A shell theory compared with the exact three dimensional theory of elasticity

Int. J. Engng

Sci.

Vol. 6, pp. 435-464.

Pergamon Press 1968.

Printed in Great Britain

A SHELL THEORY COMPARED WITH THE EXACT THREE DIMENSIONAL THEORY OF ELASTICITY? CHESTER B. SENSENlG Courant Institute of Mathematical Sciences, New York University, 25 1 Mercer Street. New York, U.S.A. Abstract - A theory for the equilibrium deformation of homogeneous isotropic shells is derived and compared with the exact theory of elasticity. The shell theory is derived axiomatically as follows. The potential energy is considered as a functional depending on displacements which are polynomials in the undeformed distance to the middle surface. The degree of the displacements tangent to the middle surface is a prescribed integer n and that of the displacement normal to the middle surface is n+ 1. The shell theory is then obtained by requiring that the potential energy be stationary with respect to all variations in this set of displacements. For each fixed n it is shown that a solution to the shell theory is an approximate solution to the exact theory at points not too near the edge of the shell provided that the displacement gradients, strains, loading, and thickness are sufficiently small. It is also shown that the agreement between the theories improves as II increases. Many of the proofs have been restricted to the case of zero body force, zero stress vector on the faces, and the strain energy density function of the linear theory, but the work has been carried out in more generality in a research report. INTRODUCTION

paper presents a theory for the equilibrium deformation of homogeneous isotropic shells, and makes a comparison between that shell theory and the exact three dimensional theory of elasticity. These are called the shell theory and the exact theory in the following. The shell theory and the exact theory depend on a strain energy density function, and the two theories can be either linear or nonlinear depending on the choice of the strain energy density function. The results stated in this introduction are valid for both the linear and the nonlinear theories using a curved undeformed middle surface, non-zero body forces, and a non-zero stress vector at the faces: however, for simplicity most of the proofs are given in this paper only for the linear version of the theories using a flat undeformed middle surface, zero body forces, and the zero stress vector at the faces. The proofs for the nonlinear theories using a curved undeformed middle surface and non-zero forces are all contained in a research report [ 11. Considering only homogeneous isotropic materials which have a strain energy density function, the exact theory may be obtained by requiring that the potential energy of the shell be stationary with respect to all variations of the displacements. To obtain the shell theory a restricted set of displacements is defined as follows. Let 0, and 8, be parameters for the undeformed middle surface, r (O,, 0,) be the position vector to the undeformed middle surface, &I= &-l&7,, (Y= 1,2, be tangent vectors, and g, = g, X g,/ lg, x g,( be a unit normal vector. In the undeformed shell let & be the distance from the middle surface along a normal line, the sign of t):,being chosen so that 0, is positive on the side of the middle surface towards which g, points and is negative on the other side. Then the displacement vector u can be expressed in the form u = Uigi (as usual, Latin indices assume the values 1,2,3; Greek indices assume the values 1,2; and summation THIS

tThis paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, with the Office of Naval Research, Contract Nonr-285(46), and the National Science Foundation, Grant NSF -GP-3465. Reproduction in whole or in part is permitted for any purpose of the United States Government. 435

4.16

C. B. SENSENIG

over repeated indices is implied). For the shell theory only those displacement vectors are admitted for which U1 and U2 are n-th degree polynomials in &, and for which U3 is an (n i- I)st degree polynomial in 0,, n = 1,2,3,. . . . For the remainder of the paper n always has the above meaning. The coefficients of the polynomials I/‘, i = 1,2,3, are functions of 8, and 13,which are to be determined. Requiring that the potential energy be stationary with respect to all variations of these 3n + 4 coefficients leads to the equilibrium equations and stress boundary conditions of the shell theory. A common procedure, in comparing a shell theory with the exact theory, is to assume that a solution to the exact theory has certain properties (such as small strains or displacements), and then to make approximations in the exact theory showing that the shell theory is approximately satisfied. An important feature of the comparison made here is that the procedure is reversed; namely, it is shown that a solution to the shell theory approximately satisfies the equilibrium equations and stress boundary conditions of the exact theory. No attempt is made, however, to determine estimates for the difference in the displacements obtained from the exact and shell theories. The shell theory is compared with the exact theory by examining the errors which result when displacements satisfying the equilibrium equations and stress boundary conditions of the shell theory are substituted into those of the exact theory. The equilibrium equations of the exact theory can be written in the form E, = 0, i = 1,2,3, the stress boundary conditions at either face can be written in the form Eif = 0, i = 1,2,3, and the stress boundary conditions at the edge can be written in the form Ei, = 0, i = 1,2,3, where Eiq, Eif and Et, are expressions depending on the displacements and their derivatives. (Since those boundary points for which e3 = &h and those on the narrow sides are frequently referred to separately, the sets of boundary points for which @,= ?h are called faces and the remaining set of boundary points is called the edge.) In particular since many proofs are carried out here using the linear theory with zero body forces, zero stress vector at the faces, and a flat undeformed middle surface, the relations Eiq = aqJc?xj,

Eif = qi3, and

Ei, = qunj - qie

will frequently be used where x,, x, and x3 are rectangular coordinates and the quantities qij, nj, and qie are the usual linear stresses, the components of the unit outer normal vector at the edge of the undeformed plate, and the components of the prescribed stress vector at the edge. If displacements satisfying the shell theory are substituted into Eiqy Ei,y and Eify they will in general have non-zero values, and here those values are called the errors which result when displacements satisfying the shell theory are substituted into the exact theory. It is shown in all cases that Eipy EC,, and their derivatives have a stated number of zeros along each curve which was a line normal to the middle surface in the undeformed shell. Also under certain hypotheses Eiq and its derivatives have small magnitude everywhere along each such curve not too near the edge, and Eti and its derivatives have small magnitude at the faces at points not too near the edge. More specifically, assuming the displacements satisfying the shell theory have as many derivatives as desired or needed, and letting Eiqy Ei,y and E,be the errors defined (Y= 1,2 (s is arc length above, it is shown that dkE,,Jdf&33paop and dkE&s kl+kraep. along the undeformed middle surface at the edge and k is any positive integer) have n -k, - 1 and n - k3 + 1 zeros respectively along each curve which was a normal line to the middle surface in the undeformed shell, while akE3,/a~%@W3p and dkE:,,/

