An approximate shell theory for unrestricted elastic deformations

hr.

J. So/ids _~rucrures.

1967. Vol.

3. pp. 905 to 925. Pergamon

Press Ltd. Printed

in Great

Bntain

AN APPROXIMATE SHELL THEORY FOR UNRESTRICTED ELASTIC DEFORMATIONS MORRIS

STERN

University of Texas, Austin Abstract-An approximate shell theory is formulated within the framework of the general theory of finite elasticity for unrestricted deformation. A deformation field is constructed throughout the shell based on the solution of the corresponding membrane problem. An additional deformation is then superposed on this “membrane state” and the resulting equations of motion for the final state linearized in the additional displacement. These equations are then “averaged” through the thickness of the shell to yield an approximate shell theory. Details are carried through for the case in which the additional deformation is itself linearized in the thickness variable so that normals to the middle surface remain straight and uniformly extended, but not necessarily normal. The resulting theory is then applied to the problem of a uniformly twisted, extended and inflated cylindrical shell.

1. INTRODUCTION A FULLY general non-linear bending theory for elastic shells within the framework of three dimensional large elastic deformation theory is not presently available. The main reason for this appears to be the absence of fully general constitutive equations within this framework relating deformation measures to stress measures for the shell. General and exact treatments of the deformation and equilibrium of shells do exist, for example those found in [l, 21, and more recently [3], where within the framework adopted (Cosserat surfaces) general constitutive equations are furnished. However their relation to the general constitutive equations of large elastic deformation theory has not as yet been fully explored except for some examples (i.e. membrane theory, Kirchhoff hypothesis). Other significant partial results also exist for non-linear material behavior. In particular, Wainwright [4] has considered such shells suffering infinitesimal deformation, and Naghdi and Nordgren [5] have constructed a theory under the Kirchhoff hypothesis. The general theory of elastic membranes (from a three dimensional point of view) may be found in [6], and Corneliussen and Shield [7] considered the addition of an infinitesimal deformation to an already finitely deformed membrane but still without bending and transverse stress effects. In the present paper a “first order” bending theory is formulated for unrestricted deformation and non-linear elastic behavior of thin shells. It is based on the general nonlinear theory of elastic membranes and the linear theory of small deformations superposed on an existing finite deformation. The development is concerned with the interior problem only ; the question of appropriate boundary conditions will be considered in a later paper. For the most part general tensor notation is used with the added convention that the range of Latin indices is 1,2, 3, while that of Greek indices is 1,2. We use freely well established results of finite elasticity and shell theory ; detailed derivations of results of the former and of membrane theory may be found in [6] and [8], of the latter in [2] from which much of the notation in the present paper is taken. 905

906

MORRIS STERN

2. BASIC EQUATIONS

OF FINITE ELASTICITY

We consider here only homogeneous isotropic elastic bodies. Take Gi and gi to be the covariant base vectors in the reference (natural) and current configurations respectively of the body referred to the convected curvilinear coordinates xk. The components of the respective metric tensors are thus G, = G, . Gj

gij = gi ’ gj

and Gij =:

COf Gij

g ij =

coigij

G G’ =

(2.1)

G’jG.

gi

=

gijgj

3

with G = det /IGij[l

g = det~lgijll.

Since the material is isotropic and elastic there is a stored energy function of the form w = WU, ,I,, 13) where I I = g,,G”” The contravariant

I, = I,g”“G,,

I, = g/G.

(2.2)

components of the stress tensor are then given by " = @G" + Y g~~(Gm#Grs- GmrGns)+ Pg’” (r

(2.3)

where (2.4) If the material is incompressible as well as elastic then W becomes a function of only I, and I,. In this case P becomes a new independent variable and we add the constraint condition I, 5%1. On the surface element whose outward normal is n = ngk we have the stress vector a(n) = d(n)g, = amkGg,.

(2.5)

In particular, on an exposed surface of the body ifs = skgkrepresents the applied surface traction there we must have a(n)=

s

or

dnkn, = 8.

(2.6)

As a consequence of the balance of momentum the stress tensor satisfies the equations of motion arslr + pP

= pf

(2.7)

where the stroke denotes covariant differentiation with respect to the metric gij, p = I;*pe is the current mass density of the material, LF = F”g, is the extrinsic body force density and f = f”g, the acceleration of the material particles.

