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cl WAStITSTV IMUEEWS Pft8gAN

ION A NONLINEAR THEORY

IOFI_ ELASTIC SHELLS

W. L.. WAINWRIGHT

,U,, AST'

t FES 8 16

APPLIED rr.r-,MECHANICS-

COFFICE oF- NAVA L

RE3EARCHPROJECT-R O64-456

TECHNICAL JANUARY 1962I REPOwr NO. I8

UNIVER31TY OF CALIFORNIA, BERKELEY

OFFICE OF NAVAL. RZEARCI

Contract Nonr-222(69)

Project NR-64- 436

Technical Report No. 18

ON A NOuLINEAR THEORY OF ELASTIC SiLLS

By

W. L. Wainwright

January 1963

Series 131, Issue 18

Institute of Engineering Research

University of California

Berkeley, California

Reproduction in whole or iii part is permitted for any purpose of the

United States Government

ON A NONLINEAR THEORY OF ELASTIC SHELLS

By

W. L. Wainwright

Sumnary

This paper is concerned with a nonlinear theory of elastic shells

with small deformations whose material response is nonlinear. The develop-

ments are carried out under the Love-Kirchhoff hypothesis. General consti-

tutive equations are derived in which the geometrical properties (due to

deformation) and the material characteristics are separable. Through this

separability, it is shown how to extract constitutive equations of pre-

determined types. Particular examples are followed by a discussion of the

membrane theory.

1. INTRODUCTION

In a recent paper, Zerna [1960] dealt with a nonlinear theory of

elastic shells which has as its basis the proposal of constitutive equa-

tions which are at once geometrically linear and physically nonlinear. The

work of Zerna [1960], however, is somewhat limited in scope since, in

particular, it contains only a third order correction to the linear theory

with no indication of how a second order or higher order corrections are

to be obtained. In addition, Zerna's constitutive equations upon complete

linearization reduce to a version of those commonly referred to as Love's

first approximation which, as brought out in more recent investigations of

Naghdi [1962], do not satisfy all requisite invariance requirements.

-1-

In the present paper, through the artifice of a series expansion for

the strain energy function, we derive in Section 3 general nonlinear con-

stitutive equations for isotropic elastic shells whose mechanical behavior is

(in the sense of Zerna) geometrically linear and physically nonlinear. The

expansion of the strain energy function just mentioned is effected with the

aid of certain results deduced in the Appendix (Section 6), where an

expression for a potential is derived relating two 3 x 3 symmetric matrices.

The notation employed throughout the paper is essentially that of the paper

by Naghdi (1962] mentioned above.

Specifically, the content of the paper is as follows: Certain pre-

liminary results which are needed subsequently are given in Section 2.

As mentioned above, Section 3 contains the derivation of the general con-

stitutive equations. Here, in addition to invoking the Love-Kirchhoff

hypothesis, it is assumed that the displacements and their gradients are

small but compatible with this the constitutive equations are fully non-

linear. Also included in Section 3 is an investigation of the invariance

requirements of the general constitutive equations. In Section 4, we first

discuss a systematic procedure for determining approximate constitutive

equations of various order in a manner which results in separation (in

product form) of the purely geometrical effects (due to deformation) and

the material response of the shell. Comparison is made with the results

of Zerna [1960] and the special cases of the first and second order theories

are discussed; the first order (or linear) theory yields the FlUgge-Lur'e-

Byrne equations. In Section 5, we specialize the nonlinear constitutive

equations to those appropriate for the membrance theory of shells. We further

show that for static problems of the membrane theory, even when the con-

stitutive equations are (physically) nonlinear, the determination of dis-

placements may be reduced to the solution of a system of linear differential

equation.

In deriving the general constitutive equations, the use of power series

form of the strain energy function offers certain advantages which should

be elaborated upon. Firstly, in order to obtain the (two-dimensional)

shell constitutive equations from the appropriate strain energy function in

3-space, the power series representation permits an explicit integration

across the thickness of the shell. Secondly, through this procedure, there

results a separation between the geometrical properties (dueto deformation)

-2-

and those representing the material response so that desired geometrical

.and physical approximations may be introduced independently of each other.

Also, it should be noted that the power series representation of the strain

energy function (discussed in general in Section 6) is independent of the

present application and may be used in other nonlinear problems.

2. Preliainary results. For later developments, it is convenient to recall

certain results concerning the geometry of the shell which here will be

referred to its deformed state. We will assume at the outset that the

coordinate system is normal relative to the m1 dle surface of the shell

and hence use freely such simplifications as evolve (see, Synge and Schild

[19491, Sec. 2.6 and Naghdi (1962), Sec. 3).