Shell theory compared with the theory of elasticity

437

aSklfk*aeF have n - k, and n - k, + 2 zeros respectivily along each such curve. These results are valid regardless of the thickness of the shell or the size of the deformation. Also, to emphasize the point, the statement about Ei, and its derivatives is valid up to and including the edge of the shell. Without further restrictions, it has not been proved that EiP, Ei,z, and their derivatives have small magnitude at all points on the curves which were normal lines to the middle surface in the undeformed shell, but, when such is the case, it is shown that i31cE,/at$1a# also has a small magnitude at the faces. To show that Ei, and its derivatives have small magnitude at all points not too near the edge, and to show that Eg and its derivatives have small magnitude at points on the faces not too near the edge, a more restricted situation is considered. First a strain energy density function is considered which agrees to lowest order terms with that of the usual linear theory. Then attention is focused on points which are not on the edge and which lie (in the undeformed shell) on a fixed line normal to the middle surface. Shells are considered which are thin relative to the distance from the fixed normal line to the edge and relative to the curvature and the derivatives of the curvature of the undeformed middle surface. Cartesian coordinates xi, i = 1,2,3, are introduced so that in the undeformed shell the fixed normal line lies on the x,-axis with the origin on the middle surface and with & = x3 on the x,-axis. The parameters 13~and e2 are chosen by letting o1 = x1 and & = x2 in a region around the x,-axis. The deformed shell along with the stress and body force vectors are moved rigidly until the displacements are zero at the origin and the matrix of displacement gradients is symmetric at the origin. If then the prescribed components of stress at the faces and their derivatives have small enough magnitudes, if the prescribed body forces and their derivatives have small enough magnitudes, and if the displacement gradients and strains have small enough magnitudes, all in a region around the x,-axis, then Eiq, Eif and their derivatives satisfy the inequalities (O-1) (to be given in a moment) on the x,-axis. In (0.1) E and d are positive parameters (E is non-dimensional and d has dimensions of length) which depend on the thickness of the undeformed shell, the distance from the x,-axis to the edge of the shell, the curvature and the derivatives of the curvature of the undeformed middle surface, the prescribed components of stress at the faces and their derivatives, the prescribed body forces and their derivatives, and the displacement gradients and strains. Letting h be half the thickness of the undeformed shell, the parameters E and h/d are small (in particular h/d < 1) under the assumptions considered here. For a complete discussion of the choice of E and d the reader is referred to section 4. For the linear version of the shell theory with zero body forces, zero stress vector at the faces, and a flat undeformed middle surfacee is simply chosen so that (d/h)e is an upper bound of the absolute values of the displacement gradients and also so that E is an upper bound of the absolute values of the strains in the region x~+x~ < 8, 1x315 h. Also in (0.1) k is any non-negative integer, Y is Young’s modulus, and K is a non-dimensional constant depending only on the strain energy density function, n, and k. K may have a different value for each inequality. -ifnisodd ’ dk+l E

-dk+l

ifn-k,-

1 L 0,

if n is even

C’. B. SENSENlCi

(0.1)

If the displacements were solutions to the exact theory, the left hand sides in (0.1) would of course be zero identically. When the displacements satisfy the linear shell theory with a flat undeformed middle surface and zero body forces and components of stress at the faces, d is the distance from the x3-axis to the undeformed edge. In that case (0.1) indicates that the left hand sides of (0.1) approach zero as h + 0 provided that E, n, and k3 are fixed and n is large enough. Furthermore the larger a is the faster the left hand sides of (0.1) approach zero as h --+ 0. For example it is sufficient to have n 3 3 and k, = 0 or 1 in the first inequality, n z I and k, = 0 in the second inequality, and n 2 1 in the last two inequalities in order that the left hand sides go to zero as h -+ 0. In the general case where the displacements satisfy the non-linear shell theory with a curved undeformed middle surface and non-zero body forces and surface tractions at the faces, the situation is more complicated. After studying the choice of E and d in section 4. it is seen that the left hand sides of (0.1) go to zero as h + 0 provided that n is large enough and E also goes to zero sufficiently rapidly as lt -+ 0. For example if E is made to go to zero so rapidly that c/h is bounded as h + 0, then there is a constant K such that (0.1) becomes

K(V/(h))“-k-JC3+1if.isodd K(~(h))“-k-“:3

if n is even

if II-~:, 2 0.

when h is sufficiently small. I? is a different constant in each inequality. The dimension of I? is such that the right hand sides of the inequalities have the same dimension as

Shell theory compared with the theory of elasticity

439

the left hand sides, and I? depends only on the strain energy density function, n, k, k,, k,, k:,, R (defined in section 4), and an upper bound for e/h. Thus, when e/h remains bounded and n is large enough, the left hand sides of (0.1) go to zero as h + 0, and the larger n is the faster the left hand sides go to zero. From section 4 it is seen that letting E * 0 so rapidly that e/h remains bounded as h * 0 means that a family of solutions to the shell theory is considered in which the displacements, body forces, and components of stress on the faces depend on h in such a way that

if k:, is even and i # 3 or if k:, is odd and i = 3 ,jkFi ,j@,a@qe$

c’

I?

.

(d(h))k-2

I? (q(h)

1fk3=oandr=3 otherwise

)”

where Ui and eij are the rectangular components of displacement and strain, @+ and @ are the components of the stress vector tangent to the undeformed surfaces C&= *h, F” are the components of the body force tangent to the undeformed middle surface, F:l is the component of body force normal to the undeformed middle surface, and I? is again a different constant in each inequality whose dimension is such that the inequalities are dimensionally correct. I? depends on the same quantities here as in the above inequalities. The above should hold for k = 0, 1, . . . up to a finite number. Notice that except for derivatives of a few low orders, the derivatives of the components of stress and body force may even go to infinity as h -+ 0. The larger n is the more slowly E need go to zero as h + 0 in order that the left hand sides of (0.1) go to zero as h + 0. To claim that the solution to the shell theory approximately satisfies the equilibrium equations and stress boundary conditions at the faces for the exact theory, it is not sufficient to show that the left hand sides of (0.1) go to zero as h and E go to zero since the left hand sides of (0.1) are a sum of terms which may themselves go to zero. However, it is not expected that the rate at which the individual terms in the sums on the left hand sides go to zero will depend significantly on n whereas the sums on the left hand sides of (0.1) do go to zero at a rate which depends significantly on n. Thus the larger n is the more subtractions are needed among the terms on the left hand sides of (0.1) in order for (0.1) to hold. In this sense (0.1) indicates that a solution to the shell theory can be made to satisfy the equilibrium equations and stress boundary conditions at the faces for the exact theory as well as one pleases by taking n sufficiently large and E and h sufficiently small. In spite of the fact that h/d < 1 it should be observed

C. B. SENSENIG

440

that (0.1) does not indicate what happens if E and h are held fixed and n increases since K depends on n in a way which has not been determined. Several inequalities of interest are derived to establish (0.1). These are listed as (0.2) and (0.3) in which the parameters appearing have the same meaning as in (0.1). Expressing Ui in the form U’ = F C/ill(0,/h)’ it follows that on the x,-axis

and ks is even [.tfif ii =# 33and k:, is odd

akui

ae:lae!pao as:I;$ae/

~ _.& hdk+

{ifi i 3 andk,.isodd if i = 3 and k:, IS even.

(0.2)

From the first two inequalities of (0.2) it follows that for the nonlinear shell theory the left hand sides again go to zero as h -+ 0 provided that 1 is large enough and E also goes to zero sufficiently rapidly. Again, if E is made to go to zero at a given rate as h + 0, the larger 1 is the faster the left hand sides go to zero. Therefore the first two inequalities give smaller upper bounds for the magnitudes of the coefficients of high degree terms in the polynomials dkUi/d8$M3~ than for the terms of low degree when E and h are sufficiently small. It is expected then that only the low degree terms will be significant in these polynomials when E and h are small. If the displacements satisfying the shell theory are substituted into the stress strain laws of the exact theory, qij denotes the Kirchoff stresses obtained (i.e. qij = aW/a (au,/axj) ) for either the linear or nonlinear theories where W is the strain energy density function and Ui is a rectangular component of displacement). Then on the x:,-axis. if k:, = 0 KYE if k:, = 1 and (i,j) = (a, 3), (3, (Y),or (3,3) dh‘ if k, is even and (i, j) = (C-Z, 0) or (3,3) a"4ii ri ifk,isoddand (i,j) = (a,3)or(3,~~) ao:laopa8pc KYE hdk-’

otherwise

(0.3)