An approximate shell theory for unrestricted elastic deformations

907

An additional deformation may now be superposed on the existing one by adding to the motion of each particle an additional displacement EU,where E is an order parameter eventually to be set equal to unity. If the additional displa~ment is presumed to be small we linearize the perturbed state equations in the added displacement Thus if $I* represents a quantity in the perturbed state whose value is I/Jin the current state, we write 9% = t/!-t&t/Y-l-O(E) and set E = 1. Then we compute

g

81 =

W.i

‘0 =

-gimginghn

&j=

&*gj+gf.g>

g‘i = g’ijgj _+gi+;

(2.8)

g’ = c!?%“Tnn Furthermore

(2.9)

and

(2.10)

where

and therefore cr‘r9

z

&?‘G” + ‘$f’g,,(G”G”

- Gm,“) -I- Pg*’

+ [Y(G”“Grs- G’“‘G”‘)- PrgQ;,.

(2.12)

For an incompressible material we note that CB,54 and d do not OCCUT whife P’ is a new independent variable and we have the addition& constraint condition I; ES0. The equations of motion in the perturbed stale are @*y3 +

p*.gw = py*s

(2.13)

where the covalent differen~ia~on is with respect to the metric sj, and the c~~~n~ of the vectors ** = P@“g: and fc =feSg: are referred to the perturbed base vectors. One

908

MORRIS

STERN

may now subtract equation (2.7) and hnearize to get (r”“,l+o”“iktrt’J,O’kjk;f’+P[(l

+p’/p)9*“-P”]

= p[(l +p’/p)f*“-f”]

(2.14)

where %,,’ = ~~pq~~6~,“+~~~“.~-~j,,,,-2~,k,~~;~l

= ~~RW[~;:;tr*,n+Rbnlm-9~n~al

(2.15)

are the additions to the Christoffel symbols. Finally since mass elements are conserved we observe that p’/p = -&/I,.

3. BASIC EQUATIONS

(2.16)

OF ELASTIC MEMBRANE THEORY

Let xa be convected coordinates in the membrane surface with a, the corresponding base vectors and aa6 the surface metric coefficients in the current state. In the reference state we denote the base vectors and metric coefficients by A, and A,, respectively. The membrane surface may be thought of as representing the middle surface of a thin shell so that we introduce a third coordinate x3 whose corresponding base vector in the current state is a unit vector a3 in the direction a, x a2. We observe that 1x3/ (: h/2 where tt is the current thickness of the shell. In the reference state we construct the unit vector A3 in the direction Al x A, and assume that the base vector corresponding to x3 in the reference state is ?.- ‘A, where i = ;I(xa) may be thought of as a uniform extension ratio through the thickness of the shell. We use the notation “1,6to denote that the quantity I,$is evaluated on the membrane surface x3 = 0. Thus we find

(3.1)

The Christoffel symbols evaluated on the membrane surface are found to be

where IY:, denotes the Christoffel symbol associated with the membrane surface metric aep, and

with

b,, = h,, = a,. a,,#

(3.3)

An approximate shell theory for unrestricted elastic deformations

the coe~cients of the second fundamental conf$uration. It now follows that

909

form for the membrane surface in the current

‘I, = A’a/A

(3.4)

so that

Og”B= “@,A”@ f “~[a,,(AP”A@ - AP”A”B)+ /2*A”fi]f ‘pa”~ oa33

ooa3

=

A*[“@+

-

0

“Ya,AuB] + “P

(3.5)

.

Next we take ca33 = 0 based on the premise that the membrane stresses in a thin shell are much more significant than the transverse normal stress. This furnishes an equation for the determination of A2 if the material is compressible, or for “P if the material is incompressible (A* = A/a in this case is determined from the condition ‘Z, = 1). Now let us restrict attention to equilibrium states so that f vanishes. We write P= @a, for the total extrinsic force measured per unit area of the current membrane surface. Then the equilibrium equations for the membrane may be written as? (3.6) where the double stroke denotes covariant differentiation with respect to the membrane surface metric u,~_ If the deformation pattern defined by membrane theory is extended through the shell thickness weobtain what might becalled the”membranestate”. Thismayalso becharacterized by the statement that the middle surface and its normals are preserved under the deformation, the material being uniformly extended along the normals. Then we can write for the membrane state configuration g, = &aP

g3 = a3

(3.7)

where (3.8) so that

(3.9) where p = Ip$ = 1-x3b;+(x3)‘1b~l

(3.10) t These equations follow from the development leading to equation (4.1) of the next section upon dropping all dependence on the thickness variable x3, or see for example [6], chapter 4.

910

MORRISSTERN

A similar treatment for the reference state yields G, = M,PA, + [email protected] ’ ),,A,

G3 = ;l-‘A,

(3.11)

where (3.12) and Bog =

B,, = ;i-‘A,.

A,,,.