Let h denote the thickness of the shell and x be the coordinate

along the normal to the middle surface. Further, with reference to

Euclidean 3-space, let ji - (6 , 43) be the base vectors of the shellh 3 h adatoeo

space (defined by its middle surface and - ; X S h ) and a those of

its middle surface. Then

Vaa (2.1)

where

"C 18 - b% (2.2)

ba is the second fundamental tensor of the surface and 8 1 is the Kro-

necker delta.

In all that follows, we assume

x < Rmin (2.3)*

For details of this restriction on the space of normal coordinatessee Naghdi [1962].

-3-

with R denoting the least radius of curvature to the middle surface. It3a

then follows immediately that the characteristic roots of x b are less

than I and hence (see Mirsky (1955] Pg. 332) " is given by the con-

vergent series

P 0

Pwhere the notation b stands for the P factors

- b •

Here, we also note that the determinant p of p is given by

1 2H x3 +Kx 3 )2p- l-2sHx K( ) ,

(2.6)2H = b K-IJbBIJ,

and that if T is an arbitrary space tensor with subtensor P , then

the corresponding surface tensor is defined through (see Naghdi (1962],

Sec. 3.3),

- , P (2.7)

which characterizes the transformation relation between space and surface

tensors.

We recall that the constitutive equations for an elastic body in the

sense of Green, i.e.,

ij - (2.8)

eij

for an isotropic material may be shown to have the form (see e.g.,

-4-

Truesdell and Toupin [1960], p. 730)

a j - o g j + $1 e + 02 a kj (2.9)

where a j and eij are, respectively, the stress and the strain tensors,

E is the strain energy and in (2.9) eAt (A - 0,1,2), are functions of

three independent invariants of eii. For the purposes of this paper,

the most convenient set of invariants is the set

Kii= i li i lie k

-ei K2 e e K 3 e' .e (2.10)

As stated previously, it is the purpose of the present paper to derive

general constitutive equations wherein such nonlinearities as exist are

purely physical and are cast into the coefficients 0A of (2.9). Accordingly,

we confine attention to linearized strain-displacement relations

2 ea = ui 1j + uj (2.11)

and in keeping with this plan set 02 - 0 in (2.9). It then follows from

(A.6) of the Appendix that E is independent of K3 so that the relevant

invariants are

1 p ua 3 1,

K2 [ UPIuI + P 13 "aIP (2.12)+ a 3 32)]

+ 4 , a+ 2(U I.)

where ui are the components of displacements and ( )i denotes co-

variant differentiation.

The analysis that follows is developed under the Love-Kirchhoff

hypothesis, although it may also be carried out so as to incorporate the

effect of transverse shear. It has been shown (Naghdi [1962]) that

-5-

under the Love-Kirchhoff hypothesis the usual displacement assumptions in

the form

- (x , x2) + X0 (x , 2)

(2.13)

u w (x, x 2)

are in fact exact. With (2.13) and the use of (2.11), we have

2 eo - 4P a Y P + X3 Kc O + 0 ( Y C + x 3 K ) * (2.14i)

"Q p( X , ap

e33 ea 3 0

and

a 0 v w ep Kaif

(2.15)

u - (w + b C ve)

where ( )II denotes coveriant derivative with respect to the first

fundsaental tensor of the surface a40.

In view of (2.15), the invariants K, and K2 take the form

(2.16)

1 .1 [ e 3 e 3 epK2 9 9P (Ac + x 5 3 + (z) z CO

where

-6-A

ep 1 7e P ep ]

Op e

Bo1 - r 7K + Ke 7+a + (Y'a K40P + K a Y'op) a I, (2.17)

ep 1 K + a epRap - t+ A K a ]

exhibit the symmetries

ep pe ep pe e, p eAaI - Ap , B -B K , % -E .-a (2.18)

3. The stress-strain relations. We have already admitted the linearized

strain-displacement relation (2.11), the displacements in the form (2.13)

which are exact under the Love-Kirchhoff hypothesis, and moreover have

required 0 2 0. On this basis, we now proceed to derive exact constitutive

equations for elastic shells which are nonlinear in the sense stated in

Section 1.

