.13) for which 

(7.14a,b,c,d) T £ (0,1), v* (T,A) £ (0,1) 

* N(v (T,A),T) < 0 , 

for A > 0 We wish to characterize where (7.13) has 

solutions satisfying (7.14) Such solutions lie in the 
52 

complement of G . (If (7.14) is not imposed, then (7.13) 

always has a solution.) 

Note that (7.14c) is automatically satisfied by 

virtue of (7.1). We now use (7.12)-(7.14) to show that 

the set of points (V,T) e: (0,1) x (0,1) at which (7.1) , 

(7.2) has solutions for (7.8), (7.9) can be quite small. 

Call this set H ()d Clearly 

(7.15) HeA) = { (V,T) e: (0,1) x (0,1) :V = V#(T,A) , 

T satisfies (7.13), T(V#(T,A),T) < 0 } 

We now ob~ain a number of inequalities that elements of 

H(A) must satisfy. 

Equation (7.3) implies that (7.14d) is satisfied 

if and only if 

(7.16) 

Equation (7.12) implies that 

(7.17) V#(T,A) > -T+ 1 + B - A B + C 

which can represent a very severe or an insignificant 

restriction depending on the relative size of A, B, C. 

Equation (7.12) and conditions (7.14a,b) imply that 

(7.18) T < 

= [2(C+A)]~ A-~ 
53 

A simple graphical analysis of (7.13) based on (7.14a,b) 

shows that 

(7.19) T ~ min( A+C 

A 

Note that if B-A and A are sufficiently large, 

then the set of TIS satisfying (7.14), (7.17), (7.18) is 

empty. If A is small enough, then (7.19) implies that 

T > 1 = 

By substituting (7.16) into (7.12) we obtain 

the interval being regarded as empty if are 

not real. If A-2B-C > 0 and if A is large enough, then 

(7.20) delivers a lower bound for T that is larger than 

that of (7.19) . 

We seek similar estimates independent of A Let 

us use ( 7 . 18 ) t t I b d f AT - (a + 1 ) . o ge a ower oun or A 

sketch shows that a lower bound for T is the solution of 

(7.21) 

Thus 

A [ 2(C+A)]-(a+1)/2 = (AT-C) 

A 
54 

(7.22) T > ~ + AA(a-1)/2 

= A [2(C+A)] (a+1)/2 

If a E (0,1), then (7.22) delivers a lower bound dependent on A , but nevertheless better than those of (7.19) 

and (7.20) when A is large enough. If a = 1 , then 

(7.22) gives bound that is clearly better than (7.19) and 

that itself has a further lower bound independent of A If 

a > 1 , then 

(7.23) 

(a+1)C(a-1)/(a+1)A2 /(a+1) 

= ~~~~--------~~----------------

(a-I) (a-I) / (a+1) 2-2 (a+2) / (a+1) (C+A) 

with x = [ 2C ]1/(a+1) (a-1)A ' 

C I; = -=-:'-:- C+A , the expressions 

on the right side of (7.23) be~ng the minimum of the right 

side of (7.22) for A E (0,00) An upper bound for the 

rightmost term of (7.23) can be obtained by the Young 

inequality. The right side of (7.23) can be made arbitrarily 

large by taking a -1 sufficiently small and positive. 

It is not difficult to obtain further estimates, e.g., 

estimates for the neighborhood discussed in the comments 

following (7.7b). We refrain from doing so. 

The basic message of this section is that the set G 

. can be very large and that precise information about it can 
55 

be gained directly from degenerate constitutive equations. 

Moreover, we would be able to strengthen the global results 

of Section 7 even more by obtaining estimates for the 

location of solutions of our boundary value problem and restricting G to refect such estimates. We do not attempt such 

estimates here. 
56 

H. CONCLUSION. 

In an unshearable shell n is constrained to be zero. 

The shear resultant H , not defined const1tut1vely, is the 

Lagrange multipl1erma1ntaining the constra1nt of unshearab1lity. It is a fundamental unknown of the constrained 

problem. The same analysis as that used above can treat 

unshearable shells if H is treated as ~ 1S. 

Some of our techniques are rem1niscent of those of 

P~tman(1978) for plates. The difference 1n boundary condit10ns forced us to give the coordinate funct10ns rand z 

a central role 1n the analysis of ~c~t1on 5, which they 

lacked 1n the plate theory. The 1ssues treated 1n Sect10n 

7 never arose for plates. Of course the nodal propert1es 

of branches for shells differ markedly from those for plates 

and for c1rcular arches (cf.Antman & Dunn, (1980).) Inc1dentally, the uniqueness theorem of Antman (1978, Sect10n 7) 1S 

faulty; it can be readily corrected by follOW1nq the development of Sect10n 6 above. 