(3.13)

Then setting M = IM,a/ = 1 -x~.B=+(x~)~~B@J a a ii;

cof M8 z--.-.-s M

(3.14)

we find

GaB = M,PM;Ap, t-(x3)2(2- *)&- “),@ G33 = 3,-2 -I

Ga3 = x3/I- “(/I- I),.

G = J-‘M2A

(3.15)

-1

G”@ -_ Mombasa (r

Ga3 = -x3&l-

i),@’

G33 = R2[1+(x~)~G@(~- ‘).,(A- ‘),p]-

On the basis of this deformation pattern and the constitutive equations for the material we may now compute the stresses throughout the shell. If one would wish to maintain the membrane state in equilibrium he would of course have to supply body forces in accordance with equation (2.7), namely pst” = -a*‘,,.

4. APPROXIMATE

BENDING

(3.16)

THEORY

Let us now consider a class of problems for which the loading consists of a prescribed body force density LF* and tractions s- and s* appfied to the inner (i.e. x3 < 0) and outer (x3 > 0) surfaces of the shell respectively. The shell is either in an equilibrium configuration or is executing small motion in the neighborhood of an equilibrium configuration. In the latter case we may wish to allow the loading to be time dependent in which event the actual loading is replaced with some time independent average loading so that an equilibrium configuration may be defined in whose neighborhood the small motion takes place. We denote by x** coordinates in the shell middle surface and x*~ is distance measured normal to the middle surface. In general an asterisk on a symbol denotes that the value of the quantity represented is to be based on the actual value in the shell referred to the x*coordinates. Thus with uSrS the components of the stress tensor throughout the shell we note that the equations of motion are simply (2.i3) The usual averaging procedure across

An approximate

shell theory

for unrestricted

elastic deformations

911

the shell thickness (see [2], Section 5 for example?) then yields the following set ofequations Nf’“B” - ,;“Q*B

+ /*a + p*’

Q;:a+b:pN*‘~+1*3+p*3 Mfiti - Q*‘+m*“+

where

s s

r*a

=

lW

=

F*3

=

~*a

:

(4.1)

h*/i

N*b

=

-

p*p;=g*B~

dX*3

he,'2

he/2

MN+

=

X*3@t)$a,+@P

dy43

-h*/2

h*/2

Q*@

=

~5.fW3 s

P

*n =

&*3

- h*/2

s s s s s h*/2

_h’,2 p*/~,8*~p*cF*~ dx*3

h’l2

P

*3

=

p*p*9*3

dX*3

(4.2)

- h*/2

he/2

r*=

=

~*~p*pp*ap*y*b

dX*3

-h*/2

h’/2

F*”

=

p*@‘p*f

*fi

dx*3

-h*/2

hei2

Fe3

=

s

j~*p*f

*3

- h*/2

dx*3

h’/2

G*=

=

x*3p’*@ap*f

*19 dX*3

-h*/2

with l* = I*k a: and m* = m*‘a,* the resultant force and couple load measured per unit area of middle surface and due to the surface tractions s+ and s-, i.e.

where n* is the unit outward normal to the appropriate shell surface. The corresponding interior membrane problem is obtained by taking the membrane loading to be precisely the total extrinsic load per unit area of the membrane surface based on the geometry of the membrane state, i.e.

P-" x'=h'2 _h,2p&~~*Pdx3+ __ n. a3 xZ=-h/2 1 h/2 s [ 1 hi2

9”

=

s

93

=

[

p~c%-*~

Ps. a3 x3=W dx3 + __ n.

-hi2

t

Although this derivation hold also in the more general

is valid only for constant case.

thickness

(4.3)

a3 x3= -h/2

shells, it is easily seen that the resulting equations

MORRIS STERN

912

where the other variables appearing are based on the resulting membrane state configuration. Furthermore, we suppose that the loading and support conditions for the shell may be phrased so that the membrane equilibrium problem has a solution. We note, as a consequence of (3X9, that (!Pafl”),,/J = -9”

hb&Y@ =

_g3

(4.4)

and in the associated membrane state configuration we observe that a@5= ocTa@ + 0(x3) a3

c7

.@ 33 =

(4.5)

0(x3)

We next superpose on the membrane state the additional the shell from the membrane state configuration to the actual without further reference linearize all subsequent expressions additional displacement. Let us therefore write this additional form

displacements which carry current configuration, and in the components of this displacement vector in the

u = zdkak(= Irka,) = 91 +x3 6u

(4.6)

where 58 = u/,3,O = U,(xP)aa-t-w(xp)a3( = ~%,+wa~)

(4.7) 6u = j,a”+i’a3(=

raa+Xa3)