First, we recall the definition of the stress resultants IPP in the

formh

-" fh 0 f dx'2 (3,1)

iidx3

I h A77 ,

2 aX

where (2.8) has been used. Observing from (2.15) that

E1 a 5Iv 5

S I( a . + 8V ) (3.2)as PV 2 J V V P

aE a

-7-

then (3.1) may be expressed as

h hIPA. d P - f p dx 3 (3.3)

h

2 Z dx , (3.4)

2

then

(3.5)

The expression (3.5) 2 involving the stress couples 03t follows from the

same procedure used to obtain (3-5). •

We now proceed to obtain a suitable representation for the scalar

potential X in terms of the invariants K., and K2 . Because of the re-

striction $2 - 0 , we have from (A.9) that C^, A -= It then follows from

(A.15) that E has the form

1 A+ 1E F " KK +2 LK . (3.6)

Consider now

a* a(3.7)

i (x) , b (z)P=O P=O

where (2..) has been used. In the last series we replace P + 1 by P and

write

l~l P-lK, 1 5 +~ 7 (X3 )p a K (x 3 )P bz (38)a La~+

-8-

so that if we define

p O, for P - 0,M p-a P (3.9)Y a K b for P l

then (3.8) becomes

3 PK, = (X)P M , (3.10)

P 0

The right hand side of (3.10) is a converging series since the series

for Ia converges. A parallel procedure gives for K2 the expression

- 0 x) ,(.I

P=O

where

Aep for P 0,Aep ,

N 2 p. bP + ,1 P for P a I , (3.12)

AP+ CepBC + , • o

In (3.12), t, B and E are given by (2.17) and

P P l

Q=O Q-o

where it should be noted that C is symetric, i.e.,

3 - C (3.1)

-9-

By a procedure similar to that used in the foregoing, it can be

shown that the powers of the invariants KI and K2 may be expressed as

.K) (i2)A D- (3.15)

P=0 P=0

where

C = . . MN. .MQ=OR=O smo

(3. 16)

P S T-s Q-R P-QD = . N N . NA A Q=X0 R=-0 S=0

The formalism of (3.16) is intended to indicate that there are A factors

and A - 1 sumations. It also follows from (3.16) that

Pr - Q P-Q

Q-OA r-

For the product (K1 )6 (K2 ) , we write

ODZ(x)z ( r . (X3) e P ^

(K3.)6 (K2 )1 x) AA r(3.18)

P PP fl P-Q 0 -A =D mC

r0a r 0 rAr L-o A-

and in order that the representation (3.18) be applicable to the special

cases A - 0 and r w 0 , we append to (3.16) the definitions

-10-

0 0 P PC -D= 1 , C - D = 0 , for P 9 1o 0 0 0 (3.19)

P PC - D 0 , for P<0

A

P

and note in passing that A is independent of x3

A rWith the aid of (3.18) and (2.6), (3.4) may now be written as

h 02 X3 3 )2 P

J- 2H + ] [ A. (x3) Afh r +1

(3.20)

1 3P

+~ A Z (x) A ]dA +1 A +1 0

Introducing the notation

T1 01)Q ( -) (3.21)

and observing that

(Q) - 0 for Q even,(3.22)

h( 1)Q-1

(Q) - K ( 21 , for Q odd,

then the integration of (3.20) yields

x - )' {I [ q (P + 1) - 21 (P+2) + K Tj (P + 3) ]P -o (3.23)

F P L P

r +1A rl A + 1 +1 0

-11-

In order to express the stress and couple resultants I and0 0

in terms of the invariants C and , we need to obtain certain additional

results. To this end, from the definitions (3.9) we obtain

0 0

PV PV(3.214.)

P P P P- 1M bLV 6 bPV (P a).

7 V V

It follows from (3.12) and (2.17) that

0 1 e ( e+7 / (3.25)

and hence,

0 02 N (pv+ as - o (3.26)C7 4V 2 v

It is convenient to introduce the quantity

X 0 a 7av bV + Ka (3.27)

in terms of which

1 1 p p .N - ( O +*) ) X (3.28)

so that

1.- bV ( 7 P + ,P)+( VP +xV) ]\,7pV 2 p.LV (3.29)

16N 1 VVIA

-12-

Similarly, for P k 2, we have

P P-l 7a,+7 a) +N - V 2 b a ( + ,

+ (P 1) x a b V + (3.30)

+Po2 °6 :"~~2q -o

so that for P 2,P

) VPP P-Q -12

+ ax bp Pp b 9 ]

Q=O

(3.,i)Ps1 (ta+ f) ) "22

PWe now turn to A in (1.i8) and proceed to establish tlu following

Aridentities:

P Z P-Z

^ I aM = AA.^r.

a r (3.31)

P Z P-ZNA I N =r AAr h r-i

To prove the first of (5.32), we first note that

Q - .PP

c = x .(3.32)

1

-13-

A

Hence,

P Z P -Z~C / M - 5z - c (3-34)