Shell theory compared with the theory of elasticity

441

@ is a linear function of e3 which equals the prescribed component of stress normal to the undeformed shell on the upper face and equals the negative of the prescribed component of stress normal to the undeformed shell on the lower face. The inequalities in (0.3) are the same as those obtained by John[2] for solutions to the exact theory except that here E and dare chosen somewhat differently than the corresponding parameters in[2] and here non-zero body forces and surface tractions at the faces are permitted. The most difficult task in showing that (0.1) holds is the derivation of the estimates (0.2) and (0.3). To do this estimates for the L, norms were obtained for the various functions, and Solobev’s inequality was then used to obtain pointwise estimates. The work of John[2] was an indispensible guide in obtaining these estimates (although the details of the calculation are quite different here), and the author is happy to acknowledge his indebtedness to that work. Appreciation is also hereby expressed to J. J. Stoker, F. John, and W. T. Koiter for valuable discussions on the work. To close this introduction, some history is given of the efforts which led to these results. Originally J. J. Stoker suggested that the shell theory, which results if one admits into the potential energy only those deformations which satisfy the geometric Kirchoff hypotheses, be studied. That is the theory which is obtained if one admits into the potential energy those deformations in which normal lines to the middle surface in the undeformed shell go into normal lines to the middle surface in the deformed shell with no change in length along those lines and then requires that the potential energy be stationary with respect to all variations of such deformations. In this study it was observed that the resulting theory did not reduce to the von Karman-Foppl theory for small displacements [4]. This was considered important since it had already been shown that the von Karman-Foppl equations were correct[3] in the sense that they are approximately satisfied by a stress function, defined in terms of a solution to the exact theory, and the vertical displacement of the middle surface in the solution of the exact theory. Of course by introducing the Kirchoff stress hypotheses in addition to the geometric ones, the von Karman-Foppl equations could be obtained, but, for that matter, so could a variety of equations have been obtained by strategic use of those mutually contradictory hypotheses. Next displacements which were n-th degree polynomials in 6J3were admitted into the potential energy. The resulting shell theory reduced to the von Karman-Foppl theory for II 2 2 but not for n = 1. Using that shell theory, a comparison was made with the exact theory, and results were obtained similar to those presented here. In the course of a discussion of the results with Professor Koiter, it was suggested that a higher degree polynomial was more important for the displacement component normal to the middle surface than for the other components of the displacement. This led to the theory presented here in which (R + 1) is the degree used for the displacement normal to the middle surface and n is the degree of the other displacements. The resulting shell theory is a better approximation to the exact theory, and the proofs are considerably simpler. In addition this last shell theory agrees to low order terms with the von Karman-Foppl theory for n 2 1, but that is not shown here.

where

1. PSEUDO-TENSOR Consider

position

a fixed rectangular

vector

IJES: Vol. 6. No. 8. I3

Cartesian

NOTATION

reference

from the origin of this reference

frame X, and let r(e,, 0,) be the frame to the undeformed middle

442

C. B. SENSENIG

surface. An undeformed middle surface is considered such that normals to it do not intersect for I&( s h, whenever h is small enough, and such that the surface and its boundary are smooth enough so that subsequent uses of the divergence theorem are valid. The undeformed shell is then defined to be the region 18:,(s h. Letting x be the position vector from the origin to an arbitrary point(x,, x2, x:,) in the shell, we also assume that the undeformed middle surface is smooth enough so that x = r + 8:,g,,

(1.1)

defines a relationship between the coordinates xi and di which has as many continuous derivatives as needed in the following, the same being true for the inverse relationship. Let gij = g, . a, (g’j) = ( gij) -1, and gi = g”gi. (1.2) Then the quantities gap and g@ are the components of the usual covariant and contravariant metric tensors for the undeformed middle surface. Also g,, = g3n = 0 and g,, = g33= 1. Finally g’j = gi . gj, gi . g j = 6/, and g, = g3. The quantities uj and Al are defined to be the components respect to the reference frame X as follows: gi = (a;, a;, a;),

of the vectors gi and g’ with

g’= (Af,A;,A;).

(1.3)

Then (uj) = (A;)-1,

gii = u;uf,

gij = A@&

(1.4)

Given any indexed set of functions associated with the X-frame, say dij (the number of indices is not important), indexed sets Dij, q, D:, and Dil are associated with the curvilinear coordinates 8i as follows: D..1J= &td. , J ,,,, 0: = ufAjd. I Al 0: = Aj&dkl,

Dij = A:.A;d,,.

When the indexed set dfj is symmetric (i.e. dij = dji), both 0,; and 0: will be denoted by dti and Dij (or Dj, Dj, D”) will be called X- and &components of the same pseudo-tensor. It is easily seen that D, = g,,Df, D” = gikgjlDk,, etc., so that the quantities gij and gij can be used to raise and lower indices of the &components of pseudo-tensors in the same way that the indices of the components of tensors are raised and lowered using a metric tensor. Observe that Sjj and g, (or g”, Sj) are X- and &components of the same pseudotensor. Also if ui denotes the X-components of displacement, then ui and U’ (where (u,, u2, u:J = UigJ are X- and &components of the same pseudo-tensor. Pseudo-tensors have a contraction principal of the same sort as ordinary tensors. Let cti and Cij (or dijk and Dijk) be X- and f&components of the same pseudo-tensors. Then cii = Ci = Ci (or diij and Dij are X- and &components of the same pseudotensor). If dij and Dij are X- and &components of a pseudo-tensor, Dij/k, Dijlk/, etc. are defined to be the &components of the pseudo-tensors having adij/axk, fPdij/axk ax,, etc. as X-components; i.e. DijJk = a~u$u~ad,,/ax,, etc. Indices of Dijlk are raised and D:. The functions

Shell theory compared with the theory of elasticity

443

lowered the same as those of the &components of other pseudo-tensors. It should be observed that quantities such as Dijlk are nx the usual covariant derivatives. It follows immediately that gijlk = 0, g”l, = 0, etc. Let baP = g, - a be the components of the second fundamental form of the undeformed middle surface (,a means a/13& and ,i will mean a/N+). For convenience let bi3 = b3$= 0 and raise the indices of the quantities bij just as if they were the &components of a pseudo-tensor. Then let (Bj) = (Sj - &$~j)-~ (observe that B$ = Bt = 0, B”,= 1). The indices of the B: are also raised and lowered just as if they were &components of a pseudo-tensor. Then H = (l/2)& is the mean curvature and K = det(b$) is the Gaussian curvature of the undeformed middle surface. For convenience let C= l-2H03+K8;. Quantities lYt are defined by gi,j = T$g,. Then also gfj = - rkjgk. The quantities I’& are the usual Christoffel symbols for the metric of the undeformed middle surface and

rg = bij, rtj= - bj, rj,= 0. In terms of the X-components

of gi and g’ one has

afJ =

rijaf;

Ak,j=

-rbAi.

(1.5)

From x = r + f&g, one obtains X,j= so

(Sq-O,by)&

that

(1.6) If di and Di are X- and &components

of the same pseudo-tensor,

= afajB:Af (AiD,,, - I&4lD,) = B;(D,,,-r&D,).

(1.7)

Similarly D’lj = (D’,, + r$D”) Bj’.

If dii and Dij are X- and &components DtiJk = (D”,l+

(1.8)

of the same pseudo-tensor,

rLlDm'+rk,Dim)Bk,

D& = (Dj,, + rj,,,Djm- rzD;n)Bi, Dij(k = (Dij,,-T$Dmj-

lY~Di,)B~,

Dfl” = (Df,,+rf,ID~-r~D~~)Bzk,

etc.

(1.9)

444

C.

7. INTRODL'(‘TION

B. SENSENIG

OF THE

EXACT

THFORY

AND

SHELL. T-HEORY

Let Ui denote the X-components of displacement. When the non-linear shell theory is being considered, let r;j = ( l/2) (?Iui/ax; C ?JU,i/axi+ (auk/ax,) (auk/ax,) ) be the strains, and let S, = pii, sz = e+ji, and s:$= e;jej/;eki be the strain invariants. For homogeneous isotropic materials and the nonlinear theory, the strain energy W per unit undeformed volume is a function of .rl, .s? and szj. In order that the strain energy density function agrees to lowest order terms with that of the linear theory of elasticity, it follows from Taylor’s theorem that w = $: + /As,-I- w,s,< + w,s,s, -t w,s; + w,s; where A and p are the Lame constants and W,, We, W:,, and W, are functions of s,, sz. and s3 which are assumed to have as many derivatives as necessary for the following. For the linear theory the strains are eij = (l/2) (dUi/dxj + duj/dxi) and W = (X/2)eiirii + /..Wijeij.