In what follows it will be important to distinguish between the x- and .u*-coordinate systems in the current configuration; the former are convected coordinates defined as “shell coordinates” in the membrane state, while the latter are “shell coordinates” with respect to the current configuration. In order to facilitate computation we suppose that to within the degree of approximation entailed in the linearization we may assume that the functions <, and 2’ depend only on the middle surface coordinates 9. Thus the middle surface in the membrane state goes into the middle surface in the perturbed state (i.e. the linearized current state) and we can, without loss of generality, take the shell coordinates to be identical to the convected coordinates on the middle surface ; that is, for the same material particle on the surface x*~= x3 = 0, we have Pa z 2’. Then one readily computes a: = Q&@ = u’ =

(4.8)

T’p 4 = =

where, in obtaining the last expression, use was made of the Gauss-Mainardi~odazzi relations baP,,,j= bsil,

R;aP = bZb,,- bjb,,

An approximate shell theory for unrestricted elastic deformations

where RF,, is the Riemann-Christoffel

913

tensor, so that uPllIcasl = 2‘R”PMu9

We have also used the convention that parentheses enclosing a pair of indices means that the term is to be symmetrized in those indices, while square brackets mean the term is antisymmetrized. Next we observe that a: must be the unit normal to the current middle surface hence a> = a+[a;

x a, +a, x a;] - &‘/u)a,

= - ( &u~ + ws,)aa. Then (4.9) and we note that (4.10) Now the integrals of equations (4.2) are evaluated along the x*~ coordinate lines in the perturbed state, for which we write the vector line element in the form dr* = dx*3a: = dx*3[a3 - (bfu, + w,Ja’]. The same material line element in the membrane state configuration may be represented as dr = dxkg, = dx3a3 + dx”&a,, . But the difference in these vector line elements is just dr* -dr

= du = dxku,, = [dx”u,,, +dxa(b;u,-+ u3,Jja3 + [dx3u,,3 + dx%= iiB- baSu3)]a”.

Upon comparing these expressions and linearizing, it follows that along the x*~ coordinate lines the convected coordinates satisfy

where

may be recognized as the relative rotation in the xPx3-plane of the normal and middle surface at the middle surface. Let us now consider integrals of the form h’/2 p

=

h”i2 i+b*(~*~)

s - h’/2

dx*3

K* =

x*3 t/%*(x*k) dx*3 s

-h*/Z

MORRIS STERN

914

where J/*(X*? = $(xk) c 41/‘(xk) with x* and x referring to the same material particle. Now along the particular x*~ coordinate line x*” = c’ we may write #(Xa; x3) = $(c” ; x3) + x%;3~$.,(c= ; x3) where x$a P zz

.X3 $“p dx3( = x3[S;+ 0(x3)]). s0

Thus we write J* = J+J’

K* = K-i-K’

with h/2

J=

s -h/2

$(xk) dx3

K= s

h/2 -h/2 x311/(xk)dx3

h/2

h/2

J’ =z l’J+sp

-h/2

s

K’

=

dx3 +

x3$@.,

$I’dx3 s

-h/2

hi2

hi2

-h/2

-h/Z

2i;‘K+EP

x3$’ dx3 s

the integrations being performed along the x3 coordinate lines in the convected system. With similar definitions for N”P, IV’@, etc., equations (4.1) can be put in the form b;Q’fl + I-fDNpa+ l-b”,N”o- b;PQfl+ p’” = F*a

Nf;-

Qiadl + b,,N’@+

I-FaQP+ b;,N”fl -+ P’~ = F*3

(4.11)

Mf; - Q’“+ rbSgMPa+ l-;;‘,M”@-I-r” = G*’