Next, from (3.17)

P P- Q P-Qc .- c M (3.35)2 Q .Q-O

so that

/ z ( P-Q Q P-Q6 c C 21M -=Q M + C z

2 QO1Q 0

(3.36)P-Z P-z

"2 M 2 C1

and by induction we arrive at

P Z P-Z6C / AM a C (3.37)A A-1

Now, since by (3.18)

P P Q P-QA c o (3.,8)

A r Q .' A r

P Qand since P as given by (3-16) is dependent only on the N's, we have in

view of (3.37)

P z Q-z P-QSA / oH M, A C D (3.39)A r X A- 1 r

If we replace Q by Q + Z, (3.39) becomes

-14-

P z P-1 Q P-Z-QAr / 6M - A C D (3.0)

A-i r

and then, through the definitions (3.19), the first of (3.32) follows.

The second of (3.32) may be established in a similar manner where in

particular it is convenient to use (3,18)3,

Explicit forms for the constitutive equations can notv be obtained

from (3.5) and (3.23) and read

00 PP- z-[I(P+l) 2Hiq(P+2) + K(P+3)] L A M 6 y

P - 0 Z-0 (3.41)

A Nr+-l F ArN-1'rI+ P. Ar A&r 6 J '

T' j (P+1) - 2HTI(P+ 2) + KTI(P + 3)] EA'&P0 Z0 P- z P-zz

Z

+ -1 r A-I+l 7K' Pa'+ Fr Ar '"

The constitutive relations (3.14) and (3.142) may be put in more explicit

forms by further substitution from results of the type (3.24), (3.26),

etc., but we delay this until the next section, where special cases are

considered.

Before closing this section, we examine the invariance properties of

the above constitutive equations.+ It is evident that the equations (3.41)

and (3.42) are tensorially invariant. Furthermore, it is easily verified

that they are also consistent with the virtual work theorem for shells.

Next, it is necessary to show that the sixth equation of equilibrium,

+ For an account of invariance requirements in shell theory, seeNaghdi (1962].

-15-

namely

f (? .- ba MVP) - 0, (343)

or equivalently

is identically satisfied. To this end, observing that X may be regardedz

as a function of A and N , by (3.44) we have

00 Z Zz zIEx/3M W- + /3N) 3

Z -o

zZ ZV axM.... _ + ( 3x/ N) a3 N~

(3.45)

Z Z Z

Z Z 0 ~ ~ i ( 6 aV K )VZ

Z

+ aX/ N N N

Z

Consider now the coefficient of a X/ a M in (3.45) in conjunction with

equations (3.214). For Z ?. 1, it is identically zero while for Z - 0, it

is symetric in a and 1 and hence vanishes. Next, consider the coefficientZ

of X / ;5 N. A little manipulation of (3.31) will show that

Z Z Z-la1 N N I---- "2 b a X 150O + b )? (3.46)

ba AV l Z a ~ p a

an expression symetric in a and A. Thus, for Z k 2, in view of the

presence of e in (3.45), this coefficient also vanishes. The vanishing

of this coefficient for Z - 0. 1, follows with the aid of equations (3.26)

-16-

and (3.29). The final invariance requirement which should be examined is

invariance under rigid rotation. It may be shown that the value of eachP P

of the invariants 14 and N is unchanged by infinitesimal rigid rotations and

thus X retains its value under such a transformation. The calculations

are somewhat lengthy and hence are not included here.

4. Approximate constitutive equations From a practical point of view, it

is of interest to consider constitutive equations which are approximations to

those given by (3.41) and (3.42). The procedure to be used in obtaining the

approximate equations will be one of simply terminating the series after a

finite number of terms and we proceed to do this in a systematic manner.

First, we recall that there are two types of suations involved in

(3.41) and (3.42), summation over the letter P and summation (implied)

over the indices A and r. These two summations are effected independently

and it is this fact which leads to the independent consideration of geo-

metric properties and material response. In order to systematize our dis-1

cussion of this matter, we will use the notation 1 to represent terms ofa R C

the type b and y to represent terms of the type Y Terms of the type

KO' and " will be regarded as Yn If we return to the definitions of

the various elements which are the constituents of (3.41) and (3.42), we1

can easily see to what power y and - occur in them. These results areRsummarized in Table I below:

TABLE I

Element Power to which PowTr to whichy occurs occurs

P14 PPN 2 P

PC A PA

PD 2r Pr

PA A+2r P

A17

-17-

1We observe from this table that the powers to which y and - occur in

Rthe various elements are independent in the sense that the former is determined

byA and r while the latter is determined by P.