It is illuminating to compare our results w1th the 

numerical results of Bauer, Reiss, & Keller (1970). Treating 

an engineerinq theory of shells they discovered the remarkable fact that all the bifurcating branches they analyzed 

are connected. Our results p01nt 1n the diametr1cally 

opposite direction. We showed that nodal properties of 

branches bifurcating from simple eigenvalues could only lose 

the1r nodal structure at certaln special places ln Solut1on-
57 

parameter space. In particular, if we could obta1n del1cate 

p01ntwise est1mates showing that v E (0,1), T E (0,1) , 

N < 0, T < 0, everywhere on a fam1ly of b1furcat1ng 

branches, then the results of Sect10ns 6 and 7 show 

that nodal properties cannot sW1tch along such branches. 

Consequently such branches could not connect b1furcation 

points with simple eigenvalues having different eigenfunctions. 

Now Section 7 shows that for fixed A the complement of G is generically a finite number of p01nts. On 

the bas1s of such genericity arguments we could expect that 

on a given b1furcating branch there would be no So E [0, Irl 

and A > 0 such that (v (s 0 ), T (s 0 ) , A) t G Such genL!rlC1ty arguments are dangerous: The nodal structure we found 

for nand w-s is not generic for pairs of functions. ) 

The chief source of the suspected var1ance between 

our results and those of Bauer, Reiss, & Keller is 11kely 

the fact that various geometrlc terms regarded as small 

were discarded in their theory ( as is typical of all tech·" 

n1cal shell theories). Consequently these terms could not 

come into play in the large solutions they computed. A 

concelvable alternative explanat10n of the connectedness 

of all the bifurcat1nq branches found by Bauer, Re1ss, & 

Keller 1S that they correspond to spur10us Solutl0n branches 

of the sort often found in numerical studIes of b1furcatlon 

problems. We tend to dismiss this explanatlon because It 
58 

does not seem compatible with theconslstent pattern they 

found. Bauer, ~eiss, & Keller dld not qlve enouah data ~or 

us to deter~lne what sort of nodal propertles their Solutlons 

possess. 

Our approach may also be compared wlth that of 

Lange & Kriegsmann (1981) who studied continuatlon properties for very thin shells by using asymptotic expansions. 

Our global blfurcation theory glves an approach to the continuation of branches that does not requlre special hypotheses on thickness,or more generally, on constltutive functions. Presumably their detalled approach could be applled 

to our geometrically exact theory for constitutive functl0ns 

corresponding to very thin shells. We could also introduce 

the thickness h as a second bifurcatlon parameter and 

study the role of thickness by using global multlparameter 

blfurcatlon theory (cf. Alexander & Antman (1981)). 

We mention that the perturbation results of 

Shih (1985) show that the local disposltl0n of b1furcatlnq 

branches depends critically on the nature of our nonllnear 

constitutive equations. In particular, the local structure 

can be quite different from that found in technical shell 

theories, which essentlally use llnear constltutlve relatl0ns. 

As we have indlcated above, the process of numerlcally 

constructing solution branches can be quite trlcky. It would 

seem that nodal propertles on Solutlon branches could glve 

an extra control on numerical computatl0n. But, to our 

knowledqe, no such use of nodd 1 propertIes lias been at tempt ed. 
59 

Acknowledgement. Part of the work presented here was developed 

by Shih (1985) for his doctoral dissertatlon In the Department of Mathematlcs of the Unlversity of Maryland. The work 

of Antman was partla11y supported by a qrant ~rom the Natlonal 

SClence Foundatlon. 
60 

§9. APPENDIX A. FORMULATION OF THE GOVERNING EQUATIONS. 

In this appendix we describe a special director theory 

for a nonlinearly elastic axisymmetric shell undergoing an 

axisymmetric deformation. Let {i,j,k} be a right-handed 

orthonormal basis for the Euclidean space E3 

In the special Cosserat theory we employ, a shell is 

regarded as a surface together with a unit vector field, 

called the director field, defined on it. For such an 

axisymmetric shell undergoing an axisymmetric deformation, the 

surface is generated by a plane curve rotated about the 

k-axis and the director field is likewise generated by 

rotating a director field coplanar with the generating curve. 

k 

= a*(s,q» 

z*(s) 

, Fig. 9-1 

Reference Configuration 

Let 

(9.1 ) el(~) = cos ~i + sin ~j 

(9.2) e2(~) = -sin ~i + cos ~J 
61 

The position ~*(s,¢) of a typical material point 

on the (mid)surface of an axisymmetric shell in its undeformed reference configuration is (see Fig. ~-1), 

(9.3) r*(s,¢) = r*(s)el(¢) + z*(s)k 

where ¢ is the meridional angle and s measures arc length 

parameter. (s,¢) identify each material point of the 

(mid) surface. We let s £ [O,L] Define 

(9.4) a*(s,¢) = cos 1jJ*(s)el(¢) + sin 1jJ*(s)k 

(9.5) b*(s,¢) = - sin 1jJ*(s)el(¢) + cos 1jJ*(s)k 

where 1jJ*(s) is the angle between the direction of the tangent vector to the curve r*(·,O) at s and that of el(¢) . 