with p’#

=I

‘p*= + I**+

-_

p*@+l*a+

NV8 _ b;Q@

[S h/2 s

~~~~~~

dx3

-h/2

=:

[p*”

+

l*”

-

$Jp”]

1 hi2

+

[I

-s h/2

hl2

(p&j&’ -h/2

b;

-

,d3

-h/2

IllJ

O~fi”)

dx3

dx3

1IIS

An approximate shell theory for unrestricted elastic deformations

915

In view of equations (4.5) we note that each of the remaining integrals above is at least 0(h3), and further because of the way in which the Pk were defined (equations 4.3) the remaining terms can be expected to be small. Once the membrane state has been determined we may compute the components of the stress tensor in this configuration from the constitutive equations and thus evaluate N”fl, Mus and Q”. Furthermore, it is assumed that enough is known about the nature of the prescribed loading so that the terms P’~and r” may also be explicitly determined. Next we take P = ii treating the membrane state as an equilibrium configuration. (In the event the actual loading is time dependent as described earlier, the varying part of the load is absorbed in plk and P.) Then f* = f *kg;

rz

ipa, = jp>‘$Tpf$3&

(4.12)

whence

f*a= j&p

f*3

and we have

s h/2

F*k =

=

ii3

s h/2

G*” =

,upiik dx3

-h/2

x3j@i” dx3

(4.13)

-h/i?

If we write (4.14) then

The only unevaluated terms remaining in equations (4.11) are those involving iV5, APB and Q’“. We define the following integrals :

s hi2

(x~)~K~(~~*~),~ dx3

-hi2

&s = P

hi2

J

dx3

(X3)kK~(&%““),,

-hi2

hi2 jy=

(~~)~pj$‘b~~ s

-h/2

P

dx3

(4.16)

hi2 $L

dx3

(x~)~~&$Y~’ s

--k/2 h!2 (~~)~jm”” -hi2

dx3

MORRIS STERN

91h

all of which may be evaluated explicitly once the membrane state configuration mined. Then

is deter-

(4.17)

where hi2

N WI _-

&P’

dxJ

-h/2

(4.18)

Q""=

h/2

pa'Q3 dX3

-h/2

are the only unevaluated terms remaining. From equations (2.8) and (4.7) we have

(4.19)

We then write equation (2.12) in the form

where

= Y rsmn

@GrSp

+

~(G’“Gk”-

-t P(grsgm"- 2g'"g'") -t

+ 2dl +

~kGs1)(2~~~~ -gktgm”)

2~.dG'"G""

3gpqGkl~~sGPq- ~p~*)~k~gm~ - gkmg’“)

2~~~,,G”“(Gk1G’“-Gk'Gf") -t-13GklG*S(gk1gm"-gkmg1n)]

+ 2%-i$grsgmn+ 2A’I 3(G”g=” -i-g’“G”“) + 2~[l,g,&“‘(Gk’GrS-

Gk’Gf”)

+ ~~Gk~~~k~g~~- gkmg’“)] is determined from the constitutivc equations in the membrane state.

(4.20)

An approximate

shell theory

for unrestricted

917

elastic deformations

Now in order that the results in the limiting case be consistent with membrane theory as described in [7], the value of A’ should be selected so that ‘a’j3 vanishes. However, this is possible only if “9’ 333 3 is never zero, a condition which is not always plausible. Indeed, if the stored energy function is linear in the invariants then a simple calculation shows that 09’3333 s 0. Some other condition is therefore needed to fix A’in such cases, a contingency which apparently was overlooked in [7]. We might note that this difficulty cannot occur in the incompressible case as was demonstrated in [7] where both R’ and P’ were eliminated with the condition “I; = 0 in addition to “g’j3 = 0. To resolve this problem in the compressible case we merely ignore any additional thickness change and take I’ = 0. Then upon writing &I”

h/Z

=

(4.21)

(x~)~/L
we find

N WJ =

[&hV_ b$+P’I~ _ b;“@‘M + ),$&P”‘](up

+&B?V_ +“I’ +

M

“4

=

[&793

same

_ as

increased

,+“P”“]

,, y _

_ b;“h’ + @$Wl’l~,,,y 3 -11

w”b

except that each overscript in value by one

is

>

(4.22)

Q”” = [&‘,’ - b$““P] (u, ,,y - b,,w)

If the material is incompressible the assumed existence of a deformed membrane state immediately leads to a contradiction, for on the one hand we are prescribing the deformation off the middle surface so that changes in area elements along a normal are in the ratio p/M, while on the other hand the normal extension ratio A is constant. This is possible in an incompressible material only if bz = BE, lb$ = jL?tI, a not very interesting case. It is therefore somewhat unrealistic to base the computation of the constitutive functions for the material on the membrane state deformation throughout the shell thickness, and the same argument can be used against incorporating the effects of changes in the transverse normal strain, in effect, A’. We therefore evaluate the material functions Q,, Y, P, d,9$, 9 only on the shell middle surface and ignore the effect of A’ in g& and gi3. However, in order to be consistent with membrane theory as developed in [7], in the limiting case, we retain (4.23)