The quantities L and F in (3.41) and (3.42) are material coefficients,A Ar

and the power to which y occurs in terms involving these coefficients is

determined solely by the value of the index of those coefficients. This

fact is seen by applying the information of Table I to (3.41) and (3.42)

and the result is summarized in Table II which, together with Table I,1

reveals the manner in which the y and - terms occur in the constitutiveR

equations.

TABLE II

Coefficient of Power to which7 occurs

L A

F A+2r +1

With the information gleaned from the above two tables, the pro-

cedure for determining various approximate constitutive equations is nowh

clear. First, we select the highest power to which we wish - to occur inR

the expansion and we then sum over P in such a manner as to obtain all

terms up to and including that power. We emphasize, however, that because

of the presence of 71 (P + 3) in (3.41) and (3.42), we cannot choose theh h

highest power of - to be less than 2. After the choice on h has beenR R

made, we are free to select the power to which 7 shall appear in the con-

stitutive equations. The term "order" will be used in this connection,

i e., equations linear in y will be called first order, those containing

both linear terms and terms of second degree in 7 will be called secondh

order, etc. It is clear that once the choices on i and 7 have been made,the determination of the specific equations is a routine matter.

* h

The sequence of choices used here in selecting the powers of - and 7may be interchanged. The one adopted here appears to be convinient.

-18-

In the examples which follow, we shall always choose the highesth

power to which - occurs to be 2. Consequently, we will first expand theR

relations (3.41) and (3.42) imposing this restriction and then, by making

use of relations of the type (3.46), obtain the desired expressions. Thus

expansion of (3.41) in the manner Just described gives

0 0 0 0

- ) o A W-FY + r '' T A- r+l Y Ar -"::

2 0 1 1 2 2

+ 1( A 06 M+ A 0 a A 0 6 aY

1 0 0 1 0 0.2H ( A ° - + A N A -- M

2 0 1 0 2+ r ; A M M.,. + A M (4.1)-t " .,,.a &- r+l P '--1 F Iq_+I 7 7 +A-laF+l 7

0 0 1 0 0-2 ( M-+ A M ) + K A 6 M

2 o 1 1 0 2A+ A 66YN.,r A r P

- 2H( A +A N ) + K r

Ar a A~ r 07O A r Y1}a

The Cayley-Hamilton theorem when applied to ba with the use of (3.24)

may be expressed as

2 1 0

, + X - - 0 (4.2)

and if we write C for the adjoint matrix of ba, we have

-19-

211 1

- 2H X -b

Further, combination of (4.3), (3.26), (3.29) and (3.31) yields

2 1 0

-'Y 2 H N+ K6

= .14 ba Xlv +bO Pv -j (Vvo+ OV)]v V V

and

1 0

-.Y 2H '_

Xo + . Y -v + 7jv)] (4.5)

Y v (b+ b5v f v) " ( 13v + 7Vr ) + (Kao + 1 ) ]

Now, with the use of (4.2) to (4.5) and with some simplification and

combination (4.1) reduces to

ho+ 0 a__0 P +Pfl° q a - A I +rr IF T 1 A-i~r+i1a l + &Ar 2 Y# I m

r a, I r(j.P+ 72)1

1 r + i Fa[ -f r+l "- '+z(4.6)

+ Far [ kr ( "0 + +) ;A(,ao + xa) - tav(yv + Yov)

+ A] ((bC X'V + bi' %QV) - g (%Vfl + OV)

-20-

and, similarly, the corresponding expression for 0 is

je, h 3 1] 1 •a 0 P

A i 0

+ F [FIA " - A (4.7)r + I Ar L-1A+l A-i r+l

+ F rP A ( 13 + Y1) + A) - V (7 + )

Equations (4.6) and (4.7) are of second degree in 1 but of arbitrary

R arbiryorder in y. Such subsequent examples as are to be given will follow directly

from these equations after a choice has been made on the order of 7. As

a first example, let us obtain the first order or linear constitutive equa-

tions. For this case, Table II instructs us to retain L, and F00 and to

set all other LA, FA r equal to zero. (In order that the state of zero

stress and zero strain coincide, we also set 10 - 0.) In fact, if we set

L, V E 00 E (4.8)"1-V2 ' +O V I -- "

we then obtain the FlUgge-Lur 'e-Byrne constitutive equations. For future

reference, we record representative examples of these equations, written

in lines of curvature coordinates and in physical components. Thus,

(- Ll+FO0 )( 11 ) + L7(22) + A (- )(K(11 ) - y(L3 '

h F fY h - 1 ) (K I(12) - 7 FOO (12)+Y(23))+ 2 " (21) i ,( j

(4.9)

2 f(L+F 0 0 ) K( 1 1 ) + LlK(2 -) + (L1+F0)(1 - ) 7(R, 9}M( h (K( +

M(12 ) 2 R( +K((3))+ (j 7 )

where the constants L, and F00 are given by (4.8).