The quantities dW 4ij

= g&i (

8Xj >

are defined as the Kirchhoff stresses. The quantities qij are always treated as functions of the displacement gradients and are defined for all displacements even if the displacements are not solutions of either the shell theory or the exact theory. Let qii+be the X-components of the stress vector (force per unit undeformed area) acting on the deformed face & = h, and let & be the corresponding components acting on the deformed face t$ = -h. Letfi be the X-components of body force (force per unit undeformed volume). Let C& be the X-components of the stress vector (force per unit undeformed area) acting at the deformed edge of the shell. Then the equilibrium equations of the exact theory are

g+./+o

(2.1)

J

and at the boundary 4<+= qijnj

for

03 = h

qi_ = qijnj

for

8:~= -11

qic = qijnj

at the edge

(2.2)

where the ni are the X-components of the unit outer normal vector of the undeformed shell (see [SD. Now let capital letters denote the O-components of pseudo-tensors whose Xcomponents are denoted by the lower case letters. Then, using the contraction principal for pseudo-tensors, the exact theory becomes Q”(jt-FL = 0, 0: = Qi3 for

8, = h

445

Shell theory compared with the theory of elasticity @=-Qi3

0:

=

for

fj3=-_h

QiaNa at the edge

(2.3)

since N, = 0, N3 = -I- 1 for 8, = k h and N:, = 0 on the edge. To obtain the shell theory, consider the potential energy E= shell

es=h

es=-h

eae

where dv is the undeformed volume element and dS is the undeformed The requirement that E is stationary becomes <= 111(

qij$-jjGi)dv-/

J qi+icidS-

shell

/I

qi_iidS-

83=-h

83=h

area element.

1 qi,icidS = 0; edge

where dots indicate variation. After using the divergence theorem, this becomes

JJ

+

( qijnj- qf-) ic,dS + J ( ( qijnj-

es=-h

SC) ic,dS.

edge

Next use is made of the contraction principle for pseudo-tensors, the fact that the quantities Ui are the &components of the pseudo-tensor whose X-components are the quantities iri, and the relations N, = 0, N:, = k 1 for e3 = +-h, and N3 = 0 on the edge. For the statement about the quantities tij and ii, recall that the displacement vector (u,, up, u,) = cl@ = Ui(A:,Ai,A$). Thus Ui = AiU.1, CJi = ajuj, and, since the quantities a( are independent of the variation parameter, rZ<= ujLij. The last equation then becomes

JJ

0=-J

(Q”jj+P)~idV+J

-

j

I

J (Qi3-@+)oidS 8.,=/l

shell (

Qi3 + &)

ci,dS + j- j- ( QiaNu-

83=-h

0;) ti,dS.

edge

But dv = C a!4 de, and dS = C dA for 8, = + h where dA is the element of area on the undeformed middle surface. Also dS = a ds d0, at the edge where a = g\/( (@- 28,bg + k’z,b’;bJ)PA,) (hug; = A,$“) is the unit tangent vector to the boundary curve on the undeformed middle surface) and s is the arc length of the boundary curve on the undeformed middle surface. In addition, for the displacements admitted into the potential energy to obtain the shell, theory, 0, = i k=O

ti,,o$,

fi, = ng fi,,e; k=O

where tiak and ir.& are arbitrary functions of 8, and f12. From this the equilibrium equations (2.4) of the shell theory are obtained together with the boundary conditions (2.5) at the edge.

446

C. B. SENSENIG

1 (Q”jlj+F3)C@d03=

[(Qa”+e”-)C@&+,,

k=

0, 1,2,.

. .,n

k=

0, 1,2,.

. .,n+

[(Q”“-&)C@]++,

-h -

[(Q3”+@)C@]+h,

1

(2.4)

/I k = 0, 1,2, . . . , n

( QaBNp - &) 6’$adt3, = 0,

/

-h

(2.5)

IL 1 -h

(

k = 0, 1,2, . . . , n + 1.

Q3PNB - @) B!$zdtI, = 0,

Letting 1 % Qi=B(Q$+&)+Z.(el;-&)

(2.6)

one has Q’ = Q$ when 13~= h and Qi = - @. when 0, = -h. Then the equilibrium equations become C@d&

k=

= [(Q”-Q”)C@]g!h,

0,1,2,.

. ., n,

-h

C@de,=

[(Q33-@)C@g]$$!h,

k=0,1,2,...,n+l.

(2.7)

-h

From (2.7) it follows that h

dk[(Qi’Ij+F’)C] -h

ae:laep

,p[(Qi3-@)C],

Pdd, =

where k is a non-negative integer and P is an arbitrary i # 3andofdegrees n+ 1 ifi= 3. Similarly

Ih

dk[(QiLIN~-Qk)aIpde

-h

polynomial

of degree 6 n if

=

0

3

dsk

(2.8)

eepaep

(2.9)

at the edge of the shell where P has the same meaning as above. 3. GENERAL

COMPARISON

OF

THE

SHELL

THEORY

AND

EXACT

THEORY

This comparison of the two theories is called a general comparison because it is valid for all solutions to the shell theory, regardless of the thickness of the shell or the size of the deformation. This comparison is contained in the following theorem. Theorem

(3.1)

Let the displacements stress boundary conditions

Vi be a solution to the equilibrium (2.5) of the shell theory. Then

equations

(2.7)

and

Shell theory

compared

with the theory

ak[(Q"ilj+Fa)C] a@Wya*

441

of elasticity

=Oatn-k,-lpointsifn-kk,--1

> 0

and

@[(Q”jI,+F3)Cl =Oatn-k,,pointsifn-kk,>O ae:la@Wy

along each curve which was a fine normal to the middle surface in the undeformed shell. Also at the edge of the shell

ak[(QdN,s- &>a1 =Oatn-kk,+lpointsifn-kk,+l asklaep

> 0

ak[(Q3’Na--@>4 a+ae,k’

> 0

and =Oatn-kk,+2pointsifn-kk,+2

along each curve which was a line normal to the undeformed middle surface. The equilibrium equations and edge boundary conditions (2.3) of the exact theory may be written as (Qi’Ij+Fi)C=O (Q”Np-Q:)u

= 0.

Thus the theorem states that if a solution to the shell theory is substituted into these equations, both the error and its derivatives are zero at a stated number of points on each curve which was a line normal to the middle surface in the undeformed shell. The proof of the statement about ak [ ( Qti )j + F3) C] /a~~16@%%I~will be given. The rest of the theorem can be proved in a similar manner. Let E = dkmka[ ( Q3j lj + F3) C]/ MklM~z. For each &, t12 it is assumed that E has isolated zeros. Otherwise it has intkitely many and the statement of the theorem for E is trivially true. First consider the case n 3 1 and let P = h2- 0:. From (2.8) one has ./ EPde, = 0. Since P is not zero for (e,] < h, it follows that E has at least one zerd!or (e3[ < h. Furthermore, since its zeros are isolated, it has at least one zero where it changes sign. Now consider the case n 2 2. Assume E has Tnly one zero where it changes sign, namely 8, = r. Let P = (8, - r) (h2 - 0;). Again J EPde, = 0. From the choice of P, the integrand does not change sign. This is a cokadiction so that E has at least two zeroes where it changes sign. Proceeding in this manner, it can be shown that E has n zeros for le,( < h. Then using the mean value theorem, the statement of the theorem about a”[ ( Q3j )j+ P’a)C]/ ae%e%e”3 f0ii0ws. 1 2 3 From (2.7) with k = 0,l it is seen that L (Qi3-@)CjBJ=-th

=;

(Qulj+

Fi)Cde3.