Deformed Configuration 

The axisymmetrically deformed configuration of an 

axisymmetric shell is determined by a pair of continuously 

differentiable functions 

(9.6) 

where 

(9.7) 

(9.8) 

[O,L] x [O,27T] s (s,¢) t+ r(s,¢), b(s,¢) 

r(s,¢) = r(s)el(¢) + z(s)k 

b(s,¢) = -sin 1jJ(s)el(¢) + cos 1jJ(s)k 
Define 

(9.9) 

(9.10) 

(9.11) 

where r. _ s (s,¢) 

62 

a(s,¢) = cos ~(s)el(¢) + sin ~(s)k 

r. _ s (s,¢) = v(s)a(s,¢) + n(s)b(s,¢) 

and are partial derivatives of 

r(s,¢) with respect to sand ¢ . (See Fig. 9-2) 

k 

\\. r, (s, ¢) \. _ S 

b(s,¢)\\ 

- \ 

\ 

z (s) 

Fig. 9-2. 

Deformed Configuration 

The strain variables ( which do not all vanish in the 

reference configuration ) cons1st of 1nvariants ( namely the 

scalar product and scalar triple products ) of 

(9.12) :: • s (s, ¢ ) = va ( s , ¢ ) + nb ( s , ¢ ) 

b. _ s (s , ¢ ) = -~. a ( s , ¢ ) 
63 

:,</>(s,</»/r*(s) = [r(s)/r*(s)]e2(</» ; 

~,¢(s,</»/r*(s) = -[sin~(s)/r*(s)]e2(¢) 

These invariants depend only on 

.. (9.13) 

where 

(9.14) 

(9.15) 

(9.16) 

w(s) = (V,T,n,lJ,a) 

T (s) = r (s) /r* (s) 

lJ(s) = ~' - (~*)' 

a (s) = sin ~ (s) /r* (s) - sin ~* (s) /r* (s) 

and the prime denotes the derivative with respect to s 

REMARKS. ( 1 ) In the case of a spherical shell of radius 1, 

where ~*(s) = s r*(s) = sin s we have v = T = 1 

n = 0 , a = lJ = 0 1n the reference configuration. Thus 

v-I is a measure of longitudinal strain, T - 1 the 

meridional strain, n the shear strain and a and lJ are 

the c1rcumferential and longitudinal bending strains respectively. Note that lJ + 1 and a + 1 are almost curvatures. 

(2) In the uniform contraction and expansion of 

a spherical shell into one of different radius, v-I and 

T - 1 change but lJ and a remain zero, so that lJ and 

a isolate the effects of bending. 
64 

EQUILIBRIUM EQUATIONS. 

We suppose the shell is subjected to a uniform external 

hydrostatic pressure of constant intensity A per unit actual 

area. Accordingly, the corresponding force per unit reference 

area of the shell is 

(9.17) 

Let 

r x r 

-,s -,4> 

r x r 

-,s -,4> 

Ir* .,.x - , '" r* I - ,4> 

= A 

r x r ,j, 

-,s -,,+, 

r* 

n1 be contact force resultant per unit reference length - acting across s = constant. 

n2 be contact force resultant per unit reference length 

acting across 4> = constant. 

~1 be contact couple resultant per un~t reference length 

acting across s = constant. 

~2 be contact couple resultant per unit reference length 

acting across 4> = constant. 

The equilibrium of total force on the material of 

; 4>1 < 4> < 4>2} implies = = 

(9.18) 

+ f

S2 

[n2 (S,4>2) - n2 (s,4>dl ds + 

Sl 

rr :'s x :'4> + A r* d4>ds = a 

4> 1 Sl r* -
65 

If n1 and n2 are continuously differentiable, we can 

differentiate (9.18) with ~espect to </>2 and S2 and then 

replace </>2 and S2 with </> and s to get 

an2 

(9.19) [r*(s)~1 (s,</»],s + a; (s,</» +A:,s(S,</»x 

x:'</> (s,</» = 0 

The equilibr1um of total moments on the same shell element 

implies 

J</>2 I

S2 {S2 (9.20) (r x ~1 + z.:tdr*(s) d</> + (: x n2 + m2) - </>1 S1 S1 

</>2 Arr ds + r x (:'s x :,</»dsd</> = 0 

</>1 </>1 S1 

Thus, if ~1 and m2 are continuously differentiable, we 

obtain 

(9.21) [r x ~1r*(s)],s + [r*(s)z.:t1]'s + (r x n2 +z.:t2)'<I> + 

+ Ar x (r, _ s x r, _ ",) 'I' = 0 

substituting (9.19) into (9.21), we get 

(9.22) 

+ z.:t 2'<1>= 0 
66 

We seek axisymmetric solutions, for which we require 

(9.23) nl(s,</» = N(s)a(s,</» + H(s)b(s,</» -. 