We now return to equations (4.6) and (4.7) and proceed to obtain the

-21-

constitutive equations for the second order theory, For this purpose,

it is convenient to introduce the notation

OPP ktpi , (4.10)

where the subscript k in each case designates that resultant which arises

from (4.6) and (4.7) from exactly those terms which are of dexree k in y.

Thus, 1 a nd 3.0 would refer to the right hand sides of the FlUgge-

Lur'e-Byrne equations (4.9). If the theory sought is Qth order, then we

have

OP te ke (4.11)k-I k-1

while for a completely general theory,

OP eo ' Z -Z k (4.12)k-i k-l

With this understanding, we need only determine 2 and 4 to complete

the description of the second order theory since 10 and 3.0 are given

by (4.9). The representative results are as follows:

+ i{L 2 (7(., 2 ) )2+ ( ) "(.)+ 2) + 1('(12) +( )

+2Nl0 ( "%(x1) 7(u) + 2 (2) (7(z2) + 7(2)) (4.11)

1 2(Y(?.) + Y(22)) + (12) + X(21) X2

+ R, - ) .( (27( + 7(22)) + (X(LI) + (22) ) ] }

-22-

2 (12) lo { ("(1) + (22)) ((12) + (21)) (4i.14i)

12~ 3(.h(1) +X'(2 2 ) )('( 12 ) +'(21) )+G(1) + Y(22)) '(23.) (IR2 - R~

and

h3 rIL L(v( 1 )+ 7(2)) 2(1.(,) + "(22))+ .1 (Y(1) + Y(22))

1 [("(I(11)y(l) + X(22)'(22))+ ('(12) + X( 2 1)) ('(12) + (21) )

(4.15)R 2 ( () 2 (22)) + 7 (Y(12) + Y(213

+ ( 1( + "(22)) 7(11) + (7(1) + 7'( )) (K (11) + 7(13)(-

2M(12 ) Flo ( 11 ) + (22)) (Y(12) + (21))

(4. 16)

+ (7(3I) + Y(22)) [ '(32) +(21) +(23) R R "

The foregoing expressions simplify somewhat for special configurations. For

example, in the case of a flat plate, the representative equations reduce to

2,I() m h fL 2 (7(t 1 )+ 7(22)2+ 110 1 2 ( 11 )+ 7(22))

+4 (7(12) + (21)) + 7(1) (7(11) + 7(22) ) ] f(-1.7)

+h 2(( +K ) + 2 2 +K+2 f ,L4 +22) l (22))

+ ((32) +K (21)) + K(I ) (K( 1) + (22))]

-23-

2 " ( 1 2 ) - 9 l 71{ ( Y( 1) + ( 2 2 ) ) ( ( 1 2 ) + ( 2 1) ) ( 1 8 )

+ 1- (1(.,.,) + IC(22 ) ) (:(.2) + (2.))l= 3

. {2L,(V(.,,+ 7(22))(K(. 1 )+ K(,)) + Fj[3 K(1 1 ) ( )+ K(.)7(22)

(4.19)

+ (K(.12 ) +K (2.)) (Y(12) + ,(21)) + (,.) K(22) + ,(22) K(22 )] },

2- p(22)) (Y(12)+

(4.20)

+ (Y(1, + Y(22)) (K(3) + K(2))}

Our purpose in recording (4,.17) to (4.20) is to observe the coupling, i.e.,

the dependence of the stress resultants on the strains K. as well as the

dependence of the stress couples on the strains Y... In contrast, it may

be recalled that when (4.9) is specialized to the case of a flat plate, no

such coupling is present.

It is of interest to compare the constitutive equations (3.41) and

(3.142) upon specialization with those obtained by Zerna (1960] whose con-

stitutive equations contain only first and third degree terms. If in

the expressions for 0 and 0 , we neglect, as Zerna does, terms involving(If with respect to the remaining ones as well as terms of the type Z

then (14.6) and (14.7) become

.h ,AO +0 -- 0r+ 0~ + i A a r _

+ F& r ,Ar '

Although Zerna [1960] dous not explicitly invoke the Love-Kirchhoffhypothesis, he neglects the effect of 7a3 and sets 033 - 0 but retainsthe effect of Y3.