(3.2)

From this it is seen that when a solution to the shell theory is substituted into the exact theory, if the errors obtained from the equilibrium equations of the exact theory and their derivatives with respect to e1 and e2 are small for ]e,] G h, then also the errors obtained from the boundary conditions at the faces and their derivatives are small.

448

C. B. SENSENIG

4.THE

PARAMETERS

USED

IN MAKING

ESTIMATES

After picking any point on the undeformed middle surface away from the edge. the X-axes can be introduced so that the chosen point will be at the origin and the equation of the undeformed middle surface will have the form x3 =~(x,.x~) in some neighborhood of the origin with f(O.0) = 0 and aj’(O, O)/ax, = 0. After an appropriate rigid transformation of the deformed shell (moving applied stress vectors and body force vectors with the deformed shell), one also has /ii (O,O, 0) = 0, and all,/a,Vj= al/j/dxi at X = (O,O,O). Let 0, =x, for the remainder of the paper and choose D so that it is less than or equal the distance from the X,-axis to the edge of the undeformed middle surface and so that det ( gij) 2 B for 0: + 19,”s D’ (observe that det (aii) = 1 at the origin). Choose R so that If&/ G I/R. j.L4,,j s I/R”, etc. for 0f + 0; s Dz and for as high an order of derivatives as needed in the following. Restrict h so that C 2 f for J&J 6 h and 0; + (9: =S D2. Consider shell thicknesses, stress vectors at the faces, body force vectors, and displacements which satisfy the equilibrium equations of the shell theory and such that

and l0,l s h andk=0,1,2,. . . up to the highest order required in the following where Y is Young’s modulus and B0is a constant which satisfies 0 < (!I,,< 1 and which depends only on the strain energy density W, and on n. Let 6 = max (h/D. g(h/R), V(E)) so that H is a function of the arguments h/D. h/R, E. For h/D and h/R fixed, 6’ and ~18 are non-decreasing as E increases. Let E be the smallest number such that

y(z)"+ 1

----E @+lhk+l if k, is even and i # 3 or k, is odd and i = 3 dkF’ ae:m$w3~

y/p+2 4 - --•ifk3=Oandi=3

V2hk+’

YfP -E

@hk+’

wherek=0,1,2,.

otherwise

. . up to the highest order required in the following.

From (4.1) it follows that V(E) < 0,, and hence 8 < 0”. Let d = (0,/O) h. Then 0 < h/d < I. From (4.2) and the definition of @.

(4.2)

449

Shell theory compared with the theory of elasticity

. ’II

if k3 is even and i # 3, or k3 is odd and i = 3

g

3°F'

ae:laepaep

s

Yhe

F”ifk,=Oandi=3

(4.3)

YE otherwise /UP

fork = 0, 1,2,. . . to the highest order needed, and for 0; + 0: c D2 and Ii&\ G h. If the left hand sides of inequalities (4.1) are much less than the right hand sides, then V(E) will be much less than &, and 0 will be much less than 8,,. Hence both E and h/d will be small. Estimates for various quantities will be given in terms of E, h, and d. All the results stated from here on will be proved only for plates using the usual linear strains and the strain energy density function of the linear theory. Also zero body forces and zero stress vectors at the faces are used. All results are proved for curved shells using the nonlinear theory with non-zero body forces and non-zero stress vectors at the faces in a research report[ 11. Since the undeformed middle surface is a plane for a plate, the coordinates &, e2, O3 as chosen in this section become rectangular coordinates in the undeformed plate. The restrictions (4.1) are not necessary when the linear version of the shell theory is used with a plate and zero surface tractions at the faces and zero body forces. Inequalities (4.1-4.3) will not be used here and are included only to show how E, h and dare chosen in the more general case. In the case for which proofs are given here the restriction h/D s 1 is made, d is defined by d = D, and E is the smallest positive number such that IUi,jI c (d/h)e and \eijI c E for e:+e’, c d” and l&l c h where now eij = (+) (ui> + u~,~) are the linear strains. The stresses qij become qfj = he&iii + 2/Lf?jj = hU~,&j +

Thus in place of (4.1-4.3)

/J

(Ufj +

Uj,i)

.

(4.4)

. h -=G 1 d

(4.5) for e;+e; c dz and le,( 6 h when the linear theory and plates are considered zero body forces and zero stress vectors at the faces.

with

450

C. B. SENSENIG

For the special case being considered uf = A; = Sj, and hence the &components pseudo-tensors are the same as theX-components. Thus (2.8) and (3.2) become

where P is an arbitrary ifi=3.

of

polynomial in (!I3of degree s II if i = 1,2 and of degree C n + 1

5. RELATIONS

BETWEEN

DERIVATIVES

In deriving estimates for the derivatives of the displacement gradients and stresses. use will be made of certain relations which exist among them. Using the relations stated in Theorem (5.1) estimates for the derivatives of all the displacement gradients and stresses are obtained as soon as estimates are obtained for a strategic few. Letak,k= 1,2,. .., be the set of all functions of the types

where for functions of the first type i f 3 and k3 is even, or i = 3 and k3 is odd; for functions of the second type l3 = 0, or 1, = 1 and at least one ofj and 1 is 3, or IS is even and neither or both of j and 1 are 3, or 1, is odd and exactly one of j and 1 are 3. The various cases mentioned are not all mutually exclusive. Letbk,k= 1,2,... be the set of all functions of the above two types with no restrictions on the indices, and let Vi = qiiJ. Observe that Vi would be zero if the displacements were a solution to the exact linear theory, see [ 61. Theorem (5.1) For k 3 2 every function in ak curt be expressed as u linear combination of the types

where lS = 0, or la is even and 1 # 3, or l3 is odd and I= 3. For k > 2 every function in b,,.can be expressed as a linear combination of the types ak-2V ak-Qj,s d m

offunctions

offunctions

“-‘qij

aepaep

ae",laepae, ’ aeyae,m*aop'

These linear combinations for functions in ak und bk can be chosen so thut the following are true. The linear combinations for ak-lqJa~lM$W~, k, z 1, involve derivatives of V, which have at most k3 - 1 dt$erentiations with respect to OS.The linear combinations for u3,3a do not involve Vt or any of its derivatives. Finally, the linear combinations for akuila0$W$W3~ involve derivatives of Vt which have at most k3 - 2

45 I

Shell theory compared with the theory of elasticity

differentiations with respect to O3 if k, > 2, and no derivatives of V, with respect to O3are involved ifk, < 2. All linear combinations referred to have constant coeficients which depend only on h and p. To prove the theorem use is made of the relation Ui,lk

A

1

=-

___ 3~+&

qii,k+qik,l-qJk,i-

zp

( qU,khj

+ %I, j&k

-

qll,iajk

>

1* (5.2)

From the definition of ee ui,jk

=

e&k +

eik,J -

ejk,i

and from (4.4) A

1 eij=G

(

4ij-3h+2F

(5.2) follows from these. From the definition of Vi 4i3.3

=

-qh,m

+

(5.3)

l/i*

From (5.2) 1 4l.m = - cc 4cdLY [

+

&Y,P - G3Y.U -

+

am3

2

k3.~‘

=

1 2~

-

quo,3

%s,a

:

2

cL(4u,vb +

qw3~ay - clrLah3Y)

1

-&-plsaLJ

[

- 433,e+-3A+A 2/~““~ 1

%a,,3

u 3,ap

-

3A

1 =

-&

[

ew3

+

q3/3>a -

q33.u

-&qu..]