gi3 = 21.’ = -2(t$-b:w) computed oar33

=

from the requirement

“r3 = 0. (The function

0.)

Thus for an incompressible

material

we write

P’ is then

determined

so that

918

MORRIS STERN

where for this case we have Y rSmn= 2~~A/~)grmgsn-t 2Y [a”@A=,&“g’”+ G’“G””- Gr”‘GS”] + 2,0z*G’“G”” + 2~g~~G~~(GkzGrs-GkrG~‘)~pq~ -gpm$“) (4.24) + 29[ GpqGrS(gPqgmn -gpmg”“) + g,,G”“(Gk’GrS- GkrG”)] where we have dropped the prescript denoting that the material eonstitutive functions are evaluated on the middle surface, this being understood in the incompressible case. We take P’ independent of x3 and hence p’ = =

_3op3mno

I g mn

oy33wt (“,ip 3333&B _

)@.dlB -

4x,++

Then (4.25)

(4.26)

s h/2

- Q”B

(x3)k@FPrs33 dx3

-hi2

we find, for an incompressible material, that equations (4.22) must be augmented by adding the term 0 (TbP~Y-b~~~p~Y)(~tl,ly-b~yw~

to N”“@and adding a similar term with the overscripts increased by one to M”*. inaddition, the term +~V(Ug[,y- b,,w)

should be added to the expression for Q”*. Finally we note that for an incompressible integrals of equation (4.14) become

material p = p. and p’ z 0, hence the

AII approximate sheli theory for unrestricted elastic deformations

919

In both cases, whether the material is compressible or incompressible* when equations (4X2), (4.19), (4.17) and whatever form off4.22) is appropriate are ~~b~~~~~~~d in equations (4.11), we obtain a system of five linear equations in the middle surface displacement components P, MIand in the relative rutation components 3. The limiting case (h -+ Of reduces to the membrane theory of (71 (except as noted in the compressible case). 5. UNIFORMLY

INRLATED, EXTENDED,

AND TWISTED CYLINDRICAL

TUBE

As an example we consider here the particular case of a circular cylindrical shell of an incompressible elastic material subjected to uniform internal pressure and to an axial tension and a twisting couple applied to the cylinder ends. The deformation of the shell middle surface, considered as a membrane, is presumed to be as indicated in the figure, In the current figuration, with respect to cylindrical polar coordinates @, 8, y), the membrane is placed at p = c and surface coordinates are chosen to be X1 = X = CO

FE.

X2 = y

1. Presumed middle surface beformatian.

Thus one computes easily a*$ = 6or@

a@ = a”@

Q:= 1

and the only no~-v~~is~~~g c=oet%cientof the second fundamental form of the surface is b,, = l?: = h” Furthermore

= -I/c*

all Christoffel symbols vanish : I-$ z 0.

Naw we suppose that the deformation of the shell from the r~f~re~~~con~~u~~~on to the membrane state is characterized by uniform expansion of the radius and length of the middle surface and of the thickness of the shell (which in the current ~on~guration is assumed to be constant at h) by the factors A,, A2and A respectively, and fmtbermore the

middle surface is twisted at a uniform rate $ referred to the current length. Thus in the reference state A,,

A,,

= 2iT2

A”

= Af+K:C2tp

= X;*+;1;*c2t,b2 A=

Alz(=

= A;

As2(=

A,,)

A”)

= -rZ,2c$ A = /i;2&2.

= &$t

Because the material is incompressible, we have =‘I, = ;i*A/a = I hence A= /q-r/1;r and B,, = -&c$2

Br, = --3,&k

&A=

&tI = 4&‘~

Next set IzI s h/2c

2 = X3/C so that for the membrane state P Z ,& z (-j&]-r

&=;:=I

= f-t-2

and all other components vanish. Then (gy

iT=g11=

=

gz2 = $2 =g33 =g33 =: 1

(l-t@

with all other components vanishing. Furthermore M = M: = (2:)-’ &f: = 2;

@=&=l

= (l+J&z)

= 0

l%i: = -(l-t

nf&s)$:

= -R&c$z

and thus GZZ = K;%-&2c2~~(1

G,, = n;z(l 44&z)* G,,(=

G = (lf~:rL&

Gar3= 0

G”(=

G,, = A$_$

G,,) = -X;2c$(1+#+$2

G” 1 = n:(t f R&z)G”)

G22 = n;

* + 1$++!2