-24-

and

12 0 nAO +r +-" 'tr 6 -1Ar +1 a(4.22)

1 _ 0o L+ FAr Ar Y + FAr A~r a

where we have written

-a . .1 ( a + 7sp, a = Kt3 + K a

Now if, as in Zerna's case, we wish to retain only the first and third degree

terms, it follows from Table II that the only nonvanishing coefficients are

L , .L 3 I FO0 , F20 , and F0l. Here expansion of (4.21) gives

IP h{ L j(_0) aaP + F 0 - ap]

+ ~ ~ _ EL(pe + 2 ~)~ )] ap (4.24)

+ [ F2 0 (_11 ) 2 + 1 F0 1 (~ VP) Y xY

which is of the same form as the corresponding expression given by Zerna.

It should be noted, however, that while the coefficients of (Y ae)(3Y P ) aale 2 -ap X P

and (' ) , are identical in (4.24), they are not the same in the

corresponding expression given by Zerna [1960]. A similar conclusion can

be arrived at upon the comparison of (4.22), when only first and third degree

terms are retained, with Zerna's expression for the stress couples after a

misprint in Zerna's equation (45) has been corrected.

5. Membrane theory. In this section, we specialize the previous results to

those appropriate for the membrance theory of shells. We recall that

for equilibrium problems of the membrane theory, where the shear stress

resultants and the stress couples are absent, the state of stress is stati-

cally determinate and the stress resultants PPP are obtained from the

equations

-25-

01 11, + p13 "

bap 1e1 + P 0, (5.1)

C - 0.

For the subsequent determination of the displacements, we turn to

the constitutive equations (4.6) and, consistent with the membrane theory,

neglect the terms with the factor (h)2 Thus, for the membrane theory,

instead of (4.6), we have0 0

II h (LA AAo + F+- , FAr a-ir+ 1 ) s

(5.2)

+ Far ^r 2 7 +'J

Because of the symmetry of the right-hand side of (5.2), it is seen

that (5.1)3 is identically satisfied. Also, in view of 3.18)2, all0 0Age appearing in (5.2) may be written in terms of N end N alone so that

I h i~a+ ..Ar +i r~r (0),&- 1 (O)r +t 1 a

+ 'A r , l)}0 (O)r I (ya Oa (5,3

0 0

Since N and N are functions of y , it is clear that the combination

of (5.3) and (2.11) constitutes a system of nonlinear differential

equations in the displacements. Since I as solutions of (5.1) can be0 0

regarded as known, it is convenient to express N and N in terms of 09.

or more specifically, in terms of the invariant& of ic0. Let

a P2a 1JPPN~a(5.4&)

and introduce further the notation

0 0F (M)A (N)r (5.5)

-26-

Contraction on (5.3) gives

P3 ( 2 + A (M)-l(')r+l 2 + } (5.6)

or

L + r + 1 r (()5 7+

- '1. 0

in terms of which becomes

- h f - ) f + ( a 3 + ) . (5.8)k2 h 2Yj

By (5.4) and (5.8), we arrive at the additional relation

'2 2l [ N - () (59)

0 0Equations (5.6) and (5.9) give M and N in terms of P, and P2 In

0the linear case,0 the solution for M is straightforward, and since in this

case FoF , M is determined by0

h (2L1 + Foo) M- P. . le (5.10)

It then follows from (5.3), after the values of FO0 and L. are supplied

from (4.8), that

1( 1 V 1 )a-] (5.11)

which is the usual result of the linear theory.

In the second order theory,

0* Foo + Flo M (5.12)

so that by (5.6) and (5.9), Pi and P2 reduce to

-27-

)2 0+O 0 0 O0P, = h f2 [L ' + L2 (M)2 + NO w] + (FO0 + lOM) N (5.13)

0 0 0P2 h2 (FO + Flo M )2 [ 4N _ (M)2 ] (5.14)

0 0Although N may be easily eliminated between these equations, giving 0in terms of P 0and P2 , the resulting equation .s of the 4t h degree in M.However, once M is known in terms of P1 and P2, (5.3) yields a system oflinear differential equations for the displacements. In fact, even 1

the general case, so long as (5.6) and (5.9) can be solved forM. and N interms of P1 and P2 , the differential equations for the displacements are

linear.

6. Appendix: Power series forms for the elastic potential.

Equations (2.9) include the most general isotropic stress-strain

relations of elasticity, but since the *A being arbitrary functions ofthe invariants of eij depend also on x3, it is desirable to express E in

(2.8) in a form which will readily lend itself to integration across the

thickness of the shell. The developments here, however, are entirely

general in that they are applicable whenever two symmetric second order

tensors are related through a potential.