4as.3

&

+

41L3b]

1 U 3&x

=

z [

1 3,33=-

U

.&

[

433,3

--

A 3* + 2p911*3

1

(5.4)

.

The statement of the theorem about the derivatives of qij in a2 and b, is seen to be true from (5.3). Substituting (5.3) into (5.4), the statement of the theorem about the derivatives of u1in a, and b, is seen to be true. Differentiating (5.3) and (5.4) once with respect to 8, or &, the statement of the theorem is obtained for those derivatives of qti and ui in aR and b, in which not all differentiations are with respect to 8,. The statement about the rest of the derivatives of qu and ui in a, and b, follow from the preceding and qo3.33

=

-4ua.#% +

va.3

4 33,33

=

-q3a,a3

v3.3

+

I

from (5.3)

C. B. SENSENIC

&

u:j,:i3:1 =

I

(%3,x,- h.%,v:r3 )

from (4.4)

qaP,:lz= &$in:&J3 + p (~,,P:c:l+ %,f%H) 42,333

A from (5.4). -!AL:1,33 - q33,u:j+ ___ 3h + 2/_lqrJ,a?II i 2E.L

=

(5.5)

The statement of the theorem is proved for those functions in a4 and 6, in not all differentiations are with respect to 0., by differentiating (5.3) and (5.4) with respect to r?I1and/or e2 and by differentiating (5.5) once with respect to 8, or get the result for those derivatives in which all differentiations are with respect (5.5) is differentiated with respect to 6,. Proceeding in this manner, Theorem proved for all values of k. 6. AN

ESTIMATE

FOR

THE

L,

NORMS

OF

V; AND

ITS

which twice 8,. To to 8,$, 5.1 is

DERIVATIVES

Before proceeding further, the reader should be familiar with the notation and results presented in the Appendix. Formula designations such as (A.k) refer to the Appendix. Let q and V denote the sets of functions {qii} and {Vi} respectively. In this section the following is established for k 3 0. /
= S[l&‘““‘ll]

ifk,=O,or if k3is even and i # 3, or if k, is odd and i = 3

llcYk’ll= [l15Li’k+“ll + 5 ,*;,=,lli $$g$]

(6.1)

3

All integrations are over the region where 5 # 0 and 18:,(s h. To establish these, use is made of the expansions in terms of Legendre polynomials of the various functions. This is convenient because of the orthogonality of the Legendre polynomials. If g(8,, 19,,19,)is defined for (&I s h, let

where P k (x)

_p_ =-1-q 2"k! dxh I)”

is the k-th Legendre polynomial. From (4.6)

fork= 0, 1,2,.. .,n. Since Pk( 1) = 1 and Pk(-1) j&v,,,

=

= (--I)“,

(Ya3)Oa=h

+

(--I

this

lc+l(

becomes

9&)0,=-h

fork=O.

l....,

n.

453

Shell theory compared with the theory of elasticity

Since the right hand side does not change ask increases by two’s, one obtains 2k+ 1

V a(k)

=

ifk+niseven,k=O,l,...,

2n+1V,n,

2k+lV 1 zn_ V a(k) -_ -

dn

_

n

ifk+nisodd,k=O,l,...,n.

1)

Since V, is an n-th degree polynomial in f&,

k=O

k=O ktn

+&‘/“(“-

even

$

1)

(2k-t

l)Pk($.

k=O k+n odd

From the identity (2kt

I)P,(x)

= P;+,(x)

k 2 1

-P;_,(x),

one has ij k+n

(2k+

1 )pk($)

=

(2k+

1)&.($)

= ff$).

P:,+I(~)

even

i k=O ktn odd

Hence V, =

.$

(6.2)

&$W=bt,(~).

k=n-1

Similarly nt1

-b ‘,= &k+l

3

k=n

Fork=

(k)P;+,

0

$f

(6.3)

.

n-1,n V atk)

=

[ 6h.P

+

p (b.33

+

=

( %rP.P

+

~~~3,3ahk)

u3,3a)

l(k))

from

(4’4)y

since u,,~~ is an (n - 2) -nd degree polynomial in e3 if n 2 2 and is zero if n = 1 (note that g,,, = 0 ifg is a polynomial in & whose degree is less thank because of the orthogonality of the Legendre polynomials). Hence

(6.4) Fork=

n,n-t

1 V 3(k)

=

[ q3PB

=

(qBP.Bhk)

+

Au P,,33+

(h+2p)~~3,331ckh

from

(4.41,

C. B. SENSENIG

454

since uc(,43and u3,33are both (n - 1) st degree polynomials in 0,. Hence (6.5) Lemma (6.6) Let A and B be finite sets offunctions

(the number depending only on n) such that

for every function a in A where each 6, is in B, pk is a k-th degree polynomial coeficients depend only on n, and m depends only on n. Then

whose

for all k, k = 0, I, 2, . . ., for which the derivatives are continuous (here for fixed k, k,, k,, k, the notation akA~aoflae!j=aO$= denotes the set of all aka/awa@=ae!p with a in A, etc.). From Bessel’s inequality the following inequality holds

for each k. Thus

so that i <“a”d&= 8( 7 ] 5”&%) -h

since 4 is independent

-h

of &, and kIa112= S(IkB11”).

Hence ((@I((= G(Il{B(() which is the desired result fork = 0. Since a, = F kzO (b&kjpk(Wh) argument. Next observe that fork z= 1

--I

h

+ then

Ilr;Adl

= ~(IIWAI) from the previous

455

Shell theory compared with the theory of elasticity

since (2k+ l)P,(x) But

= Pi+&)

-P;-&)

and P&&l)

-P&t-l)

= 0.

Then /l{A,3(l= G( Il@I,3{1)by the argument used in the first part of the lemma. The differentiations can be repeated to obtain the lemma. From (6.4), (6.5) and Lemma (6.6)

(6.7)

From Theorem (5.1)

Thus (6.1) follows from (6.7) for k3 = 0. FromTheorem(5.1)

Using (6.1) with k3 = 0, the above becomes

II

akU3,3a

5 aepaepae = 6 II@

I; I/

3II

a%3P.B

a8pat@ao 3 II

[

=

k+l)Il + CI II scib%tl] B,Y

~wf”“‘ll1

-

Equation (6.1) follows for k3 = 1 from the above and (6.7). From (5.1)

C. B. SENSENIG

Using (6.1) with k, = 0,l , the above becomes

~~t$gt&/~ = +P+“ll+

2 l154%311].

Equation (6.1) follows for k, = 2 from the above and (6.7). Proceeding (6.1) is proved for all k,. 7. L, ESTIMATES

FOR

FUNCTIONS

IN THE

in this manner,

SET il’“’

Here the following relations are established:

ll5:3;ill = 8(q))

+ d&) (7.1)

~~~r;~‘k-“~~ = ?&&~)

+ CT(&)

for

k L 4

where A, = m:X /j,$&jl, k = 1,2, 3,. . . .

(7.2)

All integrals are taken over the region where 5 f 0 and 163) s h. Hence d8,d0,de3 is omitted in the notation for all triple integrals. Using the divergence theorem, J J 1 &Wi,.i = - j J j” [&ij,j”i +

(G),&ia41+ J J G(4iG4)2 I!! ITdeldez’

From (4.6) SJ

“’ = h doId %&(qi@i)+_h

= J J j &ij,jUi.

Hence the above becomes 1 J J G4ijUi.j = - J J J ( G) x&Ii&i. Similarly

=IIS

(SK).,Yi~,p(2eip--UP,i).

Shell theory compared with the theory of elasticity

457

Using the divergence theorem and (4.6) again,

(3E> .a%B43,i = - ISJ

ISI

E(5;),&ai,iPUB + (G).nY%udQl

+ 1 J- (Sh).u(q.3.BU@);~ 1 hpwe* =-

IIS

(52> ,av%%&.