= ,I$$

-t-#?~;t,z)2

G”3

zz 0

(p

=

J-2J-2 ‘I *2

1

For the membrane problem we have Ol, = i:$A:+a.;*A;2+A;c*+*

OI2 = A;2+3L;+-IZ$;fCI;%2$2

and therefore 0633 = A;2~,2at+(~;2fX22+~;2C2~*)Y+p* With the condition 0aS3 = 0 we thus have

An approximate

shell theory

for unrestricted

921

elastic deformations

and

The constant values of the stress components are of course consistent with the membrane equilibrium equations (for P1 = 9” = 0). In terms of the applied load we have N = 27rcho~3~~ T = 2nc2hoa12 p( = S3) = (h/c)“o”

where N is the axial force extending the cylinder, T is the end couple twisting it, and p the internal pressure inflating it. We note, of course, that 9’ = P2 = 0. In the membrane state we have tJ” = (G’“-~;2;1;2grs)~++[g,n(G”“G’“-GmrG”S)-(~;2+3L;2+~;2c2II/2)g’S]Y so that

We introduce displacement :

the following direct notation 24(x,y) =

d(xa)

u(x, y) =

ax,Y) = 5W) From the incompressibility

for the components

u2(xU)

of the additional

w(x, y) = w(xU)

?(X,Y) = 52w

condition (4.23) we have

1’ = -((u,+o,+w/c) where the subscripts x and y denote partial differentiation. We also denote the relative rotation components of the normals to the middle surface by a( = 5 + w, - u/c)

I-(= q+w,).

MORRIS STERN

922

The additional

stresses a” given by equation CT‘IS

=

(4.25) are thus of the form

[(1+z)~~Sf1-~P'S33+grs(q~33333~~3311)](Ux+W/C) + [97prs22 _,yrs33 +$[(I

+f(“y3333

_ 0y3322)]~~

+Z)(.Y~S12+Y*S21)-grs(ay3312+o~4p3321)]Uy

+3[y~12+yrs21_~rs(oy3312++973321)]~~ +3[y~~l3+~~s3l]~++[yrs23+yrs32]y

+cz((l+z),4p’s’15,+,~‘s22~, +3(yrs12+~rs21)[(1+Z)5y+~,]}

with the coefficients given in equation (4.24). Before evaluating the required integrals we expand each integrand in a power series in z, and after integrating, discard all but the lowest order terms in (h/c). (This approximation is consistent with the condition that the additional displacement be linear in z.) Furthermore, while it is conceptually no more difficult to carry through these details for a general incompressible material, a considerable saving in algebraic effort is effected if attention is restricted to materials of the Mooney-Rivlin type, i.e. Q, and Y constants so that d = &? = Sr = 0. For this case then the equations of motion (4.1 l), when dynamically uncoupled, are seen to be

-

B&xx+ l$y + wxk)+ P,u,,+ m&x = I$.fpti

+~luYV+~ll~xy+~2~y/~+~~~~p'2/h 86w,,+83Wyy+B10Wxy-84WIC2-_pgU,IC -

B12~ylC

+ PLAIC

- Bzu,lc

+ %=:x

+B13Yy-~B9(~‘y+Yx)+~.I~:p’3/h

-

Yl(Wxxx

+ wxyy) -3/%o(%xy

= i:%;pti

+ N’,,, +

7w,lc2)

+Y3W,/c2+Y~U,,IC-381~U,B/C+3B~Uxx/c-2B1~Vyy/C +YhU,FIC+Y14~‘xx+Y10~‘yy+Yllrxy-3P9(~*y+rxx)

-(12c2/h2)[ifi4E/c2 -fPlo(Wx*x

+ %yy +

-&&T/c21

+ r“’

3w,Ic2) - Y2(%,y

= ,I:,&2

+ u’yyg)

+Y~~,I~2-~P1~u,,/~-~l2~yyl~+Y7~,~/c+Y8v,,/c+Yguyy/c -B12U,~/C+Y10rxx+Y12ryy+Y13~‘xy-389(~:yy+Yxy)

+(12c2/h2)[&~/c2-~13Y/c2]+r”2

= A:%:pY

An approximate

shell theory

for unrestricted

elastic deformations

923

where

y1 = (1 + $2:

+ A~/:c2~2)~

y2 = (1 + n:@B

+ (22: + n: + n:n; + 2&A+P)Y

+ (A: + 2g + $A; + ;l;c$qY

y3 = - 2(4 + 3/2:A: - 3A:A;c2ti2)Q - 2(3A: + 42; + 3$A; - 2$1, y4 = -(2+n:~:+3n:;i:-2~~n;+/I:jl:~~~~)~ + (A: - 5A: - 4n;n; ys = - (7 + 5qn:

+ 2AfA, + &%p)Y

+ 5~:~;c2lp)@

- (42: + 72: + 52;n; - 4&I, y6 = (- 7 + 3,3:A; - 2i:n;

+ 4;1;c2$2)Y

+ 3A:/I:c2~-2)@

- (8/I: + 2; + 5A;lA: - 6;1:/1, + 8;1;~~$~)Y y, = - (1 +n;n;

+21:/1;

+A:/I;c%+P)@

+ (2A: - 32; - 31;/2: + 21;c%+P)Y ys = -21:A:cD-2&q2Y yg = - (1 - n;‘n: + 21g.1: - ll:/I;c2+2p + (22: - 22’- 3;1:1; + 2&92)Y Yl#J = ~+(/2:+~:-E”:l:+~~czr1/2)Y 711 = 2Q + (22: + 1; + ;c:n; + 2/1:c%p)Y Y12 = 3Q + (3134+ 222’+ 3&24P)Y y 13 = 20 + (2: + 22: + /I;Li; + &?lp)Y y14 = 30++21:

+3/1.; +21;c2t,b2)Y

+ 3/lfc2t,b2)Y

924

MORRIS STERN