Let the symmetric tensors under discussion be Q end AiJ, where the

first is an analytic isotropic function of the second. We further assume

that a potential r (Aij) exists* such that

Qi a 6 (A.l)

Let

K -- Ai 1 Ki2 AJ2 J i'

K3 A 1 k Ai (A.2)K 3 Ak Aj i ,

The necessary and sufficient conditions for the existence of a potential rmay be found in Truesdell's (1952] paper. Truesdell's conditions, however,are given in terms of a set of invariants which are different from thoseutilized in this paper.

-28-

be selected as the three independent invariants, then

r - r (Ki, K2, K3 ). (A.3)

Our choice of the invariants (A.2) is motivated by the fact that their

derivatives have the form

__K_ i A.&.2. i __k i (A.4)-bt, k

and hence (A.1) may be written as

Qij . C 18ij + C2 AiJ + C3 Ak j , (A5)

where the coefficients

6 rCA 3 K , (A - 1,2, 3) (A.6)

are power series in the invariants KA . Since the existence of a poten-

tial r has been assumed, it follows that

S rA -(A.7)AKBdKA

or equivalently

CA . c (A.8)

which are in fact the necessary and sufficient conditions for (A.1) in

terms of K

Since the CA may be written as power series in terms of K., we write

C3 in the formC3 3 C KA Kr KA, (A.9)

A r 2

-29-

where ,, r, A - 1, 2, 3, . . and summation is intended even in those terms

(which will arise later) in which an index appears more than twice. With

Cs given by (A.9), we proceed to determine C1 and C2 with the use of (A.8).

Thus

C2 c- r CArA K r KA (A.10)

K3 K

A2

so that

C2 A_ K K2 K3 + F K1 , (A.11)A + I cAr^ A - + + ,r r

where the second term on the right-hand side represents the arbitrary

function of integration. Similarly, with the two remaining conditions of

(A.8) and with (A.9) and (A.11), we obtain

C,- A K'l K r K A+l1A + 1 ArA 1 2 3

(A. 12).AF K a-i r+l A

r + A ar2 +L,

Having determined CA , we compute r from (A.6) and obtain

r' c K,- C K2 rFK A+1A+1 A KA KA K +

(A.13)r + 1 arK 2 A + I 1 +

where the arbitrary constant of integration has been set equal to zero

without loss in generality. The choice of the set of invariants (A.2)

and the starting point (A.9) was motivated by the desire to depress the

coefficient C3. However, as should be apparent, the procedure is the

same for any choice of independent invariants such as

I = Ai * l is Ai An,i 11 = : in A n

2 jn(A.14)

1 8im p A An Ak3l= j nk i m p Ai1

-30-

If, instead of (Ao2), (A.14) is taken as independent, then different

(although equivalent) integrability conditions will result (Truesdell

[1952, p. 133]). In this case, the potential has the same form as (A.13)

except with different coefficients CAFA F~., and LA while the co-

efficients CA of (A.5) are [compare with (A.11) and (A.12)]

C, A IF 11 A + ..A... F A-l rf . A+l1iEA r A + 1 6r- A-C1 " CA II A +1 CAFA III I

+ r I A+ 1 rJ- I A+l A 1 +1 1 rAl_ A A + rI ^+ A

+ I A + Ar I

+ A - Ia-l11r+1

(A.15)

C ArI r - A 1 r-l 1 1 A+lC2 - - rA I II A + 1 - EA r A

A r

- CArA ~ II

AC1KNWQILEGDKNT

The author wishes to thank Professor P. M. Naghdi for many en-

lightening discussions and helpful criticisms during the preparation of

this paper.

-31-A

References

Miraky, L., 1955, An introduction to linear algeabra, Oxford at theClarendon Press.

Naghdi, P. K., 1962, "Foundations of elastic shell theory", Tech.Rapt. No. 15, Contract Nonr 222(69), University of California,Berkeley (January 1962); to appear in Progress in SolidMechanics.

Synge, J. L. and Schild, A., 194,9, Tensor calculus, University ofToronto Press.

Truesdell, C., 1952, "The mechanical foundations of elasticity andfluid dynamics," J. Rat'l Mech. and Analys. 1, l25-30O.

Truesdell, C. and Toupin, R. A., 1960, "the classical field theories",* Iandbuch der physik I11/1, 222-793.

Zern&, W., 1960,"Uber eine Nichtlineare Allgemeine Theorie der* Schalen' Proc. IUTAM Symposium on the Theory of Elastic Shells

(Delft 1959), North-Holland Publishing Co., 3-42.

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