Combining this with the above leads to J J J

G%d4,ja= - J f j II2(5E).&ia,L+iI3 + (4%) .uYqYa,P431~

Similarly IIS

=-I J J [2(52,).&idw%3,~+ (5%>,dhdd43,J 5%4ii,ai3”i,jd3 = - J J 1 II2(G),&ia,i3Vei/3,Y + (G),aYhu3&/3ol*

In the same way ) J

J

J

{%qij,a,ol

*... +,

~i,jm,a,..a,J

=

) J J J C2(5’k),pqia,Ba,a,...a*,eia.a,a *...uk_I o,c(,a~_.olk_:,] ) ep + (5%).ayqyu,~,a *...ak.,

. = ?j ,,&‘k-qz+L d~/je’k-3’1/2] + (+[ll
+ u(&)

of the function qlm.

II&‘“-“II = 29[~lli’k-“II] + (r[llr;$-“ll] and

Since qij = q ji, then (4.4) gives qijUi,j = qijf?ij= hf?kj&ii+ 2/.Leijf?ij3 2/&L) e I’ so that

le12=

6(q,Ui,j).

Since 141 = O( let) from (4.4), it follows that

1412 = ~(4ijUi,j). Similarly

1$‘“-“12=

g(qij,a,a,...a,~,ui,j~~~

IJES:

Vol. f,. No. 8. C

*... a,-,)

C. B. SENSENIG

45x

and hence

k=

3,45,.

(7.3)

. .

But

from (4.9, and for k 2 4 ,,$k-3),)k= ~[,,~k__B~(k-3),,] = 6(&,). Therefore

(7.1) follows from (7.3).

8. L, ESTIMATES AND POINTWISE First it is shown that

ESTIMATES

FOR FUNCTIONS

IN a, AND

bk

(8.1)

where

(8.2) and this is used to show that ,a, = “(&)

for &at k=

lb, = $-&)

1,2,3,...

(8.3)

for beb,

in a cylinder around the x,-axis. From Theorem (5.1) A3 = =

w~3iill+

~(ll1;2411+

+ 1,~3VL3ll,

using (6.1)

~(11~3~11)

using (7.1). Thus A3 = G(ed(h)/d) .42 =

k-A

and (8.1) is verified for A,. Also from Theorem (5.1)

Ik-2vll>

from (6.1). From (A.4) ll52411=

$jMl)

t- d43Gll~

=

N&h

1) +l+(~543)

=

N&h)).

Shell theory compared with the theory of elasticity

459

Thus and (8.1) is verified for AZ. Fork=4,5,6,... Ak =

~[kleP--l)lll

using Theorem (5.1) and (6.1). Then (7.1) gives

(

>

+c(Ak)

Ak = 6 ;A,_,

. By induction (8.1) follows for Ak, when k = 4,5, . . . . Next fork = 2,3,4,. . . Btc = a[

i\~k((i”k-“i\

+

2

iitk&:?i\]

(8.4)

using Theorem (5.1) and (6.1). But, from (A.8),

and from (A.6), (8.1)

andfork=4,5

,...,

(A.6)gives

(using 1 = 8({&_2) for k k 4 at points where & # 0) =

fi($k-2+$k)

=

+/th;dk_3)

using (8.1). Since also \([&‘k-“(I = B(A,) = d $$$ for k=2,3,... ( > (8.4) becomes (8.1) for BR, completing the proof of (8.1). Now let a be any function in ak, k = 2,3,. . . . Then the set of functions ii is contained in ak+2 and a ,33is an element of b,,,. Thus Sobolev’s inequality (A.7) gives %itk+2al

=

zP(A.+~+~llallk+r+~Bk+r).

C. B. SENSENIG

460

Since I = TY([~) fork > 2 at points where &.+.’# 0,

Thus

using (8.1). Thus

at points where tk # 0, completing the proof of the first part of (8.3) for k = 2,3, . . . . Similarly the second part of (8.3) is proved for k = 2,3, . . . . To obtain (8.3) for k = 1, recall that a rotation was performed so that Ui,j = uj,i at the origin. Thus lu’l = z9(lel) = g(e) at the origin. By Taylor’s theorem --and hence

in a cylinder around the x,-axis using the fact that Ju’) = 6(c) fork = 2. Since also q = 9(e) , (8.3) is verified for k = 1.

9. VERIFICATION

OF (0.1-0.3)

at the origin and (8.3)

COMPLETED

Letters a, b, c etc., will be used to denote the parts of (0.1-0.3). Thus (0.2c, d), for k 2 1, and (0.3a), for k z= 0, coincide with (8.3). Since ui = 0 at the origin, Taylor’s theorem gives 4(01, OCJ, 8,) =

~j4,j(81,&9

03).

Thus Iu,( = 6(k)

and lusl = 6( (P/~).c),

giving (0.2~44 when k = 0.

If a solution to the equilibrium equations of the nonlinear shell theory is substituted into the equilibrium equations and boundary conditions at the faces for the nonlinear exact theory, the errors Eip obtained from the equilibrium equations are Eiq=Q**Jj+Fi,

i=

1,2,3

(9.1)

Shell theory compared with the theory of elasticity and the errors

&obtained

461

from the stress boundary conditions at the faces are Ei,=

(Qk-Qi),

i= 1,293;

8,=&h.

(9.2)

For the case being proved, these reduce to i = 1, 2, 3

Eiq= qij,j,

Eif = qi3, i= 1,2,3;8,=?h.

(9.1)’ (9.2)’

From Theorem (3.1) (observe that C = 1 in the case being considered), dk&, etqa*atq hasn-k3-lzerosifn-k,-1

~=O,and a”&.

aepaopaep has n - k, zeros if n - k, > 0 along each curve which was a line normal to the middle surface in the undeformed shell. Using the mean value theorem of differential calculus

ifn-k,-1

2 0,and

if n-k, a 0. Using (8.3) and (9. l)‘, these become (0. la) and (0. lb). From (4.6), (9. l)‘, and (9.2)’ h

Ei,=

k

E&dep

-h

Using this and (0. la&), one obtains (0.1 c) and (0.16). Since

(0.2a,b) follows from (0.2c,d). To prove (0.3b), first observe from (0.1 c) that

~$!$l,,=~,=($~) forn= 1,2,%... so that from Taylor’s theorem and (0.3~)

This completes the proof of (0.3b) since qa3 = qza for the linear theory.

462

C. B. SENSENIG

Equation (0.3~) follows in the same way as (0.36) recalling that @ = 0 in the case for which proofs are given here. Finally 4XL:S = 4U,j -

q3a,a

=

-G,

-

q3a.w

Using (0. lb) and (0.3b), one obtains (0.34. REFERENCES

[II C. 8. SENSENIG, elasticity.

New

349 (I 966).

A non-linear

York University,

shell theory

compared

with the classical

Courant Institute of Mathematical

three dimensional

Sciences,

theory

of

Res. Rep. IMM-NYU

PI F. JOHN, Estimates for the derivatives of the stresses in a thin shell and interior shell equations.

Comm. Pure appl. Math. l&235-267 (I 965). 131 F. JOHN. Estimates for the error in the equations qf non-linear plate theory. New York University. Courant Institute of Mathematical Sciences. Res. Rep. IMM-NY U 308 (I 963). thin shell theory. New York University, Courant Institute of Mathc[41 C. B. SENSENIG, A Kirchhoff

matical Sciences. Res. Rep. IMM-NYU [51 C. TRUESDELI..

The Mechanical

313 (1963).

ofElasticity

Foundations

(1966). ISI I. S. SOKOLNI KOFF. MotkemuticctlTheory ofElasticity. (Received

I November

and Fluid Dynamics.