There are several things which may be noted about these equations simply from their form. First, if we consider the limiting case h/c + 0, the relative rotations E and Y tend to zero identically, and if we also remove the initial twist (c$ = 0), then the first three equations reduce identically to the membrane equations obtained by Corneliussen and Shield [7]. One might also observe that since the equations are homogeneous the membrane solution (for the uniformly extended, inflated and twisted tube) is also a solution which accounts for “first order” bending effects (of which there are none, at least in the interior of the tube). This might have been anticipated because of the high degree of symmetry in the problem. Nevertheless, these equations do represent a refinement of membrane theory in that some bending resistance is allowed in the shell, and in a forthcoming paper they are used to study the stability of the membrane solution in the presence of bending thereby generalizing the results presented in [7]. Ackno~ledRemmts~This work was supported by the National Science Foundation under grant GK-620 to the UniversityofIllinois. Most of the results in this paper were obtained while the author was visiting in the Mechanical Engineering Department at the University of Colorado. Recognition is also taken of many useful discussions with Dr. William Fourney.

REFERENCES rll _ _ J. L. ERICKSEN and C. TRUESDELL, Exact theory of stress and strain in rods and shells, Archs ration Mech. Amdvsis 1, 295 (1958). El P. M. NAGHDI. Foundations ofelastic shell theory. Proar~ss in SolidMechanics, Vol. 4. North Holland (1963). [‘3j A. E. GREEN, P. M. NAGHDI and W. L. WAINW~IGHT~A genera1 theory of a Cosserat surface. Archs ration. Mech. Analysis 20, 287 (1965). [4] W. L. WAINWRIGHT, On a nonlinear theory of elastic shells. Int. J. engng Sci. 1. 339 (1963). [5] P. M. NAGHDI and R. P. NORDGREN. On the nonlinear theory of elastic shells under the Kirchhoff hypothesis. Q. appl. Math. 21.49 (1963). [6] A. E. GREEN and J. E. ADKINS, Large Elastic Deformations. Oxford University Press (1960). [7j A. H. CORNELIUSSENand R. T. SHIELD, Finite deformation of elastic membranes with application to the stability of an inflated and extended tube. Archs ration. Mech. An&sis 7. 273 (1961). [8] A. E. GREEN and W. ZERNA, Theoretical E1asticit.v. Oxford University Press (1954).

(Receioed

I August 1966: revised 30 Nooember 1966)

Resume-Une thkorie approximative d’enveloppes est formulte dans le cadre de la thtorie g&n&-ale d’elasticitt finie pour une d&formation sans restriction. Un champ de deformation est construit dans toute I’enveloppe basi: sur la solution du probllime de membrane correspondant. Une d&formation supplementaire est alors surimposte B cet “Ctat de membrane” et les Cquations de mouvement qui resultent pour I’Ctat final IinCarisCes dans le dkplacement supplCmentaire. Ces kquations sont alors prises “en moyenne” au moyen de I’Cpaisseur de I’enveloppe pour donner une theorie d’enveloppes approximative. Les dktails sont suivis pour le cas dans lequel la d&formation supplkmentaire est elle-mCme linBaris&e dans la variable de I’epaisseur de sorte que les normales & la surface moyenne restent droites et uniformkment allongees. mais pas nbcessairement normales. La thtorie rCsultante est

An approximate shell theory for unrestricted elastic deformations

925

annshernde Schalenthoerie wird im Rahmen der allgemeinen Theorie der endlichen Elastizitlt der unbegrlnzten Deformierung formuliert. Ein Deformationsfeld wird iiber die &hale aufgebaut, das auf der L&sung des entsprechenden Membranenproblemes basiert. Eine weitere Deformierung wird dann iiber diesen ‘Membranenzustand’ gesetzt und die sich ergebenden Gleichungen des Endzustandes werden in der weiteren Verschiebung linearisiert. Diese Gleichungen werden dann ‘auf den Durchschnitt gebracht’ durch die Schalendicke und geben die entsprechende Schalenthorie. Einzelheiten werden fiir den Fall gegeben, wenn die weitere Deformierung selbst linearisiert ist sodass die Dickenvariahlen-Normale zur MitteloberRIche gerade und gleichmbsig bleibt aber nicht notwendigerweise normal. Die Theorie wird dann angewandt zur Losung des Problemes einer gieichmlssig verdrehten und vergosserten Zylinderschale. Zusammenfassung-Eine

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