(9.24) n2 (s,</» = T(s)e2 (</» 

(9.25) ml (s,</» = - M(s)e2 (</» 

(9.26) 

Thus 

m2 = (m2 • ala + (m2 • b)b with components regarded 

independent of </>. Then 

(9.27) 

= k x m2 

Thus (9.19) and (9.22) become 

A A 

(9.19a) {r*(s) [N(s)a(s,</»+H(s)b(s,</»]}, _ _ s - T(s)e2(</» + 

+ Ar(s)r(s,</», _ s x e2(</» = 0 

A A A 

(9.22a) [r*(s)M(s)], +r*(s) [vH(s)-nN(s)]-(k x m2) • e = O. s _ 

Now we set (m2· a)cos W(s)-(m2· b)sin W(s)= L(S)COS W(s) 
67 

to obtain 

(9.22b) [r*(s)M(S)] I+r*(S) (vH-nN) - L(S)COS 1/I(s) = 0 

SPECIALIZATION TO SPHERICAL SHELLS. 

Without loss of generalit~ we take the shell radius 

to be 1. Thus r*(s) = sin s, 1/I*(s) = s z*(s) = -cos s, 

L = ~ , and equi11brium equations (9.l9a) and (9.22b) 

become 

(9.28a) [N (s) sin s] 1 -T (s) cos 1/1 (s) - Arn -H (s) sin s1/l 1 (s) =0 , 

A A 

(9.28b) [H(s)sin s] '+T(s)sin 1/I(s)+ Arv +N(s)sin s1/l1 (s) = 0 , 

(9.28c) [M(s)sin S}I_L(S)cos 1/I(s)+(vH-nN)sin s = 0 

In order to understand the component L occuring in 

(9.22b) , it is convenient to have recourse to the principle 

of virtual work. 

Let y(s,¢), x(s,¢) £ C2 (n) with compact support, 

where n = [s 1 , b] x [ ¢ 1 , B], 0 ~ b ~ L, 0 < B < 2 ~ • The 

principle of virtual work requires that 

(9.29) 
68 

It ~s convenient to use the traditional notations. Let 

ov , on and o~ be in C2 (n) with compact supports and 

set 

(9.30) 

where 

(9.31) 

Thus, 

Also, we get 

(9.33) 

(9.34) 

(9.35) 

(9.36) 

or 

--r* 

ocr = 2JL cos ~ r* 

o~ = o~' 

Thus (9.29) becomes 
(9.37) 

69 

o = IB Jb(NCV+HCn+TCT+Mc~+rCa)r*dsd~ ~1 Sl 

This is the traditional form of the principle of virtual work 

with L as the generalized force corresponding to the strain 

variable a 

Note that in the expression (~2· ~)cos ~-(~2 • ~) 

sin ~= rcos~, we do not imply that L = m2 • a and that 

b • m2 = 0 
70 

9. REFERENCES 

J.C. Alexander and 5.5. Antman (1981): Global BehaVlor of 

Blfurcatlng Multidimensional Continua of Solutlons 

for Multlparameter Nonlinear Elgenvalue Problems. Arch. 

Rational Mech. Anal. 76, 339-354. 

5.5. Antman (1971): Existence and Nonuniqueness ofAxlsymmetrl( 

Equilibrium States of Nonlinearly Elastlc Shells. Arch. 

Rational Mech. Anal. iQ, 329-372. 

5.5. Antman & E.R. Carbone (1977): Shear and Necklng Instabilities in Nonlinear Elasticlty, J. Elastlclty, 2, 125-

151. 

S.S. ~ntman & S.E. Dunn (1980): nualltatlve Behavlor o~ 

Bnckled Nonllnearly Elastlc Z\rches, cT. Elastlcl t~" 1 0, 

2.i5-239. 

5.5. Antman & G. Rosenfeld (1978): Global Behavlor of Buckled 

States of Nonlinearly Elastlc Rods. SIAM. Rev. 20, 

513-566. 

L. Bauer, E.L. Reiss & H.B. Keller (1970): AXlsymmetrlc 

Buckling of Hollow Spheres and Heml-Spheres. Comm. Pure 

& Appl. Math. ~, 529-568. 

E. & F. Cosserat (1909): Theorle des Corps Deformables. 

Hermann. 

M.~. Crandall & P.H. Rablnowltz (1971): 

Simple Llocnvalucs. J. Punct. l\nal. 

BlfurcQtlon from 

8, 32]-340. 

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