Gordon and Breach

McGraw-Hill (1956). 1967)

APPENDIX Here are presented some notation and results most of which are taken from [2]. Those results taken from [2] are given here without proof.

If /3 is any set of functions of 0,, &, @,, let h be the set of a11functions b,, where b is in B, and let B’ be the set of all functions b,i where b is in B. Similarly 8 is the set of all functions b,oB, B” is the set of all functions b,. for b in B, etc. The notation tick’ denotes B with k dots, and PC’ denotes e with k primes. If B is a single function, B will also be used to denote the set whose one eiement is B. Then B, B’, etc., have the above meaning. B, is the set of all functions b,, and 8,z is the set of all functions b,,, where b is in B, etc. Let B and C be sets of functions of B,, S,, t!&BC is the set of all functions bc where b is in B and c is in C. Similarly the product of more than two sets is defined. Frequent use is made of the following functions. Let ~,=h=~~~=jl-~)?

Sk=,)-

for

e:+e;cd2

bk-:3(@+ es) 2 & ] for q+@;<$

where k=4,5,5,...

(A.11

& = 0 otherwise.

Then 5 liand its first derivatives are continuous everywhere. 5 will be used to denote any ck. Let B be any finite set of functions of 0,. B,, 0,. Then JBJ* is the sum of the squares of al1 functions in B with (B( 2 0, and 11B11: is the integral of /812 over the region where & # 0 and l&j 6 h with Jl& 2 0. The subscripton ((BJ(kis omitted if the mnge of integration is clear from the context or if ail values of k are permitted. If a,b, and c are any positive functions, the notation a = 6(b)

means there is a constant k > 0 which depends only on Wand n such that a f

kb

and a = b(b) +u(c) means that for each k > 0 there is a constant R > 0 depending only on k, W and n such that

463

Shell theory compared with the theory of elasticity Frequent use is made of the relations IBCI = s(lso+~(lcl) WI

=

tm-w + ~Wll)

where B and C are finite sets of functions. Easily established results for the functions & are

lil = a(-$

(A.2)

Let B be a finite set of functions of 0,, &, 0, defined in the region where & f 0 and I&( s h. Let [= & andB = sup IB[. Then

Iliill = d(fllBllk) + 4m

(A.3) (A.4) (A.5) (A.6) (A.7) (Sobolev’s inequality).

If in addition the functions in B are all polynomials in 0, whose degree depends only on n, then

lB.31= i+(f) for

/&I 5 h.

L4.8)

To prove (A.8), let b be a polynomial in B whose degree is WI(then m depends on n). Say b= i Letaibethevalueofbfor8:,=

b,(;)t

(i/m)h,i=O,l,...,m.Then a<=

5 b,(i) ,’ i =0, I,.

. , IYZ.

j=O

Treating the above as a system of linear equations with the b, as unknowns, it is well known that the determinant of coefficients is not zero so that the bjare linear combinations of the ai. Since the coefficients of the above equations depend only on m (and hence only on n), each bi is a linear combination of the a, with coefficients depending only on n. Hence b,= b(maxla,l) OfjStTl

But

so that (A.8) is established.

= -9(maxlbl) b’,lsh

= 9(B),

i= 0, 1,

. ., m.

464

C. B. SENSENIC;

R&urn& II est prtsente une thtorie de I’equilibre de diformation d’une enveloppe homog*ne et isotrope et on la compare k la thkorie exacte de I’elasticiG. La thtorie de I’enveloppe est d&erminee sous forme d’axiomes. de la manikre suivante. On considkre que I’energie potentielle est une fonction dtpendant des deplacemen& qui sont des polynomes sur la distance, non dtformCe, g la surface moyenne. L’ordre du deplacement. tangent g la surface moyenne, correspond h un integrant dCtermint n et celui du deplacement. normal B la surface moyenne, est )I + I. On obtient alors la thtorie de I’enveloppe en imposant que I’energie potentielle soit stationnaire pour toutes les variations pouvant se p&enter dans ces dCplacements. Pour toute valeur fix&e de n. il est montrC qu’une solution de la thCorie de I’enveloppe est une solution approximative de la thkorie exacte pour des points qui ne soient pas trop rapproches du bord de I’enveloppe et, B condition, que les gradients de dbplacement, les dCformations, la charge et les Cpaisseurs soient suffisamment faibles. On montre egalement que I’accord entre les thCories augmente avec la valeur de II. Plusieurs dCmonstrations ont et6 limit&es au cas de la force massique nulle, B un vecteur de contraintc nul sur les faces et 21la densit& de I’Cnergie de deformation fonction de la theorie lintaire. Cependant. I’ensemble du travail a ttC ttudie d’une man&e plus gCnCrale dans un rapport de recherche. Zusammenfassung- Fiir die Gleichgewichtsdeformation homogener, isotroper Schalen wird eine Theorie abgeleitet und diese mit der genauen Elastizitltstheorie verglichen. Die axiomatische Ableitung der Schalentheorie erfolgt auf folgende Weise: Die potentionelle Energie wird als ein Funktional angesehen: dieses hPngt von Verschiebungen ab, die sich bei undeformiertem Abstand von der Mittelfl5che durch polynomische Ausdriicke beschreiben lassen. Der Neigungsgrad der Verschiebungstangente zur MittelflBche ist eine vorgeschriebene ganze Zahl, n, wlhrend die Neigung der Verschiebungsnormalen zur Mittelflkhe n+ 1 betr@. Die Ableitung der Schalentheorie basiert auf der Forderung, dass die potentielle Energie in Bezug auf alle in diesem Verschiebungssystem auftretende Variationen stationsr bleiben muss. Es wird nachgewiesen. dass bei Punkten, die nicht zu nahe bei der Schalenkante liegen. die Liisung der Schalentheorie fiirjeden Festwert von 12eine Niihrungsliisung der genauen Theorie darstellt. Eine Voraussetzung ist, dass die Vershciebungstradienten und Spannungen, sowie die Beanspruchung und Dicke geniigend klein sind. Weiterhin wird gezeight. dass die beiden Theorien desto genauer iibereinstimmen, je mehr 11anwlchst. Viele der Beweise beschriinken sich auf den Fall, wo die Raumkraft und der Spannungsvektor an den FlBchen gleich Null sind und die Spannungsenergie-Dichte-Funktion der linearen Theorie gilt: es liegt judoch eine allgemeinere Ausarbeitung in der Form eines Forschungsberichts vor. Sommario-Si deriva una teoria per la deformazione d’equilibrio di gusci isotropici omogenei e la si confronta con I’esatta teoria d’elasticit8. La teoria del guscio & ricavata assiomaticamente, e precisaments: I’energia potenziale t considerata alla stregua di una funzionale a seconda degli spostamenti the sone polinomiali nella distanza non deformata sino alla superficie mediana. II grado degli spostamenti tangenti alla superficie mediana t un’integrale n prescritta e quell0 dello spostamento normale rispetto alla superficie mediana P n+ I. Si ricava quindi la teoria de1 guscio richiedendo I’immobilit8 dell’energia potenziale respetto alle variazioni in questa serie di spostamenti. Per ciascun n fisso si dimostra the una soluzione della teoria del guscio 6 data dalla soluzione approssimativa della teoria esatta in punti non troppo vicini all’orlo del guscio, sempre the le curve, le sollecitazioni. il carico e lo spessore degli spostamenti siano sufficientemente bassi. Si dimostra pure the la concordanza fra le teorie aumenta con l’aumento di n. Molte dimostrazioni sone state confinate al case di forza sero del corpo, di vettore zero delle sollecitazioni sulle facce e all’energia di deformazione (funzione della densit8) della teoria linerare, ma il lavoro 2 state svolto in mode pili generale in una relazione di ricerca. 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